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abstract: 'Hong-Ou-Mandel interference is a cornerstone of optical quantum technologies. We explore both theoretically and experimentally how the nature of unwanted multi-photon components of single photon sources affect the interference visibility. We apply our approach to quantum dot single photon sources in order to access the mean wavepacket overlap of the single-photon component - an important metric to understand the limitations of current sources. We find that the impact of multi-photon events has thus far been underestimated, and that the effect of pure dephasing is even milder than previously expected.'
author:
- 'H. Ollivier$^\P$'
- 'S. E. Thomas$^\P$'
- 'S. C. Wein'
- 'I. Maillette de Buy Wenniger'
- 'N. Coste'
- 'J. C. Loredo'
- 'N. Somaschi'
- 'A. Harouri'
- 'A. Lemaitre'
- 'I. Sagnes'
- 'L. Lanco'
- 'C. Simon'
- 'C. Anton'
- 'O. Krebs'
- 'P. Senellart'
bibliography:
- 'HOMg2.bib'
title: 'Hong-Ou-Mandel Interference with Imperfect Single Photon Sources'
---
Quantum interference of indistinguishable single photons is a critical element of quantum technologies. It allows the implementation of logical photon-photon gates for quantum computing [@KLM2001; @OBrien2007] as well as the development of quantum repeaters for secure long distance communications [@DLCZ2001; @Sangouard2011]. The development of efficient sources of single and indistinguishable photons has become a challenge of the utmost importance in this regard, with two predominant, distinct approaches. The first one is based on non-linear optical photon pair production [@Kwiat1999; @Zhong2018], and multiplexing of heralded single photon sources is being explored to overcome an intrinsic inefficiency [@Kaneda2019; @Joshi2018; @Xiong2016; @Francis-Jones:16]. The other is based on single quantum emitters such as semiconductor quantum dots [@Aharonovich2016; @Senellart2017] where ever-growing control of the solid-state emitter has enabled the combination of high efficiency and high indistinguishability [@Somaschi2016; @Wang2016; @He2019; @Wang2019].
The standard method to quantify the indistinguishability of single-photon wavepackets is to perform Hong-Ou-Mandel (HOM) interference [@HOM]. In perfect HOM interference, two indistinguishable single photons incident at each input of a 50:50 beam-splitter will exit the beam-splitter together, resulting in no two-photon coincidental detection events at both outputs. In practice, however, the two inputs only exhibit partial indistinguishability described by a non-unity mean wave-packet overlap $M_\mathrm{s}$ (also defined as the single-photon trace purity [@Fischer2018; @Trivedi2020]). Partial indistinguishability of the input states leads to coincidental detection events at the outputs and reduces the HOM interference visibility. The interference visibility $V_\mathrm{HOM}$ can therefore give direct access to the single-photon indistinguishability $M_\mathrm{s}=V_\mathrm{HOM}$ [@Trivedi2020].
For non-ideal single-photon sources, for which the photonic wavepackets present a residual multi-photon component, the HOM visibility remains the relevant quantity that determines the quality of the above-mentioned quantum operations. However, the visibility of HOM interference is reduced due to multi-photon contributions, even if $M_\mathrm{s}=1$, i.e. for an ideal single-photon indistinguishability. In most cases, the multi-photon component of the photonic wavepacket, characterized by the second order intensity autocorrelation at zero time delay ${\mathit{g}^{(2)}}(0)$, depends on the system parameters in a manner that is completely independent of the single photon indistinguishability, and it is critical to have tools to access the latter in order to understand the physics at play and improve the performance of single photon sources.
Here we explore both theoretically and experimentally HOM interference with imperfect single-photon sources. Previously, the impact of multi-photon contributions on HOM interference has been investigated in the limited case where the additional photons are in the same spectral and temporal mode as the predominant ones [@Bennett2009; @Polyakov2011; @Huber2017]. It has been shown that the visibility of HOM interference in this case is given by $V_\mathrm{HOM} = M_\mathrm{tot} - {\mathit{g}^{(2)}}(0)$ [@Uren2005; @Trivedi2020], where $M_\mathrm{tot}$ is the mean wavepacket overlap of the total input state, i.e. including the multi-photon component. Here we show that the properties of the additional or “noise" photons play a critical role in HOM interference, and that it is crucial to know the origin of the imperfections to be able to correctly extract the intrinsic single-photon indistinguishability $M_\mathrm{s}$. We validate our approach by experimentally emulating two types of imperfect sources. Finally, we investigate the case of quantum-dot based single photon sources (QDSPS) based on both neutral and charged excitons. By understanding the physical mechanisms in both cases, we are able to provide a proper way of extracting the single photon indistinguishability that accounts for the nature of the multi-photon events.
{width="\linewidth"}
We model an imperfect “single-photon" state (${\mathit{g}^{(2)}}(0) > 0$) by mixing a true single photon (${\mathit{g}^{(2)}}(0) =0$) with separable noise at a beam splitter. We limit our analysis to small ${\mathit{g}^{(2)}}(0)$ values so that the noise field itself is well-approximated by an optical field with at most one additional photon and a large vacuum contribution. This restriction to a weak, separable noise field remains relevant in practice for many situations as illustrated for QDSPSs later on.
It can be shown (see Supplementary Material), that for separable noise and a small resultant ${\mathit{g}^{(2)}}$ (typically ${\mathit{g}^{(2)}}<0.3$), the visibility of HOM interference is given by: $$V_\mathrm{HOM} = M_\mathrm{s} - \left( \frac{1 + M_\mathrm{s}}{1 + M_\mathrm{sn}} \right) g^{(2)} \label{Eqn_Model}$$
where $M_\mathrm{sn}$ is the mean wavepacket overlap between the single photon and an additional noise photon satisfying $0 \le M_\mathrm{sn} \le M_\mathrm{s}$. We have defined ${\mathit{g}^{(2)}}\equiv {\mathit{g}^{(2)}}(0)$ for simplicity.
It is instructive to consider the two limiting cases of Equation \[Eqn\_Model\]. If the additional photons are identical to the single photons, i.e. $M_\mathrm{sn} = M_\mathrm{s}$, then Equation \[Eqn\_Model\] reduces to the simple case that $ V_\mathrm{HOM} = M_\mathrm{s} - {\mathit{g}^{(2)}}$, showing that the total and single photon mean wavepacket overlap coincide, $M_\mathrm{s}=M_\mathrm{tot}$. Alternatively, if the noise has no overlap with the single photons and $M_\mathrm{sn} = 0$ , then the visibility is further reduced and given by $V_\mathrm{HOM} = M_\mathrm{s} - \left( {1 + M_\mathrm{s}} \right) g^{(2)} $. The degree to which HOM interference is affected by a non-zero ${\mathit{g}^{(2)}}$ is therefore dependent on the origin of the additional photons.
We experimentally test this model by emulating imperfect single photon sources. We prepare a train of near-optimal single photons and mix them with additional photons to controllably increase ${\mathit{g}^{(2)}}$ and measure the impact on the HOM interference. We experimentally emulate the two limiting cases outlined above: when the additional photons are completely *distinguishable* ($M_\mathrm{sn} = 0$) from our single photon input, and when they are completely *identical* ($M_\mathrm{sn} = M_\mathrm{s}$). In each case, we measure the ${\mathit{g}^{(2)}}$ and HOM interference visibility of the resultant wavepacket in an unbalanced Mach-Zehnder interferometer (see Supplementary Material for further details).
We use a state-of-the-art single photon source based on a quantum dot (QD) deterministically embedded in an electrically contacted micropillar cavity [@Somaschi2016]. The QD acts as an artificial atom which we coherently control via resonant excitation to generate single photons with high single photon purity, ${\mathit{g}^{(2)}}<0.05$, and high indistinguishability, $M_\mathrm{tot}>0.9$. The single photons are separated from the excitation laser using a cross-polarization set-up, as shown in Figure \[Fig\_LimitingCases\](a).
The experimental set-ups that enable a controlled increase of the multi-photon probability are shown in Figure \[Fig\_LimitingCases\](b) and (c) for the two limiting cases. First, to add fully distinguishable photons, we mix the single photons from the QDSPS with attenuated laser pulses at a different wavelength. A 3 ps Ti-Sapph pulsed laser centered at 925 nm is spectrally dispersed using a diffraction grating, and a narrow portion is selected to obtain a 15 ps excitation pulse resonant with the QD transition (here a charged exciton). A second, non-overlapping part of the spectrum is selected to mix with the emitted single photons. By appropriately tuning the time delay we can add synchronous spectrally-distinguishable photons to the single photon emission. The corresponding output field is then considered as an effective source, and we adjust the power of the laser beam to alter the magnitude of the two-photon component. The measured HOM visibility as a function of ${\mathit{g}^{(2)}}$ of this effective source is shown in Figure \[Fig\_LimitingCases\](d). Since the spectral overlap between the QDSPS photons and the additional laser photons is zero ($M_{\mathrm{sn}}= 0$) our model predicts that $V_\mathrm{HOM} = M_\mathrm{s} - (1 + M_\mathrm{s}){\mathit{g}^{(2)}}$, where a single parameter $M_\mathrm{s}$ accounts for both the origin of the curve at zero ${\mathit{g}^{(2)}}$ and its slope. The line in Figure \[Fig\_LimitingCases\](d) shows that this model fits the data very well with $M_\mathrm{s}=0.94 \pm 0.02 $.
To create a wavepacket where the additional photons are identical to the predominant single photon component, we build another effective source where we add a small fraction of photons from the same QDSPS generated at a later time. This is obtained by performing an unbalanced quantum interference between two photon pulses produced by the QDSPS with delay $\tau$. The first half waveplate (HWP) and polarizing beam splitter (PBS) in Figure \[Fig\_LimitingCases\](c) allow us to tune the relative intensity of the predominant single photon pulse and the additional photons. Then, a pair of QWP and HWP is used to make the polarizations of both photons identical before the 50:50 beam splitter (BS). Most of the time, only the main photon gets to the second BS and is transmitted with 50% probability. However, when this photon meets a second one generated after a $\tau$ delay and temporally overlapped, they will undergo HOM interference and exit the beam splitter in the same output port. Therefore, the output of the second BS has a higher ${\mathit{g}^{(2)}}$, since there is a small probability that some of the output pulses now contain two identical photons. By adjusting the splitting ratio at the first beam splitter, the ${\mathit{g}^{(2)}}$ of the output state can be controlled. Figure \[Fig\_LimitingCases\](d) presents the HOM visibility as we increase the ${\mathit{g}^{(2)}}$ via addition of identical photons, where a clear difference is observed compared to the previous limiting case. For $ M_\mathrm{sn} = M_\mathrm{s}$, the model predicts $V_\mathrm{HOM} = M_\mathrm{s} - {\mathit{g}^{(2)}}$, a linear dependence with slope of $-1$. The line in Figure \[Fig\_LimitingCases\](f) again demonstrates that the model gives a very good fit to the data, with an extracted $M_\mathrm{s} = 0.89 \pm 0.01 $.
We note that the extracted values of $M_\mathrm{s}$ for these two cases represent the upper and lower bound of the intrinsic single photon indistinguishability of the QDSPS used in these measurements. If the non-zero ${\mathit{g}^{(2)}}$ of the QDSPS was due to distinguishable noise then we could deduce that $M_\mathrm{s}=0.94 \pm 0.02 $. Similarly if the noise was identical then the QDSPS has a single photon indistinguishability of $M_\mathrm{s} = 0.89 \pm 0.01 $. This demonstrates that it is necessary to know the origin of the unwanted photon emission in order to be able to extrapolate the data back to ${\mathit{g}^{(2)}}= 0$.
Our study highlights the importance of determining the origin of the multi-photon component in order to properly extract the single photon indistinguishability. To the extent of our knowledge, this has so far only been done in the indistinguishable case, independent of the physical phenomena of the multi-photon components. In the following, we discuss how to properly estimate the single photon indistinguishability for the current highest performing single photon sources, i.e. QD based sources.
There are two distinct categories of QDSPS, depending on the charge state of the quantum dot: neutral excitons and charged excitons (hereafter referred to as exciton and trion states respectively). The optical selection rules and photon emission processes differ significantly between the excitons and trions [@Ollivier2020], leading to a different origin of the multi-photon component.
![ [(a) Time traces of the single photon wavepacket emitted by exciton (upper) and trion (lower) based sources. The excitation laser pulse is shown in grey. (b) Measured ${\mathit{g}^{(2)}}$ for trion (purple squares) and exciton (blue circles) based sources as a function of the excitation pulse duration [at $\pi-$pulse]{}. The error bars are within the size of the plotted points. ]{} \[Fig\_Exciton\_Trion\_Lifetime\]](Figure2.pdf){width="\linewidth"}
For an exciton, the system is described by a three level system where the excitation pulse creates a superposition of the two excitonic linear dipoles with an energy difference given by the fine-structure splitting [@Bayer2002]. This results in a time dependent phase between the two exciton eigenstates, so that the single photon emission in cross-polarization beats with a period determined by the fine structure splitting [@Lenihan2002; @Ollivier2020], as shown in Figure \[Fig\_Exciton\_Trion\_Lifetime\](a). These optical selection rules imply that the single photon emission in cross polarisation is delayed with respect to the excitation pulse. For a trion based source, the optical selection rules correspond to a four level-system with four possible linearly polarized transitions [@Xu2007]. In a cross-polarised set-up, this system behaves like an effective two-level system, and the single photon emission shows a rapid rise time and monoexponential decay as shown in Figure \[Fig\_Exciton\_Trion\_Lifetime\](a).
These optical selection rules result in very different origins of the residual two-photon component for the two types of sources. To illustrate this, we measure the ${\mathit{g}^{(2)}}$ at maximum emitted brightness for two QD sources, one exciton and one trion, whilst increasing the temporal length of the excitation pulse (Figure \[Fig\_Exciton\_Trion\_Lifetime\](b)). For the exciton source, the single photon purity remains very high for pulse durations up to 80 ps, whereas the single photon purity rapidly degrades for longer pulses for the trion based source. For the trion, the single photon emission process is fast and can occur during the laser pulse so that there is a probability that the quantum dot returns to the ground state before the end of the laser pulse and gets excited again, leading to the emission of a second photon [@Fischer2017]. For the exciton, the delayed emission in cross polarization means that the probability of collecting two photons via re-excitation is very small, and the measured value of ${\mathit{g}^{(2)}}$ for the exciton remains small for pulse durations of up to 80 ps. We notice that for both excitons and trions, the ${\mathit{g}^{(2)}}$ is higher for very short pulses. This is because the power required to reach maximum emitted brightness ($\pi$-pulse) increases as the pulse duration decreases [@Giesz2016]. This implies that in the very short pulse regime ($ < 10$ ps) , the ${\mathit{g}^{(2)}}$ is limited by imperfect suppression of the excitation laser. This remains the dominant source of an imperfect ${\mathit{g}^{(2)}}$ for exciton sources up to a pulse duration of 80 ps, whereas trion sources are limited by re-excitation for pulses longer than 15 ps.
To correctly extract the single photon indistinguishability for each type of QDSPS it is critical to account for these different origins of the multi-photon component. To do so, we experimentally increase the multi-photon component by adjusting the main parameter that is responsible for multi-photon emission in each type of source, and then measure the impact this has on the visibility of HOM interference. Specifically, we increase the probability of re-excitation for a trion source and the amount of laser photons for the exciton source.
For the exciton based source, we add laser photons to the single photon emission from the QDSPS by turning the quarter waveplate (QWP) of the excitation pulse (see Figure \[Fig\_LimitingCases\](a)). This means that the excitation pulse is no longer aligned along one of the polarisation axes of the cavity, and the light will experience a small amount of polarisation rotation due to the birefringence of the cavity. Therefore, some fraction of the excitation pulse will now be collected in the orthogonal polarisation with the single photons. By adjusting the QWP we can add more laser photons and increase the ${\mathit{g}^{(2)}}$ of this effective source and measure the corresponding impact on HOM interference, as shown in Figure \[Fig\_TrionExciton\](a). The added noise photons from the laser are separable from the single photons and therefore Equation \[Eqn\_Model\] can be used to model the data. From the time traces in Figure \[Fig\_Exciton\_Trion\_Lifetime\](b), we can calculate that there is very little overlap between the laser photons and the single photons emitted by the quantum dot so that $M_\mathrm{sn} \approx 0$. The line in Figure \[Fig\_TrionExciton\](a) corresponds to a fit using $V_\mathrm{HOM} = M_\mathrm{s} - (1 + M_\mathrm{s} ) {\mathit{g}^{(2)}}$ from which we extract as a single parameter the single photon indistinguishability $M_\mathrm{s} = 0.920 \pm 0.003 $.
![ [ (a) Measured HOM visibility as a function of the ${\mathit{g}^{(2)}}$ for a exciton source, as the suppression of the excitation laser is worsened to increase the ${\mathit{g}^{(2)}}$. (b) Measured HOM visibility as a function of the ${\mathit{g}^{(2)}}$ for a trion source as the pulse duration is increased. For longer pulse durations, there is more re-excitation and the ${\mathit{g}^{(2)}}$ is higher. Each data point was taken at “$\pi$-pulse", corresponding to maximum brightness of the source. In both plots, the solid line gives the theoretical prediction for these data. The error bars are within the size of the plotted points.]{} \[Fig\_TrionExciton\]](Figure3.pdf){width="\linewidth"}
For the trion based source, since the imperfect ${\mathit{g}^{(2)}}$ arises from re-excitation, the assumption of separable noise does not hold because the emission of the first and second photon are time correlated. However, the extra photon must be emitted during the laser pulse for re-excitation to occur [@Fischer2017], whereas the main single photon emission typically takes place after the laser pulse with the trion radiative decay time of approximately 170 ps. As a result the noise photon, while emitted by the QD, presents a very similar temporal profile to the laser photon and can be modelled as a temporally separated noise with $M_\mathrm{sn} = 0$. The validity of this analysis is verified by increasing the pulse duration to increase the probability of re-excitation. The ${\mathit{g}^{(2)}}$ and HOM visibility are measured for different pulse durations from 15 ps to 50 ps at the power that maximises the emitted count rate. The results shown in Figure \[Fig\_TrionExciton\](b) are modelled well using $V_\mathrm{HOM} = M_\mathrm{s} - (1 + M_\mathrm{s}) {\mathit{g}^{(2)}}$ with a single parameter $M_\mathrm{s} = 0.942 \pm 0.004$.
To summarize, we find that, despite their different physical origins, the multi-photon component of both exciton and trion based QDSPSs can be treated as separable distinguishable noise. In the limit of low ${\mathit{g}^{(2)}}$, the single photon indistinguishability can thus be obtained using:
$$M_\mathrm{s } = \frac{ V_\mathrm{HOM } + {\mathit{g}^{(2)}}}{1 - {\mathit{g}^{(2)}}}.\label{Eqn_CorrectionFactor}$$
This correction factor can be applied to any QDSPS with a small ${\mathit{g}^{(2)}}$ and fast excitation pulse, in order to extract the intrinsic single photon indistinguishability, $M_\mathrm{s}$, given a measurement of ${\mathit{g}^{(2)}}$ and $V_\mathrm{HOM}$. The more general case where the HOM beam splitter has an intensity reflectivity $R$ and transmission $T$ is given in the Supplementary Material.
Interestingly, Equation \[Eqn\_CorrectionFactor\] results in a higher single-photon indistinguishability than the one obtained using the identical noise model. It is therefore likely that many of the values of single photon indistinguishability that have been quoted in the literature [@Senellart2017] for QD sources are in fact an underestimate of the true value. Whilst this does not circumvent the fact that it is the overall wavepacket indistinguishability that is crucial for quantum technologies, this deepened understanding of the Hong-Ou-Mandel experiment allows for a better diagnosis regarding imperfect single photon sources. The single-photon indistinguishability, $M_\mathrm{s}$, gives the upper bound to the indistinguishability that could be achieved with an ideal experimental set–up, with no laser leakage for example, and therefore it fundamentally quantifies how temporally coherent the source itself is. Finally, we note that our simple theoretical approach is only valid for separable noise, and further studies are needed to understand the effect of a non-separable noise on the interference.
In conclusion, we have theoretically and experimentally revisited the emblematic Hong-Ou-Mandel interference. This experiment is commonly implemented to test the indistinguishability of single particles including single photons, single plasmons, single electrons or single atoms [@Martino2014; @Bocquillon1054; @Lopes2015]. We believe that the new insight brought by our study will benefit these fundamental studies as well as the development of single photon sources, allowing a better diagnosis on the current limitations.
*Acknowledgements* This work was partially supported by the ERC PoC PhoW, the French Agence Nationale pour la Recherche (grant ANR QuDICE), the IAD - ANR support ASTRID program Projet ANR-18-ASTR-0024 LIGHT, the QuantERA ERA-NET Cofund in Quantum Technologies, project HIPHOP, the French RENATECH network, a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” programme (Labex NanoSaclay, reference: ANR-10-LABX-0035). J.C.L. and C.A. acknowledge support from Marie SkłodowskaCurie Individual Fellowships SMUPHOS and SQUAPH, respectively. H. O. Acknowledges support from the Paris Ile-de-France Région in the framework of DIM SIRTEQ. S.C.W. and C.S. acknowledge support from NSERC (the Natural Sciences and Engineering Research Council), AITF (Alberta Innovates Technology Futures), and the SPIE Education Scholarships program.
HOM visibility for unentangled states
=====================================
In this section, we derive the general relationship between the total mean wavepacket overlap of two interfering states, the single-photon purity, and the HOM visibility. Suppose we have a beam splitter with input modes $\hat{a}_1$ and $\hat{a}_2$ and output modes $\hat{a}_3$ and $\hat{a}_4$ monitored by single-photon detectors. The output modes can be described by the relation $$\label{bsrelation}
\begin{pmatrix}
\hat{a}_3(t)\\
\hat{a}_4(t)
\end{pmatrix}
=
\begin{pmatrix}
\cos\theta&-e^{-i\phi}\sin\theta\\
e^{i\phi}\sin\theta&\cos\theta
\end{pmatrix}
\begin{pmatrix}
\hat{a}_1(t)\\
\hat{a}_2(t)
\end{pmatrix},$$ provided that the beam splitter interaction is constant for the relevant frequency range of the input fields. The coincident events of the detectors at the output are determined from the two-time intensity correlation between the output fields $G_{34}^{(2)}(t,\tau)=\braket{\hat{a}^\dagger_3(t)\hat{a}^\dagger_4(t+\tau)\hat{a}_4(t+\tau)\hat{a}_3(t)}$, where we restrict $\tau\geq0$ so that the detector monitoring mode $4$ clicks after the detector monitoring mode $3$. The integrated $G_{34}^{(2)}$ around zero delay normalized by the total intensity gives the probability $p_{34}$ of having a coincident count. The HOM interference visibility is then defined as $V_\text{HOM}=1-2p_{34}$. By following the methods of Ref. [@Imamoglu2004] to compute $p_{34}$ for two unentangled input states, while also keeping terms associated with multi-photon contributions, the HOM visibility is given by $$V_\text{HOM} = \frac{2RT\left[\left(1-g_1^{(2)}\right)\mu_1^2+2M_{12}\mu_1\mu_2+\left(1-g_2^{(2)}\right)\mu_2^2\right]}{(T\mu_1+R\mu_2)(T\mu_2+R\mu_1)}-1,$$ where $R=\sin^2\theta$ is the beam splitter reflectivity, $T=\cos^2\theta$ is the beam splitter transmittance, $$g_i^{(2)} = \frac{2\iint G_{ii}^{(2)}(t,\tau)d\tau dt}{\mu_i^2}=\frac{2\iint\braket{\hat{a}^\dagger_i(t)\hat{a}^\dagger_i(t+\tau)\hat{a}_i(t+\tau)\hat{a}_i(t)}d\tau dt}{\mu_i^2}$$ is the integrated intensity correlation around zero delay for input $i$ normalized by the time-integrated mean photon number $\mu_i = \int\braket{\hat{a}^\dagger_i(t)\hat{a}_i(t)}dt$ and the mean-wavepacket overlap is $$M_{ij} = \frac{2\iint\text{Re}\left(\braket{\hat{a}^\dagger_i(t+\tau)\hat{a}_i(t)}^*\braket{\hat{a}^\dagger_j(t+\tau)\hat{a}_j(t)}\right)d\tau dt}{\mu_i\mu_j}.$$
For a balanced interferometer where the input intensities of the final beam splitter are equal, we have that $\mu_1=\mu_2$ and so the relation for HOM visibility simplifies to $$V_\text{HOM} = 4RT\left(M_{12}+1-\overline{g}^{(2)}\right)-1,$$ where $\overline{g}^{(2)}=(g_1^{(2)}+g_2^{(2)})/2$ is the average $g^{(2)}$ of the interfering states. For HOM interference between identical wavepackets, $M_{12} = M_{11}=M_{22}= M_\text{tot}$ is the total mean wavepacket overlap of the source and $\overline{g}^{(2)}=g^{(2)}_1=g^{(2)}_2=g^{(2)}$ quantifies the source single-photon purity. In the ideal case where $R=T=1/2$, the above relation reduces to the equation $V_\text{HOM}=M_\text{tot}-g^{(2)}$ given in the main text.
HOM visibility in a separable noise model
=========================================
We now develop a theoretical model to describe the visibility of HOM interference for a specific type of imperfect source. This imperfect single photon source is modeled by adding noise to an ideal single photon using a beam splitter interaction, as shown in Figure \[Fig\_BS\]. Here the noise is separable and exhibits no entanglement with the single photon. For simplicity, we also model the noise by another single photon, which is valid in the limit of a weak noise field or, equivalently, when $g^{(2)}$ is small.
![ [ An imperfect photon is modelled by adding separable noise to a perfect single photon at a beam splitter ]{} \[Fig\_BS\]](FigureS1.pdf){width="0.3\linewidth"}
The initial state of the signal ($s$) and noise ($n$) is given by $\hat{\rho} =\hat{\rho}_{s} \otimes \hat{\rho}_{n} $ where $\hat{\rho}_i=p_{i,0}\ket{0}\bra{0}+p_{i,1}\hat{\rho}_{i,1}$ and $$\hat{\rho}_{i,1} = \iint \xi_i(t,t') \hat{a} ^\dagger _i (t)\ket{0} \bra{0} \hat{a} _i (t') \, \mathrm{d}t \mathrm{d}t' \nonumber,$$ where $\xi_i(t,t')$ is the normalized single-photon temporal density wavefunction for $i \in \{ s,n\} $.
The imperfect photon in the output transmission mode of the beam splitter $\hat{a}_t$ is obtained by tracing out the reflected loss mode $\hat{a}_r$ after applying the beam splitter relation, so that $\hat{\rho}_t = \mathrm{Tr}_r(\hat{\rho}_{s} \otimes \hat{\rho}_{n}) $, where $$\begin{pmatrix} \hat{a}_t(t) \\ \hat{a}_r(t) \end{pmatrix} = \begin{pmatrix} \cos\vartheta & -\sin\vartheta \\ \sin\vartheta & \cos \vartheta \end{pmatrix} \begin{pmatrix} \hat{a}_s(t) \\ \hat{a}_n(t) \end{pmatrix}.$$
The total state of the imperfect single photon can then be written as $\hat{\rho}_t = p_{0}\ket{0}\bra{0} + p_{1}\hat{\rho}_{t,1} +p_{2}\hat{\rho}_{t,2} $, where $\hat{\rho}_{t,j}$ is the density matrix for the transmitted state containing $j$ photons. The total $g^{(2)}$ and $\mu$ can be computed directly from the photon number probabilities by $\mu = p_1 + 2p_2$ and $g^{(2)}=2p_2/\mu^2$. For the average photon number, we have $\mu = p_{s,1}\cos^2\vartheta+p_{n,1}\sin^2\vartheta$ and $g^{(2)}$ is given by $$\label{separableg2}
\begin{aligned}
\mu^2g^{(2)} &= 2p_{s,1}p_{n,1}(1+M_\mathrm{sn})\cos^2\vartheta\sin^2\vartheta,
\end{aligned}$$ where $M_\mathrm{sn} = \iint \mathrm{Re} ( \xi_s (t,t') \xi_n^*(t,t') ) \mathrm{d} t \mathrm{d} t'$ is the mean wavepacket overlap of the single photon and noise, and $\vartheta$ quantifies the amount of noise.
After applying a propagation phase $\phi_i$ to each of the photon density wavefunctions, the two-time amplitude correlation of $\hat{\rho}_t$ is given by $$\braket{\hat{a}^\dagger_t(t^\prime)\hat{a}_t(t)}=p_{s,1}\cos^2\vartheta\xi_s(t,t^\prime)e^{i\phi_s(t-t^\prime)}+p_{n,1}\sin^2\vartheta\xi_n(t,t^\prime)e^{i\phi_n(t-t^\prime)}$$ and so the total mean wavepacket overlap is given by $$\label{separableoverlap}
\begin{aligned}
\mu^2M_\text{tot} &= \iint\left|\braket{\hat{a}^\dagger_t(t^\prime)\hat{a}_t(t)}\right|^2dt^\prime dt\\
&= p_{s,1}^2M_\mathrm{s}\cos^4\vartheta + p_{n,1}^2 M_n\sin^4\vartheta + 2p_{s,1}p_{n,1}M_\mathrm{sn}^\prime\cos^2\vartheta\sin^2\vartheta,
\end{aligned}$$ where $M_{s} = \iint\left|\xi_s(t,t^\prime)\right|
^2 dtdt^\prime = \text{Tr}\left[\hat{\rho}_{s,1}^2\right]$ quantifies the intrinsic single-photon indistinguishability, or single-photon trace purity [@Fischer2018; @Trivedi2020], of the source. Here we have that $$M_\mathrm{sn}^\prime = \iint \text{Re}\left(\xi_s(t,t^\prime)\xi_n^*(t,t^\prime)e^{i(\phi_s-\phi_n)(t-t^\prime)}\right) dtdt^\prime$$ is not necessarily the same as $M_\mathrm{sn}$ due to the potential relative propagation phase $\phi_s-\phi_n$.
We can reparametrize the expressions for $\mu$, $g^{(2)}$, and $M_\text{tot}$ by defining $\eta$ so that $\cos^2\!\eta=(p_{s,1}\cos^2\vartheta)/\mu$ and $\sin^2\!\eta=(p_{n,1}\sin^2\vartheta)/\mu$. The fact that this reparametrization exists stems from the independence of $M_\text{tot}$ and $g^{(2)}$ from photon loss. It also implies that the fundamental quantity affecting the photon statistics of this imperfect single photon model is $\eta$, which depends on both the beam splitter angle $\vartheta$ and the relative input intensities through $p_{s,1}$ and $p_{n,1}$. Using the relation for $V_\text{HOM}$ from the previous section, equations (\[separableg2\]), and (\[separableoverlap\]), the visibility and $g^{(2)}$ in terms of the noise parameter $\eta$ are $$\begin{aligned}
V_\text{HOM}(\eta) &= 4RT\left(1+M_\mathrm{s}\cos^4\eta+M_n\sin^4\eta -2(1+M_\mathrm{sn}-M_\mathrm{sn}^\prime)\cos^2\eta\sin^2\eta\right)-1\\
g^{(2)}(\eta) &= 2(1+M_\mathrm{sn})\cos^2\eta\sin^2\eta.
\end{aligned}$$ The value of $\eta$ can be modified by changing the intensity of the noise $p_{n,1}$ as was done for the emulated distinguishable noise source in the main text. It can also be modified by changing the relative proportions of $p_{n,1}$ and $p_{s,1}$ using an unbalanced Mach-Zehnder interferometer, as was done for the emulated identical noise source in the main text.
In our study, we are interested in the slope and intercept of the parametric curve formed by $\{g^{(2)}(\eta),V_\text{HOM}(\eta)\}$. The solution for the intercept is clear since $g^{(2)}(\eta)=0$ implies $\eta=0$ and $V_\text{HOM}(0) = 4RT(1+M_\mathrm{s})-1$. To solve for the slope at small $\eta$, we have $$\begin{aligned}
\lim_{\eta\rightarrow 0}\frac{dV_\text{HOM}(\eta)}{dg^{(2)}(\eta)} &= -4RT\left(\frac{1+M_\mathrm{s} + (M_\mathrm{sn}-M_\mathrm{sn}^\prime)}{1+M_\mathrm{sn}}\right).\end{aligned}$$
For the cases we are interested in, either distinguishable noise or if $\xi_s=\xi_n$, we have $M_\mathrm{sn}-M_\mathrm{sn}^\prime\simeq 0$. This case would also be true if $M_\mathrm{sn}$ and $M_\mathrm{sn}^\prime$ were phase averaged. Under these conditions, the HOM visibility for small $g^{(2)}$ is given by $$\label{vhomRT}
V_\text{HOM} = 4RT\left(1+M_\mathrm{s}-\left(\frac{1+M_\mathrm{s}}{1+M_\mathrm{sn}}\right)g^{(2)}\right)-1.$$
In the case where the noise is distinguishable so that $M_\mathrm{sn}=0$, the single-photon indistinguishability $M_\mathrm{s}$ can be determined by rearranging equation (\[vhomRT\]): $$M_\mathrm{s } = \frac{ V_\mathrm{HOM}+4RT\left(1+{\mathit{g}^{(2)}}\right)-1}{4RT(1 - {\mathit{g}^{(2)}})}$$
For $R=T=1/2$, we recover equations (1) and (2) presented in the main text.
Measuring ${\mathit{g}^{(2)}}$ and HOM interference
===================================================
To measure the single photon purity we perform a Hanbury Brown-Twiss experiment and measure the coincidences between the two outputs of a 50:50 beam splitter. The experimental set-up, and a typical coincidence histogram are shown in Figure \[Fig\_g2HOM\_setup\_data\](a). The second-order autocorrelation is given by ${\mathit{g}^{(2)}}= A_0 / A_\mathrm{uncor}$, where $A_0$ is the area of the coincidence peak at zero time delay and $A_\mathrm{uncor}$ is the average area of the uncorrelated peaks.
We perform a Hong-Ou-Mandel interference experiment by splitting the train of single photons at a beam splitter and delaying one arm by the pulse separation time, $\tau$. Two subsequently emitted photons then interfere at a 50:50 beam splitter, as shown in Figure \[Fig\_g2HOM\_setup\_data\](b). The visibility of HOM interference is given by $V_\mathrm{HOM} = 1 - 2 A_0 / A_\mathrm{uncor}$.
![ [ The experimental setup (left) and raw data (right) for measuring (a) ${\mathit{g}^{(2)}}$ and (b) HOM visibility. ]{} \[Fig\_g2HOM\_setup\_data\]](FigureS2.pdf){width="0.53\linewidth"}
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abstract: 'Why are classifiers in high dimension vulnerable to “adversarial" perturbations? We show that it is likely not due to information theoretic limitations, but rather it could be due to computational constraints. First we prove that, for a broad set of classification tasks, the mere existence of a robust classifier implies that it can be found by a possibly exponential-time algorithm with relatively few training examples. Then we give a particular classification task where learning a robust classifier is computationally intractable. More precisely we construct a binary classification task in high dimensional space which is (i) information theoretically easy to learn robustly for large perturbations, (ii) efficiently learnable (non-robustly) by a simple linear separator, (iii) yet is not efficiently robustly learnable, even for small perturbations, by any algorithm in the statistical query (SQ) model. This example gives an exponential separation between classical learning and robust learning in the statistical query model. It suggests that adversarial examples may be an unavoidable byproduct of computational limitations of learning algorithms.'
author:
- |
Sébastien Bubeck\
Microsoft Research\
`[email protected]`\
Eric Price\
UT Austin\
`[email protected]`\
Ilya Razenshteyn\
Microsoft Research\
`[email protected]`\
bibliography:
- 'bib.bib'
title: Adversarial examples from computational constraints
---
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---
abstract: 'With rapid adoption of deep learning in high-regret applications, the question of when and how much to trust these models often arises, which drives the need to quantify the inherent uncertainties. While identifying all sources that account for the stochasticity of learned models is challenging, it is common to augment predictions with confidence intervals to convey the expected variations in a model’s behavior. In general, we require confidence intervals to be well-calibrated, reflect the true uncertainties, and to be sharp. However, most existing techniques for obtaining confidence intervals are known to produce unsatisfactory results in terms of at least one of those criteria. To address this challenge, we develop a novel approach for building calibrated estimators. More specifically, we construct separate models for predicting the target variable, and for estimating the confidence intervals, and pose a bi-level optimization problem that allows the predictive model to leverage estimates from the interval estimator through an *uncertainty matching* strategy. Using experiments in regression, time-series forecasting, and object localization, we show that our approach achieves significant improvements over existing uncertainty quantification methods, both in terms of model fidelity and calibration error.'
author:
- 'Jayaraman J. Thiagarajan$^{\dagger}$, Bindya Venkatesh$^{\ddagger}$, Prasanna Sattigeri$^+$, Peer-Timo Bremer$^{\dagger}$ $^{\dagger}$Lawrence Livermore National Labs, $^{\ddagger}$Arizona State University , $^+$MIT-IBM Watson AI Lab'
bibliography:
- 'paper.bib'
title: Building Calibrated Deep Models via Uncertainty Matching with Auxiliary Interval Predictors
---
|
---
abstract: 'In this paper, we study the exploration / exploitation trade-off in cellular genetic algorithms. We define a new selection scheme, the centric selection, which is tunable and allows controlling the selective pressure with a single parameter. The equilibrium model is used to study the influence of the centric selection on the selective pressure and a new model which takes into account problem dependent statistics and selective pressure in order to deal with the exploration / exploitation trade-off is proposed: the punctuated equilibria model. Performances on the quadratic assignment problem and NK-Landscapes put in evidence an optimal exploration / exploitation trade-off on both of the classes of problems. The punctuated equilibria model is used to explain these results.'
author:
- |
David Simoncini, Sébastien Verel\
Philippe Collard, Manuel Clergue\
date:
title: |
Centric Selection: a Way to Tune\
the Exploration/Exploitation Trade-off
---
Introduction
============
The exploration/exploitation trade-off is an important issue in evolutionary computation. By tuning the selective pressure on the population, one can find an optimal (or near-optimal) tradeoff between exploitation and exploration. In cellular Evolutionary Algorithms (cEAs), the population is embedded on a bidimensional toroidal grid and each solution interacts with its neighbors thanks to a certain neighborhood. The convergence rate of the algorithm is then dependent of the shape and size of the grid and of the neighborhood. The smallest symetric neighborhood that can be defined is the well-known Von Neumann neighborhood of radius $1$. It guarantees a slow isotropic diffusion of genetic information through the grid. But when solving complex multimodal problems, it is necessary to slow down even more the propagation speed of the best solution because the algorithm still often converges over a local optimum.
Our goal in this paper is to establish a relation between the selective pressure on the population and the effects of recombination and mutation operators, in order to find an optimal exploration/exploitation trade-off. To do so, we propose a new selection scheme able to control the selective pressure and a theoretical model which takes into account the effects of stochastic variations on an optimization problem. In section 2 we define a selection scheme able to tune the selective pressure and present the algorithm used in the experiments. In section 3, we analyze the selective pressure with respect to the selection method and present a new model which takes into account the stochastic variations. In section 4 we present performances on Quadratic assignment problem instances and on NK-Landscapes and we explain the results with the model proposed.
Cellular Evolutionary Algorithms
--------------------------------
A cellular Evolutionary Algorithm (cEA) [@Whitley93] is an EA in which the population is embedded on a bidimensionnal toroidal grid (see figure \[cea\]). Each cell of the grid contains a solution. Embedding the solutions on a grid allows defining a neighborhood between the cells. The most commonly used one in cEAs is the Von Neumann neighborhood (shown on figure \[cea\]). At each generation, every cell on the grid is updated by selecting parents in its neighborhood and applying stochastic operators such as crossover and mutation. Several strategies exist, synchronous and asynchronous, to update the cells. The small overlapped neighborhoods guarantee the diffusion of solutions through the grid [@SpiessensM91]. Such algorithms are especially well suited for complex problems [@JongS95] and are of advantage when dealing with dynamic problems [@Tomassini05].
Selective pressure
------------------
One of the main properties that differs between EAs and cEAs is the rate of convergence (propagation speed of the best solution) : It is exponential for EAs and quadratic for cEAs. Therefore, the selective pressure on the population is weaker for a cEA than for an EA. Controlling the selective pressure is critical since it can avoid premature convergence of the algorithm when solving complex multimodal problems. Several parameters related to the selective pressure can prevent the algorithm from getting stuck in a local optimum. The topology of the grid, the local neighborhood, the properties of the selection operator are such parameters. By correctly tuning these parameters for a given problem, one can find a good exploration/exploitation tradeoff and minimize the risks of premature convergence. Sarma *et al.* established a link between the radius of the neighborhood and the radius of the grid : changing this ratio directly affects the selective pressure on the population [@SarmaJ96]. Alba *et al.* analyzed performances of cEAs with a fixed size neighborhood and different grid shapes. They arrived to the conclusion that thin grids are well-suited for complex multi modal problems and large grids are well-suited for simple problems. The main explanation is that thinner grids give lower selective pressure [@AlbaT00]. Takeover times and growth curves analysis are useful to measure the selective pressure on a population, but it is not sufficient to decide of a trade-off between exploration and exploitation: it is necessary to include effects of the stochastic variations due to the operators in the analysis. Janson *et al.* proposed a hierarchical cEA which allows achieving different levels of exploration /exploitation tradeoff in distinct zones of the population simultaneously [@Janson06].
A standard technique to study the induced selective pressure without introducing the perturbing effect of variation operators is to let selection be the only active operator, and then monitor the number of best solution copies in the population [@GoldbergD91]. The takeover time is the time needed for one single best solution to colonize the population with selection as the only active operator. Let $\lambda$ be the size of the population, $t$ the number of generations and $N(t)$ the number of best solution copies at generation $t$. The population is initialized with one solution of good fitness and $\lambda - 1$ solutions of null fitness. Since no other evolution mechanism but selection takes place, the good fitness solution spreads over the grid. The takeover time is then the smaller time $t$ such that $N(t)=\lambda$. Analysing the growth of $N(t)$ as a function of $t$ also gives an indication on the selective pressure. It shows the convergence rate of the algorithm when selection is the only active operator. When the slope of the growth curve of $N(t)$ is low, the convergence rate is low and the takeover time is high. On the other hand, a high slope of the growth curve leads to a short takeover time. In the first case, the selective pressure on the population is weaker than in the second case.
Growth curves and takeover time models
--------------------------------------
Characterizing the growth curves and the takeover time is an important issue in the study of the selective pressure [@GoldbergD91]. Many models have been proposed to define the behaviour of structured population evolutionary algorithms. Sarma and De Jong proposed a logistic model in which the coefficient of the growth curve of the best solution is shown to be an inverse exponential of the ratio between radii of the neighborhood and the underlying grid [@SarmaJ96]. This conclusion was guided by an empirical analysis of the effects on the convergence rate and the takeover time of several neighborhood sizes and shapes. Sprave proposed a hypergraph based model of population structures and a method for the estimation of growth curves and takeover times based on the probabilistic diameter of the population [@Sprave99]. Gorges-Schleuter proposed a study about takeover time and growth curves for cellular evolution strategies. She obtained a linear model for ring populations and a quadratic model for a torus population structure [@Gorges99]. Several authors wrote about theoretical or empirical models of growth curves and takeover time. Giacobini *et. al.* proposed a model for cellular evolutionary algorithms with asynchronous update policies [@Giacobini03]. He summarized his results and proposed models for synchronous updates [@Giacobini05b] that will be evoked later in this paper. Alba proposed a model for distributed evolutionary algorithms consisting in the sum of logistic definition of the component takeover regimes [@Alba04b]. In his paper, he made an interesting review of existing models and compared two of them (the logistic model and the hypergraph model) with his newly proposed one. For a detailed state of the art of cEAs, see [@alba08].
Centric selection
=================
In this section we present a new selection scheme for cEAs that allows tuning accurately the selective pressure.
**CentricSelection**[index: int, $\beta$: double]{} neighbors $\longleftarrow$ GetNeighborhood(index) $candidate_1$ $\longleftarrow$ Select(neighbors, $\beta$) $candidate_2$ $\longleftarrow$ Select(neighbors, $\beta$) Best($candidate_1$, $candidate_2$)
The centric selection (CS) idea is to change the probability of selecting the center cell of the neighborhood. This scheme allows slowing down the convergence speed while keeping an isotropic diffusion of good solutions through the grid. The CS is a determinist tournament selection. But unlike the standard deterministic tournament, cells in the neighborhood may have different probabilities of being selected for the competition. The anisotropic selection [@Gecco06] is another selection scheme which modifies the probability of selection a cell for a deterministic tournament. With the anisotropic selection, the diffusion of solutions is not isotropic, so we propose the CS which is easier to study. We have $p_c = \beta$ the probability of selecting the center cell and $p_n = p_s = p_e = p_w = \frac{1}{4} (1-\beta)$ the probability of selecting either north, south, east or west cell. When $\beta = \frac{1}{5}$, all cells have the same probability of being selected for the competition: this particular case of CS is the standard binary tournament selection. When $\beta=1$, only the center cell can be selected for the tournament: in this particular case where the same solution is selected two times, the crossover operator is not applied in the cEA. Only mutations are applied to the solution, and with an elitist replacement strategy, the algorithm behaves as the parallelisation of as many hill climbers as there are solutions in the population. The CS is described in algorithm \[centricAlgo\]. The candidates compete in a deterministic tournament returning the best one. For each cell on the grid, two parents are selected per generation, as we can see in the algorithm \[cEAlgo\]. Stochastic variations operators are applied to the parents, generating two children. The replacement strategy is elitist: the best child replaces the current solution on the grid if it has a better fitness. The use of a temporary grid is necessary for a synchronous update of the cells.
**cEA**[population: vector, $\beta$: double]{}
tempGrid: vector $parent_1$ $\longleftarrow$ CentricSelection(i, $\beta$) $parent_2$ $\longleftarrow$ CentricSelection(i, $\beta$)
($child_1$, $child_2$) $\longleftarrow$ Crossover($parent_1$,$parent_2$);
Mutate($child_1$) Mutate($child_2$)
tempGrid\[i\] $\longleftarrow$ Best($population[i]$, $child_1$, $child_2$) Replace(population, tempGrid)
Modeling cEAs
=============
In this section, we present two models of the search dynamic in cEAs. In the first one, the Equilibrium Model (EM) we consider that the optimal solution has been found and observe how it colonizes the grid. This model is classical in the studies on the selective pressure and one the exploration/exploitation trade-off. The informations given by this model are takeover times and best solution growth curves. As the stochastic variations operators are not taken into account, the same dynamic occurs in experimental runs when the recombination and mutation operators are ineffective: when the system has reached an equilibrium.
In the second one, we consider that a better solution can be found with a certain probability and observe the frequency of apparition of this new solution with respect to our algorithm’s parameters. It is a model of the transition between two periods of fitness stability. We call this new model the Punctuated Equilibria Model (PEM).
Equilibrium model
-----------------
In order to measure the selective pressure induced by the CS, we observe what happens when no more solution improvement is possible. In this case, crossover and mutation are no longer useful and the evolution process has reached an equilibrium. Hence, we observe the time needed for a single best solution to conquer the whole grid, and look at the growth curve obtained and the takeover time.
We measure the effects of CS on selective pressure by observing these growth curves and takeover times on a square grid of side $64$. Figure \[fig-tak-fuzzy\] shows the takeover time as a function of $\beta$. The takeover time is not defined for $\beta=1$. The selective pressure drops when the value of $\beta$ increases. We can see on figure \[growthPC\] the growth of the number of copies of the best solution in the population (top) and its growth rate (bottom). There are two stages in the shape of the curve. The growth rate is linear in the first part and quadratic in the second part. When using this selection scheme, the diffusion of the best solution is still isotropic. So the best solution roughly propagates describing an obtuse square as long as no side of the grid is reached. This corresponds to the first part of the growth rate curve. Once the sides are reached by some copies of the best solution, the dynamic changes as we can observe on the second part of the growth rate curves.
[c]{}\
\
Punctuated equilibria model
---------------------------
In this section, we propose a new model which will help in the understanding of the search dynamics of an Evolutionary Algorithm. This model was first designed for a cellular EA but can be easily extended to any kind of evolutionary algorithm. We consider a cEA initialized with random solutions. We make sure that the best solution in the population is unique. Our goal is to simulate an evolutionary run: We simulate recombination and mutation operators with probabilities that the mating is efficient or not (i.e. produces a new best solution). We consider three different types of matings: between two copies of the best solution (mating $11$), between one copy of the best solution and one sub-optimal solution (mating $01$) and between two sub-optimal solutions (mating $00$). We introduce probabilities $P_{11}$, $P_{01}$ and $P_{00}$ that matings of type $11$, $01$ and $00$ produce a new best solution, fitter than the previous best one. Figure \[run\] is an example of evolutionary run on some optimization problem (minimisation task). We can see that there are some stagnation periods where the best solution don’t improve. Then, an amelioration occurs and the population enters another stability period. An evolutionary run is a sum of stagnation periods and punctual improvements. Our punctuated equilibria model computes the probability of improving the best solution in the population according to the variables described above.
With this model, the probability of finding a new best solution at a given generation $t$ is : $$p(t) = 1 - (1-P_{00})^{n_{00}(t)}(1-P_{01})^{n_{01}(t)}(1-P_{11})^{n_{11}(t)}$$ where $n_{00}(t)$, $n_{01}(t)$ and $n_{11}(t)$ are the number of matings of each type for the generation $t$.
The average time to find a new best solution is given by : $$E = \sum_{t \geq 1} t p(t)$$ The performance of an algorithm can be measured by the time $E$ needed to find a new best solution but also by the probability $P$ of improvement in a preset time $T$. We have the probability of improving the best solution in $T$ generations :
$$\begin{array}{rcl}
P & = & 1 - \prod_{t=1}^T ( 1 - p(t) ) \\
P & = & 1 - ( 1 - P_{00} )^{\Sigma_{00}(T)} ( 1 - P_{01} )^{\Sigma_{01}(T)} ( 1 - P_{11} )^{\Sigma_{11}(T)} \\
\end{array}$$ with $\Sigma_{ij}(T) = \sum_{t=1}^{T} n_{ij}(t)$ the sum over $T$ of mating of each type.
The parameters $P_{ij}$ are problem dependent and the values of $\Sigma_{ij}$ are given by the selection scheme used. The selection process is usually controlled by a parameter such as the tournament size or in the case of the CS: $\beta$. This parameter should be used to maximize the probability[^1] $P$. Intuitively, the ideal selection process maximizes the $\Sigma_{ij}$ which have the higher $P_{ij}$. More precisely, assuming that the control parameter of the selection process is $\beta$, the parameter $\beta^{*}$ which maximizes the probability $P(T)$ verifies:
$$\begin{array}{rcl}
\frac{dP}{d\beta}(\beta^{*}) & = & 0 \\
\end{array}$$
which gives:
$$\label{eq-optimal}
\begin{array}{ll}
& \log ( 1 - P_{00} ) \frac{\partial \Sigma_{00}}{\partial\beta}(\beta^{*}) \\
+ & \log ( 1 - P_{01} ) \frac{\partial \Sigma_{01}}{\partial\beta}(\beta^{*}) \\
+ & \log ( 1 - P_{11} ) \frac{\partial \Sigma_{11}}{\partial\beta}(\beta^{*}) = 0 \\
\end{array}$$
If it is possible to have a model of $\Sigma_{ij}(\beta)$, it would be possible to calculate the optimal $\beta$ as a function of $P_{ij}$.
In this model, the exploration/exploitation tradeoff is given by the number of each possible matting ($00$, $01$ and $11$). The model could be used to explain the probability and the time to find a new best solution according to the selective pressure, and also to tune the value of parameters which have an impact on the selective pressure, such as $\beta$, to have the highest probability to evolve toward a new best solution. Equation \[eq-optimal\] gives precisely the best exploration and exploitation tradeoff and allows computing the optimal value of $\beta$ (in our case) for this trade-off. In the following, we will show the validity of the PEM on some optimization problems.
QAP and NK landscapes
=====================
In this section, we study the effect of selective pressure on performances through experiments of a cEA with CS on two well-known classes of problems. The optimal exploration / exploitation tradeoff found will be explained thanks to the PEM presented in the previous section.
Problems presentation
---------------------
The problems proposed, Quadratic Assignment Problem and NK landscapes, are known to be difficult to optimize. The important number of instances of the Quadratic Assignment Problem and the tunable parameters of the NK landscapes allow managing the difficulty of the problems.
### Quadratic Assignment Problem
This section presents the Quadratic Assignment Problem (QAP) which is known to be difficult to optimize. The QAP is an important problem in theory and practice as well. It was introduced by Koopmans and Beckmann in 1957 and is a model for many practical problems [@Koopmans57]. The QAP can be described as the problem of assigning a set of facilities to a set of locations with given distances between the locations and given flows between the facilities. The goal is to place the facilities on locations in such a way that the sum of the products between flows and distances is minimal. Given $n$ facilities and $n$ locations, two $n \times n$ matrices $D=[d_{kl}]$ and $F=[f_{ij}]$ where $d_{kl}$ is the distance between locations $k$ and $l$ and $f_{ij}$ the flow between facilities $i$ and $j$, the objective function is :\
$$\Phi = \sum_{i}\sum_{j}d_{p(i)p(j)}f_{ij}$$ where $p(i)$ gives the location of facility $i$ in the current permutation $p$. Nugent, Vollman and Ruml proposed a set of problem instances of different sizes noted for their difficulty [@Nugent68]. The instances they proposed are known to have multiple local optima, so they are difficult for an evolutionary algorithm. The best algorithm known is the fast hybrid evolutionary algorithm [@Mise06] which combines an evolutionary algorithm with an improvement of the fast tabu search of Taillard.
### Set up {#set-up .unnumbered}
We use a population of 400 solutions placed on a square grid ($20\times 20$). Each solution is reprensented by a permutation of $N$ where $N$ is the size of a solution. The algorithm uses a crossover that preserves the permutations:
- Select two solutions $p_1$ and $p_2$ as genitors.
- Choose a random position $i$.
- Find $j$ and $k$ so that $p_1(i) = p_2(j)$ and $p_2(i) = p_1(k)$.
- exchange positions $i$ and $j$ from $p_1$ and positions $i$ and $k$ from $p_2$.
- repeat $N/3$ times this procedure where $N$ is the size of an solution.
This crossover is an extended version of the UPMX crossover proposed in [@Migkikh]. The mutation operator consists in randomly selecting two positions from the solution and exchanging those positions. The crossover rate is 1 and we do a mutation per solution. We perform 200 runs for each tuning of the two selection operators. An elitism replacement procedure guarantees the solutions stay on the grid if they are fitter than their offspring.
### NK landscapes
The NK landscapes were proposed by Kaufmann to model the boolean network and used in optimisation in order to explore how epistasis is linked to the ruggedness of search spaces [@Kauffman93]. Epistasis corresponds to the degree of interactions between the “loci” of a solution and ruggedness is the number of local optima of the search space. The main characteristic of NK Landscapes is that they allow tuning the epistasis level with a single parameter $K$. The parameter $N$ determines the length of the solutions.
The fitness of solutions for a NK landscape is given by the function $$f : \lbrace 0,1 \rbrace ^N \rightarrow [0,1]$$ defined on binary strings of length $N$. Each binary string is a solution with $N$ locations. An *atom* with fixed epistasis level is represented by a fitness component $$f_i : \lbrace 0,1 \rbrace ^{K+1} \rightarrow [0,1]$$ associated to each bit $i$. It depends on the value of the bit $i$ and on the value of $K$ other bits of the string ($K$ must fall between $0$ and $N-1$). The fitness $f(x)$ of $x \in \lbrace 0,1 \rbrace ^N$ is the average of the values of the $N$ fitness components $f_i$ : $$f(x) = \frac{1}{N}\sum_{i=1}^N f_i(x_i,x_{i1},...,x_{ik})$$ where $ \lbrace i_1,...,i_k \rbrace \subset \lbrace 1,...,i-1,i+1,...,N \rbrace $. Many ways have been proposed to choose the $K$ other locations from the $N$ of the solutions. The mainly used ones are adjacent and random neighborhoods. With the first one, the $K$ nearest locations of the location $i$ are chosen (the solution is taken to have periodic boundaries). With the random neighborhood, $K$ locations are randomly selected from the solution. Each fitness component $f_i$ is specified by extension, *ie* a random number $y_{i,(x_i,x_{i1},...,x_{ik})}$ from $[0,1]$ is associated with each element $(x_i,x_{i1},...,x_{ik})$ from $\lbrace 0,1 \rbrace^{K+1}$. Those numbers are uniformly distributed in the interval $[0,1]$.\
### Set up {#set-up-1 .unnumbered}
The size of the population is $400$ solutions. The crossover used is a one point crossover, applied with a probability of $1$. The mutation is a bit flip applied with a probability of $\frac{1}{n}$ where $n$ is the size of a solution. We perform $200$ runs for every parameter set, and each run stops after $1500$ generations. Runs are performed on instances of sizes $N=32$, with $K \in { 2..12 }$.
Performances
------------
Figure \[nug30perf\] and table \[otherQAP\] show performances of a cEA using CS on some QAP instances of various sizes. The instance in figure \[nug30perf\] is a well-known instance of size $30$. The first fact that we notice when looking at these results is that there is an optimal setting, different from the extreme values $0$ and $1$ for the parameter $\beta$. This indicates that for a certain setting of the parameters, and thus for a certain selective pressure, the search dynamic leads to optimal results. Curves representing the instances summarized in table \[otherQAP\] have the same shape as figure \[nug30perf\]. On each instance, the optimal value of $\beta$ is around $0.85$. The performances increase up to these values and then decrease. Performances of CS are significantly better than the one obtained with a cEA using standard binary tournament selection. The standard cEA is observable on the curve at the points $\beta = 0.2$ and is reported in the table \[otherQAP\]. We can also notice in table \[otherQAP\] that the standard deviation is lower for the optimal value of $\beta$ than with a cEA with binary tournament selection.
Instance Std cGA Best avg. results Opt. $\beta$
---------- ----------------------------------- ----------------------------------- --------------
Nug30 $6178_{[ 28 ]}$ $6144_{[ 14 ]}$ $0.88$
Tai40a $3.23 \times 10^{6}_{[ 14343 ]}$ $3.21 \times 10^{6}_{[ 12000 ]}$ $0.84$
Sko42 $15969_{[ 75 ]}$ $15909_{[ 34 ]}$ $0.82$
Tai50a $5.092 \times 10^{6}_{[ 20721 ]}$ $5.080 \times 10^{6}_{[ 13372 ]}$ $0.82$
Tai60a $7429118_{[ 27760 ]}$ $7385390_{[ 19391 ]}$ $0.86$
: Avg. results and std.dev.on QAP instances[]{data-label="otherQAP"}
Figure \[nk32-10\] and table \[otherNK\] present performances of a cEA with CS on some instances of NK landscapes. Parameters of the landscapes are $N=32$ and $K=10$ for the figure \[nk32-10\] and are summarized in table \[otherNK\] for the other instances.
We can see that the shape of the performances’ curve is different from the QAP curve. The performance increases until $\beta$ reaches its maximum value. The same results are obtained for all the instances in table \[otherNK\]. The parameter $K$ tunes the difficulty of the instance. We can see that for $K=2$, there is no optimal value for $\beta$. The reason is that the optimum is always found. For $K=4$, the standard cEA sometimes get stuck in a local optimum, and with $\beta=1$ our algorithm always find the optimum. On every instance, except $K=2$, the optimal value for $\beta$ is $1$.
However, this value $\beta=1$ is a particular one, since it breaks all communications on the grid. As long as the value of the parameter increases, the chances of selecting two different solutions for recombination decrease. For $\beta = 1$, the algorithm is the parallelisation of as much hill climbers as there are cells on the grid : It constantly selects the center cells of the neighborhoods, so there is no crossover and any amelioration is due to a bit flip.
So the best setting for CS can be compared to the parallelisation of as many hill climbers as there are cells on the grid. The parallelisation of hill climbing seems to be a good algorithm for solving NK landscapes problems, which could be explained by the size of basins of attraction [@Verel08] and [@Verel08b].
$K$ Std cGA Best avg. results Best $\beta$
------ ----------------------- ------------------------ --------------
$2$ $0.734329_{[ 0 ]}$ $0.734329_{[ 0 ]}$ $[ 0,1 ]$
$4$ $0.79597_{[ 0.003 ]}$ $0.798197_{[ 0 ]}$ $1$
$6$ $0.782934_{[ 0.01 ]}$ $0.799124_{[ 0.003 ]}$ $1$
$8$ $0.771277_{[ 0.01 ]}$ $0.789103_{[ 0.004 ]}$ $1$
$10$ $0.763510_{[ 0.01 ]}$ $0.785115_{[ 0.003 ]}$ $1$
$12$ $0.750043_{[ 0.01 ]}$ $0.774479_{[ 0.009 ]}$ $1$
: Avg. performances and std.dev.on NK instances with $N=32$[]{data-label="otherNK"}
Probabilities of discovering better solutions
---------------------------------------------
In order to explain the optimal values of $\beta$ for QAP and NK-Landscape and to validate the PEM, we compute $P$, the probability of discovering a new best solution in the population taken from the PEM, with real data. We calculated it for one instance of QAP and one instance of NK-Landscapes. With this calculation we want to find the value of $\beta$ that maximizes the probability of discovering a new solution. This probability depends on the value of $\Sigma_{ij}$, and thus on time: if at a generation $t$ no new solution is discovered, the actual best solution spreads in the population according to the selective pressure. If during an interval of time corresponding to the takeover time no new solution is discovered then the population converges. We can compute the ideal $\beta$ value for a given number of generations $T$ because $\Sigma_{ij}( T )$ relies on $\beta$ and on time: after $T$ generations $\Sigma_{ij}( T )$ is different according to $\beta$, and for the optimal value of $\beta$ $\Sigma_{ij}( T )$ leads to the best probability $P$.
We estimated the $\Sigma_{ij}$ with the same experiments done to compute growth curves and takeover time. We averaged the number of matings of each type at each generation over $10^{3}$ runs. Then, we needed to know the probabilities $P_{00}$, $P_{01}$ and $P_{11}$. We estimated these probabilities using a Bayesian process during the runs. We averaged the values obtained by generations over $500$ runs. Figure \[nug30probas\] shows the result of the estimation of probabilities on the QAP instance Nug30. The ordonate scale is logarithmic because of the variations of probabilities. The curves representing the $P_{ij}$ intersect, so the value of $\beta$ which maximizes $P$ may change during a run. We computed $P$ with estimated values of $P_{ij}$ taken by steps of $50$ generations. The values of $\Sigma_{ij}$ are also generation dependent. For each value of $\beta$, we took the $\Sigma_{ij}$ value after $100$ generations: that is $\Sigma_{ij}(100)$. During a run, it would correspond to allowing a stagnation period of $100$ generations before stopping the run, which is reasonnable.
The figure \[nug30opt\] shows the optimal values of $\beta$ as a function of generations for the QAP instance Nug30. During the first $700$ generations, the optimal value is $\beta=0.2$. Then, there is a transition of approximatively $150$ generations. During this phase, the ideal value of $\beta$ grows until it reaches $1$. In our experiments, we observe optimal values of $\beta$ between $0.8$ and $0.9$ according to the QAP instance. Values of $\beta$ are constant during the runs. But the PEM shows that the selective pressure should be strong at the beginning (low values of $\beta$) and then weak (high values of $\beta$). If $\beta$ is constant, intermediate values in the range $[ 0.8, 0.9 ]$ give the best average selective pressure for QAP instances.
The figure \[nkopt\] shows the optimal values of $\beta$ as a function of generations for a NK-Landscape with $N=32$ and $K=10$. We can see that the optimal $\beta$ value increases fastly and reaches $1$ in the early generations. The ideal selective pressure is weak, and it is not surprising that the best performances are obtained when $\beta=1$ in our experiments. Figure \[nkprobas\] shows the estimation of $P_{ij}$ as a function of generations. We can see that the curve representing $P_{11}$ drops down very fast. With a negligible probability of improving the current best solution with mutations, there is no sense in spreading this solution. With $\beta=1$, the current best solution in the population cannot spread.
The PEM has been used in order to explain the exploration / exploitation trade-off on two different classes of problems. Coupled with the centric selection, it showed the ideal selective pressure along the search process. This model can be used to tune any parameter which has some influence on the number of matings of each type defined in the previous section. The computation cost is low, since the estimation of probabilities by a Bayesian process is precise: we averaged the estimation on $500$ runs but the standard deviation was low ($\approx 10^{-6}$).The $\Sigma_{ij}$ are only computed once since they are independent from the optimization problem tackled.
Conclusion
==========
The exploration/exploitation trade-off is an important issue in evolutionary algorithms. In this paper, we propose a model that takes into account stochastic variations and improvement of the quality of the solutions, the punctuated equilibria model. In order to study the exploration / exploitation trade-off we propose a tunable selection operator: the centric selection. By monitoring the probability of selecting the center cell of neighborhoods for a tournament selection, this selection operator allows tuning accurately and continuously the selective pressure with one single parameter ($\beta$). The performance results on QAP instances and NK-Landscapes showed different optimal settings of the centric selection, and thus different ideal selective pressures. Using the punctuated equilibria model, we put in evidence the optimal values of the centric selection’s control parameter observed on QAP instances and NK-Landscapes. The punctuated equilibria model also put in evidence that the ideal selective pressure is not constant during the search process in the case of QAP instances.
In this paper, we used the PEM in order to explain experimental results. In future works, we will use it in order to predict optimal exploration / exploitation trade-offs and to adapt the selective pressure during the runs. To do so, we will both estimate the $P_{ij}$ and tune the selection operator online during the search process. The centric selection will be used in auto-adaptative algorithms with the advantage of modifying the exploration / exploitation ratio with a single parameter. It will also be applied on real problems and compared to other optimization methods.
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[^1]: In the following equations, we only denote the dependance on $\beta$ for $P$ and $\Sigma_{ij}$ for readability.
|
---
abstract: 'We present a first attempt to apply the approach of deformation quantization to linearized Einstein’s equations. We use the analogy with Maxwell equations to derive the field equations of linearized gravity from a modified Maxwell Lagrangian which allows the construction of a Hamiltonian in the standard way. The deformation quantization procedure for free fields is applied to this Hamiltonian. As a result we obtain the complete set of quantum states and its discrete spectrum.'
author:
- Hernando Quevedo
- 'Julio G. Tafoya'
title: Towards the deformation quantization of linearized gravity
---
Introduction {#sec1}
============
One of the main differences between the states of a classical system and those of a quantum system consists in that the latter cannot be represented as points in phase space. In the canonical quantization approach this fact is taken into account by considering states as eigenfunctions of operators which act on Hilbert space. The eigenvalues of the operators are then interpreted as quantum observables. So, one of the main steps of the canonical quantization formalism consists basically in replacing the classical observables of the physical system by operators. Commutation relations are then imposed on the operators in order to be in agreement with Heisenberg’s uncertainty relation. Although this procedure is widely used in quantum mechanics and quantum field theory, it is still far from being completely understood. In particular, one expects that in certain limit the quantum system reduces to the classical one. This is usually done by applying the correspondence principle, according to which the quantum commutator must lead to the classical Poisson bracket when Planck’s constant vanishes. However, it is known that in general this limit is not well-defined and inconsistent [@gro].
One of the main successes of deformation quantization consists in providing the correct implementation of the correspondence principle. Indeed, deformation quantization avoids the use of operators and, instead, concentrates on the algebra of classical observables (see [@wein; @zac; @waldmann] for recent reviews, and [@hhajp] for an elementary review.) Instead of the usual (classical) point multiplication between observables, a star-product is introduced that takes into account the non-local character of quantum observables and reduces to the classical Poisson bracket in the appropriate limit. Whereas the classical observables build a commutative algebra with respect to the point product, the same set forms a noncommutative algebra with respect to the star-product. Thus, it is not necessary to introduce new entities (operators) instead of the classical observables. The quantum observables are represented by the same functions on phase space as the classical observables. The first applications of deformation quantization concerned non-relativistic quantum mechanics [@bayen], but this situation has changed dramatically in the past few years. This formalism has found many uses in perturbative and nonperturbative quantum field theory [@dito; @kon; @dut; @bhw; @cat; @hugo1; @hhqft], quantum gravity [@ant], as well as in string theory [@hugo2; @minic].
In this work we present a first attempt to apply the main concepts of deformation quantization to the case of linearized gravity. Our approach consists basically in representing Einstein’s linearized equations as a field theory of a metric on the background of Minkowski spacetime. This approach allows us to apply in this case the procedure of deformation quantization developed for free fields. Our approach is also appropriate for getting rid of the concern regarding the quantization of a quantity (the perturbation metric) which is first considered as infinitesimal in order for Einstein’s linearization to be valid. Indeed, intuitively one expects that quantum effects become important only when the gravitational field is high enough, and the perturbation metric in the standard approach does not behave this way. In the field theoretical approach the metric is arbitrary and linearized Einstein’s equations follow from a variational principle. The corresponding Lagrangian is regular and the Hamiltonian turns out to be equivalent to the Hamiltonian of an infinite sum of harmonic oscillators.
This paper is structured as follows. In Section \[sec2\] we derive Einstein’s linearized equations in the standard manner and present an alternative field theoretical approach based upon the analogy between electrodynamics and linearized gravity. In Section \[sec3\], we review the main aspects of deformation quantization and calculate explicitly the set of quantum states and the energy spectrum of linearized gravity. Finally, in Section \[conclusions\] we discuss our results.
Linearized gravity {#sec2}
==================
In this section we first review the standard approach of linearized gravity in which the field equations are obtained by linearizing Einstein’s equations. Then we present an alternative approach based on the usual variational procedure of field theory and obtain the corresponding Hamiltonian.
Linearized Einstein’s equations {#seclee}
-------------------------------
In most textbooks on general relativity, linearized gravity is considered at the level of the field equations. Indeed, in the usual approach to gravity, one starts from the Einstein-Hilbert action [@wald] S= R d\^4 x + \_m \_m d\^4x , \[eh\] where $R$ is the curvature scalar associated to the metric $g_{\alpha\beta}, \ (\alpha,\beta,... =0,1,2,3)$ of spacetime, $\alpha_m$ is a coupling constant and ${\cal L}_m$ is the Lagrangian density that represents the matter contents in spacetime. The variation of (\[eh\]) with respect to the metric yields the Einstein equations R\_ -[12]{} g\_ R = 8T\_ , \[ein\] where the energy-momentum tensor is defined in terms of the variational derivative as T\_=-[\_m8]{}[1]{} [\_m g\^]{} . \[emt\] In the weak field approximation of linearized gravity one imposes the metric $g_{\alpha\beta}=
\eta_{\alpha\beta} + h_{\alpha\beta}$ such that $h_{\alpha\beta}
<< \eta_{\alpha\beta}$ is an infinitesimal perturbation of the background Minkowski metric, $\eta_{\alpha\beta}$. In particular, one can choose a Cartesian-like coordinate system in which $\eta_{\alpha\beta}={\rm diag}(-,+,+,+)$ and $|h_{\alpha\beta}|<<1$. If we now consider the first order approximation of the left-hand side of (\[ein\]) with respect to $h_{\alpha\beta}$, and impose the Lorentz gauge condition h \^\_[ ,]{} = 0 , h\^ = h\^ - [12]{}\^ h , h=\_h\^ \[lor\] then Einstein’s equations reduce to h \_[,]{}\^[ ]{} = - 16 T\_ . \[lin\]
A field theoretical approach {#secfield}
----------------------------
An alternative approach for deriving the linearized equations (\[lin\]), in which the particular aspects of a field theory become more plausible, consists in using the analogy between Maxwell’s equations of electromagnetism and linearized Einstein’s equations. To show this analogy explicitly let us consider a non-zero vector $U^\alpha$ and define A\_=- [14]{} h\_U\^ , J\_=-T\_U\^ . \[def\] Calculating the D’Alembertian $A_{\alpha,\gamma}^{\ \ \ \gamma}$, one can see immediately that the Maxwell equations A\_[,]{}\^[ ]{} = - 4 J\_ , \[max\] are equivalent to the linearized Einstein equations (\[lin\]) if the components of the arbitrary vector $U^\alpha$ satisfy the equations h \_U\^[ ]{}\_[ ,]{} + 2 h \_[,]{} U\^[,]{} = 0 . \[con1\] On the other hand, the Lorentz gauge condition (\[lor\]) turns out to be equivalent to the condition $A^\alpha_{\ ,\alpha}=0$ if the equation h \_U\^[,]{} = 0 , \[con2\] is satisfied. In this way we see that the linearized Einstein equations can be written in the Maxwell-like form (\[max\]) (in the Lorentz gauge) by introducing the additional vector $U^\alpha$ which is required to satisfy the conditions (\[con1\]) and (\[con2\]). For a given metric $\overline h _{\alpha\beta}$, these conditions represent a system of four second order partial differential (\[con1\]) and one first order partial differential equation (\[con2\]) for the four components $U^\alpha$. So one should guarantee the existence of solutions to this system for the Maxwell-like representation (\[max\]) to be valid. Obviously, the trivial vector $U^\alpha=$ const satisfies these requirements. In principle, more general solutions might exist, but in this work we will restrict ourselves to this special case since it is sufficient for our purposes.
On the other hand, it is well known that Maxwell equations (\[max\]) can be obtained by varying the Lagrangian (density) \_[Max]{}= -[14]{} F\_F\^ + 4 A\_J\^ , F\_= A\_[,]{}-A\_[,]{} , \[lmax\] with respect to the potential $A_\alpha$. Let us now try to construct the corresponding Hamiltonian formalism. The configuration variables are given by the set of components $A_\alpha$. For the corresponding canonically conjugate momenta we obtain $\Pi_\alpha =
\partial {\cal L}_{Max} /
\partial \dot A_\alpha = F_0^{\ \alpha}$, where a dot denotes the derivative with respect to the time coordinate $x^0$. Then $\Pi_0 = 0$ and consequently we have a singular Lagrangian from which a Hamiltonian cannot be constructed. In field theory the quantization of such Lagrangians is performed by using Dirac’s method for systems with constraints (see, for instance, [@gt]). But in the context of deformation quantization this method is still under construction [@dqconst]. In the case of Maxwell’s theory, however, an alternative approach exists [@bs] that consists in modifying the original Maxwell Lagrangian according to (we take $J_\alpha=0$ for simplicity) = -[14]{} F\_F\^ -[12]{} (A\_\^[ ,]{})\^2 . \[lmod0\] The field equations are again $A_{\alpha,\beta}^{\ \ \ \beta} =0$ and the Lorentz gauge condition $A_\alpha^{\ \ ,\alpha}=0$ has to be postulated separately. After some algebraic manipulations, the Lagrangian (\[lmod0\]) can be rewritten as = -[12]{} A\_[,]{}A\^[,]{} +[12]{} \^\_[ ,]{} , \^= A\^\_[ ,]{} A\^ - A\^A\^\_[ ,]{} . \[lmod\] The second term can be neglected as it can be transformed after integration into a surface term that does not contribute to the field equations. But the main point about the Lagrangian (\[lmod\]) is that it is regular. In fact, it can easily be seen that $\Pi_\alpha
= \dot A_\alpha$ and the corresponding Hamiltonian is = [12]{} (\_\^+A\_[, i]{} A\^[, i]{} ) . \[ham\] Then the canonical variables of the phase space satisfy the canonical commutation relations with respect to the Poisson bracket: {A\_, \_}= \_ , {A\_, A\_}={\_, \_} = 0 . We will use the Hamiltonian (\[ham\]) in the context of deformation quantization in section \[sec3\]. Finally, let us mention that the Lagrangian (\[lmod\]) is invariant with respect to the transformation A\_A’\_= A\_+ \_[,]{} , \_[,]{}\^[ ]{} = 0 . \[freedom\] This is a special gauge transformation that can be used to eliminate non-physical degrees of freedom.
The main point of this alternative approach is that we now can “forget” that ${\cal L}$ is an approximate Lagrangian and proceed as in standard classical field theory. That is, (\[lmod\]) can be interpreted as the Lagrangian for the metric field $h_{\alpha\beta}$ which is defined on the Minkowski spacetime with metric $\eta_{\alpha\beta}$. So we are dealing with a standard field theory in which the background Minkowski metric does not interact with the field $h_{\alpha\beta}$ that now can be completely arbitrary, i.e. it is not necessarily an infinitesimal perturbation of the background metric. In this manner we can avoid the concern mentioned in the introduction about the quantization of an infinitesimal quantity.
Deformation quantization {#sec3}
========================
The classical description of the evolution of a physical system is usually represented in the phase space $\Gamma$, which is a manifold of even dimension. If a (non-degenerate) symplectic two-form $\alpha$ exists on $\Gamma$, then the phase space is called a symplectic space. Observables are real valued functions defined on the phase space: $f,g : \Gamma \rightarrow R$. With respect to the usual point multiplication $(fg)(x) = f(x)g(x)$, where $x =(x^1, x^2, ... x^{2n})$ is a set of coordinates on (an open subspace of) $\Gamma$, the observables build a commutative algebra. The symplectic structure $\alpha$ allows us to introduce the Poisson bracket of observables as $\{f,g\}(x) = \alpha^{ab}\partial_a f(x) \partial_b g(x)$, where $\partial_a \ (a = 1,2,... 2n)$ is the (covariant) derivative in $\Gamma$. With respect to the Poisson bracket the set of observables build a Lie-Poisson algebra. The equations of motion in phase space acquire a particular symmetric form in terms of the Poisson brackets $\dot x ^a= \{ x ^a, {\cal H}\}$, a relationship which is valid for any function of phase space coordinates.
In the canonical approach to quantization one replaces the observables by (self-adjoint) operators which act on the Hilbert space. The physical states are vectors in the Hilbert space. Poisson brackets are replaced by commutators which, when applied to the operators associated with the basic observables in phase space, satisfy the canonical commutation relations. Despite its great success especially in the perturbative approach to the physics of elementary particles, this procedure is still not completely understood. The passage from functions to operators is one important step in the canonical approach and despite many efforts done to explain it, today the best way to avoid all kind of existence proofs and mathematical difficulties is just to assume it as a postulate.
Deformation quantization is essentially an attempt to avoid the passage from functions to operators. In fact, it focuses on the algebra of observables of the phase space and replaces the usual point product of functions by a star-product. The canonical commutation relations are now a consequence of the definition of the star-product. An important advantage of this procedure is that quantum as well classical observables are functions defined on the phase space and no operators are required. From the mathematical point of view, the deformation quantization of a given classical system consists in giving an appropriate definition of the star-product which acts (on functions defined) on the phase space. In physics, however, to understand a quantum system one needs to know its quantum states and their energy spectrum. To this end, deformation quantization postulates the existence of a time-evolution function, Exp($Ht$), which satisfies the differential equation [@hhqft] i (Ht) = H \* [Exp]{}(Ht) , \[schroe\] where $H$ is the Hamiltonian of the classical system. Moreover, it is assumed that the time-evolution function allows a Fourier-Dirichlet expansion as (Ht) = \_E \_E e\^[-itE/]{} , \[exp\] where $E$ is the energy (a real number) associated with the state $\pi_E$ (distribution on the phase space), or Wigner function, which satisfies the so called \*-genvalue equation H\*\_E = E \_E . \[eigen\] The states are idempotent and complete. i.e.: \_E\*\_[E’]{} = \_[E,E’]{} \_E , \_E \_E = 1 .
As a consequence, the spectral decomposition of the Hamiltonian is give as H = \_E E \_E . Essentially, the objects that are necessary for carrying out the deformation quantization of a physical system are the [*classical*]{} Hamiltonian $H$ and the \*-product. For a given phase space it is not clear [*a priori*]{} if a consistent \*-product exists or not and, for a general phase space, this is still an open problem [@waldmann]. In the case of free (non-interacting) fields that can be considered heuristically as the sum of an infinite number of harmonic oscillators, it has been shown [@dito] that the normal star-product is the only admissible star-product. The normal \*-product between two functions $f$ and $g$ on phase space is defined by f\*\_N g = e\^[N\_[12]{}]{} f(a\^[(1)]{},a \^[(1)]{}) g(a\^[(2)]{},a \^[(2)]{})|\_[a\^[(1)]{}=a\^[(2)]{}=a]{} , N\_[12]{} = \_[ij]{}[a\_i\^[(1)]{} ]{} [a\_j\^[(2)]{} ]{} , \[norpro\] where the superscritpts (1) and (2) denote two arbitrary points in phase space and $a = (a_1, a_2, ..., a_n)$. Furthermore, an overline denotes complex conjugation. The set of phase space variables has to satisfy the standard commutation relations with respect to the Poisson bracket, i.e. $\{a_i,\overline a _j\} = \delta_{ij}, \ \{a_i, a _j\}=
\{\overline a_i,\overline a _j\}=0.$ In particular one can choose $a_j=1/\sqrt{2} (
x_j+ix_{n+j}), (j=1,...,n)$, where we are assuming that the configurational variable $x_j$ and its conjugate momentum $x_{n+j}$ have been normalized to come out with the same units.
Let us now apply this procedure to the linearized theory. According to the results given in section \[sec2\], the canonical variables in the phase space of linearized gravity are the potentials $A_\alpha=-(1/4)\overline h _{\alpha\beta} U^\beta$ and their canonical momenta $\Pi_\alpha =\dot A_\alpha$. The Hamiltonian is given by H =[12]{}d\^3x (\_\^ +A\_[,i]{}A\^[,i]{}) . \[hamc\] The main step in the quantization procedure consists in solving Eq.(\[schroe\]) by using the normal \*-product (\[norpro\]). To this end it is convenient to change from the variables of phase space $(A_\alpha, \Pi_\alpha)$ to a new set of canonically conjugate variables in which the Hamiltonian takes the simplest possible form. This procedure is very well known in field theory and consists in introducing the momentum representation of the phase variable $A_\alpha$ according to [@bs] A\_(x)= [1(2)\^[3/2]{}]{} , \[momentum\] where $k$ is a null vector $k_\mu k^\mu = -k_0^2 +{\bf k}^2 =0, $ and $kx=k_\mu x^\mu$. Then \_=A\_= [i(2)\^[3/2]{}]{}d[**k**]{} , \[pis\] and A\_[, j]{} = [i(2)\^[3/2]{}]{} k\_j . \[aaa\] From the commutation relations for $A_\alpha$ and $\Pi_\beta$ it can be shown that {a\_, a\_} = \_ , {a\_, a\_} = {a\_, a\_} = 0 , where the same value of ${\bf k}$ has been assumed in all the arguments. Introducing (\[pis\]) and (\[aaa\]) into the Hamiltonian (\[hamc\]) and performing some of the integrations we obtain H=[12]{} k\_0 \^ a\_([**k**]{})a\_([**k**]{}) d[**k**]{} \[hamas\] We see that the resulting Hamiltonian is linear in the new variables and does not contain derivatives. It can be interpreted as an infinite sum of harmonic oscillators. This is an important observation [@dito; @hugo1; @hugo2] that allows us to formally apply the normal \*-product as defined in (\[norpro\]). In fact, when going from a system with a finite number of degrees of freedom to a field theory, one only has to “replace” partial derivatives by variational derivatives. We use this fact to calculate time-evolution function as the solution of Eq.(\[schroe\]). Then we have \_N (Ht) = [1]{}( e\^[-ik\_0t]{} -1) \^ a\_([**k**]{})a\_([**k**]{}) d[**k**]{} , \[tef\] where the subscript $N$ indicates that in Eq.(\[schroe\]) the normal \*-product has been used. Using the definition of the exponential of a functional, the last expression can be written as \_N (Ht) = ( -[1]{}\^ a\_([**k**]{})a\_([**k**]{}) d[**k**]{}) \_[n=0]{}\^\^ a\^n\_([**k**]{})a\^n\_([**k**]{}) d[**k**]{} . \[tef1\] Comparing this expression with the Fourier-Dirichlet expansion (\[exp\]) we can identify the corresponding states as \^N\_[E\_0]{} = (-[1]{} \^ a\_([**k**]{})a\_([**k**]{}) d[**k**]{}) , \[state0\] \^N\_[E\_n]{} = [1n! \^n]{} \^N\_[E\_0]{} \^ a\^n\_([**k**]{})a\^n\_([**k**]{}) d[**k**]{} , \[staten\] and the energy spectrum E\_n = nk\_0 . \[spectrum\] In this manner we have arrived at the main result of quantization: The determination of the quantum states and the energy spectrum of the system. The main advantage of deformation quantization consists in achieving this goal without using the operator formalism. Now we are confronted with the problem of finding the physical significance of our results. To this end, let us remember that at the classical level we have derived Einstein’s linearized equations directly from the Lagrangian (\[lmod\]), and have noticed that the Lorentz gauge conditions have to be postulated as an additional requirement. It seems therefore natural to impose this requirement on the quantum states to find out which of them are physical. From Eq.(\[momentum\]) we find that the Lorentz gauge conditions are equivalent to A\_\^[ ,]{} = [1(2)\^[3/2]{}]{} i k\^= 0 . \[qlor\] Clearly, this condition is identically satisfied if k\^a\_([**k**]{}) = 0 , \[ort\] what implies that only three components of $a_\alpha({\bf k})$ are linearly independent. Furthermore, the gauge freedom given by Eq.(\[freedom\]) implies that the harmonic function $\Sigma(x)$ can be used to eliminate an additional component of $a_\alpha({\bf k})$. So we are left with only two true components, say, $a_1({\bf k})$ and $a_2({\bf k})$. This is in accordance with the fact that gravitational fields possess only two physical degrees of freedom [@wald]. From Eqs.(\[state0\]) and (\[staten\]) we see that all states are specified as powers of $a_1({\bf k})$ and $a_2({\bf k})$. The results should not depend on the choice of these two linear independent components, but one can use these freedom to adapt the formalism to different physical situations. For instance, in the case of the Newtonian limit it seems reasonable to choose the Newtonian potential $\phi$ and one of the “gravitomagnetic” functions [@mtw], say $\gamma$, as independent configuration variables so that $A_\alpha = -(1/4)(4\phi, \gamma, 0, 0)$. In the case of gravitational waves a more suitable choice would be $A_\alpha= -(1/4)(0, \gamma_1, \gamma_2, 0)$ where $\gamma_1$ and $\gamma_2$ are now related to the special combination of the metric components that describe gravitational waves (see, for instance, [@mtw]).
Independently of the choice of gauge, the states and spectrum are represented by Eqs.(\[state0\]), (\[staten\]) and (\[spectrum\]). The coefficients $a_\alpha({\bf k})$ can be interpreted as densities that determine distributions in phase space, i.e. as state densities. But the formalism of deformation quantization allows a transition to the operator formalism according to certain fixed rules [@hhqft]. In that case, one would expect that the operator counterparts of $a_\alpha$ and $\overline a_\alpha$ would correspond to the annihilation and creation operators of standard quantum field theory. The energy spectrum (\[spectrum\]) is discrete with vanishing zero-point energy. If we would use the Moyal product for the quantization, we would obtain a non-vanishing zero point energy and would be confronted with the problem of divergencies that commonly appears in perturbative quantum field theory [@dito].
Conclusions
===========
The aim of this work was to apply the formalism of deformation quantization to linearized Einstein’s equations. We first show two alternative ways to consider Einstein’s linearized equations in a field theoretical approach. We then use the modified Maxwell representation of linearized gravity to calculate the classical Hamiltonian of the theory, avoiding the problem of a singular Lagrangian. The Hamiltonian is one of the main ingredients necessary to carry out the deformation quantization of any physical system. We use the normal star-product to derive the commutation relations, in analogy with other free (linear) fields. The expression for the time-evolution function is found explicitly, and the Fourier-Dirichlet expansion of the Hamiltonian is used to derive the energy spectrum and the complete set of states of the system. A more detailed analysis is necessary in order to clarify further the physical meaning of the states. We have used the momentum representation in analogy with the standard methods of quantum field theory. For this reason, the results of the quantization are more adapted to a possible interpretation in terms of elementary particles and not in terms of a possible quantization of space and time. This, however, is a much more complicated problem that requires a separate and detailed treatment.
Acknowledgments {#acknowledgments .unnumbered}
===============
It is a great pleasure to dedicate this work to Alberto García on his 60-th birthday. This work was in part supported by DGAPA-UNAM grant IN112401, CONACyT grant 36581-E, and US DOE grant DE-FG03-91ER40674. H.Q. thanks UC-MEXUS for support.
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|
---
abstract: |
Fermions on the lattice have bosonic excitations generated from the underlying periodic background. These , the lattice bosons , arise near the empty band or when
the bands are nearly full. They do not depend on the nature of the interactions and exist for any fermion-fermion coupling .We discuss these lattice boson solutions for the Dirac Hamiltonian.
author:
- '$J.Chakrabarti^{a,1},A.Basu^{a} and B.Bagchi^{b,2}$'
title: Lattice Bosons
---
=-2.0cm =-0.6cm
a\. Department of Theoretical Physics\
Indian Association for the Cultivation of Science\
Calcutta-700032 INDIA\
b. Department of Applied Mathematics\
University of Calcutta\
92 Acharya Prafulla Chandra Road\
Calcutta-700009 INDIA\
Keywords: bosonisation,algebraic structure,nearly empty lattice,insulator, Dirac Hamiltonian.
PACS No: 11.10Ef , 11.15Ha , 03.65Fd , 05.30Fk
1.Electronic address: [email protected]\
2.Electronic address: [email protected]\
Introduction
============
The fermions often have boson solutions. These bosons, made usually from the fermion degrees ,exist in many systems \[1\]. We present in this work the boson solutions that arise from the background lattice.They do not arise from the interactions, but are related to the spatial geometrical features, such as the periodicities, of the underlying lattice on which the fermions reside.Because of this geometrical origin these bosons, we believe, are robust and exist for any value of coupling.These,the lattice bosons,are the subject of this work.
It is worth recalling the many-body fermion states in low dimensions are eigenstates of boson operators.These ideas of bosonisation have received wide support over the years\[2-6\].
Here, in this work , we illustrate our ideas on the one-dimensional equispaced lattice. We find the fermion algebra that is represented by this lattice.This algebra of fermions shows the curious feature that for low values of fermion filling on the lattice some of the generators behave as bosons. Interestingly, the same is true in the “insulating” region where the fermion states are nearly full.The boson interpretations in these two limits,the nearly empty lattice(nel), and the nearly filled lattice(nfl),are different.The roles of the creation and the destruction operators are reversed.These interpretations do not depend on the underlying Hamiltonian or the nature of the fermion interactions.
The lattice bosons in the nel are made of coherently superposed fermion pairs.In the “insulating ” region, i.e. in the nfl,coherently superposed hole pairs create the lattice bosons.The usual Hamiltonian of the fermions is recast in terms of these bosons in the nel and the nfl regions.The new Hamiltonian in the lattice boson variables is diagonalised to get the boson spectrum.
To begin we introduce on the one-dimensional equispaced lattice , with periodic boundary conditions,the set of generators,made of fermionic objects, that close under commutations.This algebraic structure, represented by the background lattice, arises solely from the anticommuting properties of the fermions. It does not depend on any Hamiltonian or on the interactions.We illustrate the algebra on the simple 6-point lattice. Following from this algebraic structure we identify the generators that are bosonic.We show that for low values of fermion filling a set of generators satisfy bosonic commutation rules.As the filling increases the boson approximation gets worse.Interestingly ,near maximum filling , i.e. near the “insulating” domain the boson modes return,but now with the creation and the destruction operators reversing their roles. This leads us to interpret the bosons as coherent superposition of the hole pairs. With this structure in place we take up a simple Hamiltonian and look for its lattice boson solutions. Our choice is the Dirac Hamiltonian for fermions of mass.
We set up the lattice boson algebra and solve for the boson eigenstates. These are good solutions in the nel and the nfl regions.
The Lattice Boson Algebra
=========================
Consider the lattice of 2N equispaced points with periodic boundary conditions. Let $C_{n}^{\dagger}$ and $C_{n}$ denote the creation and the annihilation operators of the fermion on site denoted by the index n.Clearly, $$\{C_{n},C_{m}^{\dagger}\}=\delta_{nm}$$
$$\{C_{n}^{\dagger},C_{m}^{\dagger}\}=\{C_{n},C_{m}\}=0$$
Consider the generators, $$e_{+l}=\sum C_{n}^{\dagger}C_{n+l}^{\dagger},$$ where l takes values 1,2,3... For the lattice of 6 points the generators read : $$\begin{aligned}
e_{+1}=c_{1}^{\dagger}c_{2}^{\dagger}+c_{2}^{\dagger}c_{3}^{\dagger}+
c_{3}^{\dagger}c_{4}{\dagger}+c_{4}^{\dagger}c_{5}^{\dagger}+c_{5}^{\dagger}
c_{6}^{\dagger}+c_{6}^{\dagger}c_{1}^{\dagger}\\
e_{+2}=c_{1}^{\dagger}c_{3}^{\dagger}+c_{2}^{\dagger}c_{4}^{\dagger}
+c_{3}^{\dagger}c_{5}^{\dagger}+c_{4}^{\dagger}c_{6}^{\dagger}
+c_{5}^{\dagger}c_{1}^{\dagger}+c_{6}^{\dagger}c_{2}^{\dagger}\\
e_{+3}=c_{1}^{\dagger}c_{4}^{\dagger}+c_{2}^{\dagger}c_{5}^{\dagger}+c_{3}
^{\dagger}c_{6}^{\dagger}+c_{4}^{\dagger}c_{1}^{\dagger}+c_{5}^{\dagger}
c_{2}^{\dagger}+c_{6}^{\dagger}c_{3}^{\dagger}\end{aligned}$$ and so on.
But ,interestingly, $e_{+3}=0$ , since the last three terms of $e_{+3}$ ,using (2), are negatives of the first three . Further $e_{+4}$ and $e_{+5}$ can also be shown to be negatives of $e_{+2}$ and $e_{+1}$ respectively . So for the lattice of 6(i.e. (2N)) points the number of independent generators($e_{+l}$) are 2 (i.e. (N-1)), corresponding to l=1 and l=2. Consider now the conjugates of $e_{+l}$ . Denoted by $e_{-l}$ ,for this case of the 6-point-lattice, these are: $$\begin{aligned}
e_{-1}=-[c_{1}c_{2}+c_{2}c_{3}+c_{3}c_{4}+c_{4}c_{5}+c_{5}c_{6}+c_{6}c_{1}]\\
e_{-2}=-[c_{1}c_{3}+c_{2}c_{4}+c_{3}c_{5}+c_{4}c_{6}+c_{5}c_{1}+c_{6}c_{2}]\end{aligned}$$ Now if we calculate the commutator of $e_{+1}$ and $e_{-1}$ ,we get: $$[e_{-1},e_{+1}]=6 - 2\sum c_{n}^{\dagger}c_{n}+ \sum (c_{n}^{\dagger}c_{n+2}
+h.c.)$$ where we have used the anticommutators ( 1 ).The quantities on the r.h.s. of (9) $$h_{0}= \sum c_{n}^{\dagger}c_{n}$$ $$h_{2}= \sum (c_{n}^{\dagger}c_{n+2}+h.c.)$$ are the fermion number operators, $h_{0}$ ,and the hopping operator, $h_{2}$.
The quantity 6(i.e.,2N) is just the total number of points on the lattice. Similarly, $$[e_{-2},e_{+1}]=-h_{1}+h_{3}$$ where the $h_{i}$ is defined as $\sum (c_{n}^{\dagger}c_{n+i}+h.c.)$; the $h_{1}$ and the $h_{3}$ are two hopping operators.It is then easy to check that : $$[e_{\pm l},h_{0}]=\mp 2e_{\pm l}$$ $$[e_{\pm 1},h_{1}]=\mp e_{\pm 2}; [e_{\pm 1},h_{2}]=\pm e_{\pm 1};
[e_{\pm 1},h_{3}]=\pm e_{\pm 2}$$ $$[e_{\pm 2},h_{1}]=\mp e_{\pm 1};[e_{\pm 2},h_{2}]=\pm e_{\pm 2};
[e_{\pm 2},h_{3}]=\pm e_{\pm 1}$$ Further, $$[e_{+i},e_{+j}]=[e_{-i},e_{-j}]=[h_{i},h_{j}]=0$$ for all i and j. Thus $e_{\pm 1},e_{\pm 2},h_{0},h_{1},h_{2},h_{3}$,form a closed algebra for the 6-point lattice. The generalisation to the case of the 2N lattice is straightforward.For this general case the number of independent $e_{+l}$ generators are (N-1). The number of independent $h_{i}$ generators are (N+1).
The structure of the algebra is: $$[e_{+l},e_{-l}]=2N-2h_{0}+h_{2l}$$ $$[e_{+l},e_{-l^{\prime}}]=\sum \alpha_{ll^{\prime}}^{j}h_{j}\:\:\:\:\:\:\:\:\:\:
(l\neq l^{\prime})$$ $$[e_{\pm l},h_{0}]= \mp 2e_{\pm l}$$ $$[e_{\pm l},h_{i}]=\sum \beta_{li}^{j}e_{\pm j}$$ $$[e_{+l},e_{+l^{\prime}}]=[e_{-l},e_{-l^{\prime}}]=[h_{i},h_{j}]=0$$
Note that the sums over j in (18) and (20) run over only a small number of j values. That is $\alpha_{ll^{\prime}}^{j}$ and $\beta_{li}^{j}$ are non-zero only for one or two values of j for given $ll^{\prime}$ and li.The fermion number h$_0$ does not appear on the rhs of (18).
The Boson Interpretation
========================
Consider now the commutator (17) of $$[e_{-l},e_{+l}]=2N-2h_{0}+h_{2l}$$ where $h_{0}$ is the fermion number operator. The first term, 2N , is the size of the lattice. For small values of fermion filling i.e. when the band is nearly empty $h_{0}$ is way small compared to 2N.
Further the operator $h_{2l}$ given by : $$h_{2l}=\sum( C_{n}^{\dagger}C_{n+2l}+ h. c.)$$ in its diagonal form reads: $$h_{2l}=\sum 2 cos 2kla C_{k}^{\dagger}C_{k}$$ Thus the value of $h_{2l}$ for the state of the single fermion is bounded by $\pm 2 $.For the small number of fermions in the lattice(nel) the value of $h_{2l}$ is again way small compared to the first term 2N. Therefore equation (22) in the nel region becomes $$[e_{-l},e_{+l}]=2N$$ For the normalised generators $\frac{1}{\sqrt 2N} e_{\pm l}$ the r.h.s. of (25) becomes 1.Consider now the commutators $[e_{-l},e_{+l'}]$ with different l and $l^{\prime}$ .From (18) we get $[e_{-l},e_{+l'}]=\sum \alpha_{ll^{\prime}}^{j}h_{j}$ (in the sum $j \neq 0$). The generators $h_{i}$ are all simultaneously diagonalisable.All of them have eigenvalues bounded by 2 for the single fermion .(See also the discussions following the eqn (21)).For the normalized generators we ,therefore, get: $$[e_{-l},e_{+l'}]=0\hspace{2mm} when\hspace{2mm} l\neq l'$$
Taken together (25) and (26) gives for all values of l and $l^{\prime}$ the relation: $$[e_{-l},e_{+l'}]=\delta _{l,l'}$$ These are therefore the creation and the destruction operators of bosons. Consider now the insulating limit , when the lattice is nearly full(nfl). In the eqn (17) the value of $h_{0}$ is about 2N, so that combining the first two terms on the r.h.s. of (17) we get $$[e_{-l},e_{+l}]=-2N$$ as for the nfl the value of the $h_{i}$ are finite ,near zero. Thus if the creation and the annihilation operators are interchanged ,the same bosonic commutator relation reappears. In the insulating limit ,therefore, the coherent superposed pairs of holes created by $e_{-l}$ on the insulator is the lattice boson creation operator.
The Dirac Hamiltonian
=====================
Consider the Dirac Hamiltonian with mass on the one-dimensional equispaced lattice of 2N sites. The continuum Hamiltonian is\[7,8\] : $$H=- i\psi^{\dagger}(\alpha.\partial)\psi$$ with $\psi$ being the two-component wave-function $( ^{\psi_{1}}_{\psi_{2}})$ and the Dirac matrices of interest are $\gamma_{0}=\sigma_{3}$ ; $\alpha=\gamma_{5}=\sigma_{1}$ given by: $$\left(\begin{array}{lr}1&0\\0&-1
\end{array}
\right)and
\left(\begin{array}{lr}0&1\\1&0
\end{array}
\right)$$ respectively. If we choose $$\psi_{\pm}=\frac{1}{2}(1\pm \gamma_{5})\psi$$ and then write $c=\psi_{1}+\psi_{2}$, $b=\psi_{1}-\psi_{2}$ and define the derivative as: $$\partial c=\frac{c_{n+1}-c_{n-1}}{2a}$$ ,a being the lattice spacing,the Dirac Hamiltonian on the lattice takes the form: $$H=H_{c}+H_{b}+H_{m}=i\sum (c_{n}^{\dagger}c_{n+1}-h.c.)-i\sum (b_{n}^{\dagger}
b_{n+1}-h.c.)+m\sum (c_{n}^{\dagger}b_{n}+h.c.)$$ where we have included the mass term as well.
In arriving at (31) we have set the lattice spacing to 1/2.
The choice of a is not important in our considerations.Our results are good for any value of a. The lattice boson generators for this Hamiltonian are now written as : $$e_{+l}^{c}=\sum c_{n}^{\dagger}c_{n+l}^{\dagger};\hspace{2mm}
e_{-l}^{c}=-\sum c_{n}c_{n+l}$$ and $$e_{+l}^{b}=\sum b_{n}^{\dagger}b_{n+l}^{\dagger};\hspace{2mm}
e_{-l}^{b}=-\sum b_{n}b_{n+l}$$ The other generators $h_{i}^{c}$ and $h_{i}^{b}$ may be constructed in analogy with h$_i$ (defined immediately following eqn (12)). In the discussions that follow it is more convenient to consider the linear combinations $$E_{\pm l}^{\pm }=\frac{1}{\sqrt{2}}(e_{\pm l}^{c}\pm e_{\pm l}^{b})$$ For the case of the Dirac Hamiltonian (31) the lattice boson generators (34) are not complete, but have to include the following further ones: $$d_{+l}^{1}=\sum c_{n}^{\dagger}b_{n+l}^{\dagger}\hspace{2mm};
d_{+l}^{2}=\sum b_{n}^{\dagger}c_{n+l}^{\dagger}$$ along with their conjugates , $d_{-l}^{1}$ and $d_{-l}^{2}$. Once again the linear combinations $$D_{\pm l}^{\pm }=\frac{1}{\sqrt{2}} (d_{\pm l}^{1} \pm d_{\pm l}^{2})$$ that turn out to be of interest.
Working through the algebra we find that $$[E_{-l}^{j},E_{+l^{\prime}}^{j^{\prime}}] = 2N \delta _{ll^{\prime}}
\delta_{jj^{\prime}}$$ $$[D_{-l}^{j},D_{+l^{\prime}}^{j^{\prime}}] = 2N \delta_{ll^{\prime}}
\delta_{jj^{\prime}}$$ $$[E_{\pm l}^{j},D_{\pm l^{\prime}}^{j^{\prime}}] = 0$$ where j,j’ take values + and - . In the nel region we are led ,therefore , to interpret the $E_{\pm l}^{j}$ and $D_{\pm l}^{j}$ as the lattice boson generators.Note the eqn (39) follows the same approximation as the eqn(26).The operators must be normalised as in (26). The analogy follows equally well for the nfl region ,where the right-hand sides of (37) and (38) reverse the signs.Thus the superposed hole pairs created on the insulating region by the actions of $E_{-l}^{j}$ and $D_{-l}^{j}$ are the lattice bosons. To solve for these lattice boson modes for the Dirac Hamiltonian let us calculate the commutators of $E_{\pm l}^{j}$ and $D_{\pm l}^{j}$ with the Hamiltonian (31).These give : $$[H_{c}+H_{b},E_{l}^{\pm}]=0$$ $$[H_{m},E_{l}^{+}]=2mD_{l}^{+}$$ $$[H_{m},E_{l}^{-}]=0$$ while $$[H_{c}+H_{b},D_{l}^{+}]=2i[D_{l+1}^{-}-D_{l-1}^{-}]$$ $$[H_{c}+H_{b},D_{l}^{-}]=2i[D_{l+1}^{+}-D_{l-1}^{+}]$$ $$[H_{m},D_{l}^{+}]=2mE_{l}^{+}$$ $$[H_{m},D_{l}^{-}]=0$$ The equivalent lattice boson Hamiltonian thus is: $$H_{B}=2i[\sum D_{+(l+1)}^{-}D_{-l}^{+}+\sum D_{+(l+1)}^{+}D_{-l}^{-}]+2m\sum
D_{+l}^{+}E_{-l}^{+}+ h.c.$$ The Hamiltonian $H_{B}$ is equivalent to (31) in the sense that they produce the same commutators for the lattice boson generators when we use the boson commutators (37-39).To diagonalize $H_{B}$ we first carry out the transform:
$$E_{\pm l}^{\pm}=\frac{1}{\sqrt{L}}\sum E_{\pm}^{\pm}(q)e^{\mp iql}$$ and
$$D_{\pm l}^{\pm}=\frac{1}{\sqrt{L}}\sum D_{\pm}^{\pm}(q)e^{\mp iql}$$
where L is the number of independent points on the l-lattice, i.e.L=(N-1). \[see the discussion below eqn. (11)\]. The transformed $H_{B}$ reads : $$H_{B}= \sum {-}4 sinq [D_{+}^{-}(q)D_{-}^{+}(q)+D_{+}^{+}(q)D_{-}^{-}(q)]
+2m\sum (D_{+}^{+}(q)E_{-}^{+}(q)+E_{+}^{+}(q)D_{-}^{+}(q))$$ The Hamiltonian matrix in the space of $E_{\pm}^{+}(q)$ and $D_{\pm}^{\pm}(q)$ reads as: $$\left( \begin{array}{lcr}0 & a & 2m \\ a & 0 & 0 \\ 2m & 0 & 0 \end{array}
\right)$$ where a=-4sinq .
The above Hamiltonian matrix (51) is diagonalized to give us the eigenstates of the lattice bosons with eigenvalues 0 and $\pm \sqrt{a^{2}+2m^{2}}$ in the nel region.In the nfl region, because of the minus sign in(28), there is an overall minus sign for $H_{B}$ of (47).Thus the eigenvalues in the nfl region just reverse their signs.
From eqn.(40) note that $E_{\pm l}^{-}$ are generators of symmetries of the massless Dirac Hamiltonian. Prior to the insertion of mass term, $E_{\pm l}^{\pm}$ are symmetries.The mass term keeps the $E_{\pm l}^{-}$ symmetries but breaks $E_{\pm l}^{+}$.The $E_{\pm l}
^{+}$ mixes with $D_{\pm l}^{+}$.The $E_{\pm l}^{-}$ generate zero mass bosonic excitations that normalises the ground state. Since the $E_{\pm l}^{-}$ modes remain decoupled they have not explicitly appeared in our equation(47). It is important to point out that the number of independent $D_{l}^{\pm}$ generators differ a bit from the number of $E_{l}^{\pm}$.
We have shown\[see below eqn.(16)\] that the number of independent $e_{+l}$ generators for the lattice of size 2N is given by N-1.For the case of the Dirac Hamiltonian of mass we have $e_{+l}^{c}$ and $e_{+l}^{b}$ making a total of 2(N-1) independent generators. There are , from (34) , 2(N-1) generators of type $E_{+l}^{\pm}$.
The cases of $d_{+l}^{1,2}$ are roughly the same,except that $d_{+0}^{1,2}$ are independent and non-zero.They add extra degrees. For the $D_{+l}^{\pm}$, made of the $d_{+l}^{1,2}$,eqn. (36), the $D_{+0}^{+}=0$. This is because $$D_{+0}^{+}=d_{+0}^{1}+d_{+0}^{2}=\sum c_{n}^{\dagger}
b_{n}^{\dagger}+b_{n}^{\dagger}c_{n}^{\dagger}=0$$ from the anticommutators $\{c_{n}^{\dagger},b_{m}^{\dagger}\}=0$ Thus the number of $D_{l}^{+}$ and $E_{l}^{+}$ are identical.
The case of $D_{+0}^{-}$ is as follows : $$D_{+0}^{-}=d_{+0}^{1}-d_{+0}^{2}=2\sum c_{n}^{\dagger}
b_{n}^{\dagger}$$ This non-zero $D_{+0}^{-}$ appears in (47 ).Thus in the hopping part of the effective boson Hamiltonian,$H_{B}$, these $D_{\pm 0}^{-}$ appear at the end point of the l-lattice for the D-operator.This end point is neglected in our diagonalisation of $H_{B}$.Adding this end point ,for reasonably large values of N, does not alter our results.
Discussions
===========
The fermions anticommute. This leads to fermion composites that behave as bosons on the lattice.The lattice background gives rise to the set of bosons that we call the lattice bosons. They exist near the empty lattice (nel) when the number of fermions are small.In the insulating region (nfl) these lattice bosons reappear as coherently superposed hole pairs on the insulator\[9\].
Since these bosons depend on the fermion algebra, they exist quite independent of the fermion-fermion interaction, or on the interactions of the fermions with the lattice background. In particular they exist even for small values of couplings.To the lowest order ,we have seen from the model of the Dirac fermions the lattice boson wavefunctions overlap,leading to the hopping over the l-lattice.This Hamiltonian is readily diagonalisable. It is to be recognised that the mass term in (50) could come from the interactions.It could arise from the fermions coupling to the Higgs boson or from parts of the QCD that lead to the dynamical mass generation and the chiral symmetry breaking \[10\].
To conclude ,we have shown that for fermion systems there are boson excitations that arise from the background lattice.
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|
---
abstract: 'The bulge is the oldest component of the Milky Way. Since numerous simulations of Milky Way formation have predicted that the oldest stars at a given metallicity are found on tightly bound orbits, the Galaxy’s oldest stars are likely metal-poor stars in the inner bulge with small apocenters (i.e., $R_{\mathrm{apo}}\lesssim4$ kpc). In the past, stars with these properties have been impossible to find due to extreme reddening and extinction along the line of sight to the inner bulge. We have used the mid-infrared metal-poor star selection of @schlaufman2014 on Spitzer/GLIMPSE data to overcome these problems and target candidate inner bulge metal-poor giants for moderate-resolution spectroscopy with AAT/AAOmega. We used those data to select three confirmed metal-poor giants ($[\mathrm{Fe/H}]=-3.15,-2.56,-2.03$) for follow-up high-resolution Magellan/MIKE spectroscopy. A comprehensive orbit analysis using Gaia DR2 astrometry and our measured radial velocities confirms that these stars are tightly bound inner bulge stars. We determine the elemental abundances of each star and find high titanium and iron-peak abundances relative to iron in our most metal-poor star. We propose that the distinct abundance signature we detect is a product of nucleosynthesis in the Chandrasekhar-mass thermonuclear supernova of a CO white dwarf accreting from a helium star with a delay time of about 10 Myr. Even though chemical evolution is expected to occur quickly in the bulge, the intense star formation in the core of the nascent Milky Way was apparently able to produce at least one Chandrasekhar-mass thermonuclear supernova progenitor before chemical evolution advanced beyond $[\mathrm{Fe/H}]\sim-3$.'
author:
- Henrique Reggiani
- 'Kevin C. Schlaufman'
- 'Andrew R. Casey'
- 'Alexander P. Ji'
bibliography:
- 'bulge.bib'
title: 'The Most Metal-poor Stars in the Inner Bulge[^1]'
---
Introduction {#intro}
============
Since galaxies form from the inside-out, the bulge is the oldest major component of the Milky Way. While there are at least six physical processes that may have contributed to the growth of the Milky Way’s bulge [e.g., @barbuy2018], it is statistically implausible that the earliest stage of Milky Way formation failed to contribute to the bulge’s stellar population at some level. Indeed, numerical simulations of Milky Way-analog formation have consistently shown that the metal-poor stars in the inner few kpc of a Milky Way-like galaxy are often the oldest stars in the dark matter halo hosting the galaxy.[^2] At the same time, the early chemical evolution of the bulge is expected to differ significantly from that of the halo and surviving dwarf galaxies [e.g., @kobayashi2006]. For these reasons, the exploration of this first stage of Milky Way formation using the chemical abundances of ancient metal-poor stars in the inner Galaxy has long been a goal of Galactic archaeology.
Some of the expected differences in the chemical evolution of the bulge and halo have already been observed [e.g., @kobayashi2006; @cunha2006; @johnson2012; @bensby2017; @gargiulo2017; @barbuy2018]. The star formation rates in the event (or events) that lead to the formation of Milky Way’s “classical bulge” component are thought to have been very high. This intense star formation in a metal-poor environment would have produced many otherwise uncommon stellar systems, perhaps including the progenitors of relatively rare classes of supernovae (e.g., hypernovae, spinstars, thermonuclear supernovae on core-collapse supernovae timescales, etc.). The existence of metal-poor stars in the inner bulge despite this intense star formation requires the accretion of unenriched gas on short timescales as expected in the denser and more gas-rich $z \gtrsim 2$ Universe. Both of these differences between the bulge and the halo/surviving dwarf galaxies—frequent contributions from rare supernovae and the signature of ongoing accretion of unenriched gas—should be apparent in comparisons of the detailed elemental abundances of the most metal-poor stars in the inner bulge, halo, and surviving dwarf galaxies.
While metal-poor stars in the inner Galaxy were historically difficult to separate from the much more numerous metal-rich stars in the bulge, significant progress has been made in the last few years. Several groups have discovered metal-poor giants in the outer bulge photometrically using ultraviolet or mid-infrared photometry or spectroscopically in multiplexed surveys [e.g., @garcia-perez2013; @ness2013; @casey2015; @howes2015; @howes2016; @lamb2017; @lucey2019; @arentsen2020]. In spite of this recent progress in the outer bulge, it has been impossible to study in detail or even find metal-poor stars with $[\mathrm{Fe/H}] \lesssim -2.0$ in the inner bulge (i.e., $|l,b| \lesssim
4^{\circ}$) due to the extreme extinction and reddening in that direction.
In this paper, we have used the infrared-only metal-poor star selection of @schlaufman2014 to discover the most metal-poor stars known in the inner bulge. Follow-up high-resolution optical spectroscopy has revealed that at $[\mathrm{Fe/H}] \sim -3$, the inner bulge has high silicon and iron-peak abundances relative to iron when compared to giant stars in the halo or surviving dwarf galaxies. We attribute these differences to the occurrence of at least one Chandrasekhar-mass thermonuclear supernova that occurred early in the bulge’s chemical evolution on a timescale comparable to core-collapse supernovae. We outline our sample selection and observations in Section \[sample\_obs\]. We describe our analyses of these data in Section \[stellar\_prop\] and report the chemical abundances we infer in Section \[chem\_abund\]. We discuss the implications of our findings in Section \[discussion\] and summarize our findings in Section \[conclusion\].
Sample Selection and Observations {#sample_obs}
=================================
As input to our candidate selection process we used the Two Micron All Sky Survey (2MASS) All-Sky Point Source Catalog (PSC) [@skrutskie2006] combined with Spitzer/IRAC Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE) II and GLIMPSE 3D catalogs [@benjamin2003; @churchwell2009]. We dereddened and extinction-corrected these photometric catalogs with the bulge-specific reddening maps from @gonzalez2011 [@gonzalez2012] assuming a @nishiyama2009 extinction law. We excluded all stars with non-zero data quality flags (indicating a possible data quality issue), neighbors within 2, or extinction-corrected apparent 2MASS $J$-band magnitudes $J_0 \geq 12.5$. The latter cut ensured that we focused our attention on giant stars on the near side of the bulge. We then applied the infrared-only “v1” metal-poor star selection from @schlaufman2014 to the dereddened and extinction-corrected 2MASS and Spitzer/IRAC photometry to generate our initial candidate list. The cuts described above resulted in a sample of 10,915 candidate metal-poor giants with $-10 \leq l \leq +10$ and $-5 \leq b \leq
+5$. To help prioritize spectroscopic follow-up, we also estimated extinction-corrected $I$-band magnitudes assuming a typical $(I - J)_0 =
0.8$ color for metal-poor giants bright enough to have $J_0 \lesssim 12.5$ at 8 kpc.
We observed several thousand of these candidate inner bulge metal-poor giants using the Anglo-Australian Telescope’s (AAT) AAOmega multiobject spectrograph fed by the 2 Degree Field (2dF) robotic fibre positioner. We used the 580V and 1700D gratings in the blue and red arms of the spectrograph, providing spectral resolution $R \approx 1,\!300$ between 370 and 580 nm in the blue and $R \approx 10,\!000$ between 845 and 900 nm in the red. We reduced these data using the standard `2dfdr` pipeline.[^3] We estimated spectroscopic stellar parameters effective temperature $T_{\mathrm{eff}}$, surface gravity $\log{g}$, and metallicity $[\mathrm{Fe/H}]$ using the `sick` package [@casey2016].[^4] We then selected the giants 2MASS J172452.74-281459.4, 2MASS J175228.08-320947.6, and 2MASS J175836.79-313707.6 for high-resolution follow-up based on their low `sick`-inferred metallicities and bright apparent magnitudes. We plot the locations of these three stars on the Gaia DR2 all-sky image of the Galaxy in Figure \[stars\_position\].
We followed up these three giants with the Magellan Inamori Kyocera Echelle (MIKE) spectrograph on the Magellan Clay Telescope at Las Campanas Observatory [@bernstein2003; @shectman2003]. We used either the 07 or 10 slits and the standard blue and red grating azimuths, yielding spectra between 335 nm and 950 nm with resolution $R \approx
40,\!000/28,\!000$ in the blue and $ R \approx 31,\!000/22,\!000$ in the red for the 07/10 slits. We collected all calibration data (e.g., bias, quartz & “milky" flat field, and ThAr lamp frames) in the afternoon before each night of observations. We present a log of these observations in Table \[obs\_log\]. We reduced the raw spectra and calibration frames using the `CarPy`[^5] software package [@kelson2000; @kelson2003; @kelson2014]. We used `iSpec`[^6] [@blanco-cuaresma2014; @blanco-cuaresma2019] to calculate radial velocities and barycentric corrections and normalized individual orders using `IRAF`[^7] [@iraf1986; @iraf1993].
[llllccD]{} J172452.74-281459.4 & 06/29/2017 & 04:21:29 & 04:32:29 & 10 & 660 & +9.23\
J175228.08-320947.6 & 06/29/2017 & 04:34:37 & 04:54:37 & 10 & 1200 & -62.52\
J175228.08-320947.6 & 07/02/2017 & 03:52:14 & 04:38:38 & 07 & 2700& -63.98\
J175836.79-313707.6 & 06/29/2017 & 02:56:51 & 04:19:21 & 10 & 4800 & -196.42\
J175836.79-313707.6 & 06/29/2017 & 04:56:40 & 05:26:40 & 10 & 1800 & -196.42
The extreme extinction and reddening towards the inner bulge strongly affected the signal-to-noise ratio S/N of our spectra blueward of 600 nm. At 400 nm near the H and K lines, our spectra have $\mathrm{S/N} \approx 3$/pixel. At 520 nm near the $b$ triplet our spectra have $\mathrm{S/N} \approx 10$/pixel, while at 660 nm near H$\alpha$ our spectra have $\mathrm{S/N} \approx 60$/pixel. Near the near-infrared triplet at 850 nm, our spectra have $\mathrm{S/N} \approx 120$/pixel. We therefore focused our absorption line measurements on the long wavelength portions of our spectra.
Stellar Properties {#stellar_prop}
==================
Stellar Parameters
------------------
We used the `isochrones`[^8] [@morton2015] package to estimate $T_{\mathrm{eff}}$ and $\log{g}$ of each star using as inputs their:
1. $g$ and $r$ magnitudes and associated uncertainties from Data Release (DR) 1.1 of the SkyMapper Southern Sky Survey [@wolf2018];
2. $J$, $H$, and $K_{\mathrm{s}}$ magnitudes and associated uncertainties from the 2MASS PSC [@skrutskie2006];
3. $W1$, $W2$, and $W3$ magnitudes and associated uncertainties from the Wide-field Infrared Survey Explorer (WISE) AllWISE Source Catalog [@wright2010; @mainzer2011];
4. prior-informed distance estimates from @bailerjones2018 based on Gaia DR2 astrometry [@gaia2016; @gaia2018; @arenou2018; @hambly2018; @lindegren2018; @luri2018].
We used `isochrones` to fit the Dartmouth Stellar Evolution Database [@dotter2007; @dotter2008] library generated with the Dartmouth Stellar Evolution Program (DSEP) to these observables using `MultiNest`[^9] [@feroz2008; @feroz2009; @feroz2019]. We restricted the Dartmouth library to $\alpha$-enhanced composition $[\alpha\mathrm{/Fe}] = +0.4$, stellar age $\tau$ in the range 10.0 Gyr $\leq \tau \leq$ 13.721 Gyr, and extinction $A_{V}$ in the range 2.0 mag $\leq A_{V} \leq$ 5.0 mag. For each star, we initially assumed the values $T_{\mathrm{eff}} =
4750 \pm 250$ K, $\log{g} = 2 \pm 1$, $[\mathrm{Fe/H}] = -3.0 \pm 1.0$ for the likelihood calculation. We limited distances $d$ considered to the range suggested by @bailerjones2018. We plot the locations of all three stars relative to isochrones in $J-K_{\mathrm{s}}$ versus $K_{\mathrm{s}}$ color–magnitude diagrams in Figure \[iso\] and give the resulting isochrone-inferred parameters $T_{\mathrm{eff}}$, $\log{g}$, $A_V$, $\tau$, stellar luminosity $L_{\ast}$, stellar mass $M_{\ast}$, and isochrone distance $d_{\mathrm{iso}}$ in Table \[stellar\_params\]. This approach is analogous to the `StarHorse` technique from @queiroz2018 [@queiroz2020].
[lcccl]{} **Astrometric and Photometric Properties** & & & &\
Gaia DR2 Source ID & 4059924905887808128 & 4043617705316006272 & 4043987927121580928 &\
Gaia DR2 R.A. $\alpha$ (J2000) & 17 24 52.7503 & 17 52 28.0857 & 17 58 36.7952 & h m s\
Gaia DR2 decl. $\delta$ (J2000) & -28 14 59.287 & -32 09 47.652 & -31 37 07.674 & d m s\
Gaia DR2 galactic longitude $l$ (J2000) & 358.1186 & 357.9902 & 359.1173 & degrees\
Gaia DR2 galactic latitude $b$ (J2000) & 4.2026 & -2.9241 & -3.7794 & degrees\
Gaia DR2 proper motion (J2000) $\mu_{\alpha} \cos{\delta}$ & $-2.72 \pm 0.10$ & $-0.84 \pm 0.09$ & $-0.81 \pm 0.16$ & mas yr$^{-1}$\
Gaia DR2 proper motion (J2000) $\mu_{\delta}$ & $-10.28 \pm 0.07$ & $-3.12 \pm 0.08$ & $-7.65 \pm 0.13$ & mas yr$^{-1}$\
Gaia DR2 parallax $\pi$ (J2000) & $0.22 \pm 0.06$ & $0.04 \pm 0.06$ & $0.15 \pm 0.07$ & mas\
SkyMapper $g$ & $15.560 \pm 0.010$ & $16.743 \pm 0.023$ & $17.261 \pm 0.027$ & AB mag\
SkyMapper $r$ & $\cdots$ & $15.384 \pm 0.007$ & $15.967 \pm 0.007$ & AB mag\
Gaia DR2 $G$ & $13.75 \pm 0.002$ & $14.95 \pm 0.002$ & $15.70 \pm 0.002$ & Vega mag\
2MASS $J$ & $10.346 \pm 0.019$ & $11.694 \pm 0.024$ & $12.843 \pm 0.042$ & Vega mag\
2MASS $H$ & $9.437 \pm 0.023$ & $10.788 \pm 0.025$ & $11.990 \pm 0.039$ & Vega mag\
2MASS $K_{\mathrm{s}}$ & $9.139 \pm 0.021$ & $10.524 \pm 0.025$ & $11.774 \pm 0.034$ & Vega mag\
WISE $W1$ & $8.936 \pm 0.023$ & $10.381 \pm 0.035$ & $\cdots$ & Vega mag\
WISE $W2$ & $8.963 \pm 0.020$ & $10.450 \pm 0.036$ & $\cdots$ & Vega mag\
WISE $W3$ & $8.995 \pm 0.041$ & $10.134 \pm 0.157$ & $\cdots$ & Vega mag\
IRAC 3.6 & $8.915 \pm 0.042$ & $10.322 \pm 0.028$ & $11.647 \pm 0.040$ & Vega mag\
IRAC 4.5 & $8.887 \pm 0.044$ & $10.309 \pm 0.029$ & $11.619 \pm 0.051$ & Vega mag\
**Isochrone-inferred Parameters** & & & &\
Effective temperature $T_{\mathrm{eff}}$ & $4360 \pm 10$ & $4760 \pm 10$ & $4900\pm 10$ & K\
Surface gravity $\log{g}$ & $0.90\pm0.01$ & $0.84\pm0.01$ & $1.72_{-0.02}^{+0.03}$ & cm s$^{-2}$\
Luminosity $L_{\ast}$ & $900_{-20}^{+10}$ & $1570 \pm 10$ & $210 \pm 10$ & $L_{\odot}$\
Radius $R_{\ast}$ & $53 \pm 1$ & $62 \pm 1$ & $20 \pm 1$ & $R_{\odot}$\
Distance $d_{\mathrm{iso}}$ & $5.6 \pm 0.1$ & $12.4 \pm 0.1$ & $7.9_{-0.3}^{+0.2}$ & kpc\
Mass $M_{\ast}$ & $0.80 \pm 0.01$ & $0.98 \pm 0.01$ & $0.79 \pm 0.01$ & $M_{\odot}$\
Age $\tau$ & $12.4 \pm 0.1$ & $10.0 \pm 0.1$ & $12.3 \pm 0.5$ & Gyr\
Extinction $A_{V}$ & $3.10 \pm 0.02$ & $3.25 \pm 0.01$ & $2.71 \pm 0.03$ & mag\
**Spectroscopy-inferred Properties and Parameters** & & & &\
Radial velocity $v_r$ & $+9.23 \pm 1$ & $-63.98 \pm 1$ & $-196.42 \pm 1$ & km s$^{-1}$\
Microturbulence $\xi$ & $2.78^{+0.03}_{-0.01}$ & $3.22^{+0.05}_{-0.04}$ & $2.69 \pm 0.04$ & km s$^{-1}$\
Metallicity $[\mathrm{Fe/H}]$ & $-2.03 \pm 0.18$ & $-2.56 \pm 0.22$ & $-3.15 \pm 0.25$ &\
Non-LTE corrected metallicity $[\mathrm{Fe/H}]_{\mathrm{NLTE}}$ & $-1.95$ & $-2.44$ & $-3.06$ &\
**Galactic Orbit Parameters** & & & &\
Total Galactic velocity $v$ & $89 \pm 3$ & $102 \pm 4$ & $220 \pm 3$ & km s$^{-1}$\
Pericenter of Galactic orbit $R_{\mathrm{peri}}$ & $0.3 \pm 0.1$ & $0.8 \pm 0.1$ & $0.1 \pm 0.1$ & kpc\
Apocenter of Galactic orbit $R_{\mathrm{apo}}$ & $2.7 \pm 0.1$ & $4.5 \pm 0.1$ & $1.5_{-0.1}^{+0.2}$ & kpc\
Eccentricity of Galactic orbit $e$ & $0.77_{-0.04}^{+0.03}$ & $0.70_{-0.03}^{+0.02}$ & $0.84_{-0.07}^{+0.03}$ &\
Maximum distance from Galactic plane $z_{\mathrm{max}}$ & $1.18_{-0.04}^{+0.05}$ & $0.77_{-0.04}^{+0.03}$ & $0.92 \pm 0.05$ & kpc
In parallel we obtained spectroscopic stellar parameter using the classical excitation/ionization balance approach. We measured the equivalent widths of and atomic absorption lines in our continuum-normalized spectra by fitting Gaussian profiles with the `splot` task in `IRAF`. We used the `deblend` task to disentangle absorption lines from adjacent spectral features whenever necessary. The atomic data for and are from `linemake` [@sneden2009; @sneden2016] maintained by Vinicius Placco and Ian Roederer[^10] as collected in Ji et al. (2020, submitted). We report our input atomic data, measured equivalent widths, and inferred abundances in Table \[measured\_ews\].
We used 1D plane-parallel $\alpha$-enhanced ATLAS9 model atmospheres [@castelli2004], the 2019 version of the `MOOG` radiative transfer code [@sneden1973], and the `q^2` `MOOG` wrapper[^11] [@ramirez2014] to calculate $T_{\mathrm{eff}}$, $\log{g}$, $[\mathrm{Fe/H}]$, and microturbulence $\xi$ by simultaneously minimizing:
1. the difference between our inferred and abundances;
2. the dependence of abundance on excitation potential;
3. the dependence of abundance on reduced equivalent width.
We initiated our optimization with reasonable guesses for $[\mathrm{Fe/H}]$ and $\xi$ plus $T_{\mathrm{eff}}$ and $\log{g}$ 500 K and 1.0 dex lower than the isochrone-inferred parameters. We find the spectroscopic stellar parameters listed in Table \[stellar\_params\_spec\]. Due to the low S/N of our spectra blueward of 600 nm, we only analyzed and lines with $\lambda \geq 500$ nm. As a result, we cannot reliably measure a large number of lines over a wide range of excitation potential. In addition, most unblended lines in the spectra of metal-poor giants have $\lambda < 500$ nm. These two issues make it difficult to infer stellar parameters in a robust way using spectroscopy alone.
[lcccccc]{} J172452.74-281459.4 & $5682.633$ & & $2.102$ & $-0.706$ & $20.70$ & $4.330$\
J172452.74-281459.4 & $5688.203$ & & $2.104$ & $-0.406$ & $36.50$ & $4.339$\
J172452.74-281459.4 & $5889.951$ & & $0.000$ & $0.108$ & $344.00$ & $4.327$\
J175228.08-320947.6 & $5682.633$ & & $2.102$ & $-0.706$ & $3.00$ & $3.728$\
J175228.08-320947.6 & $5688.203$ & & $2.104$ & $-0.406$ & $2.50$ & $3.349$\
J175836.79-313707.6 & $5889.951$ & & $0.000$ & $0.108$ & $141.00$ & $3.144$\
J172452.74-281459.4 & $5528.405$ & & $4.346$ & $-0.498$ & $124.00$ & $5.523$\
J172452.74-281459.4 & $5711.088$ & & $4.343$ & $-1.724$ & $51.20$ & $5.845$\
J175228.08-320947.6 & $5172.684$ & & $2.712$ & $-0.393$ & $245.40$ & $5.418$\
J175228.08-320947.6 & $5183.604$ & & $2.717$ & $-0.167$ & $213.00$ & $4.821$\
J175228.08-320947.6 & $5528.405$ & & $4.346$ & $-0.498$ & $132.80$ & $5.902$\
J175228.08-320947.6 & $5711.088$ & & $4.343$ & $-1.724$ & $16.20$ & $5.520$\
J175836.79-313707.6 & $5172.684$ & & $2.712$ & $-0.393$ & $160.60$ & $4.560$\
J175836.79-313707.6 & $5183.604$ & & $2.717$ & $-0.167$ & $181.70$ & $4.621$\
[lRRRR]{} J172452.74-281459.4 & 4780 240 & 1.40 0.60 & -1.90 0.20 & 3.2 0.7\
J175228.08-320947.6 & 4790 210 & 0.47 0.51 & -2.57 0.20 & 3.4 0.9\
J175836.79-313707.6 & 4820 280 & 1.94 0.86 & -3.27 0.33 & 2.5 0.5
It has long been known that spectroscopic stellar parameters inferred for metal-poor giants using the classical approach differ from those derived using photometry and parallax information [e.g., @korn2003; @frebel2013; @mucciarelli2020]. Since local thermodynamic equilibrium (LTE) is almost always assumed in the model atmospheres used to interpret equivalent width measurements, these differences are often attributed to the violation of the assumptions of LTE in the photospheres of metal-poor giants. As a result, we impose the constraints on $T_{\mathrm{eff}}$ and $\log{g}$ deduced from our isochrone analysis and use the same optimization strategy to search for a self-consistent set of spectroscopic stellar parameters $T_{\mathrm{eff}}$, $\log{g}$, $[\mathrm{Fe/H}]$, and $\xi$ using the classical excitation/ionization balance approach.
We calculated our adopted \[Fe/H\] and $\xi$ uncertainties due to our uncertain $T_{\mathrm{eff}}$ and $\log{g}$ estimates using a Monte Carlo simulation. On each iteration, we randomly sample self-consistent pairs of $T_{\mathrm{eff}}$ and $\log{g}$ from our isochrone posteriors and calculate the best \[Fe/H\] and $\xi$ using the classical excitation/ionization balance approach. After we find a converged solution, we calculate mean iron abundances using our and equivalent width measurements assuming the stellar parameters found on that iteration. We save the result of each iteration and calculate \[Fe/H\] and its uncertainty as the (16,50,84) percentiles of the resulting metallicity distribution (\[Fe/H\]$=-2.03^{+0.01}_{-0.01}$ for 2MASS J172452.74-281459.4, \[Fe/H\]$=-2.58^{+0.01}_{-0.01}$ for 2MASS J175228.08-320947.6, and \[Fe/H\]$=-3.15^{+0.02}_{-0.01}$ for 2MASS J175836.79-313707.6). We then take these uncertainties and the converged stellar parameters described in the preceding paragraph and use them to redo the isochrone calculation using the converged stellar parameters and their uncertainties in the likelihood calculation. We repeat this entire process three times to obtain our final stellar parameters presented in Table \[stellar\_params\]. The final uncertainties in Table \[stellar\_params\] are larger than those described above because the values in Table \[stellar\_params\] account for both the standard deviation in iron abundance inferred from individual lines and the uncertainties due to our imperfectly estimated stellar parameters derived from the Monte Carlo analysis. The precise $T_{\mathrm{eff}}$ and $\log{g}$ resulting from our isochrone analysis imply that the ultimate accuracy of our \[Fe/H\] estimate is limited by the uncertainties in our measured equivalent widths. As a final check, we used our measured and equivalent widths and initiated our optimization process using the final set of stellar parameters listed in Table \[stellar\_params\]. We find that the spectroscopically inferred stellar parameters that result are consistent within their uncertainties to our preferred values in Table \[stellar\_params\].
In addition, we used the [@casagrande2010] Infrared Flux Method (IRFM) to verify our isochrone-inferred $T_{\mathrm{eff}}$. We deredden the 2MASS $J-K_{\mathrm{s}}$ colors of our stars using the bulge-specific reddening maps from @gonzalez2011 [@gonzalez2012]. In the IRFM calculation itself, we used the adopted $\log{g}$ and $[\mathrm{Fe/H}]$ given in Table \[stellar\_params\]. For 2MASS J172452.74-281459.4, 2MASS J175228.08-320947.6, and 2MASS J175836.79-313707.6 we find IRFM $T_{\mathrm{eff}} \approx
4530\pm350~\mathrm{K},4780\pm400~\mathrm{K},~\mathrm{and}~4680\pm400~\mathrm{K}$ in accord with the isochrone-inferred $T_{\mathrm{eff}}$ for each star. The large uncertainties in the IRFM $T_{\mathrm{eff}}$ are due to the reddening uncertainties in the @gonzalez2011 [@gonzalez2012] map.
Finally, we calculated 1D non-LTE corrections to our individual iron line abundances using the [@amarsi2016] grid. While that grid was calculated with MARCS model atmospheres [@gustafsson2008] and our iron abundances were calculated with ATLAS9 model atmospheres, both model atmospheres are very similar and we expect any differences to have only a small effect on our abundance corrections. We find that the mean non-LTE corrections for lines in our three giant stars to be 0.18, 0.18, and 0.14 dex for 2MASS J172452.74-281459.4, 2MASS J175228.08-320947.6, and 2MASS J175836.79-313707.6. The corrections are smaller for : 0.03, 0.03, and 0.04 for 2MASS J172452.74-281459.4, 2MASS J175228.08-320947.6, and 2MASS J175836.79-313707.6. These corrections are of the magnitude expected for giant stars in this metallicity regime [e.g., HD 122563 from @amarsi2016]. We give our non-LTE \[Fe/H\] values in Table \[stellar\_params\].
Stellar Orbits
--------------
To confirm that these giants located in the bulge are indeed on tightly bound orbits, we calculated their Galactic orbits using `galpy`[^12]. We sampled 1,000 Monte Carlo realizations from the Gaia DR2 astrometric solutions for each star using the distance posterior that results from our isochrone analysis while taking full account of the covariances between position, parallax, and proper motion. We used the radial velocities derived from our high-resolution MIKE spectra and assumed no covariance between our measured radial velocity and the Gaia DR2 astrometric solution. We used each Monte Carlo realization as an initial condition for an orbit and integrated it forward 10 Gyr in a Milky Way-like potential. We adopted the `MWPotential2014` described by @bovy2015. In that model, the bulge is parameterized as a power-law density profile that is exponentially cut-off at 1.9 kpc with a power-law exponent of $-1.8$. The disk is represented by a Miyamoto–Nagai potential with a radial scale length of 3 kpc and a vertical scale height of 280 pc [@miyamoto1975]. The halo is modeled as a Navarro–Frenk–White halo with a scale length of 16 kpc [@navarro1996]. We set the solar distance to the Galactic center to $R_{0} = 8.122$, kpc, the circular velocity at the Sun to $V_{0} =
238$ km s$^{-1}$, the height of the Sun above the plane to $z_{0} =
25$ pc, and the solar motion with the respect to the local standard of rest to ($U_{\odot}$, $V_{\odot}$, $W_{\odot}$) = (10.0, 11.0, 7.0) km s$^{-1}$ [@juric2008; @blandhawthorn2016; @gravity2018]. We give the resulting Galactic orbits in Table \[stellar\_params\]. We find that the Galactic orbits of all three giant stars have apocenters $R_{\mathrm{apo}} \lesssim 4$ kpc, confirming that they are all indeed tightly bound to the Galaxy and confined to the bulge region.
Chemical Abundances {#chem_abund}
===================
We measured the equivalent widths of atomic absorption lines for , , , , , , , , , , , , , , , , , , and in our continuum-normalized spectra by fitting Gaussian profiles with the `splot` task in `IRAF`. We used the `deblend` task to disentangle absorption lines from adjacent spectral features whenever necessary. We measured an equivalent width for every transition in our line list that could be recognized as an absorption line regardless of S/N or wavelength, taking into consideration the quality of a spectrum in the vicinity of a line and the availability of alternative transitions of the same species. We employed the 1D plane-parallel $\alpha$-enhanced ATLAS9 model atmospheres and the 2019 version of `MOOG` to calculate abundances for each equivalent width. In addition, we used spectral synthesis to infer the abundance of and to confirm the equivalent-width based abundance of . We report our input atomic data from Ji et al. (2020, submitted), measured equivalent widths, and individual inferred abundances in Table \[measured\_ews\]. We present our adopted mean chemical abundances and associated uncertainties in Table \[chem\_abundances\]. The standard deviation of abundances inferred for individual lines $\sigma_{\epsilon}$ does not not take into account the uncertainties in our adopted stellar parameters. The uncertainties in individual abundances relative to iron $\sigma_{[\mathrm{X/Fe}]}$ include both the standard deviation of abundances inferred for individual lines and the spectroscopic stellar parameter uncertainties assuming local thermodynamic equilibrium. We plot these later uncertainties in Figures \[alphas\_fig\], \[light\_odd\_fig\], \[iron\_peak\_fig\], and \[neutron\_capture\_fig\].
[llrRrRRR]{} J172452.74-281459.4 & & $3$ & $4.332$ & $0.003$ & $-1.908$ & $0.124$ & $0.012$\
& $_{\mathrm{NLTE}}$ & $3$ & $4.204$ & $\cdots$ & $-2.036$ & $-0.004$ & $\cdots$\
& & $2$ & $5.684$ & $0.114$ & $-1.916$ & $0.116$ & $0.161$\
& & $2$ & $4.828$ & $0.066$ & $-1.622$ & $0.410$ & $0.093$\
& & $5$ & $5.726$ & $0.026$ & $-1.784$ & $0.248$ & $0.029$\
& & $18$ & $4.607$ & $0.081$ & $-1.733$ & $0.299$ & $0.084$\
& & $5$ & $1.250$ & $0.073$ & $-1.900$ & $0.132$ & $0.082$\
& & $42$ & $3.157$ & $0.062$ & $-1.884$ & $0.148$ & $0.065$\
& & $50$ & $2.922$ & $0.126$ & $-2.028$ & $0.004$ & $0.127$\
& $\overline{\mathrm{Ti}}$ & $\cdots$ & $3.029$ & $\cdots$ & $-1.921$ & $0.111$ & $0.106$\
& & $7$ & $3.254$ & $0.148$ & $-2.386$ & $-0.354$ & $0.161$\
& & $2$ & $4.027$ & $0.237$ & $-1.613$ & $0.419$ & $0.336$\
& & $3$ & $2.796$ & $0.014$ & $-2.634$ & $-0.602$ & $0.020$\
& & $22$ & $5.309$ & $0.174$ & $-2.191$ & $\cdots$ & $\cdots$\
& & $12$ & $5.626$ & $0.182$ & $-1.874$ & $\cdots$ & $\cdots$\
& & $5$ & $3.146$ & $0.044$ & $-1.844$ & $0.188$ & $0.050$\
& & $14$ & $4.256$ & $0.056$ & $-1.964$ & $0.068$ & $0.059$\
& & $2$ & $2.958$ & $0.117$ & $-1.602$ & $0.430$ & $0.166$\
& & $1$ & $1.124$ & $0.000$ & $-1.746$ & $0.286$ & $0.016$\
& & $2$ & $\ge-1.280$ & $\cdots$ & $\ge-4.150$ & $\ge-2.118$ & $\cdots$\
& & $3$ & $-0.076$ & $0.027$ & $-2.286$ & $-0.254$ & $0.034$\
& & $2$ & $0.131$ & $0.080$ & $-2.049$ & $-0.017$ & $0.113$\
& & $2$ & $\le-0.154$ & $\cdots$ & $\le-1.254$ & $\le0.778$ & $\cdots$\
J175228.08-320947.6 & & $2$ & $3.538$ & $0.134$ & $-2.702$ & $-0.144$ & $0.190$\
& $_{\mathrm{NLTE}}$ & $2$ & $3.445$ & $\cdots$ & $-2.795$ & $-0.236$ & $\cdots$\
& & $4$ & $5.415$ & $0.194$ & $-2.185$ & $0.373$ & $0.225$\
& & $1$ & $3.889$ & $0.000$ & $-2.561$ & $-0.003$ & $0.032$\
& & $5$ & $5.646$ & $0.061$ & $-1.864$ & $0.694$ & $0.069$\
& & $15$ & $4.053$ & $0.071$ & $-2.287$ & $0.271$ & $0.073$\
& & $5$ & $0.665$ & $0.090$ & $-2.485$ & $0.073$ & $0.101$\
& & $28$ & $3.156$ & $0.119$ & $-1.885$ & $0.673$ & $0.121$\
& & $36$ & $2.414$ & $0.088$ & $-2.536$ & $0.022$ & $0.089$\
& $\overline{\mathrm{Ti}}$ & $\cdots$ & $2.739$ & $\cdots$ & $-2.211$ & $0.347$ & $0.102$\
& & $5$ & $3.037$ & $0.190$ & $-2.603$ & $-0.045$ & $0.212$\
& & $4$ & $2.822$ & $0.109$ & $-2.818$ & $-0.260$ & $0.126$\
& & $3$ & $2.817$ & $0.138$ & $-2.613$ & $-0.055$ & $0.169$\
& & $50$ & $4.871$ & $0.218$ & $-2.629$ & $\cdots$ & $\cdots$\
& & $6$ & $5.055$ & $0.223$ & $-2.445$ & $\cdots$ & $\cdots$\
& & $4$ & $3.199$ & $0.041$ & $-1.791$ & $0.767$ & $0.048$\
& & $9$ & $3.911$ & $0.074$ & $-2.309$ & $0.249$ & $0.079$\
& & $1$ & $2.251$ & $0.000$ & $-2.309$ & $0.249$ & $0.003$\
& & $1$ & $0.898$ & $0.000$ & $-1.972$ & $0.586$ & $0.007$\
& & $1$ & $0.076$ & $0.000$ & $-2.794$ & $-0.236$ & $0.040$\
& & $2$ & $-0.392$ & $0.061$ & $-2.602$ & $-0.044$ & $0.087$\
& & $2$ & $-1.262$ & $0.077$ & $-3.442$ & $-0.884$ & $0.109$\
& & $2$ & $\le-0.318$ & $\cdots$ & $\le-1.418$ & $\le1.140$ & $\cdots$\
J175836.79-313707.6 & & $1$ & $3.144$ & $0.000$ & $-3.096$ & $0.050$ & $0.025$\
& $_{\mathrm{NLTE}}$ & $1$ & $2.778$ & $\cdots$ & $-3.462$ & $-0.312$ & $\cdots$\
& & $2$ & $4.591$ & $0.022$ & $-3.009$ & $0.137$ & $0.041$\
& & $1$ & $\le3.064$ & $\cdots$ & $\le-3.386$ & $\le-0.240$ & $\cdots$\
& & $4$ & $5.050$ & $0.224$ & $-2.460$ & $0.686$ & $0.259$\
& & $15$ & $3.529$ & $0.088$ & $-2.811$ & $0.335$ & $0.092$\
& & $3$ & $0.390$ & $0.032$ & $-2.760$ & $0.386$ & $0.042$\
& & $27$ & $3.293$ & $0.115$ & $-1.748$ & $1.398$ & $0.119$\
& & $42$ & $2.489$ & $0.133$ & $-2.461$ & $0.685$ & $0.135$\
& $\overline{\mathrm{Ti}}$ & $\cdots$ & $2.804$ & $\cdots$ & $-2.146$ & $1.000$ & $0.131$\
& & $4$ & $2.598$ & $0.014$ & $-3.042$ & $0.104$ & $0.025$\
& & $3$ & $2.926$ & $0.231$ & $-2.714$ & $0.432$ & $0.283$\
& & $3$ & $2.769$ & $0.067$ & $-2.661$ & $0.485$ & $0.083$\
& & $42$ & $4.325$ & $0.244$ & $-3.175$ & $\cdots$ & $\cdots$\
& & $6$ & $4.161$ & $0.264$ & $-3.339$ & $\cdots$ & $\cdots$\
& & $1$ & $2.561$ & $0.000$ & $-2.429$ & $0.717$ & $0.020$\
& & $6$ & $3.442$ & $0.123$ & $-2.778$ & $0.368$ & $0.136$\
& & $1$ & $1.817$ & $0.000$ & $-2.373$ & $0.773$ & $0.020$\
& & $2$ & $2.209$ & $0.016$ & $-2.351$ & $0.795$ & $0.024$\
& & $2$ & $-0.950$ & $0.058$ & $-3.820$ & $-0.674$ & $0.086$\
& & $3$ & $-0.289$ & $0.111$ & $-2.499$ & $0.647$ & $0.136$\
& & $3$ & $-1.293$ & $0.165$ & $-3.473$ & $-0.327$ & $0.202$\
& & $1$ & $\le-0.286$ & $\cdots$ & $\le-1.386$ & $\le1.760$ & $\cdots$
To serve as comparison samples, we collected chemical abundances for outer bulge stars and halo stars from the literature. Our outer bulge comparison sample comes from [@garcia-perez2013], @casey2015, @howes2015 [@howes2016], @lamb2017, and @lucey2019. Our halo comparison sample comes from @cayrel2004, @bonifacio2009, and @reggiani2017. We note that the current sample of metal-poor bulge stars shows larger abundance dispersions than the sample of well-studied halo stars for all elements. These large dispersions are most likely due to the lower S/N of the input bulge spectra combined with the lack of a large-scale homogeneous abundance analyses in the metal-poor bulge. On the other hand, it could also be that the metal-poor stars in the bulge are first-generation Population II (Pop II) stars for which the large dispersions appear because each star records the nucleosynthesis of individual Population III (Pop III) supernovae. While we regard the former as more likely, only a large-scale homogeneous abundance analysis of hundreds of metal-poor bulge stars will settle the issue.
$\alpha$ Elements {#alpha_section}
-----------------
Oxygen, magnesium, silicon, calcium, and titanium are often referred to as $\alpha$ elements. Magnesium, silicon, and calcium are formed via similar nucleosynthetic channels. Magnesium is mainly formed via carbon burning in core-collapse supernovae (thermonuclear supernovae provide an order of magnitude less). Silicon is mostly a product of oxygen burning and is itself the most abundant product of oxygen burning. Core-collapse and thermonuclear supernovae contribute to silicon production in equal proportion. Calcium is the product of both hydrostatic and explosive oxygen and silicon burning. It is mostly produced in core-collapse supernovae. Even though titanium forms either in the $\alpha$-rich freeze-out of shock-decomposed nuclei during core-collapse supernovae or in explosive $^{4}$He fusion in the envelopes of CO white dwarfs during thermonuclear supernovae [e.g., @woosley1994; @livne1995], it is often considered alongside the true $\alpha$ elements because of their correlated chemical abundances [@clayton2003].
We plot in Figure \[alphas\_fig\] our inferred $\alpha$ abundances. We find that our metal-poor giants in the inner bulge roughly track the $\alpha$ abundances observed in the outer bulge and halo comparison samples. The one exception is the high and abundances we infer for our most metal-poor star 2MASS J175836.79-313707.6. We plot in Figure \[spec\_lines\] a representative line of 2MASS J175836.79-313707.6 in comparison to the same line observed in BPS CS 30312-0059, a star from @roederer2014 with very similar spectroscopic stellar parameters ($T_{\mathrm{eff}} =
4780$ K, $\log{g} = 1.4$, and $[\mathrm{Fe/H}] = -3.3$). We will argue in Section \[discussion\] that the high titanium abundance in 2MASS J175836.79-313707.6 is best explained by explosive $^{4}$He fusion in the envelope of a CO white dwarf accreting from a helium star binary companion during a Chandrasekhar-mass thermonuclear supernova.
The silicon abundances we infer from individual transitions for our three inner bulge giants have non-negligible scatter and are affected by our stellar parameter uncertainties. Nevertheless, our inferred abundances are in accord with silicon abundance inferences in outer bulge giants with $[\mathrm{Fe/H}] \approx -3.2$ and follow the same trend observed at higher metallicities. We observe the largest silicon abundances in our most metal-poor inner bulge giant 2MASS J175836.79-313707.6. Most of the accessible silicon lines redward of 500 nm are weak in such a metal-poor giant, so we also measured two additional lines at 3906 and 4103 Å that were identifiable in its spectrum even at $\mathrm{S/N} \approx 5$/pixel at 400 nm. The apparent difference between the silicon abundances inferred for halo dwarfs and giants is usually attributed to non-LTE effects even though other factors play a role [e.g., @bonifacio2009b Amarsi et al. 2020, submitted]. According to the @amarsi2017 grid of non-LTE corrections[^13], the typical non-LTE silicon abundance correction for a giant star with parameters similar to 2MASS J175836.79-313707.6 (i.e., $T_{\mathrm{eff}}
= 4500$ K, $\log{g} = 1.5$, $[\mathrm{Fe/H}] = -3.0$, $\xi = 2$ km s$^{-1}$, and $\epsilon_{\mathrm{Si}}\approx 5.01$) is about $-0.02$ dex. Non-LTE corrections are more important for metal-poor dwarfs, as a dwarf with a similar metallicity ($T_{\mathrm{eff}} = 6500$ K, $\log{g} = 4.5$, $[\mathrm{Fe/H}] = -3.0$, $\xi = 1$ km s$^{-1}$, and $\epsilon_{\mathrm{Si}} \approx 5.01$) will have a non-LTE silicon correction of about $+0.25$ dex.
Light Odd-$Z$ Elements {#light_odd_section}
----------------------
Like magnesium, sodium is mostly produced in core-collapse supernovae via carbon burning. Unlike magnesium, the surviving fraction of sodium in supernovae ejecta depends on metallicity so it is treated as a secondary product. Sodium is also produced as a product of hydrogen and helium fusion in thermonuclear explosions, though in smaller quantities than in core-collapse supernovae. Similar to sodium, aluminum is synthesized during carbon fusion in core-collapse supernovae in a secondary reaction that is dependent on the amount of $^{22}$Ne burned (which in turn depends on the carbon and oxygen content of the star). In contrast to sodium and aluminum, scandium is formed via both oxygen burning in core-collapse supernovae and as a product of $\alpha$-rich freeze-out in the shocked region just above the rebounded core [the same region responsible for $^{44}$Ti e.g., @clayton2003].
We plot in Figure \[light\_odd\_fig\] our inferred light odd-$Z$ abundances. We find that our metal-poor giants in the inner bulge roughly track the light odd-$Z$ abundances observed in the outer bulge and halo comparison samples. The one exception is the high scandium abundance we infer for our most metal-poor star 2MASS J175836.79-313707.6. We will argue in Section \[discussion\] that the high scandium abundance in 2MASS J175836.79-313707.6 is produced by nucleosynthesis in oxygen-rich extreme Pop II stars.
Sodium abundance inferences are strongly affected by departures from LTE, as the main sodium abundance indicator in our spectra is the resonant sodium doublet at 5889/5895 Å. We corrected our abundances inferred under the assumptions of LTE using the [@lind2011] correction grid provided via the INSPECT project[^14]. To correct the abundances of 2MASS J172452.74-281459.4 and 2MASS J175228.08-320947.6, we used the correction for a star with $\log{g} = 1$ as our adopted gravities were outside the bounds of the available grid. The sodium doublet is affected by interstellar medium (ISM) absorption, and the extreme extinction along the line of sight to the inner bulge can affect both the shape and depth of the sodium doublet. As a result, we were unable to disentangle the effects of photospheric and ISM absorption for the 5895 Å line and we do not use it in our analysis. Two weaker sodium lines at 5682 and 5688 Å are available in the spectra of 2MASS J172452.74-281459.4 and 2MASS J175228.08-320947.6 though, and all abundances inferred from measured lines are in good agreement. For 2MASS J175836.79-313707.6, we only have the 5889 Å line. Its spectrum has good S/N and is clear of ISM absorption, so we believe our inferred sodium abundance is reliable. Most of the sodium abundances in our comparison outer bulge and halo samples have been corrected for departures from local thermodynamic equilibrium. [@cayrel2004] used corrections from @baumuller1998 while [@bonifacio2009] used corrections from [@andrievsky2007]. Like our sodium abundances, [@howes2015; @howes2016] and [@reggiani2017] were non-LTE corrected using the grid from @lind2011. [@casey2015] and [@lucey2019] did not account for non-LTE effects.
It was extremely difficult to infer aluminum abundances for our three stars. The best available aluminum lines in metal-poor stars are usually the 3944 and 3961 Å lines, and the spectra of our highly extincted inner bulge stars have very low S/N at $\lambda < 400$ nm. We were unable to measure the equivalent width of either line in 2MASS J172452.74-281459.4. We were only able to measure upper limits for the equivalent widths of the 3944 Å line in 2MASS J175836.79-313707.6 and the 3961 Å line in 2MASS J175228.08-320947.6. While there are two weaker aluminum lines at 6696 and 6698 Å, they were only able to provide an upper limit on the aluminum abundance of 2MASS J172452.74-281459.4. While we report aluminum abundances assuming LTE in Table \[chem\_abundances\], to fairly compare our aluminum abundances with the outer bulge and halo samples we follow @reggiani2017 and add 0.65 dex to the LTE abundances of our three stars as well as the LTE abundances in the comparison samples.
We inferred the scandium abundances of our three inner bulge giants using lines accounting for hyperfine structure (HFS) using data taken from the Kurucz compilation[^15]. Similar to what we observed with titanium, the abundance of scandium in 2MASS J175836.79-313707.6 is enhanced relative to the comparison samples.
Iron-peak Elements {#iron_peak_section}
------------------
Iron-peak elements can be formed directly or as a byproduct of explosive silicon burning, either incomplete (chromium and manganese) or complete (cobalt, nickel, and zinc). Their nucleosynthesis mainly takes place in thermonuclear supernovae [e.g., @clayton2003; @grimmett2019]. The observed increase in the abundance ratios \[Co/Fe\] and \[Zn/Fe\] with decreasing \[Fe/H\] in metal-poor stars combined with the dependence of cobalt and zinc yields on the explosion energies of core-collapse supernovae also point to contributions from hypernovae events at $[\mathrm{Fe/H}] \lesssim -3.0$ [e.g., @cayrel2004; @reggiani2017].
We plot in Figure \[iron\_peak\_fig\] our inferred iron-peak abundances. We find that our metal-poor giants in the inner bulge consistently have higher iron-peak abundances than the outer bulge and halo comparison samples. We also find a significantly supersolar manganese abundance $[\mathrm{Mn/Fe}] \approx +0.5$ for our most metal-poor star 2MASS J175836.79-313707.6. We do not correct our inferred iron-peak abundances for non-LTE effects, both because correction grids are lacking for all iron-peak elements and because the iron-peak abundances in our comparison samples have not been corrected for departures from the assumptions of LTE. We will argue in Section \[discussion\] that the supersolar manganese abundance in 2MASS J175836.79-313707.6—and indeed its entire iron-peak abundance pattern—is best explained by nucleosynthesis in a thermonuclear supernova.
We plot abundances in Figure \[iron\_peak\_fig\] despite the fact that those lines are strongly affected by departures from LTE [e.g., @bergemann2010; @reggiani2017]. We prefer to in this case because our inferred \[/Fe\] ratio in Table \[chem\_abundances\] is significantly supersolar for 2MASS J175836.79-313707.6. Chromium almost always appears in solar \[Cr/Fe\] ratios and neither core-collapse or thermonuclear supernovae produce $[\mathrm{Cr/Fe}] \gtrsim +0.3$ [e.g., @clayton2003; @grimmett2019]. We therefore suspect that our inferred abundances are affected by noise in our spectra.
We find relatively high \[Mn/Fe\] abundances in our inner bulge sample, including a significantly supersolar $[\mathrm{Mn/Fe}]
\approx +0.5$ in our most metal-poor star 2MASS J175836.79-313707.6. We plot in Figure \[spec\_lines\] three lines for 2MASS J175836.79-313707.6 in comparison to the same lines observed in the comparison star BPS CS 30312-0059. We included HFS components in our abundance inferences using data taken from the Kurucz compilation referenced above but did not correct for departures from LTE. Corrections for departures from the assumptions of LTE in metal-poor giants tend to increase \[Mn/Fe\] and would not change our conclusion about 2MASS J175836.79-313707.6 [e.g., @bergemann2019; @eitner2020]. The manganese abundance in 2MASS J175836.79-313707.6 is based on three weak manganese lines at 6013, 6016, and 6021 Å. Our spectrum of 2MASS J175836.79-313707.6 has $\mathrm{S/N} \gtrsim 50$/pixel at 600 nm and even though they are at the limit of detectability, these three manganese lines all appeared at their expected wavelengths and produce a consistent manganese abundance estimate. We therefore argue that the apparent lines are unlikely to be produced by noise in our spectrum. Although the bluer manganese lines typically analyzed in metal-poor giants are apparent in the spectrum of 2MASS J175836.79-313707.6, the abundances we infer from those lines are even higher.
We find relatively high cobalt abundances in our three metal-poor inner bulge giants, especially in the range $-2.5 \lesssim [\mathrm{Fe/H}]
\lesssim -2.0$. A comparison of cobalt lines observed in the spectra of 2MASS J175228.08-320947.6 and 2MASS J175836.79-313707.6 with lines synthesized assuming our adopted stellar parameters and cobalt abundances supports this finding (Figure \[synth\_lines\]). At the same time, our inferred zinc abundances closely track those observed in the outer bulge and halo dwarf samples. Our cobalt abundance estimates come from lines redward of 500 nm, while our zinc abundance estimates are based on lines blueward of 500 nm in the noisier parts of our spectra. We find supersolar nickel abundances in our three inner bulge stars. Like the outer bulge and halo comparison samples, we find no nickel abundance trend with metallicity. This lack of a dependence of nickel abundance on metallicity supports the idea that nickel can be used as a metallicity tracer in both the bulge and the halo [@singh2020].
Neutron-capture Elements {#neutron_capt_section}
------------------------
Elements beyond zinc are mostly produced by neutron-capture processes either “slow” or “rapid” relative to $\beta$ decay timescales. The relative contributions of these $s$- and $r$-processes to the nucleosynthesis of each element are different and functions of metallicity. Some elements like strontium, yttrium, barium, and lanthanum are more commonly used as tracers of the $s$-process. On the other hand, europium is used as tracer of $r$-process nucleosynthesis [e.g., @cescutti2006; @jacobson2013; @ji2016a]. Even though strontium and barium are usually used as tracers of the $s$-process, both can have important contributions from $r$-process nucleosynthesis at lower metallicities [e.g., @battistini2016; @casey2017; @mashonkina2019]. Indeed, isotopic analyses of very metal-poor stars have shown that up to 80% of the barium in very metal-poor stars was synthesized in the $r$-process [@mashonkina2019].
We plot in Figure \[iron\_peak\_fig\] our inferred neutron-capture element abundances. We find that our metal-poor giants in the inner bulge roughly track the neutron-capture abundances observed in the outer bulge and halo comparison samples. For strontium, we used lines to infer its elemental abundances as those lines are not significantly affected by departures from LTE [@hansen2013]. For yttrium we used HFS components from the same Kurucz compilation referenced above, though we note that the S/N of our spectra in the vicinity of the yttrium lines were not high. Although we inferred our yttrium abundances using equivalent widths, the observed spectrum of 2MASS J175836.79-313707.6 is in good agreement with a synthesized line at 5200 Å assuming our inferred abundance $\mathrm{A(Y)} = -0.29 \pm 0.2$ (Figure \[synth\_lines\]). For barium, the abundances we infer using both equivalent widths and spectral synthesis based on unblended lines in parts of our spectra with high S/N agree within about 0.2 dex. For europium and lanthanum, the low S/N of the blue parts of our spectra only allow us to infer upper limits on their abundances.
Discussion
==========
The three metal-poor giants in the inner bulge we studied have orbits that are confined to the bulge. They are therefore likely to be among the oldest stars in the Milky Way and trace the earliest stage of the formation of the Milky Way’s oldest component: the bulge. We find that the abundances of our inner bulge stars at $[\mathrm{Fe/H]} \gtrsim -3.0$ are for the most part in accord with the abundances of stars with similar metallicities in the outer bulge and halo.
The story is different for our most metal-poor inner bulge giant 2MASS J175836.79-313707.6 at $[\mathrm{Fe/H}] = -3.15$. When compared to both the outer bulge and halo comparison samples, it has high \[Ti/Fe\], \[Sc/Fe\], and iron-peak abundances combined with supersolar \[Mn/Fe\]. We propose that 2MASS J175836.79-313707.6 is an ancient third-generation star with $\alpha$ and light odd-$Z$ elements produced by massive Pop II stars that were seeded with abundant oxygen by massive Pop III stars. Unlike the progenitor(s) of the halo and the surviving dwarf galaxies, the intense star formation rate in the bulge will fully sample the stellar initial mass function and therefore produce many very massive stars. According to the Pop III supernovae yields of @heger2010, massive Pop III stars are prolific producers of oxygen relative to iron. After their supernovae, that overabundance of oxygen is transformed into an overabundance of scandium by the first generation of massive Pop II stars and injected into the interstellar medium by their supernovae. Titanium and the iron-peak elements including manganese in 2MASS J175836.79-313707.6 were simultaneously produced in the Chandrasekhar-mass thermonuclear supernova of a CO white dwarf accreting from a helium star binary companion. The combination of fast stellar evolution at low metallicities, relatively massive CO white dwarfs produced by metal-poor stars, and efficient accretion from a helium star produced a short delay time comparable to the combined lifetimes of two generations of massive stars, about 10 Myr after the onset of star formation in what would become the bulge of the Milky Way.
To verify the scenario outlined above, we evaluate the ability of models predicting the nucleosynthetic yields of core-collapse and thermonuclear supernovae to reproduce the observed abundance pattern of 2MASS J175836.79-313707.6. Oxygen-rich metal-poor stars produce more scandium relative to iron than solar-composition metal-poor stars [e.g., @woosley1995; @chieffi2004]. The Chandrasekhar-mass thermonuclear supernova of a CO white dwarf accreting from a helium star binary companion will produce large amounts of titanium and can explode with a short delay time [e.g., @woosley1994; @livne1995; @wang2009a; @wang2009b]. It is thought that near Chandrasekhar-mass thermonuclear supernovae are the only supernovae capable of producing $[\mathrm{Mn/Fe}] \gtrsim 0$, as only CO white dwarfs near the Chandrasekhar mass have densities $\rho \gtrsim 2\times10^{8}$ g cm$^{-3}$ necessary to produce large amounts of $^{55}$Co that eventually decays into manganese [e.g., @seitenzahl2013a; @yamaguchi2015; @seitenzahl2017].
In an effort to confirm the scenario outlined above, we compare the abundances of 2MASS J175836.79-313707.6 with the yields predicted by three different classes of Chandrasekhar-mass thermonuclear supernovae plus one class of sub-Chandrasekhar-mass thermonuclear supernovae:
1. Chandrasekhar-mass deflagration to detonation transition (DDT) models with fixed C/O ratios from @seitenzahl2013b and DDT models with variable C/O ratios from @ohlmann2014;
2. Chandrasekhar-mass pure (turbulent) deflagration models from @fink2014;
3. Chandrasekhar-mass gravitationally confined detonation (GCD) models from @seitenzahl2016;
4. sub-Chandrasekhar-mass models resulting from the merger of two $M_{\ast}
= 0.6~M_{\odot}$ CO white dwarfs from @papish2016.
We first fix the iron abundance predicted by each model to the metallicity of 2MASS J175836.79-313707.6 and select the model that minimizes $\chi^2$ between our observed abundances and the predicted yields. If we focus only on the iron-peak abundances of chromium, manganese, cobalt, and nickel we find that the pure deflagration model “N100Hdef” from @fink2014 provides the best match to the abundances of 2MASS J175836.79-313707.6. The observable properties of model N100Hdef do not match those of any known class of Type Ia supernovae though. If instead we consider both the abundances of silicon and the iron-peak elements chromium, manganese, cobalt, and nickel we find that the DDT model “N1600C” from @seitenzahl2013b with a compact, spherically symmetric ignition provides the best match to the abundances of 2MASS J175836.79-313707.6. In addition, model N1600C produces $[\mathrm{Si/Mg}]
\approx +0.9$ that is fully consistent with $[\mathrm{Si/Mg}] \approx
+0.6\pm0.3$ observed in 2MASS J175836.79-313707.6. Moreover, the observable properties of the DDT models from @seitenzahl2013b do seem to match the properties of ordinary Type Ia supernovae. The sub-Chandrasekhar-mass model is a poor fit to the abundances of 2MASS J175836.79-313707.6.
We also compare the abundances of 2MASS J175836.79-313707.6 to the grid of updated Pop III core-collapse supernovae “znuc2012” models from @heger2010 using STARFIT[^16]. Though the best fit model ($M_{\ast}=10.9~M_{\odot}$, $KE_{\mathrm{exp}}
= 0.6$ B, $\log{f_{\mathrm{mix}}} = -0.6$) has a similar $\chi^2$ value as our preferred Chandrasekhar-mass thermonuclear model, it has an implausibly large amount of mixing and cannot explain the chromium, manganese, cobalt, or nickel abundances of 2MASS J175836.79-313707.6. Indeed, the similar $\chi^2$ comes from the core-collapse model’s higher predicted cobalt yield that still fails to explain the cobalt abundance of 2MASS J175836.79-313707.6. Though the overall $\chi^2$ values are comparable, we prefer the Chandrasekhar-mass thermonuclear supernovae model because it can self-consistently reproduce the chromium, manganese, iron, and nickel abundances of 2MASS J175836.79-313707.6. We plot in Figure \[thermo\_model\] the best fit Chandrasekhar-mass and sub-Chandrasekhar-mass thermonuclear supernovae models along with the best-fit core-collapse supernova model.
{width="5in"}
Our scenario for the nucleosynthesis of the iron-peak elements in 2MASS J175836.79-313707.6 requires a Chandrasekhar-mass thermonuclear supernovae to occur about 10 Myr after the formation of the first stars in what would become the bulge of the Milky Way. Chandrasekhar-mass thermonuclear supernovae are often assumed to have occurred through the so-called “single degenerate” channel in which a CO white dwarf accretes material from Roche-lobe overflow or a strong wind from a main sequence, subgiant, helium star, or red giant binary companion. A single-degenerate Chandrasekhar-mass thermonuclear supernovae requires a CO white dwarf. At solar metallicity, the first CO white dwarfs appear 30 to 40 Myr after the onset of star formation when stars less massive than $8~M_{\odot}$ start to end their lives as white dwarfs. Metal-poor stars both move through their stellar evolution more quickly and produce more massive CO white dwarfs that require less accretion and therefore less time to reach the Chandrasekhar mass than solar-metallicity stars [e.g., @meng2008]. The titanium abundance of 2MASS J175836.79-313707.6 prefers accretion from a helium star companion, and such configurations have been shown to reduce the delay times of thermonuclear supernovae [e.g., @wang2009a; @wang2009b]. As a result, it seems plausible that Chandrasekhar-mass thermonuclear supernovae delay times as short as 10 Myr might be achieved in a stellar population with $[\mathrm{Fe/H}] \sim -3$.
Appealing to a thermonuclear supernova model that produces $0.5~M_{\odot}$ of iron to explain the iron-peak abundances of a star with $[\mathrm{Fe/H}] = -3.15$ requires either a significant outflow of iron or an enormous dilution by unenriched gas. The depth of the potential at the center of the nascent Milky Way combined with the presence of dense gas fueling ongoing star formation that would shock ejecta and remove its kinetic energy indicates that dilution is a better explanation. To dilute $0.5~M_{\odot}$ of iron to the metallicity of 2MASS J175836.79-313707.6 requires mixing with about $10^{6}~M_{\odot}$ of pristine gas. While this is a substantial amount of unenriched gas, the young bulge is predicted to have had high accretions rates of pristine gas due to the high gas densities and frequent mergers expected for a relatively high $\sigma$ peak in the redshift $z\gtrsim2$ Universe.
The bulge of the young Milky Way is the ideal place to form stars like 2MASS J175836.79-313707.6. Even if the progenitor system of a Chandrasekhar-mass thermonuclear supernova that explodes with a delay time of 10 Myr is intrinsically rare, the star formation rate is so high in the bulge of the young Milky Way that there are lots of chances for it to form. The high star formation rate will also fully sample the stellar initial mass function and produce the massive Pop III or extreme Pop II stars necessary to produce the oxygen that will be transformed into silicon and scandium by future massive stars and their supernovae. The gas density and frequent mergers expected above $z
\approx 2$ will provide the unenriched gas necessary to form stars with $[\mathrm{Fe/H}] \sim -3$ in a region with a high star formation rate. We therefore suggest that stars with abundance patterns like that of 2MASS J175836.79-313707.6 enriched by thermonuclear supernovae should only be observed in the halo or classical dwarf spheroidal galaxies at higher metallicities. Though uncommon, the abundance pattern of 2MASS J175836.79-313707.6 is not unique. [@reyes2020] inferred manganese abundances for 161 giants in six classical dwarf spheroidal galaxies and found \[Mn/Fe\] as high as our inferred value for 2MASS J175836.79-313707.6 in stars at approximately $-3.0 \lesssim [\mathrm{Fe/H}] \lesssim
-2.5$. They concluded that their observed abundances are indicative of Chandrasekhar-mass thermonuclear supernovae occurring at higher metallicities $-2 \lesssim [\mathrm{Fe/H}] \lesssim -1$.
Conclusion
==========
Metal-poor stars in the bulge on tightly bound orbits are thought to be the oldest stars in the Milky Way. We used the mid-infrared metal-poor star selection of @schlaufman2014 to find the three most metal-poor stars known in the inner bulge. All three stars are on tightly bound orbits and confined to the bulge region. The detailed abundances of our two inner bulge giants with $[\mathrm{Fe/H}] \gtrsim -3$ have high iron-peak abundances but are otherwise similar to metal-poor stars in the outer bulge and halo. Our most metal-poor star 2MASS J175836.79-313707.6 has high \[Ti/Fe\], \[Sc/Fe\], and iron-peak abundances. It also has supersolar \[Mn/Fe\]. We argue that it is a second-generation Pop II star that was enriched by both massive Pop III or first-generation Pop II stars and a Chandrasekhar-mass thermonuclear supernova accreting from a helium star companion that exploded with a delay time of about 10 Myr. We argue that stars like 2MASS J175836.79-313707.6 with $[\mathrm{Fe/H}]
\lesssim -3$ should be much more common in the bulge than in the halo or dwarf galaxies because of the young bulge’s high star formation rate and frequent inflows of unenriched gas.
We thank the anonymous referee for the insightful comments that helped us improve this paper. Andy Casey is the recipient of an Australian Research Council Discovery Early Career Award (DECRA 190100656) funded by the Australian Government. Alex Ji is supported by NASA through Hubble Fellowship grant HST-HF2-51393.001, awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research has made use of the NASA/IPAC Infrared Science Archive, which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology. Based in part on data acquired through the Australian Astronomical Observatory, under programs A/2015A/107 and A/2016A/103 plus Prop. ID 2016A-0086 and 2016B-0081 (PI: K. Schlaufman). We acknowledge the traditional owners of the land on which the AAT stands, the Gamilaraay people, and pay our respects to elders past and present. This paper includes data gathered with the 6.5 m Magellan Telescopes located at Las Campanas Observatory, Chile. This work has made use of data from the European Space Agency (ESA) mission [*Gaia*]{} (<https://www.cosmos.esa.int/gaia>), processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement. The national facility capability for SkyMapper has been funded through ARC LIEF grant LE130100104 from the Australian Research Council, awarded to the University of Sydney, the Australian National University, Swinburne University of Technology, the University of Queensland, the University of Western Australia, the University of Melbourne, Curtin University of Technology, Monash University and the Australian Astronomical Observatory. SkyMapper is owned and operated by The Australian National University’s Research School of Astronomy and Astrophysics. The survey data were processed and provided by the SkyMapper Team at ANU.
The SkyMapper node of the All-Sky Virtual Observatory (ASVO) is hosted at the National Computational Infrastructure (NCI). Development and support the SkyMapper node of the ASVO has been funded in part by Astronomy Australia Limited (AAL) and the Australian Government through the Commonwealth’s Education Investment Fund (EIF) and National Collaborative Research Infrastructure Strategy (NCRIS), particularly the National eResearch Collaboration Tools and Resources (NeCTAR) and the Australian National Data Service Projects (ANDS). This publication makes use of data products from the *Wide-field Infrared Survey Explorer*, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This research has made use of NASA’s Astrophysics Data System. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France [@wenger2000]. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI: 10.26093/cds/vizier). The original description of the VizieR service was published in 2000, A&AS 143, 23 [@ochsenbein2000].
[^1]: This paper includes data gathered with the 6.5-meter Magellan Telescopes located at Las Campanas Observatory, Chile.
[^2]: See for example @diemand2005, @scannapieco2006, @brook2007, @salvadori2010, @gao2010, @tumlinson2010, @ishiyama2016, @starkenburg2017b, @griffen2018, and @sharma2018
[^3]: <https://www.aao.gov.au/science/software/2dfdr>
[^4]: <https://github.com/andycasey/sick>
[^5]: <http://code.obs.carnegiescience.edu/mike>
[^6]: <https://www.blancocuaresma.com/s/iSpec>
[^7]: <https://iraf-community.github.io/>
[^8]: <https://github.com/timothydmorton/isochrones>
[^9]: <https://ccpforge.cse.rl.ac.uk/gf/project/multinest/>
[^10]: <https://github.com/vmplacco/linemake>
[^11]: <https://github.com/astroChasqui/q2>
[^12]: <https://github.com/jobovy/galpy>
[^13]: <http://www.mpia.de/homes/amarsi/index.html>
[^14]: <http://inspect.coolstars19.com/>
[^15]: <http://kurucz.harvard.edu/linelists.html>
[^16]: <http://starfit.org/>
|
---
abstract: 'LSTM-based speaker verification usually uses a fixed-length local segment randomly truncated from an utterance to learn the utterance-level speaker embedding, while using the average embedding of all segments of a test utterance to verify the speaker, which results in a critical mismatch between testing and training. This mismatch degrades the performance of speaker verification, especially when the durations of training and testing utterances are very different. To alleviate this issue, we propose the deep segment attentive embedding method to learn the unified speaker embeddings for utterances of variable duration. Each utterance is segmented by a sliding window and LSTM is used to extract the embedding of each segment. Instead of only using one local segment, we use the whole utterance to learn the utterance-level embedding by applying an attentive pooling to the embeddings of all segments. Moreover, the similarity loss of segment-level embeddings is introduced to guide the segment attention to focus on the segments with more speaker discriminations, and jointly optimized with the similarity loss of utterance-level embeddings. Systematic experiments on Tongdun and VoxCeleb show that the proposed method significantly improves robustness of duration variant and achieves the relative Equal Error Rate reduction of 50% and 11.54% , respectively.'
address: |
$^1$ National Laboratory of Patten Recognition, Institute of Automation, Chinese Academy of Sciences, China\
$^2$ School of Artificial Intelligence, University of Chinese Academy of Sciences, China\
`{bin.liu2015,shuai.nie,yaping.zhang,sliang,lwj}@nlpr.ia.ac.cn`
bibliography:
- 'strings.bib'
- 'refs.bib'
title: Deep Segment Attentive embedding for duration robust speaker verification
---
deep segment attentive embedding, speaker verification, duration robustness, LSTM
Introduction
============
\[sec:intro\] The key to speaker verification is to extract the utterance-level speaker vectors with a fixed dimension for utterances of variable duration. The extracted speaker vector is expected to be as close as possible to the same speaker while far from other speakers. It remains a challenge to extract the robust speaker vectors for utterances of variable duration, especially when the utterance duration varies greatly. The i-vector/PLDA framework [@Dehak2011Front; @Prince2007Probabilistic; @Cumani2013Probabilistic] can easily extract the fixed dimension speaker vectors for utterances of arbitrary duration using statistical modeling. But it suffers performance reduction when handling short utterances [@li2017deep; @snyder2017deep]. The reason is that i-vector is a Gaussian-based statistical feature, whose estimation need sufficient samples. And the short utterance will lead to the uncertainty in the estimated i-vector.
Deep learning based speaker embedding [@li2017deep; @Variani2014Deep; @wan2018generalized] is another mainstream approach to speaker verification, which has been extensively studied recently and achieved promising performance in short-duration text-independent task. There are two ways to extract speaker embeddings using deep models. One approach is averaging bottleneck features from frame-level speaker classification networks [@Variani2014Deep]. Another approach is directly learning utterance-level speaker embeddings with distance-based similarity loss, such as triplet loss [@li2017deep; @zhang2017end] and generalized end-to-end (GE2E) loss [@wan2018generalized].
LSTM-based speaker embedding is one of the most important deep speaker verification methods and has been demonstrated to be substantially promising [@Sainath2015Convolutional; @Heigold2015End]. Owing to the powerful ability in modeling time-series data, LSTM can effectively capture the local correlation information of speech, which is very important for speaker verification. But it is still challenging for LSTM to model the long-term dependency of utterances, especially very long utterances. In addition, in order to facilitate batch training, LSTM-based speaker verification usually uses a fixed-length local segment randomly truncated from an utterance to learn the utterance-level speaker embedding in training phase, while using the average embedding of all segments of a test utterance to verify the speaker in testing phase, which leads to a critical mismatch between testing and training. The mismatch dramatically degrades the performance of speaker verification, especially when the difference of durations between training and testing utterances is large. Many methods are proposed to handle the issue of duration variability. The attention-based pooling [@okabe2018attentive; @zhu2018self] is one of the most important technologies. But most of the attention mechanisms are performed at the frame level, which will leads to the “over-average” problem, especially when the utterance is very long.
To alleviate this issue, we propose the deep segment attentive embedding method to learn the unified speaker embeddings for utterances of variable duration. For both training and testing, we use a sliding window to divide utterances into the fixed-length segments and then use LSTM to extract the embedding of each segment. Finally, all segment-level embeddings of an utterance are pooled into a fixed-dimension vector through the segment attention, which is used as the utterance-level speaker embedding. The similarity loss of utterance-level embeddings is used to train the whole network. In addition, in order to guide the segment attention to focus on the segments with more speaker discriminations, we further incorporate the similarity loss of segment-level embeddings. With the joint optimization of the segment-level and utterance-level similarity loss, both local details and global information of utterances are taken into account. Instead of only using one local segment, we use the whole utterance to learn the utterance-level embedding, which unifies the process of training and testing and avoids the mismatch between them.
Related Work {#sec:relatedwork}
============
There are some efforts on the issue of duration variability. For example, in the conventional i-vector systems, [@Kenny2013PLDA] proposed to propagate the uncertainty relevant to the i-vector extraction process into the PLDA model, which better handled the duration variability. Moreover, in the deep learning based speaker embedding systems, the complementary center loss is proposed in [@Na2018Deep; @Nam2018Robust; @Sarthak2018Learning] in order to solve the problem of large variation in text-independent utterances, including the duration variation. It acts as a regularizer that reduces the intra-class distance variance of the final embedding vectors. However, they don’t explicitly model the duration variability of utterances and the mismatch between training and testing phase still exists.
Furthermore, attention mechanisms have been utilized to capture the long-term variations of speaker characteristics in [@okabe2018attentive; @zhu2018self]. An important metric is computed by the attention network, which is used to calculate the weighted mean of the frame-level embedding vectors. However, most of the attention mechanisms are performed at the frame level, which will leads to the “over-average" problem, especially when the utterance is very long.
Proposed Approach {#sec:Deep-Segment-Attentive-Embeddings}
=================
It is still challenging for LSTM to model the long-term dependency of utterances, especially very long utterances. And the mismatch between training and testing phase degrades the performance of speaker verification, especially when the difference of durations between training and testing utterances is large. Therefore, we propose the deep segment attentive embedding method to extract the unified speaker embeddings for utterances of variable duration.
As is shown in Fig. \[fig:System-overview\], we use a sliding window with $50\%$ overlap to divide utterances into the fixed-length segments and LSTM is used to extract the embedding of each segment. Finally, all segment-level embeddings of an utterance are pooled into a fixed-dimension utterance-level speaker embedding through the segment attention mechanism. The whole network is trained with the joint supervision of the utterance-level and segment-level similarity loss. It can extract the unified speaker embeddings for utterances of variable duration and take into account both local details and global information of utterances, especially long utterances.
Deep segment attentive embedding
--------------------------------
![System overview. For each batch training, there are $Q \times P$ utterances from $Q$ different speakers and each speaker has $P$ utterances. We only draw one utterance for simplicity.[]{data-label="fig:System-overview"}](framework8.pdf "fig:"){width="42.00000%"}\
For both training and testing, we use a sliding window with $50\%$ overlap to divide an utterance into the fixed-length segments. Supposed that we get $N$ speech segments $\mathbf{X} = \{\pmb{x}_{1}, \pmb{x}_{2}, \cdots , \pmb{x}_{N}\}$. The sliding window length $T$ is randomly chosen within $[80,120]$ frames but the length of segments in a batch is fixed. The vector $\mathbf{x}_{n}^{t}$ represents the feature of segment $n$ at frame $t$, which is fed into the network and the output is $\mathbf{h}_{n}^{t}$. The last frame of output is used as the segment representation $f(\mathbf{x}_{n};\mathbf{w})=\mathbf{h}_{n}^{T}$, where $\mathbf{w}$ represents parameters of the network. The segment-level speaker embedding is defined as the $L_2$ normalization of the segment representation: $$\label{equ_eji}
\mathbf{e}_{n} = \frac{f(\mathbf{x}_{n};\mathbf{w}) }{\left \| f(\mathbf{x}_{n};\mathbf{w}) \right \|_{2}}.$$ We compute the embedding vector of each segment according to Eq. \[equ\_eji\] $\mathbf{E} = \{\pmb{e}_{1}, \pmb{e}_{2}, \cdots , \pmb{e}_{N}\}$. Let the dimension of the segment-level speaker embedding $\pmb{e}_n$ be $d_e$.
It is often the case that some segment-level embeddings are more relevant and important for discriminating speakers than others. We therefore apply attention mechanisms to integrate the segment embeddings by automatically calculating the importance of each segment. For each segment-level embedding $\pmb{e}_n$, we could learn a score $\pmb{\alpha}_{n}$ using the segment attention mechanism. All segment-level embeddings of an utterance are pooled into a fixed-dimension utterance-level speaker embedding through the segment attention mechanism.
For each segment embedding $\pmb{e}_{n}$, we apply the multi-head attention mechanism [@lin2017structured] to learn a score $\pmb{\alpha}_{n}$ as follows:
$$\label{equ_att_ht}
\pmb{\alpha}_{n} = \text{softmax} \left( g(\pmb{e}_n \mathbf{W}_1)\mathbf{W}_2 \right),$$
where $\mathbf{W}_1$ and $\mathbf{W}_2$ are parameters of the multi-head attention mechanism; $\mathbf{W}_1$ is a matrix of size $d_e \times d_a$; $\mathbf{W}_2$ is a matrix of size $d_a \times d_r$; $d_a$ is the attention dim and $d_r$ is a hyperparameter that represents the number of attention heads; $g(\cdot)$ is the ReLU activation function [@nair2010rectified]. When the number of attention heads $d_r= 1$, it is simply a basic attention. The normalized weight $\pmb{\alpha}_{n} \in [0,1]$ is computed by the softmax function. The weight vector is then used in the attentive pooling layer to calculate the utterance-level speaker embedding $\tilde{\pmb{e}}$: $$\label{equ_e}
\tilde{\pmb{e}} = \sum_{n=1}^{N}\pmb{\alpha}_{n} \pmb{e}_n.$$
When the number of attention heads $d_r = 1$, $\tilde{\pmb{e}}$ is simply a weighted mean vector computed from $\mathbf{E}$, which is expected to reflect an aspect of speaker discriminations in the given utterance. Obviously, speakers can be discriminated along multiple aspects, especially when the utterance duration is long. By increasing $d_r$, we can easily have multiple attention heads to focus on different pattern aspects from an utterance. In order to encourage diversity in the attention vectors, [@zhu2018self] introduced a penalty term $\mathcal L_{p}$ when $d_r > 1$: $$\label{equ_lp}
\mathcal L_{p} = \left \| \mathbf{A}^T \mathbf{A}-\mathbf{I} \right \|_F^2,$$ where $\mathbf{A}=\left [ \pmb{\alpha}_{1}, \cdots , \pmb{\alpha}_{N} \right ]$ is the attention matrix; $\mathbf{I}$ is the identity matrix and $\left \| \cdot \right \|_F$ represents the Frobenius norm of a matrix. $\mathcal L_{p}$ can encourage each attention head to extract different information from the same utterance. It is similar to $L_2$ regularization and is minimized together with the original cost of the system.
Loss function
-------------
After getting the utterance-level speaker embedding, we calculate the similarity loss using the generalized end-to-end (GE2E) loss formulation [@wan2018generalized]. The GE2E loss is based on processing a large number of utterances at once to minimize the distance of the same speaker while maximizing the distance of different speakers.
For each batch training, we randomly choose $Q \times P$ utterances from $Q$ different speakers with $P$ utterances per speaker. And we calculate the utterance-level speaker embedding $\tilde{\pmb{e}}_{ji}$ based on Equations \[equ\_eji\], \[equ\_att\_ht\], \[equ\_e\] for each utterance. $\tilde{\pmb{e}}_{ji}$ represents the speaker embedding of the $j^{\text{th}}$ speaker’s $i^{\text{th}}$ utterance. And the centroid of embedding vectors from the $j^{\text{th}}$ speaker is defined: $$\label{equ_cj}
\mathbf{c}_j = \mathbf{E}_{i} \left [ \tilde{\pmb{e}}_{ji} \right ] = \frac{1}{P} \sum_{i=1}^{P} \tilde{\pmb{e}}_{ji}.$$ GE2E builds a similarity matrix $\mathbf{S}_{ji,k}$ that defines the scaled cosine similarities between each embedding vector $\tilde{\pmb{e}}_{ji}$ to all centroids $\mathbf{c}_k$ $(1 \leqslant j,k \leqslant Q \text{ and } 1 \leqslant i \leqslant P)$: $$\label{equ_sjik}
\mathbf{S}_{ji,k} = w \cdot \cos(\tilde{\pmb{e}}_{ji}, \mathbf{c}_k) + b,$$ where $w$ and $b$ are learnable parameters. The weight is constrained to be positive $w > 0$, because the scaled similarity is expected to be larger when the cosine similarity is larger.
During the training, each utterance’s embedding is expected to be similar to the centroid of that utterance’s speaker, while far from other speakers’ centroids. The loss on each speaker embedding $\tilde{\pmb{e}}_{ji}$ could be defined as: $$\label{equ_loss_e}
\mathcal L(\tilde{\pmb{e}}_{ji}) = \log \sum_{k=1}^{Q} \exp(\mathbf{S}_{ji,k}) - \mathbf{S}_{ji,j} .$$ And the utterance-level GE2E loss $\mathcal L_{u}$ is the sum of all losses over the similarity matrix, shown as: $$\label{equ_lg}
\mathcal L_{u}(\mathbf{x};\mathbf{w}) = \sum_{j,i}\mathcal L(\tilde{\pmb{e}}_{ji}).$$
For the text-independent speaker verification, each extracted segment-level embedding is expected to capture the speaker characteristics. In order to guide the segment attention to focus on the segments with more speaker discriminations, we further incorporate the similarity loss of segment-level embeddings. The segment-level GE2E loss $\mathcal L_{s}$ is similar to the utterance-level GE2E loss $\mathcal L_{u}$ except that it takes all segment-level embeddings as input, which could help the proposed model to learn more effective ways of embedding fusion and accelerate model convergence. The objective function can be formulated as: $$\label{equ_ls}
\mathcal L_{s}(\mathbf{x};\mathbf{w}) = \sum_{j,i} \sum_{n}\mathcal L(\mathbf{e}_{n}).$$
Finally, the utterance-level GE2E loss, segment-level GE2E loss and penalty loss are combined together to construct the total loss, shown as: $$\small
\label{equ_l_all}
\mathcal L = \mathcal L_{u} + \lambda_s \mathcal L_{s} + \lambda_p \mathcal L_{p}$$ The magnitude of the segment-level GE2E loss and penalty loss is controlled by hyperparameters $\lambda_s$ and $\lambda_p$. With the joint optimization of the segment-level and utterance-level GE2E loss, both local details and global information of utterances are taken into account. Our proposed method can extract the unified speaker embeddings for utterances of variable duration, which unifies the process of training and testing and avoids the mismatch between them.
Experiments {#sec:Experiments}
===========
We report speaker verification performance on Tongdun and VoxCeleb [@nagrani2017voxceleb] corpora. The proposed deep segment attentive embedding is compared with the generalized end-to-end loss based embedding as well as the traditional i-vector. We use Equal Error Rate (EER) to quantify the system performance.
Data {#ssec:Data}
----
**Tongdun**. The corpus is from the speaker verification competition held by Tongdun technology company [@kesci-web], which consists of more than $120$K utterances from $1,500$ Chinese speakers in training set and $3,000$ trial pairs are provided as test data. Most of the training data are short utterances with average duration of $3.7$s, while utterances in test set are very long and average duration is about $20$s.
**VoxCeleb**. The training set consists of more than $140$K utterances of $1,251$ speakers. And $37,720$ trial pairs from $40$ speakers are used as evaluation data for the verification process. The average duration of training and evaluation data is $8.24$s and $8.28$s, respectively.
For each speech utterance, a VAD [@Mak2014A; @yu2011comparison] is applied to prune out silence regions.
i-vector system {#ssec:i-vector-system}
---------------
The i-vector system uses $20$-dimensional MFCCs as front-end features, which are then extended to $60$-dimensional acoustic features with their first and second derivatives. Cepstral mean normalization is applied. An i-vector of $400$ dimensions is then extracted from the acoustic features using a $2048$-mixture UBM and a total variability matrix. PLDA serves as the scoring back-end. Mean subtraction, whitening, and length normalization [@Garcia2011Analysis] are applied to the i-vector as preprocessing steps, and the similarity is measured using a PLDA model with a speaker space of $400$ dimensions.
Deep speaker embedding system {#ssec:Deep-speaker-embedding-system}
-----------------------------
For deep speaker embedding systems, we take the $40$-dimensional filter-banks with $32\text{-ms}$ Hamming window and $16\text{-ms}$ frame shift as the input features, and each dimension of features is normalized to have zero mean and unit variance over the training set. A combination of $3$-layer LSTM and a linear projection layer is used to extract the speaker embeddings. Each LSTM layer contains $512$ nodes, and the linear projection layer is connected to the last LSTM layer, whose output size is $256$. Therefore, we can extract $256$-dimension speaker embeddings according to the outputs of the linear projection layer. The cosine similarity score of the pair of embedding vectors is computed to verify the speaker. According to [@wan2018generalized], the scaling factors $w$ and $b$ in Eq. \[equ\_sjik\] are initialized to $10$ and $5$, respectively.
We take the LSTM-based speaker embedding system proposed by Wan [@wan2018generalized] as the baseline, which is optimized by GE2E loss. Let us denote the baseline system as “LSTM-GE2E”. “LSTM-GE2E” uses the local segments truncated from utterances to learn the utterance-level speaker embedding. The length of segments is randomly chosen within $[80, 120]$, but all segments in a batch is fixed. In the testing phase, each utterance is segmented by a sliding window of $100$ frames with $50\%$ overlap. We extract the embedding of each segment and then average them as the speaker embedding of the utterance. The embedding of each segment is obtained by performing a frame-level attention pooling operator on the outputs of the linear projection layer.
Compared to “LSTM-GE2E”, the proposed deep segment attentive embedding system uses the whole utterance to learn the utterance-level speaker embedding by the segment attention, which is denoted as “DSAE-GE2E”. The segment attention is implemented by performing the multi-head attention pooling on the segment-level embeddings. The attention dim $d_a$ is set to $128$ and the attention head number $d_r$ is chosen from $\left [1, 2, 5 \right]$. In addition, “DSAE-GE2E” is jointly optimized by the utterance-level and segment-level GE2E losses, as shown in Eq. \[equ\_l\_all\]. The weights $\lambda_s$ and $\lambda_p$ of terms in Eq. \[equ\_l\_all\] are experimentally set to $0.2$ and $0.001$, respectively.
All deep speaker embedding models are trained from a random initialization by an Adam optimizer [@kingma2014adam]. The initial learning rate is set to $0.001$ and decayed according to the performance of the validation set. For each batch training, we randomly choose $640$ utterances of $64$ speakers with $10$ utterances per speaker. We mention that the length of segments in a batch is fixed. About $15,000$ batches are used to train the network. In addition, the $L_{2} \text{ norm}$ of gradient is clipped at $3$ to avoid gradient explosion [@pascanu2012understanding].
Results
-------
In the following results, “LSTM-GE2E” refers to the deep speaker embedding system trained with GE2E loss. “DSAE-GE2E-k” denotes the proposed deep segment attentive embedding system with the multi-head attention layer of $k$ attention heads.
[p[3.2cm]{} p[1.5cm]{}<]{} **Embedding** & **EER (%)**\
i-vector/PLDA & 3.0\
LSTM-GE2E & 2.0\
DSAE-GE2E-1 & 1.5\
DSAE-GE2E-2 & 1.3\
DSAE-GE2E-5 & 1.0\
Table \[tab:eer-Tongdun\] shows the performance on Tongdun test set. All deep learning based speaker embedding systems outperform the traditional i-vector system, which shows the effectiveness of the deep speaker embeddings. In general, the proposed “DSAE-GE2E” consistently and significantly outperform “LSTM-GE2E”. For the multi-head attention layer, more attention heads achieve greater improvement. “DSAE-GE2E-1” is $25\%$ better in EER than “LSTM-GE2E” and “DSAE-GE2E-5” outperform “LSTM-GE2E” by $50\%$. Note that the difference of durations between Tongdun training and testing utterances is very large and our systems can extract the unified utterance-level speaker embeddings for utterances of variable duration, which significantly improve the system performance. Results indicate that our proposed utterance-level speaker embedding is a duration robust representation for speaker verification.
[p[3.2cm]{} p[1.5cm]{}<]{} **Embedding** & **EER (%)**\
i-vector/PLDA &8.9\
LSTM-GE2E &6.2\
DSAE-GE2E-1 &5.8\
DSAE-GE2E-2 &5.5\
DSAE-GE2E-5 &5.2\
The performance on VoxCeleb test set is shown in Table \[tab:eer-VoxCeleb\]. Our proposed “DSAE-GE2E” also outperforms the i-vector system and “LSTM-GE2E”, which demonstrates the effectiveness of the proposed method. “DSAE-GE2E-1” is $6.5\%$ better in EER than “LSTM-GE2E” and “DSAE-GE2E-5” outperform “LSTM-GE2E” by $16.1\%$. The relative EER reduction is smaller than Tongdun corpus because there is little duration difference between VoxCeleb training and testing utterances. Our proposed method can obtain greater performance improvement when the difference of durations between training and testing utterances is larger.
Conclusions {#sec:conclusions}
===========
In this paper, we propose the deep segment attentive embedding method to learn the unified speaker embeddings for utterances of variable duration. Each utterance is segmented by a sliding window and LSTM is used to extract the embedding of each segment. Instead of only using one local segment, we use the whole utterance to learn the utterance-level embedding by applying an attentive pooling to embeddings of all segments. Moreover, the similarity loss of segment-level embeddings is introduced to guide the segment attention to focus on the segments with more speaker discriminations, and jointly optimized with the similarity loss of utterance-level embeddings. Systematic experiments on Tongdun and VoxCeleb demonstrate the effectiveness of the proposed method. In the future work, we will investigate different neural network architectures and attention strategies in order to obtain greater performance improvement.
Acknowledgements
================
This work was supported by the China National Nature Science Foundation (No. 61573357, No. 61503382, No. 61403370, No. 61273267, No. 91120303).
|
---
abstract: 'Using an array of high-purity Compton-suppressed germanium detectors, we performed an independent measurement of the $\beta$-decay branching ratio from to the second-excited state, also known as the Hoyle state, in . Our result is , which is a factor ${\mathord{\sim}}2$ smaller than the previously established literature value, but is in agreement with another recent measurement. This could indicate that the Hoyle state is more clustered than previously believed. The angular correlation of the Hoyle state $\gamma$ cascade has also been measured for the first time. It is consistent with theoretical predictions.'
author:
- 'M. Munch'
- 'M. Alcorta'
- 'H. O. U. Fynbo'
- 'M. Albers'
- 'S. Almaraz-Calderon'
- 'M. L. Avila'
- 'A. D. Ayangeakaa'
- 'B. B. Back'
- 'P. F. Bertone'
- 'P. F. F. Carnelli'
- 'M. P. Carpenter'
- 'C. J. Chiara'
- 'J. A. Clark'
- 'B. DiGiovine'
- 'J. P. Greene'
- 'J. L. Harker'
- 'C. R. Hoffman'
- 'N. J. Hubbard'
- 'C. L. Jiang'
- 'O. S. Kirsebom'
- 'T. Lauritsen'
- 'K. L. Laursen'
- 'S. T. Marley'
- 'C. Nair'
- 'O. Nusair'
- 'D. Santiago-Gonzalez'
- 'J. Sethi'
- 'D. Seweryniak'
- 'R. Talwar'
- 'C. Ugalde'
- 'S. Zhu'
bibliography:
- 'ANL1373-2.bib'
title: 'Independent measurement of the Hoyle state $\beta$ feeding from $^{12}\mathrm{B}$ using Gammasphere'
---
Introduction
============
Carbon is the fourth most abundant element in the Universe and it plays a key role in stellar nucleosynthesis. It is mainly formed in stars at a temperature of $10^{8}$-$10^{9}$ in the triple-$\alpha$ fusion reaction, which proceeds via the second-excited state, also known as the Hoyle state, at in , famously proposed by Hoyle in 1953 [@Hoyle1954].
The first attempt to theoretically explain the structure of the state was the linear alpha chain model by Morinaga in 1956 [@Morinaga1956], where he, furthermore, conjectured a $2^{+}$ state in the region. Several more sophisticated models have been developed since, as summarized in Ref. [@Freer2014]. Most of these models predict a collective $2^{+}$ excitation of the Hoyle state in the region of above it. Interestingly, the collective state increases the triple-$\alpha$ reaction rate at $T > 10^{9}\,\si{\K}$ by a factor of 5-10 compared to the results of Caughlan [*et al.* ]{}[@Caughlan1988; @Freer2009]. This makes it highly relevant for core collapse supernovae [@Tur2007; @Tur2010; @Magkotsios2011; @The1998].
Experimentally, it is challenging to investigate this energy region, since there are contributions from several broad states and from the so-called Hoyle state “ghost anomaly” [@Barker1962; @Wilkinson1963; @Selove1990]. Using inelastic proton scattering, Freer [*et al.* ]{}provided the first evidence for a broad $2^{+}$ contribution at with a width of [@Freer2009]. Itoh [*et al.* ]{}corroborated these results using inelastic $\alpha$-scattering [@Itoh2011] and a simultaneous analysis was published as well [@Freer2012]. Results from an experiment using the alternative ${\ce{^{12}C}\xspace}(\gamma,\alpha)\ce{^{8}Be}$ reaction also identified a $2^{+}$ state in this region, but at $10.13_{-0.05}^{+0.06}$ and with a much larger width of $2080^{+330}_{-260}$ [@Zimmerman2013; @Zimmerman2013phd]. The reason for this discrepancy is presently not understood. A natural explanation would be that several $2^{+}$ resonances are present in the region, and that the different reaction mechanisms populate these with different strength.
An alternative experimental probe is the $\beta$ decay of and . Due to the selection rules, decay of these $1^{+}$ systems will predominantly populate states with spin and parity $0^{+}$, $1^{+}$ or $2^{+}$ and not the $3^{-}$ state at , which is the dominant channel in inelastic scattering experiments. This technique has been used in several studies of [@Cook1957; @Alburger1977; @Fynbo2005; @Diget2005; @Hyldegaard2009; @Hyldegaard2010], but none of these has identified a $2^{+}$ state at . The $\beta$ decay populates a somewhat featureless excitation spectrum in which is analyzed with the R-matrix formalism in Ref. [@Hyldegaard2010]. This analysis identified both $0^{+}$ and $2^{+}$ resonances in the to region with recommended energies for both resonances at . The R-matrix analysis includes a large contribution from the high energy tail of the Hoyle state, which is sometimes referred to as the “ghost anomaly” [@Barker1962; @Wilkinson1963]. This contribution is strongly dependent on the branching ratio with which the Hoyle state is populated in the $\beta$ decay.
In the most recent experimental study of the $\beta$ decay, the beam was implanted in a silicon detector, which provided accurate normalization of the branching ratios, resulting in a revision of several of these [@Hyldegaard2009]. More specifically, the branching ratio to the Hoyle state from the decay of was determined to be , which is inconsistent with the previously established value of [@Selove1990; @Hyldegaard2009b] ( is listed in Ref. [@Selove1990], but this should be revised [@Hyldegaard2009b]). The reduced branching ratio for the population of the Hoyle state was used in the R-matrix analysis [@Hyldegaard2010]. Furthermore, as the $\beta$ decay to a pure $3\alpha$-cluster system is forbidden, a precise measurement of the branching ratio will provide insight into the strength of the cluster-breaking component of the Hoyle state [@Kanada-Enyo2007]. It is therefore important to provide experimental confirmation of the reduced branching ratio measured in Ref. [@Hyldegaard2009].
Here, we report on an independent measurement of this branching ratio through a measurement of the $\gamma$ decay of the Hoyle state with an array of high-purity germanium detectors. The results of a preliminary analysis have been reported in [@Alcorta2014].
Method
======
Figure \[fig:level\_scheme\] shows the lowest states in , the triple-$\alpha$ threshold and the ground state of . The first-excited state is below the $\alpha$ threshold and will only $\gamma$ decay, whereas the Hoyle state cannot $\gamma$ decay directly to the ground state as it is a $0^{+}$ level. It can, however, decay via the first-excited state by emission of a [-]{} photon.
![Level scheme of also showing the $\alpha$-threshold and the ground state. Energies are given in relative to the ground state of [@Selove1990].[]{data-label="fig:level_scheme"}](decay_scheme){width="0.935\columnwidth"}
The number of $\gamma$ decays from the Hoyle state can be determined by counting the number of [-]{} photons, and by furthermore requiring a simultaneous detection of a [[-]{} ]{}$\gamma$ ray, the background is vastly reduced. The product of the branching ratio to the Hoyle state and its relative $\gamma$ width can then be determined by normalizing to the decay of the first-excited state $$\label{eq:br}
{\textup{BR}}(7.65) \frac{\Gamma_{\gamma}}{\Gamma} = {\textup{BR}}(4.44) \frac{ N_{\gamma \gamma}}{N_{4.44} \epsilon_{3.21} C_{\theta}},$$ where $N_{\gamma \gamma}$ is the number of coincidence events, $\epsilon_{3.21}$ the efficiency for detecting a [-]{} photon and $C_{\theta}$ corrects for the angular correlation between the two photons.
The relative $\gamma$ width can be determined from all available data for the relative radiative width [@Obst1976], excluding Seeger [*et al.* ]{}[@Seeger1963], by subtracting the recommended relative pair width from [@Freer2014]; yielding $\frac{\Gamma_{\gamma}}{\Gamma} = \num{4.07(11)E-4}$. A conservative estimate of the branching ratio to the first excited state, ${\textup{BR}}(4.44) = \SI{1.23(5)}{\%}$ has been published in [@Selove1990].
Using this method, the branching ratio can be determined solely with $\gamma$-ray detectors, providing an experiment with entirely different systematic uncertainties than previous measurements based on detection of $\alpha$ or $\beta$ particles.
Experiment
==========
![(Color online) CAD drawing of the chamber manufactured from a Bonner sphere. (A) Bonner sphere, (B) vacuum flange, (C) target holder / Faraday cup, (D) electrical feed through, (E) O-rings[]{data-label="fig:bonner"}](bonner-new){width="0.466\columnwidth"}
was produced via the $\ce{^{11}B}(d, p){\ce{^{12}B}\xspace}$ reaction in inverse kinematics, using a pulsed ( on, off) [-]{} beam delivered by the Argonne Tandem-Linac Accelerator System (ATLAS) located at Argonne National Laboratory. A deuterated titanium foil (), sufficiently thick to stop the beam, was used as target. The target was manufactured according to the method discussed in Ref. [@Greene2010] and it contained approximately deuterium (estimated by weight).
Photons were detected using Gammasphere [@Lee1990], which is an array of 110 high-purity Compton-suppressed germanium detectors of which 98 were operational during the experiment. The array was operated in singles mode, where any of the detectors could trigger the data acquisition (DAQ). Data were only acquired during the beam-off period. Therefore, only delayed activity was measured (the half life of is [@Selove1990]). For each event, the time relative to beam-off as well as the energy and time for each $\gamma$ ray in the detectors were recorded.
In order to minimize bremsstrahlung caused by high-energy $\beta$ particles, a low-Z chamber was designed - see Figure \[fig:bonner\]. The chamber was manufactured from a Bonner sphere and was designed to minimize contribution from bremsstrahlung while maintaining high gamma-ray efficiency.
Analysis
========
Yield
-----
During the experiment ${\mathord{\sim}}10^9$ $\gamma$ rays were collected in 67 hours. The singles spectrum is displayed in Fig. \[fig:full\_spec\], where the transition from the first-excited state in at (A) together with the first (B) and second escape (C) peaks at lower energy are clearly seen. The insert shows the region from in which a structure around is visible, as indicated by an arrow. However, the region is dominated by a peak at .
![The entire singles spectrum acquired during the beam-off period. The [[-]{} ]{}peak (A) and its escape peaks (B,C) are clearly visible. The insert shows the region. A small structure is visible around , indicated by an arrow.[]{data-label="fig:full_spec"}](full_spec){width="\mwidth"}
The [[-]{} ]{}peak was fitted with a sum of a Gaussian distribution, a skewed Gaussian distribution, a linear background and a smoothed step function [@Debertin1988]. In order to minimize systematic effects, the fit was performed with the Poisson log likelihood ratio [@Bergmann2002] using the <span style="font-variant:small-caps;">minuit</span> minimizer [@James1975]. From this procedure, the area of the peak was determined to be $N_{4.44} = \num{9.20(2)E6}$, where the error was dominated by uncertainties in the functional form of the peak.
Coincidence spectrum {#sec:coin_spec}
--------------------
![(Color online) Coincidence spectrum, acquired by gating on the [ ]{}peak and the time difference. A clear peak centered at is consistent with the Hoyle state decaying via the first-excited state.[]{data-label="fig:coin_spec"}](coin_spec){width="\mwidth"}
To obtain a coincidence spectrum, a gate was placed on the relative time between the two $\gamma$ rays and on the energy of the [[-]{} ]{}transition. The widths of these gates were chosen to minimize any systematic effects..
The coincidence spectrum is given in Fig. \[fig:coin\_spec\], where a clear peak centered at is visible. This is consistent with a cascade decay of the Hoyle state via the first-excited level. The peak was fitted with the same functional form as in the previous section, but the parameters for the skewed Gaussian are determined from peaks I-III in Fig. \[fig:full\_spec\]. Peak I-III originate from and produced in beam by reactions with . The area of the peak, determined from the fit, is $N_{\gamma \gamma} = \num{58(9)}$.
Efficiency
----------
The relative efficiency was determined using the standard calibration sources and mounted at the target position. This provides calibration points, both at low energy and in the important [-]{} region. The absolute efficiency was calculated using the coincidence method, including a correction for random coincidence events, for both a source and , which was produced by in-beam reactions [@Siegbahn]. From this procedure, the absolute efficiency at was determined to be $\epsilon_{3.21} = \num{2.94(2)} \%$.
Angular correlation
-------------------
![(Color online) Angular correlation of the Hoyle state $\gamma$ cascade corrected for the geometric efficiency (number of detector pairs with a given angle between them). The solid line shown is the best fit to equation .[]{data-label="fig:angular"}](ang_cor){width="\mwidth"}
Due to the excellent angular coverage of Gammasphere, it is possible to measure the angular correlation of the two $\gamma$ rays, which had not been measured previously. Using the gates described above and in addition requiring the energy of the second $\gamma$ ray to be within of , it is possible to extract the true coincidence events plus some background. The shape of the background was determined by gating outside the peak, and was found to be flat.
The angular correlation, corrected for the geometric efficiency (number of detector pairs with a given angle between them), is shown in Fig. \[fig:angular\], together with the best fit to the equation $$\label{eq:angular}
W(\theta) = k\left[1+a_{2}\cos^2(\theta)+a_{4}\cos^4(\theta)\right],$$ where $\theta$ is the angle between the two $\gamma$ rays. The result of the fit is $a_{2} = \num{-3.3(7)}$ and $a_{4} = \num{4.2(9)}$, which is consistent with the theoretical expectations $a_{2} = -3$ and $a_{4} = 4$ for a $0\rightarrow2\rightarrow0$ cascade [@Brady1950].
With the theoretical angular correlation confirmed, it can be used to estimate the correction factor $C_{\theta}$ from eq. . This is done with a simple Monte Carlo simulation of the detector setup, which gives $C_{\theta} = \num{1.00(1)}$, as was expected from the large angular coverage by Gammasphere.
Extraction of branching ratio
-----------------------------
The property directly measured in this experiment is the product of the relative $\gamma$ width and the $\beta$ feeding of the Hoyle state $$\label{eq:product}
{\textup{BR}}(7.65) \frac{\Gamma_{\gamma}}{\Gamma} = \num{2.6(4)E-4}.$$ Inserting the calculated value for the relative $\gamma$ width into equation gives $$\label{eq:result}
{\textup{BR}}(7.65) = \SI{0.64(11)}{\%},$$ which is clearly inconsistent with the previous literature value of [@Selove1990; @Hyldegaard2009b], but agrees with that of found in Ref. [@Hyldegaard2009]. Therefore, the feeding of the Hoyle state from is roughly a factor of $2$ smaller than indicated by Refs. [@Selove1990; @Hyldegaard2009b].
Discussion
==========
The branching ratio from and to the Hoyle state is a sensitive way to probe the clustering of this state, as the $\beta$-decay matrix element to the pure $3\alpha$ system is exactly zero due to the Pauli principle [@Kanada-Enyo2007]. The fact that $\beta$ decay is possible means that the Hoyle state must contain some $\alpha$-cluster breaking component. Theoretically, this is obtained by mixing shell-model-like states with cluster states as it is done; e.g; in Fermionic Molecular Dynamics (FMD) [@Roth2004; @Chernykh2007] and Antisymmetrized Molecular Dynamics (AMD) approach [@Kanada-Enyo2007]. Alpha-cluster breaking was explicitly investigated in Ref. [@Suhara2015] using a hybrid shell/cluster model, where it was found that the spin-orbit force significantly changes the excited $0^{+}$ states.
Here, we compute the $\log ft$ value, which can be directly compared with these models. The available phase space ($f$ factor) for $\beta$ decay from the ground state of to the Hoyle state was computed using the method in [@Wilkinson1974], with the excitation energy and half life from [@Selove1990]. With this input our result is $$\label{eq:logft}
\log ft = \num{4.50(7)}.$$ Due to the large change of the measured branching ratio compared to previous results [@Selove1990], the theoretical prediction of the AMD model, $\log ft = \num{4.3}$ [@Kanada-Enyo2007], is no longer compatible with the experiment.
Hence, our branching ratio, together with the branching ratio for both and from Hyldegaard [*et al.* ]{}[@Hyldegaard2009], indicate that the $\alpha$ clustering of the Hoyle state is more pronounced than previously believed.
Conclusion and outlook
======================
The $\beta$-decay branching ratio from to the second-excited state of has been measured using an array of high-purity Compton-suppressed germanium detectors. The branching ratio was determined by counting the Hoyle state $\gamma$ decay, and normalizing to the decay of the first-excited state. The result is , consistent with the value found in [@Hyldegaard2009], but is a factor ${\mathord{\sim}}2$ smaller than the previously-established value from [@Selove1990]. The updated branching was used to compute $\log ft = \num{4.50(7)}$, which is not consistent with latest results from AMD calculations [@Kanada-Enyo2007]. Our results indicate that the clustering of the Hoyle state is more pronounced than previously thought.
The angular correlation between the two photons emitted in the decay of the Hoyle state has also been measured. The distribution was consistent with theoretical expectations [@Brady1950].
The errors on the present measurement are dominated by the uncertainty on the number of coincidence events, which contributes of the total error, while and come from the branching ratio to the first-excited state and the relative $\gamma$ width of the Hoyle state, respectively. Therefore, it is possible to make a ${\mathord{\sim}}\SI{6}{\%}$ measurement of either the $\gamma$ width or the $\beta$-branching ratio by increasing statistics.
During the experiment, the beam current was limited to in order to minimize neutron damage to Gammasphere. The main source for these neutrons was reactions with titanium since the beam energy is above the Coulomb barrier. Exchanging titanium with hafnium permits running with higher beam currents which, when combined with digital Gammasphere [@Anderson2012], would make it possible to accumulate sufficient statistics. Research into production of such a target is ongoing.
Acknowledgments
===============
This work is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under Contracts No. DE-AC02-06CH11357 and No. DE-FG02-04ER41320 and Grant No. DE-FG02-94ER40834. This research used resources of ANL’s ATLAS facility, which is a DOE Office of Science User facility. OSK acknowledges support from the Villum Foundation. We also acknowledge financial support from the European Research Council under ERC starting grant LOBENA, No. 307447
|
---
abstract: 'When a signal is recorded in an enclosed room, it typically gets affected by reverberation. This degradation represents a problem when dealing with audio signals, particularly in the field of speech signal processing, such as automatic speech recognition. Although there are some approaches to deal with this issue that are quite satisfactory under certain conditions, constructing a method that works well in a general context still poses a significant challenge. In this article, we propose a method based on convolutive nonnegative matrix factorization that mixes two penalizers in order to impose certain characteristics over the time-frequency components of the restored signal and the reverberant components. An algorithm for implementing the method is described and tested. Comparisons of the results against those obtained with state of the art methods are presented, showing significant improvement.'
author:
- 'Francisco J. Ibarrola [^1]'
- 'Leandro E. Di Persia $^\ast$'
- 'Ruben D. Spies [^2]'
bibliography:
- 'ref\_der.bib'
title: Mixed penalization in convolutive nonnegative matrix factorization for blind speech dereverberation
---
signal processing, dereverberation, regularization.
Introduction
============
In recent years, many technological developments have attracted attention towards human-machine interaction. Since the most natural and easiest way of human communication is trough speech, much research effort has been put into achieving the same natural interaction with machines. This effort has already generated many advances in a wide variety of fields such as automatic speech recognition ([@kim_efficient_2015]), automatic translation systems ([@yun_multilingual_2014]) and control of remote devices trough voice ([@nesselrath2016combining]), to name only a few. A significant amount of work has been recently devoted to produce robustness in speech recognition ([@DMR+2008]), resulting in several advances in the areas of speech enhancement ([@kim_efficient_2015], [@martinez_denoising_2015]), multiple sources separation ([@DMY2009a], [@di_persia_using_2016]), and particularly in dereverberation techniques ([@tsilfidis2010]), which constitute the topic of this work.
When recorded in enclosed rooms, audio signals will most certainly be affected by reverberant components due to reflections of the sound waves in the walls, ceiling, floor or furniture. This can severely degrade the characteristics of the recorded signal ([@tashev2009]), generating difficult problems for its processing, particularly when required for certain speech applications ([@huang2001]). The goal of any dereverberation technique is to remove or to attenuate the reverberant components in order to obtain a cleaner signal. The dereverberation problem is called “blind” when the available data consists only of the reverberant signal itself, and this is the problem we shall deal with in this work.
Depending on the problem, our observation might consist of a single or multi-channel signal. That is, we might have a signal recorded by one or more microphones. For the latter case, quite a few methods exist that work relatively well ([@delcroix2014], [@wisdom2014]).
For the single-channel case, we may distinguish between supervised and unsupervised approaches. The first kind refers to those that begin with a training stage that serves to learn some characteristics of the reververation conditions, while the second kind alludes to those methods that can be implemented directly over the reverberant signal. Some supervised methods ([@moshirynia2014], [@xiao2014], [@nathwani2015]) appear to perform somewhat better than unsupervised ones, but they pose the disadvantage of needing learning data corresponding to the specific room conditions, microphone and source locations, and a previous process that might take a significant amount of time.
In the context of unsupervised blind dereverberation, although some recently proposed methods ([@wisdom2014], [@kameoka2009]) work reasonably well, there is still much room for improvement. Our work is based on a convolutive non-negative matrix factorization (NMF) reverberation model, as proposed by Kameoka *et al* ([@kameoka2009]), along with a Bayesian approach for building a generalized functional that mixes two types of penalizers over the elements of the representation model. Mixed penalization approaches have been recently used and successfully applied by several authors in many areas, mainly in signal and image processing applications ([@ibarrola2017l2bva], [@lazzaro2015], [@ibarrola2017inp], [@peterson2017], [@mazzieri2015]). These techniques have shown to produce good results in terms of enhancing certain desirable characteristics on the solutions while precluding unwanted ones.
A Reverberation Model
---------------------
Let $s, x:\mathbb{R}\rightarrow\mathbb{R}$, with support in $[0, \infty)$, be the functions associated to the clean and reverberant signals, respectively. As it is customary, we shall assume that the reverberation process is well represented by a Linear Time-Invariant (LTI) system. Thus, the reverberation model can be written as $$\label{eq:cont-model}
x(t) = (h\ast s)(t),$$ where $h:\mathbb{R}\rightarrow\mathbb{R}$ is the room impulse response (RIR) signal, and “$\ast$” denotes convolution. This LTI hypothesis implies we are assuming the source and microphone positions to be static, and the energy of the signal to be low enough for the effect of the non-linear components to be relatively insignificant.
When dealing with sound signals (particularly speech signals), it is often convenient to work with the associated spectrograms rather than the signals themselves. Thus, we make use of the short time Fourier transform (STFT), defined as $$\nonumber\mathbf{x}_k(t) \doteq \int_{-\infty}^{\infty}x(u)w(u-t)e^{-2\pi i u k}du,\;\;t,k\in\mathbb{R},$$ where $w:\mathbb{R}\rightarrow\mathbb{R}^+_0$ is a compactly supported, even function such that $\|w\|_1 = 1$. This function is called *window*.
In practice, we work with discretized versions of the signals involved ($x[\cdot],h[\cdot],s[\cdot],$ and $w[\cdot]$). With this in mind, we shall define the discrete STFT as $$\nonumber
\mathbf{x}_k[n] \doteq \sum_{m =-\infty}^{\infty}x[m]w[m-n]e^{-2\pi i m k},\;\;n,k\in\mathbb{N}.$$ Denoting the STFTs of $s$ and $h$ by $ \mathbf{s}_k[n]$ and $\mathbf{h}_k[n]$, respectively, a discretized approximation of the STFT model associated to (\[eq:cont-model\]) is given by $$\mathbf{x}_k[n] \approx \tilde{\mathbf{x}}_k[n] \doteq \sum_{\tau = 0}^{N_h-1} \mathbf{s}_k[n-\tau] \mathbf{h}_k[\tau],$$ where $n=1,\ldots, N,$ is a discretized time variable that corresponds to window location, $k=1,\ldots, K,$ denotes the frequency subband and $N_h$ is a parameter of the model associated to the expected maximum duration of the reverberation phenomenon. The model is built as in [@avargel2007system], being the approximation due to the use of badn-to-band filters only. Later on, the values of $n$ will be chosen in such a way that the union of the windows’ supports contain the support of the observed signal, and the values of $k$ in such a way that they cover the whole frequency spectrum, up to half the sampling frequency.
Now, let us write $\mathbf{h}_k[\tau] = |\mathbf{h}_k[\tau]|e^{j\phi_k[\tau]}$. It is well known ([@yegnanarayana1998]) that the phase angles $\phi_k[\tau]$ are highly sensitive with respect to mild variations on the reverberation conditions. To overcome the problems derived from this, we shall proceed (see [@kameoka2009]) treating the $K\times N_h$ variables $\phi_k[\tau]$ as *i.i.d.* random variables with uniform distribution in $[-\pi,\pi)$. Denoting the complex conjugate by “$^*$” and the Kronecker delta by $\delta_{ij}$, the expected value of $|\tilde{\mathbf{x}}_k[t]|^2$ is given by $$\begin{aligned}
E|\tilde{\mathbf{x}}_k[n]|^2 &= E \sum_{\tau,\tau'} \mathbf{s}_k[n-\tau] \mathbf{s}_k^*[n-\tau'] \mathbf{h}_k[\tau] \mathbf{h}^*_k[\tau'] \nonumber\\
&= E \sum_{\tau,\tau'} \mathbf{s}_k[n-\tau] \mathbf{s}_k^*[n-\tau'] \,|\mathbf{h}_k[\tau]|\,e^{j\phi_k[\tau]}\, |\mathbf{h}_k[\tau']|\,e^{-j\phi_k[\tau']} \nonumber\\
&= \sum_{\tau,\tau'} \mathbf{s}_k[n-\tau] \mathbf{s}_k^*[n-\tau']\, |\mathbf{h}_k[\tau]| \,|\mathbf{h}_k[\tau']|\,Ee^{j(\phi_k[\tau]-\phi_k[\tau'])} \nonumber\\
&= \sum_{\tau,\tau'} \mathbf{s}_k[n-\tau] \mathbf{s}_k^*[n-\tau']\, |\mathbf{h}_k[\tau]|\, |\mathbf{h}_k[\tau']|\, \delta_{\tau \tau'} \nonumber\\
&= \sum_{\tau} |\mathbf{s}_k[n-\tau]|^2 \,|\mathbf{h}_k[\tau]|^2. \nonumber\end{aligned}$$
Note that the $[-\pi,\pi)$ interval choice for $\phi_k[\tau]$ is arbitrary, since this result holds for any $2\pi-$length interval. Finally, let us define $S_k[n] \doteq |\mathbf{s}_k[n]|^2$, $H_k[n] \doteq |\mathbf{h}_k[n]|^2$ and $X_k[n] \doteq E|\tilde{\mathbf{x}}_k[n]|^2$. Then, our model reads $$\label{eq:mod-rep}
X_k[n] = \sum_{\tau} S_k[n-\tau] H_k[\tau],$$ and the square magnitude of the observed spectrogram components can be written as $$\label{eq:mod-obs}
Y_k[n] = X_k[n] +\epsilon_k[n],$$ where $\epsilon_k[n]$ denotes the representation error. As shown in [@kameoka2009], this model is equivalent to a convolutive NMF ([@smaragdis2004]) with diagonal basis. In the next section, we derive a cost function in order to find an appropriate convolutive representation that allows us to isolate the components $S_k[n]$.
A Bayesian approach
===================
In the following, we will use a Bayesian approach to derive a cost function which we will then minimize in order to obtain our regularized solution. Let us begin by assuming, for every $k$, $\epsilon_k[n], S_k[n], H_k[n]$ are independent random variables, also independent with respect to $k$. Also, let us denote by $S,Y, X\in \mathbb{R}^{K\times N}$ and $H \in \mathbb{R}^{K\times N_h}$ the non-negative matrices whose $(k,n)$-th elements are $S_k[n],Y_k[n], X_k[n]$ and $H_k[n]$, respectively.
As it is customary ([@kameoka2009]), for the representation error, we assume $\epsilon_k[n] \sim \mathcal{N}(0,\sigma^2)$, where $\sigma>0$ is an unknown parameter, and the variables are non-correlated with respect to $n$. Hence, it follows from (\[eq:mod-obs\]) that the conditional distribution of $Y$ given $S$ and $H$ (i.e. the likelihood distribution) is given by $$\pi_{like}(Y|S,H) = \prod_{k=1}^K \prod_{n=1}^N \frac{1}{\sqrt{2\pi}\sigma} \exp \left(-\frac{(Y_k[n] - X_k[n])^2}{\sigma^2}\right). \nonumber$$
\[h\] ![Spectrograms for a clean speech signal (left) and the corresponding reverberant speech signal (right).[]{data-label="fi:clean-rev"}](r01.eps "fig:"){width="49.00000%"} ![Spectrograms for a clean speech signal (left) and the corresponding reverberant speech signal (right).[]{data-label="fi:clean-rev"}](r02.eps "fig:"){width="49.00000%"}
Let us now turn our attention to $S$. Figure \[fi:clean-rev\] depicts the $\log$-spectrograms for a clean signal and its reverberant version. As it can be observed, while the spectrogram of the clean signal is somewhat sparse, the one corresponding to the reverberant signal presents a smoother or more diffuse structure. The presence of discontinuities in the spectrogram of the clean signal can be favored by assuming $S$ follows a generalized Gaussian distribution ([@bouman1993]). Namely, $$\pi_{prior}(S) =\prod_{k=1}^K \prod_{n=1}^N \frac{1}{2\Gamma(1+1/p)b_k} \exp \left(-\frac{|S_k[n]|^p}{b_k^p}\right), \nonumber$$ where $p\in(0,2)$ is a prescribed parameter and $b_k>0$ is unknown.
\[h\] ![Log-spectrograms for a RIR signal with window length 256 and different overlappings.[]{data-label="fi:rirspecs"}](rir_ov016.eps "fig:"){width="32.00000%"} ![Log-spectrograms for a RIR signal with window length 256 and different overlappings.[]{data-label="fi:rirspecs"}](rir_ov128.eps "fig:"){width="32.00000%"} ![Log-spectrograms for a RIR signal with window length 256 and different overlappings.[]{data-label="fi:rirspecs"}](rir_ov240.eps "fig:"){width="32.00000%"}
\[h\] ![Subband signals $H_{65}[n], \; n = 1,\ldots, N$, with window length 256 and different overlappings. The signals show certain regularity, that increases with the window overlapping.[]{data-label="fi:H65s"}](H65_016.eps "fig:"){width="\textwidth"}
![Subband signals $H_{65}[n], \; n = 1,\ldots, N$, with window length 256 and different overlappings. The signals show certain regularity, that increases with the window overlapping.[]{data-label="fi:H65s"}](H65_128.eps){width="\textwidth"}
![Subband signals $H_{65}[n], \; n = 1,\ldots, N$, with window length 256 and different overlappings. The signals show certain regularity, that increases with the window overlapping.[]{data-label="fi:H65s"}](H65_240.eps){width="\textwidth"}
In regards to $H$, although no general conditions are expected on its individual components, we do expect its first order time differences to exhibit a certain degree of regularity (see Figures \[fi:rirspecs\] and \[fi:H65s\]). In fact, if windows are set close enough relative to the duration of the reverberation phenomenon, then consecutive time components of $H$ will capture overlapped information, which along with the exponential decay characteristic of the RIR ([@ratnam2003blind]) accounts for a somewhat smooth structure. Therefore, we define the time differences matrix $V\in\mathbb{R}^{K\times(N_h-1)}$, with components $V_k[n] \doteq H_k[n]-H_k\left[n-1\right]\;\forall n=1,\ldots, N_h-1,\;k=1,\ldots, K$. The regularity of these variations is contemplated by assuming $V$ follows a normal distribution: $$\pi_{prior}(V) = \prod_{k=1}^K \prod_{n=2}^{N_h} \frac{1}{\sqrt{2\pi}\eta_k }\exp \left(-\frac{V_k[n]^2}{\eta_k^2}\right). \nonumber$$
Using Bayes’ theorem, the *a posteriori* joint distribution of $S$ and $H$ conditioned to $Y$ satisfies $$\label{eq:pipost}
\pi_{post}(S,H|Y) \propto \pi_{like}(Y|S,H)\pi_{prior}(S)\pi_{prior}(H).$$
Mixed penalization
------------------
Our goal is to find $\hat{S}$ and $\hat{H}$ that are representative of the *a posteriori* distribution (\[eq:pipost\]). Although the immediate instinct might be to compute the expected value, there are quite a few other ways to proceed, with different degrees of reliability and complexity. In lights of the assumed distributions and the high dimensionality of the problem, the *maximum a posteriori* (MAP) estimator is a reasonable choice in this case. Note that maximizing (\[eq:pipost\]) is tantamount to minimizing $-\log \pi_{post}(S,H|Y)$. If we denote by $S_k,Y_k, X_k \in\mathbb{R}^N $, $H_k\in \mathbb{R}^{N_h}$ and $V_k\in \mathbb{R}^{N_h-1}$ the (transposed) rows of $S,Y, X, H$ and $V$, and define $L\in\mathbb{R}^{N_h-1\times N_h}$ in such a way that $LH_k = V_k$, then $$\begin{aligned}
\nonumber
-\log \pi_{post}(S,H|Y)
& = \sum_{k=1}^{K}\left[ \frac{1}{\sigma^2}||Y_k - X_k||_2^2 +\frac{1}{b_k^p}\sum_n|S_k[n]|^p + \frac{1}{\eta_k^2}||LH_k||_2^2 \right] +C\nonumber,\end{aligned}$$ where $C$ is a constant which does not depend on $S$ nor $H$.
Finally, the latter equation leads to the cost function $$\label{eq:cost-fun}
J(S,H) \doteq \sum_{k=1}^K\left( ||Y_k - X_k||_2^2 +\lambda_{s,k} ||S_k||_p^p + \lambda_{h,k}||LH_k||_2^2 \right),$$ which shall be minimized to find our regularized solution. In this context, $\lambda_{s,k}, \lambda_{h,k}\geq 0$ can be thought of as penalization parameters weighting both penalizers relative to the fidelity term, whereas the exponent $p\in (0,2)$ is a tunning parameter. It is timely to point out that small values of $p$ will promote sparsity, whereas values close to $2$ will promote smoothness. Since there is a clear scale indeterminacy in the representation (\[eq:mod-rep\]), we impose the (somewhat arbitrary) additional constraint $||S_k||_\infty=||Y_k||_\infty\;\forall k$, which means that the maximum values shall remain equal for every frequency.
Regularization parameters
-------------------------
As mentioned before, the parameters $\lambda_{h,k}, \lambda_{s,k},\; k=1,\ldots, K,$ weight the penalizers against the fidelity term. In this sense, the optimal weights of these regularization parameters might vary as a function of the frequency subband, and hence their proposed dependency on $k$. Since searching blindly for $2K$ parameters is non-viable in practice, we quantify this dependency by defining $\lambda_{h,k} \doteq \lambda_{h}\sum_{n=1}^N|Y_k[n]|^2$ and $\lambda_{s,k} \doteq \lambda_{s} \; \forall k = 1,\ldots,K$ (note that the relation between $S_k$ and $Y_k$ is already contemplated in the constraint that intends to avoid scale indeterminacy). This means we only need to look for two parameters ($\lambda_h, \lambda_s$) and then multiply $\lambda_{h}$ by the energy of the signal associated to each row of $Y$.
Next, we present an algorithm for approximating matrices $H$ and $S$ minimizing $J$.
Updating rules
==============
We shall build an iterative algorithm following the idea in [@kameoka2009], which is based on the auxiliary function technique.
Let $\Omega\subset\mathbb{R}$ and $f:\Omega\rightarrow\mathbb{R}_0^+$. Then, $g:\Omega\times\Omega\rightarrow\mathbb{R}_0^+$ is called an *auxiliary function* for $f$ if $$\label{eq:aux_cond}
(i)\; g(w,w) = f(w) \;\;\text{and}\;\; (ii)\; g(w,w')\geq f(w), \;\; \forall w,w'\in\Omega.$$
Let $w^0\in\Omega$ be arbitrary, and let $$\label{eq:gen-up-rule}
w^j \doteq \operatorname*{arg\,min}_wg(w,w^{j-1}).$$ With this definition, it can be shown ([@lee2001]) that the sequence $\{f(w^j)\}_j$ is non-increasing. We intend to use this property as a tool for alternatively updating the matrices $H$ and $S$. Let us begin by fixing $H=H'$, where $H'$ is an arbitrary $K\times N_h$ matrix. We will show that $$\begin{aligned}
\label{eq:aux_s}
g_s(S,S') \doteq &\sum_{k,n,\tau} \frac{S'_k[\tau]H'_k[n-\tau]}{X'_k[n]}\left( Y_k[n]-\frac{S_k[\tau]}{S'_k[\tau]}X'_k[n] \right)^2 +\sum_k\lambda_{h,k}||LH'_k||_2^2\nonumber \\ &+\sum_{k,n}\lambda_{s,k}\left(\frac{p}{2}S'_k[n]^{p-2}S_k[n]^2+|S'_k[n]|^{p} -\frac{p}{2}|S'_k[n]|^{p}\right)\end{aligned}$$ is an auxiliary function for $J$ (as defined in (\[eq:cost-fun\])) with respect to $S$. From this point on, we denote by $X'_k[n] = \sum_{\tau} S'_k[n-\tau] H'_k[\tau]$. The equality condition $(i)$ in (\[eq:aux\_cond\]) is rather straightforward. In fact, $$\begin{aligned}
g_s(S,S)
= &\sum_{k,n,\tau} \frac{S_k[\tau]H'_k[n-\tau]}{\sum_{\nu} S_k[\nu]H'_k[n-\nu]}\left( Y_k[n]-\frac{S_k[\tau]}{S_k[\tau]}\sum_{\nu} S_k[\nu]H'_k[n-\nu] \right)^2\nonumber \\
&+\sum_k\lambda_{h,k}||LH'_k||_2^2 +\sum_{k,n}\lambda_{s,k}\left(\frac{p}{2}S_k[n]^{p-2}S_k[n]^2+|S_k[n]|^{p} -\frac{p}{2}|S_k[n]|^{p}\right) \nonumber\\
=&\sum_{k,n,\tau} \frac{S_k[\tau]H'_k[n-\tau]}{\sum_{\nu} S_k[\nu]H'_k[n-\nu]}\left( Y_k[n]-\sum_{\nu} S_k[\nu]H'_k[n-\nu] \right)^2 \nonumber \\ &+\sum_k\lambda_{h,k}||LH'_k||_2^2 +\sum_{k,n}\lambda_{s,k}|S_k[n]|^{p} \nonumber \\
=&\sum_{k,n} \left( Y_k[n]-\sum_{\nu} S_k[\nu]H'_k[n-\nu] \right)^2 +\sum_k\lambda_{h,k}||LH'_k||_2^2+\sum_{k,n}\lambda_{s,k}|S_k[n]|^{p} \nonumber \\ =& J(S,H').\nonumber\end{aligned}$$
To prove condition $(ii)$ in (\[eq:aux\_cond\]) we begin by defining $$\begin{aligned}
\nonumber
P_{k,n} \doteq& \sum_{\tau} \frac{S'_k[\tau]H'_k[n-\tau]}{X'_k[n]}\left( Y_k[n]-\frac{S_k[\tau]}{S'_k[\tau]}X'_k[n] \right)^2, \nonumber\\
R_{k,n} \doteq& ( Y_k[n]-\sum_{\tau}S_k[\tau]H'_k[n-\tau] )^2, \nonumber\end{aligned}$$ and $Q:\mathbb{R}^+\rightarrow \mathbb{R}$ such that $Q(x) \doteq \frac{p}{2}x^{p-2}S_k[n]^2+x^p -\frac{p}{2}x^{p}$. With these definitions, we can write
$$\nonumber
g_s(S,S') = \sum_{k}\left(\sum_{n}(P_{k,n} +\lambda_{s,k}Q(S'_k[n]) )+\lambda_{h,k}||LH'_k||_2^2\right),$$
and $$\nonumber
J(S,H') = \sum_{k}\left(\sum_{n}(R_{k,n} +\lambda_{s,k}|S_k[n]|^p)+\lambda_{h,k}||LH'_k||_2^2\right).$$
Hence, to prove that $g_s(S,S') \geq J(S,H') \;\forall S, S'$ it is sufficient to show that $P_{k,n} \geq R_{k,n}$ and $Q(S'_k[n])\geq |S_k[n]|^p \; \forall n = 1,\ldots, N, k=1,\ldots, K$. In fact, $$\begin{aligned}
P_{k,n}-R_{k,n}
=& \sum_{\tau} \frac{S'_k[\tau]H'_k[n-\tau]}{X'_k[n]}\left( Y_k[n]-\frac{S_k[\tau]}{S'_k[\tau]}X'_k[n] \right)^2 \nonumber\\&-( Y_k[n]-\sum_{\tau}S_k[\tau]H'_k[n-\tau] )^2 \nonumber\\
=& \sum_{\tau}\frac{H'_k[n-\tau]S_k[\tau]^2X'_k[n]}{S'_k[\tau]} - \left(\sum_{\tau}S_k[\tau]H'_k[n-\tau]\right)^2\nonumber\\
=& \sum_{\tau,\nu}\frac{H'_k[n-\tau]S_k[\tau]^2H'_k[n-\nu]S'_k[\nu]}{S'_k[\tau]} - \sum_{\tau,\nu}S_k[\tau]H'_k[n-\tau]S_k[\nu]H'_k[n-\nu] \nonumber\\
=&\sum_{\tau,\nu}\left( \frac{H'_k[n-\tau]S_k[\tau]^2H'_k[n-\nu]S'_k[\nu]}{S'_k[\tau]} - S_k[\tau]H'_k[n-\tau]S_k[\nu]H'_k[n-\nu]\right) \nonumber\\
=& \sum_{\tau\neq \nu}\left( \frac{H'_k[n-\tau]S_k[\tau]^2H'_k[n-\nu]S'_k[\nu]}{S'_k[\tau]} - S_k[\tau]H'_k[n-\tau]S_k[\nu]H'_k[n-\nu]\right) \nonumber\\
=& \sum_{\tau<\nu}H'_k[n-\tau]H'_k[n-\nu]\left( \frac{S_k[\tau]^2S'_k[\nu]}{S'_k[\tau]} - 2S_k[\tau]S_k[\nu] +\frac{S_k[\nu]^2S'_k[\tau]}{S'_k[\nu]}\right) \nonumber\\
=& \sum_{\tau<\nu}\frac{H'_k[n-\tau]H'_k[n-\nu]}{S'_k[\nu]S'_k[\tau]}\left(S_k[\tau]S_k'[\nu]-S_k[\nu]S_k'[\tau]\right)^2 \geq 0. \label{eq:A_B} \nonumber\end{aligned}$$
To prove that $Q(S'_k[n])\geq |S_k[n]|^p$, we begin by noting that $Q\in\mathcal{C}^{\infty}(\mathbb{R}^+)$. Then, the first order necessary condition for $Q$ yields $$\nonumber
0 = \frac{\partial Q}{\partial x} = \frac{p(p-2)}{2}x^{p-3}S_k[n]^2+px^{p-1} -\frac{p^2}{2}x^{p-1}= \frac{p(p-2)}{2}x^{p-1}(x^{-2} S_k[n]^2-1),$$ meaning the only point at which the derivative of $Q$ equals zero is at $x=S_k[n]$. Furthermore, $\frac{\partial^2}{\partial x^2}Q( S_k[n])= S_k[n]^{p-2} (2p-p^2)>0 \;\forall p\in(0,2)$, meaning that $Q(S_k[n]) = |S_k[n]|^p$ is the global minimum of $Q$. This yields $$\begin{aligned}
\nonumber
g_s(S,S') & = \sum_{k}\left(\sum_{n}(P_{k,n} +\lambda_{s,k}Q(S'_k[n]) )+\lambda_{h,k}||LH'_k||_2^2\right) \\ \nonumber
& \geq \sum_{k}\left(\sum_{n}(R_{k,n} +\lambda_{s,k}|S_k[n]|^p)+\lambda_{h,k}||LH'_k||_2^2\right) = J(S,H').\end{aligned}$$
$\blacksquare$
In an analogous way, it can be shown that if we let $S=S'$ be fixed, where $S'$ is an arbitrary $K\times N$ matrix, then $$\begin{aligned}
\nonumber g_h(H,H') \doteq &\sum_{k,n,\tau} \frac{S'_k[n-\tau]H'_k[\tau]}{X'_k[n]}\left( Y_k[n]-\frac{H_k[\tau]}{H'_k[\tau]}X'_k[n] \right)^2 \nonumber\\
&+\sum_{k}\lambda_{s,k} ||S'_k||_p^p +\sum_{k} \lambda_{h,k}||LH_k||_2^2 \nonumber\end{aligned}$$ is an auxiliary function for $J(S',H)$ with respect to $H$.
Having defined auxiliary functions, we will use the updating rule derived from (\[eq:gen-up-rule\]) to build an algorithm for iteratively approaching matrices $S$ and $H$ minimizing $J$. Notice this requires minimizing $g_s$ and $g_h$ with respect to the updating variables, but since $g_s$ is quadratic with respect to $S$ and $g_h$ is quadratic with respect to $H$, we can simply use the first order necessary conditions in both cases. From this point on, in the context of the iterative updating process, $S'$ and $H'$ will refer not to arbitrary nonnegative matrices, but to those estimations of $S$ and $H$ obtained in the immediately previous step.
Updating rule for S
-------------------
Firstly, we shall derive an updating rule for $S_k[\tau]$. That is, we wish to minimize $g_s$ w.r.t. $S$. The first order necessary condition on $g_s$ yields
$$\begin{aligned}
0 = & \frac{\partial g_s(S,S')}{\partial S_k[\tau] } \nonumber \\
= & -2 \sum_n H'_k[n-\tau]\left(Y_k[n]-\frac{S_k[\tau]}{S_k'[\tau]}X'_k[n]\right)+\lambda_{s,k}pS_k'[\tau]^{p-2}S_k[\tau] \nonumber \\
= & -\sum_n H'_k[n-\tau]Y_k[n]+\frac{S_k[\tau]}{S_k'[\tau]}\sum_n H'_k[n-\tau] X'_k[n]+\frac{\lambda_{s,k}}{2}pS_k'[\tau]^{p-2}S_k[\tau] \nonumber \\
= & -S_k'[\tau] \sum_n H'_k[n-\tau]Y_k[n]+\left(\sum_n H'_k[n-\tau] X'_k[n] +\frac{\lambda_{s,k}}{2}pS_k'[\tau]^{p-1}\right)S_k[\tau], \nonumber\end{aligned}$$ which easily leads to the multiplicative updating rule $$\nonumber
S_k[\tau] = S_k'[\tau]\frac{\sum_n H'_k[n-\tau]Y_k[n]}{\sum_n H'_k[n-\tau]X'_k[n] \,+\,\frac{\lambda_{s,k}}{2}p|S'_k[\tau]|^{p-1}}.$$ In order to avoid the aforementioned scale indeterminacy, every updating step is to be followed by scaling $S_k$ so that its $\ell^\infty$ norm coincides with that of the observation $Y_k$.
Updating rule for H
-------------------
In order to find an updating rule for $H$, we shall write $g_h$ as a function of the transposed rows $H_k$. We begin by noting $$\begin{aligned}
\nonumber
g_h(H,H') = &\sum_{k,n,\tau} \frac{S'_k[n-\tau]H'_k[\tau]}{X'_k[n]}\left( Y_k[n]-\frac{H_k[\tau]}{H'_k[\tau]}X'_k[n] \right)^2 \nonumber \\
&+\sum_{k}\lambda_{s,k} ||S'_k||_p^p +\sum_{k} \lambda_{h,k}||LH_k||_2^2 \nonumber \\
= &\sum_{k,n,\tau}\frac{S'_k[n-\tau]H'_k[\tau]Y_k^2[n]}{X'_k[n]} -2\sum_{k,n,\tau}S'_k[n-\tau]Y_k[n]H_k[\tau] \nonumber \\ &+\sum_{k,n,\tau}\frac{S'_k[n-\tau]X'_k[n]H^2_k[\tau]}{H'_k[\tau]}\nonumber \\
&+\sum_{k}\lambda_{s,k} ||S'_k||_p^p +\sum_{k} \lambda_{h,k}||LH_k||_2^2. \nonumber\end{aligned}$$
Next, we define the diagonal matrices $A^k,B^k\in\mathbb{R}^{N_h\times N_h}$, whose diagonal elements are $A^k_{\tau,\tau} \doteq \sum_{n}S'_k[n-\tau]X'_k[n]$ and $B^k_{\tau,\tau} \doteq H'_k[\tau]$, and the vector $\zeta^k\in\mathbb{R}^{N_h}$ with components $\zeta^k_\tau=\sum_{n}S'_k[n-\tau]Y_k[n]$. With these definitions, we can write $$\begin{aligned}
\nonumber
g_h(H,H') = &\sum_{k,n,\tau} \frac{S'_k[n-\tau]H'_k[\tau]Y_k^2[t]}{X'_k[n]} -2\sum_k H_k{^{\text{\scriptsize{T}}}}\zeta^k +\sum_k H_k{^{\text{\scriptsize{T}}}}A^k(B^k)^{-1}H_k\\ \nonumber
&+\sum_{k}\lambda_{s,k} ||S'_k||_p^p +\sum_{k} \lambda_{h,k}H_k{^{\text{\scriptsize{T}}}}L{^{\text{\scriptsize{T}}}}LH_k.\end{aligned}$$
Now, the first order necessary condition for $g_h$ with respect to $H_k$ is given by $$\begin{aligned}
0 = \frac{\partial g_h(H,H')}{\partial H_k} = -2 \zeta^k +2A^k(B^k)^{-1}H_k +2 \lambda_{h,k}L{^{\text{\scriptsize{T}}}}LH_k,\end{aligned}$$ which readily leads to an updating rule consisting of solving the linear system $$\label{eq:sistH}
(A^k+\lambda_{h,k}B^kL{^{\text{\scriptsize{T}}}}L)H_k = B^k\zeta^k.$$
Let us notice that under the assumption that the diagonal elements of $A^k$ and $B^k$ are strictly positive, and since $L{^{\text{\scriptsize{T}}}}L$ is positive-semidefinite, $(B^k)^{-1}A^k +\lambda_{h,k}L{^{\text{\scriptsize{T}}}}L$ is positive-definite, and hence the linear system has a unique solution. The assumption of $A^k_{\tau,\tau}>0$ is adequate, since these elements correspond to the discrete convolution of $S'_k$ and $X'_k$. Although the validity of the hypothesis over $B^k_{\tau,\tau}$ is not so clear, in practice, the matrix in system (\[eq:sistH\]) has turned out to be non-singular. Nonetheless, $H_k$ can be computed as the best approximate solution in the least-squares sense. Then, solving this $N_h\times N_h$ linear system entails no challenge, since $N_h$ is usually chosen relatively small, depending on the window step and the reverberation time.
All the steps for the dereverberation process are stated in Algorithm \[al:mixpen\]. Note that in the initialization we define the clean spectrogram $S$ equal to the observation, which is natural since in a way they both correspond to the same signal, and $H_k$ as a vector with exponential time decay, which is an expected characteristic of a RIR. Finally, we set the stopping criterion over the decay of the norm of two consecutive approximations of $S$. This has shown to work quite well, although other stopping criteria might be considered.
Results to illustrate the performance of the algorithm are presented in the next section.
**Initializing** $S \leftarrow Y$ $H_k[n] \leftarrow \exp( -n)$ $ \forall k = 1\ldots K, \;n = 1\ldots N$ $ $
**MAIN LOOP**
$S' = S.$
$\displaystyle{X_k[n] \leftarrow \sum_{\tau} S_k[n-\tau] H_k[\tau]} \;\; \;\; \forall k = 1\ldots K, \;n = 1\ldots N$
$\displaystyle{S_k[\tau] \leftarrow S_k[\tau]\frac{\sum_n H_k[n-\tau]Y_k[n]}{\sum_n H_k[n-\tau]X_k[n] \,+\,\frac{\lambda_{s,k}}{2}p|S_k[\tau]|^{p-1}}}.$ $\displaystyle{S_k \leftarrow S_k \frac{\|Y_k\|_{\infty}}{\|S_k\|_{\infty}}.}$
$\text{Build the diagonal matrices} A^k,B^k\in\mathbb{R}^{N_h\times N_h}:$ $\;\;\; A^k_{\tau ,\tau} = \sum_{n}S_k[n-\tau]X_k[n],$ $\;\;\; B^k _{\tau ,\tau} = H_k[\tau].$ $\text{Build the vector } \zeta^k:$ $\;\;\; \zeta^k_\tau = \sum_{n}S_k[n-\tau]Y_k[n]$ $\text{Solve for } H_k:$ $\;\;\; \displaystyle{(A^k+\lambda_{h,k}B^kL{^{\text{\scriptsize{T}}}}L)H_k = B^k\zeta^k.}$
Experimental results
====================
For the experiments, we took $110$ speech signals from the TIMIT database ([@zue1990timit]), recorded at 16 KHz, and artificially made them reverberant by convolution with impulse responses generated with the software Room Impulse Response Generator[^3], based on the model in [@allen1979]. Each signal was degraded under different reverberation conditions: three different room sizes, each with three different microphone positions and four different reverberation times.[^4]
In order to avoid preprocessing, the choice of the regularization parameters was made *a priori* by means of empirical rules, based upon signals from a different database. This is supported by the fact that the parameters were observed to be rather robust with respect to variations of the reverberation conditions, and hence they were chosen simply as $\lambda_h = 1$ and $\lambda_s = 10^{-4}$. The rest of the model parameters were chosen as specified in Table \[tab:params\].
$p$ $N_h$ win. window size win. overlap. $\delta$ max. iter.
----- ------- ------ ------------- --------------- ---------- ------------
1 15 Hann 512 samples 256 samples 20
: Model parameter values[]{data-label="tab:params"}
Let us point out that the choice of $N_h$ was done as to allow $H$ to capture early reverberation while precluding overlapped representations. In the first place, it is desirable for $H$ to represent the RIR along the full Early Decay Time (EDT), the time period in which the reverberation phenomenon alters the clean signal the most, so its effect can be effectively nullified. On the other hand, if we were to choose $N_h$ too large, it might lead certain similarities in the observation $Y$ within a fixed frequency range to be represented as echoes from high energy components of $S$. It is worth mentioning, however, that the performance of our dereverberation method has shown no high sensitivity with respect to the choice of $N_h$.
In order to evaluate the performance of our model, we made comparisons against two state of the art methods that work under the same conditions. The one proposed by Kameoka *et al* in [@kameoka2009], choosing all the parameters as suggested, and the one proposed by Wisdom *et al* in [@wisdom2014], with a window length of $2048$.
To measure performance, following [@hu2008], we made use of the frequency weighted segmental signal-to-noise ratio (fwsSNR) and cepstral distance. Furthermore, we also measured the speech-to-reverberation modulation energy ratio (SRMR, [@falk2010]), which has the advantage of being non-intrusive (it does not use the clean signal as an input). The results for each performance measure are stated in Tables \[tab:results\_SNR\]-\[tab:results\_SRMR\] and depicted in Figures \[fi:fwsSNR\]- \[fi:SRMR\], classified in function of the reverberation times: $300[\text{ms}]$, $450[\text{ms}]$, $600[\text{ms}]$ and $750[\text{ms}]$. Notice that for the cases of fwsSNR and SRMR, higher values correspond to better performance, while for the cepstral distance, small values indicate higher quality.
![Mean and standard deviations of performance fwsSNR for different reverberation times.[]{data-label="fi:fwsSNR"}](results_sim_fwsSNR.eps){width="80.00000%"}
[| C[1.9cm]{} | C[2.4cm]{} | C[2.4cm]{} C[2.4cm]{} C[2.4cm]{} |]{} Rev. time \[ms\] & Rev. Signal & Kameoka der. & Wisdom der. & Mixed penalization\
300 & 8.102 (1.96)& 7.950 (1.73)& 8.262 (1.53)& **8.658** (1.59)\
450 & 4.815 (1.42)& 5.127 (1.36)& 5.771 (1.28)& **6.539** (1.56)\
600 & 3.082 (1.20)& 3.358 (1.19)& 4.140 (1.17)& **4.732** (1.43)\
750 & 1.998 (1.11)& 2.184 (1.10)& 3.013 (1.12)& **3.440** (1.31)\
![Mean and standard deviations of cepstral distance for different reverberation times.[]{data-label="fi:CD"}](results_sim_CD.eps){width="80.00000%"}
[| C[1.9cm]{} | C[2.4cm]{} | C[2.4cm]{} C[2.4cm]{} C[2.4cm]{} |]{} Rev. time \[ms\] & Rev. Signal & Kameoka der. & Wisdom der. & Mixed penalization\
300 & 3.440 (0.44)& 4.057 (0.45)& 3.908 (0.48)& **3.521** (0.35)\
450 & 4.264 (0.44)& 4.636 (0.42)& 4.511 (0.41)& **3.985** (0.39)\
600 & 4.716 (0.46)& 5.006 (0.42)& 4.860 (0.40)& **4.370** (0.40)\
750 & 5.011 (0.48)& 5.264 (0.43)& 5.089 (0.40)& **4.657** (0.41)\
![Mean and standard deviations of SRMR for different reverberation times.[]{data-label="fi:SRMR"}](results_sim_SRMR.eps){width="80.00000%"}
[| C[1.9cm]{} | C[2.4cm]{} | C[2.4cm]{} C[2.4cm]{} C[2.4cm]{} |]{} Rev. time \[ms\] & Rev. Signal & Kameoka der. & Wisdom der. & Mixed penalization\
300 & 4.297 (1.78)& 2.901 (0.92)& **5.269** (2.36)& 5.207 (1.78)\
450 & 3.020 (1.15)& 2.173 (0.64)& 3.907 (1.63)& **4.305** (1.44)\
600 & 2.378 (0.86)& 1.786 (0.51)& 3.175 (1.27)& **3.698** (1.21)\
750 & 2.003 (0.71)& 1.551 (0.44)& 2.727 (1.07)& **3.301** (1.05)\
In regard to the fwsSNR performance measure, the values in Table \[tab:results\_SNR\] (Figure \[fi:fwsSNR\]) give account of significant improvements of our proposed method with respect to the other two. This improvement becomes more evident as the reverberation time increases. As for the cepstral distance, although the results in Table \[tab:results\_CD\] (Figure \[fi:CD\]) account for a better performance of our proposed method, the quality with respect to the reverberant signal is improved only for reverberation times of 450\[ms\] or more. Finally, the SRMR also shows an improvement with respect to the other methods for reverberation times of 450\[ms\] or greater (see Table \[tab:results\_SRMR\], Figure \[fi:SRMR\]).
Conclusions
===========
In this work, a new blind dereverberation method for speech signals based on regularization over a convolutive NMF representation of the signal spectrograms was introduced and tested. Results show a significant improvement over the state of the art methods, specially for high reverberation times. There is certainly much room for improvement, e.g. finding ways of optimally choosing the regularization parameters, exploring the use of other penalizers, etc.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported in part by Consejo Nacional de Investigaciones Científicas y Técnicas, CONICET through PIP 2014-2016 N$^\text{o}$ 11220130100216-CO, the Air Force Office of Scientific Research, AFOSR/SOARD, through Grant FA9550-14-1-0130, by Universidad Nacional del Litoral, UNL, through CAID-UNL 2011 N$^\text{o}$ 50120110100519 “Procesamiento de Señales Biomédicas.” and CAI+D-UNL 2016, PIC 50420150100036LI “Problemas Inversos y Aplicaciones a Procesamiento de Señales e Imágenes”.
[^1]: Instituto de Investigación en Señales, Sistemas e Inteligencia Computacional, sinc(i), FICH-UNL/CONICET, Argentina. Ciudad Universitaria, CC 217, Ruta Nac. 168, km 472.4, (3000) Santa Fe, Argentina. ([[email protected]]{}).
[^2]: Instituto de Matemática Aplicada del Litoral, IMAL, CONICET-UNL, Centro Científico Tecnológico CONICET Santa Fe, Colectora Ruta Nac. 168, km 472, Paraje “El Pozo”, (3000), Santa Fe, Argentina and Departamento de Matemática, Facultad de Ingeniería Química, Universidad Nacional del Litoral, Santa Fe, Argentina.
[^3]: https://github.com/ehabets/RIR-Generator
[^4]: A web demo for our algorithm can be found in http://fich.unl.edu.ar/sinc/web-demo/blindder/
|
---
address:
- ' Department of Mathematics and Statistics, Sultan Qaboos University, P. O. Box 36, Al-Khod 123, Muscat, Oman '
- ' Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology, Bombay, Powai, Mumbai-400076, India '
author:
- Samir Karaa
- 'Amiya K. Pani'
title: 'A priori Error Estimates for Finite Volume Element Approximations to Second Order Linear Hyperbolic Integro-Differential Equations'
---
Introduction
============
In this paper, we discuss and analyze a finite volume element method for approximating solutions to the following class of second order linear hyperbolic integro-differential equations: $$\begin{aligned}
\label{a}
u_{tt}-\nabla\cdot\left({\mathcal{A}}(x)\nabla u+\int_{0}^{t} {\mathcal{B}}(x,t,s) \nabla u(s) \,ds \right)&=&f(x,t)
\;\;\;\;\;\mbox{in}~~\Omega\times J, \nonumber \\
u(x,t) &=&0\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{on}~\partial \Omega\times J, \\
u(x,0) &=&u_0(x)\;\;\;\;\;\;\;\mbox{in}~\Omega, \nonumber\\
u_t(x,0) &=&u_1(x)\;\;\;\;\;\;\;\mbox{in}~\Omega, \nonumber\end{aligned}$$ with given functions $u_0$ and $u_1$, where $\Omega\subset \mathbb {R}^2$ is a bounded convex polygonal domain, $J=(0,T],~T<\infty$, $u_{tt}=\partial^2 u/
{\partial t^2}$ and $f$ is given function defined on the space-time domain $\Omega\times J.$ Here, ${\mathcal{A}}=[{a_{ij}(x)}]$ and ${\mathcal{B}}=[{b_{ij}(x,t,s)}]$ are $2\times 2$ matrices with smooth coefficients. Further, assume that ${\mathcal{A}}$ is symmetric and uniformly positive definite in $\bar{\Omega}$. Problems of this kind arise in linear viscoelastic models, specially in the modelling of viscoelastic materials with memory (cf. Renardy [*et al.*]{} [@RHN]).
Earlier, the finite volume difference methods which are based on cell centered grids and approximating the derivatives by difference quotients have been proposed and analyzed, see [@EGH] for a survey. Another approach, which we shall follow in this article was formulated in the framework of Petrov-Galerkin finite element method using two different grids to define the trial space and test space. This is popularly known finite volume element methods (FVEMs). Here and also in literature, the trial space consists of $C^0$- piecewise linear polynomials on the finite element partition ${\mathcal{T}}_h$ of $\overline{\Omega}$ and the test space is piecewise constants over the control volume ${\mathcal{T}}_h^*$ to be defined in Section 2. Earlier, the FVEM has been examined by Bank and Rose [@5], Cai [@10], Chatzipantelidis [@33], Li [*et al.*]{} [@19], Ewing [*et al.*]{} [@15], etc. for elliptic problems, for parabolic and parabolic type problems by Chou [*et al.*]{} [@6], Chatzipantelidis [*et al.*]{} [@25], Ewing et al. [@50], Sinha [*et al.*]{} [@26] and for second order wave equations by Kumar [*et al.*]{} [@KNP-2008]. For a recent survey on FVEM, see, a review article by Lin [*et al.*]{} [@LLY].
For linear elliptic problems, Li [*et al.*]{} [@19] have established optimal error estimates in $H^1$ and $L^2$-norms. More precisely, for $L^2$-norm the following estimate are derived: $$\|u-u_h\|_{0} \leq Ch^2 \|u\|_{W^{3,p}(\Omega)},~~~~~~p>1,$$ where $u$ is the exact solution and $u_h$ is the finite volume element approximation of $u$. Compared to the error analysis of finite element methods, it is observed that this method is optimal in approximation property, but is not optimal with respect to the regularity of the exact solution as for $O(h^2)$ order convergence, the exact solution $u\in H^3.$ For convex polygonal domain $\Omega$, it may be difficult to prove $H^3$-regularity for the solution $u.$ Therefore, an attempt has been made in [@15] to establish optimal $L^2$ error estimate under the assumption that the exact solution $u\in H^2$ and the source term $f\in H^1.$ A counter example has also been provided in [@15] to show that if $f\in L^2 $, then FVE solution may not have optimal error estimates in $L^2$ norm. The analysis has been extended to parabolic problems in convex polygonal domain in [@25] and optimal error estimates have been derived under some compatibility conditions on the initial data. Further, the effect of quadrature, that is, when the $L^2$ inner product is replaced by numerical quadrature has been analyzed. Subsequently, Ewing et al. [@50] have employed FVEM for approximating solutions of parabolic integro-differential equations and derived optimal error estimates under $L^{\infty}(H^3)$ regularity for the exact solution and $L^2(H^3)$ regularity for its time derivative. Then on convex polygonal domain, Sinha [*et al.*]{} [@26] have examined semidiscrete FVEM and proved optimal error estimates for smooth and non smooth data. The analysis is further generalized to a second order linear wave equation defined on a convex polygonal domain and [*a priori*]{} error estimates have been established only for semidiscrete case, see, Kumar [*et al.*]{} [@50]. Further, the effect of quadrature and maximum norm estimates are proved under some additional conditions on the initial data and the forcing function. In the present article, an attempt has been made to extend the analysis of FVEM to a class of second order linear hyperbolic integro-differential equations in convex polygonal domains with minimal regularity assumptions on the initial data. Moreover, a completely discrete scheme based on a second order explicit method has been analyzed.
In order to put the present investigation into a proper perspective visa-vis earlier results, we discuss, below, the literature for the second order hyperbolic equations. Li et al. [@19] have proved an optimal order of convergence in $H^1$-norm without quadrature using elliptic projection, but the regularity of the exact solution assumed to be higher than the regularity assumed in our results even when $B=0$ for the problem (\[a\]). On a related finite element analysis for the second order hyperbolic equations without quadrature, we refer to Baker [@37] and with quadrature, see, Baker and Dougalis [@35] and Dupont [@38]. Baker and Dougalis [@35] have proved optimal order of convergence in $L^\infty( L^2)$ for the semidiscrete finite element scheme, provided the initial displacement $u_0\in H^5\cap H_0^1$ and the initial velocity $u_1\in H^4\cap H_0^1.$ Subsequently, Rauch [@36] has derived the convergence analysis for the Galerkin finite element methods when applied to a second order wave equation by using piecewise linear polynomials and established optimal $L^\infty (L^2)$ estimate with $u_0\in H^3\cap H_0^1$ and $u_1=0$ which are turned out to be the minimal regularity conditions for the second order wave equation. Subsequently, Pani [*et al.*]{} [@39] have examined the effect of numerical quadrature on finite element method for hyperbolic integro-differential equations with minimal regularity assumptions on the initial data, that is, $u_0\in H^3\cap H_0^1$ and $u_1\in H^2\cap H_0^1$. On a related article on a linear second order wave equation, we refer to Sinha [@34] and on hyperbolic PIDE, see, [@CL-89]. When FVEM is combined with quadrature for approximating solution of (\[a\]), we have, in this article, proved optimal $L^\infty (L^2)$ estimate with minimal regularity assumptions on the initial data.
The organization of the present paper is as follows: Section $2$ deals with some notations, weak formulation and the regularity results for the exact solution. Section $3$ is devoted to the primary and dual meshes for finite volume element method and semidiscrete FVE approximation to the problem (\[a\]). Section $4$ focuses on [*a priori*]{} error estimates for the semidiscrete FVE approximations and optimal order of convergence in $ L^2$ and $H^1$ norms are established under minimal regularity assumptions on the initial data. Further, quasi-optimal order of convergence in maximum norm has also been derived. Section $5$ is on completely discrete scheme which is based on a second order explicit scheme in time and [*a priori*]{} error estimates are established. Section $6$ deals with the effect of numerical quadrature and the related error estimates are derived again with minimal regularity assumption on the initial data. Finally in Section $7$, some numerical experiments are conducted which confirm our theoretical order of convergence.
Through out this paper, $C$ is a generic positive constant independent of discretising parameters $h$ and $k.$
Notation and Preliminaries.
===========================
This section is devoted to some notations and preliminary results related to the weak solution of (\[a\]).
Let $W^{m,p}(\Omega)$ denote the standard Sobolev space with the norm $$\|u\|_{m,p,\Omega}=\left(\sum_{|\alpha|\leq m
}\|D^{\alpha}u\|_{L^p(\Omega)}^p\right)^{1/p}~~~~\mbox{for} ~1\leq p<\infty,$$ and for $p=\infty$, $$\|u\|_{m,\infty,\Omega}=\sup_{|\alpha|\leq m}\|D^\alpha u\|_{L^\infty(\Omega)}.~~~~~~~~~~~~~~~~~~$$ When there is no confusion, we denote $\|u\|_{m,p,\Omega}$ by $\|u\|_{m,p}.$ For $p=2$, we simply write $W^{m,2}(\Omega)$ as $H^m(\Omega)$ and denote its norm by $\|\cdot \|_{m}.$ For a Banach space $X$ with norm $\|\cdot\|_X$ and $ 1\leq p \leq \infty,$ let $W^{m,p}(0,T; X)$ be defined by $$W^{m,p}(0,T; X):=\{v:(0,T) \longrightarrow X | \|D_t^j v\|_X \in L^p(0,T),\;\; 0\leq j \leq m\}.$$ with its norm $$\|v\|_{W{m,p} (0,T; X)}=\|u\|_{W^{m,p}(X)}:=\sum_{j=0}^{m}
\left(\int_0^T\|D^j_t v\|_{X}^p\;dt\right)^{1/p},$$ with the standard modification for $p=\infty$, see [@42]. For $m=0$, $W^{m,p}(0,T; X)$ is simply the space $L^p(X)$. Finally, let $(\cdot ,\cdot)$ and $\|\cdot\|_0$ denote, respectively, the $L^2$ inner product and its induced norm on $L^2(\Omega).$
With $H_0^1(\Omega)=\{v\in H^1(\Omega): v=0 \;\mbox{on}\; \partial \Omega \},$ define the bilinear forms $A(\cdot,\cdot)$ and $B(\cdot,\cdot)=B(t,s;\cdot,\cdot)$ on $H_0^1(\Omega)\times H_0^1(\Omega)$ by $$A(u,v)= \int_{\Omega} {\mathcal{A}}(x)\nabla u\cdot \nabla v\,dx,$$ and $$B(t,s;u(s),v)= \int_{\Omega} {\mathcal{B}}(x,t,s)\nabla u(s)\cdot \nabla v\,dx.$$ Then, the weak formulation for (\[a\]) is to seek $u:(0,T]\longrightarrow H^1_0(\Omega)$ such that $$\label{b}
(u_{tt},v)+ A(u,v)+\int_{0}^{t} B(t,s;u(s),v) \,ds = (f,v)\;\;\;
\forall v\in H_0^1(\Omega)$$ with $u(0)=u_0$ and $u_{t}(0)=u_1.$
Since ${\mathcal{A}}$ is symmetric and uniformly positive definite in $\Omega$, the bilinear form $ A(\cdot, \cdot)$ satisfies the following condition: there exist positive constants $\alpha $ and $\Lambda$ with $\Lambda \geq \alpha$ such that $$\Lambda \|v\|_1^2 \geq A(v,v) \geq \alpha \|v\|_1^2\;\;\;\forall v\in H_0^1(\Omega).
\label{eqn2.3}$$
For our subsequent use, we state without proof [*a priori*]{} estimates of the solution $u$ of the problem (\[a\]) under appropriate regularity conditions and compatibility conditions on $u_0$, $u_1$ and $f$. Its proof can be easily obtained by appropriately modified arguments in the proof of Theorem 3.1 of [@39]. For similar estimates for second order linear hyperbolic equations, see Lemma 2.1 of [@KNP-2008].
\[lem1\] Let $u$ be a weak solution of $(\ref{a})$. Then, there is a positive constant $C=C(T)$ such that the following estimates $$\begin{aligned}
\|D^{j+2}_tu(t)\|_0+\|D^{j+1}_t u(t)\|_1 +\|D^j_tu(t)\|_2
& \leq & C \Big( \|u_0\|_{j+2}+\|u_1\|_{j+1}+\\
&& \sum_{k=0}^{j}\displaystyle \|D_t^k f\|_{L^1(H^{j-k})}+ \|D_t^{j+1} f\|_{L^1(L^2)}\Big),\end{aligned}$$ hold for $j=0,1,2,$ where $D^j_t=(\partial^j/\partial t^j).$
We shall have occasion to use the following identity for $\phi \in C^1 ([0,T];X),$ where $X$ is a Banach space $$\label{phi-t}
\phi(t) = \phi(0) + \int_{0}^t \phi_t(s)\; ds.$$
Finite Volume Element Method
=============================
This section deals with primary and dual meshes on the domain $\Omega$, construction of finite dimensional spaces, finite volume element formulation and some preliminary results.
Let ${\mathcal{T}}_h$ be a family of regular triangulations of the closed, convex polygonal domain $\overline{\Omega}$ into closed triangles $K,$ and let $h=\max_{K\in {\mathcal{T}}_h}(\mbox{diam}K),$ where $h_{K}$ denotes the diameter of $K.$ Let $N_h$ be set of nodes or vertices, that is, $N_h :=\left\{P_i:P_i~~\mbox{ is a vertex of the element }~K \in
{\mathcal{T}}_h~\mbox{and}~P_i\in \overline{\Omega}\right\}$ and let $N_h^0$ be the set of interior nodes in ${\mathcal{T}}_h$ with cardinality $N$. Further, let ${\mathcal{T}}_h^*$ be the dual mesh associated with the primary mesh ${\mathcal{T}}_h,$ which is defined as follows. With $P_0$ as an interior node of the triangulation ${\mathcal{T}}_h,$ let $P_i\;(i=1,2\cdots
m)$ be its adjacent nodes (see, FIGURE \[fig:mesh\] with $m=6$ ). Let $M_i,~i=1,2\cdots
m$ denote the midpoints of $\overline{P_0P_i}$ and let $Q_i,~i=1,2\cdots
m,$ be the barycenters of the triangle $\triangle P_0P_iP_{i+1}$ with $P_{m+1}=P_1$. The [*control volume*]{} $K_{P_0}^*$ is constructed by joining successively $ M_1,~ Q_1,\cdots
,~ M_m,~ Q_m,~ M_1$. With $Q_i ~(i=1,2\cdots
m)$ as the nodes of $control~volume~$ $K^*_{p_i},$ let $N_h^*$ be the set of all dual nodes $Q_i$. For a boundary node $P_1$, the control volume $K_{P_1}^*$ is shown in the FIGURE \[fig:mesh\]. Note that the union of the control volumes forms a partition ${\mathcal{T}}_h^*$ of $\overline{\Omega}$. Assume that the partitions ${\mathcal{T}}_h$ and ${\mathcal{T}}_h^*$ are quasi-uniform in the sense that there exist positive constants $C_1$ and $C_2$ independent of $h$ such that $$\begin{aligned}
C_1\;h^2\leq |K_{Q_i}| \leq C_2\; h^2~~~~~\forall Q_i \in N_h^*,
\label{p1}
\end{aligned}$$ $$\begin{aligned}
C_1\;h^2\leq |K^*_{P_i}|\leq C_2 \;h^2~~~~~\forall P_i \in N_h,\end{aligned}$$ where $ |K|= \mbox {\; meas\;} (K).$
![[]{data-label="fig:mesh"}](figure1.eps){width="11.0cm" height="7.0cm"}
We consider a finite volume element discretization of (\[a\]) in the standard $C^0$-conforming piecewise linear finite element space $U_h$ on the primary mesh ${\mathcal{T}}_h$, which is defined by $$U_h=\{v_h\in C^0(\overline {\Omega})\;:\;v_h|_{K}\;\mbox{is linear for all}~ K\in {\mathcal{T}}_h\; \mbox{and} \; v_h|_{\partial \Omega}=0\},$$ and the dual volume element space ${U_h^*}$ on the dual mesh ${\mathcal{T}}^*_h$ given by $${U_h^*}=\{v_h\in L^2(\Omega)\;:\;v_h|_{K_{P_0}^*}\;\mbox{is constant for all}\; K_{P_0}^*\in {\mathcal{T}}_h^*\; \mbox{and}\; v_h|_{\partial \Omega}=0\}.$$ Now, $U_h=\mbox{span} \{\phi_i\;:\;P_i\in N_h^0\}$ and ${U_h^*}=\mbox{span}\{\chi_i\;:\; P_i\in N_h^0\}$, where $\phi_i$’s are the standard nodal basis functions associated with nodes $P_i$ and $\chi_i$’s are the characteristic basis functions corresponding to the control volume $K_{P_i}^*$ given by $$\chi_i(x)=\left\{\begin{array}{cc}
1,&\mbox{if}~ x\in K_{P_i}^*\\
0,& \mbox{elsewhere}.\\
\end{array}\right.$$
The semidiscrete finite volume element formulation for (\[a\]) is to seek $u_h:(0,T]\longrightarrow U_h$ such that $$\label{c}
(u_{h,tt},v_h)+ A_h(u_h,v_h)+\int_{0}^{t} B_h(t,s;u_h(s),v_h) \,ds= (f,v_h)\;\;
\forall v_h\in {U_h^*},~~~~~~~~~~~~~~~~~~~~~$$ with given initial data $u_h(0)$ and $u_{h,t}(0)$ in $U_h$ to be defined later. Here, the bilinear forms $A_h(\cdot, \cdot)$ and $B_h(t,s;\cdot, \cdot)$ are defined, respectively, by $$A_h(u_h,v_h)= -\sum_{P_i\in N_h^0} v_h(P_i) \int_{\partial K_{P_i}^*}
{\mathcal{A}}(x) \nabla u_h\cdot{\bf n}\,ds,~~~~$$ and $$B_h(t,s;u_h,v_h)= -\sum_{P_i\in N_h^0} v_h(P_i) \int_{\partial K_{P_i}^*}
{\mathcal{B}}(x,t,s) \nabla u_h\cdot{\bf n}\,ds$$ for all $(u_h,v_h)\in U_h \times {U_h^*}$, with ${\bf n}$ denoting the outward unit normal to the boundary of the control volume $K_{P_i}^*$. Notice that by taking the $L^2$ inner product of (\[a\]) with $v_h\in {U_h^*}$ and then integrating, we obtain a similar equation for $u$ as $$\label{newaa1}
(u_{tt},v_h)+ A_h(u,v_h)+\int_{0}^{t} B_h(t,s;u(s),v_h) \,ds =
(f,v_h)\;\;\;\forall v_h \in {U_h^*}.$$
For the error analysis, we first introduce two interpolation operators. Let $\Pi_h:C(\Omega)\longrightarrow U_h$ be the piecewise linear interpolation operator and $\Pi_h^*:C(\Omega)\longrightarrow {U_h^*}$ be the piecewise constant interpolation operator. These interpolation operators are defined, respectively, by $$\Pi_h u=\sum_{P_i\in N_h^0}u(P_i)\phi_i(x)\;~\mbox{and}~\; \Pi_h^*u=\sum_{P_i\in N_h^0}u(P_i)\chi_i(x).
\label{naa}$$ Now for $\psi \in H^2$, $\Pi_h$ has the following approximation property, (see, Ciarlet [@46]): $$\|\psi-\Pi_h\psi\|_0\leq Ch^2\|\psi\|_2.
\label{1.6}$$ Further, we introduce the following discrete norms $$\|v_h\|_{0,h}=\left(\sum_{K\in T_h}|v_h|_{0,h,K}^2\right)^{1/2} \;\; \mbox {and}\;\; \|v_h\|_{1,h}=\left(\|v_h\|_{0,h}^2+|v_h|_{1,h}^2\right)^{1/2},$$ where the seminorm $|v_h|_{1,h}=\left(\sum_{K\in T_h}|v_h|_{1,h,K}^2\right)^{1/2}$, and for $K=K_Q=\triangle P_1P_2P_3$, $$|v_h|_{0,h,K}=\left\{\frac {1}{3}\left(v_h(P_1)^2+v_h(P_2)^2+v_h(P_3)^2\right)\;|K|\;\right\}^{1/2}$$ $$|v_h|_{1,h,K}=\left\{(|\frac{\partial v_h}{\partial x}|^2+|\frac{\partial v_h}{\partial y}|^2
)\; |K|\;\right\}^{1/2}.$$ In the following Lemma, a relation between discrete norms and standard Sobolev norms is stated without proof. For a proof, see, [@19 pp. 124] and [@10].
\[lemma1\] For $v_h \in U_h,~ |\cdot|_{1,h}$ and $|\cdot|_1$ are identical; $\|\cdot\|_{0,h}$ and $\|\cdot\|_{1,h}$ are equivalent to $\|\cdot\|_{0}$ and $\|\cdot\|_{1}$, respectively, that is, there exist positive constants $C_3$ and $C_4>0$, independent of h, such that $$\begin{aligned}
C_3\|v_h\|_{0,h}\leq \|v_h\|_0\leq C_4\|v_h\|_{0,h}\;\;\; \forall v_h\in U_h\end{aligned}$$ and $$C_3||v_h||_{1,h}\leq ||v_h||_1\leq C_4||v_h||_{1,h}\;\;\;\;\forall v_h\in U_h.
\label{nb}$$
Note that $ \|v_h\|_{0,h} = \|\Pi_h^* v_h\|_0.$ Below, we state without proof the properties of the interpolation operator $\Pi_h^*.$ For a proof, we refer to [@19 pp. 192].
\[E\] The following statements hold true.\
$ (i) \; \mbox{For} \;\;\Pi_h^*:U_h\longrightarrow {U_h^*}\mbox{ defined in (\ref{naa}),} \qquad \qquad \qquad \qquad \qquad$ $$(\phi_h,\Pi_h^*v_h)=(v_h,\Pi_h^*\phi_h)\;\;\;\;\;\forall \phi_h,~v_h\in
U_h.
\label{nc}$$ (ii) With $\||\phi_h\||:= (\phi_h,\Pi_h^*\phi_h)^{1/2}$, the norms $\||\cdot\||$ and $\|\cdot\|_0 $ are equivalent on $ U_h$, that is, there exist positive constants $c_{eq}$ and $C_{eq}$, independent of $ h$, such that $$c_{eq}||\phi_h||_{0}\leq \||\phi_h\||\leq C_{eq}||\phi_h||_{0}\;\;\;\;\;\;\;\forall \phi_h\in U_h.
\label{nd}$$
A Priori Error Estimates
========================
This section is devoted to [*a priori*]{} error estimates of the approximation $u_h$ to the spatial semidiscrete scheme (\[c\]).
For the derivation of optimal error estimates, we split $e=u-u_h$ as $$e:=(u-{V_h}u)+({V_h}u-u_h)=:\rho+\theta,$$ where ${V_h}:L^{\infty}(H_0^1\cap H^2)\rightarrow L^{\infty}(U_h)$ is the Ritz-Volterra projection defined by $$\label{d}
A(u-{V_h}u,\chi_h)+\int_{0}^{t} B(t,s;u-{V_h}u,\chi_h)\,ds=0\;\;\;\;\forall \chi_h\in U_h.$$ With some abuse of notations, we will denote by ${V_h}u_0$ the Ritz projection of $u_0$ onto $U_h$ defined by $$A(u_0-{V_h}u_0,\chi_h)=0\;\;\;\;\forall \chi_h\in U_h.$$ For our subsequent analysis, we state without proof following error estimates for the Ritz-Volterra projection. For a proof, see, [@39], [@CL-88], [@YL], [@LTW] and [@PTW].
\[lem2\] There exist positive constants $C$, independent of $h$, such that for $j=0,1,2$, and $r=1,2$ the following estimates hold: $$\label{1Rh}
\| D_t^j\rho(t)\|_0+h\| D_t^j\rho(t)\|_1\leq C h^r
\left[\sum_{l=0}^j\| D_t^lu(t)\|_r+ \int_0^t\|u(s)\|_r\;ds\right],
$$ and $$\label{R1h}
\|\rho(t)\|_{0,\infty}\leq Ch^2 \left(\log\frac{1}{h}\right)\left(\|u(t)\|_{2,\infty}
+\int_0^t||u(s)||_{2,\infty}ds\right).$$
Now, define $$\epsilon_h(f,\chi)=(f,\chi)-(f,\Pi_h^*\chi)\;\;\;\;\forall \chi
\in U_h,~~~~~~~~~~~~~~~$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\epsilon_A(\psi,\chi)=A(\psi,\chi)-A_h(\psi,\Pi_h^*\chi)\;\;\;\;\forall \psi,~ \chi
\in U_h,$$ and $$\;\;\;\;\;\;\;\;\;\;\;\;\;\epsilon_B(t,s;\psi,\chi)=B(t,s;\psi,\chi)-
B_h(t,s;\psi,\Pi_h^*\chi)\;\;\;\;\forall \psi,~ \chi
\in U_h.$$ Then, the following lemma will be of frequent use in our analysis and the proof of which can be found in [@33].
\[A\] Assume that the coefficient matrices ${\mathcal{A}}, {\mathcal{B}}(t,s) \in W^{1+i,\infty} (\Omega; \mathbb{R}^{2\times 2})$ for $i=0,1.$ Then, there exist positive constant $C,$ independent of $h$, such that the following estimates hold for $\chi \in U_h$ and for $i,~j=0,~1$ $$\begin{aligned}
\label{f}
|\epsilon_h(f,\chi)|\leq Ch^{i+j}\|f\|_{H^i}\;\|\chi \|_{H^j}\;\;\;
\forall f\in H^i,
\label{ne}\end{aligned}$$ and for $u\in H^{1+i}\cap H_0^1$ $$\begin{aligned}
\label{ea-1}
|\epsilon_A({V_h}u,\chi)| \leq Ch^{i+j}\Big(\|u\|_{H^{1+i}} + \int_{0}^{t} \|u(s)\|_{H^{1+i}}\;ds\Big)\;\|\chi \|_{H^j}.
\label{nf}\end{aligned}$$ Moreover, $$\begin{aligned}
\label{ea-2}
|\epsilon_A(w_h,\chi)| \leq Ch\|w_h\|_{H^{1}}\;\|\chi
\|_{H^1}\;\forall w_h \in U_h.
\label{ng}\end{aligned}$$
The estimates (\[ea-1\]) and (\[ea-2\]) are also valid if $\epsilon_A$ is replaced by $\epsilon_B.$
Now, for $\psi\in H^1_0$ and for each $t\in (0,T],$ introduce a linear functional ${G}(\psi)={G}(t,\psi)$ defined on $U_h$ by $${G}(\psi)(\chi)=\epsilon_A(\psi,\chi)+\int_0^t\epsilon_B(t,s;\psi(s),\chi)\,ds,\,\,\,\chi \in U_h.$$ Notice that, by using the definition of ${G}$, (\[b\]) and (\[newaa1\]), there follows that $$\begin{aligned}
{G}(\rho)(\chi)
&=&A(u,\chi)+\int_{0}^{t} B(t,s;u(s),\chi)\,ds\nonumber \\
&&-A_h(u,\Pi_h^*\chi)-\int_{0}^{t} B_h(t,s;u(s),\Pi_h^*\chi)\,ds
-{G}({V_h}u)(\chi)\nonumber \\
&=&(f-u_{tt},\chi)-(f-u_{tt},\Pi_h^*\chi)-{G}({V_h}u)(\chi)\nonumber \\
&=&\epsilon_h(f-u_{tt},\chi)-{G}({V_h}u)(\chi).
\label{h}\end{aligned}$$ From (\[c\]) and (\[newaa1\]), we obtain the equation in $\theta$ for $v_h\in {U_h^*}$ as $$(\theta_{tt},v_h)+A_h(\theta,v_h)+\int_{0}^{t} B_h(t,s;\theta(s),v_h)\,ds=
-A_h(\rho,v_h)-\int_{0}^{t} B_h(t,s;\rho,\chi)\,ds-(\rho_{tt},v_h).$$ Choosing $v_h=\Pi_h^*\chi $ and using the definition of ${G}$ and (\[d\]), we find that $$\begin{aligned}
\label{i}
(\theta_{tt},\Pi_h^*\chi)+A(\theta,\chi)\;ds&+&\int_{0}^{t} B(t,s;\theta(s),\chi) \,ds = G(\rho)(\chi)\nonumber\\
&+& G(\theta)(\chi)-(\rho_{tt},\Pi_h^*\chi) \;\;\forall \chi \in U_h.\end{aligned}$$
For any continuous function $\phi$ in $[0,t]$, define $\hat{\phi}$ by $$\hat{\phi}(t)=\int_0^t\phi(s)\,ds.$$ Notice that $\hat{\phi}(0)=0$ and $(d\hat{\phi}/dt)(t)=\phi(t)$. Then, integrate (\[i\]) from $0$ to $t$ to obtain $$\begin{aligned}
\label{ii}
(\theta_{t},\Pi_h^*\chi)+A({\hat{\theta}},\chi)&=&
{\hat{G}}(\rho)(\chi)+{\hat{G}}(\theta)(\chi)
+(-\rho_{t},\Pi_h^*\chi)+(e_t(0),\Pi_h^*\chi)\nonumber\\
&&-\int_{0}^{t} B(s,s;{\hat{\theta}}(s),\chi)ds+
\int_{0}^{t}\int_{0}^{s}B_\tau(s,\tau;{\hat{\theta}}(\tau),\chi)d\tau ds,\end{aligned}$$ where $${\hat{G}}(\phi)(\chi)=\epsilon_A(\hat{\phi},\chi)+
\int_{0}^{t} \epsilon_B(s,s;\hat{\phi}(s),\chi)ds-\int_{0}^{t}
\int_{0}^{s}\epsilon_{B_\tau}(s,\tau;\hat{\phi}(\tau),\chi)d\tau ds.$$ For a linear functional $F$ defined on $U_h$, set $$\|F\|_{-1,h}=\sup_{0\neq \chi\in U_h}\frac{|F(\chi)|}{\|\chi\|_1}.$$ We shall need the following lemmas in our subsequent analysis.
\[lm-n\] With ${G}$ and ${\hat{G}}$ as above, there exists a positive constant $C=C(T)$ such that the following estimates $$\label{G-1}
\|D_t^j{G}({V_h}u) \|_{-1,h}\leq Ch^2 \left( \sum_{\ell=0}^{j} \|D^{\ell}_t u(t) \|_{2}+ \int_{0}^t \| u (s)\|_{2}\,ds\right),$$ and $$\label{G-2}
\|D_t^j{\hat{G}}({V_h}u) \|_{-1,h}\leq C h^2
\left(\sum_{\ell=0}^{j} \|D^j_t \hat {u} (t)\|_{2}+ \int_{0}^{t} \|\hat{u}(s)\|_{2}\; ds\right),$$ hold for $j=0,1$.
Proof. Using (\[nf\]) and the estimates in Lemma \[lem1\], we obtain $$\begin{aligned}
|{G}({V_h}u)(\chi)|&\leq&|\epsilon_A({V_h}u,\chi)|+\int_0^t|\epsilon_B(t,s;{V_h}u(s),\chi)|\,ds\\
&\leq&Ch^2\left(\|u\|_2+\int_0^t\|u(s)\|_2\,ds\right)\|\chi\|_1,
$$ and $$\begin{aligned}
|{G}_t({V_h}u)(\chi)|
&\leq&Ch^2\left(\|u_t\|_2+\|u\|_2 + \int_0^t\|u(s)\|_2\,ds\right)\|\chi\|_1\\
&\leq&Ch^2 \left(\|u_t\|_2+ \|u\|_2 + \int_0^t\|u(s)\|_2\,ds\right)\|\chi\|_1\end{aligned}$$ In a similar manner, we derive the second estimate (\[G-2\]) and this completes the rest of the proof.
In the error analysis, we shall frequently use the following inverse assumption: $$\label{inv}
\|\chi\|_1\leq C_{inv} h^{-1}\|\chi\|_0,\quad \chi \in U_h.$$
$H^1$- error estimate
----------------------
\[H2\] Let $u$ and $u_h$ be the solutions of $(\ref{a})$ and $(\ref{c}),$ respectively, and assume that $f\in L^1(H^1),\; f_t,\;f_{tt}\in L^1(L^2)$, $u_0\in H^3\cap H_0^1$ and $u_1\in H^2\cap H_0^1$. Further, assume that $u_h(0)=\Pi_hu_0$ and $u_{h,t}(0)=\Pi_hu_1$, where $\Pi_h$ is the interpolation operator defined in $(\ref{naa})$. Then, there exists a positive constant $C=C(T)$, independent of $h$, such that for $t\in (0,T]$ the following estimate $$\begin{aligned}
\|u(t)-u_h(t)\|_1 \leq C\;h \left(\|u_0\|_3+\|u_1\|_2+ \int_{0}^{t} \Big( \|f\|_{1} +\|f_t\|_0 + \|f_{tt}\|_0\Big)\,ds \right)
\end{aligned}$$ holds.
Proof. Since $u-u_h=\rho+\theta$ and estimates of $\rho$ are known from the Lemma \[lem2\], it is sufficient to estimate $\theta.$ Choose $\chi=\theta_t$ in (\[i\]) and use (\[h\]) to obtain $$\begin{aligned}
(\theta_{tt},\Pi_h^*\theta_t)+A(\theta,\theta_t)+\int_{0}^{t} B(t,s;\theta(s),\theta_t) \,ds
&=& \epsilon_h(f-u_{tt},\theta_t)
-{G}({V_h}u)(\theta_t) \nonumber\\
&&+{G}(\theta)(\theta_t)-(\rho_{tt},\Pi_h^*\theta_t).\nonumber
\label{k}\end{aligned}$$ Now use (\[nc\]) and symmetry of the bilinear form $A(\cdot,\cdot)$ to arrive at $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\Big[(\theta_t,\Pi_h^*\theta_t)+A(\theta,\theta)\Big] &=& \epsilon_h(f-u_{tt},\theta_t)
-{G}({V_h}u)(\theta_t)+{G}(\theta)(\theta_t)-(\rho_{tt},\Pi_h^*\theta_t)\\
&&-\int_{0}^{t} B(t,s;\theta(s),\theta_t(t)) \,ds.\end{aligned}$$ Integration from $0$ to $t$ yields $$\begin{aligned}
\frac{1}{2}\Big(\||\theta_t\||^2+A(\theta,\theta)\Big)&=& \frac{1}{2}\left(\||\theta_t(0)\||^2+A(\theta(0),\theta(0))\right)+
\int_0^t \epsilon_h(f-u_{tt},\theta_t)ds \nonumber \\
&&- \int_0^t {G}({V_h}u)(\theta_t) ds+
\int_0^t {G}(\theta)(\theta_t)ds + \int_0^t(-\rho_{tt},\Pi_h^*\theta_t)ds\nonumber \\
&&-\int_{0}^{t} \int_0^s B(s,\tau;\theta(\tau),\theta_t(s)) \,d\tau ds\nonumber\\
&=& J_1+J_2+J_3+J_4+J_5+J_6.
\label{1.27}\end{aligned}$$ For the first term on the right hand side of (\[1.27\]), a use of the boundedness of $A(\cdot,\cdot)$ with (\[1.6\]) and (\[nd\]) shows $$|J_1|\leq C \left(\|\theta_t(0)\|_0^2+\|\theta(0)\|_1^2\right)\leq Ch^2(\|u_1\|_1^2+\|u_0\|_2^2).
\label{1.28}$$ For estimating $J_2,$ an application of (\[ne\]) with $j=0$ implies $$\begin{aligned}
|J_2| \leq Ch
\int_0^t\left(\|f\|_1+\|u_{tt}\|_1\right)\|\theta_t\|_0\;ds.\end{aligned}$$ To estimate $J_3$, a use of the inverse inequality (\[inv\]) shows that $$|J_3|\leq C \int_{0}^{t}\|{G}({V_h}u)\|_{-1,h}\|\theta_t\|_1 \;ds \leq
Ch^{-1}\int_{0}^{t}\|{G}({V_h}u)\|_{-1,h}\|\theta_t\|_0\;ds.$$ Using the definition of $G$, (\[ng\]) and the inverse inequality, it follows that $$\begin{aligned}
\;\;\;\;\;\;\;\;\;|J_4|&\leq &
\int_0^t|{G}(\theta)(\theta_t)| \;ds \nonumber\\
&\leq& Ch\left[\int_0^t\|\theta\|_1\|\theta_t\|_1 ds +\int_0^t\int_0^s\|\theta(\tau)\|_1
\|\theta_t(s)\|_1d\tau ds\right] \nonumber\\
&\leq& C\left[\int_0^t\|\theta\|_1\|\theta_t\|_0 ds+\left(\int_0^t\|\theta\|_1 ds\right)
\left(\int_0^t\|\theta_t\|_0 ds\right)\right]. \end{aligned}$$ For $J_5$, apply the Cauchy-Schwarz inequality, $L^2$ stability of $\Pi_h^*$ and (\[1Rh\]) with $r=1$ to obtain $$|J_5|\leq \int_{0}^{t}\|\rho_{tt}\|_0\;\|\theta_t\|_0 \;ds
\leq C(T)\;h\int_{0}^{t}\Big(\|u_{tt}\|_1+ \|u_{t}\|_1+\|u\|_1\Big)\|\theta_t\|_0\; ds.
\label{1.29}$$ For the term $J_6,$ we note that an integration by parts yields $$\begin{aligned}
\label{integral-term}
\int_0^t \int_0^s B(s,\tau;\theta(\tau),\theta_t(s))\,d\tau \;ds &=&
\int_0^tB(t,s;\theta(s),\theta(t))\,ds - \int_0^tB(s,s;\theta(s),\theta(s)) \\
&&-\int_0^t\int_0^sB_s(s,\tau;\theta(\tau),\theta(s))\,d\tau \;ds, \nonumber\end{aligned}$$ and hence, deduce that $$|J_6|\leq C\left(\|\theta(t))\|_1\;\int_0^t\|\theta(s)\|_1\,\;ds
+ \int_0^t\|\theta(s)\|^2_1\,ds\right).
\label{eer}$$ Now, set ${\mathcal{E}}^2_1(t)=\|\theta_t\|^2_0+\|\theta\|^2_1$ and $${\mathcal{E}}_1(t^\ast)=\max_{0\leq \tau\leq t}{\mathcal{E}}_1(t),$$ for some $t^\ast\in [0,t]$. Then, substituting the estimates (\[1.28\])-(\[eer\]) in (\[1.27\]), using coercivity of $A(\cdot,\cdot)$, equivalence of norms $\||\cdot\||$ and $\|\cdot \|_0,$ apply standard kick back arguments to find that $$\begin{aligned}
{\mathcal{E}}_1(t^\ast) &\leq & Ch\;\Big( \|u_0\|_2+\|u_1\|_1+\int_{0}^{T} \left(\|u_{tt}(s)\|_1 + \|u_{t}(s)\|_1
+\|u(s)\|_1\right)\;ds\Big)\\
&+& C h \;\int_{0}^{T} \left( \|f(s)\|_1 + h^{-2}\|{G}({V_h}u)(s)\|_{-1,h}\right)\;ds
+ \int_{0}^{t^\ast} {\mathcal{E}}_1(s)\; ds.
\end{aligned}$$ Now replace $t^\ast$ by $t$ and apply Gronwall’s lemma with the estimate (\[G-1\]) to conclude that $${\mathcal{E}}_1(t) \leq~Ch\Big(\|u_0\|_2+\|u_1\|_1+ \int_{0}^{T}
\left(\|u\|_2 +\|u_t\|_1 +\|u_{tt}\|_1+\|f\|_1\right)\;ds\Big).$$ A use of triangle inequality with (\[1Rh\]) and the estimates from Lemma \[lem1\] completes the rest of the proof.
Optimal $L^2$- error estimates
--------------------------------
In this subsection, we shall discuss optimal $L^{\infty}(L^2)$ estimates
\[TH1\] Under the assumptions of Theorem $\ref{H2},$ there exists a positive constant $C=C(T)$, independent of $h$, such that $$\|u(t)-u_h(t)\|_0\leq C h^2 \left(\|u_0\|_3+\|u_1\|_2 + \int_{0}^{t} \Big( \|f\|_{1} +\|f_t\|_0 + \|f_{tt}\|_0\Big)\,ds \right).
$$
Proof. By setting $\chi=\theta$ in (\[ii\]), and using (\[nc\]) with symmetry of the bilinear form $A(\cdot,\cdot)$, we find that $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\left[(\theta,\Pi_h^*\theta)+A(\hat{\theta},\hat{\theta})\right]&=&
{\hat{G}}(\rho)(\theta)+{\hat{G}}(\theta)(\theta)
+(-\rho_{t},\Pi_h^*\theta)+(u_1-\Pi_hu_1,\Pi_h^*\theta)\nonumber\\
&&-\int_{0}^{t} B(s,s;{\hat{\theta}}(s),\theta(t))\;ds+
\int_{0}^{t}\int_{0}^{s}B_\tau(s,\tau;{\hat{\theta}}(\tau),\theta(t))\;d\tau \;ds.\end{aligned}$$ Integrate from $0$ to $t$ to obtain $$\begin{aligned}
\label{nn1}
\frac{1}{2}\left[ \||\theta\||^2+A(\hat{\theta},\hat{\theta})\right] &=&
\frac{1}{2}\||\theta(0)\||^2 +
\int_0^t{\hat{G}}(\rho)(\theta)\;ds+\int_0^t{\hat{G}}(\theta)(\theta)\;ds
+\int_0^t(-\rho_{t},\Pi_h^*\theta)\;ds\nonumber\\
&&+(u_1-\Pi_hu_1,\Pi_h^*\hat{\theta})
-\int_{0}^{t}\int_{0}^{s} B(\tau,\tau;{\hat{\theta}}(\tau),\theta(s))\; d\tau \;ds\nonumber\\
&&+\int_{0}^{t}\int_{0}^{s}\int_{0}^{\tau'}B_{\tau'}(\tau,\tau';{\hat{\theta}}(\tau'),\theta(s))\;d\tau'
\;d\tau \;ds\nonumber \\
&=&\frac{1}{2}\||\theta(0)\||_0^2+I_1+I_2+I_3+ I_4+I_5+ I_6.\end{aligned}$$ To estimate $I_1$, we note from (\[h\]) that $$\begin{aligned}
\label{keyy1}
{\hat{G}}(\rho)(\theta)
&=&\epsilon_h(\hat{f}-\hat{u}_{tt},\theta)-{\hat{G}}({V_h}u)(\theta)\nonumber\\
&=&\frac{d}{dt}\left (\epsilon_h(\hat{f}-\hat{u}_{tt},\hat{\theta})-{\hat{G}}({V_h}u)(\hat{\theta})\right)
-\Big(\epsilon_h({f}-{u}_{tt},\hat{\theta})-{\hat{G}}_t({V_h}u)(\hat{\theta})\Big),\end{aligned}$$ and hence, $$\begin{aligned}
I_1=\left(\epsilon_h(\hat{f}-\hat{u}_{tt},\hat{\theta})-{\hat{G}}({V_h}u)(\hat{\theta})\right)
- \int_0^t \left(\epsilon_h({f}-{u}_{tt},\hat{\theta})-{\hat{G}}_s({V_h}u)(\hat{\theta})\right) ds. \end{aligned}$$ A use of (\[ne\]) for $j=1$ shows $$\begin{aligned}
|I_1|&\leq & |\epsilon_h(\hat{f}-({u}_{t}-u_1),\hat{\theta})|+|{\hat{G}}({V_h}u)(\hat{\theta})| \nonumber \\
&&+ \int_0^t \left(|\epsilon_h({f}-{u}_{tt},\hat{\theta})|+|{\hat{G}}_s({V_h}u)(\hat{\theta})|\right)\;ds \nonumber \\
&\leq& C \left[h^2 \left( \|\hat{f}\|_1+\|u_{t}\|_1+ \|u_1\|_1\right) + \|{\hat{G}}({V_h}u)\|_{-1,h}\right]\|\hat\theta\|_1\nonumber \\
&& +C \int_0^t \left( h^2(\|f\|_1+ \|u_{tt}\|_1) +\|{\hat{G}}_s({V_h}u)\|_{-1,h}
\right) \|\hat\theta\|_1\; ds. \end{aligned}$$ Notice that $I_2$ can be written as $$\begin{aligned}
I_2&=&\int_{0}^{t}\epsilon_A(\hat{\theta},\theta)ds+
\int_{0}^{t} \int_{0}^{s}\epsilon_B(\tau,\tau;\hat{\theta}(\tau),\theta(s))\;d\tau \;ds\\
&&-\int_{0}^{t}\int_{0}^{s}\int_{0}^{\tau}
\epsilon_{B_{\tau'}}(\tau,\tau';\hat{\theta}(\tau'),\theta(s)) \;d\tau'\; d\tau \;ds\\
&=& I_{21}+I_{22}+I_{23}.\end{aligned}$$ For $I_{21}$, we apply (\[ng\]) and the inverse inequality (\[inv\]) to find that $$\label{newkeyy2}
|I_{21}|=\int_{0}^{t}|\epsilon_A(\hat{\theta},\theta)|ds\leq C h \int_{0}^{t}\|\theta\|_1\|\hat{\theta}\|_1\leq
CC_{inv}\int_{0}^{t}\|\theta\|_0\; \|\hat{\theta}\|_1.$$ In order to estimate $I_{22}$, we integrate by parts in time so that $$\begin{aligned}
|I_{22}|&=&\left|\int_0^t \epsilon_B(s,s;\hat{\theta}(s),\hat{\theta}(t))\;ds-\int_0^t
\epsilon_B(s,s;\hat{\theta}(s),\hat{\theta}(s))\;ds\right|\\
&\leq & Ch\left\{\|\hat{\theta}(t)\|_1\int_0^t\|\hat{\theta}(s)\|_1\;ds+
\int_0^t\|\hat{\theta}(s)\|_1^2\;ds\right\}.\end{aligned}$$ Similarly for $I_{23}$, we note that $$\begin{aligned}
|I_{23}|&=& \left|\int_0^t\int_0^s \epsilon_{B_{\tau}}(s,\tau;\hat{\theta}(\tau),\hat{\theta}(t))\;d\tau \;ds
-\int_0^t\int_0^s \epsilon_{B_{\tau}}(s,\tau;\hat{\theta}(\tau),\hat{\theta}(s))\;d\tau \;ds\right|\\
&\leq &
C(T)h\left\{\|\hat{\theta}(t)\|_1\int_0^t\|\hat{\theta}(s)\|_1\;ds+
\int_0^t\|\hat{\theta}(s)\|_1^2\;ds\right\}.\end{aligned}$$ Using stability of $\Pi_h^*$ (i.e., $\|\Pi_h^*\theta\|_0\leq C \|\theta\|_0$) and the Cauchy-Schwarz inequality, it follows that $$|I_3| \leq \int_{0}^{t}|(\rho_{t},\Pi_h^*\theta)|ds \leq
C \int_{0}^{t}\|\rho_t(s)\|_0\|\theta(s)\|_0 ds.$$ For $I_4$, we apply (\[1.6\]) and $\|\Pi_h^*\hat \theta\|_0\leq
C\|\hat \theta\|_1$ to obtain $$|I_4|\leq \|u_1-\Pi_hu_1\|_0\;\|\Pi_h^*\hat{\theta}\|_0 \leq C h^2 \|u_1\|_2\;\|\hat \theta\|_1.
\label{keyy2}$$ Finally, similarly for $I_{22}$ and $I_{23}$, an integration by parts leads to $$\label{ns3}
|I_5|+|I_6|\leq C(T)\left\{\|\hat{\theta}(t)\|_1\int_0^t\|\hat{\theta}(s)\|_1\;ds+
\int_0^t\|\hat{\theta}(s)\|_1^2\;ds\right\}.$$ Now, define ${\mathcal{E}}_0^2(t)=\|\theta(t)\|^2_0+\|\hat{\theta}(t)\|^2_1$ and let $t^\ast \in [0,t]$ be such that $${\mathcal{E}}_0(t^\ast)=\max_{0\leq s\leq t}{\mathcal{E}}_0(t).$$ At $t=t^\ast$, substitute the estimates (\[keyy1\])-(\[ns3\]) in (\[nn1\]) and use the equivalence of the norms $\||\cdot\||$ and $\|\cdot\|_0$ from (\[nd\]) along with the coercivity property (\[eqn2.3\]) of $A(\cdot,\cdot)$. Then a standard use of kick back arguments yields $$\begin{aligned}
{\mathcal{E}}_0(t^\ast) &\leq& C\|\theta(0)\|+
Ch^2\left[\|u_0\|_2+\|u_1\|_2
+\int_{0}^{t^\ast}\left(\|f\|_1+\|u_{tt}\|_1\right)\;ds\right]\\
&&+C\left[\|{\hat{G}}({V_h}u)\|_{-1,h}+\int_0^{t^\ast}
(\|\rho_{t}\|_0+\|{\hat{G}}_s({V_h}u)\|_{-1,h})\;ds\right]
+C\int_{0}^{t^\ast} {\mathcal{E}}_0(s)\;ds.\end{aligned}$$ Note that $\|\theta(0)\|_0\leq Ch^2 \|u_0\|_2$. Now apply Lemmas \[lm-n\], \[lem2\] along with the estimates in Lemma \[lem1\] to obtain $$\begin{aligned}
{\mathcal{E}}_0(t^\ast) &\leq&
Ch^2\left( \|u_0\|_3+\|u_1\|_2
+\int_{0}^{t^\ast}\left(\|f\|_1+\|f_{t}\|_0+\|f_{tt}\|_0\right)\;ds\right)\\
&&
+C\int_{0}^{t^\ast} {\mathcal{E}}_0(s)\;ds.\end{aligned}$$ Then replace $t^\ast$ by $t$ and use Gronwall’s lemma for $t\leq T $ to conclude that $$\|\theta(t)\|_0\leq C(T)h^2 \left(\|u_0\|_3+\|u_1\|_2+
+\int_{0}^{T}\left(\|f\|_1+\|f_{t}\|_0+\|f_{tt}\|_0\right)ds \right).$$ Finally, a use of the triangle inequality completes the proof.
Note that it is possible to choose $u_{h,t}(0)$ as the $L^2$- projection of $u_1$ onto ${U_h^*}$ and in that case, the term $(u_t(0)-u_{h,t}(0),\Pi_h^*\theta_t)$ becomes zero.
Maximum norm estimates
----------------------
In this subsection, a superconvergent result for $\|\theta\|_1$ is first derived and it is then used to analyze quasi-optimal maximum error estimates.
\[H3\] Assume that $f\in L^1(H^2),\;f_t\in L^1(H^1),\;f_{tt},\;f_{ttt}\in L^1(L^2),\; u_0\in H^4\cap H_0^1$ and $u_1\in H^3\cap H_0^1$. With $u_h(0)={V_h}u_0$ and $u_{h,t}(0)=\Pi_hu_1$, there exists a positive constant $C=C(T),$ independent of $h,$ such that the following holds for $t\in (0,T]$ $$\|\theta_t(t)\|_0+\|{\theta}(t)\|_1\leq C\;h^2 \Big(\|u_0\|_4+\|u_1\|_3+\|D_t^3f\|_{L^1(L^2)}+
\sum_{j=0}^{2} \|D_t^jf\|_{L^1(H^{2-j})}\Big).$$
Proof. We now modify the estimates of $J_1$, $J_2$, $J_3$ and $J_5$ in (\[1.27\]) of Theorem \[H2\] to obtain a superconvergence result for $\|\theta(t)\|_1$ norm. As $u_h(0)={V_h}u_0,$ it follows that $ A(\theta(0),\theta(0))=0.$ Now with $u_{h,t}(0)=\Pi_hu_1$, we obtain $$|J_1|\leq Ch^4\|u_1\|_2^2.
\label{1.31}$$ To estimate $J_2$, observe that $$\epsilon_h(f-u_{tt},\theta_t)=\frac{d}{dt}\epsilon_h(f-u_{tt},\theta)-\epsilon_h(f_t-u_{ttt},\theta),$$ and thus, rewrite $J_2$ as $$J_2= \epsilon_h(f-u_{tt},\theta)- \int_0^t \epsilon_h(f_t-u_{ttt},\theta)ds.$$ Then, a use of (\[ne\]) yields $$\begin{aligned}
|J_2| \leq Ch^2\Big[\left(\|f\|_1+\|u_{tt}\|_1\right)\|\theta\|_1+\int_{0}^{t}\left(\|f_t\|_1+\|u_{ttt}\|_1\right)\|\theta\|_1ds\Big].\end{aligned}$$ For $J_3$, rewrite $G$ term as $$G({V_h}u)(\theta_t)=\frac{d}{dt}\left\{G({V_h}u)(\theta) \right\}-G_t({V_h}u)(\theta),$$ and hence, a use of (\[nf\]) shows that $$\begin{aligned}
|J_3| &\leq & |G({V_h}u)(\theta)|+ \int_0^t|G_s({V_h}u)(\theta)|\;ds \nonumber \\&\leq&
2\left[\|G({V_h}u)\|_{-1,h}+ \int_{0}^{t}\|G_s({V_h}u)\|_{-1,h}\;ds\right]\|\theta\|_1.\end{aligned}$$ For $J_5$, apply (\[1Rh\]) to obtain $$|J_5|\leq 2 \int_{0}^{t}\|\rho_{tt}\|_0\|\theta_t\|_0 ds \leq C(T)h^2\int_{0}^{t}
\left(\|u_{tt}\|_2+|u_{t}\|_2 +\|u\|_2\right)\,\|\theta_t\|_0\,ds.
\label{1.32}$$ Substituting the estimates (\[1.31\])-(\[1.32\]) in (\[1.27\]), and apply standard kick back arguments to arrive at $$\begin{aligned}
\label{E-1}
{\mathcal{E}}_1(t) &\leq&Ch^2\Big[\|u_1\|_2+\|f\|_1+\|u_{tt}\|_1+|u_{t}\|_2+\|u\|_2 \nonumber \\
&& +\int_{0}^{t}\left(\|f_t\|_1+ \|u\|_2+\|u_t\|_2+\|u_{tt}\|_2+\|u_{ttt}\|_1\right)ds\Big] \nonumber \\
&&+C\int_{0}^{t} {\mathcal{E}}_1(s)\;ds.\nonumber\end{aligned}$$ An application of the integral identity (\[phi-t\]) shows $$\|f\|_1\leq \|f(0)\|_1+\int_{0}^{t}\|f_t\|_1ds.$$ Then using the estimates in Lemma \[lem1\] we arrive at $$\begin{aligned}
{\mathcal{E}}_1(t) &\leq&~Ch^2\displaystyle \Big(\|u_1\|_3+\|u_0\|_4+\|f(0)\|_1 \nonumber\\
&&+\int_{0}^{T}\left(\|f\|_2+\|f_t\|_1+ \|f_{tt}\|_0+\|f_{ttt}\|_0\right)ds\Big)\nonumber \\
&&+C\int_{0}^{t} {\mathcal{E}}_1(s)\;ds.\end{aligned}$$ Since $W^{1,1}([0,T];H^1)$ is continuously imbedded in $C^0([0,T];H^1)$, that is $\|f(0)\|_1\leq
C\|f\|_{W^{1,1}(H^1)}$, a use of Gronwall’s lemma completes the rest of the proof.
As a result of Lemma $\ref{H3}$, we obtain a super-convergence estimate for $\theta$ in $H^1$-norm.
For $\|\theta\|_\infty,$ a use of Sobolev inequality $$\begin{aligned}
\|\chi\|_\infty \leq C\left(\log\frac{1}{h}\right)^{1/2}\|\nabla\chi\|~~~~~~\forall \chi\in U_h
\label{NEWA1}\end{aligned}$$ with Lemma \[H3\] yields $$\begin{aligned}
\|{\theta}\|_\infty &\leq & C(T)h^2 \left(\log\frac{1}{h}\right)^{1/2} \Big[\|u_0\|_4+\|u_1\|_3 \nonumber\\
&&+\int_{0}^{T}\left(\|f\|_2+\|f_t\|_1+\|f_{tt}\|_0+\|f_{ttt}\|_0\right)ds\Big].
\label{h3}\end{aligned}$$ Below, we discuss the maximum norm estimate in form of a theorem.
\[H4n\] Let $u$ and $u_h$ be the solutions of $(\ref{a})$ and $(\ref{c})$ respectively. Further, let the assumptions of Lemma $\ref{H3}$ hold. Then, $$\begin{aligned}
\|u(t)-u_h(t)\|_\infty &\leq &C \;h^2 \left(\log\frac{1}{h}\right)\; \Big(\|u_0\|_4+\|u_1\|_3+
\|D_t^3 f\|_{L^1(L^2)}\nonumber \\
&&+ \sum_{j=0}^{2} \|D_t^j f\|_{L^1(H^{2-j})}\Big), \end{aligned}$$ where $C=C(T)$ is a positive constant independent of $h$.
Proof. By the triangle inequality $$\|u(t)-u_h(t)\|_\infty\leq \|{\theta}\|_\infty+\|{\rho}\|_\infty.$$ Now, combine the estimates obtained in (\[h3\]) and in (\[R1h\]) with Lemma \[lem1\] to obtain the required result.
Error Estimates for a Completely Discrete Scheme
================================================
In this section, we introduce further notations and formulate a completely discrete scheme by applying an explicit finite difference method to discretize the time variable of the semidiscrete system (\[c\]). Then, we discuss optimal error estimates.
Let ${k}$ $(0<{k}<1)$ be the time step, $k=T/N$ for some positive integer $N$, and $t_n=n{k}$. For any function $\phi$ of time, let $\phi^n$ denote $\phi(t_n)$. We shall use this notation for functions defined for continuous in time as well as those defined for discrete in time. Set $\phi^{n+1/2}=(\phi^{n+1}+\phi^n)/2,$ and define the following notations for the difference quotients: $${{\delta}_t}\phi^n=\frac{\phi^{n+1}-\phi^{n-1}}{2k},
\quad {{\partial}_t}\phi^{n+1/2}=\frac{\phi^{n+1}-\phi^n}{k},
\quad {{\partial}_{t}^2}\phi^n=\frac{\phi^{n+1}-2\phi^n+\phi^{n-1}}{k^2}.$$ Note that $${{\delta}_t}\phi^n=\frac{{{\partial}_t}\phi^{n+1/2}+{{\partial}_t}\phi^{n-1/2}}{2},
\qquad {{\partial}_{t}^2}\phi^n=\frac{{{\partial}_t}\phi^{n+1/2}-{{\partial}_t}\phi^{n-1/2}}{k}.$$ Then, the discrete-in-time scheme of (\[c\]) is to seek $U^{n}\in U_h$ such that for $\chi\in U_h$ $$\begin{aligned}
&&\frac{2}{{k}}({{\partial}_t}U^{{1/2}},\Pi_h^*\chi)+A_h(U^0,\Pi_h^*\chi)=(f^0+\frac{2}{{k}}u_1,\Pi_h^*\chi),
\quad \forall \chi\in U_h,\label{7-1}\\
&&({{\partial}_{t}^2}U^{n},\Pi_h^*\chi)+A_h(U^n,\Pi_h^*\chi)+
k\sum_{j=0}^{n-1}B_h(t_n,t_{j+1/2};U^{j+1/2},\Pi_h^*\chi)=(f^n,\Pi_h^*\chi),\label{7-1c}\end{aligned}$$ $n\geq 1$, with a given initial data $U^{0}$ in $U_h$. This choice of time discretization leads to a second order accuracy in ${k}$. The integral term in (\[c\]) is computed by using the second order quadrature formula $$\sigma^n(g)=k\sum_{j=0}^{n-1}g(t_{j+1/2})\approx\int_0^{t_n}g(s)\,ds,\quad
\mbox{with} \quad t_{j+1/2}=(j+1/2){k}.$$ We shall use a shorthand notation $\sigma^n(B_h^n(U,\Pi_h^*\chi))$ for $k\sum_{j=0}^{n-1}B_h(t_n,t_{j+1/2};U^{j+1/2},\Pi_h^*\chi)$. The quadrature error $q^n(g)$ is defined by $$q^n(g)=\sigma^{n}(g)-\int_0^{t_n}g(s)\,ds=
\sum_{j=0}^{n-1}\left( kg^{j+{1/2}}-\int_{t_j}^{t_{j+1}}g(s)\,ds \right).$$ Similarly, for ${\phi}\in U_h$, we define a linear functional $q_B^{n}({\phi})$ representing the error in the quadrature formula by $$q_B^{n}({\phi})({\chi})=\sigma^{n}\left(B^{n}({\phi},
{\chi})\right)-\int_0^{t_n}B(t_n,s;{\phi}(s),{\chi})\,ds.$$ Notice that $q_B^{0}({\phi})=0$.
For our future use, we state without proof the following lemma. For a proof, see, [@PTW].
\[q\] There exists a positive constant $C,$ independent of $k$ and $h,$ such that the following estimate holds: $$k\sum_{n=0}^{m}||{{\partial}_t}q_B^{n+1/2}({\phi})||_{-1,h}\leq Ck^2
\int_0^{t_{m+1}}(||{\phi}||_{1}+||{\phi}_t||_{1}+
||{\phi}_{tt}||_{1})\,ds.$$
Now, define $e^n:=u^n-U^n$. We split $e^n=\rho^n+\xi^n$ with $\rho^n=u^n-V_hu^n$ and $\xi^n=V_hu^n-U^n$. From (\[7-1\])-(\[7-1c\]) and (\[c\]), we derive equations in $e^n$ as follows $$\begin{aligned}
&&\frac{2}{{k}}({{\partial}_t}e^{{1/2}},\Pi_h^*\chi)+A_h(e^0,\Pi_h^*\chi)=(2r^0,\Pi_h^*\chi),\label{8-0-e}\\
&&({{\partial}_{t}^2}e^n, \Pi_h^*\chi)+A_h(e^n,\Pi_h^*\chi)+
\sigma^n\left(B_h^n(e,\Pi_h^*\chi\right)
=(r^n,\Pi_h^*\chi)+q_{B_h}^n(u)(\Pi_h^*\chi),\label{8-3-e}
\hspace{-2cm}\end{aligned}$$ $n\geq 1$, for all $\chi\in U_h$, where $\displaystyle r^0= \frac{1}{{k}}\left({{\partial}_t}u^{1/2}-u_1\right)-
\frac{1}{2}u_{tt}^0=\frac{1}{2k^2}
\int_{0}^{k}(t-k)^2\frac{\partial^3 u}{\partial t^3}(t)\,dt,$ and $$\label{qqw1}
r^n={{\partial}_{t}^2}u^n-u_{tt}^{0}=-\frac{1}{6k^2}\\
\int_{-k}^{k}(|t|-k)^3\frac{\partial^4 u}{\partial t^4}
(t^n+t)\,dt,\;\;n\geq 1.$$ Since estimates for $\rho$ are known from Lemma \[lem2\], it is sufficient to estimate $\xi$. From (\[8-0-e\])-(\[8-3-e\]), we obtain the following equations in $\xi^n$: $$\begin{aligned}
\frac{2}{{k}}({{\partial}_t}\xi^{1/2},\Pi_h^*\chi)&+&A(\xi^0,\chi)=
-\frac{2}{{k}}({{\partial}_t}\rho^{{1/2}},\Pi_h^*\chi)+(2r^0,\Pi_h^*\chi) \nonumber\\
&+&\epsilon_h(f^0-u_{tt}^0,\chi)-
\epsilon_A(V_hu (0),\chi),\label{8-0}\\
({{\partial}_{t}^2}\xi^n, \Pi_h^*\chi)&+&A(\xi^n,\chi)=
(r^n,\Pi_h^*\chi)-({{\partial}_{t}^2}\rho^n, \Pi_h^*\chi)+H^n(\xi)(\chi) \nonumber \\
&-&\sigma^n\left(B^n(\xi,\chi)\right)- H^n(V_hu)(\chi)+\epsilon_h(f^n-u_{tt}^n,\chi)+
q^n_B(V_hu)(\chi), \label{8-3}
$$ where $$H^n(\xi)(\chi)=\epsilon_A(\xi^n,\chi)+k\sum_{j=0}^{n-1}
\epsilon_B(t_n,t_{j+1/2};\xi^{j+1/2},\chi).$$
Below, we shall obtain $l^\infty(H^1)$-estimate for $\xi^{n+1/2}$.
\[H5\] Assume that $f\in L^1(H^2),\;f_t\in L^1(H^1),\;f_{tt},\;f_{ttt}\in L^1(L^2),\; u_0\in H^4\cap H_0^1$ and $u_1\in H^3\cap H_0^1$. Further, assume that the CFL condition $$\label{CFL}
\frac{k^2}{h^2}\leq \frac{4c_{eq}}{\Lambda C_{inv}}$$ is satisfied, where $\Lambda>0$ is the constant given in $(\ref{eqn2.3})$, $C_{inv}$ appears in the inverse inequality $(\ref{inv})$ and $c_{eq}$ is stated in the equivalence of norms as in $(\ref{nd})$. Then, with $u_h(0)={V_h}u_0$ and $u_{h,t}(0)=\Pi_hu_1,$ there exists a positive constant $C=C(T),$ independent of $h$ and $k$, such that the following estimate $$\begin{aligned}
\label{eu2-d-1}
\|{{\partial}_t}\xi^{m+1/2}\|_0+\|\xi^{m+1/2}\|_1
&\leq &
C(T)(k^2+h^2) \Big(\|u_0\|_4+\|u_1\|_3\nonumber\\
&&+\|D_t^3f\|_{L^1(L^2)}+
\sum_{j=0}^{2} \|D_t^jf\|_{L^1(H^{2-j})}\Big),\end{aligned}$$ holds for $m=0,1,\cdots,N-1$.
Proof. Choose $\chi= {{\delta}_t}\xi^n$ in (\[8-3\]) and obtain $$\begin{aligned}
\label{8-main}
\frac{1}{2} {\bar{\partial}_t}\Big(|||{{\partial}_t}\xi^{n+{1/2}}|||_0^2 &+&
A(\xi^{n+1},\xi^{n})\Big) = (r^n-{{\partial}_{t}^2}\rho^n, \delta_t
\xi^{n})+H^n(\xi)(\delta_t \xi^n)\nonumber\\
&& -\sigma^n\left(B^n(\xi,\delta_t \xi^n)\right)- H^n(V_hu)(\delta_t \xi^n)\nonumber\\
&&+\epsilon_h(f^n-u_{tt}^n,\delta_t \xi^n)+q^n_B(V_hu)(\delta_t \xi^n)\\
&=& I_1^n+ I_2^n + I_3^n+I_4^n+ I_5^n + I_6^n, \nonumber\end{aligned}$$ where ${\bar{\partial}_t}$ denotes backward differencing. Next multiply (\[8-main\]) by $2 k$ and sum the resulting one from $n=2$ to $m$ to arrive at $$\begin{aligned}
\label{8-main-1}
\frac{1}{2} \Big( |||{{\partial}_t}\xi^{m+1/2}|||_0^2+A(\xi^{m+1},\xi^{m})\Big)
&\leq& \frac{1}{2} \Big(|||{{\partial}_t}\xi^{3/2}|||_0^2+ A(\xi^{2},\xi^{1}) \Big)\nonumber\\
&&+{k}\left|\sum_{n=2}^m(I_1^n+ I_2^n + I_3^n+I_4^n+ I_5^n + I_6^n)\right|.\end{aligned}$$ Now define $$|||\xi^{n+1/2}|||_1^2=||{{\partial}_t}\xi^{n+1/2}||_0^2+||\xi^{n+1/2}||_1^2,$$ and let for some $m^{\star}$ with $ 0 \le m^{\star}\leq m,$ $$|||\xi^{m^{\star}+1/2}|||_1 =\max_{0\leq n\leq m}|||\xi^{n+1/2}|||_1.$$ To estimate the sum in $I_1^n$, an application of the Cauchy-Schwarz inequality yields $$\begin{aligned}
{k}\left|\sum_{n=2}^m I_1^n \right|
&\leq & Ck\sum_{n=2}^m\left(\|\partial^2_t \rho^n\|_0+\|r^n\|_0\right)\,
\left(\|{{\partial}_t}\xi^{n+1/2}\|_0+\|{{\partial}_t}\xi^{n-1/2}\|_0 \right)\\
&\leq & 2C{k}\sum_{n=2}^{m}\left(\|\partial^2_t \rho^n\|_0+\|r^n\|_0\right)
|||\xi^{m^{\star}+1/2}|||_1.\end{aligned}$$ For the second sum on the right hand side of (\[8-main-1\]), we use the fact that $$\begin{aligned}
\label{eq:formula}
\psi^{n}{{\delta}_t}\xi^{n}&=&{\bar{\partial}_t}(\psi^{n}\xi^{n+1/2})-{{\partial}_t}\psi^{n+1/2}\xi^{n-1/2}\end{aligned}$$ and conclude $$k\sum_{n=2}^m\epsilon_A(\xi^n,{{\delta}_t}\xi^{n})= \epsilon_A(\xi^m,\xi^{m+1/2})-
\epsilon_A(\xi^1,\xi^{1+1/2})-k\sum_{n=2}^m\epsilon_A({{\partial}_t}\xi^{n+1/2},\xi^{n-1/2}).$$ Using (\[ng\]) and the inverse inequality (\[inv\]), we obtain $$\begin{aligned}
\left|k\sum_{n=2}^m\epsilon_A(\xi^n,{{\delta}_t}\xi^{n})\right|
&\leq&
Ch\left\{\|\xi^m\|_1\|\xi^{m+1/2}\|_1+\|\xi^1\|_1\|\xi^{1+1/2}\|_1\right\}\\
&& +Chk\sum_{n=2}^m\|{{\partial}_t}\xi^{n+1/2}\|_1 \|\xi^{n-1/2}\|_1\\
&\leq& C\left\{\|\xi^m\|_0+\|\xi^1\|_0+k\sum_{n=2}^m\|{{\partial}_t}\xi^{n+1/2}\|_0\right\}
|||\xi^{m^{\star}+1/2}|||_1.\end{aligned}$$ Since $\xi^0=0$, $\xi^m=k\sum_{n=0}^{m-1} {{\partial}_t}\xi^{n+1/2}$, and it follows that $$\left|k\sum_{n=2}^m\epsilon_A(\xi^n,{{\delta}_t}\xi^{n})\right|\leq
Ck\left(\sum_{n=0}^{m-1} \|{{\partial}_t}\xi^{n+1/2}\|_0\right)
|||\xi^{m^{\star}+1/2}|||_1.$$ Similarly, we obtain $$\begin{aligned}
\left|k^2\sum_{n=2}^m\sum_{j-0}^{n-1}\epsilon_B(t_n,t_{j+1/2};\xi^{j+1/2},{{\delta}_t}\xi^{n})\right|
&\leq&k^2\sum_{n=2}^m\left(C\sum_{j=0}^{n-1}\|\xi^{j+1/2}\|_1\right)
|||\xi^{m^{\star}+1/2}|||_1\\
&\leq&CT k\left(\sum_{j=0}^{m-1}\|\xi^{j+1/2}\|_1\right)
|||\xi^{m^{\star}+1/2}|||_1,\end{aligned}$$ and hence, $${k}\left|\sum_{n=2}^m I_2^n \right|\leq
C(T)k\left(\sum_{n=0}^{m-1} |||\xi^{n+1/2}|||_1\right)
|||\xi^{m^{\star}+1/2}|||_1.$$ To estimate the sum in $I_3^n$, we again use (\[eq:formula\]) and rewrite the sum as: $$\begin{aligned}
k\sum_{n=2}^mI_3^n
&=&\sigma^m\left(B^m(\xi,\xi^{m+1/2})\right)-\sigma^1\left(B^1(\xi,\xi^{1+1/2})\right)\\
&&-k^2\sum_{n=2}^m\sum_{j=0}^{n-1}(\bar{\partial}_{t,1}B)
(t_n,t_{j+1/2};\xi^{j+1/2},\xi^{n-1/2})\\
&&+k\sum_{n=2}^mB(t_{n-1},t_{n-1/2};\xi^{n-1/2},\xi^{n-1/2}),\end{aligned}$$ where $\bar{\partial}_{t,1}B$ denotes the difference quotient of $B$ with respect to its first argument. Since, $|\bar{\partial}_{t,1}B|\leq C||B_t||_\infty< \infty$, it follows that $$\left|k\sum_{n=2}^mI_3^n\right|\leq
C(T)k\left(\sum_{j=0}^{m-1}||\xi^{j+1/2}||_1\right)
|||\xi^{m^{\star}+1/2}|||_1.$$ For the sum involving $I_4^n$, we note that $$\begin{aligned}
\left|k\sum_{n=2}^m\epsilon_A(V_hu^n,{{\delta}_t}\xi^{n})\right|&=&
\left|\epsilon_A(V_hu^m,\xi^{m+1/2})-
\epsilon_A(V_hu^1,\xi^{1+1/2})-k\sum_{n=2}^m\epsilon_A({{\partial}_t}V_hu^{n+1/2},\xi^{n-1/2})\right|\\
&\leq& Ch^2\left\{||u^m||_2+||u^1||_2+k\sum_{n=0}^m||{{\partial}_t}V_hu^{n+1/2}||_2\right\}
|||\xi^{m^{\star}+1/2}|||_1\\
&\leq& Ch^2\left\{ ||u_0||_2+||u_t||_{L^1(H^2)}\right\}
|||\xi^{m^{\star}+1/2}|||_1.\end{aligned}$$ Similarly, we have $$\left|k^2\sum_{n=2}^m\sum_{j=0}^{n-1}\epsilon_B(t_n,t_{j+1/2};V_hu^{j+1/2},{{\delta}_t}\xi^{n})\right|
\leq CTh^2\left\{ ||u_0||_2+||u_t||_{L^1(H^2)}\right\} |||\xi^{m^{\star}+1/2}|||_1.
$$ In order to estimate the sum in $I_5^n$, we repeat the previous arguments and use (\[ne\]) to arrive at $$\begin{aligned}
\left|k\sum_{n=2}^m\epsilon_h(f^n-u^n_{tt},{{\delta}_t}\xi^{n})\right|&=&
\left|\epsilon_h(f^m-u^m_{tt},\xi^{m+1/2})-
\epsilon_h(f^1-u^1_{tt},\xi^{1+1/2})\right.\\
&&\left.-k\sum_{n=2}^m\epsilon_h\left({{\partial}_t}(f^{n+1/2}-u^{n+1/2}_{tt}),\xi^{n-1/2}\right)\right|\\
&\leq& Ch^2\left\{ \|f^0-u_{tt}^0\|_1+ \|f_t-u_{ttt}\|_{L^1(H^1)}\right\}
|||\xi^{m^{\star}+1/2}|||_1.\end{aligned}$$ For the last sum, we rewrite it as $$k\sum_{n=2}^mI_6^n=q^m_B(V_hu)(\xi^{m+1/2})-q^1_B(V_hu)(\xi^{1+1/2})
-k\sum_{n=2}^m({{\partial}_t}q^{n+1/2}_B(V_hu))(\xi^{n-1/2}).$$ Since $q^0_B(V_hu)=0$, $q^m_B(V_hu)=k\sum_{n=0}^m{{\partial}_t}q^{n+1/2}_B(V_hu)$, we obtain $$k\left|\sum_{n=2}^mI_6^n\right|\leq Ck\left\{\sum_{n=0}^m
||{{\partial}_t}q^{n+1/2}_B(V_hu)||_{-1,h}\right\}|||\xi^{m^{\star}+1/2}|||_1.$$ Combining all the previous estimates, we conclude that $$\begin{aligned}
\label{8-main-2}
|||{{\partial}_t}\xi^{m+1/2}|||_0^2+A(\xi^{m+1},\xi^{m})
&\leq&|||{{\partial}_t}\xi^{3/2}|||_0^2+ A(\xi^{2},\xi^{1})+
Ck\left\{ \sum_{n=2}^{m}(\|\partial^2_t \rho^n\|_0+\|r^n\|_0)\right.\nonumber\\
&+& \left.\sum_{n=0}^m||{{\partial}_t}q^{n+1/2}_B(V_hu)||_{-1,h}+
\sum_{j=0}^{m-1}|||\xi^{j+1/2}|||_1\right\}|||\xi^{m^{\star}+1/2}|||_1\nonumber\\
&+& h^2C(T,f,u)|||\xi^{m^{\star}+1/2}|||_1,\end{aligned}$$ where $$C(T,f,u)=||u_0||_2+||u_t||_{L^1(H^2)}+
||u_{tt}(0)||_1+||u_{ttt}||_{L^1(H^1)}
+||f^0||_1+||f_t||_{L^1(H^1)}.$$ In order to estimate the first two terms on the right hand side of (\[8-main-2\]), we choose $\chi={{\partial}_t}\xi^{3/2}$ in (\[8-3\]) for $n=1$ and obtain $$\begin{aligned}
|||{{\partial}_t}\xi^{3/2}|||_0^2+A(\xi^{2},\xi^{1})
&\leq&|||{{\partial}_t}\xi^{1/2}|||_0^2+h^2\Big(\|u^1\|_2+\|u_0\|_2+k\|{{\partial}_t}u^{1/2}\|_2\Big)\\
&+&h^2\Big( \|f^0-u_{tt}^0\|_2+\|f_t-u_{ttt}\|_{L^1(0,k;H^1)} \Big) +\|{{\partial}_t}q^{1/2}_B\|_{-1,h}.\end{aligned}$$ Next, we choose $\chi={{\partial}_t}\xi^{1/2}$ in (\[8-0\]) to find that $$|||{{\partial}_t}\xi^{1/2}|||_0\leq C\left\{\|{{\partial}_t}\rho^{1/2}\|_0+k\|r^0\|_0+h^2\|f^0-u_{tt}^0\|_2+
h^2\|u_0\|_2\right\}.$$ A use of these estimates in (\[8-main-2\]) results in $$\begin{aligned}
\label{8-main-3}
|||{{\partial}_t}\xi^{m+1/2}|||_0^2&+&A(\xi^{m+1},\xi^{m})
\leq
C\left\{\|{{\partial}_t}\rho^{1/2}\|_0+k\sum_{n=1}^{m}\|\partial^2_t \rho^n\|_0+
k\sum_{n=0}^{m}\|r^n\|_0\right.\nonumber\\
&&\left.+k\sum_{n=0}^m||{{\partial}_t}q^{n+1/2}_B(V_hu)||_{-1,h}+
k\sum_{j=0}^{m-1}||\xi^{j+1/2}||_1\right\}|||\xi^{m^{\star}+1/2}|||_1\nonumber\\
&&+h^2C(T,f,u)|||\xi^{m^{\star}+1/2}|||_1.\end{aligned}$$ Note that $$A(\xi^{m+1},\xi^{m})=A(\xi^{m+1/2},\xi^{m+1/2})-
\frac{k^2}{4}A({{\partial}_t}\xi^{m+1/2},{{\partial}_t}\xi^{m+1/2}).$$ Hence, $$|||{{\partial}_t}\xi^{m+1/2}|||_0^2+A(\xi^{m+1},\xi^{m})\geq
c_{eq} \;\|{{\partial}_t}\xi^{m+1/2}\|_0^2 + \alpha\|\xi^{m+1/2}\|_1^2-
\frac{k^2}{4}A({{\partial}_t}\xi^{m+1/2},{{\partial}_t}\xi^{m+1/2}).$$ Since the CFL condition (\[CFL\]) holds, choose $k$ so that $C_\ast=\left(c_{eq}-\Lambda C_{inv} \frac{k^2}{4h^2}\right)>0$, where the constants $\Lambda$, $c_{eq}$ and $C_{inv}$ appear in (\[eqn2.3\]), (\[nd\]) and (\[inv\]), respectively. Then $$|||{{\partial}_t}\xi^{m+1/2}|||_0^2+A(\xi^{m+1},\xi^{m})\geq \min\{C_\ast,\alpha\}
|||\xi^{m+1/2}|||_1.$$ Altogether, it now results in $$\begin{aligned}
\label{8-5}
|||\xi^{m+1/2}|||_1\leq|||\xi^{m^{\star}+1/2}|||_1
&\leq&
C\left\{\|{{\partial}_t}\rho^{1/2}\|_0+k\sum_{n=1}^{m}\|\partial^2_t\rho^n\|_0+
k\sum_{n=0}^{m}\|r^n\|_0\right.\nonumber\\
&&\left.+k\sum_{n=0}^m||{{\partial}_t}q^{n+1/2}_B(V_hu)||_{-1,h}+
k\sum_{j=0}^{m-1}|||\xi^{j+1/2}|||_1\right\}\nonumber\\
&&+h^2C(T,f,u).\end{aligned}$$ To estimate the first two terms on the right hand side of (\[8-5\]), it is observed that $$\begin{aligned}
\label{8-5a}
||{{\partial}_t}\rho^{1/2}||_0\leq\frac{1}{k}\int_0^{{k}}||\rho_t(s)||_0\,ds,\end{aligned}$$ and a use of Taylor series expansion yields $$\begin{aligned}
\label{8-5b-n}
k\sum_{n=1}^{m}||{{\partial}_{t}^2}\rho^n||_0 &\leq&\frac{1}{k}\sum_{n=1}^{m}\left\{
\int_{t_n}^{t_{n+1}}(t_{n+1}-s)||\rho_{tt}(s)||_0\,ds+
\int_{t_{n-1}}^{t_n}(s-t_{n-1})||\rho_{tt}(s)||_0\,ds
\right\}\nonumber\\
&\leq& 2\int_{0}^{t_{m+1}}||\rho_{tt}(s)||_0 \,ds.\end{aligned}$$ Further, from (\[qqw1\]) it follows that $$||r^n||_0\leq Ck\int_{t_{n-1}}^{t_{n+1}}\|D_t^4u(s)\|_0\,ds,\quad n\geq 1,$$ and $$||r^0||_0\leq Ck||u_{ttt}||_{L^\infty(0,{k}/2;L^2(\Omega))}
\leq Ck\int_{0}^{t_{m+1}} \|D_t^3u(s)\|_0 \,ds.$$ Thus, we arrive at $$\label{s2}
k\sum_{n=0}^{m}||r^n||_0\leq Ck^2\int_{0}^{t_{m+1}}
\left( \|D_t^3u(s)\|_0+\|D_t^4u(s)\|_0\right)\,ds.$$ Finally, a use of Lemma \[q\] and the triangle inequality yields $$k\sum_{n=1}^{m}||{{\partial}_t}q^{n+1/2}_B(V_hu)||_{-1,h}\leq Ck^2
\sum_{j=0}^2\int_{0}^{t_{m+1}} \left(\|D_t^j u(s)\|_1+\|D_t^j \rho(s)\|_1\right)\,ds.$$ Substitute now (\[8-5a\])-(\[s2\]) in (\[8-5\]) and use the estimates in Lemmas \[lem2\] and \[lem1\]. Then, an application of the discrete Gronwall’s lemma completes the rest of the proof.
By Sobolev inequality, it follows that $$\label{s4}
\|\xi^{n+1/2}\|_\infty \leq C\left(\log \frac{1}{h}\right)^{1/2} \|\xi^{n+1/2}\|_1.$$ Using Lemma $\ref{H5}$, the triangle inequality and the estimates (\[s4\]) and (\[R1h\]), we obtain the result of th following theorem.
\[H6\] Let the assumptions of Lemma $\ref{H5}$ hold. Then, $$\begin{aligned}
\label{eu2-d-2}
\|u(t_{m+1/2})-U^{m+1/2}\|_\infty
&\leq &
C(T)\left(\log \frac{1}{h}\right)(k^2+h^2) \Big(\|u_0\|_4+\|u_1\|_3\nonumber\\
&&+\|D_t^3f\|_{L^1(L^2)}+
\sum_{j=0}^{2} \|D_t^jf\|_{L^1(H^{2-j})}\Big)\end{aligned}$$ for $m=0,1,\cdots,N-1$.
FVEM with Quadrature
=====================
In this section, we discuss the effect of numerical quadrature on FVEM, when the $L^2$ inner product $(\cdot,\cdot)$ and the bilinear forms $A_h(\cdot,\cdot)$ and $B_h(t,s;\cdot,\cdot)$ appearing in (\[c\]) are approximated by simple quadrature formulae.
For a continuous function $\phi$ on a triangle $K$, consider the quadrature formula $$\begin{aligned}
\mathcal {Q}_{K,h}(\phi)=\frac{1}{3}|K|\sum_{l=1}^3 \phi(P_l) \approx \int_K\phi(x) dx\;\;~~~~\forall K \in {\mathcal{T}}_h,
\label{Q1}\end{aligned}$$ where $P_l,~1\leq l\leq 3$ denote the vertices of the triangle $K$ and $|K|$ denotes the area of the triangle $K$. Now the quadrature formula given by (\[Q1\]) is exact for $\phi \in P_1(K) ~\forall K\in {\mathcal{T}}_h$. Using (\[Q1\]), we replace the $L^2$ inner product by the following discrete $L^2$ inner product: $$\begin{aligned}
(\chi,\Pi_h^*\psi)_h&=&\sum_{K\in {\mathcal{T}}_h}\mathcal
{Q}_{K,h}(\chi\Pi_h^*\psi) \nonumber \\
&=&\sum_{P_i\in N_h^0} \chi(P_i)\psi(P_i)|S_{K_{P_i}}^*|~~~~~~\forall \chi,~\psi \in U_h.
\label{QQ12}\end{aligned}$$ This is known as lumping of mass in the literature. Observe that $\|\chi\|_h^2=(\chi,\chi)_h~~\forall \chi \in U_h$ is a norm on $U_h,$ which is equivalent to the $L^2$ norm, i.e., there exist positive constants $C_5$ and $C_{6}$, independent of $h$, such that $$C_5\|\chi\|_0\leq\|\chi\|_h\leq C_{6}\|\chi\|_0.
\label{ap7}$$ Define quadrature error by $$\bar{\epsilon}_h(\chi,\psi)=(\chi,\Pi_h^*\psi)-
(\chi,\Pi_h^*\psi)_h.$$ Since the quadrature formula involves only the values of the functions at the interior nodes and $\Pi_h^*u_h(P_i)=u_h(P_i)\;\forall P_i\in N_h^0 \;\mbox{and}\; u_h\in U_h$, it follows that $$\begin{aligned}
(\chi,\psi )_h=(\chi,\Pi_h^*\psi )_h\; \;\;\;\forall \chi,\;\psi \in U_h.
\label{Q10}\end{aligned}$$ Below, we state the estimates related to quadrature error, whose proof can be found in [@KNP-2008].
\[L21\] For $\chi,~ \psi \in U_h$, there is a positive constant $C$, independent of $h$, such that the following estimate holds: $$\begin{aligned}
|\bar{\epsilon}_h(\chi,\psi)|\leq Ch^2\|\chi\|_1\|\psi\|_1.
\label{apnn1}\end{aligned}$$ Further, for $\chi \in H^2$ and $\psi \in U_h$, there holds: $$\begin{aligned}
|\bar{\epsilon}_h(\chi,\psi)|\leq Ch^2\|\chi\|_2\|\psi\|_1.
\label{apnn2}\end{aligned}$$
Now define the following quadrature approximation over each element $K$ by $$\begin{aligned}
\label{Q2}
\int_{\overline {M_lQ}\cap K}v(z)~ds \approx
\frac {\overline{M_lQ}}{2}\left(v(M_l)+v(Q)\right)= \tilde
{\mathcal{Q}}_{h,l}(v),\end{aligned}$$
![[]{data-label="fig:mesh-1"}](figure2.eps){width="8.0cm" height="6.0cm"}
where $M_l$ is the midpoint of $P_lP_{l+1}$ and $Q$ is the barycenter of the triangle $\triangle P_lP_{l+1}P_{l+2}$, (see FIGURE \[fig:mesh-1\] for $l=1$). Associated with (\[Q2\]), we now introduce the quadrature error as $$\mathcal E_{\overline {M_lQ}\cap K}(v)=\int_{\overline {M_lQ}\cap K}v(s)ds- \tilde {\mathcal { Q}}_{h,l}(v).$$ Then, we have the following estimate related to the above quadrature error. For a proof, see, Cai [@10 pp 732].
\[AP1\] Let $v\in W^{2,\infty}({\overline {M_lQ}\cap K}).$ Then, there is a positive constant $C,$ independent of $h_K$, such that $$\begin{aligned}
|\mathcal E_{\overline {M_lQ}\cap K}(v)|\leq Ch_K^3\|v\|_{2,\infty,\overline {M_lQ}\cap K},
\label{ni}\end{aligned}$$ where $h_K$ is the diam($K$).
Now to replace the integral in the definition of $A_h(\cdot,\cdot)$, we observe that $$\begin{aligned}
A_h(u_h,\Pi^*_hv_h)&=&-\sum_{P_l\in N_h}v_i\int_{\partial K_{P_l}^* } A\nabla
u_h.{\bf n}~ds\;\;\;\Big(v_i=v_h(P_i)\Big)\\
&=& \sum_{K}I_K (u_h, \Pi_h^*v_h),\end{aligned}$$ where $$\begin{aligned}
I_K (u_h, \Pi_h^*v_h)&=&-\sum_{{P_l}(1\leq l\leq 3)} v_l \int_{\partial K_{P_l}^*\cap K} A\nabla
u_h.{\bf n_l} ds\\
&=&\sum_{{P_l}(1\leq l\leq 3)}(v_{l+1}-v_{l})
\int_{\overline{M_lQ}\cap K} A\nabla
u_h.{\bf n_l} ds,\end{aligned}$$ $v_4=v_1$ and ${\bf n_l}$ is the outward unit normal vector to $\overline{M_lQ}$. Since $\nabla u_h.{\bf n_l}$ is constant on each element $K$, we define the quadrature rule as $$\begin{aligned}
\tilde {I}_K(u_h, \Pi_h^*v_h)=\sum_{{P_l}(1\leq l\leq 3)} \mathcal E_{\overline {M_lQ}\cap K}(A)\nabla u_h.{\bf n_l}(v_{l+2}-v_{l+1}).
\label{Q123}\end{aligned}$$ and set $${\tilde{A}_h}(\chi,\Pi_h^*\psi)= \sum_{K\in
T_h}\tilde{I}_K(\chi,\Pi_h^*\psi).$$ Note that the bilinear form $A_h(\cdot,\cdot)$ in (\[c\]) is approximated by ${\tilde{A}_h}(\cdot,\cdot).$ Simlilarly, define ${\tilde{B}_h}(\cdot,\cdot)$ as an approximation of $B_h(\cdot,\cdot).$
With the definitions as above, define quadrature error functional for the bilinear form $A_h(\cdot,\cdot)$ as $$\begin{aligned}
{{\bar\epsilon}_A}(\chi,\psi)=A_h(\chi,\Pi^*_h\psi)-{\tilde{A}_h}(\chi,\Pi^*_h\psi)\;\;\;\; \forall \chi,~\psi \in U_h.
\label{Q4}\end{aligned}$$ Below, we state without proof the estimate of (\[Q4\]) whose proof can be found in [@KNP-2008].
\[L1\] Assume that ${\mathcal{A}}\in W^{2,\infty} (\Omega; \mathbb{R}^{2\times 2}).$ Then, there exists a positive constant $C,$ independent of $h,$ such that $${{\bar\epsilon}_A}(\chi,\psi)
\leq Ch^2\|\chi\|_1\|\psi\|_1\;\;\;\;\forall \chi,~\psi \in U_h.$$
Similar results hold for ${{\bar\epsilon}_B}(t,s;\cdot,\cdot)$ which is defined as in (\[Q4\]). For the rest of our analysis, we introduce the functionals ${{S}}(t)={{S}}$ and $\hat{S}(t)=\hat{S}$ defined on $U_h$ for a given $\psi$ and $t\in (0,T]$ as $${{S}}(\psi)(\chi)={{\bar\epsilon}_A}(\psi,\chi)+\int_0^t{{\bar\epsilon}_B}(t,s;\psi(s),\chi)ds,$$ and $$\hat{S}(\psi)(\chi)={{\bar\epsilon}_A}(\hat\psi,\chi)+\int_0^t{{\bar\epsilon}_B}(s,s;\hat\psi (s),\chi)\,ds
-\int_0^t\int_0^s\bar{\epsilon}_{B_\tau}(s,\tau;\hat\psi (\tau),\chi)\,d\tau ds.$$ Then using Lemma \[L1\], we derive the following estimate for $S$ in a similar manner to those obtained in Lemma \[lm-n\] $$\|S(\psi)\|_{-1,h}\leq Ch^2 \left( \|\psi(t) \|_{2}+ \int_{0}^t \|\psi(s)\|_{2}\,ds\right).$$ Similar result can be obtain for the estimate of $\hat{S}$ again following proof of Lemma \[lm-n\].
Now the semidiscrete finite volume element method combined with quadrature is to seek $u_h:(0,T]\longrightarrow U_h$ such that $$(u_{h,tt},v_h)_h+{\tilde{A}_h}(u_h,v_h)+\int_0^t{\tilde{B}_h}(t,s;u_h(s),v_h)\,ds=(f,v_h)_h~~~~~~\forall v_h
\in {U_h^*},\label{ap4}$$ with appropriate initial data $u_h(0)$ and $u_{h,t}(0)$ in $U_h$.
Optimal error estimates
-------------------------
In this subsection, we discuss optimal estimates in $L^{\infty}(L^2)$ as well as in $L^{\infty}(H^1)$-norms and quasi-optimal estimates in $L^{\infty}(L^{\infty})$-norm.
Now replace $v_h$ by $\Pi_h^*\chi$ in (\[ap4\]) and subtract the resulting equation from (\[newaa1\]) to obtain $$\begin{aligned}
(u_{tt},\Pi_h^*\chi)-(u_{h,tt},\Pi_h^*\chi)_h&+&A_h(u,\Pi_h^*\chi)-{\tilde{A}_h}(u_h,\Pi_h^* \chi)\nonumber \\
&+&\int_0^tB_h(t,s;u,\Pi_h^*\chi)\,ds-\int_0^t{\tilde{B}_h}(t,s;u_h,\Pi_h^*\chi)\,ds \nonumber\\
&&= (f,\Pi_h^*\chi)-(f,\Pi_h^*\chi)_h \;\;\; \forall \chi \in U_h.
\label{equation1}\end{aligned}$$ Using the definitions of Ritz-Volterra projection ${V_h}u$ and ${{S}},$ we arrive at an equation in $\theta$ as $$\begin{aligned}
(\theta_{tt},\Pi_h^*\chi)_h+A(\theta, \chi)&=&
-(\rho_{tt},\Pi_h^*\chi)+{G}(\rho)(\chi)+{G}(\theta)(\chi)\nonumber \\
&&+{{S}}(\theta)(\chi)- {{S}}({V_h}u)(\chi)
+{{\bar\epsilon}_h}(f,\chi)-{{\bar\epsilon}_h}(({V_h}u)_{tt},\chi)\nonumber\\
&&-\int_0^tB(t,s;\theta,\chi)\,ds.
\label{ap5-n}\end{aligned}$$
Below, we establish $L^{\infty}(H^1)$ estimate.
\[HH1\] Let $u$ and $u_h$ be the solutions of $(\ref{a})$ and $(\ref{ap4}),$ respectively, and assume that $f\in L^1(H^1),~f_t,~f_{tt}\in L^1(L^2),~u_0\in H^3\cap H_0^1$ and $u_1\in H^2\cap H_0^1$. With $u_h(0)=\Pi_h u_0$ and $ u_{h,t}(0)=\Pi_h u_1,$ there exists a positive constant $C=C(T),$ independent of $h$, such that $$\|u(t)-u_h(t)\|_1 \leq Ch \left(\|u_0\|_3+\|u_1\|_2+ \|f\|_{L^1(H^1)}+
\sum_{j=1}^{2} \|D_t^j f\|_{L^1(L^2)}\right)$$ holds for $t\in (0,T].$
Proof. Choose $\chi=\theta_t$ in (\[ap5-n\]) so that $$\begin{aligned}
(\theta_{tt},\Pi_h^*\theta_t)_h+A(\theta,\theta_t)&=&
{G}(\rho)(\theta_t) +{G}(\theta)(\theta_t)-(\rho_{tt},\Pi_h^*\theta_t)+{{S}}(\theta)(\theta_t)\nonumber \\
&&-{{S}}({V_h}u)(\theta_t)+\bar{\epsilon}_h(f,\theta_t)-\bar{\epsilon}_h(({V_h}u)_{tt},\theta_t)\nonumber \\
&&-\int_0^tB(t,s;\theta,\theta_t)ds.
\label{nk}\end{aligned}$$ Then, use (\[Q10\]) and the symmetric property of $A(\cdot,\cdot)$ to obtain $$\begin{aligned}
\frac{1}{2} \frac{d}{dt}\left[(\theta_t,\theta_t)_h+A(\theta,\theta)\right] &=&
{G}(\rho)(\theta_t) +{G}(\theta)(\theta_t)-(\rho_{tt},\Pi_h^*\theta_t)+{{S}}(\theta)(\theta_t) \\
&&-{{S}}({V_h}u)(\theta_t)+\bar{\epsilon}_h(f,\theta_t)-\bar{\epsilon}_h(({V_h}u)_{tt},\theta_t) \\
&&-\int_0^tB(t,s;\theta,\theta_t)ds.\end{aligned}$$ Integrate from $0$ to $t$ and use the equivalence of the norms in (\[ap7\]) to find that $$\begin{aligned}
\frac{1}{2}\left[\|\theta_t\|_h^2+ A(\theta,\theta)\right]&=&\Big\{\frac{1}{2}\|\theta_t(0)\|_h^2+
\frac{1}{2}A(\theta(0),\theta(0)) +
\int_0^t\Big[{G}(\rho)(\theta_t)+{G}(\theta)(\theta_t) \nonumber \\
&&-(\rho_{tt},\Pi_h^*\theta_t)-\int_0^sB(s,\tau;\theta(\tau),\theta_t)\,d\tau\Big]\;ds\Big\} +
\int_0^t{{S}}(\theta)(\theta_t) \;ds\nonumber\\
&&- \int_0^t{{S}}({V_h}u)(\theta_t) ds
+ \int_0^t\bar{\epsilon}_h(f,\theta_t)ds-\int_0^t\bar{\epsilon}_h(({V_h}u)_{tt},\theta_t)\;ds \nonumber\\
&=&I+J_1+J_2+J_3+J_4.
\label{asp1}\end{aligned}$$ Estimates for the first term $I$ have already been derived in Theorem \[H2\]. In order to estimate $J_1$, use Lemma \[L1\] and inverse inequality (\[inv\]) to obtain $$\begin{aligned}
|J_{1}| &\leq& \int_0^t|{{S}}(\theta)(\theta_t)|\;ds \leq \int_0^t \|{{S}}(\theta)\|_{-1,h}\;\|(\theta_t)\|_1\;ds \nonumber\\
&\leq& Ch^2\left[\int_0^t\|\theta\|_1\|\theta_t\|_1 ds +\int_0^t\int_0^s\|\theta_t(\tau)\|_1
\|\theta(s)\|_1\;d\tau \;ds\right]\nonumber \\
&\leq& Ch\left[\int_0^t\|\theta\|_1\|\theta_t\|_0 ds+\left(\int_0^t\|\theta\|_1 \;ds\right)
\left(\int_0^t\|\theta_t\|_0 \; ds\right)\right].
\label{asp2}\end{aligned}$$ For $J_2$, we find that $$|J_{2}|\leq \int_0^t|{{S}}({V_h}u)(\theta_t)|\;ds\leq C h^{-1}
\int_0^t\|{{S}}({V_h}u)\|_{-1,h}\|{\theta_t}\|_0\;ds.$$ In view of Lemma \[L21\], the terms $J_3$ and $J_4$ are bounded as $$\begin{aligned}
|J_3|+|J_4|&\leq& \int_0^t|\bar{\epsilon}_h(f,{\theta_t})|ds+2 \int_0^t|\bar{\epsilon}_h(({V_h}u)_{tt},{\theta_t})|ds \nonumber \\ &\leq&
Ch^2\int_0^t(\|f\|_2+\|\rho_{tt}\|_1+\|u_{tt}\|_1)\|{\theta_t}\|_1ds\nonumber \\
&\leq& Ch\int_0^t(\|f\|_2+\|\rho_{tt}\|_1+\|u_{tt}\|_1)\|{\theta_t}\|_0ds.
\label{asp3}\end{aligned}$$ Now, substitute (\[asp2\])-(\[asp3\]) in (\[asp1\]). Use the coercivity property of the bilinear form $A(\cdot,\cdot)$ and equivalence of norms (\[ap7\]). Then, proceed as in Theorem \[H2\] to complete the rest of the proof.
In the following theorem, we prove optimal $L^{\infty}(L^2)$-estimate.
Under the assumptions of Theorem $\ref{HH1}$, there exists a positive constant $C=C(T)$, independent of $h$, such that $$\|u(t)-u_h(t)\|_0\leq Ch^2 \left(\|u_0\|_3+\|u_1\|_2+ \|f\|_{L^1(H^1)}+\sum_{j=1}^{2}
\|D_t^jf \|_{L^1(L^2)}\right)$$ holds for all $t\in (0,T].$
Proof. Integrate (\[ap5-n\]) from $0$ to $t$ to arrive at $$\begin{aligned}
(\theta_t,\Pi_h^*\chi)_h &+& A(\hat{\theta}, \chi) = -(\rho_t,\Pi_h^*\chi)+{\hat{G}}(\rho)(\chi)+{\hat{G}}(\theta)(\chi)\nonumber \\
&+&{\hat{S}}(\theta)(\chi)-{\hat{S}}({V_h}u)(\chi)
+\bar{\epsilon}_h(\hat{f},\chi)-\bar{\epsilon}_h(({V_h}u)_t,\chi)\nonumber \\
&+& (u_{t}(0),\Pi_h^*\chi)-(u_{h,t}(0),\Pi_h^*\chi)_h-\int_0^t\int_0^sB(s,\tau;\theta(\tau),\chi)\,d\tau \;ds.
\label{ap5}\end{aligned}$$ Choose $\chi=\theta$ in (\[ap5\]) and use (\[Q10\]) with the symmetry of the bilinear form $A(\cdot,\cdot)$ to obtain $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\left[(\theta,\theta)_h+A(\hat{\theta},\hat\theta)\right]&=&
I(t)+{\hat{S}}(\theta)(\theta)-{\hat{S}}({V_h}u)(\theta)
+\bar{\epsilon}_h(\hat{f},\theta)\nonumber \\
&&-\bar{\epsilon}_h(({V_h}u)_t,\theta)+(u_{t}(0),\Pi_h^*\theta)-(u_{h,t}(0),\Pi_h^*\theta)_h.
\label{bbbb1}\end{aligned}$$ where $$I(t)=-(\rho_t,\Pi_h^*\theta)+{\hat{G}}(\rho)(\theta)+{\hat{G}}(\theta)(\theta)-
\int_0^t\int_0^sB(s,\tau;\theta(\tau),\theta)\,d\tau ds.$$ Integrate (\[bbbb1\]) from $0$ to $t$ to find that $$\begin{aligned}
\frac{1}{2}\Big(\|\theta(t)\|_h^2+A(\hat{\theta},\hat\theta)\Big) &=& \frac{1}{2} \|\theta(0)\|_h^2+
\int_0^tI(s)\,ds+ \int_0^t{\hat{S}}(\theta)(\theta) \;ds \nonumber \\
&& - \int_0^t {\hat{S}}({V_h}u)(\theta)\;ds
-\int_0^t \bar{\epsilon}_h(({V_h}u)_t,\theta)\; ds \nonumber \\
&&+\int_0^t\bar{\epsilon}_h(\hat{f},\theta)\;ds+\left[(u_{t}(0),\Pi_h^*\hat \theta)-(u_{h,t}(0),\Pi_h^*\hat \theta)_h\right] \nonumber \\
&=& \frac{1}{2}\|\theta(0)\|_h^2+ \int_0^tI(s)\,ds+J_1+J_2+J_3+J_4+J_5.
\label{ap8}\end{aligned}$$ Note that estimates for the first two terms on the right hand sides of (\[ap8\]) have already been derived in Theorem \[TH1\]. For $J_1$, use the definition of $\hat{S}$ and integrate by parts to arrive at $$\begin{aligned}
J_1&=& \int_{0}^{t}{{\bar\epsilon}_A}(\hat{\theta},\theta)\;ds+
\int_{0}^{t} \int_{0}^{s}{{\bar\epsilon}_B}(\tau,\tau;\hat{\theta}(\tau),\theta(s))\;d\tau \;ds\\
&&-\int_{0}^{t}\int_{0}^{s}\int_{0}^{\tau}
\bar\epsilon_{B_{\tau'}}(\tau,\tau';\hat{\theta}(\tau'),\theta) \;d\tau' \;d\tau \;ds\\
&=& J_{11}+J_{12}+J_{13}.\end{aligned}$$ For $J_{11}$, a use of Lemma \[L1\] with the inverse inequality (\[inv\]) yields $$|J_{11}|\leq \int_0^t|\bar{\epsilon}_A(\hat{\theta},\theta)|ds \leq Ch^2\int_0^t
\|\theta\|_1\|\hat{\theta}\|_1ds \leq C h\;\int_0^t\|\theta\|_0\|\hat{\theta}\|_1\;ds.\label{ap13}$$ For $J_{12}$, an integration by parts shows $$\begin{aligned}
|J_{12}|&=&\left|\int_0^t \bar\epsilon_B(s,s;\hat{\theta}(s),\hat{\theta}(t))ds-\int_0^t
\bar\epsilon_B(s,s;\hat{\theta}(s),\hat{\theta}(s))ds\right|\\
&\leq & Ch^2\left\{\|\hat{\theta}(t)\|_1\int_0^t\|\hat{\theta}(s)\|_1ds+
\int_0^t\|\hat{\theta}(s)\|_1^2ds\right\}.\end{aligned}$$ Similarly for $J_{13}$, we have $$\begin{aligned}
|J_{13}|&=& \left|\int_0^t\int_0^s \bar\epsilon_{B_{\tau}}(s,\tau;\hat{\theta}(\tau),\hat{\theta}(t))d\tau ds
-\int_0^t\int_0^s \bar\epsilon_{B_{\tau}}(s,\tau;\hat{\theta}(\tau),\hat{\theta}(s))d\tau ds\right|\\
&\leq &
C(T)h^2\left\{\|\hat{\theta}(t)\|_1\int_0^t\|\hat{\theta}(s)\|_1ds+
\int_0^t\|\hat{\theta}(s)\|_1^2ds\right\}.\end{aligned}$$ For $J_2$, we obtain $$|J_{2}|\leq \|{\hat{S}}({V_h}u)\|_{-1,h}\|\hat{\theta}\|_1+
\int_0^t\|{\hat{S}}_s({V_h}u)\|_{-1,h}\|\hat{\theta}\|_1\;ds.
\label{ap10}$$ To bound $J_3$ and $J_4$, we integrate by parts and apply Lemma \[L21\] to arrive at $$\begin{aligned}
|J_3|&\leq & |\bar{\epsilon}_h(({V_h}u)_t,\hat{\theta})|+\int_0^t|\bar{\epsilon}_h(({V_h}u)_{tt},\hat{\theta})|ds \nonumber \\
&\leq & Ch^2 \left((\|\rho_t\|_1+\|u_t\|_1)\|\hat{\theta}\|_1 +
\int_{0}^t (\|\rho_{tt}\|_1+\|u_{tt}\|_1)\|\hat{\theta}\|_1\;ds\right)
$$ and $$\begin{aligned}
|J_{4}|&\leq& |\bar{\epsilon}_h(\hat{f},\hat{\theta})|+\int_0^t|\bar{\epsilon}_h(f,\hat{\theta})|ds \nonumber \\ &\leq&
Ch^2\Big(\|\hat{f}\|_2\|\hat{\theta}\|_1+\int_0^t\|f\|_2\|\hat{\theta}\|_1ds\Big).\end{aligned}$$ Finally, since $u_{h,t}(0)=\Pi_h u_t(0),$ we have $J_{5}=(u_{t}(0)-\Pi_h u_t(0),\Pi_h^*\hat{\theta})+\bar{\epsilon}_h(\Pi_h
u_{t}(0),\hat{\theta})$. Hence,
$$\begin{aligned}
|J_{5}|&\leq&
Ch^2\left(\|u_t(0)\|_2+\|\Pi_h u_t(0)\|_1\right)\;\|\hat{\theta}\|_1\nonumber \\
&\leq& C h^2
\|u_t(0)\|_2\;\|\hat{\theta}\|_1\leq C h^2
\|u_t(0)\|_2\|\hat{\theta}\|_1.
\label{ap12}\end{aligned}$$
Substitute (\[ap13\])-(\[ap12\]) in (\[ap8\]). We use the coercivity property of the bilinear form $A(\cdot,\cdot)$ and the equivalence of the norms, and proceed as in Theorem \[TH1\] to complete the rest of the proof.
Finally, we prove quasi-optimal maximum norm estimate.
\[thnew\] Let $u$ and $u_h$ be the solutions of $(\ref{a})$ and $(\ref{c}),$ respectively. Further, let the assumptions of Lemma $\ref{H3}$ hold. Then, $$\|u(t)-u_h(t)\|_\infty \leq C(T)h^2 \left(\log\frac{1}{h}\right) \Big(
\|u_0\|_4+\|u_1\|_3+\|D_t^3 f\|_{L^1(L^2)}
+\sum_{j=0}^{2} \|D_t^j f\|_{L^1(H^{2-j})}\Big),$$ where $C(T)$ is a positive constant, independent of $h.$
Proof: Since $u_h(0)={V_h}u_0$, it follows that $\theta(0)=0.$ Then, we modify our estimates for $J_2$ to $J_4$ in (\[asp1\]) to arrive at a superconvergence result for $\theta$ in $H^1$- norm $$\begin{aligned}
\|\theta_t\|_0+\|{\theta}\|_1&\leq& C(T)h^2 \Big(\|u_0\|_4+\|u_1\|_3+
\|D_t^3 f\|_{L^1(L^2)}\nonumber \\&&+
\sum_{j=0}^{2} \|D_t^j f\|_{L^1(H^{2-j})}\Big).
\label{NEW}\end{aligned}$$ Now, a use of (\[NEWA1\]) and (\[NEW\]) completes the rest of the proof.
Numerical Experiment
====================
In this section, we present numerical results to illustrate the performance of the finite volume element method applied to (\[a\]). Assume that ${\mathcal{T}}_h$ is an admissible regular, uniform triangulation of $\overline{\Omega}$ into closed triangles and $0=t_0<t_1<\cdots t_M=T$ is a given partition of the time interval $(0,T]$ with step length $k=\frac{T}{M}$ for some positive integer $M$. With $U^n$ denoting the approximation of $u_h$ at $t=t_n,$ consider the discrete-in-time scheme derived in Section 5, with discrete $L^2$ inner product $(\cdot,\cdot)_h$ and the bilinear forms $A_h(\cdot,\cdot)$ and $B_h(t,s;\cdot,\cdot)$ evaluated using numerical quadrature formulae. Thus, the time discretization scheme is to seek $ U^n\in U_h$ for given $U^0$, such that $$\begin{aligned}
\frac{2}{{k}}({{\partial}_t}U^{{1/2}},\Pi_h^*\chi)_h&+&{\tilde{A}_h}(U^0,\Pi_h^*\chi)=
(f^0+\frac{2}{{k}}u_1,\Pi_h^*\chi)_h, \label{9-1}\\
({{\partial}_{t}^2}U^{n},\Pi_h^*\chi)_h &+&{\tilde{A}_h}(U^n,\Pi_h^*\chi)+
k\sum_{j=0}^{n-1}{\tilde{B}_h}(t_n,t_{j+1/2};U^{j+1/2},\Pi_h^*\chi)\nonumber\\
&=&(f^n,\Pi_h^*\chi)_h,\label{9-1c}\end{aligned}$$ $n\geq 1$, for all $\chi\in U_h$. The method is explicit in time in the sense that the calculation of $U^n$ involves only the inversion of a mass-type matrix associated with the space $U_h$ and the corresponding dual volume element space $U_h^*$.
Let $\{\phi_j\}_{j=1,2,\cdots ,N}$ be the standard nodal basis functions for the trial space $U_h$ and $\{\chi_j\}_{j=1,2,\cdots ,N}$ be the characteristic basis functions corresponding to the control volumes for the test space $U_h^*$. Then, express $U^n$ as $$\displaystyle U^n=\sum_{j=1}^N{ \alpha}_j^n \phi_j(x), \;\;\mbox{where}\;{\alpha}_j^n=U^n(x_j).$$ Define now the following matrices $$\mathbb{M}=[(\phi_i,\chi_j)_h]_{N\times N},\quad
\mathbb{A}=[{\tilde{A}_h}(\phi_i,\chi_j)]_{N\times N},\quad
\mathbb{B}(t,s)=[{\tilde{B}_h}(t,s;\phi_i,\chi_j)]_{N\times N},$$ and the vector $\mathbb{F}(t)=[(f(t),\chi_j)_h]_{1\times N}$. Then, for instance, ([\[9-1c\]]{}) can be written as the following system of linear equations which can be solved for $\bar{\alpha}^{n+1}$: $$\mathbb{M}\bar{\alpha}^{n+1}=(2\mathbb{M}-k^2\mathbb{A})\bar{\alpha}^{n-1}
-\mathbb{M}\bar{\alpha}^{n-1}-k^3\sum_{l=0}^{n-1}
\mathbb{B}(t_n,t_{l+1/2})\bar{\alpha}^{l+1/2}+k^2\mathbb{F}^n,$$ where $\bar {\alpha}^n=\left(\alpha_1^n,\;\alpha_2^n,\cdots,\alpha_N^n\right)^T.$ Since we have used mass lumping for $(\cdot,\cdot)_h,$ the mass matrix $ \mathbb{M}$ is a diagonal matrix.
In order to illustrate the performance of the finite volume element method for solving (\[a\]), we consider the following test problems where the computational domain $\Omega=(0,1)\times(0,1)$ and the final time $T=1$.\
[**Example 1:**]{} We choose $u_0(x,y)=\sin(\pi x)\sin(\pi y),\; u_1(x,y)=\sin(\pi x)\sin(\pi y),\;
A=I$ and $B(t,s)=e^{(t-s)}I$. The function $f$ is chosen so that the exact solution is $$u=e^t\sin(\pi x)\sin(\pi y).$$
[**Example 2:**]{} Set $u_0(x,y)=xy(x-1)(y-1),\; u_1(x,y)=xy(x-1)(y-1),\; A=\begin{pmatrix}
1+x^2 & 0\\
0 & 1+x^2
\end{pmatrix}$ and $B(t,s)=e^{(t-s)}A$. The function $f$ is chosen in such a way that the exact solution is $$u=e^tx y (x-1)(y-1).$$
The order of convergence is computed in $L^{\infty}$ norm. In both examples, Fig \[fig1\] shows that the computed order of convergence for $\|u-u_h\|_{\infty}$ in the log-log scale matches with the theoretical order of convergence that we have derived.
![Convergence order estimate in $L^{\infty}$-norm.[]{data-label="fig1"}](fig-FVM.eps){width="10.5cm" height="7.5cm"}
[**[ Acknowledgements.]{}**]{} The two authors gratefully acknowledge the research support of the Department of Science and Technology, Government of India through the National Programme on Differential Equations: Theory, Computation and Applications vide DST Project No.SERB/F/1279/2011-2012, and the support by Sultan Qaboos University under Grant IG/SCI/DOMS/13/02.
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|
---
abstract: |
We propose a simple model with two infective classes in order to model the cholera epidemic in Haiti. We include the impact of environmental events (rainfall, temperature and tidal range) on the epidemic in the Artibonite and Ouest regions by introducing terms in the force of infection that vary with environmental conditions. We fit the model on weekly data from the beginning of the epidemic until March 2012. We then used this model to obtain epidemic projections from April 2012 through September 2013, and then modified these projections incorporating the vaccination programs that were recently undertaken in the Ouest and Artibonite regions to compare with actual cases.
Using real-time daily rainfall we found lag times between precipitation events and new cases that vary seasonably, ranging from $5.0$ to $11.8$ weeks in Artibonite, and $5.8$ to $8.5$ in Ouest. In addition, it appears that, in the Ouest region, tidal influences play a significant role in the dynamics of the disease.
Intervention efforts of all types have reduced case numbers in both regions, however, persistent outbreaks continue. In Ouest, where the population at risk seem particularly besieged and the overall population is larger, vaccination efforts seem to be taking hold more slowly than in Artibonite, where a smaller core population was vaccinated. The model implementing the vaccination program predicted that in mid March 2013, the mean number of cases in Artibonite would be about 88 hundred, and in Ouest by 34 hundred less than predicted by the model without vaccinations. The actual numbers of cholera cases were only about 18 hundred less than no-vaccine model predictions in both departments. We also found that vaccination is best when done in the early spring, and if resources are limited in scope, it may be better done in the third or fourth years of the epidemic if other interventions can be successfully applied in the interim.
. Cholera; Haiti; tides; precipitation; epidemic model; vaccination
address: 'Department of Mathematics, The George Washington University'
author:
- Stephen Tennenbaum
- Caroline Freitag
- Svetlana Roudenko
bibliography:
- 'epi.bib'
title: Modeling the Influence of Environment and Intervention on Cholera in Haiti
---
Introduction
============
On January 12, 2010, a $7.0$ magnitude earthquake struck near Haiti’s capital, Port-au-Prince [@WHO]. The poorest nation in the Western Hemisphere, the earthquake shattered Haiti’s already weak infrastructure [@WHO]. Thousands of Haitians were killed and even more were forced to flee to resettlement camps [@WHO].
In October 2010, the first case of cholera was reported in Haiti. A later UN investigation revealed the specific strain of *V. cholerae* came from South Asia [@UNOriginReport]. The UN investigation and epidemiological literature suggest that the epidemic began outside of a UN peacekeeper camp near Mirebalais in the Centre department, along the Artibonite River [@UNOriginReport; @EpiReport]. As *V. cholerae* is a waterborne pathogen, the Artibonite River is the ostensible route through which the disease spread throughout Haiti’s ten administrative regions, called departments [@MSPP].
Anecdotal news reports describe the dismal situation for thousands of Haitians that still remain displaced months after the earthquake. Sewage of millions of people flow through open ditches. Human waste from septic pits and latrines is dumped into the canals, and after it rains, ends up in the sea. Those living close to the water use over-the-sea toilets, and next to these outhouses, fishing boats unload and sell the fish from plastic buckets... [@NPR2].
Haiti’s two most populous regions, Ouest and Artibonite, were also the two regions hardest hit by the epidemic. Cases in Ouest and Artibonite account for 60% of the total burden of cholera in Haiti [@PAHO]. For this reason, we chose to focus our analysis on the Ouest and Artibonite regions. By April 7, 2012, cholera had affected $5.7\% $ of the total population in Ouest and $6.9\% $ of the population in Artibonite [@PAHO].
Previous Research
-----------------
We reviewed the literature dealing with cholera, modeling, and climatic conditions (rainfall, precipitation, and tides) in Haiti at the beginning of this study and again at the end. We did multi-database searches in the Biology, Medicine, and Health fields and found four modeling papers [@Lancet; @Tuite; @Bertuzzo; @Chao] of which the first three used variations of a basic system of differential equations proposed in 2001 by Codeço [@Codeco]. At that time, none of these models took *environmental conditions* directly into account. Two other papers [@Reiner; @Rinaldo] have since appeared that do take one of the environmental conditions, precipitation, into account in cholera. The first [@Reiner] is a spatiotemporal Markov chain model using seasonal rainfall which was outside our purview. The second [@Rinaldo] deals specifically with Haiti and was done by the same group that produced one of the earlier papers [@Bertuzzo]. In [@Rinaldo] they looked at the reliability of the earlier studies, and they found that although those models do well in capturing the early dynamics of the epidemic, they fail to track latter recurrences forced by seasonal patterns [@Rinaldo]. As a follow up, Rinaldo et al. [@Rinaldo] add a *precipitation forcing function* to their original model along with other modifications such as the river network and population mobility. These modifications produce a better fit to the observed pattern of case but predict recurring large outbreaks tracking the seasonal precipitation patterns.
The three models [@Lancet; @Tuite; @Bertuzzo] proposing a variation on the SIWR model, proposed by Codeço [@Codeco] in 2001, explain cholera transmission through susceptibles’ contact with a water reservoir, rather than susceptibles’ contact with infectious individuals. The models proposed by Tuite, et al. [@Tuite] and Bertuzzo, et al. [@Bertuzzo] both incorporated a “gravity” term to study the interaction among departments. The model proposed by Andrews and Basu [@Lancet] accounted for a bacterial “hyperinfectivity” stage, following research by Hartley, et al. in 2006 showing that *V. cholerae* initially has a higher infectivity before it decays to a lower infective rate in the aquatic reservoir. The fourth model by Chaoa, et al. [@Chao] was an agent based approach designed to look at various vaccination strategies.
In addition, all these models assessed the impact of potential intervention strategies, including vaccination. Bertuzzo found that a vaccination campaign aiming to vaccinate 150,000 people after January 1, 2011 would have little effect, in part because of the late timing and in part because of the large proportion of *asymptomatic* individuals who would need to get vaccinated [@Bertuzzo]. Both the models proposed by Tuite, et al. [@Tuite] and Andrews and Basu [@Lancet] suggest that vaccination campaigns would have modest effect. In March 2012, Partners in Health began vaccinating 100,000 individuals with Shanchol, a two-dose cholera vaccine [@NPR]. The size of the campaign was limited by the size of the global stockpile of Shanchol [@NPR]. The vaccination campaign is targeted at 50,000 individuals living in the slums of Port-au-Prince, where population density is thought to increase the rate of cholera exposure, and at 50,000 individuals living in the Artibonite River valley, where the epidemic began [@NPR]. Chao et al. [@Chao] showed that a targeted vaccination strategy would have the best results for this limited supply of vaccine, and by vaccinating 30% of the population the cases could be reduced by as much as 55%.
In 2001, Codeço proposed introducing an oscillating term to model seasonal variability [@Codeco]. However, none of the four Haiti-specific models accounted for seasonality. Haiti experienced flooding in June 2011, October 2011, and March 2012 [@ReliefWeb]. As cholera has reached an endemic state in Haiti, an analysis of cholera’s seasonality as it relates to Haiti’s rainy season is pertinent. Moreover, mathematical models should incorporate seasonality in order to more accurately predict the course of the epidemic and to simulate the effects of potential interventions. In April 2012 Rinaldo et al. [@Rinaldo] reexamined the above four models (including their own [@Bertuzzo]) and concluded that, among other factors, seasonal rainfall patterns were necessary to account for resurgences in the epidemic. They use long-term monthly averages to augment the bacterial growth term of contaminated water-bodies.
Other papers dealing with environmental factors were (1) a study of cholera in Zanzibar, East Africa that demonstrated 8 weeks fixed delay [@Reyburn] between rainfall and cholera outbreaks, and (2) a study in Bangladesh [@deMagny] that reports a somewhat shorter delay (4 weeks). Both these studies use a statistical approach with seasonal data. Both also made note of the potential influence of ocean environmental factors, and the Reyburn et al. paper included sea surface height and sea surface temperature in their analysis but failed to find any significant relationship [@Reyburn]. In a third paper Koelle et al. (2005) [@Koelle] model very long time periods, more than a year, in Bangladesh. This model also uses seasonal precipitation and models changes in the susceptible fraction of the population due to demographics and loss of immunity.
Our Model
---------
In this paper we use detailed and current *rainfall, temperature,* and *tidal records* to model cholera in the Artibonite and Ouest regions. We forego a bacterial compartment in favor of an *a posteriori* approach using climatic data directly to estimate infection rates. This has the advantage of more tractable temporal estimates without over parameterizing and including compartments that are essentially unmeasurable. This paper is also the first, that we know of, that uses *tidal range* in a model of cholera dynamics.
These long term trends and environmental influences establish the pattern of response of the epidemic in Artibonite and Ouest. Thus parameters were chosen and model calibration set prior to a vaccination program being implemented. We then used the model to evaluate the performance of the vaccination program against the backdrop of an alternative history without vaccination.
Material and Methods
====================
For Ouest and Artibonite, we investigated the correlations between reported cholera cases and rainfall, temperature, and, in the case of Ouest, tidal range. We wanted to determine
Model
-----
We take a combined mechanistic and phenomenological approach to our model.
The part is a standard SIR type model where individuals move from *Susceptible* $(S)$ to *Infected* to *Recovered* $(R)$ classes[^1]. An infectious individual may either be *symptomatic* $(I)$ or *asymptomatic* $(A)$ [@Lenhart]. The probability, $\rho,$ of asymptomatic infection is $0.79$ [@Lenhart; @Hartley]. Both *symptomatic* and *asymptomatic* individuals move to the recovered group, $R$, at a rate $\gamma$. *Symptomatic* individuals die from cholera at a rate $\mu$ (dead are denoted by $D$).
The approach comes from estimating the force of infection, $\beta\!\left( t\right)$, by fitting the number of cases predicted by the model (both incidence and cumulative cases) to the data. We chose not to incorporate water bodies and environmental bacterial populations explicitly since this requires the estimation of extra compartments and half dozen or so other parameters for which there are no data. The feedback effects, though, are included as time lags in the action of precipitation and tides. These and all other parameters are discussed below and values are given in Table \[Table 1\] and Table \[Table 2\].
This paper evaluates Artibonite and Ouest separately, in order to capture the different dynamics in each region. In each case we use the following system of non-autonomous ordinary differential equations, and the components to be fitted: $N=S+A+I+R+D$, where $N$ is the initial population of the region, $t$ is time since beginning of the epidemic (week starting 17-October-2010), and $\Delta t$ is the time interval for reporting new cases (1 week). (For the purposes, and time scale, of this study we chose to ignore the demographics of the population.)$$\begin{array}
[c]{ccc}\text{System of equations} & & \text{Variables to be fitted}\\
& & \\
\left\{
\begin{array}
[c]{l}\frac{dS}{dt}=-\beta\!\left( t\right) S \medskip\\
\frac{dA}{dt}=\rho\beta\!\left( t\right) S-\gamma A \medskip\\
\frac{dI}{dt}=(1-\rho)\beta\!\left( t\right) S-(\gamma+\mu)I \medskip\\
\frac{dR}{dt}=\gamma(A+I)
\end{array}
\right\} & \Longrightarrow & \left\{
\begin{array}
[c]{l}\text{new cases}=(1-\rho)\int_{t-\Delta t}^{t}\beta\!\left( \frak{t}\right)
S\!\left(\frak{t}\right) d\frak{t} \medskip\\
\text{total cases}=(1-\rho)\int_{0}^{t}\beta\!\left( \frak{t}\right) S\!\left(
\frak{t}\right) d\frak{t}
\end{array}
\right\}
\end{array}$$
Environmental Effects
---------------------
Significant rain events will cause overflowing of river and stream banks, which we assume will increase direct contact the bacteria via contaminated water, or indirectly via soils, vegetation, and pathogen-carrying insects, etc., that have been contaminated or have consumed cholera bacteria. We also assume that temperature plays a dual role by increasing the infection rate and decreasing the lag time between environmental events (precipitation and tidal range) and disease outbreaks. We speculate that decreases in time lags with warmer temperatures are due to more rapid growth, better survival,and more active bacteria and transmission agents. People also more frequently contact sources of contamination due to the variety of increased activities that warmer weather engenders.
Additionally, in the Ouest region there are commercial areas, tent-cities and slums that have raw waste directly discharging to Port-au-Prince Bay, or to the bay via rivers (e.g. Froide, Momance, and Grise), and numerous open sewage canals. We hypothesize that there may be additional disease generated when large tidal ranges stir up contaminated sediments or cause blooms of plankton in these coastal or estuarine waters. Again, the chain of infection may be complex and include direct contact with water, insects, plankton, benthos or consumption of contaminated seafood, etc. Why tidal range would have a significant effect and not tidal height is curious. There are a number of possible explanations. It may be that a larger bottom area is scoured by the breaking surface waters along the beaches, or more fresh and brackish water from river outflows and estuaries can contact the bottom sediment as the water falls and rises again, or a combination of actions. In any event it may warrant closer examination in the field.
To incorporate these effects, we modify the force of infection term, $\beta,$ as follows $$\beta\!\left( t\right) = u\!\left( t\right) \left[ \alpha_{p} H\!\left(
t\right) P\!\left( t-\tau_{p},\theta_{p}\right) +\alpha_{m} M\!\left(
t-\tau_{m},\theta_{m}\right) \right] , \label{betaequ}$$ where $P$ is a moving average of the amount of daily rainfall, and $M$ is a moving average of the maximum semi-diurnal tidal range, $H$ is a heat index based on mean air temperature, $\tau_{p}$, is the lag time for precipitation (which is itself affected by temperature), $\tau_{m}$ is the lag time for tides, $\theta_{p}, \text{and } \theta_{m}$ are the respective averaging periods, and $\alpha_{p},
\text{and } \alpha_{m}$ are the corresponding proportionality constants.
In the above formula \[betaequ\], we also include an expression for improvement of conditions over time, either through reducing the effective susceptibles, or reduction in the infection rate, or both. These factors would be due to increased access to clean water, increased personal hygiene, decreased contamination of the environment, vaccination, and rapid treatment of new cases, or any other means of removing risk. We model this by assuming that the force of infection is reduced by an exponential function $$u\!\left( t\right) =u_{0}+(1-u_{0})e^{-rt}.$$ In this formula $r$ is the rate of improvement in conditions, $1-u_{0}$ is the initial fraction of the population whose risk is eradicable, and $u_{0}$ is the initial fraction of the population that is chronically indigent.
This approach is tantamount to having the susceptible, or at-risk population reduced directly by public health improvements. In this interpretation the at risk population would be $S_{R} = u\!\left( t\right)S$, the $R$ subscript indicating the at-risk subgroup of the larger susceptible population[^2]. The at-risk population starts out equal to the entire susceptible population and declines asymptotically to the remaining number of indigent susceptibles as conditions improve.
Data
----
### Cholera cases.
The source of the epidemic data was *the Haitian Ministry of Public Health and Population* [@MSPP] and compiled by *the Pan American Health Organization* [@PAHO]. The available data sets from the above source, used for this study, consist of *cumulative* cholera cases, and *new* cholera cases. New cholera cases are calculated based on the difference between the latest report and the previous one. Cholera case definition also includes suspected cholera cases and deaths in addition to confirmed cases and deaths. As such, data posted on the web si te are periodically updated with minor corrections. Cases are reported on weekly basis with the reporting week beginning on Sunday.
Reported hospitalized cases and hospitalized deaths are probably more accurate, but for the purposes of this study, less useful, since we want to track the progress of the epidemic, and increases in *access* to treatment biases the data. No data is available for new or total cases $\left( TC\right) $ during the first four weeks, however hospitalized case data is available. In order to estimate the number of new and total cases during the first four weeks of the epidemic, we regressed, for the subsequent 8 weeks (14-Nov-10 through 2-Jan-11), the reported total cases against hospitalized cases $\left(HC\right) $. For the Ouest region $\ \ TC_{O}=2.2HC_{O}$ $\left(R^{2}=0.99\right) ,$ and for the Artibonite region $\ TC_{A}=2.59HC_{A}+497$ $\left( R^{2}=1.00\right) .$ Then using these formulas, we back-calculated the approximate number of new cases during the the first four weeks of the epidemic.
### Environmental data.
We compared rainfall, temperature and tide data to the pattern of new cholera cases reported each week. The rainfall data comes from NASA [@NASA], using centrally located points in each region as our datum point, given by (*latitude, longitude*). For Artibonite our datum point is $(19.125, -72.625)$ and for Ouest, it is $(18.625, -72\!\cdot
\!375)$ [@NASA]. Precipitation estimates provided in the TRMM\_3B42\_daily.007 data product are a combination of remote sensing and ground verified information reported with a spatial resolution of $0.25\times0.25$ degrees and a temporal resolution of 1 day, further details are available on the website [@NASA].
Temperature data is mean daily air temperature at Port-au-Prince, and is reported by the Weather Underground [@wunder]. We used a sine function fit to the annual cycle in simulations. This has the advantage of allowing us to extrapolate temperature patterns for model projections. Unfortunately, only temperatures from Port-au-Prince were available. The temperature index derived from Port-au-Prince data were used for both Ouest and Artibonite.
Tide data is for Port-au-Prince Bay (StationId: TEC4709) [@NOAA]. These numbers are predictions from NOAA’s tide *model* for this location and are *not* direct measurements. NOAA’s web site allows one to download tide numbers for any date starting from 2010 and extending through 2014. Again, only data from Port-au-Prince is available.
### Data analysis.
The initial modeling was done by comparing data sets in the frequency (Fourier) domain for new cases and rainfall in order to find any suggestions of matching periodicity and/or time-lags. After that code was written in Berkeley Madonna to simulate the dynamical system and study the environmental data in order to match predictions to data[^3]. Rainfall data was available only to week of 27-January-2013 (week 119) because of the lag between the timing of rainfall events and what had been processed at the time we accessed it [@NASA].
Model calibration
-----------------
All modeling and calibration was done in Berkeley Madonna. When possible we used ranges for each parameter as established by previously published research ($\rho $, $\gamma $, $N_{0}$, $S_{0}$, $I_{0}$, $A_{0}$, $R_{0}$, $D_{0}$, $t_{0}$, and $u_{0}$; see Table \[Table 1\]). Since our model contains several fitted parameters ($\theta _{p}$, $\tau _{p\_lo}$, $\tau _{p\_hi}$, $k$, $\alpha _{p}$, $\theta _{m}$, $\tau _{m}$, $\alpha _{m}$, $M_{0}$, and $r$; see Table \[Table 2\]), we needed to supply a plausible range and initial value for each parameter in order to efficiently search the parameter space for a best fit. These ranges and initial values were chosen by visually fitting the *new case* output of the model to reported new cases. The Berkeley Madonna curve fitting algorithm was then used to minimize the root mean square difference between model predictions of number of *cumulative cases* and reported cumulative cases. Confidence and prediction intervals were calculated using the delta method. [@Ramsay]
Parameters from literature
--------------------------
Table \[Table 1\] displays parameter values for each region obtained from published or online sources.
$${\small
\begin{tabular}
[c]{|c|c|c|c|c|}\hline
Parameter & Artibonite & Ouest & Units or calculation &
References\\\hline\hline
$\rho$ & $0.79$ & $0.79$ & fraction becoming asymptomatic & \cite{Lenhart,
Kaper}\\\hline
$\gamma$ & $1.4$ & $1.4$ & fraction recovered per week & \cite{Lancet,
Bertuzzo, Codeco}\\\hline
$N_{0}$ & $1,571,020$ & $3,664,620$ & $-$ & \cite{MSPP}\\\hline
$S_{0}$ & $1,534,338$ & $3,663,699$ & $N-\left( A_{0}+I_{0}+R_{0}+D_{0}\right) $ & \\\hline
$I_{0}$ & $7,653$ & $193$ & regression on early hospitalized cases &
\cite{PAHO}\\\hline
$A_{0}$ & $28,790$ & $726$ & $\frac{\rho}{1-\rho}I_{0}$ & \\\hline
$R_{0}$ & $0$ & $0$ & $-$ & \\\hline
$D_{0}$ & $239$ & $2$ & $-$ & \cite{PAHO}\\\hline
$t_{0}$ & 17-Oct.-2010 & 17-Oct.-2010 & 290$^{\text{th}}$day of the year &
\cite{PAHO}\\\hline
$u_{0}$ & $0.05$ & $0.05$ & persistant fraction of pop. at risk & \cite{OCHA}\\\hline
\end{tabular}
\ \ \ \ }$$
Environmental components of the force of infection.
---------------------------------------------------
### Artibonite
Meteorological influences on infection rate are a product of precipitation rate and a heat index. Daily precipitation $p\left( t\right) $ [@NASA] is averaged over an interval $\theta_{p}$ so the running average precipitation rate is $$P\!\left( t-\tau_{p},\theta_{p}\right) = \frac{1}{\theta_{p}}\sum_{j=0}^{\theta_{p}}p\left( t-\tau_{p}-j\right),$$ where $\tau_{p}$ is the delay in precipitation’s affect on infection rate (see below). For the delay we used a simple sine function to model a mean temperature index throughout the year. (Temperature data from [@wunder].) The mean air temperature index given by $$T_{air}\left( t\right) =\sin\left( \frac{2\pi\left( 7t+187\right)
}{365.25}\right) ,\text{ \ \ \ \ where }t\text{ is in weeks from
17-Oct.-2010}.$$ This is then used to create a delay functions that varies from a minimum of $\tau_{p\_lo}$ during summer and $\tau_{p\_hi}$ during winter $$\tau_{p}=\left( \left( \tau_{p\_hi}+\tau_{p\_lo}\right) -\left(
\tau_{p\_hi}-\tau_{p\_lo}\right) T_{air}\left( t\right) \right) /2.$$ In other words, infections follow a key set of weather and climatic variables. This would be expected since warmer temperatures mean faster growth rates for bacteria and some of their invertebrate hosts (bacterial dormancy is probably not an issue since the climate is tropical) [@deMagny; @Huq].
In addition, there is a direct influence of temperature on the infection rate. We use a heat index, $H\left( t\right)$, rather than temperature itself. This heat index is a linear function of the normalized temperature pattern with a mean (intercept) of $1$ and the slope, $k$, is a parameter to be fit. This index is used as a multiplicative factor modifying the infection rate: the mean temperature has no effect on the infection rate, low temperatures decrease the infection rate, and high temperatures increase it. Thus we have $$H\left( t\right) =1+kT_{air}\left( t\right).$$ Therefore, the infection rate is given by $$\beta_{A}\left( t\right) = u\!\left( t\right) \alpha_{p} H\!\left( t\right)
P\!\left( t-\tau_{p},\theta_{p}\right) .$$
The tide term, $\alpha_{m} M\!\left(t-\tau_{m},\theta_{m}\right)$, is not included since we had no tidal range data for the Artibonite coast, and the tidal range data we had (Port-au-Prince) was not found to explain a significant amount of variance in the number of cases in Artibonite.
### Ouest
For the Ouest region we use the same formulation as in Artibonite, however, we found that tidal *range* appeared to significantly affect infection rates as well. The maximum tidal range each day (there are two) $m\left(
t\right) $ [@NOAA] is averaged over an interval $\theta_{m}$ so the running average tidal range is $$M\!\left( t-\tau_{m},\theta_{m}\right) =\frac{1}{\theta_{m}}\sum
_{j=0}^{\theta_{m}}m\left( t-\tau_{m}-j\right) - M_{0} ,$$ where $\tau_{m}$ is the delay in the tide’s affect on infection rate. Here, $\tau_{m}$ is fixed, the effect of temperature (water or air) on lengthening or shortening the response in infection rate was not found to be sufficient to warrant adding another function and additional parameters. Thus, the overall infection rate for Ouest is$$\beta_{O}\left( t\right) =u\!\left( t\right) \left[ \alpha_{p} H\!\left(
t\right) P\!\left( t-\tau_{p},\theta_{p}\right) +\alpha_{m} M\!\left(
t-\tau_{m},\theta_{m}\right) \right] .$$
RESULTS
========
Parameter Fitting and Model Selection
-------------------------------------
Table \[Table 2\] displays parameter values for each region obtained through curve-fitting to *cumulative reported cases*.
$${\tiny{
\begin{tabular}
[c]{|c|c|c|c|c|}\hline
parameter & Artibonite & Ouest & units & description\\\hline\hline
$\theta_{p}$ & $4.61\;\left( 0.319\right) $ & $1.96\;\left( 0.526\right) $
& weeks & \multicolumn{1}{|l|}{averaging window for precip.}\\\hline
$\tau_{p\_lo}$ & $0.4019\;\left( 0.3261\right) $ & $3.800\;\left(
0.3757\right) $ & weeks & \multicolumn{1}{|l|}{minimum delay for precip.
effects}\\\hline
$\tau_{p\_hi}$ & $7.1805\;\left( 0.2056\right) $ & $6.500\;\left(
0.4240\right) $ & weeks & \multicolumn{1}{|l|}{maximum delay for precip.
effects}\\\hline
$k$ & $0.5198\;\left( 0.0177\right) $ & $0.5812\;\left( 0.03050\right) $ &
unitless & \multicolumn{1}{|l|}{temp.-precip. interaction
level}\\\hline
$\alpha_{p}$ & $6.8806\;\left( 0.1690\right) \times10^{-3}$ &
$1.1502\;\left( 0.08582\right) \times10^{-3}$ & (mm $\times$week)$^{-\text{1}}$ & \multicolumn{1}{|l|}{infection rate per mm rain}\\\hline
$\theta_{m}$ & $-$ & $1.00\;\left( 0.91\right) $ & weeks &
\multicolumn{1}{|l|}{averaging window for tidal range}\\\hline
$\tau_{m}$ & $-$ & $1.960\;\left( 0.494\right) $ & weeks &
\multicolumn{1}{|l|}{delay for tidal range effects}\\\hline
$\alpha_{m}$ & $-$ & $2.8128\;\left( 1.3360\right) \times10^{-4}$ & (cm
$\times$week)$^{-\text{1}}$ & \multicolumn{1}{|l|}{infection rate per cm
tide range}\\\hline
$M_{0}$ & $-$ & $24.791\;\left( 24.094\right) $ & cm &
\multicolumn{1}{|l|}{baseline of tidal range effect}\\\hline
$r$ & $0.06100\;\left( 0.00147\right) $ & $0.02610\;\left( 0.00139\right)
$ & week$^{-\text{1}}$ & \multicolumn{1}{|l|}{decrease in susceptibles per
week}\\\hline
\end{tabular}
\ \ \ \ }}$$
Plausible ranges for time lags were initially obtained from the Fourier analysis, then parameter ranges and initial values were further refined by visually fitting the new cases predicted by the model to the new cases data. We then used the Berkeley-Madonna curve-fitting routine to find a parameter set that minimized the sum of the square differences (SSD) between model output for cumulative cases and cumulative case data.
The Artibonite model *with* tide was not included in Table \[Table 2\] since inclusion of tide did not improve the model (see Tables \[Table 3\] and \[Table 4\]). The statistics for the model fit are given in the following two tables. Table \[Table 3\] is for cumulative cases predicted by the model compared to cumulative case data.
$${\small{
\begin{tabular}
[c]{|c|c|c|c|c|}\hline
\multicolumn{5}{|c|}{All cases}\\\hline
statistic & Artibonite (tide) & Artibonite (no tide) & Ouest (tide) & Ouest
(no tide)\\\hline\hline
data points & $74$ & $74$ & $74$ & $74$\\\hline
parameters & $10$ & $6$ & $10$ & $6$\\\hline
adj RMSD & $2210.37$ & $2214.42$ & $3708.59$ & $6336.53$\\\hline
adj R$^{2}$ & $0.9904$ & $0.9909$ & $0.9968$ & $0.9908$\\\hline
AIC & $1148.99$ & $1141.20$ & $1271.40$ & $1352.56$\\\hline
\end{tabular}
}}$$
For the full model in either region (model including tides) the parameters $\theta_{m},\tau_{m},M_{0},$ and $\alpha_{m}$ are added and the model is re-optimized. For Artibonite an $F$-test for the nested models gives the following results $F=$ $0.045$ $d.f.=\left( 4,64\right) $ and the $p$ value is $0.9960,$ indicating the inclusion of the extra detail had almost no effect and would be unjustified. For the Ouest region an $F$-test for the nested models gives the following results $F=$ $33.63$ $d.f.=\left( 4,64\right) $ and the $p$ value is $4.208\times10^{-15},$ indicating the tidal data significantly improved the model fit.
$${\small{
\begin{tabular}
[c]{|c|c|c|c|c|}\hline
\multicolumn{5}{|c|}{New cases}\\\hline
statistic & Artibonite (tide) & Artibonite (no tide) & Ouest (tide) & Ouest
(no tide)\\\hline\hline
data points & $74$ & $74$ & $74$ & $74$\\\hline
parameters & $10$ & $6$ & $10$ & $6$\\\hline
adj RMSD & $9.118$ & $8.871$ & $21.30$ & $23.08$\\\hline
adj R$^{2}$ & $0.7621$ & $0.7755$ & $0.3932$ & $0.3681$\\\hline
AIC & $336.37$ & $328.79$ & $461.92$ & $470.28$\\\hline
\end{tabular}
}}$$
Table \[Table 4\] is for new cases predicted by the model compared to new case data. The statistics in Table \[Table 4\] are done on the square roots of the values in order to produce more uniformly distributed data and stabilize the variance somewhat. The $F$-test for the Artibonite nested models using new cases gives the following results $F=$ $0.0903$ $d.f.=\left( 4,64\right) $ and the $p$ value is $0.9852.$ For Ouest the difference is again highly significant with $F=$ $3.9599$ $d.f.=\left( 4,64\right) $ and the $p$ value is $0.0062.$
Lag times
---------
The total delays in response to precipitation and tides are the sum of the averaging window and the delay function. For precipitation the minimum and maximum delays for Artibonite are $5.0$ and $11.8$ weeks, and for Ouest they are $5.7$ and $8.5$ weeks, respectively. The shorter delays are very similar in the two regions during the warmer months but during the cooler months the response time in Artibonite is 3 weeks longer. These long delays are similar in magnitude to delays reported from a study of Cholera in Zanzibar, East Africa (8 weeks fixed delay) [@Reyburn], and the shorter delays (4 weeks) to those in Bangladesh [@deMagny]. For Ouest estimated delay from response to changes from tidal range was about 3 weeks. But since influence of tidal range has not been quantitatively reported elsewhere in the literature, we have nothing to compare this number to.
Although, rainfall data was available only to week of 27-Jan-2013 (week 119), we run the simulations using contemporaneous data lag period is over (around 7 weeks). This is just before the data for new cases ends.
Vaccination
-----------
A program to vaccinate *the most at risk populations* began in the 2nd week of April and ended in mid June. Each site (Ouest and Artibonite Dept) vaccinated about 50,000 persons, and each site had about $91\%$ 2nd dose coverage. The administration of the first dose was staggered by age groups (beginning first with 10 year olds and up) because the Ministry of Health had a measles, rubella and polio vaccine catch-up campaign for children under 10 years of age that was taking place at the same time last April 2012.[^4]
In the Ouest Department, GHESKIO[^5] vaccinated adults, adolescents and children over 10 years of age from April 12-23, 2012 and children under 10 from May 26 - June 3. The first dose of vaccine was given to 52,357 persons (of which 47,520 received the second dose), living in the slums of Port-au-Prince and surrounding villages.[^6]
In the Artibonite Department, PIH[^7] vaccinated 32183 people in rural Bocozel and 13185 people in Grand Saline with $90.8\% $ of those people confirmed to get the 2nd dose (or 41194 for both locations). The campaign started April 15th 2012 and ran until June 10 2012. Here too, children under 9 year old were vaccinated in the second half of the time period because of the MMR and Polio vaccination campaign.[^8] With these basic facts we constructed a crude vaccination schedule (Table \[Table 5\]) using the following assumptions:
1\) approximately $25\%$ of the population is under 10 years old;
2\) the second dose was administered 14 days after the first dose was given [@Date];
3\) the immune response took hold about 8.5 days after the second dose was given [@Date];
4\) we used the average number of people vaccinated per day over a 12 day period for adults and 9 days for children.
$${\small{
\begin{tabular}
[c]{|l|l|l|}\hline
\multicolumn{3}{|c|}{$\text{Ouest}$}\\\hline\hline
$\text{Adult }1^{st}\text{ dose }$ & $39,268$ & $\text{April 10 - April 23}$\\\hline
$\text{Adult }2^{st}\text{ dose}$ & $35,640$ & $\text{April 26 - May 7}$\\\hline
$\text{Adult immune response}$ & $65-85\%$ & $\text{May 4 - May 15}$\\\hline\hline
$\text{Child }1^{st}\text{ dose }$ & $13,089$ & $\text{May 26 - June 3}$\\\hline
$\text{Child }2^{st}\text{ dose }$ & $11,880$ & $\text{June 9 - June 17}$\\\hline
$\text{Child immune response}$ & $65-85\%$ & $\text{June 17 - June 25}$\\\hline
\end{tabular}
}}$$ $${\small{
\begin{tabular}
[c]{|l|l|l|}\hline
\multicolumn{3}{|c|}{$\text{Artibonite}$}\\\hline\hline
$\text{Adult }1^{st}\text{ dose }$ & $34,026$ & $\text{April 15 - April 2}$6\\\hline
$\text{Adult }2^{st}\text{ dose}$ & $30,896$ & $\text{April 29 - May }$10\\\hline
$\text{Adult immune response}$ & $65-85\%$ & $\text{May 7 - May 1}$8\\\hline\hline
$\text{Child }1^{st}\text{ dose }$ & $11,342$ & $\text{May 19 - May 27}$\\\hline
$\text{Child }2^{st}\text{ dose }$ & $10,298$ & $\text{June 2 - June 1}$0\\\hline
$\text{Child immune response}$ & $65-85\%$ & $\text{June 10 - June }$18\\\hline
\end{tabular}
}}$$
We ran simulations following the above schedule as closely as the simulation would allow by subtracting the numbers of at risk persons (given below) from the susceptible compartment $(S)$ .
In Ouest the vaccination algorithm involved removing 2970 at risk persons per day starting on 4-May-2012 and ending on 15-May-2012, or 35640 total (these correspond to the vaccination of persons over age 10). then the algorithm removed 1320 at risk persons per day starting on 17-June-2012 and ending on 25-June-2012, or 11880 total (children).
In Artibonite the vaccination algorithm removed 2574.67 at risk persons per day starting on 7-May-2012 and ending on 18-May-2012, or 30896 total (over age 10), then the algorithm removed 1144.22 persons per day starting on 10-June-2012 and ending on 18-June-2012, or 10298 total (these correspond to the vaccination of the at risk children). These simulations roughly follow the actual vaccination schedules given in Table \[Table 5\].
Since the efficacy of the vaccine (oral Shanchol) is between 65 and 85% [@Date] we did three runs $65, 75,$ and $85\%$ for each department. The $75\%$ run was our mean. To get the prediction intervals on the numbers of cases in the vaccination model, the upper bound on the interval is the $95\%$ prediction interval determined for $65\%$ run and the lower bound is the $95\%$ prediction interval for the $85\%$ run.
Simulations and Projections
---------------------------
We list the time line for particular events in Table \[Table 6\]. Curve fitting (parameter estimation) was done between model output and data from week 3 to week 76,we refer to model output during this period as “predictions", simulation from week 76 through week 150 are referred to as “projections". We were able to use rainfall data that ended in week 199 for another six weeks (to week 125) due to the delay of new infections from the time of the rain fall events. By “immune response for first..." and “immune response for last..." we mean this is when we begin and end removing susceptibles from the at risk group respectively.
$${\begin{tabular}
[c]{|c|c|c|}\hline
& \multicolumn{2}{|c|}{Date (week of epidemic)}\\\cline{2-3}Event & Artibonite & Ouest\\\hline\hline
{\small Begin epidemic} & \multicolumn{2}{|c|}{{\small 17-Oct-10 \ (0)}}\\\hline
{\small Begin case data} & \multicolumn{2}{|c|}{{\small 14-Nov-10 (4)}}\\\hline
{\small End model fitting} & \multicolumn{2}{|c|}{{\small 1-Apr-12 (76)}}\\\hline
{\small Immune response for first adult vaccinated} & {\small 7-May-12 (81.1)}
& {\small 4-May-12 (80.7)}\\\hline
{\small Immune response for last adult vaccinated} & {\small 18-May-12 (82.7)}
& {\small 15-May-12 (82.3)}\\\hline
{\small Immune response for first child vaccinated} & {\small 10-Jun-12 (86)}
& {\small 17-Jun-12 (87)}\\\hline
{\small Immune response for last child vaccinated} & {\small 18-Jun-12 (87.1)}
& {\small 25-Jun-12 (88.1)}\\\hline
{\small End precipitation data} & \multicolumn{2}{|c|}{{\small 27-Jan-13
(119)}}\\\hline
{\small End precipitation data with delay} & {\small 8-Mar-13 (124.7)} &
{\small 9-Mar-13 (124.9)}\\\hline
{\small Begin random average rain fall data} & {\small 9-Mar-13 (124.9)} &
{\small 10-Mar-13 (125)}\\\hline
{\small End case data} & \multicolumn{2}{|c|}{{\small 17-Mar-13 (126)}}\\\hline
{\small End simulation} & \multicolumn{2}{|c|}{{\small 1-Sept-13 (150)}}\\\hline
\end{tabular}
}$$
### Predictions compared to observations for cumulative and new cases
The following Figures \[Figure:1\] and \[Figure:2\] show the model predictions compared to observations for cumulative number of cases in Artibonite and Ouest regions, respectively. Similarly, Figures \[Figure:3\] and \[Figure:4\] show the model predictions compared to observations for new number of cases in Artibonite and Ouest regions, respectively. Prediction intervals were calculated only for the cumulative numbers, since the final model fitting was done on these numbers. Confidence intervals are shown for incidence. The match for the trends in new cases match fairly well, the slope of the expected (model) regressed against observed (data) is nearly one in both departments (see Figures \[Figure:5\] and \[Figure:6\]) even though there is a substantial amount of unexplained variance. Whether this is due to the crude spatial resolution or other factors remains to be seen.
Prediction intervals (PIs) and confidence intervals (CIs) were calculated using the delta method adapted for differential equations (see, for example, Ramsay et al. [@Ramsay]). After 27-Jan-2013 rainfall data from NASA was unavailable, for simulations after that date we did 13 runs using rainfall patterns from each of the 13 prior years. The mean of those simulations was used and the variance of the 13 runs, at each time step, was added to the variance from the estimation procedure before computing the PIs.
![Artibonite. The predicted cumulative number of *symptomatic* individuals, against total reported cases to 1-Apr-2012. Projections are from then to end of February. Projections using the vaccination schedule (red) begin on 7-May-2012, approximately three weeks after beginning the vaccination program in Artibonite. All projections after 11-Nov-2012 are based on runs using prior 13 years of precipitation, and PI’s include the variance of those data (see text).[]{data-label="Figure:1"}](02Artibonite_cumulative){width="6in" height="4.4in"}
![Ouest. The predicted cumulative number of *symptomatic* individuals, against total reported cases to 1-Apr-2012. Projections are from then to end of February. Projections using the vaccination schedule (red) begin on 4-May-2012, approximately three weeks after beginning the vaccination program in Ouest. All projections after 11-Nov-2012 are based on runs using prior 13 years of precipitation, and PI’s include the variance of those data (see text). Note that for the Ouest region, the model begins at the *fourth week*. We assume that the low initial numbers in the first three weeks are a result of immigration of cases from the Artibonite region. The model therefore uses data for the first four weeks – assumed immigration numbers for the first three weeks and the initialization of the model from data for the fourth week.[]{data-label="Figure:2"}](03Ouest_cumulative){width="6in" height="4.4in"}
![Artibonite. The new *symptomatic* individuals, vs. time. Circles - observed; solid line - model prediction; Dashed lines - 95 percentile confidence intervals for model projections. Red line is projections with 75 percent vaccine efficacy.[]{data-label="Figure:3"}](04Artibonite_new_vs_time){width="6in" height="4.4in"}
![[]{data-label="Figure:4"}](08Ouest_new_vs_time){width="6in" height="4.4in"}
![Artibonite. The predicted new *symptomatic* individuals, against weekly reported cases to 1-Apr-2012 (square root transformed). A regression line matching the main diagonal $\left( 45^{\circ}\right) $ dashed line would show an optimal fit, the discrepancy is due in part to fitting on the cumulative numbers.[]{data-label="Figure:5"}](05Artibonite_pred_vs_data_new){width="6in" height="3.5in"}
![[]{data-label="Figure:6"}](09Ouest_pred_vs_data_new){width="6in" height="3.5in"}
### Epidemic projections for Artibonite
The model projected that by the mid March 2013, Artibonite would have seen between 120 and 128 thousand cholera cases without vaccine, and between 112 and 122 thousand with the implemented vaccination program (a decrease in about 7 thousand cases; see Figure \[Figure:7\]). The models (with and without vaccination) show a marginal difference in how well they match cumulative number of cases data. There seems to be moderate agreement between the projected and observed incidence as well (see Figure \[Figure:8\]) where the number of new cases is below what would have occurred without vaccination.
The actual number of cases is only 1,883 less than the predicted mean number of cases without vaccination. If we assume the model without vaccine is correct an average of 15,016 people would have gotten sick between 6-May-2012 and 13-Mar-2013. Over that period of time 13,143 people *actually did get sick.* This represents a $12\%$ reduction in the number of people that would have gotten cholera.
![[]{data-label="Figure:7"}](07ArtiboniteQumProject){width="6in" height="4.4in"}
![[]{data-label="Figure:8"}](06Artibonite_new_proj){width="6in" height="4.4in"}
### Epidemic projections for Ouest
For Ouest, the model projected that by mid March 2013, Ouest would have seen between 247 and 266 thousand cholera cases without vaccine, and between 244 and 264 thousand with the implemented vaccination program (a decrease in almost 3 thousand cases; see Figure 9). The *model* also predicted that with vaccination about 32,839 people would have gotten ill between the onset of the immune response (6-May-2012) and the last rainfall data week (17-Mar-2013), whereas 35,613 would have gotten sick without vaccination. This represents about a $7.8\%$ decrease in the number of people that would have gotten cholera. The different models (with and without vaccination) do not substantially differ in how well they match cumulative number of cases data. Note that this *does not mean* that the vaccination program did not have an impact, but only that the model at this point in time could not distinguish between a scenario with and without vaccination.
The actual number that got ill between 6-May-2012 and 17-Mar-2013 was 38,910. This is 3,297 *more* than the model without vaccination predicted! There was a slowdown in number of new cases in starting in mid February 2012 and running through March 2012 which dropped the cumulative case number below the mean model projections (but still within the prediction interval), however, this occurred *before* the vaccination program took place. There was another smaller slowdown in July but then new cases increased rapidly again in November (see Figure \[Figure:9\]). It would seem that these oscillations are in spite of, rather than because of vaccination, since the model predicts a steady, albeit slow, additional decline in new cases(see Figure \[Figure:10\]). It would seem that such a small percentage of the at risk population vaccinated, although probably beneficial to those individuals receiving the vaccine, may have had negligible impact in bestowing any broader community benefits.
![[]{data-label="Figure:9"}](11OuestQumProject){width="6in" height="4.4in"}
![[]{data-label="Figure:10"}](10Ouest_new_proj){width="6in" height="4.4in"}
Vaccination scenarios.
----------------------
We looked at changing the number of people vaccinated and the timing of vaccination to see if there is some optimal schedule that can be applied.
### Changing the number of people vaccinated.
The first experiment was to change the number of people vaccinated. We completed, in the model, all vaccinations within a 5 week period. The second round of vaccination was assumed to begin on epidemiological week 80 and the immune response was assumed to begin a week later on week 81 with a $75\%$ efficacy. This was to approximately match the timing of the initiation of the actual vaccination second dose. In Artibonite we varied the number vaccinated from 0 to 65,000 and in Ouest we varied the number vaccinated from 0 to 450,000. The results are illustrated in Figures \[Figure:12\], and \[Figure:13\]. Numbers vaccinated are shone on the x-axis and percent decrease in cases on the y-axis. We show the percent decrease of cases at weeks 100, 125, and 150 of the epidemic. These correspond to 20, 45, and 70 weeks after the second dose of vaccine was administered.
![[]{data-label="Figure:12"}](12Art_vacc_num){width="5.3in" height="4.0in"}
![[]{data-label="Figure:13"}](13Ouest_vacc_num){width="5.3in" height="4.0in"}
In both departments the percent decrease in number of cases increases steadily until the number vaccinated reaches the number of people remaining in the at risk group. At this point there are no more people to be vaccinated but $25\%$ of those people that were vaccinated are still susceptible. The curves level off at this point.
The curve increases almost as a straight line (almost, because the vaccination takes place over a finite period of time rather than instantaneously), indicating that the potential benefit to each person remains constant. The optimal amount to have vaccinated at 80 weeks would have been one and a half times what was done in Artibonite and nine fold greater in Ouest. However, the costs of vaccinating every person at risk is certainly not a linear function and a cost-benefit analysis would be necessary to determine if, and at what point the money and efforts would be better expended in other control measures.
### Changing the timing of vaccination.
To investigate the best timing of vaccination we ran two scenarios: the first, vaccination near the number to optimize vaccination at 80 weeks; and the second, vaccination near the numbers actually vaccinated.
We start by beginning the second round of vaccination at the 3[rd]{} and then increasing the timing to the 247[th]{} week. We then compare the decrease in cases for various weeks in the epidemic. The total numbers vaccinated in Artibonite was 60 thousand and in Ouest was 400 thousand.
![[]{data-label="Figure:14"}](14Art_vacc_time_fix_60.eps){width="5.3in" height="4.0in"}
![[]{data-label="Figure:15"}](16Ouest_vacc_time_fix_400.eps){width="5.3in" height="4.0in"}
The maximum reduction in number of cases in the 100[th]{} and 125[th]{} weeks occur when the vaccine second dose is given in the 29[th]{} week (8-May-2011) in Artibonite and 27[th]{} week (24-April-2011) in Ouest. The maximum reduction in number of cases after the 125[th]{} week occur when the vaccine second dose is given beginning in the 79[th]{} week (22-April-2012) in Artibonite and the 75[th]{} week (25-March-2012) in Ouest. However, this may belie the true best timing since the timing from vaccination to response is different for each week of the epidemic. For example for vaccine given at week 100 there is a large reduction in cases in the 225[th]{} week but no reduction at all in the 100[th]{} week since the immune response will not ever occur until the 101[st]{} week.
Alternatively, if we look at fixed intervals relative to when vaccination is done, a different picture emerges, see Figures \[Figure:16\] and \[Figure:17\]. Again there are global maximums when vaccine is given in week 29 (8-May-2011) in Artibonite and week 27 (8-May-2011) in Ouest 78 weeks (1.5 years) later. However, now there are periodic local maxima occurring each year occurring between late March and early May. At 26 weeks after vaccination (a half a year) the peak reduction in number of cases is approximately the same each spring for the first three years in Artibonite, and was greatest the first spring in Ouest.
![[]{data-label="Figure:16"}](15Art_vacc_time_rel_60.eps){width="5.3in" height="4.0in"}
![[]{data-label="Figure:17"}](17Ouest_vacc_time_rel_400.eps){width="5.3in" height="4.0in"}
The half year and one and a half year intervals match vaccination during a seasonal low with response in a seasonal high and vice versa. Thus they show the extreme seasonal pattern, whereas the one and two year offsets match low with lows and high with highs, so the percent differences display a minimal of seasonal pattern. Nevertheless, even in the whole number offsets there is still an annual periodicity where vaccination in early spring has an advantage over other times of the year. The early spring is marked by slightly cooler temperatures and less rain then later in the summer. Thus it is just before new cases start to increase in the summer months. This appears to be the optimal time of year to vaccinate.
A examination of the timing response with the number vaccinated more closely matched to the actual numbers is seen in Figures \[Figure:18\] and \[Figure:19\]. Here though the global maximums occur in the third year in Artibonite, and in the fourth or fifth year in Ouest.
![[]{data-label="Figure:18"}](19Art_vacc_time_rel_40.eps){width="5.3in" height="4.0in"}
![[]{data-label="Figure:19"}](21Ouest_vacc_time_rel_50.eps){width="5.3in" height="4.0in"}
There reason is clear when we consider again Figures \[Figure:12\] and \[Figure:13\]. Here the maximum percent decrease in cases is when the number vaccinated matches the number remaining at risk. Thus when we vaccinated at a lower level the optimal time for the this level of vaccine is at a later date, and preferably in April, when other measures have reduced the at risk population to a size comparable to the number that can be vaccinated. Thus it appears that vaccination programs, as they actually occurred, occurred at nearly the best time of year, but may have been slightly more effective this year in Artibonite, or even next year in Ouest.
Discussion
==========
Modeling the dynamics of cholera in Haiti has been hampered by the lack of easily accessible detailed historical meteorological data . We use NASA satellite data to address this problem. This study shows that with environmental data of sufficient detail and quality, projections of disease progression can be made with sufficient lead time to prepare for outbreaks. The lag times of over five weeks means that if even rudimentary but reliable meteorological and coastal records are kept, preparations and resources can be more focused. The gathering of basic weather information is simple and inexpensive and should be made standard procedure when any agency takes part in interventions, particularly when the environmental component of the epidemiology is so well established.
In addition we explored the hypothesis that, at least in the Ouest region, tidal influences play a significant role in the dynamics of the disease. It appeared that tidal range rather than the height of the tide itself had the strongest influence. Some connection to tidal influences should be expected where large populations are in close contact with bays and estuaries, and humans are consuming local seafood [@Huq; @deMagny]. It is not surprising that there was no effect of tidal range found in Artibonite since the tide model was for off the coast of Port-au-Prince. Again the lack of readily available detailed historical tide records or even a model for various regions along the coast hinders a thorough investigation of possible factors in the disease dynamics.
We also affirmed the longer time lags (8 - 10 weeks) found in previous studies from Africa [@Reyburn] and shorter ones (4 weeks) in Bangladesh [@deMagny]. Delays in the effects of precipitation on the infection rates varied for Artibonite between $5.0$ and $11.8$ weeks, and for Ouest they ranged from $5.8$ to $8.5$ weeks. The shorter delays occurring in the two regions during the warmer months but differing by up to 3 weeks during the cooler months. One possible explanation of the difference in the two regions may be due to greater sensitivity to the density of infected population in cooler months and little sensitivity in warm months. Or, to state it another way, there could be a low sensitivity to temperature changes when infected density is high, and high sensitivity to temperature changes when there are relatively less people infected. The relatively high numbers of infected in Ouest keep the lag times short no matter what the weather, whereas in Artibonite where the infection rate is a quarter to half that of Ouest cooler weather has more of an impact in slowing the disease cycle.
Over the course of the epidemic the incidence has been tapering off. There has been steady and continued effort to improve hygiene and living conditions, however, the areas where the greatest strides are made are those where people leave the camps to return to normal living conditions and employment. The declining numbers of those at risk in the overall population belie the fact that many local populations are still without basic hygienic facilities. This was reflected in the model by setting a level $u_{0}$ to 5% of the original number of susceptible. This value matches the approximate 5% of the population that still remain displaced after the 2010 earthquake [@OCHA].
On top of predicting when and how many cholera cases will increase with Haiti’s weather patterns and tides, any modeling to predict the effectiveness of interventions (such as vaccination) should consider these patterns. Considering that cholera may be maintained in the environment outside the human chain of infection is essential to planing effective prophylaxes and interventions.
Using these models we were able to assess, to some degree, the relative effectiveness of the recent vaccination program in Artibonite and Ouest. The discrepancy between the apparent effectiveness of vaccination in the two regions is perhaps not that puzzling when one considers the number vaccinated relative to the size of at risk population. In Artibonite about 41 thousand people and in Ouest about 47 thousand people received both doses of the vaccine. However, our model suggests that in Artibonite the at risk population by 6-May was about 61 thousand whereas in Ouest it was still over 428 thousand, seven times the number in Artibonite. Thus in Artibonite $67\%$ of the most at risk population apparently received the vaccine, while in Ouest only $11\%$ did. Further, in both regions 100 people receiving vaccination does not mean 100 people protected. The vaccine is about $75\%$ effective,and only $21\%$ of people who are infected with cholera show symptoms, $0.75 \times 0.21 = 0.1575$. So as a rough calculation we might expect $0.1575 \times 41,000 = 6457.5$ in Artibonite, and $0.1575 \times 47,000 = 7402.5$ people in Ouest protected directly. Any additional protection would be due to the reduction in environmental loading of cholera bacteria and the force of infection. Although ultimately more people will be protected in Ouest any of the secondary effects of vaccination will probably be lost due to dilution of the vaccinated group.
Conclusion
----------
Although progress has been made in the past two years in modeling the Haitian cholera epidemic, only one model, so far, has accounted for seasonal variation due to precipitation, and none of the model of the current Haitian epidemic have examined temperature or tidal influences. Our paper shows that with basic climatic and tidal records and a relatively unsophisticated modeling procedure, not only can these factors be accounted for but also the time lags between climatic events and outbreaks can be identified. This approach looks at specific climatic events on the scale of a week rather than just seasonal patterns. We use daily tidal range as a predictive factor for cholera epidemics for the first time in a modeling paper, and we use real time daily precipitation estimates. These provide a level of detail of environmental events in conjunction with the lag times and estimates of core population size that can help evaluate intervention (such as vaccination) and public hygiene efforts. In order to show this we examined recent vaccination efforts in the Haitian cholera epidemic. Complex environmental patterns incorporated in epidemic models allow us to remove a large source of variability and bring into relief intervention efforts by identifying deviations from the unaltered flow.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Scott Braun and George Huffman at NASA, Goddard Space Flight Center for help accessing precipitation data.
The authors are also grateful to Scott Dowell and Jordan Tappero at CDC; Louise Ivers at PIH; and Jean Pape at GHESKIO for information about the cholera vaccination programs in Haiti.
This project was partially supported by a CCFF grant of the Columbian College of Arts and Sciences, of the George Washington University. S.R. was also partially supported by the NSF CAREER grant $\# 1151618$.
appendix {#appendix .unnumbered}
========
System of equations using a reduction in the at-risk group. {#system-of-equations-using-a-reduction-in-the-at-risk-group. .unnumbered}
-----------------------------------------------------------
We have $\ u=u_{0}+\left( 1-u_{0}\right) e^{-rt}$ as before, and $\frac
{d}{dt}u=-r\left( u-u_{0}\right) $. The equation for the entire susceptible population is the same also, that is $\ \frac{d}{dt}S=-u\beta_{R}S$, but here we keep improvement factor $u$ and the crude force of infection $\beta_{R}$ separate, where $$\beta_{R}\!=\alpha_{p}H\!\left( t\right) P\!\left( t-\tau_{p},\theta
_{p}\right) +\alpha_{m}M\!\left( t-\tau_{m},\theta_{m}\right) .$$ The improvement factor, $u$, represents the fraction of the population that remains at risk at time $t$. Thus$$S_{R}=uS.$$ Differentiating with respect to time we have$$\begin{aligned}
\frac{d}{dt}S_{R} & =u\frac{d}{dt}S+S\frac{d}{dt}u\\
& =-u^{2}\beta_{R}S-r\left( u-u_{0}\right) S\\
& =-u\beta_{R}S_{R}-r\left( 1-\frac{u_{0}}{u}\right) S_{R}.\end{aligned}$$ New cases are as before $u\beta_{R}S$ $\ $which equals $\ \beta_{R}S_{R}.$ Note that the number of new cases is the same in both interpretations but the force of infection is reduced $\left(\beta = u\beta_{R}\right)$ in the original interpretation and we use the crude force of infection $\left(\beta_{R}\right)$ and the reduced at-risk susceptible population $\left( S_{R}\right)$ here.
With vaccination we have $$\begin{aligned}
\frac{d}{dt}S_{R} & =-u\beta_{R}S_{R}-r\left( 1-\frac{u_{0}}{u}\right)
S_{R}-V\\
& =-u^{2}\beta_{R}S-r\left( u-u_{0}\right) S-V,\end{aligned}$$ where $V$ is the number vaccinated per unit time. But $$\begin{aligned}
\frac{d}{dt}S & =\frac{1}{u}\left( \frac{d}{dt}S_{R}-S\frac{d}{dt}u\right)
\\
& =\frac{1}{u}\left( \left( -u^{2}\beta_{R}S-r\left( u-u_{0}\right)
S-V\right) +r\left( u-u_{0}\right) S\right) \\
& =-u\beta_{R}S-\frac{V}{u}.\end{aligned}$$ New cases are still $u\beta_{R}S$ $\ $which equals $\ \beta_{R}S_{R}$ and the $V/u$ term represents the number of all susceptibles that would have needed to have been vaccinated in the original, reduced force of infection, interpretation of the model in order to have an equivalent impact on the number of new cases.
In order to include efficacy of vaccination, $f$, in the model we need to introduce a new compartment, $Q$, for vaccinated but still susceptible. The set
of differential equations for the system then become,$$\begin{array}
[c]{l}\frac{dS}{dt}=-u\beta_{R}\!\left( t\right) S-\frac{V\!\left( t\right)}{u}\medskip\\
\frac{dQ}{dt}=-u\beta_{R}\!\left( t\right) Q+\left( 1-f\right) \frac{V\!\left( t\right)}{u}\medskip\\
\frac{dA}{dt}=\rho u\beta_{R}\!\left( t\right) \left( S+Q\right) -\gamma
A\medskip\\
\frac{dI}{dt}=(1-\rho)u\beta_{R}\!\left( t\right) \left( S+Q\right)
-(\gamma+\mu)I\medskip\\
\frac{dR}{dt}=\gamma(A+I)
\end{array}$$
New and total cases are$$\begin{array}
[c]{l}\text{new cases}=(1-\rho)\int_{t-\Delta t}^{t}u\!\left( t\right) \beta
_{R}\!\left( \frak{t}\right) \left( S\!\left( \frak{t}\right) +Q\!\left( \frak{t}\right)
\right) d\frak{t}\medskip\\
\\
\text{total cases}=(1-\rho)\int_{0}^{t}u\!\left( \frak{t}\right) \beta_{R}\!\left(
\frak{t}\right) \left( S\!\left( \frak{t}\right) +Q\!\left( \frak{t}\right) \right)
d\frak{t}\medskip
\end{array}$$
[^1]: Conceptually the underlying model is a variation on the SIWR model, which assumes that cholera is spread through susceptibles’ contact with contaminated water, food or fomites. This model uses the amount of water consumed as a proxy for all possible modes of transmission, and the concentration of bacteria in the water consumed modifies the infection rate by a dose-response expression (see [@TienandEarn] with a base model [@Codeco]).
[^2]: We will use this interpretation exclusively when we talk about vaccination. See appendix for the related system of equations using this interpretation.
[^3]: When we discuss the output of our model, will use the term “prediction" to indicate the numbers generated during the calibration phase of the model and “projection" to indicate the extrapolation of the model past cases used for the calibration.
[^4]: Communicated by Jordan Tappero, MD, MPH (CDC/CGH/DGDDER, Atlanta, GA). Nov. 29, 2012.
[^5]: Groupe Haïtien d’Étude du Sarcome de Kaposi et des Infectieuses Opportunistes
[^6]: Communicated by Jean W. Pape, MD (GHESKIO, Weill Cornell Medical College, Port-au-Prince, Haiti). Nov. 29, 2012.
[^7]: Partners in Health
[^8]: Communicated by Louise Ivers, MD (Partners in Health/ZL, Cange, Haiti). Nov. 29, 2012.
|
---
abstract: 'We show that the Hamiltonian mean field (HMF) model describes the equilibrium behavior of a system of long pendula with flat bobs that are coupled through long-range interactions (charged or self gravitating). We solve for the canonical partition function in the coordinate frame of the pendula angles. The Hamiltonian in the angles coordinate frame looks similar to the form of the HMF model but with the inclusion of an index dependent phase in the interaction term. We also show interesting non-equilibrium behavior of the pendula angles, namely that a quasistationary clustered state can exist when pendula angles are initially ordered by their index.'
author:
- Owen Myers
- Adrian Del Maestro
- Junru Wu
- 'Jeffrey S. Marshall'
bibliography:
- 'long\_range\_refs.bib'
title: 'A Simple Model for Long-Range Interacting Pendula'
---
Introduction
============
Systems with long-range interactions are a source of unique problems in the field of statistical mechanics and thermodynamics. This is due to several properties of long-range systems which fall outside of the conditions normally needing to be satisfied when applying the methodologies of thermodynamics. Simply from the words “long-range” the first infringement can be deduced, that long-range systems are not additive. If two systems with short-range interactions are brought together to form a larger system then the energy difference between the conglomerate system and the sum of its constituents is the new potential energy from the boundary between them. In the thermodynamic limit, the potential energy of the boundary is small compared to the bulk and can be neglected, making short-range systems additive. In the case of long-range interactions, one particle will feel a significant potential created by every other particle, so the additional potential energy of two systems added together does not scale as the boundary but in a more complicated way that depends on the specific nature of the interactions [@dynam_therm_intro]. Directly related to the lack of additivity is the fact that systems with long-range interactions are not extensive because their energy diverges in the thermodynamic limit [@kac1963]. Although these characteristics compel cautious use of the usual tools of statistical mechanics, they are also the source of many interesting dynamical and statistical features. Depending on the system of interest, such features include canonical and microcanonical ensemble inequivalence and related negative specific heat [@sire2002], quasistationary states (different than metastable states which lie on local extrema of equilibrium potentials) whose lifetimes increase with the number of particles [@antoniazzi2007], an interesting dependence of the largest Lyapunov exponent on particle number in a long-range Fermi-Pasta-Ulam model [@christodoulidi2014], and spontaneous creation of macroscopic structures in non-equilibrium states [@antoni1995]. In some cases, long-range interactions can greatly simplify problems. For instance, mean field models depend on one of two premises: (i) interactions are short-range but the system is embedded in a space of infinite dimension so that all bodies in the system are nearest neighbors, or (ii) interactions are infinitely long.
For some time, the primary motivation for the study of long-range interactions was to understand galaxies, galaxy clusters and the general thermodynamic properties of self-gravitating systems. Aside from mean field models, interest has further built since the observation of modified scattering lengths in Bose-Einstein Condensates (BEC) through the use of Feshbach resonances [@inouye1998]. Using this technique, a BEC can be made to be almost non-interacting by tuning the scattering length to zero. One could even tune the scattering length to a negative value, making the BEC collapse. More recently, O’dell et al. [@odell2000] has shown that it may be possible to produce an attractive $1/r$ potential between atoms in a BEC by applying an “extremely off resonant” electromagnetic field. This has opened the possibility of creating table-top methods which physically model aspects of cosmological behavior on a laboratory scale, as well as the possible development of entirely new dynamics in BEC.
The challenges in understanding long-range systems drive the development of solvable models that could help better explain some of the aforementioned phenomena. Campa et al. [@campa2009] have recently published a collection of important solvable models. One particularly significant model, which is important to this work, is the Hamiltonian Mean Field (HMF) $XY$ spin model [@antoni1995], often written in the form
$$\label{eqn:HMF}
H = \sum_{i=1}^{N} \frac{p_i^2}{2} + \frac{\gamma}{2N}\sum_{i,j=1}^{N}
\left[
1- \cos{(\theta_i-\theta_j)}
\right]
,$$
where $\theta_i$ is the angular position of the $i^{\mathrm{th}}$ particle (spin), as shown in Fig. \[fig:system\], and $p_i$ is its conjugate angular momentum. The HMF model is generally used to describe two different classes of systems: 1) a mean field $XY$ classical spin model, and 2) a one dimensional periodic system of itinerant particles with long-range interactions. Though the connection between the HMF model and the second class of systems mentioned could be thought of as contrived given the simplifications under which the model is realized, it has been shown that the model produces useful insights into how non-neutral plasmas and self gravitating systems behave [@antoni1995].
In this paper, we study the dynamics of an array of $N$ pendula with long-range interacting bobs. By considering long pendula with flat bobs undergoing small oscillations and having parallel planes of rotation, we produce a model related to the HMF model through a coordinate transformation. The transformation introduces a dependence on the indices of the particle labels. A cartoon of the physical picture is shown in Fig. \[fig:system\].
![$N$ pendulum system with parallel planes of rotation. The $i^{\mathrm{th}}$ pendulum angle at some time $t$ is $\theta_i(t)$. \[fig:system\]](system.pdf){width="8.0cm"}
The index dependence in the Hamiltonian, that will be described in detail in the next section, is a consequence of the pendula pivots being slightly offset from one another and appears as a phase in the cosine term of the HMF model. It inspires the investigation of non-equilibrium “repulsive” behavior in the angle coordinate frame where we find an interesting quasistationary state when the angles of the pendula are initially ordered according to their indices. We find the clustered positions in the usual HMF coordinate frame (biclusters), but in the angle coordinate frame clustering is only found for the initially ordered angles and, unlike the biclusters, these are clearly quasistationary states. A quasistationary state is defined as a dynamical state that can persist for a length of time which goes to infinity as the thermodynamic limit is approached [@campa2009]. In addition to discussing the clustered angle states exhibited by the system, we also solve for the canonical partition function in the pendulum angle coordinate frame, finding that in equilibrium with a heat bath, the probability distributions of the angles can be described by the original HMF model. This finding is similar to the work done by [@campa2000] on a model sometimes called the HMF $\alpha$-model. In the HMF $\alpha$ model, a $1/r_{ij}^\alpha$ dependence between the classical spins is introduced [@campa2009; @campa2003; @anteneodo1998; @tamarit2000; @cirto2014], where $r_{ij}$ is the distance between the $i^{\mathrm{th}}$ and $j^{\mathrm{th}}$ spins on a lattice. Though the physical motivations behind studying these various models can be very different, it is interesting that their equilibrium behavior is the same or nearly the same. We believe that the work in this paper further suggests that the HMF model universally describes an entire class of long-range interacting systems in equilibrium.
The Model
=========
Coordinates
-----------
In Fig. \[fig:system\], we show an array of pendula rotating in the same plane with bobs that interact through a long-range potential. If we consider the case where all the pendula only undergo small oscillations, we may write the horizontal location of the $i^{\mathrm{th}}$ particle, $x_i$, as $x_i = id +
\ell\theta_i$, where $d$ is the distance between the pivots of neighboring pendula and $\ell$ is the length of each pendulum. The small $\theta$ regime makes the problem one dimensional in $x$. We choose periodic boundary conditions and rescale the system by $2\pi/Nd$ so that $$\label{eqn:scalex}
x\rightarrow \frac{2\pi}{Nd} x$$ making the total system length a dimensionless $2\pi$ where N is the number of particles in one period. We will refer to a periodic space with length $2\pi$ as a unit circle. The position of the $i^{\mathrm{th}}$ particle (bob) is now $$\label{eqn:scaledxi}
x_i = \frac{2\pi i}{N} + \frac{2\pi}{N}\frac{\ell}{d}\theta_i
.$$ For reasonable choices of $\ell$ and $d$ ($\ell/d << N$), the second term on the RHS is suitably small such that the Hamiltonian can be written with terms that are quadratic in $\theta$. However, we are primarily interested in a regime where $\ell/d \rightarrow \infty$ as the thermodynamic limit is approached. Physically this corresponds to the small oscillations of very long pendula with suspension points that are close together compared to their lengths. In order to simplify the calculations that follow, we define $\phi_i$ to be the last term on the RHS of Eq. (\[eqn:scaledxi\]), namely $\phi_i \equiv 2\pi \ell \theta_i / Nd $. Given the choice of large $\ell/d$, $\phi_i$ can take any value in the range $[0,2\pi)$. This is only true because $\ell/d$ is large, *not* because the $\theta_i$s are. In terms of $\phi_i$, the positions can be rewritten as $$\label{eqn:simplescaledxi}
x_i = \frac{2\pi i}{N} + \phi_i
.$$
Density Approximation
---------------------
We have not yet explicitly stated the physical mechanism through which the bobs interact. Connecting the interactions with specific physical motivations should be discussed with some discretion because the development of the model leaves these motivations up to some freedom of interpretation. Imagine that the bobs all carry some charge. We will not distinguish between particles in any other way than their indices, so in the case where all particles carry the same charge, repulsive behavior is expected. On the other hand one could make the bobs attract one another, which could be thought of as the self-gravitating case. To be solvable, the model requires some simplifications. For the sake of brevity we will speak of the particle charge or mass density as the “density”.
The approximation that we invoke is similar to that used when justifying the HMF model (Eq. (\[eqn:HMF\])) to describe free particles in a one-dimensional ring [@antoni1995; @levin2014]. The distribution of the bobs is such the mass density, $\rho(x)$, is given by $$\rho(x) = \sum_{i=1}^{N}\delta(x-x_i) - \frac{1}{2\pi}
.$$ The constant $1/2\pi$ subtracted from the delta function is necessary to produce a meaningful expression for the potential $\Phi$ and corresponds to the inclusion of a neutralizing (of opposite sign) homogeneous background density. Restricting the problem further to that of solving Poisson’s equation for a one-dimensional potential physically amounts to choosing large and flat bob geometries oriented with their smallest axis parallel to the $x$ axis. Writing the delta function as a cosine Fourier series, Poisson’s equation becomes $$\label{}
\nabla^2\Phi(x) = \frac{\gamma}{\pi}\sum_{i=1}^{N}\sum_{n=1}^{\infty}
\cos{[n(x-x_i)]}
.$$ The parameter $\gamma$ contains the particle (bob) charge or mass and becomes the interaction strength in the Hamiltonian. We can see that the zeroth-order term in the Fourier series canceled the constant neutralizing background that was superficially added.
The most important simplification in this paper is truncating the sum of the Fourier coefficients used to represent the delta function after the $n = 1$ coefficient. Antoni et al. defend the truncation by asserting that the “large scale collective properties” do not greatly change when higher order terms of the sum (including interactions at the smaller length scales) are included, and discuss the consequences of the approximation in some detail [@antoni1995]. The simplification also warrants a brief discussion of the way that it could be physically interpreted. The truncation of the sum is equivalent to smearing out the density of each particle over the system so that it is peaked at its given location, $x_i$, but also having a negative density peak on the opposite side of the unit circle. This could be thought of as doubling the number of particles and enforcing that each particle has a negative partner that always remains on the opposing side of the unit circle. After this doubling, the now nebulous masses are dispersed such that a pair’s density is described by a cosine function with the positive peak centered at $x_i$.
Solving Poisson’s Equation
--------------------------
Integrating Poisson’s equation once, we obtain: $$\label{}
\nabla\Phi(x) = \frac{\gamma}{\pi}\sum_{i=1}^{N}
\left\{
\sin{(x-x_i)} + c_1
\right\}
.$$ In order to determine the constant $c_1$ from the integration, the physical picture should be examined. A sensible requirement is that when all of the bobs are hanging at their equilibrium positions, directly below their pivot (all $\phi_i=\theta_i=0$), the net force experienced by any bob is zero. This is a valid requirement if the bobs are attractive or repulsive, the only difference being that the configuration would be unstable or stable, respectively. The force that the $j^{\mathrm{th}}$ particle experiences when $\phi_j$ and all $\phi_i$ are zero is given by $$\label{}
-\nabla\Phi\left(x_j \right) = -\frac{\gamma}{\pi}\sum_{i=1}^{N}
\left\{
\sin
{
\left[
\frac{2\pi (j-i)}{N}
\right]
} + c_1
\right\}
.$$ The sum $\sum_i \sin{[2\pi (j-i)/N]}$ equals zero for any $j$, so $c_1$ must be zero. Integrating once more to obtain the potential yields $$\label{}
\Phi(x) = \frac{\gamma}{\pi}\sum_{i=1}^{N}
\left[
c_2-\cos{\left(x-\frac{2\pi i}{N} - \phi_i \right)}
\right]
.$$ To determine $c_2$ we stipulate that if all $\phi_i=0$, then $\Phi(0)=0$. Inserting Eq. (\[eqn:scaledxi\]) (or Eq. (\[eqn:simplescaledxi\])) for $x_i$ yields $$\label{}
\Phi(0) = \frac{\gamma}{\pi}\sum_{i=1}^{N}
\left[
c_2-\cos{\left(\frac{2\pi i}{N}\right)}
\right]
.$$ The sum over the cosine is zero, therefore $c_2 = 0$ and we can now write the potential energy of the $j^{\mathrm{th}}$ particle as
$$\label{}
\Phi(x_j) = -\frac{\gamma}{\pi}\sum_{i=1}^{N}
\cos{\left[\frac{2\pi (j-i)}{N}+\phi_j - \phi_i \right]}
.$$
The Hamiltonian
---------------
The Hamiltonian can be written as $$\label{}
H = H_0 + H_I
,$$ where $H_0$ is the kinetic energy piece $$\label{}
H_0 = \sum_{i=1}^{N} \frac{p_i^2}{2}
,$$ and $$\label{}
H_I = -\frac{\gamma}{2N}\sum_{i,j=1}^{N}
\cos{
\left[
\frac{2\pi (i-j)}{N}
+
\phi_i-\phi_j
\right]
}$$ is the interaction piece, so $$\label{eqn:ourH}
H = \sum_{i=1}^{N} \frac{p_i^2}{2}
-\frac{\gamma}{2N}\sum_{i,j=1}^{N}
\cos{
\left[
\frac{2\pi (i-j)}{N}
+
\phi_i-\phi_j
\right]
}
.$$ The mass of the bobs has been set to unity, $\gamma$ is the interaction strength, a factor of $1/2$ accounts for the double counting, and the $1/\pi$ coefficient in the potential energy has been absorbed into $\gamma$. The factor of $1/N$ is a rescaling of the potential energy that ensures that as the thermodynamic limit is approached, the potential energy of the system does not diverge. The $1/N$ scaling is known as the Kac prescription [@teles2012]. The Kac serves to keep both the energy and entropy of a system proportional to the number of particles in the system, an important prerequisite for phase transitions [@campa2009].
Relationship to the HMF model and the Spin Interpretation {#sub:relationship_to_the_hmf_model_and_the_spin_interpretation}
---------------------------------------------------------
Due to the simple bijective relationship between $x_i$ and $\phi_i$ one can simply solve the equations of motion for the HMF model and find the dynamics for $\phi_i$ via the coordinate transform $x_i \rightarrow \phi_i$. Previously it was mentioned that the HMF model is used to describe free particles on a ring with long-range repulsion or attraction, as well as describing a classical $XY$ spin model. The $\theta_i$ played the role of either the position of the $i^{\mathrm{th}}$ particle on the ring or the orientation of the $i^{\mathrm{th}}$ spin. Therefore, it is interesting to speculate about the type of spin system the model describes in the $\phi_i$ picture. Thus far, the rescaled angle $\phi_i = 2\pi \ell \theta_i / Nd$ describes the distance of a pendulum bob from the point directly below its pivot, but it could also be interpreted as the orientation of spin. In the spin interpretation of Eq. (\[eqn:ourH\]), the potential energy of the $i^{\mathrm{th}}$ and $j^{\mathrm{th}}$ spin pair depend on both their relative orientation as well as the difference between their indices. In the following discussion it will sometimes be convenient to speak about $\phi_i$ in the spin language.
We will prove that in the $\phi$ picture, the system in equilibrium with a heat bath is equivalent to the HMF model (the $x$ picture) in equilibrium with a heat bath by solving the partition function in the $\phi_i$ coordinate frame. In the process of simplifying the Hamiltonian to solve for the partition function, we will find expressions of the form $\cos{\phi_i}$ and $\sin{\phi_i}$ which we talk about as the horizontal and vertical components of a magnetization $\vec{m}_i = (\cos{\phi_i},\sin{\phi_i})$. It could easily be stated that in the spin analogy, the $\phi_i$ are orientations of the spins, but we should make a more concrete connection between this idea and the original presentation of the model. We would like to remind the reader that even though the angles $\theta_i$ of the pendula are small, the long suspensions of the bobs ($\ell$) allow $\phi_i$ to cover the entire system which, rescaled, has dimensionless length $2\pi$. The system is also periodic, so the bobs can be thought of as moving on a unit circle where the position of the $i^{\mathrm{th}}$ bob is $x_i=2\pi i/N + \phi_i$. In order to think of $\phi_i$ as the spin orientations, we start by considering each bob as living on its own individual unit circle. An example of these unit circles is shown in Fig. \[fig:x\_phi\_mag\],
![Example of a system of $N=8$ particles when viewed as individual spins in the a) $x$ coordinate frame, and b) the $\phi$ coordinate frame. a) In the $x$ coordinate frame the direction of the $i^{\mathrm{th}}$ spin given by the angle $x_i$ is expressed as $x_i=2\pi i/N +\phi_i$. Alternatively, $x_i$ could be thought of as the position of $i^{\mathrm{th}}$ particle on the unit circle, shown at the bottom of the figure as the projection of all positions onto the horizontal plane. The black circles on the rings in the figure mark the location of the pendulum pivots at $2\pi i/N$ in $x$. b) Twisting the column of rings in a) such that the pivots are aligned transforms the system into the $\phi_i$ coordinate frame. In this picture, the direction of the $i^{\mathrm{th}}$ spin is just given by the angle $\phi_i$. \[fig:x\_phi\_mag\]](x_phi_mag.pdf){width="8.0cm"}
a visual aid to the following. Imagine stacking horizontal circles in the vertical direction and rotating each by an angle $2\pi/N$ from the one below. The projection of these circles onto the horizontal plane would be the system viewed in $x$, i.e. the HMF model. If we twist the stack so there is no rotation between adjacent circles and then project onto the horizontal plane it creates the picture viewed in $\phi$, where the pivot points are all aligned. The reason for this artificial construction of stacked circles is partly to pictorially depict the transformation between $x$ and $\phi$ and partly to show how $\vec{m}_i$ (as defined) is just the orientation of the $i^{\mathrm{th}}$ spin in the $\phi$ picture. Said differently, each circle in the $\phi$ picture represents a spin with an orientation in the horizontal plane determined by $\phi_i$; an infinite-range classical mean field spin model described by Eq. (\[eqn:ourH\]).
Equilibrium
===========
In this section, we solve for the canonical partition function, in the $\phi$ coordinate frame using the Hamiltonian in Eq. (\[eqn:ourH\]) and show that, in equilibrium, the HMF model describes the angles of long pendula with long-range interacting bobs. In order to solve the configurational piece of the partition function the Hamiltonian must be modified. Using the cosine and sine sum and difference identities twice, the potential interaction piece of the Hamiltonian $H_I$ can be written as
$$\begin{gathered}
% \frac{-\gamma}{2} - \frac{\gamma}{2N} \sum_{i,j}
H_i =
\frac{-\gamma}{2N} \sum_{i,j}
\Bigg\{
\cos{\left[ \frac{2\pi (i-j)}{N} \right]}
[
\cos{\phi_i}\cos{\phi_j}
\\
+
\sin{\phi_i}\sin{\phi_j}
]
\\
-
\sin{\left[ \frac{2\pi (i-j)}{N} \right]}
\left[
\sin{\phi_i}\cos{\phi_j}
-
\cos{\phi_i}\sin{\phi_j}
\right]
\Bigg\}
.\end{gathered}$$
The coefficients in the Hamiltonian $\cos{[2\pi (i-j)/N]}$ and $\sin{[2\pi (i-j)/N]}$ should be thought of as matrices with components $C_{ij}$ and $S_{ij}$ respectively. The Hamiltonian can now be written in the form
$$\begin{gathered}
\label{}
\frac{\gamma}{2N}\sum_{i,j}
(
\cos{\phi_i}C_{ij} \cos{\phi_j} + \sin{\phi_i}C_{ij}\sin{\phi_j}
\\
- \sin{\phi_i}S_{ij}\cos{\phi_j} + \cos{\phi_i}S_{ij}\sin{\phi_j}
),\end{gathered}$$
which is suggestive because it can be regarded as the matrix equation
$$\begin{gathered}
\label{}
H_I = \frac{\gamma}{2N}
\Bigg[
(\cos{\phi_1},\cos{\phi_2},...,\cos{\phi_N})
C
\begin{pmatrix}
\cos{\phi_1} \\
\cos{\phi_2} \\
\vdots \\
\cos{\phi_N}
\end{pmatrix}
\\
+
(\sin{\phi_1},\sin{\phi_2},...,\sin{\phi_N})
C
\begin{pmatrix}
\sin{\phi_1} \\
\sin{\phi_2} \\
\vdots \\
\sin{\phi_N}
\end{pmatrix}
\\
-
(\sin{\phi_1},\sin{\phi_2},...,\sin{\phi_N})
S
\begin{pmatrix}
\cos{\phi_1} \\
\cos{\phi_2} \\
\vdots \\
\cos{\phi_N}
\end{pmatrix}
\\
+
(\cos{\phi_1},\cos{\phi_2},...,\cos{\phi_N})
S
\begin{pmatrix}
\sin{\phi_1} \\
\sin{\phi_2} \\
\vdots \\
\sin{\phi_N}
\end{pmatrix}
\Bigg]
.\end{gathered}$$
It is helpful to consider the particles positions on the unit circle with respect to their pivot ($\phi$) as magnetizations. Defining
$$\label{}
\vec{m}_i \equiv (\cos{\phi_i},\sin{\phi_i}),$$
and with $m^T_{\mu} = (m_{0,\mu},m_{1,\mu},...,m_{N-1,\mu})$ where $\mu$ holds the place of $x$ or $y$, the Hamiltonian becomes $$\begin{gathered}
\label{}
H_I = \frac{\gamma}{2N}
(
m^T_x C m_x + m^T_y C m_y
\\
- m^T_y S m_x + m^T_x S m_y
)
.\end{gathered}$$
A closer examination of the structure of the coefficient matrices $C$ and $S$ indicates that they take the special form of circulant matrices, and thus can be simultaneously diagonalized by a unitary matrix $U$. A circulant matrix has the form $$\label{eqn:circmatrix}
\left(
\begin{matrix}
a_1 & a_2 & a_3 & \ldots & a_{N-1} & a_N \\
a_N & a_1 & a_2 & \ldots & a_{N-2} & a_{N-1} \\
a_{N-1} & a_N & a_1 & \ldots & a_{N-3} & a_{N-2} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
a_3 & a_4 & a_5 & \ldots & a_1 & a_2 \\
a_2 & a_3 & a_4 & \ldots & a_N & a_1 \\
\end{matrix}
\right)
,$$ a special kind of Toeplitz matrix, where each subsequent row is a cyclic permutation of the row above or below it. Any matrix $A$ with elements $a_{ij}$ that can be written in terms of some function $f(i-j)$ is a circulant matrix. Because a circulant matrix is a normal matrix it can be diagonalized by a unitary matrix. We show that $C$ and $S$ are simultaneously diagonalizable by showing that they commute, i.e. $[C,S]=0$ where $[C,S] = CS-SC$. Starting with the second term, $-SC = S^TC^T=(CS)^T$ which is found by arguing that $C$ is symmetric since cosine is an even function and does not change under the exchange of $i$ and $j$, whereas $S$ is odd because sine is an odd function and does change sign under exchange of $i$ and $j$. The commutation becomes $[C,S] = CS + (CS)^T$. Also, an odd function multiplied by an even function results in an odd function so the entire matrix $CS$ is odd. Therefore $(CS)^T = -CS$ bringing us to the final expression $[C,S] = CS-CS=0$. We have shown that $C$ and $S$ can be simultaneously diagonalized by $U$. The matrix $U$ is known for circulant matrices and is called the Fourier Matrix.
The matrices $C$ and $S$ can be rewritten as $C= U^\dagger D^C U$ and $S = U^\dagger D^S U$, where $D^C$ and $D^S$ are diagonal matrices with diagonal elements that are the eigenvalues of $C$ and $S$, respectively, which we denote as $\lambda^C_i$ and $\lambda^S_i$. From here on we label the indices $i$ from $0$ to $N-1$. It is worth pointing out that due to $S$ being antisymmetric, $U$ must be complex. Equation (\[eqn:matrixCS\]) becomes $$\begin{gathered}
\label{eqn:matrixCS}
H_I = \frac{\gamma}{2N}
\Big(
m^T_x U^\dagger D^C U m_x + m^T_y U^\dagger D^C U m_y
\\
- m^T_y U^\dagger D^S U m_x + m^T_x U^\dagger D^S U m_y
\Big)
.\end{gathered}$$ We will move back to the index notation using the following relations: $$\label{}
D^{C,S} = \lambda_i^{C,S}\delta_{ij}
,$$ $$\label{}
X_j \equiv \sum_{k=1}^N U_{jk} m_k^x
,$$ and $$\label{}
Y_j \equiv \sum_{k=1}^N U_{jk} m_k^y.
,$$ where $X$ is not to be confused with $x$. Using the Kronecker delta, we set all $i=j$ since these are the only nonzero terms. The Hamiltonian is now given by $$\begin{gathered}
\label{}
H_i =
\frac{-\gamma}{2N}\sum_{j=0}^{N-1}
\Big(
\|X_j\|^2 \lambda_j^C + \|Y_j\|^2 \lambda_j^C
\\
-Y_j^* X_j \lambda_j^S + X_j^* Y_j \lambda_j^S
\Big)
,\end{gathered}$$ with $\|X\|= XX^*$ and $\|Y\|= YY^*$. The inclusion of the eigenvalues $\lambda^C$ and $\lambda^S$ simplifies the Hamiltonian further. We will now solve for $\lambda^C$ and $\lambda^S$. Looking at the form of a circulant matrix shown in Eq. (\[eqn:circmatrix\]) reminds us that the elements of a circulant matrix can be defined with a single label. We write the single labeled elements of the cosine and sine matrices respectively as $$\label{}
c_l = \cos{\frac{2\pi l}{N}}
,$$ and $$\label{}
s_l = \sin{\frac{2\pi l}{N}}
,$$ where $l=0,1,2,...,N-1$. The eigenvalues, $\lambda^A$, of a $N\times N$ circulant matrix $A$ can be written in terms of the single label elements $a_l$. The $j^{\mathrm{th}}$ eigenvalue of $A$ is known to be $\lambda^A_j=\sum_{l=0}^{N-1} a_l \exp{(2\pi i l j / N)}$, where $i$ is $\sqrt{-1}$ (not an index) and $l=0,1,2,...,N-1$. Therefore, $$\label{}
\lambda_j^C = \sum_{l=0}^{N-1} \cos{\left(\frac{2\pi l}{N}\right)} e^{2\pi i l j /N}
,$$ and $$\label{}
\lambda_j^S = \sum_{l=0}^{N-1} \sin{\left(\frac{2\pi l}{N}\right)} e^{2\pi i l j /N}
.$$ Writing cosine and sine in their exponential forms gives $$\label{}
\lambda_j^C = \frac{1}{2} \sum_{l=0}^{N-1}
\left[
e^{i 2 \pi l (j+1)/N}
+
e^{i 2 \pi l (j-1)/N}
\right]
,$$ $$\label{}
\lambda_j^S = \frac{-i}{2} \sum_{l=0}^{N-1}
\left[
e^{i 2 \pi l (j+1)/N}
+
e^{i 2 \pi l (j-1)/N}
\right]
,$$ The above representations of the eigenvalues show that $C$ and $S$ each have only two non-zero eigenvalues corresponding to $j=1,N-1$ given by $\lambda_1^C = \lambda_{N-1}^C = N/2$ and $\lambda_1^S=(\lambda_{N-1}^{S})^* = iN/2$. The Hamiltonian simplifies greatly to
$$\label{eqn:simpleH}
H_I = \frac{\gamma}{2}
-
\left(
\| X_1+iY_1 \|^2 + \| X_{N-1} - iY_{N-1} \|^2
\right).$$
The representation of $H_I$ in Eq. (\[eqn:simpleH\]) must be further modified before the partition function can be found. We do this by splitting the Fourier matrix $U$ into its real and imaginary components, $a_{ik}$ and $b_{ik}$, given by $$\label{}
a_{ik} \equiv \frac{1}{\sqrt{N}} \cos{\left(\frac{2 \pi i k}{ N} \right)},$$ and $$\label{}
b_{ik} \equiv \frac{1}{\sqrt{N}} \sin{\left(\frac{2 \pi i k }{ N} \right)}
.$$ This was done to write the absolute squares in Eq. (\[eqn:simpleH\]) in terms of the squares of $a_{ik}$ and $b_{ik}$. By noticing that $a_{1k} = a_{(N-1)k}$ and $b_{1k} = -b_{(N-1)k}$ we write the configurational partition function as
$$\begin{gathered}
\label{eqn:partition_pre_hubbard}
Z_I = A \int d^N\phi
e^
{
\frac{\beta \gamma}{2}
\left(
\sum_k
\left[
a_{1k} m_k^x - b_{1k} m_k^y
\right]
\right)^2
}
\\
\times
e^
{
\frac{\beta \gamma}{2}
\left(
\sum_k
\left[
b_{1k} m_k^x + a_{1k} m_k^y
\right]
\right)^2
}
,\end{gathered}$$
where $\beta=1/k_B T$.
The Hubbard-Stratonovich transformation is now applied twice, once to each quadratic quantity in the partition function. The integration variables introduced through this transformation are $z_1$ and $z_2$ with subscripts for first and second quadratic quantities, respectively. After after switching the order of integration, we find $$\begin{gathered}
\label{}
Z_I =
\frac{A }{2 \pi \beta \gamma}
\int_{-\infty}^{\infty} dz_1 dz_2
e^
{
-(z^2_1+z^2_2)/ 2 \beta \gamma
}
\prod_k
\\
\times
\int_{-\pi}^{\pi} d \phi_k
e^
{
(z_{1}a_{1k}+z_{2}b_{1k})\cos{\phi_k} + (z_{2}a_{1k}-z_{1}b_{1k})\sin{\phi_k}
}.\end{gathered}$$ The integration can be performed using the identity $$\label{}
\int_{-\pi}^{\pi} d\phi e^{\xi \cos{\phi} + \eta \sin{\phi}} = 2\pi I_0 \left( \sqrt{\xi^2 + \eta^2} \right)$$ where $$\label{}
\xi^2 + \eta^2 = (z_1 a_{1k} + z_2 b_{1k})^2 + (z_2 a_{1k} - z_1 b_{1k})^2$$ which simplifies when $a$ and $b$ are included to $$\begin{gathered}
\label{}
\left[
z_1 \frac{1}{\sqrt{N}}\cos{\left(\frac{2\pi k}{N}\right)}
+
z_2 \frac{1}{\sqrt{N}}\sin{\left(\frac{2\pi k}{N}\right)}
\right]^2
\\
+
\left[
z_2 \frac{1}{\sqrt{N}}\cos{\left(\frac{2\pi k}{N}\right)}
-
z_1 \frac{1}{\sqrt{N}}\sin{\left(\frac{2\pi k}{N}\right)}
\right]^2
\\
=
\frac{1}{N}(z_1^2 + z_2^2)\end{gathered}$$ It is convenient to make a change to polar coordinates by introducing $z=\sqrt{z_1^2 +z_2^2}$, following which the partition function can be written as $$\label{eqn:adZ}
Z_I =
\frac{A }{ \beta \gamma}
\int_{-\infty}^{\infty} dz
e^
{
-z/ 2 \beta \gamma
}
\prod_k
2 \pi I_0
\left(
\frac{\sqrt{z}}{\sqrt{N}}
\right)
.$$ Equation (\[eqn:adZ\]) is recognized to be an intermediate step of the solution to the canonical partition function for the HMF model. From here we jump to the main results, the details of which are included in the HMF literature [@antoni1995; @levin2014; @campa2009] .
The integration over $z$ in Eq. (\[eqn:adZ\]) can be preformed using the saddle point approximation. The rescaled free energy per particle follows as $$\label{eqn:sup}
-\beta F =
% don't have this because there is no more constant in sum in hamiltonian_I \frac{1}{2}\ln{2\pi/\beta}
- \frac{\beta}{2}
+
\inf_z
\left[
\frac{- z^2}{2\beta} + \ln{2\pi I_0(z)}
\right]$$ The expression above permits a convenient path to finding the phase transition. Solving for the minimum values of $z$ in order to satisfy the last term in Eq. (\[eqn:sup\]) results in the equation $$\label{}
\frac{z}{\beta} - \frac{I_1(z)}{I_0(z)} = 0
,$$
which can be solved self consistently for $z$ and represented graphically for different values of $\beta$ as in Fig. \[fig:graphical\_z\_bessel\].
![The solid (black) curve is the fraction of modified Bessel functions $I_1(z)/I_0(z)$, dashed (green) is $z/\beta$ for $\beta=1$, solid (red) line is $z/\beta$ for $beta=2$, dotted (blue) is $z/\beta$ for $\beta=4$, all as a function of $z$. (blue) line is $z/\beta$ for $\beta=4$. The values $\beta=1,2,4$ correspond to the pre-phase transition, critical, and post-phase transition values in that order. []{data-label="fig:graphical_z_bessel"}](graphical_z_bessel.pdf){width="8.0cm"}
The reader will see that after $\beta$ is increased passed the critical value ($\beta=2$) there are two well-defined solutions.
The Hubbard-Stratonovich transformation decouples spin-spin (squared terms in the Hamiltonian) contributions to the partition function at the price of needing to create a linear interaction between each spin with an auxiliary field $z$ [@altland2006]. Again, a more detailed procedure can be found in [@antoni1995; @levin2014; @campa2009] where discussion of the internal energy in the equilibrium state is followed by non-equilibrium behavior of the system prepared in microcanonical ensembles. Here we will simply touch on the most important point of the equilibrium behavior, being that for $\beta<2$ the system is paramagnetic but for $\beta \ge 2$ a pitchfork bifurcation occurs resulting in two stable solutions. At this point there is a discontinuity in the free energy, a second order phase transition occurs and the system can maintain finite magnetization. In this case, the order parameter is the total magnetization $\vec{M}=\frac{1}{N}\sum_{j=1}^{N} \vec{m}_i$ where $\vec{m}_i$ was defined to be $(\cos{\phi_i},\sin{\phi_i})$.
Showing that the canonical partition function in the $\phi$ coordinate frame model and the HMF model are equivalent necessitates a more detailed discussion of the equilibrum behaviors in the $\phi$ frame. Campa et al. [@campa2009], in their review of the HMF model, rigorously show ensemble equivalence between the canonical and microcanoical ensemble of the HMF model. In light of this fact, a large $N$ microcanonical simulation should be able to produce equilibrium behavior like the phase transition mentioned above. The temperature in a numerical simulation of a system with many particles would be “set” through a choice of the initial momenta distribution. In this type of simulation, it is common practice to compute the order parameter and free energy [@dynam_therm_intro; @miritello2009; @antoni1995], begging the question: does a large microcaonical simulation of Eq. (\[eqn:ourH\]) approximate the expected equilibrium behavior? Also, since the index-dependent model in equilibrium with a heat bath can be described by the HMF model, would the dynamics of such a simulation qualitatively resemble those in the HMF model? The answer to both of these questions is *no* if one were to find the equations of motions in $\phi_i$ for some large $N$ and then compare them to an HMF model or the $x$ coordinate frame. As stated, this discrepancy may appear to detract from our result. Indeed, it uncovers a conceptual omission in the model, but it is one whose rectification gives insight into the models ensemble equivalence property of the model, or lack thereof. The omission was in the arbitrary scaling of $x$ which we will now rectify.
We introduce the parameter $L$ which generalizes the scaling in Eq. (\[eqn:scalex\]) to $$\label{eqn:newscalex}
x\rightarrow \frac{2\pi L}{Nd} x
,$$ making the position of the $i^{\mathrm{th}}$ particle $$\label{eqn:newscaledxi}
x_i = \frac{2\pi L i}{N} + \frac{2\pi L}{N}\frac{\ell}{d}\theta_i
.$$ and changing the definition of $\phi_i$ to $\phi_i\equiv 2\pi L\ell\theta_i/Nd$. It can be shown that the introduction of $L$ only changes the final result of the partition function by a constant factor of $L$ due to the enlarged limits of integration. Numerically, we find is that if $L>>1$, then the simulations in $\phi$ closely reproduce the dynamics of HMF model simulations (dynamics in $x$). Therefore, for large $L$ the mirocanonical simulations can approximate equilibrium and the answers to the previous questions - does a large microcaonical simulation of Eq. (\[eqn:ourH\]) approximate the expected equilibrium behavior, and since the index-dependent model in equilibrium with a heat bath can be described by the HMF model, would the dynamics of such a simulation qualitatively resemble those in the HMF model? - becomes *yes*. Alternatively, the coordinate frame inequivelence is most extreme for small $L$. These numerical results were found using initial conditions that are randomly distributed $\phi_i$ about the domain $[-L\pi,L\pi)$. It should be stated that for the rest of this paper we work with $L=1$ becuase we are inetersed in cases where the $\phi$ coordinate frame is markedly differnet than the $x$ coordinate frame.
Non-Equilibrium Results
=======================
For a system of pendula, it is interesting to study an initial configuration where all pendula are set to random small displacements from $\phi_i=0$. Specifically we initialize the $i^{\mathrm{th}}$ pendulum angle, $\phi_i$, randomly in the range $[-\pi/N,\pi/N)$. In $x$ the indices are ordered in $x$ such that $x_1<x_2<x_3<...<x_N$ and the $i^{\mathrm{th}}$ bob is randomly distributed in the range $[2\pi i/N-\pi/N,2\pi i/N+\pi/N)$. It is possible to make some general statements about the dynamics of this configuration in $x$ using the equations of motion. Expressing the Hamiltonian with terms that are quadratic in $\phi$ yields
$$\begin{gathered}
\label{}
H_I = \frac{\gamma}{2N}\sum_{ij}
\Bigg\{
\sin
{
\left[
\frac{2 \pi (i-j)}{N}
\right]
}
(\phi_i - \phi_j)
\\
-\cos
{
\left[
\frac{2 \pi (i-j)}{N}
\right]
}
\left(
1-\frac{\phi_j^2}{2} - \frac{\phi_i^2}{2}+\phi_i \phi_j
\right)
\Bigg\}
.\end{gathered}$$
With this expression, the equations of motion for the $i^{\mathrm{th}}$ particle can be written as
$$\label{eqn:eqn_motion}
\ddot{\phi}_i = \dot{p}_i = \frac{\partial H}{\partial \phi_i},$$
from which we obtain
$$\begin{gathered}
\label{}
\ddot{\phi}_i = \frac{-\gamma}{2N}\sum_{j}
\Bigg\{
\cos{\left[\frac{2\pi(i-j)}{N}\right]}(\phi_j-\phi_i)
\\
+
\sin{\left[ \frac{2\pi (i-j)}{N} \right]}
\Bigg\}
.\end{gathered}$$
In the above equation, the last term and the $\cos{[2\pi(i-j)/N]}\phi_i$ term sum to zero, leading to
$$\label{}
\ddot{\phi}_i = \frac{-\gamma}{2N} \sum_j \cos{\left[\frac{2\pi(i-j)}{N}\right]}\phi_j
.$$
Using the difference formula, we write the acceleration as
$$\label{eqn:initF}
\ddot{\phi}_i = \frac{-\gamma}{2}
\left[
\cos{\left( \frac{2\pi i}{N} \right)} \langle \mu_1 \rangle
+
\sin{\left( \frac{2\pi i}{N} \right)} \langle \mu_2 \rangle
\right]
,$$
where $\mu_1 = \phi_j\cos{(2\pi i/N)}$ and $\mu_2 = \phi_j\sin{(2\pi i/N)}$. The mass (moment of inertia) has been set to unity so the above expression is the force as a function of index, $\ddot{\phi_i} = F(i)$. If $\langle \mu_1
\rangle$ and $\langle \mu_2 \rangle$ are known, then the initial dynamics of the system are elucidated by Eq. (\[eqn:initF\]), but in the case of randomly initialized $\phi_i$ the $\langle \mu_1 \rangle$ and $\langle \mu_2 \rangle$ are also random and can be different from one another in both magnitude and sign. However, a general description of the results can be given without exactly knowing these coefficients. Equation (\[eqn:initF\]) shows that the initial force on a given particle depends on its position because the indices are ordered in $x$. In the continuum (thermodynamic limit), the force takes the form
$$\label{eqn:initFx}
F(x) \equiv \frac{-\gamma}{2}
\left(
\langle \mu_1 \rangle \cos{x} + \langle \mu_2 \rangle \sin{x}
\right)
.$$
Therefore $\langle \mu_1 \rangle$ and $\langle \mu_2 \rangle$ partly play the role of the amplitude of this force as a function of $x$, but also can shift the $\cos{x}+\sin{x}$ spatial dependence, which is periodic over the system length. In Fig. \[fig:initF\],
![Numerical (blue) and theoretical (red) value of the $t=0$ force felt by each particle as a function of its position. This configuration was made with $N=200$ and $\gamma=10$. The initial $\phi_i$ used for the numerical calculation were chosen randomly in the range $[0,2\pi/N)$ which was restricted to positive values so that the sign of the initial $\langle \mu_1 \rangle$ and $\langle \mu_2 \rangle$ were known to be positive. This is not significantly different than when the range of $\phi_i$ is centred about 0. The theoretical curve is fitted using Eq. (\[eqn:initFx\]) with $\langle \mu_1 \rangle =
0.0109$ and $\langle \mu_2 \rangle = 0.0163$. \[fig:initF\]](initF.pdf){width="8.0"}
we show a fit of the force as a function of $x$ using Eq. (\[eqn:initFx\]) as well as the actual force calculated for an example set of initial conditions. The domain in Fig. \[fig:initF\] can be split into two pieces (independent of $\mu$)- one where the particles experience a positive force, the other in which the particles experience a negative force. As time is increased, the movements of the particles evolve the coefficients $\langle \mu_1 \rangle$ and $\langle \mu_2 \rangle$ in such a way that the magnitude of the force decreases to zero for all particles and then switches sign complementary to the original force. This results in a standing compression wave of the particles with a wavelength $2\pi$. The compression wave is not stable and eventually two clusters form about each node. These two clusters are often referred to as a “bicluster”, or the antiferromagnetic state in the HMF model, and have been explained by Barré et al. by analysing the Vlasov equation. They find that the initial compression wave (referred to by a different name) creates an effective double-well potential giving rise to the bicluster [@barre2001]. The question of the bicluster stability has not yet been definitely answered, but for a detailed discussion we refer the reader to Leyvraz et al. [@leyvraz2002]. Given the simple mapping between the $\phi$ and $x$ coordinate frames, we should also be able to show the initial form of the force in $x$ as well. As presented in Eq. (\[eqn:ourH\]), the Hamiltonian in the $x$ coordinate frame only differs from the HMF model by a constant $\gamma/2$. In $x$, $H_I$ is $$\label{}
H_I= \frac{\gamma}{2N}\sum_{i,j=1}^{N}\cos{( x_i - x_j )}.$$ Using the difference identity, we find the equations of motion for the $i^{\mathrm{th}}$ particle to be $$\label{}
\ddot{x}_i
\frac{-\gamma}{2N}
\left(
-\sin{x_i}\sum_j \cos{x_j} + \cos{x_i}\sum_j \sin{x_j}
\right).$$ The sums over cosine and sine of $x_j$ play the same role as $\langle \mu_1 \rangle$ and $\langle \mu_2 \rangle$, and the force at a given position $x_i$ is clearly of the same form as that shown in Eq. (\[eqn:initF\]). Depending on $\langle \mu_1 \rangle$ and $\langle \mu_2 \rangle$, all $\phi_i$ oscillate about zero with amplitudes and phases that depend on their location $x_i$ as discussed above. As the clustering in $x$ begins, the $\phi_i$ begin to spread out over the full domain $[0,2\pi)$ and continue to do so until it is covered. The more interesting case in $\phi$ is when all $\phi_i$ are initially randomly distributed in ranges that depend on their index, specifically when $\phi_i$ are chosen in the ranges. $[2\pi i/N-\pi/N,2\pi
i/N+\pi/N)$ so that $\phi_1<\phi_{2}<\phi_{3}<...<\phi_{N}$. It should be noted that in this new configuration the dynamics in $x$ are nearly identical to the configuration previously discussed for ordered $x_i$. The dynamics in $\phi$ differ *drastically* between the two cases though. In this ordered angle case, we find some interesting grouping of the scaled angles.
Initially the bobs oscillate with an amplitude that depends sinusoidally on their position in $\phi$, similar to the previous discussion in the $x$ picture but with four nodes where the $\phi_i$ remain relatively stationary. Once again this behavior could be thought of as a standing compression wave, but in $\phi_i$ it has a wave length of $\pi$ whereas in the $x$ picture it had a wavelength of $2\pi$. As the system evolves, all $\phi_i$ slowly begin to shift towards the nodes of this standing wave until there are four clusters of the angles. After some time, the angles begin to re-distribute themselves randomly about the domain. The distribution of $\phi_i$ in these three regimes is summarized in three histograms shown in Fig. \[fig:positionhists\].
![These three histograms are made by binning $\phi_i$ of 50 particles over three different periods of time with $\gamma=10$ . Going from top to bottom each period of time belongs to the dynamical regime of: standing “compression wave” in $\phi_i$ from initial configuration of $\phi_i \in [i-2\pi/N,i+2\pi/N)$, clustered motion about the four initial nodes of the compression wave, $\phi_i$ disordered final state. Specifically the values of $t$ are: $t_1 = 0$, $t_2 = 50$, $t_3 = 100$, $t_4 = 200$, $t_5 = 7,000$, $t_6 = 7,200$. \[fig:positionhists\]](pos_hists3.pdf){width="8.0"}
Aside from the number of clusters, there are two primary differences between the clustering in $\phi$ and the clustering in $x$: (i) The clustering in $\phi$ *only* occurs when the angles are ordered in the method described above, whereas the dynamics in $x$ look identical regardless of the distribution in $x$, presuming it is somewhat homogeneous about the domain. (ii) The clustering in $\phi$ is a quasistationary state whereas the clustering in $x$ exists for much longer times regardless of the system size. Since the clustering in $\phi$ is quasistationary, a properly prepared system could exist in the clustered angle state for an arbitrarily long time but only for large $N$. We can view the effect of increasing $N$ and therefore the lifetimes of the clustered states by observing the order of the particle index as a function of time. In Fig. \[fig:index\_color\_phi\](a)-(c),
![ Each particle is colored from blue, representing the smallest index, to red, representing the largest index. The order of the particles indices at a given moment in time is plotted along the horizontal axis. Time increases along the vertical axis.(a) $N=15$. (b) $N=30$. (c) $N=60$. (d) $N=100$, where a smaller range of time is shown in order to see the mixing of the indices as the angles begin to cluster. \[fig:index\_color\_phi\]](index_color_phi.pdf){width="8.0"}
we show that as $N$ is increased, the time it takes for particles to fully mix increases. This is shown by plotting the indices on a color scale from 0 (blue) to $N-1$ (red) along the horizontal axis as time is increased along the vertical axis. In Fig. \[fig:index\_color\_phi\](d), we show how the ordering of the particles changes at the very beginning of clustering for $N=100$. Figure. \[fig:index\_color\_phi\] also shows that the compression wave is not quasistationary since it quickly reduces to the clustered state regardless of $N$.
Conclusion
==========
Though the application of statistical mechanics and thermodynamics to systems with long-range interactions may not always be appropriate, we find that the canonical partition function improves our understanding of a system of pendula with long-range interacting bobs. Solving for the canonical partition function of the Hamiltonian in Eq. (\[eqn:ourH\]), we show that the equilibrium behavior in the $\phi$ coordinate frame is equivalent to the $x$ coordinate frame, i.e. the HMF model. As we have argued that the Hamiltonian in Eq. (\[eqn:ourH\]) describes the behavior of the angles of repulsive or attracting pendulum bobs (see Fig. \[fig:system\]), then the proven equivalence of the canonical partition function of Eq. (\[eqn:ourH\]) and the Hamiltonian mean field model suggests that the Hamiltonian mean field model sufficiently describes the angles of a system of pendula in equilibrium. Ensemble equivalence between the microcanonical ensemble and the canonical ensemble is known for the Hamiltonian mean field model model [@campa2009] and because of this, the microcanonical simulations could be used to approximate equilibrium behavior. We find numerically that in the case of large system lengths, $L$, the dynamics of the system in $\phi$ resemble the dynamics of the Hamiltonian mean field model, equivalently the behavior of the system in $x$. Therefore for large system sizes of long pendula in equilibrium, the HMF model describes their dynamics and statistics.
In this paper we also briefly discuss two particular sets of non-equilibrium results. In one case, the system is initialized with small $\phi_i$ so that $x_i$ are distributed relatively evenly throughout the $x$ domain. This initial configuration essentially gives rise to the “repulsive” low temperature HMF model which exhibits interesting non-equilibrium behavior and is described in great detail by [@dauxois2000; @barre2001]. In the second case, in which $\phi_i$ are ordered by their index $i$, we show there is a compression wave in $\phi$, followed by clustering, and finally a mixed index state displaying no apparent order or structure. This is in contrast to the dynamics produced by a randomly distributed set of initial $\phi_i$ which begins and then remains in a random disordered state. The clustering that can occur in $\phi$ is different from the clustering in $x$ because it only occurs when the angles are initially ordered and because it is quasistationary; the lifetime increases with the number of particles in the system.
We would like to acknowledge the support and generosity of Anand Sharma. This work is partially supported by a contract from NASA (NNX13AD40A).
|
---
abstract: |
We extend earlier work on nonlinear tube wave propagation in permeable formations to study, analytically and numerically, the generation and propagation of a difference frequency, $\Delta
\omega = \omega_1 - \omega_2$, due to an initial pulse consisting of carrier frequencies $\omega_1$ and $\omega_2$. Tube waves in permeable formations have very significant linear dispersion/attenuation, which is specifically addressed here. We find that the difference frequency is predicted to be rather easily measurable with existing techniques and could yield useful information about formation nonlinear properties.
author:
- Yaroslav Tserkovnyak
- David Linton Johnson
title: 'Nonlinear tube waves in permeable formations: Difference frequency generation'
---
Introduction
============
A tube wave is an acoustic normal mode in which the energy is confined to the vicinity of a fluid-filled cylinder within an elastic solid. From a practical point of view it is generally the dominant signal which appears in a typical borehole-logging measurement and thus it is important in a variety of contexts in the search for hydrocarbon sources.
One of these contexts lies in the fact that the tube wave may couple to fluid flow within the rock formation if the latter is permeable. The linearized tube wave propagation in this regime has been extensively studied both theoretically and experimentally [@Winkler89] (see also [@Pampuri98] and references therein). In the present article, we use a model of the tube wave due to [@Liu97]. According to the model, the fluid in the borehole is separated from the porous formation by an elastic membrane (mudcake) of finite thickness. As a tube wave propagates, the membrane flexes in and out of the pores, thus forcing the fluid to flow through the formation. This leads to the coupling between the tube wave and the acoustic slow wave in the formation, which in turn leads to attenuation and dispersion of the tube waves. In formations of moderate to large permeability, this mechanism is the largest known source of attenuation/dispersion of the tube wave and is the reason why it is specifically considered in the present article.
Quite apart from this effect it is also known that sedimentary rocks have very large coefficients of nonlinearity and so [@Johnson94] developed a theory for nonlinear tube waves neglecting the effects of the permeable formation. Later, [@Johnson99] combined this theory with the linearized theory of [@Liu97] to describe a situation when the two effects are simultaneously present. As a numerical demonstration of the theory, [@Johnson99] considered the propagation of a narrow-banded (long duration) pulse consisting of a single carrier frequency. He showed that for realistic system parameters, the main signal (the fundamental) quickly decays, but before completely disappearing it generates a second harmonic and a low-frequency band (the “self-demodulated" pulse) both of which are due to the nonlinearity of the problem. The second harmonic decays even faster than the carrier, with the result that the self-demodulated pulse eventually dominates the entire signal at large enough distances.
Because the second harmonic decays so fast, often it is advantageous to determine nonlinear characteristics by using pulses which consist of two different carrier frequencies, $\omega_1$ and $\omega_2$. In addition to the second harmonics (above) nonlinear effects lead to the generation of a signal centered around the difference frequency $\Delta\omega = \omega_1
- \omega_2$. This component may be reasonably energetic while at the same time it is not attenuated as much as either second harmonic, or even as much as either carrier frequency. Thus, in this article we are motivated to consider the propagation of two narrow-banded pulses whose frequency separation $\Delta\omega$ is, say, 10% of the central frequency. Moreover, because $\Delta
\omega$ is not that different from $\omega_{1,2}$, it is often possible to measure its amplitude with the same acoustic transducers as for the fundamental.
The organization of the article is as follows. First, we review the theory and derive analytical results for the nonlinear propagation of a pulse consisting of two different frequencies. We derive an approximation for the propagation of the entire signal and we find an analytical form for the energy of the band with frequency $\Delta\omega$. Next, we report results of numerical calculations for a few different parameter sets and we show a good agreement between our analytical and our numerical results. In the last section, we make a brief summary of our work.
Theory {#t}
======
The dispersion and attenuation of the linear tube wave propagation has been studied in [@Liu97]. To simplify our discussion we use an approximate form of the dispersion relation from [@Johnson99]: $$k_z(\omega)=\omega\left[S_\infty+\tilde{\Theta}(\omega)\right] ,$$ where $k_z$ is the wave vector along z-direction (the tube axis), $\omega$ is the angular frequency of the wave, $S_\infty$ is the slowness at infinite frequency and $\tilde{\Theta}$ is given by $$\tilde{\Theta}(\omega)=\frac{\rho_f}{S_\infty b \left[W_{\rm mc}+W_p(\omega)\right]} .$$ Here, $\rho_f$ is the density of the borehole fluid, $b$ is the borehole radius, $W_{\rm mc}$ is the mudcake membrane stiffness defined in [@Liu97], and $W_p$ characterizes permeability effects. $W_p$ depends on the borehole fluid viscosity, $\eta$, formation permeability, $\kappa$, and the diffusivity of the slow wave $$C_D=\frac{\kappa K_f^*}{\eta\phi}$$ through the equation $$W_p(\omega)=-\frac{\eta C_D k_{\rm Sl}H_0^{(1)}(k_{\rm
Sl}b)}{\kappa H_1^{(1)} (k_{\rm Sl}b)} .$$ Here, $k_{\rm Sl}=\sqrt{\imath\omega/C_D}$ is the wave vector of the slow-compressional wave and $H_{0,1}^{(1)}$ are Hankel functions.
Tube waves in nonlinear hyperelastic and impermeable formations have been studied in [@Johnson94]. In [@Johnson99] the effects of linear attenuation/dispersion and nonlinearity have been combined to obtain an approximate equation of motion for tube wave propagation in a realistic borehole. In the retarded time frame $\tau=t-S_\infty z$ this equation is $$\frac{\partial p(z,\tau)}{\partial z}+\frac{\partial
F(z,\tau)}{\partial \tau}- \frac{\beta S_\infty^3}{2
\rho_f}\frac{\partial p^2(z,\tau)}{\partial \tau}=0 , \label{ee}$$ where $p$ is the pressure, $\beta$ is a dimensionless parameter defined in [@Johnson99]. The function $F(z, \tau)$ is most simply related to the acoustic pressure in the Fourier transform domain: $$\tilde{F}(z,\omega)=\tilde{\Theta}(\omega)\tilde{p}(z,\omega) .$$ After performing the Fourier transform of Eq. (\[ee\]), one obtains $$\left(\frac{\partial}{\partial z}-\imath
q(\omega)\right)\tilde{p}(z,\omega)+ \frac{\imath\omega \beta
S_\infty^3}{2 \rho_f}\tilde{p^2}(z,\omega)=0 , \label{em}$$ where $$q(\omega)=\omega \tilde{\Theta}(\omega)$$ is the reduced wave number. In Eq. (\[em\]), $\tilde{p^2}(z,\omega)$ is the Fourier transform of the square of $p(z,\tau)$ ([*not*]{} the square of the Fourier transform).
In [@Johnson99], Eq. (\[em\]) has been used to study the generation of the second harmonic as well as that of a low-frequency self-demodulated signal in a situation in which initially the pressure is a narrow band/long duration pulse centered on the frequency, $\omega_1$. In the present article we consider a situation when two narrow-banded pulses are initially present. As discussed in the Introduction, for practical relevance we will take the difference in the pulse frequencies $\Delta\omega$ to be 10% of the central frequency $\omega^\prime$. The initial signal is given by $$p(z=0,\tau)=E_1(\tau)\sin[\omega_1\tau+\phi_1]+E_2(\tau)\sin[\omega_2\tau+\phi_2] .
\label{ic}$$ Here, $E_i(\tau)$ are the envelope functions, $\omega_1\equiv\omega^\prime+\Delta\omega/2$ and $\omega_2\equiv\omega^\prime-\Delta\omega/2$. For the numerical demonstration of the next section, we take the same envelope functions as in [@Johnson99]: $$E_1(\tau)=E_2(\tau)=\frac{1}{2}P_0\exp\left[-(\tau/T_W)^{10}\right] .
\label{is}$$ By setting $\omega_1=\omega_2$ and $\phi_1=\phi_2$ in Eq. (\[ic\]) one recovers the pulse considered in [@Johnson99]. In all three parameter sets used for the numerical calculations in the present article (see Table I), we take $\Delta\omega/\omega^\prime=0.1$ and $\omega^\prime\times
T_W=125\pi$ so that initially the different signals we consider all “look” the same.
There are two characteristic distances relevant to the problem: the decay length of the linearized theory, $Z_{\rm
att}=1/\gamma(\omega^\prime)$, where $$\gamma(\omega)=\omega\Im\left[\tilde{\Theta}(\omega)\right] ,$$ and the distance over which a shock front would develop, in the absence of attenuation. The latter is found in [@Hamilton98] to be $$Z_{\rm shock}=\frac{\rho_f}{\beta S_\infty^3 \omega^\prime P_0} .$$ The Gol’dberg number $$\Gamma=Z_{\rm att}/Z_{\rm shock}$$ measures the importance of nonlinear effects relative to the linear. In the three parameter sets (Table I), the amplitude $P_0$ is chosen so that $\Gamma=0.21$, i.e. the nonlinear effects are significant, but overall pulse propagation is dominated by linear dispersion/attenuation.
For $\Gamma\ll 1$, the nonlinear effects can be ignored to a first approximation. Because, by assumption, the signal consists of two narrow-band pulses one has the usual result of linear acoustics: $$\begin{aligned}
p^\prime(z,\tau)&=&E_1(\tau-\Delta S_{g1}z){\rm e}^{-\gamma_1z}\sin\left[\omega_1(\tau-\Delta S_{p1}z)+\phi_1\right] \nonumber \\
&&+E_2(\tau-\Delta S_{g2}z){\rm e}^{-\gamma_2z}\sin\left[\omega_2(\tau-\Delta S_{p2}z)+\phi_2\right] ,
\label{se}\end{aligned}$$ where $\Delta S_p$ ($\Delta S_g$) is the additional phase (group) slowness relative to $S_\infty$: $$\begin{aligned}
\Delta S_p(\omega)&=&\Re\left[\tilde{\Theta}(\omega)\right] , \nonumber\\
\Delta S_g(\omega)&=&\frac{\rm d}{{\rm d}\omega}\Re\left[\omega\tilde{\Theta}(\omega)\right] .\end{aligned}$$ Throughout this paper, the subscripts 1 and 2 correspond to frequencies $\omega_1$ and $\omega_2$, respectively. The physical meaning of Eq. (\[se\]) is transparent: The pulse envelope propagates with the group velocity and attenuates while each peak and trough travels with the phase velocity. If the envelope is broad enough, the relevant attenuation/dispersion quantities should be evaluated at the central frequency of the pulse.
If the Gol’dberg number $\Gamma$ is small compared to unity but is not completely negligible, Eq. (\[se\]) still gives a reasonable approximation for the propagation of the original pulses but, because of the quadratic nonlinearity in Eq. (\[em\]), new frequency components will be generated which are centered around $\omega=2\omega_1$, $2\omega_2$, $0$ as well as those centered around $\omega_1-\omega_2$ and $\omega_1+\omega_2$. Apart from a direct numerical solution of Eq.(\[em\]) one can develop a perturbation theory thereof:
$$p(z, \tau) = p^\prime(z,\tau) + p_0(z, \tau) + p_\Delta(z, \tau) +
p_{2\omega_1}(z, \tau) + p_{2\omega_2}(z, \tau) + p_{\omega_1 +
\omega_2}(z, \tau) + \cdots \label{pert}$$
Here, $p_0$ refers to the self-demodulated signal, $p_\Delta$ refers to the signal whose bandwidth is centered around $\Delta =
\omega_1 - \omega_2$, etc. By substitution of Eqs. (\[se\]) and (\[pert\]) into Eq.(\[em\]) one can derive an approximate equation for the evolution of these nonlinearly generated components, even in the presence of significant linear dispersion and attenuation.
In this manner the generation of the second harmonic and of the self-demodulated signal has been studied in [@Johnson99]. In the present article we focus on the component centered around the frequency $\Delta=\omega_1-\omega_2$. (The analytical expressions we derive for this mode, can easily be generalized to the $\omega=\omega_1+\omega_2$ case by redefining $\omega_2\rightarrow-\omega_2$.) We obtain ($\Delta\phi\equiv\phi_1-\phi_2$) $$p_\Delta(z,\tau)=E_1(\tau-\Delta S_{g1}z)E_2(\tau-\Delta S_{g2}z)\frac{\Delta\omega\beta S_\infty^3}{2\rho_f}\times\Re\left[{\rm e}^{-\imath(\Delta\omega\tau+\Delta\phi)}\frac{{\rm e}^{\imath(q_1-q_2^*)z}-{\rm e}^{\imath q(\Delta\omega)z}}{q(\Delta\omega)-q_1+q_2^*}\right] .
\label{pd}$$ One is reminded that this equation is valid only for narrow-banded pulses ($\Delta\omega\times T_W\gg 1$) and small nonlinear effects ($\Gamma<1$).
Next, we consider the energy of the carrier ($\omega\approx\omega^\prime$) band, ${\mathcal{E}}^\prime(z)$ as well as that of the band centered on the difference frequency $\omega\approx\Delta\omega$, ${\mathcal{E}}_\Delta(z)$. We define them by $$\begin{aligned}
{\mathcal{E}}^\prime(z)&\equiv&\int_{\omega^\prime/2}^{3\omega^\prime/2}
\left|\tilde{p}(z,\omega)\right|^2{\rm d}\omega , \label{en1}\\
{\mathcal{E}}_\Delta(z)&\equiv&\int_{\Delta\omega/2}^{3\Delta\omega/2}\left|\tilde{p}(z,\omega)\right|^2{\rm
d}\omega . \label{en2}\end{aligned}$$ Using Parseval’s theorem and Eqs. (\[se\]) and (\[pd\]), one gets $$\begin{aligned}
{\mathcal{E}}^\prime(z)/\pi&\approx&\int_{-\infty}^{+\infty}\left|\tilde{p}^\prime(z,\omega)\right|^2{\rm d}\omega \nonumber \\
&=&{\rm e}^{-2\gamma_1z}\int_{-\infty}^{+\infty}E_1^2(\tau){\rm d}\tau+{\rm e}^{-2\gamma_2z}\int_{-\infty}^{+\infty}E_2^2(\tau){\rm d}\tau
\label{ep}\end{aligned}$$ and $$\begin{aligned}
{\mathcal{E}}_\Delta(z)/\pi&\approx&\int_{-\infty}^{+\infty}\left|\tilde{p}_\Delta(z,\omega)\right|^2{\rm d}\omega \nonumber \\
&=&\left(\frac{\Delta\omega\beta S_\infty^3}{2\rho_f}\right)^2\left|\frac{{\rm e}^{\imath(q_1-q_2^*)z}-{\rm e}^{\imath q(\Delta\omega)z}}{q(\Delta\omega)-q_1+q_2^*}\right|^2\times\int_{-\infty}^{+\infty}E_1^2(\tau-\Delta S_{g1}z)E_2^2(\tau-\Delta S_{g2}z){\rm d}\tau .
\label{ed}\end{aligned}$$ In the special case that the two envelope functions are identical, $E_1(\tau)\equiv E_2(\tau)$, one can derive a simple analytic result for the nonlinearly generated signals centered around $\Delta\omega$ as well as the self-demodulated signal centered around $\omega =0$. For this, we rewrite the input signal (\[ic\]) in the form $$p(z=0,\tau)=E^\prime(\tau)\sin[\omega^\prime\tau+\phi^\prime]\nonumber ,
\label{ip}$$ where $\phi^\prime\equiv(\phi_1+\phi_2)/2$ and $$E^\prime(\tau)=2E_1(\tau)\cos\left[(\Delta\omega/2)\tau+\Delta\phi/2\right] .
\label{im}$$ $E^\prime(\tau) $ is now viewed as a narrow-banded envelope function for the $\omega^\prime$ mode (assuming that $\Delta\omega\ll\omega^\prime$). As before, the total signal can be approximated by $$p(z,\tau)\approx p_{[slow]}(z,\tau)+p^\prime(z,\tau) \label{ss}$$ where $p_{[slow]}(z,\tau) \equiv p_0(z,\tau) + p_\Delta(z,\tau)$ now includes both the self-demodulated component as well as the components centered around $\Delta\omega$. Within the context of the foregoing approximations it is given by $$p_{[slow]}(z,\tau)=\frac{\beta
S_\infty^3}{4\rho_f}\int_{-\infty}^{+\infty} \frac{-\imath\omega
\tilde{E^{\prime 2}}(\omega)}{\imath\omega\Delta S_g^\prime-
2\gamma^\prime-\imath q(\omega)}\times\left[{\rm
e}^{(\imath\omega\Delta S_g^\prime- 2\gamma^\prime)z}-{\rm
e}^{\imath q(\omega)z}\right]{\rm e}^{-\imath\omega\tau}{\rm
d}\omega . \label{so}$$ The primed quantities $\Delta S_g^\prime$ and $\gamma^\prime$ are evaluated at the frequency $\omega^\prime$.
Numerical Results {#r}
=================
For the numerical calculations, we consider three parameter sets, {A, B, C}, which are listed in Table I. These parameters are identical to those considered in [@Johnson99], where the relevance of these parameters to realistic borehole properties is discussed. There is one change in that in this article we take $T_W$ to be five times larger. This is done in order to make the band widths of the pulses so narrow that the $\Delta\omega$ mode clearly separates from the rest of the low frequency signal.
In order to solve the equation of motion (\[em\]) we use the same Lax-Wendroff algorithm as in [@Johnson99] and the initial pulse (\[ic\]) with envelope functions (\[is\]) and phases $\phi_1=\phi_2=0$. Figs. (\[a0\]), (\[b0\]), and (\[c0\]) show several snapshots of the pulse profile in both time and frequency domains for three parameter sets (Table I) A, B and C, respectively. Cases B and C have similar initial pulses, but differ somewhat in their dispersion relations, as can be seen from the corresponding parameter sets in Table I. Sample A has a center frequency two orders of magnitude lower than samples B and C, and as a consequence, the relevant modes have much longer attenuation length $Z_{\rm att}$.
It is pedagogically useful to first examine Fig. \[c0\] in detail. Considering the left column of plots, one can trace an intuitively clear sequence of events: Initially, the signal is given by two sharp pulses separated by 10% frequency difference $\Delta\omega$ (first row of plots). As this signal propagates through the borehole and gradually attenuates, modes centered at $\omega\approx\Delta\omega$ are being generated (second row). They soon start to dominate the general shape of the pulse because they attenuate less than does the carrier (third row). Concomitantly, $\omega\approx0$ modes start to appear and give a significant “background” for the $\Delta\omega$ signal (fourth row). In the end, all higher-frequency components decay and the completely self-demodulated signal $\omega\approx0$ contains most of the energy (fifth row). One has to be careful looking at the right column of the plots: The black profile of the first plot outlines an envelope Eq. (\[im\]) of the 10kHz signal, detailed structure of which can be seen only if we expand the time scale $\tau$ (second and third plots). In order to examine the $\Delta\omega\approx100$Hz components and the lower-frequency signal, we again show the entire signal in the last two plots. Although Eq. (\[ss\]) is also plotted, it is indistinguishable from the results of the full numerical calculation. For large enough distance $p(z, \tau)$ evolves to $p_{[slow]}(z,\tau)$, Eq. (\[so\]), as is indicated in the plot.
Qualitatively Fig. \[a0\] and Fig. \[b0\] look similar to Fig. \[c0\]. But in the case of sample A, because of the smaller attenuation, the $\Delta\omega$ component does not demodulate completely even after propagating a distance of $\sim40$km.
Next, for each of the three cases, A, B, and C, we calculate the energy of the $\omega=\omega^\prime$ band directly using Eq. (\[en1\]) and also using Eq. (\[ep\]). Similarly, we calculate the energy of the $\omega=\Delta\omega$ band using Eq. (\[en2\]) as well as Eq. (\[ed\]). The results are plotted in Figs. \[a1\], \[b1\] and \[c1\]. One can compare Fig. \[b1\] with Fig. 6 of [@Johnson99] which shows the energy of the second harmonic and the $\omega=0$ band for an almost monochromatic initial pulse and same system parameters. As can be intuitively expected, the $\Delta\omega$ band is intermediate in both the maximal energy it gains and the distance it propagates, as compared to the lower-frequency band and the higher harmonics. The $\Delta\omega$ component does not reach energies as high as the second harmonics do, but it propagates much further, still carrying a significant fraction of energy; it is $\sim55$dB down from the initial carrier energy, ${\mathcal{E}}^\prime(z=0)$, in our examples. On the other hand, while the lower-frequency signal persists longer than the $\Delta\omega$ modes, it is never as energetic and it would be more difficult to measure even in principle.
The intersections of the solid and dashed curves in Figs. \[a1\], \[b1\], and \[c1\] indicate the crossover region, where the $\omega=\Delta\omega$ band starts dominating over the carrier band $\omega=\omega^\prime$, as can be seen in Figs. \[a0\], \[b0\], and \[c0\], respectively.
Conclusions {#c}
===========
We have used the theory of [@Johnson99] of tube wave propagation in permeable formations to describe nonlinear interaction of two narrow-banded pulses. The theory incorporates both nonlinear effects and a realistic model for dispersion/attenuation of tube waves. We have extended this previous work on the propagation of a single narrow-banded pulse to describe the generation and propagation of the $\omega=\Delta\omega$ band when two different carrier frequencies are present in the initial pulse.
We have derived analytical results for the self-demodulated component, the $\Delta\omega$ band, and the total signal in the regime of weak nonlinearity and they are in excellent agreement with an accurate numerical calculation using three different parameter sets. Specifically, we have studied the spectral content of the signal and demonstrated that the $\Delta\omega$ band can have a potential application because of its long attenuation length and its relatively high energy content. Also, if $\Delta\omega/\omega^\prime\sim 0.1$, as we consider here, there is a practical bonus as the same transducers used for the generation of the carrier signal can, presumably, be used for the detection of the $\Delta\omega$ band.
This work was supported in part by the NSF grant DMR 99-81283.
Hamilton, M. F., and Blackstock, D. T. ([**1998**]{}). [*Nonlinear Acoustics,*]{} (Academic, New York).
Johnson, D. L. ([**1999**]{}). “Nonlinear pulse propagation in arbitrary dispersive media: Tube waves in permeable formations,” J. Acoust. Soc. Am. [**105,**]{} 3087-3096.
Johnson, D. L., Kostek, S., and Norris, A. N. ([**1994**]{}). “Nonlinear tube waves,” J. Acoust. Soc. Am., [**96,**]{} 1829-1843.
Liu, H.-L., and Johnson, D. L. ([**1997**]{}). “Effects of an elastic membrane on tube waves in permeable formations,” J. Acoust. Soc. Am. [**97,**]{} 3322-3329.
Pampuri, F., Rovellini, M., Brie, A., and Fukusima, T., ([**1998**]{}). “Effective evaluation of fluid mobility from Stoneley waves using full Biot model inversion: Two case histories," in SPE Annual Technical Conference and Exhibition, Paper 49132, New Orleans (SPE, Richardson, TX, 1998).
Winkler, K. W., Liu, H.-L., and Johnson, D. L. ([**1989**]{}). “Permeability and borehole Stoneley waves: Comparison between experiment and theory,” Geophysics [**54,**]{} 66-75.
[llcccl]{} Sample & & A & B & C &\
& $\phi$ & 0.30 & 0.30 & 0.30 &\
& $\eta$ & 0.01 & 0.01 & 0.01 & poise\
& $K_f^*$ & 2.25 & 2.25 & 2.25 & GPa\
& $b$ & 0.1 & 0.1 & 0.1 &m\
& $\rho_f$ & 1000. & 1000. & 1000. & kg/m$^3$\
Input & $S_\infty$ & 667. & 667. & 667. & $\mu$sec/m\
parameters & $\beta$ & 50.5 & 50.5 & 50.5 &\
& $W_{\rm mc}$ & 250. & 250. & 100. & GPa/m\
& $\kappa$ & 2. & 0.2 & 0.2 & $\mu$m$^2$\
& $\omega^\prime/2\pi$ & 0.1 & 10. & 10. & kHz\
& $T_W$ & 625. & 6.25 & 6.25 & msec\
& $P_0$ & 81. & 80. & 92. & kPa\
\
\
& $C$ & 15. & 1.5 & 1.5 & m$^2$/sec\
& $Z_{\rm att}$ & 278. & 2.8 & 2.4 & m\
Calculated & $Z_{\rm shock}$ & 1323. & 13.4 & 11.6 & m\
quantities & $\Gamma=Z_{\rm att}/Z_{\rm shock}$ & 0.21 & 0.21 & 0.21 &\
& $\Delta S_p(\omega^\prime)$ & 55.9 & 6.8 & 6.9 & $\mu$sec/m\
& $\Delta S_g(\omega^\prime)$ & 52.7 & 3.5 & 3.5 & $\mu$sec/m\
\[T1\]
|
---
abstract: 'The Five-hundred-meter Aperture Spherical radio Telescope (FAST) passed national acceptance and is taking pilot cycle of ‘Shared-Risk’ observations. The 19-beam receiver covering 1.05-1.45 GHz was used for most of these observations. The electronics gain fluctuation of the system is better than 1% over 3.5 hours, enabling enough stability for observations. Pointing accuracy, aperture efficiency and system temperature are three key parameters of FAST. The measured standard deviation of pointing accuracy is 7.9$''''$, which satisfies the initial design of FAST. When zenith angle is less than 26.4$^\circ$, the aperture efficiency and system temperature around 1.4 GHz are $\sim$ 0.63 and less than 24 K for central beam, respectively. The measured value of these two parameters are better than designed value of 0.6 and 25 K, respectively. The sensitivity and stability of the 19-beam backend are confirmed to satisfy expectation by spectral observations toward N672 and polarization observations toward 3C286. The performance allows FAST to take sensitive observations in various scientific goals, from studies of pulsar to galaxy evolution.'
author:
- 'Peng Jiang, Ning-Yu Tang, Li-Gang Hou, Meng-Ting Liu, Marko Kr$\rm\check{c}$o, Lei Qian, Jing-Hai Sun, Tao-Chung Ching, Bin Liu, Yan Duan, You-Ling Yue, Heng-Qian Gan, Rui Yao, Hui Li, Gao-Feng Pan, Dong-Jun Yu, Hong-Fei Liu, Di Li, Bo Peng, Jun Yan, and FAST Collaboration'
title: 'The Fundamental Performance of FAST with 19-beam Receiver at L Band '
---
Introduction {#sec:introduction}
============
The Five-hundred-meter Aperture Spherical radio Telescope (FAST) with a diameter of 500 m passed its national acceptance on January 11, 2020. It operates at frequencies ranging from 70 to 3000 MHz and is the most sensitive single dish telescope in this frequency range.
FAST commissioning began when construction was completed on September 25, 2016. The details of FAST commissioning have been introduced in Jiang et al. (2019). A lot of observations, especially pulsar related observations were taken during the commissioning phase. A series of primary results including the first detection of a pulsar by FAST have been published in the FAST issue of Science China Physics, Mechanics & Astronomy.
With the effort of the commissioning group, the performance of FAST has been improved significantly during the last two years. One example is the much improved Radio Frequency Interference (RFI) environment. In this paper we will review the current instrumental properties based on measurements with the 19 beam receiver that covers 1.05 to 1.45 GHz.
In section \[sec:19beam\_receiver\], we present the properties of the 19 beam receiver system including calibration of the noise dipole, the spatial distribution of all 19 beams, pointing accuracy, aperture efficiency and system temperature. In section \[sec:backend\], the performance of the spectral backend including stability and sensitivity, standing wave, and polarization is presented. Observation modes and status of RFI are presented in section \[sec:observation\]. Summary is presented in section \[sec:summary\].
Properties of the 19 Beam Receiver {#sec:19beam_receiver}
==================================
Measurement of the Noise Dipole {#subsec:noise_dipole}
-------------------------------
The FAST 19-beam L-band Array contains a temperature stabilized noise injection system. The noise is injected between the feed and the low noise amplifiers. The noise source is a single diode whose signal is split into each beam and polarization. The diode itself is always powered, while it can be switched in and out of the signal path using a solid state switch. This was done in order to improve stability at very high switching speeds. It takes less than 1 microsecond for the noise power to stabilize once switched on or off. The noise diode has two adjustable power output modes, but are currently kept at approximately 1.1, and 12.5K.
In order to test the performance and stability of the noise diode, we conducted a series of hot load measurements whereby the feed cabin was lowered to the ground and a foam absorber was placed directly under the feed so that it completely covered all of the beams. We then periodically measured the temperature of the absorber with a thermometer, while the noise diode was continuously switched on and off (”a winking CAL”) with a period of 1.00663296 seconds. Figure \[lowcal\] and Figure \[highcal\] show the measured noise diode temperatures (Tcal) for both the low, and high power modes with respect to frequency. The diode is tuned such as to minimize the diode temperatures near 1420.4 MHz to reduce the impact on the system temperature.
Figure \[plot3\] shows the electronics gain fluctuations of the system over several hours. This was done assuming that the noise diode temperature is absolutely constant. It shows that the electronics gain fluctuations within the FAST 19-beam receiver/backend signal path are typically on the order of a few percent over timescales of a few minutes. This implies that science projects for whom flux calibration is important should fire the noise diode at least every few minutes in order to account for typical electronics gain fluctuations. The larger fluctuations in Figure 3 are caused by RFI leaking into the signal path during the hot load measurements, and instability in the low noise amplifiers. The electronics gain fluctuations of the rest beams are presented in the Figure \[fig:electrongain\_restbeams\].
![The low power noise diode temperatures of Beam 1, Beam 5, and Beam 6. The 3 beams plotted here are representative of the rest. The data for all 19 beams and polarizations is available and is being used for subsequent calibration.[]{data-label="lowcal"}](lowcal.png){width="100.00000%"}
![The high power noise diode temperatures of Beam 1, Beam 5, and Beam 6. The 3 beams plotted here are representative of the rest. The data for all 19 beams and polarizations is available and is being used for subsequent calibration.[]{data-label="highcal"}](highcal){width="100.00000%"}
The electronics gain fluctuations in Figure \[plot3\] are mostly uncorrelated between the different beams. By taking a median value of all 19 beams, we can produce Figure \[plot4\] which yields an upper limit on the temperature fluctuations in the noise diode. While laboratory measurements of the noise diode itself yielded temperature fluctuations on the order of $\sim$0.1$\%$ over several hours, we see that once placed within the full signal path we get an upper limit of $\sim$1$\%$, implying that it is likely impossible to obtain accurate flux calibration better than $\sim$1$\%$. Considering FAST’s pointing accuracy, and beam size the best flux calibration for a point source is approximately $\sim$2$\%$, meaning that the noise diode is certainly operating within our tolerance limits. This lower limit is estimated by looking at the shape of the beam response function at 1420MHz within the pointing uncertainty limits. The effect is weaker at longer wavelengths where the pointing uncertainty is smaller compared to the beam size.
![The electronics gain fluctuations of the system over several hours. The vertical shows the ratio of fluctuations with respect to the mean temperature over time of Beams 14, 16, and 18. The three beams presented here represent the strongest electronics gain fluctuations out of all 19. []{data-label="plot3"}](plot3.png){width="100.00000%"}
![The noise diode stability. An upper limit on the noise diode stability as a function of time measured by monitoring the median value of the electronics gain in all 19 beams during hot load measurements.[]{data-label="plot4"}](plot4.png){width="100.00000%"}
Beam Properties and Pointing Accuracy {#subsec:beam_pointing}
-------------------------------------
To measure the properties of FAST’s 19-beam receiver, one method is to directly make mapping observations toward radio sources on the sky. The raster scan (or raster scanning) observation mode has been realized for FAST, which is used to map radio sources and measure the 19-beam properties.
### Observations
From November 28 to December 29, 2018, mapping observation have been conducted toward 3C380, 1902+319, 1859+129, 3C454.3, and 2023+318, etc. In the observations, a sky area centering on the target source and covering $\sim37^\prime (\rm RA)\times 37^\prime(\rm Dec)$ is mapped by a raster scan along the RA or Dec direction with the 19-beam receiver. An example of the telescope track for a raster scan along Dec is given in Fig. \[track\]. The scan velocity is 15$^{\prime}$/min. The separation between two sub-scans is 1$^\prime$, which satisfies Nyquist sampling. A total of $\sim$100 minutes is needed to acquire such a map. Observational data are simultaneously recorded with the pulsar mode, the narrow-band spectral-line mode, and also the wide-band spectral-line mode. The number of channels is 4096 for the pulsar backend and 65536 for the spectral-line backend (wideband mode), corresponding to a frequency coverage from 1000 $-$ 1500MHz, which is somewhat broader than the designed bandwidth of the 19-beam receiver (1050 $-$ 1450MHz). The sampling time, i.e., the integration time for raw data, is set to 196.608$\mu$s for the pulsar mode and 1.00663296s for the spectral-line modes. The intensity for each of the 19 beams is calibrated by the periodic injection of a high-intensity noise signal ($\sim$10K, see Sect. \[subsec:noise\_dipole\]) with a period of 0.2s.
![[*Left*]{}: The telescope track for a raster scan along the Dec direction. The covered sky area is $\sim$37$^\prime \times
37^\prime$. The target source (3C454.3) is marked by a red star in the center of the plot. The X- and Y-axes indicate the RA- and Dec-offsets relative to the equatorial coordinates of 3C454.3 (RA: 22h53m57.7479s, Dec: +16d08m53.561s, in J2000), respectively. [ *Right*]{}: distribution map of total intensity obtained by the central beam (M01) near the frequency of 1420MHz. The intensity distributions in a logarithmic-scale are indicated by different colors. The contours show the intensities with levels of 0.25, 0.5, 1.5, 15, 75 and 150 K.[]{data-label="track"}](track.png "fig:"){width="48.00000%"} ![[*Left*]{}: The telescope track for a raster scan along the Dec direction. The covered sky area is $\sim$37$^\prime \times
37^\prime$. The target source (3C454.3) is marked by a red star in the center of the plot. The X- and Y-axes indicate the RA- and Dec-offsets relative to the equatorial coordinates of 3C454.3 (RA: 22h53m57.7479s, Dec: +16d08m53.561s, in J2000), respectively. [ *Right*]{}: distribution map of total intensity obtained by the central beam (M01) near the frequency of 1420MHz. The intensity distributions in a logarithmic-scale are indicated by different colors. The contours show the intensities with levels of 0.25, 0.5, 1.5, 15, 75 and 150 K.[]{data-label="track"}](example.png "fig:"){width="48.00000%"}
### Data processing procedure
A standard pipeline for processing the mapping data of the 19-beam receiver is still under development. In the following analysis the mapping data is processed using IDL. The data recorded by the pulsar backend are adopted, which have dual linear polarizations (XX and YY) and a relatively high time-resolution. We first compress the data raw data in the time dimension to have an identical time-resolution with the periodic injected noise. The antenna temperature $T_a(\nu)$ for each of the polarization is then converted from the raw counts by using the noise diode calibrator. The observed intensity (XX or YY, in K) corresponding to a frequency can be extracted from the spectra after a removing of the radio frequency interference (RFI). Together with the data recording time given by the backend, one series of pairs of $\lbrace t_{obs}, intensity \rbrace$ can be obtained. During the raster scan observation, the phase center of the feed is measured by a real-time measurement system (see Jiang et al. 2019 for a detail), which can be converted to the telescope pointing of the central beam (M01) in the horizontal coordinate system (alt, az). The telescope pointing in the equatorial coordinates (ra, dec in J2000) are then calculated from the horizontal coordinates alt and az after considering the precession, nutation, aberration and also the refraction effect of the atmosphere. With the time information given by the measurement system, we can derive another series of pairs of $\lbrace t_{obs}, coordinates \rbrace$. By cross-matching time between the two series of flow data, a data-cube with information of RA, Dec and intensity is derived. Then the data-cube can be regridded to construct the intensity map.
An example of a total intensity map is given in Fig. \[track\]. The continuum background is calculated and removed to show the sidelobes and the weak field sources more clearly in the plots of logarithmic-scale. We compare it with the continuum map of the same sky area obtained by the NRAO/VLA Sky Survey (NVSS) [^1] at 1.4GHz, and found that the detected positions of four weak radio sources near RA- and Dec-offsets of ($-9^\prime$,+15$^\prime$), ($-$8$^\prime$,+8$^\prime$), (6.5$^\prime$, $-$16.5$^\prime$), and ($-$17$^\prime$,$-$15$^\prime$) are consistent. Then, the background radio sources are fitted by 2-D Gaussian models and removed. Similar procedure of data processing is made to the mapping data of other beams, and examples of the results are given in Fig. \[beam\]. The patterns of beam response shown in Fig. \[beam\] are smilar to the radiation patterns given by simulations (Smith et al. 2016). According to the angular distance of the beam center to that of the central beam M01, $d_{\rm M01}$, the rest 18 elements of the 19-beam receiver are distributed in three concentric ring-shaped area (see Fig. \[19beam\]): 1st-ring) M02-M07, with $d_{\rm M01} \sim
5.8$ arcmin; 2nd-ring) M09, M11, M13, M15, M17, M19, with $d_{\rm M01}
\sim 10.0$ arcmin; 3rd-ring) M08, M10, M12, M14, M16, M18, with $d_{\rm M01} \sim 11.6$ arcmin. As the increase of $d_{\rm M01}$, the sidelobes become more and more significant. The morphology of the sidelobes and their angular distances to the beam center depend on the observation frequency. Other notable features are the noncircular morphology and coma of the main beam, which are negligible for the central beam, but seem to become more and more significant as the increase of $d_{\rm M01}$ for the outer beams.
![[*Upper panels*]{}: distribution maps of total intensity obtained by raster scan observations of beam M01, M06, M17, and M18 near the frequency of 1420MHz. [*Lower panels*]{}: similar to the upper panels, but for the observation frequency near 1060MHz. The target source is 3C454.3. The observation date is December 17, 2018. In each map, the intensity distributions in a logarithmic-scale are indicated by different colors. Contours (black lines) show the intensities with levels of 3, 15, 75 and 150 K. The X- and Y-axises indicate the RA- and Dec-offsets relative to the equatorial coordinates of 3C454.3. A correction factor cos(Dec) is applied for the X-axis in each of the plots.[]{data-label="beam"}](beams.png){width="80.00000%"}
[crrrrrrrrrr]{}
& & & & &\
Scan Direction & & & & &\
& $X_c$ & $Y_c$ & $X_c$ & $Y_c$ & $X_c$ & $Y_c$ & $X_c$ & $Y_c$ & $X_c$ & $Y_c$\
Beam No. & ($^\prime$) & ($^\prime$) & ($^\prime$) & ($^\prime$) & ($^\prime$) & ($^\prime$) & ($^\prime$) & ($^\prime$) & ($^\prime$) & ($^\prime$)\
M01 & 0.00 & 0.00 & $-$0.02 & 0.01 & $-$0.06 & $-$0.05 & $-$0.06 & $-$0.02 & $-$0.04 & $-$0.02\
M02 & 5.81 & 0.03 & 5.81 & 0.03 & 5.69 & $-$0.05 & 5.73 & $-$0.04 & 5.76 & $-$0.01\
M03 & 2.89 & $-$4.97 & 2.94 & $-$4.93 & 2.81 & $-$5.06 & 2.83 & $-$4.99 & 2.86 & $-$4.98\
M04 & $-$2.84 & $-$4.99 & $-$2.81 & $-$5.00 & $-$2.95 & $-$5.03 & $-$2.95 & $-$5.02 & $-$2.89 & $-$5.01\
M05 & $-$5.75 & $-$0.03 & $-$5.74 & $-$0.01 & $-$5.82 & $-$0.04 & $-$5.80 & 0.01 & $-$5.78 & $-$0.02\
M06 & $-$2.90 & 5.00 & $-$2.91 & 5.00 & $-$2.92 & 4.94 & $-$2.94 & 4.97 & $-$2.92 & 4.98\
M07 & 2.89 & 5.01 & 2.85 & 5.03 & 2.83 & 4.90 & 2.83 & 4.95 & 2.85 & 4.97\
M08 & 11.58 & 0.01 & 11.60 & 0.05 & 11.49 & $-$0.06 & 11.53 & $-$0.07 & 11.55 & $-$0.02\
M09 & 8.67 & $-$4.96 & 8.74 & $-$4.93 & 8.56 & $-$5.07 & 8.64 & $-$5.06 & 8.65 & $-$5.01\
M10 & 5.81 & $-$9.99 & 5.89 & $-$9.94 & 5.69 &$-$10.10 & 5.70 &$-$10.04 & 5.78 &$-$10.02\
M11 & 0.08 & $-$9.99 & 0.13 & $-$9.96 & $-$0.09 &$-$10.05 & $-$0.10 &$-$10.06 & 0.00 &$-$10.04\
M12 & $-$5.69 &$-$10.06 & $-$5.65 & $-$10.10 & $-$5.90 &$-$10.07 & $-$5.90 &$-$10.07 & $-$5.78 &$-$10.07\
M13 & $-$8.57 & $-$5.07 & $-$8.60 & $-$5.05 & $-$8.75 & $-$5.05 & $-$8.74 & $-$5.04 & $-$8.67 & $-$5.05\
M14 & — & — &$-$11.58 & $-$0.05 & — & — &$-$11.64 & 0.00 & $-$11.61 & $-$0.02\
M15 & $-$8.67 & 4.93 & $-$8.69 & 5.00 & $-$8.67 & 4.96 & $-$8.72 & 4.99 & $-$8.68 & 4.99\
M16 & $-$5.81 & 10.01 & $-$5.82 & 10.05 & $-$5.81 & 10.00 & $-$5.85 & 10.02 & $-$5.83 & 10.02\
M17 & $-$0.02 & 10.01 & $-$0.02 & 10.03 & $-$0.01 & 9.95 & $-$0.07 & 9.99 & $-$0.03 & 10.00\
M18 & 5.79 & 10.04 & 5.77 & 10.06 & 5.74 & 9.94 & 5.73 & 9.96 & 5.76 & 10.00\
M19 & 8.67 & 5.01 & 8.66 & 5.07 & 8.58 & 4.92 & 8.60 & 4.94 & 8.63 & 4.98\
To fit the main-beam power pattern $P(\theta,\psi)$ with consideration of elliptcity and coma, and determine the combined response of the 19-element multi-beam receiver, we follow the definition of a ’skew Gaussian’ given by Heiles et al. (2004). Nine parameters for the $i$-th beam are the peak intensity $A_0$, beam center $X_c$ and $Y_c$, the average beamwidth $\Theta_{0}$, beam ellipticity $\Theta_{1}$, beam orientation $\phi_{\rm beam}$, coma $\alpha_{\rm coma}$ with orientation $\phi_{\rm coma}$, and a factor $\epsilon$. For the central beam M01, $X_c = (RA_{measured}-RA_{source})cos(Dec)$ and $Y_c
= Dec_{measured} - Dec_{source}$, are the pointing errors of FAST in RA and Dec, respectively. For other beams, the fitting parameter $X_c$ and $Y_c$ are taken as the angular offsets relative to the true center of beam M01. For each beam, the main-beam power pattern is parameterized in the polar coordinates ($\theta$, $\phi$) as: $$P(\theta,\phi) = A_0 exp[-\frac{\theta^2(1-min\lbrace\alpha_{\rm coma}\frac{\theta_{coma}}{\Theta_0},\epsilon\rbrace)}{\Theta^2}],
\label{beamshape}$$ where $\theta_{coma} = \theta cos(\phi - \phi_{coma})$, $\Theta =
\Theta_0 + \Theta_1cos2(\phi-\phi_{beam})$. Here, $\theta$ is the angular distance of the position to the true beam center, $\phi$ is the position angle in the equatorial coordinates (RA, Dec in J2000), and defined to be zero along the positive RA-offset axis and increases towards positive Dec-offset axis. Follow Heiles et al. (2004), we also take $\epsilon$ as a constant parameter 0.75 during the fit. Together with the term min$\lbrace\alpha_{\rm
coma}\frac{\theta_{coma}}{\Theta_0},\epsilon\rbrace$, they are used to prevent the coma term from unduly distorting the beam far from the beam center (Heiles et al. 2004). The MPFIT package[^2] is adopted to make the minimization. In the following, we give a general discussion about the combined beam response, the beamwidth and also the pointing errors of FAST, obtained by fitting the observed total intensity (XX+YY) maps.
![Combined response of the 19-beam receiver of FAST. The target object is 3C454.3. The frequency coverage used to infer the total intensity is from 1050MHz to 1450MHz. The intensity distributions in a logarithmic-scale are indicated by different colors. Contours (black lines) show the intensities with levels of 10, 75, 130 and 175 K. The X- and Y-axes indicate the RA- and Dec-offsets relative to the center of beam M01 (see Table \[beamcen\]). The numbering of the 19 beams (M01-M19) is marked in the plot.[]{data-label="19beam"}](19beams.png){width="90.00000%"}
### The Convolved Beam Pattern of the 19-beam Receiver
To measure the beam properties, we analyzed the mapping data toward 3C454.3, which is the strongest source at 1.4 GHz among the observed $37^\prime\times37^\prime$ targets (1902+319, 1859+129, 3C454.3, and 2023+318, etc) by FAST. Raster scan covering a sky area of $\sim37^\prime\times37^\prime$ were made four times along RA or Dec directions. The fitted parameters of the beam center for each of the 19 beams is given in Table \[beamcen\]. With the fitted beam center, we combine the intensity distribution maps derived by each of the 19 beams to give a composite response of the 19-beam receiver as shown in Fig. \[19beam\]. The observed layout of the 19-beam receiver is consistent with the schematic diagram given by Dunning et al. (2017) and Li et al. (2018). Around the main beams, the responses of first sidelobes for the 12 outer beams (M08-M19) are also visible in the logarithmic-scale plot. Similar results were obtained by analyzing the observation data toward other sources, e.g., 3C380, 2023+318.
It is necessary to emphasize that the intensity distributions given in Fig \[beam\] and Fig. \[19beam\] are the beam response to the point source 3C454.3 verseus the angular offsets. They are the convolution of the beam pattern with the brightness distribution on the sky.
### Beamwidth
[cccccccccccccc]{} Beam No. &\
& 1060 & 1080 & 1100 & 1120 & 1140 & 1300 & 1320 & 1340 & 1360 & 1380 & 1400 & 1420 & 1440\
M01 & 3.44$^\prime$ & 3.37$^\prime$ & 3.31$^\prime$ & 3.29$^\prime$ & 3.29$^\prime$ & 3.01$^\prime$ & 2.98$^\prime$ & 2.93$^\prime$ & 2.89$^\prime$ & 2.85$^\prime$ & 2.82$^\prime$ & 2.82$^\prime$ & 2.82$^\prime$\
M02 & 3.44$^\prime$ & 3.37$^\prime$ & 3.31$^\prime$ & 3.33$^\prime$ & 3.33$^\prime$ & 3.04$^\prime$ & 3.00$^\prime$ & 2.96$^\prime$ & 2.92$^\prime$ & 2.88$^\prime$ & 2.85$^\prime$ & 2.85$^\prime$ & 2.84$^\prime$\
M03 & 3.44$^\prime$ & 3.38$^\prime$ & 3.30$^\prime$ & 3.33$^\prime$ & 3.32$^\prime$ & 3.03$^\prime$ & 3.00$^\prime$ & 2.95$^\prime$ & 2.91$^\prime$ & 2.87$^\prime$ & 2.84$^\prime$ & 2.84$^\prime$ & 2.83$^\prime$\
M04 & 3.44$^\prime$ & 3.38$^\prime$ & 3.31$^\prime$ & 3.34$^\prime$ & 3.32$^\prime$ & 3.05$^\prime$ & 3.01$^\prime$ & 2.97$^\prime$ & 2.93$^\prime$ & 2.90$^\prime$ & 2.86$^\prime$ & 2.86$^\prime$ & 2.85$^\prime$\
M05 & 3.43$^\prime$ & 3.36$^\prime$ & 3.29$^\prime$ & 3.31$^\prime$ & 3.32$^\prime$ & 3.03$^\prime$ & 2.99$^\prime$ & 2.95$^\prime$ & 2.91$^\prime$ & 2.87$^\prime$ & 2.83$^\prime$ & 2.83$^\prime$ & 2.82$^\prime$\
M06 & 3.44$^\prime$ & 3.39$^\prime$ & 3.31$^\prime$ & 3.34$^\prime$ & 3.34$^\prime$ & 3.07$^\prime$ & 3.03$^\prime$ & 3.01$^\prime$ & 3.01$^\prime$ & 2.99$^\prime$ & 2.95$^\prime$ & 2.94$^\prime$ & 2.99$^\prime$\
M07 & 3.45$^\prime$ & 3.39$^\prime$ & 3.31$^\prime$ & 3.33$^\prime$ & 3.34$^\prime$ & 3.02$^\prime$ & 2.99$^\prime$ & 2.95$^\prime$ & 2.92$^\prime$ & 2.88$^\prime$ & 2.86$^\prime$ & 2.86$^\prime$ & 2.86$^\prime$\
M08 & 3.46$^\prime$ & 3.42$^\prime$ & 3.38$^\prime$ & 3.35$^\prime$ & 3.34$^\prime$ & 3.04$^\prime$ & 3.02$^\prime$ & 2.98$^\prime$ & 2.95$^\prime$ & 2.92$^\prime$ & 2.90$^\prime$ & 2.88$^\prime$ & 2.86$^\prime$\
M09 & 3.48$^\prime$ & 3.41$^\prime$ & 3.36$^\prime$ & 3.33$^\prime$ & 3.30$^\prime$ & 3.04$^\prime$ & 3.01$^\prime$ & 2.97$^\prime$ & 2.94$^\prime$ & 2.91$^\prime$ & 2.88$^\prime$ & 2.88$^\prime$ & 2.86$^\prime$\
M10 & 3.52$^\prime$ & 3.46$^\prime$ & 3.40$^\prime$ & 3.38$^\prime$ & 3.35$^\prime$ & 3.08$^\prime$ & 3.05$^\prime$ & 3.03$^\prime$ & 3.00$^\prime$ & 2.97$^\prime$ & 2.95$^\prime$ & 2.94$^\prime$ & 2.92$^\prime$\
M11 & 3.49$^\prime$ & 3.43$^\prime$ & 3.38$^\prime$ & 3.37$^\prime$ & 3.34$^\prime$ & 3.05$^\prime$ & 3.02$^\prime$ & 3.00$^\prime$ & 2.97$^\prime$ & 2.94$^\prime$ & 2.92$^\prime$ & 2.91$^\prime$ & 2.90$^\prime$\
M12 & 3.49$^\prime$ & 3.44$^\prime$ & 3.39$^\prime$ & 3.37$^\prime$ & 3.34$^\prime$ & 3.07$^\prime$ & 3.05$^\prime$ & 3.03$^\prime$ & 3.00$^\prime$ & 2.97$^\prime$ & 2.95$^\prime$ & 2.94$^\prime$ & 2.90$^\prime$\
M13 & 3.48$^\prime$ & 3.41$^\prime$ & 3.34$^\prime$ & 3.33$^\prime$ & 3.32$^\prime$ & 3.04$^\prime$ & 3.01$^\prime$ & 2.98$^\prime$ & 2.95$^\prime$ & 2.93$^\prime$ & 2.91$^\prime$ & 2.91$^\prime$ & 2.89$^\prime$\
M14 & 3.53$^\prime$ & 3.48$^\prime$ & 3.43$^\prime$ & 3.41$^\prime$ & 3.38$^\prime$ & 3.10$^\prime$ & 3.07$^\prime$ & 3.04$^\prime$ & 3.01$^\prime$ & 2.98$^\prime$ & 2.95$^\prime$ & 2.94$^\prime$ & 2.92$^\prime$\
M15 & 3.49$^\prime$ & 3.42$^\prime$ & 3.36$^\prime$ & 3.36$^\prime$ & 3.36$^\prime$ & 3.06$^\prime$ & 3.01$^\prime$ & 2.98$^\prime$ & 2.94$^\prime$ & 2.91$^\prime$ & 2.89$^\prime$ & 2.88$^\prime$ & 2.86$^\prime$\
M16 & 3.51$^\prime$ & 3.46$^\prime$ & 3.41$^\prime$ & 3.39$^\prime$ & 3.34$^\prime$ & 3.10$^\prime$ & 3.07$^\prime$ & 3.05$^\prime$ & 3.02$^\prime$ & 2.99$^\prime$ & 2.97$^\prime$ & 2.95$^\prime$ & 2.93$^\prime$\
M17 & 3.49$^\prime$ & 3.42$^\prime$ & 3.35$^\prime$ & 3.34$^\prime$ & 3.33$^\prime$ & 3.02$^\prime$ & 3.00$^\prime$ & 2.99$^\prime$ & 2.97$^\prime$ & 2.94$^\prime$ & 2.92$^\prime$ & 2.92$^\prime$ & 2.91$^\prime$\
M18 & 3.55$^\prime$ & 3.51$^\prime$ & 3.37$^\prime$ & 3.33$^\prime$ & 3.29$^\prime$ & 3.05$^\prime$ & 3.04$^\prime$ & 3.00$^\prime$ & 2.98$^\prime$ & 2.96$^\prime$ & 2.95$^\prime$ & 2.94$^\prime$ & 2.93$^\prime$\
M19 & 3.42$^\prime$ & 3.41$^\prime$ & 3.36$^\prime$ & 3.33$^\prime$ & 3.34$^\prime$ & 3.03$^\prime$ & 3.00$^\prime$ & 2.98$^\prime$ & 2.95$^\prime$ & 2.93$^\prime$ & 2.91$^\prime$ & 2.90$^\prime$ & 2.89$^\prime$\
The half-power beamwidth (HPBW) is related to the average beamwidth $\Theta_0$ by HPBW $= 2(ln2)^{1/2}\Theta_0$, here, $\Theta_0$ is derived by fitting the total intensity distribution maps with a ’skew Gaussian’ model as shown in Eq. \[beamshape\]. The HPBW as a function of the observation frequency for the 19 beams are given in Fig. \[beamwidth\]. In comparison to the central beam, the outer beams tend to have larger beamwidths, with a difference no more than 0.2 arcmin. The HPBW decreases with the increase of frequency. As discussed in Jiang et al. (2019), the theoretical HPBW of a telescope with diameter $D$ is HPBW $= 1.02\lambda/D$ for a uniformly illuminated aperture and HPBW $= 1.22\lambda/D$ for a cosine-tapered illuminated aperture. Here, a diameter of $D=300$m is assumed. As shown in Fig. \[beamwidth\], the fitted beamwidths of the 19-beam receiver fall within the two theoretical curves of HPBW verseus frequency, but with a less steep slope than that of a standard $\lambda/D$ proportionality.
![Half-power beamwidth (HPBW) verseus observation frequency for the 19-beam receiver of FAST. The target source is 3C454.3. The observation date is December 17, 2018. The beamwidths corresponding to the frequency range of 1160$-$1280MHz can not be well fitted, as the strong RFI appear during the observation.[]{data-label="beamwidth"}](beamwidth.png){width="50.00000%"}
![Distributions of the observed pointing calibrators in the sky coverage of FAST with full gain (zenith angle $\lesssim
26.4^\circ$). The observation period is from February 16 to March 15, 2019. In this plot, the zenith angle is from $0^\circ$ to 26.4$^\circ$ as indicated by grey circles, the azimuth angle is from $0^\circ$ to 360$^\circ$. The directions marked in the periphery of the plot are the geographic north, east, south and west at the FAST site.[]{data-label="pointingdis"}](dis.png){width="80.00000%"}
### Pointing errors
As discussed in Jiang et al. (2019), the analysis of pointing errors can be used to evaluate the pointing accuracy, improve the pointing of the telescope, and also guide the error analysis of observations. Systematic observations have been made to measure the pointing errors of FAST by raster scan observations. As we only care about the pointing errors of the central beam M01, the mapping area toward a target source is set to $\sim7^\prime\times7^\prime$. About 7 minutes is needed to acquire such a map with subscan separation of 1$^\prime$. The target sources are selected from the catalogue of pointing calibrators given by Condon & Yin (2001), which contains 3399 strong and compact radio sources with accurate positions from the NVSS, and uniformly covering the sky north of $\delta = - 40^\circ$ (J2000). The dataset used in the analysis are observed from February 16 to March 15, 2019, which include 126 raster scan observations along RA or Dec directions. As shown in Fig. \[pointingdis\], the observed pointing calibrators distribute widely in the sky coverage of FAST with full gain (zenith angle $\lesssim 26.4^\circ$).
The observation data are processed in a similar procedure as described in Sect.2.2.2. To fit the pointing errors in RA and Dec directions, a 2-D Gaussian model is adopted, which is good enough as the central beam do not present significant beam ellipticity and coma feature (Fig. \[track\] and Fig. \[beam\]). The pointing errors $[(\sigma_{\alpha}cos\delta)^2+(\sigma_\delta)^2]^{1/2}$ are typically smaller than 16$^{\prime\prime}$ as shown in Fig. \[pointingerror\]. The rms of pointing errors is 7.9$^{\prime\prime}$, which is less than one-tenth of the beamwidth near 1450MHz (beamwidth$_{1450}$/10 $\sim$ 2.8$^\prime$/10 = 16.8$^{\prime\prime}$).
![Pointing error $[(\sigma_{\alpha}cos\delta)^2+(\sigma_\delta)^2]^{1/2}$ verseus zenith angle for the 19-beam receiver of FAST. []{data-label="pointingerror"}](pointingerrors.png){width="50.00000%"}
Aperture Efficiency and System Temperature {#subsec:tsys_eff}
------------------------------------------
Aperture efficiency ($\eta$) and system temperature (T$\rm_{sys}$) are two basic parameters of FAST. Both of them are expected to vary as a function of zenith angle ($\rm\theta_{ZA}$) and frequency. The 19 beams are divided into 4 categories according to separation to the central beam. The 4 categories are : 1) Beam 1; 2) Beam 2, 3, 4, 5, 6, and 7; 3) Beam 8, 10, 12, 14, 16, and 18; 4) Beam 9, 11, 13, 15, 17 and 19. Though the performance of aperture efficiency and system temperature in each category is expected to be same, asymmetry of the receiver platform may lead to deviation. Thus we obtained the measurements toward all 19 beams.
The aperture efficiency curve was obtained by repeating observations of the calibrator 3C286 and its off position at different zenith angles. The position switch mode that is introduced in section \[sec:observation\] was adopted. This mode provides quick switch between ON and OFF source position. The separation between ON and OFF position was selected to allow for measuring aperture efficiency curve of two beams (e.g, Beam 1 and 19) simultaneously, which increases measurement efficiency.
Observations of Beam 1, 2, 8, and 19 were taken during August 7, 2019 and August 24, 2019. To make it clear, we take the aperture efficiency measurement of Beam 1 and 19 as an example. At first, the telescope tracked the calibrator 3C286 with Beam 1 (source ON for Beam 1, source OFF position for Beam 19) for 90 s, and then switched to the sky position with offset of (-8.85$^{\prime}$, -5.11$^{\prime}$). At this position, the calibrator 3C286 was OFF for Beam 1 but lay in Beam 19. After tracking for 90s, the telescope switched back to track 3C286 with Beam 1. A cycle consists of one ON and one OFF phase for each beam. A total of 90 cycles were taken during available tracking time of $\sim$ 6 hours of 3C286. Supplementary measurements of the aperture efficiency curve for the rest 15 beams were taken by tracking 3C286 during December 19, 2019 and December 25, 2019. During these observations, the total tracking time in a cycle was reduced into 60 s, including 30 s for both source on and source off. A total of 85 cycles and $\sim $ 3 hours were taken for measurement of both two beams.
The system temperature curve of 19 beams was obtained by continuously tracking a clean sky position ($\alpha_{2000}$=23$^{h}$30$^m$0$^s$.0, $\delta_{2000}$=25$^{\circ}$39$^{\prime}$10.6") from rise to set on September 14, 2019.
In the above measurements, spectral backend with sampling time of 1.00663296 s and channel width of 7.62939 kHz was adopted to record the data. The high noise signal ($\sim 11$ K) was injected at a synchronized period of 2.01326592 s. The duration time of cal on and cal off is 1.00663296 s. During calculation, channel width of the data were smoothed to 1 MHz. The aperture efficiency was derived for each observation cycle while the system temperature was derived for each sampling time.
### Aperture efficiency {#subsubsec:eta}
Based on absolute measurement of noise dipole in Section \[subsec:noise\_dipole\], the observed data were calibrated into antenna temperature $T_A$ in K . The antenna temperature of 3C286, $T_A^{3C286}$ was derived by the difference between source ON and source OFF data for each beam.
3C286 is a stable flux calibrator. The frequency dependency of the spectral flux density of 3C286 could be fitted with a polynomial function (Perley & Butler 2017),
$$log(S) = a_0 + a_1 log(\nu_G)+a_2 [log(\nu_G)]^2+a_3 [log(\nu_G)]^3,
\label{eq:polyfit}$$
where S and $\nu_G$ are the spectral flux density in Jy and the frequency respectively. We adopted the value of $a_0=1.2481$, $a_1=-0.4507$, $a_2=-0.1798$ and $a_3=0.0357$ that is valid for 3C286 at frequency range of \[0.05-50\] GHz (Perley & Butler 2017).
Measured gain ($G$) of FAST is expressed as $G_0=A_\textrm{eff}/2k$ , in which $A_\textrm{eff}$ is effective illumination area. In observation, $G$ is calculated by ratio between antenna temperature and flux density, $G = T_A^{3C286}/S_A^{3C286}$. The prefect gain $G_0$ of FAST is $G_0=A_\textrm{geo}/2k$ for single polarization, in which $A_\textrm{geo}$ and $k$ are geometric illumination area and Boltzmann constant respectively. The geometric illumination area with diameter of 300 m leads to $G_0$= 25.6 K/Jy for FAST.
The aperture efficiency $\eta$ of FAST is derived by,
$$\eta = G/G_0 = A_\textrm{eff}/A_\textrm{geo} = \frac{2k T_A^{3C286}}{A_\textrm{geo} S_A^{3C286}}$$
For the FAST system, $\eta$ is contributed by 6 main components:
1. $\eta\rm_{sf}$, the reflection efficiency of main reflector. It is described by $Ruze$ equation, $\eta_{sf}=e^{-(\frac{4\pi\varepsilon}{\lambda})^2}$, in which $\varepsilon$ and $\lambda$ are root mean square (RMS) of the surface error and observational wavelength, respectively. The RMS of the surface error is contributed by controlling accuracy of panels, which is $\sim 5$ mm at 21 cm (Jiang et al. 2019). This results in $\eta_{sf}$ of $\sim$ 91%.
2. $\eta\rm_{bl}$, the efficiency when the shielding of the feed cabin is considered. The diameter of feed cabin of $\sim 10$ m leads to $\eta\rm_{bl}$ value of 99.9%.
3. $\eta\rm_{s}$, spillover efficiency.
4. $\eta\rm_t$, illumination efficiency of the feed. It equals 76% when a -13 dB Gaussian illumination is adopted (Jiang et al. 2019).
5. $\eta\rm_{misc}$, efficiency of other aspects including the offset of the feed phase, matching loss of feed.
6. $\eta\rm_{sloss}$, the percent of effective surface area compared to a 300-m diameter paraboloid. The value is 1 when $\rm\theta_{ZA}$ is less than 26.4$^\circ$. When $\rm\theta_{ZA}$ is greater than 26.4$^\circ$, the reflection area decreases. $\eta\rm_{sloss}$ would decrease and reach $\sim$ 2/3 when $\rm\theta_{ZA}= 40^\circ$.
The aperture efficiency is a synthetical result of above effects and is described by the following equation,
$$\eta = \rm \eta_{sf}\cdot \eta_{bl} \cdot \eta_{s}\cdot \eta_t \cdot \eta_{misc} \cdot \eta_{sloss}
\label{eq:efficiency}$$
The value of $\eta\rm_{sf}$,$\eta\rm_{bl}$,$\eta\rm_{s}$,$\eta\rm_t$,and $\eta\rm_{misc}$ are almost constant. The value of $\eta$ is mainly determined by $\eta\rm_{sloss}$. As shown in Fig. \[fig:etafit\], the value $\eta$ at all frequency would keep almost constant when $\rm\theta_{ZA}$ is less than 26.4$^\circ$ and decreases linearly when $\rm\theta_{ZA}$>26.4$^\circ$. This is consistent with the fact that FAST would lose part of reflection panels when $\rm\theta_{ZA}$>26.4$^\circ$.
We fitted variation of $\eta$ as a function of $\rm\theta_{ZA}$ at specific frequency with two linear equations. The formula is shown as follows,
$$\eta = \left\{
\begin{array}{lr}
a\rm\ \theta_{ZA}+b,\ 0^\circ \leq \theta_{ZA} \leq 26.4^\circ \\
c\rm\ \theta_{ZA}+d,\ 26.4^\circ < \theta_{ZA} \leq 40^\circ.
\end{array}
\right.$$
In which $d$ satisfies the equation $d=b+26.4(a-c)$. The fitting result of parameter $a$,$b$ and $c$ for different frequencies are shown in Table \[table:etapara\]. As an example, the fitting results of $\eta$ for Beam 1, 2, 8 and 19 are shown in Fig. \[fig:etafit\]. The fitting results of rest 15 beams are shown in Fig. \[fig:etafit\_restbeams\].
The gain of FAST could be expressed with $G= \eta G_0$, in which $G_0$ = 25.6 K/Jy. Flux density (in unit of Jy) of point source is converted from antenna temperature by dividing the gain value. The averaged gain values of 19 beams within ZA of 26.4 $^{\circ}$ are shown in Table \[table:gainvalue\]. The Beam 16 has smallest gain value among 19 beams. The gain ratio of Beam 16 compared to Beam 1 reaches $0.83\pm 0.02$ at 1400 and 1450 MHz.
### System Temperature {#subsubsec:tsys}
Observation of sky position ($\alpha_{2000}$=23$^{h}$30$^m$0$^s$.0, $\delta_{2000}$=25$^{\circ}$39$^{\prime}$10.6") allows us to measure system temperature from zenith angle of 40$^{\circ}$ to nearly $0^{\circ}$. The data were calibrated into antenna temperature in K with noise curve. For a single dish telescope at L band, T$\rm_{sys}$ consists four main sources (Campbell et al. 2002):
1. Noise contribution from receiver, $T\rm_{rec}$. For 19 beam receiver of FAST, it is measured as 7-9 K, including $\sim$ 4 K from low noise amplifier (LAN) (see section 4.1.1 in Jiang et al. 2019 for details).
2. Continuum brightness temperature of the sky, $T\rm_{sky}$. This includes 2.73 K from cosmic microwave background (CMB) and non-thermal emission from the Milky Way. The value of $T\rm_{sky}$ at 1.4 GHz is $\sim 3.48$ K toward position ($\alpha_{2000}$=23$^{h}$30$^m$0$^s$.0, $\delta_{2000}$=25$^{\circ}$39$^{\prime}$10.6") (CHIPASS survey; Calabretta et al. 2014).
3. Emission from the Earth’s atmosphere $T\rm_{atm}$. This value should be in a few K.
4. Radiation from the surrounding terrain, $T_{scat}$. This contribution originates from side lobe of FAST. Unlike other single dish with fixed surface and horn, the illumination area of horn varies for different zenith angle, leading to different value of $T_{scat}$.
System temperature are synthetical result of the above components. $$\rm T_{sys}=T_{rec}+T_{sky}+T_{atm}+T_{scat}
\label{eq:tsys}$$
The variation of system temperature $T\rm_{sys}$ as a function of zenith angle could be fitted with the following formula,
$$T\rm_{sys} = P_0\arctan(\sqrt{1+{\theta_{ZA}}^n}-P_1)+P_2,
\label{eq:tsysfit}$$
which is valid for $0^\circ \leq \theta\rm_{ZA} <= 40^\circ$. The value of $P_0$, $P_1$, $P_2$ and $n$ in different frequencies are shown in Table \[table:tsys\]. An example of fitting $T\rm_{sys}$ curve is shown in Fig. \[fig:tsysfit\]. The results for the rest beams are shown in Fig. \[fig:tsysfit\_restbeams\].
A significant feature in the $T\rm_{sys}$ profile is that the $T\rm_{sys}$ value of most beams reaches its minimum at ZA range of \[10$^{\circ}$,15$^{\circ}$\]. When ZA is smaller than 10$^{\circ}$, the $T\rm_{sys}$ value would increase by a maximum value of 1 K. Though central hole of spherical surface was shielded with metal mesh during observations, the leakage of ground emission ($\sim 300$ K) from the central hole goes into the main beam and contributes $T\rm_{sys}$ increase when ZA is less than 10$^{\circ}$. When ZA gets larger than 15$^{\circ}$, the $T\rm_{sys}$ value increases by 5-7 K, which arises from emission of nearby mountain through sidelobe.
### Discussion {#subsubsec:discuss}
The observations lie in $\theta\rm_{ZA}$ range of \[4.9,40\] deg for $\eta$ curve. The fitting results are expected to be valid for $\theta\rm_{ZA}$ range of \[0,40\] degree for the following reasons. When $\theta\rm_{ZA}<4.9^{\circ}$, there is no loss of panel. In this case, $\eta$ should keep almost constant as shown in the fitting curves.
[ccccccccccc]{} & &\
& & & & & & & & & &\
M01 & a/1e-4 & 3.31 $\pm$ 2.23 & 2.54 $\pm$ 1.85 & 0.34 $\pm$ 1.83 & 5.03 $\pm$ 2.03 & 4.94 $\pm$ 2.02 & 9.83 $\pm$ 1.85 & 8.50 $\pm$ 1.93 & 7.87 $\pm$ 1.95 & 10.64 $\pm$ 1.90\
M01 & b/1e-1 & 6.19 $\pm$ 0.04 & 6.32 $\pm$ 0.03 & 6.43 $\pm$ 0.03 & 6.22 $\pm$ 0.03 & 6.22 $\pm$ 0.03 & 6.08 $\pm$ 0.03 & 6.12 $\pm$ 0.03 & 6.14 $\pm$ 0.03 & 5.98 $\pm$ 0.03\
M01 & c/1e-2 & -1.58 $\pm$ 0.02 & -1.61 $\pm$ 0.03 & -1.37 $\pm$ 0.04 & -1.40 $\pm$ 0.04 & -1.42 $\pm$ 0.03 & -1.38 $\pm$ 0.03 & -1.40 $\pm$ 0.02 & -1.34 $\pm$ 0.02 & -1.26 $\pm$ 0.02\
M02 & a/1e-4 & 9.46 $\pm$ 3.53 & 6.28 $\pm$ 3.09 & 9.01 $\pm$ 3.14 & 12.13 $\pm$ 3.12 & 10.98 $\pm$ 3.06 & 11.31 $\pm$ 2.16 & 8.71 $\pm$ 2.09 & 8.78 $\pm$ 2.06 & 13.07 $\pm$ 1.98\
M02 & b/1e-1 & 5.97 $\pm$ 0.06 & 5.97 $\pm$ 0.05 & 5.99 $\pm$ 0.05 & 5.81 $\pm$ 0.05 & 5.81 $\pm$ 0.05 & 5.65 $\pm$ 0.04 & 5.75 $\pm$ 0.03 & 5.75 $\pm$ 0.03 & 5.62 $\pm$ 0.03\
M02 & c/1e-2 & -1.54 $\pm$ 0.03 & -1.62 $\pm$ 0.03 & -1.46 $\pm$ 0.03 & -1.44 $\pm$ 0.02 & -1.42 $\pm$ 0.02 & -1.29 $\pm$ 0.02 & -1.32 $\pm$ 0.02 & -1.33 $\pm$ 0.02 & -1.21 $\pm$ 0.02\
M03 & a/1e-4 & 1.79 $\pm$ 4.21 & 4.39 $\pm$ 3.55 & 2.41 $\pm$ 4.46 & 5.95 $\pm$ 4.69 & 6.90 $\pm$ 3.21 & 7.41 $\pm$ 2.60 & 9.99 $\pm$ 2.63 & 7.66 $\pm$ 2.60 & 10.30 $\pm$ 2.72\
M03 & b/1e-1 & 6.41 $\pm$ 0.07 & 6.24 $\pm$ 0.06 & 6.42 $\pm$ 0.07 & 6.21 $\pm$ 0.07 & 6.17 $\pm$ 0.05 & 6.07 $\pm$ 0.04 & 5.91 $\pm$ 0.04 & 5.99 $\pm$ 0.04 & 5.86 $\pm$ 0.04\
M03 & c/1e-2 & -1.70 $\pm$ 0.03 & -1.80 $\pm$ 0.03 & -1.56 $\pm$ 0.03 & -1.63 $\pm$ 0.02 & -1.59 $\pm$ 0.02 & -1.56 $\pm$ 0.02 & -1.53 $\pm$ 0.02 & -1.47 $\pm$ 0.02 & -1.35 $\pm$ 0.03\
M04 & a/1e-4 & 3.57 $\pm$ 4.76 & 1.87 $\pm$ 4.08 & 3.82 $\pm$ 3.93 & 3.90 $\pm$ 4.60 & 8.46 $\pm$ 4.65 & 4.29 $\pm$ 3.22 & 2.75 $\pm$ 3.28 & 1.85 $\pm$ 3.26 & 5.10 $\pm$ 3.20\
M04 & b/1e-1 & 6.24 $\pm$ 0.08 & 6.15 $\pm$ 0.07 & 6.25 $\pm$ 0.06 & 6.05 $\pm$ 0.08 & 5.98 $\pm$ 0.08 & 5.92 $\pm$ 0.05 & 5.89 $\pm$ 0.05 & 5.89 $\pm$ 0.05 & 5.75 $\pm$ 0.05\
M04 & c/1e-2 & -1.80 $\pm$ 0.03 & -1.82 $\pm$ 0.03 & -1.65 $\pm$ 0.02 & -1.57 $\pm$ 0.06 & -1.58 $\pm$ 0.02 & -1.42 $\pm$ 0.02 & -1.45 $\pm$ 0.02 & -1.36 $\pm$ 0.02 & -1.24 $\pm$ 0.02\
M05 & a/1e-4 & -3.97 $\pm$ 2.87 & -4.25 $\pm$ 2.72 & -4.44 $\pm$ 2.72 & -7.54 $\pm$ 3.48 & -4.09 $\pm$ 4.19 & -5.41 $\pm$ 2.35 & -3.33 $\pm$ 2.26 & -4.51 $\pm$ 2.42 & -4.58 $\pm$ 2.53\
M05 & b/1e-1 & 6.45 $\pm$ 0.05 & 6.48 $\pm$ 0.04 & 6.68 $\pm$ 0.04 & 6.42 $\pm$ 0.06 & 6.39 $\pm$ 0.07 & 6.26 $\pm$ 0.04 & 6.10 $\pm$ 0.04 & 6.17 $\pm$ 0.04 & 6.11 $\pm$ 0.04\
M05 & c/1e-2 & -1.82 $\pm$ 0.03 & -1.86 $\pm$ 0.03 & -1.72 $\pm$ 0.03 & -1.56 $\pm$ 0.03 & -1.60 $\pm$ 0.03 & -1.53 $\pm$ 0.02 & -1.45 $\pm$ 0.03 & -1.44 $\pm$ 0.03 & -1.30 $\pm$ 0.03\
M06 & a/1e-4 & 1.69 $\pm$ 2.84 & 0.74 $\pm$ 2.59 & -0.00 $\pm$ 4.17 & 1.06 $\pm$ 5.18 & 0.97 $\pm$ 3.49 & 0.33 $\pm$ 1.82 & 3.11 $\pm$ 1.80 & -0.03 $\pm$ 1.76 & 1.37 $\pm$ 1.76\
M06 & b/1e-1 & 6.16 $\pm$ 0.04 & 6.22 $\pm$ 0.04 & 6.33 $\pm$ 0.07 & 6.01 $\pm$ 0.08 & 6.04 $\pm$ 0.06 & 5.92 $\pm$ 0.03 & 5.83 $\pm$ 0.03 & 5.89 $\pm$ 0.03 & 5.75 $\pm$ 0.03\
M06 & c/1e-2 & -1.74 $\pm$ 0.03 & -1.80 $\pm$ 0.03 & -1.60 $\pm$ 0.03 & -1.50 $\pm$ 0.04 & -1.44 $\pm$ 0.03 & -1.33 $\pm$ 0.02 & -1.38 $\pm$ 0.02 & -1.35 $\pm$ 0.02 & -1.22 $\pm$ 0.02\
M07 & a/1e-4 & 1.71 $\pm$ 2.43 & 0.08 $\pm$ 2.18 & 0.77 $\pm$ 2.41 & 2.03 $\pm$ 3.25 & 2.24 $\pm$ 3.64 & 3.46 $\pm$ 1.82 & 5.46 $\pm$ 1.60 & 6.70 $\pm$ 1.65 & 6.91 $\pm$ 1.49\
M07 & b/1e-1 & 6.10 $\pm$ 0.04 & 6.04 $\pm$ 0.04 & 6.18 $\pm$ 0.04 & 5.90 $\pm$ 0.05 & 6.02 $\pm$ 0.06 & 5.86 $\pm$ 0.03 & 5.77 $\pm$ 0.03 & 5.73 $\pm$ 0.03 & 5.62 $\pm$ 0.02\
M07 & c/1e-2 & -1.68 $\pm$ 0.02 & -1.64 $\pm$ 0.02 & -1.52 $\pm$ 0.02 & -1.39 $\pm$ 0.02 & -1.43 $\pm$ 0.02 & -1.32 $\pm$ 0.02 & -1.34 $\pm$ 0.02 & -1.36 $\pm$ 0.02 & -1.22 $\pm$ 0.02\
M08 & a/1e-4 & 13.91 $\pm$ 3.13 & 17.00 $\pm$ 2.98 & 14.04 $\pm$ 3.81 & 16.81 $\pm$ 4.98 & 17.71 $\pm$ 2.95 & 17.29 $\pm$ 2.75 & 16.95 $\pm$ 2.29 & 15.92 $\pm$ 2.31 & 17.76 $\pm$ 2.43\
M08 & b/1e-1 & 5.76 $\pm$ 0.05 & 5.75 $\pm$ 0.05 & 5.81 $\pm$ 0.06 & 5.62 $\pm$ 0.08 & 5.54 $\pm$ 0.05 & 5.46 $\pm$ 0.04 & 5.41 $\pm$ 0.04 & 5.34 $\pm$ 0.04 & 5.22 $\pm$ 0.04\
M08 & c/1e-2 & -1.60 $\pm$ 0.06 & -1.70 $\pm$ 0.05 & -1.52 $\pm$ 0.06 & -1.50 $\pm$ 0.06 & -1.50 $\pm$ 0.05 & -1.47 $\pm$ 0.05 & -1.45 $\pm$ 0.05 & -1.37 $\pm$ 0.05 & -1.28 $\pm$ 0.05\
M09 & a/1e-4 & -1.26 $\pm$ 2.03 & -0.43 $\pm$ 1.99 & -2.68 $\pm$ 1.94 & 0.12 $\pm$ 1.67 & 0.96 $\pm$ 1.60 & 1.53 $\pm$ 1.53 & 1.09 $\pm$ 1.36 & 1.00 $\pm$ 1.11 & 0.27 $\pm$ 0.91\
M09 & b/1e-1 & 5.96 $\pm$ 0.03 & 5.95 $\pm$ 0.03 & 5.98 $\pm$ 0.03 & 5.75 $\pm$ 0.03 & 5.77 $\pm$ 0.03 & 5.74 $\pm$ 0.02 & 5.64 $\pm$ 0.02 & 5.57 $\pm$ 0.02 & 5.53 $\pm$ 0.02\
M09 & c/1e-2 & -1.39 $\pm$ 0.04 & -1.43 $\pm$ 0.03 & -1.28 $\pm$ 0.04 & -1.24 $\pm$ 0.04 & -1.24 $\pm$ 0.03 & -1.18 $\pm$ 0.04 & -1.09 $\pm$ 0.04 & -1.06 $\pm$ 0.03 & -1.01 $\pm$ 0.03\
M10 & a/1e-4 & 14.36 $\pm$ 3.80 & 15.37 $\pm$ 3.39 & 15.65 $\pm$ 3.78 & 15.20 $\pm$ 4.85 & 18.06 $\pm$ 2.92 & 17.97 $\pm$ 2.67 & 19.34 $\pm$ 2.49 & 16.72 $\pm$ 2.52 & 18.57 $\pm$ 2.65\
M10 & b/1e-1 & 6.01 $\pm$ 0.06 & 5.98 $\pm$ 0.05 & 6.04 $\pm$ 0.06 & 5.81 $\pm$ 0.08 & 5.86 $\pm$ 0.05 & 5.75 $\pm$ 0.04 & 5.72 $\pm$ 0.04 & 5.69 $\pm$ 0.04 & 5.60 $\pm$ 0.04\
M10 & c/1e-2 & -1.79 $\pm$ 0.04 & -1.83 $\pm$ 0.04 & -1.74 $\pm$ 0.03 & -1.73 $\pm$ 0.03 & -1.78 $\pm$ 0.03 & -1.74 $\pm$ 0.03 & -1.72 $\pm$ 0.03 & -1.66 $\pm$ 0.03 & -1.58 $\pm$ 0.03\
M11 & a/1e-4 & 7.68 $\pm$ 5.51 & 2.98 $\pm$ 4.73 & 4.16 $\pm$ 4.57 & 3.38 $\pm$ 6.44 & 2.45 $\pm$ 4.86 & 4.27 $\pm$ 3.39 & 5.99 $\pm$ 3.36 & 5.03 $\pm$ 3.28 & 8.23 $\pm$ 3.05\
M11 & b/1e-1 & 5.90 $\pm$ 0.09 & 5.94 $\pm$ 0.08 & 6.00 $\pm$ 0.07 & 5.81 $\pm$ 0.10 & 5.80 $\pm$ 0.08 & 5.67 $\pm$ 0.06 & 5.66 $\pm$ 0.05 & 5.61 $\pm$ 0.05 & 5.45 $\pm$ 0.05\
M11 & c/1e-2 & -1.59 $\pm$ 0.04 & -1.59 $\pm$ 0.03 & -1.55 $\pm$ 0.03 & -1.35 $\pm$ 0.05 & -1.36 $\pm$ 0.03 & -1.31 $\pm$ 0.03 & -1.35 $\pm$ 0.02 & -1.27 $\pm$ 0.02 & -1.18 $\pm$ 0.02\
M12 & a/1e-4 & -3.00 $\pm$ 5.82 & -2.08 $\pm$ 4.75 & 1.33 $\pm$ 4.29 & 4.98 $\pm$ 5.10 & 8.50 $\pm$ 4.02 & -0.98 $\pm$ 3.17 & 2.94 $\pm$ 3.06 & 2.49 $\pm$ 2.71 & 2.77 $\pm$ 2.60\
M12 & b/1e-1 & 6.06 $\pm$ 0.09 & 5.96 $\pm$ 0.08 & 5.91 $\pm$ 0.07 & 5.82 $\pm$ 0.08 & 5.76 $\pm$ 0.06 & 5.70 $\pm$ 0.05 & 5.58 $\pm$ 0.05 & 5.53 $\pm$ 0.04 & 5.48 $\pm$ 0.04\
M12 & c/1e-2 & -1.56 $\pm$ 0.05 & -1.49 $\pm$ 0.04 & -1.45 $\pm$ 0.05 & -1.56 $\pm$ 0.05 & -1.69 $\pm$ 0.03 & -1.35 $\pm$ 0.04 & -1.40 $\pm$ 0.03 & -1.34 $\pm$ 0.03 & -1.27 $\pm$ 0.03\
M13 & a/1e-4 & -1.33 $\pm$ 4.57 & 0.41 $\pm$ 4.05 & 2.41 $\pm$ 4.07 & -1.35 $\pm$ 4.89 & 2.44 $\pm$ 3.87 & 2.60 $\pm$ 3.22 & 4.14 $\pm$ 3.11 & 5.50 $\pm$ 2.95 & 5.83 $\pm$ 2.77\
M13 & b/1e-1 & 6.18 $\pm$ 0.08 & 6.12 $\pm$ 0.07 & 6.22 $\pm$ 0.07 & 6.23 $\pm$ 0.08 & 6.04 $\pm$ 0.06 & 6.00 $\pm$ 0.05 & 5.94 $\pm$ 0.05 & 5.85 $\pm$ 0.05 & 5.71 $\pm$ 0.05\
M13 & c/1e-2 & -1.49 $\pm$ 0.04 & -1.59 $\pm$ 0.03 & -1.53 $\pm$ 0.04 & -1.54 $\pm$ 0.07 & -1.54 $\pm$ 0.03 & -1.57 $\pm$ 0.02 & -1.59 $\pm$ 0.03 & -1.53 $\pm$ 0.03 & -1.44 $\pm$ 0.03\
M14 & a/1e-4 & 0.35 $\pm$ 3.81 & -0.57 $\pm$ 3.77 & 4.27 $\pm$ 3.61 & 3.01 $\pm$ 4.84 & 7.37 $\pm$ 4.78 & 2.58 $\pm$ 2.61 & 4.43 $\pm$ 2.19 & 5.43 $\pm$ 2.22 & 5.53 $\pm$ 2.19\
M14 & b/1e-1 & 5.91 $\pm$ 0.06 & 5.98 $\pm$ 0.06 & 5.91 $\pm$ 0.06 & 5.76 $\pm$ 0.08 & 5.73 $\pm$ 0.08 & 5.78 $\pm$ 0.04 & 5.64 $\pm$ 0.04 & 5.59 $\pm$ 0.04 & 5.53 $\pm$ 0.04\
M14 & c/1e-2 & -1.56 $\pm$ 0.04 & -1.67 $\pm$ 0.04 & -1.65 $\pm$ 0.04 & -1.57 $\pm$ 0.04 & -1.64 $\pm$ 0.03 & -1.68 $\pm$ 0.02 & -1.61 $\pm$ 0.03 & -1.60 $\pm$ 0.04 & -1.51 $\pm$ 0.04\
M15 & a/1e-4 & 1.33 $\pm$ 2.89 & -0.59 $\pm$ 2.55 & -3.57 $\pm$ 3.01 & 3.71 $\pm$ 5.82 & -1.46 $\pm$ 2.36 & -2.34 $\pm$ 1.73 & -2.80 $\pm$ 1.71 & -3.55 $\pm$ 1.68 & -2.20 $\pm$ 1.64\
M15 & b/1e-1 & 5.62 $\pm$ 0.05 & 5.78 $\pm$ 0.04 & 5.83 $\pm$ 0.05 & 5.59 $\pm$ 0.09 & 5.68 $\pm$ 0.04 & 5.61 $\pm$ 0.03 & 5.53 $\pm$ 0.03 & 5.50 $\pm$ 0.03 & 5.46 $\pm$ 0.03\
M15 & c/1e-2 & -1.42 $\pm$ 0.04 & -1.53 $\pm$ 0.03 & -1.34 $\pm$ 0.04 & -1.53 $\pm$ 0.03 & -1.42 $\pm$ 0.03 & -1.32 $\pm$ 0.03 & -1.27 $\pm$ 0.03 & -1.21 $\pm$ 0.03 & -1.13 $\pm$ 0.03\
M16 & a/1e-4 & 9.81 $\pm$ 3.39 & 3.14 $\pm$ 3.06 & 0.38 $\pm$ 4.28 & 3.85 $\pm$ 3.48 & 6.30 $\pm$ 2.65 & 5.25 $\pm$ 2.13 & 2.32 $\pm$ 1.95 & 0.49 $\pm$ 1.90 & 1.14 $\pm$ 1.83\
M16 & b/1e-1 & 5.44 $\pm$ 0.06 & 5.52 $\pm$ 0.05 & 5.55 $\pm$ 0.07 & 5.47 $\pm$ 0.06 & 5.29 $\pm$ 0.04 & 5.28 $\pm$ 0.04 & 5.19 $\pm$ 0.03 & 5.17 $\pm$ 0.03 & 5.09 $\pm$ 0.03\
M16 & c/1e-2 & -1.58 $\pm$ 0.04 & -1.52 $\pm$ 0.03 & -1.46 $\pm$ 0.03 & -1.51 $\pm$ 0.03 & -1.41 $\pm$ 0.02 & -1.36 $\pm$ 0.02 & -1.28 $\pm$ 0.02 & -1.21 $\pm$ 0.02 & -1.14 $\pm$ 0.02\
M17 & a/1e-4 & -0.89 $\pm$ 3.92 & -4.49 $\pm$ 3.39 & -3.98 $\pm$ 3.71 & -7.57 $\pm$ 4.51 & -3.25 $\pm$ 2.74 & -0.30 $\pm$ 2.47 & -0.80 $\pm$ 2.36 & -1.44 $\pm$ 2.17 & -0.64 $\pm$ 2.08\
M17 & b/1e-1 & 5.81 $\pm$ 0.06 & 5.83 $\pm$ 0.06 & 5.85 $\pm$ 0.06 & 5.71 $\pm$ 0.07 & 5.65 $\pm$ 0.05 & 5.47 $\pm$ 0.04 & 5.48 $\pm$ 0.04 & 5.41 $\pm$ 0.04 & 5.28 $\pm$ 0.03\
M17 & c/1e-2 & -1.54 $\pm$ 0.02 & -1.52 $\pm$ 0.02 & -1.41 $\pm$ 0.02 & -1.25 $\pm$ 0.03 & -1.33 $\pm$ 0.02 & -1.22 $\pm$ 0.01 & -1.21 $\pm$ 0.02 & -1.16 $\pm$ 0.01 & -1.08 $\pm$ 0.02\
M18 & a/1e-4 & -1.88 $\pm$ 2.27 & -5.55 $\pm$ 2.09 & -5.94 $\pm$ 2.11 & -1.53 $\pm$ 2.53 & -2.21 $\pm$ 3.35 & -0.75 $\pm$ 1.52 & -1.19 $\pm$ 1.52 & 0.24 $\pm$ 1.51 & 0.93 $\pm$ 1.53\
M18 & b/1e-1 & 5.64 $\pm$ 0.04 & 5.68 $\pm$ 0.03 & 5.81 $\pm$ 0.03 & 5.60 $\pm$ 0.04 & 5.57 $\pm$ 0.05 & 5.46 $\pm$ 0.02 & 5.38 $\pm$ 0.02 & 5.38 $\pm$ 0.02 & 5.33 $\pm$ 0.02\
M18 & c/1e-2 & -1.37 $\pm$ 0.02 & -1.34 $\pm$ 0.02 & -1.34 $\pm$ 0.02 & -1.29 $\pm$ 0.02 & -1.20 $\pm$ 0.02 & -1.13 $\pm$ 0.02 & -1.11 $\pm$ 0.02 & -1.11 $\pm$ 0.02 & -1.05 $\pm$ 0.02\
M19 & a/1e-4 & 11.82 $\pm$ 2.94 & 6.55 $\pm$ 2.78 & 5.01 $\pm$ 3.51 & 10.73 $\pm$ 3.34 & 9.32 $\pm$ 2.79 & 8.64 $\pm$ 2.51 & 10.22 $\pm$ 2.15 & 9.32 $\pm$ 2.21 & 11.03 $\pm$ 2.24\
M19 & b/1e-1 & 5.70 $\pm$ 0.05 & 5.88 $\pm$ 0.04 & 5.92 $\pm$ 0.06 & 5.67 $\pm$ 0.05 & 5.69 $\pm$ 0.04 & 5.61 $\pm$ 0.04 & 5.48 $\pm$ 0.03 & 5.43 $\pm$ 0.04 & 5.30 $\pm$ 0.04\
M19 & c/1e-2 & -1.76 $\pm$ 0.03 & -1.75 $\pm$ 0.03 & -1.59 $\pm$ 0.03 & -1.60 $\pm$ 0.04 & -1.57 $\pm$ 0.03 & -1.50 $\pm$ 0.03 & -1.44 $\pm$ 0.03 & -1.36 $\pm$ 0.03 & -1.25 $\pm$ 0.03\
[ccccccccccc]{} & &\
& & & & & & & & & &\
M01 & P$_0$ & 4.94 $\pm$ 0.01 & 5.83 $\pm$ 0.01 & 4.36 $\pm$ 0.01 & 4.24 $\pm$ 0.03 & 4.35 $\pm$ 0.02 & 3.60 $\pm$ 0.01 & 3.10 $\pm$ 0.01 & 2.76 $\pm$ 0.01 & 2.00 $\pm$ 0.01\
M01 & P$_1$ & 9.12 $\pm$ 0.02 & 8.75 $\pm$ 0.03 & 7.53 $\pm$ 0.03 & 7.75 $\pm$ 0.08 & 6.88 $\pm$ 0.04 & 9.88 $\pm$ 0.03 & 10.03 $\pm$ 0.04 & 10.00 $\pm$ 0.04 & 11.93 $\pm$ 0.07\
M01 & P$_2$ & 26.31 $\pm$ 0.01 & 28.14 $\pm$ 0.01 & 26.89 $\pm$ 0.01 & 27.19 $\pm$ 0.02 & 25.54 $\pm$ 0.01 & 24.59 $\pm$ 0.01 & 23.91 $\pm$ 0.00 & 23.47 $\pm$ 0.00 & 22.23 $\pm$ 0.00\
M01 & n & 1.37 $\pm$ 0.00 & 1.34 $\pm$ 0.00 & 1.26 $\pm$ 0.00 & 1.27 $\pm$ 0.01 & 1.21 $\pm$ 0.00 & 1.41 $\pm$ 0.00 & 1.43 $\pm$ 0.00 & 1.42 $\pm$ 0.00 & 1.53 $\pm$ 0.00\
M02 & P$_0$ & 4.10 $\pm$ 0.00 & 4.53 $\pm$ 0.01 & 4.23 $\pm$ 0.02 & 4.17 $\pm$ 0.05 & 4.08 $\pm$ 0.02 & 3.20 $\pm$ 0.00 & 2.47 $\pm$ 0.00 & 2.29 $\pm$ 0.00 & 1.49 $\pm$ 0.00\
M02 & P$_1$ & 7.68 $\pm$ 0.01 & 7.89 $\pm$ 0.03 & 6.19 $\pm$ 0.03 & 6.32 $\pm$ 0.07 & 5.73 $\pm$ 0.03 & 8.83 $\pm$ 0.02 & 9.36 $\pm$ 0.02 & 9.32 $\pm$ 0.02 & 11.41 $\pm$ 0.06\
M02 & P$_2$ & 25.65 $\pm$ 0.00 & 26.74 $\pm$ 0.01 & 26.63 $\pm$ 0.02 & 27.53 $\pm$ 0.04 & 25.56 $\pm$ 0.02 & 24.14 $\pm$ 0.00 & 23.29 $\pm$ 0.00 & 23.23 $\pm$ 0.00 & 21.87 $\pm$ 0.00\
M02 & n & 1.25 $\pm$ 0.00 & 1.28 $\pm$ 0.00 & 1.10 $\pm$ 0.00 & 1.12 $\pm$ 0.01 & 1.07 $\pm$ 0.00 & 1.33 $\pm$ 0.00 & 1.37 $\pm$ 0.00 & 1.37 $\pm$ 0.00 & 1.48 $\pm$ 0.00\
M03 & P$_0$ & 4.89 $\pm$ 0.01 & 5.22 $\pm$ 0.01 & 4.92 $\pm$ 0.02 & 4.31 $\pm$ 0.02 & 4.70 $\pm$ 0.02 & 3.69 $\pm$ 0.01 & 3.10 $\pm$ 0.00 & 2.80 $\pm$ 0.01 & 2.01 $\pm$ 0.01\
M03 & P$_1$ & 8.16 $\pm$ 0.02 & 8.12 $\pm$ 0.03 & 6.53 $\pm$ 0.02 & 7.59 $\pm$ 0.06 & 6.29 $\pm$ 0.03 & 8.83 $\pm$ 0.03 & 9.46 $\pm$ 0.03 & 9.32 $\pm$ 0.04 & 10.91 $\pm$ 0.06\
M03 & P$_2$ & 27.87 $\pm$ 0.01 & 28.30 $\pm$ 0.01 & 27.66 $\pm$ 0.01 & 27.93 $\pm$ 0.02 & 26.41 $\pm$ 0.02 & 24.67 $\pm$ 0.01 & 23.34 $\pm$ 0.00 & 23.22 $\pm$ 0.01 & 21.86 $\pm$ 0.00\
M03 & n & 1.29 $\pm$ 0.00 & 1.29 $\pm$ 0.00 & 1.15 $\pm$ 0.00 & 1.25 $\pm$ 0.01 & 1.12 $\pm$ 0.00 & 1.33 $\pm$ 0.00 & 1.39 $\pm$ 0.00 & 1.37 $\pm$ 0.00 & 1.46 $\pm$ 0.00\
M04 & P$_0$ & 4.23 $\pm$ 0.00 & 5.00 $\pm$ 0.01 & 3.52 $\pm$ 0.01 & 3.24 $\pm$ 0.02 & 3.60 $\pm$ 0.01 & 2.70 $\pm$ 0.00 & 2.42 $\pm$ 0.00 & 2.08 $\pm$ 0.00 & 1.31 $\pm$ 0.00\
M04 & P$_1$ & 8.05 $\pm$ 0.01 & 8.10 $\pm$ 0.03 & 7.31 $\pm$ 0.03 & 7.31 $\pm$ 0.08 & 6.20 $\pm$ 0.03 & 8.88 $\pm$ 0.02 & 9.47 $\pm$ 0.02 & 9.35 $\pm$ 0.02 & 12.18 $\pm$ 0.05\
M04 & P$_2$ & 26.10 $\pm$ 0.00 & 27.88 $\pm$ 0.01 & 26.48 $\pm$ 0.01 & 27.47 $\pm$ 0.02 & 26.43 $\pm$ 0.01 & 25.13 $\pm$ 0.00 & 24.59 $\pm$ 0.00 & 23.92 $\pm$ 0.00 & 22.13 $\pm$ 0.00\
M04 & n & 1.30 $\pm$ 0.00 & 1.31 $\pm$ 0.00 & 1.26 $\pm$ 0.00 & 1.25 $\pm$ 0.01 & 1.15 $\pm$ 0.00 & 1.36 $\pm$ 0.00 & 1.40 $\pm$ 0.00 & 1.40 $\pm$ 0.00 & 1.56 $\pm$ 0.00\
M05 & P$_0$ & 4.38 $\pm$ 0.00 & 4.75 $\pm$ 0.01 & 3.43 $\pm$ 0.01 & 3.19 $\pm$ 0.03 & 3.50 $\pm$ 0.01 & 2.82 $\pm$ 0.00 & 2.47 $\pm$ 0.00 & 2.09 $\pm$ 0.00 & 1.41 $\pm$ 0.00\
M05 & P$_1$ & 8.18 $\pm$ 0.01 & 8.44 $\pm$ 0.03 & 6.79 $\pm$ 0.03 & 6.69 $\pm$ 0.09 & 6.42 $\pm$ 0.04 & 8.59 $\pm$ 0.02 & 9.29 $\pm$ 0.02 & 9.49 $\pm$ 0.02 & 11.49 $\pm$ 0.05\
M05 & P$_2$ & 26.13 $\pm$ 0.00 & 27.32 $\pm$ 0.01 & 26.06 $\pm$ 0.01 & 26.75 $\pm$ 0.02 & 25.96 $\pm$ 0.01 & 24.51 $\pm$ 0.00 & 23.88 $\pm$ 0.00 & 23.76 $\pm$ 0.00 & 22.65 $\pm$ 0.00\
M05 & n & 1.32 $\pm$ 0.00 & 1.34 $\pm$ 0.00 & 1.21 $\pm$ 0.00 & 1.20 $\pm$ 0.01 & 1.19 $\pm$ 0.00 & 1.34 $\pm$ 0.00 & 1.40 $\pm$ 0.00 & 1.41 $\pm$ 0.00 & 1.52 $\pm$ 0.00\
M06 & P$_0$ & 5.28 $\pm$ 0.01 & 6.22 $\pm$ 0.02 & 4.54 $\pm$ 0.02 & 4.29 $\pm$ 0.03 & 4.65 $\pm$ 0.02 & 3.77 $\pm$ 0.01 & 3.28 $\pm$ 0.01 & 2.99 $\pm$ 0.01 & 2.23 $\pm$ 0.01\
M06 & P$_1$ & 9.01 $\pm$ 0.03 & 8.49 $\pm$ 0.04 & 7.29 $\pm$ 0.05 & 7.96 $\pm$ 0.08 & 7.25 $\pm$ 0.04 & 9.39 $\pm$ 0.04 & 10.41 $\pm$ 0.05 & 10.16 $\pm$ 0.06 & 11.65 $\pm$ 0.09\
M06 & P$_2$ & 27.57 $\pm$ 0.01 & 29.40 $\pm$ 0.02 & 27.50 $\pm$ 0.02 & 28.62 $\pm$ 0.03 & 26.86 $\pm$ 0.02 & 25.10 $\pm$ 0.01 & 24.27 $\pm$ 0.01 & 23.75 $\pm$ 0.01 & 22.46 $\pm$ 0.01\
M06 & n & 1.35 $\pm$ 0.00 & 1.32 $\pm$ 0.00 & 1.23 $\pm$ 0.00 & 1.27 $\pm$ 0.01 & 1.22 $\pm$ 0.00 & 1.37 $\pm$ 0.00 & 1.45 $\pm$ 0.00 & 1.42 $\pm$ 0.00 & 1.50 $\pm$ 0.00\
M07 & P$_0$ & 3.72 $\pm$ 0.00 & 4.43 $\pm$ 0.02 & 3.44 $\pm$ 0.02 & 3.34 $\pm$ 0.03 & 3.38 $\pm$ 0.01 & 2.88 $\pm$ 0.00 & 2.24 $\pm$ 0.00 & 1.95 $\pm$ 0.00 & 1.25 $\pm$ 0.00\
M07 & P$_1$ & 8.78 $\pm$ 0.02 & 8.15 $\pm$ 0.05 & 6.38 $\pm$ 0.04 & 6.94 $\pm$ 0.08 & 6.41 $\pm$ 0.03 & 9.30 $\pm$ 0.02 & 10.59 $\pm$ 0.03 & 10.72 $\pm$ 0.04 & 13.50 $\pm$ 0.10\
M07 & P$_2$ & 26.36 $\pm$ 0.00 & 27.90 $\pm$ 0.01 & 26.27 $\pm$ 0.01 & 27.01 $\pm$ 0.03 & 25.08 $\pm$ 0.01 & 24.22 $\pm$ 0.00 & 23.02 $\pm$ 0.00 & 22.70 $\pm$ 0.00 & 21.36 $\pm$ 0.00\
M07 & n & 1.35 $\pm$ 0.00 & 1.30 $\pm$ 0.00 & 1.16 $\pm$ 0.00 & 1.19 $\pm$ 0.01 & 1.16 $\pm$ 0.00 & 1.38 $\pm$ 0.00 & 1.46 $\pm$ 0.00 & 1.47 $\pm$ 0.00 & 1.61 $\pm$ 0.00\
M08 & P$_0$ & 4.57 $\pm$ 0.01 & 3.93 $\pm$ 0.01 & 4.85 $\pm$ 0.04 & 4.33 $\pm$ 0.04 & 3.96 $\pm$ 0.02 & 3.20 $\pm$ 0.01 & 2.63 $\pm$ 0.01 & 2.24 $\pm$ 0.01 & 1.77 $\pm$ 0.01\
M08 & P$_1$ & 7.17 $\pm$ 0.02 & 7.87 $\pm$ 0.03 & 5.78 $\pm$ 0.02 & 6.80 $\pm$ 0.06 & 6.40 $\pm$ 0.03 & 8.69 $\pm$ 0.03 & 9.06 $\pm$ 0.05 & 9.87 $\pm$ 0.06 & 11.21 $\pm$ 0.09\
M08 & P$_2$ & 25.78 $\pm$ 0.01 & 25.29 $\pm$ 0.01 & 27.01 $\pm$ 0.04 & 27.23 $\pm$ 0.04 & 24.97 $\pm$ 0.02 & 23.97 $\pm$ 0.01 & 22.68 $\pm$ 0.01 & 22.43 $\pm$ 0.01 & 21.43 $\pm$ 0.01\
M08 & n & 1.17 $\pm$ 0.00 & 1.25 $\pm$ 0.00 & 1.02 $\pm$ 0.00 & 1.14 $\pm$ 0.01 & 1.10 $\pm$ 0.00 & 1.30 $\pm$ 0.00 & 1.34 $\pm$ 0.00 & 1.38 $\pm$ 0.00 & 1.46 $\pm$ 0.00\
M09 & P$_0$ & 3.47 $\pm$ 0.01 & 3.53 $\pm$ 0.01 & 3.28 $\pm$ 0.02 & 3.28 $\pm$ 0.02 & 3.18 $\pm$ 0.02 & 2.61 $\pm$ 0.01 & 2.07 $\pm$ 0.01 & 1.72 $\pm$ 0.00 & 1.26 $\pm$ 0.00\
M09 & P$_1$ & 8.88 $\pm$ 0.02 & 9.07 $\pm$ 0.05 & 7.01 $\pm$ 0.05 & 8.56 $\pm$ 0.10 & 7.15 $\pm$ 0.06 & 11.58 $\pm$ 0.07 & 12.20 $\pm$ 0.09 & 13.30 $\pm$ 0.11 & 17.47 $\pm$ 0.22\
M09 & P$_2$ & 25.34 $\pm$ 0.00 & 25.91 $\pm$ 0.01 & 25.78 $\pm$ 0.01 & 26.73 $\pm$ 0.02 & 25.03 $\pm$ 0.02 & 24.10 $\pm$ 0.01 & 23.22 $\pm$ 0.01 & 22.99 $\pm$ 0.00 & 21.81 $\pm$ 0.00\
M09 & n & 1.34 $\pm$ 0.00 & 1.36 $\pm$ 0.00 & 1.20 $\pm$ 0.00 & 1.33 $\pm$ 0.01 & 1.22 $\pm$ 0.01 & 1.51 $\pm$ 0.00 & 1.55 $\pm$ 0.00 & 1.60 $\pm$ 0.01 & 1.76 $\pm$ 0.01\
M10 & P$_0$ & 3.96 $\pm$ 0.00 & 3.79 $\pm$ 0.01 & 3.68 $\pm$ 0.01 & 3.49 $\pm$ 0.02 & 3.42 $\pm$ 0.01 & 2.80 $\pm$ 0.00 & 2.33 $\pm$ 0.00 & 1.93 $\pm$ 0.00 & 1.41 $\pm$ 0.00\
M10 & P$_1$ & 6.83 $\pm$ 0.01 & 7.30 $\pm$ 0.01 & 5.43 $\pm$ 0.01 & 6.64 $\pm$ 0.04 & 5.78 $\pm$ 0.02 & 7.75 $\pm$ 0.01 & 8.41 $\pm$ 0.02 & 8.59 $\pm$ 0.02 & 10.30 $\pm$ 0.03\
M10 & P$_2$ & 25.98 $\pm$ 0.00 & 26.39 $\pm$ 0.00 & 26.23 $\pm$ 0.01 & 26.51 $\pm$ 0.01 & 24.95 $\pm$ 0.01 & 23.77 $\pm$ 0.00 & 23.21 $\pm$ 0.00 & 22.64 $\pm$ 0.00 & 21.54 $\pm$ 0.00\
M10 & n & 1.18 $\pm$ 0.00 & 1.22 $\pm$ 0.00 & 1.04 $\pm$ 0.00 & 1.16 $\pm$ 0.00 & 1.08 $\pm$ 0.00 & 1.25 $\pm$ 0.00 & 1.30 $\pm$ 0.00 & 1.32 $\pm$ 0.00 & 1.43 $\pm$ 0.00\
M11 & P$_0$ & 4.91 $\pm$ 0.01 & 5.14 $\pm$ 0.01 & 4.41 $\pm$ 0.01 & 4.03 $\pm$ 0.02 & 4.02 $\pm$ 0.01 & 3.39 $\pm$ 0.01 & 3.02 $\pm$ 0.01 & 2.70 $\pm$ 0.01 & 1.95 $\pm$ 0.01\
M11 & P$_1$ & 8.77 $\pm$ 0.03 & 8.48 $\pm$ 0.04 & 7.85 $\pm$ 0.03 & 9.20 $\pm$ 0.08 & 7.58 $\pm$ 0.03 & 9.63 $\pm$ 0.04 & 10.56 $\pm$ 0.05 & 10.49 $\pm$ 0.05 & 12.10 $\pm$ 0.08\
M11 & P$_2$ & 27.18 $\pm$ 0.01 & 28.28 $\pm$ 0.01 & 27.61 $\pm$ 0.01 & 27.91 $\pm$ 0.02 & 25.57 $\pm$ 0.01 & 24.46 $\pm$ 0.01 & 24.18 $\pm$ 0.01 & 24.07 $\pm$ 0.01 & 23.14 $\pm$ 0.00\
M11 & n & 1.33 $\pm$ 0.00 & 1.31 $\pm$ 0.00 & 1.26 $\pm$ 0.00 & 1.35 $\pm$ 0.01 & 1.24 $\pm$ 0.00 & 1.38 $\pm$ 0.00 & 1.45 $\pm$ 0.00 & 1.44 $\pm$ 0.00 & 1.52 $\pm$ 0.00\
M12 & P$_0$ & 4.74 $\pm$ 0.01 & 4.62 $\pm$ 0.01 & 3.94 $\pm$ 0.01 & 3.51 $\pm$ 0.02 & 3.44 $\pm$ 0.01 & 2.90 $\pm$ 0.01 & 2.65 $\pm$ 0.00 & 2.35 $\pm$ 0.01 & 1.66 $\pm$ 0.01\
M12 & P$_1$ & 7.44 $\pm$ 0.02 & 7.82 $\pm$ 0.02 & 7.00 $\pm$ 0.02 & 8.36 $\pm$ 0.09 & 6.68 $\pm$ 0.03 & 8.30 $\pm$ 0.03 & 8.85 $\pm$ 0.03 & 8.39 $\pm$ 0.03 & 9.72 $\pm$ 0.06\
M12 & P$_2$ & 28.58 $\pm$ 0.01 & 28.71 $\pm$ 0.01 & 28.31 $\pm$ 0.01 & 28.76 $\pm$ 0.02 & 25.83 $\pm$ 0.01 & 24.49 $\pm$ 0.01 & 24.16 $\pm$ 0.00 & 23.77 $\pm$ 0.00 & 22.47 $\pm$ 0.00\
M12 & n & 1.24 $\pm$ 0.00 & 1.28 $\pm$ 0.00 & 1.20 $\pm$ 0.00 & 1.30 $\pm$ 0.01 & 1.18 $\pm$ 0.00 & 1.30 $\pm$ 0.00 & 1.35 $\pm$ 0.00 & 1.31 $\pm$ 0.00 & 1.39 $\pm$ 0.00\
M13 & P$_0$ & 5.46 $\pm$ 0.01 & 6.30 $\pm$ 0.02 & 4.78 $\pm$ 0.02 & 4.27 $\pm$ 0.02 & 4.31 $\pm$ 0.02 & 3.69 $\pm$ 0.01 & 3.38 $\pm$ 0.01 & 3.14 $\pm$ 0.01 & 2.33 $\pm$ 0.01\
M13 & P$_1$ & 8.62 $\pm$ 0.04 & 7.76 $\pm$ 0.04 & 7.53 $\pm$ 0.03 & 8.90 $\pm$ 0.08 & 7.41 $\pm$ 0.04 & 9.08 $\pm$ 0.05 & 8.92 $\pm$ 0.05 & 9.00 $\pm$ 0.05 & 10.57 $\pm$ 0.08\
M13 & P$_2$ & 27.97 $\pm$ 0.01 & 29.57 $\pm$ 0.02 & 28.34 $\pm$ 0.01 & 28.26 $\pm$ 0.02 & 25.94 $\pm$ 0.02 & 24.82 $\pm$ 0.01 & 24.14 $\pm$ 0.01 & 23.82 $\pm$ 0.01 & 22.29 $\pm$ 0.01\
M13 & n & 1.32 $\pm$ 0.00 & 1.26 $\pm$ 0.00 & 1.23 $\pm$ 0.00 & 1.34 $\pm$ 0.01 & 1.23 $\pm$ 0.00 & 1.34 $\pm$ 0.00 & 1.34 $\pm$ 0.00 & 1.34 $\pm$ 0.00 & 1.44 $\pm$ 0.01\
M14 & P$_0$ & 4.76 $\pm$ 0.01 & 5.18 $\pm$ 0.02 & 4.32 $\pm$ 0.03 & 3.33 $\pm$ 0.03 & 3.68 $\pm$ 0.02 & 2.92 $\pm$ 0.01 & 2.56 $\pm$ 0.01 & 2.39 $\pm$ 0.01 & 1.73 $\pm$ 0.01\
M14 & P$_1$ & 8.69 $\pm$ 0.03 & 8.11 $\pm$ 0.04 & 7.10 $\pm$ 0.05 & 9.13 $\pm$ 0.11 & 7.40 $\pm$ 0.05 & 9.65 $\pm$ 0.06 & 10.67 $\pm$ 0.08 & 10.62 $\pm$ 0.08 & 13.43 $\pm$ 0.18\
M14 & P$_2$ & 28.24 $\pm$ 0.01 & 29.19 $\pm$ 0.02 & 27.85 $\pm$ 0.02 & 27.87 $\pm$ 0.02 & 27.31 $\pm$ 0.02 & 25.67 $\pm$ 0.01 & 24.39 $\pm$ 0.01 & 23.82 $\pm$ 0.01 & 22.24 $\pm$ 0.01\
M14 & n & 1.32 $\pm$ 0.00 & 1.27 $\pm$ 0.00 & 1.19 $\pm$ 0.00 & 1.34 $\pm$ 0.01 & 1.20 $\pm$ 0.00 & 1.35 $\pm$ 0.00 & 1.44 $\pm$ 0.00 & 1.42 $\pm$ 0.00 & 1.55 $\pm$ 0.01\
M15 & P$_0$ & 4.13 $\pm$ 0.00 & 4.73 $\pm$ 0.01 & 4.33 $\pm$ 0.01 & 3.48 $\pm$ 0.02 & 3.26 $\pm$ 0.01 & 2.56 $\pm$ 0.00 & 2.32 $\pm$ 0.00 & 2.15 $\pm$ 0.00 & 1.39 $\pm$ 0.00\
M15 & P$_1$ & 8.49 $\pm$ 0.02 & 7.52 $\pm$ 0.02 & 5.52 $\pm$ 0.02 & 6.02 $\pm$ 0.04 & 6.46 $\pm$ 0.03 & 7.99 $\pm$ 0.02 & 7.92 $\pm$ 0.02 & 8.02 $\pm$ 0.02 & 11.21 $\pm$ 0.03\
M15 & P$_2$ & 25.79 $\pm$ 0.00 & 27.53 $\pm$ 0.01 & 27.69 $\pm$ 0.01 & 27.34 $\pm$ 0.02 & 24.98 $\pm$ 0.01 & 23.72 $\pm$ 0.00 & 23.08 $\pm$ 0.00 & 22.96 $\pm$ 0.00 & 21.65 $\pm$ 0.00\
M15 & n & 1.35 $\pm$ 0.00 & 1.27 $\pm$ 0.00 & 1.09 $\pm$ 0.00 & 1.12 $\pm$ 0.00 & 1.20 $\pm$ 0.00 & 1.29 $\pm$ 0.00 & 1.29 $\pm$ 0.00 & 1.30 $\pm$ 0.00 & 1.51 $\pm$ 0.00\
M16 & P$_0$ & 6.83 $\pm$ 0.03 & 7.12 $\pm$ 0.03 & 6.59 $\pm$ 0.03 & 5.73 $\pm$ 0.05 & 5.65 $\pm$ 0.03 & 4.64 $\pm$ 0.02 & 4.17 $\pm$ 0.02 & 4.00 $\pm$ 0.02 & 3.23 $\pm$ 0.02\
M16 & P$_1$ & 8.69 $\pm$ 0.05 & 8.50 $\pm$ 0.05 & 7.54 $\pm$ 0.04 & 8.36 $\pm$ 0.08 & 7.57 $\pm$ 0.05 & 9.72 $\pm$ 0.07 & 9.71 $\pm$ 0.08 & 9.69 $\pm$ 0.08 & 10.65 $\pm$ 0.11\
M16 & P$_2$ & 29.72 $\pm$ 0.03 & 30.43 $\pm$ 0.03 & 29.61 $\pm$ 0.03 & 29.14 $\pm$ 0.05 & 27.19 $\pm$ 0.03 & 25.36 $\pm$ 0.02 & 24.23 $\pm$ 0.02 & 24.15 $\pm$ 0.02 & 23.00 $\pm$ 0.02\
M16 & n & 1.30 $\pm$ 0.00 & 1.29 $\pm$ 0.00 & 1.21 $\pm$ 0.00 & 1.26 $\pm$ 0.01 & 1.20 $\pm$ 0.00 & 1.35 $\pm$ 0.00 & 1.36 $\pm$ 0.01 & 1.35 $\pm$ 0.01 & 1.40 $\pm$ 0.01\
M17 & P$_0$ & 4.67 $\pm$ 0.01 & 4.73 $\pm$ 0.01 & 4.10 $\pm$ 0.02 & 3.44 $\pm$ 0.03 & 3.90 $\pm$ 0.02 & 3.27 $\pm$ 0.01 & 2.67 $\pm$ 0.01 & 2.39 $\pm$ 0.00 & 1.74 $\pm$ 0.01\
M17 & P$_1$ & 7.77 $\pm$ 0.02 & 7.62 $\pm$ 0.03 & 5.93 $\pm$ 0.03 & 6.74 $\pm$ 0.07 & 6.02 $\pm$ 0.03 & 8.20 $\pm$ 0.02 & 8.84 $\pm$ 0.04 & 8.68 $\pm$ 0.03 & 9.45 $\pm$ 0.05\
M17 & P$_2$ & 27.07 $\pm$ 0.01 & 27.82 $\pm$ 0.01 & 26.74 $\pm$ 0.02 & 27.70 $\pm$ 0.03 & 26.60 $\pm$ 0.02 & 25.23 $\pm$ 0.01 & 23.98 $\pm$ 0.01 & 23.48 $\pm$ 0.00 & 22.20 $\pm$ 0.01\
M17 & n & 1.26 $\pm$ 0.00 & 1.25 $\pm$ 0.00 & 1.10 $\pm$ 0.00 & 1.15 $\pm$ 0.01 & 1.10 $\pm$ 0.00 & 1.27 $\pm$ 0.00 & 1.33 $\pm$ 0.00 & 1.32 $\pm$ 0.00 & 1.36 $\pm$ 0.00\
M18 & P$_0$ & 5.58 $\pm$ 0.01 & 5.76 $\pm$ 0.02 & 5.85 $\pm$ 0.05 & 4.83 $\pm$ 0.04 & 5.35 $\pm$ 0.03 & 4.28 $\pm$ 0.01 & 3.78 $\pm$ 0.01 & 3.31 $\pm$ 0.01 & 2.66 $\pm$ 0.01\
M18 & P$_1$ & 7.37 $\pm$ 0.02 & 7.36 $\pm$ 0.02 & 5.66 $\pm$ 0.03 & 7.19 $\pm$ 0.06 & 6.14 $\pm$ 0.02 & 7.95 $\pm$ 0.02 & 8.05 $\pm$ 0.03 & 8.52 $\pm$ 0.03 & 9.23 $\pm$ 0.04\
M18 & P$_2$ & 28.15 $\pm$ 0.01 & 29.09 $\pm$ 0.02 & 29.67 $\pm$ 0.06 & 29.79 $\pm$ 0.04 & 27.99 $\pm$ 0.03 & 25.97 $\pm$ 0.01 & 24.88 $\pm$ 0.01 & 24.46 $\pm$ 0.01 & 23.24 $\pm$ 0.01\
M18 & n & 1.19 $\pm$ 0.00 & 1.19 $\pm$ 0.00 & 1.01 $\pm$ 0.00 & 1.18 $\pm$ 0.01 & 1.07 $\pm$ 0.00 & 1.24 $\pm$ 0.00 & 1.23 $\pm$ 0.00 & 1.27 $\pm$ 0.00 & 1.31 $\pm$ 0.00\
M19 & P$_0$ & 4.56 $\pm$ 0.02 & 4.36 $\pm$ 0.02 & 4.70 $\pm$ 0.03 & 4.10 $\pm$ 0.03 & 4.13 $\pm$ 0.02 & 3.38 $\pm$ 0.01 & 2.82 $\pm$ 0.01 & 2.52 $\pm$ 0.01 & 1.96 $\pm$ 0.01\
M19 & P$_1$ & 8.61 $\pm$ 0.06 & 9.10 $\pm$ 0.05 & 7.01 $\pm$ 0.04 & 8.58 $\pm$ 0.09 & 7.50 $\pm$ 0.04 & 11.19 $\pm$ 0.07 & 11.27 $\pm$ 0.09 & 11.59 $\pm$ 0.11 & 13.00 $\pm$ 0.18\
M19 & P$_2$ & 27.90 $\pm$ 0.02 & 27.57 $\pm$ 0.02 & 28.21 $\pm$ 0.03 & 28.10 $\pm$ 0.03 & 25.91 $\pm$ 0.02 & 24.49 $\pm$ 0.01 & 23.51 $\pm$ 0.01 & 23.29 $\pm$ 0.01 & 21.95 $\pm$ 0.01\
M19 & n & 1.29 $\pm$ 0.00 & 1.32 $\pm$ 0.00 & 1.14 $\pm$ 0.00 & 1.28 $\pm$ 0.01 & 1.19 $\pm$ 0.00 & 1.44 $\pm$ 0.00 & 1.44 $\pm$ 0.01 & 1.46 $\pm$ 0.01 & 1.53 $\pm$ 0.01\
[cccccccccc]{} &\
& & & & & & & & &\
M01 & 15.98 $\pm$ 0.27 & 16.27 $\pm$ 0.21 & 16.48 $\pm$ 0.14 & 16.10 $\pm$ 0.25 & 16.12 $\pm$ 0.25 & 15.94 $\pm$ 0.28 & 15.98 $\pm$ 0.27 & 16.02 $\pm$ 0.26 & 15.71 $\pm$ 0.29\
M02 & 0.98 $\pm$ 0.03 & 0.95 $\pm$ 0.03 & 0.95 $\pm$ 0.03 & 0.95 $\pm$ 0.03 & 0.95 $\pm$ 0.03 & 0.93 $\pm$ 0.03 & 0.94 $\pm$ 0.02 & 0.94 $\pm$ 0.02 & 0.95 $\pm$ 0.03\
M03 & 1.03 $\pm$ 0.03 & 0.99 $\pm$ 0.03 & 1.00 $\pm$ 0.03 & 1.00 $\pm$ 0.04 & 1.00 $\pm$ 0.03 & 0.99 $\pm$ 0.03 & 0.97 $\pm$ 0.03 & 0.97 $\pm$ 0.03 & 0.98 $\pm$ 0.03\
M04 & 1.01 $\pm$ 0.04 & 0.97 $\pm$ 0.03 & 0.98 $\pm$ 0.03 & 0.97 $\pm$ 0.04 & 0.97 $\pm$ 0.04 & 0.96 $\pm$ 0.03 & 0.95 $\pm$ 0.03 & 0.95 $\pm$ 0.03 & 0.95 $\pm$ 0.03\
M05 & 1.02 $\pm$ 0.03 & 1.01 $\pm$ 0.02 & 1.03 $\pm$ 0.02 & 1.00 $\pm$ 0.03 & 1.00 $\pm$ 0.03 & 0.99 $\pm$ 0.02 & 0.97 $\pm$ 0.02 & 0.98 $\pm$ 0.02 & 0.98 $\pm$ 0.03\
M06 & 0.99 $\pm$ 0.03 & 0.98 $\pm$ 0.02 & 0.98 $\pm$ 0.03 & 0.96 $\pm$ 0.04 & 0.96 $\pm$ 0.03 & 0.95 $\pm$ 0.02 & 0.94 $\pm$ 0.02 & 0.94 $\pm$ 0.02 & 0.94 $\pm$ 0.02\
M07 & 0.98 $\pm$ 0.02 & 0.95 $\pm$ 0.02 & 0.96 $\pm$ 0.02 & 0.94 $\pm$ 0.03 & 0.96 $\pm$ 0.03 & 0.95 $\pm$ 0.02 & 0.94 $\pm$ 0.02 & 0.93 $\pm$ 0.02 & 0.93 $\pm$ 0.02\
M08 & 0.96 $\pm$ 0.03 & 0.94 $\pm$ 0.03 & 0.93 $\pm$ 0.03 & 0.93 $\pm$ 0.04 & 0.92 $\pm$ 0.03 & 0.92 $\pm$ 0.03 & 0.91 $\pm$ 0.03 & 0.89 $\pm$ 0.03 & 0.89 $\pm$ 0.03\
M09 & 0.95 $\pm$ 0.02 & 0.94 $\pm$ 0.02 & 0.92 $\pm$ 0.02 & 0.91 $\pm$ 0.02 & 0.92 $\pm$ 0.02 & 0.93 $\pm$ 0.02 & 0.91 $\pm$ 0.02 & 0.89 $\pm$ 0.02 & 0.90 $\pm$ 0.02\
M10 & 0.99 $\pm$ 0.03 & 0.97 $\pm$ 0.03 & 0.97 $\pm$ 0.03 & 0.96 $\pm$ 0.04 & 0.97 $\pm$ 0.03 & 0.96 $\pm$ 0.03 & 0.96 $\pm$ 0.03 & 0.95 $\pm$ 0.03 & 0.95 $\pm$ 0.03\
M11 & 0.96 $\pm$ 0.04 & 0.94 $\pm$ 0.04 & 0.94 $\pm$ 0.03 & 0.93 $\pm$ 0.05 & 0.93 $\pm$ 0.04 & 0.92 $\pm$ 0.03 & 0.92 $\pm$ 0.03 & 0.91 $\pm$ 0.03 & 0.91 $\pm$ 0.03\
M12 & 0.96 $\pm$ 0.05 & 0.93 $\pm$ 0.04 & 0.92 $\pm$ 0.03 & 0.94 $\pm$ 0.04 & 0.94 $\pm$ 0.03 & 0.91 $\pm$ 0.03 & 0.90 $\pm$ 0.03 & 0.89 $\pm$ 0.02 & 0.90 $\pm$ 0.03\
M13 & 0.99 $\pm$ 0.04 & 0.96 $\pm$ 0.03 & 0.97 $\pm$ 0.03 & 0.99 $\pm$ 0.04 & 0.96 $\pm$ 0.03 & 0.97 $\pm$ 0.03 & 0.96 $\pm$ 0.03 & 0.95 $\pm$ 0.03 & 0.95 $\pm$ 0.03\
M14 & 0.95 $\pm$ 0.03 & 0.94 $\pm$ 0.03 & 0.93 $\pm$ 0.03 & 0.92 $\pm$ 0.04 & 0.93 $\pm$ 0.04 & 0.93 $\pm$ 0.03 & 0.92 $\pm$ 0.02 & 0.91 $\pm$ 0.02 & 0.91 $\pm$ 0.02\
M15 & 0.90 $\pm$ 0.03 & 0.91 $\pm$ 0.02 & 0.90 $\pm$ 0.02 & 0.90 $\pm$ 0.05 & 0.90 $\pm$ 0.02 & 0.90 $\pm$ 0.02 & 0.88 $\pm$ 0.02 & 0.87 $\pm$ 0.02 & 0.88 $\pm$ 0.02\
M16 & 0.90 $\pm$ 0.03 & 0.88 $\pm$ 0.02 & 0.86 $\pm$ 0.03 & 0.88 $\pm$ 0.03 & 0.86 $\pm$ 0.02 & 0.86 $\pm$ 0.02 & 0.84 $\pm$ 0.02 & 0.83 $\pm$ 0.02 & 0.83 $\pm$ 0.02\
M17 & 0.93 $\pm$ 0.03 & 0.91 $\pm$ 0.03 & 0.90 $\pm$ 0.03 & 0.89 $\pm$ 0.04 & 0.89 $\pm$ 0.02 & 0.88 $\pm$ 0.02 & 0.88 $\pm$ 0.02 & 0.86 $\pm$ 0.02 & 0.86 $\pm$ 0.02\
M18 & 0.90 $\pm$ 0.02 & 0.88 $\pm$ 0.02 & 0.89 $\pm$ 0.02 & 0.89 $\pm$ 0.02 & 0.88 $\pm$ 0.03 & 0.87 $\pm$ 0.02 & 0.86 $\pm$ 0.02 & 0.86 $\pm$ 0.02 & 0.87 $\pm$ 0.02\
M19 & 0.94 $\pm$ 0.03 & 0.94 $\pm$ 0.02 & 0.93 $\pm$ 0.03 & 0.93 $\pm$ 0.03 & 0.93 $\pm$ 0.03 & 0.92 $\pm$ 0.03 & 0.90 $\pm$ 0.02 & 0.89 $\pm$ 0.02 & 0.89 $\pm$ 0.03\
![Measured $\eta$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. Fitting results when $\theta\rm_{ZA}$ $\leq$ 26.4$^\circ$ and $\theta\rm_{ZA}$ > 26.4$^\circ$ are represented with red and gold solid line, respectively.[]{data-label="fig:etafit"}](M01_eta_1400.png "fig:"){width="45.00000%"} ![Measured $\eta$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. Fitting results when $\theta\rm_{ZA}$ $\leq$ 26.4$^\circ$ and $\theta\rm_{ZA}$ > 26.4$^\circ$ are represented with red and gold solid line, respectively.[]{data-label="fig:etafit"}](M02_eta_1400.png "fig:"){width="45.00000%"} ![Measured $\eta$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. Fitting results when $\theta\rm_{ZA}$ $\leq$ 26.4$^\circ$ and $\theta\rm_{ZA}$ > 26.4$^\circ$ are represented with red and gold solid line, respectively.[]{data-label="fig:etafit"}](M08_eta_1400.png "fig:"){width="45.00000%"} ![Measured $\eta$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. Fitting results when $\theta\rm_{ZA}$ $\leq$ 26.4$^\circ$ and $\theta\rm_{ZA}$ > 26.4$^\circ$ are represented with red and gold solid line, respectively.[]{data-label="fig:etafit"}](M19_eta_1400.png "fig:"){width="45.00000%"}
![Measured $T\rm_{sys}$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. The fitting result is represented with red solid line.[]{data-label="fig:tsysfit"}](M01_tsys_1400.png "fig:"){width="45.00000%"} ![Measured $T\rm_{sys}$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. The fitting result is represented with red solid line.[]{data-label="fig:tsysfit"}](M02_tsys_1400.png "fig:"){width="45.00000%"} ![Measured $T\rm_{sys}$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. The fitting result is represented with red solid line.[]{data-label="fig:tsysfit"}](M08_tsys_1400.png "fig:"){width="45.00000%"} ![Measured $T\rm_{sys}$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. The fitting result is represented with red solid line.[]{data-label="fig:tsysfit"}](M19_tsys_1400.png "fig:"){width="45.00000%"}
Backend {#sec:backend}
=======
Stability and Sensitivity of Spectral Baseline {#subsec:spectal_sensitivity}
----------------------------------------------
Stability and sensitivity are two fundamental properties associated with spectral baseline. Stability represents fluctuation level of baseline in a time range. Sensitivity represents response capability of spectrometer. We estimated stability and sensitivity by taking observations toward the H[i]{} galaxy N672 in April 20, 2019.
A set of observation of 30 minutes without injection of noise signal was taken to test stability of the baseline. Averaged bandpass spectra per 5 minutes are shown in Fig. \[fig:spec\_stability\]. The variation of bandpass is $\sim$4% in 30 minutes.
![Variance of spectral baseline toward N672 in 30 minutes. Each spectrum is averaged in a time bin of 5 minutes.[]{data-label="fig:spec_stability"}](N672-stability.png){width="80.00000%"}
In order to measure sensitivity performance of the backend, noise signal as described in Section \[subsec:noise\_dipole\] was injected to calibrate observed spectra. Both noise signal with high and low intensity were injected for 5 minutes. The unit of spectrum was then calibrated into kelvin through the following transformation,
$$\rm T_A = T_{cal}\frac{P_{off}^{cal}}{P_{on}^{cal}-P_{off}^{cal}},$$
where $T\rm_A$ is calibrated antenna temperature. $T\rm_{cal}$ is noise diode temperature as shown in Fig.\[lowcal\] and Fig.\[highcal\]. $P\rm_{on}$ and $P\rm_{off}$ are power value when noise diode is on and off, respectively. As shown in Fig. \[fig:spec\_high\_low\], the derived continuum level under high and low noise injection are consistent within 1%.
![Calibrated H[i]{} spectrum under high power cal (blue) and low power cal (orange).[]{data-label="fig:spec_high_low"}](N672_high_low.png){width="80.00000%"}
We estimated sensitivity with root mean square (rms) of a spectrum. RMS is calculated with line-free frequency range. In the total on mode, the expected rms of an averaged spectrum of two polarizations $\sigma\rm_T$ is connected with system temperature $T\rm_{sys}$, channel resolution $\beta$ and integration time $\tau$ with $\sigma_T=T_{sys}/\sqrt{2\beta\tau}$. Periodic injection of high and low noise would lead to increase of $T_{sys}$ by 5.4 K and 0.5 K during an observation cycle (half time for cal on and half time for cal off), respectively. Comparison between obtained and theoretical rms is shown in Table \[table:sensitivity\]. It is obvious that performance of the 19 beam backend is well consistent with that of theoretical value for both low and high intensity noise injection.
[C[3cm]{} C[3cm]{} C[3cm]{} ]{} Cal Intensity & Obtained value & Theoretical value\
& (mK) & (mK)\
Low & 40.1 & 38.4\
High & 48.0 & 47.7\
Standing Waves {#subsec:standingwave}
--------------
The antenna structure and radio frequency (RF) devices of a radio telescope can cause reflection of electromagnetic (EM) wave. The coherent superposition of the received signal and its reflection wave results a periodic fluctuations in the frequency band-pass, which is called “standing wave”. This is a common phenomenon seen in the spectroscopic observations with radio telescopes [i.e. @Briggs1997; @Popping2008]. The major contribution to the standing wave for FAST is caused by the reflection between the dish and the receiver cabin. The relation between the standing wave frequency and the reflecting distance is given by Equation \[equ:sdw\]: $$D=\frac{c}{2\,\Delta f}, \label{equ:sdw}$$ where $D$ is the distance between the receiver cabin and the dish, $c$ is the speed of light, and $\Delta f$ is the width of the frequency ripple. For FAST, $D = 138$m relates $\Delta f \sim 1$MHz. At L-band, the corresponding velocity width is $\sim 200$kms$^{-1}$, which locates within the range of interest for extra-galactic observations.
Efforts have been put onto analyzing and minimizing the standing wave effects of FAST. To identify the major contribution to the FAST standing wave, we checked the band-pass during the receiver cabin rising. The width of the frequency ripple varies with the cabin height following the relation given by Equation \[equ:sdw\]. This experiment was done with the cabin dock uncovered. Thus the ground radiation passing through the central hole of the dish serves as the EM source for the standing wave. In Figure \[fig:sdw\], the figures from the upper panel shows the band-pass before the cabin dock is covered with metal mesh and those from the lower panel are after. The fluctuation declines significantly when the ground radiation is blocked out.
![The comparison of the band-pass ripples before (upper panel plots) and after (lower panel plots) the cabin dock platform is covered with metal mesh. The left side plots are with high power noise diode injection and the right side plots are with the low noise power.[]{data-label="fig:sdw"}](sdw_uncover-high.png "fig:"){width="49.00000%"} ![The comparison of the band-pass ripples before (upper panel plots) and after (lower panel plots) the cabin dock platform is covered with metal mesh. The left side plots are with high power noise diode injection and the right side plots are with the low noise power.[]{data-label="fig:sdw"}](sdw_uncover-low.png "fig:"){width="49.00000%"} ![The comparison of the band-pass ripples before (upper panel plots) and after (lower panel plots) the cabin dock platform is covered with metal mesh. The left side plots are with high power noise diode injection and the right side plots are with the low noise power.[]{data-label="fig:sdw"}](sdw_cover-high.png "fig:"){width="49.00000%"} ![The comparison of the band-pass ripples before (upper panel plots) and after (lower panel plots) the cabin dock platform is covered with metal mesh. The left side plots are with high power noise diode injection and the right side plots are with the low noise power.[]{data-label="fig:sdw"}](sdw_cover-low.png "fig:"){width="49.00000%"}
The feed leakage emission can be another EM source for the standing wave. Since the horns emit EM wave when the calibration noise diode is fired up. Figure \[fig:sdw\] shows the band-passes with high power noise diode (left column plots) and with low power noise diode (right column plots). It can be seen that the signal from the high power noise diode ($\sim 10$K) affects the band-pass fluctuation, whereas the low power signal ($\sim 1$K) does not.
The standing wave at different zenith angles (ZA) has also been checked and yet no significant difference was found. Figure \[fig:sdw-za\] shows the comparison of the spectra band-pass from ZA = 2.7 and 36.7. The data are fitted to sine function. The major fitting parameters are amplitude, phase, and period. The relative differences of those parameters are 12.4%, 10.7%, and 0.6%. The stable ripple patterns are expected since the distance between the receiver cabin and the apex of the parabolic dish should remain unchanged during observing.
The band-pass difference is smaller when the ZA difference is smaller. So ON/OFF position switch technique can be used to calibrate the baseline and to decrease the standing wave ripples. Figure \[fig:sdw-onoff\] shows the standing wave test from the ON/OFF observation towards J073631.82+383058.3. The top and middle panels are the calibrated band-passes from ON- and OFF- position. The bottom panel is the ON-OFF calibrated band-pass and a sine function fitting result, which shows the ripple amplitude of $\sim 15$ mK.
Another testing observations were obtained toward NGC2718 under ON-OFF mode. As shown in Figure \[fig:NGC2718-onoff\], the amplitude of standing wave varies as frequency.
![Band-pass ripples at different zenith angles. The blue line is the spectrum obtained with drift-scan observation at ZA = 2.7and the red line is at ZA = 36.7. The dashed lines are the sine function fitting results. A linear baseline removal was processed before hand. The intensity of the spectra are normalized and the unit is relative power.[]{data-label="fig:sdw-za"}](sdw_za.png){width="60.00000%"}
![Standing wave test from the ON/OFF observation towards J073631.82+383058.3. The top and middle panels are the calibrated band-passes from ON- and OFF- position. The bottom panel is the ON-OFF calibrated band-pass with a linear baseline removal. The sine function is fit to the spectrum, which gives the amplitude of the standing wave $\sim 15$ mK.[]{data-label="fig:sdw-onoff"}](sdw_onoff.png){width="60.00000%"}
![Bandpass spectrum at different frequency ranges during ON-OFF observations towards NGC 2718. The amplitudes of standing wave in frequency range of \[1150,1158\],\[1254,1257\], \[1367,1370\] and \[1432,1435\] MHz are 15, 11, 22, and 9.5 mK, respectively. []{data-label="fig:NGC2718-onoff"}](NGC2718_1154MHz.png "fig:"){width="48.00000%"} ![Bandpass spectrum at different frequency ranges during ON-OFF observations towards NGC 2718. The amplitudes of standing wave in frequency range of \[1150,1158\],\[1254,1257\], \[1367,1370\] and \[1432,1435\] MHz are 15, 11, 22, and 9.5 mK, respectively. []{data-label="fig:NGC2718-onoff"}](NGC2718_1254MHz.png "fig:"){width="48.00000%"} ![Bandpass spectrum at different frequency ranges during ON-OFF observations towards NGC 2718. The amplitudes of standing wave in frequency range of \[1150,1158\],\[1254,1257\], \[1367,1370\] and \[1432,1435\] MHz are 15, 11, 22, and 9.5 mK, respectively. []{data-label="fig:NGC2718-onoff"}](NGC2718_1366MHz.png "fig:"){width="48.00000%"} ![Bandpass spectrum at different frequency ranges during ON-OFF observations towards NGC 2718. The amplitudes of standing wave in frequency range of \[1150,1158\],\[1254,1257\], \[1367,1370\] and \[1432,1435\] MHz are 15, 11, 22, and 9.5 mK, respectively. []{data-label="fig:NGC2718-onoff"}](NGC2718_1432MHz.png "fig:"){width="48.00000%"}
Polarization {#subsec:polariztion}
------------
We carried out FAST polarization observations of 3C286 to measure the instrumental polarization of the telescope in Oct. 2018. 3C286 is a standard polarization calibrator with stable polarization degrees and polarization angles from 1 to 50 GHz (Perley & Butler 2013). 3C286 was drifted at parallactic angles of -60, -30, 0, 30, and 60 degrees through the central beam of the 19-beam receiver. The strength of the noise diode was set to 1K with on-off period of 0.2 sec. The four correlations of the XX, YY, XY, and YX signals were simultaneous recorded with the ROACH backends in both the spectral line modes of 500 MHz bandwidth and 32 MHz bandwidth. The data reduction including the gain and phase calibration of the system, the calibration of the four correlated spectra, and the derivation of the instrumental polarization using the data at 5 parallactic angles was carried out with the RHSTK package (Heiles et al. 2012). Fig. \[fig:pol\] shows the measurements of the Stokes Q, U, and V parameters of 3C286 at 5 parallactic angles. By fitting the sinusoidal behaviors of the Stokes parameters as functions of parallactic angles, the polarization degree and polarization angle of 3C286 are 6.6 +/- 1.5 % and 33.4 +/- 6.4 degrees, respectively. Our calibrated polarization results of 3C286 are close to the values of 9.47 +/- 0.02 % and 33 +/- 1 degrees in Perley & Butler (2013). The instrumental polarization parameters of FAST obtained from the data were close to being unitary, indicating a good isolation between the two signal paths of linear polarization feeds and a good performance of the polarization facility of FAST. We expect that with better characterization of the polarization properties including the pointing accuracy, beam width, beam squint, and beam squash of FAST, the polarization data will be calibrated to an accuracy of 0.1-0.01 %, in order to carry out scientific spectral line polarization observations in the near future.
![The measurements and the fits of the Stokes Q, U, and V parameters of 3C286 at 5 parallactic angles of the 500 MHz band (left) and the 32 MHz band (right). The red, green, and blue data sets represent the Stokes Q (XX-YY), U (2XY), and V (2YX) signals normalized with respect to the Stokes I signals.[]{data-label="fig:pol"}](FAST_pol.png){width="90.00000%"}
Observation {#sec:observation}
============
Observation Modes {#subsec:obs_mode}
-----------------
Now there are 8 observation modes in the FAST. The details are shown in the follows.
1. Drifting scan. In this mode, the position of feed cabin is fixed. To avoid pointing offset by the cooling of oil in the actuator, the actuators used for shaping the paraboloid surface are adjusted continually to keep same paraboloid in this mode. The telescope points across the sky as the earth rotates. There are four parameters in this mode, source name, source coordinate in J2000 system, observational time range, and rotating angle that represents cross angle between the line along beam 8, 2, 1, 5, and 14 and line of declination. The rotating angle has a limit of \[-80,80\] degree.
2. Total power. In this mode, the source can be tracked continuously. Tracking time for source at different latitude is shown in Fig. \[fig:trackingtime\]. Three parameters including source name, coordinate and integration time are needed. Derived rms of averaged spectrum of both polarization is estimated with,
$$\sigma = \frac{T_{sys}}{\sqrt{2\beta \tau}} \rm\ K,$$
where $\beta$=1.2B$\rm_{chan}$. B$\rm_{chan}$ is channel width of spectrometer in unit of Hz.
3. Position switch. The design of position mode is to achieve quick switch between source ON and source OFF in order to reduce baseline variation. There are five parameters in this mode.
- Coordinate of ON position.
- Coordinate of OFF position. The position of OFF source is designed to be within 1 degree from that of ON source.
- Integration time of ON source.
- Integration time of OFF source.
- Times of ON-OFF cycle.
Overhead time between ON and OFF position depends on separation of ON and OFF position, $\Delta\theta$. It is 30 s for $\Delta\theta$ < 20’ and is 60 s for $20'$ $\leq \Delta\theta$ < $60'$.
Derived rms of each spectrum with both polarizations is estimated with, $$\sigma = \frac{T_{sys}}{\sqrt{\beta \tau}} \rm \ K.$$
where $\beta$=1.2B$\rm_{chan}$. B$\rm_{chan}$ is channel width of spectrometer in unit of Hz.
4. On-the-fly (OTF) mapping. This mode is designed for mapping a sky area with Beam 1 only. Six parameters are necessary for observation: source name, source position, observational time range, sky coverage (e.g., 7$'\times$7$'$ of the mapping region), scanning separation (e.g., 1$'$) between two parallel scanning lines, and scanning direction (along RA or Dec). Scanning speed is 15 arcsec/s in default. The schematic diagram of this mode is shown in Fig. \[fig:otf\].
5. MultiBeamOTF mapping. This mode is proposed to map the sky with 19 beams simultaneously. Compared to the OTF mapping, the MultiBeamOTF mapping mode has similar scanning trajectory but a larger separation (e.g., 20 arcmin) between parallel scans. Besides, the parameter of rotation angle is available in this mode.
6. MutiBeamCalibration. In this mode, 19 beams will be switched in sequence to track the calibrator, allowing for quick calibration of the gain of 19 beams in 30 minutes. Switching time between two beams is 40 s. The integration time of each beam is a parameter for setting. The schematic diagram of this mode is shown in Fig. \[fig:multibeamcal\].
7. BasketWeaving. This mode is to scan the sky along a meridian line. Scanning speed ranges from 5 to 30 arcsec/s. Setting parameters include starting time, starting declination, ending declination, duration time.
8. Snapshot. This mode is used to fully map the sky in grid. This type of mapping is not a Nyquist sampling, but could map a region with relatively deep integration time. This is beneficial especially for pulsar searching. As shown in Fig. \[fig:snapshot\], the movement of 19 beam receiver would ensure fully cover the sky along same Galactic latitude. Necessary parameters include source name, beginning and ending coordinate (RA and Dec), observational time range, and scanning speed (less than 30$'$/s).
The above modes are available for observation at FAST now.
![Maximum tracking time for source at specific declination. Results for zenith angle of 40$^{\circ}$ and 26.4$^{\circ}$ are represented with blue and red color, respectively.[]{data-label="fig:trackingtime"}](FAST_tracktime_dec.png){width="80.00000%"}
![Schematic diagram of OTF mode. The blue line represents scanning trajectory. The top and bottom figure shows scan information along RA and Dec direction, respectively. []{data-label="fig:otf"}](OTF-pattern.png){width="80.00000%"}
![ Schematic diagram of MutiBeamCalibration mode. Green arrow line represents the beam sequence for observing the calibrator. In this plot, the []{data-label="fig:multibeamcal"}](19beam-calibration.png){width="80.00000%"}
![Schematic diagram of snapshot mode. Blue line represents moving direction of 19 beam receiver. Dashed red line represents the direction of one edge of 19 beam receiver. Galactic latitude line is represented with solid red line. The angle between dashed and solid red line is 23.4$^{\circ}$. []{data-label="fig:snapshot"}](snapshot.png){width="80.00000%"}
Effect of Radio Frequency Inference {#subsec:RFI}
-----------------------------------
The current observation hints that RFI in FAST data is mainly divided into three types: narrow-band RFI, 1 MHz wide RFI and some wider fixed frequency RFI. Every type is discussed in more detail below.
1. The narrow-band RFI has ever been ubiquitous through the FAST data. We speculate that there are many origins, like the interference from instruments or the local influence of the telescope. For the FAST spectral data with a frequency resolution of 0.48kHz (divided into one million channels in 500MHz), they are extremely narrow and mostly only appear on one or several channels. Some of the narrow-band RFI could have high strength in a short integration time, but some of the others are just like a faint bulge without the Gaussian profile. In some cases, the narrow-band RFI tend to exhibit the periodic variations in the time domain, which might be found in the pulsar data if its time resolution is less than one second. The narrow-band RFI used to bring a lot of trouble for the FAST data processing, but it has been solved now.
2. While the 1 MHz wide RFI is caused by the standing wave, which looks like regular sinusoidal wave or the single bump in FAST spectra. The big bump originates from the superpose of some standing waves with different amplitudes and periods, sometimes approaching to Gaussian profile. The typical width of this type of RFI is 1 or a few MHz. The existing L-band observation shows that their distribution in time domain and frequency domain is not regular.
3. The fixed frequency RFI is due to satellite or civil aviation from the sky. It usually have a fixed frequency in a wider distribution, and have the strongest intensity in the whole band-pass, even improve the baseline there. Its spectral profile has a complex multi-peak structure, and the width of one peak might be around 20 MHz. The profile and intensity of it varies slightly in different observation time. In addition, the existence of such a wide enough RFI may also be due to the higher sensitivity of FATS. It receives some very strong signal, which results in the frequency width of the response signal exceed the range of satellite signal on both side.
Since the successful installation of the electromagnetic shielding in April 2019, narrow-band RFI has been much reduced. Fig. \[rfi1\] shows the comparison between the spectral results before and after the installation of the electromagnetic shielding. They are observed on August 23, 2018, and April 20, 2019 in 10 minutes integration each day toward TMC-1, without periodic noise injection. These two days’ results exhibit a significant decrease in RFI, suggesting the excellent result of the electromagnetic shielding. For some narrow line-width sources, such as Taurus, their molecular line width is within several or dozens of frequency channels and their RFI problem are very confusing previously. It’s really satisfactory that the tremendous reduction provides great convenience for the spectral line observation and identification. However, the present data still exist some 1 MHz wide RFI and fixed frequency RFI.
In order to study the influence of human activities on FAST data and RFI signal, we observed 10 minutes at daytime and nighttime in one day, supposing that human activity changes with day and night. The source of daytime observations is a quasar, observed at 15:00 on June 23, 2019. The night source is a star observed at 20:00 on the same day. Considering the beam dilution toward the point source and the extremely short integration time, we can assume that no signal from the day and night sources could be received except for emission from the interstellar medium. Therefore, ignoring , almost all of the emission in the obtained spectra should be RFI. Fig. \[rfi2\] shows the comparison of the daytime RFI with nighttime RFI in L band. We can hardly see the narrow-band RFI emission in this figure. While the 1 MHz wide RFI is always existed throughout the band-pass. The total number of its signal is almost invariant, no increase or reduce. But their central frequency slightly shifts from day to night. We magnified the axises of the Fig. \[rfi2\] to check the clearer movement, showing in the Fig. \[rfi3\]. At the same time, the wider RFI from satellites and civil aviation is always located at the same frequency, just with a little variation in intensity.
Hoping to study the small frequency shift of the 1 MHz wide RFI as a result of the standing wave, we have drawn this exact movement in the Fig. \[rfi3\]. The center frequency of RFI is marked as dash dot line. It reveals that the systematic ferquency shift of every RFI emission are all approximate 1 MHz in the figure, form daytime to nighttime. However, the 1 MHz shift doesn’t apply to other FAST observation. We’ve checked the spectral data toward other sources. For the same source observed on different days, this type of RFI moves to different directions in frequency domain, but within a few MHz. While, this shift is uncorrelated with source selection, but only changes in time. The current results implies that the direction to higher or lower frequency and the amplitude of such systematic shift in total band-pass is irregular. It’s in line with our speculation: This type of RFI which looks like a single bump is caused by multiple standing waves superposition. And such a superposition would make the spectra ever-changing. We’ve used this sinusoidal superposition to fit the baseline and the RFI could be removed well. In the future, we will try to do more to solve the problem of standing wave and RFI.
Summary {#sec:summary}
=======
The FAST has achieved its designed objectives. In this paper, we have presented current status of FAST performances. They are summarized as follows.
1. The median power output of low and high noise diode are around 1.1 and 12.5 K. The measured temperature fluctuation of noise diode is $\sim$ 1%, leading to $\sim$ 2% accuracy in flux calibration when pointing accuracy and beam size are included.
2. Spatial distribution and power pattern of 19 beams within zenith angle of 26.4$^{\circ}$ were obtained by mapping observations toward 3C454.3. Beam width of 19 beams is consistent with theoretical estimation.
3. The pointing errors of 19-beam receiver in different sky position are less than 16$''$. The standard deviation of pointing errors is 7.9$''$.
4. The aperture efficiency as a function of zenith angle could be fitted with two linear lines at specific frequency. It keeps almost constant with zenith angle of 26.4$^{\circ}$ and decreases by $\sim$ 1/3 at zenith angle of 40$^{\circ}$.
5. The system temperature as a function of zenith angle could be fitted with a modified arctan function. This fitting is valid for zenith angle within 40$^{\circ}$.
6. The fluctuation of baseline is about $4$% in 30 minutes. Rms of the baseline satisfies the expected sensitivity.
7. Standing wave has an amplitude of $\sim$ 0.3% compared to continuum level. The amplitude, phase of standing wave would vary at different zenith angle during drifting. For position switch observations, standing wave of the residual ON-OFF spectrum would be suppressed with amplitude of $\sim$ 0.02 K, which varies in different frequencies.
8. Derived polarization degree and polarization angle toward 3C286 with FAST are $6.6\pm 1.5$% and 33.4$\pm$6.4 degrees, respectively. They are consistent with results from previous study.
9. Eight observation modes including drifting scan, total power, position switch, On-the-fly, MultiBeamOTF, MultiBeamCalibrtation, BasketWeaving and snapshot are available for FAST observations now.
10. RFI environment has been greatly improved in the last 18 months. Narrow RFI with several spectral channels are reduced. More effort will be done for eliminating broad RFI with width of $\sim$ 1 MHz.
![Comparison of the spectral band-pass in 1000-1500MHz (upper panel plot) and for spectrum in the frequency of 1420 MHz (lower panel plot) toward TMC-1 in different days, August 26th, 2018 (marked as blue line) and April 20th, 2019 (marked as red line).[]{data-label="rfi1"}](RFI_1.png){width="90.00000%"}
![Comparison of the band-pass in the daytime and nighttime with each 10 minutes integration. The nighttime spectrum marked as blue line and the daytime marked as red line.[]{data-label="rfi2"}](RFI_22.png){width="90.00000%"}
![Zooming in the X and Y axes in the Fig. \[rfi2\], the systemic shift of this wide RFI is plotted. Red lines represent daytime and blue lines represent nighttime. The vertical dash dot line shows the center frequency of a single RFI signal, making it easy to see the amplitude of the frequency shift from day to night.[]{data-label="rfi3"}](RFI_33.png){width="64.00000%"}
We thank the beneficial discussion with Mao Yuan. This work is supported by the National Key R & D Program of China (NO. 2017YFA0402701), the National Natural Science Foundation (NNSF) of China (No. 11803051, No. 11833009). TNY was supported by the CAS “Light of West China” program. LGH is additionally supported by the Youth Innovation Promotion Association CAS.
Jiang, P., Yue, Y. L., Gan, H. Q., Yao, R., Li, H., Pan, G. F., Sun, J. H., Yu, D. J., Liu, H. F., Tang, N. Y., Qian, L., Lu, J. G., Yan, J., Peng, B., Zhang, S. X., Wang, Q. M., Li, Q., Li, D. & FAST Collaboration, China-Phys. Mech. Astron. 62, 959502 (2019) Briggs, F. H., Sorar, E., Kraan-Korteweg, R. C., van Driel, W., , 1997, 14, 37-44 Popping, A., and Braun, R., A&A, 2008, 479, 903-913
Condon, J. J., & Yin, Q. F. 2001, , 113, 362
Heiles, C., Perillat, P., Nolan, M., et al. 2001, , 113, 1247
Li, D., Wang, P., Qian, L., et al. 2018, IEEE Microwave Magazine, 19, 112
Campbell, D. B. 2002, Single-Dish Radio Astronomy: Techniques and Applications, 278, 81
Heiles, Carl; Robishaw, Tim; Kepley, Amanda; and Furea Kiuchi. 2012. All-Stokes Single Dish Data with the RHSTK (Robishaw/Heiles SToKes) Software Package.
Perley, R. A., & Butler, B. J. 2013, , 206, 16
Perley, R. A., & Butler, B. J. 2017, , 230, 7
Calabretta, M. R., Staveley-Smith, L., & Barnes, D. G. 2014, , 31, e007
Smith, S. L., Dunning, A., Bowen, M. & Hellicar, A., 2016, IEEE International Symposium on Antennas and Propagation (APSURSI), Fajardo, pp. 383-384.
Dunning, A. et al., 2017, XXXIInd General Assembly and Scientific Symposium of the International Union of Radio Science (URSI GASS), Montreal, QC, pp. 1-4.
Electronics Gain fluctuations
==============================
![The electronics gain fluctuations of the system over several hours. The vertical shows the ratio of fluctuations with respect to the mean temperature over time of Beam 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17 and 19. []{data-label="fig:electrongain_restbeams"}](plot3_1.png "fig:"){width="50.00000%"} ![The electronics gain fluctuations of the system over several hours. The vertical shows the ratio of fluctuations with respect to the mean temperature over time of Beam 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17 and 19. []{data-label="fig:electrongain_restbeams"}](plot3_2.png "fig:"){width="50.00000%"} ![The electronics gain fluctuations of the system over several hours. The vertical shows the ratio of fluctuations with respect to the mean temperature over time of Beam 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17 and 19. []{data-label="fig:electrongain_restbeams"}](plot3_3.png "fig:"){width="50.00000%"} ![The electronics gain fluctuations of the system over several hours. The vertical shows the ratio of fluctuations with respect to the mean temperature over time of Beam 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17 and 19. []{data-label="fig:electrongain_restbeams"}](plot3_4.png "fig:"){width="50.00000%"}
Aperture Efficiency {#aperture-efficiency}
===================
{width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"}
![Measured $\eta$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18. Fitting results when $\theta\rm_{ZA}$ $\leq$ 26.4$^\circ$ and $\theta\rm_{ZA}$ > 26.4$^\circ$ are represented with red and gold solid line, respectively.[]{data-label="fig:etafit_restbeams"}](M12_eta_1400.png "fig:"){width="45.00000%"} ![Measured $\eta$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18. Fitting results when $\theta\rm_{ZA}$ $\leq$ 26.4$^\circ$ and $\theta\rm_{ZA}$ > 26.4$^\circ$ are represented with red and gold solid line, respectively.[]{data-label="fig:etafit_restbeams"}](M13_eta_1400.png "fig:"){width="45.00000%"} ![Measured $\eta$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18. Fitting results when $\theta\rm_{ZA}$ $\leq$ 26.4$^\circ$ and $\theta\rm_{ZA}$ > 26.4$^\circ$ are represented with red and gold solid line, respectively.[]{data-label="fig:etafit_restbeams"}](M14_eta_1400.png "fig:"){width="45.00000%"} ![Measured $\eta$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18. Fitting results when $\theta\rm_{ZA}$ $\leq$ 26.4$^\circ$ and $\theta\rm_{ZA}$ > 26.4$^\circ$ are represented with red and gold solid line, respectively.[]{data-label="fig:etafit_restbeams"}](M15_eta_1400.png "fig:"){width="45.00000%"} ![Measured $\eta$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18. Fitting results when $\theta\rm_{ZA}$ $\leq$ 26.4$^\circ$ and $\theta\rm_{ZA}$ > 26.4$^\circ$ are represented with red and gold solid line, respectively.[]{data-label="fig:etafit_restbeams"}](M16_eta_1400.png "fig:"){width="45.00000%"} ![Measured $\eta$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18. Fitting results when $\theta\rm_{ZA}$ $\leq$ 26.4$^\circ$ and $\theta\rm_{ZA}$ > 26.4$^\circ$ are represented with red and gold solid line, respectively.[]{data-label="fig:etafit_restbeams"}](M17_eta_1400.png "fig:"){width="45.00000%"} ![Measured $\eta$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18. Fitting results when $\theta\rm_{ZA}$ $\leq$ 26.4$^\circ$ and $\theta\rm_{ZA}$ > 26.4$^\circ$ are represented with red and gold solid line, respectively.[]{data-label="fig:etafit_restbeams"}](M18_eta_1400.png "fig:"){width="45.00000%"}
System temperature {#system-temperature}
==================
{width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"}
![Measured $T\rm_{sys}$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. The fitting result is represented with red solid line.[]{data-label="fig:tsysfit_restbeams"}](M12_tsys_1400.png "fig:"){width="45.00000%"} ![Measured $T\rm_{sys}$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. The fitting result is represented with red solid line.[]{data-label="fig:tsysfit_restbeams"}](M13_tsys_1400.png "fig:"){width="45.00000%"} ![Measured $T\rm_{sys}$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. The fitting result is represented with red solid line.[]{data-label="fig:tsysfit_restbeams"}](M14_tsys_1400.png "fig:"){width="45.00000%"} ![Measured $T\rm_{sys}$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. The fitting result is represented with red solid line.[]{data-label="fig:tsysfit_restbeams"}](M15_tsys_1400.png "fig:"){width="45.00000%"} ![Measured $T\rm_{sys}$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. The fitting result is represented with red solid line.[]{data-label="fig:tsysfit_restbeams"}](M16_tsys_1400.png "fig:"){width="45.00000%"} ![Measured $T\rm_{sys}$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. The fitting result is represented with red solid line.[]{data-label="fig:tsysfit_restbeams"}](M17_tsys_1400.png "fig:"){width="45.00000%"} ![Measured $T\rm_{sys}$ curve as a function of zenith angle $\theta\rm_{ZA}$ at 1400 MHz for Beam 1, 2, 8 and 19. The fitting result is represented with red solid line.[]{data-label="fig:tsysfit_restbeams"}](M18_tsys_1400.png "fig:"){width="45.00000%"}
[^1]: https://www.cv.nrao.edu/nvss/postage.shtml
[^2]: http://www.physics.wisc.edu/ craigm/idl/fitting.html
|
---
author:
- 'L. Ducci'
- 'P. Romano'
- 'P. Esposito'
- 'E. Bozzo'
- 'H.A. Krimm'
- 'S. Vercellone'
- 'V. Mangano'
- 'J.A. Kennea'
bibliography:
- 'igrj17354.bib'
title: '*Swift*/XRT orbital monitoring of the candidate supergiant fast X–ray transient IGR J17354–3255'
---
Introduction\[igr17354:intro\]
==============================
was discovered as a hard X-ray transient on 2006 April 21 [@Kuulkers2006:atel874], when it reached a flux of about 18 mCrab ($20-60$ keV) during two $\sim 1.8$ ks observations within the Galactic bulge monitoring program. It was first reported in the fourth IBIS/ISGRI catalog [@Bird2010:igr4cat_mn] with a 20–40 keV average flux of $1.4\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$. It is listed in the [$Swift$]{} [@Gehrels2004mn] Burst Alert Telescope [BAT, @Barthelmy2005:BAT] 58-month Hard X-ray Survey [@Baumgartner2010:BAT58mos Swift J1735.6–3255] with a 14–195 keV average flux of $2.7\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$ and in the 54-month Palermo [$Swift$]{} BAT hard X-ray catalog [@Cusumano2010:batsur_III 2PBC J1735.4–3256] with a 15–150 keV average flux of $(2.12\pm1.15)\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$.
Renewed interest in this object arose due to its positional association with the *AGILE*/GRID gamma-ray transient AGL J1734–3310 [@Bulgarelli2009:atel2017mn]. Thanks to a follow-up [$Swift$]{} target of opportunity (ToO) observation of the field on 2009 April 17, @Vercellone2009:atel2019mn detected two sources (hereafter src1 and src2) within the 4$\arcmin$ error circle [@Kuulkers2006:atel874] at a 0.2–10 keV count rate of $\sim 0.1$ counts s$^{-1}$ (src1) and $\sim 0.005$ counts s$^{-1}$ (src2). Based on a comparison with a previous observation of the same field taken on 2008 March 11 in which src1 was not detected while src2 showed the same count rate, @Vercellone2009:atel2019mn suggested that src2 was a persistent source and src1 the most likely counterpart of . observations [@Tomsick2009:cxc17354] also found two sources within the error circle. CXOU J173527.5-325554 (src1) was the brighter (unabsorbed 0.3–10 keV flux of $\sim1.3\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$), harder ($\Gamma\sim0.6$), and more heavily absorbed ($N_{\rm H}=7.5\times10^{22}$ cm$^{-2}$, $N_{\rm Gal}=1.2\times10^{22}$ cm$^{-2}$), arguing for a high-mass X-ray binary (HMXB) nature. With an unabsorbed 0.3–10 keV flux of $\sim1.4\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$, CXOU J173518.7-325428 (src2) could also contribute to the flux detected by , but @Tomsick2009:cxc17354 also suggested that CXOU J173527.5-325554 (src1) was the counterpart of .
@Dai2011:period_17354 and @Sguera2011:17354 have studied the hard X–ray properties of as they emerge from the extensive BAT and archives, respectively. They thus discovered a modulation of the light curve with a period of $8.4474\pm0.0017$ d [@Sguera2011:17354], which they interpret as the orbital period of the binary. In both works it is shown that the 2009 [$Swift$]{}/X-ray Telescope [XRT, @Burrows2005:XRTmn] observation was performed at the maximum of the folded light curve, while the 2008 one was performed at the minimum, thus strengthening the association of src1 with . Because of its hard X-ray flaring activity (mean flare flux of 20–40 mCrab with one flare peaking at 108 mCrab in the 18–60 keV band) and large dynamic range, @Sguera2011:17354 proposed as a candidate supergiant fast X-ray transient (SFXT) and investigate its association with the transient AGL J1734–3310.
The SFXTs are HMXBs firmly associated with O or B supergiant stars through optical spectroscopy [e.g., @Negueruela2006:ESASP604], which display short [just a few hours long, as observed by ; @Sguera2005; @Negueruela2006] X–ray outbursts with peak luminosities of 10$^{36}$–10$^{37}$ erg s$^{-1}$ and an overall dynamic range of 3–5 orders of magnitude. Their hard X–ray spectra during outburst resemble those of HMXBs hosting accreting neutron stars, with often heavily absorbed, hard power laws below 10 keV combined with high-energy cut-offs at $\sim 15$–30 keV. While the detailed mechanism producing the outbursts is not well established, it is probably related to either the properties of the wind from the supergiant companion [@zand2005; @Walter2007; @Negueruela2008; @Sidoli2007] or the presence of a centrifugal or magnetic barrier [@Grebenev2007; @Bozzo2008]. Recently, some evidence has been accumulating that indicates that SFXTs may be the X-ray counterparts of a new class of short Galactic transients emitting in the MeV/GeV regime [see, e.g., @Sguera2009].
In this paper we analyze all the [$Swift$]{}/XRT data collected on the region of (Sect. \[igr17354:dataredu\]). This allows us to unambiguously identify the soft X–ray counterpart of the candidate SFXT with src1, based on its timing properties (Sect. \[igr17354:results\]). Furthermore, we show the results of the most intense and complete sampling along the orbital period of the light curve of this candidate SFXT with a sensitive soft X-ray instrument and discuss the nature of the companion (Sect. \[igr17354:discuss\]).
$^{\mathrm{a}}$ Mean phase referred to $P_{\rm orb}=8.4474$ d and $T_{\rm epoch}=$ MJD 52698.205.
![Region around , as observed by [$Swift$]{}/XRT (0.3–10 keV). All data collected by XRT were used. The large circle is the 90% error box [1.4$\arcmin$, @Bird2010:igr4cat_mn]. The crosses mark the two XRT sources in the field of (src1 within the error circle, src2 outside of it). The black circle represents the position of the 2MASS counterpart of . []{data-label="igr17354:fig:map"}](figure1.eps){width="\columnwidth"}
![[$Swift$]{}/XRT 0.3–10 keV light curves of the two X-ray sources detected within the region around , during the 2012 July monitoring program. Downward-pointing arrows are 3$\sigma$ upper limits. Top: Swift J173527.7–325555 (src1, identified with the soft X-ray counterpart of ). Different colors mark different days (see Table \[igr17354:tab:xrtobs\]). Bottom: Swift J173518.7–325428 (src2). []{data-label="igr17354:fig:xrtlcv_mjd"}](figure2a.ps){width="9cm"}
![[$Swift$]{}/XRT 0.3–10 keV light curves of the two X-ray sources detected within the region around , during the 2012 July monitoring program. Downward-pointing arrows are 3$\sigma$ upper limits. Top: Swift J173527.7–325555 (src1, identified with the soft X-ray counterpart of ). Different colors mark different days (see Table \[igr17354:tab:xrtobs\]). Bottom: Swift J173518.7–325428 (src2). []{data-label="igr17354:fig:xrtlcv_mjd"}](figure2b.ps){width="9cm"}
Observations and Data Analysis \[igr17354:dataredu\]
====================================================
We first considered the hard X-ray data from the [$Swift$]{}/BAT. has never triggered BAT onboard, but it is detected in the Hard X-ray Survey[^1] at 9.6$\sigma$ confidence level. The light curve, which spans the 65 months between MJD 53373.66 and 55347.65 (2005 January 3 to 2010 May 31), shows a fairly steady level with no significant long-term variability. Two short flares are found, with fluxes consistent with -detected flares, on 2006 March 4 22:44 UT (5.6$\sigma$, lasting 1044 s), when the source reached 0.017 counts s$^{-1}$ cm$^{-2}$ (14–195 keV), and on 2006 March 25 18:29 UT (5.7$\sigma$, 1324 s), reaching 0.010 counts s$^{-1}$ cm$^{-2}$.
We also retrieved the IBIS/ISGRI light curve of from the online tool Heavens[^2], which has access to all public available data on this source, between MJD 52671.43 and 55624.28 (2003 February 1 to 2011 March 4). We selected the 17–50 keV energy band to maximize the signal-to-noise ratio.
{width="17.5cm"}
The log of the [$Swift$]{}/XRT observations used in this paper is in Table \[igr17354:tab:xrtobs\]. The [$Swift$]{} observations of during 2008 and 2009 were extensively described in @Vercellone2009:atel2019mn, @Dai2011:period_17354, and @Sguera2011:17354; here they are re-analyzed to ensure uniformity. The data collected in 2012 July were obtained as a ToO monitoring program of ten scheduled daily observations, which were each 5 ks long and equally spread in the four quarters of the day, starting on 2012 July 18. This strategy aimed at covering the light curve in all orbital phases while keeping the observing time per day reasonably short, thus not impeding observations of gamma-ray bursts (GRBs), which are the main scientific target for [$Swift$]{}. The 2012 campaign lasted 11 days divided in 22 observations for a total on-source exposure of $\sim$ 24 ks.
The XRT data were uniformly processed with standard procedures ([xrtpipeline]{} v0.12.6), as well as filtering and screening criteria by using [FTOOLS v6.12]{}. Within the XRT field of view (FOV) around , two sources were detected (see Fig. \[igr17354:fig:map\]). In both cases the source count rates never exceeded $\sim0.5$ count s$^{-1}$, so only photon-counting mode (PC) events (selected in grades 0–12) were considered and we checked that no pile-up correction was required. Source events were accumulated within a circular region with a radius of 20 pixels (1 pixel $\sim2.36$), and background events were accumulated from a nearby source-free region.
Light curves were created for both sources and corrected for point-spread function (PSF) losses, vignetting, and were background subtracted (see Fig. \[igr17354:fig:xrtlcv\_mjd\]). For src1 the light curve was binned to ensure at least 20 counts per bin, whenever the statistics allowed it. For src2 we accumulated all data within a day to obtain detections whenever possible. For our spectral analysis, we extracted events in the same regions as those adopted for the light curve creation; ancillary response files were generated with [xrtmkarf]{}. We used the latest spectral redistribution matrices in CALDB (20120713). For a more detailed discussion of the data analysis procedure, please refer to @Romano2011:sfxts_paperVI [and references therein].
All quoted uncertainties are given at 90% confidence level (c.l.) for one interesting parameter, unless otherwise stated.
Results \[igr17354:results\]
=============================
Swift J173527.7–325555 (src1) is located at RA(J$2000)=17^{\rm h}\, 35^{\rm m}\, 27\fs66$, Dec(J$2000)=-32^{\circ}\, 55^{\prime}\, 55\farcs1$ with an uncertainty radius of 20 (90% c.l.; the astrometrically corrected position was determined according to @Evans2009:xrtgrb_mn and @Goad2007:xrtuvotpostions). This position is $1\farcs1$ from the unidentified source CXOU J173527.5–325554 [@Tomsick2009:cxc17354] and $0\farcs9$ from the source 2MASS J17352760–3255544. In the following we report on src1, while the properties of src2 are described in Sect. \[igr17354:src2\].
Light curve analysis \[igr17354:timing\]
----------------------------------------
![Spectrum of the 2009 observation (grey) and 2012 campaign (black). Top: data fit with an absorbed power-law model (see Table \[igr17354:tab:xrtspec\]). Bottom: the residuals of the fits (in units of standard deviations). []{data-label="igr17354:fig:spec"}](figure4.ps){width="9cm"}
Fig. \[igr17354:fig:xrtlcv\_mjd\] (top) shows the 0.3–10 keV light curve of Swift J173527.7–325555 during the 2012 campaign. The light curve begins at phase 0.8 (see Table \[igr17354:tab:xrtobs\]) and covers the source through a little over one full period, ending at phase 0.98. Fig. \[igr17354:fig:xrtlcv\_phase\] shows the folded light curve, which includes the data collected in 2012 (black) and the 2008 and 2009 data (grey). We also include the 3-$\sigma$ upper limit obtained from a 19 ks exposure on 2011 March 6 with [@Bozzo2012:HMXBs]. The XRT count rate to flux conversion was derived from the spectral fit of the mean 2012 spectrum (see Sec. \[igr17354:spectroscopy\]). Superimposed on the long-term orbital modulation, which follows the one seen in the BAT [@Dai2011:period_17354 fig. 3c] and data [@Sguera2011:17354 fig. 4] we also observe flaring on short time scales (hundred of seconds, e.g., grey points in Fig. \[igr17354:fig:xrtlcv\_phase\]). This behavior has been observed in most SFXTs [@Sidoli2008:sfxts_paperI; @Romano2009:sfxts_paperV; @Romano2011:sfxts_paperVI], in particular those we monitored along the orbital period [@Romano2010:sfxts_18483; @Romano2012:sfxts_16418]. A noteworthy feature is the dip centered at $\phi \sim
0.7$, which starts at phase $\phi \sim 0.60$ and lasts $\Delta \phi \sim 0.2$–$0.24$. The lowest limit (as observed by XRT) was collected during the dip by combining five observations (00032513017 through 00032513021 for a total on-source exposure of 5.9 ks, shown as a larger downward-pointing arrow at $\phi\sim0.66$). These observations yielded a 3-$\sigma$ upper limit at $3.5\times10^{-3}$ counts s$^{-1}$. This corresponds to an unabsorbed 0.3–10 keV flux of $\sim 1.4 \times10^{-12}$ erg cm$^{-2}$ s$^{-1}$ and to a 2–10 keV luminosity of $8\times10^{33}$ erg s$^{-1}$ (assuming the optical counterpart distance of 8.5 kpc, as derived from the closeness to the Galactic center and the high X-ray absorption; @Tomsick2009:cxc17354). In addition, we note that the recent observation [@Bozzo2012:HMXBs], which was also obtained at phase 0.66, yielded a 3-$\sigma$ upper limit at $2\times10^{-3}$ counts s$^{-1}$ (0.5–10 keV) for the Metal Oxide Semi-conductor (MOS) cameras. This corresponds to an unabsorbed 0.3–10 keV flux of $\sim 7\times10^{-14}$ erg cm$^{-2}$ s$^{-1}$ when adopting the best-fitting spectral model for the 2012 XRT campaign (see Sect. \[igr17354:spectroscopy\]). The limit corresponds to a 2–10 keV luminosity of $4\times10^{32}$ erg s$^{-1}$.
The lowest point during the campaign [*outside*]{} the dip was recorded on MJD 56126.67 at 0.023 counts s$^{-1}$, a detection corresponding to an unabsorbed 0.3–10 keV flux of $9.1\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$ and to a luminosity of $5\times10^{34}$ erg s$^{-1}$. During the 2012 campaign, the peak count rate was $\ga 0.3$ counts s$^{-1}$ (MJD 56130.00). The highest recorded count rate, however, was achieved on 54938.33 at $\ga 0.42$ counts s$^{-1}$ or $\sim 1.7 \times10^{-10}$ erg cm$^{-2}$ s$^{-1}$ ($9\times10^{35}$ erg s$^{-1}$).
After having converted the event arrival times to the Solar system barycentric frame, we searched the XRT data for periodic pulsations longer than $\sim 5$ s by means of a Fourier transform. No significant signal was found. Because of the limited statistics and the presence of substantial noise in the time series, the upper limits on the source pulsed fraction are non constraining (being larger than 100%).
The observed orbital modulation in the folded light curves of this XRT source and , which has a dip corresponding to the minimum observed in the folded BAT and light curves (Fig. \[igr17354:fig:folded\]), allows a definitive identification of Swift J173527.7–325555 as the soft X–ray counterpart of . This identification was previously mainly based on positional association.
Spectroscopy \[igr17354:spectroscopy\]
---------------------------------------
[lcccrr]{} ObsID &$N_{\rm H}$ &$\Gamma$ &Flux$^a$ &L$^{b}$ &$\chi^{2}_{\nu}/$dof\
& (10$^{22}$ cm$^{-2}$) & & & &\
00037054002 &$7.80_{-2.16}^{+2.91}$ &$1.65_{-0.53}^{+0.63}$ &$2.0$ & 1.7 &$1.04/18$\
00032513001–25 &$10.7_{-2.1}^{+2.6}$ &$1.68_{-0.44}^{+0.48}$ &$1.0$& 0.9 &$1.17/42$\
We extracted the mean spectrum during the 2009 observation (the grey points at phase 0.17 in Fig. \[igr17354:fig:xrtlcv\_phase\]) and the mean spectrum of the 2012 campaign. The data were rebinned with a minimum of 20 counts per energy bin to allow $\chi^2$ fitting. The spectra were fit in the 0.3–10 keV energy range with a single absorbed power-law model as more complex models were not required by the data. The absorbing column is significantly in excess of the Galactic one [$1.59\times 10^{22}$ cm$^{-2}$; @LABS], while the photon index is $\Gamma\sim1.7$. The peak flux was reached in 2009 with its average 2–10 keV unabsorbed flux of $\sim 2\times 10^{-11}$ erg cm$^{-2}$ s$^{-1}$, corresponding to $\sim 1.7\times10^{35}$ erg s$^{-1}$ at 8.5 kpc). This is about a factor of 2 higher than the average spectrum obtained in 2012. The results are reported in Table \[igr17354:tab:xrtspec\] and the best fits are shown in Fig. \[igr17354:fig:spec\].
Properties of Swift J173518.7–325428 \[igr17354:src2\]
-------------------------------------------------------
Swift J173518.7–325428 (src2) is located at RA(J$2000)=17^{\rm h}\, 35^{\rm m}\, 18\fs73$, Dec(J$2000)=-32^{\circ}\, 54^{\prime}\, 27\farcs5$, with an uncertainty radius of 46 (90% c.l.). This source is clearly outside the error circle of and is thus not a viable counterpart. For completeness, we report the analysis of this source. Our observations highlight a variability in its light curve, shown in Fig. \[igr17354:fig:xrtlcv\_mjd\] (bottom), with a dynamical range of 4.5 (at a binning of $\sim 1000$ s). The mean spectrum was extracted from all the data collected in 2012 and fit with an absorbed power-law model in the 0.3–10 keV energy range with @Cash1979 statistics. The fit yielded an absorbing column $N_{\rm H}=1.33_{-0.65}^{+0.84}\times 10^{22}$ cm$^{-2}$, which is consistent with the Galactic one [$1.59\times 10^{22}$ cm$^{-2}$; @LABS], and $\Gamma=1.1_{-0.5}^{+0.6}$. The average observed and unabsorbed fluxes (0.3–10 keV) are $F_{\rm 0.3-10 keV}^{\rm obs}=7.1\times10^{-13}$ erg cm$^{-2}$ s$^{-1}$ and $F_{\rm 0.3-10 keV}=9.2\times10^{-13}$ erg cm$^{-2}$ s$^{-1}$, respectively, and are consistent with the findings of @Tomsick2009:cxc17354. Although the optical counterpart of Swift J173518.7–325428 is not detected, the observed properties of this source, as also noted by @Tomsick2009:cxc17354, are consistent with an X–ray binary.
Discussion \[igr17354:discuss\]
================================
We performed a monitoring of the FOV around , which is an transient positionally associated with the gamma-ray transient AGL J1734$-$3310. Our new XRT data represent the first soft X-ray monitoring of more than one orbital period of , allowing us to highlight conspicuous similarities between the folded light curves of the XRT source and . Consequently, they provide a definitive identification of the soft X–ray counterpart of . In contrast, only a positional association, albeit one supported by the consistency of high and low X-ray flux with the hard X-ray flux at the same orbital phase, was available before.
Based on its hard X–ray behavior, is considered a candidate SFXT by @Sguera2011:17354. They show that it is a weak persistent hard X-ray source (average out-of-outburst flux of 1.1 mCrab) with rare, fast (hr to a few days, with a peak at $\approx100$ mCrab) flares with a dynamic range $>200$. They also quote a soft X-ray dynamic range, which is based on the 2008 and 2009 XRT observations only, in excess of 300.
The X-ray flaring activity of involves intensity increases by a factor of $\sim 10 - 300$ on time scales of hundreds to thousands of seconds. These increases are hardly reconciliable with a system accreting from a disc. In fact, an accretion disc would smooth out the accretion rate variations produced by wind inhomogeneities with a time scale shorter than the viscous time scale of the accretion disc. In contrast, the observed X-ray variability, together with the broadband X-ray spectrum and the measured orbital period [@Dai2011:period_17354] is consistent with a neutron star fed by a strong stellar wind (see, e.g., @Negueruela10).
Our sensitive monitoring of this source in the soft X–ray allowed us to observe the presence of a dip at phases $0.6 \lesssim \phi \lesssim 0.85$, which correspond to a duration of $\Delta t_{\rm XRT} \lesssim 2.1$ d. This dramatic decrease in soft X-ray flux was seen during the [$Swift$]{}/XRT 2008 observation and twice in the 2012 campaign. It is also independently reported by a different soft X–ray instrument in the 2011 observation [@Bozzo2012:HMXBs]. In Fig. \[igr17354:fig:folded\]b and c we show the BAT and IBIS/ISGRI folded light curves. The dip observed in the BAT data nicely corresponds to what is observed in the XRT folded light curve. When fitted with a sinusoidal function, it has a centroid at $\phi\approx0.70$ and a width (corresponding to the portion of the sinusoidal function with counts below zero) spanning from 0.57 and 0.81 in phase. We modeled the IBIS/ISGRI light curve, which has a much larger count statistics than the BAT one, with two sinusoidal functions and obtained a minimum at $\phi \approx 0.66$ and a width spanning from 0.56 and 0.75 in phase. The soft X-ray dynamic range from XRT monitoring is $\sim 18$ outside of the dip and $\sim2400$ considering the deep *XMM-Newton* upper-limit.
![Histogram of probability distribution of luminosity of , observed with XRT. The vertical dashed line corresponds to the luminosity threshold below which the probability distribution of luminosity is not well reproduced due to the sensitivity of XRT.[]{data-label="igr17354:fig:histoobs"}](isto_obs_igrj17354.ps){width="9cm"}
Fig. \[igr17354:fig:histoobs\] shows the probability distribution histogram of luminosity of , which was obtained with the XRT observations in which the source is significantly detected. Because of the sensitivity of XRT, luminosities below $L_{\rm x}=1.6 \times 10^{35}$ erg s$^{-1}$ have been obtained with large integration times. Therefore, the luminosity distribution on the left side of the vertical dashed line of Fig. \[igr17354:fig:histoobs\] does not reproduce the real luminosity distribution of .
looks intrinsically less variable in the soft X–ray than other SFXTs monitored with XRT along their orbital periods. In the case of the confirmed SFXT IGR J18483–0311, which many consider to be an intermediate SFXT, the light curve shows a large modulation with the orbital phase [@Romano2010:sfxts_18483 fig. 1], which can be interpreted as wind accretion along a highly eccentric orbit. The dynamic range of this source, which was calculated excluding the dip in the light curve, exceeds 580 (and 1200 including the dip data). Furthermore, variability is observed on short time scales, superimposed on the long-term orbital modulation of IGR J18483–0311. It has variations by a factor of a few in count rate occurring in $\sim 1$ hr, which can be naturally explained by clumps in the accreting wind. In the case of IGR J16418–4532 [@Romano2012:sfxts_16418 fig. 4], in contrast, the X-ray light curve does not show a very strong orbital modulation, suggesting that the system has moderate eccentricity. The observed dynamical range of this source is at least 370, (1400 considering the points within the observed eclipse) and also places IGR J16418–4532 among the intermediate SFXTs.
Similar to IGR J16418–4532, has a light curve with very little orbital modulation, and the short time-scale flares we observe superimposed on it are comparatively of a lower dynamical range. Therefore, our data indicate that this source is a weak, almost persistent source in the soft X-rays (apart from the dip), which has not shown any remarkable activity during the 2012 XRT monitoring or in the previous observations.
The X-ray variability of IGR J17354–3255 is also very similar to that shown by Vela X–1, a persistent wind-fed HMXB composed of a B0.5Ib star and a pulsar with an orbital period of 8.96 d (similar to the orbital period of ) and an eccentricity of $\sim 0.09$ (e.g., @Quaintrell2003). Vela X–1 shows an X-ray variability with a dynamic range of 20–30 on time scales of a few hours and an average luminosity of $\sim 4\times 10^{36}$ erg s$^{-1}$ (@Fuerst2010; @Kreykenbohm2008). The lower luminosity of compared with Vela X–1 can be explained by reasonable values for the stellar wind parameters of the donor star.
On the basis of the above considerations, it is reasonable to assume that is a wind-fed system. In this scenario, we discuss three different hypotheses for the origin of the dip of :
1. In the framework of wind accretion along a highly eccentric orbit, the dip is due to the apastron passage of the compact object, where the faster and less dense wind reduces the amount of accreted material and consequently the X-ray luminosity.
2. The dip is caused by the onset of gating mechanisms at apastron, which become effective owing to the lower accretion rate (see, e.g., @Bozzo2008).
3. The dip is produced by an eclipse.
We discuss these three different scenarios by comparing the X-ray luminosities observed by *Swift*/XRT at different orbital phases with those calculated with a model based on the Bondi-Hoyle-Lyttleton accretion theory (@Bondi1952; @Bondi1944; @Hoyle1939, BHL hereafter). The BHL accretion theory is usually applied to X-ray binaries where the donor star produces a fast and dense stellar wind that is assumed to be homogeneous.
Since the spectral type of the donor star of is unknown, we considered both the accretion from a spherically symmetric wind (if the donor star is a supergiant or a giant/main sequence star without circumstellar disc) and the accretion from a circumstellar disc (if the donor star is a giant/main sequence). The equations that we used to compute the luminosity light curves are described in the following paragraphs. The results are discussed in Sects. \[sect. orbital modulation\] and \[sect. eclipse or c.i.\].
OB supergiant {#ob-supergiant .unnumbered}
-------------
The winds of OB supergiants are spherically symmetric with good approximation (e.g., @Kudritzki00) with a velocity law: $$\label{eq. beta-velocity law}
v(r) \simeq v_\infty \left ( 1 - \frac{R_{\rm d}}{r} \right )^\beta ,$$ called $\beta$-velocity law (@Castor1975), where $v_\infty$ is the terminal velocity, $\beta$ determines how steeply the wind velocity reaches $v_\infty$ ($0.5 \lesssim \beta \lesssim 1.5$), and $r$ is the distance from the center of the supergiant star.
The density distribution around the donor star is given by the continuity equation: $$\label{eq. continuity}
\rho(r) = \frac{\dot{M}}{4 \pi r^2 v(r)} ,$$ where $\dot{M}$ is the mass loss rate. We applied the definition of accretion radius $R_{\rm acc} $ of @Bondi1952 to find the mass accretion rate $\dot{M}_{\rm acc}$: $$\label{eq. Macc}
\dot{M}_{\rm acc} = \rho(r) v_{\rm rel}(r) \pi R_{\rm acc}^2(r) = \rho(r) v_{\rm rel}(r) \pi \left[ \frac{2GM_{\rm ns}}{v_{\rm rel}^2(r)} \right ]^2 ,$$ where $v_{rel}(r)$ is the relative velocity between the neutron star and the wind: $$\label{eq vel}
v_{\rm rel}(r) = \left \{ [v(r) - v_{\rm r}(r)]^2 + v_\phi^2(r) \right \}^{1/2} ,$$ where $v_{\rm r}$ and $v_\phi$ are the radial and tangential components of the orbital velocity and $v$ is the wind velocity (Equation \[eq. beta-velocity law\]). We thus obtain the X-ray luminosity produced by the accretion $$\label{eq Lx sgxb}
L_x \approx \frac{G M_{\rm ns}}{R_{\rm ns}} \dot{M}_{\rm acc} = \frac{(G M_{\rm ns})^3}{R_{\rm ns}} \frac{4 \pi \rho(r)}{v_{\rm rel}^3(r)}.$$
OB giants/main sequence {#ob-giantsmain-sequence .unnumbered}
-----------------------
The rapidly rotating Oe and Be main sequence/giant stars can be surrounded by a circumstellar envelope of gas confined along the equatorial plane. Most OBe/X-ray binaries are transient in X-rays, with eccentricities $e \gtrsim 0.3$. They can show periodic or quasi-periodic outbursts (called *type I*), which cover a small fraction of the orbital period and peaked at the periastron passage of the neutron star, or they can show giant outbursts (called *type II*), which last for a large fraction of the orbit and in some cases for several orbital periods. These giant outbursts have peak luminosities of $L_{\rm x} \gtrsim 10^{37}$ erg s$^{-1}$, which are larger than those observed during type I outbursts. Type I and II outbursts are believed to be due to the interaction between the neutron star and the circumstellar disc of the donor star. Persistent OBe/X-ray binaries do not display large outbursts and their X-ray luminosities are $L_{\rm x} \lesssim 10^{35}$ erg s$^{-1}$ (see, e.g., @Reig2011).
In the framework of the wind model developed by @Waters1989 to reproduce the X-ray luminosities of wind-fed neutron stars in eccentric orbits around the Oe-Be stars, we assumed a density distribution in the circumstellar disc $$\label{eq. density disc}
\rho_{\rm disc}(r) = \rho_0 \left( \frac{r}{R_{\rm d}} \right )^{-n} ,$$ where $\rho_0=10^{-11}$ g cm$^{-3}$, and $2.1<n<3.8$ [@Waters1988]. The radial wind velocity component is $$\label{eq. radial wind velocity}
v_{\rm r,w}(r) = v_0 \left ( \frac{r}{R_{\rm d}} \right )^{n-2} ,$$ where $v_0$ ranges between $2$ and $20$ km s$^{-1}$. The rotational wind velocity component is $$\label{eq rotational wind velocity}
v_{\rm rot,w} (r) = v_{\rm rot,0} \left ( \frac{r}{R_{\rm d}} \right )^{-\alpha} ,$$ where $\alpha=0.5$ for Keplerian rotation or $\alpha=1$ if the angular momentum of the outflowing matter is conserved. The relative velocity between the neutron star and the wind is $$\label{eq. rel velocity disc}
v_{\rm rel,disc}(r) = \left \{ [v_{\rm r,w}(r) - v_{\rm r}(r)]^2 + [v_{\rm rot,w}(r) - v_\phi(r)]^2 \right \}^{1/2} .$$ If the circumstellar disc is not formed (or outside of it) the wind properties are the same of those found for OB supergiants, with lower mass loss rates and terminal velocities ($10^{-10} \lesssim \dot{M} \lesssim 10^{-8}$ $M_\odot$ yr$^{-1}$; $600 \lesssim v_\infty \lesssim 1800$ km s$^{-1}$; see, e.g., @Waters1988).
{width="8cm"} {width="8cm"}
Orbital modulation {#sect. orbital modulation}
------------------
First we considered the possibility that the *XMM-Newton* dip is due to the luminosity modulation produced by a neutron star moving in a highly eccentric orbit. For this calculation we did not take into account the duration of the dip observed by XRT, which can only be considered as an upper limit.
Since the broadband spectrum is typical of accreting neutron stars in HMXBs (@Sguera2011:17354; @Dai2011:period_17354), we focused our attention on the case of a neutron star with mass $M_{\rm ns}=1.4$ $M_\odot$ and radius $R_{\rm ns}=10$ km accreting the wind material of an OB star. We calculated the X-ray luminosities as a function of the orbital phase, assuming a neutron star accreting material in a circumstellar disc or in a spherically symmetric wind and a supergiant, giant, main-sequence companion star.
We first computed $3\times 10^{6}$ light curves[^3] by varying the masses and radii of OB supergiants (using the values reported in @Martins2005 and @Searle2008) and for different values of mass loss rate $5\times 10^{-7} <\dot{M}<5\times 10^{-6}$ $M_\odot$ yr$^{-1}$, terminal wind velocity $900<v_\infty<1900$ km s$^{-1}$, and eccentricities $0<e<0.8$. We set $\beta=1$ because we ascertain that variations of this parameter do not produce appreciable variations in the X-ray luminosity.
Then, we considered the case of OB giant/main sequence stars with circumstellar disc. We calculated $2.5\times 10^{6}$ light curves by varying the masses and radii of OBV (using the mass-radius relation of @Demircan1991) and OBIII (using the Catalogue of Apparent Diameters and Absolute Radii of Stars \[CADARS\] of @Pasinetti-Fracassini2001; @Hohle2010; @Martins2005), and for different wind parameters and eccentricities: $2.1 \lesssim n \lesssim 3.8$, $2 \lesssim v_0 \lesssim 22$ km s$^{-1}$, $150 \lesssim v_{\rm rot,0} \lesssim 310$ km s$^{-1}$, $\alpha=0.5$ or $1$, $0<e<0.8$.
In the scenario of OB giant/main sequence stars without the circumstellar discs, we calculated $1.5\times 10^{6}$ light curves assuming different spectral types for the donor star, mass loss rates $10^{-10} \lesssim \dot{M} \lesssim 10^{-8}$ $M_\odot$ yr$^{-1}$, terminal velocities $600 \lesssim v_\infty \lesssim 1800$ km s$^{-1}$, eccentricities $0<e<0.8$, and $\beta=1$.
We computed the X-ray luminosities using Equation (\[eq Lx sgxb\]), with $v_{\rm rel}(r)$ and $\rho(r)$ given by Equations (\[eq vel\]) and (\[eq. continuity\]) in the spherically symmetric wind scenario and by Equations (\[eq. rel velocity disc\]) and (\[eq. density disc\]) if the donor star is an OB main sequence/giant and a circumstellar disc is present.
In all cases, no combination of the orbital and wind parameters can reproduce the observed out-of-dip ($0 < \phi < 0.6$ and $0.85 < \phi < 1$) luminosities ranging from $\sim 6 \times 10^{34}$ erg s$^{-1}$ to $\sim 3 \times 10^{36}$ erg s$^{-1}$ ($d=8.5$ kpc) and luminosities below $4 \times 10^{32}$ erg s$^{-1}$ at $\phi \approx 0.65$ (corresponding to the *XMM-Newton* observation). In fact, the observed dynamic range (considering the *XMM-Newton* upper limit) requires highly eccentric orbits. However, when $e>0$ the neutron star spends a large fraction of time in the apastron region (Kepler’s second law). Therefore, all calculated light curves reproducing the *XMM-Newton* upper limit at $\phi \approx 0.65$ show luminosities lower than $\approx 10^{34}$ erg s$^{-1}$ during orbital phases longer than the dip observed by XRT.
In conclusion, it is unlikely that the upper limit observed by *XMM-Newton* is due to the apastron passage of the neutron star.
![Histograms of the mean luminosity (averaged over the orbit) obtained with the BHL model, assuming the parameters giving the solutions of Fig. \[igr17354:fig:histo2d\], $0<e<0.64$ for an OB supergiant, $0<e<0.43$ for an OB main sequence/giant star.[]{data-label="igr17354:fig:histo1d"}](1Dsgxb.ps "fig:"){height="5.5cm"} ![Histograms of the mean luminosity (averaged over the orbit) obtained with the BHL model, assuming the parameters giving the solutions of Fig. \[igr17354:fig:histo2d\], $0<e<0.64$ for an OB supergiant, $0<e<0.43$ for an OB main sequence/giant star.[]{data-label="igr17354:fig:histo1d"}](1DOBe.ps "fig:"){height="5.5cm"}
Eclipse and gating mechanisms {#sect. eclipse or c.i.}
-----------------------------
Similarly to Sect. \[sect. orbital modulation\], we compared the observed X-ray luminosities with those obtained with a BHL model to determine the eccentricities of the system, the wind properties, and the nature of the donor star that reproduces the observed out-of-dip X-ray luminosities, which range from $\sim 6 \times 10^{34}$ erg s$^{-1}$ to $\sim 3 \times 10^{36}$ erg s$^{-1}$ ($d=8.5$ kpc).
Since the observational properties of the dip (duration and luminosity) in both the eclipse and gated mechanisms cases do not depend solely on the wind and orbital parameters[^4], we treated these two scenarios as a single one.
The 2D histograms of Fig. \[igr17354:fig:histo2d\] show the solutions that reproduce the observed X-ray luminosities obtained assuming the accretion from a spherically symmetric wind produced by a supergiant (left panel) or a giant/main sequence star (right panel). For each solution we also plotted the mean luminosity $\bar{L}_x$ (averaged over the orbit) in the histograms of Fig. \[igr17354:fig:histo1d\]. These histograms show that the X-ray luminosity produced by the accretion from a supergiant wind is on average higher than the X-ray luminosity produced by the accretion of the wind from a giant or a main sequence OB star. The allowed eccentricities (obtained by comparing the observed X-ray luminosities with those calculated with the BHL model) are $0 \lesssim e \lesssim 0.64$ in the supergiant scenario and $0 \lesssim e \lesssim 0.43$ in the giant/main sequence case.
We also considered the case of a neutron star embedded in the wind of the circumstellar disc of the donor star. We found that the observed X-ray luminosities can be reproduced assuming a main sequence donor star with mass $M_{\rm d}=8$ $M_\odot$, $R_{\rm d}=4$ $R_\odot$, $v_0=20$ km s$^{-1}$, $n=3.5$, $150<v_{\rm rot}<300$ km s$^{-1}$, $\alpha=0.5$ or $1$, $0<e<0.02$. The X-ray luminosities obtained using these parameters are about $3 \times 10^{36}$ erg s$^{-1}$. Assuming distances larger than $8.5$ kpc, more solutions are possible. Nonetheless, since the truncation of the circumstellar disc produced by the neutron star (@Reig97; @Negueruela01; @Okazaki01) is expected to be more efficient in systems with low eccentricities and narrow orbits (see @Reig2011 and references therein), it is unlikely that the neutron star of accretes the material of the circumstellar disc.
Summary {#sect. summary}
=======
We reported on *Swift*/XRT observations of the candidate SFXT , which provided for the first time a definitive identification of its soft X-ray counterpart. They also allowed us to observe the presence of a dip in the XRT light curve of folded at the orbital period. Apart from the dip, the low dynamic range observed with *Swift*/XRT indicates that is an almost persistent source in the soft X-rays.
We investigated the origin of the dip by comparing the XRT folded light curve with those calculated with models based on the BHL accretion theory. We assumed both spherical and nonspherical symmetry of the outflow from the donor star. We found that the dip cannot be explained with a luminosity modulation produced by a neutron star in a highly eccentric orbit and showed that an eclipse or the onset of a gated mechanism can explain the dip. We also determined the eccentricities of the system, the wind properties, and the nature of the donor star that reproduces the observed out-of-dip luminosities.
We thank the anomymous referee for constructive comments which helped to improve the paper. We thank the [*Swift*]{} team duty scientists and science planners. We also thank the remainder of the [*Swift*]{} XRT and BAT teams, S.D. Barthelmy, J.A. Nousek, and D.N. Burrows in particular, for their invaluable help and support of the SFXT project as a whole. We thank P.A. Evans and C. Ferrigno for helpful discussions. We acknowledge financial contribution from the contract ASI-INAF I/004/11/0. This work made use of the results of the Swift/BAT hard X-ray transient monitor: http://swift.gsfc.nasa.gov/docs/swift/results/transients/
[^1]: http://swift.gsfc.nasa.gov/docs/swift/results/bs58mon/SWIFT\_J1735.6-3255
[^2]: http://www.isdc.unige.ch/heavens
[^3]: The numbers of light curves computed in the three cases of accretion from a spherically symmetric wind with (1) supergiant, (2) giant or main sequence donor star, (3) accretion from a circumstellar disc, depend on the number of steps used for the orbital and wind parameters and their allowed ranges.
[^4]: In the eclipse scenario, the duration of the eclipse depends on the orbital separation, the radius of the donor star, and the inclination of the orbital angular momentum vector with respect to the line of sight to the Earth. In the gated mechanisms scenario, the dip can be produced in different epochs at a given orbital phase if $e > 0$ and for particular values of the spin period and magnetic field strength of the neutron star.
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---
abstract: 'For systems with only short-range forces and shallow 2-body bound states, the typical strength of any 3-body force in all partial-waves, including external currents, is systematically estimated by renormalisation-group arguments in the Effective Field Theory of Point-Like Interactions. The underlying principle and some consequences in particular in Nuclear Physics are discussed. Details and a better bibliography in Ref. [@Griesshammer:2005ga].'
author:
- 'Harald W. Grie[ß]{}hammer[^1] [^2]'
title: 'How To Classify 3-Body Forces – And Why'
---
Introduction
============
The Effective Field Theory (EFT) of Point-Like Interactions is a model-independent approach to systems without infinite-range forces in Atomic, Molecular and Nuclear Physics at very-low energies with shallow real or virtual 2-body bound-states (“dimers”), see e.g. [@bedaque_bira_review; @Braaten:2004rn] for reviews. When the size or scattering length $a$ of a 2-body system is much larger than the size (or interaction range) $R$ of the constituents, a small, dimension-less parameter $Q=\frac{R}{a}$ allows to classify the typical size of neglected corrections at $n$th order beyond leading order ([N${}^{n}$LO]{}) as about $Q^{n}$. For example, $a\approx104$ Å and $R\approx1$ Å in the ${}^4\mathrm{He}_2$ molecule, i.e. $Q\approx\frac{1}{100}$, while $a\approx4.5\;{\ensuremath{\mathrm{fm}}}$ and $R\approx1.5$ in the deuteron, i.e. $Q\approx\frac{1}{3}$ in the “pion-less” EFT, [EFT(${\pi\hskip-0.55em /}$)]{}, where pion-exchange between nucleons is not resolved as non-local. Thus, the detailed dynamics on the “high-energy” scale $R$ can vary largely: For example, attractive van-der-Waals forces $\propto\frac{1}{r^6}$ balance in ${}^4\mathrm{He}_2$ a repulsive core generated by QED; but in Nuclear Physics, one-pion exchange $\propto\frac{1}{r^{[1\dots3]}}$ is balanced by a short-range repulsion whose origin in QCD is not yet understood. It is a pivotal advantage of an EFT that it allows predictions of pre-determined accuracy without such detailed understanding – as long as one is interested in low-energy processes, i.e. Physics at the scale $a$, and not $R$. Even when possible (as in QED – at least at scales $\ge1\;{\ensuremath{\mathrm{fm}}}$), EFTs reduce numerically often highly involved computations of short-distance contributions to low-energy observables by encoding them into a few simple, model-independent constants of contact interactions between the constituents. These in turn can be determined by simpler simulations of the underlying theory, or – when they are as in QCD not (yet) tenable – by fit to data. Universal aspects of few-body systems with shallow bound-states are manifest in EFTs, with deviations systematically calculable. In 2-body scattering for example, the EFT of Point-Like Interactions reproduces the Effective-Range Expansion, but goes beyond it in the systematic, gauge-invariant inclusion of external currents, relativistic effects, etc.
Take 3-body forces (3BFs): They parameterise interactions on scales much smaller than what can be resolved by 2-body interactions, i.e. in which 3 particles sit on top of each other in a volume smaller than $R^3$. Traditionally, they were often introduced *a posteriori* to cure discrepancies between experiment and theory, but such an approach is of course untenable when data are scarce or 1- and 2-body properties should be extracted from 3-body data. But how important are 3BFs in observables? The classification in EFTs rests on the tenet that a 3BF is included if and only if necessary to cancel cut-off dependences in low-energy observables. I outline this philosophy and its results in the following, finding that – independent of the underlying mechanism – 3BFs behave very much alike in such disparate systems as molecular trimers and 3-nucleon systems, but do not follow simplistic expectations.
Construction
============
In the Faddeev equation of particle-dimer scattering without 3BFs, Fig. \[fig:kinematics\], the $\mathrm{S}$-wave 2-body scattering amplitude is given by the LO-term of the Effective-Range Expansion. This dimer and the remaining particle “interact” via ${\mathcal{K}}_l$, the one-particle propagator projected onto relative angular momentum $l$.
![Left: integral equation of particle-dimer scattering. Right: generic loop correction (rectangle) at [N${}^{n}$LO]{}. Thick line ($D$): 2-body propagator; thin line (${\mathcal{K}}_l$): propagator of the exchanged particle; ellipse: LO half off-shell amplitude $t_\lambda^{(l)}(p)$.[]{data-label="fig:kinematics"}](GriesshammerFig1.eps){width="\textwidth"}
\[fig:higherorders\]
Even for small relative on-shell momenta $k$ between dimer and particle, we need the scattering amplitude $t_\lambda^{(l)}(p)$ for *all* off-shell momenta $p$ to determine its value at the on-shell point $p=k$, and hence in particular for $p$ beyond the scale $\frac{1}{R}$ on which a description in terms of point-like constituents is tenable. It is therefore natural to demand that all low-energy observables on a scale $k\sim\frac{1}{a}$ are insensitive to derails of the amplitude at $p\gg\frac{1}{R}$, namely to form and value of the regulator, form-factor or cut-off chosen. If not, a 3BF must soak up the dependence. In an EFT, this is the fundamental tenet: Include a 3BF *if and only if* it is needed as counter-term to cancel divergences which can not be absorbed by renormalising 2-body interactions. Thus, only combinations of 2- and 3BFs are physically meaningful. With the cut-off variation of the 3BF thus fixed, the initial condition leads to one free parameter fixed from a 3-body datum or knowledge of the underlying physics. 3BFs are thus not added out of phenomenological needs but to guarantee that observables are insensitive to off-shell effects.
A Mellin transformation $t_\lambda^{(l)}(p)\propto p^{-s_l(\lambda)-1}$ solves the equation for $p\gg k,\frac{1}{a}$. The spin-content is then encoded only in the homogeneous term: $\lambda=-{\frac{1}{2}}$ for 3 nucleons with total spin $\frac{3}{2}$, or for the totally spin and iso-spin anti-symmetric part of the spin-$\frac{1}{2}$-channel; $\lambda=1$ for 3 identical spin-less bosons and the totally spin and iso-spin symmetric part of the spin-$\frac{1}{2}$-channel. The asymptotic exponent $s_l(\lambda)$ has to fulfil ${\mathrm{Re}}[s]>-1$, ${\mathrm{Re}}[s]\not={\mathrm{Re}}[l\pm2]$, and $$\label{eq:s}
1=\left(-1\right)^l\;\frac{2^{1-l}\lambda}{\sqrt{3\pi}}\;
\frac{\Gamma\left[\frac{l+s+1}{2}\right]\Gamma\left[\frac{l-s+1}{2}\right]}
{\Gamma\left[\frac{2l+3}{2}\right]}\;
{}_2F_1\left[\frac{l+s+1}{2},\frac{l-s+1}{2};
\frac{2l+3}{2};\frac{1}{4}\right].$$ This result was first derived in the hyper-spherical approach by Gasaneo and Macek [@lost][^3]. The asymptotics depends thus only but crucially on $\lambda$ and $l$. Relevant in the UV-limit are the solutions for which ${\mathrm{Re}}[s+1]$ is minimal.
At first glance, we would expect the asymptotics to be given by the asymptotics of the inhomogeneous (driving) term: $t_\lambda^{(l)}(p)\stackrel{?}{\propto}\frac{k^l}{p^{l+2}}$, i.e. $s_{l}(\lambda)\stackrel{?}{=}l+1$. However, we must sum an infinite number of graphs already at leading order. As Fig. \[fig:s-lambdafixed\] shows, this modifies the asymptotics considerably.
![The first two solutions $s_l(\lambda)$ at $\lambda=1$ (left) and $\lambda=-{\frac{1}{2}}$. Solid (dotted): real (imaginary) part; dashed: simplistic estimate. Dark/light: first/second solution. Limit cycle and Efimov effect occur only when the solid line lies below the dashed one, and ${\mathrm{Im}}[s]\not=0$.[]{data-label="fig:s-lambdafixed"}](GriesshammerFig2a.eps "fig:"){width="48.00000%"} ![The first two solutions $s_l(\lambda)$ at $\lambda=1$ (left) and $\lambda=-{\frac{1}{2}}$. Solid (dotted): real (imaginary) part; dashed: simplistic estimate. Dark/light: first/second solution. Limit cycle and Efimov effect occur only when the solid line lies below the dashed one, and ${\mathrm{Im}}[s]\not=0$.[]{data-label="fig:s-lambdafixed"}](GriesshammerFig2b.eps "fig:"){width="48.00000%"}
How sensitive are higher-order corrections to the UV-behaviour of $t_\lambda^{(l)}(p)$? The 2-body scattering-amplitude is systematically improved by including the effective range, higher partial waves etc. Corrections to 3-body observables (including partial-wave mixing) are found by perturbing around the LO solution as in Fig. \[fig:kinematics\]. Most sensitive to unphysically high momenta is each correction at [N${}^{n}$LO]{} which is proportional to the $n$th power of loop momenta. The question when it becomes cut-off sensitive is now rephrased as: When does the correction diverge as the cut-off is removed, i.e. when is its *superficial degree of divergence* non-negative? The answer by simply counting loop momenta in the diagram: $$\label{eq:twobodydivs}
{\mathrm{Re}}[n-s_l(\lambda)-s_{l^\prime}(\lambda^\prime)]\geq0\;\;.$$ We therefore find at which order the first 3BF is needed just by determining when a correction to the 3-body amplitude with only 2-body interactions becomes dependent on unphysical short-distance behaviour.
It is instructive to re-visit these findings in position space. The Schrödinger equation for the wave-function in the hyper-radial dimer-particle distance $r$, $$\label{eq:hyperradial}
\left[-\frac{1}{r}\;\frac{{\partial}}{{\partial}r}\;r\;\frac{{\partial}}{{\partial}r}+
\frac{s_l^2(\lambda)}{r^2}-ME\right] F(r)=0\;\;,$$ looks like the one for a free particle with centrifugal barrier. One would thus expect $s_l\stackrel{?}{=}l+1$ (hyper-spherical co-ordinates!). It had however already been recognised by Minlos and Faddeev that the centrifugal term is for three bosons ($\lambda=1$) despite expectations attractive, so that the wave-function collapses to the origin and seems infinitely sensitive to very-short-distance physics. In order to stabilise the system against collapse – or, equivalently, remove dependence on details of the cut-off –, a 3BF must be added – or, equivalently, a self-adjoint extension be specified at the origin i.e. a boundary condition for the wave-function must be fixed by a 3-body datum. On the other hand, 3BFs are demoted if $s_l> l+1$: The centrifugal barrier provides *more* repulsion than expected, and hence the wave-function is pushed further out, i.e. *less* sensitive to details at distances $r\lesssim R$ where the constituents are resolved as extended. Birse confirmed this recently by a renormalisation-group analysis in position-space [@Birse:2005pm].
Consequences
============
About half of the 3BFs for $l\leq 2$ are *weaker*, half *stronger* than one would expect simplistically, see Table \[tab:ordering\]. The higher partial-waves follow expectation, as the Faddeev equation is then saturated by the Born approximation.
--------------------------------------------------- ---------------------------------------------------------------------------------------------------------- -------------------------------------------------------------- ----------------------- ----------- ------------- ----------
naïve dim. analysis simplistic
bosons fermions [$\mathrm{Re}[s_l(\lambda)+s_{l^\prime}(\lambda^\prime)]$]{} [$l+l^\prime+2$]{}
[${}^{}\mathrm{S}_{}$]{}-[${}^{}\mathrm{S}_{}$]{} [${}^{2}\mathrm{S}_{}$]{}-[${}^{2}\mathrm{S}_{}$]{} [LO]{} [N${}^{2}$LO]{} [prom.]{} [$100\%$]{} ($10\%$)
[${}^{2}\mathrm{S}_{}$]{}-[${}^{4}\mathrm{D}_{}$]{} [N${}^{3.1}$LO]{} [N${}^{4}$LO]{} prom. [$3\%$]{} ($1\%$)
[${}^{}\mathrm{P}_{}$]{}-[${}^{}\mathrm{P}_{}$]{} [${}^{2}\mathrm{P}_{}$]{}-[${}^{2}\mathrm{P}_{}$]{} [N${}^{5.7}$LO]{} dem. 0.2%
[${}^{2}\mathrm{P}_{}$]{}-[${}^{2}\mathrm{P}_{}$]{}, [${}^{4}\mathrm{P}_{}$]{}-[${}^{4}\mathrm{P}_{}$]{} [N${}^{3.5}$LO]{} [N${}^{4}$LO]{} prom. [$2\%$]{} ($1\%$)
[${}^{2}\mathrm{P}_{}$]{}-[${}^{4}\mathrm{P}_{}$]{} [N${}^{4.6}$LO]{} dem. [$0.6\%$]{}
[${}^{4}\mathrm{S}_{}$]{}-[${}^{4}\mathrm{S}_{}$]{} [[N${}^{4.3+2}$LO]{}]{} [[N${}^{2+2}$LO]{}]{} [dem.]{} [$0.1\%$]{}
[${}^{4}\mathrm{S}_{}$]{}-[${}^{2}\mathrm{D}_{}$]{} [N${}^{5.0}$LO]{} dem. [$0.4\%$]{} ($1\%$)
[${}^{4}\mathrm{S}_{}$]{}-[${}^{4}\mathrm{D}_{}$]{} [N${}^{5.3}$LO]{} dem. [$0.3\%$]{}
[$\sim$ as simplistic]{} [[N${}^{l+l^\prime+2}$LO]{}]{}
--------------------------------------------------- ---------------------------------------------------------------------------------------------------------- -------------------------------------------------------------- ----------------------- ----------- ------------- ----------
: Order of the leading 3BF, indicating if actual values (\[eq:twobodydivs\]/\[eq:s\]) are stronger (“prom.”) or weaker (“dem.”) than the simplistic estimate. Last column: typical size of 3BF in [EFT(${\pi\hskip-0.55em /}$)]{}; in parentheses size from the simplistic estimate.[]{data-label="tab:ordering"}
The $\mathrm{S}$-wave 3BF of spin-less bosons is stronger, while the $\mathrm{P}$-wave 3BF is weaker.
That the first $\mathrm{S}$-wave 3BF appears already at LO leads to a new renormalisation-group phenomenon, the “limit-cycle”. It explains the Efimov and Thomas effects, and universal correlations e.g. between particle-dimer scattering length and trimer binding energy (the Phillips line). In general, it appears whenever the kernel of the integral equation not compact, i.e. ${\mathrm{Im}}[s]\not=0$ and $|{\mathrm{Re}}[s]|<{\mathrm{Re}}[l+1]$. We finally note that the power-counting requires a new, independent 3BF with $2l$ derivatives to enter at [N${}^{2l}$LO]{} and provides high-accuracy phase-shifts in atom-dimer and nucleon-deuteron scattering, and loss rates close to Feshbach resonances in Bose-Einstein condensates, see e.g. [@improve3body; @bedaque_bira_review; @Braaten:2004rn] for details.
Demotion might seem an academic dis-advantage – to include some higher-order corrections which are not accompanied by new divergences does not improve the accuracy of the calculation; one only appears to have worked harder than necessary. However, demotion is pivotal when one wants to predict the experimental precision necessary to dis-entangle 3BFs in observables, and here the error-estimate of EFTs is crucial. In ${}^4\mathrm{He}$-atom-dimer scattering, where $Q\approx\frac{1}{100}$, only high-precision experiments can however reveal contributions from 3BFs beyond the one found in the $\mathrm{S}$-wave.
On the other hand, $Q\approx\frac{1}{3}$ in [EFT(${\pi\hskip-0.55em /}$)]{}of Nuclear Physics. Now, the demotion or promotion of 3BFs makes all the difference whether an experiment to determine 3BF effects is feasible at all. For example, the quartet-$\mathrm{S}$ wave scattering-length in the neutron-deuteron system sets at present the experimental uncertainty in an indirect determination of the doublet scattering length, which in turn is well-known to be sensitive to 3BFs. Its value in [EFT(${\pi\hskip-0.55em /}$)]{}at [N${}^{2}$LO]{}, $$a({{\ensuremath{{}^{4}\mathrm{S}_{\frac{3}{2}}}}})=[
5.09(\mathrm{LO})+1.32(\mathrm{NLO})-0.06(\mathrm{{N\ensuremath{{}^{2}}LO\xspace}})]\;{\ensuremath{\mathrm{fm}}}=
[6.35\pm0.02]\;{\ensuremath{\mathrm{fm}}}\;\;,$$ converges nicely and agrees very well with experiment, $[6.35\pm0.02]\;{\ensuremath{\mathrm{fm}}}$. The theoretical accuracy by neglecting higher-order terms is here estimated conservatively by $Q\approx\frac{1}{3}$ of the difference between the NLO- and [N${}^{2}$LO]{}-result. Table \[tab:ordering\] predicts that the first 3BF enters not earlier than [N${}^{6}$LO]{}. Indeed, if the theoretical uncertainty continues to decreases steadily as from NLO to [N${}^{2}$LO]{}, an accuracy of $\pm(\frac{1}{3})^3\times0.02\;{\ensuremath{\mathrm{fm}}}\lesssim\pm0.001\;{\ensuremath{\mathrm{fm}}}$ with only 2-nucleon scattering data as input can be reached in calculations. This is comparable to the range over which modern high-precision potential-model calculations differ: $[6.344\dots6.347]\;{\ensuremath{\mathrm{fm}}}$. If the 3BF would occur at [N${}^{4}$LO]{} as simplistically expected, the error by 3BFs would be $(\frac{1}{3})^1\times0.02\;{\ensuremath{\mathrm{fm}}}\approx0.007\;{\ensuremath{\mathrm{fm}}}$, considerably larger than the spread in the potential-model predictions. Differential cross-sections and partial-waves are also in excellent agreement with much more elaborate state-of-the-art potential model calculations at energies up to $15\;{\ensuremath{\mathrm{MeV}}}$, see e.g. [@improve3body].
The cross-section of triton radiative capture $nd \to t\gamma$ at thermal energies provides another example [@withSadeghi]. Nuclear Models give a spread of $[0.49\dots0.66]\;\mathrm{mb}$, depending on the 2-nucleon potential, and how the $\Delta(1232)$ as first nucleonic excitation is included. On the other hand, a process at $0.0253$ eV \[*sic*\] incident neutron energy and less than $7\;{\ensuremath{\mathrm{MeV}}}$ photon energy should be insensitive to details of the deuteron wave-function and of a resonance with an excitation energy of $300\;{\ensuremath{\mathrm{MeV}}}$. Indeed, the power-counting of 3BFs applies equally with external currents, only that the higher-order interaction in Fig. \[fig:higherorders\] includes now also the momentum- or energy-transfer from the external source as additional low-energy scales. As no new 3BFs are needed up to [N${}^{2}$LO]{} to render cut-off independent results, the result is completely determined by simple 2-body observables as $$\sigma_\mathrm{tot}=[
0.485(\mathrm{LO})+0.011(\mathrm{NLO})+0.007(\text{{N\ensuremath{{}^{2}}LO\xspace}})
]\;\mathrm{mb}=
[0.503\pm0.003]\;\mathrm{mb}\;\;,$$ which converges and compares well with the measured value, $[0.509\pm0.015]\;\mathrm{mb}$. The cross-section relevant for big-bang nucleo-synthesis ($E_n\approx0.020\dots0.4\;\mathrm{MeV}$) is also in excellent agreement with data [@Sadeghi:2004es].
Conclusions
===========
With these findings, the EFT of 3-body systems with only contact interactions is a self-consistent, systematic field theory which contains the minimal number of interactions at each order to render the theory renormalisable. Each 3-body counter-term gives rise to one subtraction-constant which is fixed by a 3-body datum. Table \[tab:ordering\] sorts the 3BFs by their strengthes, their symmetries and the channels in which they contribute at the necessary level of accuracy. Amongst the host of applications in Nuclear Physics are triton and ${}^3$He properties, reactions in big-bang nucleo-synthesis, neutrino astro-physics, the famed nuclear $A_y$-problem, and the experimental determination of fundamental neutron properties.
The method presented here is applicable to any EFT in which an infinite number of diagram needs to be summed at LO, e.g. because of shallow bound-states. One example is Chiral EFT, the EFT of pion-nucleon interactions. Only those local $N$-body forces are added at each order which are necessary as counter-terms to cancel divergences at short distances. This mandates a careful look at the ultraviolet-behaviour of the leading-order, non-perturbative scattering amplitude. It leads at each order and to the prescribed level of accuracy to a cut-off independent theory with the smallest number of experimental input-parameters. The power-counting is thus not constructed by educated guesswork but by rigorous investigations of the renormalisation-group properties of couplings and observables using the methodology of EFT.
My thanks to the organisers for creating a highly stimulating workshop and the warm welcome. Supported in part by DFG Sachbeihilfe GR 1887/2-2, 3-1, and BMFT.
[99]{}
H. W. Grie[ß]{}hammer, Nucl. Phys. A [**760**]{} (2005) 110 \[nucl-th/0502039\]. P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part. Sci. [**52**]{} (2002) 339. E. Braaten and H. W. Hammer, cond-mat/0410417. G. Gasaneo and J. H. Macek, J. Phys. **B** 35 (2002) 2239.
M. C. Birse, nucl-th/0509031. H. W. Grie[ß]{}hammer, [[Nucl. Phys. ]{.nodecor}**A**]{}**744** (2004), 192 \[nucl-th/0404073\].
H. Sadeghi, S. Bayegan and H. W. Grie[ß]{}hammer, in preparation.
H. Sadeghi and S. Bayegan, Nucl. Phys. A [**753**]{}, 291 (2005) \[nucl-th/0411114\].
[^1]: *E-mail address:* [email protected]; present address: Universität Erlangen.
[^2]: 9th November 2005. Preprint FAU-TP3-05/8, TUM-T39-05-14, nucl-th/0511039. To appear in Few-Body Systems.
[^3]: My apologies to the authors that I found this reference only after [@Griesshammer:2005ga] was published.
|
---
abstract: 'We demonstrate dynamical control of the superradiant transition of cavity-BEC system via periodic driving of the pump laser. We show that the dominant density wave order of the superradiant state can be suppressed, and that the subdominant competing order of Bose-Einstein condensation emerges in the steady state. Furthermore, we show that additional, nonequilibrium density wave orders, which do not exist in equilibrium, can be stabilized dynamically. Finally, for strong driving, chaotic dynamics emerge.'
author:
- 'Jayson G. Cosme'
- Christoph Georges
- Andreas Hemmerich
- Ludwig Mathey
bibliography:
- 'biblio.bib'
title: 'Dynamical Control of Order in a Cavity-BEC system'
---
Recent developments in pump-probe experiments in the ultrafast regime have resulted in spectacular observations, most notably a dynamical enhancement of optical conductivity in high-$T_c$ materials, suggesting photoinduced superconductivity. This observation has been made in different materials and parameter regimes, which leads to the question if one or more mechanisms are involved in these findings. One of the observations was reported in Ref. [@Fausti2011] on pump-probe experiments in La$_{1.8-x}$Eu$_{0.2}$Sr$_x$CuO$_4$ (LESCO) at $x=1/8$ doping. Here the equilibrium material is in a charge density ordered state that strongly suppresses the superconducting dome near this commensurate doping. However, when the pump pulse is applied the superconducting response is restored. An intriguing hypothesis to explain this observation is that the pump pulse dynamically suppresses the dominant charge density wave (CDW) order allowing the next-to-leading order, i.e. superconductivity, to emerge.
We propose to test the principle of this mechanism. As a well-controlled and tunable environment [@Bloch2008], we consider a cavity-Bose-Einstein condensate (BEC) system illuminated by a transverse laser beam [@Ritsch2013; @Baumann2010; @Klinder2015]. As the intensity of the transverse laser beam is increased, the system undergoes a superradiant phase transition, at which the atoms self-organize into a density wave (DW) order, shown in Fig. \[fig:summary\](a). This DW serves as a Bragg lattice that scatters photons out of the pump laser into the cavity mode. This phase transition is related to the superradiant transition of the Dicke model [@Dicke1954; @Hepp1973]. Two experiments, performed in different parameter regimes, have observed this transition [@Baumann2010; @Klinder2015]. Theoretical [@Domokos2002; @Nagy2008; @Bakhtiari2015; @Mivehvar2017; @Mivehvar2018; @Gong2018] and experimental studies [@Black2003; @Klinder2015b; @Klinder2016; @Landig2016; @Leonard2017] on this system have been reported. At the transition, the condensate fraction of the atomic cloud drops sharply, due to the onset of the competing density order. The phase transition displays a qualitative similarity to the competition of charge density order and superconductivity in LESCO, where condensation is the analogue of superconducting order, and each of these orders competes with a density order.
![(a) For a transverse pump strength $\alpha_{p,0}$ above criticality, the system is in a DW phase. Atoms occupy the corresponding higher momentum states, and photons occupy the cavity mode. (b) By modulating the pump strength $\alpha_p(t)$, DW order is suppressed and condensation is restored. The condensate density increases, and the cavity mode population is suppressed. We modulate the pump beam by adding frequency sidebands $\pm\omega_d$, seen in the power spectrum $\mathcal{S}(\omega)$.[]{data-label="fig:summary"}](fig1){width="1.0\columnwidth"}
In this Letter, we demonstrate dynamical control of this phase transition. We show that periodic driving of the pump beam suppresses density order, and that condensation is restored, in parallel to the emergence of superconductivity due to the suppression of density order. We perform a high-frequency expansion of the Hamiltonian that demonstrates a reduction in the atom-cavity coupling parameter due to the modulation of the pump field that agrees with the numerical observation. The pump field modulation is realized by adding laser beams that are detuned from the pump beam. We emphasize that this choice of implementing the modulation leaves the magnitude of the pump laser unchanged so that the resulting control of the phase transition is purely dynamical. Furthermore, we show that nonequilibrium DW orders arise if the driving frequency is near a resonance of the frequencies of the corresponding atomic momentum states. Finally, we observe the emergence of chaotic dynamics for strong driving.
![(a) Protocol for the pump field amplitude. Dynamics of the (b) cavity mode and (c) BEC mode occupations for $\varepsilon_0/E_{\mathrm{rec}} = 2.20$, $\omega_d = 2\pi \times 6~\mathrm{kHz}$, and different strengths of the driving amplitude $f_0$. Because of periodic driving, the density order of the atoms is dynamically suppressed, and condensation is restored.[]{data-label="fig:mf_dynamics"}](fig2){width="1.0\columnwidth"}
In Fig. \[fig:summary\], we depict the cavity system, with the pump laser along the $y$ direction and the cavity axis along the $z$ direction. In the rotating frame [@Ritsch2013], we decompose the atomic field into plane waves ${e}^{i n k y}{e}^{i m k z}$, which gives $$\begin{aligned}
\label{eq:ham}
&\hat{H}=-\delta_{\mathrm{C}}\hat{\alpha}^{\dagger}\hat{\alpha}+\frac{\Delta_0}{4}\hat{\alpha}^{\dagger}\hat{\alpha}\hat{Z} +\frac{\Delta_0}{2}\hat{\alpha}^{\dagger}\hat{\alpha}\hat{N} +\omega_{\mathrm{rec}}\hat{E}\\ \nonumber
& -\frac{\omega_{\mathrm{rec}}}{2}|\alpha_p|^{2}\hat{N} - \frac{\omega_{\mathrm{rec}}}{4}|\alpha_p|^{2}\hat{Y}+\frac{\sqrt{\omega_{\mathrm{rec}}|\Delta_0|}}{4}|\alpha_p|\hat{D}\hat{J}.\end{aligned}$$ The number of atoms is $
\hat{N}=\sum \hat{\phi}^{\dagger}_{n,m}\hat{\phi}_{n,m}$ and the kinetic energy is $\hat{E}=\sum (n^2+m^2)\hat{\phi}^{\dagger}_{n,m}\hat{\phi}_{n,m}$. Momentum excitation due to the transverse pump is $\hat{Y}=\sum \left( \hat{\phi}^{\dagger}_{n+2,m}\hat{\phi}_{n,m} + \mathrm{H.c.}\right)$. The scattering of photons between the pump and the cavity fields is captured by $\hat{D}=\hat{\alpha}^{\dagger}+\hat{\alpha}$ with the momentum excitation paths $\hat{J}=\sum \left( \hat{\phi}^{\dagger}_{n,m}\left(\hat{\phi}_{n+1,m+1} + \hat{\phi}_{n+1,m-1} \right) + \mathrm{H.c.}\right)$. Excitation due to absorption and emission of cavity photons is $\hat{Z}=\sum \left( \hat{\phi}^{\dagger}_{n,m+2}\hat{\phi}_{n,m} + \mathrm{H.c.}\right)$. $\Delta_0$ is the light shift per intracavity photon, $\delta_C$ is the detuning between the pump and the cavity frequency, $\hat{\phi}_{n,m}$ ($\hat{\phi}^{\dagger}_{n,m}$) is the bosonic annihilation (creation) operator of the atomic momentum state $(n,m)\hbar k$, $\hat{\alpha}$ ($\hat{\alpha}^{\dagger}$) is the cavity mode annihilation (creation) operator, and $\alpha_p$ is the dimensionless pump strength parameter [@Cosme2018supp]. We only consider negative detuning $\delta_{\mathrm{eff}}\equiv \delta_C - (1/2)N_a\Delta_0<0$. Photons leak out of the cavity at the rate $\kappa$. We use $N_a = 60\times 10^3$ atoms, $\omega_{\mathrm{rec}}=2\pi \times 3.55~\mathrm{kHz}$, $\kappa=2\pi \times 4.50~\mathrm{kHz}$, $\Delta_0= - 2\pi \times 0.36~\mathrm{Hz}$, and $\delta_{\mathrm{eff}}=-2 \pi \times 22~\mathrm{kHz}$ from [@Klinder2015].
To elaborate on the analogy to high-$T_{c}$ materials, we consider the universal action of this system, to lowest order, analogous to [@Hayward2014; @Achkar2016]. The order parameter of condensation is $\Psi = \phi_{0,0}$, the DW order parameter is $\Phi_{a} = \phi^{*}_{0,0}(\phi_{1,1} + \phi_{1,-1} +\phi_{-1,1} + \phi_{-1,-1}) + \mathrm{c.c.}$. We include the photon field as $\Phi_{ph} = \alpha$. Including only the lowest momenta and nonlinear terms, the free energy is $F \approx s_{1}|\Psi|^{2} + s_{2} \Phi_{a}^{2} +s_{3} |\Phi_{ph}|^{2}+\nu_{1} |\Psi|^{2}|\Phi_{ph}|^{2}+\nu_{2} \Phi_{ph,r} \Phi_{a}$, with $s_{1} = - \omega_{\mathrm{rec}} |\alpha_{p}|^{2}/2$, $s_{2} = \omega_{\mathrm{rec}}$, $s_{3}= - \omega_{\mathrm{rec}}$, $\nu_{1} = \Delta_{0}/2$, $\nu_{2} = \sqrt{\omega_{\mathrm{rec}} |\Delta_{0}|}|\alpha_{p}|/2$, and $\Phi_{ph,r}=\Re{\Phi_{ph}}$. This describes a superconducting order competing with commensurate, real-valued DW order, where the atomic and photonic component of the DW have been treated explicitly. The symmetry of the system is $\mathrm{U}(1)\times \mathbb{Z}_{2}$, where the $\mathrm{U}(1)$ symmetry refers the phase invariance $\Psi \rightarrow \exp(i \theta)\Psi$, and the $\mathbb{Z}_{2}$ corresponds to the simultaneous mapping $\Phi_{ph} \rightarrow -\Phi_{ph}$ and $\Phi_{a} \rightarrow -\Phi_{a}$. If the photonic mode could be integrated out without retardation we have $\Phi_{ph}\approx \nu_{2}\Phi_{a}/\delta_{C}$, so that $F_{\mathrm{eff}} \approx s_{1}|\Psi|^{2} + s'_{2} \Phi_{a}^{2} + \nu |\Psi|^{2}\Phi_{a}^{2}$, with $s'_{2} = s_{2} + s_{3}\nu_{2}^{2}/\delta_{C}^{2} + \nu_{2}^{2}/\delta_{C}$ and $\nu = \nu_{1} \nu_{2}^{2}\delta_{C}^{2}$, which shows the competition between the BEC and DW explicitly, cf. [@Hayward2014]. We note, however, that the photonic dynamics cannot be integrated out without retardation. The cavity-BEC system is therefore a zero-dimensional analogue of the action in [@Hayward2014], but it explicitly includes the two components of the DW order, the photonic and the atomic part.
We determine the dynamics with a numerical implementation of an open system truncated Wigner (TW) approximation [@Blakie2008; @Polkovnikov2010]. For the initialization we choose $\alpha_{p}=0$, and we sample the initial state from a Wigner distribution of a coherent state for the BEC mode, with $\langle \phi_{0,0}\rangle = \sqrt{N_{a}}$, and vacuum noise in all other atomic modes and the photonic mode. We propagate an initial state according to a stochastic differential equation. The unitary evolution derives from Eq. . We include a white noise $\xi(t)$, with $\langle \xi^{*}(t) \xi(t') \rangle = \kappa \delta(t-t')$ to treat photon loss to a vacuum reservoir. We use 500 trajectories to sample the dynamics, and we include momentum modes up to $\{n,m\}\in [-6,6]$. We ramp up the driving field with a protocol, shown in Fig. \[fig:mf\_dynamics\](a). We modulate the pump field $\alpha_p$ by introducing frequency sidebands $\pm\omega_d$ detuned from the pump beam $$\label{eq:driving}
\alpha_p(t) = \sqrt{\epsilon_0}\left(1+f_0\mathrm{cos}{(\omega_d t)}\right),$$ where $f_0$ is a dimensionless driving amplitude, see also Fig. \[fig:summary\]. We emphasize that this method of driving keeps the population of the carrier frequency constant. If one would modulate the intensity $|\alpha_p(t)|^{2}$, rather than the field $\alpha_p(t)$, there would be an additional trivial suppression of the DW phase because the intensity of the carrier frequency is decreased. Experimentally, this modulation can be achieved by adding additional beams at frequencies that are detuned from the pump beam by $\pm \omega_d$. A version with a single frequency sideband is currently realized in [@Georges2018]. In Figs. \[fig:mf\_dynamics\](b) and \[fig:mf\_dynamics\]2(c) we show the cavity photon intensity and the BEC mode occupation as a function of time. After the ramp-up of the pump intensity, the system is in the DW phase, in which a sizable occupation of the cavity mode exists. When the modulation is turned on, the system relaxes to a steady state. As a crucial observation, we find that the coherent state is restored for driving amplitudes of $f_{0} \approx 0.1$, for this example.
![Comparison between the undriven and driven steady-state of (a) the cavity and (b) the BEC mode occupations and (c) the coherence decay rate as a function of the pump strength $\varepsilon_0$ (units of $E_{\mathrm{rec}}$). []{data-label="fig:shift"}](fig3){width="0.95\columnwidth"}
![Dynamical renormalization of the BEC-DW phase transition, visible in the (a) cavity mode and (b) BEC mode occupation for $w_d=2\pi\times 10~\mathrm{kHz}$. (i) Thin solid line shows the effective Hamiltonian prediction for the phase boundary, (ii) thick dashed line the TW result. The phase boundary is indicated based on $|\alpha|^2> 70$ and $n_0/N> 0.97$.[]{data-label="fig:mf_phase"}](fig4){width="1.0\columnwidth"}
Next, we vary the carrier intensity $\varepsilon_0$ for fixed driving frequency $\omega_d = 2\pi \times 10~\mathrm{kHz}$ and driving amplitude $f_0=0.20$, see Fig. \[fig:shift\]. Panel (a) and (b) show the cavity photon intensity and the BEC mode occupation, respectively, in the undriven state and the driven steady state. We observe that the transition from the BEC to the DW phase is shifted to a larger value of $\varepsilon_{0}$, which demonstrates dynamical control of the phase transition. In addition, we show the temporal correlation decay rate of the BEC mode which we determine by fitting $\langle \hat{\phi}^{\dagger}_{0,0}(t_2)\hat{\phi}_{0,0}(t_1) \rangle$ with $\sim \mathrm{exp}(-\gamma t)$. The regime of small $\gamma$ is also extended to larger $\varepsilon_{0}$, which demonstrates that coherence in the BEC mode is restored.
In Fig. \[fig:mf\_phase\], we vary both $\varepsilon_0$ and the driving amplitude $f_0$. The phase boundary between the BEC and DW phase is shifted to higher $\varepsilon_0$ with increasing $f_{0}$. We compare the numerical result to a Magnus expansion [@Hemmerich2010; @Goldman2014; @Eckardt2015; @Bukov2015; @Zhu2016] of the time-dependent Hamiltonian Eq. , at second order in $f_{0}$, which gives [@Cosme2018supp] $$\begin{aligned}
\label{eq:hameff}
&\hat{H}_\mathrm{eff}=-\delta_{\mathrm{C}}\hat{\alpha}^{\dagger}\hat{\alpha}+\frac{\Delta_0}{4}\hat{\alpha}^{\dagger}\hat{\alpha}\hat{Z} +\frac{\Delta_0}{2}\hat{\alpha}^{\dagger}\hat{\alpha}\hat{N} +\omega_{\mathrm{rec}}\hat{E}\\ \nonumber
& -\frac{\omega_{\mathrm{rec}}\epsilon_0}{2}\left(\hat{N}+\frac{f_0^2\hat{N}}{2}+\frac{\Delta_0\hat{N}}{2\omega_{\mathrm{rec}}}\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2f_0^2(\hat{D})^2 +\frac{\hat{Y}}{2} \right) \\ \nonumber
&- \frac{\omega_{\mathrm{rec}}\epsilon_0 f_0^2}{8}\hat{Y} +\frac{\sqrt{\omega_{\mathrm{rec}}|\Delta_0|\epsilon_0}}{4}\hat{D}\hat{J} \left[1-\epsilon_0\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2f_0^2\right].\end{aligned}$$ The phase boundary predicted by Eq. shows good agreement with the TW result, see Fig. \[fig:mf\_phase\]. The shift of the phase boundary is primarily due to the effective reduction of the atom-cavity coupling $\sqrt{\epsilon_0} \rightarrow \sqrt{\epsilon_0}(1-\epsilon_0 \omega^2_{\mathrm{rec}} f_0^2/ \omega_d^{2})$. A dynamical renormalization of the $c$-axis transport in high-$T_c$ superconductors has been discussed in [@Okamoto2016].
![Dynamical phase diagram as a function of $\omega_d$ in units of $2\pi\times\mathrm{kHz}$. The carrier intensity and the driving amplitude are fixed to $\varepsilon_0/E_{\mathrm{rec}} = 2.19$ and $f_0 = 0.25$, respectively. (a)-(e) Cavity and (f)-(j) BEC mode occupation dynamics, and (k)-(o) order parameter dynamics for the DW orders. For (k)-(o), each line represents the three relevant order parameters $|\Phi_{1,1}|^2$, $|\Phi_{4,0}|^2$, and $|\Phi_{1,3}|^2$. (p)-(t) Density plot on semilogarithmic scale of the momentum occupation $|\phi_{n,m}|^2$ in the steady state, as a function of the discrete momenta $k_y$ and $k_z$.[]{data-label="fig:mf_morephase"}](fig5){width="1.0\columnwidth"}
In addition to controlling the phase boundary of two equilibrium phases, we now demonstrate that we can create nonequilibrium order, see Fig. \[fig:mf\_morephase\]. These are orders that do not exist in equilibrium. In particular, we choose driving frequencies at an integer ratio to the discrete momentum $(n^2+m^2)\omega_\mathrm{rec}$, to excite new types of DW orders. In a recent work, calculations based on the Hill equation predict that parametric instabilities occur in a related system at multiples of the recoil frequency [@Molignini2017]. The associated order parameters are $\Phi_{n,m} = \mathrm{cos}(nky)\mathrm{cos}(mkz) $ as quantified by $\langle |\Phi_{n,m}|^2 \rangle$, where the DW considered above corresponds to $\Phi_{a} = \Phi_{1,1}$. We refer to this DW phase as DW$_1$. In addition to having a long-lived occupation of the cavity mode, the standard type of DW phase can also be identified by having a dominant order parameter given by $\Phi_{1,1}$ as seen in Fig. \[fig:mf\_morephase\]. We determine the additional higher order DW states, comparing the relative values of their order parameters. A new type of DW order associated with the $\phi_{\pm 4,0}$ momentum modes emerges when the driving frequency is close to half of the frequency, i.e., $2\omega_d \approx (n^2+m^2)\omega_{\mathrm{rec}} = (4^2+0^2)\omega_\mathrm{rec} = 16\omega_\mathrm{rec}$. We refer to this order DW$_4$, and note that the $\phi_{\pm 4,0}$ modes are significantly occupied, in addition to the $\phi_{\pm 2,0}$ modes, as shown for $\omega_d = 2\pi \times 28.5~\mathrm{kHz}$ in Fig. \[fig:mf\_morephase\](q). Superradiance is suppressed because the condition for Bragg scattering is not fulfilled for this type of density order. This can be seen in Fig. \[fig:mf\_morephase\](q), where the $\phi_{\pm 1,\pm 1}$ modes are depleted for the DW$_4$ phase. Furthermore, we note that the power spectrum for the DW$_4$ phase, $\mathcal{S}_{n_0}(\omega) = |\tilde{n}_0(\omega)|^2/\int d\omega |\tilde{n}_0(\omega)|^2$, where $\tilde{n}_0(\omega)$ is the Fourier transformation of $n_0(t)$, shown in Fig. \[fig:chaos\_spectrum\](a), shows a subharmonic response in the dynamics of the BEC mode. This is indicated by two prominent peaks near $\omega/\omega_d=0.5$. This is potentially related to a recent time-crystalline order proposed in Ref. [@Gong2018], but a more detailed discussion will be given elsewhere. For a driving frequency near the frequency associated with the $\phi_{\pm 1,\pm 3}$ modes, we find that its corresponding DW order, which we call DW$_3$, starts to emerge and coexist with the DW$_1$ order after transient dynamics. This intertwined order is seen for $
\omega_d = 2\pi \times 34.5~\mathrm{kHz}$ in Fig. \[fig:mf\_morephase\]. Increasing the frequency to $
\omega_d = 2\pi \times 35.5~\mathrm{kHz}$, we observe in Fig. \[fig:mf\_morephase\](e) an example for a DW$_3$ phase. Similar to the DW$_4$ phase, superradiance is suppressed as DW$_1$ order vanishes, and the order parameter for DW$_3$ becomes significant. We briefly mention that a similar emergence of metastable dynamical phases has been predicted in a periodically driven isolated Dicke model [@Bastidas2011].
![Dynamics of (a) the cavity and (b) the BEC modes. (c) Steady state momentum occupation as in Fig. \[fig:mf\_morephase\]. The driving frequency is $w_d=2\pi\times 10~\mathrm{kHz}$ with $\varepsilon_0/E_{\mathrm{rec}} = 2.17$ and $f_0 = 0.90$.[]{data-label="fig:tw_chaos"}](fig6){width="1.0\columnwidth"}
Finally, we show that for low driving frequency and large driving amplitude, the system enters a chaotic regime, as depicted in Fig. \[fig:tw\_chaos\]. This phase is characterized by sharp oscillations between vanishing and the large population of the cavity mode. Because of the large cavity mode occupation, the BEC mode is severely depleted, and higher momentum modes are populated as seen in Fig. \[fig:tw\_chaos\]. We note that, a similar dynamical phase but with regular oscillatory behavior has been discussed in [@Chitra2015; @Molignini2017]. Here, we observe the chaotic dynamics of the observables for this chaotic phase, as seen in the power spectrum of the BEC mode dynamics and phase space trajectory presented in Fig. \[fig:chaos\_spectrum\].
 and \[fig:tw\_chaos\](b). Phase space trajectory for the last 30 driving cycles for (b) chaotic regime with $f_0=0.90$ and (c) DW$_1$ with $f_0=0.25$.[]{data-label="fig:chaos_spectrum"}](fig7){width="1.0\columnwidth"}
In conclusion, we have determined and characterized the dynamical states of a periodically driven cavity-BEC system. The scenario that we have described here includes the renormalization of the phase boundary of the equilibrium orders for weak to intermediate driving strengths, the emergence of nonequilibrium orders at intermediate driving strengths and at resonant driving frequencies, and chaotic dynamics for strong driving. We derive the universal action of this system which shows that it is a paradigmatic zero-dimensional system of competing orders, featuring the competition of Bose-Einstein condensation and density wave order. The density wave order itself has both an atomic and a photonic component each of which is treated explicitly. We emphasize that a broad class of many-body systems with competing orders are of this and similar form, and our study will therefore be of guidance for dynamical control in a broad, generic class of systems. Specifically we consider the recent finding of dynamically induced superconductivity in pump-probe experiments in the high-$T_c$ superconductor LESCO at $x=1/8$ doping. For this finding it was hypothesized that the pump pulse suppresses the CDW order, and that the subdominant order of superconductivity emerges [@Patel2016]. In this Letter, we have shown that the principle of this mechanism is indeed possible, and we propose it to be tested in a cavity-BEC experiment. We find that it is crucial to separate the atomic and photonic components of the DW order [@Cosme2018supp], which suggests that, similarly, the electronic and atomic components of a CDW in a solid-state system have to be considered explicitly, for the emergence of nonequilibrium superconductivity, and more generally for the regime of ultrafast dynamics and the control of solid-state systems. Furthermore, the scenario that we have described beyond the renormalization of the equilibrium phase boundary, in particular nonequilibrium orders and chaotic dynamics, suggests further remarkable dynamical phenomena to be pursued in driven solid-state systems.
We would like to acknowledge the support from the Deutsche Forschungsgemeinschaft through the SFB 925 and the Hamburg Centre for Ultrafast Imaging. We also thank Andrea Cavalleri, Louis-Paul Henry, Jun-ichi Okamoto, and Beilei Zhu for useful discussions.
**Supplemental Materials: Dynamical control of order in a cavity-BEC system**
Jayson G. Cosme$^{1,2,3}$, Christoph Georges$^{1,2}$, Andreas Hemmerich$^{1,2,3}$, and Ludwig Mathey$^{1,2,3}$\
$^1$[*Zentrum für Optische Quantentechnologien, Universität Hamburg, 22761 Hamburg, Germany*]{}
$^2$[*Institut für Laserphysik, Universität Hamburg, 22761 Hamburg, Germany*]{}
$^3$[*The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, Hamburg 22761, Germany*]{}
Dynamical protocol
==================
The full dynamical protocol used in obtaining the mean-field results consists of two stages: (i) slow ramp towards the mean pump amplitude $\alpha_p=\sqrt{\epsilon_0}$; and (ii) the driving protocol. The exact time-dependence is shown below: $$\begin{aligned}
\label{eq:real_driving}
&\alpha_p(t)= \\ \nonumber
&\left\{
\begin{array}{ll}
\sqrt{\epsilon_0}B_1(t+T_\mathrm{r}+T_\mathrm{c},T_\mathrm{r}) &t \in [-T_\mathrm{r}-T_\mathrm{c}, \frac{-T_\mathrm{r}-2T_\mathrm{c}}{2}] \\
\sqrt{\epsilon_0}B_2(t+T_\mathrm{r}+T_\mathrm{c},T_\mathrm{r}) &t \in (\frac{-T_\mathrm{r}-2T_\mathrm{c}}{2}, -T_\mathrm{c}] \\
\sqrt{\epsilon_0} &t \in (-T_\mathrm{c}, 0] \\
\sqrt{\epsilon_0}(1+B_1(t,T_\mathrm{s})f_0\mathrm{cos}(\omega_d t)) &t \in (0, T_{\mathrm{s}}/2] \\
\sqrt{\epsilon_0}(1+B_2(t,T_\mathrm{s})f_0\mathrm{cos}(\omega_d t)) &t \in (T_{\mathrm{s}}/2, T_{\mathrm{s}}] \\
\sqrt{\epsilon_0}(1+f_0\mathrm{cos}(\omega_d t)) &t>T_{\mathrm{s}}
\end{array}
\right.\end{aligned}$$ where $$\begin{aligned}
B_1(t,T)&=\frac{2t^2}{T^2} \\ \nonumber
B_2(t,T)&=-1-\frac{2t^2}{T^2}+\frac{4t}{T}.\end{aligned}$$ Specifically, we have chosen $T_\mathrm{r}=40~\mathrm{ms}$, $T_\mathrm{c}=10~\mathrm{ms}$, and $T_\mathrm{s}=4~\mathrm{ms}$. Note that the actual experimental value for the pump beam intensity $|\varepsilon_0|$ for the setup in Ref. [@Klinder2015] can be modelled within the single mode description by an effective reduction in the coupling according to $|\varepsilon_0|/|\epsilon_0| \approx 1.44 E_{\mathrm{rec}} $ where $E_{\mathrm{rec}}$ is the recoil energy and $\epsilon_0$ is the dimensionless pump strength parameter used in the single-mode model.
![Time evolution of the pump field amplitude $\alpha_p$. []{data-label="fig:proto"}](MF_alp_driven_6kHz_eps_2p20_0p10 "fig:"){width="0.5\columnwidth"}![Time evolution of the pump field amplitude $\alpha_p$. []{data-label="fig:proto"}](MF_alp_driven_6kHz_eps_2p20_0p10_short "fig:"){width="0.5\columnwidth"}
A schematic for the time evolution of the pump field used in this work is shown in Fig. \[fig:proto\]. As seen in Fig. \[fig:proto\], the system is allowed to evolve and relax for more than $50~\mathrm{ms}$ upon reaching the desired modulation strength. The long-time average of relevant observables shown in the main text correspond to a time averaging over the final $10~\mathrm{ms}$ of the full dynamics.
![(Left) Difference between the time evolution of the pump field amplitude $\alpha_p$ for a sharp and a gradual increase in the driving amplitude $f_0$. (Right) Comparison of the corresponding dynamcis of the cavity mode between the two driving protocols. Here, we have chosen $\varepsilon_0/E_{\mathrm{rec}} = 2.24$, $\omega_d = 2\pi \times 10~\mathrm{kHz}$, and $f_0=0.15$. []{data-label="fig:protolt"}](MF_alp_sharp_v_grad "fig:"){width="0.5\columnwidth"}![(Left) Difference between the time evolution of the pump field amplitude $\alpha_p$ for a sharp and a gradual increase in the driving amplitude $f_0$. (Right) Comparison of the corresponding dynamcis of the cavity mode between the two driving protocols. Here, we have chosen $\varepsilon_0/E_{\mathrm{rec}} = 2.24$, $\omega_d = 2\pi \times 10~\mathrm{kHz}$, and $f_0=0.15$. []{data-label="fig:protolt"}](MF_cav_sharp_v_grad "fig:"){width="0.5\columnwidth"}
We show in Fig. \[fig:protolt\] a comparison for the dynamical response of the system between a sharp and a gradual switching of the driving amplitude. There, it can be seen that the two protocols only differ in the dynamical response of the system for short times but the long-time average of observables is the same in both protocols.
Advantage of double-sideband over single-sideband protocol
==========================================================
There are two obvious ways to drive the pump intensity. The first one used in the main text is generated by introducing two additional sidebands at $\pm \omega_d$. Recall that for this case, we have $$\alpha^{\{2\}}_p(t) = \sqrt{\epsilon_0}(1 + f_0 \mathrm{cos}(\omega_d t)),$$ and this creates an intensity modulation for the pump according to $$\label{eq:prot2}
|\alpha^{\{2\}}_p(t)|^2 = {\epsilon_0}\left(1 + \frac{f_0^2}{2} + \frac{f_0^2\mathrm{cos}(2\omega_d t)}{2} + 2f_0 \mathrm{cos}(\omega_d t)\right).$$ On the other hand, a second type of driving can be realized by adding just a single sideband say for example at $+ \omega_d$. This single-sideband protocol can be expressed as $$\alpha^{\{1\}}_p(t) = \sqrt{\epsilon_0}(1 + f_0 e^{i\omega_d t}),$$ which then drives the pump beam intensity given by $$\label{eq:prot1}
|\alpha^{\{1\}}_p(t)|^2 = {\epsilon_0}\left(1 + {f_0^2} + 2f_0 \mathrm{cos}(\omega_d t)\right).$$ If we compare Eqs. and , it becomes immediately obvious that the single-sideband protocol introduces a larger constant shift of $\epsilon_0f_0^2$ to the pump power as compared to the double-sideband protocol which only increases the pump intensity by a constant amount of $\epsilon_0f_0^2/2$. This becomes problematic for larger values of $\varepsilon_0$ which require stronger driving amplitude if one intends to completely wipe the DW phase. Indeed, as shown in an example presented in Fig. \[fig:proto\], the reduction in the number of cavity photons is much greater in the double-sideband protocol for a fixed value of the driving amplitude $f_0$.
![Comparison between the suppression effect of single-sideband and double-sideband protocols. Time evolution of the cavity mode occupation for $\varepsilon_0/E_{\mathrm{rec}} = 2.24$, $\omega_d = 2\pi \times 10~\mathrm{kHz}$, and $f_0=0.18$. []{data-label="fig:svd"}](MF_cav_single_v_double){width="0.45\columnwidth"}
Temporal correlation
====================
In order to obtain the dependence of the temporal correlation on the pump strength parameter shown in Fig. 3(c), we first calculate the temporal correlation according to $$G^{(1)}(t) = \frac{\left( \left(\mathrm{Re}\langle \hat{\phi}^{\dagger}_{0,0}(t)\hat{\phi}_{0,0}(t_1) \rangle\right)^2 + \left(\mathrm{Im}\langle \hat{\phi}^{\dagger}_{0,0}(t)\hat{\phi}_{0,0}(t_1) \rangle\right)^2 \right)^{1/2}}{\langle n_{0,0}(t_1) \rangle}$$ where $t_1=20~\mathrm{ms}$. The corresponding decay rates indicative of the correlation time in the system are $\gamma_u$ for the undriven case and $\gamma_d$ for the driven case. This can be extracted from fitting an exponential decay $\mathrm{exp}(-\gamma t)$ to $G^{(1)}(t)$ as exemplified in Fig. \[fig:mf\_dynamics\_N00\].
![Temporal correlation for the (Red) undriven case and the (blue) driven case. (Top to bottom) $\varepsilon_0 = \{2.15,2.17,2.23\}$. (Left) linear and (right) semi-logarithmic scale. Dashed curves correspond to the exponential fit as described in the text. []{data-label="fig:mf_dynamics_N00"}](TW_OTOC20comp_lin_N33_comp_2p15 "fig:"){width="0.45\columnwidth"}![Temporal correlation for the (Red) undriven case and the (blue) driven case. (Top to bottom) $\varepsilon_0 = \{2.15,2.17,2.23\}$. (Left) linear and (right) semi-logarithmic scale. Dashed curves correspond to the exponential fit as described in the text. []{data-label="fig:mf_dynamics_N00"}](TW_OTOC20comp_N33_comp_2p15 "fig:"){width="0.45\columnwidth"}\
![Temporal correlation for the (Red) undriven case and the (blue) driven case. (Top to bottom) $\varepsilon_0 = \{2.15,2.17,2.23\}$. (Left) linear and (right) semi-logarithmic scale. Dashed curves correspond to the exponential fit as described in the text. []{data-label="fig:mf_dynamics_N00"}](TW_OTOC20comp_lin_N33_comp_2p17 "fig:"){width="0.45\columnwidth"}![Temporal correlation for the (Red) undriven case and the (blue) driven case. (Top to bottom) $\varepsilon_0 = \{2.15,2.17,2.23\}$. (Left) linear and (right) semi-logarithmic scale. Dashed curves correspond to the exponential fit as described in the text. []{data-label="fig:mf_dynamics_N00"}](TW_OTOC20comp_N33_comp_2p17 "fig:"){width="0.45\columnwidth"}\
![Temporal correlation for the (Red) undriven case and the (blue) driven case. (Top to bottom) $\varepsilon_0 = \{2.15,2.17,2.23\}$. (Left) linear and (right) semi-logarithmic scale. Dashed curves correspond to the exponential fit as described in the text. []{data-label="fig:mf_dynamics_N00"}](TW_OTOC20comp_lin_N33_comp_2p23 "fig:"){width="0.45\columnwidth"}![Temporal correlation for the (Red) undriven case and the (blue) driven case. (Top to bottom) $\varepsilon_0 = \{2.15,2.17,2.23\}$. (Left) linear and (right) semi-logarithmic scale. Dashed curves correspond to the exponential fit as described in the text. []{data-label="fig:mf_dynamics_N00"}](TW_OTOC20comp_N33_comp_2p23 "fig:"){width="0.45\columnwidth"}\
Comparison between mean-field and truncated Wigner results
==========================================================
In order include quantum fluctuations, we have simulated the dynamics within the truncated Wigner (TW) approximation. A detailed discussion of this method and how to sample the initial quantum noise for coherent and vacuum states can be found in [@Blakie2008; @Polkovnikov2010]. In a nutshell, the TW approximation goes beyond the mean-field level by accounting for quantum fluctuations in the initial state of the system. This is done by solving the underlying mean-field equations of motion stochastically using an ensemble of initial conditions or trajectories that correctly samples the initial Wigner distribution for the available quantum states in the system. Finally, observables obtained from each trajectory are averaged over the ensemble. For the cavity-BEC system considered in this work, the corresponding set of mean-field equation reads [@Ritsch2013] $$\begin{aligned}
\label{eq:eom}
i \frac{\partial \phi_{n,m}}{\partial t} &=
\omega_{\mathrm{rec}}\left(n^2 + m^2+\frac{\Delta_0}{2\omega_{\mathrm{rec}}}|\alpha|^2-\frac{|\alpha_p(t)|^2}{2}\right)\phi_{n,m} +\frac{\Delta_0}{4}|\alpha|^2(\phi_{n,m-2}+\phi_{n,m+2})-\frac{\omega_{\mathrm{rec}}}{4}|\alpha_p(t)|^2(\phi_{n-2,m}+\phi_{n+2,m})\\ \nonumber
&+\frac{\sqrt{\omega_{\mathrm{rec}}}\sqrt{|\Delta_0|}}{2}\alpha_p(t)\mathrm{Re}({\alpha})(\phi_{n-1,m-1}+\phi_{n+1,m-1}+\phi_{n-1,m+1}+\phi_{n+1,m+1})\\ \nonumber
i \frac{\partial \alpha}{\partial t} &= \left[-\delta_{\mathrm{eff}} + \frac{1}{2}N_a\Delta_0 \sum_{n,m}\mathrm{Re}[\phi_{n,m}\phi^{*}_{n,m+2}]-i\kappa \right]\alpha + i\xi \\ \nonumber
&+\frac{N_a\sqrt{\omega_{\mathrm{rec}}}\sqrt{|\Delta_0|}}{4}\alpha_p(t)\sum_{n,m}\phi_{n,m}(\phi^{*}_{n+1,m+1}+\phi^{*}_{n+1,m-1}) + \phi^{*}_{n,m}(\phi_{n+1,m+1}+\phi_{n+1,m-1}),\end{aligned}$$ where the Gaussian noise operator $\xi$ in the cavity mode equation follows $\langle \xi(t)\xi^{\dagger}(t') \rangle= \kappa\delta(t-t')$.
![(Left) Mean-field and (Right) truncated Wigner dynamics for the (Top) cavity mode and (Bottom) BEC mode occupations for $\varepsilon_0/E_{\mathrm{rec}} = 2.20$, $\omega_d = 2\pi \times 6~\mathrm{kHz}$, and different strengths of the driving amplitude $f_0$. []{data-label="fig:mf_tw_dynamics"}](MF_TW_field_two_6kHz_driven){width="0.7\columnwidth"}
A comparison between the mean-field and TW results are shown in Fig. \[fig:mf\_tw\_dynamics\]. As seen in Fig. \[fig:mf\_tw\_dynamics\], the main difference between the mean-field and truncated Wigner simulations is the apparent earlier onset of DW formation predicted by TWA. This suggests that quantum fluctuations lower the threshold value for the phase transition from the BEC to the DW phase. A more in-depth discussion about this phenomenon and how it modifies the hysteretic dynamics observed in Ref. [@Klinder2015] will be addressed in an upcoming work [@Cosme2018futu]. Apart from this deviation, it can be seen that the ability to dynamically control the BEC and density-ordered phases in the system appears to be robust against quantum and vacuum fluctuations from the initial state.
Derivation of the Effective Time-Independent Hamiltonian
========================================================
Recall that the Hamiltonian shown in the main text reads $$\begin{aligned}
\label{eq:hamilt}
\hat{H}&= -\delta_{\mathrm{C}}\hat{\alpha}^{\dagger}\hat{\alpha} + \frac{\Delta_0}{4}\hat{\alpha}^{\dagger}\hat{\alpha}\sum_{n,m}\left( \hat{\phi}^{\dagger}_{n,m+2}\hat{\phi}_{n,m} +\hat{\phi}^{\dagger}_{n,m}\hat{\phi}_{n,m+2} \right) +\frac{\Delta_0}{2}\hat{\alpha}^{\dagger}\hat{\alpha}\sum_{n,m}\hat{\phi}^{\dagger}_{n,m}\hat{\phi}_{n,m} \\ \nonumber
&+ \omega_{\mathrm{rec}}\sum_{n,m}(n^2+m^2)\hat{\phi}^{\dagger}_{n,m}\hat{\phi}_{n,m} -\frac{\omega_{\mathrm{rec}}}{2}|\alpha_p|^{2}\sum_{n,m}\hat{\phi}^{\dagger}_{n,m}\hat{\phi}_{n,m} - \frac{\omega_{\mathrm{rec}}}{4}|\alpha_p|^{2}\sum_{n,m}\left( \hat{\phi}^{\dagger}_{n+2,m}\hat{\phi}_{n,m} +\hat{\phi}^{\dagger}_{n,m}\hat{\phi}_{n+2,m} \right) \\ \nonumber
&+\frac{\sqrt{\omega_{\mathrm{rec}}}\sqrt{|\Delta_0|}}{4}|\alpha_p|(\hat{\alpha}^{\dagger}+\hat{\alpha})\sum_{n,n}\left( \hat{\phi}^{\dagger}_{n,m}(\hat{\phi}_{n+1,m+1}+\hat{\phi}_{n+1,m-1}) + (\hat{\phi}^{\dagger}_{n+1,m+1}+\hat{\phi}^{\dagger}_{n+1,m-1})\hat{\phi}_{n,m} \right),\end{aligned}$$ For the kind of driving considered here, the pump field amplitude is driven according to $$\label{eq:side}
\alpha_p(t) = \sqrt{\epsilon_0}(1 + f_0 \mathrm{cos}(\omega_d t)),$$ which effectively drives the pump beam intensity via $$|\alpha_p(t)|^2 = {\epsilon_0}\left(1 + \frac{f_0^2}{2} + \frac{f_0^2\mathrm{cos}(2\omega_d t)}{2} + 2f_0 \mathrm{cos}(\omega_d t)\right).$$ An effective time-independent Hamiltonian can be obtained from Floquet-Magnus [@Bukov2015; @Zhu2016] or high-frequency expansion [@Hemmerich2010; @Goldman2014; @Eckardt2015]. We briefly outline the general procedure for such expansion below. To this end, it is helpful to expand the time-dependent Hamiltonian in terms of its Fourier components such that $$\hat{H}(t) \equiv \hat{H} = \sum_{m=-\infty}^{\infty} e^{im\omega_d t}\hat{H}_m.$$ The effective Hamiltonian can then be expanded as $$H_\mathrm{eff} = \sum_{n=0}^{\infty} H^{(n)}_{\mathrm{eff}}$$ where up to second-order we have [@Goldman2014; @Bukov2015; @Eckardt2015] $$\begin{aligned}
&H^{(0)}_{\mathrm{eff}} = H_0 \\ \nonumber
&H^{(1)}_{\mathrm{eff}} = \frac{1}{\omega_d}\sum_{\ell} \frac{1}{\ell}[H_{\ell},H_{-\ell}] \\ \nonumber
&H^{(2)}_{\mathrm{eff}} = \frac{1}{\omega_d^2}\sum_{\ell \neq 0} \left(\frac{[H_{-\ell},[H_{0},H_{\ell}]]}{2\ell^2} + \sum_{\ell'\neq 0,\ell} \frac{[H_{-\ell'},[H_{\ell'-\ell},H_{\ell}]]}{3\ell\ell'} \right).\end{aligned}$$ For a single frequency sideband as in Eq. , the first non-trivial correction to the time-averaged Hamiltonian $H_0$ is given by the first-order correction $H^{(1)}_{\mathrm{eff}}$ since $H_1 \neq H_{-1}$ in this case. However for the two-sideband protocol considered in this work, $H_1 = H_{-1}$ meaning the first-order correction for the effective Hamiltonian is zero, $H^{(1)}_{\mathrm{eff}}=0$. Therefore, we have the following effective time-independent Hamiltonian $$\label{eq:2Ham}
H_\mathrm{eff} = H_0 + H^{(2)}_{\mathrm{eff}}.$$ where $$\label{eq:2eff}
H^{(2)}_{\mathrm{eff}}=-\frac{1}{4\omega_d^2}\left[[H_0,A_1],A_1\right]$$ Note that in Eq. , we have introduced $$\label{eq:h0}
H_0 = -\delta_{\mathrm{C}}C + \frac{\Delta_0}{4}CZ + \omega_{\mathrm{rec}}E +\frac{\Delta_0}{2}CN -\frac{\omega_{\mathrm{rec}} \epsilon_0}{2}\left(1+\frac{f_0^2}{2}\right)N -\frac{\omega_{\mathrm{rec}} \epsilon_0}{4}\left(1+\frac{f_0^2}{2}\right)Y +\frac{\sqrt{\omega_{\mathrm{rec}}}\sqrt{|\Delta_0|} \sqrt{\epsilon_0}}{4}DJ$$ and $$\label{eq:a1}
2H_1 = 2H_{-1} \equiv A_1=-\frac{\omega_{\mathrm{rec}} (2f_0\epsilon_0)}{2}N -\frac{\omega_{\mathrm{rec}} (2f_0\epsilon_0)}{4}Y +\frac{\sqrt{\omega_{\mathrm{rec}}} \sqrt{|\Delta_0|}\sqrt{\epsilon_0} f_0}{4 }DJ.$$ For brevity we will drop the hats in the operators. Note that in Eqs and , we define the following operators: $$\begin{aligned}
C&=\alpha^{\dagger}\alpha \\ \nonumber
D&=\alpha^{\dagger}+\alpha \\ \nonumber
N&=\sum \phi^{\dagger}_{n,m}\phi_{n,m} \\ \nonumber
E&=\sum (n^2+m^2)\phi^{\dagger}_{n,m}\phi_{n,m} \\ \nonumber
Z&=\sum \left( \phi^{\dagger}_{n,m+2}\phi_{n,m} + \mathrm{h.c.}\right)\\ \nonumber
Y&=\sum \left( \phi^{\dagger}_{n+2,m}\phi_{n,m} + \mathrm{h.c.}\right)\\ \nonumber
J&=\sum \left( \phi^{\dagger}_{n,m}\left(\phi_{n+1,m+1} + \phi_{n+1,m-1} \right) + \mathrm{h.c.}\right)\\ \nonumber\end{aligned}$$ It is easy to show that the only nonzero commutator relations are $[C,D]$, $[E,J]$, $[E,Y]$, and $[E,Z]$. Then we find $$\begin{aligned}
[&[H_0,A_1],A_1] = \frac{\omega_{\mathrm{rec}}^2 (f_0\epsilon_0)^2}{16}\biggl[4\omega_{\mathrm{rec}}[[E,Y],Y] -\frac{2\sqrt{\omega_{\mathrm{rec}}}\sqrt{|\Delta_0|}}{\sqrt{\epsilon_0}}D \biggl([[E,J],Y]+[[E,Y],J]\biggr) \\ \nonumber
&-\frac{\Delta_0}{{\epsilon_0}\omega_{\mathrm{rec}}}[[C,D],D]J^2\left(-\delta_{\mathrm{C}}+\frac{\Delta_0}{2}\left(N+\frac{Z}{2}\right)\right) -\frac{\Delta_0}{\epsilon_0}D^2[[E,J],J] \biggr].\end{aligned}$$ One useful property for calculating commutators between various momentum mode operators is $$\begin{aligned}
\sum_{n,m,n',m'} &[f(n,m)\phi^{\dagger}_{n+a,m+b}\phi_{n+c,m+d},\phi^{\dagger}_{n'+a',m'+b'}\phi_{n'+c',m'+d'}] \\ \nonumber
&=\sum_{n,m}\biggl(f(n,m)\phi^{\dagger}_{n+a,m+b}\phi_{n+c+c'-a',m+d+d'-b'}-f(n+c'-a,m+d'-b) \phi^{\dagger}_{n+a',m+b'}\phi_{n+c+c'-a,m+d+d'-b} \biggr)\end{aligned}$$ Using this property, we get $$\begin{aligned}
&[[H_0,A_1],A_1] = \frac{\omega_{\mathrm{rec}}^2 (f_0\epsilon_0)^2}{16}\biggl[32{\omega_{\mathrm{rec}}}\left( \sum(\phi^{\dagger}_{n,m}\phi_{n-4,m} + \mathrm{h.c.})\right) +\frac{2\Delta_0}{{\epsilon_0}\omega_{\mathrm{rec}}}J^2\left(-\delta_{\mathrm{C}}+\frac{\Delta_0}{2}\left(N+\frac{Z}{2}\right)\right)\\ \nonumber
&-\frac{4\Delta_0}{\epsilon_0}(\alpha^{\dagger}+\alpha)^2\left(-4\sum\phi^{\dagger}_{n,m}\phi_{n,m} + \sum(\phi^{\dagger}_{n,m}(\phi_{n+2,m-2}+\phi_{n+2,m+2}) + \mathrm{h.c.}) \right) \\ \nonumber
&-16\frac{\sqrt{\omega_{\mathrm{rec}}}\sqrt{|\Delta_0|}}{\sqrt{\epsilon_0}}(\alpha^{\dagger}+\alpha)\biggl(\sum(\phi^{\dagger}_{n,m}(\phi_{n+3,m-1}+\phi_{n+3,m+1}-(\phi_{n+1,m-1}+\phi_{n+1,m+1})) + \mathrm{h.c.}) \biggr) \biggr] \end{aligned}$$ Then the first nontrivial correction to the effective Hamiltonian reads $$\begin{aligned}
\label{eq:ham2}
&H^{(2)}_{\mathrm{eff}}=-\frac{\omega_{\mathrm{rec}}^3 (f_0\epsilon_0)^2}{2\omega_d^2}\left( \sum(\phi^{\dagger}_{n,m}\phi_{n-4,m} + \mathrm{h.c.})\right) - \frac{ \omega_{\mathrm{rec}}\Delta_0 (f_0\epsilon_0)^2}{32\omega_d^2\epsilon_0}\left(-\delta_{\mathrm{C}}+\frac{\Delta_0}{2}\left(N+\frac{Z}{2}\right)\right)J^2 \\ \nonumber
&+\frac{\sqrt{\omega_{\mathrm{rec}}}\sqrt{|\Delta_0|}\sqrt{\epsilon_0} }{4}\epsilon_0\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2(f_0)^2(\alpha^{\dagger}+\alpha)\sum(\phi^{\dagger}_{n,m}(\phi_{n+3,m-1}+\phi_{n+3,m+1})+ \mathrm{h.c.}) \\ \nonumber
&+\frac{\epsilon_0}{2}\frac{\Delta_0}{8}\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2(f_0)^2(\alpha^{\dagger}+\alpha)^2\sum(\phi^{\dagger}_{n,m}(\phi_{n+2,m-2}+\phi_{n+2,m+2}) + \mathrm{h.c.}) \\ \nonumber
&-\frac{\sqrt{\omega_{\mathrm{rec}}}\sqrt{|\Delta_0|}\sqrt{\epsilon_0} }{4}\epsilon_0\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2(f_0)^2(\alpha^{\dagger}+\alpha)\sum(\phi^{\dagger}_{n,m}(\phi_{n+1,m-1}+\phi_{n+1,m+1})+ \mathrm{h.c.}) \\ \nonumber
&-\frac{\epsilon_0}{2}\frac{\Delta_0}{2}\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2(f_0)^2(\alpha^{\dagger}+\alpha)^2\sum\phi^{\dagger}_{n,m}\phi_{n,m}\end{aligned}$$ Note that the first two lines in the Eq. can be neglected *a posteriori*. In the first line, the first term can be dropped when higher momentum modes corresponding to $\{n+4,m+4\}$ for any any integer values of $n$ and $m$ have negligible occupation which is the case for all superradiant states obtained in this work as exemplified by the DW$_1$ state in Fig. 5(r). The second term, on the other hand, will have negligible contribution since $J \ll 1$ is almost zero for the BEC phase while it will be several orders of magnitude lower than the next relevant energy scale in the Hamiltonian for the self-organized phase. The second and third lines corresponding to higher-order hopping terms in momentum space can also be neglected for moderate depletion of the BEC mode such that $|\phi_{0,0}|^2+\sum_{n,m=\{\pm 1, \pm 1\}}|\phi_{n,m}|^2 \approx N_a$. This simplification is further justified in calculations considered here since we focus around the phase transition boundary where there are still relatively fewer photons occupying the cavity mode in the DW$_1$ phase. Finally, the effective time-independent Hamiltonian is given by $$\begin{aligned}
&H_{\mathrm{eff}}=-\delta_{\mathrm{C}} {\alpha}^{\dagger} {\alpha} + \frac{\Delta_0}{4} {\alpha}^{\dagger} {\alpha}\sum_{n,m}\left( {\phi}^{\dagger}_{n,m+2} {\phi}_{n,m} + \mathrm{h.c.} \right) + \omega_{\mathrm{rec}}\sum_{n,m}(n^2+m^2) {\phi}^{\dagger}_{n,m} {\phi}_{n,m}+\frac{\Delta_0}{2} {\alpha}^{\dagger} {\alpha}\sum_{n,m} {\phi}^{\dagger}_{n,m} {\phi}_{n,m} \\ \nonumber
&-\frac{\omega_{\mathrm{rec}} \epsilon_0}{2}\left[1+\frac{f_0^2}{2}+\frac{\Delta_0}{2\omega_{\mathrm{rec}}}\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2f_0^2(\alpha^{\dagger}+\alpha)^2\right] \sum_{n,m} {\phi}^{\dagger}_{n,m} {\phi}_{n,m} -\frac{\omega_{\mathrm{rec}} \epsilon_0}{4}\left(1+\frac{f_0^2}{2}\right)\sum_{n,m}\left( {\phi}^{\dagger}_{n,m} {\phi}_{n+2,m} + \mathrm{h.c.} \right) \\ \nonumber
&+\frac{\sqrt{\omega_{\mathrm{rec}}} \sqrt{|\Delta_0|}\sqrt{\epsilon_0} }{4}\left[1-\epsilon_0\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2f_0^2 \right](\alpha^{\dagger}+\alpha)\sum(\phi^{\dagger}_{n,m}(\phi_{n+1,m-1}+\phi_{n+1,m+1})+ \mathrm{h.c.})\end{aligned}$$
![Comparison of the cavity mode dynamics between the solution of the full mean-field equations and the effective time-independent Hamiltonian from the Magnus expansion. The driving frequency is set to $w_d=2\pi\times 10~\mathrm{kHz}$ and the driving amplitude is $f_0=0.152$. []{data-label="fig:mag"}](MF_cav_driven_10kHz_eps_2p20){width="0.5\columnwidth"}
Upon normalization of the momentum mode occupation $\sum \phi^*_{n,m}\phi_{n,m} = 1$, we finally obtain the corresponding mean-field equation for the effective Hamiltonian $H_{\mathrm{eff}}$ $$\begin{aligned}
\label{eq:eom_mag}
&i \frac{\partial \phi_{n,m}}{\partial t} =
\omega_{\mathrm{rec}}\left(n^2+m^2+\frac{\Delta_0}{2\omega_{\mathrm{rec}}}|\alpha|^2-\frac{\epsilon_0}{2}\left(1+\frac{f_0^2}{2}+2(\mathrm{Re}(\alpha))^2\frac{\Delta_0}{\omega_{\mathrm{rec}}}\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2f_0^2 \right) \right)\phi_{n,m} \\ \nonumber
&+\frac{\Delta_0}{4}|\alpha|^2(\phi_{n,m-2}+\phi_{n,m+2})-\frac{\omega_{\mathrm{rec}}\epsilon_0}{4}\left(1+\frac{f_0^2}{2}\right)(\phi_{n-2,m}+\phi_{n+2,m})\\ \nonumber
&+\frac{\sqrt{\omega_\mathrm{rec}}\sqrt{|\Delta_0|}\sqrt{\epsilon_0}}{2}\left[1-\epsilon_0\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2f_0^2 \right]\mathrm{Re}(\alpha)(\phi_{n-1,m-1}+\phi_{n+1,m-1}+\phi_{n-1,m+1}+\phi_{n+1,m+1}) \\ \nonumber
&i \frac{\partial \alpha}{\partial t} = \left(\left(-\delta_{\mathrm{eff}} -\frac{N_a\Delta_0 \epsilon_0}{2}\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2f_0^2 \right) +\frac{1}{2}N_a\Delta_0\sum_{n,m}\mathrm{Re}[\phi_{n,m}\phi^{*}_{n,m+2}]-i\kappa \right)\alpha \\ \nonumber
&+\frac{N_a\sqrt{\omega_{\mathrm{rec}}}\sqrt{|\Delta_0|}}{4}\sqrt{\epsilon_0}\left[1-\epsilon_0\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2f_0^2 \right]\sum_{n,m}\left(\phi_{n,m}(\phi^{*}_{n+1,m+1}+\phi^{*}_{n+1,m-1})+ \mathrm{h.c.}\right)+\frac{N_a\Delta_0 \epsilon_0}{2}\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2f_0^2\alpha^*.\end{aligned}$$ Note that we recover the mean-field equations of motion in Ref. [@Klinder2015] for the undriven case $f_0=0$. From the bracketed terms in Eq. \[eq:eom\_mag\], it is easy to see that the enhancement of the BEC phase can be explained by an effective reduction in the coupling strength of the two-photon process that scatters atom from $\phi_{0,0}$ to $\phi_{\pm 1, \pm 1}$ $$\sqrt{\epsilon_0} \xrightarrow{\text{driven}} \sqrt{\epsilon_0}\left[1-\epsilon_0\left(\frac{\omega_{\mathrm{rec}}}{\omega_d}\right)^2f_0^2 \right].$$
We numerically integrate this set of equations in order to obtain the results shown in thin solid lines in Fig. 4. We also show in Fig. \[fig:mag\] a comparison between the results of numerically integrating the full mean-field equations and those from an effective time-independent Hamiltonian according to Eq. .
![Dynamical renormalization of the BEC-DW phase transition, visible in the (top) cavity mode and (bottom) BEC mode occupation for $w_d=2\pi\times 10~\mathrm{kHz}$. (i) Thin solid line shows the effective Hamiltonian prediction for the phase boundary, (ii) thick dashed line the TW result, (ii) thick dashed-dotted line the MF result. The phase boundary is indicated based on $|\alpha|^2> 70$ and $n_0/N> 0.97$.[]{data-label="fig:mf_tw_phase"}](TW_CAV_BEC_driven_10kHz_phase_two_mf){width="0.4\columnwidth"}
For the driven case presented in Fig. \[fig:mag\], we have applied a Gaussian filter with width $\sigma=1/\omega_{\mathrm{rec}}$ to artificially remove the micromotion part of the dynamics which is inherently not captured by the effective Hamiltonian obtained here. In doing so, we can then focus on more important aspects of the dynamics including its overall trend and long-time behaviour. On one hand, we find that the effective time-independent Hamiltonian nicely captures the short time dynamics predicted by the full mean-field equations after the modulation is sharply switched on. This suggests that the driving protocol can be seen as some kind of sudden quench to an effectively weaker atom-cavity coupling. On the other hand, we find that the steady-state predictions from the effective Hamiltonian agree very well with the mean-field counterpart for a gradual ramp of the driving amplitude as exemplified in Fig. \[fig:mag\]. This is of course consistent with the good agreement for the phase boundary shown in Fig. \[fig:mf\_tw\_phase\].
Mean-field order parameters and single-particle density profiles for density-wave ordered phases
================================================================================================
Here, we present results for single trajectories in our TW simulations, which basically correspond to mean-field predictions for the dynamics. In particular, we calculate the expectation value of the dominant order parameter $\langle \Phi_{n,m} \rangle$ for the DW$_1$, DW$_4$, and DW$_3$ dynamical phases. We also obtain exemplary single-particle density (spd) profiles, $\rho(y,z)= \sum_{n,m,n',m'} \phi^{\dagger}_{n,m}\phi_{n',m'}e^{i(n-n')ky}e^{i(m-m')ky}$, in the long-time limit of each DW phases in order to gain further insights on possible symmetry breaking phenomenon. The corresponding results are shown in Fig. \[fig:mf\_op\].
For the renormalized DW$_1$ phase in the presence of driving, the original $\mathbb{Z}_2$-symmetry breaking associated to the self-organization of atoms survives as seen in the left panel of Fig. \[fig:mf\_op\]. In this case, the atoms spontaneously form one of the two possible checkerboard patterns corresponding to a positive-valued order parameter $\langle \Phi_{1,1} \rangle$. Moreover, the small temporal fluctuation of the leading order parameter suggests that the atomic ensemble essentially remains fixed in one of the symmetry broken ordered phases for long times.
In contrast to the nonequilibrium DW$_1$ phase, we find that the DW$_4$ and DW$_3$ phases exhibit strong oscillation of the dominant order parameters around zero. This physically means that the system is dynamically switching between possible symmetry broken ordered phases. This phenomenon has been also predicted for the so-called dynamical normal phase [@Chitra2015; @Molignini2017] where the atoms are dynamically switching between the even and odd checkerboard patterns. For the DW$_4$ phase shown in the middle panel of Fig. \[fig:mf\_op\], the system is oscillating between possible striped phases corresponding to density modulation along the direction of the pump beam. This is consistent with the absence of momentum excitations along the $z$-direction shown in Fig. 5(q). Similarly, the single-particle density profile for the DW$_3$ phase dynamically switches between stripe-ordered phases with additional checkerboard density modulation along the cavity axis as depicted in Fig. \[fig:mf\_op\].
![(Top) Dominant order parameters and (Bottom) exemplary single-particle density profiles within the mean-field theory for (left) DW$_1$ ($w_d=2\pi\times 34.0~\mathrm{kHz}$), (middle) DW$_4$ ($w_d=2\pi\times 28.5~\mathrm{kHz}$), and (right) DW$_3$ ($w_d=2\pi\times 35.5~\mathrm{kHz}$). The exact parameters are the same as Fig. 5 in the main text.[]{data-label="fig:mf_op"}](OP_1_wd_34p0 "fig:"){width="0.33\columnwidth"}![(Top) Dominant order parameters and (Bottom) exemplary single-particle density profiles within the mean-field theory for (left) DW$_1$ ($w_d=2\pi\times 34.0~\mathrm{kHz}$), (middle) DW$_4$ ($w_d=2\pi\times 28.5~\mathrm{kHz}$), and (right) DW$_3$ ($w_d=2\pi\times 35.5~\mathrm{kHz}$). The exact parameters are the same as Fig. 5 in the main text.[]{data-label="fig:mf_op"}](OP_2_wd_28p5 "fig:"){width="0.33\columnwidth"}![(Top) Dominant order parameters and (Bottom) exemplary single-particle density profiles within the mean-field theory for (left) DW$_1$ ($w_d=2\pi\times 34.0~\mathrm{kHz}$), (middle) DW$_4$ ($w_d=2\pi\times 28.5~\mathrm{kHz}$), and (right) DW$_3$ ($w_d=2\pi\times 35.5~\mathrm{kHz}$). The exact parameters are the same as Fig. 5 in the main text.[]{data-label="fig:mf_op"}](OP_3_wd_35p5 "fig:"){width="0.33\columnwidth"}\
Importance of the recoil resolution $\kappa$
============================================
In the case when $4\omega_{\mathrm{rec}} \ll \kappa$ just like in Ref. [@Baumann2010], the cavity mode adiabatically follows the atomic degrees of freedom such that only the dynamics of the atomic modes need to be considered explicitly. As mentioned in the main text, we find that it is important to explicitly consider the dynamics of both the atomic and cavity modes in order to mimic the dynamical suppression effect of density-wave order seen in high-$T_c$ superconductors. That is, we briefly show here the importance of having $4\omega_{\mathrm{rec}} \gg \kappa$ as in Refs. [@Klinder2015; @Klinder2015b; @Klinder2016] in the recondensation process after the modulation. To this end, we show in Fig. \[fig:tw\_kappa\] the ensuing dynamics for the BEC and cavity modes for $\kappa=\kappa_{\mathrm{expt}}=2\pi\times 4.5~\mathrm{kHz}$ and for $\kappa=10\kappa_{\mathrm{expt}}$. We adjust the mean pump strength for each case in order to fix the number of photons in the DW phase. We then choose a critical modulation amplitude $f_0$ which is just enough to completely suppress the cavity mode occupation. A higher value for $\kappa$ means that the mean pump strength needed to enter the DW phase will have to increase as well as evident from our simulation. Even though we are still able to completely suppress the DW phase for $\kappa=10\kappa_{\mathrm{expt}}$, the number of atoms that we recover back to the BEC mode is not significant in contrast to the case when $\kappa=\kappa_{\mathrm{expt}}$. Moreover, we find stronger temporal variance in the BEC mode occupation for $\kappa=10\kappa_{\mathrm{expt}}$. These observations suggest the importance of low $\kappa$ and it also emphasizes the point that the photonic and atomic degrees of freedom should be treated individually.
![Comparison of the (left) BEC and (right) cavity modes for $\kappa=\kappa_{\mathrm{expt}}$ ($\kappa=10\kappa_{\mathrm{expt}}$) with $f_0=0.12~(0.22)$, $\varepsilon_0/E_{\mathrm{rec}}=2.18~(6.93)$, and $w_d=2\pi\times 10~\mathrm{kHz}$.[]{data-label="fig:tw_kappa"}](TW_BEC_kappa_comparison "fig:"){width="0.5\columnwidth"}![Comparison of the (left) BEC and (right) cavity modes for $\kappa=\kappa_{\mathrm{expt}}$ ($\kappa=10\kappa_{\mathrm{expt}}$) with $f_0=0.12~(0.22)$, $\varepsilon_0/E_{\mathrm{rec}}=2.18~(6.93)$, and $w_d=2\pi\times 10~\mathrm{kHz}$.[]{data-label="fig:tw_kappa"}](TW_CAV_kappa_comparison "fig:"){width="0.5\columnwidth"}
|
---
abstract: 'In this paper we will first present a generalization of the wedge product of association schemes to table algebras and give a necessary and sufficient condition for a table algebra to be the wedge product of two table algebras. Then we show that if the duals of two commutative table algebras are table algebras, then the dual of their wedge product is a table algebra, and is also isomorphic to the wedge product of the duals of those table algebras in the reverse order. Some applications to association schemes are also given.'
author:
- |
Javad Bagherian\
Department of Mathematics, University of Isfahan,\
P.O. Box: 81746-73441, Isfahan, Iran,\
[email protected]
title: On the wedge product of table algebras and applications to association schemes
---
[*Key words:*]{} table algebra, wedge product, dual. 20C99; 05E30.
Introduction
============
The wreath product of table algebras provides a way to construct the new table algebras from old ones. This construction is a generalization of the wreath product of association schemes to table algebras. Moreover, if the duals of two commutative table algebras are table algebras, then the dual of their wreath product is a table algebra, and is also isomorphic to the wreath product of the duals of those table algebras in the reverse order [@xuwr]. We mention that the dual of a commutative table algebra is a C-algebra but is not a table algebra, in general. (A question in the book by Bannai and Ito [@Bannai p. 104] asks when the dual of a commutative table algebra is also a table algebra.)
Recently, the wedge product of association schemes as a generalization of the wedge product of Schur rings has been given in [@Mu]. This product of association schemes can be considered as a generalization of the wreath product of association schemes.
In this paper, we first give a generalization of the wedge product of association schemes to table algebras. Then we give a necessary and sufficient condition for a table algebra to be the wedge product of two table algebras. We can see that the wreath product of table algebras is a special case of the wedge product of table algebras. Moreover, we prove that if the duals of two table algebras are table algebras then the dual of their wedge product is a table algebra, and is also isomorphic to the wedge product of the duals of those table algebras in the reverse order. Finally, we show that the complex adjacency algebra of the wedge product of association schemes is isomorphic to the wedge product of the complex adjacency algebras of those association schemes and then we give some applications to association schemes.
Preliminaries {#2309072}
=============
In this section, we state some necessary definitions and known results about C- algebras, table algebras, association schemes, homomorphisms of table algebras and the dual of table algebras. Throughout this paper, $\C$ denotes the complex numbers, $\mathbb{R}$ the real numbers and $\mathbb{R}^+$ the positive real numbers.
C-algebras, table algebras and association schemes
--------------------------------------------------
We follow from [@EPV] for the definition of C-algebras. Hence we deal with C-algebras as the following:
(See [@EPV Difinition 3.1].) Let $A$ be a finite dimensional associative algebra over $\C$ with the identity element $1_A$ and a base $B$ in the linear space sense. Then the pair $(A,B)$ is called a C-[*algebra*]{} if the following conditions (I)-(IV) hold: $1_A \in B$ and the structure constants of ${B}$ are real numbers, i.e., for $a,b \in {B}$: $$ab = \displaystyle\sum _{c\in {B}}\lambda_{abc}c,\ \ \ \ \
\lambda_{abc} \in \mathbb{R}.$$ There is a semilinear involutory anti-automorphism (denoted by $^{*}$) of $A$ such that ${B}^{*} = {B}$. For $a,b\in B$ the equality $\lambda_{ab1_A} = \delta_{ab^*}|a|$ holds where $|a|>0$ and $\delta$ is the Kronecker symbol. The mapping $b\rightarrow |b|, b\in B$ is a one dimensional $*$-linear representation of the algebra $A$, which is called the [*degree map*]{}.
In the definition above if the algebra $A$ is commutative, then $(A,B)$ becomes a C-algebra in the sense of [@Bannai].
Let $(A,B)$ be a C-algebra. For any $x=\sum_{b\in B}x_bb\in A$ we denote by ${\rm{Supp}}(x)$ the set of all basis elements $b\in B$ such that $x_b\neq 0$. If $N_1, \ldots, N_m$ are nonempty closed subsets of $ B$, then we set $$N_1N_2\ldots N_m=\displaystyle\bigcup_{\begin{subarray}{l} b_1\in N_1\\b_2\in N_2 \\ \vdots \\ b_m\in N_m \end{subarray}}\supp(b_1\ldots b_m).$$ The set $N_1N_2\ldots N_m$ is called the [*complex product*]{} of closed subsets $N_i, 1\leq i\leq m$. If one of the factors in a complex product consists of a single element $b$, then one usually writes $b$ for $\{b\}$. A nonempty subset $N\subseteq B$ is called a [*closed subset*]{}, denoted as $N\leq B$, if ${N}^{*}{N} \subseteq {N}$, where $N^*=\{b^*|b\in N\}$. If $N$ is a closed subset of $B$, then $(<N>,N)$, where $<N>$ is the $\C$-space spanned by $N$, is a C-algebra. For every closed subset $N$ of $B$, the [*order*]{} of $N$, $o(N)$, is defined by $$o(N)=\sum_{b\in N}|b|$$ and $C^+$ is defined by $$N^+=\sum_{b\in N}b.$$
If the structure constants of a given C-algebra are nonnegative real numbers, then it is called a [*table algebra*]{} in the sense of [@Ar].
Let $(A,B)$ be a table algebra. Let $B'= \{\lambda_bb | b\in B\}$, where $\lambda_{1_A} = 1$, and $\lambda_b = \lambda_{b^*} \in \mathbb{R}^+$ for all $b\in B$. Then $(A,B')$ is also a table algebra which is called a [*rescaling*]{} of $(A,B)$.
Let $(A,B)$ be a table algebra. If $N$ is a closed subset of $B$ such that for any $b\in B$, $bN = Nb$, then $N$ is called a [*normal*]{} closed subset of $B$. It is known that if $N$ is a closed subset of $B$, then $e := o(N)^{-1}N^+$ is an idempotent of $A$, and $N$ is a normal closed subset if and only if $e$ is a central idempotent; see [@Ar Proposition 2.3(ii)].\
Let $(A,B)$ be a table algebra and $N$ be a closed subset of $B$. It follows from [@Ar Proposition 4.7] that $\{NbN \mid \ b\in {B}\}$ is a partition of ${B}$. A subset $NbN$ is called a [*$N$-double coset*]{} with respect to the closed subset $N$. Let $$b/\!\!/N := o(N)^{-1}({N}b{N})^{+} = o(N)^{-1}\displaystyle\sum_{x\in
{N}b N}x.$$ Then the following theorem is an immediate consequence of [@Ar Theorem 4.9]:
\[Arad\] Let $(A,{B})$ be a table algebra and let $N$ be a closed subset of $B$. Suppose that $\{b_1=1_A,\ldots ,b_k\}$ is a complete set of representatives of $N$-double cosets. Then the vector space spanned by the elements $b_i/\!\!/{N}, 1\leq i\leq k$, is a table algebra (which is denoted by $A/\!\!/{N}$) with a distinguished basis ${B}/\!\!/{N} = \{b_i/\!\!/{N} \mid \ 1\leq i \leq k \}.$ The structure constants of this algebra are given by the following formula: $$\gamma_{ijk}=
o(N)^{-1}\displaystyle\sum_{\begin{subarray}{l} r\in Nb_iN\\s\in Nb_jN \end{subarray}}\lambda_{rst},$$ where $t\in Nb_k N$ is an arbitrary element.
The table algebra $(A/\!\!/N,B/\!\!/N)$ is called the [*quotient table algebra*]{} of $(A,B)$ modulo $N$.
Let $(A,B)$ be a table algebra and $N$ be a closed subset of $B$. Put $e=o(N)^{-1}N^+$. Then one can see that $A/\!\!/N=eAe$ and it follows from [@Xu0] that $$\frac{|b|}{|b/\!\!/N|}b/\!\!/N=ebe,$$ for every $b\in B$.
Now we state some necessary definitions and notations for association schemes.
Let $X$ be a finite set and $G$ be a partition of $X \times X$. Then the pair $(X,G)$ is called an association scheme on $X$ if the following properties hold: $1_X \in G$, where $1_X:=\{(x,x) | x\in X\}$. For every $g\in G$, $g^*$ is also in $G$, where $g^*:=\{(x,y) |(y,x) \in g)\}$. For every $g,h,k \in G$, there exists a nonnegative integer $\lambda_{ghk}$ such that for every $(x,y)\in k$, there exist exactly $\lambda_{ghk}$ elements $z\in X$ with $(x,z)\in g$ and $(z,y)\in h$.
Let $(X,G)$ be an association scheme. For each $g\in G$, we call $n_g=\lambda_{gg^* 1_X}$ the [*valency*]{} of $g$. For any nonempty subset $H$ of $G$, put $n_H =\sum_{h\in H}n_h$. Clearly $n_G=|X|$. For every $g\in G$, let $A(g)$ be the adjacency matrix of $g$. For every nonempty subset $H$ of $G$, put $A(H):=\{A(h)| h\in H\}$ and let $\C [H]$ denote the $\C$-space spanned by $A(H)$. It is known that $(\C[G],A(G))$ is a table algebra, called the [*complex adjacency algebra*]{} of $G$. Let $(X,G)$ be an association scheme. A nonempty subset $H$ of $G$ is called a closed subset of $G$ if $A(H)$ is a closed subset of $\C[G]$. If $H$ is a closed subset of $G$, then $(\C[H], A(H)) $ is a table algebra.
Let $H$ be a closed subset of $G$. For every $h\in H$ and every $x\in X$, we define $xh=\{y\in X|(x,y)\in h\}$. Put $X/H=\{xH|x\in X\}$, where $xH=\cup_{h\in H}xh$. For $x\in X$, the [*subscheme*]{} $(X,G)_{xH}$ induced by $xH$, is an association scheme $(xH, H_{xH})$ where $H_{xH}=\{h_{xH}|h\in H\}$ and $h_{xH}=h\cap xH \times xH$. It is known that $\C [H_{xH}]\cong \C [H]$, as algebras over $\C$; see [@zi Theorem 4.4.5].
Let $(X,G)$ and $(Y,S)$ be two association schemes. A scheme epimorphism is a mapping $\varphi:(X,G)\rightarrow (Y,S)$ such that $\varphi(X)=Y$ and $\varphi(G)=S$, for every $x,y\in X$ and $g\in G$ with $(x,y)\in g$, $(\varphi(x),\varphi(y))\in \varphi(g)$.
Let $\varphi:(X,G)\rightarrow (Y,S)$ be a scheme epimorphism. The kernel of $\varphi$ is defined by $$\ker\varphi=\{g\in G\mid \varphi(g)=1_Y\}.$$ It is known that $A(\ker\varphi)$ is a closed subset of $A(G)$, but $A(\ker\varphi)$ need not be normal, in general. If $A(\ker \varphi) \unlhd A(G)$, then the scheme epimorphism $\varphi$ is called the [*normal*]{} scheme epimorphism. A scheme epimorphism with a trivial kernel is called a scheme isomorphism.
An algebraic isomorphism between two association schemes $(X, G)$ and $(Y,S)$ is a bijection $\theta:G\rightarrow S $ such that it preserves the structure constants, that is $\lambda_{ghl}=\lambda_{\varphi(g)\varphi(h)\varphi(l)}$, for every $g,h,l\in G$. It is known that if $\varphi:(X,G)\rightarrow (Y,S)$ is a scheme isomorphism, then $\varphi$ induces an algebraic isomorphism between $G$ and $S$.
Homomorphisms of table algebras
-------------------------------
Here we state some basic definitions and results of homomorphisms of table algebras from [@xus].
(See [@xus Difinition 3.1].) Let $(A,B)$ and $(C,D)$ be table algebras. A map $\varphi: A\rightarrow C$ is called the table algebra homomorphism of $(A,B)$ into $(C,D)$ if $\varphi: A\rightarrow C$ is an algebra homomorphism; and $\varphi(B):=\{\varphi(b)|b\in B\}$ consists of positive scalar multiples of elements $D$.
A table algebra homomorphism is called a monomorphism (epimorphism, isomorphism, resp.) if it is injective (surjective, bijective, resp.).
\[tr\] Let $(A,B)$ and $(C,D)$ be table algebras. Define $\varphi: (A,B)\rightarrow (C,D)$ such that $\varphi(b)=|b|1_C$. Then $\varphi$ is a table algebra homomorphism, and is called the trivial table algebra homomorphism.
\[cano\] Let $(A,B)$ be a table algebra and $N$ be a normal closed subset of $B$. It follows from [@Xu0 Theorem 2.1] that, there is a table algebra epimorphism $\pi:(A,B)\rightarrow (A/\!\!/N,B/\!\!/N)$ such that $$\pi(b)=\frac{|b|}{|b/\!\!/N|}(b/\!\!/N), ~~~\forall b\in B.$$ The table algebra epimorphism $\pi$ is called the [*canonical*]{} epimorphism from $(A,B)$ to $(A/\!\!/N,B/\!\!/N)$.
Two table algebras $(A,B)$ and $(C,D)$ are called isomorphic, denoted by $(A,B)\cong (C,D)$ or simply $B\cong D$, if there exists a table algebra isomorphism $\varphi: (A,B)\rightarrow (C,D)$.
[@xus Lemma 3.2]\[x1\] Let $(A,B)$ and $(C,D)$ be table algebras and $\varphi: (A,B)\rightarrow (C,D)$ be a table algebra homomorphism. Then the following hold: for every $b\in B$, $|\varphi(b)|=|b|$, for every $b\in B$, if $\supp(\varphi(b))=d$, then $$\varphi(b)=\frac{|b|}{|d|}d.$$
The following lemma gives some basis properties of table algebra homomorphisms.
[@xus Proposition 3.3]\[x2\] Let $(A,B)$ and $(C,D)$ be table algebras and $\varphi: (A,B)\rightarrow (C,D)$ be a table algebra homomorphism. Then the following hold: $\varphi(1_A)=1_C$, for every $b\in B$, $\varphi(b^*)={\varphi(b)}^*$, for every nonempty closed subset $N$ of $B$, $\varphi(N)=\{\supp(\varphi(b))\mid b\in N\}$ is a closed subset of $C$, for every nonempty closed subset $M$ of $D$, $\varphi^{-1}(M)=\{b\in B\mid\supp(\varphi(b))\in M\}$ is a closed subset of $B$.
(See [@xus Definition 3.4].) Let $(A,B)$ and $(C,D)$ be table algebras and $\varphi: (A,B)\rightarrow (C,D)$ be a table algebra homomorphism. Then the set $\varphi^{-1}(1_A) $ is called the kernel of $\varphi$ in $B$ and is denoted by $\ker_B \varphi$.
The next lemma will be needed later.
\[x3\] Let $(A,B)$ and $(C,D)$ be table algebras and $\varphi: (A,B)\rightarrow (C,D)$ be a table algebra homomorphism. Then the following hold: $\ker_B \varphi$ is a normal closed subset of $B$, $\varphi$ is injective if and only if $\ker_B \varphi=\{1_A\}$.
The next theorem is an isomorphism theorem for table algebras and it can be useful in the theory of table algebras.
[@xus Theorem 4.1]\[x4\] Let $(A,B)$ and $(C,D)$ be table algebras and $\varphi: (A,B)\rightarrow (C,D)$ be a table algebra homomorphism. Then $\varphi$ induces a table algebra homomorphism $$\widetilde{\varphi}:(A/\!\!/\ker_B\varphi, B/\!\!/\ker_B \varphi)\rightarrow (C,D)$$ such that $\widetilde{\varphi}(eae)=\varphi(a)$, where $e=o(\ker_B\varphi)^{-1}(\ker_B\varphi)^+.$ In particular, $$B/\!\!/\ker_B \varphi\cong \varphi(B).$$
Characters of table algebras
----------------------------
Let $(A,B)$ be a table algebra. Then $A$ is a semisimple algebra; see [@Ar Theorem 3.11]. Put $e = o(B)^{-1}B^+$. Then $Ae$ is a one dimensional $A$-module. The character of $A$ afforded by this module is called the principal character of $A$ and is denoted by $\rho$. The [*kernel* ]{} of a character $\chi$ of $A$ in $B$ is defined by $\ker_B(\chi)=\{b\in B|\chi(b)=|b|\chi(1)\}$. It follows from [@Xu0 Theorem 4.2] that $\ker_B(\chi)$ is a closed subset of $B$. Let $\operatorname{Irr}(B)$ be the set of irreducible characters of $A$ and for every closed subset $N$ of $B$, let $\operatorname{Irr}(B/\!\!/N)$ be the set of irreducible characters of $A/\!\!/N$. Then for every normal closed subset $N$ of $B$, $\operatorname{Irr}(B/\!\!/N)=\{\chi\in \operatorname{Irr}(B)|N\subseteq \ker_B(\chi)\}$; see [@Xu0 Theorem 3.6].
Let $(A,B)$ be a table algebra and $\chi, \psi\in \operatorname{Irr}(B)$. The character product of $\chi$ and $\psi$ is defined by $$\chi\psi(b)=\frac{1}{|b|}\chi(b)\psi(b),$$ (see [@BR2]). It is known that this character product need not be a character, in general. But since for every $a,b\in B$, $\chi\psi(ab)=\chi\psi(ba)$, it follows that $\chi\psi$ is a feasible trace and so $$\chi\psi=\sum_{\varphi \in \operatorname{Irr}(B)}\lambda^{\varphi}_{\chi\psi} \varphi,$$ where $\lambda^{\varphi}_{\chi\psi} \in \mathbb{C}$; see [@Hi].
Let $(A,B)$ be a table algebra. Define a linear function $\zeta$ on $A$ by $\zeta(b) = \delta_{b,1_A}o(B)$, for every $b\in B$. Then $\zeta$ is a non-degenerate feasible trace on $A$ and it follows from [@BR] that $$\label{defsta}\nonumber
\zeta=\displaystyle\sum _{\chi\in \rm{Irr(B)}}\zeta_\chi\chi,$$ where $\zeta_\chi\in \C$ and all $\zeta_\chi$ are nonzero. For every $\chi,\varphi\in {\rm Hom}_\C(A,\C)$, we define the inner product of $\chi$ and $\varphi$ as follows:
$$\begin{aligned}
\label{inner}\nonumber
[\chi,\varphi]=\frac{1}{o(B)}\displaystyle\sum_{b\in
B}\frac{1}{|b|}\chi(b)\varphi(b^*).\end{aligned}$$
Then for every $\chi, \varphi\in \operatorname{Irr}(B)$, it follows from [@BR Lemma 3.1(ii)] that $$[\chi,\varphi]=\delta_{\chi,\varphi}\frac{\chi(1)}{\zeta_\chi}.$$ So for every $\chi, \psi, \varphi\in \operatorname{Irr}(B)$ we have $$\begin{aligned}
\nonumber
[\chi\psi, \varphi]=\frac{\lambda^{\varphi}_{\chi\psi}}{\zeta_\varphi}.\end{aligned}$$ Moreover, since $$\begin{aligned}
\nonumber
[\chi\psi,\varphi]=[\overline{\chi}\varphi, \psi]=[\varphi\overline{\psi}, \chi],\end{aligned}$$ it follows that $$\begin{aligned}
\label{eq1}\nonumber
\frac{\lambda^{\varphi}_{\chi\psi}}{{\zeta_\varphi}}=\frac{\lambda^{\psi}_{\overline{\chi}\varphi}}{{\zeta_\psi}}
=\frac{\lambda^{\chi}_{\overline{\psi}\varphi}}{{\zeta_\chi}}.\end{aligned}$$
Duals of commutative table algebras
-----------------------------------
In the following, we deal with the dual of a commutative table algebra in the sense of [@Bannai].
Suppose that $(A,B)$ is a commutative table algebra of dimension $d$ with the set of primitive idempotents $\{\varepsilon_{\chi} |~ \chi \in \operatorname{Irr}(B)
\}$. Then from [@Bannai Section 2.5] there are two matrices $P =
(p_b(\chi))$ and $Q=(q_{\chi}(b))$ in $\operatorname{Mat}_d(\mathbb{C})$, where $b\in B$ and $\chi \in \operatorname{Irr}(B)$, such that $PQ = QP = o(B)I$, where $I$ is the identity matrix in $\operatorname{Mat}_d(\mathbb{C})$, and $$\begin{aligned}
\nonumber\label{25100701}
b = \displaystyle\sum_{\chi\in \operatorname{Irr}(B)}
p_b(\chi)\varepsilon_{\chi}~~~ \text{ and } ~~ \varepsilon_{\chi}
= \frac{1}{o(B)}\displaystyle\sum_ {b\in B} q_{\chi}(b)b.\end{aligned}$$ The dual of $(A,B)$ in the sense of [@Bannai] is as follows: with each linear representation $\Delta_{\chi} : b\mapsto p_b(\chi)$, we associate the linear mapping $\Delta^*_{\chi}: b \mapsto q_{\chi}(b)=\frac{\zeta_\chi \chi(b^*)}{|b|}$. Since the matrix $Q=(q_{\chi}(b))$ is non-singular, the set $\widehat{B} =
\{\Delta^*_{\chi}: \chi \in \operatorname{Irr}(B)\}$ is linearly independent and so forms a base of the set of all linear mappings $\widehat{A}$ of $A$ into $\C$. From [@Bannai Thorem 5.9] the pair $(\widehat{A},\widehat{B})$ is a C-algebra with the identity $1_{\widehat{A}} = \Delta ^*_{\rho}$, where $\rho$ is the principal character of $A$, and involutory automorphism $^*$ which maps $\Delta^*_{\chi}$ to $\Delta^*_{\overline{\chi}}$, where $\overline{\chi}$ is the complex conjugate to $\chi$. The C-algebra $(\widehat{A},\widehat{B})$ is called the [*dual*]{} of $(A,B)$. Moreover, for every $\chi,\psi\in \operatorname{Irr}(B)$ we have $$\begin{aligned}
\nonumber
\Delta^*_{\chi} \Delta^*_{\psi}=\displaystyle\sum_{\Delta^*_\varphi\in \widehat{B}} q^{\varphi}_{\chi,\psi}\Delta^*_{\varphi},\end{aligned}$$ where the structure constants $q^{\varphi}_{\chi,\psi}, \varphi \in \operatorname{Irr}(B)$, are real numbers. Furthermore, it follows from the Duality Theorem [@Bannai Theorem 5.10] that $(\widehat{\widehat{A}}, \widehat{\widehat{B}})\cong (A,B) $.
Let $(\widehat{A},\widehat{B})$ be the dual of $(A,B)$ and $N$ be a closed subset of $B$. Put $$\ker(N)=\{\Delta^*_{\chi}\in \widehat{B}~ | ~\chi(b)=|b|,~{\rm for ~every}~b\in N\}.$$ Then $\ker (N)$ is a closed subset of $\widehat{B}$; see [@b]. Moreover, since $N\subseteq \ker_{B}(\chi)$ for every $\Delta^*_{\chi}\in \ker(N)$, we conclude that $$\ker(N)=\{\Delta^*_{\chi}\in \widehat{B}~ | ~N\subseteq \ker_B(\chi)\}=\{\Delta^*_{\chi}\in \widehat{B}~ | \chi\in \operatorname{Irr}(B/\!\!/N)\}.$$
Suppose that $(A,B)$ is a table algebra such that $(\widehat{A},\widehat{B})$ is also a table algebra. Then for closed subsets $N\leq M$ of $B$, it follows from [@xueival Theorem 4.8] that $$\widehat{M/\!\!/N}\cong \ker(N)/\!\!/\ker(M).$$ In particular, we have $$\ker(N)\cong \widehat{{B/\!\!/N}},$$ and $$\begin{aligned}
\nonumber
\widehat{N} \cong \widehat{B}/\!\!/\ker(N).\end{aligned}$$
A wedge product of table algebras
==================================
In this section we will first define the wedge product of table algebras and then give a necessary and sufficient condition for a table algebra to be the wedge product of two table algebras.\
Let $(A,B)$ be a table algebra and $N$ be a closed subset of $B$. Suppose that $(C,D)$ is a table algebra and $$\begin{aligned}
\nonumber
\varphi:(C,D)\rightarrow (<N>,N)\end{aligned}$$ is a table algebra epimorphism. Put $K=\ker_D\varphi=\{d\in D|\supp(\varphi(d))=1_A\}$. Then it follows from Lemma \[x1\] and Theorem \[x3\] that for every $d\in D$ with $\supp(\varphi(d))=h$, we have $$\varphi(d)=\frac{|d|}{|h|}h.$$ In particular, for every $d\in K$, $\varphi(d)=|d|1_A$. $K\unlhd D$ and $$\begin{aligned}
\nonumber
(C/\!\!/K, D/\!\!/K)\cong (<N>,N).\end{aligned}$$
For every $1_A\neq b\in B$, put $\overline{b}=o(K)b$. Suppose that $(A,\overline{B})$ is a rescaling of $(A,B)$ where $$\overline{B}=\{1_A\}\cup\{\b\mid b\in B\setminus \{1_A\}\}.$$ Put $X=D\cup \w{B}$ and let $\widetilde{A}$ be the $\C$-space spanned by $X$. Suppose that $D=\{d_1, d_2, \ldots, d_n\}$, $B=\{b_1, b_2, \ldots ,b_m\}$ and the sets $\{\lambda_{xyz}| x,y,z\in B\}$ and $\{\mu_{xyz}| x,y,z\in D\}$ are the structure constants of $(A,B)$ and $(C,D)$, respectively. We define a multiplication “$\cdot$” on the elements of $X$ as follows: for every $d_i, d_j\in D$, $$d_i\cdot d_j=\displaystyle\sum^n_{z=1}\mu_{d_id_jd_z}d_z,$$ for every $\overline{b_i},\overline{b_j}\in \overline{B}$, $$\begin{aligned}
\nonumber
\b_i\cdot \b_j=o(K)\displaystyle\sum^m_{t=1}\lambda_{b_ib_jb_t}\b_t,\end{aligned}$$, for every $d_i\in D$ with $\supp(\varphi(d_i))=h_i$, and $\b_j\in \w{B}$, $$d_i\cdot\b_j=o(K)\varphi(d_i)b_j=\frac{|d_i|}{|h_i|}
\displaystyle\sum^m_{t=1}\lambda_{h_ib_jb_t}\b_t,$$ and similarly, $$\b_j\cdot d_i=o(K)b_j \varphi(d_i)=\frac{|d_i|}{|h_i|}
\displaystyle\sum^m_{t=1}\lambda_{b_jh_ib_t}\b_t.$$
If we extend $``\cdot$” linearly to all $\W{A}$, then it defines the structure of a $\C$-algebra on $\W{A}$.
\[l1\] $\W{A}$ is an associative $\C$-algebra.
Since $A$ and $C$ are associative $\C$-algebras, it suffices to show that: for every $d_i, d_j\in D$ and $\b_t\in \w{B}$, $(d_i\cdot d_j)\cdot\b_t=d_i\cdot(d_j\cdot\b_t)$, for every $d_i\in D$ and $\b_j,\b_t\in \w{B}$, $d_i\cdot(\b_j\cdot\b_t)=(d_i\cdot\b_j)\cdot\b_t$. To prove $(i)$, assume that $d_i, d_j\in D$ with $\supp(d_j)=h_j$ and $\b_t\in \w{B}$. Since $\varphi(d_id_j)=\varphi(d_i)\varphi(d_j)$ and $A$ is an associative algebra, we have
$$\begin{aligned}
(d_i\cdot d_j)\cdot \b_t&=&\nonumber(\displaystyle\sum^n_{s=1}\mu_{d_id_jd_s}d_s)\cdot\b_t=\displaystyle\sum^n_{s=1}\mu_{d_id_jd_s}d_s\cdot\b_t
\\&=&\nonumber o(K)\displaystyle\sum^n_{s=1}\mu_{d_id_jd_s}\varphi (d_s) b_t=o(K)\varphi(\displaystyle\sum^n_{s=1}\mu_{d_id_jd_s} d_s) b_t
\\&=&\nonumber o(K)\varphi(d_id_j)b_t=o(K)(\varphi(d_i)\varphi(d_j))b_t=
\\&=&\nonumber o(K)\varphi(d_i)(\varphi(d_j)b_t)=o(K)\varphi(d_i)(\frac{|d_j|}{|h_j|}h_jb_t)\\&=&\nonumber
\varphi(d_i)(o(K)\frac{|d_j|}{|h_j|}\displaystyle\sum^m_{r=1}\lambda_{h_jb_tb_r}b_r)=
o(K)\frac{|d_j|}{|h_j|}\displaystyle\sum^m_{r=1}\lambda_{h_jb_tb_r}\varphi(d_i)b_r
\\&=&\nonumber
\frac{|d_j|}{|h_j|}(\displaystyle\sum^m_{r=1}\lambda_{h_jb_tb_r}d_i\cdot\b_r)
=d_i\cdot(o(K)\frac{|d_j|}{|h_j|}(\displaystyle\sum^m_{r=1}\lambda_{h_jb_tb_r}b_r)\\&=&\nonumber d_i\cdot(o(K)\varphi(d_j)b_t)= d_i\cdot(d_j\cdot\b_t).\end{aligned}$$
Similarly, to prove $(ii)$, assume that $d_i\in D$ and $\b_j,\b_t\in \w{B}$. Then we have $$\begin{aligned}
d_i\cdot(\b_j\cdot\b_t)&=&\nonumber d_i\cdot(o(K)\displaystyle\sum^m_{r=1}\lambda_{b_jb_tb_r}\b_r)
=o(K)\displaystyle\sum^m_{r=1}\lambda_{b_jb_tb_r} d_i\cdot\b_r
\\&=&\nonumber o(K)^2\displaystyle\sum^m_{r=1}\lambda_{b_jb_tb_r} \varphi(d_i)b_r
=o(K)^2\varphi(d_i)(\displaystyle\sum^m_{r=1}\lambda_{b_jb_tb_r}b_r)\\&=&\nonumber o(K)^2\varphi(d_i)(b_jb_t)=o(K)^2(\varphi(d_i) b_j)b_t
\\&=&\nonumber (d_i\cdot\b_j)\cdot\b_t.\end{aligned}$$
In the rest of this paper, for convenience, we write $xy$ in place of $x\cdot y$, for every $x,y\in \widetilde{B}$.
Let $d\in D$ such that $\supp(\varphi(d))=h$. Since $\varphi(K^+)=o(K)$, we have $$\varphi(dK^+)=o(K)\varphi(d)=\frac{|d|}{|h|}\overline{h}.$$ Then for every $\b\in \w{B}$, $$(dK^+)\b=o(K)\varphi(dK^+)b=\frac{|d|}{|h|}\overline{h}~\b.$$ In particular, $$(dK^+)1_A=\frac{|d|}{|h|}\overline{h}.$$ If we identify $(dK^+)1_A$ with $dK^+$, then we can assume that $dK^+=\frac{|d|}{|h|}\overline{h}$. By this identification we can see that $\W{B}=D\cup (\w{B}\setminus \w{N})$ is a base for the algebra $\W{A}$.
Moreover, for every $\b\in \w{B}\setminus \w{N}$, we have $1_D\cdot\b=o(K)\varphi(1_D)b$. But by Lemma \[x2\], $\varphi(1_D)=1_A$. So $1_D\cdot\b=o(K)1_Ab=\b$. Similarly, $\b\cdot 1_D=\b$. Hence $1_D\in \W{B}$ is the identity element of $\W{A}$. So in the rest of this paper, we denote $1_D$ by $1_{\W{A}}$.\
In the following we will show that the pair $(\W{A},\W{B})$ is a table algebra.\
Suppose that $*_1$ and $*_2$ are semilinear involuntary anti-automorphisms of table algebras $(A,B)$ and $(C,D)$, respectively. Then we can define a semilinear involuntary anti-automorphism $*$ on $\W{B}$ as follows: for every $d\in D$, $d^*:=d^{*_2}$, for every $\b\in \w{B}$, $(\overline{b})^*:=\w{b^{*_1}}=o(K)b^{*_1}$.
Note that $(\b d)^*=o(K)(b\varphi(d))^*=o(K)\varphi(d)^{*_1}b^{*_1}=o(K)\varphi(d^{*_2})b^{*_1}=d^*(\b)^*$; see Lemma \[x2\]. Similarly, $(d\b)^*=(\b)^*d^*$.\
Moreover, if $|~|_A$ and $|~|_C$ are the degree maps of $(A,B)$ and $(C,D)$, respectively, then we can define a linear map $|~|:\W{A}\rightarrow \C$ as follows: for every $d\in D$, $|d|:=|d|_C$, for every $\b\in \w{B}$, $|\b|:=o(K)|b|_A$. We show that $|~|$ is an algebra homomorphism. To do so, suppose that $d_i, d_j\in D$ and $\b_i, b_j\in \w{B}\setminus \w{N}$. Then we have $$\begin{aligned}
|d_id_j|&=&\nonumber|\displaystyle\sum^n_{r=1}\mu_{d_id_jd_r}d_r|\\&=& \nonumber
\displaystyle\sum^n_{r=1}\mu_{d_id_jd_r}|d_r|=\displaystyle\sum^n_{r=1}\mu_{d_id_jd_r}|d_r|_C\\&=& \nonumber|d_i|_C|d_j|_C
=|d_i||d_j|\end{aligned}$$ and $$\begin{aligned}
|\b_i\b_j|&=& \nonumber o(K)|\displaystyle\sum^m_{r=1}\lambda_{b_ib_jb_r}\b_r|\\&=& \nonumber o(K)\displaystyle\sum^m_{r=1}\lambda_{b_ib_jb_r}|\b_r|
=o(K)^2(\displaystyle\sum^m_{r=1}\lambda_{b_ib_jb_r}|b_r|_A)\\&=& \nonumber o(K)^2(|b_i|_A|b_j|_A)=|\b_i|\b_j|.\end{aligned}$$ Moreover, if $\supp(d_j)=h_j$, then $$\begin{aligned}
|\b_id_j|&=& \nonumber o(K)|b_i \varphi(d_j)|=o(K)\frac{|d_j|_C}{|h_j|_A}~|b_ih_j|\\&=& \nonumber
o(K)\frac{|d_j|_C}{|h_j|_A} ~|\displaystyle\sum^m_{r=1}\lambda_{b_ih_jt_r} t_r|=
o(K)\frac{|d_j|_C}{|h_j|_A} ~ \displaystyle\sum^m_{r=1}\lambda_{b_ih_jt_r}|t_r| \\&=& \nonumber
o(K)\frac{|d_j|_C}{|h_j|_A}~ \displaystyle\sum^m_{r=1}\lambda_{b_ih_jt_r}|t_r|_A =
o(K)\frac{|d_j|_C}{|h_j|_A}||b_ih_j|_A\\&=& \nonumber o(K)\frac{|d_j|_C}{|h_j|_A}||b_i|_A|h_j|_A =(o(K)|b_i|_A)|d_j|_C
\\&=& \nonumber |\b_i||d_j|.\end{aligned}$$ Similarly, $|d_j\b_i|=|d_j||\b_i|$. So we conclude that $|~|:\W{A}\rightarrow \C$ is an algebra homomorphism. Furthermore, since $|~|_A $ and $|~|_C$ are $*$-linear representations, it follows that $|~|:\W{A}\rightarrow \C$ is also a $*$-linear representation of $\W{A}$.\
In the theorem below we show that the pair $(\W{A},\W{B})$ is a table algebra.
The algebra $\W{A}$ with the basis $\W{B}$ is a table algebra.
It follows from Lemma \[l1\] that $\W{A}$ is an associative algebra. Moreover, the identity element of $\W{B}$, $1_{\widetilde{A}}$, is in $\W{B}$ and for every $x,y\in \W{B}$, we have $$xy=\displaystyle\sum_{z\in \W{B}}\alpha_{xyz}z,$$ where $\alpha_{xyz}, z\in \W{B}$, are nonnegative real numbers. By the preceding remarks, there is a semilinear involuntary anti-automorphism $*$ on $\W{B}$ such that $(\W{B})^*=\W{B}$. Furthermore, there exists a degree map $|~|:\W{A}\rightarrow \C$ such that for every $d,c\in D$, $$\mu_{dc^*1_{\W{A}}}=\mu_{d^*c1_{\W{A}}}=\delta_{dc}|d|,$$ and for every $\overline{a}, \b \in \w{B}\setminus \w{N}$, $$\overline{a}{\b}^*=\overline{a}\overline{b^*}=o(K)^2(ab^*)=o(K)(K^+ab^*)=\delta_{ab}o(K)|b|1_{\W{A}}+o(K)\displaystyle\sum_{1_{\W{A}}\neq \overline{c}\in \W{B}}\lambda_{ab^*c}\overline{c}.$$ Note that $K^+a=\varphi(K^+)a=o(K)a$. So $|\b|=o(K)|b|=\lambda_{bb^*1_{\W{A}}}$. Thus we conclude that the pair $(\W{A},\W{B})$ is a table algebra.
With the notation above, the table algebra $(\W{A},\W{B})$ is called the wedge product of table algebras $(C,D)$ and $(A,B)$ relative to $\varphi$.
In the following we give some properties of the wedge product of table algebras.
\[kd\] Let $(\W{A},\W{B})$ be the wedge product of table algebras $(C,D)$ and $(A,B)$ relative to $\varphi$. Then $\W{B}$ contains a closed subset $K\leq D$ such that $K\unlhd \W{B}$, and for every $x\in \W{B}\setminus D$ and every $k\in K$, $kx=|k|x=x k$. In particular, for every $x\in \W{B}\setminus D$, $x K^+=o(K)x=K^+x$.
Put $K=\ker_D \varphi$. It follows from Lemma \[x3\] that $K\unlhd D$. Since for every $\b\in \w{B}\setminus \w{N}$ and $k\in K$, we have $k\b=o(K)\varphi(k)b=o(K)|k|b$, we conclude that $k\b=|k|\b=\b k$ and $K\unlhd \W{B}$. In particular, for every $\b\in \w{B}\setminus \w{N}$, $$K^+\b=o(K)\b=\b K^+.$$
\[iso\] Let $(\W{A},\W{B})$ be the wedge product of table algebras $(C,D)$ and $(A,B)$ relative to $\varphi$. Then $(\W{A}/\!\!/K,\W{B}/\!\!/K)\cong (A,B)$, where $K=\ker_{D}\varphi$.
Put $e=o(K)^{-1}K^+$. Then it follows from Lemma \[kd\] that $e$ is a central idempotent of $\W{A}$. Define $$\theta: e\W{A}e\rightarrow A$$ such that $\theta(ede)=\varphi(d)$ for every $d\in D$, and $\theta(e\b e)=o(K)b$ for every $\b\in \w{B}$. We first show that $\theta$ is an algebra homomorphism. To do so, first assume that $d_1, d_2\in D$. Then $$\begin{aligned}
\theta((ed_1e)(ed_2e))&=&\nonumber\theta(ed_1d_2e)=\theta(\displaystyle\sum_{c\in D}\mu_{d_1d_2 c}ece)\\&=&\nonumber
\displaystyle\sum_{c\in D}\mu_{d_1d_2 c}\theta(ece)=\displaystyle\sum_{c\in D}\mu_{d_1d_2 c}\varphi(c) \\&=&\nonumber
\varphi(\displaystyle\sum_{c\in D}\mu_{d_1d_2 c}c)=\varphi(d_1d_2)\\&=&\nonumber \varphi(d_1)\varphi(d_2)=
\theta(ed_1e)\theta(ed_2e).\end{aligned}$$ Similarly, for every $\b_1, \b_2\in \w{B}\setminus \w{N}$ we have $$\begin{aligned}
\theta((e\b_1e)(e\b_2e))&=&\nonumber \theta(e\b_1\b_2e)\\&=&\nonumber
o(K)\theta(\displaystyle\sum_{t\in B}\lambda_{b_1b_2 t}e\overline{t}e)\\&=&\nonumber
o(K)\displaystyle\sum_{t\in B}\lambda_{b_1b_2 t}\theta(e\overline{t}e)\\&=&\nonumber
o(K) \displaystyle\sum_{t\in B}\lambda_{b_1b_2 t}o(K)t \\&=&\nonumber
o(K)^{2}b_1b_2=\theta(e\b_1e)\theta(e\b_2e).\end{aligned}$$ Moreover, for every $d\in D$ and $\b\in \w{B}\setminus \w{N}$ with $\supp(\varphi(d))=h$, we have $$\begin{aligned}
\theta((ede)(e\b e))&=&\nonumber \theta(ed\b e)\\&=&\nonumber o(K)\theta(e\varphi(d)be)
\\&=&\nonumber o(K)\frac{|d|}{|h|}\theta(ehbe)\\&=&\nonumber
o(K)\frac{|d|}{|h|} \theta(\displaystyle\sum_{t\in B\setminus N}\lambda_{hb t}ete)
\\&=&\nonumber \frac{|d|}{|h|} \displaystyle\sum_{t\in B\setminus N}\lambda_{hb t}\theta(e\overline{t}e)
\\&=&\nonumber o(K) \frac{|d|}{|h|} \displaystyle\sum_{t\in B\setminus N}\lambda_{hb t}t
\\&=&\nonumber
o(K)\frac{|d|}{|h|}hb
\\&=&\nonumber o(K)\varphi(d) b= \theta(ede)\theta(e\b e).\end{aligned}$$ Similarly, $\theta((e\b e)(e de))=\theta(e\b e)\theta(e d e).$ So $\theta$ is an algebra homomorphism. Since $e\W{A}e=\W{A}/\!\!/K$ and for every $x\in \W{B}$, $$exe=\frac{|x|}{|x/\!\!/K|}x/\!\!/K,$$ we can define $\overline{\varphi}:(\W{A}/\!\!/K,\W{B}/\!\!/K)\rightarrow (A,B)$ such that $$\overline{\varphi}(x/\!\!/K)=\frac{|x/\!\!/K|}{|x|}\varphi(exe).$$ Since $\varphi$ is an algebra homomorphism, it follows that $\overline{\varphi}$ is a table algebra homomorphism. Now suppose that $x\in \W{B}$ such that $\supp(\overline{\varphi}(x/\!\!/K))=\{1_A\}$. If $x\in D$, then one can see that $\supp(\varphi(d))=\{1_A\}$ and so $d\in \ker_D\varphi=K$. Thus $x/\!\!/K=1_{\W{A}/\!\!/K}$. Moreover, if $x\in \w{B}\setminus \w{N}$ such that $\supp(\overline{\varphi}(x/\!\!/K))=\{1_A\}$, then $$\supp(\frac{|x/\!\!/K|}{|x|}x)=\{1_A\}$$ and hence $x=o(K)1_A$ which is a contradiction, since $x\in \w{B}\setminus \w{N}$. So we conclude that $$\ker_{\W{B}/\!\!/K}\overline{\varphi}=\{1_{\W{A}/\!\!/K}\}$$ and it follows from Lemma \[x3\] that $\overline{\varphi}$ is a table algebra monomorphism.
Moreover, since $\varphi:(C,D)\rightarrow(<N>,N)$ is a table algebra epimorphism, and for every $b\in B\setminus N$ we have $$\b/\!\!/K=o(K)^{-1}(K\b K)^+=o(K)^{-1}o(K)b=b,$$ we conclude that $\theta$ is a table algebra epimorphism. Thus $\theta$ is a table algebra isomorphism and so $(\W{A}/\!\!/K,\W{B}/\!\!/K)\cong (A,B)$.
\[main2\] Let $(A,B)$ be a table algebra and $K\leq D$ closed subsets of $B$ such that $K\unlhd B$ and for every $b\in B \setminus D$, $bK^+=o(K)b=K^+b$. Then $(A,B)$ is the wedge product of table algebras $(<D>,D)$ and $(A/\!\!/K,B/\!\!/K)$ relative to the canonical epimorphism $\pi:(<D>,D)\rightarrow (<D>/\!\!/K,D/\!\!/K)$.
Consider the table algebra $(A/\!\!/K,B/\!\!/K)$. Then $D/\!\!/K$ is a closed subset of $B/\!\!/K$ and we can define the canonical epimorphism $\pi:(<D>,D)\rightarrow (<D>/\!\!/K,D/\!\!/K)$ such that $$\pi(d)=\frac{|d|}{|d/\!\!/K|}d/\!\!/K.$$ Put $e=o(K)^{-1}K^+$. Since $e$ is a central idempotent of $A$ and $\frac{|d|}{|d/\!\!/K|}d/\!\!/K=ede$ we have $$\begin{aligned}
\ker_D\pi&=&\nonumber \{d\in D\mid\pi(d)=|d|e\}=\{d\in D\mid ede=|d|e\}
\\&=& \nonumber \{d\in D\mid de=|d|e\}=\{d\in D\mid dK^+=|d|K^+\}=K.\end{aligned}$$ Now let $(\W{A},\W{B})$ be the wedge product of $(<D>,D)$ and $(A/\!\!/K,B/\!\!/K)$ relative to $\pi$. Then $$\W{B}=D\cup \{o(K)(b/\!\!/K)\mid b/\!\!/K\in B/\!\!/K \setminus D/\!\!/K \}.$$ Since for every $b/\!\!/K\in B/\!\!/K \setminus D/\!\!/K$, we have $o(K)(b/\!\!/K)=o(K)o(K)^{-1}(KbK)^+=b$ and so $\W{B}=D\cup (B\setminus D)=B$. Moreover, for every $b\in B\setminus D$ and $d\in D$, $$\begin{aligned}
o(K) (b/\!\!/K) d&=& \nonumber o(K) (b/\!\!/K)\pi(d)=o(K) (b/\!\!/K) \frac{|d|}{|d/\!\!/K|}d/\!\!/K \\&=& \nonumber b(ede)=(be)d=(o(K)^{-1}K^+b )d \\&=& \nonumber (o(K)^{-1} o(K)b)d=bd.\end{aligned}$$ Thus we conclude that $(\W{A},\W{B})=(A,B)$ and so $(A,B)$ is the wedge product of $(<D>,D)$ and $(A/\!\!/K,B/\!\!/K)$ relative to $\pi$.
As a direct consequence of Lemma \[kd\] and Theorem \[main2\], we can give a necessary and sufficient condition for a table algebra to be the wedge product of two table algebras.
\[wg1\] Let $(A,B)$ be a table algebra and $K\unlhd D$ closed subsets of $B$. Then the following are equivalent: $K\unlhd B$, and for every $b\in B \setminus D$, $bK^+=o(K)b=K^+b$, $(A,B)$ is the wedge product of $(<D>,D)$ and $(A/\!\!/K,B/\!\!/K)$ relative to the canonical epimorphism $\pi:(<D>,D)\rightarrow (<D>/\!\!/K,D/\!\!/K)$.
$(i)\Rightarrow (ii)$ follows directly from Theorem \[main2\].
$(ii)\Rightarrow (i)$ Since $(A,B)$ is the wedge product of $(<D>,D)$ and $(A/\!\!/K,B/\!\!/K)$ we have $B=D\cup \{o(K)b/\!\!/K\mid b/\!\!/K\in B/\!\!/K\setminus D/\!\!/K\}$, and it follows from Lemma \[kd\] that $K\unlhd B$ and for every $b/\!\!/K\in B/\!\!/K\setminus D/\!\!/K$ $$K^+(o(K)b/\!\!/K)=o(K)(o(K)b/\!\!/K)=(o(K)b/\!\!/K)K^+.$$ Then for every $b/\!\!/K\in B/\!\!/K\setminus D/\!\!/K$, $$K^+(b/\!\!/K)=o(K)(b/\!\!/K)=(b/\!\!/K)K^+$$ and so $$K^+(KbK)^+=o(K)(KbK)^+=(KbK)^+K^+.$$ Thus for every $b\in B \setminus D$, $bK^+=o(K)b=K^+b$, and $(i)$ holds.
Let $(\W{A},\W{B})$ be the wedge product of table algebras $(C,D)$ and $(A,B)$ relative to $\varphi$. If $\ker_D \varphi=D$, then for every $d\in D$ and every $x\in \W{B}\setminus D$ we have $x d=|d|x=d x$. So it follows from [@b15 Definition 1.2] that $(\W{A},\W{B})$ is a wreath product $(\W{B},D)$. Thus the wreath product of table algebras is a partial case of the wedge product of table algebras whenever $\varphi$ is the trivial table algebra homomorphism; see Example \[tr\].
The dual of wedge product
=========================
In this section we first give a sufficient condition for which the dual of a commutative table algebra is also a table algebra. Then we will show that if the duals of two commutative table algebras are table algebras, then the dual of their wedge product is a table algebra.\
The following easy lemma is useful.
\[es\] Let $(A,B)$ be a commutative table algebra. Then $(\widehat{A},\widehat{B})$ is a table algebra if and only if for every $\chi, \psi\in \operatorname{Irr}(B)$, $\chi\psi$ is a linear combination of $\operatorname{Irr}(B)$ with the nonnegative real number coefficients.
For every $\chi, \psi\in \operatorname{Irr}(B)$, we have $$\Delta^{*}_{\chi} \Delta^{*}_{\psi}=\displaystyle \sum_{\varphi\in \operatorname{Irr}(B)}q^{\varphi}_{\chi\psi}\Delta^{*}_\varphi,$$ where $q^{\varphi}_{\chi\psi}, \varphi\in \operatorname{Irr}(B)$, are real numbers. One the other hand, for every $b\in B$, $$\begin{aligned}
\label{dudu0}
\Delta^{*}_{\chi} \Delta^{*}_{\psi}(b^*)=\Delta^{*}_{\chi}(b^*) \Delta^{*}_{\psi}(b^*)=\frac{\zeta_\chi \chi(b^*)}{|b|}\frac{\zeta_\psi \psi(b^*)}{|b|}=\frac{\zeta_\chi \zeta_\psi}{|b|}\chi\psi(b^*).\end{aligned}$$ Since $$\begin{aligned}
\label{mahan0}
\chi\psi=\sum_{\varphi \in \operatorname{Irr}(B)} \lambda^{\varphi}_{\chi\psi} \varphi,\end{aligned}$$ it follows from equalities (\[dudu0\]) and (\[mahan0\]) that $$\begin{aligned}
\nonumber
\Delta^{*}_{\chi} \Delta^{*}_{\psi}(b^*)=\frac{\zeta_\chi \zeta_\psi}{|b|}\sum_{\varphi \in \operatorname{Irr}(B)} \lambda^{\varphi}_{\chi\psi} \varphi(b^*)\end{aligned}$$ and hence $$\begin{aligned}
\nonumber
\Delta^{*}_{\chi} \Delta^{*}_{\psi}(b^*)=\displaystyle \sum_{\varphi\in \operatorname{Irr}(B)}\frac{\zeta_\chi \zeta_\psi}{\zeta_\varphi}\lambda^{\varphi}_{\chi\psi}\Delta^{*}_\varphi(b^*).\end{aligned}$$ This implies that $$\begin{aligned}
\nonumber
\Delta^{*}_{\chi} \Delta^{*}_{\psi}=\displaystyle \sum_{\varphi\in \operatorname{Irr}(B)}\frac{\zeta_\chi \zeta_\psi}
{\zeta_\varphi}\lambda^{\varphi}_{\chi\psi}\Delta^{*}_\varphi.\end{aligned}$$ Thus we conclude that $$\begin{aligned}
\label{a}
\frac{\zeta_\chi \zeta_\psi} {\zeta_\varphi}\lambda^{\varphi}_{\chi\psi}=q^{\varphi}_{\chi\psi}.\end{aligned}$$ Since $\frac{\zeta_\chi \zeta_\psi}{\zeta_\varphi} >0$, equality (\[a\]) shows that $q^{\varphi}_{\chi\psi}, \varphi\in \operatorname{Irr}(B)$, are nonnegative real numbers if and only if so are $\lambda^{\varphi}_{\chi\psi}, \varphi\in \operatorname{Irr}(B)$.
Let $(A,B)$ be a table algebra and $H\leq B$. For every $b\in B$, define $$\st_H(b)=\{x\in H~|~xb=|x|b=bx\},$$ and for every subset $U\subseteq B$, put $\st_H(U)=\bigcap_{b\in U}\st_H(b)$.
\[tars\] Let $(A,B)$ be a commutative table algebra. Suppose that $K\leq D$ are closed subsets of $B$ such that $K\subseteq \st_B(B\setminus D)$. Then for every $\chi\in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)$ and every $b\in B\setminus D$, $\chi(b)=0$.
Suppose $\chi\in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)$. Then there exists $k\in K$ such that $\chi(k)\neq |k|$. So for every $b\in B\setminus D$, the equality $kb=|k|b$ implies that $\chi(k)\chi(b)=|k|\chi(b)$. Thus we conclude that $\chi(b)=0$.
\[main01\] Let $(A,B)$ be a commutative table algebra. Suppose that $K$ and $D$ are closed subsets of $B$ such that $K\leq D$ and $K\subseteq \st_B(B\setminus D)$. If the duals of table algebras $(<D>,D)$ and $(A/\!\!/K,B/\!\!/K)$ are table algebras, then $(\widehat{A},\widehat{B})$ is also a table algebra.
From Lemma \[es\], it is enough to prove that for every $\chi, \psi\in \operatorname{Irr}(B)$, the coefficients $\lambda^{\varphi}_{\chi\psi}, \varphi\in \operatorname{Irr}(B)$ in the following product $$\begin{aligned}
\nonumber
\chi\psi=\sum_{\varphi \in \operatorname{Irr}(B)} \lambda^{\varphi}_{\chi\psi} \varphi,\end{aligned}$$ are nonnegative real numbers. To do this, first assume that $\chi, \psi\in \operatorname{Irr}(B/\!\!/K)$. Since $\widehat{B/\!\!/K}\cong \ker (K) $ is a closed subset of $\widehat{B}$, it follows that $$\Delta^{*}_{\chi} \Delta^{*}_{\psi}=\displaystyle \sum_{\Delta^{*}_{\varphi}\in \ker(K) }q^{\varphi}_{\chi\psi}\Delta^{*}_\varphi,$$ and so $$\begin{aligned}
\nonumber
\chi\psi=\sum_{\varphi \in \operatorname{Irr}(B/\!\!/K)} \lambda^{\varphi}_{\chi\psi} \varphi.\end{aligned}$$ But the dual of table algebra $(A/\!\!/K,B/\!\!/K)$ is a table algebra. Then it follows from Lemma \[es\] that $\lambda^{\varphi}_{\chi\psi}, \varphi\in \operatorname{Irr}(B)$, are nonnegative real numbers, as desired. So we consider the case that $\chi\in \operatorname{Irr}(B/\!\!/K)$ and $\psi\in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)$. Again, since $\widehat{B/\!\!/K}\cong \ker (K) $ is a closed subset of $\widehat{B}$, it follows that $$\Delta^{*}_{\chi} \Delta^{*}_{\psi}=\displaystyle \sum_{\Delta^{*}_{\varphi}\in \widehat{B}-\ker(K) }q^{\varphi}_{\chi\psi}\Delta^{*}_\varphi.$$ Then $$\begin{aligned}
\nonumber
\chi\psi=\sum_{\varphi \in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)} \lambda^{\varphi}_{\chi\psi} \varphi.\end{aligned}$$ But it follows from Lemma \[tars\] that for every $b\in B\setminus D$ and every $\varphi \in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K) $ where $\lambda^{\varphi}_{\chi\psi}\neq 0$, we have $\varphi(b)=0$. This implies that for every distinct irreducible characters $\varphi, \varphi'\in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)$ where $\lambda^{\varphi}_{\chi\psi}$ and $\lambda^{\varphi'}_{\chi\psi}$ are nonzero, $\varphi_D\neq \varphi'_D$, otherwise, if $\varphi_D= \varphi'_D$, then since $\varphi(b)=\varphi'(b)=0$, for every $b\in B\setminus D$, we have $\varphi=\varphi'$, a contradiction. Thus we conclude that $$\begin{aligned}
\nonumber
\chi_D\psi_D=\sum_{\varphi \in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)} \lambda^{\varphi}_{\chi\psi} \varphi_D.\end{aligned}$$ But the dual of table algebra $(<D>, D)$ is a table algebra. It follows that $\lambda^{\varphi}_{\chi\psi}\geq 0$ for every $\varphi \in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)$ and we are done.
To complete the proof it remains to consider $\chi, \psi \in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)$. Then $$\begin{aligned}
\nonumber
\chi\psi=\sum_{\varphi \in \operatorname{Irr}(B/\!\!/K)} \lambda^{\varphi}_{\chi\psi} \varphi+
\sum_{\varphi \in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)} \mu^{\varphi}_{\chi\psi} \varphi.\end{aligned}$$ Suppose that $\varphi \in \operatorname{Irr}(B/\!\!/K)$ such that $\lambda^{\varphi}_{\chi\psi}\neq 0$. Since $$\frac{\lambda^{\varphi}_{\chi\psi}}{{\zeta_\varphi}}=\frac{\lambda^{\psi}_{\overline{\chi}\varphi}}{{\zeta_\psi}},$$ we have $\lambda^{\psi}_{\overline{\chi}\varphi}\neq 0$. But since $\overline{\chi} \in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)$ and $\varphi \in \operatorname{Irr}(B/\!\!/K)$, it follows from the preceding case that $\lambda^{\psi}_{\overline{\chi}\varphi}\geq 0$. This shows that $\lambda^{\varphi}_{\chi\psi}\geq 0$. Finally, we prove that $\mu^{\varphi}_{\chi\psi} \geq 0$ for every $\varphi \in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)$. To do this, we observe that for distinct irreducible characters $\varphi, \varphi'\in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)$ such that $\mu^{\varphi}_{\chi\psi}$ and $\mu^{\varphi'}_{\chi\psi}$ are nonzero, if $\varphi_D=\varphi'_D$, then since for every $b\in B\setminus D$, $\varphi(b)=\varphi'(b)=0$, we must have $\varphi=\varphi'$, a contradiction. Thus we conclude that $$\begin{aligned}
\nonumber
\chi_D\psi_D=\sum_{\varphi \in \operatorname{Irr}(B/\!\!/K)} a^{\varphi}_{\chi\psi} \varphi_D+
\sum_{\varphi \in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)} \mu^{\varphi}_{\chi\psi} \varphi_D.\end{aligned}$$ But the dual of table algebra $(<D>, D)$ is a table algebra. So $\mu^{\varphi}_{\chi\psi}\geq 0$ for every $\varphi \in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)$ and we are done.
Let $(A,B)$ be a commutative table algebra and $K\leq D$ be closed subsets of $B$ such that the duals of $(<D>,D)$ and $(A/\!\!/K,B/\!\!/K)$ are table algebras. Since $$(<\ker(K)>,\ker(K))\cong (\widehat{A/\!\!/K},\widehat{B/\!\!/K})$$ and $$(\widehat{A}/\!\!/\ker(D),\widehat{B}/\!\!/\ker(D))\cong (\widehat{<D>},\widehat{D}),$$ it follows that $(\widehat{A},\widehat{B})$ is a C-algebra such that $(<\ker(K)>,\ker(K))$ and $(\widehat{A}/\!\!/\ker(D),\widehat{B}/\!\!/\ker(D))$ are table algebras. But $(\widehat{A},\widehat{B})$ need not be a table algebra, in general; see [@xueival Example 3.5]. So the condition $K\subseteq \st_B(B\setminus D)$ in the Theorem \[main01\] is a necessary condition.
\[aaa\] Let the commutative table algebra $(U,V)$ be the wedge product of table algebras $(C,D)$ and $(A,B)$ relative to $\varphi$. If the duals of table algebras $(C,D)$ and $(A,B)$ are table algebras, then the dual of $(U,V)$ is also a table algebra.
It follows from Lemma \[kd\] that $V$ contains the closed subset $K$ such that $K\subseteq \st_{V}(V\setminus D)$ and $V/\!\!/K\cong B$. Then since the dual of $(A,B)$ is a table algebra, the dual of $(U/\!\!/K,V/\!\!/K)$ is also a table algebra. So Theorem \[main01\] yields that the dual of $(U,V)$ is a table algebra.
\[kd2\] Let $(A,B)$ be a commutative table algebra and $K\leq D$ be closed subsets of $B$ such that $K\subseteq \st_B(B\setminus D)$. Then $\ker(D)\leq \ker(K)$ and $\ker(D)\subseteq \st_{\widehat{B}}(\widehat{B}\setminus \ker(K))$.
Since $K\leq D$, we have $$\ker(D)=\{\Delta^{*}_{\chi}|\chi\in \operatorname{Irr}(V/\!\!/D)\}\subseteq \{\Delta^{*}_{\chi}|\chi\in \operatorname{Irr}(V/\!\!/K)\}=\ker(K).$$ Moreover, since $K\subseteq \st_B(B\setminus D)$, it follows from Lemma \[tars\] that for every $b\in B\setminus D$ and every $\chi\in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)$, $\chi(b) =0$. Let $\chi\in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)$ and $\psi\in\operatorname{Irr}(B/\!\!/D)$. Since for every $b\in D$, $\psi(b)=|b|$ we have $$\chi\psi(b)=\frac{\chi(b)\psi(b)}{|b|}=\chi(b).$$ On the other hand, for every $b\in B\setminus D$, $$\chi\psi(b)=\frac{\chi(b)\psi(b)}{|b|}=0=\chi(b).$$ Thus we conclude that $\chi\psi=\chi$. This implies that for every $b\in B$, $$\Delta^{*}_{\chi}\Delta^{*}_{\psi}(b)=\Delta^{*}_{\chi}(b)\Delta^{*}_{\psi}(b)=
\frac{\zeta_\chi {\chi(b^*)}}{|b|}\frac{\zeta_\psi {\psi(b^*)}}{|b|}=\frac{\zeta_\chi\zeta_\psi\chi\psi(b^*)}{|b|}
=\frac{\zeta_\chi\zeta_\psi\chi(b^*)}{|b|}=\zeta_\psi \Delta^{*}_{\chi}(b).$$ So $\Delta^{*}_{\chi}\Delta^{*}_{\psi}=\zeta_\psi \Delta^{*}_{\chi}.$ Since $\widehat{B}\setminus \ker(K)= \{\Delta^{*}_{\chi}|\chi\in \operatorname{Irr}(B)\setminus \operatorname{Irr}(B/\!\!/K)\}$ we conclude that $$\Delta^{*}_{\psi}\in \st_{\widehat{B}}(\widehat{B}\setminus \ker(K)).$$ Hence $$\ker(D)=\{\Delta^{*}_{\psi}|\psi\in \operatorname{Irr}(B/\!\!/D)\} \subseteq \st_{\widehat{B}}(\widehat{B}\setminus \ker(K)).$$
\[wf\] Let commutative table algebra $(U,V)$ be the wedge product of table algebras $(C,D)$ and $(A,B)$ relative to $\varphi$. If the duals of table algebras $(C,D)$ and $(A,B)$ are table algebras, then $(\widehat{U},\widehat{V})$ is a table algebra, and is also the wedge product of $(<\ker(K)>,\ker(K))$ and $(\widehat{U}/\!\!/\ker(D),\widehat{V}/\!\!/\ker(D))$ relative to the canonical epimorphism $$\pi: (<\ker(K)>,\ker(K))\rightarrow (<\ker(K)>/\!\!/\ker(D),\ker(K)/\!\!/\ker(D)).$$
It follows from Corollary \[aaa\] that $(\widehat{U},\widehat{V})$ is a table algebra. Moreover, from Lemma \[kd\] we see that there exists a closed subset $K\leq D$ such that $K\subseteq \st_{V}(V\setminus D)$. Then Lemma \[kd2\] shows that $\ker(D)\leq \ker(K)$ and $\ker(D)\subseteq \st_{\widehat{V}}(\widehat{V}\setminus \ker(K))$. So the result follows from Theorem \[main2\].
\[dualwedge\] Let $(U,V)$ be a commutative table algebra. Then the following are equivalent: $(U,V)$ is the wedge product of table algebras $(C,D)$ and $(A,B)$ relative to $\varphi$, such that the duals of $(C,D)$ and $(A,B)$ are table algebras, $(\widehat{U},\widehat{V})$ is a table algebra, and is also the wedge product of $(\widehat{A},\widehat{B})$ and $(\widehat{C},\widehat{D})$ relative to $\widehat{\varphi}: (\widehat{A},\widehat{B})\rightarrow (\widehat{<N>}, \widehat{N})$.
Suppose $(i)$ holds. Then it follows from Corollary \[wf\] that $(\widehat{U},\widehat{V})$ is a table algebra, and is also the wedge product of $(<\ker(K)>,\ker(K))$ and $(\widehat{U}/\!\!/\ker(D),\widehat{V}/\!\!/\ker(D))$ relative to the canonical epimorphism $$\pi: (<\ker(K)>,\ker(K))\rightarrow (<\ker(K)>/\!\!/\ker(D),\ker(K)/\!\!/\ker(D)).$$ But from Lemma \[iso\] we have $V/\!\!/K\cong B$ and so $$\ker(K)=\widehat{V/\!\!/K}\cong \widehat{B}.$$ Then $(<\ker(K)>,\ker(K))\cong (\widehat{A},\widehat{B})$. Moreover, it follows from Lemma \[iso\] that $$(\widehat{U}/\!\!/\ker(D),\widehat{V}/\!\!/\ker(D))\cong(\widehat{C},\widehat{D}).$$ So $(\widehat{U},\widehat{V})$ is the wedge product of $(\widehat{A},\widehat{B})$ and $(\widehat{C},\widehat{D})$ relative to $$\pi: (\widehat{A},\widehat{B})\rightarrow (<\ker(K)>/\!\!/\ker(D),\ker(K)/\!\!/\ker(D)).$$ On the other hand, $\ker(K)/\!\!/\ker(D)\cong \widehat{ D/\!\!/K}$ and $D/\!\!/K\cong N$. So we deduce that $\ker(K)/\!\!/\ker(D)\cong \widehat{N}$. Hence $(\widehat{U},\widehat{V})$ is the wedge product of $(\widehat{A},\widehat{B})$ and $(\widehat{C},\widehat{D})$ relative to $$\widehat{\varphi}: (\widehat{A},\widehat{B})\rightarrow (\widehat{<N>}, \widehat{N}).$$ Now assume that $(ii)$ holds. Since $(\widehat{\widehat{A}},\widehat{\widehat{B}})\cong(A,B)$ and $(\widehat{\widehat{C}},\widehat{\widehat{D}})\cong(C,D)$ are table algebras, it follows from the first part of the proof that $(\widehat{\widehat{U}},\widehat{\widehat{V}})\cong(U,V)$ is the wedge product of table algebras $(C,D)$ and $(A,B)$ relative to $$\widehat{\widehat{\varphi}}: (\widehat{\widehat{C}},\widehat{\widehat{D}})\rightarrow (\widehat{\widehat{<N>}}, \widehat{\widehat{N}}).$$ But $(\widehat{\widehat{C}},\widehat{\widehat{D}})\cong (C,D)$ and $(\widehat{\widehat{<N>}}, \widehat{\widehat{N}})\cong (<N>,N)$. So $$\widehat{\widehat{\varphi}}=\varphi:(C,D)\rightarrow(<N>,N)$$ and thus $(i)$ holds.
Applications to association schemes
===================================
The wedge product of association schemes which provides a way to construct new association schemes from old ones, has been given by Muzychuk in [@Mu]. In the following we have a look at the wedge product of association schemes. The reader is referred to [@Mu] for more details.
Let $(X,G)$ be an association scheme and $D\leq G$. Suppose that $X/D=\{x_1D,\ldots, x_mD\}$. Put $X_i=x_iD$ and $D_i=D_{X_i}=\{d_{X_i} \mid d\in D\}$, where $d_{X_{i}}=d \cap X_i \times X_i$. Consider the bijection $\varepsilon_i:D \rightarrow D_i$ such that $\varepsilon_i(d)=d_{X_i}$. Then $\varepsilon_j\varepsilon^{-1}_i:D_i\rightarrow D_j$ is an algebraic isomorphism between association schemes $(X_i,D_i)$ and $(X_j,D_j)$. Assume that for every $i$, there exists an association scheme $(Y_i, B_i)$ and a scheme normal epimorphism $\psi_i:Y_i\cup B_i\rightarrow X_i\cup D_i$. Moreover, assume that there exist algebraic isomorphisms $\varphi_i:B_1\rightarrow B_i$ such that the diagram
$$\begin{CD}
B_1@>\varphi_i>>B_i \\
@V\psi_1V V @VV\psi_iV\\
D_1@>\varepsilon_i\varepsilon^{-1}_1 >>D_i
\end{CD}$$ is commutative for every $i$. Assume that $Y_i, 1\leq i\leq m$, are pairwise disjoint. Put $Y=Y_1 \cup \cdots \cup Y_m$, $\psi=\psi_1\cup \cdots \cup \psi_m$, $\overline{G}=\{\overline{g}\mid g\in G\}$, where $\overline{g}=\psi^{-1}(g)$, and for every $b\in B_1$, $\widetilde{b}=\cup^m_{i=1}\varphi_i(b)$. Let $\C [\widetilde{B_1}]$ be the $\C$-space spanned by $\widetilde{B_1}=\{\widetilde{b}\mid b\in B_1\}$. Then it follows from [@Mu Theorem 2.2] that $U=\C [\overline{G}] + \C [\widetilde{B_1}]$ is the adjacency algebra of an association scheme $(Y, \widetilde{B_1}\cup (\overline{C}\setminus \overline{D}))$, which is called the [*wedge*]{} product of $(Y_i,B_i),1\leq i\leq m$, and $(X,G)$.\
In the rest of this section, we show that the complex adjacency algebra of the wedge product of $(Y_i,B_i),1\leq i\leq m$, and $(X,G)$ is the wedge product of table algebras $(\C[\widetilde{B_1}], A(\widetilde{B_1}))$ and $(\C[G],A(G))$. We also study applications to association schemes.\
Put $K=\ker \psi_1$ and $V=\{A(x)\mid x\in \widetilde{B_1}\cup (\overline{C}\setminus \overline{D}) \}$. Then $(U,V)$ is a table algebra and $A(\widetilde{K})\leq A(\widetilde{B_1})$. It follows from [@Mu Theorem 2.2] that $A(\widetilde{K})\unlhd V$ and $A(\widetilde{K})\subseteq \st_{V}(V\setminus A(\widetilde {B_1}))$. Since $(\C[\widetilde{B_1}], A(\widetilde{B_1}))$ is a table algebra, it follows from Corollary \[wg1\] that $(U,V)$ is the wedge product of table algebras $(\C[\widetilde{B_1}], A(\widetilde{B_1}))$ and $(U/\!\!/A(\widetilde{K}), V/\!\!/A(\widetilde{K}))$ relative to the canonical epimorphism $$\pi:(\C[\widetilde{B_1}], A(\widetilde{B_1}))\rightarrow (\C[\widetilde{B_1}]/\!\!/A(\widetilde{K}), A(\widetilde{B_1})/\!\!/A(\widetilde{K})).$$\
Since $\psi_1:(Y_1,B_1)\rightarrow (X_1,D_1)$ is a scheme epimorphism and $$\varepsilon^{-1}_1:(\C[D_1],A(D_1))\rightarrow (\C[D],A(D))$$ is an algebra isomorphism, it follows that there is a table algebra epimorphism $$\overline{\psi_1}:(\C [B_1], A(B_1))\rightarrow (\C[D],A(D))$$ such that $$\overline{\psi_1}(A(b))=\frac{n_b}{n_{\psi_1(b)}}A(d)$$ where $\supp(\overline{\psi_1}(A(b)))=A(d)$; see Lemma \[x1\]. Then we can define a linear map $$\varphi:(\C[\widetilde{B_1}], A(\widetilde{B_1}))\rightarrow (\C[D],A(D))$$ by $\varphi(A(\widetilde{b}))=\overline{\psi_1}(A(b))$.
With the notation above, $\varphi$ is a table algebra epimorphism such that $\ker_{A(\widetilde{B_1})}\varphi=A(\widetilde{K})$.
For every $\widetilde{a}, \widetilde{b}\in \widetilde{B_1}$, it follows from [@Mu] that $A(\widetilde{a})A(\widetilde{b})=\displaystyle\sum_{c\in B_1}\lambda_{abc}A(\widetilde{c})$, where $\lambda_{abc}, c\in B_1$, are the structure constants of $(\C [B_1], A(B_1))$. Then we have $$\begin{aligned}
\varphi(A(\widetilde{a})A(\widetilde{b}))&=&\nonumber \varphi(\displaystyle\sum_{c\in B_1}\lambda_{abc}A(\widetilde{c})) \\&=&\nonumber
\displaystyle\sum_{c\in B_1}\lambda_{abc}\varphi(A(\widetilde{c}))=\displaystyle\sum_{c\in B_1}\lambda_{abc}\overline{\psi_1}(A({c}))\\&=&\nonumber
\overline{\psi_1}(\displaystyle\sum_{c\in B_1}\lambda_{abc}A({c}))=\overline{\psi_1}(A(a)A(b))\\&=&\nonumber
\overline{\psi_1}(A(a))\overline{\psi_1}(A(b))=
\varphi(A(\widetilde{a}))\varphi(A(\widetilde{b})).\end{aligned}$$ So $\varphi$ is a table algebra epimorphism. Note that $\psi_1$ is a scheme epimorphism. Moreover, we have $$\begin{aligned}
\ker_{A(\widetilde{B_1})}\varphi&=&\nonumber \{A(\widetilde{b})\in A(\widetilde{B_1})\mid \supp(\varphi(A(\widetilde{b})))=A(1_X)\}\\&=&\nonumber
\{A(\widetilde{b})\in A(\widetilde{B_1})\mid \overline{\psi_1}(A({b}))=n_bA(1_X)\}\\&=&\nonumber
\{A(\widetilde{b})\in A(\widetilde{B_1})\mid A(b)\in \ker_{A(B_1)}(\overline{\psi_1})\}\\&=&\nonumber
\{A(\widetilde{b})\in A(\widetilde{B_1})\mid b\in \ker(\psi_1)\}\\&=&\nonumber A(\widetilde{K}).\end{aligned}$$
\[note\] With the notation above, we have $n_{\widetilde{K}}=\frac{|Y|}{|X|}$.
It follows from [@Mu Proposition 2.1] that $A(\overline{i_X})A(\overline{g})=\frac{|Y|}{|X|}A(\overline{g})$. But $\overline{i_X}=\psi^{-1}(i_X)=\widetilde{K}$. So $A(\overline{i_X})=A(\widetilde{K})$ and thus $n_{\widetilde{K}}=\frac{|Y|}{|X|}$.
Now consider table algebras $(\C[\widetilde{B_1}], A(\widetilde{B_1}))$ and $(\C[G],A(G))$. Since $A(D)\leq A(G)$ and there exists the table algebra epimorphism $\varphi:(\C[\widetilde{B_1}], A(\widetilde{B_1}))\rightarrow (\C[D],A(D))$, we can construct the wedge product of $(\C[\widetilde{B_1}], A(\widetilde{B_1}))$ and $(\C[G],A(G))$ relative to $\varphi$, say $(E,F)$. Then $$F=A(\widetilde{B_1}) \cup \{n_{\widetilde{K}}A(g)\mid g\in G\setminus D\}.$$ Note that for every $A(\widetilde{b})\in A(\widetilde{B_1})$ with $\supp(\overline{\psi_1}(A(b)))=A(d)$, and $A(g)\in A(G)\setminus A(D)$ we have $$\begin{aligned}
\label{pro}
A(\widetilde{b})(n_{\widetilde{K}}A(g))=n_{\widetilde{K}}\varphi(A(\widetilde{b}))A(g)=n_{\widetilde{K}}\overline{\psi_1}(A(b))A(g)=
n_{\widetilde{K}} \frac{n_b}{n_{\psi_1(b)}}A(d)A(g),\end{aligned}$$ and similarly, $$(n_{\widetilde{K}}A(g))A(\widetilde{b})=n_{\widetilde{K}}A(g)\overline{\psi_1}(A(b))=
n_{\widetilde{K}} \frac{n_b}{n_{\psi_1(b)}}A(g)A(d).$$
The table algebras $(U,V)$ and $(E,F)$ are isomorphic.
Define $\theta:(U,V)\rightarrow (E,F)$ by $\theta(A(\widetilde{b}))=A(\widetilde{b})$ and $\theta(A(\overline{g}))=n_{\widetilde{K}}A(g)$, for every $\widetilde{b}\in \widetilde{B_1}$ and every $\overline{g}\in \overline{G}\setminus \overline{D}$. We first show that $\theta$ is a table algebra homomorphism. To do so, first assume that $\overline{g},\overline{h}\in \overline{G}\setminus \overline{D}$. Then it follows from [@Mu Proposition 2.1] that $$\begin{aligned}
A(\overline{g})A(\overline{h})&=&\nonumber \frac{|Y|}{|X|}\displaystyle\sum_{l\in G}\lambda_{ghl}A(\overline{l}).\end{aligned}$$ So we have
$$\begin{aligned}
\theta (A(\overline{g})A(\overline{h}))&=&\nonumber \theta(\frac{|Y|}{|X|}\displaystyle\sum_{l\in G}\lambda_{ghl}A(\overline{l})) \\&=&\nonumber
\frac{|Y|}{|X|}\displaystyle\sum_{l\in G}\lambda_{ghl}\theta(A(\overline{l}))=\frac{|Y|}{|X|}n_{\widetilde{K}}\displaystyle\sum_{l\in G}\lambda_{ghl}A({l})\\&=&\nonumber n_{\widetilde{K}}^2(A(g)A(h))=\theta (A(\overline{g}))\theta(A(\overline{h})).\end{aligned}$$
Note that from Lemma \[note\] we have $n_{\widetilde{K}}=\frac{|Y|}{|X|}$. Now suppose that $\widetilde{b}\in \widetilde{B_1}$ and $\overline{g}\in \overline{G}\setminus \overline{D}$. Then it follows from [@Mu Theorem 2.2] that $$\begin{aligned}
\nonumber
A(\widetilde{b})A(\overline{g})=\frac{n_b|X|}{n_{\psi_1(b)}|Y|}A(\overline{d})A(\overline{g}).\end{aligned}$$ Moreover, from [@Mu Proposition 2.1] we have $$\begin{aligned}
A(\overline{d})A(\overline{g})&=&\nonumber \frac{|Y|}{|X|}\displaystyle\sum_{l\in G}\lambda_{dgl}A(\overline{l}).\end{aligned}$$ Thus by applying equality (\[pro\]) we see that $$\begin{aligned}
\theta (A(\widetilde{b})A(\overline{g}))&=&\nonumber \frac{n_b|X|}{n_{\psi_1(b)}|Y|}\theta (A(\overline{d})A(\overline{g}))\\&=&\nonumber
\frac{n_b|X|}{n_{\psi_1(b)}|Y|}\theta(\frac{|Y|}{|X|}\displaystyle\sum_{l\in G}\lambda_{dgl}A(\overline{l}))=
\frac{n_b}{n_{\psi_1(b)}}\displaystyle\sum_{l\in G}\lambda_{dgl}\theta (A(\overline{l}))\\&=&\nonumber
\frac{n_b}{n_{\psi_1(b)}}n_{\widetilde{K}}\displaystyle\sum_{l\in G}\lambda_{dgl} A({l})=
\frac{n_b}{n_{\psi_1(b)}}n_{\widetilde{K}}A(d)A(g)\\&=&\nonumber
(\frac{n_b}{n_{\psi_1(b)}}A(d))(n_{\widetilde{K}}A(g))=
A(\widetilde{b})(n_{\widetilde{K}}A(g))\\&=&\nonumber
\theta(A(\widetilde{b}))\theta(A(\overline{g})).\end{aligned}$$ Similarly, $\theta (A(\overline{g})A(\widetilde{b}))=\theta(A(\overline{g}))\theta(A(\widetilde{b})).$ So $\theta$ is a table algebra homomorphism. But $\theta$ is also a bijection from $V$ onto $F$. Thus $\theta$ is a table algebra isomorphism, as desired.
\[wedgeass\] Let $U=\C [\overline{G}] + \C [\widetilde{B_1}]$ be the complex adjacency algebra of the wedge product of $(Y_i,B_i),1\leq i\leq m$, and $(X,G)$. Then table algebra $(U,V)$, where $V=\{A(x)\mid x\in \widetilde{B_1}\cup (\overline{C}\setminus \overline{D}) \}$, is the wedge product of table algebras $(\C[\widetilde{B_1}], A(\widetilde{B_1}))$ and $(\C[G],A(G))$ relative to $\varphi$, where $$\varphi:(\C[\widetilde{B_1}], A(\widetilde{B_1}))\rightarrow (\C[D],A(D))$$ such that $\varphi(A(\widetilde{b}))=\overline{\psi_1}(A(b))$.
By applying Theorem \[iso\] and Corollary \[wedgeass\], we can give the following result.
Let $U=\C [\overline{G}] + \C [\widetilde{B_1}]$ be the complex adjacency algebra of the wedge product of $(Y_i,B_i),1\leq i\leq m$, and $(X,G)$. Then $$U/\!\!/A(\widetilde{K})\cong A(G).$$
As a direct consequence of Theorem \[dualwedge\] and Corollary \[wedgeass\], we have the following:
Suppose that $U=\C [\overline{G}] + \C [\widetilde{B_1}]$, the complex adjacency algebra of the wedge product of $(Y_i,B_i),1\leq i\leq m$, and $(X,G)$, is commutative. Then the table algebra $(\widehat{U},\widehat{V})$, where $V=\{A(x)\mid x\in \widetilde{B_1}\cup (\overline{C}\setminus \overline{D}) \}$, is the wedge product of table algebras $(\widehat{\C[G]},\widehat{A(G)})$ and $(\widehat{\C[\widetilde{B_1}]}, \widehat{A(\widetilde{B_1})})$ relative to $\widehat{\varphi}$, where $$\widehat{\varphi}:(\widehat{\C[G]},\widehat{A(G)})\rightarrow (\widehat{\C[D]},\widehat{A(D)}).$$
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|
---
abstract: |
Slime mould *Physarum polycephalum* is a large single cell visible by unaided eye. We design a slime mould implementation of a tactile hair, where the slime mould responds to repeated deflection of hair by an immediate high-amplitude spike and a prolonged increase in amplitude and width of its oscillation impulses. We demonstrate that signal-to-noise ratio of the Physarum tactile hair sensor averages near 6 for the immediate response and 2 for the prolonged response.
*Keywords: slime mould, bionic, bioengineering, sensor*
address: 'University of the West of England, Bristol, United Kingdom'
author:
- 'Andrew Adamatzky\'
title: |
Slimy hairs:\
Hair sensors made with slime mould
---
Introduction
============
Tactile sensors are ubiquitous in robotics and medical devices [@Cutkosky_2008; @Hamed_2008; @Mukai_2008; @Lucarotti_2013] and thus development of novel types of these sensing devices remains a hot topic of engineering, material sciences and bionics. Novel designs and implementations, see overviews in [@Rocha_2008; @Lucarotti_2013], include arrays of electro-active polymers and ionic polymer metal composites [@wang_2008; @wang_2009], piezoelectric polymer oxide semiconductor field effect transistors tactile arrays [@Dahiya_2009], pressure sensitive conductive rubber [@Kato_2008; @Ohmukai_2012], flexible capacitive micro-fluidic based sensors [@Wong_2012], and patterns of micro-channels filled with eutectic gallium-indium [@Park_2010; @Park_2011]. Recently an interest in technological developments started to move away from solid materials to soft matter implementations, see overview in [@Tiwana_2012], and bio-inspired and hybrid implementations: bio-mimetic sensors which employ a conductive fluid encapsulated in elastic container and use deformation of the elastic container in transduction [@Wettels_2008], carbon nanotube filled elastomers [@Engel_2006], polymer hair cell sensors [@Engel_2006a].
Live cell sensors [@Taniguchi_2010] and bio-hybrid sensors encapsulating living fibroblasts as a part of transduction system [@Cheneler_2012] are of particular interest because they open totally new dimension in engineering sensing technologies. The living substrates used in sensor design are too dependent on a ’life-support’ system, they need supply of nutrients and removal of waste. Therefore we decided to consider an autonomous living creature which does not require sophisticate support bandwidth and can survive for a long period of time without a laboratory equipment. This is a vegetative stage, a plasmodium of acellular slime mould *Physarum polycephalum*.
![Plasmodium of *P. polycephalum* on a data set, planar configuration of oat flakes, on an agar gel in a Petri dish 8 cm diameter. (a) A virgin oat flake. (b) An oat flake colonised by the plasmodium. (c) A protoplasmic tubes. (d) An active zones, growing parts of the plasmodium. []{data-label="PhysExm"}](figs/PhysExm){width="70.00000%"}
*Physarum polycephalum* belongs to the species of order *Physarales*, subclass *Myxogastromycetidae*, class *Myxomycetes*, division *Myxostelida*. It is commonly known as a true, acellular or multi-headed slime mould. Plasmodium is a single cell with a myriad of diploid nuclei. The plasmodium is visible to the naked eye (Fig. \[PhysExm\]). The plasmodium looks like an amorphous yellowish mass with networks of protoplasmic tubes. The plasmodium behaves and moves as a giant amoeba. It feeds on bacteria, spores and other microbial creatures and micro-particles [@stephenson_2000].
Plasmodium of *P. polycephalum* consumes microscopic particles, and during its foraging behaviour the plasmodium spans scattered sources of nutrients with a network of protoplasmic tubes (Fig. \[PhysExm\]). The plasmodium optimises its protoplasmic network that covers all sources of nutrients and guarantees robust and quick distribution of nutrients in the plasmodium’s body. Plasmodium’s foraging behaviour can be interpreted as a computation: data are represented by spatial distribution of attractants and repellents, and results are represented by a structure of protoplasmic network [@adamatzky_physarummachines]. Plasmodium can solve computational problems with natural parallelism, e.g. related to shortest path [@nakagaki_2000] and hierarchies of planar proximity graphs [@adamatzky_ppl_2008], computation of plane tessellations [@shirakawa], execution of logical computing schemes [@tsuda2004; @adamatzky_gates], and natural implementation of spatial logic and process algebra [@schumann_adamatzky_2009]. In [@adamatzky_physarummachines] we experimentally demonstrated that slime mould *P. polycephalum* is a programmable amorphous biological computing substrate, capable for solving a wide range of tasks: from computational geometry and optimisation on graphs to logics and general purpose computing.
In our previous paper on slime mould’s tactile sensors [@adamatzky_2013_tactile] we experimentally studied how Physarum reacts to application of a load to its fine network of tubes or a single tube. We demonstrated impulse and pattern of oscillation responses and characterised sensorial abilities of Physarum. We found, however, that it is not practical to allow Physarum tactile sensor to be in a direct contact with a load, because the slime mould starts colonising the load and becomes attached to the load and the sensor could be damaged when the load is lifted. Thus some intermediary mechanical medium is required an object causing tactile stimulation and the Physarum. Looking for alternative solutions we encountered Engel and colleagues design [@Engel_2006a] of a spider artificial hair made of polyurethane and fixed to a flexible substrate. Being inspired by their approach we implemented a slime mould-based analog of a spider tactile hair outlined in present paper.
Methods
=======
![Scheme of experimental setup. Electrodes (r) and (e) are fixed to a bottom of polyurethane Petri dish. A hair (h) is fixed to one electrode with epoxy glue. Agar blobs (b) are placed on electrodes. The hair penetrates through the centre of one agar blob. Physarum (p) colonises both agar blobs, propagates across bare plastic substrate (s) and connects two blobs with a protoplasmic tube (t). Root of the hair is partly colonised by Physarum.[]{data-label="scheme"}](figs/scheme1){width="80.00000%"}
Plasmodium of *Physarum polycephalum* was cultivated in plastic lunch boxes (with few holes punched in their lids for ventilation) on wet kitchen towels and fed with oat flakes. Culture was periodically replanted to fresh substrate. A scheme of experimental setup is shown in Fig. \[scheme\]. A planar aluminium foil electrodes (width 5 mm, 0.04 mm thick, volume resistance 0.008 $\Omega$/cm$^2$) are fixed to a bottom of a plastic Petri dish (9 cm), see Fig. \[scheme\]er, where ’r’ is a reference electrode and ’e’ is recording electrode. Distance between proximal sites of electrodes is 10 mm. One or more hairs (either human hairs or bristles from a toothbrush), 10 mm lengths, were fixed by upright to electrode ’e’ using epoxy glue. Then 2 ml of agar was gently and slowly powered onto the electrodes to make a dome like blobs of agar (Fig. \[scheme\]b). An oat flake occupied by Physarum was placed on an agar blob, residing on a reference electrode ’r’. Another oat flake not colonised by Physarum was placed on an agar blob on electrode ’e’. Physarum exhibits chemotaxis behaviour and therefore propagates on a bare bottom of a Petri dish from blob ’r’ to blob ’e’, usually in 1-3 days. Thus in 1-3 days after inoculation of Physarum on blob ’r’, both blobs became colonised by Physarum (Fig. \[scheme\]p), and the blobs became connected by a single protoplasmic tube (Fig. \[scheme\]t). Experiments where more than one tube connected blobs with Physarum develop were usually discounted because patterns of oscillation were affected by interactions between potential waves travelling along interlinked protoplasmic tubes. Electrical activity of plasmodium was recorded with ADC-24 High Resolution Data Logger (Pico Technology, UK), a recording is taken every 10 sec (as much samples as possible are averaged during this time window). Tactile stimulation was provided by deviating whisker from its original position 30 times, two times per second; a tip of the whisker was deflected from its position by 20-30 degrees. Whiskers were deflected using a 15 cm insulator stick.
![A scheme of a spider tactile hair. Adapted from [@barth_2004].[]{data-label="schemehair"}](figs/scheme_hair){width="60.00000%"}
In our experimental setup we aimed to imitate, at least at an abstract level, a spider tactile hair, see Fig. \[schemehair\] and details in [@barth_2004]. This is a medium size seta, or hair, with slightly curved shaft. This seta is usually innervated by 1-3 neurons, one neuron is shown in Fig. \[schemehair\], which dendrites enter the shaft but may not propagate till top of the hair. We imitated seta with a natural or synthetic bristle without shaft (Fig. \[scheme\]h), rightly predicting that slime mould will climb on the bristles. Dendrite is imitated by plasmodial network on electrode ’e’, see Fig. \[scheme\]; neuron is a protoplasmic tube (Fig. \[scheme\]t) connecting the agar blobs, and axon is a plasmodial network on electrode ’r’.
Results
=======
In 1-3 days after inoculation of Physarum to an agar blob on a references electrode it propagates to and colonises agar blob on a recording electrode. A living wire — protoplasmic tube — connecting two blobs of Physarum is formed (Fig. \[photowhiskers\]a). In most cases Physarum ’climbs’ onto hairs/bristles and occupies one third to a half of their length (Fig. \[photowhiskers\]b). In many cases a sub-network of protoplasmic tubes is formed around the base of the hair/bristle (Fig. \[photowhiskers\]c). In some cases Physarum occupies the whole hair/bristle, from the bottom to the top (Fig. \[photowhiskers\]d).
An undisturbed Physarum exhibit more or less regular patterns of oscillations of its surface electrical potential (Fig. \[parameters\]a, A). The electrical potential oscillations are more likely controlling a peristaltic activity of protoplasmic tubes, necessary for distribution of nutrients in the spatially extended body of Physarum [@seifriz_1937; @heilbrunn_1939]. A calcium ion flux through membrane triggers oscillators responsible for dynamic of contractile activity [@meyer_1979; @fingerle_1982]. It is commonly acceptable is that the potential oscillates with amplitude of 1 to 10 mV and period 50-200 sec, associated with shuttle streaming of cytoplasm [@iwamura_1949; @kamiya_1950; @kashimoto_1958; @meyer_1979]. In our experiments we observed sometimes lower amplitudes because there are agar blobs between Physarum and electrodes and, also, electrodes were connected with protoplasmic tube only. Exact characteristics of electric potential oscillations vary depending on state of Physarum culture and experimental setups [@achenbach_1980].
![Physarum’s response deflecting hairs in turns. Vertical axis is an electrical potential value in mV, horizontal axis is time in seconds. (a) and (c) Hair on recording electrode is deflected 30 times. (b) and (d) Hair on reference electrode is deflected 30 times.[]{data-label="hairs(p)_030313"}](figs/AE_RE_AE_RE){width="110.00000%"}
A typical response of Physarum to stimulation is shown in Fig. \[parameters\]. The response is comprised of an immediate response (Fig. \[parameters\]a, B): a high-amplitude impulse and a prolonged response (Fig. \[parameters\]a, C). See experimental recording in Fig. \[envelopa\]a. High-amplitude impulse is always well pronounced, prolonged response oscillations can sometimes be distorted by other factors, e.g. growing branches of a protoplasmic tube or additional strands of plasmodium propagating between the agar blobs. Responses are repeatable not only in different experiments but also during several rounds of stimulation in the same experiment. An example is shown in Fig. \[envelopa\]b. Physarum responds with a high-amplitude impulse to the first package of stimulation, 890 sec, yet prolonged envelope response is not visible. Subsequent packages of hair deflection receives both immediate and well pronounced prolonged responses (Fig. \[envelopa\]b).
------- ------- ------ -------
$V$ 4.0 2.1 3.1
$V'$ 0.7 0.5 0.5
$V''$ 1.51 1.36 0.94
$D$ 135.9 72.0 129.0
$D'$ 104.7 25.9 95
$D''$ 112.9 17.9 1.21
$e$ 3.15 1.21 3
------- ------- ------ -------
: Statistics of Physarum response to deflection of hairs calculated in 25 experiments.
\[impulsestatistics\]
In 25 experiments we calculated the following characteristics, see Fig. \[parameters\]b: amplitudes of oscillations before $V'$ and after $V''$ stimulation, and of the immediate response impulse $V$, and width of impulses before $D'$ and after stimulation $D''$, and of immediate response $D$. Statistics of the characteristics is shown in Tab. \[impulsestatistics\]. In our particular setup, keep in mind that signal’s strength is reduced due to agar blobs and a single protoplasmic tube connecting electrodes, average amplitude of oscillations before stimulation is 0.7 mV and after oscillations, in the envelop of waves, is 1.51 mV. Amplitude of the immediate response is 4 mV in average. A prolonged response envelop has 3-4 waves. Width of oscillation impulses becomes slightly shorter after stimulation. Dispersion of amplitude and width values around average values are substantial (Tab. \[impulsestatistics\]). This may be because electrical activity of Physarum and its response is determined by exact topology of a protoplasmic network wrapping agar blobs, and geometry of branching of the protoplasmic tube connecting the blobs.
Signal to noise ratio (SNR) is an important characteristic of a sensor. Physarum’s electrical potential constantly oscillates. Thus we assume a ’noise’ is a background oscillatory pattern of an undisturbed Physarum and ’signal’ is an immediate response (Fig. \[parameters\]a, B) and a prolonged response, an envelop of waves (Fig. \[parameters\]a, C). Maps of SNR obtained in laboratory experiments are shown in Fig. \[SNRmaps\]. Experimental plots of immediate response’s SNR are well grouped around average SNR of amplitude 5.7 and SNR of width 1.29 (Fig. \[SNRmaps\]a). SNR of amplitude of a prolonged response varies from 1 to almost 3 with average 2.2, while SNR of width is around 1 (Fig. \[SNRmaps\]b). Analysis of SNRs shows that amplitudes of immediate and prolonged responses are robust indicators of stimulation response.
Hairy balls and slimy whiskers
==============================
Will slime mould based tactile hairs work in a less ideal, than described in the previous section, environment? To evaluate validity of the approach we assembled Physarum tactile hair setups on ping-pong balls (Fig. \[balls\]) and a rubber mouse (Fig. \[mouse\]). The balls were equipped with planar aluminium foil electrodes (the same as used in original setup) and the mouse with a hook-up wire electrodes (silver plated single core wires, cross-section area 0.23 mm$^2$, resistance 23.6 $\Omega$/1000ft). The wire electrodes were fixed to the mouse using silicon glue (Silastic medical adhesive silicon Type A, Dow Corning). Hairs on the balls were made of human hairs and whiskers on the mouse of polyurethane bristles fixed to the objects’ surface using Silastic silicon glue. We have not collected statistics but rather undertook few experiments to evaluate feasibility of the approach.
In the ping-balls setup Physarum linked blobs of agar on electrodes with a single protoplasmic tube. In experiments with mouse, tubes connecting agar blobs in the base of whiskers sometimes propagated across the mouse’s nose bridge but sometimes inframaxillary. In most experiments hairs and whiskers were partly colonised by Physarum. Degree of colonisation varied from one sevenths of a bristle to almost (Fig. \[balls\]) over the half of the hair/whisker length (Fig. \[mouse\]b)
Examples of responses to repeated deflections of hairs and whiskers are shown in Figs. \[ballsresponse\] and \[mouseresponse\]. The responses are variable. Thus, Physarum hairs on a yellow ping-pong ball (Fig. \[balls\]b) reacted with an impulse of 4.3 mV to a repeated deflection of hairs (Fig. \[ballsresponse\]a). When hairs on agar blobs on recording and reference electrodes are deflected in turns Physarum responds, with impulses of different signs but the same amplitude of 0.5 mV (Fig. \[ballsresponse\]b). In a response illustrated in Fig. \[mouseresponse\]a baseline of electrical potential drops by 0.2 mV but recovers its original value after 270-300 seconds. A response impulse shown in Fig. \[mouseresponse\]b is twice of amplitude of the background impulses.
SNRs of the responses illustrated are 15 in Figs. \[ballsresponse\]a, 2 in Figs. \[ballsresponse\]b, 0.9 in Figs. \[mouseresponse\]a and 2 in Figs. \[mouseresponse\]b. Thus in most cases response signal is well distinguishable from background impulses.
Discussion
==========
In laboratory experiments we designed a bio-hybrid system imitating some features of a tactile hair of a spider. A neuron transducing a mechanical deflection of a chair into an electrical response is physically imitated by slime mould *Physarum polycephalum*. Two types of responses are detected: immediate response with a high-amplitude impulse and a prolonged response in a form of a wave envelop. The slime mould tactile hairs show a reasonable value of a signal to noise ratio: around 6 for an immediate response and 2 for a prolonged response. We have also installed Physarum hairs and whiskers on a non-planar objects and demonstrated that a signal to noise ratio of at least 2 is reached. Thus, we can claim that Physarum based tactile hairs and whiskers might play a significant role in future designs of bio-hybrid robotic devices.
Our designs of slime mould based tactile hairs make an elegant addition to existing prototypes of bio-hybrid sensors incorporating live cells as parts of transduction system [@Taniguchi_2010; @Cheneler_2012] and open totally new dimension in engineering sensing technologies. Living substrates used for the bio-hybrid systems require connective pathways to deliver nutrients and remove products of cell metabolism [@Lucarotti_2013]. Some approaches to deal with this problem are based on micro-fabrication of vascular networks [@shin_2004]. Slime mould based sensors does not require any auxiliary ’life support’ system, the cell propagates on the electrodes, consumes nutrients from sources of food supplied and damps waste products in a substrate by itself.
Advantages of the proposed Physarum hairs are self-growth of sensors, low-power consumption, almost zero costs, reasonably good sensitivity, and a high signal to noise ratio. Disadvantages include the Physarum hairs/whiskers response dependence on a morphology of protoplasmic tubes and networks (which are in a state of continuous flux), susceptibility to temperature, light and humidity change, and relatively short period of functionality (usually 3-4 days).
The devices described in the paper are instances of *wetware of a secondary class of living technologies* [@bedau_2010]. Rephrasing, Bedau et al. [@bedau_2010], we can say that Physarum tactile hairs proposed are artificial because they are created by our intentional activities yet they are totally natural because they grow and respond to environmental stimuli by their own biological laws.
Further research in the slime mould based tactile hairs will aim to answer the following questions. What are biophysical mechanisms of hair/whisker to Physarum to electrical output transduction? Do cellular membrane and outer wall of Physarum’s protoplasmic tubes react to stretching and twisting in same manner as e.g. bacteria do [@hamil_2001]? How does a response’s strength (in amplitude and duration) depends on a number of hairs/whiskers deflected? How to keep the ’slimy whiskers’ functioning for weeks and months? What is an adaptability pattern of the Physarum hairs when they are exposes to repeated rounds of stimulation?
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|
---
author:
- 'Planck Collaboration: N. Aghanim'
- 'M. I. R. Alves[^1]'
- 'M. Arnaud'
- 'D. Arzoumanian'
- 'J. Aumont'
- 'C. Baccigalupi'
- 'A. J. Banday'
- 'R. B. Barreiro'
- 'N. Bartolo'
- 'E. Battaner'
- 'K. Benabed'
- 'A. Benoit-Lévy'
- 'J.-P. Bernard'
- 'M. Bersanelli'
- 'P. Bielewicz'
- 'A. Bonaldi'
- 'L. Bonavera'
- 'J. R. Bond'
- 'J. Borrill'
- 'F. R. Bouchet'
- 'F. Boulanger'
- 'A. Bracco'
- 'C. Burigana'
- 'E. Calabrese'
- 'J.-F. Cardoso'
- 'A. Catalano'
- 'A. Chamballu'
- 'H. C. Chiang'
- 'P. R. Christensen'
- 'S. Colombi'
- 'L. P. L. Colombo'
- 'C. Combet'
- 'F. Couchot'
- 'B. P. Crill'
- 'A. Curto'
- 'F. Cuttaia'
- 'L. Danese'
- 'R. D. Davies'
- 'R. J. Davis'
- 'P. de Bernardis'
- 'A. de Rosa'
- 'G. de Zotti'
- 'J. Delabrouille'
- 'C. Dickinson'
- 'J. M. Diego'
- 'H. Dole'
- 'S. Donzelli'
- 'O. Doré'
- 'M. Douspis'
- 'A. Ducout'
- 'X. Dupac'
- 'G. Efstathiou'
- 'F. Elsner'
- 'T. A. En[ß]{}lin'
- 'H. K. Eriksen'
- 'E. Falgarone'
- 'K. Ferrière'
- 'F. Finelli'
- 'O. Forni'
- 'M. Frailis'
- 'A. A. Fraisse'
- 'E. Franceschi'
- 'A. Frejsel'
- 'S. Galeotta'
- 'S. Galli'
- 'K. Ganga'
- 'T. Ghosh'
- 'M. Giard'
- 'E. Gjerl[ø]{}w'
- 'J. González-Nuevo'
- 'K. M. Górski'
- 'A. Gregorio'
- 'A. Gruppuso'
- 'V. Guillet'
- 'F. K. Hansen'
- 'D. Hanson'
- 'D. L. Harrison'
- 'S. Henrot-Versillé'
- 'D. Herranz'
- 'S. R. Hildebrandt'
- 'E. Hivon'
- 'M. Hobson'
- 'W. A. Holmes'
- 'A. Hornstrup'
- 'W. Hovest'
- 'K. M. Huffenberger'
- 'G. Hurier'
- 'A. H. Jaffe'
- 'T. R. Jaffe'
- 'J. Jewell'
- 'M. Juvela'
- 'R. Keskitalo'
- 'T. S. Kisner'
- 'J. Knoche'
- 'M. Kunz'
- 'H. Kurki-Suonio'
- 'G. Lagache'
- 'J.-M. Lamarre'
- 'A. Lasenby'
- 'M. Lattanzi'
- 'C. R. Lawrence'
- 'R. Leonardi'
- 'F. Levrier'
- 'M. Liguori'
- 'P. B. Lilje'
- 'M. Linden-V[ø]{}rnle'
- 'M. López-Caniego'
- 'P. M. Lubin'
- 'J. F. Macías-Pérez'
- 'B. Maffei'
- 'D. Maino'
- 'N. Mandolesi'
- 'A. Mangilli'
- 'M. Maris'
- 'P. G. Martin'
- 'E. Martínez-González'
- 'S. Masi'
- 'S. Matarrese'
- 'A. Melchiorri'
- 'L. Mendes'
- 'A. Mennella'
- 'M. Migliaccio'
- 'M.-A. Miville-Deschênes'
- 'A. Moneti'
- 'L. Montier'
- 'G. Morgante'
- 'D. Mortlock'
- 'A. Moss'
- 'D. Munshi'
- 'J. A. Murphy'
- 'P. Naselsky'
- 'F. Nati'
- 'P. Natoli'
- 'C. B. Netterfield'
- 'F. Noviello'
- 'D. Novikov'
- 'I. Novikov'
- 'N. Oppermann'
- 'L. Pagano'
- 'F. Pajot'
- 'R. Paladini'
- 'D. Paoletti'
- 'F. Pasian'
- 'G. Patanchon'
- 'O. Perdereau'
- 'V. Pettorino'
- 'F. Piacentini'
- 'M. Piat'
- 'D. Pietrobon'
- 'S. Plaszczynski'
- 'E. Pointecouteau'
- 'G. Polenta'
- 'N. Ponthieu'
- 'G. W. Pratt'
- 'G. Prézeau'
- 'S. Prunet'
- 'J.-L. Puget'
- 'R. Rebolo'
- 'M. Reinecke'
- 'M. Remazeilles'
- 'C. Renault'
- 'A. Renzi'
- 'I. Ristorcelli'
- 'G. Rocha'
- 'C. Rosset'
- 'M. Rossetti'
- 'G. Roudier'
- 'J. A. Rubiño-Martín'
- 'B. Rusholme'
- 'M. Sandri'
- 'D. Santos'
- 'M. Savelainen'
- 'G. Savini'
- 'D. Scott'
- 'J. D. Soler'
- 'L. D. Spencer'
- 'V. Stolyarov'
- 'D. Sutton'
- 'A.-S. Suur-Uski'
- 'J.-F. Sygnet'
- 'J. A. Tauber'
- 'L. Terenzi'
- 'L. Toffolatti'
- 'M. Tomasi'
- 'M. Tristram'
- 'M. Tucci'
- 'J. Tuovinen'
- 'L. Valenziano'
- 'J. Valiviita'
- 'B. Van Tent'
- 'P. Vielva'
- 'F. Villa'
- 'L. A. Wade'
- 'B. D. Wandelt'
- 'I. K. Wehus'
- 'H. Wiesemeyer'
- 'D. Yvon'
- 'A. Zacchei'
- 'A. Zonca'
---
[^1]: Corresponding author: M. I. R. Alves [[email protected]]([email protected])
|
---
abstract: 'We calculate the axial current decay constants of taste non-Goldstone pions and kaons in staggered chiral perturbation theory through next-to-leading order. The results are a simple generalization of the results for the taste Goldstone case. New low-energy couplings are limited to analytic corrections that vanish in the continuum limit; certain coefficients of the chiral logarithms are modified, but they contain no new couplings. We report results for quenched, fully dynamical, and partially quenched cases of interest in the chiral SU(3) and SU(2) theories.'
author:
- 'Jon A. Bailey'
- Weonjong Lee
- Boram Yoon
bibliography:
- 'ref.bib'
title: ' Taste non-Goldstone, flavor-charged pseudo-Goldstone boson decay constants in staggered chiral perturbation theory '
---
Introduction\[sec:intr\]
========================
The decay constants $f_\pi$ and $f_K$ parametrize hadronic matrix elements entering the leptonic decays of $\pi$ and $K$ mesons. The values of the decay constants can be combined with the leptonic decay rates from experiment to extract the CKM matrix elements $|V_{ud}|$ and $|V_{us}|$ and test first-row CKM unitarity. Tighter constraints on new physics are obtained by taking the ratio $f_K/f_\pi$ and the form factor for the semileptonic decay $K\to\pi\ell\nu$ as theoretical inputs; doing so has led to impressive agreement between the Standard Model and experiment [@Colangelo:2010et; @Beringer:2012pdg].
Staggered quarks have 4 tastes per flavor by construction [@Golterman:1984cy; @Golterman:1985dz]. The full taste symmetry group for a single massless flavor is $SU(4)_L \times SU(4)_R$ in the continuum limit ($a=0$). At finite lattice spacing, lattice artifacts of ${\mathscr{O}}(a^2)$ break the taste symmetry, and the remaining exact chiral symmetry is $U(1)_A$, which is enough to prevent the staggered quark mass from being additively renormalized. Hence, staggered fermions have an exact chiral symmetry at nonzero lattice spacing. In addition, lattice calculations with staggered fermions are comparatively fast. Staggered chiral perturbation theory (SChPT) was first developed to describe the lattice artifacts and light quark mass dependence of lattice data for the pseudo-Goldstone boson (PGB) masses [@Lee:1999zxa; @Bernard:2001yj; @Aubin:2003mg; @Aubin:2003rg]. Lattice data were extrapolated to the continuum limit and physical quark masses to determine the light quark masses, tree-level PGB mass splittings, and low-energy couplings (LECs); these served as inputs to lattice calculations of the decay constants, semileptonic form factors, mixing parameters, and other quantities [@Aubin:2004ck; @Aubin:2004fs; @Aubin:2004ej; @Aubin:2005ar; @Gray:2005ad; @Aubin:2005zv; @Okamoto:2005zg; @Dalgic:2006dt; @Aubin:2006xv; @Bernard:2008dn; @Bailey:2008wp; @Gamiz:2009ku; @Bazavov:2009bb; @Bae:2010ki; @Kim:2011qg; @Bae:2011ff; @Bazavov:2011aa; @Bailey:2012rr; @Bazavov:2012zs]. Lattice calculations of $f_\pi$ have become precise enough to use it to determine the lattice spacing [@Bazavov:2009tw].
While there have been a few attempts to calculate the decay constants for the taste non-Goldstone sectors [@Aoki:1999av; @Bernard:priv], most lattice calculations of the decay constants have been concentrated on the taste Goldstone sector associated with the exact chiral symmetry of the staggered action. In Ref. [@Aubin:2003uc], Aubin and Bernard calculated the decay constants of the taste Goldstone pions and kaons through next-to-leading order (NLO) in SChPT. Here we extend their calculation to the taste non-Goldstone pions and kaons; we find that the general results are simply related to those in the taste Goldstone case. We use the notation of Ref. [@SWME:2011aa] throughout.
In Sec. \[sec:review\] we recall the staggered chiral Lagrangian and the tree-level propagators. In Sec. \[sec:decay\_consts\] we consider the definition of the decay constants, recall the various contributions through NLO in SChPT, and write down the general results in the 4+4+4 theory. Sec. \[sec:results\] contains the results for specific cases of interest in the 1+1+1 theory, and we conclude in Sec. \[sec:conclusion\]. We use the notation of Ref. [@SWME:2011aa] throughout.
\[sec:review\]Chiral Lagrangian for staggered quarks
====================================================
In this section, we write down the chiral Lagrangian for staggered quarks. The single-flavor Lagrangian was formulated by Lee and Sharpe [@Lee:1999zxa] and generalized to multiple flavors by Aubin and Bernard [@Aubin:2003mg]. Here, we consider the 4+4+4 theory, in which there are three flavors and four tastes per flavor. The exponential parameterization of the PGB fields is a $12 \times 12$ unitary matrix, $$\begin{aligned}
\Sigma=e^{i\phi/f}\in \text{U(12)},\end{aligned}$$ where the PGB fields are $$\begin{aligned}
\phi&=\sum_a \phi^a\otimes T^a,\\
\phi^a&={
\begin{pmatrix}
U_a & \pi^+_a & K^+_a \\
\pi^-_a & D_a & K^0_a \\
K^-_a & \bar K^0_a & S_a
\end{pmatrix}},\\
T^a&\in \{\xi_5,\ i\xi_{\mu5},\ i\xi_{\mu\nu}(\mu<\nu),\ \xi_\mu, \xi_I\}.
\label{eq:T^a}\end{aligned}$$ Here $a$ runs over the 16 PGB tastes, and the $T^a$ are $4\times4$ generators of $U(4)_T$; $\xi_I$ is the identity matrix. Under a chiral transformation, $\Sigma$ transforms as $$\begin{aligned}
\text{SU}(12)_L\times \text{SU}(12)_R:\ \Sigma\rightarrow L\Sigma R^\dagger \end{aligned}$$ where $L,\ R\in \text{SU}(12)_{L,R}$.
In the standard power counting, $${{\cal O}}(p^2/\Lambda_\chi^2)
\approx {{\cal O}}(m_q/\Lambda_\chi)
\approx {{\cal O}}(a^2\Lambda_\chi^2)\,.
\label{eq:count}$$ The order of a Lagrangian operator is defined as the sum of $n_{p^2}$, $n_m$ and $n_{a^2}$, which are the number of derivative pairs, powers of (light) quark masses, and powers of the squared lattice spacing, respectively, in the operator. At leading order, the Lagrangian operators fall into three classes: $(n_{p^2}, n_m, n_{a^2}) = (1,0,0)$, $(0,1,0)$ and $(0,0,1)$, and we have $$\begin{aligned}
\label{F3LSLag}
{\mathscr{L}}_\mathrm{LO} =
&\frac{f^2}{8} {\textrm{Tr}}(\partial_{\mu}\Sigma \partial_{\mu}\Sigma^{\dagger}) -
\frac{1}{4}\mu f^2 {\textrm{Tr}}(M\Sigma+M\Sigma^{\dagger}) \nonumber\\
&+ \frac{2m_0^2}{3}(U_I + D_I + S_I)^2 + a^2 ({\mathscr{U+U^\prime}})
\,,\end{aligned}$$ where $f$ is the decay constant at leading order (LO), $\mu$ is the condensate parameter, and $M$ is the mass matrix, $$M=
\begin{pmatrix}
m_u & 0 & 0 \\
0 & m_d & 0 \\
0 & 0 & m_s
\end{pmatrix}
\otimes \xi_I.$$ The term multiplied by $m_0^2$ is the anomaly contribution [@Sharpe:2001fh], and the potentials ${\mathscr{U}}$ and ${\mathscr{U^\prime}}$ are the taste symmetry breaking potentials of Ref. [@Aubin:2003mg].
At NLO, there are six classes of operators satisfying $n_{p^2} + n_m + n_{a^2} = 2$, but only two classes contribute to the decay constants: $(n_{p^2}, n_m, n_{a^2}) = $ $(1,1,0)$ and $(1,0,1)$. Contributing operators in the former are Gasser-Leutwyler terms [@Gasser:1984gg], $$\begin{aligned}
\label{eq:GLterms}
{\mathscr{L}}_\mathrm{GL}
=& L_4\mathrm{Tr}(\partial_\mu\Sigma^\dagger\partial_\mu\Sigma)
\mathrm{Tr}(\chi^\dagger\Sigma+\chi\Sigma^\dagger) \nonumber \\
& \quad + L_5\mathrm{Tr}(\partial_\mu\Sigma^\dagger\partial_\mu\Sigma
(\chi^\dagger\Sigma+\Sigma^\dagger\chi))
\,,\end{aligned}$$ where $\chi = 2\mu M$, and contributing operators in the latter are ${\mathscr{O}}(p^2 a^2)$ terms enumerated by Sharpe and Van de Water [@Sharpe:2004is].
\[sec:decay\_consts\]Decay constants of flavor-charged pseudo-goldstone bosons
==============================================================================
For a flavor-charged PGB with taste $t$, $P^+_t$, the decay constant $f_{P_t^+}$ is defined by the matrix element of the axial current, $j_{\mu 5, t}^{P^+}$, between the single-particle state and the vacuum: $$\label{eq:def_decay_const}
\langle 0 | j_{\mu 5, t}^{P^+} | P_t^+(p) \rangle = -i f_{P_t^+} p_\mu.$$ From the LO Lagrangian, the LO axial current is $$\label{eq:axial_curr}
j_{\mu 5, t}^{P^+}
= -i\frac{f^2}{8} {\textrm{Tr}}\left[ T^{t(3)} {\mathscr{P}}^{P^+}
(\partial_\mu \Sigma \Sigma^\dag + \Sigma^\dag \partial_\mu \Sigma)
\right],$$ where $T^{a(3)} \equiv I_3 \otimes T^a$, $I_3$is the identity matrix in flavor space, and ${\mathscr{P}}^{P^+}$ is a projection operator that chooses the $P^+$ from the $\Sigma$ field. For example, for $\pi^+$ it is ${\mathscr{P}}^{\pi^+}_{ij} = \delta_{i1}\delta_{j2}$. In general, ${\mathscr{P}}^{P^+}_{ij} = \delta_{ix}\delta_{jy}$, where $x$ and $y$ are the light quarks in $P^+$. For flavor-charged states, $x\neq y$, by definition. Note that ${\mathscr{P}}^{P^+}$ and $T^{a(3)}$ commute with each other.
Expanding the exponentials $\Sigma=e^{i\phi/f}$ in the LO current gives $$\begin{aligned}
&\partial_\mu \Sigma \Sigma^\dag + \Sigma^\dag \partial_\mu \Sigma
= \frac{2i}{f} \partial_\mu \phi \nonumber \\
& \qquad - \frac{i}{3f^3}
\left(
\partial_\mu \phi \phi^2 - 2\phi \partial_\mu \phi \phi + \phi^2 \partial_\mu \phi
\right)
+ \cdots.\end{aligned}$$ The ${\mathscr{O}}(\phi)$ term of the axial current gives the LO term of the decay constants, $f$, and NLO corrections from the wavefunction renormalization. The wavefunction renormalization consists of NLO analytic terms and one-loop chiral logarithms at NLO; we denote the former by $\delta f^{\textrm{anal},Z}_{P^+_t}$ and the latter by $\delta f^Z_{P^+_t}$. The ${\mathscr{O}}(\phi^3)$ term of the axial current also gives one-loop chiral logarithms at NLO, $\delta
f^\textrm{current}_{P^+_t}$. Figs. \[fig:waveftn\_crxn\] and \[fig:current\_crxn\] show the one-loop corrections to the decay constant. In addition, there is an analytic contribution to the decay constants from the NLO current. We denote the total of the NLO analytic terms by $\delta
f^\textrm{anal}_{P^+_t}$, which consists of $\delta
f^{\textrm{anal},Z}_{P^+_t}$ and analytic terms from the NLO current. Combining $\delta f^\textrm{anal}_{P^+_t}$ with the one-loop corrections, we write the decay constants up to NLO: $$f_{P_t^+} = f\left[ 1 + \frac{1}{16\pi^2 f^2}
\left(
\delta f^Z_{P^+_t} + \delta f^\textrm{current}_{P^+_t} \right)
+ \delta f^\textrm{anal}_{P^+_t}
\right]\label{eq:ftot}.$$ In this section we outline the calculation of $\delta f^Z_{P^+_t}$, $\delta f^\textrm{current}_{P^+_t}$, and $\delta f^\textrm{anal}_{P^+_t}$ and present results for the 4+4+4 theory.
\[subsec:delta\_f\_z\]Wavefunction renormalization correction
-------------------------------------------------------------
At ${\mathscr{O}}(\phi)$ the axial current, Eq. , is $$j_{\mu 5, t}^{P^+, \phi}
= f \left( \partial_\mu \phi_{yx}^t \right),$$ where we used $\tau_{ta} = 4\delta_{ta}$, ${\mathscr{P}}^{P^+}_{ij} = \delta_{ix}\delta_{jy}$, and performed the trace over taste indices. Here $\tau_{abcd\cdots}$ is defined by $$\tau_{abcd\cdots} \equiv {\textrm{Tr}}( T^a T^b T^c T^d \cdots).$$ The contributions of the ${\mathscr{O}}(\phi)$ current to the decay constants are defined by the matrix element $$\begin{aligned}
\langle 0 | j_{\mu 5, t}^{P^+, \phi} | P_t^+(p) \rangle
&= f (-ip_\mu) \langle 0 | \phi_{yx}^t | P_t^+(p) \rangle \\
&= f (-ip_\mu) \sqrt{Z_{P_t^+}},\end{aligned}$$ where $Z_{P_t^+} = 1 + \delta Z_{P_t^+}$ is the wavefunction renormalization constant of the $\phi_{xy}^t$ field. At NLO the wavefunction renormalization corrections are $$\frac{1}{16\pi^2 f^2}\delta f^Z_{P_t^+} + \delta f^{\textrm{anal},Z}_{P_t^+}
=\frac{1}{2}\delta Z_{P_t^+}
= - \frac{1}{2} \frac{d\Sigma}{dp^2} \bigg\vert_{p^2 = - m^2_{P_t^+}},$$ where $\Sigma$ is the self-energy of $P_t^+$. Using the self-energy from Ref. [@SWME:2011aa], we find the one-loop corrections $$\begin{aligned}
\label{eq:delta_f_z}
\delta f
&^Z_{P_t^+}
= \frac{1}{24} \sum_a
\Bigg[
\sum_{Q} l(Q_a) \nonumber \\
& + 16\pi^2 \int \frac{d^4 q}{(2\pi)^4}
\left(
D_{xx}^a + D_{yy}^a - 2\theta^{at} D_{xy}^a
\right)
\Bigg],\end{aligned}$$ where $Q$ runs over the six flavor combinations $xi$ and $yi$ for $i\in\{u,d,s\}$, $a$ runs over the 16 PGB tastes in the $\mathbf{15}$ and $\mathbf{1}$ of $\text{SU}(4)_T$, $Q_a$ is the squared tree-level pseudoscalar meson mass with flavor $Q$ and taste $a$, and $\theta^{ab} \equiv \frac{1}{4}\tau_{abab}$. In Eq. (\[eq:delta\_f\_z\]), $l(Q_a)$ and $D^a_{ij}$ are chiral logarithms and the disconnected piece of the tree-level propagator, respectively [@Aubin:2003mg; @SWME:2011aa]: $$\begin{aligned}
l(X)\equiv X\Bigl(\ln X/\Lambda^2 + \delta_1(\sqrt{X}L)\Bigr),\end{aligned}$$ where $\delta_1(\sqrt{X}L)$ is the finite volume correction of Ref. [@Bernard:2001yj], and $$\begin{aligned}
D^a_{ij}&=-\frac{\delta_a}{(q^2+I_a)(q^2+J_a)}\nonumber \\
&\times\frac{(q^2+U_a)(q^2+D_a)(q^2+S_a)}{(q^2+\pi^0_a)(q^2+\eta_a)(q^2+\eta^\prime_a)}.\label{D_piece}\end{aligned}$$ Here the names of mesons denote the squares of their tree-level masses, and $$\begin{aligned}
\delta_I = 4m_0^2/3,\quad&\delta_{\mu\nu} = 0,\quad\delta_5 = 0 \\
\delta_\mu = a^2\delta_V^\prime,&\quad\delta_{\mu5} = a^2\delta_A^\prime.\label{hpV_hpA}\end{aligned}$$ For $X\in\{I,J,U,D,S\}$, $$\begin{aligned}
X_a\equiv m_{X_a}^2 = 2\mu m_x + a^2\Delta_a,\label{eq:fntm}\end{aligned}$$ where $m_x$ is the mass of the quark of flavor $x\in\{i,j,u,d,s\}$, while for $X\in\{\pi^0,\eta,\eta^\prime\}$, the squares of the tree-level meson masses are the eigenvalues of the matrix $$\label{444_mass_matrix}
\begin{pmatrix}
U_a +\delta_a & \delta_a & \delta_a \\
\delta_a & D_a +\delta_a & \delta_a \\
\delta_a & \delta_a & S_a +\delta_a
\end{pmatrix}.$$ The squared tree-level mass of a flavor-charged meson ($\pi^\pm,\ K^\pm, K^0, \bar{K}^0$) is $$\begin{aligned}
P_t^+\equiv\frac{1}{2}(X_t+Y_t)=\mu(m_x+m_y)+a^2\Delta_t,\label{eq:tm}\end{aligned}$$ where $X\ne Y\in\{U,D,S\}$ and $x\ne y\in\{u,d,s\}$. The hairpin couplings $\delta_{V,A}^\prime$ and taste splittings $\Delta_a$ are combinations of the couplings of the LO Lagrangian [@Aubin:2003mg].
We defer discussing the analytic corrections $\delta f^{\textrm{anal},Z}_{P_t^+}$ to Sec. \[subsec:analytic\_cont\].
\[subsec:delta\_f\_current\]Current correction
----------------------------------------------
The ${\mathscr{O}}(\phi^3)$ terms of the axial current are $$\begin{aligned}
\label{eq:axial_curr_phi3}
j_{\mu 5, t}^{P^+, \phi^3}
=& - \frac{1}{24f} \tau_{tabc} \Big(
\partial_\mu \phi^a_{yk} \phi^b_{kl} \phi^c_{lx} \nonumber \\
& \quad - 2 \phi^a_{yk} \partial_\mu \phi^b_{kl} \phi^c_{lx}
+ \phi^a_{yk} \phi^b_{kl} \partial_\mu \phi^c_{lx}
\Big).\end{aligned}$$ In the calculation of the matrix element defined in Eq. , each term of Eq. contributes only one contraction because the derivatively coupled fields in the current must contract with the external field to obtain a nonzero result. For example, the first term gives $$\partial_\mu \phi^a_{yk} \begC1{\phi^b_{kl}} \endC1{\phi^c_{lx}}
\quad \rightarrow \quad -ip_\mu \delta^{ta}\delta^{bc} K_{xl,lx}^b,$$ where [@Sharpe:2000bc; @Aubin:2003mg; @SWME:2011aa] $$\begin{aligned}
\label{Kintdef}
K_{ij,kl}^a & \equiv \int \frac{d^4q}{(2\pi)^4}\langle\phi_{ij}^a\phi_{kl}^a\rangle
\qquad \text{(no sum)},\\
\langle\phi^a_{ij}\phi^b_{kl}\rangle &=\delta^{ab}\left(\delta_{il}\delta_{jk}\frac{1}{q^2+\frac{1}{2}(I_a+J_a)}+\delta_{ij}\delta_{kl}D^a_{il}\right).\label{prop}\end{aligned}$$
Collecting the three contributions from Eq. , we find $$\begin{aligned}
i\frac{p_\mu}{6f} \sum_{a}
\left[
\sum_{j} \left( K^a_{xj,jx} + K^a_{yj,jy} \right)
- 2\theta^{at} K^a_{xx,yy}
\right],\end{aligned}$$ where $j$ runs over $\{u,d,s\}$. Performing the integrals over the loop momenta gives the one-loop current corrections to the decay constants: $$\begin{aligned}
\label{eq:delta_f_curr}
\delta f
&_{P_t^+} ^{\textrm{current}}
\equiv -\frac{1}{6} \sum_{a}
\Bigg[
\sum_{Q} l(Q_a) \nonumber \\
& + 16\pi^2 \int \frac{d^4q}{(2\pi)^4}
\left( D_{xx}^a + D_{yy}^a - 2\theta^{at} D_{xy}^a \right)
\Bigg].\end{aligned}$$ Note that $\delta f_{P_t^+}^{\textrm{current}}$ is proportional to the one-loop wavefunction renormalization correction, $\delta f_{P_t^+}^Z$. This was shown in the taste Goldstone case in Ref. [@Aubin:2003uc].
Next-to-leading order analytic contributions \[subsec:analytic\_cont\]
----------------------------------------------------------------------
Now we consider the NLO analytic contributions to the decay constants. They come from the ${{\mathscr{O}}}(p^2m)$ Gasser-Leutwyler Lagrangian in Eq. (\[eq:GLterms\]) and the ${{\mathscr{O}}}(p^2a^2)$ Sharpe-Van de Water Lagrangian of Ref. [@Sharpe:2004is]. Both Lagrangians contribute to wavefunction renormalization and the current.
The analytic corrections to the self-energy [@SWME:2011aa] give the wavefunction renormalization correction $$\begin{aligned}
\delta f_{P_t^+}^{\textrm{anal},Z}
=& - \frac{64}{f^2} L_4 \mu (m_u + m_d + m_s) \nonumber \\
& - \frac{8}{f^2} L_5 \mu (m_x + m_y)-\frac{8}{f^2} a^2 {{\mathscr{C}}}_t,\end{aligned}$$ while the NLO current from the Gasser-Leutwyler terms gives the current correction $$\begin{aligned}
\delta f_{P_t^+}^{\textrm{current}, \textrm{GL}}
=& \frac{128}{f^2} L_4 \mu (m_u + m_d + m_s) \nonumber \\
& \qquad + \frac{16}{f^2} L_5 \mu (m_x + m_y).\end{aligned}$$ The contributions of the ${\mathscr{O}}(p^2 a^2)$ operators coming from the Sharpe-Van de Water Lagrangian in Ref. [@Sharpe:2004is] give the current correction $$\begin{aligned}
\delta f_{P_t^+}^{\textrm{current}, \textrm{SV}} = a^2{{\mathscr{C}}}^\prime_t.\end{aligned}$$ The LECs ${{\mathscr{C}}}_t$ and ${{\mathscr{C}}}^\prime_t$ are degenerate within the irreps of the lattice symmetry group. Sharpe and Van de Water observed that contributions from the ${\mathscr{O}}(p^2 a^2)$ source operators destroy would-be relations between the SO(4)-violations in the PGB masses and the (axial current) decay constants [@Sharpe:2004is].
Collecting the analytic corrections, we have (in the 4+4+4 theory) $$\begin{aligned}
\label{eq:delta_f_anal}
\delta f_{P_t^+}^{\textrm{anal}}
=& \frac{64}{f^2} L_4 \mu (m_u + m_d + m_s) \nonumber \\
& \qquad + \frac{8}{f^2} L_5 \mu (m_x + m_y)
+ a^2 {\mathscr{F}}_t,\end{aligned}$$ where the constants ${{\mathscr{F}}}_t$ subsume the constants ${{\mathscr{C}}}^{(\prime)}_t$. Examining Eqs. (\[eq:delta\_f\_z\]), (\[eq:delta\_f\_curr\]), and (\[eq:delta\_f\_anal\]), we see that the constants ${{\mathscr{F}}}_t$ (for $t\neq 5$) are the only new LECs entering the (NLO) expressions for the decay constants, in the sense that the others are present also in the taste Goldstone case.
\[sec:results\]Results
======================
To formulate the full QCD and (partially) quenched results in rooted SChPT, we employ the replica method [@Bernard:1993sv; @Damgaard:2000gh; @Bernard:2007ma]. Rooting introduces a factor of $1/4$ in front of the explicit chiral logarithms $l(Q_a)$ in Eqs. and and in the $L_4$ term in Eq. . We must also replace the eigenvalues of the mass matrix with those of the matrix obtained by sending $\delta_a\to\delta_a/4$ there. We have $$\begin{aligned}
\label{eq:delta_f_rooted}
\delta f_{P_F^+}
&= \delta f_{P_F^+}^{Z} + \delta f_{P_F^+}^{\textrm{current}}=\delta f_{P_F^+}^{\textrm{con}} + \delta f_{P_F^+}^{\textrm{disc}}, \\
\label{eq:delta_f_anal_rooted}
\delta f_{P_t^+}^{\textrm{anal}}
&= \frac{16}{f^2} L_4 \mu (m_u + m_d + m_s) \nonumber \\
& \qquad + \frac{8}{f^2} L_5 \mu (m_x + m_y)
+ a^2 {\mathscr{F}}_t,\end{aligned}$$ where $$\begin{aligned}
\label{eq:delta_f_con}
\delta f_{P_F^+}^{\textrm{con}}
&\equiv -\frac{1}{32} \sum_{Q,B} g_B ~ l(Q_B), \\
\label{eq:delta_f_disc}
\delta f_{P_F^+}^{\textrm{disc}}
&\equiv -2\pi^2 \int \frac{d^4q}{(2\pi)^4}
\Big(
D_{xx}^I + D_{yy}^I - 2 D_{xy}^I \nonumber \\
& \qquad \qquad + 4 D_{xx}^V + 4 D_{yy}^V - 2\Theta^{VF} D_{xy}^V
\nonumber \\
& \qquad \qquad + 4 D_{xx}^A + 4 D_{yy}^A - 2\Theta^{AF} D_{xy}^A
\Big)
\,.\end{aligned}$$ In Eq. , the flavor-neutral, tree-level masses ($\pi^0_a,\ \eta_a,\ \eta^\prime_a$) appearing in $D^a_{ij}$ have been replaced with the masses obtained by sending $\delta_a\to\delta_a/4$ in the flavor-neutral meson mass matrix. In Eqs. (\[eq:delta\_f\_con\]) and (\[eq:delta\_f\_disc\]), we summed over $a$ within each SO(4) irrep in Eqs. and , $B$ and $F$ represent (taste) SO(4) irreps, $$B, F \in \{ I, V, T, A, P \},$$ $t \in F$, and $$\Theta^{BF} \equiv \sum_{a \in B} \theta^{at}, \quad
g_B \equiv \sum_{a \in B} 1.\label{eq:coeff}$$ The coefficients $\Theta^{BF}$ are given in Table \[tab:coeff\].
$B \backslash F$ $P$ $A$ $T$ $V$ $I$
------------------ ------ ------ ----- ------ ----------------
$V$ $-4$ $2$ $0$ $-2$ $\phantom{-}4$
$A$ $-4$ $-2$ $0$ $2$ $4$
The loop corrections differ from those in the taste Goldstone case only in the values of the coefficients $\Theta^{BF}$.
Eq. subsumes the NLO analytic corrections in fully dynamical and partially quenched SU(3) SChPT; in the former case, $m_x\neq m_y$ are chosen from $m_u$, $m_d$, and $m_s$. In the quenched case, the $L_4$ term is dropped. To obtain the NLO analytic corrections in SU(2) SChPT, we drop terms with the heavy quark mass(es), and the LECs become heavy quark mass dependent [@Bailey:2012wb].
Below we give the one-loop contributions to the decay constants for each of these cases. In Sec. \[subsubsec:su3\_full\] and Sec. \[subsubsec:su3\_pq\], fully dynamical and partially quenched results for the 1+1+1 and 2+1 flavor cases in SU(3) chiral perturbation theory are given. The analogous results in SU(2) chiral perturbation theory are presented in Sec. \[subsec:su2\]. In Sec. \[subsubsec:su3\_q\], we write down the results in the quenched case.
\[subsec:su3\]SU(3) chiral perturbation theory
----------------------------------------------
### \[subsubsec:su3\_full\]Fully dynamical case
In Eq. $Q$ runs over six flavor combinations, $xi$ and $yi$ for $i \in \{u, d, s\}$. Setting $xy = ud,\ us,\ ds$ gives the results for the $\pi^+$, $K^+$, and $K^0$ in the fully dynamical 1+1+1 flavor case: $$\begin{aligned}
\delta f_{\pi_F^+}^{\textrm{con}}
&= -\frac{1}{32} \sum_{B} g_B \Big( l(U_B) + 2l(\pi_B^+) \nonumber \\
& \qquad \qquad + l(K_B^+) + l(D_B) + l(K_B^0) \Big), \label{eq:f_pi_con_su3_full}\end{aligned}$$ $$\begin{aligned}
\delta f_{K_F^+}^{\textrm{con}}
&= -\frac{1}{32} \sum_{B} g_B \Big( l(U_B) + l(\pi_B^+) \nonumber \\
& \qquad \qquad + 2l(K_B^+) + l(K_B^0) + l(S_B) \Big). \label{eq:f_K_con_su3_full}\end{aligned}$$ $$\begin{aligned}
\delta f_{K_F^0}^{\textrm{con}}
&= -\frac{1}{32} \sum_{B} g_B \Big( l(\pi_B^+) + l(D_B) \nonumber \\
& \qquad \qquad + 2l(K_B^0) + l(K_B^+) + l(S_B) \Big). \label{eq:f_Kn_con_su3_full}\end{aligned}$$ In the disconnected parts, Eq. , the integrals can be performed as explained in Ref. [@Aubin:2003mg]. After performing the integrals and decoupling the $\eta_I'$ by taking $m_0^2 \rightarrow \infty$ [@Sharpe:2001fh], we find $$\begin{aligned}
\delta f
&_{\pi_F^+}^{\textrm{disc}}
= \sum_{X} \Bigg[
\frac{1}{6} \Big\{
R^{DS}_{U \pi^0 \eta} (X_I) l(X_I) \nonumber \\
& \quad + R^{US}_{D \pi^0 \eta} (X_I) l(X_I)
- 2 R^S_{\pi^0 \eta} (X_I) l(X_I)
\Big\} \nonumber \\
& + \frac{1}{4} a^2 \delta_V' \Big\{
2R^{DS}_{U \pi^0 \eta \eta'} (X_V) l(X_V) \nonumber \\
& \quad + 2R^{US}_{D \pi^0 \eta \eta'} (X_V) l(X_V)
- \Theta^{VF} R^S_{\pi^0 \eta \eta'} (X_V) l(X_V)
\Big\} \nonumber \\
& + (V \rightarrow A)
\Bigg],
\label{eq:f_pi_disc_su3_full}\end{aligned}$$ $$\begin{aligned}
\delta f
&_{K_F^+}^{\textrm{disc}}
= \sum_{X} \Bigg[
\frac{1}{6} \Big\{
R^{DS}_{U \pi^0 \eta} (X_I) l(X_I) \nonumber \\
& \quad + R^{UD}_{S \pi^0 \eta} (X_I) l(X_I)
- 2 R^D_{\pi^0 \eta} (X_I) l(X_I)
\Big\} \nonumber \\
& + \frac{1}{4} a^2 \delta_V' \Big\{
2R^{DS}_{U \pi^0 \eta \eta'} (X_V) l(X_V) \nonumber \\
& \quad + 2R^{UD}_{S \pi^0 \eta \eta'} (X_V) l(X_V)
- \Theta^{VF} R^D_{\pi^0 \eta \eta'} (X_V) l(X_V)
\Big\} \nonumber \\
& + (V \rightarrow A)
\Bigg],
\label{eq:f_K_disc_su3_full}\end{aligned}$$ $$\begin{aligned}
\delta f
&_{K_F^0}^{\textrm{disc}}
= \sum_{X} \Bigg[
\frac{1}{6} \Big\{
R^{US}_{D \pi^0 \eta} (X_I) l(X_I) \nonumber \\
& \quad + R^{UD}_{S \pi^0 \eta} (X_I) l(X_I)
- 2 R^U_{\pi^0 \eta} (X_I) l(X_I)
\Big\} \nonumber \\
& + \frac{1}{4} a^2 \delta_V' \Big\{
2R^{US}_{D \pi^0 \eta \eta'} (X_V) l(X_V) \nonumber \\
& \quad + 2R^{UD}_{S \pi^0 \eta \eta'} (X_V) l(X_V)
- \Theta^{VF} R^U_{\pi^0 \eta \eta'} (X_V) l(X_V)
\Big\} \nonumber \\
& + (V \rightarrow A)
\Bigg].
\label{eq:f_Kn_disc_su3_full}\end{aligned}$$ In the sum, $X$ runs over the subscripts of the residue, $R$, where the residues are defined by $$\begin{aligned}
R_{B_1B_2\cdots B_n}^{A_1A_2\cdots A_k}(X_F)
\equiv\frac{\prod_{A_j}(A_{jF}-X_F)}{\prod_{B_{i}\ne X}(B_{iF}-X_F)},\end{aligned}$$ where $X\in\{B_1,B_2,\dots,B_n\}$ and $F\in\{V,A,I\}$ is the $SO(4)_T$ irrep.
The results in the $2+1$ flavor case are easily obtained by setting $xy = ud,\ us$ and $m_u = m_d$. Eq. gives connected contributions for the $\pi$ and $K$: $$\begin{aligned}
\label{eq:f_pi_con_su3_full_21}
\delta f_{\pi_F}^{\textrm{con}}
&= -\frac{1}{16} \sum_{B} g_B \Big( 2l(\pi_B) + l(K_B) \Big), \\
\label{eq:f_K_con_su3_full_21}
\delta f_{K_F}^{\textrm{con}}
&= -\frac{1}{32} \sum_{B} g_B \Big( 2l(\pi_B) + 3l(K_B) + l(S_B) \Big).\end{aligned}$$ Setting $xy = ud,\ us$ and $m_u = m_d$ in Eq. gives $$\begin{aligned}
\label{eq:delta_f_pi_su3_21_a}
\delta f
&_{\pi_F}^{\textrm{disc}} =
\frac{1}{4} a^2 \delta_V'(4-\Theta^{VF}) \sum_X R^S_{\pi \eta \eta'} (X_V) l(X_V)
\nonumber \\
&\qquad\qquad + (V \rightarrow A),\end{aligned}$$ and $$\begin{aligned}
\label{eq:delta_K_pi_su3_21_a}
\delta f
&_{K_F}^{\textrm{disc}} =
\frac{1}{6}\sum_X \Big\{ R^S_{\pi \eta} (X_I) l(X_I) + R^\pi_{S \eta} (X_I) l(X_I)\Big\}
\nonumber \\
& \qquad \qquad \qquad -2 l(\eta_I)
\nonumber\\
& +\frac{1}{4} a^2 \delta_V' \sum_X \Big\{
2R^S_{\pi \eta \eta'} (X_V) l(X_V) + 2R^\pi_{S \eta \eta'} (X_V) l(X_V)
\nonumber \\
& \qquad \qquad \qquad - \Theta^{VF} R_{\eta \eta'}(X_V) l(X_V) \Big \}
\nonumber \\
& + (V \rightarrow A) \,,\end{aligned}$$ where $R_{B_1 B_2}(X_F)$ is defined by $$\begin{aligned}
R_{B_1 B_2}(X_F)
&=
\begin{cases}
\dfrac{1}{B_2 - B_1} \quad (X_F = B_1) \\
\dfrac{1}{B_1 - B_2} \quad (X_F = B_2)
\end{cases}.\end{aligned}$$ Using the tree-level masses of the taste singlet channel, one finds $$\begin{aligned}
R_{\pi\eta}^{S}(\pi_I)=\frac{3}{2},& \quad
R_{\pi\eta}^S(\eta_I)=-\frac{1}{2},\\
R_{S\eta}^\pi(S_I)=3,&\quad
R_{S\eta}^\pi(\eta_I)=-2.\end{aligned}$$ They simplify the results, Eqs. and : $$\begin{aligned}
\label{eq:delta_f_pi_su3_21_b}
\delta f
&_{\pi_F}^{\textrm{disc}} =
\frac{1}{4} a^2 \delta_V' (4-\Theta^{VF}) \Bigg[
\frac{S_V - \pi_V}{(\eta_V - \pi_V)(\eta_V' - \pi_V)} l(\pi_V)
\nonumber \\
& \qquad + \frac{S_V - \eta_V}{(\pi_V - \eta_V)(\eta_V' - \eta_V)} l(\eta_V)
\nonumber \\
& \qquad + \frac{S_V - \eta_V'}{(\pi_V - \eta_V')(\eta_V - \eta_V')} l(\eta_V')
\Bigg]
\nonumber \\
& + (V \rightarrow A),\end{aligned}$$ $$\begin{aligned}
\label{eq:delta_K_pi_su3_21_b}
\delta f
&_{K_F}^{\textrm{disc}} =
\frac{1}{12} \Big[ 3 l(\pi_I) - 5 l(\eta_I) + 6 l(S_I)
- 4 l(\eta_I) \Big]
\nonumber \\
& + \frac{1}{2} a^2 \delta_V' \Bigg[
\frac{S_V - \pi_V}{(\eta_V - \pi_V)(\eta_V' - \pi_V)} l(\pi_V)
\nonumber \\
& \qquad + \frac{(\pi_V - \eta_V)^2 + (S_V - \eta_V)^2}
{(\pi_V - \eta_V)(\eta_V' - \eta_V)(S_V - \eta_V)} l(\eta_V)
\nonumber \\
& \qquad + \frac{(\pi_V - \eta_V')^2 + (S_V - \eta_V')^2}
{(\pi_V - \eta_V')(\eta_V - \eta_V')(S_V - \eta_V')} l(\eta_V')
\nonumber \\
& \qquad + \frac{\pi_V - S_V}{(\eta_V - S_V)(\eta_V' - S_V)} l(S_V)
\nonumber \\
& \qquad - \frac{1}{2} \Theta^{VF} \frac{1}{\eta_V - \eta_V'}
\Big\{ l(\eta_V') - l(\eta_V) \Big\}
\Bigg]
\nonumber \\
& + (V \rightarrow A).\end{aligned}$$
### \[subsubsec:su3\_pq\]Partially quenched case
In the partially quenched case, the valence quark masses, $m_x$ and $m_y$ are not degenerate with the sea quark masses, $m_u$, $m_d$ and $m_s$. The connected contributions to the decay constants in the partially quenched 1+1+1 flavor case are $$\begin{aligned}
\label{eq:delta_f_con_su3_pq}
\delta f_{P_F^+}^{\textrm{con}}
= -\frac{1}{32} \sum_{Q,B} g_B ~ l(Q_B).\end{aligned}$$ Performing the integrals in Eq. keeping all quark masses distinct gives the disconnected contributions for the partially quenched 1+1+1 flavor case: $$\begin{aligned}
\label{eq:delta_f_disc_su3_pq_a}
\delta f
&_{P_F^+, m_x \neq m_y}^{\mathrm{disc}} = \sum_{Z} \Bigg[
\frac{1}{6} \Big\{
D^{UDS}_{X \pi^0 \eta, X}(Z_I) l(Z_I)
\nonumber \\
&\qquad + D^{UDS}_{Y \pi^0 \eta, Y}(Z_I) l(Z_I)
- 2 R^{UDS}_{XY \pi^0 \eta}(Z_I) l(Z_I)
\Big\} \nonumber \\
&\quad + \frac{1}{4} a^2 \delta_V' \Big\{
2D^{UDS}_{X \pi^0 \eta \eta', X}(Z_V) l(Z_V)
\nonumber \\
&\qquad + 2D^{UDS}_{Y \pi^0 \eta \eta', Y}(Z_V) l(Z_V)
\nonumber \\
&\qquad - \Theta^{VF} R^{UDS}_{XY \pi^0 \eta \eta'}(Z_V) l(Z_V)
\Big\}
+ (V \rightarrow A)
\Bigg] \nonumber \\
& + \frac{1}{6} \Big\{
R^{UDS}_{X \pi^0 \eta}(X_I) \tilde{l}(X_I)
+ R^{UDS}_{Y \pi^0 \eta}(Y_I) \tilde{l}(Y_I)
\Big\} \nonumber \\
& + \frac{1}{2} a^2 \delta_V' \Big\{
R^{UDS}_{X \pi^0 \eta \eta'}(X_V) \tilde{l}(X_V)
+ R^{UDS}_{Y \pi^0 \eta \eta'}(Y_V) \tilde{l}(Y_V)
\Big\} \nonumber \\
& + (V \rightarrow A),\end{aligned}$$ where $$\begin{aligned}
D_{B_1B_2\cdots B_n,B_i}^{A_1A_2\cdots A_k}(X_F)
\equiv -\frac{\partial}{\partial B_{iF}}
R_{B_1B_2\cdots B_n}^{A_1A_2\cdots A_k}(X_F)\end{aligned}$$ and $$\begin{aligned}
\tilde{l}(X)\equiv -\Bigl(\ln X/\Lambda^2 + 1\Bigr) + \delta_3(\sqrt{X}L).\end{aligned}$$ Here $\delta_3(\sqrt{X}L)$ is the finite volume correction defined in Ref. [@Bernard:2001yj], and $X$ and $Y$ represent the squared tree-level masses of $x\bar{x}$ and $y\bar{y}$ PGBs, respectively.
For $m_x = m_y$, we find $$\begin{aligned}
\label{eq:delta_f_disc_su3_pq_b}
\delta f
&_{P_F^+, m_x = m_y}^{\mathrm{disc}} =
\frac{1}{4} a^2 \delta_V' (4-\Theta^{VF}) \Bigg[
R^{UDS}_{X \pi^0 \eta \eta'}(X_V) \tilde{l}(X_V)
\nonumber \\
& \qquad \qquad \qquad \qquad
+ \sum_Z D^{UDS}_{X \pi^0 \eta \eta', X}(Z_V) l(Z_V)
\Bigg] \nonumber \\
&+(V \rightarrow A).\end{aligned}$$ The connected contributions in the 2+1 flavor case are obtained by setting $m_u = m_d$ in Eq. . To obtain the disconnected contributions, we perform the integrals in Eq. setting $m_u = m_d$. For $m_x \neq m_y$, we find $$\begin{aligned}
\label{eq:delta_f_xney_pq_21}
\delta f
&_{P_F^+, m_x \neq m_y}^{\mathrm{disc}} = \sum_{Z} \Bigg[
\frac{1}{6} \Big\{
D^{\pi S}_{X \eta, X}(Z_I) l(Z_I)
\nonumber \\
&\qquad + D^{\pi S}_{Y \eta, Y}(Z_I) l(Z_I)
- 2 R^{\pi S}_{XY \eta}(Z_I) l(Z_I)
\Big\} \nonumber \\
&\quad + \frac{1}{4} a^2 \delta_V' \Big\{
2D^{\pi S}_{X \eta \eta', X}(Z_V) l(Z_V)
\nonumber \\
&\qquad + 2D^{\pi S}_{Y \eta \eta', Y}(Z_V) l(Z_V)
\nonumber \\
&\qquad - \Theta^{VF} R^{\pi S}_{XY \eta \eta'}(Z_V) l(Z_V)
\Big\}
+ (V \rightarrow A)
\Bigg] \nonumber \\
& + \frac{1}{6} \Big\{
R^{\pi S}_{X \eta}(X_I) \tilde{l}(X_I)
+ R^{\pi S}_{Y \eta}(Y_I) \tilde{l}(Y_I)
\Big\} \nonumber \\
& + \frac{1}{2} a^2 \delta_V' \Big\{
R^{\pi S}_{X \eta \eta'}(X_V) \tilde{l}(X_V)
+ R^{\pi S}_{Y \eta \eta'}(Y_V) \tilde{l}(Y_V)
\Big\} \nonumber \\
& + (V \rightarrow A).\end{aligned}$$ For $m_x = m_y$, we find $$\begin{aligned}
\label{eq:delta_f_xeqy_pq_21}
\delta f
&_{P_F^+, m_x = m_y}^{\mathrm{disc}} =
\frac{1}{4} a^2 \delta_V (4-\Theta^{VF}) \Bigg[
R^{\pi S}_{X \eta \eta'}(X_V) \tilde{l}(X_V)
\nonumber \\
&\qquad \qquad \qquad \qquad
+ \sum_Z D^{\pi S}_{X \eta \eta', X}(Z_V) l(Z_V)
\Bigg] \nonumber \\
&+(V \rightarrow A).\end{aligned}$$
### \[subsubsec:su3\_q\]Quenched case
In the quenched case, there is no connected contribution, Eq. . As explained in Refs.[@Bernard:1992mk; @Bernard:2001yj; @Aubin:2003mg], quenching the sea quarks in the disconnected part can be done by replacing the disconnected propagator with $$D^{a,\mathrm{quench}}_{il}
=-\frac{\delta_a^\mathrm{quench}}{(q^2+I_a)(q^2+L_a)},\label{Dquench}$$ where $$\delta_a^\mathrm{quench}=
\begin{cases}
4(m_0^2+\alpha q^2)/3 & \text{if $a=I$}\\
\delta_a & \text{if $a\neq I$.}
\end{cases}$$ Here, note that $I_a$ and $L_a$ represent the squared tree-level masses of $i\bar{i}$ and $l\bar{l}$ PGBs, respectively, while $I$ represents the taste-singlet irrep.
Replacing $D^a_{il}$ with the quenched disconnected propagator in Eq. for $m_x \neq m_y$ gives $$\begin{aligned}
\label{eq:delta_f_xney_quench}
\delta f
&_{P_F^+, m_x \neq m_y}^{\mathrm{disc}} \nonumber \\
& =
\frac{\alpha}{6} \Big\{
\frac{Y_I+X_I}{Y_I-X_I}(l(X_I)-l(Y_I)) - X_I \tilde{l}(X_I) - Y_I \tilde{l}(Y_I)
\Big\} \nonumber \\
& \qquad + \frac{m_0^2}{6} \Big\{ \tilde{l}(X_I) + \tilde{l}(Y_I)
- 2 \frac{l(X_I) - l(Y_I)}{Y_I - X_I} \Big\}
\nonumber \\
& + \frac{1}{4}a^2 \delta_V' \Big\{ 2 \tilde{l}(X_V) + 2 \tilde{l}(Y_V)
- \Theta^{VF} \frac{l(X_V) - l(Y_V)}{Y_V - X_V} \Big\}
\nonumber \\
& + (V \rightarrow A),\end{aligned}$$ and for $m_x = m_y$, $$\begin{aligned}
\label{eq:delta_f_xeqy_quench}
\delta f
&_{P_F^+, m_x = m_y}^{\mathrm{disc}} =
\frac{1}{4} a^2 \delta_V' (4 - \Theta^{VF}) \tilde{l}(X_V)
+ (V \rightarrow A).\end{aligned}$$ Quenching the sea quarks also affects the analytic terms. In the quenched version of Eq. , there is no $L_4$ term of Eq. , which is coming from the sea quarks.
\[subsec:su2\]SU(2) chiral perturbation theory
----------------------------------------------
We obtain the SU(2) SChPT results from the SU(3) SChPT results using the prescription of Ref. [@Bailey:2012wb]. SU(2) chiral perturbation theory was developed in Ref. [@Gasser:1983yg] and applied to simulation data for the taste Goldstone decay constants in Refs. [@Bazavov:2009fk; @Bazavov:2009ir; @Bazavov:2010yq]. The results of this section extend the results of Refs. [@Du:2009ih; @Bazavov:2010yq] to the taste non-Goldstone case.
### \[subsubsec:su2\_full\]Fully dynamical case
From Eqs. , , and , we obtain the connected contributions for the fully dynamical 1+1+1 flavor case ($m_u \ne m_d \ll m_s$): $$\begin{aligned}
\label{eq:f_pi_con_su2_full}
\delta f_{\pi_F^+}^{\textrm{con}}
= -\frac{1}{32} \sum_{B} g_B \Big( l(U_B) + 2l(\pi_B^+) + l(D_B) \Big),\end{aligned}$$ $$\begin{aligned}
\label{eq:f_K_con_su2_full}
\delta f_{K_F^+}^{\textrm{con}}
= -\frac{1}{32} \sum_{B} g_B \Big( l(U_B) + l(\pi_B^+) \Big).\end{aligned}$$ $$\begin{aligned}
\label{eq:f_Kn_con_su2_full}
\delta f_{K_F^0}^{\textrm{con}}
&= -\frac{1}{32} \sum_{B} g_B \Big( l(D_B) + l(\pi_B^+) \Big).\end{aligned}$$ For the disconnected contributions, we find from Eqs. , and : $$\begin{aligned}
\delta f
& _{\pi_F^{+}}^{\textrm{disc}} = \frac{1}{2}\big(l(U_I)+l(D_I)\big)-l(\pi_I^0) \nonumber \\
& + \frac{1}{4} a^2 \delta_V' \sum_X \Big\{
2 R_{U \pi^0 \eta}^D (X_V) l(X_V) \nonumber \\
& \qquad \qquad \qquad \qquad + 2 R_{D \pi^0 \eta}^U (X_V) l(X_V) \Big\}
\nonumber \\
& \qquad
+ \frac{1}{4} a^2 \delta_V' \Theta^{VF} \frac{l(\eta_V)-l(\pi_V^0)}{\eta_V-\pi_V^0} \nonumber \\
& + (V \rightarrow A),\end{aligned}$$ $$\begin{aligned}
\delta f
& _{K_F^{+}}^{\textrm{disc}} =
\frac{1}{2}l(U_I) - \frac{1}{4}l(\pi_I^0) \nonumber \\
& + \frac{1}{2} a^2 \delta_V' \sum_X R_{U \pi^0 \eta}^D (X_V) l(X_V)
+ (V \rightarrow A),\end{aligned}$$ and $$\begin{aligned}
\delta f
& _{K_F^{0}}^{\textrm{disc}} =
\frac{1}{2}l(D_I) - \frac{1}{4}l(\pi_I^0) \nonumber \\
& + \frac{1}{2} a^2 \delta_V' \sum_X R_{D \pi^0 \eta}^U (X_V) l(X_V)
+ (V \rightarrow A).\end{aligned}$$ The connected contributions in the fully dynamical 2+1 flavor case ($m_u = m_d \ll m_s$) are $$\begin{aligned}
\delta f_{\pi_F}^{\textrm{con}}
&= -\frac{1}{8} \sum_{B} g_B l(\pi_B), \\
\delta f_{K_F}^{\textrm{con}}
&= -\frac{1}{16} \sum_{B} g_B l(\pi_B).\end{aligned}$$ For the disconnected contributions in the fully dynamical 2+1 flavor case, we find $$\begin{aligned}
\delta f_{\pi_F}^{\textrm{disc}} =
& \frac{1}{2} (4-\Theta^{VF}) \Big\{ l(\pi_V) - l(\eta_V) \Big\}
+ (V \rightarrow A), \\
\label{eq:delta_f_su2_full_21}
\delta f _{K_F}^{\textrm{disc}} =
& \frac{1}{4} l(\pi_I) + l(\pi_V) - l(\eta_V)
+ (V \rightarrow A).\end{aligned}$$
### \[subsubsec:su2\_pq\]Partially quenched case
Considering $x$ and $y$ to be light quarks ($ m_s \gg m_u, m_d, m_x, m_y$), the connected contributions to the decay constants in the partially quenched 1+1+1 flavor case can be obtained by dropping terms corresponding to strange sea quark loops from Eq. . Eqs. , and give the disconnected contributions: $$\begin{aligned}
\label{eq:delta_f_xney_su2_pq}
\delta f
&_{P_F^+, m_x \neq m_y}^{\mathrm{disc}} = \sum_{Z} \Bigg[
\frac{1}{4} \Big\{
D^{UD}_{X \pi^0, X}(Z_I) l(Z_I)
\nonumber \\
&\qquad + D^{UD}_{Y \pi^0, Y}(Z_I) l(Z_I)
- 2 R^{UD}_{XY \pi^0}(Z_I) l(Z_I)
\Big\} \nonumber \\
&\quad + \frac{1}{4} a^2 \delta_V' \Big\{
2D^{UD}_{X \pi^0 \eta, X}(Z_V) l(Z_V)
\nonumber \\
&\qquad + 2D^{UD}_{Y \pi^0 \eta, Y}(Z_V) l(Z_V)
\nonumber \\
&\qquad - \Theta^{VF} R^{UD}_{XY \pi^0 \eta}(Z_V) l(Z_V)
\Big\}
+ (V \rightarrow A)
\Bigg] \nonumber \\
& + \frac{1}{4} \Big\{
R^{UD}_{X \pi^0}(X_I) \tilde{l}(X_I)
+ R^{UD}_{Y \pi^0}(Y_I) \tilde{l}(Y_I)
\Big\} \nonumber \\
& + \frac{1}{2} a^2 \delta_V' \Big\{
R^{UD}_{X \pi^0 \eta}(X_V) \tilde{l}(X_V)
+ R^{UD}_{Y \pi^0 \eta}(Y_V) \tilde{l}(Y_V)
\Big\} \nonumber \\
& + (V \rightarrow A),\end{aligned}$$ and $$\begin{aligned}
\label{eq:delta_f_xeqy_su2_pq}
\delta f
&_{P_F^+, m_x = m_y}^{\mathrm{disc}} =
\frac{1}{4} a^2 \delta_V' (4-\Theta^{VF}) \Bigg[
R^{UD}_{X \pi^0 \eta}(X_V) \tilde{l}(X_V)
\nonumber \\
&\qquad \qquad \qquad \qquad \qquad
+ \sum_Z D^{UD}_{X \pi^0 \eta, X}(Z_V) l(Z_V)
\Bigg] \nonumber \\
&+(V \rightarrow A).\end{aligned}$$ The connected contributions to the decay constants in the partially quenched 2+1 flavor case can be obtained by setting $m_u = m_d$ and decoupling the strange quark in the 1+1+1 flavor case, Eq. . From Eqs. and , we find the disconnected contributions in the 2+1 flavor case: $$\begin{aligned}
\label{eq:delta_f_xney_su2_pq_2p1}
\delta f
&_{P^+_F, m_x \neq m_y}^{\mathrm{disc}} = \sum_{Z} \Bigg[
-\frac{1}{2}
R^{\pi}_{XY}(Z_I) l(Z_I)
\nonumber \\
&\quad + \frac{1}{4} a^2 \delta_V' \Big\{
2D^{\pi}_{X \eta, X}(Z_V) l(Z_V)
\nonumber \\
&\qquad + 2D^{\pi}_{Y \eta, Y}(Z_V) l(Z_V)
\nonumber \\
&\qquad - \Theta^{VF} R^{\pi}_{XY \eta}(Z_V) l(Z_V)
\Big\}
+ (V \rightarrow A)
\Bigg] \nonumber \\
& + \frac{1}{4} \Big\{
l(X_I) + (\pi_I-X_I)\tilde{l}(X_I) \nonumber \\
& \qquad \qquad + l(Y_I) + (\pi_I-Y_I)\tilde{l}(Y_I)
\Big\} \nonumber \\
& + \frac{1}{2} a^2 \delta_V' \Big\{
R^{\pi}_{X \eta}(X_V) \tilde{l}(X_V)
+ R^{\pi}_{Y \eta}(Y_V) \tilde{l}(Y_V)
\Big\} \nonumber \\
& + (V \rightarrow A),\end{aligned}$$ and $$\begin{aligned}
\delta f
&_{P^+_F, m_x = m_y}^{\mathrm{disc}} =
\frac{1}{4} a^2 \delta_V' (4-\Theta^{VF}) \Bigg[
R^{\pi}_{X \eta}(X_V) \tilde{l}(X_V)
\nonumber \\
&\qquad \qquad \qquad \qquad \qquad
+ \sum_Z D^{\pi}_{X \eta, X}(Z_V) l(Z_V)
\Bigg] \nonumber \\
&+(V \rightarrow A).\end{aligned}$$ Considering $x$ to be a light quark and $y$ to be a heavy quark ($m_s, m_y \gg m_u, m_d, m_x$), the connected contributions to the decay constants can be obtained by dropping terms from Eq. corresponding to strange sea quarks and $y$ valence quarks circulating in loops; [*i.e.*]{}, only the $xu$ and $xd$ terms survive in the sum over $Q$. From Eqs. , , and (or alternatively, Eqs. and ), we find the disconnected contribution for the partially quenched 1+1+1 flavor case, $$\begin{aligned}
\label{eq:delta_f_xney_su2_pq_y}
\delta f
&_{P_F^+}^{\mathrm{disc}} = \frac{1}{4} \sum_{Z} \Bigg[
D^{UD}_{X \pi^0, X}(Z_I) l(Z_I)
\nonumber \\
&\quad + 2 a^2 \delta_V'
D^{UD}_{X \pi^0 \eta, X}(Z_V) l(Z_V)
+ (V \rightarrow A)
\Bigg] \nonumber \\
& + \frac{1}{4}
R^{UD}_{X \pi^0}(X_I) \tilde{l}(X_I)
\nonumber \\
& + \frac{1}{2} a^2 \delta_V'
R^{UD}_{X \pi^0 \eta}(X_V) \tilde{l}(X_V)
+ (V \rightarrow A).\end{aligned}$$ For the 2+1 flavor case, we find $$\begin{aligned}
\label{eq:delta_f_xney_su2_pq_2+1_y}
\delta f
&_{P_F^+}^{\mathrm{disc}} = \sum_{Z} \Bigg[
\frac{1}{2} a^2 \delta_V'
D^{\pi}_{X \eta, X}(Z_V) l(Z_V)
+ (V \rightarrow A)
\Bigg] \nonumber \\
& + \frac{1}{4} \Big\{
l(X_I) + (\pi_I-X_I)\tilde{l}(X_I)
\Big\} \nonumber \\
& + \frac{1}{2} a^2 \delta_V'
R^{\pi}_{X \eta}(X_V) \tilde{l}(X_V)
+ (V \rightarrow A).\end{aligned}$$
\[sec:conclusion\]Conclusion
============================
Our results for the decay constants are given compactly by Eq. with Eqs. through ; they reduce to those of Ref. [@Aubin:2003uc] in the taste Goldstone sector. The only new LECs are those parametrizing the analytic corrections proportional to $a^2$; the SO(4)-violating contributions are independent of those in the masses. As shown in Table \[tab:coeff\], the factors $\Theta^{BF}$ multiplying the disconnected pieces of the propagators $D_{xy}^{V,A}$ differ from the coefficients in the taste Goldstone case, but no new LECs arise in the loop diagrams. In SU(2) chiral perturbation theory with a heavy valence quark, the chiral logarithms are the same in all taste channels; only the analytic ${{\mathscr{O}}}(a^2)$ corrections differ.
Results for special cases of interest can be obtained by expanding the disconnected pieces of the propagators in Eq. . For the fully dynamical case with three non-degenerate quarks, the loop corrections in the SU(3) chiral theory are in Eqs. -. Results in the isospin limit are in Eqs. -. For the partially quenched case with three non-degenerate sea quarks, loop corrections in the SU(3) chiral theory are in Eqs. -. Results in the isospin limit are in Eqs. -. For the quenched case the results are in Eqs. -. Results in SU(2) chiral perturbation theory are in Eqs. - and Eqs. -. These results can be used to improve determinations of the decay constants, quark masses, and the Gasser-Leutwyler LECs by analyzing lattice data from taste non-Goldstone channels.
W. Lee is supported by the Creative Research Initiatives program (2012-0000241) of the NRF grant funded by the Korean government (MEST). W. Lee acknowledges support from the KISTI supercomputing center through the strategic support program \[No. KSC-2011-G2-06\].
|
---
abstract: 'We present a polynomial-time algorithm that, given two independent sets in a claw-free graph $G$, decides whether one can be transformed into the other by a sequence of elementary steps. Each elementary step is to remove a vertex $v$ from the current independent set $S$ and to add a new vertex $w$ (not in $S$) such that the result is again an . We also consider the more restricted model where $v$ and $w$ have to be adjacent.'
author:
- Paul Bonsma
- Marcin Kamiński
- Marcin Wrochna
title: 'Reconfiguring Independent Sets in Claw-Free Graphs'
---
Introduction
============
[**Reconfiguration problems**]{}. To obtain a reconfiguration version of an algorithmic problem, one defines a [*reconfiguration rule*]{} – a (symmetric) [adjacency relation]{} between solutions of the problem, describing small transformations one is allowed to make. The main focus is on studying whether one given solution can be transformed into another by a sequence of such small steps. We call it a [*reachability problem*]{}. For example, in a well-studied reconfiguration version of vertex coloring [@BonsmaC09; @CerecedaHJ08; @CerecedaHJ09; @bonamy2011diameter; @bonamy2013recoloring; @bonamy2014reconfiguration; @ito2012reconfiguration], we are given two $k$-colorings of the vertices of a graph and we should decide whether one can be transformed into the other by recoloring one vertex at a time so that all intermediate solutions are also proper $k$-colorings.
A useful way to look at reconfiguration problems is through the concept of the *solution graph*. Given a problem instance, the vertices of the [solution graph]{} are all solutions to the instance, and the reconfiguration rule defines its edges. Clearly, one solution can be transformed into another if they belong to the same connected component of the solution graph. Other well-studied questions in the context of reconfiguration are as follows: can one efficiently decide (for every instance) whether the solution graph is connected? Can one efficiently find shortest paths between two solutions? Common non-algorithmic results are giving upper and lower bounds on the possible diameter of components of the solution graph, in terms of the instance size, or studying how much the solution space needs to be increased in order to guarantee connectivity.
Reconfiguration is a natural setting for real-life problems in which solutions evolve over time and an interesting theoretical framework that has been gradually attracting more attention. The theoretical interest is based on the fact that reconfiguration problems provide a new perspective and offer a deeper understanding of the solution space as well as a potential to develop heuristics to navigate that space.
Reconfiguration paradigm has been recently applied to a number of algorithmic problems: vertex coloring [@BonsmaC09; @BonsmaCHJ07; @CerecedaHJ09; @CerecedaHJ08], list-edge coloring [@ItoKD09], clique, set cover, integer programming, matching, spanning tree, matroid bases [@ItoDHPSUU08], block puzzles [@HearnD05], satisfiability [@GopalanKMP09], independent set [@HearnD05; @ItoDHPSUU08; @KaminskiMM12], shortest paths [@Bonsma12; @Bonsma13; @KaminskiMM11], and dominating set [@SuzukiMN14]; recently also in the setting of parameterized complexity [@MouawadNRSS13]. A recent survey [@Heuvel13] gives a good introduction to this area of research.
[**Reconfiguration of independent sets**]{}. The topic of this paper is reconfiguration of independent sets. An [*independent set*]{} in a graph is a set of pairwise nonadjacent vertices. We will view the elements of an independent set as tokens placed on vertices. Three different reconfiguration rules have been studied in the literature: token sliding (TS), token jumping (TJ), and token addition/removal (TAR). The reconfiguration rule in the TS model allows to slide a token along an edge. The reconfiguration rule in the TJ model allows to remove a token from a vertex and place it on another unoccupied vertex. In the TAR model, the reconfiguration rule allows to either add or remove a token as long as at least $k$ tokens remain on the graph at any point, for a given integer $k$. In all three cases, the reconfiguration rule may of course only be applied if it maintains an . A sequence of moves following these rules is called a [*, , or $k$-*]{}, respectively. Note that the TS model is more restricted than the TJ model, in the sense that any is also a . Kamiński et al. [@KaminskiMM12] showed that the TAR model generalizes the TJ model, in the sense that there exists a between two solutions $I$ and $J$ with $|I|=|J|$ if and only if there exists a $k$- between them, with $k=|I|-1$. TS seems to have been introduced by Hearn and Demaine [@HearnD05], TAR was introduced by Ito et al. [@ItoDHPSUU11] and TJ by Kamiński et al. [@IWOCA2010].
In all three models, the corresponding reachability problems are [PSPACE]{}-complete in general graphs [@ItoDHPSUU11] and even in perfect graphs [@KaminskiMM12] or in planar graphs of maximum degree $3$ [@HearnD05] (see also [@BonsmaC09]). We remark that in [@HearnD05], only the TS-model was explicitly considered, but since only maximum s are used, this implies the result for the TJ model (see Proposition \[propo:max:TSequivTJ\] below) and for the TAR model (using the aforementioned result from [@KaminskiMM12]).
[**Claw-free graphs**]{}. A *claw* is the tree with four vertices and three leaves. A graph is *claw-free* if it does not contain a claw as an induced subgraph. A claw is not a line graph of any graph and thus the class of claw-free graphs generalizes the class of line graphs. The structure of claw-free graphs is not simple but has been recently described by Chudnovsky and Seymour in the form of a decomposition theorem [@ChudnovskyS05].
There is a natural one-to-one correspondence between matchings in a graph and independent sets in its line graph. In particular, a maximum matching in a graph corresponds to a maximum independent set in its line graph. Hence, Edmonds’ maximum matching algorithm [@Edmonds1965a] gives a polynomial-time algorithm for finding maximum independent sets in line graphs. This results has been extended to claw-free graphs independently by Minty [@Minty80] and Sbihi [@sbihi1980algorithme]. Both algorithms work for the unweighted case, while the algorithm of Minty, with a correction proposed by Nakamura and Tamura in [@nakamura2001revision], applies to weighted graphs (see also [@schrijver2003combinatorial Section 69]).
A *fork* is the graph obtained from the claw by subdividing one edge. Every claw-free graph is also fork-free. Milanič and Lozin gave a polynomial-time algorithm for maximum weighted independent set in fork-free graphs [@LozinM08]. This generalizes all aforementioned results for claw-free graphs.
[**Our results**]{}. In this paper, we study the reachability problem for independent set reconfiguration, using the TS and TJ model. Our main result is that these problems can be solved in polynomial time for the case of claw-free graphs. Along the way, we prove some results that are interesting in their own right. For instance, we show that for connected claw-free graphs, the existence of a implies the existence of a between the same pair of solutions. This implies that for connected claw-free and even-hole-free graphs, the solution graph is always connected, answering an open question posed in [@KaminskiMM12].
Since claw-free graphs generalize line graphs, our results generalize the result by Ito et al. [@ItoDHPSUU11] on matching reconfiguration. Since a vertex set $I$ of a graph $G$ is an if and only if $V(G){\backslash}I$ is a vertex cover, our results also apply to the recently studied vertex cover reconfiguration problem [@MouawadNRSS13]. The new techniques we introduce can be seen as an extension of the techniques introduced for finding maximum s in claw-free graphs, and we expect them to be useful for addressing similar reconfiguration questions, such as efficiently deciding whether the solution graph is connected.
Some proof details are omitted. Statements for which further details can be found in the appendix are marked with a star.
Preliminaries {#sec:Preliminaries}
=============
For graph theoretic terminology not defined here, we refer to [@Diestel]. For a graph $G$ and vertex set $S\subseteq V(G)$, we denote the subgraph induced by $S$ by $G[S]$, and denote $G-S=G[V{\backslash}S]$. The set of neighbors of a vertex $v\in V(G)$ is denoted by $N(v)$, and the closed neighborhood of $v$ is $N[v]=N(v)\cup \{v\}$. A [*walk*]{} from $v_0$ to $v_k$ of length $k$ is a sequence of vertices $v_0,v_1,\ldots,v_k$ such that $v_iv_{i+1}\in E(G)$ for all $i\in \{0,\ldots,k-1\}$. It is a [*path*]{} if all of its vertices are distinct, and a [*cycle*]{} if $k\ge 3$, $v_0=v_k$ and $v_0,\ldots,v_{k-1}$ is a path. We use $V(C)$ to denote the vertex set of a path or cycle, viewed as a subgraph of $G$. A path or graph is called [*trivial*]{} if it contains only one vertex. Edges of a directed graph or [*digraph*]{} $D$ are called [*arcs*]{}, and are denoted by the ordered tuple $(u,v)$. A [*directed path*]{} in $D$ is a sequence of distinct vertices $v_0,\ldots,v_k$ such that for all $i\in\{0,\ldots,k-1\}$, $(v_i,v_{i+1})$ is an arc of $D$.
We denote the distance of two vertices $u,v\in V(G)$ by $\dist_G(u,v)$. By $\diam(G)$ we denote the [*diameter*]{} of a connected graph $G$, defined as $\max_{u,v\in V(G)} \dist_G(u,v)$. For a vertex set $S$ of a graph $G$ and integer $i\in \mathbb{N}$, we denote $N_i(S)=\{v\in V(G){\backslash}S : |N(v)\cap S|=i\}$.
For a graph $G$, by $\TS_k(G)$ we denote the graph that has as its vertex the set of all independent sets of $G$ of size $k$, where two independent sets $I$ and $J$ are adjacent if there is an edge $uv\in E(G)$ with $I{\backslash}J=\{u\}$ and $J{\backslash}I=\{v\}$. We say that $J$ can be obtained from $I$ by [*sliding a token from $u$ to $v$*]{}, or by the [*move $u\to v$*]{} for short. A walk in $\TS_k(G)$ from $I$ to $J$ is called a [* from $I$ to $J$*]{}. We write $I\tsr J$ to indicate that there is a from $I$ to $J$.
Analogously, by $\TJ_k(G)$ we denote the graph that has as its vertex set the set of all independent sets of $G$ of size $k$, where two independent sets $I$ and $J$ are adjacent if there is a vertex pair $u,v\in V(G)$ with $I{\backslash}J=\{u\}$ and $J{\backslash}I=\{v\}$. We say that $J$ can be obtained from $I$ by [*jumping a token from $u$ to $v$*]{}. A walk in $\TS_k(G)$ from $I$ to $J$ is called a [* from $I$ to $J$*]{}. We write $I\tjr J$ to indicate that there exists a from $I$ to $J$. Note that $\TS_k(G)$ is a spanning subgraph of $\TJ_k(G)$.
The [*reachability problem*]{} for token sliding (resp. token jumping) has as input a graph $G$ and two independent sets $I$ and $J$ of $G$ with $|I|=|J|$, and asks whether $I\tsr J$ (resp. $I\tjr J$). These problems are called and , respectively.
If $H$ is a claw with vertex set $\{u,v,w,x\}$ such that $N(u)=\{v,w,x\}$, then $H$ is called a [*$u$-claw with leaves $v,w,x$.*]{} Sets $I{\backslash}\{v\}$ and $I\cup\{v\}$ are denoted by $I-v$ and $I+v$ respectively. The symmetric difference of two sets $I$ and $J$ is denoted by $I\Delta J=(I{\backslash}J)\cup (J{\backslash}I)$. The following observation is used implicitly in many proofs: if $I$ and $J$ are independent sets in a claw-free graph $G$, then every component of $G[I\Delta J]$ is a path or an even length cycle.
By $\alpha(G)$ we denote the size of the largest independent set of $G$. An independent set $I$ is called [*maximum*]{} if $|I|=\alpha(G)$. A vertex set $S\subseteq V(G)$ is a [*dominating set*]{} if $N[v]\cap S\not=\emptyset$ for all $v\in V(G)$. Observe that a maximum independent set is a dominating set, thus the only possible token jumps from it are between adjacent vertices, and hence all are token slides:
\[propo:max:TSequivTJ\] Let $I$ and $J$ be maximum independent sets in a graph $G$. Then, $TS_k(G) = TJ_k(G)$. In particular, $I \tsr J$ if and only if $I\tjr J$.
The Equivalence of Sliding and Jumping {#sec:TSvsTJ}
======================================
In our main result (Theorem \[thm:connected\_reachability\]), we will consider equal size s $I$ and $J$ of a claw-free graph $G$, and show that in polynomial time, it can be verified whether $I\tsr J$ and whether $I\tjr J$. In this section, we show that if $G$ is connected and $G[I\Delta J]$ contains no cycles, then $I\tsr J$. From this, we will subsequently conclude that for connected claw-free graphs $I\tsr J$ holds if and only if $I\tjr J$, even in the case of nonmaximum independent sets.
\[lem:NoBadCyclesSliding\] Let $I$ and $J$ be independent sets in a connected claw-free graph $G$ with $|I|=|J|$. If $G[I\Delta J]$ contains no cycles, then $I\tsr J$.
We show that $I$ or $J$ can be modified using token slides such that the two resulting s are closer to each other in the sense that either $|I\setminus J|$ is smaller, or it is unchanged and the minimum distance between vertices $u,v$ with $u\in I\setminus J$ and $v\in J\setminus I$ is smaller. The claim follows by induction. (See the appendix for an induction statement with a bound on the length of the reconfiguration sequence.)
Suppose first that $G[I\Delta J]$ contains at least one nontrivial component $C$. Since it is not a cycle by assumption, it must be a path. Choose an end vertex $u$ of this path, and let $v$ be its unique neighbor on the path. If $u\in J$ then $N(u)\cap I=\{v\}$, so we can obtain a new $I'=I+u-v$ from $I$ using a single token slide. The new set $I'$ is closer to $J$ in the sense that $|I'{\backslash}J|<|I{\backslash}J|$, so we may use induction to conclude that $I'\tsr J$, and thus $I\tsr J$. On the other hand, if $u\in I$ then we can obtain a new $J'=J-v+u$ from $J$, and conclude the proof similarly by applying the induction assumption to $J'$ and $I$.
In the remaining case, we may assume that $G[I\Delta J]$ consists only of isolated vertices. Choose $u\in I{\backslash}J$ and $v\in J{\backslash}I$, such that the distance $d:=\dist_G(u,v)$ between these vertices is minimized. Starting with $I$, we intend to slide the token on $u$ to $v$, to obtain an $I'=I-u+v$ that is closer to $J$. To this end, we choose a shortest path $P=v_0,\ldots,v_d$ in $G$ from $v_0=u$ to $v_d=v$. If the token can be moved along this path while maintaining an throughout, then $I\tsr I'$, and the proof follows by induction as before.
So now suppose that this cannot be done, that is, at least one of the vertices on $P$ is equal to or adjacent to a vertex in $I-u$. In that case, we choose $i$ maximum such that $N(v_i)\cap I\not=\emptyset$. Using some simple observations (including the fact that $G$ is claw-free), one can now show that $N(v_i)\cap I$ consists of a single vertex $x$. By choice of $v_i$, starting with $I$, the token on $x$ can be moved along the path $x,v_i,v_{i+1},\ldots,v_d$ while maintaining an throughout. This yields an $I''=I-x+v$, with $I\tsr I''$. It can also easily be shown that $\dist_G(u,x)<\dist_G(u,v)$ and $\dist_G(x,v)<\dist_G(u,v)$. So considering the choice of $u$ and $v$, it follows that $x\in I\cap J$, and thus $|I''{\backslash}J|=|I{\backslash}J|$. Since now the pair $u\in I''{\backslash}J$ and $x\in J{\backslash}I''$ has a smaller distance $\dist_G(u,x)<\dist_G(u,v)=d$, we may assume by induction that $I''\tsr J$, and thus $I\tsr J$.
\[cor:connected:TSequivTJ\] Let $I$ and $J$ be independent sets in a connected claw-free graph $G$. Then $I \tsr J$ if and only if $I\tjr J$.
Clearly, a from $I$ to $J$ is also a . For the nontrivial direction of the proof, it suffices to show that any token jump can be replaced by a sequence of token slides. Let $J$ be obtained from $I$ by jumping a token from $u$ to $v$. Then $G[I\Delta J]$ contains only two vertices and therefore no cycles. Then Lemma \[lem:NoBadCyclesSliding\] shows that $I\tsr J$.
We now consider implications of the above corollary for graphs that are claw- and even-hole-free. A graph is [*even-hole-free*]{} if it contains no even cycle as an induced subgraph. Kamiński et al. [@KaminskiMM12] proved the following statement.
\[thm:KMM\] Let $I$ and $J$ be two s of a graph $G$ with $|I|=|J|$. If $G[I\Delta J]$ contains no even cycles, then there exists a from $I$ to $J$ of length $|I{\backslash}J|$, which can be constructed in linear time.
In particular, if $G$ is even-hole-free, then $\TJ_k(G)$ is connected (for every $k$). However, $\TS_k(G)$ is not necessarily connected (consider a claw with two tokens). This motivated the question asked in [@KaminskiMM12] whether for connected, claw-free and even-hole-free graph $G$, $\TS_k(G)$ is connected. Combining Corollary \[cor:connected:TSequivTJ\] with Theorem \[thm:KMM\] shows that the answer to this question is affirmative.
Let $G$ be a connected claw-free and even-hole-free graph. Then $\TS_k(G)$ is connected.
Nonmaximum Independent Sets {#sec:nonmaximum}
===========================
We now continue studying connected claw-free graphs. Lemma \[lem:NoBadCyclesSliding\] shows that it remains to consider the case that $G[I\Delta J]$ contains (even length) cycles. In this section, we show that when $I$ and $J$ are not maximum s of $G$, such cycles can always be resolved. This requires various techniques developed in the context of finding maximum s in claw-free graphs and the following definitions.
A vertex $v\in V(G)$ is [*free*]{} (with respect to an independent set $I$ of $G$) if $v \notin I$ and $|N(v)\cap I|\le 1$. Let $W=v_0,\ldots,v_k$ be a walk in $G$, and let $I\subseteq V(G)$. Then $W$ is called [*$I$-alternating*]{} if $|\{v_i,v_{i+1}\}\cap I|=1$ for $i=0,\dots,k-1$. In the case that $W$ is a path, $W$ is called [*chordless*]{} if $G[\{v_0,\ldots,v_k\}]$ is a path. In the case that $W$ is a cycle (so $v_0=v_k$), $W$ is called [*chordless*]{} if $G[\{v_0,\ldots,v_{k-1}\}]$ is a cycle. A cycle $W=v_0,\ldots,v_k$ is called [*$I$-bad*]{} if it is $I$-alternating and chordless. A path $W=v_0,\ldots,v_k$ with $k\ge 2$ is called [*$I$-augmenting*]{} if it is $I$-alternating and chordless, and $v_0$ and $v_k$ are both free vertices. This definition of $I$-augmenting paths differs from the usual definition, as it is used in the setting of finding [*maximum independent sets*]{}, since the chordless condition is stronger than needed in such a setting. However, we observe that in a claw-free graph $G$, the two definitions are equivalent (see Proposition \[propo:AugmPathDefEquiv\] in the appendix) so we may apply well-known statements about $I$-augmenting paths proved elsewhere. In particular, we use the following two results originally proved by Minty [@Minty80] and Sbihi [@sbihi1980algorithme] (see also [@schrijver2003combinatorial Section 69.2]).
\[thm:AugmPathPoly\] Let $I$ be an in a claw-free graph $G$. It can be decided in polynomial time whether an $I$-augmenting path between two given free vertices $x$ and $y$ exists, and if so, compute one.
\[propo:NotMaxImpliesAugmentable\] Let $I$ be a nonmaximum independent set in a claw-free graph $G$. Then $I$ is not a dominating set, or there exists an $I$-augmenting path.
We use Proposition \[propo:NotMaxImpliesAugmentable\] to handle the case of nonmaximum s. The next statement is formulated for token jumping, and (by Corollary \[cor:connected:TSequivTJ\]) implies the same result for token sliding only in the case of connected graphs.
\[lem:non\_maximum\_NEW\] Let $I$ be a nonmaximum independent set in a claw-free graph $G$. Then for any independent set $J$ with $|J|=|I|$, $I\tjr J$ holds.
By Theorem \[thm:KMM\], it suffices to consider the case where $G[I\Delta J]$ contains at least one cycle $C$. Let $C=u_1,v_1,u_2,v_2,\ldots,v_k,u_1$, so that $u_i\in I$ and $v_i\in J$ for all $i$.
Suppose first that $I$ is not a dominating set. Then we can choose a vertex $w$ with $N[w]\cap I=\emptyset$. With a single token jump, we can obtain the $I'=I+w-u_1$ from $I$. Next, apply the moves $u_k\to v_k$, $u_{k-1}\to v_{k-1}$,…, $u_2\to v_2$, in this order. (This is possible since $C$ is chordless.) Finally, jump the token from $w$ to $v_1$. It can be verified that this yields a token jumping sequence from $I$ to $I'=I\Delta V(C)$. This way, all cycles can be resolved one by one, until no more cycles remain and Theorem \[thm:KMM\] can be applied to prove the statement.
On the other hand, if $I$ is a dominating set, then Proposition \[propo:NotMaxImpliesAugmentable\] shows that there exists an $I$-augmenting path $P=v_0,u_1,v_1,\ldots,u_d,v_d$, with $u_i\in I$ for all $i$. Since $v_d$ is a free vertex, we can first apply the moves $u_d\to v_d$, $u_{d-1}\to v_{d-1}$,…$u_1\to v_1$, in this order (which can be done since $P$ is chordless), to obtain an $I'$ from $I$, with $I\tsr I'$. Then $v_0$ is not dominated by $I'$, so the previous argument can be applied to show that $I'\tjr J$, which implies $I\tjr J$.
Resolving Cycles {#sec:resolving}
================
It now remains to study the case where $G[I\Delta J]$ contains (even) cycles and both $I$ and $J$ are maximum independent sets. In this case, there may not be a from $I$ to $J$ (even though we assume that $G$ is connected and claw-free) – consider for instance the case where $G$ itself is an even cycle. In this section, we characterize the case where $I\tsr J$ holds, by showing that this is equivalent with every cycle being resolvable in a certain sense (Theorem \[thm:reachable\_iff\_all\_resolvable\] below). Subsequently, we show that resolvable cycles fall into two cases: internally or externally resolvable cycles, which are characterized next. We first define the notion of resolving a cycle.
Cycles in $G[I\Delta J]$ are clearly both $I$-bad and $J$-bad. The [*$I$-bipartition*]{} of an $I$-bad cycle is the ordered tuple $[V(C)\cap I,V(C){\backslash}I]$. We say that an $I$-bad cycle $C$ with $I$-bipartition $[A,B]$ is [*resolvable*]{} (with respect to $I$) if there exists an $I'$ such that $I\tsr I'$ and $G[I'\cup B]$ contains no cycles. A corresponding from $I$ to $I'$ is called a [*resolving sequence*]{} and is said to [*resolve $C$*]{}. By combining such a resolving sequence with a sequence of moves similar to the previous proof, and then reversing the moves in the sequence from $I'$ to $I$, except for moves of tokens on the cycle, one can show that every resolvable cycle can be ‘turned’:
\[lem:resolvable\_implies\_turnable\] Let $I$ be an independent set in a claw-free graph $G$ and let $C$ be an $I$-bad cycle. If $C$ is resolvable with respect to $I$, then $I\tsr I\Delta V(C)$.
We can now prove the following useful characterization: $I\tsr J$ if and only if every cycle in $G[I\Delta J]$ is resolvable. By symmetry, it does not matter whether one considers resolvability with respect to $I$ or to $J$.
\[thm:reachable\_iff\_all\_resolvable\] Let $I$ and $J$ be independent sets in a claw-free connected graph $G$. Then $I\tsr J$ if and only if every cycle in $G[I\Delta J]$ is resolvable with respect to $I$.
Consider an $I$-bad cycle $C$ in $G[I\Delta J]$ with $I$-bipartition $[A,B]$, and a from $I$ to $J$. Since $N_2(B)$ eventually contains no tokens, this sequence must contain a move $u\to v$ with $u\in N_2(B)$ and $v\not\in N_2(B)$. The first such move can be shown to resolve the cycle. (See Lemma \[lem:ShortResolvingSeqNEW\] in the appendix for details.)
The other direction is proved by induction on the number $k$ of cycles in $G[I\Delta J]$. If $k=0$, then by Lemma \[lem:NoBadCyclesSliding\], $I\tsr J$. If $k\ge 1$, then consider an $I$-bad cycle $C$ in $G[I\Delta J]$. Let $I'=I\Delta V(C)$. By Lemma \[lem:resolvable\_implies\_turnable\], $I\tsr I'$. The graph $G[I'\Delta J]$ has one cycle fewer than $G[I\Delta J]$. Every cycle in $G[I'\Delta J]$ remains resolvable with respect to $I'$ (one can first consider a from $I'$ to $I$, and subsequently a from $I$ that resolves the cycle). So by induction, $I'\tsr J$, and therefore, $I\tsr J$.
Finally, we show that if an $I$-bad cycle $C$ can be resolved, it can be resolved in at least one of two very specific ways. Let $[A,B]$ be the $I$-bipartition of $C$. A move $u\to v$ is called [*internal*]{} if $\{u,v\}\subseteq N_2(B)$ and [*external*]{} if $\{u,v\}\subseteq N_0(B)$. A resolving sequence $I_0,\ldots,I_m$ for $C$ is called [*internal*]{} (or [*external*]{}) if every move except the last is an internal (respectively, external) move. (Obviously, to resolve the cycle, the last move can neither be internal nor external, and can in fact be shown to always be a move from $N_2(B)$ to $N_1(B)$.) The $I$-bad cycle $C$ is called [*internally resolvable*]{} resp. [*externally resolvable*]{} if such sequences exist.
\[lem:externally\_or\_internally\] Let $I$ be an independent set in a claw-free graph $G$ and let $C$ be an $I$-bad cycle. Then any shortest that resolves $C$ is an internal or external resolving sequence.
Let $[A,B]$ be the $I$-bipartition of $C$. Since $G$ is claw-free, it follows that there are no edges between vertices in $N_2(B)$ and $N_0(B)$. This can be used to show that informally, any resolving sequence for $C$ remains a resolving sequence after either omitting all noninternal moves or omitting all nonexternal moves, while keeping the last move, which subsequently resolves the cycle.
Theorem \[thm:reachable\_iff\_all\_resolvable\] and Lemma \[lem:externally\_or\_internally\] show that to decide whether $I\tsr J$, it suffices to check whether every cycle in $G[I\Delta J]$ is externally or internally resolvable. Next we give characterizations that allow polynomial-time algorithms for deciding whether an $I$-bad cycle is internally or externally resolvable. For the external case, we use the assumption that $I$ is a maximum to show that in a [*shortest*]{} external resolving sequence $I_0,\ldots,I_m$, every token moves at most once (that is, for every move $u\to v$, both $u\in I_0$ and $v\in I_m$ hold), so these moves outline an augmenting path in a certain auxiliary graph.
\[thm:ext\_res\_augm\_path\]\[\*\] Let $I$ be a maximum in a claw-free graph $G$ and let $C$ be an $I$-bad cycle with $I$-bipartition $[A,B]$. Then $C$ is externally resolvable if and only if there exists an $(I{\backslash}A)$-augmenting path in $G-A-B$ between a pair of vertices $x\in N_0(B)$ and $y\in N_1(B)$.
For a given $I$-bad cycle $C$ with $I$-bipartition $[A,B]$, there is a quadratic number of vertex pairs $x\in N_0(B)$ and $y\in N_1(B)$ that need to be considered, and for every such a pair, testing whether there is an $(I{\backslash}A)$-augmenting path between these in $G-A-B$ can be done in polynomial time (Theorem \[thm:AugmPathPoly\]). So from Theorem \[thm:ext\_res\_augm\_path\] we conclude:
\[cor:ext\_res\_polytime\] Let $I$ be a maximum in a claw-free graph $G$, and let $C$ be an $I$-bad cycle. In polynomial time, it can be decided whether $C$ is externally resolvable.
Next, we characterize internally resolvable cycles. Shortest internal resolving sequences cannot be as easy to describe as external ones, since a token can move several times (see Figure \[fig:internal\_example\]). Nevertheless, these sequences can be shown to have a very specific structure, which can be characterized using paths in the following auxiliary digraphs.
To define these digraphs, consider an $I$-bad cycle $C=c_0,c_1,\ldots,c_{2n-1},c_0$ in $G$, with $c_i\in I$ for even $i$. Let $[A,B]$ be the $I$-bipartition of $C$. For every $i\in \{0,\ldots,n-1\}$, define the corresponding [*layer*]{} as follows: $L_i=\{v\in V(G) \mid N(v)\cap B=N(c_{2i})\cap B\}$. So when starting with $I$ and using only internal moves, it can be seen that the token that starts on $c_{2i}$ will stay in the layer $L_i$.
For such an $I$-bad cycle $C$ of length at least 8, define $D(G,C)$ to be a digraph with vertex set $V(G)$, with the following arc set. For every $i\in\{0,\ldots,n-1\}$ and all pairs $u\in L_i, v\in L_{(i+1)\bmod n}$ with $uv\not\in E(G)$, add an arc $(u,v)$. For every $i\in\{0,\ldots,n-1\}$ and $b\in N_1(B)$ with $N(b)\cap B=\{c_{(2i-1) \bmod 2n}\}$, and every $v\in L_i$ with $bv\not\in E(G)$, add an arc $(b,v)$. Also, we denote the reversed cycle by $C^{rev}=c_0,c_{2n-1},\ldots,c_1,c_0$. This defines a similar digraph $D(G,C^{rev})$ (where arcs between layers are reversed, and arcs from $N_1(B)$ go to different layers). These graphs can be used to characterize whether $C$ is internally resolvable.
\[fig:internal\_example\]
\[thm:internally\_iff\_copath\] Let $I$ be an independent set in a claw-free graph $G$. Let $C=c_0,c_1,\dots,c_{2n-1},c_{0}$ be an $I$-bad cycle ($c_0\in I$) with $I$-bipartition $[A,B]$, of length at least 8. Then $C$ is internally resolvable if and only if $D(G,C)$ or $D(G,C^{rev})$ contains a directed path from a vertex $b\in N_1(B)$ with $N(b)\cap I\subseteq A$ to a vertex in $A$.
\[cor:int\_res\_polytime\] Let $I$ be an independent set in a claw-free graph $G$ on $n$ vertices and let $C$ be an $I$-bad cycle. It can be decided in polynomial time whether $C$ is internally resolvable.
If $C$ has length at least 8, then Theorem \[thm:internally\_iff\_copath\] shows that it suffices to make a polynomial number of depth-first-searches in $D(G,C)$ and $D(G,C^{rev})$. Otherwise, let $[A,B]$ be the $I$-bipartition of $C$. $|A|\le 3$, so there are only $O(n^3)$ independent sets $I'$ with $|I'|=|I|$ and $I{\backslash}A\subseteq I'$. So in polynomial time we can generate the subgraph of $\TS_k(G)$ induced by these sets, and search whether it contains a path from $I$ to an $I^*$ with $I{\backslash}A\subseteq I^*$ where $G[B\cup I^*]$ contains no cycle. $C$ is internally resolvable if and only if such a path exists.
Summary of the Algorithm {#sec:summary}
========================
We now summarize how the previous lemmas yield a polynomial time algorithm for and in claw-free graphs.
\[thm:connected\_reachability\] Let $I$ and $J$ be independent sets in a claw-free graph $G$. We can decide in polynomial time whether $I\tsr J$ and whether $I\tjr J$.
Assume $|I|=|J|$; otherwise, we immediately return NO. We first consider the case when $G$ is connected. By Corollary \[cor:connected:TSequivTJ\], since $G$ is connected, $I \tsr J$ if and only if $I\tjr J$, thus we only need to consider the sliding model.
We test whether $I$ and $J$ are maximum s of $G$, which can be done in polynomial time (by combining Proposition \[propo:NotMaxImpliesAugmentable\] and Theorem \[thm:AugmPathPoly\]; see also [@Minty80; @sbihi1980algorithme; @schrijver2003combinatorial]). If not, then by Lemma \[lem:non\_maximum\_NEW\], $I\tjr J$ holds, and thus $I\tsr J$, so we may we return YES.
Now consider the case that both $I$ and $J$ are maximum s. Theorem \[thm:reachable\_iff\_all\_resolvable\] shows that $I\tsr J$ if and only if every cycle in $G[I\Delta J]$ is resolvable with respect to $I$. By Lemma \[lem:externally\_or\_internally\], it suffices to check for internal and external resolvability of such cycles. This can be done in polynomial time by Corollary \[cor:ext\_res\_polytime\] (since $I$ is a maximum of $G$) and Corollary \[cor:int\_res\_polytime\]. We return YES if and only if every cycle in $C$ was found to be internally or externally resolvable, and NO otherwise.
Now let us consider the case when $G$ is disconnected. Clearly tokens cannot slide between different connected components, so for deciding whether $I\tsr J$, we can apply the argument above to every component, and return YES if and only if the answer is YES for every component. If $I$ is a not a maximum then Lemma \[lem:non\_maximum\_NEW\] shows that $I\tjr J$ always holds. If $I$ is maximum, then Proposition \[propo:max:TSequivTJ\] shows that $I\tjr J$ holds if and only if $I\tsr J$.
Discussion {#sec:discussion}
==========
The results presented here have two further implications. Firstly, combined with techniques from [@Bonsma14], it follows that $I\tjr J$ can be decided for any graph $G$ that can be obtained from a collection of claw-free graphs using [*disjoint union*]{} and [*complete join*]{} operations. See [@Bonsma14] for more details.
Secondly, a closer look at constructed reconfiguration sequences (in the appendix) shows that when $G$ is claw-free, components of both $\TS_k(G)$ and $\TJ_k(G)$ have diameter bounded polynomially in $|V(G)|$. This is not surprising, since the same behavior has been observed many times. To our knowledge, the only known examples of polynomial time solvable reconfiguration problems that nevertheless require exponentially long reconfiguration sequences are on artificial instance classes, which are constructed particularly for this purpose (see e.g. [@BonsmaC09; @KMM11]).
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---
abstract: '*CausalML* is a Python implementation of algorithms related to causal inference and machine learning. Algorithms combining causal inference and machine learning have been a trending topic in recent years. This package tries to bridge the gap between theoretical work on methodology and practical applications by making a collection of methods in this field available in Python. This paper introduces the key concepts, scope, and use cases of this package.'
author:
- 'Huigang Chen\*, Totte Harinen\*, Jeong-Yoon Lee\*, Mike Yung\*, Zhenyu Zhao\*'
bibliography:
- './uplift\_references.bib'
title: 'CausalML: Python Package for Causal Machine Learning [^1]'
---
Introduction
============
*CausalML* is a Python package that provides a suite of uplift modeling and causal inference methods using machine learning algorithms based on cutting edge research. The traditional causal analysis methods, such as performing t-test on randomized experiments (a.k.a. A/B testing) can estimate the Average Treatment Effect (ATE) of the treatment or intervention. However, in many applications, it is often desired and useful to estimate these effects at a more granular scale. *CausalML* enables the end user to estimate the Conditional Average Treatment Effect (CATE), which is essentially the effect at the individual or segment level. Such estimates can unlock a large range of applications for personalization and optimization by applying different treatment to different users.
One key modeling technique enabled by *CausalML* is uplift modeling. Uplift modeling [@Grimmer2017-rl; @Guelman2015-qe; @Gutierrez2016-co; @Kunzel2017-ko; @Rzepakowski2012-br; @Soltys2015-be; @Wager2015-sd; @Zaniewicz2013-rt; @Zhao2017-kg; @nie2017quasi] is a causal learning approach to estimate the individual treatment effect for an experiment. This allows the end user to measure the incremental impact of a treatment (such as a direct marketing action) on an individual’s behaviour using experimental data. For instance, if a company is choosing between multiple product lines to up-sell / cross-sell to its customers, it can utilize *CausalML* as a recommendation engine to identify products that achieve the greatest expected lift for any given user.
It is worth mentioning that this package is not designed to replace the standard randomized experiment approach for drawing causal inference. In many scenarios, it is essential to carry out randomized experiments to evaluate the ATE for business decisions. While uplift modeling can be applied to both experimental data and observational data, the current implementation of the uplift modeling is encouraged to be applied to the data from the randomized experiment. Applications to observational data where the treatment is not assigned randomly should take extra caution. In non-randomized experiment, there is often a selection bias in the treatment assignment (a.k.a. confounding effect). One main challenge is that omitting potential confounding variables in the model can produce biased estimation for the treatment effect. On the other hand, properly randomized experiments do not suffer from such selection bias, that provides a better basis for uplift modeling to estimate the CATE (or individual level lift).
There are a few related packages. In R, the uplift, grf, and rlearner packages implement the Uplift Random Forest, Generalized Random Forest, and R-learner methods respectively. In Python, the package DoWhy is focused on structuring the causal inference problem through graphical models based on Judea Pearl’s do-calculus and the potential outcomes framework. The recently released EconML Python package implements heterogeneous treatment effect estimators from econometrics (such as instrumental variables) and machine learning methods . Another Python package Pylift implements one meta-learner for uplift modeling. The contribution of the current implementation of *CausalML* package is providing an one-stop-shop for uplift modeling techniques (8 models so far) with a set of support functions in Python. For example, according to our knowledge, the Uplift Random Forest methods and R-learner are made available in an open source Python package for the first time. In addition, we have implemented innovative methods developed in house, such as meta-learners for multiple treatment groups optimization.
Why CausalML
============
In recent years, the intersection of causal inference and machine learning has become an active area of research. Based on our experience at Uber, we believe there are going to be a number of important practical applications emerging from this research. With the *CausalML* package, we aim to make these applications accessible to a wider audience. Our ultimate goal is to build a one-stop shop for machine learning for causal inference.
The first area of focus for the package is the area of uplift modeling. We believe it is a methodology that can serve an important purpose in business, science and elsewhere. With the first version of the *CausalML* package, our goal is to democratise the uplift modelling methods that are currently available in only academic papers or in disparate statistical packages.
To do so, we offer a consistent API that makes running an uplift algorithm as easy as fitting a standard classification or regression model. Model performance can be evaluated using the included metrics and visualisation functions such as uplift curves. The first version of *CausalML* implements eight state of the art algorithms for uplift modelling (see Figure. 1).
![*CausalML* package diagram[]{data-label="fig:package_diagram"}](picture/causalml_diagram.png){width="49.00000%"}
Further, we have built the package flexible in terms of the types of outcome variables that can be modelled, covering both regression and classification type tasks. The package also contains algorithms that can be used with data from experiments with multiple treatment groups. Our hope is that these features of flexibility, generality and ease of use will make *CausalML* the tool of choice for uplift modellers in the future.
What Problems Can CausalML Solve?
=================================
*CausalML*’s use cases include, but are not limited to, targeting optimization, engagement personalization and causal impact analysis.
Targeting Optimization
----------------------
We can use *CausalML* to target promotions to those with the biggest incrementality. For example, at a cross-sell marketing campaign for existing customers, we can deliver promotions to the customers who would be more likely to use a new product specifically given the exposure to the promotions, while saving inboxes for others. An internal analysis showed that targeting only 30% of users with uplift modeling could achieve the same increase in conversion for a new product as offering the promotion to all customers.
Causal Impact Analysis
----------------------
We can also use *CausalML* to analyze the causal impact of a particular event from experimental or observational data, incorporating rich features. For example, we can analyze how a customer’s cross-sell event affects long term spending on the platform. In this case, it is impractical to set up a randomized test because we do not want to exclude any customers from being able to convert to the new product. Utilizing *CausalML*, we can run various ML-based causal inference algorithms, and estimate the impact of the cross-sell on the platform.
Personalization
---------------
*CausalML* can be used to personalize engagement. There are multiple options for a company to interact with its customers, such as different product choices in up-sell or messaging channels for communications. One can use *CausalML* to estimate the effect of each combination for each customer, and provide optimal personalized offers to customers.
Future Development
==================
*CausalML* is actively maintained and developed by the Uber development team [^2] for *CausalML*. One area of development is to further improve the computational efficiency for existing algorithms in the package. In addition, we plan to add more state-of-the-art uplift models as needed in the future. Besides uplift modeling, we are also exploring more modeling techniques in the intersection of machine learning and causal inference, with a goal of solving optimization problems.
We welcome everyone to try out *CausalML* on different use cases, and invite you to share your feedback with us. If you are interested in contributing to the development of this package, please read our code of conduct and follow contributing guidelines.
Acknowledgement
===============
We would like to extend our appreciation to our colleagues at Uber who have contributed to and supported this open source project, including Mert Bay, Fran Bell, Natalie Diao, Shuyang Du, Neha Gupta, Candice Hogan, Yiming Hu, James Lee, Paul Lo, Yuchen Luo, Lance Mack, Vishal Morde, Jing Pan, Hugh Williams, Yunhan Xu, and Yumin Zhang. We would also like to thank external contributors for this project: Peter Foley, Florian Wilhelm, Steve Yang, and Tomasz Zamacinski.
[^1]: \* Authors listed alphabetically. All authors are from Uber Technologies Inc.
[^2]: The Uber development team for *CausalML* includes Huigang Chen, Totte Harinen, Jeong-Yoon Lee, Mike Yung, and Zhenyu Zhao (alphabetically).
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---
abstract: 'The action of the mapping class group of the thrice-punctured projective plane on its ${\mathop{\rm GL}\nolimits}(2,\mathbb{C})$ character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corresponding fundamental group to prove: a sharp upper-bound for the length of its shortest geodesic, a McShane identity and the surprising result of non-polynomial growth for the number of simple closed geodesic lengths.'
address: 'Department of Mathematics and Statistics, The University of Melbourne, Australia 3010.'
author:
- Yi Huang and Paul Norbury
bibliography:
- 'Bibliography.bib'
title: Simple geodesics and Markoff quads
---
Introduction
============
Background
----------
Closed geodesics on hyperbolic surfaces have extremely rich properties, arising in geometry, topology and number theory, and this is particularly true of *simple* (that is: non-self-intersecting) closed geodesics. The central objects of our study are simple closed geodesics on 3-cusped projective planes. We describe an algorithm for generating these geodesics and their lengths, and use this algorithm to study systoles, McShane identities and simple length spectra of 3-cusped projective planes.
![Simple curves on thrice-punctured projective planes.[]{data-label="fig:borromean"}](borromean.eps)
**Systoles:** The *systole* ${\mathop{\rm sys}\nolimits}(M)$ of a Riemannian manifold $M$ is the length of its shortest essential closed curve which is necessarily a simple closed geodesic. First mentioned in [@PuPSom], Loewner’s torus inequality gives the following relationship between the systole ${\mathop{\rm sys}\nolimits}(T)$ of a Riemannian torus $T$ and its surface area ${\mathop{\rm Vol}\nolimits}(T)$: $$\begin{aligned}
{\mathop{\rm sys}\nolimits}(T)^2\leq\tfrac{2}{\sqrt{3}}\,{\mathop{\rm Vol}\nolimits}(T),\end{aligned}$$ equality is realised for tori obtained by gluing opposite sides of regular hexagons. Since then, systolic inequalities have been obtained for other surfaces and higher dimensional manifolds such as Gromov’s inequality [@GroFil] which holds for a large class of Riemannian manifolds. More recently, there has been growing interest in systolic hyperbolic geometry. For example, for orientable hyperbolic surfaces $X$ with $n\geq2$ cusps and/or boundary components [@SchCon]: $$\begin{aligned}
{\mathop{\rm sys}\nolimits}(X)\leq 4{\mathop{\rm arccosh}\nolimits}\left(\frac{3|\chi(X)|}{n}\right).\end{aligned}$$ Hyperbolic systolic inequalities do not require a volume term on the right, because the area of a hyperbolic surface $X$ is topologically determined: it is $2\pi|\chi(X)|$.**McShane identities:** A *McShane identity* may be thought of as a type of trigonometric identity for hyperbolic surfaces. It is usually a sum over functions of lengths of simple closed geodesics on a punctured/bordered hyperbolic surface. In particular, the structure of this sum is independent of the hyperbolic structure on the surface.McShane proved the first such identity in his doctoral dissertation [@McSRem]. Denote the collection of simple closed geodesics on a 1-cusped torus $X_{1,1}$ by $\mathcal{S}_{1,1}$, then: $$\begin{aligned}
1=\sum_{\gamma\in\mathcal{S}_{1,1}}\frac{2}{1+\exp\ell_\gamma},\end{aligned}$$ where $\ell_\gamma$ denotes the length of the simple closed geodesic $\gamma$. Each term in the above series corresponds to the probability that a geodesic launched from the cusp on $X_{1,1}$, up to its first point of self-intersection, does not intersect $\gamma$. McShane identities for other hyperbolic surfaces and quasi-Fuchsian representations have steadily followed [@AkiVar; @HuHIde; @HuaMcS; @LuoDil; @McSRem; @McSSim; @MirSim; @NorLen; @TWZGen].**Simple length spectra:** Given a Riemannian surface, its *length spectrum* is the multiset of lengths of closed geodesics on the surface. The Selberg trace formula interprets the length spectrum in terms of the spectrum of the Laplace-Beltrami operator [@SelHar], and can be used to show that the number of closed geodesics of length less than $L$ on an orientable hyperbolic surface grows exponentially.The *simple length spectrum* is the multiset of lengths of simple closed geodesics. Simple closed geodesics are relatively rare among closed geodesics. The growth rate of the simple length spectrum on an orientable hyperbolic surface is only polynomial in $L$. This was first proven using combinatorial arguments [@MRiSim; @RivSim]. Mirzakhani gives a novel proof of a refinement of this result in [@MirGro]: let $\mathcal{S}(X)$ denote the set of simple closed geodesics on a hyperbolic surface $X$, for $L>0$ define $$\begin{aligned}
s_{X}(L)={\mathop{\rm Card}\nolimits}\left\{\gamma\in \mathcal{S}(X) \mid \ell_{\gamma}(X)<L\right\}.\end{aligned}$$ Then, the function $\eta:\mathcal{M}(X)\rightarrow\mathbb{R}_+$ defined by taking the limit $$\begin{aligned}
\label{gropol}
\lim_{L\to\infty}\frac{s_X(L)}{ L^{\dim_{\mathbb{R}}\mathcal{M}(X)}}=\eta(X)>0\end{aligned}$$ is a continuous proper function. Mirzakhani’s proof employs the ergodicity of the mapping class group action on the space of geodesic measured laminations, as well as her calculations of moduli spaces volumes [@MirSim] — calculations utilising McShane identities.**Markoff triples:** A *Markoff triple* is a solution $(x,y,z)\in\mathbb{C}^3$ to the equation: $$\begin{aligned}
\label{triple}
x^2+y^2+z^2=xyz.\end{aligned}$$ The hypersurface in $\mathbb{C}^3$ defined by is the relative character variety of the fundamental group of the once-punctured torus. In particular, this gives a real analytic diffeomorphism between the set of positive real Markoff triples and the Fuchsian component of the character variety — a model for the Teichmüller space of hyperbolic 1-cusped tori.The following transformations: $$\begin{aligned}
\label{tripletrans}
(x,y,z)\mapsto
(x,y,xy-z)\text{ and }(x,y,z)\mapsto(y,z,x),\end{aligned}$$ take one Markoff triple to another. In particular, these transformations generate the extended mapping class group of the punctured torus, and describe its action on the corresponding relative character variety. For Fuchsian characters, any triple $(x,y,z)\in\mathbb{R}^3_+$ consists of $2\cosh(\frac{1}{2}\cdot)$ of the lengths of an ordered triple of simple closed geodesics on a particular 1-cusped torus $X$. One is thus able to generate the entire simple length spectrum of $X$ by applying sequences of the transformations in to $(x,y,z)$. This relationship between positive real Markoff triples and the simple length spectra of hyperbolic 1-cusped tori was first exploited in [@CohApp]. In [@BowPro], Bowditch uses a generalisation of this correspondence to derive a sharp systolic inequality for quasi-Fuchsian representations of 1-cusped tori, and also to establish a quasi-Fuchsian generalisation of McShane’s original identity. The length generation algorithm can also be used to prove that quasi-Fuchsian representations of the punctured torus group have $L^2$ simple length growth rates [@HuaMod].
Markoff Quads
-------------
*Markoff quads* are 4-tuples $(a,b,c,d)\in\mathbb{C}^4$ of complex numbers satisfying: $$\begin{aligned}
\label{quad}
(a+b+c+d)^2=abcd.\end{aligned}$$
The hypersurface in $\mathbb{C}^4$ defined by is the relative character variety of the fundamental group of the thrice-punctured projective plane $N_{1,3}$. In other words, we have the following bijective correspondence: $$\begin{aligned}
\left\{\text{ Markoff quads }\right\}\xleftrightarrow{1:1}&\left\{\text{ characters of }{\mathop{\rm GL}\nolimits}(2,\mathbb{C})\text{-representations of }\pi_1(N_{1,3})\right\}.\end{aligned}$$ Characters of Fuchsian and quasi-Fuchsian representations are of special importance because they respectively arise as monodromy representations of hyperbolic surfaces and hyperbolic 3-manifolds. We now describe this relationship and introduce some notation for the rest of the paper.**Fuchsian:** Given a Fuchsian representation of the fundamental group of a (possibly non-orientable) surface $S$ $$\begin{aligned}
\rho:\pi_1(S)\rightarrow{\mathop{\rm PSL}\nolimits}^{\pm}(2,\mathbb{R})={\mathop{\rm Isom}\nolimits}^{\pm}(\mathbb{H}),\end{aligned}$$ the discrete subgroup $\rho(\pi_1(S))$ acts properly discontinuously on the hyperbolic plane $\mathbb{H}$ by (possibly orientation reversing) isometries. The quotient $\mathbb{H}/\rho(\pi_1(S))$ is a complete hyperbolic surface homeomorphic to $S$, and we denote it by $X_\rho$. By identifying the universal cover of $S$ with $\mathbb{H}$, Fuchsian representations induce a homeomorphism $h_\rho:S\rightarrow X_\rho$, canonical up to homotopy.**Quasi-Fuchsian:** Similarly, for a strictly quasi-Fuchsian representation $$\begin{aligned}
\rho:\pi_1(S)\rightarrow{\mathop{\rm PSL}\nolimits}(2,\mathbb{C})={\mathop{\rm Isom}\nolimits}^+(\mathbb{H}^3), \end{aligned}$$ the discrete subgroup $\rho(\pi_1(S))$ acts properly discontinuously on $\mathbb{H}^3$ by orientation preserving isometries. The quotient space $\mathbb{H}^3/\rho(\pi_1(S))$ is an orientable complete hyperbolic 3-manifold homeomorphic to $(0,1)\times S$, and we also denote it by $X_\rho$. In analogy to the Fuchsian case, quasi-Fuchsian representations induce a canonical (up to homotopy) embedding $h_\rho: S\hookrightarrow X_\rho$. **Length functions on character varieties:** Given a simple closed curve $\gamma$ on $S$, there is a unique simple closed geodesic on $X_\rho$ homotopy equivalent to $h_\rho(\gamma)$. This allows one to define a function $\ell_{\gamma}(\cdot)$ on the space of quasi-Fuchsian representations of $\pi_1(S)$ by taking the *complex length* of the unique closed geodesic homotopy equivalent to $h_\rho(\gamma)$ in $X_\rho$. The complex length of a geodesic has real and imaginary parts respectively given by its geometric length and the angle of twisting of the normal bundle around the closed geodesic.
Main results
------------
\[th:systole\] Let $\rho$ denote a quasi-Fuchsian monodromy representation for a thrice-punctured projective plane, then $$\begin{aligned}
{\mathop{\rm sys}\nolimits}(X_\rho)\leq2{\mathop{\rm arcsinh}\nolimits}(2).\end{aligned}$$ In particular, the unique maximum of the systole function over the moduli space of all hyperbolic thrice-punctured projective planes is $2{\mathop{\rm arcsinh}\nolimits}(2)$.
The unique systolic maximum is realised by the 3-cusped projective plane with the largest isometry subgroup. This symmetric surface is doubly covered by a hyperbolic surface conformally equivalent to the unit sphere in $\mathbb{R}^3$ minus the 6 points where it meets the three axes. Its simple length spectrum can be generated from the integral Markoff quad $(4,4,4,4)$.
\[th:main\] Let $\rho$ be a quasi-Fuchsian representation of the thrice-punctured projective plane fundamental group $\pi_1(N_{1,3})$. Then, $$\begin{aligned}
\sum_{\gamma\in{\mathop{\rm Sim}\nolimits}_2(N_{1,3})}\frac{1}{1+\exp{\tfrac{1}{2}\ell_{\gamma}(\rho)}}=\frac{1}{2},\end{aligned}$$ where the sum is over the collection ${\mathop{\rm Sim}\nolimits}_2(N_{1,3})$ of free homotopy classes of essential, non-peripheral two-sided simple closed curves $\gamma$ on $N_{1,3}$.
This result is known for Fuchsian representations owing to the second author’s work in [@NorLen]. Moreover, Hu, Tan and Zhang [@HuHIde] have derived a McShane identity for solutions of Markoff-Hurwitz equations (see subsection \[markoff-hurwitz\]). In the $n=4$ case, their identity coincides with ours for quasi-Fuchsian thrice-punctured projective planes after a coordinate change. Thus, our result affirmatively answers their question of whether their identity has a geometric interpretation in the $n=4$ case.
Take note that theorem \[th:main\] is a series over ${\mathop{\rm Sim}\nolimits}_2(N_{1,3})$, the set of two-sided simple closed curves on $N_{1,3}$, rather than over the collection $\mathcal{S}_2(X_\rho)$ of two-sided simple closed *geodesics* on the hyperbolic 3-manifold $X_\rho$. This is because for non-Fuchsian representations $\rho$, the geodesic representatives of ${\mathop{\rm Sim}\nolimits}_2(N_{1,3})$ constitutes only a subset of $\mathcal{S}_2(X_\rho)$. Similarly, we consider the subset of the simple length spectrum of a quasi-Fuchsian representation $\rho$ corresponding to the set ${\mathop{\rm Sim}\nolimits}_1(N_{1,3})$ of *one-sided* simple closed curves on $N_{1,3}$, and study the growth rate of the following quantity: $$\begin{aligned}
s_\rho(L)={\mathop{\rm Card}\nolimits}\left\{
\gamma\in{\mathop{\rm Sim}\nolimits}_1(N_{1,3}) \mid |\ell_{\gamma}(\rho)|<L\right\}.\end{aligned}$$ When $\rho$ is Fuchsian, the value of $s_\rho(L)$ is equal to the number $s_{X_\rho}(L)$ of geodesics on $X_\rho$ below length $L$.
\[th:asymp\] Given a quasi-Fuchsian representation $\rho$ of the thrice-punctured projective plane $N_{1,3}$, $$\begin{aligned}
\lim_{L\to\infty}\frac{s_\rho(L)}{L^m}>0\end{aligned}$$ for some $m$ satisfying $2.430<m < 2.477$.
Acknowledgements: {#acknowledgements .unnumbered}
-----------------
The authors are grateful to Craig Hodgson for useful conversations, to Andrew Elvey-Price and Greg McShane for helping us to improve the bounds in Theorem \[th:asymp\], and to Ser Peow Tan and Hengnan Hu for conversations about their work.
Markoff Quads {#sec:mq}
=============
Consider a 4-tuple $(\alpha,\beta,\gamma,\delta)$ of distinct one-sided simple closed curves on a thrice-punctured projective plane $N_{1,3}$ that pairwise intersect once. Figure \[fig:flip\] shows two such 4-tuples $({\color{red}\alpha},{\color{red}\beta},{\color{red}\gamma},{\color{blue}\delta})$ and $({\color{red}\alpha},{\color{red}\beta},{\color{red}\gamma},{\color{blue}\delta'})$; the depicted crossed circle is a cross-cap which represents an embedded Möbius strip.
![Flipping the blue curve.[]{data-label="fig:flip"}](flip.eps)
Up to homotopy, the curves ${\color{blue}\delta}$ and ${\color{blue}\delta'}$ are the only one-sided simple closed curves that intersect each of the curves ${\color{red}\alpha},{\color{red}\beta},{\color{red}\gamma}$ exactly once. We call the process of replacing ${\color{blue}\delta}$ with ${\color{blue}\delta'}$ and vice versa, a *flip*. For any quasi-Fuchsian representation $\rho:\pi_1(N_{1,3})\rightarrow{\mathop{\rm PSL}\nolimits}(2,\mathbb{C})$, a Fricke trace identity [@MagRin] shows that $$\begin{aligned}
(a,b,c,d)=(2\sinh\tfrac{1}{2}\ell_{\alpha}(\rho),2\sinh\tfrac{1}{2}\ell_{\beta}(\rho),2\sinh\tfrac{1}{2}\ell_{\gamma}(\rho),2\sinh\tfrac{1}{2}\ell_{\delta}(\rho))\end{aligned}$$ satisfies equation and is therefore a *Markoff quad*. That is: $$\begin{aligned}
(a+b+c+d)^2=abcd.\end{aligned}$$ Just as we can flip from $(\alpha,\beta,\gamma,\delta)$ to $(\alpha,\beta,\gamma,\delta')$, a new Markoff quad may be obtained via the following transformation: $$\begin{aligned}
\label{flip}
(a,b,c,d)\mapsto(a,b,c,d'=abc-2a-2b-2c-d),\end{aligned}$$ where $d'=2\sinh\frac{1}{2}\ell_{\delta'}(\rho)$. Similarly flip $\alpha$ to $\alpha'$, $\beta$ to $\beta'$ or $\gamma$ to $\gamma'$, to correspondingly obtain three other transformations: $$\begin{aligned}
(a,b,c,d)\mapsto\arraycolsep=1.4pt\def\arraystretch{1.2}
\begin{array}{l}
(bcd-2b-2c-2d-a,b,c,d),\\
(a,acd-2a-2c-2d-b,c,d),\\
(a,b,abd-2a-2b-2d-c,d),
\end{array}\end{aligned}$$ which take $(a,b,c,d)$ to new Markoff quads. Every one-sided simple closed curve can be uniquely obtained by some sequence of flips [@SchCom], and thus gives us an algorithm for generating the (one-sided) simple length spectrum of hyperbolic 3-cusped projective planes $X_\rho$.We begin this section by considering the trace identities needed for this algorithm, before detailing how to store the combinatorics of Markoff quads (and hence the simple length spectrum) for a 3-cusped projective plane in its curve complex.
Trace identities {#sec:trid}
----------------
The fundamental group $\pi_1(N_{1,3})$ for thrice-punctured projective planes is the free froup $F_3$ of rank 3. Any representation $\rho:\pi_1(N_{1,3})\rightarrow {\mathop{\rm PSL}\nolimits}(2,\mathbb{C})$ of a free group admits a lift to a representation $$\begin{aligned}
\tilde{\rho}:\pi_1(N_{1,3})\rightarrow {\mathop{\rm SL}\nolimits}(2,\mathbb{C}).\end{aligned}$$ The character $\xi={\mathop{\rm tr}\nolimits}\tilde{\rho}$ completely determines the geometry of $X_\rho$, and so we consider the ${\mathop{\rm PSL}\nolimits}(2,\mathbb{C})$-character variety of $$\begin{aligned}
\pi_1(N_{1,3})=F_3=\left\langle\,\alpha_1,\alpha_2,\alpha_3\mid -\; \right\rangle. \end{aligned}$$ We set $A_i\in{\mathop{\rm SL}\nolimits}(2,\mathbb{C})$ to denote the matrix $\tilde{\rho}(\alpha_i)$.Goldman showed [@GolTra] that any ${\mathop{\rm SL}\nolimits}(2,\mathbb{C})$-character is determined by the values $$\begin{aligned}
({\mathop{\rm tr}\nolimits}A_1,{\mathop{\rm tr}\nolimits}A_2,{\mathop{\rm tr}\nolimits}A_3,{\mathop{\rm tr}\nolimits}A_1A_2,{\mathop{\rm tr}\nolimits}A_2A_3,{\mathop{\rm tr}\nolimits}A_3A_1,{\mathop{\rm tr}\nolimits}A_1A_2A_3)\in\mathbb{C}^7.\end{aligned}$$ Thus, the ${\mathop{\rm SL}\nolimits}(2,\mathbb{C})$-character variety for $F_3$ may be embedded as a subvariety in $\mathbb{C}^7$. In particular, the character variety is a hypersurface defined by Fricke’s relation [@MagRin] for matrices in ${\mathop{\rm GL}\nolimits}(2,\mathbb{C})$ — in fact for ${\mathop{\rm PGL}\nolimits}(2,\mathbb{C})$ since the identity is homogeneous: given three matrices $A_1,A_2,A_3\in {\mathop{\rm GL}\nolimits}(2,\mathbb{C})$, set $A_0=A_1A_2A_3$. Then, $$\begin{aligned}
\label{fricke}
4\det A_0=&
({\mathop{\rm tr}\nolimits}A_0)^2+{\mathop{\rm tr}\nolimits}A_1\cdot{\mathop{\rm tr}\nolimits}A_2\cdot{\mathop{\rm tr}\nolimits}A_3\cdot{\mathop{\rm tr}\nolimits}A_0
+{\mathop{\rm tr}\nolimits}A_1A_2\cdot{\mathop{\rm tr}\nolimits}A_2A_3\cdot{\mathop{\rm tr}\nolimits}A_3A_1\\
&+\frac{1}{2}\sum_{{\mathop{\rm sym}\nolimits}}\left\{\begin{array}{l}({\mathop{\rm tr}\nolimits}A_i)^2\cdot\det A_jA_k
-\det A_i\cdot{\mathop{\rm tr}\nolimits}A_j\cdot{\mathop{\rm tr}\nolimits}A_k\cdot{\mathop{\rm tr}\nolimits}A_jA_k\\
+\det A_i\cdot({\mathop{\rm tr}\nolimits}A_jA_k)^2-{\mathop{\rm tr}\nolimits}A_0\cdot{\mathop{\rm tr}\nolimits}A_i\cdot{\mathop{\rm tr}\nolimits}A_jA_k
\end{array}
\right\}.\nonumber\end{aligned}$$ The symmetric sum here is taken over all possible choices for $\{i,j,k\}=\{1,2,3\}$; the factor of $\tfrac{1}{2}$ in compensates for indices such as $(i,j,k)=(1,2,3)$ and $(1,3,2)$ giving repeated terms. The proof of uses the fact that it extends to a relation on $M(2,\mathbb{C})$ which is quadratic in each entry of $A_i$.For our purposes, we study the *relative character variety* consisting of *type-preserving* representations of $F_3$ (i.e. where peripheral elements are parabolic). The peripheral elements of the thrice-punctured projective plane are conjugate to $A_1A_2$, $A_2A_3$, $A_3A_1$ or their inverses, so we impose the constraints: $$\begin{aligned}
\label{eq:cuspidal}
{\mathop{\rm tr}\nolimits}A_1A_2={\mathop{\rm tr}\nolimits}A_2A_3={\mathop{\rm tr}\nolimits}A_3A_1=2.\end{aligned}$$ Moreover, instead of considering ${\mathop{\rm SL}\nolimits}(2,\mathbb{C})$ characters, we consider characters of ${\mathop{\rm SL}\nolimits}^{\pm}(2,\mathbb{C})$-representations such that $$\begin{aligned}
\label{eq:1sided}
\det A_1=\det A_2=\det A_3=-1.\end{aligned}$$ This choice may seem a little unnatural in ${\mathop{\rm GL}\nolimits}(2,\mathbb{C})$ since we can simply replace $A_k$ by $iA_k$ to recover a ${\mathop{\rm SL}\nolimits}(2,\mathbb{C})$ character. We choose $\det A_k=-1$ because it is natural when restricting to Fuchsian representations. With this normalisation, Fuchsian representations $\rho:F_3\rightarrow {\mathop{\rm PSL}\nolimits}^{\pm}(2,\mathbb{R})$ have real coefficients. Note that when lifting from ${\mathop{\rm PSL}\nolimits}^{\pm}(2,\mathbb{R})$ to ${\mathop{\rm SL}\nolimits}^{\pm}(2,\mathbb{R})$, one-sided curves necessarily have negative determinant. Since $\det A_i=-1$, then $A_i$ and $A_i^{-1}$ are not conjugate (as they would be in ${\mathop{\rm SL}\nolimits}(2,\mathbb{C})$), we need to specify orientations on the simple closed curves representing their conjugacy classes. In figure \[fig:3\],
![Representative curves for ${\color{red}A_1},{\color{blue}A_2},{\color{green}A_3}$, ${\color{red}A_1}{\color{blue}A_2}{\color{green}A_3}$ and ${\color{red}A_1}{\color{green}A_3}{\color{blue}A_2}$.[]{data-label="fig:3"}](trace.eps)
we see that there is a choice of orientation for simple close curves representing $A_1$, $A_2$ and $A_3$ (anticlockwise) so that the curves representing $A_1A_2A_3$ and $A_1A_3A_2$ are simple. These are the two choices of $A_1^{\pm 1}A_2^{\pm 1}A_3^{\pm 1}$ (up to conjugation and inversion) which are simple, and we choose $$\begin{aligned}
A_4=A_0^{-1}=(A_1A_2A_3)^{-1}\text{ and }A_4'=(A_1A_3A_2)^{-1}.\end{aligned}$$ Set $a={\mathop{\rm tr}\nolimits}A_1$, $b={\mathop{\rm tr}\nolimits}A_2$, $c={\mathop{\rm tr}\nolimits}A_3$, $d={\mathop{\rm tr}\nolimits}A_4$ and $d'={\mathop{\rm tr}\nolimits}A_4'$, then reorganises to yield : $$\begin{aligned}
(a+b+c+d)^2=abcd\text{ and }(a+b+c+d')^2=abcd'\end{aligned}$$ which means that $(a,b,c,d)$ and $(a,b,c,d')$ are Markoff quads. In addition, since $d$ and $d'$ are the roots of the polynomial $$\begin{aligned}
p(x)=x^2+(2a+2b+2c-abc)x+(a+b+c)^2=(x-d)(x-d'),\end{aligned}$$ the following identities must hold: $$\begin{aligned}
\label{eq:edge}
d+d'+2a+2b+2c=abc\text{ and }dd'=(a+b+c)^2.\end{aligned}$$ It should be noted that these are precisely the sum and product relations in [@GolTra].
In [@MPTCha], Maloni, Palesi and Tan study a different *relative character variety* of representations of $F_3$ into ${\mathop{\rm SL}\nolimits}(2,\mathbb{C})$ which arises from the four-punctured sphere — their $A_i$ are constrained to be parabolic.
Successive applications of equation enables one to generate the trace (and hence the length) of every one-sided simple closed homotopy class on $X$. We now explain how to generate the traces of all of the 2-sided simple closed homotopy classes.Any two one-sided simple closed curves ${\color{red}\gamma_i},{\color{blue}\gamma_j}$ intersecting exactly once live inside an embedded punctured Möbius strip, as depicted in Figure \[fig:mob\]. They uniquely induce a two-sided simple closed curve as a boundary component (with the other boundary component peripheral in $N_{1,3}$).
![Punctured Möbius strip[]{data-label="fig:mob"}](mobius.eps){height="2.5cm"}
The following trace identity in ${\mathop{\rm GL}\nolimits}(2,\mathbb{C})$ $$\label{eq:mobid}
{\mathop{\rm tr}\nolimits}A_iA_j+\det A_j\cdot{\mathop{\rm tr}\nolimits}A_iA_j^{-1}={\mathop{\rm tr}\nolimits}A_i\cdot{\mathop{\rm tr}\nolimits}A_j$$ relates the complex lengths of peripheral curves $\alpha$ and $\beta$ of a punctured Möbius strip to the complex lengths of ${\color{red}\gamma_i}$ and ${\color{blue}\gamma_j}$: $$\label{eq:mobident}
\cosh\left(\tfrac{1}{2}\ell_{\alpha}\right)+\cosh\left(\tfrac{1}{2}\ell_{\beta}\right)=2\sinh\left(\tfrac{1}{2}\ell_{\gamma_i}\right)\sinh\left(\tfrac{1}{2}\ell_{\gamma_j}\right).$$ Since any 2-sided geodesic on a 3-cusped projective plane necessarily bounds a pair of pants, equation allows one to obtain the length of any 2-sided simple closed geodesic.
The Curve Complex
-----------------
Equations and give us an algorithm to generate the entire length spectrum of a 3-cusped projective plane, starting from a corresponding Markoff quad. The combinatorics of this algorithm can be stored in terms of the *curve complex* of $N_{1,3}$.Consider the geometric realisation of the abstract simplicial complex $\Omega^*$ with its $n$-simplices given by subsets of $n+1$ distinct homotopy classes of one-sided simple closed curves in $N_{1,3}$ that pairwise intersect once. Identifications of simplices as the faces of higher dimensional simplicies is given by inclusion. This is a pure simplicial $3$-complex, and its $1$-skeleton has been previously described by Scharlemann [@SchCom] as being the 1-skeleton of the cell complex formed from a tetrahedron by repeated stellar subdivision of the faces, but not the edges.The curve complex $\Omega$ that we’re concerned with is the dual of $\Omega^*$. The decision to take the dual accords with Bowditch’s conventions in [@BowPro; @BowMar]. We now describe and assign notation for the cells of $\Omega$.**The vertices, or $0$-cells of $\Omega$ are:** $$\begin{aligned}
\Omega^0:=\left\{
\{\alpha,\beta,\gamma,\delta\}\
\begin{array}{|l}
\alpha,\beta,\gamma,\delta\text{ are homotopy classes of one-sided simple closed}\\
\text{curves that pairwise geometrically intersect once}
\end{array}
\right\}\end{aligned}$$ **The edges, or $1$-cells of $\Omega$ are:** $$\begin{aligned}
\Omega^1:=\left\{
\{\alpha,\beta,\gamma\}\
\begin{array}{|l}
\alpha,\beta,\gamma\text{ are homotopy classes of one-sided simple closed}\\
\text{curves that pairwise geometrically intersect once}
\end{array}
\right\}\end{aligned}$$ Observe that each edge may be interpreted as a flip from one $0$-cell to another. Hence, the 1-skeleton of $\Omega$ is a 4-regular tree (i.e. each vertex has degree 4). Further, the connectedness of this cell-complex described in [@SchCom Theorem 3.1] means that flips generate all possible $0$-cells, and hence all one-sided simple closed geodesics.**The faces, or $2$-cells of $\Omega$ are:** $$\begin{aligned}
\Omega^2:=\left\{
\{\alpha,\beta\}\
\begin{array}{|l}
\alpha,\beta\text{ are homotopy classes of one-sided simple closed curves}\\
\text{that intersect geometrically once}
\end{array}
\right\}\end{aligned}$$ It follows from the observation in the previous subsection regarding punctured Möbius strips embedded in $S$ that each face represents a unique homotopy class of essential, non-peripheral two-sided simple closed curves on $N_{1,3}$.**The 3-cells $\Omega^3$ of $\Omega$ are:** $$\begin{aligned}
\Omega^3:=\left\{\ \{\alpha\}\mid \alpha\text{ is an homotopy class of one-sided simple closed curves }\right\}\end{aligned}$$ We later sometimes denote $3$-cells by capital letters, and use: $$\begin{aligned}
\vec{\Omega}^1=\left\{ \vec{e}=\{\alpha,\beta,\gamma;\delta'\rightarrow\delta\}\mid \{\alpha,\beta,\gamma\}\in\Omega^1\right\}.\end{aligned}$$ to denote the collection of *oriented edges* of $\Omega$. In particular, $\{\alpha,\beta,\gamma;\delta'\rightarrow\delta\}$ points from $\{\delta'\}$ to $\{\delta\}$. Figure \[fig:complex\] illustrates the local geometry of an oriented edge.
![A $3$-cell (left) and an oriented edge (right).[]{data-label="fig:complex"}](complex.eps)
Markoff maps and characters
---------------------------
Given a representation $\rho:F_3\to {\mathop{\rm GL}\nolimits}(2,\mathbb{C})$ satisfying the trace condition and the determinant condition , we use Greek letters for simple closed curves and the corresponding Latin letters for the trace of the image of any homotopy class it defines. We decorate $\Omega$ with trace data by assigning to every 3-cell $\{\alpha\}\in\Omega^3$ its corresponding trace ${\mathop{\rm tr}\nolimits}\rho(\alpha)=a$ thus defining a function: $$\begin{aligned}
\phi:\Omega^3\rightarrow\mathbb{C}\text{ by }\phi(\alpha)={\mathop{\rm tr}\nolimits}\rho(\alpha).\end{aligned}$$ A Markoff map induced from $\rho$ may be thought of as the character corresponding to $\rho$ restricted to one-sided simple closed homotopy classes. Our previous discussions in subsection \[sec:trid\] mean that the data carried by a Markoff map suffices to recover the whole character.We introduce the language of Markoff maps to allude to Bowditch’s work [@BowPro; @BowMar]; lower-dimensional simplices in $\Omega$ may be interpreted as mnemonics for encoding the following relations:**Vertex relation**: for $\{\alpha,\beta,\gamma,\delta\}\in\Omega^0$, is equivalent to: $$\label{vertrel}
\frac{d}{a+b+c+d}=\frac{a+b+c+d}{abc}.$$ where $a={\mathop{\rm tr}\nolimits}\rho(\alpha)$, $b={\mathop{\rm tr}\nolimits}\rho(\beta)$, $c={\mathop{\rm tr}\nolimits}\rho(\gamma)$ and $d={\mathop{\rm tr}\nolimits}\rho(\delta)$. Note that the set of values of any four 3-cells which meet at a vertex corresponds to a Markoff quad.**Edge relation**: an edge $e=(\alpha,\beta,\gamma)\in\Omega^1$ lies in the intersection of the two 0-cells $(\alpha,\beta,\gamma,\delta)$ and $(\alpha,\beta,\gamma,\delta')$, and yields: $$\begin{aligned}
\label{edgerel}
\frac{a+b+c+d}{abc}+\frac{a+b+c+d'}{abc}=1,\end{aligned}$$ where $d'={\mathop{\rm tr}\nolimits}\rho(\delta')$ and the others are as previously defined. Since each edge joins two vertices, the edge relation therefore tells us how to flip from one Markoff quad to another.**Face relation**: given $\{\alpha,\beta\}\in\Omega^2$ and $\epsilon$ the unique non-peripheral two-sided simple closed homotopy class disjoint from $\alpha$ and $\beta$, from we have: $$\begin{aligned}
\label{facerel}
ab=e+2\end{aligned}$$ where $a={\mathop{\rm tr}\nolimits}\rho(\alpha)$, $b={\mathop{\rm tr}\nolimits}\rho(\beta)$ and $e={\mathop{\rm tr}\nolimits}\rho(\epsilon)$.We stress once again that these three relations allow us to generate the character for $\rho$ from a starting Markoff quad: the vertex and edge relations generate the traces for all the one-sided simple closed curves and the face relation then produces the traces for all of the two-sided simple closed curves.Thus, we’re led to consider general maps $\phi:\Omega^3\rightarrow \mathbb{C}$ satisfying the edge and vertex relations. We call such functions *Markoff maps*, and let $\Phi$ denote the collection of all Markoff maps. In keeping with our notation for representations, we use Greek and Latin letters respectively for 3-cells (one-sided curves) and their image under some $\phi\in\Phi$.
\[lem:markoff\] The collection of Markoff maps and the collection of characters induced from ${\mathop{\rm SL}\nolimits}^{\pm}(2,\mathbb{C})$-representations satisfying and are in canonical bijection.
The restriction of any such character $\xi:F_3\rightarrow\mathbb{C}$ to the one-sided simple closed homotopy classes may be thought of as a Markoff map. Hence, it suffices to show that every Markoff map may be induced by a ${\mathop{\rm SL}\nolimits}^{\pm}(2,\mathbb{C})$-representations satisfying and .Given a Markoff map $\phi:\Omega^3\rightarrow\mathbb{C}$, if there is a 3-cell $\{\alpha\}$ on which $\phi(\{\alpha\})=0$, fix an arbitrary 0-cell $\{\alpha,\beta,\gamma,\delta\}$ lying on the boundary of $\{\alpha\}$. Then consider the representation: $$\begin{aligned}
\rho:F_3&=\left\langle \alpha,\beta,\gamma\right\rangle\rightarrow{\mathop{\rm SL}\nolimits}^{\pm}(2,\mathbb{C})\\
\alpha&\mapsto
\left[\begin{matrix}
0 & 1\\
1 & 0
\end{matrix}\right],
\beta\mapsto
\left[\begin{matrix}
b & 1\\
1 & 0
\end{matrix}\right],
\gamma\mapsto
\left[\begin{matrix}
0 & 1\\
1 & c
\end{matrix}\right].\notag\end{aligned}$$ It is easy to check that $\rho$ satisfies the desired trace and determinant conditions and that it induces $\phi$. If $\phi$ is nowhere-zero, choose an arbitrary 0-cell $\{\alpha,\beta,\gamma,\delta\}$, since $\phi$ is nowhere-zero, $$\begin{aligned}
(a+b+c+d)^2=abcd\neq0.\end{aligned}$$ This in turn means that the following representation is well-defined: $$\begin{aligned}
\label{eq:representation}
\rho:F_3&=\left\langle \alpha,\beta,\gamma\right\rangle\rightarrow{\mathop{\rm SL}\nolimits}^{\pm}(2,\mathbb{C})\\
\alpha&\mapsto
\frac{1}{a+b+c+d}\left[\begin{matrix}
ab & b(a+c)\\
a(a+d) & a(a+c+d)
\end{matrix}\right],\notag\\
\beta&\mapsto
\frac{1}{a+b+c+d}\left[\begin{matrix}
ab & -b(b+d)\\
-a(b+c) & b(b+c+d)
\end{matrix}\right],\notag\\
\gamma&\mapsto
\frac{1}{a+b+c+d}\left[\begin{matrix}
ab+c(a+b+c+d) & b(a+c)\\
-a(b+c) & -ab
\end{matrix}\right].\notag\end{aligned}$$ We omit the check to see that $\rho$ satisfies and and induces $\phi$.
The above lemma means that the relative character variety that we wish to study may be characterised as the variety of Markoff maps. Among this collection of Markoff maps, we wish to focus on those which correspond to characters of quasi-Fuchsian representations. This leads us to define BQ-Markoff maps:For $k\geq0$ and $\phi\in\Phi$, define the set $\Omega_\phi^3(k)\subseteq \Omega^3$ by: $$\begin{aligned}
\Omega_{\phi}^3(k):=\left\{\{\alpha\}\in\Omega^3\mid |\phi(\{\alpha\})|=|a|\leq k \right\}.\end{aligned}$$ This set allows us to keep track of one-sided simple curves with trace less than $k$, and we similarly define $\Omega^2_\phi(k)\subset\Omega^2$ for two-sided simple curves. Every two-sided simple curve corresponds to a unique 2-cell $\{\alpha,\beta\}$ — the shared face of the two 3-cells $\{\alpha\}$ and $\{\beta\}$. We define: $$\begin{aligned}
\Omega^2_\phi(k):=\left\{\{\alpha,\beta\}\in\Omega^2\mid |\phi(\{\alpha\})\phi(\{\beta\})|=|ab|\leq k\right\}.\end{aligned}$$ Note that in using $|ab|$ instead of $|ab-2|$ for the conditions imposed, the set $\Omega^2_\phi(k)$ doesn’t quite correspond to the set of two-sided simple curves with trace less than $k$. Although we will find this definition more suited to our analysis. In addition, we later focus on the following collection of Markoff maps $\Phi_{BQ}\subset\Phi$: $$\begin{aligned}
\label{BQ}
\Phi_{BQ}:=\left\{
\phi\in\Phi\
\begin{array}{|l}
\Omega^2_\phi(k)\text{ is finite for any }k,\\
\text{and for any }\{\alpha,\beta\}\in\Omega^2_\phi(4),\,ab\notin[0,4]
\end{array}
\right\}.\end{aligned}$$ We show in section \[sec:mcshane\] that these are sufficient conditions to guarantee the existence of a McShane identity for a given Markoff map. These conditions are similar to Bowditch’s BQ-condition, which is a conjectural trace-based characterisation of quasi-Fuchsian representations. The following result shows that our condition is also necessary for quasi-Fuchsian representations of the thrice-punctured projective plane.
Markoff maps obtained from quasi-Fuchsian representations lie in $\Phi_{BQ}$.
Given a Markoff map $\phi$ arising from a quasi-Fuchsian representation $\rho$, consider the multiset of complex numbers obtained from evaluating $\phi$ on $\Omega^3_\phi(m)$. Since this multiset is a subset of the simple trace spectrum of $\rho$, which is obtained (up to sign) from taking $2\sinh(\tfrac{1}{2}\cdot)$ of the simple length spectrum, the discreteness of the simple length spectrum ensures that $\Omega^3_\phi(m)$ is finite. Any 2-cell in $\Omega^2_\phi(m)$ is the intersection of precisely one pair of 3-cells, hence the cardinality of $\Omega^2_\phi(m)$ is bounded by the square of the cardinality of $\Omega^3_\phi(m)$ and is finite.Next, if $ab\in[0,4]$ for some $\{a,b\}\in\Omega^2$, then there is a 2-sided non-peripheral homotopy class $\epsilon$ whose trace is $ab-2\in[-2,2]$, thus contradicting the fact that quasi-Fuchsian representations have neither parabolics nor elliptics.
Markoff-Hurwitz numbers {#markoff-hurwitz}
-----------------------
The *Markoff-Hurwitz equation* is given by $$\begin{aligned}
\label{MarkHur}
a_1^2+...+a_n^2=a_1...a_n.\end{aligned}$$ Its quadratic nature means its solution variety admits a discrete group action generated by $a_i\mapsto a_1...a_n/a_i-a_i$, i.e. one can obtain new solutions from old via flips. Previously, only the $n=3$ solutions of the Markoff-Hurwitz equation were known to have a length spectrum interpretation—for a hyperbolic punctured torus. One observation of this paper is the association of solutions of the $n=4$ solutions of the Markoff-Hurwitz equation with Markoff quads, and hence the ${\mathop{\rm SL}\nolimits}^{\pm}(2,\mathbb{C})$ relative character variety of the thrice-punctured projective plane. The substitution $a=a_1^2$, $b=a_2^2$, $c=a_3^2$, $d=a_4^2$ into defines a map between solutions $(a_1,\ldots,a_4)$ of for $n=4$ and Markoff quads. The map is a quotient by the $Z_2^3$-action $(a_1,a_2,a_3,a_4)\mapsto(\pm a_1,\pm a_2,\pm a_3,\pm a_4)$ (even number of minus signs) on solutions of the Markoff-Hurwitz equation. Flips of Markoff quads correspond to flips of solutions of the Markoff-Hurwitz equation.
Teichmüller space {#sec:teich}
-----------------
The remainder of this section deals with Markoff maps corresponding to Fuchsian representations. From the proof of lemma \[lem:markoff\], we see that these are precisely the real Markoff maps. We focus on the Teichmüller component of the real relative character variety, showing that it consists of the positive real Markoff maps.
The Teichmüller space $\mathcal{T}(S)$ of a surface $S$ encodes all the ways of assigning a complete finite-area hyperbolic metric to $S$, up to homotopy. Concretely, it may be expressed as: $$\begin{aligned}
\mathcal{T}(S):=\{\, (X,f)\mid f:S\rightarrow X\text{ is a homeomorphism }\}/\sim\end{aligned}$$ where $(X_1,f_1)\sim(X_2,f_2)$ if and only if $$\begin{aligned}
f_2\circ f_1^{-1}:X_1\rightarrow X_2\end{aligned}$$ is homotopy equivalent to a hyperbolic isometry. We denote these equivalence classes, or *marked surfaces*, by $[X,f]$.An *ideal triangulation* of $S$ is, up to homotopy, a triangulation of $S$ with vertices at the punctures of $S$. Given a marked surface $[X,f]$, the image $f(\sigma)$ of an arc $\sigma$ on $S$ pulls tight to a unique homotopy equivalent geodesic arc on $X$. Thus, any ideal triangulation on $S$ is represented by an (geodesic) ideal triangulation on $X$ — a maximal collection of simple bi-infinite geodesic arcs with both ends up cusps. For our purposes, we restrict to ideal triangulations $\triangle$ on thrice-punctured projective planes $S$ representable by paths with *distinct end points*.Horocycles of length $1$ around a cusp are always simple on a complete hyperbolic surface. Thus, given an ordered ideal triangulation $(\sigma_1,\sigma_2,\sigma_3,\tau_1,\tau_2,\tau_3)$ on $X$, we obtain lengths $(s_1,s_2,s_3,t_1,t_2,t_3)$ of these infinite geodesic arcs truncated at the three length $1$ horocycles bounding cusps $1,2,3$. The $\lambda$-lengths for $X$ with respect to this ordered ideal triangulation is then given by: $$\begin{aligned}
(\lambda_1,\lambda_2,\lambda_3,\mu_1,\mu_2,\mu_3)=(\exp\tfrac{1}{2}s_1,\exp\tfrac{1}{2}s_2,\exp\tfrac{1}{2}s_3,\exp\tfrac{1}{2}t_1,\exp\tfrac{1}{2}t_2,\exp\tfrac{1}{2}t_3).\end{aligned}$$ In [@PenDec], Penner shows that these $\lambda$-lengths form global coordinates on the Teichmüller space of any punctured surface. This is also true for the Teichmüller space of punctured non-orientable surfaces.The following lemma is a topological correspondence which is promoted to a geometric correspondence below.
\[th:bijection\] There is a natural bijection between $\Omega^0$ and $$\begin{aligned}
\left\{
\text{ the collection of ideal triangulations of }S\text{ with distinct end points }
\right\}\end{aligned}$$ given by sending $\{\alpha,\beta,\gamma,\delta\}\in\Omega^0$ to the unique (up to homotopy) ideal triangulation where each arc intersects precisely two of the geodesics in $\{\alpha,\beta,\gamma,\delta\}$.
Any essential two-sided simple closed curve is a boundary component of a thickening of a unique pair of once-intersecting one-sided simple closed curves (figure \[fig:mob\]), and a boundary component of a thickening of a unique arc joining distinct punctures. By alternately thinking of a two-sided curve as boundary components of these two thickenings, we see that pairs of intersection points between two 2-sided simple closed curves correspond to single intersection points between two one-sided simple closed curves and between two arcs joining distinct punctures (where the punctures count as single intersection points). Hence the six arcs obtained in this way are disjoint outside the punctures if and only if the homotopy classes in $\{\alpha,\beta,\gamma,\delta\}\in\Omega^0$ pairwise intersect exactly once.
![A 4-tuple of [curves]{} corresponding to a [triangulation]{}.[]{data-label="fig:triangulation"}](triangulation.eps)
The $\lambda$-lengths for an ideal triangulation $\triangle$ of $N_{1,3}$ identifies the Teichmüller space $\mathcal{T}(N_{1,3})$ as: $$\begin{aligned}
\left\{
\hspace{-1mm}\begin{array}{r|l}&\mu_1\mu_2\mu_3+\mu_1\lambda_2\lambda_3+\lambda_1\mu_2\lambda_3+\lambda_1\lambda_2\mu_3=\lambda_1\lambda_2\mu_1\mu_2\\
(\lambda_1,\lambda_2,\lambda_3,\mu_1,\mu_2,\mu_3)\in\mathbb{R}_+^6
& \mu_1\mu_2\mu_3+\mu_1\lambda_2\lambda_3+\lambda_1\mu_2\lambda_3+\lambda_1\lambda_2\mu_3=\lambda_1\lambda_3\mu_1\mu_3\\
& \mu_1\mu_2\mu_3+\mu_1\lambda_2\lambda_3+\lambda_1\mu_2\lambda_3+\lambda_1\lambda_2\mu_3=\lambda_2\lambda_3\mu_2\mu_3
\end{array}
\right\}.\end{aligned}$$
These $\lambda$-lengths of an ideal triangulation may be expressed in terms of the Markoff quad of the associated quadruple of one-sided geodesics in $\Omega^0$ corresponding to the used to the define these $\lambda$-lengths. $$\begin{aligned}
\label{quambda}
&(a,b,c,d)=
\left(\frac{\lambda_2\lambda_3}{\lambda_1}, \frac{\lambda_1\lambda_3}{\lambda_2},\frac{\lambda_1\lambda_2}{\lambda_3}, \frac{\mu_1\mu_2}{\lambda_3}=\frac{\mu_1\mu_3}{\lambda_2}=\frac{\mu_2\mu_3}{\lambda_1}\right),\\
&(\lambda_1,\lambda_2,\lambda_3,\mu_1,\mu_2,\mu_3)=(\sqrt{bc},\sqrt{ac},\sqrt{ab},\sqrt{ad},\sqrt{bd},\sqrt{cd}\,).
\notag\end{aligned}$$
Thus, we may also use positive Markoff quads to globally parametrise the Teichmüller space:
\[th:teich\] Given an ordered 4-tuple $(\alpha,\beta,\gamma,\delta)$ intersecting a fixed triangulation on $N_{1,3}$ as per figure \[fig:triangulation\], then the map $$\begin{aligned}
\mathcal{T}(N_{1,3})&\rightarrow \left\{(a,b,c,d)\in\mathbb{R}_+^4\mid(a+b+c+d)^2=abcd\right\}\\
[X,f]&\mapsto (2\sinh\tfrac{1}{2}\ell_{\alpha}(X),2\sinh\tfrac{1}{2}\ell_{\beta}(X),2\sinh\tfrac{1}{2}\ell_{\gamma}(X),2\sinh\tfrac{1}{2}\ell_{\delta}(X))\end{aligned}$$ is a real-analytic diffeomorphism, where $\ell_{\alpha}(X)$ denotes the length of the geodesic representative of $f_*(\alpha)$ on $X$. We call these global coordinates the *trace coordinates* for $\mathcal{T}(N_{1,3})$.
With a little hyperbolic trigonometry and successive applications of the ideal Ptolemy relation [@HuaMod], we can show that explicitly gives the desired diffeomorphism between the trace coordinates and the $\lambda$-coordinates for $\mathcal{T}(N_{1,3})$. The fact that this map is real-analytic is then a simple consequence of the real-analyticity of the $\lambda$-lengths.
The set of positive Markoff quads is the Teichmüller component of the real character variety.
Proposition \[th:teich\] proves that the set of positive Markoff quads is real-analytically diffeomorphic to Teichmüller space. It remains to show that it is a connected component of the real character variety. Suppose that one of the coordinates vanishes, say $d=0$. Then by , $a+b+c=0$. But if this point lies in the limit of a path in the set of *positive* Markoff quads then each of $a$, $b$ and $c$ must tend to 0 along the path. In particular, at some point on the path $abcd<256$. But this contradicts since $$\begin{aligned}
(a+b+c+d)^2\geq 16\sqrt{abcd}>abcd\end{aligned}$$ where the first inequality is the arithmetic mean-geometric mean inequality.
The mapping class group. {#sec:mcg}
------------------------
Markoff quads are points on the ${\mathop{\rm SL}\nolimits}^{\pm}(2,\mathbb{C})$ relative character variety of the thrice-punctured projective plane, which is the hypersurface in $\mathbb{C}^4$ defined by equation . The transformation , $$\begin{aligned}
(a,b,c,d)\mapsto(a,b,c,d'=abc-2a-2b-2c-d),\end{aligned}$$ combined with the following even permutations $$\begin{aligned}
(a,b,c,d)\mapsto(b,a,d,c),(c,d,a,b),(d,c,b,a)\end{aligned}$$ generate the pure mapping class group of the thrice-punctured projective plane, and specify its action on the corresponding relative character variety. *Pure* here means that we restrict to elements of the mapping class group that fix punctures.We construct an explicit homeomorphism $f_4:N_{1,3}\rightarrow N_{1,3}$ that takes $(\alpha,\beta,\gamma,\delta)$ to $(\alpha,\beta,\gamma,\delta')$. Consider the hexagonal fundamental domain of $N_{1,3}$ obtained by cutting along $\sigma_1,\sigma_2$ and $\sigma_3$.
![The map $f_4$ fixing ${\color{red}\alpha},{\color{red}\beta},{\color{red}\gamma}$, but switching ${\color{blue}\delta}$ and ${\color{blue}\delta'}$.[]{data-label="fig:flipmap"}](flipmap.eps)
From figure \[fig:flipmap\], we see that a rotation by $\pi$ of this fundamental domain fixes the labeling of the punctures and fixes each of $\alpha$, $\beta$ and $\gamma$ whilst taking $\alpha$ to $\alpha'$. The action of the mapping class $[f_4]\in\Gamma(N_{1,3})$ therefore takes the Markoff quad $(a,b,c,d)$ corresponding to a marked surface $[X,f]$ to $$\begin{aligned}
[f_4](a,b,c,d)=(a,b,c,abc-2a-2b-2c-d),\end{aligned}$$ that is: $[f_4]$ corresponds to a flip in the fourth entry. By symmetry, there are four flips $[f_1],[f_2],[f_3],[f_4]\in\Gamma(N_{1,3})$ which flip the corresponding entries of $(a,b,c,d)$. Let $F\leq\Gamma(N_{1,3})$ denote the subgroup generated by these four flips.
$F\cong\mathbb{Z}_2\ast\mathbb{Z}_2\ast\mathbb{Z}_2\ast\mathbb{Z}_2$, where each $\mathbb{Z}_2$ is generated by one of the $[f_i]$.
First observe that each $[f_i]$ is indeed order $2$. To see that there are no other relations, consider the action of a reduced string of flips on the 1-skeleton of the curve complex: since the 1-skeleton is a 4-regular tree, performing each flip in a sequence of flips necessarily takes us farther from the origin.
We now consider a different subgroup in $\Gamma(N_{1,3})$: the stabiliser of $\{\alpha,\beta,\gamma,\delta\}$. Due to lemma \[th:bijection\], this subgroup must also stabilise $\triangle$ — the triangulation corresponding to $\{\alpha,\beta,\gamma,\delta\}$.
$\mathrm{Stab}(\triangle)\cong\mathbb{Z}_2\times\mathbb{Z}_2$.
There are four triangles $T_1,T_2,T_3,T_4$ induced by the triangulation $\triangle$ on $N_{1,3}$, and any element of $\mathrm{Stab}(\triangle)$ must take $T_1$ to one these four triangles. Since there is a unique way to map $T_1$ to any of these four triangles so as to preserve puncture-labeling, knowing the image of $T_1$ determines the entire mapping class. By symmetry, these mapping classes must have the same order, hence $\mathrm{Stab}(\triangle)$ is the Klein four-group.
This stabiliser subgroup is given by: $$\begin{aligned}
\left\{[\mathrm{id}],
\begin{array}{l}
[\varphi_1]:(a,b,c,d)\mapsto (b,a,d,c),\\
{[\varphi_2]}:(a,b,c,d)\mapsto(c,d,a,b), \\
{[\varphi_3]}:(a,b,c,d)\mapsto (d,c,b,a),
\end{array}
\right\}\end{aligned}$$ when thought of as acting on the trace coordinates for $\mathcal{T}(N_{1,3})$.
The subgroups $F$ and $\mathrm{Stab}(\triangle)$ generate the whole mapping class group $\Gamma(N_{1,3})$.
Given an arbitrary element $[h]\in\Gamma(N_{1,3})$, the action of $[h]$ on $(a,b,c,d)=[X,f]\in\mathcal{T}(N_{1,3})$ produces another Markoff quad $(\bar{a},\bar{b},\bar{c},\bar{d})$ corresponding to the traces of $(h_*(\alpha), h_*(\beta),h_*(\gamma),h_*(\delta)$). Since the four flips $[f_i]$ generate all Markoff quads associated to the Fuchsian representation for $X$, there is an element $[g]\in F$ such that $[g]\circ[h]=[g\circ h]$ simply permutes $a,b,c,d$. By choosing $X$ to be a surface where there are only four simple one-sided geodesics with traces $\{a,b,c,d\}$ (e.g.: the $(4,4,4,4)$ surface), we see that $[g\circ h]\in\mathrm{Stab}(\triangle)$.
$F$ is a normal subgroup of $\Gamma(N_{1,3})$.
Note that it suffices to show that $\mathrm{Stab}(\triangle)$ preserves $\{[f_1],[f_2],[f_3],[f_4]\}$. We perform this check for $[f_1]$, the rest follow by symmetry: $$[\varphi_1]^{-1}\circ [f_1]\circ [\varphi_1]=[f_2],\,[\varphi_2]^{-1}\circ [f_1]\circ [\varphi_2]=[f_3]\text{, and }[\varphi_3]^{-1}\circ [f_1]\circ [\varphi_3]=[f_4].$$
Since $F$ and $\mathrm{Stab}$ generate $\Gamma(N_{1,3})$ and their intersection is the trivial group, we obtain the following result:
$\Gamma(N_{1,3})=F\rtimes\mathrm{Stab}(\triangle)\cong(\mathbb{Z}_2\ast\mathbb{Z}_2\ast\mathbb{Z}_2\ast\mathbb{Z}_2)\rtimes(\mathbb{Z}_2\times\mathbb{Z}_2)\cong\mathbb{Z}_2\ast(\mathbb{Z}_2\times\mathbb{Z}_2)$. In particular: $$\begin{aligned}
\Gamma(N_{1,3})&\cong
\left\langle
\begin{array}{r|l}
f_1,f_2,f_3,f_4, & f_1^2=f_2^2=f_3^2=f_4^2=g^2=h^2=1,gh=hg\\
g,h & g^{-1}f_1g=f_2,h^{-1}f_1h=f_3,g^{-1}f_3g=f_4
\end{array}
\right\rangle\\
&\cong\left\langle f,g,h\mid f^2=g^2=h^2=1,gh=hg\right\rangle.\end{aligned}$$
The moduli space
----------------
Recall that the moduli space $\mathcal{M}(N_{1,3})$ of hyperbolic structures on $N_{1,3}$ is given by $\mathcal{T}(N_{1,3})/\Gamma(N_{1,3})$. Since $F$ is a normal subgroup of $\Gamma(N_{1,3})$, the space $\mathcal{T}(N_{1,3})/F$ must be a finite cover of $\mathcal{M}(N_{1,3})$. To better see what $\mathcal{T}(N_{1,3})/F$ looks like, we first define another global coordinate chart for $\mathcal{T}(N_{1,3})$.
\[lem:horo\] The Teichmüller space $\mathcal{T}(N_{1,3})$ may be real-analytically identified with the following (open) 3-simplex: $$\begin{aligned}
\{(H_a,H_b,H_c,H_d)\in\mathbb{R}_+^4\mid H_a+H_b+H_c+H_d=1\},\end{aligned}$$ we call this the *horocyclic coordinate* for $\mathcal{T}(N_{1,3})$.
The explicit diffeomorphisms between the horocyclic coordinates and the trace coordinates is given as follows: $$\begin{aligned}
H_a=\sqrt{\frac{a}{bcd}}=\frac{a}{a+b+c+d},\,&H_b=\sqrt{\frac{b}{acd}}=\frac{b}{a+b+c+d}\\
H_c=\sqrt{\frac{c}{abd}}=\frac{c}{a+b+c+d},\,&H_d=\sqrt{\frac{d}{abc}}=\frac{d}{a+b+c+d},\end{aligned}$$ and the inverse map is given by: $$a=\sqrt{\frac{H_a}{H_bH_cH_d}},\,b=\sqrt{\frac{H_b}{H_aH_cH_d}},\,c=\sqrt{\frac{H_c}{H_aH_bH_d}},\,d=\sqrt{\frac{H_d}{H_aH_bH_c}}.$$
The horocyclic coordinates are so named because they correspond to the lengths of horocyclic segments on the length $1$ horocycles at the cusps of a marked surface $[X,f]$. Coupled with the labeling in figure \[fig:triangulation\], figure \[fig:horocycle\] illustrates this correspondence.
![Horocyclic segments (each) of length ${\color{red}H_a},{\color{blue}H_b},{\color{green}H_c},{\color{orange}H_d}$.[]{data-label="fig:horocycle"}](horocycle.eps)
\[thm:moduli\] The moduli space $\mathcal{M}(N_{1,3})$ of a thrice-punctured projective plane is homeomorphic to an open $3$-ball with an open hemisphere of order 2 orbifold points glued on, and a line of orbifold points running straight through the center of this $3$-ball — joining two antipodal points of this orbifold hemisphere. The orbifold points on this line are of order $2$, except for the very center point of the $3$-ball, which is order 4.
In the horocyclic coordinates described in lemma \[lem:horo\], the flips generating $F$ act as follows: $$\begin{aligned}
[f_1]&:(H_a,H_b,H_c,H_d)\mapsto(1-H_a,H_b\frac{H_a}{1-H_a},H_c\frac{H_a}{1-H_a},H_d\frac{H_a}{1-H_a}),\\
[f_2]&:(H_a,H_b,H_c,H_d)\mapsto(H_a\frac{H_b}{1-H_b},1-H_b,H_c\frac{H_b}{1-H_b},H_d\frac{H_b}{1-H_b}),\\
[f_3]&:(H_a,H_b,H_c,H_d)\mapsto(H_a\frac{H_c}{1-H_c},H_b\frac{H_c}{1-H_c},1-H_c,H_d\frac{H_c}{1-H_c}),\\
[f_4]&:(H_a,H_b,H_c,H_d)\mapsto(H_a\frac{H_d}{1-H_d},H_b\frac{H_d}{1-H_d},H_c\frac{H_d}{1-H_d},1-H_d).\end{aligned}$$
From this, we see that the fixed points of $[f_1],[f_2],[f_3],[f_4]$ are respectively given by imposing the following conditions on the horocyclic coordinates: $$\begin{aligned}
H_a=\tfrac{1}{2},H_b=\tfrac{1}{2},H_c=\tfrac{1}{2},H_d=\tfrac{1}{2}.\end{aligned}$$ The region in $\mathcal{T}(N_{1,3})$ enclosed by these four planes is therefore a fundamental domain for $\mathcal{T}(N_{1,3})/F$. In this case, this fundamental domain is an octahedron. Since $[f_1]$ acts by swapping the two regions separated by $H_a=\tfrac{1}{2}$, the image of these fixed points in $\mathcal{T}(N_{1,3})/F$ are order 2 (reflection) orbifold points. Similar comments hold for each of the $[f_i]$. Thus, $\mathcal{T}(N_{1,3})/F$ is an open octahedron with four triangles of order 2 orbifold points glued onto a collection of four non-adjacent sides.Finally, by noting that $\mathrm{Stab}(\triangle)$ acts on the horocyclic coordinates by: $$\begin{aligned}
[\varphi_1]&:(H_a,H_b,H_c,H_d)\mapsto(H_b,H_a,H_d,H_c),\\
[\varphi_2]&:(H_a,H_b,H_c,H_d)\mapsto(H_c,H_d,H_a,H_b),\\
[\varphi_3]&:(H_a,H_b,H_c,H_d)\mapsto(H_d,H_c,H_b,H_a).\end{aligned}$$ We obtain the desired result.
The interior $3$-ball of $\mathcal{M}(N_{1,3})$ may be geometrically interpreted as the set of 3-cusped projective planes which have a unique unordered 4-tuple of geodesics whose flips are strictly longer.
Integral Markoff quads
----------------------
To conclude this section, we characterise the positive integral Markoff quads. Positive integral Markoff triples are of importance in number theory. They arise in approximating real numbers [@CasInt], the Markoff spectrum is closely related to the Lagrange spectrum and Markoff’s theorem provides an integral Markoff triples-based characterisation of indefinite binary quadratic forms [@CohApp; @MarSur].
Every positive integer Markoff quad may be generated by a sequence of flips and coordinate permutations from precisely one of the following eight integer Markoff quads: $$\begin{aligned}
\begin{array}{cccc}
(1,5,24,30),&(1,6,14,21),&(1,8,9,18),&(1,9,10,10),\\
(2,3,10,15),&(2,5,5,8),&(3,3,6,6),&(4,4,4,4).
\end{array}\end{aligned}$$
The edge relation(s) tell us that any flip on a positive integral Markoff quad $(a,b,c,d)$ results in another positive integral Markoff quad. Thus, by applying a sequence of flips, we may assume that $(a,b,c,d)$ lies in the fundamental domain of the moduli space described in the proof of theorem \[thm:moduli\]. Furthermore, up to reordering, we may assume wlog that $0<a\leq b\leq c\leq d$. The fact that $(a,b,c,d)$ lies in this fundamental domain means that $\frac{d}{a+b+c+d}\leq\frac{1}{2}$, and hence $d\leq a+b+c$. Thus, $$\begin{gathered}
2(a+b+c)\geq a+b+c+d=\sqrt{abcd}\geq c\sqrt{ab},\\
\Rightarrow4c\geq2(a+b)\geq c(\sqrt{ab}-2),\\
\Rightarrow4\geq\sqrt{ab}-2\text{ and hence }36\geq ab.\end{gathered}$$ Moreover, if $ab\leq4$, then substituting this into equation : $$\begin{gathered}
(a+b)^2+2(a+b)(c+d)+(c+d)^2=abcd\leq4cd\\
\Rightarrow (a+b)^2+(c-d)^2\leq0.\end{gathered}$$ As this is impossible, we conclude that $5\leq ab\leq36$. If $a=1$, then $5\leq b\leq 36$ and hence $a+b\leq37$ and $c\geq d-37$. Thus: $$\begin{gathered}
(37+2d)^2\geq(a+b+c+d)^2=abcd\geq5(d-37)d.\end{gathered}$$ Solving for this quadratic over the integers shows that $1\leq d\leq 337$. This in turn also limits the possible values for $c\leq d$. Performing similar computations for $a=2,3,4$, we obtain the following cases: $$\begin{aligned}
\begin{array}{lll}
\text{if }a=1,& 5\leq b\leq 36, & 5\leq\max\{b,d-(a+b)\}\leq c\leq d\leq 337;\\
\text{if }a=2,& 3\leq b\leq 18, & 3\leq\max\{b,d-(a+b)\}\leq c\leq d\leq 101;\\
\text{if }a=3,& 3\leq b\leq 12, & 3\leq\max\{b,d-(a+b)\}\leq c\leq d\leq 40;\\
\text{if }a=4,& 4\leq b\leq 9, & 4\leq\max\{b,d-(a+b)\}\leq c\leq d\leq 26.
\end{array}\end{aligned}$$ It is unnecessary to consider the cases $a=5,6$ due to theorem \[th:systole\], specifically: the fact that $(a,b,c,d)$ lie on the fundamental domain of the moduli space means that $a$ must be the trace of the systolic homotopy class on the surface corresponding to $(a,b,c,d)$, and we know that this trace can at most be equal to $4$.Checking through these possible values for $(a,b,c,d)$ on a computer then completes this proof.
Analysis on the Curve Complex {#sec:cc}
=============================
Given a Markoff map $\phi$, for every edge $e=\{\alpha,\beta,\gamma\}$ fix an oriented edge $$\begin{aligned}
\vec{e}=\{\alpha,\beta,\gamma;\delta'\rightarrow \delta\}\text{ to satisfy }|d'|\geq |d|,\end{aligned}$$ where as usual $(a,b,c,d)=(\phi(\alpha),\phi(\beta),\phi(\gamma),\phi(\delta))$. For most edges, this choice is canonical, and for edges with equality in $|d'|=|d|$, an arbitrary orientation is chosen. This produces an orientation on $\Omega^1$, where edges may be thought of as pointing from 3-cells corresponding to longer geodesics to 3-cells corresponding to shorter geodesics. Thus, analysis of the dynamics (in terms of the directions) of these edges informs us about the behaviour of geodesic length growth for $\phi$.The following lemma gives alternative algebraic characterisations of this trace comparison.
For a Markoff quad $(a,b,c,d)\in\mathbb{C}^4$, the following conditions are equivalent: $$\begin{aligned}
\label{eq:oredge}
\mathrm{Re}\left(\tfrac{a+b+c+d'}{abc}\right)\geq\mathrm{Re}\left(\tfrac{a+b+c+d}{abc}\right)
\ \ \Leftrightarrow\ \ {\rm Re}\left(\tfrac{d'}{a+b+c+d'}\right)\geq\tfrac{1}{2}\ \ \Leftrightarrow\ \ |d'|\geq |d|.\end{aligned}$$
The edge relation proves the first equivalence. Furthermore also proves that ${\rm Im}\left(\tfrac{a+b+c+d'}{abc}\right)=-{\rm Im}\left(\tfrac{a+b+c+d}{abc}\right)$ hence the first inequality is equivalent to $$\begin{aligned}
\left|\frac{a+b+c+d'}{abc}\right|\geq\left|\frac{a+b+c+d}{abc}\right|\end{aligned}$$ which is equivalent to $$\begin{aligned}
\left|\frac{(a+b+c+d')^2}{abc}\right|\geq\left|\frac{(a+b+c+d)^2}{abc}\right|.\end{aligned}$$ By the vertex relation , this is precisely $|d'|\geq |d|$.
Local analysis
--------------
Call a vertex with all outward pointed oriented edges a *source*, a vertex with all inwardly pointed oriented edges a *sink* and a vertex with precisely one outwardly pointed oriented edge a *funnel*. The remaining two types of vertices are called *saddles*.
\[th:nosource\] There are no sources.
Given such a vertex with adjoining $3$-cells $A,B,C,D$, the vertex relation gives: $$\begin{aligned}
1=&\mathrm{Re}\left(\frac{a+b+c+d}{abc}+\frac{a+b+c+d}{abd}+\frac{a+b+c+d}{acd}+\frac{a+b+c+d}{bcd}\right)\\
\geq&\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=2\end{aligned}$$ which is a contradiction.
\[rmK:cell\] In the Teichmüller component, Markoff maps take on real positive values and the proof of Lemma \[th:nosource\] shows that any vertex has at most one outgoing edge. This means that Fuchsian Markoff maps can have at most one sink (given some choice of orientation). We later show in theorem \[th:4nonempty\] that a sink always exists, and this may be geometrically interpreted as saying that there is a unique (up to permutation) Markoff quad $(a,b,c,d)$ for any 3-cusped projective plane where the flips of $a,b,c$ of $d$ are (non-strictly) longer. However, the Markoff quad $(a,b,c,d)=\frac{i}{\sqrt{2}}(1,1,1,-2)$ is an example of a vertex with three outgoing edges.
At a sink vertex, a Markoff quad contains an element of magnitude less than or equal to 4.
Let the real parts of $$\begin{aligned}
\frac{a+b+c+d}{abc},\frac{a+b+c+d}{abd},\frac{a+b+c+d}{acd},\frac{a+b+c+d}{bcd},\end{aligned}$$ respectively be $s\leq r\leq q\leq p$.\
Then the sink part tells us that $p\leq \frac{1}{2}$ and the size ordering and the fact that $s+r+q+p=1$ tells us that $p\geq\frac{1}{4}$. And we also know that the next largest number $q\geq\frac{1}{3}(1-p)$. Therefore, $$\begin{aligned}
pq\geq\frac{1}{3}p(1-p)\geq\frac{1}{3}\times\frac{1}{4}\times\frac{3}{4}=\frac{1}{16}.\end{aligned}$$ Then, we see that: $$\begin{aligned}
\frac{1}{|cd|}=|\frac{a+b+c+d}{bcd}|\times|\frac{a+b+c+d}{acd}|\geq pq\geq \frac{1}{16}.\end{aligned}$$ Therefore, $|cd|\leq 16$ and the lesser of the magnitudes of these two traces must be less than or equal to $4$.
\[th:<2\] Given a saddle vertex $\{\alpha,\beta,\gamma,\delta\}$ with two outgoing oriented edges $$\begin{aligned}
\{\alpha,\beta,\gamma;\delta'\rightarrow \delta\}\text{ and }\{\alpha,\beta,\delta:\gamma'\rightarrow \gamma\},\end{aligned}$$ then the 2-cell $\{\alpha,\beta\}$ lies in $\Omega^2_\phi(4)$ and at least one of $\{\alpha\},\{\beta\}$ lies in $\Omega^3_\phi(2)$.
The outwards pointing condition tells us that: $$\begin{aligned}
\left|\frac{c}{a+b+c+d}\right|\geq\frac{1}{2}\text{ and }\left|\frac{d}{a+b+c+d}\right|\geq\frac{1}{2}.\end{aligned}$$ Multiplying these terms together, we have: $$\left|\frac{cd}{(a+b+c+d)^2}\right|=\frac{1}{|ab|}\geq\frac{1}{4}\Rightarrow|ab|\leq4\Rightarrow\mathrm{min}\{|a|,|b|\}\leq2.$$
Global analysis
---------------
\[th:conn\] For $k\geq2$, the cell complex comprised of all the $3$-cells in $\Omega_\phi^3(k)$ is connected.
Assume that $\Omega_\phi^3(k)$ isn’t connected and consider a shortest path of oriented edges between two distinct connected components: $$\begin{aligned}
\vec{e}_1,\vec{e}_2,\ldots,\vec{e}_{p-1},\vec{e}_p.\end{aligned}$$ Note that by assumption, any $3$-cell $X$ that contains one of these edges must satisfy $|\phi(X)|>k$.\
If $p=1$, then $\vec{e}_1=\{\alpha,\beta,\gamma;\delta\rightarrow \delta'\}$ such that $|d|,|d'|\leq k$. Then the edge relation gives $abc=(a+b+c+d)+(a+b+c+d')$ hence: $$\begin{aligned}
\sqrt{k^3}<\sqrt{|abc|}\leq&\frac{|a+b+c+d|+|a+b+c+d'|}{\sqrt{|abc|}}\\
=&\sqrt{|d|}+\sqrt{|d'|}\leq2\sqrt{k}\\
\Rightarrow k^3&\leq 4k\Rightarrow k\leq 2,\end{aligned}$$ where the first equality uses the vertex relation . This contradicts the assumption.\
On the other hand, if $p\geq2$, then $\vec{e}_1$ must point away from $\vec{e}_2$ and $\vec{e}_p$ must point away from $\vec{e}_{p-1}$. But this means that at least one of the interior vertices of the path $\{\vec{e}_n\}_{n=1,\ldots,p}$ must have two arrows pointing away from it, and hence by Lemma \[th:<2\] one of the adjacent $3$-cells $X$ of this vertex must satisfy $|\phi(X)|\leq 2$, thus contradicting the assumption.
\[ray\] Given an infinite ray of oriented edges $\{\vec{e}_n\}_{n\in{\mathbb{N}}}$ such that each $\vec{e}_n$ is directed towards $\vec{e}_{n+1}$, then this ray either:
1. eventually spirals along the boundary of some 2-cell $\{\xi,\eta\}\in\Omega^2_\phi(4)$, or
2. eventually enters and remains on the boundary of some 3-cell $\{\xi\}\in\Omega^3_\phi(2)$, or
3. there are infinitely many 3-cells in $\Omega^3_\phi(2)$.
We begin by four-colouring $\Omega^3$ with the colours $\alpha,\beta,\gamma,\delta$. In particular, we label the $3$-cells meeting $\{\vec{e}_n\}$ by $\{\alpha_i\},\{\beta_j\},\{\gamma_k\},\{\delta_l\}$ where the letter type is determined by the colour of the cell and the subscripts grow according to how early we encounter this $3$-cell as we traverse along $\{\vec{e}_n\}$.\
At each vertex along $\{\vec{e}_n\}$, we encounter six $2$-cells of different colour-types: $$\begin{aligned}
\{\alpha\beta,\alpha\gamma,\alpha\delta,\beta\gamma,\beta\delta,\gamma\delta\}.\end{aligned}$$ Now consider all the $2$-cells that we encounter as we go along $\{\vec{e}_n\}$. Since $\{\vec{e}_n\}$ does not repeat its edges, if we ever meet only finitely many $2$-cells of a certain colour-type, then $\{\vec{e}_n\}$ eventually just stays on the last $2$-cell we meet of that cell-type. This also means that we can’t meet only finitely many $2$-cells of two different colour-types because we must then stay on two distinct $2$-cells - impossible because the intersection of any two $2$-cells is either empty or consists of a single edge.\
Assume that we encounter only finitely many $2$-cells of the (wlog) $\alpha\beta$ colour-type and that the $2$-cell that we stay on is (with a little notation abuse) $\{\alpha,\beta\}$. This means that the vertices of $\{\vec{e}_n\}$ eventually take the following form: $$\begin{aligned}
\ldots,\{\alpha,\beta,\gamma_i,\delta_j\},\{\alpha,\beta,\gamma_{i+1},\delta_j\},\{\alpha,\beta,\gamma_{i+1},\delta_{j+1}\},\{\alpha,\beta,\gamma_{i+2},\delta_{j+1}\},\ldots\end{aligned}$$ and the sequences $\{|c_i|\},\{|d_j|\}$ are (monotonically) non-increasing due to the directions of the oriented edges. Alternatively, we phrase this as the statements that: $$\begin{aligned}
\mathrm{Re}\left(\frac{c_i}{a+b+c_i+d_j}\right)\geq\frac{1}{2}\text{ and } \mathrm{Re}\left(\frac{d_j}{a+b+c_{i+1}+d_j}\right)\geq\frac{1}{2},\end{aligned}$$ noting that the latter statement implies that: $$\begin{aligned}
2\sqrt{|d_j|}\geq\frac{|a+b+c_{i+1}+d_j|}{\sqrt{|d_j|}}=\sqrt{|abc_{i+1}|}.\end{aligned}$$ Now, if the sequence $\{|c_i|\}$ is bounded below by $2$, it must converge. Thus for any $\epsilon>0$, by choosing $i$ to be sufficiently large, $\sqrt{|c_{i+1}|}\leq\sqrt{|c_i|}\leq\sqrt{|c_{i+1}|}+\epsilon$. Then the edge relation for $\{\alpha,\beta,\delta_j;\gamma_i\rightarrow \gamma_{i+1}\}$ tells us that: $$\begin{aligned}
\sqrt{|abd_j|}&\leq\frac{|a+b+c_i+d_j|}{\sqrt{|abd_j|}}+\frac{|a+b+c_{i+1}+d_j|}{\sqrt{|abd_j|}}\\
&=\sqrt{|c_i|}+\sqrt{|c_{i+1}|}\leq2\sqrt{|c_{i+1}|}+\epsilon.
\end{aligned}$$ Combining this with the inequality above, we see that: $$\begin{aligned}
|ab|\leq\frac{4\sqrt{|c_{i+1}|}+2\epsilon}{\sqrt{|c_{i+1}|}}\leq4+\sqrt{2}\epsilon.\end{aligned}$$ Therefore, $|ab|\leq4$.\
We have now covered the case where we meet only finitely many 2-cells of one of the colour-types. The alternative is that we meet infinitely many 2-cells of all six colour-types, and we produce from this four sequences of 3-cells: $$\begin{aligned}
\left\{\{\alpha_i\}\},\{\{\beta_j\}\},\{\{\gamma_k\}\},\{\{\delta_l\}\right\}\end{aligned}$$ Now, the second case arises when one of these sequences is finite — that is, we stick to the surface of some 3-cell. Assume wlog that this is for the colour $\alpha$, and by truncating our ray (and abusing notation), we may take $a_j=a=\phi(\{\alpha\})$ for all $j$. Moreover, unless we’re in case $3$, we may further truncate our ray so that the non-increasing sequence $\{|b_j|\},\{|c_k|\},\{|d_l|\}$ remains bounded above $2$. Then the same analysis tells us that: $$\begin{aligned}
|ab_i|,|ac_j|,|ac_k|\rightarrow 4,\end{aligned}$$ and we can see from this that $|a|\leq2$.\
Finally, in the case that we meet infinitely many $3$-cells of every colour-type, assume that the monotonically non-increasing sequences $\{|a_i|\},\{|b_j|\},\{|c_k|\},\{|d_l|\}$ are bounded below by $2$ and hence converge. The same analysis as in case one tells us that $$\begin{aligned}
|a_i b_j|,|a_i c_k|, |a_i d_l|, |b_j c_k|, |b_j d_l|, |c_k d_l|\rightarrow 4,\end{aligned}$$ and since these numbers are the bound below by 2, we see that: $$\begin{aligned}
|a_i|,|b_j|,|c_k|,|d_l|\rightarrow2.\end{aligned}$$
Now, for the oriented edge $\{\alpha,\beta,\gamma;\delta\rightarrow \delta'\}$ sufficiently far along $\{\vec{e}_n\}$ so that $|a|,|b|,|c|,|d|,|d'|$ are each close to $2$, the edge relation $$\begin{aligned}
\frac{a+b+c+d}{abc}+\frac{a+b+c+d'}{abc}=1\end{aligned}$$ tells us that: $$\begin{aligned}
\frac{a+b+c+d}{abc},\frac{a+b+c+d'}{abc}\approx\frac{1}{2}\end{aligned}$$ By symmetry, this also holds for: $$\begin{aligned}
\frac{a+b+c+d}{abc},\frac{a+b+c+d}{abd},\frac{a+b+c+d}{acd},\frac{a+b+c+d}{bcd}\approx\frac{1}{2}.\end{aligned}$$ By mutliplying pairs of these terms and invoking the vertex relation , we obtain that: $$\begin{aligned}
ab,ac,ad,bc,bd,cd\approx 4,\end{aligned}$$ and hence either $a,b,c,d$ are approximately all $2$ or all $-2$. But the vertex relation then tells us that $$\begin{aligned}
64\approx(a+b+c+d)^2\approx abcd\approx 16,\end{aligned}$$ giving us the desired contradiction for our assumption that these sequences could be bounded below by $2$.\
In particular, this shows us that we must touch some 3-cell in $\Omega^3_\phi(2)$, and the subsequent infinitely many 3-cells of the same colour as $X$ must all be in $\Omega^3_\phi(2)$.
\[th:4nonempty\] The set of 3-cells $\Omega^3_\phi(4)$ is non-empty. Further, if $\Omega^3_\phi(2)=\varnothing$, then there is a unique sink.
If $\Omega^3_\phi(2)$ is non-empty then we’re done. But if it is empty, then lemma \[ray\] tells us that following oriented edges according to their directions must eventually result in a sink. If there are multiple sinks, they obviously cannot be distance 1 from each other. And one of the interior vertices of any path joining two sinks must have two arrows coming out of it and hence by lemma \[th:<2\], the set $\Omega^3_\phi(2)$ is non-empty.
Systolic inequality.
--------------------
\[Systolic inequality\] Let $\rho$ denote a quasi-Fuchsian representation for a thrice-punctured projective plane, then $$\begin{aligned}
{\mathop{\rm sys}\nolimits}(X_\rho)\leq2{\mathop{\rm arcsinh}\nolimits}(2).\end{aligned}$$ In particular, the unique maximum of the systole function over the moduli space of all hyperbolic thrice-punctured projective planes is $2{\mathop{\rm arcsinh}\nolimits}(2)$.
Any quasi-Fuchsian representation $\rho$ induces a BQ-Markoff map $\phi$. By Theorem \[th:4nonempty\], $\Omega^3_\phi(4)$ is non-empty: on the hyperbolic manifold $X_\rho$, there exists a one-sided simple geodesic $\gamma$ with $|{\mathop{\rm tr}\nolimits}A|=|2\sinh\frac{1}{2}\ell_{\gamma}(X)|\leq 4$ and hence $\ell_{\gamma}(X)\leq 2\hspace{.4mm} {\mathop{\rm arcsinh}\nolimits}(2)$. Thus, the maximum of the systole length function over the set of BQ-Markoff maps is less than or equal to $2\hspace{.4mm} {\mathop{\rm arcsinh}\nolimits}(2)$.To prove equality, consider the Markoff quad $(4,4,4,4)$, which we know from lemma \[lem:markoff\] arises from a Fuchsian representation. Any new Markoff quad generated from $(4,4,4,4)$ must be integral, and each entry is a positive multiple of $4$. Thus, the corresponding Markoff map has $4$ as its minimum. This in turn means that the shortest one-sided geodesic has length $2{\mathop{\rm arcsinh}\nolimits}(2)$. On the other hand, the shortest two-sided geodesic has trace $14=4\times4-2$, is of length $2{\mathop{\rm arccosh}\nolimits}(7)>2{\mathop{\rm arcsinh}\nolimits}(2)$ and hence cannot be a systolic geodesic.To prove the uniqueness of the maximum of the systole function over the moduli space $\mathcal{M}(N_{1,3})$, first recall from remark \[rmK:cell\] that for any 3-cusped projective plane, there exists a positive real Markoff quad $(a,b,c,d)\in\mathbb{R}_+^4$ such that $$\begin{aligned}
\frac{a}{a+b+c+d},\frac{b}{a+b+c+d},\frac{c}{a+b+c+d},\frac{d}{a+b+c+d}\leq\frac{1}{2}.\end{aligned}$$ If $(a,b,c,d)\neq(4,4,4,4)$ is a maximum of the systole function, we assume wlog that $4=a\leq b\leq c\leq d$ and $4<d$. Define $$\begin{aligned}
0\leq x_b=b-a\leq x_c=c-a\leq x_d=d-a\text{ and }0<x_d.\end{aligned}$$ Expanding equation in terms of these new quantities, we have: $$\begin{aligned}
\label{eq:ineq}
x_b^2+x_c^2+x_d^2=32(x_d+x_c+x_d)+14(x_bx_c+x_bx_d+x_cx_d)+4x_bx_cx_d.\end{aligned}$$ Since $x_b^2\leq x_bx_c$ and $x_c^2\leq x_cx_d$, we obtain from equation that $x_d^2\geq (32+13x_c)x_d$. Therefore: $$\begin{aligned}
d\geq 36+13x_c\geq 9\max\{c,4\}>a+b+c.\end{aligned}$$ But this contradicts the fact that $\frac{d}{a+b+c+d}\leq\frac{1}{2}$.
We can recognise the minimum of the systole geometrically due to its large symmetry group. Consider the spherical symmetric octahedron: the octahedron on the round two-sphere $S^2$ with great circle edges and full $A_4$ symmetry. Label the 6 vertices of this octahedron to get a symmetric element $\Sigma$ in the moduli space $\mathcal{M}_{0,6}$, and note that the 6 labeled points are invariant under the antipodal map. There exists a unique hyperbolic cusped surface $X$ with conformal structure $\Sigma$. And by the uniqueness of $X$, the vertex-fixing antipodal maps on $S^2$ uniformise to isometric $\mathbb{Z}_2$-actions on $X$. The 4 greater circles on $S^2$ which lie in the plane orthogonal to the vector between the centers of any two opposing faces uniformise to simple closed geodesics $\gamma_1,\gamma_2,\gamma_3,\gamma_4\in X$. By symmetry, each $\gamma_i$ is invariant under antipodal $\mathbb{Z}_2$-actions and descends to a geodesic $\overline{\gamma}_i\in X/\mathbb{Z}_2$, where $X/\mathbb{Z}_2$ is the desired 3-cusped projective plane. By symmetry, the four geodesics $\overline{\gamma}_i$ in $X/\mathbb{Z}_2$ have the same length, and hence their traces give rise to a Markoff quad $(\ell,\ell,\ell,\ell)$ which is necessarily $(4,4,4,4)$.
Fibonacci Growth {#sec:fib}
================
For any Markoff quad, solve for $d$ to get $$\begin{aligned}
d=\frac{abc}{4}\left(1\pm\sqrt{1-4\left(\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}\right)}\,\right)^2.\end{aligned}$$ For $|a|$, $|b|$, $|c|$ large, choose $d$ to be the larger of the two solutions, hence $$\begin{aligned}
\log|d|\approx \log|a|+\log|b|+\log|c|.\end{aligned}$$ In particular, $|d|$ is greater than $|a|$, $|b|$ and $|c|$. If we continue and flip from $a$ to $a'$, then $\log|a|<\log|a'|\approx \log|b|+\log|c|+\log|d|$. This gives rise to the notion of *Fibonacci growth* for Markoff quads, in keeping with Bowditch’s Fibonacci growth for Markoff triples [@BowMar].The goal of this section is to define and establish the Fibonacci growth for BQ-Markoff maps defined in , and use these growth rates to prove McShane identities and length spectrum growth rates. It should be noted that the Fibonacci growth rates that we define and prove here are strictly stronger than similar growth rates found in [@HuHIde], although only the weaker version is needed to prove theorem \[th:main\].
Fibonacci growth.
-----------------
Given an edge $e\in\Omega^1$, define the *Fibonacci function* $F_e:\Omega^3\rightarrow\mathbb{R}$ by:
- $F_e(\alpha)=1$ if $e=\{\alpha,\beta,\gamma\}$.
- For $\{\alpha,\beta,\gamma,\delta\to\delta'\}\in\vec{\Omega}^1$ oriented so that it points away from $e$ (or is either of the two possible oriented edges for $e$ itself) $$\begin{aligned}
F_e(\{\delta\})=F_e(\{\alpha\})+F_e(\{\beta\})+F_e(\{\gamma\}).\end{aligned}$$
Hence $F_e:\Omega^3\rightarrow\mathbb{R}$ takes the value of $1$ for the three $3$-cells in $\Omega^3$ that contain $e$ and subsequently define values for the rest of the tree by assigning to every hitherto unassigned $3$-cell meeting three assigned $3$-cells at some vertex the sum of the values of those already assigned $3$-cells.
\[fibound\] Given a function $f:\Omega^3\rightarrow[0,\infty)$ and $\Omega'\subset\Omega^3$, we say that $f$ has:
- a *lower Fibonacci bound* on $\Omega'$ if there’s some positive $\kappa$ such that: $$\begin{aligned}
\frac{1}{\kappa} F_e(X)\leq f(X)\text{ for all but finitely many }X\in\Omega';\end{aligned}$$
- an *upper Fibonacci bound* on $\Omega'$ if there’s some positive $\kappa$ such that: $$\begin{aligned}
f(X)\leq \kappa F_e(X)\text{ for all }X\in\Omega';\end{aligned}$$
- *Fibonacci growth* on $\Omega'$ if there’s some positive $\kappa$ such that: $$\begin{aligned}
\frac{1}{\kappa} F_e(X)\leq f(X)\leq \kappa F_e(X)\text{ for all but finitely many }X\in\Omega';\end{aligned}$$ or in other words: it has both lower and upper Fibonacci bound. We also opt to omit *“on $\Omega'$”* whenever $\Omega'=\Omega^3$.
We assumed the choice of an edge $e$ for these definitions, and now show that the existence of a $\kappa$ satisfying these conditions is independent of this choice.
\[upper\] Given some edge $e$ that is the intersection of the three $3$-cells $X_1,X_2,X_3$ and a function $f:\Omega^3\rightarrow[0,\infty)$ satisfying: $$\begin{aligned}
f(D)\leq f(A)+f(B)+f(C)+2c,\,0\leq c,\end{aligned}$$ where $A,B,C,D$ meet at the same vertex and $D$ is strictly farther from $e$ than $A,B,C$. Then: $$\begin{aligned}
f(X)\leq(M+c)F_e(X)-c,\text{ for all }X\in\Omega^3,\end{aligned}$$ where $M=\mathrm{max}\{f(X_1),f(X_2),f(X_3)\}$.
We prove this by induction on the distance of a region from $e$. The base case is due to: $$\begin{aligned}
f(X_i)\leq(\mathrm{max}\{f(X_1),f(X_2),f(X_3)\}+c)-c.\end{aligned}$$ The induction step is similarly established: $$f(D)\leq(M+c)(F_e(A)+F_e(B)+F_e(C))-3c+2c=(M+c)F_e(D)-c.$$
Note that by essentially the same proof, we obtain the following result:
Given some edge $e$ that is the intersection of the three $3$-cells $X_1,X_2,X_3$ and a function $f:\Omega^3\rightarrow[0,\infty)$ satisfying: $$\begin{aligned}
f(D)\geq f(A)+f(B)+f(C)-2c,\,0\leq c<m:=\mathrm{min}\{f(X_1),f(X_2),f(X_3)\},\end{aligned}$$ where $A,B,C,D$ meet at the same vertex and $D$ is strictly farther from $e$ than $A,B,C$. Then: $$\begin{aligned}
f(X)\geq(m-c)F_e(X)+c,\text{ for all }X\in\Omega^3.\end{aligned}$$
However, this is insufficient for our purposes. We shall require:
\[lower\] Given some oriented edge $\vec{e}$ that is the intersection of the three $3$-cells $X_1,X_2,X_3$ and a function $f:\Omega^3\rightarrow[0,\infty)$ satisfying: $$\begin{aligned}
f(D)\geq f(A)+f(B)+f(C)-2c,\,0\leq c<\mu:=\mathrm{min}\{f(X_i)+f(X_j)\}_{i\neq j},\end{aligned}$$ where $A,B,C,D$ meet at the same vertex and $D$ is strictly farther from $e$ than $A,B,C$. Then: $$\begin{aligned}
f(X)\geq(\mu-2c)F_e(X)+c,\text{ for all }X\in\Omega^3_-(\vec{e})-\Omega^3_0(\vec{e}).\end{aligned}$$
We first use induction to show that any two adjacent $3$-cells in $\Omega^3_-(\vec{e})$ satisfy: $$\begin{aligned}
f(X)+f(Y)\geq (\mu-2c)(F_e(X)+F_e(Y))+2c.\end{aligned}$$ The base case where $X$ and $Y$ are both in $\Omega_0(e)$ follows from the definition of $\mu$. We proceed by induction on the total distance of $X$ and $Y$ from $e$. Assume that $Y$ is farther than $X$ from $e$. The tree structure of $\Omega^3_-(\vec{e})$ means that there is a unique closest vertex between the edge $e$ and the face $X\cap Y$. Denote the two other $3$-cells at this vertex by $W$ and $Z$, we then have: $$\begin{aligned}
f(X)+f(Y)\geq& f(X)+f(W)+f(X)+f(Z)-2c\\
\geq&(\mu-2c)(F_e(X)+F_e(W)+F_e(X)+F_e(Z))+4c-2c\\
=&(\mu-2c)(F_e(X)+F_e(Y))+2c,\end{aligned}$$ completing the induction.Now consider a $3$-cell $D\in\Omega^3_-(\vec{e})-\Omega^3_0(e)$, and denote by $A,B,C$ the three other $3$-cells meeting $D$ at the closest vertex between $e$ and $D$. Then we have: $$\begin{aligned}
f(D)\geq&\frac{1}{2}(f(A)+f(B)+f(C)+f(D)-2c)\\
\geq&\frac{1}{2}(\mu-2c)(F_e(A)+F_e(B)+F_e(C)+F_e(D))+2c-c\\
=&(\mu-2c)F_e(D)+c.\end{aligned}$$
Since for any edge $e'$, the function $F_{e'}$ satisfies the criteria for these last two lemmas, we see that there is some $\kappa>0$ such that: $$\begin{aligned}
\frac{1}{\kappa}F_e(X)\leq F_{e'}(X)\leq\kappa F_e(X),\text{ for all }X\in\Omega^3.\end{aligned}$$ Which shows that Definition \[fibound\] is indeed independent of the choice of the edge $e$.
\[th:sumcon\] If a function $f:\Omega^3\rightarrow\mathbb{R}^+$ has a lower Fibonacci bound, then for any $\sigma>3$, the following sum converges: $$\begin{aligned}
\sum_{X\in\Omega}f(X)^{-\sigma}<\infty.\end{aligned}$$
It suffices for us to show that this sum converges for $f= F_e$. We do this by bounding the growth of the level sets of $F_e$. We will prove that: $$\label{fibtot}
{\mathop{\rm Card}\nolimits}\left\{\,X\in\Omega^3\mid\ F_e(X)=n\right\}<4J_2(n)$$ where $J_k$ is the Jordan totient function. Hence $$\begin{aligned}
\sum_{X\in\Omega}F_e(X)^{-\sigma}<\sum_{n\geq1}4J_2(n)n^{-\sigma}=\frac{4\zeta(\sigma-2)}{\zeta(\sigma)}\end{aligned}$$ for $\zeta$ the Riemann zeta function, and the sum converges for $\sigma>3$.For the remainder of this proof, we think of $F_e$ not just as a function on the $3$-cells $\Omega^3$, but also as a set-valued function on the 1-cells, where it assigns to each edge $\{\alpha,\beta,\gamma\}\in\Omega^1$ the unordered 3-tuple $\{F_e(\{\alpha\}),F_e(\{\beta\}),F_e(\{\gamma\})\}$.When $n>1$, there is a $1:3$ correspondence between $$\begin{aligned}
\{X\in\Omega^3\mid F_e(X)=n\}\text{ and }\left\{\{\alpha,\beta,\gamma\}\in\Omega^1\mid \max F_e(\{\alpha,\beta,\gamma\}) =n\right\}\end{aligned}$$ defined by assigning to $X\in\Omega^3$ the three edges closest to $e$ that lie on $X$. By uniqueness up to symmetry of values of $F_e$ on paths in $\Omega$, the preimage of any unordered triple $\{l,m,n\}$ in the image of $F_e$ has cardinality at most:
- 1, if $\{l,m,n\}=\{1,1,1\}$,
- 6, if $\{l,m,n\}=\{1,1,n\}$ and
- 12, if $\{l,m,n\}$ are all distinct integers.
Thus, the relation given by assigning to a 3-cell $X$ the unordered 3-tuples of values of $F_e$ on the edges on $X$ closest to $e$ is at most $4:1$. Any triple $\{l,m,n\}$ that is in the image of $F_e$ must be relatively prime. Otherwise, a common factor would inductively propagate back to $e$ and contradict the starting value of $\{1,1,1\}$. Thus, for $n>1$, $$\begin{aligned}
{\mathop{\rm Card}\nolimits}\left\{X\in\Omega^3\mid F_e(X)=n\right\}&\leq 4\,{\mathop{\rm Card}\nolimits}\left\{\, \{l,m,n\} \mid l,m<n\text{ and }\gcd(l,m,n)=1\right\}\\
&<4\,{\mathop{\rm Card}\nolimits}\left\{\, (l,m,n) \mid l,m\leq n\text{ and }\gcd(l,m,n)=1\right\}\\
&=4J_2(n)\text{, and \eqref{fibtot} holds as required.}\end{aligned}$$
These results enable us to conclude that: if the function $$\begin{aligned}
\log^+|\phi|:\Omega^3\rightarrow[0,\infty)\end{aligned}$$ satisfies the following inequality at every vertex $\{a,b,c,d\}\in\Omega^0$: $$\begin{aligned}
\log^+|d|\leq\log^+|a|+\log^+|b|+\log^+|c|+2\log\left(\frac{1+\sqrt{13}}{2}\right),\label{ineq}\end{aligned}$$ where $\log^+(x):=\mathrm{max}\{0,\log(x)\}$, then:
$\log^+|\phi|$ has an upper Fibonacci bound on $\Omega^3$.
By the preceding comment, we only need to show that holds. To begin with, we see that when $|d|\leq1$, the desired identity is trivially satisfied. We therefore confine ourselves to when $|d|>1$, that is: when $\log|d|=\log^+|d|$. We now assume without loss of generality that $|a|\leq|b|\leq|c|$ and case-bash the desired result.
1. If $1\leq|a|,|b|,|c|$, then: $$\begin{aligned}
\log^+|d|=\log|d|=&\log|abc|+2\log\left|\frac{1}{2}\left(1\pm\sqrt{1-4(\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc})}\right)\right| \\
\leq&\log|abc|+2\log\left|\frac{1}{2}\left(1+\sqrt{1+4(\frac{1}{|ab|}+\frac{1}{|ac|}+\frac{1}{|bc|})}\right)\right|\\
\leq&\log|a|+\log|b|+\log|c|+2\log\left(\frac{1+\sqrt{13}}{2}\right)\end{aligned}$$
2. If $|a|<1\leq|b|,|c|$, then: $$\begin{aligned}
\log^+|d|=&\log|bc|+2\log\left|\frac{1}{2}\left(\sqrt{|a|}\pm\sqrt{1-4(\frac{|a|}{ab}+\frac{|a|}{ac}+\frac{|a|}{bc})}\right)\right| \\
\leq&\log|bc|+2\log\left|\frac{1}{2}\left(\sqrt{|a|}+\sqrt{1+4(\frac{1}{|b|}+\frac{1}{|c|}+\frac{|a|}{|bc|})}\right)\right|\\
\leq&\log|b|+\log|c|+2\log\left(\frac{1+\sqrt{13}}{2}\right)\end{aligned}$$
3. And similarly, if $|a|,|b|<1\leq|c|$, then: $$\begin{aligned}
\log^+|d|=&\log|c|+2\log\left|\frac{1}{2}\left(\sqrt{|ab|}\pm\sqrt{1-4(\frac{|ab|}{ab}+\frac{|ab|}{ac}+\frac{|ab|}{bc})}\right)\right| \\
\leq&\log|c|+2\log\left|\frac{1}{2}\left(\sqrt{|ab|}+\sqrt{1+4(1+\frac{|b|}{|c|}+\frac{|a|}{|c|})}\right)\right|\\
\leq&\log|c|+2\log\left(\frac{1+\sqrt{13}}{2}\right)\end{aligned}$$
4. And finally, if $|a|,|b|,|c|<1$, then: $$\begin{aligned}
\log^+|d|=&2\log\left|\frac{1}{2}\left(\sqrt{|abc|}\pm\sqrt{1-4(\frac{|abc|}{ab}+\frac{|abc|}{ac}+\frac{|abc|}{bc})}\right)\right| \\
\leq&2\log\left(\frac{\sqrt{|ab|}+\sqrt{1+4(|a|+|b|+|c|)}}{2}\right)\leq2\log\left(\frac{1+\sqrt{13}}{2}\right)\end{aligned}$$
We now aim to show that $\log|\phi|$ has a lower Fibonacci bound, and introduce the following notation: given an oriented edge $\vec{e}$ on $\Omega$, the removal of the edge $e$ from the tree in $\Omega$ results in two connected components. We denote the collection of $3$-cells containing edges from the tree on the *head-side* of $\vec{e}$ by $$\begin{aligned}
\Omega^3_+(\vec{e})\text{; and }\Omega^3_-(\vec{e})\end{aligned}$$ for the the collection of $3$-cells containing edges from the tree on the *tail-side* of $\vec{e}$. We also use the notation $\Omega_0^3(e)=\Omega^3_+(\vec{e})\cap\Omega^3_-(\vec{e})$ to refer to the three edges containing $e$.
Given an oriented edge $\vec{e}\in\vec{\Omega}^1$ such that $\Omega^3_0(e)\cap\Omega^3_\phi(2)=\varnothing$, then $\Omega^3_\phi(2)$ lies on the head-side of $e$, that is: $$\begin{aligned}
\Omega^3_\phi(2)\subseteq\Omega^3_+(\vec{e}).\end{aligned}$$ Furthermore, all oriented edges in $\Omega^3_-(\vec{e})$ must point toward $e$.
Due to the connectedness of $\Omega^3_\phi(2)$, it must lie in either $\Omega^3_+(\vec{e})$ or $\Omega^3_-(\vec{e})$. If it lies on the tail side of $\vec{e}$ then consider a shortest path containing $\vec{e}$ and touching $\Omega^3_\phi(2)$. No $3$-cell in $\Omega^3_\phi(2)$ may be in direct contact with $\vec{e}$ as this would force the region on the other end of $\vec{e}$ be in $\Omega^3_\phi(2)$ — yielding a contradiction.\
Hence, we have a path of length at least $2$ with outwardly oriented edges at the two end of this path. Thus resulting in at least one internal vertex with two outward pointing edges and hence an adjacent result in $\Omega^3_\phi(2)$. This then contradicts the shortest assumption we placed on our path.\
We have shown that $\Omega^3_\phi(2)$ is on the head-side of $\vec{e}$ and by lemma \[th:<2\], every vertex on the tail-side of $\vec{e}$ must have three incoming edges and one outgoing edge. Then lemma \[ray\] forces all of these edges to point toward $e$.
\[tighter\] Given the hypotheses of the above result, define: $$\begin{aligned}
\mu:=\mathrm{min}\left\{\log^+|\phi(X_i)|+\log^+|\phi(X_j)|\mid X_k\in\Omega_0^3(\vec{e})\right\}>2\log2,\end{aligned}$$ then for every tail-side 3-cell $X\in\Omega_-^3(\vec{e})-\Omega^3_0(e)$, we have: $$\begin{aligned}
\log^+|\phi(X)|\geq(\mu-2\log2)F_e(X)+\log2,\end{aligned}$$ and hence $\log^+|\phi|$ has a lower Fibonacci bound over $\Omega^3_-(\vec{e})$.
Let $\{\alpha\},\{\beta\},\{\gamma\},\{\delta\}\in\Omega_-^3(\vec{e})$ be the adjacent 3-cells to an arbitrarily chosen tail-side vertex, such that $\{d\}$ is farthest from $e$. Then we know from every edge being naturally directed towards $e$ that: $$\begin{aligned}
\sqrt{\frac{|d|}{|abc|}}\geq\frac{1}{2}\Rightarrow \log|d|\geq \log|a|+\log|b|+\log|c|-2\log2.\end{aligned}$$ Since this is satisfied for every tail-side vertex, lemma \[lower\] then gives the desired conclusion.
Fix an arbitrary $2$-cell $\{\alpha,\beta\}\in\Omega^2$, its boundary is a bi-infinite path which we label by $\{e_n\}_{n\in\mathbb{Z}}$. Each edge $e_n$ is the intersection of three distinct 3-cells, two of which are $\{\alpha\},\{\beta\}$ and the last we’ll denote by $\{\gamma_n\}$.
\[growth\] Given the above setup, then:
1. If $ab\notin [0,4]$, then either $|c_n|$ grows exponentially as $n\rightarrow\pm\infty$ or $c_n=0$.
2. If $ab\in [0,4)$, then $|c_n|$ remains bounded.
3. If $ab=4$, then $c_n=A+Bn-(a+b)n^2$ for some $A,B\in\mathbb{C}$.
The edge relation then tells us that: $$\begin{aligned}
c_{n+1}+(2-ab)c_n+c_{n-1}+2(a+b)=0.\end{aligned}$$
If $ab\neq 0,4$, we may solve for this difference equation: $$\begin{aligned}
\label{eq:difference}
c_n=A\lambda^n+B\lambda^{-n}-\frac{2(a+b)}{4-ab}\text{, where }\lambda^{\pm1}=\frac{1}{2}(ab-2\pm\sqrt{ab(ab-4)}).\end{aligned}$$ **Case (1):** Assume that $ab\notin [0,4]$. Since $ab=(\lambda^{\frac{1}{2}}+\lambda^{-\frac{1}{2}})^2$, then the fact that $|\lambda|=1$ if and only $ab\in[0,4]$ means that $|\lambda|\neq1$. Thus, to show that $|c_n|$ grows exponentially, it suffices to show that neither $A$ or $B$ equals $0$. We prove this by contradiction: assume wlog that $B=0$ and that $|\lambda|>1$, then $$\begin{aligned}
c_n=A\lambda^n-\frac{2(a+b)}{4-ab}.\end{aligned}$$ Substituting this into equation and taking the limit as $n\rightarrow-\infty$, we obtain that: $$\begin{aligned}
\left(\frac{ab(a+b)}{4-ab}\right)^2=4ab\left(\frac{a+b}{4-ab}\right)^2.\end{aligned}$$ By assumption, $ab\neq 0,4$ and therefore $a+b=0$. This in turn means that $c_n=A\lambda^n$. Substuting this into equation shows that either $A=0$ or $|\lambda|=1$. Since $\lambda|>1$, we conclude that $c_n=0$ for all $n$.**Case (2):** If $ab\in(0,4)$, then $|\lambda|=1$ and by equation , $|c_n|$ is bounded above by $|A|+|B|+\left|\frac{2(a+b)}{4-ab}\right|$. If $ab=0$, we assume wlog that $b=0$. Then by the vertex relation , $c_n+c_{n-1}+a=0$ and the sequence is either constant or oscillates between two values. **Case (3):** For $ab=4$, simply solving for the edge relation as a difference equation yields the desired expression for $c_n$.
\[converge\] If $\phi\in\Phi_{BQ}$, then $\log^+|\phi|$ has Fibonacci growth.
If $\Omega^2_\phi(4)=\varnothing$, then there is a unique sink in $\Omega^0$. Otherwise, a path between two sinks would contain some vertex with at least two outward pointing oriented edges and hence be adjacent to a 2-cell in $\Omega^2_\phi(4)$ by lemma \[th:<2\]. Then apply lemma \[tighter\] to the four oriented edges pointing into this unique sink to obtain the desired Fibonacci lower bound.\
Otherwise, $$\begin{aligned}
\Omega_\phi^2(4)=\left\{\{\alpha_1,\beta_1\},\{\alpha_2,\beta_2\},\ldots,\{\alpha_l,\beta_l\}\right\}\end{aligned}$$ is finite but non-empty. Then let $T$ denote the smallest tree in $\Omega$ containing the boundaries of all the 2-cells in $\Omega^2_\phi(4)$. We claim that $T$ must contain every sink and saddle. Firstly, it’s clear from lemma \[th:<2\] that every saddle lies on the boundary of some 2-cell in $\Omega^2_\phi(4)$ and hence in $T$.\
Now take an arbitrary sink $v$. Since $\Omega^2_\phi(4)$ is non-empty, $\Omega^3_\phi(2)$ must also be non-empty. Consider the shortest path between $\Omega^3_\phi(2)$ and $v$, if the length of this path is 2 or more, then we reach a contradiction because there must be an internal vertex adjacent to a 2-cell in $\Omega^2_\phi(4)$ and hence a 3-cell in $\Omega^3_\phi(2)$. And if the length of this path is 1, then we contradict the connectedness of $\Omega^3(2)$. Hence $v$ lies on the boundary of some 3-cell $A\in\Omega^3_\phi(2)$.\
Now, thanks to the connectedness of $\Omega^3_\phi(2)$, the boundary of $A$ must contain some 2-cell in $\Omega^2_\phi(4)$. Thus the shortest path from $v$ to $\Omega^2_\phi(4)$ lies on the boundary of $A$. Note that the 3-cell at the tail of the chosen edge on this path closest to $\Omega^2_\phi(4)$ must point towards $\Omega^2_\phi(4)$ or else produce a closer 2-cell in $\Omega^2_\phi(4)$. Hence, by similar arguments as used in the previous paragraph, $v$ must lie on the boundary of a 2-cell in $\Omega^2_\phi(4)$.\
We now show that all but finitely many vertices are funnels. Observe that all but finitely many edges in $T$ lie on the boundary of some 2-cell in $\Omega^2_\phi(4)$. Then lemma \[growth\] tells us that since $|\phi|$ grows exponentially as we traverse the boundary of a 2-cell, there can only be finitely many sinks. Further observe that by lemma \[ray\], every oriented edge outside of $T$ must point into $T$. This means that along the boundary of any of the 2-cells in $\Omega^2_\phi(4)$, there must (in all but finitely many cases) be a sink in between two saddles. Hence, we see that the number of saddles is also finite.\
We now show that a Fibonacci lower bound holds over the set: $$\begin{aligned}
\Omega^3_0(T):=\{\text{ 3-cells touching }T\}.\end{aligned}$$ We know that all but finitely many of the 3-cells in $\Omega^3_0(T)$ spiral around some 2-cell in $\Omega^2_\phi(4)$. And lemma \[growth\] tells us that $\log^+|\phi|$ over each of these spirals grows linearly, and hence for the spiral $\Omega^3_0(\{\alpha_i,\beta_i\})-\{\{\alpha_i\},\{\beta_i\}\}$ around $\{\alpha_i,\beta_i\}\in\Omega^2_\phi(4)$, we have: $$\begin{aligned}
\log^+|\phi(X)|\geq\kappa_i F_e(X)+\mu_i,\end{aligned}$$ where $\kappa_i$ is a function of $|ab|$ and the minimum of $F_e$ on this spiral around $\{\alpha_i,\beta_i\}$, and $\mu_i$ may be negative. Since there are finitely many such spirals, only finitely many 3-cells in $\Omega^3_0(T)$ not on a spiral and the minimum of $\log^+|\phi|$ is greater than $0$, we see that: $$\begin{aligned}
\log^+|\phi(X)|\geq\kappa F_e(X),\text{ for all }X\in\Omega^3_0(T).\end{aligned}$$
Finally, label all the oriented edges touching but not contained in $T$ by $\{\vec{\epsilon}_i\}$ (in order of increasing distance from $e$ if you so wish), and for each $\vec{\epsilon}_i$, label the the three 3-cells in $\Omega^3_0(\epsilon_i)$ by: $$\begin{aligned}
\Omega^3_0(\epsilon_i)=\{X_i,Y_i,Z_i\},\text{ such that }\log^+|\phi(X_i)|\leq\log^+|\phi(Y_i)|\leq\log^+|\phi(Z_i)|.\end{aligned}$$ Then lemma \[tighter\] tells us that for any 3-cell $X\in\Omega^3_-(\vec{\epsilon}_i)$, $$\begin{aligned}
\log^+|\phi(X)|&\geq (\log|\phi(X_i)\phi(Y_i)|-\log(4))F_{\epsilon_i}(X),\text{ and hence }\\
&\geq \frac{\log|\phi(X_i)\phi(Y_i)|-\log(4)}{\max\{F_e(X_i),F_e(Y_i),F_e(Z_i)\}}F_e(X).\end{aligned}$$ Therefore, if we can show that $$\begin{aligned}
\inf_i\left\{\frac{\log|\phi(X_i)\phi(Y_i)|-\log(4)}{\max\{F_e(X_i),F_e(Y_i),F_e(Z_i)\}}\right\}>0\end{aligned}$$ then we’ll have shown that $\log^+|\phi|$ has a lower Fibonacci bound over all of $\Omega^3$. And to see that this holds, first notice that by going out sufficiently far from $e$, we may effectively ignore the $\log(4)$ term. Then, because $X_i$ and $Y_i$ are in $\Omega^3_0(T)$, we see that: $$\begin{aligned}
\frac{\log|\phi(X_i)\phi(Y_i)|}{\max\{F_e(X_i),F_e(Y_i),F_e(Z_i)\}}\geq\frac{\kappa(F_e(X_i)+F_e(Y_i))}{\max\{F_e(X_i),F_e(Y_i),F_e(Z_i)\}}\geq\frac{\kappa}{2},\end{aligned}$$ thus yielding the desired Fibonacci lower bound. We complete this proof by invoking lemma \[upper\] for the upper Fibonacci bound.
McShane Identity {#sec:mcshane}
----------------
Our method for proving Theorem \[th:main\] follows Bowditch [@BowPro; @BowMar]. Starting with a 4-tuple of simple closed one-sided curves $\{\alpha,\beta,{\color{red}\gamma_0},{\color{blue}\gamma_1}\}$, consider the sequence of 4-tuples produced by repeatedly flipping ${\color{red}\gamma_{2i}}$ to get ${\color{red}\gamma_{2i+2}}$ and ${\color{blue}\gamma_{2i+1}}$ to get ${\color{blue}\gamma_{2i+3}}$.
![Flipping ${\color{red}\gamma_{2i}}$ followed by ${\color{blue}\gamma_{2i+1}}$.[]{data-label="fig:doubleflip"}](doubleflip.eps)
In this manner, we obtain a sequence of Markoff quads: $$\begin{aligned}
(a,b,{\color{red}c_0},{\color{blue}c_1}),(a,b,{\color{red}c_2},{\color{blue}c_1}),(a,b,{\color{red}c_2},{\color{blue}c_3}),(a,b,{\color{red}c_4},{\color{blue}c_3}),(a,b,{\color{red}c_4},{\color{blue}c_5}),\ldots\end{aligned}$$ By comparing the vertex relation at $\{\alpha,\beta,\gamma_k,\gamma_{k+1}\}$ and the edge relation at $\{\alpha,\beta,\gamma_k\}$, we obtain that: $$\begin{aligned}
\tfrac{c_{k-1}}{a+b+c_{k-1}+c_{k}}=\tfrac{c_{k}}{a+b+c_{k}+c_{k+1}}+\tfrac{a}{a+b+c_{k}+c_{k+1}}+\tfrac{b}{a+b+c_{k}+c_{k+1}}.\end{aligned}$$ In this decomposition for $\frac{c_{k-1}}{a+b+c_{k-1}+c_{k}}$, there is another summand of the same form but with shifted indices. Thus, starting with the the vertex relation , we may iteratively decompose terms of the form $\frac{c_{k-1}}{a+b+c_{k-1}+c_k}$ to obtain: $$\begin{aligned}
1&=\tfrac{c_0}{a+b+c_0+c_1}+\tfrac{c_1}{a+b+c_0+c_1}+\tfrac{a}{a+b+c_0+c_1}+\tfrac{b}{a+b+c_0+c_1}\\
&=\tfrac{c_1}{a+b+c_1+c_2}+\tfrac{a}{a+b+c_0+c_1}+\tfrac{b}{a+b+c_0+c_1}+\tfrac{a}{a+b+c_1+c_2}+\tfrac{b}{a+b+c_1+c_2}+\tfrac{c_1}{a+b+c_0+c_1}\\
&\hspace{5em}\vdots\\
&=\tfrac{c_n}{a+b+c_n+c_{n+1}}+\sum_{i=0}^n\left(\tfrac{a}{a+b+c_i+c_{i+1}}+\tfrac{b}{a+b+c_i+c_{i+1}}\right)+\tfrac{c_1}{a+b+c_0+c_1}.\end{aligned}$$ Since the edge relation for $\{\alpha,\beta,\gamma_n\}$ is a second order difference equation, we may explicitly compute $$\begin{aligned}
\lim_{n\to\infty}\frac{c_n}{a+b+c_n+c_{n+1}}=\frac{1}{1+\lambda},\text{ where }\lambda=\frac{ab-2+\sqrt{ab(ab-4)}}{2}. \end{aligned}$$ When we indefinitely apply this splitting algorithm to every summand that arises, we might intuitively expect to derive a series that sums to one, whose summands each take the form $\frac{1}{1+\lambda}$ for some $\lambda$ — this series *is* the McShane identity.In the Fuchsian case, the summands $\Psi(\vec{e})$ are real numbers and correspond to the lengths of intervals in the length $1$ horocycle around one of the cusps on $X$. The complement of all of these intervals is, by construction, the union of a Cantor set and a countable set. To show that this Cantor set is of measure $0$ in order to conclude that the series sums to $1$, we bound the measure of this Cantor set by smaller and smaller tails of the convergent series $\frac{1}{1+\lambda}$. Convergence follows from the Fibonacci growth rates established above.Define the function $\Psi:\vec{\Omega}^1\rightarrow[0,1]$ by: $$\begin{aligned}
\Psi(\vec{e})=\Psi(\{\alpha,\beta,\gamma;\delta'\rightarrow \delta\}):=\frac{d}{a+b+c+d}=\frac{a+b+c+d}{abc}.\end{aligned}$$
Then, the edge relation becomes: $$\begin{aligned}
\Psi(\vec{e})+\Psi({\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{e}}}}}})=1,\end{aligned}$$ and the vertex relation is the following relation on four incoming oriented edges $\vec{e}_1,\vec{e}_2,\vec{e}_3,\vec{e}_4$: $$\begin{aligned}
\Psi(\vec{e}_1)+\Psi(\vec{e}_2)+\Psi(\vec{e}_3)+\Psi(\vec{e}_4)=1.\end{aligned}$$ These two properties in turn tell us that for a funnel with oriented edges $\vec{e}_1,\vec{e}_2,\vec{e}_3$ and outgoing edge $\vec{e}_4$: $$\begin{aligned}
\Psi(\vec{e}_4)=\Psi(\vec{e}_1)+\Psi(\vec{e}_2)+\Psi(\vec{e}_3),\end{aligned}$$ and so we may iteratively expand either the edge or the vertex relation into a statements about a finite collection of terms of the form $\Psi(\vec{e})$ summing to $1$. For a tree $T$ in the 1-skeleton of $\Omega$, if we use the notation $C(T)$ to denote $$\begin{aligned}
C(T):=\left\{\vec{e}\in\vec{\Omega}^1\mid\vec{e}\text{ points into, but is not contained in }T\right\},\end{aligned}$$ then we have:
\[equals1\] For any finite subtree $T$ in the 1-skeleton of $\Omega$, $$\begin{aligned}
\sum_{\vec{e}\in C(T)}\Psi(\vec{e})=1.\end{aligned}$$
Next, define the function $h:\mathbb{C}-[0,4]\rightarrow \mathbb{C}$, $$\begin{aligned}
h(x)=\frac{1}{2}(1-\sqrt{1-4/x})=\frac{2}{x(1+\sqrt{1-4/x})}.\end{aligned}$$ For an edge $e=\{\alpha,\beta,\gamma\}$, we define $$\begin{aligned}
h(e)=h(\{\alpha,\beta,\gamma\}):=h\left(\tfrac{abc}{a+b+c}\right)=h\left(\left(\tfrac{1}{ab}+\tfrac{1}{ac}+\tfrac{1}{bc}\right)^{-1}\right).\end{aligned}$$ A little algebraic manipulation shows that: $$\begin{aligned}
h(e)=\Psi(\vec{e})\text{ if and only if }\mathrm{Re}(\Psi(\vec{e}))\leq\mathrm{Re}(\Psi({\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{e}}}}}})).\end{aligned}$$ In other words, $\Psi$ of a chosen edge $\vec{e}$ is equal to $h(e)$. In fact, the main point of Theorem \[converge\] is to prove the following result:
\[converge2\] The following infinite series taken over $\Omega^2$ converges absolutely for all $s>0$, $$\sum_{\{\xi,\eta\}\in\Omega^2}|xy|^{-s}<\infty.$$
We see from Theorem \[converge\] that the following series converges (absolutely): $$\sum_{\{\xi\}\in\Omega^3}\left|\log|x|\right|^{-3}<\infty.$$ Hence, the following series converges: $$\sum_{\{\xi\}\in\Omega^3}|x|^{-\frac{s}{2}}<\infty.$$ Squaring this series, we obtain an absolutely convergent series that’s strictly greater than our desired quantity.
Before we state and prove theorem \[th:main\], we introduce one more piece of notation. Given a subset $E$ consisting of edges in the 1-skeleton of $\Omega$, we define: $$\Omega^2(E):=\left\{\{\xi,\eta\}\in \Omega^2\mid\{\xi,\eta\}\text{ contains an edge in }E\right\}.$$
\[th:bqmcshane\] If $\phi\in\Phi_{BQ}$, then $$\begin{aligned}
\sum_{\{\alpha,\beta\}\in\Omega^2} h(ab)=\frac{1}{2}.\end{aligned}$$
We first note that $h(x)$ is roughly order $O(|x|^{-1})$, and so lemma \[converge2\] tells us that: $$\begin{aligned}
\sum_{\{\alpha,\beta\}\in\Omega^2} h(ab)<\infty.\end{aligned}$$ Next, we prove an inequality of the following form: $$\begin{aligned}
|h(\{\alpha,\beta,\gamma\})-h(ab)|\leq \kappa|h(ac)+h(bc)|,\end{aligned}$$ where $\kappa>0$ is independent of $a,b$ and $c$. We begin by noting that outside of a finite set of edges $\{\alpha,\beta,\gamma\}$, either $|a|\gg0,|b|\gg0$ or $|c|\gg0$. If $|a|$ or $|b|\gg0$, then: $$\begin{aligned}
\frac{|h(\{\alpha,\beta,\gamma\})-h(ab)|}{|h(ac)+h(bc)|}\approx1,\end{aligned}$$ and if $|c|\gg0$, then: $$\begin{aligned}
\frac{|h(\{\alpha,\beta,\gamma\})-h(ab)|}{|h(ac)+h(bc)|}\approx\frac{1}{2}\left|1+\sqrt{1-\frac{4}{ab}}\right|^{-1}<\frac{1}{2}.\end{aligned}$$ Therefore, we know that there exists a $\kappa$ satisfying our requirements.In the proof of Theorem \[converge\], we construct a finite attracting tree $T$, outside of which every vertex is a funnel. Now, if we take $B_n(T)$ to be the distance $n$ neighbourhood of $T$ in the 1-skeleton of $\Omega$, then lemma \[equals1\] tells us that: $$\begin{aligned}
1=\sum_{\vec{e}\in C(B_n(T))}\Psi(\vec{e})=\sum_{\vec{e}\in C(B_n(T))} h(e).\end{aligned}$$ Given $\vec{e}=\{\alpha,\beta,\gamma;\delta'\rightarrow \delta\}\in C(B_n(T))$, and suppose that $\vec{e}$ joins directly onto the oriented edge $\{\alpha,\beta,\delta;\gamma\rightarrow \gamma'\}\in C(B_{n-1}(T))$, then of the three $2$-cells $\{\alpha,\beta\}$, $\{\alpha,\gamma\}$, $\{\beta,\gamma\}$ containing $e$, we know that $\{\alpha,\beta\}\in \Omega^2(B_{n-1}(T))$ and $\{\alpha,\gamma\}$, $\{\beta,\gamma\}\in \Omega^2_{n}(B_n(T))-\Omega^2(B_{n-1}(T))$. Hence, summing over all of $C(B_n(T))$, we obtain the following inequality: $$\begin{aligned}
\left|1-2\sum_{\{\alpha,\beta\}\in \Omega^2(B_{n-1}(T))}h(ab)\right|=&\left|\sum_{ C(B_n(T))}h(e)-2\sum_{\{\alpha,\beta\}\in \Omega^2(B_{n-1}(T))}h(ab)\right|\\
\leq&2\kappa\left|\sum_{\{\gamma,\delta\}\in \Omega^2(B_n(T))- \Omega^2(B_{n-1}(T))}h(cd)\right|,\end{aligned}$$ noting that we’d made use of the fact that any 2-cell meets either two or no edges in $C(B_n(T))$. Then, by taking $n\rightarrow\infty$ and observing that the second term tends to $0$, we obtain that: $$1=2\sum_{\{\alpha,\beta\}\in\Omega^2}h(ab).$$
The McShane identity follows as a corollary:
Let $\rho$ be a quasi-Fuchsian representation of the thrice-punctured projective plane fundamental group $\pi_1(N_{1,3})$. Then, $$\begin{aligned}
\sum_{\gamma\in{\mathop{\rm Sim}\nolimits}_2(N_{1,3})}\frac{1}{1+\exp{\tfrac{1}{2}\ell_{\gamma}(\rho)}}=\frac{1}{2},\end{aligned}$$ where the sum is over the collection ${\mathop{\rm Sim}\nolimits}_2(N_{1,3})$ of free homotopy classes of essential, non-peripheral two-sided simple closed curves $\gamma$ on $N_{1,3}$.
Given a simple closed two-sided geodesics $\gamma$, there is a unique pair of once-intersecting simple closed one-sided geodesics $\alpha,\beta$ that do not intersect $\gamma$ (and vice versa). Firstly, this bijection affords us the desired change in the summation indices. Secondly, by invoking the face relation : $$\begin{aligned}
ab=e+2=2\cosh(\tfrac{1}{2}\ell_\gamma)+2,\end{aligned}$$ $h(ab)$ yields the desired summand.
It is not yet clear to us whether Theorem \[th:main\] and Theorem \[th:bqmcshane\] are equivalent: if every BQ-Markoff map arises from a quasi-Fuchsian representation, then the two theorems are equivalent (and we’d have an algebraic characterisation for whether a representation is quasi-Fuchsian). If not, then Theorem \[th:bqmcshane\] is strictly stronger. It should be noted that it is still an open question whether Bowditch’s original BQ-conditions characterise the quasi-Fuchsian punctured torus representations [@BowMar].
Asymptotic growth of the simple length spectrum {#sec:asymp}
-----------------------------------------------
A punctured Klein bottle $K$ has a unique two-sided simple closed curve $\alpha$, and a family $\alpha_i$, $i\in\mathbb{Z}$ of one-sided simple closed curves. Set $A=2\cosh\frac{1}{2}\ell_{\alpha}(X)$ and $a_i=\sinh\frac{1}{2}\ell_{\alpha_i}(X)$ for a hyperbolic 1-cusped Klein bottle $X$. A trace identity yields: $$\label{eq:quad}
a_i^2+a_{i+1}^2-a_ia_{i+1}A=-1.$$ hence: $$\begin{aligned}
\lambda_{\pm}:=\lim_{i\to\pm\infty}\frac{a_i}{a_{i+1}}\text{ satisfy }\lambda_{\pm}^2-A\lambda_{\pm}+1=0,\text{ and }\lambda_{\pm}=\exp(\pm \tfrac{1}{2}\ell_{\alpha}).\end{aligned}$$ Thus, for $k\gg 0$, the sequence of traces for $\{\alpha_i\}$ is eventually approximated by: $$\begin{aligned}
\ldots,a_{\pm k},\ a_{\pm k}\exp\tfrac{1}{2}\ell_{\alpha},\ a_{\pm k}\exp\tfrac{2}{2}\ell_{\alpha},\ a_{\pm k}\exp\tfrac{3}{2}\ell_{\alpha},\ldots\end{aligned}$$ And since $2{\mathop{\rm arcsinh}\nolimits}(\tfrac{1}{2}\cdot)\approx2\log(\cdot)$ for large numbers, the lengths for $\{\alpha_i\}$ eventually resemble: $$\begin{aligned}
\ldots,\log(a_{\pm k}),\ \ell_{\alpha}+\log(a_{\pm k}),\ 2\ell_{\alpha}+\log(a_{\pm k}),\ 3\ell_{\alpha}+\log(a_{\pm k}),\ldots\end{aligned}$$ We see therefore that $s_X(L)$ grows linearly, $$\begin{aligned}
s_X(L) \sim \eta(X)\cdot L=\eta(X)\cdot L^{\dim\mathcal{M}(K)}\end{aligned}$$ and also holds for cusped Klein bottles. A natural question arises: *does polynomial growth still hold for non-orientable surfaces?*We show using Markoff quads that the answer is no, for 1-sided simple closed geodesics.
Given a quasi-Fuchsian representation $\rho$ of the thrice-punctured projective plane $N_{1,3}$, $$\begin{aligned}
\lim_{L\to\infty}\frac{s_\rho(L)}{L^m}>0\end{aligned}$$ for some $m$ satisfying $2.430<m < 2.477$.
Let $\phi$ denote the BQ-Markoff map induced by $\rho$ and let $\phi_0$ denote the Markoff map corresponding to the $(4,4,4,4)$ Markoff quad. Theorem \[converge\] tells us that there is a positive number $\kappa\in\mathbb{R}^+$ such that for all but finitely many $X\in\Omega^3$, $$\begin{aligned}
\frac{1}{\kappa}\log^+|\phi_0(X)|\leq\log^+|\phi(X)|\leq\kappa\log^+|\phi_0(X)|.\end{aligned}$$ Since $2\log^+(\cdot)=2\log(\cdot)\approx2{\mathop{\rm arcsinh}\nolimits}(\frac{1}{2}\cdot)$ for large inputs, this means that the growth rate of $\phi$ and $\phi_0$ are of the same order. The spectrum of $\phi_0$ consists of Markoff quads generated starting from $(4,4,4,4)$, and may be paraphrased in terms of the integral solutions of the $n=4$ Markoff-Hurwitz equation \[markoff-hurwitz\]. This is known to be between the orders $L^{2.430}$ and $L^{2.477}$ by Baragar’s work [@BarExp], and the result follows.
|
---
abstract: |
We propose a class of non-Markov population models with continuous or discrete state space via a limiting procedure involving sequences of rescaled and randomly time-changed Galton–Watson processes. The class includes as specific cases the classical continuous-state branching processes and Markov branching processes. Several results such as the expressions of moments and the branching inequality governing the evolution of the process are presented and commented. The generalized Feller branching diffusion and the fractional Yule process are analyzed in detail as special cases of the general model.
*Keywords:* Continuous-state Branching Processes, Time-change, Subordinators.
author:
- 'Luisa Andreis[^1]'
- 'Federico Polito[^2]'
- Laura Sacerdote
bibliography:
- 'paper.bib'
title: 'On a class of Time-fractional Continuous-state Branching Processes'
---
Introduction
============
Since the seminal paper of Galton-Watson [@watson1875probability], branching structures are subject to intensive theoretical and applied researches. The most studied applications of branching phenomena concern population growth models. In this context, in 1958, M. Ji[ř]{}ina [@Jir58] introduced the so-called continuous-state branching processes (shortly CSBPs) that represent a general class of linear branching processes in which jumps of any finite size and a continuous state space are permitted (see also [@Li10] and the references therein). The original definition of CSBPs is very similar to that of Lévy processes (with which they are linked by means of a random time change, the Lamperti transform). However, an alternative definition, dating back to the work of J. Lamperti [@Lam67] considers CSBPs as limit processes of sequences of rescaled Galton–Watson processes (GWPs in the following) or Markov branching processes (see also [@AlSh83; @Gri74] for further references). Due to their simple definition, generalizations of the GWPs and CSBPs have arisen in several directions, leading for example to the introduction of population-size-dependent GWPs and CSBPs [@Kleb84; @Li06], and controlled branching processes [@SeZu74], where the independence of individuals’ reproduction is modified allowing dependence on the size of the current population. In this paper we aim to extend the definition of GWPs and CSBPs in a different direction. Indeed, the Markov property characterizing these processes, although is mathematically appealing, determines a limitation for their actual application; furthermore, non-Markov branching processes would present interesting mathematical properties that constitute a reason of study by itself. Here we introduce a general class of non-Markov population models characterized by persistent memory and contructed by means of a limiting procedure on a sequence of suitably rescaled Galton–Watson processes time-changed by a specific random process. In order to clarify our approach, we briefly recall how time-changes play a fundamental role in the definition of models for anomalous diffusion. We will take inspiration from them. Roughly speaking, the basic framework is the following. Take a standard Brownian motion, say $\{B(t), t \ge 0\}$, and an independent stable subordinator $d=\{D(t), t\ge 0\}$, that is a spectrally positive increasing Lévy process with stable unilateral probability density function. Define the inverse process to $D$ as $$\mathcal{E}(t):=\inf\{u>0 \colon D(u)>t\}, \qquad t \ge 0.$$ Then, the time-changed process $\{B(\mathcal{E}(t)), t \ge 0\}$ is a non-Markov process with continuous sample paths and exhibiting a sub-diffusive behaviour. Furthermore, if $\mathbf{P}(B(\mathcal{E}(t)) \in dx)/dx = l(x,t)$ is the marginal probability density function of the time-changed Brownian motion, then $l(x,t)$ solves the fractional PDE $$\begin{aligned}
\partial_t^\beta l(x,t) = \frac{1}{2} \frac{\partial^2}{\partial x^2}l(x,t), \qquad t \ge 0, \: x \in \mathbb{R}, \: \beta \in (0,1).
\end{aligned}$$ The above operator acting on time is a non-local integro-differential operator called Džrbašjan–Caputo derivative (see Section \[back\] for prerequisites and specific information) and $\beta$ is the stability parameter. The main consequence of the presence of the fractional derivative is that, due to non-locality, it furnishes the model with a long memory.
Hence, in this paper we build via a limiting procedure and specific time-changes a large class of processes with branching structure also exhibiting non-locality and long memory. This is actually carried out in Section \[violin\]. Specific cases of interest being part of this class are, amongst others, the generalized Feller branching diffusion and the fractional Yule process.
Due to the nature of the considered problem, the paper fits exactly in-between two classical topics of probability, namely population models (processes exhibiting a branching structure) and models for anomalous diffusion (frequently associated to fractional diffusion).
The paper is organized as follows: in Section \[back\] we introduce the notation and recall the basic definitions and properties that we use in the sequel; in Section \[Limite\] we define the time-changed processes both in the discrete and the continuous setting and we prove the scaling limit; in Section \[sec\_properties\] we focus on the time-changed CSBPs with the proof of some properties and some examples.
Backgrounds {#back}
===========
The aim of this section is to give a brief overview of the processes we are interested in. We recall the definition and some basic properties of GWPs and of CSBPs; in particular the branching property is of fundamental importance. Moreover, basic information on fractional calculus and fractional diffusion is also recalled.
From GWPs to CSBPs
------------------
GWPs are classical discrete-time branching processes, where each individual of a population reproduces independently and according to the same offspring distribution $p$, see [@AtNe72] for a complete introduction. Rigorously, given a probability measure $p$ on $\mathbb{N}$, a GWP $\{Z_n\}_{n\geq0}$ with offspring distribution $p$ is the Markov chain such that, for all $n\geq 0$, $$Z_{n+1}\stackrel{d}{=}\sum_{i=1}^{Z_n}\xi_i,$$ where $\xi_i$ are i.i.d. random variables with common distribution $p$. Let us indicate with $m=\sum_{k=0}^\infty k p(k)$ the first moment of the distribution of the offspring. It classifies GWPs into three classes: subcritical if $m<1$, supercritical if $m>1$ and critical if $m=1$. The following characteristic feature of GWPs is the branching property. Let us call $_{(j)}Z$ the GWP starting with $j$ individuals, i.e. $_{(j)}Z_0=j$ almost surely. Then the GWP is the only discrete-time and discrete-space Markov process such that for all $j,k\geq 0$, $$\label{branch_property}
_{(j+k)}Z\stackrel{d}{=} {_{(j)}{Z}^{(1)}}+ {_{(k)}{Z}^{(2)}},$$ where $Z$, $Z^{(1)}$ and $Z^{(2)}$ are independent GWPs with the same offspring distribution. From a modelling point of view, this property underlines the fact that each individual in the population reproduces independently from the others according to the same offspring distribution $p$.
Since the seminal works of Jiřina and Lamperti [@Jir58; @Lam67CSBP; @Lam67], there has been interest in defining branching processes in a continuous state-space setting and in identifying them as scaling limits of GWPs. The simplest way to extend the definition of branching processes to describe the evolution in continuous time of a population with values in $\mathbb{R}^+$ is by means of the branching property. Indeed, we define the CSBPs as the continuous time-continuous space processes satisfying an analogue of the branching property as follows. Rigorously, a stochastic process $X=\{X(t):t\geq 0\}$ is a CSBP if it is a Markov process characterized by a family of transition kernels $\{P_t(x,dy), \, t\geq0, \, x \in \mathbb{R}^+\}$ satisfying, for all $t>0$ and $x,x'\in \mathbb{R}^+$ (see e.g. [@Lam67CSBP]), $$P_t(x,\cdot)*P_t(x',\cdot)=P_t(x+x',\cdot).$$ Let $\mathbb{D}(\mathbb{R}^+)$ be the set of càdlàg functions defined on $\mathbb{R}^+$ with values on $\mathbb{R}^+$, a CSBP is a random variable in $\mathbb{D}(\mathbb{R}^+)$. From now on we will consider $\mathbb{D}(\mathbb{R}^+)$ as a topological space endowed with the usual Skorokhod topology. For a complete description see [@JaSh13]. Further, we denote by $\mathbf{E}_x$ the expectation with respect to the law of the process $X$ starting from the initial value $x\in\mathbb{R}^+$. Let us underline that CSBPs are characterized by their Laplace transform, i.e. for all $\lambda>0$ we have $$\mathbf{E}_x\left[e^{-\lambda X(t)}\right]=
\int_{0}^{\infty}e^{-\lambda y}P_t(x,dy)=e^{-x \nu_t(\lambda)},$$ where $\nu_t(\lambda)$ is the unique nonnegative solution to the equation $$\label{exponent}
\nu_t(\lambda)+\int_0^t\psi(\nu_s(\lambda))ds=\lambda.$$ Here $\psi$ can be written as $$\psi(u)=bu+cu^2+\int (e^{-zu}-1+zu)m(dz),$$ where $b\in\mathbb{R}$, $c\geq0$ and $m$ is a $\sigma$-finite measure on $(0,\infty)$ such that $\int(z\wedge z^2)m(dz)<\infty$. The function $\psi$ is called the branching mechanism of the CSBP and, at the same time, it is the characteristic function of a Lévy process without negative jumps killed at the first time it becomes negative. This identifies a relationship between CSBPs and the latter class of Lévy processes that is known as Lamperti transform. Indeed, also the converse property holds true, i.e. the characteristic function $\psi$ of every Lévy process without negative jumps and killed at zero is the branching mechanism of a CSBP (see [@LeG99; @Sil68]). The branching mechanism $\psi$, in addition to the Lamperti transform, plays a role in classifying CSBPs in three categories: critical, subcritical and supercritical processes. A CSBP is supercritical when $b<0$, critical when $b=0$ and subcritical when $b>0$. Moreover, in [@Lam67] we see that the parameters of $\psi$ appear in the explicit form of the first two moments of a CSBP $X$, that is $$\begin{aligned}
\label{momenti}
\mathbf{E}_x[X(t)] & = x e^{-bt}, \\
\mathbf{E}_x[X(t)^2] & =
\begin{cases}
x^2 + x \tilde{\beta} t, & b=0, \\
x^2e^{-2bt}-\frac{\tilde{\beta}x}{b}\left( e^{-2bt}-e^{-bt}\right), & b\neq0, \notag
\end{cases}
\end{aligned}$$ where $\tilde{\beta}=\left(2c + \int_0^{\infty} u^2 m(du)\right )$. Let us mention that, despite CSBPs in general have discontinuous sample paths, the Feller branching diffusion (introduced in [@Fel51]) which is a CSBP whose branching mechanism has the form $\psi(u)=bu+cu^2$, exhibits continuous sample paths.
Results on convergence of suitably rescaled sequences of GWPs to CSBPs appeared first in [@Lam67] and, subsequently, in several other papers such as [@AlSh83; @Gri74; @Li10]. In the following we briefly state the results and the approach. Consider a sequence of GWPs $$Z^{(k)}=\{Z^{(k)}_{n}\}_{n \in\mathbb{N}}, \qquad k= 1,2,3,\dots,$$ defined through their offspring distribution $p^{(k)}$. Define a sequence of positive integers $\{c_k\}_{k\in\mathbb{N}}$, tending to infinity, and the Markov process $$\label{succlimite}
\{X_k(t)\}_{t\geq 0}=\left \{ \frac{Z^{(k)}_{\lfloor kt\rfloor}}{c_k} \right \}_{t\geq0}, \qquad
Z^{(k)}_0=c_k\, \quad a.s.,$$ where for each $y$ $\in$ $\mathbb{R}$ we denote with $\lfloor y \rfloor$ its integer part. If the sequence of processes $\{X_k\}_{k\geq0}$ has a weak limit in the sense of finite-dimensional distributions, then this limit is a CSBP. This result is extended to convergence in the Skorokhod space $\mathbb{D}(\mathbb{R}^+)$ in [@Gri74]. Briefly, let $\mu_k$ be the probability measure on $\left\{-1/c_k,0,1/c_k,2/c_k,\dots\right\}$ defined as follows: for all $n\in\mathbb{N}$, $$\mu_k\left(\frac{n-1}{c_k}\right)=p^{(k)}(n),$$ and assume that there exists a measure $\mu$ such that $(\mu_k)^{*k c_k}\rightarrow \mu$, weakly as $k\rightarrow\infty$. Then the sequence of GWPs $Z^{(k)}$ with offspring distribution $p^{(k)}$ and normalized as in , has a weak limit as a sequence of random variables on $\mathbb{D}(\mathbb{R}^+)$; this limit, say $X$, is a CSBP with initial condition $X(0)=1$ almost surely. Conversely, for every CSBP $X$ there exists a sequence of GWPs $\{Z^{(k)}\}_{k\in\mathbb{N}}$ and a sequence of positive integers $\{c_k\}_{k\in\mathbb{N}}$ such that $X$ is the limit of the sequence rescaled as in .
Random times and stable subordinators {#waiting_times}
-------------------------------------
Let us consider a sequence i.i.d. real positive random variables $J_1,J_2,\dots$ representing for us a sequence of random waiting times. We define for all $n\geq 0$ the process $T_n\colon = \sum_{i=1}^nJ_i$. Its inverse, for all $t\geq 0$, is the renewal process $$\label{N_t}
N_t\colon =\max \{n\geq0 \colon T_n\leq t\}.$$ We assume now that these waiting times belong to the strict domain of attraction of a certain completely skewed stable random variable $D$ with stability parameter $\beta\in(0,1)$. Note that due to the extended central limit theorem there exists a sequence $\{b_n\}_{n\geq0}$ such that the following convergence holds in distribution [@MeSi12]: $$b_n T_n \Rightarrow D.$$ As a consequence, the rescaled process $\left\{b_nT_{\lfloor nt \rfloor}\right\}_{t\geq0}$ converges in $\mathbb{D}(\mathbb{R}^+)$ to the stable subordinator $\{D(t)\}_{t\geq0}$ of parameter $\beta$, i.e. a Lévy process such that $D(t)\stackrel{d}{=}t^{1/\beta}D$ for all $t\geq0$ and with Laplace transform $$\begin{aligned}
\mathbf{E}[e^{-sD(t)}]=e^{-s^{\beta}t}, \qquad s>0.
\end{aligned}$$ Similarly, the scaling limit for the renewal process $\{N_t\}_{t\geq0}$ is the hitting time process of $\{D(t)\}_{t\geq0}$, that we define below. Indeed, let $\{\tilde{b}_n\}_{n\geq0}$ be a regularly varying sequence with index $\beta$ such that $\lim_{n\rightarrow\infty}n b_{\lfloor\tilde{b}_n\rfloor}=1$, then the following limit holds: $$\label{N->E}
\left\{\frac{N_{nt}}{\tilde b_n}\right\}_{t\geq0}\Rightarrow \{\mathcal{E}(t)\}_{t\geq0},$$ where the process $\{\mathcal{E}(t)\}_{t\geq0}$ is known as the inverse $\beta$-stable subordinator, defined as $$\mathcal{E}(t):=\inf\{u>0 \colon D(u)>t\}, \qquad t \ge 0.$$ The process $\{\mathcal{E}(t)\}_{t\geq 0}$ is a non-Markov process with non-decreasing continuous sample paths and plays a role in models of phenomena exhibiting long memory; for instance $\mathcal{E}$ has a fundamental importance in the study of time-fractional sub-diffusions [@MeSt13]. Let us now denote by $h(u,t)$ the probability density function of the random variable $\mathcal{E}(t)$ for a fixed time $t\geq0$. It is known that the Laplace transform of $h(u,t)$ w.r.t. variable $t$ is $$\label{laplace_h}
\mathcal{L}(h(u,t))(s)=s^{\beta-1}e^{-us^{\beta}}, \qquad s>0.$$ Furthermore, the Laplace transform w.r.t. variable $u$ reads $$\mathbf{E}[e^{-\lambda \mathcal{E}(t)}] = E_\beta(-t\lambda^\beta), \qquad \lambda > 0,$$ where $E_\nu(z)$ is the Mittag–Leffler function defined as the convergent series $$\begin{aligned}
\label{def_mittag_leffl}
E_\nu(x) = \sum_{r=0}^\infty \frac{x^r}{\Gamma(r \nu+1)}, \qquad x \in \mathbb{R}, \: \nu>0.
\end{aligned}$$ Moreover, the dynamic of this process is driven by a fractional evolution, i.e. $h(u,t)$ evolves according a governing equation involving a fractional derivative in the $t$ variable and a first order derivative in the $u$ variable. This means that $h(u,t)$, for all $t\geq0$ and $u>0$, solves the fractional PDE $$\partial_t^{\beta}h(u,t)=-\partial_u h(u,t),$$ where $\partial_t^{\beta}$ stands for the Džrbašjan–Caputo fractional derivative of order $\beta$ which is defined as follows:
\[capu\] Let $\alpha>0$, $m = \lceil \alpha \rceil$, and $f \in AC^m(0,b)$. The Džrbašjan–Caputo derivative of order $\alpha>0$ is defined as $$\label{Capu}
\partial_t^{\alpha} f(t)= \frac{1%
}{\Gamma(m-\alpha)}\int_a^{t}(t-s)^{m-1-\alpha}\frac{\textup{d}^m}{\textup{d}s^m}f(s) \, \textup{d}s.$$
Time fractional branching processes {#Limite}
===================================
Following the approach used in [@MeSc04] to define time-fractional diffusions, we introduce in this section a time-changed GWP and we prove that there exists a certain scaling such that its limit is exactly a time-changed CSBP.
Time-changed GWPs {#GW-modif}
-----------------
Let us consider a GWP $Z$, we want to define a GWP with random waiting times between successive generations. Further, let $\{J_1,J_2,\dots\}$ be a sequence of i.i.d. random variables. The time-changed GWP is defined as $$\label{time_changedGW}
\mathcal{Z}_t\colon = Z_{N_t},$$ for all $t\geq0$, where $N_t$, independent of $Z$, is the renewal process defined in . As soon as the waiting times $\{J_1,J_2,\dots\}$ are not exponentially distributed, the process $\{\mathcal{Z}_t\}_{t\geq0}$ is not a Markov process anymore. The following property holds.
\[lennovvo\] We have, for all $j,$ $k\in\mathbb{N}$ and all $\lambda\geq 0$, $$\label{Dis_branch_discrete}
\mathbf{E}_{j+k}\left[ e^{-\lambda \mathcal{Z}_t}\right]\geq \mathbf{E}_{j}\left[
e^{-\lambda \mathcal{Z}_t}\right]\mathbf{E}_{k}\left[ e^{-\lambda \mathcal{Z}_t}\right].$$
Let us consider the function $$\begin{aligned}
K_{j,k}(t)=\mathbf{E}_{j+k}\left[ e^{-\lambda \mathcal{Z}_t}\right]- \mathbf{E}_{j}\left[ e^{-\lambda \mathcal{Z}_t}\right]
\mathbf{E}_{k}\left[ e^{-\lambda \mathcal{Z}_t}\right].
\end{aligned}$$ By taking conditional expectation with respect to $N(t)$, we get $$\begin{aligned}
\label{eq2}
K_{j,k}(t)= {} & \mathbf{E}\left[\mathbf{E}_{j+k}\left[\left. e^{-\lambda Z_{N(t)}}\right | N(t)\right] \right] \\
& -\mathbf{E}\left[\mathbf{E}_{j}\left( \left.e^{-\lambda Z_{N(t)}}
\right | N(t) \right)\right]\mathbf{E}\left[\mathbf{E}_{k}\left(
\left.e^{-\lambda Z_{N(t)}} \right | N(t) \right) \right]. \notag
\end{aligned}$$ Observe that $\mathbf{E}_{x}\left[ \left.e^{-\lambda Z_{N(t)}} \right | N(t) \right]$ and $\mathbf{E}_{y}
\left[ \left.e^{-\lambda Z_{N(t)}} \right | N(t) \right]$ are positively correlated being functions of the same random variable $N(t)$. Indeed, if we denote by $f$ the generating function of the GWP $Z$ and $f_n$ its $n$-th iterate, we know that we have $$\begin{aligned}
\mathbf{E}_{x}\left[ \left.e^{-\lambda Z_{N(t)}} \right | N(t) \right]&=f_{N(t)}\left(e^{-\lambda}\right)^{x};\\
\mathbf{E}_{y}\left[ \left.e^{-\lambda Z_{N(t)}} \right | N(t) \right]&=f_{N(t)}\left(e^{-\lambda}\right)^{y}.
\end{aligned}$$ Hence $$K_{j,k}(t) = \text{Cov}(f_{N(t)}\left(e^{-\lambda}\right)^{j}, f_{N(t)}\left(e^{-\lambda}\right)^{k}).$$ Being positive powers of the same function of $N(t)$, positive correlation follows and we obtain the inequality .
In a GWP, when we start with an initial population $Z_0=j+k$, the number of individuals $Z_n$ in the $n$-th generation is the sum of two independent copies of the process with initial size equal to $j$ and $k$ respectively. By introducing a random time change between the generations, we create a positive correlation between the sizes of subgroups of a given initial population.
In the special case of deterministic time-change, that is when $\mathbb{P}(N_t=l)=1$, with $g \colon \mathbb{N}
\to \mathbb{N}$, $l=g([t])$ a suitable non decreasing function, the inequality becomes the classical equality that expresses the branching property of GWPs, i.e. $$\label{eq1}
\mathbf{E}_{j+k}\left[ e^{-\lambda Z_l}\right]=\mathbf{E}_{j}\left[
e^{-\lambda Z_l}\right]\mathbf{E}_{k}\left[ e^{-\lambda Z_l}\right],$$ for all $j,$ $k\in\mathbb{N}$ and all $\lambda\geq 0$.
Time-changed CSBP and scaling limit {#violin}
-----------------------------------
Let us consider a CSBP $X$ and an inverse $\beta$-stable subordinator $\mathcal{E}$ independent of $X$. Consider the time-changed process $$\mathcal{X}(t)\colon= X(\mathcal{E}(t)),$$ for all $t\geq0$. It is possible to show that there exists a sequence of time-changed GWPs $\{\mathcal{Z}^{(n)}_t\}_{t\geq0}$, such that, suitably rescaled, converges to the process $\{\mathcal{X}(t)\}_{t\geq0}$ in $\mathbb{D}([0,\infty))$.
\[theor\_conv\] Let $\{X(t)\}_{t\geq0}$ be a CSBP and $\{\mathcal{E}(t)\}_{t\geq0}$ be the inverse of a $\beta$-stable subordinator, $\beta\in(0,1]$, independent of $\{X(t)\}_{t \ge 0}$. Consider the process $\{\mathcal{X}(t):=X(\mathcal{E}(t))\}_{t\geq0}$; there exists a sequence of time-changed GWPs $\{\mathcal{Z}^{(n)}_t\}_{t\geq0}$ and two increasing sequences $\{\tilde{b}_n\}_{n\geq0}$ and $\{c_n\}_{n\geq0}$ with $\lim_{n\rightarrow\infty}\tilde{b}_n=\lim_{n\rightarrow\infty}c_n=\infty$, such that for $n\rightarrow\infty$ $$\label{limite_scaling}
\left\{\frac{\mathcal{Z}^{(\tilde{b}_n)}_{nt}}{c_{\tilde{b}_n}}\right\}_{t\geq0} \Longrightarrow \{\mathcal{X}(t)\}_{t\geq0},$$ where the convergence is in $\mathbb{D}([0,\infty))$.
Consider $J_1, J_2. \dots$, i.i.d. waiting times in the domain of attraction of a stable law of index $\beta$, (see Section \[waiting\_times\]). Then there exists a sequence of positive real numbers $\{\tilde b_n\}$, diverging to infinity, such that the limit holds. At the same time, we consider a sequence of GWPs $\{Z^{(k)}\}_{k\geq0}$ such that holds. Since the waiting times and the GWPs are independent, it follows that, for all $n$ $\geq 0$, $$\left( X_{\tilde{b}_n}(t), \frac{T(nt)}{b_n} \right ) =\left( \frac{Z^{(\tilde{b}_n)}(\lfloor\tilde{b}_nt\rfloor)}{c_{\tilde{b}_n}},
\frac{T(nt)}{b_n} \right) \Longrightarrow \left( X(t), D(t) \right)$$ in the product space $\mathbb{D}([0,\infty))\times \mathbb{D}([0,\infty))$, where $Z^{(k)}(0)/c_k \rightarrow x$, $X(t)$ is a [CSBP]{} with transition semigroup $P_t(x,\cdot)$ and $D(t)$ is the stable subordinator of parameter $\beta$. Let us write $\mathbb{D}_{\uparrow, u}(\mathbb{R}^+)$ for the subset of unbounded non decreasing càdlàg functions and $\mathbb{D}_{\uparrow\uparrow, u}(\mathbb{R}^+)$ for the subset of unbounded strictly increasing ones. We see that, for all $n\geq0$, the pair $$\begin{aligned}
\left( \frac{Z^{(\tilde{b}_n)}(\lfloor\tilde{b}_nt\rfloor)}{c_{\tilde{b}_n}}, \frac{T(nt)}{b_n} \right)
\end{aligned}$$ belongs to the product space $\mathbb{D}(\mathbb{R}^+)\times \mathbb{D}_{\uparrow, u}(\mathbb{R}^+)$ and the limit $\left( X(t) , D(t) \right)$ belongs to $\mathbb{D}(\mathbb{R}^+)\times \mathbb{D}_{\uparrow\uparrow, u}(\mathbb{R}^+)$. Then, following the approach in [@HeSt11], we define the function $\Psi\colon \mathbb{D}(\mathbb{R}^+)\times \mathbb{D}_{\uparrow, u}(\mathbb{R}^+) \rightarrow
\mathbb{D}(\mathbb{R}^+)\times \mathbb{D}(\mathbb{R}^+)$ mapping $(x(t),d(t))$ to $(x(e(t)), t)$, where $e(t)$ is the inverse of $d(t)$. In general the function $\Psi$ is not continuous, however, in our case, since the limit point $(X(t),D(t))$ actually belongs to $\mathbb{D}(\mathbb{R}^+)\times \mathbb{D}_{\uparrow\uparrow, u}(\mathbb{R}^+)$, as stated in [@HeSt11 Proposition 2.3], the function $\Psi$ is continuous at $(X(t),D(t))$. This implies that the following limit holds, where $\pi_1$ is the projection on the first coordinate, $$\begin{aligned}
\frac{\mathcal{Z}^{(\tilde{b}_n)}_{nt}}{c_{\tilde{b}_n}} & = X_{\tilde{b}_n}(N(nt))=\pi_1\left(\Psi\left( X_{\tilde{b}_n}(t),
\frac{T(nt)}{b_n} \right)\right) \\
& \Longrightarrow \pi_1\left(\Psi\left(X(t) , D(t) \right)\right).
\end{aligned}$$ This proves .
Some properties of the time-fractional CSBP {#sec_properties}
===========================================
In the previous section we have characterized the process $\{\mathcal{X}(t)\}_{t\geq0}$ as the limit of a rescaled sequence of time-changed GWPs, where in the discrete case the time between two generations is substituted by random variables that produce a slowed-down dynamics. In the limit this is modeled by the inverse stable subordinator. We are now interested in capturing the main features of the time-changed process $\mathcal{X}$ and in underlining the differences between it and the classical CSBP. Note that the tree structure underlying Markov branching processes and CSBPs, although randomly stretched and squashed, it is still a characterizing feature of the corresponding time-changed processes.
Branching property
------------------
Let us consider $\beta\in(0,1]$. We expect the time-changed CSBP $\{\mathcal{X}(t)\}_{t\geq0}$ to satisfy the classical branching property only when $\beta=1$. Indeed, in general, it holds $$\label{dis_branching_cont}
\mathbf{E}_{x+y}[e^{-\lambda \mathcal{X}(t)}]\geq \mathbf{E}_{x}[e^{-\lambda \mathcal{X}(t)}]
\mathbf{E}_{y}[e^{-\lambda \mathcal{X}(t)}]$$ and $$\label{lim_beta}
\lim_{\beta\rightarrow 1}\mathbf{E}_{x+y}[e^{-\lambda \mathcal{X}(t)}]=\mathbf{E}_{x}[e^{-\lambda \mathcal{X}(t)}]
\mathbf{E}_{y}[e^{-\lambda \mathcal{X}(t)}].$$ Similarly to Section \[GW-modif\], this is based on the following: $$\begin{aligned}
& \mathbf{E}_{x+y}[e^{-\lambda \mathcal{X}(t)}]- \mathbf{E}_{x}[e^{-\lambda \mathcal{X}(t)}]
\mathbf{E}_{y}[e^{-\lambda \mathcal{X}(t)}] \\
& \text{\small $= \mathbf{E}\left[\mathbf{E}_{x+y}\left(e^{-\lambda X(\mathcal{E}(t))}|\mathcal{E}(t)\right)\right]-
\mathbf{E}\left[\mathbf{E}_{x}
\left(e^{-\lambda X(\mathcal{E}(t))}|\mathcal{E}(t)\right)\right]\mathbf{E}\left[ \mathbf{E}_{y}
\left(e^{-\lambda X(\mathcal{E}(t))}|\mathcal{E}(t)\right)\right]$} \\
& = \mathbf{E}\left[e^{-(x+y)\nu_{\mathcal{E}(t)}(\lambda)}\right]
- \mathbf{E}[e^{-x\nu_{\mathcal{E}(t)}}] \mathbf{E}[e^{-y\nu_{\mathcal{E}(t)}}] \\
& = \text{Cov}(e^{-x\nu_{\mathcal{E}(t)}(\lambda)},e^{-y\nu_{\mathcal{E}(t)}(\lambda)}).
\end{aligned}$$ Then, by positive correlation, we see that $\text{Cov}(e^{-x\nu_{\mathcal{E}(t)}(\lambda)},e^{-y\nu_{\mathcal{E}(t)}(\lambda)})\geq0$. Moreover, since $\mathcal{E}(t)\rightarrow t$ in distribution as $\beta\rightarrow1$, by dominated convergence and the continuity of $\nu_{t}(\lambda)$ in $t$ (which is a consequence of ), we see that $$\begin{aligned}
\lim_{\beta\rightarrow1}\text{Cov}(e^{-x\nu_{\mathcal{E}(t)}(\lambda)},e^{-y\nu_{\mathcal{E}(t)}(\lambda)})=0,
\end{aligned}$$ proving . Note that the random time-change introduces a positive correlation between the evolution of the subgroups of the initial population that is not present in the classical CSBP. However, for any $\beta \in (0,1)$, we still have a conditional branching property, i.e. $$\begin{aligned}
\mathbf{E}[\mathbf{E}_{x+y}[e^{-\lambda X(\mathcal{E}(t))}|\mathcal{E}(t)]]= \mathbf{E}[\mathbf{E}_{x}[e^{-\lambda
X(\mathcal{E}(t))}|\mathcal{E}(t)]\ \mathbf{E}_{y}[e^{-\lambda X(\mathcal{E}(t))}|\mathcal{E}(t)]].
\end{aligned}$$
First and second moment {#momenti_sec}
------------------------
Here we obtain the expression for the first and the second moment of the process $\{\mathcal{X}(t)\}$, when they exist. To this aim, we exploit the computations in [@Lam67] for the explicit formula of first and second moment of a CSBP, see equation , and the properties of the Mittag–Leffler function defined in , see [@HaErMaOb55].
\[mom1csbpcomp\] Let $\{X(t)\}_{t\geq0}$ be a CSBP with Laplace exponent $v_t(\lambda)$ and branching mechanism $\psi(z)$ and let $\{\mathcal{E}(t)\}_{t\geq0}$ be an inverse stable subordinator with index $\beta\in(0,1)$ and with density function $h(\cdot,t)$, for every fixed time $t\geq0$. If $\frac{\partial v_t(0^+)}{\partial \lambda}$ exists finite and $\psi'(0^+)=b\geq \sigma_h$, where $\sigma_h$ is the abscissa of convergence for the Laplace transform of the function $h(\cdot,t)$, then the time-changed process $\{\mathcal{X}(t)\}_{t\geq0}$ has finite first moment that takes the form $$\label{formulamediaCSBPcomp}
\mathbf{E}_x[\mathcal{X}(t)]=x E_{\beta}(-bt^{\beta}), \qquad t\geq0.$$
For the independence of $\{\mathcal{E}(t)\}_{t\geq0}$ from $\{X(t)\}_{t\geq0}$, together with the formula for the first moment of a CSBP, we get $$\label{med}
\mathbf{E}_x[\mathcal{X}(t)]=\int_0^{\infty}\mathbf{E}_x[X(u)]h(u,t)du=\int_0^{\infty}xe^{-bu}h(u,t)du.$$ Since $b\geq\sigma_h$ the last integral is finite and it is essentially the Laplace transform of $h(u,t)$ with respect to the variable $u$. To obtain an explicit form of the integral, we apply again the Laplace transform to , this time with respect to the variable $t$, obtaining $$\mathcal{L}\left[ \mathbf{E}_x[\mathcal{X}(\cdot)]\right](\mu)= x \int_0^{\infty} e^{-bu} \int_0^{\infty} e^{-\mu t}h(u,t) dt du.$$ Formula leads to $$\begin{aligned}
\mathcal{L}\left[ \mathbf{E}_x[\mathcal{X}(\cdot)]\right](\mu)
& = x \mu^{\beta -1} \int_0^{\infty} e^{-u(b+\mu^{\beta})} du \\
& = x \frac{\mu^{\beta-1}}{\mu^{\beta} +b}.
\end{aligned}$$ Since the latter expression is the Laplace transform of the Mittag–Leffler function $E_{\beta}(-bt^\beta)$ we immediately obtain formula .
![\[fig:media\]Plots of the first moment $\mathbf{E}_1[\mathcal{X}(t)]$ of a time-changed CSBP for $t\in[0,4]$ and different values of $\beta$, from $0.2$ to $1$. The time-changed CSBP has initial condition $\mathcal{X}(0)=1$ a.s.; on the left the subcritical case with $b=1$, and on the right the supercritical case with $b=-1$.](Media_bpos.eps "fig:"){width="\linewidth"} ![\[fig:media\]Plots of the first moment $\mathbf{E}_1[\mathcal{X}(t)]$ of a time-changed CSBP for $t\in[0,4]$ and different values of $\beta$, from $0.2$ to $1$. The time-changed CSBP has initial condition $\mathcal{X}(0)=1$ a.s.; on the left the subcritical case with $b=1$, and on the right the supercritical case with $b=-1$.](Media_bneg.eps "fig:"){width="\linewidth"} ![\[fig:media\]Plots of the first moment $\mathbf{E}_1[\mathcal{X}(t)]$ of a time-changed CSBP for $t\in[0,4]$ and different values of $\beta$, from $0.2$ to $1$. The time-changed CSBP has initial condition $\mathcal{X}(0)=1$ a.s.; on the left the subcritical case with $b=1$, and on the right the supercritical case with $b=-1$.](legenda.pdf "fig:"){width="\linewidth"}
\[mom2csbpcomp\] Let $\{X(t)\}_{t\geq0}$ be a CSBP with Laplace exponent $v_t(\lambda)$ and branching mechanism $\psi(z)$ and let $\{\mathcal{E}(t)\}_{t\geq0}$ be an inverse stable subordinator with index $\beta\in(0,1)$ and with density function $h(\cdot,t)$ for every fixed time $t\geq0$. If $\frac{\partial v_t(0^+)}{\partial \lambda}$ and $\frac{\partial^2 v_t(0^+)}{\partial \lambda^2}$ exist finite and $\psi'(0^+)=b\geq\sigma_h$ as in Theorem \[mom1csbpcomp\], then the time-changed process $\{\mathcal{X}(t)\}_{t\geq0}$ has the following finite second moment: $$\label{momsecondoCSBPcomp}
\mathbf{E}_x\left[ \mathcal{X}(t)^2\right] =
\begin{cases}
x^2 + x \tilde{\beta} \frac{\Gamma(2)}{\Gamma(\beta+1)}t^{\beta}, & b=0, \\
x^2E_{\beta}(-2bt^{\beta})-\frac{\tilde{\beta}x}{b}\left( E_{\beta}(-2bt^{\beta})-E_{\beta}(-bt^{\beta})\right),
& b\neq0,
\end{cases}$$ where $\tilde{\beta}=\left(2c + \int_0^{\infty} u^2 m(du)\right )$.
Fix $t\geq0$, we divide the proof into two different cases.
- *Case $b=0$:* We know that $$\begin{aligned}
\mathbf{E}_x\left[\mathcal{X}(t)^2\right] & =\mathbf{E}\left[ \mathbf{E}_x\left[\left.
X(\mathcal{E}(t))^2\right | \mathcal{E}(t)\right] \right] \\
& = \int_0^{\infty} (x^2+ x\tilde{\beta}u) h(u, t) du
= x^2 + x \tilde{\beta} \mathbf{E}[\mathcal{E}(t)].
\end{aligned}$$ It is known, see [@MeSc04], Corollary 3.1, that the first moment of the process $\{\mathcal{E}(t)\}_{t\geq0}$, for a fixed time $t\geq0$, takes the form $$\begin{aligned}
\mathbf{E}\left[\left(\mathcal{E}(t)\right)\right]=\frac{\Gamma(2)t^{\beta}}{\Gamma(\beta+1)}.
\end{aligned}$$ Hence we obtain $$\begin{aligned}
\mathbf{E}_x\left[\mathcal{X}(t)^2\right]=x^2 + x \tilde{\beta} \frac{\Gamma(2)t^{\beta}}{\Gamma(\beta+1)}.
\end{aligned}$$
- *Case $b\neq0$:* In this case we write $$\begin{aligned}
\mathbf{E}_x\left[\mathcal{X}(t)^2\right] & = \int_0^{\infty} \left(x^2e^{-2bu}
-\frac{\tilde{\beta}x}{b}(e^{-2bu}-e^{-bu}) \right) h(u, t) du\\
& = x^2E_{\beta}(-2bt^{\beta})-\frac{\tilde{\beta}x}{b}\left( E_{\beta}(-2bt^{\beta})-E_{\beta}(-bt^{\beta})\right).
\end{aligned}$$
![\[fig:var\]Plots of $Var(\mathcal{X}(t))$ for $t\in [0,4]$ and $\beta$ from $0.2$ to $1$. The time-changed CSBP has initial condition $\mathcal{X}(0)=1$ a.s. and the pair of parameters $(b,\, \tilde \beta)$, clockwise from the upper-left, equal to $(1,\,0.1)$, $(1,\,0.5)$, $(1, \,10)$ and $(0,\,1)$, respectively.](Var_minbx.eps "fig:"){width="0.44\linewidth"} ![\[fig:var\]Plots of $Var(\mathcal{X}(t))$ for $t\in [0,4]$ and $\beta$ from $0.2$ to $1$. The time-changed CSBP has initial condition $\mathcal{X}(0)=1$ a.s. and the pair of parameters $(b,\, \tilde \beta)$, clockwise from the upper-left, equal to $(1,\,0.1)$, $(1,\,0.5)$, $(1, \,10)$ and $(0,\,1)$, respectively.](Var_b0.eps "fig:"){width="0.44\linewidth"} ![\[fig:var\]Plots of $Var(\mathcal{X}(t))$ for $t\in [0,4]$ and $\beta$ from $0.2$ to $1$. The time-changed CSBP has initial condition $\mathcal{X}(0)=1$ a.s. and the pair of parameters $(b,\, \tilde \beta)$, clockwise from the upper-left, equal to $(1,\,0.1)$, $(1,\,0.5)$, $(1, \,10)$ and $(0,\,1)$, respectively.](legenda.pdf "fig:"){width="0.1\linewidth"}
![\[fig:var\]Plots of $Var(\mathcal{X}(t))$ for $t\in [0,4]$ and $\beta$ from $0.2$ to $1$. The time-changed CSBP has initial condition $\mathcal{X}(0)=1$ a.s. and the pair of parameters $(b,\, \tilde \beta)$, clockwise from the upper-left, equal to $(1,\,0.1)$, $(1,\,0.5)$, $(1, \,10)$ and $(0,\,1)$, respectively.](Var_ugbx.eps "fig:"){width="0.44\linewidth"} ![\[fig:var\]Plots of $Var(\mathcal{X}(t))$ for $t\in [0,4]$ and $\beta$ from $0.2$ to $1$. The time-changed CSBP has initial condition $\mathcal{X}(0)=1$ a.s. and the pair of parameters $(b,\, \tilde \beta)$, clockwise from the upper-left, equal to $(1,\,0.1)$, $(1,\,0.5)$, $(1, \,10)$ and $(0,\,1)$, respectively.](Var_magbx.eps "fig:"){width="0.44\linewidth"}
Note that the Mittag–Leffler function is a generalization of the exponential function, with which it coincides for $\beta=1$. Comparing the moments of our generalized model, in and , to those of the CSBP in , it is easy to see that the Mittag–Leffler function in the generalized case plays the same role as the exponential in the classical case. See in Figure \[fig:media\] and Figure \[fig:var\] the effect of the time-change on the mean and variance of the process $\mathcal{X}$.
Some examples
-------------
In the previous sections we have described in full generality the time-changed CSBP $\{\mathcal{X}(t)\}_{t\geq0}$, now let us focus on some specific cases of interest in order to better illustrate our framework.
### Time-changed Feller branching diffusion
Consider the Feller branching diffusion [@Fel51] and recall that it is the diffusion process solving the SDE $$\label{sdefeller}
dX_t=-b X_t dt + \sqrt{2c X_t} dW_t$$ where $W_t$ is a standard Brownian motion, $b$ $\in \mathbb{R}$ and $c>0$. This is the only diffusion process in the class of CSBPs and its corresponding Fokker–Plack equation is $$\frac{\partial}{\partial t} p(y,t)=\frac{\partial }{\partial y} \left(byp(y,t)\right)
+\frac{\partial^2}{\partial y^2}\left( cyp(y,t)\right).$$
The scaling limit of GWPs that leads to Feller branching diffusion is well-known, see Pardoux [@Par08] for a nice review on it; this is one of the few cases in which this scaling scheme is known explicitly.
Consider thus a time-changed Feller branching diffusion $\{\mathcal{X}(t)\}_{t\geq0}$ with stability parameter $\beta \in (0,1)$. Since the process $\{X(t)\}_{t\geq0}$ is a diffusion, its composition with $\{\mathcal{E}(t)\}_{t\geq0}$ fits in the framework of SDE driven by time-changed Lévy processes, see [@HaKoUm12]. Therefore, it is possible to write an analogue of the Fokker–Planck equation solved by the marginal probability density function $m_x(y,t)$ of $\{\mathcal{X}(t)\}_{t\geq0}$. The following proposition shows that the equation involves Džrbašjan–Caputo derivatives of order $\beta\in(0,1)$, hence it classifies the time-changed Feller branching diffusion in the class of subdiffusions.
Let $\{\mathcal{X}(t)\}_{t\geq0}$ be a time-changed Feller branching diffusion with branching mechanism $\psi(u)=bu+cu^2$, for $b\in\mathbb{R}$ and $c>0$ and parameter $\beta \in (0,1)$. Let $\mathcal{X}(0)=x>0$ a.s. and $m_x(y,t)$ be the marginal probability density function of $\mathcal{X}(t)$, for all $t\geq0$. Then $m_x(y,t)$ satisfies the equation $$\partial^{\beta}_t m_x(y,t)=\frac{\partial }{\partial y} \left(bym_x(y,t)\right)
+\frac{\partial^2}{\partial y^2}\left( cym_x(y,t)\right),$$ where $\partial^{\beta}_t$ is the Džrbašjan–Caputo derivative.
Moreover, note that it is possible to write explicitely the SDE solved by the process $\{\mathcal{X}(t)\}_{t\geq0}$. Let $\left(\Omega, \mathcal{F},\mathbb{G}=\{\mathcal{G}_t\}_{t\geq0},\mathbf{P}\right)$ be a filtered probability space and let $D=\{D(t)\}_{t\geq0}$ be a $\mathbb{G}$-adapted stable subordinator of parameter $\beta\in(0,1)$. Furthermore, let $\mathbb{F}=\{\mathcal{F}_t\}_{t\geq0}$ be the filtration defined by means of time-change with the process $\{\mathcal{E}(t)\}_{t\geq0}$, inverse of $D$, such that, for all $t\geq0$, $\mathcal{F}_t=\mathcal{G}_{\mathcal{E}(t)}$ (see [@MR542115], page 312). Consider the filtered space $\left(\Omega, \mathcal{F},\mathbb{F},\mathbf{P}\right)$ and suppose $\{X(t)\}_{t\geq0}$ is a $\mathbb{G}$-adapted Feller branching diffusion. Then the process $\{\mathcal{X}(t)\}_{t\geq0}$ is solution of the SDE $$d\mathcal{X}(t)=-b\mathcal{X}(t)d\mathcal{E}(t)+\sqrt{2c\mathcal{X}(t)}dW_{\mathcal{E}(t)},$$ where $\{W_{\mathcal{E}(t)}\}_{t\geq0}$ is an $\mathbb{F}$-adapted time-changed Brownian motion, also known as grey Brownian motion.
### Time-changed Yule process
Let us consider a homogeneous Poisson process $\{Y(t)\}_{t\geq0}$ with rate $\theta>0$ and shifted upwards by 1. By relation , it is transformed into a CSBP $\{X(t)\}_{t\geq0}$ with Laplace exponent $$\begin{aligned}
\nu_t(\lambda)=\log(1-(1-e^{\lambda})e^{\theta t}), \qquad t\geq0 \: \lambda\geq 0.
\end{aligned}$$ This is the Laplace exponent of a Yule process $\{X(t)\}_{t\geq0}$, that is a pure birth process with linear birth rate. If $X(0)$ is supported on the strictly positive integers, then the law of $X$ at every time $t\geq0$ is a probability measure $\{p(\cdot,t)\}$ satisfying $$\begin{aligned}
\frac{\partial}{\partial t} p(n,t)=\theta (n-1) p(n-1,t)-\theta n p(n,t), \qquad n \ge 1.
\end{aligned}$$ The time-changed Yule process $\{\mathcal{X}(t)\}_{t\geq0}$ is studied in [@OrPo10], where amongst other properties it is proved that for each $t\geq0$ its law is a probability measure $p_{\beta}(\cdot, t)$ that satisfies the time-fractional difference-differential equations $$\begin{aligned}
\partial^{\beta}_t p_{\beta}(n,t)=\theta (n-1) p_{\beta}(n-1,t)-\theta n p_{\beta}(n,t), \qquad n \ge 1,
\end{aligned}$$ and whose explicit form is $$\begin{aligned}
p_{\beta}(n,t)=\sum_{j=1}^n\binom{n-1}{j-1} (-1)^{j-1}E_{\beta}(-\theta j t^{\beta}), \qquad n \ge 1,
\end{aligned}$$ that is consistent with our results in Section \[momenti\_sec\].
### Acknowledgements {#acknowledgements .unnumbered}
F. Polito and L. Sacerdote have been supported by the projects *Memory in Evolving Graphs* (Compagnia di San Paolo/Università di Torino) and by INDAM (GNAMPA/GNCS). F. Polito has also been supported by project *Sviluppo e analisi di processi Markoviani e non Markoviani con applicazioni* (Università di Torino). L. Andreis has been partially supported by Centro Studi Levi Cases (Università di Padova).
[^1]: Dipartimento di Matematica “T. L. Civita”, Università degli Studi di Padova, via Trieste 63, 35121 Padova (Italy); e-mail addresses: [email protected]
[^2]: Dipartimento di Matematica “G. Peano", Università degli Studi di Torino, via Carlo Alberto 10, 10123 Torino (Italy); e-mail address: $\{$federico.polito, laura.sacerdote$\}[email protected]
|
---
abstract: 'We present a generic model of an atom laser by including a pump and loss term in the Gross-Pitaevskii equation. We show that there exists a threshold for the pump above which the mean matter field assumes a non-vanishing value in steady-state. We study the transient regime of this atom laser and find oscillations around the stationary solution even in the presence of a loss term. These oscillations are damped away when we introduce a position dependent loss term. For this case we present a modified Thomas-Fermi solution that takes into account the pump and loss. Our generic model of an atom laser is analogous to the semi-classical theory of the laser.'
address: |
${}^1$Abteilung für Quantenphysik, Universität Ulm, D-89069 Ulm, Germany\
${}^2$Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand
author:
- 'B. Kneer,${}^{1,}$[@E.Mail] T. Wong,${}^2$ K. Vogel,${}^1$ W. P. Schleich,${}^1$ and D. F. Walls${}^2$'
date: 'Submitted to Physical Review A: June 18, 1998; revised manuscript submitted: August 20, 1998'
title: Generic model of an atom laser
---
[2]{}
Introduction
============
With the recent experiments on Bose-Einstein condensation [@BEC1; @BEC2; @BEC3; @BEC4; @BEC5; @BEC6; @BEC7; @BEC8; @BEC9; @BEC10; @BEC11; @BEC12; @reviews] an atom laser, that is a device which produces an intense coherent beam of atoms by a stimulated process [@ketterle; @wiseman], has become feasible. Already, the MIT group has realized a pulsed atom laser [@mewes] and has provided evidence for the process of coherent matter-wave amplification in the formation of a Bose condensate [@miesner].
How to describe the matter field of an atom laser? Rate equations are simple. But they cannot answer this question since they do not contain any coherence. In contrast a microscopic and fully quantum mechanical treatment can answer the question but is not easy to handle. What we need is a theory that includes coherence but is still simple. In the present paper we develop such a theory—a generic model of the atom laser.
The optical laser has three essential ingredients: (i) a resonator for the electromagnetic field, (ii) an atomic medium, and (iii) an excitation mechanism for the atoms. We start the laser cycle by preparing the atoms in an excited state using this excitation mechanism, which in general is incoherent. The radiation emitted by the atom amplifies the electromagnetic field in a mode of the resonator. To be efficient the frequency of the mode has to match appropriately the frequency of the transition. The boundary conditions set by the resonator determine via the Helmholtz equation the spatial part of the mode function. The ultimate goal is to transfer the excitation of the atom into a macroscopic excitation of the field mode. In this way we transfer the energy used to excite the atom via the gain medium into coherent excitation of the field mode. In order to make use of the radiation we have to couple it out of the resonator. We compensate for this loss by continuously re-pumping the medium.
The goal of an atom laser is completely analogous: We want to create a macroscopic coherent excitation of a mode in a resonator for atoms. Hence, in an atom laser the atoms play the role of the field excitation of the optical laser. Since atoms cannot be created or annihilated the means to achieve lasing are different. Indeed, in an atom laser there is no “real” laser medium: It is the same atom that goes through the laser process. We only manipulate the internal degrees of freedom and center-of-mass motion of the atoms. In particular we want to force their center-of-mass motion into a specific quantum state of the resonator. The resonator for the atoms is a binding potential such as provided by a trap. The spatial part of the mode function of this atomic resonator follows from the time independent Schrödinger equation [@dipdip]. Moreover, we focus on the ground state of the trap. As in the optical laser we want a macroscopic excitation of this mode, that is we strive to have as many atoms as possible in one quantum state. This is the phenomenon of Bose-Einstein condensation. As in the optical laser we need to couple the atomic wave out of the atomic resonator. In order to have a continuous wave (cw) atom laser, we have to continuously feed in more atoms.
There exist two different approaches towards a theoretical description of a cw atom laser: The first one relies on rate equations [@rateoptical1; @rateoptical2; @rateoptical3] whereas the second one derives a quantum mechanical master equation . In the present paper we suggest a third approach which makes heavily use of the close analogy between an atom laser and an optical laser. In the latter case it turned out that a classical treatment of the electromagnetic field [@haken; @lamb] was sufficient to describe many features of the laser. Can we therefore devise a semi-classical theory of the atom laser?
The semi-classical laser theory replaces the electromagnetic field operator $\hat{E}({\bf r},t)$ for the field inside the laser cavity by the expectation value ${\cal E}({\bf r},t)\equiv\langle\hat{E}({\bf r},t)\rangle$. The equation of motion for ${\cal E}$ is the wave equation of Maxwell’s electrodynamics driven by the polarization of the laser medium. An additional term introduced phenomenologically takes into account the loss of the cavity. The polarization of the laser medium follows from a microscopic, quantum mechanical description of the internal structure of the atoms. Quantum mechanics rules the atoms whereas classical Maxwell’s wave theory determines the electromagnetic field. These are the essential ideas of semi-classical laser theory.
The semi-classical laser equations do not prefer any particular phase. Nevertheless by choosing an arbitrary phase we can describe many properties of the electromagnetic field. Furthermore, we have to start with a non-vanishing seed field in order to obtain a non-vanishing solution for the electromagnetic field with a fixed phase.
In our model of an atom laser we replace the matter-wave field represented by a field operator $\hat{\Psi}({\bf r},t)$ by a scalar mean field $\psi({\bf r},t)\equiv\langle\hat{\Psi}({\bf r},t)\rangle$. This is analogous to replacing the field operator $\hat{E}$ by its expectation value ${\cal E}$ in semi-classical laser theory. The well-known Gross-Pitaevskii equation [@GPE1; @GPE2] plays now the role of Maxwell’s wave equation. It defines the equation of motion for $\psi({\bf r},t)$. Similar to the semi-classical theory of the optical laser we have to break the symmetry of the equation of motion for $\psi$ in order to have a non-vanishing value for $\psi$. For massive particles this is more problematic than for photons since they cannot be created or annihilated, and for all quantum states with fixed particle number we have $\langle\hat{\Psi}\rangle =0$. Nevertheless the concept of spontaneously broken symmetry turned out to be very useful to describe properties of a condensate, in particular interference effects. For more detailed discussions see Refs. [@wiseman; @moelmer]. In contrast to the driven electromagnetic wave equation the Gross-Pitaevskii equation does not contain a gain term analogous to the polarization. This reflects the fact that in Bose-Einstein condensation there is no “medium” in the trap. We therefore add a phenomenological pump term. Moreover, as in the electromagnetic case we have to add a loss term.
Despite the similarity there is a fundamental difference in the two equations of motion. In the absence of a medium Maxwell’s wave equation is linear. In contrast the Gross-Pitaevskii equation is non-linear. This is a manifestation of the interaction of atoms.
The crucial part of any laser is the stimulated amplification process. Different mechanisms for matter-wave amplification have been suggested and discussed: optical cooling [@rateoptical1; @rateoptical2; @rateoptical3; @masteroptical1; @masteroptical2], elastic collisions by evaporative cooling [@masterevap1; @masterevap2; @gardiner], dissociation of molecules [@masterdissoc], and cooling by a thermal reservoir [@scully]. For different schemes of pumping a condensate we refer to Refs. [@pump1; @pump2; @pump3]. Our model does not rely on a specific mechanism, but can be adapted to any mechanism where Bose enhancement is present.
The paper is organized as follows: In Sec. \[sec:model\], we generalize the Gross-Pitaevskii equation by including gain and loss terms. One important loss is due to coupling the atom wave out of the resonator. However, similar to semi-classical laser theory where usually the field inside the laser cavity is investigated, we restrict ourselves to the matter-wave field inside the resonator. We therefore do not go into details of an output coupler [@output1; @output2; @output3; @output4; @output5; @output6]. We present the stationary solutions of our equations and perform a stability analysis. We find a threshold behavior similar to the optical laser. Moreover, we calculate the time dependent solution and show how it converges to a quasi-stationary solution. In general, the quasi-stationary solution we find does not coincide with the time independent solution but shows some oscillatory behavior around it. These oscillations disappear when we modify our equations in Sec. \[sec:spaceloss\] by introducing a space dependent loss. This loss can be thought of as a consequence of collisions between condensed and un-condensed atoms at the edges of the condensate. For this improved model we present a modified Thomas-Fermi approximation for the stationary solution. Section \[sec:concludere\] summarizes our results.
Elementary model {#sec:model}
================
In this section we summarize our generic model of an atom laser.
Formulation of the model {#subsec:formulation}
------------------------
In particular, we add the phenomenological gain and loss terms to the Gross-Pitaevskii equation (GPE). In this way we couple the GPE to an equation governing the number of un-condensed atoms.
The Gross-Pitaevskii equation (GPE) for the mean field $\psi=\psi({\bf r})$ of the condensed atoms of mass $m$ in a trap potential $V({\bf r})$ reads [@BEC-GPE1; @BEC-GPE2] $$\label{GPeq}
i\hbar \frac{\partial \psi}{\partial t}
=-\frac{\hbar^2}{2m}\Delta\psi+V({\bf r})\psi+U_0|\psi|^2\psi.$$ The non-linear term $U_0|\psi|^2\psi$ with $U_0=4\pi\hbar^2 a_s/m$ takes into account two-particle interactions where $a_s$ denotes the s-wave scattering length.
The GPE has been very successful in describing the properties of Bose-Einstein condensates. However, in the present form it cannot describe the growth of or the loss of atoms out of a condensate. Indeed, the GPE keeps the number of atoms $$\label{DefNg}
N_c \equiv\int |\psi({\bf r})|^2 \, d^3r$$ constant.
In order to overcome this problem we introduce two additional terms in the GPE: a loss term and a gain term. The loss term $$\label{lossterm}
H_{{\rm loss}} \psi \equiv -\frac{i\hbar}{2}\gamma_c \psi$$ leads to an exponential decay of the number of atoms in the condensate. In contrast the gain term $$\label{gainterm}
H_{{\rm gain}} \psi \equiv \frac{i\hbar}{2}\Gamma N_u \psi$$ leads to an increase of atoms in the condensate. Here $N_u$ is the number of atoms outside the condensate, that is, the un-condensed atoms, and $\Gamma$ is the rate for the transition of these atoms into the condensate. We regard this as a generic pump mechanism [@mechanism] of an atom laser since it contains already the Bose enhancement as we will see later.
When we add the loss term (\[lossterm\]) and the gain term (\[gainterm\]) to the GPE (\[GPeq\]), we arrive at the generalized GPE $$\begin{aligned}
\label{psi}
i\hbar \frac{\partial \psi}{\partial t}
&=&-\frac{\hbar^2}{2m}\Delta\psi+V({\bf r})\psi +U_0|\psi|^2\psi
\nonumber\\
&-&\frac{i}{2}\hbar\gamma_c\psi+\frac{i}{2}\hbar\Gamma N_u\psi.\end{aligned}$$ For the number of un-condensed atoms $N_u$ we assume the rate equation $$\label{Ne}
\dot{N}_u=R_u-\gamma_u N_u-\Gamma N_c N_u.$$ The first term reflects the fact that we generate with a rate $R_u$ atoms in an un-condensed state from an infinite reservoir of atoms. The term $-\gamma_u N_u$ takes into account that atoms can escape from our system without being trapped in the condensed state. The last term describes the transition of the atoms into the condensate. Since it is proportional to the number of atoms $N_c$ in the condensate, it contains the Bose-enhancement factor.
In this way we have coupled the generalized GPE governing the condensate to the rate equation governing the number of un-condensed atoms. These two equations (\[psi\]) and (\[Ne\]) are the foundations of our model.
The number $N_c$ of condensed atoms follows from the generalized GPE (\[psi\]) with the help of the definition, Eq. (\[DefNg\]). From Eq. (\[psi\]) we obtain the rate equation $$\label{Ng}
\dot{N}_c=\Gamma N_u N_c -\gamma_c N_c$$ for the number of atoms in the condensate.
We conclude this section by noting that rate equations similar to Eqs. (\[Ne\]) and (\[Ng\]) have already been discussed in the literature [@rateoptical1; @rateoptical2]. However, in the present paper we replace the rate equation (\[Ng\]) by the generalized GPE (\[psi\]). This equation obviously contains the rate equation but in addition the coherence.
Stationary solutions and stability analysis {#subsec:stab}
-------------------------------------------
Before we turn to the stationary solution of the generalized GPE, Eq. (\[psi\]), we first discuss the stationary solutions of the rate equations (\[Ne\]) and (\[Ng\]).
### Rate equations {#subsec:rate}
The rate equations (\[Ne\]) and (\[Ng\]) have two possible stationary solutions:
The solution $$\begin{aligned}
N_c^{(s)} &\equiv &0, \nonumber\\
N_u^{(s)} &=& R_u/\gamma_u
\label{ratestatsol1}\end{aligned}$$ is reminiscent of the optical laser below threshold where the intensity of the laser vanishes.
The other stationary solution $$\begin{aligned}
N_c^{(s)}&=&\frac{R_u}{\gamma_c}-\frac{\gamma_u}{\Gamma}, \nonumber \\
N_u^{(s)}&=&\frac{\gamma_c}{\Gamma}
\label{ratestatsol2}\end{aligned}$$ corresponds to the laser above threshold.
We now perform a stability analysis of these solutions. For this purpose we introduce small deviations $$\begin{aligned}
n_u(t) &\equiv& N_u(t) - N_u^{(s)}, \nonumber \\
n_c(t) &\equiv& N_c(t) - N_c^{(s)},\end{aligned}$$ and arrive at the linearized equations $$\begin{aligned}
\dot{n}_u &=& -(\gamma_u +\Gamma N_c^{(s)}) n_u - \Gamma N_u^{(s)} n_c ,
\nonumber \\
\dot{n}_c &=& -(\gamma_c - \Gamma N_u^{(s)}) n_c + \Gamma N_c^{(s)} n_u .
\label{linearized}\end{aligned}$$ A stability analysis of these equations shows that the stationary solution in Eqs. (\[ratestatsol1\]) is stable for $$\label{threshold}
R_u < R^{th}\equiv\frac{\gamma_c\gamma_u}{\Gamma}.$$ Likewise, our stability analysis shows that Eqs. (\[ratestatsol2\]) are a stable stationary solution for $$R_u > \frac{\gamma_c\gamma_u}{\Gamma}= R^{th}.$$ Therefore, there exists a threshold $R^{th}$. When the pump rate $R_u$ is below the threshold, the number $N_c$ of atoms in the condensate vanishes. When it is above, it is non-vanishing. In that case the steady-state number of atoms in the condensate grows linearly with the pump rate $R_u$.
### Generalized GPE
We now turn to the discussion of the stationary solutions of our generic model of the atom laser, that is of the generalized GPE coupled to the rate equation for $N_u$.
The stable stationary solution of Eq. (\[psi\]) corresponding to a vanishing number of atoms in the condensate is $$\psi^{(s)} \equiv 0.$$ This is the stationary solution of the mean field below threshold.
Above threshold the stationary solution $\psi^{(s)}$ of the generalized GPE (\[psi\]) is identical to the stationary solution of the conventional GPE. Indeed, when we substitute the ansatz $\psi=\exp(-i\mu t/\hbar) \psi^{(s)}$ into Eq. (\[psi\]) and note that the gain and loss terms cancel each other as a result of the stationary solution, Eq. (\[ratestatsol2\]), we arrive at $$\label{GPeqtimeindep}
\mu\psi^{(s)} =-\frac{\hbar^2}{2m}\Delta\psi^{(s)}+
V({\bf r})\psi^{(s)}+U_0|\psi^{(s)}|^2\psi^{(s)}.$$ Here, $\mu$ denotes the chemical potential.
A stability analysis of this equation leads to collective excitations [@excitations] which in general cannot be treated analytically. For such an analysis in the case of a one-dimensional harmonic oscillator we refer to the Appendix.
Transient behavior {#subsec:num}
------------------
In the preceding section we have discussed the steady-state solutions of both the rate equations (\[Ne\]) and (\[Ng\]) and the matter-field equations (\[psi\]) and (\[Ne\]) of the atom laser. In the present section we address the question if and how the matter field evolves into a stationary state from an initial condition. Here we focus on the case above threshold. Since the equations are non-linear we solve them numerically for the case of an one-dimensional harmonic oscillator potential $$\label{harmpot}
V(x)=\frac{1}{2} m \omega^2 x^2.$$ This one-dimensional trap potential already shows the essential features of the atom laser.
We therefore analyze the one-dimensional generalized GPE $$\begin{aligned}
\label{psi1D}
i\hbar \frac{\partial}{\partial t}\psi(x,t)
&=&\left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}
+\frac{1}{2}m \omega^2 x^2 +U_x|\psi(x,t)|^2 \right.
\nonumber \\
&&-\left.\frac{i}{2}\hbar\gamma_c
+\frac{i}{2}\hbar\Gamma N_u\right]\psi(x,t),\end{aligned}$$ describing the matter field, coupled to the equation $$\label{Ne1D}
\dot{N}_u=R_u-\gamma_u N_u-\Gamma N_c N_u$$ for the number of un-condensed atoms. We use the split-operator method [@split1; @split2] to solve the generalized GPE. This technique has already been used successfully for the ordinary GPE [@munich]. Moreover, we supplement this numerical analysis by an analytical treatment of the solution in the long-time limit where we apply the method of Ref. [@coll2] to one dimension. For the details of the analytical approach we refer to the Appendix.
### Initial conditions and parameters
The semi-classical theory of the optical laser cannot explain the initial start-up of the laser since it does not contain spontaneous emission: A non-vanishing seed electromagnetic field starts the laser. Similarly, in our model of the atom laser we start from an initial condition for the mean matter field $\psi(x,t=0)$ that is non-zero. This is the seed field for our atom laser.
Let us illustrate this by considering for the moment the “natural” initial condition $\psi(x,t=0)=0$ and $N_u(0)=0$. This implies $N_c(0)=0$. We therefore start without any atoms in the system. Moreover, we consider a pump rate $R_u$ above threshold. This choice of initial conditions leads to the unstable solution $N_c=0$ and $N_u=R_u/\gamma_u $. Indeed, any small perturbation in $N_c$ leads to a completely different behavior and $N_c$ and $N_u$ approach the stable solutions $N_c^{(s)}=R_u/\gamma_c-\gamma_u/\Gamma $ and $N_u^{(s)}=\gamma_c/\Gamma $, as discussed in Sec. \[subsec:stab\]. Therefore, we use the different initial condition $N_c(0)\ll 1$ for our numerical simulations [@discnumrem]. We keep the condition $N_u(0)=0$.
For our numerical calculations we take the parameter $U_x/(\hbar\omega a) \cong 0.008$, where $a\equiv\sqrt{\hbar/(m\omega)}$ is the width of the ground state of the harmonic oscillator. As in conventional laser theory [@haken; @lamb], where one usually has a high-Q cavity, we require $\gamma_c \ll \omega$. Since in the steady state we do not want to have too many atoms in the non-condensate part, we require $\gamma_c\lesssim\Gamma$, as suggested by Eq. (\[ratestatsol2\]). In order to have a reasonably small threshold value we choose $\gamma_u$ appropriately, as indicated by Eq. (\[threshold\]). We also need to choose $R_u$ such that we have a sufficiently large number of atoms in the condensate, that is, $N_c^{(s)} = R_u/\gamma_c-\gamma_u/\Gamma \gg 1$.
### Average properties of the matter field
In Fig. \[figure1\](a) we show the time dependent solutions $N_c(t)$ and $N_u(t)$ of the rate equations (\[Ne\]) and (\[Ng\]). For very small times the atoms accumulate in the un-condensed phase: Due to the small number of atoms in the condensate the Bose enhancement is not yet effective and they cannot make a transition into the condensate. As soon as we have a significant number of atoms in the condensate, $N_u$ rapidly approaches its stationary value $N_u^{(s)}=\gamma_c/\Gamma$. Additional atoms then essentially end up in the condensate where the number of atoms slowly approaches its stationary value $N_c^{(s)}=R_u/\gamma_c - \gamma_u/\Gamma$.
For the numerical analysis of the Eqs. (\[psi1D\]) and (\[Ne1D\]) defining our generic model of the atom laser, we use the same parameters. For the initial condition $\psi(x,t=0)$ of the generalized GPE we use the quantum mechanical ground state of the trap, normalized in such a way that $N_c(0) = \int |\psi(x,t=0)|^2 \,dx$. In order to get some feeling for the accuracy of our split-operator technique for the generalized GPE we calculate $N_c(t) = \int |\psi(x,t)|^2 \,dx$ which according to Sec. \[subsec:formulation\] has to coincide with the solution of the rate equations.
We gain insight into the time dependence of $|\psi(x,t)|^2$ by calculating its first moment $$\label{firstmom}
\overline{x(t)} = \frac{\int x |\psi(x,t)|^2 \,dx}
{\int |\psi(x,t)|^2 \,dx}$$ and second moment [@noteqmexp] $$\label{secondmom}
\overline{x^2(t)} = \frac{\int x^2 |\psi(x,t)|^2 \,dx}
{\int |\psi(x,t)|^2 \,dx} .$$
We note that the symmetry of the trap and the generalized GPE ensure that the symmetry of our initial condition is preserved, that is $$\overline{x(t)}=0.$$
Hence the width $$\Delta x(t)\equiv \sqrt{\overline{x^2} - \overline{x}^2}
= \sqrt{\overline{x^2(t)}}$$ of $|\psi(x,t)|^2$ is governed by the second moment, only.
In Fig. \[figure1\](b) we plot the width $\Delta x$. We note that $\Delta x$ increases as a function of time and approaches a steady-state value for large times. Moreover, it oscillates around this steady-state value. In the inset we magnify these oscillations. We note that the oscillations do not decay [@reviv]. Therefore $\Delta x$ does not approach a time independent value and there is no steady state in a strict sense. We hence refer to this solution as the quasi-stationary solution.
In the Appendix we derive an analytical expression for the frequency $\Omega_2$ of these oscillations and find $\Omega_2= \sqrt{3}\,\omega \cong 1.73\,\omega$. From Fig. \[figure1\](b) we read off $\Omega_2 \cong 1.76\,\omega$ which is in good agreement with our prediction.
### Matter field
The oscillations in the width of the distribution are a manifestation of collective excitations [@excitations] of the condensate. Indeed, the whole distribution oscillates as shown in Fig. \[figure2\]. Here we display the mean field for a time interval where the numbers of atoms in the condensed state and the un-condensed state have already reached their steady-state value.
In order to study this oscillatory behavior in more detail, we compare in Fig. \[figure3\](a) the solution of the time independent GPE (\[GPeqtimeindep\]) in one dimension to the quasi-stationary solution of Eqs. (\[psi1D\]) and (\[Ne1D\]). Here we have depicted $|\psi(x,t)|^2$ for time moments where the width is
maximal, average, and minimal. Indeed, the solution of Eqs. (\[psi1D\]) and (\[Ne1D\]) oscillates around the solution of the time independent GPE [@statsols1; @statsols2; @statsols3].
Improved model {#sec:spaceloss}
==============
The collective excitations discussed in the preceding section depend on the initial condition $\psi(x,t=0)$. This is due to the fact that so far there is no mode selection mechanism in our model which would favor one stationary solution of the GPE over the others. This fact is similar to the optical multi-mode laser [@multimodelaser1; @multimodelaser2]. We can accomplish a single-mode atom laser when we allow for a space dependent loss. This is the topic of the present section where we formulate an improved model for an atom laser.
Formulation of the problem
--------------------------
Real losses in a trapped condensate are spatially dependent: For example, a possible loss mechanism are collisions with un-condensed atoms which are preferably located at the edge of the condensate, see Refs. [@BEC1; @BEC2; @BEC3; @BEC4; @BEC5; @BEC6; @BEC7; @BEC8; @BEC9; @BEC10; @BEC11; @BEC12]. Further, atoms can be lost at the edge of the condensate because of a finite trapping potential. Moreover, the goal of a matter-wave output-coupler is to create a directed coherent atomic beam. All of these reasons require a spatially dependent loss term. In contrast to this loss term we keep the pump term spatially independent. This reflects the fact that we assume that the condensate is pumped from a cloud of cold thermal atoms which is larger in size.
We therefore consider a modified generalized GPE $$\begin{aligned}
\label{psiSL}
i\hbar \frac{\partial \psi}{\partial t}
&=&-\frac{\hbar^2}{2m}\Delta\psi
+V({\bf r})\psi +U_0|\psi|^2\psi \nonumber \\
&-&\frac{i}{2}\hbar\gamma_c({\bf r})\psi+\frac{i}{2}\hbar\Gamma N_u\psi,\end{aligned}$$
where the decay rate $\gamma_c({\bf r})$ is spatially dependent. Hence, the rate equation (\[Ng\]) for the number $N_c$ of condensed atoms now translates into $$\label{NgSL}
\dot{N}_c=\Gamma N_u N_c
-\int\gamma_c({\bf r}) |\psi({\bf r},t)|^2 \,d^3r.$$ Note that due to the space dependent loss rate we cannot express the integral in Eq. (\[NgSL\]) by $N_c$. Therefore we do not have a rate equation for $N_c$ anymore. However, the rate equation $$\label{NeSL}
\dot{N}_u=R_u-\gamma_u N_u-\Gamma N_c N_u$$ for the number $N_u$ of un-condensed atoms remains the same.
Equations (\[psiSL\]) and (\[NeSL\]) are the two fundamental equations of our improved model of an atom laser.
Stationary solution {#subsec:statsol}
-------------------
We now study the stationary solution of Eqs. (\[psiSL\]) and (\[NeSL\]). We first derive the steady-state expressions and then discuss the laser threshold in the presence of a position dependent loss term. We conclude by introducing a modified Thomas-Fermi approximation.
### Matter field in steady-state
When we follow an analysis similar to the one of Sec. \[subsec:stab\], we find two sets of stationary solutions of the Eqs. (\[psiSL\]), (\[NgSL\]), and (\[NeSL\]). The first set with $$\begin{aligned}
N_u^{(s)} &=& \frac{R_u}{\gamma_u}, \nonumber\\
N_c^{(s)} &=& 0,\nonumber\\
\psi^{(s)} &=& 0
\label{ratesolsSL2}\end{aligned}$$ corresponds to the solution below threshold.
The second set $$\begin{aligned}
N_u^{(s)} &=&
\frac{1}{\Gamma}\int\gamma_c({\bf r})|\psi^{(s)}_1({\bf r})|^2 \,d^3r
=\frac{R_u}{\gamma_u+\Gamma N_c^{(s)}},
\label{ratesolsSL1e}\\
N_c^{(s)} &=&
\frac{R_u}{\int\gamma_c({\bf r})|\psi^{(s)}_1({\bf r})|^2 \,d^3r}
-\frac{\gamma_u}{\Gamma}
\label{ratesolsSL1g}\end{aligned}$$ corresponds to the solutions above threshold as we show now. Here, $\psi^{(s)}_1({\bf r})=[N_c^{(s)}]^{-1/2}\psi^{(s)}({\bf r})$ denotes the stationary solution of Eq. (\[psiSL\]) normalized to unity. This solution is defined by the time-independent modified generalized GPE $$\begin{aligned}
\label{time_indep}
\mu\psi^{(s)}
&=&-\frac{\hbar^2}{2m}\Delta \psi^{(s)} +V\left( {\bf r}\right)
\psi^{(s)} +U_0 |\psi^{(s)}|^2\psi^{(s)} \nonumber \\
&-&\frac{i}{2}\hbar\gamma_c\left( {\bf r}\right) \psi^{(s)}
+\frac{i}{2}\hbar\Gamma N_u\psi^{(s)} ,\end{aligned}$$ following from Eq. (\[psiSL\]) using the ansatz $\psi({\bf r},t) =\exp(-i\mu t/\hbar)\psi^{(s)}({\bf r})$ where $\mu$ denotes the “chemical potential.”
These expressions for $N_u^{(s)}$ and $N_c^{(s)}$ are not explicit since they involve the stationary solution $\psi^{(s)}({\bf r})$ of Eq. (\[time\_indep\]). In Sec. \[subsec:modTFsol\] we derive an approximate analytical expression and in Sec. \[subsec:exactsol\] we find a fully numerical solution for $\psi^{(s)}({\bf r})$.
### Lasing threshold
A stability analysis of the above solutions amounts to calculating the collective excitations of the modified generalized GPE (\[psiSL\]). This is only possible numerically. We therefore pursue a strategy where we solve numerically for the full time dependence of $\psi({\bf r},t)$. In order to gain some insight into the threshold condition we first discuss simple physical arguments.
These considerations rely on the fact that the number of atoms cannot be negative. With the help of Eq. (\[ratesolsSL1g\]) we then determine the laser threshold $$\label{SLthres0}
R^{th}
\equiv\frac{\gamma_u}{\Gamma}
\int\gamma_c({\bf r})|\psi^{(s)}_1({\bf r})|^2 \,d^3r$$ by setting $N_c^{(s)}=0$. Close to threshold the number $N_c^{(s)}$ of atoms in the condensate is close to zero. We can therefore neglect the non-linear contribution in the modified generalized GPE and arrive at a linear Schrödinger equation with pump and loss. With a position dependence of the loss term that favors the normalized ground-state energy solution $\phi_0({\bf r})$ of the linear Schrödinger equation, we can replace $\psi^{(s)}_1({\bf r})$ by $\phi_0({\bf r})$ in Eq. (\[SLthres0\]) and arrive at $$\label{SLthres2}
R^{th}
=\frac{\gamma_u}{\Gamma}\int\gamma_c({\bf r})|\phi_0({\bf r})|^2 \,d^3r.$$ When the loss shape is flat around the localization of $\phi_0({\bf r})$, that is, around the center ${\bf r}={\bf 0}$ of the trap, we can factor out $\gamma_c({\bf 0})$ from the integral in Eq. (\[SLthres2\]) and find the lasing threshold $$\label{SLthres1}
R^{th}=\frac{\gamma_u\gamma_c({\bf 0})}{\Gamma}.$$
We conclude this section by noting that in the limit of a space independent loss the above results reduce to the corresponding ones of Sec. \[subsec:rate\].
### Modified Thomas-Fermi solution {#subsec:modTFsol}
In this section we derive an approximate but analytical expression for the stationary state of the modified generalized GPE. In the case of the conventional GPE it is the so-called Thomas-Fermi (TF) approximation which describes the steady state of the condensate [@TFA]. The phase of the stationary solution of the GPE as well as the phase of the TF approximation is constant. However, for a position dependent loss it turns out that the phase of the stationary matter-wave field also depends on the position. A spatially dependent phase leads to a non-vanishing current of the condensate. We expect this feature to be important for the coherence properties of a cw atom laser. Therefore, we introduce a modified TF solution.
We start from the time independent form of the modified generalized GPE, Eq. (\[time\_indep\]). For the present problem a hydrodynamic treatment is more convenient. We therefore consider the density $$\rho({\bf r})\equiv|\psi^{(s)} ({\bf r})|^2$$ and velocity $${\bf v}({\bf r})\equiv (\hbar/m)\nabla \phi({\bf r})$$ of the condensate. Here $\phi({\bf r})$ is the phase of the mean field following from the ansatz $$\psi^{(s)} \left( {\bf r}\right)
=\sqrt{\rho \left( {\bf r}\right) }e^{i\phi\left( {\bf r}\right) }.$$ Indeed, with Eq. (\[time\_indep\]) we find the equations $$\label{real_part}
\frac{\hbar^2}{2m}\nabla^2\sqrt{\rho }
=\left[ -\mu +\frac{1}{2}m{\bf v}^2
+V\left( {\bf r}\right) +U_0\rho \right] \sqrt{\rho }$$ and $$\label{imag_part}
\frac{{\bf v}}{2}\nabla \sqrt{\rho }
=\left[ -\nabla \cdot {\bf v}-\gamma_c\left( {\bf r}\right)
+\Gamma N_u\right]\sqrt{ \rho }.$$ We make the assumption that the density profile is slowly varying. Since we are interested in loss shapes where the loss at the center of the condensate is uniform, these assumptions are reasonable at the center of the condensate. However, at the edges of the condensate where the change of the loss is large, we expect this assumption to break down. This is the idea of our modified Thomas-Fermi approximation.
This assumption allows us to neglect the terms $\nabla^2\sqrt{\rho}$ and $\nabla \sqrt{\rho }$ in the two equations. However, in order to fulfill the second equation we have to retain the derivative $\nabla\cdot {\bf v}$ in velocity. In this approximation we arrive at the approximate expression $$\label{modsqr}
\rho\left( {\bf r}\right)
\cong \frac{1}{U_0}\left[\mu
-{\small\frac{1}{2}}m {\bf v}^2\left( {\bf r}\right)
-V\left( {\bf r}\right) \right]$$ for the density and the differential equation $$\label{nablasqr_phi}
\nabla \cdot{\bf v}\left( {\bf r}\right)
\cong-\gamma_c\left( {\bf r}\right) +\Gamma N_u$$ for the velocity.
Since the density cannot be negative, these expressions are only valid for that volume ${\cal V}$ of space where $$\label{MTFradius}
\rho({\bf r})\propto \mu -\frac{1}{2}m
{\bf v}^2\left( {\bf r}\right)-V\left({\bf r}\right)\ge 0.$$ The shape of this volume ${\cal V}$ depends on the potential $V({\bf r})$ and the loss rate $\gamma_c ({\bf r})$ via the velocity ${\bf v}({\bf r})$.
Note, that the velocity ${\bf v}$ and consequently the phase $\phi$ depends on the rate $\Gamma$, the number of un-condensed atoms, and the shape of the loss rate. Indeed, it is only the spatial dependence of the loss rate that determines the spatial profile of the velocity ${\bf v}({\bf r})$. The “kinetic energy” of the condensate plays a role similar to the trap potential in shaping the density.
The expressions, Eqs. (\[modsqr\]) and (\[nablasqr\_phi\]), for the density and velocity are not explicit. Indeed, the chemical potential $\mu$ is still a free parameter. Moreover, the number of un-condensed atoms $N_u^{(s)}$ is coupled to Eqs. (\[modsqr\]) and (\[nablasqr\_phi\]) and is given by Eq. (\[ratesolsSL1e\]).
We therefore have to solve the various constraints in a self-consistent way: The number $N_c^{(s)}$ of condensed atoms reads $$\label{MTFnorm}
N_c^{(s)}=\int_{{\cal V}}\rho({\bf r}) \,d^3r.$$ On the other hand this quantity $N_c^{(s)}$ is given by Eq. (\[ratesolsSL1g\]) and reads $$\label{MTFrate}
N_c^{(s)} = \frac{R_u}{\int_{{\cal V}}
\gamma_c({\bf r})\rho_1({\bf r}) \,d^3r}-\frac{\gamma_u}{\Gamma},$$ where $\rho_1({\bf r})=[N_c^{(s)}]^{-1}\rho({\bf r})$ is the density normalized to unity. Note, that ${\cal V}$ is defined by the condition that the density is non-negative, Eq. (\[MTFradius\]).
The number of un-condensed atoms then follows from Eq. (\[ratesolsSL1e\]). Hence we have to solve the three Eqs. (\[MTFradius\]), (\[MTFnorm\]), and (\[MTFrate\]) for the three unknowns: the modified Thomas-Fermi volume ${\cal V}$, the chemical potential $\mu$, and the number $N_c^{(s)}$ of atoms in the condensate.
### Usual Thomas-Fermi solution
The most important consequence of our position dependent loss term is a non-vanishing velocity and therefore a spatially dependent phase. How does this compare to the usual Thomas-Fermi approximation? Here we consider the density $$\label{usualTFdens}
\rho({\bf r})\cong\frac{\mu-V({\bf r})}{U_0}$$ and the velocity $$\label{usualTFvelo}
{\bf v}\equiv 0.$$ We first note that this ansatz is in contradiction to Eq. (\[imag\_part\]). Nevertheless we can use it to investigate how the kinetic energy potential influences the density $\rho$. Similar to the last section we insert our ansatz into Eqs. (\[MTFradius\]) to (\[MTFrate\]) and find the Thomas-Fermi volume ${\cal V}$, the chemical potential $\mu$, and the number of atoms $N_c^{(s)}$.
### Exact solution {#subsec:exactsol}
In this section we adapt the numerical methods developed to find the ground-state solution of the ordinary GPE [@statsols1; @statsols2; @statsols3] to the present problem. This method evolves the wave function for a fixed atom number in imaginary time. The solution is normalized to unity after each time step. The evolution in imaginary time attenuates the differences between the arbitrary initial wave function and the ground-state solution of the ordinary GPE.
In our model of an atom laser, Eq. (\[psiSL\]), we have generalized the GPE by a pump and a loss term. Therefore the number of atoms is not fixed but is governed by Eq. (\[NeSL\]). In order to find the ground-state solution of our time independent Eqs. (\[ratesolsSL1g\]) and (\[time\_indep\]), we evolve an arbitrary initial wave function according to our modified generalized GPE, Eq. (\[psiSL\]), in imaginary time, using the split-operator technique [@split1; @split2]. After each time step we normalize the wave function to unity. Then we update the number of atoms with the help of Eq. (\[ratesolsSL1g\]) and use it for the next time step. We repeat this procedure until the wave function and the atom number has converged to a stationary value. This method finds the stationary state of our improved model of an atom laser in a self-consistent way. The most important result is that the pump and spatial loss gives a space dependent phase to the stationary mean-field $\psi^{(s)}({\bf r})$.
Results
-------
As in Sec. \[sec:model\] we now specify the potential $V({\bf r})$ to be the one-dimensional harmonic oscillator potential, Eq. (\[harmpot\]). Moreover, we study two models of a position dependent loss rate: a sum $$\label{gaussianloss}
\gamma_c(x)=\gamma_c'
\left( \mbox{e}^{-(x+x_0)^2/\sigma^2}
+\mbox{e}^{-(x-x_0)^2/\sigma^2} \right)$$ of two Gaussians and a sum $$\label{spatial_loss}
\gamma_c\left( x\right)=\gamma_c'\left[
\frac{\sigma^2}{\left(x+x_0\right)^2+\sigma^2}
+\frac{\sigma^2}{\left(x-x_0\right)^2+\sigma^2}\right].$$ of two Lorentzians.
In both cases $\sigma$ is a measure of the width of the corresponding distributions and $\gamma_c'$ is the maximal loss rate when the two curves have negligible overlap. In the next section we choose the locations $\pm x_0$ of the maximal loss such that they sit where the density of the condensate falls off.
### Transient behavior {#transient-behavior}
First we want to show that the collective excitations which emerged in the elementary model are damped out due to the presence of the space dependence in the loss [@damping]. For this analysis we use the Gaussian loss term, Eq. (\[gaussianloss\]). We adjust $\gamma_c'$ such that we obtain the same final number of atoms in the condensate as in the spatially independent case. The width of the loss is comparable to a typical scale, such as the width of the ground state of the harmonic oscillator.
In Fig. \[figure1\](a) we compare the time evolution of the numbers of condensed and un-condensed atoms for the position dependent loss with those of the position independent loss. We see that the overall behavior is not that different. Therefore our original rate equations of the elementary model still approximate very well the time evolution of $N_c$ and $N_u$ following from the improved model.
We then show in Fig. \[figure1\](c) the scaled width of the mean field. Indeed, the position dependent loss damps out the collective excitations, and a steady state is reached. We emphasize that this is true for any initial non-vanishing mean field.
In Fig. \[figure3\](b) we compare and contrast the stationary mean field to the numerical solution of the time independent equation. They coincide with each other. Indeed, for many initial conditions and set of parameters we have noticed that the mean field obtained by evolving the modified generalized GPE over a sufficiently long enough time such that the transient oscillations are damped out is equivalent to the time independent ground-state energy solution of the modified generalized GPE. This holds as long as the loss is confined to the edges of the condensate [@genlossrem]. This is the case in the present example as shown in Fig. \[figure3\](b) where we also display the loss function in arbitrary units. The overlap between the stationary mean field and the loss function is essential for the number of atoms in the condensate as is apparent from Eq. (\[ratesolsSL1g\]).
When we compare Fig. \[figure3\](a) and (b), we note that the modulus of the steady-state mean field in the presence of a loss located at the edges of the condensate is very similar to the ground-state energy solution of the conventional time independent GPE, Eq. (\[GPeqtimeindep\]). However, this is not true for the phase of the mean field as already discussed in Sec. \[subsec:statsol\].
A spatially dependent loss term located at the edges of the condensate damps out the collective excitations. In our theory of an atom laser this plays the role of mode selection, i.e., the ground-state energy solution survives whereas the excited states, i.e., the collective excitations, are damped away due to the fact that their overall spread in position space is greater than that of the ground state.
### Stationary solution {#stationary-solution}
[*(a) Modified Thomas-Fermi solution.*]{} We now turn to the discussion of the modified Thomas-Fermi solution derived in Sec. \[subsec:modTFsol\] for a general potential. In the present section we restrict this analysis to a one-dimensional harmonic oscillator potential, Eq. (\[harmpot\]). Moreover, we choose
[1]{}
[2]{}
the Lorentzian loss rate, Eq. (\[spatial\_loss\]). This choice is solely motivated by the fact that we can perform the resulting integral of the differential equation (\[nablasqr\_phi\]).
We start by first summarizing the Eqs. (\[MTFradius\]) to (\[MTFrate\]) for the chemical potential $\mu$, the Thomas-Fermi radius $R$, and the number of condensed atoms $N_g^{(s)}$, $$\begin{aligned}
\label{MTFradius1D}
\mu -\frac{1}{2}m\left[
v\left(R\right)\right]^2-\frac{1}{2}m \omega^2 R^2 &=& 0,\\
\label{MTFnorm1D}
N_c^{(s)}-\int_{-R}^R\rho(x) \,dx&=&0,\\
\label{MTFrate1D}
N_c^{(s)} - \frac{R_u}{\int_{-R}^R
\gamma_c(x)\rho_1(x) \,dx}-\frac{\gamma_u}{\Gamma}&=&0,\end{aligned}$$ determining the modified Thomas-Fermi solution for the one-dimensional harmonic trap and loss. From Eq. (\[modsqr\]) we obtain the one-dimensional density $$\label{density1D}
\rho\left( x\right)
= \frac{1}{U_x}\left[\mu
-{\small\frac{1}{2}}m v^2\left(x\right)
-\frac{1}{2}m \omega^2 x^2\right].$$ We find the velocity $v$ when we substitute the Lorentzian loss rate, Eq. (\[spatial\_loss\]), into Eq. (\[nablasqr\_phi\]) and integrate which yields $$\begin{aligned}
\label{velocity1D}
v\left( x\right)
&=& \Gamma N_u x-\gamma_c'\sigma \left[ \arctan \left(
\frac{x+x_0}{\sigma}\right) \right. \nonumber \\
&&\left. +\arctan \left( \frac{x-x_0}{\sigma }\right) \right].\end{aligned}$$ Here we have set $v(0)=0$ in order to preserve the symmetry of the solution of the mean field, Eq. (\[density1D\]), $\rho(x)=\rho(-x)$. The latter holds because the harmonic oscillator potential, the spatially dependent loss, and the generalized GPE show this symmetry.
[*(b) Usual Thomas-Fermi solution.*]{} In order to get a feeling how in our one-dimensional model the density $\rho$ is influenced by the velocity $v$ we also discuss the usual TF solution. For the one-dimensional harmonic trap the density, Eq. (\[usualTFdens\]), reads $$\label{usualTFdens1D}
\rho(x)\cong\frac{1}{U_x}\left[\mu-m \frac{1}{2}\omega^2 x^2\right],$$ and the velocity, Eq. (\[usualTFvelo\]), reads $$\label{usualTFvelo1D}
v(x)\equiv 0.$$ When we use this usual TF approximation, we easily find from Eq. (\[MTFradius1D\]) with $v\equiv0$ the TF radius $R=\sqrt{2\mu/m\omega^2}$. We then use Eq. (\[MTFnorm1D\]) to calculate the number of atoms in the condensate as a function of the chemical potential $\mu$. Inverting this equation we can express the chemical potential $\mu=[3 N_c^{(s)} U_x\sqrt{m}\omega/(4\sqrt{2})]^{2/3}$ as a function of the atom number $N_c^{(s)}$. Finally, we use Eq. (\[MTFrate1D\]) to determine the number $N_c^{(s)}$ of condensed atoms, the only unknown quantity. We emphasize that only this last step has to be done numerically.
[*(c) Discussion.*]{} Figure \[figure4\] shows the modified TF, the usual TF and the fully numerical solution. We have chosen two different shapes of the Lorentzian loss curve: In the right column the width of the individual Lorentzians is half the size of the one on the left column. Moreover, we note that their location is different. We depict the modified TF solution by dashed curves, the usual TF solution by dash-dotted ones, and the numerical solution by solid lines. Again we show in arbitrary units the shape of the loss $\gamma_c\left( x\right) $ by the dotted curve.
The density is shown in the upper part of Fig. \[figure4\]. We see that for the parameters of the left column both TF approximations for the density work quite well. The modified TF solution approximates the center better than the usual TF solution. However, the radius derived from the usual TF approximation is a better estimate of the edge of the condensate than the modified TF radius. For the set of parameters used in the right column of Fig. \[figure4\] both TF approximations for the density do not work that well anymore. The reason for this break-down is that the condensate reaches too far into the loss region where the density varies strongly and the derivatives neglected in the derivation of Eqs. (\[modsqr\]) and (\[nablasqr\_phi\]) become important. Nevertheless our modified TF solution shows at least qualitatively the same behavior as the fully numerical solution. Surprisingly, at regions towards the peaks of the loss where the loss rate is at its highest, the density is first decreased and then increased in the modified TF over the usual TF solution. This is understandable from Eq. (\[modsqr\]) when we consider in the lower part of Fig. \[figure4\] the current $j(x)=\rho(x) v(x)$. We note that at these regions first the current which is proportional to the velocity is at its highest and then decreases.
We see that for the current, i.e. the velocity, the modified TF solution is a very good approximation around
the central region. It gives a good estimate for the overall behavior of the current (velocity) for both sets of parameters. Around the central region of the condensate the current is approximately a linear function of the position, illustrating the flow of atoms to the ends where they are predominately lost from the peaks. At the tails of the condensate the modified TF solution even predicts a velocity changing the direction. This is in contrast to the fully numerical solution. The usual TF solution has zero velocity by default.
We solve numerically the Eqs. (\[MTFradius1D\]) to (\[MTFrate1D\]) for the modified Thomas-Fermi solution to find the condensate population for various pump strengths and display this as the solid curve in Fig. \[figure5\]. The dashed curve corresponds to the usual TF solution. We display the fully numerical solution with the help of diamonds.
Both approximate analytical curves agree quite well with the fully numerical solution over a wide region of different pump strengths. However, at pump rates just above threshold the agreement is not that good which is shown in the upper left inset of Fig. \[figure5\] . Both the usual TF and the modified TF curves lie on top of each other and over-estimate the numerical result. The fully numerical solution crosses the horizontal axis at $R_u/\omega\cong 201$ whereas the two TF solutions cross at $R_u/\omega\cong 198$. Close to threshold, the spatial extent of the two approximate wave functions is becoming smaller with decreasing atom number. They are predominately located around the relatively flat loss region and the losses appear to be position independent. Therefore the laser threshold for the two approximate solutions is predicted by Eq. (\[SLthres1\]). In contrast, the shape of the numerically calculated mean field can vary, and is in fact of Gaussian form close to threshold. Hence our approximation of the threshold, Eq. (\[SLthres2\]), agrees rather remarkably with the numerically calculated threshold. The deviation of the approximate threshold, Eq. (\[SLthres1\]), reflects the fact that even close to threshold the loss function does not appear spatially constant to the mean field for the parameters of Fig. \[figure5\]. When we recall that the TF approximation is not good at low atom numbers it is rather surprising to see such small differences between the TF solutions and the numerical solution.
The lower right inset of Fig. \[figure5\] zooms in on a region of higher pump strengths far above threshold where the spatial structure of the loss plays a role. Here the modified TF solution approximates the fully numerical solution (diamonds) better than the usual TF solution. This is understandable since the modified TF solution takes into account pump and spatially dependent loss by allowing for a spatially dependent phase, i.e. velocity.
Conclusions {#sec:concludere}
===========
In summary, we have constructed a theory of an atom laser that is analogous to semi-classical laser theory. The matter-wave equation is a generalized Gross-Pitaevskii equation with additional loss and gain terms. We derive the lasing threshold and describe the build-up of the coherent mean field of a condensate.
The elementary model uses a spatially homogeneous loss. Here we find un-damped collective excitations. Therefore the final mean field depends on the initial mean field: The known stationary ground state of the GPE which is the desired lasing mode cannot be reached in general.
The improved model has a natural mode selection built in by a space dependent loss. In this way we achieve the desired single lasing mode.
We have derived a modified Thomas-Fermi solution for the steady-state mean field. This solution takes into account the effects of a pump term and a position dependent loss term. In contrast to a constant phase of the usual Thomas-Fermi solution, the modified Thomas-Fermi solution has a spatially dependent phase, i.e. velocity, due to the permanent flow of atoms in and out of the condensate. The modified Thomas-Fermi solution is a good approximation in regions where the loss shape is slowly varying and for sufficiently large atom numbers.
We emphasize that our model of an atom laser is very simple and rather general. Therefore, we can apply it to different experimental configurations of cw atom lasers [@konstanz], or current experiments [@BEC1; @BEC2; @BEC3; @BEC4; @BEC5; @BEC6; @BEC7; @BEC8; @BEC9; @BEC10; @BEC11; @BEC12] provided the evaporative cooling process (boson amplification) and the loading of the trap is run continuously, and an output coupling mechanism is applied. We thank Eric Bolda, Michael Fleischhauer, Murray Olsen, Karl Riedel, and Janne Ruostekoski for stimulating and valuable discussions. Two of us (B.K. and K.V.) acknowledge very gratefully the warm hospitality given to them at their stay with the quantum optics group of the department of physics at the University of Auckland. This work was supported by the Deutsche Forschungsgemeinschaft, the University of Auckland Research Committee and the Marsden Fund of the Royal Society of New Zealand.
Collective excitations in one dimension {#collective-excitations-in-one-dimension .unnumbered}
=======================================
Collective excitations are usually discussed in three dimensions [@excitations]. Within the framework of the Thomas-Fermi approximation, Stringari [@coll2] has calculated analytically the excitation spectrum of a condensate in a three-dimensional isotropic harmonic trap. Since our numerical solution of Eq. (\[psi\]) is done for a one-dimensional harmonic trap, we apply Stringari’s method to a one-dimensional harmonic trap of frequency $\omega$. We substitute $$\psi(x,t) = \sqrt{\rho(x,t)} e^{i\phi(x,t)}$$ into Eq. (\[psi\]) and obtain after some algebra the hydrodynamic equations $$\frac{\partial \rho}{\partial t}
+ \frac{\partial}{\partial x} \left(\rho v\right)
+ \left(\gamma_c - \Gamma N_u \right) \rho = 0$$ and $$\begin{aligned}
m \frac{\partial v}{\partial t}
+ \frac{\partial}{\partial x} \left( \frac{m}{2} v^2
+ V + U_x \rho -\mu \right)&& \nonumber\\
- \frac{\hbar^2}{2m} \frac{\partial}{\partial x} \frac{1}{\sqrt{\rho}}
\frac{\partial^2}{\partial x^2} \sqrt{\rho} &=& 0
\label{hydro-v} \end{aligned}$$ for the “density” $$\rho(x,t) = \psi^{\ast}(x,t) \psi(x,t)$$ and the “velocity” $$v(x,t) = \frac{\hbar}{m} \frac{\partial \phi(x,t)}{\partial x} .$$ Note, that we have also introduced the chemical potential $\mu$ which is space independent.
We now consider small deviations $$\begin{aligned}
\delta \rho &\equiv& \rho - \rho^{(s)}, \nonumber\\
\delta v &\equiv& v - v^{(s)} = v, \nonumber\\
n_u &\equiv& N_u- N_u^{(s)} =N_u- \gamma_c/\Gamma\end{aligned}$$ of $\rho$, $v$, and $N_u$ from their stationary values $\rho^{(s)}$, $v^{(s)}\equiv 0$, and $N_u^{(s)} = \gamma_c/\Gamma$. Furthermore, we make the Thomas-Fermi approximation, that is, we neglect the last term in Eq. (\[hydro-v\]) and approximate the stationary solution $\rho^{(s)}(x)$ by the Thomas-Fermi solution [@TFA] $$\rho^{(s)}(x) \cong \frac{\mu - V(x)}{U_x},$$ where the chemical potential $\mu$ is defined by the normalization integral. We then arrive at the linearized equations $$\frac{\partial}{\partial t} \delta \rho
+ \frac{\partial}{\partial x} \left( \rho^{(s)} v \right)
- \Gamma \rho^{(s)} n_u = 0$$ and $$m \frac{\partial v}{\partial t}
+ U_x \frac{\partial}{\partial x}\delta \rho = 0 .$$ We combine these two equations to eliminate $v$ and arrive at $$\frac{\partial^2}{\partial t^2} \delta \rho
- \frac{U_x}{m}\frac{\partial}{\partial x}
\left( \rho^{(s)}
\frac{\partial}{\partial x} \delta \rho \right)
= \Gamma \rho^{(s)} \dot{n}_u .
\label{xxx}$$ For large times, when $N_u(t)$ and $N_c(t)$ have already reached their stationary value, we can neglect the inhomogeneous term $\Gamma \rho^{(s)} \dot{n}_u$.
Equation (\[xxx\]) cannot be solved without knowledge of the potential $V(x)$. We restrict ourselves to the case of a harmonic trap, that is, $$V(x) = \frac{1}{2}m\omega^2 x^2.$$ In order to solve Eq. (\[xxx\]), we introduce the scaled variable $\xi = x/R$, where $R=\sqrt{2\mu/m\omega^2}$ is the Thomas-Fermi radius of the condensate. Using the ansatz $$\delta \rho(x,t) = A \sin(\Omega t + \varphi) y(x/R)$$ we obtain an ordinary differential equation for $y(\xi)$ which reads $$\frac{d}{d\xi} \left[(1-\xi^2)\frac{dy(\xi)}{d\xi}\right]
+ \frac{2\Omega^2}{\omega^2} y(\xi) = 0.
\label{legendre}$$ This is the differential equation of Legendre functions which in general only has solutions that are singular at $\xi=\pm 1$ [@courant], that is at $x=\pm R$. The only exceptions are $$\frac{2\Omega_n^2}{\omega^2} = n(n+1),
\label{freq}$$ where $n$ is an integer. In this case the well-known Legendre polynomials [@courant] $$P_n(\xi) = \frac{1}{2^n n!} \frac{d^n}{d\xi^n}\left(\xi^2-1\right)^n$$ solve Eq. (\[legendre\]). Furthermore, they fulfill the orthogonality relation $$\int\limits_{-1}^{+1} P_n(\xi) P_m(\xi) \,d\xi = 0
\quad \mbox{for} \quad n \ne m .
\label{ortho}$$ The frequencies of the elementary excitations are therefore given by Eq. (\[freq\]). The solution of the homogeneous part of Eq. (\[xxx\]) reads $$\label{soldev}
\delta \rho(x,t)
= \sum_{n=1}^{\infty} A_n \sin(\Omega_n t + \varphi_n) P_n(x/R) ,$$ where $A_n$ and $\varphi_n$ follow from the initial deviation from the stationary solution.
Equation (\[soldev\]) shows that the excitations do not decay, even in the presence of pump and loss terms in our generalized Gross-Pitaevskii equation, Eq. (\[psi1D\]). However, these excitations do not grow but rather oscillate. Our numerical solution of Eq. (\[psi1D\]) in Sec. \[subsec:num\] confirms this result.
In Fig. \[figure1\](b) we do not find all frequencies allowed by Eq. (\[freq\]) since in our numerical solution of Eqs. (\[psi1D\]) and (\[Ne1D\]) we start with a symmetric initial condition $\psi(x,t=0)$. Because Eqs. (\[psi1D\]) and (\[Ne1D\]) do not destroy this symmetry, $\rho(x,t)$ as well as $\delta\rho(x,t)$, the deviation from the (symmetric) stationary solution of the generalized GPE, will always be symmetric. We therefore can only find excitations which correspond to Legendre polynomials of even order if we start with a symmetric $\psi(x,t=0)$. The corresponding frequencies are $$\Omega_{2n} = \omega \sqrt{n(2n+1)} .$$ This is also true for an anti-symmetric initial wave function since the corresponding density is symmetric.
Two other facts are worth mentioning: The excitations discussed above do not change the number of atoms. Using $P_0(\xi) = 1$ we find with the help of the orthogonality relation Eq. (\[ortho\]) $$\int \delta \rho(x,t) \,dx = 0 .$$ The second moment $$\overline{x^2(t)} = \frac{\int x^2 |\psi(x,t)|^2 \,dx}
{\int |\psi(x,t)|^2 \,dx}$$ can only oscillate with the frequency $\Omega_2=\sqrt{3}\,\omega$. This follows from the relation $$\xi^2 = \frac{2}{3} P_2(\xi) + \frac{1}{3} P_0(\xi)$$ together with the orthogonality relation, Eq. (\[ortho\]). A generalization is the statement that $$\overline{P_k(x/R)} = \frac{\int P_k(x/R) |\psi(x,t)|^2 \,dx}
{\int |\psi(x,t)|^2 \,dx}$$ only shows oscillations with frequency $\Omega_k$.
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|
---
abstract: 'We present new [*Herschel*]{}-SPIRE imaging spectroscopy (194-671 $\mu$m) of the bright starburst galaxy M82. Covering the CO ladder from [[$\rm J{\!=\!}4{\!\rightarrow\!}3$]{}]{} to [[$\rm J{\!=\!}13{\!\rightarrow\!}12$]{}]{}, spectra were obtained at multiple positions for a fully sampled $\sim$ 3 x 3 arcminute map, including a longer exposure at the central position. We present measurements of $^{12}$CO, $^{13}$CO, \[C \], \[N \], HCN, and HCO$^+$ in emission, along with OH$^+$, H$_2$O$^+$ and HF in absorption and H$_2$O in both emission and absorption, with discussion. We use a radiative transfer code and Bayesian likelihood analysis to model the temperature, density, column density, and filling factor of multiple components of molecular gas traced by $^{12}$CO and $^{13}$CO, adding further evidence to the high-J lines tracing a much warmer ($\sim$ 500 K), less massive component than the low-J lines. The addition of $^{13}$CO (and \[C \]) is new and indicates that \[C \] may be tracing different gas than $^{12}$CO. No temperature/density gradients can be inferred from the map, indicating that the single-pointing spectrum is descriptive of the bulk properties of the galaxy. At such a high temperature, cooling is dominated by molecular hydrogen. Photon-dominated region (PDR) models require higher densities than those indicated by our Bayesian likelihood analysis in order to explain the high-J CO line ratios, though cosmic-ray enhanced PDR models can do a better job reproducing the emission at lower densities. Shocks and turbulent heating are likely required to explain the bright high-J emission.'
author:
- 'J. Kamenetzky, J. Glenn, N. Rangwala, P. Maloney, M. Bradford, C.D. Wilson, G.J. Bendo, M. Baes A. Boselli, A. Cooray, K.G. Isaak, V. Lebouteiller, S. Madden, P. Panuzzo, M.R.P. Schirm, L. Spinoglio, R. Wu'
bibliography:
- 'M82.bib'
title: '[*Herschel*]{}-SPIRE Imaging Spectroscopy of Molecular Gas in M82'
---
Introduction {#sec:intro}
============
M82 is a nearly edge-on galaxy, notable for its spectacular bipolar outflow and high IR luminosity [$5.6 \times 10^{10}$ [L$_{\odot}$]{}, @Sanders:2003]. Though its high inclination angle of 77$^\circ$ makes it difficult to determine, M82 is likely a SBc barred spiral galaxy with two trailing arms [@Mayya:2005]. Its redshift-independent distance is about 3.4 $\pm$ 0.2 Mpc [@Dalcanton:2009], and after correcting the commonly cited redshift (0.000677, @deVaucouleurs:1999) with WMAP-7 parameters to the 3K CMB reference frame, we find a redshift of 0.000939. Given this distance we assume a conversion of 17 pc/[$^{\prime\prime}$]{}.
Due to its proximity, M82 is an exceptionally well-studied starburst galaxy. High star formation rates [9.8 [M$_{\odot}$]{} [yr$^{-1}$]{}, likely enhanced by interactions with M81, @Yun:1993] and a large gas reservoir produce bright molecular and atomic emission lines. Such lines can yield important information on the interaction between the interstellar medium (ISM) and star formation (SF) processes, such as the influence of SF on the ISM through photon-dominated region (PDR) or other excitation, as traced by intermediate to high-J CO rotational lines.
Ground-based studies of CO in M82 are numerous [@Wild:1992; @Mao:2000; @Weiss:2001; @Ward:2003; @Weiss:2005], examining both morphology and physical conditions of the gas. High-resolution CO maps of the 1 kpc disk indicate that the emission is largely concentrated in three areas: a northeast lobe, southwest lobe, and to a lesser extent, a central region [see Figure 1 of @Weiss:2001]. The two lobes are separated dynamically, as can be seen in position-velocity diagrams [Figure 3 of @Weiss:2001]. Outside of the disk, molecular gas emission is also detected in the halo/outflow [@Taylor:2001; @Walter:2002].
In addition to examining the morphology of molecular gas, CO emission lines can be used to determine the physical conditions of the molecular gas in galaxies. Previously, due to terrestrial atmospheric opacity, only the first few lines in the CO emission ladder could be studied. The first studies of higher-J lines (described below) have indicated that they can trace components of gas separate from those measurable with low-J lines. Many of the most interesting questions about galaxy formation, evolution, and star formation concern the balance of different energy sources, i.e. what role might cosmic rays, ultraviolet light from stars, X-rays from powerful AGN, or turbulent motion play in the star formation history of various galaxies? In what way does star formation influence the molecular gas, and vice versa? Estimating the influence of these various energy sources, however, often depends on knowing the physical conditions of the gas. We therefore model physical conditions of these high-J lines first in order to inform our later discussion on energy sources. Other molecules are also useful in this study; in a ground-based survey of 18 different molecular species, @Aladro:2011 also found that some molecules trace different temperature components than others and that the different chemical abundances in M82 and NGC253 may indicate different evolutionary stages of starbursts.
The [*Herschel Space Observatory*]{} [@Pilbratt:2010] is the unique facility that can measure the submillimeter properties of nearby galaxies in a frequency range that cannot be observed from the ground. As one of the brightest extragalactic submillimeter sources, M82 has been studied extensively with [*Herschel*]{}. For example, the imaging photometer of the Spectral and Photometric Imaging REceiver [SPIRE, @Griffin:2010] has been used to study the cool dust of M82, revealing wind/halo temperatures that decrease with distance from the center with warmer starburst-like filaments between dust spurs [@Roussel:2010]. The tidal interaction with M81 was likely very effective in removing cold interstellar dust from the disk; more than two thirds of the extraplanar dust follows the tidal streams. @Panuzzo:2010 used the SPIRE Fourier-Transform Spectrometer (FTS) to study a single spectrum of the $^{12}$CO emission from [[$\rm J{\!=\!}4{\!\rightarrow\!}3$]{}]{} to [[$\rm J{\!=\!}13{\!\rightarrow\!}12$]{}]{} to find that these higher-J CO lines likely trace a $\sim$ 500 K gas component not seen in the $\sim$ 30 K component that can be observed from ground-based studies. Also on-board [*Herschel*]{} is the Heterodyne Instrument for the Far Infrared (HIFI, @deGraauw:2010), which consists of a set of 7 heterodyne receivers with resolution of 125 kHz to 1 MHz for electronically tuneable frequency coverage of 2 x 4 GHz; it covers 480 - 1910 GHz. HIFI found ionized water absorption from diffuse gas [@Weiss:2010] and high-J transitions of the CO ladder. These CO transitions indicated a combination of one low and two high density gas components via comparison to PDR models [@Loenen:2010].
We confirm the presence of multiple molecular hydrogen thermal components in M82 by performing a more in-depth modeling analysis on a deeper dataset as part of the [*Herschel*]{} Very Nearby Galaxies Survey. We add to existing data by using the SPIRE FTS mapping mode, providing spectroscopic imaging of a region approximately 3$^{\prime}$ x 3$^{\prime}$, which helps us confirm our source-beam coupling corrections. We also present a deep pointed spectrum [64 scans vs. 10 scans in @Panuzzo:2010] in order to detect fainter lines, such as $^{13}$CO, H$_2$O, OH$^{+}$, HF, and more.
We add depth to the analysis by modeling both the cool and warm components of molecular gas, and simultaneously accounting for $^{12}$CO, $^{13}$CO, and \[C \]. We also use \[C \] emission as a separate estimate of total hydrogen mass and other absorption lines for column density estimates. We first analyze the CO excitation using likelihood analysis to determine the physical conditions, and then compare to possible energy sources. Our likelihoods test the uniqueness and uncertainty in the conditions, as has also been done in @Naylor:2010 [@Kamenetzky:2011; @Scott:2011; @Bradford:2009; @Panuzzo:2010; @Rangwala:2011].
Our observations are described in Section \[sec:obs\]. We describe the Bayesian likelihood analysis used to find the best physical properties of the molecular gas in Section \[sec:like\] and present the results in Sections \[sec:discdeep\] and \[sec:discmap\]. In the remainder of Section \[sec:disc\], we discuss absorption results that are new to this study, possible excitation mechanisms of the warm gas, and comparisons to other galaxies. Conclusions are presented in Section \[sec:concl\].
Observations with SPIRE {#sec:obs}
=======================
The SPIRE Spectrometer {#sec:spire}
----------------------
The SPIRE instrument [@Griffin:2010] is on-board the [*Herschel Space Observatory*]{} [@Pilbratt:2010]. It consists of a three-band imaging photometer (at 250, 350, and 500 $\mu$m) and an imaging Fourier-transform spectrometer (FTS). We are presenting observations from the FTS, which operates in the range of 194-671 $\mu$m (447-1550 GHz). The bandwidth is split into two arrays of detectors: the Spectrometer Long Wave (SLW, 303-671 $\mu$m) and the Spectrometer Short Wave (SSW, 194-313 $\mu$m). The SPIRE spectrometer array consists of 7 (17) operational unvignetted bolometers for the SLW (SSW) detector, arranged in a hexagonal pattern. In the SLW, the beam FWHM is about 43[$^{\prime\prime}$]{} at its largest, dropping to 30[$^{\prime\prime}$]{} and then rising again to 35[$^{\prime\prime}$]{} at higher frequency. The SSW beam is consistently around 19[$^{\prime\prime}$]{}.
Two SPIRE FTS observations from Operational Day 543 were utilized in this study: one long integration single pointing of 64 scans total (“deep", Observation ID 1342208389, 84 min \[71 min integration time\], AOR “SSpec-m82 -deep") and one fully-sampled map (“map", Observation ID 1342208388, $\sim$ 5 hrs, AOR “Sspec-m82"). The map observation was conducted in high-resolution (HR) mode and the deep observation was conducted in high+low-resolution (H+LR) mode. Both were processed in high-resolution mode ($\Delta \nu \sim$ 1.19 GHz) with the [*Herschel*]{} Interactive Processing Environment (HIPE) 7.2.0 and the version 7.0 SPIRE calibration derived from Uranus [@Swinyard:2010; @Fulton:2010].
Spectral Map Making Procedure {#sec:mapmaking}
-----------------------------
In mapping mode, the SPIRE detector arrays are moved around the sky to 16 different jiggle positions, creating 112 and 272 spectra of 16 scans each for SLW and SSW, respectively, covering an area of the sky approximately 3 x 3 arcminutes. The positions of these scans on the sky are presented in Figure \[fig:map\], with blue asterisks for SLW and red diamonds for SSW.
![Spectrometer Mapping Point Locations. SLW spectra locations are indicated by blue asterisks, SSW by red diamonds. The pixel boundaries, spaced 9.5[$^{\prime\prime}$]{} apart, are indicated by solid black lines. H$\alpha$ contours are also plotted in black to indicate the orientation of the (nearly edge-on) galactic disk, from the Mount Laguna 40 inch telescope [@Cheng:1997]. Contours are in decreasing intervals of 0.2 log(maximum), i.e. 10$^{0}$, 10$^{-0.2}$, 10$^{-0.4}$ ... 10$^{-1.2}$. The green circle indicates the size of the $^{12}$CO [[$\rm J{\!=\!}4{\!\rightarrow\!}3$]{}]{} 43[$^{\prime\prime}$]{} beam FWHM. The black X marks the position of the single pointing (“deep") spectrum.[]{data-label="fig:map"}](map.eps){width="\columnwidth"}
The recommended map making method bins the spectra into pixels approximately one half the FWHM of the beam for each detector, which are about 35[$^{\prime\prime}$]{} for SLW and 19[$^{\prime\prime}$]{} for SSW, leading to pixel sizes of 17.5[$^{\prime\prime}$]{}and 9.5[$^{\prime\prime}$]{}. We wrote a custom script to create the map, based largely on the [NaiveProjection]{} method described in the SPIRE documentation for HIPE. Each of the 256 scans per detector were processed individually. All scans for a given detector and jiggle position falling within a given pixel were then averaged and an error bar for each wavenumber bin average is determined as the standard deviation of the scans divided by the square root of number of scans. All detector averaged spectra that fall into a pixel are then averaged using a weighted mean (where the weight is the inverse of the square of the error bar). We used the same 9.5[$^{\prime\prime}$]{} grid for both bands.
Using a 9.5[$^{\prime\prime}$]{} grid introduced more blank pixels in the SLW map, because the SLW map is more sparsely sampled because of its larger beam, as can be seen in Figure \[fig:map\]. However, this enabled the comparison of the same spectral locations across both bands (where the data were available), without averaging together spatially discrepant spectra in the SSW, as would happen if pixels were made larger. We emphasize that the blank pixels are somewhat artificial; the whole galaxy has been mapped, and the pixels locations are simply meant to indicate the central location of the detectors, though the beam size is larger than the pixel boundaries.
Line Fitting and Convolution Procedure {#sec:fitting}
--------------------------------------
The mirror scan length determines the spectral resolution of the spectrum. Because the scan length is necessarily finite, the Fourier-Transformed spectrum contains ringing; therefore, the instrument’s line profile is a sinc function, as can be clearly seen in Figures \[fig:spec\] and \[fig:fitting\]. The spectrum also contains the underlying continuum which must be removed before fitting the lines, which we do sequentially rather than simultaneously (see exceptions below). We isolate $\pm$ 25 GHz around the expected line center, and mask out the $\pm$ 6 GHz around the line center. We then fit the remaining signal with a second order polynomial fit to determine the continuum shape. After subtracting this continuum fit, we then use a Levenberg-Marquardt least-squares method to fit each line as a sinc function with the following free parameters: central frequency, line width, maximum amplitude, and residual (flat) baseline value. The baseline value stays around zero because the continuum has already been subtracted. The central frequency is limited such that the line center is no greater than $\pm$ 300 [$\,\rm km\,s^{-1}$]{} from the expected frequency given the nominal redshift of M82. For comparison, the resolution varies from 230 to 810 [$\,\rm km\,s^{-1}$]{}, from the shorter to longer wavelength ends of the band.
For the deep spectrum, we detect weaker lines than in the map spectra. However, ringing from the strongest lines can interfere with the signal; therefore we first fit the strong lines ($^{12}$CO, \[C \], and \[N \]) using the procedure outlined above and subtract their fitted line profiles from the spectrum. After all of the $^{12}$CO, \[C \], and \[N \] lines are fitted and subtracted from the spectrum, we then do a second pass to fit the weaker lines. An illustration of the difference this process can make for the $^{13}$CO [[$\rm J{\!=\!}5{\!\rightarrow\!}4$]{}]{} line is in Figure \[fig:fitting\].
In general, all of the lines are fit independently, with a few exceptions: the $^{12}$CO [[$\rm J{\!=\!}7{\!\rightarrow\!}6$]{}]{} and \[C \] [[$ \rm J{\!=\!}2{\!\rightarrow\!}1$]{}]{} lines are a mere 2.7 GHz away in rest frequency, and ground state p-H$_2$O and o-H$_2$O$^+$ lines are separated by only 2 GHz. These two pairs must be fit simultaneously. Both are fit independently to supply initial guesses, which are then used to fit both lines as the sum of two sinc functions, each with their own central wavelength, width, and amplitude, but with one baseline value.
The integrated flux is simply the area under the fitted sinc function, which is proportional to the product of the amplitude and line width (converted to km/s). The error in the integrated flux is based on propagating the errors from the fitted parameters themselves. We note that the error estimation assumes all wavenumber bins are independent of one another, but that is in fact not entirely true in a FT spectrometer. Though lines that are separated spectrally do not affect one another greatly (hence why we fit most lines independently), within each line fit the data points used in the 50 GHz range around the line center are not independent.
The beam size of the SPIRE spectrometer varies between the two detector arrays. In addition, it varies across the spectral range of the SLW, as described in Section \[sec:spire\], and is not strictly proportional to wavenumber due to the presence of multiple modes in the SLW detectors [@Chattopadhyay:2003]. When examining the spectral line energy distribution (SLED), it is imperative to scale all fluxes to a single beam size. For the map observation, we first fit the spectra as they were (with no correction factor). An example of integrated flux map, prior to any convolution or beam correction, is presented in Figure \[fig:intflux1\], with all other integrated flux maps available in the online version of the Journal. For the SLW detector, we present integrated flux maps using both 9.5[$^{\prime\prime}$]{} and 17.5[$^{\prime\prime}$]{} pixels.
The integrated flux maps for each line were then convolved with a kernel that matched the beams to the beam of the $^{12}$CO [[$\rm J{\!=\!}4{\!\rightarrow\!}3$]{}]{} map, which has a FWHM of 43[$^{\prime\prime}$]{}. The kernel was created using a modified version of the procedure decribed by @Bendo:2011 [see also @Gordon:2008]. However, instead of directly applying Equation 3 from @Bendo:2011 to the images of the beams, we applied the equation to one-dimensional slices of the beams to create the radial profile of convolution kernels, which we then used to create two-dimensional kernels. The approach worked very effectively for creating kernels to match beams observed with SSW to the $^{12}$CO [[$\rm J{\!=\!}4{\!\rightarrow\!}3$]{}]{} map. However, in cases where we created convolution kernels for matching beams measured at two different wavelengths by the SLW array, we needed to manually edit individual values in the kernel radial profiles to produce effective two dimensional kernels.
After the mathematical convolution of the integrated flux map with the kernel (resampled to match our map sizes), the fluxes were all converted to the units of Jy km s$^{-1}$/beam, referring to the $^{12}$CO [[$\rm J{\!=\!}4{\!\rightarrow\!}3$]{}]{} beam. The ratio of beam areas was determined empirically by convolving the kernel with the smaller (observed) beam peak normalized to 1 and determining the ratio required to produce the larger $\Omega_{\rm CO \rm J{\!=\!}4{\!\rightarrow\!}3}$ beam with the same peak (simulating the observation of a point source of 1 Jy km s$^{-1}$). To account for blank pixels, only the portion of $\Omega_{\rm kernel}$ that encompasses data was used in the aforementioned conversion.
For the deep spectrum, we used the same source-beam coupling factor ($\eta_c(\nu)$) as in @Panuzzo:2010, which was derived by convolving the M82 SPIRE photometer 250 $\mu$m map [@Roussel:2010] with appropriate profiles to produce the continuum light distribution seen with the FTS. The deep spectrum was multiplied by this factor before the line fitting procedure. The deep spectrum is presented in four parts in Figure \[fig:spec\], and the measured lines fluxes ($\geq 3 \sigma$) are in Table \[table:fit\].
The central pixel of the convolved maps offers a direct comparison to the source-beam coupling corrected deep spectrum. In the SLW, these two SLEDS are within $\pm$ 16% of one another. Later, we assume calibration error of 20%, so these differences are within those bounds. There is greater variation in the SSW band, though this is the region in which the signal to noise of the lines greatly drops. When 20% calibration error is included, all the line measurements have overlapping error bars with the exception of $^{12}$CO [[$\rm J{\!=\!}13{\!\rightarrow\!}12$]{}]{}, where the deep spectrum measurement is more than twice that of the map. The $^{12}$CO [[$\rm J{\!=\!}13{\!\rightarrow\!}12$]{}]{} map, however, has a S/N of only 4 in the central pixel, with only 60/399 pixels having S/N greater than 3. We primarily use the deep spectrum for our likelihood analysis because of the higher S/N and access to more faint lines, but compare with using the convolved map central pixel in Section \[sec:discmap\]. The similarity of the two SLEDs, within error bars, using two independent methods (the derived $\eta_c(\nu)$ from photometry comparisons vs. map convolution) to account for source-beam coupling, indicates that both methods are robust.
Though the maps do not provide adequately high spatial/spectral resolution for a detailed study of the morphology of the emission, some qualitative assessments can be made. For example, the line centroids do trace the relative redshift/blueshift of the northeast and southwest components [$v_{hel} \sim 300$ [$\,\rm km\,s^{-1}$]{} and 160 [$\,\rm km\,s^{-1}$]{}, respectively, @Loenen:2010]. However, we do not resolve the two separate peaks in flux. The capabilities of these maps to resolve gradients in the physical parameters modeled in this work will be discussed in Section \[sec:discmap\].
[ c c c ]{} $^{12}$CO [[$\rm J{\!=\!}4{\!\rightarrow\!}3$]{}]{}& 461.041 & 85.66 $\pm$ 0.90\
$^{12}$CO [[$\rm J{\!=\!}5{\!\rightarrow\!}4$]{}]{}& 576.268 & 79.44 $\pm$ 0.94\
$^{12}$CO [[$\rm J{\!=\!}6{\!\rightarrow\!}5$]{}]{}& 691.473 & 73.54 $\pm$ 0.44\
$^{12}$CO [[$\rm J{\!=\!}7{\!\rightarrow\!}6$]{}]{}& 806.652 & 66.04 $\pm$ 0.65\
$^{12}$CO [[$\rm J{\!=\!}8{\!\rightarrow\!}7$]{}]{}& 921.800 & 58.44 $\pm$ 0.87\
$^{12}$CO [[$\rm J{\!=\!}9{\!\rightarrow\!}8$]{}]{}& 1036.912 & 42.90 $\pm$ 0.74\
$^{12}$CO [[$\rm J{\!=\!}10{\!\rightarrow\!}9$]{}]{}& 1151.985 & 29.60 $\pm$ 0.41\
$^{12}$CO [[$\rm J{\!=\!}11{\!\rightarrow\!}10$]{}]{}& 1267.014 & 19.35 $\pm$ 0.33\
$^{12}$CO [[$\rm J{\!=\!}12{\!\rightarrow\!}11$]{}]{}& 1381.995 & 13.25 $\pm$ 0.35\
$^{12}$CO [[$\rm J{\!=\!}13{\!\rightarrow\!}12$]{}]{}& 1496.923 & 10.4 $\pm$ 1.1\
$[\rm C {\sc{I}} ] \ ^3P_1 {\!\rightarrow\!}^3P_0$ & 492.161 & 23.93 $\pm$ 0.55\
$[\rm C {\sc{I}} ] \ ^3P_2 {\!\rightarrow\!}^3P_1$ & 809.342 & 38.88 $\pm$ 0.58\
$[\rm N {\sc{II}} ] \ ^3P_1 {\!\rightarrow\!}^3P_0$ & 1462.000 & 82.4 $\pm$ 1.2\
$^{13}$CO [[$\rm J{\!=\!}5{\!\rightarrow\!}4$]{}]{}& 550.926 & 3.83 $\pm$ 0.58\
$^{13}$CO [[$\rm J{\!=\!}6{\!\rightarrow\!}5$]{}]{}& 661.067 & 3.02 $\pm$ 0.15\
$^{13}$CO [[$\rm J{\!=\!}7{\!\rightarrow\!}6$]{}]{}& 771.184 & 1.84 $\pm$ 0.31\
$^{13}$CO [[$\rm J{\!=\!}8{\!\rightarrow\!}7$]{}]{} & 881.273 & 1.16 $\pm$ 0.45\
HCN [[$\rm J{\!=\!}6{\!\rightarrow\!}5$]{}]{}& 531.716 & 1.15 $\pm$ 0.42\
HCO$^+$ [[$\rm J{\!=\!}7{\!\rightarrow\!}6$]{}]{}& 624.208 & 1.08 $\pm$ 0.17\
OH$^+$ $\rm N{\!=\!}1{\!\rightarrow\!}0$, $\rm J{\!=\!}0{\!\rightarrow\!}1$ & 909.159 & -5.2 $\pm$ 1.1\
OH$^+$ $\rm N{\!=\!}1{\!\rightarrow\!}0$, $\rm J{\!=\!}2{\!\rightarrow\!}1$ & 971.805 & -8.88 $\pm$ 0.41\
OH$^+$ $\rm N{\!=\!}1{\!\rightarrow\!}0$, $\rm J{\!=\!}1{\!\rightarrow\!}1$ & 1033.118 & -9.94 $\pm$ 0.39\
HF [[$\rm J{\!=\!}1{\!\rightarrow\!}0$]{}]{}& 1232.476 & -3.64 $\pm$ 0.34\
o-H$_2$O $3_{12} {\!\rightarrow\!}3_{03}$ & 1097.365 & 1.39 $\pm$ 0.30\
o-H$_2$O $3_{12} {\!\rightarrow\!}2_{21}$ & 1153.127 & 2.57 $\pm$ 0.37\
p-H$_2$O $2_{11}{\!\rightarrow\!}2_{02}$ & 752.033 & 1.86 $\pm$ 0.36\
p-H$_2$O $2_{02} {\!\rightarrow\!}1_{11}$ & 987.927 & 3.09 $\pm$ 0.56\
p-H$_2$O $1_{11} {\!\rightarrow\!}0_{00}$ & 1113.343 & -2.69 $\pm$ 0.44\
p-H$_2$O $2_{20} {\!\rightarrow\!}2_{11}$ & 1228.789 & 2.01 $\pm$ 0.34\
o-H$_2$O$^+$ $1_{11} {\!\rightarrow\!}0_{00}$ & 1115.204 & -2.82 $\pm$ 0.42
![Example of line fit, using the $^{13}$CO [[$\rm J{\!=\!}5{\!\rightarrow\!}4$]{}]{} line. The top panel shows the continuum-subtracted spectrum (black); the ringing of the nearby $^{12}$CO [[$\rm J{\!=\!}5{\!\rightarrow\!}4$]{}]{} line is clearly visible on the right and interfering with the fit, overplotted in solid red. The middle panel shows the spectrum after subtracting out the strong $^{12}$CO, \[C \], and \[N \] lines (though only the subtraction of the $^{12}$CO [[$\rm J{\!=\!}5{\!\rightarrow\!}4$]{}]{} line is visible in this figure). In this example, the signal to noise ratio of the fit was improved from 1.1 to 6.6 (where the signal is the integrated flux and the noise is based on propagating the statistical errors of the fit parameters in the integrated flux calculation). The bottom panel is the residual of the fit. In all panels, a horizontal line indicates zero. In the top two panels, the solid vertical line indicates the expected line center, and the two dashed vertical lines demarcate the area within that was not used for fitting the continuum.[]{data-label="fig:fitting"}](fitting.eps){width="\columnwidth"}
Bayesian Likelihood Analysis {#sec:like}
============================
We follow the method described in @Ward:2003 for the $^{12}$CO [[$\rm J{\!=\!}6{\!\rightarrow\!}5$]{}]{} map of M82, used frequently in the analysis of ground based molecular observations of galaxies [e.g. the Z-spec collaboration, @Naylor:2010; @Kamenetzky:2011; @Scott:2011; @Bradford:2009] and also of the single pointing SPIRE spectrum of M82 [@Panuzzo:2010] and Arp220 [@Rangwala:2011]. The goal of our Bayesian likelihood analysis is to map the relative probabilities of physical conditions over a large parameter space; this provides a more complete statistical analysis of the physical conditions as opposed to simply finding one best-fit solution.
For each molecular species, we first calculate a grid of expected line emission temperatures for various combinations of temperature ($T_{\rm kin}$), density ($n(H_2)$), and column density per unit linewidth ($N_{^{12}CO}/dv$) using RADEX [@vanderTak:2007]. We use the uniform sphere approximation for calculating the escape probabilities; the actual morphology in M82 shows a more complex velocity structure, therefore this approximation is considered an average of the bulk properties of the gas (and the results are not sensitive to uniform sphere vs. LVG approximation). RADEX performs statistical equilibrium calculations of the level populations, including the effects of radiative trapping, for a specified gas temperature, density, and column density per unit linewidth. The resulting solutions are output in the form of background-subtracted Rayleigh-Jeans equivalent line radiation temperatures.
We use a 2.73 K blackbody to represent the cosmic microwave background (CMB). We also experimented with using the continuum flux measured in our deep spectrum as a background; we fit the continuum in Jy across both bands, masking out the lines, with a third-order polynomial. The choice of this fit was to accurately represent the continuum background as a function of frequency; it was not meant to represent any physical conditions. Because the relevant radiation background is the specific intensity (Jy/sr), we divide our continuum by the area corresponding to a 43[$^{\prime\prime}$]{} beam (as the entire deep spectrum has been corrected to that). Such a background is in fact orders of magnitude higher than the CMB at the highest frequencies that we are modeling. However, a grid with this background vs. the CMB produces the same likelihood results. This is because at a kinetic temperature of $\sim$ 500 K (the warm component we will model), collisional excitation greatly dominates over radiative excitation. In other words, at high temperatures, the modeled line intensities do not depend on the background radiation field. The cool component we will model is traced by low-J lines, whose background is not as affected, and so this component also finds the same results with either background. Therefore we are presenting results using the CMB background.
In addition to $^{12}$CO, we also model $^{13}$CO and \[C \] as a function of the same temperatures, densities, and column density of $^{12}$CO. For those molecular/atomic species, we also add the parameter of the relative abundance, e.g. \[$^{13}$CO\]/\[$^{12}$CO\] or ${X}_{^{13}CO} / {X}_{^{12}CO}$. When modeling the intensity, the column density of $^{13}$CO is simply that of $^{12}$CO times the relative abundance. \[C \] is modeled with the parameter of \[C \]/\[$^{12}$CO\]. Finally, we introduce one more parameter, the area fractional filling factor $\Phi_A$. The modeled emission may not entirely fill the beam, so the flux may be reduced by this factor. All model grid points are therefore multiplied by each value of $\Phi_A$. The ranges of parameters, as well as the number of grid points, are presented in Table \[tab:radex\].
[c c c]{} $T_{kin}$ \[K\] & $10^{0.7} - 10^{3.7}$ & 61\
$n(H_2) $ \[cm$^{-3}$\] & $10^{2} - 10^{6}$ & 41\
$N_{^{12}CO}$/$\Delta V$ \[cm$^{-2}$\] & $10^{14} - 10^{18}$ & 41\
$\Phi_A$ & $10^{-3} - 10^{0}$ & 41\
$X_{^{13}CO}/X_{^{12}CO}$ & $10^{-2} - 10^{-1}$ & 11\
$X_{[C {\sc{I}}]}/X_{^{12}CO}$ & $10^{-2} - 10^{2}$ & 11\
$\Delta V$ \[km s$^{-1}$\] & 1.0 & fixed\
$T_{background}$ \[K\] & 2.73 & fixed
The RADEX grid gives us a set of line intensities as a function of model parameters $\bm{p} = (N_{^{12}CO}/dv,n(H_2),T_{\rm kin},\Phi_A, \bm{X}_{mol} / \bm{X}_{^{12}CO})$, which we then compare to our measured intensities $\bm{x}$. To compare to column density per unit linewidth, we divide the measured intensities by 180 [$\,\rm km\,s^{-1}$]{}, so that they are also per unit linewidth. The optical depth, and in turn the RADEX results, depend only on the ratio of column density to line width. @Ward:2003 found linewidths of 180 [$\,\rm km\,s^{-1}$]{} for the NE component and 160 [$\,\rm km\,s^{-1}$]{} for the SW by resolving the structure in position-velocity diagrams for their study from CO [[$\rm J{\!=\!}1{\!\rightarrow\!}0$]{}]{} to [[$\rm J{\!=\!}7{\!\rightarrow\!}6$]{}]{}, but we do not resolve the difference between the two and so we use the larger value. The Bayesian likelihood of the model parameters given the measurements is
$$P(\bm{p} | \bm{x}) = \frac{P(\bm{p}) P(\bm{x} | \bm{p})}{P(\bm{x})},$$
where $P(\bm{p})$ is the prior probability of the model parameters (see Section \[sec:prior\]), $P(\bm{x})$ is for normalization, and $P(\bm{x} | \bm{p})$ is the probability of obtaining the observed data set given that the source follows the model described by $\bm{p}$. $P(\bm{x} | \bm{p})$ is the product of Gaussian distributions in each observation,
$$P(\bm{x} | \bm{p}) = \prod_i \frac{1}{\sqrt{2 \pi \sigma_i^2}} \rm exp \it \bigg[- \frac{(x_i - I_i (\bm{p}))^2}{2 \sigma_i^2} \bigg]$$
where $\sigma_i$ is the standard deviation of the observational measurement for transition $i$ and $I_i({\bf p})$ is the RADEX-predicted line intensity for that transition and model. For the total uncertainty, we take the statistical uncertainty in the total integrated intensity from the line fitting procedure and add 20%/10% calibration error for SSW/SLW in quadrature. To find the likelihood distribution of one parameter out of all of $\bm{p}$, we integrate over all other parameters to find, for example, $P(T_{kin})$.
Separate Components {#sec:components}
-------------------
We divide the lines fluxes into two components, one warmer and one cooler. @Panuzzo:2010 already showed that the high-J lines of $^{12}$CO trace a warmer component than those transitions available from the ground. However, some of the mid-J lines (especially CO [[$\rm J{\!=\!}4{\!\rightarrow\!}3$]{}]{}) may have significant contributions from both components. To separate the $^{12}$CO fluxes from each component, we follow an iterative procedure. We first model the lowest three $^{12}$CO lines from @Ward:2003; we take the sum of their measurements for the two observed lobes and scale the result by the ratio of their beam area to our 43[$^{\prime\prime}$]{} beam. The best-fit SLED is then subtracted from our SPIRE measurements, producing the black triangles in Figures \[fig:deep\_seds\_co\] and \[fig:deep\_seds\_multi\]. These triangles comprise the “warm component." The best fit warm SLED is then subtracted from the low-J lines, producing the asterisks in the aforementioned figure. These fluxes are refit to produce our results for the “cool component."
We present a two-component model using just $^{12}$CO, as well as one including our high-J lines of $^{13}$CO along with the warm component. We did not model the low-J $^{13}$CO lines reported in @Ward:2003 due to the uncertainties presented in their Table 2 footnotes. We instead predicted the $^{13}$CO spectrum of the cool component, given its best fit results and a $^{12}$CO/$^{13}$CO ratio of of 35 [inbetween previously found 30 to 40 @Ward:2003], and found a very small contribution to the higher-J lines we are modeling. These small contributions are subtracted from the warm component, as with $^{12}$CO in the previous paragraph.
We also sought to include \[C \], which was assumed to be associated with the cool component, due to the low excitation temperature ($\sim$ 30 K) derived from the line ratios ($n_u/n_l = g_u/g_l \exp(-\Delta T/T_{ex})$, $n_i \propto I_i {\ensuremath{\, \mathrm{[W/cm^2]}}} / (A_i \nu_i)$). Here, we use $\Delta T = 38.84$, $g_u = 5$, $g_l = 3$, $A_u=2.65 \times 10^6$ s$^{-1}$, $A_l=7.88 \times 10^8$ s$^{-1}$. See Table \[table:fit\] for the frequencies and unit conversion. However, this model produced some unphysical situations. We discuss our findings and implications of them in Section \[sec:discdeep\].
This analysis necessarily assumes that all of the line emission for a given component is coming from one portion of gas described in bulk by the model parameters. In reality, there is likely a variety of physical conditions, existing in a continuum of parameters. However, the high-J SPIRE data does not provide justification for modeling more than one warm component because the SLEDs are well-described by one component. We did attempt a procedure to model multiple warm components of CO by first fitting the highest-J lines and subtracting the predicted line fluxes for the mid-J lines. Such a procedure has been used in @Rangwala:2011, for example. However, the predictions for the mid-J lines either matched or were an overestimate of the observed fluxes, leaving no second component to be modeled. A range of conditions is definitely present in the molecular gas, yet these two (warm and cool) components are dominating the emission, within the observational and modeling uncertainties. We note that, with regards to the different molecules/atoms being modeled here, all three species have similar profiles, as shown from the HIFI (higher spectral resolution) spectra in @Loenen:2010.
Prior Probabilities {#sec:prior}
-------------------
[c c c]{} Line width & 180 & \[km s$^{-1}$\]\
Abundance ($X_{^{12}CO}/X_{H_2}$) & 3.0 $\times 10^{-4}$ &\
Angular Size Scale & 17 & \[pc/[$^{\prime\prime}$]{}\]\
Emission Size & 43.0 & \[[$^{\prime\prime}$]{}\]\
Length Limit & 900 & \[pc\]\
Dynamical Mass Limit & 2 $\times 10^{9}$ & \[[M$_{\odot}$]{}\]
We use a binary prior probability, $P(\bm{p})$, to indicate either a physically allowed scenario ($P(p)$=1) or an unphysical and thus not allowed scenario ($P(p)$=0). In other words, all combinations of parameters that are deemed physical based on the following three conditions were given equal prior probability, and all others are given zero prior probability. The conditions are as follows:
1. The total length of the column ($L_{col}$) cannot exceed the length of the entire region. This assumes the length in the plane of the sky is the same as that orthogonal to the plane of the sky; we chose an upper-limit to the length of 900 pc because of the observed size [@Ward:2003]. This is the most significant of all the priors, placing an upper limit on the column density and a lower limit on the density. This prior can be stated as
$$\frac{N_{^{12}CO}}{n(H_2) \sqrt{\Phi_A} X_{^{12}CO}} \le 900 {\ensuremath{\, \mathrm{pc}}}.$$
For the relative abundance $X_{^{12}CO}$ to molecular hydrogen, we assume 3.0 $\times 10^{-4}$ [@Ward:2003].
2. The total mass in the emission region ($M_{region}$) cannot exceed the dynamical mass of the galaxy. We use the expression
$$\label{eqn:mass}
M_{region} = \frac{A_{region} N_{^{12}CO} \Phi_A 1.5 m_{H_2}}{X_{^{12}CO}}$$
to calculate the mass in the emission region, where the 1.5 accounts for helium and other heavy elements, and $\Phi_A$ is the filling factor. We estimate the dynamical mass to be $2 \times 10^9$ [M$_{\odot}$]{} [@Ward:2003; @Naylor:2010], calculated using rotational velocity and radius. The other assumed parameters in the above expression are listed in Table \[table:likeparams\].
3. The optical depth of a line must be less than 100, as recommended by the RADEX documentation. The cloud excitation temperature can become too dependent on optical depth at high column densities, and so very high optical depths can lead to unreliable temperatures. We found that in the presence of the other priors, this limit does not affect the likelihood results.
Likelihood Analysis of the Map {#sec:maplike}
------------------------------
We run each map pixel through the aforementioned likelihood analysis independently. (Note that due to the beam size being larger than the pixels themselves, each pixel’s data are not independent of its neighbors). We only model $^{12}$CO in the spectral maps because they are of lower integration time and $^{13}$CO cannot be reliably measured. We also cannot account for cool emission at different locations in our map.
To be run through the likelihood analysis, a pixel was required to have both an SLW and SSW spectrum and at least 5 $^{12}$CO lines with S/N $\geq$ 10 (the convolution tends to increase the signal/noise of each pixel). 90 pixels met this requirement (98 pixels would meet the requirement of 5 $^{12}$CO lines with S/N $\geq$ 3, so little would be gained by going to lower S/N). Additionally, we do not find statistically significantly different results requiring only 3 lines, the minimum with which we could reasonably model the emission; to some extent, this requires at minimum the [[$\rm J{\!=\!}6{\!\rightarrow\!}5$]{}]{} transition, which means we will be tracing higher temperatures.
Modeling Results and Discussion {#sec:disc}
===============================
![Bayesian Likelihood Analysis, Spectral Line Energy Distributions, $^{12}$CO Only. Asterisks represent the cool component, with its best fit SLED (“4D Max" column in Table \[table:like1\]) shown by a dotted line. Triangles represent the warm component, with its best fit SLED shown by a dashed line. The total measurements are shown with diamonds with their associated error bars. The total of both components is the solid line.[]{data-label="fig:deep_seds_co"}](deep_sed_co.eps){width="\columnwidth"}
![Bayesian Likelihood Analysis, Spectral Line Energy Distributions, including $^{13}$CO. Each color is a separate species: black for $^{12}CO$, red for $^{13}$CO (only warm component). The total fluxes are shown by diamonds, but their separate components are in asterisks/triangles for the cool/warm components. Best fit SLEDs (“4D Max" column in Table \[table:like2\]) are shown with dotted/dashed lines for cool/warm components. The total SLED, shown with a solid line, is the sum of the two components.[]{data-label="fig:deep_seds_multi"}](deep_sed_multi.eps){width="\columnwidth"}
![Bayesian Likelihood Analysis, Primary Parameter Results, $^{12}$CO only. Each color represents a separate component; blue for cool, red for warm (see Section \[sec:like\]). Dashed/dotted vertical lines indicate the median/4D maximum of the distribution (see Table \[table:like1\]).[]{data-label="fig:deep_results1_co"}](deep_results1_co.eps){width="\columnwidth"}
![Bayesian Likelihood Analysis, Primary Parameter Results, including $^{13}$CO. Each color represents a separate component; blue for cool, red for warm (see Section \[sec:like\]). Dashed/dotted vertical lines indicate the median/4D maximum of the distribution (see Table \[table:like1\]).[]{data-label="fig:deep_results1_multi"}](deep_results1_multi.eps){width="\columnwidth"}
![Bayesian Likelihood Analysis, Secondary Parameter Results, $^{12}$CO only. Top: Each color represents a separate component; blue for cool, red for warm (see Section \[sec:like\]) Dashed/dotted vertical lines indicate the median/4D maximum of the distribution (see Table \[table:like1\]). Bottom: 2D distributions for the pairs of primary parameters from which the secondary parameters (top) were derived; colors correspond to component. Diagonal lines indicate values of the top parameters (pressure on left and beam-averaged column density on right). Contour levels are 20, 40, 60, and 80% of the maximum likelihood.[]{data-label="fig:deep_results2_co"}](deep_results2_co.eps){width="\columnwidth"}
![Bayesian Likelihood Analysis, Secondary Parameter Results, including $^{13}$CO. Top: Each color represents a separate component; blue for cool, red for warm (see Section \[sec:like\]) Dashed/dotted vertical lines indicate the median/4D maximum of the distribution (see Table \[table:like1\]). Bottom: 2D distributions for the pairs of primary parameters from which the secondary parameters (top) were derived; colors correspond to component. Diagonal lines indicate values of the top parameters (pressure on left and beam-averaged column density on right). Contour levels are 20, 40, 60, and 80% of the maximum likelihood.[]{data-label="fig:deep_results2_multi"}](deep_results2_multi.eps){width="\columnwidth"}
[c | c c c | c | c c c | c c]{} & & Cool & & Warm &\
& Median & 1 Sigma Range & 1D Max & 4D Max & Median & 1 Sigma Range & 1D Max & 4D Max &\
$T_{\rm kin}$ & 40 & 12 $-$ 472 & 13 & 63 & 414 & 335 $-$ 518 & 447 & 447 & \[K\]\
$n({\rm H_2})$ & $10^{3.23}$ & $10^{2.36} - 10^{4.81}$ & $10^{2.20}$ & $10^{3.40}$ & $10^{3.98}$ & $10^{3.53} - 10^{4.21}$ & $10^{4.10}$ & $10^{4.10}$ &\[cm$^{-3}$\]\
$N_{co}$ & $10^{19.03}$ & $10^{18.38} - 10^{19.56}$ & $10^{19.36}$ & $10^{18.56}$ & $10^{18.12}$ & $10^{17.06} - 10^{19.09}$ & $10^{18.56}$ & $10^{17.96}$ &\[cm$^{-2}$\]\
$\Phi_{\rm A}$ & $10^{-0.92}$ & $10^{-1.22} - 10^{-0.57}$ & $10^{-0.97}$ & $10^{-0.90}$ & $10^{-1.50}$ & $10^{-1.96} - 10^{-0.52}$ & $10^{-1.88}$ & $10^{-1.28}$ &\
$P$ & $10^{ 5.11}$ & $10^{ 4.55} - 10^{ 5.76}$ & $10^{ 5.24}$ & $10^{ 5.24}$ & $10^{ 6.61}$ & $10^{ 6.16} - 10^{ 6.80}$ & $10^{ 6.75}$ & $10^{ 6.75}$ &\[K cm$^{-2}$\]\
$<N_{\rm co}>$ & $10^{18.15}$ & $10^{17.63} - 10^{18.84}$ & $10^{17.78}$ & $10^{17.78}$ & $10^{16.67}$ & $10^{16.44} - 10^{17.20}$ & $10^{16.58}$ & $10^{16.58}$ &\[cm$^{-2}$\]\
Mass & $10^{ 7.67}$ & $10^{ 7.16} - 10^{ 8.36}$ & $10^{ 7.31}$ & $10^{ 7.31}$ & $10^{ 6.20}$ & $10^{ 5.97} - 10^{ 6.72}$ & $10^{ 6.11}$ & $10^{ 6.11}$ &\[M$_\odot$\]
--------------------------- -------------- --------------------------- -------------- -------------- -------------- --------------------------- -------------- -------------- -----------------
Cool Warm
Median 1 Sigma Range 1D Max 4D Max Median 1 Sigma Range 1D Max 4D Max
$T_{\rm kin}$ 35 12 $-$ 385 14 158 436 344 $-$ 548 447 501 \[K\]
$n({\rm H_2})$ $10^{3.44}$ $10^{2.48} - 10^{5.15}$ $10^{3.00}$ $10^{3.10}$ $10^{3.58}$ $10^{3.17} - 10^{3.96}$ $10^{3.80}$ $10^{3.40}$ \[cm$^{-3}$\]
$N_{co}$ $10^{19.25}$ $10^{18.69} - 10^{19.70}$ $10^{19.36}$ $10^{18.96}$ $10^{19.02}$ $10^{18.19} - 10^{19.51}$ $10^{19.16}$ $10^{19.26}$ \[cm$^{-2}$\]
$\Phi_{\rm A}$ $10^{-1.19}$ $10^{-1.52} - 10^{-0.83}$ $10^{-1.28}$ $10^{-1.35}$ $10^{-1.88}$ $10^{-2.06} - 10^{-1.50}$ $10^{-1.88}$ $10^{-1.88}$
$P$ $10^{ 5.24}$ $10^{ 4.73} - 10^{ 6.18}$ $10^{ 5.24}$ $10^{ 5.24}$ $10^{ 6.23}$ $10^{ 5.80} - 10^{ 6.60}$ $10^{ 6.61}$ $10^{ 6.61}$ \[K cm$^{-2}$\]
$<N_{\rm co}>$ $10^{18.09}$ $10^{17.59} - 10^{18.71}$ $10^{17.78}$ $10^{17.78}$ $10^{17.12}$ $10^{16.69} - 10^{17.59}$ $10^{16.76}$ $10^{16.76}$ \[cm$^{-2}$\]
Mass $10^{ 7.62}$ $10^{ 7.11} - 10^{ 8.23}$ $10^{ 7.31}$ $10^{ 7.31}$ $10^{ 6.65}$ $10^{ 6.22} - 10^{ 7.12}$ $10^{ 6.28}$ $10^{ 6.28}$ \[M$_\odot$\]
$X_{\rm 13co}/X_{\rm co}$ - - - - $10^{-1.58}$ $10^{-1.70} - 10^{-1.46}$ $10^{-1.50}$ $10^{-1.50}$
$N_{\rm 13co}$ - - - - $10^{17.44}$ $10^{16.62} - 10^{17.92}$ $10^{17.62}$ $10^{17.62}$ \[cm$^{-2}$\]
--------------------------- -------------- --------------------------- -------------- -------------- -------------- --------------------------- -------------- -------------- -----------------
Physical Conditions: Deep Spectrum {#sec:discdeep}
----------------------------------
We present two different versions of our likelihood analysis for the deep spectrum: one using only $^{12}$CO, and one using $^{12}$CO and $^{13}$CO (“multiple molecule"). The motivation behind this is to investigate two questions: does the addition of different species change the modeled parameters of the gas, and/or does it better constrain the parameters? Both versions contain a warm and cool component. The modeling assumes all of the emission in a given component is coming from the same homogeneous gas, and by comparing these models we will investigate the validity of this assumption in this subsection.
The results for each of these versions are presented in Tables \[table:like1\] and \[table:like2\] respectively. Figures \[fig:deep\_seds\_co\] and \[fig:deep\_seds\_multi\] show the input SLED as well as the best-fit model results. The primary results (temperature, density, column density, and filling factor) are displayed graphically in Figures \[fig:deep\_results1\_co\] and \[fig:deep\_results1\_multi\]. Secondary parameters, which are calculated from the aforementioned primary results, are displayed in Figures \[fig:deep\_results2\_co\] and \[fig:deep\_results2\_multi\]. These include the pressure (the product of temperature and density) and the beam-averaged column density ($\langle N_{\rm ^{12}CO} \rangle$, the product of column density and filling factor). We note that in the results, the parameters of the most likely grid point (“4D Max") is not necessarily the same as the median or the mode (“1D Max") of the integrated likelihood distributions. The “4D Max" is describing one specific point, but the median, 1D Max, and associated error range are representative of the larger likelihood across all other parameters in the grid.
We first compare our two models, $^{12}$CO only vs. multiple molecule. These are Tables \[table:like1\] vs. \[table:like2\], Figures \[fig:deep\_results1\_co\] vs. \[fig:deep\_results1\_multi\], and Figures \[fig:deep\_results2\_co\] vs. \[fig:deep\_results2\_multi\].
The addition of $^{13}$CO to the warm component does not significantly change the temperature, but it does increase the likelihood of lower densities. It also decreases the likelihood of lower column densities. The consequences of these changes can be seen in the pressure and mass distributions in Figures \[fig:deep\_results2\_co\] and \[fig:deep\_results2\_multi\]. Adding $^{13}$CO increased the likelihoods of the “shoulders" of these distributions; the lower half of pressure, and the upper half of mass. An examination of the contour plots in the bottom half of these figures illustrates why, statistically. In the $^{12}$CO only model, though column density and filling factor are not well constrained independently, they are highly correlated; their contours run along an almost constant line of beam-averaged column density. Adding $^{13}$CO introduced the $X_{^{13}CO}/X_{^{12}CO}$ parameter, which also impacts the absolute flux level of the models, like column density and filling factor. The result are likelihoods that are more constrained but not as highly correlated with one another. Therefore the mass distribution is wider. In the $^{12}$CO-only model, the mass of the warm component is about 3.4/6.3% (median/4D Max) the mass of the cool component. @Rigopoulou:2002 noted that “warm" gas is generally around 1 to 10% of total gas mass for starburst galaxies, so this is about as expected. The addition of $^{13}$CO creates somewhat overlapping likelihood distributions for mass (the 1-sigma ranges are just touching), but the median and 4D Max now warm/cool ratios of 11% and 9%, respectively. One factor that may contribute to wider distributions is overestimated error; the error bars are dominated by our 20% calibration error, not statistical error.
After this point, we compare frequently to @Panuzzo:2010. The major factor responsible for the differences, just modeling $^{12}$CO alone, is the shape of the CO SLED at those lines with upper rotational number greater than J=8. We also explicitely subtracted the cool component’s contribution from $^{12}$CO [[$\rm J{\!=\!}4{\!\rightarrow\!}3$]{}]{}, whereas @Panuzzo:2010 simply underpredicted the total flux. We will also compare with @Loenen:2010, another high-J CO study of M82, in Section \[sec:energy\].
Our results are similar to @Panuzzo:2010, who found that these high-J CO lines trace a very warm gas component that is separate from the cold molecular gas traced by those lines below [[$\rm J{\!=\!}4{\!\rightarrow\!}3$]{}]{}. Our best-fit temperature of the $^{12}$CO only model (at 447 K, Table \[table:like1\]) is close to their value at 545 K, but the overall likelihood for temperature, integrated over all parameters, yields a slightly lower 414 K. Given the size of the uncertainty (335-518 K) in the parameter, the two distributions are very similar, and therefore the result is not significantly different. Such warm gas has also been traced in the S(1) and S(2) transitions of H$_2$, at 450 K with the Infrared Space Observatory [@Rigopoulou:2002] and 536 K with the Spitzer Infrared Spectrograph [@Beirao:2008].
We do find a slightly higher density than @Panuzzo:2010, with our best-fit value of $10^{4.1}$ compared to $10^{3.7}$ cm$^{-3}$, though the integrated likelihood distributions do overlap (see Figure \[fig:deep\_results1\_co\]). However, the temperature and density are degenerate; higher temperatures and lower densities may produce the same fluxes as lower temperatures and higher densities. Their product, the pressure, is better constrained. We seem to have collapsed/constrained the pressure distribution to the upper half of that presented in @Panuzzo:2010.
The column density is not as well constrained as presented in @Panuzzo:2010; we found that they had an error in calculating the expected fluxes of the higher-J lines for lower column density values. We have recalculated the fluxes for those column densities, and we find that in fact when properly calculated these column densities have a non-zero likelihood. In the $^{12}$CO only model, the column density itself is not constrained. However, the column density and filling factor are degenerate, so it is their product (beam-averaged column density, $\langle N_{\rm co} \rangle$) that is better constrained. Our best-fit value is 10$^{16.6}$ cm$^{-2}$.
The total mass in the beam can be calculated using Equation \[eqn:mass\] (and is presented as the top y-axis in Figures \[fig:deep\_results2\_co\] and \[fig:deep\_results2\_multi\], upper right). As previously discussed, the $^{12}$CO only model produces the expected result of less gas mass in the warm component. In the cool component we find a best-fit mass of 2.0 $\times 10^7$ [M$_{\odot}$]{} (median 4.7 $\times 10^7$ [M$_{\odot}$]{}). This is smaller than the 2.0 $\times 10^8$ [M$_{\odot}$]{} traced by the LVG analysis of @Ward:2003 with lower-J CO lines. This difference is due to the fact that we subtract the contribution to the low-J flux from the warm component; in our initial modeling of the cold component, before this subtraction, our best-fit mass is 9.8 $\times 10^7$ [M$_{\odot}$]{} (with a range from 0.2 to 2.2 $\times 10^8$ [M$_{\odot}$]{}). The warm component is a smaller fraction of the gas, with a best-fit of 1.3 $\times 10^6$ [M$_{\odot}$]{}, about 6.3% the mass of the cool component.
The $^{12}$CO/$^{13}$CO relative abundance is also a free parameter in our multi-species model; we find a best-fit value of about 32, similar to the 40 and 30 found previously for the NE and SW lobes, respectively [@Ward:2003].
As mentioned in Section \[sec:components\], we also attempted to include our two \[C \] lines with the cool component. We do not present the tables for this model because the mass distributions of the warm and cool components became overlapping, indicating the same amount of mass in both components, an unphysical situation. Additionally, the derived relative abundance of \[C \] to $^{12}$CO was unusually high. We found a ratio of 0.48 to 3.3, which is higher than @White:1994 [average value $\sim 0.5$], @Schilke:1993 [0.1-0.3], and @Stutzki:1997 [0.5] using other methods. Before subtracting the warm component’s contribution to the $^{12}$CO flux (when we just fit all of the low-J $^{12}$CO flux and \[C \]), we find ratios more consistent with these values (best-fit 0.4, range 0.09 to 1.23). These two problems could be indicating that the assumption of CO and \[C \] coming from the same component is flawed. The column density, temperature, and mass developed somewhat of a double-peaked structure; specifically, the addition of \[C \] increases the likelihood of lower column densities and masses, but does not eliminiate the previous likelihood peak.
It is unclear how much of the molecular CO and atomic C are truly cospatial and therefore how physical our results for modeling them all together as one bulk gas component may be. @Papadopoulos:2004a presented results which argue that \[C \] and CO are cospatial and trace the same hydrogen gas mass (\[C \] doing so better than CO in many conditions). This conflicts with the theoretical picture of gas clouds (especially in PDRs) as a structured transition between molecular, atomic, and ionized gas, but new observational and theoretical evidence indicates the types of gas are not so distinct [see references within @Papadopoulos:2004a]. For example, @Howe:2000 and @Li:2004 have found \[C \] to trace $^{13}$CO well. If the ISM is clumpy, well mixed, and dynamic, the \[C \] and CO may be cospatial averaged over large scales. Strong stellar winds (and possibly the interaction with M81) could be contributing to the dynamic nature of the gas, so it is not unreasonable to believe that the gas has not achieved the simple layered pattern. Though the SPIRE FTS cannot resolve the two separate velocity components of M82, HIFI can, and observations of these two \[C \] lines indicate a generally similar shape to the CO lines, namely two Gaussians with the SW component demonstrating higher flux [@Loenen:2010].
@Wolfire:2010 presented a model PDR which shows a cloud layer traced by atomic (not-yet-ionized) \[C \] where the hydrogen is still molecular due to self-shielding effects. This “dark molecular gas" (called so because it is not traced by CO) would be less-shielded and at a warmer temperature than the inner-most cloud layer of CO. It is possible that \[C \] and CO are somewhat cospatial yet somewhat segregated as in the “dark molecular gas model." Our analysis is consistent with a picture in which the \[C \] and $^{12}$CO do not completely overlap spatially.
The \[C \] [[$\rm J{\!=\!}1{\!\rightarrow\!}0$]{}]{} emission can also be used to estimate the total hydrogen mass using Equation 12 of @Papadopoulos:2004a. Using the median $X_{[C {\sc{I}}]}/X_{H_2}$ of 1.5 $\times 10^{-4}$ from $X_{[C {\sc{I}}]}/X_{^{12}CO}$ = 0.5 [@White:1994; @Stutzki:1997] and our assumed $X_{^{12}CO}/X_{H_2}$, we find a total gas mass of $4.4 \times 10^7 Q_{10}^{-1}$ [M$_{\odot}$]{}. $Q_{10}$ is the ratio of the column of the [[$\rm J{\!=\!}1{\!\rightarrow\!}0$]{}]{} emission to the total \[C \] column [see Appendix A of @Papadopoulos:2004a], which depends on the excitation conditions of the gas; for $Q_{10} \sim 0.5$ [@Papadopoulos:2004b], the gas mass is $8.8 \times 10^7$ [M$_{\odot}$]{} but is uncertain by modeled uncertainty in $X_{[C {\sc{I}}]}$ alone. This method of estimating the mass is higher than total mass estimate of the cool component described earlier in this section. \[C \] may be coming from a range of temperatures, but with only two lines, we cannot sort that out.
Physical Conditions: Map {#sec:discmap}
------------------------
{width="\textwidth"}
Results for all of the same parameters for the deep spectrum were produced for each of the pixels in the map (that met the criteria in Section \[sec:maplike\]). We find that the beam size of SPIRE cannot resolve the structure in M82 as has been done with interferometric maps or high spectral resolution observations (which can resolve the velocity components of the NE and SW lobes). Because of the degeneracy between temperature/density and column density/filling factor, we present results of their products, pressure and beam-averaged column density in Figure \[fig:map\_results\]. $\langle N_{\rm ^{12}CO} \rangle$ shows a radially decreasing trend, roughly corresponding to the decrease in the beam profile (plotted with a solid black line), implying an observational effect due to source-beam coupling. We would expect an off-centered beam to be able to probe the pressure of the central region (because the relative ratios of the SLED would be preserved), so the lack of a radial trend also implies that we are not measuring different areas of M82 in each map pixel. This indicates that the SPIRE FTS map cannot resolve M82’s structure, and therefore the single “deep" spectrum is an adequate representation of the galaxy as a whole. Our results in Section \[sec:discdeep\] are descriptive of the bulk properties of the galaxy and we do not see trends on the scale of our map. This analysis is separate from the dust temperature gradients (which we also see in the continuum gradients of our map spectra) found in M82 by @Roussel:2010 and indicates that the dust and molecular gas are not coupled.
One difficulty with the map is it has lower signal/noise; it must also be convolved up the largest beam size (43[$^{\prime\prime}$]{}, like the deep spectrum). However, as mentioned in Section \[sec:discdeep\], it is the highest-J fluxes in the SSW detector that largely constrain the results of the likelihood for the deep spectrum. Therefore, we also attempted to model the map with just those lines with upper-J level of 9 or higher, without convolving. These lines all were measured with a beam FWHM of $\sim$ 19[$^{\prime\prime}$]{}, offering higher resolution. However, these lines are also weaker, and with fewer, lower signal/noise lines this method does not constrain any parameters as well.
Though the off-axis pixels may not provide new information about the physical trends of the galaxy, we can compare the deep spectrum to the center pixel of our map as a test of the source-beam coupling corrections (see Section \[sec:mapmaking\]). The results of the central pixel are very similar to those presented in Section \[sec:discdeep\], though the integrated likelihoods are generally wider (the parameters are less constrained). This is partially, but not entirely, due to larger error bars on the SSW lines.
Molecular Line Absorption {#sec:abs}
-------------------------
In addition to the Bayesian likelihood modeling, we can also briefly discuss the absorption lines presented in Table \[table:fit\].
### Hydrogen Fluoride (HF)
Hydrogen fluoride (HF) is potentially a sensitive probe of total molecular gas column, because the HF/H$_2$ ratio is more reliably constant than $^{12}$CO/H$_2$ because the formation of HF is dominated by a reaction of F with H$_2$ [@Monje:2011]. Furthermore, HF [[$\rm J{\!=\!}1{\!\rightarrow\!}0$]{}]{} is generally seen in absorption because of its high A-coefficient, 2.42 $\times 10^{-2}$ s$^{-1}$ [@Monje:2011]. Assuming all HF molecules are in the ground state [generally true in the diffuse and dense ISM, @Monje:2011], the HF [[$\rm J{\!=\!}1{\!\rightarrow\!}0$]{}]{} line yields the optical depth simply as $\tau = -ln(F_l/F_c)$, where $F_l/F_c$ is the line-to-continuum ratio. In the case of HF, we mask out a nearby water emission line (1226 to 1229 GHz), though because the signal in each wavenumber bin is not independent due to the ringing, this can introduce added uncertainty. Therefore the following discussion is meant to be approximate. We estimate the HF column density using
$$\label{eqn:column}
\int \tau dv = \frac{A_{ul} g_u \lambda^3}{8 \pi g_l} N(HF),$$
where $g_u$ = 3 and $g_l$ = 1. This implies $\int \tau dv = 4.16 \times 10^{-13} N(HF) {\ensuremath{\, \mathrm{cm^2 \ km \ s^{-1}}}}$. The HF line occurs in a part of our spectrum with some noticeable structure in the continuum (see Figure \[fig:spec\], third panel, around 1250 GHz). If we only integrate the 6 GHz surrounding the line, we find a column density of HF of 6.61 $\times 10^{13}$ cm$^{-2}$. Expanding the range over which we integrate increases the derived column density, but this may be due to other features in the spectrum, and so we consider our derived value a lower limit. Assuming a predicted abundance of HF of $3.6 \times 10^{-8}$ [@Monje:2011], this corresponds to a molecular hydrogen column density of $1.84 \times 10^{21}$ cm$^{-2}$. The column density derived from this line is similar to that of $\langle N \rangle$ of the cool component of $^{12}$CO. However, there are still some uncertainties to this calculation. We are only able to see the HF in front of the continuum emission, and therefore we are not probing the total column density. Higher spectral resolution could reveal the extent of spatial colocation of HF with CO. There are also uncertainties associated with either molecular abundance assumed and whether or not all HF molecules are truly in the ground state.
### Water and Water Ion (H$_2$O$^{(+)}$)
Water is fundamental to the energy balance of collapsing clouds and the subsequent formation of stars, planets, and life. Many [*Herschel*]{} key programs are currently studying water and chemically related molecular species in a variety of conditions. An excellent summary of water chemistry in star forming regions is available in @vanDishoeck:2011.
@Weiss:2010 studied the low-level water transitions in M82, and they detect the ground-state o-H$_2$O emission in two clearly resolved components, which we do not. With the 41[$^{\prime\prime}$]{} beam, the two components add to 370 $\pm$ 44 Jy km/s beam$^{-1}$, well below our threshold of detection, as can be seen by examining Table \[table:fit\]. Though we do not detect that line, we do detect four new water lines in addition to those presented in @Weiss:2010: two p-H$_2$O (752 and 1229 GHz) and two o-H$_2$O (1097 and 1153 GHz). Combined with their ground-state transition of o-H$_2$O, we add to the picture of the water excitation in M82.
Our ground-state lines indicate similar column densities as @Weiss:2010, within a factor of 2. Using Equation \[eqn:column\] (low-excitation approximation), we find column densities of p-H$_2$O and o-H$_2$O$^+$ of $\sim 4 \times 10^{13}$ cm$^{-2}$, whereas @Weiss:2010 finds 9.0 and 2.2 $\times 10^{13}$ cm$^{-2}$, respectively. They found that the water absorption comes from a region northeast of the central CO peak; shocks related to the bar structure of M82 could be releasing water into the gas phase at such a location. The fact that the water comes from a lower column density region seems to contradict the existence of a PDR, which would require high column densities to shield water from UV dissociation; however, the relative strength and similarity of absorption profiles of o-H$_2$O$^+$ compared to p-H$_2$O indicates some ionizing photons (see their work for complete interpretation). Though these transitions are tracing a different region than CO, they add to the picture of a complicated mix of energy sources present in the gas, as addressed in Section \[sec:energy\]. Models of water emission from shocks have been investigated by others [i.e. @Flower:2010], but detailed modeling of the water spectrum of M82 is outside the scope of this work.
Gas Excitation {#sec:energy}
--------------
At the high temperature of the warm component, the cooling will be dominated by hydrogen. @LeBourlot:1999 modeled the cooling rates for H$_2$, and made their tabulated rates available with an interpolation routine for desired values of density, temperature, ortho- to para-H$_2$ ratio, and H to H$_2$ density ratio[^1]. For our best-fit temperature and density, assuming n(H)/n(H$_2)$ = 1 [recommended by @LeBourlot:1999 for PDRs], this corresponds to a cooling rate of $10^{-22.54}$ erg s$^{-1}$ per molecule, or 3 [L$_{\odot}$]{}/[M$_{\odot}$]{}(using o/p H$_2$ = 1, though the number is only $\sim$ 3% lower for o/p H$_2$ = 3). Given the warm mass of 1.3 $\times$ 10$^{6}$ [M$_{\odot}$]{} (using the $^{12}$CO model for the rest of this section), that implies a hydrogen luminosity of 3.9 $\times 10^{6}$ [L$_{\odot}$]{}. The total observed hydrogen luminosity thus far has been higher than this number; adding the luminosities presented in @Rigopoulou:2002 and correcting for extinction as in @Draine:1989, we find a total of 1.2 $\times 10^7$ [L$_{\odot}$]{}in the (0-0)S(0)-S(3), S(5), S(7), and (1-0)Q(3) lines. However, some of these hydrogen lines are tracing lower or higher temperature gas. We note that the mass range within one standard deviation of our likelihood results for the warm component is 0.93-5.2 $\times 10^6$ [M$_{\odot}$]{}, which corresponds to a predicted luminosity of 0.28-1.6 $\times 10^7$ [L$_{\odot}$]{}, encompassing the measured hydrogen luminosity.
There are a few possibilities for the source of the excitation of the gas: X-ray photons, cosmic rays, UV excitation of PDRs and shocks/collisional excitation. Hard X-rays from an AGN have already been ruled out by others in the literature due to the lack of evidence for a strong AGN and low X-ray luminosity [1.1 $\times 10^6$ [L$_{\odot}$]{}, @Strickland:2007].
The CO emission from M82 has previously been interpreted using PDR models. @Beirao:2008 noted with the Spitzer Infrared Spectrograph that the H$_2$ emission is correlated with PAH emission, indicating that it is mainly excited by UV radiation in PDRs. @Loenen:2010 combined HIFI data with ground based detections in order to model $^{12}$CO [[$\rm J{\!=\!}1{\!\rightarrow\!}0$]{}]{} to [[$\rm J{\!=\!}13{\!\rightarrow\!}12$]{}]{} and $^{13}$CO [[$\rm J{\!=\!}1{\!\rightarrow\!}0$]{}]{} to [[$\rm J{\!=\!}8{\!\rightarrow\!}7$]{}]{}. They reproduced the measured SLEDs with one low-density (log($n(H_2)$)=3) and two high-density (log($n(H_2)$)=5,6) components with relative proportions of 70%, 29%, and 1%, respectively. The low-density component is largely responsible for the low-J emission, while the highest-density component is responsible for the highest-J emission. These high densities are not consistent with the results of our likelihood analysis detailed in Section \[sec:discdeep\]; the likelihood of solutions for the warm component at log($n(H_2)$)$>$5 is essentially zero. We note that @Loenen:2010’s Figure 3 shows the consistency between HIFI and SPIRE fluxes from @Panuzzo:2010. In other words, the difference is not due to discrepant line fluxes, but different models (PDR vs. CO likelihood analysis).
There are two major differences in the approach of this work and @Loenen:2010. First, the order in which we approach the problem is different. We first analyze the CO excitation using likelihood analysis to determine the physical conditions. Once we have these conditions, we then look to the possible energy sources *based on* the conditions we have already derived, instead of first seeing under which conditions a certain energy source fits. Second, we look beyond the one best fit solution: in addition to presenting the best-fit solution, our likelihoods analyze the relative probabilities for a larger parameter space.
We also attempted to reproduce our deep SLED with various PDR models. @Meijerink:2006 have added to their PDR and XDR models to include enhanced cosmic rays (at a rate of 5 $\times 10^{-15}$ s$^{-1}$), near our assumed rate discussed later in this section. Such models are currently available for incident flux of log(G$_0$) of 2-4 (G$_0$ = 1.6 $\times 10^{-3}$ erg cm$^{-2}$ s$^{-1}$) for log($n(H_2)$)=3 and log(G$_0$) of 3-5 for log($n(H_2)$) of 4 and 5. By examining the ratios of $^{12}$CO [[$\rm J{\!=\!}9{\!\rightarrow\!}8$]{}]{} with all higher-J lines (those largely driving the likelihood results, and also measured from similar beam sizes), none of the available 9 PDR models are an ideal match, but the PDR scenario for log($n(H_2)$)=3, log(G$_0$)=3 is the best match. Figure \[fig:PDR\] compares the predicted and observed ratios. The ratios used are without source-beam coupling correction because the SSW already has similar beam sizes, but the ratios with beam correction are within 4-8% of those presented in Figure \[fig:PDR\].
Additionally, we used a higher-resolution (in density and incident flux) grid of $^{12}$CO PDR models [@Wolfire:2010]. The same line ratios previously discussed (those shown in Figure \[fig:PDR\]) can only be reproduced by higher densities. For ([[$\rm J{\!=\!}9{\!\rightarrow\!}8$]{}]{})/([[$\rm J{\!=\!}10{\!\rightarrow\!}9$]{}]{}), the observed ratio is only found for log($n(H_2)$) $>$ 4.5, log(G$_0$) $>$ 2, and by ([[$\rm J{\!=\!}9{\!\rightarrow\!}8$]{}]{})/([[$\rm J{\!=\!}13{\!\rightarrow\!}12$]{}]{}), only for log($n(H_2)$) $>$ 5, log(G$_0$) $>$ 2.5.
![Enhanced Cosmic Ray PDR Models. The observed ratios are black diamonds. For the models, the line style indicates gas density and the line color indicates incident flux. Log(G$_0$)=2 is not plotted because the highest-J line fluxes are not reported for that model.[]{data-label="fig:PDR"}](PDR.eps){width="\columnwidth"}
To summarize, current PDR models can only explain the observed high-J $^{12}$CO emission with densities higher than those indicated by the likelihood analysis (even when all priors are excluded). The cosmic-ray enhanced PDR models, though sparse, can come closer to reproducing the high-J line ratios at a lower density, though these are also below our likelihoods. There is also evidence that shocks enhance high-J lines far more than PDRs [@Pon:2012]. In their models, almost all of the emission from the lowest-J lines came from unshocked gas, and most of the emission above [[$\rm J{\!=\!}7{\!\rightarrow\!}6$]{}]{} was from shocked gas. The combination of shocks and PDRs could be responsible for the high-J CO lines observed, while PDRs alone are adequate to explain the lower-J CO lines and PAH emission. In summary, these high-J lines are not consistent with current PDR models, but improved modeling of shocks and the effects of cosmic rays on PDRs may help explain their emission.
Cosmic rays (CRs) are another possibility for the excitation. The Very Energetic Radiation Imaging Telescope Array System (VERITAS) Collaboration recently reported that the cosmic-ray density in M82 is about 500 times the average Galactic (Milky Way) density [@Veritas:2009]. By using the cosmic ray ionization rate of the Galaxy [2-7 $\times 10^{-17}$ s$^{-1}$, @Goldsmith:1978; @vanDishoeck:1986], multiplied by 500, and then multiplied by the average energy per ionization (20 eV), one finds an energy deposition rate of 2-7 $\times 10^{-13}$ eV/s per H$_2$ molecule in M82. This implies a heating rate of 0.03 to 0.12 [L$_{\odot}$]{}/[M$_{\odot}$]{}, less than 5% of the required molecular hydrogen cooling rate. Thus, cosmic rays alone cannot excite the molecular gas.
Turbulent heating mechanisms may also play a role in M82. From @Bradford:2005, the turbulent heating per unit mass can be expressed in [L$_{\odot}$]{}/[M$_{\odot}$]{} as
$$1.10 \bigg( \frac{v_{rms}}{25 {\ensuremath{\, \mathrm{km \ s}}}^{-1}} \bigg)^3 \bigg( \frac{1 {\ensuremath{\, \mathrm{pc}}}}{\Lambda_d} \bigg)$$
where $v_{rms}$ is the turbulent velocity and $\Lambda_d$ is the typical size scale of turbulent structures. We use the Jeans length for this size scale, calculated from the parameters of the likelihood results (Section \[sec:discdeep\]), which is 0.9 pc. Given this size scale, the observed cooling rate could be replicated with a turbulent velocity of 33.7 [$\,\rm km\,s^{-1}$]{}. This would imply a velocity gradient of approximately 37.5 [$\,\rm km\,s^{-1}\,pc^{-1}$]{}. When we calculate the velocity gradient using our model results ($dv/dr = \Delta v \ n(H_2) / N(H_2)$), we find a 68% confidence lower limit of 16 [$\,\rm km\,s^{-1}\,pc^{-1}$]{}\] (4 [$\,\rm km\,s^{-1}\,pc^{-1}$]{} if we use the model with $^{13}CO$) , but the upper limit is unphysically high. The velocity required for turbulent heating seems reasonable in context of our likelihood results. @Panuzzo:2010 used their calculated velocity gradient of 35 [$\,\rm km\,s^{-1}\,pc^{-1}$]{} to determine that they could match the heating required with a sizescale of 0.3 to 1.6 pc. These velocity gradients seem large compared to Galactic star-forming sites [e.g. @Imara:2011], but M82 is known to have powerful stellar winds. According to @Beirao:2008, the starburst activity has decreased in the past few Myr, and this appears to be evidence of negative feedback (by stellar winds and supernovae), because M82 still has a large reservoir of gas available for star formation. Therefore, there is evidence for turbulent heating mechanisms being in place. Additionally, @Downes:1998 found high turbulent velocities (30-140 [$\,\rm km\,s^{-1}$]{}) in models of extreme star-forming galaxies.
None of the possibilities described seem to provide enough heating by themselves, with the exception of turbulent heating, which is based on a few approximations and assumptions. Likely, there is a combination of factors, namely PDRs and shocks/turbulent mechanisms. Such a situation has also been seen in other submillimeter-bright galaxies, discussed next. Interestingly, even a more quiescent galaxy like NGC 891 requires a combination of PDRs and shocks to explain mid-J CO transitions [[[$\rm J{\!=\!}6{\!\rightarrow\!}5$]{}]{}, [[$\rm J{\!=\!}7{\!\rightarrow\!}6$]{}]{}, @Nikola:2011].
Comparison to Other Starburst and Submillimeter Galaxies
--------------------------------------------------------
Because only the first few lines in the CO ladder are easily visible from the ground for nearby galaxies, the high-J lines detected by [*Herschel*]{} represent new territory. Therefore, while adequate diagnostics of high-J CO lines are still being developed, it is useful to compare to other submillimeter-bright galaxies.
Mrk 231 contains a luminous (Seyfert 1) AGN. It also shows a strong high-J CO ladder, such that only the emission up to [[$\rm J{\!=\!}8{\!\rightarrow\!}7$]{}]{} is explained by UV irradiation from star formation. Their high-J CO luminosity SLED however, is flat (though ours for M82 are stronger than predicted, they are still decreasing with higher-J). @vanderWerf:2010 can explain this trend with either an XDR or a dense PDR. An additional difference between M82 and Mrk 231 is that OH$^+$ and H$_2$O$^+$ are both seen in strong emission in Mrk 231 (instead of absorption), indicative of X-ray driven chemistry. Mrk 231 is also more face-on than M82.
The FTS spectrum of Arp 220 has many features not present in M82, such as strong HCN absorption, P-Cygni profiles of OH$^+$, H$_2$O$^+$ and H$_2$O, and evidence for an AGN [@Rangwala:2011]. CO modeling, similar to the procedure done in this work, also indicates that the high-J lines trace a warmer component than low-J lines ($\sim$ 1350 K). Mechanical energy likely plays a large role in the heating of this merger galaxy as well. Though M82 has an outflow, it is not detected in P-Cygni profiles of the aforementioned lines.
The redshift of HLSW-01 [@Conley:2011] allows the CO [[$\rm J{\!=\!}7{\!\rightarrow\!}6$]{}]{} to [[$\rm J{\!=\!}10{\!\rightarrow\!}9$]{}]{} lines to be observed from the ground, as has been done with Z-Spec [@Scott:2011]. Unlike M82 (and others), the known CO SLED from [[$\rm J{\!=\!}1{\!\rightarrow\!}0$]{}]{} and up can be described by a single component at 227 K ($1.2 \times 10^3$ cm$^{-3}$ density). If the velocity gradient is not constrained to be greater than or equal to that corresponding to virialized motion, the best fit solution becomes 566 K ($0.3 \times 10^3$ cm$^{-3}$ density), closer to our temperature. We chose to exclude this prior due to uncertainties in the calculation of velocity gradient related to M82’s turbulent morphology. HLSW-01 appears to be unique in that a cold gas component is not required to fit the lower-J lines of the SLED, though two-component models can find a best-fit solution with a cold component.
In summary, M82 (like Arp 220 and HLSW-01) does not have the high CO excitation dominated by an AGN as seen in Mrk 231. Therefore, in addition to distinguishing between PDRs and shocks, high-J CO lines may also be used to indicate XDRs.
Conclusions {#sec:concl}
===========
We have presented a multitude of molecular and atomic lines from M82 in the wavelength range (194-671 $\mu$m) accessible by the [*Herschel*]{}-SPIRE FTS (Table \[table:fit\]). After modeling $^{12}$CO, $^{13}$CO and \[C \], we find support for the high-temperature molecular gas component presented in the results of @Panuzzo:2010. The temperature traced by the warm component of $^{12}$CO is quite high (335-518 K), and the addition of $^{13}$CO slightly expands the likelihood ranges. The addition of \[C \] produced results that indicate that these atom is not entirely tracing the same region as $^{12}$CO. Some of the emission from these molecules (especially \[C \]) are likely tracing more diffuse gas less shielded from UV radiation.
The mapping observations did not resolve any significant gradients in physical parameters (except evidence for a slight drop-off in beam-averaged column density, consistent with the beam profile) indicating that the single “deep" spectrum is an adequate representation of the galaxy when limited by our beam size. However, the mapping observations were important in confirming the source-beam coupling factor utilized in @Panuzzo:2010 and here, because through convolution of the maps we were able to confirm the central pixel’s results matched with the deep spectrum.
Molecular absorption traces lower column regions of the disk than those traced by CO emission, but contribute to the interpretation of the molecular gas of M82 being excited by a combination of sources. Despite the enhanced cosmic ray density in M82, we do not find evidence that cosmic rays alone are sufficient to heat the gas enough to match the modeled hydrogen cooling rate. PDR models can only replicate the high-J CO line emission at high densities incompatible with those indicated by the likelihood analysis, though cosmic-ray enhanced PDRs may be a closer match at lower densities. Turbulent heating from stellar winds and supernovae likely play a large role in the heating. More specifically, shocks are required to explain bright high-J line emission [@Pon:2012].
Like other submillimeter bright galaxies, [*Herschel*]{} has opened up new opportunities and questions about molecular and atomic lines that have never been observed before. Because of this, the diagnostic power of high-J CO lines is still in development, and newer models currently being developed may be able to explain the emission seen from M82 and other extreme environments.
#### Acknowledgments
SPIRE has been developed by a consortium of institutes led by Cardiff Univ. (UK) and including: Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA). J.K. also acknowledges the funding sources from the NSF GRFP. The research of C.D.W. is supported by grants from the Natural Sciences and Engineering Research Council of Canada. Thank you to the anonymous referee for comments which significantly improved this work.
[^1]: http://ccp7.dur.ac.uk/cooling\_by\_h2/
|
---
author:
- |
Gabriel Parra\
Department of Mathematical Engineering\
Universidad de Chile\
`[email protected]`\
Felipe Tobar\
Center for Mathematical Modeling\
Universidad de Chile\
`[email protected]`\
bibliography:
- 'references.bib'
title: |
Spectral Mixture Kernels for\
Multi-Output Gaussian Processes
---
|
---
abstract: 'We theoretically study the electronic structure of small-angle twisted bilayer graphene with a large potential asymmetry between the top and bottom layers. We show that the emergent topological channels known to appear on the triangular AB-BA domain boundary do not actually form a percolating network, but instead they provide independent, perfect one-dimensional eigenmodes propagating in three different directions. Using the continuum-model Hamiltonian, we demonstrate that an applied bias causes two well-defined energy windows which contain sparsely distributed one-dimensional eigenmodes. The origin of these energy windows can be understood using a two-band model of the intersecting electron and hole bands of single layer graphene. We also use the tight-binding model to implement the lattice deformations in twisted bilayer graphene, and discuss the effect of lattice relaxation on the one-dimensional eigenmodes.'
author:
- Bonnie Tsim
- 'Nguyen N. T. Nam'
- Mikito Koshino
bibliography:
- '1d\_channel.bib'
title: 'Perfect one-dimensional chiral states in biased twisted bilayer graphene\'
---
Introduction
============
Twisted bilayer graphene (TBG) consists of two layers of graphene overlaid on top of each other with a relative twist between their crystallographic axes. A moiré interference pattern which emerges from the overlap of the two mismatched graphene lattices results in a strong modification of the electronic structure by the superlattice band folding [@lopes2007graphene; @mele2010commensuration; @trambly2010localization; @shallcross2010electronic; @morell2010flat; @bistritzer2011moirepnas; @kindermann2011local; @PhysRevB.86.155449; @moon2012energy; @de2012numerical]. The system has been shown to exhibit many interesting physical phenomena and, since the realisation of superconductivity and correlated insulating states in magic-angle TBG [@cao2018unconventional; @cao2018mott; @yankowitz2019tuning], there has been a huge surge of theoretical and experimental research in this field.
In this paper, we theoretically study the electronic structure of small-angle TBG with a large interlayer bias (i.e., potential asymmetry between the top and bottom layers), and demonstrate the formation of perfect one-dimensional (1D) states within well-defined energy windows on either side of zero energy. The effect of the interlayer bias on TBG was investigated in the previous theoretical works [@xian2011effects; @san2013helical; @moon2014optical; @ramires2018electrically; @efimkin2018helical; @fleischmann2019perfect; @walet2019emergence; @hou2019current], and it was found that a large enough bias gives rise to a network of topological channels on the domain boundaries between AB and BA stacking regions [@san2013helical; @ramires2018electrically; @efimkin2018helical; @fleischmann2019perfect; @walet2019emergence; @hou2019current]. There the electronic states at AB and BA regions are locally gapped out by the interlayer bias [@mccann2006asymmetry], and two topological modes per spin and per valley necessarily appear on each AB-BA boundary [@vaezi2013topological; @zhang2013valley; @pelc2015topologically; @ju2015topological; @yin2016direct; @lee2016zero; @li2016gate], reflecting that the two regions have different quantized values of single-valley Hall conductivity, $\pm e^2/h$. [@koshino2008electron]. In TBG, the AB and BA regions appear periodically in a hexagonal pattern [@brown2012twinning; @lin2013ac; @alden2013strain] such that the boundary channels form a triangular grid as illustrated in Fig. \[fig\_schem\_domain\](a). Recently, the network of the topological channels in TBG was experimentally probed by transport measurements [@rickhaus2018transport; @yoo2019atomic; @giant2019oscillations] and also by scanning tunneling spectroscopy [@huang2018topologically].
One may expect that the electronic transport in the topological channels of TBG could be described by a percolation model [@chalker1988percolation] on a triangular network. However, here we show that the topological modes do not form a two-dimensional network, but they are actually independent 1D eigenmodes composed of a serial connection of topological channel sections as shown in Fig. \[fig\_schem\_domain\](b). The modes along different directions are never hybridized, and therefore all these states serve as independent perfect 1D channels over the whole TBG. The result is consistent with a recent work which predicts the perfect nesting of the Fermi surface in the biased TBG [@fleischmann2019perfect].
In the following, we calculate the electronic band structure of the TBG using the continuum-model Hamiltonian and present the band structures for various twist angles and electric field dependencies. The energy band structures show that an applied bias causes two well-defined energy windows which contain sparsely distributed perfect 1D eigenmodes, separated by a cluster of nearly flat bands around the charge neutrality point. We also use the tight-binding model to implement arbitrary lattice deformations in TBG, and discuss the effect of lattice relaxation on the 1D eigenmodes. Lastly, we explain the origin of these energy windows by a perturbational approach from the small interlayer coupling limit, and also by a two-band model consisting of the intersecting electron and hole bands of single layer graphene. The tunability of the TBG energy dispersion in a perpendicular electric field means there is the potential to explore the parameter space where these 1D eigenmodes can be found in its experimental realization.
This paper is organized as follows: In Sec. II, we introduce the continuum-model Hamiltonian and describe the formation of perfect 1D eigenmodes for varying angles and biases. In Sec. III, we consider the effect of lattice relaxation on the 1D eigenmodes. Lastly, we explain the origin of 1D eigenmodes in Sec. IV, and present a brief conclusion in Sec. V.
Continuum Model
===============
We calculate the electronic band structure of the twisted bilayer graphene using the continuum model [@lopes2007graphene; @bistritzer2011moirepnas; @kindermann2011local; @PhysRevB.86.155449; @moon2013opticalabsorption; @koshino2015interlayer; @koshino2015electronic]. For a small twist angle, the Hamiltonian is given by $$\begin{aligned}
H_{\text{TBG}} =
\begin{pmatrix}
H_1 & U^\dagger \\
U & H_2
\end{pmatrix},
\label{eq_H_eff}\end{aligned}$$ where $$\begin{aligned}
& H_1 = \begin{pmatrix}
\frac{\Delta}{2} & -\upsilon\pi^{\dagger} \\
-\upsilon \pi & \frac{\Delta}{2}
\end{pmatrix},
\quad
H_2 = \begin{pmatrix}
- \frac{\Delta}{2} & -\upsilon\pi^{\dagger} \\
-\upsilon \pi & - \frac{\Delta}{2}
\end{pmatrix}
\label{eq_intra}\end{aligned}$$ and $$\begin{aligned}
U &=
u\displaystyle\sum_{j=0,1,2} e^{{{i\mkern1mu}}\Delta \mathbf{K}_j \mathbf{\cdot r}}
\begin{pmatrix}
1 & e^{-{{i\mkern1mu}}\frac{2\pi}{3}j} \\
e^{{{i\mkern1mu}}\frac{2\pi}{3}j} & 1
\end{pmatrix}.
\label{eq_interlayer_matrix}\end{aligned}$$ The Hamiltonian Eq. (\[eq\_H\_eff\]) is equivalent to the continuum-model Hamiltonian derived in [@lopes2007graphene; @bistritzer2011moirepnas; @moon2013opticalabsorption] up to a gauge transformation.[@tarnopolsky2019origin; @liu2019pseudo] The on-diagonal blocks describe the graphene layers 1 and 2 where $\pi= \hbar(\xi k_x + \text{i} k_y) $, and the valley index $\xi=\pm 1$ is used to distinguish between $K$ and $K'$ valleys. The parameter $\upsilon$ is the band velocity of monolayer graphene where $\hbar\upsilon/a = 2.1354$ eV (the lattice constant of graphene is given by $a=2.46$ ) [@moon2013opticalabsorption; @koshino2018maximally], and $\Delta$ represents the electrostatic energy shift induced by the perpendicular electric field. The off-diagonal blocks describe the moiré interlayer coupling between the two twisted layers, where the interlayer coupling strength is given by $u=0.103$ eV. The vectors $\Delta \mathbf{K}_j\,(j=0,1,2)$ account for the shift between the original Brillouin zone corners of the two layers, and are given by $$\Delta \mathbf{K}_{j} = \frac{4 \pi\theta }{3a} \Big[ -\sin\left(\frac{2\pi j}{3}\right), \cos\left(\frac{2\pi j}{3}\right) \Big],$$ where $\theta$ is the twist angle between the two layers in radians.
We calculate the energy spectrum for the $K$ and $K'$ valleys independently as intervalley coupling is negligible at small twist angles. Zone folding is used to bring the states in each valley with momenta connected by the moiré reciprocal lattice vectors, $\mathbf{G}_1 = \Delta\mathbf{K}_1 - \Delta\mathbf{K}_0$ and $\mathbf{G}_2 = \Delta\mathbf{K}_2 - \Delta\mathbf{K}_0$. The basis of $k$-states of layer 1 and 2 can be taken as $$\begin{aligned}
&\Vec{k} ^{(1)}_{m_1,m_2} = \Vec{k} + \Delta \mathbf{K}_0 + m_1 \mathbf{G}_1 + m_2 \mathbf{G}_2 \nonumber\\
&\Vec{k} ^{(2)}_{m_1,m_2} = \Vec{k} - \Delta \mathbf{K}_0 + m_1 \mathbf{G}_1 + m_2 \mathbf{G}_2,
\label{eq_k_vectors}\end{aligned}$$ respectively, where $\Vec{k}$ is the wavenumber in the first moiré Brillouin zone (mBZ) spanned by $\mathbf{G}_1$ and $\mathbf{G}_2$, and $m_1$ and $m_2$ are integers. The size of the basis is chosen such that when the Hamiltonian is numerically diagonalized, the energy bands converge up to a cut-off energy.
Figure \[fig\_band\](a) presents the electric field dependence of the TBG band structure for various twist angles, $\theta = 1\degree, 0.5\degree, 0.3\degree$ and $0.2\degree$. The band structures include energy bands from both the $K$ (black) and $K'$ (red) valleys and is shown for the path $\kappa \rightarrow \gamma \rightarrow \mu \rightarrow\kappa'$ in the mBZ illustrated in Fig. \[fig\_band\](b). The original Dirac point of layer 1 is placed at the corner of mBZ at $\kappa'$, and the original Dirac point of layer 2 is placed at $\kappa$. In increasing $\Delta$, we see that the energy bands are gradually shifted toward zero energy, forming a cluster of nearly flat bands around the charge neutrality point. At the same time, two well-defined energy windows, where energy bands are only sparsely distributed, are formed above and below the zero-energy band cluster. The size of the energy windows are not strongly affected by the size of $\Delta$ which can be seen for $\theta = 0.2\degree$ in increasing $\Delta$. Most interestingly, is the formation of 1D propagating modes inside the energy windows, which connect the zero-energy band cluster to the bulk bands outside of the energy windows. Figure \[fig\_band\](c) shows a three dimensional plot of the bands from $K$ valley calculated for $\theta = 0.5 \degree$ and $\Delta = 400$ meV, and Fig. \[fig\_band\](d) is the Fermi surface of the same system at $E_F= 50$ meV where black and red lines represent $K$ and $K'$, respectively. We see that the band dispersion of $K$ is actually composed of three intersecting planes, with band velocities parallel to $(0,-1)$, $(\sqrt{3}/2,1/2)$ and $(-\sqrt{3}/2,1/2)$ directions. The different planes are not hybridized with each other, giving nearly straight Fermi lines at the fixed energy. Such straight Fermi surfaces were also reported in the recent paper [@fleischmann2019perfect]. In the largest bias $\Delta=400$ meV, we notice some flat levels appear in the upper part of the energy window independently from the dispersive 1D states, (e.g., three horizontal lines in 50 meV $< |E| <$ 100 meV for $\theta = 0.3^\circ$), which can be interpreted as pseudo-Landau levels of the fictitious gauge field [@ramires2018electrically].
Effect of lattice relaxation
============================
The real TBG is not a simple stack of rigid graphene layers as assumed in the previous section, but it has a spontaneous lattice relaxation and resulting AB/BA domain formation [@popov2011commensurate; @brown2012twinning; @lin2013ac; @alden2013strain; @uchida2014atomic; @van2015relaxation; @dai2016twisted; @jain2016structure; @nam2017lattice; @carr2018relaxation; @lin2018shear; @yoo2019atomic; @guinea2019continuum]. Such a structural deformation modifies the electronic band structure [@nam2017lattice; @lin2018shear; @koshino2018maximally; @yoo2019atomic; @guinea2019continuum; @crucial2019lucignano; @fleischmann2019perfect; @walet2019emergence]. Here we calculate the energy band structures in the presence of the lattice strain using the tight-binding method [@nam2017lattice]. The Hamiltonian is given by $$H = -\sum_{i,j} t(\textbf{R}_i -\textbf{R}_j) |\textbf{R}_i \rangle \langle \textbf{R}_j| + \text{h.c.}$$ where $\textbf{R}_i$ is the atomic coordinate, $ |\textbf{R}_i \rangle $ is the wave function at site $i$, and $t(\textbf{R}_i -\textbf{R}_j)$ is the transfer integral between atom $i$ and $j$. We adopt the Slater-Koster type formula for the transfer integral [@slater1954simplified], $$-t (\textbf{d}) = V_{pp\pi} (d) \left[1 - \left(\frac{\textbf{d} \cdot \textbf{e}_z}{d}\right)^2 \right] +
V_{pp\sigma} (d) \left(\frac{\textbf{d} \cdot \textbf{e}_z}{d}\right)^2 ,
\label{eq_t}$$ $$\begin{aligned}
& V_{pp\pi} (d) = V_{pp\pi}^0 \exp \left(- \frac{d-a_0}{r_0} \right), \\
& V_{pp\sigma} (d) =V_{pp\sigma}^0 \exp \left(- \frac{d-d_0}{r_0} \right),\end{aligned}$$ where $\textbf{d} = \textbf{R}_i - \textbf{R}_j$ is the distance between two atoms and $\textbf{e}_z$ is the unit vector on $z$ axis. $V_{pp\pi}^0 \approx -2.7$ eV is the transfer integral between nearest-neighbor atoms of monolayer graphene which are located at distance $a_0 = a/\sqrt{3} \approx 0.142$ nm, $V_{pp\sigma}^0 \approx 0.48 $ eV is the transfer integral between two nearest-vertically aligned atoms and $d_0 \approx 0.334$nm is the interlayer spacing. The decay length $r_0$ of transfer integral is chosen as $0.184 a$ so that the next nearest intralayer coupling becomes $0.1 V_{pp\pi}^0$. At $d > \sqrt{3}a$, the transfer integral is very small and negligible. The optimized atomic positions are obtained by the method introduced in the previous work [@nam2017lattice]. Using this, we construct the tight-binding Hamiltonian of the relaxed TBG and calculate the energy bands.
Figure \[fig\_band\_relax\] compares the electronic band structure of non-relaxed (upper panels) and relaxed (lower panels) TBGs in $\theta =0.3^\circ$ and different $\Delta$’s. In the tight-binding model, the valleys are not distinguished. We see that the energy bands of the non-relaxed calculation quantitatively agree with those in the continuum method in Fig. \[fig\_band\]. In the presence of the relaxation, we confirm that the qualitative feature remains the same: we still see the energy windows and the perfect 1D eigenmodes. The major difference from the non-relaxed state is that the central pseudo-Landau levels mentioned in the previous section are completely hybridized with 1D eigenmodes, and become a part of the dispersive bands. Also we notice that the bands in the zero-energy cluster become less flat and a bit more dispersive.
Figure \[fig\_wave\] shows typical wave functions in the energy window in the non-relaxed TBG and the relaxed TBG of $\theta = 0.55^\circ$ and $\Delta = 400$ meV. Here we chose the corresponding states in non-relaxed and relaxed cases, which are connected by a continuous increase of the relaxation. The state is chosen from a 1D band of the electron side with the velocity along $k_y$ axis ($\kappa-\gamma$ direction in this figure). In each case, we observe that the wave function takes a 1D form extending along $y$ direction, while it is disconnected in the perpendicular direction. The states in the different Fermi surface branches at the same energy are obtained by $\pm 120^\circ$ rotation of this figure. We confirm that the local current density is along $-y$ direction, in accordance with the negative band velocity in the $k_y$ direction. The wave amplitude is mainly concentrated on the layer 1, while it is concentrated on layer 2 in the hole side states. In the presence of the lattice relaxation, we see that the wave function becomes more localized on the AB-BA boundary. This is natural because the AB and BA regions (where the energy band is gapped out) significantly expand under the lattice relaxation, and the wave amplitude must be confined to the narrow boundary region.[@fleischmann2019perfect; @walet2019emergence] The similar, zigzag-shaped wave function was also reported in a recent work [@fleischmann2019perfect]. Interestingly, we see that the relaxed TBG’s wave function in Fig. \[fig\_wave\] has its amplitude not on the boundary along $y$, but only on the boundary in the two other directions along $(\sqrt{3}/2, -1/2)$ and $(-\sqrt{3}/2, -1/2)$. With closer inspection, we also find that it has different structures between the boundaries along $(\sqrt{3}/2, -1/2)$ and $(-\sqrt{3}/2, -1/2)$, although the atomic structures are completely symmetric. Recalling that each single boundary has two different traveling modes (denoted as mode 1 and 2) as mentioned, this result suggests that, at every vertex of the triangular grid (AA region), mode 1 is always scattered to mode 2 in $-120^\circ$ direction, while mode 2 is always scattered to mode 1 in $+120^\circ$ direction. As a result, we have three independent, zigzag traveling modes as illustrated in Fig. \[fig\_schem\_domain\](b).
Origin of the perfect 1D eigenmodes
===================================
The origin of the energy window and the 1D eigenmodes can be intuitively understood by a perturbational approach from the small interlayer coupling limit. Figure \[fig\_band\_u\] shows the band structure of the continuum model in Eq. (\[eq\_H\_eff\]), with $\theta = 0.3 \degree$ and $\Delta=400$ meV, and with increasing interlayer coupling $u$ from zero to the actual value in TBG. With small $u$, we see that two gaps open in the electron side and the hole side, and they eventually become the window regions in the full $u$ parameter. We see that the 1D eigenmodes always remain inside the gap, preventing the spectrum from becoming fully gapped. The width of the energy window is obviously the order of $u$. The energy bands between the two gaps are squashed with increasing $u$, and finally becomes the zero-energy band cluster.
The opening of the two gaps in small $u$ can be explained by considering the following two-band model. In a large $\Delta$, the low energy region is dominated by the hole band of graphene layer 1 and the electron band of layer 2. While considering the interlayer coupling $U$, we can imagine that the two opposite conical bands of single layer graphenes are crossing with each other with a relative momentum shift of $\Delta \Vec{K}_j (j=0,1,2)$, and the band anti-crossing occurs at the cross section. Figure \[fig\_FS\](a) illustrates the actual crossing lines between the Dirac cones in the case of $\theta = 0.3 \degree$ and $\Delta=200$ meV, where three circles (red, blue and green) correspond to $j=0,1,2$.
The size of the gap is roughly proportional to the matrix element of $U$ between the two states on the crossing line. The graphene’s eigenstates are written in the $(A,B)$ spinor representation as $$|\Vec{k}, s \rangle
= \frac{1}{\sqrt{2}}
\begin{pmatrix}
1 \\
-s e^{i \theta(\Vec{k})}
\end{pmatrix},$$ where $s=\pm$ represent the conduction and valence bands, respectively, and $\theta(\Vec{k}) = {\rm arctan} (k_y/k_x)$. Now, the matrix element of $U$ from graphene 1 to graphene 2 is $$\begin{aligned}
\langle \Vec{k}+\Delta\Vec{K}_j, + | U |\Vec{k}, - \rangle \approx i u \sin \Big[\theta(\Vec{k}) - \frac{2\pi j}{3}\Big],
\label{eq_mat_approx}\end{aligned}$$ where $|\Delta\Vec{K}_j| \ll |\Vec{k}|$ is assumed. In Fig. \[fig\_FS\](b), the thickness of the crossing lines represents the amplitude of the interlayer matrix element on the Dirac cones of layer 1 and 2, respectively. We see the matrix element vanishes near $E=0$, and this is the reason why the two major gaps open above and below $E=0$. In a small $u$ limit, the number of states (per area) sandwiched by the two gaps is given by $2 n_W$ where $$n_W = g_vg_s \frac{\Delta}{4\pi \hbar v} \Delta K,$$ where $g_v=g_s=2$ are the spin and valley degeneracies, and $\Delta K = |\Delta\Vec{K}_j|=4\pi\theta/(3a)$. The $n_W$ characterizes the typical carrier density to reach the energy window of the 1D eigenmodes. For $\theta = 0.3^\circ$ and $\Delta =200$ meV, for instance, we have $n_W = 1.08\times 10^{12}$cm$^{-2}$.
The 1D eigenmodes remaining inside the energy window can be explained by the reconstruction of the Fermi surface. Figures \[fig\_FS\](c) and (d) illustrate the Fermi surfaces before introducing $u$ at $E_F=20$ meV, which is slightly below the maximum energy of the crossing rings. In panel (c), the central dashed circle represents the hole band of layer 1, and the three solid circles are the electron band of layer 2 with three momentum shifts $\Delta \Vec{K}_j \, (j=0,1,2)$. In panel (d), the Fermi surface of layer 1 is centered instead. The hybridized Fermi surfaces after the infinitesimal anti-crossing are shown in Fig. \[fig\_FS\](e). We therefore have three open Fermi surfaces that are 120 degrees apart as well as three closed pockets. By increasing $u$, the closed pockets are gapped out due to a good nesting between the electron and hole parts. On the other hand, the open Fermi surfaces remain ungapped, which explains the origin of the 1D eigenmodes filling the gap. We also see that the open Fermi surface mainly consists of the layer 1 component (solid line) which is consistent with the fact that the wave function has larger amplitude on layer 1 in Fig. \[fig\_wave\].
In this picture, we only consider the band crossing of the first order in $u$, while we actually have high-order hybridization at other crossing points. It is somewhat surprising that the 1D eigenmodes in three directions are not hybridized and remain independent even in a large $u$ beyond the pertubational regime. This is understood by the $k$-space map of the interlayer matrix element in Fig. \[fig\_network\], where open circles represent the graphene 1’s hole states at $\Vec{k} ^{(1)}_{m_1,m_2}$, filled circles the graphene 2’s electron states at $\Vec{k} ^{(2)}_{m_1,m_2}$ , and the bond thickness is proportional to the matrix element of $U$ between the two states. We can show that the 1D eigenmodes in the positive energy window are contributed by graphene’s states only in the regions I, III and V, and those in the negative energy window are by the regions II, IV and VI. This is consistent with the observation that the open Fermi surface in Fig. \[fig\_FS\](e) consists of the graphene’s Fermi surface portions in the same regions. We see that the matrix element nearly vanishes on the boundary of different regions (dashed lines) according to Eq. (\[eq\_mat\_approx\]), except for the $k$-points near the origin which do not contribute to the low-energy states. As a result, the six regions I, II, $\cdots$ VI are nearly decoupled and that is why the 1D eigenmodes running in the different directions remain independent in increasing $u$.
The perfect 1D eigenmodes in the biased TBG is analogous to those in zigzag graphene nanoribbons [@wakabayashi2007perfectly]. In a doped zigzag nanoribbon, it is known that each valley has different numbers of left-moving modes and right-moving modes at the Fermi energy; $n$ right modes and $n+1$ left modes at one valley, while $n+1$ right modes and $n$ left modes at the other valley. The excess traveling mode in each valley remains as a perfectly conducting channel even in the disordered system, as long as the impurities are long-ranged and do not mix the different valleys. The perfect 1D eigenmodes in biased TBG can be viewed as a 2D version of this, in that each single sector of I, III and V has different numbers of out-going modes and in-coming modes (with respect to the graphene’s Fermi circle), as is clear from different arc lengths of the electron and hole Fermi surfaces in Fig. \[fig\_FS\](e), and that different sectors are not hybridized by the interlayer coupling $u$. Therefore the excess modes originate from the electron Fermi surface and they remain as traveling modes in the presence of $u$.
Discussion
==========
The disorder effect on these 1D eigenmodes is an important problem when considering the electronic transport. As shown in Fig. \[fig\_schem\_domain\](b), each 1D mode is composed of straight parts on the AB-BA boundary and corner angles on AA spots. A hybridization of different 1D eigenmodes takes place only by a local mixing of the mode 1 and 2 on the AB-BA boundaries, or an irregular reflection at the AA corners. In real TBGs, the moiré structure exhibits a distorted triangular pattern with shifted AA spots and extended / shortened AB-BA boundaries [@brown2012twinning; @lin2013ac; @alden2013strain], However, we expect that such a moiré-scale distortion would not cause a strong mixing of different 1D eigenmodes, because the local atomic structures of AA and AB-BA boundary are not modified very much [@koshino2019moire], such that the hubs and the links in the triangular lattice work in the same way as in the non-distorted system. Major scatterings should be mainly caused by short-ranged disorder smaller than the local structures of AB-BA boundary and AA spots (which is about a few nm). The detailed study on the disorder scattering will be left for future works.
When the scattering can be neglected and the Fermi energy is in the energy window, the electronic transport must be dominated by the ballistic transport through the 1D eigenmodes. It is also expected that we do not have the Aharanov-Bohm (AB) oscillation in magnetic fields, because the 1D eigenmodes do not enclose triangular domains, so do not cause any interferences. Recently, transport measurements have been performed on small-angle TBGs under interlayer bias, and a significant AB oscillation was observed [@rickhaus2018transport; @yoo2019atomic; @giant2019oscillations]. We expect that magnetic oscillations take place when the perfect 1D eigenmodes are not well formed, e.g., because the bias is not enough or the Fermi energy is not in the corresponding region. To have the perfect 1D eigenmodes, it is required that the energy window $(|E_F|\lsim u)$ is dominated by the hole band of a single layer and the electron band of the other layer, and this gives a condition $\Delta/2 \gsim u$ i.e., $\Delta \gsim 200$ meV.
Conclusion
==========
We used the continuum-model and tight-binding Hamiltonians to show that TBGs with an applied bias exhibit perfect 1D eigenmodes in well-defined energy windows on either side of zero energy. We found that these states never hybridise and they propagate independently in three different directions along the domain walls separating AB and BA regions. In the presence of arbitrary lattice deformations, we show that the wave functions become even more localized on the domain boundaries. The formation of the well-defined energy windows and the origin of these states is explained by the two-band model consisting of the intersecting electron and hole bands of single layer graphene, where the 1D eigenmodes correspond to the emergent open Fermi surfaces formed by the moiré interlayer hybridization.
Acknowledgments
===============
BT acknowledges financial support from the Japan Society for the Promotion of Science as a JSPS International Research Fellow (Summer Programme 2019), EPSRC Doctoral Training Centre Graphene NOWNANO EP/L01548X/1 and the Lloyd’s Register Foundation Nanotechnology grant. MK acknowledges the financial support of JSPS KAKENHI Grant Number JP17K05496.
|
---
author:
- |
Rikkert Frederix\
Institut für Theoretische Physik, Universität Zürich, Winterthurerstrasse 190,\
CH-8057 Zürich, Switzerland
- |
Stefano Frixione [^1]\
PH Department, TH Unit, CERN, CH-1211 Geneva 23, Switzerland\
ITPP, EPFL, CH-1015 Lausanne, Switzerland
- |
Valentin Hirschi\
ITPP, EPFL, CH-1015 Lausanne, Switzerland
- |
Fabio Maltoni\
Centre for Cosmology, Particle Physics and Phenomenology (CP3)\
Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium
- |
Roberto Pittau\
Departamento de Física Teórica y del Cosmos y CAFPE, Universidad de Granada
- |
Paolo Torrielli\
ITPP, EPFL, CH-1015 Lausanne, Switzerland
bibliography:
- 'w2j.bib'
title: 'aMC@NLO predictions for $Wjj$ production at the Tevatron'
---
Introduction
============
Recently, CDF has reported [@Aaltonen:2011mk] an excess of events in two-jet production in association with a $W$ boson, in the form of a broad peak centered at $M_{jj}=144$ GeV in the dijet invariant mass. By now, i.e. with a data set corresponding to an integrated luminosity of 7.3 fb$^{-1}$, the excess has reached a statistical significance of 4.1$~\sigma$ w.r.t. the estimated Standard Model yield. In view of the possible implications for a BSM physics discovery, this anomaly has attracted a lot of attention, though it has so far failed to be confirmed by a very similar D0 analysis [@Abazov:2011af].
One of the major challenges in a measurement of this kind is posed by the need of reliable predictions and simulations of the processes that contribute to the observables of interest. In the CDF and D0 analyses, for instance, such simulations are typically performed by means of fully exclusive Monte Carlo programs based on tree-level matrix elements. In the case of multi-jet final states in association with weak bosons, a proper merging procedure [@Catani:2001cc; @Krauss:2002up; @Alwall:2007fs] between multi-parton matrix elements (which give a reliable description of large-angle and large-energy emissions) and parton shower Monte Carlo’s (PSMC’s) (which give a reliable description of small-angle or small-energy emissions) is employed that allows the generation of inclusive jet samples for all relevant multiplicities, accurate to the leading order (LO) in perturbative QCD.
Yet, the uncertainties that affect LO predictions can be very large for rates, and smaller but still discernible for differential distributions. This is the reason why parton-level NLO and, when possible, NNLO computations of infrared safe observables are used. Alternatively, and if the statistics is sufficient, control data samples are employed. For example, a theoretical analysis based on the NLO computation of the SM yield for $\ell + 2$ jets + missing transverse energy (which with the cuts used by CDF and D0 gets contributions from, in order of importance, $Wjj$, $Zjj$, $WW$, $t\bar t$, single-$t$ and $WZ$ production) has recently appeared [@Campbell:2011gp]. It has been shown that indeed the $Wjj$ process gives by far the dominant contribution, and that the NLO QCD corrections are small. Unfortunately, even though more accurate from the theoretical point of view, such small-multiplicity, parton-level calculations cannot be directly compared to experimental analyses, since this would require events with high-multiplicity, fully-fledged hadronic final states.
In order to obtain predictions that are both accurate and employable in experimental analyses, an NLO calculation needs to be consistently matched to a PSMC. This can be currently achieved with the MC@NLO [@Frixione:2002ik] or POWHEG methods [@Nason:2004rx; @Frixione:2007vw]. It is interesting to note that out of the processes listed above for the signature $\ell + 2$ jets + missing transverse energy, only the $Wjj$ and $Zjj$ contributions are not available in either of these frameworks. Given that the cross section of the latter process (within the experimental cuts adopted by CDF and D0) is smaller than that of the former by more than one order of magnitude, it is more urgent and highly desirable to have the best possible theoretical predictions for $Wjj$ production, which is a fairly challenging task. The complexity stems not only from the NLO computation itself, but also from its subsequent matching with parton showers, where the technical difficulties arise mainly from the presence of phase-space singularities at the Born level, which need to be cut-off. While this problem has already been faced in the POWHEG implementation of dijet [@Alioli:2010xa] and $Wj/Zj$ [@Alioli:2010qp] production, it is significantly simpler in these cases: a ${p_{{\scriptscriptstyle}T}}$ cut on the “recoil” system (one parton in dijet, and the vector boson in $Wj/Zj$ production) is sufficient to get rid of the divergences of the Born matrix elements. On the other hand, $Wjj$ production features a final-state three-body (of which two light partons) kinematic configuration already at the Born level, which renders the cutting-off of the singularities highly non trivial. In fact, the kinematics of $Wjj$ production is sufficiently involved to provide a proof that, if a successful matching of the NLO results with parton showers can be achieved, the same kind of matching technique can be applied to larger final-state multiplicities, without encountering any new problems of principle.
In this paper, we compute the NLO QCD corrections to the process $p\bar{p}\to\ell \nu jj$[^2] and, for the first time and in a fully-automated way, consistently match them to the [[HERWIG]{}]{} parton shower [@Marchesini:1991ch; @Corcella:2000bw; @Corcella:2002jc] according to the MC@NLO formalism [@Frixione:2002ik], as implemented in the [a[MC@NLO]{}]{} program [@amcatnlo]. One-loop corrections are obtained with [[MadLoop]{}]{} [@Hirschi:2011pa], which is based on the OPP reduction method [@Ossola:2006us] and on its implementation in CutTools [@Ossola:2007ax]. All the other contributions to the parton-level NLO cross section are dealt with by [[MadFKS]{}]{} [@Frederix:2009yq], which is based on the FKS subtraction method [@Frixione:1995ms], and takes care of determining the MC counterterms needed in the MC@NLO approach. Throughout the paper, we often refer to “the $W$ boson” or to “$Wjj$ production”; this is only for the sake of brevity, since we actually deal with the leptonic process mentioned before, and thus doing we fully retain the information on production and decay spin correlations and off-shell effects.
We begin by showing that the cutting-off of Born-level singularities (which is an arbitrary procedure) has no impact on the predictions in the kinematic regions of interest. We also show that NLO corrections are moderate, and depend mildly on the kinematics. We conclude by presenting our predictions for the dijet invariant mass, closely following the CDF analysis.
Method and validation
=====================
The $Wjj$ NLO cross section receives contributions from processes with $W\!+\!2$-parton and $W\!+\!3$-parton final states; although these diverge when independently integrated over the phase space, their combination into any infrared-safe observable is finite, thanks to the KLN and factorization theorems. For this to happen, it is a crucial condition that there be two observable jets in the final state. Although such a condition can be easily included in the definition of the short-distance cross sections, this is not the way one follows nowadays. A much-preferred option (and the only one which is viable when matching to PSMC’s) is that of imposing jet cuts at the very last step of the computation (the physics analysis), since this gives one the flexibility of e.g. using several jet-finding algorithms in parallel. It should be clear, however, that some cuts (called generation cuts henceforth) must still be imposed at the level of short-distance cross sections, which otherwise would diverge upon integration, as mentioned before. Generation cuts are therefore a technical trick that allow one to work with finite quantities; the idea is that kinematic configurations that do not pass these cuts would anyhow not contribute to the observable cross sections, which is what permits one to discard them; in other words, cross sections are not biased by generation cuts.
Unfortunately, it is not straightforward to prove that indeed physical observables are unbiased, which constitutes a necessary and very strong consistency check of one’s computation. An analytic proof not being viable, one exploits the fact that generation cuts are arbitrary. Hence, one imposes several generation cuts, and then verifies that in the kinematic regions of interest physical observables do not depend on them. This opens the question of how to define generation cuts, and it is obvious that a necessary condition is that they must be looser than the loosest of the set of cuts imposed in the physics analysis. When performing a perturbative calculation at the parton level, it is quite easy to understand whether generation cuts are sufficiently loose. This is because generation and analysis cuts are imposed on kinematic configurations that have the [*same*]{} multiplicities and particle contents. Things are significantly more complicated when one matches matrix-element computations with parton showers; the latter will in fact generally increase the final-state multiplicities w.r.t. those relevant to short-distance cross sections, and the relationship between the quantities being cut at the generation and analysis level becomes blurred. The upshot of this is the following: when considering the matching with parton showers, generation cuts are typically softer than those one would need if only performing perturbative parton-level computations, and they affect larger kinematic ranges than in the latter case.
In order to address this (among others) problem, at the LO one “merges” different parton multiplicities in a way consistent with parton showers [@Catani:2001cc; @Krauss:2002up; @Alwall:2007fs]. Although a generalization of these procedures to NLO is in its infancy [@Nagy:2005aa; @Giele:2007di; @Lavesson:2008ah; @Giele:2011cb; @Hamilton:2010wh; @Hoche:2010kg; @Alioli:2011nr], we may observe that when the merging at the LO is restricted to processes whose multiplicities differ by one unit, then one is actually dealing with (a subset of) the matrix elements used in the well-established NLO-PSMC matching procedures such as MC@NLO. Hence, one may anticipate that unphysical effects, the reduction of whose impact necessitates a merging procedure at the LO, are smaller in the context of matched NLO computations of a given multiplicity. We shall later see an explicit example of this fact.
To conclude this discussion, we mention that, although there is ample freedom in the choice of generation cuts, in practice it is convenient to employ the same jet-finding algorithm at the matrix element level as in the physics analysis, since this renders it a bit easier the task of applying generation cuts which are looser than the analysis ones.
As a technical aside, we point out that the MC@NLO formalism does not require modifications in order to be applied to processes whose Born contribution is divergent, and one simply imposes generation cuts when computing MC@NLO short-distance cross sections, fully analogously to what is done at the LO. Using the results of ref. [@Frixione:2002ik], it is easy to show [@amcatnlo] that this should be done in the following way. All contributions to ${{\mathbb S}}$ events, and the MC counterterms relevant to ${{\mathbb H}}$ events, are cut according to the corresponding Born configuration, while the real-emission contributions to ${{\mathbb H}}$ events are cut according to the corresponding fully-resolved configuration.
Our predictions are obtained with the electroweak parameters reported in table \[tab:params\]. For the (N)LO computations we use the MSTW(n)lo200868cl [@Martin:2009iq] PDFs, which also set the value of $\alpha_{{\scriptscriptstyle}S}(M_{{\scriptscriptstyle}Z})$. The renormalization and factorization scales are chosen equal to ${H_{{\scriptscriptstyle}T}}/2$, with . The sum here runs over all final-state QCD partons, and all the quantities that appear in the definition of ${H_{{\scriptscriptstyle}T}}$ are computed at the matrix-element level, i.e., before showering. We have not included the simulation of the underlying event in our predictions.
Parameter$~~~~$ value Parameter$~~~~$ value
--------------------------------------------------------- --------------------------- -------------------------------------------------------- -----------
$m_{W}$ 80.419 $\Gamma_W$ 2.0476
$G_F$ $1.16639\!\cdot\!10^{-5}$ $\alpha^{-1}$ 132.50698
$m_t$ 174.3 $m_{Z}$ 91.118
${\alpha_{{\scriptscriptstyle}S}}^{({\rm NLO})}(m_{Z})$ 0.12018 ${\alpha_{{\scriptscriptstyle}S}}^{({\rm LO})}(m_{Z})$ 0.13939
: \[tab:params\] Settings of physical parameters used in this work, with dimensionful quantities given in GeV.
We define jets by means of the anti-$k_T$ algorithm [@Cacciari:2008gp] with $R=0.4$, as implemented in FastJet [@Cacciari:2005hq]. Generation cuts are imposed by demanding the presence of at least two jets at the hard-subprocess level (hence, at this stage the inputs to the jet-finding algorithm are two- or three-parton configurations). All jets thus found are required to have either ${p_{{\scriptscriptstyle}T}}\!>\!5$ GeV or ${p_{{\scriptscriptstyle}T}}\!>\!10$ GeV. The short-distance cross sections defined with these cuts are used to obtain unweighted events as customary in MC@NLO. Such events are then showered by [[HERWIG]{}]{}, and the resulting hadronic final states are used to reconstruct about sixty observables (involving leptons, jets, lepton-jet, and jet-jet correlations) for each of the two generation ${p_{{\scriptscriptstyle}T}}$ cuts mentioned above. These observables are organized in three classes, each being associated with jets[^3] defined by imposing their transverse momenta to be larger than 10, 25, and 50 GeV; these conditions will be called analysis cuts henceforth. We finally check that the tighter the analysis cuts, the smaller the difference between the results obtained with the two generation cuts.
As an example of the outcome of this exercise, we present in fig. \[fig:all\] the transverse momentum of the hardest jet, the dijet invariant mass, and the $\Delta R$ separation between the two hardest jets. In the main frame of each plot there are six histograms: the three solid ones correspond to generation cuts ${p_{{\scriptscriptstyle}T}}=5$ GeV, while the three dashed ones correspond to generation cuts ${p_{{\scriptscriptstyle}T}}=10$ GeV. The upper (red), middle (blue), and lower (green) pairs of histograms are obtained with the analysis cuts ${p_{{\scriptscriptstyle}T}}\!=\!10$, $25$, and $50$ GeV respectively. The lower insets display three curves, obtained by taking the ratios of the ${p_{{\scriptscriptstyle}T}}\!=\!5$ GeV generation-cut results over the ${p_{{\scriptscriptstyle}T}}\!=\!10$ GeV generation-cut results, for the three given analysis cuts (in other words, these are the ratios of the solid over the dashed histograms). Fully-unbiased predictions are therefore equivalent to these ratios being equal to one in the kinematic regions of interest.
Inspection of fig. \[fig:all\], and of its analogues not shown here, allows us to conclude that the results follow the expected pattern: when one tightens the analysis cuts, the bias due to the generation cuts is reduced, and eventually disappears. Although all observables display this behaviour, the precise dependence on generation cuts is observable-specific; the three cases of fig. \[fig:all\] have been chosen since they are representative of different situations. The transverse momentum of the hardest jet shown in the upper plot of fig. \[fig:all\] is (one of) the very observable(s) on which generation cuts are imposed. Therefore, as one moves towards large ${p_{{\scriptscriptstyle}T}}$’s, one expects the bias due to generation cuts to decrease, regardless of values of the ${p_{{\scriptscriptstyle}T}}$ cut used at the analysis level. This is in fact what we see. Still, a residual dependence on generation cuts can be observed at relatively large ${p_{{\scriptscriptstyle}T}}$’s for looser analysis cuts; this could in fact be anticipated, since the events used here are $Wjj$ ones – hence, the next-to-hardest jet will tend to have a transverse momentum as close as possible to the analysis ${p_{{\scriptscriptstyle}T}}$ cut, and thus to the region affected by the generation bias in the case of looser analysis cuts. The dijet invariant mass, shown in the middle plot of fig. \[fig:all\], tells a slightly different story. Namely, the hard scale associated with this observable is not in one-to-one correspondence with that used for imposing the analysis cuts, at variance with the ${p_{{\scriptscriptstyle}T}}$ of the hardest jet discussed previously. Hence, the effects of the generation-level cuts are more evenly distributed across the whole kinematical range considered, as can be best seen from the lower inset. Essentially, the bias here amounts largely to a normalization mismatch, which disappears when tightening the analysis cuts. Finally, the $\Delta R$ distribution, presented in the lower part of fig. \[fig:all\], is representative of a case where both shapes and normalization are biased. There is a trend towards larger biases at large $\Delta R$, which is understandable since this region receives the most significant contributions from large-rapidity regions, where the transverse momenta tend to be relatively small and hence closer to the bias region.
We conclude this section with some further comments on validation exercises. Firstly, we started by testing the whole machinery in the simpler case of $Wj$ production. Although, as was discussed before, for this process generation cuts may be imposed on ${p_{{\scriptscriptstyle}T}}(W)$, we have chosen to require the presence of at least one jet with a transverse momentum larger than a given value, so as to mimic the strategy followed in the $Wjj$ case. Secondly, we have checked that we obtain unbiased results by suitably changing the jet-cone size. Thirdly, we have exploited the fact that the starting scale of the shower is to some extent arbitrary, and the dependence upon its value is very much reduced in the context of an NLO-PSMC matched computation. As was discussed in ref. [@Torrielli:2010aw], in MC@NLO the information on the starting scale is included in the MC counterterms, and the independence of the physical results of its value constitutes a powerful check of a correct implementation. We have verified that this is indeed the case.
$Wjj$ production at the Tevatron
================================
The hard events obtained with the generation cuts described above can be used to impose the selection cuts employed by the CDF collaboration [@Aaltonen:2011mk]. The latter are as follows (where with “lepton” we always mean the charged one):
- minimal transverse energy for the lepton: ${E_{{\scriptscriptstyle}T}}(\ell) > 20$ GeV;
- maximal pseudorapidity for the lepton: $|\eta(\ell)| < 1$;
- minimal missing transverse energy: $E\!\!\!\!/_{{\scriptscriptstyle}T} > 25$ GeV;
- minimal transverse $W$-boson mass: $M_{{\scriptscriptstyle}T}(\ell\nu) > 30$ GeV;
- jet definition: JetClu algorithm with 0.75 overlap and $R=0.4$;
- minimal transverse jet energy: ${E_{{\scriptscriptstyle}T}}(j) > 30$ GeV;
- maximal jet pseudorapidity: $|\eta(j)| < 2.4$;
- minimal jet pair transverse momentum: ${p_{{\scriptscriptstyle}T}}(j_1j_2) > 40$ GeV;
- minimal jet-lepton separation: $\Delta R(\ell j)>0.52$;
- minimal jet-missing transverse energy separation: $\Delta\phi(E\!\!\!\!/_{{\scriptscriptstyle}T}j)>0.4$;
- hardest jets close in pseudorapidity: ${\left|\Delta\eta(j_1j_2)\right|}<2.5$;
- lepton isolation: transverse hadronic energy smaller than 10% of the lepton transverse energy in a cone of $R=0.4$ around the lepton.
- jet veto: no third jet with ${E_{{\scriptscriptstyle}T}}(j)>30$ GeV and ${\left|\eta(j)\right|}<2.4$;
These cuts (and their analogues in the D0 analysis [@Abazov:2011af], which give very similar results in the “signal” region) are tighter than the ${p_{{\scriptscriptstyle}T}}=25$ GeV analysis cut previously discussed. Since the latter was seen to give unbiased results in the central rapidity regions relevant here, we deem our approach safe. The cuts reported above (which we dub “exclusive”) have also been slightly relaxed by CDF (see [@CDFweb]), by accepting events with three jets or more in the central and hard region – this amounts to not applying the jet-veto condition reported in the last bullet above; we call these cuts “inclusive”.
In addition to the [a[MC@NLO]{}]{} predictions, we have performed parton-level LO and NLO computations. Finally, we have showered events obtained by unweighting LO matrix elements as well. As is well known, the latter case is potentially plagued by severe double-counting effects which, although formally affecting perturbative coefficients of order higher than leading, can be numerically dominant. We have indeed found that this is the case for the cuts considered here: predictions obtained with generation cuts ${p_{{\scriptscriptstyle}T}}\!=\!5$ GeV and ${p_{{\scriptscriptstyle}T}}\!=\!10$ GeV differ by 30% or larger for total rates (shapes are in general better agreement), even for the analysis cut of ${p_{{\scriptscriptstyle}T}}\!=\!50$ GeV. We have therefore opted for using a matched LO sample, which we have obtained with Alpgen [@Mangano:2002ea] interfaced to [[HERWIG]{}]{} through the MLM prescription [@Alwall:2007fs]. In order to do this, we have generated $W+n$ parton events, with $n=1,2,3$. The dominant contribution to $Wjj$ observables is due to the $n=2$ sample, but that of $n=3$ is not negligible. The size of the $n=1$ contribution is always small, and rapidly decreasing with dijet invariant masses; it is thus fully safe not to consider $W+0$ parton events.
In figs. \[fig:CDFD01\] and \[fig:CDFD02\] we present our predictions for the invariant mass of the pair of the two hardest jets with exclusive and inclusive cuts, respectively. The three histograms in the main frames are the aMC@NLO (solid red), Alpgen+MLM (dashed blue), and NLO parton level (green symbols) predictions. The two NLO-based results are obtained with the ${p_{{\scriptscriptstyle}T}}\!=\!10$ GeV generation cuts. The Alpgen+MLM curves have been rescaled to be as close as possible to the NLO ones, since their role is that of providing a prediction for the shapes, but not for the rates (incidentally, this is also what is done in the experimental analyses when control samples are not available). The upper insets show the ratios of the Alpgen+MLM and NLO results over the aMC@NLO ones. The middle insets display the fractional scale (dashed red) and PDF (solid black) uncertainties given by [a[MC@NLO]{}]{}, computed with the reweighting technique described in ref. [@Frederix:2011ss]. The lower insets show the ratios of the aMC@NLO results obtained with the two generation cuts, and imply that indeed there is no bias due to generation cuts. We have also checked that removing the lepton isolation cut does not change the pattern of the plots, all results moving consistently upwards by a very small amount.
By inspection of figs. \[fig:CDFD01\] and \[fig:CDFD02\], we can conclude that the three predictions agree rather well, and are actually strictly equivalent, when the theoretical uncertainties affecting [a[MC@NLO]{}]{} are taken into account (i.e., it is not even necessary to consider those relevant to Alpgen+MLM and parton-level NLO). This is quite remarkable, also in view of the fact that the dominant contribution to the latter, the scale dependence, amounts to a mere $(+10\%,-15\%)$ effect. We have verified that such a dependence is in agreement with that predicted by MCFM [@Campbell:2002tg].
In spite of their being not significant for the comparison with data, it is perhaps interesting to speculate on the tiny differences between the central [a[MC@NLO]{}]{}, Alpgen+MLM, and NLO predictions. The total rates given by [a[MC@NLO]{}]{} and NLO are close but not identical; this is normal, and is a consequence of the fact that the kinematical distributions in the two computations are different, and thus differently affected by the hard cuts considered here. More interestingly, the $M_{jj}$ distribution predicted at the NLO is (very) slightly harder than that of [a[MC@NLO]{}]{}, especially in the case of exclusive cuts. This is best seen in the upper insets of figs. \[fig:CDFD01\] and \[fig:CDFD02\], and is due to the fact that the fraction of events with a third central and hard jet is larger in [a[MC@NLO]{}]{} than at the parton-level NLO. This argument applies also to the case of inclusive cuts. In fact, by requiring the two hardest jets to have a large invariant pair mass, and given the presence of a $W$ boson, one forces extra QCD radiation to be fairly soft, since relatively-hard radiation is strongly suppressed by the damping of the PDFs at large Bjorken $x$’s. This effectively imposes a veto-like condition on the events, which however, at $M_{jj}\simeq 300$ GeV, is still larger than the explicit 30 GeV one imposed by CDF; hence, NLO predictions for inclusive cuts are slightly harder than the [a[MC@NLO]{}]{} ones, but less than in the case of exclusive cuts. We point out that a veto on the third jet (be it explicit or effective) introduces a new mass scale in the problem, whose ratio over $M_{jj}$ may grow large. In such a situation, the resummation of large logarithms performed by the shower constitutes an improvement over fixed-order results. Given the level of agreement we find here, we can conclude the resummation effects are still fairly marginal.
As far as the comparison between the central [a[MC@NLO]{}]{} and Alpgen+MLM predictions is concerned, this is affected by the choice of the hard scales, which are different in the two codes: in Alpgen, the transverse $W$-boson mass is adopted (the renormalization scale is then effectively redefined through the reweighting of the matrix elements by ${\alpha_{{\scriptscriptstyle}S}}$ factors, which is specific of the merging procedure [@Catani:2001cc]). In spite of this, the agreement between the two results is quite good, with Alpgen+MLM being slightly harder than [a[MC@NLO]{}]{} (this effect being of the same order or smaller than that observed with parton-level NLO results). We have also compared Alpgen+MLM with [a[MC@NLO]{}]{}, by setting the hard scales in the latter equal to the transverse $W$-boson mass[^4]. The ratio of these two results is shown as open boxes in the upper inset of figs. \[fig:CDFD01\] and \[fig:CDFD02\], whence one sees a marginal improvement in the agreement between the two predictions w.r.t. the case corresponding to $\mu={H_{{\scriptscriptstyle}T}}/2$, which is our [a[MC@NLO]{}]{} default. We finally stress again that the MLM prescription is crucial to get rid of double-counting effects in LO samples. While double counting is guaranteed not to occur at the NLO in MC@NLO, it can still affect terms of ${\cal O}({\alpha_{{\scriptscriptstyle}S}}^4)$ and beyond. Although we did not see any evidence of these in the form of generation-cut dependence, we have also heuristically extended the MLM prescription to NLO, by requiring the two hardest jets after shower to be matched with two jets reconstructed at the hard-subprocess level (where they play the same roles as the partons in the original MLM matching). This prescription has had no visible effect on our results. Although this is a process-dependent conclusion, it confirms the naive expectation that NLO-PSMC matching is less prone to theoretical systematics than its LO counterpart, and suggests that a reduction of the dependence upon unphysical merging parameters can be achieved by extending the CKKW or MLM procedures to the NLO.
Conclusions
===========
In this paper, we have presented the automated computation of the $Wjj$ cross section to the NLO accuracy in QCD, and its matching to parton showers according to the MC@NLO formalism. This is the first time that a process of this complexity has been matched to an event generator beyond the LO. We believe this is significant not only as a phenomenological result, but also in view of the fact that it is also the first time that the MC@NLO prescription has been applied to a process that requires the presence of cutoffs at the Born level in order to prevent phase-space divergences from appearing. In fact, the structure of such divergences in $Wjj$ production is sufficiently involved to provide evidence that no new problems of principle are expected in the application of MC@NLO to processes with even larger final-state multiplicities.
We have given predictions for the dijet invariant mass in $Wjj$ events, using the same cuts as CDF and D0 in the signal region. Perturbative, parton-level results agree well with those obtained after shower, and we do not observe any significant effects in the shape of distributions due to NLO corrections, which therefore cannot be responsible for the excess of events observed by the CDF collaboration.
Acknowledgments
===============
We would like to thank Johan Alwall, Michelangelo Mangano and Bryan Webber for useful discussions. S. F. is indebted to Michelangelo Mangano for his assistance in running Alpgen. This research has been supported by the Swiss National Science Foundation (SNF) under contract 200020-138206, by the Belgian IAP Program, BELSPO P6/11-P and the IISN convention 4.4511.10, by the Spanish Ministry of education under contract PR2010-0285. F.M. and R.P. thank the financial support of the MEC project FPA2008-02984 (FALCON). R.F. and R.P. would like to thank the KITP at UCSB for the kind hospitality offered while an important part of this work was being accomplished.
[^1]: On leave of absence from INFN, Sezione di Genova, Italy.
[^2]: The mass of the charged lepton $\ell$ is set equal to zero. Furthermore, since we do not compare our predictions to data here, it is sufficient to consider only positively-charged leptons of one flavor.
[^3]: We stress that such jets are now reconstructed by clustering all stable final-state hadrons that emerge from the shower.
[^4]: Note that, since we determine the scale dependence through the reweighting technique of ref. [@Frederix:2011ss], we do not need to run [a[MC@NLO]{}]{} a second time.
|
---
abstract: 'The study of the fundamental properties of phonons is crucial to understand their role in applications in quantum information science, where the active use of phonons is currently highly debated. A genuine quantum phenomenon associated with the fluctuation properties of phonons is squeezing, which is achieved when the fluctuations of a certain variable drop below their respective vacuum value. We consider a semiconductor quantum dot in which the exciton is coupled to phonons. We review the fluctuation properties of the phonons, which are generated by optical manipulation of the quantum dot, in the limiting case of ultra short pulses. Then we discuss the phonon properties for an excitation with finite pulses. Within a generating function formalism we calculate the corresponding fluctuation properties of the phonons and show that phonon squeezing can be achieved by the optical manipulation of the quantum dot exciton for certain conditions even for a single pulse excitation where neither for short nor for long pulses squeezing occurs. To explain the occurrence of squeezing we employ a Wigner function picture providing a detailed understanding of the induced quantum dynamics.'
author:
- Daniel Wigger
- Helge Gehring
- 'V. Martin Axt'
- 'Doris E. Reiter'
- Tilmann Kuhn
title: |
Quantum dynamics of optical phonons generated by optical excitation\
of a quantum dot: A Wigner function analysis
---
Introduction {#sec:intro}
============
Phonons and their interaction with the electronic degrees of freedom are omnipresent in solid state devices. Typically associated with heat, noise or dissipation, nowadays phonons are becoming actively used, which is the foundation of the emerging field of phononics [@volz2016nan]. Examples are the use of phonons in the form of strain pulses to manipulate the lasing properties of semiconductor structures [@bruggemann2011las; @czerniuk2014las] or the application of phonons in the form of surface acoustic waves to control the dynamics in quantum dots [@stotz2005coh; @volk2010enh; @fuhrmann2011dyn; @weiss2014dyn; @gustafsson2014pro]. The generation of coherent phonons in semiconductor nanostructures has been studied [@kerfoot2014opt; @nakamura2015inf] and also phonon lasers have been proposed [@kabuss2012opt]. To explore genuine quantum features of phonons it is interesting to study their fluctuation properties and in particular the emergence of squeezing. Squeezing refers to the reduction of fluctuations of a certain variable below their vacuum level. However, one has to keep in mind that the Heisenberg uncertainty principle has to be fulfilled, which results in increased fluctuations of the conjugate variable. For photons, squeezing is well explored [@dodonov2002non; @polzik2008qua; @drummond2013qua] and is already used in applications, e.g., to detect gravitational waves at the LIGO experiment [@goda2008aqu]. The prospect of finding squeezing also in a mechanical system like phonons, and in particular in semiconductor systems, has triggered a lot of theoretical [@janszky1990squ; @hu1996squ; @sauer2010lat; @papenkort2012opt; @zijlstra2013squ] and experimental work [@garrett1997vac; @misochko2000imp; @johnson2009dir; @esposito2015pho]. In this paper, we will discuss the emergence of squeezing for phonons generated by optically exciting a semiconductor quantum dot.
To be specific, we will study the fluctuation properties of optical phonons which result from an excitation with finite pulses. We will show that the pulse duration is indeed a crucial parameter for phonon squeezing. The theoretical background is introduced in the next Sec. \[sec:theory\] and the results for various types of excitation conditions are presented in Sec. \[sec:results\]. The paper ends with some concluding remarks in Sec. \[sec:conclus\].
Theory {#sec:theory}
======
When searching for non-classical phonon states, it is convenient to use a system, where the electronic part is as simple as possible such that one can focus on the phonon properties. One such system is a self-assembled semiconductor quantum dot (QD) which, under certain conditions, constitutes an electronic two-level system. When the electronic configuration in the QD system changes, in particular by optical excitation with laser light, the lattice reacts to this change by creating phonons. We will show that this can also affect the phonon fluctuations which opens up the possibility to manipulate these fluctuations by changing the excitation conditions.
Model system
------------
In our model we treat the QD as a two-level system, which is justified in the case of a strongly confined QD excited by circularly polarized light, when only excitons with a single spin orientation can be generated. The ground state is denoted by $|g\rangle$ and the exciton state by $|x\rangle$. The states are split by the exciton energy $\hbar\Omega$. Taking the energy of the ground state as zero, the system Hamiltonian $H_{\rm sys}$ reads $$\label{eq:h_sys}
H_{\rm sys}= \hbar \Omega | x \rangle \langle x |\ .$$ The coupling to the classical light field $E(t)$ is treated in dipole and rotating wave approximation via the Hamiltonian $$H_{\rm sys-light}= -M_0 \big[E^{(-)}(t)| g \rangle \langle x |
+ E^{(+)}(t) | x \rangle \langle g | \big]\ ,$$ where $M_0$ is the dipole matrix element and $$E^{(\pm)}(t)=\frac{\hbar\Omega_{\rm R}(t)}{2M_0} e^{\mp i \omega_{\rm L} t}$$ are the positive (upper sign) and negative (lower sign) frequency component of the electric field of the laser pulse with central frequency $\omega_{\rm
L}$. Here we have expressed the pulse shape in terms of the instantaneous Rabi frequency $\Omega_{\rm R}$. Both the dipole matrix element and the Rabi frequency are taken to be real and we assume a Gaussian envelope with $$\Omega_R(t) = \frac{\Theta}{\tau \sqrt{2\pi}}\exp\left[-\frac{(t-t_0)^2}{2\tau^2}\right]\ .$$ The pulse duration is determined by $\tau$ and $\Theta$ denotes the pulse area, which is defined such that in the absence of phonons a resonant $\pi$ pulse (i.e., a pulse with $\Theta = \pi$) completely excites the exciton from the ground state.
In addition, we take into account the electron-phonon interaction. The phonon Hamiltonian is given by $$\label{eq:h_pho}
H_{\rm pho} = \sum_{\mathbf{q}} \big[\hbar\omega_{\mathbf{q}}
b_{\mathbf{q}}^{\dag} b^{}_{\mathbf{q}} + \hbar g_{\mathbf{q}}
\big(b_{\mathbf{q}}^{\dag}+ b^{}_{\mathbf{q}}\big) | x \rangle \langle x | \big]\ ,$$ where $ b_{\mathbf{q}}^{\dag}$ ($b^{}_{\mathbf{q}}$) are the creation (annihilation) operators for a phonon. For simplicity we assume that, as in the case of bulk phonons, the phonons can be classified in terms of a wave vector $\mathbf{q}$. A generalization to arbitrary phonon quantum numbers is straightforward. The first part in Eq. (\[eq:h\_pho\]) describes the free phonon part with the dispersion relation $\omega_{\mathbf{q}}$ and the second term describes the pure dephasing-type carrier-phonon interaction via the coupling matrix element $g_{\mathbf{q}}$. The Hamiltonian Eq. describes the fact that, due to the large difference between exciton and phonon energy the phonons do not induce transitions between exciton and ground state. They affect, however, the phase of the coherence between these states which, for finite pulses, also has an influence on the resulting occupation. In general, there can be different types of phonon modes (e.g., acoustic and optical, longitudinal and transverse) and different coupling mechanisms (e.g., deformation potential or polar). Here we assume that all these can be treated separately.
The operators for the phonon displacement ${\mathbf{u}}=\left<
\hat{{\mathbf{u}}}\right>$ and phonon momentum $\boldsymbol{\pi}=\left<
\hat{\boldsymbol{\pi}}\right>$ are related to the phonon mode operators $b_{\mathbf q}^{}$ and $b_{\mathbf q}^{\dagger}$ via
$$\hat{ {\mathbf{u}} }({\mathbf{r}}) = i \sum_{\mathbf{q}}
\sqrt{\frac{\hbar}{2\varrho V \omega_{\mathbf{q}}}} {\mathbf{e}}_{\mathbf{q}}
\Bigl( b_{\mathbf{q}} + b_{\mathbf{-q}}^\dag \Bigr)
e^{i{\mathbf{q}}\cdot{\mathbf{r}}} \label{eq:u}$$
and $$\hat{ {\boldsymbol{\pi}} }({\mathbf{r}}) = \sum_{\mathbf{q}}
\sqrt{\frac{\varrho\hbar\omega_{\mathbf{q}}}{2 V }}
{\mathbf{e}}_{\mathbf{q}} \Big( b_{\mathbf{q}} - b_{\mathbf{-q}}^\dag \Big)
e^{i{\mathbf{q}}\cdot{\mathbf{r}}}\ , \label{eq:pi}$$
where ${\mathbf{e}}_{\mathbf{q}}$ denotes the polarization vector of the phonon mode, $\varrho$ is the crystal density and $V$ is the normalization volume. In this paper, we are particularly interested in the fluctuation properties of the phonon displacement and momentum. These are given by $$(\Delta {\mathbf{u}})^2 =\left< \hat{{\mathbf{u}}}^2 \right>- \left< \hat{{\mathbf{u}}}\right>^2
\quad \mbox{and}
\quad (\Delta \boldsymbol{\pi})^2 =\left< \hat{\boldsymbol{\pi}}^2 \right>
- \left< \hat{\boldsymbol{\pi}}\right>^2\, .$$ Squeezing occurs, if the fluctuations fall below their respective vacuum value $(\Delta {\mathbf{u}})^2_{\rm vac} $ and $(\Delta
\boldsymbol{\pi})^2_{\rm vac}$. To simplify the discussion, we introduce the quantities
$$D_u =\frac{(\Delta {\mathbf{u}})^2 -(\Delta {\mathbf{u}})^2_{\rm vac} }{(\Delta
{\mathbf{u}})^2_{\rm vac} }$$
and $$D_{\pi} =\frac{(\Delta \boldsymbol{\pi})^2 -(\Delta \boldsymbol{\pi})^2_{\rm vac} }
{(\Delta \boldsymbol{\pi})^2_{\rm vac} } \ .$$
These definitions are particularly handy to identify squeezing, because we only need to check if these quantities, which for simplicity we will call fluctuations in the following, become negative. Thus, the presence of displacement or momentum squeezing is equivalent to $D_u<0$ or $D_{\pi}<0$, respectively.
Generating function formalism
-----------------------------
To calculate the dynamics of the system, we use generating functions which are defined as the expectation values [@vagov2002ele; @axt2005pho] $$\rho_{\nu\nu^\prime} (\{\alpha_{\mathbf{q}}\},\{\beta_{\mathbf{q}}\})=\left<
|\nu\rangle\langle\nu^\prime| e^{\sum_{\mathbf{q}} \alpha_{\mathbf{q}}
b_{\mathbf{q}}^{\dag} } \, e^{\sum_{\mathbf{q}} \beta_{\mathbf{q}} b_{\mathbf{q}}^{} }
\right> \ .$$ Here, $|\nu\rangle$ denotes the electronic state of the system, i.e., $|\nu
\rangle \in \{|g\rangle, |x\rangle \}$ and $\alpha_{\mathbf{q}}$, $\beta_{\mathbf{q}}$ are complex numbers. From the generating functions, all electronic and phononic variables can be calculated. The pure phonon variables are encoded in the function
$$\begin{aligned}
F(\{\alpha_{\mathbf{q}}\},\{\beta_{\mathbf{q}}\}) &&= \left< e^{\sum_{\mathbf{q}} \alpha_{\mathbf{q}}
b_{\mathbf{q}}^\dag} \, e^{\sum_{\mathbf{q}} \beta_{\mathbf{q}} b_{\mathbf{q}}} \right> \nonumber \\
&&= \sum_\nu \rho_{\nu\nu} (\{\alpha_{\mathbf{q}}\},\{\beta_{\mathbf{q}}\})\ ,\end{aligned}$$
while the quantities related to the occupation of the electronic levels are given by the function $C$ and those related to the interband coherence are encoded in the function $Y$ with $$\begin{aligned}
Y (\{\alpha_{\mathbf{q}}\},\{\beta_{\mathbf{q}}\}) &=& \left< |g\rangle\langle x|
\, e^{\sum_{\mathbf{q}} \alpha_{\mathbf{q}} b_{\mathbf{q}}^{\dag} }
\, e^{\sum_{\mathbf{q}} \beta_{\mathbf{q}} b_{\mathbf{q}}^{} } \right> \ , \\
C (\{\alpha_{\mathbf{q}}\},\{\beta_{\mathbf{q}}\}) &=& \left< |x\rangle\langle x|
\, e^{\sum_{\mathbf{q}} \alpha_{\mathbf{q}} b_{\mathbf{q}}^{\dag} }
\, e^{\sum_{\mathbf{q}} \beta_{\mathbf{q}} b_{\mathbf{q}}^{} } \right>\ .\end{aligned}$$
For example, for $\{\alpha_{\mathbf{q}}\}=\{\beta_{\mathbf{q}}\}=0$, we retain the occupation $f=\left< |x\rangle\langle x| \right>=C(0,0)$, while phononic and phonon assisted variables can be obtained by derivatives of the corresponding functions with respect to $\alpha_{\mathbf{q}}$ and $\beta_{\mathbf{q}}$ and setting $\{\alpha_{\mathbf{q}}\}=\{\beta_{\mathbf{q}}\}=0$ afterwards, such as
$$\begin{aligned}
\langle b_{\mathbf{q}} \rangle &=& \frac{\partial}{\partial \beta_{\mathbf{q}}}
F(\{\alpha_{\mathbf{q}}\},\{\beta_{\mathbf{q}}\})\biggl|_{\{\alpha_{\mathbf{q}}\}
=\{\beta_{\mathbf{q}}\}=0} \ , \\
\langle b_{\mathbf{q}}^{\dag} b_{\mathbf{q}}\rangle &=&
\frac{\partial^2}{\partial \beta_{\mathbf{q}}\partial \alpha_{\mathbf{q}}}
F(\{\alpha_{\mathbf{q}}\},\{\beta_{\mathbf{q}}\})\biggl|_{\{\alpha_{\mathbf{q}}\}
=\{\beta_{\mathbf{q}}\}=0} \ .\end{aligned}$$
Using Heisenberg’s equations of motion, a closed set of equations of motion for the generating functions $F$, $Y$, and $C$ can be derived [@vagov2002ele]. These are partial differential equations containing derivatives with respect to $t$, $\alpha_{\mathbf{q}}$, and $\beta_{\mathbf{q}}$. It has been shown that for an excitation with an arbitrary series of ultra short pulses an analytical solution of these equations of motion can be found, which holds for any type of dispersion relation $\omega_{\mathbf{q}}$ and coupling matrix element $g_{\mathbf{q}}$ and thus both for optical and acoustic phonons [@vagov2002ele; @axt2005pho]. The limit of ultra short pulses is reached when the pulse duration is much shorter than the characteristic phonon-induced time scale. In this case the light-induced dynamics during the pulses and the phonon-induced dynamics between and after the pulses can be separated. For longer pulses, when this separation of time scales is not anymore fulfilled, no analytical solution is known and numerical techniques have to be applied. In the following we will concentrate on the case of interaction with optical phonons, in which a numerically tractable set of equations of motion for the characteristic functions can be obtained. A detailed discussion of the phonon dynamics and phonon squeezing in the case of acoustic phonons can be found in Refs. [@wigger2013flu; @wigger2014ene; @wigger2015squ]. For acoustic phonons it has been found that squeezed single or sequences of wave packets can be generated, which travel away from the QD with the speed of sound.
Coupling to optical phonons
---------------------------
Optical phonons are typically characterized by a negligible dispersion over the range of wave vectors which are coupled to the QD exciton. Thus, they are well approximated by a constant phonon frequency $\omega_{\rm LO}$. Due to their vanishing group velocity, optical phonons do not travel but stay confined to the QD region where they are generated. Typically, longitudinal optical (LO) phonons are much more strongly coupled to the QD exciton than transverse optical (TO) phonons. Therefore in the following we will refer to LO phonons, although the formalism is the same for TO phonons.
In the case of a constant phonon frequency it is possible to rewrite the phonon modes in such a form that only a small number of modes couples to the exciton [@stauber2000ele]. We call the annihilation and creation operators in the new basis $B_{\lambda}^{}$ and $B_{\lambda}^{\dag}$. For an $N$-level system, at most $N(N+1)/2$ modes are coupled [@stauber2000ele], which in the case of a two-level system evaluates to three modes. Because we take into account only the pure dephasing mechanism only the coupling matrix element to the excited state is non-zero. Therefore, we can further reduce the number of coupled modes to a single one with the coupling strength $G=\sqrt{\sum_{\mathbf{q}} |g_{\mathbf{q}}|^2}$ [@reiter2011gen]. We further define the dimensionless coupling $\Gamma=G/\omega_{\rm LO}$. The ladder operators of the coupled mode are then given by $$B_0^{} = \sum_{\mathbf{q}} \frac{g_\mathbf{q}}{G} b_{\mathbf{q}}^{} \qquad
\mbox{and} \qquad B_0^{\dagger} = \sum_{\mathbf{q}} \frac{g_\mathbf{q}}{G}
b_{\mathbf{q}}^{\dagger} \ .$$ The other modes with $\lambda \ne 0$ are taken to be orthogonal to the coupled one. With this, the phonon Hamiltonian reads $$H_{\rm pho} = \hbar\omega_{\rm LO} \sum_{\lambda} B_{\lambda}^{\dag}
B^{}_{\lambda} + \hbar G (B_0^{\dag}+ B^{}_{0}) | x \rangle \langle x |\ .$$ To describe the coupled exciton-phonon dynamics for this system the generating functions can be reduced to the single-mode case with only a single pair of variables $\alpha$ and $\beta$ according to
$$\begin{aligned}
F(\alpha,\beta)&=&\left< e^{\alpha B^\dagger_0 }
\, e^{\beta B^{}_0 } \right> \ , \label{eq:theorie:F}\\
Y(\alpha,\beta)&=&\left< |g\rangle\langle x| e^{\alpha B^\dagger_0 }
\, e^{\beta B^{}_0 } \right> \ ,\label{eq:theorie:Y} \\
C(\alpha,\beta)&=&\left< |x\rangle\langle x| e^{\alpha B^\dagger_0 }
\, e^{\beta B^{}_0 } \right>\ . \label{eq:theorie:C}\end{aligned}$$
These generating functions satisfy the closed set of equations
$$\begin{aligned}
i \partial_t F &=& \omega_{\rm LO}\big[\beta\partial_\beta-\alpha\partial_\alpha\big]F
+G\big[\beta-\alpha\big]C \ ,\\
i \partial_t Y &=& \big[ \Omega + \omega_{\rm LO}\left(\beta\partial_\beta
-\alpha\partial_\alpha\right) + G\left(\beta+\partial_\alpha+\partial_\beta\right)
\big] Y \nonumber \\
&&\qquad - \frac{1}{2}\Omega_{\rm R}(t) \big[F-2C\big]e^{-i \omega_{\rm L} t} \ , \\
i \partial_t C &=& \big[ \omega_{\rm LO}\left(\beta\partial_\beta-
\alpha\partial_\alpha\right) + G(\beta-\alpha) \big]C \nonumber \\
&& \qquad - \frac{1}{2}\Omega_{\rm R}(t)
\big[Y^T e^{-i \omega_{\rm L} t}-Y e^{i \omega_{\rm L} t} \big] \ ,\end{aligned}$$
with $Y^T(\alpha,\beta) = Y^*(\beta^*,\alpha^*)$. To further simplify these equations, we transform the system into a rotating frame on the polaron shifted excitation frequency $\overline{\omega}_{\rm L}=\omega_{\rm L} -
\omega_{\rm LO}\Gamma^2$ resulting in the new variables
$$\begin{aligned}
\overline{F}(\alpha,\beta) &=&
F(\alpha e^{-i\omega_{\rm LO} t},\beta e^{i\omega_{\rm LO} t})\, , \\
\overline{Y}(\alpha,\beta)&=& \exp\left[{i\overline{\omega}_{\rm L}t+\beta\Gamma
e^{i\omega_{\rm LO} t}}\right] \times \nonumber \\
&& \quad Y(\alpha e^{-i\omega_{\rm LO} t}+\Gamma,\beta
e^{i\omega_{\rm LO} t}-\Gamma)\, , \\
\overline{C}(\alpha,\beta)&=& \exp\left[{\Gamma\left(\beta
e^{i\omega_{\rm LO} t}+\alpha e^{-i\omega_{\rm LO} t}\right)}\right]
\times \nonumber \\
&& \quad C(\alpha e^{-i\omega_{\rm LO} t},\beta e^{i\omega_{\rm LO} t})\end{aligned}$$
and $$\overline{\Omega}_{\rm R} = \Omega_{\rm R} e^{i(\overline{\omega}_{\rm L}-\omega_L)t}\ .$$
The notation can be further simplified by noticing that for the calculation of the relevant expectation values $\alpha$ and $\beta$ are not independent, instead it is sufficient to calculate the functions $\overline{f}(\alpha,t) =
\overline{f}(-\alpha^* ,\alpha,t)$ where $\overline{f}$ stands for $\overline{F}$, $\overline{Y}$ and $\overline{C}$, respectively. The resulting equations of motion in the case of resonant excitation read
\[eq:eom\] $$\begin{aligned}
\partial_t \overline{F}(\alpha) &=& -2i\omega_{\rm LO} \Gamma {\rm Re}
\left(\alpha e^{i\omega_{\rm LO} t}\right) \times \notag \\
&&\exp\left[{-2i\Gamma {\rm Im}\left(\alpha e^{i\omega_{\rm LO} t}
\right)}\right] \overline{C}(\alpha)\, , \\
\partial_t\overline{Y}(\alpha) &=& \frac{i}{2} \overline{\Omega}_{\rm R} \Big[
e^{\alpha\Gamma e^{i\omega_{\rm LO} t}}\overline{F}
\left(\alpha-\Gamma e^{-i\omega_{\rm LO} t}\right) \notag\\
&& - 2e^{\alpha^*\Gamma e^{-i\omega_{\rm LO} t}}\overline{C}
\left(\alpha-\Gamma e^{-i\omega_{\rm LO} t}\right) \Big]\ , \\
\partial_t\overline{C}(\alpha) &=& \frac{i}{2}e^{-\Gamma^2} \Big[
\overline{\Omega}_{\rm R} e^{\alpha\Gamma e^{i\omega_{\rm LO} t}}
\overline{Y}^* \left(\Gamma e^{-i\omega_{\rm LO} t}-\alpha\right) \notag \\
&& -\overline{\Omega}_{\rm R}^*e^{-\alpha^*\Gamma e^{-i\omega_{\rm LO} t}}
\overline{Y} \left(\Gamma e^{-i\omega_{\rm LO} t}+\alpha\right) \Big]\ .\end{aligned}$$
For the excitation with ultra short pulses, described mathematically in terms of $\delta$-functions, an analytical solution of these equations can be found [@axt1999coh; @vagov2002ele; @axt2005pho], while in the case of extended pulses, the equations cannot be solved analytically anymore. Instead, a numerical integration of the equations of motion is needed [@axt1999coh], which is feasible here since the infinite number of variables $\alpha_{\mathbf{q}}$, $\beta_{\mathbf{q}}$ in the multi-mode case has been reduced to a single complex variable $\alpha$. However, due to the shifted arguments on the right hand side of Eq. all $\alpha$-values are coupled. For the numerical integration we have used a standard fourth-order Runge-Kutta method. The initial conditions were chosen such that the system is initially in the ground state at temperature $T=0$ K resulting in $\overline{C}(t=0)= \overline{Y}(t=0)=0$ and $\overline{F}(t=0)=1$.
Having introduced a single LO phonon mode, it is possible to describe the state of the LO phonons in terms of a Wigner function in the phase space spanned by variables $U$ and $\Pi$ [@schleich2011qua]. $U$ and $\Pi$ are the phase space representations of the corresponding operators $\hat{U}$ and $\hat{\Pi}$, which are directly connected to the phonon creation and annihilation operators of the coupled mode via $$\hat{U} = B_0^{} + B_0^{\dagger} \qquad \mbox{and} \qquad
\hat{\Pi} =i( B_0^{} - B_0^{\dagger}).$$ For simplicity, although defined in a dimensionless form, we will refer to $U$ as displacement and $\Pi$ as momentum in the following.
The Wigner function is the quantum mechanical analogue of a phase-space distribution function. It is a real-valued function, however, in contrast to a classical distribution function the Wigner function can become negative. Negative values of the Wigner function therefore indicate genuine quantum mechanical behavior. From the generating phonon function the Wigner function is calculated via the characteristic Wigner function [@schleich2011qua] $$\begin{aligned}
C_{\rm W}(\alpha)&& = \left< e^{ \alpha B^\dagger_0 - \alpha^* B^{}_0 }\right>
= e^{ -\frac{1}{2}\left|\alpha\right|^2 } \left< e^{\alpha B^\dagger_0 }
\, e^{-\alpha^* B^{}_0 } \right> \nonumber \\
&& = e^{ -\frac{1}{2}\left|\alpha\right|^2 } F(\alpha,-\alpha^*).\end{aligned}$$ With this, the Wigner function is $$\begin{aligned}
W(z)\! &=&\! \frac{1}{\pi^2}\!\iint\! e^{ \alpha^\ast z - \alpha z^\ast } C_{\rm W}(\alpha)\,
{\rm d}^2\alpha \\
&=&\! \frac{1}{\pi^2}\!\iint\! e^{ \alpha^\ast z - \alpha z^\ast }
e^{ -\frac{1}{2}\left|\alpha\right|^2 } F(\alpha,-\alpha^*)\, {\rm d}^2\alpha \nonumber \\
&=&\! \frac{1}{\pi^2}\!\iint\! e^{ \alpha^\ast z e^{-i\omega_{\rm LO} t} -
\alpha z^\ast e^{i\omega_{\rm LO} t} -\frac{1}{2}\left|\alpha\right|^2 } \overline{F}(-\alpha^\ast)\, {\rm d}^2\alpha. \nonumber
\label{eq:theorie:wigner}\end{aligned}$$ Here, $z$ is a complex number and we define Re$(z)=\frac{U}{2}$ and Im$(z)=-\frac{\Pi}{2}$. Non-negative probability distributions $P(U)$ and $P(\Pi)$ can be obtained from the Wigner function by integration over $\Pi$ and $U$, respectively. From these probability distributions expectation values and fluctuations can be calculated in the standard way such as, e.g., for the operator $\hat{U}$:
$$\begin{aligned}
\langle \hat{U}\rangle &=& \int \, U \,P(U)\, {\rm d}U = \iint \, U \,
W(U,\Pi)\,{\rm d}\Pi {\rm d}U \\
\langle \hat{U}^2\rangle &=& \int \, U^2 \,P(U)\, {\rm d}U \ .\end{aligned}$$
Likewise, scaled fluctuations are introduced according to
$$D_U = \frac{(\Delta U)^2 - (\Delta U)_{\rm vac}^2}{(\Delta U)_{\rm vac}^2}$$
and $$D_\Pi = \frac{(\Delta \Pi)^2 - (\Delta \Pi)_{\rm vac}^2}{(\Delta \Pi)_{\rm vac}^2}\, .$$
Correspondingly, the phonon state is squeezed, when either $D_U<0$ or $D_{\Pi}<0$.
![Sketch of the phonon potential for the ground state and the exciton state. The phonon states are marked by the blue envelopes. []{data-label="fig:scheme"}](Figure1.pdf){width="0.5\columnwidth"}
Results {#sec:results}
=======
In the following we will discuss the phonon dynamics and in particular the possibility to achieve phonon squeezing for different excitation conditions. We will start by discussing the limiting cases of pulses which are very short or very long compared to the inverse of the phonon frequency. Then we will come to the case of comparable time scales of the light-induced and phonon-induced dynamics.
To be specific, we consider the case of a GaAs-type QD coupled via the Fr[ö]{}hlich interaction to bulk LO phonons with an energy of $\hbar\omega_{\rm LO} = 36.4$ meV, such that the phonon period is $T=2\pi/\omega_{\rm LO}=114$ fs. The typical coupling strength in such dots is rather weak with about $\Gamma=0.03$, but can be enhanced via charge separation by applying an external electric field up to $\Gamma=0.8$ [@reiter2011gen]. To facilitate the interpretation of the results, in the case of ultra short and ultra long pulses we have used an increased value of $\Gamma=2$, which could be realized in more polar materials. A comparison with values more typical for GaAs-type QDs can be found in [@reiter2011gen].
Ultra short pulses
------------------
![Snapshots of the LO phonon Wigner function for a $\pi$ pulse excitation with an ultra short pulse ($\tau\ll T$) at three different times. []{data-label="fig:limit_short"}](Figure2.pdf){width="\columnwidth"}
The phonon dynamics induced by a single or a pair of ultra short optical pulses has been analyzed in detail in Refs. [@sauer2010lat; @reiter2011gen]. Here we review the main results, which serve as a reference for the case of longer pulses discussed below.
Before discussing the generation of squeezed phonons, it is instructive to recall some features of coherent phonons. Coherent states fulfill the Heisenberg uncertainty relation between the fluctuations of the displacement $\Delta U$ and the fluctuations of the momentum $\Delta \Pi$ at its minimum value. Furthermore, in the dimensionless form of $U$ and $\Pi$ introduced above both fluctuations are equal. This condition is also realized, when the fluctuations agree with their vacuum values. In this sense, the vacuum state is a specific coherent state. Since phonons are described by bosonic ladder operators in the harmonic approximation, the vacuum state can be described by the ground state in a harmonic potential as sketched in Fig. \[fig:scheme\]. Phonons are in the vacuum state, when no phonon excitation has taken place, and in particular, when the QD is in its ground state prior to the optical excitation. In terms of the Wigner function, the vacuum state corresponds to a two-dimensional Gaussian with equal widths in $U$ and $\Pi$, which is centered at the origin as displayed in Fig. \[fig:limit\_short\] a). If the QD exciton is created, e.g., by the application of a $\pi$ pulse, the harmonic potential for the phonons shifts due to the electron-phonon interaction, thus the equilibrium position for the phonons associated with the exciton state lies at a finite value of $U$, which is determined by the coupling constant $\Gamma$. This is illustrated in Fig. \[fig:scheme\]. In the limiting case of an ultra short pulse, the optical excitation occurs so fast that the lattice ions cannot follow. Thus, the potential changes instantaneously while the form and position of the wave function are conserved. In the shifted potential, the state is now displaced with respect to the potential minimum at $U=2\Gamma$ making it a coherent state, which oscillates in time [@janszky1992inf; @vagov2002ele; @sauer2010lat]. In the Wigner function for the LO phonons shown in Fig. \[fig:limit\_short\] a)-c) at three different times this is clearly visible. The Wigner function moves on a circle around its new equilibrium position at $(2\Gamma,0)$, but keeps its form. We want to remark that here the value of the coupling constant $\Gamma$ only determines the position of the shifted equilibrium. The subsequent dynamics is not affected by this value.
![Snapshots of the LO phonon Wigner function for an excitation with two ultra short $\pi/2$ pulses with a delay of $T/2$ between them leading to the formation of two cat states.[]{data-label="fig:cat"}](Figure3.pdf){width="\columnwidth"}
Let us now turn to squeezed phonons. In the Wigner function, squeezing is seen, when the phase space distribution looks indeed squeezed, i.e., it is narrower in one direction in phase space than the Wigner function of the coherent state, hence its name. There are several ways to created squeezed states. Typically squeezing is described by the action of the a squeezing operator on a coherent state [@gerry2005int], but also special superposition states can lead to squeezing. In the Wigner function quantum mechanical effects are indicated by negative values, which can result in a reduced width. In a more strict definition, squeezing occurs, if the fluctuations of one of the variables fall below their respective vacuum fluctuations. Because of the Heisenberg uncertainty principle this always results in an increased fluctuation of the other variable and, thus, a squeezed Wigner function.
The optical excitation by a single ultra short pulse only results in a coherent state or, if the optical excitation is not complete, in a statistical mixture of vacuum state and coherent state. Thus, squeezing is never achieved in this case. However, using two ultra short pulses it has been shown that it is possible to excite squeezed phonon states [@sauer2010lat; @reiter2011gen]. The occurrence of squeezing can be explained by the build up of phononic cat states [@reiter2011gen], which are superpositions of two coherent states. One example of the phonon Wigner function for the excitation with two $\pi/2$ pulses with a delay of $T/2$ is shown in Fig. \[fig:cat\]. The two circles represent the movement of the centers of the coherent states, where the smaller circle corresponds to the exciton potential and the larger one to the ground state potential. Four coherent states emerge, which form a mixture of two cat states seen by the stripe pattern between two pairs of coherent states. For certain parameters squeezing occurs in cat states due to the quantum mechanical interference [@haroche2006exp], when the states are close enough to overlap in phase space. In Ref. [@reiter2011gen] we have shown that squeezing for a two pulse excitation can be found for a wide range of parameters considering coupling strength, phase relation between the pulses, delay and pulse areas.
Ultra long pulses
-----------------
![Snapshots of the LO phonon Wigner function for a $\pi$ pulse excitation with an ultra long pulse ($\tau\gg T$) at three different times. []{data-label="fig:limit_long"}](Figure4.pdf){width="\columnwidth"}
In the other limiting case of a pulse that is very long compared to the phonon period, the situation is different. The corresponding Wigner function is shown again at three different times in Fig. \[fig:limit\_long\] a)-c). In this case no analytical solution is possible, instead the Wigner function has been obtained from the numerical solution of the equations of motion \[Eq. (\[eq:eom\])\] for the generating functions. During the pulse the phonons are in a mixture of the phonon ground states associated with the two electronic potentials. Thus, the Wigner function changes its shape as seen in Fig. \[fig:limit\_long\] b), eventually reaching its new equilibrium depicted in Fig. \[fig:limit\_long\] c), as is expected for a quasi-adiabatic state preparation of the exciton. This new equilibrium state represents the polaron that has been built up. The final state after the pulse is therefore again a symmetric Gaussian, but now at the new equilibrium position $(2\Gamma, 0)$. In the exciton subsystem it is the vacuum state, while seen from the perspective of the electronic ground state it is displaced and thus a coherent but stationary state. Again this does not alter the fluctuation properties of the phonons.
In Ref. [@janszky1992inf] it was reported that the action of a long pulse would result in the creation of a Fock state, which is in contrast to our findings. We attribute this discrepancy to the use of perturbation theory in Ref. [@janszky1992inf], which might lose its validity in the limit of long times. Note that our generating function treatment provides a numerically complete solution of the dynamics in the considered model without further approximation except for the discretization of $t$ and $\alpha$, which is however well controlled. In particular, no perturbative approximation is made. Indeed, for the short pulse excitation, where perturbation theory typically works properly, our results and the results found in Ref. [@janszky1992inf] agree on the creation of a coherent state.
From our calculations we thus conclude that both excitations with ultra short and ultra long single pulses yield coherent phonon states after the excitation [@sauer2010lat; @reiter2011gen], in the former case oscillating around the new equilibrium position and in the latter case localized at this equilibrium position. Also during the long pulse, the fluctuations remain at or above the vacuum levels, as already implied by the stretched Wigner function in Fig. \[fig:limit\_long\] b).
Pulses with finite duration
---------------------------
We now analyze the case of excitation by pulses with pulse durations in the intermediate regime, i.e., in the regime where the pulse duration $\tau$ and the phonon oscillation period $T$ are of the same order. In this section we set $\Gamma=0.5$, which is a reasonable value for a GaAs QD in the presence of an applied electric field.
### Single pulse excitation
![a) Occupation of the exciton state $f$ and laser pulse, b) expectation values of the displacement $\langle \hat{U} \rangle$ and of the momentum $\langle \hat{\Pi}
\rangle$ and c) their fluctuations. []{data-label="fig:expec"}](Figure5.pdf){width="0.75\columnwidth"}
Figure \[fig:expec\] shows the dynamics of the LO phonon system after the excitation with a single finite pulse with a pulse area of $\Theta=2\pi$ and a pulse length of $\tau=0.2\,T$, which in the case of GaAs corresponds to a pulse with a full width at half maximum (FWHM) of about $50$ fs. In Fig. \[fig:expec\] a) the laser pulse and the occupation $f$ of the exciton state is shown, while in Fig. \[fig:expec\] b) the expectation values of the displacement $ \langle \hat{U} \rangle$ and of the momentum $\langle
\hat{\Pi} \rangle$ are displayed. Figure \[fig:expec\] c) shows the respective fluctuations for these excitation conditions. During the action of the laser pulse, which is from about $t=-0.5\,T$ to $t=0.5\,T$, the occupation cycles once through maximum and minimum, however, due to the phonon interaction, after the pulse an occupation of about $f=0.285$ remains in the system. Without the interaction one would expect a final value of $f=0$ for a pulse with $\Theta=2\pi$. Likewise the phonon expectation values and fluctuations start to oscillate. The expectation value of the displacement $\langle \hat{U} \rangle$ oscillates around a shifted mean value. This shift is due to the shifted equilibrium position in the excitonic subsystem ($2\Gamma$), which contributes to the overall mean value weighted by the occupation $f$. So $ \langle \hat{U} \rangle$ oscillates around $f\cdot2\Gamma=0.285$, marked as upper dashed line in Fig. \[fig:expec\] b). The fluctuations $D_U$ also show a periodic behavior, however, for all times the fluctuations are enhanced, such that a displacement squeezing does not take place. For the momentum, we find that the expectation value of the momentum $\langle \hat{\Pi} \rangle$ oscillates around $0$ and also the fluctuations $D_{\Pi}$ oscillate sinusoidal. Here, we see each minimum is below $0$ showing the occurrence of momentum squeezing.
![a) Occupation of the excited state, b) minimum of the fluctuations of the displacement and c) minimum of the fluctuations of the momentum. All at times after the pulse and plotted as a function of pulse area $\Theta$ and pulse width $\tau$. Note that the color scale is restricted to negative values for b) and c). []{data-label="fig:analysis"}](Figure6.pdf){width="\columnwidth"}
To analyze in more detail, whether we can have squeezing for a single pulse excitation, we have performed a systematic analysis of the fluctuations $D_U$ and $D_{\Pi}$ as function of the pulse area $\Theta$ and the pulse duration $\tau$. For this we have extracted the minimal value of $D_U$ and $D_{\Pi}$ after the action of the pulse, e.g., for $t>0.5T$ in the previous example. Because we consider LO phonons and we have neglected any phonon decay, the oscillation after the pulse is periodic with the phonon period $T$ and will go on forever. For very long times, phonon decay processes would eventually destroy the phonon signal in real systems. Furthermore, radiative decay processes would lead to a relaxation from the exciton to the ground state. These processes, however, occur on much longer time scales than considered here.
The results for the minimum values of $D_U$ and $D_\Pi$ are shown in Fig. \[fig:analysis\] together with the corresponding occupation $f$ of the exciton state. Note that for min$(D_U)$ and min$(D_{\Pi})$ we only show the negative values for clarity, i.e., it is not shown if the minimum is above zero. For an intermediate pulse length up to $\tau=0.5\,T$ we find that squeezing of both displacement and momentum can indeed occur, however the degree of squeezing is rather small with less than 5%. We find that squeezing occurs in the transition region between the limiting cases of ultra short and long pulses. Interestingly, the occurrence of squeezing is accompanied by a rather atypical behavior of the occupation. While for ultra short pulses we find the typical Rabi rotations with maxima at $\Theta=(2n+1)\pi$, for longer pulses, the phonon-induced renormalization of the Rabi rotations is clearly visible, i.e., the period of the Rabi flops gets longer. For intermediate pulse areas no clear Rabi rotations can be identified. When we look at the fluctuations, we find that exactly in the region with intermediate pulse duration, the fluctuations of the phonons fall below their vacuum value, i.e., squeezed states emerge.
![Snapshots of the Wigner function for a $2\pi$-pulse with $\tau=0.2\,T$ for different times $t/T$. []{data-label="fig:1puls"}](Figure7.pdf){width="\columnwidth"}
To understand the behavior in more detail, we look at the corresponding Wigner function plotted in Fig. \[fig:1puls\]. During the rise of the pulse up to $t=0$ the phonons do not react significantly to the change in the electronic system and essentially stay in the vacuum state. Only around $t=0$, we find that the phonons start to noticeably react to their new potential from the exciton state and move out of the center of phase space. While they start to oscillate, the electronic system already moves back to the ground state. Thus, instead of a coherent state in form of a symmetric Gaussian, a deformed shape is formed. We also find negative values of the Wigner function. This indicates that the phonon state contains features that are of genuine quantum mechanical nature. After the action of the pulse, for $t>T$, we see that the Wigner function rotates, but its form is not stable. Around the times $t=T$ and $t=3T/2$, where the momentum is squeezed, we see that the Wigner function is indeed elongated along $U$, but narrow in $\Pi$. On the other hand at time $t=5T/4$ and $t=7T/4$, where the fluctuations of the displacement have a minimum, the Wigner function is elongated along $\Pi$. Though negative parts appear in the Wigner function, it is still wider than in the vacuum case and no squeezing occurs.
Since the QD is driven by a coherent light field and the initial state is a pure state (the ground state of the QD–phonon system), the total system consisting of QD exciton and phonons remains always in a pure state. However, in general the temporal evolution results in an entanglement of the electronic and the phononic subsystem. When only one of the two subsystems is considered, the other system is traced out. This leads to a loss of coherence. If the electronic system is traced out, the phononic system falls apart into two parts: one belonging to the ground state potential and one belonging to the exciton state potential. Due to the loss of coherence, these two parts are in a statistical mixture. Note that each part on its own corresponds to a pure state. Also the Wigner function can be separated into these two parts with $W_{\rm g}(U,\Pi)$ being the Wigner function in the ground state potential and $W_{\rm x}(U,\Pi)$ for the exciton potential. The total Wigner function is the sum of both parts $W(U,\Pi)=W_{\rm
g}(U,\Pi)+W_{\rm x}(U,\Pi)$ reflecting the statistical mixture. One example for this is shown in Fig. \[fig:GX\]. For both parts $W_{\rm g}(U,\Pi)$ and $W_{\rm x}(U,\Pi)$ we find a banana shaped Wigner function consisting of a positive region that bends around a negative one. Each part now rotates with a stable shape around its respective equilibrium, i.e., $W_{\rm g}(U,\Pi)$ around $(0,0)$, while $W_{\rm x}(U,\Pi)$ moves around $(2\Gamma,0)=(1,0)$. When the two parts are summed up, the shape of the total Wigner function is not stable in time.
![Snapshot at time $t=3T/4$ of the Wigner function (left), which can be separated into the Wigner function belonging to the ground state potential $W_{\rm g}(U,\Pi)$ (middle) and the one belonging to the exciton state potential $W_{\rm x}(U,\Pi)$ (right). []{data-label="fig:GX"}](Figure8.pdf){width="\columnwidth"}
[lll]{} $n$ & $P_n^{\rm g}$ & $P_n^{\rm x}$\
0 & 0.46 & 0.09\
1 & 0.25 & 0.19\
2 & 0.005 & $7\times 10^{-4}$\
3 & $4\times 10^{-5}$ & $5\times 10^{-6}$\
The Wigner function also provides a straightforward way of expanding a given quantum state expressed in terms of a Wigner function $W(q,p)$, be it pure or mixed, into any basis $\left|\varphi_n\right>$. The probability of finding a state $\left|\varphi_n\right>$ is then given by [@schleich2011qua] $$\begin{aligned}
P_n &=& \Bigl< |\varphi_n\rangle \langle \varphi_n | \Bigr> \notag \\
&=& \pi \iint W_{\varphi_n}(q,p) W(q,p)\,{\rm d}q\,{\rm d}p\end{aligned}$$ where $W_{\varphi_n}(q,p)$ is the Wigner function representation of the projection operator $|\varphi_n\rangle \langle \varphi_n |$. Such an expansion can also be done separately for the Wigner functions in the two subspaces $W_{\rm g}(U,\Pi)$ and $W_{\rm x}(U,\Pi)$. This allows us to expand the Wigner functions of the individual subspaces into the Fock states $\left|n\right>$ corresponding to the respective potential. The Wigner functions of the Fock states in the ground state subspace are given by $$W^{\rm g}_n(U,\Pi)=\frac{(-1)^n}{2\pi}{\rm e}^{-\frac12\left(U^2+\Pi^2\right)}
L_n\left(U^2+\Pi^2\right)$$ with $L_n$ being the Laguerre polynomials. For the Wigner functions $W_n^{\rm
x}$ in the exciton system we take $U\to U-2\Gamma$.
As an example we expand the Wigner functions from Fig. \[fig:GX\], where pronounced squeezing is visible. The probabilities obtained from this expansion are listed in Table \[tab:Pn\]. Note that the sums of the probabilities in each subsystem reflect the electronic occupations. In our case, this means $\sum_n P_n^{\rm g} =
\left<\left|g\right>\left<g\right|\right>= 1-f \approx 0.715$ and $\sum_n
P_n^{\rm x}=f\approx 0.285$, as we have found in Fig. \[fig:expec\] a). Because the full state is normalized, it follows that $\sum_n P_n^{\rm
g}+\sum_n P_n^{\rm x}=1$. When looking at the actual numbers in Tab. \[tab:Pn\], we see that in each electronic subsystem only the first two Fock states, namely $n=0$ and $n=1$, contribute significantly to the phonon state. As the listed probabilities $P_1^{\rm g}$, $P_2^{\rm g}$, $P_1^{\rm x}$ and $P_2^{\rm x}$ add up to almost $1$, there are essentially no contributions from higher Fock states. The fact that the phonon state in each subsystem forms a superposition essentially of the form $$c_0\left|0\right> + c_1\left|1\right>$$ is true for a wide range of pulse areas and durations. Recently it was shown that the photons emitted from QDs can be in a superposition of the lowest two Fock states $|0\rangle$ and $|1\rangle$, and it was experimentally demonstrated that these photons exhibiting squeezing [@schulte2015qua]. Though, the photons couple in a different way to the QD and are described within the Jaynes-Cummings model, this shows that squeezing in such a superposition occurs in a wider range of systems.
### Two pulse excitation
![Minimal values of the fluctuations $D_U$ and $D_{\Pi}$ for a two pulse excitation with pulse areas $\pi/2$ each and a delay of $\Delta t=T/2$ as a function of the pulse width. a) Phase $\Phi=0$ and b) phase $\Phi=3\pi/2$.[]{data-label="fig:2puls"}](Figure9.pdf){width="0.75\columnwidth"}
For the two pulse excitation, we have already shown that squeezing can occur in the case of excitation by two ultra short pulses [@sauer2010lat; @reiter2011gen]. The strongest squeezing emerges, when the two pulses have a delay of $\tau=T/2$ and a pulse area of $\pi/2$ each. As we have seen, two pairs of phononic cat states build up, which for suitable coupling strengths (e.g., $\Gamma=0.5$) give rise to squeezing. Another crucial pulse parameter is the phase difference $\Phi$ between the two pulses. For a phase of $\Phi=0$ the state is not squeezed, while for $\Phi=3\pi/2$ the squeezing is maximal. This is in agreement with squeezing in cat states, which also depends crucially on the phase in the superposition state [@gerry2005int]. Let us briefly revisit the influence of the phase on the excitation: For the same phase, each pulse changes the occupations of the electronic states letting them perform a part of a Rabi rotation. When the phase difference is a multiple of $\pi/2$, the second pulse does not change the occupation of the states, but only influences the phase difference of the electronic states, which in turn modifies the phonon properties drastically [@reiter2011gen]. In the following we will study how squeezing prevails using finite pulses.
In Fig. \[fig:2puls\] we show the minimum of the fluctuations $D_U$ and $D_{\Pi}$ for a two pulse excitation with pulse areas $\pi/2$ each and a delay of $\Delta t=T/2$ as a function of the pulse width for two different phases $\Phi=0$ and $\Phi=3\pi/2$. For $\Phi=0$ we do not find any squeezing for ultra short pulses. For short pulses with $\tau=0.1\,T$ momentum squeezing occurs with a strength of about $5$%, similar to the case of the one pulse excitation. For $\tau>0.2\,T$, both fluctuations are larger than zero and we do not find squeezing. Interestingly, for $0.6\,T <\tau<T$, the fluctuations do not change as a function of pulse width anymore. Here, the two pulses already form a single pulse. Due to the phonon renormalization, the pulse area is slightly smaller than $\pi$. In this case a statistical mixture of two coherent states, one belonging to the ground state and one belonging to the exciton state, builds up leading to increased fluctuations in $U$, while the momentum fluctuations stay at zero. Note that these states are stationary and do not move in time.
![ Snapshot at time $t=7 T/4$ of the total Wigner function $W(U,\Pi)$ (left) and the separation into $W_{\rm g}(U,\Pi)$ (middle) and $W_{\rm x}(U,\Pi)$ (right). The pulse width is $\tau=0.1\,T$ for two pulses with pulse area $\pi/2$ each, a delay of $T/2$ and a phase $\Phi=3\pi/2$. []{data-label="fig:Wig_2puls"}](Figure10.pdf){width="\columnwidth"}
For $\Phi=3\pi/2$ we already have squeezing of about $30$% in the displacement and about $8$% in the momentum at $\tau=0$. If now the pulse width is taken to be finite, we see that the squeezing becomes even more pronounced with up to $40$% in $D_U$ at $\tau=0.1\,T$. For larger pulse width $\tau$ the squeezing in both displacement and momentum vanishes and for $\tau>0.6\,T$ also reaches a constant value almost independent of the pulse area.
It is interesting to note that also for a quantum well, i.e., for a system with a continuous electronic spectrum, a two-pulse excitation with finite pulses can lead to LO phonon squeezing under similar excitation conditions regarding the phase difference [@papenkort2012opt].
The occurrence of squeezing for small pulse areas can be nicely seen in the Wigner function as displayed in Fig. \[fig:Wig\_2puls\], where we show a snapshot at time $t=7T/4$, where the squeezing is maximal for the two pulse excitation, i.e., $D_U$ is minimal. When we separate the Wigner function into the ground state and exciton contribution with $W(U,\Pi)= W_{\rm{g}}(U,\Pi) +
W_{\rm{x}}(U,\Pi)$, we see that already in the subspaces the Wigner functions are elongated in $\Pi$-direction with banana-like shapes oriented in opposite directions. In addition, negative parts of the Wigner function are visible, indicating the non-classical character of the superposition of the cat states. When added up to the total Wigner function, as shown in the left part of Fig. \[fig:Wig\_2puls\], the Wigner function becomes squeezed.
Conclusions {#sec:conclus}
===========
In summary, we have discussed the emergence of squeezed LO phonons by the optical manipulation of a QD. After reviewing the results for ultra short excitation, we have presented new results discussing the influence of the pulse width on the creation of squeezed phonons. To this end, we have determined the solution of the equations of motion using a generating function formalism, from which the Wigner function can be calculated directly. For a single pulse, squeezing can be found if the pulse width is about $0.05-0.5$ of the LO phonon period. In these cases superposition states are created leading to reduced fluctuations. In the case of a two-pulse excitation squeezing was already found for ultra-short pulses and we have shown that a strong squeezing prevails for extended pulses up to $0.2\,T$. Our results show that for extended pulses even stronger phonon squeezing can be observed, which brings the theoretical predictions one step closer to experimental realization.
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|
---
abstract: 'An obfuscator is an algorithm that translates circuits into functionally-equivalent similarly-sized circuits that are hard to understand. Efficient obfuscators would have many applications in cryptography. Until recently, theoretical progress has mainly been limited to no-go results. Recent works have proposed the first efficient obfuscation algorithms for classical logic circuits, based on a notion of indistinguishability against polynomial-time adversaries. In this work, we propose a new notion of obfuscation, which we call partial-indistinguishability. This notion is based on computationally universal groups with efficiently computable normal forms, and appears to be incomparable with existing definitions. We describe universal gate sets for both classical and quantum computation, in which our definition of obfuscation can be met by polynomial-time algorithms. We also discuss some potential applications to testing quantum computers. We stress that the cryptographic security of these obfuscators, especially when composed with translation from other gate sets, remains an open question.'
author:
- 'Gorjan Alagic[^1], Stacey Jeffery[^2], and Stephen P. Jordan[^3]'
bibliography:
- 'obf.bib'
title: 'Partial-indistinguishability obfuscation using braids'
---
Introduction
============
Past work on circuit obfuscation
--------------------------------
Informally, an obfuscator is an algorithm that accepts a circuit as input, and outputs a hard-to-understand but functionally equivalent circuit. In this subsection, we briefly outline the state of current research in classical circuit obfuscation. To our knowledge, quantum circuit obfuscation has not been considered in any prior published work.
Methods used for obfuscating logic circuits in practice have so far been essentially ad hoc [@CT02; @Simonaire]. Until recently, theoretical progress has primarily been in the form of no-go theorems for various strong notions of obfuscation [@Barak; @GR07]. The ability to efficiently obfuscate certain circuits would have important applications in cryptography. For instance, sufficiently strong obfuscation of circuits of the form “encrypt with a hard-wired private key” could turn a private-key encryption scheme into a public-key encryption scheme. As this example illustrates, one undesirable outcome is when the input circuit can be recovered completely from the obfuscated circuit. In this case, we say that the obfuscator *completely failed* on that circuit [@Barak]. Unfortunately, every obfuscator will completely fail on some circuits (e.g., learnable circuits.) On the other hand, there are trivial obfuscators which will erase at least some information from some circuits, e.g., by removing all instances of $X^{-1}X$ for some invertible gate $X$.
In order to give a useful formal definition of obfuscation, one must decide on a reasonable definition of “hard-to-understand.” The most stringent definition in the literature demands *black-box obfuscation*, i.e., that the output circuit is computationally no more useful than a black box that computes the same function. Barak et al. [@BGIRSVY01] gave an explicit family of circuits that are not learnable and yet cannot be black-box obfuscated. They also showed that there exist (non-learnable) private-key encryption schemes that cannot be turned into a public-key cryptosystem by obfuscation. Their results do not preclude the possibility of black-box obfuscation for specific families of circuits, or of applying obfuscation to produce public-key systems from private ones in a non-generic fashion. It is an open problem whether quantum circuits can be black-box obfuscated.
A weaker but still quite natural notion is called *best-possible obfuscation*; in this case, we ask that the obfuscated circuit reveals no more information than any other circuit that computes the same function. Goldwasser and Rothblum [@GR07] showed that for efficient obfuscators, best-possible obfuscation is equivalent to *indistinguishability obfuscation*, which is defined as follows. For any circuit $C$, let $|C|$ be the number of elementary gates, and let $f_C$ be the Boolean function that $C$ computes.
\[def:indistinguishability\] A probabilistic algorithm $\mathcal O$ is an *indistinguishability obfuscator* for the collection $\mathcal C$ of circuits if the following three conditions hold:
1. (functional equivalence) for every $C \in \mathcal C$, $f_{\mathcal O (C)} = f_C$;
2. (polynomial slowdown) there is a polynomial $p$ such that $|\mathcal O(C)| \leq p(|C|)$ for every $C \in \mathcal C$;
3. (indistinguishability obfuscation) For any $C_1, C_2 \in \mathcal C$ such that $f_{C_1} =
f_{C_2}$ and $|C_1| = |C_2|$, the two distributions $\mathcal
O(C_1)$ and $\mathcal O(C_2)$ are indistinguishable.
In the third part of the above definition, one must choose a notion of indistinguishability for probability distributions. Goldwasser and Rothblum [@GR07] consider three such notions: perfect (exact equality), statistical (total variation distance bounded by a constant), and computational (no probabilistic polynomial-time Turing Machine can distinguish samples with better than negligible probability). They show that the existence of an efficient statistical indistinguishability obfuscator would result in a collapse of the polynomial hierarchy to the second level. This result also applies if the condition $|C_1| = |C_2|$ in property (3) of Definition \[def:indistinguishability\] is relaxed to $|C_1| = k|C_2|$ for any fixed constant $k$ [@GR07].
A recent breakthrough has shown that computational indistinguishability may be achievable in polynomial time. Combining a new obfuscation scheme for NC1 circuits with fully homomorphic encryption, Sahai et al. gave an efficient obfuscator which achieves the computational indistinguishability condition under plausible hardness conjectures [@GGHRSW13]. Subsequent work outlined a number of cryptographic applications of computational indistinguishability [@SahaiW13].
Outline of present work
-----------------------
### New notion of obfuscation
An exact deterministic indistinguishability obfuscator would yield a solution to the circuit equivalence problem. For general Boolean circuits, this problem is co-NP hard. Therefore, exact deterministic indistinguishability obfuscation of general Boolean circuits cannot be achieved in polynomial time under the assumption $\mathrm{P} \neq \mathrm{NP}$. We propose an alternative route to weakening the exactness condition, by pursuing a notion of “partial-indistinguishability”. In partial-indistinguishability obfuscation, we relax condition (3) so that it need only hold for $C_1$ and $C_2$ that are related by some fixed, finite set of relations on the underlying gate set.[^4]
\[def:partial-indistinguishability\] Let $G$ be a set of gates and $\Gamma$ a set of relations satisfied by the elements of $G$. An algorithm $\mathcal O$ is a *$(G, \Gamma)$-indistinguishability obfuscator* for the collection $\mathcal C$ of circuits over $G$ if the following three conditions hold:
1. (functionality) for every $C \in \mathcal C$, $f_C = f_{\mathcal
O(C)}$;
2. (polynomial slowdown) there is a polynomial $p$ such that $|\mathcal O(C)| \leq p(|C|)$ for every $C \in \mathcal C$;
3. (($G, \Gamma$)-indistinguishability) for any $C_1,
C_2 \in \mathcal C$ that differ by some sequence of applications of the relations in $\Gamma$, $\mathcal O(C_1) = \mathcal O(C_2)$.
The power of the obfuscation is now determined by the power of the relations $\Gamma$. If $\Gamma$ is a complete set of relations, generating all circuit equivalences over $G$, then a ($G, \Gamma$)-indistinguishability obfuscator is a perfect indistinguishability obfuscator according to Definition \[def:indistinguishability\]. (Complete sets of relations for $\{\mathrm{Toffoli}\}$ and $\{\mathrm{AND},\mathrm{OR},\mathrm{NOT}\}$ are given in [@Iwama; @Huntington].) If $\Gamma$ is the empty set then even the identity map fits the definition, and no obfuscation is taking place. With different sets of relations, one can interpolate between these extremes. The intermediate obfuscators form a partially ordered set, where a $(G, \Gamma')$-indistinguishability obfuscator is strictly stronger than a $(G, \Gamma)$-indistinguishability obfuscator if $\Gamma'$ is a strict superset of $\Gamma$. We remark that partial-indistinguishability is no stronger than perfect indistinguishability, and appears to be incomparable with statistical and computational indistinguishability. This is part of our motivation in considering this new definition.
In the context of quantum computation, we make only a few minor changes to Definitions \[def:indistinguishability\] and \[def:partial-indistinguishability\]. First, the obfuscators will still be classical algorithms. On the other hand, the gates will be unitary and the circuits to be obfuscated will be unitary quantum circuits. Finally, the notion of functional equivalence now simply means that the operator-norm distance between the unitary implemented by $C$ and the unitary implemented by $\mathcal O(C)$ is bounded by a small constant $\epsilon > 0$.
### Group normal forms
A finitely generated group can be specified by a presentation. This is a list of generators $\sigma_1,\ldots,\sigma_n$ and a list of relations obeyed by these generators. (A relation is simply an identity such as $\sigma_1 \sigma_3 = \sigma_3 \sigma_1$.) All group elements are obtained as products of the generators and their inverses. However, by applying the relations, we can get multiple words in the generators and their inverses that encode the same group element. A normal form specifies, for each group element, a unique decomposition as a product of generators and their inverses. For certain groups, including the braid groups, polynomial time algorithms are known which, given a product of generators and their inverses, can reduce it to a normal form. The word problem is, given two words in the alphabet $\{\sigma_1,\ldots,\sigma_n,\sigma_1^{-1},\ldots,\sigma_n\}$, to decide whether they specify the same group element. If a normal form can be computed, then this solves the word problem: just reduce both words to normal form and check whether the results are identical. However, an efficient solution for the word problem does not in general imply an efficiently computable normal form.
### Efficient constructions from group representations
In this paper, we propose a general method of designing partial-indistinguishability obfuscators based on groups with efficiently computable normal forms. If a set of gates $G$ obeys the relations $\Gamma$ of the generators of a group with an efficiently computable normal form, then the reduction to normal form is an efficient $(G,\Gamma)$-indistinguishability obfuscator. The gates may obey additional relations beyond $\Gamma$, which is why the obfuscator does not solve the circuit-equivalence problem, which is believed to be intractable for both classical and quantum circuits.
To demonstrate this method, we discuss an implementation using the braid groups $B_n$, for both classical reversible circuits and unitary quantum circuits. The number of strands $n$ in the braid group depends linearly on the number of dits or qudits on which the circuit acts. In Section \[sec:classical\], we describe a computationally universal reversible classical gate obeying the braid group relations, which was constructed in [@Mochon; @OP99; @Kitaev03] from the quantum double of $A_5$. In Section \[fibonacci\], we describe a computationally universal quantum gate obeying the braid group relations, which was constructed in [@FLW02] from the Fibonacci anyons. Our obfuscation scheme is similar in spirit to previously-proposed obfuscation schemes based on applying local circuit identities [@Simonaire], but the uniqueness of normal forms adds a qualitatively new feature. One consequence of this feature is that we can satisfy Definition \[def:partial-indistinguishability\] and guarantee the partial-indistinguishability property against computationally unbounded adversaries. The running time of the obfuscator is the same as the running time of the the normal form algorithms, which take time $O(l^2 m \log m)$ for $m$-strand braids of length $l$ [@Dehornoy08].
We remark that these gate sets that obey the braid group relations are not artificial constructions; in fact, they are the most natural choice in many contexts, some of which we list here. In the quantum case, these gates are native to certain proposed physical implementations of quantum computers [@Kitaev03], where the topological braiding property provides inherent fault-tolerance. The problem of approximating the Jones Polynomial invariant of links is complete for polynomial-time quantum computation [@AA11]; an analogous fact is true for a restricted case of quantum computations motivated by NMR implementations [@Shor_Jordan]. Both of these facts are naturally expressed in the gate set constructed from the Fibonacci representation. In the classical case, the gate set derived from quantum doubles of finite groups was recently used to show BPP-completeness for approximation of certain link invariants [@Krovi_Russell].
We remark that another potential group family for constructing partial-indistinguishability obfuscators are the mapping class groups MCG$(\Sigma_g)$ of unpunctured surfaces of genus $g$. These groups also have quantumly universal representations [@Alagic] and an efficiently solvable word problem [@Hamidi-Tehrani]. It is not known if there are also classically universal permutation representations, or if there are efficiently computable normal forms.
### Other gate sets
In some applications the native gate set will be different than the ones used in our construction. It is natural to ask if our obfuscators can be used in these settings as well. By universality (quantum or classical), one has an efficient algorithm ${\mathsf{B}}$ which translates circuits from the native gate set to the braiding gate set, as well as an efficient algorithm ${\mathsf{C}}$ for translation in the opposite direction. We also let ${\mathsf{N}}$ denote the partial-indistinguishability obfuscator. One might then attempt to obfuscate by applying the following:\
\
\[obfuscator\]
We stress that, unlike the map ${\mathsf{N}}$, the composed map ${\mathsf{N}}\circ {\mathsf{B}}$ does not necessarily satisfy Definition \[def:partial-indistinguishability\]. As we discuss in Section \[compilation\], careless choice of the map ${\mathsf{B}}$ can partially or completely break the security of the obfuscator. Finding translation algorithms securely composable with partial-indistinguishability obfuscators is an area of current investigation.
Relevant Properties of the Braid Group {#braid-groups}
======================================
The braid group $B_n$ is the infinite discrete group with generators $\sigma_1, \ldots, \sigma_{n-1}$ and relations $$\label{abstract-braids}
\begin{array}{rcll}
\sigma_i \sigma_j & = & \sigma_j \sigma_i & \forall \ |i-j| \geq 2\\
\sigma_i \sigma_{i+1} \sigma_i & = & \sigma_{i+1} \sigma_i \sigma_{i+1}
& \forall \ i.
\end{array}$$ The group $B_n$ is thus the set of all words in the alphabet $\{\sigma_1,\ldots,\sigma_{n-1},\sigma_1^{-1},\ldots,\sigma_{n-1}^{-1}\}$, up to equivalence determined by the above relations. In 1925 Artin proved that the abstract group defined above precisely captures the topological equivalence of braided strings [@Artin25], as illustrated in Fig. \[braids-example\]. A charming exposition of this subject can be found in [@Kauffman91].
![\[braids-example\] The generator $\sigma_i$ represents the (clockwise) crossing of strands $i$ and $i+1$ connecting a bottom row of “pegs” to a top row. Multiplication of group elements corresponds to composition of braids. As an example, we show the 3-strand braid $\sigma_1^{-1}\sigma_2$ (left), and the same braid composed with its inverse $\sigma_2^{-1}\sigma_1$ (middle), which is equivalent to the identity element of $B_3$ (right).](braids.pdf){width="50.00000%"}
In the word problem on $B_n$, we are given words $w$ and $z$, and our goal is to determine if they are equal as elements of $B_n$. One solution is to put both $w$ and $z$ into a *normal form*, and then check if they are equal as words. For our purposes, it is enough to describe the normal form and specify the complexity of the algorithm for computing it. The details of the algorithm, along with a thorough and accessible presentation of the relevant facts about braids, can be found in [@Dehornoy08].
We first observe that the word problem is easily shown to be decidable if we restrict our attention to an important subset of $B_n$. Note that the presentation can also be viewed as a presentation of a monoid, which we denote by $B_n^+$. The elements of $B_n^+$ are called *positive braids*, and are words in the generators $\sigma_i$ only (no inverses), up to equivalence determined by the relations in . Since all the relations of $B_n$ preserve word length, and there are only finitely many words of any given length, we can decide the word problem (albeit very inefficiently) simply by trying all possible combinations of the relations.
Building upon this, one can give an (inefficient) algorithm for the word problem on $B_n$ itself [@Gonzalez10]. First, given two elements $a, b$ of $B_n^+$, we write $a \preccurlyeq b$ if there exists $z \in B_n^+$ such that $b = az$; in this case we say that $a$ is a *left divisor* of $b$. Similarly, we write $a
\succcurlyeq b$ if there exists $y \in B_n^+$ such that $b = ya$; in this case we say that $a$ is a *right divisor*[^5] of $b$. The center of $B_n$ is the cyclic group generated by $\Delta_n^2$, where $$\Delta_n := \Delta_{n-1}\sigma_{n-1} \sigma_{n-2} \cdots \sigma_1 \in B_n^+$$ (see p.30 of [@Gonzalez10] for a simple proof). Geometrically, $\Delta_n$ implements a twist by $\pi$ in the $z$-plane as the strands move from $z=0$ to $z=1$. One can show that $\sigma_i \preccurlyeq
\Delta_n$ for all $i$, i.e. there exists $x_i \in B_n^+$ such that $\sigma_i^{-1} = x_i\Delta_n^{-1}$. Given a word $w$ in the $\sigma_i$ and their inverses, we first replace the leftmost instance of an inverse generator (say it is $\sigma_i^{-1}$) with $x_i \Delta_n^{-1}$. We then insert $\Delta_n^{-1} \Delta_n$ in front of $x_i$, and observe that conjugating a positive braid $x$ by $\Delta_n$ results in another positive braid (specifically, the rotation of $x$ by $\pi$ in the $z$-plane). In this way, we can push $\Delta_n^{-1}$ all the way to the left. We repeat this process for each inverse generator appearing in the word, resulting in a word of the form $\Delta_n^p b$ where $p
\in \mathbb{Z}$ and $b \in B_n^+$. Since we can solve the word problem in $B_n^+$, we can factor out the maximal power of $\Delta_n$ appearing as a left divisor of $b$. We thus have that, as elements of the braid group, $w = \Delta_n^{p'}b'$ with $\Delta_n$ not a left divisor of $b'$ and $p'$ unique. This solves the word problem in $B_n$.
We can make the above algorithm efficient by finding an efficiently computable normal form for a positive braid word $b$ that does not have $\Delta_n$ as a left divisor. Recall that the symmetric group $S_n$ has a remarkably similar presentation to $B_n$. Indeed, starting with , letting $\sigma_i = (i~i+1)$ and adding the relations $\sigma_i^2 = 1$ for all $i$ results in the standard presentation of $S_n$. In other words, there is a surjective homomorphism $\pi: B_n \rightarrow S_n$ with $\sigma_i \mapsto
(i~i+1)$. In terms of the geometric interpretation, a braid is mapped to the permutation on $[n]$ defined by the connections between the top and bottom “pegs,” as in Figure \[braids-example\]. For each $\sigma\in S_n$, there is a unique preimage of $\sigma$ that can be drawn so that any given pair of strands cross only in the positive direction, and at most once. We call such braids *simple braids*, and they form a subset of $B_n^+$ of size $n!$.
*p.4 of [@Dehornoy08].*
1. A sequence of simple braids $(s_1, \dots, s_p)$ is said to be *normal* if, for each $j$, every $\sigma_i$ that is a left divisor of $s_{j+1}$ is a right divisor of $s_j$.
2. A sequence of permutations $(f_1, \dots, f_p)$ is said to be *normal* if, for each $j$, $f_{j+1}^{-1}(i) > f_{j+1}^{-1}(i+1)$ implies $f_j(i) > f_j(i+1)$.
A sequence of simple braids $(s_1, \dots, s_p)$ is normal if and only if the sequence of permutations $(\pi(s_1), \dots, \pi(s_p))$ is normal. Given a permutation $f \in S_n$, let $\hat f$ denote the simple braid of $B_n$ satisfying $\pi (\hat f) = f$.
*p.4 of [@Dehornoy08] and Ch.9 of [@Epstein92].*\[thm:normal-form\]
1. Every braid $z$ in $B_n$ admits a unique decomposition of the form $\Delta_n^m s_1 \dots s_p$ with $m \in \mathbb{Z}$ and $(s_1,
\dots, s_p)$ a normal sequence of simple braids satisfying $s_1 \neq
\Delta_n$ and $s_p \neq 1$.
2. Every braid $z$ in $B_n$ admits a unique decomposition of the form $\Delta_n^m \hat f_1 \dots \hat f_p$ with $m \in \mathbb{Z}$ and $(f_1, \dots, f_p)$ a normal sequence of permutations satisfying $f_1
\neq \pi(\Delta_n)$ and $f_p \neq 1$.
The most efficient algorithms for computing the normal form of a word $w$ in the generators of $B_n$ have complexity $O(|w|^2 n
\log n)$ [@Dehornoy08].
Obfuscation of Classical Reversible Circuits {#sec:classical}
============================================
Reversible Circuits {#sec:reversible}
-------------------
In the next section, we will describe a gate $R$ which is universal for classical computation and satisfies Definition \[def:partial-indistinguishability\] when $\Gamma$ is the set of relations of the braid group. Because group elements are invertible, $R$ must be a reversible gate, that is, it must bijectively map its possible inputs to its possible outputs. We will thus work in the setting of *reversible classical circuits*. These circuits are composed entirely of reversible gates. For more background on reversible computation see [@Bennett; @Fredkin_Toffoli; @Nielsen_Chuang].
Because reversible circuits cannot erase any information, they operate using ancillary dits (“ancillas”) to store unerasable data left over from intermediate steps in the computation. A reversible circuit evaluating a function $f:\{0,\ldots,d-1\}^n \to \{0,\ldots,d-1\}^m$ thus operates on $r \geq \max(n,m)$ dits, where $r-n$ of the input dits are work dits to be initialized to some fixed value independent of the problem instance, and $r-m$ of the output dits contain unerasable leftover data, to be ignored. Efficient procedures are known for compiling arbitrary logic circuits into reversible form, e.g., by using the Toffoli (or CCNOT) gate [@Bennett; @Fredkin_Toffoli].
In adapting Definitions \[def:indistinguishability\] and \[def:partial-indistinguishability\] to reversible circuits, one is faced with two natural choices for the notion of functional equivalence. One may either demand that the original and obfuscated circuits implement the same function $f:\{0,1\}^n \to \{0,1\}^m$, ignoring the ancilla dits (*weak equivalence*), or demand that they implement the same transformation on the entire set of $r$ dits, including the ancillas (*strong equivalence*). Our constructions will satisfy the latter. Strong equivalence implies weak equivalence, so our construction proves that both possible definitions of partial-indistinguishability are polynomial-time achievable when $\Gamma$ is the set of relations of the braid group. We remark that, as with ordinary irreversible circuits, determining if two arbitrary reversible circuits are equivalent (weakly or strongly) is coNP-complete [@Jordan13].
Classical computation with braids {#sec:universal}
---------------------------------
We now briefly describe a classical reversible gate $R$ which satisfies the braid relations. The complete details of the construction and the proof of universality of $R$ are given in Appendix \[universal\]. Taken together with Theorem \[thm:normal-form\], this yields an obfuscator satisfying Definition \[def:partial-indistinguishability\].
Let $G$ be a finite group and set $d = |G|$. Consider the reversible gate $R$ that acts on pairs of dits encoding group elements by $$\label{qdouble}
R(a,b) = (b, b^{-1}ab).$$ Let $R_i$ denote $R$ acting on the $i$ and $(i+1)^{\mathrm{th}}$ wires of a circuit. By direct calculation, one can check that the set $\{R_1,\dots,R_{n-1}\}$ satisfies the braid relations, that is, $$\begin{array}{rcll}
R_i R_j & = & R_j R_i & \forall \ |i-j| \geq 2\\
R_i R_{i+1} R_i & = & R_{i+1} R_i R_{i+1}
& \forall \ i.
\end{array}$$ In 1997, Kitaev discovered that the gate set $\{R, R^{-1}\}$ is universal for classical reversible computation when $G$ is the symmetric group $S_5$ [@Kitaev03]. Ogburn and Preskill subsequently showed that the alternating group $A_5$, which is half as large as $S_5$, is already sufficient [@OP99]. The universality construction for $A_5$ was subsequently presented in greater detail and generalized to all non-solvable groups by Mochon [@Mochon]. To make our presentation more accessible and self-contained, we give in Appendix \[universal\] an explicit description of Mochon’s universality construction in the the case $G = A_5$. The construction proves computational universality by showing how to efficiently compile Toffoli circuits into $R$-circuits.
Given any $R$-circuit, we can apply the algorithm of Theorem \[thm:normal-form\] by interpreting each $R_i$ as $\sigma_i$ and each $R_i^{-1}$ as $\sigma_i^{-1}$. This leads to partial-indistinguishability obfuscation of $R$-circuits. A discussion of whether this can also yield meaningful obfuscation for classical circuits constructed from other gate sets is given in Section \[attacks\].
Quantum Circuits {#sec:quantum}
================
Quantum computation with braids {#fibonacci}
-------------------------------
In Section \[sec:universal\] and Appendix \[universal\], we discuss classical universality of circuits encoded as braids. It turns out that an analogous theory can be developed for quantum circuits, and is well-understood. The family of so-called Fibonacci representations of the braid groups have dense image in the unitary group, and there are efficient classical algorithms for translating any quantum circuit into a braid (and vice-versa) in a way that preserves unitary functionality [@FLW02]. A brief synopsis of these facts is given below. We remark that there are in fact many unitary representations of the braid groups that satisfy these properties, and which are physically motivated by the so-called fractional quantum Hall effect. In this setting, the image of these representations consists of unitary operators which describe the braiding of excitations in a 2-dimensional medium [@Kitaev03].
Approachable descriptions of the Fibonacci representation are given in [@Shor_Jordan; @Trebst]. In [@Shor_Jordan], what we call the “Fibonacci representation” here, is called the “$\star \star$” irreducible sub-representation. This is a family of representations $\rho^{(n)}_{\mathrm{Fib}}:B_n \rightarrow U(F_{n-4})$, where $F_k$ is the $k$-th Fibonacci number. For our application, the essential properties of the Fibonacci representation are *locality* and *local density*. These two properties mean that, under a certain qubit encoding, braid generators correspond to local unitaries, and local unitaries correspond to short braid words. Standard arguments from quantum computation tell us that we can achieve the latter to precision $\epsilon$ with $O(\log^{2.71}(1/\epsilon))$ braid generators by means of the Solovay-Kitaev algorithm [@Dawson_Nielsen].
A natural basis for the space of $\rho^{(n)}_{\mathrm{Fib}}$ can be identified with strings of length $n$ from the alphabet $\{\star, p\}$, which begin with $\star$, end with $p$, and do not contain “$\star \star$” as a substring[^6]. Following [@AA11][^7], for $n$ a multiple of four, we identify a particular subspace $V_n$ of $\rho^{(n)}_{\mathrm{Fib}}$ by discarding some basis elements, as follows. Partition a string $s$ into substrings of length four. If each of these substrings is equal to either $\star p
\star p$ (this will encode a $0$) or $\star p p p$ (this will encode a $1$), then the basis element corresponding to $s$ is in $V_n$; otherwise, it is not. Note that $\dim V_n = 2^{n/4}.$ The following theorem follows from [@AA11; @Dawson_Nielsen].
\[unifib\] There is a classical algorithm which, given an $(n/4)$-qubit quantum circuit $C$ and $\epsilon>0$, outputs a braid $b \in B_n$ of length $O(|C|\log^{2.71}(1/\epsilon))$ satisfying $$\left\|C - \left.\rho^{(n)}_{\mathrm{Fib}}(b)\right|_{V_n}\right\| \leq \epsilon~;$$ this algorithm has complexity $O(|b|)$.
For the opposite direction, we can identify a subspace $W_n
\subset (\mathbb C_2)^{\otimes n}$ by discarding all bitstrings except those that start with $0$, end with $1$ and do not have “$00$” as a substring. Then $\dim W_n = \dim \rho^{(n)}_{\mathrm{Fib}}$ and we have the following.
There is a classical algorithm which, given $b \in B_n$ and $\epsilon>0$, outputs a quantum circuit $C$ on $n$ qubits of length $O(|b|\log^{2.71}(1/\epsilon))$ such that $$\left\|\left.C\right|_{W_n} - \rho^{(n)}_{\mathrm{Fib}}(b)\right\| \leq \epsilon~;$$ this algorithm has complexity $O(|C|)$.
The two algorithms in the above theorems are described explicitly in [@AA11].
Obfuscating quantum computations
--------------------------------
While the state of knowledge about classical obfuscation is limited, essentially nothing is known about the quantum case. Here we discuss how to use the facts from the previous section to construct a partial-indistinguishability obfuscator for quantum circuits.
In light of Theorem \[unifib\], $\{\rho_{\mathrm{Fib}}(\sigma_1),\ldots,\rho_{\mathrm{Fib}}(\sigma_{n-1})\}$ may be regarded as a universal set of elementary quantum gates. By the homomorphism property of $\rho_{\mathrm{Fib}}$, this set satisfies the braid relations. These gates differ from conventional quantum gates in that they do not possess locality defined in terms of a strict tensor product structure. Nevertheless, as shown above, the power of unitary circuits composed from these gates is equivalent to standard quantum computation. By interpreting each $\rho_{\mathrm{Fib}}(\sigma_j)$ as a braid-group generator $\sigma_j$, we can apply the algorithm from Theorem \[thm:normal-form\] directly to circuits from this gate set, resulting in a partial-indistinguishability obfuscator satisfying Definition \[def:partial-indistinguishability\].
With the algorithms from the previous section in hand, we could also attempt to apply the obfuscation algorithm, Algorithm \[obfuscator\], directly to quantum circuits. For an input circuit $C$ on $n$ qubits, the running times of both of this algorithm is $O(|C|^2 n \cdot \text{polylog}(n, 1/\epsilon))$. The length of the output cannot be longer than the running time. We are not aware of a better upper bound for the length of the output. The security of this algorithms is questionable, and some attacks and possible countermeasures are discussed in Section \[attacks\].
Note that reduction of arbitrary quantum circuits to a normal form using a *complete* set of gate relations should not be possible in polynomial time; this would yield a polynomial-time algorithm for deciding whether a quantum circuit is equivalent to the identity, which is a coQMA-complete problem [@JWB03].
Testing claimed quantum computers with a quantum obfuscator
-----------------------------------------------------------
It is natural to consider quantum analogues of the applications of obfuscation from classical computer science. We now consider a potential application of quantum circuit obfuscation that does not fit this mold: testing claimed quantum computers. A similar proposal using a restricted class of quantum circuits has been previously made in [@SB09].
Suppose Bob claims to have access to a universal quantum computer with some fixed finite number of qubits. Alice has access to a classical computer only, as well as a classical communication channel with Bob. Can Alice determine if Bob is telling the truth? Barring tremendous advances in complexity theory, a provably correct test is unlikely;[^8] can we still design a test in which we have a high degree of confidence? Given the extensive work on classical algorithms for factoring, a reasonable idea is to simply ask Bob to factor a sufficiently large RSA number. However, Shor’s algorithm only begins to outperform the best classical algorithms when thousands of logical qubits can be employed. A much smaller universal quantum computer (e.g., a few dozen qubits) is likely to be a far simpler engineering challenge and could still be quite useful, e.g., for simulating certain quantum systems. A test that works in this case would thus be very valuable. We now outline a new proposal for such a test. Simply put, we propose asking questions that are classically easy to answer, but posing them in an obfuscated manner. In this test, Alice would repeatedly generate quantum circuits and ask Bob to run them. At least some of the circuits would in fact be quantumly-obfuscated classical reversible circuits, allowing Alice to easily check the answers. Previous work has yielded tests of quantum computers in the case that the verifier can perform some limited quantum operations [@BFK08; @ABE08].
We have considerable freedom when designing an obfuscation-based test of quantum computers. How to choose these parameters in a way that makes the test difficult to fool with a classical computer is an open question. For purposes of illustration, we give one example. Let $\mathcal O$ be the obfuscation algorithm for quantum circuits described above.
Clearly, $k$ must be chosen so that $n$ is smaller than the number of logical qubits Bob claims to control. To fool Alice, a purely classical Bob must determine the parity of $s$. The dictionary attack (*i.e.* Bob repeatedly guesses at $k$, obfuscates the corresponding circuit, and compares the result to the circuit given by Alice) is of no use provided $k$ is reasonably large, e.g., 80 bits, which can be encoded using a braid of 115 strands using the Zeckendorf encoding described in [@Shor_Jordan].
We now show that there can be no efficient general-purpose algorithm for breaking our test by detecting whether a given quantum circuit is in fact (almost) classical, and if so, simulating it.
\[def:class\] Let $c$ be a bit string specifying a quantum circuit via a standard universal set $Q$ of quantum gates, and let $U_c$ be the corresponding unitary operator. Fix some constants $r,d,a \in \mathbb{N}$, and fix a set $R$ of reversible gates. The problem $\mathrm{CLASS}(r,d,a,Q,R)$ is to find a reversible circuit of at most $r |c|^d$ gates from $R$ such that the corresponding permutation matrix $P$ satisfies $\|U_c - P \| \leq 2^{-a |c|}$.
Note that $\mathrm{CLASS}(r,d,a,Q,R)$ is not a decision problem. Thus, to formulate the question of whether this problem can be efficiently solved, we must ask not whether $\mathrm{CLASS}(r,d,a,Q,R)$ is contained in P but whether it is contained in FP. We now provide some formal evidence that this is not the case. Note that the following theorems continue to hold if we change the classicality condition in Definition \[def:class\] to $\|U_c - P \|
\leq |c|^{-a}$.
\[th:testing\] For any fixed $r,d,a \in \mathbb{N}$, any universal reversible gate set $R$, and any universal quantum gate set $Q$, if $\mathrm{CLASS}(r,d,a,Q,R) \in \mathrm{FP}$ then $\mathrm{QCMA} \subseteq \mathrm{P}^{\mathrm{NP}}$.
Note that, $\mathrm{QCMA} \subseteq \mathrm{P}^{\mathrm{NP}}$ would be very surprising because, among other things, it would imply $\mathrm{BQP} \subseteq \mathrm{PH}$, and there is evidence that this is false [@Aaronson; @Fefferman].
The standard QCMA-complete language $\mathcal{L}$ is as follows. Let $\mathcal C$ be the set of all quantum circuits (expressed as a concatenation of bitstrings that index elements of the gate set $Q$). $\mathcal C$ decomposes as the disjoint union of $\mathcal{L}$ and $\bar{\mathcal{L}}$ where $\mathcal{L}$ consists of the quantum circuits that accept at least one classical (*i.e.* computational basis state) input, and $\bar{\mathcal{L}}$ consists of the circuits that reject all inputs. Given a quantum circuit $V_1 \in \mathcal C$, (the “verifier”) we can amplify it using standard techniques [@Marriott_Watrous; @Nagaj_Wocjan_Zhang] to accept YES instances with probability at least $1-O(2^{-n})$ and accept NO instances with probability at most $O(2^{-n})$. Let $V_2$ be such an amplified verifier. Further, let $$\begin{array}{lcr}
V_3 & = &
\begin{array}{c}
\Qcircuit @C=1em @R=.5em {
& \qw & \targ & \qw & \qw \\
& \multigate{1}{V_2} & \ctrl{-1} & \multigate{1}{V_2^{-1}} & \qw \\
& \ghost{V_2} & \qw & \ghost{V_2^{-1}} & \qw
}
\end{array}
\end{array}$$ where the second-to-top qubit is the acceptance qubit of $V_2$. If $V_i \in \bar{\mathcal{L}}$ then $\| V_3 - {\mathds{1}}\| = O(2^{-n})$. By assumption, there exists a polynomial time classical algorithm for solving $\mathrm{CLASS}(r,d,a,Q,R)$. When presented with $V_3$, this algorithm will produce a polynomial-size reversible circuit $V_4$ strongly equivalent to the identity. By querying an oracle for the problem of strong equivalence of reversible circuits, one can decide whether $V_4$ is equivalent to the circuit of no gates, and hence to the identity operation. If $V_1 \in \bar{\mathcal{L}}$, this oracle will accept. If $V_1 \in \mathcal{L}$ then the algorithm for problem 1 will answer NO or produce a circuit that this oracle rejects. As shown in [@Jordan13], the problem of deciding strong equivalence of reversible circuits is contained in coNP. Thus, we can decide QCMA in $\mathrm{P}^{\mathrm{coNP}}$, which is equal to the more familiar complexity class $\mathrm{P}^{\mathrm{NP}}$.
Some Attacks {#attacks}
============
Compiler attacks {#compilation}
----------------
The security or insecurity of braid-based partial-indistinguishability obfuscation remains an area of current investigation. From a purely information-theoretic point of view, the power of this obfuscation comes from the many-to-one nature of the map ${\mathsf{N}}$ that takes arbitrary braid words to their normal form. If the initial braid words are highly structured because they are obtained by compilation from a different gate set, then this can undermine or destroy the many-to-one feature of ${\mathsf{N}}$.
In Section \[sec:universal\], we describe a reversible gate $R$ on pairs of 60-state dits, corresponding to elements of $A_5$, that obeys the relations of the braid group and can perform universal classical computation. The gate itself and the proof that it is universal come from the quantum computation literature [@Kitaev03; @OP99; @Mochon]. Appendix \[universal\] recounts the universality proof of [@Mochon], which can be viewed as a compiler ${\mathsf{B}}_{R}$ that maps circuits constructed from the well-known universal reversible Toffoli gate into circuits constructed from the $R$ gate. As a cautionary example, we now show that naively obfuscating Toffoli circuits using the composed map ${\mathsf{N}}\circ
{\mathsf{B}}_{R}$ is completely insecure.
The construction in Appendix \[universal\] gives a general mapping from a Toffoli gate to a corresponding braid. We will refer to braids obtained in this way as *Toffoli braids*. Recall that the normal form of a braid in $B_n$ has the form $\Delta_n^m s_1\dots s_p$ for a normal sequence of simple braids $(s_1,\dots,s_p)$. A Toffoli braid obtained from a Toffoli with controls $c_1$ and $c_2$ and target $t$ has normal form $$\label{eq:tof-braid}
\Delta_n^0 s_1(c_1,c_2,t)s_2s_3s_4s_5s_6s_7s_8s_9(c_1,c_2,t)s_{10}s_{11}s_{12}s_{13}(c_1,c_2,t)s_{14}(t).$$ The factors $s_2,\dots,s_8,s_{10},s_{11}$ and $s_{12}$ only depend on $n$, and not on the wires $c_1$, $c_2$ or $t$. Note that this is a positive braid — consisting only of $\sigma_1,\dots,\sigma_{n-1}$ and none of their inverses. Any product of such braids will thus also be a positive braid, so attempting to obfuscate a circuit in Toffoli gates using this construction will yield only positive braids.
Because Toffoli is a 3-bit gate, there are only $\binom{n}{3}$ ways to apply a Toffoli to $n$ bits. Thus, one may, in polynomial time, test each of these $\binom{n}{3}$ possibilities as a guess for the last gate of the obfuscated circuit. One performs the test by compiling the guessed Toffoli gate into a braid, appending the inverse of this braid to the normal form braid produced as the output the obfuscator, and then reducing the resulting braid to normal form. If the guess is correct, then the resulting braid is still a braid corresponding to a circuit — the original obfuscated circuit with its last Toffoli gate removed — and thus this will result in a positive braid. If the guess is incorrect, then appending the inverse of a positive braid, which consists entirely of $\sigma_1^{-1},\dots,\sigma_{n-1}^{-1}$, might result in a braid that is no longer positive — that is, has a negative power of $\Delta_n$, and this seems to be the case with *any* wrong guess, based on some limited tests. Furthermore, the presence of a negative power of $\Delta_n$ is efficiently recognizable, so it is immediately clear whether or not the guess was correct.
This attack is related to so-called length-based attacks. These have been introduced in the cryptanalysis of braid based key-exchanged protocols [@HS03]. In the present context, the natural length-based attack is to guess the final gate, append the inverse of the corresponding braid to the normal-form braid produced by the obfuscator, and the reduce the product braid to normal form. If the result is a shorter word in the braid-group generators than the original normal form, then this can be taken as heuristic evidence that the guess was correct. Intuitively, one expects that the longer the braid words are that implement individual gates from the original gate set, then the better such attacks should work.
One can easily propose modifications to the naive obfuscator ${\mathsf{N}}\circ {\mathsf{B}}_{R}$ that thwart guessing-based attacks such as the two attacks described above. In particular, one finds that the gate $R$ described in Appendix \[universal\] has order $60$. Hence, one can start with the positive Toffoli braid in equation and then each generator $\sigma_i$ can independently, with probability $\frac{1}{2}$, be replaced with $\sigma_i^{-59}$, without altering the functionality of the circuit. The number of generators in a Toffoli braid depends on $n$, and which wires the Toffoli acts on, but there are always at least 124. Thus, each gate will be compiled into one of $2^{124}$ braid-words uniformly at random. Thus, guessing-based attacks on the composition of this compiler with ${\mathsf{N}}$ may become impractical. Whether such a scheme is vulnerable to other attacks remains an open question for future research.
Dictionary attacks {#dictionary}
------------------
The partial-indistinguishability obfuscator described in the preceding sections deterministically maps input circuits to obfuscated circuits. This creates a potential weakness in the obfuscation. Suppose Alice wishes to run a computation $C$ on Bob’s server but does not wish Bob to know what computation she is running. Thus, she sends the obfuscated circuit $\mathcal{O}(C)$ to Bob, who executes it, and returns the result. To improve security, Alice may instead use a circuit $C'$ in which her desired input is hard-coded, and which applies a one-time pad at the end of the computation. If the obfuscation is secure, then Bob is unlikely to learn anything about $C$, the input, or the output. However, if Bob knows that the circuits Alice is likely to want to execute are drawn from some small set $S$, then Bob can simply compute $\{\mathcal{O}(s)|s \in S\}$ and identify Alice’s computation by finding it in this list. Such attacks are sometimes called “dictionary” attacks after the practice of recovering passwords by feeding all words from a dictionary into the hash function and comparing against the hashed password.
Dictionary attacks may or may not be a serious threat to our obfuscation scheme, depending on the the size of the set of likely circuits to be obfuscated. In cryptographic applications where dictionary attacks are a concern, the standard way to protect against them is to append random bits prior to encryption. (In the context of hashing passwords, this practice is called “salting”.) Such a strategy can be applied to our obfuscator, but some care must be taken in doing so. The most obvious strategy is to append a random circuit on the output ancillas prior to obfuscation. However, attackers can defeat this countermeasure by using the polynomial-time algorithms for computing left-greatest-common-divisors in the braid group [@Epstein92]. However, prior to obfuscation, one may introduce extra dits, and apply random circuits before, after, and simultaneously with the computation, in a way so as not to disrupt it. The problem of optimizing the details of this procedure so as to maximize security and efficiency is left to future work.
Future Work
===========
Classical and quantum universality
----------------------------------
It is of interest to consider other computationally universal representations of the braid group, which might provide more efficient translations from circuits to braids. One avenue for obtaining such representations is by finding other solutions to the Yang-Baxter equation, besides the operator $R$ from Appendix \[universal\]. Our investigations so far prove that no permutation matrix solution of dimension up to $16 \times 16$ is a universal gate and suggest that no permutation matrix solution of dimension $25 \times 25$ is a universal gate. In the quantum case, it has been shown that no $4 \times 4$ unitary solution is universal [@ABJ12].
More generally, one may look for other finitely-generated groups with computationally universal representations and efficiently computable normal forms. One potential candidate family are the mapping class groups MCG$(\Sigma_g)$ of unpunctured surfaces of genus $g$. These groups also have quantumly universal representations [@Alagic] and an efficiently solvable word problem [@Hamidi-Tehrani]. It is not known if there are also classically universal permutation representations, or if there are efficiently computable normal forms.
Expanding the set of indistinguishability relations
---------------------------------------------------
By [@Jordan13], achieving efficient indistinguishability obfuscation for the complete set of relations of a universal gate set is unlikely. However, it is possible that partial-indistinguishability obfuscation on $R$ gates could be achieved with a larger set of relations than the braid relations. For example, the universal reversible gate described in Appendix \[universal\] has order 60. If we add the relations $\sigma_i^{60} = {\mathds{1}}$ for $i=1,2,\ldots,n-1$ to $B_n$, we obtain a “truncated” (but still infinite for large $n$ [@Coxeter]) factor of the braid group. If a normal form can still be computed in polynomial time for this group then one could construct an efficient obfuscator using the relations of this truncated group, which would be strictly stronger than our braid group obfuscator. This approach also provides motivation for finding a complete set of relations for the gate $R$.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Anne Broadbent, Rainer Steinwandt, Scott Aaronson, Bill Fefferman, Leonard Schulman, Robert König, and Yi-Kai Liu for helpful discussions. We also thank Mariano Suárez-Alvarez and Gjergji Zaimi for leading us to reference [@Coxeter] via `math.stackexchange` and `mathoverflow`. Portions of this paper are a contribution of NIST, an agency of the US government, and are not subject to US copyright.
Classical Computation with Braids {#universal}
=================================
In this section, we present a reversible gate $R$ on pairs of 60-state dits that can perform universal computation and obeys the relations of the braid group. The universality construction for this gate comes from the quantum computation literature [@Kitaev03; @OP99; @Mochon], but we present it here in purely classical language to make it accessible to a broader audience.
Suppose we arrange $n$ dits on a line, and allow $R$ to act only on neighboring dits. Further, we do not allow $R$ to be applied “upside-down”. Then, there are $n-1$ choices for how to apply $R$. We label these $R_1, R_2, \ldots, R_{n-1}$, as illustrated in Figure \[reversible\]. Each of $R_1,\ldots,R_{n-1}$ corresponds to a $d^n \times d^n$ permutation matrix. Specifically, $R_j$ is obtained by taking the tensor product of $R$ with identity matrices according to $R_j = {\mathds{1}}_{d \times d}^{\otimes (j-1)} \otimes R \otimes {\mathds{1}}_{d \times d}^{\otimes (n-j-1)}$.
@C=1em @R=.5em [ & & & &\
& & & &\
& & & &\
& & & & ]{}
$R_1,\ldots,R_{n-1}$ generate a subgroup of $S_{d^n}$. Among others, these generators obey the relations $$R_i R_j = R_j R_i \quad \forall |i-j| \geq 2.$$ If $R$ satisfies $$\label{Yang-Baxter}
R_1 R_2 R_1 = R_2 R_1 R_2$$ then $$R_i R_{i+1} R_i = R_{i+1} R_i R_{i+1} \quad \forall i$$ and in this case the gates $R_1,\ldots,R_{n-1}$ satisfy all the relations of the braid group $B_n$. In other words, the map defined by $\sigma_i \mapsto R_i$ and $\sigma_i^{-1} \mapsto R_i^{-1}$ is a homomorphism from $B_n$ to $S_{d^n}$, *i.e.* a representation of the braid group. Note that this representation is never faithful as $B_n$ is infinite.
The condition \[Yang-Baxter\] is known as the Yang-Baxter equation[^9]. Finding all the matrices satisfying the Yang-Baxter equation at a given dimension has only been achieved at $d=2$ [@Hietarinta]. However, certain systematic constructions coming from mathematical physics can produce infinite families of solutions. In particular, let $G$ be any finite group, and let $R$ be the permutation on the set $G \times G$ defined by $$R(a,b) = (b, b^{-1}ab).$$ By direct calculation one sees that any such an $R$ satisfies the Yang-Baxter equation. (In physics language, $R$ comes from the braiding statistics of the magnetic fluxes in the quantum double of $G$.)
In 1997, Kitaev discovered that choosing $G$ to be the symmetric group $S_5$ yields an $R$ gate sufficient to perform universal reversible computation [@Kitaev03]. Ogburn and Preskill subsequently showed that the alternating group $A_5$, which is half as large as $S_5$, is already sufficient. The universality construction for $A_5$ was subsequently presented in greater detail and generalized to all non-solvable groups by Mochon [@Mochon]. In the remainder of this section we give a self-contained exposition of the universality construction from [@Mochon], shorn of physics language.
To obtain a representation of the braid group, we must strictly enforce the requirement that application of $R$ to neighboring dits on a line is the only allowed operation. In particular, we are not given as elementary operations the ability to apply $R$ upside-down, or to non-neighboring dits, or to move dits around. Thus, to prove computational universality, it is helpful to first construct a SWAP gate from $R$ gates, which exchanges neighboring dits. As is well-known, the $n-1$ swaps of nearest neighbors on a line generate the full group $S_n$ of permutations, and thus a SWAP gate enables application of $R$ to any pair of dits.
For $R$ gates of the form , two pairs of inverse group elements in the order $a,a^{-1},b,b^{-1}$ can be swapped by applying the product $R_2 R_3 R_1 R_2$. Thus, in the construction of [@OP99; @Mochon], elements of $A_5$ are always paired with their inverses. This can be regarded as a form of encoding; $|A_5| = 60$, so each 60-state dit is encoded by a corresponding pair of elements of $A_5$. We introduce the notation $\widetilde{g} \equiv
(g,g^{-1})$ for this encoding, and similarly, abbreviate the encoded swap operation as follows. $$\begin{array}{lcccr}
\begin{array}{c}
\Qcircuit @C=1em @R=.5em {
\lstick{\widetilde{a}} & \multigate{1}{S} & \rstick{\widetilde{b}} \qw \\
\lstick{\widetilde{b}} & \ghost{S} & \rstick{\widetilde{a}} \qw
}
\end{array}
& \ & \equiv & \quad &
\begin{array}{c}
\Qcircuit @C=1em @R=.5em {
\lstick{a^{\phantom{-1}}} & \qw & \multigate{1}{R} & \qw & \qw & \rstick{b} \qw\\
\lstick{a^{-1}} & \multigate{1}{R} & \ghost{R} & \qw & \multigate{1}{R} & \rstick{b^{-1}} \qw \\
\lstick{b^{\phantom{-1}}} & \ghost{R} & \qw & \multigate{1}{R} & \ghost{R} & \rstick{a} \qw\\
\lstick{b^{-1}} & \qw & \qw & \ghost{R} & \qw & \rstick{a^{-1}} \qw
}
\end{array}
\end{array}$$ Similarly, the sequence $R_2 R_3 R_3 R_2$ performs the transformation $(\widetilde{a},\widetilde{b}) \mapsto
(\widetilde{a},\widetilde{aba^{-1}})$ on a pair of encoded dits. We abbreviate this in circuit diagrams as follows. $$\begin{array}{lcccl}
\begin{array}{c}
\Qcircuit @C=1em @R=.5em {
\lstick{\widetilde{a}} & \ctrl{1} & \rstick{\widetilde{a}} \qw \\
\lstick{\widetilde{b}} & \gate{C} & \rstick{\widetilde{aba^{-1}}} \qw
}
\end{array}
& \quad \quad & \equiv & \quad &
\begin{array}{c}
\Qcircuit @C=1em @R=.5em {
\lstick{a^{\phantom{-1}}} & \qw & \qw & \qw & \qw & \rstick{a} \qw\\
\lstick{a^{-1}} & \multigate{1}{R} & \qw & \qw & \multigate{1}{R} & \rstick{a^{-1}} \qw \\
\lstick{b^{\phantom{-1}}} & \ghost{R} & \multigate{1}{R} & \multigate{1}{R} & \ghost{R} & \rstick{aba^{-1}} \qw\\
\lstick{b^{-1}} & \qw & \ghost{R} & \ghost{R} & \qw & \rstick{ab^{-1}a^{-1}} \qw
}
\end{array} \\ \\
\begin{array}{c}
\Qcircuit @C=1em @R=.5em {
\lstick{\widetilde{a}} & \ctrl{1} & \rstick{\widetilde{a}} \qw \\
\lstick{\widetilde{b}} & \gate{C^{-1}} & \rstick{\widetilde{a^{-1}ba}} \qw
}
\end{array}
& \quad \quad & \equiv & \quad &
\begin{array}{c}
\Qcircuit @C=1em @R=.5em {
\lstick{a^{\phantom{-1}}} & \qw & \qw & \qw & \qw & \rstick{a} \qw \\
\lstick{a^{-1}} & \multigate{1}{R^{-1}} & \qw & \qw & \multigate{1}{R^{-1}} & \rstick{a^{-1}} \qw \\
\lstick{b^{\phantom{-1}}} & \ghost{R^{-1}} & \multigate{1}{R^{-1}} & \multigate{1}{R^{-1}} & \ghost{R^{-1}} & \rstick{a^{-1}ba} \qw \\
\lstick{b^{-1}} & \qw & \ghost{R^{-1}} & \ghost{R^{-1}} & \qw & \rstick{a^{-1}b^{-1}a} \qw
}
\end{array}
\end{array}$$ This notation can easily be extended to provide a shorthand for the sequence of gates needed to implement a $C$ gate between non-neighboring pairs of bits, as illustrated by the following examples. $$\begin{aligned}
\begin{array}{c}
\Qcircuit @C=1em @R=1em {
& \ctrl{3} & \qw \\
& \qw & \qw \\
& \qw & \qw \\
& \gate{C} & \qw
}
\end{array}
& \equiv &
\begin{array}{c}
\Qcircuit @C=1em @R=.5em {
& \qw & \qw & \ctrl{1} & \qw & \qw & \qw \\
& \qw & \multigate{1}{S} & \gate{C} & \multigate{1}{S} & \qw & \qw \\
& \multigate{1}{S} & \ghost{S} & \qw & \ghost{S} & \multigate{1}{S} & \qw \\
& \ghost{S} & \qw & \qw & \qw & \ghost{S} & \qw
}
\end{array} \\ \\ \\
\begin{array}{c}
\Qcircuit @C=1em @R=.5em {
& \gate{C} & \qw \\
& \ctrl{-1} & \qw
}
\end{array}
& \equiv &
\begin{array}{c}
\Qcircuit @C=1em @R=.5em {
& \multigate{1}{S} & \ctrl{1} & \multigate{1}{S} & \qw \\
& \ghost{S} & \gate{C} & \ghost{S} & \qw
}
\end{array}\end{aligned}$$
Next, consider the following product of elements of $A_5$ (which should be read right-to-left). $$f(g_1,g_2) = (521)g_1(14352)g_2(124)g_1^{-1}(15342)g_2^{-1}(521)$$ One sees that $$\begin{aligned}
f((345),(345)) & = & {\mathds{1}}\\
f((345),(435)) & = & {\mathds{1}}\\
f((435),(345)) & = & {\mathds{1}}\\
f((435),(435)) & = & (12)(34)\end{aligned}$$ where ${\mathds{1}}$ denotes the identity permutation. Furthermore, conjugating $(345)$ by $(12)(34)$ yields $(435)$, and conversely, conjugating $(435)$ by $(12)(34)$ yields $(345)$. Thus, we may think of $(345)$ as an encoded zero and $(435)$ as an encoded one, and we see that $$f(g_1,g_2) g_0 f(g_1,g_2)^{-1}$$ toggles $g_0$ between one and zero if $g_1$ and $g_2$ are both encoded ones and leaves $g_0$ unchanged otherwise. Such a doubly-controlled toggling operation is known as a Toffoli gate, which is well-known to be a computationally universal reversible gate [@Fredkin_Toffoli].
As a circuit diagram, this construction can be expressed as follows.
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@C=1em @R=1.5em [ & & & & & & & & & &\
& & & & & & & & & &\
& & & & & & & & & &\
& & & & & & & & & &\
& & & & & & & & & &\
& & & & & & & & & &\
& & & & & & & & & & ]{}
----------------------------------------
Here, if $g_0,g_1,g_2$ encode bits $b_0,b_1,b_2$ then $g_0'$ encodes $b_0 \oplus b_1 \land b_2$. The four ancillary dits $\widetilde{(14352)}$, $\widetilde{(15342)}$, $\widetilde{(124)}$, and $\widetilde{(521)}$, are used to “catalytically” facilitate the construction of a Toffoli gate, and thus computations built from arbitrarily many Toffoli gates can be performed with only one copy of these four dits.
Unpacking the various shorthand notations, one sees that the above circuit represents the following braid of 132 crossings on 14 strands, which encodes a Toffoli gate with the first wire as target, and the second and third wires as controls. $$\label{explicit}
\begin{array}{ccllll}
T & = &
\sigma_{8} \sigma_{9} \sigma_{9} \sigma_{8}
& \sigma_{10} \sigma_{11} \sigma_{9} \sigma_{10}
& \sigma_{10} \sigma_{11} \sigma_{11} \sigma_{10}
& \sigma_{10} \sigma_{11} \sigma_{9} \sigma_{10} \\
& & \sigma_{2} \sigma_{3} \sigma_{1} \sigma_{2}
& \sigma_{4} \sigma_{5} \sigma_{3} \sigma_{4}
& \sigma_{6} \sigma_{7} \sigma_{5} \sigma_{6}
& \sigma_{8} \sigma_{9} \sigma_{9} \sigma_{8} \\
& & \sigma_{6} \sigma_{7} \sigma_{5} \sigma_{6}
& \sigma_{4} \sigma_{5} \sigma_{3} \sigma_{4}
& \sigma_{2} \sigma_{3} \sigma_{1} \sigma_{2}
& \sigma_{12} \sigma_{13} \sigma_{11} \sigma_{12}\\
& & \sigma_{10} \sigma_{11} \sigma_{9} \sigma_{10}
& \sigma_{10} \sigma_{11} \sigma_{11} \sigma_{10}
& \sigma_{10} \sigma_{11} \sigma_{9} \sigma_{10}
& \sigma_{12} \sigma_{13} \sigma_{11} \sigma_{12}\\
& & \sigma_{6} \sigma_{7} \sigma_{5} \sigma_{6}
& \sigma_{8} \sigma_{9} \sigma_{9} \sigma_{8}
& \sigma_{6} \sigma_{7} \sigma_{5} \sigma_{6}
& \sigma_{10} \sigma_{11} \sigma_{9} \sigma_{10}\\
& & \sigma^{-1}_{10} \sigma^{-1}_{11} \sigma^{-1}_{11} \sigma^{-1}_{10}
& \sigma_{10} \sigma_{11} \sigma_{9} \sigma_{10}
& \sigma_{4} \sigma_{5} \sigma_{3} \sigma_{4}
& \sigma_{6} \sigma_{7} \sigma_{5} \sigma_{6}\\
& & \sigma_{8} \sigma_{9} \sigma_{9} \sigma_{8}
& \sigma_{6} \sigma_{7} \sigma_{5} \sigma_{6}
& \sigma_{4} \sigma_{5} \sigma_{3} \sigma_{4}
& \sigma_{12} \sigma_{13} \sigma_{11} \sigma_{12}\\
& & \sigma_{10} \sigma_{11} \sigma_{9} \sigma_{10}
& \sigma^{-1}_{10} \sigma^{-1}_{11} \sigma^{-1}_{11} \sigma^{-1}_{10}
& \sigma_{10} \sigma_{11} \sigma_{9} \sigma_{10}
& \sigma_{12} \sigma_{13} \sigma_{11} \sigma_{12}\\
& & \sigma_{8} \sigma_{9} \sigma_{9} \sigma_{8}
\end{array}$$ Note that we take the convention that this should be read backwards compared to the way one reads English text. This is in keeping with the conventional notation for the composition of functions and our right-to-left multiplication of $R$ matrices. We have used whitespace to divide crossings into groups of four as these correspond to elementary $S$ and $R$ gates.
Given this construction of the Toffoli gate by braid crossings, it is a simple matter to “compile” any given logic circuit into a corresponding braid. $A_5$ has 60 elements. Thus, encoding a single bit into a a pair of $A_5$ elements appears somewhat wasteful. It is natural to try to find Yang-Baxter solutions acting on $d$-state dits for smaller $d$ that achieve universal classical computation. In appendix \[optimizing\], we improve upon the $A_5$-based construction to show that $d=44$ suffices. We have also used exhaustive computer search to find all permutation solutions satisfying the Yang-Baxter equation up to $d=5$ (i.e. up to $25 \times
25$ permutation matrices). Our examination of these solutions suggests that none are computationally universal. Where between $5$ and $44$ lies the minimal $d$ remains an interesting open question.
Optimizing Classical Braid Gates {#optimizing}
================================
In appendix \[universal\] we have recounted the construction of [@Mochon], which shows that the reversible gate $R$, which acts on pairs of 60-state dits and satisfies the Yang-Baxter equation, can perform universal classical computation. In this section, based on a suggestion of Robert König, we show that $R$ can be modified to obtain a gate acting on pairs of 44-state dits that satisfies the Yang-Baxter equation and can perform universal classical computation. Our computational evidence suggests that no reversible gate on $d$-state dits satisfying the Yang-Baxter equation can perform universal computation for $d \leq 5$. Where between 5 and 44 the minimal $d$ lies for which computationally universal reversible Yang-Baxter gates acting on $d$-state qudits exist remains an open question.
The universality construction of [@Mochon], recounted in appendix \[universal\], starts with all dits initialized to states from the following set. $$\begin{aligned}
S & = & \{g,g^{-1}|g \in S_0\} \\
S_0 & = & \{(14352), (15342), (124), (521), (345), (435)\}\end{aligned}$$ Here we show that the orbit of $S$ under the action of the gate $R$ is not all of $A_5$, rather the orbit has only 44 elements. Thus the restriction of the matrix $R$ onto this 44-dimensional subspace is a permutation-matrix that satisfies the Yang-Baxter equation and is capable of universal classical computation.
Recalling , one sees that the orbit $O_R$ of $S$ under $R$ is $$O_R = \{b^{-1}ab|a \in S, b \in \langle S \rangle \}$$ where $\langle S \rangle$ is the subgroup of $A_5$ generated by $S$. A simple computer algebra calculation shows that $\langle S \rangle =
A_5$, thus $O_R$ consists of exactly those elements of $A_5$ conjugate to $S$.
It is well known that the conjugacy classes of $A_5$ are as follows.
---------------------------- ---------------
1\) the identity (1 element)
2\) 3-cycles (20 elements)
3\) conjugates of (12)(34) (15 elements)
4\) conjugates of (12345) (12 elements)
5\) conjugates of (21345) (12 elements)
---------------------------- ---------------
One sees that $O_R$ contains 2), and does not contain 1) or 3). The only remaining question is whether $O_R$ contains both 4) and 5) or just one of them. A simple computer algebra calculation shows that (14352) and (15342) are non-conjugate elements of $A_5$. Hence $O_R$ must contain both 4) and 5). Therefore, $|O_R| = 44$.
[^1]: Institute for Quantum Information, California Institute of Technology, Pasadena, CA, USA. `[email protected]`
[^2]: Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada. `[email protected]`
[^3]: National Institute of Standards and Technology, Gaithersburg, MD, USA. `[email protected]`
[^4]: Our construction for satisfying this definition uses *reversible* gates. The definition of functional equivalence becomes more technical in that context, as discussed in Section \[sec:reversible\].
[^5]: The terminology is not accidental; it turns out that we can also define l.c.m.s and g.c.d.s in $B_n^+$, and that $B_n$ is the group of fractions of $B_n^+$. These facts are some of the achievements of Garside theory [@Garside69].
[^6]: In [@Shor_Jordan] the $\star \star$ subrepresentation of $B_n$ acts on strings of length $n+1$ that begin and end with $\star$. One can leave the initial and/or final $\star$ implicit as these are left unchanged by all braiding operations. We omit the final $\star$ leaving us strings of length $n$ that begin with $\star$ and end with $p$.
[^7]: Reference [@AA11] describes the basis vectors in terms of “paths”. The correspondence between the path notation and the $p \star$ notation is given in appendix C of [@Shor_Jordan].
[^8]: Notice that even a proof that BQP $\neq$ BPP would be insufficient; one would have to find specific problems and instance sizes where some quantum strategy provably beats every classical one. We are thus left with a situation analogous to the practical security guarantees of modern cryptographic systems, which tell us how many bit operations it would take to crack a given instance using the fastest known algorithms.
[^9]: Actually, two slightly different equations go by the name Yang-Baxter in the literature. Careful sources distinguish these as the algebraic Yang-Baxter equation and the braided Yang-Baxter relation (which is sometimes called the quantum Yang-Baxter equation). Equation \[Yang-Baxter\] is the latter. Furthermore, some sources treat a more complicated version of the Yang-Baxter equation in which $R$ depends on a continuous parameter. In such works equation \[Yang-Baxter\] is often referred to as the constant Yang-Baxter equation.
|
---
abstract: 'A simplified form of the time dependent evolutionary dynamics of a quasispecies model with a rugged fitness landscape is solved via a mapping onto a random flux model whose asymptotic behavior can be described in terms of a random walk. The statistics of the number of changes of the dominant genotype from a finite set of genotypes are exactly obtained confirming existing conjectures based on numerics.'
author:
- 'Clément Sire$^1$, Satya N. Majumdar$^2$ and David S. Dean$^1$'
title: 'Exact solution of a model of time-dependent evolutionary dynamics in a rugged fitness landscape '
---
In evolution, long periods of stasis or inactivity are punctuated by bursts of rapid activity. Fossil records [@evo] reveal this basic pattern in the evolution of biological species and the same behavior is observed in the development of microbial populations [@micro] and artificial life [@life]. Not surprisingly, the dynamics of genetic algorithms [@algo] also exhibits this punctuated behavior. In this paper we will show how a simple model of biological evolution can be exactly solved using a mapping onto a random flux model. The important asymptotic details of this random flux model can then be determined in terms of the first passage time distribution of a random walk.
The model we study was introduced in [@krug] as a simplified version of the quasispecies model which is used for the study of large populations of replicating macromolecules [@quasi]. In [@krug], the quasispecies model was studied in the strong selection limit where the location in the space of genotypes is defined as the genotype having the largest population. A shell model [@krug] may be derived in the strong selection limit and a further simplification of this model leads to the i.i.d. (independent and identically distributed) shell model where the natural space of genotypes, which is that of binary sequences, is replaced by a one dimensional lattice. Rather than re-derive the model we shall describe it and the reader will immediately see that it can be reinterpreted in terms of a simple evolutionary process.
We consider an ensemble of $N$ different genotypes labeled by $i=1,2,\ldots N$. The fitness of a genotype is given by its effective rate of reproduction per individual $v_i\ge 0$ and thus the size of the population at time $t$ is given by $n_i(t) =
n_i(0)\exp(v_i t)$. In terms of logarithmic variables, $y_i(t) =
\ln(n_i(t)) = \ln(n_i(0))+ v_i t$. One can interpret $y_i(t)$ as the trajectory of a particle moving ballistically with a non-negative velocity $v_i$, starting from its initial position $y_i(0)$. The i.i.d. version of the shell model [@krug], which we will call the leader model, is defined as follows: we draw $N$ velocities $\lbrace
v_i\rbrace_{1\leq i\leq N}$ independently from the same probability distribution $p(v)$ (which has positive support only). We then consider the semi-infinite lines of slope $v_i$ describing the evolution of genotype $i$ (up to an overall constant) $$y_i(t)=-i+v_i\,t.$$ At any time $t>0$, the leader is defined as the genotype $i$ having the maximum $y_i(t)$, the corresponding $i$ is thus the most populated genotype at time $t$. The choice of $y_i(0) = -i$ comes from the details of the original quasispecies model [@krug]. Thus, the evolution of the trajectories is completely deterministic, the only randomness comes from the velocities. Obviously at $t=0$, $y_1$ is the leader; however if $v_1$ is not the maximal velocity, then $y_1$ will ultimately be overtaken by a faster/fitter genotype. At each of these overtaking events the number of genotypes which have been leaders increases by one, finally the fastest genotype will become the final leader and no more leader changes will occur. In the general context of evolutionary processes these over takings correspond to punctuation events.
The total number of lead changes is denoted by $l_N$ and we denote by $w_k$ the velocity of the leading genotype after the $k$-th lead change. Clearly $l_N$ is a random variable, varying from one realization of velocities to another. Based on simulations, it was observed [@krug] that for large $N$, $\langle l_N\rangle \approx{\beta}\, \ln N$. where, remarkably, the coefficient $\beta$ is rather robust and depends only on the tails of the distribution $p(v)$. Based on numerics, Krug and Karl [@krug] made some conjectures about the value of $\beta$ and also showed how a comparison with record statistics gives the upper bound $\beta <1$. Similar logarithmic growth of the average number of lead changes has also been reported [@krap] recently in the context of growing networks where the leader is the maximally connected node.
In this letter, we present an exact solution to this problem, confirming the conjectures of [@krug]. Moreover, we calculate the variance of $l_N$ and show that $\langle (l_N-\langle
l_N\rangle)^2\rangle\approx \gamma \ln N$ for large $N$, where the coefficient $\gamma$ is calculated exactly and shown to be as robust as $\beta$. We also show that the full distribution of $l_N$ around its mean is asymptotically Gaussian. The key observation that leads to the exact solution of this model is a mapping onto a [*random flux*]{} model whose late time properties are identical to those of the original model. Here, the velocity distribution is chosen as before but instead of fixing the initial positions $y_i(0)$ of the genotype $i$ at $-i$, we chose it to be a random variable uniformly distributed on $[0,-N]$. From a coarse grained point of view, for a large number of genotypes, this difference in the initial condition is not expected to change the asymptotic properties. In the context of the quasispecies model, this random initial condition translates to having the initial population of each genotype having a probability distribution: ${\rm Prob} (n_i(0)=x)=(xN)^{-1}$, with $\exp(-N)\le
x\le 1$. An example set of trajectories for $N=4$ and where $l_N=2$ is shown in Fig. (1).
If $w_k$ is the velocity of the $k$-th leader then clearly only genotypes with velocities greater than $w_k$ can become subsequent leaders. From the rest frame of the leader, in the next time interval $\Delta t$ the genotype $i$, with velocity $v_i \ (>w_k) $, will overtake the leader if it is at a distance $\Delta x = (v_i-w_k)\Delta
t$ behind the leader. The rate at which the genotype $i$ becomes the new leader is thus given by $$r_i = (v_i-w_k)\langle \delta\left(y_k(0)- y_i(0)-(v_i-w_k)t\right)\rangle,$$ where the angled brackets indicate the average over the initial conditions. Given that $y_k(0) > y_i(0)$ the initial distance $d_{ik} =y_k(0)- y_i(0) $, between the genotypes $i$ and $k$, is a random variable also uniformly distributed over $[0, N+ y_k(0)]$ and consequently the average of the delta function in the above expressions is equal to one and independent of time. The probability that the genotype $i$ (with $v_i>w_k$) becomes the next leader is given by $r_i/\sum_j r_j$ which we write as a transition probability $$p_{k\to i} =
\frac{(v_i-w_k)\theta(v_i-w_k)}{\sum_{j=1}^N(v_j-w_k)\theta(v_j-w_k)},
\label{tp1}$$ This rather intuitive rule appears in a simple traffic model studied in [@kred], although the physics is different to that here because on catching up with a slower car the faster one then adopts the same speed. We next show that this model can be mapped onto a first-passage problem for a random process. Notice that, once the $k$-th leader is selected with velocity $w_k$, the number $N_k$ of possible future leaders is $$\frac{N_k}{N}=\frac{\sum_{j=1}^N\theta(v_j-w_k)}{N},
\label{fl1}$$ where $N$ is the total number of genotypes. In the limit of large $N$, one can replace the right hand side of Eq. (\[fl1\]) by the integral over $v$, $$\frac{N_k}{N}\to \int_{w_k}^{v_{\rm max}} p(v) \,dv = P(w_k),
\label{fl2}$$ which is exact up to $O(1/\sqrt{N})$ corrections and where $P(v)=\int_v^{v_{\rm max}} p(u)du $ is the cumulative velocity distribution. Clearly, the number of lead changes $l_N$ is the value of $k$ where $N_k=1$. This gives, $P(w_{l_N})=1/N$ and hence $$-\ln[P(w_{l_N})]= \ln N.
\label{level1}$$ We define $Y_k = -\ln[P(w_k)]$ whose evolution is given by $$Y_{k+1} = Y_{k} + \xi_k,
\label{lange1}$$ where clearly $$\xi_k = -\ln[P(w_{k+1})/P(w_k)].
\label{noise1}$$ Thus $Y_k$ can be interpreted as the position of a random walker at time $k$ and its time evolution is given by the Langevin equation (\[lange1\]) where $\xi_k$ is the noise at step $k$. This redefinition is not yet very useful since the noise at step $k$ depends on $Y_{k+1}$ and $Y_k$. However, as we will see, for large $k$ the probability distribution of the noise $\xi_k$ becomes independent of $k$ and $w_k$ and has a finite mean $\langle \xi_k \rangle = \mu$ and variance $\langle [\xi_k- \langle \xi_k \rangle]^2 \rangle =
\sigma^2$, that can be computed explicitly for arbitrary velocity distribution $p(v)$. For large $k$, Eq. (\[lange1\]) represents a discrete time random walk with a positive drift $\mu$, i.e., $$Y_{k+1}= Y_k + \mu + \sigma \eta_k
\label{lange2}$$ where $\eta_k$ is a noise with zero mean $\langle \eta_k \rangle =0$ and unit variance. We will also see that $\eta_k$’s are not only completely independent of $w_k$ for large $k$, they are also uncorrelated at different times. Thus Eq. (\[lange2\]) is a true Markovian evolution of a discrete time random walker with a positive drift $\mu$. Obviously then, by central limit theorem, $Y_k$ will have a Gaussian distribution with mean $\langle Y_k\rangle= \mu k$ and variance ${\langle Y_k^2 \rangle-\langle Y_k\rangle}^2 = \sigma^2 k$.
Once we have the Markovian random walker evolution as in Eq. (\[lange2\]), it follows from Eq. (\[level1\]) that the number of lead changes $l_N$ is just the first time the process $Y_k$ (starting at some initial value $Y_0$) hits the level $Y=\ln (N)$. Thus the distribution of $l_N$ is simply the distribution of the first-passage time to the level $Y=\ln (N)$. To compute this, it is convenient to define $Z_k=\ln N -Y_k$. Then $Z_k$’s evolve via, $Z_{k+1}= Z_k -\mu - \sigma \eta_k$ starting from $Z_0=\ln N -Y_0$. Thus $Z_k$ is the position of a random walker at step $k$ with a negative drift $-\mu$ towards the origin and $l_N$ now represents the first-passage time to the [*origin*]{} starting from the initial position $Z_0$. Now, for large $k$, the discrete-time random walker can be replaced by a continuous-time Brownian motion, $$\frac{dZ}{dt} = -\mu + \sigma \eta(t)
\label{lange4}$$ where $\eta$ is a white noise with $\langle \eta(t)\rangle=0$ and $\langle \eta(t)\eta(t')\rangle= \delta(t-t')$. For such a process, the distribution $P(t_f|Z_0)$ of the first-passage time $t_f$ to the origin is known exactly [@MC] and we can apply it here to obtain the probability that $l_N= k$ is given by $$Q(k)= \frac{\ln N}{\sigma \sqrt{2\pi k^3}}\,
\exp\left[-\frac{\mu^2}{2\sigma^2
k}\left(k - (\ln N)/\mu\right)^2\right].
\label{ld}$$ Note that this distribution of $l_N$ is non-Gaussian. However, we expect this result to be valid only in the vicinity of $k\approx \ln N/\mu$, i.e., near its mean. This can be traced back to the fact that in deriving this result we replaced a discrete-time random walk by a continuous-time Brownian process. Near its mean, using $k\approx \ln N$ in Eq. (\[ld\]), the distribution of $l_N$ becomes a Gaussian $$Q(k) \approx \frac{\mu^{3/2}}{\sigma\sqrt{2\pi \ln N}}
\exp\left[-\frac{\mu^3}{2\sigma^2 \ln
N}\left(k-(\ln N)/\mu\right)^2\right]
\label{gauss1}$$ with mean and variance (for large $N$) given by $$\begin{aligned}
\langle l_N \rangle &=& \beta\, \ln N; \quad \quad {\rm where}\quad\, \beta=\frac{1}{\mu}
\label{meanl}\\
\langle (l_N-\langle l_N\rangle)^2 \rangle &=& \gamma\, \ln N;
\quad\quad {\rm where}\quad\, \gamma=\frac{\sigma^2}{\mu^3}.
\label{varl}\end{aligned}$$ Thus, irrespective of the velocity distribution $p(v)$, the distribution of $l_N$ near its mean is is a universal Gaussian characterized by two parameters $\mu$ and $\sigma$. The only dependence on $p(v)$ appears through the two constants $\mu$ and $\sigma$.
To calculate the mean $\mu$ and the variance $\sigma^2$ of the noise $\xi_k$ defined in Eq. \[noise1\], we note that for a given $w_k$, $\xi_k$ is a random variable since $w_{k+1}$ is a random variable drawn from the distribution in Eq. (\[tp1\]). We define $$\begin{aligned}
J(v)& = & \int_v^{v_{\rm max}}P(u)\, du \label{Jv} \\
K(v)&=& \int_v^{v_{\rm max}} [P'(u)/P(u)] J(u)\, du \label{Kv} \\
L(v) & =& \int_v^{v_{\rm max}} [P'(u)/P(u)] K(u)\, du \label{Lv}. \end{aligned}$$ Using the definition in Eq. (\[noise1\]) and the transition probability in Eq. (\[tp1\]), the mean of $\xi_k$ (for a given $w_k$) is $$\langle \xi_k \rangle = -\frac{\int_{w_k}^{v_{\rm
max}}[\ln(P(v))-\ln(P(w_k))](v-w_k)p(v)\,dv}{\int_{w_k}^{v_{\rm max}} (v-w_k) p(v)\, dv}.
\label{mean1}$$ Using integration by parts, in both the numerator and denominator above we find $$\langle \xi_k \rangle = 1- \frac{K(w_k)}{J(w_k)},
\label{mean2}$$ where the function $K(v)$ is defined in Eq. (\[Kv\]). The second moment is given by $$\langle \xi_k^2 \rangle = \frac{\int_{w_k}^{v_{\rm
max}}[\ln(P(v))-\ln(P(w_k))]^2(v-w_k)p(v)\, dv}{\int_{w_k}^{v_{\rm max}} (v-w_k) p(v)\, dv},
\label{var1}$$ and a similar calculation leads to $$\langle (\xi_k-\langle \xi_k\rangle)^2 \rangle = 1+ 2\frac{L(w_k)}{J(w_k)}
-\left[\frac{K(w_k)}{J(w_k)}\right]^2,
\label{var3}$$ where the functions $J$, $K$ and $L$ are defined in Eqs. (\[Jv\]), (\[Kv\]) and (\[Lv\]) respectively.
We now consider the three classes of distributions considered by [@krug].
([*i*]{}) [**Fast decaying distribution with $v_{\rm max}=+\infty$**]{}: In this case, it is easy to see that for large $u$, $$\frac{P'(u)}{P(u)} \approx \frac{J'(u)}{J(u)}$$ Thus, using this result in the definition of $K(v)$ in Eq. (\[Kv\]) one finds that for large $w_k$ $$K(w_k)= \int_{w_k}^{\infty} \frac{P'(u)}{P(u)} J(u)\, du \approx
- J(w_k)
\label{KJ}$$ Similarly, for large $w_k$, $$L(w_k)= \int_{w_k}^{\infty} \frac{P'(u)}{P(u)} K(u) \, du \approx
J(w_k)
\label{LJ}$$ Using these results in Eqs. (\[mean2\]) and (\[var3\]) we find for large $k$ $$\begin{aligned}
\langle \xi_k \rangle & =\mu=& 2 \label{c1mean}\\
\langle (\xi_k-\langle \xi_k\rangle)^2 \rangle & = \sigma^2=& 2 \label{c1var}.\end{aligned}$$ Thus, as stated earlier, we see the variance become independent of $k$ and $w_k$.
[*(ii)*]{} [**Distribution with a finite $v_{\rm max}$, with $p(v)\sim
|\ln(v_{\rm max}-v)|^\gamma(v_{\rm max}-v)^\alpha$:**]{} In this case, for $u$ close to $v_{\rm max}$, we find $$\frac{P'(u)}{P(u)}\approx \left(\frac{1+\alpha}{2+\alpha}\right)\, \frac{J'(u)}{J(u)}.
\label{edge1}$$ and it follows that for $w_k$ close to $v_{\rm max}$ $$\begin{aligned}
K(w_k) &\approx & -\left(\frac{1+\alpha}{2+\alpha}\right)\, J(w_k) \nonumber \\
L(w_k) & \approx & \left(\frac{1+\alpha}{2+\alpha}\right)^2 J(w_k) \label{KL}\end{aligned}$$ Using these results in Eqs. (\[mean2\]) and (\[var3\]) we get $$\begin{aligned}
\langle \xi_k \rangle & =\mu=& \frac{2\alpha+3}{\alpha+2} \label{c2mean}\\
\langle (\xi_k-\langle \xi_k\rangle)^2 \rangle & = \sigma^2=&
\frac{2\alpha^2+6\alpha+5}{(\alpha+2)^2} \label{c2var}.\end{aligned}$$
[*(iii)*]{} [**Power-law decaying distribution with $v_{\rm max}=+\infty$, and $p(v)\sim \ln(v)^\gamma v^{-\alpha}$ with $\alpha>2$:**]{} In this case, for large $u$ $$\frac{P'(u)}{P(u)}\approx \left(\frac{\alpha-1}{\alpha-2}\right)\, \frac{J'(u)}{J(u)}
\label{edge2}$$ Using this result in the definition of $K(v)$ and $L(v)$ one easily finds that for large $w_k$ $$\begin{aligned}
K(w_k) &\approx & -\left(\frac{\alpha-1}{\alpha-2}\right)\, J(w_k) \nonumber \\
L(w_k) & \approx & \left(\frac{\alpha-1}{\alpha-2}\right)^2 J(w_k) \label{KL2}\end{aligned}$$ Using these results in Eqs. (\[mean2\]) and (\[var3\]) we get $$\begin{aligned}
\langle \xi_k \rangle & =\mu=& \frac{2\alpha-3}{\alpha-2} \label{c3mean}\\
\langle (\xi_k-\langle \xi_k\rangle)^2 \rangle & = \sigma^2=&
\frac{2\alpha^2-6\alpha+5}{(\alpha-2)^2} \label{c3var}.\end{aligned}$$
One can also demonstrate [@inprep] that for all these velocity distributions, and for large $k$ and $k'$ $\langle \xi_k \xi_k'\rangle
-\mu^2 \to 0$, indicating that the noise $\xi_k$’s become completely uncorrelated in time. Thus Eq. (\[lange2\]) truly represents a Markovian random walk with drift $\mu$. Knowing the exact values of $\mu$ and $\sigma$, we then find that distribution of $l_N$, near its mean, is given by the Gaussian in Eq. (\[gauss1\]) with mean and variance given by Eqs. (\[varl\]). The coefficients $\beta$ and $\gamma$ are thus calculated exactly knowing $\mu$ and $\sigma$ and are given, for each of the cases mentioned above, by $$\begin{aligned}
(i) &:&\beta = 1/2 \ ; \ \gamma = 1/4 \\
(ii) &:&\beta =\frac{\alpha+2}{2\alpha+3}
\ ; \ \gamma= \frac{(\alpha+2)(2\alpha^2+6\alpha+5)}{(2\alpha+3)^3} \\
(iii) &:&\beta= \frac{\alpha-2}{2\alpha-3}\ ;
\ \gamma = \frac{(\alpha-2)(2\alpha^2-6\alpha+5)}{(2\alpha-3)^3} \end{aligned}$$
The results for the coefficient $\beta$ are in complete agreement with those conjectured in [@krug] in all three cases and we have further verified all our results by simulating the original i.i.d. shell model with an algorithm which permits us to simulate up to $N = 10^{200}$ genotypes [@inprep]. Moreover, we have also calculated the variance exactly and shown that near its mean, the distribution of $l_N$ is a universal Gaussian. In [@krug] it was pointed ou that the variance of $l_N$ is typically smaller than the mean indicating the temporal correlation between leadership changes, this is clearly seen in our exact results. Away from its mean, one expects to see departures of the distribution of $l_N$ away from the Gaussian form. To compute the full distribution one needs to solve the first-passage problem for the discrete-time process without resorting to the continuous-time approximation. Fortunately, for our discrete-time process, this can be achieved by observing that the the evolution of $Y_k$ with $k$, though random, is actually a strictly monotonic process. This follows from Eq. (\[noise1\]) that shows that the noise $\xi_k$ is always positive. The distribution of the first-passage time $l_N$ to the level $\ln(N)$ then satisfies the identity [@inprep] $${\rm Prob}(l_N \leq k) = {\rm Prob}(Y_k \geq \ln(N)).$$ This gives $Q(k) = {\rm Prob}(l_N = k) = {\rm Prob}(l_N \leq k+1) -{\rm Prob(l_N \leq k)}
= {\rm Prob}(Y_{k+1} \geq \ln(N)) - {\rm Prob}(Y_{k} \geq \ln(N))$. Thus, a knowledge of the distribution of $Y_k$ (which is usually much simpler to compute) provides us with an exact distribution of lead changes $Q(k)$ for all $k$. For example, for an exponential velocity distribution $p(v)=e^{-v}$, the probability density function of $Y_k$ can be found explicitly for all $k$ $$\rho_k(y) = {y^{2k-1}\over (2k-1)!}\exp(-y).$$ This result is in fact asymptotically valid for any rapidly decaying distribution $p(v)$ [@inprep]. Using this result we thus obtain the full probability distribution of $l_N$ for the exponential velocity distribution as $$Q(k) = {(\ln(N))^{2k}\over N (2k)!}\left[ 1 + {\ln(N)\over 2k+1}\right]
\label{eqexp}$$ In Fig. (2) we show the predictions of Eq. (\[eqexp\]) versus the results of extensive simulations and the agreement is perfect. The above use of the monotonicity of $Y_k$ also enables one to obtain analytical results, away from the Gaussian regime, for generic fitness distributions [@inprep].
To summarize we have solved exactly the asymptotic statistics of lead changes in a quasispecies evolution model by mapping the model to a random flux model. Our results confirm previous conjectures about the mean number of leader changes. We have also computed the variance exactly and shown that the distribution is generically Gaussian in the region around the mean. Finally, we remark that the evolution time $\tau$ defined as the time when the last leader change occurs can be shown to have a distribution $q(\tau)\sim \tau^{-2}$ for large $\tau$ [@inprep], as found in more realistic models [@krug].
[**Acknowledgment**]{}: SNM and DSD would like to thank the Isaac Newton Insitute Cambridge where part of this work was carried out during the program [*Principles of the Dynamics of Non-Equilibrium Systems*]{}. The authors would also like to thank J. Krug for useful comments and suggestions.
[0]{}
|
---
author:
- Hamid Omid
date: 'May 5, 2014'
title: 'Chern-Simons terms in the 3D Weyl semi-metals'
---
ABSTRACT
========
Based on some theoretical arguments, it has been suggested that electromagnetic response of 3D Weyl semi-metals with non-zero chiral- chemical potential may have a Chern-Simons term, $\frac{1}{2}k_\mu \epsilon^{\mu\nu\rho\sigma}F_{\nu\rho}A_\sigma$, in their effective action for the gauge field. An independent numerical study has shown that such a term is absent in a similar system. In this paper, we investigate the non-equilibrium and equilibrium response of 3D Weyl semi-metals. We argue that the controversy in literature stems from the difference in response of these two distinct states. We then develop a method to deal with well-known ambiguities in quantum electrodynamics in $3$D (QED$_{3+1}$) with non-zero chiral-chemical potential and calculate the Chern-Simons term unambiguously. We find that time-like Chern-Simons term can exist in non-equilibrium conditions. We observe that there does not exist any chiral-magnetic effect in equilibrium and anomalous Hall effect replaces it.
INTRODUCTION
============
Recently, 3D Weyl semi-metals have gained much attention. 2D Weyl semi-metals became of interest by experimental fabrication of graphene[@Geim] , although they were expected to have novel properties for a while [@Gordon]. Weyl semi-metal is a phase of matter in which the valence band touches the conduction band at certain points and the dispersion around the so called ’Weyl Point’ takes the form of relativistic dispersion. Although 2D Weyl semi-metal phase is sensitive to the perturbations and can be gapped easily by breaking of underlying symmetries, 3D Weyl semi-metals show more robust behaviour[@Herring; @Murakami]. In the absence of $P$ or $T$ it can be shown that there is a finite region of internal parameters that leads to a gap-less state[@Murakami]. This can be understood by noting that Weyl nodes are associated with a pseudo-charge which are conserved. Conservation of this charge makes the gaping hard. In order to gap the spectrum, Weyl nodes that are located at different momentums need to meet or interact with each other. Based on the above consideration, there are a few suggestion for realizing 3D Weyl semi-metals experimentally [@Wan; @Burkov1].\
It has been theoretically suggested that presence of non-zero chiral- chemical potential in effective Hamiltonian of Weyl semi-metals, which violates emergent Lorentz symmetry, can result in induced Chern-Simons term in photon effective Lagrangian, $$\mathcal{L}_{\text{CS}}=\frac{1}{2}k_\mu \epsilon^{\mu\nu\rho\sigma}F_{\nu\rho}A_\sigma$$ in which $k_\mu$ is the Chern-Simons coefficient, $\epsilon^{\mu\nu\rho\sigma}$ is the anti-symmetric tensor in 3D and $F_{\nu\rho}$ is the field strength of vector potential $A_\sigma$. Chern-Simons term has the fascinating feature that in the absence of electric field there would be an electric current solely induced by a magnetic field. This feature, called ’chiral-magnetic effect’, is evident in associated current derived from gauge invariance , $$\begin{aligned}
&\rho={\bf k}.{\bf B} \nonumber \\
&{\bf j}={\bf k}\times{\bf E}-k_0{\bf B}\end{aligned}$$
Presence of Chern-Simons term in Lorentz violating QED$_{3+1}$, the effective theory describing Weyl semi-metals, is a well-known feature [@Perez; @Jackiw1; @Jackiw2]. It is believed though that $k_\mu$ is ambiguous and depends on the regulator used to regulate linear divergences of theory. It is suggested [@Marcel] that if the correct regulator gets used in calculations, chiral-magnetic effect would vanish. In this paper, we discuss a possible explanation for the discrepancy between different results. We will argue that the discrepancy stems from different linear responses associated with a system. We then try to find a proper regulator and see whether such a regulator forbids presence of chiral-magnetic effect.
THE MODEL AND REGULATOR {#model}
=======================
In the rest of the paper, we investigate the standard model describing a 3D Topological Insulator [@Qi83; @Fu105], defined by the momentum space Hamiltonian $$\begin{aligned}
\label{H_0}
H_0({\bf k})=&2\lambda \sigma_z(s_x sin(k_y)-s_y sin(k_x))+2\lambda_z\sigma_y sin(k_z)\nonumber \\&+\sigma_x M({\bf k})\end{aligned}$$ with $\sigma$ and $s$ the Pauli matrices acting in orbital and spin space respectively and $M({\bf k})=\epsilon-2t\sum_i cos(k_i)$. Without loss of generality we restrict our model to the case that $\lambda_z=\lambda$. We constrain our parameters so that the system lives in trivial phase. This can be done by implementing $\epsilon=6t$, resulting in $M({\bf k}=\Gamma)=0$ such that $\Gamma=0$ . Although this Hamiltonian realizes Weyl semi-metal phase, this phase only exist at a single point in parameter space given by $\epsilon=6t$. To realize the Weyl semi-metal phase that is stable, we add the following term, $H_1$, to our original Hamiltonian, $$H_1({\bf k})=b_0 \sigma_y s_z+{\bf b}.(-\sigma_x s_x,\sigma_x s_y,s_z)$$ $b_0$ and ${\bf b}$ terms in perturbation break $\mathcal{P}$ and $\mathcal{T}$ respectively. Violation of $\mathcal{P}$ or $\mathcal{T}$ is needed to realize Weyl semi-metal phase in a finite region of phase space[@Herring; @Murakami]. As can be expected from symmetry effect of this term, it has magnetic origin. It can be introduced by magnetic doping of a Weyl semi-metal system and has been observed experimentally to be present in topological insulators[@Grushin; @Chen].\
The low energy limit of $H_0$ can be written in familiar form of Dirac Hamiltonian, $$\label{H_02}
H_0(k)=2\lambda{\bf \a. k}+\beta m$$ with $\alpha$ defined in \[\[A1\]\] and in special case of $\epsilon=6t$, $m=0$ . In \[\[A1\]\], $H_1$ and second order contribution of $H_0$ are written in the same basis, here we recall the results, $$H_1=b_0 \gamma_5+{\bf \a.b} \gamma_5$$ There are two different linear-responses that can be obtained from this theory. These two responses have been explored in [@Marcel; @Burkov2], for example. After adding $H_1$ to the original Hamiltonian the place of Weyl nodes in momentum space changes. As a result, the electrons which have been in the ground state previously need to eventually move to a new region of phase space FIG. \[well1\]. In other words, system will form a metastable state that will eventually decay into the actual ground state. Either $H_1$ can be added to the Hamiltonian as a perturbation and the conductivity of the systems can be studied before decaying into actual ground state or the conductivity can be studied in equilibrium state. The two responses are quite different and will result in different effective actions for a coupled gauge field.\
![[The band structure of our model for $\epsilon=6t$ in units of $\lambda$ sketched at $k_y=k_z=0$ a. Doubly degenerate Dirac point for b=(0,0,0,0) b. Shifted Dirac points for b=(0,0.7,0,0) .]{}[]{data-label="well1"}](WeylPoints2.jpg)
In section \[derivation\], we study the former case in which we assume the system stays in meta-stable ground state. In this case the low-energy theory is given by the Weyl nodes around ${\bf k}=0$. Fortunately, the latter case does not need an independent calculation and can be understood without use of any regulator. As explained, the Weyl nodes move in momentum space and we need to linearize the Hamiltonian around that points. We can perform the same procedure in \[A\] and find out that the effective Hamiltonian would be given by,
$$H_0(k)=2\lambda{\bf \a. (k-k_{+}})+2\lambda{\bf \a. (k-k_{-}})$$
in which $k_{+}$ and $k_{-}$ are the location of the new Weyl points. The above Hamiltonian is the same as the Hamiltonian that is studied in Balents, et al [@Burkov1]. In this case, the chiral-chemical potential is absent which means that the theory is well-defined and there is no dependence on the regulator. Balents, et al have shown that the conductivity gets the form of an anomalous Hall effect. This shows that the chiral-magnetic effect will be absent in equilibrium and only anomalous Hall effect survives.
Regulation Problem {#divergences}
==================
It is well-known that a finite quantum field theory (QFT) with linear divergences may include ambiguities. By finite QFT that is linearly divergent, we mean a QFT that has a set of finite correlators which are superficially divergent. The presence of such ambiguities can be understood by considering elementary integrals that are linearly divergent. Consider a function $f(x)$ which is finite at infinity but non-zero. Let us consider the following integral, $$\int_0^\infty dx~(f(x+a)-f(x))\sim a~f'(x)|_0^\infty+O(a^2)$$ although the first term looks like the same as the second term with an inconsequential change of variables, the integral depends on that change of variable through its dependence on $a$. In the presence of linear divergences, although the integrals may be finite, the way that they get manipulated changes the final result. For example, a change of variable before combining individual integrals may change the final result.\
It is known [@Perez; @Jackiw1; @Jackiw2] that extended QED$_{3+1}$ with non-zero chemical potential, which breaks $\mathcal{PT}$ is a finite but lineally divergent QFT. As a result, some of the correlators, for example photon polarization tensor, depend on the way that different Feynman diagrams get combined together. To deal with this ambiguity, a regulator should be chosen so that all of the correlators become convergent and analytic. One can think of different regulators as different ways of combining the Feynman diagrams. The dependence of the results on the choice of regulator is a consequence of the importance of high-energy theory beyond its contribution to linearized action. To get an unambiguous result, one can return to the full theory that we have derived our effective theory from and perform the calculations using that theory. In this paper, we take advantage of having such a theory and find the appropriate regulator by analyzing it. In addition to finding the effective theory by finding the low-energy limit Hamiltonian, we find the next order terms which are higher in powers of momentum. This higher order term will regulate the linear divergence as it can be seen by power counting. Let us borrow the formula that we derive for two-point function in section \[derivation\] and count the powers of momentum we are integrating, $q$.
$$\begin{aligned}
\Pi_{\mu \nu}(p)&=-\frac{8i\epsilon^{\mu\nu\rho\sigma}}{(2\pi)^4}\int d^4q~\frac{-2q.b q_\rho(q+p)_\sigma+q^2b_\rho(q+p)_\sigma+m^2(q-p)_\rho b_\sigma}{(q^2-m^2)^2((q+p)^2-m^2)}\end{aligned}$$
This integral suffers from superficial linear divergence. We call it superficial as the final result for the integral is finite even without use of any regulator. It is clear that by adding a higher order term to the propagator denominator, $\frac{i}{\s{p}-m}$, the linear divergence gets removed. We use this trick to regulate our integrals.
DERIVATION OF CHERN-SIMONS TERM {#derivation}
===============================
In this section, we derive the Chern-Simons coefficient using our physical regulator. In section \[model\], we discussed the rise of QED$_{3+1}$ as the effective theory describing the quasi-particles in Topological Insulator heterostructure. The Lagrangian for QED$_{3+1}$ is given by $$\mathcal{L}_{\text{QED}}=\overline{\psi}(i\s{\partial}-m- \s{A})\psi$$ $\psi$ is the fermionic field, $A$ is the gauge field which is coupled to $\psi$ by minimal coupling resulting from Peierls substitution in equation \[H\_0\] and $m$ is the fermionic mass which we keep to regulate IR divergences and put zero at the end. We showed that presence of non-zero $b$ results in a term that breaks $\mathcal{PT}$. As it is shown in \[\[A1\]\], this term gets the form of chiral-chemical potential. The Lagrangian for this potential is given by, $$\mathcal{L}_b=-\overline{\psi}\gamma_5\s{b}\psi$$ As the last piece, we have another term in our Lagrangian that plays the role of regulator. In \[\[A1\]\], we derived the regulator and showed that it can be written as a momentum dependent mass term and has the form of, $$\mathcal{L}_{\text{regulator}}=\frac{t}{\lambda}\overline{\psi}{{\overrightarrow}{\partial}^2}\psi$$ We emphasize that this regulator breaks the Lorentz invariance manifestly, which ensures that our final result for Chern-Simons term is not an artifact of keeping Lorentz invariance in low energy theory. The Chern-Simons’s term can be calculated using the photon two-point function expansion around $p_\mu=0$. We treat the chiral-chemical potential as an extra vertex and find the two-point function order by order. The two point function can be found using usual Feynman rules. As is mentioned in [@Perez], the 1-loop calculation results in exact form of induced Chern-Simons term and we don’t need to go to higher orders to find further contributions. We use the Clifford Algebra between Dirac matrices and the identities follow from this fundamental anti-commutation relation. In particular, we use the fact that for $n~\epsilon \mathcal{N}$, $tr(\gamma_{\mu_1}...\gamma_{\mu_{2n+1}})=0$ and $tr(\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}\gamma_{5})=-4i\epsilon^{\mu\nu\rho\sigma}$. Using these identities we find an integral form for photon two-point function,
$$\begin{aligned}
\label{two-point}
\Pi_{\mu \nu}(p)&=\frac{2}{(2\pi)^4}\int d^4q~tr~(\gamma_\mu\frac{1}{\s q-m}\gamma_5\s b \frac{1}{\s q-m}\gamma_{\nu}\frac{1}{\s q+\s p-m})\nonumber \\&=-\frac{8i\epsilon^{\mu\nu\rho\sigma}}{(2\pi)^4}\int d^4q~\frac{-2q.b q_\rho(q+p)_\sigma+q^2b_\rho(q+p)_\sigma+m^2(q-p)_\rho b_\sigma}{(q^2-m^2)^2((q+p)^2-m^2)}\end{aligned}$$
in which the extra factor of two is coming from having two corresponding Feynman diagrams, FIG. \[oneloopvertex\]. In \[two-point\] the minus sign from fermion loop cancels the minus sign from the chiral-chemical potential vertex.
![[The vertex can be placed on both of the internal fermion lines resulting in two Feynman diagrams.]{}[]{data-label="oneloopvertex"}](oneloopvertex.jpg)
As we explained in section \[divergences\], the linearly divergent integrals are the source of ambiguity in Chern-Simons coefficient. The integrals that diverge slower will be independent of our regulator. We use this fact and use our regulator to find only linearly divergent terms. With our regulator turned on, all the integrals are finite and smooth, meaning that they can be Taylor expanded in terms of external momentum. From definition of Chern-Simons’s coefficient $k_\mu$, the linear part of two-point function \[two-point\] in external momentum is proportional to this coefficient, $$\begin{aligned}
\Pi^{\mu \nu}(p)=\epsilon^{\mu\nu\rho\sigma}M_{\rho}^{\lambda}b_\lambda p_\sigma~~~s.t.~~~k_\mu=-\frac{1}{2}M_{\mu}^{\lambda}b_\lambda\end{aligned}$$ in which $M_{\rho}^{\lambda}$ is a constant.\
Let us investigate the most general linearly divergent integral that may be present in our calculations and then come back to the evaluation of \[two-point\]. The most general linearly divergent integral we face can be written in the following form $$\begin{aligned}
I_{\mu\nu\rho}(p)&= \int d^4q~\frac{q_\mu q_\nu q_\rho}{(q^2-m^2)^2((q+p)^2-m^2)}~~~s.t.~~~\mu\neq\nu\nonumber \\&=
I_{\mu\nu\rho}(0)+\partial_\lambda I_{\mu\nu\rho}(0)~p^\lambda+O(p^2)\end{aligned}$$ By dimensional analysis, we can see that the second term is no longer linearly divergent, as it is proportional to $p_\mu$, as a result we only should be concerned about the first term, $I_{\mu\nu\rho}(0)$, $$\begin{aligned}
I_{\mu\nu\rho}(0)&= \int d^4q~\frac{q_\mu q_\nu q_\rho}{(q^2-m(q)^2)^3}~~~~s.t~~~~\mu\neq\nu\\end{aligned}$$
As a first observation, we note that by dimensional analysis $I_{\mu\nu\rho}(0)$ is free of IR divergences and the momentum independent part of $m$ can be set to zero . At the same time, for the general form of mass-like regulator, like what we found in \[\[A1\]\], the linear term would be absent in $m$, as we assume that the low energy theory is already chosen. Another comment is that the coefficient of the regulator($m$) won’t play a role in our calculations, and can be taken care of by a simple renaming of the momentum. Doing so, we find that the divergences are linear in that coefficient. Finally, from the symmetry of integral, we find that for symmetric regulators such that $m(p_\mu)=m(-p_\mu)$ and in particular the regulator we found in \[\[A1\]\], the integral vanishes.\
We saw that $I_{\mu\nu\rho}(0)$ vanishes for our regulator, we then only need to consider $\partial_\lambda I_{\mu\nu\rho}(0)~p^\lambda$ in order to evaluate $I_{\mu\nu\rho}(p)$. Let’s look at each linearly divergent term in \[two-point\] separately. The non-vanishing part of the first term in \[two-point\] has the form of, $$p^\lambda\partial_\lambda \int d^4q~\frac{q.b q_\rho q_\sigma}
{(q^2-m^2)^2((q+p)^2-m(q+p)^2)}|_{p=0}$$ which makes the evaluation of its contribution easy. This integral gets contracted with the Levi-Civita tensor and subsequently vanishes. The second term has a part that is linearly divergent and is given by, $$\begin{aligned}
p^\lambda&\partial_\lambda \int d^4q~\frac{q^2 q_\sigma}{(q^2-m[q]^2)^2((q+p)^2-m[q+p]^2)}|_{p=0}\nonumber \\&=-2 \int d^4q~\frac{q_\sigma q_\lambda}{(q^2-m^2)^4}q^2p^\lambda~~~~s.t~~~~\rho\neq\sigma\nonumber \\&=-2\int d^4q~\frac{1}{(q^2-m^2)^4}\frac{q^2}{4}q^2 p_\sigma
$$ Here we have used the fact that the integral is not UV divergent and only has IR divergence. As a result we can safely turn off the regulator and only keep a constant non-zero mass to regulate IR divergences(We carried out the same calculation with the UV regulator turned on and confirmed our result.). As well, we have used the fact that $<q_0^2>=-<q_i^2>$, $<q_\mu^2>$ defined as $<q_\mu^2>=\int d^4q f(q^2) q_\mu^2$ which can be proven by using Wick rotation. Let $q'_0=i q_0$, after this change of variables the integral gets an extra factor of $-i$ coming from $dq_0$ and the integration contour would rotate by $\frac{\pi}{2}$ counter-clockwise. Using the Euler’s theorem, we can replace the integral with the same integral, integrating over real momentum from minus infinity to infinity. In this new parametrization, the metric is Euclidean and it’s easy to compare the integrals. We can now use this fact, or explicitly go back to Minkowskian space and find that $<q_0^2>=-<q_i^2>=\frac{1}{4}<q^2>$.\
We can simplify our calculation by considering the $q_{\rho}$ part of last term in integral \[two-point\]. As its contribution to the coefficient of the term linear in external momentum is important for us, we can Taylor expand it in $p_\lambda$, $$\begin{aligned}
p^\lambda&\partial_\lambda \int d^4q~\frac{m^2 q_\rho}{(q^2-m[q]^2)^2((q+p)^2-m[q+p]^2)}|_{p=0} \nonumber\\&=-2 \int d^4q~\frac{q_\rho q_\lambda}{(q^2-m^2)^4}p^\lambda~~~~s.t~~~~\rho\neq\sigma\nonumber \\&=-2\int d^4q~\frac{1}{(q^2-m^2)^4}\frac{q^2}{4} m^2 p_\rho \end{aligned}$$
We now combine all of the results to find the two-point function. The two-point function simplifies to, $$\begin{aligned}
\Pi^{\mu \nu}(p)=&-\frac{8i\epsilon^{\mu\nu\rho\sigma}}{(2\pi)^4}\int d^4q~\frac{1}{(q^2-m^2)^3}~(-2q.b q_\rho p_\sigma \nonumber \\&~ +q^2b_\rho p_\sigma-m^2 p_\rho b_\sigma-\frac{1}{2}q^2p_\sigma b_\rho)\nonumber \\=&
-\frac{8i\epsilon^{\mu\nu\rho\sigma}}{(2\pi)^4}\int d^4q~\frac{1}{(q^2-m^2)^3}~(-\frac{1}{2}q^2 b_\rho p_\sigma \nonumber \\&~ +q^2b_\rho p_\sigma-m^2 p_\rho b_\sigma-\frac{1}{2}q^2p_\sigma b_\rho)
\nonumber \\=&
-\frac{8i\epsilon^{\mu\nu\rho\sigma}}{(2\pi)^4}\int d^4q~\frac{1}{(q^2-m^2)^3}~\left(-m^2 p_\rho b_\sigma\right)\end{aligned}$$ in second equality we have assumed that $m(p)=m$, is a constant, as the integrals are not linearly divergent. The remaining integral can be evaluated straightforwardly. It is given by, $$\begin{aligned}
&\int d^4q~\frac{1}{(q^2-m^2+i\xi)^3}=\frac{i\pi^2}{2m^2}\end{aligned}$$ in which $\xi$ is the Feynman regulator, regulating IR divergences.\
We finally find that the two-point function is given by,
$$\begin{aligned}
\Pi^{\mu \nu}(p)=
\frac{\epsilon^{\mu\nu\rho\sigma}}{4\pi^2}b_\rho p_\sigma\end{aligned}$$
then the Chern-Simons coefficient can be extracted,
$$\begin{aligned}
k_\mu=-\frac{1}{8\pi^2}b_\mu\end{aligned}$$
As we mentioned in last sections, we find that the space-like Chern-Simons terms can indeed be present even after regulating by a regulator that breaks emergent Lorentz symmetry manifestly. The Chern-Simons coefficient that we find is the same as the coefficient found by Perez [@Perez].
Conclusion
==========
In this work we studied in detail 3D Weyl semi-metals in presence of perturbation that break $P$, chiral-chemical potential. We argued that there can be two kinds of linear responses associated to a 3D Weyl semi-metal with broken $\mathcal{PT}$. We found that in the meta-stable scenario, there exist a Chern-Simons term in effective action of gauge field which contributes to electric conductivity. We concluded that in equilibrium there can not be any space-like Chern-Simons term. As a result, in the equilibrium case chiral-magnetic effect will be absent and the conductivity would be gain the usual Hall conductivity form. This effect still remains to be confirmed experimentally.
Acknowledgement
===============
We thank M. Franz for suggesting the problem and G.W. Semenoff, I. Affleck, M. Franz and A. Zhitnitsky for useful comments.
APPENDIX
========
EFFECTIVE HAMILTONIANIAN {#A1}
========================
In this appendix, we argue that effective Hamiltonian of a band theory without interaction between quasi-particles is given by the low energy limit of the theory. We then rewrite the low energy limit of $H_0$ and $H_1$ in relativistic notations.\
A general band theory can be written in terms of a quadratic Hamiltonian given by $\mathcal{H}=\sum_{\alpha, k} \epsilon_\alpha(k) c^\dagger_k c_k$, in which $\alpha$ indexes the bands. As a consequence of being Gaussian, path-integral formalism can be used to show that high-energy modes can be integrated out and we are left with the modes of interest. The effective theory then is given by the low-energy limit of the Hamiltonian.
Let us find the low-energy limit of our model. From equation \[H\_0\], the expansion of $H_0$ around $\bf k=0$ gets the following form $$\begin{aligned}
H_0=\lambda
\left({\scalebox{0.85}{\mbox{\ensuremath{\displaystyle
\begin{array}{cccc}
0&2 k_y+2i k_x&-2i k_z+\frac{t}{\lambda}k^2&0\\
2 k_y-2ik_x&0&0&-2i k_z+\frac{t}{\lambda} k^2\\
2i k_z+\frac{t}{\lambda} k^2&0&0&-2 k_y-2i k_x\\
0&2i k_z+\frac{t}{\lambda} k^2&-2 k_y+2ik_x&0
\end{array}}}}}
\right)\end{aligned}$$ From the definition of the lowest order term in $H_0$, in equation \[H\_02\], the $\alpha$ matrices can be extracted. For example for $\alpha_1$, we find that, $$\begin{aligned}
\alpha_1=-\sigma_zs_y=\left(\begin{array}{cccc}
0&i&0&0\\
-i&0&0&0\\
0&0&0&-i\\
0&0&i&0
\end{array}
\right)\end{aligned}$$ The rest of the matrices are $\alpha_2=\sigma_zs_x,~\alpha_3=\sigma_y1_{2\times 2},~\beta=\sigma_x1_{2\times 2}$. To extract $\beta$, which plays the role of a mass term, $H_0$ must be expanded in the region that the quasi-particles are massive($ \epsilon\neq 6t$).
The Dirac $\gamma$ matrices are as usual defined by, $$\begin{aligned}
&\gamma_i=\beta \alpha_i \nonumber \\ &
\gamma_0=\beta\end{aligned}$$ and $\gamma_5$ in 3D can be defined by, $$\begin{aligned}
\gamma_5&=i\gamma_0\gamma_1\gamma_2\gamma_3\nonumber\\ &=\sigma_ys_z\end{aligned}$$ It can be checked that $\gamma$ matrices satisfy Clifford Algebra defined by $\{\gamma_\mu,\gamma_\nu\}=2\eta_{\mu\nu}$, $\eta$ being the metric, $\eta=\text{diag}(1,-1)$. Now that we have an explicit form for $\alpha$ matrices, we can write $H_1$ and the second order part of $H_0$ in terms of these matrices. They are given by $$\begin{aligned}
H_1(k)&=b_0 \gamma_5+{\bf \a.b} \gamma_5
\nonumber \\&=b_0 \gamma_5+b_i \gamma_0 \gamma_i \gamma_5\end{aligned}$$ $$\begin{aligned}
H_{\text{regulator}}(k)&=\frac{t}{\lambda} {\bf k}^2 \beta\nonumber \\
&=\frac{t}{\lambda} {\bf k}^2 \gamma_0\end{aligned}$$
[9]{}
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|
---
abstract: 'We study the high-dimensional limit of (projective) Stiefel and Grassmann manifolds as metric measure spaces in Gromov’s topology. The limits are either the infinite-dimensional Gaussian space or its quotient by some mm-isomorphic group actions, which are drastically different from the manifolds. As a corollary, we obtain some asymptotic estimates of the observable diameter of (projective) Stiefel and Grassmann manifolds.'
address:
- 'Mathematical Institute, Tohoku University, Sendai 980–8578, JAPAN'
- 'Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Tokyo 192–0397, JAPAN'
author:
- Takashi Shioya
- Asuka Takatsu
title: 'High-dimensional metric-measure limit of Stiefel and Grassmann manifolds'
---
[^1]
Introduction
============
Gromov developed the metric geometric theory of metric measure spaces, say *mm-spaces* (see [@Gro:green; @Sy:mmg]). There he defined a concept of convergence, which we prefer to call *weak convergence*, of mm-spaces by the convergence of the sets of $1$-Lipschitz continuous functions on the spaces. This is a weaker version of measured Gromov-Hausdorff convergence. In fact, a measured Gromov-Hausdorff convergent sequence converges weakly to the same limit, however the converse does not necessarily hold. The idea of the definition of weak convergence came from the concentration of measure phenomenon due to Lévy and Milman (see [@GroMil; @Levy; @Mil:heritage; @Le]). In fact, since any function on a one-point mm-space is constant, a sequence $\{X_n\}_{n=1}^\infty$ of mm-spaces converges weakly to a one-point mm-space if and only if any $1$-Lipschitz function on $X_n$ is almost constant for all sufficiently large $n$, which is just the concentration of measure phenomenon. Lévy’s celebrated lemma [@Levy] is rephrased as that the sequence of unit spheres in Euclidean spaces converges weakly to a one-point mm-space as dimension diverges to infinity.
It is a natural problem to study the weak limit of a non-Gromov-Hausdorff precompact sequence of specific manifolds, such as homogeneous manifolds, as dimension diverges to infinity. In our main theorems, we observe that the high-dimensional weak limits of (projective) Stiefel and Grassmann manifolds are drastically different from the manifolds in the sequence. This kind of phenomenon is never seen in the Gromov-Hausdorff convergence.
For $n=1,2,\dots,\infty$, we call the mm-space $\Gamma^n := ({{\mathbb{R}}}^n,\|\cdot\|,\gamma^n)$ the *$n$-dimensional* (*standard*) *Gaussian space*, where $\|\cdot\|$ is the $l^2$ (or Euclidean) norm and $\gamma^n$ the $n$-dimensional standard Gaussian measure on ${{\mathbb{R}}}^n$. If $n = \infty$, then $\|\cdot\|$ takes values in $[\,0,+\infty\,]$. Let $F$ be one of ${{\mathbb{R}}}$, ${{\mathbb{C}}}$, and ${{\mathbb{H}}}$, where ${{\mathbb{H}}}$ is the algebra of quaternions, and let $M_{N,n}^F$ denote the set of $N \times n$ matrices over $F$. Recall that the *$(N,n)$-Stiefel manifold over $F$*, say $V^F_{N,n}$, is defined to be the submanifold of $M_{N,n}^F$ consisting of matrices with orthonormal column vectors. We equip Stiefel manifolds with the mm-structure induced from the Frobenius norm and the Haar probability measure. We set $N^F := N \cdot \dim_{{\mathbb{R}}}F$ for a number $N$. Let $\{n_N\}_{N=1}^\infty$ be a sequence of positive integers such that $n_N \le N$ for all $N$. We consider the following condition ($*$) for the sequence $\{n_N\}$. $$\text{There exists a number $c \in (0,1)$ such that}
\ \ \sup_N \left( 2\log n_N - \frac{c}{4} \sqrt{\frac{N}{n_N^3}} \right) < \infty.
\tag{$*$}$$ Note that we have ($*$) for $n_N = O(N^p)$, $0 \le p < 1/3$ (see Remark \[rem:condi\]). One of our main theorems is stated as follows.
\[thm:Stiefel\] If $\{n_N\}$ satisfies [($*$)]{}, then the $(N,n_N)$-Stiefel manifold $V_{N,n_N}^F$ with distance multiplied by $\sqrt{N^F-1}$ converges weakly to the infinite-dimensional Gaussian space $\Gamma^\infty$ as $N \to \infty$.
Note that the infinite-dimensional Gaussian space is not an mm-space in the ordinary sense and is defined as an element of a natural compactification of the space of isomorphism classes of mm-spaces. In the case where $n_N = 1$, the manifold $V_{N,n_N}^F = V_{N,1}^F$ is a sphere in a Euclidean space, for which the weak limit was obtained in [@Sy:mmg; @Sy:mmlim].
To specify the high-dimensional weak limit of Grassmann and projective Stiefel manifolds, we need some notations. Denote by $U^F(n)$ the $F$-unitary group of size $n$, i.e., $U^F(n)$ is the orthogonal group if $F = {{\mathbb{R}}}$, the complex unitary group if $F = {{\mathbb{C}}}$, and the quaternionic unitary group if $F = {{\mathbb{H}}}$. $U^F(n)$ acts on $M_{N,n}^F$ by right multiplication. We define the infinite-dimensional Gaussian measures on $M_{\infty,n}^F$ and $F^\infty$ by identifying $M_{\infty,n}^F$ and $F^\infty$ with ${{\mathbb{R}}}^\infty$ under natural isomorphisms. The *$U^F(1)$-Hopf action on $F^\infty$* is defined as the $U^F(1)$ left multiplication. Note that the *$(N,n)$-Grassmann manifold over $F$*, say $G^F_{N,n}$, is obtained as the quotient of the $(N,n)$-Stiefel manifold over $F$ by the $U^F(n)$-action, and that the *projective $(N,n)$-Stiefel manifold over $F$*, say ${\mathrm{P}V}^F_{N,n}$, is obtained as the quotient of the $(N,n)$-Stiefel manifold over $F$ by the $U^F(1)$-Hopf action. Note also that the $U^F(n)$-action and the $U^F(1)$-Hopf action on the $(N,n)$-Stiefel manifold are both mm-isomorphic. We equip Grassmann and projective Stiefel manifolds with the quotient metric of the Frobenius norm and the Haar probability measure (see Definition \[defn:quotient\]). Applying (the proof of) Theorem \[thm:Stiefel\] we prove the following theorem.
\[thm:Gr-pS\]
1. For any fixed positive integer $n$, as $N \to \infty$, the $(N,n)$-Grassmann manifold $G_{N,n}^F$ over $F$ with distance multiplied by $\sqrt{N^F-1}$ converges weakly to the quotient of $(M_{\infty,n}^F,\|\cdot\|,\gamma^\infty)$ by the $F$-unitary group $U^F(n)$ of size $n$, where $\|\cdot\|$ is the Frobenius norm.
2. If $\{n_N\}$ satisfies [($*$)]{}, then the projective $(N,n_N)$-Stiefel manifold ${\mathrm{P}V}_{N,n_N}^F$ over $F$ with distance multiplied by $\sqrt{N^F-1}$ converges weakly to the quotient of the infinite-dimensional Gaussian space $(F^\infty,\|\cdot\|,\gamma^\infty)$ by the $U^F(1)$-Hopf action.
The reason why we fix $n$ in Theorem \[thm:Gr-pS\](1) is that we do not know the weak convergence of the quotient space of $(M_{\infty,n}^F,\|\cdot\|,\gamma^\infty)$ by $U^F(n)$ as $n\to\infty$. If it converges weakly, then the $(N,n_N)$-Grassmann manifold also converges weakly to the same limit, provided $\{n_N\}$ satisfies ($*$).
In the case where $n_N = 1$, the projective $(N,1)$-Stiefel manifold over $F$ is just the projective space over $F$, for which the weak limit was obtained in [@Sy:mmlim].
Theorems \[thm:Stiefel\] and \[thm:Gr-pS\] are also true for any subsequence of $\{N\}$. The proofs are the same.
The observable diameter of an mm-space is a quantity of how much the measure of the mm-space concentrates (see Definition \[defn:ObsDiam\]). As a corollary to the above theorems, we have some asymptotic estimates for the observable diameter of (projective) Stiefel and Grassmann manifolds.
\[cor:ObsDiam\] If $\{n_N\}$ satisfies [($*$)]{}, then we have, for any $0 < \kappa < 1$, $$\begin{aligned}
\lim_{N\to\infty} \sqrt{N^F} \operatorname{ObsDiam}(V_{N,n_N}^F;-\kappa)
&= 2\,D^{-1}\left(1-\frac{\kappa}{2}\right), \tag{1}\\
\limsup_{N\to\infty} \sqrt{N^F} \operatorname{ObsDiam}(G_{N,n_N}^F;-\kappa)
&\le 2\,D^{-1}\left(1-\frac{\kappa}{2}\right), \tag{2}\\
\liminf_{N\to\infty} \sqrt{N^F} \operatorname{ObsDiam}(G_{N,n}^F;-\kappa)
&\ge e^{1/2}(1-\kappa) \qquad\text{for any fixed $n$}, \tag{3}\\
\limsup_{N\to\infty} \sqrt{N^F} \operatorname{ObsDiam}({\mathrm{P}V}_{N,n_N}^F;-\kappa)
&\le 2\,D^{-1}\left(1-\frac{\kappa}{2}\right), \tag{4}\\
\liminf_{N\to\infty} \sqrt{N^F} \operatorname{ObsDiam}({\mathrm{P}V}_{N,m_N}^F;-\kappa)
&\ge e^{1/2}(1-\kappa) \quad\text{for any $\{m_N\}$ with $m_N \le N$}, \tag{5}\end{aligned}$$ where $$D(r) := \gamma^1((\,-\infty,r\,]) = \int_{-\infty}^r \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}\;dx$$ is the cumulative distribution function of $\gamma^1$.
Some upper bounds of the observable diameter of Stiefel and Grassmann manifolds were essentially obtained by Milman [@Mil:asymp; @Mil:inf-dim] and Milman-Schechtman [@MS] formerly. Corollary \[cor:ObsDiam\](1) gives an asymptotically optimal estimate. (2) and (4) are direct consequences of (1). (3) is a nontrivial result. As far as the authors know, any lower estimate of the observable diameter of the Grassmann manifold was not known before.
The $(N,n)$-Stiefel manifold over $F$ is naturally embedded into $M_{N,n}^F$. We point out that just to compare the distance between the Haar probability measure on the $(N,n)$-Stiefel manifold and the Gaussian measure on $M_{N,n}^F$ is not enough to obtain Theorem \[thm:Stiefel\]. In fact we have the following
\[prok\] The Prohorov distance between the Haar probability measure on the $(N,n_N)$-Stiefel manifold over $F$ with distance multiplied by $\sqrt{N^F-1}$ and the Gaussian measure $\gamma^{N^F n_N}$ on $M_{N,n_N}^F$ is bounded away from zero for all $N = 1,2,\dots$.
Theorems \[thm:Stiefel\] and \[prok\] tell us that the weak convergence of mm-spaces is different from the weak convergence of measures.
The idea of the proof of Theorem \[thm:Stiefel\] is as follows. Denote by $X^F_{N,n}$ the $(N,n)$-Stiefel manifold over $F$ with distance multiplied by $\sqrt{N^F - 1}$. It suffices to prove that $$\begin{aligned}
\lim_{N\to\infty} X^F_{N,n_N} &\prec \Gamma^\infty, \tag{i}\\
\lim_{N\to\infty} X^F_{N,n_N} &\succ \Gamma^\infty, \tag{ii}\end{aligned}$$ where $\prec$ is the Lipschitz order relation (see Definition \[defn:dom\]). Note that the Lipschitz order relation naturally extends to the relation on the compactification of the space of mm-spaces.
\(ii) follows from an easy discussion. $S^n(r)$ denotes an $n$-dimensional sphere of radius $r$ in a Euclidean space. We have $$X^F_{N,n_N} \succ S^{N^F}(\sqrt{N^F -1})$$ and the Maxwell-Boltzmann distribution law implies $$\lim_{N\to\infty} S^{N^F}(\sqrt{N^F -1}) \succ \Gamma^k$$ for any $k$. Combining these leads to (ii).
For the proof of (i), we find a suitable neighborhood of $X^F_{N,n_N}$ in $M_{N,n_N}^F$ which has most of the total measure of $\gamma^{N^F n_N}$, so that the neighborhood approximates $(M_{N,n_N}^F,\|\cdot\|,\gamma^{N^F n_N})$. We estimate the Lipschitz constant of the nearest point projection from the neighborhood to $X^F_{N,n_N}$ by using the polar decomposition of a matrix in the neighborhood, where the smallest Lipschitz constant is eventually close to one. We need delicate estimates of the measure of the neighborhood and the Lipschitz constant to justify the proof of (i), in which we find out that the condition ($*$) guarantees to our estimates. Note that we do not know the weak limit without ($*$).
Theorem \[thm:Gr-pS\] is proved by using Theorem \[thm:Stiefel\]. To prove it, we need maps between $X^F_{N,n_N}$ and a finite-dimensional approximation of $\Gamma^\infty$ that are equivariant with respect to the $U^F(n)$ and $U^F(1)$-Hopf actions. We need the generalization of the Maxwell-Boltzmann distribution law due to Watson [@Watson] to obtain such maps for the case of Grassmann manifolds.
Our main theorems could be related with the infinite-dimensional analysis, such as the theory of abstract Wiener spaces. In fact, the infinite-dimensional Gaussian space $\Gamma^\infty$ is an abstract Wiener space with $l^2$ as its Cameron-Martin space. However, $\Gamma^\infty$ admits no separable Banach norm fitting to $\gamma^\infty$ and is not mm-isomorphic to any abstract Wiener space with separable Banach norm. By this reason, many useful theorems in the theory of abstract Wiener spaces cannot be applied to $\Gamma^\infty$. We conjecture that the weak limit of compact homogeneous Riemannian manifolds is not mm-isomorphic to any abstract Wiener space with separable Banach norm.
Preliminaries
=============
Metric measure geometry
-----------------------
In this subsection, we give the definitions and the facts stated in [@Gro:green]\*[§3$\frac12$]{} and [@Sy:mmg]. The reader is expected to be familiar with basic measure theory and metric geometry (cf. [@Kechris; @Bil; @Bog; @BBI]).
### mm-Isomorphism and Lipschitz order
Let $(X,d_X)$ be a complete separable metric space and $\mu_X$ a Borel probability measure on $X$. We call the triple $(X,d_X,\mu_X)$ an *mm-space*. We sometimes say that $X$ is an mm-space, in which case the metric and the measure of $X$ are respectively indicated by $d_X$ and $\mu_X$.
Two mm-spaces $X$ and $Y$ are said to be *mm-isomorphic* to each other if there exists an isometry $f : \operatorname{supp}\mu_X \to \operatorname{supp}\mu_Y$ such that $f_\#\mu_X = \mu_Y$, where $f_\#\mu_X$ is the push-forward of $\mu_X$ by $f$ and $\operatorname{supp}\mu_X$ the support of $\mu_X$. Such an isometry $f$ is called an *mm-isomorphism*. Denote by ${\mathcal{X}}$ the set of mm-isomorphism classes of mm-spaces.
Any mm-isomorphism between mm-spaces is automatically surjective, even if we do not assume it. Note that $X$ is mm-isomorphic to $(\operatorname{supp}\mu_X,d_X,\mu_X)$.
*We assume that an mm-space $X$ satisfies $$X = \operatorname{supp}\mu_X$$ unless otherwise stated.*
\[defn:dom\] Let $X$ and $Y$ be two mm-spaces. We say that $X$ (*Lipschitz*) *dominates* $Y$ and write $Y \prec X$ if there exists a $1$-Lipschitz map $f : X \to Y$ satisfying $f_\#\mu_X = \mu_Y$. We call the relation $\prec$ on ${\mathcal{X}}$ the *Lipschitz order*.
\[prop:Liporder\] The Lipschitz order $\prec$ is a partial order relation on ${\mathcal{X}}$, i.e., we have the following [(1)]{}, [(2)]{}, and [(3)]{} for any mm-spaces $X$, $Y$, and $Z$.
1. $X \prec X$.
2. If $X \prec Y$ and $Y \prec X$, then $X$ and $Y$ are mm-isomorphic to each other.
3. If $X \prec Y$ and $Y \prec Z$, then $X \prec Z$.
### Observable diameter
The observable diameter is one of the most fundamental invariants of an mm-space.
\[defn:ObsDiam\] Let $X$ be an mm-space and let $\kappa > 0$. We define the *partial diameter $\operatorname{diam}(X;1-\kappa) = \operatorname{diam}(\mu_X;1-\kappa)$ of $X$* to be the infimum of $\operatorname{diam}A$, where $A \subset X$ runs over all Borel subsets with $\mu_X(A) \ge 1-\kappa$ and $\operatorname{diam}A$ denotes the diameter of $A$. Denote by ${\mathcal{L}\mathit{ip}}_1(X)$ the set of $1$-Lipschitz continuous real-valued functions on $X$. We define the *observable diameter of $X$* to be $$\operatorname{ObsDiam}(X;-\kappa) := \sup_{f \in {\mathcal{L}\mathit{ip}}_1(X)} \operatorname{diam}(f_\#\mu_X;1-\kappa).$$
\[prop:ObsDiam-dom\] If $X \prec Y$ for two mm-spaces $X$ and $Y$, then $$\operatorname{ObsDiam}(X;-\kappa) \le \operatorname{ObsDiam}(Y;-\kappa)$$ for any $\kappa > 0$.
### Distance between measures
The *total variation distance $\operatorname{\mathit{d}_\mathrm{TV}}(\mu,\nu)$* of two Borel probability measures $\mu$ and $\nu$ on a topological space $X$ is defined by $$\operatorname{\mathit{d}_\mathrm{TV}}(\mu,\nu) := \sup_A |\,\mu(A) - \nu(A)\,|,$$ where $A$ runs over all Borel subsets of $X$.
If $\mu$ and $\nu$ are both absolutely continuous with respect to a Borel measure $\omega$ on $X$, then $$\operatorname{\mathit{d}_\mathrm{TV}}(\mu,\nu) = \frac{1}{2} \int_X \left| \frac{d\mu}{d\omega} - \frac{d\nu}{d\omega} \right| \; d\omega,$$ where $\frac{d\mu}{d\omega}$ is the Radon-Nikodym derivative of $\mu$ with respect to $\omega$.
The *Prohorov distance* $\operatorname{\mathit{d}_\mathrm{P}}(\mu,\nu)$ between two Borel probability measures $\mu$ and $\nu$ on a metric space $X$ is defined to be the infimum of $\varepsilon > 0$ satisfying $$\mu(B_\varepsilon(A)) \ge \nu(A) - \varepsilon$$ for any Borel subset $A \subset X$, where $$B_\varepsilon(A) := \{\; x \in X \mid d_X(x,A) < \varepsilon\;\}.$$
The Prohorov metric is a metrization of weak convergence of Borel probability measures on $X$ provided that $X$ is a separable metric space.
\[dpdtv\] For any two Borel probability measures $\mu$ and $\nu$ on a metric space $X$, we have $\operatorname{\mathit{d}_\mathrm{P}}(\mu,\nu) \le \operatorname{\mathit{d}_\mathrm{TV}}(\mu,\nu)$.
Let $(X,\mu)$ be a measure space and $Y$ a metric space. For two $\mu$-measurable maps $f,g : X \to Y$, we define the *Ky Fan distance ${{d_{\rm KF}^{}}}(f,g)$ between $f$ and $g$* to be the infimum of $\varepsilon \ge 0$ satisfying $$\mu(\{\;x \in X \mid d_Y(f(x),g(x)) > \varepsilon\;\}) \le \varepsilon.$$
${{d_{\rm KF}^{}}}$ is a pseudo-metric on the set of $\mu$-measurable maps from $X$ to $Y$. It follows that ${{d_{\rm KF}^{}}}(f,g) = 0$ if and only if $f = g$ $\mu$-a.e.
### Box distance and observable distance
Let $I := [\,0,1\,)$ and let $X$ be an mm-space. A map $\varphi : I \to X$ is called a *parameter of $X$* if $\varphi$ is a Borel measurable map such that $\varphi_\#{\mathcal{L}}^1 = \mu_X$, where ${\mathcal{L}}^1$ denotes the one-dimensional Lebesgue measure on $I$.
It is known that any mm-space has a parameter.
We define the *box distance $\square(X,Y)$ between two mm-spaces $X$ and $Y$* to be the infimum of $\varepsilon \ge 0$ satisfying that there exist parameters $\varphi : I \to X$, $\psi : I \to Y$, and a Borel subset $I_0 \subset I$ such that $${\mathcal{L}}^1(I_0) \ge 1-\varepsilon \quad\text{and}\quad
|\,\varphi^*d_X(s,t)-\psi^*d_Y(s,t)\,| \le \varepsilon$$ for any $s,t \in I_0$, where $\varphi^*d_X(s,t) := d_X(\varphi(s),\varphi(t))$ for $s,t \in I$.
The box metric $\square$ is a complete separable metric on ${\mathcal{X}}$.
\[prop:box-dP\] Let $X$ be a complete separable metric space. For any two Borel probability measures $\mu$ and $\nu$ on $X$, we have $$\square((X,\mu),(X,\nu)) \le 2 \operatorname{\mathit{d}_\mathrm{P}}(\mu,\nu).$$
\[defn:dconc\] For any parameter $\varphi$ of $X$, we set $$\varphi^*{\mathcal{L}\mathit{ip}}_1(X)
:= \{\;f\circ\varphi \mid f \in {\mathcal{L}\mathit{ip}}_1(X)\;\}.$$ We define the *observable distance $\operatorname{\mathit{d}_\mathrm{conc}}(X,Y)$ between two mm-spaces $X$ and $Y$* by $$\operatorname{\mathit{d}_\mathrm{conc}}(X,Y) := \inf_{\varphi,\psi} \operatorname{\mathit{d}_\mathrm{H}}(\varphi^*{\mathcal{L}\mathit{ip}}_1(d_X),\psi^*{\mathcal{L}\mathit{ip}}_1(d_Y)),$$ where $\varphi : I \to X$ and $\psi : I \to Y$ run over all parameters of $X$ and $Y$, respectively, and where $\operatorname{\mathit{d}_\mathrm{H}}$ is the Hausdorff metric with respect to the Ky Fan metric ${{d_{\rm KF}^{}}}$ for the one-dimensional Lebesgue measure on $I$. $\operatorname{\mathit{d}_\mathrm{conc}}$ is a metric on ${\mathcal{X}}$. We say that a sequence $\{X_n\}_{n=1}^\infty$ of mm-spaces *concentrates* or *converges weakly* to an mm-space $X$ if $X_n$ $\operatorname{\mathit{d}_\mathrm{conc}}$-converges to $X$ as $n\to\infty$.
\[prop:dconc-box\] For any two mm-spaces $X$ and $Y$ we have $\operatorname{\mathit{d}_\mathrm{conc}}(X,Y) \le \square(X,Y)$.
### Group action {#ssec:quotient}
Let $X$ be a metric space and $G$ a group acting on $X$ isometrically. Let $\bar{X} = X/G$ be the quotient space of $X$ by the $G$-action. Denote by $\bar{x}$ the class in $\bar{X}$ represented by a point $x \in X$. We define a pseudo-metric $d_{\bar{X}}$ on the quotient space $\bar{X}$ by $$d_{\bar{X}}(\bar{x},\bar{y}) := \inf_{g,h \in G} d_X(g\cdot x,h \cdot y),
\qquad \bar{x},\bar{y} \in \bar{X}.$$ $d_{\bar{X}}$ is a metric if every orbit of $G$ is closed in $X$.
Let $X$ and $Y$ be two metric spaces and $G$ a group acting on $X$ and $Y$ isometrically. For any $G$-equivariant map $f : X \to Y$ (i.e., $g\cdot f(x) = f(g\cdot x)$), we have a unique map $\bar{f} : \bar{X} \to \bar{Y}$ with the property that $\bar{f}(\bar{x}) = \overline{f(x)}$ for any $x \in X$. We call $\bar{f}$ the *quotient map of $f$*.
\[lem:equivdomi\] For any $L > 0$ and for any $G$-equivariant $L$-Lipschitz map $f : X \to Y$, the quotient map $\bar{f}$ is also $L$-Lipschitz.
For any $x, y \in X$, we see $$\begin{aligned}
& d_{\bar{Y}}(\bar{f}(\bar{x}),\bar{f}(\bar{y})) = d_{\bar{Y}}(\overline{f(x)},\overline{f(y)})
= \inf_{g,h \in G} d_Y(g\cdot f(x),h\cdot f(y)) \\
&= \inf_{g,h \in G} d_Y(f(g\cdot x),f(h\cdot y)) \le L \inf_{g,h \in G} d_X(g\cdot x,h\cdot y)
= L \, d_{\bar{X}}(\bar{x},\bar{y}).\end{aligned}$$ This completes the proof.
For a Borel measure $\mu$ on $X$, we denote by $\bar\mu$ the push-forward measure of $\mu$ by the natural projection $X \to \bar{X}$.
\[lem:dP-quotient\] Let $X$ be a metric space and $G$ a group acting on $X$ isometrically. Then, for any two Borel probability measures $\mu$ and $\nu$ on $X$, we have $\operatorname{\mathit{d}_\mathrm{P}}(\bar{\mu},\bar{\nu}) \le \operatorname{\mathit{d}_\mathrm{P}}(\mu,\nu)$.
\[defn:quotient\] Let $X$ be an mm-space and $G$ a group acting on $X$ isometrically such that every orbit is closed in $X$. We equip the quotient space $\bar{X}$ with $d_{\bar{X}}$ and $\mu_{\bar{X}} := \bar\mu_X$, and call it the *quotient mm-space of $X$ by the $G$-action*.
### Pyramid
\[defn:pyramid\] A subset ${\mathcal{P}}\subset {\mathcal{X}}$ is called a *pyramid* if it satisfies the following (1), (2), and (3).
1. If $X \in {\mathcal{P}}$ and if $Y \prec X$, then $Y \in {\mathcal{P}}$.
2. For any two mm-spaces $X, X' \in {\mathcal{P}}$, there exists an mm-space $Y \in {\mathcal{P}}$ such that $X \prec Y$ and $X' \prec Y$.
3. ${\mathcal{P}}$ is nonempty and $\square$-closed.
We denote the set of pyramids by $\Pi$.
For an mm-space $X$ we define $${\mathcal{P}}_X := \{\;X' \in {\mathcal{X}}\mid X' \prec X\;\},$$ which is a pyramid. We call ${\mathcal{P}}_X$ the *pyramid associated with $X$*.
We observe that $X \prec Y$ if and only if ${\mathcal{P}}_X \subset {\mathcal{P}}_Y$. It is trivial that ${\mathcal{X}}$ is a pyramid.
\[defn:w-conv\] Let ${\mathcal{P}}_n, {\mathcal{P}}\in \Pi$, $n=1,2,\dots$. We say that *${\mathcal{P}}_n$ converges weakly to ${\mathcal{P}}$* as $n\to\infty$ if the following (1) and (2) are both satisfied.
1. For any mm-space $X \in {\mathcal{P}}$, we have $\lim_{n\to\infty} \square(X,{\mathcal{P}}_n) = 0$.
2. For any mm-space $X \in {\mathcal{X}}\setminus {\mathcal{P}}$, we have $\liminf_{n\to\infty} \square(X,{\mathcal{P}}_n) > 0$.
For an mm-space $X$, a pyramid ${\mathcal{P}}$, and $t > 0$, we define $$tX := (X,t\,d_X,\mu_X) \quad\text{and}\quad
t{\mathcal{P}}:= \{\; tX \mid X \in {\mathcal{P}}\;\}.$$ The following is obvious.
\[lem:obvious\]
1. Let ${\mathcal{P}}$ and ${\mathcal{P}}_n$, $n=1,2,\dots$, be pyramids, and let $t, t_n$ be positive real numbers. If $t_n \to t$ and ${\mathcal{P}}_n$ converges weakly to ${\mathcal{P}}$ as $n\to\infty$, then $t_n{\mathcal{P}}_n$ converges weakly to $t{\mathcal{P}}$ as $n\to\infty$.
2. If $\{X_n\}_{n=1}^\infty$ is a monotone increasing sequence of mm-spaces with respect to the Lipschitz order, then ${\mathcal{P}}_{X_n}$ converges weakly to the $\Box$-closure of the union of ${\mathcal{P}}_{X_n}$.
\[thm:emb\] There exists a metric $\rho$ on $\Pi$ compatible with weak convergence and satisfying the following [(1)]{}, [(2)]{}, and [(3)]{}.
1. The map $\iota : {\mathcal{X}}\ni X \mapsto {\mathcal{P}}_X \in \Pi$ is a $1$-Lipschitz topological embedding map with respect to $\operatorname{\mathit{d}_\mathrm{conc}}$ and $\rho$.
2. \[cpt\] $\Pi$ is $\rho$-compact.
3. $\iota({\mathcal{X}})$ is $\rho$-dense in $\Pi$.
In particular, $(\Pi,\rho)$ is a compactification of $({\mathcal{X}},\operatorname{\mathit{d}_\mathrm{conc}})$.
Note that we identify $X$ with ${\mathcal{P}}_X$ in §1.
Combining Propositions \[dpdtv\], \[prop:box-dP\], \[prop:dconc-box\], and Theorem \[thm:emb\] yields the following
\[cor:rho-dTV\] For any two Borel probability measures $\mu$ and $\nu$ on a complete separable metric space $X$, we have $$\begin{aligned}
\rho({\mathcal{P}}_{(X,\mu)},{\mathcal{P}}_{(X,\nu)}) &\le \operatorname{\mathit{d}_\mathrm{conc}}((X,\mu),(X,\mu)) \le \square((X,\mu),(X,\nu)) \\
&\le 2\operatorname{\mathit{d}_\mathrm{P}}(\mu,\nu) \le 2\operatorname{\mathit{d}_\mathrm{TV}}(\mu,\nu).
\end{aligned}$$
Decompositions of real, complex, and quaternion matrices
--------------------------------------------------------
Let $F$ be one of ${{\mathbb{R}}}, {{\mathbb{C}}}$ and ${{\mathbb{H}}}$, where ${{\mathbb{H}}}$ is the non-commutative algebra ${{\mathbb{H}}}$ of quaternions, which is defined as $${{\mathbb{H}}}:=\{z:=z_0 + z_1 \i+ z_2 \j +z_3 \k \ |\ z_0, z_1,z_2, z_3 \in {{\mathbb{R}}}\}, \quad \i^2=\j^2=\k^2=\i\j\k=-1.$$ Note that ${{\mathbb{R}}}$ and ${{\mathbb{C}}}$ are naturally embedded into ${{\mathbb{H}}}$. For $z =z_0 + z_1 \i+ z_2 \j +z_3 \k \in {{\mathbb{H}}}$, we define $$\begin{gathered}
\Re(z):=z_0, \quad
z^\ast:=z_0-(z_1 \i+ z_2 \j +z_3 \k ).\end{gathered}$$ Let $M_{N,n}^F$ denote the set of all $N \times n$ matrices over $F$ and let $M_N^F := M_{N,N}^F$. For $N \in {{\mathbb{N}}}$, we set $N^F:=N \cdot \dim_{{\mathbb{R}}}F$ and sometimes identify $M_{N,n}^F$ with $(F^N)^n$ and ${{\mathbb{R}}}^{N^F n}$, where we consider $F^N$ as the space of numerical column vectors over $F$. Let $\{e_l\}_{l=1}^N$ denote the standard basis of $F^N$. The identity matrix of size $N$ is indicated by $I_{N}$ and the $N \times n$ zero matrix by $0_{N,n}$. For $Z=(z^m_l)_{1\leq m\leq N, 1\leq l\leq n} \in M_{N,n}^F$, its [*adjoint*]{} $Z^\ast$ (reap. [*transpose*]{} $\operatorname{{}^{T}\!\!}Z$) is the $n \times N$ matrix with $(l,m)$-component $(z^m_l)^\ast$ (resp. $z^m_l$). The *trace* and the *Frobenius norm* of $Z=(z^m_l)_{1\leq m, l\leq N} \in M_{N}^F$ are respectively defined as $$\operatorname{tr}(Z):=\sum_{l=1}^N z^l_l, \qquad
\|Z\|:=\sqrt{\operatorname{tr}(Z^\ast Z)}.$$ It follows that $(\operatorname{tr}(Z))^\ast=\operatorname{tr}(Z^\ast)$ and $(ZW)^\ast=W^\ast Z^\ast$ for any $Z, W\in M_{N}^F$. Define an $F$-valued function ${\left\langle{\cdot},{\cdot}\right\rangle}$ on $M_{N,n}^F \times M_{N,n}^F$ by $${\left\langle{Z},{W}\right\rangle}:=\operatorname{tr}(Z^\ast W)$$ for $Z,W \in M_{N,n}^F$. Then, $\Re{\left\langle{\cdot},{\cdot}\right\rangle}$ is an ${{\mathbb{R}}}$-inner product on $M_{N,n}^F$ and we have $$\|Z-W\|^2
=\|Z\|^2+\|W\|^2- (\operatorname{tr}(Z^\ast W)+\operatorname{tr}(W^\ast Z))
=\|Z\|^2+\|W\|^2- 2 \Re {\left\langle{Z},{W}\right\rangle}.$$ Although ${\left\langle{\cdot},{\cdot}\right\rangle}$ is not $F$-bilinear, a standard argument proves the Cauchy–Schwarz inequality $\|{\left\langle{Z},{W}\right\rangle} \| \leq \|Z\|\|W\|$ for any $Z,W \in M_{N,n}^F$.
We say that a matrix $Z \in M_{N}^F$ is [*unitary*]{} (resp. [*Hermitian*]{}) if we have $Z^\ast Z=Z Z^\ast=I_N$ (resp. $Z^\ast=Z$).
We remark that $U \in M_{N}^F$ is unitary if and only if $\|Uz\|=\|z\|$ holds for any $z \in F^N$. Let $U^F(N)$ denote the group of unitary matrices of size $N$. For any Hermitian matrix $H\in M_{N}^F$, there exist $U \in U^F(N)$ and $\{\sigma_l\}_{l=1}^N \subset {{\mathbb{R}}}$ such that $$H=U \cdot \operatorname{diag}(\sigma_1,\ldots,\sigma_N) \cdot U^\ast$$ (see [@Zhang]\*[Corollary 6.2]{}), where $\operatorname{diag}(\cdots)$ denotes the diagonal matrix. This decomposition is called the [*eigen decomposition*]{} and $\sigma_l$, $l=1,\dots,N$, are called the [*eigenvalues*]{} of $H$. It is easy to check that $Z^\ast Z$ for $Z \in M_{N,n}^F$ is Hermitian and all its eigenvalues are non-negative.
Let us introduce two matrix decompositions.
\[thm:decomp\] For any matrix $Z \in M_{N,n}^F$ with $N \geq n$, there exist a Hermitian matrix $H \in M_{n}^F$ with non-negative eigenvalues and a matrix $Q \in M_{N,n}^F$ such that $$Z=Q\cdot H, \quad Q^\ast Q=I_n.$$ In addition, there also exist $U \in U^F(N), V \in U^F(n)$ and a monotone non-increasing sequence $\{\lambda_l(Z)\}_{l=1}^n$ of non-negative numbers such that $$\begin{gathered}
Z=U \cdot \Lambda \cdot V^\ast, \quad
\Lambda:=
\begin{pmatrix}
\operatorname{diag}(\lambda_1(Z),\ldots,\lambda_n(Z))\\
0_{N-n,n}
\end{pmatrix},\\
Q= U \cdot
\begin{pmatrix}
I_n \\
0_{N-n,n}
\end{pmatrix} \cdot V^\ast,
\quad
H=V \cdot \operatorname{diag}(\lambda_1(Z),\ldots,\lambda_n(Z)) \cdot V^\ast.\end{gathered}$$
The two decompositions $Z=QH$ and $Z=U\Lambda V^\ast$ are called *polar* and *singular value decompositions*, respectively. Although the two matrix decompositions may not be unique in general, $\{\lambda_l(Z)\}_{l=1}^n$ is uniquely determined, which coincides with the positive square root of the eigenvalues of $Z^\ast Z$. We call $\lambda_l(Z)$, $l=1,\dots,n$, the [*singular values*]{} of $Z$. In the case of $\lambda_n(Z)>0$, the polar decomposition is unique and $Q,H$ are given by $$Q=ZH^{-1}, \quad
H=(Z^\ast Z)^{1/2}.$$
\[spectrum\] (1) Given any $U\in U^F(N)$ and $Z \in M_{N,n}^F$, we see that $Z=Q\cdot H$ is a polar decomposition of $Z$ if and only if so is $UZ=(UQ)\cdot H$.\
(2) We observe that the maximal singular value of a matrix coincides with its spectrum norm. The triangle inequality for the spectrum norm implies that $\lambda_1(Z+W) \leq \lambda_1(Z)+\lambda_1(W)$.
Stiefel manifold and its quotient space
---------------------------------------
For $N,n \in {{\mathbb{N}}}$ with $N \geq n$, the $(N,n)$-[*Stiefel manifold*]{} $V_{N,n}^F$ over $F$ is the set of all orthonormal $n$-frames in $F^N$, namely $$V_{N,n}^F=\{ (z_1,\ldots, z_n) \in M_{N,n}^F \ |\ {\left\langle{z_l},{z_m}\right\rangle}= \delta_{lm}, \ 1 \leq l, m \leq n \},$$ where $\delta_{lm}$ is the Kronecker delta. Denote by $\nu^{N,n,F}$ the Haar (or uniform) probability measure on $V_{N,n}^F$.
Note that $V_{N,N}^F=U^F(N)$ and $V_{N,1}^F$ is identified with the $(N^F-1)$-dimensional Euclidean unit sphere.
Let us recall a characterization of a Haar measure. Given any $U \in U^F(N)$, we define the map $\mathcal{U}^U_{m}:M_{N,m}^F \to M_{N,m}^F$ by $\mathcal{U}^U_{m}(Z)=UZ$ for $Z \in M_{N,m}^F$. The map $\mathcal{U}^U_{m}$ is isometric and its inverse map is given by $\mathcal{U}^{U^\ast}_{m}$.
\[prop:haar\] A Borel probability measure $\nu$ on $V_{N,n}^F$ coincides with $\nu^{N,n,F}$ if and only if $\nu$ is left-invariant under the $U^F(N)$-action, that is, $$\nu(\mathcal{U}_n^U(B))=\nu(B)$$ holds for any Borel set $B \subset V_{N,n}^F$ and $U \in U^F(N)$.
We consider the $(N,n)$-Stiefel manifold over $F$ as an mm-space $V_{N,n}^F=(V_{N,n}^F, \|\cdot\|,\nu^{N,n,F})$.
The unitary group $U^F(n)$ acts on $M_{N,n}^F$ by right multiplication, that is, $$M_{N,n}^F \times U^F(n) \ni (Z,U)\mapsto ZU \in M_{N,n}^F.$$ We also consider the *Hopf action* on $M_{N,n}^F$ that is the acton of the unitary group $U^F(1)=\{\;t \in F\ |\ \|t\|=1\;\}$ of size $1$ given by left multiplication, $$U^F(1) \times M_{N,n}^F \ni (t,Z)\mapsto tZ \in M_{N,n}^F.$$ Note that every orbit in $M_{N,n}^F$ of $U^F(n)$ and $U^F(1)$ is closed. We call the two quotient mm-spaces $$G_{N,n}^F := V_{N,n}^F/U^F(n) \quad\text{and}\quad {\mathrm{P}V}_{N,n}^F := U^F(1)\backslash V_{N,n}^F$$ the $(N,n)$-*Grassmann manifold* and the $(N,n)$-*projective Stiefel manifold over $F$*, respectively. The $(N,n)$-Stiefel manifold over $F$ with distance multiplied by $\sqrt{N^F - 1}$ is identified with $$\begin{aligned}
X_{N,n}^F := \{ (z_1,\ldots, z_n) \in M_{N,n}^F \ |\ {\left\langle{z_l},{z_m}\right\rangle}=(N^F-1) \delta_{lm}, \ 1 \leq l, m \leq n \}\end{aligned}$$ with the Frobenius norm and the Haar probability measure $\mu^{N,n,F}$ on $X_{N,n}^F$. Note that $\mu^{N,n,F}$ is a unique left-invariant Borel probability measure under the $U^F(N)$-action.
Gaussian space {#sec:Gauss}
--------------
For a positive integer $m$, we denote by $\gamma^m$ the (standard) [*Gaussian measure on ${{\mathbb{R}}}^m$* ]{}, which is defined for a Lebesgue measurable set $ B \subset {{\mathbb{R}}}^m$ by $$\gamma^m(B)=(2\pi)^{-m/2} \int_B \exp \left(-\frac{\|x\|^2}{2}\right) dx.$$ The mm-space $\Gamma^m:=({{\mathbb{R}}}^m, \|\cdot\|, \gamma^m)$ is called the *$m$-dimensional* (*standard*) *Gaussian space*. By Lemma \[lem:obvious\](2), as $m\to\infty$, ${\mathcal{P}}_{\Gamma^m}$ converges weakly to the $\square$-closure of the union of ${\mathcal{P}}_{\Gamma^m}$, which we call the *virtual infinite-dimensional Gaussian space* ${\mathcal{P}}_{\Gamma^\infty}$.
Let $l \le N$ and let $\pi_l^N(n):M_{N,n}^F \to M_{l,n}^F$ be the projection defined by $$\pi_l^N(n):M_{N,n}^F \ni
\begin{pmatrix}
z^1_1, &\ldots, &z^1_n\\
\vdots & &\vdots\\
z^N_1 &\ldots, &z^N_n
\end{pmatrix}
\mapsto
\begin{pmatrix}
z^1_1, &\ldots, &z^1_n\\
\vdots & &\vdots\\
z^l_1 &\ldots, &z^l_n
\end{pmatrix}
\in M_{l,n}^F.$$ We set $\pi^N_l := \pi^N_l(1) : F^N \to F^l$. The projections $\pi^N_l$ and $\pi_l^N(n)$ are both $1$-Lipschitz continuous, preserving Gaussian measures, and equivariant under the $U^F(1)$-Hopf action and the $U^F(n)$-action, respectively. We remark that the $U^F(1)$-Hopf action and the $U^F(n)$-action for $n=1$ do not coincide with each other in the case where $F = {{\mathbb{H}}}$, because of the non-commutativity of ${{\mathbb{H}}}$. We have the quotient maps $\overline{\pi}^N_l : U^F(1)\backslash F^N\to U^F(1)\backslash F^l$ and $\overline{\pi}^{N}_l(n) : M_{N,n}^F/U^F(n)\to M_{l,n}^F/U^F(n)$, which are both $1$-Lipschitz continuous by Lemma \[lem:equivdomi\]. Note that the actions of $U^F(1), U^F(n)$ on $F^N, M_{N,n}^F$ each preserve the Gaussian measure. We denote by $U^F(1)\backslash\Gamma^{N^F}$ and $\Gamma^{N^F n}/U^F(n)$ the quotient mm-spaces of $(F^N,\|\cdot\|,\gamma^{N^F})$ and $(M_{N,n}^F, \|\cdot\|,\gamma^{N^F n})$ by the $U^F(1)$-Hopf and $U^F(n)$ actions, respectively (see Definition \[defn:quotient\]). Then, the sequences $\{U^F(1)\backslash\Gamma^{N^F}\}_{N=1}^\infty$ and $\{\Gamma^{N^F n }/U^F(n)\}_{N=1}^\infty$ are both monotone increasing with respect to the Lipschitz order. Therefore, the associated pyramids converge weakly to the $\Box$-closure of the unions, $${\mathcal{P}}_{U^F(1)\backslash\Gamma^{\infty}}
:=
\overline{\bigcup_{N=1}^\infty {\mathcal{P}}_{U^F(1)\backslash\Gamma^{N^F}}}^\Box,
\qquad
{\mathcal{P}}_{\Gamma^{\infty n}/U^F(n)}:=\overline{ \bigcup_{N=1}^\infty {\mathcal{P}}_{\Gamma^{N^F n }/U^F(n)}}^\Box,$$ respectively. We remark that for each positive integer $n$ the quotient mm-space of $(M^F_{N,n},\|\cdot\|,\gamma^{N^F n})$ by the $U^F(1)$-Hopf action is mm-isomorphic to $U^F(1)\backslash \Gamma^{N^F n}$ whose associated pyramid converges weakly to ${\mathcal{P}}_{U^F(1)\backslash\Gamma^{\infty}}$ as $N \to \infty$.
Let us close this section with two approximations related to the Gaussian measure.
\[thm:MB\] For any $l,n \in {{\mathbb{N}}}$, $\lim_{m \to \infty}\operatorname{\mathit{d}_\mathrm{P}}(\pi^{m}_l(n)_\# \mu^{m,n,F},\gamma^{l^F n})=0$.
Proposition \[thm:MB\] is a generalization of the Maxwell-Boltzmann distribution law.
\[lem:stir\] Let $\Gamma$ be the Gamma function. There exists a decreasing function $\rho:(0,\infty) \to (0,\infty)$ such that $$\Gamma(x)=\frac{\Gamma(x+1)}{x}=\sqrt{ \frac{2\pi}{x}} \left(\frac{x}{e}\right)^x e^{\rho{(x)}},
\quad
\rho(x) \in\left(0, \frac{1}{12x}\right).$$
Relation between Gaussian space and Stiefel manifold
====================================================
For any $\varepsilon,r>0$ and $m \in {{\mathbb{N}}}$, we set $$A^{m}(r)_{\varepsilon}:=B_{\varepsilon r}\left(S^{m-1}(r)\right)
=\{x \in {{\mathbb{R}}}^m\ |\ (1-\varepsilon)r <\|x\|<(1+\varepsilon)r\}, \quad
A^m_\varepsilon:=A^{m}(\sqrt{m-1})_{\varepsilon}.$$ In this section, we first provide a sufficient condition for $\{\varepsilon_m\}_{m=1}^\infty$ being $\lim_{m\to\infty}\gamma^m (A^{m}_{\varepsilon_m}) =1$. Using the sufficient condition, we prove that the Prohorov distance between $\gamma^{N^F n}$ and $\mu^{N,n,F}$ does not vanish asymptotically (see Theorem \[prok\]) although $\gamma^{N^F n}$ concentrates around $X_{N,n}^F$ (see Theorem \[thm:fullmeas\]), where we regard $\mu^{N,n,F}$ as a probability measure on ${{\mathbb{R}}}^{N^F n}$ via the natural embedding $X_{N,n}^F \subset {{\mathbb{R}}}^{N^F n}$.
Behavior of Gaussian measure {#ssec:Gaussian}
----------------------------
Consider the function $$g_m(r) :
= \frac{\operatorname{vol}(S^{m-1}(1))}{(2\pi)^{m/2}}r^{m-1} e^{-r^2/2}
=\frac{2^{(2-m)/2}}{\Gamma(m/2)}r^{m-1} e^{-r^2/2},$$ which is the density of the radial distribution of the $m$-dimensional Gaussian measure. This satisfies $$g_m(r) \leq g_m(\sqrt{m-1})
=
\frac{e^{-\rho(m/2)}}{\sqrt{\pi}} e^{1/2}\left(1-\frac1m\right)^{(m-1)/2}\xrightarrow{m \to \infty}
\frac1{\sqrt{\pi}}$$ by Lemma \[lem:stir\]. Moreover, $g_m(\sqrt{m-1})$ is monotone decreasing in $m$. We directly compute $$\begin{gathered}
\label{eq:gauss}
\gamma^{m}(A^m_{\varepsilon})
=
\int_{(1- \varepsilon) \sqrt{m-1}}^{(1+ \varepsilon) \sqrt{m-1}} g_{m}(r) dr
\leq
\frac{e^{-\rho(m/2)}}{\sqrt{\pi}} e^{1/2}\left(1-\frac1m\right)^{(m-1)/2}\cdot 2\varepsilon\sqrt{m-1}, \\
\label{eq:gau}
1-\gamma^{m}(A^m_{\varepsilon})
=
\int_{0}^{(1- \varepsilon_m) \sqrt{m-1} } g_{m}(r) dr+\int_{(1+ \varepsilon_m)\sqrt{m-1}}^\infty g_{m}(r) dr.\end{gathered}$$ We provide a sufficient condition for $\{\varepsilon_m\}_{m=1}^\infty$ such that $\lim_{m \to 1} \gamma^m (A^{m}_{\varepsilon_m})=1$, based on the idea of [@Sy:mmlim Lemma 6.1].
\[lem:zero\] The zero $t_0$ of the function given by $$G(t):=e^{-t^2/2}-\int_t^\infty e^{-s^2/2} ds, \qquad t \in {{\mathbb{R}}}.$$ is unique and lies in $(0,1)$.
The lemma follows from the intermediate value theorem and the following properties $$G'(t)=e^{-t^2/2}(-t+1), \quad
G(0)=1-\sqrt{\frac{\pi}{2}}<0=\lim_{t\to \infty} G(t) <G(1).$$
\[rem:zero\] If we set $$\begin{aligned}
G_m^+(r)
&:=e^{-(r\sqrt{m-1})^2/2}-\int_{(1+r)\sqrt{m-1}}^\infty e^{-(\sqrt{m-1}-s)^2/2} ds
=e^{-(r\sqrt{m-1})^2/2}-\int_{r\sqrt{m-1}}^\infty e^{-s^2/2} ds,\\
G_m^-(r)
&:=e^{-(r\sqrt{m-1})^2/2}-\int_0^{(1-r)\sqrt{m-1}} e^{-(\sqrt{m-1}-s)^2/2} ds
=e^{-(r\sqrt{m-1})^2/2}+\int_{\sqrt{m-1}}^{r\sqrt{m-1}}e^{-s^2/2} ds,
$$ then $G_m^-(r)>G_m^+(r)=G(r\sqrt{m-1})$ always holds and $G_m^+(r) \geq 0$ if $r\sqrt{m-1} \geq t_0$.
\[lem:gauss\] If $\lim_{m \to \infty} \varepsilon_m \sqrt{m-1} =\infty$, then $\lim_{m \to \infty} \gamma^m(A^m_{\varepsilon_m})=1$.
Since we observe $$\frac{d}{dr}\log g_m(r)\big|_{r=\sqrt{m-1}}=0, \quad
\frac{d^2}{dr^2}\log g_m(r)
= -1 -\frac{m-1}{r^2} \leq -1,$$ it holds for any $r>0$ that $$\begin{aligned}
\log g_m(r)-\log g_m(\sqrt{m-1})
&=\int_r^{\sqrt{m-1}}\int_{t}^{\sqrt{m-1}} \left( \frac{d^2}{ds^2}\log g_m(s) \right) ds dt \\
&
\leq \int_r^{\sqrt{m-1}}\int_{t}^{\sqrt{m-1}} (-1) ds dt
=-\frac12 (\sqrt{m-1}-r)^2, \end{aligned}$$ providing $g_m(r) \leq g_m(\sqrt{m-1}) \exp(-(\sqrt{m-1}-r)^2/2 )$. We deduce from this and Remark \[rem:zero\] that if $\varepsilon_m \sqrt{m-1} >t_0$, then we see $$\begin{aligned}
\int_0^{(1- \varepsilon_m) \sqrt{m-1} } g_{m}(r) dr
&
\leq g_{m}(\sqrt{m-1}) \int_0^{(1- \varepsilon_m) \sqrt{m-1} } e^{- (\sqrt{m-1}-r)^2/2} dr \\
&
\leq g_{m}(\sqrt{m-1}) e^{-(\varepsilon_m \sqrt{m-1})^2/2},\\
\int_{(1+ \varepsilon_m) \sqrt{m-1} }^\infty g_{m}(r) dr
&
\leq g_{m}(\sqrt{m-1}) \int_{(1+ \varepsilon_m) \sqrt{m-1} }^\infty e^{- (\sqrt{m-1}-r)^2/2} dr \\
&\leq g_{m}(\sqrt{m-1}) e^{-(\varepsilon_m \sqrt{m-1})^2/2}.
$$ This with leads to $$\begin{aligned}
\liminf_{m \to \infty} \left( 1-\gamma^{m}(A^m_{\varepsilon_m}) \right)
\leq
\liminf_{m \to \infty}
\left\{g_{m}(\sqrt{m-1}) \cdot 2e^{-(\varepsilon_m \sqrt{m-1})^2/2}\right\}
=0.
$$ The proof is completed.
Prohorov distance between $\gamma^{N^F n}$ and $ \mu^{N,n_N,F}$
---------------------------------------------------------------
By the definition of the Prohorov distance, it holds for any $D_{N}>\operatorname{\mathit{d}_\mathrm{P}}(\gamma^{N^F n_N}, \mu^{N,n_N,F})$ that $$\begin{aligned}
1-D_N&=
\nu^{N,n_N,F}(X_{N,n_N}^{F})-D_N \\
&\leq
\gamma^{N^{F} n_N}(B_{D_N}(X_{N,n_N}^{F}))
\leq
\left( \gamma^{N^{F}}( B_{D_N}(X_{N,1}^{F})) \right)^{n_N} \\
&\leq \left\{
\frac{e^{-\rho(N^F/2)}}{\sqrt{\pi}} e^{1/2}\left(1-\frac1{N^F}\right)^{(N^F-1)/2}\cdot 2D_N \right\}^{n_N},
$$ where we use the property that $B_{D_N} (X_{N,n_N}^{F})\subset (B_{D_N}(X_{N,1}^{F}))^{n_N}$ in the second inequality and the last inequality follows from . Since $$\begin{aligned}
\log\left\{e^{1/2}\left(1-\frac1{m}\right)^{(m-1)/2}\right\}^m
= \frac{m}2\left\{1-(m-1)\left(\frac1{m}+\frac1{2m^2}+o\left(\frac1{m^2}\right)\right) \right\} = \frac{1}{4}+o(1)
$$ and $e(1-1/m)^{m-1}>1$ for any $m \in {{\mathbb{N}}}$, the property $n_N \leq N \leq N^{F}$ implies $$\begin{aligned}
1
\leq \liminf_{N\to\infty}
\left\{D_N+\gamma^{N^{F} n_N} (B_{D_N} (X_{N,n_N}^{F})) \right\}
\leq
\liminf_{N\to\infty}
\left\{D_N+ \left(\frac{2 D_N}{\sqrt{\pi}}\right)^{n_N} e^{1/4} \right\},
$$ providing $\liminf_{N\to\infty} D_N>0$. Since $D_{N}>\operatorname{\mathit{d}_\mathrm{P}}(\gamma^{N^F n_N}, \mu^{N,n_N,F})$ is arbitrary, we conclude $\liminf_{N\to\infty} \operatorname{\mathit{d}_\mathrm{P}}(\gamma^{N^F n_N}, \mu^{N,n_N,F})>0$. This completes the proof.
$\gamma^{N^F n}$ concentrates around $X_{N,n}^{F}$
--------------------------------------------------
Given a function $\theta:(0,\infty) \to (0,\infty)$ and $\varepsilon>0$, we define the *$(\varepsilon, \theta)$-approximation space $X_{N,n,\varepsilon,\theta}^F$ of $X_{N,n}^{F}$* as $$\begin{aligned}
X_{N,n,\varepsilon,\theta}^F
:=\left\{(z_1,\ldots,z_n) \in M_{N,n}^F \biggm| \|z_l\| \in A^{N^F}_\varepsilon,
\left\|{\left\langle{\frac{z_l}{\|z_l\|}},{\frac{z_m}{\|z_m\|}}\right\rangle}\right\|< \theta(\varepsilon), \ 1 \leq l<m \leq n\right\},
$$ where we identify $F^N$ with ${{\mathbb{R}}}^{N^F}$.
The purpose of this subsection is to prove the following theorem.
\[thm:fullmeas\] Suppose that a sequence $\{n_N\}_{N=1}^\infty \subset {{\mathbb{N}}}$ satisfies $$\label{eq:ass}
\sup_N \left( 2\log n_N - \frac{a'}{4} \cdot \left(\frac{N}{n_N}\right)^{1-a} \right) < \infty
\quad
\text{\ for some } a, a' \in (0,1)$$ and put $$\begin{gathered}
p_N:=\log_{(N-1)} n_N,\quad
a_N:=\frac{a}{2}(1-p_N), \quad
\varepsilon_N:=(N-1)^{-a_N}.
$$ If we choose a function $\theta:(0,\infty) \to (0,\infty)$ as $$\label{theta}
\theta(\varepsilon):= \left(\frac{5\varepsilon^2(1+\varepsilon)}{(1-\varepsilon)+5\varepsilon^2(1+\varepsilon)}\right)^{1/2},$$ then we have $$\lim_{N \to \infty}\varepsilon_N=0, \quad
\lim_{N \to \infty}\gamma^{N^F n_N}(X^F_{N,n_N,\varepsilon_{N},\theta}) = 1.$$ In the case that $$\begin{aligned}
\label{eq:condi}
\sup_N \left( 2\log n_N - \frac{a'}{4} \sqrt{\frac{N}{n_N^3}} \right) < \infty
\quad
\text{\ for some } a' \in (0,1),\end{aligned}$$ the statement also holds true if we set $a_N:=(1+p_N)/4$.
Note that coincides with ($*$) in the introduction. In the rest of this section, we always suppose or . Note that follows from (see Remark \[rem:condi\]) and $(1+p_N)/2 \leq 1-p_N$ is equivalent to $p_N \leq 1/3$. It turns out that (resp. ) is equivalent to $$\sup_N \left\{ n_N^2 \exp\left(-\frac{a'}{4}\left(\frac{N-1}{n_N}\right)^{1-a}\right)\right\}<\infty
\
\left(\text{resp.}
\sup_N \left\{n_N^2 \exp\left(-\frac{a'}{4}\sqrt{\frac{N-1}{n_N^3}}\right)\right\}<\infty
\right),$$ which yields for any $r>0$ that $$\begin{gathered}
\label{eq:lim}
\lim_{N\to \infty} (N-1)^{r(p_N-1)}=\lim_{N\to \infty} \left(\frac{n_N}{N-1}\right)^r=0 \quad
\left(
\text{resp.}\
\lim_{N\to \infty} (N-1)^{r(3p_N-1)}=0
\right).\end{gathered}$$ We may assume that $n_N \leq N-1$ without loss of generality. We in addition assume that $p_N \le 1/3$ if is satisfied. In this case, $n_N/(N-1) \leq \varepsilon_N^2 \leq 1$ holds for any $N \in {{\mathbb{N}}}$. To prove Theorem \[thm:fullmeas\], we define $b_l, \varepsilon_{N,l}, T_{N,l}$ for $1 \leq l \leq n_N-1$ by $$\begin{aligned}
b_N:&=1-\varepsilon_N, \\
\varepsilon_{N,l}
:&=1-(1-b_N\varepsilon_N) \sqrt{\frac{N-1}{(N-l)-1}} \\
&=\frac{1}{\sqrt{(N-l)-1}}\left(b_N\varepsilon_N \sqrt{N-1}-\frac{l}{\sqrt{N-1}+\sqrt{(N-l)-1}}\right),\\
T_{N,l}
:&=\left\{\frac{(N-l)-1}{l}(\varepsilon_{N}-\varepsilon_{N,l})(\varepsilon_{N}+\varepsilon_{N,l}+2) \right\}^{1/2}\\
&=\left\{\frac{(N-l)-1}{l} ((1+\varepsilon_{N})^2-(1+\varepsilon_{N,l})^2 )\right\}^{1/2}.\end{aligned}$$
\[lem:limit\] The sequence $\{\varepsilon_{N,l}\}_{l=1}^{n_N-1}$ is monotone decreasing and $\varepsilon_{N,1}<\varepsilon_{N}$. For any $1 \leq l \leq n_N-1$, we have $$\begin{gathered}
\lim_{N \to \infty}\varepsilon_{N}=0, \
\lim_{N \to \infty}\varepsilon_N \sqrt{N^F-1}=\infty, \
\lim_{N \to \infty}\varepsilon_{N,l}\sqrt{(N-l)^F-1} =\infty, \
\lim_{N \to \infty}T_{N,l}=\infty.\end{gathered}$$
The first statement follows from a direct computation as $$\begin{aligned}
&\varepsilon_{N,l-1}-\varepsilon_{N,l}
=
(1-b_N\varepsilon_N)\sqrt{N-1}\left(\frac{1}{\sqrt{(N-l)-1}}-\frac{1}{\sqrt{N-l}}\right)
> 0, \\
&\varepsilon_{N}-\varepsilon_{N,l}
=(1-b_N\varepsilon_N) \left(\sqrt{\frac{N-1}{(N-l)-1}}-1\right)+\varepsilon_N(1-b_N)> 0.
$$ It follows from that $\lim_{N \to \infty}\varepsilon_{N}=0$. We moreover have $$\begin{gathered}
\log_{(N-1)}\varepsilon_N \sqrt{N^F-1}
\geq
\log_{(N-1)}\varepsilon_N \sqrt{N-1}
=
\frac12-a_N
=
\begin{cases}
\displaystyle
\frac{1-a+ap_N}{2}
\geq\frac{1-a}{2}
&\text{for \eqref{eq:ass}}, \\
\displaystyle
\frac{1-p_N}{4}
\geq \frac16
&\text{for \eqref{eq:condi}},
\end{cases}\\
\log_{(N-1)} \frac{\varepsilon_N(N-1)}{n_N}
\geq
\log_{(N-1)} \frac{\varepsilon_N^2(N-1)}{n_N}
= 1-2a_N-p_N
=
\begin{cases}
\displaystyle
(1-a)(1-p_N)& \text{for \eqref{eq:ass}}, \\
\displaystyle
\frac{1-3p_N}{2}
& \text{for \eqref{eq:condi}}.
\end{cases}\end{gathered}$$ These and together yield that $$\lim_{N \to \infty} \varepsilon_N \sqrt{N-1}=\infty,
\quad
\lim_{N \to \infty} \frac{\varepsilon_N^2(N-1)}{n_N}=\infty,
\quad
\lim_{N \to \infty} \frac{n_N}{\varepsilon_N(N-1)}=0.$$ Since we observe that $$\begin{aligned}
\label{en}\notag
\varepsilon_{N,l}\sqrt{(N-l)^F-1}
&\geq
\varepsilon_{N,n_N-1} \sqrt{N-n_N} \\
&=\varepsilon_N \sqrt{N-1} \left(b_N-\frac{\sqrt{N-1}}{\sqrt{N-1}+\sqrt{N-n_N}}\cdot \frac{n_N-1}{\varepsilon_N(N-1)} \right),\\ \notag
T_{N,l}^2
&=\frac{(N-l)-1}{l} (\varepsilon_N-\varepsilon_{N,l}) (\varepsilon_{N}+\varepsilon_{N,l}+2) \\ \label{tn}
&\geq \frac{(N-l)-1}{l} \cdot \varepsilon_N(1-b_N) \cdot 2
\geq 2 \frac{\varepsilon_N^2(N-1)}{n_N} \cdot \frac{n_N}{n_N-1} \frac{N-n_N}{N-1},
$$ we have $\lim_{N \to \infty}\varepsilon_{N,l}\sqrt{(N-l)^F-1} =\infty$ and $\lim_{N \to \infty}T_{N,l}=\infty$. This completes the proof.
\[cor:en\] For all sufficiently large $N$, we have $$\varepsilon_{N,l} \sqrt{(N-l)^F-1} \geq a' \varepsilon_{N} \sqrt{N-1}, \quad
1 \leq l \leq n_{N}-1.$$
By , we have $$\varepsilon_{N,l} \sqrt{(N-l)^F-1}
\geq
\varepsilon_N \sqrt{N-1} \left(b_N-\frac{\sqrt{N-1}}{\sqrt{N-1}+\sqrt{N-n_N}}\cdot \frac{n_N-1}{\varepsilon_N(N-1)} \right).$$ Then the corollary follows from the fact $$\lim_{N \to \infty} \left(b_N-\frac{\sqrt{N-1}}{\sqrt{N-1}+\sqrt{N-n_N}}\cdot \frac{n_N-1}{\varepsilon_N(N-1)} \right)=1.$$
To compute $\gamma^{N^F n }(X_{N,n,\varepsilon,\theta}^{F})$, let us regard $U^F(N)$ as a subgroup of $U^{{{\mathbb{R}}}}(N^F)$ (see Lemma \[lem:unit\]) and put $$A^{N,F}_{\varepsilon, \theta}[z_1,\ldots,z_l]:=
\left\{z_{l+1} \in A^{N^F }_{\varepsilon} \ \bigg|\ \left\| {\left\langle{\frac{z_m}{\|z_m\|}},{\frac{z_{l+1}}{\|z_{l+1}\|}}\right\rangle} \right\|
< \theta(\varepsilon),\ 1 \leq m \leq l \right\}$$ for any $(z_1,\ldots,z_n) \in X_{N,n,\varepsilon,\theta}^{F}$ and $1 \le l \le n-1$. We then have $$\gamma^{N^F n }(X_{N,n,\varepsilon,\theta}^{F})
=\int_{z_1 \in A^{N^F }_{\varepsilon}}
\int_{z_{2} \in A^{N,F}_{\varepsilon,\theta }[z_1]} \cdots
\int_{z_n \in A^{N,{F}}_{\varepsilon,\theta}[z_1,\ldots,z_{n-1}]}
d\gamma^{N^F}(z_{n}) \cdots d\gamma^{N^F }(z_2) d\gamma^{N^F }(z_1).$$ Choose $U \in U^F(N)$ satisfying that ${\left\langle{e_m},{Uz_l}\right\rangle}=0$ for any pair $(m,l)$ with $m>l$. According to the $U^{{\mathbb{R}}}(N^F )$-invariance of $\gamma^{N^F }$, it holds that $$\gamma^{N^F } (A^{N,F}_{\varepsilon,\theta}[z_1,\ldots,z_l])
=\gamma^{N^F } (\mathcal{U}^{U}_1( A^{N,F}_{\varepsilon,\theta}[z_1,\ldots,z_l]))
=\gamma^{N^F } (A^{N,F}_{\varepsilon,\theta}[Uz_1,\ldots,Uz_l]).$$
\[lem:subset\] For any $1 \leq l \leq n_N-1$, if we choose $\theta$ as in , then we have $$\left(B^{F}(T_{N,l})\right)^{l} \times A^{(N-l)^{F}}_{\varepsilon_{N,l}} \subset A^{N,F}_{\varepsilon_N,\theta}[Uz_1,\ldots,Uz_l], \quad
B^{F}(T):=\{ z \in F \ |\ \|z\|<T\}.$$
Given any $$w:=(w_1,\ldots, w_l)\in \left(B^{F}(T_{N,l}) \right)^{l} \subset F^l, \quad
\zeta \in A^{(N-l)^{F}}_{\varepsilon_{N,l}} \subset F^{N-l},$$ we prove $z_{l+1}:=(w,\zeta) \in A^{N^F }_{\varepsilon_N}$ by the following computations: $$\begin{aligned}
\|z_{l+1}\|^2
&=\|w\|^2+\|\zeta\|^2
< l (T_{N,l})^2+ (1+\varepsilon_{N,l})^2((N-l)^F -1) \\
& = \left\{ \frac{l (T_{N,l} )^2}{(N-l)^F -1}+ (1+\varepsilon_{N,l} )^2 \right\}((N-l)^F -1) \\
& \leq (1+\varepsilon_N )^2 ((N-l)^F -1)
<(1+\varepsilon_N )^2(N^F -1), \\
\|z_{l+1}\|
&\geq \|\zeta\|
> (1-\varepsilon_{N,l}) \sqrt{(N-l)^{F}-1}
=(1-b_n\varepsilon_N) \sqrt{\frac{N-1}{(N-l)-1}} \cdot \sqrt{(N-l)^{F}-1}\\
&\geq(1-b_n\varepsilon_N) \sqrt{N^F-1}
>(1-\varepsilon_N) \sqrt{N^F -1}.\end{aligned}$$ Using the fact that $n_N/(N-1) \leq \varepsilon_N^2 \leq 1$, we find for $1 \leq l \leq n_N-1$ that $$\begin{aligned}
\frac{\varepsilon_N-\varepsilon_{N,l}}{1-\varepsilon_{N,l}}
&=\frac{(1-b_N\varepsilon_N) \left(\sqrt{\frac{N-l}{(N-l)-1}}-1\right)+\varepsilon_N(1-b_N)}{(1-b_N\varepsilon_N) \sqrt{\frac{N-1}{(N-l)-1}}} \\ \notag
&=\left(1-\sqrt{1-\frac{l}{N-1}}\right)+\frac{\varepsilon_N(1-b_N)}{1-b_N\varepsilon_N} \sqrt{1-\frac{l}{N-1}}\\
&
<\frac{l}{N-1} +\frac{\varepsilon_N (1-b_N)}{1-b_N\varepsilon_N}
<\frac{n_N}{N-1}+\frac43\varepsilon_N(1-b_N)
\leq\frac73\varepsilon_N^2,\\
\frac{l T_{N,l}^2}{ (1-\varepsilon_{N,l} )^2 ((N-l) -1) }
&=\frac{ (\varepsilon_N -\varepsilon_{N,l} )(\varepsilon_N +\varepsilon_{N,l} +2)}{(1-\varepsilon_{N,l} )^2}\\
&< \frac{7\varepsilon_N^2( \varepsilon_N +\varepsilon_{N,l} +2)}{3(1-\varepsilon_{N,l} )}
< \frac{ 5\varepsilon_N^2(1+\varepsilon_N )}{1-\varepsilon_{N} }
= \frac{\theta(\varepsilon_N)^2}{1-\theta(\varepsilon_N)^2}.
$$ This implies $$\begin{aligned}
\|w\|^2
<l T_{N,l}^2
&<\frac{\theta(\varepsilon_N )^2}{1-\theta(\varepsilon_N)^2} (1-\varepsilon_{N,l})^2( (N-l)-1) \\
&\leq
\frac{\theta(\varepsilon_N )^2}{1-\theta(\varepsilon_N)^2} (1-\varepsilon_{N,l})^2( (N-l)^F-1)
<\frac{\theta(\varepsilon_N )^2}{1-\theta(\varepsilon_N )^2} \|\zeta\|^2,\end{aligned}$$ which leads to $\|w\|^2< \theta(\varepsilon_N)^2 (\|w\|^2+\|\zeta\|^2)=\theta(\varepsilon_N)^2 \|z_{l+1}\|^2$. By the Cauchy–Schwarz inequality, for any $1 \leq l \leq n_N-1$, $$\begin{aligned}
\label{angle}
\left\|{\left\langle{\frac{Uz_m}{\|Uz_m\|}},{\frac{z_{l+1}}{\|z_{l+1}\|}}\right\rangle}\right\|
=\left\| {\left\langle{\frac{Uz_m}{\|Uz_m\|}},{\frac{w}{\|z_{l+1}\|}}\right\rangle}\right\|
\leq \frac{\|w\|}{\|z_{l+1}\|}< \theta(\varepsilon_N).\end{aligned}$$ We thus obtain $z_{l+1} \in A^{N,F}_{\varepsilon_N,\theta}[Uz_1,\ldots,Uz_l]$. This completes the proof.
\[lem:full\] For $0 \leq l \leq n_N-1$, we set $$v_{N,l}^F:=
\begin{cases}
1-\gamma^{N^F} \left(A^{N^F}_{\varepsilon_N}\right) &\text{\ if $l=0$}, \\
1-\gamma^{N^F} \left( B^F(T_{N,l})^l\times A^{(N-l)^{F}}_{\varepsilon_{N,l}}\right) &\text{\ if $l\neq0$},
\end{cases}\quad
v_N^F:=\max\{v_{N,l}^F \ |\ 0 \leq l \leq n_N-1 \}.$$ We then have $\lim_{N \to \infty} n_N v_N^F =0$.
Putting $$\alpha_{N,l}^F:=
\begin{cases}
\gamma^{N^F}(A_{\varepsilon_N}^{N^F}) &\text{if $l = 0$},\\
\gamma^{(N-l)^{F}}(A^{(N-l)^{F}}_{\varepsilon_{N,l}}) &\text{if $l \neq 0$},
\end{cases}
\qquad
\beta_{N,l}^F:=
\begin{cases}
0 &\text{\ if $l=0$}, \\
1-\gamma^{1^F}( B^{F}(T_{N,l}) ) &\text{\ if $l\neq0$},
\end{cases}$$ we see $$v_{N,l}^F=(1-\alpha_{N,l}^F) \left(1-\beta_{N,l}^F\right)^l+ 1-\left( 1-\beta_{N,l}^F\right)^l
\leq (1-\alpha_{N,l}^F)+l \beta_{N,l}^F$$ and it suffices to prove $$\lim_{N \to \infty} n_N (1-\alpha_{N,l}^F) =0,
\quad
\lim_{N \to \infty} n_N^2 \cdot \beta_{N,l}^F =0$$ for any $ 0\leq l \leq n_N-1$. Since we have $\lim_{N \to \infty} \alpha_{N,l}^F=1, \lim_{N \to \infty} \beta_{N,l}^F=0$ by Lemmas \[lem:gauss\] and \[lem:limit\], the lemma holds true if $\sup_N n_N <\infty$.
We consider the case where $\lim_{N \to \infty} n_N=\infty$. Set $m:=(N-l)^F$. By and the proof of Lemma \[lem:gauss\], we have $$\begin{aligned}
1-\alpha_{N,l}^F
&
\leq
g_{m}(\sqrt{m-1})\cdot 2 e^{-(\varepsilon_{N,l} \sqrt{m-1})^2/2}.\end{aligned}$$ For large enough $N$, Corollary \[cor:en\] implies $$(\varepsilon_{N,l} \sqrt{m-1})^2
\geq
a'^2 \cdot \varepsilon_{N}^2 \cdot( N-1)
=
a'^2 (N-1)^{-2a_N+1}
\geq a'^2 n_N$$ and $m=(N-l)^F \to \infty$ as $N \to \infty$, so that we have $$\begin{aligned}
n_N(1-\alpha_{N,l}^F)
&
\leq
\left\{
g_{m}(\sqrt{m-1}) \right\}
\cdot 2 (n_N e^{-a'^2n_N})
\xrightarrow{N \to \infty}
0. \end{aligned}$$ Let us show $\lim_{N \to \infty} n_N^2 \beta_{N,l}^F =0$. Assume that $N$ is so large that $$\frac{N-n_{N}}{n_{N}-1}
\geq a' \cdot \frac{N-1}{n_N}.$$ If we set $$R_N:=
\sqrt{\frac{a'(N-1)}{2n_N}}\varepsilon_N
=\sqrt{\frac{a'}{2}}(N-1)^{(1-p_N-2a_N)/2},$$ which diverges to infinity as $N \to \infty$, then we observe from that $$\begin{aligned}
\frac{T_{N,l}^2}{4}
\geq \frac14 \cdot2 \varepsilon_N^2 \cdot \frac{N-n_{N}}{n_{N}-1}
\geq R_N^2.\end{aligned}$$ Hence we conclude that $$\begin{aligned}
B^F(T_{N,l}) \supset
(B^{{\mathbb{R}}}(T_{N,l}/2))^{\dim_{{\mathbb{R}}}F}
\supset (B^{{\mathbb{R}}}(R_N))^{\dim_{{\mathbb{R}}}F},\end{aligned}$$ which implies that for $\beta:=\gamma^1(B^{{\mathbb{R}}}(R_N)) \in[0,1]$, $$\beta_{N,l}^F
\leq
1-\left(\gamma^1(B^{{\mathbb{R}}}(R_N))\right)^{\dim_{{\mathbb{R}}}F}
\leq
1-\beta^4
=(1+\beta^2)(1+\beta)(1-\beta)
\leq 4(1-\gamma^1(B^{{\mathbb{R}}}(R_N))),$$ $$\begin{aligned}
1-\gamma^1(B^{{\mathbb{R}}}(R_N))
&=
\sqrt{\frac{2}{\pi}} \int_{R_N}^\infty e^{-r^2/2} dr
=\frac{1}{\sqrt{\pi}} \int_{R_N^2/2}^\infty e^{-t} \frac{1}{\sqrt{t}} dt \\
&=-\frac{1}{\sqrt{\pi}} e^{-t} \frac{1}{\sqrt{t}}\bigg|_{t=R_N^2/2}^{t=\infty}
+\frac{1}{\sqrt{\pi}} \int_{R_N^2/2}^\infty e^{-t} \frac{-1}{2t\sqrt{t}}
\leq \sqrt{\frac{2}{\pi}}\frac1{R_N}e^{-R_N^2/2}.
$$ If the assumption holds, then we see $$n_N^2 \cdot \beta_{N,l}^F
\leq
4n_N^2 (1-\gamma^1(B^{{\mathbb{R}}}(R_N)))
\leq
4 \sqrt{\frac{2}{\pi}} \frac{1}{R_N} n_N^2 \exp\left(-\frac{a'}{4}\left(\frac{N-1}{n_{N }}\right)^{1-a}\right)
\xrightarrow{N \to \infty}
0.$$ Under the assumption , we find that $$n_N^2 \cdot \beta_{N,l}^F
\leq
4n_N^2 (1-\gamma^1(B^{{\mathbb{R}}}(R_N)))
\leq
4 \sqrt{\frac{2}{\pi}} \frac{1}{R_N} n_N^2 \exp\left(-\frac{a'}{4}\sqrt{\frac{N-1}{n_{N }^3}}\right)
\xrightarrow{N \to \infty}
0.$$ This completes the proof.
We already observe $\lim_{N \to \infty} \varepsilon_N=0$ in Lemma \[lem:limit\]. We apply Lemma \[lem:subset\] to have $$\begin{aligned}
&\gamma^{N^F n_N}(X_{N,n_N,\varepsilon_N ,\theta}^{F})\\
=&
\int_{z_1 \in A^{N^F }_{\varepsilon_N }}\cdots \int_{z_{n_N-1} \in A^{N,F}_{\varepsilon_N }[z_1,\ldots,z_{n_N-2}]}
\gamma^{N^F } (A^{N,F}_{\varepsilon_N ,\theta}[z_1,\ldots,z_{n_N-1}]) d\gamma^{N^F }(z_{n_N-1}) \cdots d\gamma^{N^F }(z_1)
\\
\geq &
\int_{z_1 \in A^{N^F }_{\varepsilon_N }}\cdots \int_{z_{n_N-2} \in A^{N,F}_{\varepsilon_N ,\theta}[z_1,\ldots,z_{n_N-3}]} \
\gamma^{N^F } (A^{n,F}_{\varepsilon_N ,\theta}[z_1,\ldots, z_{n_N-2}])
d\gamma^{N^F }(z_{n_N-2}) \cdots d\gamma^{N^F }(z_1) \\
&\quad \times
\left( \gamma^{1^F}( B^{F}(T_{N,n_N-1}^{F}) )\right)^{n_N-1}\times \gamma^{(N-(n_N-1))^{F}}(A^{(N -(n_N-1))^{F}}_{\varepsilon_{N,n_N-1}})\\
\geq&
\gamma^n(A^{N^F }_{\varepsilon_N }) \times
\prod_{l=1}^{n_N-1}\left\{ \left( \gamma^{1^F}( B^{F}(T_{N,l}^{F}) ) \right)^l\times
\gamma^{(N-l)^{F}}(A^{(N-l)^{F}}_{\varepsilon_{N,l}}) \right\}
=
\prod_{l=0}^{n_N-1} (1-v_{N,l}^F) \geq (1-v_N^F)^{n_N}. \end{aligned}$$ By Lemma \[lem:full\], we have $\lim_{N \to\infty} (1-v_N^F)^{n_N}=1$. This completes the proof of the theorem.
\[rem:angle\] If we choose a function $\theta :(0,\infty) \to (0,\infty)$ satisfying $$\lim_{N \to \infty} \gamma^{N^F n_N} (X^F_{N,n_N,\varepsilon_{N}, \theta})=1$$ with the use of Lemma \[lem:subset\], then by , the angle $\theta$ is required to satisfy $$\begin{aligned}
\theta(\varepsilon_N )^2
&> \sup\left\{ \frac{\|w\|^2}{\|(w,\zeta)\|^2}\ \Bigg|\
w\in \left(B^{F}(T_{N,l})\right)^{l},
\zeta \in A^{(N-l)^{F}}_{\varepsilon_{N,l} }
\right\}\\
&=\frac{lT_{N,l}^2}{l T_{N,l}^2+ (1-\varepsilon_{N,l} )^2((N-l)^F-1)}
=\left(1+\frac{(1-\varepsilon_{N,l} )^2((N-l)^F-1)}{lT_{N,l}^2}\right)^{-1}.
$$
\[rem:condi\] Let us comment on the conditions and .
(a) \[c1\] If (resp. ) holds, we then have $\lim_{N \to \infty}{n_N}/N^{p_\ast}=0$ for $p_\ast=1$ (resp. $p_\ast=1/3$), however the converse does not hold in general. Such an example is $$n_N:=\left[\left(\frac{N}{\log N}\right)^{p_\ast} +1 \right],$$ where $[x]$ is the largest integer not greater than $x$.\
(b) \[c3\] For $p>0$, we see that the condition $n_N=O(N^p)$ implies $\lim_{N \to \infty} p_N \leq p$. If moreover $p<1$, then is true, because there exists $c>0$ such that $$\sup_{N}\left( 2\log n_N -\frac{a'}{4} \left(\frac{N}{n_N}\right)^{1-a} \right)
\leq
\sup_{N}\left(2p\log N +2 \log c-\frac{a'}{4} \cdot c^{a-1} N^{(1-p)(1-a)} \right)<\infty$$ holds for any $a,a' \in (0,1)$. Similarly, if $p<1/3$, then is true.
(c) If $n_N=o(N (\log N)^{-A})$ for some $A>1$ , then holds true with $a=(A-1)/A$ and any $a'\in (0,1)$, because there exists $N_0 \in {{\mathbb{N}}}$ such that $$\frac{n_{N}}{N} (\log N)^A \leq \left(\frac{a'}{4} \times \frac12\right)^{A}$$ for any $N \geq N_0$, implying $$\sup_{N \geq N_0}\left( 2\log n_N -\frac{a'}{4} \left(\frac{N}{n_N}\right)^{1-a} \right)
\leq
2\sup_{N\geq N_0}\left( \log N- A\log \log N + A \log\frac{a'}{8} -\log N \right)<\infty.$$ It is impossible to reduce the condition $A>1$ to $A=1$, namely $n_N=o(N (\log N)^{-1})$. For example, if we put $$n_N:=\left[\frac{N}{\log N \log \log N} +1 \right],$$ then this satisfies $n_N=o(N (\log N)^{-1})$, but does not satisfy .
(d) We similarly derive from $n_N=o(N^{1/3} (\log N)^{-2/3} )$.
Proof of main theorems
======================
Strategy of the proof
---------------------
The idea of the proof of Theorems \[thm:Stiefel\] and \[thm:Gr-pS\] is based on that in the case of $n=1$ in [@Sy:mmlim]. The following lemma plays a crucial role in the proof.
\[keylem\] Let $\{X_N\}_{N=1}^\infty, \{Y_N\}_{N=1}^\infty$ be sequences of mm-spaces satisfying the following conditions:
1. \[ass1\] ${\mathcal{P}}_{Y_N}$ converges weakly to a pyramid ${\mathcal{P}}_\infty$ as $N \to \infty$.
2. \[ass2\] For any $N, l \in {{\mathbb{N}}}$ with $N \geq l$, there exists a $1$-Lipschitz map $p^N_l: X_N \to Y_l$ such that $\lim_{N \to \infty} \operatorname{\mathit{d}_\mathrm{P}}((p^N_l)_\# \mu_{X_N},\mu_{Y_l})=0$.
3. \[ass3\] For any $N \in {{\mathbb{N}}}$, there exists a subset $Y_N' \subset Y_N$ such that
1. \[ass31\] $\lim_{N \to \infty}\mu_{Y_N}(Y_N')=1$,
2. \[ass32\] there exists a Lipschitz map $\Phi^N: Y_N' \to X_N$ pushing $\mu_{Y'_N}:=(\mu_{Y_N}(Y'_N))^{-1}\mu_{Y_N}|_{Y'_N}$ forward to $\mu_{X_N}$ such that its smallest Lipschitz constant tends to $1$ as $N \to \infty$.
Then ${\mathcal{P}}_{X_N}$ converges weakly to ${\mathcal{P}}_\infty$ as $N \to \infty$.
Theorem \[thm:emb\] ensures the existence of a subsequence $\{X_{N_m}\}_{m=1}^\infty \subset \{X_N\}_{N=1}^\infty$ such that ${\mathcal{P}}_{X_{N_m}}$ converges weakly to a pyramid ${\mathcal{P}}$ as $m \to \infty$. It suffices to prove ${\mathcal{P}}={\mathcal{P}}_{\infty}$ for any such ${\mathcal{P}}$. We deduce (C\[ass2\]) from ${\mathcal{P}}_\infty \subset {\mathcal{P}}$.
To show the converse including relation ${\mathcal{P}}\subset {\mathcal{P}}_\infty$, let us regard $Y_N'=(Y_N', d_{Y_N}, \mu_{Y_N'})$ as an mm-space. By (C\[ass31\]), we compute $$\lim_{N \to \infty} \operatorname{\mathit{d}_\mathrm{TV}}( \mu_{Y'_N}, \mu_{Y_N} )
=\lim_{N \to \infty}\frac{1}{2} \int_{Y_N}\left| \frac{\mathbf{1}_{Y'_N}}{\mu_{Y_N}(Y_N')}-1\right| d\mu_{Y_N}
=\lim_{N \to \infty}(1-\mu_{Y_N}(Y_N'))=0,$$ where $\mathbf{1}_{Y'_N}$ stands for the indicator function of $Y_N'$. This with Corollary \[cor:rho-dTV\] implies that the pyramid associated with $Y_N'$ converges weakly to ${\mathcal{P}}_\infty$. Combining this with (C\[ass32\]) and Lemma \[lem:obvious\](1) leads to ${\mathcal{P}}\subset{\mathcal{P}}_\infty$.
\[keycor\] Let $\{X_N\}_{N=1}^\infty, \{Y_N\}_{N=1}^\infty$ be two sequences of mm-spaces as in Lemma \[keylem\]. Let $G$ be a group acting isometrically on $X_N, Y_N$ satisfying the following conditions.
1. \[asss1\] The quotient space $\bar{Y}_N$ of $Y_N$ by the $G$-action is monotone increasing in $N$ with respect to the Lipschitz order.
2. \[asss2\] $p^N_l, \Phi^N$ are both $G$-equivariant.
3. \[asss3\] $Y'_N$ is $G$-invariant [(]{}i.e., $G \cdot Y'_N = Y'_N$[)]{}.
Then, the pyramids associated with $\bar{X}_N$, $\bar{Y}_N$ both converge weakly to a common pyramid as $N\to\infty$.
Due to Lemma \[lem:obvious\](2) with (C’\[asss1\]), $\{\bar{Y}_N\}_N$ satisfies (C\[ass1\]). Combining (C’\[asss2\]) with Lemma \[lem:dP-quotient\] yields that $(\bar{p}^N_l)_\# \bar{\mu}_{X_N}$ converges weakly to $\bar{\mu}_{Y_l}$. Lemma \[lem:equivdomi\] with (C’\[asss2\]) ensures that the Lipschitz constant of $\bar{p}^N_l$ is $1$ and that of $\bar{\Phi}^N$ tends to $1$. Combining (C’\[asss2\]) with (C’\[asss3\]) leads to $\bar{\Phi}^N_\# \bar{\mu}_{Y'_N}=\bar{\mu}_{X_N}$. By (C’\[asss3\]), we have $\bar{\mu}_{Y_N}(\bar{Y'}_N)=\mu_{Y_N}(Y'_N)$. Thus $\{\bar{X}_N\}_N,\{\bar{Y}_N\}_N$ satisfies all the conditions (C\[ass1\]–\[ass3\]), which completes the proof of the corollary.
Subsection \[ssc:lip\] is devoted to construct a Lipschitz map $\Phi^{N,n,F}_{\varepsilon,\theta}:X^F_{N,n,\varepsilon,\theta} \to X^F_{N,n}$ with $\theta$ defined in , for which the smallest Lipschitz constant tends to $1$ as $N \to \infty$, with the help of the polar decomposition. In Subsection \[ssc:push\], we demonstrate that the normalized push-forward measure of $\gamma^{N^F n}|_{X^F_{N,n}, \varepsilon,\theta}$ by $\Phi^{N,n,F}_{\varepsilon,\theta}$ coincides with $\mu^{N,n,F}$. We finally apply Lemma \[keylem\] and Corollary \[keycor\] to prove Theorems \[thm:Stiefel\] and \[thm:Gr-pS\] for $$X_N=X_{N,n_N}^F, \quad
Y_N=\Gamma^{N^F n}, \quad
Y_N'=X_{N,n_N,\varepsilon_N,\theta}^F, \quad
p^N_l=\pi^N_l(n),\quad
\Phi^N=\Phi^{N,n,F}_{\varepsilon,\theta}.$$
Lipschitz map from $X_{N,n,\varepsilon, \theta}^F$ to $X_{N,n}^F$ {#ssc:lip}
-----------------------------------------------------------------
### Nearest point projection
Let us first describe the relation between singular values and polar decompositions as well as in the case of $F={{\mathbb{C}}}$ proved by Li [@Li]. In the case of $F={{\mathbb{H}}}$, we should take account of the fact that $\operatorname{tr}(ZW) \neq\operatorname{tr}(WZ)$ may happen. For $N \geq n$, set $$I^N_n:=
\begin{pmatrix}
I_n \\
0_{N-n,n}
\end{pmatrix}
\in M_{N,n}^F.$$
\[lemma:Li\] For any $Z_1, Z_2 \in M_{N,n}^F$, let $$Z_1=Q_1 H_1=U_1 \Lambda_1 V_1^\ast,\quad
Z_2=Q_2 H_2=U_2 \Lambda_2 V_2^\ast$$ be their polar and singular value decompositions. We then have $$\|Z_1-Z_2\| \geq \lambda \|Q_1-Q_2\|, \quad \lambda:=\min\{\lambda_n(Z_1), \lambda_n(Z_2)\} .$$
For $U:=U_2^\ast U_1\in U^F(N), V:=V_2^\ast V_1 \in U^F(n)$, we see that $$\begin{aligned}
\|Z_1-Z_2\|
=
\|U_1 \Lambda_1 V_1^\ast -U_2 \Lambda_2 V_2^\ast \|=
\|U \Lambda_1 - \Lambda_2 V \|, \quad
\|Q_1-Q_2\|
=\|U I^N_n - I^N_nV \|.\end{aligned}$$ So it suffices to prove $\|U \Lambda_1 - \Lambda_2 V \| \geq \lambda \|U I^N_n - I^N_nV \|$. Setting $$\begin{gathered}
\Sigma_i:=(\Lambda_i,0_{N,N-n}), \
I:=(I^N_n,0_{N,N-n}),\
A:= \Sigma_1- \lambda I, \
B:= \Sigma_2 -\lambda I \in M_{N}^F,\\
V_n:=\begin{pmatrix}
V&0_{n,N-n}\\
0_{N-n,n}&I_{N-n}
\end{pmatrix} \in U^F(N),
\quad
(\sharp):=2\Re{\left\langle{UI- I V_n },{UA-B V_n}\right\rangle}.
$$ we have $\|U I^N_n-I^N_n V\|=\|U I-I V_n\|$ and $$\|U \Lambda_1 -\Lambda_2 V \|^2
=\|U \Sigma_1 -\Sigma_2 V_n \|^2
=\lambda^2\|U I- IV_n \|^2+ \|UA-B V_n\|^2 + \lambda \times (\sharp).
$$ Thus it is enough to prove the non-negativity of $(\sharp)$, which is expressed as $$\begin{aligned}
(\sharp)
&=\operatorname{tr}\left[ (I U^\ast- V_n^\ast I) UA+AU^\ast (U I- I V_n ) \right] \\
&\quad+\operatorname{tr}\left[ (-IU^\ast+ V_n^\ast I ) B V_n + V_n^\ast B (-U I+ I V_n )\right].\end{aligned}$$ Using the assumptions that $U, V_n$ are unitary and $A,B, I$ are real diagonal, we compute $$\begin{aligned}
\operatorname{tr}(I A)=\operatorname{tr}(IU^\ast UA)=\operatorname{tr}(AU^\ast U I), \quad
\operatorname{tr}(IB)=\operatorname{tr}( V_n^\ast I B V_n )=\operatorname{tr}( V_n^\ast BI V_n ).\end{aligned}$$ If we set $X=(x^m_l)_{1\leq m,l \leq N}:= V_n^\ast I U, Y=(y^m_l)_{1\leq m,l \leq N}:=\operatorname{{}^{T}\!\!}U^\ast I\operatorname{{}^{T}\!\!}V_n$, then we observe that $$\begin{gathered}
(\operatorname{tr}(AU^\ast I V_n))^\ast
=\operatorname{tr}(V_n^\ast I UA)
=\operatorname{tr}(XA),\quad
(\operatorname{tr}( V_n^\ast B U I))^\ast
=\operatorname{tr}(IU^\ast B V_n)
=\operatorname{tr}(YB),\end{gathered}$$ which leads to $$\begin{aligned}
(\sharp)
&=\operatorname{tr}\left[(2I- (X+X^\ast))A \right]
+\operatorname{tr}\left[(2I-(Y+Y^\ast))B \right] \\
&=\sum_{l=1}^n(2-(x^l_l+(x^l_l)^\ast))(\lambda_l(Z_1)-\lambda)+\sum_{l=1}^n(2-(y^l_l+(y^l_l)^\ast))(\lambda_l(Z_2)-\lambda).\end{aligned}$$ We moreover find $\lambda_1(X) \leq 1$, which together with Remark \[spectrum\](2) implies $$\begin{gathered}
\lambda_{1}(X+X^\ast)
\leq \lambda_{1}(X)+\lambda_1(X^\ast)
\leq 2.\end{gathered}$$ Since $X+X^\ast$ is Hermitian, there exist $P=(p^m_l)_{1 \leq m,l \leq N}\in U^F(N)$ and $\{\xi_m\}_{m=1}^N, {{\mathbb{R}}}$ with $|\xi_m|=\lambda_{m}(X+X^\ast)$ such that $X+X^\ast =P^\ast \operatorname{diag}(\xi_1,\ldots,\xi_N ) P$, which implies $$2\Re(x^l_l)=x^l_l+(x^l_l)^\ast=\sum_{m=1}^N (p^m_l )^\ast \xi_m p^m_l \leq 2\sum_{m=1}^N (p^m_l )^\ast p^m_l=2.$$ The same argument proves $\Re(y^m_m) \leq 1$. This together with $\lambda_l(Z_1),\lambda_m(Z_2) \geq \lambda$ implies $$(\sharp)
= 2\sum_{l=1}^n(1-\Re(x^l_l))(\lambda_l(Z_1)-\lambda)
+ 2\sum_{l=1}^n(1-\Re(y^l_l))(\lambda_l(Z_2)-\lambda)
\geq 0.$$ This completes the proof.
We next show that the scaled polar decomposition is the nearest point projection from $M_{N,n}^F$ to $X_{N,n}^F$ even if the decomposition is not unique.
\[lem:nearest\] Let $Z=QH$ be a polar decomposition of $Z \in M_{N,n}^F$. For any $r>0$ we have $$\min_{Q' \in V_{N,n}^F}\|Z-r Q'\|=\|Z- r Q\|=\sqrt{\sum_{l=1}^n (\lambda_l(Z)-r)^2}.$$
Let $Z=QH=U \Lambda V^\ast$ be polar and singular value decompositions of $Z$. By Theorem \[thm:decomp\], we have $$\begin{aligned}
\|Z-r Q\|^2
=\|U \Lambda V^\ast-r Q\|=\|\Lambda-r U^\ast Q V\|^2
=\left\|\Lambda-r I^N_n \right\|
=\sum_{l=1}^n (\lambda_l(Z)-r)^2,\end{aligned}$$ implying the last equality in the lemma. For any $Q' \in V_{N,n}^F$, we define $X=(x^l_m)_{1 \leq m,l \leq n}\in M_{n}^F$ and $Y \in M_{N-n,n}^F$ as $$\begin{pmatrix}
X\\
Y\\
\end{pmatrix}
:=r U^\ast Q' V.$$ We then have $X^\ast X+Y^\ast Y=r^2I_n$, hence $\Re (x^l_l) \leq \|x^l_l\| \leq r$, and $$\|Z-rQ'\|
=\|\Lambda-rU^\ast Q' V\|
=
\left\|
\begin{pmatrix}
\Lambda'-X\\\
-Y\\
\end{pmatrix}
\right\|, \quad
\Lambda':=\operatorname{diag}(\lambda_1(Z),\ldots,\lambda_n(Z)).$$ A direct computation proves $$\begin{aligned}
\left\|
\begin{pmatrix}
\Lambda'-X\\\
-Y\\
\end{pmatrix}
\right\|^2
&=\|(\Lambda'-X)\|^2+\|Y\|^2\\
&=
\|(\Lambda'-r I_n)-(X-r I_n)\|^2+\operatorname{tr}\{ r^2 I_n-X^\ast X\} \\
&=\| \Lambda'-r I_n\|^2
+ \operatorname{tr}\{
\Lambda'(r I_n-X)+(r I_n-X^\ast)\Lambda'
\}\\
&=\sum_{l=1}^n (\lambda_l(Z)-r)^2+2 \sum_{l=1}^n \lambda_l(Z)(r-\Re (x^l_l))
\geq \sum_{l=1}^n (\lambda_l(Z)-r)^2,\end{aligned}$$ which completes the proof.
In the same way as for $X_{N,n}^F$, we define an $(\varepsilon, \theta)$-approximation space $U^F(n)_{\varepsilon,\theta}$ of $U^F(n)=V_{n,n}^{F}$ by $$\begin{aligned}
U^F(n)_{\varepsilon,\theta}
:=\left\{(z_1,\ldots,z_n) \in M_{n}^F \bigg| \big|\|z_l\|-1\big|<\varepsilon,
\left\|{\left\langle{\frac{z_l}{\|z_l\|}},{\frac{z_m}{\|z_m\|}}\right\rangle}\right\|< \theta(\varepsilon), \ 1 \leq l<m \leq n\right\}.
$$
\[prop:l\] For any $n \in {{\mathbb{N}}}, \varepsilon >0$ and a function $\theta:(0,\infty)\to(0,\infty)$, we set $$L(n,\varepsilon,\theta)
:=\sup_{W \in U^F(n)_{\varepsilon,\theta}}\min_{Q' \in U^F(n)}\|W-Q'\|.$$ Then, for any $N \geq n$ and $Z \in X_{N,n,\varepsilon,\theta}^F$, we have $$\lambda_n(Z) \geq \sqrt{N^F-1}\cdot (1-L(n,\varepsilon,\theta)).$$
For $Z \in X^F_{N,n,\varepsilon,\theta}$, let $Z=QH=U\Lambda V^\ast$ be polar and singular decompositions of $Z$. Set $\Lambda':=\operatorname{diag}(\lambda_1(Z), \ldots, \lambda_n(Z))$. It turns out that $$W':=\frac1{\sqrt{N^F-1}}(\Lambda' V^\ast )\in U^F(n)_{\varepsilon,\theta}$$ and $W'=V^\ast \cdot (V W')$ is a polar decomposition of $W'$. We conclude that $$\begin{aligned}
\sqrt{N^F-1}-\lambda_n(Z)
&\leq \|Z-\sqrt{N^F-1}\cdot Q\| \\
&=\left\|U \cdot\begin{pmatrix} \Lambda' V^\ast\\ 0_{N-n,n} \end{pmatrix}-\sqrt{N^F-1}\cdot U \cdot\begin{pmatrix} V^\ast\\ 0_{N-n,n} \end{pmatrix} \right\|
=\sqrt{N^F-1} \cdot \|W'- V^\ast \|\\
&
\leq
\sqrt{N^F-1} \sup_{W \in U^F(n)_{\varepsilon,\theta}}\min_{Q' \in U^F(n)}\|W-Q'\|
=\sqrt{N^F-1}\cdot L(n,\varepsilon,\theta),\end{aligned}$$ where we apply Lemma \[lem:nearest\] for $r =\sqrt{N^F-1}$ in the first inequality and for $r=1$ in the last inequality. This completes the proof.
Corollary \[prop:l\] implies that if $L(n,\varepsilon,\theta)<1$, then the map $$Q^F : X^F_{N,n,\varepsilon,\theta} \ni Z \mapsto Q^F(Z) \in V_{N,n}^F$$ is well-defined, where $Z=Q^F(Z)H$ is the polar decomposition of $Z$.
\[lem:lip\] Assume $L(n,\varepsilon,\theta)<1$. Then, the map $$\Phi^{N,n,F}_{\varepsilon,\theta}: X^F_{N,n,\varepsilon,\theta} \ni Z
\mapsto
\sqrt{N^F-1}\cdot Q^F(Z) \in X_{N,n}^F$$ has Lipschitz constant at most $(1-L(n,\varepsilon,\theta))^{-1}$.
For any $Z, W \in X^F_{N,n,\varepsilon,\theta}$, Lemma \[lemma:Li\] and Corollary \[prop:l\] together imply $$\begin{aligned}
\|\Phi^{N,n,F}_{\varepsilon,\theta}(Z)-\Phi^{N,n,F}_{\varepsilon,\theta}(W)\|
&=\sqrt{N^F-1}\cdot\|Q^F(Z)-Q^F(W)\|\\
&\le \frac{\sqrt{N^F-1}}{\min\{\lambda_n(Z), \lambda_n(W)\} }\|Z-W\|
\leq \frac{1}{1-L(n,\varepsilon,\theta)} \|Z-W\|.\end{aligned}$$
### Condition for $L(n,\varepsilon,\theta)<1$
Given any $\delta>0$, we define the positive monotone increasing functions on $[0,1)$ by $$\varphi_\delta(s):=\frac{(\delta+s)^2}{1-s}, \quad
R_\delta(s):=\frac{2(\delta+s)}{ 1-s}.$$ Moreover, for any $\sigma\in[0,1]$, we set $$\begin{gathered}
s_l=s_l(\delta):=\begin{cases}
0 & \text{\ if\ }l=0,\\
s_{l-1}+\varphi_\delta(s_{l-1}) & \text{\ if\ } 1 \leq l \leq n_\sigma,
\end{cases}\\
n_\sigma=n_\sigma(\delta):=\max\left\{ n \in {{\mathbb{N}}}\cup\{0\}\ |\ s_{n}(\delta) <\sigma \right\}+1.\end{gathered}$$ It follows that $s_1(\delta)=\delta^2$ and $s_{n_\sigma-1} < \sigma \leq s_{n_\sigma}$.
\[lem:itere\] For any $\delta>0$ and $\sigma\in[0,1)$, we have $$\log_{1+R_\delta(\sigma)}\left(1+R_\delta(\sigma)\frac{\sigma}{\delta^2} \right)
\leq
{n_\sigma}(\delta)
<
\log_{1+R_\delta(0)}\left(1+ R_\delta(0)\frac{\sigma}{\delta^2} \right)+1.$$ The last inequality also holds true for $\sigma=1$.
For $1 \leq l \leq n_\sigma-1$, we observe that $s_{l}<\sigma$ and $$\begin{gathered}
\varphi_\delta(s_{l})
=\varphi_\delta(s_{l-1}+\varphi_\delta(s_{l-1}))
=\frac{(\delta+s_{l-1}+\varphi_\delta(s_{l-1}))^2}{1-(s_{l-1}+\varphi_\delta(s_{l-1}))}
=\varphi_\delta(s_{l-1}) \left(1+ \frac{2(\delta+s_l)}{ 1-s_{l}} \right),\\
R_\delta(0)
\leq
\frac{2(\delta+s_l)}{ 1-s_{l}}
\leq
R_\delta(\sigma),\end{gathered}$$ and thereby, $$\begin{aligned}
s_{l+1}
=s_{l}+\varphi_\delta(s_{l})
&\leq
s_{l-1} + \varphi_\delta(s_{l-1}) (1+(1+R_\delta(\sigma))) \\
&\leq
s_{0} + \varphi_\delta(s_0) \sum_{m=0}^{l} (1+R_\delta(\sigma))^m
\leq \delta^2 \frac{(1+R_\delta(\sigma))^{l+1}-1}{R_\delta(\sigma)}.\end{aligned}$$ In the same way, we estimate $s_{l+1}$ from below and conclude $$\begin{aligned}
\label{eq:ratio}
\frac{(1+R_\delta(0))^{l+1}-1}{R_\delta(0)}
\leq
\frac{s_{l+1}}{\delta^2}
\leq \frac{(1+R_\delta(\sigma))^{l+1}-1}{R_\delta(\sigma)}.\end{aligned}$$ Combining this with $s_{n_\sigma-1}<\sigma \leq s_{n_\sigma}$ yields $$\begin{aligned}
\frac{(1+R_\delta(0))^{n_\sigma-1}-1}{R_\delta(0)} \leq \frac{s_{n_\sigma-1}}{\delta^2}
<
\frac{\sigma}{\delta^2}
\leq \frac{s_{n_\sigma}}{\delta^2}
\leq \frac{(1+R_\delta(\sigma))^{n_\sigma}-1}{R_\delta(\sigma)},\end{aligned}$$ which provides the desired result.
\[lem:above\] If $0\leq \sigma \leq \delta^2$, then $n_\sigma(\delta)=1$ and for $\delta^2<\sigma \leq 1$ we have $$n_{\sigma}(\delta) < 1+\frac{\sigma}{\delta^2}.$$
The first claim follows from the definition of $n_\sigma(\delta)$ and the fact $s_1=\delta^2$. In the case of $\delta^2<\sigma\leq1$, it suffices to prove by Lemma \[lem:itere\] that $$\log_{1+R_\delta(0)}\left(1+ R_\delta(0)\frac{\sigma}{\delta^2} \right)
< \frac{\sigma}{\delta^2}
\Longleftrightarrow
\left(1+ R_\delta(0)\frac{\sigma}{\delta^2} \right)^{\delta^2/\sigma R_\delta(0)}
<\left(1+R_\delta(0)\right)^{1/R_\delta(0)}.$$ This follows from the monotone increasing property of $r \mapsto (1+1/r)^r$ on $(0,\infty)$ and the condition $$\frac{\delta^2}{\sigma R_\delta(0)}<\frac{1}{R_\delta(0)}.$$ The proof is completed.
We next consider the different expression of $s_l(\delta)$. For $0 \leq l \leq n_\sigma(\delta)$ with $\sigma\in [0,1]$, we set $$c_l=c_l(\delta):=
\begin{cases}
0 & \text{\ if\ }l=0,\\
\displaystyle
\frac{1+\delta \sum_{m=0}^{l-1}c_m^2}{\sqrt{1-\delta^2 \sum_{m=0}^{l-1}c_m^2}} & \text{\ if\ } 1\leq l \leq n_1(\delta).
\end{cases}$$
\[lem:rel\] For $0 \leq l \leq n_\sigma(\delta)$ with $\sigma\in [0,1]$, we have $$\delta^2 \sum_{m=0}^{l}c_m^2=s_l(\delta).$$ In particular, the denominator of $c_l$ does not vanish and $c_l$ is well-defined.
We prove the lemma by induction on $l$. In the case of $l=0$, the statement is true since $c_0=0=s_0$. Assume that the statement is true for $l-1$. We then have $$\begin{aligned}
\delta^2 \sum_{m=0}^{l}c_m^2
&=\delta^2 \sum_{m=0}^{l-1}c_m^2 +\delta^2 c_l^2
=\delta^2 \sum_{m=0}^{l-1}c_m^2 +\frac{(\delta +s_{l-1})^2}{1-s_{l-1}}
=s_{l-1}+\varphi_\delta(s_{l-1})
=s_l,\end{aligned}$$ ensuring the statement for $l$. This completes the proof.
\[lem:1\] For any $\sigma \in [0,1], n \leq n_\sigma(\theta(\varepsilon))$, we have $$L(n,\varepsilon,\theta)^2=\sup_{W \in U^F(n)_{\varepsilon,\theta}}\min_{Q' \in U^F(n)}\|W-Q'\|^2
<
\left(n \varepsilon^2+ \frac{2n(1+\varepsilon)\sigma}{1+ \sqrt{1-\sigma}}\right).$$
For any $Z=(z_1,\ldots, z_n) \in U^F(n)_{\varepsilon, \theta}$, there exist $P \in U^F(n)$ and $(\zeta_1,\ldots, \zeta_n) \in U^F(n)_{\varepsilon,\theta}$ with $\|\zeta_l\|=1$ such that $$P z_l=\|z_l\|\zeta_l,\quad
\zeta_1=e_1, \quad
\zeta^m_l:={\left\langle{e_m},{\zeta_l}\right\rangle}=0 \quad \text{if $m>l$}.$$
For $1 \leq m\leq n_\sigma(\theta(\varepsilon))$, if $m < l \leq n_\sigma(\theta(\varepsilon))$, then $\|\zeta^m_l\|< \theta(\varepsilon) c_m(\theta(\varepsilon))$.
We prove the claim by induction on $m$. If $m=1$, we then have $c_1(\theta(\varepsilon))=1$ and, for $l \geq 2$, $$\|\zeta^1_l\|
=\|{\left\langle{e_1},{\zeta_l}\right\rangle}\|
=\left\| \left\langle \frac{z_1}{\|z_1\|}, \frac{z_l}{\|z_l\|} \right\rangle \right\|
< \theta(\varepsilon) =\theta(\varepsilon) c_1(\theta(\varepsilon)).$$ Let us assume that the claim holds true for any $1, \ldots m-1$ with $m<n_\sigma(\theta(\varepsilon))$. We derive from the assumption and Lemma \[lem:rel\] that $$\|\zeta^m_m\|^2
=\|\zeta_m\|^2-\sum_{\alpha=1}^{m-1} \|\zeta^\alpha_m\|^2
>1-\theta(\varepsilon)^2\sum_{\alpha=1}^{m-1} c_\alpha^2=1-s_{m-1} > 1-\sigma \ge 0,$$ providing $\zeta^m_m \neq 0$. Since for any $l>m$ we have $${\left\langle{\zeta_m},{\zeta_l}\right\rangle}
=\sum_{\alpha=1}^m (\zeta^\alpha_m)^\ast \zeta^\alpha_l
=\sum_{\alpha=1}^{m-1} (\zeta^\alpha_m)^\ast \zeta^\alpha_l+(\zeta^m_m)^\ast \zeta^m_l, \quad
\|{\left\langle{\zeta_m},{\zeta_l}\right\rangle}\|<\theta(\varepsilon)$$ and the inverse of $(\zeta^m_m)^\ast$ is $\zeta^m_m/\|\zeta^m_m\|^2$, we deduce from the claim for $1, \ldots, m-1$ that $$\begin{aligned}
\|\zeta^m_l\|
&=
\left\|\frac{\zeta^m_m}{\|\zeta^m_m\|^2}\cdot \left({\left\langle{\zeta_m},{\zeta_l}\right\rangle}-\sum_{\alpha=1}^{m-1} (\zeta^\alpha_m)^\ast \zeta^\alpha_l \right)\right\|
\leq
\frac{1}{\|\zeta^m_m\|} \left( \theta(\varepsilon)+ \sum_{\alpha=1}^{m-1} \|\zeta^\alpha_m\| \|\zeta^\alpha_l \| \right) \\
&<
\frac{ \theta(\varepsilon)+ \theta(\varepsilon) \sum_{\alpha=1}^{m-1} c_\alpha \|\zeta^\alpha_m\| }{\sqrt{1-\sum_{\alpha=1}^{m-1} \|\zeta^\alpha_m\|^2}}
<
\frac{ \theta(\varepsilon) \left(1 + \theta(\varepsilon) \sum_{\alpha=1}^{m-1} c_\alpha^2 \right)}{\sqrt{1- \theta(\varepsilon)^2 \sum_{\alpha=1}^{m-1} c_\alpha^2}}
=\theta(\varepsilon) c_m, \end{aligned}$$ where the last inequality follows from the monotone increasing property of each varieties of $$(r_1,\ldots,r_{m-1}) \mapsto \frac{1}{\sqrt{1-\sum_{\alpha=1}^{m-1} r_\alpha^2 }} \left(\theta(\varepsilon) +\theta(\varepsilon) \sum_{\alpha=1}^{m-1}c_\alpha r_\alpha \right).
$$ This completes the proof of the claim. It thus holds for any $1 \leq l \leq n_\sigma(\theta(\varepsilon))$ that $$\|\zeta^l_l\|
=\sqrt{1-\sum_{m=1}^{l-1} \|\zeta^m_l\|^2}>\sqrt{1-\theta(\varepsilon)^2\sum_{m=1}^{l-1} c_m^2(\theta(\varepsilon))}
=\sqrt{1-s_{l-1}(\theta(\varepsilon))}>0.$$ Setting $$P':= P^\ast \cdot \left(\frac{\zeta^1_1}{\|\zeta^1_1\|}e_1, \ldots, \frac{\zeta^n_n}{\|\zeta^n_n\|}e_n\right) \in U^{F}(n),$$ we see that $$\begin{aligned}
\min_{Q \in U^F(n)} \|Z-Q\|^2
&\leq \|Z-P'\|^2
=\sum_{l=1}^n \left\| \|z_l\| \zeta_l- \frac{\zeta^l_l}{\|\zeta^l_l\|}e_l \right\|^2
=\sum_{l=1}^n \left( \|z_l\|^2 +1-2\|z_l\| \|\zeta^l_l\| \right) \\
&<
\sum_{l=1}^n \left( (\|z_l\|-1)^2+ 2\|z_l\| -2\|z_l\| \sqrt{1-s_{l-1}(\theta(\varepsilon))} \right) \\
&< \sum_{l=1}^n \left( \varepsilon^2+2(1+\varepsilon)(1-\sqrt{1-\sigma} )\right)
= \left(n \varepsilon^2+ \frac{2n(1+\varepsilon)\sigma}{1+ \sqrt{1-\sigma}}\right).\end{aligned}$$ By the arbitrariness of $Z \in U^F(n)_{\varepsilon, \theta}$, the proof of the lemma is complete.
\[rem:orth\] In Theorem \[lem:1\], we estimate $L(n,\varepsilon,\theta)^2$ from above by the sum of two terms: the first one is due to the error of the lengths between each column vectors $Z \in U^F(n)_{\varepsilon,\theta}$ and $Q^{F}(Z)$, and the second one is due to the error of the angles between each column vectors $Z$ and $Q^{F}(Z)$. If $n_\sigma(\theta(\varepsilon)) \to \infty$ as $\varepsilon \to 0$, then we require $\theta(\varepsilon)^2<\sigma$ by Lemma \[lem:above\]. Therefore the second term is larger than the first one if $\varepsilon\leq \theta(\varepsilon)$, which holds for the $\theta$ defined in . This is according to the fact that the condition $L(n,\varepsilon,\theta)<1$ guarantees that the rank of any elements in $U^F(n)_{\varepsilon,\theta}$ equals to $n$, and the rank of a matrix is stable to the perturbation on the lengths of their column vectors, but extremely frail against the perturbation on the angles between their column vectors. Indeed, for sufficiently large $n$, there exists $\{x_l\}_{l=1}^{2n} \subset {{\mathbb{R}}}^n$ such that each angle between any two distinct vectors is close to $\pi/2$ (cf. [@CFG]\*[Theorem 6]{}).
$\mu^{N,n,F}$ as a push forward measure of $\gamma^{N^F n}$ {#ssc:push}
------------------------------------------------------------
\[lem:push\] We have $$\mu^{N,n,F}=(\Phi^{N,n,F}_{\varepsilon,\theta})_\# \omega^{N,n,F}_{\varepsilon,\theta}, \quad
\text{where}\quad\omega^{N,n,F}_{\varepsilon,\theta}:=
\frac{\gamma^{N^{F}n}|_{X^F_{N,n,\varepsilon,\theta}}}{\gamma^{N^{F}n}(X^F_{N,n,\varepsilon,\theta})}.$$
For any $U \in U^F(n)$, it is easy to check that $\mathcal{U}_{n}^U$ commutes with the ${{\mathbb{R}}}$-multiplication and we have already seen the commutativity of $\mathcal{U}_{n}^U$ with $Q^F$ in Remark \[spectrum\](1). This means that $\mathcal{U}_{n}^{U^\ast}$ commutes with $\Phi^{N,n,F}_{\varepsilon,\theta}$. In addition, the isometric property of $\mathcal{U}_m^U$ enables us to regard $\mathcal{U}_{m}^U \in U^{{\mathbb{R}}}(N^F m)$. We also have $X^F_{N,n,\varepsilon,\theta}=\mathcal{U}_{n}^U(X^F_{N,n,\varepsilon,\theta})$. These facts with the $U^{{\mathbb{R}}}(N^F n)$-invariance of $\gamma^{N^F n}$ together yield that for any Borel subset $B \subset X^F_{N,n,\varepsilon,\theta}$, $$\begin{aligned}
\gamma^{N^F n}(X^F_{N,n,\varepsilon,\theta})\cdot (\Phi^{N,n,F}_{\varepsilon,\theta})_\# \omega^{N,n,F}_{\varepsilon,\theta} (\mathcal{U}_n^U (B))
&=\gamma^{N^F n} (X^F_{N,n,\varepsilon,\theta}\cap (\Phi^{N,n,F}_{\varepsilon,\theta})^{-1} (\mathcal{U}_{n}^U(B)) \\
&= \gamma^{N^F n} (\mathcal{U}_{n}^U(X^F_{N,n,\varepsilon,\theta})\cap (\mathcal{U}_{n}^{U^\ast} \circ \Phi^{N,n,F}_{\varepsilon,\theta})^{-1} (B))\\
&=\gamma^{N^F n} (\mathcal{U}_{n}^U ( X^F_{N,n,\varepsilon,\theta} \cap (\Phi^{N,n,F}_{\varepsilon,\theta})^{-1} (B) ) ) \\
&=\gamma^{N^F n} ( X^F_{N,n,\varepsilon,\theta} \cap (\Phi^{N,n,F}_{\varepsilon,\theta})^{-1} (B) ) \\
&=\gamma^{N^F n} \cdot (\Phi^{N,n,F}_{\varepsilon,\theta})_\#\omega^{N,n,F}_{\varepsilon,\theta} (B).\end{aligned}$$ By Proposition \[prop:haar\], this completes the proof of the lemma.
Convergence of pyramids of (projective) Stiefel and Grassmann manifolds
-----------------------------------------------------------------------
\[lem:Lip-const\] Let $\theta$ be a function defined in . If $\lim_{N \to \infty}n_N/(N-1)^3=0$, then the smallest Lipschitz constant of $\Phi^{N,n_N,F}_{\varepsilon,\theta}$ tends to $1$ as $N \to \infty$.
If $L(n_{N}, \varepsilon_N, \theta) <1$, then $\Phi^{N,n,F}_{\varepsilon,\theta}$ is well-defined and has Lipschitz constant at most $(1-L(n_{N}, \varepsilon_N, \theta))^{-1}$ due to Lemma \[lem:lip\]. It thus suffices to prove $\lim_{N\to0}L(n_{N}, \varepsilon_N, \theta) =0$.
We use the same notations in Theorem \[thm:fullmeas\] as follows: $$p_N:=\log_{(N-1)}n_N, \quad
a_N:=\frac14(1+p_N),\quad
\varepsilon_N:=(N-1)^{-a_N}.$$ We moreover define $$\theta_N:=\theta(\varepsilon_N), \quad
q_N:=\frac{2}{1+p_N}\left(p_N+\frac13\right), \quad
\sigma_N:=\theta_N^{2-q_N}.$$ It then holds that $$\begin{aligned}
&\varepsilon_N \leq \theta_N \leq 3\varepsilon_N, \quad
q_N \leq 1, \quad
a_N(q_N-1)
=\frac{1}{4}\left(p_N-\frac13\right), \quad
a_N q_N=\frac{1}{2}\left(p_N+\frac13\right), \\
&
\lim_{N \to \infty} n_N \varepsilon_N^2
=\lim_{N \to \infty} (N-1)^{p_N-2a_N}
=\lim_{N \to \infty} (N-1)^{(p_N-1)/2}=0,\\
&\lim_{N \to \infty} \frac{\sigma_N}{\theta_N}
=\lim_{N \to \infty} \theta_N^{1-q_N}
\leq \lim_{N \to \infty} 3^{1-q_N} \varepsilon_N^{1-q_N}
\leq \lim_{N \to \infty} 3(N-1)^{a_N(q_N-1)}=0,\end{aligned}$$ where we use in the last inequality.
\[clm\] For large enough $N$, we have $$n_{N} \leq \frac{\sigma_{N}}{2\theta_{N}^2} \leq n_{\sigma_{N}}(\theta_{N}).$$
The first inequality follows from $$\begin{gathered}
n_N \cdot \frac{\theta_N^2}{\sigma_N}
= n_N \theta_N^{q_N}
\leq 3^{q_N} n_N \varepsilon_N^{q_N}
\leq 3(N-1)^{p_N-a_Nq_N}
= 3(N-1)^{(3p_N-1)/6},\end{gathered}$$ hence $\limsup_{N \to \infty} n_N \cdot \theta_N^2/\sigma_N =0$. We derive the second inequality from Lemma \[lem:itere\] and $$\begin{aligned}
\lim_{N \to \infty} \frac{1}{R_{\sigma_N}(\theta_N)}
&=\lim_{N \to \infty} \frac{\theta_N^2}{\sigma_NR_{\sigma_N}(\theta_N)}=\infty,\\
\lim_{N \to \infty}(1+R_{\sigma_N}(\theta_N))^{1/2R_{\sigma_N}(\theta_N)}
&=e^{1/2}<e^1
=\lim_{N \to \infty}\left(1+R_{\sigma_N}(\theta_N)\frac{\sigma_N}{\theta_N^2} \right)^{{\theta_N^2}/{\sigma_N}R_{\sigma_N}(\theta_N)}.
$$ The claim has been proved.
The claim implies that $$\begin{gathered}
\lim_{N \to \infty} n_N \sigma_N
\leq
\lim_{N \to \infty}\frac{\sigma_N^2}{2\theta_N^2}=0.\end{gathered}$$ For large enough $N$, we have, by Theorem \[lem:1\], $$L(n_{N}, \varepsilon_N, \theta)^2<
n_N \varepsilon_N^2+
\frac{2n_N(1+\varepsilon_N)\sigma_{N}}{1+\sqrt{1-\sigma_{N}}},$$ which proves $\lim_{N \to \infty} L_N=0$ as desired. This completes the proof.
We set $$X_N = X_{N,n_N}^F, \quad
Y_N = \Gamma^{N^F n_N}, \quad
Y_N' = X_{N,n_N,\varepsilon_N,\theta}^F, \quad
p^N_l = \pi^N_l(n),\quad
\Phi^N = \Phi^{N,n,F}_{\varepsilon,\theta}$$ for the $\theta$ defined in and $G=U^F(1)$ or $U^F(n)$. We already check that the pyramid ${\mathcal{P}}_{Y_N}$ converges weakly to ${\mathcal{P}}_{\Gamma^\infty}$ as $N \to \infty$. It is easy to cheek the $G$-invariance of $Y'_N$. Since $p^N_l$ is $1$-Lipschitz continuous and $G$-equivariant, the space $\bar{Y}_N$ is monotone increasing in $N$ with respect to the Lipschitz order. Proposition \[thm:MB\] implies that $\lim_{N \to \infty}\operatorname{\mathit{d}_\mathrm{P}}((p^N_l)_\# \nu_{X_N}, \nu_{Y_l})=0$. We confirm that $\Phi^N_\# \nu_{Y'_N} =\nu_{X_N}$ in Lemma \[lem:push\] and $\Phi^N$ is $G$-equivariant in Remark \[spectrum\](2). Theorem \[thm:fullmeas\] implies $\lim_{N \to \infty}\nu_{Y_N}(Y_N')=1$. The smallest Lipschitz constant of $\Phi^N$ tends to $1$ as $N \to \infty$ due to Lemma \[lem:Lip-const\]. We thus apply Lemma \[keylem\] and Corollary \[keycor\] to obtain $${\mathcal{P}}_{X^F_{N,n_N}}\to{\mathcal{P}}_{\Gamma^\infty}, \
{\mathcal{P}}_{X^F_{N,n}/U^F(n)}\to{\mathcal{P}}_{\Gamma^{\infty n}/U^F(n)}, \
{\mathcal{P}}_{U^F(1)\backslash X^F_{N,n_N}}\to{\mathcal{P}}_{U^F(1)\backslash\Gamma^\infty}, \quad
\text{as}\ N \to \infty.$$ This completes the proof of the theorems.
\(1) Under the assumption , or more weakly $\lim_{N\to\infty} n_N^3/N = 0$, we always have $\lim_{N \to \infty}n_{\sigma_N}(\theta_N)=\infty$ for $\theta$ defined in even if $\lim_{N \to \infty} n_N<\infty$. This follows from Claim \[clm\] and $$\lim_{N \to \infty} \frac{\sigma_N}{\theta_N^2}
=\lim_{N \to \infty} \theta_N^{-q_N}
\geq 3^{-q_N}\lim_{N \to \infty} \varepsilon_N^{-q_N}
\geq \frac13\lim_{N \to \infty} (N-1)^{a_Nq_N}=\infty.$$ (2) Assume $\lim_{N\to \infty} n_N =\infty$. If we construct a pair of $\varepsilon_N$ and $\theta:(0,\infty) \to (0,\infty)$ satisfying $$\lim_{N \to \infty} \varepsilon_N=0, \quad
\lim_{N \to \infty} \gamma^{N^F n} (X_{N,n,\varepsilon_N, \theta}^F)=1$$ with the help of Lemma \[lem:subset\], and we prove the convergence of ${\mathcal{P}}_{X_{N,n}^F}$ by using Lemma \[lem:lip\], Theorem \[lem:1\] and Lemma \[keylem\], where we set $\Phi^N=\Phi^{N,n,F}_{\varepsilon,\theta}$, then the assumption $$\lim_{N \to \infty} \frac{n_N^3}{N-1}=0,$$ which is slightly weaker than , is a necessary condition according the following three conditions (a), (b) and (c).
(a) \[2\] As mentioned in Remark \[rem:angle\], if we use Lemma \[lem:subset\], then we require $$\begin{gathered}
\left(B^{F}(T_{N,n_N-1})\right)^{n_N-1} \times A^{(N-n_N+1)^{F}}_{\varepsilon_{N,n_N-1} } \subset A^{N^{F}}_{\varepsilon_N },
\quad
\lim_{N\to \infty} T_{N,n_N-1}=\infty, \\
\frac{1}{\theta_N^{2}}
<
1+\frac{(1-\varepsilon_{N,n_N-1} )^2((N-n_N+1)^F-1)}{(n_N-1)T_{N,n_N-1}^2}.
$$
(b) \[4\] To use Theorem \[lem:1\], we need to assume $n_N \le n_{\sigma_{n_N}}(\theta_N)$.
(c) \[3\] By Lemma \[lem:lip\] and Theorem \[lem:1\], we need $$\lim_{N\to \infty} n_N \sigma_N = 0.$$
By , we have $$\begin{aligned}
(n_N-1)T_{N,n_N-1}^2
+(1+\varepsilon_{N,n_N-1})^2((N-n_N+1)^F-1)
<(1+\varepsilon_{N})^2(N^F-1),\end{aligned}$$ providing $$\begin{aligned}
\infty=\lim_{N \to \infty} T_{N,n_N-1}^2
=
\lim_{N \to \infty}
\frac{N-1}{n_N}.\end{aligned}$$ On one hand, if $\lim_{N\to \infty} {\sigma_N}/{\theta_N}>0$ holds, then we require, by , $$0=\lim_{N \to \infty} n_N \sigma_N=\lim_{N\to \infty} n_N \theta_N \cdot \frac{\sigma_N}{\theta_N},$$ hence $\lim_{N \to \infty} n_N\theta_N =0$. On the other hand, if we assume $\lim_{N\to \infty} {\sigma_N}/{\theta_N}=0$, then we observe from and Lemma \[lem:above\] with $\lim_{N \to \infty}n_N=\infty$ that $\theta_N^2<\sigma_N$ and $$\lim_{N \to \infty} n_N \theta_N \cdot \frac{\theta_N}{\sigma_N}
\leq
\lim_{N \to \infty} \left(1+\frac{\sigma_N}{\theta_N^2}\right) \cdot \frac{\theta_N^2}{\sigma_N}
=\lim_{N \to \infty} \frac{\theta_N^2}{\sigma_N} +1
\leq 2,$$ providing $\lim_{N \to \infty} n_N\theta_N =0$. We therefore obtain $$\begin{aligned}
\lim_{N \to \infty} \frac{n_N^3}{N-1}
=
\lim_{N \to \infty}
n_N^2 \theta^2_N
\cdot \frac{n_N}{(N-1)\theta^2_N}
=0.\end{aligned}$$
Asymptotic estimate of observable diameter
==========================================
For the proof of Corollary \[cor:ObsDiam\] we need the following
\[defn:ObsDiam-pyramid\] Let $\kappa > 0$. The *observable diameter of a pyramid ${\mathcal{P}}$* is defined to be $$\operatorname{ObsDiam}({\mathcal{P}};-\kappa)
:= \lim_{\varepsilon\to 0+} \sup_{X \in {\mathcal{P}}} \operatorname{ObsDiam}(X;-(\kappa+\varepsilon))
\quad (\le +\infty).$$
For any mm-space $X$ we have $$\operatorname{ObsDiam}({\mathcal{P}}_X;-\kappa) = \operatorname{ObsDiam}(X;-\kappa)$$ for any $\kappa > 0$ (see [@OzSy:pyramid]).
\[thm:lim\] Let ${\mathcal{P}}$ and ${\mathcal{P}}_n$, $n=1,2,\dots$, be pyramids. If ${\mathcal{P}}_n$ converges weakly to ${\mathcal{P}}$ as $n\to\infty$, then, for any $\kappa > 0$, $$\begin{aligned}
\operatorname{ObsDiam}({\mathcal{P}};-\kappa)
&= \lim_{\varepsilon\to 0+} \liminf_{n\to\infty}
\operatorname{ObsDiam}({\mathcal{P}}_n;-(\kappa+\varepsilon)) \\
&= \lim_{\varepsilon\to 0+} \limsup_{n\to\infty}
\operatorname{ObsDiam}({\mathcal{P}}_n;-(\kappa+\varepsilon)).
\end{aligned}$$
Theorem \[thm:lim\] together with [@Sy:mmg]\*[Theorem 2.21]{} (see also Corollary [@Sy:mmlim]\*[§5]{}) leads us to the following
We have $$\operatorname{ObsDiam}({\mathcal{P}}_{\Gamma^\infty};-\kappa)
= 2D^{-1}\left(1-\frac{\kappa}{2}\right)$$ for any $\kappa$ with $0 < \kappa < 1$.
We are now in a position to prove the
\(1) follows from Theorems \[thm:Stiefel\] and \[thm:lim\].
Since $G^F_{N,n_N}, \textrm{P}V^F_{N,n_N} \prec V^F_{N,n_N}$, we have (2) and (4) from (1).
We prove (3). Let $f : M^F_{1,n}/U^F(n) \to [\,0,+\infty\,)$ be the function defined by $f(\bar{z}) := \|z\|$, $\bar{z} \in M^F_{1,n}/U^F(n)$. Note that $f$ is an isometry. By using polar coordinates, we have $$d(f_\# \bar{\gamma}^{n^F})(r) = g_{n^F}(r) \; dr,$$ where $g_m$ is the function defined in §\[ssec:Gaussian\]. Since $g_m(r) \le g_2(1) = e^{-1/2}$, we have $$\begin{aligned}
\operatorname{ObsDiam}((M^F_{N,n}/U^F(n),\bar{\gamma}^{N n^F});-\kappa)
&\ge \operatorname{ObsDiam}((M^F_{1,n}/U^F(n),\bar{\gamma}^{n^F});-\kappa) \\
&= \operatorname{diam}(f_\# \bar{\gamma}^{n^F}; 1-\kappa)
\ge e^{1/2} (1-\kappa),
\end{aligned}$$ which together with Theorems \[thm:Gr-pS\](1) and \[thm:lim\] implies (3).
We prove (5). Since $\sqrt{N^F-1} \,\textrm{P}V^F_{N,m_N} = U^F(1)\backslash X^F_{N,m_N} \succ U^F(1)\backslash X^F_{N,1}$, it suffices to estimate the observable diameter of $U^F(1)\backslash X^F_{N,1}$ from below. By Theorems \[thm:Gr-pS\](2) and \[thm:lim\], $$\begin{aligned}
\lim_{N\to\infty} \operatorname{ObsDiam}(U^F(1)\backslash X^F_{N,1};-\kappa)
&= \operatorname{ObsDiam}({\mathcal{P}}_{U^F(1)\backslash \Gamma^\infty};-\kappa) \\
&\ge \operatorname{ObsDiam}((U^F(1)\backslash F,\bar\gamma^{1^F});-\kappa),
\end{aligned}$$ where the last inequality follows from ${\mathcal{P}}_{U^F(1)\backslash \Gamma^\infty} \ni (U^F(1)\backslash F,\bar\gamma^{1^F})$, $1^F = \dim_{{\mathbb{R}}}F$. We see that $(U^F(1)\backslash F,\bar{\gamma}^{1^F})$ is mm-isomorphic to $([\,0,+\infty\,), g_{1^F}(r)\,dr)$ and therefore $$\operatorname{ObsDiam}((U^F(1)\backslash F,\bar\gamma^{1^F});-\kappa) = \operatorname{diam}(g_{1^F}(r)\,dr;1-\kappa) \ge e^{1/2}(1-\kappa).$$ This completes the proof.
Corollary \[cor:ObsDiam\](2)(3) together with Theorem [@OzSy:pyramid]\*[Theorem 1.2]{} implies the following
Let $n$ be any fixed positive integer.
1. We have $t_N/\sqrt{N} \to 0$ as $N \to \infty$ if and only if $\{t_N G^F_{N,n}\}$ is a Lévy family, i.e., converges weakly to one-point mm-space.
2. We have $t_N/\sqrt{N} \to 0$ as $N \to \infty$ if and only if $\{t_N G^F_{N,n}\}$ infinitely dissipates, i.e., the associated pyramid converges weakly to ${\mathcal{X}}$.
Note that the same property for many other manifolds was already obtained in [@OzSy:pyramid].
Appendix: $U^F(N)$ as a subgroup of $U^{{\mathbb{R}}}(N^F)$
===========================================================
For $z:=z_0+z_1\i+z_2\j+z_3\k \in F^N$, we set $$\operatorname{R}(z):=z_0,\quad
\operatorname{I}(z):=z_1,\quad
\operatorname{J}(z):=z_2,\quad
\operatorname{K}(z):=z_3.$$ It follows that for $z:=z_0+z_1\i+z_2\j+z_3\k , w:=w_0+w_1\i+w_2\j+w_3\k \in F^N$, $${\left\langle{
\begin{pmatrix}
z_0\\
z_1 \\
z_2 \\
z_3
\end{pmatrix}},{\begin{pmatrix}
w_0\\
w_1 \\
w_2 \\
w_3
\end{pmatrix}}\right\rangle}
=\sum_{l=0}^3 z_lw_l
=\Re{\left\langle{z},{w}\right\rangle}.$$
\[lem:unit\] Define a map $\mathcal{O}^F: U^F(N) \hookrightarrow \mathrm{M}_{N^F}({{\mathbb{R}}})$ by $$\begin{gathered}
\mathcal{O}^{F}(U)
\begin{pmatrix}
z_0\\
z_1 \\
z_2 \\
z_3
\end{pmatrix}
:=
\begin{pmatrix}
\operatorname{R}(Uz)\\
\operatorname{I}(Uz)\\
\operatorname{J}(Uz)\\
\operatorname{K}(Uz))
\end{pmatrix},\end{gathered}$$ for $z_0,z_1,z_2,z_3 \in {{\mathbb{R}}}^N$, where $z:=z_0+z_1\i+z_2\j+z_3\k \in F^N$. Then we have for $U, V \in U^F(N)$, $$\mathcal{O}^{F}(U) \in U^{{\mathbb{R}}}(N^F), \quad
\mathcal{O}^{F}(UV)=\mathcal{O}^{F}(U)\mathcal{O}^{F}(V), \quad
\mathcal{O}^{F}(U^\ast)=\mathcal{O}^{F}(U)^\ast.$$
For any $z:=z_0+z_1\i+z_2\j+z_3\k , w:=w_0+w_1\i+w_2\j+w_3\k \in F^N$ and $U,V \in U^F(N)$, we compute that $$\begin{aligned}
{\left\langle{\mathcal{O}^{F}(U)
\begin{pmatrix}
z_0\\
z_1 \\
z_2 \\
z_3
\end{pmatrix}},{
\mathcal{O}^{F}(U)
\begin{pmatrix}
z_0\\
z_1 \\
z_2 \\
z_3
\end{pmatrix}}\right\rangle}
=\Re{\left\langle{Uz},{Uw}\right\rangle}=\Re{\left\langle{z},{w}\right\rangle}
=
{\left\langle{
\begin{pmatrix}
z_0\\
z_1 \\
z_2 \\
z_3
\end{pmatrix}},{\begin{pmatrix}
w_0\\
w_1 \\
w_2 \\
w_3
\end{pmatrix}}\right\rangle},\\
\mathcal{O}^{F}(UV)
\begin{pmatrix}
z_0\\
z_1 \\
z_2 \\
z_3
\end{pmatrix}
=
\begin{pmatrix}
\Re(UVz)\\
\operatorname{I}(UVz)\\
\operatorname{J}(UVz)\\
\operatorname{K}(UVz)
\end{pmatrix}
=
\mathcal{O}^{F}(U)
\begin{pmatrix}
\Re(Vz)\\
\operatorname{I}(Vz)\\
\operatorname{J}(Vz)\\
\operatorname{K}(Vz)
\end{pmatrix}
=
\mathcal{O}^{F}(U)
\mathcal{O}^{F}(V)
\begin{pmatrix}
z_0\\
z_1 \\
z_2 \\
z_3
\end{pmatrix},\end{aligned}$$ implying the first two claims. We also find that $\mathcal{O}^{F}(I_N)=I_{N^F}$. This implies that $$\mathcal{O}^{F}(U)^\ast \mathcal{O}^{F}(U)=I_N
= \mathcal{O}^{F}(U^\ast U)
= \mathcal{O}^{F}(U^\ast) \mathcal{O}^{F}(U),$$ hence $\mathcal{O}^{F}(U^\ast)=\mathcal{O}^{F}(U)^\ast$. This completes the proof.
[^1]: This work was supported by JSPS KAKENHI Grant Numbers 26400060, 15K17536, 15H05739.
|
---
abstract: 'We show that the usual relation between redshift and angular-diameter distance can be derived by considering light from a source to be gravitationally lensed by material that lies in the telescope beam as it passes from source to observer through an otherwise empty universe. This derivation yields an equation for the dependence of angular-diameter on redshift in an inhomogeneous universe. We use this equation to model the distribution of angular-diameter distance for redshift $z=3$ in a realistically clustered cosmology. This distribution is such that attempts to determine $q_0$ from angular-diameter distances will systematically underestimate $q_0$ by $\sim0.15$, and large samples would be required to beat down the intrinsic dispersion in measured values of $q_0$.'
author:
- |
Fedja Hadrović and James Binney\
Theoretical Physics, 1 Keble Road, Oxford OX1 3NP
title: 'Gravitational lensing and the angular-diameter distance relation'
---
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\[section\]
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Cosmology – Gravitational lensing
INTRODUCTION
============
The large-scale structure of the Universe is believed to be closely approximated by one of the cosmological models. These are characterized by the values of three parameters: the matter density, the radius of curvature of spatial sections and the cosmological constant. Determination of these values has long been considered one of the fundamental tasks of observational cosmology. In this connection a potentially key observable is the relationship between redshift $z$ and the angular-diameter distance $D(z)$, which is defined to be the ratio of the linear diameter of an object to the angular diameter that it subtends when observed at redshift $z$. There have been may attempts to determine $D(z)$ [@sandage; @kellermann; @crawford] which has recently prompted a wider discussion about the feasibility of the method [@nilsson; @dabrowski; @kantowski; @stephanas]. The form of $D(z)$ depends on the geometry of the Universe in the sense that the larger the curvature $K$ is, the smaller $D$ is at a given redshift. That is, the more positively curved the Universe is, the more slowly the angular size of an object decreases as it is moved away from the observer.
In an inhomogeneous universe deviations of the metric from the form give rise to fluctuations in measured values of $D$ at fixed $z$. Previous analyses of this effect [@zeldovich; @refsdal; @dyer-roeder; @sasaki; @kasai; @watanabe] worked directly from general relativity and produced results of considerable complexity. By delegating relativistic considerations to the theory of gravitational lensing [@schneider], we obtain a much simpler analysis and an equation (\[new:distance\]) that involves the cosmic density field rather than the cosmic metric or potential. This simplicity enables us to evaluate the effects of inhomogeneity for the case of realistic clustering, rather than the case of either weak perturbations [@sasaki] or randomly distributed point masses [@zeldovich; @refsdal; @dyer-roeder; @watanabe; @kantowski].
Over the last decade there has been a growing awareness of the importance of gravitational lensing for observations of high-redshift objects. Gravitational lensing and the dependence of $D$ on $z$ are two sides of the same coin: both phenomena are caused by the tendency of matter that lies between the observer and a distant object to focus radiation from that object, thereby increasing its apparent size and brightness. In Section 2 we demonstrate this connection quantitatively by showing that the standard formula for $D(z)$ in a universe can be obtained by applying conventional lensing theory to a Universe in which there is matter [*only*]{} in the telescope beam towards the object under study. In Section 3 we describe our model of the clustered cosmic density field, and in Section 4 we use this model to calculate probability distributions for $D(z)$ from objects of various linear sizes. Section 5 sums up.
We throughout use the convention that $\ti D$ and $D$, respectively, denote angular-diameter distance before and after lensing is taken into account.
Since luminosity distance $D_L$ is rigorously related to angular-diameter distance by $D_L=D/(1+z)^2$ [@ethering], our distributions of values of $D$ imply identical distributions of values of $D_L$.
The unit system we use is based on $G=c=H_0=1$, which significantly simplifies the equations in cosmology. All lengths quoted are scaled to $H_0=100 h\, {{\rm km}}\, {{\rm s}}^{-1}\, {{\rm Mpc}}^{-1}$.
The angular-diameter distance relation from lensing
===================================================
A sequence of gravitational lenses
----------------------------------
Fig. 1 shows a ray that is deflected through angles $\ba_1$ and $\ba_2$ by two lenses, which it passes at impact parameters $\bxi_1$ and $\bxi_2$, respectively. From the figure it is immediately apparent that \_3=[D\_3D\_1]{}\_1-(\_1)D\_[13]{} -(\_2)D\_[23]{}. The generalization of this equation to an arbitrary number of lenses is easily seen to be \[lens\_seq\] \_j=[[D\_j]{}]{}\_1 -\_[i=1]{}\^[j-1]{}[D\_[ij]{}\_i(\_i)]{}.
Light that passes at radius vector $\bxi$ through a disc of matter that has uniform surface density $\Sigma$ is deflected through an angle =4. Hence in this case the impact parameters are all parallel and satisfy \[all\_xis\] \_j=[[D\_j]{}]{}\_1 -4\_[i=1]{}\^[j-1]{}\_iD\_[ij]{}\_i. Finally, if $\bxi_i$ is a diameter of an object, then the angular-diameter distance of that object is $D_i\equiv\xi_i/(\xi_1/\ti D_1)$, so on taking the modulus of (\[all\_xis\]) and dividing through by $(\xi_1/\ti D_1)$ we conclude that true angular-diameter distances $D_i$ satisfy \[disc\_ang\_diam\] D\_j=D\_j-4\_[i=1]{}\^[j-1]{}\_iD\_iD\_[ij]{}
Application to a homogeneous universe
-------------------------------------
We now we show that equation (\[disc\_ang\_diam\]) reproduces the familiar angular-diameter distance equation for a universe. We consider an empty universe. In such a universe there is nothing to single out a unique rest frame at any given event, so redshift is not uniquely related to distance. This permits us simply to adopt the relation $s(z)$ between proper distance and redshift in a universe. We have that $s(z)$ satisfies \[Scheneider et al., eq. (4.47b)\] \[FLsz\] [[[[d]{}]{}]{}s[[[d]{}]{}]{}z]{}=(1+z)\^[-2]{}(1+z)\^[-1/2]{}.
From the gravitational-focusing equation \[Schneider et al. eq. (3.64)\] in the case of empty space (vanishing Ricci tensor and shear) we have \[focus\_e\] [[[[d]{}]{}]{}\^2D[[[d]{}]{}]{}\^2]{}=0, where $\tau$ is an affine parameter for the light beam. In terms of the wavenumber, $k$, we have ${{{\rm d}}}s/{{{\rm d}}}\tau\propto k\propto 1+z$, so we may use equation (\[FLsz\]) to eliminate $\tau$ from (\[focus\_e\]) in favour of $s$. We then find that the focusing equation states that in our empty universe, as a function of $z$, angular-diameter distance $\ti D$ satisfies $$\begin{aligned}
\label{D_empty1}
(1+z)(1+\O z)&& {\pa^2\ti D(y,z)\over\pa z^2}\nonumber\\
&&+\l(\fracj72\O z+\fracj12\O+3\r){\pa\ti D(y,z)\over\pa z}=0.
\end{aligned}$$ We also have the initial condition \[Schneider et al. eq. (4.53)\] \[D\_empty2\] [D(y,z)z]{}|\_[y=z]{}=(1+z)\^[-2]{}(1+Øz)\^[-1/2]{}.
Now we fill the telescope beam with the normal matter density of a universe and use equation (\[disc\_ang\_diam\]) to calculate the angular-diameter distance of an object at ‘redshift’ $z$. We first take the limit of equation (\[disc\_ang\_diam\]) in which there are an infinite number of discs. Since in our units the current critical density is $3/(8\pi)$, the disc that lies between $z+{{{\rm d}}}z$ and $z$ has surface density =(1+z)\^3[38]{}[[[[d]{}]{}]{}s[[[d]{}]{}]{}z]{}[[[d]{}]{}]{}z, where $\Omega$ is the usual density parameter. With equation (\[FLsz\]) this becomes (z)=[38]{}[1+z]{}[[[d]{}]{}]{}z. Inserting this expression for $\Sigma$ into equation (\[disc\_ang\_diam\]) and proceeding to the limit ${{{\rm d}}}z\to0$ we find D(z)=D(z) -32\_0\^z[[[d]{}]{}]{}y[1+y]{}D(y)D(y,z). We now convert this integral equation for $D(z)$ into a differential equation. Differentiating we find $$\begin{aligned}
{{{{\rm d}}}D\over{{{\rm d}}}z}&=&{{{{\rm d}}}\ti D\over{{{\rm d}}}z}-\fracj32\,\O\,
\int_0^z{{{\rm d}}}y\,{1+y\over\sqrt{1+\O y}}D(y)\l({\pa\over\pa z}\ti D(y,z)\r),
\nonumber\\
{{{{\rm d}}}^2D\over{{{\rm d}}}z^2}&=&{{{{\rm d}}}^2\ti D\over{{{\rm d}}}z^2}-\fracj32\,
\O\,{1+z\over\sqrt{1+\O z}}D(z)\l({\pa\over\pa z}\ti D(y,z)\r)_{y=z}\\
&&-\fracj32\,\O\,\int_0^z{{{\rm d}}}y\,{{1+y\over\sqrt{1+\O y}}D(y)
\l({\pa^2\over\pa z^2}\ti D(y,z)\r)}\nonumber.\end{aligned}$$ Combining these equations and taking advantage equations (\[D\_empty1\]) and (\[D\_empty2\]), we recover the standard equation for the angular-diameter distance in a conventional universe: \[eq:distance\] (1+z)(1+Øz)[\^2 Dz\^2]{} + ł(72Øz+ 12Ø+3)[Dz]{}+32ØD=0.
Application to an inhomogeneous universe
----------------------------------------
The most important feature of the above derivation is that it does not depend on $\O$ being constant. In the first two terms of equation (\[eq:distance\]) $\O$ appears as a result of the reparametrisation ($s\mapsto z$). It is not related to the local matter distribution and can be thought of as the averaged density parameter $\av{\O}$. The parameter $\O$ in the last term, ${3 \over 2} \O D$ [*is*]{} related to the local matter density and comes directly from the gravitational lensing calculation. Therefore, in the case of a locally inhomogeneous universe that approaches a model in the large-scale limit, we can write $$\begin{aligned}
\label{new:distance}
(1+z)(1+\av{\O} z){\pa^2 D\over\pa z^2} +
\big(\fracj72\av{\O}z+&\fracj12&\av{\O}+3\big){\pa D\over\pa z}\nonumber\\
&+&\fracj32\O(z) D=0.\end{aligned}$$ Note that $\O(z)$ describes the comoving matter density because the physical density is $\rho(z)=\fracj3{8\pi}\O(1+z)^3$.
Simply replacing $\O$ by $\av{\O}$ in all but the last term of equation (\[eq:distance\]) does not allow for a complete discussion of the effects of inhomogeneity on images: in addition to being magnified by matter within the beam, images will be distorted and may be even split into multiple images. We have neglected these potentially important effects by (i) assuming that the material that lies between redshifts $z+{{{\rm d}}}z$ and $z$ forms a uniform disc, and (ii) neglecting shear that is induced by clumps of material that lie outside the beam. Futumase & Sasaki (1989) and Watanabe & Sasaki (1990) show that as long as the scale of inhomogeneities is greater than, or equal to galactic scale, shear does not contribute significantly to focusing.
By contrast, the assumption that the beam is filled by a series of uniform-density discs constitutes a non-trivial approximation about the matter distribution in the beam, namely that we may average the density across the beam as shown in Fig. \[fig:average\].
Statistical model of the field $\O(\br)$ {#sec:matter}
========================================
We now investigate the predictions of the generalized diameter-distance equation (\[new:distance\]). For this investigation we require a statistical description of the density field along the telescope beam. This is a random field, which we think of as a function of comoving distance $x$. We assume that $\Omega(x)$ follows a log-normal distribution – see Coles & Jones (1991) for a discussion of the characteristics and advantages of the log-normal distribution in cosmology. We confine ourselves to the case of a critical-density universe: $\av{\O}=1$. With these assumptions $\O(x)$ is given by \[Ofrome\] Ø(x)=[\^[(x)]{}]{}, where $\e(x)$ is a Gaussian random field. Without loss of generality we set $\av{\e}=0$.
We define the two-point correlation function, $\xi_f$ of a field $f(x)$ by \_f(x)=[-\^2 -\^2]{}. The correlation functions of the fields $\O(x)$ and $\e(x)$ are related by \[xiO:xie\] \_Ø(x)=[(\_\^2\_(x))-1(\_\^2)-1]{}, where $\sigma_\e^2=\av{\e^2}$ is the variance of the Gaussian field.
The Gaussian field $\e(x)$ is determined by its power spectrum $P_\e(k)$, which is essentially the Fourier transform of $\xi_\e$: \[Pe:xie\] P\_(k)=[\_\^22]{}[[[d]{}]{}]{}x\^[ikx]{} \_(x). Hence, if we know $\xi_\O(x)$, we may construct realizations of $\O$ by determining $\xi_\e(x)$ from equation (\[xiO:xie\]) and then using equation (\[Pe:xie\]) to determine $P_\e(k)$.
The galaxy correlation function may be approximated by [@padmanabhan] (r)=ł([r ]{})\^[-]{}, where $\g\approx1.8$ and the correlation length is $r_c\simeq5.5h^{-1}{{\rm Mpc}}$. The correlation function of the density field is often assumed to have the same form, but a different amplitude. The bias factor is introduced by setting b=[\_\_]{}. Measurements indicate that $1\la b \la 2$. Hence we require $\xi_\O$ such that $\xi_\O(0)=1$ and \_Ø\^2\_Ø(r)b\^[-1]{}ł([r ]{})\^[-]{}. We have adopted the form \[xiofO\] \_Ø(r)=ł(1+[r\^2r\_0\^2]{})\^[-1]{} with $\s_\O r_0=b^{-1/2}r_c$. For $b=1.5$ we find $\s_\O r_0=4.5 h^{-1}
{{\rm Mpc}}$. The meaning of $r_0$ will be discussed later, but we immediately see that for small $r_0$ the model approximates the divergent galaxy correlation function better.
From equation (\[xiO:xie\]) we find \_(r)=[1\_\^2]{} ([r\_0\^2\^[\_\^2]{}+r\^2r\_0\^2+r\^2]{}). From (\[Pe:xie\]) the power spectrum is $$\begin{aligned}
\mod{P_\e(k)}^2
&=&{1\over2\pi}
\int_{-\infty}^{+\infty}{{{\rm d}}}x\,{\ln{{r_0^2\ee^{\s_\e^2}+x^2}
\over{r_0^2+x^2}}\ee^{\i kx}}\nonumber\\
&=&{1\over{k}}\l[\exp(-kr_0)-\exp(-kr_0\ee^{\s_\e^2/2})\r].\end{aligned}$$
The significance of $r_0$ now emerges: it determines how quickly $\mod{P_\e(k)}$ approaches zero at large $k$. The smaller the value of $r_0$ the larger must be the wavenumber $k_{{{\rm max}}}$ up to which we must sum the discrete Fourier transform from which we obtain realizations of $\O(x)$. Physically, we should think of $r_0$ as the scale on which the matter distribution is smoothed by the finite width of our telescope beam and the diameters of the objects we are looking at. If we take $r_0=10 h^{-1}
\kpc$, we have $\s_\e^2=12.21$.
Due to computational constraints and limitations on sampling imposed by Nyquist’s theorem, it was impracticable to generate a single random field on the range $0<z<3$. Instead, we divided this interval into 100 subintervals and create a scaled random field on each of them. This procedure destroys correlations between different intervals but these are physically unimportant because the correlation function is negligible at such large distances.
Results and discussion {#sec:discussion}
======================
Distribution of angular-diameter distances
------------------------------------------
Once a realization of $\O(x)$ has been constructed, it is straightforward to solve equation (\[new:distance\]) for $D$ at any given value of $z$. We repeated this operation for approximately 4000 realizations of $\O(x)$ to determine the distribution of angular-diameter distances at $z=3$. Fig. \[fig:dist\] shows this distribution. The distances are rescaled to the standard value D\^[FL]{}=[2 ]{}ł(1-[1]{}). An important point on the graph is the Dyer–Roeder distance corresponding to an empty light beam: D\^[DR]{}=25ł(1-[1]{}), which for $z=3$ gives $D^{\rm DR}/D^{\rm FL}=1.55$.
We see that the distribution is strongly peaked on the Dyer–Roeder side of $D^{\rm FL}$, with a long tail on the side. This is expected because regions within which the density is below average occupy the great majority of the volume of the Universe. Hence, many light paths sample only low-density regions and the distribution in Fig. \[fig:dist\] is shifted towards $D^{\rm DR}$. However, when the light beam does encounter a galaxy or other matter aggregation, it is strongly lensed. These events decrease the diameter distance and give rise to the tail on the side.
It is important to understand the impact that smoothing of the matter distribution has on our results. It is computationally convenient to investigate this for an unrealistic case: we take the correlation length to be 100 times its true value. That is, we investigate the case in which $\s_\O r_0=450 h^{-1} {{\rm Mpc}}$. Fig. \[fig:smooth\] shows our results.
For large $r_0$ the matter distribution is rather homogeneous, so the distribution of $D$ is narrow and peaked near $D^{\rm FL}$. As $r_0$ is decreased the universe becomes strongly inhomogeneous and the distribution of $D$ becomes broader. Simultaneously, its peak shifts towards the empty-beam distance $D^{\rm DR}$.
Implications for $q_0$ measurements
-----------------------------------
One of the most important undetermined quantities of cosmology is $q_0$, the deceleration parameter. For a flat universe ($K=0$), the angular-diameter distance $D$ is related to $q_0$ by \[DRz\] D(q\_0,z)=[R1+z]{}, where \[Rfromq\] R\_[1]{}\^[1+z]{}[[[[[d]{}]{}]{}u]{}]{}. Suppose we attempt to use (\[DRz\]) to determine $q_0$ from an observationally determined value of $D(z)$. We assume that the true values of the cosmic constants are those with which we have been working: $\O=1$, $\Lambda=K=0$, and thus that the true value of $q_0$ is $q_0=\fracj12$. Putting $q_0=\fracj12+\dq$ in equation (\[Rfromq\]) we find R()=\_[1]{}\^[1+z]{}[[[[[[d]{}]{}]{}u]{}]{}ł(1-)]{}. Substituting this value of $R$ into (\[DRz\]) gives \[qerror\] ł.[DD\^[FL]{}]{}|\_[z=3]{}=1-0.15. This equation relates the error, $\dq$, in the inferred value of $q_0$ to the ratio of the measured value of $D$ to the value $D^{\rm FL}$ that it would have if the Universe were homogeneous. The distribution of $D/D^{\rm
FL}$ shown in Fig. \[fig:dist\] is centred on $1.025$ and has spread $\sim\pm 0.06$. By equation (\[qerror\]) the error in $q_0$ to which this gives rise is =-0.170.4. In connection with this result three points should be made:
- We see that the conventional method of determining $q_0$ from the angular-diameter redshift relation provides a biased estimator of $q_0$ that will return significant underestimates of the true value.
- Even perfect measurements of $D(z)$ will return values of $q_0$ that are widely scattered. The breadth of this scatter is such that an accurate determination of $q_0$ would require an extremely large sample and a sophisticated statistical analysis of the data.
- The errors in $q_0$ to which inhomogeneities give rise depend on the scale of observed objects because this scale determines the effective spectrum of the inhomogeneities. Larger objects will yield smaller errors.
This last point is unfortunate because, as Kellermann (1993) has emphasized, small objects are much more likely to constitute standard measuring rods than large objects, such as giant radio sources, whose linear sizes are likely to be sensitive to the mean cosmic density.
Conclusion
==========
We have used the theory of gravitational lensing to derive the conventional relation between angular-diameter distance and redshift in a universe. The value of this derivation is that it is simple and shows that the tendency of the angular diameter of a distant object to increase with $\Omega$ arises because rays coming from the object are focused by matter that lies within the telescope beam. Hence, the angular diameter of an object is sensitive to the precise disposition of matter in the neighborhood of the telescope beam: move matter just out of the beam and the apparent size of the object will diminish. Equation (\[new:distance\]) expresses this fact mathematically.
Since the Universe is strongly inhomogeneous on small scales, telescope beams to different objects at the same redshift will contain significantly varying quantities of matter, and the apparent diameters of physically identical objects at a common redshift will vary. This variation gives rise to scatter in the angular-diameter distances $D$ of a set of objects that lie at a common redshift.
We have modelled the distribution of the values of $D$ of objects at redshift $z=3$ by assuming that the cosmic density field follows a lognormal distribution that matches the observed clustering of galaxies for bias parameter $b=1.5$. The distribution of $D$ is very skew, with its peak at a value that exceeds that associated with the corresponding homogeneous universe, $D^{\rm FL}$, and a long tail to values smaller than $D^{\rm FL}$. In consequence of this skewness, the conventional technique for measuring $q_0$ from measurements of $D(z)$ will systematically underestimate $q_0$.
The width of the distribution of $D$ at given $z$ depends upon the assumed power spectrum $P(k)$ of the cosmic density field. The true power spectrum is thought to have considerable power on small scales, and this power will generate a very broad distribution of $D$ for objects of small angular size. When the angular diameters of highly extended objects are measured, only power on scales comparable to or larger than the linear size $r_0$ of the objects will contribute to the scatter in $D$. Hence such measurements will yield less scattered values of $D$. For $r_0=10h^{-1}$kpc we estimate that $D$ will scatter by $\sim\pm6\%$ at redshift $z=3$. Unfortunately, even this small scatter will cause the derived values of $q_0$ to scatter by as much as $\pm0.4$. The scatter in values of $q_0$ that are derived from angular-diameter distances to parsec-sized objects such as those studied by Kellermann (1993), will be very much larger still.
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|
---
abstract: 'Evidence for orbital modulation of the very high energy (VHE) $\gamma$-ray emission from the high-mass X-ray binary and microquasar LS 5039 has recently been reported by the HESS collaboration. The observed flux modulation was found to go in tandem with a change in the GeV – TeV spectral shape, which may partially be a result of $\gamma\gamma$ absorption in the intense radiation field of the massive companion star. However, it was suggested that $\gamma\gamma$ absorption effects alone can not be the only cause of the observed spectral variability since the flux at $\sim 200$ GeV, which is near the minimum of the expected $\gamma\gamma$ absorption trough, remained essentially unchanged between superior and inferior conjunction of the binary system. In this paper, a detailed parameter study of the $\gamma\gamma$ absorption effects in this system is presented. For a range of plausible locations of the VHE $\gamma$-ray emission region and the allowable range of viewing angles, the de-absorbed, intrinsic VHE $\gamma$-ray spectra and total VHE photon fluxes and luminosities are calculated and compared to luminosity constraints based on Bondi-Hoyle limited wind accretion onto the compact object in LS 5039. Based on these arguments, it is found that (1) it is impossible to choose the viewing angle and location of the VHE emission region in a way that the intrinsic (deabsorbed) fluxes and spectra in superior and inferior conjunction are identical; consequently, the intrinsic VHE luminosities and spectral shapes must be fundamentally different in different orbital phases, (2) if the VHE luminosity is limited by wind accretion from the companion star and the system is viewed at an inclination angle of $i \gtrsim 40^o$, the emission is most likely beamed by a larger Doppler factor than inferred from the dynamics of the large-scale radio outflows, (3) the still poorly constrained viewing angle between the line of sight and the jet axis is most likely substantially smaller than the maximum of $\sim 64^o$ inferred from the lack of eclipses. (4) Consequently, the compact object is more likely to be a black hole rather than a neutron star. (5) There is a limited range of allowed configurations for which the expected VHE neutrino flux would actually anti-correlate with the observed VHE $\gamma$-ray emission. If hadronic models for the $\gamma$-ray production in LS 5039 apply, a solid detection of the expected VHE neutrino flux and its orbital modulation with km$^3$ scale water-Cherenkov neutrino detectors might require the accumulation of data over more than 3 years.'
author:
- Markus Böttcher
title: 'Constraints on the Geometry of the VHE Emission in LS 5039 from Photon-Photon Deabsorption'
---
\[intro\]Introduction
=====================
The recent detections of VHE ($E \gtrsim 250$ GeV) $\gamma$-rays from the high-mass X-ray binary jet sources LS 5039 with the High Energy Stereoscopic System [HESS; @aharonian05] and LS I +63$^o$303 with the Major Atmospheric Gamma-Ray Imaging Cherenkov Telescope [MAGIC @albert06] establish these sources (termed “microquasars” if they are accretion powered) as a new class of $\gamma$-ray emitting sources. These results confirm the earlier tentative identification of LS 5039 with the EGRET source 3EG J1824-1514 [@paredes00] and LSI $61^o303$ with the COS B source 2CG 135+01 [@gregory78; @taylor92] and the EGRET source 3EG J0241+6103 [@kniffen97]. Both of these objects show evidence for variability of the VHE emission, suggesting an association with the orbital period of the binary system. In the case of LS I +63$^o$303, the association with the orbital period is not yet firmly established since the MAGIC observations covered only a few orbital periods, and the orbital period [$P = 26.5$ d; @gregory02] is very close to the siderial period of the moon, which also sets a natural windowing period for VHE observations [@albert06]. In contrast, the HESS observations of LS 5039 provide rather unambiguous evidence for an orbital modulation of both the VHE $\gamma$-ray flux and spectral shape with the orbital period of $P = 3.9$ d [@aharonian06b]. Specifically, [@aharonian06b] found that between inferior conjunction (i.e., the compact object being located in front of the companion star), the VHE $\gamma$-ray spectrum could be well fitted with an exponentially cut-off, hard power-law of the form $\Phi_E \propto E^{-1.85} \, e^{-E/E_0}$ whith a cut-off energy of $E_0 = 8.7$ TeV. In contrast, the VHE spectrum at superior conjunction is well represented by a pure power-law ($\Phi_E \propto E^{-2.53}$) with a much steeper slope, but identical differential flux at $E_{\rm norm} \approx 200$ GeV. The original data from [@aharonian06b], together with these spectral fit functions, are shown in Fig. \[observed\_spectrum\].
A variety of different models for the high-energy emission from high-mass X-ray binaries have been suggested. These range from high-energy processes in neutron star magnetospheres [e.g., @moskalenko93; @moskalenko94; @bednarek97; @bednarek00; @chernyakova06; @dubus06b], via models with the inner regions of microquasar jets being the primary high-energy emission sites [e.g. @romero03; @bp04; @bosch05a; @gupta06; @db06; @gb06] and interactions of microquasar jets with the ISM [@bosch05b], to models involving particle acceleration in shocks produced by colliding stellar winds [e.g., @reimer06]. In addition to involving a variety of different leptonic and hadronic emission processes, these models also imply vastly different locations of the $\gamma$-ray production site with respect to the compact object in LS 5039 and the companion star. However, independent of the emission mechanism responsible for the VHE $\gamma$-rays, the intense radiation field of the high-mass (stellar type O6.5V) companion will lead to $\gamma\gamma$ absorption of VHE $\gamma$-rays in the $\sim 100$ GeV – TeV photon energy range. The characteristic features of the $\gamma\gamma$ absorption of VHE $\gamma$-rays by the companion star light in LS 5039 have been investigated in detail by [@bd05] and [@dubus06a]. It was found that, if the $\gamma$-ray emission originates within a distance of the order of the orbital separation of the binary system ($s \sim 2 \times
10^{12}$ cm), this effect should lead to a pronounced $\gamma\gamma$ absorption trough, in particular near superior conjunction, while $\gamma\gamma$ absorption tends to be almost negligible near inferior conjunction. In LS 5039, the minimum of the absorption trough is expected to be located around $E_{\rm min} \sim
300$ GeV and should shift from higher to lower energies as the orientation of the binary system changes from inferior to superior conjunction.
Most of the relevant parameters of LS 5039, except for the inclination angle $i$ of the line of sight with respect to the normal to the orbital plane, are rather well determined (see §\[parameters\]). This allows for a detailed parameter study of the $\gamma\gamma$ absorption effects, leaving the inclination angle and the distance of the VHE emission site from the compact object as free parameters. As will be shown in §\[cascades\], the effect of electromagnetic cascades will lead to a re-deposition of the absorbed $\gtrsim 100$ GeV luminosity almost entirely at photon energies $E \lesssim
100$ GeV. For that reason, one can easily correct for the $\gamma\gamma$ absorption effect at energies $E \gtrsim 100$ GeV by multiplying the observed fluxes by a factor $e^{\tau_{\gamma\gamma}(E)}$ (where $\tau_{\gamma\gamma}(E)$ is the $\gamma\gamma$ absorption depth along the line of sight) in order to find the intrinsic, deabsorbed VHE spectra for any given choice of $i$ and the height $z_0$ of the VHE $\gamma$-ray emission site above the compact object. Results of this procedure will be presented in §\[deabsorbed\]. In §\[constraints\], these results will then be used to estimate the total flux and luminosity in VHE $\gamma$-rays, which can be compared to limits on the available power under the assumption that the VHE emission is powered by Bondi-Hoyle limited wind accretion onto the compact object. This leads to important constraints on the location of the VHE emision site and the geometry of the system, which will be discussed in § \[summary\].
\[parameters\]Parameters of the LS 5039 System
==============================================
LS 5039 is a high-mass X-ray binary in which a compact object is in orbit around an O6.5V type stellar companion with a mass of $M_{\ast} = 23 \, M_{\odot}$, a bolometric luminosity of $L_{\ast}
= 10^{5.3} \, L_{\odot} \approx 7 \times 10^{38}$ ergs s$^{-1}$ and an effective surface temperature of $T_{\rm eff} = 39,000 \, ^o$K. The mass function of the system is $f(M) = (M_{\rm c.o.} \sin i)^3
/ (M_{\rm c.o.} + M_{\ast})^2 \approx 5 \times 10^{-3} \, M_{\odot}$. The binary orbit has an eccentricity of $e = 0.35$ and an orbital period of $P = 3.9$ d. The inclination angle $i$ is only poorly constrained in the range $13^o \lesssim i \lesssim 64^o$. Under the assumption of co-rotation of the star with the orbital motion, a preferred inclination angle of $i = 25^o$ could be inferred, leading to a compact-object mass of $M_{\rm c.o.} = 3.7^{+1.3}_{-1.0}
\, M_{\odot}$ [@casares05]. However, since there is no clear evidence for co-rotation, the inclination angle is left as a free parameter in this analysis. Within the entire range of allowed values, $13^o \le i \le 64^o$, the mass of the compact object is substantially smaller than the mass of the companion, so that the orbit can very well be approximated by a stationary companion star, orbited by the compact object for our purposes. The orbit has a semimajor axis of length $a = 2.3 \times 10^{12}$ cm, and the projection of the line of sight onto the orbital plane forms an angle of $\approx 45^o$ with the semimajor axis [see, e.g., Fig. 4 of @aharonian06b]. Consequently, the orbital separation at superior conjunction is found to be $s_{\rm s.c.} = 1.6 \times 10^{12}$ cm, while at inferior conjunction, it is $s_{\rm i.c.} = 2.7 \times 10^{12}$ cm.
The mass outflow rate in the stellar wind of the companion has been determined as $\dot M_{\rm wind} \approx 10^{-6.3} M_{\odot}$/yr, and the terminal wind speed is $v_{\infty} \approx 2500$ km s$^{-1}$ [@mg02]. EVN and MERLIN observations of the radio jets of LS 5039 suggest a mildly relativistic flow speed of $\beta \sim 0.2$ on the length scale of several hundred AU [@paredes02]. This would correspond to a bulk Lorentz factor of the flow of $\Gamma \approx 1.02$. However, it is plausible to assume that near the base of the jet, where the VHE $\gamma$-ray emission may arise (according to some of the currently most actively discussed modeles), the flow may have a substantially higher speed. Therefore, we will also consider bulk flow speeds of $\Gamma \sim 2$, more typical of the jet speeds of other Galactic microquasar jets.
For the analysis in this paper, we use a source distance of $d = 2.5$ kpc, corresponding to $4 \pi d^2 = 7.1 \times 10^{44}$ cm$^2$ [@casares05].
Inferred compact object masses and Doppler boosting factors $D =
(\Gamma [1 - \beta \cos i])^{-1}$ for representative values of $i = 20^o$, $40^o$, and $60^o$ and $\Gamma = 1.02$ and $\Gamma = 2$ are listed in Table \[parameter\_table\].
\[cascades\]The Role of Electromagnetic Cascades
================================================
The absorption of VHE $\gamma$-ray photons photons will lead to the production of relativistic electron-positron pairs, which will lose energy via synchrotron radiation and inverse-Compton scattering on starlight photons, and initiate synchrotron or inverse-Compton supported electromagnetic cascades. For photon energies of $E_{\gamma} \gtrsim
100$ GeV, the produced pairs will have Lorentz factors of $\gamma
\gtrsim 10^5$. Given the surface temperature of the companion star of $T_{\rm eff} = 39,000$ K, starlight photons have a characteristic photon energy of $\epsilon_{\ast} \equiv h \nu_{\ast} / (m_e c^2)
\sim 7 \times 10^{-6}$, electrons and positrons with energies of $\gamma \gtrsim 1/\epsilon_{\ast} \sim 1.5 \times 10^5$ will interact with these star light photons in the Klein-Nishina limit and, thus, very inefficiently. $\gamma$-rays produced through upscattering of star light photons by secondary electrons and positrons in the Thomson regime will have energies of $E_{\rm IC} \lesssim E_{\rm IC}^{\rm max}
\equiv m_e c^2 / \epsilon_{\ast} \sim 75$ GeV.
In addition to Compton scattering, secondary electron-positron pairs will suffer synchrotron losses. Magnetic fields even near the base of microquasar jets are unlikely to exceed $B \sim 10^3$ G. Consequently, along essentially the entire trajectory of VHE photons one may safely assume $B \equiv 10^3 \, B_3$ G with $B_3 \lesssim 1$. Secondary electrons and positrons traveling through such magnetic fields will produce synchrotron photons of characteristic energies $E_{\rm sy} \sim 1.2
\times B_3 \, \gamma_6^2$ GeV, where $\gamma_6 = \gamma / 10^6$ parametrizes the secondary electron’s/positron’s energy. Thus, even for extreme magnetic-field values of $B = 10^3$ G, synchrotron photons at energies $\gtrsim 100$ GeV could only be produced by secondary pairs with $\gamma \gtrsim 10^7$, resulting from primary $\gamma$-ray photons of $E_{\gamma} \gtrsim 10$ TeV, where only a negligible portion of the total luminosity from LS 5039 is expected to be liberated. Consequently, electromagnetic cascades initiated by the secondary electrons and positrons from $\gamma\gamma$ absorption in the stellar photon field will re-emit the absorbed VHE $\gamma$-ray photon energy essentially entirely at photon energies $\lesssim 100$ GeV. This conclusion is fully consistent with the more detailed consideration of the pair cascades resulting from $\gamma\gamma$ absorption of the VHE emission from LS 5039 by @aharonian06a.
As demonstrated in [@bd05], $\gamma Z$ absorption in the stellar wind of the companion as well as $\gamma\gamma$ absorption on star light photons reprocessed in the stellar wind are negligible compared to the direct $\gamma\gamma$ absorption effect for LS 5039.
Consequently, the only relevant effect modifying the VHE $\gamma$-ray spectrum of LS 5039 at energies above $\sim 100$ GeV between the emission site and the observer on Earth is $\gamma\gamma$ absorption by direct star light photons. This effect can easily be corrected for by multiplying the observed spectra by a factor $e^{\tau_{\gamma\gamma}(E)}$, where $\tau_{\gamma\gamma}(E)$ is the $\gamma\gamma$ absorption depth along the line of sight.
\[deabsorbed\]Deabsorbed VHE $\gamma$-ray spectra
=================================================
In this section, we present a parameter study of the deabsorbed photon spectra and integrated VHE $\gamma$-ray fluxes for representative values of the viewing angle of $i = 20^o$, $40^o$, and $60^o$ and a range of locations of the emission site, characterized by a height $z_0$ above the compact object in the direction perpendicular to the orbital plane. The absorption depth $\tau_{\gamma\gamma} (E)$ along the line of sight is evaluated as described in detail in [@bd05]. The observed spectra are represented by the best-fit functional forms quoted in the introduction.
Figures \[i20\] — \[i60\] show the inferred intrinsic, deabsorbed VHE $\gamma$-ray spectra from LS 5039 for a range of $10^{12} \,
{\rm cm} \le z_0 \le 1.5 \times 10^{13}$ cm. At larger distances from the compact object, $\gamma\gamma$ absorption due to the stellar radiation field becomes negligible for all inclination angles. Furthermore, models which assume a distance greatly in excess of the characteristic orbital separation (i.e., $z_0 \gtrsim 10^{13}$ cm) may be hard to reconcile with the periodic modulation of the emission on the orbital time scale since the $\gamma$-ray production site would be distributed over a large volume, with primary energy input episodes from different orbital phases overlapping and thus smearing out the original orbital modulation. As the height $z_0$ approaches values of the order of the characteristic orbital separation, $s \sim 2 \times 10^{12}$ cm, the intrinsic spectra would have to exhibit a significant excess towards the threshold of the HESS observations at $E_{\rm thr} \sim 200$ GeV in order to compensate for the $\gamma\gamma$ absorption trough with its extremum around $\sim 300$ GeV [@bd05]. At even smaller distances from the compact object, $z_0 \ll 10^{12}$ cm, the absorption features would again become essentially independent of $z_0$ since the overall geometry would not change significantly anymore with a change of $z_0$.
A first, important conclusion from Figures \[i20\] – \[i60\] is that there is no combination of $i$ and $z_0$ for which the de-absorbed VHE $\gamma$-ray spectra in the inferior and superior conjunction could be identical. Thus, the intrinsic VHE $\gamma$-ray spectra must be fundamentally different in the different orbital phases corresponding to superior conjunction (near periastron) and inferior conjunction (closer to apastron).
Second, there is a large range of instances in which the deabsorbed, differential photon fluxes at 200 GeV $\lesssim E \lesssim$ 10 TeV during superior conjunction would have to be substantially higher than during inferior conjunction, opposite to the observed trend. This could have interesting consequences for strategies of searches for high-energy neutrinos. To a first approximation, the intrinsic, unabsorbed VHE $\gamma$-ray flux is roughly equal to the expected high-energy neutrino flux [see, e.g., @lipari06]. Therefore, our results suggest that phases with lower observed VHE $\gamma$-ray fluxes from LS 5039 may actually coincide with phases of larger neutrino fluxes. This is confirmed by a plot of the integrated 0.2 – 10 TeV photon fluxes as a function of height of the emission region as shown in Figure \[flux02\]. An anti-correlation of the VHE neutrino and $\gamma$-ray fluxes would require emission region heights of $z_0 \lesssim 2.5 \times 10^{12}$ cm ($i = 20^o$), $4 \times 10^{12}$ cm ($i = 40^o$), and $7 \times 10^{12}$ cm ($i = 60^o$), respectively. As we will see in the next section, for $i = 60^o$, these configurations can be ruled out because of constraints on the available power from wind accretion onto the compact object.
The possibility of a neutrino-to-photon flux ratio largely exceeding one has also recently been discussed for the case of LSI +61$^0$303 by @th06. Those authors also compared the existing AMANDA upper limits to the MAGIC VHE $\gamma$-ray fluxes of that source. They find that a neutrino-to-photon flux ratio up to $\sim 10$ for a $\gamma\gamma$ absorption modulated photon signal would still be consistent with the upper limits on the VHE neutrino flux from that source.
In order to assess the observational prospects of detecting VHE neutrinos with the new generation of km$^3$ neutrino detectors, such as ANTARES, NEMO, or KM3Net, one can estimate the expected neutrino flux assuming a characteristic ratio of produced neutrinos to unabsorbed VHE photons at the source of $\sim 1$ [@lipari06]. Under this assumption, the resulting neutrino fluxes should be comparable to the deabsorbed photon fluxes plotted in Figures \[i20\] – \[i60\]. As a representative example of the sensitivity of km$^3$ water-Cherenkov neutrino detectors, those figures also contain the anticipated sensitivity limits of the NEMO detector for a steady point source with a power-law of neutrino number spectral index $2.5$ for 1 and 3 years of data taking, respectively [@distefano06]. Given the likely orbital modulation of the neutrino flux of LS 5039, our results indicate that neutrino data accumulated over substantially longer than 3 years might be required in order to obtain a firm detection of the neutrino signal from LS 5039 and its orbital modulation.
\[constraints\]Constraints from Accretion Power Limits
======================================================
The inferred intrinsic (deabsorbed) VHE $\gamma$-ray fluxes calculated in the previous section imply apparent isotropic 0.1 – 10 TeV luminosities in the range of $\sim 2 \times 10^{34}$ – $7 \times 10^{34}$ ergs/s ($i = 20^o$), $\sim 1.3 \times 10^{34}$ – $7 \times 10^{35}$ ergs/s ($i = 40^o$), and $\sim 1.2 \times 10^{34}$ – $\gg 10^{37}$ ergs/s ($i = 60^o$), respectively, for emission regions located at $z_0
\ge 10^{12}$ cm (see Figures \[luminosities1\] and \[luminosities2\]). These should be confronted with constraints on the available power from wind accretion from the stellar companion onto the compact object.
For this purpose, we assume that for a given directed wind velocity of $v_{\rm wind} \approx 2.5 \times 10^8$ cm/s, all matter with $(1/2) v_{\rm wind}^2
< GM_{\rm c.o.}/R_{\rm BH}$ will be accreted onto the compact object. An absolute maximum on the available power is then set by $L_{\rm max} \approx
(1/12) \, \dot M_{\rm wind} \, c^2 \, (R_{\rm BH}^2 / [4 s^2])$. We furthermore allow for Doppler boosting of this accretion power along the microquasar jet with the Doppler boosting factors $D$ listed in Table \[parameter\_table\]. This yields an available power corresponding to an inferred, apparent isotropic luminosity of
$$L_{\rm max}^{\rm iso} \approx 2.8 \times 10^{33} \, \left( {M_{\rm c.o.}
\over M_{\odot}} \right)^2 \, D^4 \, \left( {s \over 2 \times 10^{12} \,
{\rm cm}} \right)^{-2} \, \left( {v_{\rm wind} \over 2.5 \times 10^8
\, {\rm cm/s}} \right)^{-4} \; {\rm ergs / s}.
\label{Lmax}$$
The resulting luminosity limits are also included in Table \[parameter\_table\] and indicateded by the horizontal lines in Figures \[luminosities1\] and \[luminosities2\] for $\Gamma = 1.02$ and $\Gamma = 2$, respectively. Allowed configurations are those for which the inferred, apparent isotropic 0.1 – 10 TeV luminosity is substantially below the respective luminosity limit.
Additional restrictions come from the inferred luminosity of the EGRET source 3EG J1824-1514 whose 90 % confidence contour includes the location of LS 5039 [@aharonian05]. The observed $> 100$ MeV $\gamma$-ray flux from this source corresponds to an inferred isotropic luminosity of $L_{\rm EGRET} \sim 7 \times
10^{34}$ ergs s$^{-1}$. This level is also indicated in Figures \[luminosities1\] and \[luminosities2\]. However, the association of the EGRET source with LS 5039 is uncertain since the 90 % confidence contour also contains the pulsar PSR B1822-14 as a possible counterpart. Thus, even if the available power according to eq. \[Lmax\] is lower than $L_{\rm EGRET}$, this would not provide a significant problem since the EGRET flux could be provided by the nearby pulsar. However, the EGRET flux does provide an upper limit on the $\sim 30$ MeV – 30 GeV flux from LS 5039. As shown in @aharonian06a, the bulk of the VHE $\gamma$-ray flux that is absorbed by $\gamma\gamma$ absorption on companion star photons and initiates pair cascades, will be re-deposited in $\gamma$-rays in the EGRET energy range. Thus, any scenario that requires an unabsorbed source luminosity in excess of the EGRET luminosity would be very problematic. The resulting limits inferred from both eq. \[Lmax\] and the EGRET flux on the height $z_0$ are summarized in Table \[z\_limits\].
The figure indicates that for $i = 20^o$, virtually all configurations are allowed, even if the emission is essentially unboosted ($\Gamma = 1.02$) and originates very close to the compact object.
For $i = 40^o$ at the time of superior conjunction, one can set a limit of $z_0 \gtrsim 1.7 \times 10^{12}$ cm. For this viewing angle at inferior conjunction, the limit implied for $\Gamma = 1.02$ is always below the inferred value, indicating that this configuration is ruled out. This implies that, if $i \sim 40^o$, the emission at the time of inferior conjunction must be substantially Doppler enhanced.
The situation is even more extreme for $i = 60^o$. Under this viewing angle, only the $\Gamma = 1.02$ scenario leads to an allowed model and requires $z_0
\gtrsim 5 \times 10^{12}$ cm. Since there is no allowed scenario to produce the observed flux at inferior conjunction, one can conclude that a large inclination angle of $i \sim 60^o$ may be ruled out.
Note, however, that there are alternative models for the very-high-energy emission from X-ray binaries in which the source of power for the high-energy emission lies in the rotational energy of the compact object [in that case, most plausibly a neutron star, see, e.g. @chernyakova06]. The constraints discussed above will obviously not apply to such models. Instead, in the case of a rotation-powered pulsar wind/jet, one can estimate the total available power as
$$L_{\rm rot} \sim {8 \over 5} \pi^2 \, M \, R^2 \, {\dot P \over P^3} \sim
4.4 \cdot 10^{37} \, {\dot P_{-15} \over P_{-2}^3} \; {\rm ergs \; s}^{-1},
\label{L_rotation}$$
where a $1.4 \, M_{\odot}$ neutron star with $R = 10$ km is assumed and the spin-down rate and spin period are parametrized as $\dot P = 10^{-15} \,
\dot P_{-15}$ and $P = 10 \, P_{-2}$ ms, respectively. Consequently, the power requirements of even the extreme scenarios for $i = 60^o$, as discussed above, could, in principle, be met by a rotation-powered pulsar. However, such a luminosity could only be sustained over a spin-down time of
$$t_{\rm sd} \sim 3 \times 10^5 \, {P_{-2} \over \dot P_{-15}} \; {\rm years}
\label{t_spindown}$$
and should therefore be rare.
\[summary\]Summary and Discussion
=================================
In this paper, a detailed study of the intrinsic VHE $\gamma$-ray emission from the Galactic microquasar LS 5039 after correction for $\gamma\gamma$ absorption by star light photons from the massive companion star is presented. This system had shown evidence for an orbital modulation of the VHE $\gamma$-ray flux and spectral shape. A range of observationally allowed inclination angles, $13^o \le i \le 64^o$ (specifically, $i = 20^o$, $i = 40^o$, and $i = 60^o$) as well as plausible distances $z_0$ of the VHE $\gamma$-ray emission region above the compact object ($10^{12} \, {\rm cm} \le z_0 \le 1.5 \times
10^{13}$ cm) was explored. Deabsorbed, intrinsic VHE $\gamma$-ray spectra as well as integrated fluxes and inferred, apparent isotropic luminosities were calculated and contrasted with constraints from the available power from wind accretion from the massive companion onto the compact object. The main results are:
- [It is impossible to choose the viewing angle and location of the VHE emission region in a way that the intrinsic (deabsorbed) fluxes and spectra in superior and inferior conjunction are identical within the range of values of $i$ and $z_0$ considered here. For values of $z_0$ much smaller and much larger than the characteristic orbital separation ($s \sim 2 \times 10^{12}$ cm), the absorption features would become virtually independent of $z_0$. Furthermore, models assuming a VHE $\gamma$-ray emission site at a distance greatly in excess of the orbital separation might be difficult to reconcile with the orbital modulation of the VHE emission. Consequently, the intrinsic VHE luminosities and spectral shapes must be fundamentally different in different orbital phases.]{}
- [It was found that the luminosity constraints for an inclination angle of $i = 60^o$ at inferior conjunction could not be satisfied at all, and for $i = 40^o$, there is no allowed configuration in agreement with the luminosity constraint for $\Gamma = 1.02$. From this, it may be concluded that, if the VHE luminosity is limited by wind accretion from the companion star and the system is viewed at an inclination angle of $i \gtrsim 40^o$, the emission is most likely beamed by a larger Doppler factor than inferred from the dynamics of the large-scale radio outflows on scales of several hundred AU.]{}
- [Since it was found to be impossible to satisfy the luminosity constraint for inferior conjunction at a viewing angle of $i = 60^o$, one can constrain the viewing angle to values substantially smaller than the maximum of $\sim 64^o$ inferred from the lack of eclipses.]{}
- [The previous two points as well as the fact that the luminosity limits can easily be satisfied for $i = 20^o$, indicate that a rather small inclination angle $i \sim 20^o$ may be preferred. Thus, our results confirm the conjecture of [@casares05] that the compact object might be a black hole rather than a neutron star.]{}
- [Under the assumption of a photon-to-neutrino ratio of $\sim 1$ (before $\gamma\gamma$ absorption), the detection of the neutrino flux from LS 5039 and its orbital modulation might require the accumulation of data over more than 3 years with km$^3$ scale neutrino detectors like ANTARES, NEMO, or KM3Net.]{}
- [Comparing the ranges of allowed configurations from Table \[z\_limits\] to the plot of intrinsic VHE fluxes in Figure \[flux02\], one can see that there is a limited range of allowed configurations for which the expected VHE neutrino flux would actually anti-correlate with the observed VHE $\gamma$-ray emission. Specifically, for a preferred viewing angle of $i \sim 20^o$, models in which the emission originates within $z_0 \lesssim
2.5 \times 10^{12}$ cm would predict that the VHE neutrino flux at superior conjunction is larger than at inferior conjunction, opposite to the orbital modulation trend seen in VHE $\gamma$-ray photons. Thus, strategies for the identification of high-energy neutrinos from microquasars based on a positive correlation with observed VHE fluxes may fail if models with $z_0 \lesssim
2.5 \times 10^{12}$ cm apply.]{}
- [The luminosity limitations discussed above could, in principle, be overcome by a rotation-powered pulsar. However, this would require a rather short spin-down time scale of $t_{\rm sd} \lesssim 10^6$ yr. Consequently, such objects should be rare.]{}
The author thanks F. Aharonian and V. Bosch-Ramon for stimulating discussions and hospitality during a visit at the Max-Planck-Institute for Nuclear Physics, and M. De Naurois for providing the HESS data points. I also thank the referee for very insightful and constructive comments, which have greatly helped to improve this manuscript. This work was partially supported by NASA INGEGRAL Theory grant no. NNG 05GP69G and a scholarship at the Max-Planck Institute for Nuclear Physics in Heidelberg.
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![VHE $\gamma$-ray spectra of LS 5039 around superior (blue triangles) and inferior (red circles) conjunction [from @aharonian06b]. The curves indicate the best-fit straight power-law (superior) and exponentially cut-off power-law (inferior) representations of the spectra, on which the analysis in this paper is based.[]{data-label="observed_spectrum"}](fig1_color.eps){width="14cm"}
![Deabsorbed VHE $\gamma$-ray spectra of LS 5039 for $i = 20^o$ and a range of heights of $z_0$ of the emission region above the compact object. The vertical dashed line indicates the energy threshold of the HESS observations. The flat power-laws correspond to the anticipated neutrino detection sensitivities of the planned NEMO km$^3$ detector for a steady point source with an underlying power-law spectrum of neutrino number spectral index 2.5 after 1 and 3 years of data taking, respectively [@distefano06]. These are relevant for comparison to the deabsorbed photon fluxes for a characteristic ratio of unabsorbed photons to neutrinos of $\sim 1$.[]{data-label="i20"}](fig2_color.eps){width="14cm"}
![Same as Fig. \[i20\], but for $i = 40^o$.[]{data-label="i40"}](fig3_color.eps){width="14cm"}
![Same as Fig. \[i20\], but for $i = 60^o$.[]{data-label="i60"}](fig4_color.eps){width="14cm"}
![Deabsorbed, integrated 0.2 – 10 TeV $\gamma$-ray fluxes of LS 5039 as a function of height $z_0$ of the emission region above the compact object. These numbers would also approximately equal to the expected neutrino fluxes in the same energy range.[]{data-label="flux02"}](fig5_color.eps){width="14cm"}
![Inferred apparent isotropic luminosities for LS 5039 (thick curves), compared to the luminosity limits from Bondi-Hoyle limited wind accretion (horizontal lines), assuming Doppler boosting of the emission corresponding to $\Gamma = 1.02$. Allowed configurations are those for which the inferred luminosities are substantially below the respective luminosity limits.[]{data-label="luminosities1"}](fig6_color.eps){width="14cm"}
![Same as Fig. 6, but for $\Gamma = 2$.[]{data-label="luminosities2"}](fig7_color.eps){width="14cm"}
[ccccccc]{} 20 & i.c. & 4.5 & 1.21 & 2.68 & $6.7 \times 10^{34}$ & $1.6 \times 10^{36}$\
20 & s.c. & 4.5 & 1.21 & 2.68 & $1.9 \times 10^{35}$ & $4.6 \times 10^{36}$\
40 & i.c. & 2.3 & 1.13 & 1.49 & $1.3 \times 10^{34}$ & $4.0 \times 10^{34}$\
40 & s.c. & 2.3 & 1.13 & 1.49 & $3.8 \times 10^{34}$ & $1.1 \times 10^{35}$\
60 & i.c. & 1.6 & 1.09 & 0.88 & $5.6 \times 10^{33}$ & $2.4 \times 10^{33}$\
60 & s.c. & 1.6 & 1.09 & 0.88 & $1.6 \times 10^{34}$ & $6.7 \times 10^{33}$\
\[parameter\_table\]
[cccc]{} 20 & i.c. & $\ll 1$ & $\ll 1$\
20 & s.c. & $\ll 1$ & $\ll 1$\
40 & i.c. & forbidden & $\ll 1$\
40 & s.c. & $2.0$ & $1.7$\
60 & i.c. & forbidden & forbidden\
60 & s.c. & 5 & forbidden\
\[z\_limits\]
|
---
abstract: |
We use a non-Markovian approach to study the decoherence dynamics of a qubit (with and without measurement) in either a low- or high-frequency bath modeling the qubit environment. This approach is based on a unitary transformation and does not require the rotating-wave approximation. We show that without measurement, for low-frequency noise, the bath shifts the qubit energy towards higher energies (blue shift), while the ordinary high-frequency cutoff Ohmic bath shifts the qubit energy towards lower energies (red shift). In order to preserve the coherence of the qubit, we also investigate the dynamic of qubit with measurement (quantum Zeno effect) in this two cases: low- and high-frequency baths. For very frequent projective measurements, the low-frequency bath gives rise to the quantum anti-Zeno effect on the qubit. The quantum Zeno effect only occurs in the high-frequency cutoff Ohmic bath, after considering counter-rotating terms. For a high-frequency environment, the decay rate should be faster (without measurements) or slower (with frequent measurements, in the Zeno regime), compared to the low-frequency bath case. The experimental implementation of our results here could distinguish the type of bath (either a low- or high-frequency one) and protect the coherence of the qubit by modulating the dominant frequency of its environment.\
author:
- 'Xiufeng Cao[^1]'
- 'J. Q. You'
- 'H. Zheng'
- Franco Nori
title: 'Dynamics and quantum Zeno effect for a qubit in either a low- or high-frequency bath beyond the rotating-wave approximation'
---
Introduction
============
There is considerable interest in low-frequency noise (see, e.g., the review in Ref. \[\] and references therein), because this type of noise limits the coherence of qubits based on superconducting devices such as flux or phase qubits [@phys-today58-11-42; @add6]. Also, the dephasing of flux qubits is due to low-frequency flux noise with intensity comparable to the one measured in dc SQUIDs (e.g., Refs. \[\]). Under certain conditions, noise can enhance the coherence of superconduting flux qubit [@prb80]. There are also several models (e.g., Refs. \[\]) for the microscopic origin of low-frequency flux noise in Josephson circuits. Therefore, the study of low-frequency noise has becomes very important for superconductor qubits [@add1]. Moreover, the quantum Zeno effect has been proposed as a strategy to protect coherence [@A10], entanglement [@prl-100-090503; @A12], and to control thermodynamic evolutions [nature-452-724]{}. The quantum Zeno effect and anti-Zeno effect have been widely discussed [@Nature-405-546; @A11]. So it is an interesting topic to investigate the quantum Zeno effect of a qubit coupled to a low-frequency bath.
The description of low-frequency noise [@add1] (such as 1/f noise) is complicated by the presence of long-time correlations in the fluctuating environment, which prohibit the use of the Markovian approximation. In addition, the rotating wave approximation (RWA) is also unavailable in an environment with multiple modes. In the case of a time-depend external field for a qubit coupled to a thermal bath, Kofman and Kurizki developed a theory [@addKofman] which considers the counter-rotating terms of the fast modulation field through the negative-frequency part $G(\omega )$ ($\omega
<0 $) in the bath-correlation function spectrum. The dynamics in two significant models (the spin-boson model with Ohmic bath, and a qubit coupled to a bath of two level fluctuators) have been calculated within a rigorous Born approximation and without the Markovian approximation [prb-71-035318,prb-79-125317]{}. Refs. \[\] and \[\] describe the structure of the solutions in the complex plane with branch cuts and poles.
Here we present an analytical approach, based on a unitary transformation. We use neither the Markovian approximation nor the RWA in order to discuss the transient dynamics of a qubit coupled to its environment. This method has already been used [@add2] to study the decoherence of the Ohmic bath, sub-Ohmic bath, and structured bath. In this paper, we calculate the coherence dynamics of the qubit (with and without measurement) respectively in two kinds of baths. Besides producing an energy shift, the environment can change the decay rate of the qubit. To preserve the coherence, we also investigate the decay rate of the qubit subject to the quantum Zeno effect. The low-frequency noise uses a Lorentzian-type spectrum with the peak of the spectrum in the low-energy region and for the high-frequency noise we choose the ordinary Ohmic bath with Drude cutoff.
Our results show that for *low*-frequency noise, the qubit energy increases (blue shift) and an anti-Zeno effect takes place. For a *high*-frequency cutoff Ohmic bath, the qubit energy decreases (red shift) and the Zeno effect dominates. For a high-frequency environment, the decay rate should be faster (without measurements) or slower (with frequent measurements, in the Zeno regime), compared to the low-frequency bath case. The experimental implementation of our results here could distinguish the type of bath (either a low- or high-frequency one) and protect the coherence of the qubit by modulating the dominant frequency of its environment. The coherence dynamics without measurement also indicates that the coherence time of the low-frequency noise is longer than the high-frequency noise, which demonstrates the powerful temporal memory of the low-frequency bath and stem from non-Markov process of qubit-bath interaction within the bath memory time[@addnjp].
Method Beyond the RWA Based on a Unitary Transformation
=======================================================
We describe a qubit coupled to a boson bath, modeling the environment, by the Hamiltonian $$H=-\frac{1}{2}\Delta \sigma _{z}+\frac{1}{2}\sum_{k}g_{k}(a_{k}^{+}+a_{k})\sigma _{x}+\sum_{k}\omega _{k}a_{k}^{+}a_{k}.$$The boson bath and the spin-$\frac{1}{2}$ (fluctuators) bath will have the same dissipative effect on the qubit at zero temperature, $T=0,$ if both baths have the same correlation function [@U-weiss]. However, for finite temperatures, the spin bath has a smaller effect on the qubit because of the likely saturation of the populations in the spin bath. Here, for simplicity, we only consider the case of zero temperature. That is, our results are also applicable to a fluctuator-bath at zero temperature. Our approach is based on a unitary transformation and can be used for different types of environmental baths. Below we give detailed derivations for the low-frequency noise case.
The spectral density $J(\omega )$ of the environment considered here is given by $$J(\omega )=\sum_{k}g_{k}^{2}\delta (\omega -\omega _{k})=\frac{2\alpha
\omega }{\omega ^{2}+\lambda ^{2}}, \label{E2}$$where $\lambda $ is an energy lower than the qubit two-energy spacing $\Delta $ and $\alpha $ describes the coupling strength between the qubit and the environment. Choosing $\Delta $ as the energy unit, $\alpha /\Delta ^{2}$ is a dimensionless coupling strength. When $\omega \geq \lambda $, $J(\omega
)\sim 1/\omega $, corresponding to a $1/f$ noise.
We apply a canonical transformation to the Hamiltonian $H$: $H^{\prime
}=\exp \left[ S\right] H\exp \left[ -S\right] .$ Further explanations on the validity of the transformation can be found in the Appendix. Note that $\sum_{k}g_{k}^{2}\xi _{k}{}^{2}/(2\omega _{k}^{2})>0$ in Eq. (\[E9\]), so $0<\exp \left[ -\sum_{k}g_{k}^{2}\xi _{k}{}^{2}/\left( 2\omega
_{k}^{2}\right) \right] <1.$ Then the solution of $\eta $ will be in the region from $0$ to $1$. Actually, the existence and uniqueness of the solution of $\eta $ in Eq. (\[E9\]) can be used as a criterion for the validity of our method. The parameter $\eta $ can be regarded as a renormalization factor of the energy spacing $\Delta $ and is calculated as $$\eta =\exp \left\{ \alpha \frac{\pi \lambda \eta \;\Delta +\left[ -\lambda
^{2}-\eta ^{2}\Delta ^{2}+(\lambda ^{2}-\ \eta ^{2}\Delta ^{2})\log
\left\vert \frac{\lambda }{\eta \;\Delta }\right\vert \right] }{(\lambda
^{2}+\eta ^{2})^{2}}\right\} .$$Obviously, $\eta $ is determined self-consistently by the above equation.
Thus, the effective transformed Hamiltonian can be derived as $$H^{^{\prime }}\approx -\frac{1}{2}\eta \;\Delta \sigma _{z}+\sum_{k}\omega
_{k}a_{k}^{+}a_{k}+\sum_{k}V_{k}(a_{k}^{+}\sigma _{-}+a_{k}\sigma _{+}).
\label{E11}$$where$$V_{k}=\eta \;\Delta \frac{g_{k}\xi _{k}}{\omega _{k}}.$$Comparing $H^{\prime }$ in Eq. (\[E11\]) with the ordinary Hamiltonian in the rotating-wave approximation (RWA): $$H_{\mathrm{RWA}}=-\frac{1}{2}\Delta \sigma _{z}+\sum_{k}\omega
_{k}a_{k}^{+}a_{k}+\sum_{k}\frac{g_{k}}{2}(a_{k}^{+}\sigma _{-}+a_{k}\sigma
_{+}), \label{E13}$$one can see that the unitary transformation plays the role of renormalizing two parameters in the Hamiltonian, i.e., the energy spacing $\Delta $ is renormalized: $$\Delta \longrightarrow \eta \;\Delta ;$$and the coupling strength $g_{k}/2$ between the qubit and the bath is renormalized: $$\frac{g_{k}}{2}\longrightarrow \left( \frac{2\eta \;\Delta }{\omega
_{k}+\eta \;\Delta }\right) \frac{g_{k}}{2}.$$Below we study the decoherence dynamics of the qubit and the quantum Zeno effect using the transformed Hamiltonian $H^{\prime }$.
Non-measurement decoherence dynamics
------------------------------------
We diagonalize the transformed Hamiltonian $H^{\prime }$ in the ground state $\left\vert g\right\rangle =\left\vert \uparrow \right\rangle \left\vert
0_{k}\right\rangle $ and lowest excited states, $\left\vert \downarrow
\right\rangle \left\vert 0_{k}\right\rangle $ and $\left\vert \uparrow
\right\rangle \left\vert 1_{k}\right\rangle ,$ as $$H^{^{\prime }}=-\frac{1}{2}\eta \;\Delta \left\vert g\right\rangle
\left\langle g\right\vert +\sum_{E}E\left\vert E\right\rangle \left\langle
E\right\vert ,$$where $\left\vert \uparrow \right\rangle $ and $\left\vert \downarrow
\right\rangle $ are the eigenstates of $\sigma _{z}$, i.e., $\sigma
_{z}\left\vert \uparrow \right\rangle =\left\vert \uparrow \right\rangle ,$ $\sigma _{z}\left\vert \downarrow \right\rangle =-\left\vert \downarrow
\right\rangle $ and $\left\vert n_{k}\right\rangle $ denotes the state with $n$ bath excitations for mode $k$. The state $\left\vert E\right\rangle $ is $$\left\vert E\right\rangle =x(E)\left\vert \downarrow \right\rangle
\left\vert 0_{k}\right\rangle +\sum_{k}y_{k}(E)\left\vert \uparrow
\right\rangle \left\vert 1_{k}\right\rangle ,$$with $x(E)=\left[ 1+\sum_{k}\frac{V_{k}^{2}}{(E+\eta \Delta /2-\omega
_{k})^{2}}\right] ^{-1/2}$ and $y_{k}(E)=\frac{V_{k}}{E+\eta \Delta
/2-\omega _{k}}x(E).$
Here we calculate the dynamical quantity $\left\langle \sigma
_{x}(t)\right\rangle ,$ which is the analog of the population inversion $\left\langle \sigma _{z}(t)\right\rangle $ in the spin-boson model [add3]{}. Since the coupling to the environment will be always present in essentially all physically relevant situations, a natural ground state is given by the dressed state of the two-level qubit and bath. Therefore, the ground state of $H$ is $\exp
[-S]\left\vert \uparrow \right\rangle \left\vert 0_{k}\right\rangle ,$ under counter-rotating terms, and the corresponding ground-state energy is $-\eta
\;\Delta /2,$ then the ground state of $H^{^{\prime }}$ becomes $\left\vert
\uparrow \right\rangle \left\vert 0_{k}\right\rangle $ with the identical ground-state energy $-\eta \;\Delta /2$. We now prepare the initial state, which is also a dressed state of the qubit and bath, from the ground state as$$\left\vert \psi (0)\right\rangle =\frac{\left( 1+\sigma _{x}\right) }{\sqrt{2}}\exp [-S]\left\vert \uparrow \right\rangle \left\vert 0_{k}\right\rangle .$$Then the initial state in the *transformed Hamiltonian* becomes $\left\vert \psi (0)\right\rangle ^{^{\prime }}=(\left\vert \uparrow
\right\rangle +\left\vert \downarrow \right\rangle )\left\vert
0_{k}\right\rangle /\sqrt{2}$, which is the eigenstate of $\sigma _{x}$. Starting from this initial state, we obtain [@EPJB-2004-H-Zheng]$$\begin{aligned}
\left\langle \sigma _{x}(t)\right\rangle &=&\mathrm{Tr}_{B}\left\langle \psi
(t)\left\vert \sigma _{x}\right\vert \psi (t)\right\rangle \notag \\
&=&\mathrm{Tr}_{B}\left\langle \psi (0)\left\vert \exp [iHt]\sigma _{x}\exp
[-iHt]\right\vert \psi (0)\right\rangle \notag \\
&=&\frac{1}{2}\sum_{E}x(E)^{2}\exp \left[ -i\left( E+\frac{\eta \;\Delta }{2}\right) t\right] +\frac{1}{2}\sum_{E}x(E)^{2}\exp \left[ i\left( E+\frac{\eta \;\Delta }{2}\right) t\right] \notag \\
&=&\frac{1}{4\pi i}\int_{\infty }^{-\infty }dE^{^{\prime }}\exp \left[
-iE^{^{\prime }}t\right] \left( E^{^{\prime }}-\eta \;\Delta -\sum_{k}\frac{V_{k}^{2}}{E^{^{\prime }}+i0^{+}-\omega _{k}}\right) ^{-1} \notag \\
&&+\frac{1}{4\pi i}\int_{-\infty }^{\infty }dE^{^{\prime }}\exp \left[
iE^{^{\prime }}t\right] \left( E^{^{\prime }}-\eta \;\Delta -\sum_{k}\frac{V_{k}^{2}}{E^{^{\prime }}-i0^{+}-\omega _{k}}\right) ^{-1} \notag \\
&=&\mathrm{Re}\left[ \frac{1}{2\pi i}\int_{\infty }^{-\infty }\frac{\exp \left[ -iE^{^{\prime }}t\right] }{E^{^{\prime }}-\eta \;\Delta -\sum_{k}\frac{V_{k}^{2}}{E^{^{\prime }}+i0^{+}-\omega _{k}}}dE^{^{\prime }}\right] .\end{aligned}$$Here we denote the real and imaginary parts of $\sum_{k}V_{k}^{2}/\left(
\omega -\omega _{k}\pm i0^{+}\right) $ as $R(\omega )$ and $\mp \Gamma
(\omega )$, respectively. It follows that $$\begin{aligned}
R(\omega ) &=&\wp \sum_{k}\frac{V_{k}^{2}}{\omega -\omega _{k}}=(\eta
\;\Delta )^{2}\wp \int\limits_{0}^{\infty }d\omega ^{^{\prime }}\frac{J(\omega ^{^{\prime }})}{(\omega -\omega ^{^{\prime }})(\omega ^{^{\prime
}}+\eta \Delta )^{2}} \notag \\
&=&2\alpha \Delta ^{2}\eta ^{2}\frac{\omega \log \left\vert \omega
\right\vert }{(\eta \;\Delta +\omega )^{2}(\lambda ^{2}\ +\omega ^{2})}
\notag \\
&&+2\alpha \Delta ^{2}\eta ^{2}\frac{\pi \lambda \left[ \lambda ^{2}+\eta
\;\Delta \ (-\eta \;\Delta +2\omega )\right] -2\left[ \lambda ^{2}(2\eta
\;\Delta -\omega )+\eta ^{2}\Delta ^{2}\ \omega \right] \log \left\vert
\lambda \right\vert }{2(\lambda ^{2}+\eta ^{2}\Delta ^{2})^{2}(\lambda
^{2}+\ \omega ^{2})} \notag \\
&&+2\alpha \Delta ^{2}\eta ^{2}\frac{-(\eta \;\Delta +\omega )\left( \lambda
^{2}+\Delta ^{2}\eta ^{2}\right) +\left[ 2\eta ^{3}\Delta ^{3}+(-\lambda
^{2}+\eta ^{2}\Delta ^{2})\ \omega \right] \log \left\vert \eta \;\Delta
\right\vert }{(\lambda ^{2}+\eta ^{2}\Delta ^{2})^{2}(\ \eta \;\Delta
+\omega )^{2}},\end{aligned}$$and$$\Gamma (\omega )\;=\;\pi \sum_{k}V_{k}^{2}\delta (\omega -\omega
_{k})\;=\;\pi (\eta \;\Delta )^{2}\frac{J(\omega )}{(\omega +\eta \;\Delta
)^{2}},$$where $\wp $ stands for the Cauchy principal value and $J(\omega )$ is the spectral density. Then, we have $$\left\langle \sigma _{x}(t)\right\rangle =\frac{1}{\pi }\int_{0}^{\infty }\frac{\Gamma (\omega )\cos \omega t}{(\omega -\eta \;\Delta -R(\omega
))^{2}+\Gamma (\omega )^{2}}d\omega . \label{E26}$$The integration in Eq. (\[E26\]) can be calculated numerically or approximately using residual theory.
Measurement dynamics: quantum Zeno effect
-----------------------------------------
It is known that the quantum Zeno effect can effectively slow down the quantum decay rate of a quantum system. We study this effect using an approach that goes beyond the Markovian approximation and RWA. Here we consider the low-frequency bath as in Sec. II.A and derive the effective decay rate. The Hamiltonian $H^{^{\prime }}$ is given in Eq. (11), $$H^{^{\prime }}\approx -\frac{1}{2}\eta \;\Delta \sigma _{z}+\sum_{k}\omega
_{k}a_{k}^{+}a_{k}+\sum_{k}\eta \;\Delta \frac{g_{k}\xi _{k}}{\omega _{k}}(a_{k}^{+}\sigma _{-}+a_{k}\sigma _{+}).$$In this paper, the effective decay rate is defined in the same form as in Ref. \[\].
Write the wave function in the transformed Hamiltonian as $$\left\vert \Phi (t)\right\rangle ^{^{\prime }}=\chi (t)\left\vert \downarrow
\right\rangle \left\vert 0_{k}\right\rangle +\sum_{k}\beta _{k}(t)\left\vert
\uparrow \right\rangle \left\vert 1_{k}\right\rangle ,$$with probability in the excited state at initial time $\left\vert \chi
(0)\right\vert ^{2}$. Substituting $\left\vert \Phi (t)\right\rangle
^{^{\prime }}$ into the Schrödinger equation with $H^{^{\prime }}$, we have $$i\frac{d\chi (t)}{dt}=\frac{\eta \;\Delta }{2}\chi (t)+\sum_{k}V_{k}\;\beta
_{k}(t), \label{E29}$$$$i\frac{d\beta _{k}(t)}{dt}=\left( \omega _{k}-\frac{\eta \;\Delta }{2}\right) \beta _{k}(t)+\sum_{k}V_{k}\;\chi (t). \label{E30}$$When the transformations $$\chi (t)\!\!=\!\!\widetilde{\chi }(t)\exp \left[ -i\frac{\eta \;\Delta }{2}t\right] ,$$$$\beta _{k}(t)\!\!=\!\!\widetilde{\beta }_{k}(t)\exp \left[ -i\left( \omega
_{k}-\frac{\eta \;\Delta }{2}\right) t\right]$$are applied, Eqs. (\[E29\]) and (\[E30\]) can be written as $$\frac{d\widetilde{\chi }(t)}{dt}=-i\sum_{k}V_{k}\widetilde{\beta }_{k}(t)\exp \left[ -i(\omega _{k}-\eta \;\Delta )t\right] , \label{E33}$$$$\frac{d\widetilde{\beta }_{k}(t)}{dt}=-iV_{k}\widetilde{\chi }(t)\exp \left[
i(\omega _{k}-\eta \;\Delta )t\right] . \label{E34}$$Integrating Eq. (\[E34\]) and then substituting it into Eq. (\[E33\]), we obtain $$\frac{d\widetilde{\chi }(t)}{dt}=-\sum_{k}V_{k}^{2}\int\limits_{0}^{t}\widetilde{\chi }(t^{^{\prime }})\exp [-i(\omega _{k}-\eta \;\Delta
)(t-t^{^{\prime }})]\;dt^{^{\prime }}. \label{E35}$$This integro-differential equation Eq. (\[E35\]) is exactly soluble by a Laplace transformation.
As for the present study of the quantum Zeno effect, i.e., using frequent measurements, it suffices to obtain the short-time behavior and the equation can be solved iteratively. With the initial excited-state probability amplitude $\chi (0)$, in the first iteration, Eq. (\[E35\]) is solved as $$\frac{\widetilde{\chi }(t)}{\chi (0)}\simeq
1-\int\limits_{0}^{t}(t-t^{^{\prime }})\sum_{k}V_{k}^{2}\exp [-i(\omega
_{k}-\eta \;\Delta )t^{^{\prime }}]\;dt^{^{\prime }}.$$We can approximately write $\widetilde{\chi }(t)$ using an exponential form: $$\begin{aligned}
\frac{\widetilde{\chi }(t)}{\chi (0)} &=&\exp \left[ -\int\limits_{0}^{t}(t-t^{^{\prime }})\sum_{k}V_{k}^{2}\exp \left[ -i(\omega
_{k}-\eta \;\Delta )t^{^{\prime }}\right] dt^{^{\prime }}\right] \\
&=&\exp \left\{ -t\left[ -\frac{1}{t}\sum_{k}V_{k}^{2}\frac{\exp \left[
-i\left( \omega _{k}-\eta \;\Delta \right) t\right] -1+i\left( \omega
_{k}-\eta \;\Delta \right) t}{\left( \omega _{k}-\eta \;\Delta \right) ^{2}}\right] \right\} \\
&=&\exp \left\{ -t\left[ \sum_{k}V_{k}^{2}\left( \frac{2\sin \left( \frac{\omega _{k}-\eta \;\Delta }{2}t\right) ^{2}}{t\left( \omega _{k}-\eta
\;\Delta \right) ^{2}}-i\frac{\left( \omega _{k}-\eta \;\Delta \right)
t-\sin \left[ \left( \omega _{k}-\eta \;\Delta \right) t\right] }{t\left(
\omega _{k}-\eta \;\Delta \right) ^{2}}\right) \right] \right\} .\end{aligned}$$The instantaneous ideal projections are assumed to be performed at intervals $\tau $. If single measurement, the probability amplitude is $\widetilde{\chi }(t=\tau ).$ For a sufficiently large frequency of measurements, the survival population in the excited state is$$\rho _{\mathrm{ee}}(t=n\tau )=\left\vert \widetilde{\chi }(t=n\tau
)\right\vert ^{2}=\left\vert \chi (0)\right\vert ^{2}\exp [-\gamma (\tau )t],
\label{E40}$$where the subscript $\mathrm{ee}$refers to the initial and final excited state. And $\gamma (\tau )$, with projection intervals $\tau ,$ is obtained as $$\begin{aligned}
\gamma (\tau ) &=&2\pi \int_{0}^{\infty }d\omega \sum_{k}\left( \frac{g_{k}}{2}\right) ^{2}\left( \eta \;\Delta \frac{2\xi _{k}}{\omega _{k}}\right) ^{2}\frac{2\sin ^{2}(\frac{\eta \;\Delta -\omega }{2}\tau )}{\pi (\eta \Delta
-\omega )^{2}\tau }, \label{E41} \\
&=&2\pi \int_{0}^{\infty }d\omega \frac{J(\omega )}{4}\left[ 1-\frac{\omega
-\eta \;\Delta }{\omega +\eta \;\Delta }\right] ^{2}\frac{2\sin ^{2}(\frac{\eta \;\Delta -\omega }{2}\tau )}{\pi (\eta \Delta -\omega )^{2}\tau }.
\label{E42}\end{aligned}$$We now prepare the qubit to the dressed excited state $\exp [-S]\left\vert
\downarrow \right\rangle \left\vert 0_{k}\right\rangle $ $(\sigma
_{z}\left\vert \downarrow \right\rangle =-\left\vert \downarrow
\right\rangle )$ at the initial time $t=0$, which can be achieved by acting the operator $\sigma _{x}$ on the ground state $\exp [-S]\left\vert \uparrow
\right\rangle \left\vert 0_{k}\right\rangle ,$$$\left\vert \Phi (0)\right\rangle =\exp [-S]\left\vert \downarrow
\right\rangle \left\vert 0_{k}\right\rangle =\sigma _{x}\exp [-S]\left\vert
\uparrow \right\rangle \left\vert 0_{k}\right\rangle .$$In this case, the initial state in the transformed Hamiltonian is $\left\vert \downarrow \right\rangle \left\vert 0_{k}\right\rangle $ and $\chi (0)=1.$
Note that in Eq. (\[E42\]), the renormalization factor $\eta $ of the characteristic energy $\Delta $ appears in the decay rate $\gamma (\tau )$. This is different from the formulas for $\gamma (\tau )$ in Refs. \[\] and \[\]. In the case of spontaneous emission, the coupling strength between the electromagnetic field and atom is the fine structure constant $1/137$, so it belongs to the weak-coupling case and $\eta $ then becomes extremely close to $1$. Therefore, besides spontaneous emission, this result for the quantum Zeno effect can apply to other cases of strong coupling between the qubit and the bath. Under the RWA, the Hamiltonian becomes $H_{\mathrm{RWA}}$ as in Eq. ([E13]{}). We prepare the initial excited state of $H$ through the operator $\sigma _{x}$ acting on the ground state under RWA $\left\vert \uparrow
\right\rangle \left\vert 0_{k}\right\rangle ,$ $\sigma _{x}\left\vert
\uparrow \right\rangle \left\vert 0_{k}\right\rangle =\left\vert \downarrow
\right\rangle \left\vert 0_{k}\right\rangle $. Then, following the derivation in Refs. \[\], the decay rate is reduced to $$\begin{aligned}
\gamma _{\mathrm{RWA}}(\tau ) &=&2\pi \int_{0}^{\infty }d\omega
\sum_{k}\left( \frac{g_{k}}{2}\right) ^{2}\frac{2\sin ^{2}(\frac{\Delta
-\omega }{2}\tau )}{\pi (\Delta -\omega )^{2}\tau }. \\
&=&2\pi \int_{0}^{\infty }d\omega \frac{J(\omega )}{4}\frac{2\sin ^{2}(\frac{\Delta -\omega }{2}\tau )}{\pi (\Delta -\omega )^{2}\tau }.\end{aligned}$$
To compare with the high-frequency Ohmic bath, we choose the ordinary Ohmic bath with Drude cutoff:$$J^{\mathrm{oh}}(\omega )=\sum_{k}g_{k}^{2}\delta (\omega -\omega _{k})=\frac{2\alpha ^{\mathrm{oh}}\omega }{\left( \omega /\omega _{c}\right) ^{2}+1}.
\label{E46}$$This is a realistic assumption for, e.g., an electromagnetic noise. In Eq. (\[E46\]), $\alpha ^{\mathrm{oh}}$ is the coupling strength between the qubit and the Ohmic bath. The cutoff frequency $\omega _{c}$ in the spectral density $J^{\mathrm{oh}}(\omega )$ is typically assumed to be the largest frequency in the problem. As for the low-frequency noise, we use $J(\omega
)=2\alpha \omega /(\omega ^{2}+\lambda ^{2})$, which is the same as in Eq. (\[E2\]). The difference between these two baths is that $\lambda $ corresponds to an energy lower than the qubit energy. In the next section, we will show our numerical results for these two baths.
Results and Discussions
=======================
To show the effects of either a low- or a high-frequency noise on the qubit states respectively, we study the dynamical quantities $\left\langle \sigma
_{x}(t)\right\rangle $ and the quantum Zeno decay rate $\gamma (\tau )$. The energy shift and the quantum Zeno decay rate exhibit evidently different features for the low- and high-frequency noises. Thus, these two quantities (energy shift and decay rate) can be used as criteria to *distinguish the type of noise*. In Fig. 1, we show the spectral densities $J(\omega )$ and $J^{\mathrm{oh}}(\omega )$ of the two baths in the cases of both weak dissipation $\alpha /\Delta ^{2}=0.01$ ($\alpha ^{\mathrm{oh}}=0.01$) and strong dissipation $\alpha /\Delta ^{2}=0.1$ ($\alpha ^{\mathrm{oh}}=0.1$). As examples, we choose $\lambda =0.09\Delta $ and $\lambda =0.3\Delta $ in Fig. 1(a) and 1(b), respectively. Here, each value of $\lambda $ corresponds to the position of the peak in the low-frequency spectral density. For an Ohmic bath, the cutoff frequency is fixed at $\omega _{c}=10\Delta $. As expected, these very different low- and high-frequency spectral densities should give rise to different decoherence behaviors of the qubit. In Fig. 1, we also show the characteristic energy of the isolated qubit $\Delta $ (see the vertical dotted line in Fig. 1).
{width="9cm"}
Non-measurement decoherence dynamics
------------------------------------
Results from the numerical integration of Eq. (26) are shown in Fig. 2. They are qualitatively consistent with the results obtained using residual theory. This indicates that the branch cuts considered in Refs. and do not affect the oscillation frequency. The time evolution of $\left\langle \sigma
_{x}(t)\right\rangle $ is given in Fig. 2(a) for the case of weak coupling between the qubit and the bath, where $\alpha /\Delta ^{2}=0.01$ and $\lambda =0.09\Delta $ for the low-frequency noise and $\alpha ^{\mathrm{oh}}=0.01$ and $\omega _{c}=10\Delta $ for the Ohmic bath. Figure 2(b) presents the time evolution of $\left\langle \sigma _{x}(t)\right\rangle $ in the strong coupling case with the parameters $\alpha /\Delta ^{2}=0.1$ and $\lambda =0.3\Delta $ for the low-frequency noise, as well as $\alpha ^{\mathrm{oh}}=0.1$ and $\omega _{c}=10\Delta $ for the ohmic bath. As expected, the quantum oscillations of $\left\langle \sigma
_{x}(t)\right\rangle $ dampen faster in the strong-coupling case. We approximately evaluate the oscillation frequency or the effective energy of the qubit, $\omega _{0}-\eta \Delta -R(\omega _{0})=0$, using the residue theorem. The decay rate can be obtained from $\Gamma (\omega )$. We will now show the numerical values of $\eta $ in the corresponding cases. In Fig. 2(a), the renormalized factor $\eta =0.98336$, the oscillation frequency is $\omega _{0}=1.0225\Delta $ and the decay rate is $\Gamma
(\omega _{0})=0.014654\Delta $ for the low-frequency noise; while $\eta
=0.98447,$ $\omega _{0}=0.97720\Delta ,$ and $\Gamma (\omega
_{0})=0.015318\Delta $ for the Ohmic bath. In Fig. 2(b), the renormalized factor $\eta =0.91444$, the oscillation frequency is $\omega
_{0}=1.0868\Delta $ and the decay rate is $\Gamma (\omega
_{0})=0.11215\Delta $ for the low-frequency noise, while $\eta =0.84469$, $\omega _{0}=0.77221\Delta $, and $\Gamma (\omega _{0})=0.13163\Delta $ for the Ohmic bath.
![(Color online) Time evolution of the coherence $\left\langle
\protect\sigma _{x}(t)\right\rangle $ versus the time $t$ multiplied by the qubit energy spacing $\Delta$. (a) The case of *weak* interaction between the bath and the qubit, where the parameters of the low-frequency Lorentzian-type spectrum are $\protect\alpha /\Delta ^{2}=0.01, $[ ]{}$\protect\lambda =0.09\Delta $[ (red solid curve); while for the high-frequency Ohmic bath with Drude cutoff the parameters are ]{}$\protect\alpha ^{\mathrm{oh}}=0.01,$ $\protect\omega _{c}=10\Delta $ (green dashed-dotted curve)[. (b) ]{}The case of *strong* interaction between the bath and the qubit, where the parameters of the low-frequency bath are $\protect\alpha /\Delta ^{2}=0.1,$[ ]{}$\protect\lambda =0.3$[ (red solid line)]{}, and for the high-frequency Ohmic bath are[ ]{}$\protect\alpha ^{\mathrm{oh}}=0.1,$ $\protect\omega _{c}=10\Delta $ (green dashed-dotted line). These results show that the decay rate for the low-frequency bath is *shorter* than for the high-frequency Ohmic bath. This means that the coherence time of the qubit in the low-frequency bath is *longer* than in the high-frequency noise case, demonstrating the powerful temporal memory of the low-frequency bath. Also, our results reflect the structure of the solution with branch cuts [@prb-71-035318]. The oscillation frequency for the low-frequency noise is $\protect\omega _{0}>\Delta $, in spite of the strength of the interaction. This can be referred to as a blue shift. However, in an Ohmic bath, the oscillation frequency is $\protect\omega _{0}<\Delta $, corresponding to a red shift. The shifting direction of the energy is independent of the interaction strength and only determined by the spectral properties. Thus, it can be used as a *criterion for distinguishing the low- and high-frequency noises*.](Fig2.eps){width="9cm"}
The energy spectral densities in Fig. 1 and the results in Fig. 2 indicate *two opposite shifts* of the characteristic energy $\Delta $ for the two kinds of baths considered here. These opposite energy shifts are equivalent to *energy repulsion*. The energy shift is determined by the interaction term. In the transformed Hamiltonian, the interaction is $H_{1}^{\prime }$ in Eq. (\[E7\]). Also, dipolar interactions, such as $H_{I}^{JC}=g(a^{+}\sigma _{-}+a\sigma _{+})$ in the Jaynes-Cummings model, decrease the qubit’s ground-state energy and increase its excited-state energy. Thus, now we ask the following questions: how a qubit is affected by either a multimode bath or a single-mode cavity? How a qubit is influenced by these two kinds of multimode baths: low-frequency and high-frequency ones?
For a low-frequency bath, the energy peak of the bath is located between the ground-state energy and the excited-state energy, that is to say the main part of the spectrum is in the region $\omega _{k}<\Delta $. Then (as seen in Fig. 1) the interaction of the bath with the two qubit states is opposite, i.e. the ground-state energy becomes *lower* and the excited-state energy becomes *higher*. So the energy spacing for the case of a low-frequency bath exhibits a *blue shift*. This result is similar to the single-mode Jaynes-Cummings model. The energy difference of the two-state qubit is *increased* by the low-frequency bath.
For a high-frequency cutoff Ohmic bath, the energy peak of the bath is located above the excited-state energy. So the main part of the spectrum is in the region $\omega _{k}>\Delta $. The effect of the bath on the qubit mainly comes from the frequencies higher than the excited-state energy of the qubit. Then (as seen in Fig. 1) the bath repels both the excited-state and the ground-state energies to lower energies. But the effect of the high-frequency bath on the excited state is much larger than on the ground state. As a result, on the whole, the qubit energy difference in a high-frequency bath is *red shifted*. Thus, the effective energy difference of the qubit is *reduced* by the high-frequency bath.
For example, if the initial state of the qubit is an excited state, in the interaction picture, the main part of the coupling is $$\begin{aligned}
&&\!\sum_{k}g_{k}a_{k}^{+}\exp (i\omega _{k}t)\sigma _{-}\exp (-i\Delta t)
\notag \\
=\! &&\!\sum_{k}g_{k}a_{k}^{+}\sigma _{-}\left\{ \cos [(\omega _{k}-\Delta
)t]+i\sin [(\omega _{k}-\Delta )t]\right\} ,\end{aligned}$$where the real part contributes to the decay rate and the imaginary part results in the energy shift. For the low-frequency bath, the main part of the spectrum is in the region $\omega _{k}<\Delta $. Thus, the term for the energy shift is $$\sin [(\omega _{k}-\Delta )t]<0.$$However, for a high-frequency cutoff Ohmic bath, the main part of the spectrum is in the region $\omega _{k}>\Delta $, where the term for energy shift becomes $$\sin [(\omega _{k}-\Delta )t]>0.$$These results show that the energy shift for the two kinds of baths moves in opposite directions. Note that there is a minus in the interaction term in the expressions for the dynamical quantities such as Eq. (21) and Eq. (33). The above observations help us understand *why* the energy levels repel. The contribution by the real part of the interaction on the decay rate will be discussed below, when studying the quantum Zeno effect.
Measurement dynamics: quantum Zeno effect
-----------------------------------------
The quantum Zeno effect can be a useful tool to preserve the state coherence of a quantum system, with the help of repeated projective measurements. Below we investigate the quantum Zeno effect in the qubit system and *propose another criterion for distinguishing low- and high-frequency noises*. In general, without using the RWA, the effective decay rate can be obtained as $$\gamma (\tau )=2\pi \int_{0}^{\infty }d\omega J(\omega )\left( 1-\frac{\omega -\eta \;\Delta }{\omega +\eta \;\Delta }\right) ^{2}\frac{2\sin ^{2}(\frac{\eta \;\Delta -\omega }{2}\tau )}{\pi (\eta \;\Delta -\omega )^{2}\tau
}.$$This expression includes three terms, i.e., the spectral density $J(\omega )$ of the bath, the projection time modulating function $$F(\omega ,\tau )=\frac{2\sin ^{2}(\frac{\eta \;\Delta -\omega }{2}\tau )}{\pi (\eta \;\Delta -\omega )^{2}\tau },$$and the interaction contribution function of both the rotating and counter-rotating terms $$f(\omega )=\left( 1-\frac{\omega -\eta \;\Delta }{\omega +\eta \;\Delta }\right) ^{2}.$$The counter-rotating term contributing to $f(\omega )$ is $(\omega -\eta
\;\Delta )/\left( \omega +\eta \;\Delta \right) $. If the RWA is applied, $f(\omega )=1$. In Fig. 3(a), the decay rate $\gamma (\tau )/\gamma _{0}$ for the low-frequency bath is plotted in the weak-coupling case with $\alpha
/\Delta ^{2}=0.01$. Here $\gamma _{0}$ is the decay rate for $\left\vert
e\right\rangle \rightarrow \left\vert g\right\rangle $ in the long-time limit with the RWA, $$\gamma _{0}=\gamma (\tau \rightarrow \infty )=2\pi J(\Delta )/4.$$From the energy spectrum in Fig. 1(a), we can see that $\gamma _{0}$ is proportional to the spectrum density of the bath, with the magnitude corresponding to the crossing of the energy $\Delta $ of the isolated qubit and the bath spectrum. For comparison, we also plot $\gamma _{\mathrm{RWA}}(\tau )/\gamma _{0}$ as a dashed-dotted curve, with $$\gamma _{\mathrm{RWA}}(\tau )=2\pi \int_{0}^{\infty }d\omega \frac{J(\omega )}{4}\frac{2\sin ^{2}(\frac{\Delta -\omega }{2}\tau )}{\pi (\Delta -\omega
)^{2}\tau }.$$As we know, $\gamma (\tau )/\gamma _{0}<1$ means that repeated measurements slow down the decay rate $\gamma (\tau )<\gamma _{0},$ which is the quantum Zeno effect. On the contrary, $\gamma (\tau )/\gamma _{0}>1$ means an anti-Zeno effect. The curves in Fig. 3(b) show results for the Ohmic bath with $\alpha ^{\mathrm{oh}}=0.01$. In Fig. 4, we show the decay rate in the strong-coupling case for the low-frequency bath with $\alpha /\Delta
^{2}=0.1 $ and for the Ohmic bath with $\alpha ^{\mathrm{oh}}=0.1$. It can be seen that, for the low-frequency bath, the anti-Zeno effect appears as shown in Figs. 3 and 4. For the high-frequency cutoff Ohmic bath, the Zeno effect always dominates and no anti-Zeno effect occurs. Also we can see, from Figs. 3 and 4, that $\gamma (\tau )$ and$\ \gamma _{\mathrm{RWA}}(\tau
) $ approach $\gamma _{0}$ when the measurement interval $\tau \rightarrow
\infty $. In particular, if $\tau \rightarrow 0$, $F(\omega ,\tau
)\rightarrow 0$. Thus, $\gamma (\tau )\rightarrow 0$. This implies that in a sufficiently short time interval of a projective measurement, the quantum Zeno effect occurs, regardless of the bath spectrum. When the interval $\tau
$ increases, the projection interval modulation function $F(\omega ,\tau )$ displays a number of oscillations. Then, the energy peak of the bath spectrum will act on the decay rate and it is possible to implement the anti-Zeno effect. However, this result still depends on the function $f(\omega )$ and the given spectral density $J(\omega )$ of the bath. Now, we emphasize again that the second term in the bracket of the function $f(\omega )=\left[ 1-(\omega -\eta \;\Delta )/\left( \omega +\eta \;\Delta
\right) \right] ^{2}$ is due to the counter-rotating terms; when neglecting the counter-rotating terms, $f(\omega )=1$.
For low-frequency noise, the noise mainly comes from the region $\omega
<\Delta $, so $f(\omega )>1$. Here, $f(\omega )$ as well as $F(\omega ,\tau
) $ *magnify* the effect of the energy peak of the bath spectrum. Thus, the counter-rotating terms *accelerate* the decay and the anti-Zeno effect occurs.
In the high-frequency cutoff Ohmic bath, the noise mainly comes from the region $\omega >\Delta $, which leads to $f(\omega )<1$. Thus, it is mainly the counter-rotating term in $f(\omega )$ that reduces the effect of the energy peak of the bath on the decay. This *slows down the decay* and only the quantum Zeno effect can now take place. The projection-intervals-modulating function $F(\omega ,\tau )$, together with the interaction-modulating function $f(\omega )$, causes the Zeno effect to dominate in the high-frequency cutoff Ohmic bath.
{width="9cm"}
{width="9cm"}
Summary
=======
In summary, we have studied a model of a qubit interacting with its environment, modeled either as a low- or as a high-frequency bath. For each type of bath, the quantum dynamics of the qubit without measurement and the quantum Zeno effect on it are shown for the cases of weak and strong couplings between the qubit and the environment. Our results show that, for a low-frequency bath, the qubit energy increases (blue shift) and the quantum anti-Zeno effect occurs. However, for a high-frequency cutoff Ohmic bath, the qubit energy decreases (red shift) and the quantum Zeno dominates. Moreover, for a high-frequency environment, the decay rate should be faster (without measurements) or slower (with frequent measurements, in the Zeno regime), compared to the low-frequency bath case. These very different behaviors of the quantum dynamics and the Zeno effect in different baths should be helpful to experimentally distinguish the type of noise affecting the qubit and protect the coherence of the qubit through modulating the dominant frequency of its environment.
0.5cm
[[**Acknowledgements**]{}]{}
FN acknowledges partial support from the National Security Agency (NSA), Laboratory for Physical Sciences (LPS), Army Research Office (USARO), National Science Foundation (NSF) under Grant No. 0726909, and JSPS-RFBR under Contract No. 06-02-91200. X.-F. Cao acknowledges support from the National Natural Science Foundation of China under Grant No. 10904126 and Fujian Province Natural Science Foundation under Grant No. 2009J05014. J.-Q. You acknowledges partial support from the National Natural Science Foundation of China under Grant No. 10625416, the National Basic Research Program of China under Grant No. 2009CB929300 and the ISTCP under Grant No. 2008DFA01930.
Appendix: Examining the validity of the unitary transformation {#sec:appendix .unnumbered}
==============================================================
In this appendix, we show the main results of the canonical transformation to the qubit-bath Hamiltonian $H$ considered here, and prove that the contribution of $H_{2}^{^{\prime }}$ to physical quantities is of the order $\mathcal{O}$$\left( g_{k}^{4}\right) $ and higher, so we ignore $H_{2}^{^{\prime }}$ in the calculations. We now apply a canonical transformation to the Hamiltonian $H$: $$H^{\prime }=\exp \left[ S\right] H\exp \left[ -S\right] ,$$with $$S=\sum_{k}\frac{g_{k}}{2\omega _{k}}\xi _{k}(a_{k}^{+}-a_{k})\sigma _{x}.$$Here a $k$-dependent variable, $\xi _{k}=\omega _{k}/(\omega _{k}+\eta
\;\Delta ),$ is introduced in the transformation. It is clear that this canonical transformation is unitary, because $\left(
\exp \left[ S\right] \right) ^{+}=\exp \left[ -S\right] $. The transformed Hamiltonian $H^{\prime }$ can now be decomposed in three parts: $$H^{\prime }=H_{0}^{\prime }+H_{1}^{\prime }+H_{2}^{\prime },$$with $$H_{0}^{\prime }=-\frac{1}{2}\eta \;\Delta \sigma _{z}+\sum_{k}\omega
_{k}a_{k}^{+}a_{k}-\sum_{k}\frac{g_{k}^{2}}{4\omega _{k}}\xi _{k}(2-\xi
_{k}), \label{E6}$$$$H_{1}^{\prime }=\sum_{k}\eta \;\Delta \frac{g_{k}\xi _{k}}{\omega _{k}}\left( a_{k}^{+}\sigma _{-}+a_{k}\sigma _{+}\right) , \label{E7}$$$$\begin{aligned}
H_{2}^{\prime } &=&-\frac{1}{2}\Delta \sigma _{z}\left\{ \cosh \left[
\sum_{k}\frac{g_{k}}{\omega _{k}}\xi _{k}\left( a_{k}^{+}-a_{k}\right) \right] -\eta \right\} \notag \\
&&-i\frac{\Delta }{2}\sigma _{y}\left\{ \sinh \left[ \sum_{k}\frac{g_{k}}{\omega _{k}}\xi _{k}\left( a_{k}^{+}-a_{k}\right) \right] -\eta \sum_{k}\frac{g_{k}}{\omega _{k}}\xi _{k}\left( a_{k}^{+}-a_{k}\right) \right\} ,
\label{E8}\end{aligned}$$where$$\eta =\exp \left[ -\sum_{k}\frac{g_{k}^{2}}{2\omega _{k}^{2}}\xi _{k}{}^{2}\right] . \label{E9}$$ Note that no approximation was used during the transformation, so $H^{\prime }=\exp \left[ S\right] H\exp \left[ -S\right] $ is exact. Because the constant term $\sum_{k}\frac{g_{k}^{2}}{4\omega _{k}}\xi _{k}(2-\xi
_{k}) $ in Eq. (\[E6\]) has no effect on the dynamical evolution, we neglect it. Considering at low temperatures, the multiple-step process is so weak that all the higher-order terms can be neglected. In the following derivation, we will prove that the contribution of $H_{2}^{^{\prime }}$ to physical quantities is of the order $\mathcal{O}$$\left( g_{k}^{4}\right) $ and higher, so we ignore $H_{2}^{^{\prime }}$ and obtain the effective transformed Hamiltonian $H^{\prime }=H_{0}^{\prime }+H_{1}^{\prime }$.
Now let us expand the first term of $H_{2}^{^{\prime }},$ $\cosh \left[
\sum_{k}\frac{g_{k}}{\omega _{k}}\xi _{k}(a_{k}^{+}-a_{k})\right] $, as a series in $g_{k}\xi _{k}/\omega _{k}.$ We define $\chi _{k}$ as $\chi
_{k}=g_{k}\xi _{k}/\omega _{k},$ which is proportional to $g_{k}$: $$\begin{aligned}
\cosh \left[ \sum_{k}\frac{g_{k}}{\omega _{k}}\xi _{k}(a_{k}^{+}-a_{k})\right] &=&\cosh \left[ \sum_{k}\chi _{k}(a_{k}^{+}-a_{k})\right] \label{A1}
\\
&=&\frac{1}{2}\left\{ \exp \left[ \sum_{k}\chi _{k}(a_{k}^{+}-a_{k})\right]
+\exp \left[ -\sum_{k}\chi _{k}(a_{k}^{+}-a_{k})\right] \right\} .
\label{A2}\end{aligned}$$The first term in Eq. (\[A2\]) can be written as $$\exp \left[ \sum_{k}\chi _{k}(a_{k}^{+}-a_{k})\right] =\exp \left[
\sum_{k}\chi _{k}a_{k}^{+}\right] \exp \left[ \sum_{k}-\chi _{k}a_{k}\right]
\exp \left[ -\frac{1}{2}\left[ \sum_{k}\chi _{k}a_{k}^{+},\sum_{k}-\chi
_{k}a_{k}\right] _{-}\right] , \label{A3}$$where $\left[ \sum_{k}\chi _{k}a_{k}^{+},\sum_{k}-\chi _{k}a_{k}\right] _{-}$ means the commutator of two operators. Using the commutation relation $\left[
a_{k}^{+},a_{k}\right] =-1$, it is simplified to $$\exp \left[ -\frac{1}{2}\left[ \sum_{k}\chi _{k}a_{k}^{+},\sum_{k}-\chi
_{k}a_{k}\right] _{-}\right] =\exp \left[ -\frac{1}{2}\sum_{k}\chi _{k}^{2}\right] =\exp \left[ -\sum_{k}\frac{g_{k}^{2}}{2\omega _{k}^{2}}\xi
_{k}{}^{2}\right] =\eta , \label{A4}$$which is the definition of $\eta $.
Afterwards, we expand Eq. (\[A3\]) as follows,$$\begin{aligned}
\exp \left[ \sum_{k}\chi _{k}(a_{k}^{+}-a_{k})\right] &=&\eta \exp \left[
\sum_{k}\chi _{k}a_{k}^{+}\right] \exp \left[ \sum_{k}-\chi _{k}a_{k}\right]
\label{A5} \\
&=&\eta \left[ 1+\sum_{k}\chi _{k}a_{k}^{+}+\frac{\left( \sum_{k}\chi
_{k}a_{k}^{+}\right) ^{2}}{2}+\frac{\left( \sum_{k}\chi _{k}a_{k}^{+}\right)
^{3}}{3!}+...\right] \label{A6} \\
&&.\left[ 1-\sum_{k}\chi _{k}a_{k}+\frac{\left( -\sum_{k}\chi
_{k}a_{k}\right) ^{2}}{2}-\frac{\left( \sum_{k}\chi _{k}a_{k}\right) ^{3}}{3!}+...\right] \label{A7} \\
&=&\eta \left[ 1+\sum_{k}\chi _{k}a_{k}^{+}-\sum_{k}\chi
_{k}a_{k}-\sum_{k}\chi _{k}a_{k}^{+}\sum_{k}\chi _{k}a_{k}+...\right]
\label{A8}\end{aligned}$$Now we see the second-order terms in $\chi _{k}$ in Eq. (\[A8\]): $$\sum_{k}\chi _{k}a_{k}^{+}\sum_{k}\chi _{k}a_{k}=\sum_{k}\chi
_{k}^{2}a_{k}^{+}a_{k}(\mathrm{diagonal})+\sum_{k\neq k^{^{\prime }}}\chi
_{k}a_{k}^{+}\chi _{k^{\prime }}a_{k^{\prime }}(\text{off-diagonal}).
\label{A9}$$The off-diagonal terms are related to the multi-boson transition and their contributions to the physical quantities are fourth order in $\chi _{k}.$ Furthermore, the initial state $\left\vert 0_{k}\right\rangle \left\vert
\uparrow \right\rangle $ is used in the calculation, so the direct effect of the second-order diagonal term on physical quantities is zero and its effect through the interaction will also be fourth order in $\chi _{k}$. Thus we now ignore the terms higher than second order in $\chi _{k}$ in the following calculation and obtain the expression of Eq. (\[A3\]), $$\exp \left[ \sum_{k}\chi _{k}(a_{k}^{+}-a_{k})\right] \approx \eta \left[
1+\sum_{k}\chi _{k}a_{k}^{+}-\sum_{k}\chi _{k}a_{k}\right] .$$Similarly, the second term in Eq. (\[A2\]) becomes$$\begin{aligned}
\exp \left[ -\sum_{k}\chi _{k}(a_{k}^{+}-a_{k})\right] &=&\eta \exp \left[
-\sum_{k}\chi _{k}a_{k}^{+}\right] \exp \left[ \sum_{k}\chi _{k}a_{k}\right]
\\
&\approx &\eta \left[ 1-\sum_{k}\chi _{k}a_{k}^{+}+\sum_{k}\chi _{k}a_{k}\right] .\end{aligned}$$Therefore, Eq. (\[A2\]) is reduced to $$\begin{aligned}
\cosh \left[ \sum_{k}\frac{g_{k}}{\omega _{k}}\xi _{k}(a_{k}^{+}-a_{k})\right] &=&\cosh \left[ \sum_{k}\chi _{k}(a_{k}^{+}-a_{k})\right] \\
&=&\frac{1}{2}\left\{ \exp \left[ \sum_{k}\chi _{k}(a_{k}^{+}-a_{k})\right]
+\exp \left[ -\sum_{k}\chi _{k}(a_{k}^{+}-a_{k})\right] \right\} \\
&\approx &\eta .\end{aligned}$$In the same way, the third term in $H_{2}^{^{\prime }}$ is simplified to$$\sinh \left[ \sum_{k}\frac{g_{k}}{\omega _{k}}\xi _{k}(a_{k}^{+}-a_{k})\right] =\eta \left[ \sum_{k}\frac{g_{k}}{\omega _{k}}\xi
_{k}(a_{k}^{+}-a_{k})+O(\chi _{k}^{3})\right] .$$
In $H_{2}^{\prime },$ we have subtracted the terms of zero and first-order in $g_{k},$ which are included in $H_{0}^{\prime }+H_{1}^{\prime },$ and only left the terms equal to and higher than second-order in $g_{k},$ whose contribution to the physical quantities is of order $\mathcal{O}$$\left(
g_{k}^{4}\right) $ and higher. Thus, $H_{2}^{\prime }$ can be omitted.
We expanded several series in the variable $\chi _{k},$ with the real variable $\eta \Delta \chi _{k}$ less than $g_{k}$. That is: $\eta \Delta
\chi _{k}=\eta \Delta g_{k}\xi _{k}/\omega _{k}=$ $\eta \Delta g_{k}/\left(
\omega _{k}+\eta \Delta \right) <g_{k}.$ In other words, through this transformation we find a variable smaller than $g_{k}$ for the series expansion. Therefore, our method can be extended to the case of strong interaction between the qubit and the environment.
——————–
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[^1]: Email: [email protected]
|
---
abstract: 'We study the residue current $R^f$ of Bochner-Martinelli type associated with a tuple $f=(f_1,\dots,f_m)$ of holomorphic germs at $0\in\mathbf{C}^n$, whose common zero set equals the origin. Our main results are a geometric description of $R^f$ in terms of the Rees valuations associated with the ideal $(f)$ generated by $f$ and a characterization of when the annihilator ideal of $R^f$ equals $(f)$.'
address: |
Dept of Mathematics, University of Michigan, Ann Arbor\
MI 48109-1109\
USA
author:
- 'Mattias Jonsson & Elizabeth Wulcan'
date:
-
-
title: 'On Bochner-Martinelli residue currents and their annihilator ideals'
---
[^1]
Introduction {#intro}
============
Residue currents are generalizations of classical one-variable residues and can be thought of as currents representing ideals of holomorphic functions. In [@PTY] Passare-Tsikh-Yger introduced residue currents based on the Bochner-Martinelli kernel. Let $f=(f_1,\ldots,f_m)$ be a tuple of (germs of) holomorphic functions at $0\in\C^n$, such that $V(f)=\{f_1=\ldots = f_m=0\}=\{0\}$. (Note that we allow $m > n$.) For each ordered multi-index $\I=\{i_1,\ldots,i_n\}\subseteq \{1,\ldots, m\}$ let $$\label{blanc}
R^f_\I=\dbar |f|^{2\lambda}\wedge
c_n \sum_{\ell=1}^n(-1)^{\ell-1}
\frac
{\overline{f_{i_\ell}}\bigwedge_{q\neq \ell} \overline{df_{i_q}}}
{|f|^{2n}}\bigg |_{\lambda=0},$$ where $c_n=(-1)^{n(n-1)/2}(n-1)!$, $|f|^2=|f_1|^2+\ldots+|f_m|^2$, and $\alpha |_{\lambda =0}$ denotes the analytic continuation of the form $\alpha$ to $\lambda=0$. Moreover, let $R^f$ denote the vector-valued current with entries $R^f_\I$; we will refer to this as the *Bochner-Martinelli residue current* associated with $f$. Then $R^f$ is a well-defined $(0,n)$-current with support at the origin and $\overline g R^f_\I=0$ if $g$ is a holomorphic function that vanishes at the origin. It follows that the coefficients of the $R^f_\I$ are just finite sums of holomorphic derivatives at the origin.
Let $\O^n_0$ denote the local ring of germs of holomorphic functions at $0\in\C^n$. Given a current $T$ let $\ann T$ denote the (holomorphic) *annihilator ideal* of $T$, that is, $$\ann T=\{h\in \O^n_0, hT=0\}.$$ Our main result concerns $\ann R^f=\bigcap \ann R^f_\I$. Let $(f)$ denote the ideal generated by the $f_i$ in $\O^n_0$. Recall that $h\in\O^n_0$ is in the integral closure of $(f)$, denoted by $\overline{(f)}$, if $|h|\leq C|f|$, for some constant $C$. Moreover, recall that $(f)$ is a *complete intersection ideal* if it can be generated by $n=\codim V(f)$ functions. Note that this condition is slightly weaker than $\codim V(f)=n=m$.
Suppose that $f$ is a tuple of germs of holomorphic functions at $0\in\C^n$ such that $V(f)=\{0\}$. Let $R^f$ be the corresponding Bochner-Martinelli residue current. Then $$\label{eqa}
\overline{(f)^n}\subseteq\ann R^f\subseteq (f).$$ The left inclusion in is strict whenever $n\geq 2$. The right inclusion is an equality if and only if $(f)$ is a complete intersection ideal.
The new results in Theorem A are the last two statements. The left and right inclusions in are due to Passare-Tsikh-Yger [@PTY] and Andersson [@A], respectively. Passare-Tsikh-Yger defined currents $R^f_\I$ also when $\codim V(f) < n$. The inclusions hold true also in this case; one even has $\overline{(f)^{\min (m,n)}}\subseteq\ann R^f\subseteq (f)$. Furthermore, Passare-Tsikh-Yger showed that $\ann R^f=(f)$ if $m=\codim V(f)$. More precisely, they proved that in this case the only entry $R^f_{\{1,\ldots,m\}}$ of $R^f$ coincides with the classical *Coleff-Herrera product* $$R^f_{CH}=
\dbar \left [\frac{1}{f_1}\right ]\wedge\cdots\wedge
\dbar \left [\frac{1}{f_m}\right ],$$ introduced in [@CH]. The current $R^f_{CH}$ represents the ideal in the sense that $\ann R^f_{CH}=(f)$ as proved by Dickenstein-Sessa [@DS] and Passare [@P]. This so-called *Duality Principle* has been used for various purposes, see [@BGVY]. Any ideal of holomorphic functions can be represented as the annihilator ideal of a (vector valued) residue current. However, in general this current is not as explicit as the Coleff-Herrera product, see [@AW].
Thanks to their explicitness Bochner-Martinelli residue currents have found many applications, see for example [@AG], [@ASS], [@BY], and [@VY]. Even though the right inclusion in is strict in general, $\ann R^f$ is large enough to in some sense capture the “size” of $(f)$. For example (or rather the general version stated above) gives a proof of the Brian[ç]{}on-Skoda Theorem [@BS], see also [@A]. The inclusions in are central also for the applications mentioned above.
The proof of Theorem A has three ingredients. First, we use a result of Hickel [@Hi] relating the ideal $(f)$ to the Jacobian determinant of $f$. Second, we rely on a result by Andersson, which says that under suitable hypotheses, the current he constructs in [@A] is independent of the choice of Hermitian metric, see also Section \[rescurr\].
The third ingredient, which is of independent interest, is a geometric description of the Bochner-Martinelli current, and goes as follows. Let $\pi:X\to (\C^n,0)$ be a *log-resolution* of $(f)$, see Definition \[loglog\]. We say that a multi-index $\I=\{i_1,\ldots,i_n\}$ is *essential* if there is an exceptional prime $E\subseteq\pi^{-1}(0)$ of $X$ such that the mapping $[f_{i_1}\circ \pi:\ldots :f_{i_n}\circ\pi]: E\to \C\P^{n-1}$ is surjective and moreover $\vE(f_{i_k})\leq \vE(f_{\ell})$ for $1\leq k\leq n, 1\leq\ell\leq m$, see Section \[essential\] for more details. The valuations $\vE$ are precisely the *Rees valuations* of $(f)$.
Suppose that $f$ is a tuple of germs of holomorphic functions at $0\in\C^n$ such that $V(f)=\{0\}$. Then the current $R^f_\I\not\equiv 0$ if and only if $\I$ is essential.
As is well known, one can view $R^f$ as the pushforward of a current on a log-resolution of $(f)$. The support on the latter current is then exactly the exceptional components associated with the Rees valuations of $(f)$, see Section \[coeff\].
Recall that if $(f)$ is a complete intersection ideal, then $(f)$ is in fact generated by $n$ of the $f_i$. This follows for example by Nakayama’s Lemma.
Suppose that $f$ is a tuple of germs of holomorphic functions at $0\in\C^n$ such that $V(f)=\{0\}$ and such that $(f)$ is a complete intersection ideal. Then $\I=\{i_1,\ldots,i_n\}$ is essential if and only if $f_{i_1}, \ldots f_{i_n}$ generates $(f)$. Moreover $$\label{formen}
R^f_\I= C_\I ~\dbar \left [\frac{1}{f_{i_1}}\right ]\wedge\cdots\wedge
\dbar \left [\frac{1}{f_{i_n}}\right ],$$ where $C_\I$ is a non-zero constant.
Theorems B and C generalize previous results for monomial ideals. In [@W] an explicit description of $R^f$ is given in case the $f_i$ are monomials; it is expressed in terms of the Newton polytope of $(f)$. From this description a monomial version of Theorem A can be read off. Also, it follows that in the monomial case $\ann R^f$ only depends on the ideal $(f)$ and not on the particular generators $f$. This motivates the following question.
Let $f$ be a tuple of germs of holomorphic functions such that $V(f)=\{0\}$. Let $R^f$ be the corresponding Bochner-Martinelli residue current. Is it true that $\ann R^f$ only depends on the ideal $(f)$ and not on the particular generators $f$?
Computations suggest that the answer to Question D may be positive; see Remark \[Dremark\]. If $\codim V(f)<n$, then $\ann R^f$ may in fact depend on $f$ even though the examples in which this happens are somewhat pathological, see for example [@A Example 3]. A positive answer to Question D would imply that we have an ideal canonically associated with a given ideal; it would be interesting to understand this new ideal algebraically.
This paper is organized as follows. In Sections \[rescurr\] and \[reesprelim\] we present some necessary background on residue currents and Rees valuations, respectively. The proof of Theorem B occupies Section \[coeff\], whereas Theorems A and C are proved in Section \[annsection\]. In Section \[decompsection\] we discuss a decomposition of $R^f$ with respect to the Rees valuations of $(f)$. In the last two sections we interpret our results in the monomial case and illustrate them by some examples.
**Acknowledgment:** We would like to thank Mats Boij and H[å]{}kan Samuelsson for valuable discussions. This work was partially carried out when the authors were visiting the Mittag-Leffler Institute.
Residue currents {#rescurr}
================
We will work in the framework from Andersson [@A] and use the fact that the residue currents $R^f_\I$ defined by appear as the coefficients of a vector bundle-valued current introduced there. Let $f=(f_1,\ldots, f_m)$ be a tuple of germs of holomorphic functions at $0\in\C^n$. We identify $f$ with a section of the dual bundle $V^*$ of a trivial vector bundle $V$ over $\mathbb C^n$ of rank $m$, endowed with the trivial metric. If $\{e_i\}_{i=1}^m$ is a global holomorphic frame for $V$ and $\{e^*_i\}_{i=1}^m$ is the dual frame, we can write $f=\sum_{i=1}^m f_i e_i^*$. We let $s$ be the dual section $s=\sum_{i=1}^m \bar f_i e_i$ of $f$.
Next, we let $$u =
\sum_\ell\frac{s\wedge(\dbar s)^{\ell-1}}{|f|^{2\ell}},$$ where $|f|^2=|f_1|^2+\ldots + |f_m|^2$. Then $u$ is a section of $\Lambda(V\oplus T_{0,1}^*(\mathbb C^n))$ (where $e_j\wedge d\bar z_i=-d\bar z_i\wedge e_j$), that is clearly well defined and smooth outside $V(f)=\{f_1=\ldots = f_m=0\}$, and moreover $$\dbar|f|^{2\lambda}\wedge u,$$ has an analytic continuation as a current to $\Re \lambda > -\epsilon$. We denote the value at $\lambda=0$ by $R$. Then $R$ has support on $V(f)$ and $R=R_p+\ldots +R_\mu$, where $p=\codim V(f)$, $\mu=\min(m,n)$, and where $R_k\in \mathcal D'_{0,k}(\mathbb C^n,\Lambda^k V)$. In particular if $V(f)=\{0\}$, then $R=R_n$.
We should remark that Andersson’s construction of residue currents works for sections of any holomorphic vector bundle equipped with a Hermitian metric. In our case (trivial bundle and trivial metric), however, the coefficients of $R$ are just the residue currents $R^f_\I$ defined by Passare-Tsikh-Yger [@PTY]. Indeed, for $\I=\{i_1,\ldots, i_k\}\subseteq\{1,\ldots, m\}$ let $s_\I$ be the section $\sum_{j=1}^k \bar f_{i_j} e_{i_j}$, that is, the dual section of $f_\I=\sum_{j=1}^k f_{i_j} e^*_{i_j}$. Then we can write $u$ as a sum, taken over subsets $\I=\{i_1,\ldots, i_k\}\subseteq\{1,\ldots, m\}$, of terms $$u_\I=
\frac{s_\I\wedge (\dbar s_\I)^{k-1}}{|f|^{2k}}.$$ The corresponding current, $$\dbar|f|^{2\lambda}\wedge u_\I|_{\lambda=0}$$ is then merely the current $$R^f_\I:=\dbar |f|^{2\lambda}\wedge
c_k \sum_{\ell=1}^k(-1)^{\ell-1}
\frac
{\overline{f_{i_\ell}}\bigwedge_{q\neq \ell} \overline{df_{i_q}}}
{|f|^{2k}}\bigg |_{\lambda=0},$$ where $c_k=(-1)^{k(k-1)/2}(k-1)!$, times the frame element $e_\I=e_{i_k}\wedge\cdots\wedge e_{i_1}$; we denote it by $R_\I$. Throughout this paper we will use the notation $R^f$ for the vector valued current with entries $R^f_\I$, whereas $R$ and $R_\I$ (without the superscript $f$), respectively, denote the corresponding $\Lambda^n V$-valued currents.
Let us make an observation that will be of further use. If the section $s$ can be written as $\mu s'$ for some smooth function $\mu$ we have the following homogeneity: $$\label{homogen}
s\wedge (\dbar s)^{k-1}=\mu^k s'\wedge (\dbar s')^{k-1},$$ that holds since $s$ is of odd degree.
Given a holomorphic function $g$ we will use the notation $\dbar [1/g]$ for the value at $\lambda=0$ of $\dbar |g|^{2\lambda}/g$ and analogously by $[1/g]$ we will mean $|g|^{2\lambda}/g|_{\lambda=0}$, that is, the principal value of $1/g$. We will use the fact that $$\label{envar}
v^\lambda |\sigma|^{2\lambda}\frac{1}{\sigma^a}\bigg |_{\lambda=0} = \left [ \frac{1}{\sigma^a} \right ]
\quad \text{ and } \quad
\dbar(v^\lambda |\sigma|^{2\lambda})\frac{1}{\sigma^a}\bigg |_{\lambda=0} = \dbar \left [ \frac{1}{\sigma^a} \right ],$$ if $v=v(\sigma)$ is a strictly positive smooth function; compare to [@A Lemma 2.1].
Restrictions of currents and the Standard Extension Property {#pseudo}
------------------------------------------------------------
In [@AW2] the class of *pseudomeromorphic* currents is introduced. The definition is modeled on the residue currents that appear in various works such as [@A] and [@PTY]; a current is pseudomeromorphic if it can be written as a locally finite sum of push-forwards under holomorphic modifications of currents of the simple form $$[1/(\sigma_{q+1}^{a_{q+1}}\cdots \sigma_{n}^{a_{n}})]\dbar[1/\sigma_1^{a_1}]\wedge\cdots\wedge \dbar[1/\sigma_q^{a_q}]
\wedge \alpha,$$ where $\sigma_j$ are some local coordinates and $\alpha$ is a smooth form. In particular, all currents that appear in this paper are pseudomeromorphic.
An important property of pseudomeromorphic currents is that they can be restricted to varieties and, more generally, constructible sets. More precisely, they allow for multiplication by characteristic functions of constructible sets so that ordinary calculus rules holds. In particular, $$\label{mjau}
\1_V (\beta \wedge T)= \beta \wedge (\1_V T),$$ if $\beta$ is a smooth form. Moreover, suppose that $S$ is a pseudomeromorphic current on a manifold $Y$, that $\pi:Y\to X$ is a holomorphic modification, and that $A\subseteq Y$ is a constructible set. Then $$\label{joy}
\1_A(\pi_*S)=\pi_*(\1_{\pi^{-1}(A)}S).$$
A current $T$ with support on an analytic variety $V$ (of pure dimension) is said to have the so-called *Standard Extension Property (SEP) with respect to $V$* if it is equal to its standard extension in the sense of [@Bj]; this basically means that it has no mass concentrated to sub-varieties of $V$. If $T$ is pseudomeromorphic, $T$ has the SEP with respect to $V$ if and only if $\1_WT=0$ for all subvarieties $W\subset V$ of smaller dimension than $V$, see [@A6]. We will use that the current $\dbar[1/\sigma_i^a]$ has the SEP with respect to $\{\sigma_i=0\}$; in particular, $\dbar[1/\sigma_i^a]\1_{\{\sigma_j=0\}}=0$. If $S$ and $\pi$ are as above and we moreover assume that $S$ has the SEP with respect to an analytic variety $W$, then $\pi_* S$ has the SEP with respect to $\pi^{-1}(W)$.
Rees valuations {#reesprelim}
===============
The normalized blowup and Rees valuations
-----------------------------------------
We will work in a local situation. Let $\O_0^n$ denote the local ring of germs of holomorphic functions at $0\in\C^n$, and let $\m$ denote its maximal ideal. Recall that an ideal $\a\subset\O_0^n$ is *$\m$-primary* if its associated zero locus $V({\mathfrak{a}})$ is equal to the origin.
Let ${\mathfrak{a}}\subset\O_0^n$ be an ${\mathfrak{m}}$-primary ideal. The *Rees valuations* of ${\mathfrak{a}}$ are defined in terms of the normalized blowup $\pi_0:X_0\to(\C^n,0)$ of ${\mathfrak{a}}$. Since ${\mathfrak{a}}$ is ${\mathfrak{m}}$-primary, $\pi_0$ is an isomorphism outside $0\in\C^n$ and $\pi_0^{-1}(0)$ is the union of finitely many prime divisors $E\subset X_0$. The Rees valuations of ${\mathfrak{a}}$ are then the associated (divisorial) valuations $\ord_E$ on ${\mathcal{O}}_0^n$: $\ord_E(g)$ is the order of vanishing of $g$ along $E$.
The blowup of an ideal is defined quite generally in [@Ha Ch.II, §7]. We shall make use of the following more concrete description, see [@T p. 332]. Let $f_1,\dots,f_m$ be generators of ${\mathfrak{a}}$ and consider the rational map $\psi:(\C^n,0)\dashrightarrow\P^{m-1}$ given by $\psi=[f_1:\dots:f_m]$. Then $X_0$ is the normalization of the closure of the graph of $\psi$, and $\pi_0:X_0\to(\C^n,0)$ is the natural projection. Denote by $\Psi_0:X_0\to\P^{m-1}$ the other projection. It is a holomorphic map. The image under $\Psi_0$ of any prime divisor $E\subset\pi_0^{-1}(0)$ has dimension $n-1$.
Log resolutions {#logres}
---------------
The normalized blowup can be quite singular, making it difficult to use for analysis. Therefore, we shall use a *log-resolution* of ${\mathfrak{a}}$, see [@L2 Definition 9.1.12].
\[loglog\] A *log-resolution* of ${\mathfrak{a}}$ is a holomorphic modification $\pi:X\to(\C^n,0)$, where $X$ is a complex manifold, such that
- $\pi$ is an isomorphism above $\C^n\setminus\{0\}$:
- ${\mathfrak{a}}\cdot{\mathcal{O}}_X={\mathcal{O}}_X(-Z)$, where $Z=Z({\mathfrak{a}})$ is an effective divisor on $X$ with simple normal crossings support.
The simple normal crossings condition means that the exceptional divisor $\pi^{-1}(0)$ is a union of finitely many prime divisors $E_1,\dots,E_N$, called *exceptional primes*, and at any point $x\in\pi^{-1}(0)$ we can pick local coordinates $(\sigma_1,\dots,\sigma_n)$ at $x$ such that $\pi^{-1}(0)=\{\sigma_1\cdot\dots\cdot\sigma_p=0\}$ and for each exceptional prime $E$, either $x\not\in E$, or $E=\{\sigma_i=0\}$ for some $i\in\{1,\dots p\}$.
If we write $Z=\sum_{j=1}^Na_jE_j$, then the condition ${\mathfrak{a}}\cdot{\mathcal{O}}_X={\mathcal{O}}_X(-Z)$ means that (the pullback to $X$ of) any holomorphic germ $g\in\a$ vanishes to order at least $a_j$ along each $E_j$. Moreover, in the notation above, if $x\in\pi^{-1}(0)$ and $E_{j_k}=\{\sigma_k=0\}$, $1\le k\le p$ are the exceptional primes containing $x$, then there exists $g\in{\mathfrak{a}}$ such that $g=\sigma_1^{a_1}\dots \sigma_p^{a_p}u$, where $u$ is a unit in ${\mathcal{O}}_{X,x}$, that is, $u(x)\neq 0$.
The existence of a log-resolution is a consequence of Hironaka’s theorem on resolution of singularities. Indeed, the ideal ${\mathfrak{a}}$ is already principal on the normalized blowup $X_0$, so it suffices to pick $X$ as a desingularization of $X_0$. This gives rise to a commutative diagram $$\xymatrix{
&&
X
\ar[d]_{\varpi}
\ar[ddll]_{\pi}
\ar[ddrr]^{\Psi}
&&
\\
&&
X_0
\ar[dll]_(.3){\pi_0}
\ar[drr]^(.3){\Psi_0}
&&
\\
(\C^n,0)
\ar@{-->}[rrrr]^{\psi}
&&&&
\P^{m-1}
}$$ Here $\Psi:X\to\P^{m-1}$ is holomorphic.
Every exceptional prime $E$ of a log resolution $\pi:X\to(\C^n,0)$ of ${\mathfrak{a}}$ defines a divisorial valuation $\ord_E$, but not all of these are Rees valuations of ${\mathfrak{a}}$. If $\ord_E$ is a Rees valuation, we call $E$ a *Rees divisor*. From the diagram above we see:
An exceptional prime $E$ of $\pi$ is a Rees divisor of ${\mathfrak{a}}$ if and only if its image $\Psi(E)\subset\P^{m-1}$ has dimension $n-1$.
For completeness we give two results, the second of which will be used in Example \[icke-monom\].
\[intrees\] Let $E$ be an exceptional prime of a log resolution $\pi:X\to(\C^n,0)$ of ${\mathfrak{a}}$. Then the intersection number $((-Z({\mathfrak{a}}))^{n-1}\cdot E)$ is strictly positive if $E$ is a Rees divisor of ${\mathfrak{a}}$ and zero otherwise.
On the normalized blowup $X_0$, we may write ${\mathfrak{a}}\cdot{\mathcal{O}}_{X_0}={\mathcal{O}}_{X_0}(-Z_0)$, where $-Z_0$ is an *ample* divisor. Then ${\mathfrak{a}}\cdot{\mathcal{O}}_X={\mathcal{O}}_X(-Z)$, where $Z=\varpi^*Z_0$. It follows that $((-Z^{n-1})\cdot E)=((-Z_0^{n-1})\cdot\varpi_*E)$. The result follows since $-Z_0$ is ample and since $E$ is a Rees divisor if and only if $\varpi_*(E)\ne0$.
\[dimension2\] In dimension $n=2$, the Rees valuations of a product ${\mathfrak{a}}={\mathfrak{a}}_1\cdot\dots\cdot{\mathfrak{a}}_k$ of ${\mathfrak{m}}$-primary ideals is the union of the Rees valuations of the ${\mathfrak{a}}_i$.
Pick a common log-resolution $\pi:X\to(\C^n,0)$ of all the ${\mathfrak{a}}_i$. Then ${\mathfrak{a}}_i\cdot{\mathcal{O}}_X=\O_X(-Z_i)$ and ${\mathfrak{a}}\cdot{\mathcal{O}}_X=\O_X(-Z)$, where $Z=\sum_iZ_i$. Fix an exceptional prime $E$. By Proposition \[intrees\] we have $(Z_i\cdot E)\le 0$ with strict inequality if and only if $E$ is a Rees divisor of ${\mathfrak{a}}_i$. Thus $(Z\cdot E)=\sum_i(Z_i\cdot E)\le0$ with strict inequality if and only $E$ is a Rees divisor of some ${\mathfrak{a}}_i$. The result now follows from Proposition \[intrees\].
Essential multi-indices {#essential}
-----------------------
In our situation, we are given an ${\mathfrak{m}}$-primary ideal ${\mathfrak{a}}$ as well as a fixed set of generators $f_1,\dots,f_m$ of ${\mathfrak{a}}$.
Consider a multi-index ${\mathcal{I}}=\{i_1,\dots,i_n\}\subseteq\{1,\dots,m\}$. Let $\pi_{\mathcal{I}}:\P^{m-1}\setminus W_{\mathcal{I}}\to\P^{n-1}$, where $W_{\mathcal{I}}:=\{w_{i_1}=\dots=w_{i_n}=0\}\subset\P^{m-1}$, be the projection given by $[w_1:\dots:w_m]\to[w_{i_1}:\dots:w_{i_n}]$. Define $\Psi_{\mathcal{I}}:X\dashrightarrow\P^{n-1}$ by $\Psi_{{\mathcal{I}}}:=\pi_{{\mathcal{I}}}\circ\Psi$.
Let $E\subset X$ be an exceptional prime. We say that ${\mathcal{I}}$ is *$E$-essential* or that $\I$ is *essential with respect to $E$* if $\Psi(E)\not\subset W_{\mathcal{I}}$ and if $\Psi_{{\mathcal{I}}}|_E:E\dashrightarrow\P^{n-1}$ is dominant, that is, $\Psi_{\mathcal{I}}(E)$ is not contained in a hypersurface. We say that $\I$ is *essential* if it is essential with respect to at least one exceptional prime.
If ${\mathcal{I}}$ is $E$-essential, then $E$ must be a Rees divisor of ${\mathfrak{a}}$, so, in fact, $\I$ is essential if it is essential with respect to at least one Rees divisor. Conversely, if $E$ is Rees divisor of ${\mathfrak{a}}$, then there exists at least one $E$-essential multi-index ${\mathcal{I}}$. Observe, however, that $\I$ can be essential with respect to more than one $E$, and conversely that there can be several $E$-essential multi-indices; compare to the discussion at the end of Section \[monomialcase\] and the examples in Section \[teflon\].
Consider an exceptional prime $E$ of $\pi$ and a point $x\in E$ not lying on any other exceptional prime. Pick local coordinates $(\sigma_1,\dots,\sigma_n)$ at $x$ such that $E=\{\sigma_1=0\}$. We can write $f_i=\sigma_1^af'_i$, for $1\le i\le m$, where $a=\ord_E({\mathfrak{a}})$ and $f'_i\in{\mathcal{O}}_{X,x}$. The holomorphic functions $f'_i$ can be viewed as local sections of the line bundle ${\mathcal{O}}_X(-Z)$ and there exists at least one $i$ such that $f'_i(x)\ne0$.
\[bralemma\] A multi-index ${\mathcal{I}}=\{i_1,\dots,i_n\}$ is $E$-essential if and only if the form $$\label{rosa}
\sum_{k=1}^n (-1)^{k-1} f'_{i_k} df'_{i_1}\wedge\dots\wedge
\widehat{df'_{i_k}}\wedge\dots\wedge df'_{i_n}$$ is generically nonvanishing on $E$.
\[regn\] Observe in particular that $$\label{regna}
\ord_E(f_{i_1})=\ldots=\ord_E(f_{i_n})=\ord_E({\mathfrak{a}})$$ if $\I$ is $E$-essential.
Locally on $E$ (where $f_j'\neq 0$) we have that $$\Psi_\I=
\left [\frac{f_1'}{f_j'}: \ldots : \frac{f_{j-1}'}{f_j'}:\frac{f_{j+1}'}{f_j'}:
\ldots \frac{f_n'}{f_j'}
\right ].$$ Note that $\Psi_\I$ is dominant if (and only if) $\jac (\Psi_\I)$ is generically nonvanishing, or equivalently the holomorphic form $$\label{twin}
\partial \left ( \frac{f_1'}{f_j'} \right )\wedge
\ldots\wedge
\partial \left ( \frac{f_{j-1}'}{f_j'} \right )\wedge
\partial \left ( \frac{f_{j+1}'}{f_j'} \right )\wedge
\ldots\wedge
\partial \left ( \frac{f_n'}{f_j'} \right )$$ is generically nonvanishing. But is just a nonvanishing function times .
Proof of Theorem B {#coeff}
==================
Throughout this section let $\a$ denote the ideal $(f)$. Let us first prove that $R^f_\I\not\equiv 0$ implies that $\I$ is essential. Let $\pi: X\to (\C^n,0)$ be a log-resolution of $\a$. By standard arguments, see [@PTY], [@A] etc., the analytic continuation to $\lambda=0$ of $$\label{uppe}
\pi^*(\dbar|f|^{2\lambda}\wedge u)$$ exists and defines a globally defined current on $X$, whose push-forward by $\pi$ is equal to $R$; we denote this current by $\widetilde R$, so that $R=\pi_*\widetilde R$. Indeed, provided that the analytic continuation of exists, we get by the uniqueness of analytic continuation $$\begin{gathered}
\label{muck}
\pi_* \widetilde R\cdot\Phi=
\pi_* (\pi^* (\dbar |f|^{2\lambda}\wedge u))\cdot\Phi|_{\lambda=0}=\\
\pi^* (\dbar |f|^{2\lambda}\wedge u)\cdot\pi^*\Phi|_{\lambda=0}=
\dbar |f|^{2\lambda}\wedge u\cdot \Phi|_{\lambda=0}= R\cdot\Phi.\end{gathered}$$
In the same way we define currents $$\widetilde R_\I=\pi^*(\dbar|f|^{2\lambda} \wedge u_\I)|_{\lambda=0},$$ where $$u_\I=\frac{s_\I\wedge (\dbar s_\I)^{n-1}}{|f|^{2n}}.$$ Let $E$ be an exceptional prime and let us fix a chart $\U$ in $X$ such that $\U\cap E \neq \emptyset$ and local coordinates $\sigma$ so that the pull-back of $f$ is of the form $\pi^* f=\mu f'$, where $\mu$ is a monomial, $\mu=\sigma_1^{a_1}\cdots \sigma_n^{a_n}$ and $f'$ is nonvanishing, and moreover $E=\{\sigma_1=0\}$, see Section \[logres\]. Then $\pi^* s_\I=\overline \mu s'_\I$ for some nonvanishing section $s'_\I$ and $\pi^*|f|^2=|\mu|^2\nu$, where $\nu=|s'|^2$ is nonvanishing. Hence, using $$\widetilde R_\I=
\dbar(|\mu|^{2\lambda}\nu^\lambda)\frac {s'_\I\wedge (\dbar s'_\I)^{n-1}}{\mu^{n}\nu^n}\Big |_{\lambda=0}$$ which by is equal to $$\sum_{i=1}^n
\left[\frac{1}{\sigma_1^{na_1}\cdots
\sigma_{i-1}^{na_{i-1}}\sigma_{i+1}^{na_{i+1}} \cdots \sigma_n^{na_n}}\right ]\dbar\left[\frac{1}{\sigma_i^{na_i}}\right ]
\wedge\frac {s'_\I\wedge (\dbar s'_\I)^{n-1}}{\nu^n}.$$ Thus $\widetilde R$ and $\widetilde R_\I$ are pseudomeromorphic in the sense of [@AW2] and so it makes sense to take restrictions of them to subvarieties of their support, see Section \[pseudo\].
\[sing\] Let $E$ be an exceptional prime. The current $\widetilde R_\I\1_E$ vanishes unless $\I$ is essential with respect to $E$. Moreover $\widetilde R_\I\1_E$ only depends on the $f_k$ which satisfy that $\vE(f_k)=\vE(\a)$.
Recall (from Section \[pseudo\]) that $\dbar[1/\sigma_i^{a}]$ has the standard extension property with respect to $E=\{\sigma_i=0\}$. Thus $$\label{star}
\widetilde R_\I\1_E=
\left[\frac{1}{\sigma_2^{na_2}\cdots \sigma_n^{na_n}}\right ]\dbar\left[\frac{1}{\sigma_1^{na_1}}\right ]
\wedge\frac {s'_\I\wedge (\dbar s'_\I)^{n-1}}{\nu^n} \1_E.$$ It follows that $\widetilde R_\I\1_E$ vanishes unless $$s'_\I\wedge (\dbar s'_\I)^{n-1}\1_E \not \equiv 0,$$ which by Lemma \[bralemma\] is equivalent to that $\I$ is $E$-essential. Indeed, note that the coefficient of $f'\wedge (\dbar f')^{n-1}$ is $(n-1)!$ times .
For the second statement, recall that $\nu=|s'|^2=\sum |\pi^* \bar f_k/\bar\sigma_1^{a_1}|^2$. Note that $\pi^* \bar f_k/\bar\sigma_1^{a_1}\1_E=0$ if and only if $\pi^* \bar f_k/\bar\sigma_1^{a_1}$ is divisible by $\bar \sigma_1$, that is, $\vE(f_k)>\vE(\a)$. Hence $\widetilde R_\I\1_E$ only depends on the $f_k$ for which $\vE(f_k)=\vE(\a)$, compare to .
\[sepremark\] In light of the above proof, $\widetilde R\1_E$ has the SEP with respect to $E$. This follows since $\widetilde R\1_{E}$ is of the form and $\dbar[1/\sigma_1^a]$ has the SEP with respect to $E=\{\sigma_1=0\}$, see Section \[pseudo\].
Next, let us prove that $R^f_\I \not\equiv 0$ as soon as $\I$ is essential. In order to do this we will use arguments inspired by [@A2]. Throughout this section let $\widetilde M_\I$ denote the current $\widetilde R_\I \wedge \pi^* (df_\I/ (2 \pi i))^n/n!$ on $X$. Here $e^*_{i_1}\wedge\cdots\wedge e^*_{i_n}\wedge e_{i_n}\wedge\cdots\wedge e_{i_1}=e^*_\I\wedge e_\I$ should be interpreted as 1 so that in fact $\pi_*(\widetilde M_\I) =
R^f_\I \wedge df_{i_n}\wedge\cdots\wedge df_{i_1}/(2\pi i)^n$.
\[mcurrent\] The $(n,n)$-current $\widetilde M_\I$ is a positive measure on $X$ whose support is precisely the union of exceptional primes $E$ for which $\I$ is $E$-essential.
Note that Lemma \[sing\] implies that the support of $\widetilde M_\I$ is contained in the union of exceptional primes for which $\I$ is $E$-essential. Let $E$ be such a divisor and let us fix a chart $\U$ and local coordinates $\sigma$ as in the proof of Lemma \[sing\]. Then $\widetilde R_\I\1_E$ is given by . We can always write $s'_\I \wedge (\dbar s'_\I)^{n-1}$ as $$s'_\I \wedge (\dbar s'_\I)^{n-1} =
(\bar\beta
\widehat {d\bar \sigma_1} + d\bar\sigma_1 \wedge \bar \gamma)
\wedge e_\I,$$ where $\widehat{d\bar \sigma_1}$ denotes $d\bar \sigma_2\wedge\cdots \wedge d\bar \sigma_n$, $\beta$ is a holomorphic function, and $\gamma$ is a holomorphic form. Moreover, since $\I$ is $E$-essential, $s'_\I \wedge (\dbar s'_\I)^{n-1}|_E = \beta|_E \widehat {d\bar \sigma_1} \wedge e_\I$ is generically nonvanishing by Lemma \[bralemma\] (in particular, $\beta|_E$ is generically nonvanishing).
Moreover, with $\overline {e_j}$ interpreted as $e_j^*$, we have $$\begin{gathered}
\pi^*(df_\I)^n= \pi^*(\partial \bar s_\I)^n = \partial (\bar s_\I \wedge (\partial \bar s_\I)^{n-1})= \\
\partial (\sigma_1^{na_1}\cdots\sigma_n^{na_n}
(\beta
\widehat{d\sigma_1}
+ d\sigma_1 \wedge \gamma)
)\wedge e_\I^*
=\\
na_1 \sigma_1^{na_1-1}
(\sigma_2^{na_2} \cdots\sigma_n^{na_n} \beta +
\sigma_1\delta)
d \sigma
\wedge e_\I^*, \end{gathered}$$ where $\delta$ is some holomorphic function, $d \sigma$ denotes $d\sigma_1\wedge\cdots \wedge d \sigma_n$, and $e_\I^*=e_{i_1}^*\wedge\cdots\wedge e_{i_n}^*$.
Hence, using , we get $$\begin{gathered}
\label{kol}
\widetilde M_\I\1_E=
\widetilde R_\I\1_E \wedge \left ( \frac {\pi^*(df_\I)}{2 \pi i} \right )_n =
\\
\frac{1}{n!}
\left[\frac{1}{\sigma_2^{na_2}\cdots \sigma_n^{na_n}}\right ]\dbar\left[\frac{1}{\sigma_1^{na_1}}\right ]
\wedge\frac
{\beta ~ \widehat {d\bar \sigma_1}}
{|f'|^{2n}}
\1_E
\\
\wedge na_1 \sigma_1^{na_1-1}
[\sigma_2^{na_2} \cdots \sigma_n^{na_n} \beta +
\sigma_1\delta]
d \sigma
\wedge e_\I^*\wedge e_\I
= \\
\frac{na_1}{(2\pi i)^n}
\dbar\left[\frac{1}{\sigma_1}\right ]
\frac{|\beta|^2}{|f'|^{2n}}
\widehat{d\bar \sigma_1}\wedge d \sigma \1_E.\end{gathered}$$ The right hand side of is just Lebesgue measure on $E$ times a smooth, positive, generically nonvanishing function. Hence $\widetilde M_\I$ is a positive current whose support is precisely the union of exceptional primes $E$ for which $\I$ is $E$-essential.
It follows from the above proof that $\widetilde M \1_E$ is absolutely continuous with respect to Lebesgue measure on $E$.
To conclude, the only if direction of Theorem B follows immediately from Lemma \[sing\]. Lemma \[mcurrent\] implies that $\pi_*(\widetilde M_\I)=R_\I \wedge df_{i_n}\wedge\cdots\wedge df_{i_1}/(2\pi i)^n=$ is a positive current with strictly positive mass if $\I$ is essential. In particular, $R^f_\I\not\equiv 0$, which proves the if direction of Theorem B. Hence Theorem B is proved.
Annihilators {#annsection}
============
We are particularly interested in the annihilator ideal of $R^f$. Recall from Theorem B that $R^f_\I\not\equiv 0$ if and only if $\I$ is essential. Hence $$\label{irreducible}
\ann R^f=\bigcap_{\I\text{ essential}} \ann R^f_\I.$$ In this section we prove Theorem A, which gives estimates of the size of $\ann R^f$. We also prove Theorem C, which gives an explicit description of $R^f$ in case $(f)$ is a complete intersection ideal. In fact, Theorems A and C are consequences of Theorem \[annihilatorn\] and Proposition \[denandra\] below.
\[annihilatorn\] Suppose that $f=(f_1,\ldots,f_m)$ generates an $\m$-primary ideal $\a\subset \O^n_0$. Let $R^f=(R^f_\I)$ be the corresponding Bochner-Martinelli residue current. Then $\ann R^f=\a$ if and only if $\a$ is a complete intersection ideal, that is, $\a$ is generated by $n$ germs of holomorphic functions.
Moreover if $\a$ is a complete intersection ideal, then for $\I=\{i_1,\ldots,i_n\}\subseteq\{1,\ldots,m\}$ $$\label{jojojo}
R^f_\I=C_\I~
\dbar\left [\frac{1}{f_{i_1}}\right ]\wedge\cdots\wedge
\dbar\left [\frac{1}{f_{i_n}}\right ],$$ where $C_\I$ is a non-zero constant if $f_{i_1},\ldots, f_{i_n}$ generates $\a$ and zero otherwise.
For $\I=\{i_1,\ldots, i_n\}\subseteq \{1,\ldots, m\}$, let $f_\I$ denote the tuple $f_{i_1}, \ldots, f_{i_n}$, which we identify with the section $\sum_{i\in\I}f_{i}e_{i}^*$ of $V$. To prove (the first part of) Theorem \[annihilatorn\] we will need two results.
The first result is a simple consequence of Lemma \[mcurrent\]. Given a tuple $g$ of holomorphic functions $g_1,\ldots , g_n\in\O_0^n$, let $\jac (g)$ denote the Jacobian determinant $\det |\frac{\partial g_i}{\partial z_j}|_{i,j}$.
\[pink\] We have that $\jac (f_\I)\in \ann R^f_\I$ if and only if $R^f_\I\equiv 0$.
The if direction is obvious. Indeed if $R^f_\I\equiv 0$, then $\ann R^f_\I=\O_0^n$.
For the converse, suppose that $R^f_\I\not\equiv 0$. From the previous section we know that this implies that $R_\I^f\wedge df_{i_n}\wedge\cdots\wedge df_{i_1}\not\equiv 0$. However the coefficient of $df_{i_n}\wedge\cdots\wedge df_{i_1}$ is just $\pm \jac (f_\I)$ and so $\jac (f_\I)\notin \ann R^f_\I$.
The next result is Theorem 1.1 and parts of the proof thereof in [@Hi]. Recall that the socle $\soc(N)$ of a module $N$ over a local ring $(R,\m)$ consists of the elements in $N$ that are annihilated by $\m$, see for example [@BH].
\[hickel\] Assume that $g_1,\ldots,g_n$ generate an ideal $\a\subset\O_0^n$. Then $\jac(g_1,\ldots,g_n)\in\a$ if and only if $\codim V(\a) < n$.
Moreover, if $\codim V(\a)=n$, then the image of $\jac (g)$ under the natural surjection $\O_0^n \to \O_0^n/\a$ generates the socle of $\O_0^n/\a$.
\[lambi\] If $R^f_\I\not\equiv 0$ and $\codim V(f_\I)=n$, then $\ann R^f_\I\subseteq (f_\I)$.
We claim that it follows that every $\m$-primary ideal $J\subset \O_0^n$ that does not contain $\jac (f_\I)$ is contained in $(f_\I)$. Applying the claim to $\ann R^f_\I\not\ni \jac (f_\I)$ (if $R^f_\I\not\equiv 0$) proves the lemma.
The proof of the claim is an exercise in commutative algebra; however, we supply the details for the reader’s convenience. Suppose that $J\subset \O_0^n$ is an $\m$-primary ideal such that $\jac (f_\I)\notin J$, but that there is a $g\in J$ such that $g\notin (f_\I)$. The latter condition means that $0\neq \tilde g \in \tilde J$, where $\tilde g$ and $\tilde J$ denote the images of $g$ and $J$, respectively, under the surjection $\O_0^n \to \O_0^n/(f_\I)$. Then, for some integer $\ell$, $\m^\ell \tilde g \neq 0$ but $\m^{\ell+1} \tilde g =0$ in $A:= \O_0^n/(f_\I)$; in other words $\m^\ell \tilde g$ is in the socle of $A$. According to Theorem \[hickel\], the socle of $A$ is generated by $\jac (f_\I)$ and so it follows that $\jac (f_\I)\in\tilde J$. This, however, contradicts the assumption made above and the claim is proved.
We first prove that $\ann R^f=\a$ implies that $\a$ is a complete intersection ideal. Let us therefore assume that $\ann R^f = \a$.
We claim that under this assumption, $\codim V(f_\I)=n$ as soon as $\I$ is essential. To show this, assume that there exists an essential multi-index $\I=\{i_1,\ldots, i_n\} \subseteq \{1, \ldots, m\}$ such that $\codim V(f_\I)<n$. Then by Theorem \[hickel\] $\jac (f_\I)\in (f_\I)\subseteq \a$. However, by Lemma \[pink\] $\jac (f_\I)\notin \ann R^f_\I$. Thus we have found an element that is in $\a$ but not in $\ann R^f$, which contradicts the assumption. This proves the claim.
Next, let us consider the inclusion $$\label{babel}
\bigcap_{\I \text{ essential}} (f_\I)\subseteq \a.$$ Assume that the inclusion is strict. By the claim above $\codim V(f_\I)=n$ if $\I$ is essential and so by Lemma \[lambi\] $$\ann R^f = \bigcap_{\I \text{ essential}} \ann R^f_\I
\subseteq \bigcap_{\I \text{ essential}} (f_\I) \varsubsetneq \a,$$ which contradicts the assumption that $\ann R^f = \a$. Hence equality must hold in , which means that $\a$ is generated by $f_\I$, whenever $\I$ is essential. (Note that there must be at least one essential multi-index if $R^f\not\equiv 0$.) To conclude, we have proved that $\ann R^f=\a$ implies that $\a$ is a complete intersection ideal.
It remains to prove that if $\a$ is a complete intersection ideal, then $R^f_\I$ is of the form if $f_\I$ generates $\a$ and zero otherwise. Indeed, if $R^f_\I$ is given by , then $\ann R^f_\I = (f_\I)=\a$ by the classical Duality Principle; see the Introduction. This means that $\ann R^f_\I$ is either $\a$ or (if $R^f_\I\equiv 0$) $\O_0^n$ and so $\ann R^f = \bigcap \ann R^f_\I = \a$.
Assume that $\a$ is a complete intersection ideal. Then, by Nakayama’s Lemma $\a$ is in fact generated by $n$ of the $f_i$, compare to the discussion just before Theorem C. Assume that $\a$ is generated by $f_1,\ldots, f_n$; then $f_\ell=\sum_{j=1}^n \varphi_j^\ell f_j$ for some holomorphic functions $\varphi_j^\ell$. (Note that $\varphi_j^\ell=\delta_{j,\ell}$ for $\ell \leq n$.)
We will start by showing that $R^f_\I$, where $\I=\{1,\ldots,n\}$, is of the form . Recall from Section \[rescurr\] that $$\label{forstas}
R_\I=
\dbar|f|^{2\lambda} \wedge \frac{s_\I\wedge (\dbar s_\I)^{n-1}}{|f|^{2n}}\bigg |_{\lambda=0}.$$ Let us now compare with the current $R(f_\I)$, that is, the residue current associated with the section $f_\I$ of the sub-bundle $\widetilde V$ of $V$ generated by $e_1^*,\ldots,e_n^*$. Since $\codim V(f_\I)=n$, the current $R(f_\I)$ is independent of the choice of Hermitian metric on $\widetilde V$ according to [@A Proposition 2.2]. More precisely, $$R(f_\I)=
\dbar|g|^{2\lambda} \wedge \frac{\tilde s_\I\wedge (\dbar \tilde s_\I)^{n-1}}{\|f_\I\|^{2n}}\bigg |_{\lambda=0},$$ where $\|\cdot \|$ is any Hermitian metric on $\widetilde V$, $\tilde s_\I$ is the dual section of $f_\I$ with respect to $\|\cdot\|$, and $g$ is any tuple of holomorphic functions that vanishes at $\{f_\I=0\}=\{0\}$; in particular, we can choose $g$ as $f$.
Let $\Psi$ be the Hermitian matrix with entries $\psi_{i,j}=\sum_{\ell=1}^m \varphi_i^\ell \bar \varphi_j^\ell$. Then $\Psi$ is positive definite and so it defines a Hermitian metric on $\widetilde V$ by $\|\sum_{i=1}^n \xi_i e_i\|^2=\sum_{1\leq i,j \leq n}\psi_{i,j} \xi_i \bar \xi_j$. Observe that $\|f_\I\|^2= |f_1|^2+\cdots + |f_m|^2$ and moreover that $\tilde s_\I=\sum_{1\leq i,j \leq n}
\psi_{i,j} \bar f_j e_i$. A direct computation gives that $\tilde s_\I \wedge (\dbar\tilde s_\I)^{n-1}=\det(\Psi) s_\I \wedge (\dbar s_\I)^{n-1}$. It follows that $R(f_\I)= C R_\I$, where $C=\det(\Psi(0))\neq 0$. By [@A Theorem 1.7] $R(f_\I)=\dbar[1/f_1]\wedge\cdots\wedge\dbar [1/f_n]\wedge e_n\wedge\cdots\wedge e_1$, and so we have proved that $R^f_\I$ is of the form .
Next, let $\L$ be any multi-index $\{\ell_1,\ldots,\ell_n\} \subseteq \{1,\ldots,m\}$. By arguments as above $s_\L \wedge (\dbar s_\L)^{n-1}=\det(\bar \Phi_\L) s_\I\wedge (\dbar s_\I)^{n-1}$, where $\Phi_\L$ is the matrix with entries $\varphi_j^{\ell_i}$. Hence $R_\L=C_\L R^f_\I e_{\ell_n}\wedge\cdots\wedge e_{\ell_1}$, where $C_\L=\det(\bar \Phi_\L(0)) $. Note that $C_\L$ is non-zero precisely when $f_1,\ldots,f_n$ can be expressed as holomorphic combinations of $f_{\ell_1},\ldots,f_{\ell_n}$, that is, when $f_{\ell_1},\ldots,f_{\ell_n}$ generate $\a$. Hence $R_\mathcal L$ is of the form if $f_\L$ generates $\a$ and zero otherwise, and we are done.
\[denandra\] Suppose that $f=(f_1,\ldots,f_m)$ generates an $\m$-primary ideal $\a\subset \O_0^n$, where $n\geq 2$. Let $R^f$ be the corresponding Bochner-Martinelli residue current. Then the inclusion $$\overline{\a^n}\subseteq \ann R^f$$ is strict.
Observe that Proposition \[denandra\] fails when $n=1$. Then, in fact, $\a=\ann R^f=\overline \a$.
We show that $\ann R^f\setminus \overline{\a^n}$ is non-empty. Consider multi-indices $\mathcal J=\{j_1,\ldots,j_n\}, \mathcal L=\{\ell_1,\ldots,\ell_n\}\subseteq\{1,\ldots,m\}$. By arguments as in the proof of Lemma \[mcurrent\] one shows that $$df_{j_1}\wedge\cdots\wedge df_{j_n}\wedge R^f_{\mathcal L}=
\jac(f_{\mathcal J})dz_1\wedge\cdots\wedge dz_n \wedge R^f_{\mathcal L}$$ either vanishes or is equal to a constant times the Dirac measure at the origin. Thus $z_k\jac(f_{\mathcal J}) R^f_{\mathcal L}=0$ for all coordinate functions $z_k$. It follows that $\m\jac(f_\I)\subseteq\ann R^f$ for all multi-indices $\I=\{i_1,\ldots,i_n\}$.
Next, suppose that $\I=\{i_1,\ldots,i_n\}$ is essential with respect to a Rees divisor $E$ of $\a$. Then a direct computation gives that $\vE(df_{i_1}\wedge\ldots\wedge df_{i_n})=n\vE(\a)$ and $\vE(dz_1\wedge\ldots\wedge dz_n)\geq \sum_{i=1}^n\vE(z_i)-1$. Note that $\vE(z_k)\geq 1$ for $1\leq k\leq n$. Since $df_{i_1}\wedge\cdots\wedge df_{i_n}=
\jac(f_{\mathcal I})dz_1\wedge\cdots\wedge dz_n$ it follows that $$\vE(z_k\jac(f_\I))\leq n~\vE(\a)-n+1=\vE(\overline{\a^n})-n+1$$ for $1\leq k\leq n$. Hence, if $n\geq 2$, there are elements, for example $z_k \jac(f_\I)$, in $\m \jac (f_\I)$ that are not in $\overline{\a^n}$. This concludes the proof.
Theorem A is an immediate consequence of (the first part of) Theorem \[annihilatorn\] and Proposition \[denandra\].
Suppose that $(f)$ is a complete intersection ideal. Then by Theorem B and (the second part of) Theorem \[annihilatorn\] we have $$\I \text{ essential } \Leftrightarrow R^f_\I \not\equiv 0
\Leftrightarrow f_\I \text{ generates } (f).$$ Moreover Theorem \[annihilatorn\] asserts that in this case $R^f_\I$ is of the form .
\[sockelsockel\] Let us conclude this section by a partial generalization of Theorem 3.1 in [@W]. Even though we cannot explicitly determine $\ann R^f$ we can still give a qualitative description of it in terms of the essential multi-indices.
The current $R^f_\I$ is a Coleff-Herrera current in the sense of Bj[ö]{}rk [@Bj], which implies that $\ann R^f_\I$ is irreducible, meaning that it cannot be written as an intersection of two strictly bigger ideals. Thus yields an *irreducible decomposition* of $\ann R^f$, that is, a representation of the ideal as a finite intersection of irreducible ideals, compare to [@W2 Corollary 3.4]. An ideal $\a$ in a local ring $A$ always admits an irreducible decomposition and the number of components in a minimal such is unique; if $\a$ is $\m$-primary it is equal to the minimal number of generators of the socle of $A/\a$, see for example [@HRS]. In light of we see that the number of components in a minimal irreducible decomposition of $\ann R^f$ is bounded from above by the number of essential multi-indices.
In fact Lemma \[mcurrent\] gives us even more precise information: if $\I$ is essential then $\soc(\O_0^n/\ann R^f_\I)$ is generated by the image of $\jac(f_\I)$ under the natural surjection $\O_0^n\to \O_0^n/\ann R^f_\I$. It follows that $\soc(\O^n_0/\ann R^f)$ is generated by the images of $\{\jac(f_\I)\}_{\I \text{ essential}}$ under the natural surjection $\O_0^n\to\O_0^n/\ann R^f$.
A geometric decomposition {#decompsection}
=========================
In this section we will see that the current $R^f$ admits a natural decomposition with respect to the Rees valuations of $\a=(f_1,\ldots,f_m)$.
Given a log-resolution $\pi: X\to (\C^n,0)$ of $\a$, recall from Section \[coeff\] that the analytic continuation of defines a $\Lambda^n V$-valued current $\widetilde R$ on $X$, such that $\pi_* \widetilde R= R$. Let $\widetilde R^f$ denote the corresponding vector-valued current, that is, the current with the coefficients of $\widetilde R$ as entries. From Lemma \[sing\] and Remark \[sepremark\] we know that $\widetilde R^f$ has support on and the SEP with respect to the Rees divisors associated with $\a$. Hence $\widetilde R^f$ can naturally be decomposed as $\sum_{E \text{ Rees divisor}} \widetilde R^f \1_E$. Given a Rees divisor $E$ in $X$, let us consider the current $R^E:=\pi_* (\widetilde R^f\1_E)$.
\[oberoende\] The current $R^E$ is independent of the log-resolution.
Throughout this proof, given a log-resolution $\pi:X\to (\C^n,0)$, let $\widetilde R_X$ denote the current $\widetilde R$ on $X$, that is, the value of at $\lambda=0$, and let $E_X$ denote the divisor on $X$ associated with the Rees valuation $\vE$.
Any two log-resolutions can be dominated by a third, see for example [@L2 Example 9.1.16]. To prove the lemma it is therefore enough to show that $\pi_*(\widetilde R_X\1_{E_X})=\pi_*\varpi_*(\widetilde R_Y\1_{E_Y})$ for log-resolutions $$Y \stackrel{\varpi}{\longrightarrow} X \stackrel{\pi}{\longrightarrow} (\C^n,0)$$ of $\a$.
We will prove the slightly stronger statement that $\widetilde R_X\1_{E_X}=\varpi_*(\widetilde R_Y\1_{E_Y})$. Observe that $\widetilde R_X=\varpi_* \widetilde R_Y$; compare to . Moreover note that $\varpi^{-1}(E_X)=E_Y \cup \bigcup E'$, where each $E'$ is a divisor such that $\varpi(E')$ is a proper subvariety of $E_X$ (whereas $\varpi(E_Y)=E_X$). Let $A_Y = E_Y\setminus \bigcup E'$ and $A_X=\varpi(A_Y)$. Then $A_X$ and $A_Y$ are Zariski-open sets in $E_X$ and $E_Y$, respectively, and $\varpi^{-1}(A_X)=A_Y$. By Remark \[sepremark\] $\widetilde R$ has the SEP with respect to the exceptional divisors, and so, using we can now conclude that $$\widetilde R_X\1_{E_X}=
\widetilde R_X\1_{A_X}=
\varpi_*(\widetilde R_Y\1_{A_Y})=
\varpi_*(\widetilde R_Y\1_{E_Y}).$$
\[reesdec\] Suppose that $f=(f_1,\ldots,f_m)$ generates an $\m$-primary ideal $\a\subset\O^n_0$. Let $R^f$ be the corresponding Bochner-Martinelli residue current. Then $$\label{lupp}
R^f=\sum R^E,$$ where the sum is taken over Rees valuations $\vE$ of $\a$ and $R^E$ is defined as above. Moreover each summand $R^E$ is $\not\equiv 0$ and depends only on the $f_j$ for which $\vE(f_j)=\vE(\a)$.
Assume that $E$ is a Rees divisor. By Section \[essential\] there is at least one $E$-essential multi-index; let $\I$ be such a multi-index. Then, by (the proof of) Theorem B the current $\pi_* (\widetilde R_\I\1_E)\not\equiv 0$, which means that $R^E$ has at least one nonvanishing entry.
We also get that $\widetilde R^f$ has support on the union of the Rees divisors. Moreover, by Remark \[sepremark\] $\widetilde R^f\1_E$ has the SEP with respect to $E$. Thus $$\widetilde R^f=\widetilde R^f\1_{\bigcup_{E \text{ Rees divisor}} E}=
\sum_{E \text{ Rees divisor}} \widetilde R^f\1_E,$$ which proves .
The last statement follows immediately from the second part of Lemma \[sing\].
The monomial case {#monomialcase}
=================
Let $\a \subset \O_0^n$ be an $\m$-primary monomial ideal generated by monomials $z^{a^j}$, $1\leq j\leq m$. Recall that the Newton polyhedron $\np(\a)$ is defined as the convex hull in $\mathbb R^n$ of the exponent set $\{a^j\}$ of $\a$. The Rees-valuations of $\a$ are monomial and in 1-1 correspondence with the compact facets (faces of maximal dimension) of $\np(\a)$. More precisely the facet $\tau$ with normal vector $\rho=(\rho_1, \ldots, \rho_n)$ corresponds to the monomial valuation $\ord_\tau(z_1^{a_1}\cdots z_n^{a_n})=\rho_1a_1+\ldots + \rho_na_n$, see for example [@HS Theorem 10.3.5].
Let us interpret our results in the monomial case. First, consider the notion of essential multi-indices. Note that a monomial $z^{a}\in\a$ satisfies that $\ord_\tau(z^{a})=\ord_\tau(\a)$ precisely if $a$ is contained in the facet $\tau$. Thus in light of a necessary condition for $\I=\{i_1,\ldots, i_n\}\subseteq\{1,\ldots,m\}$ to be $E_\tau$-essential (if $E_\tau$ denotes the Rees divisor associated with $\tau$) is that $\{a^i\}_{i\in\I}$ are all contained in $\tau$. Moreover, for to be nonvanishing the determinant $|a^i|$ has to be non-zero; in other words $\{a^i\}_{i\in\I}$ needs to span $\R^n$. In [@W] an exponent set $\{a^i\}_{i\in\I}$ was said to be essential if all $a^i$ are contained in a facet of $\np(\a)$ and $|a^i|\neq 0$. Our notion of essential is thus a direct generalization of the one in [@W]. Moreover Theorem B can be seen as a generalization of (the first part of) Theorem 3.1 in [@W], which asserts that $R^f_\I\not\equiv 0$ precisely if $\I$ is essential. In fact, Theorem 3.1 also gives an explicit description of $\ann R^f_\I$. Moreover, Theorem \[annihilatorn\] and Proposition \[denandra\] are direct generalizations of Theorem 3.2 and Corollary 3.9, respectively, in [@W].
Concerning the decomposition in Section \[decompsection\] observe that in the monomial case each multi-index $\I$ can be essential with respect to at most one Rees divisor. Indeed, clearly a set of points in $\R^n$ cannot be contained in two different facets and at the same time span $\R^n$. Hence in the monomial case the decomposition $R^f=(R^f_\I)$ is a refinement of the decomposition ; in fact the nonvanishing entries of $R^E$ are precisely the $R^f_\I$ for which $\I$ is $E$-essential. In particular, $$\ann R =\bigcap \ann R^E \quad \text { and } \quad \ann R^E=\bigcap_{\I ~~ E-\text{essential}}\ann R^f_\I.$$ This is however not true in general. For example, if $n=m$, the set $\I=\{1,\ldots, n\}$ is essential with respect to all Rees divisors of $\a$ (and the number of Rees divisors can be $ > 1$). Also, in general, $\bigcap \ann R^E$ is strictly included in $\ann R$, see Example \[telifon\].
Examples {#teflon}
========
Let us consider some examples that illustrate the results in the paper.
\[concrete\] [@W Example 3.4] Let $\a\subset \O_0^2$ be the monomial ideal $(f_1,\ldots, f_5)=(z^8, z^6w^2, z^2w^3, zw^5, w^6)$. The exponent set of $\a$ is depicted in Figure 1, where we have also drawn $\np(\a)$.
\[lefigur\]
![The exponent set and Newton polyhedron of $\a$ in Example \[concrete\]](monomial-0)
The Newton polyhedron has two facets with normal directions $(1,2)$ and $(3,2)$ respectively. Thus there are two Rees divisors $E_1$ and $E_2$ associated with $\a$ with monomial valuations $\ord_{E_1}(z^a w^b)=a + 2 b$ and $\ord_{E_2}(z^a w^b)=3 a + 2 b$, respectively. Now the index sets $\{1,2\}$, $\{1,3\}$, and $\{2,3\}$ are essential with respect to $E_1$ whereas $\{3,5\}$ is $E_2$-essential. Thus according to Theorem B $R^f$, which a priori has one entry for each multi-index $\{i,j\}\subseteq\{1,\ldots,5\}$, has four non-zero entries corresponding to the four essential index sets. Moreover, by Lemma \[pink\] and Remark \[sockelsockel\], we have that for these index sets $\jac(f_\I)\notin \ann R^f$, whereas $\m\jac(f_\I)\subseteq\ann R^f$. For example, $\jac (z^6 w^2, z^2 w^3) = 14 z^7 w^4 \notin \ann R^f$, and thus, since $z^7 w^4 \in \a$, one sees directly that $\ann R^f\varsubsetneq \a$. Moreover $z\jac (z^6 w^2, z^2 w^3) = 14 z^8 w^4 \in \overline{\a^2}\setminus\ann R^f$.
\[icke-monom\] Let $\a\subset \O_0^2$ be the product of the ideals $\a_1=(z, w^2)$, $\a_2=(z-w,w^2)$, and $\a_3=(z+w, w^2)$, each of which is monomial in suitable local coordinates. The ideal $\a_i$ has a unique (monomial) Rees-valuation $\ord_{E_i}$, given by $\ord_{E_1}(z^a w^b) = 2a + b$, $\ord_{E_2}((z-w)^a w^b) = 2a + b$, and $\ord_{E_3}((z+w)^a w^b) = 2a + b$, respectively. By Corollary \[dimension2\] the Rees-valuations of $\a$ are precisely $\ord_{E_1}$, $\ord_{E_2}$, and $\ord_{E_3}$.
Note that after blowing up the origin once, the strict transform of $\a$ has support at exactly three points $x_1$, $x_2$, $x_3$ on the exceptional divisor; it follows that $\a$ is not a monomial ideal. A log-resolution $\pi:X\to(\C^2,0)$ of ${\mathfrak{a}}$ is obtained by further blowing up $x_1$, $x_2$ and $x_3$, thus creating exceptional primes $E_1$, $E_2$ and $E_3$.
Now $\a$ is generated by $$\{f_1,\ldots,f_4\}=
\{z(z-w)(z+w), ~z(z-w)w^2, ~z(z+w)w^2, ~(z-w)(z+w)w^2\}.$$ Observe that none of these generators can be omitted; hence $\a$ is not a complete intersection ideal. Also, note that for each Rees divisor there is exactly one essential $\I\subseteq\{1,\ldots,4\}$. For example $\ord_{E_1}(f_1)=\ord_{E_1}(f_4)=\ord_{E_1}(\a)=4$, whereas $\ord_{E_1}(f_k)>4$ for $k=2,3$, and so $\I=\{1,4\}$ is the only $E_1$-essential index set. For symmetry reasons, $\{1,3\}$ is $E_2$-essential and $\{1,2\}$ is $E_3$-essential.
Let us compute $R^f_{\{1,4\}}$. To do this, let $y\in X$ be the intersection point of $E_1$ and the strict transform of $\{z=0\}$. We choose coordinates $(\sigma,\tau)$ at $y$ so that $E_1=\{\sigma=0\}$ and $(z,w)=\pi(\sigma,\tau)=(\sigma^2\tau,\sigma)$. Then $\pi^* s_{\{1,4\}}= \bar \sigma^4 (1-\bar\sigma^2\bar\tau^2) ( \bar \tau e_1 + e_4)$ and it follows that $$\widetilde R_{\{1,4\}}=
- ~\dbar\left [\frac{1}{\sigma^8}\right ] \wedge \frac{d\bar \tau}{(1+|\tau|^2)^2}\wedge e_4\wedge e_1.$$ Let $\phi=\varphi dw\wedge dz$ be a test form at $0\in\C^n$. Near $y\in X$ we have $\pi^* dw\wedge dz = \sigma^2 d\sigma \wedge d\tau$ and so $$\begin{gathered}
R^f_{\{1,4\}}\cdot \phi=
\int
\dbar\left [\frac{1}{\sigma^6}\right ] \wedge d\sigma \wedge \frac{d\bar \tau\wedge d\tau }{(1+|\tau|^2)^2}
~\varphi(\sigma^2 \tau, \sigma)
=\\
\frac{2\pi i}{5!} ~\varphi_{0,5}(0, 0) \int_\tau
\frac{d\bar \tau\wedge d\tau}{(1+|\tau|^2)^2}
=
\frac{(2\pi i)^2}{5!} ~\varphi_{0,5}(0, 0)
=
\dbar\left [\frac{1}{z}\right ]\wedge
\dbar\left [\frac{1}{w^6}\right ]\cdot\phi.\end{gathered}$$ Hence $\ann R^f_{\{1,4\}}=(z,w^6)$. Similarly, $\ann R^f_{\{1,3\}}=(z-w,w^6)$ and $\ann R^f_{\{1,2\}}=(z+w,w^6)$, and so $$\ann R^f =
(z(z-w)(z+w), w^6).$$ Note in particular that $\ann R^f\varsubsetneq \a$ in accordance with Theorem \[annihilatorn\].
\[fragan\] Let $\a\in\O^2_0$ be the monomial ideal $(z^2, zw, w^2)$ and let $f=f(B)$ be the tuple of generators: $f=(f_1,f_2,f_3)=(z^2, zw + w^2, B w^2)$. A computation similar to the one in Example \[icke-monom\] yields that $$R^f_{\{1,2\}} =
C_0 ~\dbar\left [\frac{1}{z^3}\right ]\wedge \dbar\left [\frac{1}{w}\right ]
+ 2 ~ C_1 ~\dbar\left [\frac{1}{z^2}\right ]\wedge \dbar\left [\frac{1}{w^2}\right ],$$ where $$C_\ell=\frac{1}{2\pi i}\int\frac{|\tau|^{2\ell} d\bar\tau\wedge d\tau}
{(1+|\tau|^2|1+\tau|^2 + |B|^2|\tau|^4)^2}.$$ Note that $R^f_{\{1,2\}}$ and its annihilator ideal depend not only on $f_1$ and $f_2$ but also on $f_3$. Indeed, a polynomial of the form $D z^2-Ew$ is in $\ann R^f_{\{1,2\}}$ if and only if $D/E=2C_1/C_0$, but $2C_1/C_0$ depends on the parameter $B$.
However, $\ann R^f$ is independent of $B$. In fact, $\ann R^f_{\{1,3\}}=(z^2, w^2)$ and $\ann R^f_{\{2,3\}}=(z, w^3)$, which implies that $\ann R^f=\bigcap \ann R^f_\I=(z^3, z^2w, zw^2, w^3)$.
\[Dremark\] Example \[fragan\] shows that the vector valued current $R^f$ depends on the choice of the generators of the ideal $(f)$ in an essential way. Still, in this example $\ann R^f$ stays the same when we vary $f$ by the parameter $B$. Also, we would get the same annihilator ideal if we chose $f$ as $(z^2, zw, w^2)$, see [@W Theorem 3.1].
We have computed several other examples of currents $R^f$ in all of which $\ann R^f$ is unaffected by a change of $f$ as long as the ideal $(f)$ stays the same. To be able to answer Question D in general, however, one probably has to understand the delicate interplay between contributions to $R^f$ and $R^f_\I$ from different Rees divisors, compare to Example \[telifon\] below.
\[telifon\] Let $\a\in\O^2_{0}$ be the complete intersection ideal $(f_1,f_2)=(z^3,w^2-z^2)$. After blowing up the origin the strict transform of $\a$ has support at two points $x_1$ and $x_2$ corresponding to where the strict transforms of the lines $z=w$ and $z=-w$, respectively, meet the exceptional divisor. Further blowing up these points yields a log-resolution of $\a$ with Rees divisors $E_1$ and $E_2$ corresponding to $x_1$ and $x_2$, respectively.
A computation as in Example \[icke-monom\] yields that $$\begin{gathered}
2 R^{E_1}=
-\dbar\left [\frac{1}{z^4}\right ]\wedge \dbar\left [\frac{1}{w}\right ]
+\dbar\left [\frac{1}{z^3}\right ]\wedge \dbar\left [\frac{1}{w^2}\right ]\\
-\dbar\left [\frac{1}{z^2}\right ]\wedge \dbar\left [\frac{1}{w^3}\right ]
+\dbar\left [\frac{1}{z}\right ]\wedge \dbar\left [\frac{1}{w^4}\right ];\end{gathered}$$ $R^{E_2}$ looks the same but with the minus signs changed to plus signs. Hence $$R^f=R^{E_1}+R^{E_2}=
\dbar\left [\frac{1}{z^3}\right ]\wedge \dbar\left [\frac{1}{w^2}\right ]
+\dbar\left [\frac{1}{z}\right ]\wedge \dbar\left [\frac{1}{w^4}\right ] .$$ Note that $\ann R^f$ is indeed equal to $\a$, which we already knew by the Duality Principle. Observe furthermore that $z^3 R^{E_1}= - \dbar [1/z]\wedge\dbar [1/w]$, so that $z^3\notin\ann R^{E_1}$. Hence we conclude that in general $$\bigcap \ann R^E\varsubsetneq\ann R^f.$$
\#1\#2\#3[[\#1]{}: [*\#2*]{}, \#3.]{}
[9999]{}
[^1]: First author partially supported by the NSF and the Swedish Research Council. Second author partially supported by the Royal Swedish Academy of Sciences and the Swedish Research Council.
|
---
author:
- 'Jennifer L. Hoffman'
title: Polarized Line Profiles as Diagnostics of Circumstellar Geometry in Type IIn Supernovae
---
Introduction {#sec:intro}
============
Our understanding of supernovae (SNe) has broadened in recent years to include the recognition that both the thermonuclear and the core-collapse types of these stellar explosions are inherently aspherical phenomena [e.g., @koz05; @bur06]. Observations of net continuum polarization in both supernova (SN) types have provided key evidence of the intrinsic asymmetry of SN ejecta [e.g., @wang01; @leon06]. Many SNe also display line polarization features in addition to broadband continuum polarization; these line effects are often more complex than simple depolarization by complete scattering redistribution, and they can provide specific clues to the nature of SN ejecta and their surrounding circumstellar media [e.g., @wang04; @leon05]. However, line polarization can be produced by a combination of many different optical and geometrical effects, so its interpretation is not straightforward. Detailed radiative transfer modeling [as in @kas03] is often the best way to understand the effects that give rise to polarized lines in SN spectra. In the case of Type IIn supernovae, the situation is complicated by the presence of circumstellar material (CSM) surrounding the SN ejecta that becomes excited by the UV and X-ray photons from the SN explosion. Line polarization signatures in Type IIn SNe are superpositions of those arising from the ejecta and those arising from the CSM. Disentangling the two can be difficult but worthwhile, as it allows us to probe the nature and structure of the CSM to an extent not possible with spectroscopy alone [@leon00a; @wang01]. Since the CSM of a Type IIn supernova most likely represents material ejected by the progenitor star, studying its geometrical and optical characteristics provides a link to the mass-loss episodes and stellar winds of massive stars in their late stages of evolution.
Code {#sec:code}
====
I have developed a Monte Carlo radiative transfer code called [*SLIP*]{} (for upernova ne olarization) that simulates the ways polarized line profiles are created in Type IIn supernovae. [*SLIP*]{} uses the three-dimensional spherical polar grid structure described by @ww02; it tracks virtual photons as they arise from a model SN photosphere and scatter in a circumstellar density distribution with wavelength-dependent emission, absorption, and scattering characteristics. Similar codes have enjoyed success in analyzing related scenarios such as hot star envelopes with aspherical wind geometries [@har00] and ejecta-hole configurations in SNe Type Ia [@kas04]. Unlike most previous codes that treat SN line polarization [e.g., @hoef95], [*SLIP*]{} does not assume that line scattering is depolarizing; it also does not rely on the Sobolev approximation, but instead performs full radiative transfer in regions of high optical depth. It is thus able to probe in detail the polarized line profiles that may arise from interaction with the circumstellar material. Another advantage to this method is that it can simulate emission not only from a central source but also from extended regions such as the warm CSM (see below). However, the code is still in the early stages of development, and does not include any Doppler effects from the expanding circumstellar material; this limits the extent to which we can compare model outputs to observed line profiles, but the stationary case is a useful first approximation, especially in cases of low CSM velocity.
In the models presented here, a finite spherical source of photons at the center of the grid represents the “photosphere" of the SN ejecta, while two scattering regions surrounding the ejecta represent the warm, stationary CSM and a shock-heated region interior to the CSM, formed by its interaction with high-velocity SN material. Initially unpolarized photons are emitted from the surface of the photosphere with the synthetic H$\alpha$ P Cygni profile from a Type IIP supernova, produced with the [*PHOENIX*]{} stellar atmosphere code [@hau99]. In the hot shock region, I assume only a narrow line is emitted, with a line width of $< 80$ km/s; this is similar to the widths of the narrow hydrogen lines in the Type IIn SN 1997eg (7–40 km/s) observed by @sal02. I do not directly simulate heating of the CSM by emission from the supernova, but rather choose a CSM temperature and emit photons from the volume of the region with the expected thermal continuum and line spectra of hydrogen, assuming Case B LTE [@ost89]. All photons then scatter within the CSM via electron scattering; the optical depth of the scattering region is also chosen as an input parameter. I also implement wavelength-dependent free-free and free-bound absorption effects as in @wood96. Photons within the Doppler core of the line [@lang99] experience an additional bound-bound opacity; those absorbed by H atoms in this way are subsequently re-emitted coherently and isotropically.
When the code is run, [*SLIP*]{} emits photons sequentially from the photosphere, shock region, and CSM and follows each until it becomes absorbed or exits the system; at each scattering event, the code updates the photon’s Stokes parameters. Photons that exit the model system are binned by outgoing angle and the Stokes parameters are summed appropriately in each bin. Thus each simulation produces a full three-dimensional model whose H$\alpha$ flux and polarization spectra can be “viewed" from any desired direction.
{width="\columnwidth"}
Model Results {#sec:results}
=============
I have created a grid of 144 models spanning two CSM geometries (ellipsoids and toroids of similar sizes); CSM optical depths from 0.5 to 2; CSM luminosities from 1–20% of the photospheric luminosities; and CSM temperatures from 10,000 K to 20,000 K. Emission from the shock region was either included at 10% of the photospheric luminosity or excluded completely. In order to keep computing times reasonable, these models included only $1.8 \times 10^{7}$ photons each, enough to build up good signal in the flux spectrum but not the polarization spectrum. Since the goal was to match general features in both the flux and polarized flux, I used this grid to constrain parameter space for more computationally-intensive code runs including polarization. Figure \[fig:grid\] depicts results for nine representative models in the flux grid; it shows that significant differences in the line profiles arise for even small variations in optical depth and temperature of the scattering region.
With the flux grid complete, I compared the simulated H$\alpha$ profiles with observed H$\alpha$ profiles of Type IIn supernovae to determine which simulations to repeat for better signal in the polarized flux. This process is still underway, and the results will be published in an upcoming contribution. Here I present two representative simulations, Model A and Model B, that produce similar H$\alpha$ flux spectra but have quite different polarization behavior. Each included $1.6 \times 10^{9}$ photons, divided among 768 processors of the Seaborg parallel computing facility at the National Energy Research Scientific Computing Center (NERSC) at the Lawrence Berkeley Laboratory. Table \[tab:modpars\] compares the parameters of these two models, while Figure \[fig:flx\] compares their H$\alpha$ line profiles at a range of viewing angles with the profile observed for SN 2000P (A. V. Filippenko 2004, private communication) at day 13 post-discovery. Both models can reproduce the general observed line shape of a narrow emission “spike" superposed on a broad base (I note that due to the [*SLIP*]{} code’s limitation to stationary scattering regions, the width of the broad line in these models arises solely from the input IIP spectrum arising from the model SN photosphere). While the profiles produced by Model A (the ellipsoid) are nearly completely degenerate in viewing angle, Model B (the toroid) shows a significant variation with viewing angle, particularly in the strength of the “spike" relative to the broad base. Examination of the other models in the grid suggests that the differences between these two families of H$\alpha$ profiles are mainly due to the difference in CSM geometry between these two models.
![([*top*]{}) De-redshifted H$\alpha$ total flux profile of SN 2000P at day 13 post-discovery (A. V. Filippenko 2004, priv. comm.). ([*middle*]{}) Simulated H$\alpha$ profile arising from Model A (Table \[tab:modpars\]) at varying inclination angles from the polar axis. ([*bottom*]{}) As in the middle panel, but for Model B. All line profiles have been normalized to 1 at the rest wavelength of 6563 Å. \[fig:flx\]](hoffman-fig2.eps){width="\columnwidth"}
![([*top*]{}) De-redshifted H$\alpha$ polarized flux profile of SN 1997eg at day 44 post-discovery [@hof06]. ([*middle*]{}) Simulated H$\alpha$ polarized flux profile arising from Model A (Table \[tab:modpars\]) at varying inclination angles from the polar axis. ([*bottom*]{}) As in the middle panel, but for Model B. All polarized line profiles have been normalized to 1 at the rest wavelength of 6563 Å. \[fig:pflx\]](hoffman-fig3.eps){width="\columnwidth"}
Parameter
---------------------- ----------- ---------
CSM geometry ellipsoid toroid
CSM optical depth 1.0 2.0
$L_{CSM}/L_{phot}$ 0.01 0.1
$L_{shock}/L_{phot}$ 0.0 0.1
CSM temperature 20,000 K 15,000K
: Representative Model Parameters[]{data-label="tab:modpars"}
In Figure \[fig:pflx\] I present the H$\alpha$ line profiles of Models A and B in polarized flux and compare them with that of the Type IIn SN 1997eg [@leon00b; @hof06] at 44 days post-discovery. Recall that polarized flux is percent polarization multiplied by total flux; these profiles thus represent the spectra of the scattered light in each model. In polarized light the H$\alpha$ profiles look quite different than in direct light. The degeneracies that characterized Model A in direct light (Figure \[fig:flx\]) have been lifted, raising the possibility of using polarized line profiles to diagnose inclination angle in cases of known geometry. The two models now show line profiles quite distinct from each other; in particular, Model A produces no narrow “spike" at the rest wavelength in the polarized flux, while Model B preserves the spike. Examination of the other models in the flux grid suggests that this difference is due not to the difference in geometrical structure of the CSM between the two models, but rather to the presence of shock emission in Model B. Evidently the diffuse thermal emission from the CSM is sufficient to create a narrow spike in direct light, but in polarized light this feature requires a more directional source of narrow-line photons (the shock region). Photons from this region are more likely to scatter and become polarized while traversing the CSM than are photons that arise within its volume. However, Model B does not match the observed magnitude of polarization in SN 1997eg, most likely due to its larger CSM optical depth.
I continue to investigate these model results to pinpoint further diagnostics for geometry, temperature, and optical depth of the circumstellar material and the presence of a narrow-line “shock" region in Type IIn ejecta. More detailed analysis will be published in an upcoming contribution.
Observed H$\alpha$ Line Profiles of SN 1997eg {#sec:97eg}
=============================================
![Line profiles of the H$\alpha$ line of SN 1997eg in total flux ([*narrow smooth lines*]{}) and polarized flux ([*thick binned lines*]{}) at each of the three epochs of spectropolarimetry (days 16, 44, and 93 post-discovery). Total flux spectra have been normalized to their respective line peaks. Each polarized flux spectrum has been binned to a resolution of 10 Å and multiplied by the same normalizing factor as its corresponding total flux spectrum, then by an additional factor shown in each frame to facilitate direct comparison of the line shapes. \[fig:widths\]](hoffman-fig4.eps)
![Polarization profiles of the H$\alpha$ emission line of SN 1997eg in the [*q–u*]{} plane for each of the three epochs of spectropolarimetry (days 16, 44, and 93 post-discovery). Data have been binned to a resolution of 50 Å for clarity. Dashed lines represent negative (blueshifted) velocities; solid lines represent positive (redshifted) velocities. In each frame, the rest wavelength is shown with an open diamond and the origin of the plot is in the upper right corner. \[fig:qu\]](hoffman-fig5.eps)
As mentioned in §\[sec:code\], my current [*SLIP*]{} code has the limitation of treating only stationary scattering regions, when in fact one expects velocity effects to be quite prominent contributors to line profiles in polarized light. Here I present two examples of line polarization effects in a Type IIn supernova that cannot yet be reproduced by the code but provide key information regarding the geometry of the circumstellar material. Along with collaborators at San Diego State University and UC Berkeley, I have studied the polarization spectrum of SN 1997eg (whose H$\alpha$ polarized flux profile is shown in Figure \[fig:pflx\]); our full analysis will appear in @hof07.
Figure \[fig:widths\] compares the H$\alpha$ line profiles of SN 1997eg in total flux (nearly all direct light) and scattered light at day 16, day 44, and day 93 post-discovery. Although the continuum polarized flux is only about 2% of the total at each epoch, I have normalized both spectra to the same scale for comparison of the line profiles. At all epochs the polarized lines are broader than the unpolarized lines. This implies first that the scattering region has a different geometry from that of the broad-line emission region, and second that the scattering region is expanding at a higher velocity than the emission region. The fact that the polarized lines are broader only in the blue wing suggests the redshifted side of the scattering region may be self-occulted (or perhaps occulted by the SN ejecta) from our line of sight. If we postulate a flattened toroidal or disk-like geometry for the scattering region, this result can help place limits on the spatial inclination of the CSM configuration.
In Figure \[fig:qu\] I plot the polarized H$\alpha$ lines of the three epochs in [*q–u*]{} space, a technique that allows visualization of all polarimetric information at once. Not only does the magnitude of polarization change across the H$\alpha$ line at all epochs in SN 1997eg, but the position angle changes as well, in a manner that creates closed “loops" in the [*q–u*]{} plane (distinct from “knots," which characterize a constant polarization signal with wavelength, and from straight lines, which arise from simple envelope expansion or line depolarization). This implies that the scattering region polarizing the H$\alpha$ line has a different orientation than the one polarizing the continuum light (presumably the SN ejecta). In particular, the “loop" shape implies that the symmetry axes of the two regions are different, and that the CSM occults the SN ejecta in such a way as to create an asymmetry in the Stokes parameters across the line center. In the models of @kas03, such [*q–u*]{} loops are general features of two-axis systems; I postulate a toroidal geometry such as that shown in these authors’ Figures 14 and 15.
I note that the preceding results are both independent of interstellar polarization effects, for which the observed data have not been corrected. Similar complex behavior in the H$\beta$ and $\lambda$5876 lines in the SN 1997eg spectrum is discussed fully in the upcoming article. The results combine to suggest that SN 1997eg is characterized by ellipsoidal ejecta that polarize the continuum light and a flattened, disk-like CSM exterior to the ejecta that polarizes the hydrogen lines. The key conclusion is that the ejecta and the CSM have different axes of symmetry.
Multi-axis systems are becoming recognized as quite common in mass-loss scenarios. P Cyg [@nor01; @mea01], $\eta$ Car [@smith06], and VY CMa [@hum05] are examples of massive stellar systems in which the circumstellar material shows evidence for multiple mass-loss episodes along different axes. Our observations of polarized line profiles in SN 1997eg suggest that the geometry of an asymmetric SN explosion may, in turn, be unrelated to the geometry of the progenitor’s stellar wind or its mass eruptions. Continued refinements to the [*SLIP*]{} radiative transfer code will allow me to construct more detailed models of the CSM surrounding SN 1997eg and related objects, as well as further probing the nature both of Type II SN explosions and the eruptions that characterize their progenitor stars.
This research was supported by an NSF Astronomy & Astrophysics Postdoctoral Fellowship, AST-0302123, and by the National Energy Research Scientific Computing Center, US DOE Contract \#DE-AC03-76SF00098. I thank my primary collaborators Alex Filippenko at UC Berkeley, Peter Nugent at LBL, and Doug Leonard at San Diego State University, for their invaluable contributions.
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|
---
abstract: 'Plasmonic interferometry is a rapidly growing area of research with a huge potential for applications in terahertz frequency range. In this Letter, we explore a plasmonic interferometer based on graphene Field Effect Transistor connected to specially designed antennas. As a key result, we observe helicity- and phase-sensitive conversion of circularly-polarized radiation into dc photovoltage caused by the plasmon-interference mechanism: two plasma waves, excited at the source and drain part of the transistor interfere inside the channel. The helicity sensitive phase shift between these waves is achieved by using an asymmetric antenna configuration. The dc signal changes sign with inversion of the helicity. Suggested plasmonic interferometer is capable for measuring of phase difference between two arbitrary phase-shifted optical signals. The observed effect opens a wide avenue for phase-sensisitve probing of plasma wave excitations in two-dimensional materials.'
author:
- |
Y. Matyushkin,$^{1,2,5}$ S. Danilov,$^3$ M. Moskotin,$^{1,2}$ V. Belosevich,$^{1,2}$ N. Kaurova,$^2$\
M. Rybin,$^1$ E. Obraztsova,$^1$ G. Fedorov,$^{1,2}$ I. Gorbenko,$^4$ V. Kachorovskii,$^4$ and S. Ganichev$^3$
title: HELICITY SENSITIVE PLASMONIC TERAHERTZ INTERFEROMETER
---
Introduction
============
Interference is in heart of quantum physics and classical optics, where wave superposition plays a key role [@Scully1997; @Hariharan2007; @DemkowiczDobrzaski2015]. Besides fundamental significance, interference has very important applied aspects. Optical and electronic interferometers are actively used in modern electronics, and the range of applications is extremely wide and continuously expanding. In addition to standard applications in optics and electronics [@Scully1997; @Hariharan2007; @DemkowiczDobrzaski2015], exciting examples include multiphoton entanglement [@Pan2012], nonperturbative multiphonon interference [@ch3Ganichev86p729; @ch3Keay95p4098], atomic and molecular interferometry [@Boal1990; @Cronin2009; @Uzan2020] with recent results in cold-atoms-based precision interferometry [@Becker2018], neutron interferometry [@Danner2020], interferometers for medical purposes [@Yin2019], interference analysis of turbulent states [@Spahr2019], qubit interferometry [@Shevchenko2010] with a recent analysis of the Majorana qubits [@Wang2018], numerous amazing applications in the astronomy [@Saha2002; @Goda2008; @Adhikari2014; @Sala2019], such as interferometers for measuring of gravitational waves [@Goda2008; @Adhikari2014] and antimatter wave interferometry [@Sala2019], etc.
Recently, a new direction, plasmonic interferometry [@Gramotnev2010; @Graydon2012; @Ali2018; @Khajemiri2018; @Khajemiri2019; @Zhang2015; @Salamin2019; @Yuan2019; @Woessner2017; @Smolyaninov2019; @Hakala2018; @Dennis2015; @Haffner2015], has started to actively develop. The plasma wave velocity in 2D materials is normally an order of magnitude larger than the electron drift velocity and is much smaller than the speed of light. Hence, the plasmonic submicron sized interferometers based on 2D materials are expected to operate efficiently in the terahertz (THz) frequency range [@mittleman_2010; @Dhillon2017]. In particular, it has been predicted theoretically that a field-effect transistor (FET) can serve as a simple device for studying plasmonic interference effects [@Drexler2012; @Romanov2013; @gorbenko2018single; @Gorbenko2019]. Specifically, it was suggested that a FET with two antennas attached to the drain and source shows a dc current response to circularly polarized THz radiation which is partially driven by the interference of plasma waves and hence by helicity of incoming radiation. First experimental hint on existence of such an interference contribution was reported in Ref. \[$\!\!$\] for an industrial FET, where helicity-driven effects were obtained due to unintentional peculiarities of contact pads. Despite the first successes, creation of effective plasmonic interferometers is still a challenging task although in many aspects plasmonic-related THz phenomena are sufficiently well studied [@Dyakonov1993; @Dyakonov1996; @Knap2009; @Tauk2006; @Sakowicz2008; @JOAP_KnapKachorovskii2002; @APL_Knap2002; @Peralta2002; @Otsuji2004; @TeppeKnap2005; @Teppe2005; @Veksler2006; @Derkacs2006; @ElFatimy2006] with some commercial applications already in the market. Appearance of graphene opened rout for a novel class of active plasmonic structures [@Vicarelli2012] promising for plasmonic inteferometry due to non-parabolic dispersion of charge carriers and support of weakly decaying plasmonic excitations [@Koppens2014].
In this Letter, we explore an all-electric tunable – by the gate voltage – plasmonic interferometer based on graphene FET connected to specially designed antennas. Our interferometer demonstrates helicity-driven conversion of incoming circularly-polarized radiation into phase- and helicity-sensitive dc photovoltage signal. The effect is detected at room- and liquid helium- temperatures for radiation frequencies in the range from 0.69 to 2.54 THz. All our results show plasmonic nature of effect. Specifically, the rectification of the interfering plasma waves leads to dc response, which is controlled by the gate voltage and encodes information about helicity of the radiation and phase difference between the plasmonic signals. A remarkable feature of this plasmonic interferometer is that there is no need to create an optical delay line, which has to be comparable with the quite large wavelength of the THz signal. By contrast, in this setup, phase shift between the plasma waves excited at the source and drain electrodes of the FET is maintained by combination of the antenna geometry and the radiation helicity. It remains finite even in the limit of infinite wavelength and changes sign with inversion of the radiation helicity. The plasmonic interferometer concept realized in our work opens a wide avenue for phase-sensitive probing of plasma wave excitations in two-dimensional materials.
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Devices and measurements
========================
The single layer graphene (SLG), acting as the conducting channel of a field-effect transistor, was synthesized in a home-made cold-wall chemical vapor deposition reactor by chemical vapor deposition (CVD) on a copper foil with a thickness of 25 $\mu$m [@Rybin2016]. SLG was transferred onto an oxidized silicon wafer[@Gayduchenko2018], for more details see Appendix. The antenna sleeves were attached to the source and drain electrodes. To realize the helicity sensitive terahertz plasmonic interferometer, the antenna sleeves were bent by 45${}^\circ$ as shown in Fig. \[fig:characterization\]b. The sleeves were made using photolithographic methods and metallization sputtering (Ti/Au, 5/100 nm). At the first lithography step, contact pads to graphene channel were formed using pure Au with a thickness of 25 nm. Note that we did not use adhesion sub layer (like titan or chrome) at this step. At the next technological step, e-beam evaporator was applied to sputter a 100 nm thick layer of Al$_2$O$_3$ that acts as a gate insulator. Note that Al$_2$O$_3$ layer reduces the initially high doping level of as-transferred graphene to almost zero [@Fallahazad2012; @Kang2013]. Finally, the top gate electrode (Ti/Au, 5/200 nm) is patterned and formed using standard sputtering and lift-off techniques. The resulting structure is sketched in the Figure \[fig:characterization\]a. Two devices with channel lengths 2 $\mu$m (device 1) and 1 $\mu$m (device 2) were fabricated. Zoomed images of the channel parts are shown in insets in Figs. \[fig:characterization\]c and \[fig:characterization\]d. Note that for both devices the gates are deposited asymmetrically in respect to the channel. They cover about 75% (device 1) and 50% (device 2) of the channels and the gate stripes are located closer to the drain contact pads.
Typical transport characteristics of our graphene devices are shown in the Figs. \[fig:characterization\]c and \[fig:characterization\]d. For different directions of the gate voltage sweep as well as the sample cooldowns, the charge neutrality point (CNP) can occur at different gate voltages $U_{\rm g}$. This is well known feature and is possibly caused by the gate-sweep direction (cooldown-) dependent charge trapping. Therefore, below we indicate range of $U_{\rm g}$ corresponding to the CNP instead of providing its exact value. Comparing conductance curves of the device 1 and 2 shows that the minimal conductance $G$ scales as an inverse of the graphene channel length, indicating that most of the resistance comes from the graphene channel itself rather than from the contacts. Analysis of the channel mobilities and corresponding scattering times is presented in Appendix. Using the Drude formula generalized for graphene we estimate scattering times of the order of 10-20 fs for, e.g., device 1 at room temperature. Finally, we note that the $G(U_{\rm g})$ curves do not change as the temperature goes down (see Appendix), meaning that mobility is restricted by the defect scattering.
The experiments have been performed applying a continuous wave methanol laser operating at frequencies $f_1 = 2.54$ THz (wavelength $\lambda_1 = 118$ $\mu$m) with a power of $P \approx 20$ mW and $f_2 = 0.69$ THz (wavelength $\lambda_2 = 432$ $\mu$m) with $P \approx 2$ mW [@Ganichev2009; @Kvon2008]. The laser spot with a diameter of about 1-3 mm is substantially larger than the sample size ensuring uniform illumination of both antennas. The radiation polarization state was controllably varied by means of lambda-half plate that rotates the polarization direction of linear polarized radiation and lambda-quarter plate that transforms linearly polarized radiation into elleptically polarized one.
The helicity of the radiation is then controlled via changing the angle $\phi$ between the laser polarization and the main axes of the lambda-quarter plate, so that for $\phi = 45^\circ$ the radiation is right circularly polarized ($\sigma^+$) and for $\phi = 135^\circ$ - left circularly polarized ($\sigma^-$). The functional behavior of the Stokes parameters upon rotation of the waveplates is summarized in Appendix, see also Ref. \[$\!\!$\]. The samples were placed in an temperature variable optical cryostat and photoresponse was measured as the voltage drop $U$ directly over the sample applying lock-in technique at a modulation frequency of 75 Hz.
{width="\linewidth"}
Results
=======
The principal observation made in our experiment is that for all investigated devices the response to a circularly polarized radiation crucially depends on its helicity. Fig. \[fig:helicity\] displays the response voltage $U$ normalized by the radiation intensity as a function of the angle $\phi$ obtained under different conditions. We emphasize significant difference in the signal for $\phi = 45^\circ$ and $135^\circ$, corresponding to opposite helicities of circularly polarized light, in particular, the sign inversion observed under some conditions, see e.g. Fig. \[fig:helicity\]d. The effect is observed for radiation with frequencies 2.54 and 0.69 THz in a wide temperature range from 4.2 K to 300 K. The overall dependence of the signal on angle $\phi$ is more complex and is well described by $$\begin{aligned}
\label{eqn:eq1}
U(\phi) &=& U_{\rm C}\sin(2\phi)+U_{\rm L1}\sin(4\phi)/2+
\\ \nonumber
&+&U_{\rm L2}[\cos(4 \phi)+1]/2 + U_0
%\end{equation}
\end{aligned}$$ with $U_{\rm C} $, $U_{\rm L1}$, $U_{ \rm L2}$ and $U_0$ are fit parameters depending on gate voltage, temperature and radiation frequency. Note that trigonometric functions used for the fit are the radiation Stokes parameters describing the degree of the circular and linear polarization (see Appendix) [@bahaasaleh2019; @Belkov2005; @Weber2008]. While three last terms are insensitive to the radiation helicity the first term ($\propto U_{\rm C}$) is $\pi$-periodic and describes helicity-sensitive response: it reverses the sign upon switching from right- ($\sigma^+$) to left- ($\sigma^-$) handed circular polarization. Figure \[fig:helicity\] reveals that this term gives substantial contribution to the total signal. As we show below the $\pi$-periodic term is related to the plasma interference in the graphene-based FET channel. Measurements at room and low temperatures demonstrate that cooling the device increases the amplitude of the circular photoresponse by more than ten times, see Figs. \[fig:helicity\]a and \[fig:helicity\]b as well as \[fig:helicity\]d and \[fig:helicity\]e. The signal increase is also observed by reduction of radiation frequency, see Figs. \[fig:helicity\]b and \[fig:helicity\]c as well as \[fig:helicity\]e and \[fig:helicity\]f.
{width="\linewidth"}
Having experimentally proved the applicability of Eq. (\[eqn:eq1\]) and substantial contribution of the helicity drive signal we now concentrate on the dependence of the parameter $U_{\rm C}$ on the gate voltage that controls the type and concentration of the charge carriers in the FET channel. We use the following procedure: we obtain the gate voltage dependence of the response voltage normalized to the radiation power $P$ for two positions of the $\lambda/4$ plate $\phi = 45^\circ$ and $135^\circ$, corresponding to opposite helicities of circularly polarized light ($\sigma^+$ and $\sigma^-$). The half-difference between these two curves directly gives gate voltage dependence of the helicity sensitive photoresponse $U_{\rm C} = (U_{\sigma^+} - U_{\sigma^-})/2\,.$ \[see Eq. (\[eqn:eq1\])\] Results of these measurements, shown in Fig. \[fig:responses\], reveal that $U_{\rm C}$ is more pronounced at positive gate voltage, where the channel is electrostatically doped with electrons, and changes the sign close to the CNP. As addressed above the variation of the CNP from measurement to measurement does not allow us to allocate the exact position of the CNP for the gate voltage sweeps during the photoresponse measurements. Note that for the device 1, having the gate length twice larger as that of device 2, at large negative gate voltages the second sign inversion of the photocurrent is present. Figure \[fig:responses\]a indicates that in device 1 for the whole range of gate voltages photoresponse for $\sigma^+$- and $\sigma^-$- radiation have consistently opposite sign indicating negligible contribution of the polarization independent background. In device 2, however, the background is of the same order as the helicity sensitive response $U_{\rm C}$, see Fig. \[fig:responses\]b.
{width="\linewidth"}
Finally, we present additional data on the contributions proportional to fit parameters $U_{\rm L1}$, $U_{\rm L2}$ and $U_0$ in Eq. (\[eqn:eq1\]). As discussed above these contributions do not connected to the radiation helicity and are related to the Stokes parameters describing the degrees of linear polarization (terms $\propto U_{\rm L1}$ and $\propto U_{\rm L2}$) and radiation intensity (term $\propto U_0$). In experiments applying linearly polarized radiation with rotation of $\lambda/2$ plates Stokes parameters modifies and polarization dependence Eq. (\[eqn:eq1\]) takes a form $$U(\alpha) = U_{\rm L1}\sin(2\alpha) + U_{\rm L2}\cos(2\alpha) + U_0 .
\label{eqn:L1L2}$$ An example of the photoresponse variation upon change of the azimuth angle $\alpha$ is shown in Fig. \[fig:azimuth\]a. The data reflects the specific antenna pattern of our devices with tilted sleeves. Figure \[fig:azimuth\]b shows the gate voltage dependence of the photoresponse obtained in device 1 for $\alpha=0$. Comparing these plots with the results for circular photoresponse shows that they behave similarly: in both cases signal changes the sign close to CNP and the response for positive gate voltages is larger than that for negative $U_{\rm g}$. Transport measurements carried out parallel to photoresponse measurements show that the photosignal behaves similarly to the normalized first derivative of the conductance $G$ over $U_{\rm g}$: $(dG/dU_{\rm g})/G$, see Fig. \[fig:azimuth\]a. Note that this behavior is well known for non-coherent, phase-insensitive plasmonic detectors [@Veksler2006].
Theory and discussion
=====================
Conversion of THz radiation into dc voltage can be obtained due to several phenomena including photothermoelectric (PTE) effects [@Fedorov2013; @Gabor2011; @Fuhrer2014], rectification on inhomogeneity of carrier doping in gated structures [@Fuhrer2014; @Gayduchenko2018; @CastillaKoppens2019], photogalvanic and photon drag effects [@Karch2010; @Jiang2011; @Glazov2014] as well as rectification of electromagnetic waves in a FET channel supporting plasma waves [@Dyakonov1996]. However, in our experiment only plasmonic mechanism can yield the dc voltage whose polarity changes upon switching the radiation helicity. Indeed, PTE effects and rectification due to gradient of carrier doping in gated structures are based on inhomogeneities of either radiation heating or radiation absorption, which are helicity-insensitive [@b1]. Below, we show that the helicity-sensitive plasmonic response originates from the interference of plasmonic signals excited by the source and drain antenna sleeves. The source and drain potentials with respect to the top gate are given by $$U_{\rm A,B} (t) = U_{\rm A,B}\cos(\omega t- \varphi_{\rm A,B}),
\label{eqn:time_dep}$$ where $\omega$ is the radiation frequency. Complex amplitudes $U_{\rm A} e^{i \varphi_{\rm A}}$ and $U_{\rm B} e^{i \varphi_{\rm B}}$ of the signal at source and drain, respectively, are proportional to the electric field amplitude of the incident electromagnetic wave. Their amplitudes and the phase shift between signals ($\varphi_{\rm A}-\varphi_{\rm B}$) depend on the radiation polarization and antennas geometry, see Appendix. Design of our devices, see Fig. \[fig:interpretation\]a, ensures asymmetric coupling of radiation to the source and drain electrodes so that both amplitudes and phases of source and drain potentials are different. Most importantly, when such a bent bow-tie antenna is illuminated by circularly polarized radiation, the source- and drain-related antenna sleeves are polarized with a time delay because of rotation of the electric field vector. For circularly polarized wave, the phase shift changes sign with changing the helicity of the radiation: $ \varphi_{\rm A}-\varphi_{\rm B}=\theta_{\rm A}-\theta_{\rm B}, $ for $\omega>0$ (positive helicity) and $ \varphi_{\rm A}-\varphi_{\rm B}=-(\theta_{\rm A}-\theta_{\rm B}), $ for $\omega<0$ (negative helicity ), where $\theta_A$ and $\theta_B$ are geometrical angles of antennas sleeves. Equations describing more general case of elliptic polarization are derived in Appendix. We note that in the case of linear polarization, the phase shift is zero and the second term in Eq. (\[eqn:response1\]) is absent.
{width="\linewidth"}
Due to hydrodynamic non-linearity of plasma waves [@Dyakonov1993; @Dyakonov1996] DC voltage across the channel appears $$U = F_1(U_{\rm A}^2-U_{\rm B}^2)+F_2U_{\rm A}U_{\rm B}\cdot \sin (\varphi_{\rm A} - \varphi_{\rm B}),
\label{eqn:response1}$$ where $F_1$ and $F_2$ are gate-controlled coefficients which represent, respectively, non-coherent[@Dyakonov1996] and interference [@Drexler2012; @Romanov2013; @gorbenko2018single; @Gorbenko2019] contributions to the response. This coefficients do not depend on signal phases and amplitudes, while all information about coupling with antennas is encoded in factors $(U_{\rm A}^2-U_{\rm B}^2)$ and $U_{\rm A}U_{\rm B}\sin(\varphi_{\rm A} - \varphi_{\rm B})$.
For the case of radiation with arbitrary polarization characterized by the Stockes parameters, that are controlled by the orientation of the $\lambda/4$ plate, defined by the phase angle $\phi$, we obtain $$\begin{aligned}
\nonumber
U(\phi)&=
F_1 \left[a_0 + a_{\rm L1} \frac{\sin (4 \phi)}{2}+ a_{\rm L2}\frac{1+ \cos(4\phi)}{4}
\right] + \nonumber \\ &+
F_2~ a_{\rm C} \sin(2 \phi).
\label{eqn:response2}
\end{aligned}$$ Here $a_0$, $a_{\rm L1}$, $ a_{\rm L2}$, $a_{\rm C}$ are the geometrical factors calculated in Appendix for a simplified model. Note that these factors are independent of the gate voltage, so that the entire gate voltage dependence of the photoresponse is defined by the factors $F_1$ and $F_2$. Comparing these results with empirical Eq. (\[eqn:eq1\]) we conclude that the data shown in the Fig. \[fig:helicity\] are fully consistent with theoretical considerations discussed thus far. In particular, the interference-induced helicity-sensitive contribution, given by the last term on the right hand side of the above equation, is clearly observed in the experiment, see Fig. \[fig:helicity\].
Such interference contribution appears when source and drain electrodes “talk” to each other via exchange of plasma wave phase-shifted excitations. Hence, the characteristic length of plasma wave decay $L_*$ should not be too small as compared to channel length $L$ so that plasmons excited near source and drain electrodes could efficiently interfere within the channel, see the Fig. \[fig:interpretation\]b. As the gate voltage controls the type and concentration of the charge carriers it also controls the velocity of the plasma waves $s$ and the length $L_*.$ As a result, the second term in the Eq. (\[eqn:response1\]) should oscillate as function of the gate voltage. The general formulas for response given in Appendix can be essentially simplified for the non-resonant case $ s/L \ll \gamma, ~ \omega \ll \gamma$, which corresponds to our experimental situation [@b2]. In this case, plasma waves decay from the source and drain part of the channel within the length $ L_*= {s\sqrt 2}/{\sqrt {\omega \gamma}},$ and the response is given by Eq. (\[F\_2\]) in Appendix. The parameters $F_{1,2}$ in Eq. (\[eqn:response2\]) are given by $$F_1=\frac{1}{4U_{\rm g}}, \quad F_2= \frac{4\omega }{\gamma}~ \frac{ \sin \left({L}/{L_*} \right) e^{-L/L_*}}{U_{\rm g}}.
\label{F12}$$ In our experiment $L_*$ was essentially smaller than the device length: $L_* \sim (0.1 \div 0.3) L $. However, the interference, helicity-dependent part of the response was clearly seen in the experiment. The result of calculations are presented in Fig. \[fig:interpretation\]c. For gate voltages far from the Dirac point we obtain a qualitative agreement of the calculations and results of experiments presented in Fig. \[fig:responses\]a:
- the circular photoresponse at high positive gate voltages and for moderate negative $U_{\rm g}$ have opposite sign;
- with increase of the negative gate voltages value the response changes its sign;
- in the vicinity of the Dirac point calculations yield oscillations of the circular response.
The last statement needs a clarification. While the oscillations are not visible in the experimental magenta curve of Fig. \[fig:responses\]a, showing $U_{\rm C} = (U_{\sigma^+} - U_{\sigma^-})/2$, in individual curves obtained for left- (blue curve) and right- (red curve) they are clearly present. This difference is caused by the fact that the $U_{\sigma^+}$ and $U_{\sigma^-}$ curves represent the results of two different experiments, namely, measurements for $\sigma^+$ and $\sigma^-$ radiation. At the same time, $U_{\rm C}$ is obtained as a result of subtraction of these two curves, corresponding to different $U_{\rm g}$ sweeps. Due to the hysteresis of the $R_{xx}$ discussed above in Sec. 2, the sample parameters were slightly different for these two measurements. As a result, the oscillations present in one curve are superimposed with larger featureless signal from the other. Furthermore, Fig. \[fig:responses\]b shows that presence of the oscillations in photoresponse to circularly polarized radiation for the second device too.
Now we estimate the period of the oscillations. The dependence on the gate voltage is mostly encoded in ${L_* \propto {U_{\rm g}}^{1/4} }.$ The oscillations period can be estimated from the condition ${\delta (L/L_*) \sim 1,}$ which gives ${(L/L_*)\delta U_{\rm g}/4 U_{\rm g}\sim 1.}$ For ${U_{\rm g} \approx 2}$ V and ${L_*/L \approx 0.1,}$ we find ${\delta U_{\rm g} \approx 0.8}$ V in a good agreement with experiment. We also note that the experimentally observed oscillations (see blue curve for the device 1) decays at the same scale as an oscillation period in an excellent agreement with behavior of the function $F_2$, see Eq. (\[F12\]).
Finally, we emphasize that the presence of oscillations in the hallmark of the interference part of response. Importantly, the response to the linearly polarized radiation does not show any oscillations in the vicinity of the CNP, see Fig. \[fig:azimuth\]b. By contrast it just follows to $(dG/dU_{\rm g})/G$ — a well know behavior for Dyakonov-Shur non-coherent plasmonic detectors, see e.g. Ref. \[$\!\!$\]. Note that for linearly polarized radiation the above theory also yields this gate dependence, see structureless expression for non-coherent contribution $F_1$ in Eq. (\[F12\]).
Summary
=======
To summarize, we demonstrated that specially designed graphene-based FET can be used to study plasma wave interference effects. Our approaches can be extrapolated to other 2D materials and used as a tool to characterize optically-induced plasmonic excitations. Specifically, the conversion of the interfering plasma waves into dc response is controlled by the gate voltage and encodes information about helicity of the radiation and phase difference between the plasmonic signals. Remarkably, our work shows that CVD graphene with moderate mobility, which is compatible with most standard technological routes can be used as a material for active plasmonic devices. The suggested device design can be used for a broad-band helicity-sensitive interferometer capable of analyzing both polarization of THz radiation and geometrical phase shift caused by antennas asymmetry. By the proper choice of the antenna design and FET parameters, phase-sensitive and fast room temperature plasmonic detectors can be tuned to detect individual Stokes parameters. Hence, our work paves a novel way for developing the all-electric detectors of the terahertz radiation polarization state.
Acknowledgements
================
The work was supported by the FLAG-ERA program (project DeMeGRaS, project GA501/16-1 of the DFG), the Russian Foundation for Basic Research within Grants No. 18-37-20058 and No. 18-29-20116, the Volkswagen Stiftung Program (97738) and Foundation for Polish Science (IRA Program, grant MAB/2018/9, CENTERA). The work of V.K. was supported by the Russian Science Foundation (Grant No. 20-12-00147). The work of I.G. was supported by Russian Foundation for Basic Research (Grant No. 20-02-00490) and by Foundation for the Advancement of Theoretical Physics and Mathematics BASIS. M.M. acknowledges support of Russian Science Foundation (project No. 17-72-30036) in sample design.
Appendix
========
Devices, experimental details and fit parameters
------------------------------------------------
**Device fabrication.** The single layer graphene (SLG), acting as the conducting channel of a field-effect transistor, was synthesized in a home-made cold-wall chemical vapor deposition reactor by chemical vapor deposition (CVD) on a copper foil with a thickness of 25 $\mu$m [@Rybin2016]. SLG was transferred onto an oxidized silicon wafer[@Gayduchenko2018]. The silicon substrate consists of 480 $\mu$m thick silicon wafer covered with a 500 nm thick thermally grown SiO$_2$ layer. Note that silicon used for the substrate (with the room temperature resistivity of 10 $\Omega\cdot$cm) is transparent for the sub-THz and THz radiation. After transferring graphene onto a silicon wafer the geometry of the device is further defined by e-beam lithography using a PMMA mask and oxygen plasma etching.
At the first lithography step, contact pads to graphene channel were formed using pure Au with a thickness of 25 nm. Note that we did not use adhesion sub layer (like titan or chrome) at this step. At the next technological step, e-beam evaporator was applied to sputter a 100 nm thick layer of Al$_2$O$_3$ that acts as a gate insulator. Note that Al$_2$O$_3$ layer reduces the initially high doping level of as-transferred graphene to almost zero [@Fallahazad2012; @Kang2013].
Afterward, the antenna sleeves were attached to the source and drain electrodes. To realize the helicity sensitive terahertz plasmonic interferometer, the antenna sleeves were bent by 45${}^\circ$ as shown in Fig. \[fig:characterization\]a. The sleeves were made using photolithographic methods and metallization sputtering (Ti/Au, 5/100 nm). Finally, the top gate electrode (Ti/Au, 5/200 nm) is patterned and formed using standard sputtering and lift-off techniques. The resulting structure is sketched in Fig. \[fig:characterization\]b of the main text. Two devices with channel lengths 2 $\mu$m (device 1) and 1 $\mu$m (device 2) were fabricated. Zoomed images of the channel parts are shown in insets in Figs. \[fig:characterization\]c and d of the main text. Note that for both devices the gates are deposited asymmetrically in respect to the channel. They cover about 75% (device 1) and 50% (device 2) of the channels and the gate stripes are located closer to the drain contact pads.
**Transport characteristics and transport scattering time.** Typical transport characteristics of our graphene devices are shown in the Figs. 1c and 1d of the main text. For different directions of the gate voltage scan as well as the sample cooldowns, the charge neutrality point (CNP) can occur at different gate voltages. This is well known feature and is possibly caused by the gate-scan-direction (cooldown-) dependent charge trapping. Therefore, below we indicate range of $U_{\rm g}$ corresponding to the CNP instead of providing its exact value. Comparing conductance curves of the device 1 and 2 shows that the minimal conductance $G$ scales as an inverse of the graphene channel length, indicating that most of the resistance comes from the graphene channel itself rather than from the contacts. For that we used the transfer curves measured in two-probe configuration, since we cannot perform four contacts Hall effect measurements due to configuration of our devices.
We extract mobility and transport scattering time from the conductance measurements shown in Fig. \[Fig2\]a. The conductivity of graphene is given by $$\sigma = \frac{e^2}{\pi \hbar } \left[\frac{E \tau (E)}{\hbar} \right]_{E=E_{\rm F}}, \label{sigma}$$ where $E_{\rm F}$ is the Fermi energy and $\tau(E)$ is the energy-dependent transport scattering time. The Fermi energy depends on the concentration, $E_{\rm F}= \hbar v_{\rm F} \sqrt{\pi n},$ which is controlled by the gate voltage: $n=\varepsilon U_{\rm g}/4\pi e d,$ where $U_{\rm g}$ is the gate voltage counted from the Dirac point, $\varepsilon$ is the dielectric constant and $d$ is the width of the spacer. Here, in all calculations, we assume that $U_{\rm g}>0$ having in mind that response should change sign under replacement $U_{\rm g} \rightarrow -U_{\rm g}. $ Using Eq. (\[sigma\]) and the formula $$E_{\rm F}= \hbar v_{\rm F} \left(\frac{\varepsilon U_{\rm g}}{4 e d}\right)^{1/2},$$ one can extract from experimental data (see Fig. \[Fig2\]a) the dependence of both $\tau$ and mobility, defined as $$\mu= \frac{1}{e} \frac{d \sigma}{ dn }= \frac{1}{e} \frac{d \sigma}{dU_{\rm g} }\frac{d U_{\rm g}}{ dn }, \label{mu}$$ on $U_{\rm g}.$ The result is presented in Figs. \[Fig2\]b and c. Assuming that with approaching to the neutrality point the scattering is limited by charged impurities, for which $$\gamma= \tau^{-1} \propto 1/E \propto 1/\sqrt{U_{\rm g}},
\label{gamma}$$ we approximate the experimental dependence shown in Fig. \[Fig2\]b as $$\tau(U_{\rm g})=\tau^*~\frac{\sqrt{U_{\rm g}}}{\sqrt{U_{\rm g}} + \sqrt{U_*}}, \label{tauUg}$$ where $\tau_* = 2 \cdot 10^{-14}$s and $U_* =1.5 V.$ Equation (\[tauUg\]) was used to calculate $\Omega$ and $\Gamma$ entering Eq. (\[OmegaGamma\]). We also used the well known expression for plasma wave velocity in graphene $$s=\left( \frac{4 e^3 d v_F^2 U_{\rm g}}{\varepsilon \hbar^2 } \right)^{1/4}. \label{sUg}$$ Equations (\[tauUg\]) and (\[sUg\]) become invalid very close to the neutrality point. Finally, we note that the $G(U_{\rm g})$ curves do not change as the temperature goes down, meaning that mobility is restricted by the defect scattering.
![(Color online) Electron transport in the our CVD graphene based devices. (a) transfer characteristics of device D1 measured at different temperatures. (b) Mobility extracted from Fig. \[Fig2\]a curve using the Eq. (\[mu\]) at room temperature. (c) Scattering time extracted from the Fig. \[Fig2\]a curve using the Eq. (\[sigma\]) at room temperature. []{data-label="Fig2"}](sup_CVD_transport_python.png){width="\columnwidth"}
**Laser and experimental setup.** The experiments have been performed applying a continuous wave (CW) methanol laser operating at frequencies $f_1 = 2.54$ THz (wavelength $\lambda_1 = 118$ $\mu$m) with a power of $P \approx 20$ mW and $f_2 = 0.69$ THz (wavelength $\lambda_2 = 432$ $\mu$m) with $P \approx 2$ mW [@Ganichev2009; @Kvon2008]. The incident radiation power was monitored by a reference pyroelectric detector. The laser beam was focused onto the sample by a parabolic mirror with a focal length of 75 mm. Typical laser spot diameters varied, depending on the wavelength, from 1 to 3 mm. The spatial laser beam distribution had an almost Gaussian profile, checked with a pyroelectric camera [@Ziemann2000]. The radiation was modulated at about 75 Hz by an optical chopper. The samples were placed in an optical temperature variable cryostat and photoresponse was measured as a voltage drop using standard lock-in technique. The radiation of the laser was linearly polarized. To demonstrate sensitivity of the photovoltage to the helicity of the incoming radiation we place a $\lambda/4$ plate made of $x$-cut crystalline quartz in the incoming beam. The helicity of the radiation is then controlled via changing the angle $\phi$ between the laser polarization and the main axes (“fast direction”) rotating the plate in vertical plane. When $\varphi = 45^\circ$ the radiation incident on the device is clock wise circularly polarized (right circularly polarized radiation, $\sigma^+$) while for $\phi = 135^\circ$ one gets the opposite polarization (left circularly polarized radiation, $\sigma^-$). To study response to the linearly polarized radiation we used $\lambda/2$-plates. Rotating the plate, we rotated the radiation electric field vector $\mathbf{E}$ by an azimuth angle $\alpha$.
**Fitting parameters.** The curves in Fig. \[fig:helicity\] were obtained with fitting parameters summarized in Tab. \[table1\].
Panel in Fig. \[fig:helicity\] a) b) c) d) e) f)
-------------------------------- -------- ------- ------- -------- ------- ------
$U_{\rm C}/P$ -0.012 -0.42 1.15 0.009 0.15 2.15
$U_{\rm L1}/P$ 0.025 -0.14 -0.08 -0.007 -0.09 3.34
$U_{\rm L2}/P$ -0.039 0.49 -1.03 -0.009 0.01 0.52
$U_0/P$ -0.005 0.46 3.18 0.003 0.13 2.58
: Fitting parameters used for calculations of curves in Fig. \[fig:helicity\]. The parameters are given in mV/W. Letters on the top of the table corresponds to the labels of panels in Fig. \[fig:helicity\]. []{data-label="table1"}
Response to polarization of arbitrary polarization
---------------------------------------------------
We assume that wavelength of the radiation is much larger that the device size and that incoming beam linearly polarized along $x$ axis acquires elliptical polarization after transmission through $\lambda/4$ plate. Then, the field is described by homogeneous electric vector $\mathbf E(t) $ with the components $$\begin{aligned}
\label{Ex}
E_x(t) &=&\frac{\left( E_x e^{-i\omega t} + \rm{h.c.} \right)}{2} =E_0\cos\alpha \cos(\omega t), \\
\label{Ey}
E_y(t) &=&\frac{\left( E_y e^{-i\omega t} + \rm{h.c.} \right)}{2} =E_0\sin\alpha \cos(\omega t+\eta)\end{aligned}$$ where $$E_x=E_0\cos \alpha,\quad E_y= E_0 \sin \alpha e^{-i \eta}.
\label{Exy1}$$ We get $$\begin{aligned}
\label{Stokes0}
|E_x|^2+ |E_y|^2&=&E_0^2, \\
\label{StokestildeL}
E_x E_y^* + E_x^* E_y &=& E_0^2 ~ P_{\rm L1}, \\
\label{StokesL}
|E_x|^2- |E_y|^2 & = &E_0^2 ~P_{\rm L2},\\ \label{StokesC}
i\left( E_x E_y^* - E_x^* E_y \right) &=& - E_0^2 ~P_{\rm C}.
\end{aligned}$$
![(Color online) Sketch of the FET with two antennas[]{data-label="Fig1"}](sup_fig1.pdf){width="0.8\columnwidth"}
Here, we introduced Stokes parameters $$\begin{aligned}
\label{Stokes1tildeL}
P_{\rm L1} = & \sin(2\alpha) \cos \eta = \frac{\sin(4\phi)}{2}, \\
\label{Stokes1L}
P_{\rm L2} = & \cos(2\alpha) = \frac{1+\cos(4 \phi)}{2},\\
P_{\rm C} = & \sin(2\alpha)\sin\eta = \sin (2 \phi),
\label{Stokes1C}
\end{aligned}$$ where $\phi$ is the rotation angle of $\lambda/4$ plate with respect to $x$ axis. As seen, the angles $\phi=0, \pm 90^\circ,\pm 180^\circ$ correspond to linear polarization along $x-$axis, while angles $\phi=45^\circ$ (or $\phi=225^\circ$) and $\phi=135^\circ$ (or $\phi=315^\circ$ ) to circular polarization with right and left helicity, respectively. We note that the helicity also changes sign by inversion of the radiation frequency: $\omega \to -\omega.$ For definiteness, in all equations below we assume $\omega>0.$
The detailed analysis of time-dependent potential distribution in our device requires calculation of antennas properties, which is a challenging problem beyond the scope of the current work. However, some phenomenological properties of suggested interferometer can be understood by using a toy model, which captures basic physics of the problem. The model is illustrated in Fig. \[Fig1\], where two antennas are replaced with long thin metallic rods described by vectors $\mathbf R_{\rm A,B}$. Assuming that antennas are perfect conductors and neglecting small mutual capacitances, one can write the potentials applied to source and drain as $$\begin{aligned}
& \nonumber U_{\rm A}(t) = \mathbf E(t) \mathbf R_{\rm A}/2 =U_{\rm A}
\cos (\omega t -\varphi_{\rm A}) , \\
& \nonumber U_{\rm B}(t) = \bm E(t) \mathbf R_{\rm B} /2=U_{\rm B}\cos (\omega t -\varphi_{\rm B}), \end{aligned}$$ where parameters $U_{\rm A, B}$ and $\varphi_{\rm A, B}$ for $\omega>0$ obey \[see Eqs. (\[Ex\]) and (\[Ey\])\] $$\begin{aligned}
\label{A}
& U_{\rm A} e^{-i\varphi_{\rm A}}= \frac{E_0 R_A}{2}\left( \cos \alpha \cos \theta_{\rm A}
+ \sin \alpha \sin \theta_{\rm A}~ e^{-i \eta} \right), \\
& U_{\rm B} e^{-i\varphi_{\rm B}}= \frac{E_0 R_{\rm B}}{2}\left( \cos \alpha \cos \theta_{\rm B} + \sin \alpha \sin \theta_{\rm B} ~ e^{-i \eta} \right).
\label{B}\end{aligned}$$
In the simplest case of circular polarization considered in Refs. [@S1; @S2]($\alpha= \pm 45^\circ, \eta= 90^\circ$), equations (\[A\]) and (\[B\]) simplifies, so that $U_{\rm A,B}=E_0 R_{\rm A,B}/\sqrt{8} $ and $\varphi_{\rm A,B}=\theta_{\rm A,B}$ for $\omega>0$ (positive helicity) and $\varphi_{\rm A,B}=-\theta_{\rm A,B}$ for $\omega<0$ (negative helicity). For the case of generic elliptic polarization, it is convenient to use the Stokes parameters. After simple algebra, we find equations needed for calculation of the response according to the theory developed in Refs. [@S1; @S2] $$U_{\rm A}^2-U_{\rm B}^2 = a_0 + a_{\rm L1} P_{\rm L1}+ a_{\rm L2} P_{\rm L2} =$$ $$\label{UA}
= a_0 + a_{\rm L2} \frac{1+\cos(4 \phi)}{2} + a_{\rm L1} \frac{\sin(4\phi)}{2},$$ $$\label{UB}
U_{\rm A} U_{\rm B} \sin(\phi_{\rm A} -\phi_{\rm B}) = a_{\rm C} P_{\rm C}
= a_{\rm C} \sin(2 \phi),$$ where $$\begin{aligned}
\label{J0}
a_0 &= \frac{E_0^2(R_{\rm A}^2-R_{\rm B}^2)}{2},
\\
\label{tildeJL}
a_{\rm L1} & = \frac{E_0^2
\left[R_{\rm A}^2\sin(2 \theta_{\rm A}) -R_{\rm B}^2\sin( 2 \theta_{\rm B} \right]}{2},
\\
\label{JL}
a_{\rm L2 }& =\frac{E_0^2 \left[R_{\rm A}^2\cos(2 \theta_{\rm A})
-R_{\rm B}^2\cos( 2 \theta_{\rm B}) \right]}{2},
\\
\label{JC}
a_{\rm C} & =-\frac{E_0^2 R_{\rm A} R_{\rm B}}{2} \sin(\theta_{\rm A} -\theta_{\rm B})\end{aligned}$$ are purely geometrical factors. Although these factors can change for a more realistic models of antennas, the general structure of Eqs. \[UA\] and \[UB\] should remain the same, so that one can use $a_0, a_{\rm L1 }, a_{\rm L2 }$ and $a_{\rm C} $ as fitting parameters that do not depend on frequency and gate voltage. Importantly, terms proportional to $U_{\rm A}^2- U_{\rm B}^2$ and to $U_{\rm A} U_{\rm B}$ have different angle periodicity—$\pi/2$ and $\pi,$ respectively—that allows one to separate them in experiment. Dependence of the response $U$ on $\phi$ can be found by using results of Ref. [@S1] \[see Eqs. (14-16) in [@S1]\] : $$U(\phi)= \frac{\omega}{\sqrt{\omega^2 +\gamma^2}}\frac{\alpha (U_{\rm A}^2-U_{\rm B}^2)
+\beta U_{\rm A} U_{\rm B} \sin (\theta_{\rm A} -\theta_{\rm B})}{4 U_g \left| \sin (kL)\right|^2} ,
\label{response1}$$ $$= \frac{\omega}{4 U_{\rm g} \left| \sin (kL)\right|^2\sqrt{\omega^2 +\gamma^2}} \left\{ \displaystyle \alpha \left[a_0 + a_{\rm L1} \frac{\sin (4 \phi)}{2} \right. \right. +$$ $$+ \left. \left. a_{\rm L2}\frac{1+ \cos(4\phi)}{4} \right]+ \beta a_{\rm C} \sin(2 \phi)\right \},
\label{response2}$$ where $s$ is the plasma wave velocity, $L$ is the length of the FET channel, $$k= \frac{\sqrt{\omega(\omega+ i \gamma)}}{s} =\frac{\Omega+ i \Gamma}{ s},$$ is the plasma wave vector, $\gamma$ is the inverse momentum relaxaton time, and $\Omega$ and $\Gamma$ are effective frequency and damping rate of the plasma waves, given by $$\label{OmegaGamma} \Omega = \sqrt{\frac{\sqrt{\omega^4+ \omega^2 \gamma^2} +\omega^2 }{2}} ,
\quad \Gamma = \sqrt{\frac{\sqrt{\omega^4+ \omega^2 \gamma^2} -\omega^2 }{2}} .$$ and the coefficients $\alpha$ and $\beta$ read $$\begin{aligned}
\label{alpha}
& \alpha= \left(\! 1\!+\!\frac{\gamma \Omega\!}{ \Gamma \omega}\right) \sinh^2\!\left(\!\frac{\Gamma L}{s}\!\right)\!-\! \left( \! 1\!-\!\frac{\Gamma \gamma}{ \Omega \omega}\!\right) \sin^2\!\left(\!\frac{\Omega L}{s}\!\right),
\\
& \beta=8\sinh\!\left(\!\frac{\Gamma L}{s}\!\right)\sin\!\left(\!\frac{\Omega L}{s}\!\right).
\label{beta}
\end{aligned}$$ Equation (\[response2\]) can be written in form of Eq. (\[eqn:response2\]) with $$\begin{aligned}
\label{F1}
&F_1= \frac{ \omega~ \alpha }{4 U_{\rm g} \left| \sin (kL)\right|^2\sqrt{\omega^2 +\gamma^2}}, \\
&F_2=\frac{\omega ~\beta}{4 U_{\rm g} \left| \sin (kL)\right|^2\sqrt{\omega^2 +\gamma^2}}.
\label{F_2}\end{aligned}$$ The coefficients $U_0,~U_{\rm L1},~U_{\rm L2},$ and $U_{\rm C}$ entering Eq. (\[eqn:eq1\]) reads $$U_0= F_1 a_0 , ~U_{\rm L1}= F_1 a_{\rm L 1},~ U_{\rm L2}= F_1 a_{\rm L2},~ U_{\rm C}= F_2 a_{\rm C}.$$
Interference part of the response
---------------------------------
Eq. (\[response2\]) allows to extract experimentally interference contribution to response. We find $$U_{\rm C} = \frac{U(\phi = 45^\circ) -U(\phi = 135^\circ)}{2}= \frac{U_{\sigma^+} -U_{\sigma^-} }{2}=$$ $$\label{interference}
= \frac{\displaystyle 2\omega a_{\rm C}\sinh\!
\left({\Gamma L}/{s}\!\right)\sin\!\left({\Omega L}/{s}\right)}{ U_{\rm g} \left| \sin (kL)\right|^2
\sqrt{\omega^2 +\gamma^2}}.$$ Hence, interference term in our setup depends only on circular component of polarization. Using simplified formula for geometrical factor, $J_{\rm C},$ \[see Eq. (\[JC\])\], we rewrite Eq. (\[interference\]) as follows $$U_{\rm C}= \frac{E_0^2 R_{\rm A } R_{\rm B}}{4 U_{\rm g}} \frac{\omega}{\sqrt{\omega^2 +\gamma^2} } \times$$ $$\label{final}
\times \frac{\displaystyle \sinh\! \left({\Gamma L}/{s}\!\right)\sin\!\left({\Omega L}/{s}\right) \sin(\theta_{\rm A} -\theta_{\rm B} )}{ \left| \sin (kL)\right|^2}.$$ We see that the response is proportional to $\sin(\theta_{\rm A} -\theta_{\rm B} ).$ One can show that for device with a beam splitter and a delay line one should simply add in Eq. (\[final\]) the delay line phase shift $\delta$ to geometrical phase shift $\theta_{\rm A} -\theta_{\rm B}$: $$\sin(\theta_{\rm A} -\theta_{\rm B} ) \to \sin(\theta_{\rm A} -\theta_{\rm B} + \delta ).$$
Non-resonant regime
-------------------
Equations (\[OmegaGamma\]), (\[alpha\]), (\[beta\]), (\[F1\]) and (\[F\_2\]) simplify in the non-resonant regime, $\omega \ll \gamma, ~ s/L \ll \gamma$ under additional assumption $L \gg L_*$ (long non-resonant sample [@S1]). Here $$L_*= \frac{s\sqrt 2}{\sqrt {\omega \gamma}}
\label{eqn:L_star}$$ is the plasma wave decay length. In this regime, plasma waves, excited at the source and drain parts of the channel weakly overlap inside the transistor channel. The response becomes [@S1; @b3]$$U=\frac{U_{\rm A}^2- U_{\rm B}^2}{4 U_{\rm g}} +$$ $$+ \frac{16 U_{\rm A} U_{\rm B} e^{-L/L_*} \sin(L/L_*) (\omega/\gamma) \sin (\varphi_{\rm A}-\varphi_{\rm B}) }{4 U_{\rm g}},
\label{eqn:simple_eq}$$ while the coefficients $F_1$ and $F_2$ are given by Eqs. (\[F12\]) of the main text. Physically, last phase- and helicity-sensitive term in Eq. (\[eqn:simple\_eq\]) describes interference arising due to overlapping of the plasma waves.
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While the circular photocurrents due to the photon drag and photogalvanic effects in the bulk of graphene have been observed previously (see the Ref. \[$\!\!$\] for review), for present experimental geometry applying normal incident radiation, it is forbidden by symmetry arguments. This is because, for graphene with structure inversion asymmetry (induced by substrate and/or gate) described by C$_{6v}$ point symmetry, the circular photocurrents are allowed and experimentally demonstrated for oblique incidence only[@Karch2010; @Jiang2011; @Glazov2014].
From the experimental conductance curves the scattering rate $\gamma$ is estimated to be about 50 THz, which is much larger than the radiation frequency 2.54 THz used in our work.
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In Ref. \[$\!\!$\] Eq. (17) for non-resonant response of long sample contains typo. Correct equation is given by Eq. (24) of the arXive version of Ref. \[$\!\!$\]: I. V. Gorbenko, V. Yu. Kachorovskii, M. S. Shur, arXiv:1807.05456
|
---
author:
- Jens Marklof
- Andreas Strömbergsson
title: Generalized linear Boltzmann equations for particle transport in polycrystals
---
The linear Boltzmann equation describes the macroscopic transport of a gas of non-interacting point particles in low-density matter. It has wide-ranging applications, including neutron transport, radiative transfer, semiconductors and ocean wave scattering. Recent research shows that the equation fails in highly-correlated media, where the distribution of free path lengths is non-exponential. We investigate this phenomenon in the case of polycrystals whose typical grain size is comparable to the mean free path length. Our principal result is a new generalized linear Boltzmann equation that captures the long-range memory effects in this setting. A key feature is that the distribution of free path lengths has an exponential decay rate, as opposed to a power-law distribution observed in a single crystal.
Introduction {#sec:intro}
============
The Lorentz gas, introduced by Lorentz in the early 1900s [@Lorentz05] to model electron transport in metals, has become one of the most prominent objects in non-equilibrium statistical mechanics. It describes a gas of non-interacting point particles in an infinite array of identical spherical scatterers. Lorentz showed, by adapting Boltzmann’s classical heuristics for the hard sphere gas, that in the limit of low scatterer density (Boltzmann-Grad limit) the evolution of a macroscopic particle cloud is described by the linear Boltzmann equation. A rigorous derivation of the linear Boltzmann equation from the microscopic dynamics has been given in the seminal papers by Gallavotti [@Gallavotti69], Spohn [@Spohn78] and Boldrighini, Bunimovich and Sinai [@Boldrighini83], under the assumption that the scatterer configuration is sufficiently disordered.
For scatterer configurations with long-range correlations, such as crystals or quasicrystals, the linear Boltzmann equation fails and must be replaced by a more general transport equation that takes into account additional memory effects. The failure of the linear Boltzmann equation was first pointed out by Golse [@GolseToulouse2008] for periodic scatterer configurations. We subsequently provided a complete microscopic derivation of the correct transport equation in this setting [@partII]. An important characteristic is here that the distribution of free path lengths has a power-law tail with diverging second moment [@Bourgain98; @Caglioti03; @Boca07; @partIV] and the long-time limit of the transport problem is superdiffusive [@super]. Surveys of these and other recent advances on the microscopic justification of generalized Boltzmann equations can be found in [@GolseToulouse2008; @icmp; @ICM2014].
Independent of these developments, Larsen has recently proposed a stationary generalized linear Boltzmann equation for homogeneous media with non-exponential path length distribution, non-elastic scattering and additional source terms. We refer the reader to Larsen and Vasques [@Larsen11; @Vasques14a; @Vasques14b] and Frank and Goudon [@Frank10] for more details and applications. Prompted by a question of Larsen, the present paper aims to generalize our findings for periodic scatterer configurations [@partIII; @partI; @partII; @partIV] to polycrystals.
The approach of this study combines the methods of [@partI; @partII] with equidistribution theorems from [@union], which were originally developed to analyse unions of incommensurable lattices. We will argue that, in the Boltzmann-Grad limit, the time evolution of a particle cloud is governed by the transport equation $$\label{glB220}
\big[ \partial_t + \vecv\cdot\nabla_\vecx - \partial_\xi \big] f_t(\vecx,\vecv,\xi,\vecv_+) \\
= \int_{\S_1^{d-1}} f_t(\vecx,\vecv_0,0,\vecv) \,
p_\vecnull(\vecv_0,\vecx,\vecv,\xi,\vecv_+) \,
d\vecv_0$$ subject to the initial condition $$\label{ini0}
\lim_{t\to 0}f_t(\vecx,\vecv,\xi,\vecv_+) = f_0(\vecx,\vecv)\, p(\vecx,\vecv,\xi,\vecv_+) ,$$ where $f_0(\vecx,\vecv)$ is the particle density in phase space at time $t=0$ and $p(\vecx,\vecv,\xi,\vecv_+)$ is a stationary solution of . The variables $\xi$ and $\vecv_+$ represent the distance to the next collision and the velocity thereafter. By adapting our techniques for single crystals [@partI], we will compute the collision kernel $p_\vecnull(\vecv_0,\vecx,\vecv,\xi,\vecv_+)$ in terms of the corresponding kernel of each individual grain. The kernel yields the conditional probability measure $$p_\vecnull(\vecv_0,\vecx,\vecv,\xi,\vecv_+) \, d\xi\, d\vecv_+$$ for the distribution of $(\xi,\vecv_+)\in\RR_{>0}\times \US$ conditional on $\vecv_0,\vecx,\vecv$. If the grain diameters are sufficiently small on the scale of the mean free path length, the collision kernel has an explicit representation in terms of elementary functions. This yields particularly simple formulas in dimension $d=2$. The necessity of extending the phase space has already been observed in the case of a single crystal [@Caglioti10; @partII], finite unions [@union] and in quasicrystals [@quasi; @quasikinetic], where the collision kernel is independent of $\vecx$. If the scatterer configuration is disordered and no long-range correlations are present, the dynamics reduces in the Boltzmann-Grad limit to the classical linear Boltzmann equation [@Boldrighini83; @Gallavotti69; @Spohn78].
The generalized linear Boltzmann equation can be understood as the Fokker-Planck-Kolmorgorov equation (backward Kolomogorov equation) of the following Markovian random flight process: Consider a test particle travelling with constant speed along the random trajectory $$\label{xt}
\vecx(t) = \vecx_{\nu_t} + (t-T_{\nu_t})\vecv_{\nu_t} , \qquad \vecx(0)=\vecx_0, \qquad
\vecv(t)=\vecv_{\nu_t} , \qquad \vecv(0)=\vecv_0,$$ where $$\label{xnqn}
\vecx_n = \vecx_0+\vecq_n,\qquad \vecq_n = \sum_{j=1}^{n} \vecv_{j-1} \xi_j ,\qquad T_n := \sum_{j=1}^n \xi_j, \qquad T_0:=0,$$ are the location, displacement and time of the $n$th collision, $\vecv_n$ the velocity after the $n$th collision, and $$\label{xz}
\nu_t := \max\{ n\in\ZZ_{\geq 0} : T_n \leq t \}$$ is the number of collisions within time $t$. The above process is determined by the sequence of random variables $(\xi_j,\vecv_j)_{j\in\NN}$ and $(\vecx_0,\vecv_0)$, where $(\vecx_0,\vecv_0,\xi_1,\vecv_1)$ is distributed according to $$f_0(\vecx_0,\vecv_0)\, p(\vecx_0,\vecv_0,\xi_1,\vecv_1) \,d\vecx_0\,d\vecv_0\,d\xi_1\,d\vecv_1,$$ with $f_0$ now being an arbitrary probability density, and $(\xi_n,\vecv_n)$ is distributed according to $$p_\vecnull(\vecv_{n-2},\vecx_{n-1},\vecv_{n-1},\xi_{n},\vecv_{n})\, d\xi_{n}\,d\vecv_{n},$$ conditional on $(\xi_j,\vecv_j)_{j=1}^{n-1}$ and $(\vecx_0,\vecv_0)$.
[*Acknowledgements.*]{} JM would like to thank Martin Frank, Kai Krycki and Edward Larsen for the stimulating discussions during his visit to RWTH Aachen in June 2014, and in particular Edward Larsen for suggesting the problem of transport in polycrystals. We thank Dave Rowenhorst for providing us with the image in Fig. \[fig1\].
The setting {#sec:setting}
===========
Let $\{\scrG_i\}_i$ be a countable collection of non-overlapping convex open domains in $\RR^d$, and $\{\scrL_i\}_i$ a collection of affine lattices of covolume one. We can write each such affine lattice as $\scrL_i=({\mathbb{Z}}^d+\vecomega_i) M_i$ with row vector $\vecomega_i\in{\mathbb{R}}^d$ and matrix $M_i\in\SL(d,{\mathbb{R}})$. We define a [*polylattice*]{} $\scrP_\epsilon$ as the point set $$\label{PS}
\scrP_\epsilon = \bigcup_i \big(\scrG_i \cap \epsilon \scrL_i\big),$$ where $\epsilon>0$ is a scaling parameter. We refer to $\scrG_i$ as a [*grain*]{} of $\scrP_\epsilon$. An example for $\{\scrG_i\}_i$ is a collection of convex polyhedra that tesselate $\RR^d$, i.e., $\cup_i\overline{\scrG_i}=\RR^d$. In general we will, however, allow gaps between grains. The standing assumption in this paper is that the number of grains intersecting any bounded subset of ${\mathbb{R}}^d$ is finite. The assumption that grains are convex will allow us to ignore correlations of trajectories that re-enter the same grain without intermediate scattering. It is not difficult to extend the present analysis to include these effects. The assumption that all lattices have the same covolume is made solely to simplify the presentation and can easily be removed. Figure \[fig1\], reproduced from [@Rowenhorst10], shows the grains of an actual polycrystal sample, the $\beta$-titanium alloy Ti–21S. The $\beta$-form of titanium has a body-centered cubic lattice, which can be represented as the linear deformation $\ZZ^3 M$ of the cubic lattice $\ZZ^3$, where $$\label{BCC}
M=\begin{pmatrix}
2^{1/3} & 0 & 0 \\
0 & 2^{1/3} & 0 \\
2^{-2/3} & 2^{-2/3} & 2^{-2/3}
\end{pmatrix} .$$
![Grains in a sample of the $\beta$-titanium alloy Ti–21S. The image is reproduced from ref. [@Rowenhorst10] by Rowenhorst, Lewis and Spanos.[]{data-label="fig1"}](Figure1a){width="80.00000%"}
To model the microscopic dynamics in a polycrystal, we place at each point in $\scrP_\epsilon$ a spherical scatterer of radius $r>0$, and consider a point particle that moves freely until it hits a sphere, where it is scattered, e.g. by elastic reflection (as in the classic setting of the Lorentz gas) or by the force of a spherically symmetric potential. We denote the position and velocity at time $t$ by $\vecx(t)$ and $\vecv(t)$. Since (i) the particle speed outside the scatterers is a constant of motion and (ii) the scattering is elastic, we may assume without loss of generality $\|\vecv(t)\|=1$. The dynamics thus takes place in the unit tangent bundle $\T^1(\scrK_{\eps,r})$ where $\scrK_{\eps,r}\subset\RR^d$ is the complement of the set $\scrB^d_r + \scrP_\epsilon$. Here $\scrB^d_r$ denotes the open ball of radius $r$, centered at the origin. We parametrize $\T^1(\scrK_{\eps,r})$ by $(\vecx,\vecv)\in\scrK_{\eps,r}\times\S_1^{d-1}$, where we use the convention that for $\vecx\in\partial\scrK_{\eps,r}$ the vector $\vecv$ points away from the scatterer (so that $\vecv$ describes the velocity [*after*]{} the collision). The Liouville measure on $\T^1(\scrK_{\eps,r})$ is $d\nu(\vecx,\vecv)=d\vecx\,d\vecv$, where $d\vecx=d\!\vol_{\RR^d}(\vecx)$ and $d\vecv=d\!\vol_{\S_1^{d-1}}(\vecv)$ refer to the Lebesgue measures on $\RR^d$ and $\S_1^{d-1}$, respectively.
Free path length {#sec:Free}
================
The first collision time with respect to the initial condition $(\vecx,\vecv)\in\T^1(\scrK_{\eps,r})$ is $$\label{TAU1DEF0}
\tau_1(\vecx,\vecv) = \inf\{ t>0 : \vecx+t\vecv \notin\scrK_{\eps,r} \}.$$ Since all particles are moving with unit speed, we may also refer to $\tau_1(\vecx,\vecv)$ as the free path length. The mean free path length, i.e. the average time between collisions, is for $r\to 0$ asymptotic to $\sigmabar^{\,-1} \epsilon^d r^{-(d-1)}$ where $\sigmabar=\vol\UB$ (the total scattering cross section in units of $r$); this calculation only takes into account the time travelled inside the grains. The scaling limit we are interested in is when the typical grain size is of the order of the mean free path length. We choose (without loss of generality) $\epsilon=r^{(d-1)/d}$ and fix this relation for the rest of this paper. The mean free path length in these units is thus $\sigmabar^{\,-1}$.
Given $(\vecx,\vecv)\in\RR^d\times\US$, we call the sequence $(i_\nu)_{\nu\in\NN}$ the [*itinary of $(\vecx,\vecv)$*]{} if $i_\nu=i_\nu(\vecx,\vecv)$ is the index of the $\nu$th grain $\scrG_{i_\nu}$ traversed by the trajectory $\{(\vecx+t\vecv,\vecv) : t\geq 0\}$. We denote by $\ell_\nu^-=\ell_\nu^-(\vecx,\vecv),\ell_\nu^+=\ell_\nu^+(\vecx,\vecv)\in[0,\infty]$ the entry resp. exit time for $\scrG_{i_\nu}$. If $\vecx\in \scrG_{i_1}$, or if $\vecx\in \partial\scrG_{i_1}$ and $\vecv$ points towards the grain, we set $\ell_1^-=0$. We furthermore define the sejour time for each grain by $\ell_\nu:=\ell_\nu^+-\ell_\nu^-$. Note that, if $\vecx\in\scrG_{i_1}$ and $s$ is suffciently small so that $\vecx+s\vecv\in\scrG_{i_1}$, then $$\label{shift1}
\ell_1^-(\vecx+s\vecv,\vecv) = \ell_1^-(\vecx,\vecv)=0,\qquad
\ell_1^+(\vecx+s\vecv,\vecv) = \ell_1^+(\vecx,\vecv) - s$$ and, for all $\nu\geq 2$, $$\label{shift2}
\ell_\nu^\pm(\vecx+s\vecv,\vecv) = \ell_\nu^\pm(\vecx,\vecv) - s .$$
We now consider initial data of the form $(\vecx_{\eps,r},\vecv)=(\vecx+\epsilon \vecq+r \vecbeta(\vecv),\vecv)$, where $\vecv\in\S_1^{d-1}$ is random, $\vecx,\vecq\in\RR^d$ are fixed and $\vecbeta:\S_1^{d-1}\to\RR^{d}$ is some fixed continuous function. For $\vecx_{\eps}:=\vecx+\epsilon \vecq\notin\scrP_\epsilon$ (the particle is not $r$-close to a scatterer), the free path length $\tau_1(\vecx_{\eps,r},\vecv)$ is evidently well defined for $r$ sufficiently small. If $\vecx_{\eps}\in\scrP_\epsilon$ (the particle is $r$-close to a scatterer), we assume in the following that $\vecbeta$ is chosen so that the ray $\vecbeta(\vecv)+{\mathbb{R}}_{\geq 0}\vecv$ lies completely outside the ball $\scrB_1^d$ for all $\vecv\in\S^{d-1}_1$ (thus $r\vecbeta(\vecv)+{\mathbb{R}}_{\geq 0}\vecv$ lies outside $\scrB_r^d$ for all $r>0$).
Let $\scrS$ be the commensurator of $\SL(d,{\mathbb{Z}})$ in $\SL(d,{\mathbb{R}})$. We have $$\begin{aligned}
\scrS=
\{(\det T)^{-1/d}T{\: : \:}T\in\GL(d,{\mathbb{Q}}),\:\det T>0\},
$$ cf. [@borel Thm. 2], as well as [@studenmund Sec. 7.3]. We say that matrices $M_1,M_2,\ldots\in\SLR$ are [*pairwise incommensurable*]{} if $M_i M_j^{-1}\notin\scrS$ for all $i\neq j$. The pairwise incommensurability of $M_1,M_2,\ldots$ is equivalent to the fact that the lattices $\scrL_i= ({\mathbb{Z}}^d+\vecomega_i) M_i$ ($i=1,2,\ldots$) are pairwise incommensurable, in the sense that for any $i\neq j$, $c>0$ and $\vecomega\in{\mathbb{R}}^d$, the intersection $\scrL_i\cap(c\scrL_j+\vecomega)$ is contained in some affine linear subspace of dimension strictly less than $d$. A natural example in the present setting would be a sequence of matrices $M_i=MK_i$ with $M=1$ (or $M$ as in ) and incommensurable rotation matrices $K_i\in\SO(d)$, corresponding to (body-centered) cubic crystal grains with pairwise incommensurable orientation.
The following two theorems comprise our main results for the distribution of free path length. The first theorem deals with generic initial data, the second when the initial position is near or on a scatterer, but still generic with respect to lattices in other grains.
We will in the following use the notation $$D_\Phi(\xi)=\int_\xi^\infty \Phi(\eta) d\eta = 1-\int_0^\xi \Phi(\eta) d\eta $$ for the complementary distribution function of the probability density $\Phi$.
In the following we consider lattices $\scrL_i= \epsilon^{-1}\vecx+({\mathbb{Z}}^d+\vecomega_i) M_i$ with an additional shift by $\epsilon^{-1}\vecx$. This looks artificial but is necessary for all subsequent statements to hold. \[The problem becomes easier if we assume that $\vecomega_i$ are independent random variables uniformly distributed in the torus $\RR^d/\ZZ^d$. In this case it is fine to use $\scrL_i= ({\mathbb{Z}}^d+\vecomega_i) M_i$. All of the statements below will also hold in this case.\]
\[freeThm\] Fix $\vecx\in\RR^d$ and, for all $i\in\NN$, let $\scrL_i= \epsilon^{-1}\vecx+({\mathbb{Z}}^d+\vecomega_i) M_i$ with $\vecomega_i\in{\mathbb{R}}^d$, and $M_i\in\SL(d,{\mathbb{R}})$ pairwise incommensurable. Fix $\vecq\in{\mathbb{R}}^d$ so that $\vecomega_i-\vecq M_i^{-1}\notin\QQ^d$ for all $i$. If $(i_\nu)_{\nu\in\NN}$ is the itinary of $(\vecx,\vecv)$, then, for any Borel probability measure $\lambda$ on $\S_1^{d-1}$ and any $\xi\geq 0$, $$\label{FPL}
\lim_{r\to 0} \lambda(\{ \vecv\in\S_1^{d-1} {\: : \:}\tau_1(\vecx_{\eps,r},\vecv)\geq \xi \})
= \int_{\xi}^\infty \int_{\US} \Psi(\vecx,\vecv,\eta)\,d\lambda(\vecv)\, d\eta$$ with $$\label{limid}
\Psi(\vecx,\vecv,\xi)
=
\begin{cases}
\bigl(\prod_{\mu=1}^{\nu-1} D_\Phi(\ell_{\mu})\bigr)\, \Phi(\xi-\ell_{\nu}^-) & \text{if $\ell_{\nu}^-\leq \xi<\ell_{\nu}^+$} \\
0 & \text{otherwise,}
\end{cases}$$ where $\Phi(\xi)$ is the limit probability density of the free path length in the case of a single lattice and for generic inital data, see [@partIV Eq. (1.21)].
By [@partIV Eq. (1.23)] we have, $$\label{smallxi}
\Phi(\xi)=\sigmabar-\frac{\sigmabar^2}{\zeta(d)}\xi+O(\xi^2),$$ where $\zeta(d)$ is the Riemann zeta function and the remainder is non-negative. In dimension $d=2$ the error term in fact vanishes identically for $\xi$ sufficiently small; indeed, for $0< \xi \leq \frac12$ we have [@Boca07 Theorem 2] $$\Phi(\xi)=2-\frac{24}{\pi^2}\,\xi$$ and hence $$D_\Phi(\xi)=1-2\xi+\frac{12}{\pi^2}\,\xi^2 .$$ In dimension $d=3$ we have [@partIV Corollary (1.6)] for $0<\xi\leq\frac14$ $$\Phi(\xi)=\pi-\frac{\pi^2}{\zeta(3)}\xi
+\frac{3\pi^2+16}{2\pi\zeta(3)}\xi^2$$ and so $$D_\Phi(\xi)=1-\pi\xi+\frac{\pi^2}{2\zeta(3)}\xi^2
-\frac{3\pi^2+16}{6\pi\zeta(3)}\xi^3 .$$ This means that the limit distribution $\Psi(\vecx,\vecv,\xi)$ is completely explicit, if the diameter of each grain is bounded above by $\frac12$ in dimension $d=2$ resp. $\frac14$ in dimension $d=3$.
Let us now turn to initial data near a scatterer. Let us fix a map $K:\S_1^{d-1}\to\SO(d)$ such that $\vecv K(\vecv)=\vece_1$ for all $\vecv\in\S_1^{d-1}$ where $\vece_1:=(1,0,\ldots,0)$. We assume that $K$ is smooth when restricted to $\S_1^{d-1}$ minus one point (see [@partI footnote 3, p. 1968] for an explicit construction). We denote by $\vecx_\perp$ the orthogonal projection of $\vecx\in\RR^d$ onto the hyperplane perpendicular to $\vece_1$.
\[freeThm2\] Fix $\vecx\in\scrG_j$ for some $j\in\NN$, and, for all $i\in\NN$, let $\scrL_i= \epsilon^{-1}\vecx+({\mathbb{Z}}^d+\vecomega_i) M_i$ with $\vecomega_i\in{\mathbb{R}}^d$, and $M_i\in\SL(d,{\mathbb{R}})$ pairwise incommensurable. Fix $\vecq\in\RR^d$, such that $\vecx_\epsilon=\vecx+\epsilon\vecq\in\epsilon\scrL_j$ and such that $\vecomega_i-\vecq M_i^{-1}\notin\QQ^d$ for all $i\neq j$. If $(i_\nu)_{\nu\in\NN}$ is the itinary of $(\vecx,\vecv)$, then, for any Borel probability measure $\lambda$ on $\S_1^{d-1}$ and any $\xi\geq 0$, $$\label{FPL2}
\lim_{r\to 0} \lambda(\{ \vecv\in\S_1^{d-1} {\: : \:}\tau_1(\vecx_{\eps,r},\vecv)\geq \xi \})
= \int_\xi^\infty \int_{\US} \Psi_\vecnull(\vecx,\vecv,\eta,(\vecbeta(\vecv) K(\vecv))_\perp)\,d\lambda(\vecv) \,d\eta$$ with $\Psi_\vecnull(\vecx,\vecv,\xi,\vecw)$ defined for any $\vecx\in\cup_j \scrG_j$ by $$\label{limid2}
\Psi_\vecnull(\vecx,\vecv,\xi,\vecw)
=
\begin{cases}
\Phi_\vecnull(\xi,\vecw) & \text{if $0\leq \xi<\ell_{1}^+$} \\
\Phi(\ell_{1},\vecw) \, \bigl(\prod_{\mu=2}^{\nu-1} D_\Phi(\ell_{\mu})\bigr) \, \Phi(\xi-\ell_{\nu}^-) & \text{if $\ell_{\nu}^-\leq \xi<\ell_{\nu}^+$ ($\nu\geq2$)} \\
0 & \text{otherwise,}
\end{cases}$$ where $\Phi_\vecnull(\xi,\vecw)$ is the corresponding limit probability density in the case of a single lattice, and $\Phi(\xi,\vecw)=\int_\xi^\infty\Phi_\vecnull(\eta,\vecw)\,d\eta$.
The single-lattice density is given by $\Phi_{\mathbf{0}}(\xi,\vecw)=\int_{\scrB_1^{d-1}}\Phi_{\mathbf{0}}(\xi,\vecw,\vecz)\,d\vecz$ with $\Phi_{\mathbf{0}}(\xi,\vecw,\vecz)$ as in [@partIV Sect. 1.1] (cf. also Thm. \[exactpos12\] below). We extend the definition of $\Psi_\vecnull(\vecx,\vecv,\xi,\vecw)$ to all $\vecx\in\RR^d$ as follows. Given a grain $\scrG_j$ and $(\vecx,\vecv)$ with $\vecx\in\partial \scrG_j$, we say $\vecv$ [*is pointing inwards*]{} if there exists some $\epsilon_0>0$ such that $\{\vecx+\epsilon\vecv {\: : \:}0<\epsilon< \epsilon_0\}\subset \scrG_j$. Let $$\scrH_j:=\{ (\vecx,\vecv) \in\partial\scrG_j\times\US : \text{$\vecv$ is pointing inwards}\}$$ and $$\widehat\scrG_j:= \big(\scrG_j\times\US \big)\cup \scrH_j .$$ We now extend the definition of $\Psi_\vecnull(\vecx,\vecv,\xi,\vecw)$ to all $\vecx\in\RR^d$, $\xi>0$, by setting $$\Psi_\vecnull(\vecx,\vecv,\xi,\vecw) =
\begin{cases}
\lim_{\epsilon\to 0_+} \Psi_\vecnull(\vecx+\epsilon\vecv,\vecv,\xi-\epsilon,\vecw)
& \text{if $(\vecx,\vecv)\in \cup_j \scrH_j$} \\
0 & \text{if $(\vecx,\vecv)\notin \cup_j \widehat\scrG_j$}.
\end{cases}$$ Let us furthermore define $$\one(\vecx,\vecv) =
\begin{cases}
1 & \text{if $(\vecx,\vecv)\in\cup_i \widehat\scrG_i$,}\\
0 & \text{otherwise,}
\end{cases}$$ and the differential operator $\scrD$ by (assume $\xi>0$) $$\scrD \Psi(\vecx,\vecv,\xi) =
\lim_{\epsilon\to 0_+} \epsilon^{-1}[\Psi(\vecx+\epsilon\vecv,\vecv,\xi-\epsilon)-\Psi(\vecx,\vecv,\xi)]
.$$ Note that we have $\scrD \Psi(\vecx,\vecv,\xi)=\big[ \vecv\cdot\nabla_\vecx - \partial_\xi \big] \Psi(\vecx,\vecv,\xi)$ wherever the right-hand side is well defined (which is the case on a set of full measure). In the case of a single lattice, we have [@partIV Eq. (1.21)] $$\label{sc001}
\Phi(\xi) = \int_\xi^\infty \int_\UB \Phi_\vecnull(\eta,\vecw)\, d\vecw \,d\eta.$$ In the case of a polylattice, generalizes to $$\label{Cau1}
\begin{cases}
\scrD \Psi(\vecx,\vecv,\xi)= \int_{\UB} \Psi_\vecnull(\vecx,\vecv,\xi,\vecw)\,d\vecw & (\xi>0) \\
\Psi(\vecx,\vecv,0) = \sigmabar\, \one(\vecx,\vecv). &
\end{cases}$$ This relation follows from , and in view of the relations , .
The transition kernel {#sec:Transition}
=====================
To go beyond the distribution of free path length, and towards a full understanding of the particle dynamics in the Boltzmann-Grad limit, we need to refine the results of the previous section and consider the joint distribution of the free path length and the precise location [*on*]{} the scatterer where the particle hits.
Given initial data $(\vecx,\vecv)$, we denote the position of impact on the first scatterer by $$\vecx_1(\vecx,\vecv) := \vecx+\tau_1(\vecx,\vecv) \vecv .$$ Given the scatterer location $\vecy\in\scrP_\eps$, we have $\vecx_1(\vecx,\vecv)\in \S_r^{d-1} + \vecy$ and therefore there is a unique point $\vecw_1(\vecx,\vecv)\in \S_1^{d-1}$ such that $\vecx_1(\vecx,\vecv)=r \vecw_1(\vecx,\vecv)+\vecy$. It is evident that $-\vecw_1(\vecx,\vecv) K(\vecv)\in {{{\S'_1}^{d-1}}}$, with the hemisphere ${{{\S'_1}^{d-1}}}=\{\vecv=(v_1,\ldots,v_d)\in\S_1^{d-1} {\: : \:}v_1>0\}$. The impact parameter of the first collision is $\vecb=(\vecw_1(\vecx,\vecv) K(\vecv))_\perp$.
As in Section \[sec:Free\], we will use the initial data $(\vecx_{\eps,r},\vecv)=(\vecx+\epsilon \vecq+r \vecbeta(\vecv),\vecv)$, where $\vecv\in\S_1^{d-1}$ is random, $\vecx,\vecq\in\RR^d$ are fixed and $\vecbeta:\S_1^{d-1}\to\RR^{d}$ is some fixed continuous function.
We again have two theorems, the first for generic initial data, the second when the initial position is near or on a scatterer, but still generic with respect to lattices in other grains. Theorems \[freeThm\] resp. \[freeThm2\] follow from Theorems \[exactpos1\] resp. \[exactpos12\] below by taking the test set $\fU={{{\S'_1}^{d-1}}}$.
\[exactpos1\] Fix $\vecx\in\RR^d$ and, for all $i\in\NN$, let $\scrL_i= \epsilon^{-1}\vecx+({\mathbb{Z}}^d+\vecomega_i) M_i$ with $\vecomega_i\in{\mathbb{R}}^d$, and $M_i\in\SL(d,{\mathbb{R}})$ pairwise incommensurable. Fix $\vecq\in{\mathbb{R}}^d$ so that $\vecomega_i-\vecq M_i^{-1}\notin\QQ^d$ for all $i$. If $(i_\nu)_{\nu\in\NN}$ is the itinary of $(\vecx,\vecv)$, then for any Borel probability measure $\lambda$ on $\S_1^{d-1}$ absolutely continuous with respect to $\vol_{\S_1^{d-1}}$, any subset $\fU\subset{{{\S'_1}^{d-1}}}$ with $\vol_{\S_1^{d-1}}(\partial\fU)=0$, and any $0\leq a< b$, we have $$\begin{gathered}
\label{exactpos1eq}
\lim_{r\to 0} \lambda\bigl(\bigl\{ \vecv\in\S_1^{d-1} {\: : \:}\tau_1\in [a,b), \:
-\vecw_1K(\vecv)\in\fU \bigr\}\bigr) \\
=\int_{a}^{b} \int_{\fU_\perp} \int_{\S_1^{d-1}}
\Psi\bigl(\vecx,\vecv,\xi,\vecw)
\, d\lambda(\vecv)\, d\vecw \, d\xi,\end{gathered}$$ where $$\label{limid3}
\Psi(\vecx,\vecv,\xi,\vecw)
=
\begin{cases}
\bigl(\prod_{\mu=1}^{\nu-1} D_\Phi(\ell_{\mu})\bigr)
\, \Phi(\xi-\ell_{\nu}^-,\vecw) & \text{if $\ell_{\nu}^-\leq \xi<\ell_{\nu}^+$} \\
0 & \text{otherwise.}
\end{cases}$$
\[exactpos12\] Fix $\vecx\in\scrG_j$ for some $j\in\NN$, and, for all $i\in\NN$, let $\scrL_i= \epsilon^{-1}\vecx+({\mathbb{Z}}^d+\vecomega_i) M_i$ with $\vecomega_i\in{\mathbb{R}}^d$, and $M_i\in\SL(d,{\mathbb{R}})$ pairwise incommensurable. Fix $\vecq\in\RR^d$, such that $\vecx_\epsilon=\vecx+\epsilon\vecq\in\epsilon\scrL_j$ and such that $\vecomega_i-\vecq M_i^{-1}\notin\QQ^d$ for all $i\neq j$. If $(i_\nu)_{\nu\in\NN}$ is the itinary of $(\vecx,\vecv)$, then for any Borel probability measure $\lambda$ on $\S_1^{d-1}$ absolutely continuous with respect to $\vol_{\S_1^{d-1}}$, any subset $\fU\subset{{{\S'_1}^{d-1}}}$ with $\vol_{\S_1^{d-1}}(\partial\fU)=0$, and any $0\leq a< b$, we have $$\begin{gathered}
\label{exactpos1eq2}
\lim_{r\to 0} \lambda\bigl(\bigl\{ \vecv\in\S_1^{d-1} {\: : \:}\tau_1\in [a,b), \:
-\vecw_1K(\vecv)\in\fU \bigr\}\bigr) \\
=\int_{a}^{b} \int_{\fU_\perp} \int_{\S_1^{d-1}}
\Psi_\vecnull\bigl(\vecx,\vecv,\xi,\vecw,(\vecbeta(\vecv)K(\vecv))_\perp\bigr)
\, d\lambda(\vecv)\, d\vecw \, d\xi\end{gathered}$$ with $$\label{limid22}
\Psi_\vecnull(\vecx,\vecv,\xi,\vecw,\vecz)
=
\begin{cases}
\Phi_\vecnull(\xi,\vecw,\vecz) & \text{if $0\leq \xi<\ell_{1}^+$} \\
\Phi(\ell_{1},\vecz) \, \bigl(\prod_{\mu=2}^{\nu-1} D_\Phi(\ell_{\mu})\bigr)\, \Phi(\xi-\ell_{\nu}^-,\vecw) & \text{if $\ell_{\nu}^-\leq \xi<\ell_{\nu}^+$ ($\nu\geq2$),} \\
0 & \text{otherwise,}
\end{cases}$$ where $\Phi_\vecnull(\xi,\vecw,\vecz)$ is the transition kernel for a single lattice, cf. [@partIV Sect. 1.1].
As above, we extend the definition of $\Psi_\vecnull(\vecx,\vecv,\xi,\vecw,\vecz)$ to all $\vecx\in\RR^d$ by setting $$\Psi_\vecnull(\vecx,\vecv,\xi,\vecw,\vecz) =
\begin{cases}
\lim_{\epsilon\to 0_+} \Psi_\vecnull(\vecx+\epsilon\vecv,\vecv,\xi-\epsilon,\vecw,\vecz)
& \text{if $(\vecx,\vecv)\in \cup_j \scrH_j$} \\
0 & \text{if $(\vecx,\vecv)\notin \cup_j \widehat\scrG_j$}.
\end{cases}$$
We refer the reader to [@partI; @partIV] for a detailed study of $\Phi_{\vecnull}(\xi,\vecw,\vecz)$, $\Phi(\xi,\vecw)$ and $\Phi_{\vecnull}(\xi,\vecw)$, which are related via [@partII Eq. (6.67)], $$\label{fit}
\Phi(\xi,\vecw) =\int_{\xi}^\infty \int_{\scrB_1^{d-1}} \Phi_{\vecnull}(\eta,\vecw,\vecz)\, d\vecz\,d\eta$$ and $$\label{fit22}
\Phi_\vecnull(\xi,\vecw) =\int_{\scrB_1^{d-1}} \Phi_{\vecnull}(\xi,\vecw,\vecz)\, d\vecz .$$ We have in particular [@partIV Eq. (1.18)], $$\begin{aligned}
\label{PHI0ZEROSMALLTHMRES}
\frac{1-2^{d-1}\sigmabar \xi}{\zeta(d)}\leq\Phi_{\mathbf{0}}(\xi,\vecw,\vecz)\leq\frac{1}{\zeta(d)} ,\end{aligned}$$ that is, $\Phi_{\mathbf{0}}(\xi,\vecw,\vecz)=\zeta(d)^{-1} + O(\xi)$, and [@partIV Eq. (1.19)] $$\begin{aligned}
\Phi(\xi,\vecw)
=1-\frac{\sigmabar}{\zeta(d)}\,\xi+O(\xi^2),\end{aligned}$$ where the remainder term is everywhere non-negative, and the implied constant is independent of $\vecw$. As for the free path lengths, we have explicit expressions for these transition kernels in dimensions two and three, which will be discussed in Sections \[sec:d2\] and \[sec:d3\].
The generalization of is $$\label{fitta}
\begin{cases}
\scrD \Psi(\vecx,\vecv,\xi,\vecw)= \int_{\UB}
\Psi_\vecnull(\vecx,\vecv,\xi,\vecw,\vecz)\,d\vecz & \\
\Psi(\vecx,\vecv,0,\vecw) = \one(\vecx,\vecv) . &
\end{cases}$$ Its proof is analogous to . The single-crystal transition kernel satisfies the following invariance properties [@partI]: $$\Phi_{\vecnull}(\xi,\vecz,\vecw) = \Phi_{\vecnull}(\xi,\vecw,\vecz),$$ and for all $R\in \O(d-1)$, $$\Phi_{\vecnull}(\xi,\vecw R,\vecz R) = \Phi_{\vecnull}(\xi,\vecw,\vecz),$$ $$\Phi(\xi,\vecw R) = \Phi(\xi,\vecw), \qquad
\Phi_{\vecnull}(\xi,\vecw R) = \Phi_{\vecnull}(\xi,\vecw).$$ These relations imply $$\Psi_\vecnull(\vecx+\xi\vecv,-\vecv,\xi,\vecz,\vecw) = \Psi_\vecnull(\vecx,\vecv,\xi,\vecw,\vecz),$$ and for all $R\in \O(d-1)$, $$\Psi_\vecnull(\vecx,\vecv,\xi,\vecw R,\vecz R) = \Psi_\vecnull(\vecx,\vecv,\xi,\vecw,\vecz),$$ $$\Psi(\vecx,\vecv,\xi,\vecw R) = \Psi(\vecx,\vecv,\xi,\vecw), \qquad
\Psi_\vecnull(\vecx,\vecv,\xi,\vecw R) = \Psi_\vecnull(\vecx,\vecv,\xi,\vecw).$$
Random point processes and the proof of Theorems \[freeThm\]–\[exactpos12\]
===========================================================================
We follow the same strategy as in [@partI] but use the refined equidistribution theorems for several lattices from [@union]. Recall , namely $\scrP_\epsilon = \bigcup_i \big(\scrG_i \cap \epsilon \scrL_i\big)$, with affine lattices $\scrL_i= \epsilon^{-1}\vecx+({\mathbb{Z}}^d+\vecomega_i) M_i$. Let $$A_\epsilon = \begin{pmatrix} \epsilon & \vecnull \\ \trans\vecnull & \epsilon^{-1/(d-1)} \one_{d-1} \end{pmatrix} \in\SLR.$$ The idea is to consider the sequence of random point processes (with $\epsilon=r^{(d-1)/d}$) $$\label{RPPdef}
\begin{split}
\Theta_\epsilon(\vecx_{\epsilon,r},\vecv) & :=\epsilon^{-1} \big( [\scrP_\epsilon - (\vecx+\epsilon\vecq )]\setminus\{\vecnull\} - r \vecbeta(\vecv) \big) K(\vecv) A_\epsilon
\end{split}$$ (where $\vecv$ is distributed according to $\lambda$) and prove convergence, in finite-dimensional distribution, to a random point process as $\epsilon\to 0$. Note that the removal of the origin in has an effect only when $\vecalf_i:=\vecomega_i-\vecq M_i^{-1}\in{\mathbb{Z}}^d$ for some $i$; in fact we have $$\begin{aligned}
\Theta_\epsilon(\vecx_{\epsilon,r},\vecv)
= \bigcup_i \bigg(\epsilon^{-1}(\scrG_i-\vecx_{\epsilon,r}) \cap \big(({\mathbb{Z}}^d+\vecalf_i\setminus\{{\mathbf{0}}\})M_i-\epsilon^{1/(d-1)} \vecbeta(\vecv)\big)\bigg) K(\vecv) A_\epsilon.\end{aligned}$$
Set $G=\ASLR$, $\Gamma=\ASLZ$, and let $\mu$ be the unique $G$-invariant probability measure on $\GamG$. We let $\Omega$ be the infinite product space $\Omega=\prod_i \GamG$ (one factor for each $\scrG_i$) and let $\omega$ be the corresponding product measure $\prod_i \mu$. Let us define, for any $(g_i)\in\Omega$ and $\vecv\in\US$, $$\Theta(\vecx,\vecv,(g_i)):= \bigcup_i\Bigl[\bigl(\bigl((\scrG_i-\vecx)K(\vecv)\cap{\mathbb{R}}\vece_1\bigr)\times{\mathbb{R}}^{d-1}\bigr)
\cap\ZZ^d g_i\Bigr].$$
Theorems \[exactpos1\] and \[exactpos12\] (and thus Theorems \[freeThm\] and \[freeThm2\]) follow from the next two theorems by the same steps as in \[20, Sections 6 and 9\].
\[thm:ld1\] Fix $\vecx\in\RR^d$ and, for all $i\in\NN$, let $\scrL_i= \epsilon^{-1}\vecx+({\mathbb{Z}}^d+\vecomega_i) M_i$ with $\vecomega_i\in{\mathbb{R}}^d$, and $M_i\in\SL(d,{\mathbb{R}})$ pairwise incommensurable. Fix $\vecq\in{\mathbb{R}}^d$ so that $\vecomega_i-\vecq M_i^{-1}\notin\QQ^d$. Then for any Borel probability measure $\lambda$ on $\S_1^{d-1}$ absolutely continuous with respect to $\vol_{\S_1^{d-1}}$, any bounded sets $\scrB_1,\ldots,\scrB_k\subset\RR^d$ with boundary of measure zero, and $m_1,\ldots,m_k\in\ZZ_{\geq 0}$, $$\begin{gathered}
\lim_{\epsilon\to 0} \lambda\big(\big\{ \vecv\in\US : \, \#(\Theta_\epsilon(\vecx_{\epsilon,r},\vecv) \cap\scrB_l )=m_l
\:\: (\forall l=1,\ldots,k) \big\} \big) \\
=\int_{\US}\omega\bigl(\bigl\{(g_i)\in\Omega{\: : \:}\#(\Theta(\vecx,\vecv,(g_i))\cap\scrB_l)=m_l\:\: (\forall l=1,\ldots,k)
\bigr\}\bigr)\,d\lambda(\vecv).\end{gathered}$$
Now set $G_0=\SLR$, $\Gamma_0=\SLZ$, and let $\mu_0$ be the unique $G_0$-invariant probability measure on $\Gamma_0\backslash G_0$; then let $\widetilde\Omega^{(j)}=(\Gamma_0\backslash G_0)\times\prod_{i\neq j}\GamG$ and let $\widetilde\omega$ be the corresponding product measure $\mu_0\times\prod_{i\neq j}\mu$. Finally let us define, for any $(g_i)\in\widetilde\Omega^{(j)}$ and $\vecv\in\US$, $$\begin{aligned}
\notag
\widetilde\Theta^{(j)}(\vecx,\vecv,(g_i)):=
\Bigl[\bigl(\bigl((\scrG_j-\vecx)K(\vecv)\cap{\mathbb{R}}\vece_1\bigr)\times{\mathbb{R}}^{d-1}\bigr)\cap
\big((\ZZ^d\setminus\{{\mathbf{0}}\}) g_j-(\vecbeta(\vecv)K(\vecv))_\perp\big)\Bigr]
\hspace{20pt}
\\
\cup \:\: \bigcup_{i\neq j}\Bigl[\bigl(\bigl((\scrG_i-\vecx)K(\vecv)\cap{\mathbb{R}}\vece_1\bigr)\times{\mathbb{R}}^{d-1}\bigr)
\cap\ZZ^d g_i\Bigr].\end{aligned}$$
\[thm:ld2\] Fix $\vecx\in\scrG_j$ for some $j\in\NN$, and, for all $i\in\NN$, let $\scrL_i= \epsilon^{-1}\vecx+({\mathbb{Z}}^d+\vecomega_i) M_i$ with $\vecomega_i\in{\mathbb{R}}^d$, and $M_i\in\SL(d,{\mathbb{R}})$ pairwise incommensurable. Fix $\vecq\in\RR^d$, such that $\vecx_\epsilon=\vecx+\epsilon\vecq\in\epsilon\scrL_j$ and such that $\vecomega_i-\vecq M_i^{-1}\notin\QQ^d$ for all $i\neq j$. Then for any Borel probability measure $\lambda$ on $\S_1^{d-1}$ absolutely continuous with respect to $\vol_{\S_1^{d-1}}$, any bounded $\scrB_1,\ldots,\scrB_k\subset\RR^d$ with boundary of measure zero, and $m_1,\ldots,m_k\in\ZZ_{\geq 0}$, $$\begin{gathered}
\lim_{\epsilon\to 0} \lambda\big(\big\{ \vecv\in\US : \, \, \#(\Theta_\epsilon(\vecx_{\epsilon,r},\vecv) \cap\scrB_l
)=m_l\:\: (\forall l=1,\ldots,k) \big\} \big) \\
=\int_{\US}\widetilde\omega\bigl(\bigl\{(g_i)\in\widetilde\Omega^{(j)}{\: : \:}\#(\widetilde\Theta^{(j)}(\vecx,\vecv,(g_i))\cap\scrB_l
)=m_l\:\: (\forall l=1,\ldots,k)
\bigr\}\bigr)\,d\lambda(\vecv).\end{gathered}$$
Theorems \[thm:ld1\] and \[thm:ld2\] are implied by [@union Theorem 10] by the same arguments as in [@partI Section 6].
Tail estimates
==============
We will now show that, unlike the case of single crystals, the distribution of free path lengths, as well as the transition kernels, decay exponentially for large $\xi$. This observation relies on the following bound.
\[simpli\] For $\xi\geq 0$, $$D_\Phi(\xi) \leq \max\big( \e^{-\frac{\sigmabar}{2} \xi}, \e^{-\frac{\zeta(d)}{2}} \big).$$
Since $1-x\leq e^{-x}$ for $0\leq x <1$, we have $$D_\Phi(\xi) \leq \e^{-\int_0^\xi \Phi(\eta)\,d\eta} \leq \e^{-\int_0^\xi \max(\sigmabar-\frac{\sigmabar^2}{\zeta(d)}\eta,0) \,d\eta},$$ where the second inequality follows from the positivity of the error term in .
By [*grain diameter*]{} we mean in the following the largest distance between any two points in a single grain. We define the [*gap function*]{}, $\gap(\vecx,\vecv,\xi)$, to be the total length of the trajectory $\{ \vecx+t\vecv : 0\leq t\leq \xi \}$ that is outside $\cup_j\scrG_j$. Note that $\gap(\vecx,\vecv,\xi)\leq \xi$, and furthermore $$\gap(\vecx+s\vecv,\vecv,\xi) = \gap(\vecx,\vecv,\xi+s)-\gap(\vecx,\vecv,s)$$ for all $s\geq 0$.
\[prop:gap\] Assume that all grain diameters are uniformly bounded. Then there are constants $C,\gamma>0$ such that for all $\vecx,\vecv,\xi,\vecw,\vecz$ $$\label{eq:gap}
\Psi_\vecnull(\vecx,\vecv,\xi,\vecw,\vecz) \leq C \e^{-\gamma (\xi-\gap(\vecx,\vecv,\xi))}.$$ The same bound holds for $\Psi(\vecx,\vecv,\xi,\vecw)$, $\Psi_\vecnull(\vecx,\vecv,\xi,\vecw)$ and $\Psi(\vecx,\vecv,\xi)$.
If the grain diameters are bounded above by $\ell>0$, we have $\ell_i\leq \ell$ for all $i$, and hence in view of Lemma \[simpli\], $$D_\Phi(\ell_i)\leq \e^{-\gamma\ell_i}$$ for all $i$, where $\gamma=\min(\frac{\sigmabar}{2},\frac{\zeta(d)}{2\ell})$. The desired bound now follows from .
Therefore, if $\gap(\vecx,\vecv,\xi)\leq \delta \xi$ for some $\delta\in[0,1)$, we have exponential decay in with rate $\gamma(1-\delta)$.
Explicit formulas for the transition kernel in dimension $d=2$ {#sec:d2}
==============================================================
In dimension $d=2$ we have the following explicit formula for the transition probability [@partIII]: $$\label{Xp} \Phi_\vecnull(\xi,\vecw,\vecz)=\frac{6}{\pi^2}\Upsilon\Bigl(1+\frac{\xi^{-1}-\max(|\vecw|,|\vecz|)-1}{|\vecw+\vecz|}\Bigr)$$ with $$\Upsilon(x)=
\begin{cases}
0 & \text{if }x\leq 0\\
x & \text{if }0<x<1\\
1 & \text{if }1\leq x,
\end{cases}$$ The same formula was also found independently by Caglioti and Golse [@Caglioti10] and by Bykovskii and Ustinov [@Bykovskii09], using different methods based on continued fractions. In particular, for all $\xi\leq \frac12$, $$\Phi_\vecnull(\xi,\vecw,\vecz)=\frac{6}{\pi^2}$$ which is thus independent of $\vecw,\vecz$. We have furthermore [@partIII], again for all $\xi\leq \frac12$, $$\Phi_\vecnull(\xi,\vecw)=\frac{12}{\pi^2} ,\qquad
\Phi(\xi,\vecw)=1-\frac{12}{\pi^2} \xi.$$ Recall that in dimension $d=2$, the value $\frac12$ is precisely the mean free path length.
Explicit formulas for the transition kernel in dimension $d=3$ {#sec:d3}
==============================================================
![The functions $F(t)$ and $G(w)$.[]{data-label="fig2"}](Fplot "fig:"){width="45.00000%"} ![The functions $F(t)$ and $G(w)$.[]{data-label="fig2"}](Gplot "fig:"){width="45.00000%"}
The results in this section are proved in [@partIV]. For $0\leq t<1$ set $$\label{FDEF}
F(t)=\pi-\arccos(t)+t\sqrt{1-t^2}
=\Area\bigl(\bigl\{(x_1,x_2)\in\scrB_1^2{\: : \:}x_1<t\bigr\}\bigr),$$ see Fig. \[fig2\]. Then, for $0<\xi\leq\frac14$, $$\label{D3EXPLTHMRES}
\Phi_\vecnull(\xi,\vecw,\vecz)=\zeta(3)^{-1}\Bigl(1
-\frac6{\pi^2}F\bigl({{\textstyle \frac {1}{2}}}\|\vecw-\vecz\|\bigr)\xi\Bigr)$$ and $$\label{D3EXPLTHMCOR1RES1}
\Phi(\xi,\vecw)=1-\frac{\pi}{\zeta(3)}\xi
+\frac6{\pi^2\zeta(3)}G(\|\vecw\|)\xi^2,$$ where $G:[0,1]\to{\mathbb{R}}_{>0}$ is the function $$\label{D3EXPLTHMCOR1GDEF}
G(w)=
\pi\int_0^{1-w}F({{\textstyle \frac {1}{2}}}r)r\,dr
+\int_{1-w}^{1+w}F({{\textstyle \frac {1}{2}}}r)\arccos\Bigl(\frac{w^2+r^2-1}{2wr}\Bigr)
r\,dr,$$ cf. Fig. \[fig2\]. The function $G(w)$ is continuous and strictly increasing, and satisfies $G(0)=\frac{\pi(4\pi+3\sqrt3)}{16}$ and $G(1)=\frac5{16}\pi^2+1$.
The transport equation {#sec:GLBE}
======================
In order to prove that the dynamics of a test particle converges, in the Boltzmann-Grad limit $r\to 0$, to a random flight process $(\vecx(t),\vecv(t))$ defined in –, we require technical refinements of Theorems \[exactpos1\] and \[exactpos12\], where the convergence is uniform over a certain class of $\lambda$. This argument follows the strategy developed in [@partII] for single crystals. We will here not attempt to prove these uniform versions, but move straight to the description of the limit process which is determined by the transition kernel of Theorem \[exactpos12\].
As in the case of a single crystal [@partII], the limiting random flight process becomes Markovian on an extended phase space, where the additional variables are $$\xi(t)=T_{\nu_t+1}-t \in\RR_{>0} \qquad\text{(distance to the next collision)}$$ and $$\vecv_+(t)=\vecv_{\nu_t+1}\in\US\qquad\text{(velocity after the next collision).}$$ The continuous time Markov process $\Xi(t)=(\vecx(t),\vecv(t),\xi(t),\vecv_+(t))$ is determined by the initial distribution $f_0(\vecx,\vecv,\xi,\vecv_+)$ and the collision kernel $p_\vecnull(\vecv_{j-1},\vecx_j,\vecv_j,\xi_{j+1},\vecv_{j+1})$ which yields the probability that the $(j+1)$st collision is at distance $\xi_{j+1}$ from the $j$th collision, with subsequent velocity $\vecv_{j+1}$, given that the $j$th collision takes place at $\vecx_j$ and the particle’s velocities before and after this collision are $\vecv_{j-1}$ and $\vecv_j$. The collision kernel $p_\vecnull(\vecv_0,\vecx,\vecv,\xi,\vecv_+)$ is related to the transition kernel $\Psi_\vecnull(\vecx,\vecv,\xi,\vecw,\vecz)$ of the previous sections by $$p_\vecnull(\vecv_0,\vecx,\vecv,\xi,\vecv_+)
=\Psi_\vecnull(\vecx,\vecv,\xi,\vecb,-\vecs)\,\sigma(\vecv,\vecv_+)$$ where $\sigma(\vecv,\vecv_+)$ is the differential cross section, $\vecs=\vecs(\vecv,\vecv_0)$, $\vecb=\vecb(\vecv,\vecv_+)$ are the exit and impact parameters of the previous resp. next scattering event.
The density $f_t(\vecx,\vecv,\xi,\vecv_+)$ of the process at time $t>0$ is given by $$\int_\scrA f_t(\vecx,\vecv,\xi,\vecv_+)\, d\vecx\,d\vecv\,d\xi\,d\vecv_+ = \PP\big(\Xi(t)\in\scrA\big)$$ for suitable test sets $\scrA$. Let us write $$\label{f1}
f_t(\vecx,\vecv,\xi,\vecv_+) = \sum_{n=0}^\infty f^{(n)}_t(\vecx,\vecv,\xi,\vecv_+)$$ where $f^{(n)}_t(\vecx,\vecv,\xi,\vecv_+)$ is the density of particles that have collided precisely $n$ times in the time interval $[0,t]$. Then $$\label{f2}
f^{(0)}_t(\vecx,\vecv,\xi,\vecv_+) = f_{0}(\vecx-t\vecv,\vecv,\xi+t,\vecv_+),$$ and for $n\geq 1$ $$\begin{gathered}
\label{f3}
f^{(n)}_t(\vecx,\vecv,\xi,\vecv_+) = \int_{T_n<t} f_{0}\big(\vecx_0,\vecv_0,\xi_1,\vecv_1\big) \\ \times \prod_{j=1}^n p_\vecnull(\vecv_{j-1},\vecx_j,\vecv_j,\xi_{j+1},\vecv_{j+1}) \, d\xi_n \, d\vecv_{n-1} \cdots d\xi_1\,d\vecv_0 ,\end{gathered}$$ with $\vecv_{n+1}=\vecv_+$, $\vecv_n=\vecv$, $\xi_{n+1}=\xi+t-T_n$, $\vecx_0=\vecx-\vecq_n-(t-T_n) \vecv$, and with $\vecx_j$, $\vecq_n$, $T_n$ as in . For general densities $f_{0}(\vecx,\vecv,\xi,\vecv_+)$, relations – define a family of linear operators (for $t>0$) $$K_t^{(n)}f_0(\vecx,\vecv,\xi,\vecv_+):=f^{(n)}_t(\vecx,\vecv,\xi,\vecv_+)$$ and $$K_t f_0(\vecx,\vecv,\xi,\vecv_+):=f_t(\vecx,\vecv,\xi,\vecv_+).$$ One can show that $$\sum_{m=0}^n K_{t_2}^{(n-m)} K_{t_1}^{(m)}=K_{t_1+t_2}^{(n)},$$ which in turn implies $K_{t_1+t_2}=K_{t_1} K_{t_2}$, i.e., the operators $K_t$ form a semigroup (reflecting the fact that $\Xi(t)$ is Markovian). The proof of this is analogous to the computation in [@partII Sect. 6.2]. Hence, for $h>0$ we have $f_{t+h}=K_h f_t$. Since the probability of having more than one collision in a small time interval is negligible, we have for small $h$ (cf. [@partII Sect. 6.2]) $$f_{t+h}(\vecx,\vecv,\xi,\vecv_+) = K_h f_t(\vecx,\vecv,\xi,\vecv_+) = K^{(0)}_h f_t(\vecx,\vecv,\xi,\vecv_+) +K^{(1)}_h f_t(\vecx,\vecv,\xi,\vecv_+) +O(h^2).$$ Explicitly, we have by , , $$\begin{gathered}
f_{t+h}(\vecx,\vecv,\xi,\vecv_+) = f_t(\vecx-h\vecv ,\vecv,\xi+h,\vecv_+) \\
+\int_0^h \int_{\US} f_t(\vecx-\xi_1\vecv_0 -(h-\xi_1)\vecv ,\vecv_0,\xi_1,\vecv) \\ \times p_\vecnull(\vecv_0,\vecx-(h-\xi_1)\vecv,\vecv,\xi+h-\xi_1,\vecv_+) \,d\vecv_0 \, d\xi_1 +O(h^2) . \end{gathered}$$ Dividing this expression by $h$ and taking the limit $h\to 0$, we obtain the Fokker-Planck-Kolmogorov equation (or Kolmogorov backward equation) of the Markov process $\Xi(t)$, $$\label{glB22}
{\widetilde{\scrD}}f_t(\vecx,\vecv,\xi,\vecv_+) \\
= \int_{\S_1^{d-1}} f_t(\vecx,\vecv_0,0,\vecv) \,
p_\vecnull(\vecv_0,\vecx,\vecv,\xi,\vecv_+) \,
d\vecv_0 ,$$ where $$\begin{aligned}
{\widetilde{\scrD}}f_t(\vecx,\vecv,\xi,\vecv_+)=
\lim_{\epsilon\to 0_+} \epsilon^{-1}[f_{t+\epsilon}(\vecx+\epsilon\vecv,\vecv,\xi-\epsilon,\vecv_+)-f_t(\vecx,\vecv,\xi,\vecv_+)] .
$$ As for $\scrD$, we observe that ${\widetilde{\scrD}}=\partial_t + \vecv\cdot\nabla_\vecx - \partial_\xi$ at any point where the latter operator is well defined (which is the case for a full measure set). The physically relevant initial condition is $$\label{ini}
\lim_{t\to 0}f_t(\vecx,\vecv,\xi,\vecv_+) = f_0(\vecx,\vecv,\xi,\vecv_+)= f_0(\vecx,\vecv)\, p(\vecx,\vecv,\xi,\vecv_+)$$ with $$p(\vecx,\vecv,\xi,\vecv_+) := \Psi\big(\vecx,\vecv,\xi,\vecb \big)\, \sigma(\vecv,\vecv_+) .$$ The original phase-space density is recovered via projection, $$f_t(\vecx,\vecv)=\int_0^\infty \int_{\S_1^{d-1}} f_t(\vecx,\vecv,\xi,\vecv_+)\, d\vecv_+\, d\xi .$$ Note that implies that $f_t(\vecx,\vecv,\xi,\vecv_+)=p(\vecx,\vecv,\xi,\vecv_+)$ is the stationary solution of , corresponding to $f_0(\vecx,\vecv)=1$. Uniqueness in the Cauchy problem – follows from standard arguments, cf. [@partII Section 6.3].
The generalized linear Boltzmann equation will also hold for other grainy materials, provided different grains are uncorrelated to guarantee the factorization of the individual grain-distribution functions in . We will discuss the simplest example, grains of a disordered medium, in Section \[sec:disorder\]. Note that it is not necessary that the transition probabilities $\Phi_\vecnull(\xi,\vecw,\vecz)$ in each grain are identical—the modifications in are straightforward: replace $\Phi_\vecnull(\xi,\vecw,\vecz)$, $\Phi(\xi,\vecw)$, etc. by the grain-dependent $\Phi_\vecnull^{(i_\nu)}(\xi,\vecw,\vecz)$, $\Phi^{(i_\nu)}(\xi,\vecw)$, etc. throughout.
A simplified kernel {#sec:simple}
===================
Let us assume now that the transition kernel is given by a function $$\Psi_\vecnull(\vecx,\vecv,\xi):=\sigmabar\, \Psi_\vecnull(\vecx,\vecv,\xi,\vecw,\vecz)$$ that is independent of $\vecw,\vecz$. As we have seen in Section \[sec:d2\], this holds in the two-dimensional setting provided the grain size is less than the mean free path length. Then of course also $\Psi(\vecx,\vecv,\xi,\vecw)$ is independent of $\vecw$ and related to the distribution of free path lengths by $$\Psi(\vecx,\vecv,\xi) = \sigmabar\, \Psi(\vecx,\vecv,\xi,\vecw) .$$ Relation becomes ; that is $$\label{three8}
\begin{cases}
\scrD \Psi(\vecx,\vecv,\xi)= \sigmabar\, \Psi_\vecnull(\vecx,\vecv,\xi) & \\
\Psi(\vecx,\vecv,0) = \sigmabar\, \one(\vecx,\vecv) . &
\end{cases}$$ The ansatz $$f_t(\vecx,\vecv,\xi,\vecv_+) = \sigmabar^{\,-1} g_t(\vecx,\vecv,\xi)\, \sigma(\vecv,\vecv_+)$$ reduces equation to $$\label{glB22red}
{\widetilde{\scrD}}g_t(\vecx,\vecv,\xi) \\
= \sigmabar^{\,-1}\, \Psi_\vecnull(\vecx,\vecv,\xi) \int_{\S_1^{d-1}} g_t(\vecx,\vecv_0,0) \, \sigma(\vecv_0,\vecv)\, d\vecv_0$$ with initial condition $$\lim_{t\to 0}g_t(\vecx,\vecv,\xi) = f_0(\vecx,\vecv)\, \Psi(\vecx,\vecv,\xi) .$$ The stationary solution of corresponding to $f_0(\vecx,\vecv)=1$ is $g_t(\vecx,\vecv,\xi) =\Psi(\vecx,\vecv,\xi)$.
Disordered grains {#sec:disorder}
=================
It is instructive to contrast the case of crystal grains discussed above with grains consisting of a disordered medium. We model the medium by scatterers centred at a fixed realisation of a Poisson point process $\scrL_{\text{Poisson}}$ with intensity $1$, rescale by $\epsilon$ and intersect with the grains as in to produce the point set $$\label{PS2}
\scrP_\epsilon = \bigcup_i \big( \scrG_i \cap \epsilon \scrL_{\text{Poisson}}\big) .$$ If there are no gaps between the grains, $\scrP_\epsilon$ is precisely a fixed realisation of a Poisson process with intensity $1$, for which the convergence to the linear Boltzmann equation has been established in [@Boldrighini83]. There do not seem to be any technical obstructions in extending these results to the setting with gaps. In particular, the above results for crystal grains remain valid, if we replace the relevant single-crystal distributions by their disordered counterparts (cf. [@icmp; @ICM2014]): For the transition kernels, we have $$\Phi_\vecnull(\xi,\vecw,\vecz)=\e^{-\sigmabar \xi},\qquad \Phi(\xi,\vecw)=\e^{-\sigmabar \xi},$$ and for the distribution of free path lengths $$\Phi_\vecnull(\xi,\vecw)=\sigmabar \e^{-\sigmabar \xi}, \qquad
\Phi(\xi)=\sigmabar \e^{-\sigmabar \xi}, \qquad D_\Phi(\xi)=\e^{-\sigmabar \xi}.$$ Thus in the case of a disordered granular medium, the formulas , , and become $$\label{limidgran}
\Psi(\vecx,\vecv,\xi,\vecw)
=\sigmabar^{\,-1}\,\Psi(\vecx,\vecv,\xi)
= \e^{-\sigmabar (\xi-\gap(\vecx,\vecv,\xi))} \one(\vecx+\xi\vecv,\vecv),$$ and $$\label{limid2gran}
\Psi_\vecnull(\vecx,\vecv,\xi,\vecw,\vecz)
= \sigmabar^{\,-1}\, \Psi_\vecnull(\vecx,\vecv,\xi,\vecw)= \e^{-\sigmabar (\xi-\gap(\vecx,\vecv,\xi))} \one(\vecx,\vecv) \one(\vecx+\xi\vecv,\vecv).$$ The kernel $\Psi_\vecnull(\vecx,\vecv,\xi):=\sigmabar \, \Psi_\vecnull(\vecx,\vecv,\xi,\vecw,\vecz)$ in is evidently independent of $\vecw,\vecz$, and thus we are in the setting of Section \[sec:simple\]. We have $$\Psi_\vecnull(\vecx,\vecv,\xi) = \Psi(\vecx,\vecv,\xi) \one(\vecx,\vecv),$$ and the ansatz $g_t(\vecx,\vecv,\xi)=f_t(\vecx,\vecv)\Psi(\vecx,\vecv,\xi)$ reduces the generalised Boltzmann equation to the classical density-dependent linear Boltzmann equation $$\label{glB22red3}
\big[ \partial_t + \vecv\cdot\nabla_\vecx \big] f_t(\vecx,\vecv) \\
= \one(\vecx,\vecv) \int_{\S_1^{d-1}} \big[f_t(\vecx,\vecv_0)-f_t(\vecx,\vecv)\big] \, \sigma(\vecv_0,\vecv)\, d\vecv_0$$ with initial condition $$\lim_{t\to 0}f_t(\vecx,\vecv) = f_0(\vecx,\vecv) .$$
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---
abstract: |
We investigate the possibilities of constraining the light gravitino mass $m_{3/2}$ from future cosmic microwave background (CMB) surveys. A model with light gravitino with the mass $m_{3/2} < \mathcal{O}(10) \ \mbox{eV}$ is of great interest since it is free from the cosmological gravitino problem and, in addition, can be compatible with many baryogenesis/leptogenesis scenarios such as the thermal leptogenesis. We show that the lensing of CMB anisotropies can be a good probe for $m_{3/2}$ and obtain an expected constraint on $m_{3/2}$ from precise measurements of lensing potential in the future CMB surveys, such as the PolarBeaR and CMBpol experiments. If the gravitino mass is $m_{3/2} = 1 \ \mbox{eV}$, we will obtain the constraint for the gravitino mass as $m_{3/2}\le 3.2 \ \mbox{eV} \
\mbox{(95\% C.L.)}$ for the case with Planck+PolarBeaR combined and $m_{3/2}=1.04^{+0.22}_{-0.26} \ \mbox{eV} \ \mbox{(68\% C.L.)}$ for CMBpol. The issue of Bayesian model selection is also discussed.
---
IPMU 09-0042\
ICRR-Report-540
[ **Constraining Light Gravitino Mass from\
Cosmic Microwave Background**]{}
.45in
[Kazuhide Ichikawa$^{1}$, Masahiro Kawasaki$^{2,3}$, Kazunori Nakayama$^2$,\
Toyokazu Sekiguchi$^2$ and Tomo Takahashi$^4$ ]{}
.45in
[*$^1$Department of Micro Engineering, Kyoto University, Kyoto 606-8501, Japan\
$^2$Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan\
$^3$Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba, 277-8568, Japan\
$^4$Department of Physics, Saga University, Saga 840-8502, Japan\
*]{}
.4in
Introduction {#sec:introduction}
============
One of the most important prediction of local supersymmetry (SUSY), or supergravity, is the existence of gravitino, the spin-$3/2$ superpartner of the graviton. Although the range of the gravitino mass $m_{3/2}$ can vary from a fraction of eV up to of an order of TeV, depending on the scales of SUSY breaking, a light gravitino with $m_{3/2} < \mathcal{O}(10)$ eV is of great interest since it is free from the cosmological gravitino problem. Furthermore, for some baryogenesis scenario to work such as thermal leptgenesis, high cosmic temperature is required, which favors the range of light mass for the gravitino. Thus, in this respect, the light gravitino would be attractive.
The determination of the gravitino mass is one of the most important issues to understand how supersymmetry is broken. Some authors have discussed this issue, in particular, focusing on probing the mass of the light gravitino with LHC experiment [@Hamaguchi:2007ge]. Since gravitino with very light mass of $m_{3/2} < 100 \ \mbox{eV}$ can play a role of warm dark matter (WDM) in the universe, cosmology would be a powerful tool as well. Some authors have obtained a constraint on the light gravitino mass from Lyman-$\alpha$ forest data in combination with WMAP [@Viel:2005qj; @Boyarsky:2008xj] and its bound is $m_{3/2}<16~$ eV at 2$\sigma$ level [@Viel:2005qj][^1]. Although Lyman-$\alpha$ forest can be very useful as a cosmological probe, it may suffer from some systematic uncertainties [@Viel:2006yh; @Seljak:2006bg]. Notice that the above mentioned limit crucially depends on the usage of Lyman-$\alpha$ forest data, thus in this respect, another independent cosmological probe of the light gravitino would be of great importance. Since light gravitino with the mass of interest here acts as WDM, it affects cosmic density fluctuations through two major effects. One is the change of radiation-matter equality due to the fact that light gravitions behave as relativistic component at earlier times. The second one is the effect of free-streaming which erases density contrast on the scale under which it can free-stream. Since these two effects can make great influence on cosmic microwave background (CMB) anisotropy, CMB can be a powerful probe of its mass. But in fact, when the gravitino is very light as $m_{3/2} < \mathcal{O}(10)$ eV, which is the range of the mass of interest here, its energy density do not have a large fraction of the total one in general to affect the CMB. Thus the mass of light gravitino can be mainly probed through the latter effect, free-streaming. The ultralight gravitino is almost relativistic at the time of the recombination, its effects imprinted on temperature anisotropy is not significant enough. However, by looking at the gravitational lensing of CMB photons, we can well probe the change of the gravitational potential driven during the intermediate redshift after the recombination when light gravitino comes to act as a non-relativistic component. Thus the lensed CMB is a very useful cosmological tool for investigating the mass of light gravitino[^2], which we discuss in this paper. Since much more precise CMB experiments will be available in the near future, it is an interesting subject to investigate possible limits with such a future probe. Therefore, we in this paper study the possibilities of constraining its mass with future CMB observations without using other cosmological data. In particular, we focus on the lensing potential which can be reconstracted from CMB maps. As future CMB surveys, we consider Planck, PolarBeaR and CMBpol to discuss a possible constraint on the light gravitino mass.
The structure of this paper is as follows. We first briefly review a model which predicts light gravitino and its phenomenology in the early universe in Section \[sec:model\]. In Section \[sec:CMB\] we discuss the effects of the light gravitino on CMB anisotropy, paying particular attention to the lensing of CMB. We then present forecasts for constraints on the mass of light gravitino with future CMB surveys such as Planck, PolarBeaR and CMBpol in Section \[sec:constraints\]. In addition to the parameter estimation, we also discuss Bayesian model selection analysis for light gravitino model with future CMB surveys in Section \[sec:selection\]. The final section is devoted to summary of this paper.
Light gravitino: A model and its phenomenology in the early universe {#sec:model}
=====================================================================
A light gravitino scenario is realized in the framework of gauge-mediated SUSY breaking (GMSB) models [@Giudice:1998bp]. In GMSB models, the SUSY breaking effect in the hidden sector is transmitted to the minimal supersymmetric standard model (MSSM) sector through gauge-interactions, giving superparticles TeV scale masses. As an example, let us consider a model where the SUSY breaking field $S$ couples to $N$ pairs of messenger particles $\psi$ and $\bar
\psi$, which transform as fundamental and anti-fundamental representations of SU(5), having a superpotential $W = \lambda S \psi
\bar \psi$ with coupling constant $\lambda$ (in the following we set $\lambda = 1$ for simplicity). The superfield $S$ has a vacuum expectation value as $\langle S \rangle = M+F_S \theta^2$. Here $F_S$ gives SUSY breaking scale, which is related to the garvitino mass $m_{3/2}$ through the relation $F_S = \sqrt 3 m_{3/2} M_P$, with $M_P$ being the reduced Planck energy scale, for vanishing cosmological constant. In this model, gaugino masses $M_a$ ($a=1,2,3$ are gauge indices) and sfermion masses squared $m_{\tilde
f_i}^2$ at the messenger scale are given by $$\begin{gathered}
M_a =N\left (\frac{\alpha_a}{4\pi}\right) \Lambda_{\rm mess}, \\
m_{\tilde f_i}^2 = 2N\sum_a \left ( \frac{\alpha_a}{4\pi} \right )^2C_a^{(i)}
\Lambda_{\rm mess}^2,\end{gathered}$$ where $\alpha_a$ denotes the gauge coupling constants, $C_a^{(i)}$ are Casimir operators for the sfermion $\tilde f_i$ and the messenger scale is given by $\Lambda_{\rm mess} = F_S/M$. In order to obtain TeV scale masses, $\Lambda_{\rm mess} \sim 100$ TeV is required, but still the SUSY breaking scale $F_S$, or gravitino mass $m_{3/2}$ can take wide range of values as $1~{\rm eV}\lesssim m_{3/2} \lesssim
10$ GeV. The upper bound comes from the requirement that the gravity-mediation effect does not dominate. On the other hand, there also exists a lower bound on the gravitino mass in order not to destabilize the messenger scalar and lead to the unwanted vacuum. This requires $m_{3/2} \gtrsim \mathcal O(1)$ eV.
However, if cosmological effects of the gravitino are taken into account, not all of its mass range is favored. This is because gravitinos are efficiently produced at the reheating era and it can easily exceed the present dark matter abundance unless the reheating temperature $T_R$ is very low [@Moroi:1993mb]. This is problematic since many known leptogenesis/baryogenesis scenarios require high enough reheating temperature which may conflict with the upper bound coming from the gravitino problem. In particular, thermal leptogenesis scenario [@Fukugita:1986hr], which requires $T_R
\gtrsim 10^9~$GeV, seems to be inconsistent with the gravitino problem except for the very light gravitino mass range $m_{3/2} \lesssim
100$ eV. As we will see, gravitinos with such a small mass are thermalized in the early Universe. Thus their abundance does not depend on the reheating temperature and also it is smaller than the dark matter density for $m_{3/2}\lesssim 100$ eV. This is the reason why we pay particular attention to a light gravitino scenario.
Having described that the light gravitino scenario is appealing from the view point of cosmological gravitino problem, next we briefly discuss thermal evolution of the light gravitino in the early universe. Gravitinos are relativistic well before the recombination. In such a case, the energy density of the gravitino is parameterized by the effective number of neutrino species, and it is given by $$N_{3/2}=\frac{\rho_{3/2}}{\rho_\nu}=\left(\frac{T_{3/2}}{T_\nu} \right)^4=
\left(\frac{g_{*\nu}}{g_{*3/2}}\right)^{4/3},\label{eq:Ng}$$ where $\rho_\nu$ ($\rho_{3/2}$) and $g_{*\nu}$ ($g_{*3/2}$) are, respectively, the energy density and the effective degrees of freedom of neutrinos (gravitinos) evaluated at the epoch when neutrinos (gravitinos) have decoupled from thermal plasma while they are still relativistic. In the standard cosmology, $g_{*\nu}=10.75$. Temperatures of neutrino and gravitino are represented by $T_\nu$ and $T_{3/2}$. From Eq. (\[eq:Ng\]) we can calculate the temperature of gravitino at present: $$T_{3/2}= ( N_{3/2})^{1/4}T_\nu=1.95 (N_{3/2})^{1/4}~{\rm [K]},$$ where we have adopted the temperature of neutrino in the standard cosmology at the second equality. Eventually the gravitino loses its energy and becomes non-relativistic due to the Hubble expansion. Its present energy density is given by $$\label{eq:omega32}
\omega_{3/2} \equiv \Omega_{3/2}h^2=
0.13\left ( \frac{m_{3/2}}{100~{\rm eV}} \right )\left ( \frac{90}{g_{*3/2}} \right ).
$$ For later convenience, we also define the fraction of gravitino in the total dark matter density $\omega_{\rm dm}$ as $$f_{3/2} \equiv \frac{\omega_{3/2}}{\omega_{\rm dm}}.$$ In the following, we assume that dark matter consists two components: light gravitino, which acts as warm dark matter, and CDM. As CDM component, the Peccei-Quinn axion, a messenger baryon proposed in [@Hamaguchi:2007rb] and so on can be well-fitted into the framework of light gravitino.
Thus in order to evaluate the relic abundance of light gravitino, we must know the value of the effective degrees of freedom of relativistic particles at the freeeze-out epoch, $g_{*3/2}$. Since the production and/or destruction of the light gravitino due to scattering processes are known to be inefficient for the low temperature regime, in which we are interested, the gravitino maintains equilibrium with thermal bath through the decay and inverse-decay processes [@Moroi:1993mb; @Pierpaoli:1997im], schematically represented by $a \leftrightarrow b +\tilde G$, where $b$ is the standard model (SM) particle and $a$ is its superpartner. As the temperature decreases, particles in thermal bath ($b$ and $\tilde G$) lose an ability to create a heavy particle ($a$). Then gravitinos decouple from thermal plasma after the time when $a$ decays into $b$ and $\tilde G$ without inverse creation processes.
In order to see these processes in detail, we must solve the Boltzmann equation which governs time evolution of the system. The Boltzmann equation for the gravitino number density $n_{3/2}$ is given by $$\dot n_{3/2} +3Hn_{3/2} = \sum_{a,b}\Gamma (a \to b \tilde G)
\left \langle \frac{m_a}{E_a} \right \rangle n_a
\left (1- \frac{n_{3/2}}{n_{3/2}^{\rm (eq)}} \right ),$$ where $H$ is the Hubble expansion rate and $\langle m_a/E_a \rangle$ represents thermally averaged Lorentz factor with $m_a$ and $E_a$ being the mass and energy of the particle $a$, respectively. $\Gamma (a \to b \tilde G)$ is the decay width of $a$ into $b$ and $\tilde G$ and superscript ${\rm (eq)}$ denotes its equilibrium value. As an example, the decay rate of the stau $(\tilde
\tau)$ into tau ($\tau$) and gravitino is given by $$\Gamma (\tilde \tau \to \tau \tilde G)=\frac{1}{48\pi}
\frac{m_{\tilde \tau}^5}{m_{3/2}^2 M_P^2},$$ and similar expressions hold for other particles. By solving this equation, one obtains the final gravitino abundance which can be represented in terms of the gravitino-to-entropy ratio, $Y_{3/2}=
(n_{3/2}/s) (t\to \infty)$ and this translates into $g_{*3/2}$ through the relation $Y_{3/2} = 0.417/g_{*3/2}$. In Fig. \[fig:g\], the value of $g_{\ast 3/2}$ is shown as a function of $m_{3/2}$ for several values of $\Lambda_{\rm mess}$. We have adopted $\Lambda_{\rm
mess}=50,100,200~{\rm TeV}$ and $N$=1 and ignored running of the masses from the messenger scale down to the weak scale for simplicity.
We can understand these results intuitively. As the gravitino mass increases, the decay width becomes smaller, and hence the equilibrium lasts for rather shorter duration. This leads to higher freeze-out temperature of the gravitino, which corresponds to large $g_{*3/2}$. On the other hand, larger $\Lambda_{\rm mess}$ leads to heavier sparticle masses, which obviously makes the time of freeze-out of the gravitino earlier, and hence higher $g_{*3/2}$. However, as seen in the figure, the dependence on these parameters are not so strong and we can safely set $g_{*3/2}\simeq 90$ even if we take into account model uncertainties.
Effects of light gravitino on CMB {#sec:CMB}
=================================
In this section we discuss how light gravitino affects the structure formation and the CMB anisotropies. Since light gravitino basically acts as WDM, it is characterized by two quantities, its mass and number density. The number density is determined by the effective number of degrees of freedom at the time of the decoupling, i.e. $g_{*3/2}$. Since, as we have seen in the previous section, $g_{*3/2}$ has only mild dependence on $m_{3/2}$, we can take $g_{*3/2}=90$ as the representative value, which corresponds to $$N_{3/2} = 0.059, \label{eq:fidN}$$ from Eq. (\[eq:Ng\]). We assume Eq. (\[eq:fidN\]) as a fiducial value throughout this section, except for the last paragraph. We also assume that the universe is flat, dark energy is a cosmological constant and the primordial fluctuations are adiabatic and its power spectrum obeys a power-law without tensor perturbations. The fiducial values for cosmological parameters are adopted from the the recent result of WMAP5 [@Komatsu:2008hk], except that we consider mixed dark matter scenarios $\omega_{\rm dm} = \omega_c+\omega_{3/2}=0.1099$, instead of $\omega_c = 0.1099$ where $\omega_c$ is the density parameter for cold dark matter (CDM). By varying $m_{3/2}$ or $f_{3/2}$, we can see the effects of light gravitino on structure formation and CMB anisotropies. Moreover, we assume that neutrinos are massless in the most part of this paper. We will make some comments on the case where massive neutrinos are also included in Section \[sec:summary\].
As briefly discussed in the introduction, the effects of WDM on structure formation can be understood by considering following two main aspects: (i) the change of the energy contents of the universe, or the epoch of matter-radiation equality (unperturbed background evolutions), (ii) the erasure of perturbations on small scales via free-streaming (perturbation evolutions). The first point is due to the fact that WDM behaves as relativistic component at early times but non-relativistic one at late times. Thus it changes the time of matter-radiation equality depending on the mass. It alters the evolution of gravitational potential and drives the integrated Sachs-Wolfe (ISW) effect in the CMB temperature anisotropy. However, in the case of light gravitino in which we are interested, its abundance is so small $N_{3/2} = 0.059$ that the epoch of radiation-matter equality is scarcely affected, even when we compare the two opposite limits, $f_{3/2}=0$ ($m_{3/2}= 0$ eV) and $f_{3/2}=1$ ($m_{3/2}= 86$ eV), with fixed $\omega_{\rm dm}=0.1099$. Therefore it is almost impossible to constrain the gravitino mass from unlensed CMB anisotropies, even when the ideal observations (cosmic variance limeted survey) are available.
Possible constraints on the gravitino mass almost come from the second point. Light gravitino free-streams to erase cosmic density fluctuations while it is relativistic. On scales smaller than the free streaming length of gravitino, fluctuations of matter and hence gravitational potential are erased. To see the effect, we show matter power spectra $P(k)$ for several values of $m_{3/2}$ with $\omega_{\rm dm}$ being fixed in Fig. \[fig:mpk\]. We take $m_{3/2}=0$ eV (solid red), $m_{3/2}=1$ eV (dashed green), $m_{3/2}=10$ eV (dotted blue) and $m_{3/2}=86$ eV (dot-dashed magenta). As seen from the figure, as $m_{3/2}$ increases, the suppression of $P(k)$ becomes larger on small scales, while on large scales the amplitude of $P(k)$ is unaffected regardless of the value of $m_{3/2}$. With more careful observation we can notice that when $m_{3/2}$ is small, the suppression of the power is small, but the scale under which $P(k)$ is suppressed becomes large. On the other hand, when $m_{3/2}$ is large, the suppression is also large, however the free-streaming scale becomes small. These can be simply understood as follows. When the mass of gravitino is small, gravitino can erase cosmic structure up to large scales. However, the smallness of the mass in turn indicates that gravitino is minor component in the contents of energy density and gravitationally irrelevant. Thus density fluctuations are less suppressed. For the case of larger mass, the opposite argument holds.
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Now we move on to discuss how the suppression of matter fluctuations changes the lensed CMB anisotropy. CMB photons last-scattered at the decoupling epoch, while traveling to the present epoch, are deflected by the gravitational potential $\Phi(\mathbf{r}, \eta)$ generated by the matter fluctuations (For a recent review see e.g. [@Lewis:2006fu]). The lensing potential $\phi$ is given by $$\phi(\hat n) = -2\int^{\chi_*}_0 d\chi
\frac{\chi_*-\chi}{\chi_* \chi} \Phi(\chi \hat n, \eta(\chi)),$$ where $\chi$ is the comoving distance along the line of sight, $\chi_*$ is the comoving distance to the last scattering surface and $\eta(\chi)$ is the conformal time corresponds to the comoving distance of $\chi$. Actually, the lensing potential is not a direct observable in CMB observations, but we can reconstruct it with observed lensed CMB anisotropies. It contains much more information than the lensed power spectrum [@Smith:2006nk; @Smith:2008an], since the reconstruction is performed by making use of off-diagonal components in correlation function of lensed anisotropies [@Okamoto:2003zw]. In Fig. \[fig:phi\], we show the angular power spectra of the lensing potential and its correlation with the temperature anisotropy, $C^{\phi\phi}_\ell$ and $C^{T\phi}_\ell$, respectively. For reference, we also show expected data of future CMB experiments: Planck, PolarBeaR and CMBpol. We can see from the $C^{\phi\phi}_\ell$ in the Fig. \[fig:phi\] that the suppression of the power spectra depends on the mass of gravitino, which can be probed with future observations of CMB. When we carefully observe the power on small scales, the trend how the power is suppressed is similar to what we have seen in the matter power spectra. Therefore we can expect the mass of gravitino is constrained with reconstructed lensing potential, which would be obtained from future CMB surveys. Although cross correlation of the lensing potential with CMB temperature anisotropy $C^{T\phi}_\ell$ is affected by the mass of gravitino, the effect is much small compared with the expected errors for future CMB surveys, which can be seen from the right panel in Fig. \[fig:phi\]. Thus it is suggested that $C^{T\phi}_\ell$ has little advantage for constraining the mass of light gravitino.
Here it should be noted that some careful consideration must be given for the following fact: the heavier the light gravitino mass is, the smaller the free-streaming scale would be. Although, regarding the suppression of the power, the effects of the light gravitino is more significant for a larger mass, when a survey cannot observe up to high multipoles due to its limitation of the resolution, gravitino with lighter mass can be better probed. This is because gravitino with lighter mass can erase cosmic structure up to larger scales compared to the case with larger mass although the power suppression is milder. As a simple example, let us compare the two cases, $f_{3/2}=0$ and $f_{3/2}=1$ while keeping $\omega_{\rm dm}$ fixed. At small angular scales, the lensing potential $\phi$ for $f_{3/2}=1$ is more suppressed than that for $f_{3/2}=0$. However, for the case with $f_{3/2} = 1$ corresponding to $m_{3/2}\simeq 86$ eV, the free streaming length is small. Therefore the suppression occurs only at limited small angular scales. If the observed multipoles are limited to low $\ell$’s, where the suppression cannot be seen, gravitino with $f_{3/2}=1$ cannot leave any imprint on such a measurement, which means that we cannot differentiate models between $f_{3/2}=1$ and $f_{3/2}=0$. This makes the likelihood surface multi-modal and highly-degenerate. To break up these degeneracies, high-resolution measurement of lensing potential is needed, and currently available observations cannot suffice this requirement. In the next section, we discuss how future CMB surveys will constrain the light gravitino models.
Constraints on light gravitino mass {#sec:constraints}
===================================
Now in this section, we investigate the constraints on the light gravitino mass from future CMB surveys. As discussed in the previous section, since the current CMB surveys are not precise enough to measure the lensing potential, it is almost impossible to probe $m_{3/2}$. However, in future surveys of CMB, the measurement of lensing potential would be significantly improved. To forecast constraints on light gravitino mass from future CMB surveys, we make use of the following three surveys in this paper, the Planck [@:2006uk], PolarBeaR [@PolarBeaR] and CMBpol [@CMBpol]. The parameters for instrumental design for these surveys are summarized in Table \[table:surveys\], where $\theta_{FWHM}$ is the size of Gaussian beam[^3] at FWHM and $\sigma_T$ ($\sigma_P$) is the temperature (polarization) noise.
surveys $f_{\rm sky}$ bands \[GHz\] $\theta_{\rm FWHM}$ \[arcmin\] $\sigma_T$ \[$\mu$K\] $\sigma_P$ \[$\mu$K\]
------------------------ --------------- --------------- -------------------------------- ----------------------- -----------------------
Planck [@:2006uk] $0.65$ $100$ $9.5$ $6.8$ $10.9$
$143$ $7.1$ $6.0$ $11.4$
$217$ $5.0$ $13.1$ $26.7$
PolarBeaR [@PolarBeaR] $0.03$ $90$ $6.7$ $1.13$ $1.6$
$150$ $4.0$ $1.70$ $2.4$
$220$ $2.7$ $8.00$ $11.3$
CMBpol [@CMBpol] $0.65$ $100$ $4.2$ $0.84$ $1.18$
$150$ $2.8$ $1.26$ $1.76$
$220$ $1.9$ $1.84$ $2.60$
: Instrumental parameters for future CMB surveys used in our analysis. $\theta_{\rm FWHM}$ is Gaussian beam width at FWHM, $\sigma_T$ and $\sigma_P$ are temperature and polarization noise, respectively. For the Planck and PolarBeaR surveys, we assume 1-year duration of observation and for the CMBpol survey, we assumed 4-year duration. []{data-label="table:surveys"}
In this paper, to generate samples from the Bayesian posterior distributions of cosmological parameters, we make use of the public code [ MultiNest]{} [@Feroz:2007kg] integrated in the vastly used Monte Carlo sampling code [ COSMOMC]{} [@Lewis:2002ah]. While [COSMOMC]{} samples the posterior distributions via the Markov chain Monte Carlo (MCMC) sampling method, [MultiNest]{} is based on the different sampling called nested-sampling method [@Skilling:2004]. Use of [MultiNest]{} has several advantages in our analysis. One of the greatest advantages is that it enables efficient exploring of multi-modal/highly-degenerate likelihood surface, which is indeed the case for light gravitino models, as we have discussed in the previous section. Also it provides Bayesian evidence of a model and hence enables us to employ Bayesian model selection.
------------------- ----------------- --------------------------- -------------------
parameters fiducial values Planck/PolarBeaR/combined CMBpol
$\omega_b$ $0.02273$ $0.018 \to 0.28$ $0.021 \to 0.024$
$\omega_{\rm dm}$ $0.1099$ $0.08 \to 0.30$ $0.10 \to 0.14$
$\theta_s$ $1.0377$ $1.02 \to1.06$ $1.03 \to 1.04$
$\tau$ $0.087$ $0.01 \to 0.30$ $0.06 \to 0.14$
$N_{3/2}$ $0.059$ ($0 \to 5$) ($0 \to 2$)
$f_{3/2}$ $0.013$ $0 \to 1$ $0 \to 0.1$
$Y_p$ $0.248$ $(0.1 \to 0.5)$ ($0.2 \to 0.3$)
$n_s$ $0.963$ $0.8 \to 1.2$ $0.9 \to 1$
$\ln(10^{10}A_s)$ $3.063$ $2.8\to3.5$ $3.0 \to 3.2$
------------------- ----------------- --------------------------- -------------------
: The fiducial values and prior ranges for the parameters used in the analysis. Note that priors shown with parenthesis are imposed only when the corresponding parameters ($N_{3/2}$ and $Y_p$) are treated as free parameters and not imposed when they are fixed or derived from other parameters. For CMBpol, we take narrower range for the top priors since its accuracy is much higher than the former two surveys. Hence we do not need broad range for the priors.[]{data-label="table:priors"}
To obtain the limit for the mass of light gravitino, we can translate the constraint on the parameter $f_{3/2}= \omega_{3/2} / \omega_{\rm
dm}$ using Eq. . Since light gravitino has almost definite prediction of its abundance, we mainly report our results for the case with $N_{3/2}=0.059$ being fixed. However, in some scenario such as those with late-time entropy production, this number may be altered. In this respect, we also make analysis with $N_{3/2}$ being varied. Furthermore, regarding the treatment of the primordial abundance of $^4$He (denoted as $Y_p$), we assume two cases: treating $Y_p$ as a free parameter and fixing $Y_p$ with the derived value via the big bang nucleosynthesis (BBN) relation [@Kawanocode]. In the BBN theory, $Y_p$ is determined once baryon density $\omega_{\rm b}$ and the effective number of neutrino $N_{\rm eff}$ are given. Thus such a fixing of the value of $Y_p$ was adopted in some analysis [@Ichikawa:2006dt; @Hamann:2007sb; @Ichikawa:2007js]. Since, in the precise measurement of future CMB survey, the prior on $Y_p$ can also affect the determination of other cosmological parameters [@Hamann:2007sb; @Ichikawa:2007js], we consider the case with $Y_p$ freely varied as well.
Thus the full parameter space that we explore for light gravitino models are basically nine-dimensional: $$(\omega_b, \omega_{\rm dm}, \theta_s, \tau, N_{3/2}, f_{3/2}, Y_p, n_s, A_s),$$ where $\theta_s$ is the acoustic peak scale, $\tau$ is the optical depth of reionization and $A_s$ and $n_s$ are the amplitude and spectral index of initial power spectrum of scalar perturbations at a pivot scale $k_0 = 0.05$ Mpc$^{-1}$. In the following, we investigate four different cases: (I) fixing $N_{3/2}=0.059$ and deriving $Y_p$ via the BBN relation, (II) fixing $N_{3/2}=0.059$ and treating $Y_p$ as a free parameter, (III) treating $N_{3/2}$ as a free parameter and $Y_p$ as a derived parameter via the BBN relation, and (IV) treating $N_{3/2}$ and $Y_p$ as free parameters. The fiducial values and top-hat priors for parameter estimation are summarized in Table \[table:priors\].
The likelihood function is adopted from Ref. [@Perotto:2006rj]. We include $TT, TE, EE, \phi\phi, T\phi$ spectra for correlation of CMB anisotropy and lensing potential up to $\ell \le 2500$. We assume lensing reconstruction is performed by adopting the method based on quadratic estimator [@Okamoto:2003zw], and the expected noise in lensing potential is calculated by the publicly available [FuturCMB2]{} code developed by the authors of [@Perotto:2006rj]. Angular power spectra are calculated using the method in Ref. [@Challinor:2005jy]. For corrections for lensing potential due to nonlinear evolution of matter density perturbations, we adopt HALOFIT, which is based on the N-body simulations of CDM models [@Smith:2002dz]. Though the light gravitino model is not exactly the CDM models, we believe that the change of the nonlinear correction is negligible. This is because that gravitino has small $N_{3/2}$ and regardless of the mass of $m_{3/2}$, dark matter can be approximated by CDM very well when it begins to evolve in nonlinear regime. Furthermore, nonlinear correction changes the spectra of lensing potential by only a few percent at $\ell \le
2500$ [@Challinor:2005jy], and hence the treatment here can be justified. In addition, we also performed same analyses without including nonlinear corrections and checked that resultant constraints do not significantly change by the treatment of nonlinearity.
Now we are going to present our results. In Tables \[table:case1\]-\[table:case4\] we summarize the constraints on the cosmological parameters from Planck alone, PolarBeaR alone, Planck and PolarBeaR combined, and CMBpol alone, separately for different priors. First we discuss the case with fixed $N_{3/2}=0.059$ and the BBN relation adopted for $Y_p$. The 1d posterior distributions of cosmological parameters are shown in Fig. \[fig:1d\_fixed\]. From the posterior distributions for $f_{3/2}$ in Fig. \[fig:1d\_fixed\], we can easily see that light gravitino models are not constrained very well with Planck or PolarBeaR alone. The posterior distributions have decaying tails from the peak near $f_{3/2}=0$ to $f_{3/2}=1$. Actually, they have very smooth second peaks at around $f_{3/2}=1$. This multi-modal structure of posterior distributions comes from the degeneracies what we have discussed in Section \[sec:CMB\]. Light gravitino with a relatively large mass ($f_{3/2}\simeq 1$) can suppress the power via free-streaming only at very small scales where the Planck surveys cannot sufficiently measure the CMB. Thus a model with such a gravitino mass can fit the data from Planck alone. On the other hand, PolarBeaR has better resolution, gravitino with large mass is much constrained from observation of lensing potential at small scales. However, the sky coverage of the PolarBeaR survey is much smaller than Planck, thus the observation is worse at large angular scales. In this case, other parameters than $f_{3/2}$ are still not well-constrained from PolarBeaR alone. Therefore gravitino with relatively large mass can fit the data to some extent by tuning other cosmological parameters in this case too. To remove the degeneracy it is necessary to combine observations precise at large and small angular scales or, ultimately, use measurements precise both at large and small angular scales. With data from Planck and PolarBeaR combined, we can obtain a constraint $$f_{3/2}\le0.036 \ \mbox{(95\% C.L.)}.$$ When we use CMBpol, whose measurement is precise both on large and small scales than other two survey, this constraint can be improved as $$f_{3/2}=0.0121\pm0.0027 \ \mbox{(68\% C.L.)}.$$ These constraints are translated into the limits on the mass of light gravitino. For the case with Planck and PolarBeaR combined, the constraint is given as $$m_{3/2}\le3.2 \ \mbox{eV} \ \mbox{(95\% C.L.)}, \label{eq:combined}$$ and with CMBpol as $$m_{3/2}=1.04^{+0.22}_{-0.26} \ \mbox{eV} \ \mbox{(68\% C.L.)}.$$ Since gravitino mass should be larger than $1$eV not to destabilize the messenger scalar, CMBpol would be expected to give (counter-)evidense for existence of gravitino if its mass is (not) in the mass range considered here. In Section \[sec:selection\] we will discuss this point more quantitatively using Bayesian models selection analysis.
So far we kept assuming the energy density of gravitino fixed as $N_{3/2}=0.059$. If we loosen this assumption and take $N_{3/2}$ as a free parameter, the constraints are significantly weakened. Notice that the free-streaming scale of light gravitino is determined by its mass. Thus when $N_{3/2}$ is freely varied and hence the mass of gravitino can be large, the free-streaming scale can be shifted toward smaller scales over which PolarBeaR cannot observe for a wide range of the mass and one cannot see the damping of the power there. (On the other hand, since CMBpol is very precise on small scales, the constraint on $m_{3/2}$ from CMBpol is not affected much.) Thus we cannot obtain a meaningful constraint on $f_{3/2}$ even if we combine Planck and PolarBeaR when we take $N_{3/2}$ as a free parameter. Furthermore, the change in $N_{3/2}$ renders the shift of the radiation-matter equality which can be absorbed by tuning $\omega_{\rm dm}$ [@Hannestad:2003ye; @Crotty:2004gm; @Ichikawa:2008pz], significant degeneracies arise among $\omega_{\rm dm}$, $f_{3/2}$ and $N_{3/2}$ as shown in Fig. \[fig:2d\_deg\][^4]. For meaningful constraints we need sensitivities as good as those of CMBpol-like survey. In Fig. \[fig:1d\_free\] 1d posterior distributions for parameters including $N_{3/2}$ are shown. The 68 % limit of $f_{3/2}$ for this case is $f_{3/2}=0.0118^{+0.0032}_{-0.0031}$, which corresponds to the constraint on the gravitino mass $m_{3/2}=1.19^{+0.16}_{-0.50}$ eV.
parameters Planck PolarBeaR combined CMBpol
-------------------------- ------------------------------ ------------------------------ ------------------------------ ---------------------------------
$100~\omega_b$ $2.276^{+0.013}_{-0.012}$ $2.274^{+0.021}_{-0.022}$ $2.275^{+0.012}_{-0.009}$ $2.2739^{+0.0041}_{-0.0033}$
$\omega_{dm}$ $0.1101^{+0.0013}_{-0.0010}$ $0.1106^{+0.0022}_{-0.0022}$ $0.1100^{+0.0012}_{-0.0010}$ $0.10993^{+0.00058}_{-0.00059}$
$100~\theta_s$ $103.783^{+0.024}_{-0.026}$ $103.777^{+0.035}_{-0.031}$ $103.777^{+0.020}_{-0.019}$ $103.774^{+0.0056}_{-0.0061}$
$\tau$ $0.0871^{+0.0043}_{-0.0045}$ $0.090^{+0.009}_{-0.013}$ $0.0873^{+0.0039}_{-0.0043}$ $0.0872^{+0.0022}_{-0.0026}$
$N_{3/2}$ — — — —
$f_{3/2}$ — — $<0.036$ (95%) $0.0121^{+0.0027}_{-0.0027}$
$Y_p$ — — — —
$n_s$ $0.9622^{+0.0045}_{-0.0036}$ $0.9621^{+0.0081}_{-0.0083}$ $0.9629^{+0.0036}_{-0.0033}$ $0.9637^{+0.0017}_{-0.0017}$
$\ln(10^{10}\times A_s)$ $3.0640^{+0.0073}_{-0.0094}$ $3.077^{+0.015}_{-0.026}$ $3.0641^{+0.0063}_{-0.0094}$ $3.0637^{+0.0040}_{-0.0047}$
$m_{3/2}$ \[eV\] — — $<3.2$ (95%) $1.04^{+0.22}_{-0.26}$
: Constraints on cosmological parameters for the case with fixing $N_{3/2}=0.059$ and adopting the BBN relation to fix the value of $Y_p$ (CASE I). We basically present the mean values as well as $1\sigma$ errors. For parameters that are bounded only from one side we present 95% credible intervals.[]{data-label="table:case1"}
parameters Planck PolarBeaR combined CMBpol
-------------------------- ------------------------------ ------------------------------ ------------------------------ ---------------------------------
$100~\omega_b$ $2.278^{+0.021}_{-0.017}$ $2.272^{+0.035}_{-0.034}$ $2.274^{+0.016}_{-0.014}$ $2.2722^{+0.0055}_{-0.0047}$
$\omega_{dm}$ $0.1100^{+0.0015}_{-0.0010}$ $0.1107^{+0.0025}_{-0.0021}$ $0.1100^{+0.0011}_{-0.0011}$ $0.10990^{+0.00061}_{-0.00056}$
$100~\theta_s$ $103.788^{+0.048}_{-0.042}$ $103.775^{+0.067}_{-0.058}$ $103.775^{+0.032}_{-0.035}$ $103.771^{+0.011}_{-0.010}$
$\tau$ $0.0873^{+0.0041}_{-0.0050}$ $0.090^{+0.011}_{-0.013}$ $0.0873^{+0.0040}_{-0.0044}$ $0.0870^{+0.0020}_{-0.0028}$
$N_{3/2}$ — — — —
$f_{3/2}$ — — $<0.035$ (95%) $0.0118^{+0.0030}_{-0.0025}$
$Y_p$ $0.250^{+0.008}_{-0.011}$ $0.248^{+0.016}_{-0.016}$ $0.2486^{+0.0094}_{-0.0066}$ $0.2481^{+0.0031}_{-0.0028}$
$n_s$ $0.9627^{+0.0067}_{-0.0063}$ $0.961^{+0.012}_{-0.014}$ $0.9625^{+0.0063}_{-0.0052}$ $0.9629^{+0.0026}_{-0.0025}$
$\ln(10^{10}\times A_s)$ $3.064^{+0.008}_{-0.010}$ $3.071^{+0.018}_{-0.028}$ $3.064^{+0.006}_{-0.010}$ $3.0631^{+0.0043}_{-0.0045}$
$m_{3/2}$ \[eV\] — — $<3.1$ (95%) $1.02^{+0.26}_{-0.22}$
: Same tables as in Table \[table:case1\] but for the cases with fixing $N_{3/2}=0.059$ and treating $Y_p$ as a free parameter (CASE II).[]{data-label="table:case2"}
parameters Planck PolarBeaR combined CMBpol
-------------------------- ------------------------------ ------------------------------ ------------------------------ -------------------------------
$100~\omega_b$ $2.276^{+0.015}_{-0.015}$ $2.284^{+0.023}_{-0.030}$ $2.276^{+0.011}_{-0.014}$ $2.2735^{+0.0043}_{-0.0048}$
$\omega_{dm}$ $0.1099^{+0.0022}_{-0.0034}$ $0.1125^{+0.0036}_{-0.0060}$ $0.1097^{+0.0017}_{-0.0033}$ $0.1097^{+0.0011}_{-0.0011}$
$100~\theta_s$ $103.782^{+0.035}_{-0.029}$ $103.763^{+0.053}_{-0.038}$ $103.778^{+0.027}_{-0.027}$ $103.776^{+0.0092}_{-0.0082}$
$\tau$ $0.0873^{+0.0040}_{-0.0047}$ $0.091^{+0.008}_{-0.015}$ $0.0874^{+0.0036}_{-0.0046}$ $0.0871^{+0.0022}_{-0.0027}$
$N_{3/2}$ $<0.24$ (95%) $<0.47$ (95%) $<0.21$ (95%) $<0.10$ (95%)
$f_{3/2}$ — — — $0.0118^{+0.0032}_{-0.0031}$
$Y_p$ — — — —
$n_s$ $0.9638^{+0.0034}_{-0.0065}$ $0.968^{+0.009}_{-0.012}$ $0.9639^{+0.0038}_{-0.0054}$ $0.9635^{+0.0020}_{-0.0025}$
$\ln(10^{10}\times A_s)$ $3.064^{+0.009}_{-0.010}$ $3.076^{+0.021}_{-0.027}$ $3.064^{+0.008}_{-0.010}$ $3.0632^{+0.0051}_{-0.0049}$
$m_{3/2}$ \[eV\] — — — $1.19^{+0.16}_{-0.50}$
: Same tables as in Table \[table:case1\] but for the cases with treating $N_{3/2}$ as a free parameter and adopting the BBN relation (CASE III).[]{data-label="table:case3"}
parameters Planck PolarBeaR combined CMBpol
-------------------------- ------------------------------ ------------------------------ ------------------------------ ------------------------------
$100~\omega_b$ $2.278^{+0.017}_{-0.021}$ $2.276^{+0.030}_{-0.035}$ $2.273^{+0.012}_{-0.013}$ $2.2727^{+0.0046}_{-0.0056}$
$\omega_{dm}$ $0.1117^{+0.0023}_{-0.0056}$ $0.1176^{+0.0062}_{-0.0092}$ $0.1103^{+0.0017}_{-0.0041}$ $0.1103^{+0.0013}_{-0.0016}$
$100~\theta_s$ $103.75^{+0.10}_{-0.06}$ $103.65^{+0.14}_{-0.11}$ $103.760^{+0.068}_{-0.045}$ $103.764^{+0.027}_{-0.019}$
$\tau$ $0.0875^{+0.0042}_{-0.0049}$ $0.091^{+0.008}_{-0.015}$ $0.0872^{+0.0036}_{-0.0031}$ $0.0872^{+0.0021}_{-0.0025}$
$N_{3/2}$ $<0.47$ (95%) $<1.3$ (95%) $<0.34$ (95%) $<0.17$ (95%)
$f_{3/2}$ — — — $0.0124^{+0.0032}_{-0.0034}$
$Y_p$ $0.2458^{+0.015}_{-0.011}$ $0.230^{+0.029}_{-0.018}$ $0.246^{+0.010}_{-0.007}$ $0.2471^{+0.0043}_{-0.0043}$
$n_s$ $0.9643^{+0.0068}_{-0.0068}$ $0.965^{+0.014}_{-0.013}$ $0.9636^{+0.0047}_{-0.0045}$ $0.9632^{+0.0024}_{-0.0027}$
$\ln(10^{10}\times A_s)$ $3.067^{+0.009}_{-0.011}$ $3.082^{+0.019}_{-0.032}$ $3.0636^{+0.0079}_{-0.0077}$ $3.0641^{+0.0046}_{-0.0053}$
$m_{3/2}$ \[eV\] — — — $1.10^{+0.07}_{-0.61}$
: Same tables as in Table \[table:case1\] but for the cases with treating both $N_{3/2}$ and $Y_p$ as free parameters (CASE IV).[]{data-label="table:case4"}
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Model selection analysis on light gravitino model {#sec:selection}
=================================================
In the previous section, we have seen that future CMB surveys give rather tight constraints on mass of light gravitino, so that we can expect they would give (counter-)evidence for existence of gravitino to more or less extent. But then a question arises how strong the evidence for gravitino is. This is a kind of model selection problem in statistics theory, which has been often argued in cosmology [@Slosar:2002dc; @Beltran:2005xd; @Trotta:2005ar; @Mukherjee:2005wg; @Bridges:2005br; @Kunz:2006mc; @Magueijo:2006we; @Liddle:2006tc; @Pahud:2007gi; @Heavens:2007ka; @Bevis:2007gh; @Trotta:2008qt]. In Bayesian statistics, the natural measure for evidence of a model is [*Bayesian evidence*]{} $E$, $$E(M)=\int d\theta P(data|\theta) \pi(\theta|M),$$ where $\theta$ represents a set of parameters included in a model $M$. $P(data|\theta)$ and $\pi(\theta|M)$ are the likelihood and prior probability functions, respectively. Bayesian evidence can be efficiently calculated by nested sampling method [@Skilling:2004]. Predictiveness of a model $M_1$ against another $M_2$ can be assessed by differencing the logarithm of Bayes factors of the models, that is $$B_{12}=\ln(E(M_1)/E(M_2)),$$ which is called Bayes factor. If $B_{12}$ is positively (negatively) large, we can say the observed data can be explained well by model $M_1$ ($M_2$) compared with $M_2$ ($M_1$). As a rule of thumb the Jeffreys’ scale is often used to translate a Bayes factor into literal expression for strength of an evidence: $B_{12}<1$ is not significant, $1<B_{12}<2.5$ significant, $2.5<B_{12}<5$ strong and $5<B_{12}$ is decisive. For more details we refer to a recent review [@Trotta:2008qt] and references therein.
Now we are going to see how large the evidence is from future CMB surveys. In Table \[table:evidence\] we summarized values of obtained Bayes factor for light gravitino model with different sets of data and priors against the conventional CDM model ($f_{3/2}=N_{3/2}=0$). Here we have assumed a same fiducial model ($m_{3/2}=1$eV and $N_{3/2}=0.059$), as in the previous section.
datasets CASE I CASE II CASE III CASE IV
----------------- ---------------- ---------------- ---------------- ----------------
Planck alone $-3.36\pm0.17$ $-2.81\pm0.17$ $-5.76\pm0.17$ $-4.70\pm0.18$
PolarBeaR alone $-3.39\pm0.15$ $-2.66\pm0.18$ $-5.02\pm0.16$ $-3.62\pm0.18$
combined $-3.03\pm0.17$ $-2.86\pm0.18$ $-5.80\pm0.17$ $-5.82\pm0.18$
CMBpol alone $3.40\pm0.16$ $3.63\pm0.17$ $1.08\pm0.17$ $1.44\pm0.19$
: Bayes factors for the light gravitino model against the CDM model. Shown are the mean and standard errors from two independent samplings.[]{data-label="table:evidence"}
First of all, from Table \[table:evidence\] we can see that Planck or PolarBeaR alone and even Planck and PolarBeaR combined give only negative Bayes factor for the gravitino model against the CDM model, regardless of priors on $Y_p$ and $N_{3/2}$. This is because for most of values of added parameters $f_{3/2}$ (and $N_{3/2}$) from the CDM model, gravitino model can only marginally improve fit to the data, even though it indeed improves the fit at some values around the fiducial ones, $f_{3/2}\simeq0.013$ (and $N_{3/2}\simeq0.059$). In other words, the complexity of the gravitino model has little advantage in explaining the data.
The situation dramatically changes for the case of the CMBpol survey. From Table \[table:evidence\] we can see from CMBpol data we obtain Bayes factor for gravitino model against the CDM model as $\ln B=3.40\pm0.16$ for a case with fixing $N_{3/2}=0.059$ and using the BBN relation. This is interpreted as strong evidence in the Jeffreys’ scale, though there’s always some disagreement in that how much Bayes factor can be regarded as giving enough evidence. For the cases with treating $N_{3/2}$ as a free parameter and using the BBN relation, we obtain $\ln B\gtrsim1.08\pm 0.17$. This can be regarded as giving only marginal evidence. So it is difficult to obtain enough evidence for general WDM model whose number density is not theoretically limited in some small range. Fortunately, since gravitino has small model-dependence of $N_{3/2}$ we can take the former value of $\ln B$.
So far we have discussed the case of fiducial gravitino mass $m_{3/2}$. For larger gravitino mass, as long as it is less than the current bound ($m_{3/2}\lesssim 16$ eV [@Viel:2005qj; @Boyarsky:2008xj]), the evidence surely improves. This is because for this range of mass, the power spectra of lensing potential differ more and more from those for the CDM model as the gravitino mass increases. Since, in Section \[sec:model\] we have seen that gravitino mass is expected to be $O(1)$ eV or larger theoretically, the fiducial model of $m_{3/2}=1$ eV, which we used throughout this paper, can be supposed as a rather pessimistic case. Since we have seen that even evidence for gravitino with mass 1 eV can be probed by a CMBpol-like survey, we would expect such a survey can probe most part of theoretically-motivated range of light gravitino mass. We hope such a survey would be realized and probe light gravitino model in the near future.
Summary and discussion {#sec:summary}
======================
We investigated a possible constraint on the light gravitino mass with $m_{3/2} < 100$ eV in the light of future precise measurements of CMB. Although the effects of free-streaming of light gravitino barely leave an imprint on CMB photons at the time of last scatter, they can be deflected by the gravitational potential altered after recombination due to the free-streaming effect, which can be probed with lensed CMB. Thus we in this paper discussed the effects of the light gravitino, paying particular attention to the lensing potential, then investigated the future constraint on its mass. For this purpose, we adopt the future CMB surveys such as Planck, PolarBeaR and CMBpol and study the issue by generating posterior distributions with nested-sampling method. For a simple (but physically motivated) case, we obtained the limit on the light gravitino mass, assuming $m_{3/2}=1$ eV as a fiducial value, as $m_{3/2}\le3.2 \ \mbox{eV} \
\mbox{(95\% C.L.)}$ for the case with Planck+PolarBeaR combined and $m_{3/2}=1.04^{+0.22}_{-0.26} \ \mbox{eV} \ \mbox{(68\% C.L.)}$ for CMBpol. Thus at the time of CMBpol experiment, we can expect that the (counter-)evidence of the light gravitino can be found in cosmological observations.
In a simple case, the effective degrees of freedom at the time of the gravitino decoupling is assumed to have definite value and fixed in the analysis. However in some scenarios, this assumption may not hold, thus we have also made analysis by treating $N_{3/2}$ as a free parameter. In addition, in the future precise CMB measurements, the primordial abundance of $^4$He, which is usually not assumed as a free parameter but fixed, can also affect the determination of cosmological parameters, therefore we also performed the analysis by varying $Y_p$ too. When both or one of these parameters are varied, the constraint on the mass becomes weak. The results are summarized in Tables \[table:case1\]-\[table:case4\]. In addition, we also discussed how strong future CMB surveys can find an evidence for light gravitino by employing Bayesian model selection analysis. Even if the mass of gravitino is around 1 eV, a future CMBpol-like surveys is capable of providing some rather strong evidence for light gravitino model, which can be even stronger for larger $m_{3/2}$.
In principle the light gravitino mass of $\mathcal O(1)$ eV can be probed with the LHC experiment with the method proposed in Ref. [@Hamaguchi:2007ge]. But this requires some amount of tuning for the sparticle mass spectrum together with the gravitino mass. Thus even if the LHC will fail to determine the light gravitino mass, the future CMB experiments can do this job.
Finally we make some comment on the case where massive neutrinos are also included in the analysis. From atmospheric, solar, reactor and accelerator neutrino experiments, now we know that neutrinos have finite masses. Furthermore it has been discussed that neutrino masses can be well probed with future CMB survey [@Kaplinghat:2003bh; @Lesgourgues:2005yv; @Perotto:2006rj], which motivates us to conduct the analysis assuming that neutrinos are massive. Thus we also investigated the constraint on the light gravitino mass while the mass of neutrino is also varied. Since the effects of massive neutrino and light gravitino on the lensing potential are essentially the same, we found that strong degeneracies arise in particular, between their masses and we could not obtain any meaningful constraints on those. However, neutrino masses can also be constrained in future laboratory experiments such as tritium beta-decay and neutrinoless double beta decay. Thus in the future, we will also have some inputs from such neutrino experiments, which can remove the degeneracy in CMB survey. In light of these considerations, we can expect that cosmology and particle physics experiments will push us toward more severe constraint/precise determination of the light gravitino mass in the near future.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank Kiyotomo Ichiki and Shun Saito for useful discussions and providing data to check our numerical calculations. The authors would also like to thank Oleg Ruchayskiy for useful comments. This work is supported by Grant-in-Aid for Scientific research from the Ministry of Education, Science, Sports, and Culture, Japan, No. 14102004 (M.K.) and No. 19740145 (T.T.), and also by World Premier International Research Center Initiative, MEXT, Japan. K.N. and T.S. would like to thank the Japan Society for the Promotion of Science for financial support.
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[^1]: Recently the authors of [@Boyarsky:2008xj] also conducted a similar analysis using Lyman-$\alpha$ data combined with updated WMAP5 [@Komatsu:2008hk]. Although they did not report the constraint on the light gravitino mass, the SDSS Lyman-$\alpha$ dataset would be more effective in constraining $m_{3/2}$ than the dataset adopted in [@Viel:2005qj]. Thus an analysis optimized for the light gravitino model could give a severer constraint on $m_{3/2}$.
[^2]: A similar argument has been made for massive neutrinos on the use of CMB lensing to constrain its mass Ref. [@Kaplinghat:2003bh; @Lesgourgues:2005yv; @Perotto:2006rj].
[^3]: We assume Gaussian beam and neglect any anisotropies in beam and distortion arising from the scan strategy.
[^4]: In Figure 7 and 12 in [@Boyarsky:2008xj], a similar degeneracy can also be seen as band-like allowed region along $F_\mathrm{WDM}$ axis.
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abstract: |
In this work, a genuine mechanism of entanglement protection of a two-qubit system interacting with a dissipative common reservoir is investigated. Based on the generating of bound state for the system-reservoir, we show that stronger bound state in the energy spectrum can be created by adding other non-interacting qubits into the reservoir. In the next step, it is found that obtaining higher degrees of boundedness in the energy spectrum leads to better protection of two-qubit entanglement against the dissipative noises. Also, it is figured out that the formation of bound state not only exclusively determines the long time entanglement protection, irrespective to the Markovian and non-Markovian dynamics, but also performs the task for reservoirs with different spectral densities.\
\
[**PACS Nos:**]{}\
[**Keywords:**]{} Entanglement protection, Bound state, Additional qubits, Spectral density, Concurrence
author:
- |
N. Behzadi $^{a}$ [^1] , B. Ahansaz $^{b}$, E. Faizi $^{b}$ and H. Kasani $^{c}$\
$^a$[Research Institute for Fundamental Sciences, University of Tabriz, Tabriz, Iran,]{}\
$^b$[Physics Department, Azarbaijan Shahid Madani University, Tabriz, Iran,]{}\
$^c$[Physics Department, University of Mohaghegh Ardabili, Ardabil, Iran.]{}
title: '**Requirement of system-reservoir bound states for entanglement protection**'
---
Introduction
============
Quantum entanglement, one of the wonderful aspects of quantum mechanics which has no classical counterpart, has been considered as a main resource in understanding and development of quantum information processing protocols [@Nielsen]. However, since quantum entanglement is so fragile due to decoherence effect, it inevitably undergoes either asymptotic decay or sudden death processes [@Hornberger; @Merali]. So in this regard, unavoidable interaction between any real quantum system and its surrounding environment alters the quantum system and consequently disentanglement occurs. This is apparently a disadvantageous procedure relative to the applications of quantum information processing tasks. Therefore, it is very important to find out a mechanism whose advantage leads to an effective long-time entanglement preservation. So far, a lot of researches have been devoted to entanglement manipulation and protection, such as quantum Zeno effect (QZE) [@Maniscalco; @Mundarain; @Rossi; @Hou] and detuning modulation [@Ba; @Xiao].
It was found that the coupling between an excited atom and electromagnetic vacuum reservoir leads atom-photon bound state [@John2] which in turns exhibits a fractional steady-state atomic population on the excited state known as populating trapping [@Lambropoulos]. On the other hand, entanglement trapping due to structured environment [@Bellomo] is a direct consequence of populating trapping. Based on these finding, it was proposed a method for entanglement protection of two qubits in two uncorrelated reservoirs using the non-Markovian effect along with creating bound state for the system-environment through the manipulating of the spectral densities [@Tong].
In this paper, we give an actual mechanism for protection of two-qubit entanglement exclusively on the basis of formation of bound state for the two-qubit system and its common reservoir using additional qubits. In this approach without manipulating the spectral density of the environment, system-reservoir bound state can be generated by adding other non-interacting qubits into the reservoir. It is interesting to note that creating bound state in the energy spectrum of the total system with higher degree of boundedness is provided by inserting more additional qubits. As will be figured out, higher degree of boundedness gives the better protection of entanglement in long time limit irrespective to Markovian or non-Markovian dynamics. We also examine the performance of the approach for several different spectral densities and observe that the protection process only depends on the existence of bound state and completely independent from structure of environment.
This paper is organized as follows. In Sec. II, the model of $N$ non-interacting qubits in a common zero-temperature reservoir is introduced. We derive the condition for the formation of bound state in the spectrum of total system. In Sec. III, the entanglement dynamics of two-qubit system is studied by considering different models for the reservoir structure. Finally, a brief conclusions are given at the end of the paper.
The model
=========
In this section, we consider a system of $N$ non-interacting qubits (two-level atoms) immersed in a common zero-temperature thermal reservoir. The Hamiltonian $\hat{H}$ of the system can be written as ($\hbar=1$) $$\begin{aligned}
\hat{H}=\hat{H}_{0}+\hat{H}_{I},\end{aligned}$$ where $\hat{H}_{0}$ is the free Hamiltonian and $\hat{H}_{I}$ describes the interaction terms, $$\begin{aligned}
\hat{H}_{0}=\omega_{0} \sum_{l=1}^{N} \hat{\sigma}^{+}_{l} \hat{\sigma}^{-}_{l}+\sum_{k} \omega_{k} \hat{b_{k}}^{\dagger} \hat{b_{k}},\end{aligned}$$
$$\begin{aligned}
\hat{H}_{I}=\sum_{l=1}^{N} \sum_{k} g_{k} \hat{b_{k}} \hat{\sigma}_{l}^{+}+g_{k}^{*} \hat{b_{k}^{\dagger}} \hat{\sigma}_{l}^{-}.\end{aligned}$$
In Eqs. (2) and (3), $\hat{\sigma}_{l}^{+}$ $(\hat{\sigma}_{l}^{-})$ is the raising (lowering) operator of the $l^{th}$ qubit with transition frequency $\omega_{0}$ and $\hat{b_{k}}$ ($\hat{b_{k}}^{\dagger}$) is the annihilation (creation) operator of the $k^{th}$ field mode with frequency $\omega_{k}$. Also, the strength of coupling between the $l^{th}$ qubit and the $k^{th}$ field mode is represented by $g_{k}$. The spectrum of the Hamiltonian can be obtained by solving the following eigenvalue equation $$\begin{aligned}
\hat{H}(t) |\psi(t)\rangle=E |\psi(t)\rangle.\end{aligned}$$ Since the total Hamiltonian commutes with the number of excitations (i. e. $[(\sum_{l=1}^{N} {\hat{\sigma}_{l}^{+} \hat{\sigma}_{l}^{-}}+\sum_{k} \hat{b_{k}}^{\dagger} \hat{b_{k}}),H]=0$), therefore by considering the single excitation subspace, we have $$\begin{aligned}
|\psi(t)\rangle=\sum_{l=1}^{N}C_{l}(t)|l\rangle_{s} |0\rangle_{e} +\sum_{k}C_{k}(t)|0\rangle_{s} |1_{k}\rangle_{e}.\end{aligned}$$ where $|l\rangle_{s}=|g\rangle^{\bigotimes N}_{l^{th}\equiv e}$, which means that all of the qubits are in their respective ground states $|g\rangle$ except the $l^{th}$ qubit which is in the excited state $|e\rangle$, and $|0\rangle_{s}=|g\rangle^{\bigotimes N}=|g,g,...,g\rangle$. Also, we denote $|0\rangle_{e}$ being the vacuum state of the reservoir and $|1_{k}\rangle_{e}$ is the state for which there is only one excitation in the $k$th field mode. Substituting Eqs. (1) and (5) into Eq. (4), yields the following set of $N+1$ equations $$\begin{aligned}
\begin{array}{c}
\omega_{k} C_{k}(t)+\sum_{l=1}^{N} g_{k}^{*} C_{l}(t)=E C_{k}(t),\\\\
\omega_{0} C_{1}(t)+\sum_{k} g_{k} C_{k}(t)=E C_{1}(t),\\\\
\omega_{0} C_{2}(t)+\sum_{k} g_{k} C_{k}(t)=E C_{2}(t),\\
.\\
.\\
.\\
\omega_{0} C_{N}(t)+\sum_{k} g_{k} C_{k}(t)=E C_{N}(t).
\end{array}\end{aligned}$$ Obtaining $C_{k}(t)$ from the first equation and substituting it in the rest ones gives the following $N$ integro-differential equations $$\begin{aligned}
\begin{array}{c}
(E-\omega_{0}) C_{1}(t)=-\int_{0}^{\infty} \frac{J(\omega) d\omega}{\omega-E} \sum_{l=1}^{N} C_{l}(t),\\\\
(E-\omega_{0}) C_{2}(t)=-\int_{0}^{\infty} \frac{J(\omega) d\omega}{\omega-E} \sum_{l=1}^{N} C_{l}(t),\\
.\\
.\\
.\\
(E-\omega_{0}) C_{N}(t)=-\int_{0}^{\infty} \frac{J(\omega) d\omega}{\omega-E} \sum_{l=1}^{N} C_{l}(t).
\end{array}\end{aligned}$$ Eliminating the coefficients $C_{l}(t)$s reads $$\begin{aligned}
E=\omega_{0}-N \int_{0}^{\infty} \frac{J(\omega) d\omega}{\omega-E} \equiv y(E).\end{aligned}$$ Solving Eq. (8) gives the energy spectrum of the total system, i.e. $N$-qubit system and reservoir, which depends effectively on the spectral density of the reservoir. Note that there is a solution for Eq. (8) when the functions $y(E)$ and $y(E)=E$ are crossed with each other. Since $y(E)$ decreases monotonically with the increase of $E$ in the region $E < 0$ and $\mathrm{lim}_{E \rightarrow -\infty} y(E)=\omega_{0}$, so Eq. (8) has an isolated root in this region. The eigenstate corresponding to this isolated eigenvalue is named the bound state which exists between the $N$-qubit system and its common reservoir. It can be easily checked out that there is bound state solution for Eq. (8) when it satisfies the condition $y(0) < 0$, otherwise, there is no bound state. On the other hand, it has only complex solutions when a bound state is not formed. This means that the corresponding eigenstate experiences decay from the imaginary part of the eigenvalue during the time evolution and therefore the excited-state population approaches to zero asymptotically. However, for a bound state the population of the atomic excited state is constant in time because it has a vanishing decay rate during the time evolution. In the following, we consider some typical spectral functions such as Lorentzian, sub-Ohmic, Ohmic and super-Ohmic for the structure of the reservoir, and show how the addition of non-interacting qubits generates bound state and affects its quality.
Entanglement dynamics
=====================
To quantify the degree of entanglement between of a two-qubit system, we use concurrence as a measure of entanglement in this way [@Wootterrs]. The concurrence is defined as $$\begin{aligned}
C(\rho)=\mathrm{max} \{0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}\},\end{aligned}$$ in which $\lambda_{i}$’s are the eigenvalues, in decreasing order, of the Hermitian matrix $R=\sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}$ with $\tilde{\rho}=(\sigma_{y}^{A}\otimes \sigma_{y}^{B}) \rho^{*}(\sigma_{y}^{A}\otimes \sigma_{y}^{B})$. Here $\rho^{*}$ means the complex conjugation of $\rho$ and $\sigma_{y}$ is the Pauli matrix. Let us consider $m^{th}$ and $n^{th}$ qubits of the system with $m\neq n$, prepared initially in an EPR-type entangled state as follows $$\begin{aligned}
|\phi(0)\rangle_{m,n}=C_{m}(0)|e,g\rangle+C_{n}(0)|g,e\rangle.\end{aligned}$$ Therefore, at time $t>0$, the concurrence of the two-qubit system prepared initially in (10) is as $$\begin{aligned}
C(\rho_{m,n}(t))=2|C_{m}(t) C_{n}(t)|.\end{aligned}$$ To finding the probability amplitudes $C_{m}(t)$ and $C_{n}(t)$, we use the Schrodinger equation in the interaction picture $$\begin{aligned}
i \frac{d}{dt}|\psi(t)\rangle=\hat{H}_{I}(t) |\psi(t)\rangle,\end{aligned}$$ where the Hamiltonian in this picture is given by $\hat{H}_{I}(t)=e^{i \hat{H}_{0}t} \hat{H}_{I} e^{-i \hat{H}_{0}t}$ and substituting Eq. (5) and $\hat{H}_{I}(t)$ into Eq. (12) gives a system of $N+1$ differential equations as $$\begin{aligned}
\dot{C}_{l}(t)=-i \sum_{k} g_{k} C_{k}(t) e^{i(\omega_{0}-\omega_{k})t},\\
\dot{C}_{k}(t)=-i \sum_{l=1}^{N} g_{k}^{*} C_{l}(t) e^{-i(\omega_{0}-\omega_{k})t},\end{aligned}$$ where $l=1,2,...,N$. Integrating Eq. (14) and substituting it into the Eq. (13) gives the following set of $N$ closed integro-differential equation $$\begin{aligned}
\frac{dC_{l}(t)}{dt}=-\int_{0}^{t} f(t-t') \sum_{u=1}^{N} C_{u}(t) dt',\end{aligned}$$ where the correlation function $f(t-t')$ is related to the spectral density $J(\omega)$ of the reservoir by $$\begin{aligned}
f(t-t')=\int d\omega J(\omega) e^{i(\omega_{0}-\omega)(t-t')}.\end{aligned}$$ The solutions of Eq. (15), i.e. $C_{l}(t)$’s, depend on the particular choice of the spectral density of the reservoir, which will be considered in the following subsections.
Lorentzian spectral density
---------------------------
We first consider the Lorentzian spectral density as $$\begin{aligned}
J(\omega)=\frac{1}{2\pi} \frac{\gamma_{0} \lambda^{2}}{(\omega-\omega_{0})^2+\lambda^{2}},\end{aligned}$$ where $\omega_{0}$ is the central frequency of the reservoir equal to the transition frequency of the qubits. The parameter $\lambda$ defines the spectral width of the coupling and $\gamma_{0}$ is the coupling strength. By substituting $J(\omega)$ into Eq. (16), the correlation function $f(t-t')$ can be determined analytically. Using Laplace transform, the exact solutions of Eq. (15) can be obtained as $$\begin{aligned}
\begin{array}{c}
C_{l}(t)=e^{-\lambda t/2}\Big(\mathrm{cosh}{(\frac{Dt}{2})}+\frac{\lambda}{D} \mathrm{sinh}{(\frac{Dt}{2}})\Big) C_{l}(0)+\\\\
(\frac{(N-1) C_{l}(0)-\sum_{l'\neq l}^{N} C_{l'}(0)}{N})\times \\\\
\Big(1-e^{-\lambda t/2}\big(\mathrm{cosh}{(\frac{Dt}{2})}+\frac{\lambda}{D} \mathrm{sinh}{(\frac{Dt}{2})}\big)\Big),
\end{array}\end{aligned}$$ where $D=\sqrt{\lambda^{2}-2\gamma_{0} \lambda N}$, and by considering the point that the dynamics is Markovian in the weak coupling regime $(\gamma_{0}<\frac{\lambda}{2N})$ and non-Markovian in the strong coupling regime $(\gamma_{0}>\frac{\lambda}{2N})$.
For the Lorentzian spectral density which is physically corresponding to a single mode leaky cavity, solutions of Eq. (8) in the bound state region ($E<0$) are given in Fig. $1a$ and Fig. $1c$, corresponding to the Markovian and non-Markovian dynamics respectively. For $N=2$, corresponding to the absence of additional qubits, there is no bound state solution for Eq. (8). Absence of the bound state leads complete decay of two-qubit entanglement for both Markovian and Non-Markovian dynamics as shown in Fig. $1b$ and Fig. $1d$, respectively. For $N=8$, corresponding to the presence of six additional qubits, we have formation of the bound state for Markovian and non-Markovian regimes. In this situation, the two-qubit entanglement is suppressed from death and there is a non-zero steady value for the concurrence. As the number of additional qubits grows (for example, $N=12$) we have a bound state whose boundedness becomes stronger which consequently, irrespective to Markovian and Non-Markovian dynamics, gives a better protection of entanglement.
It is concluded that the generation of bound state for the system-environment through the addition of non-interacting qubits actually leads to the protection of two-qubit entanglement against the dissipative noises irrespective to the point that whether its dynamics is Markovian or non-Markovian.
Sub-Ohmic, Ohmic and super-Ohmic spectral density
-------------------------------------------------
To making an extension for confirming the performance of the proposed mechanism, we consider other spectral densities whose unified form is $$\begin{aligned}
J(\omega)=\frac{\gamma}{2\pi} \omega_{c}^{1-s} \omega^s e^{-\frac{\omega}{\omega_{c}}}.\end{aligned}$$ where the parameters $s=1/2$, $1$, $2$, are corresponding to sub-Ohmic, Ohmic and super-Ohmic spectral densities respectively, each of which in turns corresponds to a different physical context. It is known that the sub-Ohmic case corresponds to the type of noise appearing in solid state devices, the Ohmic case corresponds to the charged interstitials and the super-Ohmic case corresponds to a phonon bath [@Paavola]. The cut-off frequency is represented by $\omega_{c}$ and $\gamma$ is a dimensionless coupling constant. Unfortunately, there is no analytical solution for the closed integro-differential equation, i.e. Eq. (15), and so the only recourse to obtaining the concurrence evolution is numerical methods. Also according to Eq. (8), we obtain numerically the condition of existence of bound state for the sub-Ohmic, Ohmic and super-Ohmic cases to investigate how the availability of the bound state is improved by addition of other non-interacting qubits into the reservoir. As for the Lorentzian spectral density, for each of the mentioned situations, formation of bound state has a determinative role in protection of entanglement against the dissipation. Fig. 2$a$, Fig. 2$c$ and Fig. 2$e$ indicate the fruitful effects of additional qubits in creating system-environment bound state for sub-Ohmic, Ohmic and super-Ohmic spectral densities respectively. Also, it is observed that in the absence of additional qubits ($N=2$) there is no bound state in the system environment spectrum corresponding to the sub-Ohmic, Ohmic and super-Ohmic spectral densities. Therefore, under this condition, protection of entanglement is completely failed (see Fig. 2$b$, Fig. 2$d$ and Fig. 2$f$).
On the other hand, in the attendance of additional qubits into the reservoir ($N=8$), formation of bound state relative to the respective spectral densities is taken place which in turns provides the protection of entanglement from sudden death. As observed for the Lorentzian spectral density in the previous subsection, entering more additional qubits into the reservoir creates bound state with higher degree of boundedness which, in this regard, gives better protection of entanglement again.
Consequently, irrespective to the structure of environment, existence of system environment bound state completely determines the protection process of entanglement in long time limit from dissipation.
At the end, to give a reasonable physical explanation for our results in this paper, we remember that the state of whole system in Eq. (5) can be expanded in terms of eigenstates of the system-reservoir Hamiltonian as $$\begin{aligned}
|\psi(t)\rangle=D_{\mathrm{BS}}e^{-iE_{\mathrm{BS}}t}|\varphi_{\mathrm{BS}}\rangle+\sum_{j\in \mathrm{CB}}D_{\mathrm{CB}}^{j} e^{-iE_{\mathrm{CB}}^{j}t}|\varphi_{\mathrm{CB}}^{j}\rangle.\end{aligned}$$ where $|\varphi_{\mathrm{BS}}\rangle$ is the potentially formed system-reservoir bound state as an isolated eigenstate with eigenenergy $E_{\mathrm{BS}}$, $|\varphi_{\mathrm{CB}}\rangle$’s are eigenstates corresponding to the (quasi)continuous energy band, $D_{\mathrm{BS}}=\langle \varphi_{\mathrm{BS}}|\psi(0)\rangle$ and $D_{\mathrm{CB}}^{j}=\langle \varphi_{\mathrm{CB}}^{j}|\psi(0)\rangle$. All of the population terms in the summation of Eq. (20) tend to vanish due to the out-of-phase interference contributed by the (quasi)continuous spectrum. Therefore, only the first term in Eq. (20) survives in the long-time limit and plays an important role in controlling the evolution of the whole system. In the absence of additional qubits ($N=2$), the bound state $|\varphi_{\mathrm{BS}}\rangle$ is not formed (see Figs. 1(a,c) and 2(a,c,e)), so according to Eq. (20), no entanglement can be observed between the considered two qubits ($m^{th}$ and $n^{th}$ qubits) in the long-time limit. But, inserting the additional qubits into the reservoir ($N=8,12$), leads to formation of the respective bound states which can be entangled with respect to the mentioned two qubits. As a result, the time evolution of the two-qubit entanglement ultimately reaches to a non-zero steady value.
Conclusions
===========
We investigated a mechanism for two-qubit entanglement protection from dissipation caused by a common reservoir on the basis of creating bound states for the system and environment using non-interacting additional qubits. As observed for the reservoir with Lorentzian spectral density, irrespective to the dynamics of the two-qubit system from Markovian and non-Markovian point of views, stronger bound state leads to better protection of entanglement. In the next step, we examined the procedure for other reservoirs with different structures such as sub-Ohmic, Ohmic and supe-Ohmic. It was concluded that the formation of bound state has exclusively a determinative role in the long time entanglement protection irrespective to the structure of the reservoir. Finally, inspired by this mechanism, other methods for manipulating of bound states for protection of entanglement could be introduced in future. For example, entanglement protection of a two-qudit system by using this approach can be regarded as the subject of our future research. As a special case, for preservation of entanglement between two qutrits, it can be investigated that whether the addition of qubits leads to the desired result or the qutrits.
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Fig. 1. (a, c) Diagrammatic solutions of Eq. (8), and (b, d) concurrence dynamics of a two-qubit system with $N = 2$ (solid line), $N = 8$ (dashed line) and $N = 12$ (dotted line) for a Lorentizan spectral density. The initial state in Eq. (10) is determined by $C_{m}(0)=C_{n}(0)=1/\sqrt{2}$. We assume that panels (a, b) are plotted in the Markovian regime, $\lambda=15$ (in units of $\omega_{0}$) and $\gamma_{0}=0.2$ (in units of $\omega_{0}$), and panels (c, d) in the non-Markovian regime, $\lambda=0.5$ (in units of $\omega_{0}$) and $\gamma_{0}=1$ (in units of $\omega_{0}$).
a b\
[ {width="3in"} \[fig:first\_sub\] ]{}[ {width="3in"} \[fig:second\_sub\] ]{}\
c d\
[ {width="3in"} \[fig:first\_sub\] ]{}[ {width="3in"} \[fig:second\_sub\] ]{}
Fig. 2. (a, c, e) Diagrammatic solutions of Eq. (8), and (b, d, f) concurrence dynamics of a two-qubit system with $N = 2$ (solid line), $N = 8$ (dashed line) and $N = 12$ (dotted line) for $(a, b)$ sub-Ohmic $(s = 1/2)$, $(c, d)$ Ohmic $(s = 1)$ and $(e, f)$ super-Ohmic $(s = 2)$ spectral densities with fixed parameters $\omega_{c}=1$ (in units of $\omega_{0}$), $\gamma=1$ (in units of $\omega_{0}$). The initial state in Eq. (10) is determined by $C_{m}(0)=C_{n}(0)=1/\sqrt{2}$.
a b\
[ {width="3in"} \[fig:first\_sub\] ]{}[ {width="3in"} \[fig:second\_sub\] ]{}\
c d\
[ {width="3in"} \[fig:first\_sub\] ]{}[ {width="3in"} \[fig:second\_sub\] ]{}\
e f\
[ {width="3in"} \[fig:first\_sub\] ]{}[ {width="3in"} \[fig:second\_sub\] ]{}
[^1]: E-mail:[email protected]
|
---
abstract: 'We present an algorithm that covers any given rational ruled surface with two rational parametrizations. In addition, we present an algorithm that transforms any rational surface parametrization into a new rational surface parametrization without affine base points and such that the degree of the corresponding maps is preserved.'
author:
- |
J. Rafael Sendra, Carlos Villarino\
[Dept. of Physics and Mathematics, U. of Alcalá]{}\
[Ap. Correos 20]{}\
[E-28871 Alcalá de Henares (Madrid, Spain)]{}\
[$\{$Rafael.Sendra,Carlos.Villarino$\}[email protected]]{}\
\
David Sevilla\
[U. Center of Mérida, U. of Extremadura]{}\
[Av. Santa Teresa de Jornet 38]{}\
[E-06800 Mérida (Badajoz, Spain)]{}\
[[email protected]]{}
title: Covering Rational Ruled Surfaces
---
2010 Mathematics Subject Classification: 14Q10, 68W30.
Keywords: parametrization of ruled surfaces, normality, base points
Introduction
============
One of the most important features of rational varieties, at least in practice, is the possibility to choose between parametric or implicit representations depending on the nature of the problem one is dealing with; examples are the computation of intersections, plotting figures, line and surface integrals, etc. However, when using the parametric representations additional difficulties may appear, and the feasibility of the strategy is affected. In particular, if the parametrization is not surjective, some solving strategies may fail.
In [@Ricam], Example 1 illustrates a situation where the computation of the intersection of two surfaces fails when a non surjective parametrization is used. Let us see another motivating example.
The Hausdorff distance appears naturally in applications in computer aided design, pattern matching and pattern recognition (see e.g. [@Bai], [@Chen], [@Kim]), when measuring the resemblance between two geometric objects. The computation or estimation of the Hausdorff distance implies, in particular, measuring the distance of a point to a set. Let us assume that we want to measure the distance of the point $A=(4/5, 6/5, 1)$ to the surface $S$ defined by $f(x,y,z)=xy-2yz+z^2$. Applying Lagrange multipliers one gets that the distance of $A$ to $S$ is $\sqrt{2}/5=0.283\ldots$ and it is reachable at $B=(1,1,1)\in S$. Nevertheless, in general, approaching this problem using implicit equations turns to be computationally intractable. Instead, one can try to use a parametrization of the surface so that the problem reduces to a optimization problem without constrains. In our case, $S$ is rational, indeed it is rational ruled surface, and can be parametrized as $${\mathcal{P}}(s,t)=\left((s^2-1)t, s^2t, t(s^2+s)\right).$$ However, if we optimize the function $\|A-{\mathcal{P}}(s,t)\|^2$ we find that the minimum is obtained at $(0.889\ldots, -0.042\ldots, 0.155\ldots)$ and the distance is then estimated as $1.504\ldots$. The problem is that $B\in S\smallsetminus {\mathrm{Image}}({\mathcal{P}})$, so it cannot be found with the parametrization. Nevertheless, ${\mathcal{P}}$ satisfies the hypothesis in Theorem \[theorem-recta-critica\], and therefore we can determine that the $S\smallsetminus {\mathrm{Image}}({\mathcal{P}})$ is included in the line $(t,t,t)$. Thus, we now optimize the $\|A-(t,t,t)\|^2$ to get $B$ as solution.
In the case of curves, non-surjectivity is not so important since every rational proper parametrization of a curve may miss at most one point that can be easily computed (see e.g. [@AndradasRecio], [@Sendra2002a]). The situation changes when working with rational parametrizations of algebraic surfaces: the missing subset can be of dimension 1.
Some authors have addressed the problem of finding surjective parametrizations of rational surfaces; see [@Bajaj], [@Gao] for the case of quadrics or [@Ricam] for certain particular types of rational surfaces. Alternatively, one can compute finitely many rational parametrizations such that the union of their images covers the whole surface. This was done for the real general case, in [@Bajaj], by computing a cover with $2^n$ parametrizations, where $n$ is the dimension of the rational variety; i.e. in the surface case, with four pieces. In [@issac] we show that, if a surface admits a rational parametrization without projective base points, then it can be covered with at most three pieces. Continuing with this research, in this paper we analyze the problem of covering rational ruled surfaces. The next example shows that for the same surface, changing the parametrization, can make the missing subset bigger.
We consider the ruled surface $S$ given by $x^2y-2xy^2+2y^3-3y^2z+3yz^2-z^3=0$. $S$ can be parametrized in [*ruled form*]{} as $${\mathcal{P}}(s,t)= \left(\frac{(s^3-1)t}{s(s+1)}, \frac{s^2t}{s+1}, \frac{(s^2+1)t}{s+1}\right)$$ Applying the algorithm in [@Ricam] for computing the critical sets, we obtain that ${\mathcal{P}}$ covers all the surface but the three lines $\{x=2y=z\}, \{x=y=z\}, \{y=z=0\}$. However, the reparametrization $${\mathcal{P}}(s, ts(s+1))= \left( (s^2-1)t, ts^2, (s^2+s)t \right),$$ that is also in [*ruled form*]{}, only misses the line $\{x=y=z\}$ (see Theorem \[theorem-recta-critica\]).
In this paper we prove that a rational ruled surface can always be covered with two rational surface parametrizations in [*ruled form*]{}. More precisely, we prove that there always exists a rational parametrization that, at most, misses a line on the surface; then the second parametrization covers that line. In order to compute the first parametrization we need parametrizations without affine base points. Later we consider this problem in general, and we present an algorithm that transforms any rational surface parametrization into a new parametrization without affine base points.
Covering Ruled Surfaces: Main Results {#sec:covering-theory}
=====================================
In the sequel, we show that every rational ruled surface can be covered by means of, at most, two rational parametrizations.
\[def-ruled-param\] A *standardized ruled surface parametrization* of a ruled suface $S$ is a triple of rational functions that determines a dominant rational map $$\begin{array}{rccc}
{\mathcal{P}}: & k^2 & { \settowidth{\@tempdima}{$\longrightarrow$}\longrightarrow \makebox[-\@tempdima]{\hskip-1.5ex\color{white}\rule[0.5ex]{2pt}{1pt}} \phantom{\longrightarrow}}& S \\ [1em]
& (s,t) & \mapsto & \left(\displaystyle \frac{r_1(s)+t\cdot p_1(s)}{q(s)}, \frac{r_2(s)+t\cdot p_2(s)}{q(s)}, \frac{r_3(s)+t\cdot p_3(s)}{q(s)} \right)
\end{array}$$ such that those $p_i$ that are nonzero have the same degree and do not have any common root (note that not all three of them are zero).
\[remark-pi\] In a standardized ruled surface parametrization, if two of the polynomials $p_i$ are zero, then the third has to be a nonzero constant. In addition, we observe that, in that case, say $p_1=p_2=0$, then ${\mathcal{P}}(s,(-r_3+qt)/p_3)=(r_1/q,r_2/q,t)$. So, $S$ is a cylinder over the plane curve $(r_1/q,r_2/q,0)$ and hence, applying the results in [@Sendra2002a], $S$ can be parametrized surjectively.
\[lemma-stardard\] Every rational ruled surface admits a standardized ruled surface parametrization.
By [@Sonia], every rational ruled surface admits a rational parametrization of the form $${\mathcal{P}}(s,t)=\left(\frac{\alpha_1(s) +t \beta_1(s)}{\gamma(s)},\frac{\alpha_2(s) +t \beta_2(s)}{\gamma(s)}, \frac{\alpha_3(s) +t \beta_3(s)}{\gamma(s)}\right).$$ If two $\beta_i$ are zero, say $\beta_1=\beta_2=0$, then ${\mathcal{P}}(s,t/\beta_3(t))$ is standardized. Let us suppose that at least two $\beta_i$ are nonzero. Then, we can assume that those components of ${\mathcal{P}}(s,t)$ depending on $t$ also do depend on $s$; if this is not the case a suitable change of the form $(s,as+bt)$ provides a parametrization with this property. Furthermore, applying a transformation of the form $(\frac{as+b}{cs+d},t)$, we can assume that all nonzero $\beta_i$ have the same degree. It only remains to ensure that the gcd of the polynomial coefficients of $t$ are coprime. But this can be achieved by performing the transformation $(s, t/\Delta(s))$, where $\Delta$ is the gcd of the nonzero $\beta_i$.
Associated to the standardized ruled surface parametrization ${\mathcal{P}}(s,t)$, we consider the polynomials $$\label{eq-H}
\begin{array}{l} H_1=r_1(s)+t\cdot p_1(s)-x\cdot q(s), \\ H_2=r_2(s)+t\cdot p_2(s)-y\cdot q(s), \\
H_3=r_3(s)+t\cdot p_3(s)-z\cdot q(s),
\end{array}$$ as well as the polynomials $A_{ij} = p_iH_j-p_jH_i\in k[x,y,z,s]$ for $i\neq j$. We express $A_{ij}$ as $$\label{eq-A}
\begin{array}{r}
A_{12}= qp_2x -qp_1y -\alpha_{12}, \\ A_{13}= qp_3x -qp_1z -\alpha_{13}, \\
A_{23}= qp_3y -qp_2z -\alpha_{23}, \\
\alpha_{ij}=-p_{i}r_j+p_jr_i.
\end{array}$$ We have the following lemma.
\[lemma-crit\] Let ${\mathcal{P}}$ be a standardized ruled surface parametrization without affine base points of a surface $S$. Then $S\smallsetminus{\mathrm{Image}}({\mathcal{P}})$ is contained in the variety $\cal W$ defined by $\{{\rm LC}_s(A_{ij})\}_{i\neq j}$, where ${\rm LC}_s$ denotes the leading coefficient w.r.t. $s$.
In the ring $k[x,y,z,s,t,w]$ we consider the ideal $$I = (H_1(s,t,x),\ H_2(s,t,y),\ H_3(s,t,z),\ w\cdot q(s)-1).$$ Then ${\mathrm{Image}}({\mathcal{P}})=\pi(V(I))$ where $\pi(x,y,z,s,t,w)=(x,y,z)$. We will use the extension theorem (see e.g. Chp.3, Th 3, p. 115 in [@CoxLittleOshea2007a]) to determine which points $(x,y,z)\in S$ can be lifted to $V(I)$. To this end we define $$I_1 = I \cap k[x,y,z,s,t], \quad I_2 = I \cap k[x,y,z,s], \quad I_3 = I \cap k[x,y,z].$$
- Extension from $I_1$ to $I$: a point $(x_0,y_0,z_0,s_0,t_0)$ has an extension provided $q(s_0)\neq0$. But if $q(s_0)=0$ we see from the equations that $r_i(s_0)+t_0p_i(s_0)=0$ for all $i$, and $(s_0,t_0)$ would be a base point, contrary to the hypotheses.
- Extension from $I_2$ to $I_1$: in order to extend a point $(x_0,y_0,z_0,s_0)$ to the coordinate $t$ it suffices that $p_1,p_2,p_3$ do not simultaneously vanish at $s_0$. This always holds since by definition they have no common root. Note that if two of the $p_i$ are zero, the other is a nonzero constant, and the extension is possible.
- Extension from $I_3$ to $I_2$: a point $(x,y,z)$ can be extended to the coordinate $s$ if for at least one of the polynomials $A_{ij}$ the leading coefficient in $s$ does not vanish at the point.
\[lemma-recta\] The variety $\cal W$ introduced in Lemma \[lemma-crit\] is either empty or a line. Furthermore, $\mathcal{W}=\emptyset$ if and only if $\deg(\alpha_{ij})>\deg(p_k q)$, for some different $i,j\in \{1,2,3\}$ and nonzero $p_k$.
Let us assume $p_1\neq 0$. If $\deg(\alpha_{ij})>\deg(p_1 q)$, for some different $i,j\in \{1,2,3\}$, then ${\rm LC}_s(A_{ij})$ is a nonzero constant and $\mathcal{W}=\emptyset$.
If $\deg(\alpha_{ij})\leq \deg(p_1 q)$ for all $i\neq j$, we distinguish two cases. If $p_2=p_3=0$ then $A_{12}=-p_1qy-\alpha_{12}, A_{13}=-p_1qz-\alpha_{13}, A_{23}=0$. Then, $\cal W$ is defined by two linear polynomials, one depending on $y$ and the other on $z$. So $\cal W$ is a line. In the second case, let us assume that at least two $p_i$ are nonzero. Since $p_3 A_{12}-p_2 A_{13}+p_1 A_{23}=0$, then $${\rm LC}_s(p_1 A_{23})={\rm LC}_s(p_1){\rm LC}_s(A_{23})={\rm LC}_s(-p_3A_{12}+p_2 A_{13}).$$ Let us see that $${\rm LC}_s(p_1){\rm LC}_s(A_{23})=-{\rm LC}_s(p_3){\rm LC}_s(A_{12})+{\rm LC}_s(p_2){\rm LC}_s(A_{13}).$$ If either $p_2$ or $p_3$ is zero, the result is clear. So, let none of them be zero. Then, $\deg_s(p_3A_{12})=\deg_s(p_2A_{13})$. Since the leading coefficient of $p_3A_{12}$ does depend on $\{x,y\}$ and the leading coefficient of $p_2A_{13}$ does depend on $\{y,z\}$, $\deg_s(-p_3A_{12}+p_2 A_{13})=\deg_s(p_3A_{12})=\deg_{s}(p_2A_{13})$, from where the above equality on the leading coefficients follows. In this situation we get that $\cal W$ is defined by ${\rm LC}_s(A_{12}),{\rm LC}_s(A_{13})$. Now, the result follows by taking into account that the rank of the linear system $\{{\rm LC}_s(A_{12})=0={\rm LC}_s(A_{13})\}$ is 2.
Using the previous results one gets the following theorem.
\[theorem-recta-critica\] Let ${\mathcal{P}}$ be a standardized ruled surface parametrization without affine base points of a surface $S$. Then $S\smallsetminus{\mathrm{Image}}({\mathcal{P}})$ is contained in a line. Furthermore,
1. if there exists $i,j\in \{1,2,3\}$ such that $i\neq j$ and $\deg(\alpha_{ij})>\deg(p_k q)$ for nonzero $p_k$, then ${\mathcal{P}}(s,t)$ is normal.
2. if for all $i,j\in \{1,2,3\}$, with $i\neq j$, $\deg(\alpha_{ij})\leq \deg(p_k q)$ for nonzero $p_k$, then $S\smallsetminus{\mathrm{Image}}({\mathcal{P}})$ is included in the line $V({\rm LC}_s(A_{12}),{\rm LC}_s(A_{13}),{\rm LC}_s(A_{23}))$.
In Example \[ejemplo-grado-3\], one can see that the parametrization covers all the line $\cal W$ but a point, while in Example \[ejemplo-grado-5\], the parametrization only covers two points on the line $\cal W$.
In the previous theorem we have imposed the condition of not having affine base points. Let us see that this is always achievable.
\[lemma-reglada-sin-puntos-base\] Every standardized ruled surface parametrization can be reparametrized into another one without affine base points and where the degree of the induced map is preserved.
Let us assume first that all the $p_i$ are nonzero. Let $f(s)$ be a polynomial such that $f(s_1)=t_1$ for some base point $(s_1,t_1)$. With the change $(s,1/t+f(s))$ the resulting parametrization is $$\left( \frac{Q_i(s)\cdot t+p_i(s)}{t\cdot q(s)} \right)_{i=1,2,3} \qquad \mbox{where $Q_i(s)=r_i(s)+f(s)p_i(s)$}.$$ Since $s_1$ is a common root of $q$ and the $Q_i$, if we define $\widetilde{Q}=\gcd(Q_1,Q_2,Q_3,q)$, we have $\deg(\widetilde{Q})\geq1$. Now with the change $(s,1/(\widetilde{Q}t))$ we obtain the new parametrization $$\left( \frac{Q_i/\widetilde{Q} + p_i\cdot t}{q/\widetilde{Q}} \right)_{i=1,2,3}.$$ Note that this is standardized as well, but the degree of the denominator is strictly smaller than the original. Therefore repeating this procedure finitely many times we obtain a standardized parametrization without affine base points (since that is the case when the denominator is a constant). Finally, note that all transformations considered are birational, and hence the degrees of the maps are preserved.
If any $p_i=0$, the corresponding component of the parametrization does not change after the first reparametrization, resulting in $Q_i=r_i$. The second reparametrization does not change the component as well, but the common factor $\widetilde{Q}$ of $r_i$ and $q$ can be directly simplified in that fraction.
\[remark-pto-bases\] In the previous result we can have some control on the removal of base points that occurs effectively in each iteration. Namely, suppose that the zeros of $q$ are $s_1,\ldots,s_l$ where $s_1,\ldots,s_k$, $k\leq l$, are the first coordinates of the base points of ${\mathcal{P}}$. Note that for each of $s_1,\ldots,s_k$ there is exactly one base point $(s_i,t_i)$.
Let $f(s)$ be an interpolating polynomial of $$(s_1,t_1),\ldots,(s_k,t_k), (s_{k+1},0),\ldots,(s_l,0).$$ As before, we define $Q_i(s)=r_i(s)+f(s)p_i(s)$ and $\widetilde{Q}=\gcd(Q_1,Q_2,Q_3,q)$, and make the change $(s,\widetilde{Q}t+f(s))$ to obtain $${\mathcal{P}}^{(1)} = \left( \frac{r_i^{(1)} + p_i\cdot t}{q^{(1)}} \right)_{i=1,2,3} \qquad \mbox{where $r_i^{(1)}=Q_i/\widetilde{Q}$, \ $q^{(1)}=q/\widetilde{Q}$}.$$ Note that the roots of $\widetilde{Q}$ are precisely $s_1,\ldots,s_k$. We will show that ${\mathcal{P}}^{(1)}$ has at most as many base points as the number of multiple roots of $q$ among $s_1,\ldots,s_k$. To this end let $(\alpha,\beta)$ be a base point of ${\mathcal{P}}^{(1)}$. Since $\alpha$ is a root of $q^{(1)}$, it must be a multiple root of $q$, say $\alpha=s_i$. If $i>k$ then $Q_i(\alpha)=r_i(\alpha)$. Now, by definition of $\beta$, we have $\beta=-r_i^{(1)}(\alpha)/p_i(\alpha)$ for some $i$. But then the point $(\alpha,\beta\widetilde{Q}(\alpha))$ is a base point of ${\mathcal{P}}$, contradiction.
\[cor+2+1\] Every rational ruled surface can be parametrized in an standardized way that misses at most one line.
\[theorem+2+2\] Every rational ruled surface can be covered with at most two surface parametrizations.
By Lemma \[lemma-reglada-sin-puntos-base\] we can assume that we are given an standardized parametrization ${\mathcal{P}}$ without affine base points. We use for ${\mathcal{P}}$ the notation in Definition \[def-ruled-param\] and in the previous results. By Theorem \[theorem-recta-critica\], we can also assume that $\max\{\deg(\alpha_{12}),\deg(\alpha_{13}),\deg(\alpha_{23})\}\leq \deg(p_kq)$ for nonzero $p_k$.
First we assume that all $p_i(s)$ are nonzero. Consider the reparametrizations $$\mathcal{Q}(s,t)={\mathcal{P}}\left(s,\frac{qt-r_3}{p_3}\right)=\left(\frac{q p_1t+\alpha_{13}}{p_3q},\frac{qp_2t+\alpha_{23}}{p_3q},t \right)$$ and $$\mathcal{H}(s,t)=\mathcal{Q}\left(\frac{1}{s},t\right).$$ Because of our above degree assumptions, we know that the degrees in $s$ of the numerator and denominator of each (first and second) component of $\mathcal{Q}$ are equal. Therefore, $s$ is not a factor of the denominators in $\mathcal{H}(s,t)$. So, $\mathcal{H}(0,t)$ is well defined and, indeed, $$\mathcal{H}(0,t)=\left(\frac{{\rm LC}_s(q p_1t+\alpha_{13})}{{\rm LC}_s(p_3q)},\frac{{\rm LC}_s(qp_2t+\alpha_{23})}{{\rm LC}_s(p_3q)},t \right)$$ that parametrizes the line $\cal W$.
A similar argument with obvious modifications works in the case when some $p_i$ are zero.
Covering Ruled Surfaces: Algorithm and Examples {#sec:covering-algorithm}
===============================================
In order to derive an algorithm from the previous results, we need to algorithmically show how to remove the affine base points of an standardized ruled parametrization. This, essentially, requires to compute interpolation polynomials (see proof of Lemma \[lemma-reglada-sin-puntos-base\] and Remark \[remark-pto-bases\]). In the following lemma we see how to actually compute the interpolation polynomial without explicitly determining the coordinates of the base points; i.e. without approximating roots.
\[lemma-interpolation\] Let ${\mathcal{P}}(s,t)$ be an standardized ruled parametrization as in Def. \[def-ruled-param\] with affine base points. Let $I$ be the ideal generated by $\{p_1t+r_1,p_2t+r_2,p_3t+r_3,q\}$ in $k[s,t]$. Then, there exists a polynomial of the form $t-f(s)$ in $\sqrt{I}$ where $f(s)$ interpolates the affine base points of ${\mathcal{P}}(s,t)$.
As observed in Remark \[remark-pto-bases\], all affine base points of ${\mathcal{P}}$ have different $s$-coordinate. Thus, there exists an interpolating polynomial $f(s)$ passing through all base points. So, $t-f(s)$ vanishes on all the points in the variety of $I$. So, $t-f(s)\in \sqrt{I}$.
Now, we are ready to outline our algorithm.
\[alg-1\] Given a rational parametrization ${\mathcal{P}}(s,t)$ of a ruled surface $\cal S$, the algorithm computes a covering of $\cal S$.
1. If ${\mathcal{P}}$ is not of the form $((r_i(s)+p_i(s)t)/q(s))_{i=1,2,3}$ apply the algorithm in [@Sonia] and replace ${\mathcal{P}}$.
2. If ${\mathcal{P}}$ is not in standardized form (see Def. \[def-ruled-param\]) do the following
1. If some of the numerators of ${\mathcal{P}}$ does not depend on $s$, replace ${\mathcal{P}}$ by ${\mathcal{P}}(s, as +
bt)$ with $a,b\in k$.
2. If the polynomials $p_1,p_2,p_3$ do not have the same degree, replace ${\mathcal{P}}$ by ${\mathcal{P}}((as+b)/(cs+d), t)$ where $a,b,c,d\in k$ and $ad-bc\neq 0$.
3. Replace ${\mathcal{P}}$ by reparametrization ${\mathcal{P}}(s, t/\Delta(s))$ where $\Delta$ is the gcd of the nonzero $p_i$.
3. Calculate $\sqrt{I}$, where $I$ is the ideal generated by $\{p_1t+r_1,p_2t+r_2,p_3t+r_3,q\}$ in $k[s,t]$. This can be done with a relatively inexpensive Gröbner basis computation (see e.g. Ex 2.3.23 and 24 in [@Adams] and [@Seidenberg]).
4. Calculate a Gröbner basis of $\sqrt{I}$ with respect to the lexicographical ordering $t>s$.
1. If the basis does not contain a polynomial of the form $t-f(s)$, by elementary properties of Gröbner basis it follows that there is no polynomial of that form in $\sqrt{I}$, so by Lemma \[lemma-interpolation\] we know that ${\mathcal{P}}$ does not have affine base points.
2. In the other case, let $t-f(s)$ belong to the basis, do
1. Replace ${\mathcal{P}}$ by ${\mathcal{P}}(s, 1/t + f(s))$.
2. Let $\widetilde{Q}$ be the gcd of the coefficients of $t$ of the numerators of ${\mathcal{P}}$ and $q$, then replace ${\mathcal{P}}$ by ${\mathcal{P}}(s, 1/(\widetilde{Q}t))$.
3. Repeat Steps 3 and 4 while $\sqrt{I}$ has an element of the form $s-f(t)$.
5. Compute the polynomials $\alpha_{ij}$ (see (\[eq-A\])).
6. If there exist $i,j\in \{1,2,3\}$ such that $i\neq j$ and $\deg(\alpha_{ij}) > \deg(p_kq)$ for nonzero $p_k$, RETURN ${\mathcal{P}}(s,t)$.
7. Assume that $p_k\neq 0$, compute $$\mathcal{H}(s,t) = {\mathcal{P}}\left(\frac{1}{s},\frac{q(1/s)t-r_k(1/s)}{p_k(1/s)}\right)$$ and RETURN $[{\mathcal{P}}(s,t),\mathcal{H}(s,t)]$.
Remark \[remark-pto-bases\] shows that, in general, the number of iterations of the loop in Step 4 is small. Indeed it is bounded by the maximum multiplicity of the roots of the denominator $q(t)$ of ${\mathcal{P}}$.
In addition we observe that all parametrizations in the output of the algorithm are of ruled form, that is, of the form $(\alpha_1(s),\alpha_2(s),\alpha_3(s))+$$t(\beta_1(s),\beta_2(s),\beta_3(s))$.
Let us illustrate Algorithm \[alg-1\] by some examples.
\[ejemplo-grado-3\] We consider the parametrization $${\mathcal{P}}(s,t) =\left( \frac{r_1(s)+t p_1(s)}{q(s)}, \frac{r_2(s)+t p_2(s)}{q(s)}, \frac{r_3(s)+t p_3(s)}{q(s)} \right) =$$ $$\left( {\frac {t \left( {s}^{2}+s+1 \right) +s}{s \left( s-1 \right) }},{
\frac {t \left( {s}^{2}+2\,s \right) +s}{s \left( s-1 \right) }},{
\frac {t \left( {s}^{2}+1 \right) +s}{s \left( s-1 \right) }}\right).$$ It parametrizes the degree 3 ruled surface defined by
$F(x,y,z)=5\,{x}^{3}-9\,{x}^{2}y-8\,{x}^{2}z+5\,x{y}^{2}+11\,xyz+3\,x{z}^{2}-{y}
^{3}-3\,{y}^{2}z-3\,y{z}^{2}-4\,{x}^{2}+4\,xy+4\,xz-{y}^{2}-2\,yz-{z}^
{2}.$
We observe that ${\mathcal{P}}(s,t)$ is in standardized form. So, we go to Step 3 in Algorithm \[alg-1\]. $I$ is the ideal generated by $\{p_1t+r_1,p_2t+r_2,p_3t+r_3,q\}$ in $k[s,t]$. We get $\sqrt{I}=I$, and a Gröbner basis w.r.t. the lexicographic ordering $t>s$ (Step 4) is $\{s,t\}$. So, in Step 4 (b) we get that $f(s)=0$; note that the origin is the only affine base point. In Step 4 (b, i), we replace ${\mathcal{P}}(s,t)$ by ${\mathcal{P}}(s,1/t+0)$, namely $${\mathcal{P}}(s,t)=\left({\frac {{s}^{2}+st+s+1}{ts \left( s-1 \right) }},{\frac {s(s+2+t)}{ts
\left( s-1 \right) }},{\frac {{s}^{2}+st+1}{ts \left( s-1 \right) }}\right).$$ In Step 4 (b, ii), $\widetilde{Q}=\gcd(s,s,s,s(s-1))=s$. So, we replace ${\mathcal{P}}$ by ${\mathcal{P}}(s, 1/(st))$, namely $$\label{eq-ex-1}
{\mathcal{P}}(s,t)=\left( {\frac {{s}^{2}t+st+t+1}{s-1}},{\frac {{s}^{2}t+2\,st+1}{s-1}},{
\frac {{s}^{2}t+t+1}{s-1}}\right).$$ Now, the lexicographic order Gröbner basis of $\sqrt{I}$ is $\{1\}$, hence ${\mathcal{P}}$ does not have base points. In Step 5 we get $$\alpha_{12}=s-1,\alpha_{13}=-s,\alpha_{23}=2\,s-1.$$ In Step 6 the boolean conditions do not hold. In Step 7 we calculate the parametrization $$\mathcal{H}={\mathcal{P}}\left(\frac{1}{s},\frac{q(1/s)t-r_3(1/s)}{p_3(1/s)}\right)=$$ $$\left({\frac {{s}^{3}t-2\,{s}^{3}-{s}^{2}-2\,s-t}{ \left( {s}^{2}+1
\right) \left( s-1 \right) }},-{\frac {{s}^{3}-2\,{s}^{2}t+2\,{s}^{2
}+st+2\,s+t}{ \left( {s}^{2}+1 \right) \left( s-1 \right) }},{\frac {
st-2\,s-t}{s-1}}\right).$$ The algorithm returns the covering $[{\mathcal{P}}(s,t),\mathcal{H}(s,t)]$ where ${\mathcal{P}}$ is the parametrization in (\[eq-ex-1\]).
![The surface in Example \[ejemplo-grado-3\] and line $(t,t,t)$.[]{data-label="fig: ex-1"}](fig-1)
Continuing with the example, since ${\mathcal{P}}$ in (\[eq-ex-1\]) is an standardized ruled parametrization without affine base points, by Theorem \[theorem-recta-critica\], the possible missing points of ${\mathcal{P}}$ are included in the line defined by $\{x-y=0, x-z=0, y-z=0\}$, that is, the line $(t,t,t)$; see Fig. \[fig: ex-1\]. In fact, ${\mathcal{P}}$ covers all the line except the origin, by taking ${\mathcal{P}}(s,0)$. Nevertheless, ${\cal H}$ covers the whole line by taking ${\cal H}(0,t)$.
\[ejemplo-grado-5\] We consider the parametrization $${\mathcal{P}}(s,t) =\left( \frac{r_1(s)+t p_1(s)}{q(s)}, \frac{r_2(s)+t p_2(s)}{q(s)}, \frac{r_3(s)+t p_3(s)}{q(s)} \right) =$$ $$\left( {\frac {t{s}^{3}+2\,{s}^{2}+1}{{s}^{2}-1}},{\frac {t \left( {s}^{3}+2
\right) +s+1}{{s}^{2}-1}},{\frac {t \left( {s}^{3}+s+1 \right) +1}{{s
}^{2}-1}}\right).$$ It parametrizes the degree 5 ruled surface defined by
$ F(x,y,z)=9\,{x}^{5}-45\,{x}^{4}y-24\,{x}^{4}z+77\,{x}^{3}{y}^{2}+80\,{x}^{3}yz-
15\,{x}^{3}{z}^{2}-83\,{x}^{2}{y}^{3}+34\,{x}^{2}{y}^{2}z-147\,{x}^{2}
y{z}^{2}+78\,{x}^{2}{z}^{3}+66\,x{y}^{4}-60\,x{y}^{3}z-189\,x{y}^{2}{z
}^{2}+460\,xy{z}^{3}-236\,x{z}^{4}+16\,{y}^{5}-86\,{y}^{4}z+111\,{y}^{
3}{z}^{2}+118\,{y}^{2}{z}^{3}-332\,y{z}^{4}+168\,{z}^{5}-104\,{x}^{4}+
319\,{x}^{3}y+108\,{x}^{3}z-207\,{x}^{2}{y}^{2}-621\,{x}^{2}yz+452\,{x
}^{2}{z}^{2}+147\,x{y}^{3}-382\,x{y}^{2}z+1034\,xy{z}^{2}-848\,x{z}^{3
}+297\,{y}^{4}-1549\,{y}^{3}z+3390\,{y}^{2}{z}^{2}-3380\,y{z}^{3}+1344
\,{z}^{4}+304\,{x}^{3}-741\,{x}^{2}y+389\,{x}^{2}z-267\,x{y}^{2}+1761
\,xyz-2314\,x{z}^{2}-4\,{y}^{3}+922\,{y}^{2}z-1930\,{z}^{2}y+1816\,{z}
^{3}-70\,{x}^{2}+597\,yx-2060\,zx+748\,{y}^{2}-1703\,zy+2940\,{z}^{2}-
761\,x-62\,y+2085\,z+746.$
We observe that ${\mathcal{P}}(s,t)$ is in standardized form, so we go to Step 3 in Algorithm \[alg-1\]. $I$ is the ideal generated by $\{p_1t+r_1,p_2t+r_2,p_3t+r_3,q\}$ in $k[s,t]$. We get $\sqrt{I}=I$, and a Gröbner basis w.r.t. the lexicographic ordering $t>s$ (Step 4) is $\{1\}$. Thus ${\mathcal{P}}(s,t)$ does not have affine base points and we go to Step 5 to get $$\alpha_{12}=2{s}^{5}-{s}^{4}+4{s}^{2}+2 ,
\alpha_{13}=2{s}^{5}+2{s}^{3}+2{s}^{2}+s+1,
\alpha_{23}=-{s}^{4}-{s}^{2}-2s+1.$$ In Step 6 the boolean conditions do not hold. In Step 7 we get the parametrization $$\mathcal{H}={\mathcal{P}}\left(\frac{1}{s},\frac{q(1/s)t-r_3(1/s)}{p_3(1/s)}\right)=$$ $$\left(-{\frac {{s}^{5}+{s}^{4}+2\,{s}^{3}-t{s}^{2}+4\,{s}^{2}+t+2}{ \left(
{s}^{3}+{s}^{2}+1 \right) \left( {s}^{2}-1 \right) }},\right.$$$$\left. {\frac {2\,{s}^
{5}t-3\,{s}^{5}-2\,{s}^{4}-2\,t{s}^{3}-{s}^{3}+t{s}^{2}-2\,{s}^{2}-s-t
}{ \left( {s}^{3}+{s}^{2}+1 \right) \left( {s}^{2}-1 \right) }},{
\frac {t{s}^{2}-2\,{s}^{2}-t}{{s}^{2}-1}}\right).$$ The algorithm returns the covering $[{\mathcal{P}}(s,t),\mathcal{H}(s,t)]$.
![The surface in Example \[ejemplo-grado-5\] and line $(t+2,t,t)$.[]{data-label="fig: ex-2"}](fig-2)
Next, since the input parametrization ${\mathcal{P}}$ is an standardized ruled parametrization without affine base points, by Theorem \[theorem-recta-critica\], the possible missing points of ${\mathcal{P}}$ are included in the line defined by $\{x-y-2, x-z-2, y-z\}$, that is, the line $(t+2,t,t)$; see Fig. \[fig: ex-2\]. In fact, on the line, ${\mathcal{P}}$ reaches only the points $${\mathcal{P}}\left(\frac{1}{2}(1-i\sqrt{7}),\frac{1}{4}(3+i\sqrt{7})\right)=\left( {\frac {3}{32}}\,i\sqrt {7}+{\frac {65}{32}},{\frac {3}{32}}\,i\sqrt
{7}+\frac{1}{32},{\frac {3}{32}}\,i\sqrt {7}+\frac{1}{32} \right),$$ $${\mathcal{P}}\left(\frac{1}{2}(1+i\sqrt{7}),\frac{1}{4}(3-i\sqrt{7})\right)=\left(-{\frac {3}{32}}\,i\sqrt {7}+{\frac {65}{32}},-{\frac {3}{32}}\,i
\sqrt {7}+\frac{1}{32},-{\frac {3}{32}}\,i\sqrt {7}+\frac{1}{32} \right)$$ Nevertheless, ${\cal H}$ covers the whole line by taking ${\cal H}(0,t)$.
\[ejemplo-grado-4\] We consider the parametrization $${\mathcal{P}}(s,t) =\left( \frac{r_1(s)+t p_1(s)}{q(s)}, \frac{r_2(s)+t p_2(s)}{q(s)}, \frac{r_3(s)+t p_3(s)}{q(s)} \right) =$$ $$\left( {\frac {ts+ \left( -3\,s+2 \right) {s}^{5}}{ \left( s-1 \right) {s}^{
2}}},{\frac {t \left( s+1 \right) + \left( -5\,s+3 \right) {s}^{2}}{
\left( s-1 \right) {s}^{2}}},{\frac {t \left( s+2 \right) + \left( -8
\,s+5 \right) {s}^{2}}{ \left( s-1 \right) {s}^{2}}}\right).$$ It parametrizes the degree 4 ruled surface defined by
$F(x,y,z)=x{y}^{3}-3\,{y}^{2}zx+3\,xy{z}^{2}-x{z}^{3}-2\,{y}^{4}+7\,z{y}^{3}-9\,
{y}^{2}{z}^{2}+5\,y{z}^{3}-{z}^{4}-9\,{y}^{2}x+18\,zyx-9\,{z}^{2}x-5\,
{y}^{3}-9\,{y}^{2}z+16\,{z}^{2}y-5\,{z}^{3}+27\,yx-27\,zx-78\,{y}^{2}+
89\,zy-23\,{z}^{2}-27\,x-14\,y+17\,z+12.$
We observe that ${\mathcal{P}}(s,t)$ is in standardized form, so we go to Step 3 in Algorithm \[alg-1\]. $I$ is the ideal generated by $\{p_1t+r_1,p_2t+r_2,p_3t+r_3,q\}$ in $k[s,t]$. We get $\sqrt{I}\neq I$, and a Gröbner basis of $\sqrt{I}$ w.r.t. the lexicographic ordering $t>s$ (Step 4) is $\{t^2-t, -t+s\}$. So, in Step 4 (b) we get that $f(s)=s$. Note that the affine base points are $(0,0)$ and $(1,1)$ and $t=s$ is the interpolating line; observe that the corresponding Gröbner basis of $I$, $\{t^2-t, st-t, s^2-t \}$, that does not read the interpolating polynomial of minimal degree, although it contains the parabola $t=s^2$ that passes through the base points. In Step 4 (b, i), we replace ${\mathcal{P}}(s,t)$ by ${\mathcal{P}}(s,1/t+s)$, namely $${\mathcal{P}}(s,t)=\left(-{\frac {3\,{s}^{5}t-2\,{s}^{4}t-ts-1}{st \left( s-1 \right) }},-{
\frac {5\,{s}^{3}t-4\,t{s}^{2}-ts-s-1}{t \left( s-1 \right) {s}^{2}}},\right.$$$$\left.
-{\frac {8\,{s}^{3}t-6\,t{s}^{2}-2\,ts-s-2}{t \left( s-1 \right) {s}^{
2}}}\right).$$ In Step 4 (b, ii), $\widetilde{Q}=s^2-s$. So, we replace ${\mathcal{P}}$ by ${\mathcal{P}}(s, 1/((s^2-s)t))$, namely $${\mathcal{P}}(s,t)=\left( {\frac {-3\,{s}^{4}-{s}^{3}-{s}^{2}+ts-s}{s}},{\frac {ts-5\,s+t-1}{s}
},{\frac {ts-8\,s+2\,t-2}{s}}\right).$$ Now, the lexicographic order Gröbner basis of $\sqrt{I}$ is $\{s, t-1\}$, and hence ${\mathcal{P}}$ still have one base point, namely $(0,1)$. Now, the interpolation polynomial is $f(s)=1$ and $\widetilde{Q}=s$. Repeating the steps as above we reach at the end of Step 4 $$\label{eq-ejer-3}
{\mathcal{P}}(s,t)=(- (3\,{s}^{2}+s-t+1) s,ts+t-4,ts+2\,t-7)$$ In Step 5 we get $$\alpha_{12}=-3{s}^{5}-4{s}^{4}-2{s}^{3}+3{s}^{2},\alpha_{13}=-3{s}^{5}-7{s}^{4}-3{s}^{3}+5{s}^{2},\alpha_{23}=-3{s}^{2}+s$$ In Step 6 the boolean conditions do hold, and the output is the parametrization ${\mathcal{P}}$ in (\[eq-ejer-3\]) which is normal.
Removal of Base Points: the General Case
========================================
In Lemma \[lemma-reglada-sin-puntos-base\] we have seen that, for the special case of standardized ruled surfaces, one can always find a reparametrization such that the new parametrization does not have affine base points. In this section we see that the ideas applied in the proof of that lemma can be generalized to any rational parametrization. More precisely we have the following result.
\[th: remove bpt\] Let ${\mathcal{P}}\colon k^2{ \settowidth{\@tempdima}{$\longrightarrow$}\longrightarrow \makebox[-\@tempdima]{\hskip-1.5ex\color{white}\rule[0.5ex]{2pt}{1pt}} \phantom{\longrightarrow}}k^3$ be an affine rational parametrization, with nonconstant components, of a surface. Then there exists a rational reparametrization ${\mathcal{P}}\circ\psi$ without affine base points. Moreover, $\deg({\mathcal{P}})=\deg({\mathcal{P}}\circ\psi)$ as rational maps; in particular, properness is preserved.
If ${\mathcal{P}}$ has no affine base points, take as $\psi$ the identity.
We can assume without loss of generality that, after a suitable linear birational change, $${\mathcal{P}}(s,t) = \left( \frac{p_1(s,t)}{q(s,t)}, \frac{p_2(s,t)}{q(s,t)}, \frac{p_3(s,t)}{q(s,t)} \right)$$ where $\deg(p_1)=\deg(p_2)=\deg(p_3)=\deg(q)$, $\gcd(p_1,p_2,p_3,q)=1$, and the projective point $(0:1:0)$ does not belong to any of the projectivizations of the four curves determined by numerators and denominator. We also assume that there are no two affine base points with the same $s$-coordinate, since this can be achieved by composition with $(s,t)\to(s+\lambda t,t)$ for generic $\lambda$ without losing the previous assumptions.
By the last assumption, there exists an interpolation polynomial $f(s)$ for the affine base points, i.e. for every base point $(s_i,t_i)$ we have $t_i=f(s_i)$; note that the $\gcd$ condition implies finiteness of the base point set. We define the birational reparametrization $$\psi(s,t)=\left(s,\frac{1}{t}+f(s)\right)$$ and ${\widetilde{\mathcal{P}}}={\mathcal{P}}\circ\psi$. We will prove that ${\widetilde{\mathcal{P}}}$ has no affine base points. To this end we write $${\mathcal{P}}=\left(\frac{a_nt^n+a_{n-1}(s)t^{n-1}+\cdots+a_0(s)}{b_nt^n+b_{n-1}(s)t^{n-1}+\cdots+b_0(s)},\ldots,\ldots\right)$$ with $a_n,b_n\neq0$ and $\deg(a_i),\deg(b_i)\leq n-i$. This is possible by the hypothesis on the degrees of $p_1,p_2,p_3,q$, and the fact that $t^n$ appears in all of them with nonzero coefficient (equivalent to the hypothesis on $(0:1:0)$.) Then $${\widetilde{\mathcal{P}}}=\left(\frac{a_n(1+tf(s))^n+ta_{n-1}(s)(1+tf(s))^{n-1}+\cdots+t^na_0(s)}{b_n(1+tf(s))^n+tb_{n-1}(s)(1+tf(s))^{n-1}+\cdots+t^nb_0(s)},\ldots,\ldots\right).$$ This new parametrization cannot have any base points of the form $(s_0,0)$, since ${\widetilde{\mathcal{P}}}(s_0,0)=(a_n/b_n,\ldots,\ldots)$. On the other hand, if $(s_0,t_0)$ is a base point of ${\widetilde{\mathcal{P}}}$ with $t_0\neq0$, then $\psi(s_0,t_0)$ is a base point $(s_i,t_i)$ of ${\mathcal{P}}$. But this is impossible: if $\psi(s_0,t_0)=(s_i,t_i)$ then $s_0=s_i$ and $1/t_0+f(s_0)=t_i$ which imply $1/t_0=0$, contradiction. Finally, note that the previous transformations are birational, and $\psi$ is a birational map from $k^2$ on $k^2$, and hence the degree of the parametrization maps is preserved.
The reasoning in the previous proof leads to an algorithmic process to remove the affine base points of a surface parametrization. To be more precise, let $${\mathcal{P}}(s,t) = \left( \frac{p_1(s,t)}{q(s,t)}, \frac{p_2(s,t)}{q(s,t)}, \frac{p_3(s,t)}{q(s,t)} \right)$$ be the surface parametrization. First, we observe that some assumptions on the parametrization are done, namely
1. $\deg(p_1)=\deg(p_2)=\deg(p_3)=\deg(q)$, and $\gcd(p_1,p_2,p_3,q)=1$,
2. the projective point $(0:1:0)$ does not belong to any of the projectivizations of the four curves determined by numerators and denominator,
3. there are no two affine base points with the same $s$-coordinate.
Observe that, in the rational ruled case, condition 3 is satisfied while, in general, conditions 1 and 2 fail because of the particular structure of standardized form, that we wanted to be preserved. So, in Section \[sec:covering-theory\], we have developed an *ad hoc* proof for the ruled case.
Once the parametrization satisfies these conditions, one computes the interpolation polynomial $f(s)$ passing through the affine base points. Then, $\psi(s,t)=(s,\frac{1}{t}+f(s))$. We observe that condition 1 can always be achieved by a birational change of the form $$\left(\frac{a_1t+b_1s+c_1}{d_1t+e_1s+h_1}, \frac{a_2t+b_2s+c_2}{d_2t+e_2s+h_2}\right),$$ and condition 2 with a linear change $(s+\lambda t, t)$. In the following lemma we see how to check the third condition and how to actually compute the interpolation polynomial $f(s)$ without approximating roots. This result extends Lemma \[lemma-interpolation\] to the general case.
\[lemma-interpolacion-general\] Let $I$ be the ideal generated by $\{p_1,p_2,p_3,q\}$ in $k[s,t]$.
1. Condition 3 is satisfied if and only if there exists a polynomial of the form $t-g(s)$ in $\sqrt{I}$.
2. If $t-g(s)\in \sqrt{I}$, then $g(s)$ interpolates the affine base points.
If condition 3 holds, then $t-f(s)$ vanishes on all the points in the variety of $I$. So, $t-f(s)\in \sqrt{I}$. The converse is trivial, and (2) follows from (1).
\[alg-2\] The input is a rational surface parametrization with affine base points, and the output is a parametrization of the same surface without base points.
1. Reparametrize the input to satisfy conditions 1 and 2.
2. Calculate $\sqrt{I}$; see Step 3 in Algorithm \[alg-1\].
3. Calculate a Gröbner basis of $\sqrt{I}$ with respect to the lexicographical ordering $t>s$.
1. If the basis contains a polynomial of the form $t-f(s)$, then by the previous Lemma condition 3 is satisfied and we can apply the reparametrization of Theorem \[th: remove bpt\] to RETURN ${\mathcal{P}}(s,1/t+f(s))$.
2. In the negative case, by elementary properties of Gröbner bases it follows that there is no polynomial of that form in $\sqrt{I}$. Again by Lemma \[lemma-interpolacion-general\], condition 3 is not satisfied. Apply a transformation $(s+\lambda t, t)$ for random $\lambda$ in the ground field and go to step 2.
As a consequence of Theorem \[th: remove bpt\] and Algorithm \[alg-2\], the following corollaries hold.
\[cor-1\] Every rational surface over an algebraically closed field of characteristic zero can always be parametrized without affine base points.
\[cor-2\] Every rational surface parametrization can be reparametrized, without affine base points, without extending the field of coefficients and the degree as rational maps.
We illustrate the ideas of this section by an example.
\[ex-pto-base-general\] We consider the rational parametrization $${\mathcal{P}}(s,t)=\left(\frac{p_1(s,t)}{q(s,t)},\frac{p_2(s,t)}{q(s,t)}, \frac{p_3(s,t)}{q(s,t)} \right)= \left({\frac {4\,{s}^{2}-4\,st+{t}^{2}-6\,s+3\,t}{2\,{s}^{2}+8\,st+3\,{t}^{
2}-8\,s-11\,t}},\right.$$ $$\left. {\frac {{s}^{2}-6\,st-{t}^{2}+s+7\,t}{2\,{s}^{2}+8\,st
+3\,{t}^{2}-8\,s-11\,t}},{\frac {-3\,{s}^{2}+22\,st+4\,{t}^{2}-5\,s-26
\,t}{2\,{s}^{2}+8\,st+3\,{t}^{2}-8\,s-11\,t}}\right)$$ Its base points are $\{(0,0),(2,1),(1,2),(1,-1) \}$. We observe that ${\mathcal{P}}(s,t)$ satisfies conditions 1 and 2. Let $I$ be the ideal generated by $\{p_1,p_2,p_3,q\}$. A Gröbner basis of $\sqrt{I}$ w.r.t. the lexicographic order with $t>s$ is $$\{s^3-3s^2+2s, -s^2+2st+s-2t, 2s^2+t^2-4s-t \}.$$ Since there is no polynomial of the form $t-f(s)$ in the basis, condition 3 fails, and we perform a change of parameters. For example ${\mathcal{P}}$ is replaced by ${\mathcal{P}}(s+t,t)$. Applying again the Gröbner basis computation to $\sqrt{I}$ for the new ${\mathcal{P}}$, we obtain the basis $$\{s^4-2s^3-s^2+2s, 2s^3-3s^2-s+2t \}.$$ The second polynomial implies that $t-(-s^3+(3/2)s^2+(1/2)s)\in \sqrt{J}$. So condition 3 is now satisfied and $f(s)=-s^3+(3/2)s^2+(1/2)s$. Therefore, performing the transformation ${\mathcal{P}}(s,1/t+f(s))$ we get a new parametrization without affine base points, namely $$\left( {\frac {4 {s}^{6}{t}^{2}-12 {s}^{5}{t}^{2}-11 {s}^{4}{t}^{2}+42 {
s}^{3}{t}^{2}-8 {s}^{3}t+7 {s}^{2}{t}^{2}+12 {s}^{2}t-30 {t}^{2}s+
20 st-12 t+4}{52 {s}^{6}{t}^{2}-156 {s}^{5}{t}^{2}+17 {s}^{4}{t}^
{2}+226 {s}^{3}{t}^{2}-104 {s}^{3}t-69 {s}^{2}{t}^{2}+156 {s}^{2}t
-70 {t}^{2}s+100 st-76 t+52}},\right.$$ $$-2 {\frac {12 {s}^{6}{t}^{2}-36 {s
}^{5}{t}^{2}+7 {s}^{4}{t}^{2}+46 {s}^{3}{t}^{2}-24 {s}^{3}t-19 {s}
^{2}{t}^{2}+36 {s}^{2}t-10 {t}^{2}s+20 st-16 t+12}{52 {s}^{6}{t}^
{2}-156 {s}^{5}{t}^{2}+17 {s}^{4}{t}^{2}+226 {s}^{3}{t}^{2}-104 {s
}^{3}t-69 {s}^{2}{t}^{2}+156 {s}^{2}t-70 {t}^{2}s+100 st-76 t+52}
},$$ $$\left. {\frac {92 {s}^{6}{t}^{2}-276 {s}^{5}{t}^{2}+51 {s}^{4}{t}^{2}+
358 {s}^{3}{t}^{2}-184 {s}^{3}t-143 {s}^{2}{t}^{2}+276 {s}^{2}t-82
{t}^{2}s+156 st-124 t+92}{52 {s}^{6}{t}^{2}-156 {s}^{5}{t}^{2}+
17 {s}^{4}{t}^{2}+226 {s}^{3}{t}^{2}-104 {s}^{3}t-69 {s}^{2}{t}^{2
}+156 {s}^{2}t-70 {t}^{2}s+100 st-76 t+52}} \right).$$
In [@Wang], section 4.5, the author tests his implicitization algorithm with a family of rational surface parametrizations collected from different papers. For those having affine points, we apply Algorithm \[alg-2\]:
1. [Example 1 in [@Wang].]{} The parametrization is $${\mathcal{P}}=\left( {\frac {s{t}^{2}-{t}^{3}-t}{{t}^{2}-2\,t+1}},{\frac {{t}^{3}-st-{t}^{
2}+t+1}{{t}^{2}-2\,t+1}},{\frac {st-2\,t}{{t}^{2}-2\,t+1}}\right).$$ The Gröbner basis of $\sqrt{I}$ w.r.t. the lexicographical ordering $t>s$ is $\{s-2, t-1\}$; indeed ${\mathcal{P}}$ has the affine base point $(2,1)$. So the interpolating polynomial is $f(s)=1$. Therefore, ${\mathcal{P}}(s,1/t+1)$ does not have affine base points.
2. [Example 6 in [@Wang].]{} The parametrization is $${\mathcal{P}}=\left( {\frac {s \left( s+t-1 \right) }{{s}^{2}+st+{t}^{2}-1}},{\frac {t
\left( s+t-1 \right) }{{s}^{2}+st+{t}^{2}-1}},{\frac {s+t-1}{{s}^{2}+
st+{t}^{2}-1}}\right).$$ The Gröbner basis of $\sqrt{I}$ w.r.t. the lexicographical ordering $t>s$ is $\{s^2-s, s+t-1\}$; indeed ${\mathcal{P}}$ has the affine base points $(0,1),(1,0)$. So the interpolating polynomial is $f(s)=1-s$. Therefore, ${\mathcal{P}}(s,1/t+(1-s))$ does not have affine base points.
3. [Example 9 in [@Wang].]{} The parametrization is $$\begin{array}{lll}
{\mathcal{P}}&= &\left( {\dfrac {{s}^{2}t+2\,{t}^{3}+{s}^{2}+4\,st+4\,{t}^{2}+3\,s+2\,t+2}{{s}
^{3}+{s}^{2}t+{t}^{3}+{s}^{2}+{t}^{2}-s-t-1}},\right. \\ \\
& &\left. {\dfrac {-{s}^{3}-2\,s{t}
^{2}-2\,{s}^{2}-st+s-2\,t+2}{{s}^{3}+{s}^{2}t+{t}^{3}+{s}^{2}+{t}^{2}-
s-t-1}},\right.\\ \\
&& \left.{\dfrac {-{s}^{3}-2\,{s}^{2}t-3\,s{t}^{2}-3\,{s}^{2}-3\,st+2\,{
t}^{2}-2\,s-2\,t}{{s}^{3}+{s}^{2}t+{t}^{3}+{s}^{2}+{t}^{2}-s-t-1}}\right). \end{array}$$ The Gröbner basis of $\sqrt{I}$ w.r.t. the lexicographical ordering $t>s$ is $$\begin{array}{l}
\{9\,{s}^{6}+8\,{s}^{5}-12\,{s}^{4}+27\,{s}^{3}+34\,{s}^{2}-44\,s-40,\\
1665\,{s}^{5}+382\,{s}^{4}-2152\,{s}^{3}+4939\,{s}^{2}+1540\,s+3288\,t
-4268\}.
\end{array}$$ So ${\mathcal{P}}$ has 6 affine base points, and the interpolation polynomial is $$f(s)=-{\frac {555}{1096}}\,{s}^{5}-{\frac {191}{1644}}\,{s}^{4}+{\frac {269
}{411}}\,{s}^{3}-{\frac {4939}{3288}}\,{s}^{2}-{\frac {385}{822}}\,s+{
\frac {1067}{822}}.$$ Therefore, ${\mathcal{P}}(s,1/t+f(s))$ does not have affine base points.
4. [Example 10 in [@Wang].]{} The parametrization is $$\begin{array}{lll}
{\mathcal{P}}&=&\left( {\dfrac {-{s}^{4}+4\,{s}^{3}t-2\,{s}^{2}{t}^{2}+s{t}^{3}+{s}^{2}t-2\,{
t}^{3}}{-{s}^{3}t+6\,{s}^{2}{t}^{2}-3\,s{t}^{3}+{t}^{4}+{s}^{3}-2\,s{t
}^{2}}},\right.\\ \\ && {\dfrac {-{s}^{3}t-2\,{s}^{3}+{s}^{2}t+3\,s{t}^{2}-{t}^{3}}{-{s
}^{3}t+6\,{s}^{2}{t}^{2}-3\,s{t}^{3}+{t}^{4}+{s}^{3}-2\,s{t}^{2}}}, \\ \\
&&
\left.{
\dfrac {-s{t}^{3}+{s}^{3}-4\,{s}^{2}t-s{t}^{2}+6\,{t}^{3}}{-{s}^{3}t+6
\,{s}^{2}{t}^{2}-3\,s{t}^{3}+{t}^{4}+{s}^{3}-2\,s{t}^{2}}}\right).\end{array}$$ The Gröbner basis of $\sqrt{I}$ w.r.t. the lexicographical ordering $t>s$ is $$\begin{array}{l}
\{{s}^{6}-7\,{s}^{5}-20\,{s}^{4}+173\,{s}^{3}-27\,{s}^{2}+s,\\-176\,{s}^{
5}+1205\,{s}^{4}+3605\,{s}^{3}-29867\,{s}^{2}+2371\,s+703\,t\}.
\end{array}$$ So ${\mathcal{P}}$ has 6 affine base points, and the interpolation polynomial is $$f(s)={\frac {176}{703}}\,{s}^{5}-{\frac {1205}{703}}\,{s}^{4}-{\frac {3605}
{703}}\,{s}^{3}+{\frac {29867}{703}}\,{s}^{2}-{\frac {2371}{703}}\,s.$$ Therefore, ${\mathcal{P}}(s,1/t+f(s))$ does not have affine base points.
Acknowledgements
================
This work was developed, and partially supported, by the Spanish *Ministerio de Econom[í]{}a y Competitividad* under Project MTM2011-25816-C02-01; as well as Junta de Extremadura and FEDER funds (group FQM024). The first and third authors are members of the Research Group ASYNACS (Ref. CCEE2011/R34). The second author is a member of the research group GADAC (U. of Extremadura).
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|
---
abstract: 'Calculations from stellar evolutionary models of low- and intermediate-mass asymptotic giant branch (AGB) stars provide predictions of elemental abundances and yields for comparison to observations. However, there are many uncertainties that reduce the accuracy of these predictions. One such uncertainty involves the treatment of low-temperature molecular opacities that account for the surface abundance variations of C, N, and O. A number of prior calculations of intermediate-mass AGB stellar models that incorporate both efficient third dredge-up and hot bottom burning include a molecular opacity treatment which does not consider the depletion of C and O due to hot bottom burning. Here we update the molecular opacity treatment and investigate the effect of this improvement on calculations of intermediate-mass AGB stellar models. We perform tests on two masses, 5 ${\ensuremath{{\rm M}_{\odot}}}$ and 6 ${\ensuremath{{\rm M}_{\odot}}}$, and two metallicities, $Z~=~0.001$ and $Z~=~0.02$, to quantify the variations between two opacity treatments. We find that several evolutionary properties (e.g. radius, $T_{\rm eff}$ and $T_{\rm bce}$) are dependent on the opacity treatment. Larger structural differences occur for the $Z~=~0.001$ models compared to the $Z~=~0.02$ models indicating that the opacity treatment has a more significant effect at lower metallicity. As a consequence of the structural changes, the predictions of isotopic yields are slightly affected with most isotopes experiencing changes up to 60 per cent for the $Z~=~0.001$ models and 20 per cent for the $Z~=~0.02$ models. Despite this moderate effect, we conclude that it is more fitting to use variable molecular opacities for models undergoing hot bottom burning.'
author:
- |
C. K. Fishlock$^{1}$[^1], A. I. Karakas$^{1}$ and R. J. Stancliffe$^{1,2}$\
$^{1}$Research School of Astronomy & Astrophysics, Australian National University, Canberra ACT 2611, Australia\
$^{2}$Argelander-Institut für Astronomie, Auf dem Hügel 71, D-53121 Bonn, Germany
date: 'Accepted 2013 November 29. Received 2013 November 28; in original form 2013 April 3'
title: 'The effect of including molecular opacities of variable composition on the evolution of intermediate-mass AGB stars'
---
\[firstpage\]
nuclear reactions, nucleosynthesis, abundances - stars: AGB and post-AGB - stars: Population II - ISM: abundances.
Introduction
============
The asymptotic giant branch (AGB) phase is the last nuclear burning stage for low- and intermediate-mass stars with initial masses of between approximately 0.8 ${\ensuremath{{\rm M}_{\odot}}}$ to $\sim 8$ ${\ensuremath{{\rm M}_{\odot}}}$ [see @Herwig05 for a review]. During their evolution on the AGB, stars in this mass range experience mixing episodes that dredge up newly synthesised material which contains the products of He-burning and neutron-capture nucleosynthesis into the convective envelope [see @Busso99 for a review]. Combined with strong mass loss that ejects the stellar envelope into the interstellar medium, low- and intermediate-mass AGB stars are important contributors to the chemical evolution of the Galaxy [for example, @Romano10; @Kobayashi11; @RecioBlanco12]. This study is concerned with the evolution of intermediate-mass stars on the AGB with an initial mass $\gtrsim$ 4 ${\ensuremath{{\rm M}_{\odot}}}$. These stars produce substantial primary nitrogen and are predicted to produce nitrogen enhanced metal poor stars in the early Galaxy [e.g. @Pols12]. Intermediate-mass AGB stars are also implicated in the chemical evolution of globular clusters [@Gratton04; @Ventura11]. However, the modelling of the evolution and nucleosynthesis of low- and intermediate-mass AGB stars is dependent on highly uncertain input physics including convection, mass loss, reaction rates and opacity.
Briefly, the structure of an AGB star consists of an electron-degenerate CO core surrounded by a He-burning shell and a H-burning shell which are separated by the He-intershell. This is encompassed by a deep convective envelope. During the thermally-pulsing AGB (TP-AGB) phase, the star undergoes periodic thermal pulses (TPs) due to instabilities in the He-burning shell [@Herwig05; @Busso99]. During the evolution of a TP-AGB star, two known mechanisms can occur which can periodically alter the surface composition: third dredge-up (TDU) and hot bottom burning (HBB). When the He-burning shell becomes unstable it drives an expansion of the star in order to release the energy generated by He-burning. A flash driven convective zone forms in the He-intershell mixing material, mainly $^{12}$C, from the base of the He-shell throughout the intershell region. The outer layers have been cooled by the expansion and the H-burning shell has been almost extinguished which allows the outer convective envelope to move inwards. If the convective envelope reaches into the He-intershell, $^{12}$C and the products of slow neutron-capture nucleosynthesis (the $s$-process) are mixed to the surface. This process of TDU can occur multiple times and is responsible for creating carbon-rich stars with a C/O ratio greater than unity.
HBB occurs in stars with masses greater than approximately 4 ${\ensuremath{{\rm M}_{\odot}}}$ depending on metallicity and the input physics used. HBB takes place during the interpulse phase between TPs when the base of the outer convective envelope penetrates into the H-burning shell and becomes hot enough to sustain proton-capture nucleosynthesis. The CNO cycle converts $^{12}$C and $^{16}$O that has been mixed into the envelope by TDU into predominately $^{14}$N. Therefore HBB can prevent the stellar surface from becoming carbon rich by decreasing the $^{12}$C abundance in the envelope. @Frost98 presented evolutionary calculations of intermediate-mass AGB stellar models and found that the surface C/O ratio initially decreases due to HBB. However, mass loss slowly erodes the envelope causing a decrease in envelope mass. This results in a lower temperature at the base of the convective envelope which causes HBB to cease. There is also less dilution of the dredged-up material. This means that the C/O ratio starts to increase at the end of the TP-AGB phase, and in some cases to reach above unity.
Mass loss and convection have been shown to dominate modelling uncertainties in intermediate-mass AGB models [@Ventura05a; @Ventura05b; @Stancliffe07; @Karakas12]. However, other uncertainties including reaction rates and opacities have been shown to affect the stellar structure and therefore the yields [@Izzard07; @Ventura09]. In recent years there has been considerable work developing accurate low-temperature molecular opacities for stellar evolution calculations [@Marigo09]. Due to these improvements to the opacity input physics it is now possible to quantify the effect of the updated opacities on the stellar evolution calculations and yield predictions [@Ventura09; @Weiss09].
Intermediate-mass AGB stars have been shown to have effective temperatures cool enough for dust and molecule formation at solar metallicity and in the Magellanic Clouds [@GarciaHernandez09]. In particular, we also show in this study that low-metallicity AGB stars become cool enough to form molecules particularly once they become carbon rich. The opacity tables of @Alexander94, and later @Ferguson05, include a detailed treatment of the inclusion of molecules to the total opacity at low temperatures where $T \lesssim 10^4$ K. These tables, however, are only available for solar or scaled-solar composition. As previously mentioned, low- and intermediate-mass AGB stars undergo mixing episodes that alter their surface composition in a complex way. In particular, low-mass AGB stars can become carbon rich and intermediate-mass stars can display a range of behaviours for the C/O ratio. @Marigo02 shows that, at the transition point when the C/O ratio goes from below unity to above unity, the dominant source of molecular opacity changes from oxygen-bearing molecules to carbon-bearing molecules. In AGB models, this causes a sudden decrease of the effective temperature and an expansion in radius which in turn increases the rate of mass loss. It is therefore necessary to use low-temperature molecular opacities that follow the change in the C/O ratio with time. The low-temperature opacity tables of @Lederer09 only account for an enhancement in C and N compared to initial abundances whereas the ÆSOPUS opacity tables of @Marigo09 are able to account for the depletion and enhancement of C, N and the C/O ratio.
Using the ÆSOPUS molecular opacity tables, the effects of molecular opacities on the evolution of AGB stellar models have been investigated by @Ventura09 and @Ventura10. Using two different opacity treatments they determine that the yields of C, N, O and Na can be significantly altered depending on the opacity prescription used. @Ventura10 stress that it is important to use a molecular opacity treatment that accounts for variations in the surface CNO abundance when the C/O ratio exceeds unity. They find a minimum threshold mass for AGB stellar models ($\geq$ 3.5 ${\ensuremath{{\rm M}_{\odot}}}$ for $Z~=~0.001$) where the use of opacity tables that account for the variations in the surface CNO abundance becomes less important. This is because HBB prevents the C/O ratio from exceeding unity for all their models with a mass greater than 3.5 ${\ensuremath{{\rm M}_{\odot}}}$ rather than the models not being cool enough to form molecules. The models of @Ventura10 have a very efficient HBB owing to their choice of convective model, the Full Spectrum of Turbulence prescription [FST, @Canuto91] and little TDU due to how they treat the convective borders.
As intermediate-mass AGB stars can experience both TDU and HBB, the complex interplay between the two processes can cause the star to become either oxygen rich or carbon rich. Stellar models calculated with different stellar evolution codes can display varied behaviours including differences in the efficiency of TDU and HBB which can alter the surface C/O ratio allowing for either carbon- or oxygen-rich compositions. This in turn means that the effect of the molecular opacities on the stellar evolution can vary greatly depending on model assumptions. In contrast, low-mass AGB stars only experience TDU which serves to increase the surface C/O ratio.
The aim of this paper is to expand upon the work of Ventura & Marigo using intermediate-mass AGB models that show a different evolution of the surface C/O ratio. In particular, our low-metallicity models show very deep dredge up and become carbon rich during the AGB phase in contrast to the models of, for example, @Ventura10. Therefore, their conclusions are not necessarily applicable to our models. The paper is presented as follows. In Section 2 we describe the method of calculation we use to model an AGB star and detail the molecular opacities used in the investigation. In Section 3 we present the results of the stellar models and in Section 4 we present the effect on the yields between the two opacity treatments. Section 5 we compare with previous studies and summarise our results.
Numerical method
================
A two step procedure is used to calculate each AGB stellar model. Firstly, we use the Mt Stromlo stellar evolutionary code [@Karakas10 and references therein] to calculate the stellar evolutionary sequences. Each stellar model was evolved from the zero-age main sequence to near the end of the TP-AGB phase when the majority of the convective envelope is lost by strong stellar winds. For convective regions we use the standard mixing length theory [@Bohm-Vitense58] with a mixing length parameter of $\alpha= 1.86$. The details of the procedure and the evolution code is described in @Karakas10 with the following differences. We update the low-temperature molecular opacities (described in detail in Section \[sec:opacity\]) and the high-temperature radiative opacity tables. To be consistent with the molecular opacity tables, we use OPAL radiative tables [@Iglesias96] with a @Lodders03 scaled-solar abundance. We do not include convective overshoot in the formal sense in order to obtain the third dredge-up. However, we follow the method described by @Lattanzio86 and @Frost96 to determine a neutral border to each convective boundary. In this method, the ratio of the temperature gradients is linearly extrapolated from the last two convective mesh points to the first radiative point. If the extrapolated value is greater than unity then the point is determined to be in the convective zone, if less than unity the point is considered to be in the radiative zone. This means that only one point can be added, per iteration, to the convective zone.
The calculated stellar evolutionary sequences are used as input into a post-processing nucleosynthesis code [see @Cannon93; @Lugaro04]. The nuclear network we use is based on the JINA Reaclib[^2] database as of May 2012 [@Cyburt10]. It includes 589 reactions of 77 species from hydrogen to sulphur with a small group of iron-peak elements (Fe, Co and Ni). An additional ‘species’ $g$ is included in the network to account for the number of neutron captures occurring beyond $^{62}$Ni; this $g$ species simulates the $s$-process as a neutron sink. We use a scaled-solar initial composition from @Asplund09 and assume a solar global metallicity of $Z_{\odot} = 0.015$ which is comparable to the solar global metallicity recommended by @Asplund09 of $Z_{\odot} = 0.0142$. This differs from the @Lodders03 solar metallicity of 0.01321 in the opacity tables as low-temperature opacity tables using the @Asplund09 values are not available. Throughout this paper, all abundance ratios are by number whereas individual species are in mass fraction. We assume the envelope abundances are equivalent to surface abundances.
Mass loss
---------
We assume the same mass-loss formula for all our models. Mass loss prior to the AGB phase is included using the @Reimers75 formula with $\eta_R = 0.4$. This leads to very little mass being lost before the AGB phase. For example, the intermediate-mass models presented here lose less than 1 per cent of their initial mass before they reach the early AGB phase.
Mass loss is included during the AGB phase using the @Vassiliadis93 mass-loss prescription where only their equation 2 is used. The empirical formula of @Vassiliadis93 was determined using a sample of oxygen- and carbon-rich AGB stars in the Milky Way and Magellanic Clouds which cover a range of luminosities and pulsation periods.
Molecular opacities {#sec:opacity}
-------------------
Bound-bound absorption by molecules becomes an important contribution to the total opacity at temperatures below 5000 K [@Alexander94]. The abundance of carbon relative to oxygen in the stellar envelope can significantly influence the formation of molecules and has a substantial effect on the opacity at low temperatures. Changes in the molecular opacity occur when the stellar envelope transitions from an oxygen-rich (C/O $<$ 1) to a carbon-rich (C/O $>$ 1) chemical composition [@Marigo09]. This is due to the excess of carbon atoms allowing for the formation of carbon-bearing molecules such as, for example, HCN, CN, C$_2$ and SiC.
The intermediate-mass stellar models of @Karakas12 use low-temperature molecular opacity tables from @Lederer09 that account for only an increase in C due to TDU. For low-mass AGB stellar models that do not undergo HBB this method is sufficient as TDU will only increase the C abundance in the envelope. However, for intermediate-mass stars, CNO cycling during HBB causes a decrease in the C abundance (as well as the O abundance) in the envelope and it is possible that the C abundance will become lower than the initial C abundance. The subsequent decrease in O along with C can still cause the star to have a C/O ratio above unity. The opacity tables used in @Karakas12 do not account for this. In order to account for these variations we include low-temperature opacity tables to follow the decrease in C and O due to HBB.
For the stellar models presented here we use the ÆSOPUS low-temperature molecular opacity tables [@Marigo09] with a @Lodders03 scaled-solar abundance. The low-temperature opacity tables have been calculated to follow the variations in the chemical composition of C, N and O in the envelope due to TDU and HBB. The tables use Rosseland mean opacities and are a function of temperature log($T$) and log($R$) where $R=\rho/(T/10^6~{\rm K})^3$ for an arbitrary chemical composition [@Marigo09]. We use linear interpolation between tables with log($T$) from 3.2 to 4.05 in steps of 0.05 dex and log($R$) from $-$7.0 to 1.0 in steps of 0.5 dex.
To account for the changes in C, N and O in the envelope due to TDU and HBB, a variation factor $f_i$ is used,
$$X_i = f_iX_{i,{\rm ref}},$$
where $X_i$ is the current abundance of species $i$ in mass fraction and $X_{i,{\rm ref}}$ is the initial reference abundance of species $i$. A value of $f_i > 1$ indicates an enhancement in the abundance whereas $f_i < 1$ indicates a depletion in the abundance compared to the initial reference abundance. The increase (or decrease) in abundance for C, N and O can be described using the following equations from @Ventura09:
$$\left(\frac{X_C}{X_O}\right) = f_{C/O}\left(\frac{X_{C\rm{,ref}}}{X_{O\rm{,ref}}}\right),$$
$$X_C = f_C~ X_{C\rm{,ref}},$$
$$X_N = f_N~ X_{N\rm{,ref}}.$$
The helium abundance is then given by $Y = 1 - X - Z$ so that the composition conserves mass where $X$ is the hydrogen abundance and $Z$ is the global metallicity.
We perform tests using two opacity treatments: one which only accounts for the increase in C (referred to as $\kappa_{\rm C}$) and another where the increase and decrease in C and O are accounted for (referred to as $\kappa_{\rm CO}$). The $f_i$ values used in this study for C, N and C/O are detailed in Table \[tab:opacity-tab\]. We use the same scaled-solar ÆSOPUS opacity tables as @Kamath12 for the $\kappa_{\rm C}$ models. In the $\kappa_{\rm CO}$ models we incorporate two additional variation factors for C and the C/O ratio and explicitly follow the envelope C/O ratio to account for a possible reduction in C and O abundances due to HBB for the $\kappa_{\rm CO}$ models. The variations in abundance of N are treated the same for all models.
[llccccc]{} $Z$ & $X$ & log($f_{\rm C/O}$) & log($f_{\rm C}$)& log($f_{\rm N}$) & C/O\
& & & &\
0.001 & 0.5 & [**-1.00**]{} & [**-1.00**]{} & 0.00 & [**0.050**]{}\
&0.6 & [**-0.35**]{} & [**-0.35**]{} & 0.60 & [**0.224**]{}\
&0.7 & 0.00 & 0.00 & 1.20 & 0.501\
&0.8 & 0.17 & 0.17 & & 0.741\
& & 0.30 & 0.30 & & 1.000\
& & 0.35 & 0.35 & & 1.122\
&& 0.90 & 0.90 & & 3.980\
& &1.55 & 1.55 & & 17.779\
& &2.17 & 2.17 & & 74.114\
\
0.02 & 0.5 & [**-1.00**]{} & [**-1.00**]{} & 0.00 & [**0.050**]{}\
&0.6 & [**-0.35**]{} & [**-0.35**]{} & 0.60 & [**0.224**]{}\
&0.7& 0.00 & 0.00 & 0.30 & 0.501\
&0.8 & 0.17 & 0.17 & & 0.741\
&& 0.25 & 0.25 & & 0.891\
&& 0.30 & 0.30 & & 1.000\
&& 0.35 & 0.35 & & 1.122\
&& 0.50 & 0.50 & & 1.585\
&& 0.70 & 0.70& & 2.511\
\
Stellar models
==============
Stellar structure
-----------------
We model two stellar masses, namely 5 ${\ensuremath{{\rm M}_{\odot}}}$ and 6 ${\ensuremath{{\rm M}_{\odot}}}$, with two metallicities, $Z~=~0.001$ and $Z~=~0.02$, to determine the effect of the two different opacity treatments, $\kappa_{\rm C}$ and $\kappa_{\rm CO}$. All other input parameters in the stellar evolution code are kept to be the same so any differences in the stellar structure can be attributed to the different treatments of molecular opacity. There are negligible differences in the evolution of the stellar structure during the pre-AGB phase between the two opacity treatments and therefore only differences in the stellar models during the AGB phase will be discussed. Relevant properties of the stellar structure calculations for each of the models are summarised in Table \[tab:strucresults\] and includes the number of TPs calculated, final core mass (M$_{\rm core}$), final envelope mass (M$_{\rm env}$), maximum temperature reached at the base of the convective envelope ($T_{\rm bce}^{\rm max}$) and the total amount of material dredged-up due to TDU (M$_{\rm dred}^{\rm tot}$).
[ccccccccc]{} Opacity & Mass & $Z$ & TPs & Final M$_{\rm core}$& Final M$_{\rm env}$& $T_{\rm bce}^{\rm max}$ & M$_{\rm dred}^{\rm tot}$\
& (${\ensuremath{{\rm M}_{\odot}}}$) & & & (${\ensuremath{{\rm M}_{\odot}}}$) &(${\ensuremath{{\rm M}_{\odot}}}$) & ($\times 10^6$ K) & (${\ensuremath{{\rm M}_{\odot}}}$)\
$\kappa_{\rm C}$ & 5.0 & 0.001 & 108 &0.940 & 0.960 & 92.99 &0.216\
& 6.0 & 0.001 & 126 &1.018 & 0.888 & 104.91 & 0.127\
& 5.0 & 0.02 & 30 & 0.868 & 0.713& 63.29 & 0.088\
& 6.0 & 0.02 & 41 & 0.907& 1.084 & 81.59 & 0.090\
\
$\kappa_{\rm CO}$ & 5.0 & 0.001 & 96 & 0.938 & 1.035 & 92.47 & 0.192\
& 6.0 & 0.001 & 112 &1.015 & 0.662& 104.79 & 0.107\
& 5.0 & 0.02 & 29 & 0.867 & 1.013 & 63.95 & 0.083\
& 6.0 & 0.02 & 42 & 0.908 & 1.352 & 82.30 & 0.092\
\
### $Z~=~0.001$ models
Figure \[fig:Hexhaustedcore\] (panel $a$) shows the evolution of the H-exhausted core with time for the 5 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.001$ model. The H-exhausted core mass at the beginning of the TP-AGB phase is slightly higher for the $\kappa_{\rm C}$ models, around 0.001 ${\ensuremath{{\rm M}_{\odot}}}$ for the 5 ${\ensuremath{{\rm M}_{\odot}}}$ model. This can be attributed to minor differences during core He-burning evolution. It can be seen that the updated opacity treatment $\kappa_{\rm CO}$ reduces the number of TPs during the AGB phase for the $Z~=~0.001$ models. The 5 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.001$ $\kappa_{\rm CO}$ model has 12 fewer TPs while the 6 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.001$ $\kappa_{\rm CO}$ model has 14 fewer TPs when compared to the $\kappa_{\rm C}$ models. This means that less enriched material is dredged up to the surface as demonstrated in Table \[tab:strucresults\] where the total amount of He-intershell material decreases by around 15 per cent for the $\kappa_{\rm CO}$ models. The $Z~=~0.001$ $\kappa_{\rm C}$ models each dredge up approximately 0.02 ${\ensuremath{{\rm M}_{\odot}}}$ more material over the lifetime of the AGB phase compared to the $\kappa_{\rm CO}$ models.
![Evolution of the H-exhausted core (M$_{\rm H,core}$) from the start of the TP-AGB phase ($t=0$) for two of the stellar models calculated. The solid (black) lines show the $\kappa_{\rm C}$ models while the dashed (red) lines show the $\kappa_{\rm CO}$ models.[]{data-label="fig:Hexhaustedcore"}](figure1){width="\columnwidth"}
The evolution of the temperature at the base of the convective envelope for each model is shown in Figure \[fig:TatBCE\] (panels $a$ and $b$). HBB is evident as the temperature reaches higher than the $(50 - 80) \times 10^6$ K required for CNO cycling at the base of the convective envelope. Figure \[fig:TatBCE\] for the $Z~=~0.001$ models shows that, when using the updated opacity tables $\kappa_{\rm CO}$, HBB is extinguished earlier. This is because the $\kappa_{\rm CO}$ models become more extended in radius and therefore cooler. This change due to the increased molecular opacity leads to an enhanced mass-loss rate which ejects the envelope sooner.
![Evolution of the temperature at the base of the convective envelope from the start of the TP-AGB phase ($t=0$) for each stellar model calculated. The solid (black) lines show the $\kappa_{\rm C}$ models while the dashed (red) lines show the $\kappa_{\rm CO}$ models.[]{data-label="fig:TatBCE"}](figure2){width="\columnwidth"}
Figure \[fig:CO\] (panels $a$ and $b$) compares the evolution of the C/O ratio between the two opacity treatments for both $Z~=~0.001$ models. For a period during the AGB phase, the competing effects of TDU and HBB can cause the C/O ratio to fluctuate between below unity and above unity. For a brief time during this period the $^{12}$C surface abundance is below the initial $^{12}$C surface abundance while the C/O ratio is greater than unity. The molecular opacity is then underestimated in the $\kappa_{\rm C}$ model as the fact that the C/O ratio exceeds unity has not been taken into account during these conditions. This omission in the determination of the opacity results in the discrepancies seen in the stellar evolution between the $\kappa_{\rm C}$ and $\kappa_{\rm CO}$ models in Figure \[fig:CO-surf\]. Figure \[fig:CO-surf\] shows the temporal evolution of the C/O ratio, $^{12}$C surface abundance, $^{16}$O surface abundance, effective temperature and mass-loss rate for the 5 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.001$ models.
As mentioned by @Marigo02, the consequences of using opacity tables that do not correctly follow the variation of the C/O ratio when the star is carbon rich include an inaccurate effective temperature. Figure \[fig:CO-surf\] shows that the difference between the $\kappa_{\rm C}$ and $\kappa_{\rm CO}$ models is not due to a slightly different core mass at the beginning of the TP-AGB phase. The lower effective temperature (panel $d$) and higher mass-loss rate (panel $e$) for the $\kappa_{\rm CO}$ models starts to become significant when the surface C/O ratio for the models changes from below unity to above. This is where the $\kappa_{\rm CO}$ molecular opacity treatment more realistically follows the increase in the C/O ratio.
![The variation in the C/O ratio with time for each of the stellar models calculated from the start of the TP-AGB phase ($t=0$). The solid (black) lines show the $\kappa_{\rm C}$ models while the dashed (red) lines show the $\kappa_{\rm CO}$ models.[]{data-label="fig:CO"}](figure3){width="\columnwidth"}
![The 5 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.001$ stellar model showing the variation in ($a$) C/O, ($b$) $^{12}$C surface abundance, ($c$) $^{16}$O surface abundance, ($d$) log$_{10}$(T$_{\rm eff}$) and ($e$) log$_{10}$(M$_{\rm loss}$) with time from the start of the TP-AGB phase ($t=0$). The dashed (blue) line shows a C/O ratio of unity while the solid (blue) lines show the initial $^{12}$C and $^{16}$O surface abundance.[]{data-label="fig:CO-surf"}](figure4){width="\columnwidth"}
### $Z~=~0.02$ models
The effect of the different opacity treatment on the number of TPs is less pronounced for the $Z~=~0.02$ models than for the $Z~=~0.001$ models. Figure \[fig:Hexhaustedcore\] (panel $b$) shows that the difference in the evolution of the mass of the hydrogen exhausted core between the two opacity treatments is very slight for the 6 ${\ensuremath{{\rm M}_{\odot}}}$ model. The differences between the two opacity treatments are minimal and the C/O ratio of the models does not exceed unity which would result in a higher molecular opacity where carbon-bearing molecules dominate. Compared to the $Z~=~0.001$ models, there are fewer TPs with a shorter lifetime of the TP-AGB phase. The 5 ${\ensuremath{{\rm M}_{\odot}}}$ $\kappa_{\rm CO}$ model experiences one less TP while the 6 ${\ensuremath{{\rm M}_{\odot}}}$ $\kappa_{\rm CO}$ model experiences one more TP than the $\kappa_{\rm C}$ models. We conclude that the differences in the number of TPs is insignificant.
Figure \[fig:co\_m6z02\] shows the evolution of the C/O ratio, $^{12}$C surface abundance, $^{16}$O surface abundance, effective temperature and mass-loss rate for the TP-AGB phase for the 6 ${\ensuremath{{\rm M}_{\odot}}}$ model. Pre-AGB evolution mixing episodes cause the surface $^{12}$C and $^{16}$O abundances to decrease below initial values. This causes the C/O ratio to decrease with the C/O ratio being below the initial value at the start of the TP-AGB phase. HBB is only active for part of the TP-AGB evolution in the 6 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.02$ models. This is illustrated in Figure \[fig:co\_m6z02\] (panel $a$) where the C/O ratio initially increases, albeit with a shallow gradient and then decreases once the temperature at the base of the envelope becomes hot enough for proton captures onto $^{12}$C. Eventually HBB is extinguished and the C/O ratio increases to almost its initial value owing to the continuation of TDU. It is when HBB is active that the $\kappa_{\rm C}$ model does not accurately follow the variation of the C and O abundance as only the increase in the $^{12}$C abundance is taken into account.
For the $\kappa_{\rm C}$ 6 ${\ensuremath{{\rm M}_{\odot}}}$ model this situation where the current C/O, C and O abundances are never greater than the initial abundances is treated as if the surface abundance has a scaled-solar composition. The low-temperature molecular opacity does not vary for a given temperature and density with the variation factors being $f_C = f_O = 0$. Figure \[fig:opacity-m6z02\] shows the opacity below log($T$) $\lesssim$ 4 in the 6 ${\ensuremath{{\rm M}_{\odot}}}$ $\kappa_{\rm C}$ and $\kappa_{\rm CO}$ models for the same total stellar mass of approximately 2.3 ${\ensuremath{{\rm M}_{\odot}}}$. This snapshot was chosen where the opacity difference is largest. At log($T$) $\lesssim$ 3.5 near the stellar surface, the opacity is lower for the $\kappa_{\rm CO}$ model compared to the $\kappa_{\rm C}$ model. This decrease in opacity leads to a higher effective temperature (as seen in the bottom panel of Figure \[fig:co\_m6z02\]) and as a consequence the mass-loss rate is lower and HBB ceases slightly later. The sustained level of HBB causes a lower final C/O ratio compared to the $\kappa_{\rm C}$ models as seen in Figure \[fig:co\_m6z02\] (panel $a$).
![The 6 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.02$ stellar model showing the variation in ($a$) C/O, ($b$) $^{12}$C surface abundance, ($c$) $^{16}$O surface abundance, ($d$) log$_{10}$(T$_{\rm eff}$) and ($e$) log$_{10}$(M$_{\rm loss}$) with time from the start of the TP-AGB phase ($t=0$). The solid (blue) lines show the initial $^{12}$C and $^{16}$O surface abundance.[]{data-label="fig:co_m6z02"}](figure5){width="\columnwidth"}
![The difference in opacity $\kappa_R$ between the 6 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.02$ $\kappa_{\rm C}$ (solid black line) and $\kappa_{\rm CO}$ (dashed red line) models for the same total mass (approximately 2.3 ${\ensuremath{{\rm M}_{\odot}}}$). The opacity is noticeably different for log($T$) $\lesssim$ 3.5 near the stellar surface between the two opacity treatments. []{data-label="fig:opacity-m6z02"}](figure6){width="\columnwidth"}
Stellar yields
--------------
[cccccccccccc]{} Opacities & Mass & $Z$ & $^{12}$C & $^{13}$C & $^{14}$N & $^{16}$O & $^{22}$Ne & $^{23}$Na & $^{25}$Mg & $^{26}$Mg & $g$\
$\kappa_{\rm C}$ & 5.0 & 0.001 & 1.652(-3) & 5.718(-4) & 5.715(-2) & -8.273(-4) & 1.845(-3) & 4.725(-5) & 3.399(-4) & 2.866(-4) & 1.793(-6)\
& 6.0 & 0.001 & 9.784(-4) & 4.298(-4) & 3.806(-2) & -1.547(-3) & 5.404(-4) & 6.487(-6) & 2.197(-4) & 1.279(-4) & 1.241(-6)\
& 5.0 & 0.02 & 1.864(-3) & 3.476(-3) & 1.971(-2) & -2.831(-3) & 1.720(-3) & 9.558(-5) & 1.563(-4) & 8.990(-5) & 8.321(-7)\
& 6.0 & 0.02 & -1.112(-2) & 5.849(-4) & 4.305(-2) & -7.224(-3) & 1.637(-3) & 1.350(-4) & 2.039(-4) & 1.248(-4) & 1.268(-6)\
\
$\kappa_{\rm CO}$ & 5.0 & 0.001 & 2.101(-3) & 5.447(-4) & 5.030(-2) & -8.422(-4) & 1.495(-3) & 3.763(-5) & 2.991(-4) & 2.292(-4)& 1.473(-6)\
& 6.0 & 0.001 & 1.309(-3) & 3.877(-4) & 3.091(-2) & -1.621(-3) & 4.166(-4) & 2.717(-6) & 1.989(-4) & 8.397(-5) & 8.665(-7)\
& 5.0 & 0.02 & 3.780(-4) & 3.238(-3) & 2.156(-2) & -2.862(-3) & 1.691(-3) & 9.566(-5) & 1.512(-4) & 8.759(-5) & 8.183(-7)\
& 6.0 & 0.02 & -1.207(-2) & 5.589(-4) & 4.509(-2) & -7.817(-3) & 1.665(-3) & 1.332(-4) & 2.281(-4) & 1.239(-4) & 1.304(-6)\
\
In order to determine the total amount of material ejected into the interstellar medium during the lifetime of the stellar model, we calculate the net yield $M_i$ (in solar masses) of species $i$ to be,
$$M_i = \int_0^{\tau} \left [ X(i) - X_0(i) \right ] \frac{dM}{dt} dt,$$
where $dM/dt$ is the current mass-loss rate in ${\ensuremath{{\rm M}_{\odot}}}$ yr$^{-1}$, $X(i)$ and $X_0(i)$ are the current and initial mass fraction of species $i$, and $\tau$ is the total lifetime of the stellar model in years. After calculating the stellar yields we calculate the percentage change $\Delta M_i$ for each stable isotope where $M_{i,{\rm C}}$ is the yield for species $i$ using the $\kappa_{\rm C}$ models and $M_{i,{\rm CO}}$ is the yield for species $i$ using the $\kappa_{\rm CO}$ models. The percentage change in yields between the $\kappa_{\rm C}$ and $\kappa_{\rm CO}$ models for each mass and metallicity is shown in Figure \[fig:yields\]. The changes in surface abundance, and consequently the final net yield, are dependent on the amount of material dredged up from the He-intershell as well as the maximum temperature and duration of HBB. Yields for selected isotopes are presented in Table \[tab:yieldresults\] for the two opacity treatments, $\kappa_{\rm C}$ and $\kappa_{\rm CO}$.
### $Z~=~0.001$ yields
Figure \[fig:yields\] (panels $a$ and $b$) shows that the 5 ${\ensuremath{{\rm M}_{\odot}}}$ and 6 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.001$ models display a similar trend in yield differences between the two opacity treatments, $\kappa_{\rm C}$ and $\kappa_{\rm CO}$. Percentage changes of up to 30 per cent are seen for the 5 ${\ensuremath{{\rm M}_{\odot}}}$ model whereas the 6 ${\ensuremath{{\rm M}_{\odot}}}$ model shows percentage changes of up to 60 per cent in the yield. However the yield of $^{19}$F decreases by 250 per cent for the 5 ${\ensuremath{{\rm M}_{\odot}}}$ models which is not seen in the 6 ${\ensuremath{{\rm M}_{\odot}}}$ models. The reduced number of TDU episodes for the $\kappa_{\rm CO}$ influences the changes seen in the yields. A higher percentage change is seen in the yields for the 6 ${\ensuremath{{\rm M}_{\odot}}}$ model compared to the 5 ${\ensuremath{{\rm M}_{\odot}}}$ model. This is caused by a larger decrease in the number of TPs as well as the shorter duration of HBB between the $\kappa_{\rm C}$ and $\kappa_{\rm CO}$ models.
The isotopes $^{12}$C and $^{16}$O are produced in the intershell region through partial He-burning and are brought to the surface by repeated TDU. During the interpulse period, HBB produces $^{14}$N at the expense of $^{12}$C and $^{16}$O. Due to a shorter period of HBB for the $\kappa_{\rm CO}$ models, the $^{12}$C yield is higher despite these models dredging up less material (see Table \[tab:strucresults\]). Therefore the $^{14}$N yield is also lower for the $\kappa_{\rm CO}$ models for the same reason. The majority of $^{13}$C is produced during HBB through the CN cycle before being destroyed via the $^{13}$C(p,$\gamma$)$^{14}$N reaction. The $^{13}$C yield decreases for the $\kappa_{\rm CO}$ models due to a shorter period of HBB combined with the slightly lower temperature at the base of the convective envelope. Even though $^{16}$O is being dredged up to the surface the amount is much less than $^{12}$C as the intershell composition comprises of $\lesssim$ 1 per cent $^{16}$O and around 25 per cent $^{12}$C. The majority of the $^{16}$O is destroyed through HBB and the net yield of $^{16}$O is negative for both opacity treatments.
The isotope $^{22}$Ne is produced in the intershell via $\alpha$-captures onto $^{14}$N. The amount of $^{22}$Ne mixed into the envelope affects the yields of isotopes in the Ne-Na chain. For example, $^{23}$Na is formed by proton-captures onto $^{22}$Ne. As the $\kappa_{\rm CO}$ models experience fewer TDU episodes, the yields of $^{22}$Ne and $^{23}$Na are lower than for the $\kappa_{\rm C}$ models by around 20 per cent each for the 5 ${\ensuremath{{\rm M}_{\odot}}}$ model. For the 6 ${\ensuremath{{\rm M}_{\odot}}}$ $\kappa_{\rm CO}$ model the yield of $^{22}$Ne is lower by around 20 per cent and for $^{23}$Na, the yield decreases by around 60 per cent as seen in Figure \[fig:yields\]. The isotopes $^{25}$Mg and $^{26}$Mg are produced through the $^{22}$Ne($\alpha$,n)$^{25}$Mg and $^{22}$Ne($\alpha$,$\gamma$)$^{26}$Mg reactions which are activated at temperatures above $3 \times 10^8$ K. These conditions occur in the convective region that develops during a TP. Less synthesised $^{25}$Mg and $^{26}$Mg reaches the surface for the $\kappa_{\rm CO}$ models resulting in lower yields with an approximately 10 to 30 per cent decrease compared to the $\kappa_{\rm C}$ models.
Any change in the abundance of the iron group isotopes is due to neutron-capture through the $s$-process. The isotope $^{56}$Fe is used as a seed for the $s$-process during a convective TP where neutrons are released by the $^{22}$Ne($\alpha$,n)$^{25}$Mg reaction. Any neutron captures onto $^{56}$Fe or isotopes heavier than $^{56}$Fe are accounted for by the neutron sink $g$. Therefore $g$ can be thought of as the sum of abundances for the $s$-process isotopes. For the $\kappa_{\rm CO}$ models the yield of the neutron sink $g$ is lower for the $\kappa_{\rm CO}$ models compared to the $\kappa_{\rm C}$ models by 18 per cent for the 5 ${\ensuremath{{\rm M}_{\odot}}}$ model and 30 per cent for the 6 ${\ensuremath{{\rm M}_{\odot}}}$ model. These yield changes can all be attributed to the reduced number of TDU episodes.
### $Z~=~0.02$ yields
Figure \[fig:yields\] (panels *c* and *d*) shows yield differences of up to 20 per cent for lighter isotopes with an atomic mass up to about 18 including the CNO isotopes for the $Z~=~0.02$ models.. For $^{19}$F we find a difference of around 15 per cent for the 6 ${\ensuremath{{\rm M}_{\odot}}}$ model with a smaller difference for the 5 ${\ensuremath{{\rm M}_{\odot}}}$ model. For the intermediate-mass isotopes, which include here $^{22}$Ne, $^{23}$Na, $^{25}$M, $^{26}$Mg and $^{26}$Al$^g$ we find very small yield differences of less than 10 per cent for both the 5 ${\ensuremath{{\rm M}_{\odot}}}$ and 6 ${\ensuremath{{\rm M}_{\odot}}}$ models as illustrated in Figure \[fig:yields\]. There is a negligible effect on the net yield of the neutron sink $g$ with a percentage change of less than 5 per cent.
From Figure \[fig:yields\] we can see that the relative size of the yield changes between the $\kappa_{\rm C}$ and $\kappa_{\rm CO}$ models are smaller than for the $Z~=~0.001$ models. This is mainly because the $\kappa_{\rm C}$ and $\kappa_{\rm CO}$ $Z~=~0.02$ models experience almost the same number of TPs (as shown in Table \[tab:strucresults\]). In Table \[tab:yieldresults\] we see that the $^{12}$C yields of the 5 ${\ensuremath{{\rm M}_{\odot}}}$ models are positive compared to the negative $^{12}$C yields of the 6 ${\ensuremath{{\rm M}_{\odot}}}$ models. This indicates that there is an overall net production of carbon in the 5 ${\ensuremath{{\rm M}_{\odot}}}$ models. While the amount of material dredged into the envelope is similar in both the 5 ${\ensuremath{{\rm M}_{\odot}}}$ and 6 ${\ensuremath{{\rm M}_{\odot}}}$ models (e.g., see Table \[tab:strucresults\]), the maximum temperature at the base of the envelope in the 5 ${\ensuremath{{\rm M}_{\odot}}}$ is considerably lower. This means that HBB is not very efficient in the 5 ${\ensuremath{{\rm M}_{\odot}}}$ as shown in Figure \[fig:TatBCE\], where the maximum temperature is only $64 \times 10^6$ K compared to $82 \times 10^6$ K for the 6 ${\ensuremath{{\rm M}_{\odot}}}$ models and this temperature is only sustained for about 5 TPs. This means that the carbon dredged to the surface is not as efficiently burnt via the CNO cycles at the base of the envelope.
However, both $\kappa_{\rm CO}$ models have a decrease in the $^{12}$C yields compared to the $\kappa_{\rm C}$ models. In particular, the 5 ${\ensuremath{{\rm M}_{\odot}}}$ $\kappa_{\rm CO}$ model has a decrease of 80 per cent compared to the $\kappa_{\rm C}$ model. This decrease in $^{12}$C can be attributed to the longer duration, as well the higher temperatures, of HBB. This is despite the slight differences in the number of TDU episodes. The negative yield of $^{16}$O indicates that more is being destroyed through HBB than being mixed from the intershell region to the surface. The longer duration of HBB also explains the increase in the $^{14}$N yield as more $^{12}$C is burnt to produce $^{14}$N via the CNO cycles.
Discussion and Conclusions
==========================
The demand for accurate AGB models requires a more thorough understanding of the uncertainties in the input physics which play a role in determining the stellar evolution of AGB stars. Here we present new detailed evolutionary models of two masses, 5 ${\ensuremath{{\rm M}_{\odot}}}$ and 6 ${\ensuremath{{\rm M}_{\odot}}}$, and two metallicities, $Z~=~0.001$ and $Z~=~0.02$. These models are used to investigate the uncertainties in the stellar evolution when using two differing molecular opacity treatments, $\kappa_{\rm C}$ and $\kappa_{\rm CO}$, as well as investigating the effects of differing molecular opacity on stellar mass and metallicity. We determine that the AGB lifetime for the lower metallicity models is affected by the choice of the molecular opacity prescription. The increased molecular opacity has the effect of lowering the effective temperature in the low-metallicity AGB models which, in turn, has the consequence of increasing the mass-loss rate. This consequence is confirmed by @Marigo02 where the increase in the mass-loss rate due to increased molecular opacity serves to better explain the presence of carbon-rich stars in a population of Galactic giant stars. In addition, previous studies by @Ventura09 and @Ventura10 highlight the importance of using variable abundance molecular opacities only if the C/O ratio exceeds unity. For low-mass AGB models where HBB does not occur it is sufficient for the molecular opacity to only follow the increase in carbon due to TDU as with the $\kappa_{\rm C}$ opacity tables.
Previous studies, such as the stellar models by @Karakas12, calculate the evolution of intermediate-mass AGB stars using the Mt Stromlo stellar evolutionary code but utilise opacity tables that only account for an increase in carbon due to TDU. This situation has been modelled here using the $\kappa_{\rm C}$ opacity tables. For the case when the surface C/O ratio does not exceed the initial C/O ratio, the $\kappa_{\rm C}$ opacity is calculated as if scaled-solar opacity tables had been used. The $\kappa_{\rm CO}$ opacity tables are able to interpolate between lower values and as a consequence of the decreased opacity the stellar structure is affected. The 6 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.02$ model suffers from this omission of lower values below the scaled-solar composition in the low-temperature molecular opacity tables. However, as mentioned by @Marigo07, the evolution depends on the sensitivity of the mass-loss prescription to the effective temperature.
The studies by @Stancliffe04 and Pignatari et al. (2013, submitted) calculate $Z~=~0.02$ intermediate-mass AGB models which experience efficient TDU. Our study finds that with efficient TDU the surface C/O ratio for the low-metallicity models can exceed unity even with the presence of efficient HBB. The study by @Herwig04a of $Z~=~0.0001$ intermediate-mass AGB models also finds efficient TDU where the C/O ratio exceeds unity for periods of time despite efficient HBB. This is in contrast to @Ventura10 who find a threshold at $M~\geq~3.5~{\ensuremath{{\rm M}_{\odot}}}$ for a metallicity of $Z~=~0.001$ where, above this mass limit, the final C/O ratio of the models does not exceed unity. However, the Ventura & Marigo models use an $\alpha$-enhanced composition and this could affect the ability of the models to become carbon rich.
We also investigated the effect of the two different molecular opacity treatments on the stellar yield predictions for 77 species. As illustrated in Figure \[fig:yields\] the net yields of the low-metallicity models are affected by changes in the stellar evolution due to opacity. The changes in the net yields of the $Z~=~0.02$ models are negligible when using the updated $\kappa_{\rm CO}$ opacity treatment. This result indicates the possibility that the changes in yield could be more considerable at a lower metallicity than $Z~=~0.001$. The $Z~=~0.001$ models of @Marigo07 utilise synthetic TP-AGB models to investigate the effects of molecular opacity tables. @Marigo07 concludes that the use of variable molecular opacities should not considerably affect the stellar yields in massive AGB models with $Z \leq 0.001$ as effective temperatures may be so high that the molecules cannot form. We do not find this to be the case for our $Z~=~0.001$ models as the effective temperatures are comparable to low-mass AGB models at the same metallicity.
{width="2\columnwidth"}
The treatment of molecular opacity along with other input physics such as the choice of the mass-loss rate, reaction rates and treatment of convection all contribute to the uncertainty in AGB models. These uncertainties can have a larger effect at $Z~=~0.02$ when compared to the opacity treatment. @Ventura05a investigate the effect of the treatment of convection on a 5 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.001$ model. They find that the yields strongly depend on the convection model used with significant yield differences between MLT and FST. A previous study on the choice of mass-loss prescription by @Stancliffe07 finds yield changes of around 15 to 80 per cent for the light elements when investigating the effect on the yields for a 1.5 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.008$ model. These differences are comparable to the results presented in this paper for the $Z~=~0.001$ models. The investigation on mass loss by @Karakas06 finds yield changes of up to 90 per cent for a 5 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.0001$ model. When looking at uncertainties in reaction rates for the Ne-Na and Mg-Al chains, @Izzard07 find variations of up to two orders of magnitude for some species in the synthetic models. The models of @Karakas06 find maximum yield differences up to 350 per cent when looking at different $^{22}$Ne + $\alpha$ reaction rates for a 5 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.02$ model using the @Reimers75 mass-loss prescription which is different to that used in this study. The yield differences due to reaction rate uncertainties are significantly higher when compared to uncertainties found here due to the treatment of molecular opacities. @Marigo13 investigate a number of uncertainties including reaction rates, suppression of TDU and increasing the C and O abundance in the He-intershell for a 5 ${\ensuremath{{\rm M}_{\odot}}}$, $Z~=~0.001$ model. All the models tested become carbon rich and the evolution code is able to accurately follow the surface abundances of C, N and O as the molecular opacity is determined using ‘on-the-fly’ calculations rather than through the interpolation of tables. The evolution of the surface abundance of a number of light elements as well as the temperature at the base of the convective envelope is shown to be dependent on the tested input parameters. However, with all these sources of inaccuracies, mass loss and convection are thought to dominate the uncertainties in stellar modelling.
One use of stellar models is to provide theoretical predictions of chemical yields for comparison to observational data. Chemical yields are incorporated into chemical evolution models in order to understand the contribution of stellar populations to, for example, the Galaxy or globular clusters [@Kobayashi11; @Cescutti12]. We have shown that the yields of intermediate-mass AGB stars can be affected by the treatment of molecular opacity in quite a complex way with changes of up to 20 per cent for most isotopes. These results also show that the degree of the difference depends on mass and metallicity. Therefore it is more suitable to update current stellar models that experience HBB and become carbon rich to include molecular opacity tables that account for the changes in the surface abundances of C and O, as well as N, due to the CNO cycle that occurs during HBB.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are grateful to the referee for their helpful comments and careful reading of the paper. CKF thanks Anibal García-Hernández for his useful comments and suggestions. AIK is grateful for the support of the NCI National Facility at the ANU and thanks the ARC for support through a Future Fellowship (FT10100475). RJS is the recipient of a Sofja Kovalevskaja Award from the Alexander von Humboldt Foundation. This research has made use of NASA’s Astrophysics Data System.
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\[lastpage\]
[^1]: [email protected]
[^2]: https://groups.nscl.msu.edu/jina/reaclib/db/
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---
abstract: 'We prove lower bounds for the minimum distance of algebraic geometry codes over surfaces whose canonical divisor is either nef or anti-strictly nef and over surfaces without irreducible curves of small genus. We sharpen these lower bounds for surfaces whose arithmetic Picard number equals one, surfaces without curves with small self-intersection and fibered surfaces. Finally we specify our bounds to the case of surfaces of degree $d\geq 3$ embedded in $\mathbb{P}^3$.'
author:
- 'Yves Aubry, Elena Berardini, Fabien Herbaut and Marc Perret'
bibliography:
- 'references.bib'
title: Bounds on the minimum distance of algebraic geometry codes defined over some families of surfaces
---
[^1]
Introduction {#Intro}
============
The construction of Goppa codes over algebraic curves ([@Goppa]) has enabled Tsfaman, Vlăduţ and Zink to beat the Gilbert-Varshamov bound ([@Tsfasman_Vladut_Zink]). Since then, algebraic geometry codes over curves have been largely studied. Even though the same construction holds on varieties of higher dimension, the literature is less abundant in this context. However one can consult [@LittleHigher] for a survey of Little and [@soH] for an extensive use of intersection theory involving the Seshadri constant proposed by S. H. Hansen. Some work has also been undertaken in the direction of surfaces. Rational surfaces yielding to good codes were constructed by Couvreur in [@Couvreur] from some blow-ups of the plane and by Blache [*et al.*]{} in [@Blache] from Del Pezzo surfaces. Codes from cubic surfaces where studied by Voloch and Zarzar in [@Voloch_Zarzar], from toric surfaces by J. P. Hansen in [@hansen], from Hirzebruch surfaces by Nardi in [@Nardi], from ruled surfaces by one of the authors in [@Aubry] and from abelian surfaces by Haloui in [@Haloui] in the specific case of simple Jacobians of genus $2$ curves, and by the authors in [@Aubry_Berardini_Herbaut_Perret] for general abelian surfaces. Furthermore Voloch and Zarzar ([@Voloch_Zarzar], [@Zarzar]) and Little and Schenck ([@Little_Schenck]) have studied surfaces whose arithmetic Picard number is one.
The aim of this paper is to provide a study of the minimum distance $d(X, rH, S)$ of the algebraic geometry code ${\mathcal C}(X, rH, S)$ constructed from an algebraic surface $X$, a set $S$ of rational points on $X$, a rational effective ample divisor $H$ on $X$ avoiding $S$ and an integer $r>0$.
We prove in Section \[Codes\_over\_algebraic\_surfaces\] lower bounds for the minimum distance $d(X, rH, S)$ under some specific assumptions on the *geometry of the surface* itself. Two quite wide families of surfaces are studied. The first one is that of surfaces whose canonical divisor is either nef or anti-strictly nef. The second one consists of surfaces which do not contain irreducible curves of low genus. We obtain the following theorem, where we denote, as in the whole paper, the finite field with $q$ elements by $\mathbb{F}_q$ and the virtual arithmetic genus of a divisor $D$ by $\pi_D$, and where we set $m:=\lfloor 2\sqrt{q}\rfloor$.
(Theorem \[ourbound\] and Theorem \[theorem\_ell\]) Let $X$ be an absolutely irreducible smooth projective algebraic surface defined over $\mathbb{F}_q$ whose canonical divisor is denoted by $K_X$. Consider a set $S$ of rational points on $X$, a rational effective ample divisor $H$ avoiding $S$, and a positive integer $r$. In order to compare the following bounds, we set $$d^*(X, rH, S)\coloneqq \sharp S-rH^2(q+1+m)-m(\pi_{rH}-1).$$
- - If $K_X$ is nef, then $$d(X, rH, S) \geq d^*(X, rH, S).$$
- If $-K_X$ is strictly nef, then $$d(X, rH, S) \geq d^*(X, rH, S)+mr(\pi_{H}-1).$$
- If there exists an integer $\ell >0$ such that any ${\mathbb F}_q$-irreducible curve lying on $X$ and defined over ${\mathbb F}_q$ has arithmetic genus strictly greater than $\ell$, then $$d(X, rH, S) \geq d^*(X, rH, S)+\left(rH^2-\frac{\pi_{rH}-1}{\ell}\right)(q+1+m).$$
Inside both families, adding some extra geometric assumptions on the surface yields in Section \[Improvements\] to some improvements for these lower bounds. This is the case for surfaces whose arithmetic Picard number is one, for surfaces without irreducible curves defined over ${\mathbb F}_q$ with small self-intersection, so as for fibered surfaces. Theorems \[theorem\_fibration\] and \[theorem\_fibration\_two\] (that hold for fibered surfaces) improve the bounds of Theorems \[ourbound\] and \[theorem\_ell\] (that hold for the whole wide families). Indeed the bound on the minimum distance $d(X, rH, S)$ is increased by the non-negative *defect* $\delta(B) = q+1+mg_B-\sharp B({\mathbb F}_q)$ of the base curve $B$. Finally in Section \[Hypersurfaces\] we specify our bounds to the case of surfaces of degree $d\geq 3$ embedded in $\mathbb{P}^3$.
Characterizing surfaces that yield good codes seems to be a complex question. It is not the goal of our paper to produce good codes: we aim to give theoretical bounds on the minimum distance of algebraic geometry codes on general surfaces. However one can derive from our work one or two heuristics. Indeed, Theorem \[theorem\_ell\] suggests to look for surfaces with no curves of small genus and fibered surfaces provide natural examples of such surfaces (see Theorem \[theorem\_fibration\_two\]).
Background {#Backgrounds}
==========
Codes from algebraic surfaces are defined in the same way as on algebraic curves: we evaluate some functions with prescribed poles on some sets of rational points. Whereas the key tool for the study of the minimum distance in the $1$-dimensional case is the mere fact that a function has as many zeroes as poles, in the $2$-dimensional case most of the proofs rest on intersection theory.
We sum up in this section the few results on intersection theory we need. Following the authors cited in the Introduction we recall the definition of the algebraic geometry codes. We also recall quickly how the dimension of the code can be bounded from below under the assumption of the injectivity of the evaluation map. Then we prove a lemma that will be used in the course of the paper to bound from below the minimum distance of the code for several families of surfaces. Finally, we recall some results on the number of rational points on curves over finite fields.
Intersection theory {#Intersection_theory}
-------------------
Intersection theory has almost become a mainstream tool to study codes over surfaces (see [@Aubry], [@soH], [@Voloch_Zarzar], [@Zarzar], [@Little_Schenck], [@Aubry_Berardini_Herbaut_Perret]) and it is also central in our proofs. We do not recall here the classical definitions of the different equivalent classes of divisors and we refer the reader to [@Hartshorne §V] for a presentation. We denote by $\NS(X)$ the *arithmetic* Néron-Severi group of a smooth surface $X$ defined over ${\mathbb{F}_q}$ whose rank is called the arithmetic Picard number of $X$, or Picard number for short. Recall that a divisor $D$ on $X$ is said to be *nef* (respectively *strictly nef*) if $D.C\geq 0$ (respectively $D.C>0$) for any irreducible curve $C$ on $X$. A divisor $D$ is said to be *anti-ample* if $-D$ is ample, *anti-nef* if $-D$ is nef and *anti-strictly nef* if $-D$ is strictly nef. Let us emphasize three classical results we will use in this paper.
The first one is (a generalisation of) the adjunction formula (see [@Hartshorne §V, Exercise 1.3]). For any ${\mathbb F}_q$-irreducible curve $D$ on $X$ of arithmetic genus $\pi_D$, we have $$\label{adjunction_formula}
D.(D+K_X)=2\pi_D-2$$ where $K_X$ is the canonical divisor on $X$. This formula allows us to define the virtual arithmetic genus of any divisor $D$ on $X$.
The second one is the corollary of the Hodge index theorem stating that if $H$ and $D$ are two divisors on $X$ with $H$ ample, then $$\label{Hodge}
H^2D^2 \leq (H.D)^2,$$ where equality holds if and only if $H$ and $D$ are numerically proportional.
The last one is a simple outcome of Bézout’s theorem in projective spaces (and the trivial part of the Nakai-Moishezon criterion). It ensures that for any ample divisor $H$ on $X$ and for any irreducible curve $C$ on $X$, we have $H^2>0$ and $H.C>0$.
Algebraic geometry codes {#section_eval_code}
------------------------
### Definition of AG codes
We study, as in the non-exhaustive list of papers [@Aubry], [@Voloch_Zarzar], [@Couvreur], [@soH], [@Zarzar], [@Haloui], [@Little_Schenck] and [@Aubry_Berardini_Herbaut_Perret], the generalisation of Goppa algebraic geometry codes from curves to surfaces. In the whole paper we consider an absolutely irreducible smooth projective algebraic surface $X$ defined over $\mathbb{F}_q$ and a set $S$ of rational points on $X$. Given a rational effective ample divisor $G$ on $X$ avoiding $S$, the algebraic geometry code, or AG code for short, is defined by evaluating the elements of the Riemann-Roch space $L(G)$ at the points of $S$. Precisely we define the linear code $\mathcal{C} (X,G, S)$ as the image of the evaluation map $\ev : L(G) \longrightarrow \mathbb{F}^{\sharp S}_q$.
### Length and dimension of AG codes
From the very definition, the length of the code is $\sharp S$. As soon as the morphism $\ev$ is injective - see (\[injsi>0\]) for a sufficient condition - the dimension of the code equals $\ell(G)=\dim_{\mathbb{F}_q}L(G)$ which can be easily bounded from below using standard algebraic geometry tools as follows. By Riemann-Roch theorem (see [@Hartshorne V, §1]), we have $$\ell(G)-s(G)+\ell (K_X-G)=\frac{1}{2}G.(G-K_X)+1+p_a(X)$$ where $p_a(X)$ is the arithmetic genus of $X$, and where the so-called [*superabundance*]{} $s(G)$ of $G$ in $X$ is non-negative (as it is the dimension of some vector space). Now, under the assumption that $$\label{condition_dim}
K_X.A< G.A,$$ for some ample divisor $A$, we have from [@Hartshorne V, Lemma 1.7] that $\ell (K_X-G)=0$. Thus, if the evaluation map $\ev$ is injective and under assumption (\[condition\_dim\]), we get the lower bound $$\label{lowerbound_dim}
\dim\mathcal{C}(X,G, S)=\ell(G)\geq \frac{1}{2}G.(G-K_X)+1+p_a(X)$$ for the dimension of the code $\mathcal{C}(X,G, S)$.
### Toward the minimum distance of AG codes {#toward_d}
It follows that the difficulty lies in the estimation of the minimum distance $d(X,G, S)$ of the code. For any non-zero $f \in L(G)$, we introduce the number $N(f)$ of rational points of the divisor of zeroes of $f$. The Hamming weight $w(\ev(f))$ of the codeword $\ev(f)$ satisfies $$\label{weight}
w(\ev(f)) \geq \sharp S -N(f),$$ from which it follows that $$\label{d>=}
d(X,G, S) \geq \sharp S-\max_{f\in L(G)\setminus \{0\}} N(f).$$ We also deduce from (\[weight\]) that $$\label{injsi>0}
\ev \hbox{~is injective if~} \max_{f\in L(G)\setminus \{0\}} N(f) < \sharp S.$$
We now broadly follow the way of [@Haloui]. We associate to any non-zero function $f\in L(G)$ the rational effective divisor $$\label{decomposition}
D_f:=G+(f)=\sum_{i=1}^k n_i D_i \geq 0,$$ where $(f)$ is the principal divisor defined by $f$, the $n_i$ are positive integers and each $D_i$ is a reduced ${\mathbb F}_q$-irreducible curve.
Note that in this setting, the integer $k$ and the curves $D_i$’s depend on $f \in L(G)$. Several lower bounds for the minimum distance $d(X, G, S)$ in this paper will follow from the key lemma below.
\[lemma\_d\] Let $X$ be a smooth projective surface defined over $\mathbb{F}_q$, let $S$ be a set of rational points on $X$ and let $G$ be a rational effective divisor on $X$ avoiding $S$. Set $m = \lfloor 2\sqrt{q}\rfloor$ and keep the notations introduced in (\[decomposition\]). If there exist non-negative real numbers $a, b_1, b_2, c$, such that for any non-zero $f \in L(G)$ the three following assumptions are satisfied
1. \[first\] $k \leq a$,
2. \[second\] $\sum_{i=1}^k \pi_{D_i} \leq b_1+kb_2$
3. \[third\] for any $1\leq i \leq k$ we have $ \sharp D_i(\mathbb{F}_q) \leq c+m\pi_{D_i}$
then the minimum distance $d(X, G, S)$ of ${\mathcal C}(X, G, S)$ satisfies $$d(X,G,S) \geq \sharp S-a(c+mb_2)-mb_1.$$
Let us write the principal divisor $(f)=(f)_0-(f)_{\infty}$ as the difference of its effective divisor of zeroes minus its effective divisor of poles. Since $G$ is effective and $f$ belongs to $L(G)$, we have $(f)_{\infty} \leq G$. Hence, formula (\[decomposition\]) reads $G+(f)_0-(f)_{\infty}=\sum_{i=1}^k n_i D_i$, that is $$(f)_0 = \sum_{i=1}^k n_iD_i +(f)_{\infty}-G \leq \sum_{i=1}^k n_iD_i.$$ This means that any ${\mathbb F}_q$-rational point of $(f)_0$ lies in some $D_i$ so $$\label{N(f)<=}
N(f) \leq \sum_{i=1}^k \sharp D_i(\mathbb{F}_q).$$ Then it follows successively from the assumptions of the lemma that $$N(f) \leq \sum_{i=1}^k (c+m\pi_{D_i})\leq kc+m(b_1+kb_2)\leq mb_1+a(c+mb_2).$$ Finally Lemma \[lemma\_d\] follows from (\[d>=\]).
In several papers, the point of departure to estimate the minimum distance is a bound on the number of components $k$, which corresponds to the condition (\[first\]) of Lemma \[lemma\_d\] above. In the special case where $\NS(S)=\mathbb{Z}H$ and $G=rH$, for $H$ an ample divisor on $X$, Voloch and Zarzar have proven in [@Voloch_Zarzar] that $k \leq r$. In the present paper we obtain a bound on $k$ in a more general context, that is when the Néron-Severi group has rank greater than one (see for example Lemma \[k\_HodgeTGV\_bounds\], point (\[k\_bound\_ell\]) of Lemma \[lemma\_ell\] and point (\[k\_beta\]) of Lemma \[lemma\_quadra\]).
Two upper bounds for the number of rational points on curves
------------------------------------------------------------
We manage to fulfill assumption (\[third\]) in Lemma \[lemma\_d\] using the bounds on the number of rational points given in Theorem \[irreduciblecurves\] and Proposition \[covering\] below. Point $(\ref{L-S})$ of Theorem \[irreduciblecurves\] appears in the proof of Theorem 3.3 of Little and Schenck in [@Little_Schenck] within a more restrictive context, whereas point $(\ref{A-P})$ follows from [@Aubry_Perret_Weil]. We state a general theorem and give here the full proof for the sake of completeness following [@Little_Schenck].
\[irreduciblecurves\] Let $D$ be an ${\mathbb F}_q$-irreducible curve of arithmetic genus $\pi_D$ lying on a smooth projective algebraic surface. Then,
1. \[A-P\] we have $\sharp D({\mathbb F}_q) \leq q+1+m\pi_D$.
2. \[L-S\](Little-Schenck) If moreover $D$ is *not* absolutely irreducible, we have $$\sharp D({\mathbb F}_q) \leq \pi_D+1.$$
We first prove the second item, following the proof of [@Little_Schenck Th. 3.3]. Since $D$ is ${\mathbb F}_q$-irreducible, the Galois group $\Gal(\overline{\mathbb{F}}_q/\mathbb{F}_q)$ acts transitively on the set of its ${\bar{r}} \geq 1$ absolutely irreducible components $D_1,\dots,D_{\bar{r}}$. Since a ${\mathbb F}_q$-rational point on $D$ is stable under the action of $\Gal(\overline{\mathbb{F}}_q/\mathbb{F}_q)$, it lies in the intersection $\cap_{1\leq i\leq \bar{r}}D_i$. Under the assumption that $D$ is not absolutely irreducible, that is ${\bar{r}}\geq 2$, it follows that $\sharp D(\mathbb{F}_q)\leq \sharp (D_i\cap D_j)(\overline{\mathbb{F}}_q)\leq D_i.D_j$ for every couple $(i,j)$ with $i\neq j$.
As a divisor, $D$ can be written over $\overline{{\mathbb F}}_q$ as $D=\sum_{i=1}^{\bar{r}} a_iD_i$. By transitivity of the Galois action, we have $a_1=\cdots = a_{\bar{r}}=a$. Now since $D$ can be assumed to be reduced, we have $a=1$, so that finally $D=\sum_{i=1}^{\bar{r}} D_i$. Using the adjonction formula (\[adjunction\_formula\]) for $D$ and each $D_i$, and taking into account that $\pi_{D_i} \geq 0$ for any $i$, we get $$\begin{aligned}
2\pi_D-2&=(K_X+D).D\\
&=\sum_{i=1}^{\bar{r}} (K_X+D_i).D_i+\sum_{i\neq j} D_i.D_j\\
&=\sum_{i=1}^{\bar{r}}(2\pi_{D_i}-2)+\sum_{i\neq j} D_i.D_j\\
&\geq -2\bar{r}+\sum_{i\neq j} D_i.D_j.\end{aligned}$$ Since there are $\bar{r}(\bar{r}-1)$ pairs $(i, j)$ with $i\neq j$, we deduce that for at least one such pair $(i_0, j_0)$, we have $$D_{i_0}.D_{j_0} \leq \frac{2(\pi_D-1+\bar{r})}{\bar{r}(\bar{r}-1)}.$$ It is then easily checked that the left hand side of the former inequality is a decreasing function of $\bar{r} \geq 2$, so that we obtain $$\sharp D(\mathbb{F}_q)\leq D_{i_0}.D_{j_0} \leq \frac{2(\pi_D-1+2)}{2(2-1)}=\pi_D+1$$ and the second item is proved.
The first item follows from Aubry-Perret’s bound in [@Aubry_Perret_Weil] in case $D$ is absolutely irreducible, that is in case $\bar{r}=1$, and from the second item in case $D$ is not absolutely irreducible since $\pi_D+1\leq q+1+m\pi_D$.
The following bound will be usefull in Subsection \[Section\_fibration\] for the study of codes from fibered surfaces.
\[covering\] Let $C$ be a smooth projective absolutely irreducible curve of genus $g_C$ over ${\mathbb F}_q$ and $D$ be an ${\mathbb F}_q$-irreducible curve having $\bar{r}$ absolutely irreducible components $\overline{D}_1,\dots,\overline{D}_{\bar{r}}$. Suppose there exists a regular map $D \rightarrow C$ in which none absolutely irreducible component maps onto a point. Then $$|\sharp D(\mathbb{F}_q)-\sharp C(\mathbb{F}_q)|\leq (\overline{r}-1)q+m(\pi_D-g_C).$$
Since $C$ is smooth and none geometric component of $D$ maps onto a point, the map $D\rightarrow C$ is flat. Hence by [@Aubry_Perret_FFA Th.14] we have $$|\sharp D(\mathbb{F}_q)-\sharp C(\mathbb{F}_q)|\leq (\overline{r}-1)(q-1)+m\left(\sum_{i=1}^{\overline{r}} g_{\overline{D}_i} - g_C\right)+\Delta_D$$ where $\Delta_D=\sharp \tilde{D}(\overline{\mathbb{F}}_q)-\sharp D(\overline{\mathbb{F}}_q)$ with $\tilde{D}$ the normalization of $D$. The result follows from [@Aubry_Perret_FFA Lemma 2] where it is proved that $m\sum_{i=1}^{\overline{r}} g_{\overline{D}_i}+\Delta_D-\bar{r}+1\leq m\pi_D$.
The minimum distance of codes over some families of algebraic surfaces {#Codes_over_algebraic_surfaces}
======================================================================
We are unfortunately unable to fulfill simultaneously assumptions (\[first\]) and (\[second\]) of Lemma \[lemma\_d\] for general surfaces. So we focus on two families of algebraic surfaces where we do succeed. To begin with, let us fix some common notations.
We consider a rational effective ample divisor $H$ on the surface $X$ avoiding a set $S$ of rational points on $X$ and for a positive integer $r$ we consider $G=rH$. We study, in accordance to Section \[section\_eval\_code\], the evaluation code $\mathcal{C}(X,rH, S)$ and we denote by $d(X, rH, S)$ its minimum distance.
Surfaces whose canonical divisor is either nef or anti-strictly nef {#nef}
-------------------------------------------------------------------
We study in this section codes defined over surfaces such that either the canonical divisor $K_X$ is nef, or its opposite $-K_X$ is strictly nef. This family is quite large. It contains, for instance:
- surfaces whose canonical divisor $K_X$ is anti-ample.
- Minimal surfaces of Kodaira dimension $0$, for which the canonical divisor is numerically zero, hence nef. These are abelian surfaces, $K3$ surfaces, Enriques surfaces and hyperelliptic or quasi-hyperelliptic surfaces (see [@Bombieri_Mumford]).
- Minimal surfaces of Kodaira dimension $2$. These are the so called minimal surfaces of general type. For instance, surfaces in ${\mathbb P}^3$ of degree $d\geq 4$, without curves $C$ with $C^2=-1$, are minimal of general type.
- Surfaces whose arithmetic Picard number is one.
- Surfaces of degree $3$ embedded in $\mathbb{P}^3$.
The main theorem of this section (Theorem \[ourbound\]) rests mainly on the next lemma designed to fulfill assumptions (\[first\]) and (\[second\]) of Lemma \[lemma\_d\].
\[k\_HodgeTGV\_bounds\] Let $D=\sum^k_{i=1} n_iD_i$ be the decomposition as a sum of ${\mathbb F}_q$-irreducible and reduced curves of an effective divisor $D$ linearly equivalent to $rH$. Then we have:
1. \[k\_bound\] $k\leq rH^2;$
2. \[pointtwo\]
1. \[if\_nef\] if $K_X$ is nef, then $\sum_{i=1}^k \pi_{D_i} \leq \pi_{rH}-1+k$;
2. \[not\_nef\] if $-K_X$ is strictly nef, then $\sum_{i=1}^k \pi_{D_i} \leq \pi_{rH}-1 -\frac{1}{2}rH.K_X+\frac{1}{2}k$.
Using that $D$ is numerically equivalent to $rH$, that $n_i>0$ and $D_i.H>0$ for every $i=1,\dots,k$ since $H$ is ample, we prove item $(\ref{k_bound})$: $$rH.H=D.H=\sum^k_{i=1} n_iD_i.H\geq \sum^k_{i=1} D_i.H\geq k.$$
Now we apply inequality (\[Hodge\]) to $H$ and $D_i$ for every $i$, to get $D_i^2H^2\leq(D_i.H)^2$. We thus have, together with adjunction formula (\[adjunction\_formula\]) and inequality $H^2> 0$, $$\label{sum_pi}
\pi_{D_i}-1\leq (D_i.H)^2/2H^2+D_i.K_X/2.$$
To prove point \[if\_nef\] of item (\[pointtwo\]) we sum from $i=1$ to $k$ and thus obtain $$\label{demo}
\begin{split}
\sum_{i=1}^k \pi_{D_i}-k &\leq \frac{1}{2H^2}\sum_{i=1}^k(D_i.H)^2+\frac{1}{2}\sum_{i=1}^k D_i.K_X\\
&\leq \frac{1}{2H^2}\left(\sum_{i=1}^k n_iD_i.H \right)^2+\frac{1}{2}\sum_{i=1}^k n_iD_i.K_X\\
&\leq \frac{(rH.H)^2}{2H^2}+\frac{rH.K_X}{2}\\
&=\pi_{rH}-1,
\end{split}$$ where we use the positivity of the coefficients $n_i$, the numeric equivalence between $D$ and $\sum_{i=1}^k n_i D_i$, the fact that $H$ is ample and the hypothesis taken on $K_X$.
Under the hypothesis of point \[not\_nef\] we have $D_i.K_X \leq -1$. Replacing in the first line of (\[demo\]) gives $\sum_{i=1}^k \pi_{D_i}-k \leq \frac{1}{2H^2}\sum_{i=1}^k(D_i.H)^2 -\frac{k}{2} $. We conclude in the same way.
\[ourbound\] Let $H$ be a rational effective ample divisor on a surface $X$ avoiding a set $S$ of rational points and let $r$ be a positive integer. We set $$\label{d*}
d^*(X, rH, S)\coloneqq \sharp S-rH^2(q+1+m)-m(\pi_{rH}-1).$$
- If $K_X$ is nef, then $$d(X, rH, S) \geq d^*(X, rH, S).$$
- If $-K_X$ is strictly nef, then $$d(X, rH, S) \geq d^*(X, rH, S)+mr(\pi_{H}-1).$$
The theorem follows from Lemma \[lemma\_d\] for which assumptions (\[first\]) and (\[second\]) hold from Lemma \[k\_HodgeTGV\_bounds\] and assumption (\[third\]) holds from Theorem \[irreduciblecurves\].
Surfaces without irreducible curves of small genus {#Sec_without_small_genus_curves}
--------------------------------------------------
We consider in this section surfaces $X$ with the property that there exists an integer $\ell \geq 1$ such that any ${\mathbb F}_q$-irreducible curve $D$ lying on $X$ and defined over ${\mathbb F}_q$ has arithmetic genus $\pi_D\geq \ell +1$. It turns out that under this hypothesis, we can fulfill assumptions (\[first\]) and (\[second\]) of Lemma \[lemma\_d\] without any hypothesis on $K_X$ contrary to the setting of Section \[nef\].
Examples of surfaces with this property do exist. For instance:
- simple abelian surfaces satisfy this property for $\ell =1$ (see [@Aubry_Berardini_Herbaut_Perret] for abelian surfaces with this property for $\ell = 2$).
- Fibered surfaces on a smooth base curve $B$ of genus $g_B\geq 1$ and generic fiber of arithmetic genus $\pi_0 \geq 1$, and whose singular fibers are ${\mathbb F}_q$-irreducible, do satisfy this property for $\ell = \min(g_B, \pi_0)-1$.
- Smooth surfaces in ${\mathbb P}^3$ of degree $d$ whose arithmetic Picard group is generated by the class of an hyperplane section do satisfy this property for $\ell = \frac{(d-1)(d-2)}{2}-1$ (see Lemma \[genus\_on\_hypersurfaces\]).
\[lemma\_ell\] Let $X$ be a surface without ${\mathbb F}_q$-irreducible curves of arithmetic genus less than or equal to $\ell$ for $\ell$ a positive integer. Consider a rational effective ample divisor $H$ on $X$ and a positive integer $r$. Let $D=\sum^k_{i=1} n_iD_i$ be the decomposition as a sum of ${\mathbb F}_q$-irreducible and reduced curves of an effective divisor $D$ linearly equivalent to $rH$. Then we have
1. \[k\_bound\_ell\] $k\leq \frac{\pi_{rH}-1}{\ell}$;
2. \[sum\_pi\_i\_ell\] $\sum_{i=1}^k \pi_{D_i} \leq \pi_{rH}-1+k$.
In case $X$ falls in both families of Section \[nef\] and of this Section \[Sec\_without\_small\_genus\_curves\], the present new bound of the first item for $k$ is better than the one of Lemma \[k\_HodgeTGV\_bounds\] if and only if $\pi_{rH}-1<\ell rH^2$, that is if and only if $\ell>\frac{H.K_X}{2H^2}+\frac{r}{2}$. In the general setting, this inequality sometimes holds true, sometimes not. As a matter of example, supposed $K_X$ to be ample and let us consider $H=K_X$. In this setting the inequality holds if and only if $r< 2\ell - 1$.
By assumption, we have $0\leq \ell\leq \pi_{D_i}-1$ and $n_i \geq 1$ for any $1 \leq i \leq k$, hence using adjunction formula (\[adjunction\_formula\]), we have $$2\ell k \leq 2\sum_{i=1}^k (\pi_{D_i}-1)
\leq 2\sum_{i=1}^k n_i(\pi_{D_i}-1)
= \sum_{i=1}^k n_i D_i^2+\sum_{i=1}^k n_iD_i.K_X.$$ Moreover using (\[Hodge\]) and (\[decomposition\]), we get $$2\ell k
\leq \sum_{i=1}^k n_i \frac{(D_i.H)^2}{H^2}+\left(\sum_{i=1}^k n_iD_i\right).K_X
\leq \sum_{i=1}^k n_i^2 \frac{(D_i.H)^2}{H^2}+rH.K_X.$$ Since $H$ is ample, we obtain $$2\ell k
\leq \sum_{i, j=1}^k n_i n_j \frac{(D_i.H)(D_j.H)}{H^2}+rH.K_X
= \frac{(\sum_{i,=1}^k n_i D_i.H)^2}{H^2}+rH.K_X.$$ By (\[decomposition\]), we conclude that $$2\ell k\leq
\frac{(rH.H)^2}{H^2}+rH.K_X
=2(\pi_{rH}-1),$$ and both items of Lemma \[lemma\_ell\] follow.
\[theorem\_ell\] Let $X$ be a surface without ${\mathbb F}_q$-irreducible curves of arithmetic genus less than or equal to $\ell$ for $\ell$ a positive integer. Consider a rational effective ample divisor $H$ on $X$ avoiding a finite set $S$ of rational points and let $r$ be a positive integer. Then we have $$d(X, rH, S) \geq d^*(X, rH, S)+\left(rH^2-\frac{\pi_{rH}-1}{\ell}\right)(q+1+m).$$
The theorem follows from Lemma \[lemma\_d\], for which items (\[first\]) and (\[second\]) hold from Lemma \[lemma\_ell\] and item (\[third\]) holds from Theorem \[irreduciblecurves\].
Four improvements {#Improvements}
=================
In this section we manage to obtain better parameters for conditions (\[first\]), (\[second\]) or (\[third\]) of Lemma \[lemma\_d\] in four cases: for surfaces of arithmetic Picard number one, for surfaces which do not contain $\mathbb{F}_q$-irreducible curves of small self-intersection and whose canonical divisor is either nef or anti-nef, for fibered surfaces with nef canonical divisor, and for fibered surfaces whose singular fibers are $\mathbb{F}_q$-irreducible curves.
Surfaces with Picard number one {#Picard_1}
-------------------------------
As mentioned in the Introduction, the case of surfaces $X$ whose arithmetic Picard number equals one has already attracted some interest (see [@Zarzar], [@Voloch_Zarzar], [@Little_Schenck] and [@Blache]). We prove in this subsection Lemma \[bound\_picard\_rank\_one\] and Theorem \[theorem\_picard\_number\_one\] which improve, under this rank one assumption, the bounds of Lemma \[k\_HodgeTGV\_bounds\] and Theorem \[ourbound\]. These new bounds depend on the sign of $3H^2+H.K_X$, where $H$ is the ample generator of $\NS(X)$.
\[bound\_picard\_rank\_one\] Let $X$ be a smooth projective surface of arithmetic Picard number one. Let $H$ be the ample generator of $\NS(X)$ and let $r$ be a positive integer. For any non-zero function $f \in L\left( rH \right)$ consider the decomposition $D_f=\sum_{i=1}^k n_i D_i$ into $\mathbb{F}_q$-irreducible and reduced curves $D_i$ with positive integer coefficients $n_i$ as in (\[decomposition\]). Then the sum of the arithmetic genera of the curves $D_i$ satisfies:
- $\sum_{i=1}^k{\pi}_{D_i}\leq
(k-1)\pi_{H} + \pi_{(r-k+1)H}$ if $3H^2+H.K_X \geq 0$;
- $\sum_{i=1}^k{\pi}_{D_i} \leq
H^2(r-k)^2/2+H^2(r-2k)+k$ if $3H^2+H.K_X <0$.
\[proposition:maxLr\]
Note that the condition $3H^2+H.K_X\geq 0$ is satisfied as soon as $H.K_X\geq0$. It is also satisfied in the special case where $K_X=-H$ which corresponds to Del Pezzo surfaces.
In order to prove the first item, we consider a non-zero function $f \in L\left( rH \right)$ and we keep the notation already introduced in (\[decomposition\]), namely $D_f=\sum_{i=1}^k n_i D_i$. As $\NS(X)=\mathbb{Z}H$, for all $i$ we have $D_i=a_i H$ and we know by Lemma 2.2 in [@Zarzar] that $k \leq r$. Intersecting with the ample divisor $H$ enables to prove that for all $i$ we have $a_i \geq 1$ and that $\sum_{i=1}^k n_i a_i =r$. Thus to get an upper bound for $\sum_{i=1}^k \pi_{D_i}=\sum_{i=1}^k \pi_{a_i H}$, we are reduced to bounding $ \left( \sum_{i=1}^k a_i^2 \right) H^2/2 + \left( \sum_{i=1}^k a_i \right) H.K_X/2 + k $ under the constraint $\sum_{i=1}^k a_i n_i =r$. Our strategy is based on the two following arguments.
First, the condition $3H^2+H.K_X \geq 0$ guarantees that $a \mapsto \pi_{aH}$ is an increasing sequence. Indeed, for integers $a'>a\geq 1$ we have $\pi_{a' H} \geq \pi_{a H}$ if and only if $(a+a')H^2 \geq -H.K_X$, which is true under the condition above because $a+a' \geq 3$. As a consequence, if we fix an index $i$ between $1$ and $k$ and if we consider that the product $n_i a_i$ is constant, then the value of $\pi_{a_i H}$ is maximum when $a_i$ is, that is when $a_i=n_ia_i$ and $n_i=1$.
Secondly, assume that all the $n_i$ equal $1$ and that $\sum_{i=1}^k a_i=r$. We are now reduced to bounding $ \sum_{i=1}^k a_i^2 $. We can prove that the maximum is reached when all the $a_i$ equal $1$ except one which equals $r-k+1$. Otherwise, suppose for example that $2 \leq a_1 \leq a_2$. Then $a_1^2+a_2^2 < (a_1-1)^2 + (a_2+1)^2$ and $ \sum_{i=1}^k a_i^2$ is not maximum, and the first item is thus proved.
For the second item, using the adjonction formula we get $$\sum_{i=1}^k \pi_{D_i}-k \leq \frac{1}{2H^2}\sum_{i=1}^k(D_i.H)^2+\frac{1}{2}\sum_{i=1}^k D_i.K_X.$$ Again as $\NS(X)=\mathbb{Z}H$, for all $i$ we have $D_i=a_iH$. Thus we get $$\sum_{i=1}^k \pi_{D_i}-k \leq \frac{1}{2H^2}\sum_{i=1}^ka_i^2(H^2)^2+\frac{1}{2}\sum_{i=1}^k a_iH.K_X.$$ Now using that $H.K_X\leq -3H^2$ by hypothesis, that $\sum_{i=1}^k a_i \geq k$ since every $a_i$ is positive and that since $\sum_{i=1}^k a_i\leq r$ we can prove again that $\sum_{i=1}^k a_i^2\leq (r-k+1)^2+(k-1)$, we get $$\sum_{i=1}^k \pi_{D_i}-k \leq \frac{H^2}{2}((r-k+1)^2+(k-1))-\frac{3H^2}{2}k.$$ Some easy calculation shows that this is equivalent to our second statement.
\[theorem\_picard\_number\_one\] Let $X$ be a smooth projective surface of arithmetic Picard number one. Let $H$ be the ample generator of $\NS(X)$ and $S$ a finite set of rational points avoiding $H$. For any positive integer $r$, the minimum distance $d(X, rH, S)$ of the code $\mathcal{C}(X,rH,S)$ satisfies:
1. \[fab\] if $3H^2+H.K_X \geq 0$, then $$d(X, rH,S)\geq
\begin{cases}
\sharp S-(q+1+m\pi_{rH}) \text{ if } r>2(q+1+m)/mH^2,\\
\sharp S -r(q+1+m\pi_{H}) \text{ otherwise. }
\end{cases}$$
2. \[ele\] If $3H^2+H.K_X < 0$, then $$d(X, rH, S)\geq
\begin{cases}
\sharp S -(q+1+m)-mH^2(r^2-3)/2 \text{ if } r>2(q+1+m)/mH^2-3,\\
\sharp S -r(q+1+m-mH^2) \text{ otherwise. }
\end{cases}$$
For any non-zero $f \in L(rH)$, we have by (\[N(f)<=\]) and by point $(\ref{A-P})$ of Theorem \[irreduciblecurves\] the following inequality $$N(f) \leq k(q+1)+m\sum_{i=1}^k \pi_{D_i}.$$ We apply Lemma \[bound\_picard\_rank\_one\] to bound $\sum_{i=1}^k \pi_{D_i}$. We get in the first case $N(f)\leq \phi(k)$ where $\phi(k):=m\pi_{(r-k+1)H}+k(q+1+m\pi_H)-m\pi_H.$ Remark that $\pi_{(r-k+1)H}$ is quadratic in $k$ and so $\phi(k)$ is a quadratic function with positive leading coefficient. In [@Voloch_Zarzar Lemma 2.2] Voloch and Zarzar proved that if $X$ has arithmetic Picard number one then $k\leq r$. Thus $\phi(k)$ attends its maximum for $k=1$ or for $k=r$ and $N(f)\leq \max \{\phi(1),\phi(r)\}$. A simple calculus shows that $\phi(1)-\phi(r)>0$ if and only if $r>2(q+1+m)/mH^2$. Since we have $d(X, rH, S) \geq \sharp S-\max_{f\in L(rH)\setminus \{0\}} N(f)$, part \[fab\] of the theorem is proved.
The treatment of part \[ele\] is the same, except that we use Lemma \[bound\_picard\_rank\_one\] to bound $\sum_{i=1}^k \pi_{D_i}$.
\[Little\_Schenck\] Little and Schenck have given bounds in [@Little_Schenck §3] for the minimum distance of codes defined over algebraic surfaces of Picard number one. In particular, they obtain (if we keep the notations of Theorem \[theorem\_picard\_number\_one\]): $d(X, rH, S)\geq \sharp S-(q+1+m\pi_H)$ for $r=1$ ([@Little_Schenck Th. 3.3]) and $d(X, rH, S)\geq \sharp S -r(q+1+m\pi_H)$ for $r>1$ and $q$ large ([@Little_Schenck Th. 3.5]). Comparing their bounds with Theorem \[theorem\_picard\_number\_one\], one can see that when $3H^2+H.K_X \geq 0$ we get the same bound for $r=1$ and also for $r>1$ without any hypothesis on $q$. Moreover, when $3H^2+H.K_X < 0$, our bounds improve the ones given by Little and Schenck, again without assuming large enough $q$ when $r>1$.
Surfaces without irreducible curves defined over ${\mathbb F}_q$ with small self-intersection and whose canonical divisor is either nef or anti-nef {#low_selfintersections}
---------------------------------------------------------------------------------------------------------------------------------------------------
We consider in this section surfaces $X$ such that there exists some integer $\beta \geq 0$ for which any ${\mathbb F}_q$-irreducible curve $D$ lying on $X$ and defined over ${\mathbb F}_q$ has self-intersection $D^2\geq \beta$. We prove in this case Lemma \[lemma\_quadra\] below, from which we can tackle assumption (\[first\]) in Lemma \[lemma\_d\] in case $\beta >0$. Unfortunately, Lemma \[lemma\_quadra\] enables to fulfill assumption (\[second\]) of Lemma \[lemma\_d\] only in case the intersection of the canonical divisor with ${\mathbb F}_q$-irreducible curves has constant sign, that is for surfaces of Section \[nef\]. The lower bound for the minimum distance we get is better than the one given in Theorem \[ourbound\].
Let us propose some examples of surfaces with this property:
- simple abelian surfaces satisfy this property for $\beta =2$.
- Surfaces whose arithmetic Picard number is one. Indeed consider a curve $D$ defined over ${\mathbb F}_q$ on $X$, and assume that $\NS(X)={\mathbb Z}H$ with $H$ ample. Then we have that $D=aH$ for some integer $a$. Since $H$ is ample we get $1\leq D.H=aH^2$ hence $a\geq 1$ and $D^2=a^2H^2\geq H^2$.
- Surfaces whose canonical divisor is anti-nef and without irreducible curves of arithmetic genus less or equal to $\ell>0$. Indeed the adjunction formula gives $D^2=2\pi_D-2-D.K_X\geq 2\pi_D-2\geq 2\ell$.
\[lemma\_quadra\] Let $X$ be a surface on which any ${\mathbb F}_q$-irreducible curve has self-intersection at least $\beta\geq 0$. Assume that $H$ is a rational effective ample divisor on $X$ and let $r$ be a positive integer. Let $D=\sum^k_{i=1} n_iD_i$ be the decomposition as a sum of ${\mathbb F}_q$-irreducible and reduced curves of an effective divisor $D$ linearly equivalent to $rH$. Then we have
1. \[k\_beta\] if $\beta >0$ then $k \leq r\sqrt{\frac{H^2}{\beta}}$;
2. $\sum_{i=1}^k (2\pi_{D_i}-2-D_i.K_X)\leq \varphi(k)$, with $$\label{varphi(k)}
\varphi(k)\coloneqq (k-1)\beta+\left(r\sqrt{H^2}-(k-1)\sqrt{\beta}\right)^2.$$
Since by hypothesis we have $\sqrt{\beta} \leq \sqrt{D_i^2}$, we deduce that $k\sqrt{\beta} \leq \sum_{i=1}^k n_i\sqrt{D_i^2}$. By (\[Hodge\]), we get $k\sqrt{\beta} \leq \sum_{i=1}^k n_i\frac{D_i.H}{\sqrt{H^2}}= \frac{rH.H}{\sqrt{H^2}}=r\sqrt{H^2}$, from which the first item follows.
Setting $x_i\coloneqq\sqrt{2\pi_{D_i}-2-D_i.K_X}$, we have by adjunction formula $x_i = \sqrt{D_i^2}\geq \sqrt{\beta}$. Moreover the previous inequalities ensure that $\sum_{i=1}^k x_i\leq \sum_{i=1}^k n_i\sqrt{D_i^2}\leq r\sqrt{H^2}$. Then, the maximum of $\sum_{i=1}^k (2\pi_{D_i}-2-D_i.K_X)=\sum_{i=1}^k x_i^2$ under the conditions $x_i\geq\sqrt{\beta}$ and $\sum_{i=1}^kx_i \leq r\sqrt{H^2}$ is reached when each but one $x_i$ equals the minimum $\sqrt{\beta}$, and only one is equal to $r\sqrt{H^2}-(k-1)\sqrt{\beta}$, and this concludes the proof.
\[theorem\_beta\] Let $X$ be a surface on which any ${\mathbb F}_q$-irreducible curve has self-intersection at least $\beta>0$. Consider a rational effective ample divisor $H$ on $X$ avoiding a set $S$ of rational points and let $r$ be a positive integer. Then $$d(X, rH, S) \geq
\begin{cases}
\sharp S-\max\left\{\psi(1), \psi\left( r\sqrt{\frac{H^2}{\beta}}\right)\right\}
- \frac{m}{2} r\sqrt{\frac{H^2}{2\beta}} \text{ if $K_X$ is nef},\\
\sharp S-\max\left\{\psi(1), \psi\left( r\sqrt{\frac{H^2}{\beta}}\right)\right\} \text{ if $-K_X$ is nef}
\end{cases}$$ with $$\psi(k)\coloneqq\frac{m}{2}\varphi(k)+k(q+1+m),$$ where $\varphi(k)$ is given by equation (\[varphi(k)\]).
For any non-zero $f \in L(rH)$, we have by (\[N(f)<=\]) and by point $(\ref{A-P})$ of Theorem \[irreduciblecurves\] that $N(f) \leq k(q+1)+m\sum_{i=1}^k \pi_{D_i}.$ Lemma \[lemma\_quadra\] implies that $N(f) \leq k(q+1) +\frac{m}{2}\bigl(2k+\varphi(k)+\sum_{i=1}^k D_i.K_X\bigr).$ In case $K_X$ is nef, we have $\sum_{i=1}^kD_i.K_X \leq \sum_{i=1}^kn_i D_i.K_X = rH.K_X$, and in case $-K_X$ is nef, we get $\sum_{i=1}^kD_i.K_X \leq 0$, and the theorem follows.
Fibered surfaces with nef canonical divisor {#Section_fibration}
-------------------------------------------
We consider in this subsection AG codes from fibered surfaces whose canonical divisor is nef. We adopt the vocabulary of [@Silverman III, §8] and we refer the reader to this text for the basic notions we recall here. A fibered surface is a surjective morphism $\pi: X\rightarrow B$ from a smooth projective surface $X$ to a smooth absolutely irreducible curve $B$. We denote by $\pi_0$ the common arithmetic genus of the fibers and by $g_B$ the genus of the base curve $B$. Elliptic surfaces are among the first non-trivial examples of fibered surfaces. For such surfaces we have $\pi_0=1$ and the canonical divisor is always nef (see [@Bombieri_Mumford]).
We recall that on a fibered surface every divisor can be uniquely written as a sum of *horizontal* curves (that is mapped onto $B$ by $\pi$) and *fibral* curves (that is mapped onto a point by $\pi$).
\[r\_bound\] Let $\pi : X \rightarrow B$ be a fibered surface. Let $H$ be a rational effective ample divisor on $X$ and let $r$ be a positive integer. For any effective divisor $D$ linearly equivalent to $rH$, consider its decomposition $D=\sum^k_{i=1} n_iD_i$ as a sum of reduced ${\mathbb F}_q$-irreducible curves as in (\[decomposition\]). Denote by $\overline{r}_i$ the number of absolutely irreducible components of $D_i$. Then, we have $$\sum_{i=1}^{k}\overline{r}_i\leq rH^2.$$
Write $D=\sum_{i=1}^k n_i D_i=\sum_{i=1}^k n_i \sum_{j=1}^{\overline{r}_i}D_{i,j}$ where the $D_{i,j}$ are the absolutely irreducible components of $D_i$.
We use that $n_i>0$, that $D$ is numerically equivalent to $rH$ and that $D_{i,j}.H>0$ to get $$\sum_{i=1}^{k}\overline{r}_i \leq \sum_{i=1}^k \sum_{j=1}^{\overline{r}_i}D_{i,j}.H\leq \sum_{i=1}^k n_i \sum_{j=1}^{\overline{r}_i}D_{i,j}.H= \sum_{i=1}^k n_i D_i.H=rH.H,$$ which proves the lemma.
The next theorem involves the *defect* $\delta(B)$ of a smooth absolutely irreducible curve $B$ defined over $\mathbb{F}_q$ of genus $g_B$, which is defined by $$\delta(B) \coloneqq q+1+mg_B-\sharp B(\mathbb{F}_q).$$ By the Serre-Weil theorem this defect is a non-negative number. The so-called maximal curves have defect $0$, and the smaller the number of rational points on $B$ is, the greater the defect is.
\[theorem\_fibration\] Let $\pi : X \rightarrow B$ be a fibered surface whose canonical divisor $K_X$ is nef. Assume that $H$ is a rational effective ample divisor on $X$ having at least one horizontal component and avoiding a set $S$ of rational points. For any positive integer $r$ the minimum distance of $\mathcal{C}(X,rH, S)$ satisfies $$d(X,rH, S)\geq d^*(X, rH, S) + \delta(B)$$ where $d^*(X, rH, S)$ is given by formula (\[d\*\]).
Recall that the general bound we obtain in Theorem \[ourbound\] in Section \[Codes\_over\_algebraic\_surfaces\] for surfaces with nef canonical divisor is $d(X,rH, S)\geq d^*(X, rH, S)$, thus the bound from Theorem \[theorem\_fibration\] is always equal or better. Actually Theorem \[theorem\_fibration\] is surprising, since it says that the lower bound for the minimum distance is all the more large because the defect $\delta(B)$ is. Consequently it looks like considering fibered surfaces on curves with few rational points and large genus could lead to potentially good codes.
Recall that for any non-zero $f\in L(rH)$, we have $d(X,rH, S) \geq \sharp S- N(f)$, and that $N(f)\leq \sum_{i=1}^k \sharp D_i(\mathbb{F}_q)$ if we use the notation $D_f:=rH+(f)=\sum_{i=1}^k n_i D_i$ introduced in (\[decomposition\]). We again denote by $\overline{r}_i$ the number of absolutely irreducible components of $D_i$. In order to introduce the ${\mathbb F}_q$-irreducible components of $D_f$, write $k=h+v$, where $h$ (respectively $v$) is the number of horizontal curves denoted by $H_1,\ldots,H_h$, (respectively fibral curves denoted by $F_{1},\ldots,F_v$). Then we get $N(f)\leq \sum_{i=1}^h \sharp H_i(\mathbb{F}_q)+\sum_{i=1}^v \sharp F_i(\mathbb{F}_q).$ Since $B$ is a smooth curve, the morphisms $H_i\rightarrow B$ are flat. Now applying Proposition \[covering\] to horizontal curves and Theorem \[irreduciblecurves\] to fibral curves gives $$\begin{aligned}
\label{Nf_hv_0}
\begin{split}
N(f)&\leq h(\sharp B(\mathbb{F}_q)-mg_B)+m\sum_{i=1}^h \pi_{H_i}+q\sum_{i=1}^h(\overline{r}_i-1)+qv+v+m\sum_{i=1}^v \pi_{F_i}\\
&=h(\sharp B(\mathbb{F}_q)-mg_B-q)+m\sum_{i=1}^k \pi_{D_i}+q\sum_{i=1}^k \overline{r}_i+v,
\end{split}\end{aligned}$$ where we used the fact that $v\leq \sum_{i=h+1}^k \overline{r}_i$.
Since the canonical divisor of the fibered surface is assumed to be nef, Lemma \[k\_HodgeTGV\_bounds\] gives a bound for $\sum_{i=1}^k \pi_{D_i}$. We set $v=k-h$ and we use Lemma \[r\_bound\] with (\[Nf\_hv\_0\]) to obtain $$\begin{aligned}
\label{Nf_hv}
\begin{split}
N(f)&\leq h(\sharp B(\mathbb{F}_q)-mg_B-q)+m(\pi_{rH}-1)+mk+qrH^2+v\\
&=h(\sharp B(\mathbb{F}_q)-mg_B-q-1)+m(\pi_{rH}-1)+mk+qrH^2+k\\
&=-h\delta(B)+m(\pi_{rH}-1)+mk+qrH^2+k.
\end{split}\end{aligned}$$
Now, $D_f.F\equiv rH.F >0$ since $F$ is a generic fiber and $rH$ is assumed to have at least one horizontal component. Thus, $D_f$ has also at least one horizontal component, that is $h \geq 1$. Moreover, again from Lemma \[k\_HodgeTGV\_bounds\] we have $k \leq rH^2$. As the defect $\delta(B)$ is non-negative it follows that $$N(f) \leq -\delta(B)+rH^2(q+1+m)+m(\pi_{rH}-1)$$ and the theorem is proved.
Fibered surfaces whose singular fibers are irreducible.
-------------------------------------------------------
In this subsection we drop off the condition on the canonical divisor. Instead, we assume that every singular fiber on $X$ is ${\mathbb F}_q$-irreducible. To construct examples of such surfaces, fix any $d \geq 1$ and recall that the dimension of the space of degree $d$ homogeneous polynomials in three variables is $\binom{d+2}{2}$. Hence the space ${\mathcal P}_d$ of plane curves of degree $d$ is ${\mathcal P}_d={\mathbb P}^{\binom{d+2}{2}-1}$. Any curve $B$ drawn in ${\mathcal P}_d$ gives rise to a fibered surface, whose fibers are plane curves of degree $d$, that is with $\pi_0=\frac{(d-1)(d-2)}{2}$. The locus of singular curves being a subvariety of ${\mathcal P}_d$, choosing $B$ not contained in this singular locus yields to a fibered surface with smooth generic fiber. As the locus of reducible curves has high codimension in ${\mathcal P}_d$, choosing $B$ avoiding this locus yields to fibered surfaces without reducible fibers.
We consider the case where $\pi_0$ and $g_B$ are both at least $2$ and we set $\ell=\min(\pi_0,g_B)-1\geq 1$. We recall again that every divisor on $X$ can be uniquely written as a sum of horizontal and fibral curves. If we denote by $H$ an horizontal curve and by $V$ a fibral curve defined over ${\mathbb F}_q$, we have that $\pi_{H}\geq g_B$ and $\pi_{V}=\pi_0$. Therefore, in this setting, $X$ contains no $\mathbb{F}_q$-irreducible curves defined over $\mathbb{F}_q$ of arithmetic genus smaller than or equal to $\ell$. Thus Lemma \[lemma\_ell\] applies and gives the same bound for $\sum_{i=1}^k \pi_i$ as when $K_X$ is nef and the bound $k\leq (\pi_{rH}-1)/\ell$ for the number $k$ of ${\mathbb F}_q$-irreducible components of $D_f$. We consider this new bound for $k$ in the proof of Theorem \[theorem\_fibration\] and we get instead the following result.
\[theorem\_fibration\_two\] Let $\pi : X \rightarrow B$ be a fibered surface. We consider a rational effective ample divisor $H$ on $X$ having at least one horizontal component and avoiding a set $S$ of rational points. Let $r$ be a positive integer. We denote by $g_B$ the genus of $B$ and by $\pi_0$ the arithmetic genus of the fibers and we set $\ell=\min(\pi_0,g_B)-1$. Suppose that every singular fiber is ${\mathbb F}_q$-irreducible and that $\ell \geq 1$. Then the minimum distance of $\mathcal{C}(X,rH, S)$ satisfies $$d(X,rH, S)\geq d^*(X, rH, S) + \left(rH^2-\frac{\pi_{rH}-1}{\ell}\right)(q+1+m)+ \delta(B),$$ where $d^*(X, rH, S)$ is given by formula (\[d\*\]).
Naturally this bound is better than the one in Theorem \[theorem\_fibration\] if and only if $\pi_{rH}-1<\ell rH^2$. Furthermore it improves the bound of Theorem \[theorem\_ell\] by the addition of the non-negative term $\delta(B)$.
An example: surfaces in $\mathbb{P}^3$ {#Hypersurfaces}
======================================
This section is devoted to the study of the minimum distance of AG codes over a surface $X$ of degree $d\geq 3$ embedded in $\mathbb{P}^3$.
We consider the class $L$ of an hyperplane section of $X$. So $L$ is ample, $L^2=d$ and the canonical divisor on $X$ is $K_X=(d-4)L$ (see [@Shafarevich p.212]). In this setting, we fix a rational effective ample divisor $H$ and $r$ a positive integer. We apply our former theorems to this context to give bounds on the minimum distance of the code $\mathcal{C}(X,rH, S)$.
We recall that cubic surfaces are considered by Voloch and Zarzar in [@Voloch_Zarzar] and [@Zarzar] to provide computationally good codes. In Section 4 of [@Little_Schenck] Little and Schenck propose theoretical and experimental results for surfaces in $\mathbb{P}^3$ always in the prospect of finding good codes. We also contribute to this study with a view to bounding the minimum distance according to the geometry of the surface.
\[without\_hypothesis\] Let $X$ be a surface of degree $d\geq 3$ embedded in $\mathbb{P}^3$. Consider a rational effective ample divisor $H$ avoiding a set $S$ of rational points on $X$ and let $r$ be a positive integer. Then the minimum distance of the code $\mathcal{C}(X,rH, S)$ satisfies
1. if $X$ is a cubic surface, then $$d(X,rH, S)\geq d^*(X, rH, S)+mr(\pi_{H}-1).$$
2. If $X$ has degree $d\geq4$ then $$d(X,rH, S)\geq d^*(X, rH, S),$$
where $$d^*(X, rH, S)= \sharp S-rH^2(q+1+m)-m(\pi_{rH}-1)$$ is the function defined in (\[d\*\]).
Since $K_X=(d-4)L$ we have for cubic surfaces that $K_X=-L$ and thus the canonical divisor is anti-ample, while for surfaces of degree $d\geq 4$ the canonical divisor is ample or the zero divisor, thus is nef. Hence we can apply Theorem \[ourbound\] from which the proposition follows.
Surfaces in $\mathbb{P}^3$ without irreducible curves of low genus
------------------------------------------------------------------
In the complex setting, the Noether-Lefschetz theorem asserts that a general surface $X$ of degree $d\geq 4$ in $\mathbb{P}^3$ is such that $\Pic(X)=\mathbb{Z}L$, where $L$ is the class of an hyperplane section (see [@Griffith_Harris]). Here, general means outside a countable union of proper subvarieties of the projective space parametrizing the surfaces of degree $d$ in $\mathbb{P}^3$. Even if we do not know an analog of this statement in our context, it suggests us the strong assumptions we take in this subsection, namely in Lemma \[genus\_on\_hypersurfaces\] and Proposition \[hypersurfaces\_wo\_curves\_small\_genus\].
\[genus\_on\_hypersurfaces\] Let $X$ be a surface of degree $d\geq4$ in $\mathbb{P}^3$ of arithmetic Picard number one. Suppose that $\NS(X)$ is generated by the class of an hyperplane section $L$. Consider an $\mathbb{F}_q$-irreducible curve $D$ on $X$ of arithmetic genus $\pi_D$. Then $$\pi_D \geq (d-1)(d-2)/2.$$
Let $a$ be the integer such that $D=aL$ in $\NS(X)$. Since $D$ is an ${\mathbb F}_q$-irreducible curve and $L$ is ample, we must have $a>0$. Then, using the adjonction formula, we get $$\begin{aligned}
2\pi_D-2&=D^2+D.K=a^2L^2+aL.(d-4)L\\
&=a^2d+ad(d-4)\geq d+d(d-4),\end{aligned}$$ and thus $\pi_D\geq (d-1)(d-2)/2$.
By the previous lemma it is straightforward that in our context $X$ does not contain any $\mathbb{F}_q$-irreducible curves of arithmetic genus smaller than or equal to $\ell$ for $\ell= (d-1)(d-2)/2-1=d(d-3)/2$. This allows us to apply Theorem \[theorem\_ell\], and get the following proposition.
\[hypersurfaces\_wo\_curves\_small\_genus\] Let $X$ be a degree $d\geq4$ surface in $\mathbb{P}^3$ of arithmetic Picard number one whose Néron-Severi group $\NS(X)$ is generated by the class of an hyperplane section $L$. Assume that $S$ is a set of rational points avoiding $L$. For any positive integer $r$ the minimum distance of the code $\mathcal{C}(X,rL, S)$ satisfies $$d(X, rL, S) \geq d^*(X, rL, S, L)+rd\left(1-\frac{r+d-4}{d(d-3)}\right)(q+1+m)$$ where $$d^*(X, rL, S, L)= \sharp S-rd(q+1+m)-mrd(r+d-4)/2.$$
Surfaces in $\mathbb{P}^3$ of arithmetic Picard number one
----------------------------------------------------------
In this subsection we suppose that the arithmetic Picard number of $X$ is one, but we do not take the assumption that the Néron-Severi group is generated by an hyperplane section. Also in this case we can apply Theorem \[theorem\_picard\_number\_one\] which brings us to the following proposition.
\[hypersurfaces\_picard\_1\] Let $X$ be a surface of degree $d\geq4$ in $\mathbb{P}^3$. Assume that $\NS(X)=\mathbb{Z} H$ for an ample divisor $H$. Consider $L=hH$, the class of an hyperplane section of $X$, for $h$ a positive integer. Let $S$ be a set of rational points on $X$ avoiding $H$ and let $r$ be a positive integer. Then the minimum distance of the code $\mathcal{C}(X,rH, S)$ satisfies $$d(X,rH, S)\geq
\begin{cases}
\sharp S-(q+1+m)-rH^2(r+h(d-4))/2 \text{ if } r>2(q+1+m)/mH^2,\\
\sharp S-r(q+1+m)-rH^2(1+h(d-4))/2 \text{ otherwise. }
\end{cases}$$
Since we have $3H^2+H.K_X=3H^2+H.(d-4)L=3H^2+h(d-4)H^2=H^2(3+h(d-4))\geq 0$, we can apply point \[fab\] of Theorem \[theorem\_picard\_number\_one\] from which the proposition follows.
[**Acknowledgments:**]{} The authors would like to thank the anonymous referee for relevant observations.
Yves Aubry, <span style="font-variant:small-caps;">Institut de Mathématiques de Toulon - IMATH,</span>
<span style="font-variant:small-caps;">Université de Toulon and Institut de Mathématiques de Marseille - I2M,</span>
<span style="font-variant:small-caps;">Aix Marseille Université, CNRS, Centrale Marseille, UMR 7373, France</span>
*E-mail address*: `[email protected]`
Elena Berardini, <span style="font-variant:small-caps;">Institut de Mathématiques de Marseille - I2M,</span>
<span style="font-variant:small-caps;">Aix Marseille Université, CNRS, Centrale Marseille, UMR 7373, France</span>
*E-mail address*: `[email protected]`
Fabien Herbaut, <span style="font-variant:small-caps;">INSPE Nice-Toulon, Université Côte d’Azur,</span>
<span style="font-variant:small-caps;">Institut de Mathématiques de Toulon - IMATH, Université de Toulon, France</span>
*E-mail address*: `[email protected]`
Marc Perret, <span style="font-variant:small-caps;">Institut de Mathématiques de Toulouse, UMR 5219,</span>
<span style="font-variant:small-caps;">Universit' e de Toulouse, CNRS, UT2J, F-31058 Toulouse, France</span>
*E-mail address*: `[email protected]`
[^1]: Funded by ANR grant ANR-15-CE39-0013-01 “Manta"
|
---
abstract: |
In this note we revisit the SUSY effects in $R_b$ under current experimental constraints including the LHC Higgs data, the $B$-physics measurements, the dark matter relic density and direct detection limits, as well as the precision electroweak data. We first perform a scan to figure out the currently allowed parameter space and then display the SUSY effects in $R_b$. We find that although the SUSY parameter space has been severely restrained by current experimental data, both the general MSSM and the natural-SUSY scenario can still alter $R_b$ with a magnitude sizable enough to be observed at future $Z$-factories (ILC, CEPC, FCC-ee, Super $Z$-factory) which produce $10^9-10^{12}$ $Z$-bosons. To be specific, assuming a precise measurement $\delta R_b = 2.0 \times 10^{-5}$ at FCC-ee, we can probe a right-handed stop up to 530 GeV through chargino-stop loops, probe a sbottom to 850 GeV through neutralino-sbottom loops and a charged Higgs to 770 GeV through the Higgs-top quark loops for a large $\text{tan}\beta$. The full one-loop SUSY correction to $R_b$ can reach $1 \times 10^{-4}$ in natural SUSY and $2 \times 10^{-4}$ in the general MSSM.
[**Keywords:**]{} Supersymmetry, $R_b$
author:
- 'Wei Su$^{1}$'
- 'Jin Min Yang$^{1,2}$'
title: 'SUSY effects in $R_b$: revisited under current experimental constraints'
---
[1.2]{}
Introduction {#sec:introduction}
============
After the discovery of the 125 GeV Higgs boson [@Chatrchyan:2012ufa; @Aad:2012tfa], the primary task of the LHC is to hunt for new physics beyond the Standard Model (SM). Among various extensions of the SM, the low energy supersymmetry (SUSY) is the most appealing candidate[^1] since it can solve the gauge hierarchy problem, naturally explain the cosmic cold dark matter and achieve the gauge coupling unification. The search for SUSY has long been performed both directly and indirectly. On the one hand, the colliders have directly searched for the sparticle productions. On the other hand, SUSY effects have been probed indirectly through precision measurements of some low energy observables.
$R_b \equiv \Gamma(Z\to\bar bb)/\Gamma(Z \to hadrons)$ is a famous observable which is sensitive to new physics beyond the SM [@Rb]. So far the most precise experimental value $R_b^{\rm exp}= 0.21629\pm0.00066$ comes from the LEP and SLC measurements [@ALEPH:2005ab], while the SM prediction is $R_b^{\rm SM}=0.21579$ [@Freitas:2014hra]. The future $Z$-factories are expected to produce much more $Z$-bosons than the LEP experiment. For example, $10^9$, $10^{10}$ and $10^{12}$ $Z$-bosons are expected to be produced respectively at the International Linear Colldier (ILC) [@ILC], the Circular Electron-Positron Collider (CEPC) [@CEPC], the Future Circular Collider (FCC-ee) [@FCC-ee] and the Super $Z$-factory [@super-Z]. This will allow for a more precise measurement of $R_b$ [@Fan:2014axa] and help pin down the involved new physics effects.
The SUSY effects in $R_b$ were calculated and discussed many years ago [@Boulware:1991vp; @Cao:2008rc; @Garcia:1994wv; @history]. In this work we revisit these effects for two reasons: (i) The current experiments, especially the LHC experiments, have severely restrained the SUSY parameter space. It is intriguing to figure out the possible magnitude of the SUSY effects in the currently allowed parameter space; (ii) Given the possibility of some future $Z$-factories like ILC, CEPC or FCC-ee, a more precise measurement of $R_b$ will help reveal the SUSY effects although these effects may have already been restrained to be rather small by current experiments. In order to know if the SUSY effects are accessible in a future measurement of $R_b$, we must figure out their currently allowed value.
This work is organized as follows. In Sec.\[sec:formula\], we give a description of SUSY effects in $R_b$. In Sec.\[sec:result\], we scan over the SUSY parameter space and display the SUSY effects in the allowed parameter space. Finally we give our conclusion in Sec. \[sec:conclusion\].
SUSY corrections to $R_b$ {#sec:formula}
=========================
Since the SUSY effects in $R_b$ have been calculated in the literature [@Boulware:1991vp; @Cao:2008rc], here we only give a brief description. The dominant SUSY effects in $R_b$ are from the vertex corrections to $Z\to b\bar{b}$, as shown in Figs. \[fig:gluino\]-\[fig:neutralhiggs\]. These corrections come from the gluino loops, chargino loops, neutralino loops, charged Higgs loops and neutral Higgs loops.
![One-loop Feynman diagrams of gluino correction to $Z \to \bar b b$[]{data-label="fig:gluino"}](fig1.eps){width="70.00000%"}
![One-loop Feynman diagrams of\
neutralino correction to $Z \to \bar b b$[]{data-label="fig:neutralino"}](fig2.eps){width="90.00000%"}
![One-loop Feynman diagrams of\
neutralino correction to $Z \to \bar b b$[]{data-label="fig:neutralino"}](fig3.eps){width="90.00000%"}
![One-loop Feynman diagrams of\
neutral Higgs correction to $Z \to \bar b b$[]{data-label="fig:neutralhiggs"}](fig4.eps){width="90.00000%"}
![One-loop Feynman diagrams of\
neutral Higgs correction to $Z \to \bar b b$[]{data-label="fig:neutralhiggs"}](fig5.eps){width="90.00000%"}
The one-loop SUSY correction to $R_b$ can be expressed as $$\label{eq:factRb}
\delta R^{SUSY}_b \simeq \frac{R_b^{SM}(1-R_b^{SM})}{v^2_b(3-\beta^2) + 2a_b^2\beta^2}[v_b(3-\beta^2) \delta v_b+2a_b\beta^2\delta {a_b}],$$ where $v_b = 1/2-2sin\theta_w^2/3$ and $a_b = 1/2$ are respectively the vector and axial vector couplings of tree-level $Zb\bar{b}$ interaction, $\beta = \sqrt{1-4m_{b}^2/m_Z^2}$ is the velocity of bottom quark in $Z \to b\bar{b}$, and $\delta v_b$ and $\delta a_b$ are the corresponding corrections defined as [@Cao:2008rc; @Bohm:1986rj; @Hollik:1988ii] $$\label{eq:delta abvb}
\delta v_b = \frac{\delta g^b_L + \delta g^b_R}{2},\quad \delta a_b = \frac{\delta g^b_L - \delta g^b_R}{2}.$$ Here $\delta g^b_{\lambda}$ ($\lambda = L, R$) is give by $$\label{eq:delta gLR}
\delta g^b_{\lambda} = \Gamma_{f{\lambda}}(m_Z^2) - g_{\lambda}^{Zb\bar{b}}\Sigma_{b{\lambda}}(m_b^2),$$ where $\Gamma_{f{\lambda}}(m_Z^2)$ denotes the vertex loop contributions and $\Sigma_{b{\lambda}}(m_b^2)$ is the counter term from the bottom quark self-energy. We perform straightforward loop calculations and confirm the expressions in [@Cao:2008rc]. The results can be expressed as $$\begin{aligned}
\Sigma_{b_\lambda}(p_b^2) &=&
\frac{C_g}{(4\pi)^2} \biggl| g^{\bar{\psi}_j b \phi_i^\ast}_\lambda \biggr|^2
( B_0 + B_1 ) (p_b, m_{\phi_i}, m_{\psi_j}), \\
\Gamma_{b_\lambda}(q^2) &=&
-\frac{C_g}{(4\pi)^2} \Biggl\{
\biggl( g_\lambda^{\bar{\psi}_j b \phi_k^\ast} \biggr)^*
g_\lambda^{\bar{\psi}_i b \phi_k^\ast}
\biggl[
g_\lambda^{\bar{\psi}_j \psi_i Z} m_{\psi_i} m_{\psi_j} C_0
\nonumber \\
&&
+ g_{-\lambda}^{\bar{\psi}_j \psi_i Z}
\biggl(-q^2 (C_{12} + C_{23}) - 2 C_{24} + \frac{1}{2} \biggr)
\biggr] (p_{\bar{b}}, p_{b}, m_{\psi_i}, m_{\phi_k}, m_{\psi_j})
\nonumber \\
&&
- \biggl( g^{\bar{\psi}_k b \phi_i^\ast}_\lambda \biggr)^*
g^{\bar{\psi}_k b \phi_j^\ast}_\lambda
g^{\phi_i^\ast \phi_j Z}
2 C_{24}(p_{\bar{b}}, p_b, m_{\phi_j}, m_{\psi_k}, m_{\phi_i})
\Biggr\},\end{aligned}$$ where $C_g = 4/3$ for the gluino loops and $C_g=1$ for other loops, and $B_0$, $B_1$ and $C_{12}$, $C_{23}$, $C_{24}$ are Passarino-Veltman functions [@Passarino:1978jh]. The notation $(\phi,\psi)$ represents ($\tilde b, \tilde g$) for gluino loops, ($\tilde t, \tilde\chi^-$) for chargino loops, ($\tilde b, \tilde\chi^0$) for neutralino loops, ($H^-, t$) for charged Higgs loops and ($h/a/G^0$, $b$) for neutral Higgs loops.
In addition to $R_b$, we also show the SUSY effects in the forward-backward asymmetry $A^b_{FB}$ in the decay $Z \to \bar{b}b$: $$\label{eq:abfb}
\delta A^{b}_{FB}\big|_{SUSY} \simeq A^{b}_{FB}\big|_{SM}\big(\frac{v_b \delta v_b + a_b \delta a_b}{a_b v_b} - 2\frac{v_b(3-\beta^2) \delta v_b+2a_b\beta^2\delta {a_b}}{v^2_b(3-\beta^2) + 2a_b^2\beta^2}\big).$$ Its experimental value is $0.0992 \pm 0.0016$ from the LEP experiment [@ALEPH:2005ab] while its SM prediction is $0.1032 \pm 0.0004$ [@Baak:2014ora]. In the future $Z$-factories, this forward-backward asymmetry will be measured together with $R_b$, both of which will jointly allow for a revelation of SUSY effects.
Numerical calculations and results {#sec:result}
==================================
SUSY parameter space
--------------------
To clarify our numerical calculations we consider the general MSSM and the natural-SUSY scenario [@natural-susy]. From the natural-SUSY results (the natural-SUSY parameter space is much smaller than the general MSSM), we can acquire the more detailed characters of each kind of loops, while from the general MSSM results we can obtain the more general size of SUSY loop effects.
For the natural-SUSY scenario, since in this scenario only the higgsino masses and the third-generation squark masses are assumed to be light, while other sparticles are assumed to be rather heavy and thus their effects in low energy observables are decoupled, in our scan we fix the soft-breaking mass parameters in the first two generation squark sector and the slepton sector at 5 TeV, and assume $A_t = A_b$. For the electroweak gaugino masses, inspired by the grand unification relation, we take $M_1 : M_2 = 1 : 2$ and fix $M_2$ at 2 TeV. The gluino mass is fixed at 2 TeV since it is supposed to be not too far above TeV scale in natural-SUSY. Other parameters vary as follows $$\begin{aligned}
&& 1 < \tan\beta < 60, 100~{\rm GeV}< \mu < 200 ~{\rm GeV}, ~ |A_t| < 3 ~{\rm TeV}, \nonumber\\
&& 100 ~{\rm GeV} < m_{Q_3}, m_{U_3} , m_{D_3}< 2 ~{\rm TeV}.\end{aligned}$$ For the general MSSM, assuming $A_t = A_b$ and $M_1 : M_2:M_3 = 1 : 2:6$, we scan over the following parameter space $$\begin{aligned}
&& 1 < \tan\beta < 60, 100~{\rm GeV}< \mu < 1000 ~{\rm GeV}, ~ |A_t| < 3 ~{\rm TeV}, \nonumber\\
&& 100 ~{\rm GeV} < m_{Q_3}, m_{U_3} , m_{D_3}< 2 ~{\rm TeV},100~{\rm GeV}< M_2 < 20000 ~{\rm GeV}.\end{aligned}$$ In our scan we consider the following experimental constraints:
- The constraints on the Higgs sector from the LEP, Tevatron and LHC experiments. We use the package HiggsBounds-4.0.0 [@Bechtle:2011sb] to implement these constraints.
- The experimental constraints in $B$-physics. We require SUSY to satisfy various $B$-physics bounds at $2\sigma$ level with SUSY FLAVOR v2.0 [@Crivellin:2012jv], which includes $B\to X_s\gamma$, $B_s\to \mu^+\mu^-$, $B^+\to\tau^+\nu$ and so on [@Agashe:2014kda].
- The measurements of the precision electroweak observables. The SUSY predictions of $\rho_l$, $sin^2\theta_{eff}^l$ and $m_W$ are required to be within the 2$\sigma$ ranges of the experimental values [@ALEPH:2005ab].
- The dark matter constraints. We require the thermal relic density of the neutralino dark matter to be below the 2$\sigma$ upper limit of the Planck value [@Ade:2013zuv] and require the dark matter-nucleon spin-independent scattering scross section $\sigma_r^{SI}$ to satisfy the 95% C.L. limits of LUX [@Akerib:2013tjd]. We also consider the limits of spin-dependent dark matter-nucleon cross section $\sigma_r^{SD}$ from the XENON100 experiment [@Aprile:2013doa]. The relic density, $\sigma_r^{SI}$ and $\sigma_r^{SD}$ are calculated with the code MicrOmega v2.4 [@Belanger:2010gh].
About the mass bounds from the LHC direct searches, in natural SUSY the higgsinos have very weak bounds because their pair productions only give missing energy and are rather difficult to detect (a mono-jet or mono-$Z$ is needed in detection) [@mono-jet], while for the stops the right-handed one is weakly bounded (its mass can be as light as 210 GeV for higgsinos heavier than 190 GeV) [@stop-bound]. When we display the numerical results, we will not show a sharp LHC bound on stop or higgsino mass (we only consider the LEP bounds on stop and higgsinos). For each surviving sample we calculate the correction to $R_b$ and display the numerical results in the proceeding section.
Numerical results of $R_b$ and $A^{b}_{FB}$
-------------------------------------------
The results for natural-SUSY and the general MSSM are displayed in Figs.\[dia:chargino\]-\[dia:total-tan-dabfb\] and Figs.\[dia:total-tan-dabfb-general\]-\[dia:restriction\], respectively. We first show the results of different loops and then show the combined results. Finally we compare the natural-SUSY results with the general MSSM results.
About the future precision of $R_b$ measurement, the CEPC would produce $10^{10}$ $Z$-bosons and probably measure $R_b$ with an uncertainty of $1.7\times10^{-4}$ [@Fan:2014axa; @CEPC], while the FCC-ee could produce $10^{12}$ $Z$-bosons and give a much better $R_b$ measurement at $10^{-5}$ level [@FCC-ee]. In our figures, for illustration, we mark an uncertainty of $2\times10^{-5}$ [@FCC-ee; @Fan:2014axa]. The SUSY parameter space giving $\delta R_b^{\rm SUSY} > 2\times10^{-5}$ corresponds to the observable region.
![The scatter plots of the surviving samples of natural SUSY, showing the chargino one-loop effects in $R_b$.[]{data-label="dia:chargino"}](fig6a.eps "fig:"){width="45.00000%"} ![The scatter plots of the surviving samples of natural SUSY, showing the chargino one-loop effects in $R_b$.[]{data-label="dia:chargino"}](fig6b.eps "fig:"){width="45.00000%"}
![Same as Fig.\[dia:chargino\], but showing the\
gluino loop effects.[]{data-label="dia:gluino"}](fig7.eps){width="100.00000%"}
![Same as Fig.\[dia:chargino\], but showing the\
gluino loop effects.[]{data-label="dia:gluino"}](fig8.eps){width="100.00000%"}
![Same as Fig.\[dia:chargino\], but showing the\
neutral Higgs loop effects.[]{data-label="dia:neutralhiggs"}](fig9.eps){width="100.00000%"}
![Same as Fig.\[dia:chargino\], but showing the\
neutral Higgs loop effects.[]{data-label="dia:neutralhiggs"}](fig10.eps){width="100.00000%"}
![Same as Fig.\[dia:chargino-stop-dabfb\], but for the combined\
loop effects.[]{data-label="dia:total-tan-dabfb"}](fig11.eps){width="100.00000%"}
![Same as Fig.\[dia:chargino-stop-dabfb\], but for the combined\
loop effects.[]{data-label="dia:total-tan-dabfb"}](fig12.eps){width="100.00000%"}
![Same as Fig.\[dia:total-tan-dabfb\], but for the general MSSM.[]{data-label="dia:total-tan-dabfb-general"}](fig13.eps){width="50.00000%"}
![The plots of survived samples, showing the most sensitive restrictions in two scenarios. The left panel is for natural SUSY, where the samples with and without B-physics constraints are displayed (other constraints are satisfied). The right panel is for the general MSSM, where the samples with and without the dark matter-nucleon spin-independent scattering limits are displayed (other constraints are satisfied).[]{data-label="dia:restriction"}](fig14a.eps "fig:"){width="45.00000%"} ![The plots of survived samples, showing the most sensitive restrictions in two scenarios. The left panel is for natural SUSY, where the samples with and without B-physics constraints are displayed (other constraints are satisfied). The right panel is for the general MSSM, where the samples with and without the dark matter-nucleon spin-independent scattering limits are displayed (other constraints are satisfied).[]{data-label="dia:restriction"}](fig14b.eps "fig:"){width="45.00000%"}
Some discussions about the results are in order:
- From Fig.\[dia:chargino\] we see that the chargino-stop loop effects are sizable only if $\tilde t_1$ is dominated by a right-handed stop. For a left-handed stop, its coupling with higgsino and bottom $Y_b = g m_b/(\sqrt{2} m_W \cos\beta)$ is suppressed (the lightest chargino $\tilde\chi^\pm_1$ is dominated by higgsino component since the higgsino mass $\mu$ is much smaller than the gaugino masses $M_1$ and $M_2$ in natural SUSY). Only for a very large $\tan\beta$ can the coupling $Y_b$ be comparable to the corresponding right-handed stop coupling $Y_t=g m_t/(\sqrt{2} m_W \sin\beta)$. Our numerical results shows that $\tan\beta$ is smaller than 35 (so that $Y_b/Y_t<1$) for $\tilde t_1$ below 530 GeV (when $\tan\beta$ is larger, $\tilde t_1$ must be heavier to satisfy the experimental constraints). Note that, as commented in the preceding section, so far the right-handed stop mass in natural SUSY is weakly bounded by LHC experiments (its mass can be as light as 210 GeV for higgsinos heavier than 190 GeV) [@stop-bound].
- As shown in Fig.\[dia:gluino\], the gluino-sbottom loop effects are very small due to the heaviness of gluino. The loop effects of the neutralinos, charged and neutral Higgs bosons, as shown in Figs.\[dia:neturalino\], \[dia:chargedhiggs\] and \[dia:neutralhiggs\], are sensitive to $tan\beta$ and can be sizable for a large value of $\text{tan}\beta$. Our numerical results show that the neutralino loop can push the $\tilde b_1$ mass to 850 GeV when $\text{tan}\beta$ is around 32. If the $\text{tan}\beta$ is about 23, through the charged Higgs loop, $\tilde H^+$ mass less than 770 GeV is excluded. The neutral Higgs loops impose an upper bound of 46 on the value of $\text{tan}\beta$.
- From Figs.\[dia:chargino-stop-dabfb\], \[dia:total-tan-dabfb\] and \[dia:total-tan-dabfb-general\] we see that the SUSY effects in $R_b$ and $A^{b}_{FB}$ are correlated, as expected. Both observables can jointly probe the SUSY effects. While the chargino loop effects always enhance both quantities, the combined total effects of all loops can either enhance or reduce them. We also find that in the general MSSM without special naturalness requirement, both $R_b$ and $A^{b}_{FB}$ are allowed to vary in a larger region than in natural SUSY, especially when $\tan \beta$ is small.
- From Figs.\[dia:chargino\]-\[dia:total-tan-dabfb-general\] we see that in some currently allowed parameter space, the effects of natural SUSY may be accessible in the future $R_b$ measurement. If it can be measured with an uncertainty of $2\times10^{-5}$, a large part of SUSY parameter space can be covered.
- We found that for natural SUSY the most stringent limits are from B-physics, while for the general MSSM the most stringent limits are from the dark matter-nucleon spin-independent scattering limits. The results are shown in Fig.\[dia:restriction\]. Other constraints, such as the dark matter-nucleon spin-dependent scattering cross section, are also making impacts but not as stringent as these two.
Conclusion {#sec:conclusion}
==========
We revisited the SUSY effects in $R_b$ under current experimental constraints including the LHC Higgs data, the $B$-physics measurements, the dark matter relic density and direct detection limits, as well as the precision electroweak data. We scanned over the SUSY parameter space and in the allowed parameter space we displayed the SUSY effects in $R_b$. We found that although the SUSY parameter space has been severely restrained by current experimental data, SUSY can still alter $R_b$ with a magnitude sizable enough to be observed at future $Z$-factories (ILC, CEPC, FCC-ee). Assuming a precise measurement $\delta R_b = 2.0 \times 10^{-5}$ at FCC-ee, we can probe the right-handed stop to 530 GeV through the chargino-stop loops, probe the sbottom to 850 GeV through the neutralino-sbottom loops and the charged Higgs to 770 GeV through the Higgs-top quark loops for a large $\text{tan}\beta$. The full one-loop SUSY correction to $R_b$ can reach $1 \times 10^{-4}$ in natural SUSY and $2 \times 10^{-4}$ in the general MSSM.
We would like to thank Junjie Cao, Lei Wu, Yang Zhang, Mengchao Zhang for useful discussions. This work has been supported in part by the National Natural Science Foundation of China under grant Nos. 11275245 and 11135003, and by the CAS Center for Excellence in Particle Physics (CCEPP).
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[^1]: Confronted with the 125 GeV Higgs mass, the minimal SUSY model (MSSM) has a little fine-tuning while the next-to-minimal SUSY model is more favored [@mssm-nmssm].
|
---
abstract: 'As it was pointed out recently in [@Hees_PRL_17], observations of stars near the Galactic Center with current and future facilities provide an unique tool to test general relativity (GR) and alternative theories of gravity in a strong gravitational field regime. In particular, the authors showed that the Yukawa gravity could be constrained with Keck and TMT observations. Some time ago, Dadhich et al. showed in [@Dadhich_01] that the Reissner – Nordström metric with a tidal charge is naturally appeared in the framework of Randall – Sundrum model with an extra dimension ($Q^2$ is called tidal charge and it could be negative in such an approach). Astrophysical consequences of presence of black holes with a tidal charge are considerered, in particular, geodesics and shadows in Kerr – Newman braneworld metric are analyzed in [@Schee_09_IJMPD], while profiles of emission lines generated by rings orbiting braneworld Kerr black hole are considered in [@Schee_09_GRG]. Possible observational signatures of gravitational lensing in a presence of the Reissner – Nordström black hole with a tidal charge at the Galactic Center are discussed in papers [@Bin-Nun_10; @Bin-Nun_10_2; @Bin-Nun_11]. Here we are following such an approach and we obtain analytical expressions for orbital precession for Reissner – Nordström – de-Sitter solution in post-Newtonian approximation and discuss opportunities to constrain parameters of the metric from observations of bright stars with current and future astrometric observational facilities such as VLT, Keck, GRAVITY, E-ELT and TMT.'
author:
- 'Alexander F. Zakharov $^{1,2,3,4,5}$'
title: Constraints on tidal charge of the supermassive black hole at the Galactic Center with trajectories of bright stars
---
Introduction
============
The Galactic Center is a very peculiar object. A couple of different models have been suggested for it, including dense cluster of stars [@Reid_09], fermion ball [@Munyaneza_02], boson stars [@Jetzer_92; @Torres_00], neutrino balls [@DePaolis_01]. Later, some of these models have been constrained with subsequent observations [@Reid_09]. However, as it was found in computer simulations, sometimes differences for alternative models may be very tiny as it was shown in paper [@Vincent_16] where the authors discussed shadows for boson star and black hole models. The most natural and generally accepted model for the Galactic Center is a supermassive black hole (see, e.g. recent reviews [@Goddi_17; @Eckart-etal:2017:FOUNPH:; @Zakharov_MIFI_17; @Zakharov_MIFI_18]). A natural way to evaluate a gravitational potential is to analyze trajectories of photons or test particles moving in the potential. Shapes of shadows forming by photons moving around black holes were discussed in [@Bardeen_73; @Chandrasekhar_83; @Falcke_00; @Melia_01] (see also [@ZNDI_05]). Shadows (dark spots) can not be detected but theoretical models could describe a distribution of bright structures around these dark shadows. Bright structures around shadows are being observing with an improving accuracy of current and forthcoming VLBI facilities in mm-band, including the Event Horizon Telescope [@Doeleman_08; @Doeleman_08b; @Doeleman_09; @Doeleman_17].
To create an adequate theoretical model for the Galactic Center astronomers monitored trajectories of bright stars (or clouds of hot gas) using the largest telescopes VLT and Keck with adaptive optics facilities [@Ghez_00; @Ghez_03; @Ghez_04; @Ghez_05; @Weinberg_05; @Meyer_12; @Morris_12]. One could introduce a distance between observational data for trajectories of bright stars and their theoretical models. Practically, such a distance is a measure of quality for a theoretical fit. To test different theoretical models one of the most simple approach is to compare apocenter (pericenter) shifts for theoretical fits and observational data for trajectories. If an apocenter (pericenter) shifts for a theoretical fit exceed apocenter (pericenter) shifts obtained from observations one should rule out these interval for parameters for theoretical fits. Based on such an approach one could evaluate parameters of black hole, stellar cluster and dark matter cloud around the Galactic Center because if there is an extended mass distribution inside a bright star orbit in addition to black hole, the extended mass distribution causes an apocenter shift in direction which is opposite to relativistic one [@ZNDI_PRD_2007; @NDIQZ_07]. One could also check predictions of general relativity or alternative theories of gravity. For instance, one could evaluate constraints on parameters of $R^n$ theory, Yukawa gravity and graviton masses with trajectories of bright stars at the Galactic Center because in the case of alternative theories of gravity a weak gravitational field limit differs from Newtonian one, so trajectories of bright stars differ from elliptical ones and analyzing observational data with theoretical fits obtained in the framework of alternative theories of gravity one constrains parameters of such theories [@BJBZ_PRD_12; @BJBJZ_JCAP_13; @ZBBJJ_ASR_14; @ZJBBJ_JCAP_16; @Zakharov_Quarks_16; @Zakharov_JCAP_18] (see, also discussion of observational ways to investigate opportunities to find possible deviations from general relativity with observations of bright stars at the Galactic Center [@Hees_PRL_17; @Hees_17]).
In paper [@Dadhich_01] it was shown that the Reissner – Nordström metric with a tidal charge could arise in Randall – Sundrum model with an extra dimension. Braneworld black holes are considered assuming that they could substitute conventional black holes in astronomy, in particular, geodesics and shadows in Kerr – Newman braneworld metric are analyzed in [@Schee_09_IJMPD], while profiles of emission lines generated by rings orbiting braneworld Kerr black hole are considered in [@Schee_09_GRG]. Later it was proposed to consider signatures of gravitational lensing assuming a presence of the Reissner – Nordström black hole with a tidal charge at the Galactic Center [@Bin-Nun_10; @Bin-Nun_10_2; @Bin-Nun_11]. In paper [@Zakharov_PRD_2014] analytical expressions for shadow radius of Reissner – Nordström black hole have been derived while shadow sizes for Schwarzschild – de Sitter (Köttler) metric have been found in papers [@Stuchlik_83; @Zakharov_2014]. In the paper for a particle motion in Reissner – Nordström – de-Sitter metric we derive analytical expressions for orbital precession and discuss constraints on tidal charge from current and future observations of bright stars near the Galactic Center.
Basic notations {#sec1}
===============
We use a system of units where $G = c= 1$. The line element of the spherically symmetric Reissner – Nordström – de-Sitter metric is d s\^2 = - f(r) d t\^2 + f(r)\^[-1]{} d r\^2 + r\^2 d\^2 + r\^2 \^2d\^2 , \[metric\_RN\] where function $f(r)$ is defined as f(r) = 1 - + - r\^2. \[function\_f\] Here $M$ is a black hole mass, $Q$ is its charge and $\Lambda$ is cosmological constant. In the case of a tidal charge [@Dadhich_01], $Q^2$ could be negative. Similarly to [@Carter_73; @Sharp_79; @Stuchlik_83; @Stuchlik_02], geodesics could be obtained the Lagrangian $${\cal{L}}=
-\frac{1}{2} g_{\mu \nu} \frac{d x^\mu}{d \lambda} \frac{d x^\nu}{d \lambda},
\label{Lagrangian-1}$$ where $ g_{\mu \nu}$ are the components of metric (\[metric\_RN\] and $\lambda$ is the affine parameter. There are three constants of motion for geodesics which come from the metric (\[metric\_RN\]), namely $$g_{\mu \nu} \dfrac{d x^\mu}{d \lambda} \dfrac{dx^\nu}{d \lambda}=-m,
\label{mass-1}$$ which is a test particle mass and two constants connected with an independence of the metric on $\phi$ and $t$ coordinates, respectively $$g_{\phi \nu} \frac{d x^\nu}{d \lambda}=h,
\label{momentum-1}$$ and $$g_{t \nu} \frac{d x^\nu}{d \lambda}=E.
\label{Energy}$$ For vanishing $\Lambda$-term these integrals of motion ($h$ and $E$) could be interpreted as angular momentum and energy of a test particle, respectively. Geodesics for massive particles could be written in the following form $$\label{radial_coordinate_equation}
r^4 \left(\dfrac{dr}{d \lambda}\right)^2 = E^2r^4 - \Delta (m^2r^2+h^2),$$ where $$\label{Delta}
\Delta=\left(1-\frac{1}{3}\Lambda r^2\right)r^2 - 2M r+Q^2.$$ or we could write Eq. (\[radial\_coordinate\_equation\]) in the following form $$\label{radial_coordinate_equation2}
r^4 \left( \dfrac{dr}{d \tau}\right)^2 = (\hat{E}^2-1)r^4 +{2Mr^3}- Q^2r^2 +\frac{1}{3} \Lambda r^6 -\hat{h}^2(r^2-\frac{\Lambda}{3}r^4-2Mr+Q^2),$$ where $\tau=m \lambda$ is the proper time, $\hat{E}=\dfrac{E}{m}$ and $\hat{h}=\dfrac{h}{m}$. We will omit symbol $\wedge$ below. Since $$\label{radial_coordinate_equation3}
r^4 \left(\dfrac{d \phi}{d \tau}\right)^2 = h^2,$$ one could obtain $$\label{radial_coordinate_equation4}
\left(\dfrac{d r}{d \phi}\right)^2 = \frac{1}{{h}^2}({E}^2-1)r^4 +\frac{2Mr^3}{{h}^2}- \frac{Q^2r^2}{{h}^2} +\frac{1}{3h^2} \Lambda r^6 -(r^2-\frac{\Lambda}{3}r^4-2Mr+Q^2),$$ It is convenient to introduce new variable $u=1/r$. Since $$\left(\dfrac{d u}{d \lambda}\right)^2 = \left(\dfrac{d r}{d \phi}\right)^2 u^4,
\label{u_variable}$$ one obtains $$\left(\dfrac{d u}{d \lambda}\right)^2 =
\frac{1}{h^2}({E}^2-1) +\frac{2Mu}{{h}^2}- \frac{Q^2u^2}{{h}^2} +\frac{\Lambda }{3h^2 u^2} -(u^2-\frac{\Lambda}{3}-2Mu^3+Q^2u^4),
\label{u_variable2}$$ therefore, $$\dfrac{d^2 u}{d \lambda^2} +u = \frac{M}{{h}^2}+ 3Mu^2 - \frac{Q^2u}{{h}^2} -2Q^2u^3 -\frac{\Lambda }{3h^2 u^3} ,
\label{u_variable3}$$ and as it is noted in [@Adkins_07] the first term in the right hand side of Eq. (\[u\_variable3\]) corresponds to the Newtonian case, the second term corresponds to the GR correction from the Schwarzschild metric (see also book [@Anderson_67]), meanwhile one could see inspecting Eq. (\[u\_variable3\]) that third and forth term correspond to a presence of $Q$ parameter in metric (\[metric\_RN\]), the fifth term corresponds to a $\Lambda$-term presence in the metric. Assuming that second, third, forth and fifth terms in the right hand side of Eq. (\[u\_variable3\]) are small in respect to the basic Newtonian solution, one could evaluate relativistic precession for each term and after that one has to calculate an algebraic sum of all shifts induced by different terms.
Relativistic precession evaluation
==================================
An impact of non-vanishing charge in Reissner – Nordström metric on orbital precession was discussed in papers [@Burman_69; @Teli_84; @Rathod_89; @Dean_99; @Gong_09] considering perturbations of Schwarzschild metric, see for instance [@Anderson_67; @Treder_80]. However, Eq. (\[u\_variable3\]) was not considered in these papers. When people discussed astrophysical consequences of this effect they evaluated an impact of Solar charge on Mercury precession orbit [@Burman_69] and it is clear that the effect is very small due to constraints on Solar electric charge. Similarly, for astrophysical black holes including the black holes at the Galactic Center, their electric charges are expecting to be vanishing or very small. However, significant tidal charges $|Q|$ which are comparable with $M$ are discussed in the literature [@Bin-Nun_10_2; @Bin-Nun_11] where the author discussed an opportunity to evaluate a tidal charge $Q^2 \approx - 6.4 M^2$ or $Q^2 \approx 1.6 M^2$ from gravitational lensing.
An expression for apocenter (pericenter) shifts for Newtonian potential plus small perturbing function is given as a solution in the classical (L & L) textbook [@Landau_76] (see also applications of the expressions for calculations of stellar orbit precessions in presence of the the supermassive black hole and dark matter at the Galactic Center [@Dokuchaev_15; @Dokuchaev_15a]). In paper [@Adkins_07], the authors derived the expression which is equivalent to the (L & L) relation and which can be used for our needs. According to the procedure proposed in [@Adkins_07] one could re-write Eq. (\[u\_variable3\]) in the following form $$\dfrac{d^2 u}{d \tau^2} +u = \frac{M}{{h}^2}- \frac{g(u)}{{h}^2},
\label{u_variable4}$$ where $g(u)$ is a perturbing function which is supposed to be small and it could be presented as a conservative force in the following form $$g(u)=r^2 F(r)|_{r=1/u}, \quad F(r)=-\frac{dV}{dr}.
\label{u_variable5}$$ For potential $V(r)=\dfrac{\alpha_{-(n+1)}}{r^{-(n+1)}}$ (where $n$ is a natural number) one obtains [@Adkins_07] $$\Delta \theta (-(n+1))=\frac{-\pi \alpha_{-(n+1)}\chi^2_n(e)}{ML^n},
\label{Delta_negative power}$$ where $$\chi^2_n(e)=n(n+1)_2F_1\left(\frac{1}{2}-\frac{n}{2},\frac{1}{2}-\frac{n}{2},2,e^2\right),
\label{chi_n}$$ $_2F_1$ is the Gauss hypergeometrical function, $L$ is the semilatus rectum ($L=h^2/M$) and we have $L=a(1-e^2)$ ($a$ is semi-major axis and $e$ is eccentricity). An alternative approach for evaluation of pericenter advance within of Rezzolla – Zhidenko (RZ) parametrization [@Rezzolla_14] has been described in [@DeLaurentis_17] for theoretical analysis of pulsar timing in the case if pulsars are moving in the strong gravitational field of the supermassive black hole at the Galactic Center. Since pulsars are very precise and stable clocks, studies of pulsar timing gives an opportunity to investigate gravitational field in the vicinity of the supermassive black hole.
In paper [@Adkins_07] the authors obtained orbital precessions for positive powers of perturbing function $$\Delta \theta (n)=\frac{-\pi \alpha_{n}a^{n+1} \sqrt{1-e^2}\chi^2_n(e)}{M}.
\label{Delta_negative power_2}$$
For GR term in Eq. (\[u\_variable3\]) the perturbing potential is $V_{GR}(r)=-\dfrac{Mh^2}{r^3}$ and one obtains the well-known result $n=2$ (see, for instance [@Adkins_07] and textbooks on GR) $$\Delta \theta (GR) := \Delta \theta (-(3))=\frac{6\pi M}{L}.
\label{Delta_GR}$$ For the third term in Eq. (\[u\_variable3\]) one has potential $V_{RN1}(r)=\dfrac{Q^2}{2r^2}$ ($\alpha_{-2}=\dfrac{Q^2}{2}$ and $n=1$), therefore, one obtains $$\Delta \theta (RN1) := \Delta \theta (-(2))_{RN1}=-\frac{\pi Q^2}{ML}.
\label{Delta_RN1}$$ Eq. (\[Delta\_RN1\]) was derived earlier in [@Burman_69] to evaluate an impact of Solar charge on orbital precession of Mercury, however, we re-derive the Eq. (\[Delta\_RN1\]) following a procedure suggested in [@Landau_76] since this approach is more clear and it could be applied for other types of perturbing potentials. For the forth term in Eq. (\[u\_variable3\]) one has potential $V_{RN2}(r)=\dfrac{h^2Q^2}{2r^4}$ ($\alpha_{-4}=\dfrac{h^2Q^2}{2}$ and $n=3$) , therefore, one obtains $$\Delta \theta (RN2) := \Delta \theta (-(4))_{RN2}=-\frac{3\pi Q^2 (4+e^2)}{2L^2}.
\label{Delta_RN2}$$ Since according to our assumptions $M \ll L$, one has $\dfrac{Q^2}{L^2} \ll \dfrac{Q^2}{ML} $ and we ignore the apocenter (pericenter) shift which is described with Eq. (\[Delta\_RN2\]). For the fifth (de-Sitter or anti-de-Sitter) term in Eq. (\[u\_variable3\]) one has potential $V_{dS}(r)=-\dfrac{\Lambda r^2}{6}$ ($\alpha_2=-\dfrac{\Lambda}{6}$) and one has the corresponding apocenter (pericenter) shift $$\Delta \theta (\Lambda) := \Delta \theta (2)_{dS}=\frac{\pi \Lambda a^3 \sqrt{1-e^2}}{M}.
\label{Delta_dS}$$ Eq. (\[Delta\_dS\]) was derived earlier in [@Kerr_03] and re-derived in [@Adkins_07] with (L & L) approach [@Landau_76]. In paper [@Sereno_06] Eq. (\[Delta\_dS\]) was used to discuss consequences of a non-vanishing $\Lambda$-term from observations in Solar system.
Therefore, a total shift of a pericenter is $$\Delta \theta (total) := \frac{6\pi M}{L} -\frac{\pi Q^2}{ML} + \frac{\pi \Lambda a^3 \sqrt{1-e^2}}{M}.
\label{Delta_dS_2}$$ and one has a relativistic advance for a tidal charge with $Q^2 < 0$ and apocenter shift dependences on eccentricity and semi-major axis are the same for GR and Reissner – Nordström advance but corresponding factors (${6\pi M}$ and $-\dfrac{\pi Q^2}{M}$) are different, therefore, it is very hard to distinguish a presence of a tidal charge and black hole mass evaluation uncertainties. For $Q^2 > 0$, there is an apocenter shift in the opposite direction in respect to GR advance. As it was noted each term in Eq. (\[Delta\_dS\_2\]) was known earlier, but people did not consider them together perhaps because of small values electric charge and $\Lambda$-term. However, a wider range for tidal charge was considered for the black hole at the Galactic Center [@Bin-Nun_10_2; @Bin-Nun_11] and an excellent precision of astrometrical observations has been reached in last years and it gives an opportunity to evaluate parameters of alternative theories of gravity with these observations.
Estimates
=========
As it was noted by the astronomers of the Keck group [@Hees_PRL_17], pericenter shift has not be found yet for S2 star, however, an upper confidence limit on a linear drift is constrained $$| \dot{\omega}| < 1.7 \times 10^{-3} {\rm rad}/{\rm yr}.
\label{observational_constraint}$$ at 95% C.L., while GR advance for the pericenter is [@Hees_17] $$| \dot{\omega}_{GR}| = \frac{6\pi GM}{Pc^2(1-e^2)}=1.6 \times 10^{-4} {\rm rad}/{\rm yr},
\label{GR_constraint}$$ where $P$ is the orbital period for S2 star (in this section we use dimensional constants $G$ and $c$ instead of geometrical units). Based on such estimates one could constrain alternative theories of gravity following the approach used in [@Hees_PRL_17].
Estimates of (tidal) charge constraints
---------------------------------------
Assuming $\Lambda=0$ we consider constraints on $Q^2$ parameter from previous and future observations of S2 star. One could re-write orbital precession in dimensional form $$\dot{\omega}_{RN} = \frac{\pi Q^2}{PGML},
\label{RN_constraint_0}$$ where $P$ is an orbital period. Taking into account a sign of pericenter shift for a tidal charge with $Q^2 < 0$, one has $$\dot{\omega}_{RN} < 1.54 \times 10^{-3} {\rm rad}/{\rm yr} \approx 9.625~ \dot{\omega}_{GR},
\label{TRN_constraint_1}$$ therefore, $$-57.75 M^2 < Q^2 <0,
\label{TRN_constraint}$$ with $ 95 \%$ C. L. For $Q^2 >0$, one has $$| \dot{\omega}_{RN}| < 1.86 \times 10^{-3} {\rm rad}/{\rm yr} \approx 11.625~ \dot{\omega}_{GR},
\label{RN_constraint_2}$$ therefore, $$0 < |Q| < 8.3516 M,
\label{RN_constraint}$$ with $ 95 \%$ C. L. As it was noted in [@Hees_PRL_17] in 2018 after the pericenter passage of S2 star the current uncertainties of $| \dot{\omega}|$ will be improved by a factor 2, so for a tidal charge with $Q^2 < 0$, one has $$\dot{\omega}_{RN} < 6.9 \times 10^{-4} {\rm rad}/{\rm yr} \approx 4.31~ \dot{\omega}_{GR},
\label{TRN_constraint_1_18}$$ $$-25.875 M^2 < Q^2 <0,
\label{TRN_constraint_18}$$ For $Q^2 >0$, one has $$| \dot{\omega}_{RN}| < 9.1 \times 10^{-4} {\rm rad}/{\rm yr} \approx 5.69~ \dot{\omega}_{GR},
\label{RN_constraint_2_18}$$ therefore, $$0 < |Q| < 5.80 M,
\label{RN_constraint_18}$$ One could expect that subsequent observations with VLT, Keck, GRAVITY, E-ELT and TMT will significantly improve an observational constraint on $ | \dot{\omega}|$, therefore, one could expect that a range of possible values of $Q$ parameter would be essentially reduced.
As it was noted in paper [@Hees_PRL_17], currently Keck astrometric uncertainty is around $\sigma = 0.16$ mas, therefore, an angle $\delta =2 \sigma$ (or two standard deviations) is measurable with around 95% C.L. In this case $\Delta \theta (GR)_{S2}= 2.59 \delta $ for S2 star where we adopt $\Delta \theta (GR)_{S2} \approx 0.83$. Assuming that GR predictions about orbital precession will be confirmed in the next 16 years with $\delta$ accuracy (or $ \left|\dfrac{\pi Q^2}{ML}\right| \lesssim \delta$), one could constrain $Q$ parameter $$|Q^2| \lesssim 2.32 M^2,
\label{RN_constraint_34}$$ where we wrote absolute value of $Q^2$ since for a tidal charge $Q^2$ could be negative. For negative $Q^2$ this estimate is better than estimate considered in [@Bin-Nun_10_2] ($Q^2 \approx - 6.4 M^2$), however, the estimate (\[RN\_constraint\_34\]) is slightly more worse than $Q^2 \approx 1.6 M^2$.
If we adopt uncertainty $\sigma_{TMT} = 0.015$ mas for TMT-like scenario as it was used in [@Hees_PRL_17] ($\delta_{TMT}=2\sigma_{TMT}$) or in this case $\Delta \theta (GR)_{S2}= 27.67 \delta_{TMT} $ for S2 star and assuming again that GR predictions about orbital precession of S2 star will be confirmed with $\delta_{TMT}$ accuracy (or $ \left|\dfrac{\pi Q^2}{ML}\right| \lesssim \delta_{TMT}$) , one could conclude that $$|Q^2| \lesssim 0.216 M^2,
\label{RN_constraint_TMT}$$ or based on results of future observations one could expect to reduce significantly a possible range of $Q^2$ parameter in comparison with a possible hypothetical range of $Q^2$ parameter which was discussed in [@Bin-Nun_10; @Bin-Nun_10_2].
Recently the GRAVITY team reported about a discovery of post-Newtonian gravitational redshift near S2 star pericenter passage [@Gravity_18]. Assuming $f=0$ corresponds to the Newtonian case and $f=1$ corresponds to the first post-Newtonian correction of GR, the GRAVITY collaboration estimated $f$-value from observational data comparing precessions for Schwarzschild and Newtonian approaches and they concluded that the $f$-value must be much closer to GR value or more precisely $f=0.94 \pm 0.09$ [@Gravity_18] (see also discussions in [@Zakharov_Quarks_18]). If we adopt uncertainty $\sigma_{\rm GRAVITY} = 0.030$ mas of the GRAVITY facilities [@Gravity_18] and assuming again that GR predictions on orbital precession of S2 star will be confirmed with $\delta_{\rm GRAVITY}=2\sigma_{\rm GRAVITY}$ accuracy (or $ \left|\dfrac{\pi Q^2}{ML}\right| \lesssim \delta_{\rm GRAVITY}$), one could conclude that $$|Q^2| \lesssim 0.432 M^2,
\label{RN_GRAVITY}$$ or based on results of future GRAVITY observations one could expect to reduce significantly a possible range of $Q^2$ parameter in comparison with a possible range of $Q^2$ parameter constrained with current and future Keck data.
Estimates of $\Lambda$-term constraints
---------------------------------------
In this subsection we assume that $Q=0$. One could re-write orbital precession in dimensional form $$\dot{\omega}_{\Lambda} = \frac{\pi \Lambda c^2 a^3 \sqrt{1-e^2}}{PGM},
\label{DE_constraint}$$ Dependences of functions $\dot{\omega_{\Lambda}}$ and $\dot{\omega_{GR}}$ on eccentricity and semi-major axis are different and orbits with higher semi-major axis and smaller eccentricity could provide a better estimate of $\Lambda$-term (the S2 star orbit has a rather high eccentricity). However, we use observational constraints for $S2$ star. For positive $\Lambda$, one has relativistic advance and $$\dot{\omega_{\Lambda}} < 1.54 \times 10^{-3} {\rm rad}/{\rm yr} \approx 9.625~ \dot{\omega_{GR}},
\label{DE_constraint_1}$$ or $$0 < {\Lambda} < 3.9 \times 10^{-39} {\rm cm}^{-2},
\label{DE_constraint_2}$$ for $\Lambda <0$ one has $$0 < -\Lambda < 4.68 \times 10^{-39} {\rm cm}^{-2},
\label{DE_constraint_3}$$
if we use current accuracy of Keck astrometric measurements $\sigma = 0.16$ mas and monitor S2 star for 16 years and assume that additional apocenter shift ($2\sigma$)could be caused by a presence of $\Lambda$-term, one obtains $$|{\Lambda}| < 1.56 \times 10^{-40} {\rm cm}^{-2},
\label{DE_constraint_4}$$ while for TMT-like accuracy $\delta_{TMT} = 0.015$ mas one has $$|{\Lambda}| < 1.46 \times 10^{-41} {\rm cm}^{-2}.
\label{DE_constraint_5}$$ As one can see, constraints on cosmological constant from orbital precession of bright stars near the Galactic Center are much weaker than not only its cosmological estimates but also than its estimates from Solar system data [@Sereno_06].
Conclusions {#Conclusion}
===========
We consider the first relativistic corrections for apocenter shifts in post-Newtonian approximation for the case of Reissner – Nordström – de-Sitter metric. Among different theoretical models have been proposed for the Galactic Center different black hole models are rather natural. Perhaps, assumptions about a presence of electric charge in the metric do not look very realistic because a space media is usually quasi-neutral, but the charged black holes are discussed in the literature see, for instance [@Moradi_17] and references therein. Moreover, a Reissner – Nordström metric could arise in a natural way in alternative theories of gravity like Reissner – Nordström solutions with a tidal charge in Randall–Sundrum model [@Dadhich_01] (such an approach is widely discussed in the literature). Recently, it was found that Reissner – Nordström metric is a rather natural solution in Horndeski gravity [@Babichev_17] and in this case $Q^2$ parameter reflects an interaction with a scalar field and it could be also negative similarly to a tidal charge. In paper [@Babichev_17] it was expressed an opinion that the hairy black hole solutions look rather realistic and these objects could exist in centers of galaxies and if such objects (hairy black holes in Horndeski gravity) exist in nature, in particular in the Galactic Center, current and future advanced facilities such as GRAVITY [@blin15], E-ELT [@eelt14], TMT [@tmt14] etc. may be very useful to detect signatures of black hole hairs of an additional dimension. Therefore, non-vanishing (positive or negative) $Q^2$ parameter is arisen due to a presence of extra dimension or in Horndeski gravity for black holes with a scalar hair. We outline a procedure to constrain $Q^2$ parameter with current and future observations of bright stars at the Galactic Center. Even current Keck facilities could constrain $Q^2$ better ($Q^2 \approx -2.32 M^2$) than with analysis of hypothetical variations of S2 brightness as it was suggested in [@Bin-Nun_10_2].
Certainly, $\Lambda$-term should be present in the model, however, if we adopt its cosmological value it should be very tiny to cause a significant impact on relativistic precession for trajectories of bright stars. If we have a dark energy instead of cosmological constant, one should propose ways to evaluate dark energy for different cases, therefore, one could constrain $\Lambda$-term from observations as it was noted in [@Zakharov_2014] analyzing impact of $\Lambda$-term on observational phenomena near the Galactic Center (similarly to the cases where an impact of $\Lambda$-term has been analyzed for effects in Solar system [@Kagramanova_06; @Jetzer_06; @Sereno_06]).
Acknowledgments {#acknowledgments .unnumbered}
===============
The author thanks D. Borka, V. Borka Jovanović, V. I. Dokuchaev, P. Jovanović and Z. Stuchlík for useful discussions. A. F. Z. thanks PIFI grant 2017VMA0014 of Chinese Academy of Sciences at NAOC (Beijing), NSF (HRD-0833184) and NASA (NNX09AV07A) at NASA CADRE and NSF CREST Centers (NCCU, Durham, NC, USA) for a partial support.
The author thanks also an anonymous referee for useful critical remarks.
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---
abstract: 'The magnetocrystalline anisotropy energies (MAEs) of the ferromagnetic metals bcc Fe, fcc and hcp Co, and fcc Ni have been calculated by using the [*ab initio*]{} tight-binding method. Disentangling the strong correlation among the $d$ orbitals with the Hamiltonian in the local spin-density approximation, we have investigated the orbital polarizations induced by the Hubbard $U$ and Racah $B$. The experimental MAE of fcc Ni is found with the value of $U$ close to that determined from experiments and used in other theories. With the optimized values of $U$ and $J$, both the MAEs and the orbital moments for Fe and Co are in close agreement with experiment.'
author:
- Yuannan Xie
- 'John A. Blackman'
title: Magnetocrystalline anisotropy and orbital polarization in ferromagnetic transition metals
---
Obtaining the magnetocrystalline anisotropy energies (MAEs) of Fe, Co, and Ni from [*ab initio*]{} calculations within the local-spin-density approximation (LSDA) to density functional theory is of considerable current interest [@DKS90; @TJEW95; @HP98; @B98; @YSK01; @S01]. From various high-quality LSDA calculations, the best case is Fe, where the computed values differ from experiment by a factor of about 2. The result for hcp Co is far worse, and for Ni, the sign is not even correct. The effect of the so called spin-other-orbit coupling is far too small to bring theory and experiment into accord [@S01]. The discrepancy between theory and experiment, especially in the case of Ni, is usually attributed to the LSDA.
The LSDA predicted MAEs and orbital moments can be improved for Fe and Co [@TJEW95; @DKS91] by introducing the Brooks’ orbital polarization (OPB) term [@Brooks85] which mimics Hund’s second rule. However for Ni, the predicted easy axis is still wrong [@TJEW95]. In OPB, OP is driven by the Racah parameter $B$, with an energy functional related to the orbital moment $\langle\hat{L}\rangle$ given by $\Delta E_{\text{OPB}} = -\frac{1}{2}B\langle\hat{L}\rangle^2$ [@EBJ90]. It was argued that the key parameter responsible for the exchange-correlation enhancement of the orbital moments in solids is the Hubbard $U$ rather than the intra-atomic Hund’s second rule coupling [@SLT98]. Recently, the experimental MAEs of Fe and Ni have been obtained[@YSK01] in the LDA$+U$ method [@AAL97] with the noncollinearity of intra-atomic magnetization included. However, the authors found the MAE of fcc Ni to be a very rapidly varying function of $U$ (from $-50$ to $60$ $\mu$eV/atom). A slight change of the value of $U$ ($\sim0.1$eV) may predict the wrong sign. Given this sensitivity, it is highly desirable to disentangle the intra-atomic strong correlation with the Hamiltonian in the LSDA and therefore to clarify the effect of OP in first principles calculations. This is the purpose of the present paper.
The basic features of the electronic structure of Fe, Co, and Ni can be understood on the basis of their two types of valence-electron orbitals [@SAS92]. The extended $s$, $p$ (,and $f$) orbitals should be well described by the LSDA. The fairly localized $d$ orbitals, for which the electron-electron interaction is between the localized and itinerant limits, are not adequately dealt with in the LSDA [@AAL97]. In order to disentangle the strong correlation with the LSDA, we express the Hamiltonian as $\hat{H}=\hat{H}^{\text{LSDA}}+
\hat{H}^{\text{C}}+\hat{H}^{\text{SO}}$. Here $\hat{H}^{\text{LSDA}}$ is the standard LSDA Hamiltonian, $\hat{H}^{\text{C}}$ is the correlation within the $d$ orbital subspace and $\hat{H}^{\text{SO}}$ the spin-orbital interaction.
We start with the LSDA Hamiltonian in the orthogonal representation of the tight-binding linear muffin-tin orbital method in the atomic sphere approximation (TB-LMTO-ASA) [@A75], $$\hat{H}^{\text{LSDA}} = C + \sqrt{\Delta}S^{\gamma}({\bf k})\sqrt{\Delta},
\label{e:hlmto}$$ where $C$, $\Delta$, and $\gamma$ are the self-consistent standard potential parameters. Because the electron-electron interaction is included in $\hat{H}^{\text{C}}$, the on-site diagonal matrix element of the $d$ orbital is replaced with $C_d=(C^{\uparrow}_d+C^{\downarrow}_d)/2$. $S^{\gamma}({\bf k})$ is the structure constant matrix in the orthogonal representation [@A75] with ${\bf k}$ running over the Brillouin zone (BZ). $\hat{H}^{\text{LSDA}}$ is block-diagonal in the spin index $\sigma$ along the magnetization direction. The spin-orbit coupling matrix elements for $d$ orbitals are calculated in the last iteration of the self-consistent field procedure [@DKS90] and $\hat{H}^{\text{SO}}$ is treated in the usual single-site approximation [@TBF76].
Similarly to the LDA$+U$ method [@SLT98; @AAL97], we treat the screened interaction among the intra-atomic $d$-orbitals in the Hatree-Fock approximation (HFA), $$\begin{aligned}
E^{\text{ee}} = &\frac{1}{2}&\sum^{\sigma\sigma'}_{\{m\}}n^{\sigma}_{m_1m_2}
(U_{m_1m_3m_2m_4} \nonumber \\
&-&U_{m_1m_3m_4m_2}\delta_{\sigma,\sigma'})n^{\sigma'}_{m_3m_4}-E_a,
\label{e:einter}\end{aligned}$$ where $U_{m_1m_3m_2m_4}=\langle m_1,m_3|V^{\text{ee}}|m_2,m_4\rangle$ and $n^{\sigma}_{m_1m_2}$ is the on-site $d$ occupation matrix in the spin-orbital space. $E_a$ is the average interaction without spin and orbital polarization. $U_{m_1m_3m_2m_4}$ are determined by three Slater integrals $F_0$, $F_2$, and $F_4$[@T64], which are linked to three physical parameters: the on-site Coulomb repulsion $U=F_0$, exchange $J =\frac{1}{14}(F_2+F_4)$, and Racah parameter $B=\frac{1}{441}(9F_2-5F_4)$. In terms of $U$ and $J$, $E_a$ is expressed as $\frac{1}{2}Un^2_d-\frac{U+4J}{5}(\frac{n_d}{2})^2$, where $n_d$ is the on-site $d$-orbital occupation. The ratio $F_4/F_2$ is, to a good accuracy, a constant $\sim$0.625 for $d$ electrons [@AAL97], which leads to the estimation $B\approx 0.11J$. The interaction energy $E^{\text{ee}}$, which is rotationally invariant with respect to the basis, leads to an effective potential $H^{{\text{C}}}$ acting on the $d$-orbital subspace.
The spin polarization (SP) in LSDA is generically close to the Stoner concept with an energy related to spin magnetization $m$ of $\Delta E^{\text{LSDA}}_{\text{SP}}(m) = -\frac{1}{2}I(m)m^2$, where $I(m)$ is of the order of 1 eV[@G76]. In the HFA, the average spin splitting for $d$ electrons is driven by $I=\frac{1}{5}(U+4J)$, with OP determined by $U_{\text{eff}}=U-J$ [@B77]. In the limit $B=0$, $U_{m_1m_3m_2m_4}$ only involves two spherical harmonics [@T64], and $U=J=I$ is approximately equivalent to the LSDA [@MM88]. Even with $U_{\text{eff}}=0$, there is no simple relation between $E^{\text{ee}}$ and the orbital moment $\langle\hat{L}\rangle$[@SLT98]. In practice, the problem involving the OPs induced by the Hubbard $U$ and Racah $B$ can be solved numerically by working directly with the site-diagonal elements of the occupation matrix.
The MAE is calculated by taking the difference of two total energies with different directions of magnetization (MAE $= E_{111}-E_{001}$ for cubic structures and MAE $= E_{10\bar{1}0}-E_{0001}$ for hcp structure). The total energies are obtained via fully self-consistent solutions of $\hat{H}$, with the double counting corrections to the total energy included. For the ${\bf k}$-space integration, we use the special point method [@F89] with a Gaussian broadening of 50 meV [@TJEW95]. We use $100^3$ sampling points in the BZ for cubic structure and $100\times100\times56$ points for hcp structure. We have also included the occupation number broadening correction terms to the ground-state total energy [@GF98]. Numerical convergence has been tested against the number of ${\bf k}$-points and Gaussian broadening. We first calculate the electronic structure self-consistently using the scalar relativistic TB-LMTO-ASA method. Then we construct the Hamiltonian $\hat{H}$, using the spin-orbital coupling constants corresponding to the $d$ band center [@DKS90].
Considering the fact that the spin moments are well described by the LSDA, for a particular value of $U$, we have chosen the parameter $J$ such that the magnetic moment maintains the theoretical value from the LSDA without spin-orbit coupling. The calculated spin moments are almost independent of $B$. Because the strong correlation $U$ and $J$ are entangled with the LSDA potential in the LDA$+U$ method [@AAL97], the dependence of the magnetic moment on $U$ and $J$ is not clear. Moreover, the energy $E^{\text{ee}}$ defined in the LDA$+U$ method [@SLT98; @AAL97] is not zero even without any SP and OP. This may render the interpretation of the delicate MAE dependence on $U$ quite difficult. In their LDA$+U$ calculations of Fe and Ni, Yang et al. [@YSK01] scanned the ($U$,$J$) parameter space and obtained the path of $U$ and $J$ values which hold constant the theoretical magnetic moment aligned along the (001) direction. $J$ increases with $U$ in their parameter path, in contrast to the basic concept $\frac{U+4J}{5}\approx I_d$. The SP and OP are treated on the same footing in our HFA scheme. The calculated MAEs versus the values of $U$ are depicted in Fig. \[mfeconif1\]. The corresponding orbital moments are presented in Fig. \[mfeconif2\].
![MAEs of bcc Fe, fcc Co, hcp Co, and fcc Ni as a function of Hubbard $U$. Two curves in $spdf$ basis are plotted for each case, one with $B=0.11J$, the other with $B=0$. The experimental values are indicated by the horizontal dotted lines (1.4, 65, and $-2.7$ $\mu$eV/atom for bcc Fe, hcp Co and fcc Ni[@TJEW95], and 2.0 $\mu$eV/atom for fcc Co [@F98]). The MAEs calculated in $spd$ basis with $B=0.11J$ are also presented.[]{data-label="mfeconif1"}](mfeconi1.ps){width="40.00000%"}
![Orbital magnetic moment for bcc Fe, fcc Co, hcp Co, and fcc Ni along the experimental easy axis as a function of $U$. The plots are labelled the same as FIG. \[mfeconif1\].[]{data-label="mfeconif2"}](mfeconi2.ps){width="40.00000%"}
The ASA does not significantly affect the accuracy of MAE. In fact, the differences in the MAEs and orbital moments calculated by the LMTO-ASA method [@DKS90] and those calculated by the full potential (FP) LMTO method [@TJEW95] are negligible if the same partial wave expansion $l_{\text{max}}=2$ is used. In cubic structures, the difference between the MAEs with $l_{\text{max}}=3$ and $l_{\text{max}}=2$ is very small ($< 0.1$ $\mu$eV/atom)[@HP98]. However for hcp Co, the MAE changed sign when angular moment $l_{\text{max}}$ increased from 2 to 3[@DKS90]. The MAE calculated in the $spdf$ LMTOs is closer to the recent accurate result calculated from FP linearized augmented plane-wave (LAPW) method [@S01]. Because the LMTOs in the $spdf$ basis are more complete than the ones in the $spd$ basis[@A75], we regard the results in $spdf$ basis more reliable.
In the limit $B=0$ and $U\approx J$, our calculated MAEs and orbital moments (the left ends of the $B=0$ plots) lie in the range of the recent high quality LSDA values [@DKS90; @TJEW95; @HP98; @B98; @YSK01; @S01]. In particular, the predicted signs of the MAEs for hcp Co and fcc Ni are wrong in the $spdf$ basis. The calculated orbital moments are about 40% smaller than the experimental values for Fe and Co, while for Ni, it is very close to the experimental value. Since the LSDA is even poorer in Ni than in Fe and Co[@SAS92], the agreement must be accidental.
By turning on the orbital polarization induced by the Racah $B$ at $U\approx J$, the calculated MAEs for Fe and Co are in much better agreement with the experimental data. This is similar to the LSDA+OPB calculations[@TJEW95; @DKS91]. Particularly, the predicted easy axis is correct for hcp Co [@DKS91]. The calculated MAE in $spdf$ basis for hcp Co is quite close to that of the LMTO-ASA calculation with OPB[@DKS91], while our $spd$ MAE is quite close to the FP-LMTO$+$OPB result[@TJEW95]. Interestingly, as shown in Fig. [\[mfeconif2\]]{}, the enhancement of orbital moment due to OP induced by Racah $B$ is in excellent agreement with the OPB calculations [@TJEW95] despite the forms of $E^{\text{ee}}$ and $\Delta E_{\text{OPB}}$ being quite different. We suggest that the widely used OPB can be brought [*precisely into accord with*]{} the unrestricted HFA with $\Delta E_{\text{OPB}}$ replaced by $E^{\text{ee}}$ at $U=J$ ($\frac{n_d}{2}$ in $E_a$ replaced with $n^{\uparrow}_d$ and $n^{\downarrow}_d$). Only with the OP induced by $B$, the predicted sign of MAE for fcc Ni is wrong.
We now study the effect of OP induced by $U$ with $U_{\text{eff}}>0$. Similarly to the OP induced by $B$, as shown in Fig. \[mfeconif2\], the OP induced by $U$ enhances the orbital magnetic moments. For both Fe and Co, the MAEs change very smoothly and monotonically with increasing $U$. For Ni, the MAE is about zero when $U\approx 1.3$ eV. It decreases with increasing $U$, then making a flat region with MAE$\approx-1$ eV from $U=1.7$ to $U=2.5$. After the flat region, the MAE decreases with increasing $U$, reaching the experimental value at $U=2.95$ eV. The MAE predicted here changes very smoothly with increasing $U$ and with correct sign when $U>1.3$ eV, without the strong sensitivity observed in the LDA$+U$ calculations [@YSK01].
bcc Fe fcc Co hcp Co fcc Ni
------- -------- -------- -------- --------
$U$ 1.15 1.41 1.77 2.95
$J$ 0.97 0.83 0.75 0.28
$l_z$ 0.087 0.123 0.150 0.064
expt. 0.08 0.14 0.05
: $U$ and $J$ (in eV) corresponding to experimental magnetic anisotropy energy (in $spdf$ basis with $B=0.11J$). The calculated orbital moments $l_z$ ($\mu_{\text{B}}$/atom) along easy axis are compared with the experimental data [@TJEW95].[]{data-label="mfeconit1"}
When $U>2.5$ eV, the two MAE curves of $B=0$ and $B=0.11J$ are almost indistinguishable for fcc Ni. Thus we conclude that it is the Hubbard $U$ that is fully responsible for bringing theory into accord with experiment. For Fe and Co, both OPs induced by $U$ and $B$ are needed to produce the experimental MAEs. As shown in Table \[mfeconit1\], for bcc Fe and hcp Co, the optimized $U$ and $J$ almost simultaneously give the experimental MAEs and orbital moments. The predicted orbital moment for fcc Ni is slightly higher, but quite acceptable. The optimized values of $U$ and $J$ and their trend from Fe to Ni are very similar to those determined from experiments and used in other theories [@SAS92; @LKK01]. The optimized values of $U$ for fcc Co and hcp Co are close but not the same. This may be due to the fact that the experimental MAE of fcc Co was extracted from measurement on supported films [@F98] or that the calculated MAE of hcp Co is not fully converged even with $l_{\text{max}}=3$.
![Band structure of fcc Ni along $\Delta_Z=2\pi/a(0,0,l)$ with the spin quantization axis in the (001) (solid lines) and the (111) (dashed lines) directions. Here $0.4\leq l\leq 1.0$. $l=1.0$ corresponds to the $X_Z$ point.[]{data-label="mfeconif3"}](mfeconi3.ps){width="35.00000%"}
It was conjectured that the failure of the LSDA to predict the MAE of fcc Ni is related to the band structure along $\Gamma X$ direction [@DKS90]. As shown in Fig.\[mfeconif3\], similarly to the LSDA bands[@DKS90], five bands cross the Fermi energy almost at the same ${\bf k}$ point when $J\approx U=0.9$ eV and $B=0$ (The bands for $B=0.11J$ are quite similar). One of the $d$ bands is just above the Fermi level at the $X$ point and results in the appearance of small $X_2$ pocket on the Fermi surface[@DKS90; @YSK01], which has not been found experimentally. Increasing the valence electrons and thus pushing down the band corresponding to the $X_2$ pocket, Daalderop et al. [@DKS90] found the correct easy axis for fcc Ni. We find that the $X_2$ pocket disappears at $U\approx 1.5$ eV, corresponding to the start point of the correct sign of the MAE. With increasing $U$, this band is further pushed down and when $U\approx 3$ eV, it is below the Fermi energy by about 0.2 eV. In our approach, the disappearance of the $X_2$ pocket is a natural result of the OP induced by Hubbard $U$.
As a delicate property, the MAE is naturally expected to depend on the delicate changes of the band structures. We have applied the current scheme to the parametrized TB models of Fe and Ni fitting to the APW bands in the LSDA [@DAP86]. The trend and optimized values of $U$ and $J$ are found to be very similar to the [*ab initio*]{} TB calculations. This underlines the importance of how to treat the intra-atomic strong correlation.
In summary, we have calculated the MAEs of Fe, Co, and Ni from the [*ab initio*]{} tight-binding total energies. Disentangling the strong correlation among the intra-atomic $d$ orbitals with the Hamiltonian in the LSDA and therefore treating the SP and OP on the same footing, we have solved the long-standing notorious problem of the MAE of fcc Ni. The discussions on the OPs induced by Hubbard $U$ and Racah $B$ can shed light on future first principles and model TB calculations. How to calculate the interaction parameter $U$ in metallic environment directly from first principles is still an open problem.
This work was supported by the EU through the AMMARE project (Contract No. G5RD-CT-2001-00478) under the Competitive and Sustainable Growth Programme.
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|
---
abstract: 'We present exact analytical results for the statistics of nonlinear coupled oscillators under the influence of additive white noise. We suggest a perturbative approach for analysing the statistics of such systems under the action of a determanistic perturbation, based on the exact expressions for probability density functions for noise-driven oscillators. Using our perturbation technique we show that our results can be applied to studying the optical signal propagation in noisy fibres at (nearly) zero dispersion as well as to weakly nonlinear lattice models with additive noise. The approach proposed can account for a wide spectrum of physically meaningful perturbations and is applicable to the case of large noise strength.'
address:
- 'B.I. Verkin Institute for Low Temperature Physics and Engineering, NASU, 47 Lenin Av., 61103, Kharkov, Ukraine'
- 'Photonics Research Group, Aston University, Birmingham, UK, B4 7ET'
author:
- 'Jaroslaw E. Prilepsky'
- 'Stanislav A. Derevyanko'
title: 'Statistics of noise-driven coupled nonlinear oscillators: applications to systems with Kerr nonlinearity'
---
and
Stochastic dynamics, nonlinear oscillators, Fokker-Planck equation, nonlinear optics 05.10.Gg ,42.81.Dp
Introduction
============
Weakly nonlinear coupled systems (discrete self-trapping model, discrete nonlinear Shrödinger equation, etc.), belong to the universal, widely applicable, highly illustrative and thoroughly studied models of nonlinear physics. These models have applications to molecular crystals, molecular dynamics, nonlinear optics, biomolecular dynamics, and so on (see e.g. [@e90; @Elbeck85; @krb01; @Elbeckonline] and references therein). It was shown that they exhibit unusual dynamical phenomena pertaining only to the nonlinear discrete systems, like existence of intrinsic localised modes etc. [@cfk03]. Apart from the dynamical behaviour the influence of random force and statistical properties of nonlinear lattices are also subjects of keen interest [@rcj98; @mt94; @rck00; @f02]. The complex dynamics of such systems is determined by an interplay between such factors as randomness, discreetness and nonlinearity. The results concerning the statistics of coupled nonlinear oscillators driven by Gaussian white noise, given below, can be straightforwardly applied to the class of problems which are naturally discrete, e.g. for the noisy self trapping model and its modifications. The second (but not less important) purpose of the current paper is to show that the stochastic discrete models with finite number of degrees of freedom can also be applied to the *regularised stochastic continuous systems* and can have a much broader field of application. More specifically, in the spirit of the original idea of Mecozzi [@Mec1; @Mec2], we show that the model of randomly driven coupled nonlinear oscillators can be used for studying the statistics of signal propagation in a nonlinear optical fibre with inline noise sources.
A universal model describing the propagation of the wave envelope in a weakly nonlinear dispersive media (which include optical fibres) is the Nonlinear Shrödinger Equation (NLSE) and its scalar and vector modifications [@Mecozzi; @Agraval; @aa00]. The need for the profound exploration of noise-stimulated fluctuations of signal parameters in an optical fibre stems from the great practical importance of such study for immediate technical purposes [@Mecozzi; @Agraval]. The pulse propagation inside an optical fibre with inline amplifiers is described by the perturbed NLSE with additive noise, which accounts for amplifier spontaneous emission (ASE)[@Mecozzi]. It provides an illustrative example of a complicated continuous noisy system where the exact results concerning the statistics of the signal are still very scarce owing to the complexity of the problem (see for instance, review article [@Hausrev] and also [@fklt; @OSA03] for recent results). However if we assume that the noise is delta correlated in both time and space a natural regularisation occurs, which transforms our continuous system into a system of coupled nonlinear oscillators subjected to an external white noise. The statistical properties of such regularised discrete system are much easier to derive than those of the original continuous systems. In particular one can use the Fokker-Planck equation (FPE) approach to obtain the evolution of the probability density function (PDF) for all discrete components of the signal. But even the discrete system corresponding to noisy NLSE and its modifications is still very complicated because of the coupling between the oscillators, which is due to the dispersion. Therefore Mecozzi [@Mec1; @Mec2] considered propagation of an optical pulse affected by joint action of Kerr nonlinearity and additive white Gaussian noise (WGN) at zero dispersion, i.e. the dispersive (second time derivative) term in the NLSE was dropped. The resulting effective dispersion free equation (after the regularisation procedure) formally coincides with the dynamical equation for a single nonlinear oscillator driven by a noisy external force.
In the present paper we treat a zero dispersion stochastic system (a single noise-driven nonlinear oscillator) and its straightforward generalisations (nonlinearly coupled noisy oscillators) as basic systems and consider various small deterministic perturbations which can be of quite arbitrary nature: higher order nonlinearities, small residual dispersion, etc. In particular we can add a perturbation which will be the direct discrete analogue of that of the corresponding continuous optical system. Other continuous models yielding nonlinear discrete equations addressed in the current paper, include higher dimensional systems of nonlinearly coupled NLSEs. Two nonlinearly coupled oscillators with particular coupling coefficients correspond to well known Manakov equations (see e.g. [@lk97]) with zero dispersion driven by white noise. This continuous model describes the nonlinear pulse propagation in a noisy birefringent fibre at nearly zero average dispersion. Systems of more than three coupled NLSEs are also of physical relevance [@kl01]. In addition to optical communications, in the context of biophysics the case of three nonlinearly coupled NLSEs can be applied for studying the propagation of solitons along the three spines of an alpha helix in protein [@s84]. Systems involving higher numbers of coupled NLSEs have applications in the theory of optical soliton wavelength division multiplexing [@cas95], multichannel biparallel-wavelength optical fibre networks [@yb98], and so on. Hence we believe that the study of the stochastic properties of such models (which, to the best of our knowledge, is a relatively undeveloped field) is also of great interest and may have a significant practical outcome.
As mentioned earlier, we describe the system statistics in terms of the probability density function for the discrete variables and the evolution of the PDF is governed by the FPE [@Risken]. In the case of an unperturbed system of noise-driven nonlinear oscillators (which is a discrete analogue of a system of zero dispersion noisy NLSEs), the corresponding FPE can be solved analytically. The main idea of our approach is to apply the perturbation theory directly to the FPE rather than to the initial stochastic system, using analytical results for the base system. The advantage of such a method is that it allows one to obtain the PDF for the signal output directly as a series in powers of a small parameter $\varepsilon$, which is the effective “strength” of the perturbation. The perturbation theory is based on the propagator of the unperturbed FPE. Once derived, the propagator can be applied to various discrete systems, e.g. regularised systems of weakly coupled NLSE at zero dispersion, where the coupling is considered as a perturbation. This means we can incorporate the effects of coupling and consider the statistics of nonlinear lattice models in the so-called anticontinuum limit (weak intersite linear coupling). In the context of optical applications we are able to consider real polarisation mode dispersion (PMD) systems as well as systems with wavelength division-multiplexing (WDM) [@Agraval].
The paper is organised in the following way. In section \[sec:intro\] we introduce basic models of one and more noise-driven coupled nonlinear oscillators and write down the Fokker-Planck equation for each model. After that we consider noisy scalar and vector NLSE, describing the propagation of a signal in the optical fibre, and show that after the regularisation (i.e. after the introduction of a discrete time variable) the system becomes equivalent to one of the “base” perturbed systems. We also mention the stochastic discrete self-trapping model as an example of naturally discrete system which can be analysed perturbatively. In section \[sec:FP\] we proceed to find the propagators of FPEs derived for noisy oscillator systems. Using the obtained propagators in section \[sec:pertub-local\] we advance to build a perturbative expansion pf the PDF for different types of perturbations. In this section we consider non-dispersive perturbation of one and more coupled zero dispersion NLSE and provide explicit expressions for several typical example systems. In section \[sec:nonlocal\] we move to more complicated systems, i.e. to the scalar or vector NLSEs, where second dispersion is treated as a perturbation. In the conclusion we summarise the results and outline the key features and perspectives for the approach proposed.
Basic models, equations and regularisation procedure {#sec:intro}
====================================================
In first two subsections we write down the explicit form for the FPEs attached to the system of stochastic equations, governing the evolution of oscillator statistics. Then we explain how these systems can be employed in the context of optical propagation and nonlinear lattices.
Single noisy nonlinear oscillator {#sec:intro:1}
---------------------------------
By adding a white noise term to the (dimensionless) dynamical equation for a nonlinear Kerr oscillator one gains the Langevin equation for the complex field $u= x+{\mathrm{i}}y$: $$\label{1rot}
\frac{{\mathrm{d}}u}{{\mathrm{d}}\zeta} = \mathrm{i}|u|^2u+\eta(\zeta)\, .$$ In the mechanical interpretation $x$ is a position and $y$ is a velocity of the nonlinear oscillator or $x$ and $y$ are the components of a torque for the nonlinear rotator; $\zeta$ has a meaning of time and $\eta(\zeta) = \eta_1(\zeta)+{\mathrm{i}}\eta_2(\zeta)$ is the complex white Gaussian noise with the following correlation properties: $$\label{1rotn}
<\eta_i(\zeta)\eta_j(\zeta')>=2D
\delta_{ij}\,\delta(\zeta-\zeta'),$$ with $D$ being the noise strength. Equivalently one can decompose the real and imaginary parts of [(\[1rot\])]{} and obtain a system of two coupled Langevin equations: $$\label{1rotreal}
\frac{{\mathrm{d}}x}{{\mathrm{d}}\zeta} =-(x^2 + y^2) \, y + \eta_1(\zeta) \, ,
\quad \frac{{\mathrm{d}}y}{{\mathrm{d}}\zeta} = (x^2+y^2) \, x +\eta_2(\zeta) \,
.$$ Introducing the 2D vectors ${{\mbox{\boldmath$q$}}}=\{x,y \}$ and ${{\mbox{\boldmath$\eta$}}}= \{
\eta_1, \eta_2\}$ we can write equation [(\[1rotreal\])]{} in general vector form: $$\frac{{\mathrm{d}}{{\mbox{\boldmath$q$}}}}{{\mathrm{d}}\zeta}={{\mbox{\boldmath$f$}}}({{\mbox{\boldmath$q$}}})+{{\mbox{\boldmath$\eta$}}}(\zeta).
\label{Langevin}$$ Here ${{\mbox{\boldmath$f$}}}$ is a deterministic “advection vector” (note that sometimes in the literature it is written with a minus sign). This is the canonical form of the Langevin equation with additive WGN. We will be interested in the statistics of the solution of Eq.(\[Langevin\]) for a given $\zeta$. Namely, we will seek the conditional PDF of the vector ${{\mbox{\boldmath$q$}}}$, $P({{\mbox{\boldmath$q$}}},\zeta)$, provided that at the initial moment ($\zeta=0$) the PDF is given. This function (see [@Risken]) obeys the following second order partial differential equation: $$\frac{\partial P}{\partial \zeta}=\sum_i \{ - \partial_i \big[ f_i
({{\mbox{\boldmath$q$}}}) P \big] + D \, \partial^2_i P \} \, , \label{canonical}$$ which is a FPE attached to system (\[Langevin\]) with the additive white noise, given by (\[1rotn\]). Also one has to impose the normalisation condition $\int_{{{\mbox{\boldmath$q$}}}} P \, {\mathrm{d}}{{\mbox{\boldmath$q$}}}
=1$. A propagator is a special solution of [(\[canonical\])]{} with the initial condition $P({{\mbox{\boldmath$q$}}}, 0) = \delta({{\mbox{\boldmath$q$}}} - {{\mbox{\boldmath$q$}}}^0)$, i.e. it corresponds to the conditional PDF provided that at the initial moment the system is in a deterministic state ${{\mbox{\boldmath$q$}}}^0$. The explicit form of the FPE for single oscillator system [(\[1rot\])]{} is: $$\label{1rotfp} \frac{\partial P}{\partial
\zeta}=D\left(\frac{\partial^2}{\partial
x^2}+\frac{\partial^2}{\partial y^2}\right) P - (x^2+y^2)\left(x
\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}\right)P
\, .$$ The boundary conditions for [(\[1rotfp\])]{} are chosen as follows: the PDF, $P({{\mbox{\boldmath$q$}}}, \zeta)$, must have no singularities and should decrease rapidly as $|{{\mbox{\boldmath$q$}}}| \to \infty$ to provide the normalisation.
System of nonlinearly coupled oscillators {#sec:intro:2}
-----------------------------------------
Next we study the following system of $M$ coupled nonlinear oscillators: $$\begin{aligned}
\frac{{\mathrm{d}}u_1}{{\mathrm{d}}\zeta}& =& {\mathrm{i}}u_1 \sum_{i=1}^M a_{1\,i}
|u_i|^2 + \eta^1(\zeta)\nonumber \\
\vdots && \label{Krot} \\
\frac{{\mathrm{d}}u_M}{{\mathrm{d}}\zeta} & = & {\mathrm{i}}u_M \sum_{i=1}^M a_{Mi}
|u_i|^2 + \eta^M(\zeta) \, . \nonumber\end{aligned}$$ Here again $u_i = x_i + {\mathrm{i}}y_i$, are the complex fields, $a_{ij}$ are real constant coefficients and the correlation properties for complex WGNs $\eta^k=\eta^k_1+{\mathrm{i}}\eta^k_2$ are (here and further on the subindexes of complex WGNs, “1" and “2", imply the real and imaginary parts correspondingly): $$\label{2rotn}
<\eta^k_i(\zeta)\eta^l_j(\zeta')>=2D_k \delta_{kl}
\delta_{ij}\,\delta(\zeta-\zeta'),$$ (here no summation over the repeated index $k$ implied). To be more general we suppose that WGNs $\eta^k(\zeta)$ may have (possibly) different intensities $D_k$. Again we can introduce 2M-dimensional real vectors ${{\mbox{\boldmath$q$}}}=\{x_1,\ldots,x_M,y_1,\ldots
y_M\}$ and ${{\mbox{\boldmath$\eta$}}}=\{\eta_1^1,\ldots \eta_1^M, \eta_2^1, \ldots
\eta_2^M\}$, and after singling out real and imaginary parts in [(\[Krot\])]{} arrive at the generic Langevin equation [(\[Langevin\])]{}. The FPE for this system has the form $$\label{2rotfp} \frac{\partial P}{\partial
\zeta}= \sum_{i=1}^M \left( D_i \Delta_i - \sum_{j=1}^M
a_{i\,j}(x_j^2+y_j^2) \left[ x_i \frac{\partial}{\partial
y_i}-y_i\frac{\partial}{\partial x_i} \right] \right)P \, .$$ Here $\Delta_i = \partial^2/\partial x_i^2 + \partial^2/\partial
y_i^2$. We shall call the number $M$ of nonlinearly coupled equations in (\[Krot\]) the *dimensionality* of the system (the number of independent variables in the FPE (\[2rotfp\]) is then $2M$).
Regularisation procedure for optical pulse propagation: discrete time {#sec:intro:3}
---------------------------------------------------------------------
The propagation of a complex light envelope $u(\zeta,t)$ in a noisy optical fibre line with so-called weak dispersion management (see Refs. [@Mecozzi; @Tur00]) is described by the stochastic NLSE: $$\frac{\partial u}{\partial
\zeta}=-\mathrm{i}\,\frac{\bar{\beta}}{2}\, \frac{\partial^2
u}{\partial t^2}+ \mathrm{i}|u|^2u+\eta (t,\zeta).\label{nlse}$$ (Note that here $\zeta$ plays the role of coordinate in contrast to the mechanical interpretation of oscillators and “virtual” time is $t\,$.) In equation [(\[nlse\])]{} the stochastic term $\eta(t,\zeta)$ represents a complex WGN. The following correlation properties are assumed: $$\begin{split}
&<\eta(t,\zeta)>= <\eta(t,\zeta)\eta(t',\zeta')> =0, \\
&<\eta^*(t,\zeta)\eta(t',\zeta')>= 2 G \delta(\zeta-\zeta')
\delta(t-t'). \label{c1}
\end{split}$$ Here $G$ is a normalised noise strength. Equation (\[nlse\]) describes the path-averaged model of the dispersion managed optical fibre communication system and, it was shown (see [@Tur00] and references therein for details) that this model is applicable when the so-called strength of the map, proportional to the local fibre dispersion, to the dispersion oscillation period and to the squared reciprocal pulse width, is small. This strength parameter characterises the effect of the variation of local dispersion. Parameter $\bar{\beta}$ entering Eq.(\[nlse\]) is just a path-averaged dispersion coefficient. A zero dispersion NLSE occurs when the path-averaged dispersion coefficient is equal to zero, $\bar{\beta}=0$. The statistics of system (\[nlse\]) at zero dispersion point were first studied in [@Mec1; @Mec2] (see also [@PRL03]). In these Refs. an analytical formula for the conditional PDF of the output signal $u(t,L)$ given the deterministic input signal $u_0(t,0)$ was obtained.
Here we would like to consider the zero-dispersion NLSE assuming also that there is an addition corresponding to a deterministic perturbation: $$\label{nl}
\frac{\partial u}{\partial \zeta}=\mathrm{i}|u|^2u+\eta(t,\zeta) +
\varepsilon \hat N[u, \ldots] \, .$$ Here $\varepsilon \ll 1$ is intended and let $\hat N[...]$ be any sort of perturbation of the initial zero-dispersion NLSE: it may be either a “nonlocal” operator involving the time $\partial/\partial t$ derivatives $\hat N[u,u'_t,u''_{tt},
\ldots]$, or some kind of a “local” operator, $\hat N[u]$. Note that complex WGN defined by expression (\[c1\]) is equivalent to the two independent real WGN, $\eta(t,\zeta)=\eta_1(t,\zeta)+\mathrm{i}\eta_2(t,\zeta)$, with $$\begin{split}
<\eta_1(t,\zeta)>=&<\eta_2(t,\zeta)>=<\eta_1(t,\zeta)\eta_2(t',\zeta')>=0
\, ,
\\ <\eta_1(t,\zeta)\eta_1(t',\zeta')> = &<\eta_2(t,\zeta)\eta_2(t',\zeta')>=
G \delta(\zeta-\zeta') \delta(t-t') \, . \label{mean}
\end{split}$$ Because the white noise is delta correlated in both time and space, it has an infinite average power $<|\eta(t,\zeta)|^2>$. This is the consequence of the idealised character of the white noise. In practice, real noise always has small but finite correlation length $r_c$ and correlation time $\tau_c$. For ASE in optical fibre transmission links the correlation length is in the order of the average amplifier spacing $L_a$, and correlation time is inversely proportional to the bandwidth of the noise: $\tau_c
\sim B^{-1}$. In what follows we will consider our noise still delta correlated in space. However to make the correspondence between discrete and continuous systems explicit and also to get physically meaningful results we move to the context of discrete signals and introduce discrete time variable $t$. We will consider the case where both signal and noise are entirely contained to the time interval of length $T$. Then, to sample the signal effectively, we need a finite number of discrete samples, separated by an interval $\Delta t \sim \tau_c \ll T$. If we put $\Delta t = B^{-1}$, the number of samples is $N=BT$. The signal is represented as $N$-dimensional complex vector, ${{\mbox{\boldmath$u$}}}={{\mbox{\boldmath$x$}}}+\mathrm{i}{{\mbox{\boldmath$y$}}}$, in the sample space: $$u(t,\zeta) \Rightarrow {{\mbox{\boldmath$u$}}}(\zeta), \quad u_i(\zeta)=u(i\Delta
t,\zeta), \quad i=1,\ldots N \, . \label{regul}$$ Analogously one can work in the real $2N$-dimensional space of real and imaginary parts of vector ${{\mbox{\boldmath$u$}}}$: ${{\mbox{\boldmath$q$}}}=\{x_1,\ldots,x_N,y_1,\ldots,y_N\}$. After the regularisation Eq.(\[nl\]) is equivalent to the system of equations: $$\label{nl1}
\frac{{\mathrm{d}}u_i}{{\mathrm{d}}\zeta}=\mathrm{i}|u_i|^2u_i+\eta^i(\zeta) +
\varepsilon \hat N_i[{{\mbox{\boldmath$u$}}}] \, ,$$ Vector ${{\mbox{\boldmath$\eta$}}}(\zeta)={{\mbox{\boldmath$\eta$}}}_1(\zeta)+\mathrm{i}{{\mbox{\boldmath$\eta$}}}_2(\zeta)$ is a regularised white noise with the same properties as given by (\[mean\]) but with regularised temporal delta functions: $$\label{cor-r}
<\eta^i_{1}(\zeta)\eta^j_{1}(\zeta')>=
<\eta^i_{2}(\zeta)\eta^j_{2}(\zeta')>= 2D
\delta_{ij}\,\delta(\zeta-\zeta'),$$ where we set $D \equiv BG$. The operator $\hat{N}_i$ is discretised in time as well.
We call the equation, governing the unperturbed stochastic evolution of each separate sampled component, the *base system*. For all cases considered in this paper, a base system will be either of the form Eq.(\[1rot\]) or Eq.(\[Krot\]); the corresponding FPE for each base system has either a form (\[1rotfp\]) or a more general look (\[2rotfp\]), depending on the dimensionality of the base system.
One can see that each sampling component in Eq.(\[nl1\]) obeys the equation governing the noisy dynamics of a single nonlinear oscillator, Eq.(\[1rot\]): hence the dimensionality of the corresponding base system is one. The complete set of equations for all components of the sampled signal has a form of perturbed Eq.(\[Krot\]) with $M=N$ and an identity matrix of $a$-coefficients: $a_{ij}=\delta_{ij}$. As was noted before, we divide the perturbations in equation (\[nl1\]) into two categories: “local”, which depend on the value of the signal at the current moment of time only, i.e. $\hat{N}_i[{{\mbox{\boldmath$u$}}}]=\hat{N}_i[u_i]$, and “nonlocal” (as, for instance, dispersion, involving second or higher time derivatives of the signal), which comprise the field values at other sampling points.
The same regularisation procedure can be performed with other models relevant to the noisy signal propagation. For example we can consider the propagation in the birefringent fibres where instead of NLSE, under certain conditions, we have a set of Manakov equations [@lk97] perturbed by noise: $$\label{man}
\begin{split}
\frac{\partial u}{\partial
\zeta} &= -{\mathrm{i}}\frac{\bar \beta}{2} \frac{\partial^2 u}{\partial
t^2} + {\mathrm{i}}u \left( |u|^2 + |v|^2\right) +
\eta^1(\zeta,t) \, ,\\
\frac{\partial v}{\partial \zeta} &= -{\mathrm{i}}\frac{\bar \beta}{2}
\frac{\partial^2 u}{\partial t^2} + {\mathrm{i}}v \left( |u|^2 +
|v|^2\right) + \eta^2(\zeta,t) \, .
\end{split}$$ Here $u$ and $v$ stand for complex components of the left and right polarised waves and $\eta^i(\zeta,t)$ are the complex WGNs. Without noise addition this system describes the processes of the two-dimensional stationary self-focusing and one-dimensional auto-modulation of electromagnetic waves with arbitrary polarisation. Neglecting the dispersion and adding perturbation, after the regularisation one obviously gains: $$\label{man1}
\begin{split} \frac{{\mathrm{d}}u_i}{{\mathrm{d}}\zeta}&=\mathrm{i}(|u_i|^2+|v_i|^2)u_i +
\eta^1_i(\zeta) + \varepsilon \hat
N^1_i[{{\mbox{\boldmath$u$}}},{{\mbox{\boldmath$v$}}}] \, , \\
\frac{{\mathrm{d}}v_i}{{\mathrm{d}}\zeta}&=\mathrm{i}(|u_i|^2 + |v_i|^2) v_i +
\eta^2_i(\zeta) + \varepsilon \hat N^2_i[{{\mbox{\boldmath$u$}}}, {{\mbox{\boldmath$v$}}} ] \, .
\end{split}$$ In this case, for each separate sampled pair $[u_i,v_i]$ (if $\varepsilon$ is again set to zero), we arrive at the system of two nonlinearly coupled oscillators. This is just a special case of Eq.(\[Krot\]), where the number of dimensions of the base system is two. The whole system for $K=2N$ sampled components of the signal ($N$ is, as previously, the number of the sampling time-points) has the form of Eq.(\[Krot\]) as well. However, as the base system has a dimensionality two, here the coupling matrix $a_{ij}$ is no longer an identity matrix but have a $(2 \times
2)$-block diagonal form. The “local” perturbations for such systems are those which satisfy: $\hat N_i[{{\mbox{\boldmath$u$}}} , {{\mbox{\boldmath$v$}}}] = \hat
N_i[u_i,v_i]$.
In the same way one could deal with the systems involving higher number $M$ of coupled NLSEs: the number of equations would simply define the dimensionality $M$ of the corresponding base system. The regularised system for the whole set of time-sampled signal components would possess the matrix $a_{ij}$ in a block-diagonal form, where each block would be an $(M \times M)$ square matrix.
Nonlinear lattice model {#sec:intro:4}
-----------------------
The dynamics of a weakly nonlinear lattice is described by the set of differential-difference equations (the discrete self-trapping equation (DSTE), see Refs.[@e90; @Elbeck85; @Elbeckonline]): $$\label{dste}
\frac{{\mathrm{d}}u_i}{{\mathrm{d}}\zeta} = \mathrm{i}|u_i|^2u_i + {\mathrm{i}}\,
\varepsilon
\sum_{j} m_{ij} u_j +\eta^i(\zeta)\, ,$$ where we have also added the white noise terms $\eta^i(\zeta)$ at each site (with the same statistical properties as in [(\[cor-r\])]{}). In Eq.[(\[dste\])]{} $m_{ij}$ is a coupling matrix and index $j$ ranges over the whole 1D lattice containing $N$ sites. The generalisation for lattices of higher dimensions is straightforward. In physical applications $m_{ij}$ is real and symmetric; in the case of tridiagonal matrix, i.e. for the nearest neighbor coupling, we recover a particular realisation of DSTE – the discrete nonlinear Shrödinger equation [@krb01].
Equation (\[dste\]) provides an example of a naturally discrete system with Kerr nonlinearity. If we consider the case of vanishing $\varepsilon$, the so-called anticontinuum limit, we arrive at the system of uncoupled oscillators, which is obviously the unperturbed system [(\[nl1\])]{}. Here the dimensionality of the base system is one. The perturbation in Eq.[(\[dste\])]{} provides an example of a nonlocal perturbation, since it involves other lattice sites.
The list of the notations {#sec:intro:5}
-------------------------
In this subsection we summarise the notations that will be used throughout the paper. The aim of the proposed notations and indexing scheme is to demonstrate the general ideology and to underline the similarity of all final expressions: very different physical systems after the regularisation yield similar formal expansions for the PDF.
[| p[3.5cm]{} | p[9.7cm]{} |]{}
**Notation**
&
**Meaning**
\
${{\mbox{\boldmath$u$}}}$, ${{\mbox{\boldmath$v$}}}$, $u_i$, $v_i$ & The sampled complex field variables. The bold symbols indicate the whole sets ${{\mbox{\boldmath$u$}}} =
\{u_1, u_2, \ldots \}$ etc. The subscript indicates the sampling time-point number.\
$x_i$, $y_i$ and $r_i$, $\phi_i$ & Real and imaginary parts and modulus-phase representation of the complex field variables. The subscript shows the sampling number (in some examples the upper index will be used as well).\
${{\mbox{\boldmath$r$}}}$, ${{\mbox{\boldmath$\phi$}}}$, ${{\mbox{\boldmath$q$}}}$, $\tilde{{{\mbox{\boldmath$q$}}}}$ & Bold symbols denote sets ${{\mbox{\boldmath$r$}}} \equiv \{r_1, r_2, \ldots \}$, $ {{\mbox{\boldmath$\phi$}}}\equiv
\{\phi_1, \phi_2,\ldots \}$; where indicated explicitly, the subindex shows the dimensionality of the set. Vectors ${{\mbox{\boldmath$q$}}}$ and $\tilde{{{\mbox{\boldmath$q$}}}}$ are the joint sets: ${{\mbox{\boldmath$q$}}}_K \equiv \{x_1, x_2,
\ldots, x_K, y_1, y_2, \ldots, y_K \}$, and $\tilde{{{\mbox{\boldmath$q$}}}}_K
\equiv \{r_1, r_2, \ldots, r_K, \phi_1, \phi_2, \ldots, \phi_K
\}$. The superscript “0” marks the initial deterministic values. The upper index is also used to distinguish between the different sets.\
${\hat{\mathcal{L}}}_K$ & The coordinate part of the FP-operator. The subscript shows the (half)number of independent variables involved.\
$P_M({{\mbox{\boldmath$q$}}}; \zeta)$, $P^K({{\mbox{\boldmath$q$}}}; \zeta)$, ${\mathcal{P}}^K({{\mbox{\boldmath$q$}}};
\zeta)$ & $P_M({{\mbox{\boldmath$q$}}}; \zeta)$, $P^K({{\mbox{\boldmath$q$}}}; \zeta)$ are the PDFs for the base system and for the whole sampled signal. For convenience with a subscript we shall explicitly mark the dimensionality of the underlying base system $M$, i.e. for $P_M$ we have ${{\mbox{\boldmath$q$}}} = {{\mbox{\boldmath$q$}}}_M$. A superscript will be used to mark the whole PDF, and to show the total (half)number of the independent variables involved. If the dimensionality of the base system is $M$, and the number of sampling points is $N$, then $K=M \times
N$, ${{\mbox{\boldmath$q$}}} = {{\mbox{\boldmath$q$}}}_K$. The calligraphic ${\mathcal{P}}^K$ will mark the PDFs of the perturbed systems. The tilde above the PDF will mean the representation in “polar coordinates”, $\{ {{\mbox{\boldmath$x$}}} , {{\mbox{\boldmath$y$}}} \}
\to \{{{\mbox{\boldmath$r$}}} , {{\mbox{\boldmath$\phi$}}}\}$.\
[| p[3.5cm]{} | p[9.7cm]{} |]{}\
$\hat N_i[\ldots]$, $\hat{{\mathcal{N}}}[\ldots]$ & $\hat N_i[\ldots]$ and $\hat{{\mathcal{N}}}[\ldots]$ are the regularised perturbation operator and its FP-image correspondingly. The explicit form of the FP-image is derived using the general rules for the construction of the FPE [@Risken].\
${{\mbox{\boldmath$\eta$}}}= \{ \eta^1,\eta^2, \ldots \}$ & Complex regularised WGN. The superscript is used to distinguish between the different independent WGNs entering different equations.\
${\mathcal{G}}_M(\ldots)$, ${{\mbox{\boldmath${\mathcal{G}}$}}}_K(\ldots)$ & ${\mathcal{G}}_M(\ldots)$ is the propagator of the base system FPE with the coordinate part ${\mathcal{L}}_M$. Symbol ${{\mbox{\boldmath${\mathcal{G}}$}}}_K$ will mark the propagator of the whole FPE of the entire sampled system.\
Propagator functions of the unperturbed Fokker-Plank equation {#sec:FP}
=============================================================
We have just shown that after the regularisation all the models considered in the previous section yield same type of system of Langevin equations, which is merely a perturbed system of coupled nonlinear oscillators (\[Krot\]). Now consider the statistical properties of unperturbed system (\[Krot\]). The FPE attached to system [(\[Krot\])]{} is of course Eq.[(\[2rotfp\])]{}. In this section we will calculate a propagator for Eq.[(\[2rotfp\])]{} which will allow us to build an effective perturbative expansion for the PDF in the next sections.
First we note that in some examples (like for instance a scalar regularised zero dispersion NLSE [(\[nl1\])]{} with $\varepsilon=0$) the coupling matrix $a_{ij}$ is diagonal, and the dimensionality of the base system is one. In that case FPE [(\[2rotfp\])]{} admits factorised solutions in the form: $$P^K({{\mbox{\boldmath$q$}}}_K;\zeta)=P(x_1,\ldots,x_K,y_1,\ldots y_K;\zeta)=
\prod_{i=1}^{K} P_1(x_i,y_i;\zeta) \, , \label{factor}$$ where $P_1(x,y;\zeta)$ is a PDF which satisfies unperturbed FPE for a single oscillator [(\[1rotfp\])]{}. The factorised form of Eq.(\[factor\]) is a natural manifestation of the fact that for diagonal $a_{ij}$ all nonlinear oscillators in system [(\[Krot\])]{} become dynamically uncoupled and, because of [(\[2rotn\])]{}, statistically uncorrelated. Therefore in order to study the statistics of such a system it is sufficient to study the statistics of a single oscillator [(\[1rot\])]{}. Bearing this in mind we initially consider a FPE for a single oscillator [(\[1rotfp\])]{} and find its propagator, and then utilise the obtained results to find the propagator of generic system [(\[2rotfp\])]{} by virtue of Eq.(\[factor\]).
The same is true for the unperturbed systems for which a base system has a higher number of dimensions, say $M$. In such a case the coupling matrix $a_{ij}$ is $(M \times M)$-block-diagonal, and the expression for the total PDF has again a factorised form: $$P^K({{\mbox{\boldmath$q$}}}_K;\zeta)=P(x_1,\ldots,x_K,y_1,\ldots y_K;\zeta)=
\prod_{i=1}^{N} P_M( {{\mbox{\boldmath$q$}}}_M^i;\zeta) \, , \label{Kfactor}$$ where $ P_M( {{\mbox{\boldmath$q$}}}_M^i;\zeta)$ is now a PDF, which satisfies unperturbed FPE for a system of nonlinearly coupled oscillators [(\[2rotfp\])]{} and $N$ is the number of samples so that $K=N\times
M$. The factorisation in Eq.(\[Kfactor\]) is possible because each of the $N$ block-systems of $M$ nonlinear oscillators is uncoupled from all other remaining block-systems. Now to gain the statistics of this system we are to study the statistics of coupled oscillators, described by FPE [(\[2rotfp\])]{}. After it is done we employ Eq.[(\[Kfactor\])]{} to find the final answer for the statistics of the entire set of samples.
Propagator of the Fokker-Plank operator for a single oscillator
---------------------------------------------------------------
Let us study the equation for a single oscillator [(\[1rotfp\])]{}. Proceeding to the polar coordinates $x=r\cos \phi$, $y=r\sin \phi$ we introduce function $\tilde{P}_1(r,\phi;\zeta) =
P_1(x(r,\phi),y(r,\phi);\zeta)$. Note that function $\tilde{P}_1$ *is not* the PDF in polar coordinates since it does not have the Jacobian $r$ included. For the function $\tilde{P}_1$ we now obtain the following equation: $$\frac{\partial \tilde{P}_1}{\partial \zeta} +
{\hat{\mathcal{L}}}_1 \tilde{P}_1 = \frac{\partial \tilde{P}_1}{\partial \zeta} +
\left[- D \Delta^r + r^2 \frac{\partial}{\partial \phi}
\right]\tilde{P}_1 \, =0 , \label{FP1}$$ where $$\Delta^r =\frac{1}{r} \frac{\partial}{\partial r} \left(
r \, \frac{\partial}{\partial r} \right) + \frac{1}{r^2}
\frac{\partial^2}{\partial \phi^2} \, ,$$ is the Laplace operator in polar coordinates. We define a *propagator*, $\mathcal{G}_1$, as a special solution of (\[FP1\]) (as a function of $\{\zeta, r, \phi \}$), possessing the following properties: $$\label{ret2}
\begin{split}
\mathcal{G}_1(\zeta,\zeta';\phi, \phi';r,r') \Big|_{\zeta=\zeta'}
&= \delta(\phi-\phi')
\delta(r-r')/r \, ,\\
\mathcal{G}_1(\zeta,\zeta';\phi,\phi';r,r') & =0, \quad
\zeta<\zeta' \, .
\end{split}$$ Note that propagator [(\[ret2\])]{} coincides with the retarded Green function of Eq. [(\[FP1\])]{}. Since our system is uniform in $\zeta$ and $\phi$, the propagator depends only on the differences $\Delta\zeta=\zeta-\zeta'$ and $ \Delta \phi =\phi - \phi'$: $\mathcal{G}_1=\mathcal{G}_1(\Delta \zeta;\Delta \phi;r,r')$. To find the propagator we first determine eigenfunctions and eigenvalues of the Fokker-Planck operator ${\hat{\mathcal{L}}}_1$ in [(\[FP1\])]{}. The right, $\Psi$, and left, $\tilde \Psi$, eigenfunctions are calculated in the Appendix \[sec:appendix\]. Using these eigenfunctions [(\[eigensystem1\])]{}, [(\[eigensystem2\])]{}, we can write the sought propagator as an eigenvalue expansion: $$\mathcal{G}_1(\Delta \zeta;\Delta \phi;r,r')=\sum_{n,\,\nu}
\Psi_{n\nu}(r,\phi)\,\mathrm{e}^{-s_{n\nu}\Delta \zeta}
\tilde\Psi^*_{n\nu}(r',\phi').$$ Using identity [(\[ident1\])]{} the sum over $n$ can be convoluted yielding the following result: $$\begin{gathered}
{\mathcal{G}}_1(\Delta\zeta;\phi - \phi';r,r') = \frac{1}{2\pi D}
\sum_{\nu = - \infty}^{\infty} \mathrm e^{\mathrm{i}\nu(\phi -
\phi')}\, \frac{\alpha_\nu}{\sinh (2\alpha_\nu \Delta \zeta)}
\\
\times\exp \left[-\frac{\alpha_\nu(r^2+r'^2)}{2D}\, \coth
(2\alpha_\nu \Delta \zeta)\right]
{\mathrm{I}}_{|\nu|}\left(\frac{\alpha_\nu r r'}{D\sinh (2\alpha_\nu
\Delta
\zeta)}\right). \label{green}\end{gathered}$$ Here ${\mathrm{I}}_{|\nu|}$ stands for the $\nu$th order modified Bessel function of the first kind and the quantities $\alpha_{\nu}$ are defined in the Appendix \[sec:appendix\]. Formula [(\[green\])]{} is valid for $\Delta \zeta \geq 0$. For $\Delta \zeta <0$ the propagator, $\mathcal{G}_1$, must be equal to zero due to [(\[ret2\])]{}. Note that the obtained propagator is symmetric with respect to variables $(r,r')$ and is normalised to unity: $\int^{\infty}_0 {\mathrm{d}}r r \int^{2\pi}_0 {\mathrm{d}}\phi \,
\mathcal{G}_1(\Delta\zeta;\phi - \phi';r,r')=1$.
Propagator $P_1=\mathcal{G}_1(\zeta-\zeta';\phi-\phi';r,r')$ (with $r(x,y)=\sqrt{x^2+y^2}$, $\phi(x,y)=\arctan (y/x)$ and the same for primed variables) represents the solution of equation [(\[1rotfp\])]{} provided that at the point $\zeta=\zeta'$ our variables have deterministic values: $x=x', y=y'$. If we want to apply our results to the regularised scalar NLSE with additive white noise, we must use formula (\[factor\]). Function $P_1$ describing the statistics of the unperturbed NLSE at zero dispersion coincides with that obtained in [@Mec2] with a different approach (see also Ref. [@PRL03]).
Propagator for the Fokker-Planck operator for $M$ nonlinearly coupled oscillators
---------------------------------------------------------------------------------
In the same way, generalising the derivation of PDF for a single oscillator, we can obtain the PDF $\tilde{P}_M({{\mbox{\boldmath$r$}}}, {{\mbox{\boldmath$\phi$}}}, {{\mbox{\boldmath$r$}}}_0, {{\mbox{\boldmath$\phi$}}}_{0}; \zeta)$ for system of $M$ nonlinearly coupled oscillators (see Appendix \[sec:appendix\]). FPE [(\[2rotfp\])]{} in polar coordinates reads as: $$\frac{\partial \tilde{P}_M}{\partial \zeta} + {\hat{\mathcal{L}}}_M \tilde{P}_M
=\frac{\partial \tilde{P}_M}{\partial \zeta} +\sum_{i}^M \left[-
D_i \Delta^r_i + \sum_{j}^M r^2_i
a_{j\,i}\frac{\partial}{\partial \phi_j} \right]\tilde{P}_M \, =0,
\label{FP1N}$$ where $\Delta^r_i$ denotes the Laplacian in polar coordinates $\{r_i,\phi_i\}$. Similarly to the case of one oscillator, we introduce propagator $\mathcal{G}_M\,(\Delta\zeta; {{\mbox{\boldmath$\phi$}}}- {{\mbox{\boldmath$\phi$}}}';
{{\mbox{\boldmath$r$}}}, {{\mbox{\boldmath$r$}}}')$ for Eq. [(\[FP1N\])]{}, which is a solution of [(\[FP1N\])]{} for $\Delta \zeta >0$ subject to conditions: $$\label{ret3}
\begin{split}
\mathcal{G}_M(\Delta \zeta;{{\mbox{\boldmath$\phi$}}}-{{\mbox{\boldmath$\phi$}}}';{{\mbox{\boldmath$r$}}}, {{\mbox{\boldmath$r$}}}')
\Big|_{\Delta \zeta=0} &=\prod_{i=1}^M \delta(\phi_i-\phi_i')
\delta(r_i-r'_i)/r_i \, , \\
\mathcal{G}_M(\Delta \zeta;{{\mbox{\boldmath$\phi$}}}-{{\mbox{\boldmath$\phi$}}}';{{\mbox{\boldmath$r$}}}, {{\mbox{\boldmath$r$}}}') & =0, \quad
\Delta \zeta< 0 \, .
\end{split}$$ Again the propagator coincides with the retarded Green function for Eq.[(\[FP1N\])]{}. Using eigenfunctions [(\[eigensystem34\])]{} and convolution formula [(\[ident1\])]{} we obtain the following expression: $$\label{greenN}
\begin{split}
&\mathcal{G}_M\,(\Delta\zeta; {{\mbox{\boldmath$\phi$}}}- {{\mbox{\boldmath$\phi$}}}'; {{\mbox{\boldmath$r$}}}, {{\mbox{\boldmath$r$}}}') =
\frac{1}{(2 \pi )^M} \sum_{\nu_1= - \infty}^{\infty} \ldots
\sum_{\nu_M= - \infty}^{\infty} \prod_{i=1}^M \frac{1}{ D_i}\,
\mathrm e^{\mathrm{i}\nu_i(\phi_i - \phi'_i)}\,
\\ & \times \frac{\alpha_i}{\sinh (2\alpha_i \Delta \zeta)}\, \exp
\left[-\frac{\alpha_i(r_i^2+r^{\prime 2}_i)}{2D_i}\, \coth
(2\alpha_i \Delta \zeta)\right]
{\mathrm{I}}_{|\nu_i|}\left(\frac{\alpha_i r_i r'_i}{D_i\sinh (2
\alpha_i \Delta
\zeta)}\right),
\end{split}$$ where quantities $\alpha_i$ are defined in Appendix \[sec:appendix\], see Eq.[(\[cases\])]{}. Eq.[(\[greenN\])]{} is one of the major results of the current paper: it generalises the results obtained by Mecozzi [@Mec1; @Mec2] and Turitsin et al. [@PRL03] for a single oscillator. As mentioned earlier, if the matrix coefficients $a_{ij}$ are diagonal, then each coefficient $\alpha_i$ in [(\[greenN\])]{} depends on $\nu_i$ only, and propagator [(\[greenN\])]{} factorises into the product of $M$ identical functions, each having the form of Eq.[(\[green\])]{}. However, if the coupling matrix $a_{ij}$ is nondiagonal (the case of nonlinearly coupled oscillators), formula [(\[greenN\])]{} cannot be reduced to a trivial product of identical components.
Perturbational approach for zero-dispersion NLSE {#sec:pertub-local}
================================================
General remarks {#subsec:pert}
---------------
We are now in a position to build a general perturbation theory for the perturbed zero-dispersion NLSE and its generalisations. In this section we will consider perturbations $\hat{N}$ which do not contain a dispersive term. Any perturbation operator $\hat N$ in Eq.[(\[nl\])]{} after regularisation yields a Fokker-Plank image ${\hat{\mathcal{N}}}$ which will appear in the r.h.s. of Eq.[(\[FP1N\])]{}, and will retain a small perturbation parameter $\varepsilon$. Instead of equation [(\[FP1N\])]{} we arrive at the perturbed equation: $$\frac{\partial \tilde{{\mathcal{P}}}_K}{\partial \zeta} + {\hat{\mathcal{L}}}_K
\tilde{{\mathcal{P}}}_K = \varepsilon {\hat{\mathcal{N}}}\tilde{{\mathcal{P}}}_K \, ,
\label{FP1N-pert}$$ where $K=M \times N$, $M$ is the dimensionality of the base system and $N$ is the number of samples. Since the coupling matrix $a_{ij}$ is block-diagonal, we have ${\hat{\mathcal{L}}}_K = \sum_{i=1}^{N}
{\hat{\mathcal{L}}}_M^i$. When we consider local perturbations to system [(\[nl\])]{} (or regularised system [(\[nl1\])]{}), the FP-image of the perturbation also decomposes into a sum of independent operators, ${\hat{\mathcal{N}}}=\sum_{i=1}^N {\hat{\mathcal{N}}}_i \label{N-sum}$, each acting on a specific set of variables $\{ {{\mbox{\boldmath$r$}}}^i, {{\mbox{\boldmath$\phi$}}}^i \}$ belonging to a separate block. Using propagator (\[greenN\]) we arrive at the following expansion (we write only the first corrections to the unperturbed propagator): $$\label{1n1}
\tilde{{\mathcal{P}}}^K(\tilde{{{\mbox{\boldmath$q$}}}} |\, \tilde{{{\mbox{\boldmath$q$}}}}^0;\zeta) = \tilde
P^K(\tilde{{{\mbox{\boldmath$q$}}}} |\, \tilde{{{\mbox{\boldmath$q$}}}}^0;\zeta) + \varepsilon \tilde
P^K_{(1)}(\tilde{{{\mbox{\boldmath$q$}}}} |\, \tilde{{{\mbox{\boldmath$q$}}}}^0;\zeta) + {\mathcal{O}}
(\varepsilon^2) \, ,$$ where for $\tilde P^K_{(1)}$ we have: $$\begin{aligned}
\tilde P^K_{(1)}(\tilde{{{\mbox{\boldmath$q$}}}} |\, \tilde{{{\mbox{\boldmath$q$}}}}^0;\zeta) =
\intop_0^{\zeta} &{\mathrm{d}}& \zeta' \intop_{Q} {\mathrm{d}}\tilde{{{\mbox{\boldmath$q$}}}}_K' \,
{{\mbox{\boldmath${\mathcal{G}}$}}}_K(\zeta - \zeta';
\ {{\mbox{\boldmath$\phi$}}}_K- {{\mbox{\boldmath$\phi$}}}'_K;{{\mbox{\boldmath$r$}}}_K, {{\mbox{\boldmath$r$}}}'_K) \nonumber \\
& \times & {\hat{\mathcal{N}}}^{\,'} \bigl[ {{\mbox{\boldmath${\mathcal{G}}$}}}_K(\zeta'; \ {{\mbox{\boldmath$\phi$}}}'_K-
{{\mbox{\boldmath$\phi$}}}^0_K;{{\mbox{\boldmath$r$}}}^0_K, {{\mbox{\boldmath$r$}}}'_K) \bigr], \label{+psi12}\end{aligned}$$ $$\label{greenpert} {{\mbox{\boldmath${\mathcal{G}}$}}}_K(\zeta - \zeta'; \ {{\mbox{\boldmath$\phi$}}}_K -
{{\mbox{\boldmath$\phi$}}}'_K; {{\mbox{\boldmath$r$}}}_K , {{\mbox{\boldmath$r$}}}'_K) = \prod_{i=1}^N {\mathcal{G}}_{M}(\zeta -
\zeta'; \ {{\mbox{\boldmath$\phi$}}}_M^i- {{\mbox{\boldmath$\phi$}}}_M^{i \, '};{{\mbox{\boldmath$r$}}}_M^i, {{\mbox{\boldmath$r$}}}^{i \, '}_M)
\, ,$$ $$\intop_Q {\mathrm{d}}\tilde{{{\mbox{\boldmath$q$}}}}'_K \equiv
\underbrace{\intop_{0}^{\infty} {\mathrm{d}}r_1'\,r_1' \intop_{0}^{2 \pi}
{\mathrm{d}}\phi_1' \, \cdots \, \intop_{0}^{\infty} {\mathrm{d}}r_K'\,r_K'
\intop_{0}^{2 \pi} {\mathrm{d}}\phi_K'}_K \, .$$ The prime in ${\hat{\mathcal{N}}}^{\,'}$ indicates that operator ${\hat{\mathcal{N}}}$ acts on the primed variables. Note that since perturbation enters only advection terms in Langevin equation (\[Langevin\]) (or (\[nl1\])) its Fokker-Planck image has a form of divergence: ${\hat{\mathcal{N}}}=\vec{\nabla} \cdot (...)$ (see equation (\[canonical\])). Recalling also that the propagator $\mathcal{G}_M$ is normalised we can verify from equations (\[1n1\])-(\[+psi12\]) that the overall normalisation of the function $\tilde{{\mathcal{P}}}_K$ is preserved since $\int_Q {\mathrm{d}}\tilde{{{\mbox{\boldmath$q$}}}}'_K \tilde
P^K_{(1)}(\tilde{{{\mbox{\boldmath$q$}}}} |\, \tilde{{{\mbox{\boldmath$q$}}}}';\zeta) =0$. The PDF in cartesian coordinates ${\mathcal{P}}^K ({{\mbox{\boldmath$q$}}}|{{\mbox{\boldmath$q$}}}^0;\zeta)$ can be restored via the substitution $${\mathcal{P}}^K ({{\mbox{\boldmath$q$}}}|{{\mbox{\boldmath$q$}}}^0;\zeta)=\tilde{{\mathcal{P}}}^K( {{\mbox{\boldmath$r$}}}({{\mbox{\boldmath$x$}}},{{\mbox{\boldmath$y$}}}),
{{\mbox{\boldmath$\phi$}}}({{\mbox{\boldmath$x$}}},{{\mbox{\boldmath$y$}}})|\, {{\mbox{\boldmath$r$}}}^0({{\mbox{\boldmath$x$}}}^0, {{\mbox{\boldmath$y$}}}^0),{{\mbox{\boldmath$\phi$}}}^0({{\mbox{\boldmath$x$}}}^0,
{{\mbox{\boldmath$y$}}}^0);\zeta)$$ In a similar fashion, continuing this recurrent procedure we can find the higher corrections to the PDF [(\[1n1\])]{}.
Let us now consider explicitly some examples of nondispersive perturbations.
High order nonlinearities
-------------------------
First we consider the easiest case where the dimensionality of the corresponding base system is one and a perturbation is local. It seems quite natural to choose the higher nonlinear terms as a perturbation for the single zero-dispersion noisy NLSE (\[nl\]): $$\label{n1}
\hat N [u] = \mathrm{i}\sum_{k = 1}^{q} c_k \, u |u|^{2 k+2} \, ,$$ where $q$ is any positive integer number and $c_k$ are some real phenomenological coefficients. Obviously, the condition $\hat N_i
[{{\mbox{\boldmath$u$}}}] = \hat N_i [u_i]$, is fulfilled: this perturbation is local. After the regularisation the FP-image of this operator appears as a sum, ${\hat{\mathcal{N}}}= \sum_{i=1}^N {\hat{\mathcal{N}}}_i$, with $N$ being the number of samples and ${\hat{\mathcal{N}}}_i$ given by: $$\label{N1}
{\hat{\mathcal{N}}}_i = - \sum_{k=1}^q c_k \, r_i^{2k+1} \frac{\partial}{\partial
\phi_i}.$$ Up to the first order in $\varepsilon$ the solution of [(\[FP1N-pert\])]{} is then given by Eq.(\[1n1\]), where one should substitute in Eq.[(\[greenpert\])]{} the Green function of the corresponding dimensionality, Eq.[(\[green\])]{}: ${\mathcal{G}}_M \to {\mathcal{G}}_1$. Any local perturbation can be handled in the same way.
More general systems: two and more zero-dispersion NLSEs coupled via perturbations {#sec:pertub-local:mult}
----------------------------------------------------------------------------------
Now we proceed to more complicated systems, e.g. when we need to consider a set of equations (\[nl\]) with coupling perturbations. In this subsection the number of dimensions of the corresponding base systems is still one, but the perturbation now cannot be treated as local. For instance if we consider two nonlinear couplers or a single birefringent fibre, then in place of Eq.(\[nl\]) we obtain the following system: $$\label{nl2}
\begin{split}
\frac{\partial u}{\partial
\zeta}&=\mathrm{i}|u|^2u+\eta^1(t,\zeta) + \varepsilon \hat
N^1[u(\zeta),v(\zeta)] \, ,
\\ \frac{\partial v}{\partial
\zeta} & =\mathrm{i}|v|^2v+\eta^2(t,\zeta) + \varepsilon \hat
N^2[u(\zeta),v(\zeta)] \, .
\end{split}$$ Note that complex noises, $\eta^1$ and $\eta^2$, are the independent WGNs, and may possibly have different intensities $D_1$ and $D_2$. The perturbation operators, $\hat N^1[u,v]$ and (or) $\hat N^2[u,v]$, are supposed to intermix the components $u$ and $v$ because otherwise the latter would be uncorrelated and one could merely consider the first and the second equations of system (\[nl2\]) independently using the results of the previous subsection.
It is convenient to introduce a second upper index to distinguish the variables for $u$ (superscript “1”) and $v$ (superscript “2”), and opt for the following indexing rule: after the regularisation we introduce vectors ${{\mbox{\boldmath$r$}}}_{2N}=(r_1^1,\ldots,r_1^N,r_2^1, \ldots,r_2^N)$, and ${{\mbox{\boldmath$\phi$}}}_{2N}
= (\phi_1^1,\ldots, \phi_1^N,\phi_2^1,\ldots, \phi_2^N)$, where $$\begin{split}\label{sampl2}
r_1^i &=|u(i\Delta t)|,\quad r_2^i=|v(i\Delta t)|, \quad
i=1,\ldots N \, ,
\\
\phi_1^i = &\mathrm{Arg}[u(i\Delta t)], \quad
\phi_2^i=\mathrm{Arg}[v(i\Delta t)], \quad i=1,\ldots N \, .
\end{split}$$ We seek the PDF of perturbed system, Eq.[(\[FP1N-pert\])]{}: $\tilde{{\mathcal{P}}}_K({{\mbox{\boldmath$r$}}}_K,{{\mbox{\boldmath$\phi$}}}_K|\,{{\mbox{\boldmath$r$}}}_K^0,{{\mbox{\boldmath$\phi$}}}^0_K;\zeta)$, $K=2N$. The perturbation in [(\[nl2\])]{} couples the dynamics (and hence the statistics) of variables $(r_1^i,\phi_1^i)$ and $(r_2^i,\phi_2^i)$ for $i=1,\ldots,N$. Since we have only two (possibly) different noise intensities, $D_1$ and $D_2$, for this case we can rewrite general expression [(\[greenpert\])]{} in more simple explicit form: $$\begin{gathered}
\label{greencouple}
{{\mbox{\boldmath${\mathcal{G}}$}}}_K(\zeta - \zeta'; \ {{\mbox{\boldmath$\phi$}}}_K - {{\mbox{\boldmath$\phi$}}}'_K; {{\mbox{\boldmath$r$}}}_K ,
{{\mbox{\boldmath$r$}}}'_K) = \prod_{i=1}^N {\mathcal{G}}_1^{D_1}(\zeta - \zeta'; \
\phi_1^i- \phi_1^{'i}; r_1^i, r_1^{'i}) \\ \times {\mathcal{G}}_1^{D_2}(\zeta - \zeta'; \ \phi_2^i- \phi_2^{'i}; r_2^i,
r_2^{'i})\, ,\end{gathered}$$ where the superscripts of the propagators mean that one ought to insert the corresponding noise intensity in the expression for ${\mathcal{G}}_1$, Eq.[(\[green\])]{}. We can then employ the results of subsection \[subsec:pert\].
The generalisation for the case when we have $q$ zero-dispersion NLSEs coupled via the $q$-dimensional perturbation operator, $$\hat N = \big( \hat N^1, \ldots, \hat N^q \big)^T ,$$ is now straightforward. Firstly one should derive the FP-image $\hat{{\mathcal{N}}}$ for the perturbation $\hat N$. Next, by analogy with the case of two equations, one should introduce $K$-dimensional vectors ($K = q \times N$): ${{\mbox{\boldmath$r$}}}_K =(r_1,\ldots,r_K)$ and ${{\mbox{\boldmath$\phi$}}}_K = (\phi_1,\ldots,\phi_K)$. Again for the unperturbed system we have $ P^K ({{\mbox{\boldmath$r$}}}_K,\,{{\mbox{\boldmath$\phi$}}}_K | \,{{\mbox{\boldmath$q$}}}_K^0,
{{\mbox{\boldmath$\phi$}}}_K^0;\zeta)$ in the product form [(\[factor\])]{}. To obtain the expansion for function $\tilde{{\mathcal{P}}}^{K}$ we then use the formulae given in subsection \[subsec:pert\]. The expression for the propagator can be given in a simplified form similar to Eq.[(\[greencouple\])]{}: it is a product of $q$ equivalent multipliers ${\mathcal{G}}_1^{D_k}$, $k=1, \ldots, q$, where the corresponding noise intensities $D_k$ should be inserted in the expression for the propagator Eq.[(\[green\])]{}.
### Coupling perturbations of uncoupled systems
Let us now provide some examples of coupling perturbations.
- For birefringent fibres, with the coupling ($\sim
\varepsilon$) between the different polarisation components of either linearly or circularly polarised light, we have (see e.g. [@aa00]): $$\label{bf}
\hat N [u,v] = \left(\begin{array}{c}
\hat N_1[u,v] \\
\hat N_2[u,v] \end{array} \right) = {\mathrm{i}}\, \left(
\begin{array}{c}
A \,u |\, v|^2 + B \,v^2 \, u^* \\
A \,v |\, u|^2 + B \,u^2 \, v^* \end{array} \right) \, ,$$ where $A$ and $B$ are the (real) constant coefficients of cross-phase modulation and four-waves mixing. The corresponding “polar” FP-image of $\hat N$ in Eq. [(\[FP1N-pert\])]{} by virtue of indexing scheme [(\[sampl2\])]{} can be presented as a sum $\hat{{\mathcal{N}}} = \sum_{i=1}^{N} \hat{{\mathcal{N}}}_i$, where $N$ is, as usual, the number of time-samples, and for $\hat{{\mathcal{N}}}_i$ we have the following formula: $$\begin{aligned}
\label{biref}\nonumber
{\hat{\mathcal{N}}}_i &=& -\left( A + B \cos \big[ 2 (\phi_1^i - \phi_2^i)
\big]\right) \Biggl( (r_2^i)^2 \frac{\partial }{\partial \phi_1^i}
+ (r_1^i)^2 \frac{\partial}{\partial \phi_2^i} \Biggr) \\ &+& B
r_1^i \, r_2^i \sin \big[ 2(\phi_1^i-\phi_2^i)\big] \left( r_1^i
\frac{\partial}{\partial r_2^i} - r_2^i \frac{\partial}{\partial
r_1^i} \right) \, .\end{aligned}$$
- Now consider two nonlinear oscillators with small eigenfrequencies and weak linear coupling (both terms $\sim
\varepsilon$). This is a special case of self-trapping model [@kc86] in the presence of the additive noise. This system is interesting in itself: it is integrable (in the noiseless case) and possesses a finite degree of freedom analogue of soliton solution (inhomogeneous state). Therefore this system can be reckoned as one of the simplest model examples, where we can investigate the action of noise on an integrable system. The coupling operator $\hat N$ now is $$\label{rt}
\hat N [u,v] = \mathrm i \, \left( \begin{array}{c}
- \omega u + \gamma (v - u) \\
- \omega v + \gamma (u - v) \end{array} \right) \, ,$$ where $\omega$ and $\gamma$ are the real parameters. The number of independent variables is just four: $u \to \{ r_1, \phi_1\}$, $v
\to \{r_2,\phi_2\}$. The polar FP-image of $\hat N$ takes the form $$\begin{aligned}
\label{rots}\nonumber
{\hat{\mathcal{N}}}&=& (\omega + \gamma) \left( \frac{\partial}{\partial \phi_1}
+ \frac{\partial}{\partial \phi_2}\right) + \gamma \sin (\phi_2 -
\phi_1) \left( r_2 \frac{\partial}{\partial r_1} - r_1
\frac{\partial}{\partial r_2} \right) \\ &-& \gamma \cos (\phi_2 -
\phi_1) \left( \frac{r_2}{r_1} \, \frac{\partial}{\partial \phi_1}
+ \frac{r_1}{r_2} \, \frac{\partial}{\partial \phi_2} \right) \,
.\end{aligned}$$ The expression for the propagator is very simple: $$\begin{gathered}
\nonumber {{\mbox{\boldmath${\mathcal{G}}$}}}_2(\zeta - \zeta'; \ {{\mbox{\boldmath$\phi$}}}- {{\mbox{\boldmath$\phi$}}}'; {{\mbox{\boldmath$r$}}}, {{\mbox{\boldmath$r$}}}') \\ = {\mathcal{G}}_1^{D_1}(\zeta - \zeta'; \ \phi_1- \phi_1'; r_1,
r_1') \, {\mathcal{G}}_1^{D_2}(\zeta - \zeta'; \ \phi_2- \phi_2'; r_2,
r_2')\, ,\end{gathered}$$ where, as before, the expression for each ${\mathcal{G}}_1^{D_k}$ is given by Eq.[(\[green\])]{}.
- Consider a circular nonlinear fibre array comprising $q$ fibres with weak ($\sim \varepsilon$) linear coupling. Then instead of (\[nl2\]) we acquire a set of $q$ equations for $q$ functions $v_j$: ${{\mbox{\boldmath$v$}}}=(v_1, \ldots, v_q)$. The $q$-dimensional coupling operator now becomes [@aa00]: $$\label{fa}
\hat N [{{\mbox{\boldmath$v$}}} ] = \left( \begin{array}{c}
\hat N_1[{{\mbox{\boldmath$ v$}}}] \\
\hat N_2[{{\mbox{\boldmath$ v$}}}] \\
\vdots \\
\hat N_q[{{\mbox{\boldmath$ v$}}}] \end{array} \right) = {\mathrm{i}}\, C \, \left(
\begin{array}{c}
v_q + v_2 \\
v_1 + v_3 \\
\vdots \\
v_{q-1} + v_1
\end{array} \right) ,$$ with $C$ a real constant. Again it is convenient to introduce the double indexation scheme: the upper index will indicate the sample number, the lower one corresponds to the field number: $v_1(i
\Delta t) \to \{ r_1^i, \phi_1^i\}, \ldots, v_q(i \Delta t) \to \{
r_q^i, \phi_q^i \}$. Then the image of the perturbation [(\[fa\])]{} can be presented as a sum, ${\hat{\mathcal{N}}}= \sum_{i=1}^N {\hat{\mathcal{N}}}_i$ (with $N$ the number of samples), where $$\begin{aligned}
\nonumber \label{fan}
{\hat{\mathcal{N}}}_i & =& C\sum_{n=1}^{q}\left[ A_n^i ({{\mbox{\boldmath$r$}}}_q^i, {{\mbox{\boldmath$\phi$}}}_q^i) \left( \cos \phi_n^i \,
\frac{\partial}{\partial r_n^i} -\frac{\sin \phi_n^i}{r_n^i} \,
\frac{\partial}{\partial \phi_n^i}\right)+ B_n^i({{\mbox{\boldmath$r$}}}_q^i, {{\mbox{\boldmath$\phi$}}}_q^i) \right. \\
& \times & \left. \left( \sin \phi_n^i \,
\frac{\partial}{\partial r_n^i} + \frac{\cos \phi_n^i}{r_n^i} \,
\frac{\partial}{\partial \phi_n^i} \right) \right],\end{aligned}$$ and we used denotations $$\begin{split}
A_n^i ({{\mbox{\boldmath$r$}}}_q^i, {{\mbox{\boldmath$\phi$}}}_q^i) & = r_{n-1}^i \sin \phi_{n-1}^i +
r_{n+1}^i \sin
\phi_{n+1}^i \, ,\\
B_n({{\mbox{\boldmath$r$}}}_q^i,{{\mbox{\boldmath$\phi$}}}_q^i) &= - r_{n-1}^i \cos \phi_{n-1}^i -
r_{n+1}^i \cos \phi_{n+1}^i \, .
\end{split}$$ Here we have assumed that $(r_0^i,\phi_0^i) \equiv
(r_q^i,\phi_q^i)$ and $(r_{q+1}^i,\phi_{q+1}^i)\equiv(r_1^i,\phi_1^i)$. The explicit form for the propagator is now: $${{\mbox{\boldmath${\mathcal{G}}$}}}_K(\zeta - \zeta'; \ {{\mbox{\boldmath$\phi$}}}_K - {{\mbox{\boldmath$\phi$}}}'_K; {{\mbox{\boldmath$r$}}}_K ,
{{\mbox{\boldmath$r$}}}'_K) = \prod_{i=1}^N \prod_{k=1}^q {\mathcal{G}}_1^{D_k}(\zeta -
\zeta'; \ \phi_k^i - \phi_k^{'i}; r_k^i, r_k^{'i})\, ,$$ with $K = q \times N$.
### Birefringent fibre with weak four-wave mixing
So far the original unperturbed system has always been equivalent to the system of uncoupled oscillators, i.e. matrix $a_{ij}$ in [(\[2rotn\])]{}, [(\[2rotfp\])]{} and [(\[greenN\])]{} has always been an identity matrix and thus the dimensionality of the base system has been one. This is not the case for a birefringent fibre with a strong cross-phase modulation, i.e. if the inequality $A \gg
\varepsilon$ holds in Eq.[(\[bf\])]{}. Then one has to consider a more general perturbed system at zero dispersion: $$\label{nl3}
\begin{split}
\frac{\partial u}{\partial
\zeta}&=\mathrm{i}(|u|^2+A|v|^2)u+\eta^1(t,\zeta) + \varepsilon
\hat N^1[u(\zeta),v(\zeta)] \, ,
\\ \frac{\partial v}{\partial
\zeta} & =\mathrm{i}(A|u|^2+|v|^2)v+\eta^2(t,\zeta) + \varepsilon
\hat N^2[u(\zeta),v(\zeta)] \, ,
\end{split}$$ where $A$ is a real constant. Putting $A=0$ will correspond to system [(\[nl2\])]{}. We see that now the dimensionality of the base system is two, the matrix $a_{ij}$ of the base system has the elements $a_{11}=a_{22}=1$, $a_{12}=a_{21} = A$. As a simplest example of a perturbation for system [(\[nl3\])]{} we choose the four-wave mixing (a special case of [(\[bf\])]{}): $$\label{bf1}
\hat N [u,v] = {\mathrm{i}}\, B \left( \begin{array}{c}
v^2 \, u^* \\
u^2 \, v^* \end{array} \right) ,$$ with real constant $B$; the intensities of the noises, $\eta^1$ and $\eta^2$, can, of course, be different: $D_1$ and $D_2$. Unlike the case of system [(\[bf\])]{}, now the perturbation [(\[bf1\])]{} is a local one: $\hat N_i[{{\mbox{\boldmath$u$}}}, {{\mbox{\boldmath$v$}}} ] = \hat
N_i[u_i, v_i]$. The “polar” FP-image of $\hat N$ has been already derived as a part of expression [(\[biref\])]{}, responsible for the four-wave mixing, i.e. ${\hat{\mathcal{N}}}= \sum_{i=1}^{N} {\hat{\mathcal{N}}}_i$ with: $$\begin{aligned}
\label{Nbiref}\nonumber
{\hat{\mathcal{N}}}/B &=& - \cos \big[ 2 (\phi_1^i - \phi_2^i) \big] \Biggl(
(r_2^i)^2 \frac{\partial }{\partial \phi_1^i} + (r_1^i)^2
\frac{\partial}{\partial \phi_2^i} \Biggr) + r_1^i \, r_2^i \sin
\big[ 2(\phi_1^i-\phi_2^i)\big] \\ &\times& \left( r_1^i
\frac{\partial}{\partial r_2^i} - r_2^i \frac{\partial}{\partial
r_1^i} \right) \, .\end{aligned}$$ The propagator is given by $$\label{greenpert1} {{\mbox{\boldmath${\mathcal{G}}$}}}_K(\zeta - \zeta'; \ {{\mbox{\boldmath$\phi$}}}_K -
{{\mbox{\boldmath$\phi$}}}'_K; {{\mbox{\boldmath$r$}}}_K , {{\mbox{\boldmath$r$}}}'_K) = \prod_{i=1}^N {\mathcal{G}}_2(\zeta -
\zeta'; \ {{\mbox{\boldmath$\phi$}}}_2^i- {{\mbox{\boldmath$\phi$}}}_2^{' i};{{\mbox{\boldmath$r$}}}_2^i, {{\mbox{\boldmath$r$}}}^{' i}_2) \, ,$$ with the corresponding noise intensities and coupling matrix elements $a_{ij}$ inserted into general definition [(\[greenN\])]{}.
The cases of coupling (nonlocal) perturbations and the higher number of dimensions of the corresponding base system can be treated analogously.
Discrete self-trapping equation
-------------------------------
It is instructive to give explicit expressions for noisy DSTE. In this system the discreteness is postulated by the model itself, the dimensionality of the base system is one, and we can readily find the perturbation of FP operator applying the results above. Now the perturbation reads (see equation [(\[dste\])]{}): $$\label{ndste}
\hat N[{{\mbox{\boldmath$u$}}}] = {\mathrm{i}}\sum_{j=1}^N m_{nj} u_j \, ,$$ where $N$ defines the number of discrete sites in the chain (lattice). In this case the nonlocal perturbation [(\[ndste\])]{} might couple all $N$ oscillators. Its FP-image is $${\hat{\mathcal{N}}}= \sum_{n,j=1}^{N} m_{nj} \, r_{j} \Biggl[ \sin \big(
\phi_{j} - \phi_n) \, \frac{\partial}{\partial r_n}\, - \,
\frac{\cos \big(\phi_{j}-\phi_n \big )}{r_n} \,
\frac{\partial}{\partial \phi_n} \Biggr] ,$$ and the propagator of the system factorises into a product of $N$ functions ${\mathcal{G}}_1^{D_i}$, with the corresponding noise intensities. We can then apply the general formulae from subsection \[subsec:pert\] to find the perturbed PDF.
Second order dispersion as a perturbation {#sec:nonlocal}
=========================================
In this section we will consider more complicated examples of nonlocal perturbations. We will concentrate on one particular type of perturbations which is nevertheless of great practical interest: the dispersive terms in Kerr systems. Such terms can account for, for instance, a small residual dispersion in a dispersion-managed fibre link. To start we first study a scalar case of a single NLSE, and then proceed to vector generalisations, i.e. to the system of coupled NLSEs.
Scalar NLSE with weak second order dispersion. {#sec:pertub-nonlocal}
----------------------------------------------
In this subsection we will consider the perturbation in terms of weak second order dispersion (SOD) in (\[nl\]), with the dimensionality of the base system being one: $$\hat N[u]=\mathrm{i}\,\frac{d}{2}\,\frac{\partial^2 u}{\partial
t^2} . \label{sod}$$ Here $d=\pm 1$ specifies the type of the dispersion (anomalous or normal). For the sake of simplicity we consider the case $d=1$ since the choice of $d=-1$ only alters the sign in the appropriate formulae. As we work with discretely sampled data we need to write down a discrete analogue of the second derivative Eq.(\[sod\]). To do so we first define the forward finite difference as $\boldsymbol{\delta} f_n = f_{n+1} -f_n$. The arbitrary precision expansion for the 1D second derivative, $\partial^2 u(t)/\partial
\, t^2$, reads as [@a77]: $$\label{diff}
\frac{\partial^2 u}{\partial \, t^2} \Big|_{t=i\Delta t} =
\frac{1}{(\Delta t)^2} \ln^2(1 +\boldsymbol{\delta}) u_i
=\frac{1}{(\Delta t)^2} \sum_{n=2}^{\infty} b_n \,
\boldsymbol{\delta}^n u_i\, ,$$ where we have applied regularisation procedure (\[regul\]) and $\boldsymbol{\delta}^n$ is the forward difference of the $n$th order: $$\boldsymbol{\delta}^2 u_i = u_{i+1} - 2 u_i + u_{i-1}, \quad
\boldsymbol{\delta}^3 u_i = u_{i+2} -3 u_{i+1} + 3 u_i - u_{i-1}
\,, \quad {\rm etc}.$$ The squared logarithm operator in representation (\[diff\]) should be treated as a Taylor expansion in powers of $\boldsymbol{\delta}$, and the expansion coefficients $b_n$ can be easily calculated: $b_2 =1$, $b_3=-1$, $b_4=11/12$, etc.
Taking into account Eq.(\[diff\]) we now write the discretised perturbation $\hat{N}_i$ as: $$\label{ninf}
(\Delta t)^2 \hat N_i [{{\mbox{\boldmath$ u$}}}] = \mathrm i \ln^2(1 +
\boldsymbol{\delta}) u_i
= {\mathrm{i}}\sum_{n=2}^{\infty} b_n \boldsymbol{\delta}^n u_i$$ Further on it is convenient to use the explicit expression for the difference operators in terms of binomial decomposition [@a77]: $$\boldsymbol{\delta}^n u_i = \sum_{j=0}^{n}(-1)^j \binom{n}{j}
u_{i+n-j} \, .$$ The Fokker-Planck image of the perturbation ${\hat{\mathcal{N}}}$ then becomes: $$\begin{aligned}
\label{ni}\nonumber
(\Delta t)^2 {\hat{\mathcal{N}}}&=& \sum_{i=1}^{N} \sum_{n=2}^{\infty}
\sum_{j=0}^{n}(-1)^j \binom{n}{j} b_n \, r_{i+n-j} \Biggl[ \sin
\big( \phi_{i+n-j} - \phi_i) \,
\frac{\partial}{\partial r_i}\\
& - &\, \frac{\cos \big(\phi_{i+n-j}-\phi_i \big )}{r_i} \,
\frac{\partial}{\partial \phi_i} \Biggr] .\end{aligned}$$ Note that we write an infinite upper limit of summation over the index $n$. However, of course, each difference is limited by the total number of samples $N$, and all the variables outside the sampling interval should be discarded. The form of the perturbative expansion for the full PDF does not differ from that given in previous sections for the non-dispersive perturbations. The total propagator has the form of the $N$-product of functions ${\mathcal{G}}_1$, each having the same noise intensity $D$.
The same procedure may be easily developed if one wishes to account for a perturbative term in the form of a derivative of an arbitrary order with either real or complex coefficient ($\sim
\varepsilon$). For example, the dispersion of arbitrary order $\mathrm{i}^k \partial^k u/\partial t^k$ would result in the change of the power of the logarithm operator in expressions (\[diff\]),(\[ninf\]). Eventually one should arrive at the same formulae with a different set of coefficients $b_n$, which, in that case, have to be the Taylor coefficients for the expansion of the corresponding power of the logarithm operator (one will have to change the lower summation limit for $n$ from 2 to $k$ in Eqs.(\[diff\]-\[ni\]) as well). It is worth mentioning that system (\[nl1\]) with the discrete term, given by (\[ninf\]) but truncated at $n=2$, corresponds to the *weakly discrete NLSE* [@krb01] with additive noise. The increase of the number of terms in expansion (\[ninf\]) (and consequently in (\[ni\])) obviously increases the precision at the expense of the computational time.
In the same manner we could redefine the dispersion operator via the backward differences (see [@a77]) and obtain an expression very similar to [(\[ni\])]{}. If we wish to preserve the time-inversion symmetry, we might use the symmetrised form instead of Eq.[(\[diff\])]{}: ${\hat{\mathcal{N}}}= ({\hat{\mathcal{N}}}_{forward} + {\hat{\mathcal{N}}}_{backward})/2$.
Manakov equations with weak second order dispersion
---------------------------------------------------
Next we develop a perturbation theory accounting for the weak dispersion in vector systems, such as Manakov equations [(\[man\])]{} (i.e. for the regularised system [(\[man1\])]{}). Now the number of dimensions of the base system is two and we consider the nonlocal perturbation of the form: $$\label{manpert}
\hat N [u,v] = \frac{{\mathrm{i}}}{2}\left( \begin{array}{c}
\frac{\partial^2 u}{\partial t^2} \\
\frac{\partial^2 v}{\partial t^2} \end{array}\right).$$ We do not present here the explicit expressions for the PDF expansion as they are very similar to those given in the previous subsection, but outline only the general procedure. First we choose the same regularisation we used in subsection \[sec:pertub-local:mult\] for two uncoupled equations. Full coupling matrix $a_{ij}$ comprise a set of $2\times2$ blocks placed on the main diagonal, with all elements within each block equal to 1. The expression for the Fokker-Plank image of perturbation [(\[manpert\])]{} is very similar to Eq.[(\[ni\])]{}. The only difference is that now we again have to endow the variables with an additional index to distinguish between the fields: $u_i
\to (r_1^i, \phi_1^i)$, $v_i \to (r_2^i, \phi_2^i)$. Now the expression for the FP-image of perturbation [(\[manpert\])]{} consists of the two summands, ${\hat{\mathcal{N}}}_1$ and ${\hat{\mathcal{N}}}_2$, each having the form of Eq.[(\[ni\])]{}: the variables in Eq.[(\[ni\])]{} for ${\hat{\mathcal{N}}}_1$ should bear additional index “1”, and the same applies to ${\hat{\mathcal{N}}}_2$. The propagator for the entire sampled system is a direct product of ${\mathcal{G}}_2^{D_k}$, with the corresponding (possibly different) noise intensities, $D_1$ and $D_2$, and the elements of the coupling matrix for the base system, $a_{11}=a_{12}=a_{21}=a_{22}=1$. All statements from the previous subsection, concerning accounting for either the time inversion symmetry or a higher order dispersion apply to this case as well.
[*Remark on higher systems of coupled NLSEs.*]{} Following the preceding example one can build perturbation expressions for the systems comprising more than two nonlinearly coupled NLSEs. The number of coupled NLSEs, say $M$, simply defines the dimensionality of the base oscillator system. If we have $M$ coupled fields, $v_1, \ldots, v_M$, then introducing an additional index labelling the field number one finally arrives at the expression for the FP-image of a dispersive perturbation in the form of a sum, ${\hat{\mathcal{N}}}= \sum_k {\hat{\mathcal{N}}}_k$, where each ${\hat{\mathcal{N}}}_k$ has the form of Eq.[(\[ni\])]{}. The propagator for the whole sampled system is again a direct $N$-product of ${\mathcal{G}}_M^{D_k}$.
Conclusion
==========
In the current paper we have proposed an approach for studying the stochastic dynamics of a noise-driven systems with Kerr-type nonlinearity. We presented the exact expression for the PDF (and the propagator) for a system of nonlinearly coupled oscillators. Aside from the study of nonlinear lattice models with the additive WGN, we showed how to apply these results to the description of signal propagation in a weakly nonlinear dispersive media driven by WGN. We used the fact that in a system at zero dispersion the PDF for the output signal (field) can be obtained analytically. Using the propagators for a zero dispersion system we have built a perturbation theory and obtained the corrections to the unperturbed PDF in a variety of physically meaningful situations. In particular we were able to obtain the corrections to the PDF in the presence of the nonlocal perturbation, such as a second order dispersion. The knowledge of the output signal statistics allows one to calculate a multitude of important quantities like the probability of error in the fibre optics communication, or, in principle, Shannon capacity of the nonlinear communication channel, modelled by Eq.(\[nl\]).
Using the discrete time picture one can also examine the statistics of more complex systems than those, described in the current paper, like, for instance, a system of weakly coupled NLSE with (also weak) linear or nonlinear nonlocal perturbations (e.g. nonlinear dispersion). This would simply require increasing the number of dimensions in the corresponding PDF. It is also possible to consider the higher-dimensional noisy NLSE in the same fashion, e.g. the 2+1 dimensional NLSE where the (discretised) weak second order derivatives are reckoned as a perturbation.
We recognise that for the case of nonlocal perturbations, like higher order dispersion, when the perturbation operator in the FPE becomes quite involved, the corrections to the PDF can only be calculated numerically. Let us recall that the traditional way of calculating the output signal statistics numerically is to use direct Monte Carlo simulations. The latter method, however has a number of significant drawbacks. Normally the calculations are quite time consuming and cannot be used for estimation of the tails of the PDFs, something which is crucial for the performance assessment in the fibre optical communications. Therefore the perturbation technique developed here, with the iteration procedure similar to [(\[+psi12\])]{} performed numerically, may present an important method of choice for the systems where the direct Monte Carlo approach fails. The proposed method relies only on the approximate evaluation of integrals and infinite sums without direct integration of the noisy equation. And finally, the pertrubative approach devised in the present paper does not require the noise to be small - which is usually implied in the majority of the approximate methods in stochastic dynamics. Therefore it can be used to analyse systems far from the equilibrium where the impact of noise on the signal degrees of freedom becomes significant.
In conclusion we also point out that the exact analytical results concerning the statistics of nonlinear coupled systems are still few and far between. Thus we think that the exact general expression for the propagator of coupled noisy nonlinear oscillators [(\[greenN\])]{} (which coincides with a retarded Green function), is quite important in itself and can serve as a “reference point” for the studies of similar systems.
Authors would like to thank Sergei Turitsyn and Igor Yurkevich for the fruitful discussions and valuable comments. S.D. would like to acknowledge the support from the Leverhulme Trust Project F/00250/B. This work was also supported by NATO Collaborative Linkage Program (No PST.CLG.980068).
Eigenfunctions of the Fokker-Planck operator {#sec:appendix}
============================================
In this section we calculate the eigenfunctions and eigenvalues of the FPE for single oscillator [(\[FP1\])]{} and for system of nonlinearly coupled oscillators [(\[FP1N\])]{}. We start from the former case. We wish to solve the eigenvalue problem: $${\hat{\mathcal{L}}}_1 \Psi = s \Psi \label{eigen1} \, .$$ Since the obvious periodical boundary conditions in $\phi$ it is convenient to perform a Fourier transform with respect to $\phi$: $$\label{ser-phi}
\begin{split}
\Psi(r,\Delta \phi)& = \sum_{\nu = -\infty}^{\infty} \mathrm{e}^{
\mathrm{i} \nu \Delta \phi}
\Psi_\nu(r) \\
\Psi_\nu(r) & = \frac{1}{2\pi} \intop_{0}^{2 \pi}
\mathrm{e}^{-\mathrm{i} \nu \Delta \phi} \, \Psi(r,\Delta \phi)\,
{\mathrm{d}}\Delta \phi\, .
\end{split}$$ After the transform the operator ${\hat{\mathcal{L}}}_1$ reads as $$\label{ml}
{\hat{\mathcal{L}}}_1^{\nu} = -D \left( \frac{1}{r} \frac{\partial}{\partial r}
\left( r \, \frac{\partial}{\partial r} \right) -
\frac{\nu^2}{r^2} \right) + \mathrm{i} \nu r^2 \, .$$ To solve the eigenfunction equation we make the following substitution: $$\label{ef}
\Psi_\nu(y) = y^{\, |\nu| /2} \exp(-\alpha_{\nu}y/2) \varphi(y)
\,,$$ where $\alpha_{\nu} = (1+\mathrm{i}) \sqrt{\nu D/2}$ for $\nu>0$ and $\alpha_{\nu} = (1-\mathrm{i}) \sqrt{|\nu| D/2}$ for $\nu<0$, $y = r^2/D$. After all the transformations Eq.[(\[eigen1\])]{} now has the form: $$\label{eq}
y \frac{{\mathrm{d}}^2 \varphi}{{\mathrm{d}}y^2} +(1+ |\nu| - \alpha_{\nu} y)
\frac{{\mathrm{d}}\varphi}{{\mathrm{d}}y} + z_{s \nu}\varphi =0,$$ with $z_{s \nu} = (s - 2 \bigr[1 + |\nu|\bigl]\alpha_{\nu})/4$. Equation (\[eq\]) has the canonical form of the degenerated hypergeometric equation [@Gr-Ryg]. Since we want our eigenfunctions to decrease at infinity we must demand that $z_{s
\nu}/\alpha_\nu=n$, $n=0,1, \ldots $. This will determine the discrete eigenvalues $s_{n\nu}$ of [(\[eigen1\])]{}, while the corresponding eigenfunctions $\Psi_{n\nu}$ are expressed via generalised Laguerre functions ${\mathrm{L}}^{\alpha}_{\beta}(\ldots)$ as [@Gr-Ryg]: $$\label{eigensystem1}
\begin{split}
& s_{n \nu} = 2 \alpha_\nu \big( 2 n + 1 + |\nu|) \, , \\
& \Psi_{n\nu}(r,\phi)={\mathrm{e}}^{{\mathrm{i}}\nu
\phi}\,\left(\frac{\alpha_\nu}{\pi
D}\right)^{1/2}\,\left(\frac{n!}{(n+|\nu|)!}\right)^{1/2}
\,z^{|\nu|/2} \,\exp[-z/2]\,{\mathrm{L}}_n^{|\nu|}(z), \\
&z=\alpha_\nu r^2/D \, .
\end{split}$$ Operator ${\mathcal{L}}_1$ is of course not Hermitian. Therefore we need to introduce a set of left-eigenfunctions $\tilde \Psi_{n\nu}$, which are the eigenfunctions of the adjoint operator ${\mathcal{L}}_1^\dag$ with eigenvalues $\tilde s_{n\nu}$: $$\label{eigensystem2}
\begin{split}
& \tilde s_{n \nu} = s^*_{n\nu} \, ,\\
& \tilde \Psi_{n\nu}(r,\phi)={\mathrm{e}}^{{\mathrm{i}}\nu
\phi}\,\left(\frac{\alpha^*_\nu}{\pi
D}\right)^{1/2}\,\left(\frac{n!}{(n+|\nu|)!}\right)^{1/2}
\,z^{*|\nu|/2} \,\exp[-z^*/2]\,{\mathrm{L}}_n^{|\nu|}(z^*) \, .
\end{split}$$ Eigenfunctions [(\[eigensystem1\])]{},[(\[eigensystem2\])]{} form a biorthogonal system, i.e. $$\intop_0^{2\pi} d\phi \intop^{\infty}_0 dr r \tilde
\Psi^*_{n\nu}(r,\phi) \Psi_{n'\nu'}(r,\phi) =
\delta_{nn'}\,\delta_{\nu \nu'} \, . \label{bi-orthogonal}$$ Making use of the identity (see [@Gr-Ryg]): $$\sum^{\infty}_{n=0} n! \,\frac{{\mathrm{L}}^{\alpha}_n (u)\, {\mathrm{L}}^{\alpha}_n (v) \,
w^n}{\Gamma(n+\alpha+1)}=\frac{(uvw)^{-\alpha/2}}{1-w} \exp(-w
\frac{u+v}{1-w}) \, \mathrm{I}_{\alpha}\left(2
\frac{\sqrt{uvw}}{1-w}\right), \; |w|<1 \, ,\label{ident1}$$ analytically extended to the complex plane, in the limit $w \to 1$ we can obtain the *closure relation* for the left and right eigenfunctions: $$\sum_{n,\nu} \tilde \Psi^*_{n\nu}(r,\phi)\Psi_{n\nu}(r',\phi') =
r^{-1}\,\delta(r-r')\,\delta(\phi-\phi') \label{closure}$$ (factor $r^{-1}$ in the r.h.s. is merely a jacobian).
In the same way, generalising the derivation of the elementary basis [(\[eigensystem1\])]{}, [(\[eigensystem2\])]{}, we can built left- and right-eigenfunction for the general $2M$-dimensional operator ${\hat{\mathcal{L}}}_M$ in [(\[FP1N\])]{}. Using the Dirac notations one gains $$\label{eigensystem34}
\begin{split} |\, n_1, \ldots, n_M, \nu_1, \ldots, \nu_M \rangle & =
\frac{1}{\pi^{M/2}} \prod_{i=1}^{M} {\mathrm{e}}^{{\mathrm{i}}\nu_i \phi_i}
\sqrt{\frac{\alpha_i}{D_i}}\left[
\frac{n_i!}{(n_i+|\nu_i|)!}\right]^{1/2}\\
&\times z_i^{|\nu_i|/2} {\mathrm{e}}^{-z_i/2} \, {\mathrm{L}}_{n_i}^{|\nu_i|}(z_i) \, , \\
\langle \, n_1, \ldots, n_M, \nu_1, \ldots, \nu_M | & =
\frac{1}{\pi^{M/2}} \prod_{i=1}^{M} {\mathrm{e}}^{{\mathrm{i}}\nu_i \phi_i}
\sqrt{\frac{\alpha_i}{D_i}}\left[
\frac{n_i!}{(n_i+|\nu_i|)!}\right]^{1/2}\\
&\times z_i^{*|\nu_i|/2} {\mathrm{e}}^{-z^*_i/2} \,
{\mathrm{L}}_{n_i}^{|\nu_i|}(z^*_i) \, , \\
z_i&=\alpha_i r_i^2/D \, .
\end{split}$$ Here the quantities $\alpha_i(\nu_1, \ldots, \nu_M)$ are defined as: $$\label{cases} \alpha_i(\nu_1, \ldots, \nu_M) = \left\{
\begin{array}{ccc}
(1 + {\mathrm{i}}) \left(\frac{D_i}{2} \sum_{j=1 }^M a_{j\,i} \nu_j
\right)^{1/2}
& \text{for} & \sum_{j=1 }^M a_{ji} \nu_j > 0 \, , \\
(1 - {\mathrm{i}}) \left(\frac{D_i}{2} \left|\sum_{j=1 }^M a_{ji} \nu_j
\right|\right)^{1/2} & \text{for} & \sum_{j=1 }^M a_{j\,i} \nu_j<0
\, .
\end{array}
\right.$$ The corresponding (right) eigenvalues of operator ${\hat{\mathcal{L}}}_M$ are $$\label{eigen2}
\, s_{n_1, \ldots, n_M \nu_1, \ldots, \nu_M} = 2 \sum_{i=1}^{M} \alpha_i\big( 2 n_i
+ 1 + |\nu_i|).$$
Since we know the explicit expressions for eigenvalues, we know the damping eigenrates ($\gamma = |\mathrm{Re}[s]|$) for each eigenmode of FPE. These may be important for the investigation of the long scale (large values of $\zeta$) evolution of the system statistics, because for the case $\zeta \to \infty$ the statistics of the system is determined by a single eigenmode with the lowest value of $\gamma$. We can see that the damping rate for each eigenmode is proportional to $\gamma(\nu ) \sim D^{1/2} \nu^{3/2}$ for a single oscillator or, for a particular realisation $(\nu_1,
\ldots, \nu_N)$, to $\gamma(\nu_1, \ldots, \nu_N) \sim
\sum_{i,j=1}^N D_i^{1/2} \nu_i (a_{j \, i}\nu_{j})^{1/2}$ for a system of nonlinearly coupled oscillators. Note that the real parts of eigenvalues [(\[eigensystem1\])]{},[(\[eigen2\])]{} are always positive which means that equations [(\[FP1\])]{}, [(\[FP1N\])]{} do not possess a stationary solution.
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---
abstract: 'Permutation actions of simple currents on the primaries of a Rational Conformal Field Theory are considered in the framework of admissible weighted permutation actions. The solution of admissibility conditions is presented for cyclic quadratic groups: an irreducible WPA corresponds to each subgroup of the quadratic group. As a consequence, the primaries of a RCFT with an order $n$ integral or half-integral spin simple current may be arranged into multiplets of length $k^{2}$ (where $k$ is a divisor of $n$) or $3k^{2}$ if the spin of the simple current is half-integral and $k$ is odd.'
address: '[Department of Theoretical Physics,]{} Roland Eötvös University, 1117 Budapest, Hungary'
author:
- Tamas Varga
title: Simple Current Actions of Cyclic Groups
---
Introduction
============
The construction of correlation functions from conformal blocks is restricted in a Rational Conformal Field Theory by requiring the invariance of physical quantities under the action of mapping class groups [@ms; @frsch1]. The most important case is that of the torus partition function - it has to be an invariant of the representation of the modular group $SL(2,\mathbb{Z})$ on the characters of the theory. The modular representation is determined by matrices $S$ and $T$ (indexed by the primaries of the theory) corresponding to modular transformations $\tau\to-1/\tau$ and $\tau\to\tau+1$, respectively. $S$ is symmetric, unitary, its square is the charge conjugation matrix, and $T$ is a finite order diagonal matrix. Symmetries of the modular representation turned out to be the most effective tool in constructing modular invariant partition functions. One of these symmetries is provided by simple currents [@schy1; @schy2; @intril; @b-sc; @halpsc] - the corresponding simple current modular invariants yield most of the known modular invariant partition functions [@kreuzersch].
Simple currents are primary fields whose fusion with any primary field contains only one term, i.e. whose fusion matrix is a permutation matrix
$$N_{\alpha p}^{q}=\Pi(\alpha)_{p}^{q}=\delta_{\alpha p}^{q}.\label{eq:scfusmat}$$
Other equivalent definitions of simple currents could be used: they are the primaries whose fusion with their charge conjugate gives the vacuum alone, or, in unitary theories, the primaries with quantum dimension $d_{p}=S_{p0}/S_{00}=1$. It follows from the associativity and commutativity of the fusion ring that they form an abelian group under fusion, known as the the simple current group, or the center of the fusion ring. The fusion of simple currents with other primaries of the theory gives a permutation action of the simple current group on the set of primaries.
It is known from Verlinde’s theorem [@verl] that fusion matrices are diagonalized by $S$, which applied on yields
$$S_{\alpha p}^{q}=\theta(q,\alpha)S_{p}^{q},\label{eq:scsymm}$$
where the complex number $\theta(q,\alpha)$ is the (exponentialized) monodromy charge of the primary $q$ with respect to the simple current $\alpha$. The above equation is a symmetry of the $S$ matrix, which relates its rows in the same orbit of the permutation $\alpha$. In fact, it can be shown using Verlinde’s theorem and unitarity, that if $\alpha$ is a permutation satisfying , then $\alpha0$ is a simple current, where $0$ denotes the vacuum. Therefore, these symmetries of the $S$ matrix are in one-to-one correspondence with simple currents and are called simple current symmetries.
Simple currents have various other uses besides the already mentioned simple current modular invariants: they are used in the GSO projection in string theory [@fschw], through simple current extension they allow us to construct a new vertex operator algebra and modular representation [@fss; @ijmpa] from a known one, and they are also used in the determination of the projective kernel of modular representation [@b-projk]. In most of these applications the monodromy charges defined in play an important role, but hardly any more properties of the modular representation are needed. This suggests to consider simple currents and monodromy charges separately from the modular representation - that approach lead in [@b-sc] to the introduction of weighted permutation actions.
The monodromy charges can be expressed using the conformal weights $\Delta_{p}$ and the central charge of the theory $c$ as follows. Let $\omega(p)$ denote the diagonal entry of $T$ corresponding to the primary field $p$$$\omega(p)=T_{p}^{p}=exp(2\pi i(\Delta_{p}-\frac{c}{24})),$$ and let $\vartheta(\alpha)=\omega(\alpha)/\omega(0)=exp(2\pi i\Delta_{\alpha}).$ Using and the modular relation $STS=T^{-1}ST^{-1}$ one may express $\theta(p,\alpha)$ in terms of $\omega$ as
$$\theta(p,\alpha)=\frac{\omega(p)\omega(\alpha)}{\omega(\alpha p)\omega(0)}=\frac{\omega(p)}{\omega(\alpha p)}\vartheta(\alpha).\label{eq:monodr}$$
The above equation shows that the monodromies may be replaced by the weight function $\omega$, which turns out to be more suitable for the description of simple currents. It follows from that $\theta(p,\alpha\beta)=\theta(p,\alpha)\theta(p,\beta)$, and substituting the above expression for $\theta(p,\alpha)$ yields$$\frac{\omega(\alpha p)\omega(\beta p)}{\omega(p)\omega(\alpha\beta p)}=\frac{\vartheta(\alpha)\vartheta(\beta)}{\vartheta(\alpha\beta)}.\label{eq:def weight}$$ If $p$ is chosen to be a simple current $\gamma$, one obtains the following equality for $\vartheta$
$$\vartheta(\alpha\beta)\vartheta(\beta\gamma)\vartheta(\gamma\alpha)=\vartheta(\alpha)\vartheta(\beta)\vartheta(\gamma)\vartheta(\alpha\beta\gamma).\label{eq:qform1}$$
In a unitary theory one may also show that
$$\vartheta(\alpha^{n})=\vartheta(\alpha)^{n^{2}}.\label{eq:qform2}$$
These two properties define $\vartheta$ to be a *quadratic form* (written multiplicatively) on the abelian group of simple currents $G$, and the pair ${(G,\vartheta)}$ is called a *quadratic group*. A *weighted permutation action (WPA)* of a quadratic group ${(G,\vartheta)}$ is a pair ${(X,\omega)}$ such that the group $G$ acts by permutations on the finite set $X$, and the function $\omega:X\mapsto\mathbb{C}^{*}$ satisfies . $X$ is called the support of the WPA, $\omega$ its weight function, and $|X|$ the degree of the WPA.
These results give a mathematical formalization of simple current symmetries: the group of simple currents together with $\vartheta=exp(2\pi i\Delta_{\alpha})$ is a quadratic group and its permutation action on the set of primaries together with $\omega(p)$ is a weighted permutation action of it. The latter is called the simple current WPA associated to the RCFT.
Weighted permutation actions can be completely classified in terms of coset WPA-s - the details are given in the next section. Once we know the classification, it is natural to ask: what characterizes those WPA-s that are associated to some RCFT. In [@b-sc] three necessary conditions were found, and the WPA-s satisfying these were called admissible. The solution of the admissibility conditions for a given quadratic group amounts to finding a finite number of irreducible WPA-s. This can be done in principle for any quadratic group, but the complexity of the problem grows rapidly. Therefore, the solution is known only for the most trivial quadratic groups: e.g. prime order cyclic groups or some abelian groups of low order. Our aim is to present the solution for cyclic quadratic groups. Note that not all of these solutions is realized by some RCFT. Nevertheless, it provides us with non-trivial information about what the primary field content (including the differences of conformal weights in the same orbit) and simple current action *may* be in any RCFT. This would allow us, among others, to describe the possible modular invariants of cyclic simple current groups. Moreover, for some WPA-s, the corresponding simple current symmetry , together with non-linear equations like unitarity or $S^{2}=C$, completely determines the modular representation.
In the next section we shall shortly review some results of [@b-sc] in order to formulate the admissibility conditions. In we give examples of admissible WPA-s. In particular, Example provides a general construction of an irreducible WPA corresponding to each subgroup of a quadratic group. is devoted to the proof of , whose immediate consequence is that in the case of cyclic quadratic groups the irreducibles of Example exhaust the set of all irreducible WPA-s.
Admissible weighted permutation actions
=======================================
Let us review some elements of the theory of permutation actions that may be generalized for WPA-s. A WPA is called transitive if it consist of a single $G$ orbit. Given two WPA-s $(X_{1},\omega_{1})$ and $(X_{2},\omega_{2})$, it is straightforward to define their direct sum, whose support is the set $X_{1}\cup X_{2}$. Two WPA-s $(X,\omega)$ and $(X',\omega')$ are considered equivalent if they are equivalent as permutation actions, and if their weight functions differ by a factor which is locally constant on $G$ orbits - note that such rescalings are allowed by .
The radical of ${(G,\vartheta)}$ is the quadratic group ${(\sqrt{G},\sqrt{\vartheta})}$ where $\sqrt{G}=\{\alpha\in G|$ $\vartheta(\alpha,\beta)=\vartheta(\alpha)\vartheta(\beta)$ $\forall\beta\in G\}$ and $\sqrt{\vartheta}$ is the restriction of $\vartheta$ to $\sqrt{G}$. It follows from the fact that $\sqrt{\vartheta}$ is both a character and a quadratic form on $\sqrt{G}$ that the allowed values of $\sqrt{\vartheta}$ are $\pm1$. $\sqrt{\vartheta}(\alpha)=-1$ is allowed only if $\alpha$ is of even order.
Transitive WPA-s may be classified up to equivalence as follows. Let $\xi$ be a character of $\sqrt{G}$ and $H$ a subgroup of $ker(\xi\sqrt{\vartheta})$. Further, let $X$ be the coset space $X=G/H$ on which $G$ acts by left translations and $\omega(\alpha H)=\vartheta(\alpha)/\xi^{*}(\alpha)$, where $\xi^{*}$ is any character of $G$ whose restriction to $\sqrt{G}$ equals $\xi$. Then $(X,\omega)$ is a transitive WPA of ${(G,\vartheta)}$ called a coset WPA, whose equivalence class shall be denoted as ${\mathcal{W}[H,\xi]}$. The coset WPA is well defined and is determined by $\xi$ - different choices for $\xi^{*}$ lead to equivalent WPA-s. With this classification of transitive actions we may always write a WPA $\Phi=(X,\omega)$ as a direct sum of coset WPA-s $$\Phi=\bigoplus_{i\in I}n_{i}{\mathcal{W}[H_{i},\xi_{i}]}.\label{eq:gen_decomp}$$ Therefore, a WPA is determined by the non-negative integer multiplicities $n_{i}$ up to equivalence.
The coset WPA $R={\mathcal{W}[1,\xi_{0}]}$, where 1 is the trivial subgroup and $\xi_{0}$ is the trivial character of $G$, is called the regular WPA. The importance of the regular WPA lies in the fact that it represents the transitive component containing the vacuum in any simple current WPA.
There is another numerical characterization of WPA-s. To each WPA we may associate the monomial matrices
$$Y(\alpha,\beta)_{q}^{p}=\vartheta(\beta)\frac{\omega(q)}{\omega(\beta q)}\delta_{p}^{\alpha p},\label{eq:Y repr. def.}$$
where $p,q\in X$. They satisfy the multiplication rule$$Y(\alpha_{1},\beta_{1})Y(\alpha_{2},\beta_{2})=\frac{\vartheta(\alpha_{2})\vartheta(\beta_{1})}{\vartheta(\alpha_{2}\beta_{1})}Y(\alpha_{1}\alpha_{2},\beta_{1}\beta_{2}),\label{eq:Y mult. rule}$$ i.e. they form a projective representation of $G\times G$. If the WPA is associated to some RCFT, then the commutation rule of $Y(\alpha,\beta)$ with $M$ representing the $SL(2,\mathbb{Z})$ element $\left(\begin{array}{cc}
a & b\\
c & d\end{array}\right)$ on the space of genus 1 holomorphic blocks is
$$M^{-1}Y(\alpha,\beta)M=\frac{\vartheta(\alpha)^{b(c-a)}\vartheta(\beta)^{c(b-d)}}{\vartheta(\alpha\beta)^{bc}}Y(\alpha^{a}\beta^{c},\alpha^{b}\beta^{d}).\label{eq:sl(2,z) commut. rule}$$
The trace of $Y(\alpha,\beta)$ gives a useful numerical description of the equivalence classes of WPA-s
$$\Upsilon_{\Phi}(\alpha,\beta)=TrY(\alpha,\beta)=\vartheta(\beta)\sum_{p\in Fix_{\Phi}(\alpha)}\frac{\omega(p)}{\omega(\beta p)},\label{eq:chardef}$$
where $Fix_{\Phi}(\alpha)$ is the subset of the support of $\Phi$ whose elements are fixed by $\alpha$. The value of $\Upsilon_{\Phi}(\alpha,\beta)$ is zero unless $\alpha,\beta\in\sqrt{G}$, so it is in fact a function on the set $\sqrt{G}\times\sqrt{G}$. For the coset WPA $\Phi={\mathcal{W}[H,\xi]}$ it is given as$$\Upsilon_{\Phi}(\alpha,\beta)=\left\{ \begin{array}{cc}
\xi(\beta)[G:H] & \textrm{\textrm{if $\alpha\in H$}},\\
0 & \textrm{otherwise.}\end{array}\right.\label{eq:transchar}$$
For simple current WPA-s this function is related to the so-called commutator cocycle $\phi_{p}(\alpha,\beta)$ (see [@ijmpa]) as $\Upsilon(\alpha,\beta)=\sum_{p}\phi_{p}(\alpha,\beta)$. This fact, together with the commutation rule , implies that simple current WPA-s possess three additional properties, given in [@b-sc]. *Galois invariance* of $\Phi$ means that in its decomposition the multiplicities of transitives ${\mathcal{W}[H,\xi]}$ and ${\mathcal{W}[H,\xi^{l}]}$ are equal for all $l$ coprime to the exponent of $G$. *Reciprocity* requires $\Upsilon_{\Phi}(\alpha,\beta)=\Upsilon_{\Phi}(\beta,\alpha)$ for all $\alpha,\beta\in G$. Finally, *boundedness* is the property that $|\Upsilon_{\Phi}(\alpha,\beta)|\le|Fix_{\Phi}(\alpha)\cap Fix_{\Phi}(\beta)|$. WPA-s satisfying Galois invariance, reciprocity and boundedness are called admissible.
Let us mention some interesting consequences of the admissibility conditions. It follows from both Galois invariance or reciprocity of $\Phi$ that the values of $\Upsilon_{\Phi}$ are integers. Galois invariance and reciprocity implies that the expression $\vartheta(\alpha)\vartheta(\beta)\Upsilon_{\Phi}(\alpha,\beta)$ depends only on the subgroup $\left\langle \alpha,\beta\right\rangle $ generated by $\alpha$ and $\beta$ (see Proposition \[pro:subgroup property\]). Consequently, if $\Phi$ is admissible, then $\Upsilon_{\Phi}$ may be regarded as an integer valued function on the set of subgroups of $\sqrt{G}$ generated by at most two elements.
While the admissibility conditions have such non-trivial consequences, they are also simple enough for practical use - being just linear equalities and inequalities in terms of the multiplicities $n_{i}$ if $\Phi$ is written as in . Linearity implies that the direct sum of admissible WPA-s is again admissible. Thus, one defines *irreducible* WPA-s as the admissible WPA-s that cannot be written as a (non-trivial) direct sum of admissible WPA-s. The problem of finding admissible WPA-s is then reduced to the problem of finding irreducible WPA-s. Moreover, as another consequence of the linearity, there is only a finite number of irreducible WPA-s for any quadratic group ${(G,\vartheta)}$. The last important step in the classification is that the irreducible WPA-s of ${(G,\vartheta)}$ are in one-to-one correspondence with those of ${(\sqrt{G},\sqrt{\vartheta})}$. Therefore, it is enough consider only fully degenerate quadratic groups, i.e. quadratic groups with $\sqrt{G}=G$.
Although it is principle possible to find irreducible WPA-s of any quadratic group ${(G,\vartheta)}$, in practice the number of irreducible WPA-s - and thus the length of the computation - grows dramatically with the number of non-cyclic subgroups of $G$. (A possible explanation of this is given after Proposition \[pro:upsilon cycl =3D fix\].) Therefore, there are only a few examples where this has been carried out. In the next section we give a few examples of admissible WPA-s, which will aid us later in the the solution of admissibility conditions for cyclic quadratic groups.
Examples\[sec:Examples\]
========================
In the following ${(G,\vartheta)}$ denotes a completely degenerate quadratic group. Recall that in this case $\vartheta$ is a $\pm1$ valued character of $G$. Let us also introduce some notation: 1 denotes the trivial subgroup in any group, $\xi_{0}$ the trivial character, and $R$ the regular WPA ${\mathcal{W}[1,\xi_{0}]}$.
Let us first consider admissible WPA-s of lowest possible degree. The one point transitive ${M}={\mathcal{W}[G,\vartheta]}$ has, according to , $\Upsilon_{{M}}(\alpha,\beta)=\vartheta(\beta)$. The reciprocity condition is satisfied if and only if $\vartheta=\xi_{0}$. Otherwise, let $${M}={\mathcal{W}[G,\vartheta]}\oplus{\mathcal{W}[ker\vartheta,\xi_{0}]}.\label{eq:min act}$$ Using and $[G:ker\vartheta]=2$, one obtains $$\Upsilon_{{M}}(\alpha,\beta)=\left\{ \begin{array}{cc}
3 & \textrm{if }\vartheta(\alpha)=\vartheta(\beta)=1\\
-1 & \textrm{if }\vartheta(\alpha)=\vartheta(\beta)=-1\\
1 & \textrm{if }\vartheta(\alpha)\ne\vartheta(\beta).\end{array}\right.$$
Therefore, ${M}$ is an admissible WPA of degree 3 (or degree 1 if $\vartheta=\xi_{0}$), and it is called the minimal admissible WPA of ${(G,\vartheta)}$. The admissible WPA associated to the Ising model is the minimal admissible WPA of $(\mathbb{Z}_{2},\xi_{1})$, where $\xi_{1}$ is the non-trivial character of $\mathbb{Z}_{2}$. However, if $|G|>2$, the minimal admissible WPA does not contain $R$, and may not correspond to a simple current WPA.
Let us consider the group $G\times\widehat{G}$, where $\widehat{G}$ is the group of homomorphisms from $G$ to $\mathbb{C}^{*}$, and introduce the natural quadratic form$$\vartheta(\alpha,\phi)=\phi(\alpha).\label{eq:weight of g2 wpa}$$ The quadratic group $(G\times\widehat{G},\vartheta)$ is non-degenerate, $\sqrt{G\times\widehat{G}}$=1, therefore its only transitive WPA is the regular action $R$ (of degree $|G|^{2}$, with $\omega(\alpha,\phi)=\vartheta(\alpha,\phi)$), which is consequently admissible. This is, actually, the quadratic group and the simple current WPA associated to the holomorphic $G$ orbifold model [@dvvv]. It is easy to see that $\Upsilon_{R}=0$ except $\Upsilon_{R}((1,\xi_{0}),(1,\xi_{0}))=|G|^{2}$.
Since we are interested in admissible WPA-s of completely degenerate quadratic groups, let us restrict the simple current group to the subgroup $G\simeq G\times\{\xi_{0}\}$. On this subgroup the quadratic form becomes trivial, thus we arrive at the completely degenerate quadratic group $(G,\xi_{0})$. In terms of the restricted quadratic group, the regular action branches into orbits of the form $X_{\phi}=G\times\{\phi\}$. It is clear that $X_{\Phi}$ falls in the equivalence class of ${\mathcal{W}[1,\phi]}$, therefore $R$ branches into the following WPA of $(G,\xi_{0})$$$R'=\bigoplus_{\phi\in\widehat{G}}{\mathcal{W}[1,\phi]}.\label{eq:decomp}$$
Note that $\Upsilon_{R'}(\alpha,\beta)=\Upsilon_{R}((\alpha,\xi_{0}),(\beta,\xi_{0}))=|G|^{2}\delta_{\alpha,1}\delta_{\beta,1}$ since the elements we have removed from the quadratic group had no fixed points in the regular action, thus did not give a contribution to $\Upsilon_{R}$ (see \[eq:chardef\]). Therefore, $R'$ satisfies the reciprocity and boundedness conditions. Galois invariance is obvious from , so we may conclude that $R'$ is an admissible WPA. Finally, it is easy to see that the direct sum defines an admissible WPA even if the quadratic group is ${(G,\vartheta)}$, where $\vartheta$ is any fully degenerate quadratic form.
\[exa:cosetwpa\]The last example is the combination of the previous ones into a general form. For a subgroup $H<G$ let $H_{0}$ denote $H\bigcap ker\vartheta$. As a generalization of and let $${\mathcal{A}[H]}=\left(\bigoplus_{\phi\in\widehat{G},H<ker(\vartheta\phi)}{\mathcal{W}[H,\phi]}\right)\oplus\left(\bigoplus_{\phi\in\widehat{G},H_{0}<ker(\phi)}{\mathcal{W}[H_{0},\phi]}\right),\label{eq:coset wpa}$$ if $H\neq H_{0}$. If $H=H_{0}$ the two terms in are equal and only one of them is needed. With this notation Example 1 corresponds to ${\mathcal{A}[G]}$ and Example 2 to ${\mathcal{A}[1]}$. It can be shown using and that $\Upsilon_{{\mathcal{A}[H]}}(\alpha,\beta)=[G:H]^{2}\Upsilon_{{M}}(\alpha,\beta)$ if $\alpha,\beta\in H$, and is zero otherwise. The boundedness condition is satisfied by ${\mathcal{A}[H]}$ as an equality: $|\Upsilon_{{\mathcal{A}[H]}}(\alpha,\beta)|=|Fix(\alpha)\cap Fix(\beta)|$. Galois invariance can be seen from , therefore ${\mathcal{A}[H]}$ is admissible. It is called the admissible coset WPA corresponding to the subgroup $H$, since this construction is the simplest way to extend the coset WPA ${\mathcal{W}[H,\xi]}$ (with arbitrary $\xi$) to an admissible WPA. The aim of the next section is to prove , which shows the important role played by admissible coset WPA-s.
Admissible coset WPA-s\[sec:aca\]
=================================
Let us first consider properties of the function $\Upsilon_{\Phi}(\alpha,\beta)$ associated to an admissible WPA $\Phi=(X,\omega)$. Admissibility requires it to be integer valued and symmetric in $\alpha$ and $\beta$ but a stronger requirement can be given.
\[pro:subgroup property\]If $\Phi$ is a WPA of the completely degenerate quadratic group ${(G,\vartheta)}$ satisfying Galois invariance and reciprocity, then $\vartheta(\alpha)\vartheta(\beta)\Upsilon_{\Phi}(\alpha,\beta)$ depends only on the subgroup $\left\langle \alpha,\beta\right\rangle $ generated by $\alpha$ and $\beta$.
Reciprocity requires $\vartheta(\alpha)\vartheta(\beta)\Upsilon_{\Phi}(\alpha,\beta)$ to be invariant under $(\alpha,\beta)\to(\beta,\alpha)$. If one takes into account that $\Upsilon_{\Phi}(\alpha,\beta)$ is non-zero only for $\alpha,\beta\in\sqrt{G}$, then it is easy to show using that $\vartheta(\alpha)\vartheta(\beta)\Upsilon_{\Phi}(\alpha,\beta)$ is also invariant under $(\alpha,\beta)\to(\alpha,\alpha\beta)$. These transformations together generate any base change in the subgroup $\left\langle \alpha,\beta\right\rangle $.
Therefore, to an admissible WPA $\Phi$ we may associate a function $\Upsilon_{\Phi}(\left\langle \alpha,\beta\right\rangle )=\vartheta(\alpha)\vartheta(\beta)\Upsilon_{\Phi}(\alpha,\beta)$ defined on the subgroups of $\sqrt{G}$ generated by at most two elements. For the admissible coset action ${\mathcal{A}[H]}$ the form of this function depends on whether $H<ker\vartheta$:$$\Upsilon_{{\mathcal{A}[H]}}(K)=\left\{ \begin{array}{ccc}
H<ker\vartheta & H\nless ker\vartheta\\
{}[G:H]^{2} & 3[G:H]^{2} & \textrm{if }K<H_{0}\\
- & [G:H]^{2} & \textrm{if }K\nless H_{0}\textrm{ but }K<H\\
0 & 0 & \textrm{if }K\nless H,\end{array}\right.\label{eq:adm coset char}$$ where $H_{0}=H\cap ker\vartheta$.
The boundedness condition requires the absolute value of $\Upsilon_{\Phi}$ to be bounded by the fixed point function of the permutation action corresponding to the WPA. For cyclic subgroups this is a consequence of reciprocity and Galois invariance, as one may show using Proposition \[pro:subgroup property\].
\[pro:upsilon cycl =3D fix\]Let $\Phi$ be a WPA satisfying reciprocity and Galois invariance. Then $\Upsilon_{\Phi}(\left\langle \alpha\right\rangle )=\vartheta(\alpha)|Fix(\alpha)|$.
It follows from Proposition \[pro:subgroup property\] that $\Upsilon_{\Phi}(\left\langle \alpha\right\rangle )$ is meaningful. According to $$\Upsilon_{\Phi}(\left\langle \alpha\right\rangle )=\vartheta(\alpha)\Upsilon_{\Phi}(\alpha,1)=\vartheta(\alpha)\sum_{p\in Fix(\alpha)}\frac{\omega(p)}{\omega(p)}=\vartheta(\alpha)|Fix(\alpha)|.$$
This Proposition shows that independent inequalities following from the boundedness condition correspond to non-cyclic subgroups of $\sqrt{G}$ generated by two elements. If we would consider only cyclic quadratic groups then, according to the previous Proposition, we could omit the requirement of boundedness in the definition of admissible WPA-s. This simplifies the problem of finding irreducible admissible WPA-s to a great extent. However, we shall not restrict ourselves to cyclic quadratic groups yet, since it is possible to formulate our main result in a slightly more general setting. We call an admissible WPA $(X,\omega)$ of ${(G,\vartheta)}$ cyclic if its stabilizer subgroups $G_{p}<G$ $(p\in X)$ are all cyclic.
\[thm:cycl irred\]Suppose that $\Phi$ is a cyclic admissible WPA of ${(G,\vartheta)}$. Then $\Phi$ is (equivalent to) a direct sum of admissible coset actions$$\Phi=\bigoplus_{\alpha\in G}n_{\alpha}{\mathcal{A}[\left\langle \alpha\right\rangle ]}.$$
Let us consider $\Upsilon_{\Phi}(H)$ for a cyclic admissible $\Phi$. First, $Fix_{\Phi}(H)=\emptyset$ if $H$ is not cyclic, and boundedness requires that also $\Upsilon_{\Phi}(H)=0$. Together with Proposition \[pro:upsilon cycl =3D fix\] this means that for cyclic admissible WPA-s the boundedness condition is satisfied as an equality: $|\Upsilon_{\Phi}(H)|=|Fix_{\Phi}(H)|$ $(\forall H<G)$.
Now, suppose that $K<G$ is a subgroup which is maximal with the property $\Upsilon_{\Phi}(K)\ne0$, i.e. for any $H>K$ but $H\ne K$ $\Upsilon_{\Phi}(H)=0$. As we have seen, such a subgroup is necessarily cyclic, let $K=\left\langle \alpha\right\rangle $. Our aim is to show that then $\Phi=\Phi_{1}\oplus{\mathcal{A}[\left\langle \alpha\right\rangle ]}$ where $\Phi_{1}$ is admissible (and clearly also cyclic). This would prove the theorem by induction on the degree of $\Phi$.
Let us consider transitive components of $\Phi$ contributing to $\Upsilon_{\Phi}(\alpha,\cdot)$. According to a transitive component ${\mathcal{W}[H_{i},\xi_{i}]}$ contributes to $\Upsilon_{\Phi}(\alpha,\cdot)$ only if $\alpha\in H_{i}$. The maximality of $\left\langle \alpha\right\rangle $ and $|\Upsilon_{\Phi}(H)|=|Fix(H)|$ $(\forall H<G)$ implies that if $\alpha\in H_{i}$, then $H_{i}=\left\langle \alpha\right\rangle $. Thus, we may write $\Phi$ as $\Phi=\Phi_{0}\oplus\Psi$, where $\Psi=\bigoplus_{k\in K}m_{k}{\mathcal{W}[\left\langle \alpha\right\rangle ,\xi_{k}]}$ and $\Upsilon_{\Phi_{0}}(\alpha,\beta)=0$ $(\forall\beta\in G)$. Let us consider $\Psi=\bigoplus_{k\in K}m_{k}{\mathcal{W}[\left\langle \alpha\right\rangle ,\xi_{k}]}$. By the definition of transitive actions $\xi_{k}(\alpha)=\vartheta(\alpha)$ $(\forall k\in K)$. Furthermore, if $\beta\notin\left\langle \alpha\right\rangle $, then $0=\Upsilon_{\Psi}(\alpha,\beta)=\sum_{k\in K}m_{k}\xi_{k}(\beta)$, where the first equation follows from the maximality of $\left\langle \alpha\right\rangle $, and the second from . This allows us to express $m_{k}$ using the orthogonality of characters as $$m_{k}=\sum_{\beta\in G}\Upsilon_{\Psi}(\alpha,\beta)\overline{\xi_{k}(\beta)}=\sum_{\beta\in\left\langle \alpha\right\rangle }\Upsilon_{\Psi}(\alpha,\beta)\overline{\vartheta(\beta)}=:n.$$ Note that $n$ does not depend on $\xi_{k}$, thus $$\Psi=n\bigoplus_{\xi\in\widehat{G},\xi(\alpha)=\vartheta(\alpha)}{\mathcal{W}[\left\langle \alpha\right\rangle ,\xi]}.\label{eq:psidecomp}$$
In the case $\left\langle \alpha\right\rangle \in\ker\vartheta$ the above is exactly $\Psi=n{\mathcal{A}[\left\langle \alpha\right\rangle ]}$, and it only remains to show that $\Phi_{0}$ is admissible. Since $\Phi$ and $\Psi$ satisfy reciprocity and Galois invariance, also does $\Phi_{0}$. It follows from the inequality $|\Upsilon_{\Phi_{0}}(K)|\le|\Upsilon_{\Phi}(K)|+|\Upsilon_{\Psi}(K)|$ that $\Upsilon_{\Phi_{0}}(K)=0$ if $K$ is not cyclic. For cyclic subgroups the boundedness condition is satisfied by Proposition \[pro:upsilon cycl =3D fix\], therefore $\Phi_{0}$ is admissible.
It remains to finish the proof for $\left\langle \alpha\right\rangle \notin\ker\vartheta$. In that case $$n{\mathcal{A}[\left\langle \alpha\right\rangle ]}=\Psi\oplus\left(\bigoplus_{\xi\in\widehat{G},\xi(\alpha)=1}n{\mathcal{W}[ker\vartheta,\xi]}\right)=\Psi\oplus\Psi_{1}.\label{eq: decomp1}$$ Therefore, we have to show that $\Phi_{0}=\Phi_{1}\oplus\Psi_{1}$, with $\Psi_{1}$ as above and $\Phi_{1}$ admissible. The argument is similar as before. Note that we may choose $\alpha$ such that $\ker\vartheta\cap\left\langle \alpha\right\rangle =\left\langle \alpha^{2}\right\rangle $ (recall that $\vartheta$ is a $\pm1$ valued character). We may repeat the argument given at the beginning of the proof to show that the only transitive components of $\Phi_{0}$ contributing to $\Upsilon_{\Phi_{0}}(\alpha,\alpha^{2})$ are of type ${\mathcal{W}[\left\langle \alpha^{2}\right\rangle ,\xi_{k}]}$ and $\xi_{k}(\alpha^{2})=\vartheta(\alpha^{2})=1$, so either $\xi_{k}(\alpha)=1$ or $\xi_{k}(\alpha)=-1$. Similarly as before, it follows from $\Upsilon_{\Phi_{0}}(\alpha^{2},\beta)=0$ $(\forall\beta\notin\left\langle \alpha\right\rangle )$ that $$\Phi_{0}=\Phi_{1}\oplus\left(\bigoplus_{\xi\in\widehat{G},\xi(\alpha)=-1}k\cdot{\mathcal{W}[\left\langle \alpha^{2}\right\rangle ,\xi]}\right)\oplus\left(\bigoplus_{\psi\in\widehat{G},\psi(\alpha)=1}l\cdot{\mathcal{W}[\left\langle \alpha^{2}\right\rangle ,\psi]}\right).\label{eq:decomp2}$$
Note, by comparing and that what we want to show is $l\ge n$. In order to prove this we have to consider the reciprocity condition for $\Upsilon_{\Phi}(\alpha^{2},\alpha)$. It follows from that $\Upsilon_{\Psi}(\alpha^{2},\alpha)-\Upsilon_{\Psi}(\alpha,\alpha^{2})=-2n[G:\left\langle \alpha\right\rangle ]^{2}$. Then, the reciprocity of $\Phi$ implies that $\Upsilon_{\Phi_{0}}(\alpha^{2},\alpha)-\Upsilon_{\Phi_{0}}(\alpha,\alpha^{2})=\Upsilon_{\Phi_{0}}(\alpha^{2},\alpha)=2n[G:\left\langle \alpha\right\rangle ]^{2}$. With the use of and $\Upsilon_{\Phi_{1}}(\alpha^{2},\alpha)=0$ we get $\Upsilon_{\Phi_{0}}(\alpha^{2},\alpha)=2(l-k)[G:\left\langle \alpha\right\rangle ]^{2}$. Since $l$ and $k$ are positive this implies $l\geq n$, which means that $$\Phi=\Phi_{2}\oplus n\cdot{\mathcal{A}[\left\langle \alpha\right\rangle ]}.$$ We may finish the proof , by showing (similarly as before) that $\Phi_{2}$ is admissible.
The above theorem proves the irreducibility of cyclic admissible coset actions as follows. Suppose, that in contrary ${\mathcal{A}[\left\langle \alpha\right\rangle ]}=\Phi_{1}\oplus\Phi_{2}$, where $\Phi_{1}$ and $\Phi_{2}$ are admissible. Then either $\Phi_{1}$ or $\Phi_{2}$ satisfies the conditions of the theorem, with $\left\langle \alpha\right\rangle $ as a maximal subgroup, therefore equals ${\mathcal{A}[\left\langle \alpha\right\rangle ]}$. It also proves that the only irreducible admissible WPA-s among the cyclic WPA-s are the cyclic admissible coset WPA-s. The irreducibility of non-cyclic admissible coset WPA-s may be proven with a slight modification of the argument used in the proof.
As the most important application of , let us consider a completely degenerate cyclic quadratic group $(\mathbb{Z}_{n},\vartheta)$. According to its admissible WPA-s are of the form:
$$\Phi=\bigoplus_{l|n}m_{l}{\mathcal{A}[H_{l}]},\label{eq:cycl quad irr}$$
where we denoted by $H_{l}$ the order $l$ subgroup of $\mathbb{Z}_{n}$ ( where $l$ divides $n$).
The admissible coset actions in depend on the completely degenerate quadratic form $\vartheta$. This is determined by its value on any generator of $\mathbb{Z}_{n}$, let $\alpha$ denote any of them. If $\vartheta(\alpha)=1$, i.e. $\alpha$ is an integer spin simple current, then $\vartheta$ is the trivial character. In this case the admissible coset WPA ${\mathcal{A}[H_{l}]}$ is of degree $(n/l)^{2}$. Note that the only irreducible containing the regular WPA $R$ is ${\mathcal{A}[1]}$ (of degree $n^{2}$), which is Example 2 of .
The other possibility is $\vartheta(\alpha)=-1$, i.e. $\alpha$ is a half-integer spin simple current. This implies that $\alpha$ is of even order, $n=2k$, and $ker\vartheta=H_{k}$. In that case the admissible coset action ${\mathcal{A}[H_{l}]}$ is of degree $(n/l)^{2}$ if $H_{l}<H_{k}=ker\vartheta$, i.e. if $l$ divides $k$. However, if $l$ does not divide $k$ (which means that $n/l$ is odd) the admissible coset WPA ${\mathcal{A}[H_{l}]}$ is of degree $3(n/l)^{2}$ (see ). The regular WPA still appears only in ${\mathcal{A}[1]}$, except if the quadratic group is $(\mathbb{Z}_{2},\xi_{1})$ when the order 3 admissible coset WPA ${\mathcal{A}[\mathbb{Z}_{2}]}$ contains it as well.
In the general case, when $G$ is not a cyclic group, the admissible coset actions corresponding to its subgroups give only a part of its irreducible WPA-s. In that case we may apply the result obtained for cyclic groups if we restrict the quadratic group to any of its cyclic subgroups. Let us illustrate this on an example. Let ${(G,\vartheta)}$ be a (non-cyclic) fully degenerate quadratic group of order $N$ and let the exponent of $G$ be $k$. Then there exists an order $k$ cyclic subgroup of $G$ - let us denote any of these by $H$. The primaries of any RCFT whose simple current group is $G$ have to contain a $G$ orbit equivalent to the regular WPA $R={(G,\vartheta)}$. What can we say about admissible WPA-s containing $R$ in this general case? If we restrict the simple current group to $(H,\vartheta|_{H}),$ then $R$ branches into $[G:H]=N/k$ copies of the regular WPA $R'=(H,\vartheta|_{H})$. The only irreducible of $H$ containing $R'$ is the coset WPA ${\mathcal{A}[1]}$ (except if $(H,\vartheta|_{H})=(\mathbb{Z}_{2},\xi_{1})$ - we shall deal with that case later), which is of order $k^{2}$. Therefore, any admissible WPA of ${(G,\vartheta)}$ which contains the regular action should contain, if restricted to $(H,\vartheta|_{H})$, $N/k$ copies of ${\mathcal{A}[1]}$, thus it has to be at least of degree $Nk$. Finally, let us consider the case $(H,\vartheta|_{H})=(\mathbb{Z}_{2},\xi_{1})$ - then the exponent of $G$ is 2, so $G=\mathbb{Z}_{2}^{n}$, but if $n\ge2$ it is possible to choose $H$ to be a subgroup on which $\vartheta$ is the trivial character. Therefore, the only exception is $G=(\mathbb{Z}_{2},\xi_{1})$.
Summary
=======
Quadratic groups and their weighted permutation actions are the mathematical objects corresponding to simple current symmetries in RCFT. WPA-s of an arbitrary quadratic group can be classified as a direct sum of transitive coset actions. Those WPA-s that are associated to simple currents have to satisfy additional constraints: Galois- invariance, reciprocity and boundedness. To solutions of these conditions, i.e. admissible WPA-s of a given (completely degenerate) quadratic group, are generated by a finite number of irreducible WPA-s. In we have considered some examples of irreducible WPA-s - the first was motivated by the Ising model, the second by holomorphic orbifold models of abelian groups. The generalization of these, given in Example \[exa:cosetwpa\], leads to a construction that extends a transitive coset WPA to an irreducible WPA - this way one obtains an irreducible WPA corresponding to each subgroup of the quadratic group (see ). Propositions \[pro:subgroup property\] and \[pro:upsilon cycl =3D fix\] implied that the number of independent inequalities corresponding to the boundedness condition equals the number of (non-cyclic) subgroups of the quadratic group generated by two elements. This suggests that the simplest case is that of cyclic quadratic groups, or more generally, cyclic admissible WPA-s. states that these are in fact generated by admissible coset WPA-s constructed in Example \[exa:cosetwpa\]. As a special case, the admissible WPA-s of completely degenerate cyclic quadratic groups can be given in the form . This result can be applied whenever the simple current group of the RCFT contains an order $n$ simple current of integral or half-integral spin to arrange the primaries of the theory in multiplets that in general consist of several simple current orbits. As we have discussed after the length of these multiplets is $k^{2}$ (where $k$ is any divisor of $n$) if the simple current is of integer spin or $k$ is even, and $3k^{2}$otherwise. Our results can be applied even if the quadratic group is not cyclic, through considering branching rules to some cyclic subgroup. We illustrated this on the orbit corresponding to the vacuum, which led to a lower bound of the number of primaries $Nk$, where $N$ is the order and $k$ the exponent of the radical of the simple current group. Note that a degree $N^{2}$ admissible WPA always exists - it is the admissible coset WPA ${\mathcal{A}[1]}$.
I would like to thank P. Bantay for discussions.
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abstract: 'We establish existence of stochastic financial equilibria on filtered spaces more general than the ones generated by finite-dimensional Brownian motions. These equilibria are expressed in real terms and span complete markets or markets with withdrawal constraints. We deal with random endowment density streams which admit jumps and general time-dependent utility functions on which only regularity assumptions are imposed. As a side-product of the proof of the main result, we establish a novel characterization of semimartingale functions.'
address: |
Department of Mathematical Sciences\
Carnegie Mellon University\
Wean Hall 7209\
Pittsburgh, PA 15213
author:
- 'Gordan Žitkovi'' c'
title: 'Financial equilibria in the semimartingale setting: complete markets and markets with withdrawal constraints'
---
Introduction
============
#### *Existing results and history of the problem.*
The existence of financial equilibria in continuous-time financial markets is one of the central problems in financial theory and mathematical finance. Unlike the problems of utility maximization and asset pricing where the price dynamics are given, the equilibrium problem is concerned with the origin of security prices themselves. More precisely, our goal is to construct a stochastic market with the property that when the price-taking agents act rationally, supply equals demand. Of course, there are many ways to interpret the previous sentence, even in the setting of continuous-time stochastic finance - let alone broader financial theory or economics as a whole. We are, therefore, really talking about a whole class of problems.
Before delving into the specifics of our formulation, let us briefly touch upon the history of the problem. Given the amount of research published on the various facets of the financial equilibrium, we can only mention a tiny fraction of the work leading directly to the present paper. Many seminal contributions not directly related to our research are left out. The notion of competitive equilibrium prices as an expression of the basic idea that the laws of supply and demand determine prices was introduced by Leon Walras (see [@Wal74]) 130 years ago. Rigorous mathematical theory starts with [@ArrDeb54]. Continuous-time stochastic models have been investigated by [@DufHua85] and [@Duf86], among many others. The direct predecessor of this paper is the work of Karatzas, Lakner, Lehoczky and Shreve in [@KarLakLehShr91], [@KarLehShr90] and [@KarLehShr91]. A convenient exposition of the results of these papers can be found in Chapter 4. of [@KarShr98]. Recently, existence of an equilibrium functional when utilities exhibit intertemporal substitution properties has been established in [@BanRie01].
#### *Our contributions.*
The motive leading our research was to investigate how the relaxation of the assumption that the filtration is generated by a Brownian motion affects the existence theory for the financial equilibrium, and how stringent conditions on the primitives (utilities, endowments, filtration) one needs to assume in this case. We were particularly keen to impose minimal conditions on utility functions and to allow endowment density processes to admit jumps. As we are primarily concerned with the [*existence*]{} of an equilibrium market, we stress that we have not pursued in any detail the questions of uniqueness or the financial consequences of our setup. We leave this interesting line of research for the future, and direct the reader to [@Dan93] and [@DanPon92]. In the following paragraphs we describe several directions in which this work extends existing theory.
First, we start from a right-continuous and complete filtration which we [*do not*]{} require to be generated by a Brownian motion. Consequently, we look for the price processes in the set of all finite-dimensional semimartingales, thus allowing for the equilibrium prices with jumps. The conditions we impose on the filtration are directly related with the possibility of obtaining a [*finite*]{} number of assets spanning all uncertainty. In this way, virtually any complete arbitrage-free market known in the financial literature can arise as an equilibrium in our setting.
Second, we introduce a simple constraint in our model by limiting the amounts the agents can withdraw from the trading account in order to finance a consumption plan. This constraint is phrased in terms of a withdrawal-cap process, which we allow to take infinite values - effectively including the possibility of a fully complete market, with no withdrawal cap whatsoever.
Third, we relax regularity requirements imposed on the utility functions. While these are still stronger than the typical conditions found in the utility-maximization literature, we show that one can develop the theory with assumptions less stringent than, e.g. those in Chapter 4., [@KarShr98]. We also deal with utility functionals which are not necessarily Mackey-continuous due to unboundedness of the utility functions in the neighborhood of zero. Moreover, there is no need for fine growth conditions such as [*asymptotic elasticity*]{} (see [@KraSch99]) in our setting. A principal feature of our model - jumps in the endowment density processes - warrants the use and development of tools from the general theory of stochastic processes. It is in this spirit that we provide a novel characterization of semimartingale functions (the functions of both time and space arguments, that yield semimartingales when applied to semimartingales). Finally, a result due to M' emin and Shiryaev ([@MemShi79]) is used as the most important ingredient in establishing a sufficient condition on a positive semimartingale for the local martingale part in its multiplicative decomposition to be a true martingale.
Another feature in which this paper differs from the classical work (e.g. [@KarLehShr90], [@KarLakLehShr91]) is in that we do not introduce the representative agent’s utility function (which is impossible due to withdrawal constraints). Instead we use Negishi’s approach (see [@Neg60]) in the version described in [@MasZam91]. This way the proof the existence of a financial equilibrium is divided into two steps. In the first step we establish the existence of an equilibrium pricing functional (an [*abstract equilibrium*]{}). Next, we implement this pricing functional through a stochastic market consisting of a finite number of semimartingale-modeled assets.
#### *Organization of the paper and some remarks on the notation.*
After the Introduction, in Section \[sec:model\] we describe the model, state the assumptions on its ingredients and pose the central problem of this work. Section \[sec:exabs\] introduces an abstract setup and establishes the existence of a financial equilibrium there. In Section \[sec:abstosto\], we transform the abstract equilibrium into a stochastic equilibrium as defined in Section \[sec:model\]. Finally, in Appendix \[sec:semi\] we develop the semimartingale results used in Section \[sec:abstosto\]: characterization of semimartingale functions, and regularity of multiplicative decompositions. Apart from being indispensable for the main result of our work, we hope they will be of independent interest, as well.
Throughout this paper, all stochastic processes will be defined on the time horizon $[0,T]$, where $T$ is a positive constant. To relieve the notation, the stochastic process ${ ( X_t )_{t\in [0,T]}}$ will be simply denoted by $X$, and its left-limit process $(X_{t-})_{t\in
[0,T]}$, by $X_-$. Unless specified otherwise, (in)equalities between [ ]{}processes will be understood pointwise, modulo indistinguishability, i.e., $X\leq Y$ will mean $X_t\leq Y_t$, for all $t\in [0,T]$, a.s. Finally, we use both notations “$X(t)$” and “$X_t$” interchangeably, the choice depending on typographical circumstances.
The Model {#sec:model}
=========
#### *The information structure.*
We consider a stochastic economy on a finite time horizon $[0,T]$. The uncertainty reveals itself gradually and is modeled by a right-continuous and complete filtration ${ ( {{\mathcal F}}_t )_{t\in [0,T]}}$ on a probability space $(\Omega,{{\mathcal F}},{{\mathbb P}})$, where we assume that ${{\mathcal F}}_0={\left\{\emptyset,\Omega\right\}}$ mod ${{\mathbb P}}$ and ${{\mathcal F}}={{\mathcal F}}_T$. In order for the finite-dimensional stochastic process spanning all the uncertainty to exist, the size of the filtration ${ ( {{\mathcal F}}_t )_{t\in [0,T]}}$ must be restricted:
\[frp\] A filtered probability space $(\Omega,{{\mathcal F}},{ ( {{\mathcal F}}_t )_{t\in [0,T]}},{{\mathbb P}})$, with ${ ( {{\mathcal F}}_t )_{t\in [0,T]}}$ satisfying the usual conditions, is said to have the [**finite representation property**]{} if for any probability ${{\mathbb Q}}$, equivalent to ${{\mathbb P}}$, there exist a finite number $n$ of ${{\mathbb Q}}$-martingales $Y^1,\dots, Y^n$ such that
1. $Y^i$ and $Y^j$ are orthogonal for $i\not =j$, i.e., the quadratic covariation $[Y^i,Y^j]_t$ vanishes for all $t\in [0,T]$, a.s.
2. for every bounded ${{\mathbb Q}}$-martingale $M$ there exists an $n$-dimensional predictable, $(Y^1,\dots, Y^n)$-integrable process $(H^1,\dots, H^n)$ such that $$\text{$M_t={{\mathbb E}}^{{{\mathbb Q}}}[M_T]+\sum_{i=1}^n \int_0^t H^i_u\, dY^i_u,\
\text{for all $t\in [0,T]$, a.s.}$}$$
The smallest such number $n$ is called the [**martingale multiplicity**]{} of $(\Omega,{{\mathcal F}},{ ( {{\mathcal F}}_t )_{t\in [0,T]}},{{\mathbb P}})$.
The filtered probability spaces with finite representation property include $n$-dimensional Brownian filtration, filtrations generated by Poisson processes, filtrations generated by Dritschel-Protter semimartingales (see [@ProDri99]), or combinations of the above.
The notion of martingale multiplicity and the related notion of the [*spanning number of a filtration*]{} have been introduced by Duffie in [@Duf86]. Definition \[frp\] differs from Duffie’s in that we explicitly require the existence of martingales $(Y^1,\dots,Y^n)$, for [*each*]{} probability measure ${{\mathbb Q}}\sim{{\mathbb P}}$. In [@Duf85], Duffie proves that if we only considered probability measures with $\frac{d{{\mathbb Q}}}{d{{\mathbb P}}}\in{{{\mathbb L}}^{\infty}}$ in Definition \[frp\], it would be enough to postulate the existence of the processes $(Y^1,\dots,Y^n)$ under ${{\mathbb P}}$. It is an open question whether one can achieve such a simplification under less stringent conditions on ${{\mathbb Q}}$.
\[ftp\] The filtered probability space $(\Omega,{{\mathcal F}},{ ( {{\mathcal F}}_t )_{t\in [0,T]}},{{\mathbb P}})$ has the finite representation property.
The finite representation property is used to ensure that the existence of a stochastic implementation of an abstract financial equilibrium with only a finite number of assets. Without this property one could still build a financial equilibrium, but the number of assets needed to span all the uncertainty might be infinite.
#### *Random endowments.*
There are $d\in{{\mathbb N}}$ agents in our economy each of whom is receiving a [**random endowment**]{} - a bounded and strictly positive income stream, modeled by a semimartingale $e^i$. We interpret the random variable $\int_0^t e^i_u\,du$ as the total income received by agent $i$ on the interval $[0,t]$, for $t<T$. At time $t=T$ there is a lump endowment of $e^i(T)$. To simplify the notation, we introduce the measure $\kappa$ on $[0,T]$ by $d\kappa_t=dt$ on $[0,T)$ and $\kappa({\left\{T\right\}})=1$. The cumulative random endowment on $[0,t]$ can now be represented as $\int_0^t e^i_t\, d\kappa_t$, for all $t\in [0,T]$.
The results in this paper can be extended to the case where $\kappa$ is an optional random measure with $\kappa({\left\{T\right\}})>0$, a.s. We do not pursue such an extension, as it would not add to the content in any interesting way.
In order for certain stochastic exponentials to be uniformly integrable martingales, we need to impose a regularity requirement on $e^i$, $i=1,\dots, d$, described in detail in Appendix \[sub:mult\].
\[def:nnx\] For a special semimartingale $X$, let ${{\mathcal N}}(X)={\langle M,M \rangle}_T$, where $X=M+A$ is a decomposition of $X$ into a local martingale $M$ and a predictable process $A$ of finite variation, and ${\langle M,M \rangle}$ denotes the compensator of the quadratic variation $[M,M]$.
The random variable ${{\mathcal N}}(X)$ from Definition \[def:nnx\] will usually be used in requirements of the form ${{\mathcal N}}(X)\in{{{\mathbb L}}^{\infty}}$. Existence of the compensator ${\langle M,M \rangle}$ and the special semimartingale property of $X$ are tacitely assumed as parts of such requirements.
The full strength of the following assumption on random endowment processes $e^i$, $i=1,\dots,d$, is needed for the existence of a stochastic equilibrium (Theorem \[main\]), and only part 1. for the abstract equilibrium (Theorem \[exabs\]).
\[ends\] For $i=1,\dots,d$,
1. \[ends:abs\] $e^i$ is an optional process, with ${\varepsilon}\leq e^i\leq 1/{\varepsilon}$, for some ${\varepsilon}>0$,
2. \[ends:semi\] $e^i$ is a (special) semimartingale and ${{\mathcal N}}(e^i)\in {{{\mathbb L}}^{\infty}}$.
Processes $e^i$ satisfying conditions of Assumption \[ends\] include linear combinations of processes of the form $Y_t=h(t,X_t)$ where $1/{\varepsilon}\geq h \geq{\varepsilon}>0$ is a $C^{1,2}$-function, with $h_x$, and $h_{xx}$ uniformly bounded, and $X$ is a diffusion process with a bounded diffusion coefficient, or a L' evy process whose jump measure $\nu$ satisfies $\int_{{{\mathbb R}}}
x^2\, \nu(dx)<\infty$. Homogeneous and inhomogeneous Poisson processes and non-exploding continuous-time Markov chains are examples of allowable processes $X$.
#### *Utility functions.*
Apart from being characterized by the random endowment process, each agent represents her attitude towards risk by a von Neumann-Morgenstern utility function $U^i$. Before we list the exact regularity assumptions placed on $U^i$, we need the following definition:
For a continuously differentiable function $f:[x_1,x_2]\to{{\mathbb R}}$ we define the [**total convexity norm ${{||f||}}={{||f||}}_{[x_1,x_2]}$**]{} by $$\nonumber
\begin{split}
{{||f||}}_{[x_1,x_2]}={\left| f(x_1) \right|}+
{\left| f'(x_1) \right|}+{\mathrm TV}(f';\, [x_1,x_2]),
\end{split}$$ where ${\mathrm TV}(f';\, [x_1,x_2])$ denotes the total variation of the derivative $f'$ of $f$ on $[x_1,x_2]$. A function $f:[0,T]\times [x_1,x_2]\to{{\mathbb R}}$, continuously differentiable in the second variable, is said to be [**convexity-Lipschitz**]{} if there exists a constant $C$ such that, for all $t,s\in [0,T]$, we have ${{||f(t,\cdot)-f(s,\cdot)||}}\leq C {\left| t-s \right|}$. A function $f:[0,T]\times I\to{{\mathbb R}}$ (where I is a subset of ${{\mathbb R}}$) is called [**locally convexity-Lipschitz**]{} if its restriction $f|_{[0,T]\times [x_1,x_2]}$ is convexity-Lipschitz, for any compact interval $[x_1,x_2]$.
A sufficient condition for a function $f:[0,T]\times I\to{{\mathbb R}}$ to be (locally) convexity-Lipschitz is that $f(t,\cdot)\in C^2(I)$, for all $t\in [0,T]$, and $f_{xx}(x,\cdot)$ is Lipschitz, (locally) uniformly in $x$.
For each $i=1,\dots, d$, the utility function $U^i:[0,T]\times (0,\infty)\to{{\mathbb R}}$ has the following properties \[utilreg\]
1. \[utilreg:util\] $U^i(t,\cdot)$ is strictly concave, continuously differentiable and strictly increasing for each $t\in
[0,T]$. Moreover, the function $U(\cdot,x)$, is bounded for any $x\in (0,\infty)$. \[Uconc\]
2. \[utilreg:inv\] The [*inverse-marginal-utility*]{} functions $I^i:[0,T]\times
(0,\infty)\to (0,\infty)$, $I^i(t,y)=U_x(t,\cdot)^{-1}(t,y)$ are locally convexity-Lipschitz and satisfy $$\label{inadas}
\begin{split}
\lim_{y\to\infty} I^i(t,y)=0,\ \lim_{y\to 0} I^i(t,y)=\infty,\
\text{uniformly in $t\in [0,T]$.}
\end{split}$$
The most important example of a utility function satisfying Assumption \[utilreg\] is so-called [*discounted utility*]{} $U(t,x)=\exp(-\beta t) \hat{U}(x)$, where $\beta>0$ is the impatience factor, and $\hat{U}\in C^2({{\mathbb R}}_+)$ satisfies $\hat{U}'>0$ and $\hat{U}''$ is a strictly negative function of finite variation on compacts. A sufficient (but not necessary) condition for this is $\hat{U}\in C^3({{\mathbb R}}_+)$. Power utilities $\hat{U}(x)=x^p/p$, for $p\in (-\infty,1)\setminus {\left\{0\right\}}$ and $\hat{U}(x)=\log(x)$ belong to this class.
Unlike the problems of utility maximization (see [@KraSch99], e.g.) where the utility function is only required to be strictly concave and continuously differentiable, existence of financial equilibria requires a higher degree of smoothness (compare to Chapter 4., [@KarShr98], where the existence of three continuous derivatives is postulated in the Brownian setting).
Total utility accrued by an agent whose consumption equals $c_t
(\omega)$ at time $t\in [0,T]$ in the state of the world $\omega\in\Omega$, will be modeled as the aggregate of [*instantaneous utilities*]{} $U^1(t,c_t(\omega))$ in an additive way. More precisely, for each agent $i=1,\dots, d$, we define the [**utility functional**]{} ${\mathbb{U}}^i$, taking values in $[-\infty, \infty]$. Its action on an optional process $c$ is given by ${\mathbb{U}}^i(c)\triangleq {{\mathbb E}}[\int_0^T U^i(t,c(t))\,
d\kappa_t]$ when ${{\mathbb E}}[\int_0^T \min(0,U^i(t,c(t)))\,
d\kappa_t]>-\infty$ and ${\mathbb{U}}^i(c)=-\infty$, otherwise.
Due to the fact that the final time-point $t=T$ plays a special role in the definition of the endowment processes $e^i$, one would like to be able to redefine the agent’s utility quite freely there. Utility functions with virtually no continuity requirements at $t=T$ are indeed possible to include in our framework, but we decided not to go through with this in order to keep the exposition as simple as possible. It will suffice to note that most of the restrictions involving the time variable placed on the utility functions in Assumption \[utilreg\] are there to ensure that the pricing processes obtained in Theorem \[exabs\] are semimartingales and not merely optional processes. All of them superfluous at $t=T$, since the semimartingale property of a process ${ ( X_t )_{t\in [0,T]}}$ is preserved if we replace $X_T$ by another ${{\mathcal F}}_T$-measurable random variable.
#### *Investment and consumption.*
The basic premise of equilibrium analysis is that agents engage in trade with each other in order to improve their utilities. To facilitate this exchange, a stock market consisting of a finite number of risky assets $S$, and one riskless asset $B$ is to be set up. In order to have a meaningful mathematical theory, we shall require these processes to be semimartingales with respect to $(\Omega, {{\mathcal F}}, { ( {{\mathcal F}}_t )_{t\in [0,T]}}, {{\mathbb P}})$. Moreover, both the riskless asset $B$ and its left-limit process $B_-$ will be required to be strictly positive [ ]{}predictable processes of finite variation.
An agent trades in the market by dynamically readjusting the portion of her wealth kept in various risky, or the riskless asset. This is achieved by a choice of a portfolio process $H$ (in an adequate admissibility class to be specified shortly) with the same number of components as $S$. At the same time, the agent will accrue utility by choosing the consumption rate according to an optional consumption process $c$. The components of the process $H$ stand for the number of shares of each risky asset held in the portfolio. The trading is financed by borrowing from (or depositing in) the riskless asset. With that in mind, the equation governing the dynamics of the wealth ${X^{H,c,e}}$ of an agent becomes $$ \label{wealth}
\begin{split}
d{X^{H,c,e}}_t=H_t\, dS_t+\frac{({X^{H,c,e}}_{t-}-H_t S_{t-})}{B_{t-}}
\, d B_t-c(t)\, d\kappa_t+e(t)\, d\kappa_t.
\end{split}$$ We assume that the agent has no initial wealth, i.e., ${X^{H,c,e}}_0=0$ (this assumption is in place only to simplify exposition). The net effect of market involvement of the agent is a redistribution of wealth across times and states of the world. The income stream $e$ (which would have been the only possibility without the market) gets swapped for another stream - the consumption process $c$.
There are, invariably, exogenous factors which limit the scope of the market activity. In this paper we deal with one of the simplest such limitations - withdrawal constraints. After having traded for the day (with the net gain of $H_t\,
dS_t+({X^{H,c,e}}_{t-}-H_t S_{t-})/B_{t-}\, d B_t$), and having received the endowment $e_t\, d\kappa_t$, the agent decides to consume $c_t\, d\kappa_t$. If this amount is too large, it is likely to be unavailable for withdrawal from the trading account on a short notice. Therefore, a cap of $\Gamma^i$ is placed on the amount agent $i$ can consume at time $t$. We assume that $\Gamma^i,i=1,\dots,d$ are $(0,\infty]$-valued [ ]{}adapted process satisfying $\Gamma^i>e^i$. We impose no withdrawal restrictions for $t=T$, effectively requiring $\Gamma^i_T=\infty$ a.s. Moreover, an assumption analogous to Assumption \[ends\] is placed on $\Gamma^i$:
\[with\] For each $C>0$, the stochastic process $\min(\Gamma^i,C)$ is a semimartingale satisfying ${{\mathcal N}}(\min(\Gamma^i,C))\in {{{\mathbb L}}^{\infty}}$.
In addition to an abstract, exogenously given withdrawal-cap processes, in the following example we describe several other possibilities.
In all of the following examples, we set $\Gamma^i_T=\infty$:
1. [*Complete markets*]{}: $\Gamma^i_t=\infty$, $t\in [0,T]$.
2. [*Proportional constraints*]{}: For a constant $\gamma>1$, $\Gamma^i_t=\gamma e^i_t$, $t\in [0,T)$.
3. [*Constant overdraft limit*]{}: for $\delta>0$ we set $\Gamma^i_t=e^i_t+\delta$, $t\in [0,T)$.
#### *Market Equilibrium.*
Before giving a rigorous definition of an equilibrium market, we introduce the notion of affordability for a consumption process $c$. Here we assume that the market structure (in the form of the withdrawal-cap process $\Gamma$, a finite-dimensional semimartingale $S$ (risky assets), and a positive predictable [ ]{}process $B$ of finite variation (riskless asset)) and the random endowment process $e$ are given.
\[afford\] An $(S,B,e,\Gamma)$-[**affordable consumption-investment strategy**]{} is a pair $(H,c)$ of an $S$-integrable predictable [**portfolio process**]{} $H$, and an optional [**consumption process**]{} $c\geq 0$ such that
1. There exists $a\in{{\mathbb R}}$ such that $a+\int_0^t H_u\, dS_u\geq
0$, for all $t\in [0,T]$, a.s.
2. The wealth process ${ ( X_t )_{t\in [0,T]}}$, as defined in (\[wealth\]), satisfies $X_T\geq 0$, a.s.
3. The consumption process $c$ satisfies $c_t\leq
\Gamma_t$ for all $t\in [0,T]$, a.s.
\[defequ\] A pair $(S,B)$ of a finite-dimensional semimartingale $S$ and a positive predictable [ ]{}process $B$ of finite variation is said to form an [**equilibrium market**]{} if for each agent $i=1,\dots,d$ here exists an $(S,B,e^i,\Gamma^i)$-affordable consumption-investment strategy $(H^i,c^i)$ satisfying the following two conditions:
1. $\sum_i c^i_t=\sum_i e^i_t$ and $\sum_i H^i_t=0$, for all $t\in [0,T]$, a.s.
2. For each $i$, $c^i$ maximizes the utility functional ${\mathbb{U}}^i(\cdot)$ over all $(S,B,e^i,\Gamma^i)$-affordable consumption-investment strategies $(H,c)$.
Existence of an abstract equilibrium {#sec:exabs}
====================================
In this section we establish the existence of an abstract version of a market equilibrium. The notion of an abstract equilibrium encapsulates the tenet that markets in equilibrium should clear when all agents act rationally. The full-fledged stochastic market has been abstracted away in favor of a pricing functional ${{\mathbb Q}}$. ${{\mathbb Q}}$ will be an element of the topological dual ${({{{\mathbb L}}^{\infty}})^*}$ of the [*consumption space*]{} ${{{\mathbb L}}^{\infty}}$, so that the action ${\langle {{\mathbb Q}},c \rangle}$ of ${{\mathbb Q}}$ onto a consumption process $c$ has the natural interpretation of the price of the consumption stream $c$. Our setup allows for utility functions unbounded in the neighborhood of $x=0$ (in order to be able to deal with the important examples from financial theory). Even though these utilities follow the philosophy of the von Neumann - Morgenstern theory, they are [*not*]{} von Neumann - Morgenstern utilities in the sense of [@Bew72]. In fact, the corresponding utility functionals are not necessarily Mackey-continuous and thus the abstract theory pioneered by Truman Bewley and others does not apply directly to our setting. The structure of our proof of the existence of an abstract equilibrium follows the skeleton laid out in [@MasZam91]. For that reason we focus on the substantially novel parts of the proof and only outline the rest. In particular, we present a detailed proof of closedness of the set of utility vectors in Lemma \[UFclosed\], but merely refer to the corresponding parts of [@MasZam91] for the results whose derivation is a more-or-less straightforward modification of existing results.
#### *Functional-analytic setup.*
In what follows, ${{{\mathbb L}}^{\infty}}$ will denote the Banach space of ($\kappa\otimes {{\mathbb P}}$)-essentially bounded processes, measurable with respect to the $\sigma$-algebra ${{\mathcal O}}$ of ${ ( {{\mathcal F}}_t )_{t\in [0,T]}}$-optional sets. ${{{\mathbb L}}^{\infty}}_+$ will denote the positive orthant of ${{{\mathbb L}}^{\infty}}$, i.e., the set of all $(\kappa\otimes{{\mathbb P}})$-a.e. nonnegative elements in ${{{\mathbb L}}^{\infty}}$. All ${ ( {{\mathcal F}}_t )_{t\in [0,T]}}$-optional processes will be identified with the corresponding ${{\mathcal O}}$-measurable random variables without explicit mention, and the equalities and inequalities will always be understood in $(\kappa\otimes{{\mathbb P}})$-a.e. sense.
The set of all bounded consumption processes $c$ satisfying the consumption constraints introduced via cap processes $\Gamma^i$, will be denoted by ${{\mathcal A}}^i$, i.e., ${{\mathcal A}}^i = {{\left\{c\in{{{\mathbb L}}^{\infty}}_+\,:\,
c\leq \Gamma^i\right\}}}$. Also, define ${{\mathcal A}}={{\left\{{(c^i)_{i=1,\dots, d}}\,:\,c^i\in{{\mathcal A}}^i\right\}}}$, and its subset ${{{\mathcal A}}^f}$ consisting of only those allocations which can be produced by redistributing the aggregate endowment $e=\sum_i e^i$, i.e., ${{{\mathcal A}}^f}=\{{(c^i)_{i=1,\dots, d}}\in{{\mathcal A}}\,:\, \sum\nolimits_i c^i=e\}$
The topological dual ${({{{\mathbb L}}^{\infty}})^*}$ of ${{{\mathbb L}}^{\infty}}$ can be identified with the set of all finitely-additive measures ${{\mathbb Q}}$ on the $\sigma$-algebra ${{\mathcal O}}$, weakly-absolutely continuous with respect to $\kappa\otimes{{\mathbb P}}$, i.e. for $A\in{{\mathcal O}}$, ${{\mathbb Q}}[A]=0$ whenever $(\kappa\otimes{{\mathbb P}})[A]=0$.
\[Ala\]We will consider the set of finitely-additive probabilities as a subset of ${({{{\mathbb L}}^{\infty}})^*}$, supplied with the weak \* topology $\sigma({({{{\mathbb L}}^{\infty}})^*}, {{{\mathbb L}}^{\infty}})$. It is a consequence of Alaoglu’s theorem that any collection of finitely-additive probabilities is relatively $\sigma({({{{\mathbb L}}^{\infty}})^*},{{{\mathbb L}}^{\infty}})$-compact. Furthermore, the closedness of the set of finitely-additive probabilities (in the space of all finite-additive measures, and w.r.t the $\sigma({({{{\mathbb L}}^{\infty}})^*},
{{{\mathbb L}}^{\infty}})$-topology) implies that the cluster-points of nets of finitely-additive probabilities are finitely-additive probabilities themselves. In the sequel, weak \* topology will always refer to the $\sigma({({{{\mathbb L}}^{\infty}})^*}, {{{\mathbb L}}^{\infty}})$ topology of the pair $({({{{\mathbb L}}^{\infty}})^*},{{{\mathbb L}}^{\infty}})$.
We can now define the concept of an abstract equilibrium. Instead of a semimartingale price process, an abstract equilibrium requires the existence of a finitely-additive probability ${{\mathbb Q}}\in{({{{\mathbb L}}^{\infty}})^*}$ which takes the role of a pricing functional acting directly on consumption processes. Given such a finitely-additive probability ${{\mathbb Q}}$, the [**budget set**]{} $B^i({{\mathbb Q}})$ of agent $i$ is defined by $ B^i({{\mathbb Q}})={{\left\{c\in {{{\mathbb L}}^{\infty}}_+\,:\, c\in{{\mathcal A}}^i \text{\ and\
} {\langle {{\mathbb Q}},c \rangle}\leq{\langle {{\mathbb Q}},e^i \rangle}\right\}}}$.
\[absequ\] A pair $({{\mathbb Q}},{(c^i)_{i=1,\dots, d}})$ of a finitely-additive probability ${{\mathbb Q}}$ and an allocation ${(c^i)_{i=1,\dots, d}}\in{{\mathcal A}}$ is called an [**abstract equilibrium**]{} if
1. $\sum_i c^i=\sum_i e^i$, i.e., ${(c^i)_{i=1,\dots, d}}\in {{{\mathcal A}}^f}$.
2. For any $i=1,\dots,d$, $c^i\in B^i({{\mathbb Q}})$ and ${\mathbb{U}}^i(c^i)\geq {\mathbb{U}}^i(c)$ for all $c\in B^i({{\mathbb Q}})$.
#### *Existence of an abstract equilibrium.*
To simplify notation in some proofs and statements we assume that the utility functionals ${\mathbb{U}}^i$ are normalized so that ${\mathbb{U}}^i(e^i)=0$ for all $i=1,\dots, d$.
We start by introducing ${{\mathcal U}^f}$ - the set of all $d$-tuples of utilities which can be achieved by different allocations ${(c^i)_{i=1,\dots, d}}\in{{{\mathcal A}}^f}$, i.e., $$ \label{defu}
\begin{split}
{{\mathcal U}^f}={{\left\{{\left({\mathbb{U}}^1(c^1),\dots, {\mathbb{U}}^d(c^d)\right)}\,:\,{(c^i)_{i=1,\dots, d}}\in{{{\mathcal A}}^f}\right\}}},
\end{split}$$ and ${{{\mathcal U}^f}_{-}}={{\mathcal U}^f}-[0,\infty)^d$ - the set of all vectors in ${{\mathbb R}}^d$ dominated by some element in ${{\mathcal U}^f}$. The elements in ${{{\mathcal U}^f}_{-}}$ will be called [**utility vectors**]{}. Our first lemma identifies several properties of ${{{\mathcal U}^f}_{-}}$, the most important of which is closedness.
\[UFclosed\] The set ${{{\mathcal U}^f}_{-}}$ is non-empty, convex and closed.
${{{\mathcal U}^f}_{-}}$ is obviously non-empty, and its convexity follows easily from convexity of ${{{\mathcal A}}^f}$. It remains to show that it is closed. Let ${{\left\{{{\mathbf u}}_n\right\}}_{n\in{{\mathbb N}}}}$, ${{\mathbf u}}_n = ( u^1_n, u^2_n, \ldots, u^d_n )$, be a sequence in ${{{\mathcal U}^f}_{-}}$ converging to ${{\mathbf u}}= ( u^1, u^2, \ldots, u^d ) \in {{\mathbb R}}^d$. By the definition of the set ${{{\mathcal U}^f}_{-}}$, there exist two sequences ${{\mathbf c}}_n=(c^1_n,$ $c^2_n,\dots c^d_n) \in {{{\mathcal A}}^f}$ and ${{\mathbf r}}_n=(r^1_n,\dots,
r^d_n)\in{{\mathbb R}}^d_+$ such that ${\mathbb{U}}^i(c^i_n)=u^i_n+r^i_n$. Since $u^i_n\leq{\mathbb{U}}^i(c^i_n)\leq {\mathbb{U}}^i(e)<\infty$, we can assume - passing to a subsequence if necessary - that there exists a vector $\hat{{{\mathbf u}}}=(\hat{u}^1,\dots,\hat{u}^d )$ such that ${\mathbb{U}}^i(c^i_n)\to \hat{u}^i\geq u^i$.
For any $i=1\dots d$, the sequence ${{\left\{c^i_n\right\}}_{n\in{{\mathbb N}}}}$ is bounded in ${{{\mathbb L}}^{\infty}}$, and therefore also in ${{{\mathbb L}}^1}(\kappa\otimes{{\mathbb P}})$. By a simple extension of the classical Komlos’ theorem (see [@Sch86]) to the case of ${{\mathbb R}}^d$-valued random variables, there exists an infinite array of nonnegative weights $(\alpha_k^n)^{n\in{{\mathbb N}}}_{k=n,\ldots, k_n}$ and a $d$-tuple ${(c^i)_{i=1,\dots, d}}$ of nonnegative optional processes with the following properties: $\sum_{k=n}^{k_n} \alpha_k^n=1$ and $\tilde{c}^i_n=
\sum_{k=n}^{k_n} \alpha_k^n c^i_k\,\to\, c^i$, $(\kappa\otimes{{\mathbb P}})$-a.e. Consequently, $\sum_i c^i=e$ and $c^i\leq \Gamma^i$, so that ${(c^i)_{i=1,\dots, d}}\in{{\mathcal A}}^f$.
To show that ${{\mathbf u}}\in{{{\mathcal U}^f}_{-}}$, we use concavity and right-continuity of the utility functions and the Fatou Lemma (the use of which is justified by the fact that $\tilde{c}^i_n\leq e$, for all $i$ and all $n\in{{\mathbb N}}$) in the following chain of inequalities: $$\nonumber
\begin{split}
{\mathbb{U}}^i(c^i)={\mathbb{U}}^i(\lim_n \tilde{c}^i_n)\geq \operatorname*{\overline{\rm lim}}_n
{\mathbb{U}}^i(\tilde{c}^i_n)\geq
\operatorname*{\overline{\rm lim}}_n \sum_{k=n}^{k_n} \alpha_k^n
{\mathbb{U}}^i(c^i_k)=\lim_n {\mathbb{U}}^i(c^i_n)=\hat{u}^i\geq u^i,
\end{split}$$
The next task is to establish the existence of [*supporting measures*]{} for [*weakly optimal*]{} utility vectors . We start with definitions of these two concepts.
A finitely-additive probability ${{\mathbb Q}}$ is said to **support** a \[supp\] vector ${{\mathbf u}}=(u^1,\dots, u^d)$ $\in{{\mathbb R}}^d$ if for any allocation ${{\mathbf c}}={(c^i)_{i=1,\dots, d}}\in{{\mathcal A}}$ with the property that ${\mathbb{U}}^i ( c^i ) \geq u^i$ for all $i=1,\dots, d$, we have $ {\langle {{\mathbb Q}},\sum\nolimits_i c^i \rangle} \geq {\langle {{\mathbb Q}},\sum\nolimits_i
e^i \rangle}$. The set of all finitely-additive probability measures supporting a vector ${{\mathbf u}}\in{{\mathbb R}}^d$ is denoted by $P({{\mathbf u}})$.
\[wopt\] A vector ${{\mathbf u}}=(u^1,\dots, u^d)$ in ${{{\mathcal U}^f}_{-}}$ is said to be [[**]{}weakly optimal]{} if there is no allocation ${(c^i)_{i=1,\dots, d}}\in{{{\mathcal A}}^f}$ with the property that ${\mathbb{U}}^i(c^i)>u^i$ for all $i=1,\dots,d$.
\[Pnonempty\] For a weakly optimal utility vector ${{\mathbf u}}\in{{{\mathcal U}^f}_{-}}$, the set $P({{\mathbf u}})$ of finitely-additive probabilities supporting ${{\mathbf u}}$ is non-empty, convex and weak \* compact
The proof relies on a well-know separating-hyperplane-type argument. See [@MasZam91], Section 8., pp. 1859-1860 for more details.
Having established the closedness and convexity of the set ${{{\mathcal U}^f}_{-}}$ in Lemma \[UFclosed\], and the existence of supporting functionals for weakly optimal utility vectors in Lemma \[Pnonempty\], it suffices to use the proof of Theorem 7.1, p. 1856 in [@MasZam91] to establish the following abstract existence theorem:
\[exabs\]Under Assumptions \[ends\].\[ends:abs\], \[utilreg\].\[utilreg:util\] and \[utilreg\].\[utilreg:inv\], there exists an abstract equilibrium $({{\mathbb Q}},{(c^i)_{i=1,\dots, d}})$.
From abstract to stochastic equilibria {#sec:abstosto}
======================================
Our next task is to show that the abstract equilibrium obtained in the previous section can be implemented as a stochastic equilibrium. We first note that the equilibrium functional ${{\mathbb Q}}$ must be countably-additive and equivalent to $\kappa\otimes {{\mathbb P}}$. We omit the proof as it follows the argument from Theorem 8.2, p. 1863 in [@MasZam91], using the fact that $\Gamma^i>e^i$ and $\Gamma^i_T=\infty$ for all $i=1,\dots, d$.
\[countadd\] Let $({{\mathbb Q}},{(c^i)_{i=1,\dots, d}})$ be an abstract equilibrium. Then ${{\mathbb Q}}$ is countably additive and equivalent to $\kappa\otimes{{\mathbb P}}$.
In Lemma \[hasform\] we use convex duality to describe the solutions of agents’ utility-maximization problems in an equilibrium:
\[hasform\] Suppose that $({{\mathbb Q}},{(c^i)_{i=1,\dots, d}})$ is an abstract equilibrium. Then there exist constants ${\lambda}^i>0$, $i=1,\dots, d$, such that the consumption processes $c^i,i=1,\dots,d$ are of the form $$ \label{formc}
\begin{split}
c^i_t=\min(\Gamma^i_t, I^i(t,{\lambda}^i Q_t)),
\end{split}$$ where $Q={ ( Q_t )_{t\in [0,T]}}$ is the optional version of the Radon-Nikodym derivative of ${{\mathbb Q}}$ with respect to $\kappa\otimes{{\mathbb P}}$.
We prove the lemma for $i=1$. Let $N(c^1)$ be the set of all $c\in{{{\mathbb L}}^{\infty}}_+$ such that $c\leq \min(\Gamma^1,{{||c^1||}}_{{{{\mathbb L}}^{\infty}}})$. $N(c^1)$ is a $\sigma({{{\mathbb L}}^{\infty}},{{{\mathbb L}}^1})$-compact subset in ${{{\mathbb L}}^{\infty}}_+$, and by Komlos’ Lemma the restriction of ${\mathbb{U}}^1$ to $N(c^1)$ is $\sigma({{{\mathbb L}}^{\infty}},{{{\mathbb L}}^1})$-upper-semicontinuous and concave. By Lemma \[countadd\], the finitely-additive measure ${{\mathbb Q}}$ is countably-additive so the Lagrangean function $L:N(c^1)\times [0,\infty)\to
[-\infty,\infty)$, $L(c,{\lambda})={\mathbb{U}}^1(c)-{\lambda}{\langle {{\mathbb Q}},c-e^1 \rangle}$ satisfies the conditions of the Minimax theorem (see [@Sio58]).We know that the maximizer $c^1$ of the functional ${\mathbb{U}}^1$ over $B^1({{\mathbb Q}})$ trivially satisfies $c^1\leq
{{||c^1||}}_{{{{\mathbb L}}^{\infty}}}$, so $$\nonumber
\begin{split}
{\mathbb{U}}^1(c^1)&=\sup_{c\in B^1({{\mathbb Q}})\cap N(c^1)}{\mathbb{U}}^1(c) =\sup_{c\in
N(c^1)} \inf_{{\lambda}\geq
0} L(c,{\lambda}) = \inf_{{\lambda}\geq 0} \sup_{c\in N(c^1)} L(c,{\lambda})\\
&=\inf_{{\lambda}\geq 0} \Big( {\lambda}{\langle {{\mathbb Q}},e^1 \rangle}+{{\mathbb E}}\int_0^T
V(t,{\lambda}Q_t;\, m^1_t)\,d\kappa_t\Big),\\
\end{split}$$ where $m^1_t=\min(\Gamma^1_t,{{||c^1||}}_{{{{\mathbb L}}^{\infty}}})$, and the function $V:[0,T]\times[0,\infty)\times (0,\infty)\to{{\mathbb R}}$ is given by $$\nonumber
\begin{split}
V(t,{\lambda};\, \xi )\triangleq\sup_{x\in [0,\xi)} (U^1(t,x)-x{\lambda})
=\begin{cases} V(t,{\lambda};\,\infty),& {\lambda}>U^1_x(t,\xi), \\
U^1(t,\xi)-{\lambda}\xi,& {\lambda}\leq U^1_x(t,\xi).\end{cases}
\end{split}$$ $V$ is convex and nonincreasing in ${\lambda}$, and nondecreasing in $\xi$. The function $v:[0,\infty)\to [-\infty,\infty]$, where $v({\lambda})={\lambda}{\langle {{\mathbb Q}},e^1 \rangle}+{{\mathbb E}}\int_0^T V(t,{\lambda}Q_t;\, m^1_t)\,d\kappa_t$, is convex and proper, since $\inf_{{\lambda}\geq 0} v({\lambda})={\mathbb{U}}^1(c^1)\in
(-\infty,\infty)$. Furthermore, Assumption \[utilreg\].\[utilreg:util\] implies the inequality $V(t,{\lambda};\, m^1_t)\leq U^1(t,{{||c^1||}}_{{{{\mathbb L}}^{\infty}}})$ and the existence of a constant $D>0$ such that ${\mathbb{U}}^1(c^1)\leq v({\lambda})\leq
{\lambda}{\langle {{\mathbb Q}},e^1 \rangle}+D$, for all ${\lambda}>0$.
Assumption \[utilreg\].\[utilreg:inv\] ensures the existence of a constant $C>{{||c^1||}}_{{{{\mathbb L}}^{\infty}}}$ such that $I^1(t,C)<\frac{1}{2} {\langle {{\mathbb Q}},e^1 \rangle}$ for all $t\in [0,T]$. Then, for all $\xi,{\lambda},{\lambda}_0>0$ with the property that ${\lambda}>{\lambda}_0>\max(C,U^1_x(t,\xi))$, we have $$\nonumber
\begin{split}
V(t,{\lambda}; \xi)& \geq V(t,{\lambda}_0;\xi)+({\lambda}-{\lambda}_0)
V_{{\lambda}}(t,{\lambda}_0;\xi) \\
& = V(t,{\lambda}_0; \infty)-({\lambda}-{\lambda}_0) I^1(t,{\lambda}_0) \geq V(t,{\lambda}_0;
\infty)-\frac{1}{2}{\langle {{\mathbb Q}},e^1 \rangle}({\lambda}-{\lambda}_0).
\end{split}$$ Therefore, if we let $L= \operatorname*{\underline{\rm lim}}_{{\lambda}\to\infty}
\big(\frac{v({\lambda})}{{\lambda}}-{\langle {{\mathbb Q}},e^1 \rangle}\big)\in [-\infty,\infty]$, we have $$\nonumber
\begin{split}
L&=\operatorname*{\underline{\rm lim}}_{{\lambda}\to\infty}
\frac{1}{{\lambda}}{{\mathbb E}}\int_0^T{V(t,{\lambda}Q_t; m^1_t)}d\kappa_t
\geq \operatorname*{\underline{\rm lim}}_{{\lambda}\to\infty}
\frac{1}{{\lambda}} {{\mathbb E}}\int_0^T{V(t,{\lambda}Q_t; m^1_t)
{ {\mathbf 1}_{{\left\{Q_t> \frac{C}{{\lambda}_0}\right\}}}}}d\kappa_t\\ & \hspace{2em} +
\operatorname*{\underline{\rm lim}}_{{\lambda}\to\infty}
\frac{1}{{\lambda}} {{\mathbb E}}\int_0^T{V(t,{\lambda}Q_t; m^1_t)
{ {\mathbf 1}_{{\left\{Q_t\leq \frac{C}{{\lambda}_0}\right\}}}}}d\kappa_t\\
&\geq \operatorname*{\underline{\rm lim}}_{{\lambda}\to\infty} \Big(
\frac{1}{{\lambda}}{{\mathbb E}}\int_0^T{V(t,{\lambda}_0 Q_t;\, m^1_t){ {\mathbf 1}_{{\left\{Q_t>
\frac{C}{{\lambda}_0}\right\}}}}d\kappa_t-
\frac{1}{2{\lambda}}{\langle {{\mathbb Q}},e^1 \rangle}({\lambda}-{\lambda}_0)}\Big)\\
&\hspace{2em} + \operatorname*{\underline{\rm lim}}_{{\lambda}\to\infty} \frac{1}{{\lambda}}
{{\mathbb E}}\int_0^T{V(t,{\lambda}Q_t; m^1_t){ {\mathbf 1}_{{\left\{Q_t\leq
\frac{C}{{\lambda}_0} \right\}}}}}d\kappa_t\\
&\geq -\frac{1}{2}{\langle {{\mathbb Q}},e^1 \rangle}+\operatorname*{\underline{\rm lim}}_{{\lambda}\to\infty}
\frac{1}{{\lambda}}{{\mathbb E}}\int_0^T{V(t,{\lambda}Q_t; m^1_t){ {\mathbf 1}_{{\left\{Q_t\leq
\frac{C}{{\lambda}_0} \right\}}}}}d\kappa_t\geq -\frac{1}{2}{\langle {{\mathbb Q}},e^1 \rangle}.
\end{split}$$ Hence, $\lim_{{\lambda}\to\infty} v({\lambda})=\infty$ and there exists a constant ${\lambda}^1 \in [0,\infty)$ such that $v({\lambda}^1)={\mathbb{U}}^1(c^1)$, i.e., $$\nonumber
\begin{split}
{{\mathbb E}}\int_0^T U^1(t,c^1(t))\, d\kappa_t& ={{\mathbb E}}\int_0^T{\lambda}^1 Q_t
e^1_t\, d\kappa_t+{{\mathbb E}}\int_0^T V(t,{\lambda}^1
Q_t;\, m^1_t)\,d\kappa_t\\
&\geq {{\mathbb E}}\int_0^T{\lambda}^1 Q_t c^1_t\, d\kappa_t+{{\mathbb E}}\int_0^T V(t,{\lambda}^1
Q_t;\, m^1_t)\,d\kappa_t.
\end{split}$$ On the other hand, $U^1(t,x)\leq {\lambda}^1 Q_t x+V(t,{\lambda}^1 Q_t
;m^1_t)$ for all $t\in [0,T]$ and $x\in [0,m^1_t]$ (with equality only for $x=\min(m^1_t,I^1(t,{\lambda}^1 Q_t))$), so $c^1$ must be of the form (\[formc\]). To rule out the possibility ${\lambda}^1=0$, note that it would force $c^1=\Gamma^1$ and violate the budget constraint since $\Gamma^1>e^1$.
The process $Q$ has a modification which is a semimartingale, and there exists a constant ${\varepsilon}>0$ such that ${\varepsilon}\leq Q \leq
1/{\varepsilon}$.
By Lemma \[hasform\] there exists constants ${\lambda}^i > 0$ such that $e_t= \sum_i c^i_t=\sum_i \min(\Gamma^i_t, I^i(t,{\lambda}^i
Q_t))$, $\kappa\otimes{{\mathbb P}}$-a.s. Since $(\kappa\otimes{{\mathbb P}})[\sum_i \Gamma^i>e]=1$, we have $e_t=\min_{{\mathbf{b}}\in B} \big( \sum_i b_i I^i(t,{\lambda}^i Q_t)+\sum_i
(1-b_i) \Gamma^i_t\big)$ ,where $B={\left\{0,1\right\}}^d\setminus
{\left\{0,\dots,0\right\}}$.
For ${\mathbf{b}}\in B$, the function $I^{{\mathbf{b}}}$, defined by $I^{{\mathbf{b}}}(t,y)=\sum_i b_i I^i(t,{\lambda}^i y)$, is strictly decreasing in its second argument and shares the properties in Assumption \[utilreg\] with each $I^i$. Therefore, there exists a function $J^{{\mathbf{b}}}: [0,T]\times (0,\infty)\to (0,\infty)$ such that $I^{{\mathbf{b}}}(t,J^{{\mathbf{b}}}(t,x)))=x$, for all $(t,x)\in [0,T]\times
(0,\infty)$. Thus, with $\Gamma^{{\mathbf{b}}}_t=\sum_i (1-b_i)
\Gamma^i_t$, we have $$\nonumber
\begin{split}
e_t \geq x\ \Leftrightarrow
I^{{\mathbf{b}}}(t,Q_t)+\Gamma^{{\mathbf{b}}}_t\geq x,\,\forall\,{\mathbf{b}}\in B
\Leftrightarrow
Q_t \leq J^{{\mathbf{b}}}(t,x-\Gamma^{{\mathbf{b}}}_t),\,\forall\,{\mathbf{b}}\in B,
\end{split}$$ with $J(t,x-\Gamma^{{\mathbf{b}}}_t)=\infty$ for $x\leq \Gamma^{{\mathbf{b}}}_t$. Consequently, $Q_t=\min_{{\mathbf{b}}\in B} J^{{\mathbf{b}}}(t,e_t-\Gamma^{{\mathbf{b}}}_t)$. Knowing that the semimartingale property is preserved under maximization, it will be enough to prove that for each ${\mathbf{b}}\in B$, $J^{{\mathbf{b}}}$ is a semimartingale function (see Definition \[defsemi\]). By Inada conditions (\[inadas\]) - holding uniformly in $t\in [0,T]$ - $I^{{\mathbf{b}}}$ maps compact sets of the form $[0,T]\times [y_1,y_2]$ into compact intervals. The function $I^{{\mathbf{b}}}$ is locally convexity-Lipschitz, so the conclusion that $Q$ is a semimartingale follows from Proposition \[invreg\].
To show boundedness, we first set ${\mathbf{b}}_1=(1,\dots,1)$ to conclude that $Q_t\leq
J^{{\mathbf{b}}_1}(t,e_t-\Gamma^{{\mathbf{b}}_1}_t)=J^{{\mathbf{b}}_1}(t,e_t)\in{{{\mathbb L}}^{\infty}}$. On the other hand, $Q_t=\min_{{\mathbf{b}}\in B}
J^{{\mathbf{b}}}(t,e_t-\Gamma^{{\mathbf{b}}}_t)\geq \min_{{\mathbf{b}}\in B} J^{{\mathbf{b}}}(t,e_t)$ - a positive quantity, uniformly bounded from below. Therefore, the semimartingale $Q_t$ is positive and uniformly bounded from above and away from zero.
\[hasdec\] The process $Q$ admits a multiplicative decomposition $Q=\hat{Q}
\beta$ where $\hat{Q}$ is a strictly positive uniformly integrable martingale, and $\beta$ is a strictly positive [ ]{}predictable process of finite variation.
By the representation $Q_t=\min_{{\mathbf{b}}\in B}
J^{{\mathbf{b}}}(t,e_t-\Gamma^{{\mathbf{b}}}_t)$, and boundedness of $Q$ from above, there exists a constant $C>0$ such that $Q_t=\min_{{\mathbf{b}}\in B}
J^{{\mathbf{b}}}(t,\max(C,e_t-\Gamma^{{\mathbf{b}}}_t))$. Propositions \[regissemi\], \[stab\] and \[Decomp\] complete the proof.
#### *Construction of the equilibrium market.*
Thanks to Proposition \[hasdec\], there exists a measure ${\hat{{{\mathbb Q}}}}$ (with $\frac{d{\hat{{{\mathbb Q}}}}}{d{{\mathbb P}}}=\frac{\hat{Q}_T}{{{\mathbb E}}[\hat{Q}_T]}$) equivalent to ${{\mathbb P}}$ such that $$ \label{defhq}
\begin{split}
Q_t={{\mathbb E}}[\frac{d{\hat{{{\mathbb Q}}}}}{d{{\mathbb P}}}|{{\mathcal F}}_t] \beta_t,\ \text{and}\
{\langle {{\mathbb Q}},c \rangle}={{\mathbb E}}\int_0^T Q_u c_u \, d\kappa_u
= {{\mathbb E}}^{{\hat{{{\mathbb Q}}}}}\int_0^T c_u\beta_u\, d\kappa_u.
\end{split}$$ In words, the action of the pricing functional $Q$ on a consumption stream $c$ can be represented as a ${\hat{{{\mathbb Q}}}}$-expectation of a discounted version $c(u)\beta_u$ of $c$.
Let $n\in{{\mathbb N}}$ be the martingale multiplicity of the filtration ${ ( {{\mathcal F}}_t )_{t\in [0,T]}}$ under ${\hat{{{\mathbb Q}}}}$, and let $(Y_1,\dots,Y_n)$ be an $n$-dimensional positive ${\hat{{{\mathbb Q}}}}$-martingale described in Definition \[frp\]. Define the riskless asset $B$ and the stock price process $S=(S_1,\dots, S_n)$ as follows $$ \label{defeq}
\begin{split}
B(t)=1/\beta(t), S_j(t)=B(t) Y_j(t),\ t\in [0,T],\, j=1,\dots, n.
\end{split}$$
The pair $(S, B)$, defined in (\[defeq\]) is an equilibrium market.
Let $({{\mathbb Q}},{(c^i)_{i=1,\dots, d}})$ be the abstract equilibrium which produced $(S,B)$, and let the measure ${\hat{{{\mathbb Q}}}}$ be as in (\[defhq\]). For $i=1,\dots, d$, define the ${\hat{{{\mathbb Q}}}}$-martingale $\tilde{X}^i$ by $\tilde{X}^i_t={{\mathbb E}}^{{\hat{{{\mathbb Q}}}}}[\int_0^T (c^i(u)-e^i(u))\beta_u \,
du|{{\mathcal F}}_t]$. By the finite representation property (Assumption \[ftp\]), for each $i=1,\dots, d$ there exists an $S$-integrable portfolio process $H^i$ such that $\tilde{X}^i_t=\tilde{X}^i_0+\int_0^t \tilde{H}^i\, dS_u$. Moreover, the boundedness of processes $c^i$ and $e^i$ guarantees that $\tilde{H}^i$ satisfies part 1. of Definition \[afford\]. Standard calculations involving integration by parts and using the fact that $B$ is a predictable process of finite variation imply that the wealth process $X^{\tilde{H}^i,c^i,e^i}$ defined as in (\[wealth\]) is bounded and satisfies $X^{\tilde{H}^i,c^i,e^i}_T\geq 0$. Therefore, $(\tilde{H}^i,c^i)$ is an affordable consumption-investment strategy (as described in Definition \[afford\]).
Since $\sum_{i} \tilde{X}^i=0$, the mutual orthogonality of the ${\hat{{{\mathbb Q}}}}$-martingales $Y_1,\dots, Y_n$ implies that $\sum_{i=1}^d
\tilde{H}^i_j(t)=0,\ d[Y_j,Y_j]_t-\text{a.e.}$, for all $j$. In order to have markets clear for [*every*]{} $t\in [0,T]$, we define the portfolio process $H^i=(H^i_1,\dots, H^i_n)$ by $H^i_j(t)=\tilde{H}^i_j(t){ {\mathbf 1}_{{\left\{\sum_i H^i_j(t) = 0\right\}}}}$, for each $i=1,\dots, d$, so that
1. $H^i_j(t)=\tilde{H}^i_j(t)$, $d[Y_j,Y_j]$-a.e. (implying indistinguishability of the wealth processes $X^{H^i,c^i,e^i}$ and $X^{\tilde{H}^i,c^i,e^i}$) and
2. $\sum_{i} H^i_j(t)=0$, for all $t$, a.s. and all $j=1,\dots, n$.
Therefore, the $d$-tuple $(H^i,c^i)$ satisfies the part 1. of Definition \[defequ\].
It remains to show that $c^i$ maximizes ${\mathbb{U}}^i$ over all consumption process $c'$ with ${\mathbb{U}}^i(c')\in (-\infty,\infty)$ for which there exists a portfolio process $H'$ such that $(H',c')$ is $(S,B,e^i,\Gamma^i)$-affordable. We first note that each such $c'$ satisfies ${\langle {{\mathbb Q}},c' \rangle}\leq {\langle {{\mathbb Q}},e^i \rangle}$. This is due to (\[defhq\]) and the fact that the discounted wealth $X'=\beta X^{H',c',e^i}$ (which satisfies $X'_T\geq 0$) can be represented as a sum of a ${\hat{{{\mathbb Q}}}}$-martingale and a term of the form $\int_0^t \beta_u (c'(u)-e^i(u))\, d\kappa_u$. Finally, because $c'\wedge k \in B^i({{\mathbb Q}})$, for any $k\in{{\mathbb N}}$, the properties of the abstract equilibrium imply that ${\mathbb{U}}^i(c^i)\geq {\mathbb{U}}^i(c'\wedge k)$ and the Monotone Convergence Theorem yields ${\mathbb{U}}^i(c^i)\geq {\mathbb{U}}^i(c')$.
\[main\] Suppose that
1. $(\Omega, {{\mathcal F}}, { ( {{\mathcal F}})_{t\in [0,T]}},{{\mathbb P}})$ is a filtered probability space satisfying Assumption \[ftp\],
2. $(e^i)_{i=1,\dots,d}$ are random endowment processes verifying Assumption \[ends\],
3. $(U^i)_{i=1,\dots,d}$ are utility functions for which Assumption \[utilreg\] is valid, and
4. $(\Gamma^i)_{i=1,\dots,d}$ are withdrawal cap processes satisfying Assumption \[with\].
Then there exist an equilibrium market $(S,B)$ consisting of a finite-dimensional semimartingale risky-asset process $S$ and a positive predictable riskless-asset process $B$ of finite variation for which the following additional properties hold
1. The market $(S,B)$ is arbitrage free, i.e., there exists a unique measure $\hat{{{\mathbb Q}}}$ equivalent to ${{\mathbb P}}$, such that the discounted prices $S/B$ of risky assets are $\hat{{{\mathbb Q}}}$-martingales.
2. The optimal consumption densities $c^i$ in the market $(S,B)$ are uniformly bounded from above.
Semimartingale functions and multiplicative decompositions {#sec:semi}
==========================================================
In this section we provide several results which give sufficient conditions for 1) a process obtained by applying a function to a semimartingale to be a semimartingale, and 2) for a local martingale part in a multiplicative decomposition of a positive process to be a uniformly integrable martingale. These results can be improved in several directions; we are aiming for conditions easily verifiable in practice. In what follows, $I$ and $J$ will denote generic open intervals in ${{\mathbb R}}$. For a process $A$ of finite variation, ${\left| A \right|}={ ( {\left| A \right|}_t )_{t\in [0,T]}}$ will denote its total variation process.
#### *Semimartingale functions.* {#sub:semi}
A function \[defsemi\] $f:[0,T]\times I\to{{\mathbb R}}$ is called a [**semimartingale function**]{} if the process $Y$ defined by $Y_t=f(t,X_t)$, $t\in [0,T]$ is a [ ]{}semimartingale for each semimartingale $X$ taking values in $I$ and defined on an arbitrary filtered probability space $(\Omega, {{\mathcal F}}, { ( {{\mathcal F}}_t )_{t\in [0,T]}},
{{\mathbb P}})$.
In this section we provide a set of sufficient conditions for a function $f:[0,T]\times I\to{{\mathbb R}}$ to be a semimartingale function. We go beyond basic $C^{1,2}$-differentiability required by the It\^ o formula and place much less restrictive assumptions on $f$. Apart from being indispensable in Section \[sec:abstosto\], we hope that the obtained result holds some independent probabilistic interest.
\[semifun\] Suppose that a function $f:[0,T]\times I\to{{\mathbb R}}$ can be represented as $f(t,x)=f^1(t,x)-f^2(t,x)$, where for $i=1,2$,
1. $f^i$ is Lipschitz in the time variable, uniformly for $x$ in compact intervals.
2. $f^i$ is convex in the second variable.
3. The right derivative $f^i_{x+}$ is bounded on compact subsets of $[0,T]\times I$ and satisfies $f^i_{x+}(t,x)=\lim f^i_{x+}(s,x')$, when $(s,x')\to (t,x)$ and $x'\geq x$.
Then $f$ is a semimartingale function. Moreover, for a semimartingale $X$ the local martingale part $\tilde{M}$ in the semimartingale decomposition of $f(t,X_t)=f(0,X_0)+\tilde{M}_t+\tilde{A}_t$ is given by $\tilde{M}_t=\int_0^t f_{x+}(s,X_{s-})\, dM_s$, where $M$ is the local martingale part in the semimartingale decomposition $X_t=X_0+M_t+A_t$.
Before delving into the proof of Theorem \[semifun\], we recall the concept of Fatou-convergence and some useful compactness-type results related to it.
A sequence $(X^n)_{n\in{{\mathbb N}}}$ of [ ]{}adapted processes is said to [**Fatou-converge**]{} towards a [ ]{}adapted process $X$ if $$\nonumber
\begin{split}
X_t=\lim_{q\searrow t} \lim_n X^n_q,\, a.s.,
\text{\ for all $t\in [0,T)$, and}\ \ X_T=\lim_n X^n_T,\,
a.s,
\end{split}$$ where the first limit is taken over rational numbers $q>t$.
\[Fat\]
1. Let $(A^n)_{n\in{{\mathbb N}}}$ be a sequence of non-decreasing adapted [ ]{}processes taking values in $[0,\infty)$. Then there exists a sequence $(\tilde{A}^n)_{n\in{{\mathbb N}}}$ of convex combinations $\tilde{A}^n\in{\operatorname{conv}}(A^n, A^{n+1},\dots)$ and a non-decreasing [ ]{}process $\tilde{A}$ taking values in $[0,\infty]$ such that $\tilde{A}^n$ Fatou-converges to $\tilde{A}$.
2. Let $(A^n)_{n\in{{\mathbb N}}}$ be a sequence of finite-variation [ ]{}processes on $[0,T]$, with uniformly bounded total variations, i.e., $$\nonumber
\begin{split}
\text{${\left| A^n \right|}_T\leq C$ a.s., for some constant $C>0$ and all
$n\in{{\mathbb N}}$.}
\end{split}$$ Then there exists a sequence $(\tilde{A}^n)_{n\in{{\mathbb N}}}$ of convex combinations $\tilde{A}^n\in{\operatorname{conv}}(A^n, A^{n+1},\dots)$ and a [ ]{}process $\tilde{A}$ of finite variation with $|\tilde{A}|_T\leq C$ such that $\tilde{A}^n$ Fatou-converge towards $\tilde{A}$.
Part 1. is a restatement of Theorem 4.2 in [@Kra96a]. To prove part 2., note that the boundedness of total variations of processes $A^n$ implies that the increasing and decreasing parts $A^{\uparrow,n}$ and $A^{\downarrow,n}$ of $A^n$ satisfy $A^{\uparrow,n}_T+A^{\downarrow,n}_T\leq C$ a.s. for all $n$. Applying part 1. to increasing and decreasing parts and noting that the limiting processes $\tilde{A}^{\uparrow}$ and $\tilde{A}^{\downarrow}$ satisfy $\tilde{A}^{\uparrow}+\tilde{A}^{\downarrow}\leq C$ a.s., leads to the desired conclusion.
Let $X$ be a semimartingale taking values in the open interval $I$. Our goal is to prove that the process $Y$ defined by $Y_t=f(t,X_t)$ is a semimartingale. We first extend the time-domain of $X$ and $Y$ by setting $X_t=X_T$ and $Y_t=f(t,X_T)$ for $t\in (T,\infty)$. By Theorem 6, p. 54 in [@Pro04], it will be enough to find an increasing sequence $(T_n)_{n\in{{\mathbb N}}}$ of stopping times with $T_n\nearrow \infty$, a.s., such that the [*pre-stopped*]{} processes $Y^{T_n-}$ defined by $$\nonumber
\begin{split}
Y^{T_n-}_t= Y_t{ {\mathbf 1}_{{\left\{0\leq t<T_n\right\}}}}+Y_{T_n-}{ {\mathbf 1}_{{\left\{t\geq T_n\right\}}}}=f(t\wedge T_n,X^{T_n-}_t)
\end{split}$$ are semimartingales. Taking $T_n=\inf{{\left\{t\geq 0\,:\,X_t\geq
n\right\}}}\wedge n$, we reduce the problem to the case where the semimartingale $X$ takes values in a compact interval $[x_1,x_2]$, for $t\in [0,S)$, where $S=T\wedge T_n$.
Let $\eta^n: {{\mathbb R}}\times {{\mathbb R}}\to{{\mathbb R}}$ be a sequence of standard mollifier functions with supports lying in the lower half-plane and shrinking to a point, i.e.,
1. $\eta^n\in C^{\infty}({{\mathbb R}}\times {{\mathbb R}})$.
2. $\eta^n(t,x)\geq 0$, for all $t,x$ and $\int_{{{\mathbb R}}\times{{\mathbb R}}} \eta^n(t,x)\, dt\, dx=1$.
3. The supports ${{\mathcal S}}_n$ of $\eta^n$ satisfy ${{\mathcal S}}_n\subseteq {{\mathbb R}}\times (-\infty,0]$ and ${\left| t \right|}+{\left| x \right|}\leq 1/n$ for all $(t,x)\in {{\mathcal S}}_n$.
Let the functions $f^n:[0,T]\times I_n\to{{\mathbb R}}$, where $I_n={{\left\{x\in I\,:\, d(x,I^c)>1/n\right\}}}$, be the mollified versions of $f$, i.e., $$\nonumber
\begin{split}
f^n(t,x)=(\eta^n * f)(t,x)=\int_{{{\mathbb R}}\times{{\mathbb R}}} \eta^n(s,y) f(t-s, x-y)\, ds\, dy,
\end{split}$$ where we set $f(t,x)=f(T,x)$ for $t>T$ and $f(t,x)=f(0,x)$ for $t<0$. By standard arguments, the functions $f^n(t,x)$ have the following properties
1. $f^n(t,x)\to f(t,x)$ for all $(t,x)\in [0,T]\times I$, uniformly on compacts.
2. $f^n(t,x)\in C^{\infty}([0,T]\times I_n)$.
3. Let $C>0$ be a constant such that ${\left| f(t_2,x)-f(t_1,x) \right|}\leq C{\left| t_2-t_1 \right|}$, for all $t_1,t_2\in [0,T]$ and $x\in [x_1,x_2]$. Then the absolute value $|f^n_t|$ of the time derivative $f_t^n$ is bounded by the constant $C$, uniformly over $n\in{{\mathbb N}}$ and $(t,x)\in [0,T]\times [x_1,x_2]$.
4. By condition 3. in the statement of the theorem and the fact that the support ${{\mathcal S}}_n$ lies in the lower half-plane, we have $f^n_{x}(t,x)\to f_{x+}(t,x)$, for all $(t,x)\in [0,T]\times I$.
For $n\in{{\mathbb N}}$ such that $[x_1,x_2]\subseteq I_n$, the It\^ o formula applied to $f^n$ implies that $f^n(t,X_t)=f^n(0,X_0)+M^n_t+ A^n_t+B^n_t$, where $$\nonumber
\begin{split}
M^n_t&= \int_0^t f^n_x(s,X_{s-})\, dX_s,\quad
A^n_t = \int_0^t f^n_t(s,X_s)\, ds,\ \text{and}\\
B^n_t & = \frac{1}{2} \int_0^t f^n_{xx}(x,X_{s-})\, d[X,X]^c_s\\
&\qquad +
\sum_{0< s\leq t}
\big( f^n(s,X_s)-f^n(s,X_{s-})-f^n_x(s,X_{s-})\Delta X_s \big).
\end{split}$$ Note that:
1. Using properties 3. and 4. (above) of $f^n$ and the Dominated Convergence Theorem for stochastic integrals (see [@Pro04], Theorem 32, p. 174), we have $M^n_t\to M_t=\int_0^t f_{x+}(s,X_{s-})\, dX_s$, uniformly in $t\in [0,T]$, in probability. It suffices to take a subsequence to obtain convergence in the Fatou sense.
2. By convexity of $f^n$ in the second variable, the processes $B^n_t$ are non-decreasing. Thus, by Lemma \[Fat\], after a passage to a sequence of convex combinations they Fatou-converge towards a non-decreasing [ ]{}adapted process $B$ taking values in $[0,\infty]$.
3. The processes in the sequence $A^n_t$ have total variation uniformly bounded by $CT$, so by part 2. of Lemma \[Fat\], there exists a sequence of their convex combinations Fatou-converging towards a process $A$ of finite variation with the total variation bounded by the same constant $CT$.
Compounding all subsequences and sequences of convex combinations above, we obtain that $f(t,X_t)-f(0,X_0)-M_t-A_t=B_t$. We can conclude that $B_T<\infty$, a.s. and $f(t,X_t)=f(0,X_0)+M_t+A_t+B_t$ is a semimartingale.
Every locally convexity-Lipschitz function $f:[0,T]\times I\to{{\mathbb R}}$ admits a decomposition\[regissemi\] $f=f^1-f^2$, where $f^1$ and $f^2$ satisfy conditions 1.-3. of Theorem \[semifun\]. In particular, $f$ is a semimartingale function.
We shall construct the desired decomposition only on a compact interval $[x_1,x_2]$ in $I$, as the general case follows immediately.
For a fixed $t\in [0,T]$, the finite-variation function $f_{x}(t,\cdot)$ admits a decomposition into a difference of a pair $f^{\uparrow}(t,\cdot)$ and $f^{\downarrow}(t,\cdot)$ of non-increasing and non-negative functions. Lipschitz continuity of the total variation of the derivative $f_x$ implies that the functions $f^{\uparrow}$ and $f^{\downarrow}$ are Lipschitz continuous in $t$, uniformly in $x\in [x_1,x_2]$. It is now easy to check that the sought-for decomposition is $f=f^1-f^2$, where $f^1(t,x)=f(t,x_1)+\int_{x_1}^x f^{\uparrow}(t,\xi)\, d\xi$, and $f^2(t,x)=\int_{x_1}^x f^{\downarrow}(t,\xi)\, d\xi$.
\[invreg\] Let $f:[0,T]\times I\to{{\mathbb R}}$ be locally convexity-Lipschitz, with the derivative $f_x$ positive and bounded away from $0$ on compact subsets of $[0,T]\times I$. If the function $g:[0,T]\times J\to{{\mathbb R}}$ satisfies $f(t,g(t,y))=y$, for all $(t,y)\in [0,T]\times J$, then $g$ is a semimartingale function.
We note first that the assumptions of the proposition imply that both $f$ and $g$ are continuous and strictly increasing in the second argument. To simplify the proof, we shall restrict the domain of $g$ to a compact set of the form $[0,T]\times
[y_1,y_2]$, so that the range of $g$ is contained in a compact set $[x_1,x_2]\subseteq I$. The general case will follow by [*pre-stopping*]{} - the technique used in the proof of Theorem \[semifun\]. Using the relationships $0=f(t,g(t,y))-f(s,g(s,y))$ and $g_y(t,y)f_x(t,g(t,y))=1$ together with the properties of function $f$ postulated in the statement, it is tedious but straightforward to prove that both $g$ and $g_y$ are Lipschitz continuous in both variables.
Our next task is to decompose the function $g$ into a difference of two functions satisfying conditions 1.-3. in Theorem \[semifun\]. By Proposition \[regissemi\], $f$ has a decomposition $f(t,x)=f^1(t,x)-f^2(t,x)$ with properties 1.-3. from Theorem \[semifun\]. Let $h^i(t,y)$ denote the compositions $f^i_{x}(t,g(t,y))$, $i=1,2$, and let $h(t,x)=h^1(t,x)-h^2(t,x)$ so that $f_{x}(t,g(t,y))=h(t,y)$. Then, for $i=1,2$, $h^i(t,\cdot)$ is a non-decreasing function and for $y\in
[y_1,y_2]$, $$ \label{derg}
\begin{split}
g_{y}(t,y)-g_{y}(t,y_1)&=
- \int_{y_1}^y \frac{h(t,d\eta)}{(f_{x}(t,g(t,\eta)))^2} \\
&= -
\int_{y_1}^y g_{y}(t,\eta)^2 \, (h^1(t,d\eta)-h^2(t,d\eta)),
\end{split}$$ where $h^i(t,d\eta)$ stands for the Lebesgue-Stieltjes measure induced by $h^i(t,\cdot)$. With $g^1$ and $g^2$ defined as $$\nonumber
\begin{split}
g^1(t,y)& =g(t,y_1)+g_{y}(t,y_1)(y-y_1)+
\int_{y_1}^y \int_{y_1}^z g_{y}(t,\eta)^2 \, h^2(t,d\eta)\, dz,\\
g^2(t,y)&=
\int_{y_1}^y \int_{y_1}^z g_{y}(t,\eta)^2 \, h^1(t,d\eta)\, dz,
\end{split}$$ (\[derg\]) implies that $g(t,y)=g^1(t,y)-g^2(t,y)$.
What follows is the proof of Lipschitz-continuity of $g^2_y(\cdot,
y)$. A simple change of variables - valid due to the continuity of the function $g$ - yields $g^2_y(t,y)=\int_{g(t,y_1)}^{g(t,y)}
g_y(t,f(t,\xi))^2 f^1_x(t,d\xi)$, so, for $t,s\in [0,T]$ the difference $g^2_y(t,y)-g^2_y(s,y)$ can be decomposed into the sum $I_1+I_2+I_3+I_4$ where $$I_1 =\int_{g(t,y_1)}^{g(s,y_1)} g_y(s,f(s,\xi))^2 f^1_x(s,d\xi)\,
,I_2 =\int_{g(s,y)}^{g(t,y)} g_y(s,f(s,\xi))^2 f^1_x(s,d\xi),$$ $$\nonumber
\begin{split}
I_3& =\int_{g(t,y_1)}^{g(t,y)}
\big( g_y(t,f(t,\xi))^2-g_y(s,f(s,\xi))^2\big) f^1_x(t,d\xi),
\ \text{and}
\end{split}$$ $$\nonumber
\begin{split}
I_4& =\int_{g(t,y_1)}^{g(t,y)} g_y(s,f(s,\xi))^2
\big(f^1_x(t,d\xi)-f^1_x(s,d\xi)\big)
\end{split}$$ Due to boundedness of $g$ and $g_y$ and Lipschitz continuity of $g,g_y$ and $f_x$, the absolute values of the expressions $I_1$, $I_2$ and $I_3$ are easily seen to be bounded by a constant multiple of ${\left| t-s \right|}$. Additionally, the Lipschitz property of the total-variation functional allows us to conclude the same for $I_4$. Consequently, there exists a constant $C$ such that ${\left| g^2_y(t,y)-g^2_y(s,y) \right|}\leq C {\left| t-s \right|}$, for all $y\in
[y_1,y_2]$.
Finally, to show that $g$ is a semimartingale function, it suffices to check that both $g^1$ and $g^2$ satisfy conditions 1.-3. of Theorem \[semifun\]. The increase of the functions $h^1(t,\cdot)$ and $h^2(t,\cdot)$ implies that $g^1(t,\cdot)$ and $g^2(t,\cdot)$ are convex. Lipschitz-continuity of $g$ and $g^2$ in the time variable implies the same for $g^1=g+g^2$. Finally, the derivatives $g^1_y$ and $g^2_y$ are continuous due to the continuity of functions $(f^1)_x(t,\cdot)$ and $(f^2)_x(t,\cdot)$.
#### *The multiplicative decomposition of positive semimartingales.* {#sub:mult}
A key step in the transition from abstract to stochastic equilibria is the multiplicative decomposition of the pricing functional which enforces the abstract equilibrium. In this paragraph we give sufficient conditions on a positive semimartingale in order for the local martingale part in its multiplicative decomposition to be, in fact, a uniformly integrable martingale.
The following proposition establishes some useful stability properties of the condition ${{\mathcal N}}(X)\in{{{\mathbb L}}^{\infty}}$.
\[stab\]
1. Let $X^1$ and $X^2$ be semimartingales, and let $X=\min(X^1,X^2)$. If ${{\mathcal N}}(X^1)\in{{{\mathbb L}}^{\infty}}$ and ${{\mathcal N}}(X^2)\in {{{\mathbb L}}^{\infty}}$, then ${{\mathcal N}}(X)\in {{{\mathbb L}}^{\infty}}$.
2. Suppose $f:[0,T]\times I\to{{\mathbb R}}$ is a function verifying the conditions of Theorem \[semifun\], and $X$ is a bounded positive semimartingale, bounded away from $0$, such that ${{\mathcal N}}(X)\in{{{\mathbb L}}^{\infty}}$. Then the process $Y$, defined by $Y_t=f(t,X_t)$, satisfies ${{\mathcal N}}(Y)\in {{{\mathbb L}}^{\infty}}$.
<!-- -->
1. Let $X=M+A$, $X^{i}=M^{i}+A^{i}$, $i=1,2$ be the semimartingale decompositions of $X$, $X^1$ and $X^2$. The Meyer-It\^ o formula (see Theorem 70, p. 214 in [@Pro04]) states that $M_t=\int_0^t { {\mathbf 1}_{{\left\{X^1_{s-}\leq X^2_{s-}\right\}}}} dM^1_s+
\int_0^t { {\mathbf 1}_{{\left\{X^1_{s-}> X^2_{s-}\right\}}}} dM^2_s$, so ${\langle M,M \rangle}_T\leq
{\langle M^1,M^1 \rangle}_T+{\langle M^2,M^2 \rangle}_T\in{{{\mathbb L}}^{\infty}}$.
2. Assume that $X$ takes values in $[{\varepsilon},1/{\varepsilon}]$, for some ${\varepsilon}>0$. It suffices to note that the right-continuous function $f_{x+} (t,x)$ is bounded on the compact set $[{\varepsilon},1/{\varepsilon}]\times
[0,T]$, and that Proposition \[semifun\] implies that the local martingale part of the semimartingale $Y_t$ is given by $\int_0^t
f_{x+}(s,X_{s-})\, dM_s$.
\[Decomp\] Let $X$ be a positive semimartingale bounded from above and away from zero, such that ${{\mathcal N}}(X)\in {{{\mathbb L}}^{\infty}}$. Then $X$ admits a multiplicative decomposition $X={\hat{Q}}\beta$ where $\beta$ is a positive predictable process of finite variation, and ${\hat{Q}}$ is a positive uniformly integrable martingale.
Without loss of generality we assume $X_0=1$. By Theorem 8.21, p. 138 in [@JacShi03], along with the semimartingale decomposition $X=M+A$, $X$ also admits a multiplicative decomposition of the form $X={\hat{Q}}\beta$. The same theorem states that ${\hat{Q}}={{\mathcal E}}(\hat{M})$ and $\ 1/\beta={{\mathcal E}}(\hat{A})$, where $\hat{M}_t=\int_0^t H_s\, dM_s$, $\hat{A}_t=-\int_0^t H_s\, dA_s$, and $H=\frac{1}{X_{-}+\Delta A}$ is the reciprocal of the predictable projection ${}^p(X)$ of $X$. For a constant ${\varepsilon}>0$ such that ${\varepsilon}\leq X\leq 1/{\varepsilon}$ we obviously have ${\varepsilon}\leq
H\leq 1/{\varepsilon}$, a.s. Thanks to the boundedness of $H$, the compensator ${\langle \hat{M},\hat{M} \rangle}$ of the quadratic variation $[\hat{M},\hat{M}]$ satisfies ${\langle \hat{M},\hat{M} \rangle}_T=\int_0^T
H_u^2 \, d{\langle M,M \rangle}_u\leq 1/{\varepsilon}^2 {\langle M,M \rangle}_T={{\mathcal N}}(X)\in{{{\mathbb L}}^{\infty}}$. The conclusion now follows from Th[' e]{}or[\` e]{}me 1, p. 147 in [@MemShi79], aided by the fact that absolute values ${\left| \Delta M \right|}$ of the jumps of the local martingale $M$ are uniformly bounded (see Lemma 4.24, p. 44 in [@JacShi03]).
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|
---
abstract: |
We provide a lower bound showing that the $O(1/k)$ convergence rate of the NoLips method (a.k.a. Bregman Gradient or Mirror Descent) is optimal for the class of functions satisfying the $h$-smoothness assumption. This assumption, also known as relative smoothness, appeared in the recent developments around the Bregman Gradient method, where acceleration remained an open issue.
On the way, we show how to constructively obtain the corresponding worst-case functions by extending the computer-assisted performance estimation framework of Drori and Teboulle (Mathematical Programming, 2014) to Bregman first-order methods, and to handle the classes of differentiable and strictly convex functions.
address:
- 'Université Toulouse I Capitole, Toulouse, 0ex D.I. Ecole Normale Supérieure, Paris, France.'
- 'INRIA, D.I. Ecole Normale Supérieure, Paris, France'
- 'CNRS & D.I., UMR 8548,0ex École Normale Supérieure, Paris, France.'
- 'TSE (Université Toulouse 1 Capitole),0ex Toulouse, France.'
author:
- 'Radu-Alexandru Dragomir'
- 'Adrien B. Taylor'
- 'Alexandre d’Aspremont$^*$'
- 'Jérôme Bolte$^*$'
bibliography:
- 'library.bib'
title: 'Optimal Complexity and Certification of Bregman First-Order Methods'
---
Introduction {#s:intro}
============
We consider the constrained minimization problem $$\label{eq:min_problem_bpg}\tag{P}
\min_{x \in C} f(x)$$ where $f$ is a convex continuously differentiable function and $C$ is a closed convex subset of $\reals^n$. In large-scale settings, first-order methods are particularly popular due to their simplicity and their low cost per iteration.
The (projected) gradient descent (PG) is a classical method for solving , and consists in successively minimizing quadratic approximations of $f$, with $$\label{eq:grad_descent}\tag{PG}
x_{k+1} = \argmin_{u \in C} f(x_k) + \la \nabla f(x_k), u - x_k \ra + \frac{1}{2\lambda} \|u - x_k\|^2,$$ where $\|\cdot\|$ is the Euclidean norm. Although standard, there is often no good reason for making such approximations, beyond our capability of solving this intermediate optimization problem. In other words, this traditional approximation typically does not reflect neither the geometry of $f$ nor that of $C$. A powerful generalization of PG consists in performing instead a *Bregman gradient step* $$\label{eq:bregman_descent}\tag{BG}
x_{k+1} = \argmin_{u \in C} f(x_k) + \la \nabla f(x_k), u - x_k \ra + \frac{1}{\lambda} D_h(u,x_k),$$ where the Euclidean distance has been replaced by the *Bregman distance* $D_h(x,y) := h(x) - h(y) - \la \nabla h(y),x-y \ra$ induced by some strictly convex and continuously differentiable *kernel function* $h$. A well-chosen $h$ allows designing first-order algorithms adapted to the geometry of the constraint set and/or the objective function. Of course, a conflicting goal is to choose $h$ such that each iteration can be solved efficiently in practice, discarding choices such as $h=f$ (which would boil down to solve the original problem at each iteration).
Recently, Baushcke et al. [@Bauschke2017] introduced a natural condition for analyzing this scheme, which assumes that the inner objective in the iteration is an upper bound on $f$. This ensures that performing an iteration decreases the value of the function. This assumption, which we refer to as $h$-smoothness (precisely defined in Def. \[def:h-smooth\] below), generalizes the standard $L$-smoothness assumption implied by the Lipschitz continuity of $\nabla f$. The Bregman gradient algorithm, also called NoLips in the setting of [@Bauschke2017], is thus a natural extension of gradient descent to objective functions whose geometry is better modeled by a non-quadratic kernel $h$. Practical examples of $h$-smoothness arise in Poisson inverse problems [@Bauschke2017], quadratic inverse problems [@Bolte2018], rank minimization [@Dragomir2019] and regularized higher-order tensor methods [@Nesterov2018].
### Can we accelerate NoLips? {#can-we-accelerate-nolips .unnumbered}
In the Euclidean setting where $h(x) = \frac{1}{2}\|\cdot\|^2$, accelerated projected gradient methods exhibit faster convergence than the vanilla projected gradient algorithm. These methods, which can be traced back to Nesterov [@Nesterov1983], are proven to be *optimal* for $L$-smooth functions and have found a number of successful applications, in e.g. imaging [@Beck2009]. A natural question is therefore to understand whether the NoLips algorithm can be accelerated in the $h$-smooth setting. This question has been raised in several works, including that of Bauschke, Bolte and Teboulle [@Bauschke2017 Section 6], that of Lu, Freund and Nesterov [@Lu2016 Section 3.4], and the survey of Teboulle [@Teboulle2018 Section 6]. Positive answers have already been provided under somewhat strict additional regularity assumptions (see e.g., [@Auslender2006; @Bayen2015; @Hanzely2018] and discussions in the sequel), while the general case was apparently still open, and relevant in practical applications. In this work, we produce a lower complexity bound proving that NoLips is *optimal* for the general $h$-smooth setting, and therefore that generic acceleration is out of reach.
In order to do so, we adopt the standard *black-box model* used for studying complexity of first-order methods [@Nemirovski1983]. We consider that both $f$ and $h$ are described by first-order oracles, so as to obtain generic complexity results, and we look for worst-case *couples* of functions $(f,h)$ satisfying the $h$-smoothness assumption. A central idea in our approach is the fact that, when studying the worst-case behavior of Bregman methods in the $h$-smooth setting, $f$ and $h$ can get arbitrarily close to some *limiting pathological nonsmooth functions*.
The worst-case functions used for proving the lower bound were found using the recent computer-assisted analysis technique, called *performance estimation problems* (PEPs), and pioneered by [@Drori2014]. This technique consists in computing the worst-case convergence rate of a given algorithm by solving a numerical optimization problem. We rely on the approach of [@Taylor2017] and show how the PEP methodology can be adapted to the setting of Bregman methods and $h$-smooth functions. Besides discovering worst-case functions for NoLips, solving PEPs is of great interest for conjecturing (and, with some additional work, proving) new results on different settings or algorithms, as we illustrate in the sequel.
Contributions and paper organization
------------------------------------
The main contribution of this work is twofold. First, we provide a lower bound showing that it is impossible to generically accelerate Bregman gradient methods under the appropriate oracle model. More precisely, we show that the $O(1/k)$ rate on function values of NoLips is *optimal* in the $h$-smooth setting, using a family of worst-case functions that were discovered by solving a Performance Estimation Problem (PEP).
On the way, we develop PEP techniques for Bregman settings, and extend the analysis of [@Taylor2017] for handling classes of differentiable and strictly convex functions. While we present the analysis on the basic NoLips algorithm for readability purposes, our results and methodology can be applied to various Bregman methods, such as inertial variants [@Auslender2006], or the Bregman proximal point scheme for convex minimization and monotone inclusions [@Eckstein1993; @Bui2019].
The paper is organized as follows. After introducing the setup in Section \[s:setup\], we prove the optimality of NoLips in Section \[s:complexity\]. We expose the framework of computer-aided analysis of Bregman methods in Section \[s:pep\], including several applications in Section \[ss:proofs\]. We point out that Sections \[s:complexity\] and \[s:pep\] are both of independent interest and can be read separately.
Related work
------------
### Bregman methods. {#bregman-methods. .unnumbered}
The idea of using non-Euclidean geometries induced by Legendre kernels can be traced back to the work of Nemirovskii and Yudin [@Nemirovski1983]. For nonsmooth objectives, it gave birth to the mirror descent algorithm [@Ben-tal2001; @Beck2003; @Juditsky], which generalizes the subgradient method to non-quadratic geometries. It has been proven to be particularly efficient for minimization on the unit simplex, where choosing the *entropy kernel* turns out to be much more effective and scalable than the Euclidean norm. This approach has been very successful in online learning; see [@Bubeck2011 Chap. 5] and references therein. The use of Bregman distances has also been thoroughly studied for interior proximal methods [@Censor1992; @Teboulle1992; @Eckstein1993; @Auslender2006].
The introduction of the $h$-smoothness assumption in [@Bauschke2017] has provided a way to adapt the Legendre kernel to the geometry of the objective function $f$ and thus extend the domain of application of the Bregman Gradient method. Subsequent work has focused on nonconvex extensions [@Bolte2018], linear convergence rates under additional assumptions [@Lu2016; @Bauschke2019], and inertial variants [@Hanzely2018; @Mukkamala2019].
### Black-box model and lower complexity bounds. {#black-box-model-and-lower-complexity-bounds. .unnumbered}
The first-order black-box model, developed initially in the works of Nemirovskii [@Nemirovski1983] and later Nesterov [@Nesterov2004] has allowed to prove optimal complexity for several classes of problems in first-order optimization [@Drori2017]. The very related work of Guzman and Nemirovskii [@Guzman2015] studies lower bounds of first-order methods for smooth convex minimization (with a particular focus on smoothness being measured $l_p$-norms). The smoothing technique we use in the sequel is reminiscent of their technique. To the best of our knowledge, it does not contain the lower bound obtained in the sequel as a particular case.
### Performance estimation problems. {#performance-estimation-problems. .unnumbered}
The PEP methodology, proposed initially by [@Drori2014], was already used to discover optimal methods and corresponding lower bounds in other settings: for smooth convex minimization [@Drori2014; @Kim2016; @Drori2017; @Drori2019a], nonsmooth convex minimization [@Drori2016; @Drori2019a], and stochastic optimization [@Drori2019].
Notations
---------
We use $\overline{C}$ to denote the closure of a set $C$, $\interior C$ for its interior and $\partial C$ for its boundary. We denote $(e_1,\dots,e_n)$ the canonical basis of $\reals^n$, and for $p \in \{0,\dots n\}$ we write $E_p = \text{Span}(e_1,\dots,e_p)$ the set of vectors supported by the first $p$ coordinates. $\symm_n$ denotes the set of symmetric matrices of size $n$. If (P) is an optimization problem, then val(P) stands for its (possibly infinite) value.
Subscripts on a vector denotes the iteration counter, while a superscript such as $x^{(i)}$ denotes the $i$-th coordinate. The set $I = \{0,1,\dots N,*\}$ is often used to index the first $N$ iterates of an optimization algorithm as well as the optimal point: $$\{x_i\}_{i\in I} = \{x_0,x_1,\dots,x_N,x_*\}.$$ We use the standard notation $\la \cdot,\cdot\ra$ for the Euclidean inner product, and $\|\cdot\|$ for the corresponding Euclidean norm. For a vector $x \in \reals^n$, we write $\|x\|_\infty = \max_{i=1\dots n} |x^{(i)}|$ its $l_\infty$ norm. The other notations are standard from convex analysis; see e.g. [@Rockafellar2008; @Bauschke2011].
Algorithmic setup {#s:setup}
=================
In this section, we introduce the base ingredients and technical assumptions on $f$ and $h$ that are used within Bregman first-order methods. In particular, it is necessary to assume $h$ to be *Legendre* in order to have well-defined iterations of the form .
Legendre functions
------------------
Let $C$ be a closed convex subset of $\reals^n$. The first step in defining Bregman methods is the choice of a *Legendre function* $h$, or *kernel*, on $C$. In particular, when $C=\reals^n$, the technical definition below reduces to requiring $h$ to be continuously differentiable and strictly convex.
[@Rockafellar2008 Chap. 26] A function $h : \reals^n \rightarrow \reals \,\cup \, \{+\infty\} $ is called a Legendre function with zone $C$ if
1. $h$ is closed convex proper (c.c.p.),
2. $\overline{\dom h} =C$,
3. $h$ is continuously differentiable and strictly convex on $\interior \dom h \neq \emptyset$,
4. $\|\nabla h(x_k)\| \rightarrow \infty$ for every sequence $\{x_k\}_{k \geq 0}\subset \interior \dom h$ converging to a boundary point of $\dom h$ as $k \rightarrow \infty$.
A Legendre function $h$ induces a *Bregman distance* $D_h$ defined as $$D_h(x,y) = h(x) - h(y) - \la \nabla h(y), x-y \ra \quad \forall x \in \dom h, y \in \interior \dom h.$$ Note that $D_h$ is not a distance in the classical sense, however it enjoys a separation property; due to the strict convexity of $h$ we have $D_h(x,y) \geq 0 \,\,\, \forall x \in \dom h, y \in \interior \dom h$, and it is equal to zero iff $x = y$.
### Examples. {#examples. .unnumbered}
We list some of the most classical examples of Legendre functions:
- The **Euclidean kernel** $h(x) = \frac{1}{2}\|x\|^2$ with domain $\reals^n$, and for which $D_h(x,y) = \frac{1}{2}\|x-y\|^2$ is the Euclidean distance,
- The **Boltzmann-Shannon entropy** $h(x)= \sum_i x^{(i)} \log x^{(i)}$ extended to 0 by setting $0 \log 0= 0$, whose domain is thus $\reals^n_+$,
- The **Burg entropy** $h(x)= \sum_i - \log x^{(i)}$ with domain $\reals^n_{++}$,
- The **quartic kernel** $h(x) = \frac{1}{4}\|x\|^4 + \frac{1}{2}\|x\|^2$ with domain $\reals^n$ [@Bolte2018].
We refer the reader to [@Bauschke2017; @Lu2016] for more examples. It should be emphasized that, while a Legendre function $h$ is required to be differentiable on the interior of its domain, it is not differentiable on the boundary. For instance, the Boltzmann-Shannon entropy is continuous but not differentiable at $0$.
### Conjugate of a Legendre function. {#conjugate-of-a-legendre-function. .unnumbered}
We also recall that, if $h$ is a Legendre function, its convex conjugate $h^*$ defined as $$h^*(y) = \sup_{u \in \reals^n} \la u,y \ra - h(y)$$ is also Legendre [@Rockafellar2008 Thm 26.5], and that its gradient is the inverse of $\nabla h$, that is $\nabla h^* = (\nabla h)^{-1}$.
The Bregman Gradient/NoLips algorithm
-------------------------------------
We recall the framework of the NoLips algorithm described in [@Bauschke2017] for solving the minimization problem . As we are interested in studying the complexity, we focus here on the simple Bregman gradient method. Our lower bound will be *a fortiori* valid for the Bregman *proximal* gradient algorithm designed for solving composite problems [@Bauschke2017 Eq. (12)].
Let us first state our standing assumptions.
\[assumption\_bpg\]
1. \[ass:h\_leg\] $h$ is a Legendre function with zone $C$,
2. \[ass:f\_diff\] $f : \reals^n \rightarrow \reals \cup \{+\infty\}$ is a closed convex proper function such that $\dom h \subset \dom f$ and which is continuously differentiable on $\interior \dom h$,
3. \[assumption:well\_posed\] For every $\lambda > 0$, $x \in \interior \dom h$ and $p \in \reals^n$, the problem $$\min_{u \in \reals^n} \la p, u-x \ra + \frac{1}{\lambda} D_h(u,x)$$ has a unique minimizer, which lies in $\interior \dom h$,
4. \[ass:bounded\_below\] The problem is bounded from below, i.e. $f_* := \inf \,\{f(x) : x\in C\} > -\infty$,
5. \[ass:argmin\_in\_dom\] There exists at least one minimizer $x_* \in \argmin_C f$ such that $x_* \in \dom h$.
Condition \[assumption:well\_posed\] is standard and ensures that the algorithm is well-posed. It is satisfied if, for instance, $h$ is strongly convex or supercoercive [@Bauschke2017 Lemma 2]. In Condition \[ass:argmin\_in\_dom\], we make the requirement that there is a solution $x_*$ to that lies in $\dom h$. This is a nontrivial assumption and we must distinguish two cases:
- if $\dom h$ is **closed**, as for the Euclidean kernel and the Boltzmann-Shannon entropy, then $C = \dom h$ and the condition is necessarily satisfied for every minimizer.
- If $\dom h$ is **open**, like for the Burg entropy, Condition \[ass:argmin\_in\_dom\] may fail as the minimizers $x_*$ can lie on the boundary of $\dom h$, where $h$ is infinite.
In addition to these assumptions, the central property we need in order to apply the Bregman gradient method is the so-called $h$-smoothness, first introduced in [@Bauschke2017], also known as relative smoothness [@Lu2016].
\[def:h-smooth\] Let $h$ be a Legendre function with zone $C$, and $f$ a function such that $\dom h \subset \dom f$. We say that $f$ is $h$-smooth if there exists a constant $L > 0$ such that $$\label{eq:rel_smooth}\tag{LC}
Lh - f \quad \textup{is convex on}\, \dom h.$$
$h$-smoothness allows to build a simple global majorant of $f$; indeed, implies that [@Bauschke2017] $$f(x) \leq f(y) + \la \nabla f(y), x-y \ra + L D_h(x,y) \quad \forall x \in \dom h, y \in \interior \dom h,$$ and successively minimizing this upper approximation will give birth to the NoLips algorithm.
The $h$-smoothness assumption generalizes the usual smoothness assumption; in particular, when taking the Euclidean kernel $h(x) = \frac{1}{2}\|x\|^2$, reduces to standard smoothness implied by the Lipschitz continuity of $ \nabla f$. To avoid ambiguity, we will refer to this standard Euclidean smoothness as *L-smoothness*. By choosing different Legendre functions $h$, it is possible to show that holds for functions that are not $L$-smooth [@Bauschke2017; @Bolte2018].
### Remark. {#remark. .unnumbered}
A particular case of $h$-smoothness appears when $f$ has a Lipschitz continuous gradient with constant $\tilde{L}$ and the kernel $h$ is $\sigma$-strongly convex (see e.g., [@Auslender2006; @Bayen2015]), provided that the norm is Euclidean. Indeed, in this case we have $$\begin{split}
\left\{
\begin{array}{ll}
\nabla f \,\, \text{is Lipschitz continuous with constant } \tilde{L} \\
h \,\,\text{is}\,\, \sigma-\text{strongly convex}
\end{array}
\right. &\implies
\left\{
\begin{array}{ll}
\frac{\tilde{L}}{2} \|\cdot\|^2 - f \,\, \text{is convex} \\
h - \frac{\sigma}{2}\|\cdot\|^2 \,\,\text{is convex}
\end{array}
\right. \\
&\implies \frac{\tilde{L}}{\sigma}(h - \frac{\sigma}{2} \|\cdot\|^2) + (\frac{\tilde{L}}{2} \|\cdot\|^2 - f) \,\, \text{is convex} \\
&\implies \frac{\tilde{L}}{\sigma} h - f \,\, \text{is convex}
\end{split}$$ which shows that $f$ is $h$-smooth with constant $\tilde{L} /\sigma$. We use the following convenient notation to characterize functions that satisfy the assumptions for NoLips:
We say that the couple of functions $f,h : \reals^n \rightarrow \reals \cup \{+\infty\}$ is admissible for NoLips, and write $(f,h) \in \mathcal{B}_L(C)$ if
1. $f$ and $h$ satisfy Assumption \[assumption\_bpg\],
2. $Lh-f$ is convex on $C$.
Finally, let us denote by $\mathcal{B}_L$ the union of $\mathcal{B}_L(C)$ for all closed convex sets $C$: $$\mathcal{B}_L = \bigcup_{n \geq 1} \bigcup_{\substack{C \subset \reals^n \\ C \textup{ closed convex}}} \mathcal{B}_L(C)$$
With this framework, we can define the Bregman Gradient (BG)/NoLips algorithm for minimizing $f$. For simplicity, we restrict ourselves to constant step size choice.
**Input:** $(f,h) \in \mathcal{B}_L(C)$, $x_0 \in \interior \dom h$, step size $\lambda \in (0,1/L]$. $$\label{eq:iteration_nolips}
x_{k+1} = \argmin_{u \in \reals^n} \, \la \nabla f(x_k), u-x_k \ra + \frac{1}{\lambda} D_h(u,x_k)$$
Using the first-order optimality condition, the update can also be written as $$x_{k+1} = \nabla h^*\left[ \nabla h(x_k) - \lambda \nabla f(x_k) \right]$$ involving the gradient $\nabla h^*$ which we call the *mirror map*.
Convergence rate and optimality of NoLips {#s:complexity}
=========================================
In this section, we begin by recalling the $O(1/k)$ convergence rate bound for the NoLips algorithm in the setting where $(f,h) \in \mathcal{B}_L(C)$. We then proceed to prove that NoLips is an *optimal* algorithm for the class $\mathcal{B}_L(C)$, by showing that this rate is also a *lower bound* for a generic class of Bregman gradient algorithms that we define below. The key elements for proving the lower bound were discovered through the solution to a Performance Estimation Problem (PEP), which will be detailed in Section \[s:pep\].
Upper bound {#ss:upper_bound}
-----------
We first state the $O(1/k)$ convergence rate for NoLips. Comparing to previous work [@Bauschke2017], it is slightly different, as it is improved by a factor of 2 and does not involve the so-called *symmetry coefficient*.
\[thm:nolips\_bound\] Let $L > 0$, $C$ be a closed convex subset of $\reals^n$ and $(f,h) \in \mathcal{B}_L(C)$ functions admissible for NoLips. Then the sequence $\{x_k\}_{k \geq 0}$ generated by Algorithm \[algo:bpg\] with constant step size $\lambda \in (0,1/L]$ satisfies for $k \geq 0$
$$\label{eq:comp_estimate}
f(x_k) - f_* \leq \frac{D_{h}(x_*,x_0)}{\lambda \, k}$$
for $f_* = \min_{C} f$ and any $x_* \in \argmin_{C} f \cap \dom h$.
The proof, whose analytical form has been inferred from solving a PEP, is provided in Section \[sss:nolips\_pep\]. This result extends the $O(1/k)$ rate of Euclidean gradient descent for $L$-smooth functions to the $h$-smooth setting. However, unlike in the Euclidean case, we will show in the next section that this rate is actually neither improvable for NoLips, nor for other Bregman first-order methods satisfying a set of reasonable assumptions.
A lower bound for $h$-smooth Bregman optimization {#ss:lower_bound}
-------------------------------------------------
It is natural to ask whether, under the same assumptions as those of Theorem \[thm:nolips\_bound\], an *accelerated* Bregman algorithm can be obtained, with a better convergence rate than $O(1/k)$.
This has already been achieved under additional regularity assumptions, as follows
- in the Euclidean setting, when $h(x) = \frac{1}{2}\|x\|^2$ and $f$ has a Lipschitz continuous gradient, the seminal accelerated gradient method of Nesterov [@Nesterov1983] enjoys a $O(1/k^2)$ convergence rate, which is optimal for this class of functions [@Nesterov2004].
- When $h$ is a strongly convex Legendre kernel with closed domain and $f$ has a Lipschitz continuous gradient, the Improved Interior Gradient Algorithm (IGA) [@Auslender2006] also admits a $O(1/k^2)$ convergence rate, by using the same momentum technique as Nesterov-type methods.
- Recently, [@Hanzely2018] proposed an accelerated Bregman proximal gradient algorithm with rate $O(1/k^\gamma)$, where $\gamma \in [1,2]$ is determined by some crucial *triangle scaling property* of the Bregman distance, whose genericity is unclear.
However, the existence of an accelerated algorithm for the $h$-smooth setting is still an open question, and many applications [@Bauschke2017] do not satisfy the supplementary assumptions made in the works mentioned above. In this section, we prove that, up to a constant factor of $2$, the bound is not improvable, making NoLips an *optimal* algorithm in the black box setting for $(f,h)\in\mathcal{B}_L$.
More precisely, we will show in Theorem \[thm:low\_bound\] that for every $\epsilon \in (0,1)$ and number of oracle calls $N$, there is a pair of functions $(f,h) \in \mathcal{B}_L(\reals^{2N+1})$ such that for any *Bregman gradient algorithm*, the output $x_N$ returned after performing at most $N$ oracle calls satisfies $$\label{eq:low_bound_first}
f(x_N) - \min_{\reals^{2N+1}}f \geq \frac{LD_{h}(x_0, x_*)}{2N+1} \cdot(1-\epsilon).$$
But first, we need to clarify what we call a *Bregman gradient algorithm* and define the *oracle calls*.
### Defining a class of Bregman gradient methods
We adopt the first-order black-box model, where information about a function can be gained by calling an *oracle* returning the value and gradient of $f$ at a given point. In the Bregman setting, we assume that we also have access to the first-order oracles of the Legendre function $h$ and its conjugate $h^*$.
\[assumption:algorithm\] Let functions $f,h$ be in $\mathcal{B}_L(C)$ and $T \geq 0$. An algorithm $\mathcal{A}$ is called a Bregman gradient algorithm if it generates at each time step $t=0\dots T$ a set of vectors $\mathcal{V}_t$ from the following process:
1. Set $\mathcal{V}_0 = \{x_0\}$, where $x_0 \in \interior \dom h$ is some initialization point.
2. For $t =1,\dots T-1$, choose some query point $$y_t \in \textup{Span}(\mathcal{V}_t)$$ and perform one of the two following operations:
- either call the **primal oracle** $(\nabla f, \nabla h)$ at $y_t$ and update $$\mathcal{V}_{t+1} = \mathcal{V}_t \cup \{\nabla f(y_t), \nabla h(y_t)\}.$$
- Or call the **mirror oracle** $\nabla h^*$ at $y_t$ as $$\nabla h^*(y_t) = \argmin_{u \in C} h(u) - \la y_t, u\ra$$ and update $$\mathcal{V}_{t+1} = \mathcal{V}_t \cup \{\nabla h^*(y_t)\}.$$
3. Output some vector $\overline{x} \in \textup{Span}(\mathcal{V}_T)$.
This model implicitly assumes that $y_t$ is chosen in the domain of the oracle so as to guarantee the existence of the next iterate.
Such structural assumptions on the class of algorithms are classical from complexity analyses of Euclidean first order methods and are used to prove e.g. the optimality of accelerated first order methods [@Nesterov2004]. Assumption \[assumption:algorithm\] is a natural extension to the Bregman setting, allowing additional uses of the oracles associated to the Legendre function $h$. This model can often be relaxed through the use of more involved information theoretic arguments, see e.g., [@Nemirovski1983; @Guzman2015; @Drori2017].
Here, we focus on Assumption \[assumption:algorithm\] as it is general enough to encompass all Bregman-type methods that use only the oracles $\nabla f, \nabla h$, which we call the *primal oracles*, the map $\nabla h^*$, which we call the *mirror oracle*, and linear operations. One can verify that known Bregman gradient methods, including NoLips and inertial variants such as IGA [@Auslender2006] or the recent algorithm in [@Hanzely2018], fit in this model.
Note that $\mathcal{V}_t$ can contain both *points* (in the “primal” space) and *directions* (in the “dual” space), which might allow some unnatural operations (such as scaling a point), but this enables us to write a model that is simple and very general. Observe also that, as NoLips performs one primal oracle call and one mirror call per iteration, an iteration of NoLips corresponds actually to *two time steps* of the formal procedure in Assumption \[assumption\_bpg\]. This is why, in order to avoid ambiguity, we will state our lower bound as a function of the number of oracle calls.
### Proof of the lower bound
### Proof intuition. {#proof-intuition. .unnumbered}
To find a pair of functions $(f,h)$ which is a difficult instance for all Bregman methods, we use two main ideas. The first is the well-known technique used by Nesterov [@Nesterov2004] for proving that $O(1/k^2)$ is the optimal complexity for $L$-smooth convex minimization. He defines a “worst function in the world" that allows any gradient method to discover only one dimension per iteration, hence *hiding* the minimizer from the algorithm in the last dimensions explored.
The second idea is more specific to our setting, and relies on the fact that the set of admissible functions for NoLips $\mathcal{B}_L(C)$ is not closed. In particular, a limit of differentiable functions need not be differentiable. This is why, in our case, we actually have a worst-case **sequence** of differentiable functions parameterized by some parameter $\mu$, whose limit when $\mu \rightarrow 0$ is a nonsmooth pathological function. Also, it explains why the lower bound we give is not attained, but rather approached to an arbitrary precision $\epsilon$.
### Choosing the objective function. {#choosing-the-objective-function. .unnumbered}
Let us fix a dimension $n \geq 1$ and a positive constant $\eta > 0$. Define the convex function $\hat{f}$ for $x \in \reals^n$ by $$\hat{f}(x) = \max_{i=1,\dots,n} |x^{(i)} - 1-\frac{\eta}{i}| = \|x-x_*\|_\infty$$ which has an optimal value $\hat{f_*} = 0$ attained at $$x_* := (1+\eta,1+\frac{\eta}{2},\dots,1+\frac{\eta}{n}).$$ The behavior of $\hat{f}$ as a *pathological function* comes from the fact that if at least one of the coordinates of $x$ is zero, then $\hat{f}(x)-\hat{f}_* \geq 1$. Let us first prove a technical lemma about the subdifferential of $\hat{f}$.
\[lemma:subdiff\_fhat\] Let $x \in \reals^n$ and $v \in \partial \hat{f}(x)$ a subgradient of $\hat{f}$ at $x$. Then
1. \[item:first\_prop\_subgrad\] $\|v\|_\infty \leq 1$.
2. \[item:second\_prop\_subgrad\] Let $i \in \{1\dots n\}$. If $v^{(i)} \neq 0$ then $|x^{(i)}-x_*^{(i)}| = \|x-x_*\|_\infty$.
Write $\hat{f}$ as $
\hat{f}(x) = \max_{1 \leq i \leq n} \hat{f}_i(x)
$ with $\hat{f}_i(x) = |x^{(i)} - x^{(i)}_*|$. Then, by [@Nesterov2004 Lemma 3.1.10], we have $$\partial \hat{f}(x) = \text{Conv} \,\{ \partial \hat{f}_i(x) | i \in I(x)\}$$ where $I(x) = \{i \in \{1\dots n\}\,|\, \hat{f}_i(x) = \hat{f}(x)\}$. Hence, \[item:first\_prop\_subgrad\] follows immediately from the well-known property that the subgradients of the absolute value lie in $[-1,1]$. \[item:second\_prop\_subgrad\] is a consequence of the fact that if $v^{(i)} \neq 0$, then $i \in I(x)$, which means that $|x^{(i)}-x_*^{(i)}| = \|x-x_*\|_\infty$.
Note that $\hat{f}$ is nonsmooth hence does not fit in our assumptions. We approach it with a smooth function by considering its Moreau proximal enveloppe $f_\mu$ given by $$\label{eq:def_fmu}
f_\mu(x) = \min_{u \in \reals^n} \hat{f}(u) + \frac{1}{2 \mu} \| x-u \|^2$$ where $\mu \in (0,1)$ is a small parameter. $f_\mu$ is a smoothed version of $\hat{f}$, which will behave similarly to $\hat{f}$ when we choose $\mu$ small enough. Figure \[fig:f\_mu\] illustrates this phenomenon in dimension 2.
For general properties of the Moreau proximal enveloppe, we refer the reader to [@Moreau1965]. We state the properties that we will need in our analysis.
\[lemma:fmu\] $f_\mu$ is a differentiable convex function, whose minimum is the same as that of $\hat{f}$. Its gradient at a point $x \in \reals^n$ is given by $\nabla f_\mu(x) = \mu^{-1}\left(x- \textup{prox}_{\hat{f}}^\mu\left(x\right)\right)$ where
$$\textup{prox}_{\hat{f}}^\mu(x) = \argmin_{u \in \reals^n} \hat{f}(u) + \frac{1}{2\mu} \|x-u\|^2$$
is the Moreau proximal map. Moreover, $\nabla f_\mu$ is Lipschitz continuous with constant $1/\mu$.
We now prove the central property of $f_\mu$, which states that if the last $n - p$ coordinates of $x$ are small enough, then the gradient $\nabla f_\mu(x)$ is supported by the first $p+1$ coordinates. Recall that we denote $(e_1,\dots,e_n)$ the canonical basis of $\reals^n$ and write, for $p \in \{1\dots n\}$, $E_p = \text{Span}(e_1,\dots,e_p)$ and $E_0 = \{(0,\dots, 0)\}$.
\[lemma:f\_zero\_pres\] Assume that $\mu \in (0,1)$ and $\eta > 4 \mu n^2$. Let $p \in \{0\dots n-1\}$. For any vector $x \in \reals^n$ such that $$\max_{i=p+1,\dots, n} |x^{(i)}| \leq \mu$$ we have that $\nabla f_\mu(x) \in E_{p+1}$. In addition, we have $\|\nabla f_\mu(x)\|_\infty \leq 1$.
Take $x \in \reals^n$ such that $\max_{i=p+1,\dots, n} |x_i| \leq \mu$. By Lemma \[lemma:fmu\], $\nabla f_\mu$ is given by $$\label{eq:prox_grad}
\nabla f_\mu(x) = \frac{1}{\mu}(x-\text{prox}^\mu_{\hat{f}}(x))$$ Write $y = \text{prox}^\mu_{\hat{f}}(x)$. Then, the optimality condition defining the proximal map writes $$\label{eq:prox_cond}
y - x + \mu v = 0$$ where $v \in \partial \hat{f}(y)$, and therefore combining and implies $$\label{eq:grad_fmu}
\nabla f_\mu(x) = v \in \partial \hat f(y).$$
Assume by contradiction that $\nabla f_\mu(x) $ is not in $E_{p+1}$, meaning that there exists an index\
$l \in \{p+2\dots n\}$ such that $v^{(l)} \neq 0$. It follows from Lemma \[lemma:subdiff\_fhat\] that $|(y-x_*)^{(l)}| = \|y-x_*\|_\infty$. Hence we have in particular that $|y^{(l)}-x_*^{(l)}| \geq |y^{(p+1)}-x^{(p+1)}_*|$. Using Condition to replace $y$ we get $$| x^{(l)}_* + \mu v^{(l)} - x^{(l)}| \geq | x^{(p+1)}_* + \mu v^{(p+1)} - x^{(p+1)}|,$$ and recalling the definition of $x_*$ we have $$| 1 + \frac{\eta}{l} + \mu v^{(l)} - x^{(l)}| \geq | 1 + \frac{\eta}{p+1} + \mu v^{(p+1)} - x^{(p+1)}|.$$ By Lemma \[lemma:subdiff\_fhat\], $\|v\|_\infty\leq1$, so for all $i$ we have $1 + \mu v^{(i)} \geq 1 - \mu \|v\|_\infty \geq 0$. In addition, we assumed that $\max_{i=p+1,\dots, n} |x^{(i)}| \leq \mu < \frac{\eta}{4n^2}$ which implies $\frac{\eta}{i} - x^{(i)} \geq 0$ for all $i \geq p+1$. Therefore, both terms inside the absolute values are nonnegative, it follows that we can drop the absolute values and write $$\begin{split}
\mu(v^{(l)}-v^{(p+1)}) &\geq \frac{\eta}{p+1}-\frac{\eta}{l} + x^{(l)} - x^{(p+1)} \\
&\geq \eta \cdot\frac{l - (p+1)}{l(p+1)} - 2 \mu \\
&\geq \frac{\eta}{l(p+1)} - 2\mu \\
&\geq \frac{\eta}{n^2} - 2\mu \\
\end{split}$$ therefore $$v^{(l)} - v^{(p+1)} \geq \frac{\eta}{\mu n^2} - 2 > 2$$ because we assumed $\eta > 4\mu n^2$. This is a contradiction since $(v^{(l)} - v^{(p+1)}) \leq 2 \|v\|_\infty \leq 2$. Finally, the second part of the Lemma is a consequence of and the inequality $\|v\|_\infty \leq 1$.
We will also need the following Lemma which relates the values of $\hat{f}$ and $f_\mu$.
\[lemma:value\_fmu\] Let $\mu > 0$ and $x \in \reals^n$. Then $f_\mu(x) \geq \hat{f}(x) - \mu$.
Write $y = \text{prox}^\mu_{\hat{f}}(x)$. By definition of $f_\mu$ and the proximal map we have $$\begin{split}
f_\mu(x) &= \hat{f}(y) + \frac{1}{2\mu} \|y - x\|^2 \\
&\geq \hat{f}(y) \\
&= \|y-x_*\|_\infty \\
&\geq \|x - x_*\|_\infty - \|x - y\|_\infty.
\end{split}$$ Recall that the optimality condition defining the proximal map writes $$\mu^{-1}(x-y) \in \partial f(y)$$ and, since all subgradients of $\hat{f}$ have coordinates smaller than one (Lemma \[lemma:subdiff\_fhat\]), we have $\|x-y\|_\infty \leq \mu$. It follows that $
f_\mu(x) \geq \|x-x_*\|_\infty - \|x-y\|_\infty \geq \|x-x_*\|_\infty - \mu = \hat{f}(x) - \mu. $
### Choosing the kernel. {#choosing-the-kernel. .unnumbered}
As for the objective function $f_\mu$, we will also choose a of kernels $h_\mu$, whose properties will be close to the ones of a nonsmooth function as $\mu \rightarrow 0$.
Let us first define a unidimensional convex function $\phi_\mu: \reals \rightarrow \reals$ by
$$\phi_\mu(t) = \left\{
\begin{array}{ll}
t -\mu/2 & \mbox{ if } t \geq
\mu \\
\frac{1}{2\mu} t^2 & \mbox{ elsewhere}
\end{array}
\right.$$
which is convex, differentiable and continuous. Now let $d_\mu : \reals^n \rightarrow \reals$ be defined for $x \in \reals^n$ by $$\label{eq:def_dmu}
d_\mu(x) = \frac{\mu}{2}\|x\|^2 + \sum_{i=1}^n \phi_\mu(x^{(i)})$$ $d_\mu$ is a differentiable strictly convex function, whose gradient satisfies, for $x\in \reals^n$ and $i \in \{1\dots n\}$, $$\nabla d_\mu(x)^{(i)} = \mu x^{(i)} + \min(1,x^{(i)}/\mu).$$ From the expression above, we can deduce two crucial properties that we will need in the sequel: for $x \in \reals^n$ and $i \in \{1\dots n\}$ we have $$\begin{aligned}
\label{eq:cond_dmu1}
&\nabla d_\mu(x)^{(i)} = 0 \quad\text{if and only if}\quad x^{(i)} = 0,\\
\label{eq:cond_dmu2}
|&\nabla d_\mu(x)^{(i)}| \leq 1 \quad \,\,\, \text{implies} \quad \quad \, |x^{(i)}| \leq \mu.\end{aligned}$$ Now, let $L > 0$. We define the Legendre kernel $h_\mu$ for $x \in \reals^n$ as $$\label{eq:def_hmu}
h_\mu(x) = \frac{1}{L}\left(f_\mu(x) + d_\mu(x)\right).$$ By construction, $L h_\mu - f_\mu$ is convex, so the $h$-smoothness property holds. It is easy to see that Assumption \[assumption\_bpg\] is satisfied as $h_\mu$ is strongly convex, so we have $(f_\mu, h_\mu) \in \mathcal{B}_L(\reals^n)$.
### Proving the zero-preserving property of the oracles. {#proving-the-zero-preserving-property-of-the-oracles. .unnumbered}
Now that the functions are defined, we are ready to prove that all the oracles involved in the Bregman algorithm allow to discover *only one dimension per oracle call.*
\[prop:zero\_pres\] Assume that $\mu \in (0,1)$ and $\eta > 4 \mu n^2$. Let $p \in\{ 0\dots n-1\}$, and $x \in \reals^n \cap E_p$ a vector supported by the $p$ first coordinates. Then $$\nabla f_\mu(x),\nabla h_\mu(x),\nabla h_\mu^*(x) \in E_{p+1}.$$
Let $x\in E_p$. Then $x$ satisfies the assumption of Lemma \[lemma:f\_zero\_pres\] which proves that $\nabla f_\mu(x) \in E_{p+1}$. By Property of $d_\mu$ we also have that $\nabla d_\mu(x) \in E_{p}$, which allows us to conclude that $$\nabla h_\mu(x) = L^{-1}\left(\nabla f_\mu(x) + \nabla d_\mu \left(x\right) \right) \in E_{p+1}.$$ It remains to prove the result for $\nabla h_\mu^*(x)$. Write $z = \nabla h_\mu^*(x)$, which amounts to say that $\nabla h_\mu(z) = x$, that is $$\nabla f_\mu(z) + \nabla d_\mu(z) = Lx.$$ using . We have $x \in E_p$, hence the $l-th$ coordinate of $x$ is zero and $$\nabla f_\mu(z)^{(l)} + \nabla d_\mu(z)^{(l)} = 0,$$ for $l \in \{p+1\dots n\}$. Using the second part of Lemma \[lemma:f\_zero\_pres\] we have that $\|\nabla f_\mu(z)\|_\infty \leq 1$; it follows that $$| \nabla d_\mu(z)^{(l)}| \leq 1$$ which implies that $|z^{(l)}| \leq \mu$, by property of $d_\mu$. Since this holds for any $l \geq p+1$, we have established $$\max_{l=p+1,\dots,n} |z^{(l)}| \leq \mu.$$ Using Lemma \[lemma:f\_zero\_pres\] again applied to $z$, we have that $\nabla f_\mu(z) \in E_{p+1}$. Remembering that $\nabla h_\mu(z) = x \in E_p$ by construction, we get $$\nabla d_\mu(z) = L \nabla h_\mu(z) - \nabla f_\mu(z) \in E_{p+1}.$$ By Property of $d_\mu$, we conclude that $z \in E_{p+1}$, which proves the result.
We can now use Proposition \[prop:zero\_pres\] inductively to state a lower bound on the performance of any Bregman gradient algorithm applied to $(f_\mu,h_\mu)$.
\[prop:performance\] Let $N \geq 1$ and choose the dimension $n = 2N+1$. Let $\mu \in (0,1)$ and $\eta > 4 \mu n^2 $. Consider the functions $f_\mu, h_\mu : \reals^n \rightarrow \reals$ defined in and respectively. Then, for any Bregman gradient method satisfying Assumption \[assumption:algorithm\] applied to $(f_\mu,h_\mu)$ and initialized at $x_0 = (0,\dots 0)$, the output $\overx$ returned after performing at most $N$ calls to each one of the primal and mirror oracles satisfies $$f_\mu(\overline{x}) - \min_{\reals^n}f_\mu \geq \frac{LD_{h_\mu}(x_*,x_0)}{2N+1} \cdot \frac{1 - \mu}{1 + \mu + \eta + \frac{\mu}{2}(1+ \eta)^2}.$$
The zero-preserving property and the structure of Bregman gradient algorithms described in Assumption \[assumption:algorithm\] implies that the set of vectors $\mathcal{V}_t$ at time $t$ is supported by the first $t$ coordinates, i.e. $$\mathcal{V}_t \subset E_t.$$ Indeed, since we initialized $\mathcal{V}_0 = \{x_0\} \subset E_0$, this follows by induction: if at time $t$, we have $\mathcal{V}_t \subset E_t$, then the query point $y_t$ lies also in $E_t$ and thus Proposition \[prop:zero\_pres\] states that the oracle output belongs to $E_{t+1}$.
Now, because the algorithm has called at most $N$ times each oracle, it has performed at most $2N$ steps and thus the output point satisfies $\overline{x} \in E_{2N}$, which means that $\overline{x}^{(2N+1)} = 0$.
We use Lemma \[lemma:value\_fmu\] to relate $f_\mu(\overline{x})$ and $\hat{f}(\overline{x})$. Recalling that $\min f_\mu= \hat{f}_* = 0$, we get $$\label{eq:f_low_bound}
\begin{split}
f_\mu(\overline{x}) - \min_{\reals^n}f_\mu &= f_\mu(\overline{x})\\
&\geq \hat{f}(\overx)- \mu\\
&\geq | \overline{x}^{(2N+1)}- x_*^{(2N+1)}| - \mu \\
& = 1 + \frac{\eta}{2N+1} - \mu \\
&\geq 1 - \mu
\end{split}$$ where we used the definition of $\hat{f}$ and the fact that $\overline{x}^{(2N+1)} = 0$.
Let us now upper bound the initial diameter. Remembering that $Lh_\mu = f_\mu + d_\mu$ in , we have $$LD_{h_\mu}(x_*,x_0) = D_{f_\mu}(x_*,x_0) + D_{d_\mu}(x_*,x_0).$$ by definition of the Bregman distance. To deal with the first term, we recall that $f_\mu(x_*) = 0$ and write $$\begin{split}
D_{f_\mu}(x_*,x_0) &= f_\mu(x_*) - f_\mu(x_0) - \la \nabla f_\mu(x_0),x_*-x_0 \ra\\
&= - f_\mu(x_0) - \la \nabla f_\mu(x_0),x_*-x_0 \ra\\
&\leq -\hat{f}(x_0) + \mu - \la \nabla f_\mu(x_0),x_*-x_0 \ra\\
&= - 1 -\eta + \mu - \la \nabla f_\mu(x_0),x_*-x_0 \ra
\end{split}$$ where we used again Lemma \[lemma:value\_fmu\] at $x_0 = (0,\dots,0)$. Now, Lemma \[lemma:f\_zero\_pres\] applies to $x_0$ with $p = 0$ and allows to state that $\nabla f_\mu(x_0) \in E_1$ and that $\|\nabla f_\mu(x_0)\|_\infty \leq 1$. Therefore $$|\la \nabla f_\mu(x_0),x_*-x_0 \ra| =|\nabla f_\mu(x_0)^{(1)}\,(x_*^{(1)} - x_0^{(1)})| \leq |x_*^{(1)} - x_0^{(1)}| = 1 + \eta.$$ Hence we have the following upper bound $$\label{eq:bound_dfmu}
\begin{split}
D_{f_\mu}(x_*,x_0) &\leq -1 -\eta + \mu + | \la \nabla f_\mu(x_0),x_*-x_0 \ra| \\
&\leq \mu.
\end{split}$$
Now, the second term can be directly computed from the definition of $d_\mu$, recalling that $x_*^{(i)} \geq 1 \geq \mu$ for $i \in \{0\dots n\}$, $$\label{eq:dh_upp_bound}
\begin{split}
D_{d_\mu}(x_*,x_0)&= d_\mu(x_*) - d_\mu(x_0) - \la \nabla d_\mu(x_0), x_* - x_0 \ra \\
&= d_\mu(x_*)\\
&= \sum_{k=1}^{2N+1} \left[\frac{\mu}{2}(1+\frac{\eta}{k})^2 + 1 + \frac{\eta}{k} - \frac{\mu}{2}\right] \\
& \leq (2N+1) \left[ \frac{\mu}{2}(1+ \eta)^2 + \eta + 1 \right].
\end{split}$$ Combining and gives $$\begin{split}
LD_{h_\mu}(x_*,x_0) &= D_{f_\mu}(x_*,x_0) + D_{d_\mu}(x_*,x_0)\\
&\leq \mu + (2N+1) \left[ \frac{\mu}{2}(1+ \eta)^2 + \eta + 1 \right]\\
&\leq (2N+1) \left[\mu + \frac{\mu}{2}(1+ \eta)^2 + \eta + 1 \right].
\end{split}$$ This bound, along with , yields $$f_\mu(\overline{x}) - \min_{\reals^n}f_\mu\geq 1 - \mu \geq \frac{L D_{h_\mu}(x_*,x_0)}{2N+1} \cdot \frac{1 - \mu}{1 + \mu + \eta + \frac{\mu}{2}(1+ \eta)^2}$$ hence the desired result.
Since constants $\mu,\eta$ can be taken arbitrarily small, we now use Proposition \[prop:zero\_pres\] to show that the bound can be approached to any precision and thus prove our main result.
\[thm:low\_bound\] Let $N\geq 1$, a precision $\epsilon \in (0,1)$ and a starting point $x_0 \in \reals^{2N+1}$. Then, there exist functions $(f,h) \in \mathcal{B}_L(\reals^{2N+1})$ such that for any Bregman gradient method $\mathcal{A}$ satisfying Assumption \[assumption:algorithm\] and initialized at $x_0$, the output $\overx$ returned after performing at most $N$ calls to each one of the primal and mirror oracles satisfies $$f(\overline{x}) - \min_{\reals^{2N+1}}f \geq \frac{LD_{h}(x_*, x_0)}{2N+1} \cdot(1-\epsilon).$$
Consider a number $N$ of oracle calls and a target precision $\epsilon \in (0,1)$. Choose the functions $f_\mu, h_\mu$ defined respectively in Equations and on $\reals^n$ with $n = 2N+1$. These functions satisfy Assumption \[assumption\_bpg\], since their domain is $\reals^n$, they are convex, differentiable, and $h_\mu$ is strongly convex. Moreover, $h$-smoothness holds because $Lh_\mu - f_\mu = d_\mu$ is convex by construction. Hence $(f_\mu,h_\mu) \in \mathcal{B}_L(\reals^{n})$.
Because the class of functions $\mathcal{B}_L(\reals^n)$ is invariant by translation, we can assume without loss of generality that the algorithm is initialized at $x_0 = (0,\dots 0)$. Recall that the only conditions our analysis imposed on the parameters $\eta,\mu$ are that $\mu \in (0,1)$ and $\eta > 4 \mu n^2$.
We can therefore choose $\eta = \epsilon/4$ and $\mu = \eta/(5n^2) = \epsilon / (20n^2)$. Under these conditions, Proposition \[prop:performance\] applies and gives that for any point $\overline{x}$ returned by a Bregman gradient algorithm that is initialized at $x_0$ and which performs at most $N$ calls to each oracle we have
$$f_\mu(\overline{x}) - \min_{\reals^{2N+1}}f_\mu \geq \frac{LD_{h_\mu}(x_*, x_0)}{2N+1} \cdot \frac{1 - \mu}{1 + \mu + \eta + \frac{\mu}{2}(1+\eta)^2}.$$ The last term can be bounded from below, using our choice of $\mu,\eta$, and the fact that $\eta < 1$, as $$\begin{split}
\frac{1 - \mu}{1 + \eta + \mu + \frac{\mu}{2}(1+\eta)^2} &\geq \frac{1 - \mu}{1 + \eta + 3\mu }\\
&= \frac{1-\frac{\epsilon}{20n^2}}{1 + \frac{\epsilon}{4} + \frac{3\epsilon}{20n^2}}\\
&\geq \frac{1-\epsilon/2}{1+\epsilon/2}\\
& \geq 1 - \epsilon
\end{split}$$ yielding the desired result.
*Remark.* One could refine the result above in the case where the primal and mirror oracles are not used the same number of times. Indeed, if the primal oracles are called $N_1$ times and the mirror oracle is called $N_2$ times, then the same reasoning shows that the lower bound remains true by replacing $2N$ with $N_1+N_2$.
Also, our lower bound involves the $h$-smoothness constant $L$ instead of the step size $\lambda$ in , but it is equivalent (up to a factor 2) when choosing $\lambda = 1/L$, which is actually the best possible step size choice.
Computer-aided performance analyses of Bregman first-order methods {#s:pep}
==================================================================
In this section, we extend the computer-aided performance estimation framework in [@Drori2014; @Taylor2015] to the setting of Bregman methods. In short, these results show how to compute the worst-case convergence rate of a given algorithm by solving a numerical optimization problem, called performance estimation problem (PEP). Solving a PEP offers several benefits, including:
1. Computing (numerically) the *exact* worst-case complexity of an algorithm on a given class of problems after a fixed number of iterations.
2. Studying the corresponding worst-case functions.
3. Inferring an analytical proof for upper bounding this complexity through a dual PEP, whose feasible points provide combination of inequalities.
Here, we focus on inferring worst-case functions. In particular, this is how we designed the lower bound provided in Section \[ss:lower\_bound\]. However, solving the PEP is also useful for proving new convergence rates (see Section \[sss:other\_bound\]), or for getting quick numerical insights about the convergence guarantees of an algorithm, like for instance on the inertial algorithm IGA [@Auslender2006] (Section \[ss:iga\]).
To use PEPs on Bregman methods, we extend the analysis in [@Drori2014; @Taylor2015] to deal with differentiable and/or strictly convex functions. Previous works on the topic modelled differentiability through an $L$-smoothness condition, and strict convexity through strong convexity, which are assumptions that we avoid in the Bregman setting. The key difference in our work is that the classes of differentiable and/or strictly convex functions are *open* sets. Thus, the worst-case functions for this class might lie on the closure of this set and exhibit some pathological nonsmooth behavior.
This section is organized as follows. In Section \[ss:pep\_intro\], we introduce the PEP framework. Sections \[ss:interpolation\]-\[s:tightpeps\] extend PEPs to the Bregman setting. We provide in Section \[ss:proofs\] several applications, including the procedure used to find the worst-case functions involved in the proof of the general lower bound in Section \[ss:lower\_bound\].
Worst-case scenarios through optimization {#ss:pep_intro}
-----------------------------------------
We now formulate the task of finding the worst-case performance of Algorithm \[algo:bpg\] as an optimization problem. We focus on the analysis of NoLips for simplicity. However, the same ideas are directly applicable to other Bregman-type algorithms like IGA [@Auslender2006] (see Section \[ss:iga\]) or Bregman proximal point [@Eckstein1993].
Recall that we write $\mathcal{B}_L(C)$ the set of function pairs $(f,h)$ satisfying Assumption \[assumption\_bpg\], such that $Lh - f$ is convex on a convex set $C$. For simplicity, we first focus on the case where functions have full domain, i.e. $C=\reals^n$ for some $n \geq 1$. In this setting, the set $\mathcal{B}_L(\reals^n)$ can be rewritten as $$\mathcal{B}_L(\reals^n)= \left\{\begin{array}{l@{\hskip 1em}|@{\hskip 1em}l}
& f \mbox{ is convex, differentiable and has at least one minimizer,}\\
&h \mbox{ is strictly convex and differentiable,}\\
f,h : \reals^n \rightarrow \reals & Lh - f \mbox{ is convex,}\\
&\forall \lambda > 0, \,\forall x,p\in \reals^n,\mbox{ the function } u \mapsto \la p,u-x \ra + \frac{1}{\lambda} D_h(u,x)\\
&\quad \mbox{ has a unique minimizer in $u$.}\\
\end{array}\right\},$$ since all constraints in Assumption \[assumption\_bpg\] about the domains of $f$ and $h$ become unnecessary. The general case when $C$ is a convex subset of $\reals^n$ can be treated along the same approach. In fact, from the perspective of performance estimation, we can show that every problem in $\bl(C)$ can be reduced to some problem in $\bl(\reals^n)$ with equivalent convergence rate (see Appendix \[a:domh\] for details).
### Performance estimation problem. {#performance-estimation-problem. .unnumbered}
Throughout this section, we fix a number of iterations $N \geq 1$, a $h$-*smoothness* parameter $L > 0$, and a step size $\lambda > 0$. In the currently known analyses of NoLips, worst-case guarantees have the following form $$\label{eq:generic_bound}
f(x_N)-f_*\leq \theta(N,L,\lambda) D_h(x_*,x_0),$$ For instance, Theorem \[thm:nolips\_bound\] states this result with $\theta(N,L,\lambda) = 1/(\lambda N)$ when $\lambda \in (0,1/L]$. We then naturally seek the smallest $\theta(N,L,\lambda)$ such that the bound holds for any functions $(f,h) \in \mathcal{B}_L(\reals^n)$, that is, solve the optimization problem $$\label{eq:pep_nolips}\tag{PEP}
\BA{ll}
\mbox{maximize} & \big(f\left(x_N\right) - f \left(x_*\right)\big) / D_{h}(x_*,x_0)\\
\rule{0pt}{12pt}
\mbox{subject to } &(f,h) \in \mathcal{B}_L(\reals^n), \\
& x_* \mbox{ is a minimizer of $f$}, \\
& x_1,\dots,x_N \mbox{ are generated from $x_0$ by Algorithm \ref{algo:bpg} with step size $\lambda$,}\\
\EA$$ in the variables $f,h,x_0,\dots,x_N,x_*,n$. We refer to this problem as a performance estimation problem (PEP). We use the convention $0/0=0$ so that the objective is well defined when $x_* = x_0$. Optimizing over the dimension $n$ to get dimension-free bounds allows the problem to admit efficient convex reformulations, as we will see in the sequel. We seek guarantees that are independent of the kernel $h$, so $h$ is also part of the optimization variables.
We begin by simplifying the problem. First, due to the strict convexity of $h$, the NoLips iteration can be equivalently formulated via the first-order optimality condition $$\nabla h(x_{i+1}) = \nabla h(x_i) - \lambda \nabla f(x_i) \quad \forall i \in \{0\dots N-1\}$$ and, since the domain is $\reals^n$, the condition that $x_*$ minimizes $f$ reduces to requiring $\nabla f(x_*) = 0$. Second, the problem is homogeneous in $(f,h)$ (i.e., from a feasible couple $(f,h)$, take any constant $c>0$ and observe that the couple $(cf,ch)$ is also feasible with the same objective value), hence optimizing the objective function $f(x_N)-f(x_*)$ under the additional constraint $D_h(x_*,x_0)=1$ produces the same optimal value than the problem above.
Finally, we use the same argument as in [@Drori2014; @Taylor2017] and observe that the objective of and the algorithmic constraints mentioned above depend solely on the values of the first-order oracles of $f$ and $h$ at the points $x_0,\dots,x_N,x_*$. Denoting $I = \{0,1,\dots,N,*\}$ the indices associated to the points involved in the problem we proceed to write these values as $$\begin{split}
\{(f_i, g_i)\}_{i \in I} &= \left\{\big(f(x_i), \nabla f(x_i)\big)\right\}_{i \in I}, \\
\{(h_i, s_i)\}_{i \in I} &= \{\big(h(x_i), \nabla h(x_i)\big)\}_{i \in I}.
\end{split}$$ With this notation the NoLips iterations rewrite $s_{i+1} = s_i - \lambda g_i$ for $i \in \{0\dots N-1\}$, and the normalization constraint $D_h(x_*,x_0) = 1$ becomes $h_* - h_0 - \la s_0, x_* - x_0 \ra = 1$.
Using this *discrete* representation of $f$ and $h$, we can reformulate equivalently as $$\label{eq:pep_disc}\tag{PEP}
\BA{rcl}
& \mbox{maximize} & f_N - f_* \\
\rule{0pt}{11pt}
&\mbox{subject to} & f_i = f(x_i), g_i = \nabla f(x_i),\\
&& h_i = h(x_i), s_i = \nabla h(x_i),\quad \mbox{for all $ i \in I$ and some } (f,h) \in \mathcal{B}_L(\reals^n),\\
& & g_* = 0,\\
& & s_{i+1} = s_i - \lambda g_i \quad \mbox{for } i\in \{1\dots N-1\},\\
& & h_* - h_0 - \la s_0, x_* - x_0 \ra = 1,
\EA$$ in the variables $n,\{(x_i,f_i,g_i,h_i,s_i)\}_{i \in I}$. The equivalence with the initial problem is guaranteed by the first constraints which are called the *interpolation conditions*.
It turns out that interpolation conditions for the class $\mathcal{B}_L(\reals^n)$ are delicate to establish. However, there exist two classes $\blstrict(\reals^n)$ and $\blnonsmooth(\reals^n)$ for which they can be derived. The first class is a restriction of $\mathcal{B}_L(\reals^n)$ where $f$ and $Lh-f$ are both assumed to be strictly convex: $$\blstrict(\reals^n) = \mathcal{B}_L(\reals^n) \cap \{f,h:\reals^n \rightarrow \reals \,|\, f \mbox{ and } Lh-f \mbox{ are strictly convex} \}$$ whereas the second class consists in considering a relaxation with possibly nonsmooth functions: $$\blnonsmooth(\reals^n) = \{ f,h : \reals^n \rightarrow \reals \,|\, f \mbox{ and } Lh-f \mbox{ are convex}\}.$$ We then have $$\blstrict(\reals^n) \subset \mathcal{B}_L(\reals^n) \subset \blnonsmooth(\reals^n).$$ With theses classes, we can now define two easier problems. The first one is a restriction of defined on the class $\blstrict(\reals^n)$, under the additional constraint that all iterates are distinct: $$\label{eq:pep_underestimation}\tag{$\underline{\text{PEP}}$}
\BA{rcl}
&\mbox{maximize} & f_N - f_* \\
\rule{0pt}{11pt}
&\mbox{subject to} & f_i = f(x_i), g_i = \nabla f(x_i),\\
&& h_i = h(x_i), s_i = \nabla h(x_i),\quad \mbox{for all $ i \in I$ and some } (f,h) \in \blstrict(\reals^n)\\
& & g_* = 0,\\
& & s_{i+1} = s_i - \lambda g_i \quad \mbox{for } i\in \{1\dots N-1\},\\
& & h_* - h_0 - \la s_0, x_* - x_0 \ra = 1,\\
& & x_i \neq x_j \quad \mbox{for } i \neq j \in I,\\
\EA$$ in the variables $n,\{(x_i,f_i,g_i,h_i,s_i)\}_{i \in I}$. The second problem is a relaxation of , where $(f,h)\in \blnonsmooth(\reals^n)$ are possibly nonsmooth and $g_i,s_i$ are thus *subgradients*: $$\label{eq:pep_overestimation}\tag{$\overline{\text{PEP}}$}
\BA{rcl}
&\mbox{maximize} & f_N - f_* \\
\rule{0pt}{11pt}
&\mbox{subject to} & f_i = f(x_i), g_i \in \partial f(x_i),\\
&& h_i = h(x_i), s_i \in \partial h(x_i),\\
&& Ls_i - g_i \in \partial(Lh-f)(x_i) \quad \mbox{for all $ i \in I$ and some } (f,h) \in \blnonsmooth(\reals^n),\\
& & g_* = 0,\\
& & s_{i+1} = s_i - \lambda g_i \quad \mbox{for } i\in \{1\dots N-1\},\\
& & h_* - h_0 - \la s_0, x_* - x_0 \ra = 1,\\
\EA$$ in the variables $n,\{(x_i,f_i,g_i,h_i,s_i)\}_{i \in I}$. We added the technical constraint $ Ls_i - g_i \in \partial (Lh-f)(x_i)$, which is redundant for differentiable functions; but that is necessary in order to establish interpolation conditions in the nonsmooth case.
Because of the inclusions between the feasible sets of these problems, we naturally have $$\val{\eqref{eq:pep_underestimation}} \leq \val{\eqref{eq:pep_nolips}} \leq \val{\eqref{eq:pep_overestimation}}.$$ We will prove in the sequel that can be solved via a semidefinite program and that $\val\eqref{eq:pep_underestimation} = \val\eqref{eq:pep_overestimation}$ (Theorem \[thm:equiv\]), allowing to reach our claims.
Note that the relaxed problem does not correspond to any practical algorithm, as NoLips is not properly defined for nonsmooth functions $h$. However, we will see in the sequel that feasible points of this problem correspond to accumulation points of . In other words, instances of NoLips can get arbitrarily close to pathological nonsmooth functions whose behaviors are captured by .
In the following sections, we show that problems and can be cast as semidefinite programs (SDP) [@Vandenberghe96] and solved numerically using standard packages [@mosek; @yalmip]. The main ingredient consists in showing that interpolation constraints can actually be expressed using quadratic inequalities, as detailed in the next section.
Interpolation involving differentiability and strict convexity {#ss:interpolation}
--------------------------------------------------------------
In this section, we show how to reformulate interpolation constraints for and as quadratic inequalities. We start by recalling interpolation conditions for the class of $L$-smooth and $\mu$-strongly convex functions.
\[thm:interp\_mul\] Let $I$ be a finite index set, $\{(x_i,f_i,g_i)\}_{i \in I} \in (\reals^n \times \reals \times \reals^n)^{|I|}$ and $0 \leq \mu \leq L \leq +\infty$. The following statements are equivalent:
1. There exists a proper closed convex function $f:\reals^n \rightarrow \reals \cup \{+\infty\}$ such that $f$ is $\mu$-strongly convex, has a $L$-Lipschitz continuous gradient and $$f_i = f(x_i),\, g_i \in \partial f(x_i) \quad \forall i \in I.$$
2. For every $i,j \in I$ we have $$\begin{split}
f_i - f_j - \la g_j, x_i - x_j \ra &\geq \frac{1}{2L} \|g_i - g_j\|^2 + \frac{\mu }{2 (1 - \mu/L)} \|x_i - x_j - \frac{1}{L}(g_i - g_j)\|^2.
\end{split}$$
In particular, when $L=+\infty$ (meaning that we require no smoothness) and $\mu=0$, those conditions reduce to the simpler *convex interpolation* conditions, reminiscent of subgradient inequalities: $$\label{e:subgradient}
f_i - f_j - \la g_j, x_i-x_j \ra \geq 0$$ In our setting, we want to avoid working with smoothness and strong convexity, so we provide interpolation conditions for the class of differentiable strictly convex functions.
\[prop:interp\_cond\] Let $I$ be a finite index set and $\{(x_i,f_i,g_i)\}_{i \in I} \in (\reals^n \times \reals \times \reals^n)^{|I|}$. The following statements are equivalent:
1. There exists a convex function $f:\reals^n \rightarrow \reals$ such that $f$ is differentiable, strictly convex and $$f_i = f(x_i),\, g_i = \nabla f(x_i)\quad \forall i \in I.$$
2. For every $i,j\in I$ we have $$\label{eq:interp_cond}
\begin{cases}
f_i - f_j - \la g_j, x_i - x_j \ra &\geq 0,\\
f_i - f_j - \la g_j, x_i - x_j \ra & >0 \mbox{ if } x_i \neq x_j \mbox{ (strict convexity),}\\
f_i - f_j - \la g_j, x_i - x_j \ra & >0 \mbox{ if } g_i \neq g_j \mbox{ (differentiability).}
\end{cases}$$
\(i) $\implies$ (ii). Assume that (i) holds, and choose such a function $f$. The first inequality of follows from convexity of $f$. The second inequality follows directly from strict convexity when $x_i \neq x_j$. Now, to prove the third part, consider the case when we have $\nabla f(x_i) \neq \nabla f(x_j)$ for some indices $i,j$. Let us prove the result by contradiction, i.e., assume that $$\label{eq:contra_assump}
f(x_i) - f(x_j) - \la \nabla f(x_j),x_i - x_j \ra = 0.$$ Let $u \in \reals^n$, convexity implies that $$f(u) \geq f(x_j) + \la \nabla f(x_j), u-x_j \ra.$$ Combining the above inequality with gives $$f(u) \geq f(x_i) + \la \nabla f(x_j), u-x_i\ra \quad \forall u\in \reals^n$$ which shows, by definition of a subgradient, that $\nabla f(x_j) \in \partial f(x_i)$. Since $f$ is differentiable at $x_i$, we have by [@Rockafellar2008 Thm 25.1] that $\partial f(x_i) = \{\nabla f(x_i)\}$ which is a contradiction as we assumed $\nabla f(x_i) \neq \nabla f(x_j)$. Thus the third part of is proved.
\(ii) $\implies$ (i). Assume that (ii) holds. If for all $i,j\in I$, we have $g_i = g_j$ and $x_i = x_j$, then there is only one point and one subgradient to be interpolated, and the result follows immediatly from considering a well-chosen definite quadratic function. In the other case, define $$\nu = \min_{\substack{i,j \in I \\ g_i \neq g_j \text{ or } x_i \neq x_j}} f_i - f_j - \la g_j, x_i - x_j \ra.$$ Because of and the finiteness of $I$, we have that $\nu > 0$. Now, define $r$ as $$r = \max_{i,j \in I} \|g_i - g_j\|^2 + \|x_i - x_j\|^2$$ so that $r > 0$. Condition together with the definitions of $\nu$ and $r$ yield that for all $i,j \in I$ we have $$\label{eq:min_df_gj}
f_i - f_j - \la g_j, x_i - x_j \ra \geq \frac{\nu}{r} \left(\|g_i - g_j\|^2 + \|x_i-x_j\|^2 \right).\\$$ Now, let us choose two constants $0<\mu<L<+\infty$ such that $$\frac{1}{L- \mu} \leq \frac{\nu}{r}, \quad
\frac{\mu}{1 - \mu / L} \leq \frac{\nu}{r}.$$ as it suffices to take $L$ large enough and $\mu$ small enough. We now proceed to show that the interpolation conditions of Theorem \[thm:interp\_mul\] hold with the constants $\mu,L$ defined above. Using the inequality $\|u-v\|^2 \leq 2\|u\|^2 + 2\|v\|^2$ and we get that for all $i,j$, $$\begin{split}
&\frac{1}{2L} \|g_i - g_j\|^2 + \frac{\mu }{2(1 - \mu/L)} \|x_i - x_j - \frac{1}{L}(g_i - g_j)\|^2\\
& \leq \left( \frac{1}{2L}+ \frac{\mu}{L(L - \mu)} \right) \|g_i - g_j\|^2
+ \frac{\mu}{1 - \mu/L} \|x_i-x_j\|^2\\
& \leq \left( \frac{1}{L}+ \frac{\mu}{L(L - \mu)} \right) \|g_i - g_j\|^2
+ \frac{\mu}{1 - \mu/L} \|x_i-x_j\|^2\\
&= \frac{1}{L-\mu} \|g_i - g_j\|^2
+ \frac{\mu }{1 - \mu/L} \|x_i-x_j\|^2\\
& \leq \frac{\nu}{r} \|g_i - g_j\|^2 + \frac{\nu}{r} \|x_i - x_j\|^2\\
& \leq f_i - f_j - \la g_j, x_i - x_j \ra.
\end{split}$$ Under those conditions, Theorem \[thm:interp\_mul\] states that there exists a convex function $f$ that interpolates $\{(x_i,f_i,g_i)\}_{i\in I}$ which is $\mu$-strongly convex and has $L$-Lipschitz continuous gradients. A fortiori, since $\mu >0$ and $L < \infty$, $f$ is differentiable and strictly convex. Finally, $f$ is finite on $\reals^n$ since it is $L$-smooth.
*Remark.* It is easy to adapt the result of Proposition \[prop:interp\_cond\] for only one of the two conditions (strict convexity or differentiability), which amounts to choose only the corresponding inequalities in .
Using these results, we can now formulate interpolation conditions for the problems and involving the classes $\blnonsmooth(\reals^n)$ and $\blstrict(\reals^n)$ that were defined in Section \[ss:pep\_intro\].
\[cor:blnonsmooth\] Let $I$ be a finite index set and $\{(x_i,f_i,g_i,h_i,s_i)\}_{i \in I} \in (\reals^n \times \reals \times \reals^n \times \reals \times \reals^n)^{|I|}$. The following statements are equivalent.
1. There exist functions $(f,h) \in \blnonsmooth(\reals^n)$ such that $$\begin{split}
f_i = f(x_i),\, g_i \in \partial f(x_i),\\
h_i = h(x_i),\, s_i \in \partial h(x_i),\\
Ls_i - g_i \in \partial (Lh-f)(x_i).\\
\end{split}$$
2. For all $i,j \in I$ such that $i \neq j$ we have $$\label{eq:interp_blnonsmooth}
\begin{split}
f_i - f_j - \la g_j, x_i- x_j \ra \geq 0, \\
(Lh_i-f_i) - (Lh_j-f_j) - \la Ls_j - g_j, x_i- x_j \ra \geq 0.\\
\end{split}$$
\(i) $\implies$(ii) follows immediately from the definition of a subgradient applied to convex functions $f$ and $Lh-f$. Now, assume that (ii) holds. By the specialization of Theorem \[thm:interp\_mul\], conditions (ii) imply that there exist two convex functions $f,d : \reals^n \rightarrow \reals$ such that $$\begin{split}
f_i &= f(x_i), \quad\quad\quad g_i \in \partial f(x_i),\\
Lh_i - f_i &= d(x_i), \,\, L s_i - g_i \in \partial d(x_i).
\end{split}$$ Now, defining the convex function $h = (f + d)/L$, we have that $d = Lh -f $, hence $Ls_i - g_i \in \partial(Lh-f)(x_i)$. We also get $$\begin{split}
&h_i = h(x_i), \, s_i \in \partial h(x_i)
\end{split}$$ where we used the fact that $L s_i \in \partial f(x_i) + \partial d(x_i) \subset \partial (f+d)(x_i) = L \partial h(x_i)$ (see [@Rockafellar2008 Thm 23.8] for the subdifferential of a sum of convex functions). Hence (i) holds.
\[cor:blstrict\] Let $I$ be a finite index set and $\{(x_i,f_i,g_i,h_i,s_i)\}_{i \in I} \in (\reals^n \times \reals \times \reals^n \times \reals \times \reals^n)^{|I|}$. Assume that $x_i \neq x_j$ for every $i \neq j \in I$. The following statements are equivalent.
1. There exist functions $(f,h) \in \blstrict(\reals^n)$ such that $$\begin{split}
\quad f_i = f(x_i),\, g_i = \nabla f(x_i),\\
\quad h_i = h(x_i),\, s_i = \nabla h(x_i).
\end{split}$$
2. For all $i,j \in I$ such that $i \neq j$ we have $$\label{eq:interp_blstrict}
\begin{split}
f_i - f_j - \la g_j, x_i- x_j \ra &> 0, \\
(Lh_i-f_i) - (Lh_j-f_j) - \la Ls_j - g_j, x_i- x_j \ra &> 0.\\
\end{split}$$
Note that since $x_i \neq x_j$ for every $i \neq j$, interpolation conditions of Proposition \[prop:interp\_cond\] reduce to requiring the strict inequality in for every $i\neq j$. As before, define $d := Lh-f$. Then since $(f,h) \in \blstrict(\reals^n)$ the functions $f$ and $d$ are differentiable strictly convex, hence (i) $\implies$ (ii) follows simply from strict convexity of these functions.
Conversely, assume (ii). By using Proposition \[prop:interp\_cond\] again, we can interpolate differentiable strictly convex functions $f$ and $d$ and recover $h$ with $h = (f+d)/L$, thus we have naturally $Lh-f$ convex. The function $h$ is thus also differentiable and strictly convex. Moreover, it can be seen from the proof of Proposition \[prop:interp\_cond\] that the interpolating functions can actually be chosen strongly convex, hence with this choice the well-posedness condition Assumption \[assumption\_bpg\]\[assumption:well\_posed\] holds, and we can conclude that $(f,h)\in \blstrict(\reals^n)$.
Semidefinite reformulations {#ss:sdp}
---------------------------
Now that we established the interpolation conditions for and , we may use them to obtain semidefinite performance estimation formulations as in [@Drori2014; @Taylor2017]. This is made possible by observing that interpolation conditions - are quadratic inequalities in the problem variables.
Let $\{(x_i,f_i,g_i,h_i,s_i)\}_{i\in I}$ be a feasible point of one of the PEPs in dimension $n$. We write $G \in \symm_{{3(N+2)}}$ the Gram matrix that contains all dot products between $x_i,g_i,s_i$ for $i \in I$, with $$G =
\BPM
G^{xx} & G^{gx} & G^{sx}\\
G^{gx\top} & G^{gg} & G^{gs}\\
G^{sx\top} & G^{gs\top} & G^{ss}\\
\EPM
\succeq 0$$ whose size is independent of the dimension $n$, where the blocks are defined as $$G^{xx}_{ij} = \la x_i, x_j \ra,~G^{gx}_{ij} = \la g_i, x_j \ra,~G^{gs}_{ij} = \la g_i, s_j \ra, G^{gg}_{ij} = \la g_i, g_j \ra, G^{sx}_{ij} = \la s_i, x_j \ra, G^{ss}_{ij} = \la s_i, s_j \ra,\quad i,j\in I.$$ Write also $$F = (f_0,\dots,f_N,f_*) \in \reals^{N+2},\quad
H = (h_0,\dots,h_N,h_*) \in \reals^{N+2},$$ the vectors representing the function values of $f,h$ at the iterates. We now observe that all the constraints of and can be expressed using only $G$, $F$ and $H$.
For instance, interpolation conditions for $\blnonsmooth(\reals^n)$ rewrite for all $i,j \in I$ as $$\begin{split}
f_i - f_j - G^{gx}_{ji} + G^{gx}_{jj} &\geq 0,\\
(Lh_i-f_i) - (Lh_j-f_j) - L(G^{sx}_{ji} - G^{sx}_{jj}) + G^{gx}_{ji} - G^{gx}_{jj} &\geq 0,\\
\end{split}$$ This allows to reformulate the relaxation as a semidefinite program, written
$$\label{eq:pep_over_sdp}\tag{sdp-$\overline{\text{PEP}}$}
\BA{ll}
\mbox{maximize} & f_N - f_*\\
\rule{0pt}{12pt}
\mbox{subject to} & f_i - f_j - G^{gx}_{ji} + G^{gx}_{jj} \geq 0, \\
&(Lh_i-f_i) - (Lh_j-f_j) - L(G^{sx}_{ji} - G^{sx}_{jj}) + G^{gx}_{ji} - G^{gx}_{jj} \geq 0 \quad \mbox { for } i,j \in I,\\
& G^{gg}_{**} = 0,\\
& G^{sx}_{i+1,j} = G^{sx}_{ij}-\lambda G^{gx}_{ij} \quad \mbox{ for } i \in \{0\dots N-1\}, j \in I,\\
& h_* - h_0 - G^{sx}_{0*} + G^{sx}_{00} =1, \\
& G \succeq 0,
\EA$$
in the variables $G\in\symm_{{3(N+2)}}$ and $F, H \in \reals^{N+2}$.
Any feasible point of can be cast into an admissible point of by computing the semidefinite Gram matrix $G$. Conversely, if $G,F,H$ is an admissible point of , then the vectors $\{(x_i,g_i,s_i)\}_{i\in I}$ can be recovered by performing, for instance, Cholesky decomposition of $G$. Note that we expressed the algorithmic constraint $s_{i+1} = s_i - \lambda g_i$ only through scalar products with the $x_i$’s in the SDP, since only the projection of the gradients on $\text{Span}(\{x_i\}_{i \in I})$ is relevant in the PEPs. Because interpolation conditions from Corollary \[cor:blnonsmooth\] are necessary and sufficient, we conclude that the problems are equivalent, that is $$\text{val}\eqref{eq:pep_over_sdp}=\text{val}\eqref{eq:pep_overestimation}.$$
The rank of $G$ determines the dimension of the interpolated problem. If we look instead for a solution that has a given dimension $n$, this would mean imposing a nonconvex rank constraint on $G$. Our formulation, on the other side, is convex and finds the best convergence bound that is dimension-independent, which is an usual requirement for *large-scale settings*. Since $G$ has size ${3(N+2)}$, the dimension of the worst-case functions will be at most ${3(N+2)}$.
In the same way, the value of can be computed as $$\label{eq:pep_under_sdp}\tag{$\text{sdp-}\underline{\text{PEP}}$}
\BA{ll}
\mbox{maximize} & f_N - f_*\\
\rule{0pt}{12pt}
\mbox{subject to} & f_i - f_j - G^{gx}_{ji} + G^{gx}_{jj} > 0,\\
&(Lh_i-f_i) - (Lh_j-f_j) - L(G^{sx}_{ji} - G^{sx}_{jj}) + G^{gx}_{ji} - G^{gx}_{jj} > 0 \quad \mbox { for } i \neq j \in I,\\
& G^{gg}_{**} = 0,\\
& G^{sx}_{i+1,j} = G^{sx}_{ij}-\lambda G^{gx}_{ij} \quad \mbox{ for } i \in \{0\dots N-1\}, j \in I,\\
& h_* - h_0 - G^{sx}_{0*} + G^{sx}_{00} =1, \\
& G^{xx}_{ii} + G^{xx}_{jj} - 2 G^{xx}_{ij} > 0 \quad \mbox{for } i\neq j \in I,\\
& G \succeq 0,
\EA$$ in the variables $G\in\symm_{{3(N+2)}}$ and $F, H \in \reals^{N+2}$, where we used interpolation conditions for $\blstrict(\reals^n)$ from Corollary \[cor:blstrict\], since all points $\{x_i\}_{i \in I}$ are constrained to be distinct. Therefore, as above we infer that $$\text{val}\eqref{eq:pep_under_sdp}=\text{val}\eqref{eq:pep_underestimation}.$$ Recalling the hierarchy between the problems, we thus have $$\text{val}\eqref{eq:pep_under_sdp} \leq \text{val}\eqref{eq:pep_nolips} \leq \text{val}\eqref{eq:pep_over_sdp}.$$ By comparing the two semidefinite programs stated above, one can notice that the only difference is that imposes some inequalities of to be strict. In the next section, we use topological arguments to prove that the values of the two problems are actually equal. In fact, strict inequalities have little meaning in numerical optimization (the value of is actually a supremum and not a maximum); in our experiments, we will focus on as solvers usually admit only closed feasible sets.
Tightness of the approach: nonsmooth limit behaviors {#s:tightpeps}
----------------------------------------------------
We are now ready to prove the main result of this section.
\[thm:equiv\] The value of the performance estimation problem for NoLips is equal to the value of the nonsmooth relaxation , which can be computed by solving the semidefinite program .
We will show that the closure of the feasible set of is the feasible set of . We first need to prove that the strengthened problem is feasible, by finding an instance of NoLips where $f$ and $Lh-f $ are strictly convex and such that all iterates are distinct. It suffices for instance to consider two unidimensional quadratic functions. Define $f,h : \reals \rightarrow \reals$ with $$f(x) = \frac{\alpha}{2}x^2, \, h(x) = \frac{1}{2} x^2 \quad \mbox{ where } \alpha = \min \left(\frac{1}{2 \lambda}, \frac{L}{2} \right).$$ Then $f$ is strictly convex and so is $Lh-f = \frac{L-\alpha}{2}x^2$ since $L-\alpha \geq \frac{L}{2}>0$. The optimum is $x_* = 0$. Choose $$x_0 = \sqrt{2}$$ for which we have $D_{h}(x_*,x_0) = x_0^2/2 = 1$. Then, Algorithm \[algo:bpg\] is equivalent to gradient descent and the iterates satisfy $$x_N = (1 - \lambda \alpha)^Nx_0.$$ Since $\alpha \lambda \leq 1/2 < 1$, all the iterates are distinct and therefore we constructed a feasible point of . Let us therefore write $(G,F,H)$ a corresponding feasible point of , and $(\overline{G},\overline{F},\overline{H})$ a feasible point of . Define the sequence $\{(G^k,F^k,H^k)\}_{k \geq 1}$ as $$\begin{split}
G^k &= \frac{1}{k} G + (1 - \frac{1}{k}) \overline{G},\\
F^k &= \frac{1}{k} F + (1 - \frac{1}{k}) \overline{F},\\
H^k &= \frac{1}{k} H + (1 - \frac{1}{k}) \overline{H}.\\
\end{split}$$
Then, for every $k \geq 1$, $(G^k,F^k,H^k)$ is still an feasible point of , because of convexity of the constraints and the fact that adding a strict inequality to a weak inequality gives a strict inequality. Moreover, the sequence converges to the point $(\overline{G},\overline{f},\overline{h})$ when $k \rightarrow +\infty$.
Hence we proved that for any feasible point of , there is a sequence of admissible points of that converge to it. Since the objective is linear in the vector $F$ therefore continuous, we deduce that the two problems have the same value: $$\text{val}\eqref{eq:pep_under_sdp} = \text{val}\eqref{eq:pep_over_sdp},$$ which means that $\text{val}\eqref{eq:pep_underestimation} = \text{val}\eqref{eq:pep_overestimation}$. Since val lies in between these two values, we conclude that they are all equal.
Theorem \[thm:equiv\] states that the value of the original problem can be computed numerically with a semidefinite solver applied to . The result itself also helps us gain some theoretical insight: it tells us that the worst case for NoLips might be reached as $(f,h)$ approach possibly pathological limiting nonsmooth functions in $\blnonsmooth(\reals^n)$.
Oberve also that we focused on presenting the PEP for the class $\mathcal{B}_L(\reals^n)$ to avoid technicalities related to the domain of definition. However, we show in Appendix \[a:domh\] that the exact same problem also solves the performance estimation problem for NoLips on the general class $\mathcal{B}_L(C)$, for any closed convex set $C$.
Numerical evidences and computer-assisted proofs {#ss:proofs}
------------------------------------------------
We now provide several applications of the performance estimation framework that we developed for Bregman methods.
### Solving for finding the exact worst-case convergence rate of NoLips {#sss:nolips_pep}
We first start by the most direct application, that is finding exact worst-case performance of NoLips. Theorem \[thm:equiv\] states that it can be computed by solving the semidefinite program . The link to the MATLAB implementation is provided in Section \[s:conclusion\].
To simplify our setting, note that we can assume without loss of generality that the $h$-smoothness constant $L$ is $1$, since we can replace $h$ by a scaled version $Lh$. Recall that we know from Theorem \[thm:nolips\_bound\] that $$\text{val\eqref{eq:pep_nolips}} \leq \frac{1}{\lambda N}.$$ Table \[tab:pep\_nolips\] shows the result of solving for several values of $N$ up to 100, for a step size $\lambda = 1$. We observe that with high precision, val is equal to the theoretical bound $1/(\lambda N)$.
[lccc]{} N & val & Rel. error & Primal feasibility\
1 & 1.000 & 1.8e-11 & 4.3e-10\
2 & 0.500 & 1.8e-8 & 2.8e-9\
3 & 0.333 & 1.8e-8 & 2.8e-9\
4 & 0.250 & 4.9e-8 & 2.3e-8\
5 & 0.200 & 1.8e-10 & 6.4e-11\
10 & 0.100 & 6.4e-11 & 1.3e-11\
20 & 0.050 & 1.1e-8 & 1.9e-10\
50 & 0.020 & 6.5e-6 & 5.0e-7\
100 & 0.01 & 7.2e-5 & 1.6e-6\
### Other values of $\lambda$. {#other-values-of-lambda. .unnumbered}
One can wonder how the numerical value evolves when we vary the step size $\lambda$. The experimental observations are the following:
- For any $\lambda \in (0,1/L]$, val is exactly equal to the theoretical bound $1/(\lambda N)$.
- For any $\lambda > 1/L$, val $= +\infty$, hence Algorithm \[algo:bpg\] does not converge in general with these step size values. This suggests that the maximal step size value allowed for NoLips is indeed $1/L$, unlike the Euclidean setting where gradient descent can be applied with a step size that goes up to $2/L$.
While results above suggest that $1/(\lambda N)$ is the exact worst-case rate of NoLips, they provide only numerical evidence. We can however use them to deduce formal guarantees, both for proving an *upper bound* and a *lower bound*.
### Upper bound guarantee through duality. {#upper-bound-guarantee-through-duality. .unnumbered}
As noticed in previous work on PEPs [@Drori2014; @Taylor2015], solving the dual of can be used to deduce a proof. Indeed, the dual solution gives a combination of the constraints that, when transposed to analytical form, leads to a formal guarantee. This provides the following proof for the $O(1/k)$ convergence rate of Theorem \[thm:nolips\_bound\].
### Proof of Theorem \[thm:nolips\_bound\] {#proof-of-theorem-thmnolips_bound .unnumbered}
The proof relies on the fact that, since $Lh-f$ is convex we have that $\tfrac{1}{\lambda}h-f$ is convex for any $\lambda\in (0,\tfrac1L]$, and only consists in performing the following weighted sum of inequalities:
- convexity of $f$, between $x_*$ and $x_i$ ($i=0,\hdots,k$) with weights $\gamma_{*,i}=\tfrac{1}{k}$: $$f(x_*)\geq f(x_i)+ \inner{ \nabla f(x_i)}{x_*-x_i},$$
- convexity of $f$, between $x_i$ and $x_{i+1}$ ($i=0,\hdots,k-1$) with weights $\gamma_{i,i+1}=\tfrac{i}{k}$: $$f(x_i)\geq f(x_{i+1})+\inner{\nabla f(x_{i+1})}{x_i-x_{i+1}},$$
- convexity of $\tfrac{1}{\lambda}h-f$, between $x_*$ and $x_k$ with weight $\mu_{*,k}=\tfrac1k$: $$\tfrac{1}{\lambda}h(x_*)-f(x_*)\geq \tfrac{1}{\lambda}h(x_k)-f(x_k)+\inner{\tfrac{1}{\lambda}\nabla h(x_k)-\nabla f(x_k)}{x_*-x_k},$$
- convexity of $\tfrac{1}{\lambda}h-f$, between $x_{i+1}$ and $x_{i}$ ($i=0,\hdots,k-1$) with weight $\mu_{i+1,i}=\tfrac{i+1}{k}$ $$\tfrac{1}{\lambda}h(x_{i+1})-f(x_{i+1})\geq \tfrac{1}{\lambda}h(x_i)-f(x_i)+\inner{\tfrac{1}{\lambda}\nabla h(x_i)-\nabla f(x_i)}{x_{i+1}-x_i},$$
- convexity of $\tfrac{1}{\lambda}h-f$, between $x_{i}$ and $x_{i+1}$ ($i=0,\hdots,k-1$) with weight $\mu_{i,i+1}=\tfrac{i}{k}$ $$\tfrac{1}{\lambda}h(x_i)-f(x_i)\geq \tfrac{1}{\lambda}h(x_{i+1})-f(x_{i+1})+\inner{\tfrac{1}{\lambda}\nabla h(x_{i+1})-\nabla f(x_{i+1})}{x_i-x_{i+1}}.$$
The weighted sum is written as $$\begin{aligned}
0\geq &\sum_{i=0}^{k}\gamma_{*,i}\left[f(x_i)-f(x_*)+ \inner{ \nabla f(x_i)}{x_*-x_i}\right]\\
&+\sum_{i=0}^{k-1} \gamma_{i,i+1} \left[ f(x_{i+1})-f(x_i)+\inner{\nabla f(x_{i+1})}{x_i-x_{i+1}}\right]\\
&+ \mu_{*,k} \left[\tfrac{1}{\lambda}h(x_k)-f(x_k)-(\tfrac{1}{\lambda}h(x_*)-f(x_*))+\inner{\tfrac{1}{\lambda}\nabla h(x_k)-\nabla f(x_k)}{x_*-x_k} \right]\\
&+\sum_{i=0}^{k-1} \mu_{i+1,i} \left[\tfrac{1}{\lambda}h(x_i)-f(x_i)-( \tfrac{1}{\lambda}h(x_{i+1})-f(x_{i+1}))+\inner{\tfrac{1}{\lambda}\nabla h(x_i)-\nabla f(x_i)}{x_{i+1}-x_i}\right]\\
&+\sum_{i=0}^{k-1} \mu_{i,i+1} \left[\tfrac{1}{\lambda}h(x_{i+1})-f(x_{i+1})-(\tfrac{1}{\lambda}h(x_i)-f(x_i))+\inner{\tfrac{1}{\lambda}\nabla h(x_{i+1})-\nabla f(x_{i+1})}{x_i-x_{i+1}}\right],
\end{aligned}$$ By substitution of $\nabla h(x_{i+1})=\nabla h(x_{i})-\lambda \nabla f(x_{i})$ ($i=0,\hdots,k-1$), one can reformulate the weighted sum exactly as (i.e., there is no residual): $$0\geq f(x_k)-f(x_*)- \tfrac{h(x_*)-h(x_0)-\inner{\nabla h(x_0)}{x_*-x_0}}{\lambda k},$$ yielding the desired result.
### Lower bound through worst-case functions. {#lower-bound-through-worst-case-functions. .unnumbered}
As computes the *exact* worst-case performance of NoLips, experiments above suggest that $1/(\lambda N)$ is also a lower bound, meaning that for every $\epsilon > 0$, there exist functions $(f,h) \in \mathcal{B}_L$ such that the iterates of NoLips satisfy $$f(x_N) - f_* \geq \frac{D_h(x_*,x_0)}{\lambda N} - \epsilon.$$ We detail here how such functions can be constructed from the solution of . The numerical solver allow us to find a maximizer $\overline{G},\overline{F},\overline{H}$ (recall that only the relaxed problem has a maximizer as the feasible set is closed), and by factorizing the matrix $G$ as $P^T P$ we can thus recover the corresponding discrete representation $\{\overline{x}_i,\overline{g}_i,\overline{f}_i,\overline{h}_i,\overline{s}_i\}_{i \in I}$. This discretization can in turn be interpolated to get the corresponding functions $(\overline{f},\overline{h}) \in \blnonsmooth$. There are multiple ways to perform this interpolation; see [@Taylor2017 Thm. 1] for a constructive approach.
Recall that since functions $(\overline{f},\overline{h})$ are solution to , they belong to $\blnonsmooth$ and might thus form a *pathological* nonsmooth limiting worst-case. They can be approached by valid instances $(f_\mu,h_\mu) \in \mathcal{B}_L$ by performing for instance smoothing through Moreau enveloppes (as in Section \[ss:lower\_bound\]) and adding a small quadratic to $h$ to make it strictly convex.
There are however many possible maximizers of . If we seek a low-dimensional example that may be easily interpretable, we can search for a maximizer such that the Gram matrix $G$ has minimal rank. Using rank minimization heuristics, we were able to find one-dimensional worst-case functions. Fix a number of iterations $N \geq 1$, assume $\lambda = 1/L = 1$ and define $\overf,\overh:\reals \rightarrow \reals$ as $$\begin{split}
\overf(x) &= |x-1|,\\
\overh(x) &= \overf(x) + \max(- N x,0).
\end{split}$$ Then clearly $(\overf,\overh) \in \blnonsmooth(\reals)$. Figure \[fig:worstcase1d\] shows the functions $(\overf,\overh)$ as well as their smoothed versions $(f_\mu,h_\mu) \in \mathcal{B}_L(\reals)$. Note that the pathological behavior also reflects in the iterates: in the limiting instance, all iterates $\overx_0,\dots,\overx_N$ are equal. In the smoothed version, iterates are distinct (since $h_\mu$ is strictly convex), but they get closer and closer as the smoothing parameter $\mu$ goes to $0$.
The smoothed function $f_\mu$ is a Huber function, which is also the worst-case instance for Euclidean gradient descent on $L$-smooth functions described in [@Taylor2017]. This analysis could be formalized to prove the $1/k$ lower bound for NoLips; however, this bound is just a particular case of the stronger result for general Bregman gradient methods derived in Section \[ss:lower\_bound\].
### Extension to other criteria {#sss:other_bound}
In our performance estimation problem, we focused on studying bounds of the form $f(x_N)-f_*\leq \theta(N,L,\lambda) D_h(x_*,x_0)$. However, we are not limited to this criterion, and different convergence measures might be considered by changing the objective and constraints in . For instance, by adapting , we get the following new convergence result for NoLips.
\[prop:nolips\_bound2\] Let $L > 0$, $C$ be a nonempty closed convex subset of $\reals^n$ and $(f,h) \in \mathcal{B}_L(C)$ functions admissible for NoLips. Then the sequence $\{x_k\}_{k \geq 0}$ generated by Algorithm \[algo:bpg\] with constant step size $\lambda \in (0,1/L]$ satisfies for $k \geq 2$
$$\min_{1\leq i\leq k} D_h(x_{i-1},x_i) \leq \frac{2D_{h}(x_*,x_0)}{k(k-1)}$$
where $x_* \in \argmin_{C} f \cap \dom h$.
In the same way as before, the formal guarantee has been obtained by examining the dual of the corresponding PEP. The proof relies on the fact that $\tfrac{1}{\lambda}h-f$ is convex for any $\lambda\in (0,\tfrac1L]$, and only consists in performing the following weighted sum of inequalities:
- convexity of $f$, between $x_*$ and $x_i$ ($i=0,\hdots,k$) with weights $\gamma_{*,i}=\tfrac{2\lambda}{k(k-1)}$: $$f(x_*)\geq f(x_i)+ \inner{ \nabla f(x_i)}{x_*-x_i},$$
- optimality of $x_*$ for each $x_k$ with weight $\gamma_{k,*}=\tfrac{2\lambda}{k-1}$: $$f(x_k)\geq f(x_*),$$
- convexity of $\tfrac{1}{\lambda}h-f$, between $x_*$ and $x_k$ with weight $\mu_{*,k}=\tfrac{2\lambda}{k(k-1)}$: $$\tfrac{1}{\lambda}h(x_*)-f(x_*)\geq \tfrac{1}{\lambda}h(x_k)-f(x_k)+\inner{\tfrac{1}{\lambda}\nabla h(x_k)-\nabla f(x_k)}{x_*-x_k},$$
- convexity of $\tfrac{1}{\lambda}h-f$, between $x_{i+1}$ and $x_{i}$ ($i=0,\hdots,k-1$) with weight $\mu_{i+1,i}=\tfrac{2\lambda(i+1)}{k(k-1)}$ $$\tfrac{1}{\lambda}h(x_{i+1})-f(x_{i+1})\geq \tfrac{1}{\lambda}h(x_i)-f(x_i)+\inner{\tfrac{1}{\lambda}\nabla h(x_i)-\nabla f(x_i)}{x_{i+1}-x_i},$$
- definition of smallest residual among the iterates ($i=1,\hdots,k$) with weights $\tau_i=\tfrac{2(i-1)}{k(k-1)}$: $$h(x_{i-1})-h(x_i)-\inner{\nabla h(x_i)}{x_{i-1}-x_i}\geq \min_{1\leq j \leq k} \{D_h(x_{j-1},x_j)\}.$$
The weighted sum is written as $$\begin{aligned}
0\geq & \sum_{i=0}^k \gamma_{*,i} [f(x_i)-f(x_*)+ \inner{ \nabla f(x_i)}{x_*-x_i}]\\
&+ \gamma_{k,*} [f(x_*)-f(x_k)]\\
&+ \mu_{*,k} [\tfrac{1}{\lambda}h(x_k)-f(x_k)-(\tfrac{1}{\lambda}h(x_*)-f(x_*))+\inner{\tfrac{1}{\lambda}\nabla h(x_k)-\nabla f(x_k)}{x_*-x_k}]\\
&+\sum_{i=0}^{k-1} \mu_{i+1,i} [\tfrac{1}{\lambda}h(x_i)-f(x_i)-(\tfrac{1}{\lambda}h(x_{i+1})-f(x_{i+1}))+\inner{\tfrac{1}{\lambda}\nabla h(x_i)-\nabla f(x_i)}{x_{i+1}-x_i}]\\
&+ \sum_{i=1}^k \tau_i [\min_{1\leq j \leq k} \{D_h(x_{j-1},x_j)\} -( h(x_{i-1})-h(x_i)-\inner{\nabla h(x_i)}{x_{i-1}-x_i})].
\end{aligned}$$ By substitution of $\nabla h(x_{i+1})=\nabla h(x_{i})-\lambda \nabla f(x_{i})$ ($i=0,\hdots,k-1$), one can reformulate the weighted sum exactly as (i.e., there is no residual): $$0\geq \min_{1\leq j \leq k} \{D_h(x_{j-1},x_j)\} - 2\tfrac{h(x_*)-h(x_0)-\inner{\nabla h(x_0)}{x_*-x_0}}{ k(k-1)},$$ yielding the desired result.
### Beyond NoLips: inertial Bregman algorithms {#ss:iga}
table \[x=k,y=WC\_FUNC\] [Data/NoLips\_2.dat]{}; table \[x=k,y=WC\_MINGRAD\] [Data/NoLips\_2.dat]{};
table \[x=N,y=theory\] [Data/IGA.dat]{}; table \[x=N,y=pep\_iga\] [Data/IGA.dat]{};
Our approach is not limited to the NoLips algorithm. For instance, we can also solve the performance estimation problem for the inertial Bregman algorithm proposed by Auslender and Teboulle [@Auslender2006], a.k.a. the Improved Interior Gradient Algorithm (IGA). We recall its simplified formulation in Algorithm \[algo:iga\], in the case where there are no affine constraints.
[1.4]{} **Input:** Functions $f,h$, initial point $x_0 \in \interior \dom h$, step size $\lambda$.\
Set $z_0 = x_0$ and $t_0 = 1$. $y_k = (1-\frac{1}{t_k}) x_k + \frac{1}{t_k} z_k$ $
z_{k+1} = \argmin \, \{\la \nabla f(y_k), u-y_k \ra + \frac{1}{t_k \lambda}D_h(u,z_k) \,|\, u \in \reals^n\} $ $x_{k+1} = (1 - \frac{1}{t_k}) x_k + \frac{1}{t_k} z_{k+1}$ $t_{k+1} = (1 + \sqrt{1 + 4t_k^2})/2$.
In the setting where $f$ has $\tilde{L}$-Lipschitz continuous gradients and $h$ is a $\sigma$-strongly convex Legendre function, IGA with step size $\lambda = \sigma/\tilde{L}$ enjoys the following convergence rate [@Auslender2006 Thm. 5.2]: $$\label{eq:iga_bound}
f(x_N) - f_* \leq \frac{4 \tilde{L}}{\sigma N^2} \left(D_h(x_*,x_0) + f(x_0) - f_* \right).$$ Our PEP framework can be also applied to this algorithm, in order to find the smallest value of $\theta(N,\tilde{L},\sigma,\lambda)$ which satisfies $$f(x_N) - f_* \leq \theta(N,\tilde{L},\sigma,\lambda) \left(D_h(x_*,x_0) + f(x_0) - f_* \right)$$ for every instance of IGA with the supplementary assumptions made above. In this case, we use the standard interpolation conditions of Theorem \[thm:interp\_mul\] for $L$-smooth and strongly convex functions. Results are shown in Figure \[fig:pep\_nolips\]. The exact numerical worst-case performance of IGA is slightly below the theoretical bound above, since the proof in [@Auslender2006] makes some approximations.
### IGA in the general $h$-smooth case. {#iga-in-the-general-h-smooth-case. .unnumbered}
We pointed out in Section \[s:setup\] that the setting in which $f$ is $\tilde{L}$-smooth and $h$ is $\sigma$-strongly convex is a particular case of $h$-smoothness with constant $L = \tilde{L}/\sigma$. The natural question that was also raised in [@Teboulle2018 Section 6] is therefore: does IGA converge for the general class $\mathcal{B}_L(C)$ ? Solving the corresponding PEP yields the following results. For Algorithm \[algo:iga\] with the setting that $(f,h) \in \mathcal{B}_L(C)$ and several choices of step size in $(0,1/L]$, the solver states the the value of the corresponding performance estimation problem **is unbounded**, i.e., there does not exist any $\theta$ such that the bound holds for every instance $(f,h) \in \bl$. Of course, this constitutes numerical evidence and not a formal proof. Nonetheless, due to the tightness result of Theorem \[thm:equiv\], there are strong reasons to conjecture that IGA indeed does not converge in the general $h$-smooth setting.
### From worst-case functions for NoLips to a lower bound for general Bregman methods
We briefly explain how, with the PEP methodology, the worst case functions from Section \[ss:lower\_bound\] were discovered.
We described in Section \[sss:nolips\_pep\] how a one-dimensional worst-case instance $(\overf,\overh)$ for NoLips has been discovered from low-rank solutions of . However, this instance may not be difficult enough for a more generic Bregman algorithm that can use abritrary linear combinations of gradients (as in Assumption \[assumption:algorithm\], our definition of the *Bregman gradient algorithm*), and thus cannot be used to prove a general lower bound.
Our objective now is to find worst-case instances that are difficult for **any** Bregman gradient algorithm. A desirable property would be that these instances allow to explore only *one dimension* per oracle call, so that the function *hides information* in the unexplored dimensions. This similar in spirit with the so-called “worst function in the world" of Nesterov [@Nesterov2004]. In order to achieve this goal, we propose to search for functions $f$ for which all gradients $\nabla f(x_i)$ would be orthogonal, guaranteeing that one new dimension is explored at each step. Note that a similar approach has been used in some previous work on PEPs to find lower bounds or optimal methods e.g. in [@Drori2017; @Drori2019a]. This amounts to add some orthogonality constraints to and solve $$\label{eq:pep_nolips_orth}\tag{PEP-orth}
\BA{ll}
\mbox{maximize} & \big(f\left(x_N\right) - f \left(x_*\right)\big) / D_{h}(x_*,x_0)\\
\rule{0pt}{12pt}
\mbox{subject to } &(f,h) \in \mathcal{B}_L(\reals^n), \\
& x_* \mbox{ is a minimizer of $f$}, \\
& x_1,\dots,x_N \mbox{ are generated from $x_0$ by Algorithm \ref{algo:bpg} with step size $\lambda$,}\\
& \la \nabla f(x_i), \nabla f(x_j) \ra = 0 \mbox{ for } i\neq j \in I,
\EA$$ in the variables $f,h,x_0,\dots,x_N,x_*,n$.
In the same spirit as before, we were able to find a dimension-$N$ solution of . This allows us to interpolate the following worst-case pathological instance in dimension $N$: $$\begin{split}
\overline{f}(x) &= \|x-(1,\dots, 1)\|_\infty,\\
\overline{h}(x) &= \overline{f}(x) + \sum_{i=2}^N \max(-x^{(i)},0).
\end{split}$$ Again, these are nonsmooth functions and do not form valid instances of NoLips. However, they can be approached by a sequence of such functions, for instance by applying smoothing with the Moreau enveloppe, and adding a small quadratic term to make $h$ strictly convex. Along with a few tweaks, this is how we found the example that was used to prove the general lower bound for $\mathcal{B}_L$ in Section \[ss:lower\_bound\].
Conclusion {#s:conclusion}
==========
Our paper has two main contributions: proving optimality of NoLips for the general $h$-smooth setting, and developing numerical performance estimation techniques for Bregman gradient algorithms. We presented the performance estimation problem on the basic NoLips algorithm for simplicity, but our approach can be applied to different settings and various algorithms involving Bregman distances. We provided several applications illustrating how the PEP methology is an efficient tool for conjecturing and analyzing the worst-case behavior of Bregman algorithms.
There is a fundamental concept linking the two parts of the paper, which is that of *limiting nonsmooth pathological behavior*. When looking for worst-case guarantees over a class of functions that is open such as the class of differentiable convex functions, the performance estimation problem is a *supremum* and the worst-case maximizing *sequence* might approach some function that is not in this class, e.g. one that is nonsmooth in our case. This idea, observed by analyzing the equivalence between and the nonsmooth relaxation , was used in the proof of the lower bound in Section \[ss:lower\_bound\]. Moreover, the worst-case sequence of functions was directly inspired by examining particular solutions of .
It is clear that additional assumptions on functions $f$ and $h$ are needed in order to prove better bounds or devise faster algorithms than NoLips. If the usual properties of $L$-smoothness and strong convexity are too restrictive and do not hold in many applications, the future challenge is to find weaker assumptions, that define a larger class of functions where improved rates can be obtained. One other possible approach would be to find algorithms that do not fit in Assumption \[assumption:algorithm\], for instance by including second-order oracles of $h$, in the case where $h$ is simple enough.
**Code.** Experiments have been run in MATLAB, using the semidefinite solver MOSEK [@mosek] as well as the modeling toolbox YALMIP [@yalmip]. The support for Bregman methods has been added to the Performance Estimation Toolbox (PESTO, [@Taylor2018]) for which we provide some examples. The code can be downloaded from [<https://github.com/RaduAlexandruDragomir/BregmanPerformanceEstimation>]{}
**Acknowledgements.** The authors would like to thank Dmitrii Ostrovskii for helpful discussions and Edouard Pauwels for useful comments. RD would like to acknowledge support from an AMX fellowship. AT acknowledges support from the European Research Council (grant SEQUOIA 724063). AA is at CNRS & département d’informatique, École normale supérieure, UMR CNRS 8548, 45 rue d’Ulm 75005 Paris, France, INRIA and PSL Research University. AA would like to acknowledge support from the [*ML & Optimisation*]{} joint research initiative with the [*fonds AXA pour la recherche*]{} and Kamet Ventures, as well as a Google focused award. JB acknowledges the support of ANR-3IA ANITI, ANR Chess, Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant numbers FA9550-19-1-7026, FA9550-18-1-0226.
Extension of performance analysis to the case where $C$ is a general closed convex subset of $\reals^n$ {#a:domh}
=======================================================================================================
For simplicity of the presentation, we left out in Section \[s:pep\] the case where the domain $C$ is a proper subset of $\reals^n$. We show in this section that it actually corresponds to the same minimization problem .
Let us formulate the performance estimation problem for Algorithm \[algo:bpg\] in the general case. Recall that we denote $\mathcal{B}_L$ the union of $\mathcal{B}_L(C)$ for all closed convex subsets of $\reals^n$ for every $n \geq 1$. The performance estimation probem writes $$\label{eq:pep_nolips_c}\tag{PEP-C}
\BA{ll}
\mbox{maximize} & \big(f\left(x_N\right) - f \left(x_*\right)\big) / D_{h}(x_*,x_0)\\
\rule{0pt}{12pt}
\mbox{subject to } &(f,h) \in \mathcal{B}_L, \\
& x_* \mbox{ is a minimizer of $f$ on } \overline{\dom h} \mbox{ such that } x \in \dom h, \\
& x_1,\dots,x_N \mbox{ are generated from $x_0$ by Algorithm \ref{algo:bpg} with step size $\lambda$,}\\
\EA$$ in the variables $f,h,x_0,\dots,x_N,x_*,n$. Now, as is a problem that includes in the special case where $C = \reals^n$, its value is larger: $$\mbox{val\eqref{eq:pep_nolips}} \leq \mbox{val\eqref{eq:pep_nolips_c}}$$ We now proceed to show that val is upper bounded by the same relaxation val, which will allow to conclude that the values are equal. We recall that the problem can be written, using interpolation conditions of Corollary \[cor:blnonsmooth\], as $$\tag{$\overline{\text{PEP}}$}
\BA{rcl}
&\mbox{maximize} & f_N - f_* \\
\rule{0pt}{11pt}
&\mbox{subject to} & f_i - f_j - \la g_j, x_i - x_j \ra \geq 0,\\
&& (Lh_i - f_i) - (Lh_j - f_j) - \la Ls_j - g_j,x_i - x_j \ra \geq 0 \quad \mbox{ for } i,j \in I,\\
& & g_* = 0,\\
& & s_{i+1} = s_i - \lambda g_i \quad \mbox{for } i\in \{1\dots N-1\},\\
& & h_* - h_0 - \la s_0, x_* - x_0 \ra = 1,\\
\EA$$ in the variables $n,\{(x_i,f_i,g_i,h_i,s_i)\}_{i \in I}$. We show that every admissible point of can be cast into an admissible point of . This actually amounts to show that, from the point of view of performance estimation, an instance $(f,h) \in \bl(C)$ is actually equivalent to some instance in $\bl(\reals^n)$.
Let $f,h,x_0,\dots,x_N,x_*$ be a feasible point of . We distinguish two cases.
### Case 1: $x_* \in \interior \dom h$. {#case-1-x_-in-interior-dom-h. .unnumbered}
This is the simplest case, as the necessary conditions are the same as in the situation where $C = \reals^n$. Indeed, then we have $x_0,\dots,x_N,x_* \in \interior \dom h$, since $x_0$ is constrained to be in the interior and the next iterates are in $\interior \dom h$ by Assumption \[assumption\_bpg\]. Since $f$ and $h$ are differentiable on $\interior \dom h$, convexity of $f$ and $Lh-f$ imply that the first two constraints of hold for all $i,j \in I$. Finally, $g_* = 0$ follows from the fact that $x_*$ minimizes $f$ and that it lies on the interior of the domain. Hence the discrete representation satisfies the constraints of .
### Case 2: $x_* \in \partial \dom h$ {#case-2-x_-in-partial-dom-h .unnumbered}
In this case, $f$ and $h$ are not necessarily differentiable at $x_*$, but are still differentiable still at $x_0,\dots,x_N$ for the same reasons. But we can still, with a small modification at $x_*$, derive a discrete representation that fits the constraints of and whose objective is the same. Indeed, define
$$\begin{split}
(g_i,f_i,s_i,h_i) &= \left(\nabla f(x_i), f(x_i), \nabla h(x_i), h(x_i)\right) \mbox{ for } i = 0\dots N\\
(g_*,f_*,s_*,h_*) &= \left(0,f(x_*),v,h(x_*)\right)
\end{split}$$
where $v \in \reals^n$ is a vector that will be specified later. Then, for $i \in I$ and $j \in \{0\dots N\}$, convexity of $f$ and $Lh-f$ imply that the constraints $$\begin{split}
f_i - f_j - \la g_j, x_i- x_j \ra \geq 0\\
(Lh_i-f_i) - (Lh_j-f_j) - \la Ls_j - g_j, x_i- x_j \ra \geq 0
\end{split}$$ hold. It remains to verify them for $i \in \{0 \dots N\}$ and $j = *$. The first one holds because $x_*$ minimizes $f$ on $\dom h$, so with $g_* = 0$ we have $f_i - f_* \geq 0$. We now show that the second one is satisfied, i.e. that we can choose $v \in \reals^n$ so that $$(Lh_i-f_i) - (Lh_*-f_*) - \la Lv, x_i- x_* \ra \geq 0 \quad \forall i \in \{0\dots N\}$$
To this extend, we use the fact that $x_* \in \partial \dom h$ and that $x_i \in \interior \dom h$ for $i = 0 \dots N$. This means that $\{x_*\} \cap \interior \dom h = \emptyset$, and therefore by the hyperplane separation theorem [@Rockafellar2008 Thm 11.3], there exists a hyperplane that separates the convex sets $\{x_*\}$ and $\interior \dom h$ *properly*, meaning that there exists a vector $u \in \reals^n$ such that $$\la x_i - x_*, u \ra < 0 \,\,\, \forall i \in \{0 \dots N\}$$ Denote now $$\begin{split} \alpha &= \min_{i =0\dots N}\, (L h_i - f_i) - (Lh_* - f_*)\\
\beta &= \min_{i = 0\dots N} - \la x_i - x_*, u \ra > 0\end{split}$$
where $\beta > 0$ because of the separation. Choose now $s_* = v$ as $v = \frac{|\alpha|}{L \beta} u$. Then we have $$\begin{split}
(Lh_i - f_i) - (Lh_* - f_*) - \la L s_*,x_i - x_* \ra &\geq \alpha + L \frac{|\alpha|}{L \beta} \beta \\
& \geq \alpha + |\alpha| \\& \geq 0.
\end{split}$$ This achieves to show that we built an instance $\{(x_i,g_i,f_i,h_i,s_i)\}_{i \in I}$ that is admissible for .
To conclude, we proved that in both cases, an admissible point of can be turned into an admissible point of with the same objective value. Hence we have $$\mbox{val\eqref{eq:pep_nolips_c}} \leq \mbox{val\eqref{eq:pep_over_sdp}} .$$ Now, recalling that $\mbox{val\eqref{eq:pep_nolips}} \leq \mbox{val\eqref{eq:pep_nolips_c}}$ and that $\mbox{val\eqref{eq:pep_over_sdp}} = \mbox{val\eqref{eq:pep_nolips}}$ by Theorem \[thm:equiv\], we get $$\mbox{val\eqref{eq:pep_nolips_c}} = \mbox{val\eqref{eq:pep_nolips}}.$$ In other words, solving the performance estimation problem for functions with any closed convex domain is equivalent to solving the performance estimation problem restricted to functions that have full domain.
|
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abstract: 'We investigate higher order effects in electromagnetic excitation of neutron halo nuclei using a simple and realistic zero range model for the neutron-core interaction. In the sudden (or Glauber) approximation all orders in the target-core electromagnetic interaction are taken into account. Small deviations from the sudden approximation are readily calculated. We obtain very simple analytical results for the next to leading order effects, which have a simple physical interpretation. For intermediate energy electromagnetic dissociation, higher order effects are generally small. We apply our model to Coulomb dissocation of $^{19}$C at 67 A$\cdot$MeV. The analytical results are compared to numerical results from the integration of the time dependent Schrödinger equation. Good agreement is obtained. We conclude that higher order electromagnetic effects are well under control.'
address:
- |
National Superconducting Cyclotron Laboratory, Michigan State University,\
East Lansing, Michigan 48824-1321, USA
- ' Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany '
author:
- 'S. Typel'
- 'G. Baur'
title: Higher Order Effects in Electromagnetic Dissociation of Neutron Halo Nuclei
---
\#1[[$\backslash$\#1]{}]{}
Introduction
============
Electromagnetic excitation of high energy radioactive beams is a powerful method to study electromagnetic properties of loosely bound neutron rich nuclei. E.g., the low lying E1-strengths of one-neutron halo nuclei like ${}^{11}$Be and $^{19}$C have been studied in this way [@Ann; @Nak02; @Nak03]. In a similar way, two-neutron halo nuclei like $^{6}$He and $^{11}$Li were studied. Such experiments are usually analysed theoretically in first order electromagnetic perturbation theory or the equivalent photon method. In this way, the multipole (especially dipole) strength distribution is obtained. Such an analysis depends on the dominance of first order excitations. Various methods have been developed in order to consider deviations from first order perturbation theory with the usual multipole expansion of the interaction. However, a consistent picture of the importance of these approximations has still not emerged. By “higher order effects” we mean only electromagnetically induced effects on the relative momentum of the fragments. They can be studied in the semiclassical approximation. In the widest sense all effects which give rise to a deviation from the result of the traditional semiclassical first order perturbative calculation of Coulomb breakup can be summarized under this expression. In the perturbative approach higher order effects can be described as the exchange of more than one photon between the target and the projectile system. “Postacceleration” is also a higher order effect. In a classical picture it can be understood as a different acceleration of the fragments in the Coulomb field of the target which will change both the c.m. momentum and the relative momentum of the particles in the final state. In our calculations we will not treat quantal effects like diffraction or contributions to the breakup from the nuclear interaction.
There are mainly two different approaches for the investigation of higher order effects. In the semiclassical description the projecile moves on a classical trajectory (which is usually well justified) and experiences a time-dependent interaction from the target. Only the excitation of the projectile is treated quantally. In contrast to that, the total system of target, projectile and fragments, respectively, can be described by suitable wave functions in a fully quantum mechanical approach. Each of these approaches has its merits, but, at the same time, can limit the study of certain higher order effects or make it difficult to extract them by a comparison to a suitable first order calculation.
The breakup of the prototype of a loosely bound nucleus, the deuteron, has for a long time been studied in the post-form DWBA theory. Later, it has also been applied to neutron halo (core + neutron) nuclei like $^{11}$Be. In this approach, the Coulomb interaction between the target and the core is taken into account to all orders. This is done by using full Coulomb wave functions in the initial and final state. For a recent review with further references see [@TMUPROC]. A so called adiabatic breakup theory has recently been developed in [@adia]. This model is related to the post-form DWBA. It leads to a very similar formula, however, the physical interpretation is somewhat different. Without entering into the differences of the two approaches, it is clear that in these theories higher order effects are automatically included to all orders. It is therefore very interesting to note that Tostevin [@tos] claims to have found substantial higher order effects in the Coulomb breakup of $^{19}$C [@Nak03]. He compared his results from the adiabatic approach to the one using semiclassical first order theory. Both calculations have very similar relative energy spectra, but they differ by about 35 percent in absolute magnitude.
It is the purpose of this paper to investigate higher order effects in the electromagnetic excitation of neutron halo nuclei by comparing lowest order and higher order approximations [*within*]{} the same model. This is expected to give more reliable statements about the importance of these effects than the comparison of higher order calculations in [*one*]{} theory with first order calculations in [*another*]{} theory, where, e.g., the finite range effects are treated in another way or the semiclassical approximation is not applied. We will limit ourselves to the semiclassical description considering only the Coulomb interaction and will not investigate nuclear induced effects. In our approach we use a classical trajectory to describe the relative motion between the target and the projectile. It should be kept in mind that there is some ambiguity in the definition of this trajectory. The energy loss should be small compared to the total kinetic energy of the projectile and some averaging procedure can be used. We assume that the c.m. of the projectile moves on the classical trajectory (straight line or Rutherford). It has been argued that the electromagnetic interaction of the target only affects the charged core of the projectile; therefore the c.m. of the core has been used for describing the classical motion. However, for intermediate energy this effect was found to be quite small in numerical calculations in [@melbaye]. Actually, the result of a first order E1 calculation does not depend on this choice, since the dipole moment of the system does not change. There is only a change of the quadrupole moment but this has small effects since the E2 contribution to the breakup is rather small (see below). Since the c.m. trajectory is fixed, only higher order effects in the relative motion of the fragments can be handled in the semiclassical approach. Since the total momentum of the fragments is much larger than the relative momentum between them, higher order effects have a much larger effect on the relative momentum.
To some approximation the nuclear structure of neutron halo nuclei can be described by rather simple wave functions. Using these wave functions the reaction mechanism can be studied in a very transparent way and analytical results are obtained. In a later stage more refined descriptions of the nuclei can be introduced. We recall some results from [@tyba] and apply the model to the electromagnetic breakup of $^{19}$C in comparison to more accurate descriptions. In Ch. 2 the theoretical framework is given; results and comparison to experimental results [@Nak03] are presented in Ch. 3. Conclusions are given in Ch. 4.
Theoretical framework
=====================
We follow very closely the approach of [@tyba], see also [@babeka]. In this straight-line semiclassical model a projectile with charge $+Ze$ impinges on a neutron+core ($n+c$) system with impact parameter $b$ and velocity $v$. The ground state wave function of the bound $n+c$ system is given by a simple Yukawa type wave function $$\label{wfbind}
\phi =\sqrt{\frac{\eta}{2\pi}} \: \frac{\exp(-\eta r)}{r}$$ where the parameter $\eta$ is related to the binding energy $E_0$ by $E_
0= \frac{\hbar^2 \eta^2}{2 m}$ with the reduced mass $m=\frac{m_{n}m_{c}}{m_{n}+m_{c}}$ of the system. The final continuum state is given by $$\phi_q^{(-)} = \exp(i \vec{q}\cdot\vec{r})- \frac{1}{\eta-iq}
\: \frac{\exp(-iqr)}{r}$$ where the wave number $q$ is related to the relative energy by $E_{\rm rel}=\frac{\hbar^2 q^2}{2 m}$. With these wave functions the breakup probability can be calculated analytically in the sudden approximation (corresponding to the Glauber or frozen nucleus approximation) of semiclassical Coulomb excitation theory including [*all*]{} orders in the exchange of photons between the target and the projectile. But the time evolution of the sytem during the excitation is neglected and only E1 transitions are taken into account. The first approximation corresponds to an adiabaticity parameter $\xi$ of zero. This quantity is the ratio of the collision time to the nuclear interaction time and it is given by $\xi=\frac{(E_0+E_{rel})b}{\hbar v}$. The multipole response of the system is characterized by effective charges $Z_{\rm eff}^{(\lambda)}=Z_c\left(\frac{m_n}{m_n+m_c}\right)^{\lambda}$. They become very small for higher multipolarities due to the small ratio $\frac{m_n}{m_n+m_c}$. From a perturbation expansion of the excitation amplitude it can be shown that also the second order E1-E1-amplitude is much larger than the first order E2 amplitude. The ratio is given by the Coulomb (or Sommerfeld) parameter $\frac{Z Z_{c} e^{2}}{\hbar v}$ which is much larger than one for high charge numbers $Z$. Therefore we can safely neglect E2 excitation in the following. (This is, e.g., qualitatively different for p+core systems like $^8$B $\rightarrow$ $^{7}$Be + p with much larger E2 effective charges.) This can be considered as a justification of the model of [@tyba].
We expand the analytical results for the excitation probability of [@tyba] for $\xi=0$ up to second order in E1 excitation or equivalently in the the characteristic strength parameter which is given by $$\chi=\frac{2ZZ_{\rm eff}^{(1)}e^{2}}{v b \hbar k}$$ where $k=\sqrt{\eta^2+q^2}$. In leading order (LO), the sudden limit of the first order result of [@tyba] is obtained. Deviation for finite values of $\xi$ can be calculated according to [@tyba]. From eq. (12) of [@tyba] one sees that the $\xi$ dependence of the amplitudes is given by $\xi K_{1}(\xi)$ or $\xi K_{0}(\xi)$ with modified Bessel functions, respectively. For $\xi =0$ this factor (for $K_{1}$) is 1 and drops to zero exponentially for $\xi \gg 1$. The $\xi$-correction in the next-to-leading order (NLO) goes essentially like the square of this, so we can only overestimate the higher order effects in the present procedure.
Instead of using the strength parameter $\chi$ we define in the following the slightly different parameters $$y= \chi \frac{k}{\eta}$$ (independent of $q$) and $$x=\frac{q}{\eta} = \sqrt{\frac{E_{\rm rel}}{E_{0}}} \: .$$ After angular integration over the relative momentum between the fragments the LO breakup probability is found to be (see eq. 37 of [@tyba]) $$\label{dpdqlo}
\frac{d P_{LO}}{d q} = \frac{16}{3\pi \eta} y^2
\frac{x^4}{(1+x^2)^4} \: .$$ (Note that for a correct normalization of the breakup probability the results of [@tyba] have to be devided by $(2\pi)^{3}$.) The NLO contribution is proportional to $y^{4}$ and contains a piece from the second order E1 amplitude and a piece from the interference of first and third order amplitudes. Again, in terms of the variables $\eta$, $x$, and $y$ one obtains $$\label{dpdqnlo}
\frac{d P_{NLO}}{d q} = \frac{16}{3\pi \eta} y^4
\frac{x^2(5-55x^2+28x^4)}{15 (1+x^2)^6}
\: .$$ The LO-expression is directly proportional to the B(E1)-strength with its characteristic shape in the zero range model. The NLO-contribution will introduce a change of that shape. It is weighted most in collisions with the smallest possible impact parameters $b$ and can easily be evaluated. For $ 0.309 < x < 1.367$ the NLO contribution becomes negative with the largest reduction at a relative energy close to the binding energy. This is essentially due to the interference of first and third order amplitudes. The second order E1-E1 contribution is positive definite. From [@tyba] we find $$\frac{d P_{E1-E1}}{d q}=\frac{16}{3 \pi \eta} y^4
\frac{x^2 (5+5x^2+16x^4)}{15(1+x^2)^6} \: .$$ A reduction of the cross section at small relative energies is only obtained if third-order contributions in the breakup amplitude are considered, either in a perturbative treatment (cf. figs. 2 - 4 in [@tyba2]) or a full dynamical calculation (cf. figs. 5 + 7 in [@EBB]). In our analytical results we can directly see the dependence of higher order effects on the impact parameter $b$, the projectile velocity $v$ and the binding energy $E_{0}$ charactrized by $\eta$. For larger impact parameters the first order E1 contribution will dominate more and more ($y \propto b^{-1}$). Perhaps, experimental accuracy will not be high enough to see such a change of the shape of the breakup bump. The scaling variable y also displays very clearly the dependence on the binding energy, characterized by $\eta$, and the charge number $Z$. Since $\frac{Z_c}{m_c}$ is about constant for all nuclei, the breakup probabilities for heavier nuclei (like the r-process nuclei) are expected to be of the same order of magnitude as the light ones (like $^{11}$Be or $^{19}$C). This will be an interesting field for future RIA facilities, where intensive beams of medium-energy neutron-rich nuclei will become available. This is of special interest for the r-process [@erice].
The breakup cross section can be obtained by multiplying the differential breakup probability with the Rutherford scattering cross section $\frac{d\sigma_{R}}{d\Omega}$ and the density of final states. It is given for the LO approximation by $$\label{dsdo}
\frac{d^{2}\sigma_{LO}}{dE_{\rm rel}d\Omega} =
\frac{d\sigma_{R}}{d\Omega} \:
\frac{dP_{LO}}{dq} \:
\frac{m}{\hbar^{2}q}$$ and similarly for the NLO approximation. In order to have a quick estimate of higher order effects in the total breakup cross section we can integrate over the scattering angle and the breakup relative energy $$\sigma = \int dE_{\rm rel} \: \int d\Omega \:
\frac{d^{2}\sigma}{dE_{\rm rel}d\Omega}
= 2\pi \int dq \: \int\limits_{b_{\rm min}}^{b_{\rm max}} db \: b \:
\frac{dP_{LO}}{dq}$$ where we have introduced minimum and maximum impact parameters $b_{\rm min}$ and $b_{\rm max}$, respectively. We use $b_{\rm max}=\frac{\hbar v}{E_{0}+E_{\rm rel}}$ corresponding to a cutoff at an adiabaticity parameter of $\xi=1$. The integration over the impact parameter $b$ is now easily performed. Introducing the effective strength parameter $$\chi_{\rm eff} = y \eta b = \chi k b = \frac{2ZZ_{\rm eff}^{(1)}e^{2}}{\hbar v}$$ and the minimal adiabaticity parameter $$\xi_{\rm min} = \frac{E_{0}b_{\rm min}}{\hbar v}$$ we finally obtain $$\label{slo}
\sigma_{LO} = \frac{\pi}{18}
\left(\frac{\chi_{\rm eff}}{\eta}\right)^{2}
\left[ 1 - 6 \ln (4\: \xi_{\rm min})\right]$$ and $$\label{snlo}
\sigma_{NLO} = - \frac{\pi}{18} \: b_{\rm min}^{-2}
\left(\frac{\chi_{\rm eff}}{\eta}\right)^{4}
\left[ \frac{23}{40}+18 \: \xi_{\rm min}^{2}\right]
\: ,$$ i.e., a reduction of the first order result. The total cross sections can only be a rough guide because of the simple treatment of the cutoff. Modifications due to a more precise treatment of the $\xi$-dependence have usually to be introduced (see below eq. (\[phixi\])). However, the ratio gives a reasonable approximation to the higher order effects. It is a simple function depending on the characteristic parameters of the excited system and the experimental conditions.
Application to the Coulomb breakup of $^{19}$C
==============================================
In a recent experiment at RIKEN the breakup of $^{19}$C into $^{18}$C and a neutron scattered on a Pb target with a beam energy of 67 A$\cdot$MeV was studied and the binding energy of the neutron was determined to be 0.53 MeV [@Nak03]. We apply our model to this case since the high beam energy together with the simple structure and small binding energy of the neutron is favourable for a comparison. Another example would be $^{11}$Be where essentially the same considerations apply.
Besides our simple analytical model we also present results from usual first order semiclassical calculations and higher order calculation where the full time dependent Schrödinger equation is solved for the $^{19}$C system which is perturbed by the time dependent Coulomb field of the Pb nucleus. Here we have used the methods described in [@typzfn]. The neutron in the bound state of $^{19}$C was assumed to be in a $2s_{1/2}$ state as deduced by Nakamura et al. [@Nak03]. The wave function was calculated assuming a Woods-Saxon potential of radius $r=3.3$ fm and diffuseness parameter $a=0.65$ fm. The depth is adjusted to $V=-39.77$ MeV in order to get the experimentally extracted binding energy of 0.53 MeV [@Nak03]. As in the analytical model we use plane waves for the scattering wave functions in the final states for $l>0$. The wave functions are represented on a grid with exponential increasing mesh size similar to [@melbaye] with 400 points up to a maximum radius of 900 fm.
In Fig. 1 we show the double differential cross section as a function of the relative energy for three scattering angles. We have chosen $0.3^{\circ}$, $0.9^{\circ}$, and $2.7^{\circ}$, which corresponds to impact parameters of $109.7$ fm, $36.6$ fm, and $12.2$ fm, respectively. In order to compare the cross section in our model with finite-$\xi$ results of the first order semiclassical calculation we multiply the analytical cross section given in eq. (\[dsdo\]) with the shape function $$\label{phixi}
\phi(\xi) = \xi^{2} \left[ K_{0}^{2}(\xi) + K_{1}^{2}(\xi) \right]$$ of the photon spectrum and a normalization factor $N$. The function $\phi(\xi)$ gives the correct dependence on the adiabaticity parameter in first order. We have $\phi(0)=1$ and it drops to zero rapidly for $\xi > 1$. The factor $N$ accounts for finite range effects. The ground state wave function in the analytical model is a $1s_{\frac{1}{2}}$ state which has a different asymptotic normalization but the same slope as compared to the corresponding wave function from the Woods-Saxon potential. The results of the numerical calculation (dotted and dot-dashed lines) agrees very well with the $\xi$-corrected cross section in the analytical model (solid and dashed lines) for $N=2.73$. The slope of the wave function is determined by the binding energy (see eq. (\[wfbind\])) which is the same in both models. There are noticeable differences between the analytical model and the dynamical calculation only for large relative energies and scattering angles. The first order E2 contribution (multiplied with 1000) is also shown in Fig. 1 (long-dashed line). It is at least three orders of magnitudes smaller as compared to the first order E1 excitation cross section and can safely be neglected. Furthermore we observe that the cross section decreases strongly with increasing scattering angle. For small scattering angles results of first and higher order calculations are almost identical. With increasing scattering angle we notice a reduction of the cross section for small relative energies due to higher order electromagnetic effects.
In Fig. 2 we compare the ratio of higher order (i.e., all orders in the dynamical calculation or LO+NLO in the analyical model, respectively) to first order cross sections depending on the relative energy for the same scattering angles as in Fig. 1. The solid line gives the result of the analytical model. (Notice that the ratio is independent of $\phi(\xi)$ and $N$.) The dependence of the ratio on the relative energy again agrees well with the ratio in the full semiclassical model (dotted line). At small relative energies there is a reduction of the cross section (except for energies close to zero) whereas at higher relative energies we find a small increase. This behaviour can be directly understood by inspecting eqs. (\[dpdqlo\]) and (\[dpdqnlo\]). Higher order effects are largest for large scattering angles corresponding to impact parameters close to grazing scattering. A look at the breakup probabilities (\[dpdqlo\]) and (\[dpdqnlo\]) shows that higher order effects increase essentially with $b^{-2}$. The discrepancy of the two models at higher relative energy, where the exact form of the wave function for small radii in the range of the nuclear potential becomes important, is not very essential because the absolute cross sections are very small. In contrast, at small relative energies, the models agree very well since the main contribution to the matrixelements is determined by the asymptotic form of the wave function.
Integrating the double differential cross sections over scattering angles between $0^{\circ}$ and $3^{\circ}$ ($b_{\rm min}\approx 11$ fm) leads to the energy dependent cross sections in Fig. 3(a). Again we observe that both the first order and higher order calculations in the $\xi$-corrected analytical model and the full semiclassical model agree very well in the peak of the distribution. Here we find a reduction of the cross section of at most 10% at small relative energies as shown in Fig. 3(b). The spectral shape is not severely distorted. Higher order Coulomb effects cannot explain the difference of first order theoretical calculations and the experimental results with respect to the absolute magnitude and the shape of the experimental data [@Nak03]. Although they lead to a decrease of the cross section at small energies (apart from the region just above threshold) and an increase at higher energies the slope of the theoretical results is much steeper as compared to the experiment. In contrast, the position of the peak is well described since it is determined by the binding energy of the neutron in $^{19}$C.
Integrating over relative energies the effects of higher order are washed out and become even smaller in the total cross section $\sigma$. We obtain 1.44 b (1.39 b) in the first order (dynamical) semiclassical calculation and 1.49 b (1.44 b) in the LO (NL+NLO) analytical model with $\xi$-correction, respectively, for energies up to 3 MeV. Comparing to the experimental value of $\sigma=(1.34\pm 0.12)$ b [@Nak03] one has to take into account our simple nuclear model. In reality the ground state of ${}^{19}$C has a more complicated structure than a $2s_{1/2}$ single particle state. Multiplying the cross sections of our calculation with a spectroscopic factor of $0.67$ as given by Nakamura et al. [@Nak03] the total cross section would be smaller than in the experiment but the peak region in Fig. 3(a) would be well described. At higher relative energies nuclear contributions could be present in the experimental data, increasing the total cross section again. A possible Coulomb-nuclear interference effects could also lead to a change of shape of the cross section. Furthermore, the experimental data could contain contributions from final states with an excitation of the core ${}^{18}$C. Our results correspond to a reduction of the total cross section by higher order effects of $3.3\%$ in the semiclassical model and of $3.2\%$ in the $\xi$-corrected analytical model. From equations (\[slo\]) and (\[snlo\]) we predict a 2.9% reduction which is close to the more refined models. From the comparison we conclude that our simple analytical model with finite-$\xi$ correction is quite realistic in the prediction of higher order effects and gives a reliable estimate of the reduction of the total cross section. The smaller value of the reduction obtained in the simpler fully analytical model can be well understood. Without taking the adiabatic suppression correctly into account contributions to the total cross section from higher relative energies and larger impact parameters, where higher order effects are smaller, are not sufficiently reduced and lead to an underestimate. However, higher order effects in the triple differential cross section in the peak of the excitation function are well described by the simple analytical expressions.
Our results are in contrast to [@tos] where a much bigger effect of the order of 30 to 40 percent was found by comparing different models. It is difficult to assess how much of the reduction is caused essentially by higher order electromagnetic effects or by differences in these models. In both cases, the total contribution of higher orders to the cross section is negative.
Finally, let us make some remarks about post-acceleration. A semiclassical model might suggest that the parallel momentum distribution of the core is shifted towards larger values due to an “extra Coulomb push” see, e.g., [@babeka]. However, this turns out to be wrong. In the sudden approximation, the core-neutron-binding is negligible and also on its way towards the target, the core alone (and [*not*]{} the bound core-neutron system) feels the Coulomb deceleration. Formally, this is easily seen: In the sudden approximation, the momentum transfer points exactly to the direction perpendicular to the trajectory, the excitation amplitude depends only on $\vec{q} \cdot \Delta \vec{p}$ (cf. [@tyba]). This is symmetric with respect to the plane perpendicular to the beam direction. Corrections of this simple result due to small values of $\xi$ were studied in [@tyba]. They were found to depend only on the phase shift of the neutron s-wave. This phase shift is given in the analytical model by $\delta_{0} = - \arctan \frac{q}{\eta}$. It is a rather delicate quantal interference effect and even has the opposite sign to what one would have thought “intuitively”. Large values of $\xi$ correspond to large values $b$ where the strength parameter is small. Therefore, higher order effects are not so important. Indeed, in Ref. [@promptor] no effects of post-acceleration were found for the $^{11}$Be system.
Conclusions
===========
We have studied the basic example for the Coulomb dissociation of a neutron halo nucleus. From the simple zero-range wave function of a loosely bound system it becomes directly obvious that the low lying E1 strength is an immediate consequence of the halo structure. It is probably the most beautiful manifestation of the halo nature. Higher order effects can be described by analytical formulas. This allows a very transparent discussion of the effects. Our results can be easily applied to all neutron halo Coulomb dissociation experiments. They are a useful guide for the much more elaborate numerical solutions of the time-dependent Schrödinger equation. Our simple considerations are corroborated by these more sophisticated approaches. We conclude that higher order electromagnetic effects are not a significant problem in medium-energy Coulomb dissociation experiments and can be kept under control.
The authors would like to thank R. Shyam for useful discussions and B. A. Brown and P. G. Hansen for helpful remarks on the manuscript. We are grateful to T. Nakamura for providing us with the experimental data. Support for this work was provided from US National Science Foundation grant numbers PHY-0070911 and PHY-9528844.
R. Anne et al., Phys. Lett. B [**304**]{}, 55 (1993), Nucl. Phys. [**A575**]{}, 125 (1994) T. Nakamura et al., Phys. Lett. B [**331**]{}, 296 (1994) T. Nakamura et al., Phys. Rev. Lett. [**83**]{}, 1112 (1999) G. Baur, S. Typel, H. Wolter, [*Spins in Nuclear and Hadronic Reactions*]{}, Proceedings of the RCNP-TMU Symposium, edited by H. Yabu, T. Suzuki and H. Toki, World Scientific, Singapore (2000), 119 J. A. Tostevin, S. Rugmai and R. C. Johnson, Phys. Rev. C [**57**]{}, 3225 (1998) J. A. Tostevin, [*Spectroscopy of halo nuclei from reaction measurements*]{}, Proceedings of the Second International Conference on Fission and Properties of Neutron-Rich Nuclei, St. Andrews, Scotland, June 28 - July 2, 1999, edited by J. H. Hamilton, W. R. Phillips and H. K. Carter, World Scientific, Singapore (2000), 429 V. S. Melezhik and D. Baye, Phys. Rev. C [**59**]{}, 3232 (1999) S. Typel and G. Baur, Nucl. Phys. [**A573**]{}, 486 (1994) G. Baur, C. A. Bertulani and D. M. Kalassa, Nucl. Phys. [**A550**]{}, 527 (1992) S. Typel, G. Baur and H. H. Wolter, Nucl. Phys. [**A613**]{}, 147 (1997) H. Esbensen, G. F. Bertsch, C. A. Bertulani, Nucl.Phys. [**A581**]{}, 107 (1995) G. Baur, K. Hencken, D. Trautmann, S. Typel and H. H. Wolter, [*Electromagnetic Dissociation as a Tool for Nuclear Structure and Astrophysics*]{}, nucl-th/0011061, to be published in the Proceedings of the International Workshop of Nuclear Physics, 22nd course, Radioactive Beams for Nuclear and Astrophysics, Erice, Sicily, September 15th - 24th, 2000, Progress in Particle and Nuclear Physics (Elsevier Science) S. Typel and H. H. Wolter, Z. Naturforsch. [**54a**]{}, 63 (1999) J. E. Bush et al., Phys. Rev. Lett. [**81**]{}, 61 (1998)
|
---
abstract: 'Modern PC workstations often provide more CPU power than required for most control applications. On the other hand, the screen space is always in short supply. One possible solution is to use more PCs, but in fact we need only more screens, not more keyboards, mice etc. PC architecture allows using more than one videocard, and X Window protocol is also aware that there can be more than one screen. But until release of XFree86 version 4 there was no freely available server capable of driving multiple “heads”. We have been using multiheaded workstations under XFree86 in the VEPP-5 control room since early 2000 (currently 4 4-headed PCs plus several dual-headed). The “Xinerama” mode (one-large-screen) is better suited for accelerator control system than “several separate screens”. When moving to this configuration we’ve encountered a number of, mainly human-related, problems, some of which required modifications to X server Additionaly, the “style” of performing control has slightly changed.'
author:
- |
D.Yu.Bolkhovityanov, R.G.Gromov, I.L.Pivovarov, A.A.Starostenko\
The Budker Institute of Nuclear Physics, Novosibirsk, Russia
title: |
\
EXPERIENCE OF USING MULTIMONITOR WORKSTATIONS UNDER XFREE86 4.X IN VEPP-5 CONTROL ROOM
---
NEED IN MORE SCREEN SPACE
=========================
Historically automation at BINP is based on CAMAC. A home-made Odrenok [@odrenok] machines were used as both crate controllers and as the main computational power. The information was displayed via CAMAC-based display controllers, which gave color 256$\times$256 pixel picture. That allowed sufficient display space for most tasks.
On the new VEPP-5 facility the computation and high-level control was moved from crate controllers to Intel-based workstations. So, the aging CAMAC display hardware wasn’t an option.
Modern video cards and monitors have resolutions large enough to simply put contents of all 256$\times$256 displays on them.
This approach was taken by the VEPP-4 team, which exploits a large number of legacy programs using CAMAC display controllers. They made an emulation library, which redirects graphic output of such programs to X11 windows. But there was no reason for VEPP-5 to go this way.
POSSIBILITY
===========
PCI bus allows to have multiple videocards in one computer. One card is treated as primary (the one on which the boot screen appears), and others are inactive until a multihead-aware system is loaded. AGP slot looks like just one more PCI slot.
From the very beginning X theoretically allowed to use several screens on one host. These screens are referred to as [*hostname:N.0*]{}, [*hostname:N.1*]{}, etc., where [*N*]{} after semicolon is a display number (typically [*0*]{}) and [*0*]{}, [*1*]{}, etc. after dot is a screen number.
But in practice, XFree86 up to version 3.x inclusive didn’t support multihead. That capability appeared in long-awaited version 4.0, released in early 2000.
TRADITIONAL MULTIHEAD VS XINERAMA
=================================
The traditional X multihead presents each screen separately, so that when a window is created, it is placed on one of these screens and cannot span screens or be moved from one screen to another (see Fig.\[f:xinerama\],a).
![Traditional and Xinerama multihead[]{data-label="f:xinerama"}](xinerama.eps){width="\linewidth"}
On the other hand, Xinerama makes multiple physical screens behave as a single screen, transparently to the clients (see Fig.\[f:xinerama\],b). So, the windows can be freely moved between screens.
A good source of information about Xinerama is [@XINERAMAHOWTO].
Since the situation on the screen of control computer isn’t static (there’s often a need to group windows in different ways, to move more important windows to “more visible” screen), Xinerama is much better suited for use in a control room than traditional multihead.
XINERAMA PROBLEMS
=================
Technical problems
------------------
When joining screens, Xinerama leaves only depths which are common to all screens. So, it is impossible to join 16-bit screen with a 24-bit one. Additionally, the 24+8 “overlay” feature of Matrox cards is lost, since 2nd head doesn’t support it.
But the main inconvenience is that since all screens look like a single one to all clients, the window managers happily place windows between screens, maximize them on all screens, etc.
We use FVWM [@FVWM] in the VEPP-5 control room, so we invested some time to its initial xineramification, which was completed by FVWM team (now Xinerama support in FVWM is probably the most complete and configurable among all WMs). Currently, most WMs are Xinerama-aware, but some toolkits still aren’t.[^1]
![Possible Xinerama layouts[]{data-label="f:layouts"}](layouts.eps){width="\linewidth"}
And there is one more exotic problem. Technically, Xinerama makes a single large desktop with a size of a bounding rectangle of all screens. And screens themselves function as viewports to the desktop. So, Xinerama allows to place heads in many different ways (see Fig.\[f:layouts\]).
1. Heads can form a regular grid – that’s the most common case.
2. They can overlap (a so-called “clone/zoom” mode). This is used very rarely, since the position of “zoom” screen is fixed and can’t be moved e.g. following the pointer.
3. Can be disjoint[^2].
4. Or the grid can be incomplete.
In the two latter cases, there are “black holes” on the desktop, which aren’t visible on any monitor and which can’t be reached with mouse. The consequences are worst in the last case – even complete windows can disappear in the black hole.
Additionally, there are still some problems with software which either requires a direct access to framebuffer, or uses a fullscreen mode (various video capturing and movie playing programs).[^3] But, thanks to XVideo extension, these problems became very rare.
Human Problems
--------------
Some of our software developers are greedy: when they see so much display space, they say: “Hey, let’s move this and this to another screen, and my program will just fill this screen”. The common rule is “some programs tend to grow to occupy all screen space”. So, appetittes of some people need reduction.
Another problem is that mouse pointer often gets lost on large screen space. Finally a patch for X server was developed [@visxcursor], which allows to 1) doublesize the pointer and/or 2) change the default colors from black&white to something more visible. We use red doublesized pointers, which provides good visibility.
HOW MANY HEADS TO USE
=====================
The most common multihead layout in the world is two heads: side by side horizontally, or one above another (if 2nd head is used rarely).
Three heads are hard to use: the layout will be either as on Fig\[f:layouts\],d, which is inconvenient, or lined up. In the latter case it takes too much time to move the pointer between the first and the last screens. When we [*had*]{} to use three heads, we put monitors in the shape of “r” but X layout was “three heads vertically”. That setup was extremely confusing for operators.
Four heads give the best balance between “as much screen space as possible” principle and convenience of use. When used in a 2$\times$2 grid, as on Fig.\[f:layouts\],a, there are no black holes and the distance between heads is small.
HARDWARE
========
Criteria for selecting video cards
----------------------------------
First, that hardware should be multihead-capable (e.g., 3Dfx cards are known not to work in multihead mode under XFree86 at all). Second, it must have good support in XFree86 and work very stable. Third, it should produce an excellent picture and have good 2D performance (3D isn’t important). Fourth, video hardware should occupy as little PCI slots as possible.
Solution we use
---------------
There were 4 main manufacturers: ATi, Matrox, nVidia and S3 (the latter is almost dead now). We chose Matrox, because 1) it satisfied all our criteria; 2) we already had very positive experience with their products.
As to other two brands: nVidia doesn’t open specifications of their cards, so that XFree86 driver is very lacking, but instead nVidia provides binary-only driver; ATi cards are not-so-good and there are myriads of subversions, which affects stability of the driver.
We use MilleniumIIs as PCI cards, but any PCI card with 4M or more memory[^4] would do (Millenium, G100, G200).
Multiheaded videocards
----------------------
![Hardware options for 4 heads[]{data-label="f:hw"}](4heads.eps){width="\linewidth"}
One more advantage of Matrox cards is that since 1999 they have two heads on one card (G400DH, G450, G550). So, to have 4 heads, an AGP G450 plus two PCI MilleniumIIs were enough (see Fig. \[f:hw\],b), thus using only two PCI slots (which are always in deficite in control machines).
And Matrox produces a PCI version of G450, thus allowing to use only one PCI slot in addition to AGP (see Fig. \[f:hw\],c). Unfortunately, when running as non-primary card, the G450 (either AGP or PCI) requires an additional driver module, which is available as binary-only from Matrox (so-called HAL module [@HAL]). But we hope the native XFree86 support will become better soon, thus making HAL redundant.
Currently two of our 4-headed PCs are equipped with G450AGP+2$\times$MilleniumII, and two are G450AGP+G450PCI.
Theoretically there exists even a better choice – G200MMS, which supports 4 heads on one card (see Fig. \[f:hw\],d), but it exists in PCI version only, so if we need 4 heads total, it occupies the same one PCI slot as G400AGP+G450PCI. Additionally, G200MMS is almost impossible to find in Russia.
Motherboards
------------
We chose ASUS P3B-F (Intel 440BX chipset), which has 1 AGP slot, 6 PCI and one ISA (one position is shared, so you get either 6 PCI and 0 ISA or 5 PCI and 1 ISA). The main requirement was a presence of ISA slot, since we still use old ISA hardware.
FUTURE ENHANCEMENTS
===================
One feature our operators wish to have is an ability to control programs on adjacent computers with their mouse. A program [*x2x*]{} [@x2x] exists which does exactly this, but it doesn’t work with Xinerama. So, we plan to “xineramify” [*x2x*]{}.
Currently the world moves to using TFT monitors, as those are more safe for people. But most TFTs have a limited viewing angle, which is inappropriate in multiheaded system, and those TFTs which are okay (like SGI 1600SW) are too expensive. So, currently we use 17" CRT displays, but plan to replace them when affordable TFTs appear.
Finally, consistent and ergonomic placement of windows on a 4-monitor desktop is a time-consuming task. So, we are planning to implement some sort of automation for this. Currently we are experimenting with X resource database (the [*WINDOWNAME.geometry*]{} resource).
[9]{}
G.Piskunov, “CAMAC-embedded 24-bit computer”, Autometriya, N4 (1986) pp. 32-38 (in Russian).
Dennis Baker, “Using the Xinerama Extensions to MultiHead XFree86 V. 4.0+”, November 2, 2000.\
http://www.linuxdoc.org/HOWTO/Xinerama-HOWTO.html
D.Yu. Bolkhovityanov, “Visible Cursor Patch for XFree86 4.0.3”, http://www.inp.nsk.su/\~bolkhov/files/bigcursor/
Official FVWM Homepage, http://www.fvwm.org/
Matrox Graphics, Inc., “Matrox beta drivers for XFree86”, ftp://ftp.matrox.com/\
pub/mga/archive/linux/2001/beta\_133\_143/
David Chaiken, “X to X connection”,\
http://gatekeeper.dec.com/pub/DEC/SRC/x2x/
[^1]: A frequent case: a window with Yes and No buttons is centered, so that \[Yes\] goes to one monitor and \[No\] to the other one.
[^2]: That’s a pathological case; more often screens of different sizes are used (e.g. 1024$\times$768 and 800$\times$600), which has the same effect.
[^3]: Up to XFree86 4.1 Xawtv behaved very funny: the window frame could be on one screen, and undecorated video picture – on another.
[^4]: 1152$\times$864 @ 32bpp $\approx$ 4M RAM for framebuffer.
|
---
abstract: 'We obtain the optimal global upper and lower bounds for the transition density $p_n(x,y)$ of a finite range isotropic random walk on affine buildings. We present also sharp estimates for the corresponding Green function.'
address: |
Instytut Matematyczny\
Uniwersytet\
Pl. Grunwaldzki 2/4\
50-384\
Poland
author:
- Bartosz Trojan
title: |
Heat kernel and Green function\
estimates on affine buildings
---
[^1]
Introduction {#sec:1}
============
Let $\mathscr{X}$ be an irreducible locally finite regular thick affine building of rank $r$ (see Subsection \[sec:4.2\] below) and let $p(x,y)$ be the transition density of a finite range isotropic random walk on good vertices of $\mathscr{X}$ (see Subsection \[subsec:4.4\]). The main focus of the paper is to describe $p_n$ the $n$-th iteration of the transition operator, e.g. $$p_n(x, y) = \sum_{x_1, \ldots, x_{n-1}} p(x, x_1) p(x_1, x_2) \ldots p(x_{n-1}, y).$$ The continuous counterpart of $\mathscr{X}$ is a Riemannian symmetric space of noncompact type. There $h_t$ the kernel of the heat semigroup $e^{t\Delta}$ where $\Delta$ is the Laplace–Beltrami operator, is well understood. Initial results based on specific computations were established by Sawyer [@saw1; @saw2; @saw3] and Anker [@a2; @a3]. Eventually, Anker and Ji in [@aj] proved a sharp estimates on the kernel $h_t(x)$ whenever ${{\left\lVert x \right\rVert}}$ is smaller than some constant multiplicity of $1+t$. Global estimates were subsequently found by Anker and Ostellari [@ao; @ao2]. The results have important applications. Among them is the exact behaviour of the Green function which is the analytic input in the process of describing the Martin boundary [@gu; @gjt]. In [@gjt] Guivarc’h, Ji and Taylor based on [@aj] constructed Martin compactification. The authors emphasize the importance of generalizations to Bruhat–Tits buildings associated with reductive groups over $p$-adic fields all the compactification procedures. The group-theoretic part of the program has already been carried out by Guivarc’h and Rémy in [@gure]. Moreover, among the Open Problems in [@gjt], the asymptotic behaviour of the Green function of finite range isotropic random walks on affine buildings is formulated.
In the present paper we provide the answer for the posed question. We give a detailed description of the off-diagonal behaviour of $p_n$ on any affine building of reduced type. The case $\widetilde{BC}_r$ will be covered in [@sttr]. We show sharp lower and upper estimates on $p_n(x,y)$ (see Theorem \[th:4\]) uniform in the region $${\operatorname{dist}}(\delta, \partial \mathcal{M}) \geq K n^{-1/(2\eta)}$$ where $y \in V_\omega(x)$, $\delta = (n+r)^{-1}(\omega + \rho)$ and $\mathcal{M}$ is the convex envelope of the support of $p(x, \cdot)$. The definition of $V_\omega(x)$ may be found in Subsection \[subsec:4.3\]. The restriction is not a difficulty thanks to good global upper bounds on $p_n$ (see Remark \[rem:1\]). Here, we state a variant of the result convenient in most applications.
For $\epsilon > 0$ small enough [^2] $$p_n(x, y) \asymp
n^{-r/2- {{\lvert {\Phi_0^+} \rvert}}} \varrho^n e^{-n\phi(n^{-1} \omega)} P_\omega(0)$$ uniformly on $\big\{y \in V_\omega(x) \cap {\operatornamewithlimits{supp}}p_n(x, \cdot):
{\operatorname{dist}}(n^{-1} \omega, \partial\mathcal{M}) \geq \epsilon \big\}$.
In the theorem $\varrho$ is the spectral radius of $p$, $P_\omega$ Macdonald symmetric polynomial and ${{\lvert {\Phi_0^+} \rvert}}$ the number of positive root directions. The function $\phi$ is convex and satisfies $\phi(\delta) \asymp {{\left\lVert \delta \right\rVert}}^2$. If we denote by $\kappa$ the spherical Fourier transform of $p$ we can describe the asymptotic behaviour of the Green function.
\(i) If $\zeta \in (0, \varrho^{-1})$ then for all $x \neq y$ $$G_\zeta(x, y)
\asymp P_\omega(0) {{\left\lVert \omega \right\rVert}}^{-(r-1)/2 - {{\lvert {\Phi_0^+} \rvert}}} e^{-{{\langle s, \omega\rangle}}}$$ where $y \in V_\omega(x)$ and $s$ is the unique point such that $\kappa(s) = (\zeta \rho)^{-1}$ and $$\frac{\nabla \kappa(s)}{{{\left\lVert \nabla \kappa(s) \right\rVert}}} = \frac{\omega}{{{\left\lVert \omega \right\rVert}}}.$$ (ii) If $\zeta = \varrho^{-1}$ then for all $x \neq y$ $$G_\zeta(x, y) \asymp P_\omega(0) {{\left\lVert \omega \right\rVert}}^{2-r-2{{\lvert {\Phi_0^+} \rvert}}}$$ where $y \in V_\omega(x)$.
Random walks on affine buildings have already been studied for over thirty years. In 1978 Sawyer [@saw] obtained the asymptotic of $p_n(x,x)$ for homogeneous trees, i.e. affine buildings of type $\widetilde{A}_1$. The result was extended to $\widetilde{A}_r$ by Tolli [@tol], Lindlbauer and Voit [@linv] and Cartwright and Woess [@carwoe]. Eventually, Local Limit Theorem for all affine buildings was proved by Parkinson [@park1].
The off-diagonal behaviour of $p_n$ was studied only in two cases. For homogeneous trees the uniform asymptotic were obtained by Lalley [@lal1; @lal2]. It should be pointed out that Lalley considered more general random walks, e.g. not necessarily radial. For affine buildings of higher rank the first results were obtained by Anker, Schapira and the author in [@ascht] where for each building of type $\tilde{A}_r$ the distinguished averaging operator was studied.
The main tool used in the study of isotropic random walks is the spherical Fourier transform. In 1970s Macdonald [@macdo0] developed the spherical harmonic analysis for groups of $p$-adic type. However, not every affine building corresponds to a group of $p$-adic type. Later, Cartwright and [@carmlo] proposed a construction of spherical Fourier transform using geometric and combinatorial properties of buildings of type $\widetilde{A}_2$. The approach was extended by Cartwright [@car] to buildings of type $\widetilde{A}_r$ and by Parkinson [@park2] to all affine buildings.
The result of the paper is Theorem \[th:4\]. Let us give an outline of its proof: The application of spherical Fourier transform results in an oscillatory integral which is analysed by the steepest descent method. Thanks to some geometric properties of the support of spherical Fourier transform of $p(O, \cdot)$ the integral can be localized. Therefore, the proof amounts to finding the asymptotic of $$I_n(x) = \int\limits_{{{\left\lVert u \right\rVert}} \leq \epsilon} e^{n \phi_n(x, u)} f_n(x, u) du$$ uniformly with respect to $x \in \mathfrak{a}_+$ as $n$ approaches infinity. If $x$ lies on the wall of the Weyl chamber $\mathfrak{a}_+$, functions $\phi_n(x, \cdot)$ retain symmetries in the directions orthogonal to the wall. Close to the wall we take advantage of this by expanding $I_n$ into its Taylor series and using combinatorial methods we identify remaining cancellations.
In [@ascht], for each affine building of type $\widetilde{A}_r$, a particular nearest neighbour random walk was studied. Its spherical Fourier transform satisfies the key combinatorial formula that allows to avoid the analysis of cancellations (similar phenomena occurs in [@aj]).
Organization of the paper {#subsec:1.1}
-------------------------
In Section \[sec:2\] we present the definition and some estimates for the function $s$ which appears later on in the estimates for $p_n$. Next, we show two auxiliary lemmas: one analytic and one combinatorial. The later has applications beyond the subject of the paper. In Section \[sec:3\] we demonstrate the general method where we derive the optimal upper and lower bounds for a finite range random walk on a grid. Furthermore, Theorem \[th:7\] improves a classical result (see e.g [@law Theorem 2.3.11]) — to the author’s best knowledge this is a new outcome. In Section \[sec:4\] the definitions of root systems and affine buildings are recalled, and a number of spherical-analytic facts used across the paper are recollected. The main theorem is proved in Subsection \[subsec:4.5\]. As an application the optimal lower and upper bounds for the corresponding Green function are found (Theorem \[th:6\]).
We use the convention by which $C$ stands for a generic positive constant whose value can change from occurrence to occurrence.
Preliminaries {#sec:2}
=============
Function $s$
------------
Let $\mathfrak{a}$ be a $r$-dimensional real vector space with an inner product ${{\langle \cdot, \cdot\rangle}}$. By $\mathfrak{a}_\mathbb{C}$ we denote its complexification. Let $\mathcal{V}$ be a finite set spanning $\mathfrak{a}$. Given a set of positive constants $\{c_v\}_{v \in \mathcal{V}}$ satisfying $\sum_{v \in \mathcal{V}} c_v = 1$ we define a function $\kappa: \mathfrak{a}_\mathbb{C} \rightarrow \mathbb{C}$ by $$\label{eq:25}
\kappa(z) = \sum_{v \in \mathcal{V}} c_v e^{{\langle z, v\rangle}}.$$ If $x \in \mathfrak{a}$ we denote by $B_x$ a quadratic form given by $B_x(u, u) = {\operatorname{D}}_u^2 \log \kappa(x)$. Notice $$\label{eq:1}
B_x(u,u) =
\frac{1}{2} \sum_{v, v' \in \mathcal{V}} {\frac{c_{v} e^{\langle x, v\rangle}}{\kappa(x)}} {\frac{c_{v'} e^{\langle x, v'\rangle}}{\kappa(x)}} {{\langle u, v-v'\rangle}}^2.$$ Let $\mathcal{M}$ be the interior of the convex hull of $\mathcal{V}$. Then
\[th:1\] For every $\delta \in \mathcal{M}$ a function $f(\delta,\cdot): \mathfrak{a} \rightarrow
\mathbb{R}$ defined by $$f(\delta, x) = {{\langle x, \delta\rangle}} - \log \kappa(x)$$ attains its maximum at the unique point $s$ satisfying $\nabla \log \kappa(s) = \delta$.
Without loss of generality, we may assume $\nabla \kappa(0)=0$. Then, we have $$f(\delta, x) = {{\langle x, \delta\rangle}} + \mathcal{O}({{\left\lVert x \right\rVert}}^2).$$ Moreover, by the function $f(\delta, \cdot)$ is strictly concave. Given $\delta \in \mathcal{M}$ there are $v_1, \ldots, v_r \in \partial \mathcal{M}$ and $t_1,\ldots, t_r \in [0,1)$ such that $$\sum_{j=1}^r t_j v_j = \delta$$ and $\sum_{j=1}^r t_j < 1$. Since $$\sum_{j=1}^r t_j \log \kappa(x) \geq \sum_{j=1}^r t_j \big(\log c_{v_j} + {{\langle x, v_j\rangle}}\big)
=\sum_{j=1}^r t_j \log c_{v_j} + {{\langle x, \delta\rangle}}$$ we get $$f(\delta, x) \leq \Big(\sum_{j=1}^r t_j - 1\Big) \log \kappa(x) - \sum_{j=1}^r t_j \log c_{v_j}$$ what finishes the proof because there are $\eta, \xi > 0$ such that $$\log \kappa(x) \geq \eta {{\left\lVert x \right\rVert}}$$ for ${{\left\lVert x \right\rVert}} \geq \xi$.
Let us define $\phi: \mathcal{M} \rightarrow \mathbb{R}$ by $$\label{eq:2}
\phi(\delta) = {{\langle s, \delta\rangle}} - \log \kappa(s).$$ By Theorem \[th:1\], $\nabla \phi(\delta) = s$ and ${{\langle \delta, v\rangle}} = {\operatorname{D}}_v \log \kappa(s)$ for $v \in \mathfrak{a}$. Hence, for $u, v \in \mathfrak{a}$ $$\label{eq:27}
{{\langle u, v\rangle}} = {{\langle {\operatorname{D}}_u s, \nabla {\operatorname{D}}_v \log \kappa(s)\rangle}} = B_s({\operatorname{D}}_u s, v).$$ Therefore ${\operatorname{D}}_u s = B_s^{-1} u$ and so ${\operatorname{D}}^2_u \phi(\delta) = B_s^{-1}(u,u)$. In particular, $\phi$ is convex. Also for $\delta_0 = \nabla \log \kappa(0)$ we have $$\phi(\delta) = \frac{1}{2} B_0^{-1}(\delta - \delta_0, \delta-\delta_0) +
\mathcal{O}({{\left\lVert \delta-\delta_0 \right\rVert}}^3).$$ Let us fix $v_0 \in \mathcal{V}$ and consider $\delta \in \mathcal{M}$ such that ${{\langle s, v_0\rangle}} = \max \big\{{{\langle s, v\rangle}}: v \in \mathcal{V}\big\}$. Since $${{\langle s, \delta\rangle}} - {{\langle s, v_0\rangle}} =
\sum_{v \in \mathcal{V}}
\frac{c_v e^{{{\langle s, v\rangle}}} }{\kappa(s)}{{\langle s, v-v_0\rangle}} \leq 0$$ we get $$\phi(\delta) \leq {{\langle s, \delta\rangle}} - \log c_{v_0}e^{{{\langle s, v_0\rangle}}} \leq -\log c_{v_0}.$$ Therefore, $$\label{eq:33}
\phi(\delta) \asymp
{{\langle \delta-\delta_0, \delta - \delta_0\rangle}}$$ for all $\delta \in \mathcal{M}$.
\[th:2\] There are constants $\eta \geq 1$ and $C > 0$ such that for $\delta \in \mathcal{M}$ and $v \in \mathcal{V}$ $$e^{{{\langle s, v\rangle}}} \geq C \kappa(s) {\operatorname{dist}}(\delta, \partial\mathcal{M})^\eta.$$
Let $\mathcal{V} = \{v_1, \ldots, v_N\}$. Define $$\Omega = \{x \in \mathcal{S}:
{{\langle x, v_{i}\rangle}} \geq {{\langle x, v_{i+1}\rangle}} \text{ for } i=1,\ldots,N-1\}$$ where $\mathcal{S}$ is the unit sphere in $\mathfrak{a}$ centred at the origin. Suppose $\Omega \neq \emptyset$ and let $k$ be the smallest index such that points $\{v_1, \ldots, v_k\}$ do not lay on the same wall of $\mathcal{M}$. Then, there is $\epsilon>0$ such that for all $x \in \Omega$ we have $${{\langle x, v_1\rangle}} > {{\langle x, v_k\rangle}} + \epsilon.$$ Let $\mathcal{F}$ be a wall of the maximal dimension containing $\{v_1,\ldots,v_{k-1}\}$. We denote by $u$ an outward unit normal vector to $\mathcal{M}$ at $\mathcal{F}$. For $x/{{\left\lVert x \right\rVert}} \in \Omega$ and $$\delta = \sum_{v\in\mathcal{V}} {\frac{c_{v} e^{\langle x, v\rangle}}{\kappa(x)}} v$$ we have $${\operatorname{dist}}(\delta, \mathcal{F}) = {{\langle u, v_1 - \delta\rangle}}
= \sum_{v\in\mathcal{V}\setminus\mathcal{F}} {\frac{c_{v} e^{\langle x, v\rangle}}{\kappa(x)}} {{\langle u, v_1-v\rangle}}
\leq C e^{{{\langle x, v_k - v_1\rangle}}}.$$ Moreover, if $j > k$ $$e^{{{\langle x, v_j - v_1\rangle}}}
\geq \Big(e^{{{\langle x, v_k - v_1\rangle}}}\Big)^{\epsilon^{-1} {{\langle x, v_1 - v_j\rangle}}/{{\left\lVert x \right\rVert}}}
\geq C {\operatorname{dist}}(\delta, \partial\mathcal{M})^{\epsilon^{-1} {{\langle x, v_1 - v_j\rangle}}/{{\left\lVert x \right\rVert}}}.$$ Since $\kappa(x) \leq e^{{{\langle x, v_1\rangle}}}$ the theorem follows.
Analytic lemmas {#subsec:2.2}
---------------
For a multi-index $\sigma \in \mathbb{N}^r$ we denote by $X_\sigma$ a multi-set containing $\sigma(i)$ copies of $i$. Let $\Pi_\sigma$ be a set of all partitions of $X_\sigma$ and let $\{u_1,\ldots,u_r\}$ be a basis of $\mathfrak{a}$. For the convenience of the reader we recall
\[lem:1\] There are positive constants $c_\pi$, $\pi \in \Pi_\sigma$, such that for sufficiently smooth functions $f: S \rightarrow T$, $F: T \rightarrow \mathbb{R}$, $T \subset \mathbb{R}$, $S \subset \mathbb{R}^r$, we have $${\partial^{\sigma}} F(f(s)) = \sum_{\pi \in \Pi_\sigma} c_\pi \left.
\frac{{\rm d}^m }{{\rm d} t^m} \right|_{t = f(s)} F(t)
\prod_{j=1}^{m} {\partial^{B_j}} f(s)$$ where $\pi = \{B_1, \ldots, B_m\}$.
In particular, there is $C > 0$ such that for every $\sigma$ $$\label{eq:22}
\sum_{\pi \in \Pi_\sigma} c_\pi m! \prod_{j=1}^m B_j! \leq C^{{{\lvert {\sigma} \rvert}}+1} \sigma!.$$ Using Lemma \[lem:1\] one can show
\[lem:2\] Let $a_v \in \mathbb{C}$, $b_v \in \mathbb{R}_+$ for $v \in \mathcal{V}$. Then for $z = x+i\theta \in \mathfrak{a}_\mathbb{C}$, ${{\left\lVert \theta \right\rVert}} < (4 \max\{{{\left\lVert v \right\rVert}}: v \in \mathcal{V}\})^{-1}$ $$\Big| \sum_{v \in \mathcal{V}} b_v e^{{\langle z, v\rangle}} \Big|
\geq \frac{1}{\sqrt{2}} \sum_{v \in \mathcal{V}} b_v e^{{\langle x, v\rangle}}.$$ There is $C > 0$ such that for $\sigma \in \mathbb{N}^r$ $$\bigg| {\partial^{\sigma}} \bigg\{ \frac{ \sum_{v \in \mathcal{V}} a_v e^{{\langle z, v\rangle}} }
{ \sum_{v \in \mathcal{V}} b_v e^{{\langle z, v\rangle}} } \bigg\} \bigg|
\leq C^{{{\lvert {\sigma} \rvert}}} \sigma! \frac{ \sum_{v \in \mathcal{V}} {{\lvert {a_v} \rvert}} e^{{\langle x, v\rangle}} }
{ \sum_{v \in \mathcal{V}} b_v e^{{\langle x, v\rangle}} }.$$
By Lemma \[lem:2\] a function $\log \kappa(z)$ is well defined whenever ${{\left\lVert \theta \right\rVert}}$ is sufficiently small. Moreover, for $u, u' \in \mathfrak{a}$ $$\label{eq:43}
{\operatorname{D}}_{u} {\operatorname{D}}_{u'} \log \kappa(z) = \frac{1}{2} \sum_{v, v' \in \mathcal{V}}
\frac{c_v e^{{{\langle z, v\rangle}}}}{\kappa(z)} \frac{c_{v'} e^{{{\langle z, v'\rangle}}}}{\kappa(z)}
{{\langle u, v-v'\rangle}} {{\langle u', v-v'\rangle}}.$$ Therefore, by Lemma \[lem:2\] and Cauchy–Schwarz inequality, for $\sigma \succeq e_{k_1} + e_{k_2}$ we have $$\label{eq:24}
|{\partial^{\sigma}} \log \kappa(z))| \leq C^{{{\lvert {\sigma} \rvert}}+1} \sigma!
\sqrt{B_x(u_{k_1}, u_{k_1}) B_x(u_{k_2}, u_{k_2})}.$$ Let $$\label{eq:42}
\varphi(x,\theta) = -\int_0^1 (1-t) {\operatorname{D}}_\theta^2 \log \kappa(x+it\theta) dt.$$ Since for $v,v' \in \mathcal{V}$ $$\Re \bigg( \frac{e^{{\langle z, v+v'\rangle}}}{\kappa(z)^2} \bigg)
\geq \frac{1}{2} \frac{e^{{\langle x, v+v'\rangle}}}{\kappa(x)^2}$$ we get $$4 \Re \varphi(x,\theta) \leq - B_x(\theta,\theta).$$ Further application of Lemma \[lem:2\] to gives $$\label{eq:23}
|{\partial^{\sigma}}_x \varphi(x, \theta)| \leq C^{{{\lvert {\sigma} \rvert}}+1} \sigma! B_x(\theta, \theta).$$ In what follows we also need a function $$\psi(x,\theta) = -3 i \int_0^1 (1-t)^2 {\operatorname{D}}_\theta^3 \log \kappa(x+it\theta)dt.$$ Again, by and Lemma \[lem:2\], $$\label{eq:6}
| \psi(x, \theta) | \leq C {{\left\lVert \theta \right\rVert}} B_x(\theta,\theta).$$
Combinatorial lemma {#subsec:1}
-------------------
Let $\{C_1, C_2, \ldots, C_r\}$ be a fixed sequence of subsets of a finite set $X$. A multi-index $\gamma \in \mathbb{N}^r$ is called admissible if there is $\{X_j\}_{j=1}^r$ a partial partition of $X$ such that $X_j \subseteq C_j$ and ${{\lvert {X_j} \rvert}} = \gamma(j)$.
\[lem:5\] If $\gamma$ is admissible then for any partial partition $\{X_j\}_{j=1}^r$ corresponding to $\gamma$ $$\bigcup_{j \in J_\gamma} X_j = \bigcup_{j \in J_\gamma} C_j$$ where $J_\gamma = \{j: \gamma+e_j \text{ is not admissible}\}$.
Given $m \in J_\gamma$ we construct a sequence $\{I_j\}_{j = 0}^\infty$ as follows: $I_0=\{m\}$ and for $i \geq 0$ $$I_{i+1} = \{j : X_j \cap C_k \neq \emptyset \text{ for some } k \in I_i\}.$$ We notice, $I_i \subseteq I_{i+1}$. Let $I = \limsup_{i \geq 0} I_i$ and $
V = \bigcup_{j \in I} X_j.
$ Suppose, there is $$y \in \bigcup_{j \in I} C_j \cap V^c.$$ First, we observe $$y \not\in \bigcup_{j=1}^r X_j.$$ Moreover, there are sequences $\{j_i\}_{i=1}^n$ and $\{x_i\}_{i=0}^n$ of distinct elements such that $j_1 = m$, $y \in C_{j_n}$, $x_0 \in X_{j_1}$, $x_n = y$ and $$x_i \in C_{j_i} \cap X_{j_{i+1}}$$ for $i \in \{1, \ldots, n-1\}$. Setting $$Y_j =
\begin{cases}
\big(X_{j_i} \cup \{x_i\} \big)\setminus \{x_{i-1}\} & \text{ if } j = j_i \text{ for }
\in i \in \{1, \ldots, n\} \\
X_j & \text{ otherwise}
\end{cases}$$ we obtain a partial partition of $X$ corresponding to $\gamma$ such that $$x_0 \in C_m \cap \Big(\bigcup_{j=1}^r Y_j \Big)^c$$ which is not possible since $m \in J_\gamma$. Therefore, we must have $V = \bigcup_{j \in I} C_j$. In particular, $${{\lvert {V} \rvert}} = \sum_{j \in I} \gamma(j).$$ Suppose there is $k \in I \cap J_\gamma^c$. Then there exists $\{Y_j\}_{j=0}^r$ a partial partition corresponding to $\gamma$ such that $$\label{eq:31}
C_k \cap \Big(\bigcup_{j=1}^r Y_j \Big)^c \neq \emptyset.$$ Since $Y_j \subseteq C_j$ and $$\sum_{j \in I} {{\lvert {Y_j} \rvert}} = \sum_{j \in I} \gamma(j),$$ we must have $$\bigcup_{j \in I} Y_j = \bigcup_{j \in I} C_j$$ which contradicts to . Therefore, $I \subseteq J_\gamma$ and the Lemma follows.
Random Walk on $\mathbb{Z}^r$ {#sec:3}
=============================
Let $\{p(x,y)\}_{x,y \in \mathbb{Z}^r}$ be a transition probability of a random walk on $\mathbb{Z}^r$. Assume the walk is irreducible and has a finite range. We set $p_n(v) = p_n(O, v)$ where $O$ is the origin. Let $\mathcal{V} = \{v \in \mathbb{Z}^r: p(v) > 0\}$ and $c_v = p(v)$. Then $$\kappa(z) = \sum_{v \in \mathcal{V}} c_v e^{{{\langle z, v\rangle}}}$$ is the characteristic function of $p$. For $v \in {\operatornamewithlimits{supp}}p_n$ we set $\delta = n^{-1} v$ and denote by $s$ the unique solution to $\nabla \log \kappa(s) = \delta$ (see Theorem \[th:1\]). Let $$\phi(\delta) = {{\langle s, \delta\rangle}} - \log \kappa(s).$$ We have
\[th:7\] There are $\eta \geq 1$ and $K > 0$ such that $$p_n(v) \asymp \big(\! \det n B_s \big)^{-1/2} e^{-n\phi(\delta)}$$ uniformly on $\big\{v \in {\operatornamewithlimits{supp}}p_n: n {\operatorname{dist}}(\delta, \partial \mathcal{M})^{2\eta} \geq K\big\}$.
Using Fourier Inversion Formula we can write $$\label{eq:37}
p_n(v)=(2\pi)^{-r/2} \int\limits_{\mathscr{D}_r}
\kappa(i\theta)^n e^{-i{{\langle \theta, v\rangle}}} d\theta.$$ By $\mathscr{U}$ we denote a set of $\theta \in \mathscr{D}_r = [-\pi, \pi]^r$ such that ${{\lvert {\kappa(i\theta)} \rvert}} = 1$. For each $\theta_0 \in \mathscr{U}$ there is $t_0 \in [-\pi, \pi]$ such that $e^{it_0} = e^{i{{\langle \theta_0, v\rangle}}}$ for all $v \in \mathcal{V}$. Using we get $$p_n(v) = e^{i n t_0 - i {{\langle \theta_0, v\rangle}}} p_n(v)$$ thus $e^{in t_0} = e^{i{{\langle \theta_0, v\rangle}}}$. The integrand in extends to an analytic $2\pi \mathbb{Z}^r$-periodic function on $\mathfrak{a}_{\mathbb{C}}$. Therefore, changing the contour of integration we get $$p_n(v) = (2\pi)^{-r/2} e^{-n\phi(\delta)} \int\limits_{\mathscr{D}_r}
\left[ \frac{\kappa(s+i\theta)}{\kappa(s)} \right]^n e^{-i{{\langle \theta, v\rangle}}} d\theta.$$ Next $$1 - \bigg\lvert \frac{\kappa(s+i\theta)}{\kappa(s)} \bigg\rvert^2 =
2 \sum_{v, v' \in \mathcal{V}} {\frac{c_{v} e^{\langle s, v\rangle}}{\kappa(s)}} {\frac{c_{v'} e^{\langle s, v'\rangle}}{\kappa(s)}} (\sin {{\langle \theta/2, v-v'\rangle}})^2.$$ Given $0 < \epsilon < \frac{1}{2}\min\big\{{{\left\lVert \theta_0-\theta_0' \right\rVert}}:
\theta_0, \theta_0' \in \mathscr{U} \big\}$ we set $$\mathscr{D}_r^\epsilon = \bigcap_{\theta_0 \in \mathscr{U}}
\{\theta \in \mathscr{D}_r: {{\left\lVert \theta-\theta_0 \right\rVert}} \geq \epsilon\}.$$ By the irreducibility of the walk for every $v_0 \in \mathcal{V}$ and $\epsilon > 0$ there is $\xi > 0$ such that for all $\theta \in \mathscr{D}_r^\epsilon$ there is $v \in \mathcal{V}$ satisfying $$|\sin {{\langle \theta/2, v-v_0\rangle}}| \geq \xi.$$ Suppose ${{\langle s, v_0\rangle}} = \max \big\{{{\langle s, v\rangle}}: v \in \mathcal{V}\big\}$. Then, by Theorem \[th:2\], if $\theta \in \mathscr{D}_r^\epsilon$ we get $$\label{eq:4}
1 - \bigg\lvert \frac{\kappa(s+i\theta)}{\kappa(s)} \bigg\rvert^2
\geq 2 C \xi^2 {\frac{c_{v_0} e^{\langle s, v_0\rangle}}{\kappa(s)}}{\frac{c_{v} e^{\langle s, v\rangle}}{\kappa(s)}}
\geq C {\operatorname{dist}}(\delta, \partial\mathcal{M})^{\eta}.$$ Hence, we can estimate the error term $$\bigg| \int\limits_{\mathscr{D}_r^\epsilon}
\bigg[ \frac{\kappa(s+i\theta)}{\kappa(s)} \bigg]^n e^{-i{{\langle \theta, v\rangle}}} d\theta
\bigg|
\leq C' e^{-C n {\operatorname{dist}}(\delta, \partial \mathcal{M})^{\eta}}.$$ Next, we notice $$\sum_{\theta_0 \in \mathscr{U}}
\int\limits_{{{\left\lVert \theta - \theta_0 \right\rVert}} \leq \epsilon}
\kappa(s+i\theta)^n e^{-i{{\langle \theta, v\rangle}}} d\theta =
{{\lvert {\mathscr{U}} \rvert}}
\int\limits_{{{\left\lVert \theta \right\rVert}} \leq \epsilon}
\kappa(s+i\theta)^n e^{-i{{\langle \theta, v\rangle}}} d\theta.$$ If $\epsilon$ is small enough, by the inequality for ${{\left\lVert \theta \right\rVert}} \leq \epsilon$ we obtain $$\label{eq:5}
\bigg| e^{n\psi(s, \theta)} - 1 - n\psi(s, \theta) \bigg|
\leq C n^2 e^{\frac{n}{4} B_{s}(\theta, \theta)}
\big\lVert B_{s}^{-1} \big\rVert \big\lVert B_{s}^{1/2} \theta \big\rVert^6$$ thus $$\bigg\lvert \int\limits_{{{\left\lVert \theta \right\rVert}} \leq \epsilon}
e^{-\frac{n}{2} B_{s}(\theta, \theta)}
\big( e^{n\psi(s, \theta)} - 1 - n\psi(s, \theta) \big) d\theta \bigg\rvert
\leq C \big(\! \det n B_{s} \big)^{-1/2} n^{-1} \big\lVert B_{s}^{-1} \big\rVert.$$ Furthermore we have $$\label{eq:7}
\bigg| \psi(s, \theta) - \frac{D_\theta^3 \psi(s, 0)}{3!} \bigg|
\leq C \big\lVert B_{s}^{-1} \big\rVert \big\lVert B_{s}^{1/2} \theta \big\rVert^4$$ which implies $$\bigg| \int\limits_{{{\left\lVert \theta \right\rVert}} \leq \epsilon} e^{-\frac{n}{2} B_{s}(\theta, \theta)}
n \bigg( \psi(s, \theta) - \frac{D_\theta^3 \psi(s, 0)}{3!} \bigg) d\theta \bigg|
\leq C \big(\! \det n B_{s} \big)^{-1/2} n^{-1} \big\lVert B_{s}^{-1} \big\rVert.$$ On the other hand $$\int\limits_{{{\left\lVert \theta \right\rVert}} \geq \epsilon} e^{-\frac{n}{2} B_{s} (\theta, \theta)} d\theta
\leq C \big(\! \det n B_{s} \big)^{-1/2} e^{-C' n {{\left\lVert B_{s}^{-1} \right\rVert}}^{-1}},$$ which together with ${{\left\lVert B_{s}^{-1} \right\rVert}} \leq C {\operatorname{dist}}(\delta, \partial \mathcal{M})^{-\eta}$ (see Theorem \[th:2\]) concludes the proof.
Affine buildings {#sec:4}
================
Root systems {#subsec:4.1}
------------
We recall basic facts about root systems and Coxeter groups. General reference is [@bou].
Let $\Phi$ be an irreducible but not necessary reduced root system in $\mathfrak{a}$. By $Q$ we denote the root lattice, i.e. $\mathbb{Z}$-span of $\Phi$. Let $\{\alpha_i: i \in I_0\}$ where $I_0=\{1, \ldots, r\}$ be a fixed base of $\Phi$. $\Phi^+$ denotes the set of all positive roots. Let $\mathfrak{a}_+$ be the positive Weyl chamber $$\mathfrak{a}_+=\{x \in \mathfrak{a}: {{\langle \alpha, x\rangle}} > 0 \text{ for all }
\alpha \in \Phi^+\}.$$ We denote as $\tilde{\alpha}$ the highest root of $\Phi$ where numbers $m_i$ satisfy the equality $$\tilde{\alpha} = \sum_{i \in I_0} m_i \alpha_i$$ and we set $m_0 = 1$ as well as $I = I_0 \cup \{0\}$. Let us define the set $I_P$ by $$I_P=\{i \in I: m_i = 1\}.$$
The dual base to $\{\alpha_i: i \in I_0\}$ is denoted as $\{\lambda_i: i \in I_0\}$. The co-weight lattice $P$ is the ${\operatorname{\mathbb{Z}-span}}$ of fundamental co-weights $\{\lambda_i: i \in I_0\}$. A co-weight $\lambda \in P$ is called dominant if $\lambda = \sum_{i \in I_0} x_i \lambda_i$ where $x_i \geq 0$ for all $i \in I_0$. Finally, the cone of all dominant co-weights is denoted by $P^+$.
Let $H_i=\{x \in \mathfrak{a}: {{\langle \alpha_i, x\rangle}}=0\}$ for each $i \in I_0$. We denote by $r_i$ the orthogonal reflection in $H_i$, i.e. $r_i(x) = x - {{\langle \alpha_i, x\rangle}} {\check{\alpha_i}}$ for $x \in \mathfrak{a}$ where for $\alpha \in \Phi$ we put $${\check{\alpha}} = \frac{2\alpha}{{{\langle \alpha, \alpha\rangle}}}.$$ The subgroup $W_0$ of ${\operatorname{GL}}(\mathfrak{a})$ generated by $\{r_i: i \in I_0\}$ is the Weyl group of $\Phi$. Let $r_0$ be the orthogonal reflection in the affine hyperplane $H_0=\{x \in \mathfrak{a}:
{{\langle \tilde{\alpha}, x\rangle}} = 1\}$. Then the affine Weyl group $W$ of $\Phi$ is the subgroup of ${\operatorname{Aff}}(\mathfrak{a})$ generated by $\{r_i: i \in I\}$. Finally, the extended affine Weyl group of $\Phi$ is $\widetilde{W}=W_0 \ltimes P$.
Let $M=(m_{ij})_{i,j \in I}$ be a symmetric matrix with entries in $\mathbb{Z} \cup \{\infty\}$ such that for all $i,j \in I$ $$m_{ij} = \left\{
\begin{array}{ll}
\geq 2 & \text{ if } i \neq j\\
1 & \text{ if } i = j.
\end{array}
\right.$$ The Coxeter group of type $M$ is the group $W$ given by a presentation $$W=\left<r_i: (r_i r_j)^{m_{ij}}=1 \text{ for all } i,j \in I\right>.$$ For a word $f=i_1 \cdots i_k$ in the free monoid $I$ we denote by $r_f$ an element of $W$ of the form $r_f= r_{i_1} \cdots r_{i_k}$. The length of $w \in W$, denoted $l(w)$, is the smallest integer $k$ such that there is a word $f=i_1\ldots i_k$ and $w=r_f$. We say $f$ is reduced if $l(r_f) = k$.
Definition {#sec:4.2}
----------
We refer the reader to [@ron] for the theory of affine buildings.
A set $\mathscr{X}$ equipped with a family of equivalence relations $\{\sim_i:i \in I\}$ is a chamber system and the elements of $\mathscr{X}$ are called chambers. A gallery of type $f = i_1 \cdots i_k$ in $\mathscr{X}$ is a sequence of chambers $(c_0, \ldots, c_k)$ such that for all $1 \leq j \leq k$, $c_{j-1} \sim_{i_j} c_j$ and $c_{j-1} \neq c_j$. For $J \subset I$, $J$-residue is a subset of $\mathscr{X}$ such that any two chambers can be joined by a gallery of type $f = i_1\cdots i_k$ with $i_1, \ldots, i_k \in J$.
Let $W$ be a Coxeter group of type $M$. For each $i \in I$ we define an equivalence relation on $W$ by declaring $w \sim_i w'$ if and only if $w = w'$ or $w = w'r_i$. Then $W$ equipped with $ \{\sim_i\}_{i \in I}$ is a chamber system called Coxeter complex of $W$.
Let $W$ be a Coxeter group. A chamber system $\mathscr{X}$ is a building of type $W$ if
1. for all $x \in \mathscr{X}$ and $i \in I$, ${{\lvert {\{y \in \mathscr{X}: y \sim_i x\}} \rvert}} \geq 2$,
2. there is $W$-distance function $\delta: \mathscr{X} \times \mathscr{X} \rightarrow W$ such that if $f$ is a reduced word, then $\delta(x, y) = r_f$ if and only if $x$ and $y$ can be joined by a gallery of type $f$.
If $W$ is an affine Weyl group, the building $\mathscr{X}$ is called affine.
Notice, since $\delta_{W}(w, w')=w^{-1}w'$ is $W$-distance function a Coxeter complex of $W$ is a building of type $W$.
A subset $\mathcal{A} \subset \mathscr{X}$ is called an apartment if there is a mapping $\psi: W \rightarrow \mathscr{X}$ such that $\mathcal{A} = \psi(W)$ and for all $w, w' \in W$, $\delta(\psi(w), \psi(w')) = \delta_W(w, w')$.
A building $\mathscr{X}$ has a geometric realization as a simplicial complex $\Sigma(\mathscr{X})$ where a residue of type $J$ corresponds to a simplex of dimension ${{\lvert {I} \rvert}}-{{\lvert {J} \rvert}}-1$. Let $V(\mathscr{X})$ denote the set of vertices of $\Sigma(\mathscr{X})$. Define a mapping $\tau: V(\mathscr{X}) \rightarrow I$ by declaring $\tau(x)=i$ if $x$ corresponds to a residue of type $I \setminus \{i\}$.
For $x \in \mathscr{X}$ and $i \in I$, let $q_i(x)$ be equal to $$q_i(x) = {{\lvert {\{y \in \mathscr{X}: y \sim_i x\}} \rvert}} - 1.$$ A building $\mathscr{X}$ is called regular if for every $i \in I$, the numbers $q_i(x)$ are independent of $x \in V(\mathscr{X})$, locally finite if for every $i \in I$ and $x \in V(\mathscr{X})$, $q_i(x) < \infty$ and thick if for every $i \in I$ and $x \in V(\mathscr{X})$, $q_i(x) > 1$. If $\mathscr{X}$ is a regular building, we put $q_i=q_i(x)$ for $i \in I$.
To any irreducible locally finite regular affine building we associate an irreducible, but not necessary reduced, finite root system $\Phi$ (see [@park2]) such that the affine Weyl group corresponding to $\Phi$ is isomorphic to $W$ and $q_{\tau(v)} = q_{\tau(v+\lambda)}$ for all $\lambda \in P$ and $v \in \Sigma(W)$. Then the set of good vertices is defined as $$V_P = \{v \in V(\mathscr{X}): \tau(v) \in I_P\}.$$ From now on we assume the root system $\Phi$ is reduced. The case $\widetilde{BC}_r$ will be covered in [@sttr].
Spherical analysis {#subsec:4.3}
------------------
In this subsection we summarize spherical harmonic analysis on affine buildings (see [@macdo0; @park2]).
Let $\mathscr{X}$ be an irreducible locally finite regular thick affine building. Given $x \in V_P$ and $\lambda \in P^+$, let $V_\lambda(x)$ be the set of all $y \in V_P$ such that there is an apartment $\mathcal{A}$ containing $x$ and $y$, an isomorphism $\psi: \mathcal{A} \rightarrow \Sigma(W)$ and $w \in \widetilde{W}$ such that $\psi(x) = 0$ and $\psi(y) = w \lambda$. From regularity of $\mathscr{X}$ stems that $|V_\lambda(x)|$ is independent of $x$ henceforth we denote as $N_\lambda$ the common value.
For each $\lambda \in P^+$ define an operator $A_\lambda$ acting on $f \in \ell^2(V_P)$ by $$A_\lambda f(x) = \frac{1}{N_\lambda} \sum_{y \in V_\lambda(x)} f(y).$$ Then $\mathscr{A}_0={\operatorname{\mathbb{C}-span}}\{A_\lambda : \lambda \in P^+\}$ is a commutative $\star$-subalgebra of the algebra of bounded linear operators on $\ell^2(V_P)$. We denote by $\mathscr{A}_2$ its closure in the operator norm. Gelfand transform of $\mathscr{A}_2$ is defined in terms of Macdonald polynomials.
Let $\Phi_0$ be a set of indivisible roots, i.e. $\Phi_0=\{\alpha \in \Phi|\, \frac{1}{2}
\alpha \not\in \Phi\}$. For $\alpha \in \Phi_0$ put $q_\alpha = q_i$ if $\alpha \in W_0 \cdot \alpha_i$. If $\lambda \in P^+$ and $z \in \mathfrak{a}_\mathbb{C}$ let $$P_\lambda(z) = \frac{1}{W_0(q^{-1})}
\prod_{\alpha \in \Phi^+_0} q_\alpha^{-{{\langle \alpha/2, \lambda\rangle}}}
\sum_{w \in W_0} c(w \cdot z) e^{{{\langle w \cdot z, \lambda\rangle}}}$$ where $$c(z) =
\prod_{\alpha \in \Phi^+_0}
\frac{1-q_\alpha^{-1} e^{-{{\langle z, {\check{\alpha}}\rangle}}}} {1 - e^{-{{\langle z, {\check{\alpha}}\rangle}}}}$$ and $$W_0(q^{-1})=\sum_{w \in W_0} q_w^{-1}$$ with $q_w=q_{i_1} \cdots q_{i_k}$ if $w=r_f$ is a reduced expression of $w \in W_0$. Values of $P_\lambda(z)$ where the denominator of $c$-function is equal $0$ can be obtained by taking proper limits.
Given $z \in \mathfrak{a}_\mathbb{C}$ let $h_z: \mathscr{A}_0 \rightarrow \mathbb{C}$ be the linear map satisfying $h_z(A_\lambda) = P_\lambda(z)$ if $\lambda \in P^+$. Notice, $h_z = h_{z'}$ if and only if $z' \in W_0 \cdot z$. For $\theta \in \mathscr{D}_r$ where $$\mathscr{D}_r=\{\theta \in \mathfrak{a} : |{{\langle \theta, \alpha\rangle}}| \leq 2 \pi \text{ for all }
\alpha \in \Phi\}$$ the multiplicative functional $h_{i\theta}$ extends to the functional on $\mathscr{A}_2$. Moreover, every multiplicative functional on $\mathscr{A}_2$ is of that form. There is also an inversion formula. Given $A \in \mathscr{A}_2$, $x \in V_P$ and $\lambda \in P^+$, we have $$\label{eq:9}
A \delta_x(y) = \frac{1}{(2\pi)^r} \frac{W_0(q^{-1})}{|W_0|} \int\limits_{\mathscr{D}_r}
h_{i\theta}(A) \overline{P_\lambda(i\theta)} \frac{d\theta}{{{\lvert {c(i\theta)} \rvert}}^2}$$ for all $y \in V_\lambda(x)$.
Random walks {#subsec:4.4}
------------
We consider an isotropic random walk on good vertices of $\mathscr{X}$, i.e. a random walk with the transition probabilities $p(x,y)$ constant on each $\{(x,y) \in V_P \times V_P: y \in V_\lambda(x)\}$ for $\lambda \in P^+$. We denote by $A$ the corresponding operator acting on $f \in \ell^2(V_P)$ $$A f (x) = \sum_{y \in V_P} p(x, y) f(y).$$ Then $A$ belongs to the algebra $\mathscr{A}_2$ and may be expressed as $$A = \sum_{\mu\in P^+} a_\mu A_\mu$$ where $a_\mu \geq 0$ and $\sum_{\mu \in P^+} a_\mu = 1$. We assume the random walk is irreducible and has a finite range. Then $a_\mu > 0$ for finitely many $\mu \in P^+$. Let $$\varrho = h_0(A),$$ $\hat{A}(z) = h_z(A)$ and $\kappa(z) = \varrho^{-1} \hat{A}(z)$. We set $\mathcal{U}
= \{\theta \in \mathcal{D}_r: {{\lvert {\kappa(\theta)} \rvert}} = 1\}$. There are a finite set $\mathcal{V}
\subset \mathfrak{a}$ and positive real numbers $\{c_v:v \in \mathcal{V}\}$ such that (see [@park1 Remark 3.2]) $$\label{eq:10}
\kappa(z) = \sum_{v \in \mathcal{V}} c_v e^{{{\langle z, v\rangle}}}.$$ Notice, by $W_0$-invariance $\nabla \kappa(0) = 0$. If $\delta \in \mathcal{M}$ and $w \in W_0$ we can write $$w \cdot \delta = w \cdot \nabla \log \kappa(s) = \nabla \log \kappa(w \cdot s)$$ where $s = s(\delta)$. Thus Theorem \[th:1\] implies $w \cdot s(\delta) = s(w \cdot \delta)$. Since for $\alpha \in \Phi$ $$0 \leq {{\langle s, \delta\rangle}}- \log \kappa(s) - {{\langle r_\alpha s, \delta\rangle}} + \log \kappa(r_\alpha s)
={{\langle s, {\check{\alpha}}\rangle}} {{\langle \alpha, \delta\rangle}}$$ the mapping $s: \mathcal{M} \rightarrow \mathfrak{a}$ is real analytic and $s(\mathcal{M} \cap
{\operatorname{cl}}\mathfrak{a}_+) = {\operatorname{cl}}\mathfrak{a}_+$.
Heat kernel estimates {#subsec:4.5}
---------------------
Fix a good vertex $O$ and consider a vertex $v \in {\operatornamewithlimits{supp}}p_n$. We denote by $\omega$ a positive co-weight such that $v \in V_{\omega}(O)$. Let $\rho$ be the sum of fundamental co-weights $$\rho = \lambda_1 + \lambda_2 + \cdots + \lambda_r$$ and $\delta = (n+r)^{-1} (\omega + \rho)$. Since $r^{-1} \rho \in {\operatorname{cl}}\mathcal{M}$ we have $\delta \in \mathcal{M}$. We set $s = s(\delta)$ and recall $\phi(\delta) = {{\langle s, \delta\rangle}} - \log \kappa(s)$. Then
\[th:4\] There are $K > 0$ and $\eta \geq 1$ such that $$p_n(v) \asymp
\prod_{\alpha \in \Phi_0^+}
q_\alpha^{-{{\langle \alpha/2, \omega\rangle}}}
\big(\! \det n B_{s} \big)^{-1/2}
\varrho^n
e^{-n\phi(n^{-1} \omega)}
\prod_{\alpha \in \Phi_0^+} \sinh {{\langle s, {\check{\alpha}}/2\rangle}}$$ uniformly on $\big\{v \in V_{\omega}(O) \cap {\operatornamewithlimits{supp}}p_n :
n {\operatorname{dist}}(\delta, \partial \mathcal{M})^{2\eta} \geq K\big\}$.
### Proof of Theorem \[th:4\] {#theproof}
The function $1/c$ is analytic on $\{z \in \mathfrak{a}_\mathbb{C}: \Re z \in \mathfrak{b}\}$ where $\mathfrak{b} = \{x \in \mathfrak{a}: {{\langle x, {\check{\alpha}}\rangle}} > -\log q_\alpha
\text{ for all } \alpha \in \Phi_0^+ \}$. By $$p_n(v)= \frac{1}{(2\pi)^r} \frac{W_0(q^{-1})}{|W_0|} \int\limits_{\mathscr{D}_r}
\hat{A}(i\theta)^n \overline{P_\omega(i\theta)} \frac{d\theta}{{{\lvert {c(i\theta)} \rvert}}^2}.$$ Using the definition of $P_\omega$ and $W_0$-invariance of the integrand we can write $$p_n(v) = \frac{1}{(2\pi)^r}
\prod_{\alpha \in \Phi^+_0} q_\alpha^{-{{\langle \alpha/2, \omega\rangle}}} \mathcal{F}_n(\omega)$$ where $$\mathcal{F}_n(\omega) = \int\limits_{\mathscr{D}_r} \hat{A}(i\theta)^n
e^{-i {{\langle \theta, \omega\rangle}}} \frac{d\theta} {c(i\theta)}.$$ Since the integrand extends to an analytic $2\pi Q$-periodic function on $\mathfrak{b} + i\mathfrak{a}$ we can change the contour of integration and establish $$\label{eq:28}
\mathcal{F}_n(\omega)=
\hat{A}(s)^n e^{-{{\langle s, \omega\rangle}}}
\int\limits_{\mathscr{D}_r}
\bigg[ \frac{\kappa(s+i\theta)}{\kappa(s)} \bigg]^n
e^{-i {{\langle \theta, \omega\rangle}}} \frac{d\theta}{c(s+i\theta)}.$$ For $0 < \epsilon < \frac{1}{2}\min\big\{{{\left\lVert \theta_0-\theta_0' \right\rVert}}:
\theta_0, \theta_0' \in \mathscr{U} \big\}$ we set $$\mathscr{D}_r^\epsilon =
\bigcap_{\theta_0 \in \mathcal{U}}
\big\{\theta \in \mathscr{D}_r: {{\left\lVert \theta-\theta_0 \right\rVert}} \geq \epsilon\big\}.$$ By , we obtain $$\label{eq:29}
\bigg| \int\limits_{\mathscr{D}_r^\epsilon}
\bigg[ \frac{\kappa(s+i\theta)}{\kappa(s)} \bigg]^n
e^{-i {{\langle \theta, \omega\rangle}}} \frac{d\theta}{c(s+i\theta)} \bigg|
\leq C' e^{-C n {\operatorname{dist}}(\delta, \partial \mathcal{M})^\eta}.$$ Recall, for each $\theta_0 \in \mathcal{U}$ there is $t_0 \in [-\pi, \pi]$ such that $e^{it_0} = e^{i{{\langle \theta_0, v\rangle}}}$ for every $v \in \mathcal{V}$ (see [@park1 Lemma 2.12]). Hence without loss of generality we may assume $\theta_0 = 0$.
Let us consider $\{F_n\}$ a sequence of functions on $\mathfrak{b}$ defined by $$F_n(x) = \int\limits_{{{\left\lVert \theta \right\rVert}} \leq \epsilon}
e^{n\varphi(x,\theta)}
\bigg[\frac{\kappa(x)}{\kappa(x+i\theta)}\bigg]^r
\frac{e^{i{{\langle \theta, \rho\rangle}}}d\theta}{c(x+i\theta)}$$ where $\varphi$ is given by . In $\mathfrak{a}$ we fix a basis $\{\lambda_1, \ldots,
\lambda_r\}$. For $\mu, \nu \in \mathbb{N}^r$, by Lemma \[lem:1\] and formula $$\big|
{\partial^{\nu}}_x e^{n\varphi(x, \theta)}
\big|
\leq
C^{{{\lvert {\nu} \rvert}}+1} \sum_{\pi \in \Pi_\nu} c_\pi e^{-\frac{n}{4} B_x(\theta, \theta)}
(nB_x(\theta, \theta))^m \prod_{j=1}^m B_j!$$ thus, by , $$\begin{gathered}
| {\partial^{\sigma}}_x F_n(x) | \leq C^{{{\lvert {\sigma} \rvert}}+1} \big(\!\det n B_x \big)^{-1/2}
\sum_{\nu \preceq \sigma} \frac{\sigma!}{\nu!} \sum_{\pi \in \Pi_\nu}
c_\pi C^m m! \prod_{j=1}^m B_j! \\
\leq
C^{{{\lvert {\sigma} \rvert}} + 1}
\sigma!
\big(\!\det n B_x \big)^{-1/2}.\end{gathered}$$ Hence, $F_n$ is a real analytic function. Moreover, there is $\xi > 0$ such that for every $x_0 \in {\operatorname{cl}}\mathfrak{a}_+$ and $n \in \mathbb{N}$ Taylor expansion of $F_n$ at $x_0$ is convergent for ${{\left\lVert x-x_0 \right\rVert}} \leq \xi$.
We are going to describe the asymptotic behaviour of $F_n$ close to walls. The case when $x$ stays away from walls is simpler and similar to the analysis of the random walk on $\mathbb{Z}^r$.
Let $J \subset I_0$ be fixed. Then $$\Psi=\{\alpha \in \Phi: {{\langle \alpha, \lambda_j\rangle}} = 0 \text{ if and only if } j \notin J\}$$ is a root subsystem in $\mathfrak{a}_\Psi = {\operatorname{\mathbb{R}-span}}\Psi$. By $\Gamma_\Psi$ we denote a set of all multi-indices $\gamma$ such that ${\partial^{\gamma}} \Delta_\Psi \neq 0$ where $$\Delta_\Psi(x) = \prod_{\alpha \in \Psi_0^+} {{\langle \alpha, x\rangle}}.$$ We have
\[th:3\] There are $\xi, C > 0$ such that for all $x_0 \in {\operatorname{cl}}\mathfrak{a}_+ \cap
\mathfrak{a}_\Psi^\perp$ and $x \in x_0 + \{h \in \mathfrak{a}_\Psi : {{\left\lVert h \right\rVert}} \leq \xi\}$ $$\begin{gathered}
F_n(x) = \big(\! \det n B_{x_0} \big)^{-1/2}
\Delta_\Psi \big( B_{x_0}^{-1} \rho \big)
\sum_{\gamma \in \Gamma_\Psi}
\big(B_{x_0} x\big)^\gamma
n^{-{{\lvert {\Psi_0^+} \rvert}}
+ {{\lvert {\gamma} \rvert}}} A^\gamma_n(x_0, x)\\
+ E_n(x_0, x)
\end{gathered}$$ where $A^\gamma_n(x_0, x) = a_\gamma(x_0) + g_\gamma(x_0, x) + E^\gamma_n(x_0, x)$ and $$\begin{aligned}
&|a_\gamma(x_0) | \leq C,\\
&|g_{\gamma}(x_0, x)| \leq C {{\left\lVert x - x_0 \right\rVert}},\\
&|E_n^{\gamma}(x_0, x)| \leq C n^{-1} \lVert B_{x_0}^{-1} \rVert,\\
&|E_n(x_0, x)| \leq C e^{-C n {{\left\lVert B_{x_0}^{-1} \right\rVert}}^{-1}}.
\end{aligned}$$
First, we modify the basis in $\mathfrak{a}_\Psi$ by choosing $T_\Psi \lambda_j$ for $j \in J$ where $T_\Psi: \mathfrak{a} \rightarrow \mathfrak{a}$ is the orthogonal projection onto $\mathfrak{a}_\Psi$. Recall the formula for $T_\Psi$ $$T_\Psi x = x - \frac{1}{{{\lvert {W_0(\Psi)} \rvert}}} \sum_{w \in W_0(\Psi)} w \cdot x.$$ We write $$F_n(x) = \int\limits_{{{\left\lVert x \right\rVert}} \leq \epsilon} e^{n\varphi(x, \theta)} f(x, \theta)
\Delta_\Psi(x+i\theta) d\theta$$ where $$f(x, \theta) = \frac{e^{i {{\langle \theta, \rho\rangle}}}} {c(x+i\theta) \Delta_\Psi(x+i\theta)}
\bigg[\frac{\kappa(x)}{\kappa(x+i\theta)}\bigg]^r.$$ Notice, if $\epsilon > 0$ is small enough then there is $C > 0$ such for each $\mu, \nu \in \mathbb{N}^r$ $$\big\lvert {\partial^{\mu}}_x {\partial^{\nu}}_\theta f(x,\theta) \big \rvert
\leq C^{{{\lvert {\mu} \rvert}}+{{\lvert {\nu} \rvert}}+1} \mu! \nu!$$ for all $x \in {\operatorname{cl}}\mathfrak{a}_+$ and ${{\left\lVert \theta \right\rVert}} \leq \epsilon$.
Fix $x_0 \in {\operatorname{cl}}\mathfrak{a}_+ \cap \mathfrak{a}_\Psi^\perp$. We will establish the asymptotic of ${\partial^{\sigma}} F_n(x_0)$ for any $\sigma \in \mathbb{N}^J$. We have $${\partial^{\sigma}} F_n(x_0) = \sum_{\nu+\mu = \sigma} \frac{\sigma!}{\nu!\mu!}
\int\limits_{{{\left\lVert \theta \right\rVert}} \leq \epsilon} {\partial^{\nu}}_x
\big\{ e^{n\varphi(x, \theta)} f(x, \theta) \big\}_{x_0}
{\partial^{\mu}} \Delta_\Psi(i\theta) d\theta.$$ For $\mu + \nu = \sigma$, $\mu \in \Gamma_\Psi$ we set $$I_n^{\mu\nu} = \int\limits_{{{\left\lVert \theta \right\rVert}} \leq \epsilon}
{\partial^{\mu}}_\theta {\partial^{\nu}}_x \big\{ e^{n\varphi(x, \theta)} f(x, \theta) \big\}_{x_0}
\Delta_\Psi(\theta) d\theta.$$ Then Integration by Parts yields $$\begin{gathered}
\bigg| \int\limits_{{{\left\lVert \theta \right\rVert}} \leq \epsilon}
{\partial^{\nu}}_x \big\{ e^{n \varphi(x, \theta)} f(x, \theta) \big\}_{x_0}
{\partial^{\mu}} \Delta_\Psi(\theta) d\theta
- (-1)^{{{\lvert {\mu} \rvert}}} I^{\mu\nu}_n \bigg|\\
\leq C^{{{\lvert {\sigma} \rvert}} + 1} \nu! \mu! e^{-C' n \lVert B_{x_0}^{-1} \rVert^{-1}}.
\end{gathered}$$ Therefore, it is enough to find the asymptotic of $I^{\mu\nu}_n$. Let $\gamma$ denote a maximal multi-index belonging to $\Gamma_\Psi$ and such that $\mu \preceq \gamma \preceq \sigma$. We claim $$\label{eq:36}
I_n^{\mu\nu} = \big(\! \det nB_{x_0} \big)^{-1/2}
\Delta_\Psi {\big(B_{x_0}^{-1} \rho\big)} {\big(B_{x_0}^{} \rho\big)}^{\gamma} n^{- {{\lvert {\Psi_0^+} \rvert}} + {{\lvert {\gamma} \rvert}}}
A^{\mu\nu}_n(x_0)$$ where $A^{\mu\nu}_n(x_0) = a_{\mu\nu}(x_0) + E_n^{\mu\nu}(x_0)$ and $$\begin{aligned}
|a_{\mu\nu}(x_0)| &\leq& C^{{{\lvert {\sigma} \rvert}} + 1} \mu!\nu!\\
|E_n^{\mu\nu}(x_0)| &\leq& C^{{{\lvert {\sigma} \rvert}} + 1} \mu!\nu! n^{-1}
\big\lVert B_{x_0}^{-1} \big\rVert.
\end{aligned}$$ Let $\{\nu_j\}_{j = 0}^m$, $\{\mu_j\}_{j=0}^m$ be such that ${{\lvert {\mu_j} \rvert}}+{{\lvert {\nu_j} \rvert}} \geq 1$ for $j \geq 1$ and $$\mu = \sum_{j \geq 0} \mu_j, \quad \nu = \sum_{j \geq 0} \nu_j.$$ To show we need to establish the asymptotic of $$I_n = \int\limits_{{{\left\lVert \theta \right\rVert}} \leq \epsilon} e^{n \varphi(x_0, \theta)}
\Big( \prod_{j=0}^m g_j(\theta) \Big)
\Delta_\Psi(\theta) d\theta$$ where $$\nu_j! \mu_j! g_j(\theta) =
\begin{cases}
{\partial^{\mu_0}}_\theta{\partial^{\nu_0}}_x f(x_0, \theta) & \text{if } j = 0,\\
{\partial^{\mu_j}}_\theta {\partial^{\nu_j}}_x \varphi(x_0, \theta) & \text{otherwise.}
\end{cases}$$ We introduce an auxiliary root system $$\Upsilon = \{ \alpha \in \Psi: {{\langle \alpha, \lambda_j\rangle}} \neq 0 \text{ if and only if }
\gamma + e_j \in \Gamma_\Psi \}.$$ For a multi-index $\beta$ we set $$\beta'(j) =
\begin{cases}
\beta(j) & \text{if } \alpha_j \in \Upsilon,\\
0 & \text{otherwise,}
\end{cases}$$ and $\beta'' = \beta - \beta'$. Let $\Lambda_0 = \{j \geq 1: {{\lvert {\nu_j'} \rvert}}+{{\lvert {\mu_j'} \rvert}} = 0\}$.
To describe the asymptotic of $I_n$ we need to construct a sequence $\{\beta_j\}_{j=0}^m$. If $j \in \Lambda_0$ take $\beta_j \preceq \mu_j$, ${{\lvert {\beta_j} \rvert}} = \min\{2, {{\lvert {\mu_j} \rvert}}\}$ or else $\beta_j \preceq 2(\nu_j'+\mu_j')$, ${{\lvert {\beta_j} \rvert}} = 2$ and ${{\lvert {\beta_0} \rvert}} = 0$. Let $\beta = \sum_{j=0}^m \beta_j$. Since $\mu'' \preceq\gamma''$ and $\gamma' = \nu' + \mu'$ we have $$\label{eq:12}
\beta \preceq \mu'' + 2 \gamma' \preceq \gamma + \gamma'.$$ We set $k_0 = {{\lvert {\Upsilon_0^+} \rvert}} + \sum_{j \in \Lambda_0} (2-{{\lvert {\beta_j} \rvert}})$. We are going to show $$\label{eq:19}
I_n = \big(\!\det B_{x_0}\big)^{-1/2} \Delta_\Psi {\big(B_{x_0}^{-1} \rho\big)} {\big(B_{x_0}^{1/2} \rho\big)}^{\beta+\gamma''}
n^{-(k_0+{{\lvert {\Psi_0^+} \rvert}}+r)/2} A_n(x_0)$$ where $A_n(x_0) = a(x_0) + E_n(x_0)$ $$\begin{aligned}
|a(x_0)| &\leq& C^{{{\lvert {\sigma} \rvert}} + m + 1} m!\\
|E_n(x_0)| &\leq& C^{{{\lvert {\sigma} \rvert}} + m + 1} m! n^{-1} \big\lVert B_{x_0}^{-1} \big\rVert.
\end{aligned}$$ Notice, if $j \in \Lambda_0$ the function $g_j$ is $W_0(\Upsilon)$-invariant. Hence, we may write $$I_n = \frac{1}{{{\lvert {W_0(\Upsilon)} \rvert}}}
\int\limits_{{{\left\lVert \theta \right\rVert}} \leq \epsilon}
e^{n\varphi(x_0, \theta)}
G(\theta) \Delta_\Psi(\theta)d\theta$$ where $$G(\theta) = \prod_{j \in \Lambda_0} g_j(\theta) \sum_{w \in W_0(\Upsilon)} (-1)^{l(w)}
\prod_{j \notin \Lambda_0} g_j(w \cdot \theta).$$ Since the function $G$ is real analytic to analyse $I_n$ we develop $G(\theta)$ into its Taylor series about $\theta = 0$. We need the following three Propositions.
\[prop:1\] Let $\tau \in \mathbb{N}^r$, ${{\lvert {\tau} \rvert}} \geq 2$. If $\tau(k) \geq 1$ for $k \in J$ then $$| {\partial^{\tau}} \log \kappa(x_0) | \leq C^{{{\lvert {\tau} \rvert}}+1} \tau! {{\langle \alpha_k, B_{x_0} \rho\rangle}}.$$
Let $f(x) = {\partial^{\tau - e_k}} \log \kappa(x)$. Suppose that for each $e_j \preceq \tau - e_k$ we have ${{\langle \lambda_j, T_\Psi \lambda_k\rangle}} = 0$. Then ${{\langle \lambda_j, T_\Psi \lambda_k\rangle}} \neq 0$ implies $f(r_j x) = f(x)$ and $\alpha_j \in
\Psi$. Hence $$D_{\alpha_j} f(x_0) = - D_{r_j \alpha_j} f(x_0) = 0$$ and so $${\partial^{}}_k f(x_0) = \sum_{j: \alpha_j \in \Psi} {{\langle \lambda_j, T_\Psi \lambda_k\rangle}}
D_{\alpha_j} f(x_0) = 0.$$ Otherwise, there is $e_j \preceq \tau - e_k$ such that ${{\langle \lambda_j, T_\Psi \lambda_k\rangle}}
\neq 0$. Since $j \in J$ we have $$B_{x_0} T_\Psi \lambda_j = {{\langle \alpha_j, B_{x_0}\rho\rangle}} T_\Psi \lambda_j.$$ Therefore, we obtain $${{\langle \alpha_j, B_{x_0}\rho\rangle}} {{\langle T_\Psi \lambda_j, \lambda_k\rangle}}
={{\langle B_{x_0} T_\Psi \lambda_j, \lambda_k\rangle}}
={{\langle \alpha_k, B_{x_0}\rho\rangle}} {{\langle \lambda_j, T_\Psi \lambda_k\rangle}}$$ and so ${{\langle \alpha_j, B_{x_0}\rho\rangle}} = {{\langle \alpha_k, B_{x_0} \rho\rangle}}$. By the proof is finished.
\[prop:2\] Let $A$ be a self-adjoint operator commuting with $W_0(\Psi)$. If $\tau \in \mathbb{N}^r$ satisfies $$\sum_{w \in W_0(\Upsilon)} (-1)^{l(w)} (w \cdot \rho)^\tau \neq 0$$ then there is $\eta \preceq \tau$ such that $(A \rho)^\eta \Delta_\Upsilon(\rho)
= \Delta_\Upsilon(A \rho)$.
Without loss of generality, we may assume $\tau(j) = 0$ if $\alpha_j \notin \Psi$. Let $$f(x) = \sum_{w \in W_0(\Upsilon)} (-1)^{l(w)} (w \cdot x)^\tau.$$ Since $A \alpha_j = {{\langle \alpha_j, A \rho\rangle}} \alpha_j$ for $j \in J$ we have $f(A x) = (A \rho)^\tau f(x)$. Fix a multi-index $\beta$ such that ${\partial^{\beta}} f(0)
\neq 0$. Then $(A \rho)^\beta=(A \rho)^\tau$. Let $Q$ be a polynomial satisfying $f=\Delta_\Upsilon Q$. There are multi-indices $a+b = \beta$ such that ${\partial^{a}}
\Delta_\Upsilon(0) \neq 0$ and ${\partial^{b}} Q(0) \neq 0$. We choose $\eta \preceq \tau$ satisfying $$\sum_{j \in I_k} \eta(j) = \sum_{j \in I_k} a(j) \leq \sum_{j \in I_k} \beta(j)
= \sum_{j \in I_k} \tau(j)$$ where $I_k = \{j: {{\langle \alpha_j, A \rho\rangle}} = {{\langle \alpha_k, A \rho\rangle}}\}$ for $k \in J$. Then $$\Delta_\Upsilon(A \rho) = (A \rho)^a \Delta_\Upsilon(\rho)
= (A \rho)^\eta \Delta_\Upsilon(\rho)$$ what finishes the proof.
\[prop:3\] Let $A$ be a self-adjoint operator commuting with $W_0(\Psi)$. Then $$\Delta_\Upsilon(\rho) \Delta_\Psi (A \rho)
= \Delta_\Psi(\rho) \Delta_\Upsilon (A \rho) (A \rho)^{\gamma''}.$$
Let $X = \Psi_0^+$ and $C_i = \{\alpha \in \Psi_0^+: {{\langle \alpha, \lambda_i\rangle}} > 0\}$. Then $\gamma \in \Gamma_\Psi$ is admissible (see Section \[subsec:1\]). We choose any partial partition $\{X_j\}_{j \in J}$ corresponding to $\gamma$. By Lemma \[lem:5\] we obtain $$\prod_{\alpha \notin \Upsilon} {{\langle \alpha, A \rho\rangle}}
=\prod_{j: \alpha_j \notin \Upsilon} \prod_{\alpha \in X_j} {{\langle \alpha, A\rho\rangle}}
=(A \rho)^{\gamma''} \prod_{\alpha \notin \Upsilon} {{\langle \alpha, \rho\rangle}}.$$ since for $\alpha \in X_j$ we have ${{\langle A\alpha, \rho\rangle}} = {{\langle A\alpha_j, \rho\rangle}}{{\langle \alpha, \rho\rangle}}$.
We resume the analysis of $I_n$. For $j \in \Lambda_0$ let $k_j \in \mathbb{N}$ be such that $k_j + {{\lvert {\mu_j} \rvert}} \geq 2$. Then by we have $$\label{eq:14}
\lvert
{\operatorname{D}}_\theta^{k_j} g_j (0)
\rvert
\leq
C^{k_j + {{\lvert {\nu_j} \rvert}}+{{\lvert {\mu_j} \rvert}}+1} k_j!
{\big(B_{x_0}^{1/2} \rho\big)}^{\beta_j} {\big\lVert B_{x_0}^{1/2} \theta \big\rVert}^{2-{{\lvert {\beta_j} \rvert}}}
\lVert \theta \rVert^{k_j - 2 + {{\lvert {\beta_j} \rvert}}}.$$ Let us consider $\{\tau_j\}_{j \notin \Lambda_0}$ such that $$\sum_{w \in W_0(\Upsilon)} (-1)^{l(w)} \prod_{j \notin \Lambda_0}
\big(w \cdot \rho \big)^{\tau_j} \neq 0.$$ In particular, $\sum_{j \notin \Lambda_0} {{\lvert {\tau_j} \rvert}} \geq {{\lvert {\Upsilon_0^+} \rvert}}$. For $j \notin \Lambda_0$, by Proposition \[prop:1\] we obtain $$\label{eq:15}
\lvert {\partial^{\tau_j}} g_j(0) \rvert
\leq C^{{{\lvert {\tau_j} \rvert}}+{{\lvert {\nu_j} \rvert}}+{{\lvert {\mu_j} \rvert}}+1} \tau_j! {\big(B_{x_0}^{1/2} \rho\big)}^{\beta_j}.$$ Furthermore, by Proposition \[prop:2\], there is $\eta \preceq \sum_{j \notin \Lambda_0} \tau_j$ such that $$\label{eq:16}
\theta^\eta \Delta_\Upsilon(\rho) = \Delta_\Upsilon {\big(B_{x_0}^{-1/2} \rho\big)}
\big(B_{x_0}^{1/2} \theta \big)^\eta.$$ Therefore, for $k \geq {{\lvert {\Upsilon_0^+} \rvert}}$ $$\begin{gathered}
\Big\lvert
\sum_{w \in W_0(\Upsilon)} (-1)^{l(w)} {\operatorname{D}}_{w \cdot \theta}^k
\Big\{\prod_{j \notin \Lambda_0} g_j\Big\}_{\theta = 0}
\Big\rvert
\leq
C^k k!
\Delta_\Upsilon {\big(B_{x_0}^{-1/2} \rho\big)}
{\big\lVert B_{x_0}^{1/2} \theta \big\rVert}^{{{\lvert {\Upsilon_0^+} \rvert}}}\\
\times
{{\left\lVert \theta \right\rVert}}^{k - {{\lvert {\Upsilon_0^+} \rvert}}}
\prod_{j \notin \Lambda_0} C^{{{\lvert {\nu_j} \rvert}}+{{\lvert {\mu_j} \rvert}}+1}
{\big(B_{x_0}^{1/2} \rho\big)}^{\beta_j}.
\end{gathered}$$ Hence, by estimates , if $k \geq k_0$ we get $$\label{eq:17}
\lvert
{\operatorname{D}}_\theta^k G(0)
\rvert
\leq
C^{{{\lvert {\sigma} \rvert}}+k + m +1} k!
\Delta_{\Upsilon} {\big(B_{x_0}^{-1/2} \rho\big)}
{\big(B_{x_0}^{1/2} \rho\big)}^{\beta}
{\big\lVert B_{x_0}^{1/2} \theta \big\rVert}^{k_0}
{{\left\lVert \theta \right\rVert}}^{k - k_0}.$$ and ${\operatorname{D}}^k_\theta G(0) = 0$ for $k < k_0$. Taking $K \geq k_0$ and ${{\left\lVert \theta \right\rVert}} \leq \epsilon$ for $\epsilon$ small enough, we obtain $$\begin{gathered}
\label{eq:18}
\bigg\lvert
\sum_{k \geq K} \frac{{\operatorname{D}}_\theta^k G(0)}{k!}
\bigg\rvert
\leq C^{{{\lvert {\sigma} \rvert}}+K+m+1} \Delta_\Upsilon {\big(B_{x_0}^{-1/2} \rho\big)} {\big(B_{x_0}^{1/2} \rho\big)}^\beta
\big\lVert B_{x_0}^{-1} \big \rVert^{(K-k_0)/2}\\
\times
{\big\lVert B_{x_0}^{1/2} \theta \big\rVert}^{K}.
\end{gathered}$$ In particular, for ${{\left\lVert \theta \right\rVert}} \leq \epsilon$ $$\label{eq:13}
{{\lvert { G(\theta) } \rvert}} \leq
C^{{{\lvert {\sigma} \rvert}} + k_0 + m + 1} \Delta_\Upsilon {\big(B_{x_0}^{-1/2} \rho\big)} {\big(B_{x_0}^{1/2} \rho\big)}^\beta
{\big\lVert B_{x_0}^{1/2} \theta \big\rVert}^{k_0}.$$ We recall $$\varphi(x_0, \theta) = -\frac{1}{2} B_{x_0}(\theta, \theta) + \psi(x_0, \theta).$$ Assume $k_0 + {{\lvert {\Psi_0^+} \rvert}} \in 2 \mathbb{Z}$. We write [ $$\begin{aligned}
e^{n\psi(x_0, \theta)} G(\theta)
= && \big(e^{n\psi(x_0, \theta)} - 1 - n\psi(x_0, \theta) \big) G(\theta) \\
& + & n \psi(x_0, \theta) \bigg( G(\theta) -
\frac{{\operatorname{D}}_\theta^{k_0} G(0)}{k_0!} \bigg)\\
& + & G(\theta) - \frac{{\operatorname{D}}_\theta^{k_0} G(0)}{k_0!}
- \frac{{\operatorname{D}}_\theta^{k_0+1} G(0)}{(k_0+1)!}\\
& + & n \bigg(\psi(x_0, \theta) - \frac{D_\theta^3 \psi(x_0, 0)}{3!} \bigg)
\frac{{\operatorname{D}}_\theta^{k_0} G(0)}{k_0!}\\
& + & n \frac{D_\theta^3 \psi(x_0, 0)}{3!}
\frac{{\operatorname{D}}_\theta^{k_0} G(0)}{k_0!}
+\frac{{\operatorname{D}}_\theta^{k_0+1} G(0)}{(k_0+1)!}\\
& + & \frac{{\operatorname{D}}_\theta^{k_0} G(0)}{k_0!}
\end{aligned}$$]{} and split $I_n$ into six corresponding integrals. By formulas and and Proposition \[prop:3\], we obtain $$\begin{gathered}
\bigg| \int\limits_{{{\left\lVert \theta \right\rVert}} \leq \epsilon}
e^{-\frac{n}{2} B_{x_0}(\theta, \theta)}
\big(e^{n\psi(x_0, \theta)} - 1 - n\psi(x_0, \theta)\big) G(\theta) \Delta_\Psi(\theta)
d\theta \bigg|\\
\leq C^{{{\lvert {\sigma} \rvert}} + m + 1} m! \big(\!\det B_{x_0}\big)^{-1/2}
\Delta_\Psi \big( B_{x_0}^{-1} \rho \big) {\big(B_{x_0}^{1/2} \rho\big)}^{\beta+\gamma''}
n^{-(k_0 + {{\lvert {\Psi_0^+} \rvert}}+r)/2} \\
\times n^{-1} \big \lVert B_{x_0}^{-1} \big \rVert.
\end{gathered}$$ For the next three integrals we use , , and . The fifth is equal $0$. Finally, by for $k = k_0$ $$\begin{gathered}
\bigg|
\int\limits_{{{\left\lVert \theta \right\rVert}} \geq \epsilon}
e^{-\frac{n}{2} B_{x_0}(\theta, \theta)}
\frac{{\operatorname{D}}_\theta^{k_0} G(0)}{k_0!}
\Delta_\Psi(\theta)
d\theta \bigg|\\
\leq C^{{{\lvert {\sigma} \rvert}} + m + 1} m! \big(\!\det B_{x_0}\big)^{-1/2}
\Delta_\Psi \big( B_{x_0}^{-1} \rho \big) {\big(B_{x_0}^{1/2} \rho\big)}^{\beta+\gamma''}
n^{-(k_0 + {{\lvert {\Psi_0^+} \rvert}}+r)/2} \\
\times e^{-C' n \lVert B_{x_0}^{-1} \rVert^{-1}}.
\end{gathered}$$ Therefore, we establish in the case of $k_0 + {{\lvert {\Psi_0^+} \rvert}} \in 2 \mathbb{Z}$. If $k_0 + {{\lvert {\Psi_0^+} \rvert}} \notin 2\mathbb{Z}$ we write [ $$\begin{aligned}
e^{n\psi(x_0, \theta)} G(\theta)
= && (e^{n\psi(x_0, \theta)} - 1)G(\theta)\\
& + & G(\theta) - \frac{{\operatorname{D}}_\theta^{k_0} G(0)}{k_0!}\\
& + & \frac{{\operatorname{D}}_\theta^{k_0} G(0)}{k_0!}
\end{aligned}$$]{} and analogous reasoning gives $$\begin{gathered}
|I_n| \leq C^{{{\lvert {\sigma} \rvert}}+m+1} m! \big(\!\det B_{x_0}\big)^{-1/2}
\Delta_\Psi {\big(B_{x_0}^{-1} \rho\big)} {\big(B_{x_0}^{1/2} \rho\big)}^{\beta+\gamma''} n^{-(k_0+{{\lvert {\Psi_0^+} \rvert}}+r)/2} \\
\times n^{-1/2} \big\lVert B_{x_0}^{-1} \big\rVert^{1/2}.
\end{gathered}$$ Since ${{\lvert {\beta} \rvert}} + {{\lvert {\gamma''} \rvert}} = {{\lvert {\Psi_0^+} \rvert}} + 2m - k_0 \not\in 2\mathbb{Z}$, by we get ${{\lvert {\beta} \rvert}} + {{\lvert {\gamma''} \rvert}} < 2 {{\lvert {\gamma} \rvert}}$ thus $${\big(B_{x_0}^{1/2} \rho\big)}^{\beta+\gamma''} \leq {\big(B_{x_0}^{} \rho\big)}^{\gamma} \big\lVert B_{x_0}^{-1} \big\rVert^{1/2}.$$ Eventually, we obtain what finishes the proof of Theorem \[th:3\].
Before we apply Theorem \[th:3\] to $\mathcal{F}_n$ we need two lemmas.
\[lem:3\] There are $\xi, C > 0$ such that for all $x,y \in {\operatorname{cl}}\mathfrak{a}_+$, ${{\left\lVert x - y \right\rVert}} \leq \xi$ $$\big \lvert \big(\! \det B_x \big)^{-1/2} - \big(\! \det B_y \big)^{-1/2} \big\rvert
\leq C \big(\! \det B_x \big)^{-1/2} {{\left\lVert x - y \right\rVert}}.$$
Let $\{u_j\}_{j=1}^r$ be an orthonormal basis of $\mathfrak{a}$ diagonalizing $B_x$. By , for $u \in \mathfrak{a}$ and $k \in \mathbb{N}$ we have $$\big\lvert {\operatorname{D}}_u^k B_x(u_i, u_j) \big\rvert \leq C^{k+1} k!
{{\left\lVert u \right\rVert}}^k \sqrt{B_x(u_i, u_i)B_x(u_j, u_j)}.$$ Hence, $$\big\lvert {\operatorname{D}}_u^k \det B_x \big\rvert \leq C^{k+1} k! {{\left\lVert u \right\rVert}}^k \det B_x.$$ By Lemma \[lem:2\] we get $$\big\lvert {\operatorname{D}}_u^k \big\{\big(\!\det B_y\big)^{-1/2} \big\}_x \big\rvert \leq
C^{k+1} k! {{\left\lVert u \right\rVert}}^k \big(\!\det B_x\big)^{-1/2}$$ what finishes the proof.
\[lem:4\] There are $\xi, C > 0$ such that for any $x_0 \in {\operatorname{cl}}\mathfrak{a}_+ \cap
\mathfrak{a}_\Psi^\perp$ and all $x \in x_0 + \{h \in \mathfrak{a}_\Psi: {{\left\lVert h \right\rVert}} \leq \xi\}$ and $\alpha \in \Psi$ $$\vert {\operatorname{D}}_\alpha \log \kappa(x) - B_{x_0} (\alpha, x) \rvert
\leq C \lvert B_{x_0} (\alpha, x) \rvert {{\left\lVert x - x_0 \right\rVert}}.$$
Let $h = x - x_0$. Since $\log \kappa$ is real analytic in $\mathfrak{a}$, by there is $\xi$ such that if ${{\left\lVert h \right\rVert}} \leq \xi$ $${\operatorname{D}}_\alpha \log \kappa (x)
= \sum_{k \geq 0} \frac{1}{k!} {\operatorname{D}}_h^k {\operatorname{D}}_\alpha \log \kappa(x_0).$$ For $k \geq 2$, we can write ${\operatorname{D}}_h^k {\operatorname{D}}_\alpha \log \kappa(x_0)$ as $$\begin{gathered}
-{\operatorname{D}}_{h}^k {\operatorname{D}}_{r_\alpha \alpha} \log \kappa(x_0)
= -\big({\operatorname{D}}_h - {{\langle h, {\check{\alpha}}\rangle}} {\operatorname{D}}_\alpha\big)^k {\operatorname{D}}_\alpha \log \kappa(x_0)\\
= -{\operatorname{D}}_h^k {\operatorname{D}}_\alpha \log \kappa(x_0) - \sum_{j=1}^k \frac{k!}{j! (k-j)!} (-1)^j
{{\langle h, {\check{\alpha}}\rangle}}^j {\operatorname{D}}_h^{k-j} {\operatorname{D}}_\alpha^{j+1} \log \kappa(x_0).
\end{gathered}$$ Since for $k \geq j \geq 1$ we may estimate $$\lvert {\operatorname{D}}_h^{k-j} {\operatorname{D}}_\alpha^{j+1} \log \kappa(x_0) \rvert
\leq C^{k+1} (k-j)! j! B_{x_0}(\alpha, \alpha) {{\left\lVert h \right\rVert}}^{k-j} {{\left\lVert \alpha \right\rVert}}^{j-1}$$ and $$B_{x_0} \alpha = -r_\alpha \big(B_{x_0} \alpha\big)
= -B_{x_0} \alpha + B_{x_0}(\alpha, \alpha) {\check{\alpha}}$$ we get $$\lvert {\operatorname{D}}_h^k {\operatorname{D}}_\alpha \log \kappa(x_0) \rvert
\leq C^{k+1} k! \lvert B_{x_0}(\alpha,h) \rvert {{\left\lVert h \right\rVert}}^k.$$ Therefore, we obtain $$\lvert {\operatorname{D}}_\alpha \log \kappa(x) - {\operatorname{D}}_h {\operatorname{D}}_\alpha \log \kappa(x_0) \rvert
\leq C \lvert B_{x_0}(\alpha, h)\rvert {{\left\lVert h \right\rVert}}.$$
Now, we are ready to finish the proof of Theorem \[th:4\]. Let $\xi, \zeta, K > 0$ and assume $v \in V_\omega(O) \cap {\operatornamewithlimits{supp}}p_n$ is such that for $\delta = (n+r)^{-1}(\omega + \rho)$ and $s = s(\delta)$ $$\begin{cases}
{{\langle \alpha_j, s\rangle}} \leq \xi &\text{if } j \in J,\\
{{\langle \alpha_j, s\rangle}} \geq \zeta &\text{otherwise}
\end{cases}$$ and $n {\operatorname{dist}}(\delta, \partial \mathcal{M})^{2\eta} \geq K$. We set $t = (I - T_\Psi) s$. By Theorem \[th:3\], Lemma \[lem:3\] and Lemma \[lem:4\] we get $$\begin{gathered}
\label{eq:21}
\mathcal{F}_n(\omega) =
\hat{A}(s)^n e^{-{{\langle s, \omega\rangle}}}
\big(\!\det B_{s} \big)^{-1/2} \Delta_\Psi \big(B_{t}^{-1} \rho \big)
(n+r)^{-r/2-{{\lvert {\Psi_0^+} \rvert}}}\\
\times \sum_{\gamma \in \Gamma_\Psi} (\omega + \rho)^\gamma A_n^\gamma(t, s).\end{gathered}$$ We are going to calculate the leading term. For a fixed $x_0 \in {\operatorname{cl}}\mathfrak{a}_+ \cap
\mathfrak{a}_\Psi^\perp$ $$\delta_0 = \nabla \log \kappa(x_0)$$ belongs to $\mathcal{M} \cap {\operatorname{cl}}\mathfrak{a}_+$. Given $\nu \in P^+$ there is a sequence $\omega_n \in {\operatornamewithlimits{supp}}p_n$ such that $$\lim_{n \to \infty} n^{-1} \omega_n = \delta_0$$ and containing a subsequence satisfying $T_\Psi \omega_{n_k} = T_\Psi \nu$. Let $\delta_n = (n+r)^{-1} (\omega_n + \rho)$ and $s_n = s(\delta_n)$. Then we get $$\begin{gathered}
\label{eq:32}
\lim_{k \to \infty} n_k^{r/2+{{\lvert {\Psi_0^+} \rvert}}}
e^{{{\langle s_{n_k}, \omega_{n_k}\rangle}}} \hat{A}(s_{n_k})^{-n_k}
\mathcal{F}_{n_k}(\omega_{n_k})\\
=\big(\! \det B_{x_0}\big)^{-1/2} \Delta_\Psi(B_{x_0}^{-1} \rho)
\sum_{\gamma \in \Gamma_\Psi} \big(\nu + \rho\big)^\gamma a_\gamma(x_0).\end{gathered}$$ The limit in can be obtained by another approach. Let $\tilde{s}_n$ denote the unique solution to $\nabla \log \kappa(s) = (n+r)^{-1} (I-T_\Psi) (\omega_n + \varrho)$. We define $$G_n(x) =
\int\limits_{{{\left\lVert \theta \right\rVert}} \leq \epsilon}
e^{n\varphi(x, \theta)} e^{-i{{\langle \theta, T_\Psi(\nu + \rho)\rangle}}}
\bigg[\frac{\kappa(x)}{\kappa(x+i\theta)}\bigg]^r
\frac{e^{i{{\langle \theta, \rho\rangle}}}d\theta}{c(x+i\theta)}.$$ By and we get $$\bigg\lvert
\mathcal{F}_{n_k}(\omega_{n_k}) \hat{A}(\tilde{s}_{n_k})^{-n_k}
e^{{{\langle \tilde{s}_{n_k}, \omega_{n_k}\rangle}}}
- {{\lvert {\mathscr{U}} \rvert}} \cdot G_{n_k}(\tilde{s}_{n_k}) \bigg\rvert \leq C' e^{-C {n_k}}.$$ Since for a fixed $x \in {\operatorname{cl}}\mathfrak{a}_+ \cap \mathfrak{a}_\Psi^\perp$ a function $$g(x, \theta) = \frac{e^{i{{\langle \theta, (I-T_\Psi)\rho\rangle}}}}
{c(x+i\theta) c_\Psi(-i\theta) {{\lvert {\Delta_\Psi(\theta)} \rvert}}^2}
\bigg[\frac{\kappa(x)}{\kappa(x+i\theta)}\bigg]^r$$ is $W_0(\Psi)$-invariant we may write $$G_n(x) =
\int\limits_{{{\left\lVert \theta \right\rVert}} \leq \epsilon}
e^{n \varphi(x, \theta)}
Q_\nu(-i\theta) g(x, \theta) {{\lvert {\Delta_\Psi(\theta)} \rvert}}^2 d\theta$$ where $$Q_\nu(z) = \frac{1}{{{\lvert {W_0(\Psi)} \rvert}}} \sum_{w \in W_0(\Psi)} c_\Psi(w \cdot z)
e^{{{\langle w \cdot z, T_\Psi \nu\rangle}}}.$$ We have
\[prop:4\] There is $C > 0$ such that for any smooth function $f$ and $y \in \mathfrak{a}_\Psi^\perp$ $$\begin{gathered}
\label{eq:30}
\int\limits_{{{\left\lVert \theta \right\rVert}} \leq \epsilon}
e^{n\phi(y, \theta)} f(\theta) \lvert \Delta_\Psi(\theta) \rvert^2 d\theta
= \big(\! \det n B_y \big)^{-1/2} \Delta_\Psi\big(B_y^{-1} \rho \big)
n^{-{{\lvert {\Psi_0^+} \rvert}}} \\
\times \big( C_\Psi f(0) + E_n(y) \big)
\end{gathered}$$ where $$\lvert E_n(y) \rvert \leq C n^{-1} \big \lVert B_y^{-1} \big\rVert
\sum_{|\sigma| \leq 2} \sup_{{{\left\lVert \theta \right\rVert}} \leq \epsilon} \big\lvert
{\partial^{\sigma}} f(\theta) \rvert$$ and $$C_\Psi = \int\limits_{\mathfrak{a}} e^{-\frac{1}{2} {{\langle u, u\rangle}}}
\lvert \Delta_\Psi(u) \rvert^2 du.$$
The proof is left to the reader.
By we obtain $$\begin{gathered}
\label{eq:41}
\lim_{k \to \infty} n_k^{r/2+{{\lvert {\Psi_0^+} \rvert}}}
e^{{{\langle \tilde{s}_{n_k}, \omega_{n_k}\rangle}}}
\hat{A}(\tilde{s}_{n_k})^{-n_k}
\mathcal{F}_{n_k}(\omega_{n_k})\\
= {{\lvert {\mathscr{U}} \rvert}}\cdot C_\Psi \big(\! \det B_{x_0}\big)^{-1/2}
\Delta_\Psi(B_{x_0}^{-1} \rho) g(x_0, 0) Q_\nu(0).\end{gathered}$$ We claim $$\lim_{k \to \infty} n_k \big(\log \kappa(s_{n_k}) - \log \kappa(\tilde{s}_{n_k})
- {{\langle s_{n_k} - \tilde{s}_{n_k}, n_k^{-1}\omega_{n_k}\rangle}} \big) = 0.$$ Indeed, writing Taylor polynomial centred at $\tilde{s}_n$ we get $$\big \lvert \log \kappa(s_n) - \log \kappa(\tilde{s}_n) - {\operatorname{D}}_{s_n-\tilde{s}_n}
\log \kappa(\tilde{s}_n) \big \rvert \leq C \lVert s_n - \tilde{s}_n \rVert^2.$$ By Theorem \[th:1\], we can estimate $$\lvert
{\operatorname{D}}_{s_{n_k}-\tilde{s}_{n_k}} \log \kappa(\tilde{s}_{n_k}) -
{{\langle s_{n_k} - \tilde{s}_{n_k}, n_k^{-1} \omega_{n_k}\rangle}}
\rvert \leq C n_k^{-1} {{\left\lVert s_{n_k} - \tilde{s}_{n_k} \right\rVert}} {{\left\lVert \nu + \rho \right\rVert}}$$ and, by , we get $$\lVert s_n - \tilde{s}_n \rVert \leq C n^{-1} \lVert T_\Psi(\omega_n + \rho)\rVert$$ proving the claim. Now, we may compare and getting $$\label{eq:38}
\sum_{\gamma \in \Gamma_\Psi} (\nu + \rho)^\gamma a_\gamma(x_0) =
{{\lvert {\mathscr{U}} \rvert}}\cdot C_\Psi g(x_0, 0) Q_\nu(0).$$
If $\xi$ is small enough, for any $\alpha \in \Phi_0^+ \setminus \Psi_0^+$ we have $${{\langle \alpha, t\rangle}} \geq \zeta - \sum_{k \in J} {{\langle \alpha_k, s\rangle}}
{{\langle T_\Psi \alpha, \lambda_k\rangle}} \geq \zeta/2.$$ Hence, we obtain $$\label{eq:39}
C \geq g(t, 0) \geq C^{-1} \big(1 - e^{-\zeta/2}\big)^{{{\lvert {\Phi_0^+} \rvert}}-{{\lvert {\Psi_0^+} \rvert}}}.$$ Since (see [@a1; @ascht]) $$C \Delta_\Psi(\omega + \rho) \geq Q_{\omega}(0) \geq C^{-1} \Delta_\Psi(\omega + \rho),$$ by and , we get $$C \Delta_\Psi(\omega + \rho) \geq
\sum_{\gamma \in \Gamma_\Psi} (\omega + \rho)^\gamma a_\gamma(t)
\geq C^{-1} \Delta_\Psi(\omega + \rho)
\big(1 - e^{-\zeta/2}\big)^{{{\lvert {\Phi_0^+} \rvert}} - {{\lvert {\Psi_0^+} \rvert}}}.$$ Moreover, $$\bigg\lvert \sum_{\gamma \in \Gamma_\Psi} (\omega + \rho)^\gamma g_\gamma(t, s) \bigg\rvert
\leq C \xi \Delta_\Psi(\omega + \rho)$$ and $$\bigg\lvert \sum_{\gamma \in \Gamma_\Psi} (\omega + \rho)^\gamma E^\gamma_n(t, s) \bigg\rvert
\leq C K^{-1} \Delta_\Psi(\omega + \rho).$$ Therefore, taking $\xi$ small and $K$ large enough we get $$\mathcal{F}_n(\omega) \asymp
\hat{A}(s)^n e^{-{{\langle s, \omega\rangle}}}
\big(\!\det n B_{s} \big)^{-1/2}
\Delta_\Psi(B_{t}^{-1} \delta).$$ If we set $\tilde{\delta} = n^{-1} \omega$ and $\tilde{s} = s(n^{-1} \omega)$ then $$-\phi(\tilde{\delta}) \leq \log \kappa(s) - {{\langle s, \tilde{\delta}\rangle}}
+{{\langle \tilde{s}-s, \delta - \tilde{\delta}\rangle}}.$$ Since $\big \lVert \delta - \tilde{\delta} \big \rVert \leq C n^{-1}$ by and Theorem \[th:2\] $$\big\lvert
{{\langle \tilde{s} - s, \delta - \tilde{\delta}\rangle}}
\big\rvert
\leq
C n^{-2} {\operatorname{dist}}(\delta, \partial \mathcal{M})^{-\eta}.$$ Hence, $$-\phi(\tilde{\delta}) \geq \log \kappa(s) - {{\langle s, \tilde{\delta}\rangle}}
\geq -\phi(\tilde{\delta}) - C n^{-2} {\operatorname{dist}}(\delta, \partial\mathcal{M})^{-\eta}$$ what finishes the proof of Theorem \[th:4\]. As a corollary we obtain
\[th:5\] For $\epsilon > 0$ $$p_n(v) \asymp
n^{-r/2 - {{\lvert {\Phi_0^+} \rvert}}}
\varrho^n
e^{-n \phi(n^{-1} \omega)}
P_{\omega}(0)$$ uniformly on $\big\{v \in V_{\omega}(O) \cap {\operatornamewithlimits{supp}}p_n:
{\operatorname{dist}}(n^{-1} \omega, \partial \mathcal{M}) \geq \epsilon \big\}$.
\[rem:1\] There are $N > 0$ and $C > 0$ such that $$p_n(v) \leq C n^N \varrho^n e^{-n \phi(n^{-1} \omega)} P_{\omega}(0)$$ for all $v \in V_{\omega}(O) \cap {\operatornamewithlimits{supp}}p_n$.
Green function estimates {#subsec:4.6}
------------------------
In this section we show sharp estimates for Green function of the random walk with the transition probability $p$. Having Theorem \[th:4\] the proof is analogous to the proof of [@ascht Theorem 5.1] but for convenience of the reader we include the details. We recall Green function $G_\zeta$ is defined for $\zeta \in (0, \varrho^{-1}]$ and $x, y \in V_P$ by $$G_\zeta(x, y) = \sum_{n \geq 0} \zeta^n p_n(x, y).$$ Given $\zeta \in (0, \varrho^{-1})$ let $\mathcal{C} = \{x \in \mathfrak{a} : \kappa(x)
= (\zeta \varrho)^{-1}\}$. For $u \in \mathcal{S}$, the unit sphere in $\mathfrak{a}$ centred at the origin, there is the unique point $s_u \in \mathcal{C}$ such that $$\nabla \kappa(s_u) = \lVert \nabla \kappa(s_u) \rVert u.$$ We have
\[th:6\] (i) If $\zeta \in (0, \varrho^{-1})$ then for all $x \neq O$ $$G_\zeta(O, x)
\asymp P_\omega(0) {{\left\lVert \omega \right\rVert}}^{-(r-1)/2 - {{\lvert {\Phi_0^+} \rvert}}} e^{-{{\langle s_u, \omega\rangle}}}$$ where $x \in V_\omega(O)$ and $u = {{\left\lVert \omega \right\rVert}}^{-1} \omega$.
\(ii) If $\zeta = \varrho^{-1}$ then for all $x \neq O$ $$G_\zeta(O, x) \asymp P_\omega(0) {{\left\lVert \omega \right\rVert}}^{2-r-2{{\lvert {\Phi_0^+} \rvert}}}.$$ where $x \in V_\omega(O)$.
We set $t_0 = \min \{t > 0: t^{-1} u \in \mathcal{M}\}$. If $t>t_0$ we define $s_t = s(t^{-1} u)$ and $$\psi(t, u) = t (\log (\zeta \varrho) - \phi(t^{-1} u)).$$ Then $$\frac{d}{dt} \psi(t, u) = \log(\zeta \varrho) + \log \kappa(s_t), \qquad
\frac{d^2}{dt^2} \psi(t, u) = -t^{-3} B_{s_t}^{-1} (u,u).$$ Since $$\lim_{t \to t_0} \kappa(s_t) = +\infty, \qquad \lim_{t \to +\infty} \kappa(s_t) = 1$$ there is the unique $t_u > t_0$ where $\psi(\cdot, u)$ attains its maximum. Since $$0 = \frac{d}{dt} \psi(t_u, u) = \log(\zeta \varrho) + \log \kappa(s_{t_u})$$ and $\nabla \log \kappa(s_{t_u}) = t_u^{-1} u$ we get $s_u = s_{t_u}$. By compactness of $\mathcal{S}$ there is $C > 0$ such that $C^{-1} \leq t_u \leq C$ for all $u \in \mathcal{S}$. Moreover, for sufficiently small $\epsilon > 0$ there is $C > 0$ such that $$\label{eq:44}
-C^{-1} {{\lvert {t - t_u} \rvert}}^2 \leq \psi(t, u) - \psi(t_u, u) \leq -C {{\lvert {t -t_u} \rvert}}^2$$ for all $u \in \mathcal{S}$ and ${{\lvert {t-t_u} \rvert}} < \epsilon$. Also since $\psi(\cdot, u)$ is concave there is $C > 0$ $$\label{eq:45}
\psi(t, u) - \psi(t_u, u) \leq -C {{\lvert {t-t_u} \rvert}}$$ for all $u \in \mathcal{S}$ and ${{\lvert {t -t_u} \rvert}} \geq \epsilon$.
Let $\epsilon > 0$. We write $$G_\zeta(O, x) = \sum_{j=0}^\infty \sum_{n \in A_j} \zeta^n p_n(x)
+ \sum_{n \in B_+} \zeta^n p_n(x)
+ \sum_{n \in B_-} \zeta^n p_n(x)$$ where $$B_+=\big\{n \in \mathbb{N}: n \geq {{\left\lVert \omega \right\rVert}} (t_u + \epsilon) \big\},$$ $$B_-=\big\{n \in \mathbb{N}: {{\left\lVert \omega \right\rVert}}(t_u - \epsilon) \geq n
\geq {{\left\lVert \omega \right\rVert}} t_0\big\}$$ and for $j \in \mathbb{N}$ $$A_j = \big\{n \in \mathbb{N}:
n \geq {{\left\lVert \omega \right\rVert}} t_0 \text{ and } 2^{-j} \epsilon >
\big\lvert {{\left\lVert \omega \right\rVert}}^{-1} n - t_u \big\rvert
\geq 2^{-j-1} \epsilon \big\}.$$ There is $C > 0$ such that for all $\omega \neq 0$ and $n \geq {{\left\lVert \omega \right\rVert}} t_0$ $$\big\lVert n^{-1} \omega - t_u^{-1} u \big\rVert \leq
C \big\lvert n {{\left\lVert \omega \right\rVert}}^{-1} - t_u \big\rvert.$$ Therefore, if $\epsilon$ is small enough there is $\xi > 0$ such that ${\operatorname{dist}}(n^{-1} \omega, \partial \mathcal{M}) \geq \xi$ for all $n \in \bigcup_{j \geq 0} A_j$ and $\omega \neq 0$. By Theorem \[th:5\] $$\sum_{j =0}^\infty \sum_{n \in A_j} \zeta^n p_n(x) \asymp P_\omega(0)
{{\left\lVert \omega \right\rVert}}^{-r/2 - {{\lvert {\Phi_0^+} \rvert}}}
\sum_{j=0}^\infty \sum_{n \in A_j} e^{{{\left\lVert \omega \right\rVert}} \psi({{\left\lVert \omega \right\rVert}}^{-1} n, u)}.$$ Next, if $n \in A_j$, by we have $$\psi({{\left\lVert \omega \right\rVert}}^{-1} n, u) + {{\langle s_u, u\rangle}} \asymp -\epsilon^2 2^{-2j}.$$ Since ${{\lvert {A_j} \rvert}} \asymp {{\left\lVert \omega \right\rVert}} \epsilon 2^{-j}$ and for any $C > 0$ $$\sum_{j=0}^\infty {{\left\lVert \omega \right\rVert}} \epsilon 2^{-j} e^{-C {{\left\lVert \omega \right\rVert}} \epsilon^2 2^{-2j}}
\asymp {{\left\lVert \omega \right\rVert}}^{1/2},$$ we obtain $$\sum_{j=0}^\infty \sum_{n \in A_j} \zeta^n p_n(x) \asymp P_\omega(0)
e^{-{{\langle s_u, \omega\rangle}}} {{\left\lVert \omega \right\rVert}}^{-(r-1)/2 - {{\lvert {\Phi_0^+} \rvert}}}.$$ For $B_+$ we can use Theorem \[th:5\] and to get $$\sum_{n \in B_+} \zeta^n p_n(x) \leq C P_\omega(0) e^{-{{\langle s_u, \omega\rangle}}}
e^{-C\epsilon{{\left\lVert \omega \right\rVert}}} \sum_{n = 1}^\infty n^{-r/2 - {{\lvert {\Phi_0^+} \rvert}}}.$$ Finally, for $B_-$ by Remark \[rem:1\] and we may write $$\sum_{n \in B_-} \zeta^n p_n(x) \leq C P_\omega(0) e^{-{{\langle s_u, \omega\rangle}}}
{{\left\lVert \omega \right\rVert}}^{N+1} e^{-C \epsilon {{\left\lVert \omega \right\rVert}}}.$$
The proof of (ii) follows the same line. First, we write $$G_\zeta(O, x) = \sum_{j=0}^\infty \sum_{n \in A_j} \zeta^n p_n(x)
+ \sum_{n \in B} \zeta^n p_n(x)$$ where $
B = \big\{n: {{\left\lVert \omega \right\rVert}} \epsilon^{-1} \geq n \geq {{\left\lVert \omega \right\rVert}} t_0
\big\}
$ and for $j \in \mathbb{N}$ $$A_j = \big\{n: n \geq {{\left\lVert \omega \right\rVert}} t_0 \text{ and }
{{\left\lVert \omega \right\rVert}} \epsilon^{-1} 2^{j+1} > n \geq
{{\left\lVert \omega \right\rVert}} \epsilon^{-1} 2^j
\big\}.$$ By , for $n \in A_j$ $$\psi({{\left\lVert \omega \right\rVert}}^{-1} n, u) \asymp - \epsilon 2^{-j}.$$ Next, ${{\lvert {A_j} \rvert}} \asymp {{\left\lVert \omega \right\rVert}} \epsilon^{-1} 2^j$ and for every $C > 0$ $$\sum_{j = 0}^\infty e^{-C {{\left\lVert \omega \right\rVert}} \epsilon 2^{-j}}
\big({{\left\lVert \omega \right\rVert}} \epsilon^{-1} 2^j\big) ^{1 -r/2 - {{\lvert {\Phi_0^+} \rvert}}}
\asymp {{\left\lVert \omega \right\rVert}}^{2 - r - 2{{\lvert {\Phi_0^+} \rvert}}}.$$ Hence, if $\epsilon$ is small enough, by Theorem \[th:5\] $$\sum_{j=0}^\infty \sum_{n \in A_j} \zeta^n p_n(x) \asymp P_\omega(0)
{{\left\lVert \omega \right\rVert}}^{2 - r - {{\lvert {\Phi_0^+} \rvert}}}.$$ For $B$ we use Remark \[rem:1\] and and get $$\sum_{n \in B} \zeta^n p_n(x) \leq C P_\omega(0) {{\left\lVert \omega \right\rVert}}^{N+1}
e^{-C \epsilon^{-1} {{\left\lVert \omega \right\rVert}}}.$$
[heat]{}
[^1]: The author expresses his gratitude to Jean–Philippe Anker, Jacek , Tim Steger, and Ryszard Szwarc for extensive discussions, comments and moral support.
[^2]: We write $f(x) \asymp g(x)$ for $x \in A$ if and only if there is a constant $C > 0$ such that $C^{-1} g(x) \leq f(x) \leq C g(x)$ for all $x \in A$
|
---
abstract: 'We present a generalization of the classical Wang-Landau algorithm \[Phys. Rev. Lett. [**86**]{}, 2050 (2001)\] to quantum systems. The algorithm proceeds by stochastically evaluating the coefficients of a high temperature series expansion or a finite temperature perturbation expansion to arbitrary order. Similar to their classical counterpart, the algorithms are efficient at thermal and quantum phase transitions, greatly reducing the tunneling problem at first order phase transitions, and allow the direct calculation of the free energy and entropy.'
author:
- 'Matthias Troyer$^{(1,2)}$'
- 'Stefan Wessel$^{(1)}$'
- 'Fabien Alet$^{(3,1,2)}$'
title: |
Wang-Landau sampling for quantum systems:\
algorithms to overcome tunneling problems and calculate the free energy
---
Monte Carlo simulations in statistical physics now have a history of nearly half a century starting with the seminal work of Metropolis [@metropolis]. While the Metropolis algorithm has been established as the standard algorithm for importance sampling it suffers from two problems: the inability to directly calculate the partition function, free energy or entropy, and critical slowing down near phase transitions and in disordered systems.
In a standard Monte Carlo algorithm a series of configurations is generated according to a given distribution, usually the Boltzmann distribution in classical simulations. While this allows the calculation of thermal averages, it does not give the partition function, nor the free energy. They can only be obtained with limited accuracy as a temperature integral of the specific heat, or by using maximum entropy methods [@maxentcv].
The problem of critical slowing down has been overcome for second order phase transitions by cluster update schemes for classical [@swendsenwang] and quantum systems [@loop; @wiese; @sseloops; @worm]. For first order phase transitions and systems with rough free energy landscapes a decisive improvement was achieved recently by a new algorithm for classical systems due to Wang and Landau [@wanglandau]. In contrast to related methods – such as the multicanonical [@multicanonical] or the broad histogram [@broadhistogram] method – this new algorithm scales to large systems, does not suffer from systematic errors and needs no a priori knowledge. The key idea is to calculate the density of states $\rho(E)$ directly by a random walk in energy space instead of performing a canonical simulation at a fixed temperature. By visiting each energy level $E$ with a probability $1/\rho(E)$, this algorithm achieves a flat histogram and good precision over the whole energy range. Besides being efficient at first and second order phase transitions this algorithm allows the direct calculation of the free energy from the partition function $Z=\sum_E \rho(E)e^{-E/k_B T}$. The internal energy, entropy, specific heat and other thermal properties are easily obtained as well, by differentiating the free energy. Within a year of publication this algorithm has been improved using $N$-fold way [@nfold] and multibondic [@multibondic] sampling schemes, has been applied to Potts models [@potts] and generalized to reaction coordinates [@reaction], continuum models [@continuum], polymer films [@polymer], and to protein folding [@proteins].
Since simulations of quantum systems suffer from the same problems as classical simulations, in particular from long tunneling times at first order phase transitions and the inability to calculate the free energy directly, an extension of this algorithm to quantum systems is highly desired. Here we present two such algorithms. The first one is based upon a high temperature series expansion. Similarly to the classical algorithm it allows the calculation of the free energy as a function of temperature, making it ideal for the study of thermal phase transitions. The second algorithm renders the high temperature series expansion into a perturbation expansion in one of the coupling constants and is suitable for the investigation of quantum phase transitions.
Quantum Monte Carlo algorithms start by mapping the quantum system to a classical system. This can be done either through a discrete [@suzukitrotter] or continuous time [@wiese] path integral representation or by a stochastic series expansion (SSE) in the inverse temperature [@sse]. While our algorithms can be formulated in both representations, we here present the SSE version as it is the easiest and most natural representation for most problems.
We start by expressing the partition function as a high temperature expansion $$Z={\rm Tr}e^{-\beta H}=\sum_{n=0}^\infty \frac{\beta^n}{n!}{\rm
Tr}(-H)^n\equiv \sum_{n=0}^\infty g(n)\beta^n,
\label{eq:zquantum}$$ where the $n$-th order series coefficient $g(n)={\rm Tr}(-H)^n/n!$ will play the role of the density of states in the classical algorithm. The algorithm performs a random walk in the space of series expansion coefficients, achieves a flat histogram in their orders $n$ and calculates the coefficients $g(n)$. Employing Eq. (\[eq:zquantum\]) we can then calculate the free energy, internal energy, entropy and specific heat directly. Thermal averages of observables can be measured as in conventional Monte Carlo algorithms by recording a separate histogram for the expectation values at each order.
Next we note that in a computer simulation the series expansion (\[eq:zquantum\]) needs to be truncated at an order $\Lambda$. Since the orders relevant for a given inverse temperature $\beta$ are sharply peaked around $\beta |E(\beta)|$, where $E(\beta)$ is the mean energy at inverse temperature $\beta$, this cutoff does not introduce a systematic error. Its main consequence is to restrict the accessible temperature range to $\beta \lesssim \Lambda/E(\beta)$.
The next step is to introduce a complete set of basis states $\{|\alpha\rangle\}$, and to decompose the Hamiltonian $H$ into diagonal and offdiagonal bond terms $H_b^{(a)}$. For simplicity we restrict the following discussion to a spin-$1/2$ Heisenberg model where this decomposition for a bond $b=(i,j)$ reads $H_{(i,j)}^{(d)}=JS_i^zS_j^z-J/4$, and $H_{(i,j)}^{(o)}=J/2(S_i^+S_j^-+S_i^-S_j^+)$. The offset $-J/4$ is added to the diagonal part in order to render the matrix elements of $-H$ nonnegative. Using this decomposition, we can write the partition function as [@sseloops; @sse] $$Z = \sum_\alpha \sum_{\{S_\Lambda\}} \frac{\beta^n (\Lambda-n)!}{\Lambda!}
\langle\alpha| \prod_{i=0}^\Lambda (-H^{(a_i)}_{b_i}) |\alpha\rangle,
\label{eq:ssel}$$ where the operator string $S_\Lambda=((b_1,a_1),\ldots,(b_\Lambda,a_\Lambda))$ is a concatenation of $n$ bond operators and $\Lambda-n$ unit operators.
Comparing Eq. (\[eq:zquantum\]) to Eq. (\[eq:ssel\]) we see that we can obtain $g(n)$ by counting the number of times a configuration with $n$ non-unit operators is observed during a simulation at an inverse temperature $\beta=1$. As the dynamic range of $g(n)$ spans thousands of orders of magnitude \[$g(0)/g(\Lambda)$ is up to $10^{10000}$ for the examples given below\] simply collecting a histogram will not be effective. Therefore a variant of the classical Wang-Landau method [@wanglandau] will be employed: by reweighting a configuration of $n$-th order with $1/g(n)$ a flat histogram of the orders $n$ can be achieved, thus sampling all orders equally well.
Since $g(n)$ is initially unknown we start with the (bad) guess $g(n)=1$. Each time a configuration of $n$-th order is visited, $g(n)$ is multiplied by a factor $f$, i.e. $g(n)\leftarrow f g(n)$. In our implementation we store the logarithms of these quantities to avoid overflow problems. The random walk is performed until the histogram $H(n)$ – counting the number of times a configuration with $n$ operators is observed – is reasonably flat. Similar to the classical case a maximum deviation of 20% from the mean value turned out to be reasonable. The multiplicative increase of $g(n)$ is essential for the success of the algorithm. Only that way the large range of $g(n)$ can be mapped out in reasonable time, and $g(n)$ converges rapidly to a rough estimate of the true distribution. Once the histogram is flat, it is reset to zero, $f$ is decreased, in our case by $f\leftarrow\sqrt{f}$, and the process starts again, refining $g(n)$ further with smaller steps. This procedure is repeated until $f$ gets as small as $\exp(10^{-8})$, so that an accurate estimate of $g(n)$ with only negligible systematic errors will be available. The accuracy of the free energy and other calculated quantities is given by the statistical error which, as usual, scales with $1/\sqrt{N_{\rm MC}}$ where $N_{\rm MC}$ is the number of Monte Carlo steps. The overall normalization of $g(n)$ follows from $g(0)$ being equal to the dimension of the Hilbert space, which for a spin-$1/2$ Heisenberg model on a lattice with $N$ sites is $2^N$. The initial choice of $f$ is very important. Too small starting values result in long computation times, while too large values give extreme fluctuations in the initial iterations. As in the classical Wang-Landau method a good choice is to let $f^{N_{\rm MC}}$ be of the same order of magnitude as the total number of configurations, which is of order $2^NN^\Lambda$. Since usually $\Lambda\gg N$, a good initial value is $\ln f
\approx (\Lambda\ln N)/N_{\rm MC}$.
![Free energy $F$ , entropy $S$ and specific heat $C$ of an $N=10$ site antiferromagnetic Heisenberg chain. Solid lines correspond to the MC results, indistinguishable from the dotted lines for the exact results. Also shown is the relative error $\varepsilon(F$) of $F$ compared to the exact result.[]{data-label="fig:chain"}](figch.eps){width="7cm"}
To finish the description of the algorithm we discuss the update steps in more detail. Any of the known update algorithms, employing local [@sse], or cluster [@sseloops] updates can be used. The only change in the acceptance probabilities from standard SSE algorithms is to set $\beta=1$ and to include an additional factor $g(n)/g(n')$ in the acceptance probability for any move that changes the number of operators from $n$ to $n'$. As an example we discuss the Heisenberg antiferromagnet. There the optimal algorithm is the loop algorithm, which in the SSE representation consists of two parts: diagonal updates and loop updates. In a diagonal update step the operator string positions $l=1,...,\Lambda$ are traversed in ascending order. Empty and diagonal operators can be exchanged with each another. Using the notation $
|\alpha(l)\rangle = \prod_{i=1}^l H_{b_i}^{(a_i)} |\alpha\rangle
$ for the state obtained by acting on $|\alpha\rangle$ with the first $l$ bond operators, and $M$ for the total number of interacting bonds on the lattice, the update probabilities are $$\begin{aligned}
P[H^{(0)}{(l)}\rightarrow H^{(d)}_b{(l)}] &=&
\frac{g(n)M\langle\alpha(l)|H^{(d)}_b
|\alpha(l)\rangle}{g(n+1)(\Lambda-n)},\label{eq:update} \\
P[H^{(d)}_b{(l)}\rightarrow H^{(0)}{(l)}] &=&
\frac{g(n)(\Lambda-n+1)}{g(n-1)M\langle\alpha(l)|H^{(d)}_b
|\alpha(l)\rangle}\nonumber,\end{aligned}$$ where $P>1$ is interpreted as $P=1$. This choice of update probabilities requires the least changes to an existing SSE program. Alternatively the factors $M$, $\Lambda-n$ and $\Lambda-n+1$ can be dropped from the update probabilities, thus simplifying the algorithm. To correct for this omission, the obtained $g(n)$ must then be multiplied by $M^n (\Lambda-n)!/\Lambda!$. At each level $l$, independent of whether an update was performed or not (e.g. when the operator is off-diagonal) $g(n)$ for the current value of $n$ is incremented by $f$. The second part of the update cycle, the loop update, changes diagonal to off-diagonal bond operators without changing $n$ and can be performed as in standard SSE algorithms. We refer to Ref. [@sseloops] for details.
![Scaling plot of the staggered structure factor of a cubic antiferromagnet as a function of temperature, obtained from simulations at a fixed temperature for various lattice sizes. The inset shows the specific heat as a function of temperature. The cutoff $\Lambda=500 (L/4)^3$ restricts the accessible temperature range to $T\gtrsim 0.4J$.[]{data-label="fig:af"}](figaf.eps){width="7cm"}
As a first example we show in Fig. \[fig:chain\] results of calculations for the free energy $F$, entropy $S$ and specific heat $C$ of an $N=10$ site antiferromagnetic Heisenberg chain, and compare to exact results. Using $10^8$ sweeps, which can be performed in less than five hours on an 800 MHz Pentium-III CPU, the errors can be reduced down to the order of $10^{-4}$. The cutoff was set to $\Lambda=250$, restricting the accessible temperatures to $T\gtrsim 0.05J$. The sudden departure of the Monte Carlo data from the exact values below this temperature clearly shows this limit, which can be pushed lower by increasing $\Lambda$. The sudden deviation becomes even more pronounced in larger systems and provides a reliable indication for the range of validity of the results.
To illustrate the efficiency of the algorithm close to a [*thermal second order phase transition*]{}, we consider the Heisenberg antiferromagnet on a simple cubic lattice. From simulations of systems with $L^3$ sites, $L=4,6,8,12,16$, we can calculate the staggered structure factor $S(\pi,\pi,\pi)$ for any value of the temperature using the measured histograms. Figure \[fig:af\] shows the scaling plot of $S(\pi,\pi,\pi)/L^{2-\eta}$ with $\eta=0.034$. The estimate for the critical temperature $T_c=0.947 J$, obtained in only a couple of days on an 800 MHz Pentium-III CPU, compares well with earlier estimates [@3dafm].
![Average tunneling times (in units of Monte Carlo sweeps) between horizontal and vertical arrangement of stripes in a hard core boson model at a ratio $V_2/t=3$ of next nearest neighbor repulsion to kinetic energy on a $8 \times 8$ sites lattice. The solid line is obtained using the standard SSE algorithm with directed loop updates. The dashed line is obtained using our algorithm, where the temperature is defined as the lowest temperature accessible in the simulation.[]{data-label="fig:tau"}](scaling.eps){width="7cm"}
Next we demonstrate the efficiency of the algorithm at a [*first order phase transition*]{} by considering two-dimensional hardcore bosons with next-nearest neighbour interactions [@hcb]. At low temperature and half filling this model is in an insulating phase with striped charge order and provides the simplest quantum mechanical model with stripes. We are currently investigating the thermal and quantum melting transitions of this stripe phase and probe for the existence of a nematic phase [@athens]. Simulations with conventional update schemes suffer from exponentially increasing tunneling times needed to change the stripe orientation from a horizontal to a vertical arrangement. The flat histogram in the order $n$ in our algorithm reduces the tunneling times by many orders of magnitude already on small lattices (c.f. Fig. \[fig:tau\]) which demonstrates the efficiency of our algorithm at first order phase transitions.
We now turn to our second algorithm, which applies to quantum phase transitions. Instead of scanning a temperature range we vary one of the interactions at fixed temperature. Defining the Hamiltonian as $H=H_0+\lambda V$ we rewrite the partition function Eq. (\[eq:zquantum\]) as $$\label{eq:zlambda}
Z=\sum_{n=0}^\infty \frac{\beta^n}{n!}{\rm Tr}(-H_0-\lambda V)^n\equiv \sum_{n_{\lambda}=0}^\infty \tilde{g}(n_\lambda)\lambda^{n_\lambda},$$ where on the right hand side we have collected all terms associated with $\lambda^{n_\lambda}$ into $\tilde{g}(n_\lambda)$. A similar algorithm can now be devised for this perturbation expansion up to arbitrary orders by setting $\lambda=1$, replacing $M$ by $\beta M$ and $g(n)$ by $\tilde{g}(n_\lambda)$ in Eq. (\[eq:update\]). To normalize $\tilde{g}(n_\lambda)$ there are two options. If $H_0$ can be solved exactly, $\tilde{g}(0)$ can be determined directly. Otherwise, the normalization can be fixed using the first algorithm to calculate $Z(\beta)$ at any fixed value of $\lambda$. Finally, even without normalization we can still obtain entropy and energy differences.
![Scaling plot of the staggered structure factor of a Heisenberg bilayer as a function of the coupling ratio $\lambda=J/J'$. Results are shown for various linear system sizes $L$. The temperature was chosen $\beta J'=2L$, low enough to be in the scaling regime. The cutoff $\Lambda=8L^3$ was chosen large enough to cover the coupling range $J/J'\lesssim1$. The dynamical critical exponent of this model is $z=1$ and $\eta=0.034$.[]{data-label="fig:bilayer"}](figbi.eps){width="7cm"}
We consider as an example the [*quantum phase transition*]{} in a bilayer Heisenberg antiferromagnet whose ground state changes from quantum disordered to Néel ordered as the ratio $\lambda=J/J'$ of intra-plane (J) to inter-plane ($J'$) coupling is increased [@bilayer]. From the histograms generated within [*one*]{} simulation we can calculate the staggered structure factor $S(\pi,\pi)$ of the system at [*any value*]{} of $\lambda$. In Fig. \[fig:bilayer\] we show a scaling plot of $S(\pi,\pi) / L^{2-z-\eta}$ as a function of $\lambda$. In short simulations, taking only a few days on an 800 MHz Pentium-III CPU, we find the quantum critical point at $\lambda=0.396$, which again compares well with earlier results [@bilayer].
To summarize, we have presented Monte Carlo algorithms for the direct calculation of the free energy of a quantum system, based on a stochastic series expansion representation of the partition function. Our algorithms employ a variant of the Wang-Landau sampling to achieve a flat histogram and provide the free energy as well as thermodynamic averages accurately over a whole temperature or coupling range. The algorithms can be used with any of the update schemes developed for the SSE representation and require only minor modifications to existing programs. Parallelization of the algorithms can be done like in the classical case by splitting the $n$-range into multiple random walks over a shorter range.
Our algorithms are efficient not only at second order phase transitions but also at first order ones, where standard local and cluster update methods fail. The algorithms open up new possibilities for quantum Monte Carlo simulations: similar to the classical case we expect the algorithms to be efficient also for disordered systems and work is in progress to apply the methods to quantum spin glasses. Also, the ability to calculate the free energy and entropy will be useful for investigations of entropy-driven phase transitions, such as the reentrant melting transition observed in bosonic systems and anisotropic quantum magnets [@schmid].
We thank D.P. Landau for discussions and acknowledge support of the Swiss National Science Foundation.
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|
---
abstract: 'The narrow and wide-angle parabolic equations for the quasi-monochromatic sound wave packets propagating in a waveguide with a non-stationary background flow are obtained. The results of numerical simulations are presented.'
author:
- 'M. Yu. Trofimov'
date: |
Il’ichev Pacific oceanological institute FEB RAS\
Baltiyskaya St. 43, Vladivostok, 690041, Russia\
e-mail: [email protected]
title: 'Non-stationary parabolic equations for the quasi-monochromatic sound propagation in media with a non-stationary background flow'
---
Introduction
============
Parabolic equations for sound propagation in moving media were reported by many authors, but any systematic derivation of such equations on the base of some universal asymptotic method was not presented in the literature yet. In this paper we use the generalized multiple-scale method [@nay] to obtain some equations of this type. The material of this paper was presented at the seminar “Acoustics of inhomogeneous media X”, Novosibirsk, Russia, 1-6 June 2009.
Derivation of equations
=======================
We start with the equation [@ost] $$\label{nonstat}
D_t\frac{\partial}{\partial t}\left(\frac{1}{\rho c^2} D_t p\right) -
{\mathop{\rm div}\nolimits}\frac{\partial}{\partial t}\left(\frac{1}{\rho}{\mathop{\rm grad}\nolimits}p\right) +
2\cdot\sum_{ij} \frac{\partial \mathbf{V}_{i}}{\partial x_j}\frac{\partial}{\partial x_i}
\left( \frac{1}{\rho} \frac{\partial p}{\partial x_j}\right)
= 0 \,,$$ where $\mathbf{V}=(V_1,V_2,V_3)=(u,w,v)$ is the velocity vector of the background flow written on the orthogonal coordinate system $(x,y,z)$ ($z$-axis is directed upward), $D_t=(\partial/\partial t + u\partial/\partial x + w\partial/\partial y +v\partial/\partial z)$, $p$ is the acoustic pressure.
Let $\epsilon$ be a small parameter, which is the ratio of characteristic wave length to the characteristic size of the medium inhomogeneties. Following the generalized multiple-scale metod [@nay] we rewrite (\[nonstat\]) using the slow variables $T=\epsilon t$, $X = \epsilon x$, $Y = \epsilon^{1/2} y$, $Z = \epsilon^{1/2}z$ and the fast variable $\eta=(1/\epsilon)\theta(X,Y,Z,T)$. We will assume that the flow velocity is $O(\epsilon)$ and then $u=\epsilon U(X,Y,Z,T)$, $w=\epsilon^{3/2} W(X,Y,Z,T)$ and $v=\epsilon^{3/2} V(X,Y,Z,T)$, where the quantities $U$, $W$ and $V$ are$O(1)$. We introduce for convenience the quantity $n=1/c$ and postulate the expansions $$\begin{aligned}
n^2 & = & n_0^2(X,Y,T) + \epsilon \nu(X,Y,Z,T)\,,\\
p & = & p_0(X,Y,T,Z,\eta) + \epsilon p_1(X,Y,T,Z,\eta)+\ldots\,,\end{aligned}$$ The density will be assumed depending on the slow variables only, $\rho=\rho(X,Y,Z,T)$. At last, the partial derivatives are transformed by the rules $$\frac{\partial }{\partial x} \rightarrow \epsilon\frac{\partial }{\partial
X} + \theta_X\frac{\partial }{\partial \eta}\,,
\qquad
\frac{\partial }{\partial y} \rightarrow \epsilon^{1/2}\frac{\partial }{\partial
Y} + \epsilon^{-1/2}\theta_Y\frac{\partial }{\partial \eta}\,,$$ analogously for the variables $t$ è $z$. In the rewritten in such a way (\[nonstat\]) we collect terms with like degrees of $\epsilon$. The squares of the background flow velocities will be neglected as they was neglected in the derivation of (\[nonstat\]).
At $O(\epsilon^{-1}$ we obtain that the phase function $\theta$ does not depend on variables $Z$ è $Y$, $\theta=\theta(X,T)$.
At $O(\epsilon^{0}$ we obtain the Hamilton-Jacobi equation $$\label{h-j}
\left(\theta_X\right)^2 - n_0^2 \left(\theta_T\right)^2 =
\left(\theta_X + n_0 \theta_T\right) \left(\theta_X - n_0
\theta_T\right) = 0\,.$$ We shall consider the waves propagating in the positive $X$-direction and adopt as the Hamilton-Jacoby equation the first factor in (\[h-j\]).
The equation at $O(\epsilon^{1})$ does not contain $p_1$ on the strength of (\[h-j\]). Substituting in this equation the anzats $p_0 =A_0(X,Y,T,Z)\exp(\mathrm{i}\eta)$, we obtain the equation for the amplitude $A_0$ $$\label{parA}
2\mathrm{i}n_0\omega\frac{1}{\rho}\left[A_{0,X} + n_0 A_{0,T}\right] +
\left(\frac{1}{\rho}A_{0,Y}\right)_Y +
\left(\frac{1}{\rho}A_{0,Z}\right)_Z + \chi A_0 = 0\,,$$ where $\omega=-\theta_T$ is the local frequency of sound, $$\label{coef}
\chi = \left\{\left[\left(\frac{1}{\rho}n_0 \right)_X +
n_0\left(\frac{1}{\rho}n_0 \right)_T\right]\omega \mathrm{i} +
\frac{1}{\rho}\left(\nu-2 n_0^3 U\right) \omega^2\right\}\,.$$ Assuming that $$\nu=-2\frac{c_1}{c_0^3}\,,$$ where $c_0=1/n_0$, we obtain that $$\frac{1}{\rho}\omega^2\nu-2\frac{1}{\rho}\omega^2 n_0^3 U=2\frac{1}{\rho}\omega^2\frac{c_1+U}{c_0^3}\,,$$ so the potential of the parabolic equation contains the so called [*effective sound speed*]{} $c+U$, as was first shown in the paper [@nghiem]. We see that our equation is a generalization of the parabolic equation obtained in that work.
Note that the dependence on variable $Y$ in the obtained equation is just the same as on $Z$. Therefore in the sequel we shall not write the terms expressing the dependence on $Y$ and consider the $2D$ waveguide. As will be easily seen, all result can be directly transferred to the $3D$ case.
The analogous considerations at $O(\epsilon^{2})$ gives the equation for the amplitude $A_1$, $p_1 = A_1(X,Z,T)\exp(\mathrm{i}\eta)$. $$\label{parB}
2\mathrm{i}n_0\omega\frac{1}{\rho}\left[A_{1X} + n_0 A_{1T}\right] +
\left(\frac{1}{\rho}A_{1Z}\right)_Z + \chi A_1 + {\cal F} = 0\,,$$ where $$\label{parBF}
\begin{split}
{\cal F} =& \left(\frac{1}{\rho}A_{0X}\right)_X -
\left(\frac{1}{\rho}n_0^2 A_{0T}\right)_T + 2\mathrm{i}\frac{1}{\rho}\nu\omega A_{0T}
+ \mathrm{i}\left(\frac{1}{\rho}\nu\omega_T\right)_T A_{0}\\
&- \frac{\mathrm{i}}{\omega}U\left(\frac{1}{\rho}n_0^2\omega^2 A_0\right)_X -3\mathrm{i}U\frac{1}{\rho}n_0^2\omega
A_{0X} - 2\mathrm{i}U_X\frac{1}{\rho}n_0^2\omega A_0\\
&+ \mathrm{i}\left[\left(\frac{1}{\rho}n_0\right)_X- \frac{1}{\rho}n_0 n_{0T}\right]Un_0\omega A_0 -
\mathrm{i}\frac{1}{\omega}V\left(n_0^2\omega^2\frac{1}{\rho}A_0\right)_Z\\
&- \mathrm{i}\frac{1}{\rho}n_0^2\omega VA_{0Z} - Un_0\left(\frac{1}{\rho} A_{0Z}\right)_Z -
2U_Z \frac{1}{\rho} A_{0Z} \,,
\end{split}$$ It can be shown [@tr] that the system of equations (\[parA\]), (\[parB\]) generalize the known wide-angle stationary parabolic equation obtained by the factorization method with the rational-linear Padé approximation of the square root operator. Moreover, even in the stationary case our equations contains the terms that the factorization method cannot produce.
Initial-boundary value problems for the parabolic equations
===========================================================
For the simulation of the sound waves in the ocean the most interesting initial-boundary value problems for the Hamilton-Jacobi equatuion (\[h-j\]) and the parabolic equations (\[parA\]), (\[parB\]) are the problems with $X$ as the evolution variable in the domain $\Omega = \{Z_0<Z<Z_1\}\times \{T_0<T<T_1\}$.
For the energy norm $$\label{norm}
E(X) = \int_{T_0}^{T_1}\int_{Z_0}^{Z_1} \frac{1}{\rho} |\theta_X A_0|^2\,dZdT\,.$$ under some simple and natural assumptions on the boundary conditions at $\partial\Omega$, the following theorem holds
\[te2\] The energy norm of the solution of the initial-boundary value problem for the parabolic equation (\[parA\]) satisfies the inequality $$\label{enerIn}
E(X) \le E(0)\cdot\exp\left(\int_{0}^{X}\sup_T \left(\frac{n_{0,s}(s,T)}{n_0(s,T)} \right)\,ds\right)
$$ If $n_0$ can be represented in the form $n_0=\bar n_0(X) a(T)$ then the following inequality holds $$\label{enerIn1}
E(X) \le E(0)\frac{\bar n_0(X)}{\bar n_0(0)}\sup_T a(T)\,.$$
This theorem immediately implies the uniqueness of solutions of such initial-boundary value problems for (\[parA\]), (\[parB\]) in functional spaces of the type $C([0,X],L_2(\Omega))$.
Numerical simulation
====================
![Temporal variations of the transmission loss at the distance 6 km (left) and 12 km (right): with the velocity terms (dashed line), without these terms (solid line). The second mode of internal waves.[]{data-label="npa36"}](npa36.eps "fig:"){width="49.00000%"} ![Temporal variations of the transmission loss at the distance 6 km (left) and 12 km (right): with the velocity terms (dashed line), without these terms (solid line). The second mode of internal waves.[]{data-label="npa36"}](npa312.eps "fig:"){width="50.00000%"}
![Temporal variations of the transmission loss at the distance 6 km (left) and 12 km (right): with the velocity terms (dashed line), without these terms (solid line). The third mode of internal waves.[]{data-label="npa26"}](npa26.eps "fig:"){width="50.00000%"} ![Temporal variations of the transmission loss at the distance 6 km (left) and 12 km (right): with the velocity terms (dashed line), without these terms (solid line). The third mode of internal waves.[]{data-label="npa26"}](npa212.eps "fig:"){width="49.00000%"}
In the parabolic equations (\[parA\]), (\[parB\]) the influence of the background flow is taken into account not only through the terms which explicitly contain the corresponding velocities, but also through the deformation of the sound speed and density profiles. As the geometric small parameter $\epsilon$ in typical cases is much larger than the Mach number for the ocean medium, the question about essentiality of introducing the background flow velocities in the narrow-angle parabolic equation (\[parA\]) is naturally risen.
For an answer to this question the simulation of the harmonic sound wave propagation through the harmonic internal wave of a given mode was conducted, with the following parameters: the undisturbed waveguide of constant depth (100 m) was given by the parabolic sound speed profile with the minimum (1460 m/s) at the center of the waveguide and maximum (1500 m/s) at the top and bottom boundaries, the sound frequency was taken to be equal 100 Hz and the boundary conditions for the sound field was taken to be soft at the top boundary and hard at the bottom boundary. The point sound source was situated at the center of the waveguide. The density stratification was given by the constant Brent-Väsälä frequency ($1/127~\text{s}^{-1}$), for the simulation were used the five minutes period internal waves of the second mode (wavelength=214 m) and the third mode (wavelength=142 m).
The results are presented in fig. \[npa36\],\[npa26\] in the form of the temporal variations of the transmission loss at the distances of 6 km and 12 km from the source. We can conclude that the explicit introduction of the background flow velocities to the narrow-angle parabolic equation is essential.
Conclusion
==========
In this paper the system of parabolic-like equations (\[parA\]), (\[parB\]), which can be used for numerical modelling of sound propagation in waveguides with non-stationary background flow, is obtained. Some properties of these equations are established. We hope that this information will be useful for the computational acoustics community. Complete derivations, proofs and more extensive numerical modelling will be presented in the forthcoming paper.
[9]{}
Nayfeh, A. H. Perturbation methods. John Wiley & Sons, N.-Y., 1973.
Ostashev, V. E. Sound propagation in moving media. Nauka, Moscow, 1992. (in Russian)
Nghuem-Ohu, L., Tappert, F. Parabolic equation modeling of the effects of ocean currents on sound tranmission and reciprocity in the time domain. J. Acoust. Soc. Amer., v. 78, 1985, no. 2, pp. 642-648.
Trofimov, M. Yu. Time-dependent parabolic equations for two-dimensional waveguides. Technical Physics Letters v. 26, 2000 no. 9, pp. 797-798.
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author:
- '[^1]'
bibliography:
- 'pl.bib'
- 'ninaRHI.bib'
- 'selfception.bib'
title: Sensorimotor learning for artificial body perception
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Sensorimotor learning, Body perception, Hierarchical Bayesian estimation, Predictive coding, Deep learning.
Introduction
============
Artificial self-perception is the machine ability to perceive its own body, i.e., the mastery of modal and intermodal contingencies of performing an action with a specific sensors/actuators body configuration [@lanillos2016yielding]. In other words, the spatio-temporal patterns that relate its sensors (e.g. visual, proprioceptive, tactile, etc.), its actions and its body latent variables are responsible of the distinction between its own body and the rest of the world. This paper describes some of the latest approaches for modelling artificial body self-perception: from Bayesian estimation to deep learning. Results show the potential of these free-model unsupervised or semi-supervised crossmodal/intermodal learning approaches. However, there are still challenges that should be overcome before we achieve artificial multisensory body perception.
Hierarchical Bayesian models
============================
A first approach on self-perception was integrating multimodal tactile, proprioceptive and visual cues [@lanillos2016yielding] by means of Hierarchical Bayesian models and signal processing, extending [@gold2009using] and [@stoytchev2011self] ideas. Results showed that the robot was able to discern between inbody and outbody sources without using markers or simplified segmentation. Figure \[fig:self\] shows the proto-object saliency system [@lanillos2015saliency3D] used as visual input and the computed probability of the image regions belonging to the robot arm. Body perception was formalized as an inference problem while the robot was interacting with the world. In order to infer which parts of the scene belong to the robot we integrated visual and accelerometers information. We defined the visual receptive field as a grid where each node (i.e., the decimation of the pixel-wise image) should be decided whether it belongs to the body or not. For that purpose, we adapted Bayesian inference grids to estimate the probability of being its body along the time. The prediction step was computed by learning the pixel-wise velocity in four directions (i.e., up, down, left, right). Furthermore, this method was successfully applied to simplify the problem of discovering objects by interaction [@lanillos2016self].
![Self-detection combining visual attention and Bayesian filtering [@lanillos2016yielding]. (left) Saliency segmentation; (middle) inference of the body parts; and (right) inbody vs outbody sources in the visual field.[]{data-label="fig:self"}](self_result.png){width="0.90\columnwidth" height="80px"}
Predictive coding models {#sec:pc}
========================
A biologically plausible body perception model based on predictive processing [@friston2005theory] was also proposed in [@lanillos2018adaptive]. Here, body perception was transformed into approximating the latent space distribution $q(\mu)$ that defines the body schema to the real process distribution with the sensory information (posterior) $p(x|s)$. In this approach, the forward sensory model $s=g(\mu)+z$ for each modality was learnt using Gaussian process regression. Sensory fusion was computed by means of inference approximation of the body latent variables $\mu$ minimizing the free-energy bound [@friston2005theory]. Furthermore, with this model, the authors were able to replicate the proprioceptive drift pattern of the rubber-hand illusion on a robot with visual, proprioceptive and tactile sensing capabilities [@hinz2018drifting].
Generative adversarial networks (GANs)
======================================
In order to generalize the features used in the previous predictive coding approach, we investigated deep neural networks architectures for learning the generative functions and the cross-modal relations. The advantage of using GANs as a model for self-perception is that the discriminator is a potential self/other distinction mechanism learnt in a unsupervised manner.
Visual forward model learning
-----------------------------
We analysed how the forward function that relates the joint angles (body) and the visual input can be learnt using GANs. Visuomotor learning has been also approached with recurrent neural networks in [@hwang2017predictive]. Conversely, here we employ a Deep Convolutional Generative Adversarial Network (DC-GAN) [@goodfellow2014generative; @reed2016generative]. This method literally generated the arm visual shape depending on the joint angles. Figure \[fig:GAN:a\] shows the network architecture and the robot used to extract the data.
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The great challenge was to generalize the reconstruction of the arm for any background without using segmentation. For that purpose, several background images were synthetically generated and were overlaid by automated labelled masks (i.e., boolean mask of the arm in the visual field) by means of background subtraction (Fig. \[fig:GAN:b\]). An example of the results of the generated arm given the a joint angle configuration is shown in Fig. \[fig:GAN:c\]. The most right generated image shows difficulties of the model to properly reconstruct the robot arm when the majority of it is outside the field of view. Anyhow, the statistical evaluation of the network, over all experiments, showed an accuracy of 84.4% when comparing the matching between the original versus the generated image mask.
Cross-modal learning
--------------------
We further analysed self-perception from the cross-modal point of view. Instead of generating the body visual appearance from the joints angles, we extended the architecture to enable signal reconstruction from different sensor modalities using denoising autoencoders [@ngiam2011multimodal] but without shared representation. We used the iCub simulator [@tikhanoff2008open] to generate the following visuo-tactile-proprioceptive data: the left arm joint angles, the activation of the skin sensors on the left hand and left forearm and the 2D position of a red cube in the robot left eye image plane. The detection of the red cube was performed by colour blob segmentation. The skin sensors delivered a fixed length array with value 255 if there was contact or 0 otherwise. The image size was fixed to $640\times480$. We generated several left arm joints configuration, where the forearm appeared in different positions in the visual field. The red cube trajectory was then computed to touch the forearm from the hand to the elbow. After a phase of data synchronization a Wasserstein type GAN [@arjovsky2017wasserstein] was trained off-line. Figure \[fig:icubself\] shows the experimental setup of the simulation and the data reconstruction results. Convolution operators were able to extract the inherent structure of the skin data but not reliable enough to provide accurate tactile reconstruction as those operators were thought for images only.
![Learning spatio-temporal multisensory patterns using denoising autoencoders. (left) simulator used to generate the data; (right) results: columns 1-5 original multimodal data, columns 6-10 reconstructed data.[]{data-label="fig:icubself"}](results_icub.png){width="0.90\columnwidth" height="150px"}
Conclusion
==========
We have presented body learning and perception as one of the most representative and challenging cross-modal learning applications. In particular, self-perception in robots has direct applications on adaptability, safety and human-robot interaction. Through examples, we identified at least three important characteristics for modelling artificial body perception: (1) body latent space estimation through noisy sensorimotor fusion; (2) cross-modal signal recovering from multimodal information; and (3) unsupervised self-generated patterns classification. Accordingly, we have shown different techniques for partially solving the problem, such as hierarchical Bayesian models for self-detection on the visual field [@lanillos2016yielding], predictive processing with GP regression for body estimation [@lanillos2018adaptive], and deep nets for learning the forward visual-kinematics or visuo-tactile-proprioception relations. Further research will focus on developing a full cross-modal architecture able to properly tackle the nature of the different modalities and allowing sensor relevance tuning.
[^1]: This work was supported by SELFCEPTION project (www.selfception.eu) European Union Horizon 2020 Programme (MSCA-IF-2016) under grant agreement no. 741941. Workshop on Crossmodal Learning for Intelligent Robotics. IEEE Int. Conference on Intelligent Robots and Systems (IROS 2018)
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abstract: 'We study the scalarized charged black holes in the Einstein-Maxwell-Scalar (EMS) theory with scalar mass term. In this work, the scalar mass term is chosen to be $m^2_\phi=\alpha/\beta$, where $\alpha$ is a coupling parameter and $\beta$ is a mass-like parameter. It turns out that any scalarized charged black holes are not allowed for the case of $\beta \le 4.4$ because this case implies the stable Reissner-Nodström (RN) black holes. In the massless limit of $\beta\to \infty$, one recovers the case of the EMS theory. We note that the unstable RN black hole implies the appearance of scalarized charged black holes. The other unstable case of $\beta>4.4$ allows us to obtain the $n=0,1,2,\cdots$ scalaized charged black holes for $\alpha(\beta)\ge \alpha_{\rm th}(\beta)$ where $\alpha_{\rm th}(\beta)$ represents the threshold of instability for the RN black hole. Furthermore, it is shown that the $n=0$ black hole is stable against radial perturbations, while the $n=1$ black hole is unstable. This stability result is independent of the mass parameter $\beta$.'
---
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[ De-Cheng Zou$^{a,b}$[^1] and Yun Soo Myung$^a$[^2] ]{}\
[${}^a$Institute of Basic Sciences and Department of Computer Simulation, Inje University Gimhae 50834, Korea\
]{}
[${}^b$Center for Gravitation and Cosmology and College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China\
]{}
Introduction
============
Recently, introducing a scalar mass term has an effect on the bifurcation points where the scalarized black holes branch out of the Schwarzschild black hole without scalar hair in the Einstein-Gauss-Bonnet-Scalar (EGBS) theory [@Brihaye:2018grv; @Macedo:2019sem; @Doneva:2019vuh]. This theory includes a quadratic scalar term with mass as well as the scalar-Gauss-Bonnet coupling. In other words, the mass term changes the threshold for scalarization surely and it may give the black hole mass range over which scalarized black holes can exist. Moreover, it is suggested that a quartic scalar term is sufficient to make a stable $n=0$ black hole against the radial perturbations without introdcing an exponential coupling term. However, we note that this indicates a feature of the EGBS theory with quadratic coupling. In this direction, it is worth noting that the scalarized charged black holes were found from the Einstein-Maxwell-Scalar (EMS) theory [@Herdeiro:2018wub; @Fernandes:2019rez]. We would like to mention that the $n=0$ black hole are stable in the EMS theory with exponential and quadratic couplings [@Myung:2019oua].
On the other hand, it is curious to know why a single branch of the non-Schwarzschild black hole with Ricci tensor hair exists in the Einstein-Weyl (EW) gravity whose Lagrangian takes the form of ${\cal L}_{\rm EW}=\sqrt{-g}[R-C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}/2m^2_2]$ [@Lu:2015cqa]. Actually, an apparent difference implies that many branches of $\alpha_{n=0,1,2,\cdots}= \{8.019, 40.84, 99.89,\cdots\}$ for $q = 0.7$ exist in the EMS theory with exponential coupling, while a single branch of $m^2_2
= 0.767$ exists for the EW gravity [@Lu:2017kzi]. This seems to appear because an asymptotic form of Zerilli potential ($V_{\rm Z}\to m^2_2$) is different from the scalar potential ($V\to0$) in the EMS theory [@Myung:2018vug]. This means that the scalar perturbation vanishes asymptotically ($\phi_\infty\to 0$) in the EMS theory, while the $s(l = 0)$-mode of Ricci tensor perturbation takes a normalizable form ($\psi_{\infty}\to e^{-m_2 r}$) in the EW gravity. An asymptotic correspondence would be met naively when one proposes a mass term of $V_\phi=2m^2\phi^2$. In this case, a scalar potential takes the form of $V_{\rm mass}(r) = f(r)[2M/r^3+m^2-(m^2+2)Q^2/r^4]$ which shows a similar asymptote ($V_{\rm mass}\to m^2$, as $r\to \infty$) to $V_{\rm Z}(r)$. However, it turns out that for this mass term, all potentials are positive definite outside the horizon, providing the sufficient condition for stability. Therefore, this choice does not allow any scalarized charged black holes. Of course, an independent choice of mass parameter is available and it may lead to the scalarized charged black holes by analogy with the EGBS theory with scalar mass term.
In this work, we wish to investigate how the number of bifurcation points can be changed when including a specific mass term of $V_\phi=2(\alpha/\beta)\phi^2$. Here, $\alpha$ is a coupling parameter and $\beta$ is a mass parameter in the EMS theory with scalar mass term. The original motivation is mainly to explain a difference between many branches in the EMS theory and a single branch in the EW theory. However, it turns out that for $\beta>4.4$ with $q=0.7$, the number of bifurcation points remains unchanged when including such a mass term. Instead, for $\beta\le4.4$ with the same $q$, there is no unstable RN black hole and thus, one could not find any scalarized charged black holes. In the massless limit of $\beta\to \infty$, one recovers the case of the EMS theory. This implies that the role of scalar mass term provides either nothing or all bifurcation points, but it does not lead to a single branch of scalarized charged black holes. This indicates a difference between scalar and tensor hairs. Finally, we show that the $n=0$ black hole is stable against radial perturbations, while the $n=1$ black hole is unstable. This result is independent of the mass parameter $\beta$.
EMS theory
==========
The EMS theory with scalar mass term takes the form [@Herdeiro:2018wub] $$S=\frac{1}{16 \pi}\int d^4 x\sqrt{-g}\Big[ R-2\nabla_\mu \phi \nabla^\mu \phi-2m^2_\phi \phi^2-e^{\alpha \phi^2} F^2\Big],\label{Action1}$$ where the mass squared is chosen to be $m^2_\phi=\alpha/\beta>0$. Here $\alpha$($\beta$) are coupling (mass) parameters and the type of scalar coupling to the Maxwell term is exponential. The other case of $m^2_\phi<0(\beta<0)$ corresponds to a genuinely tachyonic instability and, therefore, this will be excluded from our consideration. First, we derive the Einstein equation $$\begin{aligned}
G_{\mu\nu}=2\nabla _\mu \phi\nabla _\nu \phi -\Big[(\nabla \phi)^2+\frac{\alpha}{\beta}\phi^2\Big]g_{\mu\nu}+2e^{\alpha \phi^2}T_{\mu\nu} \label{equa1}\end{aligned}$$ with $G_{\mu\nu}$ the Einstein tensor and $T_{\mu\nu}=F_{\mu\rho}F_{\nu}~^\rho-F^2g_{\mu\nu}/4$. The Maxwell equation is coupled to scalar as $$\label{M-eq}
\nabla^\mu F_{\mu\nu}-2\alpha \phi\nabla^{\mu} (\phi)F_{\mu\nu}=0.$$ We obtain the scalar field equation $$\nabla^2 \phi -\frac{\alpha}{\beta}\phi-\frac{\alpha}{2} e^{\alpha \phi^2}F^2 \phi=0 \label{s-equa}.$$
Taking into account $\bar{\phi}=0$ and electrically charged $\bar{A}_t=Q/r$, the RN solution is found when solving (\[equa1\]) and (\[M-eq\]) $$\label{ansatz}
ds^2_{\rm RN}= \bar{g}_{\mu\nu}dx^\mu dx^\nu=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2_2$$ with the metric function $$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}.$$ The outer (inner) horizon is located at $r=r_\pm=M(1\pm\sqrt{1-q^2})$ with $q=Q/M$. We stress that the RN solution (\[ansatz\]) is a black hole solution to the EMS theory with scalar mass term, being independent of $\alpha$ and $\beta$. Hereafter, we will choose a particular case of $q=0.7(M=0.5,Q=0.35)$ as a representative of non-extremal RN black holes. In this case, solving $f(r)=0$ determines the outer horizon at $r=r_+=0.857$ and the inner horizon $r=r_-=0.143$.
Finally, we would like to note that the case of $\bar{\phi}$=const may provide a different solution because their equations are given by $$\bar{G}_{\mu\nu}=-\frac{\alpha}{\beta}\bar{\phi}^2g_{\mu\nu}+2e^{\alpha \bar{\phi}^2}\bar{T}_{\mu\nu},~~\bar{\nabla}^\mu \bar{F}_{\mu\nu}=0,~~\frac{1}{\beta}=-\frac{1}{2} e^{\alpha \bar{\phi}^2}\bar{F}^2.$$ In this case, the last relation reduces to $$\frac{1}{\beta}=e^{\alpha \bar{\phi}^2}\frac{Q^2}{r^4}$$ which means that $\beta$ is not a proper coupling constant. So, we exclude the case of $\bar{\phi}$=const from our consideration.
Stability for RN black hole
============================
The linearized theory around the RN black hole could be obtained to investigate the stability analysis of a RN black hole with $q=0.7$. The perturbed fields are introduced by considering metric tensor ($g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}$), vector ($A_\mu=\bar{A}_\mu+a_\mu$), and scalar ($\phi=\bar{\phi}+\varphi$) with $\bar{\phi}=0$. We note that there are two ways to obtain the linearized theory. One way is first to bilinearize the action (\[Action1\]) and then, obtain its linearized equations by varying perturbed fields. The other is to linearize equations (\[equa1\])-(\[s-equa\]) directly. Adapting the latter leads to the linearized Einstein-Maxwell equations $$\begin{aligned}
\delta G_{\mu\nu}(h) = 2\delta T_{\mu\nu},~~
\bar{\nabla}^\mu f_{\mu\nu}=0 \label{l-eq1}\end{aligned}$$ with a decoupled scalar equation $$\Big[\bar{\nabla}^2-\frac{\alpha}{\beta}+ \alpha \frac{Q^2}{r^4}\Big] \varphi= 0. \label{l-eq2}$$ In the EMS theory, the last term in (\[l-eq2\]) develops a negative potential outside the horizon and thus, it may induce the instability. In the EMS theory with scalar mass term, however, there exists a competition between mass term and the last term to give a negative potential outside the horizon. Therefore, the instability is harder to realize for large scalar masses. Concerning the stability analysis of the RN black hole, we consider the two linearized equations in (\[l-eq1\]) first because two of metric $h_{\mu\nu}$ and vector $a_{\mu}$ are coupled to each other as in the Einstein-Maxwell theory. It is worth noting that these are exactly the same linearized equations for the Einstein-Maxwell theory. We briefly review the stability of RN black hole in the Einstein-Maxwell theory. In this case, one obtained the Zerilli-Moncrief equation describing two physical degrees of freedom (DOF) for the odd-parity perturbations [@Zerilli:1974ai; @Moncrief:1974gw], while the even-parity perturbations for two physical DOF were investigated in [@Moncrief:1974ng; @Moncrief:1975sb]. It is known that the RN black hole is stable against the tensor-vector perturbations.
Hence, the instability of RN black holes in the EMS theory with scalar mass term will be determined entirely by the linearized scalar equation (\[l-eq2\]), indicating a feature of the EMS theory with scalar mass term. Now, let us introduce the separation of variables around a spherically symmetric RN background (\[ansatz\]) $$\label{scalar-sp}
\varphi(t,r,\theta,\chi)=\frac{u(r)}{r}e^{-i\omega t}Y_{lm}(\theta,\chi).$$ Choosing a tortoise coordinate $r_*$ defined by $r_*=\int dr/f(r)$, a radial part of the scalar equation takes the form $$\label{sch-2}
\frac{d^2u}{dr_*^2}+\Big[\omega^2-V(r)\Big]u(r)=0.$$ Here the scalar potential $V(r)$ is given by $$\label{pot-c}
V(r)=f(r)\Big[\frac{2M}{r^3}+\frac{l(l+1)}{r^2}+\frac{\alpha}{\beta}-\frac{2Q^2}{r^4}-\alpha\frac{Q^2}{r^4}\Big],$$ which seems to be a complicated form. The $s(l=0)$-mode is an allowable mode for the scalar perturbation and thus, it could be used to test the instability of the RN black hole. Hereafter, we confine ourselves to the $l=0$ mode. It is interesting to note that $V(r) \to \alpha/\beta$ as $r\to \infty$, compared to the massless potential of $V_{\beta\to \infty}(r)\to 0$ in the EMS theory. From the potential (\[pot-c\]), the condition for positive definite potential which corresponds to sufficient condition for stability could be found as [@Myung:2019oua] $$V(r) \ge 0 \to \beta \le G(r,\alpha)= \frac{\alpha r^4}{Q^2(\alpha+2)-2Mr}.\label{c-po}$$ We observe the behavior of $G(r,\alpha)$ function with $M=0.5$ and $Q=0.35$ pictorially. Its minimum stays near $r=r_+$ as $\alpha$ increases for $r\in[r_+=0.857,2]$ and $\alpha \in [0.01,100]$. A minimum value of $G(r,\alpha)$ locates at 5 around $r=r_+$ for $\alpha=1000$. We read off the stability bound from $G(r,\alpha)$ as $$\label{st-con}
\beta \le G(r_+,\alpha\to \infty)=\frac{r_+^4}{Q^2}=4.4.$$ However, it is not easy to obtain the instability condition from the potential (\[pot-c\]) directly. In this direction, we need to look for the negative region of potential outside the horizon because it may indicate a signal of instability. Taking into account the stability condition (\[st-con\]), one expects that a negative region may allow for $\beta>4.4$ and $\alpha<\infty$. As an example, we wish to display the negative region of potential (\[pot-c\]) as function of $r$ and $\alpha$ for $\beta=811$ in Fig. 1(Left). We find from Fig. 1(Right) that the width and depth of negative region in $V(r,\alpha)$ increase as $\alpha$ increases.
 
It is conjectured that if the potential $V(r)$ is negative in some region, a growing perturbation may appear in the spectrum, indicating an instability of a RN black hole. However, this is not always true. A determining condition for whether a black hole is stable or not depends on whether the time-evolution of the scalar perturbation is decaying or not. The linearized scalar equation (\[sch-2\]) around RN black hole may allow an unstable (growing) mode like $e^{\Omega t}$ for a scalar perturbation and thus, it indicates the sign for instability of the black hole. Importantly, it is stated that the instability of RN black holes implies the appearance of scalarized charged black holes. Therefore, we have to solve (\[sch-2\]) after replacing $\omega =-i\Omega$ numerically by imposing boundary conditions: purely ingoing wave near the horizon and purely outgoing wave at infinity. From Fig. 2, we read off the threshold of instability \[$\alpha_{\rm th}(\beta)$\]. Hence, the instability bound can be determined numerically by $$\alpha(\beta) \ge \alpha_{\rm th}(\beta) \label{In-cond}$$ with $\alpha_{\rm th}(\beta)=\{9.345(356),~8.82(811),~8.60(1400)\}$. On the other hand, one always finds stable RN black holes for $\alpha(\beta)<\alpha_{\rm th}(\beta)$.
From (Right) Fig. 1, one finds stable RN black hole for $\alpha<\alpha_{\rm th}=8.82$ and unstable RN black holes for $\alpha\ge \alpha_{\rm th}$.
Static scalar perturbation: bifurcation points
==============================================
Now, let us check the instability bound (\[In-cond\]) again because the precise value of $\alpha_{\rm th}(\beta)$ determines the appearance of scalarized charged black holes. This can be confirmed by obtaining a static scalar solution \[scalar cloud: $\varphi(r)$\] to the linearized equation (\[sch-2\]) with $u(r)=r\varphi(r)$ and $\omega=0$ on the RN background. For a given $l=0$ and $q=0.7$, requiring an asymptotically normalizable solution ($\varphi_\infty \to e^{-\sqrt{\alpha/\beta}r}/r$) leads to the fact that the existence of a smooth scalar determines a discrete set for $\alpha_{n}(\beta)$ where $n=0,1,\cdots$ denotes the number of zero crossings for $\varphi(r)$ (or order number). The $n=0$ scalar mode represents the fundamental branch of scalarized charged black holes, while the $n=1$ scalar mode denotes $n=1$ higher branch of scalarized charged black holes. It is noted that this corresponds to finding the first two bifurcation points from the RN black hole (see Table 1).
Consequently, we confirm from Fig. 2 and Table 1 that for given $\beta$, $$\alpha_{\rm th}(\beta)=\alpha_{n=0}(\beta)$$ which states that the threshold of instability for RN black hole is precisely the appearance of $n=0$ scalarized charged black holes. We find from Fig. 3 that $\alpha_{n=0,1}(\beta)$ increases as $\beta$ decreases. The instability is therefore harder to realize for larger scalar masses (as $\beta \to 4.4$). This picture is similar to Fig. 1(Left) in Ref. [@Macedo:2019sem], where no upper limit appears because they used an independent mass term. We find that in the massless limit of $\beta\to\infty$, $\alpha_{n=0,1}(\beta)$ approaches $\alpha_{n=0,1}=\{8.019,40.84\}$ for the EMS theory. Also, we observe the other limit that $\alpha_{n=0}(\beta)\to \infty$, as $\beta \to 4.4$. In other words, we show that unstable RN black holes exist for $\beta> 4.4$ \[see the opposite bound (\[st-con\]) for stable RN black holes\]. This implies that the $n=0$ scalarized charged black hole would be found for $\alpha\ge\alpha_{n=0}(\beta)$ with $8.019\le \alpha_{n=0}(\beta) <\infty $ for $\beta\in (4.4,\infty]$, showing a significant shift of the $n=0$ scalarized charged black holes in compared to the massless case ($\alpha\ge \alpha_{n=0}(\beta \to\infty)=8.019$) in the EMS theory. Particularly, an unallowable region for scalarization is given by $0<\beta\le 4.4$ where the unstable RN black holes are never found for any $\alpha>0$. Finally, we note that the case of $m^2_\phi=2\alpha$ with $\beta=1$ corresponds to the stable RN black hole. Therefore, one could not find any scalarized black holes from this case.
Scalarized charged black holes
==============================
First of all, we would like to mention that the RN black hole is allowed for any value of $\alpha$, while a scalarized charged black hole solution may exist only for $\alpha(\beta)\ge \alpha_{\rm th}(\beta)$ and $\beta >4.4$. The threshold of instability for a RN black hole denotes an exact appearance of the $n=0$ scalarized charged black hole. So, we derive the $n=0$ scalarized RN black hole for $q=0.7$ and $\alpha(\beta=811)=8.82 \ge \alpha_{n=0}(\beta=811)=8.82$ case numerically. For this purpose, let us introduce a spherically symmetric metric ansatz as $$\begin{aligned}
\label{nansatz}
ds^2_{\rm SCBH}=-A(r)dt^2+\frac{dr^2}{B(r)}+r^2(d\theta^2+\sin^2\theta d\chi^2).\end{aligned}$$ Also, we consider the $U(1)$ potential $A_\mu=\{v(r),0,0,0\}$ and scalar $\phi(r)$. Substituting these into Eqs.(\[equa1\])-(\[s-equa\]) leads to four equations for $\{A(r),B(r),v(r),\phi(r)\}$ as $$\begin{aligned}
&&\frac{1}{r^2}+\frac{1}{B}(-\frac{1}{r^2}+\frac{\alpha \phi^2}{\beta})+\frac{A'+e^{\alpha \phi^2}r v'^2}{r A}-\phi'^2=0,\label{neom1}\\
&&-\frac{\alpha \phi^2}{\beta}+\frac{1-B-rB'}{r^2}-e^{\alpha \phi^2} \frac{ Bv'^2}{A}-B \phi'^2=0,\label{neom2}\\
&&Q+e^{\alpha \phi^2}r^2\sqrt{\frac{B}{A}}v'=0,\label{neom3}\\
&& \phi''+\Big(\frac{2}{r}+\frac{A'}{2A}+\frac{B}{2 B'}\Big)\phi'+\Big(-\frac{\alpha}{\beta B}+\frac{\alpha e^{\alpha \phi^2} v'^2}{A}\Big)\phi=0. \label{neom4}\end{aligned}$$ One finds an approximate solution to equations in the near horizon $$\begin{aligned}
&&A(r)=A_1(r-r_+)+A_2(r-r_+)^2+\ldots,\label{aps-1}\\
&&B(r)=B_1(r-r_+)+B_2(r-r_+)^2+\ldots,\label{aps-2}\\
&&\phi(r)=\phi_0+\phi_1(r-r_+)+\ldots,\label{aps-3}\\
&&v(r)=v_1(r-r_+)+v_2(r-r_+)^2+\ldots\label{aps-4}\end{aligned}$$ with the first-order three coefficients $$\begin{aligned}
&&B_1=\frac{1}{r_+}\Big(1-\frac{Q^2e^{-\alpha \phi^2_0}}{r_+^2}-\frac{\alpha r_+^2 \phi_0^2}{\beta}\Big),~~
\phi_1=\frac{\alpha(Q^2\beta-r_+^4e^{\alpha \phi^2_0})\phi_0}{Q^2r_+\beta+r_+^3(-\beta+\alpha r_+^2\phi^2_0)e^{\alpha \phi^2_0}}, \label{ncoef} \\
&&\quad v_1=-\frac{e^{-\alpha\phi_0^2}Q\sqrt{A_1}}{\sqrt{r_+(r_+^2-e^{-\alpha \phi^2_0}Q^2-\frac{\alpha r_+^4 \phi_0^2}{\beta})}}. \nonumber\end{aligned}$$ Here $A_1$ is a free parameter. $\phi_0=\phi(r_+)$ will be determined when matching (\[aps-1\])-(\[aps-4\]) with the asymptotic solutions in the far region of $r \gg r_+$ $$\begin{aligned}
&&A(r \gg r_+)=1-\frac{2M}{r}+\ldots,\quad B(r\gg r_+)=1-\frac{2M}{r}+\ldots, \nonumber \\
&& \phi(r \gg r_+)=\phi_{\rm ml}e^{-\sqrt{\frac{\alpha}{\beta}} r}+\ldots,\quad v(r \gg r_+)=\Phi+\frac{Q}{r}+\ldots,\label{asym-sol}\end{aligned}$$ where $\phi_{\rm ml}=Q_s/r$ denotes the scalar hair for the EMS theory and $\Phi=Q/r_+$ denotes the electrostatic potential. In addition, $M,~Q_s,$ and $Q$ denote the ADM mass, the scalar charge, and the electric charge, respectively. In the massless limit of $\beta \to \infty$, one recovers the asymptotic solution for the EMS theory.
 
Consequently, we obtain the $n=0$ scalarized charged black hole solution shown in Fig. 4 for $\alpha=8.82$ at $\beta=811$. The metric function $A(r)$ has a different horizon at $\ln[r]=-0.303$ in comparison to the RN horizon at $\ln[r]=-0.154$ and it approaches the RN metric function $f(r)$ as $\ln[r]$ increases. Also, $\delta(r)$ decreases as $\ln[r]$ increases, while $\delta_{\rm RN}(r)=0$ remains zero because of $B/A=1$ for the RN case. From (\[asym-sol\]), we observe a difference between $\phi(r)$ and $\phi_{\rm ml}$ for the EMS theory in the asymptotic region. The other scalarized charged black holes for $\beta=258,~356,~1400,~2000$ are found similarly.
Stability of scalarized charged black holes
===========================================
It turns out that the $n=0(\beta=\infty)$ black hole is stable, while the $n=1,2,\cdots(\beta\to\infty)$ black holes are unstable in the EMS theory with exponential and quadratic couplings [@Myung:2019oua]. Now, let us analyze the stability of $n=0,1$ black holes the EMS theory with scalar mass term. For this purpose, we choose three scalar masses of $\beta=258,~356,~811,~1400,~2000$ whose $n=0$ and $n=1$ bifurcation points are given by $\alpha_{n=0}=\{9.619,~9.345,~8.82,~8.60,~8.493\}$ and $\alpha_{n=1}=\{56.73,~54.01,~49.15,~47.03,~45.96\}$, respectively. We focus on larger $\beta$ which provides smaller scalar mass $m^2_\phi$ for computation.
For simplicity, we perform radial (spherically symmetric) perturbations by choosing three perturbations of $H_0(t,r),H_1(t,r),\delta\phi(t,r)$ as $$\begin{aligned}
ds^2_{\rm RP}&=&-A(r)\left(1+\epsilon H_0\right)dt^2+\frac{dr^2}{B(r)\left(1
+\epsilon H_1\right)}+r^2(d\theta^2+\sin^2\theta d\chi^2),\nonumber\\
\phi&=&\phi(r)+\epsilon\delta\phi, \label{pert-metric}\end{aligned}$$ where $A(r),~B(r),~\phi(r)$ denote a scalarized charged black hole and $\epsilon$ is a control parameter of perturbations. Considering the separation of variables $$\begin{aligned}
\delta\phi(t,r)=\phi_1(r)e^{\Omega t},\end{aligned}$$ we obtain the Schrödinger-type equation for scalar perturbation $$\begin{aligned}
\frac{d^2\phi_1(r)}{d r_*^2}-\Big[\Omega^2+V_{\rm SBH}(r)\Big]\phi_1(r)=0,\label{radial-pert}\end{aligned}$$ with $r_*$ is the tortoise coordinate defined by $$\begin{aligned}
\frac{dr_*}{dr}=\frac{1}{\sqrt{A(r)B(r)}}\label{tort}\end{aligned}$$ and its potential reads as $$\begin{aligned}
V_{\rm SBH}(r)&=&\frac{\alpha}{\beta}A\left(1+\alpha\phi^2-2\alpha^2\phi^4
+r\phi(4+5\alpha\phi^2)\phi'\right)\nonumber\\
&&-\frac{B'A}{2r}\left(-1-2\alpha+4\alpha^2\phi^2-10r\alpha\phi'\phi
+3r^2\phi'^2\right)\nonumber\\
&&-\frac{\alpha A}{r^2}(1-B)(1-2\alpha\phi^2+5r\alpha\phi\phi')+\frac{AB'(r^2\phi'^2-1)}{2r}\nonumber\\
&&+AB\phi'^2(-2-\alpha+2\alpha^2\phi^2-5\alpha r\phi\phi'+r^2\phi'^2).\label{potV}\end{aligned}$$
 
It is suggested from Fig. 5 that the potentials around the $n=0$ black hole indicates small negative regions around the horizon, suggesting the instability. On the other hand, the potentials around the $n=1$ black hole indicates large negative regions outside the horizon, showing the instability. However, the former case may be not true. The potential $V_{\rm SBH}$ with $\alpha=8.820(\beta=811)$ with small negative region does not imply the instability, but it might support the stability. The linearized scalar equation (\[radial-pert\]) around the $n=0,1$ scalarized charged black holes may allow either a stable (decaying) mode with $\Omega<0$ or an unstable (growing) mode with $\Omega>0$.
We solve (\[radial-pert\]) numerically with imposing a boundary condition that $\phi_1(r)$ vanishes at the horizon and at infinity. We find from Figs. 6 and 7 that the $n=0$ black hole is stable against the $l=0$ scalar mode, while the $n=1$ black hole is unstable against the $l=0$ scalar mode. Furthermore, we show that that the (in) stability of $n=0(n=1)$ black holes is independent of the mass parameter $\beta$.
Discussions
===========
One of original motivations to study this work is to understand the difference between infinite branches in the EMS theory and a single branch in the EW theory. The infinite branches of $n=0,1,2,\cdots$ scalarized charged black holes in the EMS theory are not changed for $\beta> 4.4$ even for including a scalar mass term $m^2_\phi=\alpha/\beta$, whereas these all disappear for $0<\beta \le 4.4$. This is so because the bifurcation points is determined solely by the exponential coupling to the Maxwell term in the scalar equation (\[s-equa\]). This implies that the role of scalar mass term provides either nothing or all bifurcation points, but it never lead to a single branch of scalarized charged black holes. On the other hand, the single branch is determined by the static Licherowicz-Ricci tensor equation \[$(\vartriangle_{\rm L}+m^2_2)\delta R_{\mu\nu}=0$\] where a single bifurcation point is given by $m_2^2=0.7677$ [@Lu:2017kzi]. This indicates a difference between scalar and Ricci-tensor hairs.
In this work, we have investigated the scalarized charged black holes in the EMS theory with a specific choice of scalar mass $m^2_\phi=\alpha/\beta$. The computing process is as follows: detecting instability of RN black holes$\to$ prediction of scalarized charged black holes (bifurcation points) $\to$ obtaining the $n=0,~1$ scalarized charged black holes $\to$ performing (in)stability analysis of $n=0,~1$ scalarized charged black holes.
We find that the first two bifurcation points of $\alpha_{n=0,1}(\beta)$ increases as $\beta$ decreases. The RN instability is therefore harder to realize for larger scalar masses. We found two limits. In the massless limit of $\beta\to\infty$, $\alpha_{n=0,1}(\beta)$ approaches $\alpha_{n=0,1}=\{8.019,40.84\}$ for the EMS theory. The other limit is given by $\alpha_{n=0}(\beta)\to \infty$, as $\beta \to 4.4$. In other words, we have stated that unstable RN black holes exist for $\beta> 4.4$ \[see the opposite bound (\[st-con\]) for stable RN black holes\]. This implies that the $n=0$ scalarized charged black hole was found for $\alpha\ge\alpha_{n=0}(\beta)$ with $8.019\le \alpha_{n=0}(\beta) <\infty $ for $\beta\in (4.4,\infty]$, showing a shift from $\alpha_{n=0}(\beta\to\infty)=8.019$. Also, the $n=1$ scalarized charged black hole was found for $\alpha\ge\alpha_{n=1}(\beta)$ with $40.84\le \alpha_{n=1}(\beta) <\infty $ for $\beta\in (4.4,\infty]$, showing a shift from $\alpha_{n=1}(\beta\to\infty)=40.84$. Interestingly, an unallowable region for scalarization is given by $0<\beta\le 4.4$ where the unstable RN black holes are never found for any $\alpha>0$.
Finally, we have shown that the $n=0$ black hole is stable against radial perturbations, while the $n=1$ black hole is unstable. Further, it was shown that the stability result of $n=0,~1$ black holes is independent of the mass parameter $\beta$, even though it changes the bifurcation points significantly.
[**Acknowledgments**]{}
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MOE) (No. NRF-2017R1A2B4002057).
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[^1]: e-mail address: [email protected]
[^2]: e-mail address: [email protected]
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---
abstract: |
It is explained how the time evolution of the operadic variables may be introduced. As an example, a $2$-dimensional binary operadic Lax representation of the harmonic oscillator is found.
18D50, 70G60
[**Keywords:**]{} Operad, harmonic oscillator, operadic Lax pair
author:
- Eugen Paal and Jüri Virkepu
- |
Eugen Paal and Jüri Virkepu\
\
Department of Mathematics, Tallinn University of Technology\
Ehitajate tee 5, 19086 Tallinn, Estonia\
\
E-mails: [email protected] and [email protected]
title: |
**Note on 2d binary operadic\
harmonic oscillator**
---
Introduction
============
It is well known that quantum mechanical observables are linear *operators*, i.e the linear maps $V\to V$ of a vector space $V$ and their time evolution is given by the Heisenberg equation. As a variation of this one can pose the following question [@Paal07]: how to describe the time evolution of the linear algebraic operations (multiplications) $V^{{\otimes}n}\to V$. The algebraic operations (multiplications) can be seen as an example of the *operadic* variables [@Ger; @GGS92; @KP; @KPS].
When an operadic system depends on time one can speak about *operadic dynamics* [@Paal07]. The latter may be introduced by simple and natural analogy with the Hamiltonian dynamics. In particular, the time evolution of operadic variables may be given by operadic Lax equation. In [@PV07] it was shown how the dynamics may be introduced in 2d Lie algebra. In the present paper, an operadic Lax representation for harmonic oscillator is constructed in general 2d binary algebras.
Operad
======
Let $K$ be a unital associative commutative ring, and let $C^n$ ($n\in{\mathbb{N}}$) be unital $K$-modules. For $f\in C^n$, we refer to $n$ as the *degree* of $f$ and often write (when it does not cause confusion) $f$ instead of $\deg f$. For example, $(-1)^f{\doteq}(-1)^n$, $C^f{\doteq}C^n$ and $\circ_f{\doteq}\circ_n$. Also, it is convenient to use the *reduced* degree $|f|{\doteq}n-1$. Throughout this paper, we assume that ${\otimes}{\doteq}{\otimes}_K$.
A linear (non-symmetric) *operad* with coefficients in $K$ is a sequence $C{\doteq}\{C^n\}_{n\in{\mathbb{N}}}$ of unital $K$-modules (an ${\mathbb{N}}$-graded $K$-module), such that the following conditions are held to be true.
1. For $0\leq i\leq m-1$ there exist *partial compositions* $$\circ_i\in\operatorname{Hom}(C^m{\otimes}C^n,C^{m+n-1}),\qquad |\circ_i|=0$$
2. For all $h{\otimes}f{\otimes}g\in C^h{\otimes}C^f{\otimes}C^g$, the *composition (associativity) relations* hold, $$(h\circ_i f)\circ_j g=
\begin{cases}
(-1)^{|f||g|} (h\circ_j g)\circ_{i+|g|}f
&\text{if $0\leq j\leq i-1$},\\
h\circ_i(f\circ_{j-i}g) &\text{if $i\leq j\leq i+|f|$},\\
(-1)^{|f||g|}(h\circ_{j-|f|}g)\circ_i f
&\text{if $i+f\leq j\leq|h|+|f|$}.
\end{cases}$$
3. Unit $\operatorname{I}\in C^1$ exists such that $$\operatorname{I}\circ_0 f=f=f\circ_i \operatorname{I},\qquad 0\leq i\leq |f|$$
In the second item, the *first* and *third* parts of the defining relations turn out to be equivalent.
\[HG\] Let $V$ be a unital $K$-module and ${\mathcal E}_V^n{\doteq}{{\mathcal End}}_V^n{\doteq}\operatorname{Hom}(V^{{\otimes}n},V)$. Define the partial compositions for $f{\otimes}g\in{\mathcal E}_V^f{\otimes}{\mathcal E}_V^g$ as $$f\circ_i g{\doteq}(-1)^{i|g|}f\circ(\operatorname{id}_V^{{\otimes}i}{\otimes}g{\otimes}\operatorname{id}_V^{{\otimes}(|f|-i)}),
\qquad 0\leq i\leq |f|$$ Then ${\mathcal E}_V{\doteq}\{{\mathcal E}_V^n\}_{n\in{\mathbb{N}}}$ is an operad (with the unit $\operatorname{id}_V\in{\mathcal E}_V^1$) called the *endomorphism operad* of $V$.
Therefore, algebraic operations can be seen as elements of an endomorphism operad.
Just as elements of a vector space are called *vectors*, it is natural to call elements of an abstract operad *operations*. The endomorphism operads can be seen as the most suitable objects for modelling operadic systems.
Gerstenhaber brackets and operadic Lax pair
===========================================
The *total composition* ${\bullet}{\colon}C^f{\otimes}C^g\to C^{f+|g|}$ is defined by $$f{\bullet}g{\doteq}\sum_{i=0}^{|f|}f\circ_i g\in C^{f+|g|},
\qquad |{\bullet}|=0$$ The pair $\operatorname{Com}C{\doteq}\{C,{\bullet}\}$ is called the *composition algebra* of $C$.
The *Gerstenhaber brackets* $[\cdot,\cdot]$ are defined in $\operatorname{Com}C$ as a graded commutator by $$[f,g]{\doteq}f{\bullet}g-(-1)^{|f||g|}g{\bullet}f=-(-1)^{|f||g|}[g,f],\qquad|[\cdot,\cdot]|=0$$
The *commutator algebra* of $\operatorname{Com}C$ is denoted as $\operatorname{Com}^{-}\!C{\doteq}\{C,[\cdot,\cdot]\}$. One can prove that $\operatorname{Com}^-\!C$ is a *graded Lie algebra*. The Jacobi identity reads $$(-1)^{|f||h|}[[f,g],h]+(-1)^{|g||f|}[[g,h],f]+(-1)^{|h||g|}[[h,f],g]=0$$
Assume that $K{\doteq}{\mathbb{R}}$ and operations are differentiable. The dynamics in operadic systems (operadic dynamics) may be introduced by the
Allow a classical dynamical system to be described by the evolution equations $$\dfrac{dx_i}{dt}=f_i(x_1,\dots,x_n),\quad i=1,\dots,n$$ An *operadic Lax pair* is a pair $(L,M)$ of homogeneous operations $L,M\in C$, such that the above system of evolution equations is equivalent to the *operadic Lax equation* $$\dfrac{dL}{dt}=[M,L]{\doteq}M{\bullet}L-(-1)^{|M||L|}L{\bullet}M$$ Evidently, the degree constraint $|M|=0$ gives rise to ordinary Lax pair [@Lax68; @BBT03].
Operadic harmonic oscillator
============================
Consider the Lax pair for the harmonic oscillator: $$L=\begin{pmatrix}
p&\omega q\\
\omega q &-p
\end{pmatrix},
\qquad
M=\frac{\omega}{2}
\begin{pmatrix}
0&-1\\
1&0
\end{pmatrix}$$ Since the Hamiltonian is $$H(q,p)=\frac{1}{2}(p^2+\omega^2q^2)$$ it is easy to check that the Lax equation $$\dot{L}=[M,L]{\doteq}ML - LM$$ is equivalent to the Hamiltonian system $$\dfrac{dq}{dt}=\dfrac{\partial H}{\partial p}=p,
\quad
\dfrac{dp}{dt}=-\dfrac{\partial H}{\partial q}=-\omega^2q$$ If $\mu$ is a homogeneous operadic variable one can use the above Hamilton’s equations to obtain $$\dfrac{d\mu}{dt}
=\dfrac{\partial\mu}{\partial q}\dfrac{dq}{dt}+\dfrac{\partial\mu}{\partial p}\dfrac{dp}{dt}
=p\dfrac{\partial\mu}{\partial q}-\omega^2q\dfrac{\partial\mu}{\partial p}$$ Therefore, the linear partial differential equation for the operadic variable $\mu(q,p)$ reads $$p\dfrac{\partial\mu}{\partial q}-\omega^2q\dfrac{\partial\mu}{\partial p}=M{\bullet}\mu- \mu{\bullet}M$$ By integrating one gains sequences of operations called the *operadic (Lax representations of) harmonic oscillator*.
Example
=======
Let $A{\doteq}\{V,\mu\}$ be a binary algebra with operation $xy{\doteq}\mu(x{\otimes}y)$. We require that $\mu=\mu(q,p)$ so that $(\mu,M)$ is an operadic Lax pair, i.e the operadic Lax equation $$\dot{\mu}=[M,\mu]{\doteq}M{\bullet}\mu-\mu{\bullet}M,\qquad |\mu|=1,\quad |M|=0$$ is equivalent to the Hamiltonian system of the harmonic oscillator.
Let $x,y\in V$. By assuming that $|M|=0$ and $|\mu|=1$, one has $$\begin{aligned}
M{\bullet}\mu
&=\sum_{i=0}^0(-1)^{i|\mu|}M\circ_i\mu
=M\circ_0\mu=M\circ\mu\\
\mu{\bullet}M &=\sum_{i=0}^1(-1)^{i|M|}\mu\circ_i M =\mu\circ_0
M+\mu\circ_1 M=\mu\circ(M{\otimes}\operatorname{id}_V)+\mu\circ(\operatorname{id}_V{\otimes}M)\end{aligned}$$ Therefore, one has $$\dfrac{d}{dt}(xy)=M(xy)-(Mx)y-x(My)$$ Let $\dim V=n$. In a basis $\{e_1,\ldots,e_n\}$ of $V$, the structure constants $\mu_{jk}^i$ of $A$ are defined by $$\mu(e_j{\otimes}e_k){\doteq}\mu_{jk}^i e_i,\qquad j,k=1,\ldots,n$$ In particular, $$\dfrac{d}{dt}(e_je_k)=M(e_je_k)-(Me_j)e_k-e_j(Me_k)$$ By denoting $Me_i{\doteq}M_i^se_s$, it follows that $$\dot{\mu}_{jk}^i=\mu_{jk}^sM_s^i-M_j^s\mu_{sk}^i-M_k^s\mu_{js}^i,\qquad i,j,k=1,\ldots, n$$ In particular, one has
\[lemma:first\] Let $\dim V=2$ and $
M{\doteq}(M_j^i){\doteq}\frac{\omega}{2}
\left(
\begin{smallmatrix}
0&-1\\
1&0
\end{smallmatrix}
\right)
$. Then the $2$-dimensional binary operadic Lax equations read $$\begin{cases}
\dot{\mu}_{11}^{1}=-\frac{\omega}{2}\left(\mu_{11}^{2}+\mu_{21}^{1}+\mu_{12}^{1}\right),\qquad
\dot{\mu}_{11}^{2}=\frac{\omega}{2}\left(\mu_{11}^{1}-\mu_{21}^{2}-\mu_{12}^{2}\right)\\
\dot{\mu}_{12}^{1}=-\frac{\omega}{2}\left(\mu_{12}^{2}+\mu_{22}^{1}-\mu_{11}^{1}\right),\qquad
\dot{\mu}_{12}^{2}=\frac{\omega}{2}\left(\mu_{12}^{1}-\mu_{22}^{2}+\mu_{11}^{2}\right)\\
\dot{\mu}_{21}^{1}=-\frac{\omega}{2}\left(\mu_{21}^{2}-\mu_{11}^{1}+\mu_{22}^{1}\right),\qquad
\dot{\mu}_{21}^{2}=\frac{\omega}{2}\left(\mu_{21}^{1}+\mu_{11}^{2}-\mu_{22}^{2}\right)\\
\dot{\mu}_{22}^{1}=-\frac{\omega}{2}\left(\mu_{22}^{2}-\mu_{12}^{1}-\mu_{21}^{1}\right),\qquad
\dot{\mu}_{22}^{2}=\frac{\omega}{2}\left(\mu_{22}^{1}+\mu_{12}^{2}+\mu_{21}^{2}\right)\\
\end{cases}$$
For the harmonic oscillator, define its auxiliary functions $A_\pm$ and $D_\pm$ by $$\begin{cases}
A_+^2+A_-^2=2\sqrt{2H}\\
A_+^2-A_-^2=2p\\
A_+A_-=\omega q\\
\end{cases},\qquad
\begin{cases}
D_+\doteq \frac{A_+}{2}(A_+^2-3A_-^2)\\
D_-\doteq \frac{A_-}{2}(3A_+^2-A_-^2)\\
\end{cases}$$ Then one has the following
Let $C_{{\beta}}\in\mathbb{R}$ (${\beta}=1,\ldots,8$) be arbitrary real–valued parameters, $
M{\doteq}\frac{\omega}{2}
\left(
\begin{smallmatrix}
0&-1\\
1&0
\end{smallmatrix}
\right)
$ and $$\begin{cases}
\mu_{11}^{1}(q,p)=C_5A_-+C_6A_++C_7D_-+C_8D_+\\
\mu_{12}^{1}(q,p)=C_1A_++C_2A_--C_7D_++C_8D_-\\
\mu_{21}^{1}(q,p)=-C_1A_+-C_2A_--C_3A_+-C_4A_--C_5A_++C_6A_--C_7D_++C_8D_-\\
\mu_{22}^{1}(q,p)=-C_3A_-+C_4A_+-C_7D_--C_8D_+\\
\mu_{11}^{2}(q,p)=C_3A_++C_4A_--C_7D_++C_8D_-\\
\mu_{12}^{2}(q,p)=C_1A_--C_2A_++C_3A_--C_4A_++C_5A_-+C_6A_+-C_7D_--C_8D_+\\
\mu_{21}^{2}(q,p)=-C_1A_-+C_2A_+-C_7D_--C_8D_+\\
\mu_{22}^{2}(q,p)=-C_5A_++C_6A_-+C_7D_+-C_8D_-\\
\end{cases}$$ Then $(\mu,M)$ is a $2$-dimensional binary operadic Lax pair of the harmonic oscillator.
Denote $$\begin{cases}
G_{\pm}^{\omega/2}&\doteq \dot{A}_{\pm}\pm\frac{\omega}{2}A_{\mp}\\
G_{\pm}^{3\omega/2}&\doteq \dot{D}_{\pm}\pm\frac{3\omega}{2}D_{\mp}\\
\end{cases}$$ Define the matrix $$\Gamma
=(\Gamma_{{\alpha}}^{{\beta}})\doteq\begin{pmatrix}
0 & \hphantom{-}G_+^{\omega/2} & -G_+^{\omega/2} & 0 & 0 & \hphantom{-}G_-^{\omega/2} & -G_-^{\omega/2} & 0 \\
0 & \hphantom{-}G_-^{\omega/2} & -G_-^{\omega/2} & 0 & 0 & -G_+^{\omega/2} & \hphantom{-}G_+^{\omega/2} & 0 \\
0 & 0 & -G_+^{\omega/2} & -G_-^{\omega/2} & \hphantom{-}G_+^{\omega/2} & \hphantom{-}G_-^{\omega/2} & 0 & 0 \\
0 & 0 & -G_-^{\omega/2} & \hphantom{-}G_+^{\omega/2} & \hphantom{-}G_-^{\omega/2} & -G_+^{\omega/2} & 0 & 0 \\
G_-^{\omega/2} & 0 & -G_+^{\omega/2} & 0 & 0 & \hphantom{-}G_-^{\omega/2} & 0 & -G_+^{\omega/2} \\
G_+^{\omega/2} & 0 & \hphantom{-}G_-^{\omega/2} & 0 & 0 & \hphantom{-}G_+^{\omega/2} & 0 & \hphantom{-}G_-^{\omega/2} \\
G_-^{3\omega/2} & -G_+^{3\omega/2} & -G_+^{3\omega/2} & -G_-^{3\omega/2} & -G_+^{3\omega/2} & -G_-^{3\omega/2} & -G_-^{3\omega/2} & \hphantom{-}G_+^{3\omega/2} \\
G_+^{3\omega/2} & \hphantom{-}G_-^{3\omega/2} & \hphantom{-}G_-^{3\omega/2} & -G_+^{3\omega/2} & \hphantom{-}G_-^{3\omega/2} & -G_+^{3\omega/2} & -G_+^{3\omega/2} & -G_-^{3\omega/2} \\
\end{pmatrix}$$ Then it follows from Lemma \[lemma:first\] that the $2$-dimensional binary operadic Lax equations read $$C_{{\beta}}\Gamma_{{\alpha}}^{{\beta}}=0,\qquad {\alpha}=1,\ldots,8$$ Since the parameters $C_{\beta}$ are arbitrary, the latter constraints imply $\Gamma=0$. Thus one has to consider the following differential equations $$G_{\pm}^{\omega/2}=0=G_{\pm}^{3\omega/2}$$ By direct calculations one can show that $$G_{\pm}^{\omega/2}=0
\qquad \Longleftrightarrow\qquad
\begin{cases}
\dot{p}=-\omega^{2}q\\
\dot{q}=p\\
\end{cases}\qquad \Longleftrightarrow\qquad
G_{\pm}^{3\omega/2}=0
\tag*{\qed}$$
Acknowledgement {#acknowledgement .unnumbered}
===============
The research was in part supported by the Estonian Science Foundation, Grant 6912. More expanded version of the present paper will be published in [@PV08].
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abstract: 'The predictions of Moore’s law are considered by experts to be valid until 2020 giving rise to “post-Moore’s” technologies afterwards. Energy efficiency is one of the major challenges in high-performance computing that should be answered. Superconductor digital technology is a promising post-Moore’s alternative for the development of supercomputers. In this paper, we consider operation principles of an energy-efficient superconductor logic and memory circuits with a short retrospective review of their evolution. We analyze their shortcomings in respect to computer circuits design. Possible ways of further research are outlined.'
author:
- 'I. I. Soloviev$^{1,2}$'
- 'N. V. Klenov$^{1,2,3}$'
- 'S. V. Bakurskiy$^{1}$'
- 'M. Yu. Kupriyanov$^{1,4}$'
- 'A. L. Gudkov$^{5}$'
- 'A. S. Sidorenko$^{4,6}$'
title: 'Beyond Moore’s technologies: operation principles of a superconductor alternative'
---
Introduction {#introduction .unnumbered}
============
World’s largest chipmaker Intel “has signaled a slowing of Moore’s Law” [@MLD1]. The company has decided to increase the time between future generations of chips. “A technology roadmap for Moore’s Law maintained by an industry group, including the world’s largest chip makers, is being scrapped” [@MLD].
Three years ago Bob Colwell (former Intel chief IA-32 architect on the Pentium Pro, Pentium II, Pentium III, and Pentium IV) described stagnation of semiconductor technology in the following sentences [@10]:
– Officially Moore’s Law ends in 2020 at 7 nm, but nobody cares, because 11 nm isn’t any better than 14 nm, which was only marginally better than 22 nm.
– With Dennard scaling already dead since 2004, and thermal dissipation issues thoroughly constrain the integration density – effectively ending the multicore era: “Dark Silicon” problem (only part of available cores can be run simultaneously).
The mentioned fundamental changes are most clearly manifested in supercomputer industry. Energy efficiency becomes a crucial parameter constraining its headway [@1; @2; @3]. Power consumption level of the most powerful modern supercomputer Sunway TaihuLight [@4] is as high as 15.4 MW. It corresponds to peak performance of 93 petaflops (1 petaflops is $10^{15}$ floating point operations per second). Power consumption level of the next generation – exaflops ($10^{18}$ flops) supercomputers is predicted [@5] to be in the sub-GW level. It is comparable to power generated by a small powerplant and results in an unreasonable bill of hundreds of million dollars per year.
Following roadmap [@6], goal power consumption level of exaflops supercomputer should be of the order of $\sim 20$ MW. It corresponds to energy efficiency of 20 pJ/flop or 50 Gflops/W. Unfortunately, energy efficiency of modern supercomputers is an order less than required. For example, the energy efficiency [@4] of the Sunway TaihuLight is 6 Gflops/W. It is understood that besides other issues of exaflops computer like large space and complex cooling infrastructure, the energy efficiency makes the next step in high performance computing to be extraordinarily difficult, even with planned advances in complementary metal-oxide-semiconductor (CMOS) technology [@3DP].
It is worth to mention that low energy efficiency leads to high power consumption and also limits clock frequency at the level of 4 – 5 GHz. This frequency limit occurs due to “temperature” limitations posed to integration level and switching rate of transistors. Note that cryogenic cooling of semiconductor chips will not solve the problem [@8; @9].
Future of high performance computing is most likely associated with one of alternative “Post-Moore’s” technologies where energy dissipation is drastically lower. It is expected that leader will be determined by 2030, while 2020 – 2030 will be the “decade of diversity”. In this paper, we consider one of the most promising candidates for leadership – the superconductor digital technology. Basic element switching energy here is of the order of $10^{-19}$ J, with no penalty for signal transfer. For a certain algorithm superconducting circuits were shown to be up to seven orders of magnitude more energy efficient than their semiconductor counterparts, including power required for cryogenic cooling [@AQFPrec]. Maturity level of superconductor technology can be illustrated by the notional prototype of the superconducting computer being developed under IARPA programm “Cryogenic computing complexity” [@C3]. This is a 64 bit computing machine operating at 10 GHz clock frequency with throughput of $10^{13}$ bit-op/s and energy efficiency of $10^{15}$ bit-op/J at 4 K temperature. Prospective study shows that superconductor computer could outperform its semiconductor counterparts by two orders of magnitude in energy efficiency, showing $250$ Gflops/W [@Holmes].
Purpose of this paper is a review of superconducting logic and memory circuits principles of operation and analysis of their issues in respect to computer circuits design. We certainly do not claim to be comprehensive while considering only the most common solutions. Our review contains two main parts describing logic and memory, correspondingly.
In the first part we start with examination of physical basis underlying logic circuits operation. Superconductor logics are presented by two main branches: digital single flux quantum (SFQ) logics and adiabatic superconductor logic (ASL). Basic principles of SFQ circuits operation are shown on example of the most popular rapid single flux quantum (RSFQ) logic. It’s energy efficient successors and competitor, LV-RSFQ, ERSFQ, eSFQ and reciprocal quantum logic (RQL), are considered subsequently. ASL is described in the historical context of its development for ultra energy efficient reversible computation. The modern status is presented by two implementations of this logic. It can be noted that superconducting adiabatic cells are used also in quantum computer circuits like the ones fabricated by D-Wave Systems.
The second part of the review is devoted to cryogenic memory. Four approaches are described: SQUID-based memory, hybrid Josephson-CMOS memory, Josephson magnetic random access memory (JMRAM), and orthogonal spin transfer magnetic random access memory (OST-MRAM). They are presented in historical order of their development. In the end of each part of our review we briefly discuss major challenges and directions of possible further research in the studied area.
Review {#review .unnumbered}
======
Logic
=====
Physical basis underlying logic circuits
----------------------------------------
The fundamental physical phenomena underlying superconducting logic circuits operation is the superconductivity effect, the quantization of magnetic flux and the Josephson effects. The first one enables ballistic signal transfer not limited by power necessary to charge capacitance of interconnect lines. It provides the biggest advantage in energy efficiency in comparison to conventional CMOS technology. Indeed, superconducting microstrip lines are able to transfer picosecond waveforms without distortions with speed approaching the speed of light, for distances well exceeding typical chip size, and with low crosstalk [@LSmsl]. This is the basis for fast long-range interactions in superconducting circuits.
Note that absence of resistance ($R = 0$) leads to absence of voltage ($V = 0$) in a superconducting circuit in stationary state. Superconducting current flow corresponds not to electrical potentials difference (the voltage $V = \delta \phi$) but to difference of superconducting order parameter phases $\delta\theta$, accordingly. Superconducting order parameter corresponds to superconducting electrons wave function $|\psi|e^{i\theta}$ in Ginzburg – Landau theory [@GL]. Magnetic flux $\Phi$ in a superconducting loop of inductance $L$ provides an increase of superconducting phase along the loop and results in permanent circulating current $I = \Phi/L$. This ratio is analogous to Ohm’s law $I = V/R$. It allows to write linear Kirchoff equations for superconducting circuits.
The quantization of magnetic flux introduces fundamental difference between CMOS and superconducting circuits operation. It follows from uniqueness of superconducting electrons wave function. Indeed, increase of superconducting phase along a loop corresponds to magnetic flux as $\Phi = (\Phi_0/2\pi) \oint \nabla \theta dl$ (where $\Phi_0 = h/2e \approx 2 \times 10^{-15}$ Wb is the magnetic flux quantum, $h$ is the Planck constant, and $e$ is the electron charge). Fulfillment of this relation is possible if $\oint \nabla
\theta dl = 2\pi n$ (where $n$ is integer) and therefore $\Phi =
n\Phi_0$. Magnetic flux in a superconducting loop can take only values multiple to the flux quantum, accordingly.
Physical representation of information is typically based on the quantization of magnetic flux. For example, presence or absence of SFQ in a superconducting loop can be considered as a logical unity “1” or zero “0”. Note that information is physically localized due to such representation. This is a fundamental difference compared to information representation in semiconductor circuits. The localization leads to deep analogy between superconducting logic cells and von Neuman cellular automata [@LSmsl] where short-range interactions are predominant.
The nonlinear element in superconducting circuits is the Josephson junction. It is a weak link between two superconductors, e.g., the most used superconductor-isolator-superconductor (SIS) sandwich. One of the most important Josephson junction parameters is its critical current, $I_c$. This is the maximum superconducting current capable of flowing through the junction. Josephson junction can be switched from superconducting to resistive state by increasing the current above $I_c$. Transition to resistive state allows to change magnetic flux in a superconducting loop, and hence to perform a digital logic operation.
Dynamics of SIS junction is commonly described in the frame of the resistively shunted junction model with capacitance (RSJC) [@RSJC]. This model presents Josephson junction as a parallel connection of the junction itself transmitting superconducting current, $I_s$, only, a resistor and a capacitor with corresponding currents, $I_r = V/R$ and $I_{cap} = C(\partial V/\partial t)$, where $t$ is time. The total current through the junction is the sum, $I = I_s + I_r + I_{cap}$. This model is based on DC and AC Josephson effects which determine the superconducting current $I_s$ and voltage $V$.
The DC Josephson effect describes the superconducting current-phase relation (CPR). For SIS junction it is $I_s = I_c\sin\varphi$, where $\varphi = \nabla\theta$ is the superconducting order parameter phase difference across the Josephson junction. It is called the Josephson phase. By presenting the relation between superconducting order parameter phase and magnetic flux as $\varphi = 2\pi
\Phi/\Phi_0$, we note that CPR couples current with the magnetic flux in a superconducting loop. Josephson junction acts as a nonlinear inductance in the circuits, accordingly.
The AC Josephson effect binds the voltage on Josephson junction in resistive state with the superconducting phase evolution as $V =
(\Phi_0/2\pi)[\partial\varphi/\partial t]$. According to this relation, increase of the Josephson phase in $2\pi$ is accompanied by appearance of the voltage pulse across the junction such that $\int Vdt = \Phi_0$. Therefore, a single switching of the Josephson junction into resistive state corresponds to transmission of SFQ pulse through the junction. The energy dissipated in the switching process is $E_J \approx I_c \Phi_0 \approx 2 \times 10^{-19}$ J, taking typical $I_c \approx 0.1$ mA. The typical critical current value is conditioned by working (liquid helium) temperature, $T =
4.2$ K. For proper circuits operation it should be about three orders higher than the effective noise current value, $I_T =
(2\pi/\Phi_0) k_B T \approx 0.18$ $\mu$A, where $k_B$ is the Boltzmann constant.
Characteristic frequency of the Josephson junction switching process is determined by Josephson junction parameters, $\omega_c =
(2\pi/\Phi_0) I_c R_n$, where $I_c R_n$ product is the Josephson junction characteristic voltage, and $R_n$ is the junction resistance in the normal state. Since SIS junctions possess large capacitance, they are usually shunted by external resistors to avoid $LC$-resonances. The resistance $R_n$ is approximately equal to resistance of the shunt $R_n \approx R_s$ because $R_s$ is much smaller than own tunnel junction resistance. For Nb-based junctions the characteristic frequency is of the order of $\omega_c/2\pi \sim
100 - 350$ GHz (the characteristic voltage is at the level of $\sim
0.2 - 0.7$ mV). Superconducting digital circuits are predominantly based on tunnel junctions because of high accuracy of their fabrication process and high characteristic frequencies.
By expressing the currents $I_s$, $I_r$ and $I_{cap}$ of RSJC model through the Josephson phase $\varphi$, we can present the total current flowing through the junction in the following form: $$\label{RSJC}
I/I_c = \sin\varphi + \omega_c^{-1}\dot{\varphi} +
\beta_c\omega_c^{-2}\ddot{\varphi},$$ where $\beta_c = \omega_c R_n C$ is the Stewart-McCumber parameter reflecting capacitance impact, and dot denotes time differentiation. This Equation \[RSJC\] is quite analogous to the one for mechanical pendulum with the moment of inertia $\beta_c/\omega_c^2$ (capacitance here is analogous to mass), the viscosity factor $1/\omega_c$ (resistance determines damping), and the applied torque $I/I_c$. This simple analogy allows to consider superconducting digital circuit as a net of coupled pendulums.
Pendulum $2\pi$ rotation is accompanied by subsequent oscillations around stable equilibrium point (Figure \[Fig1\]). In Josephson junction dynamics they are called “plasma oscillations”. Plasma oscillations frequency is $\omega_p = \omega_c/\sqrt{\beta_c} =
\sqrt{2\pi I_c/\Phi_0 C}$. For proper logic cell operation these oscillations should vanish before subsequent Josephson junction switching. Compliance with this requirement can be achieved with $\beta_c \approx 1$, $\omega_p \approx \omega_c$. Clock frequency is accordingly less than $\omega_c$, and is under $100$ GHz in practical circuits.
Complexity of a superconducting circuit realizable on a chip is determined by Josephson junction dimensions. Area of Josephson junction is closely related to its critical current density, $j_c$. This parameter is one of the most important in the standard Nb-based tunnel junction fabrication process. It is fixed by material properties of insulating interlayer Al$_2$O$_3$ between superconducting Nb electrodes, and its thickness $d \approx 1$ nm. The critical current density value lies typically in the range $j_c
= 10 - 100$ $\mu$A/$\mu$m$^2$. The corresponding Josephson junction specific capacitance is $c \approx 40 - 60$ fF/$\mu$m$^2$. Variation in Josephson junction critical current, $I_c = a j_c$, is obtained by variation of its area, $a$. It is accompanied by variation of Josephson junction capacitance, $C = a c$. The shunt resistance is adjusted in accordance with the mentioned condition, $\beta_c = 1$, as $R_n = \sqrt{\Phi_0/2\pi j_c c}/a$. Its area is defined by Josephson junction area $a$, minimum wiring feature size [@T; @HYPDL] ($\sim 0.5 - 1$ $\mu$m), and sheet resistance of used material ($2 - 6$ $\Omega$ per square for Mo or MoN$_x$) [@T; @HYPDL].
While own Josephson junction weak link area is typically $a \sim
1$ $\mu$m$^2$ for $j_c = 100$ $\mu$A/$\mu$m$^2$, its total area with the shunt is by an order lager. Corresponding Josephson junctions available density on a chip is $10^7/$cm$^2$. Superconducting circuit complexity becomes limited to $2.5$ million junctions per $1$ cm$^2$ nowadays under assumption that only a quarter of chip area can be occupied by Josephson junctions (with taking interconnects into account) [@T]. The circuit can be further expanded using multi-chip module (MCM) technology [@MCM1; @MCM2].
Digital SFQ logics
------------------
### SFQ circuit basic principles of operation
Data processing in SFQ circuits can be discussed on an example of RSFQ cells operation. RSFQ data bus is shown in Figure \[Fig2\]. It is a parallel array of superconducting loops composed of Josephson junctions (shown by crosses) and superconducting inductances. This structure is called the Josephson transmission line (JTL). SFQ can be transferred along this JTL by successive switchings of Josephson junctions. The switching is obtained by summing the SFQ circulating current and the applied bias current $I_b$. Josephson junction transition into resistive state leads to SFQ circulating current redistribution toward the next junction. The redistribution process ends by the next junction switching and successive returning of the current junction into the superconducting state.
This example shows the basic principle of SFQ logic cells operation. It reduces to summation of currents, which are SFQs currents and bias currents. This summation leads (or not leads) to successive Josephson junction switching resulting in reproduction (or not reproduction) of SFQ. In RSFQ convention [@LSmsl; @20] arrival of an SFQ pulse during clock period to a logic cell has a meaning of binary “1”, while absence of the one means “0”.
Figure \[Fig3\] illustrates an example of clocked readout of information from a RSFQ logic cell. Clocking is performed by means of SFQs application to the cell. Upper JTL in Figure \[Fig3\] serves for SFQ clock distribution. SFQs are allotted to the cell through extra branch coupled to the JTL as shown. Note that Josephson junction clones SFQ at the branch point. Readout operation is performed by a couple of junctions marked by dotted rectangle. This couple is commonly called the decision making pair. Existence (or absence) of an SFQ circulating current in the logic cell loop makes the lower junction to be closer (or father) to its critical current compared to the upper junction. Clocking SFQ switches the lower (or upper) junction, correspondingly. SFQ reproduction by the lower junction means logical “1” to the output, while SFQ absence from the ones means logical “0”.
One can note a couple of typical SFQ circuits features from the presented example. Considered logic cell acts as a finite state machine. Its output depends on a history of its input. This particular cell operates as a widely used D flip-flop (“D” means “data” or “delay”) – the basis of shift registers. Note that its realization is much simpler than the one of semiconductor counterparts. RSFQ basic cells are such flip-flops, and therefore RSFQ is sequential logic. This is in contrast with semiconductor logic which is combinational one (where logic cell output is a function of its present input only).
Since only one clocked operation is performed during a clock period (some operations can be performed asynchronously), a processing stage in RSFQ circuits is reduced to a few logic cells. This is also completely opposite to conventional semiconductor circuits.
### RSFQ logic
RSFQ logic dominates in superconductor digital technology since 1990-s years [@17]. Many digital and mixed signal devices like analog-to-digital converters [@OM_ADC; @OM_ADC1], digital signal and data processors [@M_DCRev] were realized on its basis.
Unfortunately, energy efficiency did not matter in the days of RSFQ development. High clock frequency was thought to be the major RSFQ advantage in the beginning. Extremely fast RSFQ-based digital frequency divider [@18] (T flip-flop) was presented just about a decade later RSFQ invention. Its clock frequency was as high as $770$ GHz. It is still among fastest ever digital circuits.
The first RSFQ basic cells were the superconducting loops with two Josephson junctions (commonly known as the superconducting quantum interference devices - SQUIDs). These cells were connected by resistors [@20; @19] (so “R” was for “resistive” in the abbreviation). Power supply bus coupling was also resistive. While resistors connecting the cells were rather quickly substituted for superconducting inductances and Josephson junctions [@RSFQind], the ones in feed lines remained until recent years, see Figure \[Fig4\]. They determined stationary energy dissipation, $P_S = I_b V_b$, where $I_b$ and $V_b$ are the DC bias current and according voltage. The bias current is typically $I_b \approx 0.75
I_c$. The bias voltage had to be an order higher than the Josephson junction characteristic voltage, $V_b \sim 10 \times I_c R_n$, to prevent the bias current redistribution. This requirement determined the bias resistors values. Typical RSFQ cell stationary power dissipation [@8] is $P_S \sim 800$ nW.
![RSFQ power supply scheme.[]{data-label="Fig4"}](Fig4.pdf){width="0.6\columnwidth"}
Another mechanism providing power dissipation corresponds to Josephson junction switching. This dynamic power dissipation is defined as $P_D = I_b\Phi_0 f$, where $f$ is the clock frequency. For a typical clock frequency of 20 GHz $P_D$ is at the level [@8] of $\sim 13$ nW. It is seen that the dynamic power dissipation is about 60 times less than the stationary one. Main efforts to increase RSFQ circuits energy efficiency were aimed at stationary energy dissipation decrease, accordingly. RSFQ energy efficient successors, LV-RSFQ, ERSFQ and eSFQ, are presented below.
### LV-RSFQ
The first step toward $P_S$ reduction was the bias voltage decrease. Bias current redistribution between neighboring cells in low-voltage RSFQ (LV-RSFQ) is damped by introduction of inductances connected in series with bias resistors in feed lines [@23; @24; @25; @26; @261].
Unfortunately, this approach limits clock frequency. Indeed, clock frequency increase is accompanied by increase of average voltage $\overline{V}$ across a cell (according to the AC Josephson effect). This in turn leads to bias current decrease proportional to $V_b -
\overline{V}$. The latter finally results in the cell malfunction [@RL]. This tradeoff with requirement of additional circuit area for inductances in feed lines practically limit application of this approach. Since static power dissipation is not eliminated, this is somewhat half-hearted solution. It was succeeded by another two RSFQ versions (ERSFQ and eSFQ, where “E/e” stays for “energy efficient”) where $P_S$ is totaly zero.
### ERSFQ
ERSFQ [@27] is the next logical step after LV-RSFQ. Resistors in feed lines are substituted for Josephson junctions limiting bias current variation in this logic, see Figure \[Fig5\]. This replacement is somewhat analogous to the one which was done for resistors connecting SQUID cells in the very first RSFQ circuits. It provides possibility for the circuits to be in pure superconducting state.
![ERSFQ power supply scheme. $L_b$ is inductance limiting bias current variation.[]{data-label="Fig5"}](Fig5.pdf){width="0.55\columnwidth"}
Main difficulty in the bias resistors elimination is formation of superconducting loops between logic cells. Generally, logic cells are switched asynchronously depending on processing data. Average voltage and total Josephson phase increment are different across them. This results in emergence of currents circulating through neighbor cells. Being added to bias current, these currents prevent correct operation of the circuits.
Imbalance of Josephson phase increment is automatically compensated by corresponding switchings of Josephson junctions placed in ERSFQ feed lines. Since these switchings are not synchronized with clock, some immediate alteration of bias current is still possible. This alteration $\Delta I \sim \Phi_0/L_b$ is limited by inductance $L_b$ connected in series with Josephson junction in the feed line. While large value of this inductance $L_b$ minimizes the bias current variation, its large geometric size increases the circuit area (similar to LV-RSFQ). Possible solutions of this problem are an increase of wiring layers number, and utilization of superconducting materials having high kinetic inductance. These materials can be also used for further miniaturization of logic cells themselves [@T].
### eSFQ
Another energy efficient logic of RSFQ family is eSFQ [@8; @28; @32; @33]. The main idea here is the “synchronous phase balancing”. Bias current is applied to decision making pair, see Figure \[Fig6\]. One Josephson junction of this pair is always switched during a clock cycle regardless data content. Therefore, average voltage and Josephson phase increment are always equal across any such pair. This prevents the emergence of parasitic circulating currents. Josephson junction in the feed line is required only for proper phase balance adjustment during power-up procedure. “It is not expected to switch during regular circuit operation” [@8].
![eSFQ power supply scheme. Dotted rectangle marks decision making pair.[]{data-label="Fig6"}](Fig6.pdf){width="0.6\columnwidth"}
Achieved phase balance allows to eliminate large inductances from ERSFQ feed lines, and so eSFQ circuits occupy nearly the same area as RSFQ ones. One should note that despite of the “synchronous” nature of this logic, a method for design of eSFQ-based asynchronous circuits was proposed in [@32], making it suitable for wave-pipelined architecture.
Since RSFQ library was designed regardless synchronous phase balancing, transition to eSFQ requires its correction. In some cases it leads to increase of Josephson junctions number. For example, JTL should be replaced by a shift register [@29] or by “Wave JTL” [@32], or by one of its asynchronous counterparts: ballistic transmission line based on unshunted Josephson junctions [@30; @31] or passive microstrip line.
Similarity of ERSFQ and eSFQ approaches allows to make an overall assessment of total increase in Josephson junctions number up to $33-40$% compared to RSFQ circuits [@8]. Inheritance of basic cells design of RSFQ by ERSFQ makes it easier to use.
### RSFQ logic family common features
Clock is effectively a part of data in ERSFQ circuits. This means that they are globally asynchronous.
Since clock frequency is determined by repetition rate of SFQs in clocking JTL, it can be adjusted “in flight” by logic cells according to processing data.
The bias voltage source can be implemented as a JTL fed by a constant bias current, for which the input signal is the SFQ clock applied from an on-chip SFQ clock generator, see Figure \[Fig7\]. Average voltage on this JTL is precisely proportional to the clock frequency, $\overline{V}_b = \Phi_0 f$, according to the AC Josephson effect. Clock control by logic cells allows to adjust this voltage or even to turn it off. The last option corresponds to circuits switching into “sleep mode” where power dissipation is totally zero. Realization of this power save mechanism at individual circuits level is possible with circuits partitioning into series connection of islands with equal bias current but different bias voltage [@T_RSFQ_SB].
![DC bias voltage source realization in RSFQ circuitry.[]{data-label="Fig7"}](Fig7.pdf){width="1\columnwidth"}
Since logic cells are fed in parallel in RSFQ logic family, total bias current increases proportional to Josephson junctions number. For 1 million Josephson junctions the bias current value could be unreasonably high $I_b \sim 100$ A. Circuits partitioning allows to keep it at acceptable level [@Jap_HS_SFQ] below $3$ A.
### RQL
RQL was proposed in about 2008. It was developed as an alternative to conventional RSFQ, and presented as “ultra-low-power superconductor logic” [@14]. Main difference between RQL and RSFQ is in the way of power supply [@RQL]. While in RSFQ it is a DC power applied to Josephson junctions in parallel through bias resistors (Figure \[Fig4\]), in RQL it is an AC power applied in series through bias transformers, see Figure \[Fig8\].
![RQL AC power supply scheme. Blue arrow shows SFQ current, violet arrows present magnetic coupling.[]{data-label="Fig8"}](Fig8.pdf){width="0.55\columnwidth"}
The proposed power supply scheme possesses some advantages. (i) No DC bias current and no bias resistors means zero static power dissipation inside cryogenic cooler. Bias current is terminated off chip at room temperature. (ii) The well known RSFQ circuits design problem is the large return bias current magnetic field affecting logic cells. It is recommended [@Jap_HS_SFQ] to keep maximum bias current below $100$ mA in RSFQ feed line. This return current is completely absent in RQL due to the mentioned off-chip bias current termination. (iii) Serial bias supply allows to keep bias current amplitude at fairy low level [@14] of the order of $I_b
\sim 1.8$ mA regardless number of Josephson junctions on a chip. There is no need for the large-scale circuit partitioning. (iv) Bias current plays a role of clock signal. There is no need for SFQ clock distribution network. (v) Clock is not affected by thermal noise.
Logical unity (zero) is presented by a pair of SFQs having opposite magnetic flux directions (or lack thereof) in RQL circuits. These SFQs can be transferred in one direction with application of inversely directed bias currents, see Figure \[Fig9\]. The SFQs are placed in positive/negative AC current wave half period, accordingly. Unfortunately, one AC bias current is insufficient for directional propagation of the SFQs. It can provide only periodic space oscillations of the flux quanta. RQL uses two AC bias currents with $\pi/2$ phase shift. RQL cells are coupled to these two feed lines in rotation (Figure \[Fig9\]). Such coupling produces space division of total bias current/clock into four windows shifted by $0$, $\pi/2$, $\pi$, and $3\pi/2$ wave period. By analogy with a car’s four-stroke engine, this four-phase bias scheme provides directionality of the SFQs propagation [@14].
![RQL transmission line with four-phase bias. $I_{b1,2}$ are AC bias currents providing power supply, and playing role of clock signal. Blue arrows show SFQ currents, violet arrows present magnetic coupling.[]{data-label="Fig9"}](Fig9.pdf){width="1\columnwidth"}
Logic elements connected to a single AC bias line within a single clock phase window form a pipeline. The pipeline in RQL can contain an arbitrary number of cells. One can increase a depth of the pipeline at the cost of clock frequency decrease. Time delay of the pipeline should be less than one-third of clock period for proper circuit operation. Circuit speed is effectively a product of clock frequency and pipeline depth. Maximum clock frequency of RQL circuits can be estimated as $f_{max} \sim 17$ GHz under assumption of the standard Josephson junction characteristic frequency $\omega_c/2\pi = 350$ GHz and $N = 8$ Josephson junctions in the pipeline [@RQL].
RQL biasing scheme provides self data synchronization. Early pulses wait at the pipeline edge for bias current rise in the next phase window. SFQ jitter is accumulated only inside one pipeline, and therefore timing errors are negligible which is in contrast to RSFQ.
RQL logic cells are state machines similar to RSFQ ones. Internal state of logic cell can be changed by SFQ propagating in front of the clock wave. Its paired SFQ with opposite polarity serves for the state resetting in the end of a clock period.
Complete set of RQL logic cells comprises just three gates which are And-Or gate, A-not-B gate and Set-Reset latch. These gates behave as combinational logic cells similar to their semiconductor counterparts [@RQL]. This makes RQL circuits design to be closer to CMOS than to RSFQ.
Particular RQL drawbacks come from power supply scheme as well as its advantages. Proper power supply requires high-frequency power splitters. These splitters often occupy quite a large area. For example, in implementation of 8-bit carry-look-ahead adder they cover area $\sim 2.5$ times larger than the adder itself [@RQL_adder]. One can note that power supply through transformers also limits the possibility for the circuits miniaturization.
Multiphase AC bias presents known difficulty for high-frequency design (clock skew etc.). This practically limits clock frequency to $10$ GHz, while RSFQ circuits routinely operate at frequency of $50$ GHz. Moreover, implementation of MCM technology becomes complicated with RQL due to possible asynchronization of chips or clock phase shift. Besides inconvenience presented by high-frequency clock supply from off-chip external source, clocking by AC bias currents eliminates possibility of clock control by logic cells. Corresponding power save mechanisms cannot be realized in RQL. In addition, one should mention RF losses in microstrip resonators which typically make up to 50% total power budget even at relatively low frequencies.
Total power dissipation of RQL and ERSFQ circuits in active mode seems similar. Static power dissipation is absent. Dynamic power dissipation is associated with Josephson junction switching in data propagation process. In RQL circuits Josephson junction is doubly switched for logical unity transfer and zero times for transfer of logical zero. In ERSFQ both unity and zero are transferred with switching of one of the Josephson junctions in decision making pair. By assuming equal number of zeros and ones in data, one comes to roughly equal estimation for energy dissipation in both RQL and ERSFQ logics [@T]. More detailed analysis shows that only adiabatic switching of logic cells improves superconducting circuits energy efficiency markedly [@T].
Adiabatic superconductor logic
------------------------------
Considered variants of superconductor logics have been proposed for non-adiabatic irreversible computation. Logical states are separated here by energy barrier $E_w \sim 10^3 - 10^4~k_B T$ ensuring proper circuit operation. Note that the energy barrier in semiconductor circuits is two to three orders higher, $E_w \sim 10^6~k_B T$. Minimal energy barrier corresponds to Landauer’s “thermodynamic limit”, [@40] $E_{min} = k_B T \ln 2$. In this limit logic states distinguishability becomes completely lost due to thermal fluctuations [@9].
Energy required to perform a non-adiabatic logic operation can be estimated as the energy of transition between logical states corresponding to $E_w$. In considered superconductor logics it is the energy of Josephson junction switching, $E_J \approx 2 \times
10^{-19}$ J. While presuming logical irreversibility, this energy can be lowered down to $E_{min} \approx 4 \times 10^{-23}$ J (at $T
= 4.2$ K) by using adiabatic switching process. Note that Landauer limit $E_{min}$ in this context reflects computing system entropy change associated with an irreversible operation [@40]. At the same time, there is no such limit for physically and logically reversible process. Therefore, energy dissipated per logical operation can approach zero in adiabatic reversible circuits.
The first ever practical reversible logic gates were realized recently [@48] on a basis of adiabatic superconductor logic. History of ASL development have begun even before RSFQ invention with proposition of “parametric quantron” [@PQ] in 1976. This cell itself was proposed even earlier in 1954 as “rf parametron” [@rfP], though for different operating regime.
It is interesting to note that the manner of parametric quantron cell operation was implemented later in a single-electron device [@17; @QCA1] in 1996. The “single-electron parametron” operation was in fact quite similar to the ones of quantum-dot cellular automata (QCA) which were proposed for computation those years [@QCA2].
### ASL circuit basic principles of operation
Parametric quantron is a superconducting loop with single Josephson junction shown in Figure \[Fig10\] leftward. Its state is conditioned by external magnetic flux, $\Phi_e$, and current, $I_e$, controlling Josephson junction critical current $I_c(I_e)$. Potential energy of this cell is a sum of Josephson junction energy, $U_J = (E_J/2\pi)[1 - \cos\varphi]$ (followed directly from the DC Josephson effect), and magnetic energy, $U_M = (E_J/2\pi)[\varphi -
\varphi_e]^2/2l$: $$\label{ParaQ}
U_{PQ} = \frac{E_J}{2\pi}\left[1 - \cos\varphi + \frac{(\varphi -
\varphi_e)^2}{2l}\right],$$ where $\varphi_e = 2\pi\Phi_e/\Phi_0$, $l = 2\pi I_c L/\Phi_0$ is the normalized loop inductance.
![Parametric quantron notional (left) and practical (right) schematic. The cell state is conditioned by bias flux $\Phi_e$ and current $I_e$ controlling Josephson junction critical current $I_c$. $L$ is the loop inductance. In practice, single Josephson junction is substituted by SQUID controlled by activation current $I_{act}$. $I_{in}$/$I_{out}$ are input/output currents.[]{data-label="Fig10"}](Fig10.pdf){width="1\columnwidth"}
It is seen that external parameters $\Phi_e$, $I_e$ control vertex (through $\varphi_e[\Phi_e]$) and slope (through $l[I_c(I_e)]$) of the potential energy parabolic term $U_M$ in Equation \[ParaQ\]. Under appropriate bias flux, $\varphi_e \approx \pi$, parametric quantron potential energy $U_{PQ}(\varphi)$ can take single-well (at $l < 1$) or double-well (at $l > 1$) shape depending on $I_e$, see Figure \[Fig11\]. Logical zero and unity can be represented by the cell states with Josephson junction phase $\varphi$ lower or higher than $\pi$. For $l > 1$ these states correspond to minima of potential wells. Physically they correspond to different magnetic flux in the loop (with current circulating in the loop in opposite directions if $\varphi \neq 2\pi n$, where $n$ is integer).
![Parametric quantron potential energy $U_{PQ}$ (\[ParaQ\]) (solid lines) and its terms: magnetic energy $U_M$ (dashed lines), and Josephson junction energy $U_J$ (dotted line).[]{data-label="Fig11"}](Fig11.pdf){width="1\columnwidth"}
Logical state transfer can be performed in array of magnetically coupled parametric quantrons biased into working point $\varphi_e =
\pi$. Current pulse $I_e$ should be applied sequentially to the cells increasing their normalized inductance one by one, see Figure \[Fig12\]. Logical state can be shared by a group of cells, wherein it is most pronounced in a cell with the largest $l$ in particular moment. Dynamics of this transfer process can be made adiabatic by adjusting the shape of the driving current pulse $I_e$. Cross-coupling of the cells enables adiabatic reversible logic operations [@LikhPQ].
![Logical state transfer in array of magnetically coupled parametric quantrons under driving current pulse $I_e$, with corresponding change of a single cell potential profile in time. Violet arrows present magnetic coupling.[]{data-label="Fig12"}](Fig12a.pdf){width="1\columnwidth"}
![Logical state transfer in array of magnetically coupled parametric quantrons under driving current pulse $I_e$, with corresponding change of a single cell potential profile in time. Violet arrows present magnetic coupling.[]{data-label="Fig12"}](Fig12b.pdf){width="1\columnwidth"}
Single Josephson junction of parametric quantron was substituted by SQUID (see Figure \[Fig10\] rightward) in practical implementations [@Convolv]. Activation current, $I_{act}$, here plays a role similar to the ones of $I_e$. It induces circulating current in activation SQUID, and therefore increases Josephson junctions phases, according to the DC Josephson effect. This in turn corresponds to increase of Josephson junctions energy which can be minimized with appearance of current circulating in the main parametric quantron loop (the loop containing inductance $L$). However, the states with both directions of the circulating current are equally favorable due to symmetry of the scheme. Choice of one of these states corresponds to direction of input current $I_{in}$ playing here the role of $\Phi_e$. Due to the fact that current-less state is unstable balance corresponding to potential energy local maximum, the current $I_{in}$ can be infinitesimally small. Parametric quantron can provide virtually infinite amplification of magnetic flux, accordingly. Since potential energy minimum is achieved with circulating currents in both: activation SQUID and main parametric quantron loop, it was noted that the roles of the activation and the input/output can be swapped [@QFP].
Unfortunately, already the first designs in mid 1980-s of physically and logically reversible parametric quantron based processor [@Convolv] showed this approach to be impractical. The reason for such conclusion was as follows. Logical reversibility can be achieved by temporary storage of all intermediate results [@BenLR]. Together with predominance of short-range interactions this produces severe hardware overhead. Indeed, realization of 8-bit 1024-points fast convolver required almost $10^7$ parametric quantrons [@Convolv]. About 90% of them were operated just as elements of shift registers, transferring data through the processor [@Convolv]. It was noted that such circuits are also featured by low speed (in comparison to RSFQ) and low tolerance to parameters variations [@17].
### Quantum flux parametron based circuits
Few years later the works on reversible circuits, the same principles of operation were utilized for development of generally non-reversible Josephson supercomputer. In this effort parametric quantron was renamed as “quantum flux parametron” (QFP) [@QFP]. The major problem of QFP-based circuits was high-frequency multi-phase AC power supply (which was later borrowed by RQL). While there were different approaches elaborated for its solution [@QFP; @QFP1], finally multi-phase AC biasing was recognized to be intractable obstacle for implementation of complex high-clock-frequency practical circuits and QFP-based approach was abandoned for some years.
Renewed interest to ASL was introduced by development of superconductor quantum computer. QFPs are utilized as qubits and couplers in adiabatic quantum optimization systems of D-Wave Systems [@DWave1; @DWave2]. Another reason for the current rise of interest to ASL is Japan JST-ALCA project “Superconductor electronics system combined with optics and spintronics” [@ALCA]. The idea of the project is the development of energy efficient supercomputer based on synergy of the technologies. Superconductor processor of the computing system is planned to be based on QFPs operated in adiabatic regime. The processor prototype has 8-bit simplified RISC architecture and is featured by $\sim 25$ thousand Josephson junctions and $\sim 10$ instructions. In this context, adiabatic operation of QFP was investigated in order to reduce its dynamic energy consumption down to the fundamental limit [@45]. Adiabatic QFP was abbreviated as AQFP in these works [@48; @45; @47; @46; @AQFPlatch; @49; @50].
AQFP-based circuits were tested experimentally [@47] at $5$ GHz clock frequency showing energy dissipation at the level of $10^{-20}$ J. Theoretical analysis reveals that AQFP can be operated with energy dissipation less than the thermodynamic limit [@46]. Product of energy dissipated per clock cycle on a cycle time could approach the quantum limit [@50] at $4.2$ K cooling temperature, with utilization of standard manufacturing processes [@54]. Comparison of AQFP-based design with design based on CMOS FPGA, on example of implementation of Collatz algorithm, showed that the first one is about seven orders of magnitude superior to its counterpart in energy efficiency, even including the power of cryogenic cooling [@AQFPrec].
AQFP-based logic cells can be implemented by combining only four building blocks: buffer, NOT, constant, and branch [@49]. Together with AQFP latch [@AQFPlatch] these blocks enable design of adiabatic circuit of arbitrary complexity. Recently, 10 thousand gate-scale AQFP circuit was reported [@AQFP10K].
Magnetic coupling of AQFP gates is performed via transformers. Current flowing through the transformer wire cannot be too small because it ought to provide appropriate bias flux to subsequent cell despite of possible technological spread of AQFP parameters. This limits maximum wire length to about $1$ mm [@49]. This length is further conditioned by trade-off with maximum clock frequency, which is limited to $5$ GHz in practical circuits [@48; @45; @47; @46; @AQFPlatch; @49; @50]. This clock frequency limitation relaxes complexity of AC bias lines design. However, with circuit scale increase, lengthy distribution of clock lines is nonetheless expected to generate a clock skew between logic cells [@AQFPrec].
While adiabatic circuits are clearly the most energy efficient ones, their operation frequency is relatively low and the latency is relatively large. However, recently it was shown that due to intrinsic periodicity of AQFP potential energy, the cell can be operated at double or even quadruple activation current frequency with an increase of the current amplitude [@MEAQFP]. This opens oportunity to speed up AQFP circuits up to $10$ GHz or even $20$ GHz clock.
### nSQUID-based circuits
Above, we already mentioned that it is possible to swap the roles of activation current and input/output in the parametric quantron. In this case, information is represented in magnetic flux of the SQUID, while its bias current flowing through the main parametric quantron loop plays the role of excitation.
It was noted that while the SQUIDs of different such cells may be coupled magnetically, their activation current pulse can be provided sequentially using a common bias bus. For better control of the SQUID state in this scheme the value of the main parametric quantron loop inductance should be minimized. In addition, it was proposed to provide negative mutual inductance between the two parts of the SQUID loop inductance [@41]. SQUID with negative mutual inductance is called “nSQUID”. Its inductance is effectively decreased for the bias current but increased for the current circulating in its loop.
Successive application of activation current pulse to nSQUIDs from a common bias bus can be realized by using an SFQ [@41; @42; @43; @44]. Note that nSQUID-based transmission line is quite similar to conventional RSFQ JTL with substitution of Josephson junctions by nSQUIDs, see Figure \[Fig13\]. Here data bit is spatially bound to SFQ. Such application of activation current pulse allowed to switch from AC to DC power supply. It was shown that it is possible to switch also from magnetic to galvanic coupling between nSQUIDs [@42].
![nSQUID-based adiabatic data bus and RSFQ data bus. Blue arrows show circulating currents, orange arrows highlight critically biased elements, violet arrows present magnetic coupling.[]{data-label="Fig13"}](Fig13.pdf){width="1\columnwidth"}
nSQUID-based circuits were successfully tested [@43] at $5$ GHz clock frequency. At lower frequency, $50$ MHz, their energy dissipation per logic operation was estimated [@8] to be close to the thermodynamic limit, $\sim 2 k_B T\ln 2$.
Since nSQUID circuits utilize SFQ clocking, the clock rate (and hence, the power dissipation) can be adjusted “in flight” like in RSFQ circuits. Note that the energy associated with SFQ creation or annihilation $E_J$ is much greater than the thermodynamic limit at $4.2$ K. SFQs are “recycled” to avoid this energy dissipation. For this purpose the circuits are made in closed loop manner as “timing belts” [@44]. Thus, total number of SFQs remains unchanged. However, this imposes certain restriction on the circuit design.
It is interesting to note that it was proposed to use nSQUID circuits for implementation of “flying qubits” transmitting quantum information [@42; @FLQubit]. Yet, this idea is not implemented experimentally.
Discussion
----------
We considered non-adiabatic and adiabatic logics which implementations are different mainly in the type of power supply, AC or DC. Each type has its own advantages and disadvantages.
The most attractive feature of AC versus DC bias is that the power is supplied in series. We should note that this feature can be utilized also for DC biased circuits by using AC-to-DC converter [@ACDCconv]. At particular frequency of AC power source the required bias voltage can be obtained by serial connection of these converters. Power supply of different parts of large scale DC biased circuit by such voltage sources could eliminate the need for the circuit partitioning.
In general, SFQ AC biased circuits are good in design of large regular structures. The largest superconductor digital circuit is AC biased shit register containing 809 thousand Josephson junctions [@RQL_SR]. It was used as a fabrication process benchmark circuit like a kind of “scan chain”.
DC power source is most convenient in terms of providing the power into the cryogenic system. Indeed, the bandwidth of microwave cables is often narrow to prevent hit inflow. In order to overcome the limitation on the maximum frequency of AC biased circuits it was proposed to use a DC-to-AC converter as on-chip power source [@DCAC]. This converter was successfully tested in experiment providing oscillation frequency of $4.4$ GHz. The output AC bias current amplitude can be tuned by varying DC bias current of the convertor. Utilization of AC-to-DC and DC-to-AC converters allows to use circuits based on different logics on a single chip, increasing the variability of design.
Physical localization of information corresponding to quantization of magnetic flux leads to another issue, especially in digital SFQ circuits. Due to low gain from Josephson junctions, the circuits are featured by low fan-out. An SFQ has to propagate through large and slow SFQ splitter tree to split information into multiple branches. The same situation is with merging of multiple outputs.
Solution of this problem can be found in utilization of magnetic control over cells by using current control line. This approach can be realized with SFQ-to-current loop converter [@CLD; @RSFQdecoder]. Similar technique can be used in merging of multiple outputs [@SQFMerg].
SFQ-to-current conversion can be realized also by Superconducting-Ferromagnetic Transistor (SFT) [@SFTNev] or by “non-Josephson” device like n-Tron [@nTron1]. The former is the three (or four) terminal device comprising two stacked Josephson junctions. One of them, “injector”, (containing ferromagnetic layer(s)) serves for injection of spin-polarized electrons in common superconducting electrode of both junctions, thus suppressing its superconductivity. This manifests itself as redistribution of superconducting current flowing through this electrode or as degradation of “acceptor” (typically SIS junction) critical current depending on configuration of the device [@SFTNev1]. While having good input/output isolation, SFT is capable of providing voltage, current, and power amplification.
n-Tron is the three terminal device comprising superconducting strip with a narrow in the middle to which the third terminal tip is connected. Current pulse from the third terminal switch off superconductivity of the nanowire, that is similar to SFT operation to some extent. Unlike Josephson junction, the nanowire in resistive state possesses several $k\Omega$ resistance which provide high output impedance and high voltage signal [@nTron2; @nTron3]. Both devices can be utilized as an interface [@SFTNev1; @nTronCMOS1; @nTronCMOSmem] between superconductor circuit and CMOS electronics or memory depending on requirements to output signal and energy efficiency.
It is well known that the major computation time and power consumption is associated with communications between logic and memory circuits [@MAGIC2]. Logic cells possessing feature of internal memory are now being considered as possible element base for development of new, more efficient computers [@Mec1; @Mec2]. Superconductor logic circuits utilizing their internal memory were named “MAGIC” (Memory And loGIC) circuits [@MAGIC2; @MAGIC1]. This concept is based on conventional ERSFQ cells involving their renaming or rewiring. It promises an increase in clock rate to above 100 GHz threshold, combined with up to ten-fold gain in functional density. In general, the mentioned quantum localization of information and high non-linearity of Josephson junctions make superconductor circuits to be ideally suit for implementation of unconventional computational paradigms like cellular automata [@CellA1; @CellA2], artificial neural networks [@ANN1; @ANN2; @ANN3] or quantum computing [@QCR1; @QCR2; @QCRIF1; @QCRIF2; @QCRIF3].
Unfortunately, the major problem of superconductor circuits does not relate to a particular logic of computation. Low integration density in all cases limits complexity, and therefore performance of modern digital superconductor device. Possible solutions here are miniaturization of existing element base and increase of its functionality.
The first one can be performed by scaling down the SIS Josephson junction [@250nmJJ], or search for other high accuracy technological processes providing nanosized junctions with high critical current density and normal-state resistance [@250nmJJ; @aSi; @aSi1]. Another direction of the research is substitution of conventional loop inductance for kinetic inductance or inductance of Josephson junction [@T]. This also allows to make the circuits more energy efficient. Indeed, Josephson junction critical current $I_c$ and loop inductance $L$ are linked for SFQ circuits. Their product should be $I_c L \approx \Phi_0$ for proper operation. While the critical current $I_c$ has to be decreased in order to improve the energy efficiency, $E_J \approx I_c \Phi_0$, this leads to increase in the inductance making the circuit to be sparse. Miniaturization of inductance weakens this problem. Unfortunately, transformer remains an inherent component of the circuits which can not be miniaturized in this way.
One should note that contrary to CMOS technology where transistor layer is implemented on a substrate, Josephson junctions can be fabricated at any layer. This provides opportunity for utilization of 3D architecture. With planned technological advances, the Josephson junction density up to $10^8/$cm$^2$ seems achievable.
Finally, Josephson junctions with unconventional current-phase relation (CPR) can be utilized in a circuit for its miniaturization. For example, the so-called “$\pi$”-junction (the junction with constant $\pi$ shift of its CPR) can be used as a “phase battery” providing constant phase shift [@pJJ1; @pJJ2] instead of conventional transformer. Control of the junction CPR phase shift [@LUTwFJJ] can provide the change in the logic cell functioning, e.g., converting AND to OR. This mechanism can be also used for implementation of memory cell [@LUTwFJJ; @NGmemcell].
Historically, the problem of element base miniaturization was first recognized in development of superconductor random access memory (RAM). Since that time, the need for dense cryogenic RAM is the major stimulus for innovative research in this area.
Memory
======
Among the many attempts to create a cryogenic memory compatible with energy-efficient superconducting electronics, we want to single out the four most productive competing directions: (A) SQUID-based memory, (B) hybrid Josephson-CMOS memory, (C) JMRAM and (D) OST-MRAM.
SQUID-based memory
------------------
The presence or absence of SFQ(s) in a superconducting loop can be the physical basis for a digital memory element. Due to high characteristic frequency of Josephson junction, SQUID-based memory cells stand out with fast (few picoseconds) [@BLZ] write/read time favorable for RAM which is indispensable for data processor. Throughout various SQUID-based RAM realizations memory element was provided with destructive [@MDRO1; @MDRO2; @MDRO3] or non-destructive [@MNDRO1; @MNDRO2; @MNDRO3] readout. Memory cell contained accordingly from two [@MDRO1] to ten [@BLZ] Josephson junctions. With Josephson junction micron-scale dimensions in the late 1990-s this resulted in memory cell area of an order of few hundreds of microns squared. While power dissipation per write/read operations was at $\mu$W level, memory chip capacity [@MNDRO3] was only up to $4$ kb. In the particular $4$ kb RAM memory [@MNDRO3] the memory drivers and sensing circuits required AC power which limited its clock frequency to $620$ MHz. Later, all-DC-powered high-speed SFQ RAM based on pipeline structure for memory cell arrays was proposed [@MNDRO4]. Estimation showed that this approach allows up to $1$ Mb memory on $2\times2$ cm$^2$ chip operated at $10$ GHz clock frequency and featuring $12$ mW power dissipation. Still, it was never realized in experiment.
Hybrid Josephson-CMOS memory
----------------------------
Low integration density of SQUID-based memory cells seemed to be significant obstacle to the development of low-temperature RAM with reasonable capacity. This approach was succeeded by hybrid Josephson-CMOS RAM where Josephson interface circuits were amended by CMOS memory chip [@HMEM1; @HMEM2; @HMEM3; @HMEM4; @HMEM5]. This combination allowed to develop $64$ kb, $4$ K temperature RAM with $400$ ps read time, and $21/12$ mW power dissipation for write/read operations, accordingly [@HMEM5]. CMOS memory cell was composed of 8 transistors. While being fabricated in a $65$ nm CMOS process, the cell size was about three orders of magnitude less than the one of its SQUID-based counterparts. Main challenge in design of this memory system was amplification of sub-mV superconductor logic signal up to the $\sim 1$ V level required by CMOS circuits. This task was accomplished in two stages. First, the signal was amplified to $60$ mV using a Suzuki stack, which can be thought as a SQUID with each Josephson junction substituted by a series array of junctions for high total resistance [@SuzukiS]. At the second step, the reached $60$ mV signal drives high sensitive CMOS comparator to produce the volt output level.
Suzuki stack [@HMEM4] and CMOS comparator [@CMOScomp] were optimized for best compromise of power and time performance. Their simulated power $\times$ delay product for read operation were $2.3$ mW $\times$ $47$ ps ($0.11$ pJ) and $6.4$ mW $\times$ $167$ ps ($1.1$ pJ). This made up $73$% and $53$% of total memory system read power and time delay, correspondingly. These results exhibit severe restriction of overall system performance by the interface circuits. Recently, it was shown that the power consumption can be significantly decreased with utilization of energy efficient ERSFQ decoders and n-Trons as high voltage drivers [@nTronCMOSmem]. This could provide the energy efficiency improvement up to $3$ times for $64$ kb, and up to $12$ times for $16$ Mb memory. In the latter case, the access time in a read operation is estimated to be $0.78$ ns.
While the hybrid memory approach showed better memory capacity, its power consumption and time requirements are still prohibitive. It was summarized that for implementation of practical low-temperature RAM one should meet the following criteria [@MRep]: (i) scale: memory element dimension $< 100$ nm ($< 200$ nm pitch); (ii) write operation: $10^{-18}$ J energy with $\sim 50 - 100$ ps time delay per cell; (iii) read operation: $10^{-19}$ J energy with $\sim 5$ ps time delay per cell. An idea to meet the requirements nowadays is to bring spintronics (including superconductor spintronics) in RAM design.
JMRAM
-----
It is possible to reduce drastically the size of superconducting memory cell by using controllable Josephson junction with magnetic interlayers instead of SQUID [@JMRAM1; @JMRAM2; @JMRAM3; @JMRAM4; @JMRAM5; @JMRAM6]. Topology of such magnetic Josephson junction (MJJ) is usually of two types: (i) sandwich topology which is well suited for CMOS-compatible fabrication technology, and (ii) the one with some heterogeneity of the junction weak-link area in-plane of the layers. Below we present MJJ valves according to this classification.
### MJJ valve of sandwich topology
Search for an optimal way of compact MJJ valve implementation remains under way now. The most obvious solution is to use two ferromagnetic layers with different magnetic rigidity in the area of the junction weak link [@JMRAM7; @JMRAM8; @JMRAM9]. Critical current of such junction is determined by effects resulting from coexistence and competition of two orderings for electron spins: “superconducting” (S) (with usually antiparallel spins of electrons in the so-called “Cooper pairs”) and “ferromagnetic” (F) (with parallel ordering of electron spins). Magnetization reversal of “weak” F-layer leads to switching between collinear and anti-collinear orientations of the F-layers magnetic moments in the bilayer. This, in turn, provides alteration in the total effective exchange energy, $E_{ex}$, and hence, in MJJ critical current effective suppression. While magnetization reversal can be executed by application of an external magnetic field [@JMRAM10], the critical current can be read-out, e.g., with MJJ inclusion into decision making pair [@JMRAMRyaz], see Figure \[Fig14\]. It is possible to trace some analogy between this effect and the phenomenon of giant magnetoresistance [@JMRAM12] which is actively used in conventional magnetic memory cells.
![Principal scheme of implementation of write and read operations in MJJ valve based circuit.[]{data-label="Fig14"}](Fig14.pdf){width="0.65\columnwidth"}
A common drawback of most MJJs is small value of their characteristic frequency ($\omega_c \sim I_c R_n$) in comparison with SIS junction. Indeed, here one has to perform the magnetization reversal of weak F-layer with relatively small exchange energy in order to manipulate the total critical current against the background of its considerable suppression by the strong ferromagnet. Low $\omega_c$ outflows in slow read operation and complicates MJJ integration in SFQ logic circuits.
There are several approaches to solve this problem. One of them is the using of noncollinearly magnetized ferromagnetic layers [@JMRAM13; @JMRAM14; @JMRAM15; @conclmem1]. In this case, the triplet superconducting correlations of electrons are formed in the junction weak-link area. A part of them are featured by collinear orientation of electron spins in Cooper pairs. They are unaffected by exchange field of the ferromagnets, thus increasing MJJ critical current while maintaining its normal state resistance. The “triplet” current can be controlled by external magnetic field through magnetization reversal [@Birgereq]. Still, this approach implies implementation of a number of additional layers (and interfaces) in the structure that reduces its critical current.
One should also note that $E_{ex}$ alteration could result in $\pi$ shift of MJJ CPR. In this case the valve can be utilized as controllable phase battery [@LUTwFJJ]. Inclusion of such MJJ into SQUID loop allows fast read-out of its state [@NGmemcell]. However, here miniaturization reduces only to replacement of the SQUID inductance by the MJJ.
Another approach is based on localization of magnetic field source outside Josephson junction weak-link area but in the nearest proximity [@FFoutJJ1; @FFoutJJ2; @FFoutJJ3; @FFoutJJ4]. For example, F-bilayer can be placed on top of SIS junction. In this case, stray magnetic field penetrating into the junction area controls its critical current. If the junction S-layer neighboring the F-bilayer is thin enough, the coupling of the vector potential of the stray magnetic field to superconducting order parameter phase could also noticeably affect Josephson phase difference across SIS junction. SIS junction is utilized here just for read-out the ferromagnet state, and therefore, its characteristic frequency remains high. Still, since the strength of magnetic field is proportional to the ferromagnet volume, a possibility of miniaturization of such memory element is doubtful.
Critical current modulation can be obtained even in the structure with a single magnetic layer by changing the value of its residual magnetization [@JMRAM16]. It is also possible to improve the characteristic frequency by the inclusion of dielectric (I) and thin superconducting (s) layers in MJJ weak-link area to increase $R_n$ and $I_c$, correspondingly [@JMRAM19; @JMRAM20; @JMRAM21; @JMRAM22; @JMRAM23; @JMRAM24; @JMRAM25; @JMRAM26]. Such SIsFS valves possess characteristics close to SIS junction [@JMRAM21]. However, compatibility with superconductors requires utilization of ferromagnets with relatively low coercive field, which are typically characterized by non-square shape of the hysteresis loop. This in turn outflows into uncertainty of MJJ critical current at zero applied magnetic field after multiple magnetization reversals. In addition, miniaturization here faces the same difficulties as in the previous approach. For these reasons, it seems especially fruitful to replace the I and F layers with two magnetic insulator IF-layers to construct a Josephson S(IF)s(IF)S valve [@JMRAM27; @JMRAM28; @JMRAM29]. Its operation relies on variable suppression of superconductivity of the middle s-layer. Yet, this promising structure is complicated for fabrication.
### MJJ valve with in-plane heterogeneity of the weak-link area
The second type of valves implies heterogeneity of the weak-link region in the junction plane providing separation of the structure into two parts. CPR of these parts can be different, e.g., the conventional CPR and the ones shifted in phase by $\pi$ [@JMRAM24; @JMRAM18]. Such MJJ may be thought as nanoSQUID with conventional and “$\pi$” lumped junctions. Its implementation may comprise a ferromagnetic interlayer and a sandwich containing the same F-layer and a normal metal (N) layer, see Figure \[Fig15\]. If F-layer magnetization is aligned perpendicular to the nanoSQUID plane, it compensates the Josephson phase gradient across the MJJ making its critical current to be high. While the magnetization is being rotated at $90^\circ$, this effect is turned off and $I_c$ becomes low. Here for proper operation of this SF-NFS-based MJJ the flux of residual magnetization must be comparable with the flux quantum $\Phi_0$.
![Cross section of SF-NFS MJJ with CPRs of its parts shifted in phase by $\pi$. Arrows show F-layer magnetization directions corresponding to the valve on/off states.[]{data-label="Fig15"}](Fig15.pdf){width="0.6\columnwidth"}
The second common problem of MJJ-based memory is a long time of write operation. Bit write is commonly performed by magnetization reversal of at least one of F-layers. For this reason, recording time is of an order of inverse frequency of ferromagnetic resonance. It is usually more than two orders of magnitude larger than the characteristic time of SIS junction switching. Thus, elimination of magnetization reversal from the valves operation is desired. It is worth noting that nano-sized trap for single Abrikosov vortex in the vicinity of Josephson junction [@JMRAM33; @JMRAM34] allows to realize fast enough write operation. However, energy dissipation associated with annihilation of such vortex ($\sim 10^{-18}$ J) may contradict with the paradigm of energy-efficiency.
This challenge can be met with MJJ having bistable Josephson potential energy. Josephson phases of its ground states could be equal to $\pm \varphi$ ($0 < \varphi < \pi$). One can realize both write and read operations with such $\varphi$-junction on picosecond timescale [@JMRAM35; @JMRAM36; @JMRAM37; @JMRAM38; @JMRAM381; @JMRAM39; @JMRAM40]. The disadvantage of this approach is the difficulties with $\varphi$-state implementation. In practice, it is possible only in the structure with heterogeneous weak-link region of a rather large size.
One more operation principle of MJJ valves relies on control of superconducting phase domains formation [@JMRAM26]. The effect can be realized in SIsFS MJJ with sFS part substituted, e.g., for heterogeneous SF-NFS combination. The middle s-layer is broken on domains with different superconducting phases if Josephson phases of the structure parts are different, and vice versa. This process can be controlled by current injection through sFS or sFNS parts. The domain formation significantly changes the MJJ critical current. This MJJ provides fast read and write operations with no need for application of external magnetic field. Still, fabrication of compact Josephson junctions having the inhomogeneous weak-link region with reproducible characteristics is a difficult technological task.
OST-MRAM
--------
The next considered type of cryogenic memory is the hybrid approach combining superconducting control circuits with spintronics memory devices. Here due to spin-based interactions between atoms in the crystal lattice and electrons, orientations of ferromagnets magnetization can determine the amount of current flow. And vice versa, spin-polarized current can affect orientations of the magnetizations. The last effect is the so-called “spin transfer torque” (STT). It was suggested as a control mechanism for magnetic memory [@COST1; @COST2; @COST3]. However, high speed and low energy of write operation can not be provided with conventional spin-valve topology with collinear orientations of ferromagnets magnetizations [@COST4].
Orthogonal spin transfer (OST) device allows to overcome the difficulties. This structure consists of an out-of-plane ferromagnetic polarizer (OPP), a free F-layer (FL), and a fixed F in-plane polarizer/analyzer (IPP), see Figure \[Fig16\]. “Write” current pulse passing through OPP provides STT effect in FL which acts to lift its magnetization out of plane. Magnetization is then rotated about the out-of-plane axis, according to Landau-Lifshitz-Gilbert equation. Current pulse applied to IPP read-out collinear or anti-collinear magnetizations of in-plane magnetized F-layers.
![Sketch of OST device. Arrows show magnetization directions in the device layers.[]{data-label="Fig16"}](Fig16.pdf){width="0.55\columnwidth"}
It is possible to obtain the necessary $180^{\circ}$ magnetization reversal with correct selection of the write pulse amplitude and duration. Quasi-static and dynamic switching characteristics of OST devices have been analyzed at cryogenic temperatures: switching between parallel and anti-parallel spin-valve states has been demonstrated for $\sim$ mA current pulses of sub-ns duration [@COST4; @COST5; @COST6].
Clear advantages of the considered approach is elimination of control lines for magnetic field application, and implementation of fast magnetization reversal at sub-ns timescale. At the same time, the problems like relatively low percent of magnetoresistance, and the ones associated with possible magnetization over-rotation still prevent its practical application. The latter one can be overcome to some extend by involving both IPP and OPP polarizers into FL switching process [@COST4].
One could note that application of STT effect in some of MJJ valves is of considerable interest. STT in voltage biased superconducting magnetic nanopillars (SFNFS and SFSFS junctions) has been studied for both equilibrium and nonequilibrium cases [@COST9; @COST10; @COST11; @COST12; @COST13]. However, rich dynamics resulting from interplay of multiple Andreev reflection, spin mixing, spin filtering, spectral dynamics of the interface states, and the Josephson phase dynamics requires further research for evaluation of STT application appropriateness in superconducting memory structures.
Discussion
----------
Lack of suitable cryogenic RAM is “... the main obstacle to the realization of high performance computer systems and signal processors based on superconducting electronics.” [@MEMDIS1] While JMRAM and OST-MRAM look as the most advanced approaches, they still require further improvement in a number of critically important areas.
Progress in considered variety of device types with no clear winner is impossible without researches on new magnetic materials like PdFe, NiFe(Nb,Cu,Mo), Co/Ru/Co, \[Co/Ni\]$_n$ etc., and novel magnetization reversal mechanisms [@MEMDIS2; @MEMDIS3; @MEMDIS4]. They can lead to development of new operation principles combining superconductivity and spintronics.
Inverse proximity effect at SF boundaries dictates utilization of pretty thin (at nm scale) magnetic layers. However, characteristics of memory devices typically depends exponentially on the F-layers thicknesses and significantly affected by interfacial roughness. This challenge can be met with further development of high-accuracy thin-film technological processes in modern fabrication technology.
Substantial part of circuit area, time delay and dissipated power in memory matrix is more likely to be associated with address lines rather than with memory cells. This makes optimization of intra-matrix interconnections and memory cell architecture of significant importance.
While we considered here only the most developed solutions for superconducting valves and memory elements, there are many other approaches to create nanosized controllable superconducting devices for applications in memory and logic. We can point out on our discretion: the nanoscale superconducting memory based on the kinetic inductance [@MEMDIS5], and the superconducting quantum interference proximity transistor [@MEMDIS6]. Such concepts could bring novel idea into nanoscale design of superconducting circuits.
Conclusion {#conclusion .unnumbered}
==========
In conclusion, we discussed different superconductor logics providing fast ($\sim 5 - 50$ GHz) and energy efficient ($10^{-19} -
10^{-20}$ J per bit) operation of circuits in non-adiabatic and adiabatic regimes. The last one allows implementation of the most energy efficient physically and logically reversible computations with no limit for minimum energy dissipation per logic operation. Possibilities to combine the schemes based on different logics as well as utilization of different (e.g. superconductor and semiconductor) technologies in a single device design are presented.
Considered physical principles underlying superconducting circuits operation provide possibility for development of devices based on unconventional computational paradigms. This could be the basis for a cryogenic cross-platform supercomputer, where each task can be executed in the most effective way. In our opinion, the development of superconducting circuits performing non-classical computations like cellular automata, artificial neural networks, adiabatic, reversible, and quantum computing is indispensable to get all the benefits of the superconductor technology.
Low integration density, and hence low functional complexity of the devices, is identified as the major problem of the considered technology. This issue can be addressed with further miniaturization of basic elements and modernization of cell libraries, including introduction of novel devices like the ones based on nanowires or magnetic Josephson junctions.
The problem of low integration density is especially acute in RAM design. We considered here four different approaches to cryogenic RAM development with no clear winner. Progress in this area now implies elaboration of new operation principles based on synergy of different physical phenomena like superconductivity and magnetism, and appearance of novel effects, as for example, triplet spin valve memory effect [@conclmem1] or superconducting control of the magnetic state [@JMRAM27]. Proposed concepts of new controllable devices could eventually change the face of superconductor technology making it universal platform of future high-performance computing.
The authors are grateful to Alexander Kirichenko and Timur Filippov for fruitful discussions. This work was supported by grant No. 17-12-01079 of the Russian Science Foundation, Anatoli Sidorenko would like to thank the support of the project of the Moldova Republic National Program “Nonuniform superconductivity as the base for superconducting spintronics” (“SUPERSPIN”, 2015-2018), partial support of the Program of Competitive Growth at Kazan Federal University is also acknowledged.
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abstract: 'It has been argued that the long gamma-ray burst (GRB) of GRB060614 without associated supernova (SN) has challenged the current classification and fuel model for long GRBs, and thus a tidal disruption model has been proposed to account for such an event. Since it is difficult to detect SNe for long GRBs at high redshift, the absence of an SN association cannot be regarded as the solid criterion for a new classification of long GRBs similar to GRB060614, called GRB060614-type bursts. Fortunately, we now know that there is an obvious periodic substructure observed in the prompt light curve of GRB060614. We thus use such periodic substructure as a potential criterion to categorize some long GRBs into new class bursts, which might have been fueled by an intermediate-mass black hole (IMBH) gulping a star, rather than a massive star collapsing to form a black hole. Therefore, the second criterion to recognize these new class bursts is if they fit the tidal disruption model. From a total of 328 *Swift* GRBs with accurate measured durations and without SN association, we find 25 GRBs satisfying the criteria for GRB060614-type bursts: 7 of them are with known redshifts, and 18 with unknown redshifts. These new bursts are $\sim 6\% $ of the total *Swift* GRBs, which are clustered into two subclasses: Type I and Type II with considerably different viscous parameters of accretion disks formed by tidally disrupting their different progenitor stars. We suggest that the two different kinds of progenitors are solar-type stars and white dwarfs: the progenitors for 4 Type I bursts with viscous parameter of around 0.1 are solar-type stars, and the progenitors for 21 Type II bursts with viscous parameter of around 0.3 are white dwarfs. Potential applications of this new class of GRBs as cosmic standard candles are discussed briefly.'
author:
- 'H. Gao, Y. Lu, S. N. Zhang'
title: 'A new class of gamma-ray bursts from stellar disruptions by intermediate mass black holes'
---
Introduction
============
Recently the peculiar long-duration GRB060614 poses a great challenge to the widely accepted concept that long gamma-ray bursts (GRBs) are the consequences of core collapse of very massive stars. GRBs are normally classified into two groups: the long-duration bursts ($T_{90} > $ 2 s) and short ones ($T_{90} <$ 2 s) [@Kou93], where $T_{90} $ is defined as the time interval in which the integrated photon counts increase from 5% to 95% of the total counts. Several nearby long GRBs were observed to be firmly associated with core-collapse supernovae (SNe), and meanwhile the “collapsar” model of massive star explosions leading to long GRBs has been well developed (@Woo06). On the other hand, short GRBs are hypothesized to be formed by the coalescence of binary compact stars, and hence with no SN connection (@Nak07). The prompt light curve of GRB060614 as detected by the *Swift* Burst Alert Telescope (BAT) instrument displayed an initial short spike lasting for $\sim 4$s followed by an extended component lasting for $\sim 100$s, the latter being softer than the former in energy. This is a long burst by the above definition. The measured redshift of its host galaxy is as low as 0.125 [@Pri06]. However, despite its promising proximity, surprisingly no SN was observed to accompany the GRB down to very deep detection limits [@Del06a; @Fyn06; @Gal06].
Various solutions have been proposed for the missing-SN puzzle of GRB060614: First, the GRB perhaps did not take place at $z\!=\!0.125$ at all, but at much higher redshift so that its associated SN was below the detection limit. @Cob06 claimed that the proximity of the GRB line of sight to the $z\!=\!0.125$ galaxy was a chance coincidence; however @Cam08 found that the chance coincidence probability is less than 0.02%, thus ruling out the chance coincidence assumption with high confidence. Alternatively, the GRB could be produced with a very faint core-collapse SN [@Tom07; @Fry07], similar to the ones described by @Tur98, @Pas04 [@Pas07] and @Val09. Finally, this long GRB was not a consequence of the core-collapse of a massive star, and hence a novel mechanism is needed. For example, @Kin07a suggested that the merger of a massive white dwarf with a neutron star can make a long-duration GRB.
Recently, [@Lu08] proposed a new mechanism to produce GRBs, namely the tidal disruption of a star by an intermediate mass black hole (IMBH). In this scenario, both the long duration and the lack of an associated SN, as observed in GRB060614, are naturally expected. Furthermore, the model can well explain a probable 9-s periodicity found in the prompt BAT light curve of GRB060614 between 7 and 50 s. Such a substructure seems difficult to fit into either the collapsar scenario or compact star mergers. It is thus natural to investigate if GRB060614 is just the first of this new class of GRBs, which is the purpose of this paper.
We focus on the periodic substructure in the prompt light curve of a burst, since for most GRBs it is impossible to tell if an SN is associated or not, due to their high redshifts. A general picture of the tidal disruption model is given in Section 2, where important physical quantities are defined. The selection criteria, and the selected GRB060614-like events in the *Swift* samples, are described in Section 3. We do statistical studies on these bursts, and combine the statistical results with the model to give predictions in Section 4. Our conclusion and discussions are given in Section 5.
The Tidal Disruption Model for gamma-ray bursts
===============================================
The model description
---------------------
Tidal disruption of a star by an IMBH is analogous to the case for a supermassive black hole. A star, which happened to be close enough to the black hole, was distorted and squashed into a pancake by the strong tidal forces of the black hole. Once the star is tidally disrupted by the black hole, the squashed debris finally falls into the black hole’s horizon, forming a transient accretion disk around the black hole. During the early high accretion rate stage (near the Eddington rate), the inner region of the disk should be dominated by radiation pressure. In this case, the disk within the spherization radius $R_{\rm sp}$ is thermally unstable [@Sha73], and the material in this inner region is likely broken into many blobs. When the blobs are dragged into the black hole, the seed magnetic field anchored in the blobs can be amplified, forming a strong and ordered poloidal field, which in turn threads the black hole with a mass-flowing ring in the inner region of the disk and extracts a large amount of rotational energy, creating two counter-moving jets along the rotation axis of the black hole [@Bla77]. As in the conventional GRB model, each jet pointed toward the observer, produces one mini-burst lasting over a blob’s free-falling timescale [@Lu08]. Consequently, many mini-bursts should be produced for the tidal disruption event. Assuming that the in-falling process of the blobs into the black hole is neither uniform nor completely unsystematic, they may fall in groups quasi-periodically and this behavior should be modulated by the Keplerian timescale, forming a periodic sub-burst. All of these mini-bursts in the sub-burst add together to form a GRB, and the duration of the GRB is determined by the time when all the blobs with the spherization radius are removed and fall into the black hole at the marginally stable radius. Note that the blobs in each group randomly fall into the black hole, so small dispersion between sub-bursts’ durations may exist.This model can reasonably explain all the observed basic features of the unusual GRB060614, including the duration, the total energy, the periodic substructure, and most importantly, the absence of SN link [@Lu08]. The general picture of the tidal disruption model for such a GRB is plotted in Figure 1.
For convenience, we introduce the following dimensionless quantities throughout this paper: $$M_5=\frac{M_{\rm bh}}{10^5M_\odot}, \dot{m}=\frac{\dot{M}}{\eta
\dot{M}_{\rm Edd}},$$ where $\dot{M}_{\rm Edd}=3\times
10^{-2}\eta_{0.1}^{-1}M_5\,M_\odot\,{\rm yr}^{-1}$ is the Eddington accretion rate, $\eta$ is the energy conversion factor and $\eta_{0.1}=\eta/0.1$. Subsequently, the key physical parameters related to the tidal disruption model derived by @Lu08 are briefly described.
The three timescales and one relation
-------------------------------------
According to @Lu08, there are three useful timescales related to explain the observations of GRB060614: the mini-burst duration, $T_{\rm pulse}$, the sub-burst period, $T_k$, and the duration of the whole GRB, **$T_{90}$**. For instance, the three timescales are marked in the prompt light curve of GRB060614 in Figure 2. They can be calculated by $$\begin{aligned}
&& T_{pulse}\simeq 3M_5\, {\rm s} \,, \\
&& T_{k}\simeq50 \hat{r}_{\rm ms}^{3/2}M_5 \, {\rm s} , \\
&& T_{90}\simeq 50\alpha^{-1}\hat{r}_{\rm ms}^{3/2}M_5\, {\rm s} \,,\end{aligned}$$ where $\alpha$ is the viscous parameter, $0\leq\alpha\leq 1$, and $\hat{r}_{\rm ms}$ is the dimensionless radius of the marginally stable circular orbit around a non-spinning in units of $6GM/c^2$, i.e., $\hat{r}_{\rm ms}=1$.
From Equations (2) and (3), we can obtain a linear relation between the duration of the bursts and the period of the substructure $$\begin{aligned}
T_k=\alpha T_{90}\,\,.\end{aligned}$$ Equation (4) shows that the slope of the linear relation corresponds to the viscous parameter of the disk. This relation could be very useful for the classification of the GRBs.
Nevertheless, the viscous parameter, $\alpha$, is uncertain, which is considered to be related to the structure and physical properties of the disk, especially the magnetic fields of the disk; recent numerical studies of the magnetorotational instability viscosity mechanism (Balbus & Hawley 1991) have shown that the value of the viscosity parameter depends upon the magnetism of the disk and sufficiently strong magnetic fields in the disk are necessary for a large viscosity parameter (e.g., $\alpha>1$; Pessah et al. 2007). Note that the disk considered here is formed through the black hole gulping a star, indicating that the disruption of a different star by black holes will give a different disk, then the different viscosity of the disk. To apply Equation (4) to certain GRBs, different values of $\alpha$ indicate different GRB subclasses, depending on the progenitors of the bursts.
The energy for a GRB
--------------------
The tidal disruption model [@Lu08] can predict the isotropic energy of a GRB, $E_{\rm iso}$, given the beaming factor of $\Gamma$, $$\begin{aligned}
E_{\rm iso} \simeq E_{\rm tot} \times \Gamma,\end{aligned}$$ where $10\leq\Gamma\leq 1000$, and $E_{\rm tot}$ is the total energy of a GRB, which is from the rotational energy of black holes extracted by the BZ process [@Bla77], $$\begin{aligned}
E_{\rm tot}\simeq 2.46\times
10^{51}\alpha^{-1}M_5\left[\frac{(2.52\dot{m})^{7/64}-1}{2.52\dot{m}-1}\right]A^2f(A)N_{\rm
tot}\,\, {\rm erg},\end{aligned}$$ where $N_{\rm tot}$ is the total number of mini-bursts in the whole GRB, $A$ is the dimensionless angular momentum of the black hole, and $f(A)=2/3$ for $A\rightarrow 0$ and $f(A)=\pi-2$ for $A\rightarrow 1$ [@Lee00]).
Substituting Equation (4) into Equation (6) to eliminate the viscous parameter $\alpha$, the expression of $E_{\rm iso}$ can be rewritten as $$\begin{aligned}
E_{\rm iso}\simeq 2.46\times
10^{51}\frac{T_{90}}{T_k}\left[\frac{(2.52\dot{m})^{7/64}-1}{2.52\dot{m}-1}\right]\Gamma
M_5A^2f(A)N_{\rm tot}\,\, {\rm erg}\,.\end{aligned}$$ Given $A^2f(A)=0.02$ and $\dot{m}=1$, @Lu08 consistently explained the properties of GRB060614. Since the purpose of this work is searching for GRBs similar to GRB060614 and then analyzing these bursts’ properties, it is reasonable to assume that the adoption of $A^2f(A)=0.02$ and $\dot{m}=1$ is still suitable. Therefore, we use these values to other bursts hereafter in this paper. Note that $E_{\rm iso}$ is the most important physical quantity for using GRBs to investigate cosmology, such as the relation of @Ama02. Equation(7) shows that the isotropic energy only depends on the prompt gamma-ray emission light curve of a burst and the beaming factor of each individual burst, if the tidal disruption model for GRB060614 can be generalized to other bursts similar to GRB060614. This is in favor of $E_{\rm iso}$ of each burst as a calibrated candle to study cosmology. We will discuss this later.
Searching for gamma-ray bursts similar to GRB060614
===================================================
The criteria and Samples
------------------------
Assuming that the tidal disruption model for GRB060614 can be generalized to investigate other gamma-ray bursts similar to GRB060614, called GRB060614-type bursts, we propose three criteria to identify this new class of bursts: (1) the bursts must be long-duration bursts ($T_{90}>2$ s) without SN association; (2) the bursts should have an obvious periodic substructure (sub-burst) in the prompt light curves of the bursts, and the periodic substructure is composed of more than three mini-bursts; and (3) the burst should satisfy the relation indicated by the tidal disruption model: $T_{k}=\alpha T_{90}$. The three criteria must be all satisfied for each burst identified.
According to the above criteria, we select the samples among the 393 GRBs discovered by the *Swift* BAT before 2008 October 1. There are 330 long-duration bursts with accurately measured $T_{\rm 90}$. For these 393 GRBs, we first exclude two GRBs with established SN associations: GRB060218 [@Pia06] and GRB050525A [@Del06b]. From the rest 328 GRBs, we identified 24 new bursts. The prompt light curve of GRB060614 is shown in Figure 2 and the light curves of the newly selected 24 samples are shown in Figure 3. All of them are from a public online database [^1]. Tables 1 and 2 list all the physical quantities related to the 25 samples (including GRB060614), where Table 1 corresponds to 7 samples with measured redshifts, and Table 2 is for the rest 18 samples with unknown redshifts. The data are derived by combining the observations and the model. The details are summarized as follows.
The observed quantities
-----------------------
\(1) From a public online database [^2], we can obtain the duration ($T_{90}$) of the 25 GRB060614-type bursts, and the redshifts ($z^{\rm obs}$) of 7 GRB060614-type bursts including GRB 060614 itself.
\(2) Based on the appearance of the observed light curves of the bursts shown in Figures 2 and 3, we extract the number, $N_{\rm
sub}$, and the duration, $T_{k}$, of the sub-bursts. First our attempt of power density spectrum (PDS) analysis failed in this process. Taking GRB070223 as an example, whose light curve (see Figure 3) shows obvious quasi-periodic substructures, we present its PDS (the analysis is based on a 64 ms BAT light curve data, 15-350 keV and $T_{\rm -20}$s to $T_{\rm +110}$s ) in Figure 4. No significant signal was found in the PDS. We then propose two explanations for this failure. First, the extreme complexity of the light curves and the sensitive dependence of PDS on the noise properties sometimes prevent the PDS analysis from detecting the obvious structures in the light curves; second, as we said in Section 2, the scatter of sub-bursts’ durations may lower the detection significance of the sub-burst structure in the PDS. Consequently, we obtain the results of sub-bursts by a visual inspection method. In the following, we describe the detailed process of identifying sub-bursts and estimating their durations from the light curves of the 25 GRB060614-type bursts by the visual inspection method. We want to stress two important features of the GRB060614-type bursts’ light curves that will be taken as our selection *priors*: (1) pulses in the light curves tend to gather into several groups which are considered as candidates for sub-bursts; and (2) quasi-periodic substructures exist in the light curves. The 25 light curves are consequently classified into three cases (details are presented in Tables 1 and 2). Case I ($2/25$): all of the candidates for sub-bursts are separated by quiescent periods (since our analysis is based on existing light curves whose backgrounds have been taken out, we define quiescent period here as the period during which no significant peak can be recognized visually). Case II ($19/25$): all of the candidates for sub-bursts join together. Case III ($4/25$): some of the candidates for sub-bursts are separated by quiescent periods and others join together. For Case I, $N_{\rm sub}$ is easily obtained. We then take the average value of sub-bursts’ durations as $T_{k}$. For Case II, we first find out the highest peak in each sub-burst candidate, and then find out the minimum of the valley between the two adjacent highest peaks. These valleys, together with the beginning of the first sub-burst candidate and the end of the last sub-burst candidate, are taken as the candidates for the beginning or end of these candidate sub-bursts, whose corresponding times are labeled as $t_i\,(i=1\ldots n+1)$, where $n$ is the number of candidate sub-bursts. Since the real sub-bursts should satisfy the quasi-periodic property requirement, the values of all $t_{i+1}-t_i$ should be approximately the same. We calculate $T=\langle
t_{i+1}-t_i\rangle$ as the first estimate to $T_{k}$. For each sub-burst candidate, if $t_{i+1}-t_i<0.5T$ or $t_{i+1}-t_i>1.5T$, we reject the sub-burst candidate corresponding to $t_{i+1}$, and repeat the above procedure with the remaining $n-1$ candidates until all the candidates satisfy the criterion. Finally, we obtain $N_{\rm
sub}$ and take the value of $T=\langle t_{i+1}-t_i\rangle$ as $T_{\rm k}$. For Case III, some sub-bursts could be first identified similar to Case I, and we take the average value of these sub-bursts’ durations $T$ as the first estimate to $T_{k}$, and then identify other sub-bursts through the same procedure for Case II. Finally we also take the average value of the durations of these sub-bursts as $T_{k}$ in Case III. We note that in all the cases, we cannot give the exact error of $T_{k}$ introduced by the visual inspection method.
\(3) Following the peak-finding algorithm proposed by @Li96, we obtain the total number of mini-bursts, $N_{\rm tot}$, and the corresponding timescale of $T_{\rm pulse}$. We use a linear function *B(t)* to fit the burst background between the pre-burst and post-burst regions, where $t$ is the time. The whole burst region is divided into many count bins. Corresponding to the mini-burst time, $t_{\rm pulse}$, there is a peak bin with a count of $C_p$. This peak bin is assumed as having more counts than the neighboring bins around it. The condition for $C_p$ is satisfied if $$\begin{aligned}
C_p - C_{\rm 1,2} \geq N_{\rm var} \sqrt{C_p}\,\,,\end{aligned}$$ where $N_{\rm var}$ is a constant parameter, and $N_{\rm var} = 5$, $C_1$ and $C_2$ are the counts in two of the neighboring bins at $t_1$ and $t_2$, where $t_1<t_{p}<t_2$ (see [@Li96] for details of the peak-finding algorithm). Searching through the whole burst light curve, all peaks can be found. Assuming that each peak count corresponds to a mini-burst, the total mini-burst number, $N_{\rm
tot}$, in each GRB can be determined immediately. Combining the assumption of the tidal disruption model, we can get $$\begin{aligned}
T_{pulse}=\frac{N_{sub}}{N_{tot}}T_k\,\,,\end{aligned}$$ where $T_k$, $N_{\rm sub}$ and $N_{\rm tot}$ are derived as in the above discussion.
\(4) The isotropic energy for seven known redshift bursts, $E^{\rm
KB08}_{\rm iso}$, are from @Koc08. This will be used as evidence to check the tidal disruption model by comparing with the model predictions.
The model quantities
--------------------
In this subsection, we address and generalize the physical quantities predicted by the tidal disruption model for the GRB060614 [@Lu08], such as, the masses of black holes, the isotropic energy, and the redshifts related to all 25 samples.
\(1) The existence of IMBHs is still hotly debated and for which astronomers are still searching for direct evidence [@Heg02; @Por04; @Mil02; @Geb05]. Recent theoretical work, however, has given their possible density and occurring rate in the Milky Way. If these predictions are confirmed by observations, there could be 1,000-10,000 IMBHs in our native Galaxy. One way to estimate the masses of IMBHs is the tidal disruption model for GRBs. From Equation (2), we find that the periodic timescale, $T_{k}$, is linearly related to the masses of black holes assuming $\hat{r}_{\rm
ms}=1$ [@Lu08]. Rewriting Equation (2), we have $$\begin{aligned}
M_{\rm 5} \simeq 0.02 T_{k}.\end{aligned}$$ Substituting $T_k$ estimated according to the burst observations (in Subsection 3.2) into Equation (10), we can immediately obtain $M_5$ for the 25 samples, which are listed in Tables 1 and 2. The results show that the masses of the black holes range from $5\times
10^3M_\odot$ to $9\times 10^4M_\odot$. These are the typical masses of IMBHs discussed by @Mil02. The distribution of $M_5$ for the 25 GRB060614-type bursts is plotted in the left panel of Figure 5.
(2)Adopting $A^2f(A)=0.02$, $\dot{m}=1$(see Section 2 for the definition of $A^2f(A)$ and $\dot{m}$), and substituting $M_5$, $T_{90}$ and $T_k$ into Equation (7), we can immediately calculate the isotropic energy, $E_{\rm iso}^{\rm pre}$ as long as we know the beaming factors for each GRB. Since we have not obtained the beaming factors for all the 25 samples, for rough estimation of $E_{\rm
iso}^{\rm pre}$, we uniformly take the average value of 500 for $\Gamma$, whose observational range is $10\leq\Gamma\leq 1000$, as the beaming factor for all the 25 samples. The results are listed in Tables 1 and 2. The middle panel of Figure 5 plots the distribution of $E_{\rm iso}^{\rm pre}$ for the 25 samples.
\(3) The redshifts for the 25 samples can also be predicted. It is known that the relation between the isotropic energy, $E_{\rm iso}$, and the fluence, $F_{\rm \gamma}$, of GRBs, satisfies $$\begin{aligned}
\label{Eiso} E_{\rm iso} = 4\pi d_{\rm L}^{\rm 2}F_{\rm \gamma},\end{aligned}$$ where $d_{\rm L}$ is the distance between a GRB and the observer, which can be calculated by [@Car92] $$\begin{aligned}
\label{dl} d_{\rm L} = \frac{1+z}{H_{\rm 0}}\int_0^z{[(1+x)^{\rm
2}(1+x\Omega_{\rm M})-x(2+x)\Omega_{\rm \Lambda}]^{-1/2}}dx,\end{aligned}$$ where $x$ is an integral variable for the redshift; $\Omega_{\rm
M}$, $\Omega_{\rm \Lambda}$ and $H_{\rm 0}$ are the cosmological constant parameters and Hubble constant, respectively. To compare with the $E_{\rm iso}^{\rm KB08}$, we adopt $\Lambda$CDM cosmology, that is: $\Omega_{\rm M} = 0.3$ , $\Omega_{\rm \Lambda} = 0.7$ and $H_{\rm 0} = 71 {\rm km s}^{\rm -1}{\rm Mpc}^{\rm -1}$. Combining Equations (11) and (12), we can numerically calculate the redshift, $z$. The distribution of the predicted redshifts for 25 samples is plotted in the right panel of Figure 5.
Statistical relations and predictions
=====================================
Based on observations and our theoretical model, Subsections 2.2 and 2.3 have considered the physical quantities related to the 25 GRB060614-type bursts. The statistical analysis and the predictions based on these quantities will be addressed in detail.
It is important to examine if the tidal disruption model, which is successful for the case of GRB 060614 [@Lu08], can be generalized to explain the other 24 bursts selected in this paper. We first consider the relation between $E_{\rm iso}^{\rm BK08}$ and $E_{\rm iso}^{\rm pre}$ for the seven known redshift samples. We plot the data of $E_{\rm iso}^{\rm BK08}$ and $E_{\rm iso}^{\rm
pre}$ in Figure 6. Using the least-squares fitting, we find a linear relation $$\begin{aligned}
\log(E_{\rm iso}^{\rm KB08})=(0.997\pm
0.03)\log(E_{\rm iso}^{\rm pre})\,\,,\nonumber\end{aligned}$$ where the adjusted $\Re$-square value is $\sim 0.999$. This indicates that the isotropic energy predicted by the model agrees well with those given by the observations [@Koc08]. We thus believe that the tidal disruption model can be successfully generalized to study the properties of the other GRB060614-type bursts. More observed isotropic energy for the rest 18 unknown redshift GRB060614-type bursts can be used to check such relation in the future if their host galaxy’s redshifts can be measured.
It is interesting to note that the isotropic energy of GRBs is one of the best rulers to measure the expansion of the universe [@Ama02]. Since $E_{\rm iso}^{\rm pre}$ predicted by the tidal disruption model is only related to $T_{90}$ and $T_k$, we can thus study cosmology using only the prompt light curves of the bursts and the redshifts of their host galaxies.
We then plot the data of $z^{\rm obs}$ and $z^{\rm pre}$ for the seven known redshift samples in Figure 7. The relation fitted by the least-squares method follows: $$\begin{aligned}
z^{\rm obs}=(1.10\pm 0.12)z^{\rm pre}\,,\nonumber\end{aligned}$$ where the adjusted $ \Re$-square is $0.93$. This again indicates that the redshifts predicted agree well with those observed, which further prove that the tidal disruption model is correct for the new class of bursts, such as GRB060614-type bursts discussed in this paper. The predictions of the 18 unknown redshift GRB060614-type bursts will be tested by the further observations.
The relations of both $E_{\rm iso}^{\rm pre}-E_{\rm iso}^{\rm KB08}$ and $z^{\rm pre}-z^{\rm obs}$ argue that the tidal disruption model can work well for the 25 new class bursts. We plot the data of $T_{k}$ and $T_{\rm 90}$ in Figure 8. We find interestingly that the relation between $T_{k}$ and $T_{\rm 90}$ for the 25 samples can be well fitted by two linear curves with different slopes: four of the samples , i.e., GRB060210, GRB060614, GRB080602, and GRB080503, follow the linear relation $$\begin{aligned}
T_{k} = (0.08\pm 0.003)~T_{\rm 90}\,,\end{aligned}$$ and the rest 21 of the samples follow $$\begin{aligned}
T_{k} = (0.27\pm 0.013)~T_{\rm 90} \,,\end{aligned}$$ respectively. The slope of the former curve is about $0.08$, and the later one is $0.27$.
Comparing with Equations (13) and (14) with Equation (4), we get two values of viscous parameters, $\alpha_1=0.08$ and $\alpha_2=0.27$, respectively. Both of them fall well in the typical range of $\alpha\sim$ 0.1-0.4 inferred from observations of unsteady accretion disks, such as in the outbursts of dwarf novae and X-ray transients [@Kin07b]. It is known that the viscous parameter depends on the detailed structure and the magnetic field of the disk, because more strongly magnetized allows more efficient angular momentum transfer in the disk, and thus results in larger values of $\alpha$ (Pessah et al. 2007). Here in our model the transient disk is formed by the IMBHs disrupting stars, thus different structure and magnetic field of the disk will be produced by the disruptions of different type of stars. If white dwarfs, instead of regular stars, are tidally disrupted by black holes [@Fro94; @Fry99; @Ros09], much more strongly magnetized disk with higher viscous parameter can be produced, because during the contraction process which forms the white dwarf, magnetic fields are significantly amplified due to magnetic flux conservation (Tout et al. 2004). As a matter of fact, white dwarfs can only be tidally disrupted by IMBHs, because they will simply fall into supermassive black holes without being tidally disrupted, due to their much smaller sizes than regular stars. Similarly, neutron stars, though have much higher surface magnetic fields than those of white dwarfs, will fall into IMBHs directly without being tidally disrupted. We thus postulate that the linear relations between $T_k$ and $T_{90}$ given in Equations (13) and (14) divide 25 GRB060614-type bursts into two subclasses: 4 of them are Type I, corresponding to IMBHs gulping to solar-type stars, and the rest 21 bursts are Type II, which are produced by the IMBHs disrupting white dwarfs, respectively. The ratio of Type I to Type II is about 1:5, suggesting that white dwarfs are much more abundant than solar-type stars around IMBHs; this possibility is discussed in the end.
Discussion and conclusion
=========================
We have identified a new class of GRBs, called GRB060614-type bursts, which is produced by IMBHs tidally disrupting stars. To select out these GRB060614-types, we have used criteria based on the observation features of GRB060614 and our tidal disruption model, and searched through all the GRBs discovered by [*Swift*]{} BAT until 2008 October 1.
\(1) We found 25 GRB060614-type bursts from the 328 [*Swift*]{} GRBs with accurately measured durations and without SN association; 4 bursts (GRB060210, GRB060614, GRB080602, and GRB080503) are Type I, and the rest 21 bursts are Type II. These new GRB060614-type bursts make $6\%$ of the total $\textit{Swift}$ GRBs, composed of $1\%$ Type I and $5\%$ Type II.
\(2) We derive the distributions of the IMBH’s masses of the 25 bursts, as well as the isotropic energies and redshifts for the 18 bursts with unknown redshifts, predicted by the tidal disruption model. The statistical studies show that the masses of the IMBHs are from $5\times 10^3M_\odot$ to $9\times 10^4M_\odot$ (in the left panel of Figure 5), the isotropic energies predicted are well consistent with those given by @Koc08, and the redshifts predicted are also agreed well with observations. These results confirm the tidal disruption model, and may possibly provide a new standard candle to study the cosmology.
\(3) We obtain statistically that the relation between the substructure period and the duration of the 25 GRB060614-type bursts is fitted by two linear curves with slopes (viscous parameters of their disks) of $0.08$ and $0.27$, respectively. This indicates that 25 GRB060614-type bursts are composed of 2 subtypes called Type I and Type II, corresponding to two different progenitors for their productions through the tidal disruption. We postulate that the progenitors for 4 Type I bursts are most likely solar-type stars, and those for the 21 Type II bursts are probably white dwarfs.
Finally, we discuss the event rate ratio between Type I and Type II bursts. Gebhardt et al.(2005) found evidence for an IMBH of mass of about $2\times 10^4M_\odot$, residing in a globular cluster G1, which makes globular cluster as the most popular candidate to probe IMBHs. In young star clusters, high-mass stars segregate through energy equipartition; as a result, the heavier stars sink to the center while the lighter stars move to the outer halo. This process is called “mass segregation” (Spitzer 1969, 1987). Through dynamical friction, most massive stars tend to concentrate toward the center and drive the system to core collapse (Gürkan et al. 2004). As shown in a number of numerical studies (e.g., Portegies Zwart et al. 2004; Gürkan et al. 2004), very high initial central densities might lead to a rapid core collapse and segregation of massive stars, and trigger a runaway merger of massive stars, leading to the formation of an IMBH. It is naturally suggested that the less massive stars, not involved in the runaway process but were also migrating toward the clusters’ centers, may eventually end as white dwarfs, neutron stars or black holes. Neutron stars and black holes, driven by dynamical frictions, will also migrate toward the center , due to their larger masses, so only white dwarfs (and some low mass stars) will remain (Heyl 2008; Heyl & Penrice 2009). On the other hand, due to the mass segregation mentioned above, those much less massive stars, such as the solar-type stars, may migrate there much later on the average. Therefore in the vicinity of IMBHs in centers of star clusters, the space density of white dwarfs should be much higher than that of solar-type stars, explaining naturally the much higher event rate of Type II bursts discussed above. We thus predict that the central region of the G1 cluster harboring an IMBH should contain many more white dwarfs than solar-type stars. However, without further extensive studies, which are beyond the scope of this present work, we cannot predict or explain quantitatively the suggested 5:1 ratio between white dwarfs and solar-type stars.
We thank J.S. Deng for the improvement on writings. This research was supported by the National Natural Science Foundation of China (grants 10573021, 10821061, 10733010 and 10725313), the Chinese Academy of Science through project no. KJCX2-YW-T03, and the 973 Program of China under grant 2009CB824800.
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[lccccccccccc]{}
050505 & 58.9 & 25 & 1.21 & 4.27 & 62 & 53.18 & 0.50 & 0.42 &53.11 & II\
051109a & 37.2 & 15 & 0.98 & 2.346 & 46 & 52.36 & 0.30 & 0.40 &52.78 & II\
060116 & 105.9 & 22 & 0.7 & 4 & 126 & 53.32 & 0.44 & 0.21 &53.67 & II\
060210 & 255 & 20 & 0.8 & 3.91 & 100 & 53.62 & 0.40 & 0.08 &53.95 & II\
060614 & 102 & 9 & 1.2 & 0.125 & 30 & 51.03 & 0.18 & 0.09 &51.75 & II\
060926 & 8 & 2.5 & 0.28 & 3.208 & 36 & 51.97 & 0.05 & 0.31 &52.00 & II\
061007 & 75.3 & 26 & 0.65 & 1.261 & 120 & 54.18 & 0.52 & 0.35 &53.50 & III\
[lccccccccccc]{}
050117 & 166.6 & 45 & 141 & 0.96 &0.90 & 0.27 &3.27 &53.92 & II\
050306 & 158.3 & 33 & 186 & 0.89 &0.66 & 0.21 &3.31 &54.01 & II\
050326 & 29.3 & 13 & 19 & 2.05 &0.26 & 0.44 &0.71 &52.29 & I\
050607 & 26.4 & 10 & 26 & 1.15 &0.20 & 0.38 &2.32 &52.38 & II\
050717 & 85 & 16 & 55 & 1.16 &0.32 & 0.19 &1.93 &53.21 & II\
060102 & 19 & 6 & 19 & 0.95 &0.12 & 0.32 &2.61 &52.10 & II\
060105 & 54.4 & 20 & 57 & 1.05 &0.40 & 0.37 &1.08 &53.04 & III\
060306 & 61.2 & 20 & 63 & 0.95 &0.40 & 0.33 &2.78 &53.13 & I\
060424 & 37.5 & 11 & 60 & 0.73 &0.22 & 0.29 &3.59 &52.90 & II\
060510a & 20.4 & 8 & 25 & 0.64 &0.16 & 0.39 &0.72 &52.25 & II\
061028 & 106.2 & 40 & 117 & 1.03 &0.80 & 0.38 &6.36 &53.64 & II\
070223 & 88.5 & 26 & 95 & 0.82 &0.52 & 0.29 &4.23 &53.47 & II\
080212 & 123 & 31 & 198 & 0.47 &0.62 & 0.25 &5.3 &53.93 & II\
080328 & 90.6 & 24 & 83 & 0.87 &0.48 & 0.26 &1.99 &53.42 & III\
080409 & 20.2 & 8 & 21 & 0.76 &0.16 & 0.40 &1.88 &52.17 & II\
080503 & 170 & 15 & 33 & 0.91 &0.30 & 0.09 &3.34 &53.29 & II\
080602 & 74 & 5 & 33 & 0.61 &0.10 & 0.07 &1.95 &52.93 & III\
080723a & 17.3 & 8 & 18 & 1.33 &0.16 & 0.46 &2.13 &52.04 & II\
{width="6in"}
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{width="6in"}
{width="6in"}
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[^1]: http://grb.physics.unlv.edu/ xrt/xrtweb/web/sum.html
[^2]: http://swift.gsfc.nasa.gov
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---
abstract: 'We investigate spin-dependent decay and intersystem crossing in the optical cycle of single negatively-charged nitrogen-vacancy (NV) centres in diamond. We use spin control and pulsed optical excitation to extract both the spin-resolved lifetimes of the excited states and the degree of optically-induced spin polarization. By optically exciting the centre with a series of picosecond pulses, we determine the spin-flip probabilities per optical cycle, as well as the spin-dependent probability for intersystem crossing. This information, together with the indepedently measured decay rate of singlet population provides a full description of spin dynamics in the optical cycle of NV centres. The temperature dependence of the singlet population decay rate provides information on the number of singlet states involved in the optical cycle.'
address: 'Kavli Institute of Nanoscience Delft, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands'
author:
- 'Lucio Robledo, Hannes Bernien, Toeno van der Sar and Ronald Hanson'
title: 'Spin dynamics in the optical cycle of single nitrogen-vacancy centres in diamond'
---
Introduction
============
Nitrogen-vacancy (NV) centres in diamond are well-defined quantum systems in the solid state, with excellent spin coherence properties [@Balasubramanian09]. Even at ambient conditions, NV centres have successfully been used in the field of quantum information processing [@Jelezko04b; @Childress06; @Dutt07; @Neumann08; @Fuchs09; @Neumann10; @Gijs], magnetic sensing [@Degen08; @Taylor08; @Maze08; @Balasubramanian08; @Gijs10b] and photonic devices [@Fu08; @Babinec10; @Hadden10; @Toeno10; @Siyushev10; @Englund10]. However, despite the rapid experimental progress the understanding of the optically-induced spin-dynamics of the NV centre is still incomplete. In particular, the spin-dependent intersystem crossing (ISC) rates as well as the number of singlet states involved in the optical cycle are still debated. These parameters are responsible for optical spin initialization and readout, and are important for a correct estimation of photon emission rates. We extract these values by a series of room-temperature experiments, where we perform spin-resolved fluorescence lifetime measurements using picosecond optical excitation pulses. The lifetime of the singlet manifold is measured by analyzing the initial fluorescence rate for consecutive microsecond optical pulses with variable delay. The temperature dependence of this lifetime yields insight into the number of involved singlet states.
Experimental setting
====================
We investigate individual NV centres contained in a high-temperature high-pressure (HTHP) grown type IIa diamond sample from Element Six ($\left\langle 111 \right\rangle$ oriented). The sample is studied in a scanning confocal microscope setup operated at $T=10 \ldots 300$K. Spin control is achieved via microwave (MW) fields applied to an Au-waveguide that is lithographically defined on the diamond surface [@Fuchs09; @Gijs]. For optical excitation we use a continuous-wave (CW) laser at $\lambda = 532$nm, equipped with an acousto-optic modulator (AOM) with 20ns rise-time, as well as a frequency-doubled diode laser at $\lambda = 532$nm with a pulse length of 62ps (max. pulse energy: 25nJ) and variable repetition rate. For photon detection we use an avalanche photo diode in the single-photon counting regime with a timing jitter of 450ps. Time-resolved data are acquired using a time-correlated single photon counting module with a jitter of 10ps, using a bin size of 512ps. An arbitrary waveform generator (channel-to-channel jitter $<100$ps) is used as timing source of the experiment.
Model
=====
The photo-dynamics of the NV centre \[figure \[fig:fig1\](a)\] are determined by six electrons, which in the ground state form a triplet occupying an orbital of $^3A_2$ symmetry. The centre can be excited via a dipole-allowed transition to a $^3E$ triplet state. This level also has a spin-dependent probability to undergo ISC to a series of singlet states [@Manson06a]. We use a five-level model to describe the spin dynamics of the NV centre \[figure \[fig:fig1\](b)\]. Spin-spin interaction splits the $^3A_2$ ground state by $D_{GS}=2.87$GHz into a state with spin projection $m_s=0$ ($\left|1\right\rangle$) and a doublet with $m_s=\pm1$ (summarized in $\left|2\right\rangle$). Correspondingly, the excited $^3E$ state is labelled $\left|3\right\rangle$ ($m_s=0$, associated with a lifetime $T_{1,3}$) and $\left|4\right\rangle$ ($m_s=\pm1$, lifetime $T_{1,4}$), and split by $D_{ES}=1.43$GHz [@Fuchs08; @Neumann]. Two singlet states with a splitting of $\Delta E = 1.189$eV have been identified experimentally [@Manson08; @Acosta], but recent theoretical studies [@Gali10] and also data obtained in this work suggest the presence of a third singlet state between $^3A_2$ and $^3E$. For the analysis of spin dynamics, however, we summarize the singlet states in $\left|5\right\rangle$, and the corresponding lifetimes are summed and denoted as $T_{1,5}$. Rates from state $\left|m\right\rangle$ to state $\left|n\right\rangle$ are denoted by $k_{mn}$, and we only consider relaxation rates indicated in figure \[fig:fig1\](b). Population in state $\left|n\right\rangle$ is denoted by $P_n$, and the spin polarization in ground and excited state are denoted by $P_{GS}=P_1/(P_1+P_2)$ and $P_{ES}=P_3/(P_3+P_4)$.
![\[fig:fig1\] [**(a)**]{} Lattice structure of the NV centre: a substitutional nitrogen atom [N]{} next to a vacancy [V]{} in the diamond lattice [C]{}. [**(b)**]{} Level structure of the NV centre: we consider spin-conserving ($k_{31}$, $k_{42}$) and spin-flip ($k_{32}$, $k_{41}$) transitions between triplets (states with spin projection $m_s = \pm 1$ are merged into states $\left|2\right\rangle$, $\left|4\right\rangle$). Spin-dependent ISC rates connect triplets to the singlet states (summarized as $\left|5\right\rangle$, and described by rates $k_{35}, k_{45}, k_{51}, k_{52}$).](fig1.eps)
Spin-dependent lifetime
=======================
Pulsed optical excitation and time-resolved detection of fluorescence provides a simple and direct way to determine the lifetime of the excited state in an optical transition. If the excitation pulse is short compared to the lifetime $T_1$, the detected time-resolved fluorescence (averaged over many excitation cycles) decays exponentially $I \propto \exp(-t/T_1)$. However, if the system under consideration is excited into a mixture of $n$ excited states with different lifetimes, the detected fluorescence decays according to a multi-exponential function $ I \propto \sum_n a_n \exp(-t/T_{1,n})$.
This situation is present in the case of NV centres in diamond, where the excited state is composed of a spin triplet. Population in this state decays radiatively to the triplet ground state with identical oscillator strength for the different spin projections [@Batalov]. Because of spin-dependent ISC [@Manson06a], the effective lifetime of the excited state is significantly different for states $\left|3\right\rangle$ and $\left|4\right\rangle$ . In literature [@Neumann; @Batalov; @Collins83] the NV centre’s lifetime generally is obtained by fitting the time-resolved fluorescence to a single-exponential decay, leading to values of $T_{1,3}\approx 12 - 13$ns and $T_{1,4}\approx 8$ns in bulk diamond. Since optically induced spin polarization in NV centres is limited [@Manson06b; @Fuchs10], in fact we expect such a lifetime measurement to yield a bi-exponential decay curve with time constants set by the sums of rates out of states $\left|3\right\rangle$ and $\left|4\right\rangle$ \[$T_{1,3}=1/(k_{31}+k_{32}+k_{35})$ and $T_{1,4}=1/(k_{41}+k_{42}+k_{45})$\], and an amplitude ratio set by the initial spin polarization. Such a bi-exponential decay has been observed in [@Fuchs10], where a polarization of $P_{ES}=0.84\pm0.08$% has been obtained. These data are based on fluorescence lifetime measurements with MW spin manipulation in the excited state, where the duration of the MW pulse was neglected, and the pulse was assumed to be perfect.
![\[fig:fig2\] [**(a)**]{} We first polarize the spin by applying a 1.3$\mu$s laser pulse at $\lambda=532$nm. After 800ns we turn on a MW field at 2.87GHz for a variable duration. 200ns after begin of the MW we excite the NV center by a 62ps laser pulse at $\lambda=532$nm and measure the time-resolved emission. The first 300ns of the subsequent polarization pulse is used for spin readout. The experiment was performed at $T=300$K. [**(b)**]{} Fluorescence decay as function of MW burst duration (fluorescence counts are encoded in a logarithmic colour scale). The decay time oscillates with the $m_s = 0$ amplitude, as confirmed by [**(c)**]{} conventional spin readout. [**(d)**]{} Fluorescence decay curve, integrated over all applied MW burst durations. A fit using a bi-exponential function, yields the lifetime of states $\left|3\right\rangle$ and $\left|4\right\rangle$. A single-exponential function (shown for comparison) cannot accurately fit the experimental data. [**(e)**]{} Degree of spin polarization $P_{ES} = P_{3}/(P_{3}+P_{4})$ with no MW applied and after a 20ns MW pulse, obtained from the relative amplitudes of a bi-exponential fit.](fig2.eps){width="15cm"}
Here we present a simple and reliable way to obtain the spin-dependent lifetimes, without any assumptions on the MW pulse and spin polarization. For that purpose we drive Rabi oscillations in a conventional fashion, i.e. we apply a 1.3$\mu$s long off-resonant laser pulse to polarize the electron spin, followed by a MW pulse of variable duration, resonant with the zero-field splitting of $D = 2.87$GHz. After the MW, we apply a ps laser pulse. The fluorescence as function of MW pulse duration is shown in figure \[fig:fig2\](b). For comparison we also plot the result of a conventional spin readout \[figure \[fig:fig2\](c)\], i.e. the intensity of the first 300ns of the subsequent polarization laser pulse [@Jelezko04]. The data in figure \[fig:fig2\](b) clearly reveal the oscillations in decay time which are in phase with oscillations of the electron spin. To accurately fit these data to a bi-exponential decay we sum over all MW pulse durations, which gives similar contribution from each spin state. From the fit we obtain the two time constants $T_{1,3} = 13.7 \pm 0.1$ns and $T_{1,4} = 7.3 \pm 0.1$ns \[figure \[fig:fig2\](d)\]. By using these values as constants for a fit to the individual decay curves, we can determine the relative contributions of states $\left|3\right\rangle$ and $\left|4\right\rangle$ to the minima and maxima of the Rabi oscillation data from the relative amplitudes of the two exponentials. For this particular centre, we find $P_{ES,max}=72.1\pm0.9\%$ and $P_{ES,min}=12.2\pm0.5\%$ \[figure \[fig:fig2\](e)\]. The value of $P_{GS,max}$ may be larger due to a nonßperfect spin conservation in optical excitation (see section \[sec:sec\_pol\_prob\]).
We note that for a spin $S=1$ system with degenerate levels $m_s=\pm1$ (i.e. for NV centres in absence of magnetic field), the effect of resonant MW driving starting from a pure $\left|m_s=0\right\rangle$ state is to cause coherent oscillations between $\left|m_s=0\right\rangle$ and the symmetric superposition $\left|m_s=+1\right\rangle + \left|m_s=-1\right\rangle$. In a more realistic scenario we need to consider an only partially polarized state, where the density matrix subspace spanned by $\left|m_s=+1\right\rangle$ and $\left|m_s=-1\right\rangle$ has equal population of the symmetric and antisymmetric superposition state. Only the symmetric state will be transferred back to $\left|m_s=0\right\rangle$, so the effect of a half-oscillation is to transfer the full $m_s=0$ population into $m_s=\pm1$, while only half of the incoherent population in $m_s=-1$ and $m_s=+1$ is transferred back into $m_s=0$. As a consequence, when driving Rabi oscillations, the maximum population in $m_s=0$ will only reach half the value of the maximum population in $m_s=\pm1$ \[figure \[fig:fig2\](e)\].
In summary, the presented method allows for accurate determination of the lifetime of the pure states $\left|3\right\rangle$ and $\left|4\right\rangle$, without the need of assumptions on the quality of spin manipulation. The knowledge of these lifetimes can then be used to quantify the spin polarization.
Temperature dependence of singlet decay
=======================================
An important parameter for the spin dynamics of the NV centre under optical excitation is the time population spends in the singlet manifold before it decays back to the triplet ground state. This time scale is long compared to the excited state lifetime, and since decay into the singlet states (ISC) is favoured for $m_s=\pm1$, this leads to reduced fluorescence for $m_s=\pm1$, a fact which is routinely used for non-resonant spin readout [@Jelezko04].
![\[fig:fig3\] Decay of population from singlet states leads to recovery of fluorescence: [**(a)**]{} individual traces and [**(b)**]{} full data set of NV fluorescence with two excitation pulses and variable delay. [**(c)**]{} Fluorescence counts integrated over first 30ns of second pulse as function of inter-pulse delay. The exponential increase is caused by decay out of singlet states. [**(d)**]{} Temperature dependence of singlet decay rate. The fit assumes a phonon-assisted decay process.](fig3.eps)
The temperature dependence of this decay rate yields information about the energy splitting involved in this relaxation process, and adds further evidence for the number of singlet states contributing to the optical cycle of NV centres. Recently an infrared (IR) emission channel was observed, and attributed to a dipole-allowed transition between two singlet levels with an energy splitting of $\Delta E = 1.189$eV [@Manson08; @Acosta]. In the same publications it was shown that the IR emission follows the same time dependence as the visible transition, implying a short lifetime of the upper singlet state. Therefore the previously observed long-lived singlet state was attributed to the ground state of this IR transition. The temperature dependence of this singlet state lifetime has been reported for an ensemble of NV centres based on IR absorption measurements [@Acosta]. Here, we present data obtained on single NV centres. Since the oscillator strength of the IR transition is very weak, we use an indirect way to obtain this timescale, following the method used in [@Manson06a]: We first apply a green pulse (1$\mu$s) to reach a steady state population in the singlet states. This population manifests itself in reduced steady-state fluorescence with respect to the beginning of the pulse. After a variable delay we apply a second 1$\mu$s pulse \[figure \[fig:fig3\](a,b)\]. During the delay between the pulses, population stored in the singlet states decays back to the triplet ground state. The fluorescence at the beginning of the second pulse is proportional to the population in the triplet ground state and thus we can attribute its dependence on inter-pulse delay to the population decay out of the singlet state. In figure \[fig:fig3\](c) we integrate the photons emitted during the first 30ns of the second pulse for each inter-pulse delay and fit these data to an exponential function. Figure \[fig:fig3\](d) summarizes the timescales we obtained in this way, for temperatures ranging from $T=13$K to $T=300$K for three different NV centres.
We model the lifetime $\tau$ of the singlet state as a combination of temperature-independent spontaneous decay rate $\tau^{-1}_{0}$ and a rate accounting for stimulated emission of phonons of energy $\Delta E$ with an occupation given by Bose-Einstein statistics: $\tau = \tau_0 [1-\exp(-\Delta E/k_BT)]$. The fit yields a spontaneous emission lifetime of $\tau = 371 \pm 6$ns and a phonon energy of $\Delta E = 16.6 \pm 0.9 $meV, in reasonable agreement with ensemble data obtained in [@Acosta].
This result suggests that, apart from the 1.189eV splitting, a third level $\Delta$E below the IR transition’s ground state is involved in the optical cycle of the NV centre. We can exclude that this third level is the triplet ground state, just 16.6meV below the lowest singlet, as this would imply phonon-assisted spin relaxation on a sub-microsecond time scale at room temperature. This scenario clearly contradicts experimentally observed spin lifetimes on a millisecond timescale [@Neumann08; @Gijs]. Consequently the third level is likely to be another singlet state. The presence of three singlet states in between the triplet $^3A_2$ and $^3E$ states was recently predicted by an ab-initio calculation of the excited states in the NV centre [@Gali10], however, there a larger energy splitting between the lowest singlet states was obtained.
\[sec:sec\_pol\_prob\]Polarization probability
==============================================
Polarizing the electron spin by off-resonant optical excitation is a key technique for room-temperature spin manipulation of NV centres. Although this effect already has been identified to be caused by a spin-dependent ISC rate [@Manson06a; @Nizovtsev03; @Harrison04; @Nizovtsev05], little is known about the relative contributions of spin-flip transitions between triplet states ($k_{32}$, $k_{41}$), and ISC rates ($k_{35}$, $k_{45}, k_{51}$, $k_{52}$). We address this question by determining the polarization change due to a single excitation cycle.
For that purpose we first initialize the NV spin by a 2$\mu$s polarization pulse. After a waiting time of 1$\mu$s we excite the NV centre by a reference ps-pulse and measure the spin polarization by analyzing the relative contributions of the amplitudes in a bi-exponential fit to the fluorescence decay curve, as outlined in the previous section. This polarization corresponds to the steady-state value after CW excitation. To determine the change in polarization per excitation cycle we now apply a MW pulse to transfer the $m_s=0$ population into the $m_s=\pm1$ states and then drive individual excitation cycles by applying 10 consecutive ps-pulses separated by 2$\mu$s. For each excitation cycle, we again determine the spin polarization \[figure \[fig:fig4\](c)\]. All these bi-exponential fits use the same two time constants, obtained from a fit to the sum of all decay curves. From a power dependence measurement of the NV fluorescence rate we determine an excitation probablity $\alpha=0.95\pm0.05$.
![\[fig:fig4\] [**(a)**]{} We first polarize the spin by means of a 2$\mu$s laser pulse at $\lambda=532$nm. After a delay of 1$\mu$s we apply a sequence of 11 pulses of 62ps duration at $\lambda=532$nm. For NV J, we apply a MW pulse 1$\mu$s after the first ps-pulse to invert the spin state (for NV C we alternately run sequences with and without MW pulse). For each ps-pulse we measure the time-resolved emission. The experiment was performed at $T=300$K. [**(b)**]{} Fluorescence decay for consecutive ps-excitation pulses. The decay follows a bi-exponential function. A reference ps-pulse after a green 2 $\mu s$ spin polarization pulse yields the initial optically induced spin polarization. A subsequent MW pulse transfers polarization to $\left|2\right\rangle$. [**(c)**]{} Change in polarization between consecutive pulses yields spin-flip probabilities $p_{12}(\left|1\right\rangle \rightarrow \left|2\right\rangle)$ and $p_{21}(\left|2\right\rangle \rightarrow \left|1\right\rangle)$ per optical excitation cycle.](fig4.eps)
The effect of a single excitation cycle on the spin polarization can be described by two counter-acting probabilities: a spin-flip from $\left|1\right\rangle$ to $\left|2\right\rangle$ ($p_{12}$) and the opposite process ($p_{21}$). Here, we consider only optically induced effects, i.e. the time between excitation pulses $\Delta_t$ is assumed to be much shorter than the spin-lattice relaxation time (this assumption is substantiated by the constant polarization $P_{ES}$ for the 10 consecutive reference pulses in case of NV C \[figure \[fig:fig4\](c)\]). For state $\left|m\right\rangle$, population just before pulse $n$ is denoted by $P_{m,n}$, and just after the excitation pulse by $P'_{m,n}$. The steady state value $P_{m,n=\infty}$ is abbreviated by $P_m$. The pulse separation $\Delta_t$ is much larger than the singlet decay time, such that $P_{1,n}+P_{2,n}=1$ and therefore $P_{GS,n} = P_{1,n}$.
Experimentally, we obtain spin polarization in the excited state. This differs from the ground state polarization due to a fraction $\epsilon = k_{23} / k_{13} = k_{14}/k_{24}$ of spin non-conserving transitions. The populations for the $(n+1)^{th}$ excitation pulse are then given by:
$$\begin{aligned}
P_{1,n+1} &= \alpha p_{21}P_{2,n} + (1 - \alpha p_{12})P_{1,n} \\
P_{2,n+1} &= \alpha p_{12}P_{1,n} + (1 - \alpha p_{21})P_{2,n} \\
P'_{3,n+1} &= \alpha \left(\frac{\epsilon}{1+\epsilon}P_{2,n+1}+\frac{1}{1+\epsilon}P_{1,n+1}\right) \\
P'_{4,n+1} &= \alpha \left(\frac{\epsilon}{1+\epsilon}P_{1,n+1}+\frac{1}{1+\epsilon}P_{2,n+1}\right).\end{aligned}$$
The asymptotic value of the polarization is $P_{GS}=p_{21}/(p_{12}+p_{21})$, and the excited state $P_{ES,n}=P'_{3,n}/(P'_{3,n}+P'_{4,n})=\left[P_{1,n}\left(1-\epsilon\right)+\epsilon\right]/\left(1+\epsilon\right) \approx P_{1,n}$ closely follows the ground state spin polarization for small $\epsilon$. From [@Manson06a] we can obtain an upper bound of $\epsilon\leq0.04$. The change in polarization per pulse \[figure \[fig:fig4\](c)\] can be fitted by an exponential function $P_{ES}(n)=P_{ES}+a\exp(-n/c)$. From the steady-state polarization $P_{ES}$ and the polarization rate $c$ we can extract the spin-flip probabilities $p_{21}$ and $p_{12}$ per optical cycle. These probabilities depend only weakly on $\epsilon$. Results are summarized in table \[tab:tab1\]:
----------------- ------------------- -------------------
NV J NV C
$p_{12}$ 0.078 $\pm$ 0.002 0.079 $\pm$ 0.004
$p_{21}$ 0.315 $\pm$ 0.011 0.372 $\pm$ 0.017
$T_{1,3} (ns)$ 13.26 $\pm$ 0.03 13.1 $\pm$ 0.1
$T_{1,4} (ns)$ 6.89 $\pm$ 0.06 7.0 $\pm$ 0.2
$p_{35}$ 0.14 $\pm$ 0.02 0.17 $\pm$ 0.03
$p_{45}$ 0.55 $\pm$ 0.01 0.56 $\pm$ 0.02
$p_{51}/p_{52}$ 1.15 $\pm$ 0.05 1.6 $\pm$ 0.4
----------------- ------------------- -------------------
: \[tab:tab1\] Summary of parameters ($T=300$K), taking $\alpha=0.95\pm0.05$ and $\epsilon = 0.01\pm0.01$. $p_{12}$ and $p_{21}$ are the total spin-flip probabilities per optical cycle. $p_{35}$, $p_{45}$, $p_{51}$ and $p_{52}$ are the spin-dependent ISC probabilities for population in state $\left|3\right\rangle$, $\left|4\right\rangle$ and $\left|5\right\rangle$.
![\[fig:fig5\] Time-resolved excitation-power dependence of NV fluorescence excited by off-resonant 1$\mu$s pulses (experimental data and model). The first pulse displays the fluorescence of NV centres that are optically polarized into $m_s=0$. The second pulse is obtained after applying a MW $\pi$-pulse, representing the fluorescence of a NV centre polarized in $m_s=\pm1$. The model shows the population in states $\left|3\right\rangle$ and $\left|4\right\rangle$. The calculations are based on parameters from NV J, assuming $\epsilon=0.01$ and $\alpha=1$ (no fit is applied). Excitation rates are taken as integer multiples of 4 MHz.](fig5.eps)
If we assume only rates indicated in figure \[fig:fig1\](b), use experimentally obtained lifetimes $T_{1,3}$, $T_{1,4}$ and take $\epsilon=0.01\pm0.01$ and $\alpha=0.95\pm0.05$, we can obtain the spin-dependent ISC probabilities of $p_{45}=k_{45}/(k_{41}+k_{42}+k_{45})$ and $p_{35}=k_{35}/(k_{31}+k_{32}+k_{35})$, and for the reverse process $p_{51}/p_{52}=k_{51}/k_{52}$, as summarized in table \[tab:tab1\]. The low values of $p_{51}/p_{52}$ are in contrast to the current picture of the singlet relaxation process, where $p_{52} = 0$ was used [@Manson06a]. This gives another hint that an additional $^1E$ state needs to be considered for the relaxation process. The temperature dependence of $p_{51}/p_{52}$ could give further insight into this topic.
We note that this set of parameters can be used to model the spin- and power dependence of time-resolved NV centre emission using $\mu$s excitation pulses at $\lambda=532$nm within the framework of this five-level model, with reasonable agreement to experimental data \[figure \[fig:fig5\]\].
Summary
=======
We have experimentally determined the spin-dependent lifetime of the NV centre’s excited state, whose difference is dominated by a spin-dependent ISC rate. Knowledge of these lifetimes allows us to determine the degree of spin polarization. In a second experiment, we identified the total lifetime of the singlet states, and, by analyzing its temperature dependence, the energy splitting of the long lived singlet transition. The measured energy of $\approx16$meV indicates that at least three singlet states are involved in the optical cycle of the NV centre. Finally, we determined the spin-dependent ISC probabilities by analyzing the change of spin polarization induced by a single excitation cycle, without making assumptions on number and nature of the singlet states. The obtained rates are consistent with spin-dependent NV fluorescence dynamics based on a five-level model.
We would like to thank V. V. Dobrovitski and L. Childress for helpful discussions and Daniel Twitchen and Paul Balog of Element Six for provision of the diamond sample. This work is supported by the Dutch Organization for Fundamental Research on Matter (FOM), the Netherlands Organization for Scientific Research (NWO) and the EU SOLID programme. L. R. acknowledges support of the European Community under a Marie-Curie IEF fellowship.
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abstract: 'A permanent electric dipole moment (EDM) of a physical system requires time-reversal ($T$) and parity ($P$) violation. Experimental programs are currently pushing the limits on EDMs in atoms, nuclei, and the neutron to regimes of fundamental theoretical interest. Here we calculate the magnitude of the $PT$-violating EDM of $^{3}$He and the expected sensitivity of such a measurement to the underlying $PT$-violating interactions. Assuming that the coupling constants are of comparable magnitude for $\pi$-, $\rho$-, and $\omega$-exchanges, we find that the pion-exchange contribution dominates. Our results suggest that a measurement of the $^{3}$He EDM is complementary to the planned neutron and deuteron experiments, and could provide a powerful constraint for the theoretical models of the pion-nucleon $PT$-violating interaction.'
address:
- 'Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A.'
- 'Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, U.S.A.'
- 'Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A.'
- 'Lawrence Livermore National Laboratory, P.O. Box 808, L-414, CA 94551, U.S.A.'
author:
- 'I. Stetcu'
- 'C.-P. Liu'
- 'J. L. Friar'
- 'A. C. Hayes'
- 'P. Navrátil'
title: 'Nuclear Electric Dipole Moment of $^{3}\mathrm{He}$'
---
,
,
,
,
Introduction
============
A permanent electric dipole moment (EDM) of a physical system would indicate direct violation of time-reversal ($T$) and parity ($P$) and thus $CP$ violation through the $CPT$ invariance. Presently there are several experimental programs pushing the limits on EDMs in atoms, nuclei, and the neutron to regimes of fundamental interest. The Standard Model (SM) predicts values for the EDMs of these systems that are too small to be detected in the foreseeable future, and hence a measured nonzero EDM in any of these systems is an unambiguous signal for a new source of $CP$ violation and for physics beyond the SM. A new experimental scheme [@Khriplovich:1998zq; @khrip2; @Farley:2003wt; @Semertzidis:2003iq; @Orlov] for measuring EDMs of nuclei (stripped of their atomic electrons) in a magnetic storage ring suggests that the EDM of the deuteron could be measured to an accuracy of better that 10$^{-27}$ $e$ cm [@Semertzidis:2003iq]. Unlike searches for $CP$-violating moments of the nucleus through measurements of atomic EDMs, a measurement for a stripped nucleus would not suffer from a suppression of the signal through atomic Schiff screening [@schiff]. For this reason, the latter could represent about an order of magnitude better sensitivity to the underlying $CP$-violating interaction than the present limit on the neutron EDM, $d_{n}$ [@khrip2]. Measurements using stripped nuclei in a magnetic storage ring are best suited to nuclei with small magnetic anomaly, making $^{3}$He an ideal candidate for a high precision measurement. Here we examine the nuclear structure issues determining the EDM of $^{3}$He and calculate the matrix elements of the relevant operators using the no-core shell model [@Navratil:2000wwNavratil:2000gs] and Podolsky’s method for implementing second-order perturbation theory [@podolskyfriar]. An approximate and incomplete calculation for the $^3$He dipole exists in the literature [@Avishai:1986dw], but here we present much more reliable calculations based on an exact solution of the three-body problem using several potential models for the nucleon-nucleon (NN) interaction, complemented with three-body forces.
Sources of Nuclear $P$,$T$ violation
====================================
A nuclear EDM consists of contributions from the following sources: (i) the intrinsic EDMs of the proton and neutron, $d_{p}$ and $d_{n}$; (ii) the polarization effect caused by the $P$-,$T$-violating (${\slashed{P}\slashed{T}}$) nuclear interaction, $H_{{\slashed{P}\slashed{T}}}$; (iii) the two-body ${\slashed{P}\slashed{T}}$ meson-exchange charge operator appropriate for $H_{{\slashed{P}\slashed{T}}}$.
The contribution due to nucleon EDMs, $D^{(1)}$, which is purely one-body, can be easily evaluated by taking the matrix element $$\begin{aligned}
D^{(1)} & =\braket{\psi|\sum_{i=1}^{A}\,\frac{1}{2}\,\left[(d_{p}+d_{n})+(d_{p}-d_{n})\,\tau_{i}^{z}\right]\,\sigma_{i}^{z}|\psi}\,,\label{eq:1-body}\end{aligned}$$ where $\ket{\psi}$ is the nuclear state that has the maximal magnetic quantum number. In the particular case of interest in this paper, $\ket{\psi}=\ket{0}$ is the ground state of $^{3}$He obtained by the diagonalization of the $P$,$T$-conserving interaction.
In perturbation theory $H_{{\slashed{P}\slashed{T}}}$ induces a parity admixture to the nuclear state $$\begin{aligned}
{\widetilde{\ket{0}}} & =\sum_{n\neq0}\,\frac{1}{E_{0}-E_{n}}\,\ket{n}\braket{n|H_{{\slashed{P}\slashed{T}}}|0}\,,\label{eq:mixture}\end{aligned}$$ where $\ket{n}$ are eigenstates of energy $E_{n}$ and opposite parity from $\ket{0}$, which are calculated with the $P$-,$T$-conserving Hamiltonian. Hence, the polarization contribution $D^{(pol)}$ can be simply calculated as $$\begin{aligned}
D^{(pol)} & =\bra{0}\,\hat{D}_{z}{\widetilde{\ket{0}}}+\mbox{c.c.}\,,\label{eq:pol}\end{aligned}$$ where $$\begin{aligned}
\hat{D}_{z} & =\frac{e}{2}\,\sum_{i=1}^{A}\,(1+\tau_{i}^{z})\, z_{i}\end{aligned}$$ is the usual dipole operator projected in the $z$-direction.
The contribution due to exchange charge, $D^{(ex)}$, is typically at the order of $(v/c)^{2}$, and explicitly evaluated to be just a few percent of the polarization contribution for the deuteron case [@Liu:2004tq]; we therefore ignore it and approximate the full two-body contribution, $D^{(2)}$, solely by the polarization term $$\begin{aligned}
D^{(2)} & =D^{(pol)}+D^{(ex)}\cong D^{(pol)}\,.\label{eq:2-body}\end{aligned}$$
Our calculation of the EDM of $^{3}$He therefore requires knowledge of both the individual EDMs of the nucleons and the ${\slashed{P}\slashed{T}}$ nuclear force. These very different quantities can only be related if some understanding exists of both the origin of the symmetry violation and its expression in strong-interaction observables. Constructing an effective field theory (EFT) that incorporates the symmetry violation, as well as the dynamics underlying the usual strong-interaction physics in nucleons and nuclei, provides a suitable framework. Chiral Perturbation Theory ($\chi$PT) supplemented with a knowledge of the symmetry violation would be the appropriate EFT. To date only a single such calculation exists [@Hockings:2005cn], and it was applied to the one-nucleon sector, although further effort is underway [@-PVTV-EFT; @Bira:PC]. The symmetry violation in that calculation was taken from the QCD $\bar{\theta}$ term, which leads to an isoscalar ${\slashed{P}\slashed{T}}$ pion-nucleon interaction in leading order, unlike the most general case that includes an isovector and an isotensor term, as well [@Barton:1969gi]. The non-analytic parts of the pion-loop diagrams [@Liu:2004tq; @Hockings:2005cn; @Crewther:1979pi; @He:1989xj] that generate nucleon EDMs then provide an appropriate estimate of the EDMs of individual nucleons. These contributions are expected to dominate in the chiral limit[@Crewther:1979pi].
In the absence of a $\chi$PT calculation of $H_{{\slashed{P}\slashed{T}}}$ we revert to a conventional formulation in terms of a one-meson-exchange model. Including $\pi$-, $\rho$-, and $\omega$-meson exchanges, [^1] the interaction is given by (see Refs. [@Liu:2004tq; @Haxton:1983dq; @Herczeg:1987gp; @Gudkov:1993yc; @Towner:1994qe]): $$\begin{aligned}
& H_{{\slashed{P}\slashed{T}}}(\bm r) =\frac{1}{2\,{m_{{\scriptscriptstyle{N}}}}}\,\bigg\{\bm\sigma_{-}\cdot\bm\nabla\left(-\bar{G}_{\omega}^{0}\, y_{\omega}(r)\right)\nonumber \\
& +\bm\tau_{1}\cdot\bm\tau_{2}\,\bm\sigma_{-}\cdot\bm\nabla\left(\bar{G}_{\pi}^{0}\, y_{\pi}(r)-\bar{G}_{\rho}^{0}\, y_{\rho}(r)\right)\nonumber \\
& +\frac{\tau_{+}^{z}}{2}\,\bm\sigma_{-}\cdot\bm\nabla\left(\bar{G}_{\pi}^{1}\, y_{\pi}(r)-\bar{G}_{\rho}^{1}\, y_{\rho}(r)-\bar{G}_{\omega}^{1}\, y_{\omega}(r)\right)\nonumber \\
& +\frac{\tau_{-}^{z}}{2}\,\bm\sigma_{+}\cdot\bm\nabla\left(\bar{G}_{\pi}^{1}\, y_{\pi}(r)+\bar{G}_{\rho}^{1}\, y_{\rho}(r)-\bar{G}_{\omega}^{1}\, y_{\omega}(r)\right)\nonumber \\
& +(3\,\tau_{1}^{z}\,\tau_{2}^{z}-\bm\tau_{1}\cdot\bm\tau_{2})\,\bm\sigma_{-}\cdot\bm\nabla\left(\bar{G}_{\pi}^{2}\, y_{\pi}(r)-\bar{G}_{\rho}^{2}\, y_{\rho}(r)\right)\bigg\}\,,\label{eq:HPVTV-ex}\end{aligned}$$ where ${m_{{\scriptscriptstyle{N}}}}$ is the nucleon mass, $\bar{G}_{x}^{T}$ is defined as the product of a ${\slashed{P}\slashed{T}}$ x-meson–nucleon coupling $\bar{g}_{x}^{T}$ (with $T$ referring to the isospin) and its associated strong one, $g_{x{\scriptscriptstyle{NN}}}$ (e.g., $\bar{G}_{\pi}^{0}=g_{\pi{\scriptscriptstyle{NN}}}\,\bar{g}_{\pi}^{0}$,where the interaction Lagrangian corresponding to these coupling constants is ${\cal{L}} = \bar{N} [i g_{\pi{\scriptscriptstyle{NN}}} \gamma_5+ \bar{g}_{\pi}^{0}] {\vec{\tau}}\cdot {\vec{\pi}}N$), $y_{x}(r)=e^{-m_{x}\, r}/(4\,\pi\, r)$ is the Yukawa function with a range determined by the mass of the exchanged $x$-meson, $\vec r=\vec r_1 -\vec r_2$, $\vec \sigma_\pm=\vec \sigma_1\pm \vec \sigma_2$, and similarly for $\vec \tau_\pm$. Unless the symmetry associated with the specific way that $P,T$ violation is generated suppresses some of the couplings, one expects (by naturalness) that these ${\slashed{P}\slashed{T}}$ meson–nucleon couplings are of similar magnitude, and this is roughly confirmed by a QCD sum rule calculation [@Pospelov:2001ys]. [^2] We note, however, that in the (purely isoscalar) $\bar{\theta}$-term model of Ref. [@Hockings:2005cn] the coupling constants $\bar{G}_{\pi}^{1}$ and $\bar{G}_{\pi}^{2}$ vanish, and the coupling constants for the short-range operators are very small compared to the pion one [@Bira:PC].
Because $H_{{\slashed{P}\slashed{T}}}$ violates parity and $^{3}$He is (largely) an $S$-wave nucleus, the matrix elements that define the EDM (see below) mostly involve $S$- to $P$-wave transitions. This has the combined effect of suppressing the short-range contributions and enhancing the long-range (i.e., pion) contribution, irrespective of the detailed nature of the force. Combined with the consideration that the short-range parameters ($\bar{G}_{\rho,\omega}$) are not much larger than the pion ones, one can roughly expect the dominance of pion exchange.
$^{3}\mathrm{He}$ in the *ab initio* No-Core Shell Model
========================================================
We solve the three-body problem in an *ab initio* no-core shell model (NCSM) framework [@Navratil:2000wwNavratil:2000gs]. The ground-state wave function is obtained by a direct diagonalization of an effective Hamiltonian in a truncated harmonic oscillator (HO) basis constructed in relative coordinates, as described in Ref. [@Navratil:1999pw]. High-precision NN interactions, such as the local Argonne $v_{18}$ [@Wiringa:1995Pieper:2001AR] and the non-local charge-dependent (CD) Bonn potential [@Machleidt:1995km] interactions, are used to derive an effective interaction in each model space via a unitary transformation [@Okubo:1954DaProvidencia:1964Suzuki:1980] in a two-body cluster approximation. The Coulomb interaction between protons is also taken into account.
In addition to the phenomenological NN interaction models cited above, we consider two- and three-body interactions derived from EFT. In a recent work [@Navratil:2007] using the NCSM, the presently available NN potential at N$^{3}$LO [@N3LO] and the three-nucleon (NNN) interaction at N$^{2}$LO [@vanKolck:1994; @Epelbaum:2002] have been applied to the calculation of various properties of $s$- and $p$-shell nuclei. In that study a preferred choice of the two NNN low-energy constants, $c_{D}$ and $c_{E}$, was found (and the fundamental importance of the chiral EFT NNN interaction was demonstrated) by reproducing the structure of mid-$p$-shell nuclei. (Note that these interactions are fitted only for a momentum cutoff of 500 MeV, and therefore we are not able at this time to demonstrate a running of the observables with the cutoff.) This Hamiltonian was then used to calculate microscopically the photo-absorption cross section of $^{4}$He [@Quaglioni:2007eg], while the full technical details on the local chiral EFT NNN interaction that was used were given in Ref. [@Navratil_FBS]. We use an identical Hamiltonian in the present work, and we compare its predictions against the phenomenological potentials.
In the NCSM the basis states are constructed using HO wave functions. Hence, all the calculations involve two parameters: the HO frequency $\Omega$ and $N_{max}$, the number of oscillator quanta included in the calculation. At large enough $N_{max}$, the results become independent of the frequency, although the rate of convergence depends on $\Omega$. Thus, for short-range operators, one can expect a faster convergence for larger values of $\Omega$, as the characteristic length of the HO is $b=1/\sqrt{{m_{{\scriptscriptstyle{N}}}}\,\Omega}$. The convergence also depends upon the $P$-,$T$-conserving interaction, $H_{0}$, used to solve the three-body problem. Thus the results obtained with Argonne $v_{18}$ show the slowest convergence, because the NN interaction has a more strongly repulsive core than the interactions obtained from EFT, which have faster convergence rates.
The nucleonic contribution $D^{(1)}$ in Eq. (\[eq:1-body\]) involves only $H_{0}$, and is easily calculated once the three-body problem is solved. We therefore concentrate on the part involving $H_{{\slashed{P}\slashed{T}}}$.
Equation (\[eq:mixture\]) suggests that in order to calculate the dipole moment one needs to obtain high-accuracy excited states of $^{3}$He in the continuum, an extremely difficult task in a NCSM framework, where the basis states are constructed using bound-state wave functions. The most straightforward technique for evaluating Eq. (\[eq:mixture\]) is to use Podolsky’s method [@podolskyfriar], in which ${\widetilde{\ket{0}}}$ is obtained as the solution of the Schrödinger equation with an inhomogeneous term $$(E_{0}-H_{0})\,{\widetilde{\ket{0}}}=H_{{\slashed{P}\slashed{T}}}\,\ket{0}\,.
\label{nonhomeq}$$ The exceptionally nice feature of this method is that continuum states do not have to be *explicitly* calculated (they are, of course, implicitly included). In this sense the technique is a relatively simple extension of bound-state methods, which have been well studied and are robust. Moreover, in this approach the convergence of the EDM reduces to a large degree to the issue of the convergence of the ground state.
We express the solution of Eq. (\[nonhomeq\]) as a superposition of a handful of vectors generated using the Lanczos algorithm [@LanczAlg1950LanczAlg]. Indeed, one can show that if we start with the inhomogeneous part of Eq. (\[nonhomeq\]) as the starting Lanczos vector $|v_{1}\rangle=H_{{\slashed{P}}{\slashed{T}}}\,|0\rangle$, the solution becomes [@LanczGreen]
$${\widetilde{\ket{0}}}\approx\sum_{k}\, g_{k}(E_{0})\,|v_{k}\rangle\,,$$
where the summation over the index $k$ runs over a finite and usually small number of iterations. The coefficients $g_{k}(E)$ are easily obtained using finite continued fractions [@LanczCoef].
We alter this approach in practice for efficiency reasons. Because Eq. (\[eq:pol\]) is symmetrical in $\hat{D}_{z}$ and $H_{{\slashed{P}}{\slashed{T}}}$, we are free to choose $|v_{1}\rangle=\hat{D}_{z}\,|0\rangle$ as the starting vector. This allows us to isolate the two isospin contributions for $H_{{\slashed{P}}{\slashed{T}}}$ in each run. Once we compute a second vector, $|v\rangle=H_{{\slashed{P}}{\slashed{T}}}^{\dagger}\,|0\rangle$, the polarization contribution to the EDM is finally evaluated as $$D^{(pol)}=2\,\sum_{k}\, g_{k}(E_{0})\,\langle v|v_{k}\rangle\,.\label{dipoleLanczos}$$ (We have verified that the altered approach gives the same results as the original one.) As a particular test case we have considered the electric polarizability
$$\alpha_{E}=\frac{1}{2\pi^{2}}\,\int\, d\omega\,\frac{\sigma(\omega)}{\omega^{2}}=2\,\alpha\,\sum_{n}\,\frac{\langle0|\hat{D}_{z}|n\rangle\langle n|\hat{D}_{z}|0\rangle}{E_{n}-E_{0}}$$
(where $\alpha$ is the fine structure constant), which reduces Eq. (\[dipoleLanczos\]) to $\alpha_{E}=-2\,\alpha\, g_{1}(E_{0})\,\langle v_{1}|v_{1}\rangle$. We estimate that the electric polarizability of the $^{3}$He nucleus is 0.183 fm$^{3}$ for the Argonne $v_{18}$ potential, compared with 0.159 fm$^{3}$ reported in Ref. [@He3alphaE] for the same interaction. The $15\%$ discrepancy is most likely the result of a difference in the theoretical approaches, as the result reported in Ref. [@He3alphaE] involves a matching of the ground-state energy to experiment (i.e., 7.72 MeV), although the calculation gives 6.88 MeV binding [@gsHe3energyAV18] in the absence of three-body forces (our converged binding energy for Argonne $v_{18}$ is 6.92 MeV). Since the electric polarizability scales roughly with the inverse of the square of the binding energy, the discrepancy between the two results is reasonable. Moreover, we have made the additional check of the Levinger-Bethe sum rule [@BetheSR], which in the case of tritium relates the total dipole strength to the charge radius, and we found it to be satisfied in all model spaces to a precision better than $10^{-5}$. Finally, the $^{3}$He polarizability calculated using the two- and three-body chiral interactions is 0.148 fm$^{3}$, compared with 0.145 fm$^{3}$ with Argonne $v_{18}$ and Urbana IX [@He3alphaE] two- and three-body forces. In both cases excellent agreement with the experimental binding energy is achieved.
In a consistent approach the same transformation used to obtain the effective interaction should be used to construct the effective operators in truncated spaces. While this has been done in the past for general one- and two-body operators [@Stetcu:2004wh], such an approach is very cumbersome for the present investigation because both the dipole transition operator and $H_{{\slashed{P}\slashed{T}}}$ change the parity of the states. We have therefore chosen not to renormalize the operators involved, except for the $P$-,$T$-conserving Hamiltonian. This problem is largely overcome by the fact that long-range operators (like the dipole) have been found to be insensitive to the renormalization in the two-body cluster approximation [@Stetcu:2004wh], which is the level of truncation for the effective interaction. We also point out that since $H_{{\slashed{P}\slashed{T}}}$ has short range, one can expect that the renormalization of this operator would improve the convergence pattern, especially for small HO frequencies. As with all operators, the effect of the renormalization decreases as the size of the model space increases, so that in large model spaces (like the ones in the present calculation) this effect can be safely neglected and good convergence of observables is achieved.
Results and Discussions
=======================
We start the discussion of our results with the one-body contribution to the EDM of $^{3}$He. In Table \[table:oneb\], we summarize the isoscalar and isovector contribution to $D^{(1)}$, which are decomposed into contributions proportional to their respective coupling constants ($d_{p}+d_{n}$ for isoscalar, and $d_{p}-d_{n}$ for isovector). All interactions give similar results, with only the Argonne $v_{18}$ result deviating more significantly from the others, albeit by less than 6%. We note that the coefficients in the upper and lower rows in Table \[table:oneb\] would be either 1/2 or -1/2 if the nuclear forces between each pair of nucleons were taken to be equal (viz., the $SU(4)$ limit, which implies that the neutron carries all of the nuclear spin).
\[table:oneb\]
--------------- ---------- ---------- ---------- ----------
CD Bonn $v_{18}$
NN NN+NNN
$d_{p}+d_{n}$ $0.430$ $0.415$ $0.437$ $0.433$
$d_{p}-d_{n}$ $-0.467$ $-0.462$ $-0.468$ $-0.468$
--------------- ---------- ---------- ---------- ----------
: The nucleonic contribution (in $e$ fm) to the $^{3}$He EDM for different potential models. We decompose our results into contributions proportional to the nucleon isoscalar ($d_{p}+d_{n}$) and isovector ($d_{p}-d_{n}$) EDMs.
In Fig. \[fig:Dpol\] we present for four HO frequencies the running with $N_{max}$ of the EDM induced by the pion-exchange part of $H_{{\slashed{P}\slashed{T}}}$. Two- and three-body EFT interactions have been used for this calculation, in order to obtain an accurate description of the ground-state of the three-body system. For the nuclear EDM we mix two types of operators: $H_{{\slashed{P}}{\slashed{T}}}$, which is short range, and $\hat{D}_{z}$, which is long range. The convergence pattern is therefore not as straightforward as presented earlier in the discussion of convergence properties of general operators. The short-range part dominates the convergence pattern, and we thus observe faster convergence for larger frequencies (smaller HO parameter length). This behavior is opposite to the convergence in the case of the electric polarizability, where we found faster convergence for smaller frequencies as expected for a long range operator. Nevertheless, just as in Fig. \[fig:Dpol\], the results become independent of the frequency at large $N_{max}$. Note in the insert the convergence behavior of the ground-state energy, which converges to the experimental value already at $N_{max}\approx22$ for most frequencies presented in the figure.
![Isoscalar, isovector, and isotensor pion-exchange contributions to the $H_{{\slashed{P}\slashed{T}}}$-induced EDM in $^{3}$He. We show four different frequencies in each case: $\Omega=10$ MeV (circles), $\Omega=20$ MeV (squares), $\Omega=30$ MeV (diamonds), and $\Omega=40$ MeV (triangles). In the insert, we present the convergence of the ground-state energy, which in the limit of large $N_{max}$ approaches the experimental value (dashed line). Both NN and NNN EFT interactions have been used for diagonalization.[]{data-label="fig:Dpol"}](pi_edm3He_2b3bEFT)
Similar convergence patterns can be observed for the other meson exchanges as well as other potential models. In Table \[table:HPTedm\] we summarize these results.
\[table:HPTedm\]
[@ l\*[4]{}[r]{}|\*[4]{}[r]{}|\*[4]{}[r]{}]{} & & &\
& CD Bonn & $v_{18}$ & & CD Bonn &$v_{18}$ & & CD Bonn & $v_{18}$ &\
& & & NN & NN+NNN & & & NN & NN+NNN & & & NN &NN+NNN\
$\bar G_{x}^0$& $0.013$ & $0.012$ & $0.015$ & $0.015$ & $-0.0012$ & $-0.0006$ & $-0.0012$ & $-0.0013$& $0.0008$ & $0.0005$ & $0.0009$ & $0.0007$\
$\bar G_{x}^1$ & $0.022$ & $0.022$ & $0.023$ & $0.023$ & $0.0011$ & $0.0009$ & $0.0013$ & $0.0012$ & $-0.0011$ & $-0.0011$ & $-0.0017$ & $-0.0018$\
$\bar G_{x}^2$ & $0.035$ & $0.034$ & $0.037$ & $0.036$ & $-0.0019$ & $-0.0015$ & $-0.0028$ & $-0.0027$ &- &- & - & -\
For pion exchange all potential models give basically the same result, as the long-range part ($r\gtrsim1/m_{\pi}$) of the $^{3}\mbox{He}$ wave function shows negligible model dependence. It is interesting to note the effect of the three-body force by comparing the results with and without NNN interactions. When only the NN EFT interaction is used, the binding energy is underestimated by about 500 keV. Therefore, since the ground-state energy is in the denominator of Eq. (\[eq:mixture\]), one could naively expect that introducing the three-body forces (which increase the binding) decreases $D^{(pol)}$. Instead we obtain nearly the same results for both isoscalar and isovector contributions. This implies that the NNN interaction reshuffles the strength to compensate for the change in binding energy, most likely at low energies. This is not a surprise, because it was already found previously that the main effect of the three-body forces for the dipole response is an attenuation of the peak region at low energies both in the three- [@He3alphaE] and four-body [@Quaglioni:2007eg] systems.
In the isoscalar pion-exchange channel our result ($0.015$) is about $50\%$ larger than an existing work [@Avishai:1986dw], which yielded $0.010$. Since the Reid soft core NN potential used in Ref. [@Avishai:1986dw] has a much more repulsive core than even Argonne $v_{18}$, that isoscalar contribution to the EDM is in line with our findings. Moreover, the approximate solution of the three-body problem (as opposed to the current work, in which the calculation is exact) can induce uncontrolled uncertainties. Finally, their calculation of the isovector term did not exhaust all the possible spin-isospin combinations, while the isotensor contribution was not computed.
For $\rho$- and $\omega$-exchanges one immediately sees that their corresponding coefficients are at most $10\%$ of the pion-exchange ones, because only the short-range wave function ($r\lesssim1/m_{\rho,\omega}$) contributes substantially. Sensitivity of the short-range ${\slashed{P}\slashed{T}}$ potentials to the short-range model dependence of the wave functions produces matrix-element variations as large as $50\%$ for some channels. While a detailed explanation for the model dependence is too intricate to be disentangled, one can roughly see the general trend that the calculation using Argonne $v_{18}$ gives consistently smaller results than ones using CD Bonn and chiral EFT, as Argonne $v_{18}$ has a harder core than the other two. The behavior of short-range ${\slashed{P}\slashed{T}}$ nuclear potentials in $\chi$PT strongly suggests that the short-range coupling constants are no larger than the pion ones, and may be significantly smaller. If this holds, pion-exchange will produce the dominant contribution to nuclear EDMs, with the heavy-meson exchanges (the short-range interaction) giving roughly a $10\%$ correction (or less) to the pion-exchange contribution. This suppression due to $P$-wave intermediate nuclear states was discussed earlier and is in accord with calculations in heavier systems [@Towner:1994qe].
Assuming the dominance of pion exchange, $D^{(2)}$ has an almost model-independent expression $$\begin{aligned}
D^{(2)} & \approx(0.015\,\bar{G}_{\pi}^{0}+0.023\,\bar{G}_{\pi}^{1}+0.036\,\bar{G}_{\pi}^{2})\, e\,\mbox{fm}.\end{aligned}$$ The single-nucleon EDMs can be estimated using the non-analytic term that results from calculating the one-pion loop diagram, which dominates in the chiral limit (see, for example, Refs. [@Liu:2004tq; @Hockings:2005cn; @Barton:1969gi; @Crewther:1979pi; @He:1989xj])
$$\begin{aligned}
d_{\stackrel{{\scriptstyle p}}{n}} & \approx\mp\frac{e}{4\,\pi^{2}\,{m_{{\scriptscriptstyle{N}}}}}\,(\bar{G}_{\pi}^{0}-\bar{G}_{\pi}^{2})\,\ln\left(\frac{{m_{{\scriptscriptstyle{N}}}}}{m_{\pi}}\right)\,,\label{eq:nucleon-EDM}\end{aligned}$$
where $e$ is the proton charge. Folding this result into $D^{(1)}$ and using the physical nucleon and pion masses ($\ln({m_{{\scriptscriptstyle{N}}}}/m_{\pi})\approx1.90$) we get $$\begin{aligned}
D^{(1)} & \approx0.009\,(\bar{G}_{\pi}^{0}-\bar{G}_{\pi}^{2})\, e\,\mbox{fm}\,.\end{aligned}$$ The total EDM of $^{3}\mbox{He}$ is therefore estimated to be $$\begin{aligned}
D & =D^{(1)}+D^{(2)}\nonumber \\
& =(0.024\,\bar{G}_{\pi}^{0}+0.023\,\bar{G}_{\pi}^{1}+0.027\bar{G}_{\pi}^{2})\, e\,\mbox{fm}\,.\end{aligned}$$
Calculating the EDMs of the neutron and deuteron [@Liu:2004tq] using Eq. (\[eq:nucleon-EDM\]) and assuming pion-exchange dominance, one can see from Table \[tab:comparison\] that an EDM measurement in $^{3}\mbox{He}$ is complementary to the former two. Assuming that similar sensitivities can be reached in these three measurements, the ${\slashed{P}\slashed{T}}$ pion-nucleon coupling constants could be well-constrained if the assumption of pion-exchange dominance holds.
[>p[0.2]{}>p[0.2]{}>p[0.2]{}>p[0.2]{}]{} & $\bar{G}_{\pi}^{0}$ & $\bar{G}_{\pi}^{1}$ & $\bar{G}_{\pi}^{2}$[\
]{} neutron & $0.010$ & $0.000$ & $-0.010$[\
]{}deuteron & $0.000$ & $0.015$ & $0.000$[\
]{}$^{3}\mbox{He}$ & $0.024$ & $0.023$ & $0.027$[\
]{}
Summary
=======
In this paper, we have calculated the nuclear EDM of $^{3}$He, which arises from the intrinsic EDMs of nucleons and the $P$-,$T$-violating nucleon-nucleon interaction. Several potential models for the $P$-,$T$-conserving nuclear interaction (including the latest-generation NN and NNN chiral EFT forces) have been used in order to obtain the solution to the nuclear three-body problem. The results obtained with these potential models agree within 25% in the ${\slashed{P}\slashed{T}}$ pion-exchange sector. Though larger spreads in ${\slashed{P}\slashed{T}}$ $\rho$- and $\omega$-exchanges are found (as the results sensitively depend on the wave functions at short range), we expect them to be non-essential as pion-exchange completely dominates the observable (unless the ${\slashed{P}\slashed{T}}$ parameters associated with heavy-meson exchanges are significantly larger than the ones for pion exchange, which is not expected). We further demonstrate that a measurement of the $^{3}\mbox{He}$ EDM would be complementary to those of the neutron and deuteron, and in combination they can be used to put stringent constraints on the three $P$-,$T$-violating pion–nucleon coupling constants. We therefore strongly encourage experimentalists to consider such a $^{3}$He measurement in a storage ring, in addition to the existing deuteron proposal [@Semertzidis:2003iq].
I.S. thanks W. Leidemann, S. Quaglioni and S. Bacca for useful discussions. J.L.F. greatly appreciates insights provided by P. Herczeg and U. van Kolck. The work of I.S., C.P.L., J.L.F. and A.C.H. was performed under the auspices of the U. S. DOE. C.P.L. acknowledges partial support from the Wisconsin Alumni Research Foundation and U.S. DOE under contract Nos. DE-FG02-08ER41531 (UWisc) and DE-AC05-00OR22725 (ORNL). Prepared by LLNL under contract No. DE-AC52-07NA27344. Support from the U.S. DOE/SC/NP (Work Proposal No. SCW0498), and from the U. S. Department of Energy Grant DE-FG02-87ER40371 is acknowledged.
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[^1]: Other mechanisms not included here are $\eta$-exchange and two-$\pi$-exchange, etc.
[^2]: In Ref. [@Pospelov:2001ys], $\bar{g}_{\rho,\omega}$ are defined differently; for conversion, see Ref. [@Liu:2004tq]
|
---
author:
- 'Eileen Giesel,'
- 'Robert Reischke,'
- 'Björn Malte Schäfer,'
- and Dominic Chia
bibliography:
- 'references.bib'
- 'My\_Library.bib'
title: Information geometry in cosmology
---
Introduction
============
Any predictive physical model contains a set of finitely many parameters which need to be determined by experiment. These parameters can be both physical parameters of interest, or nuisance parameters taking care of e.g. instrumental systematics which will eventually be marginalised. In cosmology, these physical parameters can be, for example, the matter density $\Omega_\mathrm{m}$, the Hubble-Lema[î]{}tre constant $h$, or the dark energy equation of state $w$. Depending on the construction of the model, the measurement method and the data volume, a cosmological model can be constrained to a certain domain in parameter space which might or might not be well approximated to be a linear space.
Both in a Bayesian as well as in a Frequentist framework the central object is the joint probability density function $p(\boldsymbol{x},\boldsymbol{\theta})$ of the parameters, $\boldsymbol{\theta}$, and the data $\boldsymbol{x}$. The true model is then suspected to be the one that could have brought about the data with the highest likelihood, i.e. the choice of the parameters that is able to maximise $p(\boldsymbol{x},\boldsymbol{\theta})$. Bayesian inference will interpret $p(\boldsymbol{\theta}|\boldsymbol{x})$ rather in terms of confidence in a particular set of parameters given a single realisation of the data. Cosmological inference usually follows this route since, by nature, the data is only available in a single realisation, casting a Frequentist interpretation questionable.
Values of model parameters are given with confidence intervals which describe the degree of belief that the best fit value is indeed the true value. These intervals are usually determined with two main techniques: For forecasting, where the best fit values of the parameters are assumed to be known, mainly the Fisher-matrix analysis [e.g. @tegmark_karhunen-loeve_1997] or higher order schemes [@Sellentinetal; @sellentin_fast_2015] are employed. Instead, when facing real data or for more accurate confidence contours without an assumption of (near) Gaussianity or single modality of the likelihood, one relies on Monte-Carlo Markov-chain (MCMC) methods [@skilling_nested_2006; @akeret_cosmohammer:_2013; @feroz_multinest:_2011; @audren_monte_2013; @goodman_ensemble_2010]. Approximating a likelihood as being Gaussian has tremendous computational advantages compared to MCMC-techniques, and requires only a small number of calls of the likelihood for finite differencing, at the expense of not representing the likelihood faithfully enough.
The origin of non-Gaussian features of a likelihood are non-linearities in the dependence of the model on its parameters or a non-Gaussian error process for the individual data points, giving rise as well to constraints and degeneracies that vary as a function of the fiducial model [@schafer_describing_2016]. If the data is well-constraining, these non-linear relationships can be linearised, which render the $\chi^2$-functional parabolic and cast the likelihood to be Gaussian. When a model is extended to include new parameters, the corresponding likelihood covers a larger fraction of the parameter space and a linearisation of the model might not be applicable, in which case the likelihood veers again away from the Gaussian shape. This effect can be counteracted by accumulating more data, by collecting data with smaller errors, or by combining different measurement methods with the potential of breaking degeneracies.
Whether it is possible to re-parameterise a model such that an otherwise non-Gaussian likelihood assumes a Gaussian shape is an interesting question and has a clear answer in one dimension, where this can always be achieved. For multivariate distributions, this can only be done approximatively [@box_analysis_1964; @schuhmann_gaussianization_2016], making the likelihood accessible to arguments reserved to Gaussian distributions, for instance their unbiasedness and their fulfilment of the Cram[é]{}r-Rao-bound, but also the perfect decoupling of all degeneracies by transforming the parameter space into the eigensystem of the Gaussian’s covariance matrix. Whether a perfect Gaussianisation can in principle be achieved is an interesting question on its own and it can be answered using the tools of information geometry. In information geometry, the Fisher-matrix (or, the inverse parameter covariance) takes on the role of the metric on a statistical manifold [@amari_information_2016] and can be derived from a statistical divergence defined in an axiomatic way. In this setting the parameters of the statistical model are merely a choice of coordinates [and one should be able to change the parameterisation by chart transition maps in an invertible and differentiable way]{}. [In this context,]{} Gaussian likelihoods correspond to flat manifolds [(although there might be unfortunate parameter choices where they appear to be non-Gaussian)]{} while [actual]{} non-Gaussian ones induce a non-trivial geometry. [Hence, using the concepts of information geometry leads to an understanding of non-Gaussianities as inherent geometrical properties of a non-flat manifold which is defined by a statistical model, in particular the presence of curvature. In the case of non-flat manifolds it is in general not possible to find a coordinate change to obtain a constant Fisher information corresponding to a Gaussian likelihood in the parameters. Additionally, by employing additional methods of differential geometry it could still be possible to find for instance isometries of the Fisher information and consequently integral curves in parameter space along which the metric is constant, leading to a geometric interpretation of the degeneracies of a model and indicating parameter choices where Gaussianity of the likelihood is established.]{}
In this work we will apply the concepts of information geometry to typical statistical models encountered in cosmology. As a working example we will choose the statistical manifold defined by supernova observations [@riess_observational_1998; @perlmutter_discovery_1998; @perlmutter_measurements_1999; @riess_type_2004; @riess_new_2007] and investigate its differential geometric properties in the context of a flat $w$CDM-cosmology, and restrict the inference on the matter density $\Omega_m$ and a constant dark energy equation of state $w$. Furthermore, we will discuss Gaussianisation transformations, as well as the relation between the DALI-approximation [@sellentin_fast_2015] and the Gram-Charlier series with geometric properties of statistical manifolds in the limit of weak non-Gaussianities, providing relationships between these fundamental descriptions of non-Gaussian distributions. Of course, scientists do not only intend to measure the model parameters of a single model as accurate as possible, but also look out for new phenomena beyond the accepted model. In that, we will take the point of view that a model class is already selected, for instance with concepts of Bayesian evidence or through a strong physical argument, and we are asking how well different parameter choices within this model class are compatible with data.
The paper is structured as follows: We will recapitulate the basic concepts of information geometry in \[sec:information\_geometry\]. will be devoted to the geometrical interpretation of Gaussianity and its transformation properties. In \[sec:example\] we will apply these ideas to the example of cosmological constraints from type-Ia supernovae. Finally, we summarise our results in \[sec:summary\].
Concepts of information geometry {#sec:information_geometry}
================================
In this section we will briefly summarise the basic concepts of information geometry [@amari_information_2016]. We will introduce the statistical manifold, $M$, under consideration, the divergence which itself induces a metric $g$ on $M$. Additional structure is then be provided by choosing a particular linear connection $\nabla$ such that concepts like curvature tensors and curvature scalars can be introduced on the triple $(M,g,\nabla)$. [The manifold M will turn out to be of Riemannian type equipped with a positive definite metric $g$ in contrast to relativity where a pseudo-Riemannian manifold with a metric of Lorentzian signature is considered: This is due to the fact that in relativity the concept of hyperbolic spacetimes is central along with a geometric notion of causality, both of which is irrelevant to statistical manifolds.]{}
Statistical manifold
--------------------
At the heart of inference is a statistical model which can be described as a set $$\label{eq:set_statistical_model}
M = \{p(\boldsymbol{x},\boldsymbol{\theta})\},$$ where $\boldsymbol{\theta}$ are the model parameters, $\boldsymbol{x}$ the data and $p$ is a probability or probability density, in the case of a continuum of parameters. This set has the structure of a $d$-dimensional topological manifold [@amari_information_2016], that is a paracompact Hausdorff topological ($\tau$, a suitable topology) space $(M,\tau)$ such that $\forall p\in M $ there exists an open neighbourhood $U$ with a homeomorphism $U\to U^\prime\subseteq \mathbb{R}^d$. Here, the dimensionality is given by the dimensions of the parameter space. In physics the homeomorphism is usually referred to as a coordinate system. As the statistical model is described by a set of parameters $\boldsymbol{\theta}$, they can be seen as the coordinate system parameterising the manifold. We will assume here that the manifold is smooth, i.e. that all chart transition maps in its atlas are $C^\infty$.
Divergences and invariant metric
--------------------------------
Distances between points on $M$ are described by divergences, which quantify the dissimilarity of the distributions associated with every point of the likelihood. For $p,q\in M$ we write $\boldsymbol{\theta}_i$ as the corresponding coordinate. A divergence is then given by $$\label{eq:divergence_general}
D[p:q] = D[\boldsymbol{\theta}_p,\boldsymbol{\theta}_q],$$ where the following criteria must hold:
1. $D[p:q] \geq 0$.
2. $D[p:q] = 0$ if and only if $p = q$.
3. $D[p:q]$ can be Taylor expanded in the local coordinate system if $p$ and $q$ are sufficiently close to each other: $$D[\boldsymbol{\theta}_p,\boldsymbol{\theta}_p+{\mathrm{d}}\boldsymbol{\theta}]=\frac{1}{2}{g}_{ij}(\boldsymbol{\theta}_p){\mathrm{d}}\theta^i{\mathrm{d}}\theta^j,$$ where the matrix $\boldsymbol{g}$ is positive definite.
Clearly, the squared infinitesimal distance can thus be written as $$\label{eq:squared_infi_distance}
{\mathrm{d}}s^2 =2 D[\boldsymbol{\theta}_p,\boldsymbol{\theta}_p+{\mathrm{d}}\boldsymbol{\theta}],$$ providing a Riemannian structure on $M$. The natural divergence between two points $p$ and $q$ can be derived by demanding $$\label{eq:invariance}
D[p:q] \geq D^\prime [p:q],$$ where $D^\prime$ denotes the divergence associated to another random variable $\boldsymbol{y} = \phi(\boldsymbol{x})$. The statistic $\boldsymbol{y}$ is called sufficient if the equality in (\[eq:invariance\]) holds. Clearly any one-to-one mapping provides a sufficient statistic.
The divergence between two points $p$ and $q$ measures the mutual information between the two distributions at those two points. An invariant information measure can be introduced [@1963Morimoto] called the $f$-divergence: $$\label{eq:f_divergence}
D_f[\boldsymbol{\theta}_p,\boldsymbol{\theta}_q] =
\int {\mathrm{d}}\boldsymbol{x}\:p(\boldsymbol{x}) \:f\left(\frac{q(\boldsymbol{x})}{p(\boldsymbol{x})}\right),$$ with a differentiable and convex function satisfying $f(1)=0$. This divergence can be shown to be invariant. A typical example for an $f$-divergence is the Kullback-Leibler divergence for which $f(x) =-\log(x)$. Using the properties of a divergence it is clear that the positive definite matrix $\boldsymbol{g}$ of the $f$-divergence provides a natural invariant metric on $M$. It can be seen easily that any $f$-divergence[, provided $f^\prime (1) = 0$ and $f^{\prime\prime}(1) = 1$, i.e. if $f$ is standard,]{} yields the same Riemannian metric which is the Fisher information matrix: $$\begin{split}
D_f[\boldsymbol{\theta}_p,\boldsymbol{\theta}_p, +\mathrm{d}\boldsymbol{\theta}]
= & \ \int \mathrm{d}\boldsymbol{x}\: p(\boldsymbol{x}|\boldsymbol{\theta}_p)\left[\frac{f^{\prime\prime}(1)}{2}\frac{\mathrm{d}p}{\mathrm{d}\theta^\mu}\frac{\mathrm{d}p}{\mathrm{d}\theta^\nu}\bigg|_{\mathrm{d}\boldsymbol{\theta} = 0} \frac{\mathrm{d}\theta^\mu\mathrm{d}\theta^\nu}{p(\boldsymbol{x}|\boldsymbol{\theta}_p)^2}\right], \\
= & \ \frac{1}{2}\int\mathrm{d}\boldsymbol{x}\: p(\boldsymbol{x}|\boldsymbol{\theta}_p)\left(\frac{\mathrm{d}p}{\mathrm{d}\theta^\mu}\frac{1}{p(\boldsymbol{x}|\boldsymbol{\theta}_p)}\frac{\mathrm{d}p}{\mathrm{d}\theta^\nu}\frac{1}{p(\boldsymbol{x}|\boldsymbol{\theta}_p)}\right) \\
= & \ \frac{1}{2}\left\langle \frac{\partial \mathrm{ln}p(\boldsymbol{x}|\boldsymbol{\theta}_p)}{\partial\theta^\mu}\frac{\partial \mathrm{ln}p(\boldsymbol{x}|\boldsymbol{\theta}_p)}{\partial\theta^\nu}\right\rangle \mathrm{d}\theta^\mu \mathrm{d}\theta^\nu,
\end{split}$$ Comparing this result to the third property of the divergence we will therefore write the metric as:
$$\label{eq:fisher_information_metric}
g_{ij} =
\left\langle\frac{\partial \log p(\boldsymbol{x},\boldsymbol{\theta})}{\partial\boldsymbol{\theta}^i}\frac{\partial \log p(\boldsymbol{x},\boldsymbol{\theta})}{\partial\boldsymbol{\theta}^j}\right\rangle =
\int{\mathrm{d}}\boldsymbol{x}\;p(\boldsymbol{x},\boldsymbol{\theta}) \partial_i \log p(\boldsymbol{x},\boldsymbol{\theta}) \partial_j \log p(\boldsymbol{x},\boldsymbol{\theta}).$$
More importantly this metric is unique up to a constant factor. In [@amari_information_2016] the geometrical inner product $g_{ij} = \langle \boldsymbol{e}_i, \boldsymbol{e}_j \rangle$ is identified with the statistical Fisher information such that the tangent vectors $\boldsymbol{e}_i$ can be related to the score functions $\boldsymbol{e}_i \sim \partial \log p(\boldsymbol{x},\boldsymbol{\theta})/\partial\boldsymbol{\theta}^i$ as derivative of the logarithmic likelihood. As a side remark the Riemannian structure on $M$ and hence the positive definiteness of the metric is also vital for the validity of the Cram[é]{}r-Rao inequality which states that the inverse Fisher information evaluated at the likelihoods best fit is a lower bound for the parameter covariance [e.g. @tegmark_karhunen-loeve_1997].
Connection and cubic tensor {#sec:connection_and_cubic_tensor}
---------------------------
The statistical manifold (\[eq:set\_statistical\_model\]) is equipped with a metric (\[eq:fisher\_information\_metric\]) and thus assumes the structure of a Riemannian manifold $(M,\boldsymbol{g})$. We also introduce an affine connection $\nabla$ which allows for the notion of parallel transport, geodesics and curvature on the object $(M,\boldsymbol{g},\nabla)$. It should be noted that the connection $\nabla$ is completely general. However, we will assume parallel transport to not affect the magnitude of vectors, thus restricting ourself to metric connections. [Furthermore, the connection is assumed to be symmetric [@amari_information_2016] what corresponds to the absence of torsion. If metricity is presumed one can even give an argument for the connection to be symmetric as follows: In a chart representation with $\lbrace\boldsymbol{e}_{i}\rbrace = \lbrace\partial/ \partial\theta^{i}\rbrace$ as coordinate basis of the tangent space $T_p M$ the connection coefficient functions $\Gamma^{i}_{j \, k}$ are in general defined as $\nabla_{i}\,\boldsymbol{e}_j = \Gamma^{k}_{j \, i} \boldsymbol{e}_k$ [@Nakahara_Geometry_2003 p.250], and in case of metricity this definition even simplifies to the partial derivative $\partial_i\,\boldsymbol{e}_j = \Gamma^{k}_{j \, i} \boldsymbol{e}_k $ [@Hobson_GR_2006 p.63]. For the definition of the basis vectors $\boldsymbol{e}_j$ in information geometry this becomes $\partial_{i}\,\boldsymbol{e}_j = \Gamma^{k}_{j \, i} \boldsymbol{e}_k = \partial^2 \log p(\boldsymbol{x},\boldsymbol{\theta})/\partial\boldsymbol{\theta}^i \,\partial\boldsymbol{\theta}^j $ what is clearly symmetric in the lower two indices $i$ and $j$.]{} Thus the connection is given by the torsion free Levi-Civita connection. In this latter case geodesic lines trace curves of minimal distances between two points on $M$, if they are affinely parameterised.
Indeed, the Levi-Civita connection is the unique torsion free affine connection preserving the norm of a vector under parallel transport. However, if we write the inner product of two vectors $\boldsymbol{u}$, $\boldsymbol{v}$, which are parallel transported, in the following way [@amari_information_2016]: $$\label{eq:norm_vector_parallel_transport}
\langle \boldsymbol{u},\boldsymbol{v}\rangle =
g(\boldsymbol{u},\boldsymbol{v}) =
\langle \nabla \boldsymbol{u},\nabla^*\boldsymbol{v}\rangle,$$ if the two connections $\nabla$ and $\nabla^*$ are chosen such that they do not change the inner product they are said to be dually coupled. In the case that $\nabla = \nabla^*$ the Levi-Civita connection is recovered. For the general case, one can show the following relation [@amari_information_2016]: $$\label{eq:dual_transport}
\Gamma_{ijk} = \partial_i g_{jk} -\Gamma_{ikj}^*,$$ where [$\Gamma_{ijk} =g_{il}\,\Gamma^{l}_{ij}$ and $\Gamma_{ijk}^*=g_{il}\,\Gamma^{*\,l}_{ij}$]{} are the connection coefficients associated with $\nabla$ and $\nabla^*$ respectively [with indices lowered by metric contraction]{}. Defining the tensor $$\label{eq:cubic_tensor}
T_{ijk} = \Gamma^*_{ijk} - \Gamma_{ijk},$$ [and comparing expression (\[eq:dual\_transport\]) to the first derivative of the metric expressed in terms of the Levi-Civita connection $$\label{eq:LC_transport}
\partial_i g_{jk} =\Gamma^{\mathrm{LC}}_{ijk} + \Gamma^{\mathrm{LC}}_{ikj},$$]{} the connection coefficients are obtained as $$\label{eq:connection_coefficients}
\Gamma_{ijk} =
\Gamma^{\mathrm{LC}}_{ijk} -\frac{1}{2}T_{ijk},
\quad
\Gamma_{ijk} = \Gamma^{\mathrm{LC}}_{ijk} +\frac{1}{2}T_{ijk},$$ with $\Gamma^{\mathrm{LC}}_{ijk}$ being the connection coefficients of the Levi-Civita connection, namely the Christoffel symbols. The invariant cubic tensor $T_{ijk}$ derived from an $f$-divergence is given by $$\label{eq:cubic_tensor}
T_{ijk} =
\alpha \left\langle\frac{\partial \log p(\boldsymbol{x},\boldsymbol{\theta})}{\partial\boldsymbol{\theta}^i}\frac{\partial \log p(\boldsymbol{x},\boldsymbol{\theta})}{\partial\boldsymbol{\theta}^j}\frac{\partial \log p(\boldsymbol{x},\boldsymbol{\theta})}{\partial\boldsymbol{\theta}^k}\right\rangle,$$ with $\alpha = 2f^{\prime\prime\prime}(1) + 3$. In this sense the statistical manifold (\[eq:set\_statistical\_model\]) can also be thought of as given by the triple $(M,\boldsymbol{g},\boldsymbol{T})$. [However, in special cases like for a likelihood of the form $$\label{eq:example_likelihood}
p\left(\boldsymbol{x} \vert \, \boldsymbol{\mu}\left(\boldsymbol{\theta}\right)\right) =
\frac{1}{\sqrt{\left( 2 \pi \right)^n \text{det}\boldsymbol{C}}}\, \exp^{\left(-\frac{1}{2} \boldsymbol{X}^T \, \boldsymbol{C}^{-1} \, \boldsymbol{X} \right)},$$ with a constant data covariance $ \boldsymbol{C}$ and data vector $\boldsymbol{X} \coloneqq \boldsymbol{x} - \boldsymbol{\mu}\left(\boldsymbol{\theta}\right)$ the cubic tensor vanishes. In this case, the score functions are proportional to $\boldsymbol{X}$ and odd moments of a (multivariate) Gaussian vanish due to Isserlis’ Theorem [@isserlis_theorem]. This will turn out to be important for the examples considered in \[sec:Gaussianization\].]{}
Integration on manifolds
------------------------
An invariant volume element, $\mathrm{d}\Omega_M$, on a manifold is a $d$-form: $$\mathrm{d}\Omega_M = \sqrt{\mathrm{det}\boldsymbol{g}}\; {\mathrm{d}}\theta^1\wedge\dots\wedge{\mathrm{d}}\theta^d.$$ The factor $\sqrt{\mathrm{det}\boldsymbol{g}}$ ensures that the volume element is invariant and can be directly interpreted: Consider for example the normalisation condition of a Gaussian distribution with zero mean, which does not restrict the generality of the argument, $$\int \mathrm{d}^{\,d}\theta\:\sqrt{\frac{\mathrm{det}\boldsymbol{F}}{(2\pi)^d}}\exp\left(-\frac{1}{2}\boldsymbol{\theta}^\mathrm{T}\boldsymbol{F}\boldsymbol{\theta}\right) = 1.$$ Comparing the two expressions show that effectively the co-volume factor $\sqrt{\mathrm{det}\boldsymbol{F}}$ is merged with the with the Euclidean volume element $d^d\theta$ to form the invariant volume element $\mathrm{d}\Omega_M$, such that a reparameterisation does not change the normalisation. Furthermore, $\boldsymbol{F}$ becomes the Fisher metric which is constant in case of Gaussian likelihoods. It should be noted that the averaging not only removes the [explicit]{} dependence on the data, but also ensures that a canonical volume form on the statistical manifold is given.
Gaussianisation from an information geometric viewpoint {#sec:Gaussianization}
=======================================================
As outlined , the invariant infinitesimal distance between two neighbouring points on the statistical manifold $(M,F)$ is given by $\mathrm{d}s^2= F_{ij}\mathrm{d}\theta^i\mathrm{d}\theta^j$, where the metric is derived from the divergence, \[eq:divergence\_general\]. Here, a specific coordinate system, or parameter set, $\{\theta_i \}$ has been chosen. A Gaussian distribution with respect to the parameters would correspond to the case where $\boldsymbol{F}$ is independent from the parameters, $\boldsymbol{\theta}$. This, however, is a parameter dependent statement as the components of $\boldsymbol{F}$ transform as: $$\label{eq:transformation_law_fisher_matrix}
\tensor{F}{^\prime_{ij}}(\boldsymbol{\theta}^\prime) =
\tensor{J}{^a_i}(\boldsymbol{\theta}^\prime) \tensor{J}{^b_j}(\boldsymbol{\theta}^\prime)\tensor{F}{_a_b}(\boldsymbol{\theta}),$$ with the Jacobian $J^a_b\coloneqq \partial \theta^a /\partial \theta^{\prime b}$. If we are to find any transformation \[eq:transformation\_law\_fisher\_matrix\] which leads to a globally parameter independent Fisher matrix, the manifold $(M,F)$ would be flat and there would be a global Gaussianisation transformation. Using this argument in the opposite way, even a likelihood described by a flat manifold can show non-Gaussian structure, depending on the chosen coordinate system. In particular, for an originally uncorrelated Gaussian distribution with unit variance one could generate non-Gaussianities through the transformation $$\begin{split}
\tensor{F}{^\prime_{ij,k}}(\boldsymbol{\theta}^\prime) = \tensor{J}{_{ak,i}}(\boldsymbol{\theta}^\prime)\tensor{J}{^a _j}(\boldsymbol{\theta}^\prime) + \tensor{J}{^a_{j,k}}(\boldsymbol{\theta}^\prime)\tensor{J}{_{ai}}(\boldsymbol{\theta}^\prime)\;.
\end{split}$$ Here, we denote the partial derivative as $\partial_a f \equiv f_{,a}$. The commonly used Fisher matrix approach to forecast the sensitivity of future experiments by virtue of the Cram[é]{}r-Rao bound assumes that the pair $(M,F)$ is a flat manifold, if not extended to deal with non-Gaussianities [@Sellentinetal; @sellentin_fast_2015; @schafer_describing_2016]. Going beyond the Fisher approximation thus includes terms which might be attributed to the non-vanishing curvature of the statistical manifold, and we aim to derive relations between this geometric point of view and conventional descriptions of non-Gaussianity.
Weak non-Gaussianitites in the Gram-Charlier-limit: [A first approach]{} {#sec:wng_GCL}
------------------------------------------------------------------------
Returning to the invariant infinitesimal element $\mathrm{d}s^2= F_{ij}\mathrm{d}\theta^i\mathrm{d}\theta^j$ the distance between any two points $P$ and $Q$ along a curve $c(\lambda)$ is $$\label{eq:distance_finite}
D(P,Q) \propto
\left( \int_{\lambda(P)}^{\lambda(Q)} \mathrm{d}\lambda\:\sqrt{F[\dot{c}(\lambda),\dot{c}(\lambda)]}\right)^2,$$ where the dot refers to the derivative with respect to the curve parameter $\lambda$. For $D(P,Q)$ to be the shortest distance, $c(\lambda)$ has to be a geodesic of the metric $\boldsymbol{F}$. is of particular insight when considering a Gaussian (in the data) likelihood, because in this case $D(P,Q) \propto d\chi^2(P,Q)$, thus we can stipulate that the likelihood for the parameters to be $P(\boldsymbol{\theta}) \propto \exp(-D(P,Q)/2)$, such that $\boldsymbol{\theta}$ is the image of $Q$ under some chart. The point $P$ is just for reference and can be absorbed in the proportionality constant, reflecting the fact that only differences in $\chi^2$ are of any relevance. [Since a likelihood with Gaussian distributed data is considered, we can assume that the cubic tensor (\[eq:cubic\_tensor\]) vanishes as argued in \[sec:connection\_and\_cubic\_tensor\]. Thus we can assume the Levi-Civita connection in our further calculations instead of the dually coupled ones.]{}
Going back to the definition of the distance between two points on the manifold, \[eq:distance\_finite\], one can use the expansion of the metric, $\boldsymbol{F}$ to write the distance in terms of Gaussian and non-Gaussian contributions $$D(P,Q) \approx
\left( \int_{\lambda (P)}^{\lambda(Q)} \mathrm{d}\lambda\:\sqrt{\left(\tensor{F}{^0_{ab}} +\tensor{F}{^0_{ab,g}}\Delta\theta^g +\frac{1}{2}\tensor{F}{^0_{ab,gd}}\Delta\theta^g\Delta\theta^d \right) \dot\theta^\alpha\dot\theta^b} \right)^2,$$ where the superscript $0$ denotes evaluation at the point $P$ (for instance the best-fit point) and $\Delta\theta^\alpha \coloneqq \theta^\alpha - \theta(P)^\alpha$. The latter equation can be rearranged using the inverse metric and thus to split everything into Gaussian and perturbatively non-Gaussian parts: $$\label{eq:weak_NG_integrand}
D(P,Q) \approx
\left(
\int_{\lambda (P)}^{\lambda(Q)} \mathrm{d}\lambda\:\sqrt{\tensor{F}{^0_{ib}}\dot\theta^a\dot\theta^b} \left(\delta{^i_a} + \tensor{F}{^{0ir}}\tensor{F}{^0_{ar,g}}\Delta\theta^g +\frac{1}{2}\tensor{F}{^{0ir}}\tensor{F}{^0_{ar,gd}}\Delta\theta^g\Delta\theta^d \right)^{1/2}
\right)^2.$$ This expression is of course very similar to a multidimensional Gram-Charlier series, which expands a distribution around its Gaussian part, assuming that higher order cumulants are small (compared to the variance). We now choose normal coordinates at the point $\boldsymbol{\theta}_0$. In these coordinates geodesics are again Euclidean straight lines and $\dot\theta^a = a^a$ [while the connection coefficient functions $\Gamma^{\mathrm{LC}}_{ijk}$ vanish locally [@lovelock_rund_tensors]]{}. can now be expanded further and then integrated trivially to find: $$D(P,Q) \approx \left(\tensor{F}{^0_{ab}}\Delta\theta^a\Delta\theta^b +\frac{1}{2}\tensor{F}{^0_{ab,g}}\Delta\theta^a\Delta\theta^b\Delta\theta^g +\frac{1}{6}\tensor{F}{^0_{ab,gd}}\Delta\theta^a\Delta\theta^b\Delta\theta^g \Delta\theta^d \right).$$ The remaining terms can be simplified further: the first term, [which just includes the first derivative of the metric and as such depends on the connection $\Gamma^{\mathrm{LC}}_{ijk}$ according to relation (\[eq:LC\_transport\])]{}, again vanishes in normal coordinates. For the second one, a bit more work is required: First we note that in normal coordinates we can write $$\label{eq:second_derivative_normal}
\tensor{F}{^0_{ab,g,d}} =
-\frac{1}{3}\left(R_{agbd} - R_{adbg}\right).$$ [Here $R_{agbd} = g_{ac} R^{c}_{~gbd}$ denotes the components of the Riemann curvature tensor which is here defined with respect to the Levi-Civita connection. Contracting the expression (\[eq:second\_derivative\_normal\]) with $\Delta\theta^a\Delta\theta^b\Delta\theta^g \Delta\theta^d $ can be shown to vanish in a straightforward calculation due to the symmetries of the Riemann tensor (especially $R_{agbd} = -R_{gabd}$ and $R_{agbd} = - R_{agdb}$).]{} This shows that non-Gaussianities can only play a role at second order in a suitable chosen coordinate system.
[The expansion scheme of the likelihoods’ exponent in terms of the Riemannian distance is a first effective approach. One decisive advantage is that $D(P,Q)$ is positive definite by definition and allows for a first and fast estimate of the effect of non-Gaussianity for a non constant Fisher information. In fact it was shown [@amari_information_2016] that there is a close relation between a symmetrised Kullback-Leibler divergence and Riemannian distances. However, the full Kullback-Leibler divergence is asymmetric in general. Thus we now need to turn to a different expansion scheme, which also complies with information due to this asymmetry.]{}
Curvature, Gaussianisation and the DALI expansion {#sec:curvature_gaussianization_dali}
-------------------------------------------------
There are alternative expansions of non-Gaussian likelihooods [@sellentin_fast_2015]: In particular the integrand of \[eq:distance\_finite\] can be expanded in terms of higher order derivatives of [the Fisher Information]{}. Put differently one can say that it is an expansion of the Fisher matrix around the best fit point $\boldsymbol{\theta}_0$. [However in [@sellentin_fast_2015] not the Fisher information but more generically the logarithmic likelihood $L$ itself is expanded around the best fit $\boldsymbol{\theta}_0$ and only afterwards one performs a data average over the expansion coefficients.]{} In particular one defines the flexion and the quarxion as: $$\tensor{S}{^0_{abg}} =
\left\langle \tensor{L}{_{,abg}}\right\rangle\big|_{\boldsymbol{\theta}_0},
\quad
\tensor{Q}{^0_{abgd}} =
\left\langle \tensor{L}{_{,abgd}}\right\rangle\big|_{\boldsymbol{\theta}_0},$$ respectively. Expanding the likelihood this way, allows for the calculation of the Kullback-Leibler divergence relative to the fiducial point. The likelihood for the parameters can now be written as $$\label{eq:dali_expansion}
p(\boldsymbol{\theta}) \propto
\exp\left[-\frac{1}{2}\tensor{F}{^0_{ab}}\Delta\theta^a\Delta\theta^b - \frac{1}{3!}\tensor{S}{^0_{abg}}\Delta\theta^a\Delta\theta^b \Delta\theta^g -\frac{1}{4!}\tensor{Q}{^0_{abgd}} \Delta\theta^a\Delta\theta^b \Delta\theta^g\Delta\theta^d
\right],$$ [with a logarithmic likelihood expansion up to fourth order]{}. In this sense, it is an expansion of the distribution $p(\boldsymbol{\theta})$ relative to some fiducial distribution $p(\boldsymbol{\theta}_0)$. In particular the average over the data is again necessary to measure the divergence between the two distributions. Crucially, this expansion is different from the one presented in \[sec:wng\_GCL\] since it works directly on the level of the Kullback-Leibler divergence and is there not symmetric, that is the integrand in \[eq:f\_divergence\] is expanded before the averaging is carried out. Therefore, this expansion is not necessarily symmetric. In contrast the expansion in \[sec:wng\_GCL\] relies on the expansion of a Riemannian distance measure, which of course is symmetric.
For a Gaussian sampling distribution, DALI expands the likelihood in terms of derivatives with respect to the mean or the covariance, depending which of them carries the parameter dependence. This ensures that the remaining distribution is still a proper probability distribution function. In particular the flexion and the quarxion contracted with sufficiently many $\Delta\theta^a$s can be shown to be always positive definite. Since both the flexion and the quarxion contain non-Gaussian information at third and fourth order about the likelihood a naive guess would be to relate them to the higher order cumulants of the expansion. Indeed, one can derive the following relations using a Gram-Charlier ansatz, i.e. for weak non-Gaussianities: $$\label{kumulant_relations}
\begin{split}
\kappa^a =& \ -\frac{1}{6}\tensor{S}{^0_{ijk}}\tensor{A}{^{ijka}}, \\
\kappa^{abc} =& \ -\tensor{S}{^0_{ijk}}\tensor{B}{^{ia|jb|kc}}, \\
\kappa^{ab} =& \ F^{ab}-\frac{1}{12}\tensor{Q}{^0_{ijkl}}\tensor{D}{^{ijklab}}, \\
\kappa^{abcg} =& \ -\tensor{Q}{^0_{ijkl}}\tensor{D}{^{ijklabcg}}.
\end{split}$$ Here we have introduced the following abbreviations: $$\begin{split}
A^{abcd} \coloneqq & \ F^{ab}F^{cd} + F^{ac}F^{bd} + F^{ad}F^{cb},\\
B^{ab|cd|ij} \coloneqq & \ F^{ab}F^{cd}F^{ij}, \\
D^{abcdij} \coloneqq & \ B^{ac|bi|dj} + B^{ai|bc|dj} + B^{aj|bi|cd} + F^{ci}A^{abdj},\\
E^{abcdijkl} \coloneqq & \ F^{ai}B^{bj|ck|dl}.
\end{split}$$ Thus all additional terms are proportional to the Fisher matrix and thus scale with different powers on $\sigma^{-2}$. [In appendix \[Multi\_Gram\] more details are given about the multivariate Gram-Charlier expansion and we sketch in more detail how the expressions (\[kumulant\_relations\]) can be derived from that.]{} An alternative way to arrive at these expressions is to use directly the Fa[à]{} di Bruno formula, relating flexion and quarxion directly to the moments.
We will now compare the quarxion and the flexion to differential geometric quantities on the statistical manifold. We first start from the definition of the flexion and rewrite it as $$S_{abc} =
\tensor{\Gamma}{^{0\,,\mathrm{LC}}_{cab}} +
\tensor{\Gamma}{^{0\,,\mathrm{LC}}_{acb}} +
\tensor{\Gamma}{^{0\,,\mathrm{LC}}_{bac}} +
\frac{1}{2}\left\langle L_{,a}L_{,b}L_{,c}\right\rangle,$$ where $\tensor{\Gamma}{^{0\,,\mathrm{LC}}_{cab}} = g_{ic}\tensor{\Gamma}{^{0i\,,\mathrm{LC}}_{ab}}$ are the Christoffel symbols, which are defined in terms of derivatives of the Fisher metric, and we evaluate everything again at the best fit point $\boldsymbol{\theta}_0$, where we can identify the last term as the cubic tensor. Thus, in normal coordinates, where the Christoffel symbols vanish, the flexion is completely sourced by the cubic tensor. [For a likelihood as discussed in relation (\[eq:example\_likelihood\]) in \[sec:connection\_and\_cubic\_tensor\] the cubic tensor, and hence the flexion vanishes completely in normal coordinates.]{} A similar exercise can be performed for the quarxion, where we seek a relationship with the second derivatives of the metric. In particular it can be shown that the quarxion vanishes in normal coordinates to first order in the second derivatives of the metric: $$\begin{split}
Q_{ijkl} = & -\frac{1}{3}\left(
R_{ikjl} + R_{iljk} -R_{ikjl} +R_{lkji} + R_{klij} +R_{kjil} - R_{iljk} -R_{klji} - R_{klij} +R_{ikjl}-R_{kjil} -R_{ikjl}
\right)\\
= & \ 0 + \mathcal{O}(L_{,ijk}),
\end{split}$$ where we used the symmetry properties of the Riemann tensor, which can be expressed, in normal coordinates, in terms of second derivatives of the metric for the Levi-Civita connection. Both results, for the flexion and the quarxion, apply for the dependence of the covariance on the parameters as well as for the case where the mean depends on the parameters. $(L_{,ijk})$ refers to terms which contain proper third derivatives of the quantities carrying the parameter dependence. The results presented in this section only hold for weak non-Gaussianities, i.e. up to first order. However, the curvature does not depend on derivatives of third order. For definiteness let’s assume a likelihood of the form \[eq:loglike\_sn\], where the parameter dependence is only carried by $\boldsymbol{\mu}$ such that $$\begin{split}
R_{ijkl} = & \boldsymbol{\mu}^\mathrm{T}_{\,,ik}\boldsymbol{C}^{-1}\boldsymbol{\mu}_{,jl} - \boldsymbol{\mu}^\mathrm{T}_{\,,il}\boldsymbol{C}^{-1}\boldsymbol{\mu}_{,jk}.
\end{split}$$ [Finally, using concepts from information geometry we found an interpretation of non-Gaussianities as inherent geometrical properties of a statistical manifold. These can in general only be remedied for a flat manifold by a non-linear coordinate transformation. For a non-flat manifold they only vanish in normal coordinates in first order approximation, so one can in principle not find a non-linear coordinate transformation to make them vanish completely. However, one could still search for isometries of the Fisher information as will be discussed in the next section, where we consider the example a likelihood on $\Omega_m$ and $w$ from supernova data.]{}
An illustrative example: The Supernovae likelihood {#sec:example}
==================================================
In this section we will discuss a simple cosmological example in the context of information geometry: distance measurements with supernova-observations. [We will investigate the geometrical properties of the manifold defined by the likelihood, especially geodesics and the Ricci scalar $R = g^{ik}R_{ik} = g^{ik}g^{jl}R_{ijkl}$, and investigate the Lie-derivatives to search for isometries of the Fisher information.]{}
The invariant metric
--------------------
As an example we consider the Gaussian likelihood for a supernovae measurement. It has the following simple form [@Sellentinetal] $$\label{eq:loglike_sn}
\mathcal{L}(D|\theta) \propto \exp(\boldsymbol{\mu}^T\boldsymbol{C}^{-1}\boldsymbol\mu),$$ with the vector $$\label{eq:mu_sn}
\boldsymbol{\mu} \coloneqq \boldsymbol{m} -\boldsymbol{m}_\mathrm{theory}.$$ The two vectors $\boldsymbol{m}$ and $\boldsymbol{m}_\mathrm{theory}$ bundle the observed values for the distance modulus and the corresponding model prediction for a given set of parameters $\{\theta\}$ at $n$ redshifts $z_i$ respectively. The errors of different measurements are encoded in the covariance matrix defined as $\boldsymbol{C}\coloneqq \langle\boldsymbol{\mu}\otimes\boldsymbol{\mu}\rangle$. Finally, the distance modulus is defined as $$\label{eq:distance_modulus}
m(z|\{\theta\}) = 5 \log d_\mathrm{lum}(z|\{\theta\}) +\mathrm{const},$$ where the luminosity distance can be calculated from the background cosmology as $$\label{eq:luminosity_distance}
d_\mathrm{lum}(z|\{\theta\}) = c\int_0^z\frac{(1+z^\prime)\mathrm{d}z^\prime}{H(a(z^\prime|\{\theta\})},$$ where the Hubble function $H(a)=\dot{a}/a$ is given by, $$\label{eq:Hubble_function}
\frac{H^2(a)}{H_0^2} = \frac{\Omega_\mathrm{m_0}}{a^{3}} + \frac{1-\Omega_\mathrm{m_0}}{a^{3(1+w)}},$$ for a constant equation of state function $w$ [@2001IJMPD..10..213C; @2006APh....26..102L; @2008GReGr..40..329L]. We will assume the covariance in Eq. to be diagonal and parameter independent: $$\boldsymbol{C} = \mathrm{diag}(\sigma^2_1,...,\sigma^2_n).$$ The corresponding Fisher matrix is easily derived to be $$\label{eq:fisher_matrix_sn}
F_{ab}(\{\theta\}) = \sum_{i=1}^n \frac{\partial_a m(z_i)}{\sigma_i}\frac{\partial_b m(z_i)}{\sigma_i}\bigg|_{\{\theta\}}.$$
Differential geometric quantities {#sec:differential_geometric_quantities}
---------------------------------
We will now treat the supernova likelihood as a statistical manifold $(M,\boldsymbol{g})$ with the metric given by \[eq:fisher\_matrix\_sn\]. [As discussed in \[sec:connection\_and\_cubic\_tensor\] the cubic tensor of the likelihood in our example, given by eq. (\[eq:loglike\_sn\]) vanishes, so we can assume the Levi-Civita connection.]{} For illustrative purposes we show the exact likelihood together with the the Fisher approximation in the $(\Omega_\mathrm{m},w)$-plane in \[fig:likelihood\], illustrating deviations from Gaussianity caused by the nonlinear degeneracy between $\Omega_m$ and $w$.
![Exact likelihood, \[eq:loglike\_sn\], and its Fisher approximation using \[eq:fisher\_matrix\_sn\]. The Fisher ellipse contours denote the values of the likelihood relative to its peak ($80\%$, $30\%$ and $5\%$) that the model parameters fit the data.[]{data-label="fig:likelihood"}](exact_likelihood.pdf){height="9.7cm" width="9.7cm"}
illustrates the Ricci-curvature as a function of the coordinates $\Omega_m$ and $w$ as they would result from the supernova-measurement. The nonzero values are indicative of the non-Gaussian shape of the likelihood and the failure of the Fisher-approximation.
![The Ricci scalar, i.e. the curvature scalar $R = g^{ij}R^l_{\; ilj}$, for the statistical manifold with the metric given by the supernova-measurement \[eq:fisher\_matrix\_sn\]. Since $R$ is a scalar it is invariant under a change of coordinates and therefore provides a measure of the non-Gaussianity of the underlying statistical model irrespective of the choice of parameters.[]{data-label="fig:ricci"}](ricci.pdf){height="9.7cm" width="9.7cm"}
Clearly, the Fisher approximation does not capture the complete shape of the likelihood. However, good agreement can be already achieved with the first two terms using the DALI approximation [@Sellentinetal]. In the right panel the absolute value of the Ricci scalar is shown. Notably, the scalar curvature does not vanish, showing that the non-Gaussianity of the likelihood is inherent to the manifold and is not due to a pure, although physically motivated, choice of parameters.
![Solutions to the geodesic equation on the statistical manifold of supernovae measurements, where initial velocities are chosen isotropically around the fiducial value in $\Omega_m$ and $w$. The red line represents the boundary where the geodesic distance reaches unity.[]{data-label="fig:geodesic"}](geodesicsolution.pdf){height="8cm" width="10.5cm"}
![Dependence of the Fisher-metric on the parameters $\Omega_m$ and $w$, represented as ellipses together with their eigensystems at different points. The area of the ellipses is downscaled by a factor of $0.25^2$ for a more convenient representation, and the solid lines depict the integral curves constructed from the eigenvectors.[]{data-label="fig:eigensystems"}](Integralcurves.pdf){height="6cm" width="6cm"}
Next, we solve the geodesic equation [@Nakahara_Geometry_2003] $$\ddot{\theta}^a\left(\lambda\right)+\Gamma^{{\mathrm{LC}}\,a}_{bc} \dot{\theta}^b\left(\lambda\right)\dot{\theta}^c\left(\lambda\right)=0,$$ and calculate the geodesic distance, \[eq:distance\_finite\], for the geodesic with respect to the fiducial value. The initial conditions for the geodesics are chosen such that they all start at the fiducial cosmology [with parameter values $\left(\Omega_\mathrm{m_0} = 0.28, w = -1.0\right)$]{} and the normalised initial velocities are chosen in all directions. In \[fig:geodesic\], blue dashed lines show the solution to the geodesic equation for different initial conditions. All geodesics are plotted until their geodesic distance reaches unity, this region is depicted by the red line. Clearly the geodesics trace, as expected, the exact likelihood contour. However, a direct interpretation of the probability volume in this contour is not straightforward. This has [three]{} reasons: $(i)$ the invariant metric is only determined up to a constant factor, $(ii)$ geodesics are invariant under affine change of the parametrisation, [$(iii)$ the geodesic distances are symmetric by definition, while the Kullback-Leibler divergence as a traditional measure for information difference is in general asymmetric.]{} For a Gaussian likelihood one can gain an understanding of the geodesic distance. Indeed if the geodesic distance reaches unity, one can identify this with the $1$-$\sigma$ region.
shows the metric represented as ellipses as a function of the coordinates on the statistical manifold. [The area of these ellipses is a measure for the square root of the determinant of the pointwise evaluated inverse Fisher information [@schafer_describing_2016].]{} Additionally the eigensystem of the metric is shown together with the corresponding integral curves of the eigenvectors $\boldsymbol{X}_1$ and $\boldsymbol{X}_2$ (minor and major axis respectively). We will now use these two vector fields to calculate the Lie-derivative $\mathcal{L}_{\boldsymbol{X}_n}\boldsymbol{g}$ of the metric along them [@Nakahara_Geometry_2003]: $$\label{eq:lie_derivative_metric}
\left(\mathcal{L}_{\boldsymbol{X_n}}\boldsymbol{g}\right)_{ab} =
X_n^c \partial_c g_{ab}+\partial_a X_n^c \,g_{cb}+\partial_b X_n^c \,g_{ac} =
\partial_a X_b + \partial_b X_a - 2 \Gamma^{\mathrm{LC}\,c}_{ab}X_c,$$ [where $\boldsymbol{X_n}$ is either of the two fields of eigenvectors. For the numerical evaluation the second expression in eq. (\[eq:lie\_derivative\_metric\]) was used.]{} We normalise the components of the Lie-derivative to the Frobenius matrix norm $\|\boldsymbol{g}\|^2 = \mathrm{tr}\left(\boldsymbol{g} \, \boldsymbol{g}^t \right)$ of the metric and show the components in \[fig:Lie\_derivative\]. One can clearly see that the Lie-derivative with respect to the vector field defined via the minor axis is larger than the one with respect to the major axis. This confirms the observation that the metric changes more strongly along $\boldsymbol{X}_1$. These findings indicate that a Killing vector field of this geometry should be closely aligned with the vector field generated by the major axis of the Fisher matrix. The integral curves of Killing vector fields are symmetry transformations of the metric, i.e. isometries, [meaning that the Lie-derivative of the metric along these vector fields vanishes.]{} Consequently, they give a direct indications how a non-linear transform should be structured to transform the likelihood into an approximately more Gaussian shape. However, as we have already seen earlier, there is no transformation to make the likelihood completely Gaussian globally.
Summary {#sec:summary}
=======
In this paper we have studied likelihood spaces in cosmology from a differential geometric point of view, introduced by [@amari_information_2016]. We described the methods of information geometry and used them to investigate certain approximations of likelihoods made in cosmology. In particular we looked at weak non-Gaussianities in the Gram-Charlier limit and identified non-Gaussian contributions with the Riemann curvature tensor. Furthermore, we studied the relation between the DALI expansion scheme of likelihoods [@sellentin_fast_2015] and showed the connection between its expansion coefficients and geometric objects on the statistical manifold. As an example we studied the likelihood of supernovae in a two-dimensional plane. Our main findings are the following:
i) Local non-Gaussianities in the Gram-Charlier limit can be related to local geometric properties, such as the connection coefficients and the Riemann curvature tensor.
ii) The expansion coefficients of the DALI expansion scheme are directly proportional to higher order cumulants contracted with the Riemannian metric. Furthermore, it can be shown that the flexion, i.e. the first non-trivial expansion coefficient, is related to the cubic tensor. The next order can, however, be shown to vanish in the case of weak non-Gaussianities.
iii) By applying the concepts of information geometry to the supernovae likelihood we could show that it is genuine non-Gaussian in the $(\Omega_\mathrm{m_0},w)$-plane, since the scalar curvature vanishes no-where. [We further investigated the (normalised) Lie-derivatives along the degeneracy directions of the Fisher matrices. Along the vector field generated by the directions of the major axes of the Fisher matrices the Lie-derivative was very small.]{}
The last point indicates that, if there was an isometry of the Fisher information in this specific example, the respective integral curve should be aligned to the vector field generated by the major axes of the Fisher ellipses. This could lead to an indication of non-linear coordinate transformations to achieve an approximative Gaussianisation, since a global Gaussianisation is not possible. To study this further one should however derive the Killing vector fields numerically by solving the Killing equation, i.e. find the vector fields for which the Lie-derivative of the metric vanishes identically (if a solution even exists). This could be done by finite differences methods in the parameter domain of interest, choosing appropriate boundary conditions. The partial differential equation can then be expressed in terms of a system of difference equations and solved numerically.
For future studies, further ideas to achieve an approximate Gaussianisation could be to embed the statistical manifold in a higher dimensional Euclidean space [@lee_smooth_manifolds], which however requires the use of hyper-parameters and can become arbitrarily complicated: But examples in statistics exist where extending the parameter space does provide computational advantages. Ultimately, differential geometry ensures obtaining a flat manifold in embedding at the latest when the dimensionality of the embedding is twice as high. Vice versa, one could reduce the statistical likelihood to two dimensions by marginalisation or conditionalisation and then take advantage of the fact that every two-dimensional Riemannian manifold is at least conformally flat [@Nakahara_Geometry_2003]. Then, the Fisher metric for a two dimensional statistical manifold could (in principle) be reparametrised to become constant up to a parameter dependent conformal scale factor, and it seems to us that these two avenues are the only ones where a Gaussianisation could be successful, for a genuinely curved statistical manifold. Additionally, we are curious if it was possible to derive that a flat manifold where coordinates can be chosen in a way that the Fisher-information becomes constant, by employing a variational principle: It is a well-known fact that the Shannon-entropy $S = -\int{\mathrm{d}}^n\theta\:p(\boldsymbol\theta)\ln p(\boldsymbol\theta)$ is maximised by a Gaussian distribution for a fixed variance, and this result might generalise to implying flatness as a generalisation of Gaussianity following from variation. Furthermore, a wider class of entropy measures, for instance R[é]{}nyi-entropies $S_\alpha = -\int{\mathrm{d}}^n\theta\:p(\boldsymbol\theta)\:p^{\alpha-1}(\boldsymbol\theta)/(\alpha-1)$ for $\alpha\neq 1$, can have interesting geometric implications beyond those of Shannon-entropies $S$.
Remarks on the multivariate Gram-Charlier series {#Multi_Gram}
================================================
A way to characterise the properties of a distribution is by its cumulants $\kappa_n$ which are the expansion coefficients of the logarithm of the characteristic function $K\left(t\right) = \ln \tilde{\phi}\left( t\right)= \sum_n \left( \mathrm{i}t \right)^n \kappa_n / n!$ in one dimension [@capranico_kalovidaris_schaefer_2013]. The characteristic function $\tilde{\phi}\left(t\right)$ itself is defined as the Fourier transformation of the distribution $p\left(x\right)$. For a multivariate Gaussian $G\left( \boldsymbol{x} \right)$ one can read off the cumulants, which are just the mean $\kappa^{\, \alpha}$ and the covariance $\kappa^{\, \alpha \, \beta}$, from the respective characteristic function $\tilde{\phi}_G\left(\boldsymbol{t}\right)$ as: $$\begin{aligned}
&G\left( \boldsymbol{x} \right) = \frac{\sqrt{\det \boldsymbol{C}^{-1}}}{\left(2 \pi \right)^{\,n\,/\,2} } \exp\left( -\frac{\left( x^{\alpha}-\mu^{\alpha} \right)C_{\alpha\beta} \left( x^{\beta}-\mu^{\beta} \right)}{2} \right)
\quad\text{with}\quad
C_{\alpha\beta} \coloneqq \left(C^{\, -1}\right)^{\alpha \, \beta},
\
\\
&\tilde{\phi}_G\left(\boldsymbol{t}\right)=\exp\left(\mathrm{i} \, t_{\gamma} \, \mu^{\gamma} - \frac{ t_{\alpha} C^{\alpha\beta} t_{\beta}}{2} \right)\,\text{with} \,\, \kappa^{\alpha} = \mu^{\alpha} \, \text{as mean and } \kappa^{\alpha\beta} = C^{\alpha\beta} \, \text{as covariance.}\end{aligned}$$ Here, $\boldsymbol{x}$ and $\boldsymbol{t}$ generalise to a vector in the random variable space and the respective Fourier space. If a distribution has higher order cumulants this is a clear sign of non-Gaussianity.
For instance one can quantify the asymmetry of a distribution with respect to its peak by the skewness $s \varpropto \kappa_3$, or $\kappa^{\alpha\beta\gamma}$ as multivariate expressions. Furthermore, the kurtosis excess $k \varpropto \kappa_4$, or $\kappa^{\alpha\beta\gamma\delta}$ which characterises the peak morphology is often considered. For $k>0$ the peak appears steeper compared to a Gaussian while for $k<0$ it is flattened. How one can measure these multivariate cumulants is for instance shown in [@mardia_measures_1970].
The higher order cumulants - beyond the mean and covariance - can now be introduced as small perturbations[^1] of a Gaussian characteristic function [@capranico_kalovidaris_schaefer_2013]. In the multivariate case this characteristic function with perturbations up to fourth order reads: $$\begin{split}
&\tilde{\phi}\left( \boldsymbol{t} \right) = \exp \left[ \frac{\mathrm{i}^3}{3!}t_{\alpha}t_{\beta}t_{\gamma} \, \kappa^{\alpha\beta\gamma} + \frac{\mathrm{i}^4}{4!}\, t_{\alpha}t_{\beta}t_{\gamma}t_{\delta} \, \kappa^{\alpha\beta \gamma\delta} +\mathcal{O}\left( \boldsymbol{t}^{5} \right)\right] \, \tilde{\phi}_G \left( \boldsymbol{t} \right),
\end{split}$$ We now perform a Fourier inversion to derive the multivariate Gram-Charlier series. Here the term $ \mathrm{i}^n\, t_{\alpha_1}\ldots t_{\alpha_n} \tilde{f} \left( \boldsymbol{t} \right)$ is the Fourier transformation of $(-1)^n\, \frac{\partial}{\partial x^{\alpha_1}}\ldots\frac{\partial}{\partial x^{\alpha_n}} \, f\left( \boldsymbol{x} \right)$ in complete analogy to the one-dimensional case with $f\left( \boldsymbol{x} \right)$ being some smooth function.
The multivariate Gram-Charlier series reads (truncating after fourth order in the cumulants): $$\begin{split}
p(\boldsymbol{x})
&= \exp\left((-1)^3 \frac{\kappa^{\alpha\beta\gamma}}{3!} \frac{\partial^3}{\partial x^{\alpha}\partial x^{\beta}\partial x^{\gamma}} + (-1)^4 \frac{\kappa^{\alpha\beta\gamma\delta}}{4!} \frac{\partial^4}{\partial x^{\alpha}\partial x^{\beta}\partial x^{\gamma}\partial x^{\delta}} \right)
\
\\
& \times \frac{\sqrt{\det \boldsymbol{C}^{-1}}}{\left( 2 \pi \right)^{n/2}}\exp\left(-\frac{\left(x^{\epsilon} - \mu^{\epsilon} \right)C_{\epsilon\zeta}\left(x^{\zeta} - \mu^{\zeta}\right)}{2}\right).
\end{split}$$ This expression then simplifies to: $$\label{equ:5.7}
\begin{split}
p(\boldsymbol{x})= \,&\frac{\sqrt{\det\boldsymbol{C}^{-1}}}{\left(2 \pi\right)^{n/2}}
\exp\left(-\frac{\left(x^{\epsilon} - \mu^{\epsilon} \right)C_{\epsilon\zeta} \,\left(x^{\zeta} - \mu^{\zeta} \right)}{2}\right)
\left[ 1+ \frac{\kappa^{\lambda\mu\nu}}{3!} \left(W^{-1}\right)_{\lambda\alpha}\left(W^{-1}\right)_{\mu\beta } \left(W^{-1}\right)_{\nu\gamma } H_3^{\alpha\beta\gamma} \right.
\
\\
& +\left. \frac{\kappa^{\lambda\mu\nu\xi}}{4!} \left(W^{-1}\right)_{\lambda\alpha} \, \left(W^{-1}\right)_{\mu\beta} \, \left(W^{-1}\right)_{\nu\gamma}\, \left(W^{-1}\right)_{\xi\delta} \, H_4^{\alpha\beta\gamma\delta}\right].
\end{split}$$ Here, $ H_3^{\alpha\beta\gamma}$ and $H_4^{\alpha\beta\gamma\delta}$ are mulitvariate generalisations of the Hermite polynomials of third and fourth order which are given in .
The covariance matrix can be written as $C^{\alpha\beta} \coloneqq W^{\alpha\gamma} \, W_{\gamma}^{~\beta}$ with $\boldsymbol{W}$ being defined as the matrix root, and respectively $C_{\alpha\beta} \coloneqq \left(C^{\, -1}\right)^{\alpha\beta} = \left(W^{-1}\right)_{\alpha\gamma} \,\left(W^{-1}\right)^{\gamma}_{~\beta}$ for the inverse. The relations (\[kumulant\_relations\]) between the multivariate cumulants and the non-Gaussianities of the DALI-expansion (\[eq:dali\_expansion\]) in \[sec:curvature\_gaussianization\_dali\] we can now derive using a similar calculation as for the derivation of the multivariate Gram-Charlier series under the assumption of weak non-Gaussianity. First of all we calculate the characteristic function of the expansion (\[eq:dali\_expansion\]) by employing the same techniques in terms of the Fourier transformation as for the derivation of the multivariate Gram-Charlier series, however now changing from real to Fourier space. Then the result for the characteristic function will also contain an expansion in multivariate Hermite polynomials in the exponent, which have to be written explicitly and compared to the general cumulant expansion of a multivariate characteristic function. Comparison of coefficients finally leads to the relations (\[kumulant\_relations\]).
Multivariate Hermite polynomials {#sec:app_Multivariate_Hermite_Polynomials}
================================
In a multivariate expression for the Gram-Charlier series is given in equation (\[equ:5.7\]) which contains multivariate Hermite polynomials. These can be generalised compared to the one-dimensional case as follows: $$\label{equ:B.2}
\begin{split}
H_n^{\, \alpha_1\ldots\alpha_n} = \, & (-1)^n W^{\alpha_1\beta}\ldots W^{\alpha_n\gamma}
\exp\left(
\frac{\left(x^{\epsilon} - \mu^{\epsilon} \right)\,C_{\epsilon\zeta}\,\left(x^{\zeta} - \mu^{\zeta} \right)}{2}
\right)
\
\\
& \times \frac{\partial}{\partial x^{\beta}}\ldots\frac{\partial}{\partial x^{\gamma} } \, \exp\left(-\frac{\left(x^{\varphi} - \mu^{\varphi} \right)\,C_{\varphi\chi}\,\left(x^{\chi} - \mu^{\chi} \right)}{2}\right).
\end{split}$$ Evaluation of relation (\[equ:B.2\]) up to fourth order will lead to explicit expressions for the mutlivariate Hermite polynomials: $$\begin{split}
H_{0} &= 1,
\
\\
H_1^{\,\alpha} &= \left(W^{-1}\right)^{\alpha}_{~\chi} \left(x^{\chi} - \mu^{\chi} \right),
\
\\
H_3^{\, \alpha \, \beta \, \gamma} &= \left(W^{-1}\right)^{\alpha }_{~\lambda} \, \left( x^{\lambda} -\mu^{\lambda} \right)\,\left(W^{-1}\right)^{\beta }_{~\rho} \, \left( x^{\rho} -\mu^{\rho} \right) \, \left(W^{-1}\right)^{\gamma }_{~\tau} \, \left( x^{\tau} -\mu^{\tau} \right)
\
\\
&\quad - [3]\, \delta^{\alpha\beta} \left(W^{-1}\right)^{\gamma}_{~\lambda}\left( x^{\lambda} -\mu^{\lambda} \right),
\
\\
H_4^{\alpha\beta\gamma\delta} &= \left(W^{-1}\right)^{\alpha }_{~\lambda} \left( x^{\lambda} -\mu^{\lambda} \right)\left(W^{-1}\right)^{\beta }_{~\rho} \left( x^{\rho} -\mu^{\rho} \right) \left(W^{-1}\right)^{\gamma }_{~\tau} \left( x^{\tau} -\mu^{\tau} \right)
\left(W^{-1}\right)^{\delta }_{~\eta} \left( x^{\eta} -\mu^{\eta} \right)
\
\\
&\quad -[6]\, \delta^{\alpha\beta} \left(W^{-1}\right)^{\gamma}_{~\lambda} \left(W^{-1}\right)^{\delta}_{~\eta}\left( x^{\lambda} -\mu^{\lambda} \right)\left( x^{\eta} -\mu^{\eta} \right
)+[3]\, \delta^{\alpha\beta} \, \delta^{\gamma\delta}.
\end{split}$$ Here the short-hand notation $\left[n\right]$ means that $n$ terms with permutation in indices exist, while $\delta^{\alpha\beta}$ denotes the Kronecker-Symbol.
RR acknowledges funding through the HEiKA-initative and support by the Israel Science Foundation (grant no. 1395/16 and grant no. 255/18). EG thanks the Studienstiftung des deutschen Volkes and CERN’s Wolfgang-Gentner Programme for financial support. The authors also thank Marie Teich, Rafael Arutjunjan and Nils Fischer for insightful discussion, questions and remarks.
[^1]: This means that the non-Gaussianities have to be weak. In one dimension this can be quantified as $\kappa_n / \left(\sqrt{\sigma^2}\right)^n << 1$ and generalises accordingly in the multivariate case.
|
---
abstract: 'We present a new purely equilibrium microscopic approach to the description of liquid–glass transition in terms of space symmetry breaking of three– and four– particle distribution functions in the cases of two and three dimensions, respectively. The approach has some features of spin glass theories as well as of density–functional theories of freezing.'
---
-50pt
**Classical many–particle distribution functions: some new applications**
[*Vereschagin Institute for High Pressure Physics, Russian Academy of Sciences, 142092 Troitzk, Russian Federation* ]{}
The main purpose of the report is to present a new purely equilibrium microscopic approach to the description of liquid–glass transition in terms of space symmetry breaking of three– and four– particle distribution functions in the cases of two and three dimensions, respectively. The approach has some features of the spin glass theories as well as of the density–functional theories (DFT) of freezing.
It is usually believed that there are two essential differencies between spin glasses and real structural glasses: 1) in the Hamiltonian of spin glasses there is explicit randomness from the very beginning, while in the case of real glasses there is no such randomness. 2)In experiments with spin glasses there is always the range of the concentration of magnetic impurities where nothing else that a spin glass phase appears while in the case of space glass there exists a crystalline ground state. However, in real systems one can consider these differencies simply as time scales differencies for the freezing of corresponding degrees of freedom with respect to the time scale of the real or computer experiments. In fact, there are now some indications that two possible candidates for equilibrium glasses do exist: some polydisperce hard–sphere systems and some binary mixtures of hard spheres. Even if it is not so, it seems us that one needs an “underlying” equilibrium theory of liquid–glass transition to understand what really glasses present as space symmetry breaking problem. We should mention that beautiful and fruitful time–dependent mode–coupling theory [@gotz] which describes a number of subtle experimental facts does not consider the problem of space symmetry breaking. Some other arguments can be found in the recent papers by Parisi (see, e.g. [@par] and references therein).
To describe different kinds of space symmetry breaking we use the formalism of classical many particle conditional distribution functions $$F_{s+1}({\bf r}_1|{\bf r}_1^0 ... {\bf r}_s^0)=
\frac{F_{s+1}({\bf r}_1, {\bf r}_1^0,...,{\bf r}_s^0)}
{F_s({\bf r}_1^0,...,{\bf r}_s^0)}.$$ Here $F_s({\bf r}_1,...,{\bf r}_s)$ is the $s$–particle distribution function. The functions $F_{s+1}({\bf r}_1|{\bf r}_1^0 ... {\bf
r}_s^0)$ satisfy the equation [@arin] $$\begin{aligned}
\frac{\rho F_{s+1}({\bf r}_1|{\bf r}_1^0 ... {\bf r}_s^0)}{z}& = &\exp \left\{
-\beta \sum_{k=1}^s \Phi({\bf r}_1-{\bf r}_k^0)
+\sum_{k \geq 1} \frac{\rho^k}{k!} \int \,
S_{k+1}({\bf r}_1,...,{\bf r}_{k+1}) \right. \nonumber \\
& &\left.\times F_{s+1}({\bf r}_2|{\bf r}_1^0 ... {\bf r}_s^0)...
F_{s+1}({\bf r}_{k+1}|{\bf r}_1^0 ... {\bf r}_s^0)
d{\bf r}_2... d{\bf r}_{k+1} \right\} \label{main}.\end{aligned}$$ Here $z $ is the activity, $\rho$ is the mean number density, $S_{k+1}({\bf
r}_1,...,{\bf r}_{k+1})$ is the irreducible cluster sum of Mayer functions connecting (at least doubly) $k+1$ particles, $\beta=1/k_B T$ and $T$ is the temperature.
If one takes the derivative of (\[main\]) relative to ${\bf r}_1$, one obtains the equilibrium Bogolubov hierarchy [@NNB2] along with the explicit expression for $F_{s+2}$ as the functional on $F_{s+1}$ which gives the formally exact closure. However it contains infinite series and integrals and one has to use some approximations to exploit it. The same can be said about the Eq.(\[main\]) itself.
Let us now consider the symmetry breaking of the one–particle distribution function and formulate briefly DFT of freezing (see [@dft] and the reviews [@rev]). The equation (\[main\]) for $s=0$ is the extremum condition for the free energy functional of the inhomogeneous system with the density $\rho({\bf r}) = \rho F_1(\bf r)$ and has the form: $$\begin{array}{lcl} {\cal F}/k_BT = \int\,d{\bf
r}_1\, \rho({\bf r}_1)[\ln(\lambda^d\rho({\bf r}_1)-1]-\\ -
\sum_{k \geq 1} \frac{1}{(k+1)!} \int \cdots \int\, S_{k+1}({\bf
r}_1...{\bf r}_{k+1})\rho({\bf r}_1) \cdots \rho({\bf
r}_{k+1})\,d{\bf r}_1 \cdots d{\bf r}_{k+1}.\end{array}
\label{free}$$ or $${\cal F}/k_BT =
\int\,d{\bf r}_1\, \rho({\bf r}_1)[\ln(\lambda^d\rho({\bf
r}_1)-1]-{\cal F}_{ex}[\rho({\bf r})]/ k_BT. \label{frex}$$ The excess free energy ${\cal F}_{ex}[\rho({\bf
r})]/k_BT$ is just the generating functional for direct correlation functions $$c_n({\bf r}_1...{\bf
r}_n)=\frac{\delta^n {\cal F}_{ex}[\rho({\bf r})]/k_BT} {\delta
\rho({\bf r}_1) \cdots \rho({\bf r}_n)},$$ so that Taylor expansion around the liquid can be written in the following form: $$\beta \Delta F = \int
d{\bf r} \varrho ({\bf r}) \ln \frac {\varrho ({\bf r})}
{\varrho _0} - \sum_{k \geq 2} {1 \over k!} \int c^{(n)} ({\bf
r}_1,...,{\bf r}_k) \Delta \varrho ({\bf r}_1)...\Delta\varrho
({\bf r}_k) d{\bf r}_1 ... d{\bf r}_k , \label{exfree}$$ where $$\Delta \varrho ({\bf r}) = \varrho ({\bf r}) - \varrho_l$$ is the local density difference between solid and liquid phases.
The full system of equations to be solved in DFT contains the nonlinear integral equation for the function $\rho ({\bf
r})$, obtained as the extremum condition for the free energy and the equilibrium conditions for the chemical potential and the pressure written in terms of the same functions as in (\[exfree\]). To proceed constructively in the frame of DFT we must choose a concrete form of the free energy functional – a kind of closure or truncating – and we must make an ansatz for the average density of the crystal. The importance of such an ansatz follows from the fact that we are dealing with a theory which is equivalent to Gibbs distribution and one has to break symmetry following the Bogoliubov concept of quasiaverages [@bogol1]. Now it is necessary to specify the crystal symmetry (e.g.lattice type) and to locate the freezing transition for that particular lattice type $$\begin{array}{rcl}
\Delta\rho({\bf r})&=&\rho_l\sum_{{\bf k}}\varphi_{{\bf k}}e^{i{\bf kr}}=
\rho_l\varphi_0+\rho_l\varphi({\bf r}),\\
\varphi_{{\bf k}}&=&\frac{1}{\Delta}\int_{\Delta}\,\frac{\Delta\rho({\bf r})}
{\rho_l}e^{-i{\bf kr}}d{\bf r}.\end{array} \label{233}$$ The sum is over reciprocal lattice vectors and the integral is taken over the elementary lattice cell $\Delta$ .$\varphi_{{\bf k}}$ are the order parameters of the problem. The DFT approach occurs to be very fruitful and was used to calculate a lot of melting curves for different systems.
The 3D DFT scenario of freezing is valid for some 2D systems. However, there is a number of 2D systems which melts through two continuous phase transition including intermedeate (so called hexatic) anisotropic liquid phase. The scenario for such a case of 2D melting is the well known KTNHY [@kthny] phenomenological scenario. We develop a microscopic approach to 2D melting [@ryta; @ryta1] in the spirit of 3D DFT. Our approach differs from the standard DFT theory of freezing in two main points: First, we allow the Fourier coefficients $\rho_{\bf G}({\bf
r})$ of the one-particle distribution function expanded in a Fourier series in reciprocal-lattice vectors $\{ {\bf G} \}$: $
\rho({\bf r}) = \sum_{\bf G} \rho_{\bf G}({\bf r}) e^{i{\bf G
r}} $ to fluctuate and to have amplitude and phase. Second, we allow the liquid to be anisotropic: we consider as possible the existence of a phase with constant density but angular dependent two-particle distribution function $F_2({\bf r}_1 -{\bf r}_0) \neq g(r_{10})$.
These two points of generalization define the two new order parameters: the fluctuating $\rho_{\bf G}({\bf r})$ and the Fourier coefficients characteristic for the broken symmetry of the function $F_2({\bf r}_1 -{\bf r}_0)$. Our approach again is based on the Eq.(\[main\]) but now, considering hexatic phase, we are dealing with the bifurcation of the solution for the two–particle distribution function. The relative spatial distribution of pairs of particles is characterized by the function $F_2({\bf r}_1|{\bf r}_0) = F_2({\bf r}_1 -{\bf r}_0)$. The vector ${\bf r}_1-{\bf r}_0$ defines the direction of the bond between the molecules at the points ${\bf r}_1$ and ${\bf r}_0$. In the ordinary isotropic liquid the nearest neighbouring of a given molecule (the first coordination sphere) has a definite local symmetry, which can be characterized by the set of bond directions. The local structure of the liquid in the neighbourhood of a molecule at the point ${\bf r}_0'$ is characterized by the bond directions ${\bf r}'={\bf r}_2 - {\bf r}_0'$. It occurs that if the point ${\bf r}_0'$ is at sufficiently large distance from ${\bf r}_0$ then there is no correlation between the directions ${\bf r}={\bf r}_1 - {\bf r}_0$ and ${\bf r}'={\bf r}_2 - {\bf r}_0'$. In this case after the averaging over the system as a whole the pair distribution function transforms into the RDF and the equation (\[main\]) for $s=1$ has the solution $F_2({\bf r}_1-{\bf r}_0) = g(|{\bf r}_1-{\bf r}_0|)$, which corresponds to ordinary isotropic liquid. When we approach the anisotropic liquid phase the long–ranged correlations between the bond directions ${\bf r}$ and ${\bf r}'$ do appear and the averaged two–particle distribution function depends on the bond direction now.
In the vicinity of the transition one can write $$F_2({\bf r}_1, {\bf r}_0) = g(|{\bf r}_1-{\bf r}_0|) (1+f({\bf r}_1-
{\bf r}_0)) \label{hex}$$ where $f({\bf r}_1-{\bf r}_0)$ has the symmetry of the local neighbourhood of the particle at ${\bf r}_0$. The bifurcation point is given by the linearized equation (\[main\]) for $s=1$, namely, $$f({\bf r}_1-{\bf r}_0)=\int \, \Gamma({\bf r}_1, {\bf r}_0, {\bf r}_2)
f({\bf r}_2-{\bf r}_0)\, g(|{\bf r}_2-{\bf r}_0|)d{\bf r}_2 ,
\label{eighteen}$$ where $$\begin{aligned}
\Gamma({\bf r}_1, {\bf r}_0, {\bf r}_2)=&&
\sum_{k \geq 1} \frac{\rho^{k}}{(k-1)!}\, \int\,
S_{k+1}({\bf r}_1,...,{\bf r}_{k+1})\, \nonumber\\
&&\times g(|{\bf r}_3-{\bf r}_0|)...g(|{\bf r}_{k+1}-{\bf r}_0|)\,
d{\bf r}_3...d{\bf r}_{k+1}. \label{nineteen}\end{aligned}$$
At the same time, when one approaches the line defined by the bifurcation condition, the correlation radius for the orientation fluctuations of the pair distribution function diverges. This fact can be shown with the use of the gradient expansion technique in the case of the equation (\[main\]) for $s=3$, if we write the long range part of the correlator using the principle of vanishing correlations ([@NNB2]) as: $$F_4({\bf r}_1, ..., {\bf r}_4)=g(|{\bf r}_1 - {\bf r}_2|)
g(|{\bf r}_3 - {\bf r}_4|) (1+ f_4({\bf r}_1, ..., {\bf r}_4))
\label{f4}$$ $$f_4({\bf r}_1, ..., {\bf r}_4)= f_4(r, R, \rho, \varphi_1, \varphi_2).$$ Here $\varphi_1$ is the angle between the vector ${\bf r}={\bf
r}_1-{\bf r}_2$ and the axis ${\bf R}={\bf r}_2-{\bf r}_3$, $\varphi_2$ is the angle between the vector ${\bf \rho}={\bf r}_3-{\bf r}_4$ and the same axis. We have $f_4(r, R, \rho, \varphi_1, \varphi_2) \rightarrow 0$ when $R \rightarrow \infty$ .
The microscopic expressions for the elastic moduli and Frank constant [@ryta1] enable us to understand on the microscopic level whether the 2D melting for any given potential is 3D like or whether it follows the KTHNY scenario.
Let us consider now a possible description of the liquid–glass transition in terms of space symmetry breaking for three (four) particle distribution function in 2D (3D) systems. At high temperature the nearest neighbours of a molecule can take different relative positions and there is no short–range order (SRO). At lower temperature a SRO appears which can be of different kind at different densities (for phase transitions in liquids see [@l-l]). The rotation and the translation of the clusters of prefered symmetry give rise to the fact that one-particle and two-particle distribution functions remain isotropic. If a kind of bond orientational order (BOO) appears the clusters are oriented in similar way and the two-particle distribution function becomes to be anisotropic (as in 2D hexatic phase). However, we can imagine another situation – freezing of the symmetry axes of the clusters in different position. The isotropic phase can be considered as analogous to the paramagnetic phase (of cluster symmetry axes), the BOO phase – to the ferromagnetic phase, and the mentioned freezed phase – to a spin glass phase.
Let us consider for simlicity a 2D system. In the vicinity of the transition one can write (in the superposition approximation for the liquid) $$F_3({\bf r}_1| {\bf r}_1^0, {\bf r}_2^0) =
g(|{\bf r}_1-{\bf r}_1^0|) g(|{\bf r}_1-{\bf r}_2^0|)(1+f_3({\bf
r}_1| {\bf r}_1^0, {\bf r}_2^0) \label{gla}$$ In 2D case $f_3({\bf r}_1| {\bf r}_1^0, {\bf r}_2^0)$ depends in fact on two distances and two angles $$f_3({\bf r}_1| {\bf r}_1^0, {\bf r}_2^0) =
f_3(R_0, \phi _0;R_1, \Theta _1 ),
\label{gla1}$$ where $ {\bf R}_0 = {\bf r}_2^0 - {\bf r}_1^0$, $ {\bf R}_1 = {\bf r}_1 - {\bf r}_1^0$, $ {\bf R}_2 = {\bf r}_2 - {\bf r}_1^0$ and $\phi _0$ is the angle of the vector ${\bf R}_0$ with the $z$ axis, $ \Theta _1$ – the angle between ${\bf R}_1$ and ${\bf R}_0$ and $ \Theta _2$ – the angle between ${\bf R}_2$ and ${\bf
R}_0$.
The linearization of (\[main\]) for $s=2$ gives: $$f_3(R_0, \phi _0;R_1, \Theta _1)=\int \,
\Gamma'(R_0, \phi _0;{\bf r}_2; R_1, \Theta _1)
f_3(R_0, \phi _0;R_2, \Theta _2)
g(|{\bf R}_2-{\bf R}_0|) g(R_2) d{\bf r}_2,
\label{gla2}$$ where $$\begin{aligned}
\Gamma'(R_0, \phi _0;{\bf r}_2; R_1, \Theta _1)&=&
\sum_{k \geq 1}
\frac{\rho^{k}}{(k-1)!}\, \int\, S_{k+1}({\bf r}_1,...,{\bf
r}_{k+1})g(|{\bf r}_3-{\bf r}_1^0|)\, \nonumber\\ &\times&
g(|{\bf r}_3-{\bf r}_2^0|)...
g(|{\bf r}_{k+1}-{\bf r}_1^0|)g(|{\bf r}_{k+1}-{\bf r}_2^0|) \,
d{\bf r}_3...d{\bf r}_{k+1}. \label{gla3}\end{aligned}$$ There are two kinds of angles entering the equations and two kinds of order parameters, cosequently. One angle ($\phi
_0$) fixes the position of one pair of particles of the cluster, and the other ($\Theta _i$) – the position of the third particle in the coordinate frame defined by $\phi _0$. The order parameter connected with $\Theta _i$ is the generalization of intracluster hexatic parameter for the case of different coordinate frames. The order parameter connected with $\phi _0$ is an analogue of magnetic moment and in glass–like phase one can consider an Edwards-Anderson parameter $<\cos \phi _0 (t) \cos \phi
_0(0)>$. In such a way we come to the concept of a “conditional” long range order: if we consider two pairs of particles at infinite distance from one another then there exists a preferable possibility for the relative position of the third particle near each pair. The directions of the bonds in the pairs of particles themselves are subjects to spin–glass–like order. In 3D case the rotation of clusters is given by matrices $D_{lm}^{l'm'}(\vec \omega _{0i})$ so that we obtain a kind of orientational multipole glass for the clusters.
This work was partially supported by Russian Foundation for Basic Researches (Grant No. 98-02-16805).
[99]{}
For review see, W.Götze, [*Liquid, freezing and glass transition*]{}, Les Houches (1989), J.P.Hansen, D.Levesque, J.Zinn-Justin editors, North Holland.
B.Coluzzi, G.Parisi, P.Verrocchio, [*Lennard-Jones binary mixtures: a thermodynamic approach to glass transition*]{}, cond-mat/9904124.
The first derivation of the eq. (\[main\]) by use of Bogoliubov functional method [@NNB2] was given by E.A.Arinshtain in Dokl. Akad. Nauk (USSR), [**112**]{}, 615 (1957). See also F.H.Stillinger, F.P.Buff, J.Chem.Phys., [**37**]{}, 1 (1962), and V.N.Ryzhov, thesis (JINR, Dubna, Russia, 1981).
N.N.Bogoliubov, [*Problems of dynamical theory in statistical physics*]{} (Moscow, Gostehisdat, 1946).
N.N.Bogoliubov, JINR Preprint R-1451, Dubna, 1963; Phys.Abh.S.U., [**6**]{}, 1, 113, 229 (1962).
T.V.Ramakrishnan, M.Youssouff, Phys. Rev. [**B 19**]{}, 2775 (1979); V.N.Ryzhov, E.E.Tareyeva, Phys.Lett.A, [**75**]{}, 88 (1979), Theor.Math.Phys.(Moscow), [**48**]{}, 416 (1981); A. D. J. Haymet and D. W. Oxtoby, J. Chem. Phys. [**74**]{}, 2559 (1981).
Y. Singh, Phys. Rep. [**207**]{}, 351 (1991); M. Baus, J. Phys.: Condens. Matter [**1**]{}, 3131 (1989); H.Löwen, Phys.Rep. [**237**]{}, 249 (1994).
M. Kosterlitz and D. J. Thouless, J. Phys. C [**6**]{}, 1181 (1973); B. I. Halperin and D. R. Nelson, Phys. Rev. Lett. [**41**]{}, 121 (1978); D. R. Nelson and B. I. Halperin, Phys. Rev. B [**19**]{}, 2457 (1979); A. P. Young, Phys. Rev. B [**19**]{}, 1855 (1979).
V.N.Ryzhov, E.E.Tareyeva, Theor.Math.Phys.(Moscou), [**73**]{}, 463 (1987); J. Phys. C, [**21**]{}, 819 (1988); Phys.Lett.A [**158**]{}, 321 (1991). V.N.Ryzhov, Theor.Math.Phys.(Moscou), [**80**]{}, 107 (1989); J. Phys.: Condens. Matter, [**2**]{}, 5855 (1990); Zh. Eksp. Teor. Fiz. [**100**]{}, 1627 (1991) \[Sov. Phys. JETP [**73**]{}, 899 (1991)\].
V.N.Ryzhov, E.E.Tareyeva, Theor.Math.Phys.(Moscou), [**92**]{}, 331 (1992); Theor.Math.Phys.(Moscou), [**96**]{}, 425 (1993); Phys. Rev. B [**51**]{} N 14, 8789-8794 (1995); JETP (Moscou), [**108**]{}, 2044 (1995).
See, e.g., P.G.Debenetti, [*Metastable Liquids*]{} (Princeton Univ.Press, Princeton, 1997; C.A.Angell, Science [**267**]{}, 1924 (1995); C.J.Roberts and P.G.Debenedetti, J.Chem.Phys. [**105**]{}, 658 (1996).
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---
abstract: 'We study locally conformal symplectic (LCS) structures of the second kind on a Lie algebra. We show a method to build new examples of Lie algebras admitting LCS structures of the second kind starting with a lower dimensional Lie algebra endowed with a LCS structure and a suitable representation. Moreover, we characterize all LCS Lie algebras obtained with our construction. Finally, we study the existence of lattices in the associated simply connected Lie groups in order to obtain compact examples of manifolds admitting this kind of structure.'
address: |
KU Leuven Campus Kulak Kortrijk\
E. Sabbelaan 53\
BE-8500 Kortrijk, Belgium; and FaMAF-CIEM, Universidad Nacional de Córdoba\
X5000HUA Córdoba\
Argentina
author:
- 'M. Origlia'
title: On certain class of locally conformal symplectic structures of the second kind
---
Introduction
============
A *locally conformal symplectic* structure (LCS for short) on the manifold $M$ is a non degenerate $2$-form $\omega$ such that there exists an open cover $\{U_i\}$ and smooth functions $f_i$ on $U_i$ such that $\omega_i=\exp(-f_i)\omega$ is a symplectic form on $U_i$. This condition is equivalent to requiring that $$\label{lcs}
d\omega=\theta\wedge\omega$$ for some closed $1$-form $\theta$, called the Lee form. The pair $(\omega, \theta)$ will be called a LCS structure on $M$. It is well known that if $(\omega,\theta)$ is a LCS structure on $M$, then $\omega$ is symplectic if and only if $\theta=0$. Furthermore, $\theta$ is uniquely determined by equation , but there is not an explicit formula for the Lee form. If $\omega$ is a non degenerate $2$-form on $M$, with $\dim M\ge 6$, such that holds for some $1$-form $\theta$ then $\theta$ is automatically closed and therefore $M$ is LCS.
According to Vaisman (see [@V]) there are two different types of LCS structures. If $(\omega, \theta)$ is a LCS structure on $M$, a vector field $X$ is called an infinitesimal automorphism of $(\omega,\theta)$ if $\textrm{L}_X\omega=0$, where $\textrm{L}$ denotes the Lie derivative. This implies $\textrm{L}_X\theta=0$ as well and, as a consequence, $\theta(X)$ is a constant function on $M$. We consider $\mathfrak{X}_\omega(M)=\{X\in\mathfrak{X}(M): \textrm{L}_X\omega=0\}$ which is a subalgebra of $\mathfrak{X}_\omega(M)$, then the map $\theta|_{\mathfrak{X}_\omega(M)} : \mathfrak{X}_\omega(M) \to \R$ is a well defined Lie algebra morphism called the Lee morphism. If there exists an infinitesimal automorphism $X$ such that $\theta(X)\neq 0$, the LCS structure $(\omega,\theta)$ is said to be of [*the first kind*]{}, and it is of [*the second kind*]{} otherwise. This condition is equivalent to the Lee morphism being either surjective or identically zero. In the literature, there is more information about LCS structures of the first kind, for example, in [@V] Vaisman gives relations with contact geometry and it is also proves that a manifold with a LCS structure of the first kind admits distinguished foliations. On the other hand, LCS structures of the second kind are less understood.
There is another way to distinguish LCS structures, to do this, we can deform the de Rham differential $d$ to obtain the adapted differential operator $$d_\theta \alpha= d\alpha -\theta\wedge\alpha,$$ for any differential form $\alpha \in \Omega^*(M)$. Since $\theta$ is $d$-closed, this operator satisfies $d_\theta^2=0$, thus it defines the [*Morse-Novikov cohomology*]{} $H_\theta^*(M)$ of $M$ relative to the closed $1$-form $\theta$ (see [@N1; @N2]). Note that if $\theta$ is exact then $H_\theta^*(M)\simeq H_{dR}^*(M)$. It is known that if $M$ is a compact oriented $n$-dimensional manifold, then $H_\theta^0(M)= H_\theta^n(M)=0$ for any non exact closed $1$-form $\theta$ (see for instance [@GL; @Ha]). For any LCS structure $(\omega,\theta)$ on $M$, the $2$-form $\omega$ defines a cohomology class $[\omega]_\theta\in H_\theta^2(M)$, since $d_\theta\omega=d\omega-\theta\wedge \omega=0$. The LCS structure $(\omega,\theta)$ is said to be [*exact*]{} if $\omega$ is $d_\theta$-exact or $[\omega]_\theta =0$, i.e., $\omega= d\eta-\theta\wedge \eta$ for some $1$-form $\eta$, and it is [*non-exact*]{} if $[\omega]_\theta \neq0$. It was proved in [@V] that if the LCS structure $(\omega,\theta)$ is of the first kind on $M$ then $\omega$ is $d_\theta$-exact, i.e., $[\omega]_\theta=0$. But the converse is not true. Recently in [@AOT] other cohomologies for LCS manifolds were introduced, inspired by the almost Hermitian setting. More precisely, the authors define the LCS-Bott-Chern cohomology and the LCS-Aeppli cohomology on any compact LCS manifold, and compute them for some LCS solvmanifolds in low dimensions.
In last years, special attention has been devoted to the study of left invariant LCS structures on Lie groups (see for instance [@AO1; @ABP; @AOT; @BM]), with very nice results in the case of LCS structures of the first kind. In this work we focus on LCS structures of the second kind on Lie algebras and solvmanifolds.
We recall that a LCS structure $(\omega,\theta)$ on a Lie group $G$ is called left invariant if $\omega$ is left invariant, and this easily implies that $\theta$ is also left invariant. Accordingly, we say that a Lie algebra $\g$ admits a [*locally conformal symplectic*]{} (LCS) structure if there exist $\omega\in\alt^2\g^*$ and $\theta \in \g^*$, with $\omega$ non degenerate and $\theta$ closed, such that is satisfied. As in the case of manifolds we have that a LCS structure $(\omega,\theta)$ on a Lie algebra $\g$ can be of the first kind or of the second kind. Indeed, let us denote by $\g_\omega$ the set of infinitesimal automorphisms of the LCS structure, that is, $$\label{autom}
\g_\omega = \{X\in\g: \textrm{L}_X\omega=0\} = \{X\in\g: \omega([X,Y],Z)+\omega(Y,[X,Z])=0 \, \text{for all} \, Y,Z\in\g\}$$ where $\textrm{L}$ denotes the Lie derivative. Note that $\g_\omega \subset \g$ is a Lie subalgebra, thus the restriction of $\theta$ to $\g_\omega$ is a Lie algebra morphism called the [*Lee morphism.*]{} The LCS structure $ (\omega,\theta)$ is said to be [*of the first kind*]{} if the Lee morphism is surjective, and [*of the second kind*]{} if it is identically zero.
For a Lie algebra $\g$ and a closed $1$-form $\theta\in\g^*$ we also have the Morse-Novikov cohomology $H_\theta^*(\g)$ defined by the differential operator $d_\theta$ on $\alt^* \g^*$ defined by $$d_\theta \alpha= d\alpha
-\theta\wedge\alpha.$$
As in manifolds, we have that a LCS structure $(\omega,\theta)$ on a Lie algebra is said to be exact if $[\omega]_\theta =0$ or non-exact if $[\omega]_\theta\neq0$.
It was proved in [@BM] that:
\[1kind=exact\] If the Lie algebra $\g$ is unimodular, a LCS structure on $\g$ is of the first kind if and only if it is exact.
LCS structures of the first kind on Lie algebras are better understood because they are related with other important geometric structures (see for instance [@V] or more recently [@BM]). On the other hand not much is known about Lie algebras with a LCS structure of the second kind. In [@ABP] the authors study three different type of constructions of LCS Lie algebras. One corresponds to exacts LCS Lie algebras. The second one establishes a link between cosymplectic Lie algebras in dimension $2n-1$ and non-exact LCS structure in dimension $2n$. The third one is related to the existence of Lagrangian ideals (see [@ABP] for more details). In [@AO1] we study LCS structures on almost abelian Lie groups, and we exhibit examples of solvmanifolds with LCS structure of the second kind in any dimension greater than or equal to $6$. We do not know many other explicit examples of solvmanifolds with a LCS structure of the second kind. Therefore, we consider that it would be very interesting to find new examples of Lie algebras and solvmanifolds admitting LCS structures of the second kind, hopefully they might be used to understand better this kind of structures.
In this work we deal with this problem and inspired by ideas of [@ABP] we provide another construction of LCS Lie algebras of the second kind which is different from those given in [@ABP]. After we provide this construction in Theorem \[pi=tita-ro\] two question arise naturally:
Question $1$: *Given a LCS structure of the second kind on a Lie algebra, can it be obtained from our construction?*
Question $2$: *Do there exist examples of LCS Lie algebras constructed by Theorem \[pi=tita-ro\] such that the associated simply connected Lie group admits lattices?*
We answer affirmatively question $2$. Indeed, we exhibit lattices for some associated simply connected Lie groups obtaining explicit examples of solvmanifolds admitting LCS structures of the second kind. And concerning question $1$, we give a nice characterization of Lie algebras built with our construction. Moreover, we recover most of the known examples in dimension $4$.
The outline of this article is as follows. In Section $2$ we prove that a left invariant LCS structure of the second kind on a Lie group induces a LCS structure of the second kind on any compact quotient by a discrete subgroup (see Theorem \[2tipo\_cociente\]). This allows us to study this geometric structure at the Lie algebra level. In Section $3$ we give a method to build new examples of Lie algebras admitting LCS structures of the second kind starting with a Lie algebra endowed with a LCS structure and a compatible representation (see Theorem \[pi=tita-ro\]), pointing out when the resulting Lie algebra is unimodular. We also have a converse (see Theorem \[converse\]) obtaining a nice characterization of LCS Lie algebras admitting a non degenerate abelian ideal contained in the kernel of the Lee form. In Section $4$ we show the wide range of our construction by reobtaining most of the known examples of LCS Lie algebras on the second kind in dimension $4$. We also exhibit examples of Lie algebras in higher dimension starting with a $4$-dimensional LCS Lie algebra. Moreover we give a complete list of $4$-dimensional LCS Lie algebras which can be used to produce examples in higher dimension. Finally, in Section $5$ we study the existence of lattices in the associated Lie groups and we give an explicit construction of a family of solvmanifolds admitting a LCS structure of the second kind proving that they are pairwise non homeomorphic.
Solvmanifolds with LCS structure of the second kind
===================================================
Let us consider LCS structures on Lie algebras, or equivalently left invariant LCS structures on Lie groups. If the Lie group is simply connected then any left invariant LCS structure turns out to be globally conformal to a symplectic structure, which is essentially equivalent to having a symplectic structure on the Lie group. Therefore we will study compact quotients of such a Lie group by lattices. Recall that a discrete subgroup $\Gamma$ of a simply connected Lie group $G$ is called a *lattice* if the quotient $\Gamma\backslash G$ is compact. In this case we have that $\pi_1(\Gamma\backslash G)\cong \Gamma$. The quotient $\Gamma\backslash G$ is called a solvmanifold if $G$ is solvable and it is called a nilmanifold if $G$ is nilpotent.
It is clear that a left invariant LCS structure on a Lie group $G$ induces a LCS structure on any quotient $\Gamma\backslash G$, which will be non simply connected and therefore the inherited LCS structure is “strict”.
It is important to mention that if a Lie group admits a lattice then such Lie group must be unimodular, according to [@Mi]. Besides this necessary condition, there is no general criterion to determine whether a given unimodular solvable Lie group admits a lattice. It is a very difficult problem in itself. However, there is such a criterion for nilpotent Lie groups. Indeed, Malcev proved in [@Ma] that a nilpotent Lie group admits a lattice if and only if its Lie algebra has a rational form, that is, there exists a basis of the Lie algebra such that the corresponding structure constants are all rational. More recently, Bock studied in [@B] the existence of lattices in simply connected solvable Lie groups up to dimension 6 and he gave a criterion for the existence of lattices in almost abelian Lie groups. Concerning LCS structures on a solvmanifold $\Gamma\backslash G$ which arise from a LCS structure on $\g=\text{Lie}(G)$, it is easy to see that if the LCS structure on $\g$ is of the first kind, then the induced LCS structure on the quotient $\Gamma\backslash G$ is of the first kind as well. We will prove in the next result that the same happens for LCS structures of the second kind, i.e., a LCS structure of the second kind on $\g$ induces a LCS structure of the second kind on any compact quotient $\Gamma\backslash G$.
\[2tipo\_cociente\] Let $\Gamma\backslash G$ be a solvmanifold and $\g=Lie(G)$. If $(\omega, \theta)$ is a LCS structure on $\g$ of the second kind, then the LCS structure induced on the solvmanifold $\Gamma\backslash G$ is of the second kind.
Let $(\omega,\theta)$ be a LCS structure of the second kind on $\g$. Since $\g$ is unimodular, it follows from that $(\omega,\theta)$ is not exact, i.e., $0\neq[\omega] \in H^2_\theta(\g)$. Let $(\hat\omega,\hat\theta)$ be the induced LCS structure on $\Gamma\backslash G$. According to [@K] there exists an injective map $i: H^2_\theta(\g) \to H^2_{\tilde\theta}(\Gamma\backslash G)$. This map arises from the natural inclusion of $\g$ into $\mathfrak X(\Gamma\backslash G)$. Therefore $0\neq[i(\omega)]=[\hat{\omega}] \in H^2_{\tilde\theta}(\Gamma\backslash G)$. Then $\hat\omega$ is non exact in $\Gamma\backslash G$ and it follows from [@V] that the LCS structure $(\hat\omega, \hat\theta)$ in the solvmanifold $\Gamma\backslash G$ is of the second kind.
Note that the LCS structure induced by Theorem \[2tipo\_cociente\] is indeed a non-exact LCS structure on $\Gamma\backslash G$.
Note that in general a LCS structure of the second kind on a Lie algebra induces a LCS structure on the associated simply connected Lie group which is not necessarily of the second kind.
A method to construct LCS Lie algebras of the second kind
=========================================================
In this section we give a method to build new examples of Lie algebras admitting a LCS structures of the second kind starting with a Lie algebra endowed with a LCS structure and a compatible representation (See Theorem \[pi=tita-ro\]). Then we characterize all Lie algebras built with this method in Theorem \[converse\], we also point out when the Lie algebras constructed in Theorem \[pi=tita-ro\] are unimodular which is a necessary condition to study lattices in the last section.
Let $\h$ be a Lie algebra, $(\omega,\theta)$ a LCS structure on $\h$, and let $(V,\omega_0)$ be a symplectic vector space of dimension $2n$. We consider a representation $$\pi: \h \to \operatorname{End}(V).$$ Let $\g$ be the Lie algebra given by $\g=\h\ltimes_\pi V$, equipped with the non degenerate 2-form $\tilde\omega$ given by $\tilde\omega|_\h=\omega$ and $\tilde\omega|_V=\omega_0$. In particular $\tilde\omega(X,Y)=0$ for any $X\in\h$, $Y\in V$. We define the $1$-form $\tilde\theta\in\g^*$ by $\tilde\theta|_\h=\theta$ and $\tilde\theta|_V=0$.
We determine next when the pair $(\tilde\omega, \tilde\theta)$ is a LCS structure on the Lie algebra $\g$. Computing $d\tilde\omega=\tilde\theta\wedge\tilde\omega$ we can easily see that $(\tilde\omega, \tilde\theta)$ is a LCS structure if and only if the following condition is satisfied: $$\label{cons}
-\omega_0(\pi(X)Y,Z)+\omega_0(\pi(X)Z,Y)=\theta(X)\omega_0(Y,Z),$$ for $X\in\h$ and $Y,Z\in V$.
We denote by $S$ and $\rho$ the $\omega_0$-symmetric part and $\omega_0$-skew-symmetric part of $\pi$. More precisely, for each $X\in\h$, $$\pi(X)=S(X)+\rho(X),$$ where $S(X)$ is $\omega_0$-symmetric and $\rho(X)$ is $\omega_0$-skew-symmetric with respect to the non degenerate $2$-form $\omega_0$, that is, $S(X)$ satisfies $\omega_0(S(X)Y,Z)=\omega_0(Y,S(X)Z)$ and $\rho(X)$ satisfies $\omega_0(\rho(X)Y,Z)=-\omega_0(Y,\rho(X)Z)$ for any $X\in\h$ and $Y,Z\in V$. This condition on $\rho$ is equivalent to saying that $\rho(X)\in\mathfrak{sp}(V,\omega_0)$ for any $X\in\h$. It is easy to verify that holds if and only if $-2S(X)=\theta(X)\I$ for any $X\in\h$.
With the notation above, if $\pi(X)=-\frac12\theta(X)\I+\rho(X)$ and $\rho(X)\in\mathfrak{sp}(V,\omega_0)$ for all $X\in\h$, then we say that $\pi$ is a *LCS representation*.
Note that any LCS Lie algebra $\h$ admits a LCS representation. For example, taking $\rho=0$.
Therefore we have the following result:
\[pi=tita-ro\] Let $\h$ be a Lie algebra with a LCS structure $(\omega,\theta)$, let $(V,\omega_0)$ be a $2n$-dimensional symplectic vector space and $\pi: \h \to \operatorname{End}(V)$ a representation. Let $\g=\h\ltimes_\pi V$ and $(\tilde\omega, \tilde\theta)$ given by $\tilde\omega|_\h=\omega$, $\tilde\omega|_V=\omega_0$, $\tilde\theta|_\h=\theta$ and $\tilde\theta|_V=0$. Then $(\tilde\omega, \tilde\theta)$ is a LCS structure on $\g$ if and only if $\pi$ is a LCS representation. Moreover, $\rho: \h\to \mathfrak{sp}(V,\omega_0)$ is a representation and $(\tilde\omega, \tilde\theta)$ is a LCS structure of the second kind.
As we mentioned above, $(\tilde\omega, \tilde\theta)$ is a LCS structure on $\g$ if and only if for any $X\in\h$ we have $\pi(X)=-\frac12\theta(X)\I+\rho(X)$ with $\rho(X)\in\mathfrak{sp}(V,\omega_0)$. We have to check next that $\rho(X): \h\to \mathfrak{sp}(V,\omega_0)$ is a representation. We compute $$\begin{aligned}
\pi([X,Y]) & = [\pi(X),\pi(Y)]\\
& = [S(X)+\rho(X),S(Y)+\rho(Y)]\\
& = [\rho(X),\rho(Y)],\end{aligned}$$ since $S(X)=-\frac12\theta(X)\I$. On the other hand we have that $$\begin{aligned}
\pi([X,Y]) & = S([X,Y])+\rho([X,Y])\\
& = -\frac12\theta([X,Y])+\rho([X,Y])\\
& = \rho([X,Y]),\end{aligned}$$ since $\theta$ is closed and the first part of the result follows.
Finally, we see that $(\tilde\omega, \tilde\theta)$ is a LCS structure of the second kind. Indeed, let $X\in\g_{\tilde\omega}$, $X=H+Y$ with $H\in\h$ and $Y\in V$. Then $\tilde\omega([X,Z],W)+\tilde\omega(Z,[X,W])=0$ for any $Z,W\in\g$. In particular, for any $Z,W\in V$ we have: $$\begin{aligned}
0 & = \tilde\omega([X,Z],W)+\tilde\omega(Z,[X,W])\\
& = \omega_0([H+Y,Z],W) + \omega_0(Z,[H+Y,W])\\
& = \omega_0(\pi(H)Z,W) + \omega_0(Z,\pi(H)W)\\
& = \omega_0(-\frac{1}{2}\theta(H)Z + \rho(H)Z,W) + \omega_0(Z, -\frac{1}{2}\theta(H)W + \rho(H)W)\\
& = -\frac{1}{2}\theta(H)\omega_0(Z,W) + \omega_0(\rho(H)Z,W) -\frac{1}{2}\theta(H)\omega_0(Z,W) + \omega_0(Z,\rho(H)W)\\
& = -\theta(H)\omega_0(Z,W),
\end{aligned}$$ for any $Z,W\in V$, where we used that $\rho(H)$ is $\omega_0$-skew-symmetric in the last equality. Since $\omega_0$ is non degenerate on $V$ we can choose $Z,W\in V$ such that $\omega_0(Z,W)\neq0$, and therefore we get that $\theta(H)=0$ which implies $\theta(X)=0$. Then $\g_{\tilde\omega}\subset\ker\theta$, thus $\theta|_{\g_{\tilde\omega}}\equiv0$ and therefore we have that the LCS structure $(\tilde\omega,\tilde\theta)$ is of the second kind.
provides us with a method to build new examples of Lie algebras equipped with a LCS structure of the second kind, starting with a LCS Lie algebra and a suitable representation. Note that the LCS structure on the initial Lie algebra can be of the first or of the second kind. We believe that this method is interesting because, as we mentioned before, there are not many general results about LCS structures of the second kind.
Moreover, we can see that the LCS structure in Theorem \[pi=tita-ro\] is non-exact. Indeed, suppose $(\tilde\omega, \tilde\theta)$ is an exact LCS structure. Then $\omega_0$, i.e. the restriction of $\tilde\omega$ to $\R^{2n}$, is zero, which is a contradiction since $\omega_0$ is non-degenerate.
This method can also be used to construct Lie algebras admitting other kind of structures. For example, if we start with a symplectic Lie algebra $(\h,\omega)$, that is $\theta=0$, then it is clear that we obtain a new symplectic Lie algebra $(\g,\tilde\omega)$.
The LCS Lie algebra $(\g,\tilde\omega, \tilde\theta)$ constructed in Theorem \[pi=tita-ro\] has an abelian ideal which is non degenerate with respect to the restriction of the fundamental form $\tilde\omega$, namely, $(V,\omega_0)$. Moreover, it is contained in $\ker\tilde\theta$. We show next a sort of converse of Theorem \[pi=tita-ro\]. More precisely, we prove that any LCS Lie algebra with a non degenerate abelian ideal can be constructed as in Theorem \[pi=tita-ro\].
Let $\g$ be a Lie algebra endowed with a LCS structure $(\omega', \theta')$. Let $\u$ be a non degenerate ideal, and we consider the complement $\u^\perp$ given by $\u^\perp=\{X\in \g: \omega'(X,U)=0, \forall U\in\u\}$. Since $\u$ is non degenerate we have that $\g=\u^\perp\oplus\u$ as vector spaces.
\[converse\] Let $(\g,\omega',\theta')$ be a LCS Lie algebra admitting a non degenerate ideal $\u$:
(i) if $\u\subset\ker\theta'$, then $\u^\perp$ is a subalgebra;
(ii) if, moreover, $\u$ is abelian, then $\ad: \u^\perp \to \operatorname{End}(\u)$ is a LCS representation and $(\g,\omega',\theta')$ is isomorphic to $(\u^\perp\ltimes\u,\tilde\omega,\tilde\theta)$ with the LCS structure of Theorem \[pi=tita-ro\]. In particular, $(\omega',\theta')$ is a LCS structure is of the second kind.
${{\rm (i)}}$ We show $\u^\perp$ is a subalgebra, that is $\omega'([X,Y],U)=0$ for all $U\in \u$ and $X,Y\in \u^\perp$. We compute $$\begin{aligned}
\omega'([X,Y],U) & = -d\omega'(X,Y,U) -\omega'([Y,U],X)-\omega'(U,[X,Y])\\ & = -\theta'(X)\omega'(Y,U)-\theta'(Y)\omega'(U,X)-\theta'(U)\omega'(X,Y)\\
&=0,
\end{aligned}$$ where we used the LCS condition and the fact that $\u\subset\ker\theta'$, thus ${{\rm (i)}}$ is proved.
${{\rm (ii)}}$ Let $(\omega,\theta)$ be the restriction of $(\omega',\theta')$ to $\u^\perp$. It is clear that $\omega$ is non degenerate on $\u^\perp$ and $\theta\neq0$ since $\u\subset\ker\theta'$ and $\theta'\neq0$. Therefore, $(\omega,\theta)$ satisfies the LCS condition on the subalgebra $\u^\perp$.
We can decompose $\g$ as a semidirect product $\g=\u^\perp\ltimes\u$, and we denote the restriction of $\omega'$ to the non degenerate abelian ideal $\u$ by $\omega_0$. Then, it is clear that $\omega_0$ and $\ad: \u^\perp \to \operatorname{End}(\u)$ satisfy condition , since $(\omega',\theta')$ is exactly the LCS structure $(\tilde\omega,\tilde\theta)$ constructed in Theorem \[pi=tita-ro\] with initial data $(\u^\perp, \omega, \theta)$ and $(\u, \omega_0)$.
Therefore, it follows from Theorem \[pi=tita-ro\] that $\ad: \u^\perp \to \operatorname{End}(\u)$ is a LCS representation and $(\g,\omega',\theta')$ is isomorphic to $(\u^\perp\ltimes\u,\tilde\omega,\tilde\theta)$. Moreover, $(\omega',\theta')$ is a LCS structure of the seconk kind on $\g$.
Using Theorem \[converse\] it is easy now to determine whether a given Lie algebra endowed with a LCS structure can be constructed by Theorem \[pi=tita-ro\]. This gives us a nice characterization of LCS Lie algebras admitting a non degenerate abelian ideal contained in the kernel of the Lee form. One can see Theorem \[converse\] as a kind of reduction of the LCS condition.
This construction has some similar ideas to the ones in [@ABP Proposition 1.17] since both are related with special abelian ideals contained in the kernel of the Lee form. In our work we look for a non degenerate abelian ideal instead of a Lagrangian abelian ideal.
Center of LCS Lie algebras of the second kind
---------------------------------------------
The center of a Lie algebra with a LCS structure was studied in [@ABP], in particular the authors characterized the center of a nilpotent LCS Lie algebra and they proved that the dimension of the center is at most $2$. Note that the nilpotency condition implies that the LCS structure is of the first kind. On the other hand, as a consequence of it is easy to verify that there is no restriction for the dimension of the center of a Lie algebra admitting a LCS structure of the second kind, as we show in the following example.
Consider the $4$-dimensional Lie algebra $\mathfrak r\mathfrak r_{3,\lambda}$ with structure constants $(0,-12,-\lambda 13,0)$. It means that we fix a coframe $\{e^1,e^2,e^3,e^4\}$ for $(\mathfrak r\mathfrak r_{3,\lambda})^*$ such that $de^1=0$, $de^2=-e^1\wedge e^2$, $de^3=-\lambda e^1\wedge e^3$ and $de^4=0$. According to [@ABP] this Lie algebra admits a LCS structure given by $\omega=e^{12}+ e^{34}$ with Lee form $\theta=-\lambda e^1$, where $\{e^1,e^2,e^3,e^4\}$ denotes the dual coframe. Consider now the $(2n+4)$-dimensional Lie algebra $\g=\mathfrak r\mathfrak r_{3,\lambda}\ltimes_\pi\R^{2n}$ where $\pi(e_i)=0$ for $i=2,3,4$ and $\pi(e_1)\in M(2n,\R)$ is given by $$\pi(e_1)=\begin{pmatrix}
\frac\lambda 2\I_{n\times n} & \\
&\frac\lambda 2\I_{n\times n}\\
\end{pmatrix}+
\begin{pmatrix}
\frac\lambda 2\I_{n\times n} & \\
&-\frac\lambda 2\I_{n\times n}\\
\end{pmatrix}
=\begin{pmatrix}
\lambda \I_{n\times n} & \\
&\operatorname{0}_{n\times n}\\
\end{pmatrix},$$ in a basis $\{u_1,\dots,u_n,v_1,\dots,v_n\}$ of $\R^{2n}$. More precisely, the Lie brackets on $\g$ are: $$[e_1,e_2]=e_2, \quad [e_1,e_3]=\lambda e_3, \quad [e_1,u_k]=\lambda u_k,$$ for $k=1,\dots,n$. Then $e_4\in\mathfrak z(\g)$ and $v_i \in\mathfrak z(\g)$ for $i=1,\dots,n$, therefore the dimension of $\mathfrak z(\g)$ is $n+1$. Moreover, it can be seen that $\pi$ is a LCS representation, hence according to it determines a LCS structure on the Lie algebra $\g$ given by $\tilde\omega=e^{12}+ e^{34} + \sum_{i=1}u^i\wedge v^i$ with Lee form $\tilde\theta=-\lambda e^1$.
Unimodular LCS Lie algebras of the second kind
----------------------------------------------
Since we are interested in finding examples of solvmanifolds equipped with a LCS structure of the second kind, we determine next when a Lie algebra $\g$ built in Theorem \[pi=tita-ro\] is unimodular.
\[unimodularLCS\] Let $\h$ be a Lie algebra with a LCS structure $(\omega,\theta)$, $(\pi,V)$ a $2n$-dimensional LCS representation of $\h$ and $\g=\h\ltimes_\pi V$ the Lie algebra with LCS structure $(\tilde\omega,\tilde\theta)$ built as above. Then $\g$ is unimodular if and only if $\tr(\ad_X^\h)=n\theta(X)$ for any $X\in\h$.
Given $X\in\h$, the operator $\ad_X^\g:\g\to\g$ can be written as $$\ad_X^\g=
\left(\begin{array}{c|c}
\ad_X^\h & \\
\hline
& \pi(X) \\
\end{array}\right),$$ for some bases of $\h$ and $V$. Then we have that $\g$ is unimodular if and only if $\tr(\pi(X))=-\tr(\ad_X^\h)$ for all $X\in\h$, and using the characterization in this happens if and only if $$\tr(\ad_X^\h)= -\tr\left(-\frac12\theta(X)\I+\rho(X)\right)=n\theta(X),$$ for any $X\in\h$.
In particular we have the following corollary, which will be used later:
Let $\h$ be a Lie algebra with a LCS structure $(\omega,\theta)$. If there exists $n \in \N$ such that $$\label{unimodular-condition}
\tr(\ad_X^\h)=n\theta(X)$$for all $X\in\h$, then for any LCS representation $(\pi, V)$ with $\dim V=2n$, the LCS Lie algebra $\g=\h\ltimes_\pi V$ is unimodular.
Therefore the method of Theorem \[pi=tita-ro\] together with condition allow us to build unimodular Lie algebras admitting a LCS structure of the second kind starting from a non unimodular Lie algebra with a LCS structure.
Examples of Lie algebras with LCS structures of the second kind
===============================================================
In this section we show first that our construction is quite general. Indeed, we can reobtain with this construction most of the known examples of Lie algebras admitting a LCS structure of the second kind. More precisely, we see that every unimodular $4$-dimensional Lie algebra admitting a LCS structure of the second kind has a non degenerate abelian ideal contained in the kernel of the Lee form, and then by Theorem \[converse\] they can be obtained with our construction. We also use Theorem \[pi=tita-ro\] to construct new examples of unimodular Lie algebras admitting LCS structures of the second kind in higher dimension.
Dimension $4$
-------------
We start by recalling in Table \[4uniLCS\] the unimodular $4$-dimensional Lie algebras admitting a LCS structure of the second kind (see [@ABP]).
We explain in details the first case in Table \[4uniLCS\]. Let $\g=\mathfrak r\mathfrak r_{3,-1}$ with the LCS structure $\omega=e^{12}+e^{34}, \theta= e^1$. It is clear that $\u=\text{span}\{e_3,e_4\}$ is a non degenerate abelian ideal of $\g$. It follows from Theorem \[converse\] that $\g=\u^\perp\ltimes\u$ where $\u^\perp=\text{span}\{e_1,e_2\}$ is isomorphic to the $2$-dimensional non abelian Lie algebra $\mathfrak{aff}(\R)$. The LCS structure on $\u^\perp$ is given by $\omega=e^{12}, \theta= e^1$, note that since $\dim\u^\perp=2$, this structure is in fact a symplectic structure. Therefore $\mathfrak r\mathfrak r_{3,-1}\simeq\mathfrak{aff}(\R)\ltimes_{\pi_1} \R^{2}$ with $\pi_1(e_1)=\operatorname{diag}(-1,0)=\operatorname{diag}(-\frac12,-\frac12)+\operatorname{diag}(-\frac12,\frac12)$ and $\pi_1(e_2)=0$. Clearly, $\pi_1$ is an LCS representation.
It can be easily seen that for any of the Lie algebras and any of the LCS structures of Table \[4uniLCS\] we can proceed in the same way with the exception of $\mathfrak d_4$ with the LCS structure $\omega=e^{12}-e^{34}+e^{24}$ and $\theta= e^4$. Indeed, this is the only LCS structure of the second kind on a $4$-dimensional unimodular Lie algebra which does not satisfy the condition of Theorem \[converse\], that is, it does not have a non degenerate abelian ideal contained in $\ker\theta$.
To summarize we have the following result:
Any unimodular $4$-dimensional LCS Lie algebra of the second kind, with the only exception of $(\mathfrak d_4, \omega=e^{12}-e^{34}+e^{24}, \theta=e^4)$, can be reobtained by Theorem \[pi=tita-ro\] for a suitable representation.
According to [@ABP] the simply connected Lie groups associated with the Lie algebras of Table \[4uniLCS\] (for a countable set of parameters $\alpha$ and $\delta$) admit lattices (see also [@B]). Then we have that:
Any unimodular $4$-dimensional LCS Lie algebra of the second kind (except for $(\mathfrak d_4, e^{12}-e^{34}+e^{24}, e^4)$) associated with a compact solvmanifold can be obtained by Theorem \[pi=tita-ro\] for a suitable representation.
Moreover, it follows from Theorem \[2tipo\_cociente\] that the induced LCS structures on any quotient are LCS structures of the second kind.
Higher dimension
----------------
We focus now on provide examples in higher dimension. In order to build examples of unimodular Lie algebras admitting LCS structures we need to start with a non unimodular Lie algebra with a LCS structure in lower dimension. In [@ABP] the authors classify the $4$-dimensional solvable Lie algebras admitting LCS structures up to automorphism of the Lie algebra. Using their classification we show in Table \[extensibles\] all the $4$-dimensional solvable Lie algebras with their associated LCS structure satisfying condition , which means that they can be extended to a higher dimensional unimodular Lie algebra admitting a LCS structure given by .
We explain in details how to extend one example.
\[Ejemplo r’2\] Let $\mathfrak r'_2$ be the Lie algebra with structure constants $(0,0,-13+24,-14-23)$. According to [@ABP] this Lie algebra admits 4 non equivalent LCS structures up to Lie algebra automorphism $$\left\{\begin{array}{l}
\theta = \sigma e^1+\tau e^2 \\
\omega = e^{13} - \tau e^{14} -\frac{1+\tau^2}{1+\sigma} e^{24} \\
\text{with } \sigma\neq -1,0, \quad \tau>0
\end{array}\right. \quad \quad \;\;
\left\{\begin{array}{l}
\theta = -2 e^1 \\
\omega = \sigma e^{12} + e^{34} \\
\text{with } \sigma\neq 0
\end{array}\right.$$ $$\quad \left\{\begin{array}{l}
\theta = \tau e^2 \\
\omega = e^{13} - \tau e^{14} -(1+\tau^2) e^{24} \\
\text{with } \tau>0
\end{array}\right. \quad
\left\{\begin{array}{l}
\theta = \sigma e^1 \\
\omega = e^{13} -\frac{1}{1+\sigma} e^{24} \\
\text{with } \sigma\neq -1,0
\end{array}\right.$$
We verify next if each LCS structure satisfies condition . If we consider the cases $\theta =\sigma e^1+\tau e^2 $, $\theta = \tau e^2$ or $\theta = -2 e^1$ the condition does not hold, then the possible LCS extension will not be unimodular. Finally we consider the LCS structure with Lee form $\theta = \sigma e^1$ and $\sigma\neq -1,0$. In this case is satisfied for $2=n\sigma$. Therefore for any $n\in \N$ and $\sigma=\frac 2n$, the LCS structure on $\mathfrak r'_2$ given by $$\left\{\begin{array}{l}
\theta = \sigma e^1 \\
\omega = e^{13} -\frac{1}{1+\sigma} e^{24} \\
\text{with } \sigma\neq -1,0
\end{array}\right.$$ can be extended to the $(2n+4)$-dimensional unimodular Lie algebra $$\g=\mathfrak r'_2\ltimes_\pi \R^{2n},$$ where $\pi$ is a suitable LCS representation. It follows from that $\g$ admits a LCS structure $(\tilde\omega,\tilde\theta)$ given by $\tilde\omega|_\h=\omega$, $\tilde\omega|_{\R^{2n}}=\omega_0$, $\tilde\theta|_\h=\theta$ and $\tilde\theta|_{\R^{2n}}=0$, where $\omega_0$ is any symplectic form on $\R^{2n}$. To be more specific we consider $n=2$, then $\sigma=1$. Let $\pi:\mathfrak r'_2\to\mathfrak{gl}(4,\R)$ be the representation given by $$\pi(e_1)=\begin{pmatrix}
0 &&& \\
&-1&&\\
&&-1&\\
&&&0
\end{pmatrix}=-\frac12\I+\begin{pmatrix}
\frac12 &&& \\
&-\frac12&&\\
&&-\frac12&\\
&&&\frac12
\end{pmatrix},$$ and $\pi(e_2)=\pi(e_3)=\pi(e_4)=0$, in a basis $\{e_5,e_6,e_7,e_8\}$ of $\R^4$ with $\omega_0=e^{56}+e^{78}$. It is easy to see that $\pi$ is a LCS representation. Then the $8$-dimensional Lie algebra $\g=\mathfrak r'_2\ltimes_\pi \R^4$ has the following Lie brackets $$[e_1,e_3]=e_3, \quad [e_2,e_3]=e_4, \quad [e_1,e_6]=-e_6,$$ $$[e_1,e_4]=e_4, \quad [e_2,e_4]=-e_3, \quad [e_1,e_7]=-e_7,$$ and the LCS structure is given by $$\left\{\begin{array}{l}
\tilde\omega=e^{13}-\frac{1}{2}e^{24}+e^{56}+e^{78}\\
\tilde\theta= e^1.
\end{array}\right.$$
Note that this example is not covered by the construction given in [@ABP Proposition 1.8].
Examples of solvmanifolds with LCS structures of the second kind
================================================================
In the section we use Theorem \[pi=tita-ro\] and Theorem \[2tipo\_cociente\] to construct a family of solvmanifolds $\Gamma_m\backslash G$ admitting a LCS structure of the second kind, where $G$ is the simply connected Lie group associated to the Lie algebra considered in the Example \[Ejemplo r’2\]. This Lie algebra can be decomposed as $\g=\R e_2\ltimes\R e_1\ltimes \R^{6}$ where the adjoint actions of $e_1$ and $e_2$ are given by $$\ad_{e_1}=\begin{pmatrix}
1 &&&&& \\
&-1&&&&\\
&& 0&&&\\
&&&1 && \\
&&&&-1&\\
&&&&&0
\end{pmatrix}, \quad
\ad_{e_2}=\begin{pmatrix}
&&&& 0&&&\\
&&&&-1&&&\\
&&&& 0&&&\\
&&&& 0&&& \\
0&1&0& 0&0&0&0&\\
&&&& 0&&& \\
&&&& 0&&&
\end{pmatrix}$$ in the reordered bases $\{e_3,e_6,e_5,e_4,e_7,e_8\}$ and $\{e_1,e_3,e_6,e_5,e_4,e_7,e_8\}$ respectively. The simply connected Lie group associated to $\g$ is $$G=\R e_2\ltimes_\psi(\R e_1\ltimes_\varphi \R^{6}),$$ where $\varphi(t)=\exp(t\ad_{e_1})$ and $\psi(t)=\exp(t\ad_{e_2})$. Next we build a lattice in $G$, to do this we start considering the Lie subgroup $H:=\R e_1\ltimes_\varphi \R^{6}$. Note that $H$ is an almost abelian Lie group. According to [@B] the Lie group $H$ admits a lattice if and only if there exists $t_0\in\R$, $t_0\neq0$, such that $\varphi(t_0)$ is conjugated to an integer matrix.
We can write the matrix $\varphi(t)$ in the basis $\{e_3,e_6,e_5,e_4,e_7,e_8\}$ as $$\varphi(t)=\begin{pmatrix}
e^t &&&&& \\
&e^{-t} &&&&\\
&& 1&&&\\
&&&e^t && \\
&&&&e^{-t} &\\
&&&&&1
\end{pmatrix}.$$ We consider only the block $$M=\begin{pmatrix}
e^t && \\
&e^{-t} &\\
&& 1
\end{pmatrix}.$$ The characteristic polynomial of the matrix $M$ is $$p(x)=(x-1)(x-e^t)(x-e^{-t}).$$ Fixing $m\in\mathbb{N}$, $m>2$, we define $t_m=\operatorname {arccosh}(\frac{m}{2})$, $t_m>0$, and we have that $(x-e^{t_m})(x-e^{-t_m})=x^2-mx+1\in\Z[x]$. Then the characteristic polynomial of $M$ for $t=t_m$ can be written as $p(x)=x^3-(m+1)x^2+(m+1)x-1$. Therefore, it is easy to see that $M$ is conjugated to the companion matrix $C_m$ of the polynomial $p$, that is, $M=Q_mC_mQ_m^{-1}$ where $$C_m=\begin{pmatrix}
0&0&1\\
1&0&-(1+m) \\
0&1&1+m
\end{pmatrix} \quad \text{and} \quad
Q_m=\begin{pmatrix}
1&e^{t_m}&e^{2t_m}\\
1&e^{-t_m}&e^{-2t_m} \\
1&1&1
\end{pmatrix}.$$ We can copy this process in the second block of $\varphi(t)$ and we easily can check that $$\varphi(t_m)=P_mD_mP_m^{-1},$$ where $$D_m=\begin{pmatrix}
C_m&0\\
0&C_m
\end{pmatrix}\quad \text{and}\quad
P_m=\begin{pmatrix}
Q_m&0\\
0&Q_m
\end{pmatrix}.$$ Therefore $\varphi(t_m)$ is conjugate to the integer matrix $D_m$. It follows from [@B] that $H$ admits a lattice $$\Gamma_m=t_m\Z\ltimes P_m\Z^6.$$
We now write the matrix $\psi(t)$ in the basis $\{e_1,e_3,e_6,e_5,e_4,e_7,e_8\}$ as $$\psi(t)=\begin{pmatrix}
1 &0&0&0&0&0&0 \\
0&\cos(t) &0&0&-\sin(t)&0&0 \\
0&0&1&0&0&0&0\\
0&0&0&1&0&0&0\\
0&\sin(t) &0&0&\cos(t)&0&0 \\
0&0&0&0&0&1&0\\
0&0&0&0&0&0&1\\
\end{pmatrix}.$$ Clearly, $\psi(2\pi)$ preserves the lattice $\Gamma_m$. Thus $$\Lambda_m=2\pi\Z\ltimes\Gamma_m=2\pi\Z\ltimes (t_m\Z\ltimes P_m\Z^6)$$ is a lattice in $G$ for any $m>2$. Therefore we obtain an explicit construction of examples of solvmanifolds $\Lambda_m\backslash G$ admitting a LCS structure of the second kind.
Since $\psi(2\pi)=\I$, we have that $\Lambda_m\simeq 2\pi\Z\times\Gamma_m$, and therefore it can be considered as a lattice in $G'=\R\times H$. Then, according to [@R Theorem 3.6] the corresponding solvmanifolds $M_m:=\Lambda_m\backslash G$ and $M_m':=\Lambda_m\backslash G'$ are diffeomorphic. Note that $M'$ is diffeomorphic to the product of $S^1$ and the solvmanifold $\Gamma_m\backslash H$. We also note that $G'$ can be seen as an almost abelian Lie group, more explicity, $G'=\R \ltimes_\rho \R^7$, where $$\rho(t)=\begin{pmatrix}
1 &&&&& \\
&e^{t} &&&&& \\
&&e^{-t} &&&&\\
&&& 1&&&\\
&&&&e^{t} && \\
&&&&&e^{-t} &\\
&&&&&&1
\end{pmatrix}.$$ It is easy to see that $\rho(t_m)$ is conjugated to the integer matrix $$R_m=\begin{pmatrix}
1&0\\
0&P_m
\end{pmatrix}.$$ Using this identification between $M_m$ and $M'_m$ we can prove that:
The solvmanifolds $\Lambda_m\backslash G$ are pairwise non homeomorphic.
We assume that $M_m$ and $M_n$ are homeomorphic, then $M'_m$ and $M'_n$ are homeomorphic as well, and therefore their fundamental groups $\pi_1(M'_m)$ and $\pi_1(M'_n)$ are isomorphic. Since $G'$ is simply connected, we have that these fundamental groups are isomorphic to the lattices, and therefore $\Lambda_m\cong\Lambda_n$. Since $G'$ is completely solvable, we can use the Saito’s rigidity theorem [@Sai] to extend this isomorphism to an automorphism of $G'$. Since the lattices differ by an automorphism of $G'$, it follows from [@Hu Theorem 3.6] that the integer matrix $R_m$ is conjugated either to $R_n$ or to $R^{-1}_n$. Finally comparing the eigenvalues of $R_m$ and $R_n$ we obtain that this happens if and only if $m=n$.
We study now the de Rham and the Morse-Novikov cohomology of the solvmanifolds $\Lambda_m\backslash G\cong\Lambda_m\backslash G'$. We note first that the Lie algebra $\g'$ of $G'$ is given by $\g'=\R e_1\ltimes \R^{7}$ with $$\ad_{e_1}=\begin{pmatrix}
0 & &&&&&\\
&1 &&&&& \\
&&-1&&&&\\
&&& 0&&&\\
&&&&1 && \\
&&&&&-1&\\
&&&&&&0
\end{pmatrix}$$ in the basis $\{e_2,e_3,e_6,e_5,e_4,e_7,e_8\}$ of $\R^{7}$. Therefore the Lie group $G'$, unlike $G$, is completely solvable, that is, $\ad_X$ has real eigenvalues for all $X \in \g'$. Since $G'$ is completely solvable these cohomologies can be computed in terms of the Lie algebra cohomology. Indeed, it follows from [@Hat] (see also [@AO1]) that $H^*_{dR}(\Lambda_m\backslash G')\cong H^*(\g') $ and $H^*_\theta(\Lambda_m\backslash G')\cong H^*_\theta(\g')$ for any $m\in\N$.
According to [@Sa], the $k^{th}$ Betti number of $\g'$, $\beta_k=\dim H^k(\g')$, can be computed in terms of the dimension of $Z^j(\g')=\{\alpha\in\alt^j \g^* : d\alpha=0\}$ as follows: $$\label{betti}
\beta_k=\dim H^k(\g')=\dim Z^k(\g')+\dim Z^{k-1}(\g')-\binom{8}{k-1},$$ for $k>2$. Note that $\beta_0=1$ and $\beta_1=\dim(\g'/[\g',\g'])=4$. Using equation it can be seen that $\beta_2=10$, $\beta_3=20$ and $\beta_4=26$. Finally, due to Poincaré duality, we obtain that $\beta_5=20$, $\beta_6=10$, $\beta_7=4$ and $\beta_8=1$.
For the Morse-Novikov cohomology, the corresponding Betti numbers $\beta^\theta_k=\dim H_\theta^k(\g')$ satisfy a similar equation. Indeed, setting $Z^j_{\theta}(\g)=\{\alpha\in\alt^j \g^* : d_{\theta}\alpha=0\}$ we have $$\label{bettitilde}
\beta^ \theta_k=\dim H^k_{\theta}(\g')=\dim Z^k_{\theta}(\g')+\dim
Z^{k-1}_{\theta}(\g')-\binom{8}{k-1},$$ for $k>2$. It is easy to see that $\beta^ \theta_0=0$ and $\beta^ \theta_1=2$. Then, after some computations and using , it can be seen that $\beta^ \theta_2=8$, $\beta^ \theta_3=14$, $\beta^ \theta_4=16$, $\beta^ \theta_5=14$, $\beta^ \theta_6=8$, $\beta^ \theta_2=2$ and $\beta^ \theta_8=0$.
[*Acknowledgement.*]{} I would like to thank A. Andrada for a careful reading of the first draft of the paper and useful remarks to improve the final version. Also thanks to D. Angella for his useful comments during his stay at Universidad Nacional de Córdoba. Finally, I would like to thanks I. Dotti for her useful suggestions to improve the exposition of this paper.
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abstract: 'A light collective $\theta^+$ baryon state (with strangeness +1) was predicted via rigid-rotor collective quantization of SU(3) chiral soliton models. This paper explores the validity of this treatment. A number of rather general analyses suggest that predictions of exotic baryon properties based on this approximation do not follow from large $N_c$ QCD. These include an analysis of the baryon’s width, a comparison of the predictions with general large $N_c$ consistency conditions of the Gervais-Sakita-Dashen-Manohar type; an application of the technique to QCD in the limit where the quarks are heavy; a comparison of this method with the vibration approach of Callan and Klebanov; and the $1/N_c$ scaling of the excitation energy. It is suggested that the origin of the problem lies in an implicit assumption in the that the collective motion is orthogonal to vibrational motion. While true for non-exotic motion, the Wess-Zumino term induces mixing at leading order between collective and vibrational motion with exotic quantum numbers. This suggests that successful phenomenological predictions of $\theta^+$ properties based on rigid-rotor quantization were accidental.'
author:
- 'Thomas D. Cohen'
title: 'The $\theta^+$ baryon in soliton models: large $N_c$ QCD and the validity of rigid-rotor quantization'
---
introduction\[Intro\]
=====================
A narrow baryon resonance with a strangeness of +1 ([*i.e.*]{} containing one excess strange antiquark) has recently been identified by a number of experimental groups[@exp]. Such a state is exciting: it is unambiguously exotic in the sense that it cannot be a simple three-quark state. These experiments have spurred considerable theoretical activity. Much of this activity has been aimed at understanding the structure of this exotic state. The most common treatment of this problem has been based on variants of the quark model; the new baryon is identified as a pentaquark[@qm]. Other approaches treat the $\theta^+$ in terms of meson-baryon binding[@IKOR; @KK] or as a kaon-pion nucleon state[@LOM] or are based on on QCD sum-rules [@sumrules]. While these approaches are all interesting, they are are also all highly model dependent and it is difficult to assess in an [*a priori*]{} way their validity. A discussion of some the issues raised by various models may be found in [@JM].
The analysis based on the SU(3) chiral soliton model treated with rigid-rotor quantization [@Pres; @DiaPetPoly; @WalKop; @BorFabKob; @Kim] appears different from other treatments of the $\theta^+$ structure in a number of ways : i) The approach was used to predict the state and its properties [@Pres; @DiaPetPoly]. In contrast most of the other treatments were subsequent to the experimental discoveries. ii) The predictions of the mass were very accurate[@Pres; @DiaPetPoly]; the width was predicted to be narrow[@DiaPetPoly], which is consistent with the widths presently observed [@Nus]. iii) The prediction of the mass is totally insensitive to the dynamical details of the model. The analysis holds regardless of the detailed shape of the profile function that emerges from the dynamics.
While the first two points are certainly of interest, the third point is of far greater significance from the point of view of theory. It was noticed quite some time ago by Adkins and Nappi that a number of relationships between observables in chiral soliton models depend on the structure of the models and their symmetries, but do not depend on the dynamical details such as the parameters in the lagrangian [@AdkNap]. Moreover, these relations were derived in a fully model-independent way directly from large $N_c$ using consistency relations[@GS; @DM; @Jenk; @DJM].
In light of this, it is natural to conjecture that the relationships between masses underlying the analysis in refs. [@Pres; @DiaPetPoly; @WalKop; @BorFabKob; @Kim] which do not depend on the dynamical details of the model are similarly model-independent consequences of large $N_c$ QCD. If this is true, it greatly aids in our understanding of the $\theta^+$: modulo corrections due to $1/N_c$ effects of higher order effects in SU(3), the $\theta^+$ seems essentially understood. However, it remains quite controversial as to whether this is indeed the case.
In a previous paper[@Coh03], it was argued that the prediction of properties of exotic states based on rigid-rotor quantization are [*not*]{} generic predictions of large $N_c$ QCD. Before discussing the content of this paper, a linguistic issue should be addressed to avoid possible confusion. The analysis is done in a $1/N_c$ expansion so one must be able to consider systems with varying $N_c$. Accordingly throughout this paper the phrases “exotic state” or “exotic baryon” refer to states which are exotic for the given value of $N_c$ which is relevant (and not necessarily for $N_c=3$); [*i.e.,*]{} those baryons which cannot be described in a quark model with $N_c$ quarks. Clearly states with positive strangeness are exotic for any $N_c$.
Two basic arguments were presented in ref. [@Coh03] which suggest that the rigid-rotor quantization is not valid for exotic states. The first was based on large $N_c$ consistency rules of the sort discussed above. [*All*]{} of the previously known model independent properties of baryons were derived from such relations. However, such relations do not predict the existence of collective exotic states at large $N_c$. The only collective states predicted are precisely those with the same quantum numbers as those which emerge in a large $N_c$ quark model with all quarks in the lowest s-wave orbital. The second argument was that the excitation energies of the exotic state of order $N_c^0$ is inconsistent with the type of scale separation needed to justify collective quantization.
Itzhaki, Klebanov, Ouyang and Rastelli (IKOR)[@IKOR] reached similar conclusions about the validity of the rigid rotor approximation for exotic states from a rather different starting point: the treatment of SU(3) symmetry breaking. It is clear that for large symmetry breaking fluctuations into the strangeness direction are of a vibrational character and the appropriate formalism to describe these are the Callan-Klebanov “bound state” approach[@CK]. For the case of non-exotic baryons it can be shown that as the SU(3) symmetry breaking goes to zero, the predicted energies and spatial distributions in the Callan-Klebanov approach match those of the rigid-rotor quantization[@CK]; the two approaches are compatible for non-exotic states in the regime where both should work. In contrast, at small but nonzero SU(3) symmetry breaking the excitation energy for exotic states such as the $\theta^+$ as calculated via the rigid-rotor treatment does not match smoothly onto the Callan-Klebanov value. IKOR took this as evidence that rigid-rotor treatment, while valid for non-exotic states, gives spurious results for exotic states.
Pobylitsa[@Pob] also concluded that rigid-rotor quantization failed for exotic states based on a study of an exactly solvable toy model of the Lipkin-Meshkov-Glick type [@LMG].
Despite the arguments in refs. [@Coh03; @IKOR; @Pob], it is not universally accepted that rigid-rotor collective quantization of chiral soliton models is invalid for exotic baryon properties. Diakonov and Petrov[@DP] have recently argued that that rigid-rotor collective quantization is accurate for such states up to $1/N_c$ corrections, and have specifically attempted to rebut the argument in ref. [@Coh03]. Both a general analysis as to why the authors believe the argument of ref. [@Coh03] is incorrect and a toy model to illustrate the point are given. However, as will be discussed below, the general arguments are flawed and the toy model is not anlagous to the problem of interest.
It is hoped that the present paper will end help this controversy by giving convincing evidence that the rigid-rotor collective quantization is not valid at large $N_c$. Given this goal, it is important to state precisely what is meant by this claim so as to minimize possible misunderstandings. The claim is formal. If rigid-rotor quantization were valid at large $N_c$, then predictions based on it would become arbitrarily accurate as $N_c
\rightarrow \infty$ ([*i.e.,*]{} there are no corrections due to other effects which survive at large $N_c$). Arguments are given here that this is false: even at large $N_c$ there are nonzero corrections. This is in contrast to properties of non-exotic baryons for which calculations based on rigid-rotor quantization do become arbitrarily accurate.
It is important to make quite clear the fundamental nature of the difficulty with the rigid-rotor collective quantization. The issue is not related to whether this treatment based on large $N_c$ analysis is a good approximation to the physical world of $N_c=3$, but to the nature of exotic states in the large $N_c$ world.
This paper presents evidence for the invalidity of the rigid-rotor quantization for exotic baryons. Several general arguments are presented showing that the rigid-rotor quantization leads to predictions of exotic baryon properties which are inconsistent with known large $N_c$ results. These general arguments strongly suggest an inconsistency between various predictions of exotic baryon properties at large $N_c$ as based on rigid-rotor quantization and the known behavior at large $N_c$. If correct, they imply a flaw in the original derivation of the rigid-rotor collective quantization of refs. [@SU3Quant] for these exotic states. Such a flaw must give spurious results for exotic states while being valid for non-exotic states. An analysis of the derivation indicates where such a flaw may lie. The derivation implicitly assumes the orthogonality of collective and vibrational motion. However, the Wess-Zumino term induces a coupling between vibrational modes and the collective motion associated with exotic excitations which spoils this orthogonality.
This paper is organized as follows: In the following section the salient features of the treatment of exotic baryons based on the rigid-rotor collective quantization[@Pres; @DiaPetPoly; @WalKop; @BorFabKob; @Kim] is briefly presented. The next section discusses the large $N_c$ scaling behavior of the mass splittings between exotic states and the ground state. Following this, a series of arguments arguments are presented that indicate that the collective quantization treatment yields results in conflict to what one expects from general large $N_c$ considerations. A section detailing a possible flaw in the original derivation of rigid-rotor quantization as applied to exotic states is next, followed by a discussion of the claims of ref. [@DP] which purports to demonstrate the validity of rigid rotor. The final section discusses the significance of these results in light of the recent experimental reports of exotic s=1 baryons.
Rigid-rotor predictions for exotic baryons \[Rigid\]
====================================================
There are a number of important assumptions which go into the predictions of exotic baryon states in refs. [@Pres; @DiaPetPoly; @WalKop; @BorFabKob; @Kim]. These include the assumption that low-order perturbation theory in the strange quark mass is justified for real world values, the assumption that $1/N_c$ expansion truncated at low order is justified for these observables for $N_c=3$ as well as the assumption that rigid-rotor quantization is valid for exotic states. This paper focuses on the issue of the validity of rigid-rotor quantization. It is worth bearing in mind, however, that these other assumptions are not totally innocuous. For example, prior to the experimental observation of the $\theta^+$, Weigel[@Weigel] observed that effects which were higher-order in the strange quark mass induced mixing between the vibrational and collective modes which had nontrivial effects on predictions of the properties of the exotic states. However, the central questions of principle addressed in this paper are seen at leading order in $1/N_c$ and in the exact SU(3) limit and we review the leading order treatment below.
The analysis is based on a standard treatment of SU(3) chiral soliton models developed in the mid-1980s[@SU3Quant]. The starting point in the analysis is a classical static “hedgehog” configuration in an SU(2) subspace (which for convenience one may take to be the u-d subspace). The profile function of this hedgehog is obtained by minimizing the action for a static configuration subject to the constraint that the baryon number (which is taken to be the topological winding number for the chiral field) is unity. The detailed shape of the profile function depends on the model—the types of couplings included in the values of the parameters, and so on. However, the general structure of the theory is completely model independent. As noted above, for the present purpose it is sufficient to consider the exact SU(3) symmetric limit of the theory. In the absence of symmetry breaking effects there are eight collective (rotational) variables. That is, there are eight flat directions in which one can rotate to the classical configuration to obtain a new classical configuration which also corresponds to static solutions of the classical equations of motion.
These collective variables are then quantized semi-classically using an SU(3) generalization[@SU3Quant] of the usual SU(2) collective quantization scheme[@ANW]. If the classical collective motion is fully decoupled from the internal motion of the hedgehog shape, then one can quantize the two motions separately. Assuming this to be true, the dynamics of the collective variables are expressed in terms of time-dependent SU(3) rotations on the hedgehog shape $$U(\vec{x},t) = R(t) U_0(\vec{x}) R^{\dagger}(t) \label{ut}$$ where $R(t)$ is a global (space-independent) time-dependent SU(3) rotation, and $U_0$ is the hedgehog solution. Inserting this equation into the full lagrangian yields a collective lagrangian whose variables are $R$ and $\dot{R}$. By a standard Legendre transformation this can be converted into a collective Hamiltonian. This collective Hamiltonian is given by $$H_{\rm rot} = M_0+ \frac{1}{2 I_1} \sum_{A=1}^3 {\hat{J}_A'}{}^2
\, + \, \frac{1}{2 I_2} \sum_{A=4}^7 {\hat{J}_A'}{}^2 \; ,
\label{collective}$$ where $M_0$ is the mass of the static soliton and $I_1$ ($I_2$) is the moment of inertia within (out of) the SU(2) subspace, and $\hat{J}_A'$ are generators of SU(3) in a body-fixed (co-rotating) frame. The moments of inertia are computed in the standard way: $$\begin{aligned}
\hat{J}_A' &=& I_1 \dot{\theta}_A' \; \; \; {\rm for} \; \; \; A=1,2,3
\nonumber\\
\hat{J}_A' &=& I_2 \dot{\theta}_A' \; \; \; {\rm for} \; \; \;
A=4,5,6,7\; .\end{aligned}$$ The numerical values of the moments of inertia are model dependent but the structure of the collective Hamiltonian is not.
Before proceeding, a few comments on the generality of this procedure is in order. While it is easy to see that the precise procedure outlined above is only applicable to models in which the chiral field is the only degree of freedom, the ultimate result of any analysis valid at large $N_c$ will yield a collective Hamiltonian precisely of the form of eq. (\[collective\]) [*provided the assumption that the collective and vibrational modes decouple is valid*]{}. In particular, modifications to this procedure are needed for models which have first time derivatives of fields in the Lagrangian. For example, in models with explicit quark degrees of freedom, one has a nonzero quark contribution to the moments of inertia which can be computed via a standard “cranking” procedure borrowed from many-body physics[@crank], and introduced into chiral soliton physics in refs. [@CB1; @CB2]. The method was first used for the SU(3) quark-soliton mode by McGovern and Birse[@MB]. With such a treatment the quark contributions to the moments of inertia are given by $$\begin{aligned}
I_{1}^{\rm quark} &=& \frac{N_c}{2} \sum_{i} \frac{|\langle i |
\lambda^A | 0
\rangle|^2}{\epsilon_i -\epsilon_0} \; \; A=1,2,3 \nonumber\\
I_{2}^{\rm quark} &=& \frac{N_c}{2} \sum_{i} \frac{|\langle i | \lambda^A | 0
\rangle|^2}{\epsilon_i -\epsilon_0} \; \; A=4,5,6,7
\label{quarkcont} \end{aligned}$$ where $|i \rangle$ ($\epsilon_i$) are the single-particle quark eigenstates (eigenenergies) for quarks propagating in the static background of the hedgehog fields. The state $i=0$ corresponds to the quark ground state; the factor of $N_c$ reflects the fact that there are $N_c$ quarks in the system, each in the ground state in a mean-field treatment. However, for the present purpose the key point is that although the quantization requires a treatment somewhat more sophisticated than that used in eq. (\[ut\]), the sole effect of this added sophistication is a change in the numerical value of the moments of inertia; the structure of the collective Hamiltonian in eq. (\[collective\]) remains valid provided the central assumption that the collective and rotation degrees of freedom decouple is correct.
In quantizing the collective Hamiltoninan in eq. (\[collective\]), a constraint plays an essential role: $${J'_8}=-\frac{N_c B }{2\sqrt{3}} \; ,\label{quantcond}$$ where $B$ is the baryon number. In the context of Skyrme-type models this quantization condition is deduced from the topology of the Wess-Zumino term [@SU3Quant]. As noted by Witten[@Wit1], this constraint can be understood in analogy to the constraint on the body-fixed angular momentum which arises when quantizing a charged particle in the field of a magnetic monopole.
The masses which emerge from eqs. (\[collective\]) and (\[quantcond\]) may be found easily. Using the fact that $$\sum_{A=1}^8 (\hat{J}'_A){}^2 = \sum_{A=1}^8 (\hat{J}_A){}^2 = C_2$$ where $C_2$ is the quadratic Casimir operator and $\hat{J}_A$ is a generator in the space-fixed frame, one can rewrite the collective Hamiltonian as $$H_{\rm rot} = M_0 + \frac{1}{2 I_2} \sum_{A=1}^8 \hat{J}_A{}^2 \,
+ \, \frac{I_2-I_1}{2 I_1 I_2} \sum_{A=1}^3 \hat{J}_A'{}^2 -
\frac{1}{2 I_2} \hat{J}'_8{}^2 \; .$$ Equation (\[quantcond\]) can be used to replace the last term. Moreover, the intrinsic SU(2) subspace satisfies the usual SU(2) soliton rule that $I=J$. Together these relations allow one to express the eigenstates of $H_{\rm rot}$, [*i.e.,*]{} the physical masses: $$\begin{aligned}
M & = & M_0 + \frac{C_2}{2 I_2} + \frac{(I_2 -
I_1) J (J+1) }{2 I_1 I_2} - \frac{N_c^2}{24 I_2} \; , \nonumber \label{mass} \\
{\rm with} \; \; &C_2 & = \left( p^2 + q^2 + p q + 3(p +q)\right
)/3 \; ,
\end{aligned}$$ where $C_2$, the quadratic Casimir, is expressed in terms of the traditional labels $p,q$ which denote the SU(3) representation. The quantization condition in eq. (\[quantcond\]) greatly limits the possible SU(3) representations which can be associated with physical states: those SU(3) representations which do not contain states with hypercharge equal to $N_c /3$ are clearly unphysical: if the hypercharge in a body-fixed frame satisfies eq. (\[quantcond\]), then that representation will of necessity include a state with that hypercharge. Angular momentum also limits the physically allowed representations. In the body-fixed frame the SU(2) manifold has $I=J$ and $S=0$, which implies that the number of angular momentum states associated with representation, (2 J+1), must equal the number of states in the representation with $S=0$. This whole procedure is rigid-rotor collective quantization. The moments of inertia are treated as constants independent of the rotational state of the system and in that sense corresponds to a rigid rotor.
There is a practical issue about how one one chooses to implement this procedure. One natural approach would be to choose to quantize the theory at large $N_c$ and then to treat systematically all $1/N_c$ corrections. An alternative approach would be to fix $N_c=3$ at the outset when implementing the quantization condition of eq. (\[quantcond\]). If the approach is valid and if $N_c=3$ can be considered large, it ought not make any difference which of these approaches is used. The choice of taking $N_c=3$ at the outset has been the one typically made[@SU3Quant]. Making this choice, it is straightforward to see that the lowest-lying states are: $$\begin{aligned}
J=1/2 \; \; \; (p,q) &=& (1,1) \; \; \;({\rm octet}) \nonumber \\
J=3/2 \; \; \; (p,q) &=& (3,0) \; \; \;({\rm decuplet}) \nonumber \\
J=1/2 \; \; \; (p,q) &=& (0,3) \; \; \;({\rm anti-decuplet}) \;
.\label{multi}\end{aligned}$$ Equation (\[mass\]) can be used to find the mass splitting of the decuplet and the anti-decuplet relative to the octet: $$\begin{aligned}
M_{10} - M_{8} & = & \frac{3}{2 I_1} \; ,\label{10-8} \\
M_{\overline{10}} - M_{8} & = & \frac{3}{2 I_2} \; .
\label{10bar-8}\end{aligned}$$
The prediction of an anti-decuplet representation is at the heart of the issue. The anti-decuplet contains a state with s=+1 (which has been identified with the $\theta^+$). Such a state is necessarily exotic, even in the large $N_c$ limit.
In outlining the rigid-rotor quantization procedure, large $N_c$ QCD considerations appeared explicitly only when discussing the quantization constraint of eq. (\[quantcond\]). In fact, large $N_c$ considerations are at the core of the method and have been used implicitly throughout in two essential ways. In the first place, large $N_c$ is necessary for the justification of the classical static hedgehog configurations in an underlying quantum theory. Standard large $N_c$ scaling rules for couplings ensure that effects of quantum fluctuations around the hedgehogs are suppressed by $1/N_c$. Large $N_c$ also plays a central role in justifying the semi-classical treatment in the rigid rotor collective quantization which requires the decoupling of the collective motion from the vibrational motion around the static of the hedgehog. This is also suppressed by $1/N_c$, at least in certain situations. It should be clear from the previous comment, however, that the validity of the rigid-rotor collective approach depends on restricting its application to those modes which decouple from the vibrational ones. This issue is at the heart of the present paper.
Most treatments of SU(3) solitons identify the octet and decuplet states with the known $N_c=3$ octets and decuplets. Until fairly recently, the anti-decuplet was often assumed to be a large artifact of large $N_c$ QCD and hence ignored for essentially the same reason that I=J=5/2 baryons are ignored in SU(2) soliton models [@ANW]. The central point of ref. [@DiaPetPoly] is that the anti-decuplet should be taken seriously. The authors of ref. [@DiaPetPoly] distinguish the situation of anti-decuplet for SU(3) solitons from the J=I=5/2 baryons in SU(2) in terms of their widths. The J=I=5/2 baryon width is predicted to be so wide with real world parameters that the state can not be observed[@CohGri], while the computed $\theta^+$ width turns out to be quite small.
Large $N_c$ Scaling of Mass Splittings \[LargeN\]
=================================================
The analysis presented above is based on making the choice to fix $N_c=3$ at the outset when implementing the constraint of eq. (\[quantcond\]). This is not a reasonable way of doing phenomenology but may obscure the large $N_c$ scaling of the system. Consider the scaling of the splitting as given in eq. (\[10bar-8\]). Both $I_1\sim N_c$ and $I_2 \sim N_c$, eq. (\[10bar-8\]) and this appears to imply that $(M_{10} -
M_{8}) \sim 1/N_c$ and $(M_{\overline{10}} - M_{8}) \sim 1/N_c$. Thus the exotic states appear to behave similarly with the non-exotic states at large $N_c$. However, this is misleading. It has been known for some time that the exotic states have mass splittings relative to the ground state of order $N_c^0$ and not $N_c^{-1}$[@KlebKap; @Coh03; @DP]. To see how this arises we explore the implementation of eqs. (\[quantcond\]) and (\[mass\]) for $N_c$ arbitrary and large.
In studying large $N_c$ baryons it is useful to restrict attention to the case of odd $N_c$; this ensures that the baryons are fermions. The lowest-lying representation for odd $N_c$ consistent with the quantization condition in eq. (\[quantcond\]) is $ \left (p,q \right ) = \left( 1,
\frac{N_c-1}{2} \right) $; this representation can easily be shown to have $J=1/2$ using the method described in sect.
\[Rigid\]. The Young tableau for this representation is given in diagram a) of fig. \[young\]. Note that this representation does not correspond to [*any*]{} of the usual representations at $N_c=3$, and, in particular, is not an octet. However, the states in this representation do include those in the usual octet. Accordingly it is natural to take this representation to be the large $N_c$ generalization of the octet. This representation may be denoted “8” ; the quotation marks act as reminders that this is not the octet but its large $N_c$ generalization.
Similarly, the next representation has $ \left ( p,q \right ) =
\left( 3,\frac{N_c-3}{2} \right ) $ which has $J=3/2$. The Young tableau for this representation is given in diagram b) of fig. \[young\]. This representation contains all the states in the usual decuplet and can be regarded as the large $N_c$ generalization of the decuplet; accordingly, this representation is denoted by “10”. The mass relation in eq. (\[mass\]) then gives the mass splitting of the “10” from the “8”: $$M_{ ``10"} - M_{``8"} = \frac{3}{2 I_1}
\label{108quote}$$ The splitting obtained at large $N_c$ is thereby identical to the analogous result for the decuplet-octet splitting in eq. (\[10-8\]) which was obtained with the assumption $N_c=3$. The $N_c$ scaling of this splitting is found to scale as $N_c^{-1}$ (since $I_1$ scales as $N_c$). Thus the representations become degenerate as $N_c$ goes to infinity. This is for a deep reason: all of these non-exotic collective states are part of one contracted SU(2 $N_f$) representation which emerges at large $N_c$ (as will be discussed in subsection \[cons\]).
![ Young tableau for arbitrary but large $N_c$: a) the “8” representation with $(p,q)=\left( 1,\frac{N_c-1}{2} \right
)$; b)the “10” representation with $(p,q)=\left(
3,\frac{N_c-3}{2} \right )$: c) the “$\overline{10}$” representation with $(p,q)=\left( 0,\frac{N_c+3}{2} \right )$. The Young tableau in a) and b) have $N_c$ boxes; the tableau in c) has $N_c+3$ boxes []{data-label="young"}](young.eps)
To study of exotic states we need the large $N_c$ generalization of the $\overline{10}$ representation. The key feature of the $\overline{10}$ is that it is the lowest-lying representation that contains a state with strangeness +1. Accordingly, its large $N_c$ analog should be the lowest-lying representation that includes an exotic state with strangeness +1. This is readily seen to be $ \left ( p,q \right ) = \left( 0, \frac{N_c+3}{2}
\right ) $ and has $J=1/2$. This representation is associated with the Young tableau c) in fig. \[young\] and is denoted as “$\overline{10}$”. The excitation energy of this representation is obtained via eq. (\[mass\]): $$M_{``\overline{10}"} - M_{``8"} = \frac{3 + N_c}{4 I_2} \; .
\label{10bar8quote}$$
By construction, eq. (\[10bar8quote\]) agrees with eq. (\[10bar-8\]) when $N_c=3$. However, there is an explicit $N_c$ in the numerator of the right-hand side while the denominator is proportional to $I_2$ which scales as $N_c$. Thus, at large $N_c$, the scaling is given by $$M_{``\overline{10}"} - M_{``8"} \sim N_c^0 \; \; .
\label{scaling}$$ In the large $N_c$ limit the “$\overline{10}$” does not become degenerate with the “8”. It is easy to see that this behavior is generic for exotic representations. That is, any representation which contains at least one manifestly exotic state will have a splitting from the ground state which is finite as $N_c \rightarrow \infty$. It is interesting to note that this behavior is characteristic of typical vibrational excitations[@Wit3].
The study of the excitation energies as calculated via rigid-rotor quantization reveals a fundamental difference between exotic and non-exotic representations and the same fundamental difference is also seen via large $N_c$ consistency rules. The non-exotic representations become degenerate with the ground state at large $N_c$ while the exotic representations do not—they remain split from the ground state at leading order in the $1/N_c$ expansion.
Rigid-rotor quantization versus large $N_c$ QCD\[Args\]
=======================================================
This section demonstrates that the predictions of rigid-rotor quantization do not appear to follow from the known behaviors of such states in large $N_c$ QCD. This is seen from a wide variety of perspectives. Five rather general arguments are given all of which imply that the results of rigid-rotor quantization do not follow with the known behavior of large $N_c$ QCD.
Exotic baryon widths \[width\]
------------------------------
This subsection focuses on the $N_c$ scaling of the width of the exotic resonance. The essence of this argument is quite simple and requires two things be demonstrated. The first is that rigid-rotor quantization, if valid, must predict widths which vanish at large $N_c$; the second, that the width of exotic states with rigid-rotor quantization is, in fact, of order $N_c^0$ implying that it is invalid. Before demonstrating both of these points, it should be noted that this argument is [*formal*]{}. The issue of relevance is not whether the width of the $\theta^+$ is [*numerically*]{} large or small, but how it scales with $N_c$. The key point is that if the width does not vanish as $N_c \rightarrow
\infty$, as a matter of principle, the quantization prescription is presumably wrong.
Let us begin with the first point. As noted in the introduction, the question of whether the rigid-rotor quantization is valid at large $N_c$ has a clear formal definition. It is valid, if and only if, the properties computed via rigid-rotor quantization become exact as $N_c \rightarrow \infty$. The excitation energy of the exotic state at large in rigid-rotor quantization is given by eq. (\[10bar8quote\]); the issue is whether this becomes exact as $N_c \rightarrow \infty$. It is not exact if that there is a non-zero width in the large $N_c$ limit. This can be understood in two complementary ways. One way is simply to note that unless the width goes to zero at large $N_c$, the state is not well defined at large $N_c$—it does not correspond to an asymptotic state of the theory; it has no well-defined mass and eq. (\[10bar8quote\]) does not give the exact value for the mass at large $N_c$. An complimentary perspective is to assign the width to be an imaginary part of the mass. However, if the width remains finite at large $N_c$, then the mass has a non-vanishing imaginary part at large $N_c$, in which case eq. (\[10bar8quote\]) does not give the exact result for the mass. From either perspective, eq. (\[10bar8quote\]) is not exact at large $N_c$ unless the width vanishes.
Note, for comaprison that for non-exotic collective baryons such as the $\Delta$ , the widths do go to zero at large $N_c$ and therefore the prediction of eq. (\[108quote\]) can indeed become exact. To see this, one simply notes that the non-exotic collective excitation energies as given by eq. (\[108quote\]) scale as $1/N_c$ while meson masses scale as $N_c^0$. Thus, at large $N_c$ non-exotic collective space have no phase-space for decay and, hence, have zero width.
Let us now turn to the case of the exotic collective baryons which is quite different. Recently, Praszalowicz demonstrated that the width of the $\theta^+$ is of order $N_c^0$ [@Pras]. His result is summarized below.
Equation (\[scaling\]) implies that the excitation energy of the “$\overline{10}$” representation does not vanish at large $N_c$. Thus, in general phase space it does not inhibit decay. Of course, phase space is not the only reason the width can go to zero. It is also logically possible that the coupling between the initial exotic baryon and the final state of meson plus baryon could vanish at large $N_c$. The issue boils down to whether or not this happens.
As shown by Praszalowicz [@Pras], in the context of rigid-rotor quantization, there are two possible structures for the coupling of collective baryon states to mesons which contribute in leading order to the width: $$\hat{O}_\kappa = -i \frac{3}{2 M_B} \left (G_0
\hat{O}^{0}_{\kappa j} + G_1 \hat{O}^{1}_{\kappa j} \right ) p_j
\label{cupop}$$ with $$G_0 \hat{O}^{0}_{\kappa j} = D^{8}_{\kappa j} \; \; \; {\rm and}
\; \; \; G_1 \hat{O}^{0}_{\kappa j} = d_{j k l} D^{8}_{\kappa k}
\hat{S}_l$$ where $\kappa$ indicates the meson species and $\hat{S}$ is the spin operator. While the values of the coupling constants, $G_0$ and $G_1$, are model dependent, their $N_c$ scaling is known: $$G_0 \sim N_c^{3/2} \; \; \; G_1 \sim N_c^{1/2} \; .\label{Gscale}$$ Using the collective SU(3) wave functions for the “8” and “$\overline{10}$” representations one finds:
$$\left |\langle \theta^+,s=\downarrow|\hat{O}_{K 3}|N,
s=\downarrow \rangle \right |^2 = \frac{3}{\left (M_N +
M_{\theta^+} \right )^2} \frac{3 (N_c+1)}{(N_c+3)(N_c+7)} \left
[G_0 - \frac{N_c+1}{4} G_1 \right ]^2 p^2. \label{MEsq}$$
The $G_1$ term enhanced by a factor of $N_c$ (coming from the integral over the collective wave function) compensates for the fact that it is characteristically smaller than $G_0$ by a factor of $N_c$. Using the scaling in eq. (\[Gscale\]) and the fact the phase space implies $p \sim N_c^0$, one sees the matrix element is of order $N_c^0$. Combining this with the phase space one sees that $$\Gamma_{\theta^+ }\sim N_c^0 \label{widtheq}.$$
The scaling in eq. (\[widtheq\]) completes the demonstration. The width as computed using states from rigid-rotor quantization shows that the rigid rotor mass formula of eq. (\[10bar8quote\]) does not correctly give the mass at large $N_c$: rigid-rotor quantization does not appear to be self-consistent in large $N_c$ QCD .
Time scales for collective and vibrational motion \[tscales\]
-------------------------------------------------------------
The key to the success of collective quantization is a clear separation of time scales between the collective motion and the intrinsic motion. As a general principle, to quantize the theory one must first enumerate the degrees of freedom. The easiest way to do this is via the study of small classical fluctuations around the classical soliton: each classical mode associated with small amplitude fluctuations is a degree of freedom which can be quantized. In studying these small amplitude fluctuations one finds that although the typical modes have a vibrational frequency which scales as $N_c^0$, of necessity there will also be zero frequency modes. Such zero modes arise because symmetries associated with the underlying theory are broken by the soliton configurations.
Once these modes have been identified at the classical level, they can be canonically quantized. As a practical matter, a parametric expansion is used as a systematic way to organize things[@sol]. It can be shown self-consistently that for properties of low-lying states the neglect of these couplings leads to a $1/N_c$ correction to the soliton mass as compared to a leading order value which goes as $N_c$, and a correction due to the quantized zero modes which goes as $N_c^0$. The quantization of the collective degrees of freedom associated with the zero modes can be handled separately for exactly the same reason—the effects of the coupling of the collective to the vibrational degrees of freedom on the soliton mass is suppressed by $1/N_c$.
The collective degrees of freedom are those which at the classical level correspond to zero modes; modes having a finite frequency are quantized as vibrations. As will be discussed in sect. \[revisit\], the modes associated with the exotic excitations turn out not to be collective in this sense; they are associated with modes which have finite frequencies at small amplitude and, thus, act as vibrations.
The effect of the quantization of the collective degrees of freedom on the soliton mass is typically of order $1/N_c$. At the classical level the degrees of freedom are flat and have no natural time scale—the classical motion associated with them can occur arbitrarily slowly. The quantization of these collective modes keeps them slow—the typical time scales are of order $N_c$. The characteristic angular speed for non-exotic collective coordinates is $1/N_c$—it goes as the inverse of the moment of inertia since the associated angular momentum is typically of order unity and the motion is of large amplitude (order $N_c^0$ typically circumnavigating the collective space), so the characteristic time, $\tau$, is of order $ N_c$. In semi-classical motion one has a general energy-time uncertainty relation: $$\Delta E \sim 1/\tau \; \label{tau},$$ where $\Delta E$ is the typical splitting between levels. This naturally gives energy splittings of order $1/N_c$. It is straightforward to see that this behavior reproduces what is seen in eq. (\[108quote\]).
The time scale for the non-exotic collective motion ($N_c$) scales in a qualitative different way from that of the internal vibrational motion (shown below to scale as $N_c^0$). This difference is critical for the success of the quantization program, particularly for the separate treatment of the collective and vibrational degrees of freedom. Note that there exist vibrational modes with the same quantum numbers as those of the collective degrees of freedom. One generally expects degrees of freedom with identical quantum numbers to couple. However, two degrees of freedom with identical quantum numbers naturally remain weakly coupled if the characteristic times of the two degrees of freedom are widely separated. In this case there is a Born-Oppenheimer type separation of the dynamics of the modes. For the case of collective rotational degrees of freedom this is precisely what happens and the modes are effectively decoupled for low-lying states.
Let us now turn to the behavior of the exotic states. It is apparent from eq. (\[10bar8quote\]) that excitations with exotic quantum numbers have excitation energies of order $N_c^0$, implying that even for the lowest-lying exotic state, $\Delta E
\sim N_c^0$. This in turn implies that $\tau \sim N_c^0$, where $\tau$ is the characteristic time scale of the motion. This is highly problematic from the viewpoint of collective motion. In the first place, this is manifestly inconsistent with the motion being collective in the sense it corresponds to large amplitude motion. Equation (\[collective\]) implies that that excitation energies of order $N_c^0$ correspond to motion with ${J'_A}^2/I_2
\sim N_c^0$ (with A=4,5,6,7). This, along with the fact that $I_c\sim N_c$, further implies $J'_A \sim N_c^{1/2}$, and the relevant angular velocities are of order $N_c^{-1/2}$. Therefore the motion is slow—the angular velocity goes to zero as $N_c$ goes to infinity. However, we also know that the characteristic time for this motion is of order $N_c^0$. Clearly this is not possible unless the typical angular displacement associated with this motion is of order $N_c^{-1/2}$. Thus, this motion is confined to a region of angular space which goes to zero at large $N_c$: the motion does not explore the full collective space. In this sense the motion is clearly not collective. This should be contrasted to the case of low-lying non-exotic states which have a characteristic time scale of $N_c^1$, a characteristic angular velocity of $N_c^{-1}$ and, hence, a characteristic angular displacement of $N_c^0$, which is of large amplitude and subsequently collective.
Given the fact that motion is not collective, a collective quantization as in the rigid-rotor approach is presumably invalid. Moreover, it is easy to understand why it can fail. The time scale for even the lowest-lying exotic states, is of order $N_c^0$; this is the as for vibrational motion. In the absence of such a scale separation, there is nothing to prevent the ostensibly collective mode from mixing strongly with vibrational motion with the same quantum numbers and [*a priori*]{} that is exactly what one expects to happen. When this occurs the rigid-rotor approximation based on the separate dynamical treatment of the two types of motion clearly fails.
Predictions from large $N_c$ consistency rules\[cons\]
------------------------------------------------------
The introduction stressed the remarkable fact that the prediction of the mass in refs. [@Pres; @DiaPetPoly; @WalKop; @BorFabKob; @Kim] was independent of dynamical details of the model. The key model independent relations were that the SU(3) symmetry breaking used for the exotic states was identical to that used for the ground state band. It was largely because of this model independence that the prediction should be taken seriously. Model-independent predictions are special; the very fact that they do not depend on model details suggests that they may well be reflecting the underlying structure of large $N_c$ QCD.
Of course, just because a particular relation obtained in a chiral soliton model does not depend on the detailed dynamics of the model, need not imply the result is truly model independent results of large $N_c$ QCD. There is strong evidence, however, these relations are, in fact, truly model independent. [*All*]{} of these relations seen to date ([*with the exception of those involving exotic baryons computed via rigid-rotor quantization*]{}), are known to be correct model-independent predictions of large $N_c$ QCD[@GS; @DM; @Jenk; @DJM]. These include relations of standard static observables (such as magnetic moments or axial couplings) as considered in ref. [@AdkNap], as well as the rather esoteric quantities, such as the non-analytic quark mass dependence of observables near the chiral limit[@nc], or relations between meson-baryon scattering observables in different spin and flavor channels [@MatPes; @CohLeb]. All of these can similarly be derived in a “model-independent” manner in the chiral soliton model and are also derivable by a truly model-independent way.
The basis for demonstrating these model-independent predictions is the use of large $N_c$ consistency rules[@GS; @DM; @Jenk; @DJM]: the large $N_c$ predictions for various related quantities are not self consistent unless various relations are imposed. For example, according to Witten’s large $N_c$ counting rules generically meson-baryon coupling constants scale as $N_c^{1/2}$ [@Wit3]. Thus, two insertions of coupling constants will yield a contribution to meson-baryon scattering of order $N_c^1$, but unitarity ) requires the scattering to be of order $N_c^0$. This can only be satisfied if the baryons form towers of states which are degenerate at large $N_c$ with the values of the coupling constants related to one another by geometrical factors (up to $1/N_c$ corrections)[@GS; @DM; @Jenk; @DJM]. These factors turn out to be precisely the ones found in the chiral soliton models. Other quantities can be derived via similar means.
The results of this type of analysis are well known: A contracted SU(2$N_f$) symmetry emerges in the large $N_c$ limit. Baryon states fall into multiplets of this contracted SU(2$N_f$), and the low-lying states in these multiplets are split from the ground state by energies of order $1/N_c$—these excitations with the SU(2$N_f$) multiplets are collective. In the space of these collective states all operators can ultimately be expressed in terms of generators of the group and from this, relations can be obtained.
The key issue here is simply that the multiplet of low-lying baryons has been explicitly constructed—it coincides exactly with the low spin states of a quark model with $N_c$ quarks confined to a single s-wave orbital[@DJM]. It is well known that at large $N_c$ there are no low-lying collective baryon states ([*i.e,*]{} states with excitation energies of order $1/N_c$
) with exotic quantum numbers. This neatly mirrors the analysis of sect. \[LargeN\]: exotic states have excitation energies of order $N_c^0$.
This situation is quite problematic, however, if one wishes to assert that relations for the exotic states are truly model independent. The difficulty is simply that the exotic states are not in the same contracted SU(2$N_f$) multiplet as the ground band baryons. This means that group theory alone cannot relate matrix elements in the ground band to matrix elements involving exotic states: the standard large $N_c$ consistency rules do not allow one to relate any property of the exotic states to the ground band states.
At a minimum this implies that the apparently model-independent predictions of exotic state properties seen from rigid-rotor quantization have not been shown to be truly model independent. This is characteristically different from the properties of non-exotic states which are related to one another by large $N_c$ consistency rules. Of course, just because the large $N_c$ consistency conditions do not give the relations seen for exotic states in rigid rotor computation does not by itself mean that these relations are wrong; it merely means we do not know them to be correct. However, it does mean that the principle reason to take the predictions of the chiral soliton model seriously—their apparently model-independent status—has no known basis in QCD.
Solitons for large $N_c$ in a world with heavy quarks
-----------------------------------------------------
The predictions in refs. [@DiaPetPoly] were based on chiral soliton models. However, this is somewhat misleading. The basic arguments of rigid-rotor quantization depends critically on the SU(3) flavor symmetry of the underlying theory and the fact that the classical solution breaks both rotational and flavor symmetries in a correlated way resulting in a hedgehog configuration. However chiral symmetry plays role only through the chiral anomalies encoded in the Wess-Zumino term which leads to the constraint in eq. (\[quantcond\]). However, as shown in appendix \[const\], eq. (\[quantcond\]) can be derived directly at the quark level with no reference to anomalies. Thus, the logic underlying collective quantization applies to all SU(3) symmetric theories which have hedgehog mean-field solutions. In particular, it applies to models of QCD with three degenerate flavors in the limit where all quark masses are heavy; $m_u = m_d
= m_d \equiv m_q$ with $m_q>> \Lambda$ where $\Lambda$ is the QCD scale (provided that such models produce a hedgehog at the mean-field level). In this section it is shown that the application of the rigid-rotor quantization in such a regime gives results which are manifestly wrong suggesting that something is wrong with the underlying logic.
In fact, in this regime one need not consider a model of QCD, but rather QCD itself. Recall that in Witten’s original derivation of baryon properties in large $N_c$ QCD the case of heavy quarks was considered for simplicity (it was later argued that the conclusions are valid for the case of light quarks)[@Wit3]. The derivation is based on the fact that in this limit the quarks are non-relativistic and can be described via the many-body potentials arising from gluon exchange. Witten then demonstrated that the Hartree mean-field approximation becomes valid in the large $N_c$ limit, that all $N_c$ quarks are in the same single particle wave function (modulo their color degree of freedom) and that the size of this orbital is independent of $N_c$. In doing this analysis, the role of flavor and spin degrees of freedom was not highlighted.
Consider the role played by spin and flavor in this system. From the analysis of Dashen, Jenkins and Manohar we know that for states with $J,I \sim N_c^0$ at leading order in the $1/N_c$ expansion the only interactions which contribute are either spin and isospin independent or both spin dependent and isospin dependent[@DJM]. The spin dependence enters solely through the magnetic coupling of the gluons to the quarks. Note, however, that the underlying magnetic interaction is small for large quark masses since the quark magnetic moment goes as $1/m_q$. Thus all spin-flavor dependent interactions are small in the combined large $N_c$ and heavy quark limits.
The heavy quark limit implies that the spatial shape of the single-particle levels does not depend on spin and flavor. Assuming that rotational symmetry is not broken at this order, the orbitals will be s-wave. However, the Hartree state is highly degenerate at this order: any spin and flavor orientation is equivalent. If one one includes the leading $1/m_q$ correction, this degeneracy is broken (but at this order the spatial part of the wave functions are unchanged) and one simply chooses the single-particle state to have a spin-flavor orientation which minimizes the spin-flavor interaction at order $1/m_q$. The precise form of this spin-flavor orientation depends on the sign of the interaction which either favors a ground of the baryon with maximal spin or minimal spin. If it favors maximal spin, then all quarks will have the same well-defined spin and flavor projections. In contrast, if it favors minimal spin (as is believed to happen in nature, which will be assumed here) the spin-flavor part of the state takes the conventional hedgehog form: $$|h \rangle = 2^{-1/2} \left ( |\uparrow d \rangle - |\downarrow u
\rangle \right ) \label{h}$$ where the arrows indicate spin projection and the letters indicate flavor. It is assumed for simplicity that the hedgehog is in the u-d subspace with grand-spin of zero.
It is easy to see that there are low-lying single-particle excited states for quarks propagating in the background of the Hartree potential generated by the hedgehog state. There are six distinct spin-flavor states, one of which, the state, $|h\rangle$, is, by construction, the lowest-lying state for the Hartree potential. These states differ from the lowest lying state in energy by an amount of order $\Lambda^2/m_q$, where the $1/m_q$ is due to the spin dependence as noted above and the factors of $\Lambda^2$ follow from dimensional analysis. Furthermore, all flavor generators (except for $\lambda_8$), when acting on the hedgehog, will produce a superposition of these low-lying excited single-particle states. Thus, if one computes the moments of inertia using eqs. (\[quarkcont\]) one has the following scaling in $N_c$ and $m_q$: $$I_1^{-1} \sim \frac{N_c \Lambda^2}{m_q} \; \; I_2^{-1} \sim
\frac{N_c \Lambda^2}{m_q} \; . \label{hqmi}$$
Equation (\[hqmi\]) has profound implications for excitation energies assuming that rigid-rotor quantization is legitimate. Consider first the behavior of the non-exotic states, such as those in the “10” representation. Equations (\[108quote\]) along with eq. (\[hqmi\]) imply that scaling of the excitation energy with $m_q$ and $N_c$ goes as $$M_{``10"}-M_{``8"} \sim \frac{\Lambda^2}{m_q N_c} \; .$$ This behavior is precisely what one expects. It is suppressed by $1/N_c$ since it involves the excitation of a single quark and it goes as $1/m_q$ since it relies on a magnetic gluon-quark coupling.
In a similar way one can compute the scaling of the excitation energy of the exotic states such as the “$\overline{10}$”. Equations (\[10bar8quote\]) and eq. (\[hqmi\]) imply that this excitation scales as $$M_{``\overline{10}"}-M_{``8"} \sim \frac{\Lambda^2}{m_q} \; .
\label{es}$$ However, this scaling is quite problematic on physical grounds. To construct an exotic state such as the “$\overline{10}$” one must include an extra quark-antiquark pair relative to the “8”. Recall that by assumption the quarks are heavy enough to be non-relativistic; the binding energies of the quarks are much smaller than the mass. The excitation energy of the exotic state must then be $2
m_q$ plus small corrections; it grows with $m_q$. This is in contradiction with eq. (\[es\]) which has the excitation energy decreasing with increasing $m_q$ assuming that rigid-rotor quantization is valid. The contradiction implies that the assumption is false: rigid-rotor quantization is not valid for the exotic states. As noted above, the logic underlying rigid-rotor quantization does not depend on the quark mass and the failure for the present case indicates that the logic is flawed.
As pointed out recently by Pobyltisa[@Pob], the issue is even more stark if presented in the context of an SU(3) symmetric non-relativistic quark model. In this case the model space can be constructed to have only quark—and no anti-quark—degrees of freedom. However, the interactions in such a theory can be chosen to reproduce the same $N_c$ scaling seen in QCD. Again, as $N_c$ becomes large the Hartree approximation becomes increasingly well justified and again the Hartree minimum will be of a hedgehog configuration (provided that the ground state orbitals turn out to be s-waves). The justification for doing rigid-rotor quantization is identical to that in the chiral soliton case. In such a model both $I_1$ and $I_2$ can be computed using the standard expressions in eq. (\[quarkcont\]) and rigid-rotor quantization can then be implemented. Such a quantization implies the existence of exotic states with an excitation energy of order $N_c$, [*yet by construction such states are not in the Hilbert space from which the model was constructed*]{}. The rigid-rotor quantization is wrong for exotic states in such models. Of course, one might argue that such a model does not represent QCD in a particularly realistic way. However, that should be irrelevant to the central issue of the justification of rigid-rotor quantization; the derivation of this approach has always been based on general considerations and not on the detailed structure of QCD.
The Callan-Klebanov approach at zero SU(3) breaking
---------------------------------------------------
The discussion in this paper has focused on states as computed in the limit of zero SU(3) symmetry breaking. As was argued above, the fundamental issues of principle interest here can be most cleanly addressed if detailed numerical questions, such as whether the SU(3) breaking effects are too large to justify perturbation theory, aside. One powerful way to focus on the problem is to consider how to treat the problem in the presence of SU(3) symmetry breaking and then consider the limit as this breaking goes to zero.
The basic formalism of how to treat chiral solitons in the presence of SU(3) breaking was developed long ago by Callan and Klebanov[@CK]. The logic underlying this approach is very simple. If $m_s - m_q$ is greater than zero (where $m_q$ is the light quark mass), then at the classical level the minimum energy configuration is a hedgehog in the u-d subspace. A flavor rotation in the 4,5,6 or 7 directions will yield a classical configuration with higher energy. Thus these directions are not flat collective ones but are unambiguously of a vibrational character. The formalism is based on the straightforward quantization of these vibrational degrees of freedom along with the collective quantization of the SU(2) degrees of freedom in the presence of the Wess-Zumino term. It is often called the “bound state approach” since states with negative strangeness are viewed as a bound state of an SU(2) skyrmion with an anti-kaon. Here it will be referred to as the Callan-Klebanov approach in order to avoid confusion when discussing states of positive strangeness which are unbound. It should be noted that a principal reason why this approach was introduced was to deal with circumstances where SU(3) symmetry breaking was large enough so that simple perturbative treatments were potentially unreliable. But, as stressed in ref. [@IKOR], the formalism should be valid regardless of the size of the symmetry breaking and, in particular, holds as $ m_s - m_q \rightarrow 0$.
The key issue here was discussed in the recent paper by IKOR[@IKOR]. Consider the Callan-Klebanov formalism as $ m_s -
m_q \rightarrow 0$. For states with non-exotic quantum numbers it is possible to show analytically that as this limit is approached, the excitation energy of the bound state goes to zero (as it is in rigid-rotor quantization) and the structure of the vibrational mode goes over precisely to the spatial distribution seen in the collective moment of inertia [@CK; @IKOR]. This supports the view that the Callan-Klebanov formalism applies regardless of the size of SU(3) symmetry breaking and remains valid to the exact SU(3) limit, and that rigid-rotor quantization is valid for non-exotic collective states.
However, the situation is radically different in the case of exotic s=+1 excitations. In that case, there is no analytic demonstration that the results of the Callan-Klebanov treatment goes over to those of rigid-rotor quantization as the SU(3) limit is approached. Of course, it is logically possible that the two approaches are, in fact, equivalent in the SU(3) limit but that a mathematical demonstration of this equivalence has not yet been found. However, there is strong numerical evidence in ref. [@IKOR] that this is not the case. In particular, for small values of $m_K$ ([*i.e*]{}, for small SU(3) symmetry breaking), if the two approaches were equivalent there ought to be a resonant vibrational mode whose frequency is near to that predicted in rigid-rotor quantization for all Skyrme-type models. In fact, no such mode is seen. Indeed, for the standard Skyrme model, there are no exotic resonances at all. This strongly suggests that as the exact SU(3) limit is approached the Callan-Klebanov approach does not become equivalent to the rigid-rotor quantization as $ (m_s - m_q) \rightarrow 0$.
If one accepts that the standard derivation of the Callan-Klebanov formalism is valid for this problem, then the inequivalence between rigid-rotor quantization and the Callan-Klebanov method for exotic states implies that the rigid-rotor quantization is not valid. There is a mathematical subtlety associated with this: We know the derivation of Callan and Klebanov in ref. [@CK] is valid when SU(3) breaking is large enough so that motion in the strange direction is of a vibrational nature. As a formal matter, this is guaranteed to happen in the large $N_c$ limit for any finite value of $(m_s-m_q)$. As long as the physical quantities of interest have a uniform limit as $N_c \rightarrow \infty$ and $(m_s-m_q) \rightarrow 0$, then the Callan-Klebanov derivation is automatically valid in the large $N_c$ limit for exact SU(3) symmetry. However, if the two limits do not commute, then taking the SU(3) limit prior to the large $N_c$ limit (as is implicitly done in rigid-rotor quantization ) would give results which differ from those when taking the large $N_c$ limit first (as is implicitly done in the Callan-Klebanov approach). If this is the case, then it remains possible mathematically that both approaches are valid in particular regimes but that their domains of validity do not overlap.
However, on physical grounds this mathematical possibility seems quite unlikely. In the first place, it seems implausible [*a priori*]{} that the large $N_c$ and SU(3) limits commute for non-exotic states and then fail to exotic states. More importantly, there is considerable experience with cases where the large $N_c$ limit does not commute with some other limit, and in these cases the lack of commutativity can be traced to a clear physical origin which is apparent at the hadronic level. For example, it is well known that the chiral limit and the large $N_c$ limit do not commute for baryon quantities which diverge in the chiral limit (such as isovector charge or magnetic radii) [@nc]. In these cases the physical origin is easily traced to the role of the $\Delta$ resonance whose excitation energy is anomalously low parametrically (it scales as $1/N_c$) compared to typical excited baryons and which therefore leads to a class of infrared enhancements in loop graphs. Thus, it is the interplay between the two light scales in the problem—the chiral scale, $m_\pi$ and the light scale induced at large $N_c$, $M_\Delta-M_N$—which leads to the non-commuting behavior for problems that are infrared singular in the combined limit. There is nothing analogous to this when considering exotic baryons near the SU(3) limit: the SU(3) symmetry breaking scale is the only relevant low scale in the problem since the low-lying decuplet type excitations play no special role. Given these physical arguments it is highly unlikely that the two limits do not commute; as note above, this implies that rigid-rotor quantization is invalid.
Apart from the physical grounds discussed in the previous paragraph there is a clear mathematical way to understand why numerical work of IKOR fails to find a vibrational mode whose properties match those predicted in rigid-rotor quantization for exotic states. This is the mixing between the “collective” mode in rigid-rotor quantization and ordinary vibrational modes which is discussed in detail in sect. \[revisit\]. This mixing occurs at leading order in both the large $N_c$ expansion and in SU(3) breaking regardless of the ordering of limits, and thus demonstrates mathematically that the physical arguments given above are correct: the two limits do commute, but the rigid-rotor quantization is not valid.
Rigid-Rotor quantization Revisited \[revisit\]
==============================================
General considerations
----------------------
The previous section provides strong evidence that rigid-rotor quantization is not valid for the description of states with exotic quantum numbers. This can be seen as somewhat paradoxical since the derivation of rigid-rotor quantization[@SU3Quant] closely paralleled the derivation for SU(2) skyrmions, by Adkins, Nappi and Witten (ANW)[@ANW], which is generally agreed to be correct. How can this method work for SU(2) solitons and for non-exotic states in SU(3) solitons, yet fail for exotic states?
It is important to begin by recalling the fundamental assumption made in the derivation in sect. \[Rigid\]; namely, that the collective motion and the intrinsic motion are dynamically separate. The issue is whether this is true; as will be seen in this section, it does not appear to be true generally for exotic motion even at large $N_c$.
The ANW procedure amounts to putting an ansatz for a class of allowed motion into the Lagrangian thereby obtaining a proposed collective Lagrangian. This is done in eq. (\[ut\]) where the ansatz made is that the motion corresponds to an overall time-dependent rotation of the static soliton. As a general rule, the insertion of an ansatz into a Lagrangian yields a legitimate collective Lagrangian if, and only if, all classical solutions of the collective Lagrangian so obtained are also solutions of the full equations of motion. Only after the collective Lagrangian has been isolated at the classical level can the collective Hamiltonian be found and then quantized.
As alluded to in sec. \[Rigid\], the ANW treatment is strictly valid at large $N_c$ only for models whose Lagrangians have no first derivatives in time (such as the original Skyrme model). The reason for this is that the ansatz in eq. (\[ut\]) only corresponds to an approximate solution of the full equations of motion in such cases. It has long been known that the method needs to be modified for models where first derivatives in time are present (such as in soliton models with explicit quark degrees of freedom)[@CB1; @CB2]. For the SU(2) model with explicit quarks one can find an appropriate ansatz which corresponds to a solution of the mean-field (classical) equations; the cranking equations provide such an ansatz. The reason this issue becomes central here is the role of the Wess-Zumino term. This term has an explicit time derivative, and [*a priori*]{} one ought not to expect the ANW method to work without modification.
First consider models such as the original SU(2) Skyrme model, which only has pion degrees of freedom and has no first derivatives in time. In this case it is easy to find such families of solutions which become exact at large $N_c$[@CB2]. In particular, $$U(\vec{r},t) = A e^{i \vec{\lambda} \cdot \vec{\tau} t/2}
U_0(\vec{r}) e^{-i \vec{\lambda} \cdot \vec{\tau} t /2} A^\dagger
\label{td}$$ is an approximate time-dependent solution of the classical equations of motion provided that $U_0$ is a static solution and $\vec{\lambda}$ is an angular velocity which is small at large $N_c$ (typically going as $N_c^{-1}$). The parameters that specify the motion are the initial angles given in $A$ and the angular velocities in $\vec{\lambda}$. This is an allowable approximate time-dependent solution since the effect of the second derivative with respect to time on the field configuration (which is neglected when using a rotating soliton) is of order $ \sim 1/N_c^2$ down relative to contributions to the static solution. Thus, the neglected shifts in the fields are of relative order $N_c^{-2}$. This in turn implies a neglected shift in the angular momentum of order $N_c^{-1}$ (since the angular momentum is intrinsically of order $N_c$); this implies that the neglected shift in the moment of inertia is of order $N_c^0$ and may be neglected at large $N_c$ compared to the leading order contribution of order $N_c^1$.
The ANW ansatz of eq. (\[ut\]) contains all the solutions of the form of eq. (\[td\]). Moreover [*all*]{} solutions of the classical equations of motion which emerge from the collective Lagrangian are of this form. Thus, the ANW ansatz gives a legitimate collective Lagrangian on the SU(2) Skyrmion. This Lagrangian can then be quantized.
The situation is quite different if there are first-order time derivatives. In that case the neglected effect on the fields is proportional to $\lambda \sim N_c^{-1/2}$ yielding a neglected shift in the fields proportional of relative order $N_c^{-1/2}$ which in turn implies that the neglected shift in the angular momentum is of order $N_c^{1/2}$. The neglected shift in the moment of inertia is then of order $N_c$. These effects cannot be neglected since the neglected contribution is of the same order as the contribution which is kept. There is a simple way to incorporate these effects in SU(2) solitons containing quarks. In that case, the key point is that one needs an ansatz for a time-dependent solution which corresponds to the rotating soliton. If such a corresponding solution exists, it is equivalent to a static solution calculated in a rotating frame and one obtains the cranking result of eq. (\[quarkcont\]).
To summarize the general situation, ANW quantization, while agreeing with the generally correct method for the case where it was introduced, it does not directly apply to models with first order time derivatives. In these cases one needs to find families of approximate classical time-dependent solutions which are decoupled from the remaining degrees of freedom.
In the case of exotic motion in SU(3) solitons the Wess-Zumino term plays a dynamical role and [*a priori*]{} there is no reason to believe it should be valid in such a case. In contrast, for non-exotic motion the Wess-Zumino term is inert (as can be seen in eq. (\[quantcond\])). In principle, it is sufficient to stop the argument here—rigid-rotor quantization has never been correctly derived for exotic motion. The fact that the derivation was not shown to be correct does not logically mean the result is wrong. However, the fact that rigid-rotor quantization gives inconsistent results as seen in the previous section indicates that the result is, in fact, not correct.
It is useful to understand [*why*]{} the approach fails in a bit more dynamical detail. The key point is that the there is no family of classical solutions corresponding to the exotic collective as given by the ansatz.
Before discussing the full problem it is useful to gain some intuition about how things work by considering a couple of “toy” problems.
A charged particle in the field of a magnetic monopole \[mono\]
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To begin consider the following simple toy model: the non-relativistic motion of a charged particle of mass $m$ and charge $q$ confined on a sphere of radius $R$ with a magnetic monopole of strength $g$ at its center. This problem was introduced by Witten [@Wit1] to illustrate the role of the Wess-Zumino term and was used to motivate the treatment of rigid-rotor quantization for SU(3) solitons [@SU3Quant]. The monopole case is essentially similar to the Wess-Zumino case in two fundamental aspects: a) its effect is first order in time derivatives, and b) it is essentially topological in nature imposing toplogical quantization rules (the Dirac condition for monopoles, the quantization of the Wess-Zumino term for chiral soliton models[@Wit2]). To simulate the case of SU(3) solitons at large $N_c$ one has the following scaling rules: $$q \sim N_c^0 \; \; \; \; R \sim N_c^0 \; \; \; \; g \sim N_c^1 \;
\; \; \; m \sim N_c^1 \; ;\label{toyscale}$$ the Dirac quantization condition implies that $2 q g$ is an integer. This problem is exactly solvable. The key physical point is that in addition to the usual kinetic contribution to the moment of inertia, there is an angular momentum associated with the magnetic field energy which is given by $\vec{L}_{field} = q g \hat{r}$. Mathematically, the key issue is the unphysical nature of a rotation about the axis linking the charge and the monopole which then imposes the Dirac condition and the restriction $L \ge |q
g|$. The wave functions may be expressed in terms of three Euler angles; the independence of the the physical results on the third angle is what gives rise to the Dirac quantization condition.
The spectrum for this problem is $$E = \frac{L(L+1) - (q g)^2}{2 m R^2} \; \; {\rm with} \; \; L \ge
|q g| \; .\label{toyspec}$$
The analogy with the SU(3) soliton is the following. The smallest allowable $L$ corresponds to the non-exotic baryons with the (2L+1) allowable $m$’s corresponding to the different non-exotic baryon states. The various states with $L > q g$ correspond to exotic states. To make this manifest we follow Diakonov and Petrov[@DP] and express the energy in terms of an “exoticness”: $$E = \frac{e^2 + e (2 |q g| +1) + |q g|}{2 m R^2} \; \; \; {\rm
with} \; \; \; e \equiv L - |q g|$$ If one focuses on the low-lying exotic states (those states for which $e \sim
N_c^0$) which are the states of interest in hadronic physics one finds the excitation energies are given by $$E_e - E_0 = e \frac{ |q g|}{ m R^2} + {\cal O}(N_c^{-1})
\label{toyspeca}$$ where the $N_c$ scaling is fixed from eq. (\[toyscale\]).
Conventional treatments of rigid-rotor quantization model their treatment of the Wess-Zumino term on this exact treatment of this toy problem [@SU3Quant]. The present purpose is different: Here the goal is to understand which collective degrees of freedom can be isolated in a treatment which will ultimately be semi-classical in nature. Of course, this toy problem has no internal degrees of freedom—by construction it is rigid-rotor quantization. However, it is useful to illuminate the underlying physics of this toy problem in a manner which transparently can be used in the case where collective and vibrational modes mix. In particular, it is quite instructive to derive the result in eq. (\[toyspeca\]) from a semi-classical treatment which can then be generalized.
We need to first describe the classical motion of the particle. Let us consider the particle at rest at the north pole and we can describe all solutions of the equations of motion relative to this. Because this problem has spherical symmetry, there are “zero modes”. One can rotate the charge from the north pole and leave it in a displaced static position. A static rotation of this type is equivalent in the soliton case to a non-exotic “excitation” (although in this toy model all non-exotic states are degenerate with the ground state so “excitation” is a bit of a misnomer).
Classical behavior associated with the exotic degrees of freedom necessarily involves motion. Were there no velocity dependent forces present due to the monopole, the existence of flat directions would imply the existence of dynamical rotational modes with the system slowly rotating around the entire sphere. However, the magnetic monopole fundamentally alters this. As soon as the particle starts moving, the magnetic force acts to bend the particle into a curved orbit and this curvature can be on a scale much smaller than the radius of our sphere $R \sim N_c^0$; indeed one can see self-consistently that the characteristic size of such an orbit for states with small “exoticness”(where $e = L
-|q g|$) will scale as $N_c^{-1/2}$ and, hence, does not go fully around the sphere. If the orbit is localized in a region of this size then at large $N_c$ it effectively stays in a region small compared to $R$, the curvature of the sphere (which is order $N_c^0$). In this case the classical motion is effectively that of a charged particle in a magnetic field, $\vec{B} = \hat{z}
g/R^2$, moving on a plane. Consider an orbit centered around the north pole (one can always rotate your coordinate labels to do this). The position of the particle is then given approximately by $$x \approx r \cos (\omega t + \delta) \; \; \; y \approx r
\sin(\omega t + \delta) \; \; \; \omega \approx \frac{q g}{m R}
\label{classsol}$$ where $r$ and $\delta$ are fixed from the initial conditions and the corrections are of order $1/N_c$.
Now suppose one wishes to quantize this classical circular motion. This is the familiar problem of Landau levels. The energy spectrum of such a system is of precisely the form of a harmonic oscillator[@Landau]: $$E = (n + 1/2) \frac{| q B |}{m } = (n + 1/2) \frac{|q g|}{m R^2}
\label{Landau}$$ Equation (\[Landau\]) can be derived using the Bohr-Sommerfeld formula. In doing this one finds that the radius of the orbits are quantized to have $$r^2 \approx \frac{n+1/2}{|q B|} \approx
\frac{(n+1/2)R^2}{|q g|} \; ; \label{rtoy}$$ the scaling rules in eq. (\[toyscale\]) then imply that the radius scales as $r \sim N_c^{-1/2}$. This in turn, justifies treating the problem as motion in the plane. Let us now look at the excitation energies predicted from this semi-classical quantization: $$E_n -E_0 = n \frac{|q g|}{m R^2}\label{toysc} \; .$$ The significant point is that eq (\[toysc\]) gives the same excitation spectrum as in eq. (\[toyspeca\]) provided one identifies the “exoticness”, $e$, with the index $n$.
Of course, this toy model corresponds to rigid-rotor quantization since there is a fixed moment of inertia. It nevertheless teaches us several things of importance which can be generalized to situations where the moment of inertia is dynamical. i) The classical motion associated with the exotic excitation is not a zero mode; no matter how slow the velocity the period of the orbit is of order $N_c^0$. ii) The classical motion is bounded; for energies corresponding to low exoticness ($e \sim N_c^0$) the typical size of the collective orbit is $N_c^{-1/2}$. iii) In the large $N_c$ limit the excitation spectrum can be understood by purely semi-classical means. Points i) and ii) indicate that as far as the scales are concerned this problem “looks like” vibrational motion.
One aspect of point i) ought to be stressed. The fact that this dynamical mode is not a zero mode is essential. An apparently odd feature is that the problem has two flat directions and one might naively think there ought to be two zero modes. Where then does the nonzero mode come from given the fact that there are only two degrees of freedom in the problem? Actually, the issue is largely semantic. Consider a typical harmonic vibrational mode in one dimension associated with the equation of motion $\ddot{x}= -
\omega_0^2 x $. The motion is given by $x=\alpha \cos(\omega_0 t)
+ \beta \sin(\omega_0 t)$ and is parameterized by two numbers which are fit by two initial conditions, $\alpha =x(0)$ and $\beta =\dot{x}(0)/\omega_0$.
One typically describes this as one mode of oscillation despite the need for two parameters to specify the motion since one can always rewrite it as $x= A \cos (\omega_0 t
+ \delta)$, with $A=\sqrt{\alpha^2+\beta^2}$ and $\delta =
\tan^{-1}(\beta/\alpha)$. Thus, the path followed depends only on one parameter $A$, and the second parameter merely serves to induce a phase shift in the single mode of motion. One might make the following, somewhat pedantic, description: the motion consists of two distinct modes; the general motion is then the superposition of these two distinct modes. From this perspective one can write the motion as two coupled first-order differential equations for $x$ and $\dot{x}$ and define modes as solutions where $x$ and $\dot{x}$ each evolve as $e^{ i \omega t}$ with the same omega. By defining modes this way it can be seen that there are two modes in the harmonic problem given above—one with $\omega=\omega_0$ and the other with $\omega=-\omega_0$—which form a pair. For the problem of a single harmonic oscillator, calling this a pair of modes with equal and opposite frequencies may seem a bit artificial. However, in the toy problem of a particle on a spherical shell moving in the field of a magnetic monopole, this distinction is important. There are two degrees of freedom and thus, in the sense given above, one expects four modes to come in two pairs of equal and opposite $\omega$. This is precisely what happens. One pair of modes corresponds to fully static configurations (one moves the charge in either the $x$ or $y$ direction to a new position and leaves it there at rest) and these correspond to two zero modes which we view as a single pair of modes. The other pair of modes correspond to the charge orbiting either clockwise or counterclockwise corresponding to $\omega=\pm \frac{|q g|}{m R^2}$. However, while we still have two pairs of modes the existence of a velocity dependent force means that the pairs do not correspond to a mode associated with initial displacement in a direction paired with an initial velocity in the same direction. This means that although this problem has two zero modes associated with displacements, it does not have two pairs of zero modes but only one.
Coupled particles in a monopole field\[coupled\]
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While the problem in the previous section gives some insights into the scales of the problem and the nature of modes and semi-classical quantization, it cannot answer the fundamental question about whether rigid-rotor quantization is generically valid since by construction the problem is a rigid rotor. Accordingly we need a model where the underlying dynamics is not rigid; [*i.e.,*]{} a model where the analog of the soliton has some nontrivial internal dynamics. Moreover, because the central concern is mixing between collective and internal modes, it is important that the internal degrees of freedom have excitations which have the same quantum number numbers as the exotic motion. The simplest model of this sort is a generalization of the model studied in the previous section. Consider a non-relativistic theory of two particles confined to the surface of a sphere. For simplicity we will take them to have equal mass, $m$. At the center of the sphere is a magnetic monopole of strength $g$. The two particles have different charges—which we will take to be $q$ and zero—so they interact differently with the monopole. Finally, the two particles interact with each other via a quadratic potential $V_{\rm int} = k |\vec{r_1} -\vec{r_2}|^2/2$, where $k$ acts as a spring constant. To create an analogy with SU(3) solitons at large $N_c$ the following scaling rules must be imposed for the parameters: $$q \sim N_c^0 \; \; \; \; R \sim N_c^0 \; \; \; \; g \sim N_c^1 \;
\; \; \; m \sim N_c^1 \; \; \; k \sim N_c^1 \; .\label{toyscale2}$$
The analog of the classical soliton is a static configuration which solves the classical equations of motion. The solution is simple: the particles are on top of each other and we can take their position to be at the north pole. The rigid-rotor quantization of this system is quite trivial. The classical “soliton” is constrained to move coherently with the two particles in their classical ground state ([*i.e.,*]{} with the two particles on top of each other). This reduces immediately to the classical motion of a single particle of mass $2m$ and charge $q$ and one can immediately read off the excitation by making the appropriate substitutions in eq. (\[toyspeca\]): $$E_e^{\rm rigid} - E_0 = e \omega_r \; \; \; {\rm with} \; \; \; \omega_r \equiv \frac{ |q
g|}{ 2 m R^2} \label{toyrigid} \; ,$$ where as before $e$ is a non-negative integer.
On the other hand, we can follow the correct full procedure of finding classical time-dependent solutions and then quantize this motion semi-classically. All small amplitude motion can easily be found via a linearization of the equation of motion; this is sufficient provided the quantization keeps the motion within the regime of validity of linear response. As before, this can be checked [*a posteriori*]{}. If one is in the small amplitude regime the problem again reduces to motion on a plane in the presence of a magnetic field. Thus we can parameterize the two degrees of freedom for each particle by its $x$ and $y$ coordinates. Following the discussion in the previous subsection, it is useful to write first-order equations of motion for the particles and their associated velocities:
$$\frac{d}{d t} \left( \begin{array}{c} x_1\\ y_1\\ x_2 \\
y_2\\\dot{x}_1\\\dot{y}_1\\\dot{x}_2\\\dot{y}_2 \end{array}
\right ) \, = \, \left ( \begin{array}{c c c c c c c c}
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\frac{-\omega_v^2}{2} & 0 & \frac{\omega_v^2}{2}& 0 & 0 & 2 \omega_r & 0 & 0 \\
0 & \frac{-\omega_v^2}{2} & 0 & \frac{\omega_v^2}{2} & 2 \omega_r & 0 & 0 & 0 \\
\frac{\omega_v^2}{2} & 0 & \frac{-\omega_v^2}{2}& 0 & 0 & 0 & 0 & 0 \\
0 & \frac{\omega_v^2}{2} & 0 & \frac{-\omega_v^2}{2} & 0 & 0 & 0 & 0 \end{array} \right )
\left( \begin{array}{c} x_1\\ y_1\\ x_2 \\
y_2\\\dot{x}_1\\\dot{y}_1\\\dot{x}_2\\\dot{y}_2 \end{array}
\right )\; \; \; {\rm with} \; \; \; \omega_v=\sqrt{\frac{2
k}{m}} \; \; \; {\rm and} \; \; \; \omega_r=\frac{g q}{2 m R^2} \;
\; . \label{m}$$
The quantities $\omega_v$ and $\omega_r$ have simple physical interpretations: $\omega_v$ is the vibrational frequency of the two particles when the monopole field is turned off and $\omega_r$ is the Landau frequency assuming that rigid-rotor quantization is valid. From the scaling rules in eq. (\[toyscale2\]), ones sees that $\omega_r \sim N_c^0$ and $\omega_v \sim N_c^0$. The normal mode frequencies are $-i$ times eigenvalues of the matrix on the right-hand side of eq. (\[m\]).
The matrix can be diagonalized explicitly and it is found that, as expected, the eight modes group into four sets of pairs of modes with equal and opposite frequencies. There is one pair of zero modes which corresponds to static configurations in which both particles are displaced by the same amount. Analytic expressions can be obtained for the frequencies of the three pairs of non-zero modes. However, these expressions are quite cumbersome. Since the modes depend only on $\omega_v$ and $\omega_r$, it is useful to express the frequencies as multiples of $\omega_r$ and to express results as a function of $\frac{\omega_v}{\omega_r}$. A plot of the three (positive) frequencies for these pairs is given in fig. \[omega\]. These modes can be semi-classically quantized and the lowest-lying excitation associated with the motion is $\omega$ above the ground state. As before, the quantized orbits correspond to velocities of order $N_c^{-1/2}$ and displacements of order $N_c^{-1/2}$ which self-consistently justifies the neglect of the curvature at large $N_c$.
![Non-vanishing normal mode frequencies for the model in subsection \[coupled\] in units of of $\omega_r$ as a function of the ratio $\frac{\omega_v}{\omega_r}$ with $\omega_r$ and $\omega_v$ defined in eq. (\[m\]). The strongly coupled limit corresponds to $\frac{\omega_v}{\omega_r} \rightarrow \infty$. The dashed line represents the prediction for the lowest excitation energy in rigid-rotor quantization. This prediction agrees with the full calculation only in the strong coupling regime. []{data-label="omega"}](omega.eps)
The results in fig. (\[omega\]) are quite striking. One sees that the rigid-rotor result is reproduced for one of the modes only in the limit where $\frac{\omega_v}{\omega_r} \rightarrow
\infty$. This can be verified explicitly: expanding the analytic expression for the appropriate eigenvalue which we denote as $\omega_c$ as a series in $\frac{\omega_r}{\omega_v}$ yields $$\omega_c = \omega_r \left( 1 - \left(\frac{\omega_r}{\omega_v}
\right )^2 + \left(\frac{\omega_r}{\omega_v} \right )^4 - 3
\left(\frac{\omega_r}{\omega_v} \right )^8 \, + \, \cdots \right
) \; \; . \label{modeex}$$ Clearly this series converges onto $\omega_r$ as $\frac{\omega_v}{\omega_r} \rightarrow \infty$ and deviates from this asymptotic value at finite values of the ratio. This result is very easy to understand physically. The limit $\frac{\omega_v}{\omega_r} \rightarrow \infty$ corresponds to tight binding; the ratio diverges as $k $, the strength of the inter-particle interaction goes to $ \infty$ with all other parameters held fixed. Of course, in the tight binding limit the system will act like a single coherent particle of mass $2 m$ and charge $q$ and one recovers the rigid-rotor result.
Moving away from the tight binding limit the particles no longer move coherently. Suppose one were to provide an initial condition in which both the particles were given an equal kick so that at $t=0$ they both had the same initial velocity and had no initial separation. As they move, they feel different magnetic forces due to the differing charges and hence begin to move apart. At this point the Hook’s law potential between the particle adds a new restoring force and the “collective” and vibrational motion now mix and the modes differ from the rigid-rotor modes. Indeed, as one moves to the opposite limit the result is also easy to understand analytically. As $k \rightarrow 0$ with all other parameters held fixed the two particles only weakly interact with each other; the charged particle is expected to make Landau oscillations essentially unencumbered by the other particle. Since the single-particle mass is exactly half the mass of the coherent system in the tight binding limit the frequency of the Landau oscillation is exactly double that case. As can be seen from fig. \[omega\], this is precisely what happens: one of the modes goes to 2 $\omega_r$ in this extreme weak binding limit. The other frequencies become small due to the lack of a significant restoring force.
The important thing to note is that scaling rules in eq. (\[toyscale2\]) imply the ratio $\frac{\omega_v}{\omega_r} $ is generically of order $N_c^0$. Thus, the large $N_c$ does not automatically force one into the tight binding limit. This in turn means that large $N_c$ by itself does not push one into the regime of validity of the rigid-rotor quantization. We see explicitly that rigid-rotor quantization is not automatically justified at large $N_c$.
It is reassuring that the analysis of this simple toy model reproduces the general arguments about time scales discussed in subsection \[tscales\]. The rigid-rotor quantization is seen to be justified only when $\omega_v \gg \omega_r$. This is the situation when there is a scale separation between the collective and vibrational motion and Born-Oppenheimer reasoning applies. However, as just noted, large $N_c$ QCD does not imply that the dynamics is in this regime.
This model is also useful in clarifying some key issues. It is well known in typical soliton models that zero modes associated with broken symmetries are orthogonal to vibrational degrees of freedom at small amplitude and do not mix[@sol]. It was precisely this fact that was used to justify rigid-rotor quantization. However, we see that for this toy problem the rigid-rotor quantization fails. The reason for this failure is easy to isolate. Although there are two flat directions, the presence of velocity-dependent forces induced by the monopole implies that there is only one pair of zero modes and not two. This second “would be” zero mode scales with $N_c$, an ordinary vibrational mode, and acts as an ordinary vibrational mode. Nothing prevents it from mixing with other vibrational modes and indeed it does. This can be seen explicitly by looking at one of the eigenvectors of the “would be” zero mode. Again, the expression is long and cumbersome in general, but can be written in a Taylor series in $\frac{\omega_r}{\omega_v}$. To second order, the motion associated with this normal mode is given by
$$\left( \begin{array}{c} x_1\\ y_1\\ x_2 \\
y_2\\\dot{x}_1\\\dot{y}_1\\\dot{x}_2\\\dot{y}_2 \end{array}
\right ) = \Re{\rm e} \left ( A \exp(i \omega_c t) \left \{
\left( \begin{array}{c} 1\\ i\\ 1 \\
i \\0\\0\\0\\0 \end{array}
\right ) + \left( \frac{\omega_r}{\omega_v} \right ) \left( \begin{array}{c} 0\\ 0\\ 0 \\
0\\ i\\ -1\\ -i\\ -1\end{array}
\right ) + \left ( \frac{\omega_r}{\omega_v}\right )^2 \left( \begin{array}{c} -1\\ i\\ 1 \\
-i\\0\\0\\0\\0 \end{array} \right ) \, + \, \cdots \right \}
\right ) \label{modemix}$$
where $\Re{\rm e} $ indicates the real part, $A$ is a complex constant fixed by initial conditions and $\omega_c$ is the normal mode frequency given approximately in eq. (\[modeex\]). Note that although the leading term in the expansion has particles one and two moving together in a coherent manner (making a circular orbit), the correction terms do not. Again, we should recall that the large $N_c$ limit does not drive one to the limit where $\frac{\omega_r}{\omega_v}$ goes to zero so that large $N_c$ does not force these correction terms to be small: at leading order in a $1/N_c$ expansion, the collective motion has mixed with vibrational motion.
Returning to the general considerations at the beginning of this section, we see that for models such as this one which contain first order time derivatives, the classical intrinsic vibrational motion can mix with the collective rotational motion. This in turn implies that the collective motion can not be isolated and quantized separately.
SU(3) soliton models
--------------------
The preceding subsection shows explicitly why rigid-rotor quantization fails for that toy model. But we are interested in SU(3) soliton models. The issue is whether the same type of behavior is seen in SU(3) models. The important point is that the behavior seen in the toy model should be generic. Any model with “exotic” motion of order $N_c^0$ and vibrational motion of the same order can be expected to mix in the absence of some symmetry keeping them distinct.
In fact, we know from the analysis of IKOR that this same behavior is seen for SU(3) solitons[@IKOR]. They study the classical motion around the soliton at quadratic order. At large $N_c$ this quadratic order is sufficient since all nonzero kaon modes when quantized have $K/f_\pi \sim N_c^{-1/2}$ corresponding to localized and nearly harmonic motion. While for the non-exotic states at large $N_c$ and exact SU(3) symmetry, they find a rotational zero mode as seen in collective quantization, they also note that there is no exotic mode with a frequency equal to the excitation given in rigid-rotor quantization. However, if the classical motion really had separated into collective and intrinsic motion as assumed in rigid-rotor quantization there would have been a classical mode at the rigid-rotor frequency. Thus the SU(3) solitons do behave in the same way as the toy model.
The critique of the time scale argument
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As noted in the introduction, the conclusion that the rigid-rotor quantization for exotic states is not justified by large $N_c$ QCD is not universally accepted. Recently Diakonov and Petrov (DP) [@DP]criticized the argument given in ref. [@Coh03] (and presented here in subsection \[tscales\] of this paper)—[*i.e.,*]{} the argument based on time scales; using the logic of this critique they concluded that rigid-rotor quantization was valid for “exoticness” of order $N_c^0$ and only breaks down when the “exoticness” is large enough to substantially alter the moment of inertia (which occurs at order $N_c^1$)[@DP]. In this section we will discuss this critique and argue that it is erroneous.
The critique has three parts: i) An argument that relevant time scale for exotic excitations is of order $N_c^{1}$ and not order $N_c^0$; ii) a general treatment of rotational-vibrational mixing in SU(3) solitons in which it is asserted that the mixing is small at large $N_c$; and iii) a toy model to illustrate the argument.
Let us begin with the discussion of point i). DP argue correctly that in the rigid-rotor quantization the characteristic angular velocity is given by $$v_{char} \sim \frac{\sqrt{\sum_{A=4}^7 {\hat{J}_A'}{}^2}}{ I_2}
\sim {\cal O}(\sqrt{\frac{e}{N_c}}) \label{vchar}$$ where $e$ is the “exoticness” . This is slow at large $N_c$ when $e \sim N_c^0$. From the behavior in eq. (\[vchar\]), it is asserted that the characteristic time scale is of order $N_c^{-1/2}$[@DP], in contradiction to the claims in ref. [@Coh03] and subsection \[tscales\] that the time scale is of order $N_c^0$. This assertion is, on its face, paradoxical. If true, general semi-classical considerations as seen in Bohr’s correspondence principle would imply that the quantal excitation energies associated with this be of order $1/N_c$ and not $N_c^0$, as, in fact, they are . DP attempt to resolve this paradox by stating that the rotation is not semi-classical because of the quantization condition in eq. \[quantcond\] which arises from the Wess-Zumino term; since the Wess-Zumino term is a full derivative term it should be thrown out classically and is thus not suitable to semi-classical arguments. However, this resolution does not hold up. As seen explicitly in subsection \[mono\] the problem of “exotic” motion of a charged particle in a monopole field can be quantized explicitly by quantizing the Landau orbits and the semi-classical result agrees with the exact answer at the analog of large $N_c$.
Given the problem with the resolution of this paradox by DP, how is one to reconcile eq. (\[vchar\]) with a characteristic time of order $N_c^0$? The answer is trivial and was discussed in detail in subsection \[tscales\]: the velocity can be of order $N_c^{-1/2}$ and the time scale $N_c^0$ provided the motion is localized to a region of order $N_c^{1/2}$. This is precisely what happens and this is verified in the model in subsection \[mono\]. In summary, part i) of this critique appears to be without foundation.
Next consider part ii). The analysis here closely parallels the original derivation of rigid-rotor quantization. Let us recapitulate the salient points.
The analysis is based on a soliton which corresponds to the local minimum of an effective action $S_{\rm eff}[\pi(x)]$ (where $\pi$ represents a dimensionless pion field—namely, the usual pion field divided by $f_\pi$); the action is proportional to $N_c$. The classical configuration $\pi_{\rm class}(x)$ which minimizes the effective action gives the soliton profile and the moments of inertia $I_{1,2}$ are computed at this minimum. One finds that the classical soliton mass, ${\cal M}_0$, and the moments of inertia, $I_{1,2}$, are all proportional to $N_c$.
The effective action may be expanded about the classical minimum at second order in the fields; it is given by: $${\cal E}_{\rm eff}[\pi_{\rm class}+\delta\pi] =
{\cal M}_0+\frac{1}{2}\delta\pi\,W[\pi_{\rm
class}]\,\delta\pi+\ldots \label{S2}$$ where $W$ is an operator for any given external field $\pi_{\rm
class}$. Since the dimensionless fields scale as $N_c^0$, $W$ is of the same order $S_{\rm eff}$, namely, $N_c^1$. This in turn implies that the harmonic fluctuations scale as $\delta\pi(x)
=O(1/\sqrt{N_c})$. The spectrum of $W$ and its eigenmodes both scale as $N_c^0$. Clearly $W$ has zero modes which are related to symmetry breaking in the classical solution. For these models this includes both translations and rotations.
Up to this point the analysis seems reasonable, although one might quibble that the object on the left-hand side of eq. (\[S2\]) should be considered an effective Lagrangian rather than an energy function since no Legendre transformation has yet been made. In any event, the next steps [@DP] are based on a set of assumptions which are quite problematic: “ [*The quantization of rotations (which are large fluctuations as they occur in flat zero-mode directions) leads to the rotational spectrum discussed in the previous section [\[that is rigid-rotor quantization\]]{}. The vibrational modes are orthogonal to those zero modes.*]{}” Given theses assumptions, it is shown that vibration-rotational coupling only becomes important when the moment of inertia changes substantially which occurs at $e \sim N_c^1$.
However, the assumption that the exotic motion is associated with zero modes and thus do not mix with vibrational motion due to orthogonality was assumed without proof to be true in analogy with problems without a Wess-Zumino term. This assumption doe not appear to be correct. In the first place, the classical motion associated with exotic motion is not associated with a zero mode due to the presence of the Wess-Zumino term. This was shown in subsection \[tscales\] where it was seen that classical zero modes are associated with quantum states that have level spacing which go to zero as the large $N_c$ (classical) limit is taken. Exotic states are not in this class. Moreover, in subsection \[mono\] it was shown explicitly that the classical motion associated with exotic states were not zero modes; rather, they were Landau orbits with a nonzero frequency. Secondly, for exotic quantum numbers, the collective motion does mix with the vibrational motion at leading order as is shown explicitly in eq. (\[modemix\]). Thus, the assumption underlying this general part of the critique seems to be wrong.
Finally, we consider part iii) of the critique. This consists of studies with a toy model with both vibrational and rotational degrees of freedom and with an analog of a Wess-Zumino term. The model considered was a charged particle moving in the field of a magnetic monopole and subject to a spherically symmetric potential with a minimum at $r=R$. In this model, it is shown that rigid-rotor quantization only becomes inaccurate when the moment of inertia is altered substantially—an effect which occurs when the exoticness is of order $N_c$. However, this model is a very poor analog of the problem of interest. Recall that the danger posed to rigid-rotor quantization is the mixing of the collective modes with vibrational modes [*which carry the same quantum number*]{}. Note that in the toy model considered here the exotic motion carries angular momentum (also note that the exotic states correspond to states with different J) while the only vibrations allowed in this model are radial vibrations which do not carry angular momentum. In this model there is nothing for the collective modes to couple to at lowest order and rigid-rotor quantization works. However, as soon as the model is rich enough to include vibrational degrees of freedom with the same quantum numbers as the collective motion (as, for example, in the model of subsection \[coupled\]), the rigid-rotor quantization fails.
To summarize this section, DP make a three part critique at the analysis of time scales given in subsection \[tscales\] and in ref. [@Coh03]. However, the first two parts of the critique are based on faulty assumptions, while the third part is based on a model which is not an analog of the relevant problem.
Discussion
==========
This paper presents strong evidence that rigid-rotor quantization is not justified on the basis of large $N_c$ considerations. The important issue is what this tells us about the nature of the $\theta^+$ and other possible exotic states.
One possibility is that the analysis based on the rigid-rotor quantization is, in fact, well justified despite the arguments given in this paper. Note that this paper [*does not*]{} show that rigid-rotor quantization is necessarily invalid but rather that it is not justified due to large $N_c$ QCD. It remains possible that it is justified due to some other reason. (For example, in the toy model in subsection \[coupled\] rigid-rotor quantization was justified if the parameters of the model had $\omega_v \gg \omega_r$). This would be most satisfactory in that the very successful prediction of the phenomenology based on rigid-rotor quantization would remain. However, if correct, it raises a very important theoretical question: namely, what justifies the rigid-rotor quantization? A second possibility is that the rigid-rotor quantization is not justified. In this case, the accurate prediction of the mass and the prediction of a narrow width in rigid-rotor quantization would have to simply be dismissed as fortuitous.
In either case, large $N_c$ QCD by itself does not appear to allow one to understand the structure of this state. In this respect, the $\theta^+$ is quite unlike the $\Delta$ and is like the more typical excited baryons. As with such baryons it may well be that the best phenomenological treatments may be based on models whose connections to QCD are quite tenuous.
The lack of validity of the rigid-rotor quantization does have one important phenomenological consequence. If the exotic states were of a collective character one could not justify treating meson-baryon scattering in exotic channels via a simple linear response theory in the context of chiral soliton models. After all, one cannot use linear response to describe pion-nucleon scattering at the $\Delta$ resonance in the Skyrme model[@KarMat; @MatPes]. However, because these exotic resonances are of a vibrational character, linear response is justified. This is useful in and of itself to describe scattering. Moreover, by imposing the $I=J$ rule on such scattering one can predict that the $\theta^+$ has low-lying partners (at least) at large $N_c$[@CohLebTheta]. These partners are related in much the same way as the $\Delta$ is to the nucleons except that both states have widths which are of order $N_c^0$. The quantum numbers of such partner states are enumerated in ref. [@CohLebTheta].
The author acknowledges Rich Lebed and Pavel Poblytisa for insightful suggestions about this work. The support of the U.S. Department of Energy for this research under grant DE-FG02-93ER-40762 is gratefully acknowledged.\
Quark Based Derivation of the Wess-Zumino Constriant \[const\]
==============================================================
In Skyrme-type models eq. (\[quantcond\]), which constrains the allowable representations follows directly from the Wess-Zumino term (which topology fixes to have a strength which is integer and can be identfied identified with $N_c$ [@Wit2]). At a more pedestrian level, it can also be easily understood at the quark level for models with explicit quark degrees of freedom. In the body-fixed frame the baryon number of the unrotated hedgehog is associated with the SU(2) sub-manifold. The body-fixed hypercharge is also associated with this sub-manifold. One can relate the body-fixed hypercharge to the body-fixed SU(3) generator as usual so that $Y'= -2 {J'}_8/ \sqrt{3}$. The baryon number, hypercharge and strangeness are related linearly. The appropriate relation for arbitrary $N_c$ is $$Y = \frac{N_c B}{3} + S \label{hyper} .$$ Note that eq. (\[hyper\]) does not coincide with the familiar relation $Y=B+S$ except for $N_c=3$. For arbitrary $N_c$ eq. (\[hyper\]) may be obtained from the known hypercharges of up, down and strange quarks: $$Y_u = 1/3 \; \;Y_d =1/3 \; \; Y_s=-2/3\; .$$ (These are the standard hypercharge assignments for quarks at $N_c=3$. It is straightforward to see that these assignments must hold for any $N_c$ provided hypercharge is isosinglet and traceless in SU(3) and has the property that the hypercharge of mesons is equal to the strangeness.) Each quark carries a baryon number of $1/N_c$ while the strangeness is zero for u and d quarks and -1 for s quarks. The combination of hypercharge, strangeness and baryon number assignments can only be satisfied if eq. (\[hyper\]) holds. Finally, observe that in the body-fixed frame, the SU(2) sub-manifold by construction has zero strangeness; thus, eq. (\[hyper\]) requires that $Y' = N_c B/3$ and the quantization condition in eq. (\[quantcond\]) immediately follows.
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---
abstract: 'We propose a probabilistic framework for modelling and exploring the latent structure of relational data. Given feature information for the nodes in a network, the scalable deep generative relational model (SDREM) builds a deep network architecture that can approximate potential nonlinear mappings between nodes’ feature information and the nodes’ latent representations. Our contribution is two-fold: (1) We incorporate high-order neighbourhood structure information to generate the latent representations at each node, which vary smoothly over the network. (2) Due to the Dirichlet random variable structure of the latent representations, we introduce a novel data augmentation trick which permits efficient Gibbs sampling. The SDREM can be used for large sparse networks as its computational cost scales with the number of positive links. We demonstrate its competitive performance through improved link prediction performance on a range of real-world datasets.'
author:
- Xuhui Fan
- Bin Li
- 'Scott A. Sisson'
- Caoyuan Li
- Ling Chen
bibliography:
- 'StochasticPartitionProcessBase.bib'
title: |
Scalable Deep Generative Relational Models\
with High-Order Node Dependence
---
Introduction
============
Bayesian relational models, which describe the pairwise interactions between nodes in a network, have gained tremendous attention in recent years, with numerous methods developed to model the complex dependencies within relational data; in particular, probabilistic Bayesian methods [@nowicki2001estimation; @kemp2006learning; @airoldi2009mixed; @miller2009nonparametric; @pmlr-v89-fan18a; @NIPS2018_RBP]. Such models have been applied to community detection [@nowicki2001estimation; @karrer2011stochastic], collaborative filtering [@porteous2008multi; @Li_transfer_2009], knowledge graph completion [@hu2016non] and protein-to-protein interactions [@huopaniemi2010multivariate]. In general, the goal of these Bayesian relational models is to discover the complex latent structure underlying the relational data and predict the unknown pairwise links [@xuhui2016OstomachionProcess; @pmlr-v84-fan18b].
Despite improving the understanding of complex networks, existing models typically have one or more weaknesses: (1) While data commonly exhibit high-order node dependencies within the network, such dependencies are rarely modelled due to limited model capabilities; (2) Although a node’s feature information closely informs its latent representation, existing models are not sufficiently flexible to describe these (potentially nonlinear) mappings well; (3) While some scalable network modelling techniques (e.g. Ber-Poisson link functions [@Ber_poisson_rai_1; @zhou2015infinite]) can help to reduce the computational complexity to the number of positive links, they require the elements of latent representations to be independently generated and cannot be used for modelling dependent variables (e.g. membership distributions on communities).
In order to address these challenges, we develop a probabilistic framework using a deep network architecture on the nodes to model the relational data. The proposed scalable deep generative relational model (SDREM) builds a deep network architecture to efficiently map the nodes’ feature information to their latent representations. In particular, the latent representations are modelled via Dirichlet distributions, which permits their interpretation as membership distributions on communities. Based on the output latent representations (i.e. membership distributions) and an introduced community compatibility matrix, the relational data is modelled through the Ber-Poisson link function [@Ber_poisson_rai_1; @zhou2015infinite], for which the computational cost scales with the number of positive links in the network.
We make two novel contributions: First, as the nodes’ latent representations are Dirichlet random variables, we incorporate the full neighbourhood’s structure information into its concentration parameters. In this way, high-order node dependence can be modelled well and can vary smoothly over the network. Second, we introduce a new data augmentation trick that enables efficient Gibbs sampling on the Ber-Poisson link function due to the Dirichlet random variable structure of the latent representations. The SDREM can be used to analyse large sparse networks and may also be directly applied to other notable models to improve their scalability (e.g. the mixed-membership stochastic blockmodel (MMSB) [@airoldi2009mixed] and its variants [@koutsourelakis2008finding; @doi:10.1080/01621459.2012.682530; @conf/icml/KimHS12]).
In comparison to existing approaches, the SDREM has several advantages. (1) [*Modelling high-order node dependence*]{}: Propagating information between nodes’ connected neighbourhoods can improve information sharing and dependence modelling between nodes. Also, it can largely reduce computational costs in contrast to considering all the pairwise nodes’ dependence, as well as avoid spurious or redundant information complications from unrelated nodes. Moreover, the non-linear real-value propagation in the deep network architecture can help to approximate the complex nonlinear mapping between the node’s feature information and its latent representations. (2) [*Scalable modelling on relational data*]{}: Our novel data augmentation trick permits an efficient Gibbs sampling implementation, with computational costs scaling with the number of positive network links only. (3) [*Meaningful layer-wise latent representation*]{}: Since the nodes’ latent representations are generated from Dirichlet distributions, they are naturally interpretable as the nodes’ memberships over latent communities.
In our analyses on a range of real-world relational datasets, we demonstrate that the SDREM can achieve superior performance compared to traditional Bayesian methods for relational data, and perform competitively with other approaches. As the SDREM is the first Bayesian relational model to use neighbourhood-wise propagation to build the deep network architecture, we note that it may straightforwardly integrate other Bayesian methods for modelling high-order node dependencies in relational data, and further improve relationship predictability.
Scalable Deep Generative Relational Models (SDREMs)
===================================================
The relational data in the SDREM is represented as a binary matrix $\pmb{R}\in\{0, 1\}^{N\times N}$, where $N$ is the number of nodes and the element $R_{ij}$ ($\forall i,j$) indicates whether node $i$ relates to node $j$ ($R_{ij}=1$ if the relation exists, otherwise $R_{ij}=0$), with the self-connection relation $R_{ii}$ not considered here. The matrix $\pmb{R}$ can be symmetric (i.e. undirected) or asymmetric (i.e. directed). The network’s feature information is denoted by a non-negative matrix $\pmb{F}\in\{\mathbb{R}^+\cup 0\}^{N\times D}$, where $D$ denotes the number of features, and where each element $F_{id}$ ($\forall i,d$) takes the value of the $d$-th feature for the $i$-th node. The deep network architecture of the SDREM is controlled by two parameters: $L$, representing the number of layers, and $K$, denoting the length of the nodes’ latent representation in each layer. The latent representation $\pmb{\pi}_i^{(l)}$ of node $i$ in the $l$-th layer is a Dirichlet random variable (i.e. a normalised vector with $(K-1)$ active elements). In this way, $\pmb{\pi}_i^{(l)}$, which we term the “membership distribution”, is interpretable as node $i$’s community distribution, where $K$ communities are modelled and ${\pi}_{ik}^{(l)}$ denotes node $i$’s interaction with the $k$-th community in the $l$-th layer. The deep network architecture of the SDREM is composed of three parts: (1) The input layer feeding the feature information; (2) The hidden layers modelling high-order node dependences; (3) The output layer of the relational data model. These component parts are detailed below.
![[Illustration and visualization of a SDREM on a $5$-node (i.e. $A, B, C, D, E$) directed network. Left: the graphical model of a $3$-layer SDREM modelling $R_{BA}, R_{ED}$. Shaded nodes (i.e. $F_{\cdot}, R_{\cdot}$) denote variables with known values, unshaded nodes denote latent variables. Right top: the generative process of a SDREM. Right bottom: the directed connection types of all $5$ nodes.]{}[]{data-label="fig:generative_process"}](GNN_Uncertainty_GM.pdf){width="\linewidth"}
![[Illustration and visualization of a SDREM on a $5$-node (i.e. $A, B, C, D, E$) directed network. Left: the graphical model of a $3$-layer SDREM modelling $R_{BA}, R_{ED}$. Shaded nodes (i.e. $F_{\cdot}, R_{\cdot}$) denote variables with known values, unshaded nodes denote latent variables. Right top: the generative process of a SDREM. Right bottom: the directed connection types of all $5$ nodes.]{}[]{data-label="fig:generative_process"}](generative_process.pdf){width="1.0\linewidth"}
![[Illustration and visualization of a SDREM on a $5$-node (i.e. $A, B, C, D, E$) directed network. Left: the graphical model of a $3$-layer SDREM modelling $R_{BA}, R_{ED}$. Shaded nodes (i.e. $F_{\cdot}, R_{\cdot}$) denote variables with known values, unshaded nodes denote latent variables. Right top: the generative process of a SDREM. Right bottom: the directed connection types of all $5$ nodes.]{}[]{data-label="fig:generative_process"}](GNN_Uncertainty_connection.pdf){width="0.6\linewidth"}
Feeding the feature information
-------------------------------
When nodes’ feature information is available, we introduce a feature-to-community transition coefficient matrix $\pmb{T}\in(\mathbb{R}^+)^{D\times K}$, where $T_{dk}$ indicates the activity of the $d$-th feature in contributing to the $k$-th latent community. The linear sum of the transition coefficients $\pmb{T}$ and feature $\pmb{F}$ forms the prior for the nodes’ first layer membership distribution $$\begin{aligned}
\label{eq:feature_information}
T_{dk}\sim\text{Gam}(\gamma^{(1)}_{d}, \frac{1}{c^{(1)}})\quad \forall d, k;\quad
\pmb{\pi}_i^{(1)}\sim \text{Dirichlet}(\pmb{F}_{i}\pmb{T}+\alpha)\quad \forall i. \end{aligned}$$ where $\text{Gam}(\gamma, 1/c)$ denotes a gamma random variable with mean $\gamma/c$ and variance $\gamma/c^2$; $\{\gamma^{(1)}_{d}\}_d$ and $c^{(1)}$ are the hyper-parameters for generating $\{T_{dk}\}_{d,k}$. From Eq. (\[eq:feature\_information\]), nodes with close feature information have similar prior knowledge and similar generated membership distributions. A supplementary contribution $\alpha$ is included in case that a node has no feature information available. For node $i$ without feature information, we have $\pmb{\pi}_i^{(1)}\sim\text{Dirichlet}(\alpha\cdot{\pmb{1}}^{1\times K})$, which is a common setting in Bayesian relational data modelling.
Modelling high-order node dependence
------------------------------------
High-order node dependence is modelled within the deep network architecture of the SDREM. In general, node $i$’s membership distribution $\pmb{\pi}_i^{(l)}$ is conditioned on the membership distributions at the $(l-1)$-th layer via an information propagation matrix $\pmb{B}^{(l-1)}\in\{\mathbb{R}^+\cup 0\}^{N\times N}$: $$\begin{aligned}
{B}_{i'i}^{(l-1)}\left\{\begin{array}{ll}
\sim\text{Gam}(\gamma^{(l)}_{1}, \frac{1}{c^{(l)}}) & \mbox{if }R_{i'i}=1; \\
\sim\text{Gam}(\gamma^{(l)}_0, \frac{1}{c^{(l)}}) & \mbox{if }i'=i; \\
=0 & \text{otherwise},
\end{array}\right.
\quad
\pmb{\pi}_i^{(l)}\sim \text{Dirichlet}((\pmb{B}^{(l-1)}_{\cdot i})^{\top}\cdot\pmb{\pi}_{1:N}^{(l-1)}),\end{aligned}$$ Following [@zhao2018dirichlet], we set the hyper-parameter distribution as $\gamma_{1}^{(l)}, \gamma_{0}^{(l)}\sim\text{Gam}({e_0^{(l)}}, \frac{1}{f_0^{(l)}}), c^{(l)}\sim\text{Gam}(g_0, \frac{1}{h_0})$. $B_{i'i}^{(l-1)}$ denotes node $i'$’s influence on node $i$ from the $(l-1)$-th to the $l$-th layer (e.g. larger values of $B_{i'i}^{(l-1)}$ will make $\pmb{\pi}_i^{(l)}$ more similar to $\pmb{\pi}_{i'}^{(l-1)}$) and $\pmb{\pi}_{1:N}^{(l)}\in\{\mathbb{R}^+\}^{N\times K}$ denotes the matrix of $N$ nodes’ membership distributions at the $l$-th layer. When there is no direct connection from node $i'$ to node $i$ (i.e. $i'\neq i\cap R_{i'i}=0$), we restrict the corresponding information propagation coefficients $B_{i'i}$ at all layers to be $0$; otherwise, we generate $B_{i'i}^{(l-1)}$ either from a node and layer-specified Gamma distribution (when $R_{i'i}=1$) or a layer-specified Gamma distribution (when $i'=i$). This can produce various benefits. On one hand, it promotes the sparseness of $\pmb{B}^{(l)}$ and reduces the cost of calculating $\pmb{B}^{(l)}$ from $\mathcal{O}(N^2)$ to the scale of the number of positive network links. On the other hand, since the SDREM uses a Dirichlet distribution (parameterised by the linear sum of node $i$’s neighbourhoods’ membership distributions at the $(l-1)$-th layer) to generate $\pmb{\pi}_i^{(l)}$, all the nodes’ membership distributions are expected to vary smoothly over the connected graph structure. That is, connected nodes are expected to have more similar membership distributions than unconnected ones.
**Flexibility in modelling variance and covariance in membership distributions** Neighbourhood-wise information propagation allows for more flexible modelling than the extreme case of independent propagation whereby $\pmb{\pi}_i^{(l)}$ is conditioned on $\pmb{\pi}_i^{(l-1)}$ only (i.e. $\{\pmb{B}^{(l)}\}_l$ is a diagonal matrix). Under independent propagation, the expected membership distribution at each layer does not change: $\mathbb{E}[\pmb{\pi}^{(l)}_{1:N}] = \pmb{\pi}^{(1)}_{1:N}$. In the SDREM, we have $\mathbb{E}[\pmb{\pi}^{(l)}_{1:N}] = [\prod_{l'=1}^{l-1}(D^{(l')})^{-1}(\pmb{B}^{(l')})^{\top}]\pmb{\pi}^{(1)}_{1:N}$, where $D^{(l)}$ is a level $l$ diagonal matrix with $D_{ii}^{(l)}=\sum_{i'}B_{i'i}^{(l)}$, $\forall i$. Based on different choices for $\{\pmb{B}^{(l)}\}_l$, the expected mean of each node’s membership distribution can incorporate information from other nodes’ input layer. In terms of variance and covariance within each $\pmb{\pi}_i^{(l)}$, independent propagation is restricted to inducing a larger variance in $\pi_{ik}^{(l)}$ and smaller covariance between $\pi_{ik_1}^{(l)}$ and $\pi_{ik_2}^{(l)}$ due to the layer stacking architecture (this can be easily verified through the law of total variance and the law of total covariance). In contrast, for the SDREM, these variances and covariances can be made either large or small depending on the choices of $\{\pmb{B}^{(l)}\}_l$ through the deep network architecture.
The Dirichlet distribution models the membership distribution $\{\pmb{\pi}_i^{(l)}\}_{i,l}$ in a non-linear way. As non-linearities are easily captured via deep learning, it is expected that the deep network architecture in the SDREM can approximate the complex nonlinear mapping between the nodes’ feature information and membership distributions sufficiently well. Further, the technique of propagating real-valued distributions through different layers might be a promising alternative to sigmoid belief networks [@gan2015learning; @NIPS2015_5655; @deep_lfrm], which mainly propagate binary variables between different layers.
#### Comparison with spatial graph convolutional networks:
Propagating information through neighbourhoods works in a similar spirit to the spatial graph convolutional network (GCN) [@atwood2016diffusion; @duvenaud2015convolutional; @hamilton2017inductive; @dai2018learning] in a frequentist setting. In addition to providing variability estimates for all latent variables and predictions, the SDREM may conveniently incorporate beliefs on the parameters and exploit the rich structure within the data. Beyond the likelihood function, the SDREM uses a Dirichlet distribution as the activation function, whereas GCN algorithms usually use the logistic function. The resulting membership distribution representation of the SDREM may provide a more intuitive interpretation than the node representation (node embedding) in the GCN.
Scalable relational data modelling
----------------------------------
We model the final-layer relational data via the Ber-Poisson link function [@Ber_poisson_rai_1; @zhou2015infinite], $R_{ij}\sim\text{Bernoulli}(1-e^{-\sum_{k_1k_2}X_{ik_1}\Lambda_{k_1k_2} X_{jk_2}})$, where $X_{ik}$ is the latent count of node $i$ on community $k$ and ${\Lambda}_{k_1k_2}\in \mathbb{R}^+$ is a compatibility value between communities $k_1$ and $k_2$. In existing work with the Ber-Poisson link function, all of the $\{X_{ik}\}_{i,k}$ terms are required to be independently generated (either from a Gamma [@zhou2015infinite; @leveraging_node_attributes] or Bernoulli distribution [@deep_lfrm]) to allow for efficient Gibbs sampling. However, in the SDREM, the elements of the output latent representation $(\pi_{i1},\ldots, \pi_{iK})$ are jointly generated from a Dirichlet distribution. These normalised elements are dependent on each other and it is not easy to enable Gibbs sampling for each individual element $\{\pi_{ik}\}_k$.
To address this problem, we use a decomposition strategy to isolate the elements $\{\pi_{ik}\}_k$. We use multinomial distributions, with $\{\pmb{\pi}_i\}_i$ as event probabilities, to generate $K$-length counting vectors $\{\pmb{X}_i\}_i$. Each $\pmb{X}_i$ can be regarded as an estimator of $\pmb{\pi}_i$. Since the sum of the $\{{X}_{ik}\}_k$ is fixed as the number of trials (denoted as $M_i$) in the multinomial distribution, we further let $M_i$ be generated as $M_i\sim\text{Poisson}(M)$. Based on the Poisson-Multinomial equivalence [@dunson2005bayesian], each $X_{ik}$ is then equivalently distributed $X_{ik}\sim\text{Poisson}(M\pi_{ik})$.
Following the settings of Ber-Poisson link function, a [latent integer matrix]{} $\pmb{Z}_{ij}\in\mathbb{N}^{K\times K}$ is introduced, where the $(k_1, k_2)$-th entry is $Z_{ij, k_1k_2}\sim\text{Poisson}(X_{ik_1}\Lambda_{k_1k_2}X_{jk_2})$. $R_{ij}$ is then generated by evaluating the degree of positivity of the matrix $Z_{ij}$. That is, $\forall (i,j), k_1, k_2$: [$$\begin{aligned}
& M_{i}\sim\text{Poisson}(M),\quad (X_{i1}, \ldots, X_{iK})\sim\text{Multi}(M_i;\pi_{i1}^{(L)}, \ldots, \pi_{iK}^{(L)}),\quad \Lambda_{k_1k_2}\sim\text{Gam}(k_{\Lambda}, \frac{1}{\theta_{\Lambda}}), \nonumber \\
& Z_{ij, k_1k_2}\sim \text{Poisson}(X_{ik_1}\Lambda_{k_1k_2}X_{jk_2})\quad\mbox{and}\quad R_{ij}=\pmb{1}(\sum_{k_1,k_2} Z_{ij,k_1k_2}>0).\end{aligned}$$]{}Here, the prior distribution for generating $X_{ik}$ and the likelihood based on $X_{ik}$ are both Poisson distributions. Consequently, we may implement posterior sampling by using Touchard polynomials [@roman2005umbral] (details in Section \[section:inference\]).
To model binary or count data, the Ber-Poisson link function [@Ber_poisson_rai_1; @zhou2015infinite] decomposes the latent counting vector $\pmb{X}_i$ into the latent integer matrix $\pmb{Z}_{ij}$. An appealing property of this construction is that we do not need to calculate the latent integers $\{z_{ij,k_1k_2}\}_{k_1,k_2}$ over the $0$-valued $R_{ij}$ data as they are equal to $0$ almost surely. Hence, the focus can be on the positive-valued relational data. This is particularly useful for real-world network data as usually only a small fraction of the data is positive. Hence, the computational cost for inference scales only with the number of positive relational links.
When nodes’ feature information is not available (i.e. $\pmb{F}=0^{N\times D}$) and $L=1$, the SDREM reduces to the same settings as the MMSB [@airoldi2009mixed]. In particular, the membership distributions of both the MMSB and the SDREM follow the same Dirichlet distribution $\{\pmb{\pi}_i\}_i\sim\text{Dirichlet}({\alpha}^{1\times K})$. As the MMSB and its variants [@koutsourelakis2008finding; @doi:10.1080/01621459.2012.682530; @conf/icml/KimHS12] introduce pairwise latent labels for all the relational data (both $1$ and $0$-valued data), it requires a computational cost of $\mathcal{O}(N^2)$ to infer all latent variables. In contrast, our novel data augmentation trick can be straightforwardly applied in these models (by simply replacing the Ber-Beta likelihood [@nowicki2001estimation; @kemp2006learning] with Ber-Poisson link function) and reduce their computational cost to the scale of the number of positive links. We show in Section \[sec:experiments\] that we can also get better predictive performance with this strategy.
Inference {#section:inference}
=========
The joint distribution of the relational data and all latent variables in the SDREM is: [$$\begin{aligned}
&P(\{\pmb{\pi}_i^{(l)}\}_{i, l},
\{\pmb{B}^{(l)}\}_l,\pmb{\Lambda}, \{Z_{ij, k_1k_2}\}_{i,j,k_1,k_2}, \{R_{ij}\}_{i,j}, \{X_{ik}\}_{i,k}, \pmb{T}|\pmb{F}, \pmb{\gamma}, \pmb{c}, \alpha, M, k_{\Lambda}, \theta_{\Lambda})\nonumber \\
=&\left[\prod_{i=1}^nP(\pmb{\pi}_i^{(1)}|\alpha, \pmb{F}_i, \pmb{T})\right]\nonumber \prod_{l=1}^{L-1}\left[P(\pmb{B}^{(l)}\vert \gamma_{i}^{(l)}, c^{(l)})\prod_{i=1}^nP(\pmb{\pi}_i^{(l+1)}|\{\pmb{\pi}_{i'}^{(l)}\}_{i':R_{i'i}=1},\pmb{\pi}_i^{(l)},\pmb{B}^{(l)})\right]P(\pmb{\Lambda}\vert k_{\Lambda}, \theta_{\Lambda})\nonumber \\
&\times\left[\prod_{i,k}P(X_{ik}|{\pi}_{ik}^{(L)}, M)\right]\left[\prod_{(i,j)|R_{ij}=1,k_1,k_2}P(Z_{ij, k_1k_2}|X_{ik_1}, X_{jk_2}, {\Lambda}_{k_1k_2})\right]\left[\prod_{f,k}P(T_{dk}\vert \gamma_{f}^{(1)}, c^{(1)})\right].\end{aligned}$$]{} By introducing auxiliary variables, all latent variables can be sampled via efficient Gibbs sampling. This section focuses on inference for $\{X_{ik}\}_{i,k}$, which is the key variable involving the data augmentation trick. Sampling the membership distributions $\{\pmb{\pi}_i^{(l)}\}_{i,l}$ is as implemented in Gamma Belief Networks [@JMLR:v17:15-633] and Dirichlet Belief Networks [@zhao2018dirichlet], which mainly use a bottom-up mechanism to propagate the latent count information in each layer. As sampling the other variables is trivial, we relegate the full sampling scheme to the Supplementary Material (Appendix A).
#### Sampling $\{X_{ik}\}_{i,k}$:
From the Poisson-Multinomial equivalence [@dunson2005bayesian] we have $M_{i}\sim\text{Poisson}(M_i)$, $$(X_{i1}, \ldots, X_{iK})\sim\text{Multi}(M_i;\pi_{i1}^{(L)}, \ldots, \pi_{iK}^{(L)})\overset{d}{=} X_{ik}\sim\text{Poisson}(M\pi_{ik}^{(L)}), \forall k. \nonumber
$$ Both the prior distribution for generating $X_{ik}$ and the likelihood parametrised by $X_{ik}$ are Poisson distributions. The full conditional distribution of $X_{ik}$ (assuming $z_{ii,\cdot\cdot}=0, \forall i$) is then [$$\label{HERE}
P(X_{ik}\vert M_i,\pmb{\pi}, \pmb{\Lambda}, \pmb{Z})\propto \frac{\left[M_i\pi_{ik}^{(L)}e^{-\sum_{j\neq i,k_2}X_{jk_2}(\Lambda_{kk_2}+\Lambda_{k_2k})}\right]^{X_{ik}}}{X_{ik}!} \left(X_{ik}\right)^{\sum_{j_1 ,k_2}Z_{ij_1,kk_2}+\sum_{j_2,k_1}Z_{j_2i,k_1k}}. $$]{} This follows the form of Touchard polynomials [@roman2005umbral], where $1=\frac{1}{e^{x}T_n(x)}\sum_{k=0}^{\infty}\frac{x^kk^n}{k!}$ with $T_n(x)=\sum_{k=0}^n\{
\begin{matrix}
n \\
k
\end{matrix}\}x^k$ and where $\{
\begin{matrix}
n \\
k
\end{matrix}\}$ is the Stirling number of the second kind. A draw from is then available by comparing a $\text{Uniform}(0,1)$ random variable to the cumulative sum of $\{\frac{1}{e^{x}T_n(x)}\cdot\frac{x^kk^n}{k!}\}_k$.
Related Work
============
There is a long history of using Bayesian methods for relational data. Usually, these models build latent representations for the nodes and use the interactions between these representations to model the relational data. Typical examples include the [stochastic blockmodel]{} [@nowicki2001estimation; @newman2004finding; @kemp2006learning] (which uses latent labels), the mixed-membership stochastic blockmodel (MMSB) [@airoldi2009mixed; @koutsourelakis2008finding] (which uses membership distributions) and the latent feature relational model (LFRM) [@miller2009nonparametric; @PalKnoGha12] (which uses binary latent features). As most of these approaches are constructed using shallow models, their modelling capability is limited.
The Multiscale-MMSB [@doi:10.1080/01621459.2012.682530] is a related model, which uses a nested-Chinese Restaurant Process to construct hierarchical community structures. However, its tree-type structure is quite complicated and hard to implement efficiently. The [Nonparametric Metadata Dependent Relational]{} model (NMDR) [@conf/icml/KimHS12] and the Node Attribute Relational Model (NARM) [@leveraging_node_attributes] also use the idea of transforming nodes’ feature information to nodes’ latent representations. However, because of their shallow latent representation, these methods are unable to describe higher-order node dependencies.
The hierarchical latent feature model (HLFM) [@deep_lfrm] may be the closest model to the SDREM, as they each build up deep network architecture to model relational data. However, the HLFM uses a sigmoid belief network, and does not consider high-order node dependencies, so that each node only depends on itself through layers. Finally, feature information enters in the last layer of the deep network architecture, and so the HLFM is unable to sufficiently describe nonlinear mappings between the feature information and the latent representation.
Recent developments [@gan2015learning; @NIPS2015_5655] in Poisson matrix factorisation also try to build deep network architecture for latent structure modelling. Since these mainly use sigmoid belief networks, the way of propagating binary variables is different from our real-valued distributions propagation. Information propagation through Dirichlet distributions in the SDREM follows the approaches of [@JMLR:v17:15-633][@zhao2018dirichlet]. However, their focus is on topic modelling and no neighbourhood-wise propagation is discussed in these methods.
Our SDREM shares similar spirit of the Variational Graph Auto-Encoder (VGAE) [@kipf2016variational; @mehta2019stochastic] algorithms. Both of the algorithms aim at combining the graph convolutional networks with Bayesian relational methods. However, VGAE has a larger computational complexity ($\mathcal{O}(N^2)$). It uses parameterized functions to construct the deep network architecture and the probabilistic nature occurs in the output layer as Gaussian random variables only. In contrast, SDREM constructs multi-stochastic-layer architectures (with Dirichlet random variables at each layer). Thus, SDREM would have better model interpretations (see Figure \[fig:latent\_feature\_vis\]). We note that recent work [@zhang2018bayesian] also claims to estimate uncertainty in the graph convolutional neural networks setting. This work uses a two-stage strategy: it firstly takes the observed network as a realisation from a parametric Bayesian relational model, and then uses Bayesian Neural Networks to infer the model parameters. The final result is a posterior distribution over these variables. Unlike the SDREM, this work performs the inference in two stages and also lacks inferential interpretability.
#### Computational complexities
The computational complexity of the SDREM is $\mathcal{O}(NDK+(NK+N_E)L+N_EK^2)$ and scales to the number of positive links, $N_E$. In particular, $\mathcal{O}(NDK)$ refers to the feature information incorporation in the input layer, $\mathcal{O}((NK+N_E)L)$ refers to the information propagation in the deep network architecture and $\mathcal{O}(N_EK^2)$ refers to the relational data modelling in the output layer. The SDREM’s computational complexity is comparable to that of the HLFM, which is $\mathcal{O}(NDK+NKL+N_EK^2)$, and the NARM, which is $\mathcal{O}(NDK+N_EK^2)$ [@leveraging_node_attributes] and is significantly less than that of the MMSB-type algorithms.
Experiments {#sec:experiments}
===========
Dataset $N$ $N_E$ $D$ F.D. Dataset $N$ $N_E$ $D$ F.D.
--------- ---------- ---------- ---------- ---------- --------- --------- ----------- --------- -----------
Citeer $3,312 $ $4,715$ $3,703 $ $0.86\%$ Cora $2,708$ $5,429$ $1,433$ $1.27\%$
Pubmed $2,000 $ $17,522$ $500 $ $1.80\%$ PPI $4,000$ $105,775$ $50$ $10.20\%$
: [Dataset information. $N$ is the number of nodes, $N_E$ is the number of positive links, $D$ is the number of features, F.D.$=\#$ nonzeros entries$/\#$ total entries in $F$ and it refers to the density of features.]{}[]{data-label="dataset_information"}
#### Dataset Information
In the following, we examine four real-world datasets: three standard citation networks ([*Citeer*]{}, [*Cora*]{}, [*Pubmed*]{} [@citation_dataset] and one protein-to-protein interaction network ([*PPI*]{}) [@protein_dataset]. Summary statistics for these datasets are displayed in Table \[dataset\_information\]. In the citation datasets, nodes correspond to documents and edges represent citation links. A node’s features comprise the documents’ bag-of-words representations. In the protein-to-protein dataset, we use the pre-processed feature information provided by [@hamilton2017inductive].
#### Evaluation Criteria
We primarily focus on link prediction and use this to evaluate model performance. We use AUC (Area Under ROC Curve) and Average Negative-Log-likelihood on test relational data as the two comparison criteria. The AUC value represents the probability that the algorithm will rank a randomly chosen existing-link higher than a randomly chosen non-existing link. Therefore, the higher the AUC value, the better the predictive performance. For hyper-parameters we specify $M\sim\text{Gam}(N, 1)$ for all datasets, and $\{\gamma^{(1)}_d\}_d, \{\gamma_1^{(l)}, \gamma_0^{(l)}\}_{l}, \{c^{(l)}\}_l$ are all given $\text{Gam}(1, 1)$ priors. Each reported criteria value is the mean of $10$ replicate analyses. Each replicate uses $2000$ MCMC iterations with the first $1000$ discarded as burn-in. Unless specified, reported AUC values are obtained by using $90\%$ (per row) of the data as training data and the remaining $10\%$ as test data. The testing relational data are not used when constructing the information propagation matrix (i.e. we set $\{\beta_{i'i}^{(l)}\}_l=0$ if $R_{i'i}$ is testing data).
#### Validating the data augmentation trick:
We first evaluate the effectiveness of the data augmentation trick through comparisons with the MMSB [@airoldi2009mixed]. To make a fair comparison, we specify the SDREM as $\pmb{F}=0^{N\times 1}, L=1,K=20$, so that the membership distributions in each model follow the same Dirichlet distribution $\{\pmb{\pi}_i\}_i\sim\text{Dirichlet}({\alpha}\cdot\pmb{1}^{1\times 20})$. Figure \[fig:different\_parameter\_settings\] (left panel) displays the mean AUC and per iteration running time for these two models. It is clear that the AUC values of the simplified SDREM are always better than those of the MMSB, and the time required for one iteration in the SDREM is substantially lower (at least two orders of magnitude lower) than that of the MMSB. Note that the running time of the SDREM is highest for the PPI dataset, since it contains the largest number of positive links and the computational cost of the SDREM scales with this value.
#### Different settings of $K$ and $L$:
We evaluate the SDREM’s behaviour under different architecture settings, through the influence of two parameters: $K$, the length of the membership distributions, and $L$, the number of layers. When testing the effect of different values of $K$ we fixed $L=3$, and when varying $L$ we fixed $K=20$. Figure \[fig:different\_parameter\_settings\] (right panel) displays the resulting mean AUC values under these settings. As might be expected, the SDREM’s AUC value increases with higher model complexity (i.e. larger values of $K$ and $L$). The worst performance occurs with $L=1$ layer as it has the least flexible modelling capability. Considering the computational complexity and modelling power, we set $K=20$ and $L=4$ for the remaining analyses in this paper.
#### Deep network architecture:
We evaluate the advantage of using neighbourhood connections to propagate layer-wise information. Three different deep network architectures are compared: (1) [*Plain-SDREM.*]{} We assume the nodes’ feature information is unavailable and use an identity matrix to represent the features (i.e. $\pmb{F}=I_{N\times N}$) (we tried two cases, $\pmb{F}=0^{N\times 1}$ and $\pmb{F}=I_{N\times N}$ and found the latter to perform better). (2) [*Fully-connected-SDREM (Full-SDREM).*]{} The propagation coefficient $B_{i'i}^{(l)}$ is not restricted to be $0$ when $R_{i'i}=0$ and instead a hierarchical Gamma process is specified as a sparse prior on all the propagation coefficients. (3) [*Independent-SDREM (Inde-SDREM).*]{} This assumes each node propagates information only to itself and does not exchange information with other nodes in the deep network architecture (i.e. each $\{\pmb{B}^{(l)}\}_l$ is a diagonal matrix).
Figure \[fig:different\_network\_configuration\] shows the performance of each of these different configurations against the non-restricted SDREM. It is clear that the non-restricted SDREM achieves the best performance in both mean AUC and negative-Log-Likelihood among all network configurations. The Full-SDREM consistently performs the worst among all configurations. This suggests that the fully connected architecture is a poor candidate, and the sampler may become easily be trapped in local modes.
#### Performance in the presence of feature information:
We compare the SDREM with several alternative Bayesian methods for relational data and one Graph Convolutional Network model. We examine: the Hierarchical Latent Feature Relational Model (HLFM) [@deep_lfrm], the Node Attribute Relational Model (NARM) [@leveraging_node_attributes], the Hierarchical Gamma Process-Edge Partition Model (HGP-EPM) [@zhou2015infinite] and a graph convolutional neural network (GCN) [@kipf_semi_supervised]. The NARM, HGP-EPM and GCN methods are executed using their respective authors’ implementations, under their default settings. The HLFM is implemented to the best of our abilities and we set the same number of layers and length of latent binary representation as the SDREM. For the GCN, the AUC value is calculated based on the pairwise similarities between the node representations and the ground-truth relational data and the Negative Log-Likelihood is unavailable due to its frequentist setting.
Figure \[fig:different\_methods\_comparison\] shows the performance of each method on the four datasets, under different ratios of training data ($x$-axis). In terms of AUC, the SDREM performs the best among all the methods when the proportion of training data ratio is larger than $0.5$. However, the performance of the SDREM is not outstanding when the training data ratio is less than $0.5$. This may partly be due to there being insufficient relational data to effectively model the latent counts. Since the SDREM and the HLFM are the best performing two algorithms in most cases, this confirms the effectiveness of utilising a deep network architecture. Similarly conclusions can be drawn based on the negative log-likelihood: the SDREM and the HLFM are the best performing two algorithms. **Comparison with Variational Graph Auto-Encoder** We also make brief comparisons with the Variational Graph Auto-Encoder (VGAE) [@kipf2016variational]. Taking $90\%$ of the data as training data and the remaining as testing data, the average AUC scores of $16$ random VGAE runs for these datasets are: [*Citeseer*]{} (0.863), [*Cora*]{} (0.854), [*Pubmed*]{} (0.921) and [*PPI*]{} (0.934). Considering the attributes of these datasets, we find that VGAE obtains a better performance than our SDREM in the datasets with sparse linkages, whereas their performance in other types of datasets are competitive. This phenomenon might be caused by two reasons: (1) due to the inference nature (backward latent counts propagating and forward variable sampling), our SDREM propagates less counting information (see Table \[table:latent\_counts\]) to higher layers. The deep hierarchical structure might be less powerful in sparse networks; (2) the Sigmod and ReLu activation functions might be more flexible than the Dirichlet distribution for the case of sparse networks. We will keep on investigating this issue in the future work.
Dataset Layer $3$ Layer $2$ Layer $1$ Dataset Layer $3$ Layer $2$ Layer $1$
--------- ----------- ----------- ----------- --------- ----------- ----------- -----------
Citeer $533.7 $ $7.8$ $2.5 $ Cora $290.1$ $7.0$ $2.3$
Pubmed $292.4 $ $24.8$ $10.1 $ PPI $65.6$ $20.1$ $12.7$
: [Average latent counts (per node) in different layers.]{}[]{data-label="table:latent_counts"}
#### Latent structure visualization:
We also visualize the latent structures of the model to get further insights in Figure \[fig:latent\_feature\_vis\]. According to the left panel, we can see that the membership distributions gradually become more distinguished along with the layers. The less distinguished membership distributions might indicate higher abstraction of the latent features. In particular, the normalized latent counting vector ($\pmb{X}$) looks to be identical to the output membership distribution $\pmb{\pi}^{(3)}$. This verifies that our introduction of $\pmb{X}$ seems to successfully pass the information to the latent integers variable $\pmb{Z}$. In the right panel of information propagation matrix, we can see that the neighbourhood-wise information seems to become weaker from the input layer to the output layer.
Conclusion
==========
We have introduced a Bayesian framework by using deep latent representations for nodes to model relational data. Through efficient neighbourhood-wise information propagation in the deep network architecture and a novel data augmentation trick, the proposed SDREM is a promising approach for modelling scalable networks. As the SDREM can provide variability estimates for its latent variables and predictions, it has the potential to be a competitive alternative to frequentist graph convolutional network-type algorithms. The promising experimental results validate the effectiveness of the SDREM’s deep network architecture and its competitive performance against other approaches. Since the SDREM is the first work to use neighbourhood-wise information propagation in Bayesian methods, combining this with other Bayesian relational models and other applications with pairwise data (e.g. collaborative filtering) would be interesting future work.
Acknowledgements {#acknowledgements .unnumbered}
================
Xuhui Fan and Scott A. Sisson are supported by the Australian Research Council through the Australian Centre of Excellence in Mathematical and Statistical Frontiers (ACEMS, CE140100049), and Scott A. Sisson through the Discovery Project Scheme (DP160102544). Bin Li is supported by Shanghai Municipal Science & Technology Commission (16JC1420401) and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning.
Inference algorithm
===================
Back-propagate the hidden counts from the output layer to the input layer
-------------------------------------------------------------------------
We first back-propagate the hidden counts from the output layer to the input layer sequentially. In the output layer, $\{\pmb{X}_{i}\}_i$ are regarded as the hidden counts and we denote as $\pmb{X}_{i}=\pmb{m}_i^{(L)}, \forall i$. For layer $l$ ($l=1, \ldots, L-1$), we may first integrate $\pmb{\pi}_i^{(l)}$ and obtain the likelihood term for $\pmb{\psi}_i^{(l)}$ as: [ $$\begin{aligned}
\label{eq:counts_origin}
\mathcal{L}(\pmb{\psi}_i^{(l)})=\frac{\Gamma(\sum_k\psi_{ik}^{(l)})}{\Gamma(\sum_k\psi_{ik}^{(l)}+\sum_km_{ik}^{(l)})}\cdot \prod_k\frac{\Gamma(\psi_{ik}^{(l)}+m_{ik}^{(l)})}{\Gamma(\psi_{ik}^{(l)})}\end{aligned}$$]{}where $\Gamma(\cdot)$ is a Gamma function, $m_{ik}^{(l)}$ refers to the “hidden counts” for the $l$-th layer. Introducing two auxiliary variables $q_i^{(l)}, \{y_{ik}^{(l)}\}_k$ [@zhao2018dirichlet] helps to further augment the likelihood of $\pmb{\psi}_i^{(l)}$ in Eq. (\[eq:counts\_origin\]) as: $$\begin{aligned}
\label{eq:auxiliary_variable}
\mathcal{L}(\pmb{\psi}_i^{(l)}, {q}_i^{(l)}, \pmb{y}_{i}^{(l)})\propto\prod_k\left(q_i^{(l)}\right)^{\psi_{ik}^{(l)}}\left(\psi_{ik}^{(l)}\right)^{y_{ik}^{(l)}}\end{aligned}$$ where $q_i^{(l)}\sim\text{Beta}(\sum_k\psi_{ik}^{(l)}, \sum_km_{ik}^{(l)}), y_{ik}^{(l)}\sim\text{CRT}(m_{ik}^{(l)}, \psi_{ik}^{(l)})$, $\text{CRT}(\cdot)$ is a Chinese Restaurant Table distribution. As a result, $\pmb{y}_i^{(l)}$ can be defined as the latent count vector from the input count vector $\pmb{m}_{i}^{(l)}$.
While $\psi_{ik}^{(l)}=\sum_{i'}{\pi}_{i'k}^{(l-1)}B_{i'i}^{(l-1)}$, the latent count $y_{ik}^{(l)}$ on $\psi_{ik}^{(l)}$ to the previous $(l-1)$-th layer can be generated as: [$$\begin{aligned}
\label{eq:latent_counts_z_ikl_bottom_layer}
\left(h_{1ik}^{(l)}, \ldots, h_{Nik}^{(l)}\right)\sim\text{Multi}\left(y_{ik}^{(l)}; \frac{{\pi}_{1k}^{(l-1)}B_{1i}^{(l-1)}}{\psi_{ik}^{(l)}}, \ldots, \frac{{\pi}_{Nk}^{(l-1)}B_{Ni}^{(l-1)}}{\psi_{ik}^{(l)}}\right)\end{aligned}$$]{} The latent count of the $(l-1)$-th layer can be summarized as $$\begin{aligned}
\label{eq:sum_labelss}
m_{i'k}^{(l-1)}=\sum_{i}h_{i'ik}^{(l)}\end{aligned}$$ to represent the “hidden counts” in the $(l-1)$-th layer.
Posterior sampling in a top-down manner
---------------------------------------
#### Sampling $\{T_{dk}\}_{d, k}$
As $\pmb{T}$ is in the shape of $D\times K$, Eq. (\[eq:auxiliary\_variable\]) may be modified as: $$\begin{aligned}
\mathcal{L}\left(\{\pmb{\psi}_i^{(1)}, {q}_i^{(1)}, \pmb{y}_{i}^{(1)}\}_i\right)\propto\prod_{i=1}^N\prod_{k=1}^K\left(q_{i}^{(1)}\right)^{\psi_{ik}^{(1)}}\left(\psi_{ik}^{(1)}\right)^{y_{ik}^{(1)}}\end{aligned}$$ where $q_i^{(1)}\sim\text{Beta}(\sum_k\psi_{ik}^{(1)}, \sum_km_{ik}^{(1)}), y_{ik}^{(1)}\sim\text{CRT}(m_{ik}^{(1)}, \psi_{ik}^{(1)})$. The latent count $y_{ik}^{(1)}$ on $\psi_{ik}^{(1)}$ to the input layer $k'$ can be defined as: $$\begin{aligned}
\label{eq:latent_counts_z_ifl_top_layer}
\left(h_{i1k}^{(1)}, \ldots, h_{iDk}^{(1)}, h_{i\alpha k}^{(1)}\right)\sim\text{Multi}\left(y_{ik}^{(1)}; \frac{{F}_{i1}T_{1k}}{\psi_{ik}^{(1)}}, \ldots, \frac{{F}_{iD}T_{Dk}}{\psi_{ik}^{(1)}}, \frac{\alpha}{\psi_{ik}^{(1)}}\right)\end{aligned}$$
Replacing $\psi_{ik}^{(L)}=\sum_{d}F_{id}T_{dk}+\alpha$, we get the likelihood of $T_{dk}$ as: $$\begin{aligned}
\mathcal{L}(T_{dk})\propto e^{T_{dk}(\sum_iF_{id}\log q_{i}^{(L)})}\left(T_{dk}\right)^{\sum_i h_{idk}^{(L)}}\end{aligned}$$ $T_{dk}$’s posterior distribution is $$\begin{aligned}
\label{eq:posterior_t_fk}
T_{dk}\sim \text{Gam}(k_T+\sum_{i}h_{idk}^{(L)}, \frac{1}{\theta_T-\sum_iF_{id}\log q_i^{(L)}})\end{aligned}$$
#### Sampling $\{\pmb{\pi}_i^{(l)}\}_{i, l}$
After obtaining the latent counts for each layer, the posterior inference in $\pmb{\pi}_i^{(l)}$ can be proceeded as: $$\begin{aligned}
\label{eq:posterior_pi_i}
\pmb{\pi}_i^{(l)}\sim\text{Dirichlet}(\psi_{i1}^{(l)}+m_{i1}^{(l)}, \ldots, \psi_{iK}^{(l)}+m_{iK}^{(l)})\end{aligned}$$
#### Sampling $\{\pmb{B}_{i'i}^{(l)}\}_{i', i, l}$
For $\pmb{B}_{i'i}^{(l)}$, the likelihood can be represented as: $$\begin{aligned}
\mathcal{L}(B_{i'i}^{(l)})\propto e^{\log q_{i'}^{(l)}B_{i'i}^{(l)}}\left(B_{i'i}^{(l)}\right)^{\sum_k h_{i'ik}^{(l)}}\end{aligned}$$
For $R_{i'i}\neq 0\cap i'\neq i$, the prior for $B_{i'i}^{(l)}$ is $\text{Gam}(\gamma^{(l)}_{1}, \frac{1}{c^{(l)}})$, the posterior distribution is $$\begin{aligned}
\label{eq:posterior_beta_ii}
B_{i'i}^{(l)}\sim \text{Gam}(\gamma_{1}^{(l)}+\sum_{k}h_{i'ik}^{(l)}, \frac{1}{c^{(l)}-\log q_{i'}^{(l)}})\end{aligned}$$
For $i'= i$, the prior for $B_{ii}^{(l)}$ is $\text{Gam}(\gamma_0^{(l)}, \frac{1}{c^{(l)}})$, the posterior distribution is $$\begin{aligned}
\label{eq:posterior_beta_ii_1}
B_{ii}^{(l)}\sim \text{Gam}(\gamma_0^{(l)}+\sum_{k}h_{iik}^{(l)}, \frac{1}{c^{(l)}-\log q_i^{(l)}})\end{aligned}$$
#### Sampling $\{X_{ik}\}_{i,k}$:
From the Poisson-Multinomial equivalence [@dunson2005bayesian] we have $M_{i}\sim\text{Poisson}(M)$, $$(X_{i1}, \ldots, X_{iK})\sim\text{Multi}(M_i;\pi_{i1}^{(L)}, \ldots, \pi_{iK}^{(L)})\overset{d}{=} X_{ik}\sim\text{Poisson}(M\pi_{ik}^{(L)}), \forall k. \nonumber
$$ Both the prior distribution for generating $X_{ik}$ and the likelihood parametrised by $X_{ik}$ are Poisson distributions. The full conditional distribution of $X_{ik}$ (assuming $z_{ii,\cdot\cdot}=0, \forall i$) is then [$$\label{HERE}
P(X_{ik}\vert M,\pmb{\pi}, \pmb{\Lambda}, \pmb{Z})\propto \frac{\left[M\pi_{ik}^{(L)}e^{-\sum_{j\neq i,k_2}X_{jk_2}(\Lambda_{kk_2}+\Lambda_{k_2k})}\right]^{X_{ik}}}{X_{ik}!} \left(X_{ik}\right)^{\sum_{j_1 ,k_2}Z_{ij_1,kk_2}+\sum_{j_2,k_1}Z_{j_2i,k_1k}}. $$]{} This follows the form of Touchard polynomials [@roman2005umbral], where $1=\frac{1}{e^{x}T_n(x)}\sum_{k=0}^{\infty}\frac{x^kk^n}{k!}$ with $T_n(x)=\sum_{k=0}^n\{
\begin{matrix}
n \\
k
\end{matrix}\}x^k$ and where $\{
\begin{matrix}
n \\
k
\end{matrix}\}$ is the Stirling number of the second kind. A draw from is then available by comparing a $\text{Uniform}(0,1)$ random variable to the cumulative sum of $\{\frac{1}{e^{x}T_n(x)}\cdot\frac{x^kk^n}{k!}\}_k$.
#### Sampling $\{Z_{ij, k_1k_2}\}_{i,j,k_1,k_2}$
We first sample $Z_{ij,\cdot \cdot}$ from a Poisson distribution with positive support: $$\begin{aligned}
\label{eq:posterior_Z_ij_1}
Z_{ij, \cdot \cdot}\sim \text{Poisson}_+(\sum_{k_1,k_2}X_{ik_1}X_{jk_2}\Lambda_{k_1k_2}), \text{where } Z_{ij, \cdot\cdot} = 1, 2, 3,\ldots\end{aligned}$$ Then, $\{Z_{ij, k_1k_2}\}_{k_1, k_2}$ can be obtained through the Multinomial distribution as: $$\begin{aligned}
\label{eq:posterior_Z_ij_2}
(\{Z_{ij, k_1k_2}\}_{k_1, k_2})\sim\text{Multinomial}\left(Z_{ij, \cdot\cdot}; \left\{\frac{X_{ik_1}X_{jk_2}\Lambda_{k_1k_2}}{\sum_{k_1,k_2}X_{ik_1}X_{jk_2}\Lambda_{k_1k_2}}\right\}_{k_1, k_2}\right)\end{aligned}$$
#### Sampling $\{\Lambda_{k_1k_2}\}_{k_1, k_2}$
For $\Lambda_{k_1k_2}$’s posterior distribution, we get $$\begin{aligned}
P(\Lambda_{k_1k_2}\vert -)\propto \exp{\left(-\Lambda_{k_1k_2}(\sum_{i,j}X_{ik_1}X_{jk_2})\right)}\Lambda_{k_1k_2}^{\sum_{i,j}Z_{ij, k_1k_2}}\cdot \exp{(-\Lambda_{k_1k_2}\theta_{\Lambda})}\Lambda^{k_{\Lambda}-1}\end{aligned}$$ Thus, we get $$\begin{aligned}
\label{eq:posterior_Lambda}
\Lambda_{k_1k_2}\sim\text{Gam}\left(\sum_{i,j}Z_{ij, k_1k_2}+k_{\Lambda}, \frac{1}{\theta_{\Lambda}+\sum_{i,j}X_{ik_1}X_{jk_2}}\right)\end{aligned}$$
#### Sampling $M$
$M_i$’s posterior distribution is: $$\begin{aligned}
P(M\vert -) = M^{k_{M}-1}\exp(-\theta_{M} M)\prod_{i,k}\left(\exp(-M\pi_{ik}^{(L)})\right)M^{\sum_{i,k}X_{ik}}\end{aligned}$$ Thus, we sample $M$ from: $$\begin{aligned}
\label{eq:posterior_M}
M\sim\text{Gam}\left(k_{M}+\sum_{i,k}X_{ik}, \frac{1}{\theta_M+N}\right)\end{aligned}$$
#### Sampling $\alpha$
Similarly, $\alpha$’s posterior distribution is $$\begin{aligned}
\label{eq:posterior_alpha}
\alpha\sim \text{Gam}(k_{\alpha}+\sum_{i,k}h_{i\alpha k}^{(1)}, \frac{1}{\theta_{\alpha}-\sum_{i,d}F_{id}\log q_i^{(1)}})\end{aligned}$$
#### Sampling hyper-parameters of $\pmb{\Lambda}$
We set the following distributions for the hyper-parameters: $$\begin{aligned}
&k_{\Lambda}\sim\text{Gam}({k_2}, \frac{1}{\theta_2}), \theta_{\Lambda}\sim\text{Gam}(k_{3}, \frac{1}{\theta_{3}})\end{aligned}$$ The posterior distribution of these hyper-parameters are: $$\begin{aligned}
& l_{k_1k_2}\sim\sum_{t=1}^{\sum_{i,j}Z_{ij, k_1k_2}}\text{Ber}\left(\frac{k_{\Lambda}}{k_{\Lambda}+t-1}\right),\quad k_{\Lambda}\sim\text{Gam}(k_2+\sum_{k_1,k_2} l_{k_1k_2}, \frac{1}{\theta_2-\sum_{k_1,k_2} \log(1-p'_{k_1k_2})})\nonumber \\
& \theta_M\sim\text{Gam}(k_3+K^2k_{\lambda}, \frac{1}{\theta_3+\sum_{k_1,k_2} \Lambda_{k_1k_2}})\end{aligned}$$ where $p'_{k_1k_2}=\frac{\sum_{i,j}X_{ik_1}X_{jk_2}}{\theta_{\Lambda}+\sum_{i,j}X_{ik_1}X_{jk_2}}$.
#### Sampling hyper-parameters of $\pmb{\beta}$
We set the following distributions for the hyper-parameters: $$\begin{aligned}
&\gamma_{1}^{(l)}, \gamma_{0}^{(l)}\sim\text{Gam}({e_0^{(l)}}, \frac{1}{f_0^{(l)}}), c^{(l)}\sim\text{Gam}(g_0, \frac{1}{h_0})\end{aligned}$$
The posterior distribution of these hyper-parameters are: $$\begin{aligned}
J_{i'i}^{(l)} & \sim\text{CRT}(\sum_{k}h_{i'ik}^{(l)}, \gamma_{1}^{(l)}), \forall (i',i)|R_{i'i}=1\cap i'\neq i \\
J_{ii}^{(l)} & \sim\text{CRT}(\sum_{k}h_{iik}^{(l)}, \gamma_{0}^{(l)}), \forall i \\
n_{1}^{(l)} & = \sum_{(i, i')| i\neq i'\cap R_{i'i}=1} \log\frac{c^{(l)}-\log q_{i}^{(l)}}{c^{(l)}}, n_0^{(l)} = \sum_{i} \log\frac{c^{(l)}-\log q_{i}^{(l)}}{c^{(l)}} \\
\gamma_{1}^{(l)} & \sim \text{Gam}({e_0}+\sum_{i\neq i'}J_{i'i}^{(l)}, \frac{1}{f_0+n_{1}^{(l)}}) \\
\gamma_0^{(l)} & \sim \text{Gam}(e_0+\sum_{i'}J_{i'i'}^{(l)}, \frac{1}{f_0+n_0^{(l)}}) \\
c^{(l)} & \sim\text{Gam}(g_0+N\gamma_0^{(l)}+\gamma_{1}^{(l)}\sum_{i\neq i'}\pmb{1}(R_{ii'}=1), \frac{1}{h_0+\sum_{i,i'}\beta_{i'i}^{(l)}}) \end{aligned}$$
relational data $\{R_{ij}\}_{i,j=1}^N$, nodes’ feature information $\pmb{F}\in(\mathbb{R}^+\cup 0)^{N\times D}$, iteration time $T$ $\{\pmb{\pi}_{i}^{(l)}\}_{i,l}, \{\pmb{B}^{(l)}\}_{l=1}^{L-1}, \{\pmb{X}_i\}_i, \{\Lambda_{k_1k_2}\}_{k_1, k_2}, \pmb{T}, \alpha, M$ // Update the latent counts in a bottom-up manner Update latent count vector $y_{ik}^{(l)}\sim\text{CRT}(m_{ik}^{(l)}, \psi_{ik}^{(l)})$ Update the latent count on the $l$-layer Eq. (\[eq:latent\_counts\_z\_ikl\_bottom\_layer\]) Summarize the input $m_{ik}^{(l-1)}$ for $(l-1)$-th layer Eq. (\[eq:sum\_labelss\]), $\forall i,k$ Update latent count vector $y_{ik}^{(1)}\sim\text{CRT}(m_{ik}^{(1)}, \psi_{ik}^{(1)})$ Update the latent count on the $1$st-layer Eq. (\[eq:latent\_counts\_z\_ifl\_top\_layer\]) // Update $\{\pmb{\pi}_i^{(l)}\}_{i,l}$ and $\{B_{i'i}^{(l)}\}_{i,l}$ from the input layer to the output layer Update $\{T_{dk}\}_{d,k}$ according to Eq. (\[eq:posterior\_t\_fk\]) Update membership distribution $\pmb{\pi}_i^{(l)}$ Eq. (\[eq:posterior\_pi\_i\]) Update coefficients $B_{i'i}^{(l)}$ according to Eq. (\[eq:posterior\_beta\_ii\])(\[eq:posterior\_beta\_ii\_1\]), $\forall i',i$ // Update relational data generation structure Update latent counts $X_{ik}$ according to Eq. (\[HERE\]) Update latent representation $\{Z_{ij,k_1k_2}\}$ according to Eq. (\[eq:posterior\_Z\_ij\_1\])(\[eq:posterior\_Z\_ij\_2\]) Update compatibility value $\{\Lambda_{k_1k_2}\}$ according to Eq. (\[eq:posterior\_Lambda\]) // Update variables $\alpha, M$ Update $\alpha, M$ according to Eq. (\[eq:posterior\_alpha\])(\[eq:posterior\_M\]) Update hyper-parameters Update hyper-parameters of $\pmb{\Lambda}, \pmb{\beta}$ according to Eq. $(22)\sim (31)$
Latent feature visualization for the datasets of Citeer, Pubmed and PPI
=======================================================================
We provide the visualizations on latent features for the datasets of Citeer, Pubmed and PPI in Figure \[fig:latent\_feature\_vis\]. Similar conclusions (as mentioned in the main paper) can be obtained.
We also provide visualization on the compatibility matrix $\pmb{\Lambda}$.
|
---
abstract: 'The long-term dynamics of Oort cloud comets are studied under the influence of both the radial and the vertical components of the Galactic tidal field. Sporadic dynamical perturbation processes are ignored, such as passing stars, since we aim to study the influence of just the axisymmetric Galactic tidal field on the cometary motion and how it changes in time. We use a model of the Galaxy with a disc, bulge and dark halo, and a local disc density, and disc scale length constrained to fit the best available observational constraints. By integrating a few million of cometary orbits over 1 Gyr, we calculate the time variable flux of Oort cloud comets that enter the inner Solar System, for the cases of a constant Galactic tidal field, and a realistically varying tidal field which is a function of the Sun’s orbit. The applied method calculates the evolution of the comets by using first-order averaged mean elements. We find that the periodicity in the cometary flux is complicated and quasi-periodic. The amplitude of the variations in the flux are of order 30 %. The radial motion of the Sun is the chief cause of this behaviour, and should be taken into account when the Galactic influence on the Oort cloud comets is studied.'
author:
- |
E. Gardner$^1$[^1], P. Nurmi$^1$, C. Flynn$^2$ and S. Mikkola$^1$\
$^1$Tuorla Observatory, Department of Physics and Astronomy, University of Turku, Väisäläntie 20, FI-21500, Piikkiö, Finland\
$^2$Finnish Centre for Astronomy with ESO (FINCA), University of Turku, Väisäläntie 20, FI-21500, Piikkiö, Finland
bibliography:
- 'Oort\_cloud\_dynamics\_in\_MW.bib'
date: 'Accepted 2010 September 16. Received 2010 September 16; in original form 2010 June 24'
title: 'The effect of the Solar motion on the flux of long-period comets'
---
\[firstpage\]
methods: numerical – celestial mechanics – comets: general – Solar system : general – stars: kinematics and dynamics.
Introduction
============
Strong observational evidence supports the idea that the inner Solar system is subject to a steady flux of ‘new’ comets which originate from the ‘Oort cloud’ [@Oort50]. The semi-major axes of these comets are thought to evolve under the influence of external forces such as the Galactic tidal field and passing stars. The comets evolve from a typical $a_{orig} \simeq 3 \times 10^4$ AU to either hyperbolic orbits or to larger binding energies, depending on the orbital evolution during the cometary encounters with the Solar System planets. During one orbital evolution, comets typically experience perturbations that change the comet’s orbit into a short-period or hyperbolic orbit, which leads to a subsequent ejection into interstellar space [@Nurmi2001].
The Galactic potential exerts a force on Oort cloud comets, and is important for the steady state flux of comets with $a > 20000$ AU, e.g. [@Byl1983; @Heisler86; @Matese89]. The Galactic tidal force has a dominant component that is perpendicular to the Galactic plane; the radial component is 10 times weaker than the perpendicular component [@Heisler86]. For this reason, many studies (e.g. @Matese1995 [@Wick2008]) have assumed that the radial Galactic tidal field component is negligible. Other studies have included the radial component, for more accurate modelling of cometary motion, such as in [@Matese96; @Brasser01]. The radial component of the tide has been found to have an effect on the long-term evolution of the comets’ perihelia, on the distribution of the longitudes of the perihelion [@Matese96], and the origin of chaos in the cometary motion [@Breiter2008]. In recent large scale simulations, [@Rickman08] found that a fundamental role is played by perturbations due to passing stars on comets, contrary to the investigations during the previous two decades, starting with [@Heisler86]. The stellar perturbations, do of course act together with the Galactic tide.
The topic of this paper is the effect on comets due to the Galactic tide alone. We use a realistic model of the local Galaxy, which is well constrained by observations, which contains a disc, bulge and halo. We follow the orbit of the Sun in this model using recent constraints on the Solar motion. This motion allows us to compute the change in the vertical and radial components of the Galactic tide with time, and the change in the tidal force due to the radial motion of the Sun is fully accounted for.
Qualitatively, the tidal effect on the cometary orbits can be evaluated by studying the change in angular momentum averaged over one orbit. The Galactic tidal force periodically changes the angular momentum ($J=\sqrt{GM_\odot a (1-e^2)}$) of the Oort cloud comets [@Fernandez91]. The angular momentum of the comets changes as [@Heisler86]:
$$\label{DJ}
\frac{dJ}{dt}=-\frac{5\pi \rho_0}{G
\mathrm{M}_{\odot}^{2}}L^2(L^2-J^2)[1-(J_{z}^{2}/J^2)]\sin2\omega_g ~.$$
Here, $\omega_g$ is the Galactic argument of perihelion, $J_{z}$ is the $z$-component of angular momentum, and is perpendicular to the Galactic plane, $L = \sqrt{\mu a}$, $\rho_0$ is the local mass density, and $G$ is the gravitational constant. Note that this equation is valid for first order mean elements under the action of axially symmetric disc tides. The periodically changing angular momentum causes variations mainly in perihelion distance $q$, for comets in near-parabolic orbits, since $q \approx
J^2/(2G\mathrm{M}_{\odot})$. The typical assumptions in the cometary flux calculations due to the Galactic tidal force suppose that the local tidal field is axisymmetric, perpendicular to the mid-plane, and adiabatically changing [@Matese1992].
The aim of many of these studies is to correlate the motion of the Sun in the Galaxy with phenomena on the Earth, such as mass extinctions of species, the cratering record and climate change. An extensive review of this topic has been made recently by [@Bailer-Jones09].
For example, early studies have shown that the ages of well dated impact craters on Earth are not distributed randomly, but that there is a possible 28 Myr [@Alvarez1984] or 30 $\pm$ 1 Myr periodicity in crater ages over the past 250 Myr [@Rampino1984]. Since then, several authors have claimed that there is a significant periodic signal present, but the periods differ quite a lot from study to study. The signal is the most prominent for 40 large, well-dated craters that are up to 250 Myr old [@Napier2006], but the period is difficult to measure, with estimates of between 24-26 Myr [@Napier2006], 30 Myr [@Napier2006; @Stothers2006], 36 Myr [@Napier2006; @Stothers2006], 38 Myr [@Yabushita2004; @Wick2008] and 42 Myr [@Napier2006].
In some studies, the reliability of the signal is questioned altogether, based on the inaccuracy of the age estimates of the impact craters, possible biases caused by rounding the ages of craters, and the small number of craters [@Grieve1996; @Jetsu2000].
By critically reviewing many studies that have tried to connect the Solar motion and periodicity in terrestrial phenomena such as biodiversity, impact cratering and climate change, [@Bailer-Jones09] has concluded that there is little evidence to support these connections. By studying the artificial cratering data [@Lyytinen2009] came to the same conclusion, that the reliable detection of any periodicity is currently impossible with the existing cratering data.
In this study, we statistically analyse how the Galactic tidal force changes cometary orbits over 1 Gyr, using numerical simulations. The 1 Gyr time-scale is long enough to observe changes in the Galactic tide due to both the radial and vertical motion of the Sun. A simple axisymmetric Galactic potential is adopted. To our knowledge, there has been no study to date, in which the effect of both radial and vertical components, in a time varying Galactic potential (via the variation in mass density $\rho$), has been analysed in detail. Our purpose is to study the statistical effects of the complete Galactic potential to the Oort cloud comets in detail, especially concentrating on the comets that enter the Solar System ($q<30$ AU). In particular, we analyse the differences in cometary motion for when the tidal field is constant, and when it varies as the Sun moves in a realistic orbit in a fairly realistic Galactic potential. We find that it is important to include the radial motion of the Sun in the calculations, since the local density varies significantly as the Sun moves towards and away from the Galactic centre.
Methods
=======
The method of simulation requires two, traditionally separate, components. The first is to simulate the motion of the Sun around the Galaxy. The second involves the evolution of the orbits of comets in the Oort cloud. We integrate the motion of the Sun in an axisymmetric Galactic potential. The method of integration of the comets is described in [@Mikkolanurmi]. The method calculates the evolution of the comets by using first-order averaged mean elements. We do not include the random perturbations caused by the planets, since the aim is to identify the tidal effects of the Galactic potential on the flux of comets reaching the inner Solar System.
The Galactic potential
----------------------
The Galactic potential consists of a disc, bulge, and dark halo, and is partially described in [@Gardner10]. The model used here differs most notably from [@Gardner10] in the treatment of the disc, as well as a slightly modified dark halo. We noticed that the vertical density profile of the disc, from [@Gardner10], does not accurately reproduce the observational profile from [@Holmberg04]. We proceeded to modify our disc-model by changing a few of the model’s parameters, and adding three more Miyamoto-Nagai potentials to the model, to emulate a very thin layer of gas in the disc. The equation for the full potential is: $$\Phi = \Phi_H + \Phi_C + \Phi_D + \Phi_g~,$$ $$\Phi_H = \frac{1}{2}V_h^2 {\rm ln}(r^2 + r_0^2)~ ,$$ $$\Phi_C = - \frac{GM_{C_1}}{ \sqrt{r^2+r^2_{C_1}} } -
\frac{GM_{C_2}}{\sqrt{r^2+r^2_{C_2}}}~ ,~ \\$$ $$\Phi_D = \sum_{i=1}^3~~\frac{-GM_{d_i}}{\sqrt{(R^2+(a_{d_i}+\sqrt{(z^2+b^2)})^2)}}~ ,\\$$ $$\Phi_g = \sum_{n=1}^3~~\frac{-GM_{g_n}}{\sqrt{R^2+[a_{d_n}+\sqrt{(z^2+b_g^2)}]^2}} ~,$$
where $G$ is the gravitational constant, $R$ is the Galactocentric radius, and $z$ is the height. The modified parameters are in Table \[newparameters\]. The vertical density profile of the model can be seen in Fig. \[heightprofile\].
Property value Unit
----------- ------------------------ --------------------
$V_h$ 220 km s$^{-1}$
$r_0$ 10 kpc
$b$ 0.45 kpc
$b_g$ 0.12 kpc
$r_{C_1}$ 2.7 kpc
$r_{C_2}$ 0.42 kpc
$a_{d_1}$ 5.81 kpc
$a_{d_2}$ 17.43 kpc
$a_{d_3}$ 34.86 kpc
$M_{C_1}$ 3 10$^9$ M$_\odot$
$M_{C_2}$ 16 10$^{9}$ M$_\odot$
$M_{d_1}$ 66.06$\times 10^9$ M$_\odot$
$M_{d_2}$ $-$59.05 $\times 10^9$ M$_\odot$
$M_{d_3}$ 22.97$\times 10^9$ M$_\odot$
$M_{g_1}$ 18.63$\times 10^9$ M$_\odot$
$M_{g_2}$ $-$16.66$\times 10^9$ M$_\odot$
$M_{g_3}$ 6.48$\times 10^9$ M$_\odot$
: Parameters of the full potential.[]{data-label="newparameters"}
The mass density at the Sun is $0.11$ M$_\odot/$pc$^3$, consistent with observational constraints (0.10 $\pm$ 0.01 M$_\odot/$pc$^3$ @Holmberg00 and 0.105 $\pm$ 0.005 M$_\odot/$pc$^3$ [@Korchagin03]). This corresponds to a nominal T$_z$=$83\times10^6$ year period for small amplitude simple harmonic motion in the vertical direction. The surface density of disc matter in the model is 54.9 M$_\odot/$pc$^2$, compared with a measured disc surface density of $56 \pm 6 $M$_\odot/$pc$^2$ [@Holmberg04]. The adopted current position of the Sun in the model, $(R,z)_\odot$, is (8,0) kpc. The local circular velocity of the model is 221 km s$^{-1}$.
### Density and rotation curve
Fig. \[discprofile\] shows the surface density of the disc with radius ($R$), the change in density with height ($z$) at the Sun ($R=
8 kpc$) (Fig. \[heightprofile\]), and the rotation curve (Fig. \[vcirc\]), as these are the two factors that have the most impact on the orbits of the comets. The disc has a scale-length of 3 kpc, and a local scale-height of 0.24 kpc, consistent with recent measurements by [@Juric08] using the Sloan Digital Sky Survey (SDSS).
![The surface density of the disc component as a function of Galactocentric radius. The dashed line corresponds to an exponential density falloff of 3 kpc, which is a good fit to the model over a wide range of radii. Note that the density truncates strongly at 18 kpc.[]{data-label="discprofile"}](disk-density.eps){width="84mm"}
![Vertical density of the model, at the Sun ($R$ = 8 kpc). The dotted line represents the baryonic contribution in the model, the dashed line the dark matter contribution, and the dashed-dotted line the total density of the model at a certain height ($z$). The solid line corresponds to an exponential fit to the baryonic component of the model of 0.25 kpc.[]{data-label="heightprofile"}](height-density.eps){width="\columnwidth"}
![Rotation curve for the model Milky Way (dotted), and the different contributions of the disc (dashed), bulge (long dashed), and dark halo (solid).[]{data-label="vcirc"}](circularvelocity.eps){height="84mm"}
The potential is axisymmetric, so we do not model effects such as molecular clouds, spiral arms, and bubbles in the interstellar matter and passing stars. Our interest here is on the global Galactic effects of the Solar motion on the comets.
### Motion of the Sun
The orbit of the Sun, using the Solar motion measured by [@Schonrich10], where ($U, V, W$)=(11.1, 12.24, 7.25) km s$^{-1}$, and assuming ($R,z$)$_\odot$ to be (8,0) kpc, is shown in the ($R,z$)-plane in Fig. \[orbit\]. $U$ is the velocity towards the Galactic centre, $V$ is the velocity along rotational direction, and $W$ is the velocity perpendicular to the Galactic plane. Properties of this orbit, such as eccentricity, radial and vertical period and maximum $z$ height, are shown in Table \[sunmotion\]. The eccentricity of the Solar orbit ($e$) is defined as: ${(R_{\mathrm{max}}-R_{\mathrm{min}})}/{(R_{\mathrm{max}}+R_{\mathrm{min}})}$.
Property value Unit
-------------------- ------------------- ------
$e$ 0.059 $\pm$ 0.003
$z_{\mathrm{max}}$ 0.102 $\pm$ 0.006 kpc
$T_R$ 149 $\pm$ 1 Myr
$T_z$ 85 $\pm$ 4 Myr
: Eccentricity, $e$, maximum vertical height, $z_{\mathrm{max}}$, radial oscillation period, $T_R$, and vertical oscillation period, $T_z$, of the Sun in the adopted potential. The adopted solar motion is that of [@Schonrich10].[]{data-label="sunmotion"}
![Motion of the Sun in the model potential for 1 Gyr, in radial $R$ and vertical $z$ components. The initial position of the Sun is $(R,z)_0
= (8,0)$ kpc.[]{data-label="orbit"}](3d-orbit.eps){height="84mm"}
Variation of the tidal parameters caused by the motion of the Sun {#motion-section}
-----------------------------------------------------------------
Our integration method for cometary motion is based on the computation of secular motion of the comet in quadratic perturbation by the Galactic potential. The method used is presented in @Mikkolanurmi. As the comet orbits the Sun under the Galactic potential it experiences a force $\bmath{F}$ per unit mass:
$$\bmath{F}=-\frac{G
M_{\odot}}{r^3}\bmath{r}-G_1x\bmath{x}-G_2y\bmath{y}-G_3z\bmath{z},$$
where $G_1=-(A-B)(3A+B)$, $G_2=(A-B)^2$, and $G_3=4 \pi G \rho(R,z)
-2(B^2-A^2)$. Here $r$ is the Sun-Comet distance, $\rho(R,z)$ is the local mass density, and $G$ is the gravitational constant [@Heisler86]. $A$ and $B$ are the Oort constants, and are obtained from the Galactic model. Usually the radial components ($\bmath{x},\bmath{y})$ are neglected, so that $G_1 = G_2 =
0$.
We will examine the effect on comets in two particular cases. Firstly we study what we call the ’dynamic’ case, in which the Galactic tide changes realistically along the Solar orbit, for the case that the Sun oscillates both vertically and radially in the potential. Secondly, we assume that there is a ’constant’ tidal field (i.e. the tidal field does not change as the Sun moves around the Galaxy). In the constant case the Sun moves on a flat, circular orbit.
We integrate the Solar orbit in the ’dynamic’ case for 1 Gyr, sampling the values of the $G$-parameters every 100 kyr: these are shown in Fig. \[gs\]. The changes in $G_3$ are dominated by the changes in local density during the orbit, since the changes in $A$ and $B$ over the orbit are not very large. Fig. \[gs\] (top and centre panel) shows how the changing values of the Oort constants, $A$ and $B$, affect the values of $G_1$ and $G_2$.
Fig. \[dynamic-changes\] shows the combined effects of the radial ($R$) and vertical motion ($z$) of the orbit, on $G_3$. It is clear that $G_3$ increases in two situations: when the radial position is the closest to the Galactic centre, and when the vertical motion crosses the mid-plane. Due to the slightly eccentric motion of the orbit, it is the radial component of the motion which dominates the changes in local density ($\rho(R,z)$), rather than the vertical motion. This is partly because we adopt the Solar motions of [@Schonrich10], which produces a mildly eccentric orbit for the Sun ($e=0.059\pm0.003$): it oscillates between Galactocentric radii of 7.9 and 8.9 kpc. As such, the evolution of $G_3$ depends on the radial motion and the vertical motion, both being of equal magnitude (Fig. \[dynamic-changes\]).
\
\
![Galactocentric radius $R$, vertical height $z$ and $G_3$ for the ’dynamic’ case of the Solar motion over 1 Gyr in the Galactic adopted model. Upper panel: the radial motion. Middle panel: vertical motion. Lower panel: $G_3$.[]{data-label="dynamic-changes"}](changes.eps){height="84mm"}
In the second case studied, the Sun is set on a perfectly circular and flat orbit, so that the local mass density does not change with time. For the ’constant’ case, the values of $G_i$ are the mean values from the ’dynamical’ case, and have the values $G_1 = -7.00897 \times 10^{-16}$, $G_2 =
7.27613 \times 10^{-16}$, and $G_3 = 4.55749 \times 10^{-15} \mathrm{yr}^{-2}$.
Simulations of the comets
=========================
Observations support the idea that the *in situ* flux of new comets is roughly linear with respect to heliocentric distance, up to the distance of Jupiter [@Hughes2001]. Comets with highly eccentric orbits with randomised directions of motion have a $q$-distribution that is flat [@Opik66]. If a long period comet has a perihelion distance between $\sim$ 10$-$15 AU, it is quickly ejected to interstellar space or perturbed so that it becomes a short-period comet [@Wiegert99]. From the perspective of the Oort cloud, the comets are removed from the Oort cloud and from the loss cone to the orbital parameter distribution [@Fernandez81]. The loss cone (lc) is the population of comets that have orbits that will allow them to penetrate the planetary system, making it possible to observe them [@Hills81]. Planetary perturbations move comets efficiently from $q$ values less than $q_\mathrm{lc} \simeq 15$ AU into either hyperbolic orbits or into orbits that are more tightly bound to the solar system [@Hills81]. In the steady state situation, the new comets are distributed uniformly to the perihelion distances of $q \leq q_\mathrm{lc}$, for $a>30000$ AU, while $q>q_\mathrm{lc}$ comets come also from the inner region. For this reason, in all of our simulations we have assumed that initial perihelion distances are distributed uniformly outside the loss cone. The inclination distribution of the outer Oort cloud of comets is isotropic (uniform in $\cos i$). All the other angular elements are uniformly distributed between $[0-2\pi]$.
Structure of the Oort cloud
---------------------------
An important issue in studying the structure of the Oort cloud, is what energy distribution to adopt for the comets. We assume that the density of comets between 3000 AU and 50000 AU in the Oort cloud is proportional to $1/r^{3.5 \pm 0.5}$, so that the number of comets $N$ is $dN \propto 1/r^{\alpha} dr$, where $\alpha = 1.5 \pm 0.5$ [@Duncan87]. The conclusions of this paper are not particularly sensitive to the adopted value of $\alpha$. The existence of an inner Oort cloud has been speculated upon in many studies although there is no direct evidence for it [@Hills81]. An inner Oort cloud is the extension of the Kuiper belt, filling the gap between the Kuiper belt and the outer Oort cloud. Semi-major axes in the inner Oort cloud are typically between 50$-$15000 AU [@Leto2008]. The steady state flux from the inner Oort cloud cannot be uniformly distributed in perihelion distance, since the planetary perturbations move comets efficiently from $q$ values less than $q_{lc} \simeq 15$ AU to either hyperbolic orbits or into orbits that are more tightly bound to the solar system [@Hills81]. In the steady state situation, the new comets come uniformly to perihelion distances $q \leq q_{lc}$ when $a>30000$ AU, while the $q>q_{lc}$ comets also come from the inner region.
Simulation parameters of the comets
-----------------------------------
Due to this complicated picture, we study different semi-major axes in separate simulations, and evaluate the efficiency of tidal injection in each simulation. We chose initial sample conditions for the Oort cloud comets, setting the semi-major axis to be 10000, 20000, 30000, 40000, 50000, and 60000 AU. We choose the comet’s eccentricity ($e$) randomly, so that the resulting values of $q$ would be uniformly distributed from 35 to the value of the semi-major axis ($a$). All the other parameters, $\cos(i)$, $\Omega$, $\omega$, and the initial eccentric anomaly were all chosen randomly in appropriate intervals. The number of comets in each simulation is $10^6$ at each of the sampled semi-major axes. For the computation of the numbers of comets reaching the inner Solar System, the results from each semi-major axis are normalised to the adopted number density law.
Simulations of the Galactic tide
--------------------------------
To study the effect of the Galactic potential on cometary motion, we chose two hypothetical solar systems: a ‘constant’ background density, where the Sun would be on a pure circular orbit with no vertical motion, and a realistic ’dynamic’ Solar orbit. In all systems, we analyse the flux of Oort cloud comets into the Solar System. This means that we consider a comet to have been detected in the inner Solar System when its $q$ is within 30 AU, and it has a heliocentric radius of less than 1000 AU. The last criterion is important, as the osculating elements of comets can evolve to have $q$ $\leq$ 30 AU, far away from the Sun, and evolve to more than 30 AU, without ever entering the inner Solar System. Since the minimum of the osculating $q$ is at the aphelion, the criterion $q \leq$ 30 AU is approximate. For this reason, we also check if the comet is actually approaching the aphelion, by choosing $r < 1000$ AU. This requires use of the mean anomaly, which is approximately calculated from the derived time co-ordinate. Comets which have been detected are removed from the simulation. We do not replace them with new comets.
Results
=======
Our aim is to determine the effect of the Galactic tide on Oort cloud comets by separately adopting a constant local density and a varying local density. Examining the flux of comets into the inner Solar System (Table \[fluxes\]), we find that there is no significant difference between the two models. Most of the detected comets come from the middle (30$-$40 kAU) range of semi-major axes. Likewise, the resulting distribution of orbital elements, for comets coming into the Solar System, is the same in both the constant and dynamic cases. Fig. \[elements\] shows example distributions for the simulation cases where the semi-major axis is 30000 AU. The gap at $\omega = 0$ and 180 is caused by the $\sin 2\omega_g$ term going to 0 in Equation \[DJ\]. Similarly, the factor $1 - (J_{z}^{2}/J^2)$ goes to zero at $i$=0 or 180, since $J_z=J \cos i$. The $q$-distribution of incoming comets shows no significant difference between the two cases, as seen in Fig. \[qs\]. The high peak at $q=30$ AU, in Fig. \[qs\] is caused by the slow evolution of $q$, at $a=30000$ AU. Finally, the total flux of comets entering the inner-Solar system is not significantly different between the two cases. For the ’constant’ case, from a source pool of 1.8 million comets in the simulated Oort cloud, we find that approximately 120 comets per Myr reach the inner Solar System (which here means comets with $q<30$ AU). Assuming that there are about $10^{12}$ comets in the Oort cloud [@Wiegert99], this corresponds to about 70 comets per year with $q<30$ AU. This is a bit higher than the observed number of new comets with $q<30$ AU, which is about 20 per year, assuming a cometary flux of 0.65 $\pm$ 0.18 yr$^{-1}$ AU$^{-1}$ [@Fernandez10].
Semi-major axis (AU) constant dynamic
---------------------- ---------- ---------
10000 0.0729 0.0732
20000 0.1803 0.1804
30000 0.2405 0.2405
40000 0.2381 0.2369
50000 0.1504 0.1507
60000 0.1179 0.1184
: Relative fraction of comets entering the inner Solar System, from each of the simulated semi-major axis values, for the ’constant’ and ’dynamic’ cases.[]{data-label="fluxes"}
![Distribution of the orbital elements $i$, $\Omega$, and $\omega$ for $a = 30000$. The solid column represents constant case, and the shaded column the dynamic case. There is no significant difference between the two distributions.[]{data-label="elements"}](30k-i.eps "fig:"){width="84mm"}\
![Distribution of the orbital elements $i$, $\Omega$, and $\omega$ for $a = 30000$. The solid column represents constant case, and the shaded column the dynamic case. There is no significant difference between the two distributions.[]{data-label="elements"}](30k-Omega.eps "fig:"){width="84mm"}\
![Distribution of the orbital elements $i$, $\Omega$, and $\omega$ for $a = 30000$. The solid column represents constant case, and the shaded column the dynamic case. There is no significant difference between the two distributions.[]{data-label="elements"}](30k-omega.eps "fig:"){width="84mm"}
![Distribution of the perihelion distance $q$ for the incoming comets, for $a = 30000$. The solid column represents the constant case, and the shaded column the dynamic case. There is no significant difference between the two distributions.[]{data-label="qs"}](30k-q.eps){width="84mm"}
Time evolution of the cometary flux {#timeevolution-section}
-----------------------------------
There is a clear difference between the two models when we look at the temporal evolution of the cometary flux. The top panel of Fig. \[ts-dynamic\] shows, for the ’dynamic’ case (and after a relaxation time of about 100 Myr) that the mean cometary flux (shown as a 10 Myr moving average) is well correlated with the changes in $G_3$ (dashed line). The resulting fluxes from the simulation have been weighted according to their relative number-density ($a^{-1.5}$), the resulting total amount of comets in the simulation, by using this weighting system is 1.8 $\times 10^6$. From Section \[motion-section\], there are two main causes for the evolution of the flux, the major being the radial motion, the minor being the vertical motion. The top panel of Fig \[ts-dynamic\] shows the major trend clearly following the radial motion. The vertical motion is also followed, as is clear in the 10 Myr moving average (which smooths out some of the Poisson noise in the individual 1 Myr samples). We also ran a separate simulation with a circular orbit, and with the vertical component intact, the resulting fluxes followed perfectly the vertical motion, as was expected due to the changes in $G_3$ being solely contributed by the vertical component. In the case where the values for $G_i$ have been kept constant (i.e. corresponding to a completely circular orbit with no vertical motion), there is no evidence of any evolution, as seen in the bottom panel of Fig. \[ts-dynamic\].
Simple Fourier-analysis of the dynamic flux (in the top panel of Fig. \[ts-dynamic\]) finds two distinct periods in the cometary flux. The strongest signal is produced by a period of 143$-$167 Myr, and an equal signal from a period of 41$-$45 Myr. The former corresponds to the radial period (152 Myr) and the latter to the half-period of the vertical period (43 Myr). The vertical signal is found to lie in the range 41$-$45 Myr, and is quasi-periodic, as has been seen earlier by [@Matese2001].
![The time evolution of the relative flux of comets in the dynamic case (top panel), and the constant case (bottom panel). The solid line represents the relative flux of comets in 1 Myr bins, the white line a 10 Myr moving average. The dashed line in the top panel represents the evolution of the $G_3$-parameter.[]{data-label="ts-dynamic"}](timeseries.eps "fig:"){width="84mm"}\
![The time evolution of the relative flux of comets in the dynamic case (top panel), and the constant case (bottom panel). The solid line represents the relative flux of comets in 1 Myr bins, the white line a 10 Myr moving average. The dashed line in the top panel represents the evolution of the $G_3$-parameter.[]{data-label="ts-dynamic"}](timeseries-static.eps "fig:"){width="84mm"}
Comparison with earlier results {#flux-section}
-------------------------------
[@Matese1995] derived various periods for the vertical Solar oscillation, depending on the adopted model of the disc. For example, their ‘No-Dark-Disk Model’ has a crossing-period very close to ours (43 Myr), while their ‘Best-fit Model’ produces a much lower period of 33 Myr. They assumed a $W$-velocity of 7.5 km s$^{-1}$, compared to ours of 7.25 km s$^{-1}$, and does not cause much qualitative difference in the vertical period of the orbit of the Sun. The lower period comes from assuming a considerably higher mid-plane density ($\rho \approx 0.13$ M$_\odot$/pc$^3$) than ours.
[@Matese2001] found that the radial period of Solar motion modulates the vertical period. However, they used very high local densities of matter in the disc, so that the vertical oscillations dominated the radial oscillations. This meant that the flux of comets into the inner Solar System with each passage through the disc was greatly amplified compared to our simulations. We find that the cometary flux varies with an amplitude of about 20%, whereas [@Matese2001] find flux variations of about a factor of two. The much smaller amplitude in the signal which we advocate, even taking into account the radial oscillations, would make finding a period in the scant cratering record very difficult.
[@Fouchard06] calculated that a local mass density of $\rho = 0.1$ M$_\odot/$pc$^3$ corresponds to a cometary flux of around 10$^4$ for their first 500 Myr interval for a source population of 10$^6$ comets, where observed comets have $q \le$ 15 AU. Assuming that there are 10$^{12}$ comets in the Oort cloud. This produces a flux of $\sim$20 comets/yr, which is a factor of two less than our estimate.
The Galactic tide case in [@Rickman08] gives a flux of 100 comets per 50 Myr, from a source population of 10$^6$ comets, where observed comets have evolved from $q >15 AU$ to $q <$ 5 AU. This corresponds to a flux of 2 comets/yr, again assuming an Oort cloud with 10$^{12}$ comets. Using similar analysis we find that we get a flux of 290 comets per 50 Myr, corresponding to a factor of three larger than [@Rickman08].
A review by [@Bailer-Jones09] correlates terrestrial events with cometary signals. One of the more interesting ones in the review is the 140 $\pm$ 15 Myr period found by [@Rohde05] in the number of known marine animal genera as a function of time. While this could correspond to the Sun’s radial period, [@Bailer-Jones09] considers that it has not been significantly detected, the main problem being that the entire time-span of the data covers no more than three oscillations. Many other periods, proxies, and studies are mentioned in [@Bailer-Jones09], although the conclusion is that there is no proven impact on biodiversity as a result of the orbital motion of the Sun.
Discussion and conclusions
==========================
We have studied the long-term dynamics of Oort cloud comets under the influence of both the radial and the vertical components of the Galactic tidal field. Other perturbing forces on the comets, such as passing stars or passage through spiral arms are ignored, since we aim to study the influence of just the axisymmetric Galactic tidal field on the cometary motion.
We use an axisymmetric model of the Galaxy, and a recently revised value for the Solar motion by [@Schonrich10]. This leads to vertical oscillations of the Sun with an amplitude of about 100 pc, and radial oscillations over about 1 kpc. The changing tidal forces on the Oort cloud are computed as the Sun orbits for 1 Gyr in this potential, and the flux of comets entering the inner Solar System is computed in simulations.
As expected, the cometary flux is strongly coupled to the $G_3$-parameter in the tidal forces, which is dominated by the local mass density seen along the Solar orbit. Both the radial and vertical motions of the Sun can be seen in the cometary flux, although the amplitude of the variations is small, implying that detecting such a signal from the small number of age-dated craters would be very difficult. This agrees with the recent review of the detectability of the Solar motion in terrestrial proxies [@Bailer-Jones09].
As $G_3$ is directly coupled to local density, it is easily affected by the non-axisymmetric components in the motion of the Sun in the Galaxy. This implies that spiral arms, a Galactic bar, giant molecular clouds, or any other intermittently encountered structure should have an effect on the flux.
Acknowledgements {#acknowledgements .unnumbered}
================
PN wants to thank the Academy of Finland for the financial support in this work. EG acknowledges the support of the Finnish Graduate School in Astronomy and Space Physics.
\[lastpage\]
[^1]: E-mail: [email protected]
|
---
abstract: 'Cold atoms bring new opportunities to study quantum magnetism, and in particular, to simulate quantum magnets with symmetry greater than $SU(2)$. Here we explore the topological excitations which arise in a model of cold atoms on the triangular lattice with $SU(3)$ symmetry. Using a combination of homotopy analysis and analytic field–theory we identify a new family of solitonic wave functions characterised by integer charge , with . We use a numerical approach, based on a variational wave function, to explore the stability of these solitons on a finite lattice. We find that solitons with charge $\mbox{${\bf Q} = (1,1,-2)$}$ spontaneously decay into a pair of solitons with elementary topological charge, and emergent interactions. This result suggests that it could be possible to realise a new class of interacting soliton, with no classical analogue, using cold atoms. It also suggests the possibility of a new form of quantum spin liquid, with gauge–group U(1)$\times$U(1).'
author:
- 'Hiroaki T. Ueda$^{1,3}$[^1], Yutaka Akagi$^{2,3}$, and Nic Shannon$^3$'
title: Quantum solitons with emergent interactions in a model of cold atoms on the triangular lattice
---
While many aspects of quantum systems can be understood at a local level, it is their non–local, topological properties which offer the deepest and most surprising insights. Topology underlies our understanding of such highly correlated systems as $^3$He and the fractional quantum Hall effect [@thouless98], and has come to play an important role in the theory of metals [@nagaosa10; @xu15], superconductors [@qi11], and even systems as seemingly conventional as band insulators [@hasan10]. The study of topological excitations in magnets has also enjoyed a recent renaissance, as it has become possible to study the interplay between topological excitations, such as skyrmions, and itinerant electrons [@nagaosa13].
At the same time, cold atoms have brought an opportunity to study quantum many-body physics in a new context [@jaksch98; @bloch08]. The phases realized include analogues of both magnetic metals and magnetic insulators [@joerdens08; @schneider08], with the exciting new possibility of extending spin symmetry from the familiar SU(2) to SU(N) [@wu03; @honerkamp04; @gorelik09; @cazalilla09; @gorshkov10; @taie12; @bonnes12]. Spin models with enlarged symmetry bring with them the possibility to study new kinds of topological excitation, but to date, these remain relatively unexplored [@cazalilla09].
In this Communication, we consider the topological excitations which arise in an SU(3) antiferromagnet that could be realised quite naturally using cold atoms. Starting from the most general model for SU(3) spins on the triangular lattice, we use a combination of field–theory and homotopy analysis to categorise stable topological defects. We find a new kind of stable lump soliton with 2$^{nd}$ homotopy group , characterized by the integer charges $(Q_A,Q_B,Q_C)$, with $Q_A+Q_B+Q_C=0$, and obtain an analytic wave function for solitons with charge $(-Q,Q,0)$. We then study these solitons numerically, introducing a new quantum variational approach. The numerical analysis confirms the stability of the soliton with elementary topological charge $(-1,1,0)$. We find that the soliton for higher topological charge with $(1,1,-2)$ decompose into solitons with elementary charge, with emergent repulsive interactions. An example of a soliton with elementary charge ${\bf Q} = (-1,1,0)$ is shown in Fig. \[fig:1\]
![(Color online). Illustration of the spin configuration of a soliton of elementary charge ${\bf Q} = (-1,1,0)$ on the triangular lattice, decomposed into each of the three sublattices. The new soliton is composed of orthogonal CP$^2$ solitons on the A and B sublattices, while the C sublattice remains topologically trivial. The probability–surface for each spin-1 moment (defined in the supplemental materials), is rendered in blue, while the color underlay shows the dipole moment, $S^z$, induced by the soliton. Results are taken from the exact, analytic, wave function Eq. (\[eq:analytic-wave-function\]). The reference state ${| r \rangle}$, Eq. (\[eq:reference-state\]), is shown as an inset. []{data-label="fig:1"}](fig1.pdf){width="1.0\columnwidth"}
The model we consider is the SU(3)–symmetric generalisation of the Heisenberg model on a triangular lattice $$\begin{aligned}
\mathcal{H}^{\sf exchange}_{\text{SU(3)}}
= J \sum_{{\langle {\mathbf{l}},{\mathbf{m}}\rangle}} {\mathcal P}^{\text{SU(3)}}_{{\mathbf{l}},{\mathbf{m}}}
\label{eq:HeisenbergSU3}\end{aligned}$$ where ${\mathcal P}^{\text{SU(3)}}_{{\mathbf{l}},{\mathbf{m}}}$ is a permutation operator exchanging the states of the atoms on sites ${\mathbf{l}}$ and ${\mathbf{m}}$. Magnetism of this type can be realised using the hyperfine multiplets of repulsively–interacting Fermi atoms [@honerkamp04; @gorelik09; @bauer12], an approach which has already been shown to work for SU(2) Mott Insulators [@joerdens08; @schneider08], and was recently also demonstrated in experiment for an SU(6) Mott insulator [@taie12] — indeed SU(N) systems with $N > 2$ may have advantages for cooling [@taie12; @bonnes12].
We consider the fundamental representation of SU(3), for which $$\begin{aligned}
\mathcal{H}^{\sf exchange}_{\sf SU(3)}
= J \sum_{{\langle {\mathbf{l}},{\mathbf{m}}\rangle}}
{\text{\bf T}}_{\mathbf{l}}\cdot {\text{\bf T}}_{\mathbf{m}}\; ,\end{aligned}$$ where ${\text{\bf T}}$ is an 8–component vector, comprising the 8 independent generators of SU(3) [@peskin95]. These generators can be expressed terms of a quantum spin–1 [@papanicolaou88; @batista02; @batista04; @penc11-book.chapter; @smerald13-PRB88], with $$\begin{aligned}
(T^1,T^2,T^3) = (S^x,S^y,S^z) \; ,\end{aligned}$$ while the remaining five components of ${\text{\bf T}}$ are given by the quadrupole moments $$\begin{aligned}
\left(
\begin{array}{c}
T^4 \\
T^5 \\
T^6 \\
T^7 \\
T^8
\end{array}
\right)
=
\left(
\begin{array}{c}
(S^{\sf x})^2 - (S^{\sf y})^2 \\
\frac{1}{\sqrt{3}}[2(S^{\sf z})^2-(S^{\sf x})^2 - (S^{\sf y})^2] \\
S^{\sf x}S^{\sf y}+S^{\sf y}S^{\sf x} \\
S^{\sf y}S^{\sf z}+S^{\sf z}S^{\sf y} \\
S^{\sf x}S^{\sf z}+S^{\sf z}S^{\sf x}
\end{array}
\right) \; ,\end{aligned}$$ familiar from the theory of liquid crystals [@degennes95]. Viewed this way, the SU(3) symmetry takes on a clear physical meaning — spin quadrupoles and dipoles enter into Eq. (\[eq:HeisenbergSU3\]) on an equal footing. In addition, SU(3) rotations permit dipole moments to be transformed continuously into quadrupole moments, and vise–versa [@batista02; @batista04; @penc11-book.chapter; @smerald13-PRB88]. This is a process which has no analogue in classical magnets or liquid crystals, and has vital implications for topological defects.
In addition to providing a natural description of an SU(3)–symmetric Mott insulator, Eq. (\[eq:HeisenbergSU3\]) can also be thought of as a SU(3)–symmetric limit of the spin–1 bilinear biquadratic (BBQ) model [@papanicolaou88], a model which may also be realised using cold atoms [@imambekov03; @forgesdeparny14-PRL113]. The BBQ model has been widely studied on the triangular lattice, where it supports both conventional magnetic ground states, and quadrupolar phases analogous to liquid crystals [@tsunetsugu06; @bhattacharjee06; @laeuchli06; @tsunetsugu07; @stoudenmire09; @penc11-book.chapter; @grover11; @kaul12; @smerald13-PRB88; @smerald13-book; @voell15]. These, in turn, play host to a rich variety of topological excitations [@belavin75; @kawamura84; @ivanov03; @ivanov07; @grover11; @xu12; @galkina15; @yutaka-unpub]. In particular, unconventional solitons have been shown to arise in the SU(3)–symmetric Heisenberg ferromagnet, i.e. Eq. (\[eq:HeisenbergSU3\]), for $J < 0$ [@ivanov08]. As yet, however, little is known about the topological defects of Eq. (\[eq:HeisenbergSU3\]) for [*antiferromagnetic*]{} interactions $J>0$, the case which arises most naturally for cold atoms [@honerkamp04; @gorelik09; @bauer12].
The topological excitations of a given state follow from the structure of its ground–state manifold [@mermin79]. The ground state of Eq. (\[eq:HeisenbergSU3\]) on the triangular lattice, for $J>0$, is known to break spin–rotation symmetry, and to have 3–sublattice order [@laeuchli06; @penc11-book.chapter; @bauer12]. However, since the SU(3) symmetry permits rotations between quadrupole and dipole moments of spin, this ordered state does not correspond to any single 3–sublattice dipolar or quadrupolar state, but rather a continuously connected manifold [@smerald13-PRB88; @smerald13-book]. In what follows we construct a representation of this ground state manifold and use it to classify the topological excitations of Eq. (\[eq:HeisenbergSU3\]). While these results are completely general, they can most easily be understood through a mean–field description of Eq. (\[eq:HeisenbergSU3\]).
Following [@laeuchli06; @smerald13-PRB88], we write $$\mathcal{H}^{\sf MFT}_{\text{SU(3)}}
= 2J \sum_{{\langle {\mathbf{l}},{\mathbf{m}}\rangle}} |{\mathbf{d}}_{\mathbf{l}}\cdot\bar{{\mathbf{d}}}_{\mathbf{m}}|^2
+ \text{const} \; ,
\label{eq:H.MFT}$$ where ${\mathbf{d}}$ is a complex vector with unit norm, expressing the most general wave function for a quantum spin–1 $${| {\mathbf{d}}\rangle} = d^x {| x \rangle} + d^y {| y \rangle} + d^z {| z \rangle}\ ,$$ in terms of a basis of orthogonal spin quadrupoles $${| x \rangle} \! = \! i \frac{{| 1 \rangle} \! - \! {| -1 \rangle}}{\sqrt{2}}\ ,\
{| y \rangle} \! = \! \frac{{| 1 \rangle} \! + \! {| -1 \rangle}}{\sqrt{2}}\ ,\
{| z \rangle} \! = \! -i{| 0 \rangle}\ ,
\label{ketxyz}$$ and ${| 1 \rangle}$ is the state with $S^z = 1$, etc. For $J > 0$, Eq. (\[eq:H.MFT\]) supports a manifold of 3–sublattice ground states satisfying the orthogonality condition $${\mathbf{d}}_\lambda \cdot \bar{{\mathbf{d}}}_{\lambda'} = \delta_{\lambda \lambda'}
\label{eq:orthogonality.condition}$$ where $\lambda , \lambda' = \text{\{A, B, C\}}$. The simplest wave function satisfying Eq. (\[eq:orthogonality.condition\]) is the 3–sublattice antiferroquadrupolar (AFQ) state $${\mathbf{d}}_{A} = (1,0,0) \; , \;
{\mathbf{d}}_{B} = (0,1,0) \; , \;
{\mathbf{d}}_{C} = (0,0,1) \; ,
\label{eq:reference-state}$$ illustrated in the inset of Fig. \[fig:1\]. We take this a reference state, and denote it ${| r \rangle}$.
We are now in a position to determine the symmetry of the ground state manifold, and the topological excitations which follow from it. The universal covering group, G [@mermin79], is given by a global SU(3) rotation acting on ${| r \rangle}$. However the order parameters ${\langle {\text{\bf T}}_A \rangle}$, ${\langle {\text{\bf T}}_B \rangle}$, ${\langle {\text{\bf T}}_C \rangle}$ are unchanged if the following matrices act on ${| r \rangle}$: $$f=\left( \begin{array}{ccc}
e^{i\theta_1} & 0 & 0 \\
0 & e^{-i\theta_1} & 0 \\
0 & 0 & 1
\end{array} \right)\times
\left( \begin{array}{ccc}
e^{i\theta_2} & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & e^{-i\theta_2}
\end{array} \right)\ ,\
\label{isotropy1}$$ where $\theta_1$ and $\theta_2$ are two freely–chosen phases. It follows that the isotropy subgroup of our model is $$\begin{aligned}
H = U(1) \times U(1) \; ,\end{aligned}$$ where each U(1) refers to a phase $0 \leq \theta < 2\pi$. Hence, the ground state manifold of Eq. (\[eq:HeisenbergSU3\]), for $J > 0$, is given by $\text{G/H} = \text{SU(3)} /( \text{U(1)} \times \text{U(1)} )$. The topological excitations supported by Eq. (\[eq:HeisenbergSU3\]) follow directly from this result, and a standard application of homotopy theory [@mermin79] gives $$\begin{aligned}
&\pi_1( \text{SU(3)}/(\text{U(1)} \times \text{U(1)} ) ) = 0 \; , \\
&\pi_2( \text{SU(3)}/(\text{U(1)} \times \text{U(1)} ) ) = \mathbb{Z} \times \mathbb{Z} \; ,
\label{eq:homotopy_AFQ_SU3}\end{aligned}$$ where the trivial first homotopy group $\pi_1$ implies the absence of point–like defects, while the non–zero second homotopy group $\pi_2$ implies the existence of a stable soliton classified by two integers. This result should be contrasted with that obtained for [*ferromagnetic*]{} interactions, $J < 0$, in which case , and stable solitons are classified by a single integer $\mathbb{Z}$ [@dadda78; @ivanov08].
We can gain more understanding of this new class of solitons by constructing explicit wave function describing them. To this end, we consider the continuum limit of Eq. (\[eq:H.MFT\]) and, following Refs. [@smerald13-PRB88; @smerald13-book], write $$\begin{aligned}
\mathcal{H}^{\sf eff}_{\text{SU(3)}}
&=& \frac{2J}{\sqrt{3}} \int d{\bf x} \sum_{\mu = x, y}
\left( | {\mathbf{d}}^\ast_A \cdot \partial_\mu {\mathbf{d}}_B |^2
+ | {\mathbf{d}}^\ast_B \cdot \partial_\mu {\mathbf{d}}_C |^2 \right. \nonumber\\
&& \left. \qquad \qquad + \ | {\mathbf{d}}^\ast_C \cdot \partial_\mu {\mathbf{d}}_A |^2 \right) \; .
\label{eq:Heff}\end{aligned}$$ Building on previous work on CP$^2$ solitons [@dadda78], we introduce a real scalar field $$A_\mu^{\lambda}
= \frac{1}{2} i \left[ {\mathbf{d}}^\ast_\lambda \cdot \partial_\mu {\mathbf{d}}_\lambda
- ( \partial_\mu {\mathbf{d}}^\ast_\lambda ) \cdot {\mathbf{d}}_\lambda \right] \; ,$$ and use identities of the form $$\begin{split}
& |{\mathbf{d}}^\ast_B \cdot \partial_\mu {\mathbf{d}}_A |^2
+ |{\mathbf{d}}^\ast_C \cdot \partial_\mu {\mathbf{d}}_A |^2\\
& = | \langle {\mathbf{d}}_B |\partial_\mu {| {\mathbf{d}}_A \rangle} |^2
+| \langle {\mathbf{d}}_C |\partial_\mu {| {\mathbf{d}}_A \rangle}|^2\\
& = (\partial_\mu {\langle {\mathbf{d}}_A |})\partial_\mu {| {\mathbf{d}}_A \rangle}
- |\langle {\mathbf{d}}_A|\partial_\mu {| {\mathbf{d}}_A \rangle}|^2 \; , \end{split}$$ where , to express Eq. (\[eq:Heff\]) as $$\begin{aligned}
\label{eq:elegant.form.of.H}
\mathcal{H}^{\sf eff}_{\text{SU(3)}}
&=& \sum_{\lambda = A,B,C}
\frac{J}{\sqrt{3}}
\int d{\bf x}
\sum_{\mu = x, y}
| D^\lambda_\mu {\mathbf{d}}_\lambda |^2 \; , \\
D^\lambda_\mu &=& \partial_\mu + i A^\lambda_\mu \; .\end{aligned}$$ Here $A^\lambda_\mu$ transforms as under the gauge transformation ${\mathbf{d}}^\prime (x) = e^{i\Lambda}{\mathbf{d}}(x)$. We note that the orthogonality condition, Eq. (\[eq:orthogonality.condition\]), implies that any two of the vectors ${\mathbf{d}}_\lambda$ uniquely determine the third, leaving only phase degrees of freedom.
Solitonic solutions of Eq. (\[eq:elegant.form.of.H\]) are characterised by a finite topological charge. In order to parameterise this, we consider the based, second homotopy group $$\begin{aligned}
\pi_2(SU(3)/(U(1)\times U(1)) \; , \; b)
\nonumber\end{aligned}$$ where the base–point $b$ is given by $$\begin{aligned}
b = f {| r \rangle} \; .\end{aligned}$$ Following Ref. [@mermin79], this is determined by the [*first*]{} homotopy group of the isotropy subgroup $\pi_1(H) = {\mathbb Z} \times {\mathbb Z}$ \[cf. Eq. (\[eq:homotopy\_AFQ\_SU3\])\]. The two independent integers $\mathbb Z$ distinguish the different solitons which are possible within this order–parameter space and, thereby, their topological charge.
We can evaluate the topological charge associated with a given soliton by considering how the vectors ${\mathbf{d}}_\lambda$ evolve on a closed path $C(l)$ for $0\leq l \leq 1$ (with $C(0)=C(1)$), which encloses the soliton in the two–dimensional lattice space. For practical purposes, this closed path could be the boundary of a finite–size cluster. The simplest example is a soliton characterised by the integers , in which case the state on the path $C(l)$ is given by $$f_0(l){| r \rangle}=
\left( \begin{array}{ccc}
e^{-i2\pi l} & 0 & 0 \\
0 & e^{i2\pi l} & 0 \\
0 & 0 & 1
\end{array} \right){| r \rangle}\ ,\
\label{eq:loopf1}$$ where $0\leq l \leq 1$ and $C(0)=C(1)$. The contribution to the topological charge from each sublattice can be identified with the winding number associated with the diagonal elements of $f_0$. In the case of the A–sublattice $${\mathbf{d}}_A (l) = f_0(l) {\mathbf{d}}_A = (e^{-i2\pi l},0,0) \; ,$$ and it follows that the winding number on the path $C(l)$ is unity, and the associated topological charge is given by $Q_A = 1$. More formally, we can calculate this winding number as $$\begin{aligned}
&& Q_A = \frac{i}{2\pi} \oint_{C(l)} d{\mathbf l} \cdot
\left[ {\mathbf{d}}^\ast_A({\mathbf{x}}) {\bf \nabla} {\mathbf{d}}_A ({\mathbf{x}}) \right] \nonumber\\
&& = \frac{i}{2\pi} \int d{\bf x}\ (\partial_x {\mathbf{d}}^\ast_A({\mathbf{x}}))(\partial_y {\mathbf{d}}_A({\mathbf{x}}))
- (\partial_y {\mathbf{d}}^\ast_A({\mathbf{x}}))(\partial_x {\mathbf{d}}_A({\mathbf{x}})) \nonumber\\
&& = \frac{i}{2\pi} \int d{\bf x}\
\epsilon_{\mu\nu}\ (D_\mu^A {\mathbf{d}}_A)^\ast
\cdot D^A_\nu {\mathbf{d}}_A = 1 \; ,\end{aligned}$$ where the two–dimensional integral $\int d{\bf x}$ is carried out over the area enclosed by the path $C(l)$. By inspection of Eq. (\[eq:loopf1\]), the contribution of the B–sublattice is $Q_B = - Q_A = -1$, while the contribution from the (topologically–trivial) vanishes. Therefore, for this example, .
This approach to evaluating the topological charge remains valid, regardless of the base point, and can be applied to any spin configuration. So quite generally, we can write $$Q_\lambda = \frac{i}{2\pi} \int d{\bf x}\
\epsilon_{\mu\nu}\ (D_\mu^\lambda {\mathbf{d}}_\lambda ) ^\ast
\cdot D^\lambda_\nu {\mathbf{d}}_\lambda \; ,
\label{eq:topological.charge}$$ subject to the constraint $$Q_A + Q_B + Q_C = 0 \; .$$ This constraint follows from the structure of the isotropy subgroup $H$, i.e. the fact that there are only two undetermined angles in Eq. (\[isotropy1\]). Thus, while it is covenant to quote the topological charge as a vector ${\mathbf Q}$ with three components, there are only ever two independent degrees of freedom. Ultimately, this reflects the orthogonality condition on the three vectors ${\mathbf d}_{\lambda=A,B,C}$, as defined in Eq. (8). We note that our definition of the topological charge, Eq. (\[eq:topological.charge\]), differs by a sign convention from that definition used by in Ref. [@ivanov08].
Viewed this way, the continuum theory, Eq. (\[eq:elegant.form.of.H\]), comprises three copies of a CP$^2$ nonlinear sigma model, linked by the orthogonality condition, Eq. (\[eq:orthogonality.condition\]). The Cauchy–Schwartz inequality [@dadda78] enables us to place a lower bound on the energy of a soliton, based on its charge $$E_{\bf Q}
\geq \frac{2\pi J}{\sqrt{3}} \left( |Q_A|+|Q_B|+|Q_C| \right) \; .
\label{eq:bound}$$ Equality in Eq. (\[eq:bound\]), also known the as the Bogomol’nyi–Prasad–Sommerfield (BPS) bound [@manton04], is achieved where $$D_\mu^\lambda {\mathbf{d}}_\lambda
= \pm\ i \epsilon_{\mu\nu}\ D_\nu^\lambda {\mathbf{d}}_\lambda
\quad \forall \quad \lambda = A,B,C \; .
\label{eq:constraint}$$ We have been able to construct an exact wave function satisfying the Eq. (\[eq:constraint\]), for the special case . This corresponds to Q pairs of orthogonal CP$^2$ solitons [@dadda78; @ivanov08], found on two of the three sublattices, while the third remains topologically trivial. Specifically, we find $$\begin{split}
{\mathbf{d}}_A (z)
& = \frac{\xi {\mathbf{u}}_1 + \left[ \Pi_{k=1}^Q (z - z_k) \right] {\mathbf{v}}_1 }
{\sqrt{\xi^2 + \Pi_{k=1}^Q | z - z_k |^2)}} \; ,\\
{\mathbf{d}}_B (z)
& =\frac{-\xi {\mathbf{v}}_1 + \left[ \Pi_{k=1}^Q (z^\ast - z^\ast_k) \right] {\mathbf{u}}_1}
{\sqrt{(\xi^2 + \Pi_{k=1}^Q | z^\ast - z^\ast_k |^2)}} \; ,\\
{\mathbf{d}}_C (z)
& = {\mathbf{y}}_1\ \; .
\end{split}
\label{eq:analytic-wave-function}$$ where $z = x+iy$ is a complex coordinate, ${\mathbf{u}}_1$, ${\mathbf{v}}_1$ and ${\mathbf{y}}_1$ are complex orthonormal vectors, taken to be $$\begin{aligned}
{\mathbf{v}}_1=(1,0,0) \; , \; {\mathbf{u}}_1=(0,1,0) \; , \; {\mathbf{y}}_1=(0,0,1) \; .\end{aligned}$$ The coordinate $z_k$ specifies the positions of each soliton. Since the energy of this family of solitons is completely determined by its charge \[cf. Eq. (\[eq:bound\])\], it is independent of their size, which is set by the real parameter $\xi$. The wave function, Eq. (\[eq:analytic-wave-function\]), for a soliton with elementary charge ${\bf Q} = (-1,1,0)$, is illustrated in Fig. \[fig:1\].
![(Color online). Illustration of how repulsive interactions cause a single soliton with charge ${\bf Q} = (1,1,-2)$ to break into two separate pieces. (a) Quadrupole moment $Q^{x^2-y^2}_C$ on the C sublattice associated with the trial wave function Eq. (\[eq:trial\_wave\_function\]). (b) Quadrupole moment after numerical minimisation of a variational wave function. []{data-label="fig:2"}](fig2.pdf){width="0.95\columnwidth"}
{width="1.8\columnwidth"}
In order to gain more insight into solitons with general charge, for which no closed–form solution exists, we now switch to numerical analysis. We adopt a variational approach, based on a general product wave function, and minimise the energy of this wave function using simulated annealing [@kirkpatrick83], as described in the supplemental materials. Simulations were carried out for hexagonal clusters, and seeded with a trial wave function with definite topological charge ${\bf Q}$.
In the simplest case, a soliton of elementary charge ${\bf Q} = (-1,1,0)$, we can use the BPS bound Eq. (\[eq:analytic-wave-function\]) as a trial wave function, and simulations converge on a state like that shown in Fig. \[fig:1\], confirming the stability of the analytic solution on a finite lattice [@supplemental]. We next consider the soliton with charge . A suitable trial wave function is $$\begin{aligned}
\begin{split}
&{\mathbf{d}}_C(z) = \frac{{\mathbf{u}}_1 + c_1 z {\mathbf{v}}_1 + c_2 z^2 {\mathbf{y}}_1}
{\sqrt{1 + |c_1 z|^2 + |c_2 z|^2}}\ ,\\
&{\mathbf{d}}_B(z) = \frac{- c_1 z^\ast {\mathbf{u}}_1 + {\mathbf{v}}_1}
{\sqrt{1 + |c_1 z|^2}}\ ,\\
&{\mathbf{d}}_A(z) \perp {\mathbf{d}}_B(z)\ ,\ {\mathbf{d}}_C(z)\ ,
\label{eq:trial_wave_function}
\end{split}\end{aligned}$$ where we impose the boundary conditions , , , at the edges of the cluster, with $\theta_z = \arg (z)$. The coefficients $c_{1,2}$ are chosen so as to create a single soliton, localized at the origin, as illustrated in Fig. \[fig:2\](a). This state has a higher energy than the BPS bound, Eq. (\[eq:bound\]), and in numerical simulations, decays into two separate solitons of elementary charge, ${\bf Q} = (0,1,-1)$ and ${\bf Q} = (1,0,-1)$, as shown in Fig. \[fig:2\](b) and Fig. \[fig:3\]. In an infinitely extended system, these elementary charges could satisfy the BPS bound by separating entirely. However in our simulations, solitons also interact with the boundary of the cluster, and so only separate to finite distance.
We interpret these results as follows. The model, Eq. (\[eq:HeisenbergSU3\]) supports six different types of “elementary” soliton, with charge ${\bf Q} = \pm (0,1,-1)$, etc. Solitons with like charge do not interact, while solitons with unlike charge interact repulsively. We have also carried out preliminary simulations for solitons with higher topological charge, including ${\bf Q} = (0, 2,-2)$, ${\bf Q} = (1,2,-3)$ and ${\bf Q} = (2,2,-4)$, which confirm that this picture holds for more general topological charge. These results will be reported elsewhere.
In conclusion, we have explored the topological excitations which arise in an SU(3)–symmetric antiferromagnet on a triangular lattice. We find that this model supports a new class of stable solitons, with second homotopy group . These solitons can be characterised by integer topological charge , with . In the case of solitons with charge ${\bf Q} = (-Q, Q, 0)$, we are able to construct an exact wave function satisfying the BPS bound, comprising orthogonal CP$^{2}$ solitons on two of the three sublattices A, B, C \[cf. Fig. \[fig:1\]\]. Numerical simulations were used to confirm the stability of these solutions, and to explore the structure of solitons with more general topological charge. We find that solitons with charge spontaneously decay into solitons with “elementary” charge and \[cf. Fig. \[fig:3\]\]. We infer that solitons with different elementary charge interact repulsively.
To the best of our knowledge, these results represent the first example of quantum solitons characterised by two integers, with emergent repulsive interactions. The model solved has direct application to experiments on cold atoms in an optical lattice, where quantum magnets with SU(3) symmetry arise quite naturally [@honerkamp04; @gorelik09; @bauer12]. It is also interesting to speculate that these new solitons might survive as dynamical excitations in a magnet where SU(3) symmetry was broken, as argued for the CP$^2$ solitons found in the SU(3)–symmetric Heisenberg ferromagnet [@ivanov08]. And, since topological defects determine the type of quantum spin liquid which follows when classical order melts [@chubukov94; @grover11; @xu12], these results suggest the possibility of a new class of quantum spin liquid with an underlying U(1)$\times$U(1) gauge structure. We hope that this may help to shed light on the quantum spin–liquids found in two–dimensional quantum magnets which might otherwise support 3–sublattice order [@ishida97; @lee08; @cheng11; @xu12prl; @bieri12prb].
[*Acknowledgments :*]{} The authors are pleased to acknowledge helpful conversations with C. D. Batista, M. Kobayashi, T. Momoi, Y. Motome, T. Nitta, K. Penc, A. Smerald, K. Totsuka, and D. Yamamoto, and are grateful to F. Mila for a careful reading of the manuscript. This work was supported by the Okinawa Institute of Science and Technology Graduate University, and by JSPS KAKENHI Grant No. 26800209.
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Supplemental material for : Quantum solitons with emergent interactions in a model of cold atoms on the triangular lattice
Pictorial representation of wave functions
------------------------------------------
It is convenient to represent the wave function of the solitons studied in this Rapid Communication pictorially. The state of quantum spin–1 can conveniently be represented as a probability surface $$P(\theta,\phi) = | {\langle {\mathbf{d}}|} \boldsymbol{\Omega} \rangle |^2
\label{eq:prob.surface}$$ through its projection onto a spin coherent state $${| \boldsymbol{\Omega} \rangle} = \mathcal{R}(\theta,\phi) {| 1 \rangle}$$ where ${| 1 \rangle}$ is the eigenstate with $S^z = +1$, and $\mathcal{R}(\theta,\phi)$ is an SU(2) rotation matrix. A pictorial representation of a soliton with elementary charge ${\bf Q} = (-1,1,0)$ is shown in Fig. \[fig:1\_suppl\], below. The results are taken from numerical simulation, as described below, and in the main text.
In many cases it is also interesting to examine the quadrupole, or dipole, moments of spin which are induced by the soliton, relative to the reference state ${| r \rangle}$ \[cf. Eq. (9) of the main text\]. These can be calculated directly from the complex vector ${\mathbf{d}}$ [@ivanov08; @penc11-book.chapter; @Toth12] — for example the dipole moment associated with ${| {\mathbf{d}}\rangle}$ is given by $$\begin{aligned}
{\bf S} = 2 {\bf u} \times {\bf v} \; ,
\label{eq:Suv}\end{aligned}$$ where $$\begin{aligned}
{\bf d} = {\bf u} + i {\bf v} \; .\end{aligned}$$
To illustrate how this works, in Fig. \[fig:1\_suppl\] we show the results of numerical simulations for a soliton with elementary charge ${\bf Q} = (-1,1,0)$. Here, the probability surface for a quantum spin-1 at each site \[cf. Eq. (\[eq:prob.surface\])\], has been rendered in blue, while red, green and blue bars show the orientation of the quadrupole moment on each of the three sublattices $A$, $B$, and $C$. The dipole moment $S^z$, which vanishes in the reference state, has been represented as a color underlay.
This soliton comprises orthogonal $CP^2$ solitons on two of the three sublattices of the triangular lattice. Each of these $CP^2$ solitons can be thought of as a localised “twist” in the phase of the vector [**d**]{}, as described in Eq. (27) of the main text. This twist can be smoothly matched to ferroquadrupolar ($FQ$) order at infinity. However it induces a $\pi$ rotation of the associated director, accompanied by a finite dipole moment, in the vicinity of the soliton.
More detailed description of individual $CP^2$ solitons can be found in Ref. [@ivanov08] and Ref. [@dadda78].
Details of numerical simulations
--------------------------------
In our numerical studies we consider the most general class of product wave function for a quantum spin–1 $${| \{ {\mathbf{d}}_{\mathbf{l}}\} \rangle} = \prod_{\mathbf{l}}{| {\mathbf{d}}_{\mathbf{l}}\rangle} \; ,$$ where the product on ${\mathbf{l}}= 1 \ldots N$ runs over all sites of a finite–size cluster, and ${| {\mathbf{d}}_{\mathbf{l}}\rangle}$ is defined through Eq. (6) of the main text. Since ${\mathbf{d}}$ has unit norm, and the physical properties of ${| {\mathbf{d}}_{\mathbf{l}}\rangle}$ are invariant under a change of phase, $${| {\mathbf{d}}_{\mathbf{l}}\rangle} \rightarrow e^{i\theta} {| {\mathbf{d}}_{\mathbf{l}}\rangle} \; ,$$ a wave function of this form has $4^N$ variational parameters. These parameters are determined by simulated annealing [@kirkpatrick83], starting from an initial “guess” at the soliton wave function, with definite topological charge ${\bf Q}$.
In this approach, simulations are carried out at a temperature $T$, which is decreased gradually, and at each temperature the wave function is updated using Markov–chain Monte Carlo sampling. Monte Carlo updates involve the random re–orientation of the vector ${\mathbf{d}}$ for each spin in turn, following the standard Metropolis algorithm. Temperatures are reduced following a geometrical progression $$T_{k+1} = \alpha\ T_k \quad , \quad 0 < \alpha < 1 \; ,$$ where $T_k$ is the temperature of the $k^{th}$ of $N_{\sf annealing}$ annealing steps. Typical values are $$\begin{aligned}
\begin{split}
N_{\sf annealing} & = 300 & \\
\alpha & = 0.95 & \\
T_{\sf initial} &= 0.1\ J & \\
T_{\sf final} & \simeq 2.1 \times 10^{-8}\ J &
\end{split}\end{aligned}$$
We consider hexagonal clusters with the full symmetry of the triangular lattice, and impose a boundary condition at the edges of the cluster consistent with the reference state $| r \rangle$ \[Eq. (9) of main text\].
In addition, to ensure that simulations are carried out at fixed topological charge, we impose a “smoothness condition” $$| \partial_\mu {\mathbf{d}}_\lambda |^2 \leq 1 \quad , \quad \mu = x,y \quad , \quad \lambda =A, B, C \; .$$
{width="1.95\columnwidth"}
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[^1]: E-mail address: [email protected]
|
---
abstract: 'Rumour stance classification, defined as classifying the stance of specific social media posts into one of supporting, denying, querying or commenting on an earlier post, is becoming of increasing interest to researchers. While most previous work has focused on using individual tweets as classifier inputs, here we report on the performance of sequential classifiers that exploit the discourse features inherent in social media interactions or ‘conversational threads’. Testing the effectiveness of four sequential classifiers – Hawkes Processes, Linear-Chain Conditional Random Fields (Linear CRF), Tree-Structured Conditional Random Fields (Tree CRF) and Long Short Term Memory networks (LSTM) – on eight datasets associated with breaking news stories, and looking at different types of local and contextual features, our work sheds new light on the development of accurate stance classifiers. We show that sequential classifiers that exploit the use of discourse properties in social media conversations while using only local features, outperform non-sequential classifiers. Furthermore, we show that LSTM using a reduced set of features can outperform the other sequential classifiers; this performance is consistent across datasets and across types of stances. To conclude, our work also analyses the different features under study, identifying those that best help characterise and distinguish between stances, such as supporting tweets being more likely to be accompanied by evidence than denying tweets. We also set forth a number of directions for future research.'
address:
- 'University of Warwick, Coventry, UK'
- 'Alan Turing Institute, London, UK'
- 'University of Sheffield, Sheffield, UK'
- 'University of Melbourne, Melbourne, Australia'
- 'University of Copenhagen, Copenhagen, Denmark'
author:
- Arkaitz Zubiaga
- Elena Kochkina
- Maria Liakata
- Rob Procter
- Michal Lukasik
- Kalina Bontcheva
- Trevor Cohn
- Isabelle Augenstein
bibliography:
- 'sdqc.bib'
title: 'Discourse-Aware Rumour Stance Classification in Social Media Using Sequential Classifiers'
---
stance classification, social media, breaking news, veracity classification
Introduction
============
Social media platforms have established themselves as important sources for learning about the latest developments in breaking news. People increasingly use social media for news consumption [@hermida2012share; @mitchell2015millennials; @zubiaga2015real], while media professionals, such as journalists, increasingly turn to social media for news gathering [@zubiaga2013curating] and for gathering potentially exclusive updates from eyewitnesses [@diakopoulos2012finding; @tolmie2017microblog]. Social media platforms such as Twitter are a fertile and prolific source of breaking news, occasionally even outpacing traditional news media organisations [@kwak2010twitter]. This has led to the development of multiple data mining applications for mining and discovering events and news from social media [@dong2015multiscale; @stilo2016efficient]. However, the use of social media also comes with the caveat that some of the reports are necessarily rumours at the time of posting, as they have yet to be corroborated and verified [@mendoza2010twitter; @procter2013readingb; @procter2013readinga]. The presence of rumours in social media has hence provoked a growing interest among researchers for devising ways to determine veracity in order to avoid the diffusion of misinformation [@derczynski2015pheme].
Resolving the veracity of social rumours requires the development of a rumour classification system and we described in [@zubiaga2017detection], a candidate architecture for such a system consisting of the following four components: (1) detection, where emerging rumours are identified, (2) tracking, where those rumours are monitored to collect new related tweets, (3) stance classification, where the views expressed by different tweet authors are classified, and (4) veracity classification, where knowledge garnered from the stance classifier is put together to determine the likely veracity of a rumour.
In this work we focus on the development of the third component, a stance classification system, which is crucial to subsequently determining the veracity of the underlying rumour. The stance classification task consists in determining how individual posts in social media observably orientate to the postings of others [@walker2012stance; @qazvinian2011rumor]. For instance, a post replying with “no, that’s definitely false” is *denying* the preceding claim, whereas “yes, you’re right” is *supporting* it. It has been argued that aggregation of the distinct stances evident in the multiple tweets discussing a rumour could help in determining its likely veracity, providing, for example, the means to flag highly disputed rumours as being potentially false [@mendoza2010twitter]. This approach has been justified by recent research that has suggested that the aggregation of the different stances expressed by users can be used for determining the veracity of a rumour [@derczynski2015pheme; @liu2015realtime].
In this work we examine in depth the use of so-called sequential approaches to the rumour stance classification task. Sequential classifiers are able to utilise the discursive nature of social media [@tolmie2017microblog], learning from how ‘conversational threads’ evolve for a more accurate classification of the stance of each tweet. The use of sequential classifiers to model the conversational properties inherent in social media threads is still in its infancy. For example, in preliminary work we showed that a sequential classifier modelling the temporal sequence of tweets outperforms standard classifiers [@lukasik2016hawkes; @zubiaga2016coling]. Here we extend this preliminary experimentation in four different directions that enable exploring further the stance classification task using sequential classifiers: (1) we perform a comparison of a range of sequential classifiers, including a Hawkes Process classifier, a Linear CRF, a Tree CRF and an LSTM; (2) departing from the use of only local features in our previous work, we also test the utility of contextual features to model the conversational structure of Twitter threads; (3) we perform a more exhaustive analysis of the results looking into the impact of different datasets and the depth of the replies in the conversations on the classifiers’ performance, as well as performing an error analysis; and (4) we perform an analysis of features that gives insight into what characterises the different kinds of stances observed around rumours in social media. To the best of our knowledge, dialogical structures in Twitter have not been studied in detail before for classifying each of the underlying tweets and our work is the first to evaluate it exhaustively for stance classification. Twitter conversational threads are identifiable by the relational features that emerge as users respond to each others’ postings, leading to tree-structured interactions. The motivation behind the use of these dialogical structures for determining stance is that users’ opinions are expressed and evolve in a discursive manner, and that they are shaped by the interactions with other users.
The work presented here advances research in rumour stance classification by performing an exhaustive analysis of different approaches to this task. In particular, we make the following contributions:
- We perform an analysis of whether and the extent to which use of the sequential structure of conversational threads can improve stance classification in comparison to a classifier that determines a tweet’s stance from the tweet in isolation. To do so, we evaluate the effectiveness of a range of sequential classifiers: (1) a state-of-the-art classifier that uses Hawkes Processes to model the temporal sequence of tweets [@lukasik2016hawkes]; (2) two different variants of Conditional Random Fields (CRF), i.e., a linear-chain CRF and a tree CRF; and (3) a classifier based on Long Short Term Memory (LSTM) networks. We compare the performance of these sequential classifiers with non-sequential baselines, including the non-sequential equivalent of CRF, a Maximum Entropy classifier.
- We perform a detailed analysis of the results broken down by dataset and by depth of tweet in the thread, as well as an error analysis to further understand the performance of the different classifiers. We complete our analysis of results by delving into the features, and exploring whether and the extent to which they help characterise the different types of stances.
Our results show that sequential approaches do perform substantially better in terms of macro-averaged F1 score, proving that exploiting the dialogical structure improves classification performance. Specifically, the LSTM achieves the best performance in terms of macro-averaged F1 scores, with a performance that is largely consistent across different datasets and different types of stances. Our experiments show that LSTM performs especially well when only local features are used, as compared to the rest of the classifiers, which need to exploit contextual features to achieve comparable – yet still inferior – performance scores. Our findings reinforce the importance of leveraging conversational context in stance classification. Our research also sheds light on open research questions that we suggest should be addressed in future work. Our work here complements other components of a rumour classification system that we implemented in the PHEME project, including a rumour detection component [@zubiaga2016learning; @zubiaga2017exploiting], as well as a study into the diffusion of and reactions to rumour [@zubiaga2016analysing].
Related Work
============
Stance classification is applied in a number of different scenarios and domains, usually aiming to classify stances as one of “in favour” or “against”. This task has been studied in political debates [@hasan2013extra; @walker2012your], in arguments in online fora [@hasan2013stance; @sridhar2014collective] and in attitudes towards topics of political significance [@augenstein2016stance; @mohammad2016semeval; @Augenstein2016SemEval]. In work that is closer to our objectives, stance classification has also been used to help determine the veracity of information in micro-posts [@qazvinian2011rumor], often referred to as *rumour stance classification* [@lukasik2015classifying; @lukasik2016hawkes; @procter2013readinga; @zubiaga2016coling]. The idea behind this task is that the aggregation of distinct stances expressed by users in social media can be used to assist in deciding if a report is actually true or false [@derczynski2015pheme]. This may be particularly useful in the context of rumours emerging during breaking news stories, where reports are released piecemeal and which may be lacking authoritative review; in consequence, using the ‘wisdom of the crowd’ may provide a viable, alternative approach. The types of stances observed while rumours circulate, however, tend to differ from the original “in favour/against”, and different types of stances have been discussed in the literature, as we review next.
Rumour stance classification of tweets was introduced in early work by Qazvinian et al. [@qazvinian2011rumor]. The line of research initiated by [@qazvinian2011rumor] has progressed substantially with revised definitions of the task and hence the task tackled in this paper differs from this early work in a number of aspects. Qazvinian et al. [@qazvinian2011rumor] performed 2-way classification of each tweet as *supporting* or *denying* a long-standing rumour such as disputed beliefs that *Barack Obama is reportedly Muslim*. The authors used tweets observed in the past to train a classifier, which was then applied to new tweets discussing the same rumour. In recent work, rule-based methods have been proposed as a way of improving on Qazvinian et al.’s baseline method; however, rule-based methods are likely to be difficult – if not impossible – to generalise to new, unseen rumours. Hamidian et al. [@hamidian2016rumor] extended that work to analyse the extent to which a model trained from historical tweets could be used for classifying new tweets discussing the same rumour.
The work we present here has three different objectives towards improving stance classification. First, we aim to classify the stance of tweets towards rumours that emerge while breaking news stories unfold; these rumours are unlikely to have been observed before and hence rumours from previously observed events, which are likely to diverge, need to be used for training. As far as we know, only work by Lukasik et al. [@lukasik2015classifying; @lukasik2016using; @lukasik2016hawkes] has tackled stance classification in the context of breaking news stories applied to new rumours. Zeng et al. [@zeng2016unconfirmed] have also performed stance classification for rumours around breaking news stories, but overlapping rumours were used for training and testing. Augenstein et al. [@augenstein2016stance; @Augenstein2016SemEval] studied stance classification of unseen events in tweets, but ignored the conversational structure. Second, recent research has proposed that a 4-way classification is needed to encompass responses seen in breaking news stories [@procter2013readinga; @zubiaga2016analysing]. Moving away from the 2-way classification above, which [@procter2013readinga] found to be limited in the context of rumours during breaking news, we adopt this expanded scheme to include tweets that are *supporting*, *denying*, *querying* or *commenting* rumours. This adds more categories to the scheme used in early work, where tweets would only support or deny a rumour, or where a distinction between querying and commenting is not made [@augenstein2016stance; @mohammad2016semeval; @Augenstein2016SemEval]. Moreover, our approach takes into account the interaction between users on social media, whether it is about appealing for more information in order to corroborate a rumourous statement (*querying*) or to post a response that does not contribute to the resolution of the rumour’s veracity (*commenting*). Finally – and importantly – instead of dealing with tweets as single units in isolation, we exploit the emergent structure of interactions between users on Twitter, building a classifier that learns the dynamics of stance in tree-structured conversational threads by exploiting its underlying interactional features. While these interactional features do not, in the final analysis, map directly onto those of conversation as revealed by Conversation Analysis [@sacks1974simplest], we argue that there are sufficient relational similarities to justify this approach [@tolmie2017ugc]. The closest work is by Ritter et al. [@ritter2010unsupervised] who modelled linear sequences of replies in Twitter conversational threads with Hidden Markov Models for dialogue act tagging, but the tree structure of the thread as a whole was not exploited.
As we were writing this article, we also organised, in parallel, a shared task on rumour stance classification, RumourEval [@derczynski2017semeval], at the well-known natural language processing competition SemEval 2017. The subtask A consisted in stance classification of individual tweets discussing a rumour within a conversational thread as one of *support*, *deny*, *query*, or *comment*, which specifically addressed the task presented in this paper. Eight participants submitted results to this task, including work by [@kochkina2017turing] using an LSTM classifier which is being also analysed in this paper. In this shared task, most of the systems viewed this task as a 4-way single tweet classification task, with the exception of the best performing system by [@kochkina2017turing], as well as the systems by [@wang2017ecnu] and [@singh2017iitp]. The winning system addressed the task as a sequential classification problem, where the stance of each tweet takes into consideration the features and labels of the previous tweets. The system by Singh et al. [@singh2017iitp] takes as input pairs of source and reply tweets, whereas Wang et al. [@wang2017ecnu] addressed class imbalance by decomposing the problem into a two step classification task, first distinguishing between comments and non-comments, to then classify non-comment tweets as one of support, deny or query. Half of the systems employed ensemble classifiers, where classification was obtained through majority voting [@wang2017ecnu; @garcialozano2017mama; @bahuleyan2017uwaterloo; @srivastava2017dfki]. In some cases the ensembles were hybrid, consisting both of machine learning classifiers and manually created rules with differential weighting of classifiers for different class labels [@wang2017ecnu; @garcialozano2017mama; @srivastava2017dfki]. Three systems used deep learning, with [@kochkina2017turing] employing LSTMs for sequential classification, Chen et al. [@chen2017ikm] used convolutional neural networks (CNN) for obtaining the representation of each tweet, assigned a probability for a class by a softmax classifier and García Lozano et al. [@garcialozano2017mama] used CNN as one of the classifiers in their hybrid conglomeration. The remaining two systems by Enayet et al. [@enayet2017niletmrg] and Singh et al. [@singh2017iitp] used support vector machines with a linear and polynomial kernel respectively. Half of the systems invested in elaborate feature engineering, including cue words and expressions denoting Belief, Knowledge, Doubt and Denial [@bahuleyan2017uwaterloo] as well as Tweet domain features, including meta-data about users, hashtags and event specific keywords [@wang2017ecnu; @bahuleyan2017uwaterloo; @singh2017iitp; @enayet2017niletmrg]. The systems with the least elaborate features were Chen et al. [@chen2017ikm] and García Lozano et al. [@garcialozano2017mama] for CNNs (word embeddings), Srivastava et al. [@srivastava2017dfki] (sparse word vectors as input to logistic regression) and Kochkina et al. [@kochkina2017turing] (average word vectors, punctuation, similarity between word vectors in current tweet, source tweet and previous tweet, presence of negation, picture, URL). Five out of the eight systems used pre-trained word embeddings, mostly Google News word2vec embeddings[^1], whereas [@wang2017ecnu] used four different types of embeddings. The winning system used a sequential classifier, however the rest of the participants opted for other alternatives.
To the best of our knowledge Twitter conversational thread structure has not been explored in detail in the stance classification problem. Here we extend the experimentation presented in our previous work using Conditional Random Fields for rumour stance classification [@zubiaga2016coling] in a number of directions: (1) we perform a comparison of a broader range of classifiers, including state-of-the-art rumour stance classifiers such as Hawkes Processes introduced by Lukasik et al. [@lukasik2016hawkes], as well as a new LSTM classifier, (2) we analyse the utility of a larger set of features, including not only local features as in our previous work, but also contextual features that further model the conversational structure of Twitter threads, (3) we perform a more exhaustive analysis of the results, and (4) we perform an analysis of features that gives insight into what characterises the different kinds of stances observed around rumours in social media.
Research Objectives
===================
The main objective of our research is to analyse whether, the extent to which and how the sequential structure of social media conversations can be exploited to improve the classification of the stance expressed by different posts towards the topic under discussion. Each post in a conversation makes its own contribution to the discussion and hence has to be assigned its own stance value. However, posts in a conversation contribute to previous posts, adding up to a discussion attempting to reach a consensus. Our work looks into the exploitation of this evolving nature of social media discussions with the aim of improving the performance of a stance classifier that has to determine the stance of each tweet. We set forth the following six research objectives:
**RO 1.** *Quantify performance gains of using sequential classifiers compared with the use of non-sequential classifiers.*
Our first research objective aims to analyse how the use of a sequential classifier that models the evolving nature of social media conversations can perform better than standard classifiers that treat each post in isolation. We do this by solely using local features to represent each post, so that the analysis focuses on the benefits of the sequential classifiers.
**RO 2.** *Quantify the performance gains using contextual features extracted from the conversation.*
With our second research objective we are interested in analysing whether the use of contextual features (i.e. using other tweets surrounding in a conversation to extract the features of a given tweet) are helpful to boost the classification performance. This is particularly interesting in the case of tweets as they are very short, and inclusion of features extracted from surrounding tweets would be especially helpful. The use of contextual features is motivated by the fact that tweets in a discussion are adding to each other, and hence they cannot be treated alone.
**RO 3.** *Evaluate the consistency of classifiers across different datasets.*
Our aim is to build a stance classifier that will generalise to multiple different datasets comprising data belonging to different events. To achieve this, we evaluate our classifiers on eight different events.
**RO 4.** *Assess the effect of the depth of a post in its classification performance.*
We want to build a classifier that will be able to classify stances of different posts occurring at different levels of depth in a conversation. A post can be from a source tweet that initiates a conversation, to a nested reply that occurs later in the sequence formed by a conversational thread. The difficulty increases as replies are deeper as there is more preceding conversation to be aggregated for the classification task. We assess the performance over different depths to evaluate this.
**RO 5.** *Perform an error analysis to assess when and why each classifier performs best.*
We want to look at the errors made by each of the classifiers. This will help us understand when we are doing well and why, as well as in what cases and with which types of labels we need to keep improving.
**RO 6.** *Perform an analysis of features to understand and characterise stances in social media discussions.*
In our final objective we are interested in performing an exploration of different features under study, which is informative in two different ways. On the one hand, to find out which features are best for a stance classifier and hence improve performance; on the other hand, to help characterise the different types of stances and hence further understand how people respond in social media discussions.
Rumour Stance Classification
============================
In what follows we formally define the rumour stance classification task, as well as the datasets we use for our experiments.
Task Definition
---------------
The rumour stance classification task consists in determining the type of orientation that each individual post expresses towards the disputed veracity of a rumour. We define the rumour stance classification task as follows: we have a set of conversational threads, each discussing a rumour, $D = \{C_1, ..., C_n\}$. Each conversational thread $C_j$ has a variably sized set of tweets $|C_j|$ discussing it, with a source tweet (the root of the tree), $t_{j,1}$, that initiates it. The source tweet $t_{j,1}$ can receive replies by a varying number $k$ of tweets $Replies_{t_{j,1}} = \{t_{j,1,1}, ..., t_{j,1,k}\}$, which can in turn receive replies by a varying number $k$ of tweets, e.g., $Replies_{t_{j,1,1}} = \{t_{j,1,1,1}, ..., t_{j,1,1,k}\}$, and so on. An example of a conversational thread is shown in Figure \[fig:example\].
The task consists in determining the stance of each of the tweets $t_j$ as one of $Y = \{supporting, denying, querying, commenting\}$.
Dataset
-------
As part of the PHEME project [@derczynski2015pheme], we collected a rumour dataset associated with eight events corresponding to breaking news events [@zubiaga2016analysing].[^2] Tweets in this dataset include tree-structured conversations, which are initiated by a tweet about a rumour (source tweet) and nested replies that further discuss the rumour circulated by the source tweet (replying tweets). The process of collecting the tree-structured conversations initiated by rumours, i.e. having a rumour discussed in the source tweet, and associated with the breaking news events under study was conducted with the assistance of journalist members of the Pheme project team. Tweets comprising the rumourous tree-structured conversations were then annotated for stance using CrowdFlower[^3] as a crowdsourcing platform. The annotation process is further detailed in [@zubiaga2015crowdsourcing].
The resulting dataset includes 4,519 tweets and the transformations of annotations described above only affect 24 tweets (0.53%), i.e., those where the source tweet denies a rumour, which is rare. The example in Figure \[fig:example\] shows a rumour thread taken from the dataset along with our inferred annotations, as well as how we establish the depth value of each tweet in the thread.
*\[depth=0\]* **u1:** These are not timid colours; soldiers back guarding Tomb of Unknown Soldier after today’s shooting \#StandforCanada –PICTURE– **\[support\]**
[0pt]{} *\[depth=1\]* **u2:** @u1 Apparently a hoax. Best to take Tweet down. **\[deny\]**
[0pt]{} *\[depth=1\]* **u3:** @u1 This photo was taken this morning, before the shooting. **\[deny\]**
[0pt]{} *\[depth=1\]* **u4:** @u1 I don’t believe there are soldiers guarding this area right now. **\[deny\]**
[0pt]{} *\[depth=2\]* **u5:** @u4 wondered as well. I’ve reached out to someone who would know just to confirm that. Hopefully get response soon. **\[comment\]**
[0pt]{} *\[depth=3\]* **u4:** @u5 ok, thanks. **\[comment\]**
One important characteristic of the dataset, which affects the rumour stance classification task, is that the distribution of categories is clearly skewed towards *commenting* tweets, which account for over 64% of the tweets. This imbalance varies slightly across the eight events in the dataset (see Table \[tab:dataset-stats\]). Given that we consider each event as a separate fold that is left out for testing, this varying imbalance makes the task more realistic and challenging. The striking imbalance towards *commenting* tweets is also indicative of the increased difficulty with respect to previous work on stance classification, most of which performed binary classification of tweets as supporting or denying, which account for less than 28% of the tweets in our case representing a real world scenario.
Event Supporting Denying Querying Commenting Total
------------------- ------------- ------------ ------------ --------------- -------
charliehebdo 239 (22.0%) 58 (5.0%) 53 (4.0%) 721 (67.0%) 1,071
ebola-essien 6 (17.0%) 6 (17.0%) 1 (2.0%) 21 (61.0%) 34
ferguson 176 (16.0%) 91 (8.0%) 99 (9.0%) 718 (66.0%) 1,084
germanwings-crash 69 (24.0%) 11 (3.0%) 28 (9.0%) 173 (61.0%) 281
ottawashooting 161 (20.0%) 76 (9.0%) 63 (8.0%) 477 (61.0%) 777
prince-toronto 21 (20.0%) 7 (6.0%) 11 (10.0%) 64 (62.0%) 103
putinmissing 18 (29.0%) 6 (9.0%) 5 (8.0%) 33 (53.0%) 62
sydneysiege 220 (19.0%) 89 (8.0%) 98 (8.0%) 700 (63.0%) 1,107
Total 910 (20.1%) 344 (7.6%) 358 (7.9%) 2,907 (64.3%) 4,519
: Distribution of categories for the eight events in the dataset.[]{data-label="tab:dataset-stats"}
Classifiers
===========
In this section we describe the different classifiers that we used for our experiments. Our focus is on sequential classifiers, especially looking at classifiers that exploit the discursive nature of social media, which is the case for Conditional Random Fields in two different settings – i.e. Linear CRF and tree CRF – as well as that of a Long Short-Term Memory (LSTM) in a linear setting – Branch LSTM. We also experiment with a sequential classifier based on Hawkes Processes that instead exploits the temporal sequence of tweets and has been shown to achieve state-of-the-art performance [@lukasik2016hawkes]. After describing these three types of classifiers, we outline a set of baseline classifiers.
Hawkes Processes
----------------
One approach for modelling arrival of tweets around rumours is based on point processes, a probabilistic framework where tweet occurrence likelihood is modelled using an intensity function over time. Intuitively, higher values of intensity function denote higher likelihood of tweet occurrence. For example, Lukasik et al. modelled tweet occurrences over time with a log-Gaussian Cox Process, a point process which models its intensity function as an exponentiated sample of a Gaussian Process [@lukasik15_dynamics; @lukasik15_tweetarrival; @lukasik16_conv]. In related work, tweet arrivals were modelled with a Hawkes Process and a resulting model was applied for stance classification of tweets around rumours [@lukasik2016hawkes]. In this subsection we describe the sequence classification algorithm based on Hawkes Processes.
#### Intensity Function
The intensity function in a Hawkes Process is expressed as a summation of base intensity and the intensities corresponding to influences of previous tweets,
\[eq:HawkesCI2\] \_[y,m]{}(t) = \_y + \_[t\_< t]{} (m\_= m) \_[y\_,y]{} (t - t\_),
where the first term represents the constant base intensity of generating label $y$. The second term represents the influence from the previous tweets. The influence from each tweet is modelled with an exponential kernel function $\kappa(t - t_\ell) = \omega \exp(-\omega (t - t_\ell))$. The matrix $\alpha$ of size $|Y| \times |Y|$ encodes how pairs of labels corresponding to tweets influence one another, e.g. how a *querying* label influences a *rejecting* label.
#### Likelihood function
The parameters governing the intensity function are learnt by maximising the likelihood of generating the tweets:
\[eq:factorizedCL\] L(, , , ) = \_[n=1]{}\^N p(\_n | y\_n) p(E\_T),
where the likelihood of generating text given the label is modelled as a multinomial distribution conditioned on the label (parametrised by matrix $\beta$). The second term provides the likelihood of occurrence of tweets at times $t_1, \ldots , t_n$ and the third term provides the likelihood that no tweets happen in the interval $[0,T]$ except at times $t_1, \ldots, t_n$. We estimate the parameters of the model by maximising the log-likelihood. As in [@lukasik2016hawkes], Laplacian smoothing is applied to the estimated language parameter $\beta$.
In one approach to $\mu$ and $\alpha$ optimisation (*Hawkes Process with Approximated Likelihood*, or *HP Approx.* [@lukasik2016hawkes]) a closed form updates for $\mu$ and $\alpha$ are obtained using an approximation of the log-likelihood of the data. In a different approach (*Hawkes Process with Exact Likelihood*, or *HP Grad.* [@lukasik2016hawkes]) parameters are found using joint gradient based optimisation over $\mu$ and $\alpha$, using derivatives of log-likelihood[^4]. L-BFGS approach is employed for gradient search. Parameters are initialised with those found by the *HP Approx.* method. Moreover, following previous work we fix the decay parameter $\omega$ to $0.1$.
We predict the most likely sequence of labels, thus maximising the likelihood of occurrence of the tweets from Equation (\[eq:factorizedCL\]), or the approximated likelihood in case of *HP Approx.* Similarly as in [@lukasik2016hawkes], we follow a greedy approach, where we choose the most likely label for each consecutive tweet.
Conditional Random Fields (CRF): Linear CRF and Tree CRF
--------------------------------------------------------
We use CRF as a structured classifier to model sequences observed in Twitter conversations. With CRF, we can model the conversation as a graph that will be treated as a sequence of stances, which also enables us to assess the utility of harnessing the conversational structure for stance classification. Different to traditionally used classifiers for this task, which choose a label for each input unit (e.g. a tweet), CRF also consider the neighbours of each unit, learning the probabilities of transitions of label pairs to be followed by each other. The input for CRF is a graph $G = (V, E)$, where in our case each of the vertices $V$ is a tweet, and the edges $E$ are relations of tweets replying to each other. Hence, having a data sequence $X$ as input, CRF outputs a sequence of labels $Y$ [@lafferty2001conditional], where the output of each element $y_i$ will not only depend on its features, but also on the probabilities of other labels surrounding it. The generalisable conditional distribution of CRF is shown in Equation \[eq:crf\] [@sutton2011introduction].
$$p(y|x) = \frac{1}{Z(x)} \prod_{a = 1}^{A} \Psi_a (y_a, x_a)
\label{eq:crf}$$
where Z(x) is the normalisation constant, and $\Psi_a$ is the set of factors in the graph $G$.
We use CRFs in two different settings.[^5] First, we use a linear-chain CRF (Linear CRF) to model each branch as a sequence to be input to the classifier. We also use Tree-Structured CRFs (Tree CRF) or General CRFs to model the whole, tree-structured conversation as the sequence input to the classifier. So in the first case the sequence unit is a branch and our input is a collection of branches and in the second case our sequence unit is an entire conversation, and our input is a collection of trees. An example of the distinction of dealing with branches or trees is shown in Figure \[fig:tree-and-branches\]. With this distinction we also want to experiment whether it is worthwhile building the whole tree as a more complex graph, given that users replying in one branch might not have necessarily seen and be conditioned by tweets in other branches. However, we believe that the tendency of types of replies observed in a branch might also be indicative of the distribution of types of replies in other branches, and hence useful to boost the performance of the classifier when using the tree as a whole. An important caveat of modelling a tree in branches is also that there is a need to repeat parts of the tree across branches, e.g., the source tweet will repeatedly occur as the first tweet in every branch extracted from a tree.[^6]
{width="80.00000%"}
To account for the imbalance of classes in our datasets, we perform cost-sensitive learning by assigning weighted probabilities to each of the classes, these probabilities being the inverse of the number of occurrences observed in the training data for a class.
Branch LSTM
-----------
{width="70.00000%"}
Another model that works with structured input is a neural network with Long Short-Term Memory (LSTM) units [@hochreiter1997long]. LSTMs are able to model discrete time series and possess a ‘memory’ property of the previous time steps, therefore we propose a *branch-LSTM* model that utilises them to process branches of tweets.
Figure \[fig:LSTMbranchio\] illustrates how the input of the time step of the LSTM layer is a vector that is an average of word vectors from each tweet and how the information propagates between time steps.
The full model consists of several LSTM layers that are connected to several feed-forward ReLU layers and a softmax layer to obtain predicted probabilities of a tweet belonging to certain class. As a means for weight regularisation we utilise *dropout* and *l2-norm*. We use categorical cross-entropy as the loss function. The model is trained using mini-batches and the Adam optimisation algorithm [@kingma2014adam].[^7]
The number of layers, number of units in each layer, regularisation strength, mini-batch size and learning rate are determined using the Tree of Parzen Estimators (TPE) algorithm [@bergstra2011algorithms][^8] on the development set.[^9]
The *branch-LSTM* takes as input tweets represented as the average of its word vectors. We also experimented with obtaining tweet representations through per-word nested LSTM layers, however, this approach did not result in significantly better results than the average of word vectors.
Extracting branches from a tree-structured conversation presents the caveat that some tweets are repeated across branches after this conversion. We solve this issue by applying a mask to the loss function to not take repeated tweets into account.
Summary of Sequential Classifiers
---------------------------------
All of the classifiers described above make use of the sequential nature of Twitter conversational threads. These classifiers take a sequence of tweets as input, where the relations between tweets are formed by replies. If C replies to B, and B replies to A, it will lead to a sequence “A $\rightarrow$ B $\rightarrow$ C”. Sequential classifiers will use the predictions on preceding tweets to determine the possible label for each tweet. For instance, the classification for B will depend on the prediction that has been previously made for A, and the probabilities of different labels for B will vary for the classifier depending on what has been predicted for A.
Among the four classifiers described above, the one that differs in how the sequence is treated is the Tree CRF. This classifier builds a tree-structured graph with the sequential relationships composed by replying tweets. The rest of the classifiers, Hawkes Processes, Linear CRF and LSTM, will break the entire conversational tree into linear branches, and the input to the classifiers will be linear sequences. The use of a graph with the Tree CRF has the advantage of building a single structure, while the rest of the classifiers building linear sequences inevitably need to repeat tweets across different linear sequences. All the linear sequences will repeatedly start with the source tweet, while some of the subsequent tweets may also be repeated. The use of linear sequences has however the advantages of simplifying the model being used, and one may also hypothesise that inclusion of the entire tree made of different branches into the same graph may not be suitable when they may all be discussing issues that differ to some extent from one another. Figure \[fig:tree-and-branches\] shows an example of a conversation tree, how the entire tree would make a graph, as well as how we break it down into smaller branches or linear sequences.
Baseline Classifiers
--------------------
**Maximum Entropy classifier (MaxEnt).** As the non-sequential counterpart of CRF, we use a Maximum Entropy (or logistic regression) classifier, which is also a conditional classifier but which will operate at the tweet level, ignoring the conversational structure. This enables us to directly compare the extent to which treating conversations as sequences instead of having each tweet as a separate unit can boost the performance of the CRF classifiers. We perform cost-sensitive learning by assigning weighted probabilities to each class as the inverse of the number of occurrences in the training data.
**Additional baselines.** We also compare two more non-sequential classifiers[^10]: Support Vector Machines (SVM), and Random Forests (RF).
Experiment Settings and Evaluation Measures
-------------------------------------------
We experiment in an 8-fold cross-validation setting. Given that we have 8 different events in our dataset, we create 8 different folds, each having the data linked to an event. In our cross-validation setting, we run the classifier 8 times, on each occasion having a different fold for testing, with the other 7 for training. In this way, each fold is tested once, and the aggregation of all folds enables experimentation on all events. For each of the events in the test set, the experiments consist in classifying the stance of each individual tweet. With this, we simulate a realistic scenario where we need to use knowledge from past events to train a model that will be used to classify tweets in new events.
Given that the classes are clearly imbalanced in our case, evaluation based on accuracy arguably cannot suffice to capture competitive performance beyond the majority class. To account for the imbalance of the categories, we report the macro-averaged F1 scores, which measures the overall performance assigning the same weight to each category. We aggregate the macro-averaged F1 scores to get the final performance score of a classifier. We also use the McNemar test [@mcnemar1947note] throughout the analysis of results to further compare the performance of some classifiers.
It is also worth noting that all the sequential classifiers only make use of preceding tweets in the conversation to classify a tweet, and hence no later tweets are used. That is the case of a sequence *$t_1$, $t_2$, $t_3$* of tweets, each responding to the preceding tweet. The sequential classifier attempting to classify $t_2$ would incorporate $t_1$ in the sequence, but $t_3$ would not be considered.
Features
========
While focusing on the study of sequential classifiers for discursive stance classification, we perform our experiments with three different types of features: local features, contextual features and Hawkes features. First, local features enable us to evaluate the performance of sequential classifiers in a comparable setting to non-sequential classifiers where features are extracted solely from the current tweet; this makes it a fairer comparison where we can quantify the extent to which mining sequences can boost performance. In a subsequent step, we also incorporate contextual features, i.e. features from other tweets in a conversation, which enables us to further boost performance of the sequential classifiers. Finally, and to enable comparison with the Hawkes process classifier, we describe the Hawkes features.
Table \[tab:features\] shows the list of features used, both local and contextual, each of which can be categorised into several subtypes of features, as well as the Hawkes features. For more details on these features, please see \[ap:features\].
[0.66]{}[l | l]{}\
& Word embeddings\
& POS tags\
& Negation\
& Swear words\
& Tweet length\
& Word count\
& Question mark\
& Exclamation mark\
& URL attached\
\
& Word2Vec similarity wrt source tweet\
& Word2Vec similarity wrt preceding tweet\
& Word2Vec similarity wrt thread\
& Is leaf\
& Is source tweet\
& Is source user\
& Has favourites\
& Has retweets\
& Persistence\
& Time difference\
\
& Bag of words\
& Timestamp\
Experimental Results {#sec:experimental-results}
====================
Evaluating Sequential Classifiers (RO 1) {#sec:eval-sequential}
----------------------------------------
First, we evaluate the performance of the classifiers by using only local features. As noted above, this enables us to perform a fairer comparison of the different classifiers by using features that can be obtained solely from each tweet in isolation; likewise, it enables us to assess whether and the extent to which the use of a sequential classifier to exploit the discursive structure of conversational threads can be of help to boost performance of the stance classifier while using the same set of features as non-sequential classifiers.
Therefore, in this section we make use of the local features described in Section \[ssec:local-features\]. Additionally, we also use the Hawkes features described in Section \[ssec:hawkes-features\] for comparison with the Hawkes processes. For the set of local features, we show the results for three different scenarios: (1) using each subgroup of features alone, (2) in a leave-one-out setting where one of the subgroups is not used, and (3) using all of the subgroups combined.
Table \[tab:results-sequence\] shows the results for the different classifiers using the combinations of local features as well as Hawkes features. We make the following observations from these results:
- LSTM consistently performs very well with different features.
- Confirming our main hypothesis and objective, sequential classifiers do show an overall superior performance to the non-sequential classifiers. While the two CRF alternatives perform very well, the LSTM classifier is slightly superior (the differences between CRF and LSTM results are statistically significant at $p < 0.05$, except for the LF1 features). Moreover, the CRF classifiers outperform their non-sequential counterpart MaxEnt, which achieves an overall lower performance (all the differences between CRF and MaxEnt results being statistically significant at $p < 0.05$).
- The LSTM classifier is, in fact, superior to the Tree CRF classifier (all statistically significant except LF1). While the Tree CRF needs to make use of the entire tree for the classification, the LSTM classifier only uses branches, reducing the amount of data and complexity that needs to be processed in each sequence.
- Among the local features, combinations of subgroups of features lead to clear improvements with respect to single subgroups without combinations.
- Even though the combination of all local features achieves good performance, there are alternative leave-one-out combinations that perform better. The feature combination leading to the best macro-F1 score is that combining lexicon, content formatting and punctuation (i.e. LF123, achieving a score of 0.449).
Summarising, our initial results show that exploiting the sequential properties of conversational threads, while still using only local features to enable comparison, leads to superior performance with respect to the classification of each tweet in isolation by non-sequential classifiers. Moreover, we observe that the local features combining lexicon, content formatting and punctuation lead to the most accurate results. In the next section we further explore the use of contextual features in combination with local features to boost performance of sequential classifiers; to represent the local features, we rely on the best approach from this section (i.e. LF123).
[l || l l l l l l l l l l ]{}\
& HF & LF1 & LF2 & LF3 & LF4 & LF123 & LF124 & LF134 & LF234 & LF1234\
SVM & 0.336 & 0.356 & 0.231 & 0.258 & 0.313 & 0.403 & 0.365 & 0.403 & 0.420 & 0.408\
Random Forest & 0.325 & 0.308 & 0.276 & 0.267 & **0.437\*** & 0.322 & 0.310 & 0.351 & 0.357 & 0.329\
MaxEnt & 0.338 & 0.363 & 0.272 & 0.263 & 0.428 & 0.415 & 0.363 & 0.421 & 0.427 & 0.422\
Hawkes-approx & 0.309 & – & – & – & – & – & – & – & – & –\
Hawkes-grad & 0.307 & – & – & – & – & – & – & – & – & –\
Linear CRF & **0.362\*** & 0.357 & 0.268 & 0.318 & 0.317 & 0.413 & 0.365 & 0.403 & 0.425 & 0.412\
Tree CRF & 0.350 & **0.375\*** & 0.285 & 0.221 & 0.217 & 0.433 & 0.385 & **0.413** & **0.436\*** & 0.433\
LSTM & 0.318 & 0.362 & **0.318\*** & **0.407\*** & 0.419 & **0.449\*** & **0.395\*** & 0.412 & 0.429 & **0.437\***\
Exploring Contextual Features (RO 2) {#sec:eval-contextual}
------------------------------------
The experiments in the previous section show that sequential classifiers that model discourse, especially the LSTM classifier, can provide substantial improvements over non-sequential classifiers that classify each tweet in isolation, in both cases using only local features to represent each tweet. To complement this, we now explore the inclusion of contextual features described in Section \[ssec:contextual-features\] for the stance classification. We perform experiments with four different groups of features in this case, including local features and the three subgroups of contextual features, namely relational features, structural features and social features. As in the previous section, we show results for the use of each subgroup of features alone, in a leave-one-out setting, and using all subgroups of features together.
Table \[tab:results-contextual\] shows the results for the classifiers incorporating contextual features along with local features. We make the following observations from these results:
- The use of contextual features leads to substantial improvements for non-sequential classifiers, getting closer to and even in some cases outperforming some of the sequential classifiers.
- Sequential classifiers, however, do not benefit much from using contextual features. It is important to note that sequential classifiers are taking the surrounding context into consideration when they aggregate sequences in the classification process. This shows that the inclusion of contextual features is not needed for sequential classifiers, given that they are implicitly including context through the use of sequences.
- In fact, for the LSTM, which is still the best-performing classifier, it is better to only rely on local features, as the rest of the features do not lead to any improvements. Again, the LSTM is able to handle context on its own, and therefore inclusion of contextual features is redundant and may be harmful.
- Addition of contextual features leads to substantial improvements for the non-sequential classifiers, achieving similar macro-averaged scores in some cases (e.g. MaxEnt / All vs LSTM / LF). This reinforces the importance of incorporating context in the classification process, which leads to improvements for the non-sequential classifier when contextual features are added, but especially in the case of sequential classifiers that can natively handle context.
[l || l l l l l l l l l ]{}\
& LF & R & ST & SO & LF+R+ST & LF+R+SO & LF+ST+SO & R+ST+SO & All\
SVM & 0.403 & **0.335\*** & **0.318** & 0.260 & 0.429 & 0.347 & 0.388 & 0.295 & 0.375\
Random Forest & 0.322 & 0.325 & 0.269 & 0.328 & 0.356 & 0.358 & 0.376 & **0.343\*** & 0.364\
MaxEnt & 0.415 & 0.333 & **0.318** & 0.310 & 0.434 & **0.447** & 0.447 & 0.318 & **0.449**\
Linear CRF & 0.413 & 0.318 & **0.318** & **0.334\*** & 0.424 & 0.431 & 0.431 & 0.342 & 0.437\
Tree CRF & 0.433 & 0.322 & 0.317 & 0.312 & 0.425 & 0.429 & 0.430 & 0.232 & 0.433\
LSTM & **0.449\*** & 0.318 & **0.318** & 0.315 & **0.445\*** & 0.436 & **0.448** & 0.314 & 0.437\
Summarising, we observe that the addition of contextual features is clearly useful for non-sequential classifiers, which do not consider context natively. For the sequential classifiers, which natively consider context in the classification process, the inclusion of contextual features is not helpful and is even harmful in most cases, potentially owing to the contextual information being used twice. Still, sequential classifiers, and especially LSTM, are the best classifiers to achieve optimal results, which also avoid the need for computing contextual features.
Analysis of the Best-Performing Classifiers
-------------------------------------------
Despite the clear superiority of LSTM with the sole use of local features, we now further examine the results of the best-performing classifiers to understand when they perform well. We compare the performance of the following five classifiers in this section: (1) LSTM with only local features, (2) Tree CRF with all the features, (3) Linear CRF with all the features, (4) MaxEnt with all the features, and (5) SVM using local features, relational and structural features. Note that while for LSTM we only need local features, for the rest of the classifiers we need to rely on all or almost all of the features. For these best-performing combinations of classifiers and features, we perform additional analyses by event and by tweet depth, and perform an analysis of errors.
### Evaluation by Event (RO 3)
The analysis of the best-performing classifiers, broken down by event, is shown in Table \[tab:results-events\]. These results suggest that there is not a single classifier that performs best in all cases; this is most likely due to the diversity of events. However, we see that the LSTM is the classifier that outperforms the rest in the greater number of cases; this is true for three out of the eight cases (the difference with respect to the second best classifier being always statistically significant). Moreover, sequential classifiers perform best in the majority of the cases, with only three cases where a non-sequential classifier performs best. Most importantly, these results suggest that sequential classifiers outperform non-sequential classifiers across the different events under study, with LSTM standing out as a classifier that performs best in numerous cases using only local features.
[l || l l l l l l l l ]{}\
& CH & Ebola & Ferg. & GW crash & Ottawa & Prince & Putin & Sydney\
SVM & 0.399 & 0.380 & 0.382 & 0.427 & **0.492** & 0.491 & 0.509 & 0.427\
MaxEnt & 0.446 & 0.425 & **0.418** & 0.475 & 0.468 & **0.514** & 0.381 & 0.443\
Linear CRF & 0.443 & 0.619 & 0.380 & 0.470 & 0.412 & 0.512 & **0.528** & **0.454**\
Tree CRF & 0.457 & 0.557 & 0.356 & 0.523 & 0.441 & 0.505 & 0.491 & 0.426\
LSTM & **0.465** & **0.657** & 0.373 & **0.543** & 0.475 & 0.379 & 0.457 & 0.446\
### Evaluation by Tweet Depth (RO 4)
The analysis of the best-performing classifiers, broken down by depth of tweets, is shown in Table \[tab:results-depth\]. Note that the depth of the tweet reflects, as shown in Figure \[fig:example\], the number of steps from the source tweet to the current tweet. We show results for all the depths from 0 to 4, as well as for the subsequent depths aggregated as 5+.
Again, we see that there is not a single classifier that performs best for all depths. We see, however, that sequential classifiers (Linear CRF, Tree CRF and LSTM) outperform non-sequential classifiers (SVM and MaxEnt) consistently. However, the best sequential classifier varies. While LSTM is the best-performing classifier overall when we look at macro-averaged F1 scores, as shown in Section \[sec:eval-contextual\], surprisingly it does not achieve the highest macro-averaged F1 scores at any depth. It does, however, perform well for each depth compared to the rest of the classifiers, generally being close to the best classifier in that case. Its consistently good performance across different depths makes it the best overall classifier, despite only using local features.
[l || l l l l l l ]{}\
& 0 & 1 & 2 & 3 & 4 & 5+\
Counts & 297 & 2,602 & 553 & 313 & 195 & 595\
\
& 0 & 1 & 2 & 3 & 4 & 5+\
SVM & 0.272 & 0.368 & 0.298 & 0.314 & 0.331 & 0.274\
MaxEnt & 0.238 & 0.385 & 0.286 & 0.279 & **0.369** & **0.290**\
Linear CRF & **0.286** & 0.394 & **0.306** & 0.282 & 0.271 & 0.266\
Tree CRF & 0.278 & **0.404** & 0.280 & **0.331** & 0.230 & 0.237\
LSTM & 0.271 & 0.381 & 0.298 & 0.274 & 0.307 & 0.286\
### Error Analysis (RO 5)
To analyse the errors that the different classifiers are making, we look at the confusion matrices in Table \[tab:confusion\]. If we look at the correct guesses, highlighted in bold in the diagonals, we see that the LSTM clearly performs best for three of the categories, namely *support*, *deny* and *query*, and it is just slightly behind the other classifiers for the majority class, *comment*. Besides LSTM’s overall superior performance as we observed above, this also confirms that the LSTM is doing better than the rest of the classifiers in dealing with the imbalance inherent in our datasets. For instance, the *Deny* category proves especially challenging for being less common than the rest (only 7.6% of instances in our datasets); the LSTM still achieves the highest performance for this category, which, however, only achieves 0.212 in accuracy and may benefit from having more training instances.
We also notice that a large number of instances are misclassified as *comments*, due to this being the prevailing category and hence having a much larger number of training instances. One could think of balancing the training instances to reduce the prevalence of *comments* in the training set, however, this is not straightforward for sequential classifiers as one needs to then break sequences, losing not only some instances of *comments*, but also connections between instances of other categories that belong to those sequences. Other solutions, such as labelling more data or using more sophisticated features to distinguish different categories, might be needed to deal with this issue; given that the scope of this paper is to assess whether and the extent to which sequential classifiers can be of help in stance classification, further tackling this imbalance is left for future work.
[l || l l l l ]{}\
& Support & Deny & Query & Comment\
Support & **0.657** & 0.041 & 0.018 & 0.283\
Deny & 0.185 & **0.129** & 0.107 & 0.579\
Query & 0.083 & 0.081 & **0.343** & 0.494\
Comment & 0.150 & 0.075 & 0.053 & **0.723**\
\
& Support & Deny & Query & Comment\
Support & **0.794** & 0.044 & 0.003 & 0.159\
Deny & 0.156 & **0.130** & 0.079 & 0.634\
Query & 0.088 & 0.066 & **0.366** & 0.480\
Comment & 0.152 & 0.074 & 0.048 & **0.726**\
\
& Support & Deny & Query & Comment\
Support & **0.603** & 0.048 & 0.013 & 0.335\
Deny & 0.219 & **0.140** & 0.050 & 0.591\
Query & 0.071 & 0.095 & **0.357** & 0.476\
Comment & 0.139 & 0.072 & 0.062 & **0.726**\
\
& Support & Deny & Query & Comment\
Support & **0.552** & 0.066 & 0.019 & 0.363\
Deny & 0.145 & **0.169** & 0.081 & 0.605\
Query & 0.077 & 0.081 & **0.401** & 0.441\
Comment & 0.128 & 0.074 & 0.068 & **0.730**\
\
& Support & Deny & Query & Comment\
Support & **0.825** & 0.046 & 0.003 & 0.127\
Deny & 0.225 & **0.212** & 0.125 & 0.438\
Query & 0.090 & 0.087 & **0.432** & 0.390\
Comment & 0.144 & 0.076 & 0.057 & **0.723**\
Feature Analysis (RO 6)
-----------------------
To complete the analysis of our experiments, we now look at the different features we used in our study and perform an analysis to understand how distinctive the different features are for the four categories in the stance classification problem. We visualise the different distributions of features for the four categories in beanplots [@kampstra2008beanplot]. We show the visualisations pertaining to 16 of the features under study in Figure \[fig:features\]. This analysis leads us to some interesting observations towards characterising the different types of stances:
- As one might expect, *querying tweets* are more likely to have question marks.
- Interestingly, *supporting tweets* tend to have a higher similarity with respect to the source tweet, indicating that the similarity based on word embeddings can be a good feature to identify those tweets.
- *Supporting tweets* are more likely to come from the user who posted the source tweet.
- *Supporting tweets* are more likely to include links, which is likely indicative of tweets pointing to evidence that supports their position.
- Looking at the delay in time of different types of tweets (i.e., the *time difference* feature), we see that *supporting*, *denying* and *querying tweets* are more likely to be observed only in the early stages of a rumour, while later tweets tend to be mostly comments. In fact, these suggests that discussion around the veracity of a rumour occurs especially in the period just after it is posted, whereas the conversation then evolves towards comments that do not discuss the veracity of the rumour in question.
- *Denying tweets* are more likely to use negating words. However, negations are also used in other kinds of tweets to a lesser extent, which also makes it more complicated for the classifiers to identify denying tweets. In addition to the low presence of denying tweets in the datasets, the use of negations also in other kinds of responses makes it more challenging to classify them. A way to overcome this may be to use more sophisticated approaches to identify negations that are rebutting the rumour initiated in the source tweet, while getting rid of the rest of the negations.
- When we look at the extent to which users persist in their participation in a conversational thread (i.e., the *persistence* feature), we see that users tend to participate more when they are posting *supporting tweets*, showing that users especially insistent when they support a rumour. However, we observe a difference that is not highly remarkable in this particular case.
The rest of the features do not show a clear tendency that helps visually distinguish characteristics of the different types of responses. While some features like swear words or exclamation marks may seem indicative of how they orient to somebody else’s earlier post, there is no clear difference in reality in our datasets. The same is true for social features like retweets or favourites, where one may expect, for instance, that denying tweets may attract more retweets than comments, as people may want to let others know about rebuttals; the distributions of retweets and favourites are, however, very similar for the different categories.
One possible concern from this analysis is that there are very few features that characterise *commenting tweets*. In fact, the only feature that we have identified as being clearly distinct for *comments* is the *time difference*, given that they are more likely to appear later in the conversations. This may well help classify those late *comments*, however, early comments will be more difficult to be classified based on that feature. Finding additional features to distinguish *comments* from the rest of the tweets may be of help for improving the overall classification.
![Distributions of feature values across the four categories: Support, Deny, Query and Comment.[]{data-label="fig:features"}](features.pdf){width="\textwidth"}
Conclusions and Future Work
===========================
While discourse and sequential structure of social media conversations have been barely explored in previous work, our work has performed an analysis on the use of different sequential classifiers for the rumour stance classification task. Our work makes three core contributions to existing work on rumour stance classification: (1) we focus on the stance of tweets towards rumours that emerge while breaking news stories unfold; (2) we broaden the stance types considered in previous work to encompass all types of responses observed during breaking news, performing a 4-way classification task; and (3) instead of dealing with tweets as single units in isolation, we exploit the emergent structure of interactions between users on Twitter. In this task, a classifier has to determine if each tweet is supporting, denying, querying or commenting on a rumour’s truth value. We mine the sequential structure of Twitter conversational threads in the form of users’ replies to one another, extending existing approaches that treat each tweet as a separate unit. We have used four different sequential classifiers: (1) a Hawkes Process classifier that exploits temporal sequences, which showed state-of-the-art performance [@lukasik2016hawkes]; (2) a linear-chain CRF modelling tree-structured conversations broken down into branches; (3) a tree CRF modelling them as a graph that includes the whole tree; and (4) an LSTM classifier that also models the conversational threads as branches. These classifiers have been compared with a range of baseline classifiers, including the non-sequential equivalent Maximum Entropy classifier, on eight Twitter datasets associated with breaking news.
While previous stance detection work had mostly limited classifiers to looking at tweets as single units, we have shown that exploiting the discursive characteristics of interactions on Twitter, by considering probabilities of transitions within tree-structured conversational threads, can lead to substantial improvements. Among the sequential classifiers, our results show that the LSTM classifier using a more limited set of features performs the best, thanks to its ability to natively handle context, as well as only relying on branches instead of the whole tree, which reduces the amount of data and complexity that needs to be processed in each sequence. The LSTM has been shown to perform consistently well across datasets, as well as across different types of stances. Besides the comparison of classifiers, our analysis also looks at the distributions of the different features under study as well as how well they characterise the different types of stances. This enables us both to find out which features are the most useful, as well as to suggest improvements needed in future work for improving stance classifiers.
To the best of our knowledge, this is the first attempt at aggregating the conversational structure of Twitter threads to produce classifications at the tweet level. Besides the utility of mining sequences from conversational threads for stance classification, we believe that our results will, in turn, encourage the study of sequential classifiers applied to other natural language processing and data mining tasks where the output for each tweet can benefit from the structure of the entire conversation, e.g., sentiment analysis [@kouloumpis2011twitter; @tsytsarau2012survey; @saif2016contextual; @liu2016identifying; @vilares2017supervised; @pandey2017twitter], tweet geolocation [@han2014text; @zubiaga2017towards], language identification [@bergsma2012language; @zubiaga2016tweetlid], event detection [@srijith2017sub] and analysis of public perceptions on news [@reis2015breaking; @an2011media] and other issues [@pak2010twitter; @bian2016mining].
Our plans for future work include further developing the set of features that characterise the most challenging and least-frequent stances, i.e., denying tweets and querying tweets. These need to be investigated as part of a more detailed and interdisciplinary, thematic analysis of threads [@tolmie2017microblog; @housley2017digitizing; @housley2017membership]. We also plan to develop an LSTM classifier that mines the entire conversation as a single tree. Our approach assumes that rumours have been already identified or input by a human, hence a final and ambitious aim for future work is the integration with our rumour detection system [@zubiaga2016learning], whose output would be fed to the stance classification system. The output of our stance classification will also be integrated with a veracity classification system, where the aggregation of stances observed around a rumour will be exploited to determine the likely veracity of the rumour.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been supported by the PHEME FP7 project (grant No. 611233), the EPSRC Career Acceleration Fellowship EP/I004327/1, Elsevier through the UCL Big Data Institute, and The Alan Turing Institute under the EPSRC grant EP/N510129/1.
References {#references .unnumbered}
==========
Features {#ap:features}
========
Local Features {#ssec:local-features}
--------------
Local features are extracted from each of the tweets in isolation, and therefore it is not necessary to look at other features in a thread to generate them. We use four types of features to represent the tweets locally.
**Local feature type \#1: Lexicon.**
- *Word Embeddings:* we use Word2Vec [@mikolov2013distributed] to represent the textual content of each tweet. First, we trained a separate Word2Vec model for each of the eight folds, each having the seven events in the training set as input data, so that the event (and the vocabulary) in the test set is unknown. We use large datasets associated with the seven events in the training set, including all the tweets we collected for those events. Finally, we represent each tweet as a vector with 300 dimensions averaging vector representations of the words in the tweet using Word2Vec.
- *Part of speech (POS) tags:* we parse the tweets to extract the part-of-speech (POS) tags using Twitie [@bontcheva2013twitie]. Once the tweets are parsed, we represent each tweet with a vector that counts the number of occurrences of each type of POS tag. The final vector therefore has as many features as different types of POS tags we observe in the dataset.
- *Use of negation:* this is a feature determining the number of negation words found in a tweet. The existence of negation words in a tweet is determined by looking at the presence of the following words: not, no, nobody, nothing, none, never, neither, nor, nowhere, hardly, scarcely, barely, don’t, isn’t, wasn’t, shouldn’t, wouldn’t, couldn’t, doesn’t.
- *Use of swear words:* this is a feature determining the number of ‘bad’ words present in a tweet. We use a list of 458 bad words[^11].
**Local feature type \#2: Content formatting.**
- *Tweet length:* the length of the tweet in number of characters.
- *Word count:* the number of words in the tweet, counted as the number of space-separated tokens.
**Local feature type \#3: Punctuation.**
- *Use of question mark:* binary feature indicating the presence or not of at least one question mark in the tweet.
- *Use of exclamation mark:* binary feature indicating the presence or not of at least one exclamation mark in the tweet.
**Local feature type \#4: Tweet formatting.**
- *Attachment of URL:* binary feature, capturing the presence or not of at least one URL in the tweet.
Contextual Features {#ssec:contextual-features}
-------------------
**Contextual feature type \#1: Relational features.**
- *Word2Vec similarity wrt source tweet:* we compute the cosine similarity between the word vector representation of the current tweet and the word vector representation of the source tweet. This feature intends to capture the semantic relationship between the current tweet and the source tweet and therefore help inferring the type of response.
- *Word2Vec similarity wrt preceding tweet:* likewise, we compute the similarity between the current tweet and the preceding tweet, the one that is directly responding to.
- *Word2Vec similarity wrt thread:* we compute another similarity score between the current tweet and the rest of the tweets in the thread excluding the tweets from the same author as that in the current tweet.
**Contextual feature type \#2: Structural features.**
- *Is leaf:* binary feature indicating if the current tweet is a leaf, i.e. the last tweet in a branch of the tree, with no more replies following.
- *Is source tweet:* binary feature determining if the tweet is a source tweet or is instead replying to someone else. Note that this feature can also be extracted from the tweet itself, checking if the tweet content begins with a Twitter user handle or not.
- *Is source user:* binary feature indicating if the current tweet is posted by the same author as that in the source tweet.
**Contextual feature type \#3: Social features.**
- *Has favourites:* feature indicating the number of times a tweet has been favourited.
- *Has retweets:* feature indicating the number of times a tweet has been retweeted.
- *Persistence:* this feature is the count of the total number of tweets posted in the thread by the author in the current tweet. High numbers of tweets in a thread indicate that the author participates more.
- *Time difference:* this is the time elapsed, in seconds, from when the source tweet was posted to the time the current tweet was posted.
Hawkes Features {#ssec:hawkes-features}
---------------
- *Bag of words:* a vector where each token in the dataset represents a feature, where each feature is assigned a number pertaining its count of occurrences in the tweet.
- *Timestamp:* The UNIX time in which the tweet was posted.
[^1]: <https://github.com/mmihaltz/word2vec-GoogleNews-vectors>
[^2]: The entire dataset included nine events, but here we describe the eight events with tweets in English, which we use for our classification experiments. The ninth dataset with tweets in German was not considered for this work.
[^3]: https://www.crowdflower.com/
[^4]: For both implementations we used the ‘seqhawkes’ Python package: <https://github.com/mlukasik/seqhawkes>
[^5]: We use the PyStruct to implement both variants of CRF [@muller2014pystruct].
[^6]: Despite this also leading to having tweets repeated across branches in the test set and hence producing an output repeatedly for the same tweet with Linear CRF, this output does is consistent and there is no need to aggregate different outputs.
[^7]: For implementation of all models we used Python libraries Theano [@bastien2012theano] and Lasagne [@lasagne].
[^8]: We use the implementation in the hyperopt package [@bergstra2013making].
[^9]: For this setting, we use the ‘Ottawa shooting’ event for development.
[^10]: We use their implementation in the scikit-learn Python package, using the *class\_weight=“balanced”* parameter to perform cost-sensitive learning.
[^11]: <http://urbanoalvarez.es/blog/2008/04/04/bad-words-list/>
|
---
abstract: |
The observable gravitational and electromagnetic parameters of an electron: mass $m$, spin $J=\hbar/2$, charge $e$ and magnetic moment $ea = e\hbar /(2m)$ indicate unambiguously that the electron should had the Kerr-Newman background geometry – exact solution of the Einstein-Maxwell gravity for a charged and rotating black hole. Contrary to the widespread opinion that gravity plays essential role only on the Planck scales, the Kerr-Newman gravity displays a new dimensional parameter $a
=\hbar/(2m),$ which for parameters of an electron corresponds to the Compton wavelength and turns out to be very far from the Planck scale. Extremely large spin of the electron with respect to its mass produces the Kerr geometry without horizon, which displays very essential topological changes at the Compton distance resulting in a two-fold structure of the electron background. The corresponding gravitational and electromagnetic fields of the electron are concentrated near the Kerr ring, forming a sort of a closed string, structure of which is close to the described by Sen heterotic string. The indicated by Gravity stringlike structure of the electron contradicts to the statements of Quantum theory that electron is pointlike and structureless. However, it confirms the peculiar role of the Compton zone of the “dressed” electron and matches with the known limit of the localization of the Dirac electron. We discuss the relation of the Kerr string with the low energy string theory and with the Dirac theory of electron and suggest that the predicted by the Kerr-Newman gravity closed string in the core of the electron, should be experimentally observable by the novel regime of the high energy scattering – the Deeply Virtual (or “nonforward”) Compton Scattering".
address: 'Lab. of Theor. Phys. , NSI, Russian Academy of Sciences, B. Tulskaya 52 Moscow 115191 Russia'
author:
- Alexander Burinskii
title: 'Gravity vs. Quantum theory: Is electron really pointlike?'
---
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Introduction
============
Modern physics is based on Quantum theory and Gravity. The both theories are confirmed experimentally with great precision. Nevertheless, they are conflicting and cannot be unified in a whole theory. General covariance is main merit of General Relativity and the main reason of misinterpretation. The freedom of coordinate transformations is one of the source of the conflict. One of the source of the problems is the absence of the usual plane waves in general relativity, which causes the conflict with the Fourier transform and prevents expansion of the quantum methods to the curved spacetimes.
The analogs of the plane waves in gravity are the pp-wave solutions which are singular, either at infinity or at some lightlike (twistor) line, forming a singular ray similar to the laser beam [@HorSt]. The pp-wave singular beams are modulated by the usual plane waves and form singular strings, which are in fact the fundamental strings of the low energy string theory. Moreover, it turns out that the pp-waves don’t admit $\alpha '$ stringy corrections [@HorSt; @Tseyt; @Coley], and therefore they are exact solutions to the full string theory.
The null Killing direction of the pp-waves, $k_\m , \quad (k_\m k^\m = 0 ) ,$ is adapted to the Kerr-Schild (KS) form of the metric $g^\mn
=\eta^\mn + 2H k^\m k^\n $ which is rigidly related with the auxiliary Minkowski background geometry $\eta^\mn$. The KS class of metrics is matched with the light-cone structure of the Minkowski background which softens conflict with quantum theory. In spice of the extreme rigidity, the Kerr-Schild coordinate system allows one to describe practically all the physically interesting solutions of General Relativity, for example:
- rotating black holes and the sources without horizon,
- de Sitter and Anti de Sitter spaces, and their rotating analogues,
- combinations of a black hole inside the de Sitter or AdS background spacetime,
- the opposite combinations: dS or AdS spaces as regulators of the black hole geometry
- charged black holes and rotating stars, and so on.
In particular, the Kerr-Schild metric describes the Kerr-Newman (KN) solution for a charged and rotating black hole, which for the case of very large angular momentum may be considered as a model of spinning particles.
In 1968 Carter obtained, [@Car; @DKS], that the KN solution has the gyromagnetic ratio $g = 2$ as that of the Dirac electron, which initiated a series of the works on the KN electron model [@Car; @DKS; @BurSol; @BurTwi; @BurKN; @Isr; @BurGeonIII; @Bur0; @IvBur; @Lop; @BurSen; @BurStr; @BurBag; @Arc; @Dym; @TN].
In this paper we discuss one of the principal contradictions between Quantum theory and Gravity, the question on the shape and size of the electron, believing that the nature of the electron is principal point for understanding of Quantum Theory. Quantum theory states that electron is pointlike and structureless. In particular, Frank Wilczek writes in [@FWil]: “...There’s no evidence that electrons have internal structure (and a lot of evidence against it)”, while the superstring theorist Leonard Susskind notes that electron radius is “...most probably not much bigger and not much smaller than the Planck length..”, [@LSuss]. This point of view is supported by the experiments with high energy scattering, which have not found the electron structure down to $10^{-16} cm .$
On the other hand, the experimentally observable parameters of the electron: angular momentum $J$, mass $m$, charge $e$ and the magnetic moment $ \mu $ indicate unambiguously that its background geometry should be very close to the corresponding Kerr-Newman (KN) solution of the Einstein-Maxwell field equations. The observed parameters of the electron $J,m,e, \mu $ determine also the corresponding parameters of the KN solution: the mass $m ,$ charge $e ,$ and the new dimensional parameter $a =L/m $ which fixes radius of the Kerr singular ring. The fourth observable parameter of the electron, magnetic momentum $\mu ,$ conform to the KN solution automatically as consequence of the above discussed specific gyromagnetic ratio of the Dirac electron coinciding with that of the KN solution. As a result, the KN solution indicates a characteristic radius of the electron as the Compton one $a=\hbar/(2m) ,$ corresponding to the radius of the Kerr singular ring. Therefore, contrary to Quantum theory, the KN gravity predicts the ring like structure of the electron and its Compton size.
The metric of the Kerr-Newman spacetime has the form g\_=\_+ 2 H k\_k\_, \[ksm\] where H= , \[H\] and the electromagnetic (EM) vector potential of the KN solution is \^\_[KN]{} = Re e [r+ia ]{} k\^\[ksGA\], where $r$ and $\theta$ are Kerr’s oblate spheroidal coordinates which are related to the Cartesian coordinates as follows
x+iy &=& (r + ia) e\^[i]{}\
z &=&r. \[oblate\]
The metric and EM field are aligned with the null vector field $k^\m$ forming a Principal Null Congruence (the ‘Kerr congruence’), see Figure.1. The Kerr congruence is determined by [*the Kerr Theorem*]{} in twistor terms (each line of the congruence is a twistor null line). Although the KN spacetime is curved, the Kerr congruence foliates it into a family of the flat complex twistor null planes, which allows one to use in the Kerr-Schild spaces a twistor version of the Fourier transform, which forms a holographic bridge between the classical Kerr-Schild gravity and Quantum theory, [@BurExa; @BurPreQ].
![\[label\]Oblate coordinate system $r, \ \theta $ with focal points at $r=\cos\theta = 0$ forms a twofold analytic covering: for $r>0$ and $r<0.$ ](Singcong.eps){width="17pc"}
![\[label\]Oblate coordinate system $r, \ \theta $ with focal points at $r=\cos\theta = 0$ forms a twofold analytic covering: for $r>0$ and $r<0.$ ](Oblate.eps){width="15pc"}
The KN gravitational and EM fields are concentrated near the Kerr singular ring, which appears in the rotating BH solutions instead of the pointlike Schwarzschild singularity. One sees that radius of the ring $a =J/m$ increases for the small masses and is proportional to the spin $J.$ Therefore, contrary to the characteristic radius of the Schwarzschild solution (related with position of the BH horizon, $ r_g=2m $), the characteristic extension of the KN gravitational field turns out to be much beyond the Planck length, and corresponds to the Compton length, $r_{compt}= a =\hbar/(2m),
$ or to the radius of a “dressed” electron. In the units $c=\hbar
=G=1 , $ mass of the electron is $m\approx 10^{-22},$ while $ \
a=J/m \approx 10^{22} .$ Therefore, $a>>m ,$ and the black hole horizons disappear, showing that the Kerr singular ring is naked. In this case the Kerr spacetime turns out to be twosheeted, since the Kerr ring forms its branch line creating a twosheeted topology. The relations (\[ksm\]) and (\[ksGA\]) show that the gravitational and electromagnetic fields of the KN solution are concentrated in a thin vicinity of the Kerr singular ring $r=\cos\theta=0,$ forming a type of “gravitational waveguide”, or a closed string, [@IvBur]. The Kerr string takes the Compton radius, corresponding to the size of a “dressed” electron in QED and to the limit of localization of the electron in the Dirac theory [@BjoDr].
There appear two questions:
\(A) How does the KN gravity know about one of the principal parameters of Quantum theory? and
\(B) Why does Quantum theory works successfully on the flat spacetime, ignoring the stringy defect of the background geometry?
A small and slowly varying gravitational field could be neglected, however the stringlike KN singularity forms a branch-line of the KS spacetime, and such a topological defect cannot be ignored. A natural resolution of this trouble could be the assumption that there is an underlying theory providing the consistency of quantum theory and gravity. In this paper we suggest a rather unexpected resolution of this puzzle, claiming that underlying theory is the low energy string theory, in which the closed string is created by the KN gravity related with twistorial structure of the Kerr-Schild pre-quantum geometry [@BurExa; @BurPreQ]. The Kerr singular ring is generated as a caustic of the Kerr twistor congruence and forms a closed string on the boundary of the Compton area of the electron. The KN gravity indicates that this string should represent a principal element of the extended electron structure.
If the closed Kerr string is really formed on the boundary of the Compton area, it should be experimentally observable. There appears the question while it was not obtained earlier in the high energy scattering experiments. We find some explanation to this fact and arrive at the conclusion that the KN string should apparently be detected by the novel experimental regime of the high energy scattering which is based on the theory of Generalized Parton Distributions (GPD), and corresponds to a “non-forward Compton scattering” [@Rad; @Ji], suggested recently for tomography of the particle images [@Hoyer].
Twosheetedness of the Kerr-Geometry
===================================
In the KS representation [@DKS], a few coordinate systems are used simultaneously. In particular, [*the null Cartesian coordinates*]{} $$\z = (x+iy)/\sqrt 2 , \
\Z = (x-iy)/\sqrt 2 , \ u = (z - t)/\sqrt 2 , \
v = (z + t)/\sqrt 2$$ are used for description of the Kerr congruence in the differential form k\_dx\^= P\^[-1]{}( du + |Y d + Y d |- Y |Y dv), \[kY\]via the complex function $Y(x)=e^{i\phi} \tan \frac \theta 2 ,$ which is a projective angular coordinate on the celestial sphere, Y(x)=e\^[i]{} 2 . \[Y\]
The Kerr Theorem
----------------
Kerr congruence (PNC) is controlled by [*THE KERR THEOREM:*]{}
The geodesic and shear-free Principal null congruences (type D metrics) are determined by holomorphic function $Y(x)$ which is analytic solution of the equation F (T\^a) = 0 , where $F$ is an arbitrary analytic function of the projective twistor coordinates T\^a ={ Y,- Y v, u + Y } .
The Kerr theorem is a practical tool for the obtaining the exact Kerr-Schild solutions. The following sequence of steps is assumed: $$F (T^a) =0 \Rightarrow F (Y, x^\m) = 0 \Rightarrow Y(x^\m)
\Rightarrow k^\m (x) \Rightarrow g^\mn$$ For the Kerr-Newman solution function $F$ is quadratic in $Y ,$ which yields TWO roots $Y^\pm(x)$ corresponding to two congruences!
As a result the obtained two congruences (IN and OUT) determine two sheets of the Kerr solution: the “negative (–)" and “positive (+)" sheet, where the fields change their directions. In particular, two different congruences $ k^{\m(+)} \ne
k^{\m(-)}$ determine two different KS metrics $ g_\mn^{(+)} \ne
g_\mn^{(-)} $ on the same Minkowski background. As it shows the Fig.1, the Kerr congruence propagates analytically from IN to OUT- sheet via the disk $r=0 ,$ and therefore, the two KS sheets are linked analytically. The twosheeted KN space is parametrized by the oblate spheroidal coordinate system $r, \ \theta, \phi ,$ which tends asymptotically, by $r\to \infty ,$ to the usual spherical coordinate system. Twosheetedness is the long-term mystery of the Kerr solution! For the multiparticle Kerr-Schild (KS) solutions, [@Multiks], the Kerr theorem yields many roots $Y^i, \ i=1,2,...$ of the Kerr equation $F(Y)=0 ,$ and the KS geometry turns out to be [*multivalued*]{} and [*multisheeted*]{}.
The extremely simple form of the Kerr-Schild metric (\[ksm\]) is related with complicate form of the Kerr congruence, which represents a type of deformed (twisted) hedgehog. In the rotating BH solutions the usual pointlike singularity inside the BH turns into a *a closed singular ring,* which is interpreted as a closed string in the corresponding models of the spinning elementary particles [@BurStr]. The KN twosheetedness was principal puzzle of the Kerr geometry for four decades and determined development of the KN electron models along two principal lines of investigation: I) the bubble models, and II) the stringlike models.
I. – In 1968 Israel suggested to truncate negative KN sheet, $r<0 ,$, and replace it by the [*rotating disklike source*]{} ( $r=0 $) spanned by the Kerr singular ring of the Compton radius $a=\hbar/2m ,$ [@Isr]. Then, Hamity obtained in [@Ham] that the disk has to be rigidly rotating, which led to a reasonable interpretation of the matter of the source as an exotic stuff with zero energy density and negative pressure. The matter distribution appeared singular at the disk boundary, forming an additional closed string source, and López suggested in [@Lop] to regularize this source, covering the Kerr singular ring by a disklike ellipsoidal surface. As a result, the KN source was turned into a rotating and charged oblate bubble with a flat interior, and further it was realized as a regular soliton-like bubble model [@BurSol], in which the boundary of the bubble is formed by a domain wall interpolating between the external KN solution and a flat pseudovacuum state inside the bubble.
II\. The stringlike models of the KN source retain the twosheeted topology of the KN solution, forming a closed ’Alice’ string of the Compton size [@IvBur; @BurKN; @BurAxi]. The Kerr singular ring is considered as a waveguide for electromagnetic traveling waves generating the spin and mass of the KN solution in accordance with the old Wheeler’s “geon” model of ‘mass without mass’ [@Wheel; @BurGeonIII; @Bur0; @BurGeon0]. In this paper we concentrate on the stringlike model of the electron, which displays close relations to the low energy string theory. The bubble model of regularization of the KN solution is discussed briefly in sec.5 along our previous papers.
The Kerr singular ring as a closed string
=========================================
Exact [*non-stationary*]{} solutions for electromagnetic excitations on the Kerr-Schild background, [@BurAxi; @BurA; @BurExa], showed that there are no smooth harmonic solutions. The typical exact electromagnetic solutions on the KN background take the form of singular beams propagating along the rays of PNC, contrary to smooth angular dependence of the wave solutions used in perturbative approach!
Position of the horizon for the excited KS black holes solutions is determined by function $H $ which has for the exact KS solutions the form, [@DKS], H = , \[Hpsi\] where $\psi(x)$ is related to the vector potential of the electromagnetic field =\_dx\^\
= -12 Re \[() e\^3 + d \], = 2(1+Y)\^[-2]{} dY , \[alpha\] which obeys the alignment condition \_k\^=0 . The equations (\[ksm\])and (\[Hpsi\]) display compliance and elasticity of the horizon with respect to the electromagnetic field.
The Kerr-Newman solution corresponds to $\psi=q=const.$. However, any nonconstant holomorphic function $\psi(Y) $ yields also an exact KS solution, [@DKS]. On the other hand, any nonconstant holomorphic functions on sphere acquire at least one pole. A single pole at $Y=Y_i$ \_i(Y) = q\_i/(Y-Y\_i) produces the beam in angular directions Y\_i=e\^[i\_i]{} \[Yi\] .
The function $\psi(Y)$ acts immediately on the function $H$ which determines the metric and the position of the horizon. The analysis showed, [@BurA], that electromagnetic beams have very strong back reaction to metric and deform topologically the horizon, forming the holes which allows matter to escape interior (see fig.3).
The exact KS solutions may have arbitrary number of beams in different angular directions $Y_i=e^{i\phi_i} \tan \frac
{\theta_i}{2}.$ The corresponding function (Y) = \_i , \[psiY\]leads to the horizon with many holes. In the far zone the beams tend to the known exact singular pp-wave solutions. The considered in [@Multiks] multi-center KS solutions showed that the beams are extended up to the other matter sources, which may also be assumed at infinity.
The stationary KS beamlike solutions may be generalized to the time-dependent wave pulses, [@BurAxi], which tend to exact solutions in the low-frequency limit.
Since the horizon is extra sensitive to electromagnetic excitations, it may also be sensitive to the vacuum electromagnetic field which is exhibited classically as a Casimir effect, and it was proposed in [@BurExa] that the vacuum beam pulses shall produce a fine-grained structure of fluctuating microholes in the horizon, allowing radiation to escape interior of black-hole, as it is depicted on Fig.3.
The function $\psi(Y,\t) ,$ corresponding to beam pulses, has to depend on retarded time $\t $ and satisfy to the obtained in [@DKS] nonstationary Debney-Kerr-Schild (DKS) equations leading to the extra long-range radiative term $\gamma(Y,\t)Z .$ The expression for the null electromagnetic radiation take the form, [@DKS], $F^\mn = Re \cF _{31} e^{3\m}\wedge e^{1\n},$ where \_[31]{}=Z - (AZ),\_1 , $Z=P/(r+ia\cos
\theta), \quad P=2^{-1/2}(1+\Y\bar \Y) ,$ and the null tetrad vectors have the form $e^{3\m}= Pk^\m, \ e^{1\m} = \d_\Y e^{3\m}.$
The long-term attack on the DKS equations has led to the obtained in [@BurExa] time-dependent solutions which revealed a holographic structure of the fluctuating Kerr-Schild spacetimes and showed explicitly that the electromagnetic radiation from a black-hole interacting with vacuum contains two components:
a\) a set of the singular beam pulses (determined by function $\psi(Y,\t) ,$) propagating along the Kerr PNC and breaking the topology and stability of the horizon;
b\) the regularized radiative component (determined by $\gamma_{reg}(Y,\t)$) which is smooth and, similar to that of the the Vaidya ‘shining star’ solution, determines evaporation of the black-hole, m = - 12 P\^2<\_[reg]{}|\_[reg]{}> .
The mysterious twosheetedness of the KS geometry plays principal role in the holographic black-hole spacetime [@BurExa], allowing one to consider action of the electromagnetic in-going vacuum as a time-dependent process of scattering. The obtained solutions describe excitations of electromagnetic beams on the KS background, the fine-grained fluctuations of the black-hole horizon, and the consistent back reaction of the beams to metric [@BurA; @BurExa]. The holographic space-time is twosheeted and forms a fluctuating pre-geometry which reflects the dynamics of the singular beam pulses. This pre-geometry is classical, but has to be still regularized to get the usual smooth classical space-time. In this sense, it takes an intermediate position between the classical and quantum gravity.
Meanwhile, the function $Z =P/(r+ia \cos\theta)$ tends to infinity near the Kerr ring, indicating that any electromagnetic excitation of the KN geometry should generates the related singular traveling waves along the Kerr singular ring, and therefore, the ‘axial’ singular beams turn out to be topologically coupled with the ‘circular’ traveling waves, see Fig.4. The both these excitations travel at the speed of light, and the ‘axial’ beams tend asymptotically (by $r\to \infty$) to the pp-wave (plane fronted wave) solutions, for which the vector $k_\m$ in (\[ksm\]) forms a covariantly constant Killing direction.
![Skeleton of the Kerr geometry [@BurAxi; @BurExa] formed by the topologically coupled ‘circular’ and ‘axial’ strings.](Vacbh.eps){width="15pc"}
![Skeleton of the Kerr geometry [@BurAxi; @BurExa] formed by the topologically coupled ‘circular’ and ‘axial’ strings.](Kout.eps){width="17pc"}
The pp-waves take very important role in superstring theory, forming the singular classical solutions to the low-energy string theory [@HorSt]. The string solutions are compactified to four dimensions and the singular pp-waves are regarded as the massless fields around a lightlike fundamental string. It is suspected that the singular source of the string will be smoothed out in the full string theory, taking into account all orders in $\alpha '.$ In the nonperturbative approach based on analogues between the strings and solitons, the pp-wave solutions are considered as fundamental strings [@DabhGibHarvRR]. The pp-waves may carry traveling electromagnetic and gravitational waves which represents propagating modes of the fundamental string [@Garf]. In particular, the generalized pp-waves represent the singular strings with traveling electromagnetic waves [@BurAxi; @Tseyt]. It has been noticed that the field structure of the Kerr singular ring is similar to a closed pp-wave string [@Bur0; @IvBur]. This similarity is not incidental, since many solutions to the Einstein-Maxwell theory turn out to be particular solutions to the low energy string theory with a zero (or constant) axion and dilaton fields. Indeed, the bosonic part of the action for the low-energy string theory takes after compactification to four dimensions the following form, [@ShapTriWilc], S = d\^4 x ( R -2 ()\^2 - e\^[-2]{} F\^2 - 12 e\^[4]{}(a)\^2 - a FF) \[Seff\], which contains the usual Einstein term $S_g = \int d^4 x \sqrt {-g} R $ completed by the kinetic term for dilaton field $-2 (\d \phi)^2$ and by the scaled by $\phi$ electromagnetic field. The last two terms are related with axion field $a$ and represent its nonlinear coupling with dilaton field $- \frac 12 e^{4\phi}(\d a)^2$ and interaction of the axion with the dual electromagnetic field $ \tilde F_\mn =
\epsilon_\mn ^{\lambda \rho} F_{\lambda \rho}.$
It follows immediately that [*any solution of the Einstein gravity, and in particular the Kerr solution, is to be exact solution of the effective low energy string theory*]{} with a zero (or constant) axion and dilaton fields. Situation turns out to be more intricate for the Einstein-Maxwell solutions since the electromagnetic invariant $F^2$ plays the role of the source of dilaton field. Similarly, the term $F\tilde F$ turns out to be the source of the axion field. The stringy analog to the Kerr-Newman solution with nontrivial axion and dilaton fields was obtained by Sen [@Sen], and it was shown in [@BurStr], that the field around the singular string in the ‘axidilatonic’ Kerr-Sen solution is very similar to the field around a heterotic string. This proximity of the pure gravitational strings [@IvBur] to the low-energy string theory allows us to consider the Kerr singular ring as a closed heterotic string and the corresponding traveling waves as its lightlike propagating modes. The axidilaton field is related with the string tension, and therefore, the nontrivial solutions to the low energy string theory should be very important, allowing one to estimate the mass-energy of the excited string states. The structure of the Lagrangian (\[Seff\]) shows that the axion field involves the dual magnetic field, and therefore, the complex axidilaton combination may generate the duality rotation and create an additional twist of the electromagnetic traveling waves. However, the exact solutions of this type are so far unknown. Note also that the axidilaton field appears naturally in the based on the special-Kähler geometry 4D models of black holes in supergravity, which may have important consequences for the models of regularized KN solution [@BurSol].
Assuming that the lightlike string forms a core of the electron structure, we have to obtain a bridge to the one-particle quantum theory. Traveling waves along the KN closed string generate the spin and mass of the stringlike particle. Physically, it is equivalent to the original Wheeler’s model of ‘mass without mass’ [@Wheel]. In the next section we show emergence of the Dirac equation from this physical picture.
Mass without mass
=================
The puzzle of “zitterbewegung” and the known processes of annihilation of the electron-positron pairs brought author in 1971 to the Wheeler “geon” model of the “mass without mass” [@Wheel]. In [@BurGeon0] we considered a massless particle circulating around z-axis. Its local 4-momentum is lightlike, p\_x\^2 + p\_y\^2 + p\_z\^2 = E\^2 \[Ephot\] ,while the effective mass-energy was created by an averaged orbital motion, +<p\_y\^2> = m\^2 \[mPxy\] .Averaging (\[Ephot\]) under the condition (\[mPxy\]) yields = m\^2 +p\_z\^2 = E\^2 \[mPzE\] .Quantum analog of this model corresponds to a wave function $\psi(\vec x,t) $ and operators, $
\vec p \to \hat {\vec p} = -i\hbar \nabla , \quad \hat E= i \hbar
\d_t .$ From (\[Ephot\]) and (\[mPxy\]) we obtain the D’Alembert equation $\d^\m \d_\m \psi =0$ and the constraint $(\d_x^2 + \d_y^2)\psi=0 ,$ which for the chosen coordinate system are reduced to the equations (\_x\^2 + \_y\^2)= m\^2 = (\_t \^2 - \_z\^2)\[msep\] ,and may be separated by the ansatz =(x,y)\_0 (z,t) \[ans\]. The RHS of (\[msep\]) yields the usual equation for a massive particle, $ (\d_t ^2 - \d_z^2)\Psi_0
=\tilde m^2 \Psi_0 ,$ and the corresponding (de Broglie) plane wave solution \_0 (z,t) = , \[deBr\] while the l.h.s. determines the “internal” structure factor \_=\_( ) {i} \[MHan\], in polar coordinates $\rho, \phi ,$ where ${\cal{H}}_\n ( \frac {\tilde m } \hbar \rho) $ are the Hankel functions of index $\n.$ ${\cal{M}}_\n$ are eigenfunctions of operator $\hat J_z = \frac \hbar i \d_\phi $ with eigenvalues $J_z=
\n\hbar .$ For electron we have $J_z= \pm \hbar/2, \quad \n=\pm
1/2 ,$ and the factor \_[1/2]{}= \^[-1/2]{}{ i ( 12 )} \[M12\] creates a singular ray along $z$-axis, which forms a branch line, and the wave function is twovalued.
There are diverse generalizations of this solution. First of all, there may be obtained the corresponding wave functions based on the eigenfunctions of the operator of the total angular momentum and simultaneously of the spin projection operator. Next, the treatment may be considered in a Lorentz covariant form for arbitrarily positioned and oriented wave functions. And finally, the corresponding spinor models, together with all the corresponding spinor solutions may also be obtained (see [@Beyond]).
Principal peculiarity of the obtained massless model is that the usual plane wave functions are replaced by the vortex waves generating the spin and mass of the particle-like solutions of the [*massless*]{} equations, and therefore, the contradiction between the massive wave equation and the lightlike zitterbewegung (determined by the Dirac operators $\alpha$) disappears. On the other hand, the wave functions (\[ans\]) are factorized into the usual plane waves $\Psi_0 (z,t)$ and the string-like singular factors ${\cal{M}}(x,y)$ playing the role of singular carriers of de Broglie waves, which reproduces de Broglie’s wave-pilot conception, which is however principally different from the corresponding Bohm model.
It should also be mentioned that the characteristic spinor twovaluedness appears also for scalar waves, as a consequence of the topological twosheetedness generated by the singular branch line. In the Kerr geometry this ‘axial’ branch line is linked with the Kerr ‘circular’ branch line (Figure 1.), forming a topologically nontrivial spacetime structure of the KN geometry, (Figure 2.).
Regularization: Electron as a gravitating soliton
=================================================
The experimentally indicated KN background of the electron exhibits the closed singular string which contradicts to the Quantum assumptions that the gravity is negligible and the background is flat. As a result, the justification of the Dirac electron theory and QED requires [*regularization of the KN metric,*]{} which should be performed with invariability of its asymptotic form. Similar regularization of the singular strings of the low-energy string theory is assumed in the full string theory [@BBS]. For the KN solution this problem is close related with general problem of the regularization of the black hole singularity [@Dym0] and with the old problem of the regular source of the KN solution [@BurBag; @BEHM; @GG].
Gravitational aspect
--------------------
The used by Israel truncation of the negative sheet of the KN solution [@Isr] led to the disklike model of the KN source which retained the Kerr singular ring. It was replaced by López by the regular model of a rigidly rotating charged bubble with a flat interior [@Lop], which was a prototype of the gravitating soliton model [@BurSol]. The singular region of the KN solution is rejected in the López model and replaced by the flat space-time, forming a bubble with flat interior. One should retain the asymptotic form of the external KN solution and provide a smooth matching of the external metric with the flat bubble interior. It is achieved by the special chose of the bubble boundary $r_{b}$ which is determined by the condition $H(r_{b})=0
.$ From (\[H\]) one obtains r\_b =r\_e = e\^2/(2m) ,\[rb\] where $r_b$ is the Kerr ellipsoidal radial coordinate, (\[oblate\]). As a result, the regular KN source takes the form of an oblate disk of the Compton radius $r_c \approx a
=\hbar/(2m)$ with the thickness $r_e = e^2/(2m) $ corresponding to the known ’classical size’ of the electron. One sees that the consistent regularization needs an extension up to the Compton distance. The electromagnetic field is also regularized, and the López bubble model represents a charged and rotating singular shell. The corresponding smooth and regular rotating sources of the Kerr-Schild class were considered in [@BurBag; @BEHM] on the base of the generalized KS class of metrics suggested by Grses and Gürsey in [@GG]. The function $H $ in the generalized KS form of metric $$g_\mn = \eta_\mn + 2 H
k_\m k_\n ,$$ is taken to be H=f(r)/(r\^2 + a\^2 \^2) , \[HGG\]where the function $f(r)$ interpolates between the inner regular metric and the external KN solution. By such a deformation, the Kerr congruence, determined by the vector field $k^\m( m) \in M^4 ,$ should retain the usual KS form (\[kY\]).
It allows one to suppress the Kerr singular ring ($r=\cos \theta =0$) by a special choice of the function $f(r).$
The regularized solutions have tree regions:
i\) the Kerr-Newman exterior, $r>r_0 $, where $f(r)=mr -e^2/2,$
ii\) interior $r<r_0-\delta $, where $f(r) =f_{int}$ and function $f_{int}=\alpha r^n ,$ and $n\ge 4$ to suppress the singularity at $r=0,$ and provide the smoothness of the metric up to the second derivatives.
iii\) intermediate region providing a smooth interpolation between i) and ii).
Material aspect
---------------
To remove the Kerr-Newman singularity, one has to set for the internal region $$f_{int}=\alpha r^4 .$$ In this case, the Kerr singularity is replaced by a regular rotating internal space-time with a constant curvature, $ R=-24
\alpha $ [@BurBag; @BEHM].
The functions $$D= - \frac{f^{\prime\prime}} {\Sigma}, \quad G=
\frac{f'r-f}{\Sigma^2} \label{Gt}.$$ determine stress-energy tensor in the orthonormal tetrad $\{u,l,m,n\}$ connected with the Boyer-Lindquist coordinates, $$T_{ik} = (8\pi)^{-1} [(D+2G) g_{ik} - (D+4G) (l_i l_k - u_i
u_k)]. \label{Tt}$$ In the above formula, $u^i$ is a timelike vector field given by $$u^i=\frac 1{\sqrt{\Delta\Sigma}}(r^2+a^2,0,0,a) .$$ This expression shows that the matter of the source is separated into ellipsoidal layers corresponding to constant values of the coordinate $r$, each layer rotates with angular velocity $\omega(r)= \frac {u^{\phi}}{u^0}=a/(a^2+r^2)$. This rotation becomes rigid only in the thin shell approximation $r=r_0$. The linear velocity of the matter w.r.t. the auxiliary Minkowski space is $v=\frac {a \sin \theta}{\sqrt {a^2 + r^2}}$, so that on the equatorial plane $\theta =\pi /2$, for small values of $r$ ($r\ll
a $), one has $v \approx c=1$, that corresponds to an oblate, relativistically rotating disk.
The energy density $\rho$ of the material satisfies to $T^i_ku^k=-\rho u^i$ and is, therefore, given by $$\rho = \frac{1}{8\pi} 2G. \label{rhot}$$ Two distinct spacelike eigenvalues, corresponding to the radial and tangential pressures of the non rotating case are $$p_{rad} = -\frac{1}{8\pi} 2G=-\rho, \label{pradt}$$ $$p_{tan} = \frac{1}{8\pi}(D+ 2G)=\rho +\frac{D}{8\pi}. \label{prtt}$$ In the exterior region function $f$ must coincide with Kerr-Newman solution, $f_{KN} = mr -e^2/2$.
There appears a transition region placed in between the boundary of the matter object and the de Sitter core. This transition region has to be described by a smooth function $f(r)$ which interpolates between the functions $f_{int}(r)$ and $f_{KN}(r)$. Graphical analysis allows one to determine position of the bubble boundary $r_b ,$ [@BEHM; @RenGra]. For the López model of the flat interior $f_{int}=0$ and $r_b =r_e =e^2/(2m).$ The case $\alpha
>0$ corresponds to de Sitter interior and uncharged source. There is only one intersection between $f_{int}(r)=\alpha r^4$ and $f_{KN}(r)=mr$. The position of the transition layer will be $r_b
=(m/\alpha)^{-1/3}$. The second derivative of the corresponding interpolating function will be negative at this point, yielding an extra contribution to the positive tangential pressure in the transition region.
Chiral field model and the Higgs field
--------------------------------------
In accordance with the Einstein equations, the considered smooth and regular metric should be generated by a system of the matter fields forming a classical source of the vacuum bubble. In the suggested in [@BurSol] soliton model, the smooth phase transition from the external KN solution to the internal ‘pseudovacuum state’ is generated by a supersymmetric set of chiral fields $\Phi^i, \quad i=1,2,3 \ ,$ [@BurSol; @BurBag; @BurCas] controlled by the suggested by Morris [@Mor] super-potential W= Z(|-\^2) + (cZ+ ) |,where $c, \ \m, \ \eta, \
\lambda$ are the real constants, and we have set $\Phi^1 =\Phi, \
\Phi^2=Z $ and $\Phi^3 =\Sigma .$ The potential is determined by the usual relations of the supersymmetric field theory [@WesBag] V(r)=\_i |\_i W|\^2 ,where $ \d_1 =
\d_\Phi , \ \d_2 = \d_Z , \ \d_3 = \d_\Sigma .$ The vacuum states are determined by the conditions $\d_i W =0 $ which yield $V=0$
i\) for ‘false’ vacuum ($r<r_0$): $Z=- \m/c; \Sigma=0; |\Phi|=
\eta\sqrt{\lambda/c},$ and also
ii\) for ‘true’ vacuum ($r>r_0$) : $ Z=0; \Phi=0; \Sigma=\eta .$
which provides a phase transition from the external KN ‘vacuum state’, $ V_{ext}=0 ,$ to a flat internal ‘pseudovacuum’ state, $ V_{int}=0 ,$ providing regularization of the Kerr singular ring, [@BurBag; @BurCas; @BurSol]. One of the chiral fields, $\Phi^1 ,$ is set as the Higgs field $\Phi^1 \equiv \Phi =
\Phi_0 \exp(i\chi ).$ As a result of the phase transition, the Higgs field $\Phi_0 \exp(i\chi )$ with a nonzero vev $\Phi_0$ and the phase $\chi$ fills interior of the bubble and regularizes the electromagnetic Kerr-Newman field by the Higgs mechanism of broken symmetry.
Regularization of the electromagnetic KN field
----------------------------------------------
The electromagnetic KN field inside the bubble interacts with the Higgs field in agreement with Landau-Ginzburg type field model with the Lagragian \_[NO]{}= -14 F\_F\^+ 12 (\_)(\^)\^\* + V(r), \[LNO\]where $ \cD_\m
= \nabla_\m +ie \alpha_\m $ are to be covariant derivatives. This model was used by Nielsen and Olesen [@NO] for the obtaining the string-like solutions in superconductivity. The model determines the current I\_= 12 e ||\^2 (,\_+ e \_) as a source of the Maxwell equations $F^{\n\m}_{
\ \ ;\n}=I^\m .$ This current should vanish inside the bubble, which sets a relation between incursion of the phase of the Higgs field and the value of the vector potential of the KN solution on the boundary of the bubble ,\_=- e \_\^[(str)]{} . \[Iinside\]The maximal value of the regularized vector potential $\alpha_\m^{(str)}$ is reached on the boundary of the bubble. In the agreement with (\[rb\]) and (\[ksGA\]) we have the vector relation \_\^[(str)]{} = \_(r\_[b]{}) = 2m/e , \[Amax\]which results in two very essential consequences [@BurSol]:
- the Higgs field (matched with the regularized KN electromagnetic field) forms a coherent vacuum state oscillating with the frequency $\omega=2m ,$ which is a typical feature the “oscillon” soliton models.
- the regularized KN electromagnetic potential $\alpha^{(str)}_\m$ forms on the boundary of the bubble a closed Aharonov-Bohm-Wilson quantum loop $ \oint e\alpha^{(str)}_\phi
d\phi=-4\pi ma \label{WL} ,$ which determines quantized spin of the soliton, $J=ma=n/2, \ n=1,2,3,...$
Does the KN model of electron contradict to Quantum Theory? It seems “yes”, if one speaks on the “bare” electron. However, in accordance with QED, vacuum polarization creates in the Compton region a cloud of virtual particles forming a “dressed” electron. This region gives contribution to electron spin, and performs a procedure of renormalization, which determines physical values of the electron charge and mass. Therefore, speaking on the “dressed” electron, one can say that the real contradiction between the KN model and the Quantum electron is absent.
Note that dynamics of the virtual particles in QED is chaotic and can be conventionally separated from the “bare”electron. In the same time, the vacuum state inside the Kerr-Newman soliton forms a [*coherent oscillating state*]{} joined with a closed Kerr string. It represents an [*integral whole of the extended electron,*]{} its ‘internal’ structure which cannot be separated from a “bare” particle. In any case, the Kerr string appears as an analogue of the pointlike bare electron.
Conclusion
==========
We have showed that gravity definitely indicates presence of a closed string of the Compton radius $a=\hbar/(2m)$ in the electron background geometry. This string has gravitational origin and is close related with the fundamental closed strings of the low energy string theory. Corpuscular aspect of the traveling waves along the Kerr string allows us to ‘derive’ the Dirac equation. The original Dirac theory is modified in this case: the wave functions are factorized and acquire the singular stringlike carriers. As a result, the new wave functions turn out to be propagating along the ‘axial’ singular strings, which is reminiscent of the de Broglie wave-pilot conjecture. Therefore, the gravitational KN closed string represents a bridge between gravity, superstring theory and the Dirac quantum theory towards the consistency of these theories. We arrive at the extremely unexpected conclusion that Gravity, as a basic part of the superstring theory, may lie beyond Quantum theory and play a fundamental role in its ‘emergence’.
The observable parameters of the electron determine unambiguously the Compton size of the Kerr string. This size is very big with respect to the modern scale of the experimental resolutions, and it seems, that this string should be experimentally detected. However, the high-energy scattering detects the pointlike electron structure down to $10^{-16} cm $. One of the explanations of this fact, given in [@BurTwi], is related with the assumption that interaction of the KN particles occurs via the lightlike KN ‘axial’ strings. Just such a type of the ‘direct’ lightlike interaction follows from the analysis of the Kerr theorem for multiparticle Kerr-Schild solutions, [@Multiks]. So far as the KN circular string is also lightlike, the lightlike photon can contact it only at one point. The resulting scattering of the Kerr string by the *real* photons of high energy can exhibit only the pointlike interaction, and neither form of the string, nor its extension cannot be recognized. To recognize the shape of the string as a whole, it is necessary two extra conditions:
a\) a [*relative low-energy*]{} resonance scattering with the wavelengths comparable with extension of the string. It means that there must be a scattering with a low-energy momentum transfer, i.e. with a small Bjorken parameter $x= q_t /P .$
b\) simultaneously, to avoid the pointlike contact interaction, the scattering should be deeply virtual, i.e. the square of the transverse four-momentum transfer should satisfy $Q^2 >> m^2.$
Both these conditions correspond to the novel tool – the Deeply Virtual Compton Scattering (DVCS) described by the theory of Generalized Parton Distribution (GPD) [@Rad; @Ji], which allows one to probe the shape of the elementary particles by the “non-forward Compton scattering” [@Hoyer]. If the predicted Kerr-Newman string will be experimentally recognized in the core of electron structure, it could be great step in understanding Quantum theory towards to Quantum Gravity.
Acknowledgments
---------------
Author thanks M. Bordag, A. Efremov, D. Stevens, O. Teryaev and F. Winterberg for conversations and useful references, and also Yu. Danoyan, A. Efremov, D. Gal’tsov, A. Khrennikov, T. Nieuwenhuizen, Yu. Obukhov, O.Selyugin, K. Stepaniants and G. ‘t Hooft for useful discussions, and also especially thankful to A. Radyushkin for discussions concerning the GPD application.
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|
---
abstract: 'In the present paper we discuss how trans-Planckian physics affects the inflationary vacuum fluctuation or the primordial density perturbation and leaves a specific imprint on the cosmic microwave background (CMB). The inflationary trans-Planckian problem has been under debate and these is a variety of conclusions in literature. Here we consider this problem by using the two-point correlation function $\left< { \delta \phi }^{ 2 } \right>$ of the non-minimally coupled scalar fields and constructing the effective potential in curved spacetime. We point out that in the UV regime the quantum fluctuation does not drastically change even in de Sitter spacetime and the trans-Planckian corrections can be embedded in effective potential. Thus, the UV sensitivity on the primordial density perturbation is sufficiently sequestered and we only receive a tiny trans-Planckian imprint on the CMB in contrast with previous suggestions. However, the trans-Planckian corrections give a strong impact on the fine-tuning problems of the inflaton potential and also inflationary vacuum fluctuation. We simply obtain a inflationary conjecture $\Lambda_{\rm UV} \ll H/g^{1/2}$ where $g$ is the interaction coupling at the UV scale $\Lambda_{\rm UV}$.'
author:
- Hiroki Matsui
bibliography:
- 'trans-Planckian.bib'
nocite: '\nocite{}'
title: 'Trans-Planckian quantum corrections and inflationary vacuum fluctuations of non-minimally coupled scalar fields'
---
Preprint numbers: KEK-TH-1966
Introduction
============
Recently an intriguing possibility has been discussed in the literature [@Niemeyer:2000eh; @Brandenberger:2000wr; @Martin:2000xs; @Tanaka:2000jw; @Hui:2001ce; @Niemeyer:2001qe; @Kaloper:2002uj; @Kaloper:2002cs; @Burgess:2002ub; @Brandenberger:2002hs; @Elgaroy:2003gq; @Greene:2004np; @Greene:2005wk; @Danielsson:2002kx; @Danielsson:2002mb; @Danielsson:2002qh; @Danielsson:2005cc; @Danielsson:2006gg; @Goldstein:2002fc; @Easther:2002xe; @Chung:2003wn; @Kaloper:2003nv; @Burgess:2003hw; @Alberghi:2003am; @Martin:2003kp; @Meerburg:2010rp; @Kundu:2011sg; @Groeneboom:2007rf; @Ashoorioon:2013eia; @Ashoorioon:2014nta; @Ashoorioon:2017toq; @Broy:2016zik] where inflation can provide important clues about ultraviolet (UV) physics or trans-Planckian physics. In the standard paradigm of the inflation, quantum fluctuations are assumed not to be modified up to the infinitely short length. However, it is not realistic because new physics is to be expected below the Planck scale and this simple assumption could not be correct. There are so many discussions about how the UV physics or trans-Planckian physics modify the inflationary fluctuation and leave a specific imprint on the cosmic microwave background (CMB). The inflationary trans-Planckian problem has been under debate and a variety of the literature have reached a range of conclusions.
In the literature [@Danielsson:2002kx; @Danielsson:2002mb; @Danielsson:2002qh; @Danielsson:2005cc], the trans-Planckian problem has been discussed from the viewpoint of the choice of the initial vacuum. Taking the initial vacuum as $\alpha$-vacua defined at the finite time [@Allen:1985ux; @Mottola:1984ar; @Danielsson:2002mb], the Bogoliubov coefficients ${ \alpha }_{ k }$ and ${ \beta }_{ k }$ are constrained by the following relation $$\begin{aligned}
{ \beta }_{ k }=\frac { i{ e }^{ -2ik{ \eta }_{ 0 } } }{ 2k{ \eta }_{ 0 }+i } { \alpha }_{ k },\end{aligned}$$ where $\eta_{0}$ is the conformal initial time and $k$ is the wave mode. Note that the Bunch-Davies vacuum is restored in the infinite past (${ \eta }_{ 0 }\rightarrow -\infty $, ${ \alpha }_{ k }=1$ and ${ \beta }_{ k }=0$). Initial condition should be imposed when the wavelength crosses to some fundamental length scale. Therefore, the initial condition could be imposed at the $k$-dependent initial time $\eta_{0}
= -\Lambda_{\rm UV}/Hk$ [@Danielsson:2002kx] where $\Lambda_{\rm UV}$ is the UV cut-off scale and $H$ is the Hubble constant during inflation. In this condition, the quantum fluctuation $\left< { \delta \phi^{2} } \right>$ can be written as follows: $$\begin{aligned}
\left< { \delta \phi^{2} } \right>=
\int { { d }^{ 3 }k{ \left| \delta { \phi }_{ k }\left( \eta ,x \right) \right| }^{ 2 } }
=\int{ \frac { dk }{ k } }{ P }_{\delta { \phi } }\left( \eta, k \right)\label{eq:hfhfhedg},\end{aligned}$$ where ${ P }_{\delta { \phi } }\left( \eta, k \right)$ is the power spectrum of the quantum fluctuation given by $$\begin{aligned}
{ P }_{\delta { \phi } }\left( \eta, k \right)={ \left( \frac { H }{ 2\pi } \right) }^{ 2 }
\left( 1-\frac { H }{ \Lambda_{\rm UV} } \sin { \left( \frac { 2\Lambda_{\rm UV} }{ H }
\right) } \right) ,\end{aligned}$$ which shows a optimistic size of the UV corrections as $\mathcal{O}\left( H/\Lambda_{\rm UV} \right)$ and provides a window towards physics beyond the UV cut-off scale, e.g. the Planck scale or the string scale. On the other hand, in the literature [@Kaloper:2002uj; @Kaloper:2002cs] based on the effective field theory, the inflationary quantum fluctuation can be given as follows: $$\begin{aligned}
\left< { \delta \phi }^{ 2 } \right> \Bigr|_{k \approx H} = { H }^{ 2 }+{ c }_{ 1 }{ H }^{ 2 }\left( { H }^{ 2 }/ \Lambda_{\rm UV}^{ 2 } \right)
+{ c }_{ 2 }{ H }^{ 2 }{ \left( { H }^{ 2 }/\Lambda_{\rm UV}^{ 2 } \right) }^{ 2 }
+{ c }_{ 3 }{ H }^{ 2 }{ \left( { H }^{ 2 }/ \Lambda_{\rm UV}^{2} \right) }^{ 3 }+\cdots,\end{aligned}$$ where the coefficients ${ c }_{ i }$ are determined by the cut-off scale $ \Lambda_{\rm UV}$. The above estimates suggest that the contributions from UV physics can not be larger than $\mathcal{O}\left( H^{2}/\Lambda_{\rm UV}^{2}\right)$ and have been criticized as being too pessimistic. At least there are two competing estimates of the UV corrections to the inflationary quantum fluctuation or the CMB power spectrum in the literature and no consensus has been reached.
In this paper we discuss how the UV or trans-Planckian corrections modify the inflationary vacuum fluctuation based on the standard analysis of the quantum field theory (QFT) in curved spacetime. Here we deal with this problem by using the two-point correlation function $\left< { \delta \phi }^{ 2 } \right>$ of the non-minimally coupled scalar fields which can be strictly calculated under the QFT in curved spacetime [^1]. As a consequence, we clearly show that the correction of the UV or trans-Planckian physics can be embedded in effective potential, and the UV sensitivity on the primordial density perturbation is sufficiently sequestered and we only receive a tiny imprint on the CMB at the short distance rather than $\mathcal{O}\left( H/\Lambda_{\rm UV} \right)$ or $\mathcal{O}\left( H^{2}/\Lambda_{\rm UV}^{2}\right)$. However, the trans-Planckian corrections give non-negligible impact on the fine-tuning problems of the inflation itself and we obtain an upper-bound of the UV or trans-Planckian physics which can be a window towards the high-scale physics.
The renormalization of quantum fluctuation {#sec:renormalization}
===========================================
The quantum fluctuation necessarily causes a problem about renormalization. As well-known facts in QFT, the two-point correlation function $\left<{ \delta \phi }^{ 2 } \right>$ which express the quantum fluctuation have the UV (quadratic and logarithmic) divergences and therefore some regularization or renormalization methods are required. In flat spacetime the quantum fluctuation with the divergences can be eliminated by the bare parameters of the Lagrangian thorough standard renormalization technique. But in curved spacetime [@birrell1984quantum] due to the quantum particle creations the representation of the quantum fluctuation and the renormalization has some ambiguity which complicates the trans-Planckian problem [@Niemeyer:2000eh; @Brandenberger:2000wr; @Martin:2000xs; @Tanaka:2000jw; @Hui:2001ce; @Niemeyer:2001qe; @Kaloper:2002uj; @Kaloper:2002cs; @Burgess:2002ub; @Brandenberger:2002hs; @Elgaroy:2003gq; @Greene:2004np; @Greene:2005wk; @Danielsson:2002kx; @Danielsson:2002mb; @Danielsson:2002qh; @Danielsson:2005cc; @Danielsson:2006gg; @Goldstein:2002fc; @Easther:2002xe; @Chung:2003wn; @Kaloper:2003nv; @Burgess:2003hw; @Alberghi:2003am; @Martin:2003kp; @Meerburg:2010rp; @Kundu:2011sg; @Groeneboom:2007rf; @Ashoorioon:2013eia; @Ashoorioon:2014nta; @Ashoorioon:2017toq; @Broy:2016zik]. In this section, let us consider carefully the renormalization of the quantum fluctuation in de-Sitter spacetime using the adiabatic regularization [@bunch1980adiabatic; @Parker:1974qw; @Fulling:1974pu; @Fulling:1974zr; @birrell1978application; @Anderson:1987yt; @Haro:2010zz; @Haro:2010mx; @Kohri:2017iyl] which is a powerful method to remove the divergences. We clearly show that the inflationary fluctuation or quantum particle creation are sequestered from the cut-off sensitivity of the UV or trans-Planckian physics under reasonable assumptions.
Through this paper, we consider a spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime, $$\begin{aligned}
ds^{2}:=-dt^{2}+a^{2}\left(t\right)\delta_{ij}dx^{i}dx^{j},\end{aligned}$$ where $a\left(t\right)$ is the scale factor and the scalar curvature is give as $R=6{ \left( { \dot { a } }/{ a } \right) }^{ 2 }+6\left( { \ddot { a } }/{ a } \right)
=6\left({a''}/{a^{3}}\right)$ where $\eta$ is the conformal time and defined by $d\eta=dt/a$. The scale factor becomes $a\left(t\right) =e^{Ht}$ and the scalar curvature is expressed as $R=12H^{2}$ in de Sitter spacetime. For simplicity, we consider two nonminimal coupled scalar fields with the interaction coupling $g$. The bare (unrenormalized) Lagrangian is defined by, $$\begin{aligned}
\begin{split}
\mathcal{L}\left[ \phi, S \right] &:=
\frac { 1 }{ 2 } { g }^{ \mu \nu }
{ \partial }_{ \mu }\phi {\partial }_{ \nu }\phi+\frac{1}{2}\left(m^{2}_{\phi}+\xi_{\phi} R\right)\phi^{2}+\frac{\lambda}{4}\phi^{4} \\
&+\frac { 1 }{ 2 } { g }^{ \mu \nu }{ \partial }_{ \mu }S{\partial }_{ \nu }S
+\frac{1}{2}\left(m^{2}_{S}+\xi_{S} R\right)S^{2}
+\frac{g}{2}\phi^{2}S^{2} \label{eq:dddgedg},
\end{split}\end{aligned}$$ where the inflaton field is $\phi$, the massive scalar field is $S$ with $m_{S} \simeq \Lambda_{\rm UV } \gg m_{\phi}$ and $\xi_{\phi},\xi_{S}$ are the nonminimal curvature couplings. The Klein-Gordon equations for these two scalar field are given as follows: $$\begin{aligned}
&\Box \phi+m^{2}_{\phi}\phi+\xi_{\phi} R\phi
+\lambda\phi^{3}+{g}S^{2}\phi =0 \label{eq:dsssssdg}, \\
&\Box S+m^{2}_{S}S+\xi_{S} R S+{g}\phi^{2}S =0 \label{eq:dssdssdg},\end{aligned}$$ where $\Box =g^{\mu\nu}{ \nabla }_{ \mu }{ \nabla }_{ \nu }=1/\sqrt { -g }
{ \partial }_{ \mu }\left( \sqrt { -g } { \partial }^{ \mu } \right) $ express generally covariant d’Alembertian operator.
Next we treat the scalar fields $\phi $, $S$ as the field operators acting on the ground states and these scalar fields $\phi $, $S$ are decomposed into the classic parts and the quantum parts: $\phi =\phi \left(\eta,x \right)+\delta \phi \left(\eta ,x \right)$, $S =S \left(\eta,x \right)+\delta S\left(\eta ,x \right) $ and we assume $\left< 0 \right| { \delta\phi \left(\eta ,x\right) }
\left| 0 \right>=\left< 0 \right| { \delta S \left(\eta ,x\right) }\left| 0 \right>=0$. By introducing the renormalized parameters and the counterterms to be $m_{\phi}^{2}=m^{2}_{\phi}\left(\mu\right)+\delta m^{2}_{\phi}$, $\xi_{\phi}=\xi_{\phi}\left(\mu\right)+\delta \xi_{\phi}$, $\lambda=\lambda\left(\mu\right)+\delta \lambda$ and $g=g\left(\mu\right)+\delta g$, the one-loop Klein-Gordon equations of the inflaton field are given by $$\begin{aligned}
&\Box \phi +\left(m^{2}_{\phi}\left(\mu\right)+\delta m^{2}_{\phi}\right)\phi +\left(\xi_{\phi}\left(\mu\right)
+\delta \xi_{\phi}\right)R\phi \nonumber \\ &+3\left(\lambda\left(\mu\right)+\delta \lambda \right)
\left< \delta { \phi }^{ 2 } \right>\phi +\left(\lambda\left(\mu\right)+\delta \lambda \right)\phi^{3}
+\left(g\left(\mu\right)+\delta g \right)\left< \delta { S }^{ 2 } \right>\phi=0, \label{eq:lklkedg} \\
&\left(\Box +m^{2}_{\phi}\left(\mu\right)+\xi_{\phi}\left(\mu\right) R
+3\lambda\left(\mu\right)\phi^{2}+g\left(\mu\right)S^{2} \right)\delta \phi =0, \nonumber\end{aligned}$$ where the first equation shows the dynamics of the average inflaton field whereas the second equation shows the quantum fluctuation of the inflaton field. The quantum field $\delta \phi$ can be decomposed into each $k$ modes by $$\begin{aligned}
\delta \phi \left( \eta ,x \right) =
\int { { d }^{ 3 }k\left( { a }_{ k }\delta { \phi }_{ k }\left( \eta ,x \right)
+{ a }_{ k }^{ \dagger }\delta { \phi }_{ k }^{ * }\left( \eta ,x \right) \right) }
\label{eq:ddfkkfledg},\end{aligned}$$ where the creation and annihilation operators of $\delta { \phi }_{ k }$ are required to satisfy the standard commutation relations $\bigl[ { a }_{ k },{ a }_{ k' }\bigr] =\bigl[ { a }_{ k }^{ \dagger },
{ a }_{ k' }^{ \dagger } \bigr] =0$ and $\bigl[ { a }_{ k },{ a }_{ k' }^{ \dagger } \bigr] ={ \delta }\left( k-k' \right)$. The in-vacuum state $\left| 0 \right>$ is defined by $a_{k}\left| 0 \right>=0 $ and corresponds to the initial conditions of the mode functions of $\delta { \phi }_{ k }$. The quantum fluctuation $\left<{ \delta \phi }^{ 2 } \right>$ of the inflaton field can be written by $$\begin{aligned}
\left< 0 \right| { \delta \phi^{2} }\left| 0 \right>
=\int { { d }^{ 3 }k{ \left| \delta { \phi }_{ k }\left( \eta ,x \right) \right| }^{ 2 } }
=\frac { 1 }{ 2{ \pi }^{ 2 }a^{2}\left( \eta \right) } \int _{ 0 }^{ \infty } { dk { k }^{ 2 }{ \left| \delta { \chi }_{ k }\left( \eta \right) \right| }^{ 2 } } \label{eq:xkkgdddgedg},\end{aligned}$$ where we introduce the rescaled mode functions $\delta { \chi }_{ k }\left( \eta \right)$ as ${ \delta \phi }_{ k }\left( \eta ,x \right)=
e^{ik\cdot x}\delta { \chi }_{ k }\left( \eta \right)/{ \left( 2\pi \right) }^{ 3/2 }
a\left( \eta \right)$. From Eq. (\[eq:lklkedg\]), the Klein-Gordon equation for the quantum rescaled field $\delta \chi$ is given by $$\begin{aligned}
{ \delta \chi}''_{ k }\left( \eta \right)+{ \Omega }_{ k }^{ 2 }
\left( \eta \right) { \delta \chi }_{ k }\left( \eta \right)=0 \label{eq:dlksdsg},\end{aligned}$$ where: $$\begin{aligned}
{ \Omega }_{ k }^{ 2 }\left( \eta \right) ={ k }^{ 2 }
+a^{2}\left( \eta \right) \left( { m }_{\phi}^{ 2 }
+gS^{2}+3\lambda \phi^{2}+\left( \xi_{\phi} -1/6 \right) R \right). \nonumber \end{aligned}$$ Now, we rewrite the rescaled mode function $\delta \chi \left( \eta \right)$ by the Bogoliubov coefficients ${ \alpha }_{ k }\left( \eta \right)$, ${ \beta }_{ k }\left( \eta \right) $ as $$\begin{aligned}
\delta { { \chi }_{ k } }\left( \eta \right) =\frac { 1 }{ \sqrt { 2{ \Omega }_{ k }
\left( \eta \right) } } \left\{ { \alpha }_{ k }\left( \eta \right)
{ \delta \varphi }_{k}\left( \eta \right) +{ \beta }_{ k }\left( \eta \right)
{ \delta \varphi }_{k}^{*}\left( \eta \right) \right\} \label{eq:dldfghdhg},\end{aligned}$$ where ${ \alpha }_{ k }\left( \eta \right)$, ${ \beta }_{ k }\left( \eta \right) $ satisfy the Wronskian condition: ${ \left| { \alpha }_{ k }\left( \eta \right) \right| }^{ 2 }
-{ \left| { \beta }_{ k }\left( \eta \right) \right| }^{ 2 }=1$. The initial conditions for ${ \alpha }_{ k }\left( \eta_{0} \right) $, ${ \beta }_{ k }\left( \eta_{0} \right) $ are equivalent to the choice of the in-vacuum state. From Eq. (\[eq:dldfghdhg\]) the quantum fluctuation $\left<{ \delta \phi }^{ 2 } \right>$ can be given by $$\begin{aligned}
\left<{ \delta \phi }^{ 2 } \right>
=\frac { 1 }{ 4{ \pi }^{ 2 }a^{2}\left( \eta \right) }\int _{ 0 }^{ \infty }dk{ k }^{ 2 }
{ \Omega }_{ k }^{ -1 }\biggl\{ 1+2{ \left| { \beta }_{ k } \right| }^{ 2 }
+{ \alpha }_{ k }{ \beta }_{ k }^{ * }{ \delta \varphi }_{k}^{2}
+{ \alpha }_{ k }^{ * }{ \beta }_{ k }{ \delta \varphi^{*}_{k} }^{2} \biggr\}. \end{aligned}$$ For convenience, we introduce the following quantities $n_{k}={ \left| { \beta }_{ k } \right| }^{ 2 }$ and $z_{k}={ \alpha }_{ k }{ \beta }_{ k }^{ * }{ \delta \varphi }_{k}^{2}$ where $n_{k}={ \left| { \beta }_{ k }\left( \eta \right) \right| }^{ 2 }$ can be interpreted as the particle number density created in curved spacetime. By using $n_{k}$ and $z_{k}$, we obtain the following expression of the quantum fluctuation of the inflaton field as $$\begin{aligned}
\left<{ \delta \phi }^{ 2 } \right>=\left<{ \delta \phi }^{ 2 } \right>^{({\rm q})}+\left<{ \delta \phi }^{ 2 } \right>^{({\rm c})} \label{eq:adbatigudf},\end{aligned}$$ where: $$\begin{aligned}
\left<{ \delta \phi }^{ 2 } \right>^{( {\rm q} )}=
\frac { 1 }{ 4{ \pi }^{ 2 }a^{2}\left( \eta \right) }
\int _{ 0 }^{ \infty }{ dk{ k }^{ 2 }{ \Omega }_{ k }^{ -1 } },\quad
\left<{ \delta \phi }^{ 2 } \right>^{( {\rm c} )} =
\frac { 1 }{ 4{ \pi }^{ 2 }a^{2}\left( \eta \right) }
\int _{ 0 }^{ \infty }{ dk{ k }^{ 2 }{ \Omega }_{ k }^{ -1 }
\left\{ 2n_{k}+2{\rm Re}z_{k} \right\} } ,\end{aligned}$$ where $\left<{ \delta \phi }^{ 2 } \right>^{({\rm c})}$ can be regarded as the classic field fluctuations and expresses finite particle creations in curved spacetime [@birrell1984quantum], whereas $\left<{ \delta \phi }^{ 2 } \right>^{({\rm q})}$ obviously diverges as with the flat spacetime. Thus, we regularize the divergences of $\left<{ \delta \phi }^{ 2 } \right>^{({\rm q})}$ by the cut-off or dimensional regularization, and cancel them by the counterterms of the couplings. By using the dimensional regularization, we obtain the following expression, $$\begin{aligned}
\left<{ \delta \phi }^{ 2 } \right>^{({\rm q})}
=\frac { { M }^{ 2 }\left( \phi \right) }{ 16{ \pi }^{ 2 } }
\left[ \ln { \left( \frac { { M }^{ 2 }\left( \phi \right) }{ { \mu }^{ 2 } } \right) }
-N_{\epsilon}-\frac { 3 }{ 2 } \right] \label{eq:rehldg},\end{aligned}$$ where: $$\begin{aligned}
{ M }^{ 2 }\left( \phi \right) ={ m }^{ 2 }_{\phi}\left(\mu\right) +g\left(\mu\right)S^{2}
+3\lambda\left(\mu\right) \phi^{2}+\left( \xi_{\phi}\left(\mu\right) -1/6 \right) R,\end{aligned}$$ where $N_{\epsilon}=-1/{ \epsilon } -\log { 4\pi }-\gamma$ which is offset by the coupling counterterms, $\gamma$ is the Euler-Mascheroni constant and $\mu$ is the renormalization parameter. From Eq. (\[eq:rehldg\]) the quantum fluctuation in de-Sitter spacetime can be written as $$\begin{aligned}
\left<{ \delta \phi }^{ 2 } \right>
=\left<{ \delta \phi }^{ 2 } \right>^{({\rm q})}
+ \left<{ \delta \phi }^{ 2 } \right>^{({\rm c})}
= \frac { { M }^{ 2 }\left( \phi \right) }{ 16{ \pi }^{ 2 } }
\left[ \ln { \left( \frac { { M }^{ 2 }\left( \phi \right) }{ { \mu }^{ 2 } } \right) }
-N_{\epsilon}-\frac { 3 }{ 2 } \right]
+ \left<{ \delta \phi }^{ 2 } \right>^{({\rm c})}. \label{eq:xvxcldg}\end{aligned}$$ By using Eq. (\[eq:xvxcldg\]) the one-loop effective potential in de-Sitter spacetime can be given as follows [@Ringwald:1987ui; @Kohri:2017iyl]: $$\begin{aligned}
\label{eq:tutusssdg}
V_{\rm eff}\left( \phi \right)& =\ \frac{1}{2}m^{2}_{\phi}\left(\mu\right) \phi^{2}
+\frac{1}{2}\xi_{\phi}\left(\mu\right) R \phi^{2}
+\frac{\lambda\left(\mu\right)}{4}\phi^{4} \nonumber \\ &
+\frac{3\lambda\left(\mu\right)}{2}
\left<{ \delta \phi }^{ 2 } \right>^{({\rm c})}\phi^{2} +\frac { { M }^{ 4 }\left( \phi \right) }
{ 64{ \pi }^{ 2 } } \left[ \ln { \left( \frac { {M}^{ 2 }
\left( \phi \right) }{ { \mu }^{ 2 } } \right) } -\frac { 3 }{ 2 } \right]
+\mathcal{O}\left( \left\{\left<{ \delta \phi }^{ 2 } \right>^{({\rm c})}\right\}^2\right),\end{aligned}$$ where the effective potential is closely related with the energy density $\rho$ and pressure $p$ to be $$\begin{aligned}
\begin{split}
\rho&=\frac{1}{2}{ \dot { \phi } }^{ 2 }+\frac { 1 }{ 2 } \nabla { \phi }^{ 2 }
+V_{\rm eff}\left( \phi \right), \\
p&=\frac{1}{2}{ \dot { \phi } }^{ 2 }-\frac { 1 }{ 6 } \nabla { \phi }^{ 2 }
-V_{\rm eff}\left( \phi \right).
\end{split}\end{aligned}$$ Thus, the UV divergences can be eliminated by the renormalization parameters and the radiative corrections of the UV or trans-Planckian physics are sequestered form the cosmological observations as with the flat spacetime. On the other hand, primordial density perturbations originate from $\left<{ \delta \phi }^{ 2 } \right>^{({\rm c})} $ corresponding to the quantum particle creations and one is troubled with whether $\left<{ \delta \phi }^{ 2 } \right>^{({\rm c})} $ has the sensitivity of the UV or trans-Planckian physics.
However, the sensitivity of the cut-off scale is also sufficiently sequestered from the inflationary fluctuations. Let us discuss the issues using the adiabatic regularization method. The adiabatic regularization is a powerful method to calculate the vacuum fluctuation or the energy density from the quantum particle creation in curved spacetime, and proceed the regularization through subtracting $\left<{ \delta \phi }^{ 2 } \right>^{({\rm q})} $ from $\left<{ \delta \phi }^{ 2 } \right>$. Using the method, the vacuum fluctuation for the inflaton field can be given by $$\begin{aligned}
\begin{split}\label{eq:ohsdedg}
{ \left< { \delta \phi }^{ 2 } \right>}^{({\rm c})}
&=\int{ \frac { dk }{ k } }{ P }_{\delta { \phi } }\left( \eta, k \right)\\
&={ \left< { \delta \phi }^{ 2 } \right>}-{ \left< { \delta \phi }^{ 2 } \right>^{({\rm q})} }
=\frac { 1 }{ 4{ \pi }^{ 2 }a^{2}\left( \eta \right) }
\int _{ 0 }^{ \infty }{ dk{ k }^{ 2 }{ \Omega }_{ k }^{ -1 }\left\{2n_{k}+2{\rm Re}z_{k} \right\} }
\\
&=\frac { 1 }{ 4{ \pi }^{ 2 }a^{2}\left( \eta \right) } \left[\int _{ 0 }^{ \infty }{ dk2{ k }^{ 2 }{ \left| \delta { \chi }_{ k } \right| }^{ 2 } }-
\int _{ 0 }^{ \infty }{ dk{ k }^{ 2 }{ \Omega }_{ k }^{ -1 } } \right] ,
\end{split}\end{aligned}$$ where we must choose an appropriate initial vacuum and determine the mode function of $\delta \chi \left( \eta \right)$. In the discussion from now on, let us consider the massive non-minimally coupled case (for the details, see Ref.[@Haro:2010zz; @Haro:2010mx; @Kohri:2017iyl]) where the mode function $\delta { \chi }_{ k }\left( \eta \right) $ is given by $$\begin{aligned}
\delta { \chi }_{ k }\left( \eta \right) =\sqrt{\frac{\pi}{4}}\eta^{1/2}\left\{ { \alpha }_{ k }{ H}_{ \nu }^{(2)}\left( k\eta \right)
+{ \beta }_{ k }{ H}_{ \nu }^{(1)}\left( k\eta \right) \right\} \label{eq:ohghggedg},\end{aligned}$$ with $$\begin{aligned}
\nu \equiv \sqrt{\frac{9}{4}-\frac{{ M }^{ 2 }\left( \phi \right)}{H^{2}}}\simeq \frac{3}{2}-\frac{{ M }^{ 2 }\left( \phi \right)}{3H^{2}},\end{aligned}$$ ${ H}_{ \nu }^{(1, 2 )}\left( k\eta \right)$ are the Hankel functions. For simplicity we assume the specific universe from the radiation-dominated stage to the de Sitter stage and require the matching conditions at $\eta =\eta_{0}$ to determine the Bogoliubov coefficients $$\begin{aligned}
\alpha_{k}=&\frac { 1 }{ 2i } \sqrt { \frac { \pi k{ \eta }_{ 0 } }{ 2 } } \biggl( \left( -i+\frac { H }{ 2k } \right) { H }_{ \nu }^{ (1)}\left( k{ \eta }_{ 0 } \right)
-{ H }_{ \nu }^{ (1)' }\left( k{ \eta }_{ 0 } \right) \biggr) { e }^{ ik/H } \label{eq:ksfdedg}, \\
\beta_{k}=&-\frac { 1 }{ 2i } \sqrt { \frac { \pi k{ \eta }_{ 0 } }{ 2 } } \biggl( \left( -i+\frac { H }{ 2k } \right) { H }_{ \nu }^{ (2) }\left( k{ \eta }_{ 0 } \right)
-{ H }_{ \nu }^{ (2)' }\left( k{ \eta }_{ 0 } \right) \biggr) { e }^{ ik/H } \label{eq:kgdggedg}.\end{aligned}$$ From Eq. (\[eq:ohsdedg\]) the vacuum fluctuations are given as follows: $$\begin{aligned}
{ \left< { \delta \phi }^{ 2 } \right> }^{({\rm c})}
=&\lim _{ \Lambda \rightarrow \infty }\frac { 1 }{ 4{ \pi }^{ 2 }C\left( \eta \right) }
\Biggl[ \int _{ 0 }^{ \Lambda }{ 2k^{2}{ \left| \delta { \chi }_{ k } \right| }^{ 2 }dk }
-\int _{ \sqrt{2}/\left| \eta \right| }^{ \Lambda }{ dk{ k }^{ 2 }{ \Omega }_{ k }^{ -1 }} \Biggr] \nonumber \\
=&\frac { \eta^{2}H^{2} }{ 2{ \pi }^{ 2 } } \int _{ 0 }^{ H }{ k^{2}{ \left| \delta { \chi }_{ k } \right| }^{ 2 }dk }
+\frac { \eta^{2}H^{2} }{ 2{ \pi }^{ 2 }} \int _{ H }^{ \sqrt{2}/\left| \eta \right| }{ k^{2}{ \left| \delta { \chi }_{ k } \right| }^{ 2 }dk }
\label{eq:kdddegedg}.\end{aligned}$$ The divergence parts exactly cancel as follows, $$\begin{aligned}
\label{eq:dffsedg}
\lim _{ \Lambda \rightarrow \infty }\frac { 1 }{ 4{ \pi }^{ 2 }C\left( \eta \right) }
\Biggl[ \int _{ \sqrt{2}/\left| \eta \right|}^{ \Lambda }{ 2k^{2}{ \left| \delta { \chi }_{ k } \right| }^{ 2 }dk }
-\int _{ \sqrt{2}/\left| \eta \right| }^{ \Lambda }{ dk{ k }^{ 2 }{ \Omega }_{ k }^{ -1 }} \Biggr] ,\end{aligned}$$ where we take the adiabatic mode cut-off as $k> \sqrt{2-{ M }^{ 2 }\left( \phi \right)/H^{2}}/\left|\eta \right|
\simeq \sqrt{2}/\left|\eta \right|\simeq \sqrt{2}aH$. Beyond the mode cut-off the mode function does not drastically change against the evolution of the universe. In this sense the mode cut-off can be recognized as the UV cut-off of the inflationary quantum fluctuations and therefore the corrections of the trans-Planckian physics are sequestered as long as the Hubble parameter is smaller than the Planck scale.
By using the formula of the Hankel functions $$\begin{aligned}
{ H }_{ \nu }^{ (1,2)' }\left( k{ \eta }_{ 0 } \right)=
{ H }_{ \nu-1 }^{ (1,2) }\left( k{ \eta }_{ 0 } \right)-\frac{\nu}{k{ \eta }_{ 0 }}{ H }_{ \nu }^{ (1,2) }\left( k{ \eta }_{ 0 } \right),\end{aligned}$$ and the Bessel function of the first kind defined by $J_{\nu}=( { H }_{ \nu }^{ (1) }+{ H }_{ \nu }^{ (2) } )/2$, we can obtain the expression $$\begin{aligned}
\left| { \alpha }_{ k }-{ \beta }_{ k } \right| =\sqrt { \frac { \pi k }{ 2H } } \left| { J }_{ \nu -1 }\left( k{ \eta }_{ 0 } \right) +\left( i-\frac { H }{ 2k } +\frac { \nu H }{ k } \right) { J }_{ \nu }\left( k{ \eta }_{ 0 } \right) \right| .\end{aligned}$$ For small $k$ modes, the the Bessel function and the Hankel function asymptotically behave as $$\begin{aligned}
J_{\nu}\left(k\eta_{0}\right)&\simeq
\frac { 1 }{ \Gamma \left( \nu +1 \right) }
{ \left( \frac { k{ \eta }_{ 0 } }{ 2 } \right) }^{ \nu },\\
{ H }_{ \nu }^{ (2) }\left( k{ \eta }_{ 0 } \right)&
\simeq -{ H }_{ \nu }^{ (1) }\left( k{ \eta }_{ 0 } \right)
\simeq \frac { i }{ \pi } \Gamma \left( \nu \right)
{ \left( \frac { k{ \eta }_{ 0 } }{ 2 } \right) }^{ -\nu }.\end{aligned}$$ Thus, we can obtain the following expression of the mode function, $$\begin{aligned}
{ \left| \delta { \chi }_{ k } \right| }^{ 2 } &\simeq
\frac { \pi }{ 4 } \left| \eta \right| { \left| { \alpha }_{ k }-{ \beta }_{ k } \right| }^{ 2 }{ { \left| { H }_{ \nu }^{ (2) }\left( k{ \eta }\right) \right| }^{ 2 } }
\simeq
\frac{2}{9k}\left(H\left| \eta \right|\right)^{1-2\nu}\quad\ \left( 0\le k \le H \right)\label{eq:kddffsedg}.\end{aligned}$$ For large $k$ modes, we approximate the Bogoliubov coefficients to be $\alpha_{k} \simeq1$ and $\beta_{k} \simeq 0$ and evaluate the mode function as $$\begin{aligned}
\delta { \chi }_{ k }\left( \eta \right)
\simeq \sqrt{\frac{\pi}{4}}\eta^{1/2}{ H}_{ \nu }^{(2)}\left( k\eta \right) .\end{aligned}$$ Thus, we can get the following expression $$\begin{aligned}
{ \left| \delta { \chi }_{ k } \right| }^{ 2 } \simeq \frac{\left| \eta \right|}{16}
\left(\frac{k\left| \eta \right|}{2}\right)^{-2\nu}\quad \left( H\le k \le \sqrt { 2 } /\left| \eta \right| \right).\label{eq:kdyydg}\end{aligned}$$ From Eq. (\[eq:kddffsedg\]) and Eq. (\[eq:kdyydg\]), the vacuum fluctuations are written as $$\begin{aligned}
\begin{split}
{ \left< { \delta \phi }^{ 2 } \right> }^{({\rm c})}
&\simeq \frac { \left(H\left| \eta \right|\right)^{3-2\nu} }{ 9{ \pi }^{ 2 } } \int _{ 0 }^{ H }{ k dk } +\frac { H^{2}\left| \eta \right|^{3-2\nu} }{ 4{ \pi }^{ 2 }\cdot2^{3-2\nu}} \int _{ H }^{ \sqrt{2}/\left| \eta \right| }{ k^{2-2\nu}dk }\\
&\simeq \frac { H^{2} }{ 18{ \pi }^{ 2 } }e^{-\frac{2{ M }^{ 2 }\left( \phi \right)t}{3H}}
+\frac { 3H^{4} }{ 8{ \pi }^{ 2 }{ M }^{ 2 }\left( \phi \right)}
\left(1-e^{-\frac{2{ M }^{ 2 }\left( \phi \right)t}{3H}}\right)
\label{eq:kfhshggdg}.
\end{split}\end{aligned}$$ For late cosmic-time ($N_{\rm tot}=Ht\gg H^{2}/{ M }^{ 2 }\left( \phi \right)$), the vacuum fluctuations ${ \left< { \delta \phi }^{ 2 } \right> }^{({\rm c})}$ in de Sitter background are approximately written as [^2] $$\begin{aligned}
{ \left< { \delta \phi }^{ 2 } \right> }^{({\rm c})}
\simeq \frac { 3H^{4} }{ 8{ \pi }^{ 2 }{ M }^{ 2 }\left( \phi \right)} , \quad\ \left( { M }\left( \phi \right)\ll H \right)
\label{eq:klkddussg}.\end{aligned}$$ which modify the effective potential of Eq. (\[eq:tutusssdg\]) in de Sitter spacetime and provides primordial density perturbations. Now, we found out that the UV divergences are safely sequestered and the sensitivity of the trans-Planckian physics is negligible. This conclusion is consistent with the approach of the effective field theories [@Kaloper:2002uj; @Kaloper:2002cs] but the quantum effects of the UV or trans-Planckian physics only changes the effective potential through the radiative corrections. Note that the effective mass ${ M }\left( \phi \right)$ of the inflaton should be smaller than the Hubble scale if not the inflationary vacuum fluctuations are strongly suppressed as ${ \left< { \delta \phi }^{ 2 } \right> }^{({\rm c})}\rightarrow0$ for ${ M }\left( \phi \right)\gg H$ [@Mottola:1984ar], and therefore, the radiative corrections of the UV or trans-Planckian physics to the inflaton mass must be hardly small and this fact is consistent with the fine-tuning of the inflaton potential. Precisely, however, ${ \left< { \delta \phi }^{ 2 } \right> }^{({\rm c})}$ has the mode dependence and might leave a tiny trans-Planckian imprint on the CMB at the short distance.
The UV completion and inflationary quantum fluctuation {#sec:renormalization}
=======================================================
Next, let us consider the UV corrections of the massive scalar field $S$ to the inflationary vacuum fluctuation and construct the effective potential $V_{\rm eff}\left( \phi \right)$ in curved spacetime. By using $\left<{ \delta \phi }^{ 2 } \right>$ and $\left<{ \delta S }^{ 2 } \right>$ we obtain the one-loop effective potential in de-Sitter spacetime as follows: $$\begin{aligned}
V_{\rm eff}\left( \phi \right)& =\ \frac{1}{2}m^{2}_{\phi}\left(\mu\right) \phi^{2}
+\frac{1}{2}\xi_{\phi}\left(\mu\right) R \phi^{2}+\frac{\lambda\left(\mu\right)}{4}\phi^{4}+\frac{3\lambda\left(\mu\right)}{2}
\left<{ \delta \phi }^{ 2 } \right>^{({\rm c})}\phi^{2}+\mathcal{O}\left( \left\{\left<{ \delta \phi }^{ 2 } \right>^{({\rm c})}\right\}^2\right) \nonumber \\
&+\frac { { M }^{ 4 }\left( \phi \right) }{ 64{ \pi }^{ 2 } } \left[ \ln { \left( \frac { {M}^{ 2 }\left( \phi \right) }{ { \mu }^{ 2 } } \right) }
-\frac { 3 }{ 2 } \right] +\frac{g}{2}\phi^{2}S^{2} +\frac { { M }^{ 4 }\left( S \right) }{ 64{ \pi }^{ 2 } } \left[ \ln { \left( \frac { { M }^{ 2 }\left( S \right) }{ { \mu }^{ 2 } } \right) }
-\frac { 3 }{ 2 } \right] +\cdots, \label{eq:tutssdg}\end{aligned}$$ with $$\begin{aligned}
{ M }^{ 2 }\left( S \right)={ m }^{ 2 }_{S}\left(\mu\right) +g\left(\mu\right)\phi^{2}
+\left( \xi_{S}\left(\mu\right) -1/6 \right) R, \nonumber \end{aligned}$$ where $\left<{ \delta S }^{ 2 } \right>^{({\rm c})}$ is sufficiently suppressed and can be negligible. By using Eq. (\[eq:tutssdg\]), we can read off the $\mu$ dependence of these couplings and the one-loop $\beta$ function of ${ m }^{ 2 }_{\phi}$ can be given as follows: $$\begin{aligned}
\beta_{ m^{2}_{\phi} }= {\mu} \frac { \partial }{ \partial { \mu } }{ { m }^{ 2 }_{\phi} } =\frac { 6{ \lambda }m^{2}_{\phi}
+2gm_{S}^{2} }{ { \left( 4\pi \right) }^{ 2 } }
+\cdots.
\end{aligned}$$ Now we can simply understand that the UV corrections of the massive scalar field $S$ drastically changes the effective potential of the inflaton and the fine-tuning matters appear to realize the inflation. These issues can be interpreted by the inflationary vacuum fluctuation of Eq. (\[eq:klkddussg\]). To improve the expression of Eq. (\[eq:klkddussg\]) we shifts the inflaton field to be $$\begin{aligned}
\phi^{2} \rightarrow \phi^{2} + \left<{ \delta \phi }^{ 2 } \right>^{({\rm c})}
=\phi^{2} +\frac { { M }^{ 2 }\left( \phi \right) }{ 16{ \pi }^{ 2 } }
\left[ \ln { \left( \frac { { M }^{ 2 }\left( \phi \right) }{ { \mu }^{ 2 } } \right) }
-N_{\epsilon}-\frac { 3 }{ 2 } \right]
+ \left<{ \delta \phi }^{ 2 } \right>^{({\rm c})},\end{aligned}$$ which is reasonable to include quantum backreactions in perturbative contexts and we can get the following expression of the inflationary vacuum fluctuation $$\begin{aligned}
\left< { \delta \phi }^{ 2 } \right>^{({\rm c})}_{\rm eff}\simeq
\frac{3{ H }^{ 4}}{8{ \pi }^{ 2 }{ M }^{ 2 }_{\rm eff}\left( \phi \right)}, \quad\ \left( { M }_{\rm eff}\left( \phi \right)\ll H \right)
\label{eq:klkkdjfdssg},\end{aligned}$$ where ${ M }^{ 2 }\left( \phi \right)$ on Eq. (\[eq:klkddussg\]) is replaced by ${ M }_{\rm eff}\left( \phi \right)$ including quantum radiative corrections as follows: $$\begin{aligned}
{ M }^{ 2 }_{\rm eff}\left( \phi \right) &= { m }^{ 2 }_{\phi}\left(\mu\right) +g\left(\mu\right)S^{2}
+3\lambda\left(\mu\right) \phi^{2}
+\left( \xi_{\phi}\left(\mu\right) -1/6 \right) R
\nonumber \\
&+ \frac { 3\lambda\left(\mu\right){ M }^{ 2 }\left( \phi \right) }{ 16{ \pi }^{ 2 } } \left[ \ln { \left( \frac { {M}^{ 2 }\left( \phi \right) }{ { \mu }^{ 2 } } \right) }
-\frac { 3 }{ 2 } \right]
+ \frac { g\left(\mu\right){ M }^{ 2 }\left( S \right) }{ 16{ \pi }^{ 2 } } \left[ \ln { \left( \frac { {M}^{ 2 }\left( S \right) }{ { \mu }^{ 2 } } \right) }
-\frac { 3 }{ 2 } \right]+\cdots \label{eq:ksjdjrujfdssg}.\end{aligned}$$ In $m_{S} \simeq \Lambda_{\rm UV} \gg m_{\phi}$, the quantum radiative corrections are approximately order of the UV scale as ${ M }_{\rm eff}\left( \phi \right) \simeq g^{1/2}m_{S}
\simeq g^{1/2}\Lambda_{\rm UV}$. Therefore, the inflationary vacuum fluctuation can be simplified as follows: $$\begin{aligned}
\left< { \delta \phi }^{ 2 } \right>^{({\rm c})}_{\rm eff}\simeq
\frac{3{ H }^{ 4}}{8{ \pi }^{ 2 }\bigl( { m }^{ 2 }_{\phi}+ g \Lambda_{\rm UV}^{2} \bigr)},
\quad\ \left( { m }_{\phi}+ g^{1/2}\Lambda_{\rm UV} \ll H \right)
\label{eq:klkdslfssg},\end{aligned}$$ where the UV corrections are of order $\mathcal{O}\left(1\right)$ rather than $\mathcal{O}\left( H/\Lambda_{\rm UV} \right)$ or $\mathcal{O}\left( H^{2}/\Lambda_{\rm UV}^{2}\right)$ in comparison with ${ m }_{\phi}$, and therefore, the inflationary vacuum fluctuation or the primordial CMB perturbation are naively depend on the UV contributions. In this sence, for the very massive case ($ { m }_{\phi}+ g^{1/2}\Lambda_{\rm UV} \gg H $), the slow-roll condition of the inflation violates or the inflationary fluctuation breaks the scale invariance of the spectrum of CMB perturbations and are sufficiently suppressed (see e.g. Ref.[@Linde:1982uu; @Starobinsky:1982ee; @Vilenkin:1983xp]). Therefore, we can impose a tight constraint on the UV physics as follows: $$\begin{aligned}
\Lambda_{\rm UV} \ll H/g^{1/2}.\end{aligned}$$
Now, we can obtain an upper bound of the interaction coupling to be $g \ll \mathcal{O}(10^{-10})$ if we assume the Planck scale cut-off as $\Lambda_{\rm UV} \simeq M_{\rm Pl} \simeq 10^{18}\ {\rm GeV}$ and take the current upper value of the Hubble parameter $H \simeq 10^{13}\ {\rm GeV}$ [@Ade:2015lrj; @Ade:2015tva]. Note that in the same way we obtain the constraint of the existence of the coherent fields $\phi$, $S$ as $\phi,S \ll H/g$ from Eq. (\[eq:ksjdjrujfdssg\]). From these points, we found out that the inflaton sector should be decoupled with the high energy physics like the Planck or string physics. From this standpoint, the Starobinsky inflation [@Starobinsky:1980te] or the Higgs inflation [@Bezrukov:2007ep; @Barvinsky:2008ia] are attractive [^3] due to the non-requirement for the new physics in comparison with any other inflation models [@Martin:2013tda; @Martin:2013nzq; @Martin:2013gra; @Ijjas:2013vea], and furthermore, matched with the current constraints of the CMB observations. These things from Eq. (\[eq:ksjdjrujfdssg\]) are consistent with the fine-tuning problems of the inflaton potential, and therefore, if we neglect the fine-tuning for inflation the quantum effects of the UV or trans-Planckian physics on the primordial density perturbation is sufficiently sequestered. The reason for this is that in the UV regime the quantum fluctuation does not drastically change described by Eq. (\[eq:dffsedg\]) and we only receive a tiny imprint on the CMB at the short distance rather than $\mathcal{O}\left( H/\Lambda_{\rm UV} \right)$ or $\mathcal{O}\left( H^{2}/\Lambda_{\rm UV}^{2}\right)$ as previously discussed. However, the trans-Planckian corrections give non-negligible impact on the inflaton potential and the inflationary vacuum fluctuation as the fine-tuning issues of the inflation itself.
Conclusion and Summary {#sec:relaxation}
======================
In this paper, we have discussed how the UV or trans-Planckian physics affect the inflationary vacuum fluctuation and the primordial density perturbation from the rigid perspective of the QFT in curved spacetime. Here we have dealt with this problem by using the two-point correlation function $\left< { \delta \phi }^{ 2 } \right>$ of the non-minimally coupled scalar fields and constructing the effective potential in curved spacetime. Clearly we have shown that in the UV regime the quantum fluctuation does not drastically change as described by Eq. (\[eq:dffsedg\]) and the UV or trans-Planckian corrections can be embedded in the effective potential, and the UV sensitivity on the primordial density perturbation is sufficiently sequestered. However, the trans-Planckian corrections give a strong impact on the fine-tuning problems of the inflation itself and we have obtained a inflationary conjecture which can be a window towards the UV or trans-Planckian physics.
[*Acknowledgments*]{}: I would like to thank Kazunori Kohri for numerous helpful discussions and collaboration.
[^1]: In principle, these problems should be discussed in the framework of quantum gravity (QG) theory. However, owing essentially to the non-renormalizable and non-unitary properties, we do not have even a consistent theory for QG yet [@DeWitt:2007mi; @Smolin:2003rk; @Giddings:2011dr], and furthermore, it is well-known that the primordial density perturbation can be successfully described by a semiclassical approach to gravity. In this sense, our goal will be sufficiently achieved by simply adopting QFT in curved spacetime [@birrell1984quantum; @fulling1989aspects; @parker2009quantum; @DeWitt:1975ys; @Bunch:1979uk; @buchbinder1980effective; @Shapiro:2008sf] which is a completely self-consistent theory.
[^2]: In this case, the power spectrum on super-horizon scale ($\left| k\eta \right| \ll1$) can be approximately written by [@Riotto:2002yw] $$\begin{aligned}
{ P }_{\delta { \phi } }\left(\eta,k\right)={ \left( \frac { H }{ 2\pi } \right) }^{ 2 }{ \left( \frac { k }{ aH } \right) }^{ 3-2\nu }\end{aligned}$$ where the inflationary vacuum fluctuation with non-vanishing mass (${ M }\left( \phi \right) \ll H)$ has a tiny $k$-dependence, i.e scale invariance.
[^3]: From the viewpoint of the quantum radiative corrections on the curved spacetime, there is a strong correspondence between the Starobinsky inflation ($a_{1} R^{2}$) and the Higgs inflation ($\xi_{H}H^{\dagger}H$). In fact, the one-loop $\beta$-function of the gravitational coupling ${a_{1}}$ can be written as follows [@Shapiro:2008sf; @Salvio:2015kka; @Calmet:2016fsr]: $$\begin{aligned}
\beta_{a_{1}}={\mu} \frac { \partial }{ \partial { \mu } }{ a_{1} }
= \frac{\left(\xi_{H}-1/6\right)^{2}}{2\left(4\pi \right)^{2}},\end{aligned}$$ where $a_{1}\approx 10^{9}$ for the Starobinsky inflation or $\xi_{H}\approx10^{4}$ for the Higgs inflation are compatible with the latest Planck data.
|
---
abstract: 'We prove rigorously that the ferromagnetic Ising model on any nonamenable Cayley graph undergoes a continuous (second-order) phase transition in the sense that there is a unique Gibbs measure at the critical temperature. The proof of this theorem is quantitative and also yields power-law bounds on the magnetization at and near criticality. Indeed, we prove more generally that the magnetization $\langle \sigma_o \rangle_{\beta,h}^+$ is a locally Hölder-continuous function of the inverse temperature $\beta$ and external field $h$ throughout the non-negative quadrant $(\beta,h)\in [0,\infty)^2$. As a second application of the methods we develop, we also prove that the free energy of Bernoulli percolation is twice differentiable at $p_c$ on any transitive nonamenable graph.'
author:
- '[**Tom Hutchcroft**]{}'
title: '**Continuity of the Ising phase transition on nonamenable groups**'
---
Introduction {#sec:intro}
============
It has been known since the 19th century that the magnetic properties of certain metals such as iron, cobalt, and nickel undergo a qualitative change as they pass through a certain critical temperature, now known as the *Curie temperate*[^1] of the metal: Below the critical temperature the metal is *ferromagnetic*, meaning that it will remain permanently magnetized after temporary exposure to an external magnetic field, while above the critical temperature the metal is *paramagnetic*, meaning that it will become magnetized in the presence of an external magnetic field but will revert back to being unmagnetized when the external field is removed. The magnetization that remains when the external field is removed is referred to as the *spontaneous magnetization*: it is positive in the ferromagnetic regime and zero in the paramagnetic regime.
The *Ising model* is a mathematical model that attempts to describe this phase transition. It was introduced in 1920 by Wilhelm Lenz, who suggested the model to his student Ernst Ising as a thesis subject [@ising1925beitrag]. The model attracted widespread attention following the 1936 work of Peierls [@peierls1936ising], who argued that the model does indeed undergo a phase transition on Euclidean lattices of dimension at least two. The Ising model remains today arguably the most famous and intensively studied model in statistical mechanics, with a vast literature devoted to it, and is now used to model many other ‘cooperative’ phenomena in statistical mechanics beyond magnetism. See e.g. [@MR3752129; @1707.00520] for introductions to the Ising model for mathematicians, [@MR1446000] for a more physical introduction, and [@brush1967history] for a history.
Although the Ising model has traditionally been studied primarily in the setting of Euclidean lattices, there has more recently been substantial interest among both mathematicians and physicists in determining the model’s behaviour in other geometric settings, such as hyperbolic spaces. A natural level of generality at which to study the model is that of *(vertex-)transitive graphs*, that is, graphs for which any vertex can be mapped to any other vertex by a symmetry of the graph. The resulting literature is now rather extensive, and includes e.g.numerical and non-rigorous studies of critical behaviour [@breuckmann2020critical; @serina2016free; @benedetti2015critical; @gendiar2014mean; @iharagi2010phase], rigorous analysis of critical behaviour for some examples [@MR1413244; @MR1798548; @MR1833805; @1712.04911; @1606.03763], and analysis of the set of Gibbs measures at low temperature [@gandolfo2015manifold; @MR1768240; @series1990ising; @MR1684757]. Moreover, it is now known that the Ising model has a non-trivial phase transition on any infinite transitive graph that has superlinear volume growth (i.e., is not one-dimensional) [@1806.07733].
Once non-triviality of the phase transition has been established, it becomes of great interest to understand the model *at the critical temperature*, where it is expected to display various interesting behaviours. Perhaps the most basic question one can ask about the critical model is whether it belongs to the ferromagnetic or paramagnetic regime. Mathematically, this amounts to asking if the spontaneous magnetization of the model vanishes at the critical temperature, in which case we say that the Ising model undergoes a *continuous phase transition*. It is widely believed that the Ising phase transition should be continuous in most cases that it is non-trivial, although this is known to be *false* for certain long-range models in one dimension [@MR939480].
The primary goal of this paper is to prove that the Ising model undergoes a continuous phase transition on any *nonamenable, unimodular transitive graph*. Here, we recall that a graph $G=(V,E)$ is said to be **nonamenable** if $\inf\left\{ |\partial_E W|/\sum_{v\in W} \deg(v) : W \subseteq V \text{ finite} \right\}>0$, where $\partial_E W$ is the set of edges with one endpoint in $W$ and the other not in $W$. *Unimodularity* is a technical condition that holds in most natural examples, including in every Cayley graph of a finitely generated group and every transitive amenable graph [@MR1082868]; see \[subsec:unimodularity\_background\] for background. The theorem applies in particular to the Ising model on tessellations of $d$-dimensional hyperbolic space $\mathbb{H}^d$ with $d\geq 2$, for which the result was only previously known under perturbative hypotheses [@MR1798548; @MR1413244; @MR1833805].
\[thm:main\_simple\] Let $G$ be a connected, locally finite, transitive, unimodular, nonamenable graph. Then the phase transition of the Ising model on $G$ is continuous: at the critical temperature the spontaneous magnetization is zero and there is a unique Gibbs measure.
We will in fact prove more general and quantitative versions of this theorem, \[thm:main,thm:main\_continuity\], which establish continuity of the model at *all* temperatures, as well as power-law bounds on the magnetization at and near the critical temperature under the same hypotheses.
Let us now briefly outline how our results relate to previous work. For the hypercubic lattice $\Z^d$, continuity of the phase transition is well understood: The case $d=2$ was settled by Yang in 1952 [@yang1952spontaneous], who built upon the works of Onsager [@MR0010315] and Kaufman [@kaufman1949crystal], the case $d\geq 4$ was settled by Aizenman and Fernandez in 1986 [@MR857063], while the case $d=3$ was settled relatively recently by Aizenman, Duminil-Copin, and Sidoravicius in 2015 [@MR3306602]. Some aspects of each of these proofs are rather specific to the hypercubic case and do not generalize to other Euclidean lattices, let alone arbitrary transitive graphs. While various subsequent works have extended these results to several other Euclidean models [@sakai2007lace; @MR3898174; @MR1896880; @MR4026609], the rigorous understanding of the critical Ising model beyond the Euclidean setting has remained somewhat limited. In our context, the most significant progress was due to Schonmann [@MR1833805 Theorem 1.9] who proved (among many other things) that the Ising model undergoes a continuous phase transition with mean-field critical exponents on certain ‘highly nonamenable’ Cayley graphs. Similar results in the more specific setting of hyperbolic lattices have been obtained by Wu [@MR1798548; @MR1413244]. The arguments of Schonmann and Wu are of a perturbative nature (that is, they require some parameter associated to the graph to be small), and cannot be used to treat arbitrary nonamenable transitive Cayley graphs. Aside from the classical case of trees, we are only aware of two previous works establishing non-perturbative results in the non-Euclidean context: Our earlier paper [@1712.04911], in which we established continuity of the phase transition for products of regular trees of degree at least three, and the work of Raoufi [@1606.03763], who combined the methods of [@Hutchcroft2016944] and [@MR3306602] to prove that the Ising model undergoes a continuous phase transition on any *amenable* transitive graph of exponential volume growth. Raoufi’s argument relies on amenability in a crucial way and cannot be used to analyze nonamenable examples.
Our techniques draw heavily on the machinery that has been developed to understand *Bernoulli percolation* in the same context [@bperc96; @1808.08940; @BLPS99b]. Indeed, the central technical contribution of our paper is a new method, based on the spectral theory of automorphism-invariant processes, that allows the machinery of [@1808.08940] to be applied to certain models that are not positively associated. This new method can be applied to prove that the *double random current model* does not have any infinite clusters at criticality, from which \[thm:main\_simple\] can be deduced by the methods of Aizenman, Duminil-Copin, and Sidoravicius [@MR3306602]. A detailed overview of this new method and how it compares to existing techniques is given in \[subsec:intro\_overview\].
We hope that this paper will be of value and interest both to experts on percolation and the Ising model who know relatively little group theory and to experts on group theory who know relatively little about the Ising model; we have included a detailed discussion of background material with the aim of making the paper accessible to both communities.
Definitions and statement of results {#subsec:intro_definitions}
------------------------------------
Let us now define the Ising model formally. Further background on the Ising model may be found in e.g. [@MR3752129; @1707.00520]; see also [@Pete Section 13.1] and [@MR1757952] for background on aspects specific to the nonamenable case. We will take the approach of [@1901.10363], which allows for a unified treatment of short- and long-range models. We define a **weighted graph** $G=(V,E,J)$ to be a countable graph $(V,E)$ together with an assignment of positive **coupling constants** $\{J_e : e \in E\}$ such that for each vertex $v$ of $G$, the sum of the coupling constants $J_e$ over all edges $e$ adjacent to $v$ is finite. Locally finite graphs can be considered as weighted graphs by setting $J_e \equiv 1$. A graph automorphism of $(V,E)$ is a weighted graph automorphism of $(V,E,J)$ if it preserves the coupling constants, and a weighted graph $G$ is said to be **transitive** if for every two vertices $x$ and $y$ in $G$ there exists an automorphism of $G$ sending $x$ to $y$. A weighted graph $G=(V,E,J)$ is said to be **nonamenable** if $\inf\{\sum_{e\in \partial_E K} J_e / \sum_{e \in E(K)} J_e : K \subseteq V$ finite$\}>0$, where $E(K)$ denotes the set of edges with at least one endpoint in $K$.
Let $G=(V,E,J)$ be a weighted graph with $V$ finite, so that $\sum_{e\in E} J_e <\infty$. For each $\beta\geq 0$ and $h\in \R$ we define the **Ising measure** ${\mathbf{I}}_{\beta,h}={\mathbf{I}}_{G,\beta,h}$ to be the probability measure on $\{-1,1\}^V$ given by $${\mathbf{I}}_{G,\beta,h}(\{\sigma\}) \propto \exp\left[ \beta \sum_{e\in E} J_e \sigma_e + \beta \sum_{v\in V} h \sigma_v \right] \qquad \text{ for each $\sigma \in \{-1,1\}^V$}$$ where for each edge $e\in E$ with endpoints $x$ and $y$ we define $\sigma_e=\sigma_x\sigma_y \in \{-1,1\}$. The parameters $\beta$ and $h$ are known as the **inverse temperature** and **external field** respectively. The quantity $\sigma_v\in \{-1,1\}$ is known as the **spin** at $v$. Thus, the measure favours configurations in which the spins of adjacent vertices are aligned with each other and with the external field.
Now suppose that $G=(V,E,J)$ is an *infinite* weighted graph. For each $\beta\geq 0$ and $h\in \R$ we define $\cG_{\beta,h}$ to be the set of **Gibbs measures** for the Ising model on $G$, that is, the set of probability measures $\mu$ on $\{-1,1\}^V$ satisfying the **Dobrushin, Lanford, and Ruelle (DLR) equations** $$\mu\Bigl(\sigma|_A = \xi|_A \Bigm \vert \sigma|_{V\setminus A} = \xi|_{V\setminus A}\Bigr) = \frac{1}{Z(\xi|_{V\setminus A})} \exp\left[ \beta \sum_{e\in E(A)} J_e \xi_e + \beta \sum_{v\in A} h\xi_v \right]$$ for every $A\subseteq V$ finite and $\xi \in \{-1,1\}^V$, where $E(A)$ denotes the set of edges that have at least one endpoint in $A$ and $Z(\xi|_{V\setminus A})$ is a normalizing constant. Note that Gibbs measures need not in general be invariant under the automorphisms of $G$. A central problem in the study of the Ising model is is to understand the structure of the set of Gibbs measures $\cG_{\beta,h}$, and in particular how this structure depends on $\beta$ and $h$. The **critical inverse temperature** $\beta_c$ is defined by $\beta_c=\inf\bigl\{\beta\geq 0 : |\cG_{\beta,0}| >1\bigr\}$. We now introduce three particularly important Gibbs measures for the Ising model: the free, plus, and minus measures. Let $G=(V,E,J)$ be an infinite, connected, weighted graph, and let $(V_n)_{n\geq 1}$ be an **exhaustion** of $V$, that is, an increasing sequence of finite subsets of $V$ with $\bigcup_{n \geq 1} V_n = V$. For each $n\geq 1$, let $G_n$ be the weighted subgraph of $G$ induced by $V_n$. (That is, $G_n$ has vertex set $V_n$, edge set equal to the set of all edges of $G$ with both endpoints in $V_n$, and edge weights inherited from $G$.) For each $\beta \geq 0$ and $h\in \R$, we define the **free Ising measure** ${\mathbf{I}}_{\beta,h}^f={\mathbf{I}}_{G,\beta,h}:={\mathop{\operatorname{w-lim}}}_{n\to\infty}{\mathbf{I}}_{G_n,\beta,h}$ to be the weak limit of the measures ${\mathbf{I}}_{G_n,\beta,h}$, so that $${\mathbf{I}}_{G,\beta,h}^f (\sigma_a =\kappa_a \text{ for every $a\in A$}) = \lim_{n\to \infty} {\mathbf{I}}_{G_n,\beta,h}(\sigma_a =\kappa_a \text{ for every $a\in A$})$$ for every finite set $A \subseteq V$ and $\kappa\in \{-1,1\}^A$. See e.g. [@MR3752129 Exercise 3.16] for a proof that this limit exists, belongs to $\cG_{\beta,h}$, and does not depend on the choice of exhaustion. For each $n\geq 1$ we also define $G_n^*$ to be the finite weighted graph obtained from $G$ by contracting every vertex in $V\setminus V_n$ into a single vertex $\partial_n$ and deleting all self-loops from $\partial_n$ to itself. For each $\beta \geq 0$ and $h\in \R$, the **plus** and **minus Ising measures** ${\mathbf{I}}_{\beta,h}^+={\mathbf{I}}_{G,\beta,h}^+$ and ${\mathbf{I}}_{G,\beta,h}^-={\mathbf{I}}_{\beta,h}^-$ on $G$ are defined to be the weak limits of the *conditional* measures of the Ising model on $G_n^*$ given that the boundary spin is $+1$ or $-1$ as appropriate. In particular, $$\begin{aligned}
{\mathbf{I}}_{G,\beta,h}^+ (\sigma_a =\kappa_a \text{ for every $a\in A$}) &= \lim_{n\to \infty} {\mathbf{I}}_{G_n^*,\beta,h}(\sigma_a =\kappa_a \text{ for every $a\in A$} \mid \sigma_{\partial_n}=1)
$$ for every finite set $A \subseteq V$ and $\kappa\in \{-1,1\}^A$. The fact that these weak limits exist and do not depend on the choice of exhaustion is a consequence of the Holley inequality [@MR3752129 Theorem 3.17]. The measures ${\mathbf{I}}_{\beta,h}^+$ and ${\mathbf{I}}_{\beta,h}^-$ are maximal and minimal elements of $\cG_{\beta,h}$ with respect to the partial ordering of stochastic domination: If $\mu$ is any element of $\cG_{\beta,h}$ then $\mu$ stochastically dominates ${\mathbf{I}}_{\beta,h}^-$ and is stochastically dominated by ${\mathbf{I}}_{\beta,h}^+$ [@MR3752129 Lemma 3.23]. It follows in particular that $|\cG_{\beta,h}|=1$ if and only if ${\mathbf{I}}_{\beta,h}^+={\mathbf{I}}_{\beta,h}^-$ if and only if ${\mathbf{I}}_{\beta,h}^+={\mathbf{I}}_{\beta,h}^f$. Note also that the measure ${\mathbf{I}}_{\beta,h}^\#$ is invariant under all automorphisms of $G$ for every $\beta\geq 0$, $h\in \R$, and $\#\in\{f,+,-\}$; this follows from the fact that the limits defining these measures do not depend on the choice of exhaustion [@MR3752129 Theorem 3.17 and Exercise 3.16].
If $G$ is transitive and amenable then ${\mathbf{I}}_{\beta,h}^-={\mathbf{I}}_{\beta,h}^f={\mathbf{I}}_{\beta,h}^+$ for every $\beta >0$ and $h \neq 0$ [@MR1684757 Section 3.2], so that the question of uniqueness of Gibbs measures is only interesting in the case $h=0$. See also [@MR3752129 Section 3.7.4]. This is no longer true when $G$ is nonamenable. Indeed, it is a theorem of Jonasson and Steif [@MR1684757] that if $G$ is a nonamenable, bounded degree graph then there exists $h_0>0$ such that if $|h| \leq h_0$ then there exists $\beta_c(h)<\infty$ such that $|\cG_{\beta,h}|>1$ for all $\beta > \beta_c(h)$. (The statement they give is different since their definition of the Ising model with external field follows different conventions to ours.) Intuitively, the difference between these two theorems stems from the fact that boundary effects are always negligible compared with bulk effects in the amenable setting, while the two effects can be of the same order in the nonamenable setting.
It is traditional to denote expectations taken with respect to the measures ${\mathbf{I}}_{\beta,h}^f$, ${\mathbf{I}}_{\beta,h}^+$ and ${\mathbf{I}}_{\beta,h}^-$ using the notation $\langle \cdots \rangle_{\beta,h}^f$, $\langle \cdots \rangle_{\beta,h}^+$, and $\langle \cdots \rangle_{\beta,h}^-$ respectively, so that, for example, $$\langle \sigma_x \sigma_y \rangle_{\beta,h}^f = {\mathbf{I}}_{\beta,h}^f[\sigma_x\sigma_y]$$ denotes the expectation of the product of the spins $\sigma_x$ and $\sigma_y$ under the measure ${\mathbf{I}}_{\beta,h}^f$ for each $x,y\in V$. We will use both notations throughout the paper as is convenient.
Now suppose that $G=(V,E,J)$ is a transitive weighted graph and let $o$ be a fixed root vertex of $G$. For each $\beta\geq 0$, $h\in \R$, and $\#\in\{f,+,-\}$ we define the **magnetization** $$m^\#(\beta,h)=m_G^\#(\beta,h) = \langle \sigma_o \rangle_{\beta,h}^\#.$$ Note that $m^f(\beta,0)=0$ for every $\beta \geq 0$ by symmetry. For each $\beta \geq 0$, the **spontaneous magnetization** is defined by $m^*(\beta):= m^+(\beta,0)$ The spontaneous magnetization is a quantitative measure of how much the measures ${\mathbf{I}}_{\beta,0}^+$ and ${\mathbf{I}}_{\beta,0}^f$ differ, and we have in particular that $$m^*(\beta)=0 \iff {\mathbf{I}}^f_{\beta,0} = {\mathbf{I}}^+_{\beta,0} \iff {\mathbf{I}}^+_{\beta,0} = {\mathbf{I}}^-_{\beta,0} \iff |\cG_{\beta,0}|=1
$$ for every $\beta \geq 0$. Thus, we can express the critical inverse temperature $\beta_c$ equivalently as $\beta_c=\inf\bigl\{\beta \geq 0 : |\cG_{\beta,0}| >1\bigr\}=\inf\bigl\{\beta \geq 0 : m^*(\beta)>0 \bigr\} =\inf\bigl\{\beta \geq 0 : {\mathbf{I}}^f_{\beta,0} \neq {\mathbf{I}}^+_{\beta,0}\bigr\}$.
The following theorem strengthens and generalizes \[thm:main\_simple\].
\[thm:main\] Let $G=(V,E,J)$ be a connected, nonamenable, transitive, unimodular weighted graph. Then there exist positive constants $C$ and $\delta$ such that $$|m^\#(\beta,h)| \leq C \left( |h| + \max\{ \beta-\beta_c,0\} \right)^\delta$$ for every $\beta \geq 0$, $h\in \R$, and $\# \in \{f,+,-\}$. In particular, $m^*(\beta_c)=0$ and $|\cG_{\beta_c,0}|=1$.
In fact, our proof establishes more generally that the spontaneous magnetization is continuous not just at $\beta_c$, but for *all* non-negative $\beta$. The following theorem provides a strong quanitative statement to this effect which implies \[thm:main\]. Recall that if $\alpha>0$ and $X$ is a locally compact metric space then a function $f:X\to \R$ is said to be **locally $\alpha$-Hölder continuous** if for every compact set $K \subseteq X$ there exists $C<\infty$ such that $|f(x)-f(y)|\leq C d(x,y)^\alpha$ for every $x,y\in K$.
\[thm:main\_continuity\] Let $G=(V,E,J)$ be a connected, nonamenable, transitive, unimodular weighted graph. Then there exists $\delta>0$ such that if $F:\{-1,1\}^V \to \R$ is any function depending on at most finitely many vertices then $\langle F(\sigma) \rangle^+_{\beta,h}$ is a locally $\delta$-Hölder continuous function of $(\beta,h)\in [0,\infty)^2$. In particular, the plus Ising measure ${\mathbf{I}}_{\beta,h}^+$ is a weakly-continuous function of $(\beta,h)\in [0,\infty)^2$.
It is a theorem of Raoufi [@MR4089494 Theorem 1 and Corollary 1] that if $G$ is an *amenable* transitive weighted graph then the plus and free Ising measures ${\mathbf{I}}_{\beta,0}^+$ and ${\mathbf{I}}_{\beta,0}^f$ are equal and depend continuously on $\beta$ throughout $[0,\beta_c)\cup(\beta_c,\infty)$. In fact, [@MR4089494 Theorem 1] together with the uniqueness of the Gibbs measure in non-zero external field [@MR1684757 Section 3.2] imply more generally that the plus Ising measure ${\mathbf{I}}_{\beta,h}^+$ depends continuously on $(\beta,h)$ throughout $[0,\infty)^2 \setminus \{(\beta_c,0)\}$ for every amenable transitive weighted graph. Combining this result with \[thm:main\_continuity\], we deduce that this conclusion holds for *all* unimodular transitive weighted graphs, and in particular for all Cayley graphs.
\[cor:general\_continuity\] Let $G=(V,E,J)$ be an infinite, connected, transitive, unimodular weighted graph. Then the plus Ising measure ${\mathbf{I}}_{\beta,h}^+$ is a weakly-continuous function of $(\beta,h)$ on $[0,\infty)^2 \setminus \{(\beta_c,0)\}$.
We show in \[subsec:discontinuity\] that there exist nonamenable Cayley graphs for which the *free* Ising measure ${\mathbf{I}}^f_{\beta,0}$ is weakly *discontinuous* at some $\beta>\beta_c$. Thus, \[thm:main\_continuity,cor:general\_continuity\] cannot be extended to the free Ising measure in general.
\[thm:main,thm:main\_continuity\] have various consequences for the Ising model on transitive nonamenable *planar* graphs with transitive dual, which are discussed in \[subsec:planar\]. In particular, applying the results of [@MR1894115], we obtain that for any such graph there is a non-trivial interval of $\beta$ for which the free and plus Ising measures are distinct.
The proofs of \[thm:main,thm:main\_continuity\] are effective, and can be used to give explicit estimates on the constants $C$ and $\delta$ depending only on a few important parameters associated to the graph, such as the spectral radius and the value of $\beta_c$. It is strongly believed that the Ising model on any transitive nonamenable graph should be governed by the mean-field critical exponents $$\langle \sigma_o \rangle_{\beta_c,h}^+ \asymp h^{1/3} \qquad \text{ and } \qquad \langle \sigma_o \rangle_{\beta_c+{\varepsilon},0}^+ \asymp {\varepsilon}^{1/2}.$$ See [@MR1833805; @MR857063] for further discussion. It seems unlikely that our methods can be used to establish this conjecture, and the exponent $\delta$ that we obtain will be very small in general. See [@1808.08940] for a detailed discussion of related issues in the context of Bernoulli percolation.
All the results of this paper should generalize unproblematically to *quasi-transitive* weighted graphs. We restrict attention to the transitive case to clarify the exposition.
Overview of previous work {#subsec:intro_overview}
-------------------------
In this section we outline previous work on critical statistical mechanics models beyond $\Z^d$, describing in particular the strengths and limitations of existing methods in the context of the Ising model. We also take the opportunity to define the random cluster model and briefly explain its connection to the Ising model via the Edwards–Sokal coupling [@edwards1988generalization].
*Bernoulli bond percolation* is by far the most-studied statistical mechanics model outside of the Euclidean context, with an extensive literature stemming from the seminal 1996 work of Benjamini and Schramm [@bperc96]; see [@LP:book Chapters 7 and 8] and references therein for background. The study of the Ising model and of Bernoulli percolation are closely analogous, and techniques developed to study one model can often (but not always) be applied to study the other. The analogue of \[thm:main\_simple\] for Bernoulli percolation was established in the milestone work of Benjamini, Lyons, Peres, and Schramm [@BLPS99b], who proved that critical percolation on any unimodular transitive graph has no infinite clusters almost surely. This result was extended to transitive graphs of *exponential growth* by the author [@Hutchcroft2016944]. More recently, a new and more quantitative method of proof was developed in [@1808.08940], which allowed us to prove in particular that the the tail of the volume of the cluster of the origin in critical percolation satisfies power-law upper bounds on any unimodular transitive graph of exponential growth. These methods were pushed further to handle certain graphs of *subexponential* volume growth in joint work with Hermon [@HermonHutchcroftIntermediate].
The methods of both [@BLPS99b] and [@1808.08940] are not particularly specific to Bernoulli percolation and can both be modified to establish various more general results. The proof of [@BLPS99b], which relies only on soft properties of percolation, can be generalized to show in particular that if $G=(V,E)$ is a unimodular nonamenable transitive graph and $(\omega_p)_{p\in [0,1]}$ is a family of random subsets of $E$ such that
1. The law of $(\omega_p)_{p\in [0,1]}$ is invariant and ergodic under the automorphisms of $G$,
2. $\omega_p$ is contained in $\omega_{p'}$ for every $p' \geq p$ almost surely,
3. $\omega_p$ is insertion-tolerant for each $p>0$, and
4. $\omega_p = \lim_{{\varepsilon}\downarrow 0} \omega_{p-{\varepsilon}}$ almost surely for each $p>0$
then the set $\{p\in [0,1] : \omega_p$ has no infinite clusters almost surely$\}$ is a closed interval [@LP:book Theorem 8.23]. See also [@AL07] for extensions of the results of [@BLPS99b] to the setting of *unimodular random rooted graphs*. The minimum hypotheses needed to apply the methods of [@1808.08940] are a little less clear. One rather general general statement that these methods can be used to prove is as follows: Let $G=(V,E)$ be a unimodular transitive graph of exponential growth and suppose that $(\mu_n)_{n \geq 1}$ is a sequence of automorphism-invariant probability measures on $\{0,1\}^E$ converging weakly to some probability measure $\mu$. Suppose further that the following hold:
1. each of the measures $\mu_n$ is positively associated,
2. the expected size of the cluster of the origin in $\mu_n$ is finite for each $n\geq 1$, and
3. each of the measures $\mu_n$ may be written as ‘percolation in random environment’, where the conditional probability of an edge being open given the environment is bounded away from zero by some positive constant that does not depend on $n$.
Then $\mu$ is supported on configurations in which there are no infinite clusters, and the tail of the volume of the cluster of the origin in $\mu$ satisfies a power-law upper bound. If $G$ is taken to be nonamenable, the hypothesis II.ii above may be replaced with the weaker assumption that each of the measures $\mu_n$ is supported on configurations with no infinite clusters. See \[sec:free\_energy\] for various precise statements. While it may seem that these conditions are much more restrictive than the conditions I.i–iv required to implement the proof of [@BLPS99b], we note that, crucially, we do *not* require a monotone coupling of the measures $(\mu_n)_{n\geq 1}$. Both methods can, with work, be applied to the *random cluster model* (a.k.a. FK-percolation) with $q\geq 1$ and *free boundary conditions*; This was done for the method of [@BLPS99b] by Häggström, Jonasson, and Lyons [@MR1913108; @MR1894115]. Let us now quickly recall the definition of this model and its relation to the Ising model, referring the reader to e.g. [@1707.00520; @GrimFKbook] for further background. Let $G=(V,E,J)$ be a weighted graph with $V$ finite so that $\sum_{e\in E} J_e <\infty$. ($E$ may be finite or infinite.) For each $q>0$ and $\beta,h\geq 0$, we define the **random cluster measure** $\phi_{q,\beta,h}=\phi_{G,q,\beta,h}$ to be the purely atomic probability measure on $\{0,1\}^E \times \{0,1\}^V$ given by[^2] $$\phi_{q,\beta,h}(\{\omega\}) \propto q^{k(\omega)} \prod_{e\in E} (e^{2\beta J_e}-1)^{\omega(e)}\prod_{v\in V} (e^{2\beta h}-1)^{\omega(v)}$$ for each $\omega \in \{0,1\}^E\times\{0,1\}^V$, where $k(w)$ is the number of clusters (i.e., connected components) of the subgraph of $G$ spanned by $\{e:\omega(e)=1\}$ that do not contain a vertex $v$ with $\omega(v)=1$. Note that if $q=1$ and $J_e \equiv 1$ then the measure $\phi_{q,\beta,0}$ is simply the law of Bernoulli bond percolation on $G$ with retention probability $p=(e^{2\beta}-1)/e^{2\beta}=1-e^{-2\beta}$.
Now suppose that $G=(V,E,J)$ is an infinite connected weighted graph, let $(V_n)_{n\geq 1}$ be an exhaustion of $G$, and let $(G_n)_{n\geq 1}$ and $(G_n^*)_{n\geq 1}$ be defined as in \[subsec:intro\_definitions\]. For each $q\geq 1$ and $\beta,h\geq 0$ we define the **free** and **wired** random cluster measures $\phi^f_{q,\beta,h}$ and $\phi^w_{q,\beta,h}$ to be $$\begin{aligned}
\phi^f_{q,\beta,h}=\phi^f_{G,q,\beta,h}:= {\mathop{\operatorname{w-lim}}}_{n\to\infty} \phi_{G_n,q,\beta,h} \qquad \text{ and } \qquad \phi^w_{q,\beta,h} = \phi^w_{G,q,\beta,h}:= {\mathop{\operatorname{w-lim}}}_{n\to\infty}\phi_{G^*_n,q,\beta,h}.
$$ Both of these weak limits exist and do not depend on the choice of exhaustion. Indeed, it is a consequence of the Holley inequality [@1707.00520 Theorem 1.6 and Proposition 1.8] that if $A\subseteq E \cup V$ is finite and $n_0$ is such that every vertex in $A$ and every edge touching $A$ belongs to $V_{n_0}$ then $$\begin{aligned}
\phi^f_{q,\beta,h}(\omega(x)=1 \text{ for every $x\in A$}) &= \sup_{n\geq n_0} \phi_{G_n,q,\beta,h}(\omega(x)=1 \text{ for every $x\in A$}) \qquad \text{and}\\
\phi^w_{q,\beta,h}(\omega(x)=1 \text{ for every $x\in A$}) &= \inf_{n\geq n_0} \phi_{G_n^*,q,\beta,h}(\omega(x)=1 \text{ for every $x\in A$})\end{aligned}$$ for every $\beta,h \geq 0$, so that $\phi^\#_{q,\beta,h}(\omega(x)=1 \text{ for every $x\in A$})$ depends on $(\beta,h)$ lower semicontinuously when $\#=f$ and upper semicontinuously when $\# = w$. A further consequence of the Holley inequality is that $\phi_{q,\beta,h}^w$ stochastically dominates $\phi_{q,\beta,h}^f$ for each fixed $\beta,h\geq 0$ and $q\geq 1$ and that $\phi_{q_1,\beta_1,h_1}^\#$ stochastically dominates $\phi_{q_2,\beta_2,h_2}^\#$ for each $\# \in \{f,w\}$, $q_2 \geq q_1 \geq 1$, $0 \leq \beta_2 \leq \beta_1$, and $0 \leq h_2 \leq h_1$ [@GrimFKbook Theorem 3.21]. Putting these two facts together, it follows that $\phi^f_{q,\beta,h}$ is weakly left-continuous in $\beta$ and that $\phi^w_{q,\beta,h}$ is weakly right-continuous in $\beta$ [@GrimFKbook Proposition 4.28].
For each $q \geq 1$ and $\# \in \{f,w\}$ we define the critical inverse temperature $\beta_c^\#(q)= \sup\{ \beta\geq 0: \phi^\#_{q,\beta,0}$ is supported on configurations with no infinite clusters$\}$. In \[sec:free\_energy\] we extend the analysis of [@1808.08940] to the random cluster model, proving the following.
\[thm:free\_random\_cluster\] Let $G=(V,E)$ be a locally finite, transitive, unimodular graph of exponential growth, and let $q \geq 1$. Then there exist positive constants $\delta$ and $C$ such that $$\phi_{q,\beta_c^f(q),0}^f(|K_o| \geq n) \leq C n^{-\delta} \qquad \text{for every $n\geq 1$.}$$
The Ising model and the $q=2$ random cluster model, a.k.a. the **FK-Ising model**, are related by the **Edwards–Sokal coupling** [@GrimFKbook Section 1.4]. We describe this coupling in the wired/plus case, which is the only case we will use; a similar coupling holds in the free case. Let $G=(V,E,J)$ be an infinite connected weighted graph with $V$ finite, let $\beta,h\geq 0$, and let $\omega$ a random variable with law $\phi_{2,\beta,h}^w$. Given $\omega$, we assign a value of $+1$ or $-1$ to each *cluster* of $\omega$ as follows:
1. If the cluster is infinite or intersects the set $\{v:\omega(v)=1\}$, we assign it the value $+1$.
2. Otherwise, the cluster is finite and does not intersect the set $\{v:\omega(v)=1\}$, in which case we assign it a value from $\{-1,+1\}$ uniformly at random, where the choices of signs for different clusters are made independently given $\omega$.
Finally, let $\sigma_v$ be equal to the value assigned to the cluster of $v$ for each $v\in V$. Then the resulting random variable $\sigma=(\sigma_v)_{v\in V}$ has law ${\mathbf{I}}^+_{\beta,h}$. It follows in particular that $m^*(\beta_c)=0$ if and only if the *wired* FK-Ising model has no infinite clusters at $\beta_c$ [@MR3752129 Exercise 3.77]. (In \[prop:betafbetaw\] we show that $\beta_c=\beta_c^f(2)=\beta_c^w(2)$ on any transitive weighted graph.)
Unfortunately, the methods of [@BLPS99b] and [@1808.08940] cannot be used to say anything about the *wired* random cluster model at criticality since the measure $\phi_{q,\beta}^w$ need not be weakly left-continuous in $\beta$. This is not merely a technical obstacle, as it is expected that the random cluster model undergoes a *discontinuous* phase transition on nonamenable transitive graphs when $q>2$. See [@bollobas1996random; @laanait1991interfaces; @duminil2016discontinuity; @ray2019short; @MR1894115; @MR1833805] and references therein for related results. As such, any proof of continuity of the phase transition for the Ising model or FK-Ising model must use some property that distinguishes between the cases $q=2$ and $q>2$, and it is unclear how this could be done within the frameworks of [@BLPS99b] or [@1808.08940].
A similar obstacle was overcome in the *amenable* setting by Aizenman, Duminil-Copin, and Sidoravicius [@MR3306602], who proved in particular that if $G$ is an amenable transitive graph such that the *free* FK-Ising model on $G$ has no infinite clusters at criticality, then the free and wired FK-Ising models coincide at criticality and the spontaneous magnetization of the Ising model vanishes at criticality. It will be informative for later developments for us to briefly outline their argument, which was based on the analysis of the *random current[^3] model*. This is an alternative graphical representation of the Ising model that was introduced by Griffiths, Hurst, and Sherman [@griffiths1970concavity] and developed extensively by Aizenman [@MR678000]; see \[subsec:random\_currents\] for further background and definitions. Although the random current model is in many ways a much less well-behaved object than FK-percolation (it is not positively associated or deletion-tolerant, but is insertion tolerant), it has many very interesting features which, roughly speaking, allow it to communicate information between Ising models with different parameters and boundary conditions. In particular, it is established in [@MR3306602] that the Ising model on a transitive graph undergoes a continuous phase transition if and only if a certain system of two independent random currents on the graph has no infinite clusters at the critical temperature, where one random current is taken with free boundary conditions and the other with wired. On the other hand, the probability that any two vertices are connected in this duplicated system of random currents is bounded by the probability that they are connected in the *free* FK-Ising model.
To conclude, the authors of [@MR3306602] applied the classical theorem of Burton and Keane [@burton1989density] (as generalized by Gandolfi, Keane, and Newman [@gandolfi1992uniqueness]) to deduce that, *in the amenable case*, the duplicated system of random currents has at most one infinite cluster. Thus, the existence of an infinite cluster is incompatible with connection probabilities between the origin and a distant vertex tending to zero, and the proof of their theorem may easily be concluded. Note that the last part of this argument is very specific to the amenable setting and cannot be used in the nonamenable case where the Burton–Keane theorem does not hold. Note also that this proof is not quantitative, and does not lead to any explicit control of the magnetization near $\beta_c$.
Overview of the proof and applications to the random cluster model
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In order to prove \[thm:main\_simple,thm:main,thm:main\_continuity\], we develop a new variation on the methods of [@1808.08940] that can be applied to certain models that are *not* positively associated. More specifically, we argue that this hypothesis may be replaced by the assumption that the measures in question have a *spectral gap*; see \[subsec:spectral\_background\] for definitions. The fact that the random cluster measure on a nonamenable transitive graph has such a spectral gap follows from the results of [@MR1913108]. This new method also allows us to study the *finite* clusters in supercritical models, leading to the following theorem which is new even in the case of Bernoulli percolation. (Note that the collection of finite clusters in the random cluster model is itself an automorphism-invariant percolation model, but is not positively associated in the supercritical regime.)
\[thm:finite\_clusters\] Let $G=(V,E,J)$ be an infinite transitive nonamenable unimodular weighted graph and let $q \geq 1$. Then there exist positive constants $C$ and $\delta$ such that $$\phi_{q,\beta,h}^\#(n \leq |K_o| < \infty) \leq C n^{-\delta}$$ for every $n\geq 1$, $\beta,h \geq 0$, and $\#\in \{f,w\}$.
In order to prove our main theorems, we argue that this new method can also be applied to obtain uniform polynomial tail bounds on the finite clusters in a certain variation on the double random current model in which the two currents can have different values of $\beta$ and $h$. See \[prop:finite\_clusters\_mismatched\] for a precise statement. To do this we must first bound the spectral radius of the random current model, which we do in \[thm:spectralradius\]. In \[subsec:mismatched,subsec:mainproof\] we use our new construction of double random currents with mismatched temperatures to develop a quantitative version of the arguments of [@MR3306602]. This lets us deduce the Hölder continuity claimed in \[thm:main\_continuity\] from the uniform control of finite clusters for the double random current and FK-Ising models provided by \[prop:finite\_clusters\_mismatched\] and \[thm:finite\_clusters\]. Once this is done, \[thm:main\_simple,thm:main\] are easily deduced from \[thm:main\_continuity\].
Our proof also yields the following analogue of \[thm:main\_continuity\] for the FK-Ising model.
\[thm:main\_continuity\_FK\] Let $G=(V,E,J)$ be a connected, nonamenable, transitive, unimodular weighted graph. Then the wired FK-Ising measure $\phi_{2,\beta,h}^w$ is a weakly-continuous function of $(\beta,h)\in [0,\infty)^2$. Moreover, there exists $\delta>0$ such that if $F:\{0,1\}^{E \cup V} \to \R$ is any function depending on at most finitely many edges and vertices of $G$ then $\phi_{2,\beta,h}^w[F(\omega)]$ is a locally $\delta$-Hölder continuous function of $(\beta,h)\in [0,\infty)^2$.
We show in \[subsec:discontinuity\] that the *free* FK-Ising measure $\phi_{2,\beta,0}^f$ can be weakly *discontinuous* in $\beta$ under the same hypotheses.
Corollaries for the percolation free energy
-------------------------------------------
We now briefly discuss an interesting application of \[thm:finite\_clusters\] to Bernoulli percolation. Let $G$ be a connected, locally finite, transitive graph, and write ${\mathbf E}_p$ for expectations taken with respect to Bernoulli-$p$ bond percolation on $G$. For each $p\in [0,1]$, the **free energy** (a.k.a. open-clusters-per-vertex) $\kappa(p)$ of Bernoulli-$p$ percolation is defined to be $$\kappa(p) := {\mathbf E}_p \frac{1}{|K_o|}.$$ It has historically been a problem of great interest, motivated in part by the non-rigorous work of Sykes and Essam [@sykes1964exact], to determine the location and nature of the singularities of this function. See [@grimmett2010percolation Chapter 4] for further background. In the nonamenable context, it follows from the results of [@HermonHutchcroftSupercritical] that $\kappa(p)$ is an analytic function of $p$ on $[0,p_c)\cup(p_c,1]$. See [@georgakopoulos2018analyticity; @georgakopoulos2020analyticity] for analogous results in the Euclidean context. On the other hand, the nature of the singularity at $p_c$ (and indeed the question of whether or not there is such a singularity) remains open, even in the nonamenable context:
\[conj:free\_energy\] Let $G$ be an infinite, connected, locally finite, transitive graph with $p_c<1$. Then the percolation free energy $\kappa(p)$ is twice differentiable but *not* thrice differentiable at $p_c$.
See [@grimmett2010percolation Chapter 4 and Proposition 10.20] for an overview of progress on this conjecture. Our results lead to the following partial progress on this conjecture in the nonamenable setting.
Let $G$ be a connected, locally finite, transitive nonamenable graph. Then the percolation free energy $\kappa(p)$ is twice continuously differentiable at $p_c$.
Aizenman, Kesten, and Newman [@MR901151 Proposition 3.3] proved that if there exists ${\varepsilon}>0$ such that the truncated $\log^{1+{\varepsilon}}$-moment ${\mathbf E}_p \left[\mathbbm{1}(|K_o| <\infty)\log^{1+{\varepsilon}} |K_o|\right]$ is bounded in a neighbourhood of $p_c$ then the free energy $\kappa(p)$ is twice continuously differentiable in a neighbourhood of $p_c$. The $q=1$ case of \[thm:finite\_clusters\] (see also \[thm:finite\_percolation\]) is easily seen to imply that this criterion holds when $G$ is unimodular. On the other hand, it follows from [@hutchcroft2020slightly Theorem 1.1] that this criterion holds whenever critical percolation on $G$ satisfies the *$L^2$ boundedness condition*, which is always the case when $G$ is nonunimodular by the results of [@Hutchcroftnonunimodularperc].
Background
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Unimodularity and the mass-transport principle {#subsec:unimodularity_background}
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We now briefly review the notions of unimodularity and the mass-transport principle. See e.g. [@LP:book Chapter 8] and [@MR1082868] for further background.
Let $\Gamma$ be a locally compact Hausdorff topological group. Recall that a Radon measure $\nu$ on $\Gamma$ is said to be a **left Haar measure** if it is non-zero, locally finite, and left-invariant in the sense that $\nu(\gamma A)=\nu(A)$ for every Borel set $A \subseteq \Gamma$ and $\gamma \in \Gamma$. Similarly, $\nu$ is said to be a **right Haar measure** if it is locally finite and right-invariant in the sense that $\nu(A \gamma) = \nu(A)$ for every Borel set $A \subseteq \Gamma$ and $\gamma \in \Gamma$. Haar’s Theorem states that every locally compact Hausdorff topological group has a left Haar measure that is unique up to multiplication by a positive scalar. (Similar statements hold for right Haar measures by symmetry.) The group $\Gamma$ is said to be **unimodular** if its left Haar measures are also right Haar measures. Note that every countable discrete group is unimodular since the counting measure is both left- and right-invariant.
Let $G=(V,E,J)$ be a transitive weighted graph, and let ${\operatorname{Aut}}(G)$ be the group of automorphisms of $G$, which is a locally compact Hausdorff topological group when equipped with the product topology (i.e., the topology of pointwise convergence). The weighted graph $G$ is said to be **unimodular** if ${\operatorname{Aut}}(G)$ is unimodular. It follows from [@LP:book Proposition 8.12] that if $\Gamma$ is a closed, transitive, unimodular subgroup of ${\operatorname{Aut}}(G)$ then every intermediate closed subgroup $\Gamma \subseteq \Gamma' \subseteq {\operatorname{Aut}}(G)$ is unimodular also. In particular, if $G$ is a Cayley graph of a finitely generated group $\Gamma$ then $\Gamma$ can also be thought of as a discrete unimodular transitive subgroup of ${\operatorname{Aut}}(G)$, so that $G$ is unimodular [@MR1082868]. Note that if $\Gamma \subseteq {\operatorname{Aut}}(G)$ is a closed subgroup of ${\operatorname{Aut}}(G)$ then the stabilizer ${\operatorname{Stab}}(v)=\{\gamma \in \Gamma : \gamma v = v\}$ of each vertex $v$ of $G$ is a compact subgroup of $\Gamma$, and in particular has finite Haar measure. If $\Gamma$ is transitive then the subgroups $\{{\operatorname{Stab}}(v) :v\in V\}$ are all conjugate to each other, so that if $\Gamma$ is unimodular and $\nu$ is a Haar measure on $\Gamma$ then $\nu({\operatorname{Stab}}(u))=\nu({\operatorname{Stab}}(v))$ for every $u,v\in V$. It follows that if $\Gamma$ is transitive and unimodular then there exists a unique Haar measure $\nu$ such that $\nu({\operatorname{Stab}}(v))=1$ for every $v\in V$, which we call the **unit Haar measure**.
Let $G=(V,E,J)$ be a connected transitive weighted graph, let $\Gamma$ be a closed transitive unimodular subgroup of ${\operatorname{Aut}}(G)$, and let $o$ be an arbitrary root vertex of $G$. The **mass-transport principle** [@LP:book Eq. 8.4] states that for every function $F:V^2\to [0,\infty]$ that is diagonally-invariant in the sense that $F(\gamma u, \gamma v)=F(u,v)$ for every $u,v \in V$ and $\gamma \in \Gamma$, we have that $$\label{eq:MTP}
\sum_{v\in V} F(o,v) = \sum_{v\in V} F(v,o).$$ As in [@1808.08940], we will also use a version of the mass-transport principle indexed by oriented edges rather than vertices. Write $E^\rightarrow$ for the set of oriented edges of $G$, where an oriented edge $e$ is oriented from its tail $e^-$ to its head $e^+$ and has reversal $e^\leftarrow$. Let $\eta$ be chosen at random from the set of oriented edges of $G$ emanating from $o$ with probability proportional to $J_e$, so that $\eta$ has the law of the first edge crossed by a random walk started at $o$. Then for every function $F:E^\rightarrow \times E^\rightarrow \to [0,\infty]$ that is diagonally-invariant in the sense that $F(\gamma e_1, \gamma e_2)=F(e_1,e_2)$ for every $e_1,e_2\in E^\rightarrow$ and $\gamma \in \Gamma$, we have that $$\E\sum_{e\in E^\rightarrow} J_e F(\eta,e) = \E\sum_{e\in E^\rightarrow} J_e F(e,\eta),
\label{eq:edgeMTP}$$ where the expectation is taken over the random oriented edge $\eta$. This equality follows by applying to the function $\tilde F(u,v) = \sum_{e_1^-=u} \sum_{e_2^-=v} J_{e_1} J_{e_2} F(e_1,e_2)$. Moreover, the equality also holds for *signed* diagonally-invariant functions $F:E^\rightarrow\times E^\rightarrow \to\R$ satisfying the absolute integrability condition $$\label{eq:integrability}
\E\sum_{e\in E^\rightarrow}J_e |F(\eta,e)|<\infty.$$ This follows by applying separately to the positive and negative parts of $F$, which are defined by $F^+(e_1,e_2)=0\vee F(e_1,e_2)$ and $F^-(e_1,e_2)= 0\vee (-F(e_1,e_2))$.
The spectral theory of automorphism-invariant processes {#subsec:spectral_background}
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In this section we review some notions from the spectral theory of group-invariant processes that will be used in the proofs of our main theorems. Everything we discuss in this section is likely to be known to some experts, but we have given a fairly detailed and self-contained account since we expect it to be unfamiliar to many of our readers and the required material is spread over several papers and not always written in the form that we wish to apply it. Good resources for further background on this material from a probabilistic perspective include [@MR3729654; @MR1082868; @MR2825538].
We begin by quickly recalling the definition of the spectral radius of a weighted graph. Let $G=(V,E,J)$ be a connected, transitive weighted graph. The random walk on $G$ is defined to be the reversible Markov chain on $V$ which, at each time step, chooses a random oriented edge $e$ emanating from its current location with probability proportional to $J_e$ and then crosses this edge, independently of everything it has done previously. We write ${\mathbf E}_v$ for the law of the random walk $X=(X_n)_{n\geq 0}$ on $G$ started at $v$. The **Markov operator** $P:L^2(V)\to L^2(V)$ is defined by $$Pf(v)={\mathbf E}_v f(X_1)$$ for each $v\in V$. The operator $P$ is clearly self-adjoint, while Jensen’s inequality implies that $P$ is bounded with operator norm $\|P\|\leq 1$. It is a well-known theorem of Kesten [@Kesten1959b] (see also [@LP:book Chapter 6.2]) that the strict inequality $\|P\|<1$ holds if and only if $G$ is nonamenable, and moreover that $$\|P\| = \lim_{n\to\infty} p_{2n}(v,v)^{1/2n} =\limsup_{n\to\infty} p_n(u,v)^{1/n}$$ for every $u,v\in V$. The norm $\|P\|$ is known as the **spectral radius** of $G$ and is also denoted $\rho(G)$.
**Random walks on graphs vs. random walk on groups.** When $G$ is a Cayley graph of a finitely generated group $\Gamma$, we can always think of the random walk on $G$ as a random walk on the group and hence as a random walk on (a subgroup of) ${\operatorname{Aut}}(G)$. This duality between random walks on graphs and on their automorphism groups comes with some subtleties, however. Let $(Z_n)_{n\geq 0}$ be a sequence of i.i.d. $\Gamma$-valued random variables such that $Z_n$ is distributed as the first step of a random walk on $G$ started from the identity and define $X=(X_n)_{n\geq0}$ by $X_0=\mathrm{id}$ and $X_n=X_{n-1}Z_n$ for every $n\geq 1$ then $X$ is distributed as a random walk on $G$ started at the identity. Moreover, if we let $\Gamma$ act on $L^2(V) \cong L^2(\Gamma)$ by $\gamma f(v) = f(\gamma^{-1} v)$ then we can define a bounded self-adjoint operator $\hat P$ on $L^2(V)$ by $$\hat P f(v) = \E \left[X_1 f(v)\right] = \E\left[ f(X_1^{-1} v)\right].$$ It follows by induction on $n \geq 1$ that $$\hat P^n f(v) = \E\left[ Z_1 \hat P^{n-1} f(v) \right] = \E\left[ Z_1 \E \left[ Z_2 \cdots Z_n f(v)\right]\right] = \E\left[ X_n f(v)\right]$$ for every $v\in V$, $n\geq 1$, and $f\in L^2(V)$, where we used that $X_{n-1}=Z_1\cdots Z_{n-1}$ and $Z_2 \cdots Z_n$ have the same distribution in the central equality. Note however that $X_n^{-1}v$ does *not* in general have the same distribution as the $n$th step of a random walk on $G$ started at $v$, so that the operators $P$ and $\hat P$ are not generally the same. Nevertheless, the *spectral radii* $\|P\|$ and $\|\hat P\|$ are always the same, for the simple reason that $\hat P$ is the Markov operator for the random walk on the *left* Cayley graph of $\Gamma$ with respect to the same generating set as $G$, which is isomorphic to $G$. More concretely, if we consider the isometric involution $\operatorname{Inv}$ of $L^2(V) \cong L^2(\Gamma)$ defined by $$\begin{aligned}
\operatorname{Inv} f(\gamma) = f(\gamma ^{-1}) \qquad \text{ for all $f\in L^2(\Gamma)$ and $\gamma \in \Gamma$}\end{aligned}$$ then $\hat P = \operatorname{Inv} P \operatorname{Inv}$ and $P = \operatorname{Inv} \hat P \operatorname{Inv}$, so that all the spectral properties of the two operators $P$ and $\hat P$ are the same.
It will be useful to have a similar duality in place for general transitive graphs. Much of this duality was developed by Soardi and Woess [@MR1082868], although we will follow some slightly different conventions. Let $G=(V,E,J)$ be a connected transitive weighted graph, let $o$ be a fixed root vertex of $G$, and let $\Gamma \subseteq {\operatorname{Aut}}(G)$ be a unimodular closed transitive group of automorphisms. As above, $\Gamma$ acts on $L^2(V)$ by $$\gamma f(v) = f(\gamma^{-1}v) \text{ for every $\gamma\in \Gamma$, $f\in L^2(V)$, and $v\in V$.}$$ Let $X=(X_n)_{n\geq 0}$ be a random walk on $G$ started at $o$. Conditional on $X$, let $\hat X=(\hat X_{n})_{n\geq 0}$ be drawn independently at random from the normalized Haar measures on the compact sets of automorphisms $\{\gamma \in \Gamma : \gamma o = X_n\}$. Meanwhile, let $\nu$ be the law of $\hat X_1$, let $(Z_n)_{n\geq 1}$ be i.i.d. random variables each with law $\nu$, and let $\hat Y = (\hat Y_n)_{n\geq 0}$ be the random walk on $\Gamma$ defined by $\hat Y_0=\mathrm{id}$ and $\hat Y_n = \hat Y_{n-1} Z_n = Z_1 \cdots Z_n$ for every $n\geq 1$. The following lemma is an easy consequence of the fact that the Haar measure on the unimodular group $\Gamma$ is both left- and right-invariant.
\[lem:groupsvsgraphs\] Let $G=(V,E,J)$ be a connected transitive weighted graph, let $o$ be a fixed root vertex of $G$, and let $\Gamma \subseteq {\operatorname{Aut}}(G)$ be a unimodular closed transitive group of automorphisms. Then the two processes $\hat X$ and $\hat Y$ we have just defined have the same distribution.
Thus, we may think of $\hat X$ as a random walk on $\Gamma \subseteq {\operatorname{Aut}}(G)$. We define the associated Markov operator $\hat P : L^2(\Gamma) \to L^2(\Gamma)$ by $$\hat P f(\gamma) = \E \left[ Z_1 f(\gamma)\right] = \E\left[f( Z_1^{-1}\gamma)\right]$$ for every $f\in L^2(\Gamma)$ and a.e. $\gamma\in \Gamma$. Similarly to above, $\hat P$ is bounded, self-adjoint, and satisfies $$\hat P^n f(\gamma) = \E \left[Z_1 \hat P_{n-1} f(\gamma)\right] = \E\left[ Z_1 \E \left[ Z_2 \cdots Z_n f(\gamma)\right]\right] = \E\left[ \hat X_n f(\gamma)\right]$$ for every $n\geq 0$, $f\in L^2(V)$, and a.e. $\gamma \in \Gamma$. The Markov operator $\hat P$ can be related to the usual Markov operator $P$ as follows: Let $\lambda$ be the unit Haar measure on $\Gamma$ and consider the three operators $$\begin{aligned}
\operatorname{Proj} : L^2(\Gamma) &\to L^2(V) \qquad &\operatorname{Proj} f(v) &= \int_\Gamma f (\gamma) \mathbbm{1}(\gamma o = v) \dif \lambda (\gamma),\\
\operatorname{Inj} : L^2(V) &\to L^2(\Gamma) \qquad & \operatorname{Inj} f(\gamma) &= f(\gamma o),
&\text{ and }\\
\operatorname{Inv} : L^2(\Gamma) &\to L^2(\Gamma) \qquad & \operatorname{Inv} f(\gamma) &= f(\gamma^{-1}),\end{aligned}$$ each of which is easily seen to be bounded with norm $1$. The self-adjoint operator $\operatorname{Inv}$ is an isometric involution of $L^2(\Gamma)$, while $\operatorname{Proj}$ and $\operatorname{Inj}$ are adjoints of each other. The two Markov operators $P$ and $\hat P$ satisfy the congruence-type relation $$\label{eq:congruence}
P = \operatorname{Proj} \operatorname{Inv} \hat P \operatorname{Inv} \operatorname{Inj} \qquad \text{ and } \qquad \hat P = \operatorname{Inv} \operatorname{Inj} P \operatorname{Proj} \operatorname{Inv},$$ which implies in particular that $\|P\|=\|\hat P\|=\rho(G)$. The equation , which is easily verified directly, is essentially equivalent to [@MR1082868 Proposition 1].
**Spectral radii of automorphism-invariant processes.** We now define the spectral radius of an *automorphism-invariant stochastic process* on $G$. Let $G=(V,E,J)$ be a connected transitive weighted graph, let $o$ be a fixed root vertex of $G$, and let $\Gamma \subseteq {\operatorname{Aut}}(G)$ be a unimodular closed transitive group of automorphisms. Let $X=(X_n)_{n\geq 0}$ be a random walk started at $o$ on $G$ and let $\hat X= (\hat X_n)_{n\geq 0}$ be the associated random walk on $\Gamma$ as above. Let $\mathbb{X}_V$ and $\bbX_E$ be Polish spaces, which we will usually take to be either $\{\emptyset\}$, $\{0,1\}$, $[0,1]$, or $\N_0 = \{0,1,2,\ldots\}$, and let $\Omega$ be the product space[^4] $\Omega=\bbX_V^V \times \bbX^E_E$. The group $\Gamma$ acts on $\Omega$ by $$\gamma \omega(x) = \omega(\gamma^{-1} x) \qquad \text{ for each $\gamma \in \Gamma$, $\omega \in \Omega$, and $x\in V \cup E$,}$$ and a probability measure $\mu$ on $\Omega$ is said to be **$\Gamma$-invariant** if $\mu(A)=\mu(\gamma^{-1}A)$ for every Borel set $A \subseteq \Omega$ and $\gamma \in \Gamma$. Given an automorphism-invariant probability measure $\mu$ on $\Omega$, we define the Markov operator $\hat P_\mu$ on $L^2(\Omega,\mu)$ by $$\hat P_\mu f(\omega) = {\mathbf E}\left[\hat X_1 f( \omega) \right] = {\mathbf E}\left[f(\hat X_1^{-1} \omega) \right] \qquad \text{ for every $f\in L^2(\Omega,\mu)$ and $\omega\in \Omega$,}$$ which is bounded and self-adjoint with norm $\|\hat P_\mu\| = 1$. Let $L^2_0(\Omega,\mu)=\{f \in L^2(\Omega,\mu): \mu(f)=0\}$. The Markov operator $\hat P_\mu$ fixes $L_0^2(\Omega,\mu)$, and can therefore also be seen as a bounded self-adjoint operator on $L^2_0(\Omega,\mu)$. We define $$\rho(\mu) = \rho(\mu,\Gamma)= \sup\left\{ \frac{\|\hat P_{\mu} f\|_2}{\|f\|_2} : f\in L^2_0(\Omega,\mu) \setminus \{0\} \right\} =
\sup\left\{ \frac{|(\hat P_{\mu} f,f)|}{|(f,f)|} : f\in L^2_0(\Omega,\mu) \setminus \{0\} \right\}$$ to be the spectral radius of the Markov operator $\hat P_{\mu}$ on $L^2_0(\Omega,\mu)$. We say that $\mu$ has a **spectral gap** if $\rho(\mu)<1$.
The spectral radius may also be expressed probabilistically as follows. Let $\bbX_V$ and $\bbX_E$ be Polish spaces and let $\varphi=(\varphi_x)_{x\in V \cup E}$ be a random variable taking values in $\Omega= \bbX_V^V \times \bbX_E^E$ whose law $\mu$ is $\Gamma$-invariant and let the processes $X$ and $\hat X$ and as be defined as above and independent of $\varphi$. The definition of the spectral radius may be rewritten probabilistically as $$\begin{aligned}
\label{eq:covariance}
\rho(\mu) = \rho(\mu,\Gamma) &=
\sup\left\{ \frac{\bigl|{{\mathrm{Cov}}}\bigl( F(\varphi),F(\hat X_{1}^{-1}\varphi)\bigr)\bigr|}{{{\mathrm{Var}}}(F(\varphi))}: F\in \R^\Omega, 0<{{\mathrm{Var}}}(F(\varphi)) <\infty
\right\}
$$ where we write ${{\mathrm{Var}}}$ and ${{\mathrm{Cov}}}$ for variances and covariances taken with respect to the joint law of the random variables $\varphi$ and $\hat X$.
Note that we will typically be interested in random fields that are indexed only by the edge set or by the vertex set. Such fields are easily included within this formalism by setting $\bbX_V=\{\emptyset\}$ or $\bbX_E=\{\emptyset\}$ as appropriate and setting the random field to be constantly equal to $\emptyset$ over the irrelevant indices, and we will apply the results and terminology of this section to such fields without further comment in the remainder of the paper.
\[example:FiniteClusters\] Let $G=(V,E,J)$ be a connected, transitive weighted graph, let $\Gamma$ be a closed transitive unimodular subgroup of ${\operatorname{Aut}}(G)$ and let $o$ be a fixed root vertex of $G$. Let $\mu$ be an automorphism-invariant probability measure on $\{0,1\}^E$ and let $\omega \in \{0,1\}^E$ be a random variable with law $\mu$. Let $X=(X_n)_{n\geq 0}$ be a random walk on $G$ started at $o$ and let $\hat X$ be the associated random walk on $\Gamma$, where we take $X$ and $\hat X$ to be independent of $\omega$. Letting $K_v$ be the cluster of $v$ in $\omega$ for each vertex $v$ of $G$, we have that $\hat X_k\mathbbm{1}(n\leq |K_o| < \infty)= \mathbbm{1}(n\leq |K_{X_k}|<\infty)$ for every $k,n\geq 0$, so that $$\begin{gathered}
|\P( n \leq |K_{o}|,|K_{X_{k}}| < \infty)-\P( n \leq |K_{o}|< \infty)^2| \\\leq \rho(\mu)^{k} |\P( n \leq |K_{o}| < \infty)-\P( n \leq |K_{o}|< \infty)^2| \leq \rho(\mu)^k\end{gathered}$$ for every $n,k\geq 0$ by definition of $\rho(\mu)$. This inequality will play a central role in the proofs of our main theorems.
We next discuss some useful properties of the spectral radius that we will use in the proofs of our main theorems.
**The limit formula.** We first recall some standard facts about self-adjoint operators on Hilbert spaces that will help us to compute spectral radii in examples. Let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. Cauchy-Schwarz gives that $$\| T^{n+1} x\|^4 = (T^nx,T^{n+2}x)^2 \leq \|T^n x\|^2\|T^{n+2}x\|^2$$ for every $x \in H$ and $n\geq 0$, which implies that if $x\in H$ is such that $Tx \neq 0$ then $T^n x \neq 0$ for all $n\geq 0$ and that $\|T^{n+1}x\|/\|T^n x\|$ is an increasing function of $n\geq 0$. This is easily seen to imply that $\lim_{k\to\infty} \|T^{k} x\|^{1/k}$ exists for every $x\in H$ and that $$\label{eq:norm_limit_bound}
\frac{\|T x\|}{\|x\|} \leq
\lim_{k\to\infty} \|T^{k} x\|^{1/k} \leq \|T\|$$ for every $x\in H\setminus \{0\}$. Moreover, we have by the triangle inequality that $$\lim_{k\to\infty} \Bigl\|T^k\sum_{i=1}^m a_i x_i\Bigr\|^{1/k}
\leq \lim_{k\to\infty} \left(\sum_{i=1}^m |a_i| \|T^k x_i\| \right)^{1/k} =
\max \Bigl\{ \lim_{k\to\infty} \|T^kx_i\|^{1/k} : 1\leq i \leq m, a_i \neq 0\Bigr\}$$ for every $x_1,\ldots,x_m \in H$ and $a_1,\ldots a_m \in \R$. It follows that if $A$ is a subset of $H$ with dense linear span $S(A)$ then $$\begin{aligned}
\|T\|
=\sup\Bigl\{ \lim_{k\to\infty} \|T^kx\|^{1/k} : x\in S(A)\Bigr\}
=\sup\Bigl\{ \lim_{k\to\infty} \|T^kx\|^{1/k} : x\in A\Bigr\}.
\label{eq:norm_limit_subset}\end{aligned}$$ Translating this into probabilistic notation, the formula yields in the context of that $$\begin{aligned}
\rho(\mu) &=
\sup\left\{ \lim_{k\to\infty} {{\mathrm{Cov}}}\left( F(\varphi),F(\hat X_{2k}^{-1}\varphi)\right)^{1/2k} : F\in A
\right\}
\label{eq:rho_limit_covariance}\end{aligned}$$ for every set of functions $A \subseteq L^2(\Omega,\mu)$ that has dense linear span in $L^2(\Omega,\mu)$.
**Spectral radii of i.i.d. processes.** Let $G=(V,E,J)$ be a connected transitive weighted graph and let $\Gamma \subseteq {\operatorname{Aut}}(G)$ be a closed group of automorphisms. A **Bernoulli process** on $G$ is a family of independent random variables $(\varphi_x)_{x\in E \cup V}$ taking values in a Polish space of the form $\bbX_V^V \times \bbX_E^E$ such that $\varphi_x$ and $\varphi_{\gamma x}$ have the same distribution for every $x\in V \cup E$ and $\gamma \in \Gamma$. The law of a Bernoulli process is called a **Bernoulli measure**. We say that a Bernoulli measure is **non-trivial** if it is not concentrated on a single point. The following theorem is folklore.
\[thm:Bernoulli\_radius\] Let $G=(V,E,J)$ be an infinite, connected, transitive weighted graph and let $\Gamma \subseteq {\operatorname{Aut}}(G)$ be a unimodular, closed, transitive group of automorphisms. If $\mu$ is a non-trivial Bernoulli measure on $G$ then $\rho(\mu)=\rho(G)$.
See [@MR2825538 Theorem 2.1 and Corollary 2.2] for stronger results in the case that $G$ is a Cayley graph.
Let $\varphi$ be a random variable with law $\mu$, let $X$ be a random walk started from the origin on $G$, and let $\hat X$ be the associated random walk on $\Gamma$, where we take $\varphi$ and $\hat X$ to be independent. Observe that functions of the form $\mathbbm{1}(\varphi|_A \in \mathscr{A})$ where $A \subseteq V \cup E$ is finite and $\sA \subseteq \bbX^{A \cap V}_V \times \bbX_E^{A \cap E}$ is Borel have dense linear span in $L^2(\Omega,\mu)$. Fix one such pair of sets $A$ and $\sA$ and let $V(A)$ be the set of vertices that either belong to $A$ or are the endpoint of an edge belonging to $A$. We have by independence that $$\begin{aligned}
\lim_{k\to\infty} {{\mathrm{Cov}}}\left(\mathbbm{1}(\varphi|_A \in \mathscr{A}),\hat X_{2k} \mathbbm{1}(\varphi|_A \in \mathscr{A})\right)^{1/2k} &\leq \lim_{k\to\infty}\P(\hat X_{2k}^{-1} A \cap A \neq \emptyset)^{1/2k} \\
&\leq \lim_{k\to\infty} (\hat P^{2k} \mathbbm{1}_{V(A)}, \mathbbm{1}_{V(A)})^{1/2k} \leq \|\hat P\|=\|P\|=\rho(G),
$$ and it follows from that $\rho(\mu)\leq \rho(G)$. The matching lower bound (which we will not use) follows by similar reasoning, using the assumption that $\mu$ is non-trivial, and is left as an exercise to the reader.
**Monotonicity under factors.** Let $G=(V,E,J)$ be a transitive connected weighted graph and let $\Gamma$ be a closed unimodular transitive subgroup of ${\operatorname{Aut}}(G)$. Let $\bbX_V$, $\bbX_E$, $\bbY_V$, and $\bbY_E$ be Polish spaces, and suppose that $\mu$ and $\nu$ are $\Gamma$-invariant probability measures on the product spaces $\Omega_1 = \bbX_V^V \times \bbX_E^E$ and $\Omega_2=\bbY_V^V \times \bbY_E^E$ respectively. We say that $\nu$ is a $\Gamma$-**factor** of $\mu$ if there exists a measurable function $\pi :\Omega_1 \to \Omega_2$ such that $\mu(\pi^{-1}(A))=
\nu(A)$ for every measurable set $A \subseteq \Omega_2$ — this means that if $\varphi=(\varphi_x)_{x\in V \cup E}$ is a random variable with law $\mu$ then $\pi(\varphi)=(\pi(\varphi)_x)_{x\in V \cup E}$ has law $\nu$ — and that is $\Gamma$-equivariant in the sense that $$\label{eq:intertwining}
\gamma \pi \omega_1 = \pi \gamma \omega_1 \qquad \text{ for $\mu$-a.e.\ $\omega_1 \in \Omega_1$ for each $\gamma \in \Gamma$}.$$ In this case we say that $\nu$ is a $\Gamma$-factor of $\mu$ with **factor map** $\pi$. We say that a probability measure $\mu$ on a product space $\bbX_V^V \times \bbX_E^E$ is a $\Gamma$**-factor of i.i.d.** if it is a $\Gamma$-factor of a Bernoulli measure.
Observe that if $\pi:\Omega_1\to \Omega_2$ is such a factor map then $\pi_* L^2(\Omega_2,\nu) := \{f \in L^2(\Omega_1,\mu) : f = g \circ \pi$ for some $g \in L^2_0(\Omega_2,\nu)\}$ is a closed linear subspace of $L^2(\Omega_1,\mu)$ that is naturally identified with $L^2(\Omega_2,\nu)$ via the linear isometry $$\begin{aligned}
\label{eq:pi*identification}
\pi_*:L^2(\Omega_2,\nu) \to \pi_* L^2(\Omega_2,\nu) \qquad \qquad g \mapsto g \circ \pi.\end{aligned}$$ Moreover, it follows by $\Gamma$-equivariance that the Markov operator $\hat P_{\nu}$ coincides with the restriction of $\hat P_{\mu}$ to $\pi_* L^2(\Omega_2,\nu)$ under the identification . A simple consequence of this is that $$\begin{aligned}
\rho(\nu,\Gamma) &= \sup\left\{ \frac{\|\hat P_{\nu} f\|_2}{\|f\|_2} : f\in L^2_0(\Omega_2,\nu) \setminus \{0\} \right\}
=
\sup\left\{ \frac{\|\hat P_{\mu} f\|_2}{\|f\|_2} : f\in \pi_* L^2_0(\Omega_2,\nu) \setminus \{0\} \right\}
\nonumber
\\&\leq
\sup\left\{ \frac{\|\hat P_{\mu} f\|_2}{\|f\|_2} : f\in L^2_0(\Omega_1,\mu) \setminus \{0\} \right\} = \rho(\mu,\Gamma)
\label{eq:factor_rho}\end{aligned}$$ whenever $\nu$ is a $\Gamma$-factor of $\mu$ with factor map $\pi$: *the spectral radius is decreasing under factors*. To apply these results in our setting, we will use the fact, originally due to Häggström, Jonasson, and Lyons [@MR1913108], that the Ising model and random cluster models can often be expressed as factors of i.i.d. The strongest and most general versions of these theorems are due to Harel and Spinka [@harel2018finitary], who study the Gibbs measures of a very general class of positively associated models. The following theorem is an immediate consequence of [@harel2018finitary Theorem 7] together with . See also [@ray2019finitary; @MR3603969] for further related results.
\[thm:Ising\_factor\] Let $G=(V,E,J)$ be a connected transitive weighted graph and let $\Gamma$ be a closed, transitive, unimodular subgroup of ${\operatorname{Aut}}(G)$. Then the following hold:
1. The free and wired random cluster measures $\phi^f_{q,\beta,h}$ and $\phi^w_{q,\beta,h}$ on $G$ are $\Gamma$-factors of i.i.d. for every $q\geq 1$ and $\beta,h\geq 0$, so that $$\rho\bigl(\phi^\#_{q,\beta,h}\bigr) \leq \rho(G)$$ for every $q \geq 1$, $\beta,h \geq 0$, and $\#\in\{f,w\}$.
2. The plus Ising measure $\mathbf{I}^+_{\beta,h}$ on $G$ is a $\Gamma$-factor of i.i.d. for every $\beta\geq 0$ and $h\geq 0$, and therefore satisfies $\rho({\mathbf{I}}_{\beta,h}^+) \leq \rho(G)$ for every $\beta > 0$ and $h\geq 0$.
Note that item $1$ of this theorem does not imply that the free Ising measure is a factor of i.i.d. when $\beta>\beta_c$, since in this case we do not know that the Edwards–Sokal coupling can be implemented as a factor of the random cluster measure and a Bernoulli measure. This is related to several very interesting problems regarding the regimes in which the free Ising model is a factor of i.i.d. that remain open in the nonamenable case, even when the underlying graph is a regular tree; see [@MR3603969] and references therein. In \[subsec:free\_spectral\_radius\], we show that the free *gradient* Ising measure always has spectral radius at most $\rho(G)$.
Bounds on the volume of finite clusters without FKG {#sec:free_energy}
===================================================
Let $G=(V,E,J)$ be a countable weighted graph. Suppose that $\mu$ is a probability measure on $[0,1]^E$, and let ${\mathbf{p}}=({\mathbf{p}}_e)_{e\in E}$ be a $[0,1]^E$-valued random variable with law $\mu$. Let $(U_e)_{e\in E}$ be i.i.d. Uniform$[0,1]$ random variables independent of $\mathbf{p}$ and let $\omega=\omega({\mathbf{p}},U)$ be the $\{0,1\}^E$-valued random variable defined by $$\omega(e) = \mathbbm{1}(U_e \leq {\mathbf{p}}_e) \text{ for each $e\in E$.}$$ We write ${\mathbf P}_\mu$ for the law of the pair of random variable $({\mathbf{p}},\omega)$ and $\P_\mu$ for the joint law of $({\mathbf{p}},\omega)$ and an independent random oriented root edge $\eta$ defined as in \[subsec:unimodularity\_background\]. We say that the random variable $\omega$ is distributed as **percolation in random environment** on $G$ with **environment distribution** $\mu$. Note that every random variable $\omega$ on $\{0,1\}^E$ can trivially be represented as percolation in random environment by taking the environment ${\mathbf{p}}_e=\omega(e)$; we will be interested in less degenerate random environments in which at least some of the probabilities ${\mathbf{p}}_e$ do not belong to $\{0,1\}$. (We shall see that edge probabilities close to zero are far more problematic than edge probabilities close to $1$ as far as our methods are concerned.)
In this section we show how the methods of [@1808.08940] can be extended to percolation in random environment models that have a spectral gap but are not necessarily positively associated.
The two-ghost inequality
------------------------
We begin by proving a generalization of the *two-ghost inequality* of [@1808.08940] that applies to (possibly long-range) percolation in random environment models. The proof of this inequality is based ultimately on the methods of Aizenman, Kesten, and Newman [@MR901151], who implicitly proved a related inequality in the course of their proof that Bernoulli percolation on $\Z^d$ has at most one infinite cluster almost surely. See [@MR3395466] and the introduction of [@1808.08940] for further discussion of inequalities derived from the Aizenman-Kesten-Newman method and their applications.
Let $G=(V,E,J)$ be a connected, transitive weighted graph and let $\Gamma\subseteq {\operatorname{Aut}}(G)$ be a closed transitive group of automorphisms. Let $\mu$ be a $\Gamma$-invariant probability measure on $[0,1]^E$, let ${\mathbf{p}}$ be a random variable with law $\mu$ and let $\omega$ be the associated percolation in random environment process as above. Let $h>0$. Given the environment ${\mathbf{p}}$, let $\cG \in \{0,1\}^E$ be a random subset of $E$ where each edge $e$ of $E$ is included in $\cG$ independently at random with probability $1-e^{-hJ_e}$ of being included, and where we take $\cG$ and $\omega$ to be conditionally independent given ${\mathbf{p}}$. Following [@aizenman1987sharpness], we call $\cG$ the **ghost field** and call an edge **green** if it is included in $\cG$. We write ${\mathbf P}_{\mu,h}$ and ${\mathbf E}_{\mu,h}$ for probabilities and expectations taken with respect to the joint law of ${\mathbf{p}}$, $\omega$, and $\cG$. Similarly, we write $\P_{\mu,h}$ and $\E_{\mu,h}$ for probabilities and expectations taken with respect to the joint law of ${\mathbf{p}}$, $\omega$, $\cG$, and $\eta$, where $\eta$ is the random oriented root edge of $G$ defined as in \[subsec:unimodularity\_background\], which is taken to be independent of $({\mathbf{p}},\omega,\cG)$. The density of $\cG$ is chosen so that ${\mathbf P}_{\mu,h}(A \cap \cG \neq \emptyset \mid {\mathbf{p}}) = \exp\left[-h|A|_{J}\right]$ for every finite set $A \subseteq E$, where we write $|A|_J=\sum_{e\in A} J_e$.
Define $\sT_e$ to be the event that $e$ is closed in $\omega$ and that the endpoints of $e$ are in distinct clusters of $\omega$, each of which touches some green edge, and at least one of which is finite. The primary purpose of this section is to prove the following inequality.
\[thm:two\_ghost\] Let $G=(V,E,J)$ be a connected transitive weighted graph and let $\Gamma \subseteq {\operatorname{Aut}}(G)$ be a closed transitive unimodular subgroup of automorphisms. If $\mu$ is a $\Gamma$-invariant probability measure on $[0,1]^E$ then the inequality $$\begin{aligned}
\label{eq:two_ghost}
\E_{\mu,h}\left[\mathbbm{1}(\sT_\eta) \sqrt{\frac{{\mathbf{p}}_\eta}{(1-{\mathbf{p}}_\eta)J_\eta}}\right] \leq 21 \sqrt{h}\end{aligned}$$ holds for every $h>0$, where we take $\mathbbm{1}(\sT_\eta) \sqrt{\frac{{\mathbf{p}}_\eta}{(1-{\mathbf{p}}_\eta)J_\eta}}=0$ when ${\mathbf{p}}_\eta=1$.
Note that it is not obvious *a priori* that that the left hand side of is finite.
\[thm:two\_ghost\] has the following corollary which does not refer to the ghost field. For each $e\in E$ and $\lambda>0$, let ${\mathscr{S}}_{e,\lambda}$ be the event that $e$ is closed in $\omega$ and that the endpoints of $e$ are in distinct clusters $K_1$ and $K_2$ of $\omega$, each of which has $|E(K_i)|_J \geq \lambda$ and at least one of which is finite. The deduction of \[cor:two\_ghost\_S\] from \[thm:two\_ghost\] is similar to the proof of [@1808.08940 Corollary 1.7] and is omitted.
\[cor:two\_ghost\_S\] Let $G=(V,E,J)$ be a connected transitive weighted graph and let $\Gamma \subseteq {\operatorname{Aut}}(G)$ be a closed transitive unimodular subgroup of automorphisms. If $\mu$ is a $\Gamma$-invariant probability measure on $[0,1]^E$ then the inequality $$\begin{aligned}
\E_{\mu}\left[\mathbbm{1}({\mathscr{S}}_{\eta,\lambda}) \sqrt{\frac{{\mathbf{p}}_\eta}{(1-{\mathbf{p}}_\eta)J_\eta}}\right] \leq \frac{42}{\sqrt{\lambda}}\end{aligned}$$ holds for every $\lambda >0$, where we take $\mathbbm{1}({\mathscr{S}}_{\eta,\lambda}) \sqrt{\frac{{\mathbf{p}}_\eta}{(1-{\mathbf{p}}_\eta)J_\eta}}=0$ when ${\mathbf{p}}_\eta=1$.
We now begin to work towards the proofs of \[thm:two\_ghost,cor:two\_ghost\_S\]. We will first prove these results under the additional assumptions that ${\mathbf{p}}_e \in (0,1)$ for every $e\in E$ a.s. and that $$\E_{\mu}\left[\sqrt{\frac{{\mathbf{p}}_\eta(1-{\mathbf{p}}_\eta)}{J_\eta}} \right]<\infty$$ and then show that both assumptions can be removed via a limiting argument.
Let $G=(V,E,J)$ be a connected transitive weighted graph and let $\Gamma$ be a closed transitive subgroup of automorphisms of $G$. For each environment ${\mathbf{p}}\in (0,1)^E$ and subgraph $H$ of $G$, we define the **fluctuation** of $H$ to be $$\begin{gathered}
h_{{\mathbf{p}}}(H):= \sum_{e \in E(H)} \sqrt{J_e} \left[\sqrt{\frac{{\mathbf{p}}_e}{1-{\mathbf{p}}_e}}\mathbbm{1}\left(e\in \partial H\right)-\sqrt{\frac{1-{\mathbf{p}}_e}{{\mathbf{p}}_e}} \mathbbm{1}\left(e\in E_o(H)\right) \right]\\
=\sum_{e \in E(H)} \sqrt{\frac{J_e {\mathbf{p}}_e}{1-{\mathbf{p}}_e}} \frac{{\mathbf{p}}_e-\mathbbm{1}(e\in E_o(H))}{{\mathbf{p}}_e} \end{gathered}$$ where $E(H)$ denotes the set of (unoriented) edges that *touch* $H$, i.e., have at least one endpoint in the vertex set of $H$, $\partial H$ denotes the set of (unoriented) edges of $G$ that touch the vertex set of $H$ but are not included in $H$, and $E_\circ(H)$ denotes the set of (unoriented) edges of $G$ that are included in $H$, so that $E(H)=\partial H \cup E_o(H)$. This quantity is defined so that $h_{{\mathbf{p}}}(K_v)$ and $|E(K_v)|_{J}$ are the final value and total quadratic variation of a certain martingale that arises when exploring the cluster $K_v$ of $v$ in $\omega$ in an edge-by-edge manner.
\[lem:AKN\] Let $G=(V,E,J)$ be a connected transitive weighted graph and let $\Gamma \subseteq {\operatorname{Aut}}(G)$ be a closed transitive unimodular subgroup of automorphisms. Let $\mu$ be a $\Gamma$-invariant probability measure on $(0,1)^E$. If $$\E_{\mu}\left[\sqrt{\frac{{\mathbf{p}}_\eta(1-{\mathbf{p}}_\eta)}{J_\eta}} \right]<\infty$$ then the inequality $$\begin{aligned}
\E_{\mu,h}\left[\mathbbm{1}(\sT_\eta) \sqrt{\frac{ {\mathbf{p}}_\eta}{J_\eta(1-{\mathbf{p}}_\eta)}}\right] \leq 2{\mathbf E}_{\mu,h}\left[\frac{|h_{{\mathbf{p}}}(K_{o})|}{|E(K_{o})|_{J}} \mathbbm{1}\bigl(|K_o| < \infty \text{ and } E(K_{o}) \cap \cG \neq \emptyset \bigr)\right]
\label{eq:AKN_main_estimate}\end{aligned}$$ holds for every $p\in (0,1]$ and $h>0$.
Let $\sF_e$ be the event that every cluster touching $e$ is finite, so that $\sT_e \cap \sF_e$ is the event that the endpoints of $e$ are in distinct finite clusters each of which touches $\cG$, and let $\sG_e$ be the event that there exists a finite cluster touching $e$ and $\cG$. For each edge $e$ of $G$ we can verify that $$\mathbbm{1}(\sT_e \cap \sF_e) = \mathbbm{1}(\omega(e)=0)\cdot \#\{\text{finite clusters touching $e$ and $\cG$}\}
\\- \mathbbm{1}\bigl(\{\omega(e)=0\}\cap \sG_e),$$ and hence that $$\begin{gathered}
\label{eq:unimodghost1}
{\mathbf P}_{\mu,h}(\sT_e \cap \sF_e \mid {\mathbf{p}}\,) = {\mathbf E}_{\mu,h}\left[\mathbbm{1}(\omega(e)=0)\cdot\#\{\text{finite clusters touching $e$ and $\cG$}\} \mid {\mathbf{p}}\,\right]
\\- {\mathbf P}_{\mu,h}\bigl(\{\omega(e)=0\} \cap \sG_e \mid {\mathbf{p}}\,\bigr). \end{gathered}$$ The event $\sF_e \cap \sG_e$ is conditionally independent of the value of $\omega(e)$ given ${\mathbf{p}}$, so that $$\begin{gathered}
{\mathbf P}_{\mu,h}\bigl(\{\omega(e)=0\} \cap \sF_e \cap \sG_e \mid {\mathbf{p}}\, \bigr)
= \frac{1-{\mathbf{p}}_e}{{\mathbf{p}}_e} {\mathbf P}_{\mu,h}\bigl(\{\omega(e)=1\} \cap \sF_e \cap \sG_e \mid {\mathbf{p}}\, \bigr).
\\
= \frac{1-{\mathbf{p}}_e}{{\mathbf{p}}_e} {\mathbf P}_{\mu,h}\bigl(\{\omega(e)=1\} \cap \sG_e \mid {\mathbf{p}}\,\bigr).
\label{eq:unimodghost2}\end{gathered}$$ Putting together and yields that $$\begin{gathered}
\label{eq:AKNmainstep}
{\mathbf P}_{\mu,h}(\sT_e \cap \sF_e \mid {\mathbf{p}}\,) = {\mathbf E}_{\mu,h}\left[\mathbbm{1}(\omega(e)=0)\cdot \#\{\text{finite clusters touching $e$ and $\cG$}\} \mid {\mathbf{p}}\,\right]
\\- \frac{1-{\mathbf{p}}_e}{{\mathbf{p}}_e}{\mathbf P}_{\mu,h}(\{\omega(e)=1\} \cap \sG_e \mid {\mathbf{p}}\,) - {\mathbf P}_{\mu,h}\bigl(\{\omega(e)=0\} \cap \sG_e \setminus \sF_e \mid {\mathbf{p}}\,\bigr).\end{gathered}$$ Finally, observe that $\{\omega(e)=0\} \cap \sG_e \setminus \sF_e$ and $\sT_e \cap \sF_e$ are disjoint and that $\sT_e$ coincides with $(\sT_e \cap \sF_e) \cup (\{\omega(e)=0\} \cap \sG_e \setminus \sF_e)$ up to a null set, so that implies that $$\begin{gathered}
{\mathbf P}_{\mu,h}(\sT_e \mid {\mathbf{p}}\,) = {\mathbf E}_{\mu,h}\left[\mathbbm{1}(\omega(e)=0)\cdot \#\{\text{finite clusters touching $e$ and $\cG$}\} \mid {\mathbf{p}}\right]
\\- \frac{1-{\mathbf{p}}_e}{{\mathbf{p}}_e}{\mathbf P}_{\mu,h}(\{\omega(e)=1\} \cap \sG_e \mid {\mathbf{p}}\,).\end{gathered}$$ This equality can be written more concisely as $$\label{eq:AKNmainstep2}
{\mathbf P}_{\mu,h}(\sT_e \mid {\mathbf{p}}\,)=
{\mathbf E}_{\mu,h}\left[\frac{{\mathbf{p}}_e-\omega(e)}{{\mathbf{p}}_e} \cdot \#\{\text{finite clusters touching $e$ and $\cG$}\} \;\Bigm|\; {\mathbf{p}}\;\right].$$ Note that we have not yet used any assumptions on the weighted graph $G$ or the group $\Gamma$. We will now apply the assumption that the group $\Gamma$ is transitive and unimodular. Define a mass-transport function $F:E^\rightarrow\times E^\rightarrow \to \R$ by $$F(e_1,e_2) =\\ {\mathbf E}_{\mu,h}\sum\left\{\frac{1}{2|E(K)|_{J}} \left[\frac{{\mathbf{p}}_{e_1}-\omega(e_1)}{{\mathbf{p}}_{e_1}}\right] \sqrt{\frac{{\mathbf{p}}_{e_1}}{(1-{\mathbf{p}}_{e_1})J_{e_1}}} : \begin{array}{l}\text{$K$ is a finite cluster}\\ \text{of $\omega$ touching $e_1,e_2$, and $\cG$}\end{array}\right\},$$ where we write $\sum\{x(i) :i\in I\} = \sum_{i\in I} x(i)$ and where we include the factor of $1/2$ to account for the fact that each edge in $E(K)$ can be oriented in two directions. The multiset of numbers being summed over has cardinality either $0,1,$ or $2$, and we can therefore compute that $$\E\sum_{e\in E^\rightarrow} J_e |F(\eta,e)| \leq 2 \E_{\mu,h}\left[ \frac{|{\mathbf{p}}_{\eta}-\omega(\eta)|}{{\mathbf{p}}_{\eta}} \sqrt{\frac{{\mathbf{p}}_{\eta}}{(1-{\mathbf{p}}_{\eta})J_{\eta}}}\right] = 4 \E_{\mu,h} \left[\sqrt{\frac{{\mathbf{p}}_\eta(1-{\mathbf{p}}_\eta)}{J_\eta}} \right] < \infty,$$ where the final inequality is by the hypotheses of the lemma. Thus, we may safely apply the mass-transport principle together with to deduce that $$\begin{aligned}
\E_{\mu,h}\left[\mathbbm{1}(\sT_\eta)\sqrt{\frac{{\mathbf{p}}_{\eta}}{(1-{\mathbf{p}}_{\eta})J_\eta}}\right] &= {\mathbf E}_{\mu,h}\left[ \frac{{\mathbf{p}}_\eta-\omega(\eta)}{{\mathbf{p}}_\eta} \sqrt{\frac{{\mathbf{p}}_{\eta} }{(1-{\mathbf{p}}_{\eta})J_{\eta}}}\cdot\#\{\text{finite clusters touching $e$ and $\cG$}\} \right]\nonumber
\\&=\E \sum_{e\in E^\rightarrow}J_e F(\eta,e) = \E \sum_{e\in E^\rightarrow}J_e F(e,\eta)\nonumber \\&=
\E_{\mu,h} \sum\left\{\frac{h_{{\mathbf{p}}}(K)}{|E(K)|_{J}} : \begin{array}{l}\text{$K$ is a finite cluster}\\ \text{of $\omega$ touching $\eta$ and $\cG$}\end{array}\right\}.\end{aligned}$$ For each vertex $v$ of $G$, let $\sO_v$ be the event that the cluster $K_v$ is finite and touches $\cG$. Then we deduce from the above that $$\begin{gathered}
\E_{\mu,h}\left[\mathbbm{1}(\sT_\eta)\sqrt{\frac{{\mathbf{p}}_{\eta}}{(1-{\mathbf{p}}_{\eta})J_\eta}}\right]
\leq \E_{\mu,h} \sum\left\{\frac{|h_{{\mathbf{p}}}(K)|}{|E(K)|_{J}} : \begin{array}{l}\text{$K$ is a finite cluster}\\ \text{of $\omega$ touching $\eta$ and $\cG$}\end{array}\right\}\\
\leq\E_{\mu,h}\left[\frac{|h_{\mathbf{p}}(K_{\eta^-})|}{|E(K_{\eta^-})|}\mathbbm{1}\bigl(\sO_{\eta^-} \bigr)+ \frac{|h_{\mathbf{p}}(K_{\eta^+})|}{|E(K_{\eta^+})|}\mathbbm{1}\bigl(\sO_{\eta^+} \bigr)\right]
=2{\mathbf E}_{\mu,h}\left[\frac{|h_{\mathbf{p}}(K_{o})|}{|E(K_{o})|}\mathbbm{1}\bigl(\sO_{o} \bigr)\right]\end{gathered}$$ as claimed, where the final equality follows by transitivity.
As in [@1808.08940], we will now bound the right hand side of using maximal inequalities for martingales[^5]. Since the martingale we have here is a little more complicated than that of [@1808.08940], we will need to introduce some more machinery before doing this. In particular, we will employ the following simple variation on Doob’s $L^2$ maximal inequality, which is inspired by Freedman’s maximal inequality [@MR0380971]. It seems unlikely that this inequality is new, but we are not aware of a reference. Let $X=(X_n)_{n\geq0}$ be a real-valued martingale with respect to the filtration $\cF=(\cF_n)_{n\geq 0}$, and suppose that $X_0=0$. The **quadratic variation process** $Q=(Q_n)_{n\geq 0}$ associated to $(X,\cF)$ is defined by $Q_0=0$ and $$Q_n = \sum_{i=1}^n\E\left[ |X_i - X_{i-1}|^2 \mid \cF_{i-1} \right]$$ for each $n\geq 1$. Note that $Q$ is *predictable*, that is, $Q_n$ is $\cF_{n-1}$-measurable for every $n\geq 1$.
\[lem:martingale\_stuff\] Let $(X_n)_{n\geq0}$ be a martingale with respect to the filtration $(\cF_n)_{n\geq 0}$ such that $X_0=0$, and let $(Q_n)_{n\geq 0}$ be the associated quadratic variation process. Then $$\E\Bigl[ \sup\bigl\{X_n^2 : n\geq 0,\, Q_n \leq \lambda \bigr\} \Bigr] \leq 4\lambda$$ for every $\lambda \geq 0$.
(This lemma holds vacuously if the increments of $X$ have infinite conditional variance a.s.)
Fix $\lambda \geq 0$ and let $\tau=\sup\{k\geq 0: Q_k \leq \lambda\}=\inf\{k\geq 0 : Q_k > \lambda\}-1$, which may be infinite. Since $Q_n$ is $\cF_{n-1}$-measurable for every $n\geq 0$, $\tau$ is a stopping time and $X_{n\wedge \tau}$ is a martingale. Thus, we have by the orthogonality of martingale increments that $$\begin{aligned}
\E\left[X^2_{n\wedge \tau}\right] &= \sum_{i=1}^n\E\left[ (X_{i\wedge \tau}-X_{(i-1)\wedge \tau})^2\right]
= \sum_{i=1}^n\E\left[ \E\left[(X_{i\wedge \tau}-X_{(i-1)\wedge \tau})^2\mid \cF_{i-1} \right]\right]\\
&=\sum_{i=1}^n\E\left[ \E\left[(X_{i}-X_{i-1})^2\mid \cF_{i-1} \right] \mathbbm{1}(i \leq \tau)\right] = \E\left[ Q_{n \wedge \tau}\right] \leq \lambda\end{aligned}$$ for every $n\geq 1$. The claim follows by applying Doob’s $L^2$ maximal inequality to $(X_{n\wedge \tau})_{n\geq 0}$.
We next apply this lemma to prove a generalized version of the martingale estimate appearing in the proof of [@1808.08940 Theorem 1.6]. (Note that $Q_n$ is increasing in $n$, so that $Q_\infty$ is well-defined as an element of $[0,\infty]$ and the case $T=\infty$ does not cause us any problems.)
\[lem:martingale\_stuff2\] Let $(X_n)_{n\geq0}$ be a martingale with respect to the filtration $(\cF_n)_{n\geq 0}$ such that $X_0=0$, and let $(Q_n)_{n\geq 0}$ be the associated quadratic variation process. Then $$\label{eq:martingale_stuff2}
\E\left[ \frac{\sup_{0 \leq n \leq T}|X_n|}{Q_T} (1-e^{-h Q_T})\mathbbm{1}(0<Q_T < \infty) \right] \leq 2\sqrt{eh}\sum_{k=-\infty}^\infty \frac{1-e^{-e^{k}}}{e^{k/2}} \leq \frac{21}{2} \sqrt{h}$$ for every stopping time $T$ and every $h> 0$.
Write $M_n=\max_{0\leq m \leq n} |X_n|$ for each $n\geq 0$. Since $(1-e^{-hx})/x$ is a decreasing function of $x>0$, we may write $$\E\left[ \frac{M_T}{Q_T}\bigl(1-e^{-hQ_T}\bigr)\mathbbm{1}(0<Q_T<\infty) \right]
\leq h\sum_{k=-\infty}^\infty \frac{1-e^{-e^{k}}}{e^k} \E\left[ M_T \mathbbm{1}(e^k \leq h Q_T \leq e^{k+1})\right].
\label{eq:scales}$$ \[lem:martingale\_stuff\] and Jensen’s inequality let us bound each summand $$\begin{aligned}
\label{eq:martingale_Jensen_not_optimized}
\E\left[ M_T \mathbbm{1}(e^k \leq hQ_T \leq e^{k+1})\right] &\leq \E\Bigl[ \max\bigl\{X_n^2 : n\geq 0,\, hQ_n \leq e^{k+1} \bigr\} \Bigr]^{1/2}\leq \sqrt{\frac{4e^{k+1}}{h}}\end{aligned}$$ for each $k\in \Z$, so that $$\begin{aligned}
\E\left[ \frac{M_T}{Q_T}\bigl(1-e^{-hQ_T}\bigr)\mathbbm{1}(0<Q_T<\infty) \right]
\leq 2\sqrt{e h}\sum_{k=-\infty}^\infty \frac{1-e^{-e^{k}}}{e^{k/2}}\end{aligned}$$ as claimed. This series is easily seen to converge. Moreover, the constant appearing here can be evaluated numerically as $2\sqrt{e}\sum_{k=-\infty}^\infty \frac{1-e^{-e^{k}}}{e^{k/2}}=10.47\ldots$, which we bound by $21/2$ for simplicity.
The inequality can be improved if one knows something about the tail of $Q_T$ by using Cauchy-Schwarz instead of Jensen in . This eventually leads to better bounds on the exponents appearing in \[thm:main,thm:free\_random\_cluster,thm:finite\_clusters\]. We do not pursue this further here, but similar considerations for Bernoulli percolation are discussed in detail in [@1808.08940 Section 6].
As discussed above, we will first prove the theorem under the additional assumption that ${\mathbf{p}}\in (0,1)^E$ almost surely and that $$\label{eq:assumption}
\E_{\mu}\left[\sqrt{\frac{{\mathbf{p}}_\eta(1-{\mathbf{p}}_\eta)}{J_\eta}} \right]<\infty;$$ We will then show that these assumptions can be removed via a limiting argument.
To this end, let $\mu$ be a $\Gamma$-invariant probability measure on $(0,1)^E$ and let $({\mathbf{p}},\omega)$ be random variables with law ${\mathbf P}_\mu$. Write $K=K_o$ for the cluster of $o$ in $\omega$. Fix an enumeration $E=\{e_1,e_2,\ldots\}$ of the edge set of $G$, and let ${\preccurlyeq}$ be the associated well-ordering of $E$, so that $e_i {\preccurlyeq}e_j$ if and only if $i \leq j$. After conditioning on the environment ${\mathbf{p}}$, we will explore $K$ one edge at a time and define a martingale in terms of this exploration process. At each stage of the exploration we will have a set of vertices $U_n$, a set of revealed open edges $O_n$, and a set of revealed closed edges $C_n$. We begin by setting $U_0=\{o\}$ and $C_0=O_0=\emptyset$. Let $n\geq 1$. Given everything that has happened up to and including step $n-1$ of the exploration, we define $(U_n,O_n,C_n)$ as follows: If every edge touching $U_{n-1}$ is included in $O_{n-1}\cup C_{n-1}$, we set $(U_n,O_n,C_n)=(U_{n-1},O_{n-1},C_{n-1})$. Otherwise, we take $E_n$ to be the ${\preccurlyeq}$-minimal element of the set of edges that touch $U_{n-1}$ but are not in $O_{n-1}$ or $C_{n-1}$. If $E_n$ is open in $\omega$, we set $O_n=O_{n-1}\cup\{E_n\}$, $C_n=C_{n-1}$, and set $U_{n}$ to be the union of $U_n$ with the set of endpoints of $E_n$. Otherwise, $E_n$ is closed in $\omega$ and we set $O_n=O_{n-1}$, $C_n =C_{n-1} \cup \{E_n\}$, and $U_n = U_{n-1}$. Let $\cF_0$ be the $\sigma$-algebra generated by the environment ${\mathbf{p}}$ and let $(\cF_n)_{n\geq 0}$ be the filtration generated by this exploration process and the environment ${\mathbf{p}}$.
Let $T = \inf\{n \geq 0: E(U_n) \subseteq O_n \cup C_n\}$ be the first time that there are no unexplored edges touching $U_n$, setting $T=\infty$ if this never occurs, and observe that $(U_T,O_T,C_T,T)$ is equal to $(K,E_\circ(K),\partial K,|E(K)|)$. Let the process $(Z_n)_{n\geq 0}$ be defined by $Z_0=0$ and $$Z_n = \sum_{i=1}^{n\wedge T}
\sqrt{J_{E_i}}
\left[\sqrt{\frac{{\mathbf{p}}_{E_i}}{1-{\mathbf{p}}_{E_i}}} \mathbbm{1}(\omega(E_i)=0) - \sqrt{\frac{1-{\mathbf{p}}_{E_i}}{{\mathbf{p}}_{E_i}}} \mathbbm{1}(\omega(E_i)=1)\right]
$$ for each $n\geq 1$. The process $Z$ is a martingale with respect to the filtration $(\cF_n)_{n\geq 0}$ satisfying $Z_T = h_{{\mathbf{p}}}(K)$. Moreover, the quadratic variation process $Q_n=\sum_{i=1}^n {\mathbf E}_{\mu}[(Z_{i+1}-Z_i)^2 \mid \cF_i]$ satisfies $$Q_n = \sum_{i=1}^{n\wedge T} {\mathbf E}_{\mu}\left[J_{E_i}
\left[\frac{{\mathbf{p}}_{E_i}}{1-{\mathbf{p}}_{E_i}} \mathbbm{1}(\omega(E_i)=0) + \frac{1-{\mathbf{p}}_{E_i}}{{\mathbf{p}}_{E_i}} \mathbbm{1}(\omega(E_i)=1)\right] \Biggm| \cF_{n-1}\right] = \sum_{i=1}^{n\wedge T} J_{E_i}$$ for every $n\geq 0$, so that $Q_T = |E(K)|_{J}$. It follows from \[lem:AKN\] and \[lem:martingale\_stuff2\] that if holds then $$\begin{aligned}
\label{eq:converttoMTG}
\E_{\mu,h}\left[\mathbbm{1}(\sT_\eta) \sqrt{\frac{{\mathbf{p}}_\eta}{(1-{\mathbf{p}}_\eta)J_\eta}}\right] \leq 2 {\mathbf E}_{p}\left[\frac{|h_{{\mathbf{p}}}(K)|}{|E(K)|_{J}} (1-e^{-h |E(K)|_{J}}) \mathbbm{1}\bigl(|E(K)|_{J}<\infty\bigr)\right]
\nonumber
\\ = 2 {\mathbf E}_p\left[ \frac{|Z_T|}{Q_T}\bigl(1-e^{-h Q_T}\bigr)\mathbbm{1}(0<Q_T<\infty) \right] \leq 21 \sqrt{h}.\end{aligned}$$ This establishes the claim in the case that ${\mathbf{p}}\in(0,1)^E$ almost surely and holds.
Now suppose that ${\mathbf{p}}\in (0,1]^E$ almost surely and that does not necessarily hold. Let ${\mathbf{p}}$ be a random environment with law $\mu$ and for each $n\geq 1$ let ${\mathbf{p}}^n \in (0,1)^E$ be the environment defined by $${\mathbf{p}}^n_e = \min\left\{{\mathbf{p}}_e,nJ_e,e^{-1/n}\right\} \qquad \text{ for each $e\in E$.}$$ We couple percolation in the random environments $({\mathbf{p}}^n)_{n\geq 1}$ and ${\mathbf{p}}$ in the standard monotone way by letting $(U_e)_{e\in E}$ be i.i.d. Uniform$[0,1]$ random variables independent of ${\mathbf{p}}$ and setting $$\omega(e)=\mathbbm{1}(U_i \leq {\mathbf{p}}_e) \quad \text{ and } \quad \omega^n(e)=\mathbbm{1}(U_i \leq {\mathbf{p}}_e^n) \quad \text{ for each $n\geq 1$ and $e\in E$,}$$ so that $\omega^n$ converges to $\omega$ pointwise from below almost surely. Write $\E_h$ for expectations taken with respect to the joint law of ${\mathbf{p}}$, $\omega$, $(\omega^n)_{n\geq 1}$, the independent ghost field $\cG$, and the independent root edge $\eta$. Let $\sT^n_e$ be the event that $e$ is closed in $\omega^n$ and that the endpoints of $e$ are in distinct clusters of $\omega^n$, at least one of which touches some green edge and at least one of which is finite. The law of ${\mathbf{p}}^n$ is clearly $\Gamma$-invariant, and since $$\E_{\mu}\left[ \sqrt{\frac{{\mathbf{p}}^n_\eta (1-{\mathbf{p}}^n_\eta)}{J_\eta}}\right] \leq \E_\mu\left[ \sqrt{n(1-{\mathbf{p}}^n_\eta)}\right]< \infty,$$ we may apply the inequality to deduce that $$\begin{aligned}
\label{eq:AKN_Fatou}
\E_h\left[\mathbbm{1}(\sT_\eta^n) \sqrt{\frac{ {\mathbf{p}}_\eta^n}{(1-{\mathbf{p}}_\eta^n)J_\eta}}\right] \leq 21 \sqrt{h}\end{aligned}$$ for every $n\geq 1$ and $h>0$. Since $\omega^n$ converges to $\omega$ pointwise from below, if $\sT_\eta$ holds then $\sT^n_\eta$ holds for all $n$ sufficiently large almost surely. It follows that $$\label{eq:Fatou}
\mathbbm{1}(\sT_\eta) \sqrt{\frac{{\mathbf{p}}_\eta}{(1-{\mathbf{p}}_\eta)J_\eta}} \leq \liminf_{n\to\infty} \mathbbm{1}(\sT_\eta^n) \sqrt{\frac{ {\mathbf{p}}_\eta^n}{(1-{\mathbf{p}}_\eta^n)J_\eta}}$$ almost surely, and Fatou’s lemma implies that $$\label{eq:ppositive}
\E_{\mu,h}\left[\mathbbm{1}(\sT_\eta) \sqrt{\frac{{\mathbf{p}}_\eta}{(1-{\mathbf{p}}_\eta)J_\eta}}\right] \leq 21 \sqrt{h}$$ for every $h>0$ under the assumption that ${\mathbf{p}}\in (0,1]^E$ almost surely. (Eq. is an inequality rather than an equality since we might have that $\eta$ is incident to a finite cluster in $\omega^n$ for every $n\geq 1$ without this being true in $\omega$.)
It remains to consider the case in which edge probabilities may be zero. Let ${\mathbf{p}}$ be a random environment with law $\mu$ and for each $n\geq 1$ let ${\mathbf{p}}^n \in (0,1]^E$ be the environment defined by $${\mathbf{p}}^n_e = \max\left\{{\mathbf{p}}_e,\min\left\{1,\frac{J_e}{n}\right\}\right\} \qquad \text{ for each $e\in E$.}$$ Similarly to before, we can couple the associated percolation processes $\omega$ and $(\omega^n)_{n\geq 1}$ so that $\omega^n$ tends to $\omega$ pointwise from above. Since $\sum_{e\in E^\rightarrow_v} J_e<\infty$ for every $v\in V$, we have for every finite set $A \subseteq V$ there exists an almost surely finite random $N_A$ such that $\{e \in E(A) : \omega^n(e)=1\}=\{e \in E(A) : \omega(e)=1\}$ for every $n\geq N_A$. It follows easily that $$\mathbbm{1}(\sT_\eta) \sqrt{\frac{{\mathbf{p}}_\eta}{(1-{\mathbf{p}}_\eta)J_\eta}} = \lim_{n\to\infty} \mathbbm{1}(\sT_\eta^n) \sqrt{\frac{{\mathbf{p}}_\eta^n}{(1-{\mathbf{p}}_\eta^n)J_\eta}}$$ almost surely, and the claim follows from and Fatou’s lemma as before.
Finite clusters in Bernoulli percolation
----------------------------------------
We now apply the two-ghost inequality to study finite clusters in percolation in random environment models under a spectral gap condition. We begin with the case of Bernoulli percolation on a locally finite graph so that we can present the basic method in the simplest possible setting.
\[thm:finite\_percolation\] Let $G$ be a connected, locally finite, nonamenable, transitive unimodular graph with spectral radius $\rho<1$ and let $o$ be a vertex of $G$. Then there exist positive constants $C=C(\deg(o),\rho)$ and $\delta=\delta (\deg(o),\rho)$ such that $${\mathbf P}_p(n\leq |K_o| <\infty) \leq C n^{-\delta}$$ for every $n\geq 1$ and $p\in [0,1]$.
The proof will apply the following general fact about percolation on nonamenable graphs, which is a version of *Schramm’s Lemma*. A similar lemma for Bernoulli percolation (with a very different proof) first arose in unpublished work of Schramm; see [@kozma2011percolation] for a detailed discussion and [@1712.04911 Section 3] for further related results.
\[prop:Schramm\] Let $G=(V,E,J)$ be a connected, nonamenable, transitive, weighted graph, let $o$ be a vertex of $G$, and let $\Gamma \subseteq {\operatorname{Aut}}(G)$ be a closed unimodular transitive subgroup of automorphisms. Suppose that $\omega \in \{0,1\}^E$ is a random variable whose law is invariant under $\Gamma$ and that $(X_n)_{n\geq 0}$ is an independent random walk on $G$ started at $X_0=o$. Then $$\P(X_0 \text{ and } X_n \text{ both belong to the same finite cluster of $\omega$}) \leq \rho(G)^n$$ for every $n\geq 0$.
Let $P:L^2(V)\to L^2(V)$ be the Markov operator on $G$, so that $$\P(X_0 \text{ and } X_n \text{ both belong to the same finite cluster of $\omega$})
= \E \left[\langle P^n \mathbbm{1}_{K_o},\mathbbm{1}_{o}\rangle \mathbbm{1}(|K_o|<\infty)\right].$$ We have by the mass-transport principle that $$\E \left[\langle P^n \mathbbm{1}_{K_o},\mathbbm{1}_{o}\rangle \mathbbm{1}(|K_o|<\infty)\right] = \E \left[\frac{\langle P^n \mathbbm{1}_{K_o},\mathbbm{1}_{K_o}\rangle}{|K_o|} \mathbbm{1}(|K_o|<\infty)\right] \leq \|P\|^n \cdot \P(|K_o|<\infty)$$ which implies the claim.
First note that if $p \leq 1/2\deg(o)$ then counting paths gives that ${\mathbf E}_p |K_o| \leq \sum_{i=0} p^i \deg(o)^i \leq 2$, so that the claim is trivial in this case. We may therefore assume throughout the proof that $p \geq p_0:= 1/2\deg(o)$.
Fix $p \geq p_0$ and let $\omega$ be an instance of Bernoulli-$p$ bond percolation on $G$. Let $X$ be a random walk on $G$ started at $o$ and independent of the percolation configuration $\omega$, and let $X_{i,i+1}$ be the edge crossed by $X$ between times $i$ and $i+1$ for each $i\geq 0$. For each $i\geq 0$, let $\omega^i$ be obtained from $\omega$ by setting $\omega^0=\omega$ and $$\omega^i(e) =\begin{cases} 1 & e \in \{X_{j,j+1} : 0 \leq j \leq i-1\}\\
\omega(e) & e \notin \{X_{j,j+1} : 0 \leq j \leq i-1\}
\end{cases}$$ for each $i\geq 1$ and $e\in E$. For each $n,m \geq 1$ let $\sA_{n,m}$ be the event that the cluster of $X_0=o$ in $\omega$ is finite and that $X_0$ and $X_m$ are in distinct clusters of $\omega$ each of which touches at least $n$ edges. For each $n,m \geq 0$ and $1\leq i \leq m$, let $\sB_{n,m,i}$ be the event that the following hold:
1. $X_0$ and $X_m$ are in distinct clusters of $\omega^{i-1}$ each of which touches at least $n$ edges,
2. the cluster of $X_0$ is finite in $\omega^{i-1}$, and
3. either $X_0$ and $X_m$ are connected in $\omega^i$ or the cluster of $X_0$ is infinite in $\omega^i$.
On the event $\sA_{n,m}$ the vertices $X_0$ and $X_m$ are connected in $\omega^m$ and not connected in $\omega^0$, and since $\omega^i$ is monotone increasing in $i$ it follows that $$\label{eq:ABsetinclusion}
\sA_{n,m} \subseteq \bigcup_{i=1}^m \sB_{n,m,i}$$ for every $n,m\geq 1$. Now, for each $n,i\geq 1$ let $\sC_{n,i}$ be the event that the cluster of $X_{i-1}$ in $\omega$ is finite and that $X_{i-1}$ and $X_{i}$ are in distinct clusters of $\omega$ each of which touches at least $n$ edges. Observe that $\sC_{n,i} \supseteq \sB_{n,m,i} \cap \{\omega(X_{j,j+1})=1 \text{ for every $0\leq j \leq i-2$}\}$ for every $n,m\geq 1$ and $1\leq i \leq m$. Moreover, these two events are conditionally independent given the random walk $X$, and we deduce that $$\P\bigl(\sC_{n,i}) \geq \E \left[\P(\omega(X_{j,j+1})=1 \text{ for every $0\leq j \leq i-2$}\mid X) \P\bigl( \sB_{n,m,i}\mid X)\right]\geq p^{i-1} \P(\sB_{n,m,i})
$$ for every $n,m\geq 1$ and $1\leq i \leq m$. Applying and \[cor:two\_ghost\_S\] we deduce that $$\label{eq:Bernoulli_surgery}
\P(\sA_{n,m}) \leq \sum_{i=1}^m p^{-i+1} \P\bigl(\sC_{n,i})= \P\bigl({\mathscr{S}}_{\eta,n}) \sum_{i=1}^m p^{-i+1}
\leq \frac{p_0^{-m+1}}{1-p_0} \sqrt{\frac{1-p}{p}} \frac{42}{ \sqrt{n}}
\leq \frac{42}{ p_0^m\sqrt{n}}
$$ for every $n,m\geq 1$, where we used transitivity in the central equality. On the other hand, we trivially have that $$\P(\sA_{n,m}) \geq \P(n \leq |K_{X_0}|,|K_{X_m}| <\infty) \\- \P(\text{$X_0$ and $X_m$ belong to the same finite cluster of $\omega$})$$ for every $n,m\geq 1$. Write $\rho=\rho(G)$. Using \[thm:Bernoulli\_radius\] as in \[example:FiniteClusters\] to bound the first term and \[prop:Schramm\] to bound the second gives that $$\begin{aligned}
\P(\sA_{n,m})&\geq {\mathbf P}_p(n \leq |K_{o}| <\infty)^2 - \rho^m \left[ {\mathbf P}_p(n \leq |K_{o}| <\infty)-{\mathbf P}_p(n \leq |K_{o}| <\infty)^2\right] - \rho^m
\nonumber
\\
&\geq {\mathbf P}_p(n \leq |K_{o}| <\infty)^2 - 2 \rho^m,
\label{eq:using_rho}\end{aligned}$$ and hence by that $${\mathbf P}_p(n \leq |K_{o}| <\infty)^2 \leq 2\rho^m + \frac{42}{ p_0^m\sqrt{n}}$$ for every $n,m\geq 1$. The claim follows easily by taking $m=\lceil c \log n \rceil$ for an appropriate choice of constant $c=c(\deg(o),\rho)$; we omit the details.
Finite clusters in the random cluster model {#subsec:finite_FK}
-------------------------------------------
We now prove \[thm:finite\_clusters\], which concerns finite clusters in the random cluster model on transitive weighted graphs that are not necessarily locally finite.
Let us first discuss how the random cluster model may be represented as a percolation in random environment model via a non-integer version of the Edwards–Sokal coupling. This representation was first used by Bollobás, Grimmett, and Janson in the context of the complete graph [@bollobas1996random Section 3]. Let $q \geq 1$ and $\beta \geq 0$ and let $\omega$ be a sample of the random cluster measure $\phi_{q,\beta,0}$ on a weighted graph $G=(V,E,J)$ with $V$ finite. Given $\omega$, colour each *cluster* of $\omega$ *red* or *white* independently at random with probability $1/q$ to be coloured red, let $R$ be the set of vertices belonging to a red cluster, and let $\omega' \in \{0,1\}^E$ be defined by $\omega'(e)=\omega(e)\mathbbm{1}($both endpoints of $e$ belong to $R)$. Note that when $q \in \{2,3,\ldots\}$, the set $R$ has the same distribution as set of vertices that have some particular colour in the Potts model. For each set $A \subseteq V$, let $E(A)$ be the set of edges touching $A$, let $E_o(A)$ be the set of edges with both endpoints in $A$, and let $\overline{A}$ be the subgraph of $G$ with vertex set $V \setminus A$ and edge set $E \setminus E(A)$, where edges inherit their weights from $G$. It is shown in [@GrimFKbook Eq. (3.76)] that $$\begin{aligned}
\label{eq:red_percolation}
\P(R=A, \omega' = \xi)
&= \frac{Z_{\overline{A}}(q-1,\beta,0)}{Z_G(q,\beta,0)} \prod_{e\in E_o(A)} (e^{2\beta J_e}-1)^{\xi(e)} \end{aligned}$$ for every $A\subseteq V$ and $\xi \in \{0,1\}^{E}$ such that $\xi(e)=0$ for every edge $e \notin E_o(A)$, where $Z_G(q,\beta,h)$ is the partition function for the random cluster model on $G$. Since this expression depends on $\xi$ only through the product $\prod_{e\in E_o(A)} (e^{2\beta J_e}-1)^{\xi(e)}$, it follows that the conditional distribution of $\omega'$ given $R$ coincides with that of the Bernoulli bond percolation process on $E_o(R)$ in which each edge of $E_o(R)$ is included independently at random with inclusion probability $(e^{2\beta J_e}-1)/e^{2\beta J_e}=1-e^{-2\beta J_e}$. This allows us to think of the restriction of the random cluster model to the (random) set of red vertices as a percolation in random environment model.
(We note that for the FK-Ising model there is an alternative percolation in random environment representation, due to Lupu and Werner [@MR3485382], in which Bernoulli edges are added to the loop $O(1)$ model. See \[subsec:mainproof\] for further discussion. This representation could also be used to prove \[thm:finite\_clusters\] in the case $q=2$. In fact, using this representation makes the proof somewhat simpler in this case since the edge-inclusion probabilities ${\mathbf{p}}_e \geq \sinh(\beta J_e)/\cosh^2(\beta J_e)$ are bounded away from zero for each $e\in E$ almost surely.)
Let us now discuss how this representation extends to the infinite volume case and to models with non-zero external field. Let $G=(V,E,J)$ be an infinite, connected, weighted graph. Let $q\geq 1$, $\beta,h\geq 0$, and $\# \in \{f,w\}$, and let $\omega \in \{0,1\}^{E \cup V}$ be a random variable with law $\phi_{q,\beta,h}^\#$. Given $\omega$, we colour the clusters of $\omega$ red or white as follows:
1. Colour each cluster of $\omega$ intersecting the set $\{v:\omega(v)=1\}$ red.
2. If $\#=w$, colour each infinite cluster of $\omega$ red.
3. Choose to colour each remaining cluster of $\omega$ red or white independently at random, with probability $1/q$ to be coloured red.
Let $R$ be the set of vertices that are coloured red. It follows from \[eq:red\_percolation\] and a straightforward limiting argument that, conditional on $R$, the restriction of $\omega$ to $E_o(R) \cup R$ is a product measure in which $\P(\omega(x)=1\mid R)= (e^{2\beta J_x}-1)/e^{2\beta J_x}=1-e^{-2\beta J_x}$ for every $x \in E_o(R) \cup R$, where we write $J_v = h$ for every $v\in R$.
By scaling, we may assume without loss of generality that $\sum_{e\in E^\rightarrow_o} J_e=1/2$. Since the the restriction of $\phi^\#_{q,\beta,h}$ is stochastically dominated by the product measure $\phi^\#_{1,\beta,h}$ [@GrimFKbook Theorem 3.21], a simple counting argument as before yields that $\phi^\#_{q,\beta,h} |K_o| \leq \phi^\#_{1,\beta,h} |K_o| \leq 2$ for every $\beta \leq 1/2$, $h \geq 0$, $q\geq 1$, and $\# \in \{f,w\}$. This concludes the proof in this case, so that it suffices to consider the case $\beta \geq 1/2$. Fix $\beta \geq 1/2$, $h \geq 0$, $q\geq 1$, and $\# \in \{f,w\}$. Let $\omega$ be a random variable with law $\phi^\#_{q,\beta,h}$, let $R$ be the random subset of $V$ defined by colouring the clusters of $\omega$ red or white as above, and let $\omega_R \in \{0,1\}^E$ be defined by $\omega_R(e)=\omega(e) \mathbbm{1}($both endpoints of $e$ belong to $R)$. Thus, as discussed above, $\omega_R$ may be thought of as a percolation in random environment model in which the environment ${\mathbf{p}}$ is given by $${\mathbf{p}}_e = (1-e^{-2\beta J_e})\mathbbm{1}\left(e \in E_o(R)\right).$$ For a more general percolation in random environment model, the fact that these probabilities can be zero could be problematic. In our case, however, there is enough independence to pull the proof through with care.
Let $X$ be a random walk on $G$ started at $o$ and independent of $(\omega,R)$, let $X_{i,i+1}$ be the edge crossed by $X$ between times $i$ and $i+1$ for each $i\geq 0$, and let $J_i$ be the weight of the edge $X_{i,i+1}$ for each $i\geq 0$. For each $m\geq1$, let $\sR_m$ be the event that $X_i \in R$ for every $0 \leq i \leq m$. The definitions ensure that $$\label{eq:Rprob}
\P(\sR_m \mid \omega, X) \geq q^{-m-1}$$ for every $m \geq 1$. For each $i\geq 0$, let $\omega^i_R$ be obtained from $\omega_R$ by setting $\omega^0_R=\omega_R$ and $$\omega^i_R(e) =\begin{cases} 1 & e \in \{X_{j,j+1} : 0 \leq j \leq i-1\}\\
\omega_R(e) & e \notin \{X_{j,j+1} : 0 \leq j \leq i-1\}
\end{cases}$$ for each $i\geq 1$ and $e\in E$. For each $m \geq 1$ and $\lambda>0$, let $\sA_{\lambda,m}$ be the event that the cluster of $X_0=o$ in $\omega$ is finite and that $X_0$ and $X_m$ are in distinct clusters of $\omega$ each of which touches a set of edges with total weight at least $\lambda$. For each $m \geq 1$, $\lambda>0$, and $1\leq i \leq m$, let $\sB_{\lambda,m,i}$ be the event that the following hold:
1. $X_0$ and $X_m$ are in distinct clusters of $\omega^{i-1}_R$ each of which touches a set of edges with total weight at least $\lambda$,
2. the cluster of $X_0$ is finite in $\omega^{i-1}_R$, and
3. either $X_0$ and $X_m$ are connected in $\omega^i_R$ or the cluster of $X_0$ is infinite in $\omega^i_R$.
On the event $\sA_{\lambda,m} \cap \sR_m$ the vertices $X_0$ and $X_m$ are connected in $\omega^m_R$ and not connected in $\omega^0_R$, and since $\omega^i_R$ is monotone increasing in $i$ it follows that $$\label{eq:ABsetinclusionFK}
\sA_{\lambda,m} \cap \sR_m \subseteq \bigcup_{i=1}^m \sB_{\lambda,m,i} \cap \sR_m$$ for every $m\geq 1$ and $\lambda>0$. Now, for each $i\geq 1$ and $\lambda>0$ let $\sC_{\lambda,i}$ be the event that the cluster of $X_{i-1}$ in $\omega_R$ is finite and that $X_{i-1}$ and $X_{i}$ are in distinct clusters of $\omega_R$ each of which touches a set of edges with total weight at least $\lambda$. Observe that $\sC_{\lambda,i} \cap \sR_m \supseteq \sB_{\lambda,m,i} \cap \{\omega_R(X_{j,j+1})=1 \text{ for every $0\leq j \leq i-2$}\} \cap \sR_m$ for every $n,m\geq 1$ and $1\leq i \leq m$. The events $\sB_{\lambda,m,i}$ and $\{\omega_R(X_{j,j+1})=1$ for every $0\leq j \leq i-2\}$ are conditionally independent given the random walk $X$ and the set $R$, and we deduce that $$\begin{aligned}
\P\bigl(\sC_{\lambda,i} \cap \sR_m \mid X,R) &\geq \mathbbm{1}(\sR_m)\P(\omega_R(X_{j,j+1})=1 \text{ for every $0\leq j \leq i-2$}\mid X,R) \P\bigl( \sB_{\lambda,m,i} \mid X,R)
\nonumber
\\&\geq \mathbbm{1}(\sR_m) \P(\sB_{\lambda,m,i} \mid X,R) \prod_{j=0}^{m-2} (1-e^{-2\beta J_j})
\label{eq:FK_R_Surgery}
$$ for every $m\geq 1$, $\lambda>0$, and $1\leq i \leq m$.
To proceed, we will first complete the proof under the additional assumption that there exists $\alpha<1$ such that $\sum_{e \in E^\rightarrow_o} J_e^{\alpha}<\infty$, which holds trivially in the locally finite case, before explaining how this assumption can be removed. Under this assumption we have that $\E J_\eta^{-(1-\alpha)} = \E e^{-(1-\alpha)\log J_\eta} < \infty$ and since $\beta \geq 1/2$ and $1-e^{-x} \geq x/2$ for every $x \in [0,1]$ we deduce that $$\begin{gathered}
\E \exp\left( -(1-\alpha) \log (1-e^{-2\beta J_\eta}) \right) \leq
\E \exp\left( -(1-\alpha) \log (1-e^{-J_\eta}) \right) \\ \leq 2 \E \exp\left( -(1-\alpha) \log J_\eta \right)<\infty.
$$ Since the random variables $(J_i)_{i\geq 0}$ are i.i.d., we have by a Chernoff bound that there exists a finite constant $C_1$ such that $$\begin{aligned}
\P\left(\prod_{j=0}^{m-2} (1-e^{-2\beta J_j}) \leq e^{-C_1 (m-1)} \right)
&=
\P\left(\sum_{j=0}^{m-2} -\log (1-e^{-2\beta J_j}) \geq C_1 (m-1) \right)
\\&\leq e^{-C_1(1-\alpha)(m-1)}\left[2 \E \exp\left( -(1-\alpha) \log J_\eta \right)\right]^{m-1} \leq q^{-2(m-1)}\end{aligned}$$ for every $m \geq 1$. For each $m \geq 1$, let $\sW_{m}$ be the event that $\prod_{j=0}^{m-2} (1-e^{-2\beta J_j}) \geq e^{-C_1 (m-1)}$. It follows from that $$\begin{aligned}
\P(\sB_{\lambda,m,i} \cap \sR_m \mid X,R) \leq e^{C_1 (m-1)} \P\bigl(\sC_{\lambda,i} \cap \sR_m \mid X,R) + \mathbbm{1}(\sW_m)
$$ for every $m\geq 1$, $\lambda>0$, and $1\leq i \leq m$. Taking expectations and applying we deduce that $$\begin{aligned}
\label{eq:Bernoulli_surgeryFK}
\P(\sA_{\lambda,m} \cap \sR_m) &\leq m q^{-2(m-1)} + e^{C_1 (m-1)} \sum_{i=1}^m \P\bigl(\sC_{\lambda,i} \cap \sR_m).\end{aligned}$$ for every $m\geq 1$ and $\lambda>0$. There is easily seen to exist a positive constant $c_1$ such that ${\mathbf{p}}_e / (1-{\mathbf{p}}_e) J_e \geq c_1 \mathbbm{1}(e\in E_o(R))$ for every $\beta \geq 1/2$. Thus, \[cor:two\_ghost\_S\] yields that there exists a constant $C_2 = 42/c_1$ such that $$\P\bigl(\sC_{\lambda ,i} \cap \sR_m ) \leq \frac{C_2}{\sqrt{\lambda}}$$ for every $m\geq 1$, $\lambda>0$, and $1 \leq i \leq m$ and hence that $$\P(\sA_{\lambda,m} \cap \sR_m) \leq m q^{-2(m-1)} + m e^{C_1 (m-1)} \frac{C_2}{\sqrt{\lambda}}$$ for every $m\geq 1$ and $\lambda>0$. Since $\sA_{\lambda,m}$ is measurable with respect to the $\sigma$-algebra generated by $\omega$ and $X$, it follows from that there exist finite constants $C_3$ and $C_4$ such that $$\begin{gathered}
\P(\sA_{\lambda,m})= \frac{\P(\sA_{\lambda,m} \cap \sR_m)}{\P(\sR_m \mid \sA_{\lambda,m})} \leq q^{m+1}\P(\sA_{\lambda,m} \cap \sR_m) \\\leq m q^{-m+3} + m e^{C_1 m} q^{m+1}\frac{C_2}{\sqrt{\lambda}}
\leq m q^{-m+3} + \frac{C_3 e^{C_4 m}}{\sqrt{\lambda}}
\label{eq:FK_surgery}\end{gathered}$$ for every $m\geq 1$ and $\lambda>0$.
We may now conclude the proof in an essentially identical way to the proof of \[thm:finite\_percolation\]. Indeed, we have trivially have that $$\begin{gathered}
\P(\sA_{\lambda,m}) \geq \P(\lambda \leq |E(K_{X_0})|_J,|E(K_{X_m})|_J <\infty) \\- \P(\text{$X_0$ and $X_m$ belong to the same finite cluster of $\omega$})\end{gathered}$$ for every $m\geq 1$ and $\lambda>0$. We deduce from \[thm:Ising\_factor,prop:Schramm\] that $$\begin{aligned}
\P(\sA_{\lambda,m})
&\geq \P(\lambda \leq |E(K_{X_0})|_J <\infty)^2 - 2 \rho(G)^m,
\label{eq:using_rho_FK}\end{aligned}$$ and hence by that $$\P(\lambda \leq |E(K_{X_0})|_J <\infty)^2 \leq 2\rho(G)^m + m q^{-m+3} + \frac{C_3 e^{C_4 m}}{\sqrt{\lambda}}$$ for every $m\geq 1$ and $\lambda>0$. As before, the claim follows easily by taking $m=\lceil c \log \lambda \rceil$ for an appropriate choice of constant $c$.
Let us now briefly indicate how the assumption that $\sum_{e\in E^\rightarrow_o} J_e^\alpha < \infty$ for some $\alpha<1$ can be removed; if the reader is only interested in the locally finite case they may safely skip this paragraph. First, we easily verify from the definitions that the connected, transitive weighted graph $G'=(V,E,J^2)$ is nonamenable if and only if $G=(V,E,J)$ is. Moreover, any automorphism-invariant percolation process on $G$ may also be thought of as an automorphism-invariant percolation process on $G'$, and this change in perspective does not affect whether or not the process is a factor of i.i.d. Applying \[thm:Ising\_factor\], it follows in particular that the random cluster model on $G$ has spectral radius at most $\rho(G')<1$ when considered as a percolation process on $G'$. Moreover, if $\eta'$ is a random edge emanating from $o$ chosen with probability proportional to $J_e^2$ then we have that $\E J_{\eta'}^{-1} = (\sum_{e\in E^\rightarrow_o} J_e)/(\sum_{e\in E^\rightarrow_o} J_e^2)<\infty$ and hence that $$\E \exp\left( -\log \frac{e^{2\beta J_{\eta'}}-1}{e^{2\beta J_{\eta'}}} \right) \leq
\E \exp\left( - \log \frac{e^{J_{\eta'}}-1}{e^{J_{\eta'}}} \right) \leq 2 \E \exp\left( - \log J_{\eta'} \right)<\infty
$$ for every $\beta \geq 1/2$. These observations allow us to straightforwardly extend the above analysis to arbitrary connected, nonamenable, transitive weighted graphs by considering the random walk on $G'$ instead of $G$; we omit the details.
Fix $q \geq 1$. It is proven in [@1901.10363] that $\phi^\#_{q,\beta,0} |K_o| < \infty$ for every $\beta<\beta_c^\#(q)$ and $\# \in \{f,w\}$. As in [@Hutchcroft2016944], the FKG inequality implies that the sequence $$\kappa_{q,\beta}^\#(m):=\inf\left\{\phi^f_{q,\beta,0}(x \leftrightarrow y) : x,y \in V, d(x,y) \leq m \right\}
$$ is supermultiplicative in the sense that $\kappa_{q,\beta}^\#(n+m)\geq \kappa_{q,\beta}^\#(n)\kappa_{q,\beta}^\#(m)$ for every $n,m\geq 1$, $\beta \geq 0$, and $\#\in \{f,w\}$, and it follows from Fekete’s lemma [@grimmett2010percolation Appendix II] that $$\sup_{m\geq 1} \kappa_{q,\beta}^\#(m)^{1/m} = \lim_{m\to \infty} \kappa_{q,\beta}^\#(m)^{1/m} \leq \liminf_{m\to\infty} \left(\frac{\phi^\#_{q,\beta,0} |K_o|}{|B(o,m)|} \right)^{1/m} = \frac{1}{\operatorname{gr}(G)}$$ for every $\# \in \{f,w\}$ and $0\leq \beta < \beta_c^\#(q)$, where $\operatorname{gr}(G)=\limsup_{n\to\infty} |B(o,n)|^{1/n}$ is the exponential growth rate of $G$. Following a very similar argument to that of \[thm:finite\_clusters\] but using a geodesic between two points $x$ and $y$ with $d(x,y)=m$ minimizing $\phi^f_{q,\beta,0}(x \leftrightarrow y)$ instead of a random walk and using the FKG inequality instead of spectral considerations in yields that there exist constants $C$ and $\delta$ (depending on $\operatorname{gr}(G)$, $\deg(o)$, and $q$) such that $$\phi^\#_{q,\beta,0}(|K_o|\geq n) \leq C n^{-\delta}$$ for every $\# \in \{f,w\}$, $n\geq 1$, and $0\leq \beta<\beta_c^f$. We conclude by taking $\beta \uparrow \beta_c^f(q)$ and using left-continuity of the free random cluster measure $\phi^f_{q,\beta,0}$.
A general continuity theorem
----------------------------
To illustrate the flexibility of the method of proof developed here, and for possible future applications, let us also make note of the following very general theorem for percolation in random environment models with all edge probabilities positive. This theorem is not needed for the proofs of our main results.
\[thm:general\_AKN\] Let $G=(V,E,J)$ be a connected, transitive, nonamenable weighted graph and let $\Gamma \subseteq {\operatorname{Aut}}(G)$ be a closed transitive unimodular subgroup of automorphisms. Let $\cM$ be a tight family of $\Gamma$-invariant probability measures on $(0,1]^E$ with $\sup_{\mu \in \cM} \rho(\mu) < 1$. Then there exists a decreasing function $f:\N\to [0,1]$ such that $\lim_{n\to\infty} f(n)=0$ and $${\mathbf P}_{\mu}(n \leq |K_o| < \infty) \leq f(n)$$ for every $n\geq 1$ and $\mu \in \cM$. In particular, $\cM_\infty=\{\mu \in \cM : {\mathbf P}_\mu$ is supported on configurations with no infinite clusters$\}$ is a weakly closed subset of $\cM$.
Note that the assumption that $\sup_{\mu \in \cM} \rho(\mu) < 1$ can be replaced by the assumption that $\cM=\overline{\cM}_\infty$ and that every measure in $\cM$ is positively associated.
By scaling, we may assume without loss of generality that $\sup_e J_e \leq 1$. The assumption that $\cM$ is tight on $(0,1]^E$ is equivalent to the assertion that there exists an increasing function $g:(0,1]\to [0,1]$ with $\lim_{{\varepsilon}\downarrow 0}g({\varepsilon})=0$ such that $$\P_\mu({\mathbf{p}}_\eta \leq {\varepsilon}) \leq g({\varepsilon})$$ for every $\mu \in \cM$ and ${\varepsilon}>0$.
Let $\mu \in \cM$, let $({\mathbf{p}},\omega)$ be drawn from ${\mathbf P}_\mu$, and let $X=(X_m)_{m\geq 0}$ be an independent random walk on $G$ independent of $({\mathbf{p}},\mu)$. Let the modified configurations $(\omega^i)_{i\geq 0}$ be defined as in the proof of \[thm:finite\_percolation\]. Similarly, for each $\lambda>0$, $m\geq 1$, and $1 \leq i \leq m$, let the events $\sA_{\lambda,m}$, $\sB_{\lambda,m,i}$, and $\sC_{\lambda,i}$ be defined as in the proof of \[thm:finite\_percolation\] but replacing each instance of the phrase ‘touches at least $n$ edges’ with ‘touches a set of edges of total weight at least $\lambda$’. Thus, we have as before that $$\sA_{\lambda,m} \subseteq \bigcup_{j=1}^m\sB_{\lambda,m,j} \quad \text{ and } \quad \sC_{\lambda,i} \supseteq \sB_{\lambda,m,i} \cap \{\omega(X_j,X_{j+1})=1 \text{ for every $0 \leq j \leq i-2$}\}$$ for every $\lambda>0$, $m\geq 1$, and $1\leq i \leq m$. For each ${\varepsilon}>0$ and $i\geq 1$ let $\sD_{{\varepsilon},i}$ be the event that ${\mathbf{p}}_{X_{j,j+1}} \geq {\varepsilon}$ for every $0\leq j \leq i-2$. The events $\sB_{\lambda,m,i}$ and $\{\omega(X_j,X_{j+1})=1$ for every $0 \leq j \leq i-2\}$ are conditionally independent given $X$ and ${\mathbf{p}}$, so that $$\begin{aligned}
\P\bigl(\sC_{\lambda,i} \cap \sD_{{\varepsilon},i}) &\geq \E \left[ \mathbbm{1}(\sD_{{\varepsilon},i})\P(\omega(X_{j,j+1})=1 \text{ for every $0\leq j \leq i-2$}\mid {\mathbf{p}},X) \P\bigl( \sB_{\lambda,m,i}\mid {\mathbf{p}},X)\right]\\
&
\geq {\varepsilon}^{i-1} \P(\sB_{\lambda,m,i} \cap \sD_{{\varepsilon},i}) \geq {\varepsilon}^{i-1} \P(\sB_{\lambda,m,i})-{\varepsilon}^{i-1}\P(\sD_{{\varepsilon},i}^c)
\geq {\varepsilon}^{i-1} \P(\sB_{\lambda,m,i})-{\varepsilon}^{i-1}(i-1)g({\varepsilon})
$$ for every $\lambda,{\varepsilon}>0$ and $m \geq i \geq 1$, where the final two inequalities follow by union bounds. Meanwhile, \[cor:two\_ghost\_S\] and the assumption that $\sup_e J_e \leq 1$ imply that $$\P(\sC_{\lambda,i} \cap \sD_{{\varepsilon},i}) \leq 42 \sqrt{\frac{1-{\varepsilon}}{{\varepsilon}}} \frac{1}{\sqrt{\lambda}}$$ for every $\lambda>0$ and $i\geq 1$. Rearranging, we deduce that $$\begin{gathered}
\P(\sA_{\lambda,m}) \leq \sum_{i=1}^m \left[{\varepsilon}^{-i+1}\P(\sC_{\lambda,i} \cap \sD_{{\varepsilon},i}) + (i-1)g({\varepsilon})\right]
\leq
\sum_{i=1}^m \left[42{\varepsilon}^{-i+1} \sqrt{\frac{1-{\varepsilon}}{{\varepsilon}}}\frac{1}{\sqrt{\lambda}} + (i-1)g({\varepsilon})\right]
\\
\leq 42 {\varepsilon}^{-m} \sqrt{\frac{{\varepsilon}}{1-{\varepsilon}}}\frac{1}{\sqrt{\lambda}} + \binom{m}{2} g({\varepsilon})\end{gathered}$$ for every $\lambda,{\varepsilon}>0$ and $m\geq 1$. On the other hand, using \[prop:Schramm\] and the definition of the spectral radius as in yields that $$\P(\sA_{\lambda,m}) \geq \P(\lambda \leq |E(K_o)|_J<\infty)^2 - \rho(\mu)^m-\rho(G)^m$$ and hence that $$\P_\mu(\lambda \leq |E(K_o)|_J<\infty)^2 \leq \rho(\mu)^m+\rho(G)^m+42 {\varepsilon}^{-m} \sqrt{\frac{{\varepsilon}}{1-{\varepsilon}}}\frac{1}{\sqrt{\lambda}} + \binom{m}{2} g({\varepsilon})$$ for every $\lambda,{\varepsilon}>0$ and $m\geq 1$.
The claim now follows by appropriate choice of ${\varepsilon}>0$ and $m\geq 1$: For each ${\varepsilon}>0$ let $m({\varepsilon})$ be maximal such that $\binom{m}{2} \leq g({\varepsilon})^{-1/2}$ and for each $\lambda>0$ let ${\varepsilon}(\lambda)>0$ be minimal such that $42 {\varepsilon}^{-m({\varepsilon})} \sqrt{\frac{{\varepsilon}}{1-{\varepsilon}}} \leq \lambda^{1/4}$. Then we have that $\lim_{{\varepsilon}\downarrow 0} m({\varepsilon}) = \infty$ and $\lim_{\lambda \uparrow \infty} {\varepsilon}(\lambda)=0$. Thus, if we define $f:(0,\infty)\to (0,\infty)$ by $$f(\lambda)^2 = \sup_{\nu \in \cM} \rho(\nu)^{m({\varepsilon}(\lambda))} + \rho(G)^{m({\varepsilon}(\lambda))} + \lambda^{-1/4} + \sqrt{g({\varepsilon}(\lambda))}$$ for every $\lambda>0$ then $f$ is decreasing, $\lim_{\lambda\uparrow \infty}f(\lambda)=0$, and $$\P_\mu(\lambda \leq |E(K_o)|_J<\infty) \leq f(\lambda)$$ for every $\lambda>0$. The first claim follows since $\mu\in \cM$ was arbitrary. It follows in particular that $$\P_\mu( |E(K_o)|_J\geq \lambda) \leq f(\lambda)$$ for every $\mu\in \cM_\infty$ and $\lambda>0$. The portmanteau theorem implies that the same estimate holds for every $\mu \in \overline{\cM_{\infty}}$ and $\lambda>0$, completing the proof.
Analysis of the Ising model
===========================
In this section we apply the technology developed in \[sec:free\_energy\] to prove our main theorems, \[thm:main,thm:main\_simple,thm:main\_continuity,thm:main\_continuity\_FK\].
It will be notationally convenient throughout this section for us to consider both the Ising measures ${\mathbf{I}}_{\beta,h}^\#$ and the **gradient Ising measures** ${\mathbf{G}}_{\beta,h}^\#$, defined as follows. Let $G=(V,E,J)$ be an infinite, connected weighted graph. For each $\sigma \in \{-1,1\}^V$ and $h\geq 0$, we define the **gradient** $\nabla_h \sigma \in \R^{E \cup V}$ by $\nabla_h \sigma (e) = J_e \sigma_e = J_e \sigma_x \sigma_y$ for each $e\in E$ with endpoints $x$ and $y$ and $\nabla_h \sigma (v) = h \sigma_v$ for each $v\in V$. For each $\beta,h \geq 0$ we define ${\mathbf{G}}_{\beta,h}^f$ and ${\mathbf{G}}_{\beta,h}^w$ to be the push-forwards of ${\mathbf{I}}_{\beta,h}^f$ and ${\mathbf{I}}_{\beta,h}^+$ through the gradient $\nabla_h$. That is, if $\sigma$ is a random variable with law ${\mathbf{I}}_{\beta,h}^+$ then the random variable $\nabla_h \sigma$ has law ${\mathbf{G}}_{\beta,h}^w$, with a similar statement holding in the free case. The gradient Ising measure ${\mathbf{G}}_{\beta,h}$ on a finite weighted graph is defined similarly. When $h>0$ we can trivially recover $\sigma$ from $\nabla_h \sigma$, so that the two measures are just different ways of thinking about the same object. On the other hand, when $h=0$, $\nabla_0 \sigma$ only retains the even information about the configuration $\sigma$, so that the difference is more genuine, and the two measures can have rather different properties. For example, it is possible for ${\mathbf{G}}^f_{\beta,0}$ to be a factor of i.i.d. in situations when ${\mathbf{I}}^f_{\beta,0}$ is not even ergodic [@ray2019finitary].
Double random currents and the loop $O(1)$ model {#subsec:random_currents}
------------------------------------------------
We now introduce the (double) random current and loop $O(1)$ models, referring the reader to [@MR3890455] for further background. The random current model was introduced by Griffiths, Hurst, and Sherman [@griffiths1970concavity] and developed extensively by Aizenman [@MR678000]. It has been of central importance to most modern work on the Ising model, with notable recent applications including [@MR4026609; @MR4089494; @aizenman2019marginal; @MR4072233; @duminil2015new].
Let $G=(V,E,J)$ be a weighted graph with $V$ finite. A **current** $\mathbf{n}=(\mathbf{n}_e)_{e\in E}=(\mathbf{n}(e))_{e\in E}$ on $G$ is an assignment of non-negative integers to the edges of $G$. We write $\Omega_G$ for the set of currents on $G$. A vertex $v$ of $G$ is said to be a **source** of the current $\mathbf{n}$ if $\sum_{e\in E^\rightarrow_v} \mathbf{n}_e$ is odd, and the set of sources of $\mathbf{n}$ is denoted $\partial \mathbf{n}$. For each current $\mathbf{n}$ and $\beta >0$, we define $$w_\beta(\mathbf{n}) = \prod_{e\in E} \frac{(\beta J_e)^{\mathbf{n}_e}}{n_e!}.
$$ Many quantities of interest for the Ising model can be expressed in terms of sums over currents. For example, if $G=(V,E,J)$ is a weighted graph with $V$ finite and $x$ and $y$ are vertices of $G$ then $$\label{eq:firstrandomcurrent}
\langle \sigma_x \sigma_y \rangle_{G,\beta,0} = \frac{\sum_{\mathbf{n}\in \Omega_G:\partial \mathbf{n}=\{x,y\}} w_\beta(\mathbf{n})}{\sum_{\mathbf{n}\in \Omega_G:\partial \mathbf{n}=\emptyset} w_\beta(\mathbf{n})}$$ for every $\beta\geq 0$. This formula becomes much more useful when combined with the following fundamental lemma of Griffiths, Hurst, and Sherman [@griffiths1970concavity], known as the *switching lemma*. We write ‘$x \xleftrightarrow{{\mathbf{n}}_1+{\mathbf{n}}_2} y$ in $H$’ to mean that there exists a path connecting $x$ to $y$ in $H$ all of whose edges $e$ have ${\mathbf{n}}_1(e)+{\mathbf{n}}_2(e)\geq 1$. The following statement of the switching lemma is adapted from [@MR3306602 Lemma 2.2][^6].
\[lem:switching\] Let $G$ be a weighted finite graph, let $H$ be a subgraph of $G$, let $x$ and $y$ be vertices of $H$, and let $A$ be a set of vertices of $G$. Then $$\begin{gathered}
\sum_{\substack{{\mathbf{n}}_1 \in \Omega_H : \partial {\mathbf{n}}_1 =\{x,y\}\\{\mathbf{n}}_2 \in \Omega_G : \partial{\mathbf{n}}_2 =A}} F({\mathbf{n}}_1+{\mathbf{n}}_2) w_\beta({\mathbf{n}}_1)w_\beta({\mathbf{n}}_2)
\\=
\sum_{\substack{{\mathbf{n}}_1 \in \Omega_H : \partial {\mathbf{n}}_1 =\emptyset\\{\mathbf{n}}_2 \in \Omega_G : \partial{\mathbf{n}}_2 =A\Delta\{x,y\}}} F({\mathbf{n}}_1+{\mathbf{n}}_2) w_\beta({\mathbf{n}}_1)w_\beta({\mathbf{n}}_2) \mathbbm{1}\left(x \xleftrightarrow{{\mathbf{n}}_1+{\mathbf{n}}_2}y \text{ \emph{in} $H$}\right)\end{gathered}$$ for every $F:\Omega_G\to [0,\infty]$ and $\beta\geq 0$, where $ A \Delta B = A \cup B \setminus A \cap B$ denotes the symmetric difference of two sets.
It follows in particular that if $G$ is a finite graph and $H$ is a subgraph of $G$ then $$\begin{gathered}
\langle \sigma_x \sigma_y \rangle_{H,\beta,0}\langle \sigma_x \sigma_y \rangle_{G,\beta,0}=\frac{\sum_{\mathbf{n_1}\in \Omega_H:\partial \mathbf{n_1}=\{x,y\}} w_\beta(\mathbf{n_1})}{\sum_{\mathbf{n_1}\in \Omega_H:\partial \mathbf{n_1}=\emptyset} w_\beta(\mathbf{n_1})}
\frac{\sum_{\mathbf{n_2}\in \Omega_G:\partial \mathbf{n_2}=\{x,y\}} w_\beta(\mathbf{n_2})}{\sum_{\mathbf{n_2}\in \Omega_G:\partial \mathbf{n_2}=\emptyset} w_\beta(\mathbf{n_2})}
\\
=\frac{\sum_{\mathbf{n_1}\in \Omega_H:\partial \mathbf{n_1}=\emptyset}\sum_{\mathbf{n_2}\in \Omega_G:\partial \mathbf{n_2}=\emptyset} w_\beta(\mathbf{n_1})w_\beta(\mathbf{n_2})\mathbbm{1}(x \xleftrightarrow{\mathbf{n_1}+{\mathbf{n}}_2}y \text{ in $H$})}
{\sum_{\mathbf{n}\in \Omega_H:\partial \mathbf{n_1}=\emptyset} w_\beta(\mathbf{n_1})\sum_{\mathbf{n_2}\in \Omega_G:\partial \mathbf{n_2}=\emptyset} w_\beta(\mathbf{n_2})}
\label{eq:correlations_to_DRC0}\end{gathered}$$ for every two vertices $x$ and $y$ of $H$ and every $\beta\geq 0$. This formula motivates the definitions of the *random current* and *double random current* models. Given a finite graph $G$ and $\beta \geq 0$, we define the **random current measure** $\mathbf{C}_{G,\beta}$ on $\Omega_G$ by setting $$\mathbf{C}_{G,\beta}(\{\mathbf{n}\}) = \frac{w_\beta(\mathbf{n})\mathbbm{1}(\partial {\mathbf{n}}= \emptyset)}{\sum_{{\mathbf{m}}\in \Omega_G}w_\beta(\mathbf{m})\mathbbm{1}(\partial {\mathbf{m}}= \emptyset)}$$ for each current ${\mathbf{n}}\in \Omega_G$. The equality can be rewritten succinctly in terms of this measure as follows: If $G$ is a finite graph and $H$ is a subgraph of $G$ then $$\begin{aligned}
\langle \sigma_x \sigma_y \rangle_{H,\beta,0}\langle \sigma_x \sigma_y \rangle_{G,\beta,0}&=\mathbf{C}_{H,\beta}\otimes\mathbf{C}_{G,\beta}
\bigl(\bigl\{
({\mathbf{n}}_1,{\mathbf{n}}_2) : x \xleftrightarrow{{\mathbf{n}}_1+{\mathbf{n}}_2} y
\text{ in $H$}
\bigr\}\bigr)
\label{eq:correlations_to_DRC}\end{aligned}$$ for every two vertices $x$ and $y$ of $H$ and every $\beta\geq 0$. This equality leads us naturally to consider the double random current model (i.e., the measure $\mathbf{C}_{H,\beta}\otimes\mathbf{C}_{G,\beta}$) as a percolation model on $H$, where an edge is open if it takes a positive value in at least one of the two currents.
Correlations of the Ising model in non-zero external field may also be expressed in terms of the random current model as follows. Let $G=(V,E,J)$ be a weighted graph with $V$ finite, let $h>0$, and let $\overline{G}_h$ be the graph obtained from $G$ by the addition of a special vertex $\partial$ that is connected to every vertex of $G$ by a single edge of weight $h$, so that the edge set of $\overline{G}_h$ is naturally identified with $V \cup E$. We observe that the gradient Ising measure with zero external field on $\overline{G}_h$ coincides with the gradient Ising measure on $G$ with external field $h$, and define the random current measure ${\mathbf{C}}_{G,\beta,h}={\mathbf{C}}_{\overline{G}_h,\beta,0}$ for each $\beta\geq 0$, which we consider as a probability measure on $\N_0^E \times \N_0^V$. Since every subgraph $H$ of $G$ is also a subgraph of $\overline{G}_h$, we deduce from that $$\begin{aligned}
\langle \sigma_x \sigma_y \rangle_{H,\beta,0}\langle \sigma_x \sigma_y \rangle_{G,\beta,h}&=\mathbf{C}_{H,\beta,0}\otimes\mathbf{C}_{G,\beta,h}
\bigl(\bigl\{
({\mathbf{n}}_1,{\mathbf{n}}_2) : x \xleftrightarrow{{\mathbf{n}}_1+{\mathbf{n}}_2} y
\text{ in $H$}
\bigr\}\bigr).
\label{eq:correlations_to_DRC2}\end{aligned}$$ for every $\beta,h \geq 0$ and every two vertices $x$ and $y$ of $H$. For consistency, we will from now on consider ${\mathbf{C}}_{G,\beta,0}$ as a probability measure on $\N_0^E \times \N_0^V$ that is supported on configurations in which ${\mathbf{n}}_v=0$ for every $v\in V$.
**Relation to the loop $O(1)$ model.** Let $G=(V,E,J)$ be a weighted graph with $V$ finite and let $\beta, h\geq 0$. To ease notation, we will write $J_v=h$ for each vertex $v$ of $G$. It follows from the definitions that if ${\mathbf{n}}=({\mathbf{n}}_x)_{x\in V \cup E}$ is a random variable with law $\mathbf{C}_{G,\beta,h}$ then the values of ${\mathbf{n}}$ are conditionally independent given the the field of parities $(\mathbbm{1}({\mathbf{n}}_x \text{ odd}))_{x\in V \cup E}$. Indeed, if we let $(\mathsf{Odd}_x)_{x\in V \cup E}$ and $(\mathsf{Even}_x)_{x\in V \cup E}$ be non-negative integer-valued random variables, independent of ${\mathbf{n}}$, such that all the random variables $(\mathsf{Odd}_x)_{x\in V \cup E}$ and $(\mathsf{Even}_x)_{x\in V \cup E}$ are mutually independent with laws $$\label{eq:odd_even_distribution}
\P(\mathsf{Odd}_x = n) = \frac{(\beta J_x)^n}{n! \sinh(\beta J_x)}\mathbbm{1}(n \text{ odd}) \quad \text{ and } \quad \P(\mathsf{Even}_x = n) = \frac{(\beta J_x)^n}{n! \cosh(\beta J_x)}\mathbbm{1}(n \text{ even})$$ then $$({\mathbf{n}}_x)_{x\in V \cup E} \quad \text{ has the same distribution as } \quad \Bigl(\mathsf{Odd}_x\mathbbm{1}({\mathbf{n}}_x \text{ odd})+\mathsf{Even}_x\mathbbm{1}({\mathbf{n}}_x \text{ even})\Bigr)_{x\in V \cup E}.$$ This leads to a special role for the sign field $(\mathbbm{1}({\mathbf{n}}_x \text{ odd}))_{x\in V\cup E}$. The law of this random variable under $\mathbf{C}_{G,\beta,h}$ is denoted by $\mathbf{L}_{G,\beta,h}$ and is known as the **loop $O(1)$ measure** on the graph $G$. The loop $O(1)$ measure on the weighted graph $G=(V,E,J)$ with $V$ finite can also be defined explicitly as the unique purely atomic probability measure on $\{0,1\}^{E \cup V}$ satisfying $$\mathbf{L}_{G,\beta,h}(\{\omega\}) \propto \mathbbm{1}(\partial \omega = \emptyset)\prod_{v\in V} \tanh(\beta h)^{\mathbbm{1}(\omega(v)=1)} \prod_{e\in E} \tanh(\beta J_e)^{\mathbbm{1}(\omega(e)=1)}$$ for each $\omega \in \{0,1\}^{V \cup E}$, where $\partial \omega = \{v\in V: \omega(v)+\sum_{e\in E^\rightarrow_v} \omega(e)$ is odd$\}$.
A further very useful expression for the distribution of the loop $O(1)$ model in terms of the (gradient) Ising model, proven in [@MR3306602 Equations 2.10-2.12], states that if $G=(V,E,J)$ is a weighted graph with $V$ finite and $\beta,h \geq 0$ then $$\begin{aligned}
\nonumber
\mathbf{L}_{G,\beta,h} (\{ \omega : \omega(x) = 0 \text{ for every } x\in A\}) &= \Bigl\langle \exp\Bigl(-\beta \sum_{x\in A} J_x \sigma_{x}\Bigr) \Bigr\rangle_{G,\beta,h} \prod_{x\in A} \cosh(\beta J_x) \\
&=C_{\beta,h}(A) \mathbf{G}_{G,\beta,h}\left[e^{\beta H_A}\right]
\label{eq:LoopO(1)Ising}\end{aligned}$$ for every finite set $A \subseteq E \cup V$, where we set $H_A=H_A(\sigma)=-\sum_{x\in A} J_x \sigma_{x}$ and $C_{\beta,h}(A) = \prod_{x\in A} \cosh(\beta J_x)$ and recall that we write $\sigma_e=\sigma_x \sigma_y$ for an edge $e$ with endpoints $x$ and $y$. Note that the equation completely characterizes the measure $\mathbf{L}_{G,\beta,h}$ since, by inclusion-exclusion, the family of indicator functions $\{\mathbbm{1}(\omega(x) = 0 \text{ for every } x\in A) : A \subseteq E \cup V\}$ has linear span equal to the space of all functions from $\{0,1\}^{E \cup V}$ to $\R$ depending on at most finitely many edges and vertices.
As observed in [@MR3306602], the equation allows us to deduce various statements about infinite volume limits of the loop $O(1)$ and random current models from the corresponding statements concerning the (gradient) Ising model. Indeed, let $G=(V,E,J)$ be an infinite connected weighted graph, let $(V_n)_{n\geq 0}$ be an exhaustion of $G$, and let $(G_n)_{n\geq 0}$ and $(G_n^*)_{n\geq 0}$ be defined as in \[subsec:intro\_definitions\]. It follows from together with the corresponding statements for the Ising model that the weak limits $$\begin{aligned}
\mathbf{L}_{G,\beta,h}^f &= {\mathop{\operatorname{w-lim}}}_{n\to\infty} \mathbf{L}_{G_n,\beta,h}, &\mathbf{L}_{G,\beta,h}^w &= {\mathop{\operatorname{w-lim}}}_{n\to\infty} \mathbf{L}_{G_n^*,\beta,h},\\
\mathbf{C}_{G,\beta,h}^f &= {\mathop{\operatorname{w-lim}}}_{n\to\infty} \mathbf{C}_{G_n,\beta,h}, \text{ and } &\mathbf{C}_{G,\beta,h}^w &= {\mathop{\operatorname{w-lim}}}_{n\to\infty} \mathbf{C}_{G_n^*,\beta,h}\end{aligned}$$ all exist and do not depend on the choice of exhaustion. Moreover, these infinite volume loop $O(1)$ measures are related to the infinite volume Ising measures by the relations $$\begin{aligned}
\mathbf{L}_{G,\beta,h}^\# (\{ \omega : \omega(x) = 0 \text{ for every } x\in A\}) =C_{\beta,h}(A)\mathbf{G}_{\beta,h}^\#\left[e^{\beta H_A}\right]
\label{eq:LoopO(1)Ising_infinite},
$$ which holds for every finite set $A \subseteq V \cup E$, $\beta,h\geq 0$, and $\# \in \{f,w\}$. As in the finite case, the equation completely determines the measures $\mathbf{L}_{G,\beta}^\#$ since, by inclusion-exclusion, the family of indicator functions $\{\mathbbm{1}(\omega(x) = 0 \text{ for every } x\in A) : A \subseteq V \cup E \text{ is finite}\}$ has linear span equal to the set of all functions depending on at most finitely many edges.
It follows by taking limits over exhaustions that, as in the finite case, we may obtain a random variable with law ${\mathbf{C}}_{\beta,h}^\#$ as follows: Fix $\beta >0, h\geq 0$, and $\#\in \{f,w\}$. Let $L=(L_x)_{x\in V \cup E}$ be a random variable with law ${\mathbf{L}}_{\beta,h}^\#$. Independently of $L$, let $\mathsf{Odd}=(\mathsf{Odd}_x)_{x\in V \cup E}$ and $\mathsf{Even}=(\mathsf{Even}_x)_{x\in V \cup E}$ be independent random variables with distributions given as in . Then the random variable $${\mathbf{n}}=({\mathbf{n}}_x)_{x\in V \cup E}= \Bigl(\mathsf{Odd}_x\mathbbm{1}(L_x=1)+\mathsf{Even}_x\mathbbm{1}(L_x=0)\Bigr)_{x\in V \cup E}$$ has law ${\mathbf{C}}_{\beta,h}^\#$. It follows in particular that if $G$ is transitive then ${\mathbf{C}}_{\beta,h}^\#$ may be expressed as an ${\operatorname{Aut}}(G)$-factor of ${\mathbf{L}}_{\beta,h}^\# \otimes \mu$ where $\mu$ is an appropriately chosen Bernoulli measure on $(\{0,1,\ldots\}^2)^{V \cup E}$. Moreover, this representation allows us to consider the random subgraph of $G$ spanned by those edges with a non-zero current as a percolation in random environment model, where the environment ${\mathbf{p}}$ is defined in terms of the loop $O(1)$ configuration $\omega$ by $${\mathbf{p}}_e = \mathbbm{1}(\omega(e)=1) + \frac{\cosh(\beta J_e)-1}{\cosh(\beta J_e)} \mathbbm{1}(\omega(e)=0).$$ This representation will allow us to apply the machinery of \[sec:free\_energy\] to the random current and double random current models once we have bounded their spectral radius in \[subsec:randomcurrentspectrum\].
**The Lupu–Werner coupling**. In addition to the indirect connection between the random current model and the FK-Ising model via the Ising model and the Edwards–Sokal coupling, there is also a direct probabilistic connection between these two models due to Lupu and Werner [@MR3485382]. They proved that for each $\beta,h\geq 0$ and $\#\in \{f,w\}$ it is possible to obtain a sample of $\phi_{2,\beta,h}^\#$ as the union of a sample of the random current model ${\mathbf{C}}_{\beta,h}^\#$ with an independent Bernoulli process in which $x \in E \cup V$ is included independently at random with inclusion probability $1-e^{-\beta J_x}$, where we set $J_x=h$ for every $x\in V$. Combining this relationship with the relationship between the random current and loop $O(1)$ models discussed above, it follows that for each $\beta,h\geq 0$ and $\#\in \{f,w\}$ it is possible to obtain a sample of $\phi_{2,\beta,h}^\#$ as the union of a sample of the loop $O(1)$ model ${\mathbf{L}}_{\beta,h}^\#$ with an independent Bernoulli process with inclusion probabilities $$1 - \frac{1}{\cosh(\beta J_x)}+\frac{1-e^{-\beta J_x}}{\cosh(\beta J_x)}=\tanh(\beta J_x).
$$ Together with the formula , this implies that $$\begin{aligned}
\phi^\#_{2,\beta,h}(\omega(x)=0 \text{ for all $x\in A$}) &= {\mathbf{L}}^\#_{\beta,h}\bigl(\omega(x)=0 \text{ for all $x\in A$}\bigr)\prod_{x \in A} \left( 1-\tanh(\beta J_x)\right)
\nonumber
\\
&={\mathbf{G}}^\#_{\beta,h} \left[e^{\beta H_A} \right]\prod_{x \in A} e^{-\beta J_x} =
{\mathbf{G}}^\#_{\beta,h} \left[e^{-\beta \sum_{x \in A} J_x(\sigma_x + 1) } \right]
\label{eq:FK_Gradient_Formula}\end{aligned}$$ for every finite set $A \subseteq E \cup V$, where we write $J_v=h$ for each $v\in V$ and recall that $H_A =H_A(\sigma)= -\sum_{x\in A} J_x \sigma_x$ for each finite set $A \subseteq E \cup V$. This formula, which can also be proven using the percolation-in-random-environment representation of the random cluster model used in \[subsec:finite\_FK\], establishes a relationship between the gradient Ising and FK-Ising models that has better continuity properties than the Edwards–Sokal coupling, and will be very useful throughout our analysis.
Since $\phi_{2,\beta,h}^\#$ is stochastically dominated by the product measure $\phi_{1,\beta,h}^\#$ [@GrimFKbook Theorem 3.21], it follows from the Lupu–Werner coupling that the measures ${\mathbf{L}}_{\beta,h}^\#$ and ${\mathbf{C}}_{\beta,h}^\#$ are stochastically dominated by this product measure also. We deduce in particular that $$\label{eq:bounding_current_by_FK1}
{\mathbf{L}}_{\beta,h}^\#(e \text{ open})\leq {\mathbf{C}}_{\beta,h}^\#({\mathbf{n}}(e)>0) \leq \phi_{2,\beta,h}^\#(e \text{ open}) \leq \frac{e^{2\beta J_e}-1}{e^{2\beta J_e}} \leq 2\beta J_e$$ for every $e\in E$, $\beta,h\geq 0$, and $\#\in \{f,w\}$, and similarly that $$\label{eq:bounding_current_by_FK2}
{\mathbf{C}}_{\beta,h}^\#({\mathbf{n}}(v)>0) \leq \phi_{2,\beta,h}^\#(\omega(v)=1) \leq \frac{e^{2\beta h}-1}{e^{2\beta h}} \leq 2\beta h$$ $v\in V$, $\beta,h\geq 0$, and $\#\in \{f,w\}$.
The spectral radius of the random current model {#subsec:randomcurrentspectrum}
-----------------------------------------------
We now apply to bound the spectral radii of the loop $O(1)$ and random current models on a transitive weighted graph. Although very little is known about whether or not the loop $O(1)$ and random current models are factors of i.i.d. (see [@harel2018finitary] for some discussion), the connections between the (gradient) Ising model and these models are strong enough to carry through bounds on the spectral radius without needing to express anything as a factor.
\[thm:spectralradius\] Let $G=(V,E,J)$ be a connected transitive weighted graph and let $\Gamma$ be a closed unimodular transitive group of automorphisms. Then $$\rho(\mathbf{L}^\#_{\beta,h}) \leq \max\left\{\rho(\mathbf{G}^\#_{\beta,h}),\rho(G)\right\}$$ for every $\beta,h \geq 0$ and $\# \in \{w,f\}$.
Fix $\beta,h\geq 0$, and $\# \in \{f,w\}$. To ease notation, we write ${\mathbf{L}}={\mathbf{L}}_{\beta,h}^\#$ and ${\mathbf{G}}={\mathbf{G}}_{\beta,h}^\#$. Let $X=(X_n)_{n\geq 0}$ be the random walk on $G$ and let $\hat X = (\hat X_n)_{n\geq 0}$ be the associated random walk on $\Gamma$ as defined in \[subsec:spectral\_background\]. We write ${\mathbf E}$ for expectations taken with respect to the law of $\hat X$ and write $\E$ for expectations taken with respect to the product measure ${\mathbf E}\otimes {\mathbf{G}}$. Given $A \subseteq V \cup E$ and $\omega \in \{0,1\}^{V \cup E}$, we write $A \perp \omega$ to mean that $\omega(x)=0$ for every $x\in A$. Inclusion-exclusion implies that events of the form $\{\omega : A \perp \omega \}$ with $A$ finite have dense linear span in $L^2(\{0,1\}^{V \cup E},{\mathbf{L}})$. Thus, by , it suffices to prove that $$\limsup_{k\to\infty} \left|{\mathbf E}\left[{\mathbf{L}}(\{\omega : A \perp \omega \text{ and } \hat X_{2k}^{-1} A \perp \omega\}) \right] -{\mathbf{L}}(\{\omega : A \perp \omega\})^2 \right|^{1/2k} \\\leq \max\{\rho({\mathbf{G}}),\rho(G)\}
\label{eq:spectralradiusclaim}$$ for every finite set $A \subseteq V \cup E$. Fix one such finite set $A \subseteq E$ and write $A_k=\hat X_{k}^{-1} A$ for every $k\geq 0$. Then we have by that $$\begin{aligned}
{\mathbf E}\left[{\mathbf{L}}(\{\omega : A \perp \omega\})\right]
&= \E\left[C_{\beta,h}(A ) \exp\left(\beta H_{A}\right)\right] &&\text{ and}
\label{eq:spectral-2}
\\
{\mathbf E}\left[{\mathbf{L}}(\{\omega : A \perp \omega \text{ and } \hat X_{2k}^{-1} A \perp \omega\})\right]
&= \E\left[C_{\beta,h}(A \cup A_{2k}) \exp\left(\beta H_{A\cup A_{2k}}\right)\right]&&
\label{eq:spectral-1}
\end{aligned}$$ for every $k\geq 0$ and hence that $$\begin{aligned}
{\mathbf E}\left[{\mathbf{L}}(\{\omega : A \perp \omega \text{ and } \hat X_{2k}^{-1} A \perp \omega\})\right]
=\E\left[ C_{\beta,h}(A) \exp\left(\beta H_{A}\right) C_{\beta,h}(A_{2k}) \exp\left(\beta H_{A_{2k}}\right)\right]\phantom{.}
\nonumber
\\
+ \E\left[ C_{\beta,h}(A \cup A_{2k}) \exp\left(\beta H_{A\cup A_{2k}}\right) \mathbbm{1}(A \cap A_{2k} \neq \emptyset)\right]\phantom{.}
\nonumber
\\
-\E\left[ C_{\beta,h}(A) \exp\left(\beta H_{A}\right) C_{\beta,h}(A_{2k}) \exp\left(\beta H_{A_{2k}}\right)\mathbbm{1}(A \cap A_{2k} \neq \emptyset)\right].
$$ (Note that $C_{\beta,h}(A)=C_{\beta,h}(A_{2k})$ is a constant.) Since the random variables $ e^{\beta H_{A\cup A_{2k}}} C_\beta(A \cup A_{2k})$ and $ e^{\beta H_{A}} C_\beta(A) e^{\beta H_{A_{2k}}} C_\beta(A_{2k})$ are both bounded between two positive constants (depending on $A$, $\beta$, and $h$ but not $k$), the second and third terms satisfy $$\begin{gathered}
\limsup_{k\to\infty} \E\left[ C_{\beta,h}(A \cup A_{2k}) \exp\left(\beta H_{A\cup A_{2k}}\right) \mathbbm{1}(A \cap A_{2k} \neq \emptyset)\right]^{1/2k} \\ = \limsup_{k\to\infty} \P\left(\hat X_{2k}^{-1} A \cap A \neq \emptyset\right)^{1/2k}\leq \|\hat P\|= \rho(G)
\label{eq:spectral1}\end{gathered}$$ and $$\begin{gathered}
\label{eq:spectral2}
\limsup_{k\to\infty}\E\left[ C_{\beta,h}(A) \exp\left(\beta H_{A}\right) C_{\beta,h}(A_{2k}) \exp\left(\beta H_{A_{2k}}\right)\mathbbm{1}(A \cap A_{2k} \neq \emptyset)\right]^{1/2k} \\ = \limsup_{k\to\infty} \P\left(\hat X_{2k}^{-1} A \cap A \neq \emptyset\right)^{1/2k}\leq \|\hat P\|=\rho(G).\end{gathered}$$ Meanwhile, we also have by definition of the spectral radius that $$\begin{gathered}
\Bigl|\E\left[ C_{\beta,h}(A) \exp\left(\beta H_{A}\right) C_{\beta,h}(A_{2k}) \exp\left(\beta H_{A_{2k}}\right)\right]
\\-
\E\left[ C_{\beta,h}(A) \exp\left(\beta H_{A}\right)\right]^2\Bigr| \leq \rho({\mathbf{G}})^{2k} {{\mathrm{Var}}}\left(C_{\beta,h}(A)e^{\beta H_A}\right)\end{gathered}$$ for each $k\geq 0$ and hence that $$\begin{gathered}
\limsup_{k\to\infty}\Bigl|\E\left[ C_{\beta,h}(A) \exp\left(\beta H_{A}\right) C_{\beta,h}(A_{2k}) \exp\left(\beta H_{A_{2k}}\right)\right]
-
\E\left[ C_{\beta,h}(A) \exp\left(\beta H_{A}\right)\right]^2\Bigr|^{1/2k} \leq \rho({\mathbf{G}}).\end{gathered}$$ Applying this estimate together with those of and in light of and yields the claimed inequality .
Since ${\mathbf{C}}_{\beta,h}^\#$ may be expressed as an ${\operatorname{Aut}}(G)$-factor of ${\mathbf{L}}_{\beta,h}^\# \otimes \mu$ where $\mu$ is an appropriately chosen Bernoulli measure on $(\{0,1,\ldots\}^2)^{V \cup E}$, the following corollary follows immediately from \[thm:Ising\_factor,thm:spectralradius,thm:Bernoulli\_radius\].
\[cor:current\_radius\] Let $G=(V,E,J)$ be a connected transitive weighted graph and let $\Gamma$ be a closed unimodular transitive group of automorphisms. Then $$\rho(\mathbf{C}^w_{\beta,h})\leq \rho(G) \qquad \text{ and } \qquad \rho(\mathbf{L}^w_{\beta,h})\leq \rho(G)$$ for every $\beta,h\geq 0$.
In \[subsec:free\_spectral\_radius\], we use to show that the conclusions of \[cor:current\_radius\] can in fact be extended to the case of free boundary conditions. With slightly more work one can show that the inequalities in \[thm:spectralradius,cor:current\_radius,thm:free\_spectral\_radius\] are equalities when $\beta>0$.
Double random currents with mismatched temperatures {#subsec:mismatched}
---------------------------------------------------
In this section we discuss how the switching lemma can be used to study pairs of random currents with *different* values of the inverse temperature $\beta$ and external field $h$. The resulting tools, which appear to be new, lead to quantitative versions of the arguments of [@MR3306602] that will be used to control the effect of changing $\beta$ and $h$ on the Ising model in the proofs of our main theorems. Let $G=(V,E,J)$ be a connected weighted graph with $V$ finite. For each $\theta \in (0,1)$, let $\tilde G_\theta$ be the weighted graph obtained from $G$ by replacing each edge $e$ of $G$ by two edges $e_1$ and $e_2$ in parallel, where $e_1$ has coupling constant $J_{e_1}:=(1-\theta)J_e$ and $e_2$ has coupling constant $J_{e_2}:=\theta J_e$. Observe from the definitions that $$\label{eq:theta_Ising}
{\mathbf{I}}_{\tilde G_\theta,\beta,h} = {\mathbf{I}}_{G,\beta,h}$$ for every $\theta\in (0,1)$ and $\beta,h\geq 0$. There is also a simple probabilistic relationship between the random current models on $G$ and $\tilde G_\theta$: Let $\mathbf{m}=(\mathbf{m}_x)_{x\in V \cup E}$ be a random variable with law ${\mathbf{C}}_{G,\beta,h}$. Conditional on $\mathbf{m}$, for each $e \in E$ let $\mathbf{n}_{e_1}$ be a binomial random variable with distribution $\operatorname{Binom}(1-\theta,\mathbf{m}_e)$ and let $\mathbf{n}_{e_2}=\mathbf{m}_e - \mathbf{n}_{e_1}$, where we take the random variables $(\mathbf{n}_{e_1})_{e\in E}$ to be conditionally independent given $\mathbf{m}$. Finally, set $\mathbf{n}_v=\mathbf{m}_v$ for each $v\in V$. It follows from the identity $$\label{eq:binomial_identity}
\sum_{k=0}^n \frac{((1-\theta)\beta)^k}{k!}\frac{(\theta \beta)^{n-k}}{(n-k)!} =
\frac{\beta^n}{n!}\sum_{k=0}^n \binom{n}{k}(1-\theta)^k\theta^{n-k}
= \frac{\beta^n}{n!}$$ that $\mathbf{n}$ has law ${\mathbf{C}}_{\tilde G_\theta, \beta, h}$. Now consider the subgraph $H_\theta$ of $\tilde G_\theta$ spanned by the edges $\{e_1:e\in E\}$, so that $H_\theta$ is isomorphic to the weighted graph obtained from $G$ by multiplying all coupling constants by $(1-\theta)$. Observe that if we identify the edge set of $H_\theta$ with that of $G$ then we have the equalities $${\mathbf{I}}_{H_\theta,\beta,h} = {\mathbf{I}}_{G,(1-\theta)\beta,(1-\theta)^{-1}h}
\qquad \text{ and } \qquad
{\mathbf{C}}_{H_\theta,\beta,h} = {\mathbf{C}}_{G,(1-\theta)\beta,(1-\theta)^{-1}h}$$ for every $\beta,h \geq 0$. Since $H_\theta$ is a subgraph of $\tilde G_\theta$, this construction will therefore allow us to apply the switching lemma \[lem:switching\] to study the Ising model at two different values of the inverse temperature. We now introduce an infinite-volume version of this construction in the wired case, in which we will also allow ourselves to change the strength of the external field. Let $G=(V,E,J)$ be an infinite, connected, weighted graph and let $\beta_2 \geq \beta_1 \geq 0$ and $h_2 \geq h_1 \geq 0$. Let ${\mathbf{n}}_1=({\mathbf{n}}_1(x))_{x\in E \cup V}$ and ${\mathbf{m}}_2=({\mathbf{m}}_2(x))_{x\in E\cup V}$ be independent random variables with laws ${\mathbf{C}}^w_{\beta_1,h_1}$ and ${\mathbf{C}}^w_{\beta_2,h_2}$ respectively. Let $\theta=1-\beta_1/\beta_2$ and let $\phi=1-(\beta_1 h_1)/(\beta_2h_2)$, so that $\theta,\phi\in [0,1]$. Conditional on ${\mathbf{n}}_1$ and ${\mathbf{m}}_2$, let ${\mathbf{n}}_2 =({\mathbf{n}}_2(x_i))_{x\in E \cup V, i \in \{0,1\}}$ be defined as follows:
1. For each $e\in E$, let ${\mathbf{n}}_2(e_1)$ be a $\operatorname{Binom}(1-\theta,{\mathbf{m}}_2(e))$ random variable and let ${\mathbf{n}}_2(e_2)={\mathbf{m}}_2(e)-{\mathbf{n}}_2(e_1)$.
2. For each $v\in V$, let ${\mathbf{n}}_2(v_1)$ be a $\operatorname{Binom}(1-\phi,{\mathbf{n}}_2(v))$ random variable and let ${\mathbf{n}}_2(v_2)={\mathbf{m}}_2(v)-{\mathbf{n}}_2(v_1)$.
We take the random variables $({\mathbf{n}}_2(x_1):x \in E \cup V)$ to be conditionally independent of each other and of ${\mathbf{n}}_1$ given ${\mathbf{m}}_2$. Let ${\mathbf{Q}}_{\beta_1,h_1,\beta_2,h_2}$ denote the law of the resulting pair of random variables $({\mathbf{n}}_1,{\mathbf{n}}_2)$, which we may think of as a measure on the product space $
\N_0^E \times \N_0^V \times (\N_0^2)^E \times (\N_0^2)^V \cong (\N_0^3)^E \times (\N_0^3)^V$.
We now relate the dependence on $\beta$ and $h$ of correlations in the Ising model to the percolative properties of ${\mathbf{Q}}_{\beta_1,h_1,\beta_2,h_2}$. Let $({\mathbf{n}}_1,{\mathbf{n}}_2)$ be a pair of random variables with law ${\mathbf{Q}}_{\beta_1,h_1,\beta_2,h_2}$, and let $u$ and $v$ be vertices of $G$. We say that an edge or vertex $x$ of $G$ is $1$**-open** if ${\mathbf{n}}_1(x)+{\mathbf{n}}_2(x_1)>0$ and that $x$ is $2$**-open** if ${\mathbf{n}}_1(x)+{\mathbf{n}}_2(x_1)+{\mathbf{n}}_2(x_2)>0$. In particular, every $1$-open $x$ is also $2$-open. For each vertex $v$ of $G$ and $i\in \{1,2\}$, we define $K^i_v$ to be the set of vertices that are connected to $v$ by an $i$-open path, so that $K^1_v \subseteq K^2_v$ for every $v\in V$. We say that a set $A \subseteq V$ is **$i$-infinite** if it is infinite or contains an $i$-open vertex.
For each $e\in E$, $v \in V$, and $i\in \{1,2\}$, let $K^1_{v,e}$ be the set of vertices that are connected to $v$ by an $i$-open path that does not include the edge $e$. Let $x$ and $y$ be the endpoints of $e$ and define $\sA_e$ to be the event that the following hold:
1. ${\mathbf{n}}_1(e)=0$ and ${\mathbf{n}}_2(e_1)=1$,
2. $x$ and $y$ are not connected by any $1$-open path that does not include $e$ and the clusters $K^1_{x,e}$ and $K^1_{y,e}$ are not both $1$-infinite.
3. At least one of the following hold:
1. ${\mathbf{n}}_2(e_2)>0$,
2. $x$ and $y$ are connected by a $2$-open path that does not include the edge $e$, or
3. $K^2_{x,e}$ and $K^2_{y,e}$ are both $2$-infinite.
Intuitively, $\sA_e$ is the event that $x$ and $y$ are connected ‘through infinity’ off of $e_1$ by $2$-open edges and vertices but are not connected through infinity off of $e$ by $1$-open edges and vertices. Similarly, for each $v\in V$ we define $\sA_v$ to be the event that the following hold:
1. ${\mathbf{n}}_1(v)=0$ and ${\mathbf{n}}_2(v_1)=1$,
2. $K_v^1 \setminus \{v\}$ is not $1$-infinite.
3. Either ${\mathbf{n}}_2(v_2)>1$ or $K^1_v \setminus \{v\}$ is $2$-infinite.
Note that if $G$ is locally finite then the set $\sA_x$ is closed for every $x\in E \cup V$.
\[cor:gradient\] Let $G=(V,E,J)$ be an infinite, connected, transitive weighted graph. Then $$\beta_1 J_x \left(\langle \sigma_x \rangle_{\beta_2,h_2}^+ - \langle \sigma_x \rangle_{\beta_1,h_1}^+\right) \leq {\mathbf{Q}}_{\beta_1,h_1,\beta_2,h_2}(\sA_x)$$ for each $\beta_2 \geq \beta_1 \geq 0$, $h_2 \geq h_1 \geq 0$, and $x\in E \cup V$, where we set $J_x=h_1$ if $x\in V$.
With a little more work this inequality can be shown to be an equality.
We will deduce this proposition from the following general lemma, which is similar to [@1707.00520 Lemma 4.5] and [@MR3306602 Eq. 3.10].
\[lem:general\_gradient\] Let $G=(V,E,J)$ be a weighted graph with $V$ finite, let $H$ be a subgraph of $G$, and let $e$ be an edge of $H$ with endpoints $x$ and $y$. Then $$\beta J_e \left(\langle \sigma_e \rangle_{G,\beta,0}- \langle \sigma_e \rangle_{H,\beta,0}\right) = \mathbf{C}_{H,\beta}\otimes\mathbf{C}_{G,\beta}
\bigl(\sB_e\bigr)$$ for every $\beta \geq 0$, where $\sB_e$ is the set of pairs $({\mathbf{n}}_1,{\mathbf{n}}_2)$ such that ${\mathbf{n}}_1(e)=0$, ${\mathbf{n}}_2(e)=1$, and $x$ and $y$ are connected to each other by an $({\mathbf{n}}_1+{\mathbf{n}}_2)$-open path in $G \setminus \{e\}$ but not in $H\setminus \{e\}$.
We may assume that $\beta>0$, the claim being trivial otherwise. We have by that $$\begin{aligned}
\langle \sigma_x \sigma_y \rangle_{G,\beta,0}- \langle \sigma_x \sigma_y \rangle_{H,\beta,0} &=
\frac{\sum_{\mathbf{n_2}\in \Omega_G:\partial \mathbf{n_2}=\{x,y\}} w_\beta(\mathbf{n_2})}{\sum_{\mathbf{n_2}\in \Omega_G:\partial \mathbf{n_2}=\emptyset} w_\beta(\mathbf{n_2})}
-
\frac{\sum_{\mathbf{n_1}\in \Omega_H:\partial \mathbf{n_1}=\{x,y\}} w_\beta(\mathbf{n_1})}{\sum_{\mathbf{n_1}\in \Omega_H:\partial \mathbf{n_1}=\emptyset} w_\beta(\mathbf{n_1})}
\\
&=
\frac{
\sum_{
\substack{
\mathbf{n_1}\in \Omega_H:\partial \mathbf{n_1}=\emptyset
\\
\mathbf{n_2}\in \Omega_G:\partial \mathbf{n_2}=\{x,y\}
}}
w_\beta({\mathbf{n}}_1)w_\beta({\mathbf{n}}_2)
-
\sum_{
\substack{
\mathbf{n_1}\in \Omega_H:\partial \mathbf{n_1}=\{x,y\}
\\
\mathbf{n_2}\in \Omega_G:\partial \mathbf{n_2}=\emptyset
}}
w_\beta(\mathbf{n_1})w_\beta({\mathbf{n}}_2)
}
{\sum_{\mathbf{n_1}\in \Omega_H:\partial \mathbf{n_1}=\emptyset} \sum_{\mathbf{n_2}\in \Omega_G:\partial \mathbf{n_2}=\emptyset} w_\beta(\mathbf{n_2}) w_\beta(\mathbf{n_1})},\end{aligned}$$ and applying the switching lemma to the second term yields that $$\begin{aligned}
\langle \sigma_x \sigma_y \rangle_{G,\beta,0}- \langle \sigma_x \sigma_y \rangle_{H,\beta,0}&=
\genfrac{}{}{}{}{\raisebox{1.1em}{$\sum_{
\substack{
\mathbf{n_1}\in \Omega_H:\partial \mathbf{n_1}=\emptyset
\\
\mathbf{n_2}\in \Omega_G:\partial \mathbf{n_2}=\{x,y\}
}}
w_\beta({\mathbf{n}}_1)w_\beta({\mathbf{n}}_2)
\mathbbm{1}\Bigl(x { \mathrel{\tikz[baseline=-.7ex] \path node[slash underlined,draw,<->,anchor=south] {\(\scriptstyle {\mathbf{n}}_1+{\mathbf{n}}_2\)} node[anchor=north] {\(\scriptstyle \)};}} y \text{ in $H$}\Bigr)
$}}{\raisebox{-0em}{$\sum_{\mathbf{n_1}\in \Omega_H:\partial \mathbf{n_1}=\emptyset} \sum_{\mathbf{n_2}\in \Omega_G:\partial \mathbf{n_2}=\emptyset} w_\beta(\mathbf{n_2}) w_\beta(\mathbf{n_1})$}}.\end{aligned}$$ (This equality holds for all vertices $x$ and $y$.) The constraint “$\partial {\mathbf{n}}_2 = \{x,y\}$” forces $x$ and $y$ to be connected in ${\mathbf{n}}_2$, while the constraint “$x { \mathrel{\tikz[baseline=-.7ex] \path node[slash underlined,draw,<->,anchor=south] {\(\scriptstyle {\mathbf{n}}_1+{\mathbf{n}}_2\)} node[anchor=north] {\(\scriptstyle \)};}} y$ in $H$" forces ${\mathbf{n}}_1(e)={\mathbf{n}}_2(e)=0$. Thus, if we consider the two sets $$A = \{({\mathbf{n}}_1,{\mathbf{n}}_2) \in \Omega_H \times \Omega_G : \partial {\mathbf{n}}_1 = \emptyset,\, \partial {\mathbf{n}}_2 = \{x,y\},\, \text{ and } x { \mathrel{\tikz[baseline=-.7ex] \path node[slash underlined,draw,<->,anchor=south] {\(\scriptstyle {\mathbf{n}}_1+{\mathbf{n}}_2\)} node[anchor=north] {\(\scriptstyle \)};}} y \text{ in $H$}\}$$ and $$\begin{gathered}
B = \{({\mathbf{n}}_1,{\mathbf{n}}_2) \in \Omega_H \times \Omega_G : \partial {\mathbf{n}}_1 = \emptyset,\, \partial {\mathbf{n}}_2 = \emptyset,\, x { \mathrel{\tikz[baseline=-.7ex] \path node[slash underlined,draw,<->,anchor=south] {\(\scriptstyle {\mathbf{n}}_1+{\mathbf{n}}_2\)} node[anchor=north] {\(\scriptstyle \)};}} y \text{ in $H \setminus \{e\}$},\\ x \xleftrightarrow{{\mathbf{n}}_1+{\mathbf{n}}_2} y \text{ in $G \setminus \{e\}$},\, {\mathbf{n}}_1(e)=0, \text{ and } {\mathbf{n}}_2(e)=1\}\end{gathered}$$ then we have a bijection $A \to B$ given by incrementing the value of ${\mathbf{n}}_2(e)$ from $0$ to $1$. This increment changes the weight $w_\beta({\mathbf{n}}_2)$ by a factor of $\beta J_e$, so that $$\begin{aligned}
\langle \sigma_x \sigma_y \rangle_{G,\beta,0}- \langle \sigma_x \sigma_y \rangle_{H,\beta,0}&=
\frac{\sum_{{\mathbf{n}}_1,{\mathbf{n}}_2 \in A}
w_\beta({\mathbf{n}}_1)w_\beta({\mathbf{n}}_2)}
{\sum_{\mathbf{n_1}\in \Omega_H:\partial \mathbf{n_1}=\emptyset} \sum_{\mathbf{n_2}\in \Omega_G:\partial \mathbf{n_2}=\emptyset} w_\beta(\mathbf{n_2}) w_\beta(\mathbf{n_1})}\\
&=
\frac{\sum_{{\mathbf{n}}_1,{\mathbf{n}}_2 \in B}
w_\beta({\mathbf{n}}_1)w_\beta({\mathbf{n}}_2)}
{\beta J_e\sum_{\mathbf{n_1}\in \Omega_H:\partial \mathbf{n_1}=\emptyset} \sum_{\mathbf{n_2}\in \Omega_G:\partial \mathbf{n_2}=\emptyset} w_\beta(\mathbf{n_2}) w_\beta(\mathbf{n_1})}.\end{aligned}$$This is equivalent to the claim.
We may assume that $\beta>0$, the claim being trivial otherwise. We prove the formula in the case that $x$ is an edge of $G$, the case that $x$ is a vertex being similar. We will also assume for convenience that $h_2>h_1$ and $\beta_2>\beta_1$; the case of equality is similar but requires one to define the graphs $\tilde G_\theta$ and $H_\theta$ differently to avoid having edges of weight zero, which were not allowed by the definition of a weighted graph. (This does not cause any actual problems in the proof.)
Fix $e\in E$, $\beta_2 > \beta_1 > 0$, and $h_2 > h_1 \geq 0$. Let $\theta=1-\beta_1/\beta_2$, let $\phi=1-\beta_1h_1/\beta_2h_2$, and let ${\mathbf{Q}}={\mathbf{Q}}_{\beta_1,h_1,\beta_2,h_2}$. Let $\tilde G_\theta$ be the weighted graph obtained from $G$ by replacing each edge $e$ of $G$ by parallel edges $e_1$ and $e_2$, where $e_1$ has coupling constant $J_{e_1}:=(1-\theta)J_e$ and $e_2$ has coupling constant $J_{e_2}:=\theta J_e$, and let $H_\theta$ be the subgraph of $G_\theta$ spanned by the edges $\{e_1 :e \in E\}$. As above, it follows from the definitions and that a random variable $({\mathbf{n}}_1,{\mathbf{n}}_2)$ with law ${\mathbf{Q}}_{\beta_1,h_1,\beta_2,h_2}$ can be obtained from a random variable $({\mathbf{n}}_1,{\mathbf{m}}_2)$ with law ${\mathbf{C}}^w_{H_\theta,\beta_2,(1-\phi) h_2} \otimes {\mathbf{C}}^w_{G_\theta,\beta_2,h_2}$ by setting ${\mathbf{n}}_2(e_i)={\mathbf{m}}_2(e_i)$ for each $e\in E$ and $i\in \{1,2\}$, taking $({\mathbf{n}}_2(v_1))_{v\in V}$ to be independent Binomial random variables conditioned on $({\mathbf{n}}_1,{\mathbf{m}}_2)$, and setting ${\mathbf{n}}_2(v_2)={\mathbf{m}}_2(v)-{\mathbf{n}}_2(v_1)$ for each $v\in V$.
Let $(V_n)_{n\geq 1}$ be an exhaustion of $V$ such that both endpoints of $x$ belong to $V_n$ for every $n\geq 1$, and let $(G_n^*)_{n\geq 1}$ be formed by contracting each vertex in $V\setminus V_n$ into a single vertex $\delta_n$ as in \[subsec:intro\_definitions\]. For each $n\geq 1$, let the weighted graph $\tilde G_n$ be obtained from $G_n^*$ as follows:
1. Replace each edge $e$ of $G_n^*$ with two parallel edges $e_1$ and $e_2$ with coupling constants $(1-\theta)J_e$ and $\theta J_e$.
2. Add an additional vertex $\delta$ distinct from $\delta_n$.
1. If $h_1=0$, attach each vertex $v$ of $G_n^*$ (including $\delta_n$) to $\delta$ by a single edge of weight $h_2$. Call this edge $g_2(v)$.
2. If $h_1>0$, attach each vertex $v$ of $G_n^*$ (including $\delta_n$) to $\delta$ by two edges in parallel. The first edge is called $g_1(v)$ and is given weight $(1-\phi)h_2=\beta_1 h_1/\beta_2$, while the second edge is called $g_2(v)$ and is given weight $\phi h_2$.
If $h_1>0$ we define $H_n$ be the subgraph of $\tilde G_n$ spanned by the edges $\{e_1: e$ an edge of $G_n\} \cup \{g_2(v) : v\in V_n \cup \{\delta_n\} \}$. Otherwise, $h_1=0$ and we define $H_n$ to be the subgraph of $\tilde G_n$ spanned by the edges $\{e_1: e$ an edge of $G_n\}$. These weighted graphs are defined so that the gradient Ising models at inverse temperature $\beta$ and zero external field on $\tilde G_n$ and $H_n$ are equivalent to the gradient Ising models on $G_n^*$ with inverse temperatures $\beta_2$ and $\beta_1$ and external fields $h_2$ and $h_1$ respectively. Let $({\mathbf{m}}_{n,1},{\mathbf{m}}_{n,2})$ have law ${\mathbf{C}}_{H_n,\beta_2,0}\otimes {\mathbf{C}}_{\tilde G_n,\beta_2,0}$ and let ${\mathbf{Q}}_n$ be the law of the random variable $({\mathbf{n}}_{n,1},{\mathbf{n}}_{n,2}) \in (\N_0^3)^{E \cup V}$ defined by $${\mathbf{n}}_{n,1}(x) = \begin{cases} {\mathbf{m}}_{n,1}(x_0)\mathbbm{1}(\text{$x$ has an endpoint in $V_n$}) & x \text{ is an edge}\\
{\mathbf{m}}_{n,1}(x_0)\mathbbm{1}(x \in V_n) & x \text{ is a vertex}
\end{cases}$$ for each $x\in E \cup V$ and $${\mathbf{n}}_{n,2}(x_i) = \begin{cases} {\mathbf{m}}_{n,2}(x_i)\mathbbm{1}(\text{$x$ has an endpoint in $V_n$}) & x \text{ is an edge}\\
{\mathbf{m}}_{n,2}(g_i(x))\mathbbm{1}(x \in V_n) & x \text{ is a vertex, $h>0$}\\
{\mathbf{m}}_{n,2}(g_i(x))\mathbbm{1}(x \in V_n, i=2) & x \text{ is a vertex, $h=0$}
\end{cases}$$ for each $x\in E \cup V$ and $i\in \{1,2\}$. It follows from the definitions and from \[eq:LoopO(1)Ising,eq:LoopO(1)Ising\_infinite\] that ${\mathbf{Q}}={\mathop{\operatorname{w-lim}}}_{n\to\infty} {\mathbf{Q}}_n$. Since $H_n$ is a subgraph of $\tilde G_n$ for each $n\geq 1$, we may apply \[lem:general\_gradient\] to deduce that $$\begin{aligned}
\nonumber
\beta_1 J_x \langle \sigma_x \rangle_{G_n^*,\beta_2,h_2} - \langle \sigma_x \rangle_{G_n^*,\beta_1,h_1}
&=\beta_2 (1-\theta) J_x \left[\langle \sigma_x \rangle_{\tilde G_n,\beta_2,0} - \langle \sigma_x \rangle_{H_n,\beta_2,0}\right]
\\&= {\mathbf{C}}_{H_n,\beta_2,0}\otimes {\mathbf{C}}_{\tilde G_n,\beta_2,0}
\bigl(\sB_{x,n}\bigr)
= {\mathbf{C}}_{H_n,\beta_2,0}\otimes {\mathbf{C}}_{\tilde G_n,\beta_2,0}
\bigl(\sB_{x,n}\bigr)
\label{eq:Gradient_to_Bxn}
$$ where $\sB_{x,n}$ is the set of pairs $({\mathbf{m}}_{n,1},{\mathbf{m}}_{n,2})$ such that ${\mathbf{m}}_{n,1}(x_1)=0$, ${\mathbf{m}}_{n,2}(x_1)=1$, and the endpoints $u$ and $v$ of $x$ are connected to each other by an $({\mathbf{m}}_{n,1}+{\mathbf{m}}_{n,2})$-open path in $\tilde G_n \setminus \{x_1\}$ but not in $H_n\setminus \{x_1\}$. Let $u$ and $v$ be the endpoints of $x$. For each $n\geq 1$, let $\sA_{x,n}$ be the set of pairs $({\mathbf{n}}_{1},{\mathbf{n}}_{2}) \in \N_0^{E \cup V} \times (\N_0^2)^{E \cup V}$ such that the following hold:
1. ${\mathbf{n}}_1(x)=0$ and ${\mathbf{n}}_2(x_1)=1$.
2. the endpoints $u$ and $v$ of $x$ are not connected by any $1$-open path that does not include $x$, $K^1_{u,x}({\mathbf{n}}_1,{\mathbf{n}}_2)$ and $K^1_{v,x}({\mathbf{n}}_1,{\mathbf{n}}_2)$ do not both intersect the set $V \setminus V_n$, and do not both intersect the set $\{ w \in V: {\mathbf{n}}_1(w)+{\mathbf{n}}_2(w_1)>0 \}$.
3. At least one of the following conditions hold:
1. ${\mathbf{n}}_2(x_1)>0$,
2. $u$ and $v$ are connected by a $2$-open path that does not include the edge $x$, or
3. $K^2_{u,x}({\mathbf{n}}_1,{\mathbf{n}}_2)$ and $K^2_{v,x}({\mathbf{n}}_1,{\mathbf{n}}_2)$ either both intersect the set $V \setminus V_n$ or both intersect the set $\{ w \in V: {\mathbf{n}}_1(w)+{\mathbf{n}}_2(w_1)+{\mathbf{n}}_2(w_2)>0 \}$.
Here, as before $K^i_{w,x}({\mathbf{n}}_1,{\mathbf{n}}_2)$ denotes the set of vertices that are connected to $w \in V$ by an $i$-open path in $G \setminus \{x\}$, where an edge $e$ is said to be $1$-open if ${\mathbf{n}}_1(e)+{\mathbf{n}}_2(e_1)>0$ and $2$-open if ${\mathbf{n}}_1(e)+{\mathbf{n}}_2(e_1)+{\mathbf{n}}_2(e_2)>0$. It follows from the definitions that ${\mathbf{C}}_{H_n,\beta_2,0}\otimes {\mathbf{C}}_{\tilde G_n,\beta_2,0}
\bigl(\sB_{x,n}\bigr) = {\mathbf{Q}}_n(\sA_{x,n})$ and that $$\sA_x = \limsup_{n\to\infty} \sA_{x,n} = \liminf_{n\to\infty} \sA_{x,n}.$$ If $G$ is locally finite then $\sA_x$ is closed in $\N_0^{E \cup V} \times (\N_0^2)^{E \cup V}$ and it follows by and the portmanteau theorem that $${\mathbf{Q}}(\sA_x) \geq \lim_{n\to\infty} {\mathbf{Q}}_n (\sA_{x,n}) = \beta_1 J_x \lim_{n\to\infty} \left(\langle \sigma_x \rangle_{G_n^*,\beta_2,h_2} - \langle \sigma_x \rangle_{G_n^*,\beta_1,h_1} \right)
=
\beta_1 J_x \left(\langle \sigma_x \rangle_{\beta_2,h_2}^+ - \langle \sigma_x \rangle_{\beta_1,h_1}^+ \right),$$ completing the proof in this case.
Let us now briefly discuss how the proof can be extended to the case that $G$ is not locally finite. The problem in this case is that $\sA_x$ need not be closed in the product topology on $\N_0^{E \cup V} \times (\N_0^2)^{E \cup V}$. (Indeed, the set of pairs $({\mathbf{n}}_1,{\mathbf{n}}_2)$ for which $K_{u,x}^2$ contains more than one vertex is not closed.) This can be remedied as follows: Let $X$ be the subset of $\N_0^{E \cup V} \times (\N_0^2)^{E \cup V}$ such that $\sum_{e\in E^\rightarrow_w} {\mathbf{n}}_1(e)+{\mathbf{n}}_2(e_1)+{\mathbf{n}}_2(e_2)<\infty$ for every $w\in V$, and endow $X$ with the weakest topology that makes the families of functions $$\begin{aligned}
({\mathbf{n}}_1,{\mathbf{n}}_2) &\mapsto ({\mathbf{n}}_1(x),{\mathbf{n}}_2(x_1),{\mathbf{n}}_2(x_2)) &&: x\in E \cup V \qquad \text{ and }
\\ ({\mathbf{n}}_1,{\mathbf{n}}_2) &\mapsto \sum_{e\in E^\rightarrow_w} {\mathbf{n}}_1(e)+{\mathbf{n}}_2(e_1)+{\mathbf{n}}_2(e_2) &&: w \in V\end{aligned}$$ continuous. Note that this topology is stronger than the product topology on $X$ and that $\sA_x \cap X$ is a closed subset of $X$ with respect to this topology. It follows easily from the Lupu–Werner coupling and the domination of the FK-Ising model by Bernoulli percolation that the measures $({\mathbf{Q}}_n)_{n\geq 1}$ are all supported on $X$ and are tight with respect to this topology. Since the probability measures ${\mathbf{Q}}_n$ weakly converge to the probability measure ${\mathbf{Q}}$ with respect to the product topology on $X$, it follows by tightness that they also converge to ${\mathbf{Q}}$ with respect to the stronger topology introduced above. The claim now follows from the portmanteau theorem as before.
Hölder continuity of the plus Ising measure {#subsec:mainproof}
-------------------------------------------
In this section we complete the proof of \[thm:main,thm:main\_simple,thm:main\_continuity,thm:main\_continuity\_FK\]. We begin by applying the methods of \[sec:free\_energy\] to the percolation model considered in \[subsec:mismatched\].
\[prop:finite\_clusters\_mismatched\] Let $G=(V,E,J)$ be a connected, unimodular, transitive, nonamenable weighted graph and let $o$ be a vertex of $G$. There exist positive constants $C$ and $\delta$ such that $${\mathbf{Q}}_{\beta_1,h_1,\beta_2,h_2} \left( n \leq |K_o^1| < \infty \right) \leq C n^{-\delta}$$ for every $n\geq 1$, $\beta_2 \geq \beta_1 \geq 0$ and $h_2 \geq h_1 \geq 0$.
By scaling, we may assume without loss of generality that $\sum_{e\in E^\rightarrow_o} J_e=1$. Let $({\mathbf{n}}_1,{\mathbf{n}}_2)$ be distributed according to ${\mathbf{Q}}_{\beta_1,\beta_2,h}$, let $\omega\in \{0,1\}^E$ be defined by $\omega(e)=\mathbbm{1}({\mathbf{n}}_1(e)+{\mathbf{n}}_2(e_1) >0)$, and let $\mu_{\beta_1,\beta_2,h}$ be the law of $\omega$. It follows from the Lupu-Werner coupling that the sets $\{e:{\mathbf{n}}_1(e)>0\}$ and $\{e:{\mathbf{n}}_2(e_1)>0\}$ are each stochastically dominated by the Bernoulli percolation process with edge inclusion probabilities $1-e^{-2\beta_1 J_e} \leq 2\beta_1 J_e$ and hence that $\omega$ is stochastically dominated by the Bernoulli percolation process with edge inclusion probabilities $1-e^{-4\beta_1 J_e} \leq 4\beta_1 J_e$. It follows by a counting argument that ${\mathbf{Q}}|K_o^1| \leq 2$ when $\beta_1 \leq 1/8$, so that it suffices to consider the case $\beta \geq 1/8$.
The connection between the loop $O(1)$ model and the random current model allows us to consider $\omega$ as a percolation in random environment model where the environment ${\mathbf{p}}$ satisfies ${\mathbf{p}}_e \geq (\cosh(\beta_1 J_e)-1)/\cosh(\beta_1 J_e)$ almost surely for every $e\in E$, and it follows by calculus that there exists a positive constant $c$ such that ${\mathbf{p}}_e \geq (\cosh(\beta_1 J_e)-1)/\cosh(\beta_1 J_e) \geq c J_e^2$ for every $e\in E$ and $\beta_1 \geq 1/8$. Moreover, \[thm:spectralradius\] implies that $\rho(\mu_{\beta_1,\beta_2,h})\leq \rho({\mathbf{Q}}_{\beta_1,\beta_2,h}) \leq \rho(G)$. With these ingredients in place, the claim follows by a similar (and slightly simpler) proof to that of \[thm:finite\_clusters\], and we omit the details.
We next apply \[prop:finite\_clusters\_mismatched,cor:gradient\] to control the change in the expected degree in the FK-Ising model as we increase $\beta$ or $h$.
\[cor:change\_in\_correlations\] Let $G=(V,E,J)$ be a connected, unimodular, transitive, nonamenable weighted graph, and let $o$ be a vertex of $G$. There exists $\delta>0$ such that $$\beta h \langle \sigma_o \rangle_{\beta,h}^+, \quad \sum_{e\in E^\rightarrow_o} \beta J_e \langle \sigma_e \rangle_{\beta,h}^+, \quad \phi_{2,\beta,h}^w\bigl(\omega(o)=1\bigr), \; \text{ and } \; \sum_{e\in E^\rightarrow_o} \phi_{2,\beta,h}^w\bigl(\omega(e)=1\bigr)$$ are locally $\delta$-Hölder continuous functions of $(\beta,h)\in [0,\infty)^2$.
Throughout the remainder of this section, we will use $\asymp$, $\preceq,$ and $\succeq$ to denote equalities and inequalities that hold to within positive multiplicative constants depending on the weighted graph $G$ but not any further parameters.
By scaling we may assume without loss of generality that $\sum_{e\in E^\rightarrow_o} J_e=1$. We first prove local Hölder continuity of $\beta h\langle \sigma_o \rangle_{
\beta,h}^+$ and $\sum_{e\in E^\rightarrow_o} J_e \langle \sigma_e \rangle_{\beta,h}^+$. Since $\langle \sigma_x \rangle_{\beta,h}^+$ is increasing in $\beta$ and $h$ for each $x\in E \cup V$, it suffices to prove that there exists $\delta>0$ such that $$\begin{aligned}
\beta_1 h_1 \left[\langle \sigma_o \rangle_{\beta_2,h_2}^+-\langle \sigma_o \rangle_{\beta_1,h_1}^+\right] &\preceq \left(\beta_2h_2 -\beta_1 h_1 + \beta_2-\beta_1\right)^{\delta} \qquad \text{ and }
\label{eq:HolderClaim11}
\\
\sum_{e\in E^\rightarrow_o} \beta_1 J_e \left[ \langle \sigma_e \rangle_{\beta_2,h_2}^+ - \langle \sigma_e\rangle_{\beta_1,h_1}^+\right] &\preceq (\beta_1^{1/2} \vee \beta_1) \left(\beta_2h_2 -\beta_1 h_1 + \beta_2-\beta_1\right)^{\delta}
\label{eq:HolderClaim12}\end{aligned}$$ for every $\beta_2 \geq \beta_1 \geq 0$ and $h_2 \geq h_1 \geq 0$, the claim then following by a little elementary analysis. Fix one choice of these parameters $\beta_1,\beta_2,h_1,h_2$, and let ${\mathbf{Q}}={\mathbf{Q}}_{\beta_1,h_1,\beta_2,h_2}$. All the implicit constants appearing in this proof will depend on $G$ but not on the choice of these parameters. We have by \[cor:gradient\] that $$\beta_1 J_x \left[\langle \sigma_x \rangle_{\beta_2,h_2}^+ - \langle \sigma_x \rangle_{\beta_1,h_1}^+\right]
\leq
{\mathbf{Q}}(\sA_x)$$ for each $x\in E \cup V$, where $\sA_x$ is the event defined just before the statement of that proposition and where we set $J_v=h_1$ for each $v\in V$. Let $\cB$ be the set of vertices $v$ of $G$ such that either ${\mathbf{n}}_2(v_2)>0$ or there exists an edge $e$ of $G$ touching $v$ such that ${\mathbf{n}}_2(e_2)>0$, and let $\sB$ be the event that $K_o^1$ is finite and that $K_o^1\cap \cB \neq \emptyset$. Observe that $$\sA_x \subseteq \sB \cap \{{\mathbf{n}}_2(x_1)>0\}$$ for every $x \in E^\rightarrow_o \cup \{o\}$, so that $$\beta_1 h_1 \left[\langle \sigma_o \rangle_{\beta_2,h_2}^+-\langle \sigma_o \rangle_{\beta_1,h_1}^+\right] \leq {\mathbf{Q}}(\sB)$$ and similarly that $$\begin{aligned}
\sum_{e\in E^\rightarrow_o} \beta_1 J_e \left[\langle \sigma_e \rangle_{\beta_2,h_2}^+ - \langle \sigma_e \rangle_{\beta_1,h_1}^+\right]
&\leq \sum_{e\in E^\rightarrow_o}
{\mathbf{Q}}(\sA_e) \leq {\mathbf{Q}}\left[ \mathbbm{1}(\sB) \cdot \#\{e\in E^\rightarrow_o : {\mathbf{n}}_2(e_1)>0 \} \right]
\nonumber
\\
&
\leq \sqrt{{\mathbf{Q}}\left(\sB\right) {\mathbf{Q}}\left[\#\{e\in E^\rightarrow_o : {\mathbf{n}}_2(e_1)>0 \}^2 \right]},
\label{eq:degree_2nd_moment}\end{aligned}$$ where we used Cauchy-Schwarz in the second line. (One can improve the exponent obtained here by using Hölder instead of Cauchy-Schwarz.) As discussed in \[subsec:random\_currents\], it is a consequence of the Lupu-Werner coupling that the set $\{e:{\mathbf{n}}_2(e_1)>0\}$ is stochastically dominated by the random cluster model on $G$ and hence also by a Bernoulli bond percolation process on $G$ in which each edge $e$ of $G$ is included independently at random with probability $(e^{\beta_1 J_e}-1)/e^{\beta_1 J_e} \leq \beta_1 J_e$. It follows easily that $${\mathbf{Q}}\left[\#\{e\in E^\rightarrow_o : {\mathbf{n}}_2(e_1)>0 \}^2 \right]
\leq \sum_{e\in E^\rightarrow_o} \beta_1 J_e + 2\sum_{e \neq e' \in E^\rightarrow_o} \beta_1^2 J_e J_{e'}
\preceq \beta_1 \vee \beta_1^2$$ and hence that $$\begin{aligned}
\sum_{e\in E^\rightarrow_o} \beta_1 J_e \left[\langle \sigma_e \rangle_{\beta_2,h_2}^+ - \langle \sigma_e \rangle_{\beta_1,h_1}^+\right]
&
\preceq (\beta_1^{1/2} \vee \beta_1) {\mathbf{Q}}\left(\sB\right)^{1/2}.
\label{eq:degree_2nd_moment2}\end{aligned}$$ Thus, to prove the claimed inequalities and , it suffices to prove that there exists a constant $\delta>0$ such that $$\label{eq:HolderClaim2}
{\mathbf{Q}}(\sB) \preceq \left(\beta_2h_2 -\beta_1 h_1 + \beta_2-\beta_1\right)^{\delta}.$$
To this end, we consider the union bound $$\label{eq:Qunion}
{\mathbf{Q}}(\sB)
\\\leq {\mathbf{Q}}(n<|K_o^1|<\infty) + {\mathbf{Q}}(\sB \cap \{|K_o^1| \leq n\}),$$ which holds for every $n\geq 1$. It follows from \[prop:finite\_clusters\_mismatched\] that there exists a positive constant $\delta_1$ such that the first term satisfies $$\label{eq:Qlargen}
{\mathbf{Q}}(n<|K_o^1|<\infty) \preceq n^{-\delta_1}$$ for every $n\geq 1$. We now bound the second term. For each $n\geq 1$ define a mass-transport function $F_n:V^2 \to [0,\infty]$ by $$F_n(u,v) = {\mathbf{Q}}\left(|K_u^1| \leq n, v \in K_u \cap \cB \right),$$ so that $$\begin{gathered}
{\mathbf{Q}}(\sB \cap \{|K_o^1| \leq n\}) \leq \sum_{v\in V} F_n(o,v) = \sum_{v\in V} F_n(v,o)\\ = {\mathbf{Q}}\left[|K_o^1| \mathbbm{1}\left(|K_o^1|\leq n, o \in \cB\right) \right]\leq n {\mathbf{Q}}(o \in \cB).\end{gathered}$$ (The final inequality here is presumably rather wasteful.) Let $\theta=1-\beta_1/\beta_2$ and $\phi=1-(\beta_1h_1)/(\beta_2h_2)$. It follows straightforwardly from and and the definition of ${\mathbf{Q}}$ that $$\begin{aligned}
\label{eq:Qsmalln}
{\mathbf{Q}}(o \in \cB) &\leq {\mathbf{Q}}({\mathbf{n}}_2(o_2)>0)+\sum_{e\in E^\rightarrow_o}{\mathbf{Q}}({\mathbf{n}}_2(e_2)>0)
\\&\leq \phi \beta_2 h_2 + \sum_{e\in E^\rightarrow_o} \theta \beta_2 J_e = \beta_2h_2-\beta_1h_1 + \beta_2-\beta_1.\end{aligned}$$ Putting together , , and yields that $${\mathbf{Q}}(\sB)
\\\preceq n^{-\delta_1}+ n \left(\beta_2h_2-\beta_1h_1 + \beta_2-\beta_1\right)$$ for every $n\geq 1$, and taking $n=\lceil (\beta_2h_2-\beta_1h_1 + \beta_2-\beta_1)^{-1/(1+\delta_1)} \rceil$ yields the claimed inequality with the exponent $\delta=\delta_1/(1+\delta_1)$. This completes the proof of local Hölder continuity of $\beta h\langle \sigma_o \rangle_{
\beta,h}^+$ and $\sum_{e\in E^\rightarrow_o} J_e \langle \sigma_e \rangle_{\beta,h}^+$.
We now deduce local Hölder continuity of $\sum_{e\in E^\rightarrow_o} \phi_{2,\beta,h}^w\bigl(\omega(e)=1\bigr)$ and $\phi_{2,\beta,h}^w\bigl(\omega(o)=1\bigr)$ from local Hölder continuity of $\sum_{e\in E^\rightarrow_o} \beta J_e \langle \sigma_e \rangle_{\beta,h}^+$ and $\beta h\langle \sigma_o \rangle_{\beta,h}^+$. The equality implies that $$\phi_{2,\beta,h}^w\bigl(\omega(x)=1\bigr)
= 1 - e^{-\beta J_x}\langle e^{-\beta J_x \sigma_x}\rangle_{\beta,h}^+.$$ Using the identity $e^{-\beta J_x \sigma_x} = \cosh(\beta J_x) - \sigma_x \sinh(\beta J_x)$ (which holds since $\sigma_x\in \{-1,+1\}$) yields that $$\begin{aligned}
\phi_{2,\beta,h}^w\bigl(\omega(x)=1\bigr) &= 1- e^{-\beta J_x} \cosh(\beta J_x) + e^{-\beta J_x} \sinh(\beta J_x)\langle \sigma_x \rangle_{\beta,h}^+
\label{eq:phisigma_formula}\end{aligned}$$ for every $x\in E \cup V$ and $\beta,h \geq 0$. Since the functions $1-e^{-\beta J_x} \cosh(\beta J_x)$ and $e^{-\beta J_x}\sinh(\beta J_x)/\beta J_x$ are both locally Lipschitz and since local Hölder continuity is preserved under sums and products, we deduce immediately that $\phi_{2,\beta,h}^w(\omega(o)=1)$ is locally $\delta$-Hölder continuous as claimed. The proof that $\sum_{e\in E^\rightarrow_o} \phi_{2,\beta,h}^w\bigl(\omega(e)=1\bigr)$ is locally $\delta$-Hölder continuous for some $\delta>0$ is similar and we omit the details.
The $\beta$-derivatives of $\sum_{e\in E^\rightarrow_o} J_e \langle \sigma_e \rangle_{\beta,0}^+$ and $\sum_{e\in E^\rightarrow_o} \phi_{2,\beta,0}^w\bigl(\omega(e)=1\bigr)$ are closely related to the *specific heat* of the Ising model. It is conjectured, and known in some cases, that these derivatives are bounded (but not necessarily continuous) in high-dimensional models and unbounded in low-dimensional models; see e.g. [@MR3162481; @MR678000; @MR857063; @MR588470] for detailed discussions.
We are now ready to complete the proof of \[thm:main\_continuity\]. (Note that local Hölder continuity of the magnetization $\langle \sigma_o \rangle_{\beta,h}^+$ is *not* implied by \[cor:change\_in\_correlations\], which does not give any control of $\langle \sigma_o \rangle_{\beta,0}^+$.) The proof will use the fact that there exists an automorphism-invariant monotone coupling, sometimes known as the **Grimmett coupling** [@MR1379156], between the two random cluster measures $\phi^w_{q,\beta_1,h_1}$ and $\phi^w_{q,\beta_2,h_2}$ whenever $q\geq 1$, $\beta_2 \geq \beta_1 >0$ and $h_2 \geq h_1 \geq 0$. That is, for each such $q,\beta_1,\beta_2,h_1,h_2$, there exists a pair of random variables $(\omega_1,\omega_2)$ whose law is invariant under the automorphisms of $G$ such that the marginal law of $\omega_1$ is $\phi^w_{q,\beta_1,h_1}$, the marginal law of $\omega_2$ is $\phi^w_{q,\beta_2,h_2}$, and $\omega_1(x) \leq \omega_2(x)$ for every $x\in V \cup E$ almost surely. The existence of such a coupling is essentially due to Häggström, Jonasson, and Lyons [@MR1913108], who proved that Grimmett’s monotone coupling of the random cluster model at different temperatures [@MR1379156] can be extended to infinite graphs in an automorphism-invariant way; while that paper considers only locally finite models in zero external field, the proof generalizes straightforwardly to possibly long-range models in non-negative external field.
By inclusion-exclusion, it suffices to prove that there exists $\delta>0$ such that ${\mathbf{I}}^+_{\beta,h}(\sigma(v)=1 \text{ for every $v\in A$})$ is a locally $\delta$-Hölder continuous function of $(\beta,h)\in [0,\infty)^2$ for each finite $A \subseteq V$. Given the configuration $\omega$, we say that a set $A \subseteq V$ is $w$**-finite** if it is finite and does not intersect the set $\{w\in V: \omega(w)=1\}$. The Edwards–Sokal coupling implies that $$\label{eq:ES_cylinder}
{\mathbf{I}}^+_{\beta,h}(\sigma(v)=1 \text{ for every $v\in A$}) = \phi^w_{2,\beta,h}\left[2^{-\#\{w\text{-finite clusters intersecting }A\}} \right]$$ for every $\beta,h \geq 0$ and every finite set $A \subseteq V$. Since the right hand side is increasing in $\beta$ and $h$, it suffices to prove that for each $M<\infty$ there exists a constant $C=C(M)$ such that $$\begin{gathered}
{\mathbf{I}}^+_{\beta_2,h_2}(\sigma(v)=1 \text{ for every $v\in A$})
-
{\mathbf{I}}^+_{\beta_1,h_1}(\sigma(v)=1 \text{ for every $v\in A$})
\\
\hspace{3.5cm}\leq C |A| (\beta_2-\beta_1 + h_2-h_1)^\delta\end{gathered}$$ for every $A \subseteq V$, $0\leq \beta_1 \leq \beta_2 \leq M$ and $0\leq h_1 \leq h_2 \leq M$. Fix $M$ and one such choice of $M \geq \beta_2 \geq \beta_1 \geq 0$, $M \geq h_2 \geq h_1 \geq 0$. We write $\preceq_M$ for an inequality that holds to within a positive multiplicative constant depending on $M$ but not any further parameters. Let $(\omega_1,\omega_2)$ be an automorphism-invariant monotone coupling of $\phi^w_{2,\beta_1,h_1}$ and $\phi^w_{2,\beta_2,h_2}$ as above. We write $\P$ for probabilities taken with respect to the joint law of $(\omega_1,\omega_2)$, and write $K^1_v$ and $K^2_v$ for the clusters of $v$ in $\omega_1$ and $\omega_2$ respectively for each $v\in V$. We say that a set $W \subseteq V$ is $w_i$**-finite** if it is finite and does not intersect the set $\{w\in V: \omega_i(w)=1\}$. Let $\sA$ be the event that there are fewer $w_2$-finite clusters intersecting $A$ in $\omega_2$ than there are $w_1$-finite clusters intersecting $A$ in $\omega_1$. The equality implies that $$\begin{aligned}
{\mathbf{I}}^+_{\beta_2,h_2}(\sigma(v)=1 \text{ for every $v\in A$})
-
{\mathbf{I}}^+_{\beta_1,h_1}(\sigma(v)=1 \text{ for every $v\in A$})
\leq \P(\sA).\end{aligned}$$ Let $\cP$ be the set of vertices $v\in V$ such that either $\omega_2(v)=1$ and $\omega_1(v)=0$ or there exists $e\in E^\rightarrow_v$ such that $\omega_2(e)=1$ and $\omega_1(e)=0$. We have from the definitions that $$\sA \subseteq \bigcup_{v\in A} \{ K^1_v \text{ is $w_1$-finite and } K^1_v \cap \cP \neq \emptyset\}
\subseteq \bigcup_{v\in A} \{ |K^1_v|<\infty \text{ and } K^1_v \cap \cP \neq \emptyset\}$$ so that transitivity and a union bound give that $$\begin{gathered}
{\mathbf{I}}^+_{\beta_2,h_2}(\sigma(v)=1 \text{ for every $v\in A$})
-
{\mathbf{I}}^+_{\beta_1,h_1}(\sigma(v)=1 \text{ for every $v\in A$})\\
\leq \P(\sA) \leq |A| \P\bigl(|K_o^1|<\infty \text{ and } K_o^1 \cap \cP \neq \emptyset\bigr).\end{gathered}$$ Thus, to conclude the proof it suffices to prove that there exists a positive constant $\delta$ such that $$\label{eq:HolderClaim4}
\P\bigl(|K_o^1|<\infty \text{ and } K_o^1 \cap \cP \neq \emptyset\bigr) \preceq_M (\beta_2-\beta_1 + h_2-h_1)^\delta.$$
We prove by following a similar strategy to the proof of but using \[thm:finite\_clusters\] and \[cor:change\_in\_correlations\] instead of \[prop:finite\_clusters\_mismatched\] and \[eq:bounding\_current\_by\_FK1,eq:bounding\_current\_by\_FK2\]. We begin by writing down for each $n\geq 1$ the union bound $$\P(|K_o^1|<\infty \text{ and } K_o^1 \cap \cP \neq \emptyset) \leq \P(n \leq |K_o^1| < \infty) + \P(|K_o^1| \leq n \text{ and } K_o^1 \cap \cP \neq \emptyset).$$ \[thm:free\_random\_cluster\] implies that there exists a constant $\delta_1>0$ such that $\P(n \leq |K_o^1| <\infty) \preceq n^{-\delta_1}$. Meanwhile, as in the proof of \[cor:change\_in\_correlations\], we may apply the mass-transport principle to bound $$\P(|K_o^1| \leq n \text{ and } K_o^1 \cap \cP \neq \emptyset) \leq n \, \P(o \in \cP)$$ for every $n\geq 1$. Another union bound then implies that $$\begin{aligned}
\P(o \in \cP) &\leq \P(\omega_2(o) = 1)-\P(\omega_1(o) = 1) +
\sum_{e\in E^\rightarrow_o} \left[\phi_{2,\beta_2,h_2}^w\bigl(\omega(e)=1\bigr)-\phi_{2,\beta_1,h_1}^w\bigl(\omega(e)=1\bigr) \right],\end{aligned}$$ and applying \[cor:change\_in\_correlations\] yields that there exists a positive constant $\delta_2$ such that $$\P(o \in \cP) \preceq_M (\beta_2-\beta_1 + h_2-h_1)^{\delta_2}.$$ It follows that $$\P\bigl(|K_o^1|<\infty \text{ and } K_o^1 \cap \cP \neq \emptyset\bigr) \preceq_M n^{-\delta_1} + n \left(\beta_2-\beta_1+h_2-h_1\right)^{\delta_2}$$ for each $n\geq 1$, and taking $n=\left\lceil \left(\beta_2-\beta_1+h_2-h_1\right)^{-\delta_2/(1+\delta_1)} \right\rceil$ implies the claimed inequality . This completes the proof.
It follows immediately from \[thm:main\_continuity\] that $$m^*(\beta_c)=m^+(\beta_c,0)=\lim_{\beta \uparrow \beta_c} m^+(\beta,0)=0,$$ establishing \[thm:main\_simple\]. Similarly, \[thm:main\] follows from \[thm:main\_continuity\] together with the fact that $$m^\#(\beta,h) \leq m^+(\beta \vee \beta_c,|h|) = m^+(\beta \vee \beta_c,|h|) - m^+(\beta_c,0)$$ for every $\beta \geq 0$, $h\in \R$, and $\#\in \{f,+,-\}$.
It remains to deduce \[thm:main\_continuity\_FK\] from \[thm:main\_continuity\].
By \[thm:main\_continuity\], there exists $\delta>0$ such that if $F:\{0,1\}^V \to \R$ depends on at most finitely many vertices then ${\mathbf{I}}^+_{\beta,h} [F(\sigma)]$ is a locally $\delta$-Hölder continuous function of $(\beta,h)\in [0,\infty)^2$. It follows easily that ${\mathbf{G}}^w_{\beta,h} \left[e^{\beta H_A} \right]$ is locally $\delta$-Hölder continuous for each finite $A \subseteq E \cup V$, and hence by that $\phi^w_{2,\beta,h}(\omega(x)=0 \text{ for all $x\in A$})$ is locally $\delta$-Hölder continuous for each finite $A \subseteq E \cup V$. The claim now follows by inclusion-exclusion.
We note that the local Hölder continuity provided by \[thm:main\_continuity\] is presumably very far from optimal when $\beta \neq \beta_c$. We conjecture that the following much stronger statement holds. This conjecture is most interesting in the case that $h=0$ and $\beta>\beta_c$.
Let $G$ be an infinite Cayley graph. Then the magnetization $m^+(\beta,h)$ is an infinitely differentiable function of $(\beta,h) \in [0,\infty)^2 \setminus \{(\beta_c,0)\}$.
See [@HermonHutchcroftSupercritical; @georgakopoulos2018analyticity; @georgakopoulos2020analyticity] for related results for Bernoulli percolation.
Closing remarks and open problems
=================================
Equality of critical parameters for the FK-Ising model {#subsec:betafbetaw}
------------------------------------------------------
Let $G=(V,E,J)$ be a connected, transitive, weighted graph. Recall that we define $\beta_c^\#(q)= \sup\{ \beta \geq 0: \phi^\#_{q,\beta,0}$ is supported on configurations with no infinite clusters$\}$ for each $q \geq 1$ and $\#\in \{f,w\}$. When $G$ is amenable and $q\geq 1$, it is a classical theorem [@GrimFKbook Theorem 4.63] that $\phi^f_{q,\beta,0} \neq \phi^w_{q,\beta,0}$ for at most countably many values of $\beta$ and hence that $\beta_c^f(q)=\beta_c^w(q)$ for every $q\geq 1$. On the other hand, when $G$ is nonamenable, Jonasson [@MR1671859] proved that there is a strict inequality $\beta_c^w(q)<\beta_c^f(q)$ between these two critical parameters for all sufficiently large values of $q$ [@MR1671859]. For a regular tree, strict inequality holds if and only if $q>2$ [@MR1373377]. We now show that equality always holds when $q=2$; we believe that this result is new in the nonamenable case.
\[prop:betafbetaw\] Let $G=(V,E,J)$ be a connected, transitive, weighted graph. Then the critical inverse temperatures for the Ising model, the wired FK-Ising model, and the free FK-Ising model coincide. That is, $\beta_c=\beta_c^f(2)=\beta_c^w(2)$.
It follows by sharpness of the Ising phase transition [@MR894398; @duminil2015new] that $$\sum_{x\in V} \langle \sigma_o \sigma_x \rangle_{\beta,0}^f < \infty \iff \sum_{x\in V} \langle \sigma_o \sigma_x \rangle_{\beta,0}^+ < \infty \iff
\text{$\beta < \beta_c$}.$$ See in particular [@duminil2015new Theorem 2.1 and Section 2.2, Remark 4]. On the other hand, we have by the Edwards–Sokal coupling that $$\langle \sigma_o \sigma_x \rangle_{\beta,0}^f = \phi^f_{2,\beta,0}(o \leftrightarrow x) \qquad \text{ and } \qquad \langle \sigma_o \sigma_x \rangle_{\beta,0}^+ = \phi^w_{2,\beta,0}(o \leftrightarrow x \text{ or } |K_o|=|K_x|=\infty),$$ so that $\phi_{2,\beta,0}^\#[|K_o|]<\infty$ if and only if $\beta<\beta_c$ for each $\# \in \{f,w\}$. The claimed equality $\beta_c^f(2)=\beta_c^w(q)=\beta_c$ follows from the sharpness of the phase transition for the random cluster model [@MR3898174; @1901.10363], and in particular from [@1901.10363 Theorem 1.5] which states that $\phi_{q,\beta,0}^\#[|K_o|] < \infty$ for every $\beta < \beta_c^\#(q)$ and hence that $\beta_c^\#(q)=\sup\{\beta \geq 0 : \phi_{q,\beta,0}^\#[|K_o|] < \infty\}$ for every $q\geq 1$ and $\# \in \{f,w\}$.
Consequences for planar graphs {#subsec:planar}
------------------------------
We now explain how our results interact with those of Häggström, Jonasson, and Lyons [@MR1894115] to deduce further consequences in the planar case. Let $G=(V,E)$ be a transitive nonamenable graph. For each $q \geq 1$ and $\#\in \{f,w\}$ we define $$\beta_u^\#(q) = \inf \bigl\{ \beta \geq 0 : \phi^\#_{q,\beta,0} \text{ is supported on configurations with a unique infinite cluster}\bigr\}.$$ It follows from a theorem of Lyons and Schramm [@LS99 Theorem 4.1] that if $G$ is unimodular then $\phi^\#_{q,\beta,0}$ is supported on configurations with a unique infinite cluster if and only if $\inf_{u,v\in V} \phi^\#_{q,\beta,0}(u \leftrightarrow v) > 0$, so that $\phi^\#_{q,\beta,0}$ is supported on configurations with a unique infinite cluster for every $\beta>\beta_u^\#(q)$. The relationships between $\beta_c^f,\beta_c^w,\beta_u^f,$ and $\beta_u^w$ are discussed in detail in [@MR1894115 Section 3].
Let $G$ be a unimodular, quasi-transitive, nonamenable proper plane graph with locally finite quasi-transitive dual $G^\dagger$. Let $\omega$ be a random variable with law $\phi^\#_{G,q,\beta,0}$ for some $q\geq 1$, $\beta > 0$, and $\# \in \{f,w\}$. It is well-known (see e.g. [@MR1894115 Proposition 3.4]) that the dual configuration $\omega^\dagger = \{e^\dagger : e\notin \omega\}$ has law $\phi^{\#^\dagger}_{G^\dagger,q,\beta^\dagger,0}$, where $w^\dagger=f$, $f^\dagger = w$, and $\beta^\dagger=\beta^\dagger(q,\beta)>0$ is the unique solution to $$(e^{2\beta}-1)(e^{2\beta^\dagger}-1)=q.$$ Let $k$ and $k^\dagger$ be the number of infinite clusters of $\omega$ and $\omega^\dagger$ respectively. Proposition 3.5 of [@MR1894115] generalizes an argument of Benjamini and Schramm [@BS00] to show that $$\label{eq:duality_clusters}
(k,k^\dagger) \in \bigl\{(0,1),(1,0),(\infty,\infty)\bigr\} \qquad \text{almost surely.}$$ It follows in particular that $\beta_c^\#(G^\dagger,q) = \beta_u^{\#^\dagger}(G,q)^\dagger$ for every $q \geq 1$ [@MR1894115 Corollary 3.6]. When $q=2$ and $G^\dagger$ is transitive we have that $\beta_c^w(G^\dagger)=\beta_c^f(G^\dagger)$ by \[prop:betafbetaw\] and hence that $\beta_u^w(G)=\beta_u^f(G)$ also. Since there are a.s. no infinite clusters in the critical free random-cluster model on $G^\dagger$, it follows from that there is a unique infinite cluster in the wired random cluster model on $G$ at $\beta_u^w$ [@MR1894115 Corollary 3.7]. Combining these facts with \[thm:main,thm:main\_continuity\] yields the following corollary.
\[cor:planar\] Let $G=(V,E)$ be a unimodular, transitive, nonamenable proper plane graph with locally finite transitive dual $G^\dagger$, and consider the FK-Ising model on $G$. Then the following hold:
1. The parameters $\beta_u=\beta_u^f=\beta_u^w$ coincide and satisfy $\beta_u>\beta_c$.
2. The free and wired FK-Ising measures on $G$ coincide at $\beta_u$ and are both supported on configurations with a unique infinite cluster.
3. The free FK-Ising measure $\phi^f_{2,\beta,0}$ is weakly continuous in $\beta$.
4. For each $\beta \geq 0$, ${\mathbf{G}}^w_{\beta,0} \neq {\mathbf{G}}^f_{\beta,0}$ if and only if $\phi^w_{2,\beta,0} \neq \phi^f_{2,\beta,0}$ if and only if $\beta_c < \beta <\beta_u$.
The analogous results for Bernoulli percolation are due to Benjamini and Schramm [@BS00]. The condition that the dual of $G$ is transitive should not really be necessary, since \[thm:main,thm:main\_continuity\] should both extend to quasi-transitive nonamenable graphs.
The identity $\beta_u=\beta_u^f=\beta_u^w$ follows from \[prop:betafbetaw\] and duality as explained above. \[thm:main\_continuity\_FK\] implies that the free and wired FK-Ising models on $G^\dagger$ coincide at $\beta_c$, so that item $2$ follows by planar duality together with the corresponding fact for the free measure at $\beta_c$ [@MR1894115 Theorem 3.1]. Since the free and wired FK-Ising measures both have no infinite clusters at $\beta_c$ and a unique infinite cluster at $\beta_u$, it follows that $\beta_u>\beta_c$, completing the proof of item $1$. Item $3$ follows immediately from \[thm:main\_continuity\_FK\] applied to $G^\dagger$ together with planar duality. Item $4$ follows from [@MR1894115 Proposition 3.8] and the formula .
Discontinuity of the free Ising measure {#subsec:discontinuity}
---------------------------------------
It is natural to wonder whether an analogue of \[thm:main\_continuity\] also holds for the *free* Ising measure with zero external field; we now argue that this is not the case in general. In the previous subsection we saw examples of Cayley graphs in which there is a unique infinite cluster in the free FK-Ising model at $\beta_u^f$. It is also possible for there to be infinitely many infinite clusters at $\beta_u^f$. Indeed, it follows from the proof of [@LS99 Corollary 6.6] that if $G$ is a Cayley graph of an infinite Kazhdan group then $\beta_f^u<\infty$ and the number of infinite clusters in the free FK-Ising model at $\beta_c^f$ is either $0$ or $\infty$. The perturbative criteria of [@MR1833805] imply that every nonamenable group has a Cayley graph for which $\beta_u^f > \beta_c$ (see [@MR1756965] for a similar result for Bernoulli percolation), and it follows that there exists a nonamenable Cayley graph such that the free FK-Ising model has infinitely many infinite clusters at the uniqueness threshold $\beta_u^f$.
We claim that if $\beta_u^f>\beta_c$ and there is non-uniqueness at $\beta_u^f$ then there exist vertices $u$ and $v$ such that $\langle \sigma_u \sigma_v \rangle^f_{\beta,0} =\phi^f_{2,\beta,0}(u \leftrightarrow v)$ is discontinuous at $\beta_u^f$. Indeed, under this assumption, we have by the aforementioned theorem of Lyons and Schramm [@LS99 Theorem 4.1] that there exist $u,v \in V$ such that $$\phi^f_{2,\beta_u^f,0}(u \leftrightarrow v) \leq \frac{1}{2}\phi^f_{2,\beta_u^f,0}(o \to \infty)^2.$$ On the other hand, for each $\beta>\beta_u^f$ we have by FKG that $$\phi^f_{2,\beta,0}(u \leftrightarrow v) \geq \phi^f_{2,\beta,0}(u \to \infty \text{ and } v\to \infty) \geq \phi^f_{2,\beta_u^f,0}(o \to \infty)^2,$$ which yields the desired discontinuity. It follows that the measures ${\mathbf{I}}^f_{\beta,0}$ and ${\mathbf{G}}_{\beta,0}^f$ are both weakly discontinuous in $\beta$ at $\beta_u^f$, so that \[thm:main\_continuity\] cannot be extended to the case of free boundary conditions. Similar phenomena for Bernoulli percolation are discussed in [@HermonHutchcroftSupercritical Section 5.1].
In fact, the free FK-Ising measure $\phi^f_{2,\beta,0}$ is also weakly discontinuous in $\beta$ at $\beta_u^f$ in the same class of examples. This follows from the following general proposition, which shows that the gradient Ising model and FK-Ising model always have the same continuity properties.
\[prop:continuity\_equivalence\] Let $G=(V,E)$ be an infinite connected weighted graph, let $\# \in \{f,w\}$, and let $((\beta_n,h_n))_{n\geq1}$ be a sequence in $[0,\infty)^2$ converging to some $(\beta,h)$. Then $${\mathbf{G}}^\#_{\beta,h} = {\mathop{\operatorname{w-lim}}}_{n\to\infty} {\mathbf{G}}^\#_{\beta_n,h_n} \iff \phi^\#_{\beta,h} = {\mathop{\operatorname{w-lim}}}_{n\to\infty} \phi^\#_{\beta_n,h_n}.$$
A similar proof extends this equivalence to the random current and loop $O(1)$ models. It is also possible to prove a similar statement for the random cluster model with $q\in \{3,4,\ldots\}$ and the Potts model by using the fact that the random cluster model can be represented as Bernoulli percolation on the colour clusters to deduce a formula analogous to \[eq:FK\_Gradient\_Formula\].
The claim is trivial when $\beta=0$ so we may suppose that $\beta>0$. The implication $\Rightarrow$ follows immediately from . To deduce the implication $\Leftarrow$ from $\eqref{eq:FK_Gradient_Formula}$, it suffices to prove that
- the functions $\{e^{-\beta \sum_{e\in A} J_e \sigma_e} : A \subseteq E \text{ finite}\}$ have dense linear span in $C(\{-1,1\}^E,\| \cdot \|_\infty)$ for each $\beta>0$, and that
- the functions $\{e^{-\beta \sum_{e\in A \cap E} J_e \sigma_e - \beta \sum_{v\in A \cap V} h\sigma_v } : A \subseteq E \cup V \text{ finite}\}$ have dense linear span in $C(\{-1,1\}^{E\cup V},\| \cdot \|_\infty)$ for each $\beta,h>0$.
In both cases, the set $A$ is allowed to be empty. We prove the first claim, the proof of the second being similar. Fix $\beta>0$. Using the identity $e^{-\beta \sum_{e\in A} J_e \sigma_e} =\prod_{e\in A} (\cosh(\beta J_e)-\sigma_x \sinh(\beta J_e))$ one can prove by induction on $|B|$ that $\prod_{e\in B} \sigma_e$ belongs to the linear span of $\{e^{-\beta \sum_{e\in A} J_e \sigma_e} : A \subseteq E \text{ is finite}\}$ for every finite (possibly empty) set $B \subseteq E$. The linear span of $\{\prod_{e\in B} \sigma_e : B \subseteq E$ finite$\}$ is an algebra that separates points and is therefore dense in $C(\{-1,1\}^E,\| \cdot \|_\infty)$ by Stone-Weierstrass, concluding the proof.
At this point, the reader may be wondering where the *proof* of \[thm:main\_continuity,thm:main\_continuity\_FK\] breaks down in the case of free boundary conditions. Here is a short answer: The free version of \[cor:gradient\] in zero external field does not allow connections through infinity. This means that one must also consider infinite clusters when applying the free version of this proposition to control the change in the edge marginals of the FK-Ising model as in \[cor:change\_in\_correlations\], and the free version of \[prop:finite\_clusters\_mismatched\] does not suffice to do this. A similar problem also arises when attempting to apply the free version of .
The spectral radius of the free gradient Ising and random current models {#subsec:free_spectral_radius}
------------------------------------------------------------------------
We now note that, although the free gradient Ising measure ${\mathbf{G}}^f_{\beta,0}$ is not known to be a factor of i.i.d. when $\beta > \beta_c$, it always has spectral radius at most that of the graph. Note that, in contrast, the free Ising measure ${\mathbf{I}}^f_{\beta,0}$ on a $k$-regular tree has spectral radius strictly greater than that of the tree when $\beta$ is sufficiently large [@MR3603969].
\[thm:free\_spectral\_radius\] Let $G=(V,E,J)$ be a connected transitive weighted graph and let $\Gamma$ be a closed unimodular transitive group of automorphisms. Then $\rho(\mathbf{G}^f_{\beta,h})$, $\rho(\mathbf{C}^f_{\beta,h})$, and $\rho(\mathbf{L}^f_{\beta,h})$ are all at most $\rho(G)$ for every $\beta,h\geq 0$.
Note that when $G$ is a tree, the gradient Ising measure $\mathbf{G}^f_{\beta,0}$ is equivalent to Bernoulli bond percolation on $G$, so that this inequality is trivial.
The claim is trivial when $\beta=0$, so suppose $\beta>0$. By \[thm:spectralradius\], it suffices to prove the claim for the gradient Ising measure $\mathbf{G}^f_{\beta,h}$. Let $X=(X_n)_{n\geq 0}$ be the random walk on $G$ and let $\hat X = (\hat X_n)_{n\geq 0}$ be the associated random walk on $\Gamma$ as defined in \[subsec:spectral\_background\], and let $\sigma$ be a random variable with law ${\mathbf{G}}^f_{\beta,h}$ that is independent of $\hat X$. Recall from that $$\begin{aligned}
\phi^f_{2,\beta,h}(\omega(x)=0 \text{ for all $x\in A$})
&={\mathbf{G}}^f_{\beta,h} \left[e^{\beta H_A} \right]\prod_{x \in A} \left( 1-\tanh(\beta J_x)\right)\end{aligned}$$ for every finite set $A \subseteq E \cup V$, where we write $J_v=h$ for each $v\in V$ and recall that $K_A = \sum_{x\in A} J_x \sigma_x$. Thus, it follows by a similar analysis to the proof of \[thm:spectralradius\] that if we set $F_A(\sigma)=e^{\beta H_A(\sigma)}$ for each finite $A \subseteq E \cup V$ (so that $F_\emptyset \equiv 1$) then $$\limsup_{k\to\infty} {{\mathrm{Cov}}}\left(F_A(\sigma),F_A(\hat X_{2k}^{-1} \sigma) \right)^{1/2k} \\\leq \max\{\rho(\phi^f_{\beta,h}),\rho(G)\} = \rho(G),$$ for every finite set $A \subseteq E \cup V$, where the final inequality follows from \[thm:Ising\_factor\]. The proof of \[prop:continuity\_equivalence\] implies that the set $\{F_A : A \subseteq E \cup V$ is finite$\}$ has dense linear span in $L^2({\mathbf{G}}^\#_{\beta,h})$, so that the claim follows from .
This theorem raises the following natural question. See [@ray2019finitary; @MR3603969; @harel2018finitary] for related results.
Let $G=(V,E,J)$ be a connected transitive weighted graph. For what values of $\beta$, $h$, and $\#$ can $\mathbf{G}^\#_{\beta,h}$, $\mathbf{C}^\#_{\beta,h}$, and $\mathbf{L}^\#_{\beta,h}$ be expressed as factors of i.i.d.?
Other graphs
------------
We remark that the methods used to prove \[thm:finite\_clusters,thm:free\_random\_cluster\] can easily be combined with the methods of [@HermonHutchcroftIntermediate] to prove the following theorem, which applies in particular to certain Cayley graphs of intermediate volume growth.
Let $G=(V,E)$ be a connected, locally finite, unimodular transitive graph, and suppose that there exist constants $c>0$ and $\gamma > 1/2$ such that the return probabilities for simple random walk on $G$ satisfy $p_n(o,o) \leq e^{-cn^\gamma}$ for every $n\geq 1$. Then for each $q\geq 1$ the critical free random cluster measure $\phi^f_{q,\beta_c^f(q),0}$ is supported on configurations with no infinite clusters.
Applying the results of [@MR3306602], we immediately deduce the following corollary.
Let $G=(V,E)$ be an *amenable*, connected, locally finite, unimodular transitive graph, and suppose that there exist constants $c>0$ and $\gamma > 1/2$ such that the return probabilities for simple random walk on $G$ satisfy $p_n(o,o) \leq e^{-cn^\gamma}$ for every $n\geq 1$. Then the Ising model on $G$ has a continuous phase transition in the sense that $m^*(\beta_c)=0$.
Our results naturally raise the following interesting problem.
Extend \[thm:main\_continuity,thm:main\_continuity\_FK\] to nonunimodular transitive graphs.
One approach to this problem, which may be very challenging, would be to attempt to extend the analysis of [@Hutchcroftnonunimodularperc] from Bernoulli percolation to the Ising model. This would have the added benefit of giving a very complete description of the Ising model at and near criticality on such graphs, or more generally on graphs with a nonunimodular transitive subgroup of automorphisms such as $T \times \Z$, going far beyond the conclusions of \[thm:main\_continuity,thm:main\_continuity\_FK\].
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Jonathan Hermon for making us aware of Freedman’s work on maximal inequalities for martingales [@MR0380971], which inspired \[lem:martingale\_stuff,lem:martingale\_stuff2\]. We also thank Hugo Duminil-Copin, Geoffrey Grimmett, and Russ Lyons for helpful comments on an earlier version of the manuscript.
[^1]: The Curie temperature is named after Pierre Curie, who carried out a detailed study of this phase transition in his 1895 doctoral thesis. The fact that such a transition occurs was, however, known well before the work of P. Curie, with credit due most appropriately to Pouillet and Faraday; see [@jossang1997monsieur] for details. We thank Geoffrey Grimmett for making us aware of this.
[^2]: Using $2\beta$ instead of $\beta$ in the definition of $\phi_{q,\beta,h}$ is not standard, but makes the relationship between the Ising model and random cluster models simpler to state.
[^3]: In this context, a *current* on a graph is an $\N$-valued function on the edge set. Any measure on currents defines a measure on subgraphs by taking an edge $e$ to be open if and only if the current takes a positive value on $e$.
[^4]: We have chosen to restrict to product spaces of this form to help make the resulting theory more intuitive to probabilists; one could just as well consider arbitrary actions of $\Gamma$ on probability spaces by measure preserving transformations (a.k.a. pmp actions of $\Gamma$), as is standard in other parts of the literature.
[^5]: The original paper of Aizenman, Kesten and Newman [@MR901151] used large deviations estimates rather than maximal inequalities. The idea of using maximal inequalities instead, which leads to cleaner proofs and sharper inequalities, first arose in discussions with Vincent Tassion in 2018.
[^6]: NB: The names of $G$ and $H$ are switched in this reference.
|
---
author:
- 'Uffe Haagerup and Agata Przybyszewska[^1]'
bibliography:
- 'midtvej\_references.bib'
title: 'Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces.'
---
[**Abstract**]{}
Introduction
============
We consider a special class of metrics on second countable, locally compact groups, namely proper left invariant metrics which generate the given topology on $G$, and we will denote such a metric a [[**plig**]{} metric]{}.
It is fairly easy to show, that if a locally compact group $G$ admits a [[**plig**]{} metric]{}, then $G$ is second countable. Moreover, any two [[**plig**]{} metrics]{}$d_1$ and $d_2$ on $G$ are coarsely equivalent, ie. the identity map on $G$ defines a coarse equivalence between $(G, d_1)$ and $(G, d_2)$ in the sense of [@roe].
In [@lubotzky_etal pp. 14-16] Lubotzky, Moser and Ragunathan shows, that every compactly generated second countable group has a [[**plig**]{} metric]{}. Moreover, Tu shoved in [@tu lemma 2.1], that every contable discrete group has a [[**plig**]{} metric]{}.
The main result of this paper is that every [ locally compact, second countable]{}group $G$ admits a [[**plig**]{} metric]{}. Moreover a [[**plig**]{} metric]{}$d$ can be chosen, such that the $d$-balls have exponentially controlled growth in the sense that $$\mu( B_d(e,n)) \leq \beta\cdot e^{\alpha n}, \quad n \in {{\mathbb N}},$$ for suitable constants $\alpha$ and $\beta$. Here $\mu$ denotes the Haar measure on $G$.
In [@brown_guentner], Brown and Guentner proved that for every contable discrete group $\Gamma$ there exists a sequence of numbers $p_n\in (1,
\infty)$ converging to $\infty$ for $n\rightarrow\infty$, such that $\Gamma$ has a proper affine action on the reflexive and strictly convex Banach space $$X_0 = \bigoplus^\infty_{n=1} l^{p_n}(\Gamma),$$ where the direct sum is in the $l^2$-sense.
By similar methods, we prove that every second countable group has a proper affine action on the reflexive and strictly convex Banach space $$X = \bigoplus^\infty_{n=1}\quad L^{2n}(G, d\mu),$$ where the sum is taken in the $l^2$-sense. However, in order to prove, that the exponents $(p_n)^\infty_{n=1}$ can be chosen to be $p_n=2n$, it is essential to work with a [[**plig**]{} metric]{}on $G$ for which the $d$-balls have exponentially controlled growth. As a corollary we obtain, that a second countable locally compact group $G$ has a uniform embedding in the above Banach space $X$
Note, that the Banach spaces $X$ and $X_0$ above are not uniformly convex. Kasparov and Yu have recently proved, that the Novikov conjecture holds for every discrete countable group, which has a uniform embedding in a uniformly convex Banach space (cf. [@kasparov_yu]).
Acknowledgments {#acknowledgments .unnumbered}
---------------
We would like to thank Claire Anantharaman-Delaroche, George Skandalis, Alain Valette and Guoliang Yu for stimulating mathematical conversations.
Coarse geometry and [[**plig**]{} metrics]{}
============================================
First, let us fix notation and basic definitions:
Let $G$ be a topological group, ie. a group equipped with a Hausdorff topology, such that the map $ (x,y) \rightarrow x\cdot y^{-1}$ is continuous. If $G$ is equipped with a metric $d$, we put $ B_d(x, R) = \{ y\in G: d(x,y)<R\}$ and $D_d(x, R) = \{ y\in G: d(x,y)\leq R\}$.
- $G$ is [*locally compact*]{} if every $x\in G$ has a relative compact neighbourhood.
- We say that the metric $d$ [*induces the topology*]{} of $G$ if the topology generated by the metric $\tau_d$ coincides with the original topology $\tau$.
- The metric $d$ is said to be [*left invariant*]{} if for all $g, x, y \in G$ we have that $$d(x, y) = d(g\cdot x, g\cdot y).$$
- Following [@roe p. 10], a metric space is called [ *proper*]{} if all closed bounded sets are compact. When $G$ is a group, this reduces by the left invariance of the group metric to the requirement, that for every $M>0$ all the closed balls $$D(e, M) = \{ h\in G: d(e, h)\leq M \}$$ are compact.
We will work with a special class of metrics on [ locally compact, second countable]{}groups, which we define here:
Let $G$ be a topological group. A [[**plig**]{} metric]{}$d$ on $G$ is a metric on $G$, which is [ **P**]{}roper, [**L**]{}eft [**I**]{}nvariant and [**G**]{}enerates the topology.
M. Gromov started investigating asymptotic invariants of groups, particularly fundamental groups of manifolds.
This lead to the development of coarse geometry - or large scale geometry . Coarse geometry studies global properties of metric spaces, neglecting small (bounded) variations of these spaces. The properties and invariants in coarse geometry are treated in the limit at infinity, as opposed to the traditional world of topology, which focuses on the local structure of the space.
Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces.
- A map $f:X\rightarrow Y$ is called [ *uniformly expansive*]{} if $$\forall_{ R>0} \exists_{S>0} \quad\mbox{ such that }\quad
d_X(x,x')\leq R\Rightarrow d_Y(f(x), f(x'))\leq S.$$
- A map $f:X\rightarrow Y$ is called [ *metrically proper*]{} if $$\forall_{B\subset Y} \qquad B\mbox{ is bounded }\Rightarrow f^{-1}(B)\subset
X \mbox{ is bounded}.$$
- A map $f:X\rightarrow Y$ will by called a [*coarse map*]{} if it is both metrically proper and uniformly expansive.
- Two coarse maps $h_0, h_1:X\rightarrow Y$ are said to be [*coarsely equivalent*]{} when $$\exists_{C>0}\forall_{x\in X}:\quad d_Y(h_0(x), h_1(x))<C.$$ We denote the relation of coarse equivalence by ’$\sim_c$’.
- The spaces $X$ and $Y$ are said to be [*coarsely equivalent*]{} if there exist coarse maps $f:X\rightarrow Y$, and $ g:Y\rightarrow X$ such that $$f\circ g \sim_c
{\operatorname{Id}}_Y\mbox{ and } g\circ f\sim_c{\operatorname{Id}}_X.$$
- The [*coarse structure*]{} of $X$ means the coarse equivalence class of the given metric, $[d_X]_{\sim}$.
It is well known that $$(0,1) \sim_h {{\mathbb R}}\quad\mbox{ and }\quad
{{\mathbb Z}}\not\sim_h{{\mathbb R}},$$ where $\sim_h$ means that the two spaces in question are homeomorphic.
It is easy to see, that in the coarse case we have: $$(0,1) \not\sim_c {{\mathbb R}}\quad\mbox{ and }\quad
{{\mathbb Z}}\sim_c{{\mathbb R}}.$$
The reason for working with coarse geometry is, that many geometric group theory properties are coarse invariants. Examples of coarse invariants are: property A [@higson_and_roe], asymptotic dimension [@bell_drani_asdim] and change of generators for a finitely generated group.
M. Gromov has suggested in [@gromov_novikov] to solve the Novikov conjecture by considering classes of groups admitting uniform embeddings into Banach spaces with various restraints.
G. Yu proved in [@yu:uniform_embedding], that the Novikov Conjecture is true for a space that admits a uniform embedding into a Hilbert space. This result was strenghtened in [@kasparov_yu], where it is shown that the Novikov Conjecture is true for a space that admits a uniform embedding into a uniformly convex Banach space.
Together with the fact, that exact groups admit a uniform embedding into a Hilbert space, see [@kaminker_and_guentner], makes uniform embedding extremely interesting to study.
The idea of a uniform embedding is to map a metric space $(X, d_X)$ into a metric space $(Y, d_Y)$ in such a way that the large-scale geometry of $X$ is preserved.
This means for instance that we are not allowed to “squease” unbounded segments into a point, and we are not allowed to “blow up” bounded segments to unbounded - the limits at infinity must be preserved.
A map $f:X\rightarrow Y$ will by called a [*uniform embedding*]{} if there exist non-decreasing functions $\rho_-,
\rho_+:[0,\infty)\rightarrow [0, \infty)$ such that $$\lim_{t\rightarrow\infty} \rho_-(t) = \lim_{t\rightarrow\infty}\rho_+(t) = \infty$$ and $$\label{eq:df_ue_1}
\rho_-(d_X(x,y)) \leq d_Y(f(x), f(y)) \leq \rho_+(d_X(x,y))$$
When $f:X\rightarrow Y$ is a coarse map, we will denote a map $\phi:f(X)\rightarrow X$ a section of $f$ if it fullfills that $$f\circ \phi = id_{f(x)}.$$ The set of sections of the map $f$ will be denoted by ${{\operatorname{Inv}}}(f)$.
It is easy to see that we can not have a uniform embedding of the free group $F_2$ in any ${{\mathbb R}}^n$. On the other hand it was shown in [@haagerup], [@harpe_valette p. 63] that $F_2$ has a uniform embedding in the infinite dimensional Hilbert space ${{\mathbb H}}$.
The following is a folklore lemma, as different definitions of uniform embedding are used in the litterature, see [@gromov], [@higson_and_roe] and [@kaminker_and_guentner], see [@przybysz_thesis] for a detailed proof.
\[l:coarse\_eq\] Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces, and $f:X\rightarrow Y$ a map. The following are equivalent:
1. $f:X\rightarrow Y$ is a uniform embedding.
2. (Guentner and Kaminker version) $f$ is uniformly expansive, and $$\label{eq:kaminker}
\forall_{S>0}\exists_{R>0}\quad d_X(x,y)\geq R\Rightarrow d_Y(f(x),
f(y))\geq S.$$
3. (Higson and Roe version) The map $f$ is uniformly expansive, and so is any $\phi\in{{\operatorname{Inv}}}(f)$.
4. $f$ is a coarse equivalence between $X$ and $f(X)$ and any section $\phi\in{{\operatorname{Inv}}}(f)$ is a coarse map.
\[thm:plig\_coarse\_eq\] Let $G$ be a locally compact, second countable group. Assume that $d_1$ and $d_2$ are [[**plig**]{} metrics]{}on $G$.
Then the identity map $$Id:(G, d_1)\rightarrow (G,d_2)$$ is a coarse equivalence.
To establish the coarse equivalence of the metric spaces, it is by the 3rd case of lemma [(\[l:coarse\_eq\])]{}, enough to show that $ {\operatorname{Id}}:(G,
d_1)\rightarrow (G, d_2)$ and $ {\operatorname{Id}}:(G, d_2)\rightarrow (G, d_1)$ are both uniformly expansive, that is: $$\begin{aligned}
\forall_{ R>0} \exists_{S>0} \qquad
d_1(x,y)\leq R\Rightarrow d_2(x, y)\leq S.\\
\forall_{ R>0} \exists_{S>0} \qquad
d_2(x,y)\leq R\Rightarrow d_1(x, y)\leq S.\\
\end{aligned}$$
Since both $d_1$ and $d_2$ generates the topology of $G$, the identity map $$Id:(G, d_1)\rightarrow (G,d_2)$$ is a homeomorphism, and the maps $\phi_i: (G, d_i) \rightarrow {{\mathbb R}}_+$ given by $$\phi_i(x) = d_i(e, x), \quad x\in G$$ are continuous.
Let $R>0$. Since the closed $d_1$-ball $$D_1(x, R) = \{ x\in G: d_1(x,e)\leq R \}$$ is compact, $\phi_2$ attains a maximum values $S$ on $D_1(x, R)$. Moreover $$d_1( e, x)\leq R \Rightarrow d_2(e,x)\leq S.$$
Hence, by the left invariance of $d_1$ and $d_2$ $$d_1(x,y) \leq R \Rightarrow d_2(x,y) \leq S, \quad \forall_{x,y\in G}.$$
Since $R$ was arbitary, this shows the uniform expansiveness of ${\operatorname{Id}}$.
By reversing the roles of $d_1$ and $d_2$ in the last argument, we also obtain that the inverse map is uniformly expansive.
In the special case of a countable discrete group, Theorem [(\[thm:plig\_coarse\_eq\])]{} was proved by J. Tu (cf. the uniqueness part of [@tu lemma 2.1]).
Bounded geometry on locally compact groups.
===========================================
The purpose of this section is to show, that a [[**plig**]{} metric]{}implies bounded geometry on a [ locally compact, second countable]{}group. Let us begin by defining bounded geometry, which is a concept from the world of coarse geometry.
Following [@roe p. 13], the metric space $(G, d)$ is said to have [*bounded geometry*]{} if it is coarsely equivalent to a discrete space $(Q, d_Q)$, where for every $M>0$ there exists constants $\Gamma_M$ such that $$\forall_{q\in Q}\quad |D(q, M)| = |\{ p\in Q: d(q, p)\leq M
\}| \leq \Gamma_M .$$
Note that $(G,d)$ is a finitely generated group, and $d$ a word length metric. Then $G$ has bounded geometry.
Let $(Y,d)$ be a metric space. We say that a discrete space $X\subset Y$ is a [*coarse lattice*]{}, if $$\exists_{\lambda\in{{\mathbb R}}} \forall_{y\in Y}\exists_{x_y\in X}\quad d(x_y,y) \leq \lambda.$$
\[lem:proper\_gives\_bdg\] Let $G$ be a locally compact group, and let $d$ be a [[**plig**]{} metric]{}on $G$. Then $(G, d)$ is second countable and has bounded geometry.
First we observe that the conditions on $(G,d)$ imply, that $G$ is second countable. We can write $G$ as a union of compact metric spaces, namely: $$G=\bigcup^\infty_{n=1}D_d(e,n),$$ where we have that each $D_d(e,n)$ is a compact metric space, since $d$ is proper.
Now, since every compact metric space is seperable, see [@engelking Theorem 4.1.17, page. 297], we can conclude that every $D_d(e,n)$ has a countable dense subset. Hence $G$ is seperable, and since for any metric space second countability is equivalent to seperability, see [@willard Theorem 16.11, page 112], it follows that $G$ is second countable.
We will now show that $G$ has bounded geometry by constructing a countable coarse lattice $X\subset G$ such that $X$ has bounded geometry.
Let $X= \{x_i\}_{i\in I}$ be a maximal family of elements from $G$, such that $d(x_i, x_j)\geq 1, i\neq j$. Since $G$ is seperable, the index set $I$ is at most countable.
By maximality of $X$, we have that $$G = \bigcup_{i\in I} B_d(x_i,1).$$ If we had that $|I|<\infty$, then $G = \cup_{i\in I}
\overline{B_d(x_i,1)} $ would be a finite union of compact sets, and therefore compact, and hence bounded. Therefore $G$ is coarsely equivalent to $\{\bullet\}$ if $I$ is finite, and therefore $G$ has bounded geometry.
Let us therefore assume, that $|I|=\infty$, and we can use ${{\mathbb N}}$ instead of the index set $I$. Let us construct the coarse equivalences between $X$ and $G$:
Start by setting $$\begin{aligned}
A_1 &=& B_d(x_1, 1)\\
A_2 &=& B_d(x_2, 1)\setminus A_1\\
&\ldots&\\
A_n &=& B_d(x_n, 1)\setminus \bigcup^{n-1}_{i=1} A_i
\end{aligned}$$
We have now that the family $\{A_n\}_{n=1}^\infty$ in $G$ fullfills the following: $$\begin{cases}
A_n \cap A_m = \emptyset & \mbox{ if }m\neq n\\
x_n\in A_n\subset B_d(x_n, 1)&\mbox{for all }n\in{{\mathbb N}}\\
\cup_{n=1}^\infty A_n = G\\
\end{cases}$$
Now equip the set $X = \{x_i\}_{i\in I}$ with the metric inherited from $(G,d)$. Define: $$\begin{aligned}
\phi:G\rightarrow X,&\mbox{ by }& \phi(x) = x_n\quad\mbox{when }x\in A_n\\
\psi:X\rightarrow G,&\mbox{ by }& \phi(x_n) = x_n\quad\mbox{for } n\in{{\mathbb N}}\end{aligned}$$
Remark that both $\psi$ and $\phi$ are coarse maps. We have from the construction of $\phi$ and $\psi$ that
$$\phi\circ\psi = {\operatorname{Id}}_X$$ and $$\forall_{x\in G}\qquad d( \psi\circ\phi(x), x) \leq 1$$ and therefore we see, that the spaces $X$ and $G$ are coarsely equivalent.
Now, we have to show that the set $X$ indeed has bounded geometry. Let $M>0$ be given, and let us look at the disks of radius $M$ in $X$: $$D_X(q, M) = \{ x_n\in X: d(q, x_n) \leq M, \quad n\in {{\mathbb N}}\}.$$ For every $M>0$ we need to find a constant $\Gamma_M$ such that $$\sup_{q\in X}|D_X(q, M) | \leq \Gamma_M.$$
Since $d(x_n, x_m)\geq 1$ when $n\neq m$, the balls $B(x_n, \frac{1}{2})$ are disjoint. Moreover, we have that $$\bigcup_{x_n\in D_X(q, M) } B(x_n, \frac{1}{2}) \subset B(q, M+\frac{1}{2}).$$
Let $\mu$ denote the Haar measure on $G$, then we have that $$\sum_{x_n\in D_X(q, M)} \mu( B(x_n, \frac{1}{2})) \leq \mu( B(q, M+\frac{1}{2}))$$
Since the number of terms in the sum above is equal to $|D(q,M)|$, we get by the left invariance of the Haar measure, that $$|D(q, M)|\cdot\mu(B(e, \frac{1}{2})) \leq \mu(B(e,M+\frac{1}{2})).$$ Hence $$\sup_{q\in X} |D(q, M)|\leq \frac{ \mu(B(e,M+\frac{1}{2}))}{
\mu(B(e,\frac{1}{2}))} < \infty.$$ Therefore we see that $(X, d)$ has bounded geometry, and we can conclude that $(G,d)$ also has bounded geometry.
\[ex:prop\_not\_bdg\] Remark, that lemma [(\[lem:proper\_gives\_bdg\])]{} is not true for general metric spaces, as we can find an example of a metric space $X$ that is proper, but does not have bounded geometry:
Consider the triple $(D_n, d_n, x_n)$, where $D_n$ is the discrete space with $n$ points, equipped with the discrete metric: $$d_n(x, y) = \begin{cases} 1 & x\neq y\\
0 & x= y\end{cases},$$ and where $x_n \in D_n$.
Let $X = \coprod_{n\in {{\mathbb N}}} D_n$, and equip $X$ with the following metric: $$d(z, y) =
d_{j(z)}(z, x_{j(z)})+ |j(z)-j(y)|+ d_{j(y)}(y, x_{j(y)}) \mbox{ where } j(x) = n \Leftrightarrow x\in D_n.$$ It is easy to see that $X$ is proper, but on the other hand $X$ does not have bounded geometry since the number of elements in $D_d(x_n, 1)$ tends to infinity for $n\rightarrow\infty$.
Construction of a [[**plig**]{} metric]{}on $G$.
=================================================
In this and the section, we will prove that every [ locally compact, second countable]{}group has a [[**plig**]{} metric]{}$d$. Together with theorem [(\[thm:plig\_coarse\_eq\])]{} we have therefore that a [ locally compact, second countable]{}group has exactly one coarse equivalence class of [[**plig**]{} metrics]{}.
We will call a map $l:G\rightarrow {{\mathbb R}}_+$ for a [ *length function*]{}, if it satisfies the following conditions [(\[eq:wbh1\])]{}–[(\[eq:wbh3\])]{}.
$$\begin{aligned}
\label{eq:wbh1}
\forall_{g\in G}\quad
l(g) = 0 &\Leftrightarrow& g = e.\\\label{eq:wbh2}
\forall_{g\in G}\quad l(g)& = &l(g^{-1}).\\
\label{eq:wbh3}
\forall_{g, h\in G}\quad l(g\cdot h) &\leq& l(g)+l(h).
\end{aligned}$$
\[lem:length\_cond\]
1. If $l:G\rightarrow {{\mathbb R}}_+$ is a length function, then $$\label{eq:u11}
d(x,y) = l(y^{-1}x), \quad x,y\in G$$ is a left invariant metric on $G$.
2. Conversily, if $d: G\times G\rightarrow {{\mathbb R}}_+$ is a left invariant metric on $G$, then $$l(x) = d(x,e), \quad x\in G$$ is a length function on $G$, and $d$ is the metric obtained from $l$ by [(\[eq:u11\])]{}.
Moreover, if $l$ is a length function on $G$ and $d(x,y) = l(y^{-1}x)$ is the associated left invariant metric, then $d$ generates the topology on $G$ if and only if $$\label{eq:wbh5}
\{l^{-1}[0, r):\quad r>0\}\mbox{ is a basis for the neighborhoods of }\ e\in G_0.$$ Moreover, $d$ is proper if and only if: $$\label{eq:wbh4}
\forall_{r>0}\quad l^{-1}([0,r])\quad \mbox{ is compact.}$$
The proof is elementary, and will be left to the reader.
\[rem:4.3.5\] If $l$ is a length function on $G$ satisfying [(\[eq:wbh5\])]{}, then the associated metric $$d(x,y) = l( y^{-1}x), \quad x,y \in G$$ generates the given topology on $G$. Therefore $$l^{-1}([0,r]) = \{ x\in G:\quad d(x,e)\leq r \}$$ is closed in $G$. Hence if we assume [(\[eq:wbh5\])]{}, then [(\[eq:wbh4\])]{} is equivalent to that $l^{-1}([0,r])$ is relatively compact for all $r>0$, which again is equialent to that $$B_d(e,n) = l^{-1}([0,n))$$ is relatively compact for all $n\in {{\mathbb N}}$.
\[lem:ball\_growth\_limit\] Let $G$ be a [ locally compact, second countable]{}group. Assume that the topology on $G$ is generated by a left invariant metric $\delta$, for which $$\begin{aligned}
U = B_\delta(e,1)&\mbox{is relatively compact}\label{eq:cptl_gen}\\
G = \bigcup^\infty_{k=1} U^k\label{eq:ass_cpct_gen}
\end{aligned}$$
Then $G$ admits a left invariant metric $d$ generating the topology on $G$, for which $B_d(e,1) =U$, and $$\label{eq:ball_growth_limit}
\forall_{n\in{{\mathbb N}}}\quad B(e,n) \subset B(e,1)^{2n-1}.$$
Moreover $d$ is a [[**plig**]{} metric]{}on $G$.
Let $l_\delta(g) = \delta(g,e)$ be the length function associated to $\delta$, and define a function $l:G\rightarrow{{\mathbb R}}_+$ by $$l(g) = \inf\bigg\{
\sum_{i=1}^k l_\delta(g_i):\quad g = g_1, \ldots, g_k, \mbox{
where }g_i\in U, \quad i =1, \ldots k, k\in{{\mathbb N}}\bigg\}$$
Clearly, $l(g)\geq 0$ for all $g\in G$, and by the assumption [(\[eq:ass\_cpct\_gen\])]{}, we have that $l(g)<\infty$. Moreover, one checks easily that $$l(g\cdot h) \leq l(g) + l(h), \quad g,h\in G.$$
Since $U = U^{-1}$, we have also that $$l(g^{-1}) = l(g), \quad g\in G.$$
Moreover $$\label{eq:uffe19}
l(g) \geq l_\delta(g),\quad g\in G$$ and $$\label{eq:uffe20}
l(g) = l_\delta(g),\quad g\in U$$
By [(\[eq:uffe19\])]{} and [(\[eq:uffe20\])]{}, we have that $$l(g) = 0 \Leftrightarrow g= e.$$
Hence by lemma [(\[lem:length\_cond\])]{}, $l$ is a length function on $G$. Let $$\label{eq:uffe21}
d(g,h) = l(g^{-1}\cdot h),\quad g,h\in G$$ be the associated left invariant metric on $G$. By [(\[eq:uffe20\])]{} and [(\[eq:uffe21\])]{} we have that $$\label{eq:uffe22}
B_d(x,r) = B_\delta(x,r),\quad r\leq 1$$ for all $g\in G$. Hence $d$ generates the same topology on $G$ as $\delta$ does. Moreover, we have that $$B_d(e,1) = B_\delta(e,1) =U.$$
We next turn to the proof of [(\[eq:ball\_growth\_limit\])]{}. Let $n\in{{\mathbb N}}$, and let $g\in B_d(e,n)$. Then there exists a $k\in{{\mathbb N}}$, and $g_1, \ldots,
g_k\in U$ such that $$g = g_1\ldots g_k\quad\mbox{ and }\quad\sum_{i=1}^kl_\delta(g_i) < n.$$ Further we may assume, that $k\in{{\mathbb N}}$ is minimal among all the numbers for which such a representation is possible.
We claim, that in this case $$\label{eq:uffe23}
l_\delta(g_i) + l_\delta(g_{i+1}) \geq 1,\quad i=1,\ldots, k-1.$$
Assume namely, that $ l_\delta(g_i) + l_\delta(g_{i+1}) < 1$ for some $i$, where $1\leq i\leq k-1$, then we have $$l_\delta (g_i\cdot g_{i+1}) \leq l_\delta(g_i) + l_\delta(g_{i+1}) < 1,$$ and thus we have that $g_i\cdot g_{i+1}\in U$. Hence $g$ can be written as a product of $k-1$ elements from $U$:
$$g = g_1\ldots g_i\cdot g_{i-1}(g_i\cdot g_{i+1})g_{i+2}\ldots g_k$$
for which $$\sum_{j=1}^{i-1} l_\delta(g_j) + l_\delta( (g_i\cdot g_{i+1}) +
\sum_{j=i+2}^{k} l_\delta(g_j) \leq
\sum_{j=1}^k l_\delta(g_j) < n.$$
This contradicts the minimality of $k$, and hence [(\[eq:uffe23\])]{} must hold.
Let $\lfloor r\rfloor$ as usual denote the largest integer such that $$\lfloor r\rfloor \leq r
$$ >From [(\[eq:uffe23\])]{} we get that
$$\lfloor \frac{k}{2}\rfloor \leq
\sum^{\lfloor \frac{k}{2}\rfloor}_{j=1}( l_\delta(g_{2j-1}) +
l_\delta(g_{2j}))
\leq
\sum_{j=1}^k l_\delta(g_j) < n$$
Since both $\lfloor \frac{k}{2}\rfloor$ and $n$ are integers, we have that $ \lfloor \frac{k}{2}\rfloor \leq n-1$, and therefore $$k \leq 2\cdot \lfloor \frac{k}{2}\rfloor+1 \leq 2n-1,$$ and hence we get that $$g\in U^k \subset U^{2n-1} = B_d(e,1)^{2n-1},$$ which prooves [(\[eq:ball\_growth\_limit\])]{}.
Since $ U^{2n-1} \subset \overline{U}^{2n-1}$, where the latter set is compact by assumption [(\[eq:cptl\_gen\])]{}, we have that $B_d(e,n)$ is relatively compact for all $n\in{{\mathbb N}}$. Hence, by lemma [(\[lem:length\_cond\])]{} and remark [(\[rem:4.3.5\])]{} $d$ is a [[**plig**]{} metric]{}on $G$.
We are now ready toprove, that there exists a [[**plig**]{} metric]{}on every [ locally compact, second countable]{}group.
\[thm:plig\_on\_lcsc\] Every [ locally compact, second countable]{}group $G$ has a [[**plig**]{} metric]{}$d$.
Let $G$ be [ locally compact, second countable]{}group. By Remark [@willard Theorem 1.22,page 34], we can choose a left invariant metric $\delta_0$ on $G$, which generates the topology on $G$. Moreover, since $G$ is locally compact,there exists an $r>0$ such that the open ball $B_{\delta_0}(e,r)$ is relatively compact. Put now $\delta = \frac{1}{r}\delta_0$. Then $\delta$ is again a left invariant metric on $G$, which generates the topology. Moreower, $$U = B_\delta(e,1)$$ is relatively compact.
Put $$G_0 = \bigcup^\infty_{k=1} U^k.$$
Then $G_0$ is an open and closed subgroup of $G$. Since $G$ is second countable, it follows that the space $Y = G/G_0$ of left $G_0$-cosets in $G$ is a countable set. In the following we will assume that $|Y| =
\infty$. The proof in the case $|Y| < \infty$ can be obtained by the same method with elementary modifications.
We can choose a sequence $\{ x_n\}^\infty_{n=1} \subset G$, such that $x_0=e$ and such that is a disjoint union of cosets: $$G = \bigcup^\infty_{n=0} x_n\cdot G_0.$$
By lemma [(\[lem:ball\_growth\_limit\])]{}, there is a [[**plig**]{} metric]{}$d_0$ on $G_0$, such that $$\begin{aligned}
\label{eq:uffe25}
B_{d_0}(e,1) &=& U\\
B_{d_0}(e,n)& \subset & U^{2n-1}.\label{eq:uffe26} \end{aligned}$$
In particular, $d_0$ is proper. Let $$\label{eq:uffe27}
l_0(h) = d_0(h,e),\quad h\in G_0$$ be the length function associated with $d_0$.
Put $$S = \{ x_1, x_2, x_3 \ldots \} \subset G$$ and define $l_1:S\rightarrow {{\mathbb N}}$ by $$\label{eq:uffe27,5}
l_1(x_n) = n.$$
Define furthermore a function $\tilde{l}: G\rightarrow [0, \infty [$ by setting $$\label{eq:uffe28}
\tilde{l}(g) = \inf\bigg\{
l_0(h_0) + \sum^{k}_{i=1} (l_1(s_i) + l_0(h_i))
\bigg\},$$ where the infimum is taken over all the representations of $g$ of the form $$\label{eq:uffe29}
\begin{cases}
g = h_0\cdot s_1 \cdot h_1 \cdot s_2\cdot h_2\cdots h_k\\
k\in {{\mathbb N}}\cup\{ 0 \}, h_0, \ldots, h_k\in G_0\\
s_1, \ldots, s_k\in S
\end{cases}$$
Note that $$G = G_0 \cup \bigcup^\infty_{n=1} x_n\cdot G_0 = G_0\cup S\cdot G_0
\subset G_0 \cup G_0\cdot S\cdot G_0,$$ so that every $g\in G$ has a representation of the form [(\[eq:uffe29\])]{} with $k=0$ or $k=1$.
Next, put $$\label{eq:uffe30}
l(g) = \max \bigg\{
\tilde{l}(g), \tilde{l}(g^{-1})
\bigg\}, \quad g\in G$$
We will show, that $l(g)$ is a length function on $G$, and that the associated metric $$\label{eq:uffe31}
d(g,h) = l(g^{-1}\cdot h),\quad g,h\in G$$ is a [[**plig**]{} metric]{}on $G$.
It is easily checked that $$\tilde{l}(g\cdot h) \leq \tilde{l}(g)+ \tilde{l}(h),\quad g,h\in G$$
and hence also $$\label{eq:uffe32}
l(g\cdot h) \leq l(g) + l(h), \quad g,h\in G.$$
Moreover, by [(\[eq:uffe29\])]{} $$\label{eq:uffe33}
l(g^{-1}) = l(g), \quad g\in G.$$
If $g\in G$ and $l(g) < 1$, then $\tilde{l}(g) \leq l(h) < 1$. Thus for every $\epsilon>0$, $g$ has a representation of the form [(\[eq:uffe29\])]{}, such that $$l_0(h_0) + \sum^k_{i=1} (l_1(s_i) + l_0(h_i)) < l(g) + \epsilon$$ and for sufficiently small $\epsilon$, we have that $$l(g)+\epsilon < 1,$$ which implies that $k=0$, because $$\forall_{s\in S}\quad l_1(s) \geq 1.$$
Hence $g=h_0 \in G_0$, and $$l_0(g) = l_0(h_0) < l(g) +\epsilon.$$
Since $\epsilon$ can be chosen arbitrarily small, we have shown that $$\label{eq:uffe34}
g\in G\mbox{ and }l(g)<1 \Rightarrow g\in G_0 \mbox{ and } l_0(g) \leq l(g).$$
In particular, $l(g) =0 $ implies that $g=e$, which together with [(\[eq:uffe32\])]{} and [(\[eq:uffe33\])]{} shows, that $l$ is a length function on $G$. Hence by lemma [(\[lem:length\_cond\])]{} $$d(g,h) = l(g^{-1}\cdot h), \quad g,h\in G$$ is a left invariant metric on $G$.
>From [(\[eq:uffe34\])]{} we have $$\label{eq:uffe35}
g\in B_d(e,1) \Rightarrow g\in G_0\quad\mbox{ and }\quad l_0(g)\leq l(g).$$
Conversely, if $g\in G_0$, then using [(\[eq:uffe28\])]{} with $k=0$ and $h_0=g$, we get that $$\tilde{l}(g) \leq l_0(g)$$ and therefore $$\label{eq:uffe36}
l(g) \leq \max \bigg\{
l_0(g), l_0(g^{-1})
\bigg\} = l_0(g), \quad g\in G_0.$$
By [(\[eq:uffe35\])]{} and [(\[eq:uffe36\])]{}, we have $$\label{eq:uffe36,5}
B_d(e,r) = B_{d_0}(e,r), \quad 0<r\leq 1,$$ and since $G_0$ is open in $G$, the sets $$B_{d_0}(e,r), \quad 0<r\leq 1$$ form a basis of neighbourhoods for $e$ in $G$. Hence $d$ generates the original topology on $G$.
It remains to be proved, that $d$ is proper, ie. that $B_d(e,r)$ is relatively compact for all $r>0$. Note that it is sufficient to consider the case, where $r= n \in {{\mathbb N}}$.
Let $n\in {{\mathbb N}}$, and let $g\in B_d(e,n)$. Then we have that $$\tilde{l}(g) \leq l(g) < n.$$
Hence by [(\[eq:uffe29\])]{}, we see that $$\label{eq:uffe37}
g = h_0\cdot s_1 \cdot h_1 \cdot s_2\cdot h_2\cdots s_k\cdot h_k,$$ where $$\begin{cases}
k\in {{\mathbb N}}\cup\{ 0 \}\\
h_0, \ldots, h_k\in G_0\\
s_1, \ldots, s_k\in S\\
l_0(h_0) + \sum^k_{i=1} (l_1(s_i) + l_0(h_i)) < n \label{eq:uffe38}
\end{cases}$$
Since $$l_1(s) \geq 1\quad \forall_{s\in S},$$ we have that $k\leq n-1$. Moreover, since $l_1:S\rightarrow {{\mathbb N}}$ is defined by $$l_1(x_m) = m,\quad m=1, 2, \ldots$$ we have $$\label{eq:uffe39}
s_i \in \{ x_1, x_2, \ldots, x_{n-1} \}, \quad 1\leq i \leq k.$$
Moreover $$\label{eq:uffe40}
h_i \in B_{d_0}(e,n), \quad 0 \leq i \leq k$$ because $l_0(h_i) < n $ by [(\[eq:uffe38\])]{}. Put $$T(n) = \{ x_1, \ldots, x_{n-1} \} \cup \{ e \}$$
Then by [(\[eq:uffe37\])]{}, [(\[eq:uffe38\])]{}, [(\[eq:uffe39\])]{} and [(\[eq:uffe40\])]{} we have $$g \in B_{d_0}(e,n)\bigg( T(n)\cdot B_{d_0}(e,n)\bigg)^k
\subset \bigg( T(n)\cdot B_{d_0}(e,n)\bigg)^{k+1}
\subset \bigg( T(n)\cdot B_{d_0}(e,n)\bigg)^n$$
where the last inclusion follows from the inequality $k\leq n-1$. Since $g\in B_d(e,n)$ was chosen arbitrarily, we have shown that $$B_d(e,n) \subset \bigg( T(n)\cdot B_{d_0}(e,n)\bigg)^n.$$
But $d_0$ is a proper metric on $G_0$, and since $T(n)$ is a finite set, it follows, that $$\bigg( T(n)\cdot \overline{B_{d_0}(e,n)}\bigg)^n$$ is compact.
Hence $B_d(e,n)$ is relatively compact for all $n\in {{\mathbb N}}$, and therefore $d$ is a proper metric on $G$, cf. remark [(\[rem:4.3.5\])]{}.
As mentioned in the introduction, Theorem [(\[thm:plig\_on\_lcsc\])]{} has previously been obtained in two important special cases, the compactly generated case [@lubotzky_etal] and the countable, discrete case [@tu].
In this example, we give an explicit formula for a [[**plig**]{} metric]{}on . The same formula will also define a [[**plig**]{} metric]{}on every closed subgroup of . Recall that $${\mbox{$GL(n, {\mathbb R})$}}= \{ A \in M_n({{\mathbb R}}): \det(A)\neq 0 \}$$ and the topology on is inherited from the topology of $M_n({{\mathbb R}})\simeq
{{\mathbb R}}^{n^2}$. We equip $M_n({{\mathbb R}})$ with the operator norm $$||A|| = \sup \{ ||Ax|| : x\in {{\mathbb R}}^n, ||x|| \leq 1 \},$$ where $||x|| = \sqrt{x_1^2 + \cdots + x^2_n }$ is the Euclidian norm on ${{\mathbb R}}^n$.
Define a function on $GL(n, {{\mathbb R}})$ by $$\label{eq:lgth_on_gl}
l(A) =\max \{ \ln(1+ ||A-I||), \ln(1+||A^{-1}-I|| \}.$$
We claim that $l$ is a length function on , and that the associated metric $$d(A, B) = l(B^{-1}A), \quad A, B\in {\mbox{$GL(n, {\mathbb R})$}}$$ is a [[**plig**]{} metric]{}on . We prove first, that $l$ is a length function. Clearly, $l(A) = l(A^{-1})$ and $l(A)= 0
\Leftrightarrow A = I$.
Let $A, B \in {\mbox{$GL(n, {\mathbb R})$}}$. Then $$|| A-I|| \leq e^{l(A)} -1, ||B-I||\leq e^{l(B)}-1.$$
Put $ X = A- I$ and $Y = B-I$. Then $$\begin{gathered}
|| AB -I || = || XY + X + Y || \leq ||X||\cdot||Y||+||X||+||Y||\\ =
(||X||+1)(||Y||+1)-1
\leq e^{l(A)} e^{l(B)} -1\end{gathered}$$ and hence $$\label{eq:u60}
\ln(1+ (|| AB -I ||) \leq l(A)+ l(B).$$
Substituting $(A, B)$ with $(B^{-1}, A^{-1})$ in this inequality, we get $$\label{eq:u61}
\ln(1+ (|| (AB)^{-1} -I ||) \leq l(B^{-1}) + l(A^{-1}) = l(A) + l(B),$$ and by [(\[eq:u60\])]{} and [(\[eq:u61\])]{} it follows that $l(A+B) \leq
l(A)+l(B)$. Hence $l$ is a length function on .
To prove that $d$ is a [[**plig**]{} metric]{}on , it suffices to check, that the conditions [(\[eq:wbh4\])]{} and [(\[eq:wbh5\])]{} in lemma [(\[lem:length\_cond\])]{} are fullfilled.
Since $A \rightarrow A^{-1}$ is a homeomorphism of onto itself, [(\[eq:wbh5\])]{} is clearly fullfilled. To prove [(\[eq:wbh4\])]{} we let $r\in (0,
\infty)$ and put $M = e^r$. Since $||C|| \leq 1+ || C-I||$ for $C\in{\mbox{$GL(n, {\mathbb R})$}}$, we see that $l^{-1}([0,r])$ is a closed subset of $$K = \{ A\in{\mbox{$GL(n, {\mathbb R})$}}: ||A||\leq M, ||A^{-1}||\leq M \}.$$
Denote $$L = \{ (A,B)\in M_n({{\mathbb R}})\times M_n({{\mathbb R}}): AB = BA = I, ||A||\leq M, ||B||\leq M \},$$ then $L$ is a compact subset of $M_n({{\mathbb R}})^2$, and $K$ is the range of $L$ by the continuous map $\pi: (A, B) \rightarrow A$ of $M_n({{\mathbb R}})^2$ onto $M_n({{\mathbb R}})$. Hence $K$ is compact, and therefore $l^{-1}([0,r])$ is also compact. This proves [(\[eq:wbh4\])]{}, and therefore $d$ is a [[**plig**]{} metric]{}on .
Let $G$ be a connected Lie group. Then we can choose a left invariant Riemannian structure on $G$. Let $(g_p)_{p\in G}$ denote the corresponding inner product on the spaces $(T_p)_{p\in G}$. The path length metric on $G$ corresponding to the Riemannian structure is $$d(g,h) = \inf_\gamma L(\gamma)$$ where $$L(\gamma) = \int^b_a \sqrt{g_{\gamma(t)}( \gamma'(t), \gamma'(t))}dt$$ is the path length of a piecewise smooth path $\gamma$ in $G$, and where the infimum is taken over all such paths, that starts in $\gamma(a)=g$ and ends in $\gamma(b)= h$. Then $d$ is a left invariant metric on $G$ which induces the given topology on $G$, cf. [@helgason:diff_geom p.51-52].
We claim that $d$ is a proper metric on $G$. To prove this, it is sufficient to prove, that $B_d(e,r)$ is relatively compact for all $r>0$. Let $r>0$, and let $g\in B(e,r)$. Then $e$ and $g$ can be connected with a piecewise smooth path $\gamma$ of length $L(\gamma) <r$.
Now $\gamma$ can be divided in two paths each of length $\frac{1}{2}L(\gamma)$. Let $h$ denote the endpoint of the first path. Then $$d(e,h) \leq \frac{L(\gamma)}{2}\quad\mbox{and}\quad d(h,g)\leq \frac{L(\gamma)}{2}.$$
Hence $g = h(h^{-1}g)$, where $d(e,r) < \frac{r}{2}$ and $d(h^{-1}g, e) =
d(g,h) < \frac{r}{2}$. This shows, that $$B(e,r)\subset B(e,\frac{r}{2})^2$$ and hence $$B(e,r) \subset B(e,r\cdot 2^{-k})^{2^k}$$ for all $k\in{{\mathbb N}}$. Since $G$ is locally compact, we can choose a $k\in{{\mathbb N}}$ such that $B(e, r2^{-k})$ is relatively compact. Hence $B(e,r)$ is contained in the compact set $$\overline{ B(e, r2^{-k})}^{2^k}.$$
This shows that $d$ is proper, and therefore $d$ is a [[**plig**]{} metric]{}on $G$.
Exponentially controlled growth of the $d$-balls
================================================
Let $(G,d)$ be a [ locally compact, second countable]{}group with a [[**plig**]{} metric]{}, and let $\mu$ denote the Haar measure on $G$. Then we say that [ *[the $d$-balls have exponentially controlled growth]{}*]{} if there exists constants $\alpha, \beta>0$, such that $$\label{eq:sphere_growth}
\mu (B_d(e,n))\leq \beta e^{\alpha n}, \quad\forall_{n\in {{\mathbb N}}}.$$
Note, that if [(\[eq:sphere\_growth\])]{} holds, then $$\label{eq:expgrow_general}
\mu( B_d(e,r)) \leq \beta'\cdot e^{c_2\cdot r},\quad r\in [1, \infty),$$ where $\beta' = \beta\cdot e^{\alpha}$.
We now turn to the problem of constructing a [[**plig**]{} metric]{}on $G$, for which [the $d$-balls have exponentially controlled growth]{}. We first prove the following simple combinatorial lemma:
\[lem:combinatorial\] Let $n\in {{\mathbb N}}$, and let $k\in \{ 1, \ldots, n \}$.
Put $$N_{n,k} = \bigg\{
(n_1, n_2, \ldots, n_k)\in {{\mathbb N}}^k:\quad \sum^k_{i=1} n_i \leq n
\bigg\}$$
Then the number of elements in $N_{n,k}$ is $$|N_{n,k}| = \binom{n}{k}.$$
The map $$(n_1, \ldots, n_k) \rightarrow \{ n_1, n_1+n_2, \ldots, n_1+n_2+\cdots
+ n_k \}$$ is a bijection from $N_{n,k}$ onto the set of subsets of $\{ 1, \ldots, n
\}$ with $k$ elements, and the latter set has of course $\binom{n}{k}$ elements.
Having established lemma [(\[lem:combinatorial\])]{}, we can now turn to giving a proof of the main theorem of this section:
\[thm:lcsc\_has\_expgrow\] Every [ locally compact, second countable]{}group $G$ has a [[**plig**]{} metric]{}$d$, for which [the $d$-balls have exponentially controlled growth]{}.
The result is obtained, by modifying the construction of a [[**plig**]{} metric]{}on $G$ from the proof of theorem [(\[thm:plig\_on\_lcsc\])]{}.
Let $U, G_0, d_0, d$ and $S= \{ x_1, x_2, \ldots \}$ be as in the proof of theorem [(\[thm:plig\_on\_lcsc\])]{}, and note that by [(\[eq:uffe36,5\])]{} we have $$B_d(e,1) = B_{d_0}(e,1) = U.$$
For each $i\in {{\mathbb N}}$ the set $\overline{U}\cdot x_i$ is compact in $G$, and can therefore be covered by finitely many left translates of $U$: $$\label{eq:uffe41}
U\cdot x_i \subset \overline{U}\cdot x_i \subset \bigcup^{p(i)}_{j=1}
y_{i,j}\cdot U.$$
Define $l^*_1:S\rightarrow [1,\infty[$ by $$\label{eq:uffe42}
l_1^*(x_i) = i+ \log_2(p(i)),$$ and note that $$l^*_1(x_i) \geq i = l_1(x_i), \quad x_i\in S,$$ where $l_1:S\rightarrow {{\mathbb N}}$ is the map defined in [(\[eq:uffe27,5\])]{}.
We will now repeat the construction of the left invariant metric $d$ in the proof of theorem [(\[thm:plig\_on\_lcsc\])]{}, with $l_1$ replaced by $l^*_1$, ie. we first define a function $\tilde{l}^*:G\rightarrow [0,\infty[$ by $$\label{eq:uffe43}
\tilde{l}^*(g) = \inf \bigg\{
l_0(h_0) + \sum^k_{i=1} (l^*_1(s_i) + l_0(h_i))
\bigg\}$$ where the infimum is taken over all representations of $g$ of the form $$\label{eq:uffe44}
\begin{cases}
g = h_0\cdot s_1\cdot h_1 \cdot s_2\cdot h_2 \cdots s_k\cdot h_k\\
k\in {{\mathbb N}}\cup \{ 0 \}\\
h_0, \ldots, h_k \in G_0\\
s_1, \ldots, s_k \in S
\end{cases}$$
Next, we put $$\label{eq:uffe44.1}
l^*(g) = \max \bigg\{
\tilde{l}^*(g), \tilde{l}^*(g^{-1})
\bigg\}, \quad g\in G$$ and $$\label{eq:uffe44.2}
d^*(g,h) = l^*(g^{-1}\cdot h),\quad g,h\in G.$$
Then, exactly as for the metric $d$ in the proof of theorem [(\[thm:plig\_on\_lcsc\])]{} we get that $d^*$ is a left invariant metric on $G$, which generates the given topology on $G$, and which satisfies the following $$B_{d^*}(e,1) = B_{d_0}(e,1) = U,$$ (cf. the proof of theorem [(\[thm:plig\_on\_lcsc\])]{}).
Moreover, since $l_1^* \geq l_1$, we have that $d^*\geq d$, so the properness of $d$ implies, that $d^*$ is also proper.
Let $n\in {{\mathbb N}}$. Since $$l^*(g) \leq \tilde{l}^*(g), \quad g\in G,$$ we get from [(\[eq:uffe43\])]{} and [(\[eq:uffe44\])]{} that $$B_{d^*}(e,n) \subset \bigg\{
g\in G:\quad \tilde{l}^*(g) < n
\bigg\}$$
Hence every $g\in B_{d^*}(e,n) $ can be written on the form [(\[eq:uffe44\])]{} with $$l_0(h_0) + \sum^k_{i=1} (l^*_1(s_i) + l_0(h_i)) < n.$$
Note, that since $$l^*_1(s) \geq l_1(s) \geq 1, \quad s\in S,$$ we have that $k\leq n-1$.
Choose next natural numbers $m_0, \ldots, m_k$, such that $$l_0(h_i) < m_i \leq l_0(h_i) +1, \quad i= 0, \ldots, k.$$
Then we have that $$h_i \in B_{d^*}(e,m_i), \quad i = 0, \ldots, k,$$ and $$m_0 + \sum^k_{i=1} (l_1^*(s_i) + m_i) < n + (k+1) \leq 2n+1.$$
Hence $$\label{eq:uffe45}
B_{d^*}(e,n) \subset
\bigcup_M\quad B_{d_0}(e, m_o )\cdot x_{n_1}\cdot B_{d_0}(e, m_1 )\cdot x_{n_2}\cdots
x_{n_k}\cdot B_{d_0}(e, m_k),$$ where $M$ is the set of tuples $$\label{eq:uffe46}
\bigg(
k, m_0, m_1, \ldots, m_k, n_1, \ldots, n_k
\bigg)$$ for which $$\begin{cases}
k\in \{ 0, \ldots, n-1 \}\\
n_1, \ldots, n_k, m_0, \ldots, m_k \in {{\mathbb N}}\\
\sum_{i=0}^k m_i + \sum_{i=1}^k l^*_1(x_{n_i}) < 2n +1
\end{cases}$$
By [(\[eq:uffe42\])]{}, the latter condition can be rewritten as $$\label{eq:uffe47}
\sum_{i=0}^k m_i + \sum_{i=1}^k ( n_i + log_2(p(n_i)) < 2n +1,$$ where $p(n_i)\in {{\mathbb N}}$ are given by formula [(\[eq:uffe41\])]{}. Since $m_i, n_i
\in {{\mathbb N}}$ and $p(n_i)\geq 1$, it follows that $$\sum_{i=0}^k m_i + \sum_{i=1}^k n_i \leq 2n.$$
Hence $M \subset \bigcup^{n-1}_{k=0} M_k$, where $M_k$ is the set of $2k+1$-tuples $(m_0, \ldots, m_k, n_1, \ldots, n_k)$ of natural numbers for which $$\sum_{i=0}^k m_i + \sum_{i=1}^k n_i \leq 2n.$$
Therefore, by lemma [(\[lem:combinatorial\])]{} we have that $|M_k| =
\binom{2n}{2k+1}$, and thus $$\label{eq:uffe48}
|M| \leq \sum^{n-1}_{k=0}\binom{2k}{2k+1} \leq
\sum^{2n}_{j=0}\binom{2n}{j} = 2^{2n}.$$
>From [(\[eq:uffe26\])]{}, we have $$B_{d_0}(e, m_i) \subset U^{2m_i-1} \subset U^{2m_i}.$$
Hence by [(\[eq:uffe45\])]{}, we have $$\label{eq:uffe49}
B_{d^*}(e,n) \subset
\bigcup_M\quad U^{2m_o}\cdot x_{n_1}\cdot U^{2m_1} \cdot x_{n_2}\cdots
x_{n_k}\cdot U^{2m_k},$$ where $|M| \leq 2^{2n}$ and where [(\[eq:uffe47\])]{} holds for each $(k, m_0,
\ldots, m_k, n_1, \ldots, n_k) \in M$. Since $\overline{U}^2$ is compact, it can be covered by finitely many left translates of $U$, ie. $$U^2 \subset \overline{U}^2 \subset \bigcup^q_{i=1} z_i\cdot U,\quad z_1,
\ldots z_q \in G.$$
It now follows that for every $k\in {{\mathbb N}}$, the set $U^k$ can be covered by $q^{k-1}$ translates of $U$, namely $$\label{eq:uffe50}
U^k \subset \bigcup^q_{i_1 = \cdots = i_{k-1} = 1}\quad z_{i_1}\cdots
z_{i_{k-1}}\cdot U, \quad z_1, \ldots, z_q\in G.$$
We can now use [(\[eq:uffe41\])]{} and [(\[eq:uffe50\])]{} to control the Haar measure of each of the sets $$\label{eq:uffe51}
U^{2m_o}\cdot x_{n_1}\cdot U^{2m_1} \cdot x_{n_2}\cdots
x_{n_k}\cdot U^{2m_k},$$ from [(\[eq:uffe49\])]{}. By [(\[eq:uffe50\])]{} we see that $U^{2m_0}$ can be covered by $q^{2m_0-1}$ left translations of $U$.
Combined with [(\[eq:uffe41\])]{}, we get that $U^{2m_0}\cdot x_{n_1}$ can be covered by $q^{2m_0-1}\cdot p(n_1)$ left translates of $U$. Hence $$U^{2m_0}x_{n_1}U^{2m_1} = \bigcup_{w\in A_1}w\cdot U^{2m_1+1},$$ where $|A_1| \leq 2^{2m_0 -1}p(n_1)$.
Again by [(\[eq:uffe50\])]{}, we see that $U^{2m_1+1}$ can be covered by $q^{2m_1}$ left translations of $U$, so altogether we see that the set $$U^{2m_0}x_{n_1}U^{2m_1}$$ can be covered by $q^{2m_0 + 2m_1-1}p(n_1)$ left translations of $U$. Continuing this procedure, we get that the set in [(\[eq:uffe51\])]{} can be covered by $$q^{2m_0 + 2m_1 + \cdots + 2m_k -1}p(n_1)\cdot p(n_2)\cdots p(n_k)$$ left translates of $U$, and hence the Haar measure of the set satisfies that $$\begin{gathered}
\mu\bigg(
U^{2m_o}\cdot x_{n_1}\cdot U^{2m_1} \cdot x_{n_2}\cdots
x_{n_k}\cdot U^{2m_k}
\bigg) \\
\leq
q^{2\cdot \sum^k_{i=0} m_i}\cdot \prod^k_{i=1} p(n_i)\mu(U)
\\
\leq
q^{2\cdot \sum^k_{i=0} m_i}\cdot 2^{ \sum^k_{i=1} \log_2(p(n_i))}\mu(U)\end{gathered}$$
By [(\[eq:uffe47\])]{}, we have that $$\begin{cases}
\sum^k_{i=0} m_i \leq 2n+1 \\
\sum^k_{i=1} \log_2(p(n_i)) \leq 2n+1
\end{cases}.$$
Hence, we have that $$\mu\bigg(
U^{2m_o}\cdot x_{n_1}\cdot U^{2m_1} \cdot x_{n_2}\cdots
x_{n_k}\cdot U^{2m_k}
\bigg) \leq (2q^2)^{2n+1}\cdot \mu(U).$$
This holds for all tupples $(k, m_0, \ldots, m_k, n_1, \ldots, n_k) \in M$, and since we have shown that $|M| \leq 2^{2n}$, we get from [(\[eq:uffe45\])]{} that $$\mu( B_{d^*}(e,n)) \leq (4q^2)^{2n+1}\mu(U), \quad n\in{{\mathbb N}},$$ which shows that the $d^*$-balls have exponentially controlled growth.
\[thm:discrete\_countable\_gp\]
In this example, we will give a more direct proof of Theorem [(\[thm:lcsc\_has\_expgrow\])]{} in the case of a countable discrete group $\Gamma$. If $\Gamma$ is finitely generated, it is elementary to check, that the word length metric $d$ is proper and that the $d$-balls have exponentially controlled growth, so we can assume that $\Gamma$ is generated by an infite, symmetric set $S$, such that $e\notin S$.
We can write $S$ as a disjoint union $$S = \bigcup^\infty_{n=1} Z_n,$$ where each $Z_n$ is of the form $\{ x_n, x_n^{-1} \}$. Note that $|Z_n|= 2$ if $x_n \neq x_n^{-1}$, and $|Z_n|= 1$ if $x_n = x_n^{-1}$. Define a function $$l_0: S\rightarrow {{\mathbb N}}$$ by $$l_0(x_n) = l_0(x_n^{-1}) = n.$$
Next, define a function $$l: \Gamma\rightarrow {{\mathbb N}}\cup\{ 0 \}$$ by $$\label{eq:length_df}
l(g) =
\begin{cases}
\inf\{ \sum_{k=1}^n l(g_k) \} & g\neq e \\
0& g= e
\end{cases}$$ where the infimum is taken over all representations of $g$ of the form $$g = g_1\cdots g_n, \quad g_i\in S, n\in {{\mathbb N}}.$$ Then it is easy to check, that $l$ is a length function on $\Gamma$, and since $$l(g) \geq 1\quad\mbox{for}\quad g\in\Gamma\setminus\{ e \}$$ the associated left invariant metric $$d(g,h) = l(g^{-1}h), \quad g,h\in \Gamma$$ generates the discrete topology on $\Gamma$. Put $$D(e,n) = \{ g\in \Gamma: d(g, e) \leq n \}.$$
We will next show, that $$\label{eq:u76}
|D(e,n) | \leq 3^n, \quad n\in{{\mathbb N}},$$ which clearly implies, that [the $d$-balls have exponentially controlled growth]{}. In order to prove [(\[eq:u76\])]{}, we will show by induction in $n\in{{\mathbb N}}$, that the set $$\partial(e, n) = \{ g\in \Gamma: d(g,e) = n \}$$ satisfies $$\label{eq:u77}
|\partial(e,n)| \leq 2\cdot 3^{n-1}, \quad n\in{{\mathbb N}}.$$
Since $l_0(s)\geq 2$ for $s\in S\setminus\{ x_1, x_1^{-1} \}$, we have for $g\in\Gamma$, that $$l(g) = 1 \Leftrightarrow g \in \{ x_1, x_1^{-1} \}.$$
Hence $$| \partial(e, 1) | = |\{ x_1, x_1^{-1} \}| \leq 2$$ which proves [(\[eq:u77\])]{} for $n=1$. Let now $n\geq 2$ and assume as induction hyphotesis, that $$\label{eq:u78}
|\partial(e,i)| \leq 2\cdot 3^{i-1}, \quad i = 1, \ldots, n-1.$$
We shall then show, that $$|\partial(e,n)| \leq 2\cdot 3^{n-1}.$$
We claim, that $$\label{eq:u79}
\partial(e,n) \subset \bigcup^n_{k=1} Z_k\cdot \partial(e,n-k)$$ To prove [(\[eq:u78\])]{}, let $g\in \partial(e,n)$. Then there exists a $m\in {{\mathbb N}}$ and $g_1, \ldots, g_m\in S$ such that $$g = g_1\cdots g_m$$ and $$\sum^m_{i=1} l_0 (g_i) = n$$
Put $k = l_0(g_1)$. Then $k\in{{\mathbb N}}$ and $k\leq n$. Now $$g = g_1\cdot (g_2\cdots g_m ),$$ where $$\begin{aligned}
\label{eq:u80}
l(g_1) \leq l_0( g_1) = k\\
\label{eq:u81}
l(g_2\cdots g_m) \leq \sum^m_{i=2} l_0(g_i) = n-k.\end{aligned}$$
But, since $$n = l(g) \leq l(g_1) + l(g_2 \cdots g_m)$$ equality holds in both [(\[eq:u80\])]{} and [(\[eq:u81\])]{}. Hence $g_2\cdots
g_m\in \partial(e, n-k)$, which proves [(\[eq:u79\])]{}. By [(\[eq:u79\])]{} we have
$$|\partial(e, n)| \leq
\sum_{k=1}^{n} |Z_k|\cdot|\partial(e, n-k)| \leq 2\sum_{k=1}^{n}
|\partial(e, n-k)| = 2\sum_{i=1}^{n-1} |\partial(e, i)|,$$
Since $|\partial(e,0)| = |\{
e\}| =1 $, we get by the induction hypothesis [(\[eq:u78\])]{}, that $$\begin{gathered}
|\partial(e, n)| \leq \sum_{i=0}^{n-1} 2|\partial(e, i)|
\leq 2(1+ \sum_{i=1}^{n-1} 2\cdot 3^{i-1})
= 2(1+3^{n-1}-1)) = 2\cdot 3^{n-1}.\end{gathered}$$ This completes the proof of the induction step. Hence [(\[eq:u77\])]{} holds for all $n\in{{\mathbb N}}$. Since $l$ only takes integer values, we have $$D(e,n) = \bigcup_{i=0}^n \partial(e, i).$$ Therefore $$|D(e, n)| = \sum_{i=0}^n |\partial(e, i)| \leq 1 + \sum_{i=1}^{n}
2\cdot 3^{i-1} = 3^n.$$
This proves [(\[eq:u76\])]{}, and it follows that [the $d$-balls have exponentially controlled growth]{}.
Affine actions on Banach spaces.
================================
We have shown in theorem \[thm:plig\_on\_lcsc\] that for any [ locally compact, second countable]{}group $G$ there exists a [[**plig**]{} metric]{}$d$, and we have shown in theorem \[thm:lcsc\_has\_expgrow\] that $d$ can be chosen so that [the $d$-balls have exponentially controlled growth]{}. We will now construct an affine action of $G$ on the reflexive seperable strictly convex Banach space $${\mbox{$\bigoplus^\infty_{n=1} L^{2n}(G, \mu)$ (in the $l^2$ sense)}}.$$
Gromov suggested in [@gromov_novikov], that it is purposeful to attack the Baum Connes Conjecture by considering proper affine isometric actions on various Banach spaces.
It was shown by N. Higson and G. Kasparov in [@higson_kasparov] that the Baum-Connes Conjecture holds for discrete countable groups that admit a proper affine isometric action on a Hilbert space. In particular, this holds for all discrete amenable groups. Moreover, Yu proved in [@yu_hyperbolic], that a word hyperbolic group $\Gamma$ has a proper affine action on the uniform convex Banach space $l^p(\Gamma\times \Gamma)$ for some $p\in [2, \infty)$.
Therefore, it is interesting to study what kind of proper affine isometric actions on Banach spaces a given [ locally compact, second countable]{}group admits.
The group of affine actions on $G$: Let $X$ be a normed vector space, then the affine group of $X$ is: $${\operatorname{Aff}}(X) = \{ \phi:X\rightarrow X\quad : \phi(x) = Ax+b; A\in GL(X), b\in X \}.$$ We say that [*$G$ has an affine action on $X$*]{}, if there exists a group homomorphism of $G$ on ${\operatorname{Aff}}(X)$, ie: $$\alpha:G\circlearrowright {\operatorname{Aff}}(X),\label{eq:affine_action}$$ such that $$\forall_{g,h\in G}\quad\alpha(g\cdot h) = \alpha(g)\circ
\alpha(h).\label{eq:cont_action}$$ Let $\pi_g$ denote the linear part of $\alpha(g)$, and denote the translation part by $b(g)$. We say that $\alpha(g)$ is isometric if the linear part $\pi_g$ is isometric, i.e: $$\forall_{\xi\in X}\quad ||\pi_g\xi|| = ||\xi|| .$$
Moreover, we say that the action is proper, if $$\forall_{\xi\in X}\quad \lim_{g\rightarrow\infty} ||\alpha(g)\xi|| = \infty.$$
Since $\alpha$ is a homomorphism of the group $G$ into ${\operatorname{Aff}}(X)$, we have that: $$\begin{aligned}
\label{eq:cocycle_cond}
\forall_{\xi\in X}& \quad \alpha(st)\xi &= \alpha(s)(\alpha(t)\xi)
\Leftrightarrow\notag\\
\forall_{\xi\in X}&\quad\pi_{st}\xi +b(st) &= \pi_s\pi_t\xi + \pi_sb(t)+b(s) \Leftrightarrow \notag\\
&\pi_{st} = \pi_s\circ \pi_t&\mbox{ and }\quad b(st) =
\pi_s(b(t)) + b(s).
\end{aligned}$$ The formula for $b(st)$ is called the [*cocycle condition with respect to $\pi$*]{}.
And we also need to know what a strictly convex space is – and we will use the opportunity to define a uniformly convex space as well:
Let $X$ be a normed vector space, denote the unit ball by $S_X$.
1. The following two conditions are equivalent (see [@megginson Prop. 5.1.2]). If $X$ satisfies any of them, it is called [*strictly convex*]{}.
1. $$\label{eq:rotund_1}
\forall_{x\neq y\in S_X, 1>t>0}\quad ||tx + (1-t)y|| <1.$$
2. $$\label{eq:rotund}
\forall_{x\neq y\in S_X}\quad ||\frac{1}{2}(x+y)|| < 1.$$
2. $X$ is called [*uniformly convex*]{} if $$\label{eq:uniformly_rotund}
\forall_{\epsilon>0}\exists_{\delta>0}\forall{ x,y\in
S_X}\quad ||x-y|| \geq
\epsilon\Rightarrow ||\frac{1}{2}(x+y) || \leq 1-\delta$$
Every space that is uniformly convex is also strictly convex (see [@megginson Proposition 5.2.6]). Examples of uniformly convex Banach spaces include $$l_p, l_p^n, \quad \infty>p>1, n\geq 1$$ (this follows from Milman-Pettis theorem, see [@megginson Theorem 5.2.15]).
A uniformly convex Banach space is necessarily reflexive (see [@megginson Theorem ]). There are spaces that are strictly convex, but not uniformly convex, and also spaces that are strictly convex and not reflexive. An example of a strictly convex but not uniformly convex Banach space is: $$\bigoplus_{i=1}^\infty l^n_{p_n},\quad\mbox{ where } p_n = 1+\frac{1}{n}$$ (with $l^2$ norm on the direct sum).
As an application of the construction of a [[**plig**]{} metric]{}with [the $d$-balls have exponentially controlled growth]{}on a given [ locally compact, second countable]{}group in theorem [(\[thm:lcsc\_has\_expgrow\])]{}, we will construct a proper isometric action on the Banach space $${\mbox{$\bigoplus^\infty_{n=1} L^{2n}(G, \mu)$ (in the $l^2$ sense)}}.$$
In [@brown_guentner] a proper isometric action is constructed for a discrete group $\Gamma$ into the Banach space $\bigoplus_{p=1}^\infty L^{p_n}(G, d\mu),$ where $p_n$ is an unbounded sequence. We have generalized this result as follows:
\[thm:affine\_action\] Let $G$ be a [ locally compact, second countable]{} group, and let $\mu$ denote the Haar measure. Then there exists a proper affine isometric action $\alpha$ of $G$ on the seperable, strictly convex, reflexive Banach space $$X = {\mbox{$\bigoplus^\infty_{n=1} L^{2n}(G, \mu)$ (in the $l^2$ sense)}}.$$
Let $G$ be as in the statement of the theorem, then according to theorem [(\[thm:plig\_on\_lcsc\])]{} and theorem [(\[thm:lcsc\_has\_expgrow\])]{} we can choose a [[**plig**]{} metric]{}$d$ on $G$ where [the $d$-balls have exponentially controlled growth]{}, ie. $$\label{eq:uffe2.6enhalv}
\exists_{\alpha>0}\quad \mu(B_d(e,n) \leq \beta\cdot e^{\alpha n},$$ for some constants $\alpha, \beta > 0$. We can without loss of generality assume that $\beta\geq 1$.
Consider the functions $phi^n_g: G -> R$ given by: $$\label{eq:lip_fn}
\phi^n_x(y) =
\begin{cases}
1 - \frac{d(x,y)}{n}&\mbox{when }d(x,y)\leq n\\
0 &\mbox{when } d(x,y)\geq n
\end{cases}$$
It is easy to check, that $phi^n_g$ is $\frac{1}{n}$-Lipschitz: $$\label{eq:uffe2.8}
|\phi^n_x(y)-\phi^n_x(z)|\leq \frac{d(y,z)}{n}.$$
Assume that $x\in B_d(e, \frac{n}{2})$: $$\label{eq:bound_by_charac}
\phi^n_e(x) = 1 - \frac{d(x,e)}{n} \geq 1 - \frac{\frac{n}{2}}{n} =
\frac{1}{2}\cdot 1_{ B_d(e, \frac{n}{2})}(x),$$
Let ${\operatorname{C_{ucb}}}(G)$ denote the set of uniformly continuous bounded functions from $G$ to ${{\mathbb R}}$. Define $b^n:G\rightarrow {\operatorname{C_{ucb}}}(G)$ by: $$\label{eq:b_n}
b^n(g) = \lambda(g)\phi^n_e - \phi^n_e \Rightarrow b^n(g)(h) = \phi^n_e(g^{-1}h)-\phi^n_e(h).$$
Since $d(g,e) = d(g^{-1},e)$, we have that $$\phi^n_e(g) = \phi^n_e(g^{-1}),
\quad g\in G.$$
Hence, by [(\[eq:b\_n\])]{} and [(\[eq:uffe2.8\])]{} we have $$\label{uffe:2.11}
|b^n(g)| \leq \big| \phi^n_e(g^{-1}h)-\phi^n_e(h) \big| =
\big| \phi^n_e(h^{-1}g)-\phi^n_e(h^{-1}) \big|
\leq
\frac{d(h^{-1}g,h^{-1}) }{n} \leq \frac{d(e,g)}{n}$$
Since $b_n(g) =0$, when $x\not\in B_d(e,n)\cup B_d(g,n)$, it follows that $$| b^n(g)| \leq \frac{d(e,g)}{n}\cdot 1_{ B_d(e, n)\cup B_d(g, n)} .$$
Hence $b^n(g) \in L^2(G, \mu)$ and $$|| b^n(g)||^{2n}_{2n} \leq \big( \frac{d(e,g)}{n}\big)^{2n} \big(\mu( B_d(e, n)) +
\mu(B_d(g, n))\big).$$
Therefore, by [(\[eq:uffe2.6enhalv\])]{} and the left invariance of $\mu$, we have that $$|| b^n(g)||^{2n}_{2n} \leq \big( \frac{d(e,g)}{n}\big)^{2n}\cdot
2\beta e^{\alpha n}.$$
Using now, that $\beta\geq 1$ and $\sum^\infty_{n=1} \frac{1}{n^2} =
\frac{\pi^2}{6} < 2$, we get $$\sum^\infty_{n=1} || b^n(g)||^{2}_{2n} \leq \sum^\infty_{n=1}
\frac{d(e,g)^2}{n^2}(2\beta)^{\frac{1}{n}}e^\alpha
\leq 4\beta e^\alpha d(g,e)^2$$
Hence $$b(g) = \oplus^\infty_{n=1} b^n(g) \in X$$ and $$||b(g)||_X \leq 2\sqrt{\beta} e^\frac{\alpha}{2} d(g,e).$$
Let $\tilde{\lambda}$ denote the left regular representation of $G$ on $X =
{\mbox{$\bigoplus^\infty_{n=1} L^{2n}(G, \mu)$ (in the $l^2$ sense)}}$. Clearly $\tilde{\lambda}(g)$ is an isometry of $X$ for every $g\in G$. We show next, that $b(g)$ fulfills the cocycle condition $$\label{eq:2.12}
b(st) = \tilde{\lambda}(s)b(t) + b(s), \quad s,t\in G$$ and [(\[eq:2.12\])]{} follows from $$\begin{gathered}
b^n(st) = \lambda(st)\phi^n_e - \phi^n_e \\
=
\lambda(s)(\lambda(t)\phi^n_e - \phi^n_e ) + (\lambda(s)\phi^n_e -
\phi^n_e )
= \lambda(s)b^n(t) + b^n(s), \quad s,t\in G, \end{gathered}$$ for alle $n\in {{\mathbb N}}$. By [(\[eq:2.12\])]{} we can define a continuous affine action $\alpha$ of $G$ on $X$ by $$\alpha(g)\xi = \tilde{\lambda}(g)\xi + b(g), \quad x\in X, g\in G.$$
The last thing to show is that the action is metrically proper. For $\xi\in
X$ and $g\in G$, we have $$|| \alpha(g)\xi|| = || \tilde{\lambda}(g)\xi + b(g) || \geq
|| b(g)|| - || \tilde{\lambda}(g)\xi|| = ||b(g)|| - ||\xi||.$$
Hence, we only have to check, that $$||b(g)||\rightarrow \infty\quad\mbox{when}\quad d(g,e)\rightarrow\infty.$$
Let $g\in G$ and assume that $d(g,e)>2$. Moreover, let $N(g)\in {{\mathbb N}}$ denote the integer for which $$\frac{d(g,e)}{2} -1 \leq N(g) < \frac{d(g,e)}{2}.$$
For $ n = 1\ldots, N(g)$, we have that $$d(g,e) > 2N(g) \geq 2n.$$
Hence $$\overline{B(e,n)} \cap \overline{B(g,n)} = \emptyset,$$ which implies that $\phi^n_e$ and $\phi^n_g$ have disjoint supports. Therefore $$|| b^n(g)||^{2n}_{2n} = || \phi^n_g - \phi^n_e||^{2n}_{2n} = || \phi^n_g ||^{2n}_{2n} + ||
\phi^n_e ||^{2n}_{2n} \geq || \phi^n_e ||^{2n}_{2n}$$
Since we have that $$\phi^n_e \geq \frac{1}{2}\cdot 1_{B(e,\frac{n}{2})}$$ it follows that $$|| b^n(g)||^{2n}_{2n} \geq 2^{-2n}\mu(B(e,\frac{n}{2})) \geq 2^{-2n}\mu(B(e,\frac{1}{2})).$$
Hence $$\begin{gathered}
||b(g)||^2 \geq \sum^{N(g)}_{n=1} || b^n(g) ||^{2}_{2n} \geq
\frac{1}{4} \sum^{N(g)}_{n=1} \mu(B(e, \frac{1}{2}))^{\frac{1}{n}} \\
\geq
\frac{N(g)}{4}\cdot\min\big\{ \mu(B(e,\frac{1}{2})), 1 \big\}.\qquad\end{gathered}$$
Since $$N(g) \geq \frac{d(g,e)}{2}-1$$ it follows that $$|| b(g) || \rightarrow \infty\mbox{ for }\quad d(g,e)\rightarrow\infty.$$
Let $G$ be a locally compact, 2-nd countable group. Then $G$ has a uniform embedding into the seperable, strictly convex Banach space $${\mbox{$\bigoplus^\infty_{n=1} L^{2n}(G, \mu)$ (in the $l^2$ sense)}}.$$
We will show, that the map $b:G\rightarrow X$ constructed in the proof of theorem [(\[thm:affine\_action\])]{} is a uniform embedding. By the proof of theorem [(\[thm:affine\_action\])]{}, we have that $$\label{eq:100}
c_1\sqrt{d(g,e)} \leq ||b(g) ||_X \leq c_2 d(g,e),$$ when $d(g,e) \geq c_3$ for some positive constants $c_1, c_2, c_3$. By the cocycle condition [(\[eq:cocycle\_cond\])]{}, we get for $g, h\in G$ that $$b(g) = b(h(h^{-1}g)) = \tilde{\lambda}(h)b(h^{-1}g) + b(h).$$
Hence $$|| b(g) - b(h) ||_X = || \tilde{\lambda}(h)b(h^{-1}g)||_X = ||b(h^{-1}g)||_X.$$
Since $d(h^{-1}g, e) = d(g,h)$, we obtain by applying [(\[eq:100\])]{} to $b(h^{-1}g)$, that $$c_1\sqrt{d(g,h)} \leq ||b(g) - b(h)||_X \leq c_2 d(g,h),$$ when $d(g,h) \geq c_3$. This proves, that $b$ is a uniform embedding.
[^1]: Supported by the Danish National Research Foundation
|
---
abstract: |
We investigate the expressive power of ${\mathsf{MATLANG}}$, a formal language for matrix manipulation based on common matrix operations and linear algebra. The language can be extended with the operation ${\mathsf{inv}}$ of inverting a matrix. In ${\mathsf{MATLANG}}+{\mathsf{inv}}$ we can compute the transitive closure of directed graphs, whereas we show that this is not possible without inversion. Indeed we show that the basic language can be simulated in the relational algebra with arithmetic operations, grouping, and summation. We also consider an operation ${\mathsf{eigen}}$ for diagonalizing a matrix, which is defined so that different eigenvectors returned for a same eigenvalue are orthogonal. We show that ${\mathsf{inv}}$ can be expressed in ${\mathsf{MATLANG}}+{\mathsf{eigen}}$. We put forward the open question whether there are boolean queries about matrices, or generic queries about graphs, expressible in ${\mathsf{MATLANG}}+
{\mathsf{eigen}}$ but not in ${\mathsf{MATLANG}}+{\mathsf{inv}}$. The evaluation problem for ${\mathsf{MATLANG}}+
{\mathsf{eigen}}$ is shown to be complete for the complexity class $\exists \mathbf{R}$.
author:
- Robert Brijder
- Floris Geerts
- Jan Van den Bussche
- Timmy Weerwag
bibliography:
- 'database.bib'
title: On the expressive power of query languages for matrices
---
Introduction
============
Data scientists often use matrices to represent their data, as opposed to using the relational data model. These matrices are then manipulated in programming languages such as R or . These languages have common operations on matrices built-in, notably matrix multiplication; matrix transposition; elementwise operations on the entries of matrices; solving nonsingular systems of linear equations (matrix inversion); and diagonalization (eigenvalues and eigenvectors). Providing database support for matrices and multidimensional arrays has been a long-standing research topic [@rusu_survey], originally geared towards applications in scientific data management, and more recently motivated by machine learning over big data [@systemml; @olteanu_regression; @naughton_la; @ngolteanu_learning].
Database theory and finite model theory provide a rich picture of the expressive power of query languages [@ahv_book; @kolaitis_expressivepower]. In this paper we would like to bring matrix languages into this picture. There is a lot of current interest in languages that combine matrix operations with relational query languages or logics, both in database systems [@hutchison] and in finite model theory [@dawar_linearalgebra; @dghl_rank; @holm_phd]. In the present study, however, we focus on matrices alone. Indeed, given their popularity, we believe the expressive power of matrix sublanguages also deserves to be understood in its own right.
The contents of this paper can be introduced as follows. We begin the paper by defining the language ${\mathsf{MATLANG}}$ as an analog for matrices of the relational algebra for relations. This language is based on five elementary operations, namely, the one-vector; turning a vector in a diagonal matrix; matrix multiplication; matrix transposition; and pointwise function application. We give examples showing that this basic language is capable of expressing common matrix manipulations. For example, the Google matrix of any directed graph $G$ can be computed in ${\mathsf{MATLANG}}$, starting from the adjacency matrix of $G$.
Well-typedness and well-definedness notions of ${\mathsf{MATLANG}}$ expressions are captured via a simple data model for matrices. In analogy to the relational model, a schema consists of a number of matrix names, and an instance assigns matrices to the names. Recall that in a relational schema, a relation name is typed by a set of attribute symbols. In our case, a matrix name is typed by a pair $\alpha \times \beta$, where $\alpha$ and $\beta$ are size symbols that indicate, in a generic manner, the number of rows and columns of the matrix.
In Section \[secsum\] we show that our language can be simulated in the relational algebra with aggregates [@klug_agg; @libkin_sql], using a standard representation of matrices as relations. The only aggregate function that is needed is summation. In fact, ${\mathsf{MATLANG}}$ is already subsumed by aggregate logic with only three nonnumerical variables. Conversely, ${\mathsf{MATLANG}}$ can express all queries from graph databases (binary relational structures) to binary relations that can be expressed in first-order logic with three variables. In contrast, the four-variable query asking if the graph contains a four-clique, is not expressible.
In Section \[secinv\] we extend ${\mathsf{MATLANG}}$ with an operation for inverting a matrix, and we show that the extended language is strictly more expressive. Indeed, the transitive closure of binary relations becomes expressible. The possibility of reducing transitive closure to matrix inversion has been pointed out by several researchers [@laubner_phd; @schwentick_reachdynfo; @sato_ladatalog]. We show that the restricted setting of ${\mathsf{MATLANG}}$ suffices for this reduction to work. That transitive closure is not expressible without inversion, follows from the locality of relational algebra with aggregates [@libkin_sql].
Another prominent operation of linear algebra, with many applications in data mining and graph analysis [@hms_miningbook; @ullman_mining], is to return eigenvectors and eigenvalues. There are various ways to define this operator formally. In Section \[seceigen\] we define the operation ${\mathsf{eigen}}$ to return a basis of eigenvectors, in which eigenvectors for a same eigenvalue are orthogonal. We show that the resulting language ${\mathsf{MATLANG}}+ {\mathsf{eigen}}$ can express inversion. The argument is well known from linear algebra, but our result shows that it can be carried out in ${\mathsf{MATLANG}}$, once more attesting that we have defined an adequate matrix language. It is natural to conjecture that ${\mathsf{MATLANG}}+ {\mathsf{eigen}}$ is actually strictly more powerful than ${\mathsf{MATLANG}}+ {\mathsf{inv}}$ in expressing, say, boolean queries about matrices. Proving this is an interesting open problem.
Finally, in Section \[seceval\] we look into the evaluation problem for ${\mathsf{MATLANG}}+ {\mathsf{eigen}}$ expressions. In practice, matrix computations are performed using techniques from numerical mathematics [@golub]. It remains of foundational interest, however, to know whether the evaluation of expressions is effectively computable. We need to define this problem with some care, since we work with arbitrary complex numbers. Even if the inputs are, say, 0-1 matrices, the outputs of the ${\mathsf{eigen}}$ operation can be complex numbers. Moreover, until now we have allowed arbitrary pointwise functions, which we should restrict somehow if we want to discuss computability. Our approach is to restrict pointwise functions to be semi-algebraic, i.e., definable over the real numbers. We will observe that the input-output relation of an expression $e$, applied to input matrices of given dimensions, is definable in the existential theory of the real numbers, by a formula of size polynomial in the size of $e$ and the given dimensions. This places natural decision versions of the evaluation problem for ${\mathsf{MATLANG}}+ {\mathsf{eigen}}$ in the complexity class $\exists \mathbf{R}$ (combined complexity). We show moreover that there exists a fixed expression (data complexity) for which the evaluation problem is $\exists \mathbf{R}$-complete, even restricted to input matrices with integer entries. It also follows that equivalence of expressions, over inputs of given dimensions, is decidable.
${\mathsf{MATLANG}}$
====================
We assume a sufficient supply of *matrix variables*, which serve to indicate the inputs to expressions in ${\mathsf{MATLANG}}$. Variables can also be introduced in let-constructs inside expressions. The syntax of ${\mathsf{MATLANG}}$ expressions is defined by the grammar: $$\begin{aligned}
e &::= {M}&& \text{(matrix variable)} \\
&\mid\quad {{\mathsf}{let}\ {M}=e_1\ {\mathsf}{in}\ e_2} && \text{(local binding)} \\
&\mid\quad {e^*} && \text{(conjugate transpose)} \\
&\mid\quad {\mathbf{1}}(e) && \text{(one-vector)} \\
&\mid\quad \operatorname{\mathsf{diag}}(e) && \text{(diagonalization of a vector)} \\
&\mid\quad e_1 \cdot e_2 && \text{(matrix multiplication)} \\
&\mid\quad \operatorname{\mathsf{apply}}[f](e_1, \ldots, e_n) && \text{(pointwise
application, $f \in \Omega$)}\end{aligned}$$ In the last rule, $f$ is the name of a function $f : {\mathbf{C}}^n \to
{\mathbf{C}}$, where ${\mathbf{C}}$ denotes the complex numbers. Formally, the syntax of ${\mathsf{MATLANG}}$ is parameterized by a repertoire $\Omega$ of such functions, but for simplicity we will not reflect this in the notation.
\[exex\] Let $c \in {\mathbf{C}}$ be a constant; we also use $c$ as a name for the constant function $c : {\mathbf{C}}\to {\mathbf{C}}: z \mapsto
c$. Then $${{\mathsf}{let}\ N={{\mathbf{1}}(M)^*}\ {\mathsf}{in}\ \operatorname{\mathsf{apply}}[c]({\mathbf{1}}(N))}$$ is an example of an expression. At this point, this is a purely syntactical example; we will see its semantics shortly. The expression is actually equivalent to $\operatorname{\mathsf{apply}}[c]({\mathbf{1}}({{\mathbf{1}}(M)^*}))$. The let-construct is useful to give names to intermediate results, but is not essential for now. It will become essential later, when we enrich ${\mathsf{MATLANG}}$ with the ${\mathsf{eigen}}$ operation.
In defining the semantics of the language, we begin by defining the basic matrix operations. Following practical matrix sublanguages such as R or , we will work throughout with matrices over the complex numbers. However, a real-number version of the language could be defined as well.
Transpose:
: If $A$ is a matrix then ${A^*}$ is its conjugate transpose. So, if $A$ is an $m \times n$ matrix then ${A^*}$ is an $n \times m$ matrix and the entry $A^*_{i,j}$ is the complex conjugate of the entry $A_{j,i}$.
One-vector:
: If $A$ is an $m \times n$ matrix then ${{\mathbf{1}}}(A)$ is the $m \times 1$ column vector consisting of all ones.
Diag:
: If $v$ is an $m \times 1$ column vector then $\operatorname{\mathsf{diag}}(v)$ is the $m \times m$ diagonal square matrix with $v$ on the diagonal and zero everywhere else.
Matrix multiplication:
: If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix then the well known matrix multiplication $A B$ is defined to be the $m \times p$ matrix where $(AB)_{i,j} = \sum_{k=1}^n A_{i,k}B_{k,j}$. In ${\mathsf{MATLANG}}$ we explicitly denote this as $A \cdot B$.
Pointwise application:
: If $A^{(1)},\dots,A^{(n)}$ are matrices of the same dimensions $m \times p$, then $\operatorname{\mathsf{apply}}[f](A^{(1)},\dots,A^{(n)})$ is the $m \times p$ matrix $C$ where $C_{i,j} = f(A^{(1)}_{i,j},\dots,A^{(n)}_{i,j})$.
$$\begin{array}{l@{\hspace{2em}}l}
\displaystyle
\begin{pmatrix} 0 & 1+i \\ 2 & 3-i \\ 4 +4i & 5
\end{pmatrix}^* =
\begin{pmatrix} 0 & 2 & 4-4i \\ 1-i & 3+i & 5 \end{pmatrix}
&
\displaystyle
{{\mathbf{1}}}\begin{pmatrix} 2 & 3 & 4 \\ 4 & 5 & 6 \end{pmatrix} =
\begin{pmatrix} 1 \\ 1 \end{pmatrix}
\\[4.5ex]
\begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 &
6 \end{pmatrix} \cdot \begin{pmatrix} 6 & 5 & 4 & 3 \\ 2 & 1 & 0
& -1 \end{pmatrix} = \begin{pmatrix} 10
& 7 & 4 & 1 \\ 26 & 19 & 12 & 5 \\ 42 & 31 & 20 & 9
\end{pmatrix}
&
\displaystyle
\operatorname{\mathsf{diag}}\begin{pmatrix} 6 \\ 7 \end{pmatrix} =
\begin{pmatrix} 6 & 0 \\ 0 & 7 \end{pmatrix}
\end{array}$$ $$\operatorname{\mathsf{apply}}[{\mathbin{\dot-}}](\begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 &
0 & 0 \end{pmatrix},
\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1
\end{pmatrix}) =
\begin{pmatrix} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$$
The operations are illustrated in Figure \[figops\]. In the pointwise application example, we use the function ${\mathbin{\dot-}}$ defined by $x {\mathbin{\dot-}}y
= x - y$ if $x$ and $y$ are both real numbers and $x \geq y$, and $x {\mathbin{\dot-}}y = 0$ otherwise.
Formal semantics
----------------
The formal semantics of expressions is defined in a straightforward manner, as shown in Figure \[fig:bigstep\]. An *instance* $I$ is a function, defined on a nonempty finite set $\operatorname{var}(I)$ of matrix variables, that assigns a matrix to each element of $\operatorname{var}(I)$. Figure \[fig:bigstep\] provides the rules that allow to derive that an expression $e$, on an instance $I$, successfully evaluates to a matrix $A$. We denote this success by ${e(I)=A}$. The reason why an evaluation may not succeed can be found in the rules that have a condition attached to them. The rule for variables fails when an instance simply does not provide a value for some input variable. The rules for $\operatorname{\mathsf{diag}}$, $\operatorname{\mathsf{apply}}$, and matrix multiplication have conditions on the dimensions of matrices, that need to be satisfied for the operations to be well-defined.
\[exscalar\] The expression from Example \[exex\], regardless of the matrix assigned to $M$, evaluates to the $1 \times 1$ matrix whose single entry equals $c$. We introduce the shorthand $c$ for this constant expression. Obviously, in practice, scalars would be built in the language and would not be computed in such a roundabout manner. In this paper, however, we are interested in expressiveness, so we start from a minimal language and then see what is already expressible in this language.
\[exscalmul\] Let $A$ be any matrix and let $C$ be a $1 \times 1$ matrix; let $c$ be the value of $C$’s single entry. Viewing $C$ as a scalar, we define the operation $C\odot A$ as multiplying every entry of $A$ by $c$. We can express $C \odot A$ as $${{\mathsf}{let}\ M={\mathbf{1}}(A) \cdot C \cdot {{\mathbf{1}}({A^*})^*}\ {\mathsf}{in}\ \operatorname{\mathsf{apply}}[\times](M,A)}.$$ If $A$ is an $m \times n$ matrix, we compute in variable $M$ the $m
\times n$ matrix where every entry equals $c$. Then pointwise multiplication is used to do the scalar multiplication.
\[ex:google-matrix\] Let $A$ be the adjacency matrix of a directed graph (modeling the Web graph) on $n$ nodes numbered $1,\dots,n$. Let $0 < d < 1$ be a fixed “damping factor”. Let $k_i$ denote the outdegree of node $i$. For simplicity, we assume $k_i$ is nonzero for every $i$. Then the Google matrix [@pagerank; @bonato_webgraph] of $A$ is the $n \times n$ matrix $G$ defined by $$G_{i,j} =
d \frac{A_{ij}}{k_i} + \frac{1-d}{n}.$$ The calculation of $G$ from $A$ can be expressed in ${\mathsf{MATLANG}}$ as follows:
let $J = {{\mathbf{1}}}(A) \cdot {{\mathbf{1}}}(A)^*$ in\
let $K = A \cdot J$ in\
let $B = \operatorname{\mathsf{apply}}[/](A,K)$ in\
let $N = {{\mathbf{1}}}(A)^* \cdot {{\mathbf{1}}}(A)$ in\
$\operatorname{\mathsf{apply}}[+]( d \odot B, (1-d) \odot \operatorname{\mathsf{apply}}[1/x](N) \odot J)$
In variable $J$ we compute the $n \times n$ matrix where every entry equals one. In $K$ we compute the $n \times n$ matrix where all entries in the $i$th row equal $k_i$. In $N$ we compute the $1 \times 1$ matrix containing the value $n$. The pointwise functions applied are addition, division, and reciprocal. We use the shorthand for constants ($d$ and $1-d$) from Example \[exscalar\], and the shorthand $\odot$ for scalar multiplication from Example \[exscalmul\].
\[exmin\] Let $v=(v_1,\dots,v_n)^*$ be a column vector of real numbers; we would like to extract the minimum from $v$. This can be done as follows:
let $V = v \cdot {{\mathbf{1}}}(v)^*$ in\
let $C = \operatorname{\mathsf{apply}}[\leq](V,V^*) \cdot {{\mathbf{1}}}(v)$ in\
let $N = {{\mathbf{1}}}(v)^* \cdot {{\mathbf{1}}}(v)$ in\
let $S = \operatorname{\mathsf{apply}}[=](C,{{\mathbf{1}}}(v) \cdot N)$ in\
let $M = \operatorname{\mathsf{apply}}[1/x](S^* \cdot {{\mathbf{1}}}(v))$ in\
$M \cdot v^* \cdot S$
The pointwise functions applied are $\leq$, which returns 1 on $(x,y)$ if $x \leq y$ and $0$ otherwise; $=$, defined analogously; and the reciprocal function. In variable $V$ we compute a square matrix holding $n$ copies of $v$. Then in variable $C$ we compute the $n \times 1$ column vector where $C_i$ counts the number of $v_j$ such that $v_i \leq v_j$. If $C_i=n$ then $v_i$ equals the minimum. Variable $N$ computes the scalar $n$ and column vector $S$ is a selector where $S_i=1$ if $v_i$ equals the minimum, and $S_i=0$ otherwise. Since the minimum may appear multiple times in $v$, we compute in $M$ the inverse of the multiplicity. Finally we sum the different occurrences of the minimum in $v$ and divide by the multiplicity.
Types and schemas {#sectypes}
-----------------
We have already remarked that, due to conditions on the dimensions of matrices, ${\mathsf{MATLANG}}$ expressions are not well-defined on all instances. For example, if $I$ is an instance where $I(M)$ is a $3 \times 4$ matrix and $I(N)$ is a $2 \times 4$ matrix, then the expression $M \cdot N$ is not defined on $I$. The expression $M \cdot N^*$, however, is well-defined on $I$. We now introduce a notion of schema, which assigns types to matrix names, so that expressions can be type-checked against schemas.
Our types need to be able to guarantee equalities between numbers of rows or numbers of columns, so that $\operatorname{\mathsf{apply}}$ and matrix multiplication can be typechecked. Our types also need to be able to recognize vectors, so that $\operatorname{\mathsf{diag}}$ can be typechecked.
Formally, we assume a sufficient supply of *size symbols*, which we will denote by the letters $\alpha$, $\beta$, $\gamma$. A size symbol represents the number of rows or columns of a matrix. Together with an explicit 1, we can indicate arbitrary matrices as $\alpha \times \beta$, square matrices as $\alpha \times \alpha$, column vectors as $\alpha \times 1$, row vectors as $1 \times \alpha$, and scalars as $1 \times 1$. Formally, a *size term* is either a size symbol or an explicit 1. A *type* is then an expression of the form $s_1
\times s_2$ where $s_1$ and $s_2$ are size terms. Finally, a *schema* ${\mathcal{S}}$ is a function, defined on a nonempty finite set $\operatorname{var}({\mathcal{S}})$ of matrix variables, that assigns a type to each element of $\operatorname{var}({\mathcal{S}})$.
The typechecking of expressions is now shown in Figure \[fig:matlang-type-rules\]. The figure provides the rules that allow to infer an output type $\tau$ for an expression $e$ over a schema ${\mathcal{S}}$. To indicate that a type can be successfully inferred, we use the notation ${{\mathcal{S}}\vdashe:\tau}$. When we cannot infer a type, we say $e$ is not well-typed over ${\mathcal{S}}$. For example, when ${\mathcal{S}}(M)=\alpha \times \beta$ and ${\mathcal{S}}(N) = \gamma \times
\beta$, then the expression $M \cdot N$ is not well-typed over ${\mathcal{S}}$. The expression $M \cdot N^*$, however, is well-typed with output type $\alpha \times \gamma$.
To establish the soundness of the type system, we need a notion of conformance of an instance to a schema.
Formally, a *size assignment* $\sigma$ is a function from size symbols to positive natural numbers. We extend $\sigma$ to any size term by setting $\sigma(1) = 1$. Now, let ${\mathcal{S}}$ be a schema and $I$ an instance with $\operatorname{var}(I) =
\operatorname{var}({\mathcal{S}})$. We say that $I$ is an instance of ${\mathcal{S}}$ if there is a size assignment $\sigma$ such that for all ${M}\in \operatorname{var}({\mathcal{S}})$, if ${\mathcal{S}}({M}) = s_1 \times s_2$, then $I({M})$ is a $\sigma(s_1) \times \sigma(s_2)$ matrix. In that case we also say that $I$ *conforms* to ${\mathcal{S}}$ by the size assignment $\sigma$.
We now obtain the following obvious but desirable property.
\[prop:matlang-safety\] If ${\mathcal{S}}\vdash e : s_1 \times s_2$, then for every instance $I$ conforming to ${\mathcal{S}}$, by size assignment $\sigma$, the matrix $e(I)$ is well-defined and has dimensions $\sigma(s_1) \times \sigma(s_2)$.
Expressive power of ${\mathsf{MATLANG}}$ {#secsum}
========================================
It is natural to represent an $m \times n$ matrix $A$ by a ternary relation $${\mathit{Rel}}_2(A) := \{(i,j,A_{i,j}) \mid i \in \{1,\dots,m\}, \ j \in
\{1,\dots,n\}\}.$$ In the special case where $A$ is an $m \times 1$ matrix (column vector), $A$ can also be represented by a binary relation ${\mathit{Rel}}_1(A) :=
\{(i,A_{i,1}) \mid i \in \{1,\dots,m\}\}$. Similarly, a $1
\times n$ matrix (row vector) $A$ can be represented by ${\mathit{Rel}}_1(A)
:= \{(j,A_{1,j}) \mid j \in \{1,\dots,n\}\}$. Finally, a $1
\times 1$ matrix (scalar) $A$ can be represented by the unary singleton relation ${\mathit{Rel}}_0(A) := \{(A_{1,1})\}$.
Note that in ${\mathsf{MATLANG}}$, we perform calculations on matrix entries, but not on row or column indices. This fits well to the relational model with aggregates as formalized by Libkin [@libkin_sql]. In this model, the columns of relations are typed as “base”, indicated by ${\mathbf{b}}$, or “numerical”, indicated by ${\mathbf{n}}$. In the relational representations of matrices presented above, the last column is of type ${\mathbf{n}}$ and the other columns (if any) are of type ${\mathbf{b}}$. In particular, in our setting, numerical columns hold complex numbers.
Given this representation of matrices by relations, ${\mathsf{MATLANG}}$ can be simulated in the relational algebra with aggregates. Actually, the only aggregate operation we need is summation. We will not reproduce the formal definition of the relational algebra with summation [@libkin_sql], but note the following salient points:
- Expressions are built up from relation names using the classical operations union, set difference, cartesian product ($\times$), selection ($\sigma$), and projection ($\pi$), plus two new operations: *function application* and *summation*.
- For selection, we only use equality and nonequality comparisons on base columns. No selection on numerical columns will be needed in our setting.
- For any function $f:{\mathbf{C}}^n \to {\mathbf{C}}$, the operation $\operatorname{\mathsf{apply}}[f;i_1,\dots,i_n]$ can be applied to any relation $r$ having columns $i_1$, …, $i_n$, which must be numerical. The result is the relation $\{(t,f(t({i_1}),\dots,t({i_n}))) \mid t \in
r\}$, adding a numerical column to $r$. We allow $n=0$, in which case $f$ is a constant.
- The operation $\operatorname{\mathsf{sum}}[i;i_1,\dots,i_n]$ can be applied to any relation $r$ having columns $i$, $i_1$, …, $i_n$, where column $i$ must be numerical. In our setting we only need the operation in cases where columns $i_1$, …, $i_n$ are base columns. The result of the operation is the relation $$\{(t(i_1),\dots,t(i_n),\sum_{t' \in {\mathsf{group}}[i_1,\dots,i_n](r,t)}
t'(i)) \mid t \in r\},$$ where $${\mathsf{group}}[i_1,\dots,i_n](r,t) = \{t' \in r \mid t'(i_1)=t(i_1)
\land \cdots \land t'(i_n)=t(i_n)\}.$$ Again, $n$ can be zero, in which case the result is a singleton.
From ${\mathsf{MATLANG}}$ to relational algebra with summation
--------------------------------------------------------------
To state the translation formally, we assume a supply of *relation variables*, which, for convenience, we can take to be the same as the matrix variables. A *relation type* is a tuple of ${\mathbf{b}}$’s and ${\mathbf{n}}$’s. A *relational schema* ${\mathcal{S}}$ is a function, defined on a nonempty finite set $\operatorname{var}({\mathcal{S}})$ of relation variables, that assigns a relation type to each element of $\operatorname{var}({\mathcal{S}})$.
One can define well-typedness for expressions in the relation algebra with summation, and define the output type. We omit this definition here, as it follows a well-known methodology [@crash] and is analogous to what we have already done for ${\mathsf{MATLANG}}$ in Section \[sectypes\].
To define relational instances, we assume a countably infinite universe ${\mathbf{dom}}$ of abstract atomic data elements. It is convenient to assume that the natural numbers are contained in ${\mathbf{dom}}$. We stress that this assumption is not essential but simplifies the presentation. Alternatively, we would have to work with explicit embeddings from the natural numbers into ${\mathbf{dom}}$.
Let $\tau$ be a relation type. A *tuple of type $\tau$* is a tuple $(t(1),\dots,t(n))$ of the same arity as $\tau$, such that $t(i) \in {\mathbf{dom}}$ when $\tau(i) = {\mathbf{b}}$, and $t(i)$ is a complex number when $\tau(i) = {\mathbf{n}}$. A *relation of type $\tau$* is a finite set of tuples of type $\tau$. An *instance* of a relational schema ${\mathcal{S}}$ is a function $I$ defined on $\operatorname{var}({\mathcal{S}})$ so that $I(R)$ is a relation of type ${\mathcal{S}}(R)$ for every $R \in \operatorname{var}({\mathcal{S}})$.
We must connect the matrix data model to the relational data model. Let $\tau = s_1\times s_2$ be a matrix type. Let us call $\tau$ a *general type* if $s_1$ and $s_2$ are both size symbols; a *vector type* if $s_1$ is a size symbol and $s_2$ is 1, or vice versa; and the *scalar type* if $\tau$ is $1\times 1$. To every matrix type $\tau$ we associate a relation type $${\mathit{Rel}}(\tau) := \begin{cases}
({\mathbf{b}},{\mathbf{b}},{\mathbf{n}}) & \text{if $\tau$ is general;} \\
({\mathbf{b}},{\mathbf{n}}) & \text{if $\tau$ is a vector type;} \\
({\mathbf{n}}) & \text{if $\tau$ is scalar.} \end{cases}$$ Then to every matrix schema ${\mathcal{S}}$ we associate the relational schema ${\mathit{Rel}}({\mathcal{S}})$ where ${\mathit{Rel}}({\mathcal{S}})(M) = {\mathit{Rel}}({\mathcal{S}}(M))$ for every $M \in \operatorname{var}({\mathcal{S}})$. For each instance $I$ of ${\mathcal{S}}$, we define the instance ${\mathit{Rel}}(I)$ over ${\mathit{Rel}}({\mathcal{S}})$ by $${\mathit{Rel}}(I)(M) = \begin{cases}
{\mathit{Rel}}_2(I(M)) & \text{if ${\mathcal{S}}(M)$ is a general type;} \\
{\mathit{Rel}}_1(I(M)) & \text{if ${\mathcal{S}}(M)$ is a vector type;} \\
{\mathit{Rel}}_0(I(M)) & \text{if ${\mathcal{S}}(M)$ is the scalar type.}
\end{cases}$$ Here we use the relational representations ${\mathit{Rel}}_2$, ${\mathit{Rel}}_1$ and ${\mathit{Rel}}_0$ of matrices introduced in the beginning of Section \[secsum\].
\[sumtheorem\] Let ${\mathcal{S}}$ be a matrix schema, and let $e$ a ${\mathsf{MATLANG}}$ expression that is well-typed over ${\mathcal{S}}$ with output type $\tau$. Let $\ell=2$, $1$, or $0$, depending on whether $\tau$ is general, a vector type, or scalar, respectively.
1. There exists an expression ${\mathit{Rel}}(e)$ in the relational algebra with summation, that is well-typed over ${\mathit{Rel}}({\mathcal{S}})$ with output type ${\mathit{Rel}}(\tau)$, such that for every instance $I$ of ${\mathcal{S}}$, we have ${\mathit{Rel}}_\ell(e(I)) = {\mathit{Rel}}(e)({\mathit{Rel}}(I))$.
2. The expression ${\mathit{Rel}}(e)$ uses neither set difference, nor selection conditions on numerical columns.
3. The only functions used in ${\mathit{Rel}}(e)$ are those used in pointwise applications in $e$; complex conjugation; multiplication of two numbers; and the constant functions $0$ and $1$.
We only give a few representative examples.
- If $M$ is of type $\alpha \times \beta$ then ${\mathit{Rel}}(M^*)$ is $\operatorname{\mathsf{apply}}[\overline z;3] \, \pi_{2,1,3}(M)$, where $\overline z$ is the complex conjugate. If $M$ is of type $\alpha \times 1$, however, ${\mathit{Rel}}(M^*)$ is $\operatorname{\mathsf{apply}}[\overline z;2](M)$.
- If $M$ is of type $1 \times \alpha$ then ${\mathit{Rel}}({{\mathbf{1}}}(M))$ is $\pi_3(\operatorname{\mathsf{apply}}[1;2](M))$. Here, $1$ stands for the constant $1$ function.
- If $M$ is of type $\alpha \times 1$ then ${\mathit{Rel}}(\operatorname{\mathsf{diag}}(M))$ is $$\sigma_{\$1=\$2}(\pi_1(M) \times M) \cup
\operatorname{\mathsf{apply}}[0;\,] \, \sigma_{\$1\neq \$2}(\pi_1(M) \times \pi_1(M)).$$
- If $M$ is of type $\alpha \times \beta$ and $N$ is of type $\beta
\times \gamma$, then ${\mathit{Rel}}(M \cdot N)$ is $$\operatorname{\mathsf{sum}}[7;1,5] \,
\operatorname{\mathsf{apply}}[\times;3,6] \, \sigma_{\$2=\$4}(M \times N).$$ If, however, $M$ is of type $\alpha \times 1$ and $N$ is of type $1
\times 1$, then ${\mathit{Rel}}(M \cdot N)$ is $$\pi_{1,4} \, \operatorname{\mathsf{apply}}[\times;2,3](M
\times N).$$ We use pointwise multiplication.
- If $M$ and $N$ are of type $1 \times \beta$ then ${\mathit{Rel}}(\operatorname{\mathsf{apply}}[f](M,N))$ is $\pi_{1,5} \, \operatorname{\mathsf{apply}}[f;2,4] \,
\sigma_{\$1=\$3}(M \times N)$.
We may ignore the let-construct as it does not add expressive power.
The different treatment of general types, vector types, and scalar types is necessary because in our version of the relational algebra, selections can only compare base columns for equality; in particular we can not select for the value 1.
We can sharpen the above theorem a bit if we work in the relational calculus with aggregates. Every ${\mathsf{MATLANG}}$ expression can already be expressed by a formula in the relational calculus with summation that uses only three distinct base variables (variables ranging over values in base columns). The details are given in the Appendix.
Expressing graph queries
------------------------
So far we have looked at expressing matrix queries in terms of relational queries. It is also natural to express relational queries as matrix queries. This works best for binary relations, or graphs, which we can represent by their adjacency matrices.
Formally, define a *graph schema* to be a relational schema where every relation variable is assigned the type $({\mathbf{b}},{\mathbf{b}})$ of arity two. We define a *graph instance* as an instance $I$ of a graph schema, where the active domain of $I$ equals $\{1,\dots,n\}$ for some positive natural number $n$. The assumption that the active domain always equals an initial segment of the natural numbers is convenient for forming the bridge to matrices. This assumption, however, is not essential for our results to hold. Indeed, the logics we consider do not have any built-in predicates on base variables, besides equality. Hence, they view the active domain elements as abstract data values.
To every graph schema ${\mathcal{S}}$ we associate a matrix schema ${\mathit{Mat}}({\mathcal{S}})$, where ${\mathit{Mat}}({\mathcal{S}})(R) = \alpha \times \alpha$ for every $R \in \operatorname{var}({\mathcal{S}})$, for a fixed size symbol $\alpha$. So, all matrices are square matrices of the same dimension. Let $I$ be a graph instance of ${\mathcal{S}}$, with active domain $\{1,\dots,n\}$. We will denote the $n \times n$ adjacency matrix of a binary relation $r$ over $\{1,\dots,n\}$ by ${\mathit{Adj}}_I(r)$. Now any such instance $I$ is represented by the matrix instance ${\mathit{Mat}}(I)$ over ${\mathit{Mat}}({\mathcal{S}})$, where ${\mathit{Mat}}(I)(R) =
{\mathit{Adj}}_I(I(R))$ for every $R \in \operatorname{var}({\mathcal{S}})$.
A *graph query* over a graph schema ${\mathcal{S}}$ is a function that maps each graph instance $I$ of ${\mathcal{S}}$ to a binary relation on the active domain of $I$. We say that a ${\mathsf{MATLANG}}$ expression $e$ *expresses* the graph query $q$ if $e$ is well-typed over ${\mathit{Mat}}({\mathcal{S}})$ with output type $\alpha \times \alpha$, and for every graph instance $I$ of ${\mathcal{S}}$, we have ${\mathit{Adj}}_I(q(I)) =
e({\mathit{Mat}}(I))$.
We can now give a partial converse to Theorem \[sumtheorem\]. We assume active-domain semantics for first-order logic [@ahv_book]. Please note that the following result deals only with pure first-order logic, without aggregates or numerical columns. The proof, while instructive, has been relegated to the Appendix.
\[fo3\] Every graph query expressible in $\rm FO^3$ (first-order logic with equality, using at most three distinct variables) is expressible in ${\mathsf{MATLANG}}$. The only functions needed in pointwise applications are boolean functions on $\{0,1\}$, and testing if a number if positive.
We can complement the above theorem by showing that the quintessential first-order query requiring four variables is not expressible. The proof is given in the Appendix.
\[4clique\] The graph query over a single binary relation $R$ that maps $I$ to $I(R)$ if $I(R)$ contains a four-clique, and to the empty relation otherwise, is not expressible in ${\mathsf{MATLANG}}$.
Matrix inversion {#secinv}
================
Matrix inversion (solving nonsingular systems of linear equations) is an ubiquitous operation in data analysis. We can extend ${\mathsf{MATLANG}}$ with matrix inversion as follows. Let ${\mathcal{S}}$ be a schema and $e$ be an expression that is well-typed over ${\mathcal{S}}$, with output type of the form $\alpha \times \alpha$. Then the expression $e^{-1}$ is also well-typed over ${\mathcal{S}}$, with the same output type $\alpha
\times \alpha$. The semantics is defined as follows. For an instance $I$, if $e(I)$ is an invertible matrix, then $e^{-1}(I)$ is defined to be the inverse of $e(I)$; otherwise, it is defined to be the zero square matrix of the same dimensions as $e(I)$. The extension of ${\mathsf{MATLANG}}$ with inversion is denoted by ${\mathsf{MATLANG}}+ {\mathsf{inv}}$.
Recall Example \[ex:google-matrix\] where we computed the Google matrix of $A$. In the process we already showed how to compute the $n \times n$ matrix $B$ defined by $B_{i,j} =
A_{i,j}/k_i$, and the scalar $N$ holding the value $n$. So, in the following expression, we assume we already have $B$ and $N$. Let $I$ be the $n \times n$ identity matrix, and let ${{\mathbf{1}}}$ denote the $n \times 1$ column vector consisting of all ones. The PageRank vector $v$ of $A$ can be computed as follows [@delcorso]: $$v = \frac{1-d}n(I - dB)^{-1} {{\mathbf{1}}}.$$ This calculation is readily expressed in ${\mathsf{MATLANG}}+ {\mathsf{inv}}$ as $$(1-d) \odot \operatorname{\mathsf{apply}}[1/x](N) \odot \operatorname{\mathsf{apply}}[-](\operatorname{\mathsf{diag}}({{\mathbf{1}}}(A)),d
\odot B)^{-1} \cdot {{\mathbf{1}}}(A).$$
\[tc\] We next show that the reflexive-transitive closure of a binary relation is expressible in ${\mathsf{MATLANG}}+
{\mathsf{inv}}$. Let $A$ be the adjacency matrix of a binary relation $r$ on $\{1,\dots,n\}$. Let $I$ be the $n \times n$ identity matrix, expressible as $\operatorname{\mathsf{diag}}({{\mathbf{1}}}(A))$. From earlier examples we know how to compute the scalar $1 \times
1$ matrix $N$ holding the value $n$. The matrix $B = \frac 1{n+1} A$ has 1-norm strictly less than 1, so $S = \sum_{k=0}^\infty B^k$ converges, and is equal to $(I - B)^{-1}$ [@golub Lemma 2.3.3]. Now $(i,j)$ belongs to the reflexive-transitive closure of $r$ if and only if $S_{i,j}$ is nonzero. Thus, we can express the reflexive-transitive closure of $r$ as $$\operatorname{\mathsf{apply}}[\neq 0] \bigl (
\operatorname{\mathsf{apply}}[-](\operatorname{\mathsf{diag}}({{\mathbf{1}}}(A)) ,
\operatorname{\mathsf{apply}}[1/(x+1)](N) \odot A)^{-1} \bigr ),$$ where $x \neq 0$ is $1$ if $x \neq 0$ and $0$ otherwise. We can obtain the transitive closure by multiplying the above expression with $A$.
By Theorem \[sumtheorem\], any graph query expressible in ${\mathsf{MATLANG}}$ is expressible in the relational algebra with aggregates. It is known [@hlnw_aggregate; @libkin_sql] that such queries are local. The transitive-closure query from Example \[tc\], however, is not local. We thus conclude:
${\mathsf{MATLANG}}+ {\mathsf{inv}}$ is strictly more powerful than ${\mathsf{MATLANG}}$ in expressing graph queries.
Once we have the transitive closure, we can do many other things such as checking bipartiteness of undirected graphs, checking connectivity, checking cyclicity. ${\mathsf{MATLANG}}$ is expressive enough to reduce these queries to the transitive-closure query, as shown in the following example for bipartiteness. The same approach via $\rm FO^3$ can be used for connectedness or cyclicity.
To check bipartiteness of an undirected graph, given as a symmetric binary relation $R$ without self-loops, we first compute the transitive closure $T$ of the composition of $R$ with itself. Then the $\rm FO^3$ condition $\neg \exists x \exists y (R(x,y)
\land T(y,x))$ expresses that $R$ is bipartite (no odd cycles). The result now follows from Theorem \[fo3\].
Using transitive closure we can also easily compute the number of connected components of a binary relation $R$ on $\{1,\dots,n\}$, given as an adjacency matrix. We start from the union of $R$ and its converse. This union, denoted by $S$, is expressible by Theorem \[fo3\]. We then compute the reflexive-transitive closure $C$ of $S$. Now the number of connected components of $R$ equals $\sum_{i=1}^n 1/k_i$, where $k_i$ is the degree of node $i$ in $C$. This sum is simply expressible as ${{\mathbf{1}}}(C)^*
\cdot \operatorname{\mathsf{apply}}[1/x](C \cdot {{\mathbf{1}}}(C))$.
Eigenvalues {#seceigen}
===========
Another workhorse in data analysis is diagonalizing a matrix, i.e., finding a basis of eigenvectors. Formally, we define the operation ${\mathsf{eigen}}$ as follows. Let $A$ be an $n \times n$ matrix. Recall that $A$ is called diagonalizable if there exists a basis of ${\mathbf{C}}^n$ consisting of eigenvectors of $A$. In that case, there also exists such a basis where eigenvectors corresponding to a same eigenvalue are orthogonal. Accordingly, we define ${\mathsf{eigen}}(A)$ to return an $n \times n$ matrix, the columns of which form a basis of ${\mathbf{C}}^n$ consisting of eigenvectors of $A$, where eigenvectors corresponding to a same eigenvalue are orthogonal. If $A$ is not diagonalizable, we define ${\mathsf{eigen}}(A)$ to be the $n \times n$ zero matrix.
Note that ${\mathsf{eigen}}$ is nondeterministic; in principle there are infinitely many possible results. This models the situation in practice where numerical packages such as R or return approximations to the eigenvalues and a set of corresponding eigenvectors, but the latter are not unique. Hence, some care must be taken in extending ${\mathsf{MATLANG}}$ with the ${\mathsf{eigen}}$ operator. Syntactically, as for inversion, whenever $e$ is a well-typed expression with a square output type, we now also allow the expression ${\mathsf{eigen}}(e)$, with the same output type. Semantically, however, the rules of Figure \[fig:bigstep\] must be adapted so that they do not infer statements of the form $e(I)=B$, but rather of the form $B \in e(I)$, i.e., $B$ is a possible result of $e(I)$. The let-construct now becomes crucial; it allows us to assign a possible result of ${\mathsf{eigen}}$ to a new variable, and work with that intermediate result consistently.
In this and the next section, we assume notions from linear algebra. An excellent introduction to the subject has been given by Axler [@ladoneright].
We can easily recover the eigenvalues from the eigenvectors, using inversion. Indeed, if $A$ is diagonalizable and $B \in
{\mathsf{eigen}}(A)$, then $\Lambda=B^{-1} A B$ is a diagonal matrix with all eigenvalues of $A$ on the diagonal, so that the $i$th eigenvector in $B$ corresponds to the eigenvalue in the $i$th column of $\Lambda$. This is the well-known eigendecomposition. However, the same can also be accomplished without using inversion. Indeed, suppose $B = (v_1,\dots,v_n)$, and let $\lambda_i$ be the eigenvalue to which $v_i$ corresponds. Then $A B =
(\lambda_1v_1,\dots,\lambda_nv_n)$. Each eigenvector is nonzero, so we can divide away the entries from $B$ in $A B$ (setting division by zero to zero). We thus obtain a matrix where the $i$th column consists of zeros or $\lambda_i$, with at least one occurrence of $\lambda_i$. By counting multiplicities, dividing them out, and finally summing, we obtain $\lambda_1$, …, $\lambda_n$ in a column vector. We can apply a final $\operatorname{\mathsf{diag}}$ to get it back into diagonal form. The ${\mathsf{MATLANG}}$ expression for doing all this uses similar tricks as those shown in Examples \[ex:google-matrix\] and \[exmin\].
The above remark suggests a shorthand in ${\mathsf{MATLANG}}+ {\mathsf{eigen}}$ where we return both $B$ and $\Lambda$ together: $$\mathsf{let} \
(B,\Lambda) = {\mathsf{eigen}}(A) \ \mathsf{in} \ \dots$$ This models how the ${\mathsf{eigen}}$ operation works in the languages R and . We agree that $\Lambda$, like $B$, is the zero matrix if $A$ is not diagonalizable.
\[exrank\] Since the rank of a diagonalizable matrix equals the number of nonzero entries in its diagonal form, we can express the rank of a diagonalizable matrix $A$ as follows: $$\mathsf{let} \ (B,\Lambda) ={\mathsf{eigen}}(A) \ \mathsf{in} \
{{\mathbf{1}}}(A)^* \cdot \operatorname{\mathsf{apply}}[\neq 0](\Lambda) \cdot {{\mathbf{1}}}(A).$$
A well-known heuristic for partitioning an undirected graph without self-loops is based on an eigenvector corresponding to the second-smallest eigenvalue of the Laplacian matrix [@ullman_mining]. The Laplacian $L$ can be derived from the adjacency matrix $A$ as . (Here $D$ is the degree matrix.) Now let $(B,\Lambda) \in
{\mathsf{eigen}}(L)$. In an analogous way to Example \[exmin\], we can compute a matrix $E$, obtained from $\Lambda$ by replacing the occurrences of the second-smallest eigenvalue by 1 and all other entries by 0. Then the eigenvectors corresponding to this eigenvalue can be isolated from $B$ (and the other eigenvectors zeroed out) by multiplying $B \cdot E$.
It turns out that ${\mathsf{MATLANG}}+ {\mathsf{inv}}$ is subsumed by ${\mathsf{MATLANG}}+ {\mathsf{eigen}}$. The proof is in the Appendix.
\[inv2eigen\] Matrix inversion is expressible in ${\mathsf{MATLANG}}+ {\mathsf{eigen}}$.
A natural question to ask is if ${\mathsf{MATLANG}}$ with ${\mathsf{eigen}}$ is strictly more expressive than ${\mathsf{MATLANG}}$ with ${\mathsf{inv}}$. In a noninteresting sense, the answer is affirmative. Indeed, when evaluating a ${\mathsf{MATLANG}}+ {\mathsf{inv}}$ expression on an instance where all matrix entries are rational numbers, the result matrix is also rational. In contrast, the eigenvalues of a rational matrix may be complex numbers. The more interesting question, however, is: *Are there graph queries expressible deterministically in ${\mathsf{MATLANG}}+ {\mathsf{eigen}}$, but not in ${\mathsf{MATLANG}}+ {\mathsf{inv}}$?* This is an interesting question for further research. The answer may depend on the functions that can be used in pointwise applications.
The stipulation *deterministically* in the above open question is important. Ideally, we use the nondeterministic ${\mathsf{eigen}}$ operation only as an intermediate construct. It is an aid to achieve a powerful computation, but the final expression should have only a single possible output on every input. The expression of Example \[exrank\] is deterministic in this sense, as is the expression for inversion described in the proof of Theorem \[inv2eigen\].
The evaluation problem {#seceval}
======================
The evaluation problem asks, given an input instance $I$ and an expression $e$, to compute the result $e(I)$. There are some issues with this naive formulation, however. Indeed, in our theory we have been working with arbitrary complex numbers. How do we even represent the input? For practical applications, it is usually sufficient to support matrices with rational numbers only. For ${\mathsf{MATLANG}}+ {\mathsf{inv}}$, this approach works: when the input is rational, the output is rational too, and can be computed in polynomial time. For the basic matrix operations this is clear, and for matrix inversion we can use the well known method of Gaussian elimination.
When adding the ${\mathsf{eigen}}$ operation, however, the output may become irrational. Much worse, the eigenvalues of an adjacency matrix (even of a tree) need not even be definable in radicals [@godsil_radicals]. Practical systems, of course, apply techniques from numerical mathematics to compute rational approximations. But it is still theoretically interesting to consider the exact evaluation problem.
Our approach is to represent the output symbolically, following the idea of constraint query languages [@kkr_cql; @cdbbook]. Specifically, we can define the input-output relation of an expression, for given dimensions of the input matrices, by an existential first-order logic formula over the reals. Such formulas are built from real variables, integer constants, addition, multiplication, equality, inequality ($<$), disjunction, conjunction, and existential quantification.
\[exeigenformula\] Consider the expression ${\mathsf{eigen}}(M)$ over the schema consisting of a single matrix variable $M$. Any instance $I$ where $I(M)$ is an $n \times n$ matrix $A$ can be represented by a tuple of $2 \times n \times n$ real numbers. Indeed, let $a_{i,j} = \Re A_{i,j}$ (the real part of a complex number), and let $b_{i,j} = \Im A_{i,j}$ (the imaginary part). Then $I(M)$ can be represented by the tuple $(a_{1,1},b_{1,1},a_{1,2},b_{1,2},\dots,a_{n,n},b_{n,n})$. Similarly, any $B \in {\mathsf{eigen}}(A)$ can be represented by a similar tuple. We introduce the variables $x_{M,i,j,\Re}$, $x_{M,i,j,\Im}$, $y_{i,j,\Re}$, and $y_{i,j,\Im}$, for $i,j \in \{1,\dots,n\}$, where the $x$-variables describe an arbitrary input matrix and the $y$-variables describe an arbitrary possible output matrix. Denoting the input matrix by $[\bar x]$ and the output matrix by $[\bar y]$, we can now write an existential formula expressing that $[\bar y]$ is a possible result of ${\mathsf{eigen}}$ applied to $[\bar x]$:
- To express that $[\bar y]$ is a basis, we write that there exists a nonzero matrix $[\bar z]$ such that $[\bar y] \cdot [\bar z]$ is the identity matrix. It is straightforward to express this condition by a formula.
- To express, for each column vector $v$ of $[\bar y]$, that $v$ is an eigenvector of $[\bar x]$, we write that there exists $\lambda$ such that $[\bar x] \cdot v = \lambda[\bar x]$.
- The final and most difficult condition to express is that distinct eigenvectors $v$ and $w$ that correspond to a same eigenvalue are orthogonal. We cannot write $\exists \lambda ([\bar x]\cdot v =
\lambda v \land [\bar x]\cdot w = \lambda w) \to v^* \cdot w =
0$, as this is not a proper existential formula. (Note though that the conjugate transpose of $v$ is readily expressed.) Instead, we avoid an explicit quantifier and rewrite the antecedent as the conjunction, over all positions $i$, of $v_i
\neq 0 \neq w_i \to
([\bar x] \cdot v)_i / v_i =
([\bar x] \cdot w)_i / w_i$.
- A final detail is that we should also be able to express that $[\bar x]$ is not diagonalizable, for in that case we need to define $[\bar y]$ to be the zero matrix. Nondiagonalizability is equivalent to the existence of a Jordan form with at least one 1 on the superdiagonal. We can express this as follows. We postulate the existence of an invertible matrix $[\bar z]$ such that the product $[\bar z] \cdot [\bar x] \cdot [\bar z]^{-1}$ has all entries zero, except those on the diagonal and the superdiagonal. The entries on the superdiagonal can only by 0 or 1, with at least one 1. Moreover, if an entry $i,j$ on the superdiagonal is nonzero, the entries $i,i$ and $j,j$ must be equal.
The approach taken in the above example leads to the following general result. The operations of ${\mathsf{MATLANG}}$ are handled using similar ideas as illustrated above for the ${\mathsf{eigen}}$ operation, and are actually easier. The let-construct, and the composition of subexpressions into larger expression, are handled by existential quantification.
\[existsrtheorem\] An input-sized expression consists of a schema ${\mathcal{S}}$, an expression $e$ in ${\mathsf{MATLANG}}+ {\mathsf{eigen}}$ that is well-typed over ${\mathcal{S}}$ with output type $t_1 \times t_2$, and a size assignment $\sigma$ defined on the size symbols occurring in ${\mathcal{S}}$. There exists a polynomial-time computable translation that maps any input-sized expression as above to an existential first-order formula $\psi$ over the vocabulary of the reals, expanded with symbols for the functions used in pointwise applications in $e$, such that
1. Formula $\psi$ has the following free variables:
- For every $M \in \operatorname{var}({\mathcal{S}})$, let ${\mathcal{S}}(M)=s_1\times s_2$. Then $\psi$ has the free variables $x_{M,i,j,\Re}$ and $x_{M,i,j,\Im}$, for $i=1,\dots,\sigma(s_1)$ and $j=1,\dots,\sigma(s_2)$.
- In addition, $\psi$ has the free variables $y_{i,j,\Re}$ and $y_{i,j,\Im}$, for $i=1,\dots,\sigma(t_1)$ and $j=1,\dots,\sigma(t_2)$.
The set of these free variables is denoted by ${\mathrm{FV}}({\mathcal{S}},e,\sigma)$.
2. Any assignment $\rho$ of real numbers to these variables specifies, through the $x$-variables, an instance $I$ conforming to ${\mathcal{S}}$ by $\sigma$, and through the $y$-variables, a $\sigma(t_1)\times \sigma(t_2)$ matrix $B$.
3. Formula $\psi$ is true over the reals under such an assignment $\rho$, if and only if $B \in e(I)$.
The existential theory of the reals is decidable; actually, the full first-order theory of the reals is decidable [@arnon; @basu_algorithms]. But, specifically the class of problems that can be reduced in polynomial time to the existential theory of the reals forms a complexity class on its own, known as $\exists \mathbf R$ [@schaefer_existsR; @schaefer_nash]. The above theorem implies that the *partial evaluation problem for ${\mathsf{MATLANG}}+ {\mathsf{eigen}}$* belongs to this complexity class. We define this problem as follows. The idea is that an arbitrary specification, expressed as an existential formula $\chi$ over the reals, can be imposed on the input-output relation of an input-sized expression.
The *partial evaluation problem* is a decision problem that takes as input:
- an input-sized expression $({\mathcal{S}},e,\sigma)$, where all functions used in pointwise applications are explicitly defined using existential formulas over the reals;
- an existential formula $\chi$ with free variables in ${\mathrm{FV}}({\mathcal{S}},e,\sigma)$ (see Theorem \[existsrtheorem\]).
The problem asks if there exists an instance $I$ conforming to ${\mathcal{S}}$ by $\sigma$ and a matrix $B \in e(I)$ such that $(I,B)$ satisfies $\chi$.
For example, $\chi$ may completely specify the matrices in $I$ by giving the values of the entries as rational numbers, and may express that the output matrix has at least one nonzero entry.
An input $({\mathcal{S}},e,\sigma,\chi)$ is a yes-instance to the partial evaluation problem precisely when the existential sentence $\exists {\mathrm{FV}}({\mathcal{S}},e,\sigma) (\psi \land \chi)$ is true in the reals, where $\psi$ is the formula obtained by Theorem \[existsrtheorem\]. Hence we can conclude:
\[upperbound\] The partial evaluation problem for ${\mathsf{MATLANG}}+ {\mathsf{eigen}}$ belongs to $\exists \mathbf R$.
Since the full theory of the reals is decidable, our theorem implies many other decidability results. We give just two examples.
The equivalence problem for input-sized expressions is decidable. This problem takes as input two input-sized expressions $({\mathcal{S}},e_1,\sigma)$ and $({\mathcal{S}},e_2,\sigma)$ (with the same ${\mathcal{S}}$ and $\sigma$) and asks if for all instances $I$ conforming to ${\mathcal{S}}$ by $\sigma$, we have $B \in e_1(I) \; \Leftrightarrow \; B
\in e_2(I)$.
Note that the equivalence problem for ${\mathsf{MATLANG}}$ expressions on arbitrary instances (size not fixed) is undecidable by Theorem \[fo3\], since equivalence of $\rm FO^3$ formulas over binary relational vocabularies is undecidable [@gor_un2].
The determinacy problem for input-sized expressions is decidable. This problem takes as input an input-sized expression $({\mathcal{S}},e,\sigma)$ and asks if for every instance $I$ conforming to ${\mathcal{S}}$ by $\sigma$, there exists at most one $B \in e(I)$.
Corollary \[upperbound\] gives an $\exists \mathbf R$ upper bound on the combined complexity of query evaluation [@vardi_comp]. Our final result is a matching lower bound, already for data complexity alone. The proof is in the Appendix.
\[hardness\] There exists a fixed schema ${\mathcal{S}}$ and a fixed expression $e$ in ${\mathsf{MATLANG}}+ {\mathsf{eigen}}$, well-typed over ${\mathcal{S}}$, such that the following problem is hard for $\exists \mathbf R$: Given an integer instance $I$ over ${\mathcal{S}}$, decide whether the zero matrix is a possible result of $e(I)$. The pointwise applications in $e$ use only simple functions definable by quantifier-free formulas over the reals.
Our proof of Theorem \[hardness\] relies on the nondeterminism of the ${\mathsf{eigen}}$ operation. Coming back to our remark on determinacy at the end of the previous section, it is an interesting question for further research to understand not only the expressive power but also the complexity of the evaluation problem for *deterministic* ${\mathsf{MATLANG}}+ {\mathsf{eigen}}$ expressions.
Conclusion
==========
There is a commendable trend in contemporary database research to leverage, and considerably extend, techniques from database query processing and optimization, to support large-scale linear algebra computations. In principle, data scientists could then work directly in SQL or related languages. Still, some users will prefer to continue using the matrix sublanguages they are more familiar with. Supporting these languages is also important so that existing code need not be rewritten.
From the perspective of database theory, it then becomes relevant to understand the expressive power of these languages as well as possible. In this paper we have proposed a framework for viewing matrix manipulation from the point of view of expressive power of database query languages. Moreover, our results formally confirm that the basic set of matrix operations offered by systems in practice, formalized here in the language ${\mathsf{MATLANG}}+ {\mathsf{inv}}+ {\mathsf{eigen}}$, really is adequate for expressing a range of linear algebra techniques and procedures.
In the paper we have already mentioned some intriguing questions for further research. Deep inexpressibility results have been developed for logics with rank operators [@pakusa_phd]. Although these results are mainly concerned with finite fields, they might still provide valuable insight in our open questions. Also, we have not covered all standard constructs from linear algebra. For instance, it may be worthwhile to extend our framework with the operation of putting matrices in upper triangular form, with the Gram-Schmidt procedure (which is now partly hidden in the ${\mathsf{eigen}}$ operation), and with the singular value decomposition.
Finally, we note that various authors have proposed to go beyond matrices, introducing data models and algebra for tensors or multidimensional arrays [@rusu_survey; @kim_tensordb; @sato_tensors]. When moving to more and more powerful and complicated languages, however, it becomes less clear at what point we should simply move all the way to full SQL, or extensions of SQL with recursion.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank Bart Kuijpers for telling us about the complexity class $\exists \mathbf R$. We thank Lauri Hella and Wied Pakusa for helpful discussions, and Christoph Berkholz and Anuj Dawar for their help with the proof of Proposition \[4clique\].
Appendix {#appendix .unnumbered}
========
It is known [@tarskigivant; @marxvenema_multi] that $\rm FO^3$ graph queries can be expressed in the algebra of binary relations with the operations ${\mathit{all}}$, identity, union, set difference, converse, and relational composition. These operations are well known, except perhaps for ${\mathit{all}}$, which, on a graph instance $I$, evaluates to the cartesian product of the active domain of $I$ with itself. Identity evaluates to the identity relation on the active domain of $I$. Each of these operations is easy to express in ${\mathsf{MATLANG}}$. For ${\mathit{all}}$ we use ${{\mathbf{1}}}(R) \cdot {{\mathbf{1}}}(R)^*$, where for $R$ we can take any relation variable from the schema. Identity is expressed as $\operatorname{\mathsf{diag}}({{\mathbf{1}}}(R))$. Union $r \cup s$ is expressed as $\operatorname{\mathsf{apply}}[x \lor y](r,s)$, and set difference $r - s$ as $\operatorname{\mathsf{apply}}[x
\land \neg y](r,s)$. Converse is transpose. Relational composition $r \circ s$ is expressed as $\operatorname{\mathsf{apply}}[x>0](r \cdot s)$, where ${x>0} = 1$ if $x$ is positive and $0$ otherwise.
#### The relational calculus with aggregates. {#the-relational-calculus-with-aggregates. .unnumbered}
In this logic, we have base variables and numerical variables. Base variables can be bound to base columns of relations, and compared for equality. Numerical variables can be bound to numerical columns, and can be equated to function applications and aggregations. We will not recall the syntax formally [@libkin_sql]. The advantage of the relational calculus is that variables, especially base variables, can be repeated and reused. For example, matrix multiplication $M \cdot N$ with $M$ of type $\alpha \times
\beta$ and $N$ of type $\beta \times \gamma$ can be expressed by the formula $$\varphi(i,j,z) \equiv z = \operatorname{\mathsf{sum}}k,x,y . (M(i,k,x)
\land N(k,j,y), x \times y).$$ Here, $i$, $j$ and $k$ are base variables and $x$, $y$ and $z$ are numerical variables. Only two base variables, $i$ and $j$, are free; in the subformula $M(i,k,x)$ only $i$ and $k$ are free, and in $N(k,j,y)$ only $k$ and $j$ are free. So, if $M$ or $N$ had been a subexpression involving matrix multiplication in turn, we could have reused one of the three variables. The other operations of ${\mathsf{MATLANG}}$ need only two base variables. We conclude:
\[calc3\] Let ${\mathcal{S}}$, $e$, $\tau$ and $\ell$ as in Theorem \[sumtheorem\]. For every ${\mathsf{MATLANG}}$ expression $e$ there is a formula $\varphi$ over ${\mathit{Rel}}({\mathcal{S}})$ in the relational calculus with summation, such that
1. If $\tau$ is general, $\varphi(i,j,z)$ has two free base variables $i$ and $j$ and one free numerical variable $z$; if $\tau$ is a vector type, we have $\varphi(i,z)$; and if $\tau$ is scalar, we have $\varphi(z)$.
2. For every instance $I$, the relation defined by $\varphi$ on ${\mathit{Rel}}(I)$ equals ${\mathit{Rel}}_\ell(e(I))$.
3. The formula $\varphi$ uses only three distinct base variables.
To prove Proposition \[4clique\] we state a lemma, which refines Proposition \[calc3\] in the setting of graph queries.
\[lemmapsi\] If a graph query $q$ is expressible in ${\mathsf{MATLANG}}$, then $q$ is expressible by a formula $\psi(i,j)$ in the relational calculus with summation, where $i$ and $j$ are base variables, and $\psi$ uses at most three distinct base variables.
Let $e$ be a ${\mathsf{MATLANG}}$ expression that expresses $q$. Let $\varphi(i,j,z)$ be the formula given by Proposition \[calc3\]. Let $\varphi'(i,j,z)$ be the formula obtained from $\varphi$ as follows. We replace each atomic formula of the form $R(i',j',x)$, where $i'$ and $j'$ are base variables and $x$ is a numerical variable, by $((x=1 \land R(i',j')) \lor (x=0 \land \neg R(i',j'))$. Now $\psi$ can be obtained as $\exists z(z=1 \land \varphi')$.
We can now give the
Let $e$ be a ${\mathsf{MATLANG}}$ expression expressing some graph query $q$. Let $\psi$ be the formula given by Lemma \[lemmapsi\]. It is known [@hlnw_aggregate; @libkin_sql] that every formula in the relational calculus with aggregates can be equivalently expressed by a formula in infinitary logic with counting, where the only variables in the latter formula are the base variables in the original formula. Hence, $q$ is expressible in $C^3$, infinitary counting logic with three distinct variables.
The four-clique query, however, is not expressible in $C^3$. In proof, consider the four-clique graph $G$, to which we apply the Cai-Fürer-Immerman construction [@cfi; @otto_bounded], yielding graphs $G^0$ and $G^1$ which are indistinguishable in $C^3$. This construction is such that $G^0$ contains a “four-clique formed by paths of length three”: four nodes such that there is a path of length three between any two of them. The graph $G^1$, however, does not contain four such nodes.
Now suppose, for the sake of contradiction, that there would be a sentence $\varphi$ in $C^3$ expressing the existence of a four-clique. We can replace each atomic formula $R(x,y)$ by $\exists z(R(x,z) \land \exists
x(R(z,x) \land R(x,y)))$. The resulting $C^3$ sentence looks for a four-clique formed by paths of length three, and would distinguish $G^0$ from $G^1$, which yields our contradiction.
We describe a fixed procedure for determining $A^{-1}$, for any square matrix $A$. Let $S = A^*A$. Then $A$ is invertible if and only if $S$ is. Let us assume first that $S$ is indeed invertible.
Since $S$ is self-adjoint, ${\mathbf{C}}^n$ has an orthogonal basis consisting of eigenvectors of $S$. Eigenvectors of a self-adjoint operator that correspond to distinct eigenvalues are always orthogonal. Hence, ${\mathsf{eigen}}(S)$ always returns an orthogonal basis of ${\mathbf{C}}^n$ consisting of eigenvectors of $S$. Let $(B,\Lambda) \in {\mathsf{eigen}}(S)$ (using the shorthand introduced before Example \[exrank\]). We can normalize the columns of $B$ in ${\mathsf{MATLANG}}$ as $$\operatorname{\mathsf{apply}}[x/\sqrt y](B,{{\mathbf{1}}}(B) \cdot (B^*\cdot B
\cdot {{\mathbf{1}}}(B))^*).$$ (This expression works because the columns in $B$ are mutually orthogonal.) So, we may now assume that $B$ contains an orthonormal basis consisting of eigenvectors of $S$. In particular, $B^{-1}=B^*$, and $S = B \Lambda B^*$.
Since we have assumed $S$ to be invertible, none of the eigenvalues is zero. We can invert $\Lambda$ simply by replacing each entry on the diagonal by its reciprocal. Thus, $\Lambda^{-1}$ can be computed from $\Lambda$ by pointwise application.
Now $A^{-1}$ can be computed by the expression $C=B \Lambda^{-1} B^* A^*$. To see that $C$ indeed equals $A^{-1}$, we calculate $CA =
B \Lambda^{-1} B^* A^* A = B \Lambda^{-1} B^* S = B \Lambda^{-1}
B^* B \Lambda B^*$ which simplifies to the identity matrix.
When $S$ is not invertible, we should return the zero matrix. In ${\mathsf{MATLANG}}$ we can compute the matrix $Z$ that is zero if one of the eigenvalues is zero, and the identity matrix otherwise. We then multiply the final expression with $Z$. A final detail is to make the computation well-defined in all cases. Thereto, in the pointwise applications of $x/\sqrt y$ and the reciprocal, we extend these functions arbitrarily to total functions.
The feasibility problem [@schaefer_nash] takes as input an equation $p=0$, with $p$ a multivariate polynomial with integer coefficients, and asks whether the equation has a solution over the reals. We may assume that $p$ is given in “standard form”, as a sum of terms of the form $a\mu$ where $a$ is an integer and $\mu$ is a monomial [@matousek_existsr]. The feasibility problem is known to be complete for $\exists \mathbf R$. We will design a schema ${\mathcal{S}}$ and an expression $e$ so that the feasibility problem reduces in polynomial time to our problem.
We use a construction by Valiant [@valiant_algebra]. This construction converts any $p$ as above, in polynomial time, to a directed, edge-weighted graph $G$. The fundamental property of Valiant’s construction is that the determinant of the adjacency matrix $A$ of $G$ equals $p$. The edge weights in $G$ are coefficients or variables from $p$, or the value 1. The entries in $A$ are zero or edge weights from $G$. We now observe that the construction has a specific property: when $p$ is given in standard form, with an explicit coefficient before each monomial (even if it is merely the value 1), each row of $A$ contains at most one variable. This property is important for the expression $e$, specified below, to work.
Assume $G$ has nodes $1$, …, $n$, and let the variables in $p$ be $x_1$, …, $x_k$. We represent $A$ by three integer matrices ${\mathit{Coef}}$, ${\mathit{Vars}}$, and ${\mathit{Enc}}$. Matrix ${\mathit{Coef}}$ is the $n \times n$ matrix obtained from $A$ by omitting the variable entries (these are set to zero). On the other hand, ${\mathit{Vars}}$, also $n \times n$, is obtained from $A$ by keeping only the variable entries, but setting them to 1. All other entries are set to zero. Finally, ${\mathit{Enc}}$ encodes which variables are represented by the one-entries in ${\mathit{Vars}}$. Specifically, ${\mathit{Enc}}$ is the $n \times k$ matrix where $E_{i,j}=1$ if the $i$th row of $A$ contains variable $x_j$, and zero otherwise.
We thus reduce an input $p=0$ of the feasibility problem to the instance $I$ consisting of the matrices ${\mathit{Coef}}$, ${\mathit{Vars}}$, ${\mathit{Enc}}$. Additionally, for technical reasons, $I$ also has the $k \times
1$ column vector $F$, which has value 1 in its first entry and is zero everywhere else. Formally, this instance is over the fixed schema ${\mathcal{S}}$ consisting of the matrix variables $M_{{\mathit{Coef}}}$, $M_{{\mathit{Vars}}}$, $M_{{\mathit{Enc}}}$, and $M_F$, where the first two variables have type $\alpha \times \alpha$; the third variable has type $\alpha \times \beta$; and $M_F$ has type $\beta \times 1$. To reduce clutter, however, in what follows we will write these variables simply as ${\mathit{Coef}}$, ${\mathit{Vars}}$, ${\mathit{Enc}}$ and $F$.
We must now give a expression $e$ that has the zero matrix as possible result of $e(I)$ if and only if $p=0$ has a solution over the reals. For any $k \times 1$ vector $v$ of real numbers, let $A^{(v)}$ denote the matrix $A$ where we have substituted the entries of $v$ for the variables $x_1$, …, $x_k$. By Valiant’s construction, the expression $e$ should return the zero matrix as a possible result, if and only if there exists $v$ such that $A^{(v)}$ has determinant zero, i.e., is not invertible.
The desired expression $e$ works as follows. By applying ${\mathsf{eigen}}$ to the $k\times k$ zero matrix $O$, and selecting the first column, we can nondeterministically obtain all possible nonzero $k \times 1$ column vectors. Taking only the real part ($\Re$) of the entries, we obtain all possible real column vectors $v$. Then the matrix $A^{(v)}$ is assembled (in matrix variable $AA$) using the matrices ${\mathit{Coef}}$, ${\mathit{Vars}}$ and ${\mathit{Enc}}$. Finally, we apply ${\mathsf{inv}}$ to $AA$ so that the zero matrix is returned if and only if $AA$ has determinant zero.
In conclusion, expression $e$ reads as follows:
let $O = \operatorname{\mathsf{apply}}[0](F \cdot F^*)$ in\
let $B = {\mathsf{eigen}}(O)$ in\
let $v = \operatorname{\mathsf{apply}}[\Re](B \cdot F)$ in\
let $AA = \operatorname{\mathsf{apply}}[+]({\mathit{Coef}},\operatorname{\mathsf{apply}}[g]({\mathit{Vars}},{\mathit{Enc}}\cdot v
\cdot {{\mathbf{1}}}({\mathit{Coef}})^*))$ in\
${\mathsf{inv}}(AA)$
Here, in the last expression, $g(x,y)=y$ if $x=1$, and zero otherwise.
|
---
abstract: 'Chiral symmetry of the 2-dimensional chiral Gross-Neveu model is broken explicitly by a bare mass term as well as a splitting of scalar and pseudo-scalar coupling constants. The vacuum and light hadrons — mesons and baryons which become massless in the chiral limit — are explored analytically in leading order of the derivative expansion by means of a double sine-Gordon equation. Depending on the parameters, this model features new phenomena as compared to previously investigated 4-fermion models: spontaneous breaking of parity, a non-trivial chiral vacuum angle, twisted kink-like baryons whose baryon number reflects the vacuum angle, crystals with alternating baryons, and appearance of a false vacuum.'
author:
- Christian Boehmer
- 'Michael Thies[^1]'
title: |
Competing mechanisms of chiral symmetry breaking\
in a generalized Gross-Neveu model
---
Consider the Lagrangian density of $N$ species of massive Dirac fermions in 1+1 dimensions with attractive, U($N$) invariant scalar and pseudoscalar interactions, $${\cal L} = \bar{\psi} \left( {\rm i} \gamma^{\mu} \partial_{\mu}-m_0 \right) \psi + \frac{g^2}{2} (\bar{\psi}\psi)^2
+ \frac{G^2}{2} (\bar{\psi}{\rm i}\gamma_5 \psi)^2.
\label{A1}$$ Flavor indices are suppressed ($\bar{\psi}\psi = \sum_{k=1}^N \bar{\psi}_k \psi_k$ etc.) and the large $N$ limit will be assumed. This 3-parameter field theoretic model generalizes the (massive) chiral Gross-Neveu (GN) model [@1] to two different coupling constants. Its massless 2-parameter version is related to the early work of Klimenko [@1a] and has only recently been investigated comprehensively [@2]. Our main motivation for considering the Lagrangian (\[A1\]) is to study the competition of two different mechanisms of explicit chiral symmetry breaking, both of which are well understood in isolation. The first one is kinematical and familiar from gauge theories — the bare fermion mass. The second one is dynamical — breaking chiral symmetry through the interaction term while preserving parity. This seems to have no analogue in pure gauge theories. In the present work we do not attempt a complete solution of the model (\[A1\]) which would require extensive numerical computations. To get a first overview of its physics content, we focus on the vicinity of the chiral limit at zero temperature, where everything can be done in closed analytical form.
Following ’t Hooft [@3], the large $N$ limit is implemented by letting $N\to \infty$ while keeping $Ng^2$ and $NG^2$ constant. As is well known, this justifies the use of semiclassical methods [@1; @4]. Thereby the Euler-Lagrange equation of the Lagrangian (\[A1\]) gets converted into the Dirac-Hartree-Fock equation, $$\left({\rm i} \gamma^{\mu} \partial_{\mu} -S-{\rm i}\gamma_5 P\right)\psi=0,
\label{A2}$$ where the scalar and pseudo-scalar mean fields are related to condensates (ground state expectation values) through $$\begin{aligned}
S & = & - g^2 \langle \bar{\psi}\psi \rangle + m_0,
\nonumber \\
P & = & - G^2 \langle \bar{\psi} {\rm i} \gamma_5 \psi \rangle.
\label{A3}\end{aligned}$$ Further simplifications arise if we concentrate on static problems in the vicinity of the chiral limit, where the potentials are slowly varying in space. This allows us to invoke a systematic expansion in derivatives of $S$ and $P$ without assuming that the potentials are weak [@5; @6]. As a result, we arrive at an effective bosonic field theory in which the Hartree-Fock potentials appear as complex scalar field (written here in polar coordinates), $$S-{\rm i}P = \rho {\rm e}^{{\rm i} \theta}.
\label{A4}$$ Note that this method can only handle full occupation of single particle levels at present. It was pioneered in Ref. [@7] and applied systematically to two variants of the Lagrangian (\[A1\]), the massive chiral GN model ($g^2=G^2$, Ref. [@6]) and the massless generalized GN model ($m_0=0$, Ref. [@2]). Since the form of the Hartree-Fock equation is the same in all of these cases, the problem at hand differs from previous ones only through the form of the double counting correction to the energy density, $$\begin{aligned}
{\cal E}_{\rm d.c.} &=& \frac{(S-m_0)^2}{2 Ng^2}+\frac{P^2}{2 N G^2}
\nonumber \\
& = & \frac{\rho^2 \cos^2 \theta}{2Ng^2} - \frac{m_0 \rho \cos \theta}{Ng^2}+ \frac{\rho^2 \sin^2 \theta}{2NG^2}.
\label{A5}\end{aligned}$$ An irrelevant term $\sim m_0^2$ has been dropped. Regularization and renormalization require only a straightforward extension of previous works. We replace the 3 bare parameters ($m_0,g^2,G^2$) by physical parameters ($\xi_1,\xi_2,\eta$) via $$\begin{aligned}
\frac{\pi}{Ng^2} & = & \ln \Lambda + \xi_1,
\nonumber \\
\frac{\pi}{NG^2} & = & \ln \Lambda + \xi_2,
\nonumber \\
\frac{\pi m_0}{Ng^2} & = & \eta.
\label{A6}\end{aligned}$$ The $\ln \Lambda$ dependence is mandatory to ensure that the ultraviolet divergence in the sum over single particle energies is cancelled by the double counting correction. In the last line of Eq. (\[A6\]), we avoid the use of the standard confinement parameter [@8] $$\gamma = \frac{\pi m_0}{Ng^2 m} = \frac{\eta}{m}
\label{A7}$$ at this stage. This is done in order not to mix the parameters of the model with dynamical quantities, which may lead to confusion in the present 3-parameter model. Restricting ourselves to the leading order of the derivative expansion, we assume furthermore that the radius $\rho$ is fixed at the dynamical fermion mass and that the chiral angle field $\theta$ is slowly varying. These assumptions can be justified by looking at higher order terms of the derivative expansion, but they also have a very simple physical basis: Close to the chiral limit, the would-be Goldstone field $\theta$ (the “pion" field) is the only one which can be modulated at low cost of energy [@7]. The renormalized ground state energy density (including the vacuum contribution) corresponding to Lagrangian (\[A1\]) then reads $$\begin{aligned}
2 \pi {\cal E} &=& \rho^2 \left( \ln \rho - \frac{1}{2} \right) + \frac{1}{4} \rho^2 (\theta')^2 - 2 \eta \rho \cos \theta
\nonumber \\
& & + \xi_1 \rho^2 \cos^2 \theta + \xi_2 \rho^2 \sin^2 \theta .
\label{A8}\end{aligned}$$ Fermion number is given by the winding number of the chiral field [@6] $$N_f = \frac{N}{2\pi} \int_{-\infty}^{\infty} {\rm d}x \theta' = \frac{N}{2\pi} \left[ \theta(\infty)- \theta(-\infty) \right].
\label{A9}$$ All we have to do is to minimize the energy $\int {\rm d}x {\cal E}$ classically. As a result, we will get information on the vacuum and its symmetries, as well as on light mesons and baryons in the vicinity of the chiral limit. For a homogeneous vacuum, the truncated derivative expansion is exact since the condensates are spatially constant. Hence our results for the vacuum may be taken as the large $N$ limit without any further approximation. Light hadrons are those which become massless in the chiral limit. Here the derivative expansion can be viewed as a kind of chiral perturbation theory, reliable close to the chiral limit. The expression for the pion mass for example is of the type of the Gell-Mann, Oakes, Renner (GOR) relation [@9] in the real world. The fact that baryons emerge from a non-linear theory for the pion field with the baryon number as topological winding number is of course reminiscent of the Skyrme model in 3+1 dimensions [@10; @11].
We first determine the vacuum. To this end, we minimize $2\pi {\cal E}$, Eq. (\[A8\]), with respect to ($x$-independent) $\rho$ and $\theta$ – the dynamical fermion mass and chiral vacuum angle. This yields the transcendental equations $$\begin{aligned}
0 & = & \ln \rho +\xi_1 \cos^2 \theta +\xi_2 \sin^2 \theta - \frac{\eta}{\rho} \cos \theta,
\nonumber \\
0 & = & \sin \theta \left( \cos \theta - \frac{\eta}{\rho(\xi_1-\xi_2)}\right).
\label{A10}\end{aligned}$$ Their solution requires a case differentiation. To understand qualitatively what to expect, let us temporarily choose units such that the dynamical fermion mass is 1 ($\rho=1$) and focus on the $\theta$-dependent part of the vacuum energy density, $$2\pi \tilde{\cal E}(\theta) = -2\gamma \cos \theta - \frac{1}{2} \xi \cos (2\theta), \qquad \xi=\xi_2-\xi_1.
\label{A11}$$ We have used the fact that the distinction between $\eta$ and the confinement parameter $\gamma$ disappears in these units, cf. Eq. (\[A7\]). Note also that depending on the sign of $\xi$, either the scalar coupling (for $\xi>0$) or the pseudoscalar coupling (for $\xi<0$) dominates.
A survey of the $\theta$-dependence of this effective potential in the ($\xi, \gamma$) half plane ($\gamma \geq 0$) reveals a rich landscape (see Fig. \[fig1\]): At the origin ($\xi=0,\gamma=0$), the potential is identically zero (not shown in Fig. 1) and the vacuum infinitely degenerate. This is the U(1) chirally symmetric point. Along the $\gamma$ axis there is a minimum at $\theta=0$ and a maximum at $\theta=\pi$ — the massive chiral GN model. Along the $\xi$ axis, there are two degenerate minima separated by two degenerate maxima — the massless generalized GN model. As discussed in Ref. [@2], the minima can be identified with $0$ and $\pi$ for both $\xi>0$ and $\xi<0$ by means of a global chiral rotation, so that the positive and negative $\xi$ half-axes are in fact equivalent. What happens in the parameter region away from the $\gamma$- and $\xi$-axes depends on the sign of $\xi$. If $\xi>0$, the quadratic maximum becomes a quartic maximum at $\gamma=\xi$; for larger values of $\xi$, a false vacuum develops at $\theta=\pi$. In the limit $\gamma \to 0$ the two minima become degenerate. If $\xi<0$ on the other hand, the quadratic minimum becomes quartic when crossing the critical line $\gamma=-\xi$. This is indicative of a pitchfork bifurcation with two symmetric, degenerate minima present for $\gamma < -\xi$. A non-trivial vacuum angle signals a non-vanishing pseudoscalar condensate and hence a breakdown of parity. This breakdown of parity is spontaneous, but induced by the explicit breaking of chiral symmetry.
With this overall picture in mind, we return to Eqs. (\[A10\]) and solve them in two distinct cases:
- Unbroken parity (phase I) $$\begin{aligned}
\theta_{\rm vac} & = & 0
\nonumber \\
\rho_{\rm vac} & = & \frac{\eta}{W(\eta {\rm e}^{\xi_1})},
\nonumber \\
2 \pi {\cal E}_{\rm vac} & = & - \frac{\eta^2}{2} \left( \frac{1 + 2 W(\eta {\rm e}^{\xi_1})}{W^2(\eta {\rm e}^{\xi_1})}\right).
\label{A12}\end{aligned}$$
- Broken parity (phase II) $$\begin{aligned}
\theta_{\rm vac} & = & \pm \arccos \frac{\eta {\rm e}^{\xi_2}}{\xi_1-\xi_2},
\nonumber \\
\rho_{\rm vac} & = & {\rm e}^{- \xi_2},
\nonumber \\
2 \pi {\cal E}_{\rm vac} & = & - \frac{1}{2} {\rm e}^{-2\xi_2}- \frac{\eta^2}{\xi_1-\xi_2}.
\label{A13}\end{aligned}$$
In Eqs. (\[A12\]) we have introduced the Lambert $W$ function with the defining property $$x=W(x){\rm e}^{W(x)}.
\label{A14}$$ The vacuum energy in the parity broken phase II is lower than in the symmetric phase I. However, phase II only exists if $\theta_{\rm vac}$ is real or, equivalently, $$\xi_1-\xi_2 \geq W(\eta {\rm e}^{\xi_1}).
\label{A15}$$
The next steps can be further simplified as follows. After minimization and determining the phase on the basis of Eq. (\[A15\]), we normalize the radius of the chiral circle (the physical fermion mass) to 1 by a choice of units, $$\rho=\rho_{\rm vac} = 1.
\label{A16}$$ Then $\eta$ may be identified with the confinement parameter (\[A7\]) familiar from the standard massive GN models, $$\eta = \rho \gamma \to \gamma.
\label{A17}$$ In phase I, the condition $\rho=1$ implies $$\xi_1 = \gamma.
\label{A18}$$ The vacuum energy density becomes $${\cal E}_{\rm vac}^I = - \frac{1}{4\pi} - \frac{\gamma}{2\pi},
\label{A19}$$ in agreement with the standard massive GN models. The $\theta$-dependent part of the energy density will be needed for the analysis of light mesons and baryons; in phase I it is given by $$2\pi {\cal E}_{\theta}^I = \frac{1}{4} (\theta')^2 - 2 \gamma \cos \theta -\frac{1}{2} (\xi_2 - \gamma) \cos (2\theta).
\label{A20}$$ In phase II, the condition $\rho=1$ implies $$\xi_2 = 0,
\label{A21}$$ whereas the vacuum energy density assumes the form $${\cal E}_{\rm vac}^{II}=-\frac{1}{4\pi} - \frac{\gamma^2}{2\pi \xi_1}.
\label{A22}$$ In this phase, the $\theta$-dependent part of the energy density reads $$2\pi {\cal E}_{\theta}^{II} = \frac{1}{4} (\theta')^2 - 2 \gamma \cos \theta +\frac{1}{2} \xi_1 \cos (2\theta) .
\label{A23}$$ Eqs. (\[A20\],\[A23\]) can be treated simultaneously by setting $$2\pi {\cal E}_{\theta} = \frac{1}{4} (\theta')^2 - 2 \gamma \cos \theta - \frac{1}{2} \xi \cos (2\theta)
\label{A24}$$ with the definition $$\xi = \xi_2 - \xi_1 = \left\{
\begin{array}{ll} \xi_2- \gamma & ({\rm phase}\ I, \xi>-\gamma) \\
- \xi_1 & ({\rm phase}\ II, \xi<-\gamma) \end{array} \right.
\label{A25}$$ We have traded the original bare parameters $g^2, G^2, \eta$ against two dimensionless parameters $\gamma, \xi$ and one scale, the dynamical fermion mass $\rho=1$. The notation is chosen so as to agree with previous results for the massive chiral GN model [@6] for $\xi=0$ and the massless generalized GN model [@2] for $\gamma=0$.
Next consider the light meson mass in both phases. Expanding expression (\[A24\]) around the vacuum angle $\theta_{\rm vac}$ to 2nd order in $\vartheta= \theta-\theta_{\rm vac}$, we can simply read off the pion mass as follows:
- Phase I ($\xi>-\gamma$) $$\begin{aligned}
\theta_{\rm vac} & = & 0
\nonumber \\
2\pi {\cal E} & \approx & \frac{1}{4} (\vartheta')^2 + (\gamma+\xi)\vartheta^2 + {\rm const.}
\nonumber \\
m_{\pi}^2 & = & 4 (\gamma+\xi)
\label{A26}\end{aligned}$$
- Phase II ($\xi<-\gamma$) $$\begin{aligned}
\theta_{\rm vac} & = & \pm \arccos \left( - \frac{\gamma}{\xi}\right) \ = \ \pm 2 \arctan \sqrt{\frac{\xi + \gamma}{\xi - \gamma}}
\nonumber \\
2\pi {\cal E} & \approx & \frac{1}{4} (\vartheta')^2 + \left( \frac{\gamma^2-\xi^2}{\xi}\right) \vartheta^2 + {\rm const.}
\nonumber \\
m_{\pi}^2 & = & 4 \left(\frac{\gamma^2-\xi^2}{\xi}\right)
\label{A27}\end{aligned}$$
The last lines of Eqs. (\[A26\],\[A27\]) may be regarded as the generalized GOR relations in our model.
It is amusing that a well-known mechanical system is closely analogue to the present problem in the case $\xi<0$, cf. Fig. \[fig2\]: A bead (mass $m$) is sliding without friction on a circular hoop (radius $R$) in a homogeneous gravitational field. The hoop rotates with constant angular velocity $\omega$ around a vertical axis through its center. The Lagrangian reads $$L = \frac{1}{2} m R^2 \left( \dot{\theta}^2+ \omega^2 \sin^2 \theta\right) + mgR\cos \theta.
\label{A28}$$ Denote the pendulum frequency by $\omega_0=\sqrt{g/R}$. At $\omega=0$, there is a unique stable minimum at $\theta=0$, accompanied by small oscillations of frequency $\omega_0$. If one increases $\omega$, this minimum stays stable at first, but the frequency decreases like $\sqrt{\omega_0^2-\omega^2}$ until it vanishes at the critical value $\omega=\omega_0$. At this point, two symmetric stable minima at $\theta=\pm {\rm arccos}\,( \omega_0^2/\omega^2)$ develop, a textbook example of a pitchfork bifurcation [@12]. Beyond this point, the frequency of small oscillations is replaced by $\sqrt{\omega^4-\omega_0^4}/\omega$. The gravitational field and the uniform rotation are two distinct mechanisms of breaking the original SO(2) symmetry of the circle. The mapping of this mechanical problem onto our field theory model is obvious: U(1) chiral symmetry corresponds to the rotational symmetry of the circle, the bare mass plays the role of gravity, the difference in coupling constants corresponds to the uniform rotation, the pion masses to the frequencies of small oscillations. We only have to identify $ \gamma=\omega_0^2/4, \xi=- \omega^2/4$ to map the two problems onto each other quantitatively. In principle, the regime $\xi>0$ could also be modeled by assuming that the particle is charged and invoking an additional constant magnetic field, but in the absence of a phase transition this is less instructive.
Let us now turn to baryons and baryon crystals. Here we need large amplitude solutions of the equation $$\theta'' = 4 \gamma \sin \theta + 2 \xi \sin 2 \theta.
\label{A29}$$ For small values of the parameters $\xi,\eta$ the kink-like soliton solutions of this equation are slowly varying so that the derivative expansion is applicable. The same is true for periodic soliton crystal solutions at sufficiently low density. However there is no restriction on the ratio $\xi/\gamma$, so that the full phase structure shown in Fig. \[fig1\] is accessible in the vicinity of the chiral limit. Since Eq. (\[A29\]) has no explicit $x$-dependence, it can be integrated once, $$\frac{1}{2} (\theta')^2+ 4 \gamma \cos \theta + \xi \cos (2\theta) = {\rm const.}
\label{A30}$$ The second integration is then carried out by separation of variables.
The mechanical interpretation of the kinks is well-known: If we interpret $x$ as time coordinate, Eq. (\[A29\]) describes motion of a classical particle in a potential inverted as compared to the potentials shown in Fig. \[fig1\]. The kink-like tunneling solutions between different vacua in field theory go over into classical paths joining two degenerate maxima in the mechanics case. In this classical mechanics interpretation, Eq. (\[A30\]) expresses conservation of the Hamilton function. As a matter of fact, Eq. (\[A29\]) is nothing but the double sine-Gordon equation, a widely used generalization of the sine-Gordon equation to which it reduces if either $\gamma$ or $\xi$ vanishes. Its solutions can be found in the literature, see e.g. [@13], so that we refrain from giving any details of the derivation. Since $\theta$ is an angular variable, kinks do exist everywhere in the ($\xi,\gamma$) half-plane. Inspection of the effective potentials of Fig. \[fig1\] then helps to understand the following results:
For $\xi>-\gamma$ (phase I) there is only one kink solution $$\theta_{\rm kink} = - 2 \arctan \frac{\sqrt{\xi + \gamma}}{\sqrt{\gamma} \sinh(2\sqrt{\xi+\gamma}x)}.
\label{A31}$$ We define the branch of the arctan such that $\theta$ goes from 0 to $2\pi$ along the $x$ axis. For $\xi<-\gamma$ (phase II) there are two different kinks depending on how one connects the minima along the chiral circle, $$\begin{aligned}
\theta_{\rm large} & = & - 2 \arctan \left[ \sqrt{\frac{\xi+\gamma}{\xi-\gamma}}\coth \left( \sqrt{\frac{\gamma^2-\xi^2}{\xi}}x\right)\right],
\nonumber \\
\theta_{\rm small} & = & + 2 \arctan \left[ \sqrt{\frac{\xi+\gamma}{\xi-\gamma}}\tanh \left( \sqrt{\frac{\gamma^2-\xi^2}{\xi}}x\right)\right].
\nonumber \\
\label{A32}\end{aligned}$$ Here, our choice of the branch of arctan is such that $\theta$ goes from $- \theta_{\rm vac}$ to $\theta_{\rm vac}$ for the small kink and from $\theta_{\rm vac}$ to $2\pi - \theta_{\rm vac}$ for the large kink. The baryon numbers $B=N_f/N$ are $$\begin{aligned}
B_{\rm kink} & = & 1,
\nonumber \\
B_{\rm large} & = & 1- \frac{\theta_{\rm vac}}{\pi},
\nonumber \\
B_{\rm small} & = & \frac{\theta_{\rm vac}}{\pi},
\label{A33}\end{aligned}$$ with $\theta_{\rm vac}$ from Eq. (\[A27\]) with the + sign. The terms small and large refer to the chiral twist of the two kinks which in turn is reflected in the baryon number. The baryon numbers of a small and a large kink add up to 1 simply because these kinks correspond to the 2 possibilities of travelling from one minimum to the other one along a circle. Eqs. (\[A31\]-\[A33\]) refer to kinks with positive baryon number. By changing the sign of $\theta$, these can be converted into antikinks with opposite baryon number.
In Figs. \[fig3\] and \[fig4\], we illustrate the scalar and pseudoscalar potentials for the small and large kinks in the parity broken phase II. To understand these graphs, we recall that the two vacua are characterized by the chiral angles $\pm \theta_{\rm vac}$, Eq. (\[A27\]). The parity even, scalar vacuum condensate ($\cos \theta_{\rm vac}$) is the same in both vacua, the parity odd, pseudoscalar condensate ($-\sin \theta_{\rm vac}$) has opposite sign. This is reflected in the asymptotic behavior of $S$ and $P$ for the kinks which connect these two vacua. To contrast this behavior with baryons in phase I (unbroken parity, $\xi>-\gamma$), we show in Fig. \[fig5\] the kink baryon from Eq. (\[A31\]) where now both $S$ and $P$ are periodic. For the parameters chosen here, it resembles closely the standard sine-Gordon kink.
Note the following limits:
- Massive NJL model ($\gamma>0, \xi=0$): There is a unique minimum at $\theta=0$. We recover previous (sine-Gordon) results [@6; @7] with the help of the identity $$\theta = \mp 2 \arctan \frac{1}{\sinh (2\sqrt{\gamma}x)} = \pm 4 \arctan {\rm e}^{2 \sqrt{\gamma}x}.
\label{A34}$$
- Massless generalized GN model ($\gamma=0, \xi>0$): There are 2 degenerate minima at $\theta=0,\pi$ and correspondingly 2 kink baryons with baryon number 1/2. The limit is singular (see Fig. \[fig6\]): As $\gamma \to 0$, the kink develops a plateau which becomes infinitely wide at $\gamma=0$. The kink decouples into 2 half-kinks each carrying baryon number 1/2 [@2]. As one sees in Fig. \[fig1\], this happens when the maxima in the inverted potential become degenerate or, equivalently, the false vacuum and the true vacuum in the original potential become equal.
It is worth mentioning that there is yet another solitonic solution of some physics relevance: If one is interested in the decay of the false vacuum, one has to consider tunneling through the barrier. This in turn is related to the kink-antikink which starts from the lower maximum, is reflected at the barrier and returns to the starting point (the bounce [@14]). Since we are mainly interested in the vacuum and low-lying hadrons here, we do not go further into this problem.
We now turn to a useful test of the consistency of our results, following Ref. [@15]. Consider the divergence of the axial current as obtained from the Euler-Lagrange equations for the Lagrangian (\[A1\]), $$\partial_{\mu} j_5^{\mu} = 2 \bar{\psi} {\rm i} \gamma_5 \psi \left[ m_0-(g^2-G^2)\bar{\psi}\psi \right].
\label{A35}$$ The right-hand side exhibits the 2 sources of chiral symmetry breaking, the bare fermion mass and the splitting of the coupling constants. The self-consistency conditions (\[A3\]) and the renormalization scheme (\[A6\]) can be used to rewrite Eq. (\[A35\]) as $$\partial_{\mu} j_5^{\mu} = - \frac{2NP}{\pi}\left[ \eta - (\xi_1-\xi_2)S\right]
\label{A36}$$ or, in units $\rho=1$ and with the notation of Eqs. (\[A17\],\[A25\]), $$\partial_{\mu} j_5^{\mu} = - \frac{2NP}{\pi}\left( \gamma + \xi S\right).
\label{A37}$$ Taking the expectation value of this equation in a time-independent state and remembering that $j_5^1 = j^0$ in 1+1 dimensions, we arrive at the following expression for the fermion density, $$j^0(x) = - \frac{2N}{\pi} \int_{-\infty}^x {\rm d}x' P(x') \left[ \gamma + \xi S(x')\right],
\label{A38}$$ and, after another integration, the sum rule $$N_f = \frac{2N}{\pi} \int_{-\infty}^{\infty} {\rm d}x x P(x) \left[ \gamma + \xi S(x)\right].
\label{A39}$$ The last equation in particular provides us with a non-trivial way of testing the baryon potentials. By inserting $S=\cos \theta$ and $P=-\sin \theta$ into the sum rule with $\theta$ from Eqs. (\[A31\],\[A32\]), we indeed reproduce the baryon numbers (\[A33\]). Notice also that the expectation value of Eq. (\[A36\]) for the divergence of the axial current, $$(j^0)'(x) = - \frac{2NP(x)}{\pi}\left[ \gamma+ \xi S(x)\right],
\label{A40}$$ reduces to the double sine-Gordon equation, Eq. (\[A29\]), if we insert $$j^0(x) = \frac{N}{2\pi} \theta'(x)
\label{A41}$$ and express $S,P$ in terms of the chiral angle $\theta$. This points to an alternative derivation of the basic equation (\[A29\]) which would not even require the derivative expansion, at least to leading order considered here.
It is straightforward to compute the baryon masses by integrating the energy density and subtracting the vacuum contribution, $$\begin{aligned}
2\pi M & = & \int {\rm d}x \left\{ \frac{1}{4} [\theta'(x)]^2 - 2\gamma (\cos \theta(x)-\cos \theta_{\rm vac}) \right.
\nonumber \\
& & \left. - \frac{1}{2}\xi \left[ \cos (2\theta(x))- \cos(2\theta_{\rm vac})\right] \right\}.
\label{A42}\end{aligned}$$ One finds $$\begin{aligned}
M_{\rm kink} & = & \frac{2\sqrt{\gamma+\xi}}{\pi} + \frac{\gamma}{\pi \sqrt{\xi}} \ln \left( \frac{\sqrt{\gamma+\xi}+\sqrt{\xi}}{\sqrt{\gamma+\xi}-
\sqrt{\xi}}\right),
\nonumber \\
M_{\rm large} & = & \frac{1}{\pi} \sqrt{\frac{\xi^2-\gamma^2}{-\xi}}+ \frac{2\gamma}{\pi \sqrt{-\xi}}\arctan \sqrt{\frac{\xi-\gamma}{\xi +\gamma}},
\nonumber \\
M_{\rm small} & = & \frac{1}{\pi} \sqrt{\frac{\xi^2-\gamma^2}{-\xi}} - \frac{2\gamma}{\pi \sqrt{-\xi}}\arctan \sqrt{\frac{\xi+\gamma}{\xi -\gamma}},
\nonumber \\
\label{A43}\end{aligned}$$ and the same results for the corresponding antikinks. These expressions are of course known from studies of the classical double sine-Gordon equation.
Finally, consider baryonic matter at low density. The pertinent solutions of the double sine-Gordon equation are kink crystals which can be evaluated analytically in terms of Jacobi elliptic functions. Since we work only to lowest order of the derivative expansion in the present study, we bypass the complicated exact solution by simply gluing together kink solutions. This is adequate in the low density limit. In the parity preserving phase I, the basic building block is $\theta_{\rm kink}$, Eq. (\[A31\]). Let us denote the separation between two kinks (i.e., the lattice constant) by $d$, so that the baryon density is $\rho_B=1/d$. A dilute periodic array of kinks is then well approximated by $$\theta_{\rm crystal}^I = \theta_{\rm kink}(x-nd)+2\pi n \ {\rm for} \ x\in [nd-d/2,nd+d/2].
\label{A44}$$ For sufficiently large $d$ this yields a smooth staircase curve which solves the double sine Gordon equation exactly except at the gluing points $x=(n+1/2)d$. There the error can be made arbitrarily small for large $d$. The energy density in the dilute limit is just $M_{\rm kink}\rho_B$ with the kink mass from Eq. (\[A43\]). In phase II, we have to proceed slightly differently. Obviously one can only glue together the small and large kinks in an alternating way, see Figs. \[fig3\],\[fig4\]. We therefore first construct a unit cell of the crystal by joining one small and one large kink, $$\tilde{\theta}_{\rm kink}(x) = \left\{ \begin{array}{lll}
\theta_{\rm small}(x+d/4) & {\rm for} & -d/2 <x<0 \\
\theta_{\rm large}(x-d/4) & {\rm for} & 0 <x< d/2 \end{array} \right.
\label{A45}$$ This carries baryon number 1 and is periodic modulo $2\pi$, so that the unit cells can now be assembled into a crystal in the same way as in phase I, Eq. (\[A44\]), $$\theta_{\rm crystal}^{II} = \tilde{\theta}_{\rm kink}(x-nd)+2\pi n \ {\rm for} \ x\in [nd-d/2,nd+d/2].
\label{A46}$$ The energy density in the low density limit of phase II becomes $${\cal E} = (M_{\rm small} + M_{\rm large})\rho_B
\label{A47}$$ where the sum of the kink masses from Eqs. (\[A43\]) can be simplified to $$M_{\rm small} +M_{\rm large} = \frac{2}{\pi} \sqrt{\frac{\xi^2-\gamma^2}{-\xi}} + \frac{2\gamma}{\pi \sqrt{-\xi}}\arctan \frac{\gamma}{\sqrt{\xi^2-\gamma^2}}.
\label{A48}$$ An example for a unit cell is shown in Fig. \[fig7\] with the same ratio $\xi/\gamma$ and hence the same shape of the small and large kinks as in Figs. \[fig3\],\[fig4\].
Summarizing, we have investigated a 3-parameter generalization of the U(1) chirally symmetric GN model. The two dimensionless parameters $\gamma$ and $\xi$ stem from two different mechanisms of breaking chiral symmetry explicitly, the bare mass term and the difference between scalar and pseudoscalar couplings. Close to the chiral limit, the leading order derivative expansion has revealed the following scenario. If the scalar coupling dominates, we find in general a unique vacuum with scalar condensate, light pions and kink-like baryons with baryon number 1. In the region $\xi>\gamma$ a false vacuum shows up in the form of a second local minimum. If the pseudoscalar coupling dominates, at first nothing changes. Starting from a critical strength of the coupling ($\xi<-\gamma$), two symmetric minima appear together with scalar and pseudoscalar condensates; parity is spontaneously broken. The mechanical model of a particle on a rotating circle in the gravitational field illustrates nicely the concomitant pitchfork bifurcation. The two ways of connecting two minima along the chiral circle are reflected in two baryons whose baryon numbers add up to 1. These chirally twisted baryons are mathematically well known from studies of the double sine-Gordon equation and quite different from another type of twisted bound state specific for the chiral limit [@16; @17]. In our case, the baryons are stabilized by topology. Shei’s bound state is stabilized by partially filling the valence level and does not carry baryon number as a result of a cancellation with induced fermion number [@15]. In many respects the limits $\gamma\to 0$ and $\xi\to 0$ are atypical so that previously explored 2-parameter versions of the present model cannot convey the full picture of chiral symmetry breaking in 4-fermion models. In view of the rich structure of the 3-parameter model, it seems worthwhile to pursue its study, in particular to explore the fate of the symmetries at finite temperature and chemical potential.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work has been supported in part by the DFG under grant TH 842/1-1.
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[^1]: [email protected]
|
---
abstract: 'Formation of galaxy clusters corresponds to the collapse of the largest gravitationally bound overdensities in the initial density field and is accompanied by the most energetic phenomena since the Big Bang and by the complex interplay between gravity–induced dynamics of collapse and baryonic processes associated with galaxy formation. Galaxy clusters are, thus, at the cross-roads of cosmology and astrophysics and are unique laboratories for testing models of gravitational structure formation, galaxy evolution, thermodynamics of the intergalactic medium, and plasma physics. At the same time, their large masses make them a useful probe of growth of structure over cosmological time, thus providing cosmological constraints that are complementary to other probes. In this review, we describe our current understanding of cluster formation: from the general picture of collapse from initial density fluctuations in an expanding Universe to detailed simulations of cluster formation including the effects of galaxy formation. We outline both the areas in which highly accurate predictions of theoretical models can be obtained and areas where predictions are uncertain due to uncertain physics of galaxy formation and feedback. The former includes the description of the structural properties of the dark matter halos hosting cluster, their mass function and clustering properties. Their study provides a foundation for cosmological applications of clusters and for testing the fundamental assumptions of the standard model of structure formation. The latter includes the description of the total gas and stellar fractions, the thermodynamical and non-thermal processes in the intracluster plasma. Their study serves as a testing ground for galaxy formation models and plasma physics. In this context, we identify a suitable radial range where the observed thermal properties of the intra-cluster plasma exhibit the most regular behavior and thus can be used to define robust observational proxies for the total cluster mass. Finally, we discuss the formation of clusters in non-standard cosmological models, such as non-Gaussian models for the initial density field and models with modified gravity, along with prospects for testing these alternative scenarios with large cluster surveys in the near future.'
address:
- |
$^1$Department of Astronomy & Astrophysics, Kavli Institute for Cosmological Physics, The University of Chicago, Chicago, IL 60637\
email: [[email protected]]{}
- |
$^2$Dipartimento di Fisica dell’Università di Trieste, Sezione di Astronomia, I-34131 Trieste, Italy;\
INAF – Osservatorio Astronomico di Trieste, Italy;\
and INFN – Istituto Nazionale di Fisica Nucleare, Trieste, Italy;\
email: [[email protected]]{}
author:
- 'Andrey V. Kravtsov$^1$ and Stefano Borgani$^2$'
bibliography:
- 'araa\_final.bib'
title: Formation of Galaxy Clusters
---
epsf.tex psfig.sty
Cosmology, galaxy clusters, intra-cluster medium
Introduction
============
Tendency of nebulae to cluster has been ddiscovered by Charles Messier and William Herschel, who have constructed the first systematic catalogs of these objects. This tendency has become more apparent as larger and larger samples of galaxies were compiled in the 19th and early 20th centuries. Studies of the most prominent concentrations of nebulae, the clusters of galaxies, were revolutionized in the 1920s by Edwin Hubble’s proof that spiral and elliptical nebulae were bona fide galaxies like the Milky Way located at large distances from us [@hubble25; @hubble26], which implied that clusters of galaxies are systems of enormous size. Just a few years later, measurements of galaxy velocities in regions of clusters made by @hubble_humason31 and assumption of the virial equilibrium of galaxy motions were used to show that the total gravitating cluster masses for the Coma [@zwicky33 and see also @zwicky37] and Virgo clusters [@smith36] were enormous as well.
The masses implied by the measured velocity dispersions were found to exceed combined mass of all the stars in clusters galaxies by factors of $\sim 200-400$, which prompted Zwicky to postulate the existence of large amounts of “dark matter” (DM), inventing this widely used term in the process. Although the evidence for dark matter in clusters was disputed in the subsequent decades, as it was realized that stellar masses of galaxies were underestimated in the early studies, dark matter was ultimately confirmed by the discovery of extended hot intracluster medium (ICM) emitting at X–ray energies by thermal bremsstrahlung that was found to be smoothly filling intergalactic space within the Coma cluster [@gursky_etal71; @meekins_etal71; @kellogg_etal72; @forman_etal72; @cavaliere_etal71]. The X–ray emission of the ICM has not only provided a part of the missing mass [as was conjectured on theoretical grounds by @limber59; @van_albada60], but also allows the detection of clusters out to $z>1$ [@rosati_etal02]. Furthermore, measurement of the ICM temperature has provided an independent confirmation that the depth of gravitational potential of clusters requires additional dark component. It was also quickly realized that inverse Compton scattering of the cosmic microwave background (CMB) photons off thermal electrons of the hot intergalactic plasma should lead to distortions in the CMB spectrum, equivalent to black body temperature variations of about $10^{-4}$–$10^{-5}$ \[the Sunyaev–Zel’dovich (SZ) effect;@sz70 [@sz72; @sz80]\]. This effect has now been measured in hundreds of clusters [e.g., @carlstrom_etal02].
Given such remarkable properties, it is no surprise that the quest to understand the formation and evolution of galaxy clusters has become one of the central efforts in modern astrophysics over the past several decades. Early pioneering models of collapse of initial density fluctuations in the expanding Universe have shown that systems resembling the Coma cluster can indeed form [@van_albada60; @van_albada61; @peebles70; @white76]. @gott_gunn71 [see also @sunyaev_zeldovich72] showed that hot gas observed in the Coma via X-ray observations can be explained within such a collapse scenario by heating of the infalling gas by the strong accretion shocks. Subsequently, emergence of the hierarchical model of structure formation [@press_schechter74; @gott_rees75; @white_rees78], combined with the cold dark matter (CDM) cosmological scenario [@bond_etal82; @blumenthal_etal84], provided a powerful framework for interpretation of the multi-wavelength cluster observations. At the same time, rapid advances in computing power and new, efficient numerical algorithms have allowed fully three-dimensional [*ab initio*]{} numerical calculations of cluster formation within self-consistent cosmological context in both dissipationless regime [@klypin_shandarin83; @efstathiou_etal85] and including dissipational baryonic component [@evrard88; @evrard90].
In the past two decades, theoretical studies of cluster formation have blossomed into a vibrant and mature scientific field. As we detail in the subsequent sections, the standard scenario of cluster formation has emerged and theoretical studies have identified the most important processes that shape the observed properties of clusters and their evolution, which has enabled usage of clusters as powerful cosmological probes [see, e.g., @allen_etal11 for a recent review]. At the same time, observations of clusters at different redshifts have highlighted several key discrepancies between models and observations, which are particularly salient in the central regions (cores) of clusters.
In the current paradigm of structure formation clusters are thought to form via an hierarchical sequence of mergers and accretion of smaller systems driven by gravity and DM that dominates the gravitational field. Theoretical models of clusters employ a variety of techniques determined by a particular aspect of cluster formation they aim to understand. Many of the bulk properties of clusters are thought to be determined solely by the initial conditions, dissipationless DM that dominates cluster mass budget, and gravity. Thus, cluster formation is often approximated in models as DM–driven dissipationless collapse from cosmological initial conditions in an expanding Universe. Such models are quite successful in predicting the existence and functional form of correlations between cluster properties, as well as their abundance and clustering, as we discuss in detail in Section \[sec:clusform\]. One of the most remarkable models of this kind is a simple self-similar model of clusters [@kaiser86 see § \[sec:selfsimilar\] below]. Despite its simplicity, the predictions of this model are quite close to results of observations and have, in fact, been quite useful in providing baseline expectations for evolution of cluster scaling relations. Studies of abundance and spatial distribution of clusters using dissipationless cosmological simulations show that these statistics retain remarkable memory of the initial conditions.
The full description of cluster formation requires detailed modeling of the non–linear processes of collapse and the dissipative physics of baryons. The gas is heated to high, X-ray emitting temperatures by adiabatic compression and shocks during collapse and settles in hydrostatic equilibrium within the cluster potential well. Once the gas is sufficiently dense, it cools, the process that can feed both star formation and accretion onto supermassive black holes (SMBHs) harbored by the massive cluster galaxies. The process of cooling and formation of stars and SMBHs can then result in energetic feedback due to supernovae (SNe) or active galactic nuclei (AGN), which can inject substantial amounts of heat into the ICM and spread heavy elements throughout the cluster volume.
Galaxy clusters are therefore veritable crossroads of astrophysics and cosmology: While abundance and spatial distribution of clusters bear indelible imprints of the background cosmology, gravity law, and initial conditions, the nearly closed–box nature of deep cluster potentials makes them ideal laboratories to study processes operating during galaxy formation and their effects on the surrounding intergalactic medium.
In this review we discuss the main developments and results in the quest to understand the formation and evolution of galaxy clusters. Given the limited space available for this review and the vast amount of literature and research directions related to galaxy clusters, we have no choice but to limit the focus of our review, as well as the number of cited studies. Specifically, we focus on the most basic and well-established elements of the standard paradigm of DM-driven hierarchical structure formation within the framework of $\Lambda$CDM cosmology as it pertains to galaxy clusters. We focus mainly on the theoretical predictions of the properties of the total cluster mass distribution and properties of the hot intracluster gas, and only briefly discuss results pertaining to the evolution of stellar component of clusters, understanding of which is still very much a work in progress. Comparing model predictions to real clusters, we mostly focus on comparisons with X-ray observations, which have provided the bulk of our knowledge of ICM properties so far. In § \[sec:cosmo\], we briefly discuss the differences in formation of clusters in models with the non-Gaussian initial conditions and modified gravity. Specifically, we focus on the information that statistics sensitive to the cluster formation process, such as cluster abundance and clustering, can provide about the primordial non-Gaussianity and possible deviations of gravity from General Relativity. We refer readers to recent extensive reviews on cosmological uses of galaxy clusters by @allen_etal11 and @weinberg_etal12 for a more extensive discussion of this topic.
The observed properties of galaxy clusters {#sec:obsprop}
==========================================
Observational studies of galaxy clusters have now developed into a broad, multi-faceted and multi-wavelength field. Before we embark on our overview of different theoretical aspects of cluster formation, we briefly review the main observational properties of clusters and, in particular, the basic properties of their main matter constituents.
Figure \[fig:a1689\] shows examples of the multiwavelength observations of two massive clusters at two different cosmic epochs: the Abell 1689 at $z=0.18$ and the SPT-CL J2106-5844 at $z = 1.133$. It illustrates all of the main components of the clusters: the luminous stars in and around galaxies (the intracluster light or ICL), the hot ICM observed via its X-ray emission and the Sunyaev-Zel’dovich effect and, in the case of Abell 1689, even the presence of invisible DM manifesting itself through gravitational lensing of background galaxies distorting their images into extended, cluster-centric arcs [@bartelmann10 and references therein]. At larger radii, the lensing effect is weaker. Although not easily visible by eye, it can still be reliably measured by averaging the shapes of many background galaxies and comparing the average with the expected value for an isotropic distribution of shapes. The gravitational lensing is a direct probe of the total mass distribution in clusters, which makes it both extremely powerful in its own right and a very useful check of other methods of measuring cluster masses. The figure shows several bright elliptical galaxies that are typically located near the cluster center. A salient feature of such central galaxies is that they show little evidence of ongoing star formation, despite their extremely large masses.
The diffuse plasma is not associated with individual galaxies and constitutes the intra-cluster medium, which contains the bulk of the normal baryonic matter in massive clusters. Although the hot ICM is not directly associated with galaxies, their properties are correlated. For example, Fig. \[fig:mgms\] shows the mass of the ICM gas within the radius $R_{500}$, defined as the radius enclosing mean overdensity of $\Delta_c=500\rhoc$, versus stellar mass in galaxies within the same radius for a number of local ($z\lesssim 0.1$) and distant ($0.1<z<0.6$) clusters [@lin_etal12]. Here $\rhoc(z)=3H(z)^2/(8\pi G$ is the critical mean density of the Universe, defined in terms of the Hubble function $H(z)$. The figure shows a remarkably tight, albeit non-linear, correlation between these two baryonic components. It also shows that the gas mass in clusters is on average about ten times larger than the mass in stars, although this ratio is systematically larger for smaller mass clusters, ranging from $M_{\ast}/M_{\rm g}\approx 0.2$ to $\approx 0.05$, as mass increases from group scale ($M_{500}\sim {\rm few\ }\times 10^{13}\rm\
M_{\odot}$) to massive clusters ($M_{500}\sim 10^{15}\rm\ M_{\odot}$).
The temperature of the ICM is consistent with velocities of galaxies and indicates that both galaxies and gas are nearly in equilibrium within a common gravitational potential well. The mass of galaxies and hot gas is not sufficient to explain the depth of the potential well, which implies that most of the mass in clusters is in a form of DM. Given that hydrogen is by far the most abundant element in the Universe, most of the plasma particles are electrons and protons, with a smaller number of helium nuclei. There are also trace amounts of heavier nuclei some of which are only partially ionized. The typical average abundance of the heavier elements is about one-third of that found in the Sun or a fraction of one per cent by mass; it decreases with increasing radius and can be quite inhomogeneous, especially in merging systems [@werner_etal08 for a review].
Thermodynamic properties of the ICM are of utmost importance, because comparing such properties to predictions of baseline models without cooling and heating can help to isolate the impact of these physical processes in cluster formation. The most popular baseline model is the self-similar model of clusters developed by @kaiser86, which we consider in detail in Section \[sec:selfsimilar\] below. In its simplest version, this model assumes that clusters are scaled versions of each other, so that gas density at a given fraction of the characteristic radius of clusters, defined by their mass, is independent of cluster mass. Figure \[fig:obsICM\] shows the electron density in clusters as a function of ICM temperature (and hence mass) at different radii. It is clear that density is independent of temperature only outside cluster core at $r\sim R_{500}$, although there is an indication that density is independent of temperature at $r=R_{2500}$ for $k_BT\gtrsim 3$ keV. This indicates that processes associated with galaxy formation and feedback affect the properties of clusters at $r\lesssim R_{2500}$, but their effects are mild at larger radii.
During the past two decades, it has been established that the core regions of the relaxed clusters are generally characterized by a strongly peaked X–ray emissivity, indicating efficient cooling of the gas [e.g., @fabian94]. Quite interestingly, spectroscopic observations with the Chandra and XMM–Newton satellites have demonstrated that, despite strong X-ray emission of the hot gas, only a relatively modest amount of this gas cools down to low temperatures [e.g., @peterson_etal01; @boehringer_etal01]. This result is generally consistent with the low levels of star formation observed in the brightest cluster galaxies [BCGs; e.g., @mcdonald_etal11]. It implies that a heating mechanism should compensate for radiative losses, thereby preventing the gas in cluster cores to cool down to low temperature. The presence of cool cores is also reflected in the observed temperature profiles [e.g. @leccardi_molendi08; @pratt_etal07; @vikhlinin_etal06 see also Figure \[fig:Tprofs\]], which exhibit decline of temperature with decreasing radius in the innermost regions of relaxed cool–core clusters.
One of the most important and most widely studied aspects of ICM properties are correlations between its different observable integrated quantities and between observable quantities and total mass. Such [*scaling relations*]{} are the key ingredient in cosmological uses of clusters, where it is particularly desirable that the relations are characterized by small scatter and are independent of the relaxation state and other properties of clusters. Although clusters are fascinatingly complex systems overall, they do exhibit some remarkable regularities. As an example, Figure \[fig:ly\_rexcess\] shows the correlation between the bolometric luminosity emitted from within $R_{500}$ and the $Y_{\rm X}$ parameter defined as a product of gas mass within $R_{500}$ and ICM temperature derived from the X–ray spectrum within the radial range $(0.15-1)R_{500}$ [@kravtsov_etal06] for theRepresentative XMM–Newton Cluster Structure Survey (REXCESS) sample of clusters studied by @pratt_etal08. Different symbols indicate clusters in different states of relaxation, whereas clusters with strongly peaked central gas distribution (the cool core clusters) and clusters with less centrally concentrated gas distribution are shown with different colors. The left panel shows total luminosity integrated within radius $R_{500}$, wheareas the right panel shows luminosity calculated with the central region within $0.15R_{500}$ excised. Quite clearly, the core-excised X-ray luminosity exhibits remarkably tight correlation with $Y_{\rm X}$, which, in turn, is expected to correlate tightly with total cluster mass [@kravtsov_etal06; @stanek_etal10; @fabjan_etal11]. This figure illustrates the general findings in the past decade that clusters exhibit strong regularity and tight correlations among X-ray observable quantities and total mass, provided that relevant quantities are measured after excluding the emission from cluster cores.
Understanding the formation of galaxy clusters {#sec:clusform}
==============================================
Initial density perturbation field and its linear evolution {#sec:collbasics}
-----------------------------------------------------------
In the currently standard hierarchical structure formation scenario, objects are thought to form via gravitational collapse of peaks in the initial primordial density field characterized by the density contrast (or overdensity) field: $\delta({\bf x})=(\rho({\bf
x})-\bar{\rho}_{\rm m})/{\bar{\rho}_{\rm m}}$, where $\bar{\rho}_{\rm m}$ is the mean mass density of the Universe. Properties of the field $\delta({\bf x})$ depend on specific details of the processes occurring during the earliest inflationary stage of evolution of the Universe [@guth_pi82; @starobinsky82; @bardeen_etal83] and the subsequent stages prior to recombination [@peebles82; @bond_efstathiou84; @bardeen_etal86; @eisenstein_hu99]. A fiducial assumption of most models that we discuss is that $\delta({\bf x})$ is a homogeneous and isotropic Gaussian random field. We briefly discuss non-Gaussian models in Section \[sec:nong\].
Statistical properties of a uniform and isotropic Gaussian field can be fully characterized by its power spectrum, $P(k)$, which depends only on the modulus $k$ of the wavevector, but not on its direction. A related quantity is the variance of the density contrast field [ *smoothed*]{} on some scale $R$: $\delta_{R}({\bf x})\equiv \int
\delta({\bf x}-{\bf r})W({\bf r}, R)d^3r$, where \_[R]{}\^2\^2(R)=P(k)([**k**]{},R)\^2d\^3k, \[eq:variance\] where $\tilde{W}({\bf k},R)$ is the Fourier transform of the window (filter) function $W({\bf r},R)$, such that $\delta_{R}({\bf
k})=\delta({\bf k})\tilde{W}({\bf k},R)$ \[see, e.g., [@zentner07] or [@mo_etal10] for details on the definition of $P(k)$ and choices of window function\]. For the cases, when one is interested in only a narrow range of $k$ the power spectrum can be approximated by the power–law form, $P(k)\propto k^n$, and the variance is $\sigma^2(R)\propto R^{-(n+3)}$.
At a sufficiently high redshift $z$, for the spherical top–hat window function mass and radius are interchangeable according to the relation $M=4\pi/3\rho_{\rm m}(z)R^3$. We can think about the density field smoothed on the scale $R$ or the corresponding mass scale $M$. The characteristic amplitude of peaks in the $\delta_R$ (or $\delta_{M}$) field smoothed on scale $R$ (or mass scale $M$) is given by $\sigma(R)\equiv\sigma(M)$. The smoothed Gaussian density field is, of course, also Gaussian with the probability distribution function (PDF) given by p(\_[M]{})=. \[eq:pGaussian\]
During the earliest linear stages of evolution in the standard structure formation scenario the initial Gaussianity of the $\delta({\bf x})$ field is preserved, as different Fourier modes $\delta({\bf k})$ evolve independently and grow at the same rate, described by the [*linear growth factor*]{}, $D_+(a)$, as a function of expansion factor $a=(1+z)^{-1}$, which for a $\Lambda$CDM cosmology is given by [@heath77]: (a)D\_+(a)=E(a)\_0\^a, \[eq:Dgrowth\] where $E(a)$ is the normalized expansion rate, which is given by E(a)=\^[1/2]{}, \[eq:Ea\] if the contribution from relativistic species, such as radiation or neutrinos, to the energy-density is neglected. Growth rate and the expression for $E(a)$ in more general, homogeneous dark energy (DE) cosmologies are described by @percival05. Note that in models in which DE is clustered [@alimi_etal10] or gravity deviates from General Relativity (GR, see Section \[sec:nongr\]), the growth factor can be scale dependent.
Correspondingly, the linear evolution of the root mean square (rms) amplitude of fluctuations is given by $\sigma(M,a)=\sigma(M,a_i)D_+(a)/D_+(a_i)$, which is often useful to recast in terms of linearly extrapolated rms amplitude $\sigma(M,a=1)$ at $a=1$ (i.e., $z=0$): (M,a)=(M,a=1)D\_[+0]{}(a), [where ]{} D\_[+0]{}(a)D\_+(a)/D\_+(a=1).
Once the amplitude of typical fluctuations approaches unity, $\sigma(M,a)\sim 1$, the linear approximation breaks down. Further evolution must be studied by means of nonlinear models or direct numerical simulations. We discuss results of numerical simulations extensively below. However, we consider first the simplified, but instructive, spherical collapse model and associated concepts and terminology. Such model can be used to gain physical insight into the key features of the evolution and is used as a basis for both definitions of collapsed objects (see Section \[sec:massdef\]) and quantitative models for halo abundance and clustering (Section \[sec:mf\] and \[sec:bias\]).
Non–linear evolution of spherical perturbations and non–linear mass scale {#sec:sphcoll}
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The simplest model of non–linear collapse assumes that density peak can be characterized as constant overdensity spherical perturbation of radius $R$. Despite its simplicity and limitations discussed below, the model provides a useful insight into general features and timing of non–linear collapse. Its results are commonly used in analytic models for halo abundance and clustering and motivate mass definitions for collapsed objects. Below we briefly describe the model and non–linear mass scale that is based on its predictions.
### Spherical collapse model.
The [*spherical collapse model*]{} considers a spherically-symmetric density fluctuation of initial radius $R_{\rm
i}$, amplitude $\delta_{\rm i}>0$, and mass $M=(4\pi/3)(1+\delta_{\rm i})\bar{\rho}R_{\rm i}^3$, where $R_{\rm i}$ is physical radius of the perturbation and $\bar{\rho}$ is the mean density of the Universe at the initial time. Given the symmetry, the collapse of such perturbation is a one-dimensional problem and is fully specified by evolution of the top-hat radius $R(t)$ [@gunn_gott72; @lahav_etal91]. It consists of an initially decelerating increase of the perturbation radius, until it reaches the maximum value, $R_{\rm ta}$, at the turnaround epoch, $t_{\rm ta}$, and subsequent decrease of $R(t)$ at $t>t_{\rm ta}$ until the perturbation collapses, virializes, and settles at the final radius $R_{\rm f}$ at $t=t_{\rm coll}$. Physically, $R_{\rm f}$ is set by the virial relation between potential and kinetic energy and is $R_{\rm f}=R_{\rm ta}/2$ in cosmologies with $\Oml=0$. The turnaround epoch and the epoch of collapse and virialization are defined by initial conditions.
The final mean internal density of a collapsed object can be estimated by noting that in a $\Oml=0$ Universe the time interval $t_{\rm
coll}-t_{\rm ta}=t_{\rm ta}$ should be equal to the free-fall time of a uniform sphere $t_{\rm ff}=\sqrt{3\pi/(32G\rho_{\rm ta})}$, which means that the mean density of perturbation at turnaround is $\rho_{\rm ta}=3\pi/(32Gt^2_{\rm ta})$ and $\rho_{\rm coll}=8\rho_{\rm
ta}=3\pi/(Gt_{\rm coll}^2)$. These densities can be compared with background mean matter densities at the corresponding times to get mean internal density contrasts: $\Delta=\rho/\bar{\rho}_{\rm m}$. In the Einstein-de Sitter model ($\Omm=1$, $\Oml=0$), background density evolves as $\bar{\rho}_{\rm m}=1/(6\pi G t^2)$, which means that density contrast after virialization is \_[vir]{}=18\^2=177.653.
For general cosmologies, density contrast can be computed by estimating $\rho_{\rm coll}$ and $\bar{\rho}_m(t_{\rm coll})$ in a similar fashion. For lower $\Omm$ models, fluctuation of the same mass $M$ and $\delta$ has a larger initial radius and smaller physical density and, thus, takes longer to collapse. The density contrasts of collapsed objects therefore are larger in lower density models because the mean density of matter at the time of collapse is smaller. Accurate (to $\lesssim 1\%$ for $\Omm=0.1-1$) approximations for $\Delta_{\rm vir}$ in open ($\Oml=0$) and flat $\Lambda$CDM ($1-\Oml-\Omm=0$) cosmologies are given by @bryan_norman98 [their equation 6]. For example, for the concordance $\Lambda$CDM cosmology with $\Omm=0.27$ and $\Oml=0.73$ [@komatsu_etal11], density contrast at $z=0$ is $\Delta_{\rm vir}\approx 358$.
Note that if the initial density contrast $\delta_{\rm i}$ would grow only at the linear rate, $D_+(z)$, then the density contrast at the time of collapse would be more than a hundred times smaller. Its value can be derived starting from the density contrast linearly extrapolated to the turn around epoch, $\delta_{\rm ta}$. This epoch corresponds to the time at which perturbation enters in the non–linear regime and detaches from the Hubble expansion, so that $\delta_{\rm ta}\sim 1$ is expected. In fact, the exact calculation in the case of $\Omm(z)=1$ at the redshift of turn–around gives $\delta_{ta}=1.062$ [@gunn_gott72]. Because $t_{\rm coll}=2t_{\rm ta}$, further linear evolution for $\Omm(z)=1$ until the collapse time gives $\dc=\delta_{\rm ta} D_+(t_c)/D_+(t_{\rm ta})\approx 1.686$. In the case of $\Omm\ne 1$ we expect that $\delta_{\rm ta}$ should have different values. For instance, for $\Omm< 1$ density contrast at turn–around should be higher to account for the higher rate of the Hubble expansion. However, linear growth from $t_{\rm ta}$ to $t_{\rm
coll}$ is smaller due to the slower redshift dependence of $D_+(z)$. As a matter of fact, these two factors nearly cancel, so that $\delta_c$ has a weak dependence on $\Omm$ and $\Oml$ [e.g., @percival05]. For the concordance $\Lambda$CDM cosmology at $z=0$, for example, $\delta_c\approx 1.675$.
Additional interesting effects may arise in models with DE characterized by small or zero speed of sound, in which structure growth is affected not only because DE influences linear growth, but also because it participates non-trivially in the collapse of matter and may slow down or accelerate the formation of clusters of a given mass depending on DE equation of state [@abramo_etal07; @creminelli_etal10]. DE in such models can also contribute non-trivially to the gravitating mass of clusters.
### The nonlinear mass scale $\Mnl$. {#sec:Mnl}
The linear value of the collapse overdensity $\delta_c$ is useful in predicting whether a given initial perturbation $\delta_i\ll 1$ at initial $z_{\rm i}$ collapses by some later redshift $z$. The collapse condition is simply $\delta_iD_{+0}(z)\geq \delta_c(z)$ and is used extensively to model the abundance and clustering of collapsed objects, as we discuss below in § \[sec:mf\]. The distribution of peak amplitudes in the initial Gaussian overdensity field smoothed over mass scale $M$ is given by a Gaussian PDF with a rms value of $\sigma(M)$ (Equation \[eq:pGaussian\]). The peaks in the initial Gaussian overdensity field smoothed at redshift $z_i$ over mass scale $M$ can be characterized by the ratio $\nu=\delta_i/\sigma(M,z_i)$ called the [*peak height*]{}. For a given mass scale $M$, the peaks collapsing at a given redshift $z$ according to the spherical collapse model have the peak height given by: . \[eq:nu\] Given that $\delta_c(z)$ is a very weak function of $z$ (changing by $\lesssim 1-2\%$ typically), whereas $\sigma(M,z)=\sigma(M,z=0)D_{+0}(z)$ decreases strongly with increasing $z$, the peak height of collapsing objects of a given mass $M$ increases rapidly with increasing redshift.
Using Equation \[eq:nu\] we can define the characteristic mass scale for which a typical peak ($\nu=1$) collapses at redshift $z$: (,z)=(,z=0)D\_[+0]{}(z)=\_c(z). \[eq:Mnl\] This [*nonlinear mass*]{}, $\Mnl(z)$, is a key quantity in the self-similar models of structure formation, which we consider in Section \[sec:selfsimilar\].
Nonlinear collapse of real density peaks {#sec:realcoll}
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The spherical collapse model provides a useful approximate guideline for the time scale of halo collapse and has proven to be a very useful tool in developing approximate statistical models for the formation and evolution of halo populations. Such a simple model and its extensions (e.g., ellipsoidal collapse model) do, however, miss many important details and complexities of collapse of the real density peaks. Such complexities are usually explored using three-dimensional numerical cosmological simulations. Techniques and numerical details of such simulations are outside the scope of this review and we refer readers to recent reviews on this subject [@bertschinger98; @dolag_etal08b; @norman10; @borgani_kravtsov11]. Here, we simply discuss the main features of gravitational collapse learned from analyses of such simulations.
Figure \[fig:clevol\] shows evolution of the DM density field in a cosmological simulation of a comoving region of $15h^{-1}$ Mpc on a side around cluster mass-scale density peak in the initial perturbation field from $z=3$ to the present epoch. The overall picture is quite different from the top-hat collapse. First of all, real peaks in the primordial field do not have the constant density or sharp boundary of the top-hat, but have a certain radial profile and curvature [@bardeen_etal86; @dalal_etal08]. As a result, different regions of a peak collapse at different times so that the overall collapse is extended in time and the peak does not have a single collapse epoch [e.g., @diemand_etal07]. Consequently, the distribution of matter around the collapsed peak can smoothly extend to several virial radii for late epochs and small masses [@prada_etal06; @cuesta_etal08]. This creates ambiguity about the definition of halo mass and results in a variety of mass definitions adopted in practice, as we discuss in Section \[sec:massdef\].
Second, the peaks in the smoothed density field, $\delta_{\rm R}({\bf
x})$, are not isolated but are surrounded by other peaks and density inhomogeneities. The tidal forces from the most massive and rarest peaks in the initial density field shepherd the surrounding matter into massive filamentary structures that connect them [@bond_etal96]. Accretion of matter onto clusters at late epochs occurs preferentially along such filaments, as can be clearly seen in Figure \[fig:clevol\].
Finally, the density distribution within the peaks in the actual density field is not smooth, as in the smoothed field $\delta_{\rm
R}({\bf x})$, but contains fluctuations on all scales. Collapse of density peaks on different scales can proceed almost simultaneously, especially during early stages of evolution in the CDM models when peaks undergoing collapse involve small scales, over which the power spectrum has an effective slope $n\approx -3$. Figure \[fig:clevol\] shows that at high redshifts the proto-cluster region contains mostly small-mass collapsed objects, which merge to form a larger and larger virialized system near the center of the shown region at later epochs. Nonlinear interactions between smaller-scale peaks within a cluster-scale peak during mergers result in relaxation processes and energy exchange on different scales, and mass redistribution. Although the processes accompanying major mergers are not as violent as envisioned in the violent relaxation scenario [@valluri_etal07], such interactions lead to significant redistribution of mass [@kazantzidis_etal06] and angular momentum [@vitvitska_etal02], both within and outside of the virial radius.
Equilibrium {#sec:hse}
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Following the collapse, matter settles into an equilibrium configuration. For collisional baryonic component this configuration is approximately described by the hydrostatic equilibrium (HE hereafter) equation, in which the pressure gradient $\nabla p(\bx)$ at point $\bx$ is balanced by the gradient of local gravitational potential $\nabla\phi(\bx)$: $\nabla\phi(\bx) = -\nabla
p(\bx)/\rho_{\rm g}(\bx)$, where $\rho_{\rm g}(\bx)$ is the gas density. Under the further assumption of spherical symmetry, the HE equation can be written as $\rho_{\rm g}^{-1}dp/dr=-GM(<r)/r^2$, where $M(<r)$ is the mass contained within the radius $r$. Assuming the equation of state of ideal gas, $p=\rho_{\rm g}k_BT/\mu m_p$ where $\mu$ is the mean molecular weight and $m_p$ is the proton mass; cluster mass within $r$ can be expressed in terms of the density and temperature profiles, $\rho_{\rm g}(r)$ and $T(r)$, as M\_[HE]{}( <r)=-[rk\_BT(r)Gm\_p]{} . \[eq:m\_he\] Interestingly, the slopes of the gas density and temperature profiles that enter the above equation exhibit correlation that appears to be a dynamical attractor during cluster formation [@juncher_etal12].
For a collisionless system of particles, such as CDM, the condition of equilibrium is given by the Jeans equation [e.g., @binney_tremaine08]. For a non–rotating spherically symmetric system, this equation can be written as M\_J(<r)=-[r\_r\^2G]{}, \[eq:j\_sphe2\] where $\beta=1-{\sigma_t^2\over 2\sigma_r^2}$ is the orbit anisotropy parameter defined in terms of the radial ($\sigma_r$) and tangential ($\sigma_t$) velocity dispersion components ($\beta=0$ for isotropic velocity field). We consider equilibrium density and velocity dispersion profiles, as well as anisotropy profile $\beta(r)$ in § \[sec:veldisp\]. Equation \[eq:j\_sphe2\] is also commonly used to describe the equilibrium of cluster galaxies. Although, in principle, galaxies in groups and clusters are not strictly collisionless, interactions between galaxies are relatively rare and the Jeans equation should be quite accurate.
Note that the difference between equilibrium configuration of collisional ICM and collisionless DM and galaxy systems is significant. In HE, the iso-density surfaces of the ICM should trace the iso-potential surfaces. The shape of the iso-potential surfaces in equilibrium is always more spherical than the shape of the underlying mass distribution that gives rise to the potential. Given that the potential is dominated by DM at most of the cluster-centric radii, the ICM distribution (and consequently the X-ray isophotes and SZ maps) will be more spherical than the underlying DM distribution.
As we noted in the previous section, the gravitational collapse of a halo is a process extended in time. Consequently, a cluster may not reach complete equilibrium over the Hubble time due to ongoing accretion of matter and the occurrence of minor and major mergers. The ICM reaches equilibrium state following a major merger only after $\approx
3-4$ Gyr [e.g., @nelson_etal12]. Deviations from equilibrium affect observable properties of clusters and cause systematic errors when equations \[eq:m\_he\] and \[eq:j\_sphe2\] are used to estimate cluster masses [e.g., @rasia_etal04; @nagai_etal07b; @ameglio_etal09; @piffaretti_valda08; @lau_etal09].
Internal structure of cluster halos {#sec:structure}
-----------------------------------
Relaxations processes establish the equilibrium internal structure of clusters. Below we review our current undertstanding of the equilibrium radial density distribution, velocity dispersion, and triaxiality (shape) of the cluster DM halos.
### Density Profile. {#sec:denpro}
Internal structure of collapsed halos may be expected to depend both on the properties of the initial density distribution around collapsing peaks [@hoffman_shaham85] and on the processes accompanying hierarchical collapse [e.g., @syer_white98; @valluri_etal07]. The fact that simulations have demonstrated that the characteristic form of the spherically averaged density profile arising in CDM models, characterized by the logarithmic slope steepening with increasing radius [@dubinski_carlberg91; @katz91; @navarro_etal95; @navarro_etal96], is virtually independent of the shape of power spectrum and background cosmology [@katz91; @cole_lacey96; @navarro_etal97; @huss_etal99b] is non trivial. Such a generic form of the profile also arises when small-scale structure is suppressed and the collapse is smooth, as is the case for halos forming at the cut-off scale of the power spectrum [@moore_etal99; @diemand_etal05; @wang_white09] or even from non-cosmological initial conditions [@huss_etal99].
The density profiles measured in dissipationless simulations are most commonly approximated by the “NFW” form proposed by @navarro_etal95 based on their simulation of cluster formation: \_[NFW]{}(r)=, xr/r\_s, where $r_s$ is the scale radius, at which the logarithmic slope of the profile is equal to $-2$ and $\rho_s$ is the characteristic density at $r=r_s$. Overall, the slope of this profile varies with radius as $d\ln\rho/d\ln r=-[1+2x/(1+x)]$, i.e., from the asymptotic slope of $-1$ at $x\ll 1$ to $-3$ at $x\gg 1$, where the enclosed mass diverges logarithmically: $M(<r)=M_{\Delta}f(x)/f(c_{\Delta})$, where $M_{\Delta}$ is the mass enclosing a given overdensity $\Delta$, $f(x)\equiv \ln(1+x)-x/(1+x)$ and $c_{\Delta}\equiv R_{\Delta}/r_s$ is the concentration parameter. Accurate formulae for the conversion of mass of the NFW halos defined for different values of $\Delta$ are given in the appendix of @hu_kravtsov03.
Subsequent simulations [@navarro_etal04; @merritt_etal06; @graham_etal06] showed that the @einasto65 profile and other similar models designed to describe de-projection of the Sérsic profile [@merritt_etal06] provide a more accurate description of the DM density profiles arising during cosmological halo collapse, as well as profiles of bulges and elliptical galaxies [@cardone_etal05]. The Einasto profile is characterized by the logarithmic slope that varies as a power law with radius: \_[ E]{}(r)=\_s, xr/r\_s, where $r_s$ is again the scale radius at which the logarithmic slope is $-2$, but now for the Einasto profile, $\rho_s\equiv\rho_{\rm E}(r_s)$, and $\alpha$ is an additional parameter that describes the power-law dependence of the logarithmic slope on radius: $d\ln\rho_{\rm E}/d\ln r=-2x^{\alpha}$.
Note that unlike the NFW profile and several other profiles discussed in the literature, the Einasto profile does not have an asymptotic slope at small radii. The slope of the density profile becomes increasingly shallower at small radii at the rate controlled by $\alpha$. The parameter $\alpha$ varies with halo mass and redshift: at $z=0$ galaxy-sized halos are described by $\alpha\approx 0.16$, whereas massive cluster halos are described by $\alpha\approx
0.2-0.3$; these values increase by $\sim 0.1$ by $z\approx 3$ [@gao_etal08]. Although $\alpha$ depends on mass and redshift (and thus also on the cosmology) in a non-trivial way, @gao_etal08 [and see also @duffy_etal08 [-@duffy_etal08]] showed that these dependencies can be captured as a universal dependence on the peak height $\nu=\delta_c/\sigma(M,z)$ (see Section \[sec:Mnl\] above): $\alpha=0.0095\nu^2+0.155$. Finally, unlike the NFW profile, the total mass for the Einasto profile is finite due to the exponentially decreasing density at large radii. A number of useful expressions for the Einasto profile, such as mass within a radius, are provided by @cardone_etal05, @mamon_lokas05, and @graham_etal06.
The origin of the generic form of the density profile has recently been explored in detail by @lithwick_dalal11, who show that it arises due to two main factors: [*(a)*]{} the density and triaxiality profile of the original peak and [*(b)*]{} approximately adiabatic contraction of the previously collapsed matter due to deepening of the potential well during continuing collapse. Without adiabatic contraction the profile resulting from the collapse would reflect the shape of the initial profile of the peak. For example, if the initial profile of mean linear overdensity within radius $r$ around the peak can be described as $\bar{\delta}_{\rm L}\propto r^{-\gamma}_{\rm L}$, it can be shown that the resulting differential density profile after collapse without adiabatic contraction behaves as $\rho(r) \propto
r^{-g}$, where $g=3\gamma/(1+\gamma)$ [@fillmore_goldreich84]. Typical profiles of initial density peaks are characterized by shallow slopes, $\gamma\sim
0-0.3$ at small radii, and very steep slopes at large radii [e.g., @dalal_etal08], which means that resulting profiles after collapse should have slopes varying from $g\approx
0-0.7$ at small radii to $g\approx 3$ at large radii.
However, @lithwick_dalal11 showed that contraction of particle orbits during subsequent accretion of mass interior to a given radius $r$ leads to a much more gradual change of logarithmic slope with radius, such that the regime within which $g\approx 0-0.7$ is shifted to very small radii ($r/r_{\rm vir}\lesssim 10^{-5}$), whereas at the radii typically resolved in cosmological simulations the logarithmic slope is in the range of $g\approx 1-3$, so that the radial dependence of the logarithmic slope $g(r)=d\ln\rho/d\ln r$ is in good qualitative agreement with simulation results. This contraction occurs because matter that is accreted by a halo at a given stage of its evolution can deposit matter over a wide range of radii, including small radii. The orbits of particles that accreted previously have to respond to the additional mass, and they do so by contracting. For example, for a purely spherical system in which mass is added slowly so that the adiabatic invariant is conserved, radii $r$ of spherical shells must decrease to compensate an increase of $M(<r)$. This model thus elegantly explains both the qualitative shape of density profiles observed in cosmological simulations and their universality. The latter can be expected because the contraction process crucial to shaping the form of the profile should operate under general collapse conditions, in which different shells of matter collapse at different times.
Although the model of @lithwick_dalal11 provides a solid physical picture of halo profile formation, it also neglects some of the processes that may affect details of the resulting density profile, most notably the effects of mergers. Indeed, major mergers lead to resonant dynamical heating of a certain fraction of collapsed matter due to the potential fluctuations and tidal forces that they induce. The amount of mass that is affected by such heating is significant [e.g., @valluri_etal07]. In fact, up to $\sim 40\%$ of mass within the virial radii of merging halos may end up outside of the virial radius of the merger. This implies, for example, that virial mass is not additive in major mergers. Nevertheless, in practice the merger remnant retains the functional form of the density profiles of the merger progenitors [@kazantzidis_etal06], which means that major mergers do not lead to efficient violent relaxation.
Although the functional form of the density profile arising during halo collapse is generic for a wide variety of collapse conditions and models, initial conditions and cosmology do significantly affect the physical properties of halo profiles such as its characteristic density and scale radius [@navarro_etal97]. These dependencies are often discussed in terms of halo concentrations, $c_{\Delta}\equiv R_{\Delta}/r_s$. Simulations show that the scale radius is approximately constant during late stages of halo evolution [@bullock_etal01; @wechsler_etal02], but evolves as $r_s=c_{\rm
min}\,R_{\Delta}$ during early stages, when a halo quickly increases its mass through accretion and mergers [@zhao_etal03b; @zhao_etal09]. The minimum value of concentration is $c_{\rm min}=\rm const\approx 3-4$ for $\Delta=200$. For massive cluster halos, which are in the fast growth regime at any redshift, the concentrations are thus expected to stay approximately constant with redshift or may even increase after reaching a minimum [@klypin_etal11; @prada_etal11].
The characteristic time separating the two regimes can be identified as the formation epoch of halos. This time approximately determines the value of the scale radius and the subsequent evolution of halo concentration. The initial conditions and cosmology determine the formation epoch and the typical mass accretion histories for halos of a given mass [@navarro_etal97; @bullock_etal01; @zhao_etal09], and therefore determine the halo concentrations. Although these dependences are non-trivial functions of halo mass and redshift, they can also be encapsulated by a universal function of the peak height $\nu$ [@zhao_etal09; @prada_etal11].
Baryon dissipation and feedback are expected to affect the density profiles of halos appreciably, although predictions for these effects are far less certain than predictions of the DM distribution in the purely dissipationless regime. The main effect is contraction of DM in response to the increasing depth of the central potential during baryon cooling and condensation, which is often modelled under the assumption of slow contraction conserving adiabatic invariants of particle orbits [e.g., @zeldovich_etal80; @barnes_white84; @blumenthal_etal86; @ryden_gunn87]. The standard model of such [*adiabatic contraction*]{} assumes that DM particles are predominantly on circular orbits, and for each shell of DM at radius $r$ the product of the radius and the enclosed mass $rM(r)$ is conserved [@blumenthal_etal86]. The model makes a number of simplifying assumptions and does not take into account effects of mergers. Nevertheless, it was shown to provide a reasonably accurate description of the results of cosmological simulations [@gnedin_etal04]. Its accuracy can be further improved by relaxing the assumption of circular orbits and adopting an empirical ans[ä]{}tz, in which the conserved quantity is $rM(\bar{r})$, where $\bar{r}$ is the average radius along the particle orbit, instead of $rM(r)$ [@gnedin_etal04]. At the same time, several recent studies showed that no single set of parameters of such simple models describes all objects that form in cosmological simulations equally well [@gustafsson_etal06; @abadi_etal10; @tissera_etal10; @gnedin_etal11].
A more subtle but related effect is the increase of the overall concentration of DM within the virial radius of halos due to re-distribution of binding energy between DM and baryons during the process of cluster assembly [@rudd_etal08]. The larger range of radii over which this effect operates makes it a potential worry for the precision constraints from the cosmic shear power spectrum [@jing_etal06; @rudd_etal08]. This effect depends primarily on the fraction of baryons that condense into the central halo galaxies and may be mitigated by the blow-out of gas by efficient AGN or SN feedback [@van_daalen_etal11]. The effects of baryons on the overall concentration of mass distribution in clusters are thus uncertain, but can potentially increase halo concentration and thereby significantly enhance the cross section for strong lensing [@puchwein_etal05; @rozo_etal08; @mead_etal10] and affect statistics of strong lens distribution in groups and clusters [e.g., @more_etal11].
A number of studies have derived observational constraints on density profiles of clusters and their concentrations [@pointecouteau_etal05; @vikhlinin_etal06; @schmidt_allen07; @buote_etal07; @mandelbaum_etal08; @wojtak_lokas10; @okabe_etal10; @ettori_etal10; @umetsu_etal11a; @umetsu_etal11b; @sereno_zitrin11]. Although most of these studies find that the concentrations of galaxy clusters predicted by $\Lambda$CDM simulations are in the ballpark of values derived from observations, the agreement is not perfect and there is tension between model predictions and observations, which may be due to effects of baryon dissipation [e.g., @rudd_etal08; @fedeli12].
Some studies do find that the concordance cosmology predictions of the average cluster concentrations are somewhat lower than the average values derived from X-ray observations [@schmidt_allen07; @buote_etal07; @duffy_etal08]. Moreover, lensing analyses indicate that the slope of the density profile in central regions of some clusters may be shallower than predicted [@tyson_etal98; @sand_etal04; @sand_etal08; @newman_etal09; @newman_etal11], whereas concentrations are considerably higher than both theoretical predictions and most other observational determinations from X-ray and WL analyses [@comerford_natarajan07; @oguri_etal09; @oguri_etal11; @zitrin_etal11].
At this point, it is not clear whether these discrepancies imply serious challenges to the $\Lambda$CDM structure formation paradigm, unknown baryonic effects flattening the profiles in the centers, or unaccounted systematics in the observational analyses [e.g., @dalal_keeton03; @hennawi_etal07]. When considering such comparisons, it is important to remember that density profiles in cosmological simulations are always defined with respect to the center defined as the global density peak or potential minimum, whereas in observations the corresponding location is not as unambiguous as in simulations and the choice of center may affect the derived slope.
It should be noted that improved theoretical predictions for cluster-sized systems generally predict larger concentrations for the most massive objects than do extrapolations of the concentration-mass relations from smaller mass objects [@zhao_etal09; @prada_etal11; @bhattacharya_etal11a]. In addition, as we noted above, the evolution predicted for the concentrations of these rarest objects is much weaker than $c\propto (1+z)$ found for smaller mass halos, so rescaling the concentrations of high-redshift clusters by $(1+z)$ factor, as is often done, could lead to an overestimate of their concentrations.
### Velocity dispersion profile and velocity anisotropy. {#sec:veldisp}
Velocity dispersion profile is a halo property related to its density profile. Simulations show that this profile generally increases from the central value to a maximum at $r\approx r_s$ and slowly decreases outward [e.g., @cole_lacey96; @rasia_etal04]. One remarkable result illustrating the close connection between density and velocity dispersion is that for collapsed halos in dissipationless simulations the ratio of density to the cube of the rms velocity dispersion can be accurately described by a power law over at least three decades in radius [@taylor_navarro01]: $Q(r)\equiv \rho/\sigma^3\propto
r^{-\alpha}$ with $\alpha\approx 1.9$.
An important quantity underlying the measured velocity dispersion profile is the profile of the mean velocity, and the mean radial velocity, $\bar{v}_{\rm r}$, in particular. For a spherically symmetric matter distribution in HE, we expect $\bar{v}_{\rm
r}=0$. Therefore, the profile of $\bar{v}_{\rm r}$ is a useful diagnostic of deviations from equilibrium at different radii. Simulations show that clusters at $z=0$ generally have zero mean radial velocities within $r\approx R_{\rm vir}$ and turn sharply negative between $1$ and $\approx 3R_{\rm vir}$, where density is dominated by matter infalling onto cluster [@cole_lacey96; @eke_etal98; @cuesta_etal08].
The distinguishing characteristic between gas and DM is the fact that gas has an isotropic velocity dispersion tensor on small scales, whereas DM in general does not. On large scales, however, both gas and DM may have velocity fields that are anisotropic. The degree of velocity anisotropy is commonly quantified by the anisotropy profile, $\beta(r)$ (see § \[sec:hse\]). DM anisotropy is mild: $\beta\approx 0-0.1$ near the center and increases to $\beta\approx
0.2-0.4$ near the virial radius [@cole_lacey96; @eke_etal98; @colin_etal00; @rasia_etal04; @lemze_etal11]. Interestingly, velocities exhibit substantial tangential anisotropy outside the virial radius in the infall region of clusters [@cuesta_etal08; @lemze_etal11]. Another interesting finding is that the velocity anisotropy correlates with the slope of the density profile [@hansen_moore06], albeit with significant scatter [@lemze_etal11].
The gas component also has some residual motions driven by mergers and gas accretion along filaments. Gas velocities tend to have tangential anisotropy [@rasia_etal04], because radial motions are inhibited by the entropy profile, which is convectively stable in general.
### Shape. {#sec:shape}
Although the density structure of mass distribution in clusters is most often described by spherically averaged profiles, clusters are thought to collapse from generally triaxial density peaks [@doroshkevich70; @bardeen_etal86]. The distribution of matter within halos resulting from hierarchical collapse is triaxial as well [@frenk_etal88; @dubinski_carlberg91; @warren_etal92; @cole_lacey96; @jing_suto02; @kasun_evrard05; @allgood_etal06], with triaxiality predicted by [*dissipationless*]{} simulations increasing with decreasing distance from halo center [@allgood_etal06]. Triaxiality of halos decreases with decreasing mass and redshift [@kasun_evrard05; @allgood_etal06] in a way that again can be parameterized in a universal form as a function of peak height [@allgood_etal06]. The major axis of the triaxial distribution of clusters is generally aligned with the filament connecting a cluster with its nearest neighbor of comparable mass [e.g., @west_etal00; @lee_etal08], which reflects the fact that a significant fraction of mass and mergers is occurring along such filaments [e.g., @onuora_thomas00; @lee_evrard07].
@jing_suto02 showed how the formalism of density distribution as a function of distance from cluster center can be extended to the density distribution in triaxial shells. Accounting for such triaxiality is particularly important in theoretical predictions and observational analyses of weak and strong lensing [@dalal_keeton03; @clowe_etal04; @oguri_etal05; @corless_king07; @hennawi_etal07; @becker_kravtsov11]. At the same time, it is important to keep in mind that, as with many other results derived mainly from dissipationless simulations, the physics of baryons may modify predictions substantially.
The shape of the DM distribution in particular is quite sensitive to the degree of central concentration of mass. As baryons condense towards the center to form a central galaxy within a halo, the DM distribution becomes more spherical [@dubinski94; @evrard_etal94; @tissera_etal98; @kazantzidis_etal04]. The effect increases with decreasing radius, but is substantial even at half of the virial radius [@kazantzidis_etal04]. The main mechanism behind this effect lies in adiabatic changes of the shapes of particle orbits in response to more centrally concentrated mass distribution after baryon dissipation [@dubinski94; @debattista_etal08].
In considering effects of triaxiality, it is important to remember that triaxiality of the hot intracluster gas and DM distribution are different [gas is rounder, see, e.g., discussion in @lau_etal11 and references therein]. This is one of the reasons why mass proxies defined within spherical aperture using observable properties of gas (see § \[sec:regul\] below) exhibit small scatter and are much less sensitive to cluster orientation.
The observed triaxiality of the ICM can be used as a probe of the shape of the underlying potential [@lau_etal11] and as a powerful diagnostic of the amount of dissipation that is occurring in cluster cores [@fang_etal09] and of the mass of the central cluster galaxy [@lau_etal12].
Mass definitions {#sec:massdef}
----------------
As we discussed above, the existence of a particular density contrast delineating a halo boundary is predicted only in the limited context of the spherical collapse of a density fluctuation with the top-hat profile (i.e., uniform density, sharp boundary). Collapse in such a case proceeds on the same time scale at all radii and the collapse time and “virial radius” are well defined. However, the peaks in the initial density field are not uniform in density, are not spherical, and do not have a sharp boundary. Existence of a density profile results in different times of collapse for different radial shells. Note also that even in the spherical collapse model the virial density contrast formally applies only at the time of collapse; after a given density peak collapses its internal density stays constant while the reference (i.e., either the mean or critical) density changes merely due to cosmological expansion. The actual overdensity of the collapsed top-hat initial fluctuations will therefore grow larger than the initial virial overdensity at $t>t_{\rm collapse}$.
The triaxiality of the density peak makes the tidal effects of the surrounding mass distribution important. Absence of a sharp boundary, along with the effects of non-uniform density, triaxiality and nonlinear effects during the collapse of smaller scale fluctuations within each peak, result in a continuous, smooth outer density profile without a well-defined radial boundary. Although one can identify a radial range, outside of which a significant fraction of mass is still infalling, this range is fairly wide and does not correspond to a single well-defined radius [@eke_etal98; @cuesta_etal08]. The boundary based on the virial density contrast is, thus, only loosely motivated by theoretical considerations.
The absence of a well-defined boundary of collapsed objects makes the definition of the halo boundary and the associated enclosed mass ambiguous. This explains, at least partly, the existence of various halo boundary and mass definitions in the literature. Below we describe the main two such definitions: the Friends-of-Friends (FoF) and spherical overdensity [SO, see also @white01]. The FoF mass definition is used almost predominantly in analyses of cosmological simulations of cluster formation, whereas the SO halo definition is used both in observational and simulation analyses, as well as in analytic models, such as the Halo Occupation Distribution (HOD) model. Although other definitions of the halo mass are discussed, theoretical mass determinations often have to conform to the observational definitions of mass. Thus, for example, although it is possible to define the entire mass that will ever collapse onto a halo in simulations [@cuesta_etal08; @anderhalden_diemand11], it is impossible to measure this mass in observations, which makes it of interest only from the standpoint of the theoretical models of halo collapse.
### The Friends-of-Friends mass. {#sec:fof}
Historically, the FoF algorithm was used to define groups and clusters of galaxies in observations [@huchra_geller82; @press_davis82; @einasto_etal84] and was adopted to define collapsed objects in simulations of structure formation [@einasto_etal84; @davis_etal85]. The FoF algorithm considers two particles to be members of the same group (i.e., “friends”), if they are separated by a distance that is less than a given linking length. Friends of friends are considered to be members of a single group – the condition that gives the algorithm its name. The linking length, the only free parameter of the method, is usually defined in units of the mean interparticle separation: $b=l/\bar{l}$, where $l$ is the linking length in physical units and $\bar{l}=\bar{n}^{-1/3}$ is the mean interparticle separation of particles with mean number density of $\bar{n}$.
Attractive features of the FoF algorithm are its simplicity (it has only one free parameter), a lack of any assumptions about the halo center, and the fact that it does not assume any particular halo shape. Therefore, it can better match the generally triaxial, complex mass distribution of halos forming in the hierarchical structure formation models.
The main disadvantages of the FoF algorithm are the difficulty in theoretical interpretation of the FoF mass, and sensitivity of the FoF mass to numerical resolution and the presence of substructure. For the smooth halos resolved with many particles the FoF algorithm with $b=0.2$ defines the boundary corresponding to the fixed [*local*]{} density contrast of $\delta_{\rm FoF}\approx 81.62$ [@more_etal11]. Given that halos forming in hierarchical cosmologies have concentrations that depend on mass, redshift, and cosmology, the [*enclosed*]{} overdensity of the FoF halos also varies with mass, redshift and cosmology. Thus, for example, for the current concordance cosmology the FoF halos (defined with $b=0.2$) of mass $10^{11}-10^{15}\ \Msun$ have enclosed overdensities of $\sim
450-350$ at $z=0$ and converge to overdensity of $\sim 200$ at high redshifts where concentration reaches its minimum value of $c\approx
3-4$ [@more_etal11]. For small particle numbers the boundary of the FoF halos becomes “fuzzier” and depends on the resolution (and so does the FoF mass). Simulations most often have fixed particle mass and the number of particles therefore changes with halo mass, which means that properties of the boundary and mass identified by the FoF are mass dependent. The presence of substructure in well-resolved halos further complicates the resolution and mass dependence of the FoF-identified halos [@more_etal11]. Furthermore, it is well known that the FoF may spuriously join two neighboring distinct halos with overlapping volumes into a single group. The fraction of such neighbor halos that are “bridged” increases significantly with increasing redshift.
### The Spherical Overdensity mass. {#sec:so}
The spherical overdensity algorithm defines the boundary of a halo as a sphere of radius enclosing a given density contrast $\Delta$ with respect to the reference density $\rho$. Unlike the FoF algorithm the definition of an SO halo also requires a definition of the halo center. The common choices for the center in theoretical analyses are the peak of density, the minimum of the potential, the position of the most bound particle, or, more rarely, the center of mass. Given that the center and the boundary need to be found simultaneously, an iterative scheme is used to identify the SO boundary around a given peak. The radius of the halo boundary, $\Rdc$, is defined by solving the implicit equation M(<r)=(z) r\^3, \[eq:mdef\] where $M(<r)$ is the total mass profile and $\rho(z)$ is the reference physical density at redshift $z$ and $r$ is in physical (not comoving) radius.
The choice of $\Delta$ and the reference $\rho$ may be motivated by theoretical considerations or by observational limitations. For example, one can choose to define the enclosed overdensity to be equal to the “virial” overdensity at collapse predicted by the spherical collapse model, $\Delta\rho=\Delta_{\rm vir,c}\rho_{\rm crit}$ (see Section \[sec:sphcoll\]). Note that in $\Omm\ne 1$ cosmologies, there is a choice for reference density to be either the critical density $\rhoc(z)$ or the mean matter density $\rhom(z)$ and both are in common use. The overdensities defined with respect to these reference densities, which we denote here as $\Dc$ and $\Dm$, are related as $\Dm=\Dc/\Omm(z)$. Note that $\Omm(z)=\Omega_{\rm
m0}(1+z)^3/E^2(z)$, where $E(z)$ is given by Equation \[eq:Ea\]. For concordance cosmology, $1-\Omm(z)<0.1$ at $z\geq 2$ and the difference between the two definitions decreases at these high redshifts. In observations, the choice may simply be determined by the extent of the measured mass profile. Thus, masses derived from X-ray data under the assumption of HE are limited by the extent of the measured gas density and temperature profiles and are therefore often defined for the high values of overdensity: $\Delta_{\rm c}=2,500$ or $\Delta_{\rm c}=500$.
The crucial difference from the FoF algorithm is the fact that the SO definition forces a spherical boundary on the generally non-spherical mass definition. In addition, spheres corresponding to different halos may overlap, which means that a certain fraction of mass may be double counted [although in practice this fraction is very small, see, e.g., discussion in § 2.2 of @tinker_etal08].
The advantage of the SO algorithm is the fact that the SO-defined mass can be measured both in simulations and observational analyses of clusters. In the latter the SO mass can be estimated from the total mass profile derived from the hydrostatic and Jeans equilibrium analysis for the ICM gas and galaxies, respectively (see Section \[sec:hse\] above), or gravitational lensing analyses [e.g., @vikhlinin_etal06; @hoekstra07]. Furthermore, suitable observables that correlate with the SO mass with scatter of $\mincir 10\%$ can be defined (see § \[sec:regul\] below), thus making this mass definition preferable in the cosmological interpretation of observed cluster populations. The small scatter shows that the effects of triaxiality is quite small in practice. Note, however, that the definition of the halo center in simulations and observations may not necessarily be identical, because in observations the cluster center is usually defined at the position of the peak or the centroid of X-ray emission or SZ signal, or at the position of the BCG.
Abundance of halos {#sec:mf}
------------------
Contrasting predictions for the abundance and clustering of collapsed objects with the observed abundance and clustering of galaxies, groups, and clusters has been among the most powerful validation tests of structure formation models [e.g., @press_schechter74; @blumenthal_etal84; @kaiser84; @kaiser86].
Although real clusters are usually characterized by some quantity derived from observations (an [*observable*]{}), such as the X-ray luminosity, such quantities are generally harder to predict [*ab initio*]{} in theoretical models because they are sensitive to uncertain physical processes affecting the properties of cluster galaxies and intracluster gas. Therefore, the predictions for the abundance of collapsed objects are usually quantified as a function of their mass, i.e., in terms of the mass function $dn(M,z)$ defined as the comoving volume number density of halos in the mass interval $[M,M+dM]$ at a given redshift $z$. The predicted mass function is then connected to the abundance of clusters as a function of an observable using a calibrated mass-observable relation (discussed in § \[sec:regul\] below). Below we review theoretical models for halo abundance and underlying reasons for its approximate universality.
### The mass function and its universality.
The first statistical model for the abundance of collapsed objects as a function of their mass was developed by @press_schechter74. The main powerful principle underlying this model is that the mass function of objects resulting from nonlinear collapse can be tied directly and uniquely to the statistical properties of the initial [*linear*]{} density contrast field $\delta({\bx})$.
Statistically, one can define the probability $F(M)$ that a given region within the initial overdensity field smoothed on a mass scale $M$, $\delta_{\rm M}({\bf x})$, will collapse into a halo of mass $M$ or larger: F(M)=\^\_[-1]{}p()C\_[coll]{}() d, \[eq:FM\] where $p(\delta)d\delta$ is the PDF of $\delta_{\rm M}({\bf x})$, which is given by Equation \[eq:pGaussian\] for the Gaussian initial density field, and $C_{\rm coll}$ is the probability that any given point ${\bf x}$ with local overdensity $\delta_{\rm M}({\bf x})$ will actually collapse. The mass function can then be derived as a fraction of the total volume collapsing into halos of mass $(M,M+dM)$, i.e., $dF/dM$, divided by the comoving volume within the initial density field occupied by each such halo, i.e., $M/\bar{\rho}_{\rm}$: =. \[eq:dndMdef\]
In their pioneering model, @press_schechter74 have adopted the ans[ä]{}tz motivated by the spherical collapse model (see § \[sec:collbasics\]) that any point in space with $\delta_{\rm M}({\bf x})D_{+0}(z)\geq \delta_c$ will collapse into a halo of mass $\geq M$ by redshift $z$: i.e., $C_{\rm coll}(\delta)=\Theta(\delta-\delta_c)$, where $\Theta$ is the Heaviside step function. Note that $\delta_{\rm M}({\bf x})$ used above is not the actual initial overdensity, but the initial overdensity evolved to $z=0$ with the linear growth rate. One can easily check that for a Gaussian initial density field this assumption gives $F(M)=\frac{1}{2}{\rm erfc}[\delta_c/(\sqrt{2}\sigma(M,z))]=F(\nu)$. This line of arguments and assumptions thus leads to an important conclusion that [*the abundance of halos of mass $M$ at redshift $z$ is a universal function of only their peak height*]{} $\nu(M,z)\equiv\delta_c/\sigma(M,z)$. In particular, the fraction of mass in halos per logarithmic interval of mass in such a model is: $$\frac{dn(M)}{d\ln M}=\frac{\bar{\rho}_{\rm m}}{M}\left\vert\frac{dF}{d\ln M}\right\vert=\frac{\bar{\rho}_{\rm m}}{M}\left\vert\frac{d\ln\nu}{d\ln M}\frac{\partial F}{\partial\ln\nu}\right\vert\equiv\frac{\bar{\rho}_{\rm m}}{M}\left\vert\frac{d\ln\nu}{d\ln M}\right\vert\,g(\nu)\equiv \frac{\bar{\rho}_{\rm m}}{M}\,\psi(\nu).
\label{eq:PSmf}$$
Clearly, the shape $\psi(\nu)$ in such models is set by the assumptions of the collapse model. Numerical studies based on cosmological simulations have eventually revealed that the shape $\psi_{\rm PS}(\nu)$ predicted by the @press_schechter74 model deviates by $\gtrsim 50\%$ from the actual shape measured in cosmological simulations [e.g., @klypin_etal95; @gross_etal98; @tormen98; @lee_shandarin99; @sheth_tormen99; @jenkins_etal01].
A number of modifications to the original ans[ä]{}tz have been proposed, which result in $\psi(\nu)$ that more accurately describes simulation results. Such modifications are based on the collapse conditions that take into account asphericity of the peaks in the initial density field [@monaco95; @audit_etal97; @lee_shandarin98; @sheth_tormen02; @desjacques08] and stochasticity due to the dependence of the collapse condition on peak properties other than $\nu$ or shape [e.g., @desjacques08; @maggiore_riotto10; @desimone_etal11; @ma_etal11; @corasaniti_achitouv10]. The more sophisticated excursion set models match the simulations more closely, albeit at the expense of more assumptions and parameters. There may be also inherent limitations in the accuracy of such models given that they rely on the strong assumption that one can parameterize all the factors influencing collapse of any given point in the initial overdensity field in a relatively compact form. In the face of complications to a simple picture of peak collapse, as discussed in Section \[sec:realcoll\], one can indeed expect that the excursion set ansätze are limited in how accurately they can ultimately describe the halo mass function.
### Calibrations of halo mass function in cosmological simulations.
An alternative route to derive predictions for halo abundance accurately is to calibrate it using large cosmological simulations of structure formation. Simulations have generally confirmed the remarkable fact that the abundance of halos can be parameterized via a universal function of peak height $\nu$ [@sheth_tormen99; @jenkins_etal01; @evrard_etal02; @white02; @warren_etal06; @reed_etal07; @lukic_etal07; @tinker_etal08; @crocce_etal10; @courtin_etal11; @bhattacharya_etal11b]. Note that in many studies the linear overdensity for collapse is assumed to be constant across redshifts and cosmologies and the mass function is therefore quantified as a function of $\sigma^{-1}$ – the quantity proportional to $\nu$. However, as pointed out by @courtin_etal11 it is necessary to include the redshift and cosmology dependence of $\delta_c(z)$ for an accurate description of the mass function across cosmologies. Even though $\delta_c$ varies only by $\sim 1-2\%$, it enters into halo abundance via an exponent and such small variations can result in variations in the mass function of several per cent or more.
The main efforts with simulations have thus been aimed at improving the accuracy of the $\psi(\nu)$ functional form, assessing systematic uncertainties related to the mass definition, and quantifying deviations from the universality of $\psi(\nu)$ for different redshifts and cosmologies. The mass function, and especially its exponential tail, is sensitive to the specifics of halo mass definition, a point emphasized strongly in a number of studies [@jenkins_etal01; @white02; @tinker_etal08; @cohn_white08; @klypin_etal11]. Thus, in precision cosmological analyses using an observed cluster abundance, care must be taken to ensure that the cluster mass definition matches that used in the calibration of the halo mass function.
Predictions for the halo abundance as a function of the SO mass for a variety of overdensities used to define the SO boundaries, accurate to better than $\approx 5-10\%$ over the redshift interval $z=[0,2]$, were presented by @tinker_etal08. In Figure \[fig:fnu\] we compare the form of the function $\psi(\nu)$ calibrated through simulations by different researchers and compared to $\psi(\nu)$ predicted by the Press-Schechter model, and to the calibration of the functional form based on the ellipsoidal collapse ans[ä]{}tz by @sheth_etal01.
These calibrations of the mass function through N–body simulations provide the basis for the use of galaxy clusters as tools to constrain cosmological models through the growth rate of perturbations (see the recent reviews by @allen_etal11 and @weinberg_etal12). As we discuss in Section \[sec:cosmo\] below, similar calibrations can be extended to models with non-Gaussian initial density field and models of modified gravity.
Future cluster surveys promise to provide tight constraints on cosmological parameters, thanks to the large statistics of clusters with accurately inferred masses. The potential of such surveys clearly requires a precise calibration of the mass function, which currently represents a challenge. Deviations from universality at the level of up to $\sim 10\%$ have been reported [@reed_etal07; @lukic_etal07; @tinker_etal08; @cohn_white08; @crocce_etal10; @courtin_etal11]. In principle, a precise calibration of the mass function is a challenging but tractable technical problem, as long as it only requires a large suite of [*dissipationless*]{} simulations for a given set of cosmological parameters, and an optimal interpolation procedure [e.g., @lawrence_etal10].
A more serious challenge is the modelling of uncertain effects of baryon physics: baryon collapse, dissipation, and dynamical evolution, as well as feedback effects related to energy release by the SNe and AGN, may lead to subtle redistribution of mass in halos. Such redistribution can affect halo mass and thereby halo mass function at the level of a few per cent [@rudd_etal08; @stanek_etal09; @cui_etal11], although the exact magnitude of the effect is not yet certain due to uncertainties in our understanding of the physics of galaxy formation in general, and the process of condensation and dynamical evolution in clusters in particular.
Clustering of halos {#sec:bias}
-------------------
Galaxy clusters are clustered much more strongly than galaxies themselves. It is this strong clustering discovered in the early 1980s [@klypin_kopylov83; @bahcall_soneira83] that led to the development of the concept of bias in the context of Gaussian initial density perturbation field [@kaiser84]. Linear bias of halos is the coefficient between the overdensity of halos within a given sufficiently large region and the overdensity of matter in that region: $\delta_{\rm h}=b\delta_{\rm}$, with $b$ defined as the bias parameter. For the Gaussian perturbation fields, local linear bias is independent of scale [@scherrer_weinberg98], such that the large–scale power spectrum and correlation function on large scales can be expressed in terms of the corresponding quantities for the underlying matter distribution as $P_{\rm
hh}(k)=b^2P_{\rm mm}(k)$ and $\xi_{\rm hh}(r)=b^2\xi_{\rm mm}(r)$, respectively. As we discuss in Section \[sec:cosmo\], this is not true for non-Gaussian initial perturbation fields [@dalal_etal08] or for models with scale–dependent linear growth rate [@parfrey_etal11], in which cases the linear bias is generally scale-dependent.
In the context of the hierarchical structure formation, halo bias is closely related to the overall abundance of halos discussed above, as illustrated by the “peak-background split” framework [@kaiser84; @cole_kaiser89; @mo_white96; @sheth_tormen99], in which the linear halo bias is obtained by considering a Lagrangian patch of volume $V_0$, mass $M_0$, and overdensity $\delta_0$ at some early redshift $z_0$. The bias is calculated by requiring that the abundance of collapsed density peaks within such a patch is described by the same function $\psi(\nu_p)$ as the mean abundance of halos in the Universe, but with peak height $\nu_p$ appropriately rescaled with respect to the overdensity of the patch and relative to the rms fluctuations on the scale of the patch. Thus, the functional form of the bias dependence on halo mass, $b_h(M)$, depends on the functional form of the mass function explicitly in this framework. Simulations show that the peak-background split model provides a fairly accurate (to $\sim
20\%$) prediction for the linear halo bias [@tinker_etal10].
Another line of argument illustrating the close connection between the halo abundance and bias is the fact that if one assumes that all of the mass is in the collapsed halos, as is done for example in the halo models [@cooray_sheth02], the requirement that matter in the Universe is not biased against itself implies that $\int b(\nu)g(\nu)d\ln\nu=1$ [e.g., @seljak00], where $g(\nu)\equiv \vert d\ln\nu/d\ln M\vert^{-1}\psi(\nu)$ (see eq. \[eq:PSmf\]). This integral constraint requires that the form of the bias function $b(\nu)$ is changed whenever $\psi(\nu)$ changes. Incidentally, the close connection between $b(\nu)$ and $\psi(\nu)$ implies that if $\psi(\nu)$ is a universal function, then the bias $b(\nu)$ should be a universal function as well.
The function $b(\nu)$ recently calibrated for the SO-defined halos of different overdensities using a suite of large cosmological simulations with accuracy $\lesssim 5\%$ and satisfying the integral constraint [@tinker_etal10] is shown in Figure \[fig:bnu\] for halos defined using an overdensity of $\Delta=200$ with respect to the mean. This calibration of the bias is compared to the corresponding prediction of the @press_schechter74 and the @sheth_etal01 ansätze. The figure shows that $b(\nu)$ is a rather weak function of $\nu$ at $\nu<1$, but steepens substantially for rare peaks of $\nu>1$. It also shows that the rarest clusters ($\nu \sim 5$) in the Universe can have the amplitude of the correlation function or power spectrum that is two orders of magnitude larger than the clustering amplitude of the galaxy-sized halos ($\nu\lesssim 1$).
Self–similar evolution of galaxy clusters {#sec:selfsimilar}
-----------------------------------------
In the previous sections we have considered processes that govern the collapse of matter during cluster formation, the transition to equilibrium and the equilibrium structure of matter distribution in collapsed halos. In the following sections, we consider baryonic processes that shape the observable properties of clusters, such as their X-ray luminosity or the temperature of the ICM. However, before we delve into the complexities of such physical processes, it is instructive to introduce the simplest models based on assumptions of self-similarity, in which the number of control parameters is minimal. We discuss the assumptions and predictions of the self-similar model in some detail because parametric scalings that it predicts are in wide use to interpret results from both cosmological simulations of cluster formation and observations.
### Self-similar model: assumptions and basic expectations. {#sec:ssm}
The self-similar model developed by @kaiser86 makes three key assumptions. The first assumption is that clusters form via gravitational collapse from peaks in the initial density field in an Einstein–de-Sitter Universe, $\Omega_m=1$. Gravitational collapse in such a Universe is scale-free, or [*self-similar*]{}. The second assumption is that the amplitude of density fluctuations is a power-law function of their size, $\Delta(k)\propto k^{3+n}$. This means that initial perturbations also do not have a preferred scale (i.e., they are scale-free or self-similar). The third assumption is that the physical processes that shape the properties of forming clusters do not introduce new scales in the problem. With these assumptions the problem has only two control parameters: the normalization of the power spectrum of the initial density perturbations at an initial time, $t_{\rm i}$, and its slope, $n$. Properties of the density field and halo population at $t>t_{\rm i}$ (or corresponding redshift $z<z_{\rm i}$), such as typical halo masses that collapse or halo abundance as a function of mass, depend only on these two parameters. One can choose any suitable variable that depends on these two parameters as a characteristic variable for a given problem. For evolution of halos and their abundance, the commonly used choice is to define the characteristic nonlinear mass, $\Mnl$ (see Section \[sec:collbasics\]), which encapsulates such dependence. The halo properties and halo abundance then become universal functions of $\mu\equiv M/\Mnl$ in such model. Thus, for example, clusters with masses $M_1(z_1)$ and $M_2(z_2)$ that correspond to the same ratio $M_1(z_1)/\Mnl(z_1)=M_2(z_2)/\Mnl(z_2)$ have the same dimensionless properties, such as gas fraction or concentration of their mass distribution.
As we have discussed above, in more general cosmologies the halo properties and mass function should be universal functions of the peak height $\nu$, which encapsulates the dependence on the shape and normalization of the power spectra for general, non power-law shapes of the fluctuation spectrum.
### The Kaiser model for cluster scaling relations. {#sec:ssrel}
In the this Section, we define cluster mass to be the mass within the sphere of radius $R$, encompassing characteristic density contrast, $\Delta$, with respect to some reference density $\rho_r$ (usually either $\rhoc$ or $\rhom$): $M=(4\pi/3)\Delta\rho_r R^3$. In this definition, radius and mass are directly related and interchangeable. The model assumes spherical symmetry and that the ICM is in equilibrium within the cluster gravitational potential, so that the HE equation (eq. \[eq:m\_he\]) holds. The mass $M(<R)$ derived from the HE equation is proportional to $T(R)R$ and the sum of the logarithmic slopes of the gas density and temperature profiles at $R$. [*In addition*]{} to the assumptions about self-similarity discussed above, a key assumption made in the model by @kaiser86 is that these slopes are independent of $M$, so that T(\_r)\^M\^. \[eq:TMo\] Note that formally the quantity $T$ appearing in the above equation is the temperature measured at $R$, whereas some average temperature at smaller radii is usually measured in observations. However, if we parameterize the temperature profile as $T(r)=\Ts\tT(x)$, where $\Ts$ is the characteristic temperature and $\tT$ is the dimensionless profile as a function of dimensionless radius $x\equiv r/R$, and we assume that $\tT(x)$ is independent of $M$, any temperature averaged over the same fraction of radial range $[x_1,x_2]$ will scale as $\propto \Ts\propto T(R)$. The latter is not strictly true for the “spectroscopic” temperature, $\Tx$, derived by fitting an observed X–ray spectrum to a single–temperature bremsstrahlung model [@mazzotta_etal04; @vikhlinin06], although in practice deviations of $\Tx$ from the expected behavior for $\Ts$ are small.
The gas mass within $R$ can be computed by integrating over the gas density profile, which, by analogy with temperature, we parameterize as $\rhog(r)=\rhogs\trhog(x)$, where $\rhogs$ is the characteristic density and $\trhog$ is the dimensionless profile. The gas mass within $R$ can then be expressed as (<R)=4R\^3\^[1]{}\_0x\^2(x)dx=3M I\_M(<R). \[eq:MgMo\] The latter proportionality is assumed in the @kaiser86 model, which means that $\rhogs$ and $I_{\rhog}$ are assumed to be independent of $M$. Note that $\rhogs\propto \Delta\rho_r$, so $\Delta\rho_r$ does not enter the $\Mg$-$M$ relation.
Using the scalings of $\Mg$ and $T$ with mass, we can construct other cluster properties of interest, such as the luminosity of ICM emitted due to its radiative cooling. Assuming that the ICM emission is due to the free-free radiation and neglecting the weak logarithmic dependence of the Gaunt factor on temperature, the bolometric luminosity can be written as [e.g., @sarazin86]: \^2T\^V T\^\^M\^. \[eq:LbolM\] We omit $\rho_r$ in these equations for clarity; it suffices to remember that $\rho_r$ enters into the scaling relations exactly as $\Delta$. Note that the bolometric luminosity of a cluster is not observable directly, and the X-ray luminosity in soft band (e.g., $0.5-2$ keV), $\Lsx$, is frequently used. Such soft band X-ray luminosity is almost insensitive to temperature at $T>2$ keV [@fabricant_gorenstein83 as can be easily verified with a plasma emission code], so that its temperature dependence can be neglected. $\Lsx$ then scales as: \^2 VM. At temperatures $T<2$ keV temperature dependence is more complicated both for the bolometric and soft X-ray emissivity due to significant flux in emission lines. Therefore, strictly speaking, for lower mass systems the above $L-M$ scaling relations are not applicable, and scaling of the emissivity with temperature needs to be calibrated separately taking also into account the ICM metallicity. The same is true for luminosity defined in some other energy band or for the bolometric luminosity.
Another quantity of interest is the ICM “entropy” defined in X-ray analyses as K\_[X]{}\^[-2/3]{}T\^[-1/3]{}M\^[2/3]{}. \[eq:KMo\] where $n_e$ is the electron number density. Finally, the quantity, $Y=M_{\rm g}T$ where gas mass and temperature are measured within a certain range of radii scaled to $R_\Delta$, is used to characterize the ICM in the analyses of SZ and X-ray observations. This quantity is expected to be a particularly robust proxy of the cluster mass [e.g., @dasilva_etal04; @motl_etal05; @nagai06; @kravtsov_etal06; @fabjan_etal11 see also discussion in Section \[sec:regul\]] because it is proportional to the global thermal energy of ICM. Using Equations \[eq:MgMo\] and \[eq:TMo\] the scaling of $Y$ with mass in the self-similar model is YT\^[1/3]{}M\^[5/3]{}. \[eq:YMo\]
Note that the redshift dependence in the normalization of the scaling relations introduced above [*is due solely to the particular SO definition of mass*]{} and associated redshift dependence of $\Delta\rho_r$. In $\Omm\ne 1$ cosmologies, there is a choice of either defining the mass relative to the mean matter density or critical density (Section \[sec:so\]). This specific, [ *arbitrary*]{} choice determines the specific redshift dependence of the observable-mass relations. It is clear that this evolution due to $\Delta(z)$ factors has no deep physical meaning. However, the [*absence*]{} of any additional redshift dependence in the normalization of the scaling relations is just the consequence of the assumptions of the @kaiser86 model and is a physical reflection of these assumptions.
Extra evolution can, therefore, be expected if one or more of the assumptions of the self–similar model is violated. This can be due to either actual physical processes that break self-similarity [*or*]{} the fact that some of the model assumptions are not accurate. We discuss physical processes that lead to the breaking of self-similarity in subsequent sections. Here below we first consider possible deviations that may arise because assumptions of the Kaiser model do not hold exactly, i.e. deviations not ascribed to physical processes that explicitly violate self-similarity.
### Extensions of the Kaiser model.
Going back to equations \[eq:TMo\] and \[eq:MgMo\], we note that the specific scaling of $T\propto M^{2/3}$ and $\Mg\propto M$ will only hold, if the assumption that the dimensionless temperature and gas density profiles, $\tT(x)$ and $\trhog(x)$, are independent of $M$ holds. In practice, however, some mass dependence of these profiles is expected. For example, if the concentration of the gas distribution depends on mass similarly to the concentration of the DM profile [@ascasibar_etal06], the weak mass dependence of DM concentration implies weak mass dependence of $\trhog(x)$ and $\tT(x)$. Indeed, concentration depends on mass even in purely self-similar models [@cole_lacey96; @navarro_etal97]. These dependencies imply that predictions of the Kaiser model may not describe accurately even the purely self-similar evolution. This is evidenced by deviations of scaling relation evolution from these predictions in hydrodynamical simulations of cluster formation even in the absence of any physical processes that can break self-similarity [e.g., @nagai06; @stanek_etal10].
In addition, the characteristic gas density, $\rhogs$, may be mildly modified by a mass-dependent, non self-similar process during some early stage of evolution. If such a process does not introduce a pronounced mass scale and is confined to some early epochs (e.g., owing to shutting off of star formation in cluster galaxies due to AGN feedback and gas accretion suppression), then subsequent evolution may still be described well by the self-similar model. The Kaiser model is just the simplest [*specific*]{} example of a more general class of self-similar models, and can therefore be extended to take into account deviations described above.
For simplicity, let us assume that the scalings of gas mass and gas mass fraction against total mass can be expressed as a power law of mass: =M\^[1+]{}, f\_[g]{}=M\^=\^\^. \[eq:fga\] This does not violate the self-similarity of the problem per se, as long as dimensionless properties of an object, such as $\fgas$, remain a function of only the dimensionless mass $\mu(z)\equiv M/\Mnl(z)$. This means that the normalization of the $\Mg-M$ relation must scale as $\Cg\propto \Mnl^{-\ag}$ during the self-similar stages of evolution, such that =\^[-]{}(z)M\^[1+]{}=M\^. \[eq:MgMa\] Note that self-similarity requires that the slope $\ag$ does not evolve with redshift.
By analogy with the $\Mg-M$ relation, we can assume that the $T-M$ relation can be well described by a power law of the form T=\^M\^[+]{}, \[eq:TME\] where $\aT$ describes mild deviation from the scaling due, e.g., to mild dependence of gas and temperature profile slopes in the HE equation. The dimensionless quantities involving temperature $T$ can be constructed using ratios of $T$ with $T_{\rm NL}=(\mu m_p/k_B)G\Mnl/R_{\rm NL}\propto \Delta^{1/3}\Mnl^{2/3}$. As before for the gas fraction, requirement that such dimensionless ratio depends only on $\mu$ requires $\CT\propto\Mnl^{-\aT}$ so that T=C\_[To]{}\^[-]{}\^M\^[+]{}=C\_[To]{}\^M\^\^, \[eq:TME2\]
Other observable-mass scaling relations can be constructed in the manner similar to the derivation of the original relations above. These are summarized below for the specific choice of $\Delta\rho_r\equiv \Dc\rhoc(z)\propto h^2E^2(z)$: $$\begin{aligned}
\Mg&\propto& \Mnl^{-\ag}(z)\Mdc^{1+\ag},\label{eq:MgMEa}\\
T&\propto& E(z)^{2/3}\Mnl^{-\aT}\Mdc^{2/3+\aT},\label{eq:TMEa}\\
\Lbol&\propto& E(z)^{7/3}\Mnl^{-2\ag-\aT/2}(z)\Mdc^{4/3+2\ag+\frac{\aT}{2}},\label{eq:LbolMEa}\\
K&\propto&E^{-2/3}(z)\Mnl^{\frac{2}{3}\ag-\aT}(z)\Mdc^{\frac{2}{3}(1-\ag)+\aT},\label{eq:SMEa}\\
Y&\propto& E(z)^{2/3}(z)\Mnl^{-\ag-\aT}(z)\Mdc^{5/3+\ag+\aT}.
\label{eq:YMEa}\end{aligned}$$ In all of the relations one can, of course, recover the original relations by setting $\ag=\aT=0$. The observable-mass relations can be used to predict observable-observable relations by eliminating mass from the corresponding relations above.
Note that the evolution of scaling relations in this extended model arises both from the redshift dependence of $\Delta(z)\rho_r(z)$ and from the extra redshift dependence due to factors involving $\Mnl$. The practical implication is that if measurements show that $\ag\ne 0$ and/or $\aT\ne 0$ at some redshift, the original Kaiser scaling relations are not expected to describe the evolution, even if the evolution is self-similar. Instead, relations given by Equations \[eq:MgMEa\]-\[eq:YMEa\] should be used. Note that at $z\approx 0$, observations indicate that within the radius $r_{500}$ enclosing overdensity $\Delta_c=500$, $\ag\approx 0.1-0.2$, while $\aT\approx 0.-0.1$. Therefore, the extra evolution compared to the Kaiser model predictions due to factors involving $\ag$ and, to a lesser degree, factors involving $\aT$ is expected. Such evolution, consistent with predictions of the above equations, is indeed observed both in simulations [see, e.g., Fig. 10 in @vikhlinin_etal09a] and in observations [@lin_etal12 although see @boehringer_etal11 [-@boehringer_etal11]].
In practice, evolution of the scaling relations can be quite a bit more complicated than the evolution predicted by the above equations. The complication is not due to any deviation from self-similarity but rather due to specific mass definition and the fact that cluster formation is an extended process that is not characterized by a single collapse epoch. Some clusters evolve only mildly after their last major merger. However, the mass of such clusters will change with $z$ even if their potential does not change, simply because mass definition is tied to a reference density that changes with expansion of the Universe and because density profiles of clusters extends smoothly well beyond the virial radius. Any observable property of clusters that has radial profile differing from the mass profile, but which is measured within the same $R_{\Delta}$, will change differently than mass with redshift. As a simplistic toy model, consider a population of clusters that does not evolve from $z=1$ to $z=0$. Their X-ray luminosity is mostly due to the ICM in the central regions of clusters and it is not sensitive to the outer boundary of integration as long as it is sufficiently large. Thus, X-ray luminosity of such a non-evolving population will not change with $z$, but masses $M_{\Delta}$ of clusters will increase with decreasing $z$ as the reference density used to define the cluster boundary decreases. Normalization of the $L_{\rm X}-M_{\Delta}$ relation will thus decrease with decreasing redshift simply due to the definition of mass. The strength of the evolution will be determined by the slope of the mass profile around $R_{\Delta}$, which is weakly dependent on mass. Such an effect may, thus, result in the evolution of both the slope and normalization of the relation. In this respect, quantities that have radial profiles most similar to the total mass profile (e.g., $\Mg$, $Y$) will suffer the least from such spurious evolution.
Finally, we note again that in principle for general non power-law initial perturbation spectra of the CDM models the scaling with $M/\Mnl$ needs to be replaced with scaling with the peak height $\nu$. For clusters within a limited mass range, however, the power spectrum can be approximated by a power law and thus a characteristic mass similar to $\Mnl$ can be constructed, although such mass should be within the typical mass range of the clusters. The latter is not true for $\Mnl$, which is considerably smaller than typical cluster mass at all $z$.
### Practical implications for observational calibrations of scaling relations.
In observational calibrations of the cluster scaling relations, it is often necessary to rescale between different redshifts either to bring results from the different $z$ to a common redshift, or because the scaling relation is evaluated using clusters from a wide range of redshifts due to small sample size. It is customary to use predictions of the Kaiser model to carry out such rescaling to take into account the redshift dependence of $\Delta(z)$. In this context, one should keep in mind that these predictions are approximate due to the approximate nature of some of the assumptions of the model, as discussed above. Inaccuracies introduced by such scalings may, for example, then be incorrectly interpreted as intrinsic scatter about the scaling relation.
In addition, because the $\Delta(z)$ factors are a result of an arbitrary mass definition, they should not be interpreted as physically meaningful factors describing evolution of mass. For example, in the $T$-$M$ relation, the $\Delta^{1/3}$ factor in Equation \[eq:TMo\] arises due to the [*dimensional*]{} $M/R$ factor of the HE equation. As such, this factor does not change even if the power-law index of the $T$-$M$ relation deviates from $2/3$, in which case the relation has the form $T\propto \Delta^{1/3}M^{2/3+\alpha_T}$. In other words, if one fits for the parameters of this relation, such as normalization $A$ and slope $B$, using measurements of temperatures and masses for a sample of clusters spanning a range of redshifts, the proper parameterization of the fit should be =A\^[1/3]{}()\^B, =A\^[1/3]{}()\^[2/3+B]{}, where $T_{\rm p}$ and $M_{\rm p}$ are appropriately chosen pivots. The parameterization $T/T_{\rm p}=A(\Delta^{1/3}M/M_{\rm p})^B$ that is sometimes adopted in observational analyses [*is not correct*]{} in the context of the self-similar model. In other words, only the observable quantities should be rescaled by the $\Delta\rho_r$ factors, and not the mass. Likewise, only the $\Delta\rho_r$ factors actually predicted by the Kaiser model should be present in the scalings. For example, no such factor is predicted for the $\Mg$-$M$ relation and therefore the gas and total masses of clusters at different redshifts should not be scaled by $\Delta\rho_r$ factors in the fits of this relation.
Finally, we note that observational calibrations of the observable-mass scaling relations generally depend on the distances to clusters and are therefore cosmology dependent. Such dependence arises because distances are used to convert observed angular scale to physical scale within which an “observable” is defined, $R=\theta d_A(z)\propto
\theta h^{-1}$, or to convert observed flux $f$ to luminosity, $L=4\pi fd_L(z)$, where $d_A(z)$ and $d_L(z)=d_A(z)(1+z)^2$ are the angular diameter distance and luminosity distance, respectively. Thus, if the total mass $M$ of a cluster is measured using the HSE equation, we have $M_{\rm HE}\propto TR\propto d_A\propto h^{-1}$. The same scaling is expected for the mass derived from the weak lensing shear profile measurements.
If the gas mass is measured from the X-ray flux from a volume $V\propto R^3\propto \theta^3d_A^3$, which scales as $f=\Lx/(4\pi
d_L^2)\propto \rhog^2 V/d_L^2\propto \Mg^2/(Vd_L^2)\propto
\Mg^2/(\theta d_L^2 d_A^3)$ and where $f$ and $\theta$ are observables, gas mass then scales with distance as $\Mg\propto d_L
d_A^{3/2}\propto h^{5/2}$. This dependence can be exploited to constrain cosmological parameters, as in the case of X–ray measurements of gas fractions in clusters [@ettori_etal03; @allen_etal04; @laroque_etal06; @allen_etal08; @ettori_etal09] or abundance evolution of clusters as a function of their observable [e.g., @vikhlinin_etal09b]. In this respect, the $\Mg-M$ relation has the strongest scaling with distance and cosmology, whereas the scaling of the $T-M$ relation is the weakest [e.g., see discussion by @vikhlinin_etal09a].
Cluster formation and Thermodynamics of the Intra-cluster gas {#sec:thermo}
-------------------------------------------------------------
Gravity that drives the collapse of the initial large-scale density peaks affects not only the properties of the cluster DM halos, but also the thermodynamic properties of the intra-cluster plasma. The latter are also affected by processes related to galaxy formation, such as cooling and feedback. Below, we discuss the thermodynamic properties of the ICM resulting from gravitational heating, radiative cooling, and stellar and AGN feedback during cluster formation.
### Gravitational collapse of the intra-cluster gas.
The diffuse gas infalling onto the DM-dominated potential wells of clusters converts the kinetic energy acquired during the collapse into thermal energy via adiabatic compression and shocks. As gas settles into HE, its temperature approaches values close to the virial temperature corresponding to the cluster mass. In the spherically symmetric collapse model of [@1985ApJS...58...39B], supersonic accretion gives rise to the expanding shock at the interface of the inner hydrostatic gas with a cooler, adiabatically compressed, external medium. Real three-dimensional collapse of clusters is more complicated and exhibits large deviations from spherical symmetry, as accretion proceeds both in a quasi-spherical fashion from low-density regions and along relatively narrow filaments. The gas accreting along the latter penetrates much deeper into the cluster virial region and does not undergo a shock at the virial radius (see Fig. \[fig:shocks\]). The strong shocks are driven not just by the accretion of gas from the outside but also “inside-out” during major mergers [e.g., @poole_etal07].
The shocks arising during cluster formation can be classified into two broad categories: strong external shocks surrounding filaments and the virialized regions of DM halos and weaker internal shocks, located within the cluster virial radius [e.g., @pfrommer_etal06; @skillman_etal08; @vazza_etal09]. The strong shocks arise in the high-Mach number flows of the intergalactic gas, whereas weak shocks arise in the relatively low-Mach number flows of gas in filaments and accreting groups, which was pre-heated at earlier epochs by the strong shocks surrounding filaments or external groups. The left panel of Figure \[fig:shocks\] shows these two types of shocks in a map of the shocked cells identified in a cosmological adaptive mesh refinement simulation of a region surrounding a galaxy cluster [from @vazza_etal09], along with the gas velocity field. This map highlights the strong external shocks, characterized by high Mach numbers ${\mathcal M}> 30$, surrounding the cluster at several virial radii from the cluster center, and weaker internal shocks, with ${\mathcal M}\mincir 2-3$. The cluster is shown at the epoch immediately following a major merger, which generated substantial velocities of gas within virial radius. As we discuss in § \[sec:regul\] below, incomplete thermalization of these gas motions is one of the main sources of non–thermal pressure support in the ICM.
The right panel of Figure \[fig:shocks\] shows the distribution of the kinetic energy processed by shocks, as a function of the local shock Mach number, for different redshifts [@skillman_etal08]. The figure shows that a large fraction of the kinetic energy is processed by weak internal shocks and this fraction increases with decreasing redshift as more and more of the accreting gas is pre-heated in filaments. Yet, the left panel of Fig. \[fig:shocks\] highlights that large–${\mathcal M}$ shocks surround virialized halos in such a way that most gas particles accreted in a galaxy cluster must have experienced at least one strong shock in their past.
Becasuse gravity does not have a characteristic length scale, we expect the predictions of the self–similar model, presented in Section \[sec:selfsimilar\], to apply when gravitational gas accretion determines the thermal properties of the ICM. The scaling relations and their evolution predicted by the self-similar model are indeed broadly confirmed by the non-radiative hydrodynamical simulations that include only gravitational heating [e.g., @navarro_etal95; @eke_etal98; @nagai_etal07b], although some small deviations arising due to small differences in the dynamics of baryons and DM were also found [@ascasibar_etal06; @nagai06; @stanek_etal10].
As discussed in Section \[sec:obsprop\], observations carried out with the [*Chandra*]{} and [*XMM*]{}–Newton telescopes during the past decade showed that the outer regions of clusters ($r\gtrsim
r_{2500}$) exhibit self-similar scaling, whereas the core regions exhibit strong deviations from self-similarity. In particular, gas density in the core regions of small-mass clusters is lower than expected from self-similar scaling of large-mass systems. These results indicate that some additional non-gravitational processes are shaping properties of the ICM. We review some of these processes studied in cluster formation models below.
### Phenomenological pre-heating models.
The first proposed mechanism to break self–similarity was high-redshift ($z_h\magcir 3$) pre–heating by non–gravitational sources of energy, presumably by a combined action of the AGN and stellar feedback [@kaiser91; @evrard_henry91]. The specific extra heating energy per unit mass, $E_h$, defines the temperature scale $T^*\propto E_h/k_B$, such that clusters with virial temperature $T_{\rm vir}>T^*$ should be left almost unaffected by the extra heating, whereas in smaller clusters with $T_{\rm vir}<T^*$ gas accretion is suppressed. As a result, gas density is relatively lower in lower massive systems, especially at smaller radii, while their entropy will be higher.
Both analytical models [e.g. @tozzi_norman01; @babul_etal02; @voit_etal03] and hydrodynamical simulations [e.g. @bialek_etal01; @borgani_etal02; @muanwong_etal02] have demonstrated that with a suitable pre-heating prescription and typical heating injection of $E_h\sim 0.5$–1 keV per gas particle self–similarity can be broken to the degree required to reproduce observed scaling relations. Studies of the possible feedback mechanisms show that such amounts of energy cannot be provided by SNe [e.g., @kravtsov_yepes00; @renzini00; @borgani_etal04; @kay_etal07; @henning_etal09], and must be injected by the AGN population [e.g., @wu_etal00; @lapi_etal05; @bower_etal08] or by some other unknown source.
However, regardless of the actual sources of heating, strong widespread heating at high redshifts would conflict with the observed statistical properties of the Lyman–$\alpha$ forest [@shang_etal07; @borgani_viel09]. Moreover, hydrodynamical simulations have demonstrated that simple pre–heating models predict large isentropic cores [e.g., @borgani_etal05; @younger_bryan07] and shallow pressure profiles [@kay_etal12]. This is at odds with the entropy and pressure profiles of real clusters which exhibit smoothly declining entropy down to $r\sim 10-20$ kpc [e.g., @cavagnolo_etal09; @arnaud_etal10].
### The role of radiative cooling.
The presence of galaxies in clusters and low levels of the ICM entropy in cluster cores are a testament that radiative cooling has operated during cluster formation in the past and is an important process shaping thermodynamics of the core gas at present. Therefore, in general radiative cooling cannot be neglected in realistic models of cluster formation. Given that cooling generally introduces new scales, it can break self–similarity of the ICM even in the absence of heating [@voit_bryan01]. In particular, cooling removes low–entropy gas from the hot ICM phase in the cluster cores, which is replaced by higher entropy gas from larger radii. Somewhat paradoxically, the cooling thus leads to an entropy increase of the hot, X-ray emitting ICM phase. This effect is illustrated in Figure \[fig:entrmaps\], which shows the entropy maps in the simulations of the same cluster with and without cooling. In the absence of cooling (left panel), the innermost region of the cluster is filled by low–entropy gas. Merging substructures also carry low–entropy gas, which generates comet–like features by ram–pressure stripping, and is hardly mixed in the hotter ambient of the main halo. In the simulation with radiative cooling (right panel), most of the low–entropy gas associated with substructures and the central cluster region is absent, and most of the ICM has a relatively high entropy.
A more quantitative analysis of the entropy distribution for these simulated clusters is shown in Figure \[fig:simul\_profs\], in which the entropy profiles of clusters simulated with inclusion of different physical processes are compared with the baseline analytic spherical accretion model; this model predicts the power-law entropy profile $K(r)\propto r^{1.1}$ [e.g. @tozzi_norman01; @voit05]. The figure shows that the entropy profile in the simulation with radiative cooling is significantly higher than that of the non-radiative simulation. The difference in entropy is as large as an order of magnitude in the inner regions of the cluster and is greater by a factor of two even at $r_{500}$.
Interestingly, the predicted level of entropy at $r\sim
r_{2500}-r_{500}$ in the simulations with cooling (but no significant heating) is consistent with the ICM entropy inferred from X-ray observations. However, this agreement is likely to be spurious because it is achieved with the amount of cooling that results in conversion of $\approx 40\%$ of the baryon mass in clusters into stars and cold gas, which is inconsistent with observational measurements of cold fraction varying from $\simeq 20-30\%$ for small-mass, X-ray-emitting clusters to $\mincir 10\%$ for massive clusters (see § \[sec:obsprop\]).
Finally, note that inclusion of cooling in simulations with pre-heating discussed above usually results in problematic star-formation histories. In fact, if pre-heating takes place at a sufficiently high redshift, clusters exhibit excessive cooling at lower redshifts, as pre-heated gas collapses and cools at later epochs compared to the simulations without pre-heating [e.g. @tornatore_etal03]. These results highlight the necessity to treat cooling and heating processes simultaneously using heating prescriptions that can realistically reproduce the heating rate of the ICM gas as a function of cosmic time. We discuss efforts in this direction next.
### Thermodynamics of the intracluster medium with stellar and active galactic nuclei feedback. {#sec:feedb}
The results discussed above strongly indicate that, in order to reproduce the overall properties of clusters, cooling should be modelled together with a realistic prescription for non-gravitational heating. This is particularly apparent in the cluster cores, where a steady heating is required to offset the ongoing radiative cooling observed in the form of strong X-ray emission [see, e.g., @peterson_fabian06]. Studies of the feedback processes in clusters is one of the frontiers in cluster formation modelling. Although we do not yet have a complete picture of the ICM heating, a number of interesting and promising results have been obtained.
In Figure \[fig:simul\_profs\], the solid line shows the effect of the SN feedback on the entropy profile. In these simulations, the kinetic feedback of SNe is included in the form of galactic winds carrying the kinetic energy comparable to all of the energy released by Type-II SNe expected to occur according to star formation in the simulation. This energy partially compensates for the radiative losses in the central regions, which leads to a lower level of entropy in the core. However, the core ICM entropy in these simulations is still considerably higher than observed [e.g., @sun_etal09]. The inefficiency of the SN feedback in offsetting the cooling sufficiently is also evidenced by temperature profiles.
Figure \[fig:Tprofs\] [from @leccardi_molendi08] compares the observed temperature profiles of a sample of local clusters with results from simulations that include the SN feedback. The figure shows that simulations reproduce the observed temperature profile at $r\magcir 0.2r_{180}$. The overall shape of the profile at these large radii is reproduced by simulations including a wide range of physical processes, including non-radiative simulations [e.g., @loken_etal02; @borgani_etal04; @nagai_etal07b]. At large radii, however, the observed and predicted profiles do not match: The profiles in simulated clusters continue to increase to the smallest resolved radii, whereas the observed profiles reach a maximum temperature $T_{\rm max}\approx 2T_{180}$ and then decrease with decreasing radius to temperatures of $\sim 0.1-0.3T_{\rm max}$. The high temperatures of the central gas reflects its high entropy and is due to the processes affecting the entropy, as discussed above.
Another indication that the SN feedback alone is insufficient is the fact that the stellar mass of the BCGs in simulations that include only the feedback from SNe is a factor of two to three larger than the observed stellar masses. For example, in the simulated clusters shown in figure \[fig:entrmaps\], the baryon fraction in stars within $r_{500}$ decreases from $\simeq 40\%$ in simulations without SN feedback to $\simeq 30\%$, which is still a factor of two larger than observational measurements. The overestimate of the stellar mass is reflected in the overestimate of the ICM metallicity in cluster cores [e.g., @borgani_etal08a and references therein].
Different lines of evidence indicate that energy input from the AGN in the central cluster galaxies can provide most of the energy required to offset cooling [see @mcnamara_nulsen07 for a comprehensive review]. Because the spatial and temporal scales resolved in cosmological simulations are larger than those relevant for gas accretion and energy input, the AGN energy feedback can only be included via a phenomenological prescription. Such prescriptions generally model the feedback energy input rate by assuming the Bondi gas accretion rate onto the SMBHs, included as the sink particles, and incorporate a number of phenomenological parameters, such as the radiative efficiency and the feedback efficiency, which quantify the fraction of the radiated energy that thermally couples to the surrounding gas [e.g., @springel_etal05b]. The values of these parameters are adjusted so that simulations reproduce the observed relation between black hole mass and the velocity dispersion of the host stellar bulge [e.g., @marconi_hunt03]. An alternative way of implementing the AGN energy injection is the AGN–driven winds, which shock and heat the surrounding gas [e.g., @omma_etal04; @dubois_etal11; @gaspari_etal11].
In general, simulations of galaxy clusters based on different variants of these models have shown that the AGN feedback can reduce star formation in massive cluster galaxies and reduce the hot gas content in the poor clusters and groups, thereby improving agreement with the observed relation between X–ray luminosity and temperature [e.g., @sijacki_etal07; @puchwein_etal08]. Figure \[fig:abund\_match\] (from @martizzi_etal11) shows that simulations with the AGN feedback results in stellar masses of the BCGs that agree with the masses required to match observed stellar masses of galaxies and masses of their DM halos predicted by the models. The figure also shows that stellar masses are over-predicted in the simulations without the AGN feedback. Incidentally, the large-scale winds at high redshifts and stirring of the ICM in cluster cores by the AGN feedback also help to bring the metallicity profiles into cluster simulations in better agreement with observations [@fabjan_etal10; @mccarthy_etal10].
Although results of simulations with the AGN feedback are promising, simulations so far have not been able to convincingly reproduce the observed thermal structure of cool cores. As an example, Figure \[fig:simul\_profs\] shows that the entropy profiles in such simulations still develop large constant entropy core inconsistent with observed profiles. Interestingly, the adaptive mesh refinement simulations with jet-driven AGN feedback by @dubois_etal11 reproduce the monotonically decreasing entropy profiles inferred from observations. However, such agreement only exists if radiative cooling does not account for the metallicity of the ICM; in simulations that take into account the metallicity dependence of the cooling rates the entropy profile still develosp a large constant entropy core.
The presence of a population of relativistic particles in AGN–driven high–entropy bubbles has been suggested as a possible solution to this problem [@guo_oh08; @sijacki_etal08]. A relativistic plasma increases the pressure support available at a fixed temperature and can, therefore, help to reproduce the observed temperature and entropy profiles in core regions. However, it remains to be seen whether the required population of the cosmic rays is consistent with available constraints inferred from $\gamma$ and radio observations of clusters [e.g., @brunetti11 and references therein]. A number of additional processes, such as thermal conduction [e.g., @narayan_medvedev01] or dynamical friction heating by galaxies [@elzant_etal04] have been proposed. Generally, these processes cannot effectively regulate cooling in clusters by themselves [e.g., @dolag_etal04; @conroy_ostriker08], but they may play an important role when operating in concert with the AGN feedback [@voit11] or instabilities in plasma [e.g., @sharma_etal12].
In summary, results of the theoretical studies discussed above indicate that the AGN energy feedback is the most likely energy source regulating the stellar masses of cluster galaxies throughout their evolution and suppressing cooling in cluster cores at low redshifts. The latter likely requires an interplay between the AGN feedback and a number of other physical processes: e.g., injection of the cosmic rays in the high–entropy bubbles, buoyancy of these bubbles stabilized by magnetic fields, dissipation of their mechanical energy through turbulence, thermal conduction, and thermal instabilities. Although details of the interplay are not yet understood, it is clear that it must result in a robust self-regulating feedback cycle in which cooling immediately leads to the AGN activity that suppresses further cooling for a certain period of time.
Regularity of the cluster populations {#sec:regul}
=====================================
Processes operating during cluster formation and evolution discussed in the previous section are complex and nonlinear. However, it is now also clear that most of the complexity is confined to cluster cores and affects a small fraction of volume and mass of the clusters. In this regime, clusters’ observational properties exhibit strong deviations from the self–similar scalings described in Section \[sec:selfsimilar\] (see also @voit05). At larger radii, ICM is remarkably regular. In this section, we discuss the origins of such highly regular behaviour and the range of radii where it can be expected. We argue that the existence of this radial range allows us to define integral observational quantities, which have low scatter for clusters of a given mass that are not sensitive to the astrophysical processes operating during cluster formation and evolution. This fact is especially important for the current and future uses of clusters as cosmological probes [@allen_etal11; @weinberg_etal12].
Characterizing regularity {#sec:chareg}
-------------------------
A number of observational evidences, based on X–ray measurements of gas density [e.g. @croston_etal08] and temperature profiles [@vikhlinin_etal06; @pratt_etal07; @leccardi_molendi08], and the combination of the two in the form of entropy profile [@cavagnolo_etal09], demonstrate that clusters have a variety of behaviors in central regions, depending on the presence and prominence of cool cores. As discussed in Section \[sec:obsprop\], outside of core regions, clusters behave as a more homogeneous population and obey assumptions and expectations of the self–similar model (discussed above in \[sec:ssm\]). For instance, Figure \[fig:obsICM\] shows that the ICM density is nearly independent of temperature once measured outside of core regions $r\magcir r_{2500}$, at least for relatively hot systems with $T\magcir 3$ keV. Quite remarkably, observed and simulated temperature profiles agree with each other within this same radial range (see Figure \[fig:Tprofs\]).
A good illustration of the regularity of the ICM properties is represented by the pressure profiles shown in Figure \[fig:press\_arnaud\] (from @arnaud_etal10, but see also @sun_etal11) rescaled to the values of radius and pressure at $r_{500}$. The perfectly regular, self-similar behavior would correspond to a single line in this plot for clusters of all masses. The pressure profiles shown in this figure are derived from X–ray observations and are defined as the product of electron number density and temperature profiles. Similar profiles are now derived from SZ observations, which probe pressure more directly [e.g., @bonamente_etal12]. Quite remarkably, the observed pressure profiles at $r\magcir 0.2r_{500}$ follow a nearly universal profile [see also @nagai_etal07b], exhibiting fractional scatter of $\lesssim 30\%$ at $r\sim 0.2r_{500}$ and even smaller scatter of $\sim 10-15\%$ at $r\sim 0.5r_{500}$. At smaller radii the scatter of pressure profiles is much larger, with steep profiles corresponding to the cool core clusters and flatter profiles for disturbed clusters. Figure \[fig:press\_arnaud\] shows that simulated and observed pressure profiles agree well with each other for $r\magcir 0.2r_{500}$, i.e., in the regime where the cluster population has a more regular behaviour. At smaller radii the profiles from simulations are on average steeper than observed, and exhibit a lower degree of diversity between cool core and non-cool core clusters.
The scatter in the cluster radial profiles can be used to define the following three radial regimes.
- Cluster cores, $r\lesssim r_{2500}$, which exhibit the largest scatter and where scaling with mass differs significantly from the self-similar scaling expectation. We do not yet have a complete and adequate theoretical understanding of the observed properties of the ICM and their diversity in the cluster cores. This is one of the areas of active ongoing theoretical and observational research.
- Intermediate radii, $r_{2500}\lesssim r\lesssim r_{500}$, which exhibit the smallest scatter and scaling with mass close to the self-similar scaling. Although the processes affecting thermodynamics of these regions are not yet fully understood, the simple scaling and regular behavior make observable properties of clusters at these radii useful for connecting them to the total cluster mass.
- Cluster outskirts $r>r_{500}$, where scatter is increasing with radius and scaling with mass can be expected to be close to self-similar on theoretical grounds, but have not yet been constrained observationally. In this regime, clusters are dynamically younger, characterized by recent mergers, departures from equilibrium, and a significant degree of gas clumping. Significant progress is expected in the near future due to a combination of high–sensitivity SZ and X–ray observations using the next generation of instruments.
The physical origin of the regular scaling with mass is the fact that cluster mass is the key control variable of cluster formation, which sets the amount of gas mass, the average temperature of the ICM, etc. It is important to note that the close to self-similar scaling with mass outside the cluster core does not imply that the non-gravitational physical processes are negligible in this regime. For instance, [@sun_etal09] showed that entropy measured at $r_{500}$ has a scaling with temperature quite close to the self–similar prediction, yet its level is higher than expected from a simple model in which only gravity determines the evolution of the intra–cluster baryons. This implies that whatever mechanism one invokes to account for such an entropy excess, it must operate in such a way as to not violate the self–similar scaling. The scatter around the mean profile exhibited by clusters at different radii can be due to a number of factors. In particular, the small scatter at intermediate radii is a non-trivial fact, given that different clusters of the same mass are in different stages of their dynamical evolution and physical processes affecting their profiles may have operated differently due to different formation histories.
One of the interesting implications of the small scatter in the pressure profiles is that it provides an upper limit on the contribution of non–thermal pressure support or, at least, on its cluster-by-cluster variation. A well-known source of non–thermal pressure is represented by residual gas motions induced by mergers, galaxy motions, and gas inflow along large-scale filaments. Cosmological hydrodynamical simulations of cluster formation have been extensively used to quantify the pressure support contributed by gas motions and the corresponding level of violation of HE [e.g., @rasia_etal04; @nagai_etal07b; @ameglio_etal09; @piffaretti_valda08; @lau_etal09; @biffi_etal11a]. All these analyses consistently found that ICM velocity fields contribute a pressure support of about $5\%$ to $15\%$ per cent of the thermal one, the exact amount depending on the radial range considered (being larger at larger radii) and on the dynamical state of the clusters. Currently, there are only indirect indications for turbulent motions in the ICM of the real clusters from fluctuations of gas density measured in X–ray observations [e.g., @schuecker_etal04; @churazov_etal12]. Direct measurements or upper limits on gas velocities and characterization of their statistical properties should be feasible with future high–resolution spectroscopic and polarimetric instruments on the next-generation X–ray telescopes [e.g., @inogamov_sunyaev03; @zhuravleva_etal10].
The galaxies and groups orbiting or infalling onto clusters not only stir the gas, but also make the ICM clumpier. The dense inner regions of clusters ram-pressure strip the gas on a fairly short time scale, so that the clumping is fairly small near cluster cores. However, it is substantial in the outskirts in cluster simulations where orbital times are longer and accretion of new galaxies and groups is ongoing. Given that the X-ray emissivity scales as the square of the local gas density, the clumpiness can bias the measurement of gas density from X–ray surface brightness profiles toward higher values if it is not accounted for. Because clumping is expected to increase with increasing cluster-centric radius, the inferred slope of gas density profiles can be underestimated, thus affecting the resulting pressure profile and hydrostatic mass estimates. Furthermore, gas clumping also affects X–ray temperature which is measured by fitting the X–ray spectrum to a single–temperature plasma model [@mazzotta_etal04; @vikhlinin06]. Clumping can therefore contribute to the scatter of pressure profiles at large radii, especially at $r>r_{500}$ [e.g., @nagai_lau11].
Indirect detections of gas clumping through X–ray observations out to $r_{200}$ have been recently claimed, based on [*Suzaku*]{} observations of a flattening in the X–ray surface brightness profiles at such large radii (@simionescu_etal11). However, these results are prone to significant systematic uncertainties [@ettori_molendi11]. Independent analyses based on the ROSAT data [e.g. @eckert_etal11]) show the surface brightness profiles steepens beyond $r_{500}$ [see also @vikhlinin_etal99; @neumann_etal05], which is inconsistent with the degree of gas clumping inferred from the [ *Suzaku*]{} data, but consistent with predictions of hydrodynamical simulations.
Clearly, the clumpiness of the ICM depends on a number of uncertain physical processes, such as efficient feedback, which removes gas from merging structures, or thermal conduction, which homogenizes the ICM temperatures [e.g., @dolag_etal04]. The degree of gas clumping in density and temperature is therefore currently uncertain in both theoretical models and observations. Future high-sensitivity SZ observations of galaxy clusters with improved angular resolution will allow a direct measurement of projected pressure profiles. Their comparison with X–ray derived profiles will help in understanding the impact of gas clumping on the thermal complexity of the ICM.
Additional non-thermal pressure support can be provided by the magnetic fields and relativistic cosmic rays, the presence of which in the ICM is demonstrated by radio observations of the radio halos: diffuse and faint radio sources filling the central Mpc$^3$ region of many galaxy clusters [e.g., @giovannini_etal09; @venturi_etal08] arising due to the synchrotron emission of highly relativistic electrons moving in the ICM magnetic fields. The origin of these relativistic particles still needs to be understood, although several models have been proposed. Shocks and turbulence associated with merger events are expected to compress and amplify magnetic fields and accelerate relativistic electrons (see, e.g., @ferrari_etal08 and @dolag_etal08 for reviews). Numerical simulations including injection of cosmic rays from accretion shocks and SN explosions [e.g., @pfrommer_etal07; @vazza_etal12] indicate that cosmic rays contribute a pressure support, which can be as high as $\sim
10\%$ for relaxed clusters and $\sim 20\%$ for unrelaxed clusters at the outskirts. At smaller radii, the pressure contribution of cosmic rays in these models becomes small ($\lesssim 3\%$ at $r\lesssim 0.1r_{\rm
vir}$), which is consistent with the upper limits from $\gamma$–ray observations by the [*Fermi Gamma-ray Telescope*]{} [e.g., @ackermann_etal10].
The role of intracluster magnetic fields have been investigated in a number of studies using cosmological simulations [see @dolag_etal08 for a review]. The general result is that pressure support from magnetic fields should be limited to $\mincir 5\%$, which is consistent with observational constraints on the magnetic field strength ($\sim \mu G$) [e.g., @vogt_ensslin05; @govoni_etal10] and upper limits on the contribution of magnetic fields to non-thermal pressure support [e.g. @lagana_etal10].
As a summary, the scatter in cluster profiles in the cluster cores is mainly driven by differences in the physical processes such as cooling and heating by AGN feedback and different merger activity that different clusters experienced during their evolution. At intermediate radii, the scatter is small because the ICM is generally in good HE within cluster gravitational potential and because processes that shaped its thermodynamic processes have not introduced new mass scale so that self-similar scaling is not broken. In the cluster outskirts, the scatter is expected to be driven by deviations from HE and other sources of non-thermal pressure support such as the cosmic rays, as well as by a rapid increase of ICM clumpiness with increasing radius.
Scaling relations {#sec:scal}
-----------------
Existence of the radial range, where ICM properties scale with mass similarly to the self-similar expectation with a small scatter, implies that we can define integral observable quantities within this range that will obey tight scaling relations among themselves and with the total cluster mass. Furthermore, these scaling relations are also expected to be weakly sensitive to the cluster dynamical state, given that relaxed and unrelaxed clusters have similar profiles at these intermediate radii. Indeed, as we showed in § \[sec:obsprop\] (see Fig. \[fig:ly\_rexcess\]), X–ray luminosity measured within the radial range $[0.15-1]r_{500}$ exhibits a tight scaling against the total ICM thermal content measured by the $Y_X$ parameter, with relaxed and unrelaxed clusters following the same relation. Here $Y_X$ is defined as the product of gas mass and X–ray temperature, both measured within $r_{500}$ but like $L_{\rm X}$ temperature is measured after excising the contribution from $r<0.15r_{500}$.
As discussed in § \[sec:selfsimilar\], the gas temperature $T$, gas mass $M_{\rm gas}$ and total thermal content of the ICM $Y=M_{\rm
gas}T$, are commonly used examples of integral observational quantities whose scaling relations with cluster mass are predicted by the self–similar model and for which calibrations based on X–ray and SZ observations (or their combinations) and simulations are available. For example, Figure \[fig:yx\_k06\] shows the scaling relation between $Y_X$, and $M_{500}$ for simulated clusters and for a set of clusters with detailed Chandra observations from a study by [@kravtsov_etal06], where $Y_{\rm X}$ was introduced and defined specifically to use the temperature estimated only at $0.15r_{500}<r<r_{500}$ in order to minimize the scatter. The relation of $Y_{\rm X}$ with $M_{500}$ in simulations has scatter of only $\approx 8\%$ when both relaxed and unrelaxed clusters are included and evolution of its normalization with redshift is consistent with expectations of the self–similar model. The insensitivity of the relation the dynamical state of clusters is not trivial and is due to the fact that during mergers clusters move almost exactly along the relation [e.g., @poole_etal07; @rasia_etal11]. In addition, the slope and normalization of the $Y_{\rm X}-M_{500}$ relation is also not sensitive to specific assumptions in modelling cooling and feedback heating processes in simulations [@stanek_etal10; @fabjan_etal11], which makes them more robust theoretically.
The $Y_{SZ}$–$M$ relation also exhibits a comparably low–scatter and the slope and evolution of normalization are close to the predictions of the self–similar model [@dasilva_etal04; @motl_etal05], which is not surprising given the similarity between the $Y_{\rm X}$ and the integrated $Y_{\rm SZ}$ measured from SZ observations. Its normalization changes by up to 30–40% depending on the interplay between radiative cooling and feedback processes included in the simulations [e.g. @nagai06; @bonaldi_etal07; @battaglia_etal11 and references therein]. At the same time, simulation analysis is also shedding light on the effect of projection [e.g. @kay_etal12] and mergers [e.g. @krause_etal12] on the scatter in the $Y_{SZ}$–$M$ scaling.
The tight relation of integral quantities such as $Y_{\rm X}$, $Y_{\rm
SZ}$, $\Mg$, or core-excised X-ray luminosity with the total mass makes them good proxies for observational estimates of cluster mass, which can be used at high redshifts even with a relatively small number of X-ray photons. For instance, integral measurements of gas mass or temperature requires $\sim 10^3$ photons, which is feasible for statistically complete cluster samples out to $z\sim 1$ [e.g., @maughan07; @vikhlinin_etal09; @mantz_etal10a] or even beyond. This makes these integral quantities very useful as “mass proxies” in cosmological analyses of cluster populations [e.g., @allen_etal11]. Clearly, the relation of such mass proxies with the actual mass needs to be calibrated both via detailed observations of small controlled cluster samples and in cosmological simulations of cluster formation.
The potential danger of relying on simulations for this calibration is that results could be sensitive to the details of the physical processes included. This implies that a mass proxy is required to have not only a low scatter in its scaling with mass, but also to be robust against changing the uncertain description of the ICM physics. As we noted above, $Y_{\rm X}$ is quite robust to changes within a wide range of assumptions about cooling and heating processes affecting the ICM. This is illustrated in Figure \[fig:scal\_f11\] (taken from @fabjan_etal11) which shows how the normalization and slope of the scaling relation of gas mass and $Y_X$ versus $M_{500}$ change with the physical processes included. The evolution of the $Y_{\rm X}-M_{500}$ relation with redshift is also consistent with self-similar expectations for different models of cooling and feedback. Other quantities, such as $\Mg$, often exhibit a similar or even smaller degree of scatter compared to $Y_{\rm X}$ but are more sensitive to the choice of physical processes included in simulations. An additional practical consideration is that theoretical models should consider observables derived from mock observations of simulated clusters that take into account instrumental effects of detectors and projection effects [e.g., @rasia_etal06; @nagai_etal07a; @biffi_etal11b].
Ultimately, calibration of mass proxies for precision use should be obtained via independent observational mass measurements, using the weak lensing analysis, HE, or velocity dispersions of member galaxies. The combination of future large, wide-area X–ray, SZ, and optical/near-IR surveys should provide a significant progress in this direction.
Cluster formation in alternative cosmological models {#sec:cosmo}
====================================================
In previous sections, we have discussed the main elements of cluster formation in the standard $\Lambda$CDM cosmology. Although this model is very successful in explaining a wide variety of observations, some of its key assumptions and ingredients are not yet fully tested. This provides motivation to explore different assumptions and alternative models.
As discussed in Section \[sec:mf\], the halo mass function for a Gaussian random field is uniquely specified by the peak height $\nu=\delta_c/\sigma(R,z)$, where $R$ is the filtering scale corresponding to the cluster mass scale $M$. For sufficiently large mass, that is rare peaks with $\nu\gg 1$, the mass function becomes exponentially sensitive to the value of $\nu$. At the same time, the mass function also determines the halo bias (see Section \[sec:bias\]). Again, for $\nu \gg 1$ and Gaussian perturbations, the bias function scales as $b(\nu)\sim
\nu^2/\delta_c=\nu/\sigma(R,z)$. Therefore, the cluster 2–point correlation function can be written as $\xi_{\rm
cl}(r)=\nu^2(\xi_R(r)/\sigma_R^2)$, where $\xi_R(r)$ is the correlation function of the smoothed fluctuation field (see Section \[sec:collbasics\]). Once the peak height $\nu$ is constrained by requiring a model to predict the observed cluster abundance, the value of the cluster correlation function at a single scale $r$ provides a measurement of the shape of the power spectrum through the ratio of the clustering strength at the scale $r$ and at the cluster characteristic scale $R$. These predictions are only valid under two assumptions, namely Gaussianity of primordial density perturbations and scale independence of the linear growth function $D(z)$, as predicted by the standard theory of gravity. Therefore, the combination of number counts and large–scale clustering studies offers a powerful means to constrain the possible violation of either one of these two assumptions that hold for the $\Lambda$CDM model.
In this section, we briefly review the specifics of cluster formation in models with non-Gaussian initial density field and with non-standard gravity, the most frequently discussed modifications to the standard structure formation paradigm.
Mass function and bias of clusters in non-Gaussian models {#sec:nong}
---------------------------------------------------------
One of the key assumptions of the standard model of structure formation is that initial density perturbations are described by a Gaussian random field (see Section \[sec:collbasics\]). The simplest single-field, slow-roll inflation models predict nearly Gaussian initial density fields. However, deviations from Gaussianity are expected in a broad range of inflation models that violate slow-roll approximation, and have multiple fields, or modified kinetic terms [see @bartolo_etal04 for a review]. Given that there is no single preferred inflation model, we do not know which specific form of non-Gaussianity is possibly realized in nature. Deviations from Gaussianity are parameterized using a heuristic functional form. One of the simplest and most common choices for such a form is the local non-Gaussian potential given by $\Psi_{\rm NG}(\bx) =
-(\phi_{\rm G}(\bx) +f_{\rm NL}[\phi_{\rm G}(\bx)^2 - \langle\phi_{\rm
G}^2\rangle])$, where $\Psi_{\rm NG}$ is the usual Newtonian potential, $\phi_{\rm G}$ is the Gaussian random field with zero mean, and the parameter $f_{\rm NL}=\rm const$ controls the degree and nature of non–Gaussianity [e.g., @salopek_bond90; @matarrese_etal00; @komatsu_spergel01]. The simplest inflation models predict $f_{\rm NL}\approx 10^{-2}$ [e.g., @maldacena03], but a number of models that predict much larger degree of non-Gaussianity exist as well [@bartolo_etal04]. The current CMB constraint on scale-independent non-Gaussianity is $f_{\rm NL}=30\pm 20$ at the 68% confidence level [e.g., @komatsu10] and there is thus still room for existence of sizable deviations from Gaussianity.
The non-Gaussian fields with $f_{\rm NL}<0$ have a PDF of the potential field that is skewed toward positive values and the abundance of peaks that seed the collapse of halos is reduced compared to Gaussian initial conditions. Conversely, the PDF of the potential field in models with $f_{\rm NL}>0$ has negative skewness, and hence an increased number of potential minima (density peaks). This would result in an enhanced abundance of rare objects, such as massive distant clusters, relative to the Gaussian case [see, e.g., figure 1 in @dalal_etal08 for an illustration of the effect of $f_{\rm NL}$ on the large-scale structure that forms]. The suppression or enhancement of abundance of halos increases with increasing peak height.
The mass functions resulting from non-Gaussian initial conditions have been studied both analytically [e.g., @chiu_etal98; @matarrese_etal00; @loverde_etal08; @afshordi_tolley08] and using cosmological simulations [@grossi_etal07; @dalal_etal08; @loverde_etal08; @loverde_smith11; @wagner_verde11]. These studies showed that accurate formulae for the halo abundance from the initial linear density field exist for the non-Gaussian models as well. The general result is that the fractional change in the abundance of the rarest peaks is of order unity for the initial fields with $\vert f_{\rm NL}\vert\sim 100$. The abundance of clusters is thus only mildly sensitive to deviations of Gaussianity within the currently constrained limits [@scoccimarro_etal04; @sefusatti_etal07; @sartoris_etal10; @cunha_etal10]. In contrast, primordial non-Gaussianity may also leave an imprint in the spatial distribution of clusters in the form of a scale-dependence of large-scale linear bias.
As was discovered by @dalal_etal08 and confirmed in subsequent analytical [@matarrese_verde08; @mcdonald08; @afshordi_tolley08; @taruya_etal08; @slosar_etal08] and numerical studies [@desjacques_etal09; @pillepich_etal10; @grossi_etal09; @shandera_etal11], the linear bias of collapsed objects in the models with [*local*]{} non-Gaussianity can be described as a function of wavenumber $k$ by $b_{\rm NG}=b_{\rm G}+ f_{\rm NL}\times{\rm\, const}/k^2$, where $b_{\rm G}$ is the linear bias in the corresponding cosmological model with the Gaussian initial conditions discussed in § \[sec:bias\]. This scale dependence arises because in the non-Gaussian models the large-scale modes that boost the abundance of peaks are correlated with the peaks themselves, which enhances (or suppresses) the peak amplitudes by a factor proportional to $f_{\rm
NL}\phi\propto f_{\rm NL}\delta/k^2$. Because this effect of modulation increases with increasing peak height, $\nu=\delta_c/\sigma(M,z)$, the scale–dependence of bias increases with increasing halo mass. This unique signature can be used as a powerful constraint on deviations from Gaussianity (at least for models with [*local*]{} non-Gaussianity) in large samples of clusters in which the power spectrum or correlation function can be measured on large scales.
Formation of clusters in modified gravity models {#sec:nongr}
------------------------------------------------
Recently, there has been a renewed interest in modifications to the standard GR theory of gravity [e.g., see @capozziello_delaurentis11; @durrer_maartens08; @silvestri_trodden09 for recent reviews]. These models have implications not only for cosmic expansion, but also for the evolution of density perturbations and, therefore, for the formation of galaxy clusters.
For instance, in the class of the $f(R)$ models, cosmic acceleration arises from a modification of gravity law given by the addition of a general function $f(R)$ of the Ricci curvature scalar $R$ in the Einstein-Hilbert action [see, e.g., @sotiriou_faraoni10; @jain_khoury10 for recent reviews]. Such modifications result in enhancements of gravitational forces on scales relevant for structure formation in such a way that the resulting linear perturbation growth rate $D$ becomes scale dependent; whereas on very large scales gravity behaves similarly to GR gravity, on smaller scales it is enhanced compared to GR and the rate of structure formation is thereby also enhanced. The nonlinear halo collapse and growth are also faster in $f(R)$ models, which leads to enhanced abundance of massive clusters [@schmidt_etal09a; @ferraro_etal11; @zhao_etal11] compared to the predictions of the models with GR gravity and identical cosmological parameters. Likewise, the peaks collapsing by a given $z$ have lower peak height $\nu$ in the modified gravity models compared to the peak height in the standard gravity model. This results in the reduced bias of clusters of a given mass compared to the standard model. Furthermore, the scale dependence of the linear growth also induces a scale dependence of bias, thus offering another route to detect modifications of gravity [@parfrey_etal11]. Qualitatively similar effects on cluster abundance and bias are expected in the braneworld-modified gravity models based on higher dimensions, such as the Dvali-Gabadadze-Porrati [DGP, @DGP] gravity model [@schaefer_koyama08; @khoury_wyman09; @schmidt09; @schmidt_etal10] and its successors with similar LSS phenomenology consistent with current observational constraints, such as models of ghost-free massive gravity [@derham_etal11; @damico_etal11].
A general consequence of modifying gravity is that the Birkhoff theorem no longer holds, which does not allow a straightforward extension of the spherical collapse model described in Section \[sec:sphcoll\] to a generic model of modified gravity. Nevertheless, numerical calculations of spherical collapse have been presented for a number of specific models [e.g., @schaefer_koyama08; @schmidt_etal09a; @schmidt_etal10; @martino_etal09]. For both the $f(R)$ and the DGP classes of models, the results of simulations obtained so far suggest that halo mass function and bias can still be described by the universal functions of peak height, in which the threshold for collapse and the linear growth rate are modified appropriately from their standard model values [@schmidt_etal09a; @schmidt_etal10]. This implies that it should be possible to calibrate mass function and bias of halos in the modified gravity models with the accuracy comparable to that in the standard structure formation models.
Summary and outlook
===================
All of the main elements of the overall narrative of how clusters form and evolve discussed in this review have been established over the past four decades. The remarkable progress in our understanding of cluster formation has been accompanied by great progress in multi-wavelength observations of clusters and our knowledge of the properties of the main mass constituents of clusters: stars, hot intracluster gas, and gravitationally dominant DM.
Formation of galaxy clusters is a complicated, non-linear process accompanied by a host of physical phenomena on a wide range of scales. Yet, some aspects of clusters exhibit remarkable regularity, and their internal structure, abundance, and spatial distribution carry an indelible memory of the initial linear density perturbation field and the cosmic expansion history. This is manifested both by tight scaling relations between cluster properties and the total mass, as well as by the approximate universality of the cluster mass function and bias, when expressed as a function of the peak height $\nu$.
Likewise, there is abundant observational evidence that complex processes – in the form of a non-linear, self-regulating cycle of gas cooling and accretion onto the SMBHs and associated feedback – have been operating in the central regions of clusters. In addition, the ICM is stirred by continuing accretion of the intergalactic gas, motion of cluster galaxies, and AGN bubbles. Studies of cluster cores provide a unique window into the interplay between the evolution of the most massive galaxies, taking place under extreme environmental conditions, and the physics of the diffuse hot baryons. At the same time, processes accompanying galaxy formation also leave a mark on the ICM properties at larger radii. In these regions, the gas entropy measured from observations is considerably higher than predicted by simple models that do not include such processes, and the ICM is also significantly enriched by heavy elements. This highlights that the ICM properties are the end product of the past interaction between the galaxy evolution processes and the intergalactic medium. Nevertheless, at intermediate radii, $r_{2500}\lesssim r\lesssim r_{500}$, the scaling of the radial profiles of gas density, temperature, and pressure with the total mass is close to simple, self-similar expectations for clusters of sufficiently large mass (corresponding to $kT\gtrsim 2-3$ keV). This implies that the baryon processes affecting the ICM during cluster formation do not introduce a new mass scale. Such regular behaviour of the ICM profiles provides a basis for the definition of integrated quantities, such as the core-excised X-ray luminosity and temperature, gas mass, or integrated pressure, which are tightly correlated with each other and with the total cluster mass.
The low-scatter scaling relations are used to interpret abundance and spatial distribution of clusters and derive cosmological constraints (see @allen_etal11 and @weinberg_etal12 for recent reviews). Currently, cluster counts measured at high redshifts provide interesting constraints on cosmological parameters complementary to other methods [e.g., @vikhlinin_etal09b; @mantz_etal10b; @rozo_etal10] and a crucial test of the entire class of $\Lambda$CDM and quintessence models [e.g., @jee_etal11; @benson_etal11; @mortonson_etal11]. Although the statistical power of large future cluster surveys will put increasingly more stringent requirements on the theoretical uncertainties associated with cluster scaling relations and mass function [@cunha_evrard10; @wu_etal10], future cluster samples can provide competitive constraints on the non-Gaussianity in the initial density field and deviations from GR gravity.
A combination of cluster abundance and large–scale clustering measurements can be used to derive stringent constraints on cosmological parameters and possible deviations from the standard $\Lambda$CDM paradigm. As an example, Figure \[fig:nong\] shows the constraints on the normalization of the power spectrum and the $f_{\rm
NL}$ parameter, [from @sartoris_etal10] expected for a future high–sensitivity X-ray cluster survey. It shows that future cluster surveys can achieve a precision of $\sigma_{f_{\rm NL}}\approx
5-10$ [see also @cunha_etal10; @pillepich_etal12], thus complementing at smaller scales constraints on non–Gaussianity, which are to be provided on larger scales by observations of CMB anisotropies from the Planck satellite.
Although a variety of methods will provide constraints on the equation of state of DE and other cosmological parameters [e.g., @weinberg_etal12], clusters will remain one of the most powerful ways to probe deviations from the GR gravity [e.g., @lombriser_etal09]. Even now, the strongest constraints on deviations from the GR on the Hubble horizon scales are derived from the combination of the measured redshift evolution of cluster number counts and geometrical probes of cosmic expansion [@schmidt_etal09]. Figure \[fig:modgrav\] illustrates the potential constraints on the linear rate of perturbation growth that can be derived from a future high–sensitivity X–ray cluster survey using similar analysis. The figure shows that a sample of about 2000 clusters at $z< 2$ with well-calibrated mass measurements would allow one to distinguish the standard $\Lambda$CDM model from a braneworld–modified gravity model with the identical expansion history at a high confidence level.
The construction of such large, homogeneous samples of clusters will be aided in the next decade by a number of cluster surveys both in the optical/near-IR (e.g., DES, PanSTARRS, EUCLID) and X-ray (e.g., eROSITA, WFXT) bands. At the same time, the combination of higher resolution numerical simulations including more sophisticated treatment of galaxy formation processes and high–sensitivity multi-wavelength observations of clusters should help to unveil the nature of the physical processes driving the evolution of clusters and provide accurate calibrations of their masses. The cluster studies thus will remain a vibrant and fascinating area of modern cosmology for years to come.
We are grateful to Brad Benson, Klaus Dolag, Surhud More, Piero Rosati, Elena Rasia, Ming Sun, Paolo Tozzi, Alexey Vikhlinin, Mark Voit, and Mark Wyman for useful discussions and comments, and to John Carlstrom for a careful reading of the manuscript. We thank Dunja Fabjan and Barbara Sartoris for their help in producing Fig. \[fig:scal\_f11\] and Fig. \[fig:nong\], respectively. The authors wish to thank the Kavli Institute for Theoretical Physics (KITP) in Santa Barbara for hospitality during the early phase of preparation of this review and participants of the KITP workshop “Galaxy clusters: crossroads of astrophysics and cosmology” for many stimulating discussions. AK was supported by NSF grants AST-0507596 and AST-0807444, NASA grant NAG5-13274, and by the Kavli Institute for Cosmological Physics at the University of Chicago through grants NSF PHY-0551142 and PHY-1125897. SB acknowledges partial support by the European Commission’s FP7 Marie Curie Initial Training Network CosmoComp (PITN-GA-2009-238356), by the PRIN-INAF09 project “Towards an Italian Network for Computational Cosmology”, by the PRIN-MIUR09 “Tracing the growth of structures in the Universe” and by the PD51 INFN grant.
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abstract: |
To deal with uncertainty in reasoning, interval-valued logic has been developed. But uniform intervals cannot capture difference in degrees of belief for different values in the interval. To salvage the problem triangular and trapezoidal fuzzy numbers are used as set of truth values along with traditional intervals. Preorder-based truth and knowledge ordering are defined over the set of fuzzy numbers defined over $[0,1]$. Based on this enhanced set of epistemic states, an answer set framework is developed, with properly defined logical connectives. This type of framework is efficient in knowledge representation and reasoning with vague and uncertain information under nonmonotonic environment where rules may posses exceptions.
**Keywords** Fuzzy numbers, Interval valued fuzzy sets, Preorder-based triangle, Answer Set Programming.
author:
- |
Sandip Paul\
ECSU,Indian Statistical Institute, Kolkata
- |
Kumar Sankar Ray\
ECSU,Indian Statistical Institute, Kolkata
- |
Diganta Saha\
CSE Department\
Jadavpur University
bibliography:
- 'biblist3.bib'
title: Modeling Uncertainty and Imprecision in Nonmonotonic Reasoning using Fuzzy Numbers
---
Introduction
============
Modern applications of artificial intelligence in decision support systems, plan generation systems require reasoning with imprecise and uncertain information. Logical frameworks based on bivalent reasoning are not suitable for such applications, because the set $\{0,1\}$ cannot capture the vagueness or uncertainty of underlying proposition. Though fuzzy logic-based systems can represent imprecise linguistic information by ascribing membership values to attributes (or truth values to propositions) taken from the interval \[0,1\], but this graded valuation becomes inadequate if the precise membership can not be determined due to some underlying uncertainty. This uncertainty may arise from lack of complete information or from lack of reliability of source of information or lack of unanimity among rational agents in a multi-agent reasoning system or from many other reasons. This uncertainty with respect to the assignment of membership degrees is captured by assigning a range of possible membership values, i.e. by assigning an interval. In other words by replacing the crisp {0,1} set by the set of sub-intervals of \[0,1\]. The intuition of such interval-valued system is that the actual degree, though still unknown, would be some value within the assigned interval and all the values in the interval are *equally-likely* to be the actual one.
However, there may be situations where all the values of an interval are not equally likely, rather, the information in hand suggests that some values are more plaussible. For instance, consider the motivating example presented in [@bauters2010towards]. It states that, “if the tumor suppressing genes(TSG) are lost due to mutation during cell division and chromosomal instability (CIN) is activated, a reasonably large tumor will grow”. In the proposed approach, this single information is represented by four rules and the resultant valuation assigned to the fact *tumor* is given by $\{tumor^{0.8:0.4}, tumor^{0.6:0.6}, tumor^{0.4:0.8}, tumor^{0.2:1}\}$. This representation is very inefficient and the number of rules and number of elements in the valuation would grow proportionately to the number of truth degrees considered within $[0,1]$. This example denotes that in real-world applications assignment of uniform intervals is inadequate. Instead, if arbitrary distributions over the interval \[0,1\] are allowed for truth values of propositions that would hugely increase the expressibility of the system and reduce the number of necessary rules in the logic program. Therefore, instead of assigning a sub-interval of \[0,1\] as the epistemic state to some vague, uncertain proposition, a *fuzzy number* defined over \[0,1\] would be a better choice, since, fuzzy numbers precisely allow to specify a membership distribution over \[0,1\].
Specifying the set of epistemic states is not enough, there has to be some underlying algebraic structure for ordering the values with respect to their degree of truth and degree of certainty(or uncertainty). For uniform interval-valued case Bilattice-based triangle structure were proposed [@cornelis2007uncertainty]. However later it is demonstrated [@ray2018preorder] that bilattice-based ordering is not suitable for belief revision in nonmonotonic reasoning and a preorder-based algebraic structure was constructed. Similar type of ordering has to be extended over the fuzzy numbers defined on $[0,1]$.
The main contributions of this work are as follows:
$\bullet$ The set of fuzzy numbers defined on \[0,1\] is considered as the set truth values for nonmonotonic reasoning with vague and uncertain information. In this work uniform, triangular and trapezoidal fuzzy numbers are considered only.
$\bullet$ Truth ordering and knowledge ordering over the set are defined (section 3) to construct the underlying preorder-based algebraic structure (section 4).
$\bullet$ This approach is used for answer set programming (section 5) to demonstrate the advantage.
Fuzzy Numbers
=============
This section provides necessary preliminary concepts.
**Definition 1:**
A fuzzy set $A$ over some $X \subseteq \mathbb{R}$ is called a fuzzy number if
1\. $A$ is convex, i.e.,
$\mu_{A}(\lambda x_1 + (1-\lambda) x_2) \geq min(\mu_A(x_1), \mu_A(x_2))$
where, $x_1, x_2 \in X$ and $\lambda \in [0,1]$.
2\. $A$ is normalised, i.e. $\max \mu_A(x) = 1$.
3\. There is some $x \in X$ such that $\mu_A(x) = 1$.
4\. $\mu_A(x)$ is piecewise continuous.
Triangular and Trapezoidal Fuzzy Number
---------------------------------------
The membership function of a triangular fuzzy number $TFN(a,b,c)$ for $a,b,c \in X $ and $a \leq b \leq c$ is specified as:
$\mu_{(a,b,c)}(x) = \begin{cases} 0, & x < a \\ \frac{x-a}{b-a}, & x \in [a,b] \\ \frac{c-x}{c-b}, & x \in [b,c] \\ 0, & x > c \end{cases}$
The membership function of a trapezoidal fuzzy number $TrFN(a,b,c,d)$ for $a,b,c,d \in X $ and $a \leq b \leq c \leq d$ is specified as:
$\mu_{(a,b,c,d)}(x) = \begin{cases} 0, & x < a \\ \frac{x-a}{b-a}, & x \in [a,b] \\ 1, & x \in [b,c] \\ \frac{d-x}{d-c}, & x \in [c,d] \\ 0, & x > d \end{cases}$
The uniform interval is a special case of $TrFN(a,b,c,d)$ when $a = b, c = d$, i.e., $TrFN(a,a,d,d)$ can be thought of as an interval $[a,d]$ so that all the values within the range has membership value 1. In this work an interval $[a,b]$ will be denoted as $IFN(a,b)$ to keep parity with the other two notations.
$\alpha$-cut decomposition of fuzzy numbers
-------------------------------------------
Another way of specifying a fuzzy number is by computing $\alpha$-cuts for $\alpha \in [0,1]$. For any fuzzy number $x$ and any specific value of $\alpha$, the $\alpha$-cut produces an interval of the form $x_{\alpha} = [\underline{x}_{\alpha}, \overline{x}_{\alpha}]$, where $\underline{x}_{\alpha}$ and $\overline{x}_{\alpha}$ are the intersection values with the left and right segment of $x$. The $\alpha$-cuts for a specific $\alpha$ for a TFN and TrFN are shown in Figure 1. $x_{\alpha}$ for $\alpha = 0$, will be referred to as *base-range* of $x$ ($x_0$).
(0,0) – (4,0); (0,0) – (0,3) node\[left\] [$\mu$]{}; (0.5,0) – (1.5,2) – (2.5,0); (0,1.0) – (2.5,1.0); (1.5,0) – (1.5,2); at (-0.2,0) [0]{}; at (-0.2,2) [1]{}; at (0.5,-0.2) [a]{}; at (1.5,-0.2) [b]{}; at (2.5,-0.2) [c]{}; at (-.2,1) [$\alpha$]{}; at (0.9, 1.2) [$\underline{x}_{\alpha}$]{}; at (2.2, 1.2) [$\overline{x}_{\alpha}$]{}; at (1.5,-1) [x = TFN(a,b,c)]{};
(0,0) – (4.5,0); (0,0) – (0,3) node\[left\] [$\mu$]{}; (0,1.0) – (4.5,1.0); (1,0) – (1,2); (2.5,0) – (2.5,2); at (-0.2,0) [0]{}; at (-0.2,2) [1]{}; at (-.2,1) [$\alpha$]{}; (0.2,0) – (1,2) – (2.5,2) – (4,0); at (0.2,-0.2) [a]{}; at (1,-0.2) [b]{}; at (2.5,-0.2) [c]{}; at (4,-0.2) [d]{}; at (0.5, 1.2) [$\underline{y}_{\alpha}$]{}; at (3.5, 1.2) [$\overline{y}_{\alpha}$]{};
at (1.5,-1) [y = TrFN(a,b,c,d)]{};
Analytically the $\alpha$-cut for the fuzzy numbers can be specified as follows:
$\bullet$ For $x = TFN(a,b,c)$; $x_{\alpha} = [a + \alpha(b-a), c - \alpha(c-b)]$;
$\bullet$ For $y = TrFN(a,b,c,d)$; $y_{\alpha} = [a + \alpha(b-a), d - \alpha(d-c)]$;
$\bullet$ For $z = IFN(a,d)$; $z_{\alpha} = [a,d]$.
Since, $IFN(a,d)$ is a special case of $TrFN(a,b,c,d)$ $z_{\alpha}$ can be obtained from $y_{\alpha}$ by setting $b = a$ and $c = d$. Similarly if the condition $b=c$ is imposed on $TrFN (a,b,c,d)$ a $TFN$ is obtained. Hence both are special cases of $TrFN$. Therefore, in later sections some concepts will be explained in terms of $TrFN$s only because same will be applicable for $IFN$ and $TFN$ by imposing the aforementioned conditions.
Fuzzy numbers as truth assignment and their truth and knowledge ordering:
=========================================================================
It is already demonstrated by means of an example that specifying an interval of real numbers from \[0,1\] is not sufficient to express the epistemic state of propositions in real life reasoning with vague and uncertain information. Now, general fuzzy numbers can be used as truth assignment of a proposition to capture various degrees of belief over the range of $[0,1]$. However, just specifying fuzzy numbers as the set of epistemic states is not enough, there must be some ordering to order two such epistemic states with respect to the degree of truth (truth ordering) and degree of certainty (knowledge ordering). Instead of considering any general type of fuzzy numbers, here, only the three types, that are specified in Section 2 (i.e., IFN, TFN and TrFN), are considered as truth assignments.
**Definition 2:**
A $TrFN(a,b,c,d)$ is said to be *restricted* if $0 \leq a,b,c,d \leq 1$, i.e., the base-range $x_0 \subseteq [0,1]$. . Similarly *restricted* versions of $IFN$ and $TFN$ are defined.
A $TrFN(a,b,c,d)$ is *semi-restricted* if $b,c \in [0,1]$ and $a < 0 \ \text{or} \ d > 1$ or both $a,d \notin [0,1]$. A $TFN(a,b,c)$ is *semi-restricted* if $b \in [0,1]$ and any or both of $a$ and $c \notin [0,1]$.
Construction of the Set of Epistemic States:
--------------------------------------------
In this section the set of truth assignments $\mathscr{T}$ is constructed so that any element from $\mathscr{T}$ can be assigned to some proposition to express its degree of belief. $\mathscr{T}$ is constructed from following conditions:
1\. All restricted $TrFN$, $TFN$ and $IFN$ are member of $\mathscr{T}$.
2\. For a semi-restricted $TrFN(a,b,c,d)$ its truncated version $[TrFN(a,b,c,d)]$ confined within \[0,1\] is included in $\mathscr{T}$. Thus,
$[TrFN(a,b,c,d)] = \begin{cases} TrFN(a,b,c,d), & x \in [0,1] \\ 0, & \text{otherwise} \end{cases}$
Here some intuitive aspects are explained to justify the necessity of $\mathscr{T}$ by means of examples.
**Example 1:** The case described in the introduction section can be re-considered. The epistemic state of the fact $tumor$ that was specified by $\{tumor^{0.8:0.4}, tumor^{0.6:0.6},\\ tumor^{0.4:0.8}, tumor^{0.2:1}\}$ can be approximately represented by $TFN(0.4,0.4,1.5)$. This assignment(shown in Figure 2a) is more compact representation.
(0,0) – (4,0); (0,0) – (0,3) node\[left\] [$\mu$]{}; (0.8,0) – (0.8,2) – (2,0.9); (2,0.9) – (3,0); (2,0) – (2,0.9); (2,0.9) – (2,2.0); at (-0.2,0) [0]{}; at (-0.2,2) [1]{}; at (2,-0.2) [1]{}; at (0,-0.2) [0]{};
(0.8,0) – (0.8,2) – (2,0.9) – (2,0);
at (1.5,-1) [(a) TFN(0.4,0.4,1.5)]{};
(0,0) – (3,0); (0,0) – (0,3) node\[left\] [$\mu$]{}; at (-0.2,0) [0]{}; at (-0.2,2) [1]{};
(0.8,0) – (1.6,2) – (2,2) – (2,0); at (0,-0.2) [0]{}; at (2,-0.2) [1]{};
at (1.5,-1) [(b) TrFN(0.4,0.8,1,1)]{};
**Example 2:** Suppose a group of agents with different degree of expertise is asserting their degree of belief about some proposition $P$ under uncertainty. They all agree that $P$ is not false and has moderate to high degree of truth. The most reliable experts tend to ascribe very high degree of truth, which shows that they believe $P$ will be true. This scenario can be expressed by using a trapezoidal fuzzy number $TrFN(0.4,0.8,1,1)$, as shown in Figure 2b.
**Example 3:** If nothing is known about a proposition then $IFN(0,1)$ is assigned. If a proposition is known to be True, with absolute certainty, then $IFN(1,1)$ is assigned.
Bimodal or Multi-modal distributions can not be expressed using $\mathscr{T}$.
Truth ordering and knowledge ordering of restricted $TrFN$s and restricted $TFN$s:
----------------------------------------------------------------------------------
Now that the set of epistemic states $\mathscr{T}$ is specified and intuitively justified, elements of $\mathscr{T}$ are to be ordered with respect to their degree of truth and certainty. These orderings play crucial role in revising beliefs during nonmonotonic reasoning. For instance, suppose, based on available knowledge the truth status of certain proposition has been determined. Now, some additional information becomes available and based on the new enhanced information set, the proposition is re-evaluated. In such a scenario, it becomes important to compare the two new assignment with the previous one with respect to degree of truth and degree of certainty. If some contradiction arises, some previously known facts or rules are to be withdrawn and this withdrawal procedure mandates ordering various rules or facts with respect to their degree of certainty. It is demonstrated in [@ray2018preorder], for $IFN$s preorder-based ordering is more intuitive and suitable for performing nonmonotonic reasoning with imprecise and uncertain information.
**Definition 3:** For any two $IFN$, $[x_1,x_2]$ and $[y_1,y_2] \in \mathscr{T}$ the truth ordering($\leq_{t_p}$) and knowledge ordering($\leq_{k_p}$), defined in [@ray2018preorder], are as follows:
$[x_1,x_2] \leq_{t_p}[y_1,y_2]\Leftrightarrow \frac{x_1+x_2}{2} \leq \frac{y_1+y_2}{2}$.
$[x_1,x_2] \leq_{k_p}[y_1,y_2]\Leftrightarrow (x_2-x_1) \geq (y_2-y_1)$.
The truth ordering ($\leq_{t_p}$) and the knowledge ordering ($\leq_{k_p}$) are preorders and combined they give rise to a **preorder-based triangle**. These definitions are generalized for $TFN$s and $TrFN$s in the next subsections.
### Truth-ordering
The intuition of assigning a fuzzy number for the epistemic state of a proposition, $p$ is that, due to uncertainty the actual truth assignment for $p$ (say, $\hat{p}$) is unknown, and hence is approximated by the assigned fuzzy number. If $\hat{p}$ is approximated by $IFN(a,b)$, then every value within the interval $[a,b]$ is equally probable to be $\hat{p}$. If $TFN(a,b,c)$ is assigned to $p$, then it signifies, in the range $[a,c]$, $b$ has a higher chance of being the actual truth status ($\hat{p}$) of $p$. Assignment of a $TrFN(a,b,c,d)$ can be interpreted similarly.
[l\*2[C]{}@]{} Assigned Fuzzy Number to $p$ & Equivalent probability density function of $\hat{p}$\
$IFN(a,b)$ &
$P_{IFN}(\hat{p}) = \begin{cases} \frac{1}{b-a}, & a \leq \hat{p} \leq b \\ 0, & \text{otherwise} \end{cases}$\
$TFN(a,b,c)$ &
$P_{TFN}(\hat{p}) = \begin{cases} 0, & \hat{p} < a \\ \frac{2(\hat{p}-a)}{(c-a)(b-a)}, & \hat{p} \in [a,b] \\ \frac{2(c-\hat{p})}{(c-1)(c-b)}, & \hat{p} \in [b,c] \\ 0, & \hat{p} > c \end{cases}$
\
$TrFN(a,b,c,d)$ &
$P_{TrFN}(\hat{p}) = \begin{cases} 0, & \hat{p} < a \\ \frac{2(\hat{p}-a)}{(d+c-b-a)(b-a)}, & \hat{p} \in [a,b] \\ \frac{2}{d+c-b-a}, & \hat{p} \in [b,c] \\ \frac{2(d-\hat{p})}{(d+c-b-a)(d-c)}, & \hat{p} \in [c,d] \\ 0, & \hat{p} > d \end{cases}$\
\[The Table\]
If we perform a random experiment, where an agent guesses the actual truth value of proposition $p$, then $\hat{p}$ can be thought of as a random variable, which follows a probability distribution. Now given the information in hand, assigning an epistemic state for $p$ is same as assigning an equivalent probability distribution over the random variable $\hat{p}$. So, for any restricted fuzzy number in $\mathscr{T}$, an equivalent probability distribution can be defined (as shown in Table 1).
For two propositions $p$ and $q$ with truth assignments $IFN(p_1,p_2)$ and $IFN(q_1,q_2)$ from $\mathscr{T}$, the intuition for ordering the truth assignments, with respect to the degree of truth is [@deschrijver2009generalized]
$IFN(p_1,p_2) \leq_{t_p} IFN(q_1,q_2)$ iff $Prob(\hat{p} \leq \hat{q}) \geq Prob(\hat{p} \geq \hat{q})$
where, $\hat{p}$ and $\hat{q}$ stands for the actual (yet unknown) truth status of propositions $p$ and $q$ respectively.
Now following this intuition we intend to extend the truth ordering from Definition 3 to ordering $TFN$s and $TrFN$s.
As explained above, for two propositions $p$ and $q$, $\hat{p}$ and $\hat{q}$ can be thought of as two random variables. In order to calculate $Prob(\hat{p} \leq \hat{q})$ or $Prob(\hat{p} \geq \hat{q})$ another random variable $\hat{r}$ is defined as:
$\hat{r} = \hat{p} - \hat{q}$.
Then, $Prob(\hat{p} \leq \hat{q}) = Prob(\hat{r} \leq 0)$ and $Prob(\hat{p} \geq \hat{q}) = Prob(\hat{r} \geq 0)$. Moreover the expectations(or means) of the random variables are related by $E(\hat{r}) = E(\hat{p}) - E(\hat{q})$.
Now, if probability distributions of $\hat{p}$ and $\hat{q}$ are chosen so that $E(\hat{p}) = E(\hat{q})$, then $E(\hat{r}) = 0$. This makes, $Prob(\hat{r} \leq 0) = Prob(\hat{r} \leq E(\hat{r})) = Prob(\hat{r} \geq E(\hat{r})) = Prob(\hat{r} \geq 0)$. Thus, $Prob(\hat{p} \leq \hat{q}) = Prob(\hat{p} \geq \hat{q})$. Since, the truth ordering is a total preorder, this would signify that propositions $p$ and $q$ have *same* degree of truth. This occurs irrespective of the chosen probability distribution of $\hat{p}$ and $\hat{q}$.
**Definition 4:** For any member $\mathscr{P} \in \mathscr{T}$, its **equivalent-interval**$(Eq-int)$ is any restricted $IFN(a,b)$ so that mean value of the equivalent probability distribution of $\mathscr{P}$ is equal to $\frac{a+b}{2}$, i.e., the expected value of a random variable $X$ that follows the probability density function $P_{IFN(a,b)}$. Therefore, any $IFN \in \mathscr{T}$ centered around the value $\frac{a+b}{2}$ is an equivalent-interval to $\mathscr{P}$.
The truth ordering defined over $IFN$s (from Definition 3) can be extended for ordering restricted $TrFn$s and $TFN$s using their equivalent-intervals.
**Theorem 1:** For any members $\mathscr{P}_1, \mathscr{P}_2 \in \mathscr{T}$,
$\mathscr{P}_1 \leq_{t_p} \mathscr{P}_2$ iff $E(X_{\mathscr{P}_1}) \leq E(X_{\mathscr{P}_2})$.
where, $X_{\mathscr{P}_1}$ and $X_{\mathscr{P}_2}$ are random variables following probability density functions equivalent to $\mathscr{P}_1$ and $\mathscr{P}_2$(as specified in Table 1) respectively.
**Proof:** If $\mathscr{P}_1$ and $\mathscr{P}_2$ are $IFN$s then the theorem directly follows from Definition 3, as $E(X_{IFN(a,b)}) = \frac{a+b}{2}$.
Suppose, $\mathscr{P}_1$ and $\mathscr{P}_2$ are respectively $Trapz_1 = TrFN(a_1,b_1,c_1,d_1)$ and $Trapz_2 = TrFN(a_2,b_2,c_2,d_2)$, and their corresponding equivalent-intervals are $Eq-int_1$ and $Eq-int_2$ respectively. Following the aforementioned rationale $Trapz_1$ and $Eq-int_1$ have same degree of truth and same holds for $Trapz_2$ and $Eq-int_2$. The two $IFN$s, $Eq-int_1$ and $Eq-int_2$ can be ordered with respect to $(\leq_{t_p})$ following Definiton 3.Thus,
$TrFN(a_1,b_1,c_1,d_1) \leq_{t_P} TrFN(a_2,b_2,c_2,d_2)$ iff $Eq-int_1 \leq_{t_p} Eq-int_2$.
In other words,
$Trapz_1 \leq_{t_P} Trapz_2$ iff $E(X_{Eq-int_1}) \leq E(X_{Eq-int_2})$,
$\Rightarrow Trapz_1 \leq_{t_P} Trapz_2$ iff $E(X_{Trapz_1}) \leq E(X_{Trapz_2})$
Since, following Definition 4, $E(X_{Trapz_1}) = E(X_{Eq-int_1})$ and $E(X_{Trapz_2}) = E(X_{Eq-int_2})$.
$TFN$s being special cases of $TrFN$s the theorem can similarly be proved if $\mathscr{P}_1$ and $\mathscr{P}_2$ are $TFN$s, or if $\mathscr{P}_1$ is an $IFN$ and $\mathscr{P}_2$ is a $TFN$ or a $TrFN$ as well. (**Q.E.D**)
Theorem 1 essentially gives the definition of preorder-based truth ordering ($\leq_{t_p}$) of restricted fuzzy numbers of $\mathscr{T}$. Therefore, for any restricted fuzzy numbers $\mathscr{P}_1, \mathscr{P}_2 \in \mathscr{T}$;
$\bullet$ $\mathscr{P}_1 = IFN(a_1,b_1), \mathscr{P}_2 = IFN(a_2,b_2)$;
$\mathscr{P}_1 \leq_{t_p} \mathscr{P}_2$ iff $\frac{a_1+b_1}{2} \leq \frac{a_2+b_2}{2}$.
$\bullet$ $\mathscr{P}_1 = TFN(a_1,b_1,c_1), \mathscr{P}_2 = TFN(a_2,b_2,c_2)$;
$\mathscr{P}_1 \leq_{t_p} \mathscr{P}_2$ iff $\frac{a_1+b_1+c_1}{3} \leq \frac{a_2+b_2+c_2}{3}$.
$\bullet$ $\mathscr{P}_1 = TrFN(a_1,b_1,c_1,d_1), \mathscr{P}_2 = TrFN(a_2,b_2,c_2,d_2)$;
$\mathscr{P}_1 \leq_{t_p} \mathscr{P}_2$ iff $\frac{1}{3(d_1+c_1-b_1-a_1)}(\frac{d_1^3-c_1^3}{d_1-c_1} - \frac{b_1^3-a_1^3}{b_1-a_1}) \leq \frac{1}{3(d_2+c_2-b_2-a_2)}(\frac{d_2^3-c_2^3}{d_2-c_2} - \frac{b_2^3-a_2^3}{b_2-a_2})$.
$\bullet$ $\mathscr{P}_1 = TFN(a_1,b_1,c_1), \mathscr{P}_2 = TrFN(a_2,b_2,c_2,d_2)$;
$\mathscr{P}_1 \leq_{t_p} \mathscr{P}_2$ iff $\frac{a_1+b_1+c_1}{3} \leq \frac{1}{3(d_2+c_2-b_2-a_2)}(\frac{d_2^3-c_2^3}{d_2-c_2} - \frac{b_2^3-a_2^3}{b_2-a_2})$.
**Example 4:** This example analytically validates Theorem 1. Consider two truth assignments $\mathscr{P} = IFN(a,d)$ and $\mathscr{Q} = TrFN(a,b,c,d)$, with $\hat{p}$ and $\hat{q}$ being their actual truth values approximated by $\mathscr{P}$, $\mathscr{Q}$ respectively. The actual truth status $\hat{p}$ and $\hat{q}$ are independent random variables, that follow a uniform and a trapezoidal probability density functions $P_{IFN(a,b)}$ and $P_{TrFN(a,b,c,d)}$ respectively. So, $P_{\mathscr{P}} = P_{IFN(a,d)}$ and $P_{\mathscr{Q}} = P_{TrFN(a,b,c,d)}$. The joint probability density function $f_{\mathscr{P}\mathscr{Q}} = P_{\mathscr{P}}P_{\mathscr{Q}}$.
$Prob(\hat{p} \leq \hat{q}) = \int_{a}^{d} \int_{\hat{p}}^{d} f_{\mathscr{P}\mathscr{Q}}(\hat{p},\hat{q}) d\hat{q}d\hat{p}$
$=\int_{a}^{d} \int_{\hat{p}}^{d} P_{\mathscr{P}}(\hat{p}).P_{\mathscr{Q}}(\hat{q}) d\hat{p}d\hat{q}$,
$=\frac{1}{d-a} \int_{a}^{d} \int_{\hat{p}}^{d} P_{\mathscr{Q}}(\hat{q}) d\hat{q}d\hat{p}$,
$= \frac{1}{d-a} \int_{a}^{b} \int_{\hat{p}}^{d} P_{\mathscr{Q}}(\hat{q})d\hat{q}d\hat{p} + \frac{1}{d-a} \int_{b}^{c} \int_{\hat{p}}^{d} P_{\mathscr{Q}}(\hat{q})d\hat{q}d\hat{p} + \frac{1}{d-a} \int_{c}^{d} \int_{\hat{p}}^{d} P_{\mathscr{Q}}(\hat{q})d\hat{q}d\hat{p}$,
$= \frac{1}{d-a} \int_{a}^{b}[1-\frac{(\hat{p}-a)^2}{(d-a+c-b)(b-a)}]d\hat{p} + \frac{1}{d-a} \int_{b}^{c}[\frac{2(c-\hat{p})}{d-a+c-b} + \frac{d-c}{d-a+c-b}]d\hat{p} + \frac{1}{d-a} \int_{c}^{d} \frac{d-\hat{p}}{(d-a+c-b)(d-c)}d\hat{p}$,
$=\frac{3(d-a+c-b)(b-a) - (b-a)^2}{3(d-a+c-b)(d-a)} + \frac{(c-b)(d-b)}{(d-a+c-b)(d-a)} + \frac{(d-c)^2}{3(d-a+c-b)(d-a)}$,
$= \frac{-ab-b^2+cd-3ad+2a^2-3ac+3ab+d^2+c^2}{3(d-a+c-b)(d-a)}$.
Now, $Prob(\hat{p} \leq \hat{q}) \geq Prob(\hat{p} \geq \hat{q})$
$\Rightarrow Prob(\hat{p} \leq \hat{q}) \geq \frac{1}{2}$,
$\Rightarrow \frac{-ab-b^2+cd-3ad+2a^2-3ac+3ab+d^2+c^2}{3(d-a+c-b)(d-a)} \geq \frac{1}{2}$
$\Rightarrow a^2-2b^2+2c^2-d^2+ad-3ac-cd+3bd \geq 0$
$\Rightarrow 2d^2 + 2cd + 2c^2 - 2b^2 - 2ab - 2a^2 \geq 3ac - 3a^2 - 3ab + 3d^2 + 3cd - 3bd$,
$\Rightarrow 2(\frac{d^3-c^3}{d-c} - \frac{b^3 - a^3}{b-a}) \geq 3a(d+c-b-a) + 3d(d-a+c-b)$,
$\Rightarrow \frac{1}{3(d+c-b-a)}(\frac{d^3-c^3}{d-c} - \frac{b^3 - a^3}{b-a}) \geq \frac{a+d}{2}$,
$\Rightarrow E(\hat{q}) \geq E(\hat{p}).$
As a special case, having $b=c$ in $TrFN(a,b,c,d)$ gives $\mathscr{Q} = TFN(a,b,d)$. Putting this condition in the above derivation gives,
$Prob(\hat{p} \leq \hat{q}) \geq \frac{1}{2}$
$\Rightarrow a^2 - d^2 - 2ab + 2bd \geq 0$
$\Rightarrow 2b(d-a) - (a+d)(d-a) \geq 0$
$\Rightarrow b \geq \frac{a+d}{2}$
$\Rightarrow a+b+d \geq \frac{3(a+d)}{2}$
$\Rightarrow \frac{a+b+d}{3} \geq \frac{a+d}{2}$
$\Rightarrow E(\hat{q}) \geq E(\hat{p})$.
Consider three propositions $p$, $q_1$ and $q_2$, ascribed with $\mathscr{P} = IFN(0.3,0.7)$, $\mathscr{Q}_1 = TrFN(0.3,0.3,0.5,0.7)$ and $\mathscr{Q}_2 = TrFN(0.3,0.5,0.7,0.7)$.
$E(\hat{p}) = 0.5$, $E(\hat{q}_1) = 0.455$, $E(\hat{q}_2) = 0.56$. It can be seen, $E(\hat{p}) \geq E(\hat{q}_1)$ and $E(\hat{p}) \leq E(\hat{q}_2)$. Also, $Prob(\hat{p} \leq \hat{q}_2) = 0.617 > 0.5$ and $Prob(\hat{p} \leq \hat{q}_1) = 0.388 < 0.5$.
So, $\mathscr{P} \leq_{t_p} \mathscr{Q}_2$ and $\mathscr{Q}_1 \leq_{t_p} \mathscr{P}$.
### Knowledge-ordering
As evident from Definition 3, the knowledge ordering is based on the length of $IFN$s, i.e., more is the length more is the underlying uncertainty. Therefore, the length of an $IFN$ identifies its level of uncertainty.
**Uncertainty degree of TFN:**
The concept of length is not so obvious for $TFN$ as it is for $IFN$s. To do so, the $\alpha$-cut decomposition of $TFN$ is used.
For $x = TFN(a,b,c)$ the $\alpha$-cut for any any value of $\alpha$ is an $IFN$ given as $x_{\alpha} = [\underline{x}_{\alpha}, \overline{x}_{\alpha}] = [a + \alpha(b-a), c - \alpha(c-b)]$. Now $x_{\alpha}$ being an $IFN$, the degree of its uncertainty can be evaluated to be:
$k_{x_{\alpha}} = [\overline{x}_{\alpha} - \underline{x}] = (c-a) - \alpha(c-b+b-a) = (c-a)-\alpha(c-a)$.
$k_{x_{\alpha}}$ varies with different values of $\alpha$ in \[0,1\]. Hence the average uncertainty(or length) is obtained as:
$k_x = \int_{0}^{1}[(c-a)-\alpha(c-a)] d\alpha = \frac{(c-a)}{2}$
Thus for two $TFN$s in $\mathscr{T}$, namely $\mathscr{P}_1 = TFN(a_1,b_1,c_1)$ and $\mathscr{P}_2 = TFN(a_2,b_2,c_2)$; it can be said,
$TFN(a_1,b_1,c_1) \leq_{k_p} TFN(a_2,b_2,c_2) \Leftrightarrow \frac{(c_1-a_1)}{2} \geq \frac{(c_2-a_2)}{2}$
**Uncertainty degree of TrFN:**
For $y = TrFN(a,b,c,d)$ and for some $\alpha$ in $\in [0,1]$; $y_{\alpha} = [a + \alpha(b-a), d - \alpha(d-c)]$ and $k_{y_{\alpha}} = (d-a) - \alpha(d-c+b-a)$. Therefore,
$k_y = \int_{0}^{1}[(d-a)-\alpha(d-c+b-a)] d\alpha = \frac{d+c-b-a}{2}$.
Hence, for $TrFN(a_1,b_1,c_1,d_1)$ and $TrFN(a_2.b_2,c_2,d_2)$,
$TrFN(a_1,b_1,c_1,d_1) \leq_{k_p} TrFN(a_2,b_2,c_2,d_2) \Leftrightarrow \frac{(d_1+c_1-b_1-a_1)}{2} \geq \frac{(d_2+c_2-b_2-a_2)}{2}$.
In a nutshell,
$\bullet$ uncertainty degree of $IFN(a,b)$, $k_{IFN} = b-a$;
$\bullet$ uncertainty degree of $TFN(a,b,c)$, $k_{TFN} = \frac{c-a}{2}$;
$\bullet$ uncertainty degree of $TFN(a,b,c,d)$, $k_{TrFN} = \frac{d+c-b-a}{2}$.
For any restricted fuzzy numbers $\mathscr{P}_1, \mathscr{P}_2 \in \mathscr{T}$,
$\mathscr{P}_1 \leq_{k_p} \mathscr{P}_2$ iff $k_{\mathscr{P}_1} \geq k_{\mathscr{P}_2}$.
**Example 5:** Consider $\mathscr{P}_1$, $\mathscr{P}_2$, $\mathscr{P}_3 \in \mathscr{T}$ and $\mathscr{P}_1 = IFN(a,d)$, $\mathscr{P}_2 = TFN(a,b,d)$ and $\mathscr{P}_3 = TrFN(a,c,e,d)$.
Now, $d-a \geq \frac{d+e-c-a}{2} \geq \frac{d-a}{2}$, i.e., $k_{\mathscr{P}_1} \geq k_{\mathscr{P}_3} \geq k_{\mathscr{P}_2}$. Therefore,
$\mathscr{P}_1 \leq_{k_p} \mathscr{P}_3 \leq_{k_p} \mathscr{P}_3$.
This is intuitive, since in case of $IFN(a,d)$ all values in $[a,d]$ are equally probable, whereas for $TFN(a,b,d)$, $b$ is more likely than any other value in $[a,d]$; which means $TFN(a,b,d)$ provides more information about the truth status of the underlying proposition than $IFN(a,d)$. $TrFN(a,c,e,d)$ lies in between.
**Note**: One notable point is that the uncertainty degrees of a fuzzy number as calculated is actually equal to the underlying area of the membership function of the fuzzy number. When there is no uncertainty and a specific membership value is assigned then the uncertainty degree is zero and so is the area under the curve of the form $IFN(a,a)$, for some $a \in [0,1]$. This can be utilised for calculating the uncertainty degree of semi-restricted $TFN$s and $TrFN$s.
Truth ordering and knowledge ordering of truncated semi-restricted $TrFN$s and $TFN$s in $\mathscr{T}$:
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The notion of truth and knowledge ordering, as defined over restricted fuzzy numbers of $\mathscr{T}$, can be extended to every pair of members of $\mathscr{T}$.
### Uncertainty degree and knowledge ordering:
For semi-restricted fuzzy numbers their truncated versions within the interval \[0,1\] are considered. Therefore, the expressions for uncertainty degree as specified in section 3.2.2 is no longer valid if the base-range of the fuzzy number exceeds \[0,1\]. However, from the notion developed in previous subsection, the uncertainty degree can be easily calculated from evaluating the area underlying the curve in the interval \[0,1\]. The more general expressions for uncertainty degree of elements of $\mathscr{T}$ are presented here.
For $[TFN(a,b,c)]$,
$k_{[TFN]} = \begin{cases} \frac{c-a}{2} - \frac{a^2}{2(b-a)}, & a < 0, c \in [0,1] \\ \frac{c-a}{2} - \frac{(c-1)^2}{2(c-b)}, & a \in [0,1], c > 1 \\ \frac{c-a}{2} - \frac{a^2}{2(b-a)} - \frac{(c-1)^2}{2(c-b)}, & a < 0, c > 1 \end{cases}$
For $[TrFN(a,b,c,d)]$,
$k_{[TrFN]} = \begin{cases} \frac{d+c-a-b}{2} - \frac{a^2}{2(b-a)}, & a < 0, d \in [0,1] \\ \frac{d+c-a-b}{2} - \frac{(d-1)^2}{2(d-c)}, & a \in [0,1], d > 1 \\ \frac{d+c-a-b}{2} - \frac{a^2}{2(b-a)} - \frac{(d-1)^2}{2(d-c)}, & a < 0, d > 1 \end{cases}$
In general, for any $TFN$ in $\mathscr{T}$, the knowledge degree can be specified as:
$k_{\Delta} = \frac{c-a}{2} - \frac{(min(a,0))^2}{2(b-a)} - \frac{(max(c,1)-1)^2}{2(c-b)}$.
For any $TrFN \in \mathscr{T}$;
$k_{\Box} = \frac{d+c-b-a}{2} - \frac{(min(a,0))^2}{2(b-a)} - \frac{(max(d,1)-1)^2}{2(d-c)}$
Based on the uncertainty degree the knowledge ordering can be induced in the same way as mentioned in the previous subsection.
**Note:** $\Delta$ and $\Box$ notations are used to denote both restricted and truncated semi-restricted triangular or trapezoidal fuzzy numbers respectively in general.
### Equivalent probability distribution and Truth ordering:
Suppose a fuzzy number $\mathscr{P}_g$ (which may be restricted or truncated semi-restricted) is assigned as epistemic state to some proposition $p$, then the underlying probability density function of the actual and unknown truth degree $\hat{p}$ can be defined as follows:
$\bullet$ If $\mathscr{P}_g = \Delta(a,b,c)$, then
$P_{\Delta(a,b,c)}(\hat{p}) = \begin{cases} h_{\Delta}\frac{\hat{p}-a}{b-a}, & max(0,a) \leq \hat{p} \leq b \\ h_{\Delta}\frac{c-\hat{p}}{c-b}, & b \leq \hat{p} \leq min(c,1) \\ 0, & \text{otherwise} \end{cases}$
where, $h_{\Delta} = \frac{1}{k_{\Delta}}$.
$\bullet$ If $\mathscr{P}_g = \Box(a,b,c,d)$, then
$P_{\Box(a,b,c,d)}(\hat{p}) = \begin{cases} h_{\Box}\frac{\hat{p}-a}{b-a}, & max(0,a) \leq \hat{p} \leq b \\ h_{\Box}, & b \leq \hat{p} \leq c \\ h_{\Box}\frac{d-\hat{p}}{d-c}, & c \leq \hat{p} \leq min(d,1) \\ 0, & \text{otherwise} \end{cases}$
where, $h_{\Box} = \frac{1}{k_{\Box}}$.
For specifying the truth-ordering Theorem 1 is used. The expected values for random variables following the above-mentioned probability density functions can be calculated as:
$\bullet$ If $\hat{p}$ follows $P_{\Delta(a,b,c)}$, then
$E(\hat{p}) = \int_{max(0,a)}^{b} \frac{\hat{p}(\hat{p}-a)h_{\Delta}}{(b-a)}d\hat{p} + \int_{b}^{min(c,1)} \frac{\hat{p}(c-\hat{p})h_{\Delta}}{(c-b)} d \hat{p}$.
$\bullet$ If $\hat{p}$ follows the pdf $P_{\Box(a,b,c,d)}$, then
$E(\hat{p}) = \int_{max(a,0)}^b \frac{h_{\Box}}{(\hat{p}-a)\hat{p}}{(b-a)} d\hat{p} + \int_{b}^{c} h_{\Box} d\hat{p} + \int_{c}^{min(d,1)} \frac{\hat{p}(d-\hat{p})h_{\Box}}{d-c} d\hat{p}$.
Once the mean value is calculated the truth ordering can be induced based on Theorem 1.
**Example 6:** The fuzzy number $[TFN](0.4,0.4,1.5)$ shown in Figure 2(a) is a semi-restricted element of $\mathscr{T}$, assigned to a proposition, say $p$.
Now, the uncertainty degree $k_{[TFN]} = \frac{(1.5-.4)}{2} - \frac{(1.5-1)^2}{2(1.5-0.4)} = 0.436$; thus $h_{[TFN]} = \frac{1}{0.436} = 2.292$.
Truth degree = $E(\hat{p}) = $
$\int_{0.4}^{1} 2.083\hat{p}(1.5 - \hat{p}) d\hat{p} = 0.574$.
Preorder-based Triangle for $\mathscr{T}$:
==========================================
Each element $\mathscr{P} \in \mathscr{T}$ can be seen as a pair $(t_{\mathscr{P}}, k_{\mathscr{P}})$, where, $t_{\mathscr{P}}$ is the truth degree of $\mathscr{P}$, which is the expected value (mean) of a random variable following the probability density function $P_{\mathscr{P}}$ and $k_{\mathscr{P}}$ is the uncertainty degree.
For any two members $\mathscr{P}_1, \mathscr{P}_2 \in \mathscr{T}$:
$\mathscr{P}_1 \leq_{t_p} \mathscr{P}_2 \ \text{iff} \ t_{\mathscr{P}_1} \leq t_{\mathscr{P}_2}$
$\mathscr{P}_1 \leq_{k_p} \mathscr{P}_2 \ \text{iff} \ k_{\mathscr{P}_1} \geq k_{\mathscr{P}_2}$
These two orderings imposed on the elements of $\mathscr{T}$ give rise to a *Preorder-based triangle* for $\mathscr{T}$, **P**($\mathscr{T}$), which can be thought as an extension of the structure developed in [@ray2018preorder].
**Example 7:** Consider a lattice **L**$ = [\{0,\frac{1}{3}, \frac{2}{3}, 1\}, \leq]$. Let $\mathscr{T}_R(\textbf{L})$ be the set of $IFN$s and restricted $TFN$s and $TrFN$s constructed from **L**. The elements of $\mathscr{T}_R(L)$ are shown in table
[|c|c?c|c?c|c|]{}
**IFN** & $(t_{I},k_{I})$ & **TFN** & $(t_{\Delta},k_{\Delta})$ & **TrFN** & $(t_{\Box},k_{\Box})$\
1\. \[0,0\] & (0,0) & 1.(0,$\frac{1}{3}$,1) & ($\frac{4}{9}$,$\frac{1}{2}$) & 1.(0,$\frac{1}{3}$,$\frac{2}{3}$,1) & ($\frac{1}{2}$, $\frac{2}{3}$)\
2. \[$\frac{1}{3}$, $\frac{1}{3}$\] & ($\frac{1}{3}$,0) & 2. (0,$\frac{1}{3}$,$\frac{2}{3}$) & ($\frac{1}{3}$,$\frac{1}{3}$) & 2. (0,0,$\frac{1}{3}$,$\frac{2}{3}$) & ($\frac{7}{27}$, $\frac{1}{2}$)\
3. \[$\frac{2}{3}$, $\frac{2}{3}$\] & ($\frac{2}{3}$,0) & 3. (0,$\frac{2}{3}$,1) & ($\frac{5}{9}$,$\frac{1}{2}$) & 3. (0,0,$\frac{2}{3}$,1) & ($\frac{19}{45}$, $\frac{5}{6}$)\
4. \[1,1\] & (1,0) & 4. (0,0,$\frac{2}{3}$) & ($\frac{2}{9}$,$\frac{1}{3}$) & 4. ($\frac{1}{3}$,$\frac{1}{3}$,$\frac{2}{3}$,1) & ($\frac{16}{27}$, $\frac{1}{2}$)\
5. \[0,$\frac{1}{3}$\] & ($\frac{1}{6}$,$\frac{1}{3}$) & 5. (0,0,1) & ($\frac{1}{3}$,$\frac{1}{2}$) & 5. (0,$\frac{1}{3}$,1,1) & ($\frac{26}{45}$, $\frac{5}{6}$)\
6. \[0,$\frac{2}{3}$\] & ($\frac{1}{3}$,$\frac{2}{3}$) & 6. ($\frac{1}{3}$,1,1) & ($\frac{7}{9}$,$\frac{1}{3}$) & 6. (0,$\frac{2}{3}$,1,1) & ($\frac{23}{36}$, $\frac{2}{3}$)\
7. \[0,1\] & ($\frac{1}{2}$,1) & 7. (0,1,1) & ($\frac{2}{3}$,$\frac{1}{2}$) & 7. ($\frac{1}{3}$,$\frac{2}{3}$,1,1) & ($\frac{20}{27}$, $\frac{1}{2}$)\
8. \[$\frac{1}{3}$,$\frac{2}{3}$\] & ($\frac{1}{2}$,$\frac{1}{3}$) & 8. ($\frac{1}{3}$,$\frac{2}{3}$,1) & ($\frac{2}{3}$,$\frac{1}{3}$) & 8. (0,$\frac{1}{3}$,$\frac{2}{3}$,$\frac{2}{3}$) & ($\frac{11}{27}$, $\frac{1}{2}$)\
9. \[$\frac{1}{3}$,1\] & ($\frac{2}{3}$,$\frac{2}{3}$) & 9. ($\frac{1}{3}$,$\frac{1}{3}$,1) & ($\frac{5}{9}$,$\frac{1}{3}$) & 9. (0,0,$\frac{1}{3}$,1) & ($\frac{13}{36}$, $\frac{2}{3}$)\
10. \[$\frac{2}{3}$,1\] & ($\frac{5}{6}$,$\frac{1}{3}$) & 10. (0,$\frac{2}{3}$,$\frac{2}{3}$) & ($\frac{4}{9}$,$\frac{1}{3}$) & &\
![Preorder-based Triangle for $\mathscr{T}_R(L)$[]{data-label="fig:tri"}](triangle1.eps){width="100mm"}
Figure 3 shows the preorder-based triangle for $\mathscr{T}_R(L)$ constructed from the truth ordering($\leq_{t_p}$) and knowledge ordering($\leq_{k_p}$) as specified. However, for clarity, all the comparibility connections are not shown in the figure. If truncated semi restricted fuzzy numbers were included in $\mathscr{T}_R(L)$ it would not change the “triangular” nature of the algebraic structure, rather that would increase the number of available epistemic states in $\mathscr{T}_R(L)$.
Evidently, the extension of set of truth values from just the set of sub-intervals of $[0,1]$ to $\mathscr{T}$ introduces many new epistemic states.
Answer set programming using $\mathscr{T}$ as set of truth values:
==================================================================
Answer Set Programming (ASP) [@lifschitz2008answer; @baral2003knowledge] is a nonmonotonic reasoning framework which is hugely used in declarative problem solving and reasoning with rules having exceptions. Numerous Fuzzy [@janssen2009general; @janssen2012core; @blondeel2014complexity; @mushthofa2014finite] and possibilistic [@bauters2012possibilistic; @bauters2015characterizing] extensions of ASP are developed for dealing with real-life problems that encounter imprecise and uncertain information.
In [@bauters2010towards] a possibilistic fuzzy ASP framework is proposed. However, as mentioned earlier, the knowledge representation of the proposed framework is inefficient. Moreover, the approach is developed only for programs with positive rules and no negation has been introduced in the system. A more concrete and intuitive approach, namely Unified Answer set Programming has been reported [@paul2019unified], that uses interval-valued fuzzy sets (IVFS) defined over the unit interval $[0,1]$, as the set of truth values. Replacing IVFS with $\mathscr{T}$, as developed in the previous section, would enrich the Unified Answer Set Programming framework in terms of its expressing ability and intuitive knowledge representation.
Answer set programming with $\mathscr{T}$ (defined over some lattice **L**) as the truth value space is described in this section.
Logical Operators:
------------------
The logical operators are defined based on the traditional operations defined over \[0,1\] in Fuzzy logic and the algebraic operations on fuzzy numbers. Traditional fuzzy t-(co)norms, negation and the operators defined in [@paul2019unified] can be obtained as special cases of the operators defined here.
### Negation:
For a $\mathscr{P} = \Box(a,b,c,d) \in \mathscr{T}$, its negation is defined as:
$\neg \mathscr{P} = \neg \Box(a,b,c,d) = \Box(1-d,1-c,1-b,1-a)$.
This negation is involutive. This is the generalised version of the standard negator defined over IVFS [@cornelis2007uncertainty], which is $\neg[a,b] = [1-b,1-a]$. Negation of a certain assertion, denoted by an exact interval $IFN(x,x)$ for $x \in [0,1]$ becomes $1-x$, which is compatible with the standard negator defined for fuzzy logic.
The negation doesn’t change the uncertainty degree of the element, rather it can be viewed as a rotation of the fuzzy number in $\mathscr{T}$ with respect to the line $x = 0.5$.
For example,
$\neg TFN(0.2,0.6,0.7) = TFN(0.3,0.4,0.8)$ and $\neg[TrFN(-2,0.3,0.9,3)] = [TrFN(-2,0.1,0.7,3)]$.
### Negation-as-failure($not$):
Negation-as-failure($not$) is crucial to capture the nonmonotonicity of answer set programming. The significance of $not$ is that, it enables syntactical representation of incompleteness of knowledge in logic programs.
For a proposition $p$, $not \ p$ is to be true if nothing is known about $p$. It is notable that unlike $\neg p$, the truth of $not \ p$ doesn’t require evidential refutation of $p$, rather we can perform reasoning even if the acquired knowledge about $p$ is incomplete.
When nothing is known about $p$, the epistemic state $IFN(0,1)$ (or $TrFN(0,0,1,1)$) is assigned to $p$, which has uncertainty degree of $1$. For that, $not \ p$ is True, i.e. assigned with $IFN(1,1)$(or $TrFN(1,1,1,1)$). Hence, $not \ IFN(0,1) = not \ TrFN(0,0,1,1) = IFN(1,1)$. When some knowledge about $p$ is available, the epistemic state of $not \ p$ depends upon the degree of truth and uncertainty degree of $p$. Thus, the truth assignment of $not \ p$, is a *meta-level* assertion, depending on the epistemic state of $p$, which is already determined. Thus, there is no inherent uncertainty in the epistemic state of $not \ p$ and because of this, the epistemic state of $not \ p$ would be an exact interval of the form $IFN(x,x)$, for $x \in [0,1]$. The negation-as-failure can be defined as:
$not \ IFN(a,b) = IFN(1-a,1-a)$.
$not \ \Box(a,b,c,d) = IFN(1-b,1-b) = not \ \Delta(a,b,c)$.
### Conjunction and Disjunction:
T-representable product t-norm is used as conjunctor here. For two $IFN$s their product can be defined using interval algebra [@gao2009multiplication] as follows:
$[a_1,a_2] \odot [b_1,b_2] = [min(a_1b_1, a_1b_2, a_2b_1, a_2b_2), max(a_1b_1, a_1b_2, a_2b_1, a_2b_2)]$
When $a_1,a_2,b_1,b_2$ are positive real $\in \Re^+$ then $[a_1,a_2] \odot [b_1,b_2] = [a_1a_2, b_1b_2]$.
The product t-norm of two restricted elements of $\mathscr{T}$ is defined from the standard approximated product of two fuzzy numbers [@giachetti1997analysis; @dubois1993fuzzy; @kaufmann1988fuzzy].
$\bullet \ IFN(a_1,b_1) \wedge IFN(a_2,b_2) = [a_1a_2,b_1b_2]$;
$\bullet \ TFN(a_1,b_1,c_1) \wedge TFN(a_2,b_2,c_2) = TFN(a_1a_2, b_1b_2, c_1c_2)$;
$\bullet \ TrFN(a_1,b_1,c_1,d_1) \wedge TrFN(a_2,b_2,c_2,d_2) = TrFN(a_1a_2, b_1b_2, c_1c_2, d_1d_2)$.
In case of semi-restricted fuzzy numbers the tnorms will be:
$\bullet \ [TFN(a_1,b_1,c_1)] \wedge [TFN(a_2,b_2,c_2)] = [TFN(min(a_1a_2, a_1c_2, c_1a_2, c_1c_2), b_1b_2,$
$max(a_1a_2, a_1c_2, c_1a_2, c_1c_2))]$.
$\bullet \ [TrFN(a_1,b_1,c_1,d_1)] \wedge [TrFN(a_2,b_2,c_2,d_2)] = [TrFN(min(a_1a_2, a_1d_2, d_1a_2, d_1d_2),$
$ min(b_1b_2, b_1c_2, c_1b_2, c_1c_2), max(b_1b_2, b_1c_2, c_1b_2, c_1c_2), max(a_1a_2, a_1d_2, d_1a_2, d_1d_2))]$.
The disjunction of $\mathscr{P}_1, \mathscr{P}_2 \in \mathscr{T}$ can be obtained from the standard negator($\neg$) and $\wedge$ by means of De Morgan’s Law as follows:
$\mathscr{P}_1 \vee \mathscr{P}_2 = \neg [(\neg \mathscr{P}_1) \wedge (\neg \mathscr{P}_2)]$.
### Knowledge aggregation operator ($\otimes_k$)
Apart from the aforementioned connectives, another connective is introduced for non-monotonic reasoning, which is the knowledge aggregation operator $\otimes_k$. For two elements $\mathscr{P}_1, \mathscr{P}_2 \in \mathscr{T}$, $\otimes_k$ is defined as follows:
$\mathscr{P}_1 \otimes_k \mathscr{P}_2 = \begin{cases} \mathscr{P}_1 & k_{\mathscr{P}_1} \geq k_{\mathscr{P}_2} \\ \mathscr{P}_2 & k_{\mathscr{P}_1} \leq k_{\mathscr{P}_2} \end{cases}$
Thus, $\otimes_k$ chooses which among the two truth assignments is more certain.
Syntax:
-------
The language consists of infinitely many variables, finitely many constants and predicate symbols and no function symbol. For a predicate symbol $p$ of arity $n$, $p(t_1, t_2,...,t_n)$ is an atom, where $t_1,..t_n$ are variables or constants or an element of $\mathscr{T}$. A grounded atom contains no variable. A literal is a positive atom or its negation. For a literal $l$, $not \ l$ is a naf-literal.
An UnASP program consists of weighted rules of the form:
$r:a \stackrel{\alpha_r}{\longleftarrow} b_1 \wedge ... \wedge b_k \wedge not \ b_{k+1} \wedge ... \wedge not \ b_n$
where, $\alpha_r \in \mathscr{T}$ is the weight of the rule, which denotes the epistemic state of the consequent or head ($a$) of the rule, when the antecedent or body ($b_1 \wedge ... \wedge b_k \wedge not \ b_{k+1} \wedge ... \wedge not \ b_n$) of the rule is true, i.e., has the truth status $IFN(1,1)$.
$a,b_1,...,b_n$ are positive or negative literals or elements of $\mathscr{T}$. For simplicity the body of the rule will be denoted by ’,’ separated literals instead of using the $\wedge$ symbols. A rule is said to be a **fact** if $b_i, 1 \leq i \leq n$ are elements of $\mathscr{T}$.
The rule weight $\alpha_r$ may denote the inherent uncertainty of the rule, or the degree of reliability or priority of the source of the rule. Even $\alpha_r$ can be used to denote that the rule ’$r$’ is a disposition, i.e. a proposition with exceptions.
Declarative Semantics:
----------------------
The semantics is similar to the UnASP framework, proposed in [@paul2019unified]; hence here it is specified briefly.
$\mathscr{L}$ be the set of literals (excluding naf-literals). An **interpretation**, $I$, is a set $\{a:\mathscr{P}_a|\mathscr{P}_a \in \mathscr{T}\}$, which specifies the epistemic states of the literals in the program.
**Definition 5:** An interpretation $I$ is inconsistent if there exists an atom $a$, such that, $a:\mathscr{P}_a \in I$ and $\neg a: \mathscr{P}_{\neg a} \in I$ and $k_{\mathscr{P}_a} = k_{\mathscr{P}_{\neg a}}$ but $t_{\mathscr{P}_a} \neq 1-t_{\mathscr{P}_{\neg a}}$.
In other words, an inconsistent interpretation assigns contradictory truth status to two complemented literals with same confidence.
The set of interpretations can be ordered with respect to the uncertainty degree by means of the knowledge ordering ($\leq_{k_p}$). For two interpretations $I$ and $I^*$, $I \leq_k I*$ iff $\forall a \in \mathscr{L}, I(a) \leq_{k_p} I^*(a)$. An interpretation $I_k$ is the **k-minimal** interpretation of a set of interpretations $\Gamma$, iff for no interpretation $I^* \in \Gamma$; $I^* \leq_{k_p} I_k$. If for any $\Gamma$, $I_k$ is unique then it is *k-least*.
**Definition 6:** An interpretation $I$ *satisfies* a rule $r$ if for every ground instance of $r$ of the form $r_g:head \stackrel{\alpha_r}{\longleftarrow} body$, $I(head) = (I(body)\wedge \alpha_r)$ or $I(head) >_{k_p} (I(body) \wedge \alpha_r)$ or $I(head) >_{t_P} (I(body) \wedge \alpha_r)$. $I$ is said to be a *model* of a program $P$, if $I$ satisfies every rule of $P$.
**Definition 7:** A model of a program $P$, $I_m$, is said to be **supported** iff:
1\. For every grounded rule $r_g: a \stackrel{\alpha_r}{\leftarrow} b$, such that $a$ doesn’t occur in the head of any other rule, $I_m(a) = I_m(b)$.
2\. For grounded rules $\{ a \stackrel{\alpha_1}{\leftarrow} b_1, a$ $\stackrel{\alpha_2}{\leftarrow} b_2,..,a \stackrel{\alpha_n}{\leftarrow} b_n\} \ \in P$ having same head $a$, $I_m(a) = (I_m(b_1) \wedge \alpha_1) \vee ... \vee (I_m(b_n) \wedge \alpha_n)$.
3\. For literal $l \in \mathscr{L}$, and grounded rules $r_l: \ l \longleftarrow b_l$, and $\ r_{\neg l}: \neg l \longleftarrow \ b_{\neg l}$, in $P$, $I_m(l) = I_m(b_l) \otimes_K \neg I_m(b_{\neg l})$ and $I_m(b_l) \otimes_k \neg I_m(b_{\neg l})$ exists in $\mathscr{T}$.
The first condition of supportedness guarantees that the inference drawn by a rule is no more certain and no more true than the degree permitted by the rule body and rule weight. The second condition specifies the optimistic way of combining truth assertions for an atom coming from more than one rule. The third condition captures the essence of nonmonotonicity of reasoning. For an atom $a$, rules with $a$ in the head are treated as evidence in favour of $a$ and rules with $\neg a$ in the head stands for evidence against $a$. In such a scenario, the conclusion having more certainty or reliability is taken as the final truth status of $a$.
**Definition 8:** The **reduct** of a program $P$ with respect to an interpretation $I$ is defined as:
$P^I = \{r_I:a \stackrel{\alpha_r}{\longleftarrow} b_1 \wedge ... \wedge b_k \wedge not \ I(b_{k+1}) \wedge ... \wedge not \ I(b_n) \ | \ r \in P\}$.
$P^I$ doesn’t contain any naf-literal in any rule. For a positive program $P$ (with no rules ontaining $not$), $P^I = P.$
**Definition 9:** For any UnASP program $P$, an interpretation $I$ is an **answer set** if $I$ is an k-minimal supported model of $P^I$. For a positive program the k-minimal model is unique.
**Example 8:** The motivating example described in [@bauters2010towards] is considered here.
$P=\{ r1: tumor \xleftarrow{[TFN(0.4,0.4,1.5)]} cin_{on} \wedge tsg_{off}$
$r2: tumor \xleftarrow{TFN(0.1,0.1,0.5)} tsg_{off}$
$r3: tsg_{off} \xleftarrow{IFN(0.6,1)} cin_{on} \}$
These rules describe the same information described there by means of 9 rules in a lot more brief and intuitive way. When $tsg_{off}$ and $cin_{on}$ both holds rule $r1$ infers $tumor:[TFN(0.4,0.4,1.5)]$. This essentially a similar truth assertion as derived in [@bauters2010towards] as $\{tumor^{0.8:0.4}, tumor^{0.6:0.6}, tumor^{0.4:0.8}, tumor^{0.2:1}\}$. Rule $r3$ signifies that when $cin_{on}$ holds there is a chance that $tsg_{off}$ holds; the underlying uncertainty is depicted by the interval $[0.6,1]$. This example illustrates the effectiveness of the developed framework.
Conclusion:
===========
This paper explores the feasibility of considering fuzzy numbers as truth values of propositions. Unlike a uniform interval, fuzzy numbers defined over the interval $[0,1]$ can be viewed as an interval with a membership distribution defined over it. Here, mainly triangular and trapezoidal membership distributions are considered. Using them as the set of truth values greatly enhance the expressive power of a logical framework. The truth values or epistemic states are ordered with respect to degree of truth and degree of certainty by means of a preorder-based algebraic structure. The truth and knowledge orderings defined here are intuitive and also enable performing nonmonotonic reasoning using uncertain and imprecise information. To demonstrate the effectiveness of the modified truth value space an answer set programming framework is developed over it. This type of framework can be utilized in decision support systems or as the logical system underlying a semantic web to represent the underlying uncertainty and vagueness.
Using the fundamental idea behind defining truth and knowledge ordering, other membership distributions, like *sigmoid*, *gaussian* defined over the interval $[0,1]$, may be fitted in $\mathscr{T}$, if necessary. Even *bimodal* or *multi-modal* distributions can be used and ordered using $\leq_{t_p}$ and $\leq_{k_p}$ as defined here. Thus more accurate representation of various real life situations is attainable.
**Acknowledgment:** The first author acknowledges the scholarship obtained from Department of Science and Technology, Government of India, in the form of INSPIRE Fellowship.
|
---
abstract: 'We describe the structure of geometric quotients for proper locally triangulable $\mathbb{G}_{a}$-actions on locally trivial $\mathbb{A}^{3}$-bundles over a nœtherian normal base scheme $X$ defined over a field of characteristic $0$. In the case where $\dim X=1$, we show in particular that every such action is a translation with geometric quotient isomorphic to the total space of a vector bundle of rank $2$ over $X$. As a consequence, every proper triangulable $\mathbb{G}_{a}$-action on the affine four space $\mathbb{A}_{k}^{4}$ over a field of characteristic $0$ is a translation with geometric quotient isomorphic to $\mathbb{A}_{k}^{3}$.'
address:
- |
Adrien Dubouloz\
CNRS\
Institut de Mathématiques de Bourgogne\
Université de Bourgogne\
9 Avenue Alain Savary\
BP 47870\
21078 Dijon Cedex\
France
- |
David R. Finston\
Mathematics Department\
Brooklyn College, CUNY\
2900 Bedford Avenue\
Brooklyn, NY 11210
- |
Imad Jaradat\
Department of Mathematical Sciences\
New Mexico State University\
Las Cruces, New Mexico 88003
author:
- 'Adrien Dubouloz, David R. Finston, and Imad Jaradat'
title: 'Proper triangular $\mathbb{G}_{a}$-actions on $\mathbb{A}^{4}$ are translations'
---
[^1]
Introduction {#introduction .unnumbered}
============
The study of algebraic actions of the additive group $\mathbb{G}_{a}=\mathbb{G}_{a,\mathbb{C}}$ on complex affine spaces $\mathbb{A}^{n}=\mathbb{A}_{\mathbb{C}}^{n}$ has a long history which began in 1968 with a pioneering result of Rentschler [@Ren68] who established that every such action on the plane $\mathbb{A}^{2}$ is triangular in a suitable polynomial coordinate system. Consequently, every fixed point free $\mathbb{G}_{a}$-action on $\mathbb{A}^{2}$ is a translation, in the sense that the geometric quotient $\mathbb{A}^{2}/\mathbb{G}_{a}$ is isomorphic to $\mathbb{A}^{1}$ and that $\mathbb{A}^{2}$ is equivariantly isomorphic to $\mathbb{A}^{2}/\mathbb{G}_{a}\times\mathbb{G}_{a}$ where $\mathbb{G}_{a}$ acts by translations on the second factor.
Arbitrary $\mathbb{G}_{a}$-actions turn out to be no longer triangulable in higher dimensions [@Bass84]. But the question whether a fixed point free $\mathbb{G}_{a}$-action on $\mathbb{A}^{3}$ is a translation or not was settled affirmatively, first for triangulable actions by Snow [@Snow88] in 1988, then by Deveney and the second author [@DevFin94a] in 1994 under the additional assumption that the action is proper and then in general by Kaliman [@Kal04] in 2004. The argument for triangulable actions depends on their explicit form in an appropriate coordinate system which is used to check that the algebraic quotient $\pi:\mathbb{A}^{3}\rightarrow\mathbb{A}^{3}/\!/\mathbb{G}_{a}={\rm Spec}(\Gamma(\mathbb{A}^{3},\mathcal{O}_{\mathbb{A}^{3}})^{\mathbb{G}_{a}})$ is a geometric quotient and that $\mathbb{A}^{3}/\!/\mathbb{G}_{a}$ is isomorphic to $\mathbb{A}^{2}$. For proper actions, the properness implies that the geometric quotient $\mathbb{A}^{3}/\mathbb{G}_{a}$, which a priori only exists as an algebraic space, is separated whence a scheme by virtue of Chow’s Lemma. This means equivalently that the $\mathbb{G}_{a}$-action is not only locally equivariantly trivial in the étale topology but in fact locally trivial in the Zariski topology, i.e. that $\mathbb{A}^{3}$ is covered by invariant Zariski affine open subsets of the from $V_{i}=U_{i}\times\mathbb{G}_{a}$ on which $\mathbb{G}_{a}$ acts by translations on the second factor. Since $\mathbb{A}^{3}$ is factorial, the open subsets $V_{i}$ can even be chosen to be principal, which implies in turn that $\mathbb{A}^{3}/\mathbb{G}_{a}$ is a quasi-affine scheme, in fact an open subset of $\mathbb{A}^{3}/\!/\mathbb{G}_{a}\simeq\mathbb{A}^{2}$ with at most finite complement. The equality $\mathbb{A}^{3}/\mathbb{G}_{a}=\mathbb{A}^{3}/\!/\mathbb{G}_{a}$ ultimately follows by comparing Euler characteristics. Kaliman’s general proof proceeds along a completely different approach, drawing on topological arguments to show directly that the algebraic quotient morphism $\pi:\mathbb{A}^{3}\rightarrow\mathbb{A}^{3}/\!/\mathbb{G}_{a}$ is a locally trivial $\mathbb{A}^{1}$-bundle.
Kaliman’s result can be reinterpreted as the striking fact that the topological contractiblity of $\mathbb{A}^{3}$ is a strong enough constraint to guarantee that a fixed point free $\mathbb{G}_{a}$-action on it is automatically proper. This implication fails completely in higher dimensions where non proper fixed point free $\mathbb{G}_{a}$-actions abound, even in the case of triangular actions on $\mathbb{A}^{4}$ as illustrated by Deveney-Finston-Gehrke in [@DevFinGe94]. Starting from dimension $5$, it is known that properness and triangulability are no longer enough to imply global equivariant triviality or at least local equivariant triviality in the Zariski topology, as shown by examples of triangular actions on $\mathbb{A}^{5}$ with either strictly quasi-affine geometric quotients or with geometric quotients existing only as separated algebraic spaces constructed respectively by Winkelmann [@Win90] and Deveney-Finston [@DevFin95].\
But the question whether a proper $\mathbb{G}_{a}$-action on $\mathbb{A}^{4}$ is a translation or is at least locally equivariantly trivial in the Zariski topology remains open. Very little progress had been made in the study of these actions during the last decades, and the only existing partial results so far concern triangular actions: Deveney, van Rossum and the second author [@DevFinvR04] established in 2004 that a Zariski locally equivariantly trivial triangular $\mathbb{G}_{a}$-action on $\mathbb{A}^{4}$ is a translation. The proof depends on the finite generation of the ring of invariants for such actions established by Daigle-Freudenburg [@DaiFreu01] and exploits the very particular structure of these rings. Incidentally, it is known in general that local triviality for a proper action on $\mathbb{A}^{n}$ follows from the finite generation and regularity of the ring of invariants. But even knowing the former for triangular actions on $\mathbb{A}^{4}$, a direct proof of the latter condition remains elusive. The second positive result concerns a special type of triangular $\mathbb{G}_{a}$-actions generated by derivations of $\mathbb{C}[x,y,z,u]$ of the form $r(x)\partial_{y}+q(x,y)\partial_{z}+p(x,y)\partial_{u}$ where $r(x)\in\mathbb{C}[x]$ and $p(x,y),q(x,y)\in\mathbb{C}[x,y,]$. To insist on the fact that $p(x,y)$ belongs to $\mathbb{C}[x,y]$ and not only to $\mathbb{C}[x,y,z]$ as it would be the case for a general triangular situation, these derivations (and the $\mathbb{G}_{a}$-actions they generate) were named *twin-triangular* in [@DevFin02]. The case where $r(x)$ has simple roots was first settled in 2002 by Deveney and the second author in *loc. cit.* by explicitly computing the invariant ring $\mathbb{C}[x,y,z,u]^{\mathbb{G}_{a}}$ and investigating the structure of the algebraic quotient morphism $\mathbb{A}^{4}\rightarrow\mathbb{A}^{4}/\!/\mathbb{G}_{a}=\mathrm{Spec}(\mathbb{C}[x,y,z_{1},z_{2}]^{\mathbb{G}_{a}})$. The simplicity of the roots of $r(x)$ was crucial to achieve the computation, and the generalization of the result to arbitrary twin-triangular actions obtained in 2012 by the first two authors [@DubFin11] required completely different methods which focused more on the nature of the corresponding geometric quotients $\mathbb{A}_{\mathbb{C}}^{4}/\mathbb{G}_{a}$. The latter a priori exist only as separated algebraic spaces and the crucial step in *loc. cit.* was to show that for twin-triangular actions they are in fact schemes, or, equivalently that proper twin-triangular $\mathbb{G}_{a}$-actions on $\mathbb{A}^{4}$ are not only locally equivariantly trivial in the étale topology but also in the Zariski topology. This enabled in turn the use of the aforementioned result of Deveney-Finston-van Rossum to conclude that such actions are indeed translations.
One of the main obstacles to extend the above results to arbitrary triangular actions comes from the fact that in contrast with fixed point freeness, the property for a triangular $\mathbb{G}_{a}$-action on $\mathbb{A}^{4}$ to be proper is in general subtle to characterize effectively in terms of its associated locally nilpotent derivation. A good illustration of these difficulties is given by the following family of fixed point free $\mathbb{G}_{a}$-actions $$\sigma_{r}:\mathbb{G}_{a}\times\mathbb{A}^{4}\rightarrow\mathbb{A}^{4},\;\left(t,(x,y,z,u)\right)\mapsto(x,y+tx^{2},z+2yt+x^{2}t^{2},u+(1+x^{r}z)t+x^{r}yt^{2}+\frac{1}{3}x^{r+1}t^{3})\quad r\geq1,$$ generated by the triangular derivations $\delta_{r}=x^{2}\partial_{y}+2y\partial_{z}+(1+x^{r}z)\partial_{u}$ of $\mathbb{C}[x,y,z,u]$, which are either non proper if $r=1,2$ or translations otherwise. The fact that $\sigma_{r}$ is a translation for every $r\geq4$ follows immediately from the observation that $\delta_{r}$ admits the variable $s=u-x^{r-2}yz+\frac{2}{3}x^{r-4}y^{3}$ as a global slice. The case $r=3$ is slightly more complicated: one can first observe that $\delta_{3}$ is conjugated via the triangular change of variable $\tilde{u}=u-x^{r-2}yz$ to the twin-triangular derivation $x^{2}\partial_{y}+2y\partial_{z}+(1-2xy^{2})\partial_{\tilde{u}}$ of $\mathbb{C}[x,y,z,\tilde{u}]$. The projection $\mathrm{pr}_{x,y,\tilde{u}}:\mathbb{A}^{4}\rightarrow\mathbb{A}^{3}$ is then equivariant for the fixed point free $\mathbb{G}_{a}$-action on $\mathbb{A}^{3}$ generated by the triangular derivation $x^{2}\partial_{y}+(1-2xy^{2})\partial_{\tilde{u}}$ of $\mathbb{C}[x,y,\tilde{u}]$ and it descends to a locally trivial $\mathbb{A}^{1}$-bundle $\rho:\mathbb{A}^{4}/\mathbb{G}_{a}\rightarrow\mathbb{A}^{3}/\mathbb{G}_{a}\simeq\mathbb{A}^{2}$ between the respective geometric quotients. Since $\mathbb{A}^{2}$ is affine and factorial, $\rho$ is a trivial $\mathbb{A}^{1}$-bundle and hence the $\mathbb{G}_{a}$-action generated by $\delta_{3}$ is a translation. On the other hand, the non properness of $\sigma_{2}$ can be seen quickly via the invariant hypersurface method outlined in [@DubFin11], namely, one checks in this case by a direct computation that the induced $\mathbb{G}_{a}$-action on the invariant hypersurface $H=\left\{ x^{2}z=y^{2}-\frac{3}{2}\right\} \subset\mathbb{A}^{4}$ is not proper, with non separated geometric quotient $H/\mathbb{G}_{a}$ isomorphic to the product of the affine line $\mathbb{A}^{1}$ with the affine line with a double origin. The failure of properness in the case where $r=1$ is even more subtle to analyze since in contrast with the previous case, the induced action on every invariant hypersurface of the form $H_{\lambda}=\left\{ x^{2}z=y^{2}-\lambda\right\} $, $\lambda\in\mathbb{C}$, turns out to be proper. Going back to the definition of the properness for the action $\sigma_{1}$, which says that the morphism $\Phi=(\mathrm{pr}_{2},\sigma_{1}):\mathbb{G}_{a}\times\mathbb{A}^{4}\rightarrow\mathbb{A}^{4}\times\mathbb{A}^{4}$ is proper, one can argue that the union of the following sequence of points $$(p_{n},q_{n})=(p_{n};\mu_{1}(\sqrt{n^{3}},p_{n}))=((\frac{\sqrt[3]{6}}{n},-\frac{\sqrt[3]{36}}{2\sqrt{n}},\frac{1}{\sqrt[3]{6}\sqrt{n}},0);(\frac{\sqrt[3]{6}}{n},\frac{\sqrt[3]{36}}{2\sqrt{n}},\frac{1}{\sqrt[3]{6}\sqrt{n}},1))\in\mathbb{A}^{4}\times\mathbb{A}^{4},\quad n\in\mathbb{N}$$ and its limit $(p_{\infty};q_{\infty})=(p_{\infty},\mu_{1}(1,p_{\infty}))=\left((0,0,0,0);(0,0,0,1)\right)$ is a compact subset of $\mathbb{A}^{4}\times\mathbb{A}^{4}$ equipped with the analytic topology whose inverse image by $\Phi$ is unbounded. So $\Phi$ is not proper as an analytic map between the corresponding varieties equipped with their respective underlying structures of analytic manifolds and hence is not proper in the algebraic category either.\
In this article, we reconsider proper triangular actions on $\mathbb{A}^{4}$ in broader framework and we develop new techniques to overcome the above difficulties. These enable in turn to completely solve the question of global equivariant triviality for such actions. Since a triangular $\mathbb{G}_{a}$-action on $\mathbb{A}^{4}=\mathrm{Spec}(\mathbb{C}[x,y,z,u])$ preserves the variable $x$, it can be considered as an action of the additive group scheme $\mathbb{G}_{a,\mathbb{C}[x]}=\mathbb{G}_{a}\times_{\mathrm{Spec}(\mathbb{C})}\mathrm{Spec}(\mathbb{C}[x])$ on the affine $3$-space $\mathbb{A}_{\mathbb{C}[x]}^{3}$ over $\mathrm{Spec}(\mathbb{C}[x])$ so that the setup is in fact $3$-dimensional over a parameter space. The properties for a $\mathbb{G}_{a,\mathbb{C}[x]}$-action on $\mathbb{A}_{\mathbb{C}[x]}^{3}$ to be proper or triangulable being both local on the parameter space, a cost free generalization is obtained by replacing $\mathrm{Spec}(\mathbb{C}[x])$ by an arbitrary nœtherian normal scheme $X$ defined over a field of characteristic zero and the trivial $\mathbb{A}^{3}$-bundle $\mathrm{pr}_{x}:\mathbb{A}_{\mathbb{C}[x]}^{3}\rightarrow\mathrm{Spec}(\mathbb{C}[x])$ of $\mathrm{Spec}(\mathbb{C}[x])$ by a Zariski locally trivial $\mathbb{A}^{3}$-bundle $\pi:E\rightarrow X$. Our main result then reads as follows:
Let $X$ be a nœtherian normal scheme defined over a field of characteristic zero, let $\pi:E\rightarrow X$ be a Zariski locally trivial $\mathbb{A}^{3}$-bundle equipped with a proper locally triangulable $\mathbb{G}_{a,X}$-action and let $\mathrm{p}:\mathfrak{X}=E/\mathbb{G}_{a,X}\rightarrow X$ be the geometric quotient taken in the category of algebraic $X$-spaces. Then there exists an open sub-scheme $U$ of $X$ with $\mathrm{codim}_{X}(X\setminus U)\geq2$ such that $\mathfrak{X}_{U}=\mathrm{p}^{-1}(U)\rightarrow U$ has the structure of a Zariski locally trivial $\mathbb{A}^{2}$-bundle.
Note in particular that since in the original problem, the base $X=\mathrm{Spec}(\mathbb{C}[x])$ is $1$-dimensional, this Theorem and an appeal to the aforementioned result [@DevFinvR04] are enough to settle the question for $\mathbb{A}_{\mathbb{C}}^{4}$. The conclusion of the above Theorem is essentially optimal. Indeed, in the example due to Winkelmann [@Win90], one has $X=\mathrm{Spec}(\mathbb{C}[x,y])$, $\pi=\mathrm{pr}_{x,y}:\mathbb{A}_{X}^{3}=\mathrm{Spec}(\mathbb{C}[x,y][u,v,w])\rightarrow X$ equipped with the proper triangular $\mathbb{G}_{a,X}$-action generated by the $\mathbb{C}[x,y]$-derivation $\partial=x\partial_{u}+y\partial_{v}+(1+xv-yu)\partial_{w}$ of $\mathbb{C}[x,y][u,v,w]$, and the geometric quotient $\mathrm{p}:\mathfrak{X}=\mathbb{A}_{X}^{3}/\mathbb{G}_{a,X}\rightarrow X$ is the strictly quasi-affine complement of the closed subset $\left\{ x=y=z=0\right\} $ in the $4$-dimensional smooth affine quadric $Q\subset\mathbb{A}_{X}^{3}$ with equation $xt_{2}+yt_{1}=z(z+1)$. The structure morphism $\mathrm{p}:\mathfrak{X}\rightarrow X$ is easily seen to be an $\mathbb{A}^{2}$-fibration, which restricts to a locally trivial $\mathbb{A}^{2}$-bundle over the open subset $U=X\setminus\{(0,0)\}$. However, there is no Zariski or étale open neighborhood of the origin $(0,0)\in X$ over which $\mathrm{p}:\mathfrak{X}\rightarrow X$ restricts to a trivial $\mathbb{A}^{2}$-bundle for otherwise $\mathrm{p}:\mathfrak{X}\rightarrow X$ would be an affine morphism and so $\mathfrak{X}$ would be an affine scheme. The situation for the $\mathbb{C}[x,y]$-derivation $\partial=x\partial_{u}+y\partial_{v}+(1+xv^{2})\partial_{w}$ of $\mathbb{C}[x,y][u,v,w]$ constructed by Deveney-Finston [@DevFin95] is very similar: here the geometric quotient $\mathfrak{X}=\mathbb{A}_{X}^{3}/\mathbb{G}_{a,X}$ is a separated algebraic space which is not a scheme and the structure morphism $\mathrm{p}:\mathfrak{X}\rightarrow X$ is again an $\mathbb{A}^{2}$-fibration restricting to a Zariski locally trivial $\mathbb{A}^{2}$-bundle over $U=X\setminus\{(0,0)\}$ but whose restriction to any Zariski or étale open neighborhood of the origin $(0,0)\in X$ is nontrivial.\
In contrast, in the case of a $1$-dimensional affine base, we can immediately derive the following Corollaries:
Let $\pi:E\rightarrow S$ be a rank $3$ vector bundle over an affine Dedekind scheme $S=\mathrm{Spec}(A)$ defined over a field $k$ of characteristic $0$. Then every proper locally triangulable $\mathbb{G}_{a,S}$-action on $E$ is equivariantly trivial with geometric quotient $E/\mathbb{G}_{a,S}$ isomorphic to a vector bundle of rank $2$ over $S$, stably isomorphic to $E$.
By the previous Theorem, the geometric quotient $\mathrm{p}:E/\mathbb{G}_{a,S}\rightarrow S$ has the structure of a Zariski locally trivial $\mathbb{A}^{2}$-bundle, hence is a vector bundle of rank $2$ by [@BCW77]. In particular, $E/\mathbb{G}_{a,S}$ is affine which implies in turn that $\rho:E\rightarrow E/\mathbb{G}_{a,S}$ is a trivial $\mathbb{G}_{a,S}$-bundle. So $E\simeq E/\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{1}$ as vector bundles over $S$.
Let $S=\mathrm{Spec}(A)$ be an affine Dedekind scheme defined over a field of characteristic $0$. Then every proper triangular $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{3}$ is a translation.
By the previous Corollary, $\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ is a stably trivial vector bundle of rank $2$ over $S$, whence is isomorphic to the trivial bundle $\mathbb{A}_{S}^{2}$ over $S$ by virtue of [@Bas68 IV 3.5].
Coming back to the original problem for triangular $\mathbb{G}_{a,k}$-actions on $\mathbb{A}_{k}^{4}$, the previous Corollary does in fact eliminate the need for [@DevFinvR04] hence the dependency on the fact that the corresponding rings of invariants are finitely generated:
If $k$ is a field of characteristic $0$, then every proper triangular $\mathbb{G}_{a,k}$-action on $\mathbb{A}_{k}^{4}$ is a translation.
Letting $\mathbb{A}_{k}^{4}=\mathrm{Spec}(k[x,y,z,u])$, we may assume that the action is generated by a $k$-derivation of the form $\partial=r(x)\partial_{y}+q(x,y)\partial_{z}+p(x,y,z)\partial_{u}$. As explained above, the latter can be considered as a triangular $k[x]$-derivation of $k[x][y,z,u]$ generating a proper $\mathbb{G}_{a,k[x]}$-action on $\mathbb{A}_{k}^{4}=\mathbb{A}_{k[x]}^{3}$ which is, by the previous Corollary, a trivial $\mathbb{G}_{a}$-bundle over its geometric quotient $\mathbb{A}_{k}^{4}/\mathbb{G}_{a,k}\simeq\mathbb{A}_{k[x]}^{3}/\mathbb{G}_{a,k[x]}\simeq\mathbb{A}_{k[x]}^{2}\simeq\mathbb{A}_{k}^{3}$.
Let us now briefly explain the general philosophy behind the proof. After localizing at codimension $1$ points of $X$, the Main Theorem reduces to the statement that a proper $\mathbb{G}_{a,S}$-action $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ on the affine affine space $\mathbb{A}_{S}^{3}=\mathrm{Spec}(A[y,z,u])$ over the spectrum of a discrete valuation ring, generated by a triangular $A$-derivation $\partial=a\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u}$ of $A[y,z,u]$, where $a\in A\setminus\left\{ 0\right\} $, $q(y)\in A[y]$ and $p(y,z)\in A[y,z]$, is a translation. Triangularity immediately implies that the restriction of $\sigma$ to the generic fiber of $\mathrm{pr}_{S}:\mathbb{A}_{S}^{3}\rightarrow S$ is a translation with $a^{-1}y$ as a global slice. This reduces the problem to the study of neighborhoods of points of the geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ supported on the closed fiber of the structure morphism $\mathrm{p}:\mathfrak{X}\rightarrow S$. A second feature of triangularity is that $\sigma$ commutes with the action $\tau:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ generated by the $A$-derivation $\partial_{u}$ which therefore descends to a $\mathbb{G}_{a,S}$-action $\overline{\tau}$ on the geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$. On the other hand, $\sigma$ descends via the projection $\mathrm{pr}_{y,z}:\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{2}=\mathrm{Spec}(A[y,z])$ to the action $\overline{\sigma}$ on $\mathbb{A}_{S}^{2}$ generated by the $A$-derivation $\overline{\partial}=a\partial_{y}+q(y)\partial_{z}$ of $A[y,z]$. Even though $\overline{\sigma}$ and $\overline{\tau}$ are no longer fixed point free in general, if we take the quotient of $\mathbb{A}_{S}^{2}$ by the action $\overline{\sigma}$ as an algebraic stack $[\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}]$ we obtain a cartesian square $$\xymatrix{ \mathbb{A}^3_S \ar[d]_{\mathrm{pr}_{y,z}} \ar[r] & \mathfrak{X}=\mathbb{A}^3_S/\mathbb{G}_{a,S} \ar[d] \\ \mathbb{A}^2_S \ar[r] & [\mathbb{A}^2_S/\mathbb{G}_{a,S}]}$$which simultaneously identifies the quotient stacks $[\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}]$ for the action $\overline{\sigma}$ and $[\mathfrak{X}/\mathbb{G}_{a,S}]$ for the action $\overline{\tau}$ with the quotient stack of $\mathbb{A}_{S}^{3}$ for the $\mathbb{G}_{a,S}^{2}$-action defined by the commuting actions $\sigma$ and $\tau$. In this setting, the global equivariant triviality of the action $\sigma$ becomes equivalent to the statement that a separated algebraic $S$-space $\mathfrak{X}$ admitting a $\mathbb{G}_{a,S}$-action whose algebraic stack quotient $[\mathfrak{X}/\mathbb{G}_{a,S}]$ is isomorphic to that of a triangular $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{2}$ is an affine scheme.
While a direct proof of this reformulation seems totally out of reach with existing methods, it turns out that its conclusion holds over a certain $\mathbb{G}_{a,S}$-invariant principal open subset $V$ of $\mathbb{A}_{S}^{2}$ which dominates $S$ and for which the algebraic stack quotient $[V/\mathbb{G}_{a,S}]$ is in fact represented by a locally separated algebraic sub-space of $[\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}]$. This provides at least an affine open subscheme $V\times_{S}\mathbb{A}_{S}^{1}/\mathbb{G}_{a,S}$ of $\mathfrak{X}$ dominating $S$, and leaves us with a closed subset of codimension at most $2$ of $\mathfrak{X}$, supported on the closed fiber of $\mathrm{p}:\mathfrak{X}\rightarrow S$, in a neighborhood of which no further information is a priori available to decide even the schemeness of $\mathfrak{X}$. But similar to the argument in [@DubFin11], this situation can be rescued for twin-triangular actions: the fact that for such actions $\partial u=p(y,z)$ is actually a polynomial in $y$ only enables the same reasoning with respect to the other projection $\mathrm{pr}_{y,u}:\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{2}=\mathrm{Spec}(A[y,u])$, yielding a second affine open sub-scheme $V'\times_{S}\mathbb{A}_{S}^{1}/\mathbb{G}_{a,S}$ of $\mathfrak{X}$ dominating $S$. This implies at least the schemeness of $\mathfrak{X}$, provided that the open subsets $V$ and $V'$ can be chosen so that the union of the corresponding open subschemes of $\mathfrak{X}$ covers the closed fiber of $\mathrm{p}:\mathfrak{X}\rightarrow S$.\
The scheme of the article is the following. The first two sections recall basic notions and discuss a couple of preliminary technical reductions. The third section is devoted to establishing an effective criterion for non properness of fixed point free triangular actions from which we deduce the intermediate fact that every proper triangular action is twin-triangulable. Then in the next section, we establish that proper twin-triangular actions are indeed translations. Here, in contrast with the proof for the complex case given in [@DubFin11], our argument is independent of finite generation of rings of invariants and reduces the systematic study of algebraic spaces quotients to a minimum thanks to an appropriate Sheshadri cover trick [@Sesh72].
Recollection on proper, fixed point free and locally triangulable $\mathbb{G}_{a}$-actions
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Proper versus fixed point free actions
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Recall that an action $\sigma:\mathbb{G}_{a,S}\times_{S}E\rightarrow E$ of the additive group scheme $\mathbb{G}_{a,S}=\mathrm{Spec}_{S}(\mathcal{O}_{S}[t])=S\times_{\mathbb{Z}}\mathrm{Spec}(\mathbb{Z}[t])$ on an $S$-scheme $E$ is called proper if the morphism $\Phi=(\mathrm{pr}_{2},\sigma):\mathbb{G}_{a,S}\times_{S}E\rightarrow E\times_{S}E$ is proper.
If $S$ is moreover defined over a field $k$ of characteristic zero, then the fact that $\mathbb{G}_{a,k}$ is affine and has no nontrivial algebraic subgroups implies that properness is equivalent to $\Phi$ being a closed immersion. In particular, a proper $\mathbb{G}_{a,S}$-action is in this case fixed point free and as such, is equivariantly locally trivial in the étale topology on $E$. That is, there exists an affine $S$-scheme $U$ and a surjective étale morphism $f:V=U\times_{S}\mathbb{G}_{a,S}\rightarrow E$ which is equivariant for the action of $\mathbb{G}_{a,S}$ on $U\times_{S}\mathbb{G}_{a,S}$ by translations on the second factor. This implies in turn the existence of a geometric quotient $\rho:E\rightarrow\mathfrak{X}=E/\mathbb{G}_{a,S}$ in the form of an étale locally trivial principal $\mathbb{G}_{a,S}$-bundle over an algebraic $S$-space $\mathrm{p}:\mathfrak{X}\rightarrow S$ (see e.g. [@LMB00 10.4]). Informally, $\mathfrak{X}$ is the quotient of $U$ by the étale equivalence relation which identifies two points $u,u'\in U$ whenever there exists $t,t'\in\mathbb{G}_{a,S}$ such that $f(u,t)=f(u',t')$.
\[par:Properness\_charac\] Conversely, a fixed point free $\mathbb{G}_{a,S}$-action is proper if and only if the geometric quotient $\mathfrak{X}=E/\mathbb{G}_{a,S}$ is a separated $S$-space. Indeed, by definition $\mathrm{p}:\mathfrak{X}\rightarrow S$ is separated if and only if the diagonal morphism $\Delta:\mathfrak{X}\rightarrow\mathfrak{X}\times_{S}\mathfrak{X}$ is a closed immersion, a property which is local on the target with respect to the fpqc topology [@Knu71 II.3.8] and [@SGA1 VIII.5.5]. Since $\rho:E\rightarrow\mathfrak{X}$ is a $\mathbb{G}_{a,S}$-bundle, taking the fpqc base change by $\rho\times\rho:E\times_{S}E\rightarrow\mathfrak{X}\times_{S}\mathfrak{X}$ yields a cartesian square $$\xymatrix{\mathbb{G}_{a,S} \times_S E \ar[r]^{\Phi} \ar[d]_{\rho \circ \mathrm{pr}_2} & E \times_S E \ar[d]^{\rho\times\rho} \\ \mathfrak{X} \ar[r]^-{\Delta} & \mathfrak{X} \times_S \mathfrak{X} }$$from which we see that $\Delta$ is a closed immersion if and only if $\Phi$ is.
Locally triangulable actions
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Given an affine scheme $S=\mathrm{Spec}(A)$ defined over a field of characteristic zero, an action $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{n}\rightarrow\mathbb{A}_{S}^{n}$ generated by a locally nilpotent $A$-derivation $\partial$ of $\Gamma(\mathbb{A}_{S}^{n},\mathcal{O}_{\mathbb{A}_{S}^{n}})$ is called *triangulable* if there exists an isomorphism of $A$-algebras $\tau:\Gamma(\mathbb{A}_{A}^{n},\mathcal{O}_{\mathbb{A}_{A}^{n}})\stackrel{\sim}{\rightarrow}A[x_{1},\cdots,x_{n}]$ such that the conjugate $\delta=\tau\circ\partial\circ\tau^{-1}$ of $\partial$ is triangular with respect to the ordered coordinate system $(x_{1},\ldots,x_{n})$, i.e. has the form $$\delta=p_{0}\frac{\partial}{\partial x_{1}}+\sum_{i=1}^{n}p_{i-1}(x_{1},\ldots,x_{i-1})\frac{\partial}{\partial x_{i}}$$ where $p_{0}\in A$ and where for every $i=1,\ldots,n$, $p_{i-1}(x_{1},\ldots,x_{i-1})\in A[x_{1},\ldots,x_{i-1}]\subset A[x_{1},\ldots,x_{n}]$. By localizing this notion over the base $S$, we arrive at the following definition:
Let $X$ be a scheme defined over a field of characteristic zero and let $\pi:E\rightarrow X$ be a Zariski locally trivial $\mathbb{A}^{n}$-bundle over $X$. An action $\sigma:\mathbb{G}_{a,X}\times_{X}E\rightarrow E$ of $\mathbb{G}_{a,X}$ on $E$ is called *locally triangulable* if there exists a covering of $\mathrm{Spec}(A)$ by affine open sub-schemes $S_{i}=\mathrm{Spec}(A_{i})$, $i\in I$, such that $E\mid_{S_{i}}\simeq\mathbb{A}_{S_{i}}^{n}$ and such that the $\mathbb{G}_{a,S_{i}}$-action $\sigma_{i}:\mathbb{G}_{a,S_{i}}\times_{S_{i}}\mathbb{A}_{S_{i}}^{n}\rightarrow\mathbb{A}_{S_{i}}^{n}$ on $\mathbb{A}_{S_{i}}^{n}$ induced by $\sigma$ is triangulable.
A Zariski locally trivial $\mathbb{A}^{1}$-bundle $\pi:E\rightarrow X$ equipped with a fixed point free $\mathbb{G}_{a,X}$-action is nothing but a principal $\mathbb{G}_{a,X}$-bundle. As mentioned in the introduction, the nature of fixed point free locally triangulable $\mathbb{G}_{a,X}$-actions on Zariski locally trivial $\mathbb{A}^{2}$-bundles $\pi:E\rightarrow X$ is classically known. Namely, we have the following generalization of the main theorem of [@Snow88]:
\[prop:Rank2-bundle\] Let $X$ be a nœtherian normal scheme defined over a field of characteristic $0$ and let $\pi:E\rightarrow X$ be a Zariski locally trivial $\mathbb{A}^{2}$-bundle equipped with a fixed point free locally triangulable $\mathbb{G}_{a,X}$-action. Then the geometric quotient $\mathrm{p}:E/\mathbb{G}_{a,X}\rightarrow X$ has the structure of a Zariski locally trivial $\mathbb{A}^{1}$-bundle over $X$.
The assertion being local on the base $X$, we may assume that $X=\mathrm{Spec}(A)$ is the spectrum of a normal local domain containing a field of characteristic $0$ and that $E=\mathbb{A}_{X}^{2}=\mathrm{Spec}(A[y,z])$ is equipped with the $\mathbb{G}_{a,X}$-action generated by a triangular derivation $\partial=a\partial_{y}+q(y)\partial_{z}$ of $A[y,z]$, where $a\in A$ and $q(y)\in A[y]$. The fixed point freeness hypothesis is equivalent to the property that $a$ and $q(y)$ generate the unit ideal in $A[y,z]$. So $q(y)$ has the form $q(y)=b+c\tilde{q}(y)$ where $b\in A$ is relatively prime with $a$, $c\in\sqrt{aA}$ and $\tilde{q}(y)\in A[y]$. Letting $Q(y)=\int_{0}^{y}q(\tau)d\tau=by+c\int_{0}^{y}\tilde{q}(\tau)d\tau$, the polynomial $v=az-Q(y)\in A[y,z]$ belongs to the kernel $\mathrm{Ker}\partial$ of $\partial$ hence defines a $\mathbb{G}_{a,X}$-invariant morphism $v:E\rightarrow\mathbb{A}_{X}^{1}=\mathrm{Spec}(A[t])$. Since $a$ and $b$ generate the unit ideal in $A$, it follows from the Jacobian criterion that $v:E\rightarrow\mathbb{A}_{X}^{1}$ is a smooth morphism. Furthermore, the fibers of $v$ coincide precisely with the $\mathbb{G}_{a,X}$-orbits on $E$. Indeed, over the principal open subset $X_{a}=\mathrm{Spec}(A_{a})$ of $X$, $\partial$ admits $a^{-1}y$ as a slice and we have an equivariant isomorphism $E\mid_{X_{a}}\simeq\mathrm{Spec}(A[a^{-1}v,a^{-1}y])\simeq\mathbb{A}_{X_{a}}^{1}\times_{X}\mathbb{G}_{a,X}$ where $\mathbb{G}_{a,X}$ acts by translations on the second factor. On the other hand, the restriction $E\mid_{Z}$ of $E$ over the closed subset $Z\subset X$ with defining ideal $\sqrt{aA}\subset A$ is equivariantly isomorphic to $\mathbb{A}_{Z}^{2}$ equipped with the $\mathbb{G}_{a,Z}$-action generated by the derivation $\overline{\partial}=\overline{b}\partial_{z}$ of $(A/\sqrt{aA})[y,z]$, where $\overline{b}\in(A/\sqrt{aA})^{*}$ denotes the residue class of $b$. The restriction of $v$ to $E\mid_{Z}$ coincides via this isomorphism to the morphism $\mathbb{A}_{Z}^{2}\rightarrow\mathbb{A}_{Z}^{1}$ defined by the polynomial $\overline{v}=\overline{b}y\in(A/\sqrt{aA})[y,z]$ which is obviously a geometric quotient. The above properties imply that the morphism $\tilde{v}:E/\mathbb{G}_{a,X}\rightarrow\mathbb{A}_{X}^{1}$ induced by $v$ is smooth and bijective. Since it admits étale quasi-sections, $\tilde{v}$ is then an isomorphism locally in the étale topology on $\mathbb{A}_{X}^{1}$ whence an isomorphism.
preliminary reductions
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Reduction to a local base
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The statement of the Main Theorem can be rephrased equivalently as the fact that a proper locally triangulable $\mathbb{G}_{a,S}$-action on a Zariski locally trivial $\mathbb{A}^{3}$-bundle $\pi:E\rightarrow S$ is a translation in codimension $1$. This means that for every point $s\in S$ of codimension $1$ with local ring $\mathcal{O}_{S,s}$, the fiber product $E\times_{S}S'\simeq\mathbb{A}_{S'}^{3}$ of $E\rightarrow S$ with the canonical immersion $S'=\mathrm{Spec}(\mathcal{O}_{S,s})\hookrightarrow S$ equiped with the induced proper triangular action of $\mathbb{G}_{a,S'}=\mathbb{G}_{a,S}\times_{S}S'$ is equivariantly isomorphic to the trivial bundle $\mathbb{A}_{S'}^{2}\times_{S'}\mathbb{G}_{a,S'}$ over $S'$ equipped with the action of $\mathbb{G}_{a,S'}$ by translations on the second factor.
\[par:local\_notation\] So we are reduced to the case where $S$ is the spectrum of a discrete valuation ring $A$ containing a field of characteristic $0$, say with maximal ideal $\mathfrak{m}$ and residue field $\kappa=A/\mathfrak{m}$, and where $\pi=\mathrm{pr}_{S}:E=\mathbb{A}_{S}^{3}=\mathrm{Spec}(A[y,z,u])\rightarrow S=\mathrm{Spec}(A)$ is equipped with a proper triangulable $\mathbb{G}_{a,S}$-action $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$. Letting $x\in\mathfrak{m}$ be uniformizing parameter, every such action is equivalent to one generated by an $A$-derivation $\partial$ of $A[y,z,u]$ of the form $$\partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u}$$ where $n\geq0$, $q(y)\in A[y]$ and $p(y,z)=\sum_{r=0}^{\ell}p_{r}(y)z^{r}\in A[y,z]$, the fixed point freeness of $\sigma$ being equivalent to the property that $x^{n}$, $q(y)$ and $p(y,z)$ generate the unit ideal in $A[y,z,u]$.
\[sub:Reduction-to-Affineness\] Reduction to proving the affineness of the geometric quotient
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With the notation of §\[par:local\_notation\], we can already observe that if $n=0$ then $y$ is an obvious global slice for $\partial$ and hence that the action is globally equivariantly trivial with geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}\simeq\mathbb{A}_{S}^{2}$. Similarly, if the residue class of $q(y)$ in $\kappa[y]$ is a non zero constant then the action $\sigma$ is a translation. Indeed, in this case, the $\mathbb{G}_{a,S}$-action $\overline{\sigma}:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{2}\rightarrow\mathbb{A}_{S}^{2}$ on $\mathbb{A}_{S}^{2}=\mathrm{Spec}(A[y,z])$ generated by the $A$-derivation $\overline{\partial}=x^{n}\partial_{y}+q(y)\partial_{z}$ of $A[y,z]$ is fixed point free hence globally equivariantly trivial with geometric quotient $\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}\simeq\mathbb{A}_{S}^{1}$ by virtue of Proposition \[prop:Rank2-bundle\]. On the other hand, the $\mathbb{G}_{a,S}$-equivariant projection $\mathrm{pr}_{y,z}:\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{2}$ descends to a locally trivial $\mathbb{A}^{1}$-bundle between the geometric quotients $\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ and $\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}$, and since $\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}\simeq\mathbb{A}_{S}^{1}$ is affine and factorial, it follows that $\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}\simeq\mathbb{A}_{S}^{2}/\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{1}\simeq\mathbb{A}_{S}^{2}.$ The affineness of $\mathbb{A}_{S}^{2}$ implies in turn that the quotient morphism $\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ is the trivial $\mathbb{G}_{a,S}$-bundle whence that $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ is a translation. Alternatively, one can observe that a global slice $s\in A[y,z]$ for the action $\overline{\sigma}$ is also a global slice for $\sigma$ via the inclusion $A[y,z]\subset A[y,z,u]$
More generally, the following Lemma reduces the question of global equivariant triviality with geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ isomorphic to $\mathbb{A}_{S}^{2}$ to showing that $\mathfrak{X}$, which a priori only exists as an algebraic $S$-space, is an affine $S$-scheme:
\[lem:Reduction\_to\_affineness\] A fixed point free triangular action $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ is a translation if and only if its geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ is an affine $S$-scheme.
One direction is clear, so assume that $\mathfrak{X}$ is an affine $S$-scheme. It suffices to show that the structure morphism $\mathrm{p}:\mathfrak{X}\rightarrow S$ is an $\mathbb{A}^{2}$-fibration, i.e. a faithfully flat morphism with all its fibers isomorphic to affine planes over the corresponding residue fields. Indeed, if so, the affineness of $\mathfrak{X}$ implies on the one hand that $\mathfrak{X}$ is isomorphic to the trivial $\mathbb{A}^{2}$-bundle $\mathbb{A}_{S}^{2}$ by virtue of [@Sat83] and on the other hand that $\rho:\mathbb{A}_{S}^{3}\rightarrow\mathfrak{X}$ is isomorphic to the trivial $\mathbb{G}_{a,S}$-bundle $\mathfrak{X}\times_{S}\mathbb{G}_{a,S}$ over $S$, which yields $\mathbb{G}_{a,S}$-equivariant isomorphisms $\mathbb{A}_{S}^{3}\simeq\mathfrak{X}\times_{S}\mathbb{G}_{a,S}\simeq\mathbb{A}_{S}^{2}\times_{S}\mathbb{G}_{a,S}$.
To see that $\mathrm{p}:\mathfrak{X}\rightarrow S$ is an $\mathbb{A}^{2}$-fibration, recall that $\mathrm{pr}_{S}:\mathbb{A}_{S}^{3}\rightarrow S$ and the quotient morphism $\rho:\mathbb{A}_{S}^{3}\rightarrow\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ are both faithfully flat, so that $\mathrm{p}:\mathfrak{X}\rightarrow S$ is faithfully flat too ([@Knu71 II.3.2] and [@EGA4 Corollaire 2.2.13(iii)]). Letting $\mathfrak{m}$ and $\xi$ be the closed and generic points of $S$ respectively, the fibers $\mathrm{pr}_{S}^{-1}(\mathfrak{m})\simeq\mathbb{A}_{\kappa}^{3}$ and $\mathrm{pr}_{S}^{-1}(\xi)\simeq\mathbb{A}_{\kappa(\xi)}^{3}$ coincide with the total spaces of the restriction of the $\mathbb{G}_{a,S}$-bundle $\rho:\mathbb{A}_{S}^{3}\rightarrow\mathfrak{X}$ over the fibers $\mathfrak{X}_{\mathfrak{m}}=\mathrm{p}^{-1}(\mathfrak{m})$ and $\mathfrak{X}_{\xi}=\mathrm{p}^{-1}(\xi)$ respectively. Since the $\mathbb{G}_{a,\kappa(\xi)}$-action induced by $\sigma$ on $\mathrm{pr}_{S}^{-1}(\xi)$ admits $x^{-n}y$ as a global slice, it is a translation with geometric quotient $\mathbb{A}_{\kappa(\xi)}^{3}/\mathbb{G}_{a,\kappa(\xi)}\simeq\mathbb{A}_{\kappa(\xi)}^{2}$ and so $\mathfrak{X}_{\xi}\simeq\mathbb{A}_{\kappa(\xi)}^{2}$. On the other hand, we may assume in view of the above discussion that $n\geq1$ so that the $\mathbb{G}_{a,\kappa}$-action on $\mathrm{pr}_{S}^{-1}(\mathfrak{m})\simeq\mathbb{A}_{\kappa}^{3}$ induced by $\sigma$ coincides with the fixed point free action generated by the $\kappa[y]$-derivation $\overline{\partial}=\overline{q}(y)\partial_{z}+\overline{p}(y,z)\partial_{u}$ of $\kappa[y][z,u]$, where $\overline{q}(y)$ and $\overline{p}(y,z)$ denote the respective residue classes of $q(y)$ and $p(y,z)$ modulo $x$. By virtue of Proposition \[prop:Rank2-bundle\], the geometric quotient $\mathbb{A}_{\kappa}^{3}/\mathbb{G}_{a,\kappa}$ has the structure of a Zariski locally trivial $\mathbb{A}^{1}$-bundle over $\mathbb{A}_{\kappa}^{1}=\mathrm{Spec}(\kappa[y])$ hence is isomorphic to $\mathbb{A}_{\kappa}^{2}$. This implies that $\mathfrak{X}_{\mathfrak{m}}\simeq\mathbb{A}_{\kappa}^{3}/\mathbb{G}_{a,\kappa}\simeq\mathbb{A}_{\kappa}^{2}$ as desired.
By exploiting the fact that arbitrary $\mathbb{G}_{a,S}$-actions on the affine $3$-space $\mathbb{A}_{S}^{3}$ over the spectrum $S$ of a discrete valuation ring $A$ containing a field of characteristic $0$ have finitely generated rings of invariants [@BhaDa09], one can derive the following stronger characterization: a fixed point free action $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ is either a translation or its geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ is an algebraic space which is not a scheme.
Indeed, the quotient morphism $\rho:\mathbb{A}_{S}^{3}\rightarrow\mathfrak{X}$ is again an $\mathbb{A}^{2}$-fibration thanks to [@DaiKal09 Theorem 3.2] which asserts that for every field $\kappa$ of characteristic $0$ a fixed point free action of $\mathbb{G}_{a,\kappa}$-action on $\mathbb{A}_{\kappa}^{3}$ is a translation, and so the assertion is equivalent to the fact that a Zariski locally equivariantly trivial action $\sigma$ has affine geometric quotient $\mathfrak{X}$. This can be seen in a similar way as in the proof of Theorem 2.1 in [@DevFinvR04]. Namely, by hypothesis we can find an open covering of $\mathbb{A}_{S}^{3}$ by finitely many invariant affine open subsets $U_{i}$ on which the induced $\mathbb{G}_{a,S}$-action is a translation with affine geometric quotient $U_{i}/\mathbb{G}_{a,S}$, $i=1,\ldots,n$. Since $U_{i}$ and $\mathbb{A}_{S}^{3}$ are affine, $\mathbb{A}_{S}^{3}\setminus U_{i}$ is a $\mathbb{G}_{a,S}$-invariant Weil divisor on $\mathbb{A}_{S}^{3}$ which is in fact principal as $A$, whence $A[y,z,u]$, is factorial. It follows that there exists invariant regular functions $f_{i}\in A[y,z,u]^{\mathbb{G}_{a}}\simeq\Gamma(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})$ such that $U_{i}=\mathrm{Spec}(A[x,y,z]_{f_{i}})$ coincides with the inverse image by the quotient morphism $\rho:\mathbb{A}_{S}^{3}\rightarrow\mathfrak{X}$ of the principal open subset $\mathfrak{X}_{f_{i}}$ of $\mathfrak{X}$, $i=1,\ldots,n$. Since $\rho:\mathbb{A}_{S}^{3}\rightarrow\mathfrak{X}$ is a $\mathbb{G}_{a,S}$-bundle and $U_{i}\simeq U_{i}/\mathbb{G}_{a,S}\times_{S}\mathbb{G}_{a,S}$ by assumption, we conclude that $\mathfrak{X}$ is covered by the principal affine open subsets $\mathfrak{X}_{f_{i}}\simeq U_{i}/\mathbb{G}_{a,S}$, $i=1,\ldots,n$, whence is quasi-affine. Now since by the aforementioned result [@BhaDa09], $A[y,z,u]^{\mathbb{G}_{a}}$ is an integrally closed finitely generated $A$-algebra, it is enough to check that the canonical open immersion $j:\mathfrak{X}\rightarrow X=\mathrm{Spec}(\Gamma(\mathfrak{X},\mathcal{O}_{\mathfrak{X}}))\simeq\mathrm{Spec}(A[y,z,u]^{\mathbb{G}_{a}})$ is surjective. The surjectivity over the generic point of $S$ follows immediately from the fact the kernel of a locally nilpotent derivation derivation of a polynomial ring in three variables over a field $K$ of characteristic $0$ is isomorphic to a polynomial ring in two variables over $K$ (see e.g. [@Miy85]). So it remains to show that the induced open immersion $j_{\mathfrak{m}}:\mathfrak{X}_{m}\simeq\mathbb{A}_{\kappa}^{2}\hookrightarrow X_{\mathfrak{m}}=\mathrm{Spec}(A[y,z,u]^{\mathbb{G}_{a}}\otimes_{A}A/\mathfrak{m})$ between the corresponding fibers over the closed point $\mathfrak{m}$ of $S$ is surjective, in fact, an isomorphism. Since $x\in A[y,z,u]^{\mathbb{G}_{a}}$ is prime, $X_{\mathfrak{m}}\simeq\mathrm{Spec}(A[y,z,u]^{\mathbb{G}_{a}}/(x))$ is an integral $\kappa$-scheme of finite type and Corollary 4.10 in [@BhaDa09] can be interpreted more precisely as the fact that $X_{\mathfrak{m}}\simeq C\times_{\kappa}\mathbb{A}_{\kappa}^{1}$ for a certain $1$-dimensional affine $\kappa$-scheme $C$. This implies in turn that $j_{\mathfrak{m}}$ is an isomorphism. Indeed, since $C$ is dominated via $j_{\mathfrak{m}}$ by a general affine line $\mathbb{A}_{\kappa}^{1}\subset\mathbb{A}_{\kappa}^{2}$, its normalization $\tilde{C}$ is isomorphic to $\mathbb{A}_{\kappa}^{1}$ and so $j_{\mathfrak{m}}$ factors through an open immersion $\tilde{j}_{\mathfrak{m}}:\mathbb{A}_{\kappa}^{2}\hookrightarrow\tilde{C}\times_{\kappa}\mathbb{A}_{\kappa}^{1}\simeq\mathbb{A}_{\kappa}^{2}$. The latter is surjective for otherwise the complement of its image would be of pure codimension $1$ hence a principal divisor $\mathrm{div}(f)$ for a non constant regular function $f$ on $\tilde{C}\times_{\kappa}\mathbb{A}_{\kappa}^{1}$. But then $f$ would restrict to a non constant invertible function on the image of $\mathbb{A}_{\kappa}^{2}$ which is absurd. Thus $\tilde{j}_{\mathfrak{m}}:\mathbb{A}_{\kappa}^{2}\hookrightarrow\tilde{C}\times_{\kappa}\mathbb{A}_{\kappa}^{1}\simeq\mathbb{A}_{\kappa}^{2}$ is an isomorphism and since the normalization morphism $\tilde{C}\times_{\kappa}\mathbb{A}_{\kappa}^{1}\rightarrow C\times_{\kappa}\mathbb{A}_{\kappa}^{1}$ is finite whence closed it follows that $j_{\mathfrak{m}}:\mathbb{A}_{\kappa}^{2}\hookrightarrow C\times_{\kappa}\mathbb{A}_{\kappa}^{1}$ is an open and closed immersion hence an isomorphism.
Reduction to extensions of irreducible derivations
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In view of the discussion at the beginning of subsection \[sub:Reduction-to-Affineness\], we may assume for the $A$-derivation $$\partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u}$$ that $n>0$ and that the residue class of $q(y)$ in $\kappa[y]$ is either zero or not constant. In the first case, $q(y)\in\mathfrak{m}A[y]$ has the form $q(y)=x^{\mu}q_{0}(y)$ where $\mu>0$ and where $q_{0}(y)\in A[y]$ has non zero residue class modulo $\mathfrak{m}$, so that the derivation $\overline{\partial}=x^{n}\partial_{y}+q(y)\partial_{z}$ induced by $\partial$ on the sub-ring $A[y,z]$ is reducible. On the other hand, the fixed point freeness of the $\mathbb{G}_{a,S}$-action $\sigma$ generated by $\partial$ implies that up to multiplying $u$ by an invertible element in $A$, one has $p(y,z)=1+x^{\nu}p_{0}(y,z)$ for some $\nu>0$ and $p_{0}(y,z)\in A[y,z]$.
If $\mu\geq n$, then letting $Q_{0}(y)=\int_{0}^{y}q_{0}(\tau)d\tau\in A[y]$, the $\mathbb{G}_{a,S}$-invariant polynomial $z_{1}=z-x^{\mu-n}Q_{0}(y)$ is a variable of $A[y,z,u]$ over $A[y,u]$, and so $\partial$ is conjugate to the derivation $x^{n}\partial_{y}+p(y,z_{1}+x^{\mu-n}Q_{0}(y))\partial_{u}$ of the polynomial ring in two variables $A[z_{1}][y,u]$ over $A[z_{1}]$. Since $\sigma$ is fixed point free, Proposition \[prop:Rank2-bundle\] implies that it is equivariantly trivial with geometric quotient isomorphic to the total space of the trivial $\mathbb{A}^{1}$-bundle over $\mathbb{A}_{S}^{1}=\mathrm{Spec}(A[z_{1}])$ whence to $\mathbb{A}_{S}^{2}$.
Otherwise, if $\mu<n$, then the $\mathbb{G}_{a,S}$-action $\tilde{\sigma}:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ on $\mathbb{A}_{S}^{3}=\mathrm{Spec}(A[\tilde{y},\tilde{z},\tilde{u}])$ generated by the $A$-derivation $$\tilde{\partial}=x^{n-\mu}\partial_{\tilde{y}}+q_{0}(\tilde{y})\partial_{\tilde{z}}+(1+x^{\nu}p_{0}(\tilde{y},\tilde{z}))\partial_{\tilde{u}}$$ is again fixed point free, hence admits a geometric quotient $\tilde{\rho}:\mathbb{A}_{S}^{3}\rightarrow\tilde{\mathfrak{X}}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ in the form of an étale locally trivial $\mathbb{G}_{a,S}$-bundle over a certain algebraic $S$-space $\tilde{\mathfrak{X}}$.
The quotient spaces $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ and $\tilde{\mathfrak{X}}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ for the $\mathbb{G}_{a,S}$-actions $\sigma$ and $\tilde{\sigma}$ on $\mathbb{A}_{S}^{3}$ generated by $\partial$ and $\tilde{\partial}$ respectively are isomorphic. In particular $\sigma$ is proper (resp. equivariantly trivial) if and only if $\tilde{\sigma}$ is proper (resp. equivariantly trivial).
Letting $\tilde{\rho}_{i}:V_{i}=\mathbb{A}_{S}^{3}\rightarrow\tilde{\mathfrak{X}}_{i}=V_{i}/\mathbb{G}_{a,S}$, $i=0,\ldots,\mu$, denote the geometric quotient of $V_{i}=\mathrm{Spec}(A[\tilde{y}_{i},\tilde{z}_{i},\tilde{u}_{i}])$ for the fixed point free $\mathbb{G}_{a,S}$-action $\tilde{\sigma}_{i}$ generated by the $A$-derivation $$\tilde{\partial}_{i}=\left(1+x^{\nu}p_{0}(\tilde{y}_{i},\tilde{z}_{i})\right)\partial_{\tilde{u}_{i}}+x^{\mu-i}q_{0}(\tilde{y}_{i})\partial_{\tilde{z}_{i}}+x^{n-i}\partial_{\tilde{y}_{i}},$$ the first assertion will follow from the more general fact that $\tilde{\mathfrak{X}}_{i}\simeq\tilde{\mathfrak{X}}_{i+1}$ for every $i=0,\ldots,\mu-1$. Indeed, we first observe that since $\tilde{u}_{i}$ is a slice for $\tilde{\partial}_{i}$ modulo $x$, $\tilde{\mathfrak{X}}_{i,\mathfrak{m}}=\tilde{\mathfrak{X}}_{i}\times_{S}\mathrm{Spec}(\kappa)$ is isomorphic to $\mathbb{A}_{\kappa}^{2}=\mathrm{Spec}((A/\mathfrak{m})[\tilde{y}_{i},\tilde{z}_{i}])$ and the restriction of $\tilde{\rho}_{i}$ over $\tilde{\mathfrak{X}}_{i,\mathfrak{m}}$ is isomorphic to the trivial bundle $\mathrm{pr}_{1}:\tilde{\mathfrak{X}}_{i,\mathfrak{m}}\times_{\kappa}\mathrm{Spec}(\kappa[\tilde{u}_{i}])\rightarrow\tilde{\mathfrak{X}}_{i,\mathfrak{m}}$. Now let $\beta_{i}:V_{i+1}\rightarrow V_{i}$ be the affine modification of the total space of $\tilde{\rho}_{i}:\mathbb{A}_{S}^{3}\rightarrow\tilde{\mathfrak{X}}_{i}$ with center at the zero section of the induced bundle $\mathrm{pr}_{1}:\tilde{\mathfrak{X}}_{i,\mathfrak{m}}\times_{\kappa}\mathrm{Spec}(\kappa[\tilde{u}_{i}])\rightarrow\tilde{\mathfrak{X}}_{i,\mathfrak{m}}$ and with principal divisor $x$. In view of the previous description, $\beta_{i}:V_{i+1}\rightarrow V_{i}$ coincides with the affine modification of $\mathrm{Spec}(A[\tilde{y}_{i},\tilde{z}_{i},\tilde{u}_{i}])$ with center at the ideal $(x,\tilde{u}_{i})$ and principal divisor $x$, that is, with the birational $S$-morphism induced by the homomorphism of $A$-algebra $$\beta_{i}^{*}:A[\tilde{y}_{i+1},\tilde{z}_{i+1},\tilde{u}_{i+1}]\rightarrow A[\tilde{y}_{i},\tilde{z}_{i},\tilde{u}_{i}],\;(\tilde{y}_{i+1},\tilde{z}_{i+1},\tilde{u}_{i+1})\mapsto(\tilde{y}_{i},\tilde{z}_{i},x\tilde{u}_{i}).$$ By construction, $\beta_{i}$ is equivariant for the $\mathbb{G}_{a,S}$-actions $\tilde{\sigma}_{i+1}$ and $\overline{\sigma}_{i}$ generated respectively by the locally nilpotent $A$-derivations $\tilde{\partial}_{i+1}$ of $A[\tilde{y}_{i+1},\tilde{z}_{i+1},\tilde{u}_{i+1}]$ and $\overline{\partial}_{i}=x\tilde{\partial}_{i}$ of $A[\tilde{y}_{i},\tilde{z}_{i},\tilde{u}_{i}]$. Furthermore, since $\tilde{\rho}_{i}:V_{i}\rightarrow\tilde{\mathfrak{X}}_{i}$ is also $\mathbb{G}_{a,S}$-invariant for the action $\overline{\sigma}_{i}$, the morphism $\tilde{\rho}_{i}\circ\beta_{i}:V_{i+1}\rightarrow\tilde{\mathfrak{X}}_{i}$ is $\mathbb{G}_{a,S}$-invariant, whence descends to a morphism $\tilde{\beta}_{i}:\tilde{\mathfrak{X}}_{i+1}\rightarrow\tilde{\mathfrak{X}}_{i}$. Since the latter restricts to an isomorphism over the generic point of $S$, it remains to check that it is also an isomorphism in a neighborhood of every point $p\in\tilde{\mathfrak{X}}_{i}$ lying over the closed point $\mathfrak{m}$ of $S$. Let $f:U=\mathrm{Spec}(B)\rightarrow\tilde{\mathfrak{X}}_{i}$ be an affine étale neighborhood of such a point $p\in\tilde{\mathfrak{X}}_{i}$ over which $\tilde{\rho}_{i}:V_{i}\rightarrow\tilde{\mathfrak{X}}_{i}$ becomes trivial, say $V_{i}\times_{\tilde{\mathfrak{X}}_{i}}U\simeq\mathbb{A}_{U}^{1}=\mathrm{Spec}(B[\tilde{v}_{i}])$. The $\mathbb{G}_{a,S}$-action on $V_{i}$ generated by $\overline{\partial}_{i}$ lifts to the $\mathbb{G}_{a,U}$-action on $\mathbb{A}_{U}^{1}$ generated by the locally nilpotent $B$-derivation $x\partial_{\tilde{v}_{i}}$ and since $\beta_{i}:V_{i+1}\rightarrow V_{i}$ is the affine modification of $V_{i}$ with center at the zero section of the restriction of $\tilde{\rho}_{i}:V_{i}\rightarrow\tilde{\mathfrak{X}}_{i}$ over the closed point of $S$, we have a commutative diagram $$\xymatrix@R=12pt@C=11pt{ & V_{i+1} \ar'[d][dd]_(.3){\tilde{\rho}_{i+1}} \ar[dl]_{\beta_i} && \mathbb{A}^1_U \ar[ll] \ar[dd]^{\mathrm{pr}_U} \ar[dl]_{\delta_i} \\ V_i \ar[dd]_{\tilde{\rho}_i} && \mathbb{A}^1_U \ar[ll] \ar[dd]^(.3){\mathrm{pr}_U} \\ & \tilde{\mathfrak{X}}_{i+1} \ar[dl]_{\tilde{\beta}_i} && U \ar@{=}[dl] \ar'[l][ll] \\ \tilde{\mathfrak{X}}_i && U \ar[ll]_{f} }$$in which the top and front squares are cartesian, and where the morphism $\delta_{i}:\mathbb{A}_{U}^{1}=\mathrm{Spec}(B[\tilde{v}_{i+1}])\rightarrow\mathbb{A}_{U}^{1}=\mathrm{Spec}(B[\tilde{v}_{i}])$ is defined by the $B$-algebras homomorphism $B[\tilde{v}_{i}]\rightarrow B[\tilde{v}_{i+1}]$, $\tilde{v}_{i}\mapsto x\tilde{v}_{i+1}$. The latter is equivariant for the action on $\mathrm{Spec}(B[\tilde{v}_{i+1}])$ generated by the locally nilpotent $B$-derivation $\partial_{\tilde{v}_{i+1}}$ and we conclude that $\mathrm{pr}_{2}:\mathbb{A}_{U}^{1}\simeq\mathbb{A}_{U}^{1}\times_{V_{i}}V_{i+1}\rightarrow V_{i+1}$ is an étale trivialization of the $\mathbb{G}_{a,S}$-action induced by $\tilde{\sigma}_{i+1}$ on the open sub-scheme $(\tilde{\rho}_{i}\circ\beta_{i})^{-1}(f(U))$ of $V_{i+1}$. This implies in turn that $U\times_{\tilde{\mathfrak{X}}_{i}}\tilde{\mathfrak{X}}_{i+1}\simeq U$, whence that $\tilde{\beta}_{i}:\tilde{\mathfrak{X}}_{i+1}\rightarrow\tilde{\mathfrak{X}}_{i}$ is an isomorphism in a neighborhood of $p\in\tilde{\mathfrak{X}}_{i}$ as desired.
The second assertion is a direct consequence of the fact that properness and global equivariant triviality of $\sigma$ and $\tilde{\sigma}$ are respectively equivalent to the separatedness and the affineness of the geometric quotients $\mathfrak{X}\simeq\tilde{\mathfrak{X}}$.
Summing up, we are now reduced to proving that a proper $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{3}$ generated by an $A$-derivation $$\partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u}$$ of $A[y,z,u]$, such that $n>0$ and $q(y)\in A[y]$ has non constant residue class in $\kappa[y]$, has affine geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$. This will be done in two steps in the next sections: we will first establish that a proper $\mathbb{G}_{a,S}$-action as above is conjugate to one generated by a special type of $A$-derivation called *twin-triangular.* Then we will prove in section \[sec:Twin-Triviality\] that proper twin-triangular $\mathbb{G}_{a,S}$-actions on $\mathbb{A}_{S}^{3}$ do indeed have affine geometric quotients.
Reduction to twin-triangular actions
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We keep the same notation as in §\[par:local\_notation\] above, namely $A$ is a discrete valuation ring containing a field of characteristic $0$, with maximal ideal $\mathfrak{m}$, residue field $\kappa=A/\mathfrak{m}$, and uniformizing parameter $x\in\mathfrak{m}$. We let again $S=\mathrm{Spec}(A)$.
We call an $A$-derivation $\partial$ of $A[y,z,u]$ *twin-triangulable* if there exists a coordinate system $(y,z_{+},z_{-})$ of $A[y,z,u]$ over $A[y]$ in which the conjugate of $\partial$ is *twin-triangular*, that is, has the form $x^{n}\partial_{y}+p_{+}(y)\partial_{z_{+}}+p_{-}(y)\partial_{z_{-}}$ for certain polynomials $p_{\pm}(y)\in A[y]$. This section is devoted to the proof of the following intermediate characterization of proper triangular $\mathbb{G}_{a,S}$-actions:
\[prop:Main-Local\] With the notation above, let $\partial$ by an $A$-derivation of $A[y,z,u]$ of the form $$\partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u}$$ where $n>0$ and where $q(y)\in A[y]$ has non constant residue class in $\kappa[y]$. If the $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{3}=\mathrm{Spec}(A[y,z,u])$ generated by $\partial$ is proper, then $\partial$ is twin-triangulable.
The proof given below proceeds in two steps: we first construct a coordinate $\tilde{u}$ of $A[y,z,u]$ over $A[y,z]$ with the property that $\partial\tilde{u}=\tilde{p}(y,z)$ is either a polynomial in $y$ only or its leading term $\tilde{p}_{\ell}(y)$ as a polynomial in $z$ has a very particular form. In the second case, we exploit the properties of $\tilde{p}_{\ell}(y)$ to show that the $\mathbb{G}_{a,S}$-action generated by $\partial$ is not proper.
The $\sharp$-reduction of a triangular $A$-derivation
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The conjugate of an $A$-derivation $\partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u}$ of $A[y,z,u]$ as in Proposition \[prop:Main-Local\] by an isomorphism of $A[y,z]$-algebras $\psi:A[y,z][\tilde{u}]\stackrel{\sim}{\rightarrow}A[y,z][u]$ is again triangular of the form $$\psi^{-1}\partial\psi=x^{n}\partial_{y}+q(y)\partial_{z}+\tilde{p}(y,z)\partial_{\tilde{u}}$$ for some polynomial $\tilde{p}(y,z)\in A[y,z]$. In particular, we may choose from the very beginning a coordinate system of $A[y,z,u]$ over $A[y,z]$ with the property that the degree of $\partial u\in A[y,z]$ with respect to $z$ is minimal among all possible conjugates $\psi^{-1}\partial\psi$ of $\partial$ as above. In what follows, we will say for short that such a derivation $\partial$ is *$\sharp$-reduced* with respect to the coordinate system $(y,z,u)$. Letting $Q(y)=\int_{0}^{y}q(\tau)d\tau\in A[y]$, this property can be characterized effectively as follows:
Let $\partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u}$ be a $\sharp$-reduced derivation of $A[y,z,u]$ as in Proposition \[prop:Main-Local\]. If $\partial$ is not twin-triangular $($i.e. $p(y,z)=p_{0}(y)\in A[y]$$)$ then the leading term $p_{\ell}(y)$, $\ell\geq1$, of $p(y,z)$ as a polynomial in $z$ is not congruent modulo $x^{n}$ to a polynomial of the form $q(y)f(Q(y))$ for some $f(\tau)\in A[\tau]$.
Suppose that $p(y,z)=\sum_{r=0}^{\ell}p_{r}(y)z^{r}$ with $\ell\geq1$ and that $p_{\ell}(y)=q(y)f(Q(y))+x^{n}g(y)$ for some polynomials $f(\tau),g(\tau)\in A[\tau]$. Then letting $G(y)=\int_{0}^{y}g(\tau)d\tau$ and $$\tilde{u}=u-G(y)z^{\ell}-\sum_{k=0}^{\deg f}\frac{(-1)^{k}}{\prod_{j=0}^{k}(\ell+1+j)}f^{(k)}(Q(y))x^{kn}z^{\ell+1+k},$$ one checks by direct computation that $$\partial\tilde{u}=\sum_{r=0}^{\ell-2}p_{r}(y)z^{r}+\left(p_{\ell-1}(y)-G(y)q(y)\right)z^{\ell-1}.$$ Thus $(y,z,\tilde{u}$) is a coordinate system of $A[y,z,u]$ over $A[y,z]$ in which the image of $\tilde{u}$ by the conjugate of $\partial$ has degree $\leq\ell-1$, a contradiction to the $\sharp$-reducedness of $\partial$.
To prove Proposition \[prop:Main-Local\], it remains to show that a proper $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{3}$ generated by $\sharp$-reduced $A$-derivation of $A[y,z,u]$ is twin-triangular. This is done in the next sub-section.
A non-valuative criterion for non-properness
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To disprove the properness of an algebraic action $\sigma:\mathbb{G}_{a,S}\times_{S}E\rightarrow E$ of $\mathbb{G}_{a,S}$ on an $S$-scheme $E$, it suffices in principle to check that the image of $\Phi=(\mathrm{pr}_{2},\sigma):\mathbb{G}_{a}\times_{S}E\rightarrow E\times_{S}E$ is not closed. However, this image turns out to be complicated to determine in general, and it is more convenient for our purpose to consider the following auxiliary construction: letting $j:\mathbb{G}_{a,S}\simeq\mathrm{Spec}(\mathcal{O}_{S}[t])\hookrightarrow\mathbb{P}_{S}^{1}=\mathrm{Proj}(\mathcal{O}_{S}[w_{0},w_{1}])$, $t\mapsto[t:1]$ be the natural open immersion, the properness of the projection $\mathrm{pr}_{E\times_{S}E}:\mathbb{P}_{S}^{1}\times_{S}E\times_{S}E\rightarrow E\times_{S}E$ implies that $(\mathrm{p}_{2},\sigma)$ is proper if and only if $\varphi=(j\circ\mathrm{pr}_{1},\mathrm{pr}_{2},\sigma):\mathbb{G}_{a,S}\times_{S}E\rightarrow\mathbb{P}_{S}^{1}\times_{S}E\times_{S}E$ is proper, hence a closed immersion. Therefore the non properness of $\sigma$ is equivalent to the fact that the closure of $\mathrm{Im}(\varphi)$ in $\mathbb{P}_{S}^{1}\times_{S}E\times_{S}E$ intersects the “boundary” $\{w_{1}=0\}$ in a nontrivial way.
Now let $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ be the $\mathbb{G}_{a,S}$-action generated by a non twin-triangular $\sharp$-reduced $A$-derivation $\partial=x^{n}\partial_{y}+q(y)\partial_{z}+p(y,z)\partial_{u}$ of $A[y,z,u]$ and let $$\varphi=(j\circ\mathrm{pr}_{1},\mathrm{pr}_{2},\mu):\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}=\mathrm{Spec}(A[t][y,z,u])\rightarrow\mathbb{P}_{S}^{1}\times_{S}\mathbb{A}_{S}^{3}\times_{S}\mathbb{A}_{S}^{3}$$ be the corresponding immersion. To disprove the properness of $\sigma$, it is enough to check that the image by $\varphi$ of the closed sub-scheme $H=\left\{ z=0\right\} \simeq\mathrm{Spec}(A[t][y,u])$ of $\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}$ is not closed in $\mathbb{P}_{S}^{1}\times_{S}\mathbb{A}_{S}^{3}\times_{S}\mathbb{A}_{S}^{3}$. After identifying $A[y,z,u]\otimes_{A}A[y,z,u]$ with the polynomial ring $A[y_{1},y_{2},z_{1},z_{2},u_{1},u_{2}]$ in the obvious way, the image of $H$ by $(\mathrm{pr}_{1},\mathrm{pr}_{2},\sigma):\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{1}\times_{S}\mathbb{A}_{S}^{3}\times_{S}\mathbb{A}_{S}^{3}$ is equal to the closed sub-scheme of $\mathrm{Spec}(A[t][y_{1},y_{2},z_{1},z_{2},u_{1},u_{2}])$ defined by the following system of equations $$\begin{cases}
y_{2} & =y_{1}+x^{n}t\\
z_{1} & =0\\
z_{2} & =x^{-n}(Q(y_{1}+x^{n}t)-Q(y_{1}))=(y_{1}-y_{2})^{-1}(Q(y_{2})-Q(y_{1}))t\\
u_{2} & =u_{1}+x^{-n}\int_{0}^{t}p(y_{1}+x^{n}\tau)(Q(y_{1}+x^{n}\tau)-Q(y_{1})))d\tau.
\end{cases}$$ Letting $p(y,z)=\sum_{r=0}^{\ell}p_{r}(y)z^{r}$ with $\ell\geq1$ and $$\Gamma_{r}(y_{1},y_{2})=\int_{y_{1}}^{y_{2}}p_{r}(\xi)(Q(\xi)-Q(y_{1}))^{r}d\xi\in A[y_{1},y_{2}],\quad r=0,\ldots,\ell,$$ the last equality can be re-written modulo the first ones in the form $$\begin{aligned}
u_{2} & = & u_{1}+\sum_{r=0}^{\ell}x^{-nr}\int_{0}^{t}p_{r}(y_{1}+x^{n}\tau)(Q(y_{1}+x^{n}\tau)-Q(y_{1}))^{r}d\tau\\
& = & u_{1}+t(y_{2}-y_{1})^{-1}\sum_{r=0}^{\ell}x^{-nr}\int_{y_{1}}^{y_{2}}p_{r}(\xi)(Q(\xi)-Q(y_{1}))^{r}d\xi\\
& = & u_{1}+\sum_{r=0}^{\ell}\left((y_{2}-y_{1})^{-r-1}\Gamma_{r}(y_{1},y_{2})\right)t^{r+1}.\end{aligned}$$ It follows that the closure $V$ of $\varphi(H)$ is contained in the closed sub-scheme $W$ of $\mathbb{P}_{S}^{1}\times_{S}\mathbb{A}_{S}^{3}\times_{S}\mathbb{A}_{S}^{3}$ defined by the equations $z_{1}=0$ and $$\begin{cases}
(y_{2}-y_{1})w_{1}-x^{n}w_{0} & =0\\
w_{1}z_{2}-(y_{2}-y_{1})^{-1}(Q(y_{2})-Q(y_{1}))w_{0} & =0\\
w_{1}^{\ell+1}(u_{2}-u_{1})-\sum_{r=0}^{\ell}\left((y_{2}-y_{1})^{-r-1}\Gamma_{r}(y_{1},y_{2})\right)w_{0}^{r+1}w_{1}^{\ell-r} & =0.
\end{cases}$$ We further observe that $W$ is irreducible, whence equal to $V$, provided that $\Gamma_{\ell}(y_{1},y_{2})\in A[y_{1},y_{2}]$ does not belong to the ideal generated by $x^{n}$ and $Q(y_{2})-Q(y_{1})$. If so, then $W=V$ intersects $\{w_{1}=0\}$ along a closed sub-scheme $Z$ isomorphic to the spectrum of the following algebra:
$$\left(A[y_{1},y_{2}]/(x^{n},(y_{2}-y_{1})^{-1}(Q(y_{2})-Q(y_{1})),(y_{2}-y_{1})^{-\ell-1}\Gamma_{\ell}(y_{1},y_{2}))\right)[z_{2},u_{1},u_{2}].$$ Since by virtue of the $\sharp$-reducedness assumption $p_{\ell}(y)$ is not of the form $q(y)f(Q(y))+x^{n}g(y)$, the non properness of $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ is then a consequence of the following Lemma which guarantees precisely that $\Gamma_{\ell}(y_{1},y_{2})\not\in(x^{n},Q(y_{2})-Q(y_{1}))A[y_{1},y_{2}]$ and that $Z$ is not empty.
Let $q(y)\in A[y]$ be a polynomial with non constant residue class in $\kappa[y]$ and let $Q(y)=\int_{0}^{y}q(\tau)d\tau$. For a polynomial $p(y)\in A[y]$ and an integer $\ell\geq1$, the following holds:
a\) The polynomial $\Gamma_{\ell}(y_{1},y_{2})=\int_{y_{1}}^{y_{2}}p(y)(Q(y)-Q(y_{1}))^{\ell}dy$ belongs to the ideal $(x^{n},Q(y_{2})-Q(y_{1}))$ if and only if $p(y)$ can be written in the form $q(y)f(Q(y))+x^{n}g(y)$ for certain polynomials $f(\tau),g(\tau)\in A[\tau]$.
b\) The polynomial $(y_{2}-y_{1})^{-\ell-1}\Gamma_{\ell}(y_{1},y_{2})$ is not invertible modulo the ideal $(x^{n},(y_{2}-y_{1})^{-1}(Q(y_{2})-Q(y_{1})))$.
For the first assertion, a sequence of $\ell$ successive integrations by parts shows that $$\begin{aligned}
\Gamma_{\ell}(y_{1},y_{2}) & = & \left[E_{1}(y)(Q(y)-Q(y_{1}))^{\ell}\right]_{y_{1}}^{y_{2}}-\ell\int_{y_{1}}^{y_{2}}E_{1}(y)q(y)(Q(y)-Q(y_{1}))^{\ell-1}dy\\
& = & S(y_{1},y_{2})+(-1)^{\ell}\ell!\int_{y_{1}}^{y_{2}}E_{\ell}(y)q(y)dy\\
& = & S(y_{1},y_{2})+(-1)^{\ell}\ell!(E_{\ell+1}(y_{2})-E_{\ell+1}(y_{1}))\end{aligned}$$ where $E_{k}$ is defined recursively by $E_{1}(y)=\int_{0}^{y}p(\tau)d\tau$ and $E_{k+1}(y)=\int_{0}^{y}E_{k}(\tau)q(\tau)d\tau$, and where $S(y_{1},y_{2})\in(Q(y_{2})-Q(y_{1}))A[y_{1},y_{2}]$. So $\int_{y_{1}}^{y_{2}}p(y)(Q(y)-Q(y_{1}))^{r}dy$ belongs to $(x^{n},Q(y_{2})-Q(y_{1}))A[y_{1},y_{2}]$ if and only if $E_{\ell+1}(y_{2})-E_{\ell+1}(y_{1})$ belongs to this ideal.
Since the residue class of $Q(y)\in A[y]$ in $\kappa[y]$ is not constant, it follows from the local criterion for flatness that $A[y]$ is a faithfully flat algebra over $A[Q(y)]$. By faithfully flat descent, this implies in turn that the sequence $$A[Q(y)]\hookrightarrow A[y]\stackrel{\cdot\otimes1-1\otimes\cdot}{\longrightarrow}A[y]\otimes_{A[\tau]}A[y]$$ is exact whence, using the natural identification $A[y]\otimes_{A[\tau]}A[y]\simeq A[y_{1},y_{2}]/(Q(y_{2})-Q(y_{1}))$, that a polynomial $F\in A[y]$ with $F(y_{2})-F(y_{1})$ belonging to the ideal $(Q(y_{2})-Q(y_{1}))A[y_{1},y_{2}]$ has the form $F(y)=G(Q(y))$ for a certain polynomial $G(\tau)\in A[\tau]$. Thus $E_{\ell+1}(y_{2})-E_{\ell+1}(y_{1})$ belongs to $(x^{n},Q(y_{2})-Q(y_{1}))A[y_{1},y_{2}]$, if and only if $E_{\ell+1}(y)$ is of the form $G(Q(y))+x^{n}R_{\ell+1}(y)$ for some $G(\tau),R_{\ell+1}(\tau)\in A[\tau]$. This implies in turn that $E_{\ell}(y)q(y)=G'(Q(y))q(y)+x^{n}R_{\ell+1}'(y)$ whence, since $q(y)\in A[y]\setminus\mathfrak{m}A[y]$ is not a zero divisor modulo $x^{n}$, that $E_{\ell}(y)=G'(Q(y))+x^{n}R_{\ell}(y)$ for a certain $R_{\ell}(\tau)\in A[\tau]$. We conclude by induction that $E_{1}(y)=G^{(\ell+1)}(Q(y))+x^{n}R_{1}(y)$ and finally that $p(y)=G^{(\ell+2)}(Q(y))q(y)+x^{n}R(y)$ for a certain $R(\tau)\in A[\tau]$. This proves a).
The second assertion is clear in the case where $p(y)\in\mathfrak{m}A[y]$. Otherwise, if $p(y)\in A[y]\setminus\mathfrak{m}A[y]$ then reducing modulo $x$ and passing to the algebraic closure $\overline{\kappa}$ of $\kappa$, it is enough to show that if $q(y)\in\overline{\kappa}[y]$ is not constant and $p(y)\in\overline{\kappa}[y]$ is a nonzero polynomial then for every $\ell\geq1$, the affine curves $C$ and $D$ in $\mathbb{A}_{\overline{\kappa}}^{2}=\mathrm{Spec}(\overline{\kappa}[y_{1},y_{2}])$ defined by the vanishing of the polynomials $\Theta(y_{1},y_{2})=(y_{2}-y_{1})^{-\ell-1}\int_{y_{1}}^{y_{2}}p(y)(Q(y)-Q(y_{1}))^{\ell}dy$ and $R(y_{1},y_{2})=(y_{2}-y_{1})^{-1}\int_{y_{1}}^{y_{2}}q(y)dy$ respectively always intersect each other. Suppose on the contrary that $C\cap D=\emptyset$ and let $m=\deg q\geq1$ and $d=\deg p\geq0$. Then the closures $\overline{C}$ and $\overline{D}$ of $C$ and $D$ respectively in $\mathbb{P}_{\overline{\kappa}}^{2}=\mathrm{Proj}(\overline{\kappa}[y_{1},y_{2},y_{3}])$ intersect each others along a closed sub-scheme $Y$ of length $\deg\overline{C}\cdot\deg\overline{D}=m(d+\ell m)$ supported on the line $\{y_{3}=0\}\simeq\mathrm{Proj}(\overline{\kappa}[y_{1},y_{2}])$. By definition, up to multiplication by a nonzero scalar, the top homogeneous components of $R$ and $\Theta$ have the form $\prod_{i=1}^{m}(y_{2}-\zeta^{i}y_{1})$, where $\zeta\in\overline{\kappa}$ is a primitive $(m+1)$-th root of unity, and $(y_{2}-y_{1})^{\ell-1}\int_{y_{1}}^{y_{2}}y^{d}(y^{m+1}-y_{1}^{m+1})^{\ell}dy$ respectively. But on the other hand, we have for every $i=1,\ldots,m$
$$\overline{\kappa}[y_{2}]/(y_{2}-\zeta^{i},(y_{2}-1)^{-r-1}\int_{1}^{y_{2}}y^{d}(y^{m+1}-1)^{r}dy)\simeq\overline{\kappa}[y_{2}]/(y_{2}-\zeta^{i},(\zeta^{i}-1)^{-r-1}\int_{1}^{\zeta^{i}}\tau^{d}(\tau^{m+1}-1)^{r}d\tau),$$ and hence the length of the above algebra is either $1$ or $0$ depending on whether $\int_{1}^{\zeta^{i}}\tau^{d}(\tau^{m+1}-1)d\tau\in\overline{\kappa}$ is zero or not. This implies that the length of $Y$ is at most equal to $m$ and so the only possibility would be that $d=0$ and $\ell=m=1$, i.e. $C$ and $D$ are parallel lines in $\mathbb{A}_{\overline{\kappa}}^{2}$. But since $\int_{1}^{-1}(\tau^{2}-1)d\tau\neq0$, this last possibility is also excluded.
\[sec:Twin-Triviality\] Global equivariant triviality of twin-triangular actions
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By virtue of Proposition \[prop:Main-Local\], every proper triangular $\mathbb{G}_{a,S}$-action on $\mbox{\ensuremath{\sigma}:}\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ on $\mathbb{A}_{S}^{3}$ is conjugate to one generated by a twin-triangular $A$-derivation $\partial$ of $A[y,z_{+},z_{-}]$ of the form $$\partial=x^{n}\partial_{y}+p_{+}(y)\partial_{z_{+}}+p_{-}(y)\partial_{z_{-}}$$ for certain polynomials $p_{\pm}(y)\in A[y]$. So to complete the proof of the Main Theorem, it remains to show the following generalization of the main result in [@DubFin11]:
\[prop:Twin-Loc-trivi\] Let $S$ be the spectrum of discrete valuation $A$ containing a field of characteristic $0$. Then a proper twin-triangular $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{3}$ has affine geometric quotient $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$.
The principle of the proof given below is the following: we exploit the twin triangularity to construct two $\mathbb{G}_{a,S}$-invariant principal open subsets $W_{\Gamma_{+}}$ and $W_{\Gamma_{-}}$ in $\mathbb{A}_{S}^{3}$ with the property that the union of corresponding principal open subspaces $\mathfrak{X}_{\Gamma_{\pm}}=W_{\Gamma_{\pm}}/\mathbb{G}_{a,S}$ of $\mathfrak{X}$ covers the closed fiber of the structure morphism $\mathrm{p}:\mathfrak{X}\rightarrow S$. We then show that $\mathfrak{X}_{\Gamma_{+}}$ and $\mathfrak{X}_{\Gamma_{-}}$ are in fact affine sub-schemes of $\mathfrak{X}$. On the other hand, since $\partial$ admits $x^{-n}y$ as a global slice over $A_{x}$, the generic fiber of $\mathrm{p}$ is isomorphic to the affine plane over the function field $A_{x}$ of $S$. So it follows that $\mathfrak{X}$ is covered by three principal affine open sub-schemes $\mathfrak{X}_{\Gamma_{+}}$, $\mathfrak{X}_{\Gamma_{-}}$ and $\mathfrak{X}_{x}$ corresponding to regular functions $x$, $\Gamma_{+}$, $\Gamma_{-}$ which generate the unit ideal in $\Gamma(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})\simeq A[y,z_{+},z_{-}]^{\mathbb{G}_{a,S}}\subset A[y,z_{+},z_{-}]$, whence is an affine scheme.
The fact that the affineness of $\mathrm{p}:\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}\rightarrow S=\mathrm{Spec}(A)$ is a local property with respect to the fpqc topology on $S$ [@SGA1 VIII.5.6] enables a reduction to the case where the discrete valuation ring $A$ is Henselian or complete. Since it contains a field of characteristic zero, an elementary application of Hensel’s Lemma implies that a maximal subfield of such a local ring $A$ is a field of representatives, i.e. a subfield which is mapped isomorphically by the quotient projection $A\mapsto A/\mathfrak{m}$ onto the residue field $\kappa=A/\mathfrak{m}$. This is in fact the only property of $A$ that we will use in the sequel. So from now on, $(A,\mathfrak{m},\kappa)$ is a discrete valuation ring containing a field $\kappa$ of characteristic $0$ and with residue field $A/\mathfrak{m}\simeq\kappa$.
Twin-triangular actions in general position and associated invariant covering
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Here we construct a pair of principal $\mathbb{G}_{a,S}$-invariant open subsets $W_{\pm}=W_{\Gamma_{\pm}}$ of $\mathbb{A}_{S}^{3}$ associated with a twin-triangular $A$-derivation of $A[y,z_{+},z_{-}]$ whose geometric quotients will be studied in the next sub-section. We begin with a technical condition which will be used to guarantee that the union of $W_{+}$ and $W_{-}$ covers the closed fiber of the projection $\mathrm{pr}_{S}:\mathbb{A}_{S}^{3}\rightarrow S$.
\[def:general\_position\] Let $(A,\mathfrak{m},\kappa)$ be a discrete valuation valuation ring containing a field of characteristic $0$ and let $x\in\mathfrak{m}$ be a uniformizing parameter. A twin-triangular $A$-derivation $\partial=x^{n}\partial_{y}+p_{+}(y)\partial_{z_{+}}+p_{-}(y)\partial_{z_{-}}$ of $A[y,z_{+},z_{-}]$ is said to be in *general position* if it satisfies the following properties:
a\) The residue classes $\overline{p}_{\pm}\in\kappa[y]$ of the polynomials $p_{\pm}\in A[y]$ modulo $\mathfrak{m}$ are both non zero and relatively prime,
b\) There exist integrals $\overline{P}_{\pm}\in A[y]$ of $\overline{p}_{\pm}$ with respect to $y$ for which the inverse images of the branch loci of the morphisms $\overline{P}_{+}:\mathbb{A}_{\kappa}^{1}\rightarrow\mathbb{A}_{\kappa}^{1}$ and $\overline{P}_{-}:\mathbb{A}_{\kappa}^{1}\rightarrow\mathbb{A}_{\kappa}^{1}$ are disjoint.
\[lem:Bad-Plane-Removal\] With the notation above, every twin-triangular $A$-derivation $\partial$ of $A[y,z_{+},z_{-}]$ generating a fixed point free $\mathbb{G}_{a,S}$-action on $\mathbb{A}_{S}^{3}$ is conjugate to one in general position.
A twin-triangular derivation $\partial=x^{n}\partial_{y}+p_{+}(y)\partial_{z_{+}}+p_{-}(y)\partial_{z_{-}}$ generates a fixed point free $\mathbb{G}_{a,S}$-action if and only if $x^{n}$, $p_{+}(y)$ and $p_{-}(y)$ generate the unit ideal in $A[y,z_{+},z_{-}]$. So the residue classes $\overline{p}_{+}$ and $\overline{p}_{-}$ of $p_{+}$ and $p_{-}$ are relatively prime and at least one of them, say $\overline{p}_{-}$, is nonzero. If $\overline{p}_{+}=0$ then $p_{-}$ is necessarily of the form $p_{-}(y)=c+x\tilde{p}_{-}(y)$ for some $c\in A^{*}$ and so changing $z_{+}$ for $z_{+}+z_{-}$ yields a twin-triangular derivation conjugate to $\partial$ for which the corresponding polynomials $p_{\pm}(y)$ both have non zero residue classes modulo $x$. More generally, changing $z_{-}$ for $az_{-}+bz_{+}$ for general $a\in A^{*}$ and $b\in A$ yields a twin-triangular derivation conjugate to $\partial$ and still satisfying condition a) in Definition \[def:general\_position\]. So it remains to show that up to such a coordinate change, condition b) in the Definition can be achieved.
This can be seen as follows : we consider $\mathbb{A}_{\kappa}^{2}$ embedded in $\mathbb{P}_{\kappa}^{2}={\rm Proj}(\kappa[u,v,w])$ as the complement of the line $L_{\infty}=\left\{ w=0\right\} $ so that the coordinate system $\left(u,v\right)$ on $\mathbb{A}^{2}$ is induced by the projections from the $\kappa$-rational points $\left[0:1:0\right]$ and $\left[1:0:0\right]$ respectively. We let $C$ be the closure in $\mathbb{P}^{2}$ of the image of the morphism $j=(\overline{P}_{+},\overline{P}_{-}):\mathbb{A}_{\kappa}^{1}={\rm Spec}(\kappa[y])\rightarrow\mathbb{A}_{\kappa}^{2}$ defined by the residue classes $\overline{P}_{+}$ and $\overline{P}_{-}$ in $\kappa[y]$ of integrals $P_{\pm}(y)\in A[y]$ of $p_{\pm}(y)$, and we denote by $Z\subset C$ the image by $j$ of the inverse image of the branch locus of $\overline{P}_{+}:\mathbb{A}_{\kappa}^{1}\rightarrow\mathbb{A}_{\kappa}^{1}$. Note that $Z$ is a finite subset of $C$ defined over $\kappa$. Since the condition that a line through a fixed point in $\mathbb{P}_{\kappa}^{2}$ intersects transversally a fixed curve is Zariski open, the set of lines in $\mathbb{P}_{\kappa}^{2}$ passing through a point of $Z$ and tangent to a local analytic branch of $C$ at some point is finite. Therefore, the complement of the finitely many intersection points of these lines with $L_{\infty}$ is a Zariski open subset $U$ of $L_{\infty}$ with the property that for every $q\in U$, the line through $q$ and every arbitrary point of $Z$ intersects every local analytic branch of $C$ transversally at every point. By construction, the rational projections from $\left[0:1:0\right]$ and an arbitrary $\kappa$-rational point in $U\setminus\{\left[0:1:0\right]\}$ induce a new coordinate system on $\mathbb{A}_{\kappa}^{2}$ of the form $\left(u,av+bu\right)$, $a\neq0$, with the property that $Z$ is not contained in the inverse image of the branch locus of the morphism $a\overline{P}_{-}+b\overline{P}_{+}:\mathbb{A}_{\kappa}^{1}\rightarrow\mathbb{A}_{\kappa}^{1}$. Changing $z_{-}$ for $az_{-}+bz_{+}$ for a pair $(a,b)\in\kappa^{*}\times\kappa\subset A^{*}\times A$ corresponding to a general point in $U$ yields a twin-triangular derivation conjugate to $\partial$ and satisfying simultaneously conditions a) and b) in Definition \[def:general\_position\].
\[par:finite\_etale\_restriction\] Now let $\partial=x^{n}\partial_{y}+p_{+}(y)\partial_{z_{+}}+p_{-}(y)\partial_{z_{-}}$ be a twin-triangular $A$-derivation of $A[y,z_{+},z_{-}]$ generating a proper whence fixed point free $\mathbb{G}_{a,S}$-action $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$. By virtue of Lemma \[lem:Bad-Plane-Removal\] above, we may assume up to a coordinate change preserving twin-triangularity that $\partial$ is in general position. Property a) in Definition \[def:general\_position\] then guarantees in particular that the triangular derivations $\partial_{\pm}=x^{n}\partial_{y}+p_{\pm}(y)\partial_{z_{\pm}}$ of $A[y,z_{\pm}]$ are both irreducible. Furthermore, given any integral $P_{\pm}(y)\in A[y]$ of $p_{\pm}(y)$, the morphism $\overline{P}_{\pm}:\mathbb{A}_{\kappa}^{1}\rightarrow\mathbb{A}_{\kappa}^{1}$ obtained by restricting $P_{\pm}:\mathbb{A}_{S}^{1}={\rm Spec}(A[y])\rightarrow\mathbb{A}_{S}^{1}={\rm Spec}(A[t])$ to the closed fiber of $\mathrm{pr}_{S}:\mathbb{A}_{S}^{3}\rightarrow S$ is not constant. The branch locus of $\overline{P}_{\pm}$ is then a principal divisor $\mathrm{div}(\alpha_{\pm}(t))$ for a certain polynomial $\alpha_{\pm}(t)\in\kappa[t]\subset A[t]$ generating the kernel of the homomorphism $\kappa[t]\rightarrow\kappa[y]/(\overline{p}_{\pm}(y))$, $t\mapsto\overline{P}_{\pm}(y)+(\overline{p}_{\pm}(y))$. Property b) in Definition \[def:general\_position\] guarantees that we can choose $P_{+}$ and $P_{-}$ in such a way that the polynomial $\alpha_{+}(\overline{P}_{+}(y))$ and $\alpha_{-}(\overline{P}_{-}(y))$ generate the unit ideal in $\kappa[y]$. Up to a diagonal change of coordinates on $A[y,z_{+},z_{-}]$, we may further assume without loss of generality that $\overline{P}_{+}$ and $\overline{P}_{-}$ are monic.
\[par:open\_cover\_def\] We let $R_{\pm}=A[t]_{\alpha_{\pm}}$ and we let $U_{\pm}=\mathrm{Spec}(R_{\pm})$ be the principal open subset of $\mathbb{A}_{S}^{1}=\mathrm{Spec}(A[t])$ where $\alpha_{\pm}$ does not vanish. The polynomial $\Phi_{\pm}=-x^{n}z_{\pm}+P_{\pm}(y)\in A[y,z_{+},z_{-}]$ belongs to the kernel of $\partial$ hence defines a $\mathbb{G}_{a,S}$-invariant morphism $\Phi_{\pm}:\mathbb{A}_{S}^{3}=\mathrm{Spec}(A[y,z_{+},z_{-}])\rightarrow\mathbb{A}_{S}^{1}=\mathrm{Spec}(A[t])$. We let $$\begin{aligned}
W_{\pm} & = & \Phi_{\pm}^{-1}(U_{\pm})\simeq\mathrm{Spec}(R_{\pm}[y,z_{+},z_{-}]/(-x^{n}z_{\pm}+P_{\pm}(y)-t))\end{aligned}$$ Note that $W_{\pm}$ is a $\mathbb{G}_{a,S}$-invariant open subset of $\mathbb{A}_{S}^{3}$ which can be identified with the principal open subset where the $\mathbb{G}_{a,S}$-invariant regular function $\Gamma_{\pm}=\alpha_{\pm}\circ\Phi_{\pm}$ does not vanish. Since $\alpha_{+}(\overline{P}_{+}(y))$ and $\alpha_{-}(\overline{P}_{-}(y))$ generate the unit ideal in $\kappa[y]$, it follows that the union of $W_{+}$ and $W_{-}$ covers the closed fiber of the projection $\mathrm{pr}_{S}:\mathbb{A}_{S}^{3}\rightarrow S$.
Affineness of geometric quotients
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With the notation of §\[par:open\_cover\_def\] above, the geometric quotient $\mathfrak{X}_{\pm}=W_{\pm}/\mathbb{G}_{a,S}$ for the action induced by $\sigma:\mathbb{G}_{a,S}\times_{S}\mathbb{A}_{S}^{3}\rightarrow\mathbb{A}_{S}^{3}$ can be identified with the principal open sub-space $\mathfrak{X}_{\Gamma_{\pm}}$ of $\mathfrak{X}=\mathbb{A}_{S}^{3}/\mathbb{G}_{a,S}$ where the invariant function $\Gamma_{\pm}\in A[y,z_{+},z_{-}]^{\mathbb{G}_{a,S}}\simeq\Gamma(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})$ does not vanish. The properness of $\sigma$ implies that $\mathfrak{X}$, whence $\mathfrak{X}_{+}$ and $\mathfrak{X}_{-}$, are separated algebraic spaces, and the construction of $W_{+}$ and $W_{-}$ guarantees that the closed fiber of the structure morphism $\mathrm{p}:\mathfrak{X}\rightarrow S$ is contained in the union of $\mathfrak{X}_{+}$ and $\mathfrak{X}_{-}$. So to complete the proof of Proposition \[prop:Twin-Loc-trivi\], it remains to show that $\mathfrak{X}_{\pm}$ is an affine scheme. In fact, since $\mathfrak{X}_{\pm}$ is by construction an algebraic space over the affine scheme $U_{\pm}=\mathrm{Spec}(R_{\pm})$, its affineness is equivalent to that of the structure morphism $q_{\pm}:\mathfrak{X}_{\pm}\rightarrow U_{\pm}$, a property which can be checked locally with respect to the étale topology on $U_{\pm}$.
In our situation, there is a natural finite étale base change $\varphi_{\pm}:\tilde{U}_{\pm}\rightarrow U_{\pm}$ which is obtained as follows: By construction, see §\[par:finite\_etale\_restriction\] above, the morphism $\overline{P}_{\pm}:\mathbb{A}_{\kappa}^{1}=\mathrm{Spec}(\kappa[y])\rightarrow\mathrm{Spec}(\kappa[t])$, restricts to a finite étale covering $h_{0,\pm}:C_{1,\pm}={\rm Spec}(\kappa[y]_{\alpha_{\pm}(\overline{P}_{\pm}(y))})\rightarrow C_{\pm}={\rm Spec}(\kappa[t]_{\alpha_{\pm}(t)})$ of degree $r_{\pm}=\deg_{y}(\overline{P}_{\pm}(y))$. Letting $\tilde{C}_{\pm}=\mathrm{Spec}(B_{\pm})$ be the normalization of $C_{\pm}$ in the Galois closure $L_{\pm}$ of the field extension $i_{\pm}:\kappa(t)\hookrightarrow\kappa(y)$, the induced morphism $h_{\pm}:\tilde{C}_{\pm}\rightarrow C_{\pm}$ is an étale Galois cover with Galois group $G_{\pm}=\mathrm{Gal}(L_{\pm}/\kappa(t))$, which factors as $$h_{\pm}:\tilde{C}_{\pm}=\mathrm{Spec}(B_{\pm})\stackrel{h_{1,\pm}}{\longrightarrow}C_{1,\pm}={\rm Spec}(\kappa[y]_{\alpha_{\pm}(\overline{P}_{\pm}(y))})\stackrel{h_{0,\pm}}{\longrightarrow}C_{\pm}={\rm Spec}(\kappa[t]_{\alpha_{\pm}(t)})$$ where $h_{1,\pm}:\tilde{C}_{\pm}\rightarrow C_{1,\pm}$ is an étale Galois cover for a certain subgroup $H_{\pm}$ of $G_{\pm}$ of index $r_{\pm}$. Letting $\tilde{R}_{\pm}=A\otimes_{\kappa}B_{\pm}\simeq A[t]_{\alpha_{\pm}(t)}\otimes_{\kappa[t]_{\alpha_{\pm}(t)}}B_{\pm}$ and $\tilde{U}_{\pm}=\mathrm{Spec}(\tilde{R}_{\pm})$, the morphism $\varphi_{\pm}=\mathrm{pr}_{1}:\tilde{U}_{\pm}\simeq U_{\pm}\times_{C_{\pm}}\tilde{C}_{\pm}\rightarrow U_{\pm}$ is an étale Galois cover with Galois group $G_{\pm}$, in particular a finite morphism. Since $\mathfrak{X}_{\pm}$ is separated, the algebraic space $\tilde{\mathfrak{X}}_{\pm}=\mathfrak{X}_{\pm}\times_{U_{\pm}}\tilde{U}_{\pm}$ is separated and, by construction, isomorphic to the geometric quotient of the scheme $$\begin{aligned}
\tilde{W}_{\pm}=W_{\pm}\times_{U_{\pm}}\tilde{U}_{\pm} & \simeq & \mathrm{Spec}(\tilde{R}_{\pm}[y,z_{+},z_{-}]/(-x^{n}z_{\pm}+P_{\pm}(y)-t))\end{aligned}$$ by the proper $\mathbb{G}_{a,\tilde{U}_{\pm}}$-action generated by the locally nilpotent $\tilde{R}_{\pm}$-derivation $x^{n}\partial_{y}+p_{+}(y)\partial_{z_{+}}+p_{-}(y)\partial_{z_{-}}$ of $\tilde{R}_{\pm}[y,z_{+},z_{-}]//(-x^{n}z_{\pm}+P_{\pm}(y)-t)$, which commutes with the action of $G_{\pm}$. The following Lemma completes the proof of Proposition \[prop:Twin-Loc-trivi\] whence of the Main Theorem.
The geometric quotient $\tilde{\mathfrak{X}}_{\pm}=\tilde{W}_{\pm}/\mathbb{G}_{a,\tilde{U}_{\pm}}$ is an affine $\tilde{U}_{\pm}$-scheme.
Since $\tilde{U}_{\pm}$ is affine, the assertion is equivalent to the affineness of $\tilde{\mathfrak{X}}_{\pm}$. From now on, we only consider the case of $\tilde{\mathfrak{X}}_{+}=\tilde{W}_{+}/\mathbb{G}_{a,\tilde{U}_{+}}$, the case of $\tilde{\mathfrak{X}}_{-}$ being similar. To simplify the notation, we drop the corresponding subscript “$+$”, writing simply $\tilde{W}=\mathrm{Spec}(\tilde{R}[y,z,z_{-}]/(-x^{n}z+P(y)-t))$. We denote $x\otimes1\in\tilde{R}=A\otimes_{\kappa}B$ by $x$ and we further identify $B$ with a sub-$\kappa$-algebra of $\tilde{R}$ via the homomorphism $1\otimes\mathrm{id}_{B}:B\rightarrow\tilde{R}$ and with the quotient $\tilde{R}/x\tilde{R}$ via the composition $1\otimes\mathrm{id}_{B}:B\rightarrow A\otimes_{\kappa}B\rightarrow A\otimes_{\kappa}B/((x\otimes1)A\otimes_{\kappa}B)=\kappa\otimes_{\kappa}B\simeq B$.
By construction of $B$, the monic polynomial $\overline{P}(y)-t\in B\left[y\right]$ splits as $\overline{P}(y)-t=\prod_{\overline{g}\in G/H}(y-t_{\overline{g}})$ for certain elements $t_{\overline{g}}\in B$, $\overline{g}\in G/H$, on which the Galois group $G$ acts by permutation $g'\cdot t_{\overline{g}}=t_{\overline{(g')^{-1}\cdot g}}$. Furthermore, since $h_{0}:C_{1}\rightarrow C$ is étale, it follows that for distinct $\overline{g},\overline{g}'\in G/H$, $t_{\overline{g}}-t_{\overline{g'}}\in B$ is an invertible regular function on $\tilde{C}$ whence on $\tilde{U}=S\times_{\mathrm{Spec}(\kappa)}\tilde{C}$ via the identifications made above. This implies in turn that there exists a collection of elements $\sigma_{\overline{g}}\in\tilde{R}$ with respective residue classes $t_{\overline{g}}\in B=\tilde{R}/x\tilde{R}$ modulo $x$, $\overline{g}\in G/H$, on which $G$ acts by permutation, a $G$-invariant polynomial $S_{1}\in\tilde{R}\left[y\right]$ with invertible residue class modulo $x$ and a $G$-invariant polynomial $S_{2}\in\tilde{R}\left[y\right]$ such that in $\tilde{R}\left[y\right]$ one can write $$P(y)-t=S_{1}(y)\prod_{\overline{g}\in G/H}(y-\sigma_{\overline{g}})+x^{n}S_{2}(y).$$ Concretely, the elements $\sigma_{\overline{g}}=\sigma_{\overline{g},n-1}\in\tilde{R}$, $\overline{g}\in G/H$, can be constructed by induction via a sequence of elements $\sigma_{\overline{g},m}\in\tilde{R}$, $\overline{g}\in G/H$, $m=0,\ldots,n-1$, starting with $\sigma_{\overline{g},0}=t_{\overline{g}}\in B\subset\tilde{R}$ and culminating in $\sigma_{\overline{g},n-1}=\sigma_{\overline{g}}$, and characterized by the property that for every $m=0,\ldots,n-1$, there exists $\mu_{\overline{g},m}\in\tilde{R}$ such that $P(\sigma_{\overline{g},m})-t=x^{m+1}\mu_{\overline{g},m}$, $\overline{g}\in G/H$. Indeed, writing $P(y)-t=\prod_{\overline{g}\in G/H}(y-t_{\overline{g}})+x\tilde{P}(y)$ for a certain $\tilde{P}(y)\in\tilde{R}[y]$ and assuming that the $\sigma_{\overline{g},m}$, $\overline{g}\in G/H$, have been constructed up to a certain index $m<n-1$, we look for elements $\sigma_{\overline{g},m+1}\in\tilde{R}$ written in the form $\sigma_{\overline{g},m}+x^{m+1}\lambda_{\overline{g}}$ for some $\lambda_{\overline{g}}\in\tilde{R}$. For a fixed $\overline{g}_{0}\in G/H$, the conditions impose that
$$\begin{aligned}
P(\sigma_{\overline{g}_{0},m+1})-t & = & \prod_{\overline{g}\in G/H}(\sigma_{\overline{g}_{0},m}+x^{m+1}\lambda_{\overline{g}_{0}}-t_{\overline{g}})+x\tilde{P}(\sigma_{\overline{g}_{0},m}+x^{m+1}\lambda_{\overline{g}_{0}})\\
& = & x^{m+1}\lambda_{\overline{g}_{0}}\prod_{\overline{g}\in(G/H)\setminus\{\overline{g}_{0}\}}(t_{\overline{g}_{0}}-t_{\overline{g}})+P(\sigma_{\overline{g}_{0},m})-t+x^{m+2}\nu_{\overline{g}_{0},m}\\
& = & x^{m+1}\lambda_{\overline{g}_{0}}\prod_{\overline{g}\in(G/H)\setminus\{\overline{g}_{0}\}}(t_{\overline{g}_{0}}-t_{\overline{g}})+x^{m+1}\mu_{\overline{g}_{0,}m}+x^{m+2}\nu_{\overline{g}_{0},m}\end{aligned}$$
for some $\nu_{\overline{g}_{0},m}\in\tilde{R}$, and since $\prod_{\overline{g}\in(G/H)\setminus\{\overline{g}_{0}\}}(t_{\overline{g}_{0}}-t_{\overline{g}})\in\tilde{R}^{*}$, we conclude that $$\lambda_{\overline{g}_{0}}=\frac{\mu_{\overline{g}_{0},m}}{\prod_{\overline{g}\in(G/H)\setminus\{\overline{g}_{0}\}}(t_{\overline{g}_{0}}-t_{\overline{g}})}\quad\textrm{and}\quad\mu_{\overline{g}_{0},m+1}=\nu_{\overline{g}_{0},m}.$$ A direct computation shows further that $g'\cdot\sigma_{\overline{g},m+1}=\sigma_{\overline{(g')^{-1}\cdot g},m+1}$ and that $g'\cdot\mu_{\overline{g},m+1}=\mu_{\overline{(g')^{-1}\cdot g},m+1}$. Iterating this procedure $n-1$ times yields the desired collection of elements $\sigma_{\overline{g}}=\sigma_{\overline{g},n-1}\in\tilde{R}$. By construction, $\prod_{\overline{g}\in G/H}(y-\sigma_{\overline{g}})\in\tilde{R}[y]$ is then an invariant polynomial which divides $P(y)-t$ modulo $x^{n}\tilde{R}$, which implies in turn the existence of the $G$-invariant polynomials $S_{1}(y),S_{2}(y)\in\tilde{R}[y]$.
The closed fiber of the induced morphism $\tilde{W}\rightarrow S$ consists of a disjoint union of closed sub-schemes $D_{\overline{g}}\simeq\mathrm{Spec}(\tilde{R}[z,z_{-}])\simeq\mathbb{A}_{\tilde{C}}^{2}$ with defining ideals $(x,y-\sigma_{\overline{g}})$, $\overline{g}\in G/H$. The open sub-scheme $\tilde{W}_{\overline{g}}=\tilde{W}\setminus\bigcup_{\overline{g}'\in(G/H)\setminus\{\overline{g}\}}D_{\overline{g}'}$ of $\tilde{W}$ is $\mathbb{G}_{a,\tilde{U}}$-invariant and one checks using the above expression for $P(y)-t$ that the rational map $$\tilde{W}\dashrightarrow\mathrm{Spec}(\tilde{R}[u_{\overline{g}},z_{-}]),\quad(y,z,z_{-})\mapsto(u_{\overline{g}},z_{-})=(\frac{y-\sigma_{\overline{g}}}{x^{n}}=\frac{z-S_{2}(y)}{S_{1}(y)\prod_{\overline{g}'\in(G/H)\setminus\{\overline{g}\}}(y-\sigma_{\overline{g}'})},z_{-})$$ induces a $\mathbb{G}_{a,\tilde{U}}$-equivariant isomorphism $\tau_{g}:\tilde{W}_{\overline{g}}\stackrel{\sim}{\rightarrow}\mathbb{A}_{\tilde{U}}^{2}=\mathrm{Spec}(\tilde{R}[u_{\overline{g}},z_{-}])$ for the $\mathbb{G}_{a,\tilde{U}}$-action on $\mathbb{A}_{\tilde{U}}^{2}$ generated by the locally nilpotent $\tilde{R}$-derivation $\partial_{u_{\overline{g}}}+p_{-}(x^{n}u_{\overline{g}}+\sigma_{\overline{g}})\partial_{z_{-}}$ of $\tilde{R}[u_{\overline{g}},z_{-}]$. The latter is a translation with $u_{\overline{g}}$ as a global slice and with geometric quotient $\tilde{W}_{\overline{g}}/\mathbb{G}_{a,\tilde{U}}$ isomorphic to $\mathrm{Spec}(\tilde{R}[v_{\overline{g}}])$ where $$v_{\overline{g}}=z_{-}-x^{-n}(P_{-}(x^{n}u_{\overline{g}}+\sigma_{\overline{g}})-P_{-}(\sigma_{\overline{g}}))\in\tilde{R}[u_{\overline{g}},z_{-}]^{\mathbb{G}_{a,\tilde{U}}}.$$ By construction, for distinct $\overline{g},\overline{g}'\in G/H$, the rational functions $\tau_{\overline{g}}^{*}v_{\overline{g}}$ and $\tau_{\overline{g}'}^{*}v_{\overline{g}'}$ on $\tilde{W}$ differ by the addition of the element $$f_{\overline{g},\overline{g}'}=x^{-n}(P_{-}(\sigma_{\overline{g}})-P_{-}(\sigma_{\overline{g}'}))\in\tilde{R}_{x}\in\Gamma(\tilde{W}_{\overline{g}}\cap\tilde{W}_{\overline{g}'},\mathcal{O}_{\tilde{W}}).$$ This implies that $\tilde{\mathfrak{X}}=\tilde{W}/\mathbb{G}_{a,\tilde{U}}$ is isomorphic to the $\tilde{U}$-scheme obtained by gluing $r$ copies $\tilde{\mathfrak{X}}_{g}=\mathrm{Spec}(\tilde{R}[v_{\overline{g}}])$ of $\mathbb{A}_{\tilde{U}}^{1}$ along the principal open subsets $\tilde{\mathfrak{X}}_{\overline{g},x}\simeq\mathrm{Spec}(\tilde{R}_{x}[v_{\overline{g}}])$ via the isomorphisms induced by the $\tilde{R}_{x}$-algebra isomorphisms $$\xi_{\overline{g},\overline{g}'}^{*}:\tilde{R}_{x}[v_{\overline{g}}]\rightarrow\tilde{R}_{x}[v_{\overline{g}'}],v_{\overline{g}}\mapsto v_{\overline{g}'}+f_{\overline{g},\overline{g}'},\quad\overline{g},\overline{g}'\in G/H,\;\overline{g}\neq\overline{g}'.$$ Since by assumption $\tilde{\mathfrak{X}}$ is separated, it follows from [@EGA1 I.5.5.6] that for every pair of distinct elements $\overline{g},\overline{g}'\in G/H$, the sub-ring $\tilde{R}[v_{\overline{g}'},f_{\overline{g},\overline{g}'}]$ of $\tilde{R}_{x}[v_{\overline{g}'}]$ generated by the union of $\tilde{R}[v_{\overline{g}'}]$ and $\xi_{\overline{g},\overline{g}'}^{*}(\tilde{R}[v_{\overline{g}}])$ is equal to $\tilde{R}_{x}[v_{\overline{g}'}]$. This holds if and only if $\tilde{R}[f_{\overline{g},\overline{g}'}]=\tilde{R}_{x}$ whence if and only if $f_{\overline{g},\overline{g}'}\in\tilde{R}_{x}$ has the form $f_{\overline{g},\overline{g}'}=x^{-m_{\overline{g},\overline{g}'}}F_{\overline{g},\overline{g}'}$ for a certain $m_{\overline{g},\overline{g}'}>1$ and an element $F_{\overline{g},\overline{g}'}\in\tilde{R}$ with invertible residue class modulo $x$.
This additional information enables a proof of the affineness of $\tilde{\mathfrak{X}}$ by induction on $r$ as follows: given a pair of distinct elements $\overline{g},\overline{g}'\in G/H$ such that $m_{\overline{g},\overline{g}'}=m>0$ is maximal, we let $\theta_{\overline{g}}=0$ and $\theta_{\overline{g}''}=x^{m-m_{\overline{g},\overline{g}''}}F_{\overline{g},\overline{g}''}\in\tilde{R}$ for every $\overline{g}''\in(G/H)\setminus\{\overline{g}\}$. The choice of the elements $\theta_{\overline{g}''}\in\tilde{R}$ guarantees that the local sections $$\psi_{\overline{g}''}=x^{m}v_{\overline{g}''}+\theta_{\overline{g}''}\in\Gamma(\tilde{\mathfrak{X}}_{\overline{g}''},\mathcal{O}_{\tilde{\mathfrak{X}}}),\quad\overline{g}''\in G/H$$ glue to a global regular function $\psi\in\Gamma(\tilde{\mathfrak{X}},\mathcal{O}_{\tilde{\mathfrak{X}}})$. Since $\theta_{\overline{g}'}=F_{\overline{g},\overline{g}'}$ is invertible modulo $x$, the regular functions $x$, $\psi$ and $\psi-\theta_{\overline{g}'}$ generate the unit ideal in $\Gamma(\tilde{\mathfrak{X}},\mathcal{O}_{\tilde{\mathfrak{X}}})$. The principal open subset $\tilde{\mathfrak{X}}_{x}$ of $\tilde{\mathfrak{X}}$ is isomorphic to $\tilde{\mathfrak{X}}_{\overline{g},x}\simeq\mathrm{Spec}(\tilde{R}_{x}[v_{\overline{g}}])$ for every $\overline{g}\in G/H$, hence is affine. On the other hand, $\tilde{\mathfrak{X}}_{\psi}$ and $\tilde{\mathfrak{X}}_{\psi-\theta_{\overline{g}'}}$ are contained respectively in the open sub-schemes $\tilde{\mathfrak{X}}(\overline{g})$ and $\tilde{\mathfrak{X}}(\overline{g}')$ obtained by gluing only the $r-1$ open subsets $\tilde{\mathfrak{X}}_{\overline{g}''}$ corresponding to the elements $\overline{g}''$ in $\left(G/H\right)\setminus\{\overline{g}\}$ and $\left(G/H\right)\setminus\{\overline{g}'\}$ respectively. By the induction hypothesis, the latter are both affine and hence $\tilde{\mathfrak{X}}_{\psi}$ and $\tilde{\mathfrak{X}}_{\psi-\theta_{\overline{g}'}}$ are affine as well. This shows that $\tilde{\mathfrak{X}}$ is an affine scheme and completes the proof.
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[^1]: Research supported in part by NSF Grant OISE-0936691 and ANR Grant “BirPol” ANR-11-JS01-004-01.
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---
abstract: 'We report strong electron-phonon coupling in magic-angle twisted bilayer graphene (MA-TBG) obtained from atomistic description of the system including more than 10 000 atoms in the moiré supercell. Electronic structure, phonon spectrum, and electron-phonon coupling strength $\lambda$ are obtained before and after atomic-position relaxation both in and out of plane. Obtained $\lambda$ is very large for MA-TBG, with $\lambda > 1$ near the half-filling energies of the flat bands, while it is small ($\lambda \sim 0.1$) for monolayer and unrotated bilayer graphene. Significant electron-hole asymmetry occurs in the electronic structure after atomic-structure relaxation, so $\lambda$ is much stronger with hole doping than electron doping. Obtained electron-phonon coupling is nearly isotropic and depends very weakly on electronic band and momentum, indicating that electron-phonon coupling prefers single-gap $s$-wave superconductivity. Relevant phonon energies are much larger than electron energy scale, going far beyond adiabatic limit. Our results provide a fundamental understanding of the electron-phonon interaction in MA-TBG, highlighting that it can contribute to rich physics of the system.'
author:
- Young Woo Choi and Hyoung Joon Choi
date: 'September 22, 2018'
title: |
Strong electron-phonon coupling, electron-hole asymmetry, and nonadiabaticity\
in magic-angle twisted bilayer graphene
---
Interplay between the interlayer coupling and the rotational mismatch between two graphene layers in bilayer graphene results in flattening of Dirac cones at certain special twist angles $\theta_M$, called magic angles [@Magaud:2010; @ref2; @ref3]. Recently, correlated insulator behavior and superconductivity are experimentally observed near the first magic angle $\theta_M=1.08^\circ$, demonstrating rich physics induced by the presence of the flat bands [@Cao:2018a; @Cao:2018b]. In this regard, more detailed characterizations for the magic-angle twisted bilayer graphene (MA-TBG) are attracting great interest [@p1; @p2; @p3; @p4; @p5; @p6; @p7; @p8; @p10; @p11; @a1; @a2; @a4; @a5; @a7; @a8; @a9; @a10; @a11; @a12; @a15; @a16; @a17; @a18; @a19; @a20; @a21; @a22; @a23; @a24; @a27; @a29; @a30; @a31; @a32; @a33; @a34; @a35; @a37; @a38; @a39; @a40; @a41; @a42; @a43; @a44].
In addition to the exotic electronic properties, it has been observed that low-angle bilayer graphene exhibits atomic-scale reconstruction [@Yoo:2018]. The essential effect of the lattice relaxation is such that the area of the AA stacking region becomes smaller, while the AB stacking region is larger, and this effect becomes more important as the twist angle gets smaller. Also, it is suggested that the lattice relaxation can affect the electronic structure, opening superlattice-induced energy gaps at the band edges on both electron and hole sides [@Nam:2017]. Since these gaps are clearly observed in the experiments [@Cao:2018a; @Cao:2016], it is necessary to consider the lattice relaxation when studying TBG in the low-angle regime.
As the electron-phonon coupling strength $\lambda$ in simple monolayer and unrotated bilayer graphene is too weak, superconductivity in MA-TBG is suspected to be originated from the electron correlation. [*Ab initio*]{} calculations found that $\lambda$ of monolayer and unrotated bilayer graphene is less than 0.1 near the charge-neutral Fermi level [@Park:2008]. If $\lambda$ has a similar value in MA-TBG, it cannot account for the observed superconducting transition temperature $T_c~\sim 1$ K.
However, since $\lambda$ is proportional to the electron density of states, $\lambda$ of AB-stacked bilayer graphene (AB-BLG), for example, can be as large as 0.28 when the Fermi level is tuned to near the van Hove singularity points. This suggests that $\lambda$ is likely to be further enhanced in low-angle twisted bilayer graphene where the flattening of Dirac cones brings large enhancements of the electron density of states. Thus, quantitative estimation of $\lambda$ in low-angle twisted bilayer graphene can provide a valuable insight into the nature of superconductivity.
In this work, we investigate the electron-phonon interaction in MA-TBG with atomistic description of the system including more than 10 000 atoms needed for the moiré supercell. We use a tight-binding approach for electrons and atomic force constants for phonons. We find that the electron-phonon coupling strength $\lambda$ in MA-TBG is almost directly proportional to the electron density of states and becomes greater than 1 near the half-filling energies of the flat bands. It is shown that the lattice relaxations can bring electron-hole asymmetry to the electron density of states and, as a result, the hole-side flat bands have much stronger $\lambda$ than the electron-side. We also find that the electron-phonon coupling depends very weakly on the direction and magnitude of the electronic crystal momentum. We discuss implications of our results for superconductivity in MA-TBG.
Although the electron-phonon interaction can be, in principle, obtained accurately by self-consistent density functional perturbation theory (DFPT), the large number ($\sim$$10^4$) of atoms in the moiré supercell is a practical barrier making DFPT calculations very difficult to achieve. In addition, considering correlation effects between electrons in atomistic description also requires challenging development due to the large number of atoms. In our present work, we employ a tight-binding approach with one $p$ orbital per carbon atom and atomic force constants for atomic vibrations without considering correlation effects between electrons. Our results provide a fundamental understanding of the electron-phonon interaction in the system obtained from atomistic description of electrons and phonons.
A moiré supercell of twisted bilayer graphene is constructed by rotating each layer of AA-stacked bilayer graphene by $\theta/2$ and $-\theta/2$, respectively. The resulting atomic structure has sixfold rotation symmetry axis around the $z$ axis, and three twofold rotation symmetry axes that are perpendicular to the $z$ axis, which swap two graphene layers as a result.
Preserving the crystal symmetry of nonrelaxed structure, we determine the equilibrium atomic positions by minimizing the total energy $U$ that is the sum of in-plane strain energy and interlayer binding energy,
$$\begin{aligned}
U =\; & \frac{1}{2} \sum_{l=1}^{2} \sum_{p\kappa\alpha,p'\kappa'\beta}
C^\text{MLG}_{p\kappa\alpha,p'\kappa'\beta} \;
\Delta\tau^{l}_{p\kappa\alpha} \Delta\tau^{l}_{p'\kappa'\beta}\\
& + \sum_{p\kappa,p'\kappa'}
V_\text{KC} ( \bm{\tau}^{1}_{p\kappa}-\bm{\tau}^{2}_{p'\kappa'})~.
\end{aligned}
\label{eq:etot}$$
Here $\tau^{l}_{p\kappa\alpha}$ is the $\alpha$ ($\alpha=x,y,z$) component of the position of the $\kappa$th atom in layer $l$ located at the $p$th moiré supercell of TBG, $\Delta \bm{\tau^{l}}_{p\kappa} =
\bm{\tau}^{l}_{p\kappa}-\bm{\tilde{\tau}}^{l}_{p\kappa}$ is the deviation from the nonrelaxed position $\bm{\tilde{\tau}}^{l}_{p\kappa}$, and $C^\text{MLG}_{p\kappa\alpha,p'\kappa'\beta}$ are force constants between two atoms in the same layer up to fourth-nearest neighbors, taken from Ref. [@Wirtz:2004], which are obtained by fitting to the [*ab initio*]{} phonon dispersion calculations of monolayer graphene. The interlayer binding energy is calculated using Kolmogorov-Crespi (KC) potential $V_\text{KC}$ that depends on interlayer atomic registry [@Kolmogorov:2005]. Without the interlayer binding energy, our total energy function has its minimum, by construction, at the atomic positions of the rigidly rotated two graphene layers. With the interlayer binding energy, the equilibrium atomic positions show that the area of AA-stacked regions is shrunk, while AB-stacked regions expanded, and interlayer distances in AA-stacked regions become larger than AB-stacked regions [@Yoo:2018; @Nam:2017; @Tadmor:2017; @Tadmor:2018].
Figure \[fig:relax\] shows the atomic displacements due to the relaxation at $\theta=1.08^\circ$. We find that maximal out-of-plane displacements are about two times maximal in-plane displacements. Out-of-plane displacements are largest at the AA-stacked region, and also noticeable at the AB/BA domain boundary. The existence of locally confined strains at AB/BA domain boundaries is one of the most important consequences of the lattice relaxations in low angle TBG. Our results are consistent with previous studies on the lattice relaxations in low-angle TBG.
To investigate electronic structures of TBG in both nonrelaxed and relaxed structure, we employ a single-orbital tight-binding approach where the electronic Hamiltonian is $$\label{h_tb}
\hat{H} = \sum_{p\kappa,p'\kappa'}
t(\bm{\tau}_{p\kappa}-\bm{\tau}_{p'\kappa'})
|\phi_\kappa;\bm{R}_p\rangle \langle\phi_{\kappa'};\bm{R}_{p'}|~,$$ where $|\phi_\kappa; \bm{R}_p\rangle$ is a carbon $p_z$-like orbital at $\bm{\tau}_{p\kappa}$. Here we drop the layer index on $\bm{\tau}_{p\kappa}$, $\kappa$ sweeps all atoms in both layers, and $\bm{\tau}_{p\kappa}=\bm{\tau}_{0\kappa}+\bm{R}_p$ for the $p$th moiré supercell at $\bm{R}_p$. We use the Slater-Koster-type hopping integral, $$\begin{aligned}
t(\bm{d}) &=& V_{pp\pi}^0 e^{-(d-a_0)/\delta}
\{ 1-(d_z/d)^2 \} \nonumber \\
&& + \; V_{pp\sigma}^0 e^{-(d-d_0)/\delta}(d_z/d)^2~,\end{aligned}$$
![ Magnitude of (a) in-plane and (b) out-of-plane displacements of the upper layer after the structural relaxation in the TBG at $\theta=1.08^\circ$. Red circular arrow denotes the directions of in-plane displacements. The other layer has a similar displacement pattern, except that the directions are opposite. The stacking pattern of two graphene layers varies within the moiré supercell of TBG. AA-, AB-, and BA-type stacking regions are denoted by AA, AB, and BA, respectively. []{data-label="fig:relax"}](figure1.eps){width="8.8cm"}
where $\bm{d}$ is the displacement vector between two orbitals. The hopping energy $V^0_{pp\pi}=-2.7\,\text{eV}$ is between in-plane nearest neighbors separated by $a_0=a/\sqrt{3}=1.42\,\text{\AA}$, and $V^0_{pp\sigma}=0.48\,\text{eV}$ is between two veritcally aligned atoms at the distance $d_0 = 3.35\,\text{\AA}$. Here $\delta = 0.184 a$ is chosen to set the magnitude of the next-nearest-neighbor hopping amplitude to be $0.1V^0_{pp\pi}$ [@Moon:2012; @Ando:2000]. We use the cutoff distance $d_c = 5\,\text{\AA}$, beyond which the hopping integral is negligible.
Figure \[fig:tb\](a) shows the band structures for TBLG at $\theta=1.08^\circ$ in the nonrelaxed and relaxed structure. One of the most noticeable effects of the lattice relaxation is the opening of the gaps at the edges of the flat bands. Furthermore, the electron-side and hole-side flat bands become significantly asymmetric due to the relaxation. The hole side gets much narrower than the electron side so that the peak height of the density of states \[Fig. \[fig:tb\](b)\] in the hole side is more than twice the electron side. The gap opening and the electron-hole asymmetry are consistent qualitatively with previous results considering in-plane relaxation only [@Nam:2017].
Figures \[fig:tb\](c) and (d) show Fermi surfaces at energies where the hole-side flat bands are half-filled for the nonrelaxed and relaxed structures, respectively. At these energies, Fermi surfaces become more complicated than those near the charge-neutral energy, where only circular Fermi sheets originating from the Dirac cones are located at Brillouin zone corners. At half-filling energies, Fermi sheets at the zone corners become similar to triangles, and the additional $\Gamma$-centered Fermi sheets appear.
![Electronic structure of MA-TBG. Tight-binding (a) band structure and (b) density of states per spin per moiré supercell for nonrelaxed (blue) and relaxed (red) structures at $\theta=1.08^\circ$. Vertical dashed lines show hole-side half-filling energies. (c),(d) Fermi surfaces at energies denoted by dashed lines in (b) for nonrelaxed and relaxed structures, respectively. []{data-label="fig:tb"}](figure2.eps){width="9.0cm"}
Phonons in twisted bilayer graphene are calculated using atomic force constants $C_{p\kappa\alpha,p'\kappa'\beta} =
\partial^2 U / \partial \tau_{p\kappa\alpha} \partial \tau_{p' \kappa'\beta}$, where $U$ is given by Eq. (\[eq:etot\]). Since we treat the in-plane strain energy with the harmonic approximation, the in-plane force constants are unaltered by the lattice relaxation. The interlayer force constants, however, are evaluated at relaxed atomic positions because the KC potential [@Kolmogorov:2005] is not harmonic. Our approach is similar to Ref. [@Cocemasov:2013], except that the Lennard-Jones interlayer potential between two graphene layers is replaced by the KC potential which can account for registry-dependent energy differences in TBG. From the force constants, we obtain the dynamical matrix $D_{\kappa\alpha,\kappa'\beta}(\bm{q})= \sum_{p}
e^{i\bm{q}\cdot\bm{R}_p} \; C_{0\kappa\alpha, p\kappa'\beta}/M_C $ for phonon wave vector $\bm{q}$, where $M_C$ is the mass of a carbon atom. Then, we solve the phonon eigenvalue problems $\omega^2_{\bm{q}\nu} \; e_{\bm{q}\nu,\kappa\alpha} =
\sum_{\kappa',\beta} D_{\kappa\alpha,\kappa'\beta}(\bm{q}) \;
e_{\bm{q}\nu,\kappa'\beta}$ at the irreducible Brillouin zone of TBG for the energy $\omega_{\bm{q}\nu}$ and polarization vector $\bm{e}_{\bm{q}\nu,\kappa}$ of the $\nu$th phonon mode. The phonons in the rest of the Brillouin zone are obtained from the symmetry relations [@Maradudin:1968]. We considered all phonon modes in the moiré supercell to obtain unbiased results for electron-phonon interaction.
Figure \[fig:ep\](a) shows phonon density of states $F(\omega)$ for $\theta=1.08^\circ,1.12^\circ,\mathrm{and}\;1.16^\circ$ as well as AB-BLG. Phonon spectra are nearly insensitive to small twist-angle differences. So a tiny difference is that, compared to AB-BLG, interlayer breathing modes near $\omega\sim11\;\mathrm{meV}$ are slightly softened in TBG. \[see Fig. S1(a) in the Supplemental Material [@SM] for phonon dispersions in AB-BLG\].
![ (a) Phonon density of states for AB-BLG (dashed black), TBG at $\theta=1.08^\circ$ (solid red), $\theta=1.12^\circ$ (solid green), and $\theta=1.16^\circ$ (solid blue). Phonons are insensitive to the small twist-angle differences between those angles. The inset shows the frequency range of the interlayer shear and breathing modes, which are softened by the twist. (b) Total electron-phonon coupling strength $\lambda$ in TBG as a function of the Fermi energy ($E_F$). The vertical red dashed line denotes the energy where hole-side flat bands in $\theta=1.08^\circ$ are half-filled. (c) Eliashberg function $\alpha^2F(\omega)$, shown in red, at the half-filling energy in the hole side. The dashed black line denotes $\lambda(\omega)=2\int^{\omega}_0 \alpha^2 F(\omega')/\omega' d\omega'$. The inset shows the low-frequency range of $\alpha^2F(\omega)$. Phonon modes at this range contribute to about 30% of the total coupling strength. (d) Distribution of band- and momentum-resolved coupling strength $\lambda_{n\bm{k}}$ of Eq. (\[eq:4a\]) at the hole-side half-filling energy.[]{data-label="fig:ep"}](figure3.eps){width="9.0cm"}
Now, we calculate the standard electron-phonon coupling strengths defined as
$$\begin{aligned}
\label{eq:4a}
\lambda_{n\bm{k}} &=& 2 N_F \sum_{m\bm{q}\nu}
\frac{|g_{mn\nu}(\bm{k},\bm{q})|^2}{\omega_{\bm{q}\nu}} W_{m\bm{k+q}}, \\
\lambda &=& \sum_{n\bm{k}} \lambda_{n\bm{k}} W_{n\bm{k}},
\label{eq:4b} \end{aligned}$$
where $N_F$ is the electron density of states per spin at the Fermi level $E_F$, and $W_{n\bm{k}}=\delta(E_F - \varepsilon_{n\bm{k}})/N_{F}$ is the partial weight of the density of states. Here, $\varepsilon_{n\bm{k}}$ is the electron energy of the $n$th band with wavevector $\bm{k}$, and $W_{n\bm{k}}$ is obtained by the linear tetrahedron method [@Bloechl:1994]. The electron-phonon matrix elements $g_{mn\nu}(\bm{k},\bm{q})=\langle m\bm{k+q} |\delta_{\bm{q}\nu} \hat{H}
|n\bm{k}\rangle$ couple the electronic states $|n\bm{k}\rangle$ and $|m\bm{k+q}\rangle$, where $\delta_{\bm{q}\nu} \hat{H}$ is the change in $\hat{H}$ due to phonon mode $(\bm{q}\nu)$. The electron-phonon matrix elements in localized orbital basis can be expressed in terms of the changes in the hopping matrix elements due to the atomic displacements of phonon modes [@Giustino:2007; @Gunst:2016; @Bernadi:2018], $$\begin{aligned}
g_{mn\nu}(\bm{k},\bm{q})
&= & l_{\bm{q}\nu}\sum_{\kappa \alpha} e_{\bm{q}\nu,\kappa \alpha}
\sum_{pp',ij}
e^{-i(\bm{k+q})\cdot\bm{R}_{p'}} e^{i\bm{k}\cdot\bm{R}_p} \nonumber \\
&& \times c^*_{m\bm{k+q},j} c_{n\bm{k},i}
\langle\phi_j; \bm{R}_{p'}|
\frac{\partial \hat{H}}{\partial \tau_{0 \kappa \alpha}}
|\phi_i; \bm{R}_p\rangle,\end{aligned}$$ where $l_{\bm{q}\nu}=\sqrt{\hbar/(2M_C\omega_{\bm{q}\nu})}$ is the length scale of phonon mode $(\bm{q}\nu)$, and $c_{n\bm{k},i}$ is the coefficient of the electron wavefunctions in local orbital basis, i.e., $c_{n\bm{k},i}e^{i\bm{k}\cdot\bm{R}_p}=
\sqrt{N}\langle\phi_i;\bm{R}_p|n\bm{k}\rangle$. Here, $N$ is the total number of unit cells over which the electronic states are normalized. Thus, in our tight-binding approach, we obtain the electron-phonon matrix elements $$\begin{aligned}
g_{mn\nu}(\bm{k},\bm{q})
&= & l_{\bm{q}\nu} \sum_{\kappa \alpha}
e_{\bm{q}\nu,\kappa \alpha} \sum_{p,i}
\frac{\partial}{\partial x_\alpha}
t(\bm{\tau}_{0\kappa} - \bm{\tau}_{pi}) \nonumber \\
&& \times \{
e^{i\bm{k}\cdot\bm{R}_p} c^*_{m\bm{k+q},\kappa} c_{n\bm{k},i} \\
&& +e^{-i(\bm{k+q})\cdot\bm{R}_p} c^*_{m\bm{k+q},i} c_{n\bm{k},\kappa} \}.\end{aligned}$$ When we apply our method to calculate $\lambda$ for simple monolayer graphene and AB-BLG, $\lambda$ is less than 0.1 near the charge-neutral energy but it increases up to $0.2-0.3$ in proportion to the density of states when the chemical potential is varied \[Fig. S1(b) [@SM]\]. This is consistent with the previous studies for monolayer and bilayer graphene [@Park:2008; @epc1; @epc2].
Figure \[fig:ep\](b) shows calculated electron-phonon coupling strength as a function of the Fermi energy ($E_F$) for the three twist angles. $\bm{k}$ and $\bm{q}$ grids of $30\times30$ in the moiré Brillouin zone are used for electrons and phonons. Due to the large density of states of the flat bands, $\lambda$ becomes extremely large as $\theta$ approaches $1.08^\circ$, where the Dirac cones are nearly flat. The average interaction between electronic states, $\lambda/N_F$, for $\theta = 1.08^\circ$ is approximately twice that for $\theta = 0$. Furthermore, as the lattice relaxation brings electron-hole asymmetry in the density of states, the maximum value of $\lambda$ in the hole-side flat bands is almost twice that in the electron side for $\theta = 1.08^\circ$.
Figure \[fig:ep\](c) shows the isotropic Eliashberg function $ \alpha^2F(\omega) = \frac{1}{N_F}\sum_{nm\nu\bm{k}\bm{q}}
|g_{mn\nu}(\bm{k},\bm{q})|^2 \delta(E_F-\varepsilon_{n\bm{k}})
\delta(E_F-\varepsilon_{m\bm{k+q}}) \delta(\omega-\omega_{\bm{q}\nu})$ at the half-filling energy of the hole-side flat bands in $\theta=1.08^\circ$. With $\alpha^2F(\omega)$, $\lambda$ of Eq. (\[eq:4b\]) is equal to $\lambda=2\int^{\infty}_0 \alpha^2 F(\omega)/\omega d\omega$. In Fig. \[fig:ep\](c), in-plane optical modes generate strong peaks at 150 and 200 meV in $\alpha^2F(\omega)$, contributing to about 70% of $\lambda$. Although the interlayer shear ($\sim$2 meV) and breathing modes ($\sim$11 meV) have an order of magnitude smaller values of $\alpha^2F(\omega)$ than the in-plane optical modes, they have significant contributions to $\lambda$ due to their low-phonon energies. We also find $\lambda_{n\bm{k}}$ is nearly isotropic and depends very weakly on the electronic band and momentum \[Fig. \[fig:ep\](d)\], which indicates the electron-phonon coupling prefers single-gap $s$-wave superconductivity [@single_gap].
In conventional phonon-mediated superconductors, transition temperature can be reliably calculated from the Migdal-Eliashberg equations [@Migdal:1958; @Eliashberg:1960]. But the validity of the Migdal-Eliashberg theory depends on the existence of small parameter $\omega_{\mathrm{ph}}/E_F\ll1$ where $\omega_{\mathrm{ph}}$ is relevant phonon energy scale. This condition is obviously violated in magic-angle twisted bilayer graphene. For instance, while $E_F \approx 1 \; \mathrm{meV}$ near the half-fillings of the flat bands, $\omega_{\mathrm{ph}} \approx 2\sim11 \; \mathrm{meV}$ for the interlayer shear and breathing modes, and $\omega_{\mathrm{ph}} \approx 150\sim200 \; \mathrm{meV}$ for the in-plane optical modes. In this sense, MA-TBG systems are close to the antiadiabatic limit $\omega_{\mathrm{ph}}/E_F\gg1$.
In the antiadiabatic limit, $T_c$ was studied in several literatures [@Eagles:1967; @Ikeda:1992; @Gorkov:2016a; @Gorkov:2016b; @Sadoskii:2018], where the prefactor of $T_c$ is determined by $E_F$ instead of the phonon energy [@Gorkov:2016a; @Sadoskii:2018], that is, $$T_c\sim E_F \exp(-1/\lambda).$$ In our calculations for $\theta=1.08^\circ$, $\lambda$ = 3.6 (0.56) at $E_F$ = 0.86 (1.02) meV when the hole-side (electron-side) flat bands are half-filled. These values give $T_c\sim 7.5\;\mathrm{K}$ for the hole side and $T_c \sim 1.9\;\mathrm{K}$ for the electron side. Although our estimation is crude for direct comparison with experiments, the order of magnitude is close to the experimentally observed $T_c\sim1.7\;\mathrm{K}$ in the hole side. Since our estimation did not consider the effect of Coulomb interaction, which can reduce $T_c$, we expect that calculating $T_c$ including the Coulomb effect can give more consistent results to the experimental situations. Also, the rapid energy dependence of the electronic density of states can play an important role in determining $T_c$ [@Allen:1982].
In conclusion, we have calculated the electron-phonon coupling strength in the magic-angle twisted bilayer graphene using atomistic description of electrons and phonons. Obtained $\lambda$ in MA-TBG becomes almost an order of magnitude larger than that in simple monolayer or unrotated bilayer graphene. For $\theta=1.08^\circ$, the electron-hole asymmetry arises from atomic-structure relaxation due to interlayer interaction so that the electron-phonon coupling is stronger in the hole-side flat bands. The obtained electron-phonon interaction is almost isotropic and depends very weakly on the electronic band and momentum, which indicates the electron-phonon coupling prefers single-gap $s$-wave superconductivity. We also found that MA-TBG is in the antiadiabatic limit where the electron energy scale is much smaller than the phonon energy scale. Although the $T_c$ formula in the antiadiabatic limit produces values of $T_c$ comparable to the experiments, theory of $T_c$ of the system may require including Coulomb interaction and rapid energy dependence of the electronic density of states as well as electron correlation and any possible presence of magnetic fluctuations. Our results provide a fundamental understanding of the electron-phonon interaction in MA-TBG obtained from an atomistic description of electrons and phonons, highlighting that it can contribute to rich physics of the system.
This work was supported by National Research Foundation of Korea (Grant No. 2011-0018306). Y.W.C. acknowledges support from National Research Foundation of Korea (Global Ph.D. Fellowship Program NRF-2017H1A2A1042152). Computational resources have been provided by KISTI Supercomputing Center (Projects No. KSC-2017-C3-0079).
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**Supplemental Material:\
Strong electron-phonon coupling, electron-hole asymmetry, and nonadiabaticity\
in magic-angle twisted bilayer graphene**\
Young Woo Choi and Hyoung Joon Choi$^*$\
[*Department of Physics, Yonsei University, Seoul 03722, Republic of Korea\
*]{} (Dated: September 22, 2018)\
Figure S1(a) shows that the phonon dispersions of AB-stacked bilayer graphene obtained by our method, which are in good agreement with those from [*ab initio*]{} density functional perturbation theory (DFPT). Figure S1(b) shows the electron-phonon coupling strengths for monolayer graphene and AB-stacked bilayer graphene obtained by our method.
{width="13cm"}
$^*$ [email protected]
|
---
author:
- 'Ian Heywood, Hans-Rainer Klöckner and Steve Rawlings'
title: 'Equatorial Imaging with e-MERLIN Including the Chilbolton Antenna'
---
Introduction
============
The facilities at Chilbolton Observatory in Hampshire, UK, include a fully steerable 25-metre parabolic antenna which is mainly used for meteorological Doppler-polarisation radar. The use of this antenna in the MERLIN array (Thomasson, 1986) has been discussed for many years, and with the provision of a fast fibre link it would become a prime candidate for inclusion in the e-MERLIN[^1] array.
Figure \[fig:merlin\] is a version of the diagram presented on the online MERLIN user guide[^2] which has been modified in order to show the location of the Chilbolton antenna in addition to the locations of the existing MERLIN stations. Antenna numbers for the existing MERLIN stations have also been added. As can be seen, inclusion of the Chilbolton antenna will boost the number of long and intermediate-length baselines, as well as extending the north-south span of the array, facilitating the sampling of a much greater range of spatial frequencies. This is particularly crucial for ‘snapshot’ mode observations, and large-scale survey programmes where on-source time may necessarily be rather brief.
In this article we demonstrate the improved $uv$-plane sampling of an e-MERLIN+Chilbolton array for both snapshot and full synthesis equatorial observations. We present simulations demonstrating the response of both arrays to an unpolarized point source, and discuss the implications of the inclusion of the Chilbolton antenna when carrying out equatorial surveys.
All observations are simulated using AIPS with standard imaging procedures. All simulations are at L-band, using 100 frequency channels of 5 MHz each to simulate a 1.3 - 1.8 GHz contiguous band, assuming a spectrally flat source. Each antenna has an efficiency of 0.8 and system temperatures in the range 20 - 33 K. The 76-metre Lovell telescope is not included in the simulated array.
Increased $uv$-plane coverage
=============================
Figure \[fig:uv\_plots\] shows the $uv$-plane coverage for the e-MERLIN array plus the Chilbolton antenna for a 5-minute snapshot observation (left panel) and a 12-hour track (right panel). The additional visibilities arising from the inclusion of the Chilbolton antenna are shown in light grey.
Each of these simulations is an observation of an equatorial source (with a Right Ascension and Declination of zero). The snapshot observation has an on-source time of 5 minutes at an hour angle of zero and the 12-hour track has an hour angle range of -6 to +6. The latter scenario gives the best possible $uv$-coverage for a source at this position due to the opacity of the earth. The integration time assumed in both cases is 60 seconds.
The key feature to note in both of these plots is that the additional Chilbolton baselines greatly reduce the east-west bias of the array and increase the number of long and intermediate-length baselines. The image-plane implications of this modified $uv$-plane coverage are demonstrated in Section 3.
Imaging considerations
======================
Simulated, zero-noise dirty images of a 5 minute snapshot observation of an equatorial 1-mJy point source are presented in Fig. \[fig:beams\]. These images correspond to a 6-antenna e-MERLIN array without (left) and with (right) the Chilbolton antenna, observing with the 1.3-1.8 GHz contiguous band. Since these are dirty images and, since they are observations of a point source, they represent the point spread function of the array. A Gaussian fitted to the half power level of the central lobe is plotted in the lower left of each panel. The dimensions of these ellipses in arcseconds are 0.42 $\times$ 0.10 arcsec$^{2}$ (position angle = 20.4$^{o}$, without Chilbolton) and 0.24 $\times$ 0.10 arcsec$^{2}$ (position angle = 33.3$^{o}$, with Chilbolton).
The benefits of the additional baselines to the Chilbolton station are immediately apparent. The dominant, near vertical structure and strength of the sidelobes in the beam pattern are significantly reduced with the addition of the Chilbolton antenna. The central lobe of the beam is also much closer to circularity.
Figure \[fig:snapshots\] shows cleaned, simulated snapshot images without (left) and with (right) Chilbolton, with Gaussian noise added, the level of which is calculated by the AIPS task UVCON according to the antenna and observation parameters. These simulations obviously assume perfect phase and amplitude calibration which is never the case in practice. Similarly, Fig. \[fig:12hours\] shows cleaned images corresponding to a 12-hour track. The contour levels are adjusted for these images in order to display the $\sim\mu$Jy-level background noise. Note that phase and amplitude errors would yield additional symmetric and anti-symmetric (relative to the peak) noise patterns, similar to those seen in Fig. \[fig:beams\].
Relevant parameters for these four images are listed in Table \[tab:stats\]. The values and uncertainties in the peak fluxes and integrated flux densities are determined by fitting Gaussians to the centre of the image using the AIPS task IMFIT. The background RMS values in the image are returned by isolating the background region using TVWIN and executing the AIPS verb IMSTAT.
Note that in the case of Fig. \[fig:snapshots\] there is no improvement at all in the RMS achieved despite the increased collecting area. The decrease in RMS expected because of the $\sim$10% increase in collecting area is masked by sources of systematic error.
------------------- ------------------- ------------------- ---------------- ----------- ----------- -------
Simulation Peak flux Integrated flux Background RMS B$_{maj}$ B$_{min}$ BPA
(mJy/beam) density (mJy) (mJy/beam) (arcsec) (arcsec) (deg)
Snapshot ($eM$) 0.903 $\pm$ 0.044 1.081 $\pm$ 0.085 0.056 0.422 0.104 20.4
Snapshot ($eM+C$) 0.936 $\pm$ 0.063 1.024 $\pm$ 0.115 0.058 0.241 0.108 33.3
12 hour ($eM$) 1.006 $\pm$ 0.007 1.025 $\pm$ 0.012 0.009 0.297 0.138 22.5
12 hour ($eM+C$) 1.013 $\pm$ 0.007 1.009 $\pm$ 0.011 0.007 0.174 0.122 31.9
------------------- ------------------- ------------------- ---------------- ----------- ----------- -------
Assuming that the sidelobe structure can be successfully removed by whatever means, then when an extended Gaussian-like radio source is observed its brightness distribution is essentially convolved with the central lobe of the beam. This convolution will ‘smear’ the flux density out over an area depending on the size of the source and the size of the beam, the result being another Gaussian[^3]. Given that a radio source will generally have a fixed brightness over the course of an observation, the observed peak brightness (i.e. the ‘height’ of the resulting Gaussian) will be dependent on the size of the central lobe. Using a more compact beam, as in the case where the Chilbolton antenna is included, reduces the broadening and increases the peak brightness, allowing a flux limit to be reached faster in certain cases.
MERLIN observations of the Hubble deep and flanking fields have shown that a typical micro-Jansky radio source has a characteristic size of $\sim$0.6 arcsec (Muxlow et al., 2005). The major axes of the snapshot beams are $\sim$0.4 arcseconds (without Chilbolton) and $\sim$0.2 arcseconds (with Chilbolton).
With the above considerations in mind, we present quantitative estimates of the increased mapping speed provided by the inclusion of the Chilbolton antenna in Table \[tab:maptimes\]. These values have been calculated by convolving two-dimensional Gaussians, measuring the difference in the peak value for the two arrays, and assuming the sensitivity has an inverse dependence on the square root of the on-source time. Two models consistent with the full-synthesis beam shapes with and without Chilbolton, and two source models with major axes of 0.6 and 0.3 arcsec (with minor axes half this value) are used. The calculation is performed with both parallel and orthogonal alignment of the major axes of the source and beam, representing the best and worst case respectively for a random distribution of source alignments.
Simulation Factor
------------------------ --------
0.6 arcsec, aligned 85%
0.6 arcsec, orthogonal 68%
0.3 arcsec, aligned 64%
0.3 arcsec, orthogonal 46%
: Increase in mapping speed due to the inclusion of the Chilbolton antenna, expressed as a percentage of the observing time required for full synthesis observations using the regular e-MERLIN array. This is presented for sources with major axes of 0.6 and 0.3 arcseconds, aligned both parallel and orthogonal to the major axis of the beam. \[tab:maptimes\]
Conclusions
===========
The geographical location of the Chilbolton antenna introduces baselines which complement the existing e-MERLIN array. This is particularly evident in the case of equatorial observations, where the dominant east-west layout of the existing array biases the $uv$ coverage, resulting in strong linear structure in the sidelobes of a synthesised beam which is highly eccentric.
Putting aside the increased sensitivity due to the $\sim$10% increase in collecting area with the additional antenna, the synthesised beam is much more compact and circular for equatorial observations, facilitating more efficient detection of characteristic $\mu$Jy radio sources, reducing the mapping speed by up to a factor of two (see Table \[tab:maptimes\] for favourably oriented sources.
The addition of the Chilbolton antenna would significantly enhance the power of e-MERLIN. There are many experiments that would benefit significantly from this enhancement, such as high-resolution radio surveys complementing optical and infrared deep-fields such as those to be undertaken by VISTA, and high-resolution radio observations of galactic and extra-galactic radio sources targetted with ALMA. Both e-MERLIN and ALMA naturally complement each other due to their similar sub-arcsecond resolutions, despite their operating frequencies differing by a factor of $\sim$100.
The authors would like to thank Simon Garrington of Jodrell Bank Observatory and Ken Craig of Rutherford Appleton Laboratory.
Muxlow, T.W.B., et al., 2005, MNRAS, 358, 1159
Thomasson, P., 1986, RASQJ, 27, 413
[^1]:
[^2]:
[^3]: The convolution of any two Gaussians is another Gaussian, since *(i)* the product of any two co-centred Gaussians is a Gaussian, *(ii)* the convolution of any two functions is equal to the Fourier transform of their product, and *(iii)* the Fourier transform of a Gaussian is another Gaussian.
|
---
abstract: 'We investigate the stability of Quantum Critical Points (QCPs) in the presence of two competing phases. These phases near QCPs are assumed to be either classical or quantum and assumed to repulsively interact via square-square interactions. We find that for any dynamical exponents and for any dimensionality strong enough interaction renders QCPs unstable, and drives transitions to become first order. We propose that this instability and the onset of first-order transitions lead to spatially inhomogeneous states in practical materials near putative QCPs. Our analysis also leads us to suggest that there is a breakdown of Conformal Field Theory (CFT) scaling in the Anti de Sitter models, and in fact these models contain first-order transitions in the strong coupling limit.'
author:
- 'Jian-Huang She$^{1,2}$, Jan Zaanen$^2$, Alan R. Bishop$^1$ and Alexander V. Balatsky$^1$'
bibliography:
- 'strings.bib'
- 'refs.bib'
date: ' \[file: \]'
title: Stability of Quantum Critical Points in the Presence of Competing Orders
---
Introduction
============
Quantum criticality is an important concept that has dominated the landscape of modern condensed matter physics for the last decade [@Sachdev99]. The idea behind quantum criticality is simple and powerful. Imagine competing interactions that typically drive the transitions between different phases. Logically one has to allow for the possibility that the relative strength of these competing interactions is tunable as a function of the external control parameters such as pressure, magnetic field, doping: we deliberately omit temperature as a control parameter since quantum phase transitions (QPTs) will occur at T=0. The simplest route to arrive at a QPT is to consider a line of finite temperature phase transition as a function of some control parameter, such as pressure $P$, magnetic field $B$ or doping $x$. At $T=0$ this line will indicate a critical value of the control parameter. This specific value of the control parameter, where one expects a precise balance between tendency to different phases or states, is called a quantum critical point (QCP). Near this point, competing interactions nearly compensate each other. It is often asserted that it is the physics of frustration and competition, which leads to the finite temperature transition, that also controls and enables the interesting properties of materials that are brought to the $T=0$ QCP.
Much of the attention on quantum criticality has been focused on the finite temperature scaling properties [@Sachdev99; @Continentino01; @Lohneisen07]. Temperature is the only relevant scale in the quantum critical region above the QCP, bounded by the crossover line $T^*\sim |r|^{\nu z}$. The parameter $r$ measures the distance to the QCP, $\nu$ is the correlation-length exponent in $\xi\sim r^\nu$ and $z$ is the dynamical exponent in $\xi_{\tau}\sim\xi^z$. With the correlation length $\xi$ and correlation time $\xi_\tau$ much larger than any other scale of the system, power law behavior is expected for many physical observables, e. g. the specific heat, magnetic susceptibility, and most notably resistivity. Clear deviations from the Fermi liquid predictions are experimentally detected, and these phases are commonly termed non-Fermi liquids. In many systems, the anomalous finite temperature scaling properties are asserted to result from the underlying zero temperature QCPs.
In this paper, we would like to emphasize another aspect of quantum criticality, namely that it serves as a driving force for new exotic phenomena at extremely low temperatures and in extremely clean systems. One possibility is the appearance of new phases around the QCPs. It has been found in numerous experiments as one lowers temperature, seemingly inevitably in all the systems available, new phases appear near the QCP. Most commonly observed to date is the superconducting phase. The phenomenon of a superconducting dome enclosing the region near the QCP is quite general (see Fig. 1). It has been identified in many heavy fermion systems [@Lonzarich98; @Stewart06; @Gegenwart08], plausibly also in cuprates [@Sachdev03], even possibly in pnictides [@Zhao08; @XHChen08; @XHChen09; @Fisher09; @Canfield08; @Ning09], and probably in organic charge-transfer salts [@Brown00; @Itoi07; @Itoi0702]. Other examples include the nematic phase around the metamagnetic QCP in the bilayer ruthenate $\rm Sr_3Ru_2O_7$ [@Mackenzie01; @Mackenzie04; @Mackenzie07; @Mackenzie09], the origin of which is still under intense debate [@Kee05; @Simons09; @Kivelson09; @Wu09; @Sigrist10]. The emerging quantum paraelectric - ferroelectric phase diagram is also very reminiscent [@Millis03; @Rowley09], as is the disproportionation-superconducting phase in doped bismuth oxide superconductors [@Cava88; @Johnson88; @Pei90; @Goodenough90; @Allen97; @Allen02].
![Illustration of the competing phases and superconducting dome. Here for concreteness, we consider the ordered phase to be an antiferromagnetic phase. $x$ is the tuning parameter. It can be pressure, magnetic field or doping. The superconducting temperature usually has the highest value right above the QCP.[]{data-label="figqcp"}](qcp.eps){width="7cm"}
It has also been discovered recently that, as samples are becoming cleaner, on the approach to QCP we encounter first order transitions, and the new phases near the QCP are usually inhomogeneous and exhibit finite wavevector orderings (see [@Pfleiderer05; @Pfleiderer09; @Rosenbaum09] and references therein). For example, the heavy fermion compound $\rm CeRhIn_5$ orders antiferromagnetically at low temperature and ambient pressure. As pressure increases, the Neel temperature decreases and at some pressure the antiferromagnetic phase is replaced by a superconducting phase through a first-order phase transition. There are also evidences for a competitive coexistence of the two phases within the antiferromagnetic phase, as in some organic charge-transfer superconductor precursor antiferromagnetic phases. Such coexistence was also observed in $\rm Rh$-doped $\rm CeIrIn_5$. The heavy fermion superconductor $\rm CeCoIn_5$ has the unusual property that when a magnetic field is applied to suppress superconductivity, the superconducting phase transition becomes first-order below $T_0\simeq 0.7 K$. For the superconducting ferromagnet $\rm UGe_2$, where superconductivity exists within the ferromagnetic state, the two magnetic transitions (ferromagnetic to paramagnetic and large-moment ferromagnetic to small-moment ferromagnetic) are both first order [@Flouquet00; @Huxley01; @Huxley02]. Other examples of continuous phase transitions turning first-order at low temperatures include $\rm CeRh_2Si_2$ [@Roman97; @Roman96], $\rm CeIn_3$ [@Onuki08], $\rm URhGe$ [@Levy05], $\rm ZrZn_2$ [@Uhlarz04] and $\rm MnSi$ [@Pfleiderer07]. The prevailing point of view seems to be that this happens only in a few cases and these are considered exceptions. Yet we are facing a rapidly growing list of these “exceptions”, and we take the view here that they rather represent a general property of QCPs.
The point is that, on approach to the QCP, an interaction that was deemed irrelevant initially, takes over and dominates. For example it has been proposed recently that the superconducting instability, which is marginal in the usual Fermi liquids, becomes relevant near the QCP and leads to a high transition temperature [@She09]. Actually these instabilities are numerous and can vary, depending on the system at hand. However there seems to be a unifying theme of those instabilities. We suggest that QCPs are unstable precisely for the reasons we are interested in these points: extreme softness and extreme susceptibility of the system in the vicinity of QCPs. We regard the recently discovered first order transitions as indicators of a more fundamental and thus powerful phsyics. We are often prevented from reaching quantum criticality, and often the destruction is relatively trivial and certainly not as appealing and elegant as quantum criticality. We can draw an analogy from gravitational physics, where the naked singularities are believed to be prevented from happening due to many kinds of relevant instabilities. This is generally known as the “cosmic censorship conjecture” [@Penrose69]. The recently proposed AdS/CFT correspondence [@Maldacena97; @Witten98; @Gubser98], which maps a non-gravitational field theory to a higher dimensional gravitational theory, adds more to this story. Here researchers have begun to realize that the Reissner-Nordstrom black holes in AdS space, which should have a macroscopic entropy at zero temperature, are unstable to the spontaneous creation of particle-antiparticle pairs, and tend to collapse to a state with lower entropy [@Hartnoll08; @Zaanen0902; @Sachdev10].
There have appeared in the literature scattered examples of first-order quantum phase transitions at the supposed-to-be continuous QCPs [@Belitz05; @Continentino01; @Continentino04; @Continentino0402; @Continentino0403; @Continentino05; @Qi09; @Millis10], however it appears that the universality of this phenomenon is not widely appreciated. This universality is the main motivation for our paper. We will systematically study the different possibilities for converting a continuous QPT to first order.
The first striking example how fluctuations of one of the order parameters can qualitatively change the nature of the transition comes from the Coleman-Weinberg model [@Coleman73], where they showed how gauge fluctuations of the charged field introduce a first order transition. In this work it was shown that in dimension $d= 3$, for any weak coupling strength, one develops a logarithmic singularity, and therefore the effective field theory has a first-order phase transition. Subsequently, this result was extended to include classical gauge field fluctuations by Halperin, Lubensky and Ma [@Halperin74], where a cubic correction to the free energy was found. Nontrivial gradient terms can also induce an inhomogeneous phase and/or glassy behavior [@Sasha09].
A prototypical example for the competing phases and superconducting dome is shown schematically in Fig.1. Below, we apply the renormalization group (RG) and scaling analysis to infer the stability if the QCP as a result of competition. We find in our analysis that the QCP is indeed unstable towards a first order transition as a result of competition. Obviously details of the collapse of a QCP and the resulting phase diagram depend on details of the nature of the fluctuating field and details of the interactions. We find that the most relevant parameters that enter into criterion for stability of a QCP are the strength of interactions between competing phases: we take this interaction to be repulsive between squares of the competing order parameters. When the two order parameters break different symmetries, the coupling will be between the squares of them. Another important factor that controls the phase diagram is the dynamical exponents $z$ of the fields. The nature of the competition also depends on the classical or quantum character of the fields. Here by classical we do not necessarily mean a finite temperature phase transition, but rather that the typical energy scale is above the ultraviolet cutoff, and the finite frequency modes of the order parameters can be ignored, so that a simple description in terms of free energy is enough to capture the physics. We analyzed three possibilities for the competing orders:
i\) *classical* + *classical*. Here we found that interactions generally reduce the region of coexistence, and when interaction strength exceeds some critical value, the second-order phase transitions become first order.
ii\) *classical* + *quantum*. Here the quantum field is integrated out, giving rise to a correction to the effective potential of the classical order parameter. For a massive fluctuating field with $d+z\leqslant 6$, or a massless one with $d+z\leqslant 4$, the second-order quantum phase transition becomes first order.
iii\) *quantum* + *quantum*. Here RG analysis was employed, and we found that in the high dimensional parameter space, there are generally regions with runaway flow, indicating a first-order quantum phase transition.
It has been proposed recently that alternative route to the breakdown of quantum criticality is through the basic collapse of Landau-Wilson paradigm of conventional order parameters and formation of the deconfined quantum critical phases ([@Senthil04; @Senthil0402]). This is a possibility that has been discussed for specific models and requires a different approach than the one taken here. We are not addressing this possibility.
The plan of the paper is as follows. In section II, we consider coupling two classical order parameter fields together. Both fields are characterized by their free energies and Landau mean field theory will be used. In section III, we consider coupling a classical order parameter to a quantum mechanical one, which can have different dynamical exponents. The classical field is described by its free energy and the quantum field by its action; the latter is integrated out to produce a correction to the effective potential for the former. In section IV, we consider coupling two quantum mechanical fields together. With both fields described by their actions, we use RG equations to examine the stability conditions. In particular, we study in detail the case where the two coupled order parameters have different dynamical exponents, which, to our knowledge, has not been considered previously. In the conclusion section, we summarize our findings.
Two competing classical fields
==============================
We consider in this section two competing classical fields. Examples are the superconducting order and antiferromagnetic order in $\rm CeRhIn_5$ and $\rm Rh$-doped $\rm CeIrIn_5$, and the superconducting order and ferromagnetic order near the large-moment to small-moment transition in $\rm UGe_2$. We will follow the standard textbook approach, and this case is presented as a template for the more complex problems studied later on.
We first study the problem at zero temperature. For simplicity, both of them are assumed to be real scalars. The free energy of the system consists of three parts, the two free parts $F_{\psi}, F_M$ and the interacting part $F_{\rm int}$: $$\begin{split}
F=&F_{\psi}+F_M+F_{\rm int};
\\F_{\psi}=&\frac{\rho}{2}(\nabla \psi)^2-\alpha \psi^2+\frac{\beta}{2}\psi^4;
\\ F_M=&\frac{\rho_M}{2}(\nabla M)^2-\alpha_M M^2+\frac{\beta_M}{2}M^4;
\\F_{\rm int}=&\gamma \psi^2M^2 .
\end{split}
\label{freeenergy}$$ Here, by changing $\alpha, \alpha_M$, the system is tuned through the phase transition points. When the two fields are decoupled, with $\gamma=0$, there will be two separated second-order phase transitions. Assume the corresponding values of the tuning parameter $x$ at these two transition points are $x_1$ and $x_2$, we can parameterize $\alpha, \alpha_M$ as $\alpha=a(x-x_1)$ and $\alpha_M=a_M(x_2-x)$, where $a,a_M$ are constants.
![(Color online) Illustration of the mean field phase diagram for two competing orders. Here for concreteness we consider antiferromagnetic and superconducting orders. The two orders coexist in the yellow region, whose area shrinks as the coupling increases from left to right. The left figure has $\gamma=0$, the central one has $0<\gamma<\sqrt{\beta\beta_M}$, and the right one has $\gamma>\sqrt{\beta\beta_M}$. When $\gamma$ exceeds the critical value $\sqrt{\beta\beta_M}$, the two second-order phase transition lines merge and become first order (the thick vertical line).[]{data-label="fig1"}](cc1.eps "fig:"){width="5.2cm"} ![(Color online) Illustration of the mean field phase diagram for two competing orders. Here for concreteness we consider antiferromagnetic and superconducting orders. The two orders coexist in the yellow region, whose area shrinks as the coupling increases from left to right. The left figure has $\gamma=0$, the central one has $0<\gamma<\sqrt{\beta\beta_M}$, and the right one has $\gamma>\sqrt{\beta\beta_M}$. When $\gamma$ exceeds the critical value $\sqrt{\beta\beta_M}$, the two second-order phase transition lines merge and become first order (the thick vertical line).[]{data-label="fig1"}](cc2.eps "fig:"){width="5.2cm"} ![(Color online) Illustration of the mean field phase diagram for two competing orders. Here for concreteness we consider antiferromagnetic and superconducting orders. The two orders coexist in the yellow region, whose area shrinks as the coupling increases from left to right. The left figure has $\gamma=0$, the central one has $0<\gamma<\sqrt{\beta\beta_M}$, and the right one has $\gamma>\sqrt{\beta\beta_M}$. When $\gamma$ exceeds the critical value $\sqrt{\beta\beta_M}$, the two second-order phase transition lines merge and become first order (the thick vertical line).[]{data-label="fig1"}](cc3.eps "fig:"){width="5.2cm"}
We would like to know the ground state of the system. Following the standard procedure, we first find the homogeneous field configurations satisfying $\frac{\partial F}{\partial \psi}=\frac{\partial F}{\partial M}=0$, and then compare the corresponding free energy. It is easy to see that the above equations have four solutions, with $(|\psi|, |M|)=(0,0), (0,\sqrt{\alpha_M/\beta_M}), (\sqrt{\alpha/\beta},0),(\psi_*,M_*)$, where $$\begin{split}
\alpha \psi_*^2=&\frac{\gamma'-\beta'_M}{{\gamma'}^2-\beta'\beta'_M},
\\ \alpha_M M_*^2=&\frac{\gamma'-\beta'}{{\gamma'}^2-\beta'\beta'_M},
\end{split}
\label{psiM}$$ and the rescaled parameters are $\gamma'=\gamma/\alpha\alpha_M, \beta'=\beta/\alpha^2,\beta'_M=\beta_M/\alpha_M^2$. When $\gamma=0$, the fourth solution reduces to $(\psi_*, M_*)=(\sqrt{\alpha/\beta},\sqrt{\alpha_M/\beta_M})$, with the two orders coexisting but decoupled. We are interested in the case where the two orders are competing, thus a relatively large positive $\gamma$.
For $x_1<x<x_2$, we have $\alpha>0, \alpha_M>0$. The necessary condition for the existence of the fourth solution is $\gamma'>\beta', \beta'_M, \sqrt{\beta'\beta'_M}$ or $\gamma'<\beta', \beta'_M, \sqrt{\beta'\beta'_M}$. In this case, the configuration $(0,0)$ has the highest free energy $F[0,0]=0$. For the configuration $(\psi_*,M_*)$ with coexisting orders to have lower free energy than the two configurations with single order, one needs to have $\gamma'<\sqrt{\beta'\beta'_M}$, which reflects the simple fact that when the competition between the two orders is too large, their coexistence is not favored. Thus the condition for the configuration $(\psi_*,M_*)$ to be the ground state of the system is $\gamma'<\beta'$ and $\gamma'<\beta'_M$. If $\gamma'>{\rm min}\{\beta',\beta'_M\}$, one of the fields has to vanish.
Next we observe that, for $x$ near $x_1$, $\beta'_M$ remains finite, $\alpha \sim (x - x_1)$, and $\gamma'$ diverges as $1/(x-x_1)$, while $\beta'$ diverges as $1/(x-x_1)^2$. So the lowest energy configuration is $\psi=0$, $|M|=\sqrt{\alpha_M/\beta_M}$. Similarly, near $x_2$, the ground state is $(\sqrt{\alpha/\beta},0)$. The region with coexisting orders shrinks to $$\frac{\gamma a_Mx_2+\beta_M ax_1}{\gamma a_M+\beta_M a}<x<\frac{\gamma ax_1+\beta a_Mx_2}{\gamma a+\beta a_M}.$$ For $\gamma<\sqrt{\beta\beta_M}$, this region has finite width. In this region, $(0,0)$ is the global maximum of the free energy, $(0,\sqrt{\alpha_M/\beta_M}), (\sqrt{\alpha/\beta},0)$ are saddle points, and $(\psi_*, M_*)$ is the global minimum. The phase with coexisting order is sandwiched between the two singly ordered phases, and the two phase transitions are both second-order. The shift in spin-density wave ordering and Ising-nematic ordering due to a nearby competing superconducting order has been studied recently by Moon and Sachdev [@Moon09; @Moon10], where they found that the fermionic degrees of freedom can play important roles.
For $\gamma>\sqrt{\beta\beta_M}$, this intermediate region with coexisting orders vanishes, and the two singly ordered phases are separated by a first-order quantum phase transition. The location of the phase transition point is determined by equating the two free energies at this point, $$F\left[\sqrt{\frac{\alpha(x_c)}{\beta}},0\right]=F\left[0,\sqrt{\frac{\alpha_M(x_c)}{\beta_M}}\right],$$ which gives $x_c=(x_2+Ax_1)/(1+A)$, with $A=(a/a_M)\sqrt{\beta_M/\beta}$. The slope of the free energy changes discontinuously across the phase transition point, with a jump $$\delta F^{(1)}\equiv\left\vert\left(\frac{dF}{dx}\right)_{x_c^+}-\left(\frac{dF}{dx}\right)_{x_c^-}\right\vert=\frac{aa_M}{\sqrt{\beta\beta_M}}(x_2-x_1).$$
The size of a first-order thermal phase transition can be characterized by the ratio of latent heat to the jump in specific heat in a reference second-order phase transition [@Halperin74]. A similar quantity can be defined for a quantum phase transition, where the role of temperature is now played by the tuning parameter $x$. We choose as our reference point $\gamma=0$, where the two order parameters are decoupled. For $x<x_1$, one has $d^2F/dx^2=-a_M^2/\beta_M$; for $x>x_2$, one has $d^2F/dx^2=-a^2/\beta$; and $d^2F/dx^2=-a_M^2/\beta_M-a^2/\beta$ for $x_1<x<x_2$. We take the average of the absolute value of the two jumps to obtain $$\delta F^{(2)}=\frac{1}{2}(a_M^2/\beta_M+a^2/\beta).$$ So the size of this first-order quantum phase transition is $$\delta x=\frac{\delta F^{(1)}}{\delta F^{(2)}}=\frac{2\sqrt{\tilde{\beta}\tilde{\beta}_M}}{\tilde{\beta}+\tilde{\beta}_M}(x_2-x_1),$$ with $\tilde{\beta}=\beta/a^2$ and $\tilde{\beta}_M=\beta_M/a_M^2$. It is of order $x_2-x_1$, when $\tilde{\beta}$ and $\tilde{\beta}_M$ are not hugely different.
The above consideration can be generalized to finite temperature, by including the temperature dependence of all the parameters. Specially, there exists some temperature $T^*$, where $x_1(T^*)=x_2(T^*)$. In this way we obtain phase diagrams similar to those observed in experiments (see Fig. 1).
Effects of quantum fluctuations
===============================
In this section, we consider coupling an order parameter $\psi$ to another field $\phi$, which is fluctuating quantum mechanically. The original field $\psi$ is still treated classically, meaning any finite frequency modes are ignored. For the quantum fields, in the spirit of Hertz-Millis-Moriya [@Hertz76; @Millis93; @Moriya85], we assume that the fermionic degrees of freedom can be integrated out, and we will only deal with the bosonic order parameters. This model may, for example, explain the first-order ferromagnetic to paramagnetic transition in $\rm UGe_2$, where the quantum fluctuations of the superconducting order parameter are coupled with the ferromagnetic order parameter, which can be regarded as classical near the superconducting transition point.
We will integrate out the quantum field to obtain the effective free energy of a classical field. The partition function has the form $$Z[\psi(\mathbf r)]=\int {\cal D}\phi({\mathbf r},\tau) \exp\left({-\frac{{\cal F}_{\psi}}{T}-S_{\phi}-S_{\psi\phi}}\right).$$ The free energy is of the same form as in the previous section with ${\cal F}_{\psi}=\int d^d{\mathbf r}F_{\psi}$. Thus, in the absence of coupling to other fields, the system goes through a second-order quantum phase transition as one tunes the control parameter $x$ across its critical value. We consider a simple coupling $$S_{\psi\phi}=g\int d^d{\mathbf r}d\tau \psi^2\phi^2.$$ The action of the $\phi$ field depends on its dynamical exponent $z$. We notice that such classical + quantum formalism has been used to investigate the competing orders in cuprates in [@Sachdev02].
The saddle point equation for $\psi$ reads $$\frac{\delta\ln Z[\psi(\mathbf r)]}{\delta\psi(\mathbf r)}=0,$$ which gives $$\left[-\alpha+\beta\psi^2(\mathbf r)-\frac{\rho}{2}\nabla^2+g\langle\phi^2(\mathbf r)\rangle\right]\psi(\mathbf r)=0.$$ Here we have defined the expectation value, $$\langle\phi^2(\mathbf r)\rangle=\frac{1}{\beta}\int {\cal D}\phi({\mathbf r'},\tau') \int_0^{\beta}d\tau\phi^2({\mathbf r},\tau) \exp\left({-S_{\phi}-S_{\psi\phi}}\right).$$ It can also be written in terms of the different frequency modes, $$\langle\phi^2(\mathbf r)\rangle=T\sum_{\omega_n}\langle\phi(\mathbf r,\omega_n)\phi(\mathbf r,-\omega_n)\rangle=T\sum_{\omega_n}\int {\cal D}\phi({\mathbf r'},\nu_s)\phi(\mathbf r,\omega_n)\phi(\mathbf r,-\omega_n) \exp\left({-S_{\phi}-S_{\psi\phi}}\right).$$ The quadratic term in $S_{\phi}$ is of the form $$S_{\phi}^{(2)}=\sum_{\nu_s}\int d^d{\mathbf r}' \int d^d{\mathbf r}'' \phi({\mathbf r'},\nu_s)\chi_0^{-1}({\mathbf r'},{\mathbf r''},\nu_s)\phi(\mathbf r'',-\nu_s),$$ or more conveniently, in terms of momentum and frequency, $$S_{\phi}^{(2)}=\sum_{\nu_s}\int \frac{d^d\mathbf k}{(2\pi)^d} \phi({\mathbf k},\nu_s)\chi_0^{-1}({\mathbf k},\nu_s)\phi(-\mathbf k,-\nu_s).$$ So in the presence of translational symmetry, we find $$\langle\phi^2\rangle=T\sum_{\omega_n}\int \frac{d^d\mathbf k}{(2\pi)^d} \frac{1}{\chi_0^{-1}({\mathbf k},\omega_n)+g\psi^2}.$$ This leads to the 1-loop correction to the effective potential for $\psi$, determined by $$\frac{\delta V_{\rm eff}^{(1)}[\psi]}{\delta\psi}=2g\langle\phi^2\rangle\psi.$$
So far we have been general in this analysis. Further analysis requires us to make more specific assumptions about the dimensionality and dynamical exponents.
Fluctuations with $d=3, z=1$
----------------------------
When the $\phi$ field has dynamical exponent $z=1$, its propagator is of the form $$\chi_0({\mathbf k},\omega_n)=\frac{1}{\omega_n^2+k^2+\xi^{-2}}.$$ A special case is a gauge boson, which has zero bare mass, and thus $\xi\to\infty$. This problem has been studied in detail by Halperin, Lubensky and Ma [@Halperin74] for a classical phase transition (see also [@Lubensky78]), and by Coleman and Weinberg [@Coleman73] for relativistic quantum field theory. Other examples are critical fluctuations associated with spin-density wave transitions and superconducting transitions in clean systems. We also note that Continentino and collaborators have used the method of effective potential to investigate some special examples of the fluctuation-induced first order quantum phase transition [@Continentino01; @Continentino04; @Continentino0402; @Continentino0403; @Continentino05].
![(Color online) Schematic illustration of the fluctuation-induced first-order phase transition. Here, for concreteness, we consider ferromagnetic and superconducting orders. The ferromagnetic order is regarded as classical, while the superconducting one as quantum mechanical. At low temperatures, the second-order ferromagnetic to paramagnetic phase transition becomes first order (the thick vertical line), due to fluctuations of the superconducting order parameter.[]{data-label="fig2"}](cq1.eps "fig:"){width="6cm"} ![(Color online) Schematic illustration of the fluctuation-induced first-order phase transition. Here, for concreteness, we consider ferromagnetic and superconducting orders. The ferromagnetic order is regarded as classical, while the superconducting one as quantum mechanical. At low temperatures, the second-order ferromagnetic to paramagnetic phase transition becomes first order (the thick vertical line), due to fluctuations of the superconducting order parameter.[]{data-label="fig2"}](cq2.eps "fig:"){width="6cm"}
Let us consider $T=0$, for which the summation $T\sum_{\omega_n}$ can be replaced by the integral $\int d\omega/(2\pi)$. We then get for the one-loop correction to the effective potential $$\frac{\delta V_{\rm eff}^{(1)}[\psi]}{\delta\psi}=2g\psi\int \frac{d\omega}{2\pi}\int \frac{d^d{\mathbf k}}{(2\pi)^d}\frac{1}{\omega^2+k^2+\xi^{-2}+g\psi^2}.$$ Carrying out the frequency integral, we obtain for $d=3$, $$\frac{\delta V_{\rm eff}^{(1)}[\psi]}{\delta\psi}=\frac{g\psi}{2\pi^2}\int_0^{\Lambda}dk\frac{k^2}{\sqrt{k^2+\xi^{-2}+g\psi^2}},$$ where an ultraviolet cutoff is imposed. Integrating out momentum gives $$\frac{\delta V_{\rm eff}^{(1)}[\psi]}{\delta\psi}=\frac{g\psi}{4\pi^2}\left[\Lambda\sqrt{\Lambda^2+\xi^{-2}+g\psi^2}-(\xi^{-2}+g\psi^2)\ln\left(\frac{\Lambda+\sqrt{\Lambda^2+\xi^{-2}+g\psi^2}}{\sqrt{\xi^{-2}+g\psi^2}}\right)\right],$$ which can be simplified as $$\frac{\delta V_{\rm eff}^{(1)}[\psi]}{\delta\psi}=\frac{g\psi}{4\pi^2}\left[\Lambda^2+\frac{1}{2}(\xi^{-2}+g\psi^2)-(\xi^{-2}+g\psi^2)\ln\left(\frac{2\Lambda}{\sqrt{\xi^{-2}+g\psi^2}}\right)\right].$$ Combined with the bare part, $$V_{\rm eff}^{(0)}(\psi)=-\alpha\psi^2+\frac{1}{2}\beta\psi^4,$$ we get the effective potential to one-loop order, $$V_{\rm eff}(\psi)=-{\hat \alpha}\psi^2+\frac{1}{2}{\hat \beta}\psi^4-\frac{1}{16\pi^2}(\xi^{-2}+g\psi^2)^2\ln\left(\frac{2\Lambda}{\sqrt{\xi^{-2}+g\psi^2}}\right),$$ with the quadratic and quartic terms renormalized by ${\hat \alpha}=\alpha-g(4\Lambda^2+\xi^{-2})/(32\pi^2)$ and ${\hat \beta}=\beta+3g/(32\pi^2)$. When $\phi$ field is critical with $\xi\to\infty$, the third term is of the well-known Coleman-Weinberg form $\psi^4\ln(2\Lambda/\sqrt{g\psi^2})$, which drives the second-order quantum phase transition to first order.
For $\xi$ large but finite, we can expand the third term as a power series in $\xi^{-2}/(g\psi^2)$, and the effective potential is of the form $$V_{\rm eff}(\psi)=-{\bar \alpha}\psi^2+\frac{1}{2}{\bar \beta}\psi^4-\frac{1}{16\pi^2}(2\xi^{-2}g\psi^2+g^2\psi^4)\ln\frac{2\Lambda}{\sqrt{g\psi^2}}.$$ In addition to the Coleman-Weinberg term, there is another term of the form $\psi^2\ln\psi$, and again we have also a first-order phase transition.
To study the generic case where the $\phi$ field is massive, we rescale the $\psi$ field and cutoff, defining $$u^2\equiv \frac{g\psi^2}{\xi^{-2}} ,~~~~ {\tilde \Lambda}\equiv\frac{2\Lambda}{\xi^{-1}}.$$ The rescaled effective potential takes the form $${\tilde V}_{\rm eff}(u)=-{\tilde A}u^2+\frac{1}{2}{\tilde B}u^4-(1+u^2)^2\ln\left(\frac{\tilde \Lambda}{\sqrt{1+u^2}}\right),$$ which can be further simplified as $${\hat V}_{\rm eff}(u)=-Au^2+\frac{1}{2} Bu^4+(1+u^2)^2\ln(1+u^2).$$ The above potential is plotted in Fig. 4. We notice that with large enough cutoff $\Lambda$, one generally has $B={\tilde B}-\ln{\tilde \Lambda}$ large and negative. For $A<1$, $u=0$ is a local minimum. There are also another two local minima with $u^2\equiv y$ a positive solution of equation $$2(1+y)\ln(1+y)+(1+B)y+1-A=0.$$ So we generally have a first-order quantum phase transition in this case (see Fig. 3 for a schematic picture).
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Fluctuations with $d=3, z=2$
----------------------------
With dynamical exponent $z=2$, the propagator of $\phi$ field is $$\chi_0({\mathbf k},\omega_n)=\frac{1}{|\omega_n|\tau_0+k^2+\xi^{-2}}.$$ Examples are charge-density-wave and antiferromagnetic fluctuations. In the presence of dissipation, superconducting transitions also have dynamical exponent $z=2$.
{width="0.38\linewidth"} {width="0.38\linewidth"} {width="0.38\linewidth"} {width="0.38\linewidth"}
So the one-loop correction to the effective potential at zero temperature becomes $$\frac{\delta V_{\rm eff}^{(1)}[\psi]}{\delta\psi}=2g\psi\int \frac{d\omega}{2\pi}\int \frac{d^d{\mathbf k}}{(2\pi)^d}\frac{1}{|\omega|\tau_0+k^2+\xi^{-2}+g\psi^2}.$$ The momentum integral is cutoff at $|\mathbf k|=\Lambda$, and correspondingly the frequency integral is cutoff at $|\omega|\tau_0=\Lambda^2$. First, we integrate out frequency to obtain $$\frac{\delta V_{\rm eff}^{(1)}[\psi]}{\delta\psi}=\frac{g\psi}{\pi^3\tau_0}\int_0^{\Lambda}dkk^2\ln\left(1+\frac{\Lambda^2}{k^2+\xi^{-2}+g\psi^2}\right),$$ and then integrate out momentum, with the final result $$\begin{split}
\frac{\delta V_{\rm eff}^{(1)}[\psi]}{\delta\psi}=\frac{g\psi}{3\pi^3\tau_0}\left[ \Lambda^3\ln\left(\frac{\xi^{-2}+g\psi^2+2\Lambda^2}{\xi^{-2}+g\psi^2+\Lambda^2}\right) +2\Lambda^3+2(\xi^{-2}+g\psi^2)^{3/2} \arctan\frac{\Lambda}{\sqrt{\xi^{-2}+g\psi^2}}\right. \\\left.-2(\xi^{-2}+g\psi^2+\Lambda^2)^{3/2} \arctan\frac{\Lambda}{\sqrt{\xi^{-2}+g\psi^2+\Lambda^2}}\right].
\end{split}$$ Up to order $\Lambda^0$, this is $$\frac{\delta V_{\rm eff}^{(1)}[\psi]}{\delta\psi}=\frac{g\psi}{3\pi^3\tau_0}\left[ \Lambda^3\left(2+\ln 2-\frac{\pi}{2}\right)+\frac{3\pi}{4}\Lambda(\xi^{-2}+g\psi^2) +\pi(\xi^{-2}+g\psi^2)^{3/2} \right].$$ The first two terms just renormalize the bare $\alpha$ and $\beta$. When the $\phi$ field is critical, $\xi\to\infty$, the third term becomes of order $\psi^5$, and is thus irrelevant. When $\xi$ is large but not infinite, we get the effective potential $$V_{\rm eff}(\psi)=-{\bar \alpha}\psi^2+\frac{1}{2}{\bar \beta}\psi^4+\frac{g^{3/2}\xi^{-2}}{15\pi^2\tau_0}|\psi|^3+\frac{g^{5/2}}{15\pi^2\tau_0}|\psi|^5.$$ In addition to the $\psi^5$ term there is another term of order $\psi^3$, which may drive the second-order quantum phase transition to first order.
Let us consider a massive $\phi$ field. Carrying out the same rescaling as we made for $z=1$, we get the rescaled effective potential of the form $${\hat V}_{\rm eff}(u)=-Au^2+\frac{1}{2}Bu^4+(1+u^2)^{5/2}.$$ For large negative $B$, we obtain a first-order quantum phase transition (see Fig. 5(a)).
Fluctuations with $d=3, z=3$
----------------------------
When the $\phi$ field has dynamical exponent $z=3$, e.g. for ferromagnetic fluctuations, its propagator is $$\chi_0({\mathbf k},\omega_n)=\frac{1}{\gamma\frac{|\omega_n|}{k}+k^2+\xi^{-2}}.$$ Thus the one-loop correction to the effective potential at $T=0$ is determined from $$\frac{\delta V_{\rm eff}^{(1)}[\psi]}{\delta\psi}=2g\psi\int \frac{d\omega}{2\pi}\int \frac{d^d{\mathbf k}}{(2\pi)^d}\frac{1}{\gamma\frac{|\omega|}{k}+k^2+\xi^{-2}+g\psi^2},$$ with a momentum cutoff at $|\mathbf k|=\Lambda$, and a frequency cutoff at $\gamma|\omega|=\Lambda^3$. The frequency integral gives $$\frac{\delta V_{\rm eff}^{(1)}[\psi]}{\delta\psi}=\frac{g\psi}{4\pi^4\gamma}\int_0^{\Lambda}dkk^3\ln\left[1+\frac{\Lambda^3}{k^3+k(\xi^{-2}+g\psi^2)}\right],$$ and the momentum integral further leads to the result $$V_{\rm eff}(\psi)=-{\bar \alpha}\psi^2+\frac{1}{2}{\bar \beta}\psi^4+\frac{1}{96\pi^4\gamma}(\xi^{-2}+g\psi^2)^3\ln(\xi^{-2}+g\psi^2).$$ When $\phi$ is critical, $\xi\to\infty$, the third term is of the form $\psi^6\ln\psi$, which is irrelevant. For finite $\xi$, there is also a term of the form $\psi^4\ln\psi$, which will drive the second-order quantum phase transition to first order.
For general $\xi$, the rescaled effective potential reads $${\hat V}_{\rm eff}(u)=-Au^2+\frac{1}{2} Bu^4+(1+u^2)^3\ln(1+u^2),$$ which can lead to a first-order quantum phase transition (see fig. 5(b)).
Fluctuations with $d=3, z=4$
----------------------------
For a dirty metallic ferromagnet, the dynamical exponent is $z=4$. In this case, with the propagator $$\chi_0({\mathbf k},\omega_n)=\frac{1}{\gamma'\frac{|\omega_n|}{k^2}+k^2+\xi^{-2}},$$ the rescaled effective potential reads $${\hat V}_{\rm eff}(u)=-Au^2+\frac{1}{2} Bu^4-(1+u^2)^{7/2}.$$ Higher order terms need to be included at large $u$ to maintain stability. When the $\phi$ field is critical, the third term is of order $\phi^7$, which is irrelevant. When the $\phi$ field is massive but light, there will also be a term of order $\phi^5$ which is again irrelevant. For general $\phi$, in order for $u=0$ to be a local minimum, we need to have $A<-7/2$. In this case, ${\hat V}'_{\rm eff}(u)=0$ has only one positive solution. Thus we have a second-order quantum phase transition.
Fluctuations in $d=2$ and $d=1$
-------------------------------
We can calculate the fluctuation-induced effective potential in other dimensions in the same way as above. For $d=2, z=1$, and also for $d=1, z=2$, with the rescaled field defined by $u^2\equiv \frac{g\psi^2}{\xi^{-2}}$, the rescaled effective potential is of the form $${\hat V}_{\rm eff}(u)=-Au^2+\frac{1}{2}Bu^4-(1+u^2)^{3/2}.$$ When the $\phi$ field is critical, the third term becomes of order $-|\psi|^3$, of the Halperin-Lubensky-Ma type, thus the quantum phase transition is first-order. Generally when $A<-1.5, AB>-0.5, B(A+B)>-0.25$, $u=0$ will be a local minimum of the rescaled effective potential ${\hat V}_{\rm eff}$, and there are two other local minima at nonzero $u$. Hence there is again a first-order quantum phase transition (see Fig. 5(c)). Otherwise there will be a second-order phase transition.
The effective potential in the case with $d=2, z=2,$ and $d=1, z=3$ turns out be of the same form as that of $d=3, z=1$, as expected from the fact that both cases have the same effective dimension $d+z=4$. The case $d=2, z=3$ is the same as $d=3, z=2$.
For $d=1, z=1$, the effective potential takes the form $${\hat V}_{\rm eff}(u)=-Au^2+\frac{1}{2}Bu^4-(1+u^2)\ln(1+u^2),$$ which leads to a first-order phase transition for $B<1$ (see Fig. 5(d)). The third term reduces to $\psi^2\ln\psi$ when $\phi$ is critical. In this case the quantum phase transition is always first order for any positive value of $B$.
Summary of the classical + quantum cases
----------------------------------------
In the table below, we list the most dangerous terms generated from integrating out the fluctuating fields. The second row in the table corresponds to the case where $\phi$ is critical or massless, and the third row has $\phi$ massive.
$d+z$ 2 3 4 5 6 7
---------- --------------------- --------------- ----------------- ---------- ----------------- ----------
massless $\psi^2\ln\psi$ $\psi^3$ $\psi^4\ln\psi$ $\psi^5$ $\psi^6\ln\psi$ $\psi^7$
massive $(\psi^2+1)\ln\psi$ $\psi^3+\psi$ $\psi^2\ln\psi$ $\psi^3$ $\psi^4\ln\psi$ $\psi^5$
One can clearly see that in the massless case, the fluctuations are irrelevant when $d+z\geqslant 5$, while in the massive case, they are irrelevant for $d+z\geqslant 7$. Otherwise the second-order quantum phase transition can be driven to first order. The order of the correction is readily understood from the general structure of the integrals. With effective dimension $d+z$, in the massless case one has $\delta V/\delta\psi\sim \psi\int d^{d+z}k(1/k^2)$. Since $k^2\sim \psi^2$, this gives the correct power $\delta V\sim \psi^{d+z}$. Replacing $g\psi^2$ by $g\psi^2+\xi^{-2}$ and then carrying out the expansion in $\xi^{-2}/g\psi^2$, one gets for the massive case a reduction by $2$ in the power. We also notice the even/odd effect in the effective potential: for $d+z$ even, there are logarithmic corrections. The case $d+z=4$ can be easily understood, as the system is in the upper critical dimension, and logarithmic corrections are expected. We still do not have a simple intuitive understanding of the logarithm for $d+z=2, 6$.
Two fluctuating fields
======================
We consider in this section the case where the two coupled quantum fields are both fluctuating substantially. The partition function now becomes $$Z=\int {\cal D}\psi({\mathbf r},\tau)\int {\cal D}\phi({\mathbf r},\tau) \exp\left({-S_{\psi}-S_{\phi}-S_{\psi\phi}}\right).$$ We will use RG equations to determine the phase diagram of this system. When there is no stable fixed point, or the initial parameters lie outside the basin of attraction of the stable fixed points, the flow trajectories will show runaway behavior, which implies a first-order phase transition [@Fisher77; @Lubensky78; @Rudnick78; @Amit81; @Cardy96]. The spin-density-wave transitions in some cuprates and pnictides fall in this category [@Tranquada95; @Ando02; @Tailefer07; @Kivelson98; @Fradkin09; @Dai08; @Xu08; @Fang08; @Huang08; @Krellner08; @Yan08; @WangDai08; @McQueeney08; @Vicari06].
![Illustration of the fluctuation-induced first-order phase transition in the case of two quantum fields. Here for concreteness we consider the antiferromagnetic order and superconducting order. At low temperatures, the phase transitions may become first order (the thick vertical lines), due to fluctuations.[]{data-label="fig3"}](qq1.eps "fig:"){width="6cm"} ![Illustration of the fluctuation-induced first-order phase transition in the case of two quantum fields. Here for concreteness we consider the antiferromagnetic order and superconducting order. At low temperatures, the phase transitions may become first order (the thick vertical lines), due to fluctuations.[]{data-label="fig3"}](qq2.eps "fig:"){width="6cm"}
We have considered in the previous sections coupling two single component fields, having in mind that this simplified model captures the main physics of competing orders. However, we will see below that when the quantum fluctuations of both fields are taken into account, the number of components of the order parameters do play important roles. So from now on we consider explicitly a $n_1$-component vector field $\boldsymbol \psi$ and a $n_2$-component vector field $\boldsymbol \phi$ coupled together. When both fields have dynamical exponent $z=1$, the action reads $$\begin{split}
S_{\psi}=&\int d^d{\mathbf r}d\tau\left[-\alpha_1 |{\boldsymbol \psi}|^2+\frac{1}{2}\beta_1 |{\boldsymbol \psi}|^4+\frac{1}{2}|\partial_\mu \boldsymbol \psi|^2\right],
\\S_{\phi}=&\int d^d{\mathbf r}d\tau\left[-\alpha_2 |\boldsymbol \phi|^2+\frac{1}{2}\beta_2 |\boldsymbol \phi|^4+\frac{1}{2}|\partial_\mu \boldsymbol \phi|^2\right],
\\S_{\psi\phi}=&g \int d^d{\mathbf r}d\tau |\boldsymbol\psi|^2 |\boldsymbol \phi|^2,
\end{split}
\label{action1}$$ where $\mu=0, 1, \cdots, d$. This quantum mechanical problem is equivalent to a classical problem in one higher dimension. Then one can follow the standard procedure of RG: first decompose the action into the fast-moving part, the slow-moving part and the coupling between them. The Green’s functions are $G_{\psi}=1/(-2\alpha_1+k^2+\omega^2)$ and $G_{\phi}=1/(-2\alpha_2+k^2+\omega^2)$. The relevant vertices are $\beta_1\psi_s^2\psi_f^2, \beta_2\phi_s^2\phi_f^2, g\psi_s^2\phi_f^2, g\psi_f^2\phi_s^2, g\psi_s\psi_f\phi_s\phi_f$. To simplify the notation we rescale the momentum and frequency according to ${\mathbf k}\to {\mathbf k}/\Lambda, \omega\to\omega/\Lambda$, so that they lie in the interval $[0, 1]$. The control parameters and couplings are rescaled according to $\alpha_{1,2}\to\alpha_{1,2}\Lambda^2$, $\beta_{1,2}\to\beta_{1,2}\Lambda^{3-d}, g\to g\Lambda^{3-d}$. Afterwards we integrate out the fast modes with the rescaled momentum and frequency in the range $[b^{-1},1]$. Finally, we rescale the momentum and frequency back to the interval $[0, 1]$, thus ${\mathbf k}\to b{\mathbf k}, \omega\to b\omega$, and the fields are rescaled accordingly with $\psi\to b^{(d-1)/2}\psi, \phi\to b^{(d-1)/2}\phi$. Using an $\epsilon$-expansion, where $\epsilon=3-d$, one obtains the set of RG equations to one-loop order, $$\begin{split}
\frac{d\alpha_i}{dl}=&2\alpha_i-\frac{1}{8\pi^2}[(n_i+2)\beta_i(1+2\alpha_i)+n_jg(1+2\alpha_j)],
\\ \frac{d\beta_i}{dl}=&\epsilon\beta_i-\frac{1}{4\pi^2}[(n_i+8)\beta_i^2+n_jg^2],
\\ \frac{dg}{dl}=&g\left(\epsilon-\frac{1}{4\pi^2}\left[(n_1+2)\beta_1+(n_2+2)\beta_2+4g\right]\right),
\end{split}
\label{rg1}$$ with index $i, j=1, 2$, and $i\neq j$. These equations are actually more general than considered above. They also apply to generic models where two fields with the same dynamical exponent $z$ are coupled together. Generally one has $\epsilon=4-d-z$, thus a quantum mechanical model with dynamical exponent $z$ is equivalent to a classical model in dimension $d+z$.
![(Color online) Plot of the RG trajectories in the $\beta_1-\beta_2$ plane for two quantum fields with the same dynamical exponent below the upper critical dimension. Here we have chosen $\epsilon=4-d-z=0.1$. The RG trajectories have been projected onto a constant $g$ plane with $g=g^*$, and $g^*$ the value of the coupling strength at the stable fixed point. (a) corresponds to the case $n_1=n_2=1$, where the fixed point is at $\beta_1^*=\beta_2^*=g^*= 4\pi^2\epsilon/(n_1+n_2+8)\simeq 0.3948$. (b) corresponds to the case $n_1=2, n_2=3$, where the fixed point is at $(\beta_1^*, \beta_2^*, g^*)=4\pi^2\epsilon(0.0905, 0.0847, 0.0536)\simeq (0.3573, 0.3344, 0.2116)$. In both cases we found that, above some curve (the dashed lines), the RG trajectories flow to the corresponding stable fixed point, while below this curve, the RG trajectories show runaway behavior.[]{data-label="fig9"}](FL11.eps "fig:"){width="7cm"} ![(Color online) Plot of the RG trajectories in the $\beta_1-\beta_2$ plane for two quantum fields with the same dynamical exponent below the upper critical dimension. Here we have chosen $\epsilon=4-d-z=0.1$. The RG trajectories have been projected onto a constant $g$ plane with $g=g^*$, and $g^*$ the value of the coupling strength at the stable fixed point. (a) corresponds to the case $n_1=n_2=1$, where the fixed point is at $\beta_1^*=\beta_2^*=g^*= 4\pi^2\epsilon/(n_1+n_2+8)\simeq 0.3948$. (b) corresponds to the case $n_1=2, n_2=3$, where the fixed point is at $(\beta_1^*, \beta_2^*, g^*)=4\pi^2\epsilon(0.0905, 0.0847, 0.0536)\simeq (0.3573, 0.3344, 0.2116)$. In both cases we found that, above some curve (the dashed lines), the RG trajectories flow to the corresponding stable fixed point, while below this curve, the RG trajectories show runaway behavior.[]{data-label="fig9"}](FL23.eps "fig:"){width="7cm"}
It is known that the above equations have six fixed points [@Fisher76], four of which have the two fields decoupled, i.e., $g^*=0$. They are the Gaussian-Gaussian point at $(\beta_1^*, \beta_2^*)=(0, 0)$, the Heisenberg-Gaussian point at $(\beta_1^*, \beta_2^*)=(4\pi^2\epsilon/(n_1+8), 0)$, the Gaussian-Heisenberg point at $(\beta_1^*, \beta_2^*)=(0, 4\pi^2\epsilon/(n_2+8))$, and the decoupled Heisenberg-Heisenberg point at $(\beta_1^*, \beta_2^*)=(4\pi^2\epsilon/(n_1+8), 4\pi^2\epsilon/(n_2+8))$. The isotropic Heisenberg fixed point is at $\beta_1^*=\beta_2^*=g^*= 4\pi^2\epsilon/(n_1+n_2+8), \alpha_1^*=\alpha_2^*=\epsilon(n_1+n_2+2)/4(n_1+n_2+8)$. Finally there is the biconical fixed point with generally unequal values of $\beta_1^*$, $\beta_2^*$, and $g^*$. In the case, with $n_1=n_2=1$, this is at $(\beta_1^*, \beta_2^*, g^*)=2\pi^2\epsilon/9(1,1,3)$. For $n_1=2, n_2=3$, one has $(\beta_1^*, \beta_2^*, g^*)=4\pi^2\epsilon(0.0905, 0.0847, 0.0536)$.
We find that there is always just one stable fixed point for $d+z<4$, below the upper critical dimension [@Fisher76]. The isotropic Heisenberg fixed point is stable when $n_1+n_2<n_c=4-2\epsilon+O(\epsilon^2)$, the biconical fixed point is stable when $n_c<n_1+n_2<16-n_1n_2/2+O(\epsilon)$, and when $n_1n_2+2(n_1+n_2)>32+O(\epsilon)$, the decoupled Heisenberg-Heisenberg point is the stable one. When the initial parameters are not in the basin of attraction of the stable fixed point, one obtains runaway flow, strongly suggestive of a first-order phase transition. Consider for example $n_1=2, n_2=3$, where the biconical fixed point is stable. For two critical points not too separated, that is, $|\alpha_1-\alpha_2|$ not too large, when $g>\sqrt{\beta_1\beta_2}$ the RG flow shows runaway behavior, and one gets a first-order quantum phase transition. The corresponding classical problem has been discussed in [@Nagaosa00]. We notice the difference from the case with two competing classical fields, where one also obtains the same condition for the couplings $\gamma>\sqrt{\beta_1\beta_2}$ in order to have a first-order phase transition. There, the two ordered phases are required to overlap in the absence of the coupling, in other words, one needs to have $x_1<x_2$. However, in the quantum mechanical case we are considering here, this is not necessary. We plot in Fig. 7 the RG trajectories for two cases (a) $n_1=n_2=1$ and (b) $n_1=2, n_3=3$, where in both cases, below some curve, runaway behavior in the RG trajectories is found.
![(Color online) Plot of the RG trajectories in the $\beta_1-\beta_2$ plane for two quantum fields with the same dynamical exponent in and above the upper critical dimension. The RG trajectories have been projected to a constant $g$ plane. And we have chosen $n_1=n_2=3$. (a) corresponds to the case exactly at the critical dimension with $\epsilon=4-d-z=0$. In this case there is only one fixed point with $\beta_1^*=\beta_2^*=g^*=0$, the Gaussian fixed point, which is unstable. We found runaway flows everywhere. (b) corresponds to the case above the critical dimension, where the Gaussian fixed point is the stable one. Here we have chosen $\epsilon=4-d-z=-0.1$. We found, below some curve (the dashed line), that the RG trajectories show runaway behavior.[]{data-label="fig10"}](mar.eps "fig:"){width="7cm"} ![(Color online) Plot of the RG trajectories in the $\beta_1-\beta_2$ plane for two quantum fields with the same dynamical exponent in and above the upper critical dimension. The RG trajectories have been projected to a constant $g$ plane. And we have chosen $n_1=n_2=3$. (a) corresponds to the case exactly at the critical dimension with $\epsilon=4-d-z=0$. In this case there is only one fixed point with $\beta_1^*=\beta_2^*=g^*=0$, the Gaussian fixed point, which is unstable. We found runaway flows everywhere. (b) corresponds to the case above the critical dimension, where the Gaussian fixed point is the stable one. Here we have chosen $\epsilon=4-d-z=-0.1$. We found, below some curve (the dashed line), that the RG trajectories show runaway behavior.[]{data-label="fig10"}](irr.eps "fig:"){width="7cm"}
When $d+z=4$, all the other fixed points coalesce with the Gaussian point, forming an unstable fixed point, thus leading to a first-order phase transition (see Fig. 8(a) ). A similar model with an extra coupling and $n_1=n_2=3$ has been discussed by Qi And Xu [@Qi09], where runaway flows were also identified. Another similar problem with $d=2, z=2$ was studied by Millis recently [@Millis10], where a fluctuation-induced first-order quantum phase transition was shown to occur. We also notice that in some situations, including fluctuations of the order parameter itself may drive the supposed-to-be first-order transitions to second order for both classical and quantum phase transitions [@Parisi81; @Belitz02; @Jakubczyk09; @Metzner0902].
For $d+z>4$, the stabilities are interchanged. The Gaussian fixed point becomes the most stable one. So the basin of attraction of the stable fixed point changes. We found numerically that for a given coupling strength $g$, in the $\beta_1-\beta_2$ plane, the RG trajectories show runaway behavior when the initial points lie below some curve (see Fig. 8(b)). That is, when the coupling between the two fields is strong enough, the QPTs become first order. Just above these curves, we found that the RG trajectories will enter the domain with negative $\beta_1$ or negative $\beta_2$, and then converge to the Gaussian fixed point. For $\beta_1, \beta_2$ large enough, the RG trajectories just converge to the Gaussian fixed point without entering the negative domain.
Competing orders with different dynamical exponents
---------------------------------------------------
We consider next coupling a $z=1$ field to another field with dynamical exponent $z=z_1\geqslant 2$. To our knowledge, such models of two competing order parameters with different dynamical exponents have not been studied previously. The action now takes the form $$\begin{split}
S_{\psi}=&\int d^d{\mathbf k}d\omega \left( -\alpha_1+\frac{k^2}{2}+\frac{\gamma_1}{2}\frac{|\omega|}{k^{z_1-2}} \right)|{\boldsymbol \psi}|^2+\int d^d{\mathbf r}d\tau\frac{1}{2}\beta_1 |{\boldsymbol \psi}|^4,
\\S_{\phi}=&\int d^d{\mathbf r}d\tau\left[-\alpha_2 |\boldsymbol \phi|^2+\frac{1}{2}\beta_2 |\boldsymbol \phi|^4+\frac{1}{2}|\partial_\mu \boldsymbol \phi|^2\right],
\\S_{\psi\phi}=&g \int d^d{\mathbf r}d\tau |\boldsymbol\psi|^2 |\boldsymbol \phi|^2.
\end{split}
\label{action2}$$ The new parameter $\gamma_1$ has dimension $[\gamma_1]=L^{1-z}$, and its one-loop RG equation is simply $$\frac{d\gamma_1}{dl}=(z-1)\gamma_1.
\label{rgg}$$ The Green’s function for the $\psi$ field becomes $G_{\psi}=1/(-2\alpha_1+k^2+\gamma_1|\omega|/k^{z_1-2})$. The RG equations for the other parameters are modified accordingly, $$\begin{split}
\frac{d\alpha_1}{dl}=&2\alpha_1-\frac{\Omega_{d}}{\pi\gamma_1}(n_1+2)\beta_1(\ln 2+2\alpha_1)-\Omega_{d+1}n_2g(2+2\alpha_2),
\\ \frac{d\alpha_2}{dl}=&2\alpha_2-\Omega_{d+1}(n_2+2)\beta_2(2+2\alpha_2)-\frac{\Omega_{d}}{\pi\gamma_1}n_1g(\ln 2+2\alpha_1),
\\ \frac{d\beta_1}{dl}=&\epsilon\beta_1-\frac{2\Omega_{d}}{\pi\gamma_1}(n_1+8)\beta_1^2-2\Omega_{d+1}n_2g^2,
\\ \frac{d\beta_2}{dl}=&\epsilon\beta_2-2\Omega_{d+1}(n_2+8)\beta_2^2-\frac{2\Omega_{d}}{\pi\gamma_1}n_1g^2,
\\ \frac{dg}{dl}=&g\left(\epsilon-\frac{2\Omega_{d}}{\pi\gamma_1}(n_1+2)\beta_1-2\Omega_{d+1}(n_2+2)\beta_2-8\frac{\Omega_d}{2\pi}\frac{ 2 \gamma_1\ln{\gamma_1} +\pi }{1+\gamma_1^2}
g\right),
\end{split}
\label{rgab}$$ where $\epsilon=3-d$ and $\Omega_{d}=2\pi^{d/2}/(2\pi)^{d}\Gamma[d/2]$ is the volume of the $d$-dimensional unit sphere. The derivation of the above RG equations is included in the appendix. We notice from the above procedure that when the two fields have the same dynamical exponent $z>1$, one can rescale the couplings to ${\tilde\beta}_1=\beta_1/\gamma, {\tilde\beta}_2=\beta_2/\gamma, {\tilde g}=g/\gamma$, and these new parameters satisfy the RG equations (\[rg1\]) with ${\tilde\epsilon}=4-d-z$.
The presence of two different dynamical exponents obviously complicates the problem. It is generally expected that the modes with a larger dynamical exponent dominates the specific heat of the system, since they have a large phase space, while the modes with a smaller dynamical exponent may produce infrared singularities, since they have a smaller upper critical dimension [@Peter09]. In the absence of the coupling between the two fields, we have the RG equations $$\begin{split}
\\ \frac{d\tilde{\beta}_1}{dl}=&(4-d-z)\tilde{\beta}_1-\frac{2\Omega_{d}}{\pi}(n_1+8)\tilde{\beta}_1^2,
\\ \frac{d\beta_2}{dl}=&(3-d)\beta_2-2\Omega_{d+1}(n_2+8)\beta_2^2.
\end{split}
\label{rg3}$$ For $d=3$, $\beta_2$ is marginal with an unstable fixed point, while $\tilde{\beta}_1$ is irrelevant and its Gaussian fixed point is stable.
![(Color online) Plot of the RG trajectories in the $\beta_1-\beta_2$ plane for two coupled quantum fields with different dynamical exponents. The RG trajectories have been projected to a constant $g$ plane with $g=1$. We have chosen the spatial dimension to be $d=3$, the dynamical exponents $z_1=2, z_2=1$ and the number of field components $n_1=n_2=3$. (a) shows the RG trajectories originating from the region below the dashed line, which flow to negative $\beta_1$ or negative $\beta_2$ regions. (b) shows the RG trajectories originating from the region above the dashed line, and those flow to the stable points on the positive axes of $\beta_1$, the location of which is sensitive to the initial value of the parameters..[]{data-label="fig11"}](rgd12.eps "fig:"){width="7cm"} ![(Color online) Plot of the RG trajectories in the $\beta_1-\beta_2$ plane for two coupled quantum fields with different dynamical exponents. The RG trajectories have been projected to a constant $g$ plane with $g=1$. We have chosen the spatial dimension to be $d=3$, the dynamical exponents $z_1=2, z_2=1$ and the number of field components $n_1=n_2=3$. (a) shows the RG trajectories originating from the region below the dashed line, which flow to negative $\beta_1$ or negative $\beta_2$ regions. (b) shows the RG trajectories originating from the region above the dashed line, and those flow to the stable points on the positive axes of $\beta_1$, the location of which is sensitive to the initial value of the parameters..[]{data-label="fig11"}](rgu12.eps "fig:"){width="7cm"}
Generally, for $z>1$, if the initial value of $\gamma_1$ is nonzero, the absolute value of $\gamma_1$ will increase exponentially. The RG equation for $\beta_2$ becomes independent of other parameters, $$\frac{d\beta_2}{dl}=\epsilon\beta_2-2\Omega_{d+1}(n_2+8)\beta_2^2.$$ We are interested in the case $\epsilon=0$, for which $\beta_2$ is readily solved to be $$\beta_2(l)=\frac{1}{\bar{\beta}_2^{-1}+ 2\Omega_{4}(n_2+8)(l-l_{\rm cr})},$$ with $\bar{\beta}_2$ taken at the crossover scale $l_{\rm cr}$ at which the $\beta_2^2$ term begins to dominate the $g^2$ term. Only the sign of $\bar{\beta}_2$ matters. If $\bar{\beta}_2>0$, as $l$ increases, $\beta_2$ will decay to zero, flowing to its Gaussian fixed point. From the simplified RG equations for $g$, $$\\ \frac{dg}{dl}=-2\Omega_4(n_2+2)g\beta_2,
\label{rg22}$$ one can see that with a lower power in $\beta_2$, $g$ drops to zero even more quickly than $\beta_2$. Taking $\beta_2$ as quasi-static when considering the evolution of $g$, one notices that $g$ decays exponentially as $g(l)\sim \exp(-2\Omega_4(n_2+2)\beta_2l)$. So $d\beta_1/dl$ also decays exponentially, and before $\beta_2$ goes to zero, $\beta_1$ already stabilizes to a finite value $\beta_1^*$, which depends on the initial value of $\beta_1$. Actually from the simplified RG equations for $\beta_1, \beta_2, g$ with $1/\gamma_1$ set to zero, one can see directly that the fixed points are at $\beta_2^*=g^*=0$, with $\beta_1^*$ any real number: we have a line of fixed points. When $\beta_1^*>0$, there will be a second-order phase transition. When $\beta_1^*<0$, the transition becomes first order (see Fig. 9).
If $\bar{\beta}_2<0$, the absolute value of $\beta_2$ will increase without bound. Subsequently $g$ and $\beta_1$ also diverge, leading to runaway flows.
Conclusion
==========
Quantum criticality in the presence of competing interactions is an important guiding concept that allows us to organize a framework for emergent states near QCPs. Here we investigated the stability of a quantum critical point in the presence of competing orders. We focused on a simple quadratic-quadratic interaction, where coupling between two competing phases is assumed to be of $g \psi^2 \phi^2$ form. We find that QCPs are often unstable and transform into first order lines of transitions. The detailed scenario on how the instability develops depends on the precise nature of the competing interactions, dynamical exponents and strength of the coupling. The general trend we observe is that competing interactions, be they classical or quantum, often lead to the instability of QCPs. This instability in fact always occurs, in the cases we have investigated, if the coupling $g$ is strong enough. We thus conclude that breakdown of QCPs is a ubiquitous phenomenon. The magnitude of the specific heat jump in some first order transitions (the *classical* + *classical* case) is of the same order as the specific heat released in a second order transition and these first order transitions are strong, and not weakly first order as found in Halperin-Lubensky-Ma. An immediate consequence of this breakdown is that we can expect spatially modulated inhomogeneous phases to be present near QCPs, given their propensity to turn into first order transitions. The wide likelihood identified here of first order transitions preempting a QCP leads us to anticipate the nucleation and metastability phenomena associated with such transitions [@Hohenberg95]. Additionally, proximity to first-order transitions makes auxiliary fields (e. g. magnetic field, strain) and disorder very important over substantial parameter regions [@Alan93].
The broad similarities we pointed out between QCPs and AdS/CFT models offers an interesting possibility that in fact AdS models are also spatially inhomogeneous. More detailed analysis that allows breakdown of scaling, specific for AdS/CFT is suggested.
We derived the renormalization group equations for two coupled order parameters with different dynamical exponents. We found that there are a line of fixed points, which is quite different from the case where two order parameters have the same dynamical exponent. Very recently, there have appeared some interesting reports [@Peter09; @Chubukov10] investigating the effects of the presence of two order parameters with different dynamical exponents near the Pomeranchuk instability [@Pomeranchuk58], as examples of multiscale quantum criticality. It would be interesting to see how the presence of two different dynamical exponents, and the coupling between the corresponding order parameters, affect the scaling of resistivity, especially whether a linear-resistivity is possible, overcoming the “no-go” theorem for single parameter scaling [@Phillips05].
In this paper, we have confined ourselves to the framework of Hertz-Millis-Moriya [@Hertz76; @Millis93; @Moriya85], considering only the interplay of bosonic order parameters. It would also be interesting to study the electronic instabilities, to see whether the superconducting instabilities and Pomeranchuk instabilities are enhanced in fermionic quantum critical states. Fermi liquids, even with repulsive interactions, are unstable towards forming a superconducting state, due to the Kohn-Luttinger effect [@Kohn65] resulting from the presence of a sharp Fermi surface. For the fermionic quantum critical states, the momentum distribution function may have only higher order singularities [@Senthil08]. It would be interesting to check whether the Kohn-Luttinger effect is still active in this case.
Acknowledgments
===============
We are grateful to G. Aeppli, E. D. Bauer, Y. Dubi, P. Littlewood, F. Ronning, T. Rosenbaum and P. W$\rm\ddot{o}$lfle for stimulating discussions over the years about stability of QCPs. J. S. and J. Z. thank K. Schalm and V. Juricic for helpful discussions on AdS/CFT correspondence and the RG results. This work was supported by US DoE, BES and LDRD. J. S. and J. Z. are supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) via a Spinoza grant.
Appendix: Derivation of the RG equations for two fields with different dynamical exponents
==========================================================================================
In this appendix, we will derive the RG equations of two competing orders with different dynamical exponents. We follow the notation of [@Altland]. Our starting point is the action (\[action2\]). First we count the dimensions of the field operators and all the parameters: $$\begin{split}
\\ [r]=[\tau]=&L,
\\ [k]=[\omega]=&L^{-1},
\\ [\psi]=[\phi]=&L^{(1-d)/2},
\\ [\alpha_1]=[\alpha_2]=&L^{-2},
\\ [\beta_1]=[\beta_2]=&L^{d-3},
\\ [g]=&L^{d-3},
\\ [\gamma_1]=&L^{1-z}.
\end{split}
\label{dim1}$$ Then we decompose the action into slow and fast modes. The action for the slow modes reads $$\begin{split}
\\ S^{(s)}=&S^{(s)}_{\psi}+S^{(s)}_{\phi}+S^{(s)}_{\psi\phi},
\\S^{(s)}_{\psi}=&\int d^d{\mathbf k}d\omega \left( -\alpha_1+\frac{k^2}{2}+\frac{\gamma_1}{2}\frac{|\omega|}{k^{z_1-2}} \right)|{\boldsymbol \psi}_s|^2+\int d^d{\mathbf r}d\tau\frac{1}{2}\beta_1 |{\boldsymbol \psi}_s|^4,
\\S^{(s)}_{\phi}=&\int d^d{\mathbf r}d\tau\left[-\alpha_2 |\boldsymbol \phi_s|^2+\frac{1}{2}\beta_2 |\boldsymbol \phi_s|^4+\frac{1}{2}|\partial_\mu {\boldsymbol \phi}_s|^2\right],
\\S^{(s)}_{\psi\phi}=&g \int d^d{\mathbf r}d\tau |\boldsymbol\psi_s|^2 |\boldsymbol \phi_s|^2.
\end{split}
\label{actions1}$$ Since we will only consider RG to one-loop order, the interaction terms in the fast modes, the contraction of which leads to second-order diagrams, can be ignored. Thus we obtain the action for the fast modes, $$\begin{split}
\\ S^{(f)}=&S^{(f)}_{\psi}+S^{(f)}_{\phi},
\\S^{(f)}_{\psi}=&\int d^d{\mathbf k}d\omega \left( -\alpha_1+\frac{k^2}{2}+\frac{\gamma_1}{2}\frac{|\omega|}{k^{z_1-2}} \right)|{\boldsymbol \psi}_f|^2,
\\S^{(f)}_{\phi}=&\int d^d{\mathbf r}d\tau\left[-\alpha_2 |\boldsymbol \phi_f|^2+\frac{1}{2}|\partial_\mu {\boldsymbol \phi}_f|^2\right],
\end{split}
\label{actions2}$$ from which one can easily identify the Green’s functions as $$\begin{split}
\\ G^{f}_{ij}[\psi]=&\frac{\delta_{ij}}{-2\alpha_1+k^2+\gamma_1\frac{|\omega|}{k^{z_1-2}}},
\\ G^{f}_{ij}[\phi]=&\frac{\delta_{ij}}{-2\alpha_2+k^2+\omega^2}.
\end{split}
\label{green2}$$ The coupling between the slow modes and fast modes takes the form $$S_{c}=\int d^d{\mathbf r}d\tau\left[ \sum_{ijkl}F_{ijkl}\left(3\beta_1 \psi_f^i \psi_f^j \psi_s^k \psi_s^l+3\beta_2 \phi_f^i \phi_f^j \phi_s^k \phi_s^l \right) +g |\boldsymbol \psi_s|^2 |\boldsymbol \phi_f|^2 +g |\boldsymbol \psi_f|^2 |\boldsymbol \phi_s|^2 +4g (\boldsymbol \psi_s\cdot\boldsymbol \psi_f ) (\boldsymbol\phi_s\cdot\boldsymbol \phi_f) \right],
\label{actioncoup}$$ with the tensor $F_{ijkl}=\frac{1}{3}(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$.
Now we can integrate out the fast modes and see how the different parameters change accordingly. The effective action of the slow modes is determined by $$\exp\left[-{S^{(s)}_{\rm eff}}\right]=\exp\left[-S^{(s)}\right]\exp\left[ -\langle S_c\rangle_f +\frac{1}{2}\langle S_c^2\rangle_f^{\rm con} \right].$$ In the $S_c^2$ term we take a connected average, thus the superscript “$\rm con$”. The coefficients in the RG equations will depend on the different renormalization schemes. Here we will use the procedure that is most convenient for the problem at hand, similar in spirit to what was outlined in [@Herbut07]. We integrate over the momentum interval $\Lambda/b<k<\Lambda$, which after rescaling $k\to k/\Lambda$, gives $b^{-1}<k<1$. The frequency part is more complicated. We will introduce a cutoff when it is necessary, otherwise just integrate over the whole range $-\infty<\omega<\infty$. The main reason for us to choose this RG scheme is that in calculating the third correction to the coupling $g$, the two internal lines come from order parameters with different dynamical exponents, thus the two frequencies scale differently with momentum, and this RG scheme offers a simple and self-consistent treatment of the cutoffs.
$\gamma_1$ does not receive corrections up to first-order.
First Order Corrections to $\alpha_1$
-------------------------------------
![One-loop diagrams contributing to the first order correction of $\alpha_1$. The solid lines represent the $\psi$ fields, and the dashed lines represent the $\phi$ fields. The external lines are slow modes, and the internal lines are fast modes.[]{data-label="aaaa1"}](a11.eps "fig:"){width="4cm"} ![One-loop diagrams contributing to the first order correction of $\alpha_1$. The solid lines represent the $\psi$ fields, and the dashed lines represent the $\phi$ fields. The external lines are slow modes, and the internal lines are fast modes.[]{data-label="aaaa1"}](a12.eps "fig:"){width="4cm"}
Two terms in the action (\[actioncoup\]) contribute to the first-order corrections of $\alpha_1$. The coupling $ \psi_s^2 \psi_f^2$ leads to the correction $$\delta^{(1)}S[\alpha_1]=3\beta_1\sum_{ijkl}F_{ijkl}\int_f \frac{d^{d+1}\mathbf q'}{(2\pi)^{d+1}} \langle\psi_f^i(\mathbf q') \psi_f^j(-\mathbf q') \rangle \int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}} \psi_s^k(\mathbf q) \psi_s^l(-\mathbf q).$$ Using the identity, $$\sum_{i}F_{iikl}=\frac{n_1+2}{3}\delta_{kl},$$ one obtains $$\delta^{(1)}S[\alpha_1]=(n_1+2)\beta_1\frac{\Omega_d}{2\pi} \int d\omega \int_{b^{-1}}^1dkk^{d-1} \frac{1}{-2\alpha_1+k^2+\gamma_1\frac{|\omega|}{k^{z_1-2}}} \int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}}\boldsymbol \psi_s(\mathbf q)\cdot \boldsymbol \psi_s(-\mathbf q).$$ Assuming that the $\psi$ field is near its critical point, thus $\alpha_1$ is a small parameter, the Green’s function can be expanded in terms of $-2\alpha_1$. The correction term can be written as $$\delta^{(1)}S[\alpha_1]=(n_1+2)\beta_1\frac{\Omega_d}{2\pi} (I_1+2\alpha_1I_2) \int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}}\boldsymbol \psi_s(\mathbf q)\cdot \boldsymbol \psi_s(-\mathbf q),$$ where we have defined the series of functions $$I_n= \int d\omega \int_{b^{-1}}^1dkk^{d-1} \frac{1}{\left(k^2+\gamma_1\frac{|\omega|}{k^{z_1-2}}\right)^n}.$$ Let us first calculate $I_1$. The frequency integral requires a cutoff. From dimensional analysis, we choose to integrate over the region $-1<\gamma_1\omega<1$, and obtain the result $$I_1= \frac{2}{\gamma_1} \int_{b^{-1}}^1dkk^{d+z_1-3} \ln \left( \frac{1+k^{z_1}}{k^{z_1}}\right).
\label{caliii}$$ To proceed further, we are required to specify the dimension and dynamical exponent. Consider $d=3, z_1=2$, where one has $$I_1=\frac{2}{3\gamma_1}\left[ \ln2-b^{-3}\ln \left( \frac{1+b^{-2}}{b^{-2}}\right)+2(1-b^{-1})-2\arctan 1+ 2\arctan b^{-1} \right].$$ Expanded to first order in $(1-b^{-1})$, it is simply $$I_1=\frac{2\ln 2}{\gamma_1}(1-b^{-1}).
\label{iii}$$ For $d=2, z_1=2$, we obtain $$I_1=\frac{1}{\gamma_1}\left[ 2\ln2-(1+b^{-2})\ln (1+b^{-2})+b^{-2}\ln b^{-2} \right],$$ which leads to the same result (\[iii\]) when expanded to first order in $(1-b^{-1})$. This result can also be obtained more crudely by setting $k=1$ in the integrand of (\[caliii\]). $I_2$ can be calculated similarly, with the result $$I_2=\frac{2}{\gamma_1}(1-b^{-1}).$$ So the one-loop correction to $\alpha_1$ coming from the coupling $\psi_s^2 \psi_f^2$ is $$\delta^{(1)}S[\alpha_1]=(n_1+2)\beta_1\frac{\Omega_d}{\pi\gamma_1} (1-b^{-1})(\ln2+2\alpha_1) \int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}}\boldsymbol \psi_s(\mathbf q)\cdot \boldsymbol \psi_s(-\mathbf q),$$
We next calculate contributions from the coupling $ \psi_s^2 \phi_f^2$, which takes the form $$\delta^{(2)}S[\alpha_1]=g\sum_{ijkl}F'_{ijkl}\int_f \frac{d^{d+1}\mathbf q'}{(2\pi)^{d+1}} \langle\phi_f^i(\mathbf q') \phi_f^j(-\mathbf q') \rangle \int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}} \psi_s^k(\mathbf q) \psi_s^l(-\mathbf q),$$ with $F'_{ijkl}=\delta_{ij}\delta_{kl}$. So we have simply the identity $$\sum_{i}F'_{iikl}=n_2\delta_{kl},$$ which gives $$\delta^{(2)}S[\alpha_1]=n_2 g\frac{\Omega_d}{2\pi} \int d\omega \int_{b^{-1}}^1dkk^{d-1} \frac{1}{-2\alpha_2+k^2+\omega^2} \int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}}\boldsymbol \psi_s(\mathbf q)\cdot \boldsymbol \psi_s(-\mathbf q).$$ Defining the new set of functions $$I'_n= \int d\omega \int_{b^{-1}}^1dkk^{d-1} \frac{1}{\left(k^2+\omega^2\right)^n},$$ one obtains $$\delta^{(2)}S[\alpha_1]=n_2 g\frac{\Omega_d}{2\pi} (I'_1+2\alpha_2 I'_2) \int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}}\boldsymbol \psi_s(\mathbf q)\cdot \boldsymbol \psi_s(-\mathbf q).$$ Here we integrate over frequencies in the range $-\infty<\omega<\infty$, and get $$I'_1=\pi\int^1_{b^{-1}}dk k^{d-2},$$ which is, to first order in $(1-b^{-1})$, $$I'_1=\pi(1-b^{-1}).$$ Similarly for $I'_2$ we have $$I'_2=\frac{\pi}{2}\int^1_{b^{-1}}dk k^{d-4},$$ thus $$I'_2=\frac{\pi}{2}(1-b^{-1}).$$ Near $d=3$, one has $\Omega_d/4\simeq\Omega_{d+1}$. Grouping all these together, we obtain the second term in the correction to $\alpha_1$ as $$\delta^{(2)}S[\alpha_1]=n_2 g \Omega_{d+1} (1-b^{-1}) (2+2\alpha_2) \int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}}\boldsymbol \psi_s(\mathbf q)\cdot \boldsymbol \psi_s(-\mathbf q).$$
First Order Corrections to $\alpha_2$
-------------------------------------
![One-loop diagrams contributing to the first order correction of $\alpha_2$. The solid lines represent the $\psi$ fields, and the dashed lines represent the $\phi$ fields. The external lines are slow modes, and the internal lines are fast modes.[]{data-label="aaaa2"}](a21.eps "fig:"){width="4cm"} ![One-loop diagrams contributing to the first order correction of $\alpha_2$. The solid lines represent the $\psi$ fields, and the dashed lines represent the $\phi$ fields. The external lines are slow modes, and the internal lines are fast modes.[]{data-label="aaaa2"}](a22.eps "fig:"){width="4cm"}
The calculation of the first order corrections to $\alpha_2$ is quite similar to that of $\alpha_1$. There are again two terms contributing. The coupling $\psi_f^2 \phi_s^2$ gives rise to a term of the form $$\delta^{(1)}S[\alpha_2]=g\sum_{ijkl}F'_{ijkl}\int_f \frac{d^{d+1}\mathbf q'}{(2\pi)^{d+1}} \langle\psi_f^i(\mathbf q') \psi_f^j(-\mathbf q') \rangle \int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}} \phi_s^k(\mathbf q) \phi_s^l(-\mathbf q),$$ Summing over the field indices, $$\sum_{i}F'_{iikl}\phi_s^k \phi_s^l=n_1 |\boldsymbol \phi_s|^2,$$ we obtain $$\delta^{(1)}S[\alpha_2]=n_1g\frac{\Omega_d}{2\pi} \int d\omega \int_{b^{-1}}^1dkk^{d-1} \frac{1}{-2\alpha_1+k^2+\gamma_1\frac{|\omega|}{k^{z_1-2}}} \int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}}\boldsymbol \phi_s(\mathbf q)\cdot \boldsymbol \phi_s(-\mathbf q),$$ which can be expanded as $$\delta^{(1)}S[\alpha_2]=n_1g\frac{\Omega_d}{2\pi} (I_1+2\alpha_1I_2) \int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}}\boldsymbol \phi_s(\mathbf q)\cdot \boldsymbol \phi_s(-\mathbf q).$$ The result is $$\delta^{(1)}S[\alpha_2]=n_1g \frac{\Omega_d}{\pi\gamma_1} (1-b^{-1})(\ln2+2\alpha_1)
\int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}}\boldsymbol \phi_s(\mathbf q)\cdot \boldsymbol \phi_s(-\mathbf q).$$
The other term comes from the coupling $\phi_f^2 \phi_s^2$. It has the form $$\delta^{(2)}S[\alpha_2]=3\beta_2\sum_{ijkl}F_{ijkl}\int_f \frac{d^{d+1}\mathbf q'}{(2\pi)^{d+1}} \langle\phi_f^i(\mathbf q') \phi_f^j(-\mathbf q') \rangle \int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}} \phi_s^k(\mathbf q) \phi_s^l(-\mathbf q).$$ We first sum over the field indices, $$\sum_{i}F_{iikl}\phi_s^k \phi_s^l=\frac{n_2+2}{3} |\boldsymbol \phi_s|^2,$$ resulting in $$\delta^{(2)}S[\alpha_2]=(n_2+2) \beta_2\frac{\Omega_d}{2\pi} \int d\omega \int_{b^{-1}}^1dkk^{d-1} \frac{1}{-2\alpha_2+k^2+\omega^2} \int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}}\boldsymbol \phi_s(\mathbf q)\cdot \boldsymbol \phi_s(-\mathbf q).$$ Expanding to first order in $\alpha_2$, one has $$\delta^{(2)}S[\alpha_2]=(n_2+2) \beta_2\frac{\Omega_d}{2\pi}(I'_1+2\alpha_2 I'_2)
\int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}}\boldsymbol \phi_s(\mathbf q)\cdot \boldsymbol \phi_s(-\mathbf q),$$ and the final result is $$\delta^{(2)}S[\alpha_2]=(n_2+2) \beta_2 \Omega_{d+1} (1-b^{-1}) (2+2\alpha_2)
\int_s \frac{d^{d+1}\mathbf q}{(2\pi)^{d+1}}\boldsymbol \phi_s(\mathbf q)\cdot \boldsymbol \phi_s(-\mathbf q),$$
First Order Corrections to $\beta_1$
------------------------------------
![One-loop diagrams contributing to the first order correction of $\beta_1$. The solid lines represent the $\psi$ fields, and the dashed lines represent the $\phi$ fields. The external lines are slow modes, and the internal lines are fast modes.[]{data-label="bbbb1"}](b11.eps "fig:"){width="4cm"} ![One-loop diagrams contributing to the first order correction of $\beta_1$. The solid lines represent the $\psi$ fields, and the dashed lines represent the $\phi$ fields. The external lines are slow modes, and the internal lines are fast modes.[]{data-label="bbbb1"}](b12.eps "fig:"){width="4cm"}
The first order correction to $\beta_1$ comes from two one-loop diagrams, one with two internal $\psi_f$ lines, the other with two $\phi_f$ lines. The dependence of the internal lines on the external momenta and frequencies can be ignored here, since they are of higher order.
The first term with $\psi_f$ internal lines is of the form $$\begin{split}
\\ \delta^{(1)}S[\beta_1]=&-(3\beta_1)^2\sum_{k_1k_2l_1l_2}\sum_{i_1i_2j_1j_2}F_{i_1j_1k_1l_1}F_{i_2j_2k_2l_2}\int_f \frac{d^{d+1}\mathbf q'}{(2\pi)^{d+1}} \langle\psi_f^{i_1}(\mathbf q') \psi_f^{i_2}(-\mathbf q') \rangle \langle\psi_f^{j_1}(\mathbf q') \psi_f^{j_2}(-\mathbf q') \rangle \\&\times\int d^d{\mathbf r}d\tau \psi_s^{k_1} \psi_s^{k_2} \psi_s^{l_1} \psi_s^{l_2}
+~2 ~{\rm permutations}.
\end{split}$$ Using the identity, $$\sum_{ij}F_{ijk_1l_1}F_{ijk_2l_2}=\frac{1}{9}\left[(n_1+4)\delta_{k_1l_1}\delta_{k_2l_2}+2\delta_{k_1k_2}\delta_{l_1l_2}+2\delta_{k_1l_2}\delta_{k_2l_1} \right],$$ combined with the 2 other permutations of the external lines, the part containing the field component indices can be simplified as $$\sum_{k_1k_2l_1l_2}\sum_{ij}F_{ijk_1l_1}F_{ijk_2l_2} \psi_s^{k_1} \psi_s^{k_2} \psi_s^{l_1} \psi_s^{l_2} +~2 ~{\rm permutations}= \frac{n_1+8}{9}|\boldsymbol \psi_s|^4.$$ Thus the first correction to $\beta_1$ reads $$\delta^{(1)}S[\beta_1]=-(n_1+8)\beta_1^2 \frac{\Omega_d}{2\pi} \int d\omega \int_{b^{-1}}^1dkk^{d-1} \frac{1}{(-2\alpha_1+k^2+\gamma_1\frac{|\omega|}{k^{z_1-2}})^2} \int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^4,$$ which is, to leading order of $\alpha_1$, $$\delta^{(1)}S[\beta_1]=-(n_1+8)\beta_1^2 \frac{\Omega_d}{2\pi} I_2 \int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^4.$$ Substituting the explicit expression for $I_2$, we get the result $$\delta^{(1)}S[\beta_1]=-(n_1+8)\beta_1^2 \frac{\Omega_d}{\pi\gamma_1} (1-b^{-1}) \int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^4.$$
The second term has two $\phi_f$ internal lines, and takes the form $$\begin{split}
\\ \delta^{(2)}S[\beta_1]=&-g^2 \sum_{i_1i_2j_1j_2}\sum_{k_1k_2l_1l_2} F'_{i_1j_1k_1l_1}F'_{i_2j_2k_2l_2}\int_f \frac{d^{d+1}\mathbf q'}{(2\pi)^{d+1}} \langle\phi_f^{i_1}(\mathbf q') \phi_f^{i_2}(-\mathbf q') \rangle \langle\phi_f^{j_1}(\mathbf q') \phi_f^{j_2}(-\mathbf q') \rangle\\&\times \int d^d{\mathbf r}d\tau \psi_s^{k_1} \psi_s^{k_2} \psi_s^{l_1} \psi_s^{l_2}
+~2 ~{\rm permutations}.
\end{split}$$ The part with the field component indices gives $$\sum_{k_1k_2l_1l_2}\sum_{ij}F'_{ijk_1l_1}F'_{ijk_2l_2} \psi_s^{k_1} \psi_s^{k_2} \psi_s^{l_1} \psi_s^{l_2} +~2 ~{\rm permutations}=n_2 |\boldsymbol \psi_s|^4,$$ which further leads to the result $$\delta^{(2)}S[\beta_1]=-n_2g^2 \frac{\Omega_d}{2\pi} \int d\omega \int_{b^{-1}}^1dkk^{d-1} \frac{1}{(-2\alpha_2+k^2+\omega^2)^2} \int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^4.$$ To leading order in $\alpha_2$, it is $$\delta^{(2)}S[\beta_1]=-n_2g^2 \frac{\Omega_d}{2\pi} I'_2 \int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^4,$$ or more explicitly, $$\delta^{(2)}S[\beta_1]=-n_2g^2 \Omega_{d+1}(1-b^{-1}) \int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^4.$$
First Order Corrections to $\beta_2$
------------------------------------
![One-loop diagrams contributing to the first order correction of $\beta_2$. The solid lines represent the $\psi$ fields, and the dashed lines represent the $\phi$ fields. The external lines are slow modes, and the internal lines are fast modes.[]{data-label="bbbb2"}](b21.eps "fig:"){width="4cm"} ![One-loop diagrams contributing to the first order correction of $\beta_2$. The solid lines represent the $\psi$ fields, and the dashed lines represent the $\phi$ fields. The external lines are slow modes, and the internal lines are fast modes.[]{data-label="bbbb2"}](b22.eps "fig:"){width="4cm"}
The first order correction to the $\beta_2$ term also comes from two diagrams. The first one has two $\psi_f$ internal lines, and is of the form $$\begin{split}
\\ \delta^{(1)}S[\beta_2]=&-g^2\sum_{i_1i_2j_1j_2}\sum_{k_1k_2l_1l_2}F'_{i_1j_1k_1l_1}F'_{i_2j_2k_2l_2}\int_f \frac{d^{d+1}\mathbf q'}{(2\pi)^{d+1}} \langle\psi_f^{i_1}(\mathbf q') \psi_f^{i_2}(-\mathbf q') \rangle \langle\psi_f^{j_1}(\mathbf q') \psi_f^{j_2}(-\mathbf q') \rangle\\&\times \int d^d{\mathbf r}d\tau \phi_s^{k_1} \phi_s^{k_2} \phi_s^{l_1} \phi_s^{l_2}
+~2 ~{\rm permutations}.
\end{split}$$ Summing over different field components, where one has $$\sum_{k_1k_2l_1l_2}\sum_{ij}F'_{ijk_1l_1}F'_{ijk_2l_2} \phi_s^{k_1} \phi_s^{k_2} \phi_s^{l_1} \phi_s^{l_2} +~2 ~{\rm permutations}=n_1 |\boldsymbol \phi_s|^4,$$ the first correction to the $\beta_2$ term is $$\delta^{(1)}S[\beta_2]=-n_1g^2 \frac{\Omega_d}{2\pi} \int d\omega \int_{b^{-1}}^1dkk^{d-1} \frac{1}{(-2\alpha_1+k^2+\gamma_1\frac{|\omega|}{k^{z_1-2}})^2} \int d^d{\mathbf r}d\tau |\boldsymbol \phi_s|^4.$$ To first order in $\alpha_1$, it is simply $$\delta^{(1)}S[\beta_2]=-n_1g^2 \frac{\Omega_d}{2\pi}I_2 \int d^d{\mathbf r}d\tau |\boldsymbol \phi_s|^4,$$ which can be written as $$\delta^{(1)}S[\beta_2]=-n_1g^2 \frac{\Omega_d}{\pi\gamma_1} (1-b^{-1}) \int d^d{\mathbf r}d\tau |\boldsymbol \phi_s|^4.$$
The second diagram contains two $\phi_f$ internal lines, thus the correction reads $$\begin{split}
\\ \delta^{(2)}S[\beta_2]=&-(3\beta_2)^2\sum_{k_1k_2l_1l_2}\sum_{i_1i_2j_1j_2}F_{i_1j_1k_1l_1}F_{i_2j_2k_2l_2}\int_f \frac{d^{d+1}\mathbf q'}{(2\pi)^{d+1}} \langle\phi_f^{i_1}(\mathbf q') \phi_f^{i_2}(-\mathbf q') \rangle \langle\phi_f^{j_1}(\mathbf q') \phi_f^{j_2}(-\mathbf q') \rangle \\&\times\int d^d{\mathbf r}d\tau \phi_s^{k_1} \phi_s^{k_2} \phi_s^{l_1} \phi_s^{l_2}
+~2 ~{\rm permutations}.
\end{split}$$ The summation over the field indices gives $$\sum_{k_1k_2l_1l_2}\sum_{ij}F_{ijk_1l_1}F_{ijk_2l_2} \phi_s^{k_1} \phi_s^{k_2} \phi_s^{l_1} \phi_s^{l_2} +~2 ~{\rm permutations}= \frac{n_2+8}{9}|\boldsymbol \phi_s|^4.$$ Thus the second correction to $\beta_2$ reads $$\delta^{(2)}S[\beta_2]=-(n_2+8)\beta_2^2 \frac{\Omega_d}{2\pi} \int d\omega \int_{b^{-1}}^1dkk^{d-1} \frac{1}{(-2\alpha_2+k^2+\omega^2)^2} \int d^d{\mathbf r}d\tau |\boldsymbol \phi_s|^4.$$ When the $\phi$ field is near its critical point, the above expression can be simplified to be $$\delta^{(2)}S[\beta_2]=-(n_2+8)\beta_2^2 \frac{\Omega_d}{2\pi} I'_2 \int d^d{\mathbf r}d\tau |\boldsymbol \phi_s|^4,$$ which is $$\delta^{(2)}S[\beta_2]=-(n_2+8)\beta_2^2 \Omega_{d+1}(1-b^{-1}) \int d^d{\mathbf r}d\tau |\boldsymbol \phi_s|^4.$$
First Order Corrections to $g$
------------------------------
![One-loop diagrams contributing to the first order correction of $g$. The solid lines represent the $\psi$ fields, and the dashed lines represent the $\phi$ fields. The external lines are slow modes, and the internal lines are fast modes.[]{data-label="gggg"}](g1.eps "fig:"){width="4cm"} ![One-loop diagrams contributing to the first order correction of $g$. The solid lines represent the $\psi$ fields, and the dashed lines represent the $\phi$ fields. The external lines are slow modes, and the internal lines are fast modes.[]{data-label="gggg"}](g2.eps "fig:"){width="4cm"} ![One-loop diagrams contributing to the first order correction of $g$. The solid lines represent the $\psi$ fields, and the dashed lines represent the $\phi$ fields. The external lines are slow modes, and the internal lines are fast modes.[]{data-label="gggg"}](g3.eps "fig:"){width="4cm"}
There are three diagrams contributing to the first order corrections of the coupling $g$ between the squares of the two fields. The first diagram has two $\psi_f$ fields as internal lines. This term takes the form $$\begin{split}
\delta^{(1)}S[g]=&-\frac{1}{2}\times 2\times 2(3\beta_1)g\sum_{k_1k_2l_1l_2}\sum_{i_1i_2j_1j_2}F_{i_1j_1k_1l_1}F'_{i_2j_2k_2l_2}\int_f \frac{d^{d+1}\mathbf q'}{(2\pi)^{d+1}} \langle\psi_f^{i_1}(\mathbf q') \psi_f^{i_2}(-\mathbf q') \rangle \langle\psi_f^{j_1}(\mathbf q') \psi_f^{j_2}(-\mathbf q') \rangle\\&\times \int d^d{\mathbf r}d\tau \psi_s^{k_1} \psi_s^{l_1} \phi_s^{k_2} \phi_s^{l_2}.
\end{split}$$ The $1/2$ comes from $(1/2)S_c^2$, and the two $2$ factors come from the expansion in $S_c^2$ and the number of contractions in $\langle \psi_f\psi_f(x)\psi_f\psi_f(y) \rangle$. We first sum over the field indices, $$\sum_{k_1k_2l_1l_2}\sum_{ij}F_{ijk_1l_1}F'_{ijk_2l_2} \psi_s^{k_1} \psi_s^{l_1} \phi_s^{k_2} \phi_s^{l_2}=\frac{n_1+2}{3} |\boldsymbol \psi_s|^2 |\boldsymbol \phi_s|^2,$$ and then substitute the Green’s functions, $$\delta^{(1)}S[g]=-2\beta_1g (n_1+2) \frac{\Omega_d}{2\pi} \int d\omega \int_{b^{-1}}^1dkk^{d-1} \frac{1}{(-2\alpha_1+k^2+\gamma_1\frac{|\omega|}{k^{z_1-2}})^2}
\int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^2 |\boldsymbol \phi_s|^2.$$ Keeping only the leading order term, $$\delta^{(1)}S[g]=-2\beta_1g (n_1+2) \frac{\Omega_d}{2\pi} I_2
\int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^2 |\boldsymbol \phi_s|^2,$$ one arrives at the result, $$\delta^{(1)}S[g]=-2\beta_1g (n_1+2) \frac{\Omega_d}{\pi\gamma_1} (1-b^{-1})
\int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^2 |\boldsymbol \phi_s|^2.$$
The internal lines of the second diagram are two $\phi_f$ fields. The corresponding correction term is now $$\begin{split}
\delta^{(2)}S[g]=&-\frac{1}{2}\times 2\times 2(3\beta_2)g\sum_{k_1k_2l_1l_2}\sum_{i_1i_2j_1j_2}F_{i_1j_1k_1l_1}F'_{i_2j_2k_2l_2}\int_f \frac{d^{d+1}\mathbf q'}{(2\pi)^{d+1}} \langle\phi_f^{i_1}(\mathbf q') \phi_f^{i_2}(-\mathbf q') \rangle \langle\phi_f^{j_1}(\mathbf q') \phi_f^{j_2}(-\mathbf q') \rangle\\&\times \int d^d{\mathbf r}d\tau \phi_s^{k_1} \phi_s^{l_1} \psi_s^{k_2} \psi_s^{l_2}.
\end{split}$$ The summation over field indices is similar to the first term, $$\sum_{k_1k_2l_1l_2}\sum_{ij}F_{ijk_1l_1}F'_{ijk_2l_2} \phi_s^{k_1} \phi_s^{l_1} \psi_s^{k_2} \psi_s^{l_2}=\frac{n_2+2}{3} |\boldsymbol \psi_s|^2 |\boldsymbol \phi_s|^2.$$ Thus the correction to the action is also similar, $$\delta^{(2)}S[g]=-2\beta_2g (n_2+2) \frac{\Omega_d}{2\pi} \int d\omega \int_{b^{-1}}^1dkk^{d-1} \frac{1}{(-2\alpha_2+k^2+\omega^2)^2}
\int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^2 |\boldsymbol \phi_s|^2,$$ which is, to leading order in $\alpha_2$, $$\delta^{(2)}S[g]=-2\beta_2g (n_2+2) \frac{\Omega_d}{2\pi} I'_2 \int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^2 |\boldsymbol \phi_s|^2,$$ or $$\delta^{(2)}S[g]=-2\beta_2g (n_2+2) \Omega_{d+1}(1-b^{-1}) \int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^2 |\boldsymbol \phi_s|^2.$$
The third diagram has one $\phi_f$ internal line, and one $\psi_f$ internal line. The correction takes the form $$\begin{split}
\delta^{(3)}S[g]=&-\frac{1}{2}(4g)^2\sum_{k_1k_2l_1l_2}\sum_{i_1i_2j_1j_2}F'_{i_1j_1k_1l_1}F'_{i_2j_2k_2l_2}\int_f \frac{d^{d+1}\mathbf q'}{(2\pi)^{d+1}} \langle\psi_f^{i_1}(\mathbf q') \psi_f^{i_2}(-\mathbf q') \rangle \langle\phi_f^{j_1}(\mathbf q') \phi_f^{j_2}(-\mathbf q') \rangle\\&\times \int d^d{\mathbf r}d\tau \psi_s^{k_1} \phi_s^{l_1} \psi_s^{k_2} \phi_s^{l_2}.
\end{split}$$ With the summation $$\sum_{k_1k_2l_1l_2}\sum_{ij}F'_{ik_1jl_1}F'_{ik_2jl_2} \psi_s^{k_1} \phi_s^{l_1} \psi_s^{k_2} \phi_s^{l_2}= |\boldsymbol \psi_s|^2 |\boldsymbol \phi_s|^2,$$ we obtain $$\delta^{(3)}S[g]=-8g^2 \frac{\Omega_d}{2\pi} \int d\omega \int_{b^{-1}}^1dkk^{d-1} \frac{1}{-2\alpha_2+k^2+\omega^2} \frac{1}{-2\alpha_1+k^2+\gamma_1\frac{|\omega|}{k^{z_1-2}}}
\int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^2 |\boldsymbol \phi_s|^2.$$ Assuming both fields are near their critical points, the above equation is approximately $$\delta^{(3)}S[g]=-8g^2 \frac{\Omega_d}{2\pi}I''
\int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^2 |\boldsymbol \phi_s|^2,$$ with the new function $I''$ defined as $$I''=\int _{-\infty}^{\infty} d\omega\int_{b^{-1}}^1dkk^{d-1} \frac{1}{k^2+\omega^2} \frac{1}{k^2+\gamma_1\frac{|\omega|}{k^{z_1-2}}} .$$ As mentioned before, here in our RG scheme, frequency is integrated over the whole real axes. In the two propagators, frequency scales differently with momentum. For the $\psi$ field, $\gamma_1\omega\sim k^{z_1}$; for the $\phi$ field, $\omega\sim k$. And a finite cut-off in frequency would lead to inconsistencies for such cases with miscellaneous dynamical exponents. Here the frequency integral gives $$I''=\int_{b^{-1}}^1dkk^{d+z_1-3}\frac{1}{\gamma_1^2k^2+k^{2z_1}}\left( -\gamma_1\ln\frac{k^{2z_1-2}}{\gamma_1^2} +\pi k^{z_1-1} \right).$$ To leading order in $(1-b^{-1})$, we have $$I''=\frac{1}{1+\gamma_1^2}(1-b^{-1})\left( 2 \gamma_1\ln{\gamma_1} +\pi \right).$$ This leads to the third term in the correction to the $g$ term $$\delta^{(3)}S[g]=-8g^2 \frac{\Omega_d}{2\pi}\frac{ 2 \gamma_1\ln{\gamma_1} +\pi }{1+\gamma_1^2}(1-b^{-1})
\int d^d{\mathbf r}d\tau |\boldsymbol \psi_s|^2 |\boldsymbol \phi_s|^2,$$
Rescaling of the Parameters
---------------------------
Now combining all the above results for the corrections of the different parameters, and carrying out the rescaling $$\begin{split}
\\ {\mathbf r}\to & {\mathbf r}/b,
\\ \tau\to&\tau/b,
\\ \boldsymbol \psi\to& b^{(d-1)/2}\boldsymbol \psi,
\\ \boldsymbol \phi\to& b^{(d-1)/2}\boldsymbol \phi,
\end{split}
\label{dim2}$$ we obtain the RG equations $$\begin{split}
\\ \gamma_1\to & b^{z-1}\gamma_1,
\\ -\alpha_1\to&b^{2}\left[ -\alpha_1+(n_1+2)\beta_1\frac{\Omega_d}{\pi\gamma_1} (1-b^{-1})(\ln2+2\alpha_1) +n_2 g \Omega_{d+1} (1-b^{-1}) (2+2\alpha_2) \right],
\\ -\alpha_2\to&b^{2}\left[ -\alpha_2+ (n_2+2) \beta_2 \Omega_{d+1} (1-b^{-1}) (2+2\alpha_2) +n_1g \frac{\Omega_d}{\pi\gamma_1} (1-b^{-1})(\ln2+2\alpha_1) \right],
\\ \frac{\beta_1}{2} \to & b^\epsilon \left[ \frac{\beta_1}{2}- (n_1+8)\beta_1^2 \frac{\Omega_d}{\pi\gamma_1} (1-b^{-1}) -n_2g^2 \Omega_{d+1}(1-b^{-1}) \right],
\\ \frac{\beta_2}{2} \to & b^\epsilon \left[ \frac{\beta_2}{2}-n_1g^2 \frac{\Omega_d}{\pi\gamma_1} (1-b^{-1}) -(n_2+8)\beta_2^2 \Omega_{d+1}(1-b^{-1}) \right],
\\ g\to & b^\epsilon \left[ g-2\beta_1g (n_1+2) \frac{\Omega_d}{\pi\gamma_1} (1-b^{-1})-2\beta_2g (n_2+2) \Omega_{d+1}(1-b^{-1}) -8g^2 \frac{\Omega_d}{2\pi}\frac{ 2 \gamma_1\ln{\gamma_1} +\pi }{1+\gamma_1^2} (1-b^{-1})\right],
\end{split}$$ the differential form of which has been presented in equations (\[rgg\], \[rgab\]).
|
---
abstract: 'Motivated by the proposed experiment $^{14}N(d,{^2He})^{14}C$, we study the final states which can be reached via the allowed Gamow-Teller mechanism. Much emphasis has been given in the past to the fact that the transition matrix element from the $J^{\pi}=1^+~T=0$ ground state of $^{14}N$ to the $J^{\pi}=0^+~T=1$ ground state of $^{14}C$ is very close to zero, despite the fact that all the quantum numbers are right for an allowed transition. We discuss this problem, but, in particular, focus on the excitations to final states with angular momenta $1^+$ and $2^+$. We note that the summed strength to the $J^{\pi}=2^+~T=1$ states, calculated with a wide variety of interactions, is significantly larger than that to the $J^{\pi}=1^+~T=1$ final states.'
author:
- |
S. Aroua$^1$, P. Navrátil$^2$, L. Zamick$^3$, M.S. Fayache$^1$, B. R. Barrett$^4$, J.P. Vary$^5$, N. Smirnova$^6$ and K. Heyde$^6$\
(1) Département de Physique, Faculté des Sciences de Tunis,\
Université de Tunis El-Manar, Tunis 1060, Tunisia\
(2) Lawrence Livermore National Laboratory, L-414,\
P. O. Box 808, Livermore, CA 94551\
(3) Department of Physics and Astronomy, Rutgers University\
Piscataway, New Jersey 08855\
(4) Department of Physics, P.O. Box 210081, University of Arizona,\
Tuscon, Arizona 85721.\
(5) Department of Physics and Astronomy, Iowa State University\
Ames, Iowa 50011\
(6) Vakgroep Subatomaire en Stralingsfysica, University of Gent\
Proeftuinstraat, 86 B-9000 Gent, Belgium\
title: 'Allowed Gamow-Teller Excitations from the Ground State of $^{14}N$'
---
Introduction
============
Much attention has been given over the past several decades to the fact that the Gamow-Teller ($GT$) matrix element between the $J^{\pi}=1^+~T=0$ ground state of $^{14}N$ and the $J^{\pi}=0^+~T=1$ ground state of $^{14}C$ (or that of its mirror nucleus $^{14}O$) is very close to zero, despite the fact that all the quantum numbers are right for an allowed Gamow-Teller transition. Of particular interest is the early work of Inglis [@inglis] who showed that in the simplest shell model space (2 holes in the $0p$ shell), it is $not$ possible to get this $GT$ matrix element to vanish if the residual nucleon-nucleon ($NN$) interaction consists of only a central part and a spin-orbit part. Inglis then commented upon the possibility that $A(GT)$ might vanish with only these two interactions if higher shells were included. He himself did not carry out such a calculation, but an attempt to do so was made a few years later by Baranger and Meshkov [@bar]. They concluded that it was possible that configuration mixing was the sole agent to cause $A(GT)$ to vanish; however, they had to speculate on the signs of certain matrix elements.
Following Inglis’ work, Jancovici and Talmi [@talmi] showed that if one also had a tensor component present in the interaction one $could$ get the $GT$ matrix element to vanish. Thus, the $A=14$ system affords us one of the few instances where one can study the elusive effects of the tensor interaction in nuclear structure [@zheng; @prep]. Visscher and Ferrel [@viss] plotted the strength parameters of the spin-orbit and tensor interactions for which one could get the $GT$ matrix element to vanish. They noted that if the spin-orbit interaction is too weak then they cannot get the $GT$ matrix element to vanish for $any$ value of the tensor interaction strength parameter.
Zamick showed, in Ref. [@larry], that the $GT$ matrix element comes out too large when one uses the non-relativistic $G-$matrix elements which Kuo [@kuo] obtained from the Hamada-Johnston interaction [@hamada]. However, if the spin-orbit interaction was increased by about 50%, he could get this matrix element to vanish.
Whereas most early calculations were carried out in the model space of two holes in the $0p$ shell, more recently, Fayache, Zamick and Müther have reconsidered this issue by performing no-core shell model calculations ($NCSM$) in the larger model space ($[(0p)^{-2}]$ + 2$\hbar \omega$)[@a14]. First they used an interaction previously constructed by Zamick and Zheng [@ann]:
$$V_{zz} = V_{c}+xV_{so}+yV_{t}, \label{eq1}$$
where $c$=central, $so$=two-body spin-orbit, and $t$=tensor. For $x$=$y$=1, the matrix elements of $V_{zz}$ are in approximate agreement with those of the non-relativistic OBE potential Bonn A of [@mach2]. They then studied the effects of the spin-orbit and tensor components of the $NN$ interaction on the $GT$ matrix element by varying the strength parameters $x$ and $y$. They found the interesting result that in the small model space (2 holes in the $0p$ shell) they could (for the standard value $x=1$ of the spin-orbit interaction strength parameter) find a value of $y$ for which the $GT$ matrix element vanishes. However, in a larger model space which also included 2$\hbar \omega$ excitations and still using the standard value $x=1$, they could not get the $GT$ matrix element to vanish for $any$ value of the tensor interaction strength parameter $y$. Thus they reached the opposite conclusion to that of Baranger and Meshkov. However, if the spin-orbit interaction was enhanced by 50% (to $x=1.5$), then they could find a reasonable value of $y$ for which the $GT$ matrix element vanishes (but not for $y=0$). Furthermore, using a relativistic Bonn A $G-$matrix with a Dirac effective mass $m_D=0.6m$ ($m$ being the mass of the free nucleon), they found that they can make the $GT$ matrix element vanish in both the small and large model spaces. It is known that the spin-orbit interaction gets enhanced by a factor $m/m_D$ in relativistic calculations [@dirac].
In the above discussion, we have focused on the ground-state-to-ground-state transition $^{14}N~(J^{\pi}=1^+_1,~T=0)~\rightarrow~^{14}C~(J^{\pi}=0^+_1,
~T=1)$. But in a proposed experiment ($^{14}N(d,{^2He})^{14}C$) [@De; @Frenne], one can reach excited $T=1$ states as well with spins $J^{\pi}=0^+,~1^+$ and $2^+$ via the allowed $GT$ mechanism. In section II, we shall present the results of theoretical calculations of the $GT$ reduced transition probability
$$B(GT)=(\frac{g_A}{g_V})^2\frac{1}{2J_i+1}|A(GT)|^2,$$
as well as the summed strengths $\sum B(GT)$ to these states. In Eq. 2, $\frac{g_A}{g_V}=1.251$ is the ratio of the Gamow-Teller to Fermi coupling constants introduced here for convenience [@gagv]. The $GT$ matrix element itself, denoted as $A(GT)$, is given by the expression
$$|A(GT)|^2=\sum_{M_i,M_f,\mu}\langle \psi_f^{J_f,M_f,T_f,T_{fz}}
| \sum_{k=1}^A \sigma_{\mu}(k) t_+(k)~| \psi_i^{J_i,M_i,T_i,T_{iz}} \rangle^2.$$
We perform our calculations with a variety of realistic interactions in both the small and large model spaces. In light of the fact that in Ref. [@a14] there were such drastic differences between the results obtained in the small and large model spaces for the $GT$ transition from $J^{\pi}=1_1^+$ to $J^{\pi}=0_1^+$, we should also investigate the effects of going from small to large model spaces for the other allowed transitions, namely from $J^{\pi}=1_1^+$ to $J^{\pi}=1^+$ and to $J^{\pi}=2^+$. This is one of the main points of the present work. Furthermore, we will do these calculations using both a phenomenological approach as in [@a14] and a purely theoretical one in which a modern realistic $N-N$ effective interaction is used in larger and larger model spaces.
The results of our calculations are presented in section II. Section III deals with the interpretation of the results, followed by concluding remarks.
Results of the Calculations
===========================
We first show in Table I results of calculations of the $GT$ ground-state-to-ground-state transition strength $B(GT):~^{14}N~(J^{\pi}=
1^+_1,~T=0)~\rightarrow~^{14}C~(J^{\pi}=0^+_1,~T=1)$, followed by the $summed$ strengths of the $GT$ transition from the ground state of $^{14}N$ to the $J^{\pi}=0^+,~1^+$ and $2^+$ ($T=1$) states of $^{14}C$, using the interaction $V_{zz}$ (Eq. 1) of Zamick and Zheng [@ann]. Note that we never introduce a single-particle spin-orbit term, since in our case the average one-body spin-orbit interaction is implicitly generated by our two-body spin-orbit interaction $xV_{so}$ in our no-core shell-model ($NCSM$) calculations, which we performed using the nuclear shell model code $OXBASH$ [@oxbash].
First, let us compare the small- and large-space results for $J^{\pi}=0^+$ final states obtained with the standard two-body spin-orbit strength $x=1$. As we vary the strength parameter $y$ of the tensor interaction in the small model space calculation, we see that for $y=0.5$ $B(GT)$ becomes vanishingly small. Indeed, for $y=1.0~x=1$ $A(GT)$ has an opposite sign to that for $y=0~x=1$. This verifies the contention of Jancovici and Talmi that one can get $A(GT)$ to vanish with a suitable tensor interaction.
However, when we go to the $large$ (0+2)$\hbar \omega$ model space, we see that $B(GT)$ for $J^{\pi}=0^+_1$ does $not$ go to zero for any value of $y$ when $x=1$, and we no longer get the Jancovici-Talmi behaviour.
The situation is restored if a combination of a weaker strength of the tensor interaction and an enhanced strength of the spin-orbit interaction ($i.e.~x=1.5$ and $y=0.75$) is applied. In that case we get $B(GT)$ to vanish in both the $small$ and $large$ model spaces.
We next come to one of the main points of the paper: a comparison of the $GT$ summed strengths to the $J^{\pi}=1^+$ and $J^{\pi}=2^+$ ($T=1$) final states in $^{14}C$. We see consistently that the excitation strengths to the $J^{\pi}=2^+$ states are much larger than to the $J^{\pi}=1^+$ states. For example, in the large space with $x=1.5~y=0.75$, the values of the summed strength to the $1^+$ states is only 0.193, but to the $2^+$ states it is 3.113. We will discuss this further in the next section.
Table II presents results of calculations done with the relativistic Bonn A interaction of Müther $et.~al.$ [@dirac]. In this approach, one has a Dirac effective mass $m_D$ such that $m_D/m$ is typically less than one with $m_D/m=1$ corresponding to the non-relativistic limit. In our case, the value of $m_D/m=0.6$ seems to work best in as far as achieving a vanishing $GT$ transition between the ground states of the $A=14$ system. This is true in both the small and the large model spaces.
In Table III, we present the results of calculations done with the Argonne V8’ effective interaction in four model spaces: 0$\hbar \omega$, (0+2)$\hbar \omega$, (0+2+4)$\hbar \omega$ and (0+2+4+6)$\hbar \omega$, all performed with the Many-Fermion Dynamics code of [@MFD]. For this set of calculations, we followed the procedure described in Refs [@Petr96; @PRLC00] in order to construct the two-body effective interaction. Note that in Tables I-III we give the ground-state-to-ground-state transition $B(GT):~^{14}N~(J^{\pi}=1^+_1,~T=0)~\rightarrow~^{14}C~(J^{\pi}=0^+_1,~T=1)$, as well as the summed strength $\sum B(GT)$, $i.e.$ summing the $B(GT)$ values starting from the $^{14}N~(J^{\pi}=1^+_1,~T=0)$ ground state to all final $0^+,~1^+$ and $2^+$ $T=1$ states in $^{14}C$. In the smallest model space, the Argonne V8’ interaction gives a poor result for the ground-state-to-ground-state $B(GT)$, a value of 2.518 which is far from the desired result of $zero$. When the model space is enlarged to (0+2)$\hbar \omega$, the $B(GT)$ to the $0^+_1$ state obtained with the Argonne V8’ interaction goes down to 1.403, then it goes further down to 0.430 in the larger model space (0+2+4)$\hbar \omega$, and in the yet larger model space (0+2+4+6)$\hbar \omega$ it goes way down to 0.164. This shows that the results are quite sensitive to the model space used, but overall they are rather encouraging in the sense that one seems to be converging to the desired result that the ground-state-to-ground-state $B(GT)$ vanishes in the limit that the model space becomes sufficiently large. Indeed, it is clear that the many-body correlations in the large model spaces are causing the decrease in the transitions to the $0^+$ states and their increase for $2^+$ states.
Interpretation of the Results
=============================
The $L-S$ picture
-----------------
We can make sense of the results obtained in the small model space 0 $\hbar \omega$ by following the approach of Zheng and Zamick [@ann] and use an $LS$ representation ($^{2S+1}L_J$) for the two-hole $A=14$ system. For instance, the ground state ($J^{\pi}=1_1^+,~T=0$) wavefunction of $^{14}N$ ($i.e.$ the initial state) is represented as follows:
$$\psi_i = C_i^S\; |{}^3S_1\rangle + C_i^P\; |{}^1P_1\rangle\;
+ C_i^D\; |{}^3D_1\rangle \;,$$
whereas for final $J^{\pi}=0^+,~T=1$ states the wavefunctions are of the form
$$\psi_f = C_f^S\; |{}^1S_0\rangle + C_f^P\; |{}^3P_0\rangle\;.$$
The expression for the transition amplitude $A(GT)$ (see Eq. 3) is then
$$A(GT)=\sqrt{6}[C_f^SC_i^S-C_i^PC_f^P/\sqrt{3}].$$
It should be noted that if the $^{14}N$ ground-state wavefunction had a pure $^3D_1$ configuration then the Gamow-Teller transition amplitude $A(GT)$ to $J^{\pi}=0^+$ and $1^+$ states would vanish. The reason for this, of course, is that the $GT$ operator $\sum_k \sigma_{\mu}(k)
t_+(k)$ cannot change the orbital quantum number $L$. But from the above expression for $A(GT)$, it is not a necessary condition to have $C_i^D=1$ and $C_i^S=C_i^P=0$ in order for $A(GT)$ to vanish. Interference from the $L=1$ contributions can and does make $A(GT)$ vanish before $C_i^D=1$. Nevertheless, $C_i^D$ is very close to one at the point where $A(GT)$ vanishes.
In Table IV, we present the values of the coefficients $C_i^S$, $C_i^P$ and $C_i^D$ related to the 0$\hbar \omega$ model space calculations done with various interactions considered earlier in Tables I, II and III, as well as the corresponding $A(GT)$. By comparing the values of the $LS$ coefficients shown in the upper half of this table to those in the lower one, it becomes clear that the argument presented above, about the crucial role of the $^3D_1$ component in the $J^{\pi}=1^+~T=0$ $^{14}N$ ground-state wavefunction in insuring the vanishing of $A(GT)$, holds for the other interactions as well.
We can also see why the $2^+$ final states are more strongly excited than the $J^{\pi}=1^+$ final states. In the two-hole model space, and by virtue of the generalized Pauli exclusion principle, there is only $one$ $J^{\pi}=1^+,~T=1$ final state, corresponding to $L=1~S=1~T=1$ and denoted by $^3P_1$. It can be excited by the $GT$ mechanism only via the $^1P_1$ component of the $^{14}N$ $J^{\pi}=1_1^+~T=0$ ground-state wavefunction. We see from Table IV that the $^1P_1$ component $C_i^P$ is rather small when $A(GT)$ vanishes. It is possible, however, to form two $J^{\pi}=2^+~T=1$ states in the two-hole model space (corresponding to $L=2~S=0$ and $L=1~S=1$), and the first one of these two configurations ($^1D_2$) will carry most of the strength of the $GT$ excitation emanating from the dominantly $^3D_1$ $^{14}N$ ground-state wavefunction ($C_i^D \ge 0.96$ when $A(GT)$ vanishes).
Renormalization of the spin-orbit interaction
---------------------------------------------
As mentioned in the introduction, it has been noted in the past [@viss; @larry; @a14; @ann] that the vanishing of the ground-state-to-ground state $GT$ matrix element in the $A=14$ system as calculated in the valence space ($i.e.$ 0$\hbar \omega$ model space) requires either an enhancement of the two-body spin-orbit interaction and/or a weakening of the tensor interaction -see the lower half of table IV. In a different but somewhat related context, Fayache $et.~al.$ had come to a similar conclusion in their study of $M1$ excitation rates in the $0p$ and $1d-0d$ shells [@npa]. It is useful to note here, as pointed out by Wong [@wong], that the tensor interaction in an [*open-shell*]{} acts to some extent like a spin-orbit interaction of the [*opposite*]{} sign of the basic spin-orbit interaction, so that these two types of adjustments to the $N-N$ interaction are really equivalent for our purpose.
In the present work, we have shown in table III that, using a modern realistic effective $N-N$ interaction and performing no-core shell-model calculations with it in progressively larger and larger model spaces, we were able to obtain the desired vanishing of the ground-state-to-ground-state $GT$ matrix element in a natural way, $i.e.$ without having to adjust any parameters. This suggests that some renormalization of the effective spin-orbit interaction coupling strength (in the sense of an enhancement of the latter relative to its strength in the 0$\hbar \omega$ model space) must be taking place as one works in larger and larger model spaces.
In Table V, we present results of calculations that further corroborate the interpretation just given. Loosely speaking, the $J^{\pi}=2^+_1$ state is mainly a $(p^{-1}_{1/2})(p^{-1}_{3/2})$ two-hole state, so that its excitation energy scales mainly as the spin-orbit splitting $E({3/2}^-)-E({1/2}^-)$ in the $A=15$ system. Evidently, the latter can be thought of as a measure of the strength of the effective two-body spin-orbit interaction as calculated in a given model space. A striking systematics then emerges when one combines the results of tables III and V. Clearly, there is a one-to-one correlation between the re-distribution of the $GT$ strength in the $A=14$ system (table III) and the effective spin-orbit strength as the size of the shell-model space is varied (table V). Indeed, there is a clear trend taking place in the sense that, as the size of the shell-model space gets larger, the calculated excitation energy $E_x(2^+_1)$ in $^{14}C$ as well as the calculated energy splitting $E({3/2}^-)-
E({1/2}^-)$ in $^{15}N$ are increasing, and all the while a re-distribution of the $A=14$ $GT$ strength is taking place, with all the combined results becoming in better agreement with experiment.
Concluding Remarks
==================
We have performed theoretical calculations of the allowed Gamow-Teller transitions from the ground state of $^{14}N$ to the lowest lying states in $^{14}C$ in anticipation of a proposed experiment involving the reaction $^{14}N(d,{^2He})^{14}C$. We discussed the problem of the near vanishing of the $GT$ transition to the $J^{\pi}=0^+_1,~T=1$ ground state of $^{14}C$, but principally focused on the transitions to the final states with angular momenta $1^+$ and $2^+$.
In calculations limited to a 0 $\hbar \omega$ model space, it is necessary to effect a phenomenological enhancement of the $N-N$ two-body spin-orbit interaction in order to obtain a vanishing ground-state-to-ground-state $GT$ matrix element in the $A=14$ system. We have found that this in turn results in the $GT$ strength going overwhelmingly to the lowest $2^+$ state. Such a result can be easily accounted for by the fact that the $^{14}N$ $J^{\pi}=
1_1^+~T=0$ ground state wavefunction is predominantly composed of an $LS$ component $^3D_1$.
Using an effective interaction theoretically derived from the realistic $N-N$ Argonne V8’ interaction, and performing shell-model calculations in progressively larger and larger model spaces (with up to 6 $\hbar \omega$ excitations), we were able to achieve a similar degree of success in agreement with experiment as that obtained phenomenologically earlier in the 0$\hbar \omega$ model space calculations, but this time without any adjustments of parameters. We have interpreted this as an indication of a natural renormalization of the effective two-body spin-orbit interaction affected by the many-body correlations taking place in the larger model spaces.
In concluding, we note that, when only the charge-symmetry-conserving strong interactions are taken into account (as we did in this paper), the matrix element for the transition from $^{14}N$ to $^{14}C$ is the same as that from $^{14}N$ to $^{14}O$. However, since the transitions to the $J=0^+$ states are strongly suppressed, large charge-symmetry effects can be induced by the Coulomb interaction. Indeed the $ft$ values in the decays of $^{14}C$ and $^{14}O$ to the ground state of $^{14}N$ are respectively $1.1~10^9$ and $2~10^7$, quite different values indeed. Talmi [@talmi2] was able to verify that indeed the Coulomb interaction could explain this difference to a large extent. It will be interesting in the near future to see the effects of other charge-symmetry-breaking interactions.
Acknowledgements
================
K. Heyde thanks D. De Frenne for discussions on the relevance of the $(d,{^2He})$ reaction in order to study the Gamow-Teller strength in light nuclei. L. Zamick ackowledges support by DOE Grant No. DE-FG02-95ER-40940. M.S. Fayache and B.R. Barrett acknowledge partial support from NSF Grants No. PHY0070858 and INT-0096785. B.R. Barrett thanks I. Talmi for useful discussions. This work was performed in part under the auspices of the U. S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. P.Navratil received support from LDRD contract 00-ERD-028. N. Smirnova and K. Heyde thank the DWTC for the grant IUAP \#P5/07 and the “FWO-Vlaanderen” for financial support. Finally, M.S. Fayache is grateful for the hospitality of the High-Energy Section of the Abdus Salam International Centre for Theoretical Physics, where part of this work was done.
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----- ------ --------- ------- ------- ------- --------- ------- ------- -------
$x$ $y$ $0^+_1$ $0^+$ $1^+$ $2^+$ $0^+_1$ $0^+$ $1^+$ $2^+$
1 0 0.738 0.792 0.280 2.617 1.235 1.345 0.276 2.030
1 0.5 0.002 0.092 0.148 3.187 1.682 2.158 0.009 0.933
1 0.75 0.668 1.047 0.017 2.100 1.555 1.985 0.005 1.091
1 1.0 0.991 1.427 0.003 1.707 1.532 1.929 0.004 1.132
1.5 0 0.359 0.364 0.345 3.111 0.448 0.456 0.351 2.970
1.5 0.5 0.161 0.169 0.317 3.278 0.093 0.119 0.287 3.230
1.5 0.75 0.062 0.099 0.281 3.312 0.003 0.126 0.193 3.113
1.5 1.0 0.004 0.097 0.230 3.263 0.165 0.422 0.103 2.709
----- ------ --------- ------- ------- ------- --------- ------- ------- -------
: Ground-state-to-ground-state Gamow-Teller strength $B(GT)$ (denoted by $0^+_1$ in columns three and seven) and summed strengths ($\sum B(GT)$) from the $J^{\pi}=1^+_1~T=0$ ground state of $^{14}N$ to the $J^{\pi}=0^+,~1^+$ and $2^+$ states in $^{14}C$, using the ($x,y$) interaction.
--------- --------- ------- ------- ------- --------- ------- ------- -------
$m_D/m$ $0^+_1$ $0^+$ $1^+$ $2^+$ $0^+_1$ $0^+$ $1^+$ $2^+$
1.0 1.765 2.144 0.004 0.990 0.779 1.064 0.013 2.001
0.75 0.051 0.194 0.124 3.061 0.043 0.168 0.119 2.986
0.60 0.033 0.085 0.255 3.300 0.001 0.078 0.180 3.115
--------- --------- ------- ------- ------- --------- ------- ------- -------
: Same as Table I, but using Müther’s relativistic Bonn A interaction, characterized by the ratio of the nucleon’s Dirac mass $m_D$ to its free mass $m$.
Model Space $0^+_1$ $0^+$ $1^+$ $2^+$
------------------------- --------- ------- -------- --------
0 $\hbar \omega$ 2.518 2.905 0.0262 0.2511
(0+2)$\hbar \omega$ 1.403 1.839 0.0004 1.230
(0+2+4)$\hbar \omega$ 0.430 0.799 0.0291 2.264
(0+2+4+6)$\hbar \omega$ 0.164 0.480 0.318 3.081
: Same as Table I, but using two-body effective interactions derived from the Argonne V8’ NN potential without Coulomb. The HO frequency of $\hbar\omega=14$ MeV was employed. The calculated $6\hbar\omega$ binding energies of $^{14}$N is 110.52 MeV. The binding energy is expected to decrease with a further enlargement of the basis size.
Interaction $C_i^S$ $C_i^P$ $C_i^D$ $A(GT)$
---------------------- --------- --------- --------- ---------
$x$=1.0, $y$=1.0 0.675 0.032 0.737 1.378
Bonn A ($m_D/m=1$) 0.827 -0.037 0.561 1.839
Argonne $V8'$ 0.963 -0.093 0.255 2.197
$x$=1.0, $y$=0.49 0.086 0.224 0.971 0.000
$x$=1.44, $y$=1.0 0.116 0.256 0.960 0.000
Bonn A ($m_D/m=0.6$) 0.057 0.364 0.930 0.250
: The $LS$-representation coefficients of the $J^{\pi}=1^+_1~T=0$ ground state wavefunction of $^{14}N$ (Eq. 4), and the Gamow-Teller amplitude $A(GT)$ to the $J^{\pi}=0_1^+~T=1$ ground state of $^{14}C$ (Eq. 6) for various interactions considered in the previous tables. For the AV8$^\prime$ interaction, the $0\hbar\omega$ basis results are shown.
Model Space $A=14$ $E_x(2^+_1)$ Expt $A=15$ $E({3/2}^-)-E({1/2}^-)$ Expt
------------------------- --------------------- ------ -------------------------------- -------
0 $\hbar \omega$ 3.152 6.59 3.314 6.176
(0+2)$\hbar \omega$ 4.854 ” 5.366 ”
(0+2+4)$\hbar \omega$ 5.564 ” 6.326 ”
(0+2+4+6)$\hbar \omega$ 5.874 ” 6.731 ”
: Calculated excitation energy of the $2^+_1$ state in the $A=14$ system and calculated spin-orbit splitting in the $A=15$ system (in MeV) in various model spaces, using two-body effective interactions derived from the Argonne V8’ NN potential without Coulomb. The HO frequency of $\hbar\omega=14$ MeV was used. The calculated $6\hbar\omega$ binding energies are 108.65 MeV and 126.73 MeV for $^{14}$C and $^{15}$N, respectively. The binding energy is expected to decrease with a further enlargement of the basis size. The experimental values of $E_x(2^+_1)$ in $^{14}O$ and of $E({3/2}^-)-E({1/2}^-)$ in $^{15}O$ given under Expt are from Nuclear Data Retrieval ($http://www.nndc.bnl.gov$).
|
---
author:
- 'C. Ngeow'
- 'S. M. Kanbur'
title: Large Magellanic Cloud Distance from Cepheid Variables using Least Squares Solutions
---
Distance to the Large Magellanic Cloud (LMC) is determined using the Cepheid variables in the LMC. We combine the individual LMC Cepheid distances obtained from the infrared surface brightness method and a dataset with a large number of LMC Cepheids. Using the standard least squares method, the LMC distance modulus can be found from the ZP offsets of these two samples. We have adopted both a linear P-L relation and a “broken” P-L relation in our calculations. The resulting LMC distance moduli are $18.48\pm0.03$ mag and $18.49\pm0.04$ mag (random error only), respectively, which are consistent to the adopted $18.50$ mag in the literature.
Introduction {#sec:1}
============
Recently, [@gie05] (hereafter G05) has used the infrared surface brightness method to obtain the individual distances to 13 LMC Cepheids with an averaged LMC distance modulus of $18.56\pm0.04$ mag (random error only). However, LMC hosts more than $600$ Cepheids with data available from the Optical Gravitational Lensing Experiment (OGLE) [@uda99]. A linear least squares solution (LSQ) will allow a simultaneous determination for [*both*]{} of the LMC distance and the P-L relation (see Figure \[fig:1\]).
![Illustration of the LSQ. Introducing an offset, $\Delta \mu$, between the calibrated Cepheids and the other LMC Cepheids will allow us to simultaneously solve for the LMC distance modulus and the P-L relation with LSQ.[]{data-label="fig:1"}](CNgeow.fig1.eps){height="4cm"}
Data, Method and Results {#sec:2}
========================
The data include the absolute magnitudes for the 13 Cepheids with individual distance measurements from G05 and the apparent magnitudes (after extinction correction) for $\sim630$ LMC Cepheids from [@kan06], which is based on the OGLE database. For both datasets, we fit the following regression using the LSQ: $x=\alpha \Delta \mu + a + b \log(P)$ where $\alpha=0$ if $x=M$ (for G05 data) or $\alpha=1$ if $x=m$ (for OGLE data). The results of the LSQ are: $M^V=-2.76\pm0.04\log(P)-1.36\pm0.07$ with $\Delta \mu(V)=18.47\pm0.06$, and $M^I=-2.98\pm0.02\log(P)-1.86\pm0.05$ with $\Delta \mu(I)=18.48\pm0.04$. The weighted average of these $\Delta \mu$ is $18.48\pm0.03$ mag (random error only).
Since the recent studies have strongly suggested the LMC P-L relation is not linear ([@kan06; @kan04; @nge05; @san04]), we also use the following “broken” regression to fit the data:
$$x = \alpha \Delta \mu + a_S + \beta b_S \log(P) + \gamma (a_L-a_S) + \epsilon b_L \log(P),
\ \alpha = \left\{ \begin{array}{ll}
0 & \mathrm{if}\ x = M \\
1 & \mathrm{if}\ x = m
\end{array}
\right . \nonumber$$
where subscripts $_S$ and $_L$ refer to the short and long period Cepheids, respectively, and $\beta =1,\ \gamma=0,\ \epsilon=0,\ \mathrm{for}\ \log(P) < 1.0$; $\beta = 0,\ \gamma = 1,\ \epsilon = 1,\ \mathrm{for}\ \log(P) \geq 1.0$. The results are: $M^V_L=-2.84\pm0.16\log(P)-1.23\pm0.21;\ M^V_S=-2.94\pm0.06\log(P)-1.28\pm0.07$ with $\Delta \mu(V)=18.49\pm0.06$, and $M^I_L=-3.09\pm0.11\log(P)-1.69\pm0.14;\ M^I_S=-3.09\pm0.04\log(P)-1.80\pm0.05$ with $\Delta \mu(I)=18.49\pm0.04$. The weighted average of the distance moduli in both bands is $18.49\pm0.04$ mag (random error only). $F$-test ([@kan04; @nge05]) is also applied to examine if the data is more consistent with a single-line regression (the null hypothesis) or a two-lines regression (the alternate hypothesis). For our data, we obtain $F(V)=7.3$ and $F(I)=6.9$, where $F\sim3$ at 95% confident level. This suggested the null hypothesis can be rejected and the data is more consistent with the broken P-L relation.
|
---
abstract: 'The one-dimensional Kondo lattice model is investigated by means of Wegner’s flow equation method. The renormalization procedure leads to an effective Hamiltonian which describes a free one-dimensional electron gas and a Heisenberg chain. The localised spins of the effective model are coupled by the well-known RKKY interaction. They are treated within a Schwinger boson mean field theory which permits the calculation of static and dynamic correlation functions. In the regime of small interaction strength static expectation values agree well with the expected Luttinger liquid behaviour. The parameter $K_\rho$ of the Luttinger liquid theory is estimated and compared to recent results from density matrix renormalization group studies.'
author:
- 'T. Sommer'
title: 'Flow equations for the one-dimensional Kondo lattice model: Static and dynamic ground state properties'
---
Introduction {#sec:introduction}
============
The fascinating subject of heavy fermion physics in rare-earth and actinide systems has been a challenge for theoretical and experimental investigations for decades [@fulde]. The intriguing properties of these materials are far from being understood and still give us a lot of puzzles to solve. Theoretical studies of the heavy fermion materials are based on several models like the periodic Anderson model (PAM) [@hewson]. Another generic model is the Kondo lattice model (KLM) which describes a noninteracting electron gas coupled to localised spin moments via a Heisenberg spin interaction. The Hamiltonian reads
$${\cal H} = \sum_{k \sigma} \varepsilon_k \; c^\dagger_{k \sigma}
c_{k \sigma} + \frac{J}{2} \, \sum_{i \, \alpha \beta}
{\bf S}_i \, c^\dagger_{i \alpha} {\boldsymbol \sigma}_{\alpha \beta}
\, c_{i \beta},
\label{eq:kondo_hamiltonian}$$
where $\varepsilon_k = - \sum_{ij} t_{ij} \, \text{e}^{ik(R_i-R_j)}$ is the dispersion relation for the electrons on the lattice, $t_{ij}$ being the hopping integrals. The parameter $J$ is the exchange integral of the local spin interaction, the so called Kondo exchange.
We want to consider here the one-dimensional case which has been the subject of numerous numerical and analytical investigations. Numerical studies were based on the Quantum Monte Carlo (QMC) method [@troyer_wuertz], exact diagonalization (ED) studies [@tsunetsugu_rmp; @tsunetsugu_prb], the density matrix renormalization group (DMRG) [@moukouri_caron; @caprara_rosengren; @shibata_jp; @xavier_1; @shibata_tsunetsugu] or the numerical renormalization group (NRG) method [@yu_white]. Analytical approaches comprised the bosonisation technique [@honner_gulacsi; @mcculloch] or the renormalization group (RG) theory [@pivovarov_si].
The phase diagram of the one-dimensional KLM as a function of the Kondo coupling $J$ and the band filling $n_c$ of the conduction electrons is quite accurately known. In higher dimensions the KLM is believed to show the well-known Doniach phase diagram [@doniach]. In contrast to the latter the one-dimensional model does not exhibit a magnetically ordered phase in the parameter regime of small interaction strengths $J$. In this parameter regime the corresponding phase diagram is governed by a paramagnetic metallic phase [@tsunetsugu_rmp]. There, the model is assumed to belong to the universality class of the so called Luttinger liquids [@luttinger] which possess gapless charge and spin excitations resulting in an algebraic decay of the corresponding correlation functions. The asymptotic form for density-density- and spin-spin-correlations are [@voit] $$\begin{aligned}
\left< \delta n(x) \delta n(0) \right> = \frac{K_\rho}{(\pi \, x)^2}
&+ A_1 \cos(2 k_F x) \, x^{-1-K_\rho} \notag \allowdisplaybreaks \\
&+ A_2 \cos(4 k_F x) \, x^{-4 K_\rho} \allowdisplaybreaks \\[10pt]
\left< {\bf S}(x) \cdot {\bf S}(0) \right> = \frac{1}{(\pi \, x)^2}
&+ B_1 \cos(2 k_F x) \, x^{-1-K_\rho}.\end{aligned}$$ The parameter $K_\rho$ is a model dependent constant which determines the low-energy physics. Apart from the paramagnetic metallic phase for small $J/t$ the phase diagram further comprises a ferromagnetic ordered phase for large $J$ and a spin liquid insulator phase at half-filling, $n_c = 1$. There are two limiting cases in which the ground state has been proven to be ferromagnetic [@tsunetsugu_rmp]. Firstly, the limit of vanishing electron density $n_c \to 0$, secondly the case of infinite coupling strength $J/t \to
\infty$. The situation at half filling is special in the sense that it exhibits finite gaps for spin and charge excitations at any finite coupling $J$.
The KLM can be understood as an effective Hamiltonian of the above mentioned PAM. It is connected to the PAM by a Schrieffer-Wolff transformation [@schrieffer_wolff]. This property naturally raises the question whether the localised spins in the KLM participate in the formation of the Fermi surface, or in other words: Does $k_F = n_c
\pi/2$ or $k_F = (n_c+1) \pi/2$ hold? The size of the Fermi surface can be read from the positions of singularities in certain correlation functions. Recent results seemed to confirm the picture of a small Fermi surface with $k_F = n_c \pi/2$ [@xavier_1]. However, a more careful analysis which has recently been performed by Shibata *et al.* [@shibata_new] supports a large Fermi surface.
In this paper we shall apply the analytical method of continuous unitary transformations (flow equations) proposed by Wegner [@wegner] and Głazek/Wilson [@glazek_wilson] to the one-dimensional KLM. It was first applied to this model in arbitrary dimensions by Stein [@stein]. He derived an analytical expression for the RKKY interaction.
In Sec. \[sec:flow\_equations\] we shall give a short introduction into the flow equation method. In Sec. \[sec:flow\_klm\] the method will be applied to the one-dimensional KLM. By integrating out the Kondo coupling between the conduction electrons and localised spins we arrive at a decoupled system of a renormalized noninteracting one-dimensional electron gas and a renormalized spin chain. In the latter the spins interact via an effective spin exchange. Within the framework of the flow equation method it is straightforward to find expectation values and correlation functions, if the eigenvalue problem of the effective model is known. In Sec. \[sec:results\] we shall show how the method can be used in order to verify the expected characteristic behaviour of a Luttinger liquid. Previous investigations of the one-dimensional KLM have mainly focused on static properties like the momentum distribution or spin and charge correlation functions. In this work we shall put special emphasis on the investigation of dynamic properties and extend already existing results for the dynamics.
Flow equation method {#sec:flow_equations}
====================
To begin with we would like to sketch the concept of the flow equation method which was independently developed by Wegner [@wegner] and Głazek/Wilson [@glazek_wilson] in 1994. Since then the method has successfully been applied to a great number of problems, *e.g.* the electron-phonon-problem [@electron_phonon], one-dimensional interacting fermion systems [@1d_systeme] or the spin-boson-problem [@spin_boson].
The basic idea of the flow equation method is the application of a continuous set of unitary transformations to a given Hamiltonian $${\cal H}(l) = {\cal U}(l) \, {\cal H} \, {\cal U}^\dagger(l).$$ Here $l$ means the continuous flow parameter. The purpose of this procedure is that one wishes to diagonalize or at least simplify the Hamiltonian. Thereby the parameters of the Hamiltonian become renormalized. This treatment is translated into the language of differential equations by using the expression $$\begin{aligned}
\eta(l) = \frac{d \,{\cal U}(l)}{dl} \, {\cal U}^\dagger(l)\end{aligned}$$ for the antihermitean generator $\eta(l) = -\eta^\dagger(l)$ of the unitary transformation. The differential equation for the Hamiltonian takes the simple form $$\frac{d \, {\cal H}(l)}{d l} = \left[\eta(l) , {\cal H}(l)\right].
\label{eq:definition_flussgleichungen}$$ The generator has to be suitably chosen. Wegner’s approach starts from a decomposition of the Hamiltonian into an unperturbed part ${\cal
H}_0$, whose eigenvalue problem is assumed to be known, and a perturbation ${\cal H}_1$. Wegner’s generator is given by $$\eta(l) = \left[{\cal H}_0(l),{\cal H}(l)\right]
\label{eq:wegner_generator}$$\
which is simply the commutator between the unperturbed part ${\cal
H}_0(l)$ and the perturbation ${\cal H}_1(l)$. This generator integrates out all interaction terms except for possible degenerations [@wegner]. It finally leads to a diagonal or block-diagonal effective Hamiltonian.
Flow equations for the Kondo lattice model {#sec:flow_klm}
==========================================
We now turn to the derivation of the flow equations for the parameters of the Hamiltonian. With this in mind we proceed as follows. Firstly, we give the flow invariant Hamiltonian which includes new generated, effective interactions. The flow invariant Hamiltonian then leads us to the specification of the generator. Thereby we shall introduce some of the necessary approximations within our approach.
Flow equations for the Hamiltonian {#subsec:hamiltonian_flow}
----------------------------------
The first step in deriving the flow equations is the determination of the generator $\eta(l)$. In a first step we wish to integrate out the Kondo coupling between the conduction electrons and the localised spins, so the most simple generator is $$\eta(l) = \frac{1}{2N} \, \sum_{ikq \, \alpha \beta}
\eta^J_{kq}(l) \, {\bf S}_i \cdot {\boldsymbol \sigma}_{\alpha \beta}
\, c^\dagger_{k \alpha} c_{q \beta} \; \text{e}^{i (k-q) R_i} =:
\eta^J(l),
\label{eq:eta_0}$$\
where the coefficients $\eta^J_{kq}(l)$ are still unspecified. They depend on the concrete choice of the generator. Wegner’s approach starts out from the generalised form of Eq. (\[eq:wegner\_generator\]). If we take only the conduction electrons to be ${\cal H}_0$, we obtain $\eta_{kq}(l) = (\varepsilon_k-\varepsilon_q) J_{kq}(l)$ for the coefficients of the generator. The commutator between the generator (\[eq:eta\_0\]) and the Hamiltonian (\[eq:kondo\_hamiltonian\]) gives rise to new, effective interactions. Using Wegner’s approach they enter the generator and are eventually integrated out. We shall introduce a more general form of $\eta(l)$ below.
In order to see what kind of effective interactions emerge, let us commute the initial Hamiltonian (\[eq:kondo\_hamiltonian\]) and the generator of Eq. (\[eq:eta\_0\]). After some calculation we obtain the following Hamiltonian
$$\begin{aligned}
{\cal H}(l) &= \sum_{k\sigma} \varepsilon_k(l) :c^\dagger_{k\sigma}
c_{k\sigma}: + \frac{1}{2N} \, \sum_{kq} \chi_{kq}(l) \, :{\bf
S}_{k-q} \cdot {\bf S}_{q-k}: + E_c(l)
\notag \allowdisplaybreaks \\%[10pt]
&+\frac{1}{2N} \, \sum_{kq \, \alpha \beta} J_{kq}(l) \, {\bf S}_{k-q}
\cdot {\boldsymbol \sigma}_{\alpha \beta} \, :c^\dagger_{k \alpha}
c_{q \beta}: + \frac{1}{4 N^2} \sum_{kpq \sigma} M_{kpq}(l) :{\bf
S}_{k-p} \cdot {\bf S}_{p-q}: \, :c^\dagger_{k \sigma} c_{q \sigma}:
\notag \allowdisplaybreaks \\%[10pt]
&+ \frac{1}{4 N^2} \sum_{kpq \alpha \beta} i D_{kpq}(l) \, ( {\bf
S}_{k-p} \times {\bf S}_{p-q} ) \cdot {\boldsymbol \sigma}_{\alpha
\beta} :c^\dagger_{k \alpha} c_{q \beta}:
\notag \allowdisplaybreaks \\[10pt]
&= {\cal H}_e (l) + {\cal H}_S (l) + E_c (l) + {\cal H}_J (l) + {\cal
H}_M (l) + {\cal H}_D(l),
\label{eq:flow_hamiltonian}\end{aligned}$$
where $:{\cal X}:$ denote operators resulting from a decoupling scheme which we shall discuss later.
Before we proceed let us take a closer look at equation (\[eq:flow\_hamiltonian\]). The first line represents the block diagonal part of the Hamiltonian since electron and spin operators are decoupled. It contains a complicated RKKY-like spin interaction term between the local moments. The second and third line comprise the nondiagonal or interaction part. Aside from the Kondo coupling we get interactions between the local moments which are either symmetric or antisymmetric with respect to interchange of the sites. Correspondingly, the first one couples to the electronic charge density, whereas the second one couples to the electronic spin density. We restrict ourselves to these terms because they are the most important ones in the regime of small interaction strength $J$. That way the above Hamiltonian becomes flow invariant and Eq. (\[eq:flow\_hamiltonian\]) is valid for all flow parameters $l$. For $l = 0$ it represents the initial Hamiltonian (\[eq:kondo\_hamiltonian\]). This implies the following initial values of the parameters $$\begin{aligned}
\varepsilon_k(l = 0) &= \varepsilon_k \, , \qquad J_{kq}(l = 0) = J
\notag \allowdisplaybreaks \\[10pt]
\chi_{kq}(l = 0) &= 0 \, , \qquad M_{kpq}(l = 0) = 0
\notag \allowdisplaybreaks \\[10pt]
D_{kpq}(l = 0) &= 0 \, , \qquad E_c(l = 0) = \sum_k \varepsilon_k
n_k.\end{aligned}$$
The prefactor of any operator term of Eq. (\[eq:flow\_hamiltonian\]) controls the strength of the respective operator. Within the framework of the flow equation method they are determined by corresponding differential equations. With the choice of the generator $\eta(l)$ we can control which of these terms are kept and which are to be vanished. Since the aim of our renormalization procedure is a blockdiagonal Hamiltonian in which electron and spin operators are decoupled, we have to remove all terms describing interactions between electron and spin operators. The generator $\eta(l)$ of the continuous unitary transformation has to be chosen appropriately.
With this in mind we can now write down the generator $\eta(l)$. Using Wegner’s approach we have to take into account the generated, effective interactions. The generator reads
$$\begin{aligned}
\eta(l) = \eta^J(l) + \eta^M(l) + \eta^D(l)
&= \frac{1}{2N} \sum_{kq \alpha \beta} \eta^J_{kq}(l) \; {\bf S}_{k-q}
\cdot {\boldsymbol \sigma}_{\alpha \beta} \, :c^\dagger_{k \alpha}
c_{q \beta}: + \frac{1}{4 N^2} \sum_{kpq \sigma} \eta^M_{kpq}(l) :{\bf
S}_{k-p} \cdot {\bf S}_{p-q}: \, :c^\dagger_{k \sigma} c_{q \sigma}:
\notag \\[10pt]
&+ \frac{1}{4 N^2} \sum_{kpq \alpha \beta} \eta^D_{kpq}(l) \; i ( {\bf
S}_{k-p} \times {\bf S}_{p-q} ) \cdot {\boldsymbol \sigma}_{\alpha
\beta} \, :c^\dagger_{k \alpha} c_{q \beta}:
\label{eq:generator}\end{aligned}$$
and the prefactors $\eta^J_{kq}(l)$, $\eta^D_{kpq}(l)$ and $\eta^M_{kpq}(l)$ are determined by Eq. (\[eq:wegner\_generator\]).
After the transformation, *i.e.* in the limit $l \to \infty$, only the first line of Eq. (\[eq:flow\_hamiltonian\]) remains. It represents the diagonal part ${\cal H}_0(l)$. This effective Hamiltonian can be used to easily calculate physical properties. The nondiagonal part ${\cal H}_1(l)$ vanishes for $l \to \infty$ and the effective Hamiltonian $\tilde{\cal H} := {\cal H}(l = \infty)$ then reads $$\begin{aligned}
\tilde{\cal H} &= \sum_{k\sigma}
\tilde{\varepsilon}_k :c^\dagger_{k\sigma} c_{k\sigma}:
+ \frac{1}{2N} \, \sum_{kq} \tilde{\chi}_{kq} \, :{\bf S}_{k-q} \cdot
{\bf S}_{q-k}: + \tilde{E}_c
\notag \\ %\allowdisplaybreaks \\
&= \tilde{\cal H}_e + \tilde{\cal H}_S + \tilde{E}_c.
\label{eq:eff_modell}\end{aligned}$$ In the following we shall denote all renormalized variables by a tilde. As Eq. (\[eq:eff\_modell\]) tells us the effective model will consist of a one-dimensional noninteracting electron gas and a Heisenberg spin chain with renormalized parameters.
We now have all ingredients needed to derive the flow equations for the parameters of the Hamiltonian. Before doing this we want to look at the approximations that have to be done. Firstly, we neglect interactions of order ${\cal O}(J^3)$ and higher in the Hamiltonian (\[eq:flow\_hamiltonian\]). Secondly, we decouple higher operator products in order to reduce them to those appearing in ${\cal H}(l)$. This gives rise to operator expressions of the form $:{\cal X}:$. They refer to fluctuation operators and mean either a normal order product of fermionic operators or a Hartree-Fock-decoupling scheme of spin operator products $$\begin{aligned}
:c^\dagger_{k \sigma} c_{k \sigma}: &= c^\dagger_{k \sigma} c_{k
\sigma} - \langle c^\dagger_{k \sigma} c_{k \sigma} \rangle,
\allowdisplaybreaks \\[10pt]
:{\bf S}_{k-q} \cdot {\bf S}_{q-k}: &= {\bf S}_{k-q} \cdot {\bf
S}_{q-k} - \langle {\bf S}_{k-q} \cdot {\bf S}_{q-k} \rangle.\end{aligned}$$ The thermodynamic average will here be taken with respect to the effective model $\tilde{\cal H}$, Eq. (\[eq:eff\_modell\]), which describes a decoupled system of a simple Fermi gas (electrons) and a Heisenberg spin chain with long-range interactions. These expectation values are therefore $l$-independent. The decoupling leads to a formal temperature dependence of the flow equations. Here we consider only the ground state properties, *i.e.* $T = 0$. For the sake of simplicity we introduce the abbreviation $S(k-q) := \langle {\bf S}_{k-q} \cdot {\bf S}_{q-k} \rangle$ for the spin correlation function. One may also think of other expectation values like $\langle {\bf S}_{k-q} \times {\bf S}_{q-k} \rangle$ or $\langle {\bf S}_q
\rangle$. Since we consider the limit of small Kondo coupling $J$, the system is in the paramagnetic phase, where no symmetry is broken. Therefore these expectation values vanish.
Evaluating the commutator between the generator (\[eq:generator\]) and the Hamiltonian (\[eq:flow\_hamiltonian\]) we arrive at the flow equations for the parameters of the Hamiltonian. For the sake of clarity the $l$-dependence of all parameters is dropped. The electronic single particle energies $\varepsilon_k$ obey the following differential equation $$\frac{d \varepsilon_k}{d l} = \frac{1}{2 N} \sum_q S(k-q) \,
\eta^J_{kq} \, J_{qk}.
\label{eq:dgl_ek}$$ Here $S(k-q)$ is the local moment’s spin correlation function which has to be evaluated with respect to the renormalized Hamiltonian $\tilde{\cal H}$. It is therefore $l$-independent. As the effective model is not known before the end of the transformation we have to solve all flow equations self-consistently.
For the paramter $\chi_{kq}$ of the effective spin interaction we obtain the following flow equation $$\begin{aligned}
\frac{d \chi_{kq}}{d l} &= (n_k - n_q) \, \eta^J_{kq} \, J_{qk}.
\label{eq:dgl_rkky}\end{aligned}$$ The occupation numbers $n_k$ which enter the above equation are again formed with respect to the effective model. The constant $E_c$ of ${\cal H}_0(l)$ follows $$\begin{aligned}
\frac{d E_c}{d l} &= \frac{1}{N} \sum_{kq} (n_k - n_q) \, S(k-q) \,
\eta^J_{kq} \, J_{qk}.
\label{eq:dgl_const}\end{aligned}$$ We restrict the renormalization of the effective interaction terms to contributions of order ${\cal O}(J^2)$. Therefore both coupling parameters $D_{kpq}$ and $M_{kpq}$ obey the same flow equation $$\begin{aligned}
\frac{d D_{kpq}}{d l} &= \frac{1}{2} (\eta^J_{kp} J_{pq} + \eta^J_{qp}
J_{kp}) -(\varepsilon_k - \varepsilon_q) \, \eta^D_{kpq}.\end{aligned}$$ The first term is responsible for the generation of the effective coupling while the second contribution, which is always negative, ensures the vanishing at the end of the renormalization procedure. Finally for the flow equation of the Kondo coupling we find $$\begin{aligned}
\frac{d J_{kq}}{d l} &= -(\varepsilon_k - \varepsilon_q) \eta^J_{kq}
\notag \allowdisplaybreaks \\
&+ \frac{1}{N} \sum_p \left( n_p - \frac{1}{2} \right) (\eta^J_{kp}
J_{pq} + \eta^J_{qp} J_{pk})
\notag \allowdisplaybreaks \\
&+ \frac{3}{8N} \sum_p (\eta^J_{kp} D_{pkq} + \eta^J_{qp} D_{pqk})
\notag \allowdisplaybreaks \\
&+ \frac{3}{8N} \sum_p (\eta^D_{kqp} J_{pk} + \eta^D_{qkp} J_{pk})
\notag \allowdisplaybreaks \\
&- \frac{1}{8N} \sum_p (\eta^J_{kp} D_{p,p+q-k,q} + \eta^J_{qp}
D_{p,p+k-q,k})
\notag \allowdisplaybreaks \\
&- \frac{1}{8N} \sum_p (\eta^D_{k,p+q-k,p} J_{pk} +
\eta^D_{q,p+k-q,p} J_{pk}),
\label{eq:dgl_kondo}\end{aligned}$$ where we have taken into account correction terms up to order ${\cal O}(J^3)$. Therefore we expect to find reasonable results only in the parameter regime of small coupling strength $J/t$. As this ratio increases further correction terms have to be included. The flow equations (\[eq:dgl\_ek\]) to (\[eq:dgl\_kondo\]) represent a closed system of first order differential equations, whose solution can only be found by numerical integration.
Approximations for the effective model {#subsec:sbmft}
--------------------------------------
In the preceeding subsection we have derived flow equations for the parameters of the Hamiltonian. As to solve them we still need an analytical expression for the spin correlation function $S(k-q)$. As it describes spin correlations of the effective model, we are dealing here with a one-dimensional Heisenberg chain with long-range interactions whose exact solution is not known. Hence, we have to resort to further approximations. We stress here that this is the most crucial approximation within our approach because it strongly affects all renormalized quantities. Since the spin interaction is the result of the continuous unitary transformation it is not known until the transformation is completely performed.\
As our approach is only valid for small $J/t$, *i.e.* for the paramagnetic metallic phase with no broken symmetry, we use the Schwinger boson formalism to describe the spin system [@arovas_auerbach]. It preserves the rotational invariance of the spin Hamiltonian. The spin operators are expressed in terms of Schwinger bosons $a_{i \sigma}$ and $a^\dagger_{i \sigma}$ according to $$S^\gamma_i = \frac{1}{2} \; \sum_{\sigma \sigma'} a^\dagger_{i \sigma}
\, \sigma^\gamma_{\sigma \sigma'} a_{i \sigma'},$$ where $\sigma^\gamma_{\sigma \sigma'}$ stands for the Pauli spin matrix. Since the occupation number for bosons is not restricted, a local constrained of the form $\sum_\sigma \; a^\dagger_{i \sigma} a_{i \sigma} = 2 S$ must be enforced.
We follow here the procedure of Trumper *et al.* [@trumper_manuel_gazza_ceccatto] or of Ceccatto *et al.* [@ceccatto_gazza_trumper] and introduce two fields $$\begin{aligned}
A_{ij} &= \frac{1}{2} \; \sum_\sigma \sigma \, a_{i \sigma} a_{j
\bar{\sigma}} = -A_{ji}\\
\intertext{and}
B_{ij} &= \frac{1}{2} \; \sum_\sigma a^\dagger_{i \sigma} a_{j \sigma}
= B^\dagger_{ji}\end{aligned}$$ describing antiferro- and ferromagnetic correlations, respectively ($\bar{\sigma} = -\sigma$). This yields to the following Hamiltonian $$\tilde{\cal H}_S = \sum_{ij} \; J_{ij} \, {\cal N} (B^\dagger_{ij}
B_{ij}) - A^\dagger_{ij} A_{ij}.$$ The expression ${\cal N} ({\cal O})$ stands for a normal order product of boson operators. The Hamiltonian is now biquadratic with respect to the Schwinger boson operators. We use a mean field theory in order to decouple the biquadratic terms. By using the mean field parameters $\langle B_{ij} \rangle$ and $\langle A_{ij} \rangle$ and replacing the local constrained by a global one we obtain a Hamiltonian which can easily be diagonalized via a Bogolubov transformation. Introducing new boson operators $\alpha_{k \sigma} = u_k \,
a_{k \sigma} + i \sigma v_k \, a^\dagger_{-k \bar{\sigma}}$ we obtain $$\tilde{\cal H}_S = \sum_{q \sigma} \, \omega_q \, \alpha^\dagger_{q
\sigma} \alpha_{q \sigma} + \frac{1}{2} \sum_{q \sigma} \omega_q,
\label{eq:sbmf_hamiltonian_operatorform_diagonal}$$ with $\omega(q) = \sqrt{(\gamma_B(q)-\lambda)^2-\gamma^2_A(q)}$ representing the energies of the elementary excitations $\alpha^\dagger_{q \sigma}$ of the spin system. Here the quantities $\gamma_A(q) = \frac{i}{2} \, \sum_{R_{ij}}
J_{ij} \langle A_{ij} \rangle \, \text{e}^{i q R_{ij}}$ and $\gamma_B(q) =
\frac{1}{2} \sum_{R_{ij}} J_{ij} \langle B_{ij} \rangle \, \text{e}^{i q
R_{ij}}$ are used. The mean field parameters $\langle B_{ij} \rangle$ and $\langle A_{ij} \rangle$ and the Lagrange parameter $\lambda$ have to be determined selfconsistently by solving the corresponding saddle point equations.
Finally we find an analytic expression for the spin correlation function which for $T = 0$ reads $$S(q)_{T = 0} = \frac{1}{4 N} \sum_k \left( \cosh\left[ 2(\theta_k -
\theta_{k+q}) \right] - 1 \right),
\label{eq:sq_0}$$ with $\theta_k = - \frac{1}{2} \tanh^{-1} \left(
\frac{\gamma_A(k)}{\gamma_B(k)-\lambda} \right)$.\
Compared to methods like the Bethe ansatz for the nearest-neighbour Heisenberg chain the approximative Schwinger boson treatment discussed above has the advantage that as many interaction terms as possible can be taken into account. With the approximation for the effective model we are able to describe the one-dimensional KLM consistently within the framework of the flow equation method. Any physical quantity we are interested in can be evaluated within the present approach. Especially, we emphasise that nothing has to be put in by hand.
Expectation values and correlation functions {#subsec:correlation_functions}
--------------------------------------------
We now turn to the calculation of expectation values and correlation functions. In this subsection we give the essentials for the derivation of certain important expectation values and correlation functions. We shall discuss the results in Sec. \[sec:results\].
The retarded Green’s function between operators $A$ and $B$ is in general defined as the following commutator or anticommutator relation $$G_{AB}(t) = -i \theta(t) \langle \langle A(t); B \rangle \rangle =
-i \theta(t) \langle [A(t), B]_\pm \rangle,$$ depending on the statistics under consideration. The thermodynamic average and the time-dependence have to be taken with respect to the full Hamiltonian. One can exploit the invariance of the trace under unitary transformations and obtains $$G_{AB}(t) = -i \theta(t) \langle \langle \tilde{A}(t); \tilde{B}
\rangle \rangle_{\tilde{\cal H}}.$$ Now the thermodynamic average and the time-dependence are taken with respect to the effective model. According to the transformation of the Hamiltonian we also have to transform the operators. They obey a similar flow equation as the Hamiltonian $$\frac{dA(l)}{dl} = \left[\eta(l), A(l) \right].
\label{eq:operatortrafo}$$
The commutation between $\eta(l)$ of Eq. (\[eq:generator\]) and the electron operator $c_{k \sigma}$ leads to the following operator structure $$\begin{aligned}
c_{k \sigma} (l) = \alpha_k (l) \, c_{k \sigma}
&+ \frac{1}{N} \sum_{q} \sigma \, \gamma_{kq} (l) \,
S^z_{k-q} c_{q \sigma} \notag \\
&+ \frac{1}{N} \sum_{q} \gamma_{kq} (l) \,
S^{-\sigma}_{k-q} c_{q \bar{\sigma}},
\label{eq:operator_e}\end{aligned}$$ where we have taken only the correction terms into account that couple to one local moment. The initial conditions of the parameters are $\alpha_k(l = 0) =
1$ and $\gamma_{kq}(l = 0) = 0$. We transform the spin operator according to $$\begin{aligned}
S^z_i (l) = \beta (l) S^z_i &+ \frac{1}{N} \sum_{kq
\sigma} \, \zeta_{kq} (l) \; \sigma S^\sigma_i
\text{e}^{i(k-q)R_i} \, c^\dagger_{k \bar{\sigma}} c_{q \sigma}
\label{eq:operator_sz_l} \\
S^\sigma_i (l) = \beta (l) S^\sigma_i &+
\frac{1}{N} \sum_{kq \sigma'} \, \zeta_{kq} (l) \; \sigma
S^\sigma_i \, \text{e}^{i(k-q)R_i} \, \sigma' \, c^\dagger_{k \sigma'} c_{q
\sigma'} \notag \\
&+ \frac{2}{N} \sum_{kq} \zeta_{kq} (l) \; \sigma \, S^z_i \,
\text{e}^{i(k-q)R_i} \, c^\dagger_{k \sigma} c_{q \bar{\sigma}}.
\label{eq:operator_ssigma_l}\end{aligned}$$ Here the initial parameters are $\beta(l = 0) = 1$ and $\zeta_{kq} (l = 0) =
0$.
In order to derive the flow equations for the parameters of the operator transformations we have to use an equivalent decoupling scheme as for the Hamiltonian. We finally arrive at the following differential equations $$\begin{aligned}
\frac{d \alpha_k}{dl} &= \frac{1}{2 N} \sum_q S(k-q) \, \eta^J_{kq}
\gamma_{qk}
\label{eq:flussgleichung_alpha} \\[10pt]
\frac{d \gamma_{kq}}{dl} &= \frac{1}{2} \, \eta^J_{qk} \, \alpha_{k}
\label{eq:flussgleichung_gamma}\end{aligned}$$ for the parameters of the electron operator transformation and $$\begin{aligned}
\frac{d \beta}{dl} &= -\frac{2}{N^2} \, \sum_{kq} \eta^J_{kq} \,
\zeta_{kq} \, n_k ( 1 - n_q )
\label{eq:flussgleichung_beta} \\[10pt]
\frac{d \zeta_{kq}}{dl} &= \frac{1}{2} \, \beta \, \eta^J_{qk}
\label{eq:flussgleichung_zeta}\end{aligned}$$ for the parameters of the spin operator transformations. We notice that the spin correlation function $S(k-q)$ of the effective model enters the flow equation of $\alpha_k(l)$ whereas the occupation numbers $n_k$ govern the flow equation of $\beta(l)$. We restrict the flow equations for the correction terms to first order contributions in the Kondo coupling. Going beyond this approximation could bring us up against the violation of certain summation rules which have to be fullfilled. We can combine the above equations to obtain flow invariant expressions. The expectation values $S(k-q)$ and $n_k$ are taken with respect to the effective Hamiltonian and are therefore $l$-independent. We arrive at $$\alpha^2_k(l) + \frac{1}{N}\sum_{q} S(k-q) \gamma^2_{kq}(l) = 1
\label{eq:const_el}$$ and $$\beta^2 (l) + \frac{4}{N^2} \sum_{kq} \zeta^2_{kq} (l) \,
n_k ( 1 - n_q ) = 1,
\label{eq:const_spin}$$ which displays the unitarity of the transformation.
After determining the operator transformation we are now able to calculate static and dynamic correlation functions that characterise the ground state properties of the one-dimensional KLM. One of the most important quantities is the momentum distribution $n(k)$ which reads $$\begin{aligned}
n(k) = \langle c^\dagger_{k\sigma} c_{k\sigma} \rangle =
\tilde{\alpha}^2_k \, n_k + \frac{1}{N} \sum_q \,
\tilde{\gamma}^2_{kq} \, S(k-q) \, n_q.
\label{eq:nk}\end{aligned}$$ For a Luttinger liquid we expect a continuous behaviour with respect to the momentum $k$ and a power law singularity at the Fermi momentum. The position of this singularity fixes the size of the Fermi surface.
The static correlation function of the local moments $S_{ff}(q)$ indicates the phase transition from the paramagnetic phase into the ferromagnetic phase on increasing the Kondo coupling $J$. Within our approach it is given by $$\begin{aligned}
S_{ff}(q) &= \langle {\bf S}_q \cdot {\bf S}_{-q} \rangle \notag \\
&= \tilde{\beta}^2 \, S(q) + \frac{4}{N^2} \, \sum_{kp}
\tilde{\zeta}^2_{kp} \, S(k-p+q) \, n_k (1-n_p).
\label{eq:Sff}\end{aligned}$$ We can also evaluate the static charge correlation function $C(q)$ and the static spin correlation function $S_{cc}(q)$ of the electrons. Their rather lengthy expressions are given in the appendix.
The flow equation formalism allows us to calculate dynamic quantities. The first quantities we look at are the one-particle spectral functions $A_\pm (k,
\omega)$ of the conduction electrons which measure occupied and empty states of the conduction electrons. $$\begin{aligned}
A_+ (k, \omega) &= \int_{-\infty}^{\infty} dt \langle c_{k \sigma} (t)
\, c^\dagger_{k \sigma} \rangle \, \text{e}^{i \omega t}
\notag \\[10pt]
&= \tilde{\alpha}^2_k \, (1-n_k) \, \delta(\omega -
\tilde{\varepsilon}_k)
\notag \\
&+ \frac{1}{2 N^2} \, \sum_{qp} \, \tilde{\gamma}^2_{kq} \, ( u_p \,
v_{k+p-q} - v_p \, u_{k+p-q} )^2 \notag \\
&\hspace{1.75cm} \times (1-n_q) \, \delta( \omega -
\tilde{\varepsilon}_q - \tilde{\omega}_p - \tilde{\omega}_{k+p-q})
\allowdisplaybreaks \\[10pt]
A_- (k, \omega) &= \int_{-\infty}^{\infty} dt \langle c^\dagger_{k
\sigma} (t) \, c_{k \sigma} \rangle \, \text{e}^{i \omega t}
\notag \\[10pt]
&= \tilde{\alpha}^2_k \, n_k \, \delta(\omega - \tilde{\varepsilon}_k)
\notag \\
&+ \frac{1}{2 N^2} \, \sum_{qp} \, \tilde{\gamma}^2_{kq} \, ( u_p \,
v_{k+p-q} - v_p \, u_{k+p-q} )^2 \notag \\
&\hspace{2.5cm} \times n_q \, \delta( \omega - \tilde{\varepsilon}_q +
\tilde{\omega}_p + \tilde{\omega}_{k+p-q}).\end{aligned}$$ Here $u_k$ and $v_k$ are the coefficients of the Bogolubov transformation used to diagonalize the Schwinger boson mean field Hamiltonian (\[eq:sbmf\_hamiltonian\_operatorform\_diagonal\]). The spectral functions $A_\pm (k, \omega)$ comprise two contributions. The first term ($\sim
\tilde{\alpha}^2$) represents a coherent quasiparticle excitation. The second term is an incoherent background. It is important to note that the elementary excitations of the spin system of the effective Hamiltonian $\tilde{\omega}_q$ enter the latter contribution. The electronic density of states defined by $$\rho(\omega) = - \frac{1}{N} \sum_k \; \frac{1}{\pi} \; \text{Im} \,
G(k,\omega),
\label{eq:rho}$$ with $G(k,\omega)$ being the electronic Green’s function, can also be calculated.\
Another important quantity is the dynamic spin structure factor $S_{ff}(q,\omega)$ of the local moments
$$\begin{aligned}
S_{ff}(q,\omega) = \int_{-\infty}^{\infty} dt \, \langle {\bf S}_q (t)
\cdot {\bf S}_{-q} \rangle \, \text{e}^{i \omega t} &= \frac{1}{2 N}
\, \sum_{p} \, \tilde{\beta}^2 \, ( u_p \, v_{p+q} - v_p \, u_{p+q}
)^2 \, \delta(\omega - \tilde{\omega}_p - \tilde{\omega}_{p+q})
\allowdisplaybreaks \notag \\
&+ \frac{2}{N^3} \sum_{kpp'} \tilde{\zeta}^2_{kp} (u_{p'} v_{p'+k-p+q}
- v_{p'} u_{p'+k-p+q} )^2 \, n_k \, (1-n_p) \notag \\
& \hspace{4cm} \times \delta(\omega - \tilde{\omega}_{p'} -
\tilde{\omega}_{p'+k-p+q} + \tilde{\varepsilon}_k -
\tilde{\varepsilon}_p).
\label{eq:S_qw}\end{aligned}$$
The first line describes only the spin excitations of the effective model in terms of Schwinger bosons. The second line of Eq. (\[eq:S\_qw\]) results from the coupling of the local moments to electronic particle-hole excitations of the effective Hamiltonian. In addition, we can also calculate the dynamic spin structur factor of the conduction electrons $S_{cc}(q,\omega)$ which is found in the appendix.
Results {#sec:results}
=======
After having derived theoretical expressions for various correlation functions from the flow equation approach we now turn to present the outcome of the numerical solution of the flow equations (\[eq:dgl\_ek\]) - (\[eq:dgl\_kondo\]) and (\[eq:flussgleichung\_alpha\]) - (\[eq:flussgleichung\_zeta\]) . We start with the result for the parameters of the Hamiltonian and subsequently show our findings for static and dynamic correlation functions. We shall show to what extend the statics reflects the expected Luttinger liquid behaviour. We also clarify the possibility of the approach to describe the quantum phase transition on increasing coupling strength.
Parameters of the Hamiltonian {#subsec:parameters}
-----------------------------
In order to solve Eqs. (\[eq:dgl\_ek\]) - (\[eq:dgl\_kondo\]) we used a Runge Kutta algorithm. The complexity of the differential equation restricted our system size to $N = 120$. Remember that the spin correlation function $S(k-q)$ which enters the flow equations has to be calculated with respect to the effective model (\[eq:eff\_modell\]). Therefore the parameters of the Hamiltonian had to be determined self-consistently.
The spin correlation function $S(q)$ of the effective model plays an important role. We therefore start our discussion with $S(q)$ which is shown in Fig. \[fig:Sq\]. The main feature is the dominant peak that shows up exactly at the wave vector $q = 2 k_F^c = n_c \pi$, where $k_F^c$ is the Fermi momentum of the conduction electrons. As we shall see later the pronounced structure has severe consequences for other quantities that are related to $S(q)$. The pronounced peak is due to the special excitation spectrum of the Schwinger bosons. The other main property of $S(q)$ is the vanishing ferromagnetic component ($q = 0$) which can easily be understood from Eq. (\[eq:sq\_0\]).
The elementary excitations $\tilde{\omega}_q$ of the spin system of $\tilde{\cal H}$ are shown in Fig. \[fig:wq\]. They exhibit a small but finite gap at $q = k_F^c \mod \pi$. This small gap is responsible for the strong peak in $S(q)$. It manifests the rotational invariance of the ground states and is an artifact of the Schwinger boson approach as we are dealing here with half integer spins ($S = 1/2$) which may have a gapless excitation spectrum. Nevertheless, within the Schwinger boson approach a vanishing gap would give rise to a ground state with broken symmetry that contradicts the assumption of a rotational invariant paramagnetic phase. However, the important point is the position of this gap. It determines the maximum of the spin correlation function which turns out to be at the expected position. Therefore we assume that a description in terms of spinons should not change these results decisively. Looking at Eq. (\[eq:sq\_0\]) we see that always pairs of excitations enter the equation for $S(q)$ so that the maximum of the spin correlation function is found at $q = 2 k_F^c = n_c
\pi$.
At this point we add that we found solutions for the saddle point equations of the SBMFT only in the parameter regime $1/2 < n_c < 1$. The case of half filling is special in the sense that there exists a gapped spin liquid phase. It remains an open question whether the present approach can also be used to describe this phase. Below $n_c = 1/2$ the dominance of the ferromagnetic components in the RKKY coupling ${\cal J}_{ij}$ prevents a solution of the saddle point equations of the SBMFT.
Finally we discuss the renormalized electronic single-particle energies. The dispersion relation $\tilde{\varepsilon}_k$ is presented in Fig. \[fig:ek\]. We assume the unrenormalized single-particle energies to follow a tight binding dispersion $\varepsilon_k = -2t(\cos k-1)$, where we set the bottom of the band equal to zero and $t = 1/2$. We recognise two basic features for $\tilde{\varepsilon}_k$. The first is a broadening of the band. The effective band width is enlarged compared with the original band width $W = 4t$. The other one is a decreasing density of states at the Fermi momentum $k = k_F^c$ and at $k = \pi-k_F^c$. This property is mainly due to the dominant peak structure in the spin correlation function $S(q)$ at $q = 2
k_F^c$. The wave vector $q = 2 k_F^c$ connects the two points of the Fermi surface. Therefore the energies near the Fermi surface become more strongly renormalized than energies near the band edge. The pseudo-gap like structure at $k = k_F^c$ is thus due to the strong spin fluctuations at $q = 2
k_F^c$. As we are going to see this behaviour shall have consequences for the electronic density of states $\rho(\omega)$. As the Luttinger liquid theory expects $\rho(\omega)$ to vanish at $\omega = 0$ a decreasing density of states in the renormalized electron spectrum is reasonable. In order to resolve the observed structures in the renormalized electron spectrum we need to examine larger systems.
Static properties {#subsec:statics}
-----------------
Let us now study the static correlation functions calculated in the last section. The first quantity we want to consider is the momentum distribution function $n(k)$ which is shown in Fig. \[fig:nk\]. We obtain meaningful results only for couplings up to $J/t \approx 1$. This signals a breakdown of the flow equation treatment. In order to get better results for larger ratios $J/t$ we need to go beyond the third order corrections in the flow equations. Looking at $n(k)$ we notice that it is smeared out around $k_F^c = n_c \pi / 2$. However, we can not decide whether these results support the expected Luttinger liquid picture or not. The special behaviour of $n(k)$ at $k = k_F^c$ may be due to the dominant peak structure of $S(q)$. Since it is difficult to resolve the sharp peak of $S(q)$ appropriately for a finite system, we are not sure whether the artifact at $k
= k_F^c$ is due to the finite system size or the approximations. Nevertheless, the shape of the momentum distribution function tends to support a small Fermi surface, because there is no feature at $k =
\left(n_c+1\right)\pi/2$. Additionally, one may question if the effective model (\[eq:eff\_modell\]) is capable of describing a large Fermi surface. The system of conduction electrons within the effective model has a Fermi momentum $n_c \pi/2$. Therefore a singularity in the momentum distribution function is likely to be expected only at the point $k = k_F^c$.
We can get further information from the charge correlation function $C(q)$. The results are depicted in Fig. \[fig:Cq\]. As we expect for small couplings $J/t$ the function $C(q)$ takes the form of a noninteracting one-dimensional electron gas with a kink at $q = 2 k_F^c = n_c
\pi$. Increasing $J/t$ leads to a cusp-like behaviour of $C(q)$ at $q = 2
k_F^c$. In addition, the slope at $q = 0$ drops with growing interaction stregth $J/t$. The results displayed in Fig. \[fig:Cq\] agree qualitatively with the findings from numerical treatments [@tsunetsugu_rmp; @moukouri_caron] in the examined parameter regime ($J/t \lesssim 1$). This supports the Luttinger liquid picture of our description.
The charge correlation function gives us the possibility to derive the parameter $K_\rho$ of the Luttinger liquid theory. This parameter is connected to the slope of $C(q)$ at $q = 0$ via the relation [@daul_noack] $$K_\rho = \pi \, \frac{\partial C(q)}{\partial q} \, \bigg|_{q=0}.$$
The outcome is depicted in Fig. \[fig:K\_r\_J\] as a function of the Kondo coupling $J/t$. As we have already mentioned before the slope of $C(q)$ at $q
= 0$ decreases with growing coupling strength (up to the allowed value of $J/t
\lesssim 1$). For vanishing interaction strength $K_\rho \to 1$ corresponding to a noninteracting electron gas. This can be understood from the equation for $C(q)$ given in the appendix. Due to the flow equation (\[eq:flussgleichung\_gamma\]) for the parameter $\gamma_{kq}$ of the electron operator transformation all terms vanish which represent corrections to the charge correlation function of independent electrons. Our findings are in qualitative agreement with recent numerical results from DMRG calculations [@xavier_miranda]. Xavier and Miranda find a minimum of $K_\rho(J)$ at $J/t \approx 1.5$. Remember that our largest possible coupling is smaller than $1.5$. Quantitatively our results are always considerably smaller than the values found in ref. [@xavier_miranda]. Another work by Shibata *et al.* [@shibata_jp] gives results for large $J/t$. In contrast to our findings and to those of Xavier and Miranda [@xavier_miranda] these authors expect $K_\rho \to 0$ in the limit $J/t \to 0$.
We also considered the dependence of $K_\rho$ on the band filling $n_c$. This is depicted in Fig. \[fig:K\_r\_nu\]. The lower possible value of the band filling is $n_c = 1/2$ as we do not obtain a solution of the flow equations below this value within the present approach. At small values $J/t$ we find a monotonic decrease by lowering $n_c$. Again we find qualitative agreement with Xavier and Miranda [@xavier_miranda]. As we already mentioned in the last discussion our values for $K_\rho$ are considerably smaller compared to the numerical data. For larger $J/t$ the behaviour deviates even qualitatively from the numerical DMRG data. Whereas in ref. [@xavier_miranda] for all values of $J/t$ a monotonic increase was obtained on increasing $n_c$, we find a maximum in the function $K_\rho(n_c)$.
The magnetic properties of the one-dimensional KLM are significant for the determination of the phase transition from the paramagnetic metallic phase into the ferromagnetic phase. The spin correlation function for the conduction electrons $S_{cc}(q)$ as well as for the local moments $S_{ff}(q)$ show a characteristic increase of the ferromagnetic component $q = 0$ on approaching the quantum phase transition.
The spin correlation function of the electrons $S_{cc}(q)$ is shown in Fig. \[fig:Scc\]. The strong peak at $q = 2 k_F^c$ results from the sharp maximum of the spin correlation function $S(q)$ of the effective model. This can easily be seen from the expression of $S_{cc}(q)$ given in the appendix. Another characteristic is the finite weight of the ferromagnetic component $q = 0$ which is directly connected with the occurence of the quantum phase transition. On approaching the critical $J/t$ the maximum of $S_{cc}(q)$ at $q = 2 k_F^c$ loses weight in favour of the ferromagnetic component. This behaviour marks the phase transition [@tsunetsugu_rmp]. As we already mentioned the present approach is restricted to values of $J/t \lesssim 1$. These values are too small compared to the value at the transition point which is $J/t \lesssim 2.5$ for $n_c =
2/3$ [@shibata_jp]. Nevertheless, we observe some tendency towards the magnetic phase transition. As in the case of the charge correlation function we compare our results with numerical data from [@moukouri_caron]. One can clearly see the qualitative agreement between the two approaches, although our findings tend to be smaller than the DMRG results. This is important if one considers the points at $q = 0$. The increase of the ferromagnetic component, which signals the tendency towards the quantum phase transition, turns out to be comparably weak.
The situation we have just described is also characteristic for the spin correlation function $S_{ff}(q)$ of the local moments which is drawn in Fig. \[fig:Sff\]. For small couplings we see that the corrections to the spin correlation function of the effective model $S(q)$ are negligibly small. Even for larger values of $J/t$ we find only small corrections. The vicinity of the ferromagnetic component $q = 0$ is shown in the inset. Nevertheless, the qualitative behaviour is once again in agreement with numerical results [@tsunetsugu_rmp] though the values are somewhat larger. Again, the ferromagnetic component gets an increasing weight while the $q = 2 k_F^c$ component is suppressed.
Dynamic properties {#subsec:dynamics}
------------------
In the last section we have presented the results for static expectation values and correlation functions. We have found that our results are in qualitative agreement with numerical data for small couplings $J/t$. The flow equation method sets us in the position to calculate not only static but also dynamic correlation functions. Within our approach the dynamics of the KLM is described in terms of the effective model (\[eq:eff\_modell\]). Since $\tilde{\cal H}$ is blockdiagonal the dynamics for electrons and local spin moments seperate. The SBMFT allows us, at least approximately, to characterise the excitations of the spin system. The excitations of the KLM are determined by a noninteracting Fermi gas (conduction electrons) and the Schwinger bosons. In this section we shall add new aspects to the results obtained by [@shibata_tsunetsugu].
We start with the dynamic properties of the conduction electrons. The first quantities we want to consider are the electronic spectral functions $A_\pm (k,
\omega)$ which can be measured in XPS and inverse XPS experiments. The outcome is shown in Fig. \[fig:dynamik\_el\]. The energy is measured with respect to the Fermi-energy of the conduction electrons $\varepsilon^c_F = n_c \pi/2$. As we have already mentioned in the last section both functions consist of two parts. A coherent quasiparticle-like contribution embodied by the finite peak which has a weight $\tilde{\alpha}_k$. Its position is simply given by the renormalized single-particle energies $\tilde{\varepsilon}_k$. The incoherent background contains pairs of elementary excitations of the spin system of $\tilde{\cal H}$. This follows directly from Eq. (\[eq:operator\_e\]) since within the Schwinger boson approach for the effective spin system the corrections to the spectral functions $A_\pm (k, \omega)$ are always connected to the creation (annihilation) of pairs of bosons. The coupling to the continuum of Schwinger boson excitations and the results for $\tilde{\gamma}_{kq}$ give rise to the two maxima around the quasiparticle-like peak.
The electronic density of states $\rho(\omega)$ is an important quantity which shows a characteristic behaviour for Luttinger liquids. It is drawn in Fig. \[fig:rho\]. Again, the energy is measured with respect to $\varepsilon^c_F = n_c \pi/2$. We find two minima, one in the vicinity of the Fermi level of the conduction electrons, $\omega = 0$, the other above the Fermi level. This behaviour follows from the pseudogap-like behaviour of the renormalized single-particle energies $\tilde{\varepsilon}_k$. In contrast to our findings DMRG studies from Shibata and Tsunetsugu [@shibata_tsunetsugu] do not yield a minimum but rather a peak structure just below $\omega = 0$ indicating the development of a pseudo gap. However, their results were performed at finite temperatures. The Luttinger liquid theory predicts a density of states following $\rho(\omega) \sim
|\omega|^\alpha$, $0 < \alpha < 1$ in the vicinity of $\omega = 0$. As can be seen from Fig. \[fig:rho\] there is no real vanishing of $\rho(\omega)$ at $\omega = 0$. As we are dealing here with a finite system size we are not able to resolve $\rho(\omega)$ near $\omega = 0$ and to verify the expected behaviour.
Let us now turn to the magnetic properties. We want to present the results for the dynamic spin structure factors of the electrons $S_{cc}(q, \omega)$ and the local moments $S_{ff}(q, \omega)$. They describe the magnetic excitations of the coupled system and can be measured by inelastic neutron scattering experiments.
We begin with the electronic dynamic spin structure factor $S_{cc}(q, \omega)$ which consists of a low- and a high-energy part. Both are discussed seperately. The low energy sector, left panel of Fig. \[fig:Sqw\_el\], is characterised by the spin part of the effective model $\tilde{\cal{H}}_S$, *i.e.* the continuum of pair excitations of the Schwinger bosons. The dominant contribution is therefore found at $q = 2 k_F^c$. It is multiplied by a factor $1/4$ for a better comparison. We also see that there are regions where no excitations are possible. Furthermore, the gap in the spectrum of the elementary excitations $\tilde{\omega}_q$ leads to a gap in the low-energy part of $S_{cc}(q, \omega)$. The high-energy sector of $S_{cc}(q, \omega)$ is shown in the right panel of Fig. \[fig:Sqw\_el\]. The spectral weights are about 10 times smaller compared to the weights of the low-energy part. The main contribution arises from electronic particle-hole excitations of the effective model. The specific form of this contribution shows therefore the characteristics of a one-dimensional electron gas: a gapless excitation at $q
= 2 k_F^c$ and regions between $0 < q < 2 k_F^c$ where no excitations are possible. In addition to the terms describing pure particle-hole excitations there are also terms involving the elementary excitations $\tilde{\omega}_q$ of the effective spin system. They are responsible for the broadening of the structures in the high energy sector of $S_{cc}(q, \omega)$.\
The DMRG calculations of Shibata *et al.* for $S_{cc}(\omega) = \int
\frac{dq}{2\pi} S_{cc}(q, \omega)$ showed a small peak at very low energies and a larger double peak structure at higher energies [@shibata_tsunetsugu]. We obtain a similar peak structure, but in contrast to the results of [@shibata_tsunetsugu] the spectral weight of the low energy part is much larger than the spectral weight of the high energy part. This does not agree with the picture of an exhaustion of the electronic low-energy spin degrees of freedom due to singlet formation described by [@shibata_tsunetsugu].
Finally we want to discuss the magnetic excitations of the system of local spin moments described by the dynamic spin structure factor $S_{ff}(q,
\omega)$. As in the case of the electronic spin structure factor $S_{cc}(q,
\omega)$ this function comprises a low- and a high-energy part. The first one is again determined by the elementary excitations $\tilde{\omega}_q$ of the spin part of the effective model $\tilde{{\cal H}}_S$. It is depicted in Fig. \[fig:S\_0\_loc\] and possesses the same features as the low-energy part of $S_{cc}(q, \omega)$. From this picture we can clearly see the influence of the low-energy spin excitations. The distinct structure at $q = 2 k_F^c$ gives rise to the pronounced peak in the static spin correlation function $S(q)$. Once again we point out that the energy scale of these excitations are quite small compared with the effective band width of the electrons. The DMRG results of Shibata *et al.* for $S_{ff}(\omega) = \int \frac{dq}{2\pi}
S_{ff}(q, \omega)$ show a large peak structure at very small energies [@shibata_tsunetsugu]. They assume that this is due to collective spin excitations of the Luttinger-liquid. In our approach the low-energy peak is the result of the continuum of elementary excitations of the effective spin system, which we described in terms of Schwinger bosons.
The high-energy part of $S_{ff}(q, \omega)$ is shown in Fig. \[fig:Sqw.loc\]. As in the case of $S_{cc}(q, \omega)$ it is mainly determined by particle-hole excitations of the Fermi sea. At larger couplings $J/t$ the elementary spin excitations $\tilde{\omega}_q$ lead to the broadening of the peak structure. Shibata *et al.* obtain a second peak in the high-energy sector of $S_{ff}(\omega)$ [@shibata_tsunetsugu]. Our approach yields a similar structure in the local spin dynamics, although the spectral weight of the high-energy part is much smaller than the spectral weight of the low-energy part.\
We further note that $S_{ff}(q, \omega)$ exhibits a finite gap. This is an artifact and due to the approximations that we have made for the spin operator transformation. By taking into account higher correction terms the transformed spin operator of the local spin moment couples to the spin operator of the conduction electrons. This gives rise to a gapless mode in $S_{ff}(q,
\omega)$.
conclusion {#sec:conclusion}
==========
In summary, we used the method of continuous unitary transformations (flow equation method) to examine the one-dimensional KLM. The renormalization procedure was employed to integrate out the coupling between conduction electrons and local spin operators. In that way we derived an effective Hamiltonian which consists of an one-dimensional noninteracting electron gas and a Heisenberg chain interacting via an RKKY-like coupling. In order to treat the spin chain we used a Schwinger boson mean field theory (SBMFT). Thereby we were able to calculate static and dynamic correlation functions. The investigation of the electronic momentum distribution revealed a small Fermi surface. We gave arguments, referring to the effective model, why we were not able to obtain the large Fermi surface scenario in our approach. Nevertheless, the static spin and charge correlation functions of the electrons agreed qualitatively with numerical results. In addition we obtained the parameter $K_\rho$ of the Luttinger liquid theory and found also qualitative agreement with recent DMRG calculations. The present approach was restricted to parameter regimes $J/t \lesssim 1$. Although the quantum phase transition from the paramagnetic metallic into the ferromagnetic phase takes place at larger values, we observed some tendency to a stronger ferromagnetic component in the static spin correlation functions. The new aspect of this work was the extension of calculations for dynamic properties by means of the flow equation’s method. We showed that the electronic spectral functions comprised a coherent quasiparticle-like peak determined by the renormalized electronic dispersion relation. The coupling to the low-energy excitation of the effective spin model gave an incoherent background comprising two maxima near the quasiparticle-like peaks. Finally, we also computed the magnetic excitations of both the electrons and the local spins. The corresponding spin structure factors always consisted of a low-energy part, determined by the Schwinger boson pair excitations, and a high-energy part, mostly determined by electronic particle-hole excitations. The latter therefore showed the special features of the one-dimensional Fermi surface. The electronic spin structure factor exhibited a gapless mode at $q = 2 k_F^c$. Our results for the electronic spin dynamics did not agree with the exhaustion picture described by [@shibata_tsunetsugu]. The gapless mode at $q = 2 k_F^c$ should also be seen in the spin structure factor of the local moments. There we argued that further corrections in the spin operator transformation would lead to a gapless mode.
The author would like to thank K. W. Becker, D. Efremov and K. Meyer for helpful discussions and hints. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the research programme SFB 463, Dresden.
correlation functions in the flow equation approach
===================================================
In this appendix we give the rather lengthy expressions for the static and dynamic correlation functions omitted in the text. These are the static charge correlation function $C(q)$ which reads
$$\begin{aligned}
C(q) &= \frac{1}{N} \sum_{k k' \sigma \sigma'} \langle
c^\dagger_{k+q\sigma} c_{k\sigma} c^\dagger_{k'-q\sigma'}
c_{k'\sigma'} \rangle
\allowdisplaybreaks \notag \\%[10pt]
&= \frac{2}{N} \sum_k \, \tilde{\alpha}^2_k \, \tilde{\alpha}^2_{k+q}
\, n_{k+q} (1-n_k)
\allowdisplaybreaks \notag \\%[10pt]
&+ \frac{4}{N^2} \sum_{kp} \, \tilde{\alpha}_k \,
\tilde{\alpha}_{k+q} \, \tilde{\gamma}_{p,k+q} \,
\tilde{\gamma}_{p-q,k} \, S(k-p-q) \, n_{k+q} (1-n_k)
%\allowdisplaybreaks \notag \\%[10pt]
%&
+ \frac{2}{N^2} \sum_{kp} \, \tilde{\alpha}^2_k \,
\tilde{\gamma}^2_{k-q,p} \, S(k-p-q) \, n_k (1-n_p)
\allowdisplaybreaks \notag \\%[10pt]
&+ \frac{2}{N^2} \sum_{kp} \, \tilde{\alpha}^2_k \,
\tilde{\gamma}^2_{k+q,p} \, S(k-p+q) \, n_p (1-n_k)
%\allowdisplaybreaks \notag \\%[10pt]
%&
+ \frac{4}{N^2} \sum_{kp} \, \tilde{\alpha}_k \,
\tilde{\alpha}_{p} \, \tilde{\gamma}_{p+q,k} \,
\tilde{\gamma}_{k-q,p} \, S(k-p-q) \, n_k ( 1 - n_p )
\label{eq:C_q}\end{aligned}$$
The dynamic spin structure factor $S_{cc}(q,\omega)$ of the electrons takes the form $$\begin{aligned}
S_{cc}(q,\omega) &= \int_{-\infty}^{\infty} dt \,
\langle {\bf s}_q (t) \cdot {\bf s}_{-q} \rangle \, \text{e}^{i \omega t}
\allowdisplaybreaks \notag \\%[10pt]
&=\frac{3}{2 N} \sum_k \, \tilde{\alpha}^2_k \,
\tilde{\alpha}^2_{k+q} \, n_{k+q} (1-n_k) \,
\delta(\omega-\tilde{\varepsilon}_k+\tilde{\varepsilon}_{k+q})
\allowdisplaybreaks \notag \\%[10pt]
&- \frac{1}{N^2} \sum_{kp} \, \tilde{\alpha}_k \, \tilde{\alpha}_{k+q}
\, \tilde{\gamma}_{p-q,k} \, \tilde{\gamma}_{p,k+q} \, S(k-p-q) \,
n_{k+q} (1 - n_k) \,
\delta(\omega-\tilde{\varepsilon}_k+\tilde{\varepsilon}_{k+q})
\allowdisplaybreaks \notag \\%[10pt]
&- \frac{1}{2 N^3} \sum_{kpp'} \, \tilde{\alpha}_k \, \tilde{\alpha}_{p}
\, \tilde{\gamma}_{p+q,k} \, \tilde{\gamma}_{k-q,p} \, (u_{p'}
v_{p'+k-p-q}-v_{p'} u_{p'+k-p-q})^2 \, n_k ( 1 - n_p ) %\notag \\
%& \hspace{7cm} \times
\delta(\omega-\tilde{\omega}_{p'}-\tilde{\omega}_{p'+k-p-q}-
\tilde{\varepsilon}_k+\tilde{\varepsilon}_p)
\allowdisplaybreaks \notag \\%[10pt]
&+ \frac{3}{4 N^3} \sum_{kpp'} \tilde{\alpha}^2_k \,
\tilde{\gamma}^2_{k+q,p} \, (u_{p'} v_{p'+k-p+q}-v_{p'} u_{p'+k-p+q})^2
\, n_p (1 - n_k) %\notag \\
%& \hspace{7cm} \times
\delta(\omega-\tilde{\omega}_{p'}-\tilde{\omega}_{p'+k-p+q}-
\tilde{\varepsilon}_p+\tilde{\varepsilon}_k)
\allowdisplaybreaks \notag \\%[10pt]
&+ \frac{3}{4 N^3} \sum_{kpp'} \tilde{\alpha}^2_k \,
\tilde{\gamma}^2_{k-q,p} \, (u_{p'} v_{p'+k-p-q}-v_{p'} u_{p'+k-p-q})^2
\, n_k (1 - n_p) %\notag \\
%& \hspace{7cm} \times
\delta(\omega-\tilde{\omega}_{p'}-\tilde{\omega}_{p'+k-p-q}-
\tilde{\varepsilon}_k+\tilde{\varepsilon}_p)
\allowdisplaybreaks \notag \\%[10pt]
&+ \frac{1}{2 N^3} \sum_{kpp'} \, \tilde{\alpha}_k \, \tilde{\alpha}_{p}
\, (\tilde{\gamma}_{k+q,k} \, \tilde{\gamma}_{p-q,p} +
\tilde{\gamma}_{k-q,k} \, \tilde{\gamma}_{p+q,p})
n_k \, n_p \, (u_{p'} v_{p'+q}-v_{p'} u_{p'+q})^2 \,
\delta(\omega-\tilde{\omega}_{p'}-\tilde{\omega}_{p'+q})
\allowdisplaybreaks \notag \\%[10pt]
&+ \frac{1}{2 N^3} \sum_{kpp'} \, \tilde{\alpha}_k \, \tilde{\alpha}_{p}
\, (\tilde{\gamma}_{k-q,k} \, \tilde{\gamma}_{p-q,p} +
\tilde{\gamma}_{k+q,k} \, \tilde{\gamma}_{p+q,p})
n_k \, n_p \, (u_{p'} v_{p'+q}-v_{p'} u_{p'+q})^2 \,
\delta(\omega-\tilde{\omega}_{p'}-\tilde{\omega}_{p'+q})
\label{eq:Scc_qw}\end{aligned}$$ Here, it can clearly be seen that the second and third line involves only particle hole excitations of the Fermi sea of the effective model. The last two lines represent the low energy sector of $S_{cc}(q, \omega)$ as they include only pair excitations of Schwinger bosons. On integrating over the energy $\omega$ one obtains the expression for the static spin correlation function $S_{cc}(q)$.
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|
---
abstract: |
The Hardy–Littlewood inequality for complex homogeneous polynomials asserts that given positive integers $m\geq2$ and $n\geq1$, if $P$ is a complex homogeneous polynomial of degree $m$ on $\ell_{p}^{n}$ with $2m\leq
p\leq\infty$ given by $P(x_{1},\ldots,x_{n})=\sum_{|\alpha|=m}a_{\alpha
}\mathbf{{x}^{\alpha}}$, then there exists a constant $C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq1$ (which is does not depend on $n$) such that $$\left( {\sum\limits_{\left\vert \alpha\right\vert =m}}\left\vert a_{\alpha
}\right\vert ^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}}\leq
C_{\mathbb{C},m,p}^{\mathrm{pol}}\left\Vert P\right\Vert ,$$ with $\Vert P\Vert:=\sup_{z\in B_{\ell_{p}^{n}}}|P(z)|$. In this short note, among other results, we provide nontrivial lower bounds for the constants $C_{\mathbb{C},m,p}^{\mathrm{pol}}$. For instance we prove that, for $m\geq2$ and $2m\leq p<\infty$, $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq2^{\frac{m}{p}}$$ for $m$ even, and $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq2^{\frac{m-1}{p}}$$ for $m$ odd. Estimates for the case $p=\infty$ (this is the particular case of the complex polynomial Bohnenblust–Hille inequality) were recently obtained by D. Nuñez-Alarcón in 2013.
address:
- |
Departamento de Matemática\
Universidade Federal da Paraíba\
58.051-900 - João Pessoa, Brazil.
- |
Departamento de Matemática\
Universidade Federal da Paraíba\
58.051-900 - João Pessoa, Brazil.
author:
- Gustavo Araújo
- Daniel Pellegrino
title: 'Lower bounds for the complex polynomial Hardy–Littlewood inequality'
---
Introduction
============
Let $\mathbb{K}$ denote the field of real or complex scalars. Given $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in{\mathbb{N}}^{n}$, define $|\alpha|:=\alpha_{1}+\cdots+\alpha_{n}$ and $\mathbf{x}^{\alpha}$ stands for the monomial $x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}$ for $\mathbf{x}=(x_{1},\ldots,x_{n})\in{\mathbb{K}}^{n}$. The polynomial Bohnenblust–Hille inequality (see [@alb; @bh] and the references therein) ensures that, given positive integers $m\geq2$ and $n\geq1$, if $P$ is a homogeneous polynomial of degree $m$ on $\ell_{\infty}^{n}$ given by $P(x_{1},...,x_{n})=\sum
_{|\alpha|=m}a_{\alpha}\mathbf{{x}^{\alpha}}$, then $$\left( {\sum\limits_{\left\vert \alpha\right\vert =m}}\left\vert a_{\alpha
}\right\vert ^{\frac{2m}{m+1}}\right) ^{\frac{m+1}{2m}}\leq B_{\mathbb{K},m}^{\mathrm{pol}}\left\Vert P\right\Vert$$ for some constant $B_{\mathbb{K},m}^{\mathrm{pol}}\geq1$ which does not depend on $n$ (the exponent $\frac{2m}{m+1}$ is optimal), where $\Vert P\Vert
:=\sup_{z\in B_{\ell_{\infty}^{n}}}|P(z)|$.
The search of precise estimates of the growth of the constants $B_{\mathbb{K},m}^{\mathrm{pol}}$ is fundamental for different applications and remains an important open problem (see [@bps] and the references therein). For real scalars it was shown in [@camposjimenezrodriguezmunozfernandezpellegrinoseoanesepulveda2014] that $$\left( 1.17\right) ^{m}\leq B_{\mathbb{R},m}^{\mathrm{pol}}\leq
C(\varepsilon)\left( 2+\varepsilon\right) ^{m},$$ where $C(\varepsilon)\left( 2+\varepsilon\right) ^{m}$ means that given $\varepsilon>0$, there is a constant $C\left( \varepsilon\right) >0$ such that $B_{\mathbb{R},m}^{\mathrm{pol}}\leq C(\varepsilon)\left( 2+\varepsilon
\right) ^{m}$ for all $m$. In other words, for real scalars the hypercontractivity of $B_{\mathbb{R},m}^{\mathrm{pol}}$ is optimal. For complex scalars the behavior of $B_{\mathbb{K},m}^{\mathrm{pol}}$ is still unknown. The best information we have thus far about $B_{\mathbb{C},m}^{\mathrm{pol}}$ are due D. Núñez-Alarcón [@nunez] (lower bounds) and F. Bayart, D. Pellegrino and J.B. Seoane-Sepúlveda [@bps] (upper bounds) $$\begin{array}
[c]{l}B_{\mathbb{C},m}^{\mathrm{pol}}\geq\left\{
\begin{array}
[c]{lcl}\displaystyle\left( 1+\frac{1}{2^{m-1}}\right) ^{\frac{1}{4}} & & \text{for
}m\text{ even};\vspace{0.2cm}\\
\displaystyle\left( 1+\frac{1}{2^{m-1}}\right) ^{\frac{m-1}{4m}} & &
\text{for }m\text{ odd};
\end{array}
\right. \\
B_{\mathbb{C},m}^{\mathrm{pol}}\leq C(\varepsilon)\left( 1+\varepsilon
\right) ^{m}.
\end{array}$$
The natural extension to $\ell_{p}$ spaces of the polynomial Bohnenblust–Hille inequality is called polynomial Hardy–Littlewood inequality (see [@n; @hardy; @pra] and the references therein). More precisely, given positive integers $m\geq2$ and $n\geq1$, if $P$ is a homogeneous polynomial of degree $m$ on $\ell_{p}^{n}$ with $2m\leq
p\leq\infty$ given by $P(x_{1},\ldots,x_{n})=\sum_{|\alpha|=m}a_{\alpha
}\mathbf{{x}^{\alpha}}$, then there exists a constant $C_{\mathbb{K},m,p}^{\mathrm{pol}}\geq1$ (which does not depend on $n$) such that $$\left( {\sum\limits_{\left\vert \alpha\right\vert =m}}\left\vert a_{\alpha
}\right\vert ^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}}\leq
C_{\mathbb{K},m,p}^{\mathrm{pol}}\left\Vert P\right\Vert ,$$ with $\Vert P\Vert:=\sup_{z\in B_{\ell_{p}^{n}}}|P(z)|$. Using the generalized Kahane–Salem–Zygmund inequality (see, for instance, [@alb]) we can verify that the exponents $\frac{2mp}{mp+p-2m}$ are optimal for $2m\leq
p\leq\infty$. When $p=\infty$, since $\frac{2mp}{mp+p-2m}=\frac{2m}{m+1}$, we recover the polynomial Bohnenblust–Hille inequality. In a more genreal point of view this kind of results can be seen as coincidence results of the theory of absolutely summing operators (see [@die]).
Very recently, the authors in collaboration with P. Jiménez-Rodriguez, G.A. Muñoz-Fernández, D. Núñez-Alarcón, J.B. Seoane-Sepúlveda and D. M. Serrano-Rodríguez (see [@ajmnpss]) proved that for real scalars and $m\geq2$, the constants of the polynomial Hardy–Littlewood inequality has at least an hypercontractive growth. More precisely, it was proved that, for all positive integers $m\geq2$ and all $2m\leq p<\infty$, $$\left( \sqrt[16]{2}\right) ^{m}\leq2^{\frac{m^{2}p+10m-p-6m^{2}-4}{4mp}}\leq
C_{\mathbb{R},m,p}^{\mathrm{pol}}\leq C_{\mathbb{R},m,p}^{\mathrm{mult}}\frac{m^{m}}{\left( m!\right) ^{\frac{mp+p-2m}{2mp}}},$$ where $C_{\mathbb{R},m,p}^{\mathrm{mult}}$ are the constants of the real case of the multilinear Hardy-Littlewood inequality (for estimates of these constants see [@ap; @aps2014]).
In the case of complex scalars (and concerning upper bounds) similar results were proved (see [@ajmnpss]): $$1\leq C_{\mathbb{C},m,p}^{\mathrm{pol}}\leq C_{\mathbb{C},m,p}^{\mathrm{mult}}\frac{m^{m}}{\left( m!\right) ^{\frac{mp+p-2m}{2mp}}}.$$ However, there are no lower bounds for $C_{\mathbb{C},m,p}^{\mathrm{pol}}$ that gives us nontrivial information. In this note we provide nontrivial lower bounds for the constants of the complex case of the polynomial Hardy–Littlewood inequality. More precisely we prove that, for $m\geq2$ and $2m\leq p<\infty$, $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq\displaystyle2^{\frac{m}{p}}$$ for $m$ even, and $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq2^{\frac{m-1}{p}}$$ for $m$ odd. For instance, $$\sqrt{2}\leq C_{\mathbb{C},2,4}^{\mathrm{pol}}\leq3.1915.$$
The result
==========
Let $m\geq2$ be an even positive integer and let $p\geq2m$. Consider the $2$–homogeneous polynomials $Q_{2}:\ell_{p}^{2}\to\mathbb{C}$ and $\widetilde{Q_{2}}:\ell_{\infty}^{2}\rightarrow\mathbb{C}$ both given by $(z_{1},z_{2})\mapsto z_{1}^{2}-z_{2}^{2}+cz_{1}z_{2}$. We know from [@aronklimek2001; @camposjimenezrodriguezmunozfernandezpellegrinoseoanesepulveda2014] that $$\Vert\widetilde{Q_{2}}\Vert=\left( 4+c^{2}\right) ^{\frac{1}{2}}.$$
If we follow the lines of [@nunez] and we define the $m$–homogeneous polynomial ${Q_{m}}:\ell_{p}^{m}\rightarrow\mathbb{C}$ by ${Q_{m}}(z_{1},...,z_{m})=z_{3}\ldots z_{m} Q_{2}(z_{1},z_{2})$ we obtain $$\Vert{Q_{m}}\Vert\leq2^{-\frac{m-2}{p}}\Vert{Q_{2}}\Vert\leq2^{-\frac{m-2}{p}}\Vert\widetilde{Q_{2}}\Vert=2^{-\frac{m-2}{p}}\left( 4+c^{2}\right)
^{\frac{1}{2}},$$ where we use the obviuos inequality $$\Vert Q_{2}\Vert\leq\Vert\widetilde{Q_{2}}\Vert.$$ Therefore, for $m\geq2$ even and $c\in\mathbb{R}$, from the polynomial Hardy–Littlewood inequality it follows that $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq\frac{\left( 2+|c|^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}}}{2^{-\frac{m-2}{p}}\left( 4+c^{2}\right)
^{\frac{1}{2}}}.$$
If $$c>\left( \frac{2^{\frac{2p+4-2m}{p}}-2^{\frac{mp+p-2m}{mp}}}{1-2^{-\frac
{2m-4}{p}}}\right) ^{\frac{1}{2}},$$ it is not to difficult to prove that $$2^{-\frac{m-2}{p}}\left( 4+c^{2}\right) ^{\frac{1}{2}}<\left( \left(
2^{\frac{mp+p-2m}{2mp}}\right) ^{2}+c^{2}\right) ^{\frac{1}{2}},$$ i.e.,$$2^{-\frac{m-2}{p}}\left( 4+c^{2}\right) ^{\frac{1}{2}}<\left\Vert \left(
2^{\frac{mp+p-2m}{2mp}},c\right) \right\Vert _{2}.$$ Since $\frac{2mp}{mp+p-2m}\leq2$, we know that $\ell_{\frac{2mp}{mp+p-2m}}\subset\ell_{2}$ and $\Vert\cdot\Vert_{2}\leq\Vert\cdot\Vert_{\frac
{2mp}{mp+p-2m}}$. Therefore, for all $$c>\left( \frac{2^{\frac{2p+4-2m}{p}}-2^{\frac{mp+p-2m}{mp}}}{1-2^{-\frac
{2m-4}{p}}}\right) ^{\frac{1}{2}},$$ we have $$\begin{aligned}
2^{-\frac{m-2}{p}}\left( 4+c^{2}\right) ^{\frac{1}{2}} & <\left\Vert
\left( 2^{\frac{mp+p-2m}{2mp}},c\right) \right\Vert _{2}\\
& \leq\left\Vert \left( 2^{\frac{mp+p-2m}{2mp}},c\right) \right\Vert
_{\frac{2mp}{mp+p-2m}}\\
& =\left( 2+c^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}},\end{aligned}$$ from which we conclude that $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq\frac{\left( 2+c^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}}}{2^{-\frac{m-2}{p}}\left( 4+c^{2}\right)
^{\frac{1}{2}}}>1.$$
If $m\geq3$ is odd, since $\Vert Q_{m}\Vert\leq\Vert Q_{m-1}\Vert$, then we have $\Vert Q_{m}\Vert\leq2^{-\frac{m-3}{p}}\left( 4+c^{2}\right) ^{\frac
{1}{2}}$ and thus we can now proceed analogously to the even case and finally conclude that for $$c>\left( \frac{2^{\frac{2p+6^{-}2m}{p}}-2^{\frac{mp+p-2m}{mp}}}{1-2^{-\frac{2m-6}{p}}}\right) ^{\frac{1}{2}}$$ we have $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq\frac{\left( 2+c^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}}}{2^{-\frac{m-3}{p}}\left( 4+c^{2}\right)
^{\frac{1}{2}}}>1.$$
So we have:
\[777\]Let $m\geq2$ be a positive integer and let $p\geq2m$. Then, for every $\epsilon>0$, $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq\frac{\left( 2+\left( \left(
\frac{2^{\frac{2p+4-2m}{p}}-2^{\frac{mp+p-2m}{mp}}}{1-2^{-\frac{2m-4}{p}}}\right) ^{\frac{1}{2}}+\epsilon\right) ^{\frac{2mp}{mp+p-2m}}\right)
^{\frac{mp+p-2m}{2mp}}}{2^{-\frac{m-2}{p}}\left( 4+\left( \left(
\frac{2^{\frac{2p+4-2m}{p}}-2^{\frac{mp+p-2m}{mp}}}{1-2^{-\frac{2m-4}{p}}}\right) ^{\frac{1}{2}}+\epsilon\right) ^{2}\right) ^{\frac{1}{2}}}>1\ \ \ \text{ if }m\text{ is even}$$ and $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq\frac{\left( 2+\left( \left(
\frac{2^{\frac{2p+6^{-}2m}{p}}-2^{\frac{mp+p-2m}{mp}}}{1-2^{-\frac{2m-6}{p}}}\right) ^{\frac{1}{2}}+\epsilon\right) ^{\frac{2mp}{mp+p-2m}}\right)
^{\frac{mp+p-2m}{2mp}}}{2^{-\frac{m-3}{p}}\left( 4+\left( \left(
\frac{2^{\frac{2p+6^{-}2m}{p}}-2^{\frac{mp+p-2m}{mp}}}{1-2^{-\frac{2m-6}{p}}}\right) ^{\frac{1}{2}}+\epsilon\right) ^{2}\right) ^{\frac{1}{2}}}>1\ \ \ \text{ if }m\text{ is odd}.$$
Howerver we have another approach to the problem, which is surprisingly simpler than the above approach and still seems to give best (bigger) lower bounds for the constants of the polynomial Hardy–Littlewood inequality.
\[main\] Let $m\geq2$ be a positive integer and let $p\geq2m$. Then $$C_{\mathbb{C},m,p}^{\mathrm{pol}} \geq\left\{
\begin{array}
[c]{lcl}\displaystyle 2^{\frac{m}{p}} & & \text{for } m \text{ even}; \vspace
{0.2cm}\\
\displaystyle 2^{\frac{m-1}{p}} & & \text{for } m \text{ odd};
\end{array}
\right.$$
Consider $P_{2}:\ell_{p}^{2}\rightarrow\mathbb{C}$ the $2$–homogeneous polynomial given by $\mathbf{z}\mapsto z_{1}z_{2}$. Observe that $$\Vert P_{2}\Vert=\sup_{|z_{1}|^{p}+|z_{2}|^{p}=1}|z_{1}z_{2}|=\sup_{|z|\leq
1}\left\vert z\right\vert (1-\left\vert z\right\vert ^{p})^{\frac{1}{p}}=2^{-\frac{2}{p}}.$$
More generally, if $m\geq2$ is even and $P_{m}$ is the $m$–homogeneous polynomial given by $\mathbf{z}\mapsto z_{1}\cdots z_{m}$, then $$\Vert P_{m}\Vert\leq2^{-\frac{m}{p}}.$$ Therefore, from the polynomial Hardy–Littlewood inequality we know that $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq\frac{\left( \displaystyle\sum
_{|\alpha|=m}|a_{\alpha}|^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}}}{\Vert P_{m}\Vert}\geq\frac{1}{2^{-\frac{m}{p}}}=2^{\frac{m}{p}}.$$
If $m\geq3$ is odd, we define again the $m$–homogeneous polynomial $P_{m}$ given by $\mathbf{z}\mapsto z_{1}\cdots z_{m}$ and since $\Vert P_{m}\Vert
\leq\Vert P_{m-1}\Vert$, then we have $\Vert P_{m}\Vert\leq2^{-\frac{m-1}{p}}$ and thus $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq\frac{1}{2^{-\frac{m-1}{p}}}=2^{\frac{m-1}{p}}.$$
Comparing the estimates
=======================
The estimates of Theorem \[777\] seems to become better when $\epsilon$ grows (this seems to be a clear sign that we should avoid the terms $z_{1}^{2}$ and $z_{2}^{2}$ in our approach). Making $\epsilon\rightarrow\infty$ in Theorem \[777\] we obtain$$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq\left\{
\begin{array}
[c]{lcl}\displaystyle2^{\frac{m-2}{p}} & & \text{for }m\text{ even};\vspace{0.2cm}\\
\displaystyle2^{\frac{m-3}{p}} & & \text{for }m\text{ odd},
\end{array}
\right.$$ which are slightly worse than the estimates from Theorem \[main\].
The case $m<p<2m$
=================
For the case $m<p<2m$, there is also a version of the polynomial Hardy–Littlewood inequalities (see [@dimant]): there exists a constant $C_{\mathbb{K},m,p}^{\mathrm{pol}}\geq1$ such that, for all positive integers $n$ and all continuous $m$–homogeneous polynomial $P:\ell_{p}\rightarrow
\mathbb{K}$ given by $P(x_{1},...,x_{n})=\sum_{|\alpha|=m}a_{\alpha
}\mathbf{{x}^{\alpha}}$ we have$$\left( \sum_{\left\vert \alpha\right\vert =m}^{n}\left\vert a_{\alpha
}\right\vert ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\leq C_{\mathbb{K},m,p}^{\mathrm{pol}}\left\Vert T\right\Vert \label{ohl}$$ and the exponent $\frac{p}{p-m}$ is optimal. Using a polarization argument (as, for instance in [@camposjimenezrodriguezmunozfernandezpellegrinoseoanesepulveda2014]), but this procedure is essencially folklore, we have:
\[pro:first\_approach\] If $P$ is a homogeneous polynomial of degree $m$ on $\ell_{p}^{n}$ with $m<p<2m$ given by $P(x_{1},\ldots,x_{n})=\sum_{|\alpha
|=m}a_{\alpha}\mathbf{{x}^{\alpha}}$, then $$\left( {\sum\limits_{\left\vert \alpha\right\vert =m}}\left\vert a_{\alpha
}\right\vert ^{\frac{p}{p-m}}\right) ^{\frac{p-2}{p}}\leq C_{\mathbb{K},m,p}^{\mathrm{pol}}\left\Vert P\right\Vert$$ with $$C_{\mathbb{K},m,p}^{\mathrm{pol}}\leq C_{\mathbb{K},m,p}^{\mathrm{mult}}\frac{m^{m}}{\left( m!\right) ^{\frac{p-m}{p}}},$$ where $C_{\mathbb{K},m,p}^{\mathrm{mult}}$ are the constants of the multilinear Hardy-Littlewood inequality.
With the same argument used in the proof of Theorem \[777\] we obtain similar estimates for the case $m<p<2m,$ i.e., $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq\left\{
\begin{array}
[c]{lcl}\displaystyle2^{\frac{m}{p}} & & \text{for }m\text{ even};\vspace{0.2cm}\\
\displaystyle2^{\frac{m-1}{p}} & & \text{for }m\text{ odd.}\end{array}
\right.$$
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|
---
abstract: 'The aim of this paper is to exhibit a necessary and sufficient condition of optimality for functionals depending on fractional integrals and derivatives, on indefinite integrals and on presence of time delay. We exemplify with one example, where we find analytically the minimizer.'
author:
- 'Ricardo Almeida[^1]'
title: |
Fractional variational problems\
depending on indefinite integrals and with delay
---
49K05, 49S05, 26A33, 34A08.
calculus of variations, fractional calculus, Caputo derivatives, time delay.
Introduction {#sec:intro}
============
In this paper we proceed the work started in [@Almeida5], where the authors studied fractional variational problems with the Lagrangian containing not only fractional integrals and fractional derivatives, but an indefinite integral as well. With this approach, we tried not only to obtain new results but also generalize some already known. The novelty of this paper is that we consider dependence on time delay in the cost functional. Since fractional derivatives are characterized by retaining memory, it is natural to state the system at an earlier time and many phenomena have time delays inherent in them. This is a field under strong research, namely for optimal control problems, differential equations, biology, etc (see e.g. [@Chen; @Dehghan; @Liu0; @Liu; @Mo; @Udaltsov; @Xu; @Zhu]). For some literature on what this paper concerns, we suggest the reader to [@AGRA1; @Almeida; @Baleanu1; @Bhrawy; @Chen2; @Gastao0; @Loghmani; @Malinowska; @Mozyrska; @Yueqiang] for fractional variational problems dealing with Caputo derivative, in [@Almeida1] for Lagrangians depending on fractional integrals, and in [@Gregory; @Nat] when presence of indefinite integrals. For a standard variational approach to systems in presence of time delay or more general topics, we suggest the interested reader to the papers [@AGRA0; @Rosenblueth1; @Rosenblueth2; @Wang], and for the fractional approach to [@Baleanu; @Jarad].
The paper is organized in the following way. For the reader’s convenience, in section \[sec:frac\] we recall some definitions and results on fractional calculus; namely the definitions of fractional integral and fractional derivative, and some fractional integration by parts formulas. Section \[sec:ELequation\] is the main core of the paper: we exhibit a necessary and sufficient condition of optimality for the functional that we purpose to study in this paper.
Review on fractional calculus {#sec:frac}
=============================
Let us now explain the notation used. For more, see e.g. [@Kilbas; @Miller; @samko].
Given a function $f:[a,b]\to\mathbb{R}$, $\alpha\in(0,1)$ and $\beta>0$, the left and right fractional integrals of order $\beta$ of $f$ are respectively $${_aI_x^\beta}f(x)=\frac{1}{\Gamma(\beta)}\int_a^x (x-t)^{\beta-1}f(t)dt,$$ and $${_xI_b^\beta}f(x)=\frac{1}{\Gamma(\beta)}\int_x^b(t-x)^{\beta-1} f(t)dt.$$ The left and right Riemann–Liouville fractional derivatives of order $\alpha$ of $f$ are respectively $${_aD_x^\alpha}f(x)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dx}\int_a^x(x-t)^{-\alpha}f(t)dt$$ and $${_xD_b^\alpha}f(x)=\frac{-1}{\Gamma(1-\alpha)}\frac{d}{dx}\int_x^b (t-x)^{-\alpha} f(t)dt.$$ The left and right Caputo fractional derivatives of order $\alpha$ of $f$ are respectively $${_a^CD_x^\alpha}f(x)=\frac{1}{\Gamma(1-\alpha)}\int_a^x (x-t)^{-\alpha}\frac{d}{dt}f(t)dt$$ and $${_x^CD_b^\alpha}f(x)=\frac{-1}{\Gamma(1-\alpha)}\int_x^b(t-x)^{-\alpha}\frac{d}{dt} f(t)dt.$$ It is obvious that these operators are linear, and in some sense fractional differentiation and fractional integration are inverse operations.
Caputo fractional derivative seems to be more natural than the Riemann-Liouville fractional derivative. There are two main reasons for that. The first one is that the Caputo derivative of a constant is zero, while the Riemann-Liouville derivative of $f(x)=C$ is $C(x-a)^{-\alpha}/\Gamma(1-\alpha)$. The second one is that the Laplace transform of the Caputo derivative depends on the derivative of integer order of the function $$(\mathcal{L}\,{_0^CD^\alpha_s}f)(s)=s^\alpha (\mathcal{L}\,f)(s)-\sum_{k=0}^{n-1}s^{\alpha-k-1}\frac{d^kf}{ds^k}(0),$$ in opposite to the Riemann-Liouville derivative that uses fractional integrals evaluated at the initial value.
A basic result needed to apply variational methods is the integration by parts formula, that in case for fractional integrals is $$\label{Int2}\displaystyle\int_{a}^{b} g(x) \cdot {_aI_x^\beta}f(x)dx=\int_a^b f(x) \cdot {_x I_b^\beta} g(x)dx \, ,$$ and for Caputo fractional derivatives, we have $$\label{Int}\int_{a}^{b}g(x)\cdot {_a^C D_x^\alpha}f(x)dx=\int_a^b f(x)\cdot {_x D_b^\alpha} g(x)dx+\left[f(x)\cdot{_xI_b^{1-\alpha}}g(x)\right]_a^b.$$
Formula can be generalized in a way to include the case where the lower bound of the integral is distinct of the lower bound of the Caputo derivative.
\[LemmaInt\] Let $f$ and $g$ be two functions of class $C^1$ on $[a,b]$, and let $r\in(a,b)$. Then $$\begin{gathered}
\label{GenInt}\int_{r}^{b}g(x)\cdot {_a^C D_x^\alpha}f(x)dx=\int_r^b f(x)\cdot {_x D_b^\alpha} g(x)dx\\
-\int_a^r\frac{f(x)}{\Gamma(1-\alpha)}\, \frac{d}{dx}\left(\int_r^b (t-x)^{-\alpha}g(t)\,dt\right)dx-
\frac{f(a)}{\Gamma(1-\alpha)}\int_r^b(t-a)^{-\alpha}g(t)dt.\end{gathered}$$
It follows due the next relations: $$\begin{array}{ll}
\displaystyle \int_{r}^{b}g(x)\cdot {_a^C D_x^\alpha}f(x)dx & =\displaystyle \int_{a}^{b}g(x)\cdot {_a^C D_x^\alpha}f(x)dx-\int_{a}^{r}g(x)\cdot {_a^C D_x^\alpha}f(x)dx\\
& = \displaystyle\int_{a}^{b}f(x)\cdot {_x D_b^\alpha}g(x)dx+ \left[f(x) \cdot{_xI_b^{1-\alpha}}g(x) \right]_a^b\\
&\quad - \displaystyle \int_{a}^{r}f(x)\cdot {_x D_r^\alpha}g(x)dx-\left[f(x)\cdot {_xI_r^{1-\alpha}}g(x) \right]_a^r\\
&=\displaystyle\int_{r}^{b}f(x)\cdot {_x D_b^\alpha}g(x)dx+\int_{a}^{r}f(x)\cdot \left({_x D_b^\alpha}g(x)-{_x D_r^\alpha}g(x)\right)dx\\
& \displaystyle\quad +\left[f(x)\cdot {_xI_b^{1-\alpha}}g(x) \right]_a^b-\left[f(x)\cdot {_xI_r^{1-\alpha}}g(x) \right]_a^r\\
&=\displaystyle\int_r^b f(x)\cdot {_x D_b^\alpha} g(x)dx\\
&\displaystyle-\int_a^r\frac{f(x)}{\Gamma(1-\alpha)}\, \frac{d}{dx}\left(\int_r^b (t-x)^{-\alpha}g(t)\,dt\right)dx-
\frac{f(a)}{\Gamma(1-\alpha)}\int_r^b(t-a)^{-\alpha}g(t)dt.
\end{array}$$
The Euler-Lagrange equation {#sec:ELequation}
===========================
\[sec:ELequation\]
The cost functional that we will study is given by the expression $$\label{funct}
J(y)=\int_a^b L(x,y(x),{^C_aD_x^\alpha}y(x),{_aI_x^\beta}y(x),z(x), y(x-\tau), y'(x-\tau))dx,$$ defined on $C^1[a-\tau,b]$, where $$\left\{
\begin{array}{l}
\tau>0, \mbox{ and } \tau<b-a,\\
\alpha\in(0,1) \mbox{ and }\beta>0,\\
z(x)=\int_a^x l(t,y(t),{^C_aD_t^\alpha}y(t),{_aI_t^\beta}y(t))dt,\\
L=L(x,y,v,w,z,y_\tau,v_\tau) \mbox{ and } l=l(x,y,v,w) \mbox{ are of class } C^1
\end{array}\right.$$ and the admissible functions are such that $$\left\{
\begin{array}{l}
{^C_aD_x^\alpha}y(x) \mbox{ and } {_aI_x^\beta}y(x) \mbox{ exist and are continuous on } [a,b],\\
y(b)=y_b\in \mathbb R,\\
y(x)=\phi(x), \mbox{ for all } x\in [a-\tau,a], \, \phi \mbox{ a fixed function.}
\end{array}\right.$$ The set of variation functions of $y$ that we will consider are those of type $y+\epsilon h$, such that $|\epsilon| \ll1$ and $h\in C^1[a-\tau,b]$ with $$\left\{
\begin{array}{l}
h(b)=0,\\
h(x)=0, \mbox{ for all } x\in [a-\tau,a].
\end{array}\right.$$
An important result in variational calculus is the so called du Bois-Reymond Theorem:
\[dubois\] (see e.g. [@Brunt]) Let $f:[a,b]\to\mathbb R$ be a continuous functions, and suppose that the relation $$\int_a^b f(x)h(x)dx=0$$ holds for every $h\in C^k[a,b]$, with $k\geq 0$. Then $f(x)=0$ on $[a,b]$.
Theorem \[dubois\] still holds if we impose the auxiliary conditions $h(a)=h(b)=0$.
From now on, to simplify writing, by $[y](x)$ and $\{y\}(x)$ we denote the vectors $$[y](x)=(x,y(x),{^C_aD_x^\alpha}y(x),{_aI_x^\beta}y(x),z(x), y(x-\tau), y'(x-\tau))\quad \mbox{and}\quad \{y\}(x)=(x,y(x),{^C_aD_x^\alpha}y(x),{_aI_x^\beta}y(x)).$$ Let $y$ be a minimizer or maximizer of $J$ as in . As it is known, at the extremizers of the functional we have $$\frac{d}{d\epsilon}J(y+\epsilon h)=0,$$ where $y+\epsilon h$ is any variation of $y$. Proceeding with the necessary calculations, we deduce that $$\begin{gathered}
\int_a^b \left[ \frac{\partial L}{\partial y}[y](x)h(x)
+ \frac{\partial L}{\partial v}[y](x){^C_aD^\alpha_x}h(x)
+ \frac{\partial L}{\partial w}[y](x){_aI^\beta_x}h(x)\right.\\
\left.+\frac{\partial L}{\partial z}[y](x)\int_a^x\left(
\frac{\partial l}{\partial y}\{y\}(t)h(t)
+\frac{\partial l}{\partial v}\{y\}(t){^C_aD^\alpha_t}h(t)
+\frac{\partial l}{\partial w}\{y\}(t){_aI^\beta_t}h(t)\right)dt\right.\\
\left. +\frac{\partial L}{\partial y_\tau}[y](x)h(x-\tau)
+ \frac{\partial L}{\partial v_\tau}[y](x)h'(x-\tau)\right]dx=0.\end{gathered}$$ Next, we use the following relations
*R1:* $$\int_a^b \frac{\partial L}{\partial y}[y](x)h(x)dx=
\int_a^{b-\tau} \frac{\partial L}{\partial y}[y](x)h(x)dx+\int_{b-\tau}^b \frac{\partial L}{\partial y}[y](x)h(x)dx;$$
*R2:* Since $h(a)=0$, and using formulas and $$\begin{aligned}
\int_a^b \frac{\partial L}{\partial v}[y](x){^C_aD^\alpha_x}h(x) dx&=\int_a^{b-\tau}
\frac{\partial L}{\partial v}[y](x){^C_aD^\alpha_x}h(x) dx
+\int_{b-\tau}^b \frac{\partial L}{\partial v}[y](x){^C_aD^\alpha_x}h(x) dx\\
&=\int_a^{b-\tau} {_x D_{b-\tau}^\alpha} \left(\frac{\partial L}{\partial v}[y](x) \right)h(x)dx
+\left[{_x I_{b-\tau}^{1-\alpha}} \left(\frac{\partial L}{\partial v}[y](x) \right)h(x)\right]_a^{b-\tau}\\
&\quad +\int_{b-\tau}^b {_x D_{b}^\alpha} \left(\frac{\partial L}{\partial v}[y](x) \right)h(x)dx
-\frac{h(a)}{\Gamma(1-\alpha)}\int_{b-\tau}^b(t-a)^{-\alpha}\frac{\partial L}{\partial v}[y](t)dt \\
&\quad -\int_a^{b-\tau}\frac{h(x)}{\Gamma(1-\alpha)}\, \frac{d}{dx}
\left( \int_{b-\tau}^b(t-x)^{-\alpha} \frac{\partial L}{\partial v}[y](t)dt\right) dx\\
&=\int_a^{b-\tau} \left[{_x D_{b-\tau}^\alpha} \left(\frac{\partial L}{\partial v}[y](x)\right)-
\frac{1}{\Gamma(1-\alpha)}\, \frac{d}{dx}\left( \int_{b-\tau}^b(t-x)^{-\alpha} \frac{\partial L}{\partial v}[y](t)dt\right)\right]h(x)dx\\
&\quad + \int_{b-\tau}^b {_x D_{b}^\alpha} \left(\frac{\partial L}{\partial v}[y](x) \right)h(x)dx;\\\end{aligned}$$
*R3:* Using equation and Lemma 2(b) of [@Baleanu], $$\begin{aligned}
\int_a^b \frac{\partial L}{\partial w}[y](x){_aI^\beta_x}h(x) dx&=
\int_a^{b-\tau} \frac{\partial L}{\partial w}[y](x){_aI^\beta_x}h(x) dx+\int_{b-\tau}^b \frac{\partial L}{\partial w}[y](x){_aI^\beta_x}h(x) dx\\
&= \int_a^{b-\tau} {_xI^\beta_{b-\tau}}\left(\frac{\partial L}{\partial w}[y](x)\right)h(x) dx\\
&\quad +\int_{b-\tau}^b {_xI^\beta_b}\left(\frac{\partial L}{\partial w}[y](x)\right)h(x) dx
+\frac{1}{\Gamma(\beta)}\int_a^{b-\tau}h(x)\left( \int_{b-\tau}^b (t-x)^{\beta-1} \frac{\partial L}{\partial w}[y](t)dt \right)dx\\
&= \int_a^{b-\tau} \left[ {_xI^\beta_{b-\tau}}\left(\frac{\partial L}{\partial w}[y](x)\right) +
\frac{1}{\Gamma(\beta)} \left( \int_{b-\tau}^b (t-x)^{\beta-1} \frac{\partial L}{\partial w}[y](t)dt \right) \right]h(x) dx\\
&\quad +\int_{b-\tau}^b {_xI^\beta_b}\left(\frac{\partial L}{\partial w}[y](x)\right)h(x) dx;\end{aligned}$$
*R4:* Using standard integration by parts, $$\begin{aligned}
\int_a^b & \frac{\partial L}{\partial z}[y](x)\left(\int_a^x\frac{\partial l}{\partial y}\{y\}(t)h(t) dt \right) dx
=\int_a^b \left( -\frac{d}{dx}\int_x^b\frac{\partial L}{\partial z}[y](t)dt \right)
\left( \int_a^x \frac{\partial l}{\partial y}\{y\}(t)h(t) dt \right) dx\\
&= \int_a^b \left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt \right)
\frac{\partial l}{\partial y}\{y\}(x)h(x) \, dx\\
&= \int_a^{b-\tau} \left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt \right)\frac{\partial l}{\partial y}\{y\}(x)h(x) dx
+ \int_{b-\tau}^b \left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt \right)\frac{\partial l}{\partial y}\{y\}(x)h(x) dx;\end{aligned}$$
*R5:* Using standard integration by parts, formulas and and since $h(a)=0$, $$\begin{aligned}
&\int_a^b \frac{\partial L}{\partial z}[y](x)\left(\int_a^x\frac{\partial l}{\partial v}\{y\}(t){^C_aD^\alpha_t}h(t) dt \right) dx
= \int_a^b \left( -\frac{d}{dx}\int_x^b\frac{\partial L}{\partial z}[y](t)dt \right)\left(\int_a^x \frac{\partial l}{\partial v}\{y\}(t){^C_aD^\alpha_t}h(t)dt\right)dx\\
&= \int_a^b \left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt \right)\frac{\partial l}{\partial v}\{y\}(x){^C_aD^\alpha_x}h(x) dx\\
&= \int_a^{b-\tau} \left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt \right)\frac{\partial l}{\partial v}\{y\}(x){^C_aD^\alpha_x}h(x)dx
+\int_{b-\tau}^b \left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt \right)\frac{\partial l}{\partial v}\{y\}(x){^C_aD^\alpha_x}h(x)dx\\
&= \int_a^{b-\tau} {_xD^\alpha_{b-\tau}}\left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt\frac{\partial l}{\partial v}\{y\}(x)\right)h(x)dx
+\left[ {_xI^{1-\alpha}_{b-\tau}}\left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt\frac{\partial l}{\partial v}\{y\}(x) \right) h(x) \right]_a^{b-\tau}\\
& \quad +\int_{b-\tau}^b {_xD^\alpha_b}\left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt\frac{\partial l}{\partial v}\{y\}(x)\right)h(x)dx
-\frac{h(a)}{\Gamma(1-\alpha)}\int_{b-\tau}^b(t-a)^{-\alpha}\int_t^b \frac{\partial L}{\partial z}[y](k)dk \frac{\partial l}{\partial v}\{y\}(t)dt \\
&\quad -\int_a^{b-\tau}\frac{h(x)}{\Gamma(1-\alpha)}\, \frac{d}{dx}
\left( \int_{b-\tau}^b(t-x)^{-\alpha}\int_t^b \frac{\partial L}{\partial z}[y](k)dk \frac{\partial l}{\partial v}\{y\}(t)dt \right)dx\\
&=\int_a^{b-\tau} \left[{_xD^\alpha_{b-\tau}}\left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt\frac{\partial l}{\partial v}\{y\}(x)\right)
-\frac{1}{\Gamma(1-\alpha)} \frac{d}{dx}
\left( \int_{b-\tau}^b(t-x)^{-\alpha}\int_t^b \frac{\partial L}{\partial z}[y](k)dk \frac{\partial l}{\partial v}\{y\}(t)dt \right)\right]h(x)dx\\
&\quad + \int_{b-\tau}^b {_xD^\alpha_b}\left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt\frac{\partial l}{\partial v}\{y\}(x)\right)h(x)dx\end{aligned}$$
*R6:* Using standard integration by parts, equation and Lemma 2(b) of [@Baleanu], $$\begin{aligned}
&\int_a^b \frac{\partial L}{\partial z}[y](x)\left(\int_a^x\frac{\partial l}{\partial w}\{y\}(t){_aI^\beta_t}h(t) dt \right) dx=
\int_a^b \left( -\frac{d}{dx}\int_x^b\frac{\partial L}{\partial z}[y](t)dt\right)\left(\int_a^x\frac{\partial l}{\partial w}\{y\}(t){_aI^\beta_t}h(t) dt \right) dx\\
&=\int_a^b \left( \int_x^b\frac{\partial L}{\partial z}[y](t)dt\right) \frac{\partial l}{\partial w}\{y\}(x){_aI^\beta_x}h(x) dx\\
&=\int_a^{b-\tau} \left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt\right) \frac{\partial l}{\partial w}\{y\}(x){_aI^\beta_x}h(x) dx
+\int_{b-\tau}^b \left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt\right) \frac{\partial l}{\partial w}\{y\}(x){_aI^\beta_x}h(x) dx\\
&=\int_a^{b-\tau}\left[ {_xI^\beta_{b-\tau}}\left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt\frac{\partial l}{\partial w}\{y\}(x)\right)
+\frac{1}{\Gamma(\beta)}\left( \int_{b-\tau}^b (t-x)^{\beta-1}\int_t^b \frac{\partial L}{\partial z}[y](k)dk \, \frac{\partial l}{\partial w}\{y\}(t)dt\right)
\right]h(x) dx \\
&+ \int_{b-\tau}^b {_xI^\beta_b}\left(\int_x^b\frac{\partial L}{\partial z}[y](t)dt\frac{\partial l}{\partial w}\{y\}(x)\right)h(x)dx.\end{aligned}$$
*R7:* Since $h(x)=0$ for all $x\in[a-\tau,a]$, $$\int_a^b \frac{\partial L}{\partial y_\tau}[y](x)h(x-\tau)dx=\int_{a-\tau}^{b-\tau} \frac{\partial L}{\partial y_\tau}[y](x+\tau)h(x)dx
=\int_a^{b-\tau} \frac{\partial L}{\partial y_\tau}[y](x+\tau)h(x)dx$$
*R8:* Since $h(x)=0$ for all $x\in[a-\tau,a]$, using standard integration by parts, we have $$\int_a^b \frac{\partial L}{\partial v_\tau}[y](x)h'(x-\tau)dx=\int_a^{b-\tau} \frac{\partial L}{\partial v_\tau}[y](x+\tau)h'(x)dx=
\frac{\partial L}{\partial v_\tau}[y](b)h(b-\tau)-\int_a^{b-\tau} \frac{d}{dx}\left(\frac{\partial L}{\partial v_\tau}[y](x+\tau)\right)h(x)dx$$
We are now in position to obtain a necessary condition of optimality when in presence of the time delay $\tau>0$.
\[Teo1\] If $y$ is a minimizer or maximizer of $J$ as in , then $y$ is a solution of the system of equations
1. $\displaystyle \frac{\partial L}{\partial v_\tau}[y](b)=0$;
2. for every $x\in[a,b-\tau]$, $$\begin{aligned}
&\frac{\partial L}{\partial y}[y](x)+{_xD^\alpha_{b-\tau}}\left( \frac{\partial L}{\partial v}[y](x) \right)
- \frac{1}{\Gamma(1-\alpha)}\, \frac{d}{dx}\left( \int_{b-\tau}^b(t-x)^{-\alpha} \frac{\partial L}{\partial v}[y](t)dt\right)\\
&\quad +{_xI_{b-\tau}^\beta}\left(\frac{\partial L}{\partial w}[y](x)\right)
+\frac{1}{\Gamma(\beta)} \left( \int_{b-\tau}^b (t-x)^{\beta-1} \frac{\partial L}{\partial w}[y](t)dt \right)
+\int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial y}\{y\}(x)\\
&\quad+{_xD^\alpha_{b-\tau}}\left( \int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial v}\{y\}(x)\right)
-\frac{1}{\Gamma(1-\alpha)} \frac{d}{dx}
\left( \int_{b-\tau}^b(t-x)^{-\alpha}\int_t^b \frac{\partial L}{\partial z}[y](k)dk \frac{\partial l}{\partial v}\{y\}(t)dt \right)\\
&\quad +{_xI^\beta_{b-\tau}}\left( \int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial w}\{y\}(x) \right)
+\frac{1}{\Gamma(\beta)}\left( \int_{b-\tau}^b (t-x)^{\beta-1}\int_t^b \frac{\partial L}{\partial z}[y](k)dk \, \frac{\partial l}{\partial w}\{y\}(t)dt\right)\\
& \quad +\frac{\partial L}{\partial y_\tau}[y](x+\tau) -\frac{d}{dx}\frac{\partial L}{\partial v_\tau}[y](x+\tau)=0;\end{aligned}$$
3. for every $x\in[b-\tau,b]$, $$\begin{aligned}
&\frac{\partial L}{\partial y}[y](x)+{_xD^\alpha_b}\left( \frac{\partial L}{\partial v}[y](x) \right)+{_xI_b^\beta}\left(\frac{\partial L}{\partial w}[y](x)\right)\\
&\quad +\int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial y}\{y\}(x)
+{_xD^\alpha_b}\left( \int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial v}\{y\}(x)\right)
+{_xI^\beta_b}\left( \int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial w}\{y\}(x) \right)=0.\end{aligned}$$
If follows combining relations *R1*-*R8*, the arbitrariness of $h$ and from Theorem \[dubois\].
\[example2\] Consider the function $$y_\alpha(x)=\left\{
\begin{array}{lll}
0&\mbox{ if }& x\in[-1,0]\\
x^{\alpha+1}&\mbox{ if }& x\in[0,2].\\
\end{array}\right.$$ Then $${^C_0D_x^\alpha}y_\alpha(x)=\Gamma(\alpha+2)x.$$ For the cost functional, let $$\label{example}
J(y)=\int_0^2 ({^C_0D_x^\alpha}y(x)-\Gamma(\alpha+2)x)^2+z(x)+(y'(x-1)-y'_\alpha(x-1))^2dx,$$ where $$z(x)=\int_0^x (y(t)-t^{\alpha+1})^2 \, dt,$$ defined on the set $C^1[-1,2]$, under the constraints $$\left\{
\begin{array}{l}
y(2)=2^{\alpha+1},\\
y(x)=0, \mbox{ for all } x\in [-1,0].
\end{array}\right.$$ Since $J(y)\geq0$ for all admissible functions $y$, and $J(y_\alpha)=0$, we have that $y_\alpha$ is a minimizer of $J$ and zero is its minimum value. Equations *1-3* of Theorem \[Teo1\] applied to $J$ read as
1. $\displaystyle \left[y'(x-1)-y'_\alpha(x-1)\right]_{x=2}=0$;
2. for every $x\in[0,1]$, $$\begin{aligned}
&{_xD_1^\alpha}({^C_0D_x^\alpha}y(x)-\Gamma(\alpha+2)x)
- \frac{1}{\Gamma(1-\alpha)}\,\frac{d}{dx}\left( \int_{1}^2(t-x)^{-\alpha}({^C_0D_t^\alpha}y(t)-\Gamma(\alpha+2)t)dt\right)\\
&\quad +\int_x^21dt \, (y(x)-x^{\alpha+1})-\frac{d}{dx}\left(y'(x)-y'_\alpha(x)\right)=0\\\end{aligned}$$
3. for every $x\in[1,2]$, $$\begin{aligned}
&{_xD_2^\alpha}({^C_0D_x^\alpha}y(x)-\Gamma(\alpha+2)x)+\int_x^21dt \, (y(x)-x^{\alpha+1})=0.\\\end{aligned}$$
Obviously, $y_\alpha$ is a solution for the three previous conditions 1–3.
In [@Jarad] fractional variational problems in presence of Caputo derivatives and delays are considered. Since the variational functions $h$ are chosen in such a way that take the value zero at the extrema, the Caputo and the Riemann-Liouville derivative of these functions are equal. Using a general integration by parts formula of [@Baleanu] similar to our Lemma \[LemmaInt\], but for Riemann-Liouville derivative, the problem of [@Jarad] is solved for Caputo derivative. Here we choose to obtain the equivalent formula of [@Baleanu] for the Caputo derivative.
Consider the case when $\alpha$ goes to 1 and $\beta$ goes to zero. If so, we obtain the standard functional derived from : $$\label{funct2}
J(y)=\int_a^b L(x,y(x),y'(x),z(x), y(x-\tau), y'(x-\tau))dx,$$ where $$z(x)=\int_a^x l(t,y(t),y'(t))dt,$$ defined for $y\in C^1[a-\tau,b]$ satisfying the boundary conditions $$\left\{
\begin{array}{l}
y(b)=y_b\in \mathbb R,\\
y(x)=\phi(x), \mbox{ for all } x\in [a-\tau,a].
\end{array}\right.$$ If $y$ is a minimizer or maximizer of $J$ as in , then $y$ is a solution of the system of equations
1. $\displaystyle \frac{\partial L}{\partial v_\tau}[y](b)=0$;
2. for every $x\in[a,b-\tau]$, $$\begin{aligned}
&\frac{\partial L}{\partial y}[y](x)-\frac{d}{dx}\left( \frac{\partial L}{\partial v}[y](x) \right)
+\int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial y}\{y\}(x)\\
&\quad -\frac{d}{dx}\left(\int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial v}\{y\}(x)\right)
+\frac{\partial L}{\partial y_\tau}[y](x+\tau) -\frac{d}{dx}\frac{\partial L}{\partial v_\tau}[y](x+\tau)=0;\end{aligned}$$
3. for every $x\in[b-\tau,b]$, $$\frac{\partial L}{\partial y}[y](x)-\frac{d}{dx}\left(\frac{\partial L}{\partial v}[y](x) \right)
+\int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial y}\{y\}(x)
-\frac{d}{dx}\left( \int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial v}\{y\}(x)\right)=0.$$
This result is apparently new also.
Theorem \[Teo1\] can be generalized for functionals with several dependent variables. Let us consider $$\label{funct3}
J(y_1,\ldots,y_n)=\int_a^b L(x,y_1(x),\ldots,y_n(x),{^C_aD_x^{\alpha_1}}y_1(x),\ldots,{^C_aD_x^{\alpha_n}}y_n(x),$$ $${_aI_x^{\beta_1}}y_1(x),\ldots,{_aI_x^{\beta_n}}y_n(x),z(x), y_1(x-\tau_1),\ldots,y_n(x-\tau_n), y_1'(x-\tau_1),\ldots, y_n'(x-\tau_n))dx,$$ defined on $C^1 \prod^{n}_{i=1} [a-\tau_i,b]$, where for all $i\in\{1,\ldots,n\}$, $$\left\{
\begin{array}{l}
\tau_i>0, \mbox{ and } \tau_i<b-a,\\
\alpha_i\in(0,1) \mbox{ and }\beta_i>0,\\
z(x)=\int_a^x l(t,y_1(t),\ldots,y_n(t),{^C_aD_t^{\alpha_1}}y_1(t),\ldots,{^C_aD_t^{\alpha_n}}y_n(t),{_aI_t^{\beta_1}}y_1(t),\ldots,{_aI_t^{\beta_n}}y_n(t))dt,\\
L=L(x,y_1,\ldots,y_n,v_1,\ldots,v_n,w_1,\ldots,w_n,z,y_{\tau_1},\ldots,y_{\tau_n},v_{\tau_1},\ldots,v_{\tau_n}) \mbox{ and } \\
l=l(x,y_1,\ldots,y_n,v_1,\ldots,v_n,w_1,\ldots,w_n) \mbox{ are of class } C^1
\end{array}\right.$$ and the admissible functions are such that $$\left\{
\begin{array}{l}
{^C_aD_x^{\alpha_i}}y_i(x) \mbox{ and } {_aI_x^{\beta_i}}y_i(x) \mbox{ exist and are continuous on } [a,b],\\
y_i(b)=y_{b_i}\in \mathbb R,\\
y_i(x)=\phi_i(x), \mbox{ for all } x\in [a-\tau_i,a], \, \phi_i \mbox{ a fixed function.}
\end{array}\right.$$ If the *n*-uple function $(y_1,\ldots,y_n)$ is a minimizer or maximizer of $J$ as in , then $(y_1,\ldots,y_n)$ is a solution of the system of equations similar to the ones of *1-3* of Theorem \[Teo1\], replacing the variables $$y\to y_i,\quad v\to v_i,\quad w\to w_i,\quad y_\tau\to y_{\tau_i},\quad v_\tau\to v_{\tau_i},\quad \alpha\to\alpha_i,\quad \beta\to\beta_i,\quad\tau\to\tau_i,$$ for all $i\in\{1,\dots,n\}$.
Sufficient condition {#sec:SufConditions}
====================
Assuming some convexity conditions on the Lagrangian $L$ and on the supplementary function $l$, we may present a necessary condition that guarantees the existence of minimizers for the problem. For convenience, recall the definition of convex and concave function. Given $k\in\{1,\ldots,n\}$ and $f:D\subseteq\mathbb{R}^n\to \mathbb{R}$ a function differentiable with respect to $x_k,\ldots,x_n$, we say that $f$ is convex (resp. concave) in $(x_k,\ldots,x_n)$ if $$f(x_1+c_1,\ldots,x_n+c_n)-f(x_1,\ldots,x_n)\geq \, (resp. \leq) \, \sum_{i=k}^n\frac{\partial f}{\partial x_i}(x_1,\ldots,x_n)c_i,$$ for all $(x_1,\ldots,x_n),(x_1+c_1,\ldots,x_n+c_n)\in D$.
Let $y$ be a function satisfying conditions 1–3 of Theorem \[Teo1\]. If $L$ is convex in $(y,v,w,z,y_\tau,v_\tau)$ and one of the two following conditions are met
1. $l$ is convex in $(y,v,w)$ and $\frac{\partial L}{\partial z}[y](x) \geq 0$ for all $x \in [a,b]$,
2. $l$ is concave in $(y,v,w)$ and $\frac{\partial L}{\partial z}[y](x) \leq 0$ for all $x \in [a,b]$,
then $y$ is a minimizer of the functional $J$ as in .
Let $y+h$ be a variation of $y$. Using relations *R1*-*R8* of Section \[sec:ELequation\], we have that $$\begin{array}{ll}
J(y+h) - J(y) & \displaystyle\geq \int_a^{b-\tau}\left[ \frac{\partial L}{\partial y}[y](x)+{_xD^\alpha_{b-\tau}}\left( \frac{\partial L}{\partial v}[y](x) \right)
- \frac{1}{\Gamma(1-\alpha)}\, \frac{d}{dx}\left( \int_{b-\tau}^b(t-x)^{-\alpha} \frac{\partial L}{\partial v}[y](t)dt\right)\right.\\
&\displaystyle\quad+{_xI_{b-\tau}^\beta}\left(\frac{\partial L}{\partial w}[y](x)\right)+\frac{1}{\Gamma(\beta)} \left( \int_{b-\tau}^b (t-x)^{\beta-1} \frac{\partial L}{\partial w}[y](t)dt \right)\\
&\displaystyle\quad+\int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial y}\{y\}(x)
+{_xD^\alpha_{b-\tau}}\left( \int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial v}\{y\}(x)\right)\\
&\displaystyle\quad-\frac{1}{\Gamma(1-\alpha)} \frac{d}{dx}
\left( \int_{b-\tau}^b(t-x)^{-\alpha}\int_t^b \frac{\partial L}{\partial z}[y](k)dk \frac{\partial l}{\partial v}\{y\}(t)dt \right)\\
&\displaystyle\quad+{_xI^\beta_{b-\tau}}\left( \int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial w}\{y\}(x) \right)
+\frac{1}{\Gamma(\beta)}\left( \int_{b-\tau}^b (t-x)^{\beta-1}\int_t^b \frac{\partial L}{\partial z}[y](k)dk \, \frac{\partial l}{\partial w}\{y\}(t)dt\right)\\
&\displaystyle\left.\quad +\frac{\partial L}{\partial y_\tau}[y](x+\tau) -\frac{d}{dx}\frac{\partial L}{\partial v_\tau}[y](x+\tau)\right] h(x)dx\\
&\displaystyle\quad+ \int_{b-\tau}^b\left[ \frac{\partial L}{\partial y}[y](x)+{_xD^\alpha_b}\left( \frac{\partial L}{\partial v}[y](x) \right)+{_xI_b^\beta}\left(\frac{\partial L}{\partial w}[y](x)\right)+\int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial y}\{y\}(x)\right.\\
&\displaystyle\quad\left. +{_xD^\alpha_b}\left( \int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial v}\{y\}(x)\right)
+{_xI^\beta_b}\left( \int_x^b \frac{\partial L}{\partial z}[y](t)dt \frac{\partial l}{\partial w}\{y\}(x) \right)\right] h(x)dx\\
&\displaystyle\quad+ \frac{\partial L}{\partial v_\tau}[y](b)h(b-\tau)=0.\\
\end{array}$$
For example, report to the example \[example2\]. For this case, $$L(x,y,v,w,z,y_\tau,v_\tau) =(v-\Gamma(\alpha+2)x)^2+z+(y_\tau-(\alpha+1)(x-1)^\alpha)^2\mbox{ and }l(x,y,v,w)=(y-x^{\alpha+1})^2$$ are both convex, and $\frac{\partial L}{\partial z}[y](x) =1$. Observe that $y_\alpha$ is a solution of equations *1–3* of Theorem \[Teo1\], and in fact is a minimizer of $J$.
Conclusion
==========
The aim of the paper is to generalize the main result of [@Almeida5], by considering delays in our system. Necessary conditions are proven in case the Lagrange function depends on fractional derivatives and on indefinite integral as well. For future work, we will study numerical tools to solve directly these kind of problems, avoiding to solve analytically fractional differential equations.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT–-Fundação para a Ciência e a Tecnologia”), within project PEst-OE/MAT/UI4106/2014.
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|
---
abstract: |
We obtain the following two results through foliation theoretic approaches including a review of Lawson’s construction of a codimension-one foliation on the $5$-sphere:\
1) The standard contact structure on the $5$-sphere deforms to ‘Reeb foliations’.\
2) We define a $5$-dimensional Lutz tube which contains a plastikstufe. Inserting it into any contact $5$-manifold, we obtain a contact structure which violates the Thurston-Bennequin inequality for a convex hypersurface with contact-type boundary.
address: 'Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan'
author:
- Atsuhide MORI
title: 'Reeb Foliations on $S^5$ and Contact $5$-Manifolds Violating the Thurston-Bennequin Inequality'
---
Introduction and preliminaries
==============================
The first aim of this paper is to show that the standard contact structure ${\mathcal{D}}_0$ on $S^5$ deforms via contact structures into spinnable foliations, which we call Reeb foliations (§2). Here a spinnable foliation is a codimension-one foliation associated to an open-book decomposition whose binding is fibred over $S^1$. In 1971, Lawson[@Lawson] constructed a spinnable foliation on $S^5$ associated to a Milnor fibration. We construct such a spinnable foliation on $S^5$ as the limit ${\mathcal{D}}_1$ of a family $\{{\mathcal{D}}_t\}_{0\le t<1}$ of contact structures. Since $S^5$ is compact, the family $\{{\mathcal{D}}_t\}_{0\le t<1}$ can be traced by a family of diffeomorphisms $\varphi_t:S^5\to S^5$ wth $\varphi_0=\mathrm{id}$ and $(\varphi_t)_*{\mathcal{D}}_0={\mathcal{D}}_t$ (Gray’s stability).
The second aim is to show that any contact $5$-manifold admits a contact structure which violates the Thurston-Bennequin inequality for a [*convex*]{} hypersurface (§3). We define a $5$-[*dimensional Lutz tube*]{} and explain how to insert it into a given contact $5$-manifold to violate the inequality. Moreover a $5$-dimensional Lutz tube contains a [*plastikstufe*]{}, which is an obstruction to symplectic fillability found by Niederkrüger[@Niederkruger] and Chekanov. A different [*Lutz twist*]{} on a contact manifold $(M^{2n+1},\alpha)$ was recently introduced in Etnyre-Pancholi[@Etnyre] as a modification of the contact structure ${\mathcal{D}}=\ker\alpha$ near an $n$-dimensional submanifold. Contrastingly, the core of our Lutz tube is a codimension-two contact submanifold. We change the standard contact structure on $S^5$ by inserting a Lutz tube along the binding of the open-book decomposition of a certain Reeb foliation.
The author[@MoriTB] also showed that any contact manifold of dimension$>3$ violates the Thurston-Bennequin inequality for a [*non-convex*]{} hypersurface. However he conjectures that the inequality holds for any [*convex*]{} hypersurface in the standard $S^{2n+1}$. See §4 for related problems.
The rest of this section is the preliminaries.
Thurston-Bennequin inequality
-----------------------------
A [*positive*]{} (resp. [*negative*]{}) [*contact manifold*]{} consists of an oriented $(2n+1)$-manifold $M^{2n+1}$ and a $1$-form $\alpha$ on $M^{2n+1}$ with $\alpha\wedge(d\alpha)^n>0$ (resp. $\alpha\wedge(d\alpha)^n<0$). The (co-)oriented hyperplane distribution ${\mathcal{D}}=\ker\alpha$ is called the [*contact structure*]{} on the contact manifold $(M^{2n+1},\alpha)$. In the case where $(M^{2n+1},\alpha)$ is positive, the symplectic structure $d\alpha|\ker\alpha$ on the oriented vector bundle $\ker\alpha$ is also positive, i.e., $(d\alpha)^n|\ker\alpha>0$. Hereafter we assume that all contact structures and symplectic structures are positive.
In this subsection, we assume that any compact connected oriented hypersurface $\Sigma$ embedded in a contact manifold $(M^{2n+1},\alpha)$ tangents to the contact structure $\ker\alpha$ at finite number of interior points. Note that the hyperplane field $\ker\alpha$ is maximally non-integrable. Let $S_+(\Sigma)$ (resp. $S_-(\Sigma)$) denote the set of the positive (resp. negative) tangent points, and $S(\Sigma)$ the union $S_+(\Sigma)\cup S_-(\Sigma)$. The sign of the tangency at $p\in S(\Sigma)$ coincides with the sign of $\{(d\alpha|\Sigma)^n\}_p$. Considering on $(\ker\alpha,d\alpha|\ker\alpha)$, we see that the symplectic orthogonal of the intersection $T\Sigma\cap \ker\alpha$ forms an oriented line field $L$ on $\Sigma$, where the singularity of $L$ coincides with $S(\Sigma)$.
The singular oriented foliation ${\mathcal{F}}_\Sigma$ defined by $T{\mathcal{F}}_\Sigma=L$ is called the [*characteristic foliation*]{} on $\Sigma$ with respect to the contact structure $\ker\alpha$.
Put $\beta=\alpha|\Sigma$ and take any volume form $\nu$ on $\Sigma$. Then we see that the vector field $X$ on $\Sigma$ defined by $\iota_X\nu=\beta\wedge(d\beta)^{n-1}$ is a positive section of $L$. Moreover, $$\iota_X\{\beta\wedge(d\beta)^{n-1}\}=
-\beta\wedge\iota_X(d\beta)^{n-1}=0,\quad
\beta\wedge(d\beta)^{n-1}\ne 0
\quad\Longrightarrow
\quad
\beta\wedge\iota_Xd\beta=0.$$ Thus the flow generated by $X$ preserves the conformal class of $\beta$. Since $\nu$ is arbitrary, we may take $X$ as any positive section of $L$. Therefore the $1$-form $\beta$ defines a holonomy invariant transverse contact structure of the characteristic foliation ${\mathcal{F}}_\Sigma$.
On the other hand, for any volume form $\mu(\ne\nu)$ on $\Sigma$, we see that the sign of $\mathrm{div}\, X=({\mathcal{L}}_X\mu)/\mu$ at each singular point $p\in S(\Sigma)$ coincides with the sign of $p$. Thus ${\mathcal{F}}_\Sigma$ contains the information about the sign of the tangency to the contact structure $\ker\alpha$. We also define the [*index*]{} $\mathrm{Ind}\,p=\mathrm{Ind}_X p$ of a singular point $p\in S(\Sigma)$ by using the above vector field $X$.
Suppose that the boundary $\partial \Sigma$ of the above hypersurface $\Sigma$ is non-empty, and the characteristic foliation ${\mathcal{F}}_\Sigma$ is positively (i.e., outward) transverse to $\partial\Sigma$. Then we say that $\Sigma$ is a hypersurface with [*contact-type*]{} boundary. Note that $\beta|\partial\Sigma=\alpha|\partial\Sigma$ is a contact form.
The [*Liouville vector field*]{} $X$ on a given exact symplectic manifold $(\Sigma, d\lambda)$ with respect to a primitive $1$-form $\lambda$ of $d\lambda$ is defined by $\iota_X d\lambda=\lambda$. If $X$ is positively transverse to the boundary $\partial \Sigma$, then $(\partial\Sigma,\lambda|\partial\Sigma)$ is called the contact-type boundary. The above definition is a natural shift of this notion into the case of hypersurfaces in contact manifolds.
Let $D^2$ be an embedded disk with contact-type boundary in a contact $3$-manifold. We say that $D^2$ is [*overtwisted*]{} if the singularity $S(D^2)$ consists of a single sink point. Note that a sink point is a negative singular point since it has negative divergence. A contact $3$-manifold is said to be [*overtwisted*]{}, or [*tight*]{} depending on whether there exists an overtwisted disk with contact-type boundary in it, or not. We can show that the existence of an overtwisted disk with contact-type boundary is equivalent to the existence of an [*overtwisted disk with Legendrian boundary*]{}, which is an embedded disk $D'$ similar to the above $D^2$ except that the characteristic foliation ${\mathcal{F}}_{D'}$ tangents to the boundary $\partial D'$, where $\partial D'$ or $-\partial D'$ is a closed leaf of ${\mathcal{F}}_{D'}$.
Let $\Sigma$ be [*any*]{} surface with contact-type (i.e., transverse) boundary embedded in the standard $S^3$. Then Bennequin[@Bennequin] proved the following inequality which implies the tightness of $S^3$:
$
\displaystyle \sum_{p\in S_-(\Sigma)}\mathrm{Ind}\,p \leq 0.
$
Eliashberg proved the same inequality for symplectically fillable contact $3$-manifolds ([@Eliashberg2]), and finally for all tight contact $3$-manifolds ([@Eliashberg3]). Recently Niederkrüger[@Niederkruger] and Chekanov found a $(n+1)$-dimensional analogue of an overtwisted disk with Legendrian boundary | a [*plastikstufe*]{} which is roughly the trace $K^{n-1}\times D^2$ of an overtwisted disk $D^2$ with Legendrian boundary travelling along a closed integral submanifold $K^{n-1}\subset M^{2n+1}$. However, in order to create some meaning of the above inequality in higher dimensions, we need a $2n$-dimensional analogue of an overtwisted disk with contact-type boundary.
The Thurston-Bennequin inequality can also be written in terms of relative Euler number: The vector field $X\in T\Sigma \cap \ker\alpha$ is a section of $\ker\alpha|\Sigma$ which is canonical near the boundary $\partial\Sigma$. Thus under a suitable boundary condition we have $$\langle e(\ker\alpha),\,[\Sigma,\partial\Sigma]\rangle=
\sum_{p\in S_+(\Sigma)}\mathrm{Ind}\,p-\sum_{p\in S_-(\Sigma)}\mathrm{Ind}\,p.$$ Then the Thurston-Bennequin inequality can be expressed as $$-\langle e(\ker\alpha),\,[\Sigma,\partial\Sigma]\rangle \leq -\chi(\Sigma).$$ There is also an absolute version of the Thurston-Bennequin inequality for a closed hypersurface $\Sigma$ with $\chi(\Sigma)\leq 0$, which is expressed as $|\langle e(\ker\alpha),\,[\Sigma]\rangle|\leq -\chi(\Sigma)$, or equivalently $$\sum_{p\in S_-(\Sigma)}\mathrm{Ind}\,p \leq 0\quad\quad
\textrm{and}\quad\quad
\sum_{p\in S_+(\Sigma)}\mathrm{Ind}\,p \leq 0.$$ The absolute version trivially holds if the Euler class $e(\ker \alpha)\in H^{2n}(M; {\mathbb{Z}})$ is a torsion. Note that the inequality and its absolute version can be defined for any oriented plane field on an oriented $3$-manifold $M^3$ (see Eliashberg-Thurston[@EliashbergThurston]). They are originally proved for a foliation on $M^3$ without Reeb components by Thurston (see [@Thurston]).
Convex hypersurfaces
--------------------
In this subsection we explain Giroux’s convex hypersurface theory outlined in [@GirouxConvex] and add a possible relative version to it.
A vector field $Y$ on a contact manifold $(M,\alpha)$ which satisfies $\alpha\wedge{\mathcal{L}}_Y\alpha=0$ is called a [*contact vector field*]{}. Let $V_\alpha$ denote the set of all contact vector fields on $(M,\alpha)$. It is well-known that the linear map $\alpha(\cdot):V_\alpha\to C^\infty(M)$ is an isomorphism.
1. For a given contact vector field $Y$ on a contact manifold $(M,\alpha)$, the function $H=\alpha(Y)\in C^\infty(M)$ is called the [*Hamiltonian function*]{} of $Y$. Conversely for a given function $H\in C^\infty(M)$, the unique contact vector field $Y$ with $\alpha(Y)=H$ is called the [*contact Hamiltonian vector field*]{} of $H$. The contact Hamiltonian vector field of $1$ lies in the degenerate direction of $d\alpha$ and is called the [*Reeb field*]{} of $\alpha$.
2. A closed oriented hypersurface $\Sigma$ embedded in a contact manifold $(M,\alpha)$ is said to be [*convex*]{} if there exists a contact vector field transverse to $\Sigma$.
Let $Y$ be a contact vector field positively transverse to a closed convex hypersurface $\Sigma$, and $\Sigma\times(-{\varepsilon},{\varepsilon})$ a neighbourhood of $\Sigma=\Sigma\times\{0\}$ with $Y=\partial/\partial z$ ($z\in(-{\varepsilon},{\varepsilon})$). We may assume that the contact form $\alpha$ is $Y$-invariant after rescaling it by multiplying a suitable positive function. Note that this rescaling does not change the level set $\{\alpha(Y)=0\}$. By perturbing $Y$ in $V_\alpha$ if necessary, we can modify the Hamiltonian function $H=\alpha(Y)$ so that the level set $\{H=0\}$ is a regular hypersurface of the form $\Gamma\times(-{\varepsilon},{\varepsilon})$ in the above neighbourhood $\Sigma\times(-{\varepsilon},{\varepsilon})$, where $\Gamma\subset M$ is a codimension-$2$ submanifold. Put $h=H|\Sigma$.
The submanifold $\Gamma=\{h=0\} \subset \Sigma$ is called the [*dividing set*]{} on $\Sigma$ with respect to $Y$. $\Gamma$ divides $\Sigma$ into the [*positive region*]{} $\Sigma_+=\{h\ge 0\}$ and the [*negative region*]{} $\Sigma_-=-\{h\le 0\}$ so that $\Sigma=\Sigma_+\cup (-\Sigma_-)$. We orient $\Gamma$ as $\Gamma=\partial \Sigma_+=\partial \Sigma_-$.
Note that $\pm Y|\{\pm H>0\}$ is the Reeb field of $\alpha/|H|=\beta/|H|\pm dz$, where $\beta$ is the pull-back of $\alpha|\Sigma$ under the projection along $Y$. Since the $2n$-form $$\Omega=(d\beta)^{n-1}\wedge (Hd\beta+n\beta dH)$$ satisfies $\Omega\wedge dz=\alpha\wedge(d\alpha)^n>0$, the characteristic foliation ${\mathcal{F}}_\Sigma$ is positively transverse to the dividing set $\Gamma$. Thus $\Gamma$ is a positive contact submanifold of $(M,\alpha)$. The open set $U=\{|H|<{\varepsilon}'\}$ is of the form $(-{\varepsilon}',{\varepsilon}')\times\Gamma\times(-{\varepsilon},{\varepsilon})$ for sufficiently small ${\varepsilon}'>0$. Let $\rho(H)>0$ be an even function of $H$ which is increasing on $H>0$, and coincides with $1/|H|$ except on $(-{\varepsilon}',{\varepsilon}')$. Then we see that $d(\rho\alpha)|\mathrm{int}\,\Sigma_\pm$ are symplectic forms.
On the other hand, let $(\Sigma_\pm,d\lambda_\pm)$ be compact exact symplectic manifolds with the same contact-type boundary $(\partial\Sigma_\pm, \mu)$, where we fix the primitive $1$-forms $\lambda_\pm$ and assume that $\mu=\lambda_\pm | \partial\Sigma_\pm$. Then $\lambda_i+dz$ is a $z$-invariant contact form on $\Sigma_i\times{\mathbb{R}}$ ($i=+$ or $-$, $z\in{\mathbb{R}}$).
The contact manifold $(\Sigma_i\times{\mathbb{R}},\lambda_i+dz)$ is called the [*contactization*]{} of $(\Sigma_i,d\lambda_i)$. Take a collar neighbourhood $(-{\varepsilon}',0]\times\partial\Sigma_i\subset \Sigma_i$ such that $$\lambda_i+dz|((-{\varepsilon}',0]\times\partial\Sigma_i\times{\mathbb{R}})=e^s\mu+dz\quad (s\in(-{\varepsilon}',0]).$$ We modify $\lambda_i+dz$ near $(-{\varepsilon}',0]\times\partial\Sigma_i\times{\mathbb{R}}$ in a canonical way into a contact form $\alpha_i$ with $$\alpha_i|((-{\varepsilon}',0]\times\partial\Sigma_i\times{\mathbb{R}})=e^{-s^2/{\varepsilon}'}\mu-\frac{s}{{\varepsilon}'}dz.$$ We call the contact manifold $(\Sigma_i\times{\mathbb{R}},\alpha_i)$ the [*modified contactization*]{} of $(\Sigma_i,d\lambda_i)$.
The above symplectic manifold $(\Sigma_i,d\lambda_i)$ can be fully extended by attaching the half-symplectization $({\mathbb{R}}_{\ge0}\times\partial\Sigma_i,d(e^s\mu))$ to the boundary. The interior of the modified contactization is then contactomorphic to the contactization of the fully extended symplectic manifold.
The modified contactizations $\Sigma_+\times{\mathbb{R}}$ and $\Sigma_-\times{\mathbb{R}}'$ match up to each other to form a connected contact manifold $((\Sigma_+\cup(-\Sigma_-))\times {\mathbb{R}},\alpha)$ where ${\mathbb{R}}'=-{\mathbb{R}}$. Indeed, $\alpha$ can be written near $\Gamma\times {\mathbb{R}}=\partial\Sigma_+\times{\mathbb{R}}=\partial\Sigma_-\times(-{\mathbb{R}}')$ as $$\alpha|(-{\varepsilon}',{\varepsilon}')\times\Gamma\times{\mathbb{R}}=e^{-s^2/{\varepsilon}'}\mu-\frac{s}{{\varepsilon}'}dz\quad
(s\in(-{\varepsilon}',{\varepsilon}'),\,z\in{\mathbb{R}}).$$
The contact manifold $((\Sigma_+\cup(-\Sigma_-))\times {\mathbb{R}},\alpha)$ is called the [*unified contactization*]{} of $\Sigma=\Sigma_+\cup(-\Sigma_-)$.
Since $(-\Sigma)_+=\Sigma_-$ and $(-\Sigma)_-=\Sigma_+$, the unified contactization of $-\Sigma=\Sigma_-\cup(-\Sigma_+)$ can be obtained by turning the unified contactization of $\Sigma_+\cup(-\Sigma_-)$ upside-down. Note that $-Y\in V_\alpha$. Clearly, a small neighbourhood of any convex hypersurface $\Sigma_+\cup(-\Sigma_-)$ is contactomorphic to a neighbourhood of $(\Sigma_+\cup(-\Sigma_-))\times\{0\}$ in the unified contactization.
Conceptually, a convex hypersurface in contact topology play the same role as a contact-type hypersurface in symplectic topology | both are powerful tools for cut-and-paste because they have canonical neighbourhoods modeled on the unified contactization and the symplectization. Further Giroux[@GirouxConvex] showed that any closed surface in a contact $3$-manifold is smoothly approximated by a convex one. This fact closely relates contact topology with differential topology in this dimension. On the other hand, there exists a hypersurface which cannot be smoothly approximated by a convex one if the dimension of the contact manifold is greater than three (see [@MoriTB]).
A compact connected oriented embedded hypersurface $\Sigma$ with non-empty contact-type boundary in a contact manifold $(M,\alpha)$ is said to be [*convex*]{} if there exists a contact vector field $Y$ such that $\alpha(Y)|\partial\Sigma>0$ and $Y$ is transverse to $\Sigma$.
Put $h=\alpha(Y)|\Sigma$ after perturbing $Y$. Then the dividing set $\Gamma=\{h=0\}$ divides $\Sigma$ into the positive region $\Sigma_+=\{h\ge 0\}$ and the (possibly empty) negative region $\Sigma_-=-\{h\le 0\}$ so that $$\Sigma=\Sigma_+\cup(-\Sigma_-)\quad\textrm{and}\quad
\partial\Sigma=\partial \Sigma_+ \setminus \partial \Sigma_-\ne\emptyset.$$ Note that the above definition avoids touching of $\Gamma$ to the contact-type boundary $\partial\Sigma$. Now the Thurston-Bennequin inequality can be written as
$
\displaystyle \chi(\Sigma_-)\leq 0\quad(\textrm{or}\quad\Sigma_-=\emptyset).
$
Suppose that there exists a convex disk $\Sigma=D^2$ with contact-type boundary in a contact $3$-manifold which is the union $\Sigma_+\cup(-\Sigma_-)$ of a negative disk region $\Sigma_-$ and a positive annular region $\Sigma_+$. Then the convex disk $\Sigma$ violates the Thurston-Bennequin inequality and is called a [*convex overtwisted disk*]{} ($\chi(\Sigma_-)=1>0$). Conversely, it is clear that any overtwisted disk with contact-type boundary is also approximated by a convex overtwisted disk.
A [*convex overtwisted hypersurface*]{} is a connected convex hypersurface $\Sigma_+\cup(-\Sigma_-)$ with contact-type boundary which satisfies $\chi(\Sigma_-)>0$.
Note that any convex overtwisted hypersurface $\Sigma$ contains a connected component of $\Sigma_+$ whose boundary is disconnected. This relates to Calabi’s question on the existence of a compact connected exact symplectic $2n$-manifold ($n>1$) with disconnected contact-type boundary. McDuff[@McDuff] found the first example of such a manifold. Here is another example:
(Mitsumatsu[@Mitsumatsu], Ghys[@Ghys] and Geiges[@Geiges]) To obtain a symplectic $4$-manifold with disconnected contact-type boundary, we consider the mapping torus $T_A=T^2\times [0,1]/A \ni ((x,y),z)$ of a linear map $A\in SL( 2;{\mathbb{Z}})$ $(A:T^2\times \{1\} \to T^2\times \{0\})$ with $\mathrm{tr}\,A>2$. Let $dvol_{T^2}$ be the standard volume form on $T^2={\mathbb{R}}^2/{\mathbb{Z}}^2$ and $v_\pm$ eigenvectors of $A$ which satisfy $$Av_\pm=a^{\pm 1} v_\pm,\quad \textrm{where}\quad
a>1 \quad \textrm{and}\quad
dvol_{T^2}(v_+,v_-)>0.$$ In general, a cylinder $[-1,1]\times M^3$ admits a symplectic structure with contact-type boundary if $M^3$ admits a co-orientable Anosov foliation (Mitsumatsu[@Mitsumatsu]). In the case where $M^3=T_A$, the $1$-forms $\beta_\pm=\pm a^{\mp z}dvol_{T^2}(v_\pm,\cdot)$ define Anosov foliations. Then the cylinder $$(W_A=[-1,1]\times T_A, d(\beta_++s\beta_-))\quad(s\in [-1,1])$$ is a symplectic manifold with contact-type boundary $(-T_A) \sqcup T_A$.
\[1.2\]
Using the above cylinder $W_A$, we construct a convex overtwisted hypersurface in §3.
Convergence of contact structures to foliations
===============================================
First we define a supporting open-book decomposition on a closed contact manifold.
Let $(M^{2n+1},\alpha)$ be a closed contact manifold and $\mathcal{O}$ an open-book decomposition on $M^{2n+1}$ by pages $P_\theta$ $(\theta\in{\mathbb{R}}/2\pi{\mathbb{Z}})$. Suppose that the binding $(N^{2n-1}=\partial P_\theta,\alpha|N^{2n-1})$ of $\mathcal{O}$ is a contact submanifold. Then if there exists a positive function $\rho$ on $M^{2n+1}$ such that $$d\theta\wedge \{d(\rho\alpha)\}^n>0\quad \textrm{on}\quad
M^{2n+1}\setminus N^{2n-1},$$ the open-book decomposition $\mathcal{O}$ is called a [*supporting open-book decomposition*]{} on $(M^{2n+1},\alpha)$.
The function $\rho$ can be taken so that $\rho\alpha$ is axisymmetric near the binding. Precisely, we can modify the function $\rho$ near a tubular neighbourhood $N^{2n-1}\times D^2$ except on the binding $N^{2n-1}\times\{0\}$, if necessary, so that with respect to the polar coordinates $(r,\theta)$ on the unit disk $D^2$
1. the restriction $\rho\alpha|(N^{2n-1}\times D^2)$ is of the form $f(r)\mu+g(r)d\theta$,
2. $\mu$ is the pull-back $\pi^*(\rho\alpha|N^{2n-1})$ under the projection $\pi:N^{2n-1}\times D^2\to N^{2n-1}$,
3. $f(r)$ is a positive function of $r$ on $N^{2n-1}\times D^2$ with $f'(r)<0$ on $(0,1]$,
4. $g(r)$ is a weakly increasing function with $g(r)\equiv r^2$ near $r=0$ and $g(r)\equiv 1$ near $r=1$.
Next we prove the following theorem.
Let $\mathcal{O}$ be a supporting open-book decomposition on a closed contact manifold $(M^{2n+1},\alpha)$ of dimension greater than three $(n>1)$. Suppose that the binding $N^{2n-1}$ of $\mathcal{O}$ admits a non-zero closed $1$-form $\nu$ with $\nu \wedge \{d(\rho\alpha|N^{2n-1})\}^{n-1} \equiv 0$ where $\rho$ is a function on $M^{2n+1}$ satisfying all of the above conditions. Then there exists a family of contact forms $\{\alpha_t\}_{0\le t<1}$ on $M^{2n+1}$ which starts with $\alpha_0=\rho\alpha$ and converges to a non-zero $1$-form $\alpha_1$ with $\alpha_1\wedge d\alpha_1\equiv 0$. That is, the contact structure $ker\alpha$ then deforms to a spinnable foliation. \[convergence\]
Take smooth functions $f_1(r)$, $g_1(r)$, $h(r)$ and $e(r)$ of $r\in [0,1]$ such that
1. $f_1\equiv 1$ near $r=0$,$f_1\equiv 0$ on $[1/2,1]$,$f_1'\le 0$ on $[0,1]$,
2. $g_1\equiv 1$ near $r=1$,$g_1\equiv 0$ on $[0,1/2]$, $g_1'\ge 0$ on $[0,1]$,
3. $h\equiv 1$ on $[0,1/2]$,$h\equiv 0$ near $r=1$,
4. $e$ is supported near $r=1/2$,and $e(1/2)\ne 0$.
Put $f_t(r)=(1-t)f(r)+tf_1(r)$, $g_t(r)=(1-t)g(r)+tg_1(r)$ and $$\alpha_t|(N\times D^2)=f_t(r)\{(1-t)\mu+th(r)\nu\}+g_t(r)d\theta+te(r)dr,$$ where $\nu$ also denotes the pull-back $\pi^*\nu$. We extend $\alpha_t$ by $$\alpha_t|(M\setminus (N\times D^2))=\tau\rho\alpha+(1-\tau)d\theta
\quad\textrm{where}\quad
\tau=(1-t)^2.$$ Then we see from $d\nu\equiv 0$ and $\nu \wedge (d\mu)^{n-1} \equiv 0$ that $\alpha_t\wedge (d\alpha_t)^n$ can be written as $$nf_t^{n-1}(1-t)^n(g_t'f_t-f_t'g_t)
\mu\wedge (d\mu)^{n-1}\wedge dr\wedge d\theta \quad
\textrm{on} \quad N\times D^2 \quad \textrm{and}$$ $$\tau^{n+1}\rho^{n+1}\alpha\wedge(d\alpha)^n+\tau^n(1-\tau)
d\theta\wedge\{d(\rho\alpha)\}^n \quad
\textrm{on} \quad M\setminus (N\times D^2).$$ Therefore we have $$\alpha_t\wedge (d\alpha_t)^n>0\quad (0\le t<1), \quad
\alpha_1\wedge d\alpha_1 \equiv 0 \quad \textrm{and} \quad
\alpha_1\ne 0.$$ This completes the proof of Theorem \[convergence\].
1. A similar result in the case where $n=1$ is contained in the author’s paper[@Mori]: Any contact structure $\ker\alpha$ on a closed $3$-manifold deforms to a spinnable foliation.
2. The orientation of the compact leaf $\{r=1/2\}$ depends on the choice of the sign of the value $e(1/2)$. Here the choice is arbitrary.
We give some examples of the above limit foliations which relate to the following proposition on certain $T^2$-bundles over the circle.
Let $T_{A_{m,0}}$ denote the mapping torus $T^2\times [0,1]/A_{m,0}$ $\ni\left((x,y),z\right)$ of the linear map $
A_{m,0}=
\left(
\begin{array}{cc}
1 & 0 \\
m & 1
\end{array}
\right): T^2\times\{1\} \to T^2\times\{0\}$ $(m\in {\mathbb{N}})$. Then $\ker(dy+mzdx)$ is the unique Stein fillable contact structure on $T_{A_{m,0}}$ [(]{}up to contactomorphism[)]{}. Moreover it admits a supporting open-book decomposition $\mathcal{O}_{m,0}$ such that
1. the page is a $m$-times punctured torus, and
2. the monodromy is the right-handed Dehn twist along [(]{}the disjoint union of $m$ loops parallel to[)]{} the boundary of the page.
\[nil\]
Let ${\mathbb{C}}^3$ be the $\xi\eta\zeta$-space, and $\pi_\xi,\pi_\eta$ and $\pi_\zeta$ denote the projections to the axes.
The link $L$ of the singular point $(0,0,0)$ of the complex surface $\{\xi^3+\eta^3+\zeta^3=0\}\subset {\mathbb{C}}^3$ is diffeomorphic to the $T^2$-bundle $T_{A_{3,0}}$. (To see this, consider the projective curve $\{\xi^3+\eta^3+\zeta^3=0\}\subset {\mathbb{C}}P^2$ diffeomorphic to $T^2$. Since $L$ is the union of the Hopf fibres over this torus, it is also a $T^2$-bundle over the circle.) Moreover, since $L$ is Stein fillable, it is contactomorphic to $(T_{A_{3,0}},\mu=dy+3zdx)$. Indeed the open-book decomposition $\mathcal{O}_{3,0}$ in Proposition \[nil\] is equivalent to the supporting open-book decomposition $\{\arg (\pi_\xi|L)=\theta\}_{\theta\in {\mathbb{R}}/2\pi{\mathbb{Z}}}$ on $L$. (To see this, regard $\xi$ as a parameter and consider the curve $C_\xi=\{\eta^3+\zeta^3=-\xi^3\}$ on the $\eta\zeta$-plane, which is diffeomorphic to $T^2\setminus \{\textrm{three points}\}$. Then we can see that the fibration $\displaystyle \left\{(\{\xi\}\times C_\xi)\cap B^6\right\}_{|\xi|={\varepsilon}}$ $(0<{\varepsilon}\ll 1)$ is equivalent to the page fibration of $\mathcal{O}_{3,0}$, where $B^6$ denotes the unit hyperball of ${\mathbb{C}}^3\approx{\mathbb{R}}^6\ni(x_1,y_1,\dots,x_3,y_3)$.) We put $$\Lambda=\sum_{i=1}^3(x_{i}dy_{i}-y_{i}dx_{i})
\quad \textrm{and} \quad
V_{{\varepsilon},\theta}=\{\xi^3+\eta^3+\zeta^3={\varepsilon}e^{\sqrt{-1}\theta}\}\cap B^6\quad
(\theta\in{\mathbb{R}}/2\pi{\mathbb{Z}}).$$ Then Gray’s stability implies that $\partial V_{{\varepsilon},\theta}\subset (S^5,\Lambda|S^5)$ is contactomorphic to $L$. Since $\xi^3+\eta^3+\zeta^3$ is a homogeneous polynomial, the $1$-form $\Lambda|V_{{\varepsilon},\theta}$ is conformal to the pull-back of the restriction of $\Lambda$ to $\Sigma_{{\varepsilon},\theta}=\{\rho p\,|\,
\rho>0, p\in V_{{\varepsilon},\theta}\}\cap S^5
\subset\{\arg(\xi^3+\eta^3+\zeta^3)=\theta\}$ under the central projection. Indeed $\rho x_id(\rho y_i)-\rho y_id(\rho x_i)=\rho^2(x_idy_i-y_idx_i)$ holds for any function $\rho$. Thus the fibration $\{\Sigma_{{\varepsilon},\theta}\}_{\theta\in{\mathbb{R}}/2\pi{\mathbb{Z}}}$ extends to a supporting open-book decomposition on the standard $S^5$. We put $\nu=dz$ and apply Theorem \[convergence\] to obtain a limit foliation ${\mathcal{F}}_{3,0}$ of the standard contact structure. This is the memorable first foliation on $S^5$ discovered by Lawson[@Lawson]. \[Lawson\]
For other examples, we need the following lemma essentially due to Giroux and Mohsen.
Let $f:{\mathbb{C}}^n \to {\mathbb{C}}$ be a holomorphic function with $f(0,\dots,0)=0$ such that the origin $(0,\dots,0)$ is an isolated critical point. Take a sufficiently small hyperball $B_{\varepsilon}=\{|z_1|^2+\dots+|z_n|^2={\varepsilon}^2\}$. Then there exists a supporting open-book decomposition on the standard $S^{2n-1}$ such that the binding is contactomorphic to the link $\{f=0\}\cap \partial B_{\varepsilon}$ and the page fibration is equivalent to the fibration $\{\{f=\delta\}\cap B_{\varepsilon}\}_{|\delta|={\varepsilon}'}$ $(0<{\varepsilon}'\ll{\varepsilon})$. \[sfl\]
Take the hyperball $B'_{\varepsilon}=\{|z_1|^2+\dots+|z_{n+1}|^2={\varepsilon}^2\}$ on ${\mathbb{C}}^{n+1}$ and consider the complex hypersurface $\Sigma_k=\{z_{n+1}=kf(z_1,\dots,z_n)\}\cap B'_{\varepsilon}$ with contact-type boundary $\partial\Sigma_k$ ($k\ge 0$). Then Gray’s stability implies that $\partial\Sigma_k$ is contactomorphic to $\partial\Sigma_0(=\partial B_{\varepsilon})$. From $dz_{n+1}|\Sigma_\infty=0$ and $(x_{n+1}dy_{n+1}-y_{n+1}dx_{n+1})
(-y_{n+1}\partial/\partial x_{i+1}+x_{n+1}\partial/\partial y_{n+1})\ge 0$, we see that $\{\arg(f|\partial \Sigma_k)=\theta\}_{\theta\in S^1}$ is a supporting open-book decomposition of $\partial\Sigma_k$ equivalent to $\{\{f=\delta\}\cap B_{\varepsilon}\}_{|\delta|={\varepsilon}'}$ if $k$ is sufficiently large and ${\varepsilon}'>0$ is sufficiently small.
Consider the polynomials $f_1=\xi^6+\eta^3+\zeta^2$ and $f_2=\xi^4+\eta^4+\zeta^2$. Then the link $L_m$ of the singular point $(0,0,0)\in \{f_m=0\}$ is contactomorphic to the above $T^2$-bundle $T_{A_{m,0}}$ with the contact form $\mu=dy+mzdx$ ($m=1,2$). Indeed $\mathcal{O}_{m,0}$ is equivalent to the supporting open-book decomposition $\{\arg (\pi_\xi|L_m)=\theta\}_{\theta\in {\mathbb{R}}/2\pi{\mathbb{Z}}}$ on $L_m$. (To see this, regard $\xi$ as a parameter and consider $C_\xi=\{f_m=0\}$ on the $\eta\zeta$-plane, which is diffeomorphic to $T^2\setminus\{m\, \textrm{points}\}$. Then we can see that the fibration $\displaystyle \left\{(\{\xi\}\times C_\xi)\cap B^6\right\}_{|\xi|={\varepsilon}}$ is equivalent to the page fibration of $\mathcal{O}_{m,0}$.) On the other hand, Lemma \[sfl\] sais that there exists a supporting open-book decomposition on the standard $S^5$ which is equivalent to the Milnor fibration with binding $L_m$. We put $\nu=dz$ and apply Theorem \[convergence\] to obtain a limit foliation ${\mathcal{F}}_{m,0}$ $(m=1,2)$. \[eg\]
1. Let $f:{\mathbb{C}}^{n+1} \to {\mathbb{C}}$ be a holomorphic function with $f(0,\dots,0)=0$ such that the origin is an isolated critical point or a regular point of $f$. If the origin is singular, the Milnor fibre has the homotopy type of a bouquet of $n$-spheres. Suppose that the Euler characteristic of the Milnor fibre is positive, that is, the origin is regular if $n$ is odd. Then we say that the Milnor fibration is [*PE*]{} (=positive Euler characteristic).
2. Let $\mathcal{O}$ be a supporting open-book decomposition of the standard $S^{2n+1}$. Suppose that the binding is the total space of a fibre bundle $\pi$ over ${\mathbb{R}}/{\mathbb{Z}}\ni t$, and the Euler characteristic of the page is positive. Then if $\nu=\pi^*dt$ satisfies the assumption of Theorem \[convergence\], the resultant limit foliation is called a [*Reeb foliation*]{}.
The above ${\mathcal{F}}_{m,0}$ ($m=1,2,3$) are Reeb foliations associated to PE Milnor fibrations. To obtain other examples of foliations associated to more general Milnor fibrations, Grauert’s topological characterization of Milnor fillable $3$-manifolds is instructive ([@Grauert], see also [@CNP]).
Five-dimensional Lutz tubes
===========================
In this section, we define a $5$-dimensional Lutz tube by means of an open-book decomposition whose page is a convex hypersurface. We insert the Lutz tube along the binding of a certain supporting open-book decomposition on the standard $S^5$. Then we obtain a new contact structure on $S^5$ which violates the Thurston-Bennequin inequality for a convex hypersurface. We also show that the $5$-dimensional Lutz tube contains a plastikstufe.
Convex open-book decompositions
-------------------------------
We explain how to construct a contact manifold with an open-book decomposition by convex pages.
Let $(\Sigma_\pm,d\lambda_\pm)$ be two compact exact symplectic manifolds with contact-type boundary. Suppose that there exists an inclusion $\iota:\partial \Sigma_-\to
\partial\Sigma_+$ such that $\iota^*(\lambda_+|\partial\Sigma_+)=\lambda_-|\partial\Sigma_-$. Let $\varphi$ be a self-diffeomorphism of the union $\Sigma=\Sigma_+\cup_\iota (-\Sigma_-)$ supported in $\mathrm{int}\,\Sigma_+\sqcup\mathrm{int}(-\Sigma_-)$ which satisfies $$(\varphi|\Sigma_\pm)^\ast\lambda_\pm-\lambda_\pm=dh_\pm$$ for suitable positive functions $h_\pm$ on $\Sigma_\pm$. We choose some connected components of $\partial\Sigma_+\setminus \iota(\partial\Sigma_-)$ and take their disjoint union $B$. Then there exists a smooth map $\Phi$ from the unified contactization $\Sigma\times{\mathbb{R}}$ to a compact contact manifold $(M,\alpha)$ such that
1. $\Phi|(\Sigma\setminus B)\times{\mathbb{R}}$ is a cyclic covering which is locally a contactomorphism,
2. $P_0=\Phi(\Sigma\times\{0\})\approx \Sigma$ is a convex page of an open-book decomposition $\mathcal{O}$ on $(M,\alpha)$,
3. $\Phi(B\times{\mathbb{R}})\approx B$ is the binding contact submanifold of $\mathcal{O}$,and
4. $\varphi$ is the monodromy map of $\mathcal{O}$.
\[ob\]
In the case where $\Sigma_-=\emptyset$ and $B=\partial\Sigma_+$, this proposition was proved in Giroux[@Giroux] (essentially in Thurston-Winkelnkemper[@TW]). Then $\mathcal{O}$ is a supporting open-book decomposition. In general, let $\Sigma\times{\mathbb{R}}$ be the unified contactization, i.e., the union of the modified contactizations of $\Sigma_\pm$ by the attaching map $(\iota, -\mathrm{id}_{{\mathbb{R}}'}):\partial \Sigma_-\times {\mathbb{R}}'
\to \partial \Sigma_+\times {\mathbb{R}}(=-{\mathbb{R}}')$. Consider the quotient $\Sigma\times({\mathbb{R}}/2\pi c{\mathbb{Z}})$ ($c>0$), and cap-off the boundary components $B\times({\mathbb{R}}/2\pi c{\mathbb{Z}})$ by replacing the collar neighbourhood $(-{\varepsilon}',0]\times B\times({\mathbb{R}}/2\pi c{\mathbb{Z}})$ with $(B\times D^2,\,(\lambda_+ | B)+r^2d\theta)$ where $\theta=z/c$. Adding constants to $h_\pm$ if necessary, we may assume that $h_\pm$ are the restrictions of the same function $h$. We change the identification $(x,z+2\pi c)\sim (x,z)$ to $(x,z+h)\sim(\varphi(x),z)$ before capping-off the boundary $B\times S^1$. This defines the map $\Phi$ and completes the proof.
Giroux and Mohsen proved that any symplectomorphism supported in $\mathrm{int}\,\Sigma_+$ is isotopic via such symplectomorphisms to $\varphi$ with $\varphi^*\lambda_+-\lambda_+=dh_+$ ($\exists h_+>0$). They also proved that there exists a supporting open-book decomposition on any closed contact manifold by interpreting the result of Ibort-Martinez-Presas[@IMP] on the applicability of Donaldson-Auroux’s asymptotically holomorphic methods to complex functions on contact manifolds (see [@Giroux]).
Definition of Lutz tubes
------------------------
Let $W_A$ be the symplectic manifold with disconnected contact-type boundary $(-T_A) \sqcup T_A$ in Example \[1.2\]. Then, by a result of Van Horn[@VH], each of the boundary component is a Stein fillable contact manifold. Note that $\mathrm{tr}(A^{-1})>2$ and $(-T_A) \approx T_{A^{-1}}$. Precisely, there exists a supporting open-book decomposition on $T_A$ described as follows.
1. [(Honda[@HondaII], see also Van Horn[@VH])]{} Any element $A\in SL(2;{\mathbb{Z}})$ with $\mathrm{tr}\,A>2$ is conjugate to at least one of the elements $$A_{m,k}=
\left(
\begin{array}{cc}
1 & 0 \\
1 & 1
\end{array}
\right)
\left(
\begin{array}{cc}
1 & k_1 \\
0 & 1
\end{array}
\right)
\dots
\left(
\begin{array}{cc}
1 & 0 \\
1 & 1
\end{array}
\right)
\left(
\begin{array}{cc}
1 & k_m \\
0 & 1
\end{array}
\right)\in SL(2;{\mathbb{Z}}),$$ where $m\in {\mathbb{Z}}_{>0}$, $k=(k_1,\dots,k_m) \in ({\mathbb{Z}}_{\ge0})^m$ and $k_1+\dots+k_m>0$.
2. [(Van Horn[@VH])]{} The contact manifold $(T_{A_{m,k}},(\beta_-+\beta_+)|T_{A_{m,k}})$ $(k\ne 0)$ admits a supporting open-book decomposition which is determined up to equivalence by the following data:
Page
: The page $P$ is the $m$-times punctured torus $\bigcup_{i\in {\mathbb{Z}}_m} P_i$, where $P$ is divided into three-times punctured spheres $P_i$ by mutually disjoint loops $\gamma_i$ with $P_i\cup P_{i+1}=\gamma_i$.
Monodromy
: The monodromy is the composition $\displaystyle \tau(\partial P)\circ \prod_{i=1}^m \{\tau(\gamma_i)\}^{k_i}$, where $\tau(\gamma)$ denotes the right-handed Dehn twist along $\gamma$.
\[sol\]
These data determines a PALF (=positive allowable Lefschetz fibration) structure of the canonical Stein filling $V$ of the contact manifold $T_{A_{m,k}}$ if $m\ge 2$ (see Loi-Piergallini [@LP] and Giroux [@Giroux]). Here we see through the PALF structure that attaching $1$-handles to the page corresponds to attaching $1$-handles to the canonical Stein filling, and adding right-handed Dehn twists along non-null-homologous loops to the monodromy corresponds to attaching $2$-handles to the canonical Stein filling. Thus we have $$\chi(V)=1_{(=\#\{\textrm{0-handle}\})}-
(m+1)_{(=\#\{\textrm{1-handles}\})}+
(m+k_1+\dots+k_m)_{(=\#\{\textrm{2-handles}\})}>0.$$ In the case where $m=1$, let $\ell_1$ and $\ell_2$ denote simple loops on $P$ which intersect transversely at a single point. It is well-known that $\tau(\partial P)$ is isotopic to $\left(\tau(\ell_1)\circ\tau(\ell_2)\right)^6$. This new expression determines a PALF structure with $12+k_1$ singular fibres on the canonical Stein filling $V$ of $T_{A_{1,(k_1)}}$. Then we have $\chi(V)=1-2+12+k_1>11$. Thus the following corollary holds.
The contact manifold $(T_A,(\beta_-+\beta_+)|T_A)$ admits a Stein filling $V$ with $\chi(V)>0$.
Then the unified contactization of $W_A\cup(-V)$ under the natural identification $\{1\}\times T_A\sim \partial V$ contains a convex overtwisted hypersurface $(W_A\cup(-V))\times\{0\}$.
Now we define a $5$-dimensional Lutz tube.
Putting $\Sigma_+=W_A$, $\Sigma_-=\emptyset$, $B=-(\{-1\}\times T_{A})\approx T_{A^{-1}}$ and $\varphi=\mathrm{id}_\Sigma$, we apply [Proposition \[ob\]]{} to obtain a contact manifold $T_{A^{-1}}\times D' \approx T_A\times D^2$ $(D'=-D^2)$, which we call the [*$5$-dimensional Lutz tube*]{} associated to $A$ ($\mathrm{tr}\, A>2$).
The next proposition explains how to insert a Lutz tube.
Let $(V,d\lambda)$ be an exact strong symplectic filling of $T_A$ with $\mathrm{tr}\,A>2$, $\psi:V\to V$ a diffeomorphism supported in $\mathrm{int}\,V$. Suppose that $\chi(V)>0$ and $\psi^\ast\lambda-\lambda=dh$ $(\exists h>0)$. Putting $\Sigma_+=V$, $\Sigma_-=\emptyset$, $B=\partial V$ and $\varphi=\psi$, we apply [Proposition \[ob\]]{} to obtain a closed contact manifold $(M^5,\alpha)$ with an open-book decomposition whose binding is $T_A$. Next we consider the union $W_A\cup(-V)$ with respect to the natural identification $\iota: \partial V\to \{1\}\times T_A\subset W_A$. Again putting $\Sigma_+=W_A$, $\Sigma_-=V$, $B=\partial W_A\setminus \iota(\partial V)$ and $\varphi=($the trivial extension of $\psi^{-1})$, we apply [Proposition \[ob\]]{} to obtain another contact manifold $(M',\alpha')$ with an open-book decomposition, where the page is a convex overtwisted hypersurface and the binding is $T_{A^{-1}}$. Then there exists a diffeomorphism from $M^5$ to $M'$ which preserves the orientation and sends $T_A$ to $-T_{A^{-1}}$. We may consider that $(M',\alpha')$ is obtained by inserting the $5$-dimensional Lutz tube associated to $A$ along the binding $T_A$ of a supporting open-book decomposition on $(M^5,\alpha)$. \[def\]
1. Similarly, we can insert a $5$-dimensional Lutz tube along [*any*]{} codimension-$2$ contact submanifold with trivial normal bundle which is contactomorphic to $T_A$ ($\mathrm{tr}A>2$). Particularly, we can insert the Lutz tube associated to $A^{-1}$ along the core of the Lutz tube associated to $A$. Then we obtain a $5$-dimensional analogue of the full Lutz tube.
2. We may consider the original $3$-dimensional (half) Lutz tube as a trivial open-book decomposition by positive annuli whose binding is a connected component of the boundary. That is, starting with the exact symplectic annulus $([-1,1]\times S^1, sd\theta)$ ($s\in[-1,1], \theta\in S^1$) we can construct the $3$-dimensional Lutz tube by Proposition \[ob\] ($\varphi=\mathrm{id}$).
3. Geiges[@Geiges] constructed an exact symplectic $6$-manifold $[-1,1]\times M^5$ with contact-type boundary, where $M^5$ is a certain $T^4$-bundle over the circle. From his example, we can also construct a $7$-dimensional Lutz tube. The author suspects that this Lutz tube enables us to change not only the contact structure but also the homotopy class of the almost contact structure of a given contact $7$-manifold. See Question 5.3 in Etnyre-Pancholi[@Etnyre].
Exotic contact structures on $S^5$
----------------------------------
We can insert a Lutz tube into the standard $S^5$. Namely,
In the case where $m\le2$ and $k\ne0$, $T_{A_{m,k}}$ is contactomorphic to the link of the isolated singular point $(0,0,0)$ of the hypersurface $\{f_{m,k}=0\} \subset {\mathbb{C}}^3$, where $$f_{1,(k_1)}=(\eta-2\xi^2)(\eta^2+2\xi^2\eta+\xi^4-\xi^{4+k_1})+\zeta^2
\quad \textrm{and}$$ $$f_{2,(k_1,k_2)}=\{(\xi+\eta)^2-\xi^{2+k_1}\}\{(\xi-\eta)^2+\xi^{2+k_2}\}+\zeta^2.$$ \[main\]
Let $\mathcal{O}_{m,k}$ denote the Milnor fibration of the singular point $(0,0,0)\in\{f_{m,k}=0\}$.
From [Theorem \[convergence\]]{} and [Lemma \[sfl\]]{} we obtain a Reeb foliation ${\mathcal{F}}_{m,k}$ associated to $\mathcal{O}_{m,k}$.
In order to prove Theorem \[main\] we prepare an easy lemma.
1. The complex curve $$C=\{\zeta^2=-(\eta-p_1)\cdots(\eta-p_{m+2})\} \quad (m=1,2)$$ on the $\eta\zeta$-plane ${\mathbb{C}}^2$ is topologically an $m$-times punctured torus in ${\mathbb{R}}^4$ if the points $p_i$ are mutually distinct. These points are the critical values of the branched double covering $\pi_\eta|C$, where $\pi_\eta:{\mathbb{C}}^2\to{\mathbb{C}}$ denotes the projection to the $\eta$-axis.
2. Let $B: p_1=p_1(\theta),\dots,p_{m+2}=p_{m+2}(\theta)$ be a closed braid on ${\mathbb{C}}\times S^1$ $(\theta\in S^1)$. Then the above curve $C=C_\theta$ traces a surface bundle over $S^1$. Fix a proper embedding $l:{\mathbb{R}}\to {\mathbb{C}}$ into the $\eta$-axis such that $l(1)=p_1(0),\dots,l(m+2)=p_{m+2}(0)$. Suppose that the closed braid $B$ is isotopic to the geometric realization of a composition $$\prod_{j=1}^J\{\sigma_{i(j)}\}^{q(j)}\quad
(q(j)\in{\mathbb{Z}},\,i(j)\in\{1,\dots,m+1\}),$$ where $\sigma_i:{\mathbb{C}}\to{\mathbb{C}}$ denotes the right-handed exchange of $p_i$ and $p_{i+1}$ along the arc $l([i,i+1])$ $(i=1,\dots,m+1)$. Then the monodromy of the surface bundle $C_\theta$ is the composition $$\displaystyle \prod_{j=1}^J\{\tau(\ell_{i(j)})\}^{q(j)}\quad
\textrm{where} \quad
\ell_i=(\pi_\eta|C)^{-1}(l([i,i+1])).$$
\[easy\]
Regard $\xi\ne 0$ as a small parameter and take the branched double covering $\pi_\eta|C_\xi$ of the curve $C_\xi=\{f_{m,k}=0,\xi=\mathrm{const}\}\cap B^6$. Then the critical values of $\pi_\eta|C_\xi$ are $$p_1,p_2=-\xi^2\{1-(\xi^{k_1})^{1/2}\}
\quad \textrm{and}
\quad p_3=2\xi_2\quad
\textrm{in the case where}
\quad m=1$$ $$(\textrm{resp.}\quad
p_1,p_2=-\xi\{1-(\xi^{k_1})^{1/2}\},\,\,
p_3,p_4=\xi\{1-(-\xi^{k_2})^{1/2}\}\,\, \textrm{in the case where} \,\,
m=2).$$ As $\xi$ rotates along a small circle $|\xi|={\varepsilon}$ once counterclockwise, the set $\{p_1,\dots,p_{m+2}\}$ traces a closed braid, which is clearly a geometric realization of the composition $$(\sigma_1\circ\sigma_2)^6(\sigma_1)^{k_1}\qquad(\textrm{resp.}\quad
(\sigma_1\circ\sigma_2\circ\sigma_3)^4(\sigma_1)^{k_1}(\sigma_3)^{k_2}).$$ From Lemma \[easy\] and the well-known relation $$\tau(\partial C_\xi)\simeq
(\tau(\ell_1)\circ \tau(\ell_2))^6
\quad (\textrm{resp.}\quad
\tau(\partial C_\xi)\simeq
(\tau(\ell_1)\circ \tau(\ell_2) \circ \tau(\ell_3))^4),$$ we see that the link of the singular point $(0,0,0)\in\{f_{m,k}=0\}$ admits the supporting open-book decomposition in Proposition \[sol\] 2). This completes the proof of Theorem\[main\].
If we insert the Lutz tube associated to $A_{m,k}$ ($m=1,2,\,k\ne 0$) along the binding of the supporting open-book decomposition equivalent to $\mathcal{O}_{m,k}$, we obtain a contact structure $\ker(\alpha_{m,k})$ on $S^5$. Then the page becomes a convex overtwisted hypersurface. The following theorem can be proved in a similar way to the proof of Lemma \[convergence\].
The contact structure $\ker(\alpha_{m,k})$ $(m=1,2,\,k\ne 0)$ deforms via contact structures into a foliation which is obtained by cutting and turbulizing the page leaves of the Reeb foliation $\mathcal{F}_{m,k}$ along the hypersurface corresponding to the boundary of the Lutz tube.
Let $(M',\alpha')$ be the contact connected sum of any contact $5$-manifold $(M^5,\alpha)$ with the above exotic $5$-sphere $(S^5,\alpha_{m,k})$. Then we see that the contact manifold $(M'\approx M^5,\alpha')$ contains a convex overtwisted hypersurface. Namely,
Any contact $5$-manifold admits a contact structure which violates the Thurston-Bennequin inequality for a convex hypersurface with contact-type boundary.
Plastikstufes in Lutz tubes
---------------------------
We show that there exists a plastikstufe in any $5$-dimensional Lutz tube. First we define a plastikstufe in a contact $5$-manifold.
Let $(M^5,\alpha)$ be a contact $5$-manifold and $\iota : T^2\to M^5$ a Legendrian embedding of the torus which extends to an embedding $\tilde{\iota}: D^2\times S^1 \to M^5$ of the solid torus. Then the image $\tilde{\iota}(D^2\times S^1)$ is called a [*plastikstufe*]{} if there exists a function $f(r)$ such that $$(r^2d\theta+f(r)dr)\wedge(\tilde{\iota}^\ast\alpha)\equiv 0,\quad
\lim_{r\to 0}\frac{f(r)}{r^2}=0\quad\textrm{and}
\quad\lim_{r\to 1}|f(r)|=\infty,$$ where $r$ and $\theta$ denote polar coordinates on the unit disk $D^2$.
Consider the contactization $$({\mathbb{R}}\times T_A\times {\mathbb{R}}(\ni t),\, \alpha=\beta_++s\beta_-+dt)$$ of the exact symplectic manifold $({\mathbb{R}}(\ni s)\times T_A, d(\beta_++s\beta_-))$, where $\beta_\pm$ are the $1$-forms described in Example \[1.2\]. Take coordinates $p$ and $q$ near the origin on $T^2={\mathbb{R}}^2/{\mathbb{Z}}^2$ such that $p=q=0$ at the origin, $\partial/\partial p=v_+$ and $\partial/\partial q=v_-$. Then for small ${\varepsilon}>0$, the codimension-$2$ submanifold $$\mathcal{P}=\{p={\varepsilon}a^{-z}g(s), q={\varepsilon}a^zsg(s)\}\subset {\mathbb{R}}\times T_A\times{\mathbb{R}}$$ is compactified to a plastikstufe on the Lutz tube $T_A\times D^2$, where $g(s)$ is a function with $$g(s)\equiv 0\quad \textrm{on}\quad (-\infty,1],\quad
\textrm{and} \quad
g(s)\equiv \frac{1}{s\log s}\quad \textrm{on}\quad [2,\infty).$$ Note that the boundary $\{(\infty,(0,0,z),t))\,|\,z\in S^1, t\in S^1\}\approx T^2$ of the plastikstufe is a Legendrian torus, and on the submanifold $\mathcal{P}$ the contact form $\alpha$ can be written as $$\alpha|\mathcal{P}=a^{-z}dq+sa^zdp+dt=
{\varepsilon}\left( g(s)+2sg'(s) \right)ds+dt.$$ Indeed, as $s\to\infty$, $$g(s)\to 0,\quad sg(s)\to 0\quad \textrm{and}\quad
\int_{2}^s(g(s)+2sg'(s))ds\to-\infty.$$
As ${\varepsilon}\to 0$, the above plastikstufe converges to a solid torus $S^1\times D^2$ foliated by $S^1$ times the straight rays on $D^2$, i.e., the leaves are $\{t=\mathrm{const}\}$.
The following theorem is proved in the above example.
There exists a plastikstufe in any $5$-dimensional Lutz tube. \[plastikstufe\]
[(Niederkrüger-van Koert[@NK].)]{} Any contact $5$-manifold $(M^5,\alpha)$ admits another contact structure $\ker\alpha'$ such that $(M^5,\alpha')$ contains a plastikstufe.
Topology of the pages
---------------------
We decide the Euler characteristic of the page of the open-book decomposition given in Theorem \[main\] which is diffeomorphic to $$F=\{f_{m,k}(\xi,\eta,\zeta)=\delta\}\cap
\{|\xi|^2+|\eta|^2+|\zeta|^2\le {\varepsilon}\},$$ where $\delta \in{\mathbb{C}}, {\varepsilon}\in {\mathbb{R}}_{>0}, 0<|\delta|\ll {\varepsilon}\ll 1$. Let $\pi_\xi,\pi_\eta$ and $\pi_\zeta$ denote the projections to the axes.
In the case where $m=1$, the critical values of $\pi_\xi|F$ are the solutions of the system $$f_{1,(k_1)}-\delta=0,\quad
\frac{\partial}{\partial \eta} f_{1,(k_1)}=0,\quad
\frac{\partial}{\partial \zeta} f_{1,(k_1)}=2\zeta=0\quad
\textrm{and} \quad |\xi|\ll {\varepsilon}.$$ Therefore, for each critical value $\xi$ of $\pi_\xi|F$, we have the factorization $$(\eta-2\xi^2)(\eta^2+2\xi^2\eta+\xi^4-\xi^{4+k_1})-\delta = (\eta-a)^2(\eta+2a)$$ of the polynomial of $\eta$, where the parameter $a \in{\mathbb{C}}$ depends on $\xi$. By comparison of the coefficients of the $\eta^1$-terms and the $\eta^0$-terms we have $$-4\xi^4+\xi^4-\xi^{4+k_1}=a^2-2a^2-2a^2
\quad \textrm{and}\quad -2\xi^6+2\xi^{6+k_1}-\delta=2a^3.$$ Eliminating the parameter $a$, we obtain the equation $$4\xi^{12+k_1}(9-\xi^{k_1})^2=
108\xi^6(1-\xi^{k_1})\delta+27\delta^2.$$ Then we see that $\pi_\xi|F$ has $12+k_1$ critical points, which indeed satisfy $a\ne -2a$, i.e., the map $\pi_\xi|F$ defines a PALF structure on $F$ with $12+k_1$ singular fibres. Thus we have $$\chi(F)=1-2+12+k_1=11+k_1.$$
In the case where $m=2$, we have the factorization $$\{(\xi+\eta)^2-\xi^{2+k_1}\}\{(\xi-\eta)^2+\xi^{2+k_2}\}-\delta =
(\eta-a)^2(\eta+a-b)(\eta+a+b)
\quad (a,b \in {\mathbb{C}}).$$ By comparison of the coefficients we have $$\left\{
\begin{array}{l}
\xi^2(2+\xi^{k_1}-\xi^{k_2})=2a^2+b^2\\
\xi^3(\xi^{k_1}+\xi^{k_2})=ab^2\\
\xi^4(1-\xi^{k_1})(1+\xi^{k_2})-\delta=a^2(a^2-b^2)
\end{array}
\right..$$ In order to eliminate $a,b$, we put $a=u+v$ and $\xi^2(2+\xi^{k_1}-\xi^{k_2})=6uv$. Then we have $$\left\{
\begin{array}{l}
6uv-2(u+v)^2=b^2\\
\xi^3(\xi^{k_1}+\xi^{k_2})=-2(u^3+v^3)\\
\xi^4(1-\xi^{k_1})(1+\xi^{k_2})-\delta=(u+v)^4+2(u^3+v^3)(u+v)
\end{array}
\right..$$ Further we put $$p=uv,\quad q=u^3+v^3\quad \textrm{and}\quad
r=(u+v)^4+2(u^3+v^3)(u+v).$$ Then $p,q$ and $r$ are polynomials of $\xi$. Eliminating $a$ from $$q(=q(p,a))=3pa-a^3\quad \textrm{and}\quad r(=r(p,a))=-6pa^2+3a^4,$$ we obtain $$(27q^4-r^3)+54(prq^2-p^3q^2)+18p^2r^2-81p^4r=0,$$ which is a polynomial equation of $\xi$. As $\delta\to0$, the left hand side converges to $$\xi^{12+k_1+k_2}
\left\{1-\frac{\xi^{k_1}-\xi^{k_2}}{2}+\frac{(\xi^{k_1}+\xi^{k_2})^2}{16}
\right\}^2.$$ Therefore $\pi_\xi|F$ has $12+k_1+k_2$ critical points, which indeed satisfy $4a^2\ne b^2$ and $b\ne 0$, i.e., the map $\pi_\xi|F$ defines a PALF structure on $F$ with $12+k_1+k_2$ singular fibres. Thus we have $$\chi(F)=1-3+12+k_1+k_2=10+k_1+k_2.$$
Symplectic proof
----------------
In this subsection we sketch another proof of the following theorem, which is slightly weaker than Theorem \[main\] and the result of the previous subsection.
The contact manifold $T_{A_{m,k}}$ $(m=1,2)$ is contactomorphic to the binding of a supporting open-book decomposition on the standard $S^5$ whose page $P_{m,k}$ satisfies $\chi(P_{m,k})=11+k_1$ in the case where $m=1$ and $\chi(P_{m,k})=10+k_1+k_2$ in the case where $m=2$. \[sproof\]
We start with the following observation.
1. We consider the fibre $$V_\delta=\{\xi^2+\eta^2+\zeta^2=\delta\}\subset {\mathbb{C}}^3$$ of the singular fibration $f(\xi,\eta,\zeta)=\xi^2+\eta^2+\zeta^2:
{\mathbb{C}}^3\to{\mathbb{C}}(\ni\delta)$, which we call the first fibration. If $\delta\ne 0$, the restriction $\pi_\xi|V_\delta$ is a singular fibration over the $\xi$-axis, which we call the second fibration. The fibre of the second fibration is $$F_\xi=(\pi_\xi|V_\delta)^{-1}(\xi)=\{\eta^2+\zeta^2=\delta-\xi^2\}.$$ If $\xi^2\ne \delta$, the restriction $\pi_\eta|F_\xi$ has critical values $\pm\gamma=(\delta-\xi^2)^{1/2}$. That is, the second fibre $F_\xi$ is a double cover of the $\eta$-axis branched over $\pm\gamma$. We call $\pi_\eta|F_\xi$ the third fibration. Then the line segment between $\pm\gamma$ lifts to the vanishing cycle $$\{(0,\gamma x,\gamma y)\,|\, (x,y)\in S^1\subset{\mathbb{R}}^2 \}
(\approx S^1)\subset F_\xi$$ which shrinks to the singular points $(\delta^{1/2},0,0)$ on the fibres $F_{\delta^{1/2}}$. On the other hand, the line segment between $\delta^{1/2}$ on the $\xi$-axis lifts to the vanishing Lagrangian sphere $$L=\{(\delta^{1/2} x,\delta^{1/2} y,\delta^{1/2} z)\,|\,
(x,y,z)\in S^2\subset{\mathbb{R}}^3\}(\approx S^2)\subset V_\delta$$ which shrinks to the singular point $(0,0,0)$ on $V_0$. The monodromy of the first fibration around $\delta=0$ is the Dehn-Seidel twist along the Lagrangian sphere $L$ (see [@Giroux]).
2. Next we consider the [*tube*]{} $$B'=\{|\xi+\eta^2+\zeta^2|\le{\varepsilon}\}\cap B^6$$ of the regular fibration $g(\xi,\eta,\zeta)=\xi+\eta^2+\zeta^2: {\mathbb{C}}^3 \to {\mathbb{C}}$. Let $V'_\delta$ denote the fibre $g^{-1}(\delta)$ ($|\delta|\le{\varepsilon}$) of the first fibration $g$ and $F'_\xi$ the fibre of the second fibration $\pi_\xi|V'_\delta$. The third fibration $\pi_\eta|F'_\xi$ has two critical points $(\delta-\xi)^{1/2}$ on the $\eta$-axis. Then the line segment between $(\delta-\xi)^{1/2}$ lifts to the vanishing circle $$\{(\xi,(\delta-\xi)^{1/2}x,(\delta-\xi)^{1/2}y)\,|\,
(x,y)\in S^1\subset{\mathbb{R}}^2\}(\approx S^1)\subset F'_\xi$$ which shrinks to the singular point $N=(\delta,0,0)$ on $F'_\delta$. By attaching a symplectic $2$-handle to $B'$, we can simultaneously add a singular fibre to each second fibration $\pi_\xi|V'_\delta$ so that the above vanishing cycle shrinks to another singular point $S$ than $N$. Here the vanishing cycle traces a Lagrangian sphere from the north pole $N$ to the south pole $S$. (The attaching sphere can be considered as the equator.) The symplectic handle body $B'\cup$(the $2$-handle) can also be realized as a regular part $$f^{-1}(U)\cap B^6\quad (\exists U\approx D^2, U\not\ni 0)$$ of the singular fibration $f$ in the above $1)$. Thus we can add a singular fibre $V_0\cap B^6$ to it by attaching a symplectic $3$-handle. That is, the tube $\{|f|\le{\varepsilon}\}\cap B^6$ of the singular fibration $f$ can be considered as the result of the cancellation of the $2$-handle and the $3$-handle attached to the above tube $B'$ of the regular fibration $g$. Note that such a cancellation preserves the contactomorphism-type of the contact-type boundary.
Take the tubes $\{|h_m|\le{\varepsilon}\}\cap B^6$ ($m=1,2$) of the regular fibrations $$h_1(\xi,\eta,\zeta)=\xi+\eta^3+\zeta^2\quad
\textrm{and}\quad
h_2(\xi,\eta,\zeta)=\xi+\eta^4+\zeta^2.$$ Let $F_{m,\xi}$ denote the fibre of the second fibration $\pi_\xi|h_m^{-1}(\delta)$. Then the third fibration $\pi_\eta|F_{m,\xi}$ has $m+2$ singular fibres ($m=1,2$). We connect the corresponding critical values on the $\eta$-axis by a simple arc consisting of $m+1$ line segments $\sigma_1,\dots,\sigma_{m+1}$, which lift to vanishing cycles $\ell_1,\dots,\ell_{m+1}$ on the fibre $F_{m,\xi}$. Then we can attach a symplectic $2$-handle to the tube along one of the vanishing cycles $\ell_1,\dots,\ell_{m+1}$, and cancel it by attaching a $3$-handle as is described in the above observation. Iterating this procedure, we can obtain a symplectic filling of the standard $S^5$ as the total space of a symplectic singular fibration over $D^2$ whose regular fibre is an exact symplectic filling of $T_{A_{m,k}}$. Then the Euler characteristic of the fibre is $11+k_1\,(m=1)$ or $10+k_1+k_2\,(m=2)$. Indeed, the relation $$\tau(\partial F_{1,\xi})\simeq (\tau(\ell_1)\circ \tau(\ell_2))^6
\quad(\textrm{resp.} \quad
\tau(\partial F_{2,\xi})\simeq (\tau(\ell_1)\circ \tau(\ell_2)
\circ \tau(\ell_3))^4)$$ implies that we can attach $12+k_1$ (resp. $12+k_1+k_2$) pairs of $2$- and $3$-handles to the tube $\{|h_1|\le{\varepsilon}\}\cap B^6$ (resp. $\{|h_2|\le{\varepsilon}\}\cap B^6$) to obtain the above fibration. This completes the proof of [Theorem \[sproof\]]{}
1. Giroux and Mohsen further conjectured that, for any supporting open-book decomposition on a contact manifold $(M^{2n+1},\alpha)$, we can attach a $n$-handle to the page to produce a Lagrangian $n$-sphere $S^n$, and then add a Dehn-Seidel twist along $S^n$ to the monodromy to obtain another supporting open-book decomposition on $(M^{2n+1},\alpha)$ ([@Giroux]). We did a similar replacement of the supporting open-book decomposition in the above proof of [Theorem \[sproof\]]{} by means of symplectic handles.
2. Take a triple covering from the three-times punctured torus to the once punctured torus $F_{1,\xi}$ such that $\ell_2$ lifts to a long simple closed loop. Then from the relations $$\tau(\partial F_{1,\xi})\simeq(\tau(\ell_1)\circ \tau(\ell_2))^6\simeq
(\tau(\ell_1)\circ \tau(\ell_2)^3)^3$$ we see that the Dehn twist along the boundary of the three-times punctured torus is also isotopic to a composition of Dehn twists along non-separating loops. On the other hand, take a double covering from the four-times punctured torus to the twice punctured torus $F_{2,\xi}$ such that $\ell_2$ lifts to a long simple closed loop. Then from the relations $$\begin{aligned}
\tau(\partial F_{2,\xi})
& \simeq &
\{\tau(\ell_1)\}^{-1}\circ\{\tau(\ell_1)\circ\tau(\ell_3)
\circ\tau(\ell_2)\}^4\circ\tau(\ell_1) \\
& \simeq & \{\tau(\ell_2)\circ\tau(\ell_3)
\circ\tau(\ell_2)^2\circ\tau(\ell_1)\circ\tau(\ell_2)\}^2\end{aligned}$$ we see that the Dehn twist along the boundary of the four-times punctured torus is also isotopic to a composition of Dehn twists along non-separating loops.
Can we generalize Theorem \[sproof\] to the case where $m=3$ or $4$?
Further discussions
===================
A (half) Lutz twist along a Hopf fibre in the standard $S^3$ produces a basic overtwisted contact manifold $\overline{S^3}$ diffeomorphic to $S^3$. This overtwisted contact structure is supported by the negative Hopf band. Indeed any overtwisted contact manifold $\overline{M^3}$ is a connected sum with $\overline{S^3}$ (i.e., $\overline{M^3}=\exists M^3\#\overline{S^3}$). Moreover a typical supporting open-book decomposition on $\overline{M^3}$ is the Murasugi-sum (=plumbing) with a negative Hopf band (see [@Giroux]). | The author’s original motivation was to find various Lutz tubes in a given overtwisted contact $3$-manifold or simply in $\overline{S^3}$. (See [@Mori] for the first model of a Lutz tube by means of a supporting open-book decomposition.) Since the binding of the trivial supporting open-book decomposition on $S^3$ is a Hopf fibre, the above Lutz twist inserts a Lutz tube along the binding. Then the Lutz twist produces a non-supporting trivial open-book decomposition $\mathcal{O}$ by convex overtwisted disks. In $5$-dimensional case, the Lutz tube (i.e., the neighbourhood of the binding of $\mathcal{O}$) is replaced by a $5$-dimensional Lutz tube which contains a plastikstufe, and the convex overtwisted disk (i.e., the page) by a convex overtwisted hypersurface violating the Thurston-Bennequin inequality. The idea of placing a Lutz tube around the binding of a non-supporting open-book decomposition is also found in the recent work of Ishikawa[@Ishikawa]. However, in general, the insertion of a $5$-dimensional Lutz tube requires only the normal triviality of the contact submanifold $T_A$ in the original contact manifold.
Suppose that a contact $5$-manifold $(M^5,\alpha)$ contains a Lutz tube. Then does it always contain a convex overtwisted hypersurface?
We also have the basic exotic contact $5$-manifold $\overline{S^5}$ which is diffeomorphic to $S^5$ and supported by the $5$-dimensional negative Hopf band. Here the negative Hopf band is the mirror image of the positive Hopf band which is (the page of) the Milnor fibration of $(0,0,0)\in\{\xi^2+\eta^2+\zeta^2=0\}$. Thus the monodromy of the negative Hopf band is the inverse of the Dehn-Seidel twist (see Obsrvation 1) in §3.6). The fundamental problem is
Does $\overline{S^5}$ contains a Lutz tube or a plastikstufe? Could it be that $\overline{S^5}$ is contactomorphic to $(S^5,\ker(\alpha_{m,k}))$? Note that almost contact structures on $S^5$ are mutually homotopic.
The next problem can be considered as a variation of Calabi’s question (see §1).
Does the standard $S^{2n+1}$ ($n>1$) contains a convex hypersurface with disconnected contact-type boundary?
If there is no such hypersurfaces, the following conjecture trivially holds.
The Thurston-Bennequin inequality holds for any convex hypersurface with contact-type boundary in the standard $S^{2n+1}$.
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abstract: 'The goal of this preliminary note is to introduce and study a conjectural picture on lower bounds of Seshadri contants of indecomposable polarized abelian varieties. This is inspired by some work of Debarre on the subject together with the author study [@Loz18] of syzygies of abelian threefolds using the convex geometry of Newton–Okounkov bodies.'
address:
- 'Victor Lozovanu, Universitá degli Studi di Genova, Dipartimento di Matematica, Via Dodecaneso 35, 16146, Genova, Italy. *Email address*: []([email protected])'
- ' Leibniz Universität Hannover, Institut für Algebraische Geometrie, Welfengarten 1, 30167, Hannover, Germany. *Email address*: []([email protected])'
author:
- Victor Lozovanu
title: Seshadri constants of indecomposable polarized abelian varieties
---
Introduction
============
Let $X$ be a smooth complex projective variety and $L$ an ample line bundle on $X$. To measure the positivity of $L$ at a point $x\in X$, Demailly defines the *Seshadri constant* of this data to be $$\epsilon(L;x) \ {\ensuremath{\stackrel{\textrm{def}}{=}}}\ \inf_{x\in C\subseteq X}\Big\{\frac{(L\cdot C)}{\operatorname{mult}_0(C)}\Big\} \ ,$$ where the infimum is taken over all reduced and irreducible curves $C$ on $X$ containing $x$. In some sense this invariant encodes from numerical and infinitesimal perspectives all the “minimal curves” through $x$ (see [@PAG Section 5] for a nice introduction in the field).
Two lines of research became prominent in the area. In [@EL93; @EKL95; @N96] differentiation techniques lead to strong lower bounds on Seshadri constant, when $x\in X$ is very general. Second, it turns out that these invariants are connected to other fields of geometry. They appear in Kähler geometry [@Ny15; @Ny18] and diophantine approximation problems [@MR15], are linked to convex geometry [@KL17] and positivity issues of abelian varieties [@LPP11; @Loz18; @KL19; @Loz20].
The main goal of this note is to give credence to a conjectural picture on lower bounds of Seshadri constants on abelian manifolds. So, let $(A,L)$ be a $g$-dimensional abelian variety. Due to the group structure on $A$, the Seshadri constant of $L$ doesn’t depend on the base point. Denoting it by $\epsilon(L)$, then differentiation techniques, see [@ELN94], lead Nakamaye [@N96] to prove that $$\epsilon(L) \ {\geqslant}\ 1.$$ Equality holds if and only if $(A,L)$ is a product of an abelian subvariety and an elliptic curve.
With this in hand, we propose and give some credence to the following conjecture:
\[conj:main1\] Let $(A,L)$ be a $g$-dimensional indecomposable polarized abelian variety, that is not the Jacobian $(J_C,\Theta_C)$ of a smooth hyperelliptic curve $C$ of genus $g$. Then $\epsilon(L){\geqslant}2$.
Inspired by a classical conjecture of van Geemen and van der Geer [@GG86], Debarre [@D04] introduces this statement for theta-divisors. He shows that for the Jacobian of hyper-elliptic curves one has $\epsilon(L)=\frac{2g}{g+1}$. Moreover, Debarre’s initial conjecture holds for $g=3$ by [@BS01] and $g=4$ by [@I95]. In higher dimensions it is still an open question.
Our first goal is to give some credence to Conjecture \[conj:main1\]. We start with a proposition, that states in a sense that this generalized form of Debarre’s conjecture follows from the one on theta divisors.
\[prop:main1\] Suppose Conjecture \[conj:main1\] holds for irreducible theta divisors. Then for any pair $(A,L)$ that is not the Jacobian of a hyperelliptic curve, we have $\epsilon(L){\geqslant}\frac{8}{5}$.
Based on what was said above, this proposition provides the first non-trivial bound on the Seshadri constant for indecomposable polarized abelian manifolds of small dimension, i.e. $g=3,4$.
The main reason for this statement is the “minimality principle” for theta divisors. More precisely, [@BL04 Proposition 4.1.2] yields that for any ample line bundle $L$ on $A$, there is an étale map $$f: (A,L)\ \longrightarrow \ (A_L,\Theta_L) \ ,$$ such that $f^*(\Theta_L)=L$ and $\Theta_L$ is a theta divisor on $A_L$. As a consequence, one gets $\epsilon(L){\geqslant}\epsilon(\Theta_L)$. Furthermore, by Debarre’s conjecture, the bad case happens when $L$ contains a decomposable divisor, coming from $\Theta_L$. But then Bézout’s theorem and an inductive argument will do the trick, as long as we have good lower bounds in low dimensions.
In this respect we then prove the following theorem:
\[thm:main1\] Let $(A,L)$ be an indecomposable polarized abelian variety.
1. If $\textup{dim}(A)=2$, then Conjecture \[conj:main1\] holds.
2. If $\textup{dim}(A)=3$ and $(A,L)$ is not the Jacobian of a hyper-elliptic curve, then $\epsilon(L){\geqslant}\frac{12}{7}$.
When $g=2$ this follows from [@EL93] and Hodge index theorem and should be known to the experts. When $g=3$ the problem is more difficult. In the following we explain the main ideas.
Based on the proof of Proposition \[prop:main1\] and [@BS01], we need to deal with the case when the pair $(A_L,\Theta_L)$ is the Jacobian of a hyperelliptic curve $C$, where we know $\epsilon(\Theta_L)=\frac{6}{4}$. Turning our attention to the blow-up $\pi:\overline{A}\rightarrow A$ of the origin $0\in A$, the goal is to study the class $$B_t\ :=\ \pi(L)- tE \ ,\textup{ for any }t{\geqslant}0 \ ,$$ where $E\simeq {\ensuremath{\mathbb{P}}}^{2}$ is the exceptional divisor, from a numerical perspective.
In order to do so we use the theory of restricted volumes and their connection to intersection numbers as developed in [@ELMNP09] and [@LM09]. Basically we have the following formula $$\label{eq:main}
k \ {\ensuremath{\stackrel{\textrm{def}}{=}}}\ \textup{deg}(f)\ = \ \frac{L^3}{6}\ = \ \int_{0}^{\infty} \frac{\textup{vol}_{\overline{A}|E}(B_t)}{2} dt \ ,$$ where $\textup{vol}_{\overline{A}|E}(B)$ encodes asymptotically the dimension of global sections of powers of the class $B_t$ that can be restricted non-trivially to the exceptional divisor $E$.
Now, assume $\epsilon(L)<2$. Then the results in [@BS01] and the geometry of $\Theta_L=C-C$ imply that both $\epsilon(L)$ and $\epsilon(\Theta_L)$ are defined by the same numerical data. Going forward, the differentiation techniques from [@CN14] and [@Loz18] lead to strong upper bounds on $\textup{vol}_{\overline{A}|E}(B_t)$, by making use of the geometry of the blow-up of ${\ensuremath{\mathbb{P}}}^2$ at $4$ general points. When $k{\geqslant}3$, applying these bounds to $(\ref{eq:main})$, leads to a contradiction. The same happens in the case when $k=2$ by using in addition some special features of the surface $f^*(\Theta_L)$ and Zariski’s decomposition.
In order to prove Conjecture \[conj:main1\] for $g=3$, it remains to tackle the case when the pair $(A_L,\Theta_L)$ is the Jacobian of a non-hyperelliptic curve $C$. Most techniques above can be translated in this setup. But the geometry of the surface $C-C$ seems here more complex. In higher dimensions it’s not yet clear if these methods work. Still the current bounds play an important role on the author’s recent work on singularities of irreducible theta divisors in any dimension [@Loz20].
Acknowledgements {#acknowledgements .unnumbered}
----------------
Special thanks for the amazing support to all the members of Institut für Algebraische Geometrie at Leibniz Universität Hannover. The author is grateful to Víctor Gonzalez-Alonso, whose expertise on abelian varieties played an important role during this project. Also, many thanks are due to M. Fulger and A. Küronya for helpful discussions about some of the ideas in this article.
Notations
=========
In this article we work over the complex numbers ${\ensuremath{\mathbb{C}}}$. A pair $(A,L)$ stands for a $g$-dimensional abelian variety together with an ample polarization $L$. The pair $(A,L)$ is said to be *indecomposable* if it’s not isomorphic to the product of two polarized abelian varieties.
Most of the time our problems are local. So, we will be translating them to the blow-up at the origin of $A$. To use this infinitesimal perspective we fix some notation. We set $\pi:\overline{A}\rightarrow A$ to be the blow-up of the origin $0\in A$ with the exceptional divisor $E\simeq{\ensuremath{\mathbb{P}}}^{g-1}$. We denote by $$B_t \ {\ensuremath{\stackrel{\textrm{def}}{=}}}\ \pi^*(L)-tE, \textup{ for any } t{\geqslant}0 \ .$$ By [@PAG Proposition 5.1.5], then the Seshadri constant can be defined as follows $$\epsilon(L)\ = \ \max\{t{\geqslant}0\ | \ B_t\textup{ is nef}\}\ .$$ Finally we define the *infinitesimal width* of $L$ as follows $$\mu(L) \ {\ensuremath{\stackrel{\textrm{def}}{=}}}\ \max\{t{\geqslant}0\ | \ B_t\textup{ is pseudo-effective}\} \ .$$ This is the maximum multiplicity at the origin of a ${\ensuremath{\mathbb{Q}}}$-effective divisor in the class of $L$.
Seshadri constants on abelian surfaces.
=======================================
In this section we study the Seshadri constant on polarized abelian surfaces. The main result is probably known to experts, i.e. see [@BS98]. We still include it here, for the benefit of the reader and also to contrast it to how more difficult the higher-dimensional case is.
\[prop:surface\] Let $(S,L)$ be an indecomposable polarized abelian surface that is not principle. Then $\epsilon(L){\geqslant}2$.
\[rem:smooth\] If $L=\Theta_S$ is an irreducible theta divisor then $\epsilon(L)=\frac{4}{3}$ by [@St98]. Moreover $\Theta_S$ is smooth, as otherwise the condition $\Theta_S^2=2$ forces $\epsilon(\Theta_S;0){\leqslant}1$. By applying then [@N96] this contradicts that $\Theta_S$ is irreducible.
\[rem:surface\] Looking more carefully at the proof and making use of [@N96], we deduce that for a $(S,L)$ a polarized abelian surface that is not principle, then either $\epsilon(L)=1$ and there exists an elliptic curve $F\subseteq S$ with $(L\cdot F)=1$ or $\epsilon(L){\geqslant}2$.
Based on asymptotic Riemann-Roch and Remark \[rem:smooth\], we can assume that $(L^2)=2k{\geqslant}4$. Now, let $\epsilon(L)<2$, and the goal is to get a contradiction.
By Nakai-Moishezon criterion, used on the blow-up at the origin, the condition that $L^2{\geqslant}4$ implies the existence of an irreducible curve $C\subseteq S$ with $q=\operatorname{mult}_0(C)$, $p=(L\cdot C)$, and $$1 \ < \ \epsilon(L;0)\ = \ \frac{p}{q}\ < \ 2 \ ,$$ where the first inequality is due to [@N96]. Furthermore, [@KSS09], yields $C^2{\geqslant}q^2-q+2$. Applying this statement together with Hodge index, we then get the following string of inequalities $$4 \ {\leqslant}\ 2k=(L^2) \ {\leqslant}\ \frac{(L\cdot C)^2}{C^2}\ {\leqslant}\ \epsilon(L;0)^2\cdot \frac{q^2}{q^2-q+2} \ = \ \frac{p^2}{q^2-q+2} \ .$$ As $p$ is a positive integer and $\frac{p}{q}<2$, then $p{\leqslant}2q-1$. Plugging this upper bound into the expression on the right leads to an inequality that doesn’t hold for any $q{\geqslant}1$. This finishes the proof.
Conjectural lower bounds in any dimension
=========================================
In this section we assume Debarre’s original conjecture and provide a non-trivial lower bound on Seshadri constants of arbitrary polarization. In particular, we prove the following proposition:
\[prop:specialcase\] Suppose Conjecture \[conj:main1\] holds for any principle polarization. Then for an indecomposable polarized abelian variety $(A,L)$ one has $$\epsilon(L)\ {\geqslant}\ \frac{3}{2} \ .$$
The proof is done by induction. Proposition \[prop:surface\] dealt with the case $g=2$. So, we assume the statement holds in any dimension at most $g-1$ and prove it in dimension $g{\geqslant}3$.
[@BL04 Proposition 4.1.2] constructs an étale map of degree $k$ between abelian manifolds $$f \ : \ (A,L) \ \rightarrow \ (A_L,\Theta_L)$$ where $\Theta_L$ is a theta divisor and $L=f^*(\Theta_L)$, where $k=L^g/g!$.
We will assume $\epsilon(L)<1.5$ and the goal is to get a contradiction. Due to the definition of Seshadri constants, there is an irreducible curve $F\subseteq A$, satisfying the following inequalities $$\label{eq:seshadri1}
1\ < \ \epsilon(\Theta_L)\ {\leqslant}\ \epsilon(L) \ {\leqslant}\ \frac{(L\cdot F)}{\operatorname{mult}_0(F)} \ < \ 1.5 \ .$$ The first one is due to the main result of [@N96] and the indecomposability assumption. The second inequality from the left follows from [@MR15 Lemma 8.1], as $f$ is an étale map. The proof of this last statement is not hard. The basic idea is to translate the problem to the blow-up of the origin, chase diagrams, and obtain it as a consequence that the pull-back of a nef divisor remains nef.
With this inequality in hand, Debarre’s conjecture for theta divisors forces the pair $(A,\Theta_L)$ to be decomposable. Hence there is an isomorphism $$(A_L,\Theta_L)\ \simeq \ (A^L_1,\Theta^L_1)\ \times\ \ldots \ \times (A^L_r,\Theta^L_r)\ .$$ for some positive integer $r{\geqslant}2$. Furthermore, by Künneth’s formula, each divisor $\Theta^L_i$ is a theta divisor and without loss of generality we can assume that this divisor is also irreducible, by taking into account [@BL04 Theorem 4.3.1].
Moving forward, notice that choosing a smooth point $x_i\in \textup{Supp}(\Theta^L_i)$ for each $i=1,\ldots ,r$, we then automatically have the following numerical equality $$L \ \equiv_{\textup{num}} \ \sum_{i=1}^{i=r}\overbrace{f^*\big(A^L_1\times \ldots A^L_{i-1}\times(\Theta^L_i-x_i)\times A^L_{i+1}\times \ldots \times A^L_r\big)}^{D^L(x_i)} \ .$$ In particular, if there exists two smooth points $x_i\in \textup{Supp}(\Theta^L_i)$ and $x_j\in \textup{Supp}(\Theta^L_j)$, so that $$F\nsubseteq \textup{Supp}(D^L(x_i)) \textup{ and } F\nsubseteq \textup{Supp}(D^L(x_j)) \ ,$$ then Bézout’s theorem leads to the following inequality $$(L\cdot F)\ {\geqslant}\ \operatorname{mult}_{0}(D^L_{i})\operatorname{mult}_{0}(F) \ + \ \operatorname{mult}_{0}(D^L_{i})\operatorname{mult}_{0}(F) \ = \ 2\operatorname{mult}_{0}(F) \ .$$ This contradicts the upper-bound on the Seshadri constant we assumed in $(\ref{eq:seshadri1})$.
It remains to deal with the case when $F$ is contained in the support of at least $r-1$ of these divisors no matter which smooth points we are taking on the respective theta divisors. Without loss of generality we can assume that $$F \ \subseteq \textup{Supp}\Big(f^*\big(A^L_1\times \ldots A^L_{i-1}\times(\Theta^L_i-x_i)\times A^L_{i+1}\times\ldots \times A^L_r\big)\Big),$$ for any smooth point $x_i\in\textup{Supp}(\Theta^L_i)$ and each $i=1,\ldots ,r-1$.
Under these assumptions, we now use the fact that $\Theta^L_i$ is an irreducible divisor on $A^L_i$. With this in hand, [@BL04 Proposition 4.4.1] yields that the image of the Gauß map defined by $\Theta^L_i$ is not contained in a hyperplane. In particular, by semi-continuity this implies that $$\bigcap_{x_i\in \textup{Supp}(\Theta^L_i) \textup{-smooth pt.}} \textup{Supp}(\Theta^L_i-x_i)\ = \ 0_{A^L_i}\ .$$ So, going back to our curve $F$, then this equality forces the following inclusion $$F \ \subseteq \ f^{-1}\big(\overbrace{\{0_{A^L_1}\}\times \ldots \ \{0_{A^L_{r-1}}\}\times A^L_r}^{A_L'}\big) \ .$$ Since $A_L'$ is abelian then the inverse image $f^{-1}(A_L')$ remains abelian. Choose a component of this preimage that contains the curve $F$ and denote by $A'$. Then the restriction map $$f|_{A'} \ : \ A'\ \rightarrow \ A_L',$$ satisfies the property that $L|_{A'} =f^*(\Theta_L')$. Now, denote by $g'$ the dimension of the abelian subvariety $A'\subseteq A$. By [@DH07 Lemma 1], the condition that $(A,L)$ is not decomposable would imply that $L|_{A'}$ is not a theta divisor.
With the data from the last paragraph we deduce the inequality in the statement based on how big is $g'$ and Debarre’s conjecture for theta divisors. So, first we consider the case when $g'=1$. Then $F=A'$ in which case we get a contradiction to $(\ref{eq:seshadri1})$ since our curve $F$ is not smooth.
Second, let $g'=2$. In this case note that the curve $F\subseteq A'$. But then the inequalities in $(\ref{eq:seshadri1})$ imply that $\epsilon(L|_{A'})<1.5$ for a polarization $L|_{A'}$ that is not principle. Hence, Corollary \[rem:surface\] will then imply the existence of an elliptic curve $F'\subseteq A'$ with $(L\cdot F')=1$. Applying [@DH07 Lemma 1] this forces $(A,L)$ to be decomposable, contradicting the assumption in the statement.
When $g'{\geqslant}3$, we apply Debarre’s conjecture to the irreducible principle polarized abelian variety $(A',\Theta_L')$ together with [@MR15 Lemma 8.1] to obtain the following string of inequalities $$\epsilon(L) \ {\geqslant}\ \epsilon(\Theta_L')\ {\geqslant}\ \frac{2g'}{g'+1} \ {\geqslant}\ \frac{3}{2} \,$$ The latter follows easily as $g'{\geqslant}3$. This finishes the proof.
This is an application of the ideas already developed in the proof of Proposition \[prop:specialcase\]. The only difference is the last paragraph in that proof. So, for $g'=3$, we use the bound from Theorem \[thm:main1\], to say that in this case we have $$\epsilon(L)\ {\geqslant}\ \frac{12}{7} \ > \ \frac{8}{5} .$$ When $g'{\geqslant}4$, the same ideas as above then imply the inequalities $$\epsilon(L) \ {\geqslant}\ \epsilon(\Theta_L')\ {\geqslant}\ \frac{2g'}{g'+1} \ {\geqslant}\ \frac{8}{5} \ .$$ This finishes the proof.
Seshadri constants on abelian threefolds.
=========================================
In this section we give a proof of Theorem \[thm:main1\] for abelian three-fold. Since the geometry of principally polarized abelian three-folds plays an important role, we present first the main properties, classical by now, we will make use later. Then we proceed with the proof of the main result. We will try as much as possible to explain in details the tools we use. For a complete understanding though, the reader is advised to look at [@Loz18].
Infinitesimal data on the Jacobian of a hyper-elliptic curve.
-------------------------------------------------------------
We start with a short review of the infinitesimal picture for the Jacobian of a hyper-elliptic curve of genus three. The material is mostly classical and is inspired by [@BL04], [@BS01] and [@L96].
Let $C$ be a hyper-elliptic curve of genus $g=3$. Let $(J_C,\Theta_C)$ be the associated Jacobian three-fold. Since $C$ is hyperelliptic the canonical divisor $K_C$ defines a finite map $$\phi_{K_C} \ : \ C \ \xlongrightarrow{2:1} \ Q\ \subseteq \ {\ensuremath{\mathbb{P}}}^2,$$ where $Q\simeq {\ensuremath{\mathbb{P}}}^1$ is a planar smooth quadric. This map is defined by the natural involution $\sigma:C\rightarrow C$.
There is a natural embedding $C\subseteq J_C$, based on which we can assume without loss of generality that the theta divisor $S:=\Theta_C=C-C$. By [@BL04 Theorem 11.2.5] we know further that this divisor is smooth everywhere with one exception. At the origin $\operatorname{mult}_0(S)=2$, corresponding to the unique $g_2^1$, given by the map $\phi_{K_C}$. Based on this information we define the difference map $$\partial \ : \ C\times C\ \xlongrightarrow\ S=C-C, \textup{ where } (x,y)\rightarrow x-y ,$$ which contracts the diagonal $\Delta$. Denote also by $F_1$ and $F_2$ the corresponding fibers.
Now, consider the blow-up $\pi:\overline{J}_C\rightarrow J_C$ of the origin, where as usual $E\simeq {\ensuremath{\mathbb{P}}}^2$ is the exceptional divisor. Then we have the following diagram $$\begin{tikzcd}
C\times C \arrow{r}{\overline{q}} \arrow[swap]{d}{\textup{id}\times \sigma} \arrow[rr, bend left=25, "\partial"] & \overline{S} \arrow{r}[swap]{\pi|_{\overline{S}}} \arrow{d}{\sim} & S\\
C\times C \arrow{r}{q}& \textup{Sym}^2(C)&
\end{tikzcd}$$ where $q$ is the natural quotient map, which is generically $2:1$, and $\overline{S}$ is the proper transform of $S$ through $\pi$.
First note that the right vertical map is an isomorphism. Second, looking at the differential of $\phi_{K_C}$ (see [@G84] or [@BL04 Proposition 11.1.4]), we have the following identification $$\overline{S}\cap E=Q\ \subseteq \ E\simeq \ {\ensuremath{\mathbb{P}}}^2 \ .$$ Finally, we can also describe the pull-back $\partial^*(\Theta_C|_S)=2F_1+2F_2+\Delta$.
With this in hand, we denote by $\Gamma\subseteq C\times C$ the graph of the involution $\sigma$. It is not hard to show that $\Gamma\in |2F_1+2F_2-\Delta|$. Furthermore, the degree $2$ map $\overline{q}:C\times C\rightarrow \overline{S}$ is not ramified over the proper transform $\overline{F}_C$ of $F_C=\partial(\Gamma)$.
Going back to the Seshadri constant, by [@BS01], we know that $$\epsilon(\Theta_C;0) \ = \ \frac{(\Theta_C\cdot F_C)}{\operatorname{mult}_0(F_C)} \ = \ \frac{3}{2} \ ,$$ where $(\Theta_C\cdot F_C)=6$ and $\operatorname{mult}(F_C)=4$. Furthermore, the intersection $\overline{F}_C\cap E$ consists of four different points lying on the quadric $Q$, i.e. these are the ramification points of the canonical map. As a consequence these four points are in general position, i.e. no three lie on a line.
The final property is the “minimality” of $F_C$ in a numerical sense as a curve on $C\times C$.
\[lem:hyperelliptic\] For any irreducible curve $F\neq F_C\subseteq J_C$ the following inequality holds: $$\frac{(\Theta_C\cdot F)}{\operatorname{mult}_0(F)}\ {\geqslant}\ 2 \ .$$
As $\operatorname{mult}_0(S)=2$, note that Bézout’s theorem yields the inequality $$(\Theta_C\cdot F)\ {\geqslant}\ \operatorname{mult}_0(S)\cdot \operatorname{mult}_0(F)\ = \ 2\operatorname{mult}_0(F) \ ,$$ whenever $F\nsubseteq S$. Thus it remains to deal with the case when $F\subseteq S$.
The idea is to transfer the data on $C\times C$ making use of the difference map. Based on the proof of Proposition \[prop:etale\], even if the difference map is not étale, one can easily show that $$\big(2F_1+2F_2+\Delta\cdot \partial^*(F)\big)\ = \ 2(\Theta_C\cdot F) \textup{ and } \big(\partial^*(F)\cdot \Delta\big)\ = \ 2\operatorname{mult}_0(F) \ .$$ Since we can assume already that $\Gamma\nsubseteq \textup{Supp}(\partial^*(F))$, then automatically the inequalities hold $$\big(\partial^*(F)\cdot (2F_1+2F_2+\Delta)\big)\ = \ \big(\partial^*(F)\cdot \Gamma\big)+2\big(\partial^*(F)\cdot \Delta\big)\ {\geqslant}\ 2\big(\partial^*(F)\cdot\Delta\big) \ .$$ This finishes the proof.
Seshadri constant for polarized abelian threefolds.
---------------------------------------------------
We already know the behaviour of the Seshadri constant for a principle polarized abelian three-fold by [@BS01]. Combining this with the following statement, gives automatically a proof of Theorem \[thm:main1\].
\[thm:threefold\] Let $(A,L)$ be an indecomposable polarized abelian threefold that is not principle. Then $$\epsilon(L)\ {\geqslant}\ \frac{12}{7} \ .$$
For a product of polarized varieties $(X_1\times X_2,L_1\boxplus L_2)$ and a point $(x_1,x_2)\in X_1\times X_2$, [@MR15 Proposition 3.4] yields the following: $$\epsilon(L_1\boxplus L_2) \ = \ \min\{\epsilon(L_1,x_1),\epsilon(L_2,x_2)\} \ .$$ Now, suppose $(A,L)$ is a decomposable abelian three-fold that is not principle. Then Proposition \[prop:surface\] and Remark \[rem:surface\] imply that $\epsilon(L)$ is either $1$,$\frac{4}{3}$, or at least $2$.
Assume the statement doesn’t hold and our goal is to find a contradiction. By the definition of Seshadri constant, this implies that there exists a curve $F\subseteq A$ with $$\label{eq:cond}
1\ <\ \epsilon(L)\ {\leqslant}\ \frac{(L\cdot F)}{\operatorname{mult}_0(F)} \ < \ \frac{12}{7} \ .$$ Our first step is to show that $F$ must be non-degenerate. If $F$ is elliptic, so it’s smooth, and automatically does not satisfy the inequality. When $F$ is contained in an abelian surface $S\subseteq X$, the above inequality holds also by the pair $(S,L|_S)$. So, by Proposition \[prop:surface\] yields that $L|_S$ must be principal. But this cannot be true, as [@DH07 Lemma 1] would imply that $L$ is decomposable and cContradicting one of the conditions in the statement.
For our second step we go back to the proof of Proposition \[prop:specialcase\]. By asymptotic Riemann-Roch, since $L$ is not principle, then $L^3=6k$ for some $k{\geqslant}2$. Applying [@BL04 Proposition 4.1.2], there exists then an étale map $$f:(A,L) \ \rightarrow \ (J_C,\Theta_C)$$ of degree $k$ and $L=f^*(\Theta_C)$.
Based on the description of the Seshadri constant for principally polarized abelian three-folds from [@BS01] and [@MR15 Lemma 8.1], we deduce that $(\ref{eq:cond})$ forces us to deal with two cases. The first one is when the pair $(J_C,\Theta_C)$ is decomposable. But this has been dealt already in the proof of Proposition \[prop:specialcase\], leading to a contradiction of $(\ref{eq:cond})$.
In the rest of the proof we deal with the second case and that is when $(J_C,\Theta_C)$ is the Jacobian of a hyperelliptic curve $C$. Under these assumptions, our first step is to show that $f(F)=F_C$ and $f^*(F_C)$ is $k$ copies of $F$, no two intersecting. Without any confusion we will keep here the notation from the previous subsection on the infinitesimal picture of hyper-elliptic Jacobians.
In order to prove the first step, we make the following notation $$F\cap f^{-1}(0)\ = \ \{x_1=0_X,\ldots ,x_d\} \ , \textup{ for some} \ d{\leqslant}k \ .$$ Applying now Proposition \[prop:etale\], $(\ref{eq:cond})$ the following inequalities hold $$2 \ > \ \frac{12}{7} \ > \ \frac{(L\cdot F)}{\operatorname{mult}_0(F)} \ = \ \frac{\textup{deg}\big(F\rightarrow f(F)\big)\cdot (\Theta_C\cdot f(F))}{\textup{deg}\big(F\rightarrow f(F)\big)\cdot \operatorname{mult}_{0}(f(F))-\sum_{i=2}^{i=d}\operatorname{mult}_{x_i}(F)} \ {\geqslant}\ \frac{(\Theta_C\cdot f(F))}{\operatorname{mult}_0(f(F))} \ .$$ Together with Lemma \[lem:hyperelliptic\], this then forces $f(F)=F_C$.
Finally, $f:A\rightarrow J_C$ is a local analytical isomorphism around the origin. So, $\operatorname{mult}_{0}(F){\leqslant}\operatorname{mult}_0(F_C)=4$. When either this last inequality is strict or $\textup{deg}(F\rightarrow f(F)){\geqslant}2$, then $$\frac{(L\cdot F)}{\operatorname{mult}_0(F)} \ {\geqslant}\ \frac{6\cdot \textup{deg}\big(F\rightarrow f(F)\big)}{\operatorname{mult}_0(F_C)}{\geqslant}2 \ ,$$ contradicting our assumption in $(\ref{eq:cond})$. This implies that the map $F\rightarrow f(F)$ is birational and $\operatorname{mult}_0(F)=4$. The latter data forces then $d=1$. In particular, $f^*(F_C)$ is $k$ distinct copies of $F$.
As a consequence of the first step, we know the following numerical data about the curve $F$ on $A$ $$\label{eq:number}
(L\cdot F)\ = \ 6, \operatorname{mult}_{0}(F)=4 \textup{ and } \epsilon(L)=\frac{3}{2}.$$ Next we turn our focus to the surface $S_A{\ensuremath{\stackrel{\textrm{def}}{=}}}f^*(S)$. Since $S=\Theta_C$ is singular only at the origin with multiplicity two and the map $f$ is étale, then $S_A$ is irreducible on $X$. Furthermore, $S_A$ is smooth at all points with the exception of those in $f^{-1}(0)$ where it has multiplicity $2$.
As usual let $\pi_A:\overline{A}\rightarrow A$ be the blow-up of $A$ at the origin. Let $\overline{F}$ and $\overline{S}_A$ be the proper transform through $\pi_A$ of the curve $F$ and respectively surface $S_A$. Now, taking into account the infinitesimal picture of $(J_C,\Theta_C)$ from Section 3.2, and the fact that $f$ is a local isomorphism around the origin imply that $\overline{S}_A\cap E_A=Q_A\subseteq E_A\simeq{\ensuremath{\mathbb{P}}}^2$ is a smooth quadric and $\overline{F}\cap E_A=\{P_1,P_2,P_3,P_4\}$ are four distinct points sitting on $Q_A$, and thus no three of them sitting on a line.
Let $B_t=\pi_A^*(L)-tE_A$ for any $t{\geqslant}0$. With all this data in hand our first claim is to prove that $$\label{eq:assumption}
\mu(L) \ {\leqslant}\ 2.5 \ , \textup{ when } k=2 \ .$$ Assume the opposite that $\mu(L)>2.5$. Thus there exists a rational number $t_0>2.5$, such that the divisor class $B_{t_0}$ is actually big. Since $\overline{S}_A\equiv B_2$, then the last assumption automatically yields that $\overline{S}_A\nsubseteq {\ensuremath{\textbf{\textup{B}}_{+} }}(B_{t})$ for any $t{\leqslant}t_0$. Making use now of the universal property of blow-up, we then have the the following equalities of divisors $$B_t|_{\overline{S}_A}\ = \ \big(\pi^*_A(L) \ - \ tE_A\big)|_{\overline{S}_A} \ = \ \pi_A^*(L|_{S_A})-tQ_A, \textup{ for any } t>0 \ .$$ In order to get a contradiction the main idea is to understand better the behaviour of the multiplicity function of the class $B_t|_{\overline{S}_A}$ along the curve $\overline{F}$.
When $t\in[0,1.5)$, this class is ample as $B_t$ is so. When $t\in (1.5,2)$, it is not hard to see that $\overline{F}$ is the only one curve intersecting the class $B_t|_{\overline{S}_A}$ negatively. This can be seen by reducing the problem on $\overline{A}$ and then the exact same ideas as above, used to obtain $(\ref{eq:number})$, imply the uniqueness of $\overline{F}$ satisfying the property that $(\overline{F}\cdot B_t|_{\overline{S}_A})<0$.
With this latter property in hand, applying the first step in the Zariski decomposition algorithm, as described in [@B01 Theorem 14.14], we can write $$B_t|_{\overline{S}_A} \ = \ P_t\ + \ (2t-3)\overline{F} \ , \textup{ for any } t\in (1.5,2) \ ,$$ where $P_t$ might not be nef but it is a big class. The equality was deduced using the fact that $(\overline{F}^2)_{\overline{S}_A}=-2$. Finally, it is worth poiting out that $\overline{S}_A$ is not a smooth surface, so we might get in trouble with intersection theory, applied for the algorithm above. The correct way is to do all of these computations on the proper transform of $S_A$ through the blow-up of $A$ at all the points in $f^{-1}(0)$, as this surface is smooth. But due to the form of all the divisors involved, as real Cartier ones, and the fact that $F$ contains only the origin from the points in $f^{-1}(0)$, it is clear that these computations are the same.
In particular, the convexity property for multiplicity [@Loz18 Lemma 2.5] yields then $$\operatorname{mult}_{\overline{F}}\big(||B_t|_{\overline{S}_A}||\big) \ {\geqslant}\ 2t-3 \ ,$$ for any $t{\geqslant}2$, as long as the class on $\overline{S}_A$ is pseudo-effective.
With this in hand we turn out attention to a very general choice of an effective ${\ensuremath{\mathbb{Q}}}$-divisor $\overline{D}\equiv B_{t_0}$. By above $\overline{S}_A\nsubseteq \textup{Supp}(\overline{D})$ and thus the restriction $\overline{D}|_{\overline{S}_A}$ makes sense as an effective divisor. Applying the inequality above yields $$\operatorname{mult}_{\overline{F}}(\overline{D}|_{\overline{S}_A}) \ {\geqslant}\ 2t_0-3\ > \ 2 \ .$$ So, taking the push-forward divisor $D=\pi_*(\overline{D})$, then this inequality considered on $A$ implies $$6k \ = \ L^3 \ = \ (L\cdot S_A\cdot D)\ > \ 2(L\cdot F) \ = \ 12 \ .$$ We obtain a contradiction and thus $(\ref{eq:assumption})$ holds whenever $k=2$.
The rest of the proof is inspired by the ideas developed in [@Loz18]. Our first step is to find upper bounds on the restricted volume of the class $B_t$ along the exceptional divisor $E_A$. So, we consider initially the case when $t\in [0,1.5]$, where the class $B_t$ is ample. But here [@ELMNP09 Corollary 2.17] yields easily the following equality $$\label{eq:volumeone}
\textup{vol}_{\overline{A}|E_A}(B_t)\ = \ (B_t^2\cdot E_A) \ = \ t^2 \ .$$ We turn next our attention to the interval $[1.5,\mu(L)]$. The problem here is much harder, as the class $B_t$ is not ample anymore. But [@N05 Lemma 1.3] yields the following inequality: $$\operatorname{mult}_{\overline{F}}(||B_t||) \ {\geqslant}\ t-\frac{3}{2}\ .$$ So, taking an effective divisor $\overline{D}\equiv B_t$, which does not contain $E_A$ in its support, then its restriction to it makes sense and so the above inequality yields $$\operatorname{mult}_{P_i}(\overline{D}|_{E_A})\ {\geqslant}t-\frac{3}{2}, \textup{ for any } i=1,2,3,4 \ .$$ Taking this into account, the definition of restricted volumes from [@ELMNP09] implies the following upper bound: $$\begin{aligned}
\textup{vol}_{\overline{A}|E_A}(B_t)\ & = \ \limsup_{m\rightarrow\infty}\frac{\textup{dim}_{{\ensuremath{\mathbb{C}}}}\big(\textup{Im}\big(H^0(\overline{A},{\ensuremath{\mathscr{O}}}_{\overline{A}}(mB_t))
\rightarrow H^0(E_A,{\ensuremath{\mathscr{O}}}_{E_A}(mt))\big)\big)}{m^2/2} \\ &{\leqslant}\ \limsup_{m\rightarrow \infty}\frac{\textup{dim}_{{\ensuremath{\mathbb{C}}}}\big(\{P\in H^0({\ensuremath{\mathbb{P}}}^2,{\ensuremath{\mathscr{O}}}_{{\ensuremath{\mathbb{P}}}^2}(mt)\big)\ | \ \operatorname{mult}_{P_i}(P){\geqslant}m(t-1.5)\}\big)}{m^2/2}\ .
\end{aligned}$$ The importance of this upper bound lies in the fact that it reduces the problem to one on ${\ensuremath{\mathbb{P}}}^2$.
So, let $\phi: S'\rightarrow {\ensuremath{\mathbb{P}}}^2$ be the blow-up of ${\ensuremath{\mathbb{P}}}^2$ at the points $P_1, P_2,P_3,P_4$, with $E_1,E_2,E_3,E_4$ the corresponding exceptional divisors that are all $(-1)$ rational curves. It is then not hard to see that the previous inequality can be translated to the following one $$\textup{vol}_{\overline{A}|E_A}(B_t)\ {\leqslant}\ \textup{vol}_{S'}(R_t), \ \textup{where } R_t\ {\ensuremath{\stackrel{\textrm{def}}{=}}}\ \phi^*({\ensuremath{\mathscr{O}}}_{{\ensuremath{\mathbb{P}}}^2}(t))-(t-1.5)\sum_{i=1}^{i=4} E_i \ .$$ In order to find an upper bound on the volume on the right, we use the basic properties of the surface $S'$. As we said above, the four points $P_1,\ldots ,P_4\in Q_A$ are distinct and no three lie on a line. Furthermore, the proper transform of $Q_A$ on $S'$ is a nef class, which is not ample, as its self-intersection is zero. Using this class and its relation to $R_t$, it is not then hard to prove that the class $R_t$ is nef, whenever $t\in [1.5,3]$, and not pseudo-effective if $t>3$. Hence, $\textup{vol}_{S'}(R_t)=(R_t^2)$. In particular, this provides upper-bounds for our initial restricted volume $$\label{eq:volumetwo}
\textup{vol}_{\overline{A}|E_A}(B_t)\ {\leqslant}\begin{cases}
t^2-4(t-1.5)^2, \textup{ if } t\in [1.5,3]\\
0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textup{if } t>3.
\end{cases}$$ Furthermore, this forces $\mu(L){\leqslant}3$.
Our final step is to use [@LM09 Corollary C], based on the theory of Newton–Okounkov bodies. This provides us with the following formula: $$\frac{L^3}{6} \ = \ \frac{\textup{vol}_{\overline{A}}(\pi^*(L))}{6}\ = \ \int_{0}^{\mu(L;0)}\frac{\operatorname{vol}_{\overline{A}|E_A}(B_t)}{2} \ dt \ .$$ Plugging in $(\ref{eq:volumeone})$ and $(\ref{eq:volumetwo})$ into it, yields the following the inequality $$k \ {\leqslant}\ \int_{0}^{1.5}\frac{t^2}{2} \ dt\ + \ \int_{1.5}^{3}\frac{t^2-4(t-1.5)^2}{2} \ dt \ = \ \frac{13.5}{6}$$ In particular, we get a contradiction when $k{\geqslant}3$.
When $k=2$, we have a slightly better upper-bound on the infinitesimal width, given by $(\ref{eq:assumption})$. So, applying the same algorithm as above, we get the following inequality: $$k\ = \ 2 \ {\leqslant}\ \int_{0}^{1.5}\frac{t^2}{2} \ dt\ + \ \int_{1.5}^{2.5}\frac{t^2-4(t-1.5)^2}{2} \ dt \ = \ \frac{11.625}{6} \ ,$$ which again forces a contradiction. In particular, $(\ref{eq:cond})$ cannot hold and this finishes the proof of the main statement.
Appendix: multiplicity under étale maps
=======================================
The goal of this section is to describe o formula for the behaviour under étale maps of the multiplicity at a point of a subvariety. This formula most surely is known to the experts but for completeness we include here its proof.
\[prop:etale\] Let $f:X\rightarrow Y$ be an étale map of degree $d{\geqslant}2$ between two smooth varieties. Let $V\subseteq X$ an irreducible subvariety passing through the point $x_1\in X$, and let $x_1,\ldots x_d$ be the points in the fiber $f^{-1}(f(x))$. Then we have the following inequality $$\textup{deg}\big(V\rightarrow f(V)\big)\cdot \operatorname{mult}_{f(x)}(f(V))\ = \ \sum_{i=1}^{i=d}\cdot \operatorname{mult}_{x_i}(V) \ .$$
Let $W\subseteq Y$ be a subvariety passing through some point $y\in Y$. Denote by $\pi_Y:\overline{Y}\rightarrow Y$ the blow-up of $Y$ at the point $y$ with $E_Y$ the exceptional divisor and by $\overline{W}$ be the proper transform of $W$ through the blow-up map $\pi_Y$. Under this notation we can translate the multiplicity as an intersection number on the blow-up borrow the following [@F84 p.79]: $$\label{eq:mult}
\operatorname{mult}_y(W) \ =\ -\overline{W}\cdot (-E_Y)^{\textup{dim}(W)}\ .$$ Taking this into account, denote by $y=f(x)$ and let $W=f(V)$. With this in hand we will be using the following commutative diagram: $$\begin{tikzcd}
\overline{X} \arrow{r}{\overline{f}} \arrow[swap]{d}{\pi_X} & \overline{Y} \arrow{d}{\pi_Y} \\ X \arrow{r}{f}& Y
\end{tikzcd}$$ Here the map $\pi_X$ is the blow-up of $X$ at the points $x_1,\ldots ,x_d$ with the exceptional divisors $E_1,\ldots ,E_d$. Due to the fact that $f$ is an étale map of degree $d$, then we know for sure that $$\overline{f}^*(E_Y)\ = \ E_1\ + \ \ldots\ +\ E_d \ .$$ Let $\overline{W}\subseteq \overline{Y}$ be the proper transform of $W$ through $\pi_Y$ and $\overline{V}$ of $V$ through $\pi_X$.
With these in hand we proceed to prove the main equality. Since no two divisors from $E_1,\ldots ,E_d$ intersect and making use of $(\ref{eq:mult})$ , we then deduce the following list of equalities $$\big(-\overline{V}\cdot \big(-\overline{f}^*(E)\big)^{\textup{dim}(\overline{V})} \big) \ = \ \big(-\overline{V}\cdot \big(-\sum_{i=1}^{i=d} E_i\big)^{\textup{dim}(\overline{V})} \big) \ =\ \sum_{i=1}^{i=d}\big(-\overline{V}\cdot \big(-E_i)\big)^{\textup{dim}(\overline{V})} \big) \ = \ \sum_{i=1}^{i=d} \operatorname{mult}_{x_i}(V) \ .$$ Now projection formula for intersection numbers and the fact that $\overline{Y}=\overline{f}(\overline{V})$ yield also $$\big(-\overline{V}\cdot \big(-\overline{f}^*(E)\big)^{\textup{dim}(\overline{V})} \big) \ = \ \textup{deg}\big(\overline{V}\rightarrow \overline{W}\big)\cdot \big(-\overline{W}\cdot \big(-E\big)^{\textup{dim}(\overline{W})} \big) \ = \ \textup{deg}\big(V\rightarrow W\big)\cdot \operatorname{mult}_y(W)\ .$$ Finally putting together the last two sequences of equalities implies easily the statement and this finishes the proof.
[RWY]{}
|
---
abstract: 'In two recent papers, O. Entin–Wohlman [*et al.*]{} studied the question: “Which physical information is carried by the transmission phase through a quantum dot?” In the present paper, this question is answered for an islolated Coulomb–blockade resonance and within a theoretical model which is more closely patterned after the geometry of the actual experiment by Schuster [*et al.*]{} than is the model of O. Entin–Wohlman [*et al.*]{}. We conclude that whenever the number of leads coupled to the Aharanov–Bohm interferometer is larger than two, and the total number of channels is sufficiently large, the transmission phase does reflect the Breit–Wigner behavior of the resonance phase shift.'
address: 'Max-Planck-Institut für Kernphysik, D-69029 Heidelberg, Germany'
author:
- 'H. A. Weidenmüller'
title: 'Transmission Phase of an Isolated Coulomb–Blockade Resonance'
---
[2]{}
Introduction {#int}
============
In 1997, Schuster [*et al.*]{} [@sch97] reported on a measurement of the transmission phase through a quantum dot (QD). These authors used an Aharanov–Bohm (AB) interferometer with the QD embedded in one of its arms. The device is schematically shown in Figure \[fig1\]. The current through the device is made up of coherent contributions from both arms and is, therefore, a periodic function of the magnetic flux $\phi$ through the AB interferometer. A sequence of Coulomb–blockade resonances in the QD was swept by adjusting the plunger gate voltage $V_g$ on the QD. (The plunger gate is not shown in the Figure). The phase shift $\delta \phi$ of the oscillatory part of the current was measured as a function of $V_g$. We refer to this phase shift as to the transmission phase through the QD. As expected, $\delta \phi$ showed an increase by $\pi$ over each Coulomb–blockade resonance.
The AB interferometer of Ref. [@sch97] was attached to six external leads. The complexity of this arrangement was caused by the failure of an earlier two–lead experiment [@yac95] also aimed at measuring $\delta \phi$. Instead of a smooth rise by $\pi$ over the width of each resonance, a sudden jump by $\pi$ near each resonance was observed. This feature was later understood to be caused by a special symmetry property of the two–lead experiment [@bue86]: The conductance $g(\phi)$ and, therefore, the current are symmetric functions of $\phi$ and, hence, even in $\phi$. Thus, $g$ is a function of $\cos ( \phi )$ only, and the apparent phase jump of $\delta \phi$ by $\pi$ is actually due to the vanishing at some value of $V_g$ near resonance of the coefficient multiplying $\cos ( \phi
)$.
[2]{}
Following the work of Ref. [@sch97], theoretical attention was largely focused on the least expected feature of the data of Ref. [@sch97]: A sequence of Coulomb–blockade resonances displayed very similar behavior regarding the dependence of both, the conductance and the transmission phase, on $V_g$. In particular, $\delta \phi$ displayed a rapid drop between neighboring Coulomb–blockade resonances. For references, see Ref. [@bal99]. It is only recently that Entin–Wohlman, Aharony, Imry, Levinson, and Schiller [@ent01; @ent02] drew attention to the behavior of $\delta
\phi$ at a [*single*]{} resonance. In Ref. [@ent01] the first four authors consider a three–lead situation. Two leads are attached to the AB interferometer in a fashion analogous to Figure \[fig1\]. The third lead connects directly to the QD. The authors take the arms of the AB interferometer and the three external leads as one–dimensional wires. They consider an isolated resonance due to a single state on the QD. Solving this model analytically, they come up with a disturbing result: The transmission phase increases by $\pi$ over an energy interval given by that part of the width of the QD which is due to its coupling to the third lead. As this coupling is gradually turned off, the rise by $\pi$ of $\delta \phi$ becomes ever more steep, and eventually becomes a phase jump by $\pi$ as the coupling to the third lead vanishes. In Ref. [@ent02], the model is extended to include additional one–dimensional wires directly attached to the AB ring and coupled to it in a special way. Again, it is found that the transmission phase reflects the resonance phase shift only “for specific ways of opening the system”[@ent02]. These results immediately poses the following questions: What is the behavior of $\delta \phi$ for a single resonance in the six–lead case and, more generally, for any number of leads in a geometry like the one shown in Figure \[fig1\]? Does the rise of $\delta \phi$ by $\pi$ occur over an energy interval given by the actual width of the resonance or only by part of that width? And what happens to $\delta
\phi$ as the number of leads is gradually reduced to two? It is the purpose of the present paper to answer these questions within a theoretical framework which is more closely patterned after the experimental situation than is the work of Ref. [@ent01]. In particular, our work differs from that of Refs. [@ent01; @ent02] in the following respects: We do not assume that the QD is directly coupled to a lead, we do not make any specific assumptions about the way in which the leads are coupled to the AB ring, and we allow for an arbitrary number of leads (as long as this number is at least equal to two) and of channels in each lead.
Model
=====
In one of the first theoretical papers [@hack97] addressing the data of Ref [@sch97], the transmission phase $\delta \phi$ was calculated in the framework of a model designed in Ref. [@hack96], and for the geometry shown in Figure \[fig1\]. Inspection of the curves published in Ref. [@hack97] shows that the transmission phase rises roughly by $\pi$ over an energy interval roughly equal to the width of each Coulomb–blockade resonance. The curves shown in Ref. [@hack97] were, however, calculated in the framework of specific assumptions on a number of parameters and do not, therefore, constitute a general answer to the questions raised at the end of Section \[int\]. Still, it is useful to employ again the model used in these calculations. As we shall see, the model yields a completely general expression for the conductance and its dependence upon $V_g$ and $\phi$ in the framework of the geometry displayed in Figure \[fig1\].
Starting point for the study of a case with $R$ leads where $R$ is integer and $R \geq 2$ is the Landauer–Büttiker formula $$I_r = \sum_{s = 1}^R G_{rs} V_s \ .
\label{eq0}$$ The formula connects the current $I_r$ in lead $r$, $r = 1,\ldots,R$ with the voltages $V_s$ applied to leads $s$. The conductance coefficients $G_{rs}$ are given by $$G_{rs} = \frac{e^2}{h} T^{rs} = \frac{e^2}{h} \sum_{ab} \int {\rm
d}{\cal E} \biggl ( \frac{{\rm d}F({\cal E})} {{\rm d}{\cal E}}
\biggr ) [ |S^{rs}_{ab}({\cal E})|^2 - \delta_{ab} ] \ .
\label{eqg}$$ Here $S^{rs}_{ab}({\cal E})$ are the elements of the scattering matrix $S$ connecting channel $a$ in lead $r$ with channel $b$ in lead $s$ at an energy ${\cal E}$ of the electron. We have used the terminology of scattering theory and identified the transverse modes of the electron in each lead with the channels. The function $F({\cal E})$ is the Fermi function. We simplify our reasoning by considering very low temperatures where the integral in Eq. (\[eqg\]) can be replaced by $[ |S^{rs}_{ab}(E_F)|^2 - \delta_{ab} ]$, identifying the Fermi energy parametrically with the plunger gate voltage $V_g$. We do so because this brings out the energy dependence of the transmission phase most clearly. Subsequent averaging over temperature does not change the essential aspects. We recall that the Landauer–Büttiker formula is restricted to the case of independent electrons. This is the approximation used throughout the paper.
To proceed, we must introduce a model for the scattering matrix $S(E_F)$. We consider a geometrical arrangement of the type shown schematically in Figure \[fig1\] without, however, limiting ourselves to six channels. We emphasize that this geometry differs from the one considered in Ref.[@ent01] where, as mentioned above, the QD is directly coupled to one of the leads. The electrons move independently in the two–dimensional area defined by the leads, the AB interferometer, and the QD. A homogeneous magnetic field is applied perpendicularly to the plane of Figure \[fig1\].
The two–dimensional configuration space is divided into disconnected subspaces defined by the interior of the QD, of the AB interferometer without the QD, and by each of the leads. The free scattering states in lead $r$ carry the labels $E$ for energy and $a$ for the channel, with $c^{r \dagger}_{aE}$ and $c^r_{aE}$ the corresponding creation and destruction operators. The bound states in the closed AB ring have energies $\varepsilon_i$, $i = 1,2,\ldots$ and associated operators $d^{\dagger}_i$ and $d_i$. The QD supports a single bound state with energy $E_0$ and associated operators $q^{\dagger}$ and $q$. This last simplification is introduced because we wish to investigate the behavior of $\delta \phi$ at an [*isolated*]{} Coulomb–blockade resonance. At the expense of an increase of the number of indices, this assumption can easily be removed, see Refs. [@hack96; @hack97].
The single–particle Hamiltonian $H$ is accordingly written as the sum of two terms, $$H = H_0 + H_1 \ .
\label{eq1}$$ Here, $H_0$ describes free electron motion in each of the disconnected subspaces, $$H_0 = \sum_{r,a} \int {\rm d}E \ E c^{r \dagger}_{aE} c^r_{aE} +
\sum_i \varepsilon_i d^{\dagger}_i d_i + E_0 q^{\dagger} q \ .
\label{eq2}$$ The coupling between the various subspaces, and the influence of the magnetic field are contained in the coupling Hamiltonian $$\begin{aligned}
H_1 &=& \sum_{r,a,i} \int {\rm d}E \biggl ( W^r_{a i}(E) c^{r
\dagger}_{a E} d_i + h. c. \biggr ) \nonumber \\
&&\qquad + \sum_{i p} \biggl ( V^p_i q^{\dagger} d_i + h. c. \biggr )
\ .
\label{eq3}\end{aligned}$$ The matrix elements $W^r_{a i}(E)$ describe the coupling between channel $a$ in lead $r$ and the state $i$ in the AB ring. There are no barriers separating the leads from the AB ring. Therefore, the coupling to the leads will change the states $i$ into strongly overlapping resonances. We will accordingly assume later that the resulting terms depend smoothly on energy $E$. We also assume that on the scale of the mean level spacing in the AB ring, the energy dependence of the $W$’s is smooth and, in effect, negligible. The matrix elements $V^p_i$ describe the coupling between the states $i$ and the state in the QD with energy $E_0$. The upper index $p$ with $p
= 1,2$ distinguishes the two barriers which separate the QD from the AB ring, see Figure \[fig1\]. In our model, the topology of the AB ring enters via the occurrence of two independent amplitudes for decay of the state with energy $E_0$ on the QD into each of the states $i$ of the AB ring. Because of gauge invariance, the entire dependence of $H$ on the applied magnetic field can, without loss of generality, be put into one of the matrix elements $V$. We accordingly assume that the amplitudes $W^r_{a i}(E)$ and $V^1$ are real and write $V^2$ in the form $$V^2_i \exp(+i \phi) = (V^{2}_i)^* \exp(-i \phi) = v^{(2)}
\label{eq4}$$ where $v^{(2)}$ is real. The phase $\phi$ is given by $2 \pi$ times the magnetic flux through the AB ring in units of the elementary flux quantum. Eqs. (\[eq3\],\[eq4\]) imply that the electron picks up the phase factor $\exp (+i \phi )$ as it leaves the QD through barrier $2$. Here and likewise in Section \[sca\], we neglect all other effects that the magnetic field may have on the motion of the electron, and take account of the Aharanov–Bohm phase only.
The Hamiltonian used in Ref. [@hack96] differs from our $H_0$ in that it also contains the Coulomb interaction between electrons within the QD in a mean–field approximation. This interaction is known [@bal99] to be important for the behavior of the phase of the transmission amplitude through the QD between resonances. It is not expected, however, to affect this phase in the domain of an isolated Coulomb–blockade resonance, or the width of such a resonance.
Scattering Matrix {#sca}
=================
One may wonder whether the model defined in Eqs. (\[eq1\]) to (\[eq4\]) is sufficiently general to give a completely satisfactory answer to the questions posed at the end of Section \[int\]. It is for this reason that we now derive the form of the scattering matrix from very general principles. These are unitarity, time–reversal invariance, the topology of the AB interferometer, gauge invariance, and the single–level approximation for the passage of electrons through the QD. As remarked above, the last of these assumptions can easily be lifted. At the end, it will turn out that the scattering matrix determined in this way has indeed the same form as the one calculated from Eqs. (\[eq1\]) to (\[eq4\]).
The total scattering matrix $S$ for the passage of electrons through the AB interferometer is the sum of two terms. We consider first the contribution $S^{(0)}$ from that arm of the interferometer which does not contain the QD. (In the scheme of Figure \[fig1\], the total scattering matrix $S(E)$ would become equal to $S^{(0)}(E)$ if the barriers $1$ and $2$ separating the QD from the AB ring were closed). We neglect the dependence of $S^{(0)}$ on energy over an interval given by the width of the resonance due to the single level in the QD in the other arm and, therefore, consider $S^{(0))}$ as independent of energy. Since $S$ is unitary, and since the contribution from the other arm vanishes for energies far from the resonance, $S^{(0)}$ must also be unitary. Moreover, $S^{(0)}$ is not affected by the presence of the magnetic field. (As in the model of Eqs. (\[eq1\]) to (\[eq4\]), the entire dependence on the magnetic field will be contained in the amplitude coupling the QD to the AB ring through barrier $2$, see Eq. (\[eq13\]) below). Hence, $S^{(0)}$ is time–reversal invariant and, thus, symmetric. As is the case for every unitary and symmetric matrix, $S^{(0)}$ can be written in the form [@nis85] $$S^{(0)} = U U^T \ .
\label{eq11}$$ The symbol $T$ denotes the transpose of a matrix, and the matrix $U$ is unitary. It is the product of an orthogonal matrix which diagonalizes $S^{(0)}$, and of a diagonal matrix. The elements of the latter have the form $\exp(i \delta)$ where the $\delta$’s are the eigenphaseshifts of $S^{(0)}$. We now use the more explicit notation introduced in Eq. (\[eq2\]) to write $S^{(0)}$ as $(S^{(0)})^{rs}_{ab}$, and $U$ as $U^r_{a \alpha}$. With $N_r$ the number of channels in lead $r$ and $N = \sum_r N_r$ the total number of channels, the index $\alpha$ runs from $1$ to $N$. The matrix $U$ represents a rotation in the space of channels from the physical channels $(r,a)$ to the eigenchannels $\alpha$ of $S^{(0)}$.
Using Eq. (\[eq11\]), we write the total $S$–matrix $S$ in the form $$S^{rs}_{ab} = \sum_{\alpha \beta} U^r_{a \alpha} \bigl (
\delta_{\alpha \beta} - i \frac{x_{\alpha \beta}}{E - E_0 + (i/2)
\Gamma} \bigr ) U^s_{b \beta} \ .
\label{eq12}$$ The first term in brackets on the right–hand side of Eq. (\[eq12\]) represents $S^{(0)}$ and the second, the contribution of the single resonance due to the QD. This contribution is written in Breit–Wigner form. The numerator $x_{\alpha \beta}$ has the form $$\begin{aligned}
x_{\alpha \beta} &=& z^{(1)}_{\alpha} z^{(1)}_{\beta} +
z^{(2)}_{\alpha} z^{(2)}_{\beta} \nonumber \\
&&\qquad + z^{(1)}_{\alpha} z^{(2)}_{\beta} \exp (i \phi) +
z^{(2)}_{\alpha} z^{(1)}_{\beta} \exp( - i \phi) \ .
\label{eq13}\end{aligned}$$ The amplitudes $z^p_{\alpha}$ with $p = 1,2$ and $\alpha = 1,\ldots,N$ are the amplitudes for decay of the Breit–Wigner resonance into the eigenchannels $\alpha$ through the first or the second barrier, respectively, see Figure \[fig1\]. The entire magnetic–field dependence is contained explicitly in the phase factors. Therefore, the amplitudes $z^p_{\alpha}$ can be chosen real. The four terms on the right–hand side of Eq. (\[eq13\]) account for the four ways in which the Breit–Wigner resonance contributes to the scattering process, see Figure \[fig1\]: Formation and decay of the resonance through barrier one, formation and decay through barrier two, formation through barrier one and decay through barrier two, and formation through barrier two and decay through barrier one, respectively. Thus, the form of Eq. (\[eq13\]) accounts for the topology of the AB ring and for gauge invariance. In writing Eq. (\[eq13\]), we have assumed that passage through the QD is possible only via intermediate formation of the resonance.
The matrix $S$ in Eq. (\[eq12\]) must be unitary. This condition is met if the total width $\Gamma$ obeys the equation $$\begin{aligned}
\Gamma &=& \sum_{\alpha} \sum_p (z^{(p)}_{\alpha})^2 + 2 \cos ( \phi )
\sum_{\alpha} z^{(1)}_{\alpha} z^{(2)}_{\alpha} \nonumber \\
&&\qquad = \sum_{\alpha} | z^{(1)}_{\alpha} + z^{(2)}_{\alpha} \exp( i
\phi ) |^2 \ .
\label{eq14}\end{aligned}$$ The last form of Eq. (\[eq14\]) shows that the amplitude for decay of the resonance into channel $\alpha$ is the sum of two terms, $z^{(1)}_{\alpha}$ and $z^{(2)}_{\alpha} \exp ( i \phi )$. Again, this reflects the topology of the AB interferometer. We note that the Breit–Wigner term in Eq. (\[eq12\]) describes both, single and multiple passage of the electron through the QD, the latter possibly in conjunction with multiple turns around the AB ring. This is seen by expanding the Breit–Wigner denominator in Eq. (\[eq12\]) in powers of $\Gamma$, using Eq. (\[eq14\]), and identifying the $n^{\rm th}$ power of pairs of coefficients $z^{(1)}, z^{(2)}$ with the $n$–fold passage of the electron through the QD. The expansion gives rise, among many others, for instance to the term $$\begin{aligned}
&& (\frac{- i}{E - E_0})^n (1/2)^{n-1} \nonumber \\
&& \times (z^{(1)}_{\alpha} [ \sum_{\gamma} z^{(2)}_{\gamma} \exp ( +i
\phi ) z^{(1)}_{\gamma} ]^{n-1} z^{(2)}_{\beta} \exp ( +i \phi ) \ .
\label{eq30}\end{aligned}$$ With $(- i)$ the propagator in each of the eigenchannels and $(E -
E_0)^{-1}$ the propagator through the resonance on the QD, this term describes an electron passing $n$ times through the QD and circling the AB ring $(n-1)$ times counter–clockwise before leaving the AB device. (The factor $(1/2)^{(n-1)}$ is a matter of convention).
According to Eq. (\[eq14\]), the width $\Gamma$ depends upon the magnetic flux $\phi$. In the context of the question addressed in the present paper, this fact is worrysome. Indeed, what is the meaning of the question “Does the transmission phase increase by $\pi$ over the width of the resonance?” if that width changes with the applied magnetic field? We now show that the dependence of $\Gamma$ on $\phi$ becomes negligible when the total number $N$ of channels becomes large, $N \gg 1$. The amplitudes $z^{(p)}_{\alpha}$ are real but may be positive or negative. For a rough estimate, we assume that the $z$’s are Gaussian–distributed random variables with zero mean value and a common variance $z^2$. Then the mean value of $\Gamma$ is easily seen to be independent of $\phi$ and given by $2 N z^2$. The variance of $\Gamma$, on the other hand, is given by $[4 N^2 + 4 N (1 + \cos^2
( \phi ))] (z^2)^2$. This establishes our claim: The dependence of $\Gamma$ on $\cos ( \phi )$ is small of order $1/\sqrt{N}$. To simplify the discussion, we will assume in the sequel that $\Gamma$ is independent of $\phi$.
Inspection shows that the matrix $S$ obeys the identity $$S ( \phi ) = S^T ( - \phi ) \ .
\label{eq15}$$ This equation expresses time–reversal invariance in the presence of a magnetic field.
For later use, we write, with $f(E)$ real, the Breit–Wigner denominator in the form $$\frac{1}{E - E_0 + \frac{i}{2} \Gamma} = f(E) \exp ( i \xi(E) ) \ .
\label{eq16}$$ Here, $\xi(E)$ is the resonance phase shift. It increases by $\pi$ over the width $\Gamma$ of the resonance. We recall that we assume $\Gamma$ to independent of $\phi$. The questions raised at the end of Section \[int\] amount to asking: “What is the connection between the transmission phase and the resonance phase shift $\xi(E)$?” We will turn to this question presently.
It is useful to introduce the amplitudes $$\gamma^r_a = \sum_{\alpha} U^r_{a \alpha} z^{(1)}_{\alpha}\ , \
\delta^r_a = \sum_{\alpha} U^r_{a \alpha} z^{(2)}_{\alpha}\ .
\label{eq17}$$ The symbol $\delta^r_a$ should not be confused with the eigenphaseshift $\delta_{\alpha}$ of $S^{(0)}$. After multiplication with the matrix $U$, the numerator of the Breit–Wigner term takes the form $$\begin{aligned}
\gamma^r_a \gamma^s_b &+& \delta^r_a \delta^s_b + [\gamma^r_a \delta^s_b
+ \delta^r_a \gamma^s_b] \cos ( \phi ) \nonumber \\
&&\qquad + i [\gamma^r_a \delta^s_b - \delta^r_a \gamma^s_b] \sin (
\phi ) \ .
\label{eq18}\end{aligned}$$ The total scattering matrix is given by $$\begin{aligned}
S^{rs}_{ab}(E) &=& (S^{(0)})^{rs}_{ab} - i f(E) \exp ( i \xi(E) ) [
\gamma^r_a \gamma^s_b + \delta^r_a \delta^s_b \nonumber \\
&&+ [\gamma^r_a \delta^s_b + \delta^r_a \gamma^s_b] \cos (
\phi ) \nonumber \\
&&\qquad + i [\gamma^r_a \delta^s_b - \delta^r_a \gamma^s_b] \sin (
\phi ) ] \ .
\label{eq18a}\end{aligned}$$ The width $\Gamma$ can also be expressed in terms of the amplitudes $\gamma^r_a$ and $\delta^r_a$, $$\Gamma = \sum_{ra} |\gamma^r_a + \delta^r_a \exp ( i \phi ) |^2 \ .
\label{eq18b}$$
As announced above, we have shown that the scattering matrix $S$ can indeed be constructed from the requirements of unitarity, time–reversal invariance, the topology of the AB interferometer, gauge invariance, and the single–level approximation. Explicit construction of the scattering matrix from the Hamiltonian formulated in Eqs. (\[eq1\]) through (\[eq4\]) as done in Ref. [@hack96] yields an expression which is identical in form to our $S$ in Eq. (\[eq18a\]). This shows that our formal construction possesses a dynamical content. Conversely, this result shows that our model Hamiltonian in Eqs. (\[eq1\]) to (\[eq4\]) leads to the most general form of the scattering matrix which is consistent with the requirements just mentioned. We recall that our construction is strictly based upon a single–particle picture and does not account for interactions between electrons beyond the mean–field approximation.
The Transmission Phase {#tra}
======================
Equipped with an explicit expression for $S$, we return to the transmission phase. It should first be noted that different experiments may determine different combinations of the conductance coefficients $G_{rs}$ introduced in Eq. (\[eq0\]). As shown in Ref. [@hack97], the relevant quantity for the experiment of Schuster [*et al.*]{} [@sch97] is $T^{41} /(T^{44} - N_4)$. Here, the indices $1$ and $4$ label the source and the collector, respectively, for the electrons in the six–lead experiment. We will show presently that $T^{rr}$ with $r = 1,\ldots,R$ is an even function of the phase $\phi$ and, therefore, depends only upon $\cos ( \phi)$. A non–trivial dependence on $\phi$ involving both $\cos (\phi)$ and $\sin (\phi)$ and, thus, a trigonometric dependence on $(\phi \pm
\xi(E_F))$, arises only from the terms $T^{rs}$ with $r \neq s$. Without loss of generality we, therefore, focus attention on $T^{12}$ and, thus, on $\sum_{ab} | S^{12}_{ab}(E_F) |^2$, see Eq. (\[eq0\]). This quantity is expected to display a non–trivial dependence on $\phi$. We expect that the transmission phase $\delta \phi$ increases by $\pi$ as the Fermi energy sweeps the Coulomb–blockade resonance. We ask how this increase depends on the width $\Gamma$ of the resonance and on the number $R$ of leads.
It is useful to address these questions by using the unitarity relation. We write $$\sum_{ab} | S^{12}_{ab} |^2 = N_1 - \sum_{ab} | S^{11}_{ab} |^2 -
\sum_{s \geq 3} \sum_{ab} | S^{1s}_{ab} |^2 \ .
\label{eq19}$$ The advantage of Eq. (\[eq19\]) is that the sum over $s$ vanishes when there are only two leads. Thus, the influence of the number of leads is made explicit. We now discuss the dependence of the terms on the right–hand side of Eq. (\[eq19\]) on the resonance phase shift $\xi(E)$.
Each of the terms $\sum_{ab} | S^{1s}_{ab} |^2$ with $s =
1,3,4,\ldots$ in Eq. (\[eq19\]) is the sum of three contributions, involving $|S^{(0)}|^2$, the square of the Breit–Wigner contribution, and the interference term between $S^{(0)}$ and the Breit–Wigner term. The contributions from $\sum_{ab} | (S^{(0)})^{1s}_{ab} |^2$ are independent of both, energy and magnetic field and, therefore, without interest. These terms only provide a smooth background. The squares of the Breit–Wigner terms are each proportional to $f^2(E_F)$ and are independent of the resonance phase shift $\xi(E_F)$. This is expected. Each such term depends on the magnetic flux $\phi$ in two distinct ways. The squares of the terms in Eq. (\[eq18a\]) involving either $\cos ( \phi )$ or $\sin ( \phi )$, and the product of these two terms yield a dependence on $\phi$ which is periodic in $\phi$ with period $\pi$. Such terms can easily be distinguished experimentally from terms which are periodic in $\phi$ with period $2 \pi$. As we shall see, it is some of these latter terms which carry the resonance phase shift $\xi(E_F)$. Therefore, we confine attention to terms of this latter type. A contribution of this type arises from the square of the Breit–Wigner term via the interference of that part of the resonance amplitude which is independent of $\phi$ with the terms proportional to either $\cos ( \phi )$ or $\sin ( \phi )$. The sum of all such contributions (from values of $s = 1$ and $s = 3,4,\ldots,R$) has the form $$f^2(E_F) A \cos ( \phi + \alpha_0 )
\label{eq19a}$$ where $A$ and $\alpha_0$ are constants wich depend on $R$ but not on $E_F$ or $\phi$. We note that the constant $A$ is of fourth order in the decay amplitudes $\gamma^r_a$ and $\delta^r_a$. Isolated resonances in quantum dots require high barriers, i.e., small values of these decay amplitudes. Therefore, the term (\[eq19a\]) may be negligible. We continue the discussion under this assumption but note that there is no problem in taking account of this term if the need arises.
We turn to the interference terms. We first address the case $s = 1$ which differs from the cases with $s \geq 3$. We observe that $(S^{(0)})^{11}_{ab}$ is even in $a,b$. Therefore, multiplication of $S^{(0)}$ with the Breit–Wigner term and summation over $a$ and $b$ will cancel those parts of the Breit–Wigner term which are odd under exchange of $a$ and $b$. Inspection of Eq. (\[eq18\]) shows that these are the terms proportional to $\sin ( \phi )$. As a consequence, the interference term for $s = 1$ is even in $\phi$ and a function of $\cos ( \phi )$ only. Explicit calculation shows that the term proportional to $\cos ( \phi )$ can be written in the form $$4 f(E_F) x^{(1)} \cos ( \phi ) \sin ( \xi(E_F) + \zeta^{(1)} ) \ .
\label{eq20}$$ Here, $x^{(1)}$ and $\zeta^{(1)}$ are independent of energy and explicitly given by $x^{(1)} \exp ( i \zeta^{(1)} ) = \sum_{ab}
\gamma^{(1)}_a \delta^{(1)}_b ((S^{(0)})^{11}_{ab})^*$. We observe that as a function of energy, $\sin ( \xi(E_F) + \zeta )$ always has a zero close to the resonance energy $E_0$. When only two channels are open, the entire dependence on $\phi$ which has period $2 \pi$ resides in this term. The term does not display the resonance phase shift except through the zero near $E_0$. It is symmetric in $\phi$ about $\phi = 0$. These facts are well known [@bue86], of course, and are reproduced here for completeness only. The form (\[eq20\]) of the interference term was responsible for the failure of the experiment in Ref. [@yac95] to measure the transmission phase.
For the interference terms with $s \geq 3$, the matrix $(S^{(0)})^{1s}_{ab}$ is not symmetric in $a,b$. (It is symmetric only with respect to the simultaneous interchange of $1,s$ [*and*]{} $a,b$). Therefore, the terms proportional to $\sin ( \phi )$ in Eq. (\[eq18a\]) do not cancel, and the interference terms acquire a genuine joint dependence on both, the resonance phase shift $\xi(E_F)$ and the phase $\phi$ of the magnetic flux. Proceeding as in the previous paragraph, we introduce the constants $x \exp ( i \zeta ) =
\sum_{s \geq 3} \sum_{ab} \gamma^{(1)}_a \delta^{(s)}_b
((S^{(0)})^{1s}_{ab})^*$ and $y \exp ( i \theta ) = \sum_{s \geq 3}
\sum_{ab} \gamma^{(s)}_a \delta^{(1)}_b ((S^{(0)})^{1s}_{ab})^*$. The $\phi$–dependent part of the sum of the interference terms with $s
\geq 3$ takes the form $$2 f(E_F) [ x \sin (\phi + \xi(E_F) + \zeta) + y \sin(-\phi + \xi(E_F)
+ \theta) ] \ .
\label{eq21}$$ This expression depends on $\xi(E_F)$ in the expected way.
We are now in a position to answer the questions raised at the end of Section \[int\]. Whenever the total number $N$ of channels coupled to the AB device is sufficiently large, the resonance width $\Gamma$ becomes independent of magnetic flux. This property can be checked experimentally. It is only in this limit that the statement “The resonance phase shift increases by $\pi$ over the width of the resonance” acquires its full meaning. The limit $N \gg 1$ may, of course, be realised even when the number $R$ of leads is small. We turn to the behavior of the transmission phase. We have shown that there are terms proportional to $f^2(E_f)$ which depend upon $\cos (
2 \phi )$ but not on $\xi(E_F)$. The form of these terms was discussed above. For a quantum dot with high barriers, it is expected that these terms are small. The terms periodic in $\phi$ with period $2 \pi$ are listed in Eqs. (\[eq19a\]) to (\[eq21\]). For a quantum dot with high barriers, we expect that the contribution (\[eq19a\]) is small. We focus attention on the remaining terms. These depend on the value of $R$. For $R = 2$, the phase dependence is given by the term (\[eq20\]). This term is even in the magnetic flux $\phi$ and has a zero close to the resonance energy $E_0$. It does not, however, display the smooth increase of the resonance phase phase shift $\xi(E_F)$ over the width $\Gamma$ of the resonance. If, on the other hand, the number of leads $R$ is large compared to unity, then it is reasonable to expect that the terms in Eq. (\[eq21\]) are large compared to the term in Eq. (\[eq20\]). This is because the number of contributions to the terms in Eq. (\[eq21\]) is proportional to $R - 2$. In this case, the transmission phase faithfully reflects the energy dependence of the resonance phase shift $\xi(E_F)$. Deviations from this limit are of order $1/(R - 2)$. As we gradually turn off the coupling to the leads $s$ with $s \geq 3$, the terms (\[eq21\]) gradually vanish. Nevertheless, it is possible — within experimental uncertainties that become ever more significant as the terms (\[eq21\]) become smaller — even in this case to determine the resonance phase shift $\xi(E_F)$ from the data on the transmission phase. We propose the following procedure. Add formulas (\[eq20\]) and (\[eq21\]) and fit the resulting expression to the data. This should allow a precise determination of $\xi(E_F)$ and of $\Gamma$ also in cases where the coupling to the leads $s$ with $s \geq 3$ is small. This statement holds with the proviso that the number of channels must be large enough to allow us to consider the total width $\Gamma$ as independent of $\phi$. Whenever the coupling to the leads $s$ with $s \geq 3 $ is small, the energy dependence of the transmission phase is quite different from that of the resonance phase shift. Nevertheless, the transmission phase $\delta \phi$ does reflect the energy dependence of the resonance phase shift $\xi$ whenever the number of leads is larger than two. In particular, this energy dependence is governed by the total width $\Gamma$.
In conclusion, we have seen that in a theoretical model which is more closely patterned after the geometry of Figure \[fig1\] than is the model of Ref.[@ent01], the transmission phase does reflect the value of the total width. This statement applies whenever the number of leads exceeds the value two, and whenever the total number of channels is large compared to one. Both conditions are expected to be met in the experiment of Schuster [*et al.*]{} [@sch97].
[*Acknowledgment.*]{} I am grateful to O. Entin–Wohlman, A. Aharony, and Y. Imry for informative discussions which stimulated the present investigation. I thank O. Entin–Wohlman for a reading of the paper, and for useful comments. This work was started when I was visiting the ITP at UCSB. The visit was supported by the NSF under contract number PHY99-07949.
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author:
- 'Tapan Mukhopadhyay$^1$, Joydev Lahiri$^2$ and D. N. Basu$^3$'
title: |
Reply to the comments by I. Angeli and J. Csikai on :\
‘Cross sections of neutron-induced reactions’
---
The first equation of Eqs.(3) in [@Mu10] was used to describe the mass number and energy dependence of experimental total neutron cross sections for the first time in [@An70], while the second and third ones for scattering and reaction cross sections in [@An74]. We are sorry for the omission of these two references which were not in our knowledge. In fact we derived these equations and Eq.(4) of Ref.\[12\] \[J.D. Anderson and S.M. Grimes, Phys. Rev. [**C 41**]{}, 2904 (1990) [@An90]\] of our paper [@Mu10] as follows. From partial wave analysis of scattering theory, we know the standard expressions for scattering $\sigma_{sc}$ and reaction $\sigma_r$ cross sections as
$$\sigma_{sc}=\frac{\pi}{k^2} \Sigma_l~(2l+1)|1-\eta_l|^2,~~~~\sigma_r=\frac{\pi}{k^2} \Sigma_l~(2l+1)[1-|\eta_l|^2]
\label{seqn5}$$
where the quantity $\eta_l = e^{2i\delta_l}$. With the assumption that the phase shift $\delta_l$ is independent of $l$ and the summation over partial waves $l$ is upto $kR$ only, it follows that $\sigma_{sc}=\pi(R+\lambdabar)^2(1+\alpha^2-2\alpha \cos\beta)$, $\sigma_r=\pi(R+\lambdabar)^2(1-\alpha^2)$ and $\sigma_{tot}= \sigma_{sc} + \sigma_r = 2\pi(R+\lambdabar)^2 (1-\alpha \cos\beta)$ where $\lambdabar=1/k$, $R$ is the channel radius beyond which partial waves do not contribute, $\beta=2 {\rm Re}\delta_l = 2 {\rm Re}\delta $, $\alpha = e^{-2 {\rm Im}\delta_l} = e^{-2 {\rm Im}\delta}$ and summing over $l$ from 0 to $kR$ yielded $\Sigma_l~(2l+1) = (kR+1)^2$.\
We used the name ‘nuclear Ramsauer model’ from Ref.\[12\] of our paper [@Mu10]. Carpenter [@Ca59] was the first to call the structure found in total neutron cross sections as nuclear Ramsauer effect. This name was adopted by subsequent authors although the nature of the oscillation in fast neutron cross sections is essentially different from that observed for slow electrons by Ramsauer. In other works the name ‘semiclassical optical model’ [@An74] or ‘diffraction effect’ [@La53] were used which are more appropriate.\
In fact the radius of the potential well is just $r_0 A^\frac{1}{3}=r_1 A^{\frac{1}{3}+\gamma}$ and $r_1$= constant. The parameter $\gamma$ is a very small number (0.00793) compared to $\frac{1}{3}$ needed for fine tuning. It should, therefore, be emphasized that, as mentioned in our paper [@Mu10], it is $r_0$ which is used for fixing $\beta_0$. It is the channel radius which is energy dependent. Channel radius is the radius \[appearing in Eqs.(3) of our paper\] beyond which no partial waves contribute. It is well known from R-matrix theory that the channel radius is less than the nuclear (potential) radius which is precisely the case here.\
The drawback of Peterson’s derivation is that the neutron (although massive) is treated like a photon and as its velocity inside nucleus and vacuum are proportional to $\sqrt{E+V}$ and $\sqrt{E}$, respectively, it would result in bending of the ray (as in optics) away from the normal inside nucleus (in fig.16 of Peterson’s paper [@Pe62], just the opposite was shown) where velocity is more. This would lead to the existence of critical angle $sin^{-1}\sqrt{E/(E+V)}$ beyond which there is no transmission (even in an attractive potential of $-V$!) and a refractive index less than vacuum for the nuclear medium which are physically unacceptable. Even then if one sticks to Peterson’s assumption of a light ray, the average chord length inside nucleus, with ray bending away from the normal, turns out to be less than our result of $4R/3$ as opposed to greater than $4R/3$ as derived in Peterson’s paper where $R$ is nuclear radius. However, his result [@Pe62] goes over to our result of $4R/3$ asymptotically at energies higher than magnitude $V$ of the real part of the nuclear potential.\
Obviously, these omissions do not affect the results and conclusion in the original manuscript [@Mu10].\
We thank Drs. I. Angeli and J. Csikai for bringing this matter to our attention.
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J. D. Anderson and S. M. Grimes, Phys. Rev. [**C 41**]{}, 2904 (1990).
S. G. Carpenter and R. Wilson, Phys. Rev. 114 510 (1959).
J. D. Lawson, Phil. Mag. [**44**]{}, 102 (1953).
J. M. Peterson, Phys. Rev. [**125**]{}, 955 (1962).
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abstract: 'A theory of systems with long-range correlations based on the consideration of *binary N-step Markov chains* is developed. In the model, the conditional probability that the $i$-th symbol in the chain equals zero (or unity) is a linear function of the number of unities among the preceding $N$ symbols. The correlation and distribution functions as well as the variance of number of symbols in the words of arbitrary length $L$ are obtained analytically and numerically. A self-similarity of the studied stochastic process is revealed and the similarity group transformation of the chain parameters is presented. The diffusion Fokker-Planck equation governing the distribution function of the $L$-words is explored. If the persistent correlations are not extremely strong, the distribution function is shown to be the Gaussian with the variance being nonlinearly dependent on $L$. The applicability of the developed theory to the coarse-grained written and DNA texts is discussed.'
author:
- 'O. V. Usatenko , V. A. Yampol’skii'
- 'K. E. Kechedzhy, S. S. Mel’nyk'
title: 'Symbolic Stochastic Dynamical Systems Viewed as Binary $N$-Step Markov Chains'
---
Introduction
============
The problem of systems with long-range spatial and/or temporal correlations (LRCS) is one of the topics of intensive research in modern physics, as well as in the theory of dynamical systems and the theory of probability. The LRC-systems are usually characterized by a complex structure and contain a number of hierarchic objects as their subsystems. The LRC-systems are the subject of study in physics, biology, economics, linguistics, sociology, geography, psychology, etc. [@stan; @prov; @mant; @kant]. At the present time, there is no generally accepted theoretical model that adequately describes the dynamical and statistical properties of the LRC-systems. Attempts to describe the behavior of the LRCS in the framework of the Tsalis non-extensive thermodynamics [@tsal; @abe] were undertaken in Ref. [@den]. However, the non-extensive thermodynamics is not well-grounded and requires the construction of the additional models which could clarify the properties of the LRC-systems.
One of the efficient methods to investigate the correlated systems is based on a decomposition of the space of states into a finite number of parts labelled by definite symbols. This procedure referred to as coarse graining is accompanied by the loss of short-range memory between states of system but does not affect and does not damage its robust invariant statistical properties on large scales. The most frequently used method of the decomposition is based on the introduction of two parts of the phase space. In other words, it consists in mapping the two parts of states onto two symbols, say 0 and 1. Thus, the problem is reduced to investigating the statistical properties of the symbolic binary sequences. This method is applicable for the examination of both discrete and continuous systems.
One of the ways to get a correct insight into the nature of correlations consists in an ability of constructing a mathematical object (for example, a correlated sequence of symbols) possessing the same statistical properties as the initial system. There are many algorithms to generate long-range correlated sequences: the inverse Fourier transform [@czir], the expansion-modification Li method [@li], the Voss procedure of consequent random addition [@voss], the correlated Levy walks [@shl], etc. [@czir]. We believe that, among the above-mentioned methods, using the Markov chains is one of the most important. We would like to demonstrate this statement in the present paper.
In the following sections, the statistical properties of the *binary many-steps Markov chain* is examined. In spite of the long-time history of studying the Markov sequences (see, for example, [@kant; @nag; @trib] and references therein), the concrete expressions for the variance of sums of random variables in such strings have not yet been obtained. Our model operates with two parameters governing the conditional probability of the discrete Markov process, specifically with the memory length $N$ and the correlation parameter $\mu$. The correlation and distribution functions as well as the variance $D$ being nonlinearly dependent on the length $L$ of a word are derived analytically and calculated numerically. The nonlinearity of the $D(L)$ function reflects the existence of strong correlations in the system. The evolved theory is applied to the coarse-grained written texts and dictionaries, and to DNA strings as well.
Some preliminary results of this study were published in Ref. [@prl].
Formulation of the problem
==========================
Markov Processes
----------------
Let us consider a homogeneous binary sequence of symbols, $a_{i}=\{0,1\}$. To determine the $N$-*step Markov chain* we have to introduce the conditional probability $P(a_{i}\mid a_{i-N},a_{i-N+1},\dots ,a_{i-1})$ of occurring the definite symbol $a_i$ (for example, $a_i =0$) after symbols $a_{i-N},a_{i-N+1},\dots ,a_{i-1}$. Thus, it is necessary to define $2^{N}$ values of the $P$-function corresponding to each possible configuration of the symbols $a_{i-N},a_{i-N+1},\dots ,a_{i-1}$. We suppose that the $P$-function has the form, $$P(a_{i}=0\mid a_{i-N},a_{i-N+1},\dots ,a_{i-1})$$ $$=\frac{1}{N} \sum\limits_{k=1}^{N}f(a_{i-k},k). \label{1}$$ Such a relation corresponds to the additive influence of the previous symbols on the generated one. The homogeneity of the Markov chain is provided by the independence of the conditional probability Eq. (\[1\]) of the index $i$.
It is reasonable to assume the function $f$ to be decreasing with an increase of the distance $k$ between the symbols $a_{i-k}$ and $a_{i}$ in the Markov chain. However, for the sake of simplicity we consider here a step-like memory function $f(a_{i-k},k)$ independent of the second argument $k$. As a result, the model is characterized by three parameters only, specifically by $f(0)$, $f(1)$, and $N$: $$P(a_{i}=0\mid a_{i-N},a_{i-N+1},\dots ,a_{i-1})$$ $$=\frac{1}{N} \sum\limits_{k=1}^{N}f(a_{i-k}). \label{2}$$ Note that the probability $P$ in Eq. (\[2\]) depends on the numbers of symbols 0 and 1 in the $N$-word but is independent of the arrangement of the elements $a_{i-k}$. We also suppose that $$f(0)+f(1)=1. \label{2a}$$ This relation provides the statistical equality of the numbers of symbols zero and unity in the Markov chain under consideration. In other words, the chain is non-biased. Indeed, taking into account Eqs. (\[2\]) and (\[2a\]) and the sequence of equations, $$P(a_{i} = 1|a_{i-N},\dots ,a_{i-1})=1-P(a_{i}=0|a_{i-N},\dots
,a_{i-1})$$ $$=\frac{1}{N}\sum\limits_{k=1}^{N}f(\tilde{a}_{i-N})= P(a_{i}=0\mid
\tilde{a} _{i-N},\dots ,\tilde{a}_{i-1}), \label{2b}$$ one can see the symmetry with respect to interchange $\tilde{a}_{i}\leftrightarrow a_{i}$ in the Markov chain. Here $\tilde{a}_{i}$ is the symbol opposite to $a_{i}$, $\tilde{a}_{i}=1-a_{i}$. Therefore, the probabilities of occurring the words $(a_{1},\dots ,a_{L})$ and $(\tilde{a}_{1},\dots ,\tilde{a}_{L})$ are equal to each other for any word length $L$. At $L=1$ this yields equal average probabilities that symbols $0$ and $1$ occur in the chain.
Taking into account the symmetry of the conditional probability $P$ with respect to a permutation of symbols $a_{i}$ (see Eq. (\[2\])), we can simplify the notations and introduce the conditional probability $p_{k}$ of occurring the symbol zero after the $N$-word containing $k$ unities, e.g., after the word $\underbrace{(11...1}_{k}\;\underbrace{00...0}_{N-k})$, $$p_{k}=P(a_{N+1}=0\mid \underbrace{11\dots
1}_{k}\;\underbrace{00\dots 0} _{N-k})$$ $$=\frac{1}{2}+\mu (1-\frac{2k}{N}), \label{14}$$ with the correlation parameter $\mu $ being defined by the relation $$\mu =f(0)-\frac{1}{2}. \label{3}$$
We focus our attention on the region of $\mu $ determined by the persistence inequality $0 < \mu <1/2$. In this case, each of the symbols unity in the preceding N-word promotes the birth of new symbol unity. Nevertheless, the major part of our results is valid for the anti-persistent region $-1/2<\mu <0$ as well.
Asimilarrulefortheproductionofan$N$-word\
$(a_{1},\dots,a_{N})$ that follows after a word $(a_{0},a_1,\dots
,a_{N-1})$ was suggested in Ref. [@kant]. However, the conditional probability $p_k$ of occurring the symbols $a_N$ does not depend on the previous ones in the model [@kant].
Statistical characteristics of the chain
----------------------------------------
In order to investigate the statistical properties of the Markov chain, we consider the distribution $W_{L}(k)$ of the words of definite length $L$ by the number $k$ of unities in them, $$k_{i}(L)=\sum\limits_{l=1}^{L}a_{i+l}, \label{5}$$ and the variance of $k$, $$D(L)=\overline{k^{2}}-\overline{k}^{2}, \label{7}$$ where $$\overline{f(k)}=\sum\limits_{k=0}^{L}f(k)W_{L}(k). \label{8}$$ If $\mu =0,$ one arrives at the known result for the non-correlated Brownian diffusion, $$D(L)=L/4. \label{6}$$ We will show that the distribution function $W_{L}(k)$ for the sequence determined by Eq. (\[14\]) (with nonzero but not extremely close to 1/2 parameter $\mu $) is the Gaussian with the variance $D(L)$ nonlinearly dependent on $L$. However, at $\mu \rightarrow 1/2$ the distribution function can differ from the Gaussian.
Main equation
-------------
ForthestationaryMarkovchain,theprobability\
$b(a_{1}a_{2}\dots a_{N})$ of occurring a certain word $(a_{1},a_{2},\dots ,a_{N})$ satisfies the condition of compatibility for the Chapmen-Kolmogorov equation (see, for example, Ref. [@gar]): $$b(a_{1}\dots a_{N})$$ $$=\sum_{a=0,1}b(aa_{1}\dots a_{N-1})P(a_{N}\mid a,a_{1},\dots
,a_{N-1}). \label{10}$$ Thus, we have $2^{N}$ homogeneous algebraic equations for the $2^{N}$ probabilities $b$ of occurring the $N$-words and the normalization equation $\sum b=1$. In the case under consideration, the set of equations can be substantially simplified owing to the following statement.
**Proposition** $\spadesuit$: *The probability $b(a_{1}a_{2}\dots a_{N})$ depends on the number $k$ of unities in the $N$-word only*, i. e., it is independent of the arrangement of symbols in the word $(a_{1},a_{2},\dots ,a_{N})$.
[![The probability $b$ of occurring a word $(a_1, a_2,
\dots , a_ N)$ vs its number $z$ expressed in the binary code, $z=\sum_{i=1}^N a_i \cdot 2^{i-1}$, for $N=8$, $\mu=0.4$.[]{data-label="f1"}](f1.eps "fig:"){width="45.00000%" height="35.00000%"}]{}
This statement illustrated by Fig. 1 is valid owing to the chosen simple model (\[2\]), (\[14\]) of the Markov chain. It can be easily verified directly by substituting the obtained below solution (\[b\]) into the set (\[10\]). Note that according to the Markov theorem, Eqs. (\[10\]) do not have other solutions [@kat].
Proposition $\spadesuit$ leads to the very important property of isotropy: any word $(a_{1},a_{2},\dots ,a_{L})$ appears with the same probability as the inverted one, $(a_{L},a_{L-1},\dots ,a_{1})$.
LetusapplythesetofEqs. (\[10\])totheword\
$(\underbrace{11\dots 1}_{k}\;\underbrace{00\dots 0}_{N-k})$: $$b(\underbrace{11\dots 1}_{k}\;\underbrace{00\dots 0}_{N-k}) =b(0
\underbrace{11\dots 1}_{k}\;\underbrace{00\dots 0}_{N-k-1})p_{k}$$ $$\label{13}
+b(1\underbrace{11\dots 1}_{k}\;\underbrace{00\dots
0}_{N-k-1})p_{k+1}.$$ Thisyieldstherecursionrelationfor$b(k)=$\
$b(\underbrace{11...1}_{k}\; \underbrace{00...0}_{N-k})$, $$b(k)=\frac{1-p_{k-1}}{p_{k}}b(k-1)$$ $$=\frac{N-2\mu (N-2k+2)}{N+2\mu (N-2k)} b(k-1). \label{15}$$ The probabilities $b(k)$ for $\mu>0$ satisfy the sequence of inequalities, $$b(0)=b(N)>b(1)=b(N-1)>...>b(N/2), \label{15b}$$ which is the reflection of persistent properties for the chain. At $\mu=0$ all probabilities are equal to each other.
The solution of Eq. (\[10\]) is $$\label{b}
b(k)=A\cdot \Gamma ( n+k) \Gamma ( n+N-k)$$ with the parameter $n$ defined by $$\label{18a}
n= \frac{N(1-2\mu)}{4\mu}.$$ The constant $A$ will be found below by normalizing the distribution function. Its value is, $$A=\frac{4^n }{2\sqrt{\pi}}
\frac{\Gamma(1/2+n)}{\Gamma(n)\Gamma(2n+N)}.\label{17a}$$
Distribution function of $L$-words
==================================
In this section we investigate the statistical properties of the Markov chain, specifically, the distribution of the words of definite length $L$ by the number $k$ of unities. The length $L$ can also be interpreted as the number of jumps of some particle over an integer-valued 1-D lattice or as the time of the diffusion imposed by the Markov chain under consideration. The form of the distribution function $W_{L}(k)$ depends, to a large extent, on the relation between the word length $L$ and the memory length $N$. Therefore, the first thing we will do is to examine the simplest case $L = N$.
Statistics of $N$-words
-----------------------
The value $b(k)$ is the probability that an $N$-word contains $k$ unities with a *definite* order of symbols $a_i$. Therefore, the probability $W_{N}(k)$ that an $N$-word contains $k$ unities with *arbitrary* order of symbols $a_i$ is $b(k)$ multiplied by the number $\mathrm{C}_{N}^{k}=N!/k!(N-k)!$ of different permutations of $k$ unities in the $N$-word, $$W_{N}(k)=\text{C}_{N}^{k}b(k). \label{19}$$ Combining Eqs. (\[b\]) and (\[19\]), we find the distribution function, $$W_{N}(k)= W_{N}(0)\text{C}_{N}^{k}\frac{\Gamma ( n+k) \Gamma (
n+N-k) }{\Gamma (n ) \Gamma (n+N)}. \label{18}$$ The normalization constant $W_{N}(0)$ can be obtained from the equality $\sum\limits_{k=0}^{N}W_{N}(k)=1$, $$W_N(0)=\frac{4^n }{2\sqrt{\pi}}
\frac{\Gamma(n+N)\Gamma(1/2+n)}{\Gamma(2n+N)}.\label{17}$$ Comparing Eqs. (\[b\]), (\[19\])-(\[17\]), one can get Eq. (\[17a\]) for the constant $A$ in Eq. (\[b\]).
Note that the distribution $W_{N}(k)$ is an even function of the variable $\kappa =k-N/2$, $$W_{N}(N-k)=W_{N}(k). \label{19b}$$ This fact is a direct consequence of the above-mentioned statistical equivalence of zeros and unities in the Markov chain being considered. Let us analyze the distribution function $W_{N}(k)$ for different relations between the parameters $N$ and $\mu$.
### Limiting case of weak persistence, $n \gg 1$
In the absence of correlations, $n \rightarrow \infty$, Eq. (\[18\]) and the Stirling formula yield the Gaussian distribution at $k,\, N,\, N-k \gg 1$. Given the persistence is not too strong, $$\label{19c}
n \gg 1,$$ one can also obtain the Gaussian form for the distribution function, $$W_{N}(k)=\frac{1}{\sqrt{2\pi D(N)}}\exp \left\{
-\frac{(k-N/2)^{2}}{2D(N)} \right\} , \label{27}$$ with the $\mu$-dependent variance, $$D(N)=\frac{N(N+2n)}{8n}=\frac{N}{4(1-2\mu )}. \label{28}$$ Equation (\[27\]) says that the $N$-words with equal numbers of zeros and unities, $k=N/2$, are most probable. Note that the persistence results in an increase of the variance $D(N)$ with respect to its value $N/4$ at $\mu =0$. In other words, the persistence is conductive to the intensification of the diffusion. Inequality $n \gg 1$ gives $D(N) \ll N^{2}$. Therefore, despite the increase of $D(N)$, the fluctuations of $(k-N/2)$ of the order of $N$ are exponentially small.
### Intermediate case, $n \gtrsim 1$
If the parameter $n$ is an integer of the order of unity, the distribution function $W_{N}(k)$ is a polynomial of degree $2(n-1)$. In particular, at $n=1$, the function $W_{N}(k)$ is constant, $$W_{N}(k)=\frac{1}{N+1}. \label{24}$$ At $n\neq 1,$ $W_{N}(k)$ has a maximum in the middle of the interval $[0,N]$.
### Limiting case of strong persistence
If the parameter $n$ satisfies the inequality, $$\label{24a}
n \ll \ln^{-1}N,$$ one can neglect the parameter $n$ in the arguments of the functions $\Gamma (n+k)$, $\Gamma (n+N)$, and $\Gamma
(n+N-k)$ in Eq. (\[18\]). In this case, the distribution function $W_{N}(k)$ assumes its maximal values at $k=0$ and $k=N$, $$W_{N}(1)=W_{N}(0)\frac{nN}{N-1} \ll W_{N}(0). \label{20}$$ Formula (\[20\]) describes the sharply decreasing $W_{N}(k)$ as $k$ varies from $0$ to $1$ (and from $N$ to $N-1$). Then, at $1<k<N/2$, the function $W_{N}(k)$ decreases more slowly with an increase in $k$, $$W_{N}(k)=W_{N}(0)\frac{nN}{k(N-k)}. \label{21}$$ At $k=N/2,$ the probability $W_{N}(k)$ achieves its minimal value, $$W_{N}\left(\frac{N}{2}\right)= W_{N}(0)\frac{4n}{N}. \label{22}$$
It follows from normalization (\[17\]) that the values $W_{N}(0)=W_N (N)$ are approximatively equal to $1/2$. Neglecting the terms of the order of $n^2$, one gets $$W_{N}(0)=\frac{1}{2} ( 1 - n \ln N ). \label{22a}$$ In the straightforward calculation using Eqs. (\[7\]) and (\[21\]) the variance $D$ is $$D(N)=\frac{N^2}{4} -\frac{nN(N-1)}{2}. \label{22b}$$
Thus, the variance $D(N)$ is equal to $N^2 /2$ in the leading approximation in the parameter $n$. This fact has a simple explanation. The probability of occurrence the $N$-word containing $N$ unities is approximatively equal to $1/2$. So, the relations $\overline{k^{2}} \approx N^2/2 $ and $\overline{k}^{2}=N^2/4$ give (\[22b\]). The case of strong persistence corresponds to the so-called ballistic regime of diffusion: if we chose randomly some symbol $a_i$ in the sequence, it will be surrounded by the same symbols with the probability close to unity.
The evolution of the distribution function $W_N(k)$ from the Gaussian form to the inverse one with a decrease of the parameter $n$ is shown in Fig. 2. In the interval $\ln^{-1}N
< n < 1 $ the curve $W_{N}(k)$ is concave and the maximum of function $W_{N}(k)$ inverts into minimum. At $N \gg 1 $ and $\ln^{-1}N < n < 1 $, the curve remains a smooth function of its argument $k$ as shown by curve with $n=0.5$ in Fig. 2. Below, we will not consider this relatively narrow region of the change in the parameter $n$.
Formulas (\[27\]), (\[28\]), (\[21\]), (\[22a\]) and (\[22b\]) describe the statistical properties of $L$-words for the fixed ”diffusion time” $L=N$. It is necessary to examine the distribution function $W_{L}(k)$ for the general situation, $L\neq
N$. We start the analysis with $L<N$.
[![The distribution function $W_N(k)$ for $N$=20 and different values of the parameter $n$ shown near the curves.[]{data-label="f2"}](f2.eps "fig:"){width="45.00000%" height="35.00000%"}]{}
Statistics of $L$-words with $L<N$
----------------------------------
### Distribution function $W_{L}(k)$
The distribution function $W_{L}(k)$ at $L<N$ can be given as $$W_{L}(k)=\sum\limits_{i=k}^{k+N-L}b(i)\text{C}_{L}^{k}\text{C}_{N-L}^{i-k}.
\label{29}$$ This equation follows from the consideration of $N$-words consisting of two parts, $$(\underbrace{a_{1},\dots ,a_{N-L},}_{i-k\text{
unities}}\;\underbrace{a_{N-L+1},\dots ,a_{N}}_{k \text{
unities}}). \label{29b}$$ The total number of unities in this word is $i$. The right-hand part of the word ($L$-sub-word) contains $k$ unities. The remaining ($i-k$) unities are situated within the left-hand part of the word (within $(N-L)$-sub-word). The multiplier $\mathrm{C}_{L}^{k}\mathrm{C}_{N-L}^{i-k}$ in Eq. (\[29\]) takes into account all possible permutations of the symbols ”1” within the $N$-word on condition that the $L$-sub-word always contains $k$ unities. Then we perform the summation over all possible values of the number $i$. Note that Eq. (\[29\]) is a direct consequence of the proposition $\spadesuit$ formulated in Subsec. C of the previous section.
The straightforward summation in Eq. (\[29\]) yields the following formula that is valid at any value of the parameter $n$: $$\label{W(L)}
W_L(k)=W_L(0)\text{C}_{L}^{k}\frac{\Gamma(n+k) \Gamma
(n+L-k)}{\Gamma (n) \Gamma (n+L)}$$ where $$\label{W(0)}
W_L(0)=\frac{4^n }{2\sqrt{\pi}}\frac{\Gamma(1/2+n)\Gamma
(n+L)}{\Gamma (2n+L)}.$$
It is of interest to note that the parameter of persistence $\mu$ and the memory length $N$ are presented in Eqs. (\[W(L)\]), (\[W(0)\]) via the parameter $n$ only. This means that the statistical properties of the $L$-words with $L<N$ are defined by this single “combined” parameter.
In the limiting case of weak persistence, $n\gg 1$, at $k,\;L-k
\gg 1$, Eq. (\[W(L)\]) along with the Stirling formula give the Gaussian distribution function, $$W_{L}(k)=\frac{1}{\sqrt{2\pi D(L)}}\exp \left\{
-\frac{(k-L/2)^{2}}{2D(L)} \right\} \label{31}$$ with the variance $D(L)$, $$D(L)=\frac{L}{4}\left(1+\frac{L}{2n}\right)=
\frac{L}{4}\left[1+\frac{2\mu L}{N(1-2\mu )}\right]. \label{32}$$
In the case of strong persistence (\[24a\]), the asymptotic expression for the distribution function Eq. (\[W(L)\]) can be written as $$\label{45f}
W_{L}(k)=W_{L}(0)\frac{nL}{k(L-k)}, \,\,\, k\neq 0,\,\, k\neq L,$$ $$W_{L}(0)=W_{L}(L)=\frac{1}{2} ( 1 - n \ln L ). \label{45b}$$ Both the distribution $W_{L}(k)$ (\[45f\]) and the function $W_{N}(k)$ (\[21\]) has a concave form. The former assumes the maximal value (\[45b\]) at the edges of the interval $[0, L]$ and has a minimum at $k=L/2$.
### Variance $D(L)$
Using the definition Eq. (\[7\]) and the distribution function Eq. (\[W(L)\]) one can obtain a very simple formula for the variance $D(L)$, $$\label{D(L)}
D(L)=\frac{L}{4}[1+m(L-1)],$$ with $$\label{m}
m=\frac{1}{1+2n}= \frac{2\mu}{N-2\mu(N-1)}.$$ Eq. (\[D(L)\]) shows that the variance $D(L)$ obeys the parabolic law independently of the correlation strength in the Markov chain.
In the case of weak persistence, at $n\gg 1$, we obtain the asymptotics Eq. (\[32\]). It allows one to analyze the behavior of the variance $D(L)$ with an increase in the “diffusion time” $L$. At small $mL \ll 1$, the dependence $D(L)$ follows the classical law of the Brownian diffusion, $D(L)\approx L/4$. Then, at $mL\sim 1$, the function $D(L)$ becomes super-linear.
For the case of strong persistence, $n \ll 1$, Eq. (\[D(L)\]) gives the asymptotics, $$D(L)=\frac{L^2}{4} - \frac{nL(L-1)}{2}. \label{45c}$$ The ballistic regime of diffusion leads to the quadratic law of the $D(L)$ dependence in the zero approximation in the parameter $n \ll 1$.
The unusual behavior of the variance $D(L)$ raises an issue as to what particular type of the diffusion equation corresponds to the nonlinear dependence $D(L)$ in Eq. (\[32\]). In the following subsection, when solving this problem, we will obtain the conditional probability $p^{(0)}$ of occurring the symbol zero after a given $L$-word with $L<N$. The ability to find $p^{(0)}$, with some reduced information about the preceding symbols being available, is very important for the study of the self-similarity of the Markov chain (see Subsubsec. 4 of this Subsection).
### Generalized diffusion equation at $L<N$, $n \gg 1$
It is quite obvious that the distribution $W_{L}(k)$ satisfies the equation $$W_{L+1}(k)=W_{L}(k)p^{(0)}(k)+W_{L}(k-1)p^{(1)}(k-1). \label{33}$$ Here $p^{(0)}(k)$ is the probability of occurring ”0” after an average-statistical $L$-word containing $k$ unities and $p^{(1)}(k-1)$ is the probability of occurring ”1” after an $L$-word containing $(k-1)$ unities. At $L<N$, the probability $p^{(0)}(k)$ can be written as $$p^{(0)}(k)=\frac{1}{W_L(k)}
\sum\limits_{i=k}^{k+N-L}p_{i}b(i)\mathrm{C}_{L}^{k}\mathrm{C}_{N-L}^{i-k}.
\label{34}$$ The product $b(i)\mathrm{C}_{L}^{k}\mathrm{C}_{N-L}^{i-k}$ in this formula represents the conditional probability of occurring the $N$-word containing $i$ unities, the right-hand part of which, the $L$-sub-word, contains $k$ unities (compare with Eqs. (\[29\]), (\[29b\])).
The product $b(i)\mathrm{C}_{N-L}^{i-k}$ in Eq. (\[34\]) is a sharp function of $i$ with a maximum at some point $i=i_0$ whereas $p_{i}$ obeys the linear law (\[14\]). This implies that $p_{i}$ can be factored out of the summation sign being taken at point $i=i_0$. The asymptotical calculation shows that point $i_0$ is described by the equation, $$i_{0}=\frac{N}{2}-\frac{L/2}{1-2\mu (1-L/N)}\left(
1-\frac{2k}{L}\right). \label{35}$$ Expression (\[14\]) taken at point $i_0$ gives the desired formula for $p^{(0)}$ because $$\sum\limits_{i=k}^{k+N-L}b(i)\mathrm{C}_{L}^{k}\mathrm{C}_{N-L}^{i-k}$$ is obviously equal to $W_L(k)$. Thus, we have $$p^{(0)}(k)=\frac{1}{2}+\frac{\mu L}{N-2\mu (N-L)}\left(
1-\frac{2k}{L}\right). \label{36}$$
Let us consider a very important point relating to Eq. (\[35\]). If the concentration of unities in the right-hand part of the word (\[29b\]) is higher than $1/2$, $k/L >1/2$, then the most probable concentration $(i_0-k)/(N-L)$ of unities in the left-hand part of this word is likewise increased, $(i_0-k)/(N-L)>1/2$. At the same time, the concentration $(i_0-k)/(N-L)$ is less than $k/L$, $$\label{36b}
\frac{1}{2} <\frac{i_0-k}{N-L}<\frac{k}{L}.$$ This implies that the increased concentration of unities in the $L$-words is necessarily accompanied by the existence of a certain tail with an increased concentration of unities as well. Such a phenomenon is referred by us as the *macro-persistence*. An analysis performed in the following section will indicate that the correlation length $l_c$ of this tail is $\gamma N $ with $\gamma \geq 1$ dependent on the parameter $\mu$ only. It is evident from the above-mentioned property of the isotropy of the Markov chain that there are two correlation tails from both sides of the $L$-word.
Note that the distribution $W_L(k)$ is a smooth function of arguments $k$ and $L$ near its maximum in the case of weak persistence and $k, L-k\gg 1$. By going over to the continuous limit in Eq. (\[33\]) and using Eq. (\[36\]) with the relation $p^{(1)}(k-1)=1-p^{(0)}(k-1)$, we obtain the diffusion Fokker-Planck equation for the correlated Markov process, $$\frac{\partial W}{\partial L}=\frac{1}{8}\frac{\partial
^{2}W}{\partial \kappa ^{2}}-\eta(L)\frac{\partial }{\partial
\kappa }( \kappa W), \label{39}$$ where $\kappa =k-L/2$ and $$\label{39b}
\eta(L)=\frac{2\mu}{(1-2\mu )N+2\mu L}.$$ Equation (\[39\]) has a solution of the Gaussian form Eq. (\[31\]) with the variance $D(L)$ satisfying the ordinary differential equation, $$\frac{\mathrm{d}D}{\mathrm{d}L}=\frac{1}{4}+2\eta(L)D. \label{40}$$ Its solution, given the boundary condition $D(0)=0$, coincides with (\[32\]).
### Self-similarity of the persistent Brownian diffusion
In this subsection, we point to one of the most interesting properties of the Markov chain being considered, namely, its self-similarity. Let us reduce the $N$-step Markov sequence by regularly (or randomly) removing some symbols and introduce the decimation parameter $\lambda$, $$\lambda =N^{\ast }/N \leq 1. \label{41}$$ Here $N^{\ast }$ is a renormalized memory length for the reduced $N^{\ast }$-step Markov chain. According to Eq. (\[36\]), the conditional probability $p_{k}^{\ast }$ of occurring the symbol zero after $k$ unities among the preceding $N^{\ast }$ symbols is described by the formula, $$p_{k}^{\ast }=\frac{1}{2}+\mu ^{\ast }\left( 1-\frac{2k}{N^{\ast
}}\right), \label{42}$$ with $$\mu ^{\ast }=\mu \frac{\lambda }{1-2\mu (1-\lambda )}. \label{43}$$ The comparison between Eqs. (\[14\]) and (\[42\]) shows that the reduced chain possesses the same statistical properties as the initial one but it is characterized by the renormalized parameters ($N^{\ast }$, $\mu ^{\ast }$) instead of ($N$, $\mu $). Thus, Eqs. (\[41\]) and (\[43\]) determine the one-parametrical renormalization of the parameters of the stochastic process defined by Eq. (\[14\]).
The astonishing property of the reduced sequence consists in that *the variance $D^{\ast }(L)$ is invariant with respect to the one-parametric decimation transformation* (\[41\]), (\[43\]). In other words, it coincides with the function $D(L)$ for the initial Markov chain: $$\label{44}
D^{\ast }(L) = \frac{L}{4}[1+m ^{\ast } (L-1)] = D(L), \qquad
L<N^{\ast }.$$ Indeed, according to Eqs. (\[41\]), (\[43\]), the renormalized parameter $m ^{\ast }=2\mu ^{\ast}/[N^{\ast} - 2\mu ^{\ast}
(N^{\ast} -1)]$ of the reduced sequence coincides exactly with the parameter $m =2\mu/[N - 2\mu (N-1)]$ of the initial Markov chain. Since the shape of the function $W_L(k)$ Eq. (\[W(L)\]) is defined by the invariant parameter $n=n^{\ast}$, the distribution $W_L(k)$ is also invariant with respect to the decimation transformation.
The transformation ($N$, $\mu $) $\rightarrow$ ($N^{\ast }$, $\mu ^{\ast }$) (\[41\]), (\[43\]) possesses the properties of semi-group, i. e., the composition of transformations ($N$, $\mu $) $\rightarrow$ ($N^{\ast }$, $\mu ^{\ast }$) and ($N^{\ast }$, $\mu ^{\ast }$) $\rightarrow$ ($N^{\ast \ast }$, $\mu ^{\ast \ast}$) with transformation parameters $\lambda_1$ and $\lambda_2$ is likewise the transformation from the same semi-group, ($N$, $\mu$) $\rightarrow$ ($N^{\ast \ast }$, $\mu ^{\ast \ast}$), with parameter $\lambda = \lambda_1 \lambda_2$.
The invariance of the function $D(L)$ at $L<N$ was referred to by us as the phenomenon of *self-similarity*. It is demonstrated in Fig. 3 and is accordingly discussed below, in Sec. IV A.
It is interesting to note that the property of self-similarity is valid for any strength of the persistency. Indeed, the result Eq. (\[36\]) can be obtained directly from Eqs. (\[b\])-(\[17a\]), and (\[34\]) not only for $n\gg 1$ but also for the arbitrary value of $n$.
[![The dependence of the variance $D$ on the tuple length $L$ for the generated sequence with $N=100$ and $\mu=0.4$ (solid line) and for the decimated sequences (the parameter of decimation $\lambda =0.5$). Squares and circles correspond to the stochastic and deterministic reduction, respectively. The thin solid line describes the non-correlated Brownian diffusion, $D(L)=L/4$.[]{data-label="f3"}](f3.eps "fig:"){width="45.00000%" height="35.00000%"}]{}
Long-range diffusion, $L>N$
---------------------------
Unfortunately, the very useful proposition $\spadesuit$ is valid for the words of the length $L\leq N$ only and is not applicable to the analysis of the long words with $L>N$. Therefore, investigating the statistical properties of the long words represents a rather challenging combinatorial problem and requires new physical approaches for its simplification. Thus, we start this subsection by analyzing the correlation properties of the long words ($L>N$) in the Markov chains with $N\gg 1$. The two first subsubsections of this subsection mainly deal with the case of relatively weak correlations, $n \gg 1$.
### Correlation length at weak persistence
Let us rewrite Eq. (\[14\]) in the form, $$<a_{i+1}>=\frac{1}{2}+\mu \left(
\frac{2}{N}\sum_{k=i-N+1}^{i}<a_{k}>-1 \right). \label{46}$$ The angle brackets denote the averaging of the density of unities in some region of the Markov chain for its definite realization. The averaging is performed over distances much greater than unity but far less than the memory length $N$ and correlation length $l_c$ (see Eq. (\[50b\]) below). Note that this averaging differs from the statistical averaging over the ensemble of realizations of the Markov chain denoted by the bar in Eqs. (\[7\]) and (\[8\]). Equation (\[46\]) is a relationship between the average densities of unities in two different macroscopic regions of the Markov chain, namely, in the vicinity of $(i+1)$-th element and in the region $(i-N,\,\,i)$. Such an approach is similar to the mean field approximation in the theory of the phase transitions and is asymptotically exact at $N\rightarrow \infty$. In the continuous limit, Eq. (\[46\]) can be rewritten in the integral form, $$<a(i)>=\frac{1}{2}+\mu \left(
\frac{2}{N}\int_{i-N}^{i}<a(k)>\textrm{d}k-1\right). \label{47}$$ It has the obvious solution, $$<a(i)-\frac{1}{2}>=<a(0)-\frac{1}{2}>\exp \left(-i/\gamma
N\right), \label{49}$$ where the parameter $\gamma $ is determined by the relation, $$\gamma \left( \exp \left( \frac{1}{\gamma }\right) -1\right)
=\frac{1}{2\mu}. \label{50}$$ A unique solution $\gamma $ of the last equation is an increasing function of $\mu \in(0, 1/2)$.
Formula (\[49\]) shows that any fluctuation (the difference between $<a(i)>$ and the equilibrium value of $\overline{a_i}=1/2$) is exponentially damped at distances of the order of the *correlation length* $l_c$, $$\label{50b}
l_{c}=\gamma N.$$ Law (\[49\]) describes the phenomenon of the *persistent macroscopic correlations* discussed in the previous subsection. This phenomenon is governed by both parameters, $N$ and $\mu$. According to Eqs. (\[50\]), (\[50b\]), the correlation length $l_c$ grows as $\gamma=
1/4 \delta$ with an increase in $\mu$ (at $\mu \rightarrow
1/2$) until the inequality $\delta \gg 1/N$ is satisfied. Here $$\label{delta}
\delta = 1/2-\mu.$$ Let us note that the inequality $\delta \gg 1/N$ defining the regime of weak persistence can be rewritten in terms of $\gamma$, $\gamma \ll N/4$. At $\delta \approx 1/N$, the correlation length $l_c$ achieves its maximum value $N^2/4$. With the following increase of $\mu$, the diffusion goes to the regime of strongly correlated diffusion that will be discussed in Subsubsec 3 of this Subsection.
At $\mu \rightarrow 0$, the macro-persistence is broken and the correlation length tends to zero.
### Correlation function at weak persistence
Using the studied correlation properties of the Markov sequence and some heuristic reasons, one can obtain the correlation function ${\cal K}(r)$ being defined as, $${\cal K}(r)=\overline{a_{i}a_{i+r}}-\overline{a_{i}}^2, \label{51}$$ and then the variance $D(L)$. Comparing Eq. (\[51\]) with Eqs. (\[5\]), (\[7\]) and taking into account the property of sequence, $\overline{a_{i}}=1/2$, it is easy to derive the general relationship between functions ${\cal
K}(r)$ and $D(L)$, $$\label{51c}
D(L)=\frac{L^{2}}{4}+4\sum_{i=1}^{L-1}\sum_{r=1}^{L-i}{\cal K}(r).$$ Considering (\[51c\]) as an equation with respect to ${\cal
K}(r)$, one can find its solution, $${\cal K}(1) = \frac{1}{2}D(2)-\frac{1}{4}, \quad {\cal K}(2) =
\frac{1}{2}D(3)-D(2)+\frac{1}{8},$$ $$\label{51e}
{\cal K}(r)=\frac{1}{2}\left[D(r+1) -2D(r) +D(r-1)\right], \quad
r\geq 3.$$ This solution has a very simple form in the continuous limit, $$\label{51f}
{\cal K}(r) = \frac{1}{2}\frac{{\textrm d}^2 D(r)}{{\textrm
d}r^2}.$$
Equations (\[51e\]) and (\[D(L)\]) give the correlation function at $r<N$, $n\gg 1$, $${\cal K}(r)=C_{r}m,$$ with $$C_{1}=1/2, \qquad C_{2}=1/8, \qquad C_{3\leq r\leq N}=1/4,$$ and $m$ determined by Eq. (\[m\]). In the continuous approximation, the correlation function is described by the formula, $${\cal K}(r)=\frac{m}{4 }, \qquad r \leq N. \label{54b}$$ The independence of the correlation function of $r$ at $r<N$ results from our choice of the conditional probability in the simplest form (\[14\]). At $r>N$, the function ${\cal K}(r)$ should decrease because of the loss of memory. Therefore, using Eqs. (\[49\]) and (\[50b\]), let us prolongate the correlator ${\cal K}(r)$ as the exponentially decreasing function at $r>N$, $${\cal K}(r)=\frac{m}{4}\cases {1,\;\qquad \;\;\;\;\;\qquad r\leq
N, \cr \exp \left(-\frac{r-N}{l_{c}}\right ), \;\;r>N.} \label{55}$$ The lower curve in Fig. \[f4\] presents the plot of the correlation function at $\mu =0.1$.
[![The dependence of the correlation function $K$ on the distance $r$ between the symbols for the sequence with $N=20$. The dots correspond to the generated sequence with $\mu=0.1$ and $\mu=50/101$. The lower line is analytical result (\[55\]) with $l_c=\gamma N$ and $\gamma=0.38$.[]{data-label="f4"}](f4.eps "fig:"){width="45.00000%" height="35.00000%"}]{}
According to Eqs. (\[51f\]), (\[55\]), the variance $D(L)$ can be written as $$\label{56}
D(L)=\frac{L}{4}\left(1+m F(L)\right)$$ with $$\label{57}
F(L)= \cases {L, \qquad \qquad \qquad \qquad \qquad \qquad L<N,
\cr2(1+ \gamma)N - (1+2\gamma ) \frac{N^2}{L} \cr -
2\gamma^{2}\frac{N^2}{L} \left[1-\exp \left(
-\frac{L-N}{l_c}\right) \right], \, \, L>N.}$$
[![The numerical simulation of the dependence $D(L)$ for the generated sequence with $N=100$ and $\mu=0.4$ (circles). The solid line is the plot of function Eq. (\[56\]) with the same values of $N$ and $\mu$.[]{data-label="f5"}](f5.eps "fig:"){width="45.00000%" height="35.00000%"}]{}
As an illustration of the result Eq. (\[56\]), we present the plot of $D(L)$ for $N=100$ and $\mu=0.4$ by the solid line in Fig. \[f5\]. The straight line in the figure corresponds to the dependence $D(L)=L/4$ for the usual Brownian diffusion without correlations (for $\mu=0$). It is clearly seen that the plot of variance (\[56\]) contains two qualitatively different portions. One of them, at $L\lesssim N$, is the super-linear curve that moves away from the line $D=L/4$ with an increase of $L$ as a result of the persistence. For $L\gg N$, the curve $D(L)$ achieves the linear asymptotics, $$\label{58}
D(L)\cong \frac{L}{4}\left ( 1+ \frac{4\mu
(1+\gamma)}{1-2\mu}\right).$$ This phenomenon can be interpreted as a result of the diffusion in which every *independent* step $\sim
\sqrt{D(L)}$ of wandering represents a path traversed by a particle during the characteristic “fluctuating time” $L
\sim (N+l_c)$. Since these steps of wandering are quasi-independent, the distribution function $W_L(k)$ is the Gaussian. Thus, in the case of relatively weak persistence, $n \gg 1$, $W_L(k)$ is the Gaussian not only at $L<N$ (see Eq. (\[31\])) but also for $L>N, \, l_c$.
Note that the above-mentioned property of the self-similarity is valid only at the portion $L<N$ of the curve $D(L)$. Since the decimation procedure leads to the decrease of the parameter $\mu$ (see Eq. (\[43\])), the plot of asymptotics (\[58\]) for the reduced sequence at $L\gg N^{\ast}$ goes below the $D(L)$ plot for the initial chain.
### Statistics of the $L$-words for the case of strong persistence, $n \ll \ln^{-1}N $
In this subsection, we study the statistical properties of long words ($L>N$) in the sequences of symbols with strong correlations. It is convenient to rewrite formula (\[14\]) for the conditional probability of occurring the symbol zero after the $N$-word containing $k$ unities in the form, $$p_{\nu}=\delta+2\mu \frac{\nu}{N}, \label{58b}$$ where $\nu$ is the number of zeros in the precedent $N$-word, $\nu=N-k$.
In the case of strong persistence, $n \ll \ln^{-1}N $, the parameter $\delta =1/2-\mu$ is much smaller than $1/N$. Therefore, the probability $p_{\nu}$ can be written as $$p_{\nu} \approx \cases {\delta,\;\qquad \,\,\,\,\,\,\,\,\,\,
\nu=0, \cr \nu /N, \;\qquad \, \,\nu \neq0,\, \,\, \nu \neq
N, \cr 1-\delta, \;\qquad \, \nu=N.} \label{58d}$$ It is seen that the probability of occurring the symbol zero after the $N$-word which contains only unities ($\nu =0$) represents very small value $\delta$ and it increases significantly if $\nu \neq 0$. This situation differs drastically from the case of weak persistency. At $n \gg 1$, the parameter $\delta$ exceeds noticeably the value $1/N$, and the probability $p_\nu$ does not actually change with an increase in the number of zeros in the preceding $N$-word.
The analysis of the symbol generation process in the Markov chain in the case of strong persistence gives the following picture of the fluctuations. There exist the entire portions of the chain consisting of the same symbols, say unities. The characteristic length of such portions is $1/\delta \gg
N$. These portions are separated by one or more symbols zero. The number of such packets of the same symbols in one fluctuation zone is about $N$. Thus, the characteristic correlation distance at which the $N$-word containing the same symbols converts into the $N$-word with $\nu =N/2$ is about $N/\delta$, $$l_c\approx \frac{N}{\delta}. \label{58e}$$
The described structure of the fluctuations defines the statical properties of the $L$-words with $L>N$ in the case of strong persistence. The distribution function differs significantly from the Gaussian and is characterized by a concave form at $L\lesssim l_c \sim N/\delta $. As $L$ increases, the correlations between different parts of the $L$-words get weaker and the $L$-word can be considered as consisting of a number of independent sub-words. So, according to the general mathematical theorems [@nag; @ibr], the distribution function takes on the usual Gaussian form. Such an evolution of the distribution function is depicted in Fig. \[f6\].
[![The distribution function $w(k/L)=LW_L(k)$ for $N$=8 and $\delta=1/150$. Different values of the length $L$ of words is shown near the curves.[]{data-label="f6"}](f6.eps "fig:"){width="45.00000%" height="35.00000%"}]{}
The variance $D(L)$ follows the quadratic law $D=L^2/4$ (see Eq. (\[45c\])) up to the range of $L\lesssim l_c \sim N/\delta$ and then approaches to the asymptotics $D(L) = B L$ with $B \sim
N/4\delta$ (see Fig. \[f7\]).
[![The dependence of the variance $D$ on the word length $L$ for the sequence with $N=20$ and $\mu=50/101$ (solid line). The thin solid line describes the non-correlated Brownian diffusion, $D(L)=L/4.$ []{data-label="f7"}](f7.eps "fig:"){width="45.00000%" height="35.00000%"}]{}
The upper curve in Fig. \[f4\] presents the correlation function for the case of strong persistence ($\mu = 50/101$, $N=20$).
Results of numerical simulations and applications
=================================================
In this section, we support the obtained analytical results by numerical simulations of the Markov chain with the conditional probability Eq. (\[14\]). Besides, the properties of the studied binary $N$-step Markov chain are compared with those for the natural objects, specifically for the coarse-grained written and DNA texts.
Numerical simulations of the Markov chain
-----------------------------------------
The first stage of the construction of the $N$-step Markov chain was a generation of the initial non-correlated $N$ symbols, zeros and unities, identically distributed with equal probabilities 1/2. Each consequent symbol was then added to the chain with the conditional probability determined by the previous $N$ symbols in accordance with Eq. (\[14\]). Then we numerically calculated the variance $D(L)$ by means of Eq. (\[7\]). The circles in Fig. \[f5\] represent the calculated variance $D(L)$ for the case of weak persistence ($n=12.5 \gg 1$). A very good agreement between the analytical result (\[56\]) and the numerical simulation can be observed. The case of strong persistence is illustrated by Figs. \[f6\] and \[f7\] where the distribution function $W_L(k)$ and the variance $D(L)$ are calculated numerically for $n=4/37$ and $n=0.1$, respectively. The dots on the curves in Fig. \[f4\] represent the calculated results for the correlation function ${\cal K}(r)$ for $n=0.1$ (the upper curve) and $n=40$ (the lower curve).
The numerical simulation was also used for the demonstration of the proposition $\spadesuit$ (Fig. 1) and the self-similarity property of the Markov sequence (Fig. 3). The squares in Fig. 3 represent the variance $D(L)$ for the sequence obtained by the stochastic decimation of the initial Markov chain (solid line) where each symbol was omitted with the probability 1/2. The circles in this figure correspond to the regular reduction of the sequence by removing each second symbol.
And finally, the numerical simulations have allowed us to make sure that we are able to determine the parameters $N$ and $\mu$ of a given binary sequence. We generated the Markov sequences with different parameters $N$ and $\mu$ and defined numerically the corresponding curves $D(L)$. Then we solved the inverse problem of the reconstruction of the parameters $N$ and $\mu$ by analyzing the curves $D(L)$. The reconstructed parameters were always in good agreement with their prescribed values. In the following subsections we apply this ability to the treatment of the statistical properties of literary and DNA texts.
Literary texts
--------------
It is well-known that the statistical properties of the coarse-grained texts written in any language exhibit a remarkable deviation from random sequences [@schen; @kant]. In order to check the applicability of the theory of the binary $N$-step Markov chains to literary texts we resorted to the procedure of coarse graining by the random mapping of all characters of the text onto the binary set of symbols, zeros and unities. The statistical properties of the coarse-grained texts depend, but not significantly, on the kind of mapping. This is illustrated by the curves in Fig. \[f8\] where the variance $D(L)$ for five different kinds of the mapping of Bible is presented. In general, the random mapping leads to nonequal numbers of unities and zeros, $k_1$ and $k_0$, in the coarse-grained sequence. A particular analysis indicates that the variance $D(L)$ (\[32\]) gets the additional multiplier, $$\frac{4k_0 k_1}{(k_0+k_1)^2},$$ in this biased case. In order to derive the function $D(L)$ for the non-biased sequence, we divided the numerically calculated value of the variance by this multiplier.
[![The dependence $D(L)$ for the coarse-grained text of Bible obtained by means of five different kinds of random mapping.[]{data-label="f8"}](f8.eps "fig:"){width="45.00000%" height="35.00000%"}]{}
The study of different written texts has suggested that all of them are featured by the pronounced persistent correlations. It is demonstrated by Fig. \[f9\] where the five variance curves go significantly higher than the straight line $D=L/4$. However, it should be emphasized that regardless of the kind of mapping the initial portions, $L<80$, of the curves correspond to a slight anti-persistent behavior (see insert to Fig. \[f10\]). Moreover, for some inappropriate kinds of mapping (e.g., when all vowels are mapped onto the same symbol) the anti-persistent portions can reach the values of $L\sim 1000$. To avoid this problem, all the curves in Fig. \[f9\] are obtained for the definite representative mapping: (a-m) $\rightarrow$ 0; (n-z) $\rightarrow$ 1.
[![The dependence $D(L)$ for the coarse-grained texts of collection of works on the computer science ($m=2.2\cdot 10^{-3}$, solid line), Bible in Russian ($m=1.9\cdot 10^{-3}$, dashed line), Bible in English ($m=1.5\cdot 10^{-3}$, dotted line), “History of Russians in the 20-th Century” by Oleg Platonov ($m=6.4\cdot
10^{-4}$, dash-dotted line), and “Alice’s Adventures in Wonderland” by Lewis Carroll ($m=2.7\cdot 10^{-4}$, dash-dot-dotted line).[]{data-label="f9"}](f9.eps "fig:"){width="45.00000%" height="35.00000%"}]{}
Thus, the persistence is the common property of the binary $N$-step Markov chains that have been considered in this paper and the coarse-grained written texts at large scales. Moreover, the written texts as well as the Markov sequences possess the property of the self-similarity. Indeed, the curves in Fig. \[f10\] obtained from the text of Bible with different levels of the deterministic decimation demonstrate the self-similarity. Presumably, this property is the mathematical reflection of the well-known hierarchy in the linguistics: *letters $\rightarrow$ syllables $\rightarrow$ words $\rightarrow$ sentences $\rightarrow$ paragraphs $\rightarrow$ chapters $\rightarrow$ books*.
[![The dependence of the variance $D$ on the tuple length $L$ for the coarse-grained text of Bible (solid line) and for the decimated sequences with different parameters $\lambda$: $\lambda
= 3/4 $ (squares), $\lambda = 1/2 $ (stars), and $\lambda = 1/256
$ (triangles). The insert demonstrates the anti-persistent portion of the $D(L)$ plot for Bible.[]{data-label="f10"}](f10.eps "fig:"){width="45.00000%" height="35.00000%"}]{}
All the above-mentioned circumstances allow us to suppose that our theory of the binary $N$-step Markov chains can be applied to the description of the statistical properties of the texts of natural languages. However, in contrast to the generated Markov sequence (see Fig. \[f5\]) where the full length $\mathcal{M}$ of the chain is far greater than the memory length $N$, the coarse-grained texts described by Fig. \[f9\] are of relatively short length $\mathcal{M}\lesssim N$. In other words, the coarse-grained texts are similar not to the Markov chains but rather to some non-stationary short fragments. This implies that each of the written texts is correlated throughout the whole of its length. Therefore, as fae as the written texts are concerned, it is impossible to observe the second portion of the curve $D(L)$ parallel (in the log-log scale) to the line $D(L)=L/4$, similar to that shown in Fig. \[f5\]. As a result, one cannot define the values of both parameters $N$ and $\mu$ for the coarse-grained texts. The analysis of the curves in Fig. 6 can give the combination $m=2\mu/N(1-2\mu)$ only (see Eq. (\[32\])). Perhaps, this particular combination is the real parameter governing the persistent properties of the literary texts.
We would like to note that the origin of the long-range correlations in the literary texts is hardly related to the grammatical rules as is claimed in Ref. [@kant]. At short scales $L\leq 80$ where the grammatical rules are in fact applicable the character of correlations is anti-persistent (see the insert in Fig. \[10\]) whereas semantic correlations lead to the global persistent behavior of the variance $D(L)$ throughout the entire of literary text.
The numerical estimations of the persistent parameter $m$ and the characterization of the languages and different authors using this parameter can be regarded as a new intriguing problem of linguistics. For instance, the unprecedented low value of $m$ for the very inventive work by Lewis Carroll as well as the closeness of $m$ for the texts of English and Russian versions of Bible are of certain interest.
It should be noted that there exist special kinds of short-range correlated texts which can be specified by both of the parameters, $N$ and $\mu$. For example, all dictionaries consist of the families of words where some preferable letters are repeated more frequently than in their other parts. Yet another example of the shortly correlated texts is any lexicographically ordered list of words. The analysis of written texts of this kind is given below.
Dictionaries
------------
As an example, we have investigated the statistical properties of the coarse-grained alphabetical (lexicographically ordered) list of the most frequently used 15462 English words. In contrast to other texts, the statistical properties of the coarse-grained dictionaries are very sensitive to the kind of mapping. If one uses the above-mentioned mapping, (a-m) $\rightarrow$ 0; (n-z) $\rightarrow$ 1, the behavior of the variance $D(L)$ similar to that shown in Fig. \[f9\] would be obtained. The particular construction of the dictionary manifests itself if the preferable letters in the neighboring families of words are mapped onto the different symbols. The variance $D(L)$ for the dictionary coarse-grained by means of such mapping is shown by circles in Fig. \[f11\]. It is clearly seen that the graph of the function $D(L)$ consists of two portions similarly to the curve in Fig. \[f5\] obtained for the generated $N$-step Markov sequence. The fitting of the curve in Fig. \[f11\] by function (\[56\]) (solid line in Fig. \[f11\]) yielded the values of the parameters $N=180$ and $\mu =0.44$.
[![The dependence $D(L)$ for the coarse-grained alphabetical list of 15462 English words (circles). The solid line is the plot of function Eq. (55) with the fitting parameters $N=180$ and $\mu=0.44$.[]{data-label="f11"}](f11.eps "fig:"){width="45.00000%" height="35.00000%"}]{}
DNA texts
---------
It is known that any DNA text is written by four “characters”, specifically by adenine (A), cytosine (C), guanine (G), and thymine (T). Therefore, there are three nonequivalent types of the DNA text mapping onto one-dimensional binary sequences of zeros and unities. The first of them is the so-called purine-pyrimidine rule, {A,G} $\rightarrow$ 0, {C,T} $\rightarrow$ 1. The second one is the hydrogen-bond rule, {A,T} $\rightarrow$ 0, {C,G} $\rightarrow$ 1. And, finally, the third is {A,C} $\rightarrow$ 0, {G,T} $\rightarrow$ 1.
[![The dependence $D(L)$ for the coarse-grained DNA text of *Bacillus subtilis, complete genome*, for three nonequivalent kinds of mapping. Solid, dashed, and dash-dotted lines correspond to the mappings {A,G} $\rightarrow$ 0, {C,T} $\rightarrow$ 1 (the parameter $m=4.1\cdot 10^{-2}$), {A,T} $\rightarrow$ 0, {C,G} $\rightarrow$ 1 ($m=2.5\cdot 10^{-2}$), and {A,C} $\rightarrow$ 0, {G,T} $\rightarrow$ 1 ($m=1.5\cdot
10^{-2}$), respectively.[]{data-label="f12"}](f12.eps "fig:"){width="45.00000%" height="35.00000%"}]{}
Bywayofexample,thevariance$D(L)$forthecoar-\
se-grainedtextof*Bacillussubtilis,completegenome*\
(ftp:$//$ftp.ncbi.nih.gov$/$genomes$/$bacteria$/$bacillus\_subti-\
lis$/$NC\_000964.gbk) is displayed in Fig. \[f12\] for all possible types of mapping. One can see that the persistent properties of DNA are more pronounced than for the written texts and, contrary to the written texts, the $D(L)$ dependence for DNA does not exhibit the anti-persistent behavior at small values of $L$. In our view, the noticeable deviation of different curves in Fig. \[f12\] from each other demonstrates that the DNA texts are much more complex objects in comparison with the written ones. Indeed, the different kinds of mapping reveal and emphasize various types of physical attractive correlations between the nucleotides in DNA, such as the strong purine-purine and pyrimidine-pyrimidine persistent correlations (the upper curve), and the correlations caused by a weaker attraction A$\leftrightarrow$T and C$\leftrightarrow$G (the middle curve).
It is interesting to compare the correlation properties of the DNA texts for three different species that belong to the major domains of living organisms: the Bacteria, the Archaea, and the Eukarya [@mad]. Figure \[f13\] shows the variance $D(L)$ for the coarse-grained DNA texts of Bacillus subtilis (the Bacteria), Methanosarcina acetivoransthe (the Archaea), and Drosophila melanogaster - fruit fly - (the Eukarya) for the most representative mapping {A,G} $\rightarrow$ 0, {C,T} $\rightarrow$ 1. It is seen that the $D(L)$ curve for the DNA text of Bacillus subtilis is characterized by the highest persistence. As well as for the written texts, the $D(L)$ curves for the DNA of both the Bacteria and the Archaea do not contain the linear portions given by Eq. (\[58\]). This suggests that their DNA chains are not stationary sequences. In this connection, we would like to point out that their DNA molecules are circular and represent the collection of extended coding regions interrupted by small non-coding regions. According to Figs. \[f12\], \[f13\], the non-coding regions do not disrupts the correlation between the coding areas, and the DNA systems of the Bacteria and the Archaea are fully correlated throughout their entire lengths. Contrary to them, the DNA molecules of the Eukarya have the linear structure and contain long non-coding portions. As evident from Fig. \[f13\], the DNA sequence of the representative of the Eukarya is not entirely correlated. The $D(L)$ curve for the X-chromosome of the fruit fly corresponds qualitatively to Eqs. (\[56\]), (\[57\]) with $\mu \approx 0.35$ and $N \approx 250$. If one draws an analogy between the DNA sequences and the literary texts, the resemblance of the correlation properties of integral literary novels and the DNA texts of the Bacteria and Archaea are to be found, whereas the DNA texts of the Eukarya are more similar to the collections of 10$^4$–10$^5$ short stories.
[![The dependence $D(L)$ for the coarse-grained DNA texts of *Bacillus subtilis, complete genome*, the Bacteria, (solid line); *Methanosarcina acetivorans, complete genome*, the Archaea, (dashed line); *Drosophila melanogaster chromosome X, complete sequence*, the Eukarya, (dotted line) for the mapping {A,G} $\rightarrow$ 0, {C,T} $\rightarrow$ 1.[]{data-label="f13"}](f13.eps "fig:"){width="45.00000%" height="35.00000%"}]{}
Conclusion
==========
Thus, we have developed a new approach to describing the strongly correlated one-dimensional systems. The simple, exactly solvable model of the uniform binary $N$-step Markov chain is presented. The memory length $N$ and the parameter $\mu$ of the persistent correlations are two parameters in our theory. The correlation function ${\cal K}(r)$ is usually employed as the input characteristics for the description of the correlated random systems. Yet, the function ${\cal K}(r)$ describes not only the direct interconnection of the elements $a_i$ and $a_{i+r}$, but also takes into account their indirect interaction via other elements. Since our approach operates with the “original” parameters $N$ and $\mu$, we believe that it allows us to reveal the intrinsic properties of the system which provide the correlations between the elements.
We have demonstrated the applicability of the developed theoretical model to the different kinds of relatively weakly correlated stochastic systems. Perhaps, the case of strong persistency is also of interest from the standpoint of possible applications. Indeed, the domain structure of the symbol fluctuations at $n\ll 1$ is very similar to the domains in magnetics. Thus, an attempt to model the magnetic structures by the Markov chains with strongly pronounced persistent properties can be appropriate.
We would like to note that there exist some features of the real correlated systems which cannot be interpreted in terms of our two-parametric model. For example, the interference of the grammatical anti-persistent and semantic persistent correlations in the literary texts requires more than two parameters for their description. Obviously, more complex models should be worked out for the adequate interpretation of the statistical properties of the DNA texts and other real correlated systems. In particular, the Markov chains consisting of more than two different elements (non-binary chains) can be suitable for modelling the DNA systems.
We acknowledge to Yu. L. Rybalko and A. L. Patsenker for the assistance in the numerical simulations, M. E. Serbin and R. Zomorrody for the helpful discussions.
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---
author:
- 'Peter Haïssinsky, Pierre Mathieu, and Sebastian Müller[^1]'
bibliography:
- 'bib.bib'
title: Renewal theory for random walks on surface groups
---
\
renewal theory, surface groups, central limit theorem, analyticity 60G50, 60F05, 60B15
Introduction
============
The idea and motivation behind a renewal theory for random walks on groups is to find a decomposition of the trajectory of the walk into aligned pieces in such a way that these pieces are identically and independently distributed. The main result of this paper is the construction of such a renewal theory for random walks on an important class of hyperbolic groups.
Analogous renewal structures have been developed for random walks on free groups and trees with finitely many cone types, [@NW], free products of groups [@G:07], and regular languages [@G:08]. It is also a common technique used in the study of random walks in random environment in order to prove laws of large numbers and [annealed]{} central limit theorems, *e.g.,* see [@Z]. However, to the best of our knowledge the renewal structure given in this paper constitutes the first example on one-ended groups beyond ${\mathbb{Z}}^{d}$.
We invite the reader to consider the following simple but instructing example: nearest neighbor random walk on the free group $\mathbb{F}_{2}=\langle a,b\rangle$ with two generators. In order to define a suitable renewal structure we recall the definition of cone types after Cannon, *e.g.*, see [@ECHLPT]. A cone $C(x)$ consists of all vertices $y$ such that $x$ lies on the geodesic from the group identity $e$ to $y$. A cone type describes the way one *can look to infinity*, *i.e.*, $T(x)=\{y:~xy\in C(x)\}$. It turns out that there are five different cone types, say $\textbf{e},\textbf{a},\textbf{a}^{-},\textbf{b},$ and $\textbf{b}^{-}$. The cone type $\textbf{e}$ corresponds to the cone of $e$ and $\textbf{x}$ corresponds to the one where an $x$ edge *leads back* to the identity. The cone types $\textbf{a},\textbf{a}^{-},\textbf{b},$ and $\textbf{b}^{-}$ have the important property that a cone of one type contains cones of the other three types. Furthermore, any irreducible random walk is transient and hence has a positive probability to stay in a cone $C(x)$ for all times. So, let us fix a cone type, say $\textbf{a}$, and define the renewal times $(R_{n})_{n\geq 1}$ as follows. Let $R_{1}$ be the first time that the walk visits a cone of type $\textbf{a}$ that it will never leave again. Inductively, we define $R_{n+1}$ as the first time after $R_{n}$ that the walk visits a cone of type $\textbf{a}$ that it will never leave again. One can check that $(R_{n+1}-R_{n})_{n\geq 1}$ is indeed an i.i.d. sequence of random variables. Furthermore, we have (using the definition of the cones) that $$d(Z_{R_{n}},e)=d(Z_{R_{1}},e)+\sum_{i=1}^{n-1}d(Z_{R_{i+1}}, Z_{R_{i}}).$$ Non-amenability of the free group implies that the random variables in the above equation all have some exponential moments. It is now standard to deduce a law of large number and a central limit theorem for the distance to the origin. Moreover, the renewal structure enables us to describe the rate of escape and the asymptotic variance, in terms of first and second moments of random variables with exponential moments. This fact allows a very good control of the regularity of these two quantities.
The main technical difficulties that arise when developing the above sketch into a mathematical proof are due to the fact that the random times $R_{n}$ are not stopping times but depend on future events of the walk. Hence, conditioning on events described by $R_{n}$ destroys the Markovian structure of the random walk. Furthermore, for general hyperbolic groups it is not known wether the Cannon automaton gives rise to cones with as nice properties as the ones in the free group. However, we shall prove that one can get a nice renewal structure for random walks on surface groups.
The next Subsection contains a short introduction on central limit theorems for random walks on groups and Subsection \[intr:anal\] concerns recent results on analyticity of the rate of escape. Section \[sec:not\] prepares the ground for the main results in giving the necessary notation, definitions and preliminary results. In Section \[sec:ren\] the renewal structure is formally defined and the main results are proven.
Central limit theorem {#intr:clt}
---------------------
Let $(X_{i})_{i\geq 0}$ be i.i.d random variables taking values in ${\mathbb{Z}}^{d}$. Under a second moment condition we have the classical central limit theorem (CLT) $$\frac{\sum_{i=1}^{n} X_{i}-nv}{\sqrt n } \stackrel{\mathcal{D}}{\longrightarrow} \mathcal{N}(0,\sigma^{2}),$$ where $v={\mathbb{E}}[X_{1}]$ is the rate of escape (or drift) and $\sigma^{2}$ the asymptotic variance. A natural question, which goes back to Bellman [@bellman] and Furstenberg and Kesten [@FK], is to which extent this phenomenon generalizes to $(X_{i})_{i\geq 1}$ taking values in some finitely generated group $\Gamma$. Let $d(\cdot,\cdot)$ be a left invariant metric on $\Gamma$ and $\mu$ a probability measure whose support generates the group $\Gamma$. Let $(X_{i})_{i\geq 1}$ be i.i.d. random variables with distribution $\mu$ and define the random walk $Z_{n}= X_{1} X_{2} \cdots X_{n} .$ Then, if $d(X_{1},e)$ has a finite first moment, Kingman’s subadditive ergodic theorem ensures that $$\lim_{n\to\infty} \frac1n d(Z_{n},e)=:v$$ exists in the almost sure and $L^1$ senses and is deterministic. In other words, there is a law of large numbers for random walks on groups. Moreover, Guivarc’h [@Gui:80] proved that if $\Gamma$ is a non-amenable finitely generated group then any random walk with a finite first moment has positive rate of escape with respect to any word metric. However, it turns out that a central limit theorem can not be stated in this general setting. As described in [@bjorklund] one can use the result of Erschler ([@E:99], [@E:01]) to construct the following counterexample. Let $\Gamma= ({\mathbb{Z}}\wr {\mathbb{Z}}) \times \mathbb{F}_{2}$ where $\wr$ is the wreath product and $\mathbb{F}_{2}$ denotes the free group on two generators. There exists a symmetric probability measure $\mu$ with finite support on $\Gamma$ and a word metric $d$ such that the fluctuations around the linear (positive) drift are of order $n^{\frac34}$.
However, there are several situations where central limit theorems are established. Sawyer and Steger in [@ST] studied the case of the free group $\mathbb{F}_{d}$ with $d$ standard generators and the corresponding word distance. Under technical moment conditions they prove that $(d(Z_{n},e)-nv)/\sqrt{n}$ converges in law to some non-degenerated Gaussian distribution. While their proof uses analytic extensions of Green functions, another proof was given by Lalley [@La:93] using algebraic function theory and Perron–Frobenius theory. A geometric proof was later presented by Ledrappier [@Le:01]. A generalization for trees with finitely many cone types can be found in Nagnibeda and Woess [@NW]. Recently, Björklund [@bjorklund] proved a central limit theorem on hyperbolic groups with respect to the Green metric. The proof in [@bjorklund] is based on the identification of the Gromov boundary with the horofunction boundary. This fact enables to prove the CLT using a martingale approximation. However, the CLT for the Green metric does not seem to imply directly the central limit theorem for the drift with respect to any word metric on $\Gamma$.
One of the main objectives of this paper is to demonstrate that a CLT for one-ended groups, Theorem \[thm:CLT\_planar\], can also be obtained by using a renewal structure. We have managed to do so for random walks on surface groups. However, we believe that our approach should work in the general setting of hyperbolic groups, see Section \[sec:discussion\] for a short discussion.
\[thm:CLT\_planar\] Let $\Gamma$ be a surface group with standard generating set $S$ and corresponding word metric $d$. Furthermore, let $\mu$ be a driving measure with exponential moments whose support contains the generating set $S$. Then $$\frac{d(Z_{n}, e) -nv }{\sqrt{n}} \stackrel{{\mathcal{D}}}{\longrightarrow} {\mathcal{N}}(0, \sigma^2),$$ with $$v=\frac{{\mathbb{E}}[ d(Z_{ R_{2}}, Z_{ R_{1}})]}{{\mathbb{E}}[ R_{2}- R_{1}]} \mbox{ and }
\sigma^{2}=\frac{{\mathbb{E}}[(d(Z_{R_{2}}, Z_{R_{1}}) - (R_{2}- R_{1})v)^{2}]}{{\mathbb{E}}[R_{2}-R_{1}]}.$$
An analogous result holds true for random walks on groups with infinitely many ends. In fact, by applying Stalling’s splitting theorem, one can check that all arguments work fine for amalgamated free products and HNN-extensions over finite groups with a suitable choice of generators, see also Remark \[rem:inftyends\].
Let us note here that other results in this direction are known for actions of linear semigroups on projective spaces by Le Page [@LePage] and Guivarc’h and le Page [@GLeP:04]. We also want to mention earlier works of Tatubalin [@Tutubalin:65] and [@Tutubalin:68] on random walks on hyperbolic space.
More recently, Pollicot and Sharp [@PS:10] use a thermodynamical formalism to prove limit theorems for matrix groups acting cocompactly on the hyperbolic group. Calegari gives generalizations to actions on general hyperbolic groups in the survey paper [@Calegari Section 3].
Analyticity of the rate of escape and asymptotic variance {#intr:anal}
---------------------------------------------------------
Fix a group $\Gamma$, a finite generating set $S$, and a probability measure $\mu$ on $\Gamma$. Let $v_{\mu}$ be the drift corresponding to $\mu$ with respect to the word metric induced by the generating set. A natural question asks whether $v_{\mu}$ (and the asymptotic entropy) depends continuously on $\mu$. Continuity of the rate of escape (and the asymptotic entropy) is known on hyperbolic groups under the more general condition of having a finite first moment, see Kaimanovich and Erschler [@EK][^2]. Analyticity of the rate of escape and of the asymptotic entropy on free groups was proven by Ledrappier in [@Le:10]. More recently, Ledrappier [@Le:11] proves Lipschitz continuity for the rate of escape and asymptotic entropy for random walks on Gromov hyperbolic groups.
Moreover, analyticity of the rate of escape also follows in certain cases where explicit formulæ for the rate of escape are known, see Mairesse and Mathéus [@MM:07b] and Gilch, [@G:07] and [@G:08]. Mairesse and Mathéus [@MM:07] show that the rate of escape for some random walks on the Braid group $B_{3}=\langle a,b | aba = bab\rangle$ is continuous but not(!) differentiable. Finally, we refer to the recent survey of Gilch and Ledrappier [@GL] on results on the regularity of drift and entropy of random walks on groups.
The central limit theorem, Theorem \[thm:CLT\_planar\], provides formulæ for the drift $v$ and asymptotic variance $\sigma^{2}$ in terms of renewal times and hence offers a new approach in order to study analyticity. Moreover, this approach allows to consider random walks with infinite support.
Fix a driving measure $\nu$ of a random walk with exponential moments, *i.e.*, ${\mathbb{E}}[\exp(\lambda d(X_{1},e))]<\infty$ for some $\lambda>0$. Let $B$ be a finite subset of the support of $\nu$, *i.e.,* $B \subseteq supp(\nu)$. Let $\Omega_{\nu} (B)$ be the set of probability measures that give positive weight to all elements of $B$ and coincide with $\nu$ outside $B$. The set $\Omega_{\nu} (B)$ can be identified with an open bounded convex subset in ${\mathbb{R}}^{|B|-1}$. For each $\mu\in \Omega_{\nu}(B)$ we define the functions $v_{\mu}$ and $\sigma_{\mu}$ as the rate of escape and the asymptotic variance for the random walk with law $\mu$.
\[thm:analytic\] Let $\Gamma$ be a surface group with standard generating set $S$ and let $\nu$ be a driving measure with exponential moments whose support contains the generating set $S$. Then, for all $B$ such that $ B \subseteq supp(\nu)$ the functions $\mu\mapsto v_{\mu}$ and $\mu\mapsto \sigma_{\mu}$ are real analytic on $\Omega_{\nu}(B)$.
Notation and Preliminaries {#sec:not}
==========================
Cone types, geodesic automata of hyperbolic groups
--------------------------------------------------
Let $\Gamma$ be a finitely generated group and let $S$ be a symmetric and finite generating set. For sake of brevity we speak just of the group $(\Gamma,S)$ instead of the group $\Gamma$ together with a finite generating set $S$. The Cayley graph $X$ associated with $S$ is the graph whose vertex set is the set of all group elements and whose edge set consists of all pairs $({\gamma},{\gamma}')\in\Gamma\times\Gamma$ such that ${\gamma}^{-1}{\gamma}'\in S$. Endowing $X$ with the length metric which makes each edge isometric to the segment $[0,1]$ defines the [*word metric $d(\cdot,\cdot)$ associated with $S$*]{}. This metric turns $X$ into a geodesic proper metric space on which $\Gamma$ acts geometrically by left-translation.
Let $B_{n}(x)$ be the ball of radius $n$ around $x$; set for brevity $B_{n}=B_{n}(e)$. The neighborhood relation is written as $\sim$, *i.e.*, $x\sim y$ if $d(x,y)=1$, or equivalently $x^{-1}y\in S$. A path is a sequence of adjacent vertices in $X$ and is denoted by $\langle \cdot \rangle$. Let ${\mathcal{A}}=(V_{{\mathcal{A}}}, E_{{\mathcal{A}}}, s_*)$ be a finite directed graph with distinguished vertex $s_*$ together with a labeling $\alpha: E_{{\mathcal{A}}} \to S$ of the edges. Vertices and edges of ${\mathcal{A}}$ will be denoted using bold fonts. The vertex set $V_{{\mathcal{A}}}$ will often be identified with ${\mathcal{A}}$, *i.e.*, ${\textbf{x}}\in {\mathcal{A}}$ means a vertex ${\textbf{x}}\in V_{{\mathcal{A}}}$.
Denote the set $${\mathcal{P}}:=\{ \mbox{finite paths in } {\mathcal{A}}\mbox{ starting in } s_*\}.$$ For $m\in{\mathbb{N}}\cup\{\infty\}$, each path $\gamma=\langle {\textbf{x}}_{1},\ldots, {\textbf{x}}_{m}\rangle \in {\mathcal{P}}$ gives rise to a path in $\Gamma$ starting from $e$. Denote by ${\textbf{e}}_{1}$ the edge between $s_{*}$ and ${\textbf{x}}_{1}$ and by ${\textbf{e}}_{i}$ the edge between ${\textbf{x}}_{i-1}$ and ${\textbf{x}}_{i}$ for $i\geq 2$. The path corresponding to $\gamma$ is then defined by $$\alpha(\gamma) =\langle e, \alpha({\textbf{e}}_{1}),\alpha({\textbf{e}}_{1})\alpha({\textbf{e}}_{2}),\ldots, \prod_{i=1}^{m} \alpha({\textbf{e}}_{i})\rangle.$$
An *automatic structure* for a group $(\Gamma,S)$ is given by a finite state automaton ${\mathcal{A}}$ and a labeling $\alpha$ which satisfy the following properties:
- no edge in $E_{{\mathcal{A}}}$ ends at $s_*$,
- every vertex $v\in V_{{\mathcal{A}}}$ is accessible from $s_*$,
- for every path $\gamma\in {\mathcal{P}}$, the path $\alpha(\gamma)$ is a geodesic path in $\Gamma$,
- the mapping $\alpha^{*}$ from ${\mathcal{P}}$ to $\Gamma$ which associates the endpoint of the geodesic is surjective.
We talk of a *strongly automatic structure* if $\alpha^{*}$ defines a bijection between ${\mathcal{P}}$ and $\Gamma$.
For hyperbolic groups, the existence of a (strongly) automatic is due to Cannon and based on the definition of cones for any choice of generating set. The *cone* after Cannon of a group element $x$ is defined as $$C(x):=\{ xy:~y\in \Gamma,~d(e,xy)=d(e,x)+d(x,xy)\}.$$ We say $x$ is the *root* of $C(x)$. The *cone type* is defined as $$T(x):=\{ y\in \Gamma:~d(e,xy)=d(e,x)+d(x,xy)\} = x^{-1}C(x).$$
Cannon’s fundamental result, see *e.g.,* [@ECHLPT], is that a hyperbolic group has only finitely many cone types. Furthermore, we may thus associate a directed graph ${\mathcal{A}}_C=(V_{{\mathcal{A}}}, E_{{\mathcal{A}}}, s_*)$ with distinguished vertex $s_*$ together with a labeling $\alpha: E_{{\mathcal{A}}} \to S$ of the edges as follows. The set of vertices $V_{{\mathcal{A}}}$ is the set of cone types, and $s_*=T(e)$ is the cone type of the neutral element; there is a directed edge ${\textbf{e}}=(T_1,T_2)$ labeled by $s$ ($\alpha({\textbf{e}})=s$) between two cone types if there is an element $x\in\Gamma$ such that $T(x)= T_1$, $T(xs)=T_2$ and $s\in T(x)$. This structure $({\mathcal{A}}_C,\alpha)$ is by definition the [*Cannon automaton*]{} of $(\Gamma,S)$. We may obtain a strongly automatic structure from ${\mathcal{A}}_C$ by choosing a lexicographic ordering of the cone types; see [@ECHLPT] for details.
Furthermore, any strongly automatic structure ${\mathcal{A}}$ defines *cones* $C_{\mathcal{A}}$ and *cone types* $T_{{\mathcal{A}}}$ as follows. Given $x\in\Gamma$ and a path $\gamma_x\subset{\mathcal{A}}$ representing $x$, we let $C_\mathcal{A}(x)$ denote the set of all points of $\Gamma$ which are represented by paths with $\gamma_x$ as prefix. The [cone]{} $C_\mathcal{A}(x)$ is well defined since its construction does not depend on the choice of the representing path $\gamma_{x}$. We say that $x$ is the *root* of the cone $C_\mathcal{A}(x)$. Moreover, we can define [cone types]{} as $T_\mathcal{A}(x)=x^{-1}C_\mathcal{A}(x)$ which has the neutral element $e$ as root.
A vertex ${\textbf{y}}$ (or cone type) is *accessible* from ${\textbf{x}}$ if there is a path from ${\textbf{x}}$ to ${\textbf{y}}$. In this case we write ${\textbf{x}}\to{\textbf{y}}$. A vertex ${\textbf{x}}\in {\mathcal{A}}$ is *recurrent* if ${\textbf{x}}\to{\textbf{x}},$ otherwise it is called *transient*. The set of recurrent vertices $\mathcal{R}$ induces a subgraph ${\mathcal{A}}_{\mathcal{R}}$ of ${\mathcal{A}}$, *i.e.,* the graph whose vertex set equals to $\mathcal{R}$ and two vertices ${\textbf{x}}$ and ${\textbf{y}}$ are joint by an edge if only if they are neighbors in ${\mathcal{A}}$. By extension and abuse of standard notation, we will say $x\in\Gamma$ is *recurrent* if its cone type is recurrent. Recall that a (directed) graph is *strongly connected* if every vertex is reachable from any other vertex by following the directions.
\[ass:1\] There exists an automatic structure ${\mathcal{A}}$ associated to $S$ such that the subgraph ${\mathcal{A}}_{\mathcal{R}}$ is strongly connected.
This assumption is verified for non-exceptional Fuchsian groups with particular generating sets as shown is [@Se:82]. In particular, surface groups with standard generating sets satisfy Assumption \[ass:1\]. Not astonishingly, it also holds for groups with infinitely many ends for a suitable choice of generators. In fact this is a consequence of Stalling’s splitting theorem; any finitely generated group $\Gamma$ has more than one end if and only if the group splits as an amalgamated free product or an HNN-extension over a finite subgroup of $\Gamma$.
In the sequel we need the following definitions. Let us say a cone type ${\textbf{T}}$ is [*large*]{} if it is a neighborhood in $\Gamma\cup \partial\Gamma$ of a boundary point of $\partial \Gamma$. Any cone type containing a large cone type is again large. Moreover, we have the following fact.
\[lem:intnonempty\] Let $(\Gamma,S)$ be a non-elementary hyperbolic group and $\mathcal{A}$ an automatic structure satisfying Assumption \[ass:1\]. If there exists at least one large recurrent cone type, then all recurrent cone types are large.
We shall say that a cone type ${\textbf{T}}$ is [*ubiquitous*]{} if there exists some $R$ such that any ball $B_R(x)$ in $X$ contains a vertex $y$ with $T(y)={\textbf{T}}$. A ubiquitous cone type is recurrent, and under Assumption \[ass:1\] every recurrent cone type is ubiquitous. We define the (inner) boundary of a cone $C_\mathcal{A}(x)$ as $$\partial_{\Gamma} C_\mathcal{A}(x):=\{y\in C_\mathcal{A}(x):~ \exists z\in \Gamma\setminus C_\mathcal{A}(x)~\mbox{such that}~z\sim y\}$$ and $\partial_{\infty} C_\mathcal{A}(x)$ as the closure of $C_\mathcal{A}(x)$ at infinity, *i.e.*, in the Gromov hyperbolic compactification. Let $\gamma=\langle x_{1}, x_{2}, \ldots\rangle $ be a geodesic, we also denote by $\gamma$ the set $\{ x_{1}, x_{2}, \ldots\}$.
Use of constants
----------------
Constants in capital letters are chosen *sufficiently large* and constants in small letters stand for positive constants that are *sufficiently small.* Constants without any label, *e.g.,* $C$, are considered to be *local*, *i.e.*, their values may change from line to line. Labelled constants, *e.g.,* $C_{h}$, are defined *globally* and their values do not change as the paper goes along.
Random walks on groups
----------------------
Let $\Gamma$ be a finitely generated group and $S$ a symmetric and finite generating set. Let $\mu$ be a probability measure on $\Gamma$ with support generating $\Gamma$ as a semigroup. By definition, the random walk associated with $\mu$ is the Markov chain with state space $\Gamma$ and transition probabilities $p(x,y)=\mu(x^{-1}y)$ for $x,y\in \Gamma$. The measure $\mu$ is called the driving measure of the random walk. We shall use the notation ${\mathcal{T}}= \Gamma^{\mathbb{N}}$ for the path space and $Z_n$ for the position of the walk at time $n$ and $X_n:=Z_{n-1}^{-1}Z_n$ for its increment.
Let ${\mathbb{P}}_{x}$ denote the distribution of the random walk $(Z_{n})_{n\geq 0}$ when started at $x\in\Gamma$, and write ${\mathbb{P}}$ for ${\mathbb{P}}_{e}$. Observe that ${\mathbb{P}}_x$ is also the unique probability measure on ${\mathcal{T}}$ under which $Z_0=x$ and the $X_n$’s are i.i.d. random variables with law $\mu$. On the set of trajectories ${\mathcal{T}}$ we will also make use the shift map $\theta:{\mathcal{T}}\to{\mathcal{T}}$ defined by $\theta[ (z_n)_{n\ge 0}] = (z_{n+1})_{n\ge 0}$.
An elementary hyperbolic group is either finite or has two ends. Random walks on non-elementary hyperbolic groups are transient. As soon as the law $\mu$ has a finite first moment, *i.e.,* ${\mathbb{E}}[d(e,X_{1})]<\infty$, the random walk $Z_{n}$ converges ${\mathbb{P}}$-a.s. to some point $Z_{\infty}$ in the Gromov hyperbolic boundary $\partial\Gamma$, see Theorem 7.3 in [@Kai:00].
The harmonic measure $\nu$ is defined as the law of $Z_{\infty}$. In other words, it is the probability measure on $\partial \Gamma$ such that $\nu(A)={\mathbb{P}}[Z_{\infty}\in A]$ for $A\subset\partial\Gamma$. Since $\Gamma$ is non-elementary and the random walk is assumed to be irreducible we have that $\nu(\xi)=0$ for all $\xi\in\partial \Gamma$ and $\nu(O)>0$ for any open set $O\subset \partial \Gamma$.
\[lem:stayincone\] Let $(\Gamma, S)$ be a non-elementary hyperbolic group and $\mathcal{A}$ a corresponding automatic structure. Let $\mu$ be a driving measure whose support generates $\Gamma$ as a semigroup. If ${\textbf{T}}$ is a large cone type in $\mathcal{A}$ then for all $x$ with $T_\mathcal{A}(x)={\textbf{T}}$ we have that $${\mathbb{P}}_{x}[Z_{n}\in C_\mathcal{A}(x) \mbox{ for all but finitely many } n\geq 0]>0.$$
Let $O$ be an open subset of $\partial_{\infty}C_\mathcal{A}(x)$. On the event that $Z_{\infty}\in O$, at some moment, the random walk $(Z_n)$ enters $C_\mathcal{A}(x)$ and never leaves it afterwards.
Recall that our aim is to define a sequence of renewal times that corresponds to a sequence of cones in which the random walks stays forever. Therefore, we need the statement of Lemma \[lem:stayincone\] to hold for all $n\in{\mathbb{N}}$. The next assumption is made to ensure this; however we see in Section \[sec:bypass\] how to bypass this assumption.
\[ass:2\] The support of the driving measure $\mu$ contains the generating set $S$ of the group $\Gamma$.
\[lem:stayinconegen\] Let $(\Gamma, S)$ be a non-elementary hyperbolic group and $\mathcal{A}$ a corresponding automatic structure. Let $\mu$ be a driving measure that satisfies Assumption \[ass:2\]. Then, there exists some $c>0$ such that for all $x$ such that $T(x)$ is large we have that $${\mathbb{P}}_{x}[Z_{n}\in C_\mathcal{A}(x) \mbox{ for all } n\geq 0]>c.$$
The event $\{Z_{0}=x, Z_{n}\in C_\mathcal{A}(x) \mbox{ for all } n\geq 0\}$ consists only of trajectories that stay inside the cone $C_\mathcal{A}(x)$. Hence, invariance of the walk implies that for $x,y$ such that $T_\mathcal{A}(x)=T_\mathcal{A}(y)$ we have $${\mathbb{P}}_{x}[Z_{n}\in C_\mathcal{A}(x) \mbox{ for all } n\geq 0]={\mathbb{P}}_{y}[Z_{n}\in C_\mathcal{A}(y) \mbox{ for all } n\geq 0].$$ Since there is only a finite number of cone types it suffices to prove that the latter probability is positive for vertices of large cone types. Let $x$ be such that $T(x)$ is large. Now, Lemma \[lem:stayincone\] implies that there exists some $y\in C_\mathcal{A}(x)$ such that ${\mathbb{P}}_{y}[Z_{n}\in C_\mathcal{A}(x) \mbox{ for all } n\geq 0]>0.$ Assumption \[ass:2\] guarantees that there exists $n$ such that ${\mathbb{P}}_{x}[Z_{n}=y,~Z_{k}\in C_\mathcal{A}(x)~\forall k\leq n]>0$. The claim now follows by applying the law of total probability and the Markov property of the random walk.
Surface groups
--------------
In general, the geometry of cone types is hardly understood. In order to avoid artificial conditions, we will focus on hyperbolic surface groups. A surface group is the fundamental group of a closed and orientable surface of genus $2$ or more. The standard presentation for an (orientable) surface group of genus $g$ is $$\langle a_{1}^{\pm 1},b_{1}^{\pm 1},\ldots, a_{g}^{\pm 1}, b_{g}^{\pm 1} \mid \prod_{i=1}^{g} a_{i}b_{i}a_{i}^{-1}b_{i}^{-1}\rangle.$$ Its Cayley $2$-complex is the $2$-complex such that the one-skeleton is given by the Cayley graph $X$, and the $2$-cells are bounded by loops in $X$ labeled by the relations. A surface group with standard presentation is planar, *i.e.,* its 2-complex is homeomorphic to the hyperbolic disc. A strongly automatic structure of a surface group can be given explicitly, *e.g.,* see [@GL:11 Section 5.2], and in particular there exists an automatic structure associated to the standard generating set that satisfies Assumption \[ass:1\]. The planarity of the Cayley $2$-complex allows moreover a neat description of the cones and their boundaries.
\[lem:coneshape\] Let $(\Gamma,S)$ be surface group with standard generating set. Then, there exists an automatic structure $\mathcal{A}$ that satisfies Assumption \[ass:1\]. Moreover, any cone type of ${\mathcal{A}}$ is large and is bounded by two geodesic rays starting from the neutral element.
We refer to [@GL:11 Section 5.2] for the fact that there exists an automatic structure that satisfies Assumption \[ass:1\]. Let $x\in \Gamma\setminus\{e\}$ and $C_\mathcal{A}(x)$ its cone defined by the automaton $\mathcal{A}$. Since the 2-complex is homeomorphic to the plane, it can be endowed with an orientation. Let $r_1,r_2:{\mathbb{R}}_+\to X$ be two infinite rays going through $x$ and which coincide up to $x$; let $c_1,c_2$ be the geodesic rays extracted from $r_1,r_2$ starting at $x$. Let $V$ be a component of $X\setminus(c_1\cup c_2)$ which does not contain $e$. Let us prove that $V$ is contained in $C_\mathcal{A}(x)$: let $y\in V$, and let us consider a segment $c_y$ joining $e$ to $y$. Since the 2-complex is planar, Jordan’s theorem implies that $c_y$ has to intersect $\partial V$ at a point $z$, hence $c_1$ or $c_2$ beyond $x$. Let us assume that it intersects $c_1$. Since $c_1$ is geodesic, we may replace the portion of $c_y$ before $z$ by $c_1$: it follows that the concatenation of $c_1$ up to $z$ and $c_y$ from $z$ to $y$ is geodesic; this implies that $y\in C_\mathcal{A}(x)$.
By Arzela-Ascoli’s theorem and the planarity of the graph, we may find two rays $c_{\ell}$ and $c_{r}$ going through $x$ such that $C_\mathcal{A}(x)$ is the union of those rays with all the components of their complement which do not contain $e$.
Renewal structure and applications {#sec:ren}
==================================
The construction
----------------
In this section we assume that $\Gamma$ is a non-elementary hyperbolic group endowed with a finite generating set $S$ such that there is a ubiquitous large cone type ${\textbf{T}}$. The aim of the following part is to construct a sequence of renewal times $R_{n}$ on which the random walk visits the root of a cone of type ${\textbf{T}}$ that it will never leave again.
The main idea behind the construction is quite natural and is first sketched informally; we also refer to Figures \[fig:first\] and \[fig:nth\] for an illustration. The trajectory of the walk will be decomposed into parts of two different types: the “[**e**]{}xploring” and the “[**d**]{}eciding” parts. Though, let us fix a large ubiquitous cone type ${\textbf{T}}$ and start a random walk in the origin $e$. After some random time $E$ the random walk will visit a vertex of type ${\textbf{T}}$. At this point the walk may stay in this cone forever or may leave it after some finite random time $D$. In the first case, we set $E$ to be the first renewal time. In the second case, the random walk after having left the cone at time $D$ will explore the underlying group in order to find another vertex of type $\bf T$. This procedure continues until the walk decides to stay eventually in one cone of type ${\textbf{T}}$ and hence the first renewal point and renewal time are fixed. The construction of the subsequent renewal points is analogous. Eventually, this procedure decomposes the trajectory into aligned and independently distributed pieces. However, the distribution of the first piece differs from the distributions of the subsequent ones, since the law of these latter pieces is given by the law conditioned to stay in the cones of the previous renewal points.
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(0,-1.764145)(9.502,2.8464108) (3.96,-2.69)[3.0]{}[18.217094]{}[75.25644]{} (3.0,2.67)[3.0]{}[-54.344673]{}[2.7591076]{} (6.62,-2.67)[3.0]{}[18.217094]{}[75.25644]{} (5.66,2.69)[3.0]{}[-54.344673]{}[2.7591076]{} (4.76,0.23) (7.42,0.25) (4.91,-1.045)[$Z_{R_{n-1}}$]{} (4.3,-0.81)(4.7,0.11) (7.5,-0.83)(7.42,0.07) (7.78,-1.065)[$Z_{R_n}$]{}
We now present all the details of the construction. Let $$E=\inf\{n\geq 0\,;\,T_{\mathcal{A}}(Z_{n})={\textbf{T}}\}$$ the first time the random walk visits a vertex of type ${\textbf{T}}$. The random variable $E$ is a stopping time and a priori takes values in ${\mathbb{N}}\cup\{\infty\}$. However, since ${\textbf{T}}$ is ubiquitous it can be shown that $E$ is almost surely finite, see proof of Lemma \[lem:R\_1finite\]. Recall that $\theta$ is the canonical shift on the space of trajectories $\mathcal{T}$ and thus $$E\circ \theta^{k}=\inf\{n\geq 0\,;\,T_{\mathcal{A}}(Z_{n+k})={\textbf{T}}\}.$$ Define the stopping time $$D=\inf\{n\geq 1\,;\,Z_{n}\notin C_{\mathcal{A}}(Z_{0})\}$$ and consider its shifted versions $$D\circ \theta^{k}=\inf\{n\geq 1\,;\,Z_{n+k}\notin C_{\mathcal{A}}(Z_{k})\}.$$ Observe that the random variables $D\circ \theta^{k}$ might be finite or infinite, see Lemma \[lem:stayinconegen\]. We define $$R_{0}=0.$$ In order to define the subsequent renewal times we introduce a sequence of stopping times $(S_{k}^{(1)})_{k\geq 0}$: $$S_{0}^{(1)}=E~\mbox{and inductively}~S_{k+1}^{(1)}=S_{k}^{(1)}+ D_{k}^{(1)} + E_{k}^{(1)}\leq \infty,$$ where $$D_{k}^{(1)} = D\circ \theta ^{{S^{(1)}_{k}}}\mbox{ and } E_{k}^{(1)}= E\circ\theta^{{S^{(1)}_{k}}+D_{k}^{(1)} }.$$ Letting $$\label{eq:K}
K^{(1)}=\inf\{k\geq 0\,;\,S_{k}^{(1)}<\infty, S_{k+1}^{(1)}=\infty\}\leq\infty$$ we define the first renewal time $$R_{1}=S_{K^{(1)}}^{(1)}\leq \infty.$$ Equivalently, this renewal time can be written as $$R_{1}=\inf\{{k\geq 0}\,;\,Z_{i}\in C_{\mathcal{A}}(Z_{k})~\forall i\geq k,~T_{\mathcal{A}}(Z_{k})={\textbf{T}}\}.$$ In words, $R_{1}$ is the first time the random walk hits the root of a cone of type ${\textbf{T}}$ that it never leaves afterwards. Note that $R_{1}$ is not a stopping time.
Inductively, we define the $n$th renewal time. Provided that $R_{n-1}<\infty$ we define as above: $$S^{(n)}_{0}=R_{n-1}+ 1+ E\circ \theta^{R_{n-1}+1}~\mbox{and inductively}~S_{k+1}^{(n)}=S_{k}^{(n)}+ D_{k}^{(n)} + E_{k}^{(n)}\leq \infty,$$ where $$D_{k}^{(n)} = D\circ \theta ^{{S^{(n)}_{k}}}\mbox{ and } E_{k}^{(n)}= E\circ\theta^{{S^{(n)}_{k}}+D_{k}^{(n)} }.$$ Letting $K^{(n)}=\inf\{k\geq 0\,;\,S^{(n)}_{k}<\infty, S^{(n)}_{k+1}=\infty\}\leq\infty$ we can define the $n$th renewal time $$R_{n}=S^{(n)}_{K^{(1)}}\leq \infty,$$ which is the same as $$R_{n}=\inf\{k> R_{n-1}\,;\,Z_{i}\in C_{\mathcal{A}}(Z_{k})~\forall i\geq k,~T_{\mathcal{A}}(Z_{k})={\textbf{T}})\}.$$
Without any further assumption, we have the following basic result.
\[thm:reg\] Let $(\Gamma, S)$ be a non-elementary hyperbolic group and $\mathcal{A}$ a corresponding automatic structure with a large ubiquitous cone type ${\textbf{T}}$. Let $\mu$ be a driving measure satisfying Assumption \[ass:2\]. Then, the renewal times $R_{n}$ are almost surely finite and $d(e,Z_{R_{n}})=\sum_{i=1}^{n} d(Z_{R_{i-1}}, Z_{R_{i}})$, where $d((Z_{R_{i-1}}, Z_{R_{i}}))_{i\geq 2}$ are i.i.d. random variables.
In order to prove Theorem \[thm:reg\] we first prove two Lemmata.
\[lem:R\_1finite\] Under the assumption of Theorem \[thm:reg\] the random variable $R_1$ is almost surely finite under ${\mathbb{P}}_{x}$ for any $x\in\Gamma$.
By the irreducibility of the random walk and the ubiquity of ${\textbf{T}}$ there exist $c>0$ and $m\in{\mathbb{N}}$ such that $${\mathbb{P}}_{y}[\exists n\in [0, m-1]:~T(Z_{n})={\textbf{T}}]>c>0 \mbox{ for all } y\in \Gamma.$$ Hence, by the Markov property we have $$\label{eq:Eexpmom}
{\mathbb{P}}_{y}[E\geq N m] \leq (1-c)^{N},$$ and hence that ${\mathbb{P}}_{y}[E<\infty]=1$ for all $y\in\Gamma$. Fix $x\in\Gamma$. Since we are dealing with stopping times, for any $y\in\Gamma$, the law of $E_k^{(1)}$ conditioned on $\{S_{k}^{(1)}<\infty, D_{k}^{(1)}<\infty, Z_{S_{k}^{(1)} + D_{k}^{(1)}}=y\}$ is the law of $E$ under ${\mathbb{P}}_{y}$. Therefore $${\mathbb{P}}_{x}[E_{k}^{(1)}<\infty\mid S_{k}^{(1)}<\infty, D_{k}^{(1)}<\infty]=1.$$ Since the law of $D_{k}^{(1)}$ conditioned on $\{S_{k}^{(1)}<\infty, Z_{S_{k}^{(1)}}\}$ is the law of $D$ under ${\mathbb{P}}_{y}$ for $y$ such that $T(y)={\textbf{T}}$ we obtain using Lemma \[lem:stayinconegen\] that $${\mathbb{P}}_{x}[ D_{k}^{(1)}<\infty\mid S_{k}^{(1)}<\infty]\leq 1-c.$$ Therefore, $${\mathbb{P}}_{x}[S_{k+1}^{(1)}<\infty \mid S_{k}^{(1)}<\infty ] = {\mathbb{P}}_{x}[D_{k}^{(1)}<\infty \mid S_{k}^{(1)}<\infty ] \leq 1-c.$$ Hence, by the strong Markov property we obtain for all $N\in{\mathbb{N}}$ $$\label{eq:lem:Rfinite2}
{\mathbb{P}}_{x}[S^{(1)}_{k}<\infty~\forall k\leq N] \leq (1-c)^{N}.$$ and hence ${\mathbb{P}}_{x}[R_{1}=\infty]={\mathbb{P}}_{x}[K^{(1)}=\infty]=0$.
A main feature of the definition of the cones is the following property: for any $x,y\in\Gamma$ with same cone type and any $A\subset \mathcal{T}$ we have that $${\mathbb{P}}_x[ (x^{-1}Z_{n})_{n\in {\mathbb{N}}}\in A \mid D=\infty]={\mathbb{P}}_{y} [(y^{-1}Z_{n})_{n\in {\mathbb{N}}}\in A \mid D=\infty].$$ Therefore, we may introduce a new probability measure: for $A\subset \mathcal{T}$ let $${\mathbb{Q}}_{{\textbf{T}}}[ (Z_{n})_{n\in{\mathbb{N}}}\in A]= {\mathbb{P}}_{x}[ (x^{-1}Z_{n})_{n\in {\mathbb{N}}}\in A \mid D=\infty],$$ where $x$ is of cone type ${\textbf{T}}$. We write ${\mathbb{E}}_{{\textbf{T}}}$ for the corresponding expectation.
Define the $\sigma$-algebras $${\mathcal{G}}_{n}=\sigma(R_{1},\ldots, R_{n}, Z_{0}, \ldots, Z_{R_{n}})\quad n\geq 1.$$ Although the $R_{n}$ are not stopping times we have the following “Markov property”.
\[lem:markov\] Let ${\textbf{T}}$ be some ubiquitous large cone type. Then, for all $n\geq 1$ we have that $R_{n}$ is almost surely finite and for any measurable set $A\subset{\mathcal{T}}$ and any $y\in\Gamma$ $${\mathbb{P}}_{y}[ (Z_{R_{n}}^{-1}Z_{R_{n}+k})_{k\in{\mathbb{N}}}\in A\mid {\mathcal{G}}_{n}] = {\mathbb{Q}}_{{\textbf{T}}} [ (Z_{k})_{k\in{\mathbb{N}}}\in A].$$
Without loss of generality let us assume that $y=e$. Besides the finiteness of the $R_{n}$ we have to check the definition of the conditional expectation: for all bounded ${\mathcal{G}}_n$-measurable function $H$ and all measurable set $A\subset {\mathcal{T}}$ it holds that $${\mathbb{E}}[H {\textbf{1}}_{(Z_{R_n}^{-1}Z_{R_n+k})_{k}\in A} ]= {\mathbb{Q}}_{{\textbf{T}}}[(Z_k)_k\in A]\cdot {\mathbb{E}}[H].$$
We will proceed by induction. So let us consider the case $n=1$. Lemma \[lem:R\_1finite\] implies that ${\mathbb{P}}[R_{1}<\infty]=1$. Now, we observe that $\{R_1= S_l\}=\{S_l <\infty\}\cap \{D\circ{\theta^{S_l}}=\infty\}$. Let $l\in{\mathbb{N}}$ and $x\in\Gamma$. Then, there exists (due to ${\mathcal{G}}_{1}$-measurability) some random variable $H_{x,l}$ measurable with respect to $\{Z_{i}\}_{i\leq S_{l}}, S_{l}\}$ such that $H=H_{x,l}$ on the event $\{R_{1}=S_{l}, Z_{S_{l}}=x\}$. Therefore, we may write $$\begin{aligned}
{\mathbb{E}}[H {\textbf{1}}_{(Z_{R_1}^{-1}Z_{R_1+k})_k\in A}]
&=& \sum_{l\geq 1} \sum_{x\in\Gamma} {\mathbb{E}}[{\textbf{1}}_{S_{l}<\infty} {\textbf{1}}_{D\circ \theta^{S_{l}}=\infty} {\textbf{1}}_{Z_{S_{l}}=x} {\textbf{1}}_{{(Z_{S_l}^{-1}Z_{S_l+k})_k\in A}} H_{x,l} ] \cr
&=& \sum_{l\geq 1}\sum_{x\in\Gamma \atop T(x)={\textbf{T}}} {\mathbb{E}}[{\textbf{1}}_{S_{l}<\infty} {\textbf{1}}_{Z_{S_{l}}=x} H_{x,l} ] {\mathbb{E}}_{x}[{\textbf{1}}_{D=\infty} {\textbf{1}}_{(x^{-1}Z_k)_k \in A}] \cr
&=& {\mathbb{Q}}_{{\textbf{T}}}[(Z_k)_k\in A] \sum_{l\geq 1} \sum_{x\in\Gamma \atop T(x)={\textbf{T}}} {\mathbb{E}}[{\textbf{1}}_{S_{l}<\infty} {\textbf{1}}_{Z_{S_{l}}=x} H_{x,l} ] {\mathbb{P}}_x[D=\infty]\,.\end{aligned}$$ In the second equality we have used the strong Markov property since $S_l$ is a stopping time, and in the third we have applied the definition of ${\mathbb{Q}}_{\textbf{T}}$ as a conditional probability. Substituting in the above a trivial $A$ we have $${\mathbb{E}}[H] = \sum_{l\geq 1} \sum_{x\in\Gamma \atop T(x)={\textbf{T}}} {\mathbb{E}}[{\textbf{1}}_{S_{l}<\infty} {\textbf{1}}_{Z_{S_{l}}=x} H_{x,l} ] {\mathbb{P}}_x[D=\infty]$$ which shows that $$\begin{aligned}
{\mathbb{E}}[H {\textbf{1}}_{(Z_{R_1}^{-1}Z_{R_1+k})_k\in A}]
& = & {\mathbb{Q}}_{{\textbf{T}}}[(Z_k)_k\in A]\cdot {\mathbb{E}}[H ].
\end{aligned}$$ This concludes the proof for $n=1$ and implies the finiteness of $R_{2}$ since now ${\mathbb{P}}[R_{2}<\infty]={\mathbb{Q}}_{{\textbf{T}}}[R_{1}<\infty]$ and due to Lemma \[lem:R\_1finite\] the latter probability is equal to one.
The induction proceeds similarly. If $H$ is ${\mathcal{G}}_n$-measurable, then, there exists some random variable $H_{x,l}$ measurable with respect to ${\mathcal{G}}_{n-1}$ such that $H=H_{x,l}$ on the event $\{R_{n}=S^{(n)}_{l}, Z_{S^{(n)}_{l}}=x\}$, to which we may apply the induction hypotheses. The computations are left to the reader.
(Theorem \[thm:reg\]) By Lemma \[lem:markov\] the renewal times $R_{n}$ are almost surely finite and, by construction, all renewal points $Z_{R_{n}}$ lie on one geodesic. Hence, $d(Z_{R_{n}},e)=\sum_{i=1}^{n} d(Z_{R_{i}}, Z_{R_{i-1}})$. Eventually, Lemma \[lem:markov\] implies that $(d(Z_{R_{i}}, Z_{R_{i-1}}))_{i\geq 2}$ all have the same distribution and are independent.
The renewal structure yields an alternative construction of the law of the walk. Let $Q_0$ be the law of $(Z_n;n\leq R_{1})$ under ${\mathbb{P}}_x$ and let $Q$ be the law of $(Z_{R_1}^{-1}Z_{(R_1+n)}; n\leq R_2)$. We can obtain the measure ${\mathbb{P}}_x$ by choosing a path according to $Q_0$ and concatenate it with an i.i.d. sequence sampled from $Q$.
Surface groups
==============
While the results in the previous sections are valid for random walks with finite first moments, we need some additional assumptions in order to prove a central limit theorem and the analyticity of the rate of escape and of the asymptotic variance.
We say a real valued random variable $Y$ has exponential moments if ${\mathbb{E}}[\exp(\lambda Y)]<\infty$ for some $\lambda>0$, or equivalently, if there exist positive constants $C$ and $c<1$ such that ${\mathbb{P}}[Y=n]\leq C c^{n} $ for all $n\in {\mathbb{N}}$. The random variables appearing in Theorem \[thm:reg\] do in general not have exponential moments. However, this is the case under the following assumption.
\[ass:3\] The driving measure $\mu$ has exponential moments, *i.e.*, ${\mathbb{E}}[\exp(\lambda_{\mu} d(X_{1},e))]<\infty$ for some $\lambda_{\mu}>0$.
In the sequel of this section we will only consider surface groups with standard generating sets. Lemma \[lem:coneshape\] assures the existence of an automatic structure $\mathcal{A}$ with a ubiquitous large cone type ${\textbf{T}}$. The latter allows the construction of the renewal points and times, see Section \[sec:ren\], and this construction depends on the choices of $\mathcal{A}$ and ${\textbf{T}}$. However, in order to facilitate the reading, we formulate the statements without specifying the structure $\mathcal{A}$ nor the type ${\textbf{T}}$.
\[lem:momentbounds\] Let $(\Gamma,S)$ be a surface group with standard generating set. Under Assumption \[ass:3\] the renewal times $R_{1}$ and $(R_{i+1}-R_{i})$ for $i\geq 1$ have exponential moments. The same holds true for $d(Z_{R_{1}},e)$ and $d(Z_{R_{i+1}},Z_{R_{i}})$ for $i \geq 1$.
Let us first prove that $R_{1}$ has exponential moments. In Equation (\[eq:Eexpmom\]) we have established that $E$ has uniform exponential moments: there are some constants $\lambda_E>0$ and $C_E<\infty$ such that for all $x\in\Gamma$ we have ${\mathbb{E}}_x[\exp(\lambda_E E)]\le C_E$.
In order to control the moments of $D$ we make use of the non-amenability and the planarity of $\Gamma$. Let us recall some well-known facts, *e.g.,* see [@woess]. Irreducibility of the walk implies the existence of the spectral radius $$\rho(\mu):=\limsup_{n\to\infty} \left({\mathbb{P}}_{x}(Z_{n}=y)\right)^{1/n},~x,y\in \Gamma.$$ Kesten’s amenability criterion implies that $\rho(\mu)<1$. Moreover, there exists $C>0$ such that, for all $x,y\in\Gamma$ and all $n\ge 1$, $$\label{eq:rho}
{\mathbb{P}}_x[Z_n=y] \le C \rho(\mu)^n.$$
We proceed with the tails of ${\mathbb{P}}_{x}[D=n]$ for $x$ such that $T(x)={\textbf{T}}$. Let $\delta>0$ to be chosen later, then $$\label{eq:lem:momentbound:1}
{\mathbb{P}}_{x}[D=n+1]\leq {\mathbb{P}}_{x}[d(Z_{n},x)\leq \delta n, D=n+1]+ {\mathbb{P}}_{x}[d(Z_{n},x)\geq \delta n, D=n+1].$$ The second summand is controlled by using the Chebyshev inequality: $$\begin{aligned}
{\mathbb{P}}_{x}[d(Z_{n},x) \geq \delta n, D=n+1] & \leq & {\mathbb{P}}\left[\sum_{i=1}^{n} d(X_{i},e)\geq \delta n\right]\\
& \leq & \frac{{\mathbb{E}}[\exp(\lambda_{\mu} \sum_{i=1}^{n} d(X_{i},e))] }{\exp(\lambda_{\mu}\delta n)}\\
& = & \frac{{\mathbb{E}}[\exp(\lambda_{\mu} d(X_{1},e))] ^{n}}{\exp(\lambda_{\mu}\delta n)}.\end{aligned}$$ Since $\mu$ has exponential moments we can choose $\delta$ sufficiently large such that the latter term converges exponentially fast to $0$.
In order to treat the first summand of Equation (\[eq:lem:momentbound:1\]) we make use of Lemma \[lem:coneshape\]. Let $\gamma$ be a geodesic. We define the $m$-tube of $\gamma$ as $\gamma^{(m)}:=\bigcup_{x\in \gamma} B(x,m)$. Let $\gamma_{\ell}, \gamma_{r}$ be the two geodesics such that $\partial_{\Gamma} C(x) = \gamma_{\ell}\cup \gamma_{r}$, then we define the $m$-tube of $\partial_{\Gamma} C(x)$ as $\partial^{(m)} C:= \gamma_{\ell}^{(m)}\cup \gamma_{r}^{(m)}$. Now we obtain, using mainly Equation (\[eq:rho\]) and the linear growth of the boundary of cones, that for all ${\varepsilon}>0$: $$\begin{aligned}
{\mathbb{P}}_{x}\left[d(Z_{n},x) \leq \delta n, D=n+1\right]
&\leq & {\mathbb{P}}_{x}[ d(Z_{n},x) \leq \delta n, Z_{n}\in \partial^{({\varepsilon}n)} C(x)] \cr
& &+ {\mathbb{P}}_{x}[ D=n+1, Z_{n}\notin \partial^{({\varepsilon}n)} C(x)] \cr
&\leq & C \rho(\mu)^{n} |\partial^{({\varepsilon}n)} C(x)\cap B(x,\delta n)| + {\mathbb{P}}[d(Z_{n+1},Z_{n})>{\varepsilon}n]\cr
&\leq& C \rho(\mu)^{n} (2\delta n) |S|^{{\varepsilon}n} + {\mathbb{P}}[d(X_{1},e)>{\varepsilon}n].
\end{aligned}$$ Choose eventually ${\varepsilon}>0$ sufficiently small so that $\rho(\mu) |S|^{{\varepsilon}}<1$. Since for ${\varepsilon}$ fixed the probability ${\mathbb{P}}[d(X_{1},e)>{\varepsilon}n]$ decays exponentially in $n$, there are some constants $\lambda_D>0$ and $C_D<\infty$ such that $$\label{eq:PzD}
{\mathbb{E}}_{x}[\exp(\lambda_D D) {\textbf{1}}_{\{D<\infty\}}] \le C_D\quad \forall x: T(x)={\textbf{T}}.$$
Now, recall that $$\label{eq:R1}
R_{1}= E+\sum_{k=1}^{K^{(1)}} (D_{k}^{(1)}+E_{k}^{(1)}),$$ where $K^{(1)}$ is the smallest time $k$ such that $D^{(i)}_{k+1}=\infty$.
Recall that $K^{(1)}$ has exponential moments, see (\[eq:lem:Rfinite2\]), so that there exist constants $\lambda_K>0$ and $C_K<\infty$ such that ${\mathbb{P}}[K^{(1)}=k] \le C_K \exp(-\lambda_{K}k)$. We can decompose $${\mathbb{P}}_x[n\leq D+E\circ\theta^D<\infty] \leq {\mathbb{P}}_{x}[n/2\leq D <\infty] + \sum_{k=1}^{n/2}{\mathbb{P}}_x[ E\circ\theta^k \geq n/2] .$$ Hence, we may find constants $C>0$, $\lambda>0$ such that $${\mathbb{E}}_x[\exp(\lambda(D+ E\circ\theta^D)){\textbf{1}}_{\{D <\infty\}}] \le C \quad \forall x: T(x)={\textbf{T}}.$$ Therefore, we may choose $\lambda_1$ small enough such that $${\mathbb{E}}_{x}[\exp(\lambda_1(D+ E\circ\theta^D)) | D<\infty] \le \exp(\lambda_{K} /2)\quad \forall x: T(x)={\textbf{T}}.$$ Eventually, using the strong Markov property, $$\begin{aligned}
{\mathbb{E}}[\exp(\lambda_{1} R_{1})] & = &\sum_{k=1}^{\infty}
{\mathbb{E}}\left[\exp\left(\lambda_{1}\left(E+\sum_{i=1}^{k} (D_{i}^{(1)}+E_{i}^{(1)})\right)\right)\mid K^{(1)}=k\right] {\mathbb{P}}[K^{(1)}=k]\cr
& \leq & C_K\sum_{k=1}^{\infty} {\mathbb{E}}[\exp{\lambda_{1} E}] (\exp(\lambda_{K}/2))^{k}\exp(-\lambda_K k)\cr
&\leq& C \sum_{k=1}^{\infty} \exp(-\lambda_{K}k/2) <\infty.\end{aligned}$$
The proof for $R_{i+1}-R_{i},$ $i\geq 1$, is analogous since the laws of the different $D_k^{(i+1)}$ are independent of $R_i$. Moreover, Equation (\[eq:Eexpmom\]) implies exponential moments for $E\circ \theta^{R_{i}}$ and $E_k^{(i+1)}$ as well.
We turn to the exponential moments of the distances between two successive renewal points. Let $\delta>0$ to be chosen later. Then, since $R_{1}$ has exponential moments, $$\begin{aligned}
{\mathbb{P}}[ d(Z_{R_{1}},e)\geq k] &\leq & {\mathbb{P}}[d(Z_{R_{1}},e)\geq k, R_{1}\geq k\delta] + {\mathbb{P}}[d(Z_{R_{1}},e)\geq k, R_{1}\leq k\delta]\cr
&\leq & C e^{-c k \delta} + {\mathbb{P}}\left[\sum_{i=1}^{k\delta} d(X_{i},e)\geq k\right]\end{aligned}$$ and hence, using again Chebyshev’s inequality, we see that for suitable $\delta$ the last term decays exponentially fast. The proof for $d(Z_{R_{i}}, Z_{R_{i+1}})$, $i\geq 1$, is in the same spirit: $$\begin{aligned}
{\mathbb{P}}[ d(Z_{R_{i+1}}, Z_{R_{i}})\geq k] &\leq & {\mathbb{P}}[ R_{i+1}-R_{i}\geq k\delta]
+ {\mathbb{P}}[d(Z_{R_{i+1}}, Z_{R_{i}})\geq k, R_{i+1}-R_{i}\leq k\delta]\cr
&\leq & C e^{-ck\delta} + {\mathbb{P}}[d(Z_{R_{i+1}}, Z_{R_{i}})\geq k, R_{i+1}-R_{i}\leq k\delta].\end{aligned}$$ Using Lemma \[lem:markov\] the last summand becomes $${\mathbb{P}}[d(Z_{R_{i+1}}, Z_{R_{i}})\geq k, R_{i+1}-R_{i}\leq k\delta] = {\mathbb{Q}}_{{\textbf{T}}}[d(Z_{R_1},Z_{0})\geq k, R_{1}\leq k\delta]$$ Once again, an application of the exponential Chebyshev inequality yields that $\delta$ can be chosen such that ${\mathbb{Q}}_{{\textbf{T}}}\left[\sum_{i=1}^{n} d(X_{i},x)\geq \delta n\right]$ decays exponentially fast and the claim follows as above.
\[rem:inftyends\] In the case of hyperbolic groups with infinitely many ends, it follows from Stalling’s splitting theorem that the boundaries of the cones are finite if we choose the generators accordingly. Hence the proof of Lemma \[lem:momentbounds\] applies to this setting.
\[cor:overshoot\] Set $$M_{k}=\sup\{d(Z_{n},Z_{R_{k}}), R_{k}\le n \leq R_{k+1}\}, ~k\geq 1,$$ and $$k(n)=\sup\{k:~R_{k}\le n\}\,.$$ Under the assumptions of Lemma \[lem:momentbounds\], $(M_{k})_{k\geq 1}$ is an i.i.d. sequence with exponential moments and $$\frac{n}{k(n)} \stackrel{a.s}{\longrightarrow} {\mathbb{E}}[R_{2}-R_{1}]<\infty.$$
The proof that $(M_k)$ are i.i.d. follows from Lemma \[lem:markov\], as in the proof of Theorem \[thm:reg\]. The fact that $M_k$ have exponential moments can either be seen as in Lemma \[lem:momentbounds\] or as follows. Let $\delta>0$ (to be chosen later). Then, by the law of total probability, for $m\in{\mathbb{N}}$ $$\begin{aligned}
{\mathbb{P}}[M_{k}\geq m] &\leq& {\mathbb{P}}[R_{k+1}-R_{k}\geq \delta m] + {\mathbb{P}}[\sup\{ d(Z_{R_{k}},Z_{n}), R_{k}\leq n\leq R_{k}+\delta m\}\geq m]\cr
& \leq & {\mathbb{P}}[R_{k+1}-R_{k}\geq \delta m] + {\mathbb{Q}}_{T}\left[ \sum_{i=1}^{\delta m} d(e,X_{i})\geq m\right].\end{aligned}$$ Since $R_{k+1}-R_{k}$ and $\mu$ have exponential moments, yet another application of Chebyshev’s inequality shows that we can choose $\delta$ sufficiently small such that ${\mathbb{P}}[M_{k}\geq m]$ decays exponentially fast to $0$. Concerning $k(n)$, we write $$\frac{n}{k(n)}= \frac{n}{R_{k(n)}}\frac{R_{k(n)}}{k(n)}\,.$$ The second factor tends a.s. to ${\mathbb{E}}[R_2-R_1]$ by the strong law of large numbers (since $k(n)$ tends to infinity). For the first factor we observe that $R_{k(n)}\le n \le R_{k(n)+1}$, hence $$\limsup_{n\to\infty} \frac{R_{k(n)}}{n}\le 1\,.$$ On the other hand, since $n\ge k(n)$ and $(R_{k(n)}-R_{k(n)+1})$ have finite moments, $$\lim_{n\to\infty} \frac{R_{k(n)}-R_{k(n)+1}}{n}=0\quad \hbox{a.s.}$$ and hence $$\liminf_{n\to\infty}\frac{R_{k(n)}}{n}\ge \liminf_{n\to\infty}\left( \frac{R_{k(n)}-R_{k(n)+1}}{n}\right)+ \frac{R_{k(n)+1}}{n}\ge 1\,.$$
Limit Theorems
--------------
The existence of the law of large numbers (LLN) is a direct consequence of Kingman’s subadditive ergodic theorem. Moreover, it was proven by Guivarc’h [@Gui:80] that for non-amenable graphs the speed is positive. We give a formula for the speed in terms of the renewal structure and recover the above results without using Kingman’s theorem for driving measures with exponential moments.
\[thm:LLN\_planar\] Let $(\Gamma,S)$ be a surface group with standard generating set and assume the driving measure $\mu$ to have exponential moments. Then, $$\label{eq:v}
\frac1n d(Z_{n},e) \stackrel{a.s}{\longrightarrow} v= \frac{{\mathbb{E}}[ d(Z_{ R_{2}}, Z_{ R_{1}})]}{{\mathbb{E}}[ R_{2}- R_{1}]}>0~\mbox{ as } n\to\infty.$$
The law of large numbers for i.i.d. sequences applied to $(R_{k+1}-R_k)_k$ and $(d(Z_{R_k}, Z_{R_{k+1}}))_k$, tells us that $$\frac{R_{k}}k \stackrel{a.s}{\longrightarrow} {\mathbb{E}}[R_{2}-R_{1}] \mbox{ and }
\frac{d(Z_{R_{k}},e)}{k} \stackrel{a.s}{\longrightarrow} {\mathbb{E}}[d(Z_{R_{2}},Z_{R_{1}})].$$ With $k(n)=\max\{k:~R_{k}\leq n\}$ we have $k(n)/n \to 1/{\mathbb{E}}[R_{2}-R_{1}]$ a.s. by Corollary \[cor:overshoot\]. Moreover, the latter also implies that $$\lim_{n\to\infty}\frac{d(Z_{n},e)-d(Z_{R_{k(n)}},e)}n \le \lim_{k\to\infty}\frac{M_k}k=0 \quad\hbox{a.s.}\,.$$ Hence $$\begin{aligned}
\frac{d(Z_{n},e)}n& =& \frac{d(Z_{n},e)-d(Z_{R_{k(n)}},e)}n + \frac{d(Z_{R_{k(n)}},e)}{k(n)} \frac{k(n)}{n}\cr
& \stackrel{a.s}{\longrightarrow} & 0 + \frac{{\mathbb{E}}[ d(Z_{ R_{2}}, Z_{ R_{1}})]}{{\mathbb{E}}[ R_{2}- R_{1}]}.\end{aligned}$$ The strict positiveness of $v$ follows from the fact that ${\mathbb{E}}[ R_{2}- R_{1}]<\infty.$
Proof of Theorem \[thm:CLT\_planar\]
------------------------------------
Consider the following sequence of real valued random variables: $$\xi_{i}=d(Z_{R_{i+1}}, Z_{R_{i}}) - (R_{i+1}- R_{i})v\quad i\geq 1.$$ According to Theorem \[thm:reg\] this is a sequence of centered i.i.d. random variables. Moreover, $\Sigma={\mathbb{E}}[\xi_{1}^2]>0$ since ${\mathbb{P}}( |\xi_{i}|>k)>0$ for some $k\ge 0$. Let $$S_{n}=\sum_{i=1}^n \xi_{i}\mbox{, and } \Sigma={\mathbb{E}}[\xi_{1}^2],$$ The sequence $(S_{n})_n$ does not only satisfy a central limit theorem, *i.e.*, $S_{n}/\sqrt{n}\stackrel{{\mathcal{D}}}{\longrightarrow} {\mathcal{N}}(0, \Sigma)$, but also an invariance principle, *i.e.*, $\frac1{\Sigma\sqrt{n}} S_{\lfloor nt\rfloor}$ converges in distribution to a standard Brownian motion (*e.g.,* see Donsker’s Theorem 14.1 in [@billingsley]). Let $$k(n)=\max\{k:~\sum_{i=1}^k (R_{i}-R_{i-1})<n\}$$ as in Corollary \[cor:overshoot\].
As the invariance principle is preserved under change of time (*e.g.,* see Theorem 14.4 in [@billingsley]) the sequence $\frac1{\Sigma\sqrt{k(n)}} S_{\lfloor k(n)t\rfloor}$ also converges in distribution to a standard Brownian motion. Choosing $t=1$ yields in particular $$S_{k(n)}/\Sigma\sqrt{k(n)}\stackrel{{\mathcal{D}}}{\longrightarrow} {\mathcal{N}}(0, 1)\,.$$ From Corollary \[cor:overshoot\], since $(n/k(n))$ tends to a constant almost surely, we get that $$S_{k(n)}/\Sigma\sqrt{n}\stackrel{{\mathcal{D}}}{\longrightarrow}
{\mathcal{N}}(0, \sigma^2),$$ where $$\sigma^2= 1 / {\mathbb{E}}[R_{2}-R_{1}].$$ Corollary \[cor:overshoot\] also ensures that the random variables $$M_{k}=\sup\{d(Z_{n},Z_{R_{k}}), R_{k}\le n \leq R_{k+1}\}, ~k\geq 1,$$ form an i.i.d. sequence with exponential moments. Now, for any positive $\eta$ $$\begin{aligned}
\label{eq:thm:clt:final1}
{\mathbb{P}}\left[| S_{k(n)}- (d(Z_{n},e)-nv)| > \eta \sqrt{n}\right]
& \leq & {\mathbb{P}}\left[ d(Z_{R_{k(n)+1}},e)-d(Z_{n},e) \geq \frac\eta{2} \sqrt{n}\right] \cr
& & + {\mathbb{P}}\left [ v(n-R_{k(n)+1}) \geq \frac\eta{2} \sqrt{n}\right].\end{aligned}$$ We start by treating the first summand in (\[eq:thm:clt:final1\]). Let $M_{0}=\sup\{d(Z_{R_{1}},Z_{0}), n \leq R_{1}\}$, then $$\begin{aligned}
{\mathbb{P}}\left[ d(Z_{R_{k(n)+1}},e)-d(Z_{n},e) \geq \frac\eta{2} \sqrt{n}\right]
& \leq & {\mathbb{P}}\left[\exists k\le n+1:~M_{k} > \frac\eta{2} \sqrt{n}\right] \cr
& \leq & {\mathbb{P}}\left[\exists 1\leq k\le n+1:~M_{k} > \frac\eta{2} \sqrt{n}\right]+ {\mathbb{P}}\left[M_{0} > \frac\eta{2} \sqrt{n}\right] \cr
& \leq & n {\mathbb{P}}\left[ M_{1}>\frac\eta{2} \sqrt{n}\right] + {\mathbb{P}}\left[M_{0} > \frac\eta{2} \sqrt{n}\right]\cr
& \stackrel{\longrightarrow}{_{n\to\infty}} & 0+0.\end{aligned}$$ Here we used for the second summand that $M_{0}$ is almost surely finite and for the first summand we applied once again the existence of exponential moments and the Chebyshev inequality: $$n{\mathbb{P}}\left[ M_{1}> \frac{\eta\sqrt{n}}{2}\right] \leq C n \frac{{\mathbb{E}}[\exp(\delta M_{1})] }{\exp(\delta \sqrt{n})} \stackrel{\longrightarrow}{_{n\to\infty}} 0,$$ for some $\delta>0$. The treatment of the second summand in (\[eq:thm:clt:final1\]) is analogous by noting that $${\mathbb{P}}\left[ v(n-R_{k(n)+1}) \geq \frac\eta{2} \sqrt{n}\right] \leq
{\mathbb{P}}\left[ \exists k\leq n:~ R_{k+1}-R_{k} \geq \frac\eta{2v} \sqrt{n}\right].$$ Altogether, the term in (\[eq:thm:clt:final1\]) tends to $0$ for all $\eta>0$; this finishes the proof of Theorem \[thm:CLT\_planar\].
Analyticity of $v$ and $\sigma^{2}$
-----------------------------------
Let $\nu$ be a driving measure of a random walk with exponential moments, *i.e.*, ${\mathbb{E}}[\exp(\lambda d(X_{1},e))]<\infty$ for some $\lambda>0$. Furthermore, let $B$ be a finite subset of the support of $\nu$, *i.e.,* $B \subseteq supp(\nu)$. Denote by $\Omega_{\nu} (B)$ the set of probability measures that give positive weight to all elements of $B$ and coincide with $\nu$ outside $B$. The set $\Omega_{\nu} (B)$ can be identified with an open bounded convex subset in ${\mathbb{R}}^{|B|-1}$. We say, by abuse of notation, that ${\mathcal{O}}_{\mu}\subset \Omega_{\nu} (B) $ is an open neighborhood of $\mu\in \Omega_{\nu} (B)$ if its restriction to $B$ is an open neighborhood of the restriction of $\mu$ to $B$.
For each $\mu\in \Omega_{\nu}(B)$ we define the functions $v_{\mu}$ and $\sigma_{\mu}$ as the rate of escape and the asymptotic variance for the random walk with driving measure $\mu$, compare with Theorem \[thm:CLT\_planar\]. We write ${\mathbb{P}}^{\mu}$ (resp. ${\mathbb{E}}^{\mu)}$) for the probability measure (resp. expectation) corresponding to $\mu$ of the random walk.
### Preparations {#preparations .unnumbered}
In order to show analyticity of $v_{\mu}$ and $\sigma_{\mu}$ we make the following preparations. Define $D_{x}=\inf\{n\geq 1\,;\,Z_{n}\notin C_{\mathcal{A}}(x)\}$ and for $z\in C_{\mathcal{A}}$ let $h^{\mu}(z)=h^{\mu}_{x}(z)={\mathbb{P}}_z^{\mu}[D_{x}=\infty].$
In this section we consider surface groups with standard generating sets and assume $\nu$ to verify Assumption \[ass:2\]. Under this assumptions one verifies, as in the proof of Lemma \[lem:stayinconegen\], that $h^\mu(z)>0$ for all $z\in C_{{\mathcal{A}}}(x)$.
\[lem:claim\] Let $n\geq 1$, $x$ of type ${\textbf{T}}$ and $z_{0}, z_{1},\ldots, z_{n}\in C_{{\mathcal{A}}}(x)$. Then, $${\mathbb{E}}^{\mu}_{z_{0}}[{\textbf{1}}_{Z_{1}=z_{1}}\cdots {\textbf{1}}_{Z_{n}=z_{n}} \mid D_{x}=\infty]=
{\mathbb{E}}^{\mu}_{z_{0}}\left[{\textbf{1}}_{Z_{1}=z_{1}} \cdots {\textbf{1}}_{Z_{n}=z_{n}}\prod_{i=1}^{n} \frac{h^{\mu}(Z_{i})}{h^{\mu}(Z_{i-1})}\right].$$
The proof of the claim is a straightforward application of the Markov property; we just write it for $n=1$: $$\begin{aligned}
{\mathbb{E}}^{\mu}_{z_{0}}[{\textbf{1}}_{Z_{1}=z_{1}}\mid D_{x}=\infty]
&= & \frac{{\mathbb{P}}^{\mu}_{z_{0}}[Z_{1}=z_{1}, D_{x}=\infty]}{{\mathbb{P}}^{\mu}_{z_{0}}[D_{x}=\infty]}\cr
& = &{\mathbb{P}}^{\mu}_{z_{0}}[Z_{1} = z_{1}]\frac{h^{\mu}(z_{1})}{h^{\mu}(z_{0})}
={\mathbb{E}}^{\mu}_{z_{0}}\left[{\textbf{1}}_{Z_{1}=z_{1}} \frac{h^{\mu}(Z_{1})}{h^{\mu}(Z_{0})}\right].\end{aligned}$$
\[lem:hanalytic\] The function $\mu\mapsto h^{\mu}(z)$ in analytic on $\Omega_{\nu} (B)$ for any $z\in C_{{\mathcal{A}}}(x)$.
It follows from Equation (\[eq:PzD\]) that ${\mathbb{P}}_{z}^{\mu}(D_{x}=k)\leq C_{D} c_{D}^{k}$ for all $z\in C_{{\mathcal{A}}}(x)$ and some constants $C_{D}$ and $c_{D}<1$ that do not depend on $z$. Now, choose an open neighborhood ${\mathcal{O}}_{\mu}$ of $\mu$ such that there exists some $d\in[1,1/c_{D})$ such that $$\max_{x\in B}\left\{\frac{\tilde \mu(x)}{\mu(x)}\right\}\leq d$$ for all $\tilde \mu\in {\mathcal{O}}_{\mu}$. For each $\tilde \mu\in {\mathcal{O}}_{\mu}$ we have $$1-h^{\tilde \mu}(z)={\mathbb{P}}_{z}^{\tilde \mu}[D_{x}<\infty]=\sum_{k} {\mathbb{P}}_{z}^{\tilde\mu}[D_{x}=k]= \sum_{k} {\mathbb{E}}_{z}^{\mu}[\prod_{i=1}^{k} \frac{\tilde\mu (X_{i})}{\mu(X_{i})} {\textbf{1}}_{D_{x}=k}]\,.$$ Define $\gamma_{h}^{(k)}$ as the set of all paths $(z_{0},\ldots, z_{k})$ of length $k$ starting at $z$ and leaving $ C_{{\mathcal{A}}}(x)$ for the first time at time $k$, *i.e.*, $z_{0}=z$, $z_{i}\in C_{{\mathcal{A}}}(x)~\forall 1\leq i \leq k-1,$ and $z_{k}\notin C_{{\mathcal{A}}}(x)$. We write $x_{1},\ldots,x_{k}$ for the increments of a path in $\gamma^{(k)}_{h}$. The crucial observation now is that $${\mathbb{P}}_{z}^{\tilde\mu}[D_{x}=k]=
\sum_{(x_{1},\ldots x_{k})\in\gamma^{(k)}_{h}}\prod_{i=1}^{k} \tilde\mu (x_{j})$$ is a polynomial of degree at most $k$ and therefore is analytic. Furthermore, $$\label{eq:lem:hanalytic:uniform}
|{\mathbb{E}}_{z}^{\mu}[\prod_{i=1}^{k} \frac{\tilde\mu (X_{i})}{\mu(X_{i})} {\textbf{1}}_{D_{x}=k}]|=|\sum_{(x_{1},\ldots x_{k})\in\gamma_{h}^{(k)}}\prod_{i=1}^{k} \frac{\tilde\mu (x_{i})}{\mu(x_{i})}\prod_{i=1}^{k} \mu(x_{i})|\leq d^{k} {\mathbb{P}}_{z}^{\mu}(D_{x}=k)\leq C_{D} (dc_{D})^{k}.$$ Eventually, $1-h^{\tilde \mu}(z)$ is given locally as a uniform converging series of analytic functions and therefore is analytic on $\Omega_{\nu}(B).$
### Proof of Theorem \[thm:analytic\] {#proof-of-theorem-thmanalytic .unnumbered}
We have to prove that $\mu\mapsto v=\frac{{\mathbb{E}}^{\mu}[ d(Z_{ R_{2}}, Z_{ R_{1}})]}{{\mathbb{E}}^{\mu}[ R_{2}- R_{1}]}$ is analytic. We will only prove that the denumerator is an analytic function; the proof of the analyticity of the numerator is then a straightforward adaptation. For the sake of simplicity we write $S_{k}, E_{k},$ and $D_{k}$ for $S_{k}^{(2)}, E_{k}^{(2)},$ and $D_{k}^{(2)}$. Moreover, we define $D_{0}=0$ and $E_{0}=E$. We have $$R_{2}-R_{1}= 1 + E\circ \theta^{R_{1}+1} + \sum_{k=1}^{\infty} (S_{k}-S_{k-1}) {\textbf{1}}_{S_{1}<\infty,\ldots, S_{k}<\infty}.$$ Therefore, the denumerator can be written as $$\label{eq:vfrac}
{\mathbb{E}}^{\mu}[ R_{2}- R_{1}]={\mathbb{E}}_{{\textbf{T}}}^{\mu}[1 + E\circ \theta^{1}]+ \sum_{k=1}^{\infty} {\mathbb{E}}_{{\textbf{T}}}^{\mu}[(S_{k}-S_{k-1}) {\textbf{1}}_{S_{1}<\infty,\ldots, S_{k}<\infty}].$$ Since $S_{k}-S_{k-1}=D_{k-1}+E_{k-1}$ we first prove that ${\mathbb{E}}_{z}^{\mu}[E\mid D_{x}=\infty]$ and ${\mathbb{E}}_{z}^{\mu}[D\mid D_{x}=\infty]$ are analytic in $\mu$.
Due to Equations (\[eq:PzD\]) and (\[eq:lem:hanalytic:uniform\]) we can choose an open neighborhood ${\mathcal{O}}^{(h)}_{\mu}$ of $\mu$ and some $c_{h}>0$ such that $h^{\tilde \mu}(z)\geq c_{h} $ for all $\tilde\mu\in{\mathcal{O}}_{\mu}$ and all $z\in C_{{\mathcal{A}}}(x)$. Recall, that $E$ has exponential moments, *i.e.*, there exist constants $C_{E}$ and $c_{E}<1$ such that ${\mathbb{P}}_{z}^{\mu}[E=k]\leq C_{E} c_{E}^{k}$ for all $z\in C_{{\mathcal{A}}}(x)$. Now, choose an open neighborhood ${\mathcal{O}}_{\mu}\subset {\mathcal{O}}^{(h)}_{\mu}$ of $\mu$ such that, for all $\tilde{\mu}\in {\mathcal{O}}_{\mu}$, $$\max_{x\in B}\left\{\frac{\tilde \mu(x)}{\mu(x)}\right\}\leq d<1/c_{E}.$$ For each $\tilde\mu\in{\mathcal{O}}_{\mu}$ we have
$${\mathbb{E}}_{z}^{\tilde \mu}[E\mid D_{x}=\infty]=\sum_{k=0}^{\infty} k {\mathbb{E}}_{z}^{\tilde\mu}[ {\textbf{1}}_{E=k}\mid D=\infty].$$ Consider $$f_{k}(\tilde\mu)= k {\mathbb{E}}_{z}^{\tilde\mu}[{\textbf{1}}_{E=k}\mid D=\infty] = {\mathbb{E}}_{z}^{\tilde\mu}[k \prod_{j=1}^{k}\frac{h^{\tilde\mu}(Z_{j})}{h^{\tilde\mu}(Z_{j-1})} {\textbf{1}}_{E=k} ]
= \sum_{(z_{1},\ldots z_{k})\in\gamma^{(k)}_{E}}k \frac{h^{\tilde \mu}(z_{k})}{h^{\tilde\mu}(z)}\prod_{i=1}^{k} {\tilde\mu (x_{j})} ,$$ where $\gamma^{(k)}_{E}$ is the set of all paths of length $k$ corresponding to the event $\{E=k\}$. Hence, each $f_{k}({\mu})$ is analytic. Furthermore, $$|f_{k}(\tilde \mu)|\leq C k d^{k} c_{E}^{k} \frac1{1-c_{h}}$$ and hence ${\mathbb{E}}_{z}^{ \mu}[E\mid D_{x}=\infty]$ is given locally as a uniform converging series of analytic functions and therefore is analytic on $\Omega_{\nu}.$ The proof of the analyticity of ${\mathbb{E}}_{z}^{ \mu}[D\mid D_{x}=\infty]$ is similar and therefore omitted.
Let us return to the denumerator in Equation (\[eq:vfrac\]) that can we written as $${\mathbb{E}}^{\mu}[ R_{2}- R_{1}]=\sum_{k=0}^{\infty} g_{k}(\mu)$$ where $$g_{0}(\mu)={\mathbb{E}}_{{\textbf{T}}}^{\mu}[1 + E\circ \theta^{1}]\mbox{ and } g_{k}(\mu)={\mathbb{E}}_{{\textbf{T}}}^{\mu}[S_{k}-S_{k-1} {\textbf{1}}_{S_{1}<\infty,\ldots, S_{k}<\infty}]\quad k\geq1.$$
The same arguments as above imply that $g_{k}(\mu)$ is an analytic function for all $k$. Hence, in order to prove the analyticity of ${\mathbb{E}}^{\mu}[ R_{2}- R_{1}]$ it suffices to show that $\sum_{k} |g_{k}(\mu)|$ converges uniformly in some open neighborhood of $\mu$. Recall that we have on $\{S_{1}<\infty,\ldots, S_{k}<\infty\}$ that $$S_{k}-S_{k-1}= D_{k-1} +E_{k-1},$$ and hence $$g_{k}(\tilde\mu) = {\mathbb{E}}_{{\textbf{T}}}^{\tilde \mu}[(D_{k-1} +E_{k-1}){\textbf{1}}_{D_{1}<\infty,\ldots, D_{k-1}<\infty}].$$ Observe that we already know that ${\mathbb{E}}_{z}^{ \mu}[D+E\circ \theta^{D}\mid D_{x}=\infty]$ is analytic. Furthermore we have, $$\begin{aligned}
{\mathbb{E}}_{z}^{\tilde \mu}[(D_{k}+E_{k}){\textbf{1}}_{D_{1}<\infty,\ldots, D_{k}<\infty}\mid D_{x}=\infty] &= &{\mathbb{E}}^{\tilde \mu}_{z}[(D_{k}+E_{k}) {\textbf{1}}_{D_{1}<\infty,\ldots, D_{k}<\infty} \prod_{i=1}^{D_{k}}\frac{h^{\tilde \mu}(Z_{i})}{h^{\tilde \mu}(Z_{0})}]\cr
&\leq& C {\mathbb{E}}_{z}^{\mu}[(D_{k}+E_{k}) {\textbf{1}}_{D_{1}<\infty,\ldots, D_{k}<\infty}],\end{aligned}$$ using the fact that $h^{\mu}$ is analytic and applying Equation (\[eq:PzD\]). Hence, there exists some open neighborhood ${\mathcal{O}}_{\mu}$ of $\mu$ such that $${\mathbb{E}}_{z}^{\tilde \mu}[(D_{k}+E_{k}) {\textbf{1}}_{D_{1}<\infty,\ldots, D_{k}<\infty}\mid D_{x}=\infty]\leq C \mbox{ and } {\mathbb{P}}^{\tilde \mu}_{z}[S_{1}<\infty \mid D_{x}=\infty]\leq 1-c$$ for all $k\geq 1$, all $z\in C_{{\mathcal{A}}}(x)$ and all $\tilde\mu\in{\mathcal{O}}_{\mu}$.
Altogether, for $x$ such that $T(x)={\textbf{T}}$ and all $\tilde\mu\in {\mathcal{O}}_{\mu}$ we obtain that $$g_{k}(\tilde \mu)\leq C {\mathbb{E}}^{\tilde \mu}_{x}[(D_{k-1} +E_{k-1}){\textbf{1}}_{D_{1}<\infty,\ldots, D_{k-1}<\infty}] \leq C (1-c)^{k-2}\quad \forall k.$$ This finishes the proof of the analyticity of the rate of escape. The proof of the analyticity of the asymptotic variance is a straightforward adaption of the proof above.
Bypassing of Assumption \[ass:2\] {#sec:bypass}
---------------------------------
Let $\Gamma$ be a surface group and $\mu$ be a probability measure on $\Gamma$ with a finite exponential moment and whose support generates $\Gamma$ as a semigroup. Due to the irreducibility there exists some $\ell\in {\mathbb{N}}$ such that $$\bar\mu:=\frac1\ell\sum_{i=1}^{\ell} \mu^{(\ell)}$$ fulfills Assumption \[ass:2\]. Let $(\bar X_{j})_{j\geq 1}$ be a sequence of i.i.d. random variables with law $\bar\mu$ and $\bar Z_{n}=\prod_{j=1}^{n}\bar X_{j}$ the corresponding random walk. Due to its construction the variables $\bar X_{j}$ can be seen as the result of a *two-step probability event*: let $(U_{j})_{j\geq 1}$ be i.i.d. random variables (independent of $(X_{i})_{i\geq 1}$ with uniform distribution on $\{1,\ldots,\ell\}$ then $\bar X_{j}\stackrel{\mathcal{D}}{=} Z_{U_{j}}$ for all $j\geq 1$ on an appropriate joint probability space. Define $T_{n}=\sum_{j=1}^{n} U_{j}$ then $$\bar Z_{n}\stackrel{\mathcal{D}}{=} Z_{T_{n}}.$$ The proof of Theorem \[thm:LLN\_planar\] can now be adapted as follows. Denote $\bar R_{n}$ the renewal times corresponding to $\bar Z_{n}$. Define $k(n)=\max\{k: T_{\bar R_{k}<n}\}$. As $T_{n}/n\to (\ell+1)/2$ a.s. and $\bar R_{k}/k \to {\mathbb{E}}[\bar R_{2}-\bar R_{1}]$ a.s. we have that $$\frac{k(n)}n\to \frac{\ell+1}2 {\mathbb{E}}[\bar R_{2}-\bar R_{1}]~\mbox{a.s.}.$$ Eventually, we obtain $$\label{eq:v:general}
\frac1n d(Z_{n},e) \stackrel{}{\longrightarrow}
\frac{\ell+1}2 \frac{{\mathbb{E}}[ d(\bar Z_{ \bar R_{2}}, \bar Z_{ \bar R_{1}}])}{{\mathbb{E}}[ \bar R_{2}- \bar R_{1}]}~\mbox{a.s.}.$$ In the same spirit one can adjust the proof of Theorem \[thm:CLT\_planar\] and obtains a formula for the asymptotic variance
$$\sigma^{2}=\frac{2 {\mathbb{E}}[(d(\bar Z_{\bar R_{2}}, \bar Z_{\bar R_{1}}) - (\bar R_{2}- \bar R_{1})v)^{2}]}{(\ell+1)
{\mathbb{E}}[\bar R_{2}-\bar R_{1}]}.$$
Due to the above formulæ for the rate of escape and asymptotic variance the results on analyticity also hold without Assumption \[ass:2\].
Discussion {#sec:discussion}
==========
The key ingredient that we used for the renewal theory is that for all $x\in \Gamma$ $${\mathbb{P}}_{x}[Z_{n}\in C(x) \mbox{ for almost all } n]>0.$$ This fact does not hold in general as Example \[ex:counter1\] shows.
\[ex:counter1\] Let $\Gamma$ be a hyperbolic group with generating set $S$ and neutral element $e$. We set $\Gamma'=\Gamma\times({\mathbb{Z}}/2{\mathbb{Z}})$ with generating set $S'=\{(s,0),~s\in S\}\cup \{(e,1)\}$. So the Cayley graph of $\Gamma'$ consists of two copies of the Cayley graph of $\Gamma$ that are connected by edges between $(x,0)$ and $(x,1)$ for all $x\in \Gamma$. Observe that every geodesic starting from the origin $(e,0)$ that goes through a point $(x,1)$ will never visit the $0$-level afterwards. Eventually, while the cones types of the level $1$ may be ubiquitous they are not large. In particular we have for all $x$ on level $1$ that ${\mathbb{P}}_{x}[Z_{n}\in C(x) \mbox{ for almost all } n]=0.$
This fact was the motivation of the definition of large cone types. Indeed, the existence of large cone types is almost necessary to the renewal structure we have defined. Let $\Gamma$ be a non-amenable finitely generated group endowed with a word metric and a probability measure $\mu$ whose support generates $\Gamma$ as a semigroup. Let $A\subset\Gamma$ and let us consider $D_A=\inf\{n\geq 0:\ Z_n\notin A\}$. If ${\mathbb{P}}_x[D_A=\infty]>0$ then it is straightforward to show that $${\mathbb{P}}_x\left[ \lim_{n\to\infty} d(Z_n,^c\!A)=\infty| D_A=\infty\right]=1\,.$$
Let us therefore end with two questions.
Does any hyperbolic (automatic) group have large cone types for some (all) finite generating set ?
Does any hyperbolic (automatic) group have ubiquitous cone types for some (all) finite generating set ?
-------------------------- -- -------------------------------------
Peter Haïssinsky Pierre Mathieu and Sebastian Müller
IMT LATP
Université Paul Sabatier Aix-Marseille Université
118 route de Narbonne 39, rue F. Joliot Curie
31062 Toulouse Cedex 9 13453 Marseille Cedex 13
France France
-------------------------- -- -------------------------------------
[^1]: This work was supported by the ESF grant PIEF-2009-235688.
[^2]: The authors learnt about this reference in [@Le:10] but have no version at their disposal.
|
---
author:
- 'Johan M. M. van Rooij[^1]'
- 'Hans L. Bodlaender'
- 'Erik Jan van Leeuwen[^2]'
- 'Peter Rossmanith[^3]'
- Martin Vatshelle
title: 'Fast Dynamic Programming on Graph Decompositions[^4]'
---
Introduction
============
Width parameters of graphs and their related graph decompositions are important concepts in the theory of graph algorithms. Many investigations show that problems that are [$\mathcal{NP}$]{}-hard on general graphs become polynomial or even linear-time solvable when restricted to the class of graphs in which a given width parameter is bounded. However, the constant factors involved in the upper bound on the running times of such algorithms are often large and depend on the parameter. Therefore, it is often useful to find algorithms where these factors grow as slow as possible as a function of the width parameter $k$.
In this paper, we consider such algorithms involving three prominent graph-width parameters and their related decompositions: *treewidth* and *tree decompositions*, *branchwidth* and *branch decompositions*, and *cliquewidth* and *$k$-expressions* or *clique decompositions*. These three graph-width parameters are probably the most commonly used ones in the literature. However, other parameters such as rankwidth [@OumS06] or booleanwidth [@Bui-XuanTV09] and their related decompositions also exist.
Most algorithms solving combinatorial problems using a graph-width parameter consist of two steps:
1. Find a graph decomposition of the input graph of small width.
2. Solve the problem by dynamic programming on this graph decomposition.
In this paper, we will focus on the second of these steps and improve the running time of many known algorithms on all three discussed types of graph decompositions as a function of the width parameter. The results have both theoretical and practical applications, some of which we survey below.
We obtain our results by using variants of the [*covering product*]{} and the [*fast subset convolution*]{} algorithm [@BjorklundHKK07] in conjunction with known techniques on these graph decompositions. These two algorithms have been used to speed up other dynamic programming algorithms before, but not in the setting of graph decompositions. Examples include algorithms for [Steiner Tree]{} [@BjorklundHKK07; @Nederlof09], graph motif problems [@BetzlerFKN08], and graph recolouring problems [@PontaHN08]. An important aspect of our results is an implicit generalisation of the fast subset convolution algorithm that is able to use multiple states. This contrasts to the set formulation in which the covering product and subset convolution are defined: this formulation is equivalent to using two states (in and out). Moreover, the fast subset convolution algorithm uses ranked Möbius transforms, while we obtain our results by using transformations that use multiple states and multiple ranks. It is interesting to note that the state-based convolution technique that we use reminds of the technique used in Strassen’s algorithm for fast matrix multiplication [@Strassen69].
Some of our algorithms also use *fast matrix multiplication* to speed up dynamic programming as introduced by Dorn [@Dorn06]. To make this work efficiently, we introduce the use of asymmetric vertex states. We note that matrix multiplication has been used for quite some time as a basic tool for solving combinatorial problems. It has been used for instance in the [All Pairs Shortest Paths]{} problem [@Seidel95], in recognising triangle-free graphs [@ItaiR78], and in computing graph determinants. One of the results of this paper is that (generalisations of) fast subset convolution and fast matrix multiplication can be combined to obtain faster dynamic programming algorithms for many optimisation problems.
#### Treewidth-Based Algorithms.
Tree-decomposition-based algorithms can be used to effectively solve combinatorial problems on graphs of small treewidth both in theory and in practice. Practical algorithms exist for problems like partial constraint satisfaction [@KosterHK02]. Furthermore, tree-decomposition-based algorithms are used as subroutines in many areas such as approximation algorithms [@DemaineH08; @Eppstein00], parameterised algorithms [@CyganNPPRW11; @DemaineFHT05; @MolleRR08; @ThilikosSB05], exponential-time algorithms [@FominGSS09; @ScottS07; @vanRooijND09], and subexponential-time algorithms [@BodlaenderR10; @CyganNPPRW11; @FominT04].
Many [$\mathcal{NP}$]{}-hard problems can be solved in polynomial time on a graphs whose treewidth is bounded by a constant. If we assume that a graph $G$ is given with a tree decomposition $T$ of $G$ of width $k$, then the running time of such an algorithm is typically polynomial in the size of $G$, but exponential in the treewidth $k$. Examples of such algorithms include many kinds of vertex partitioning problems (including the $[\rho,\sigma]$-domination problems) [@TelleP97], edge colouring problems such as [Chromatic Index]{} [@Bodlaender90], or other problems such as [Steiner Tree]{} [@KorachS90].
Concerning the first step of the general two-step approach above, we note that finding a tree decomposition of minimum width is [$\mathcal{NP}$]{}-hard [@ArnborgCP87]. For fixed $k$, one can find a tree decomposition of width at most $k$ in linear time, if such a decomposition exists [@Bodlaender96]. However, the constant factor involved in the running time of this algorithm is very high. On the other hand, tree decompositions of small width can be obtained efficiently for special graph classes [@Bodlaender98], and there are also several good heuristics that often work well in practice [@BodlaenderK10].
Concerning the second step of this two-step approach, there are several recent results about the running time of algorithms on tree decompositions, with special considerations for the running time as a function of the width of the tree decomposition $k$. For several vertex partitioning problems, Telle and Proskurowski showed that there are algorithms that, given a graph with a tree decomposition of width $k$, solve these problems in ${\ensuremath{\mathcal{O}}}(c^k n)$ time [@TelleP97], where $c$ is a constant that depends only on the problem at hand. For [Dominating Set]{}, Alber and Niedermeier gave an improved algorithm that runs in ${\ensuremath{\mathcal{O}}}(4^k n)$ time [@AlberN02]. Similar results are given in [@AlberBFKN02] for related problems: [Independent Dominating Set]{}, [Total Dominating Set]{}, [Perfect Dominating Set]{}, [Perfect Code]{}, [Total Perfect Dominating Set]{}, [Red-Blue Dominating Set]{}, and weighted versions of these problems.
If the input graph is planar, then other improvements are possible. Dorn showed that [Dominating Set]{} on planar graphs given with a tree decomposition of width $k$ can be solved in $O^*(3^k)$ time [@Dorn10]; he also gave similar improvements for other problems. We obtain the same result without requiring planarity.
In this paper, we show that the number of dominating sets of each given size in a graph can be counted in ${\ensuremath{\mathcal{O}^*}}(3^k)$ time. After some modifications, this leads to an ${\ensuremath{\mathcal{O}^*}}(nk^23^k)$-time algorithm for [Dominating Set]{}. We also show that one can count the number of perfect matchings in a graph in ${\ensuremath{\mathcal{O}^*}}(2^k)$ time, and we generalise these results to the $[\rho,\sigma]$-domination problems, as defined in [@TelleP97].
For these $[\rho,\sigma]$-domination problems, we show that they can be solved in ${\ensuremath{\mathcal{O}^*}}(s^k)$ time, where $s$ is the natural number of states required to represent partial solutions. Here, $\rho$ and $\sigma$ are subsets of the natural numbers, and each choice of these subsets defines a different combinatorial problem. The only restriction that we impose on these problems is that we require both $\rho$ and $\sigma$ to be either finite or cofinite. That such an assumption is necessary follows from Chappelle’s recent result [@Chapelle10]: he shows that $[\rho,\sigma]$-domination problems are [$\mathcal{W}$]{}\[1\]-hard when parameterised by the treewidth of the graph if $\sigma$ is allowed to have arbitrarily large gaps between consecutive elements and $\rho$ is cofinite. The problems to which our results apply include [Strong Stable Set]{}, [Independent Dominating Set]{}, [Total Dominating Set]{}, [Total Perfect Dominating Set]{}, [Perfect Code]{}, [Induced $p$-Regular Subgraph]{}, and many others. Our results also extend to other similar problems such as [Red-Blue Dominating Set]{} and [Partition Into Two Total Dominating Sets]{}.
Finally, we define families of problems that we call $\gamma$-clique covering, $\gamma$-clique packing, or $\gamma$-clique partitioning problems: these families generalise standard problems like [Minimum Clique Partition]{} in the same way as the $[\rho,\sigma]$-domination problems generalise [Domination Set]{}. The resulting families of problems include [Maximum Triangle Packing]{}, [Partition Into $l$-Cliques]{} for fixed $l$, the problem to determine the minimum number of odd-size cliques required to cover $G$, and many others. For these $\gamma$-clique covering, packing, or partitioning problems, we give ${\ensuremath{\mathcal{O}^*}}(2^k)$-time algorithms, improving the straightforward ${\ensuremath{\mathcal{O}^*}}(3^k)$-time algorithms for these problems.
#### Branchwidth-Based Algorithms.
Branch decompositions are closely related to tree decompositions. Like tree decompositions, branch decompositions have shown to be an effective tool for solving many combinatorial problems with both theoretical and practical applications. They are used extensively in designing algorithms for planar graphs and for graphs excluding a fixed minor. In particular, most of the recent results aimed at obtaining faster exact or parameterised algorithms on these graphs rely on branch decompositions [@Dorn06; @DornFT08; @FominT04; @FominT06]. Practical algorithms using branch decompositions include those for ring routing problems [@CookS93], and tour merging for the [Travelling Salesman Problem]{} [@CookS03].
Concerning the first step of the general two-step approach, we note that finding the branchwidth of a graph is [$\mathcal{NP}$]{}-hard in general [@SeymourT94]. For fixed $k$, one can find a branch decomposition of width $k$ in linear time, if such a decomposition exists, by combining the results from [@Bodlaender96] and [@BodlaenderT97]. This is similar to tree decompositions, and the constant factors involved in the running time of this algorithms are very large. However, in contrast to tree decompositions for which the complexity on planar graphs is unknown, there exists a polynomial-time algorithm that computes a branch decomposition of minimal width of a planar graph [@SeymourT94]. For general graphs several useful heuristics exist [@CookS93; @CookS03; @Hicks02].
Concerning the second step of the general two-step approach, Dorn has shown how to use fast matrix multiplication to speed up dynamic programming algorithms on branch decompositions [@Dorn06]. Among others, he gave an ${\ensuremath{\mathcal{O}^*}}(4^k)$-time algorithm for the [Dominating Set]{} problem. On planar graphs, faster algorithms exist using so-called sphere-cut branch decompositions [@DornPBF05]. On these graphs, [Dominating Set]{} can be solved in ${\ensuremath{\mathcal{O}^*}}(3^{\frac{\omega}{2}k})$ time, where $\omega$ is the smallest constant such that two $n \times n$ matrices can be multiplied in ${\ensuremath{\mathcal{O}}}(n^\omega)$ time. Some of these results can be generalised to graphs that avoid a minor [@DornFT08]. We obtain the same results without imposing restrictions on the class of graphs to which our algorithms can be applied.
In this paper, we show that one can count the number of dominating sets of each given size in a graph in ${\ensuremath{\mathcal{O}^*}}(3^{\frac{\omega}{2}k})$ time. We also show that one can count the number of perfect matchings in a graph in ${\ensuremath{\mathcal{O}^*}}(2^{\frac{\omega}{2}k})$ time, and we show that the $[\rho,\sigma]$-domination problems with finite or cofinite $\rho$ and $\sigma$ can be solved in ${\ensuremath{\mathcal{O}^*}}(s^{\frac{\omega}{2}k})$ time, where $s$ is again the natural number of states required to represent partial solutions.
#### Cliquewidth Based Algorithms.
The notion of cliquewidth was first studied by Courcelle et al. [@CourcelleER93]. The graph decomposition associated with cliquewidth is a $k$-expression, which is sometimes also called a clique decomposition. Similar to treewidth and branchwidth, many well-known problems can be solved in polynomial time on graphs which cliquewidth is bounded by a constant [@CourcelleMR00].
Whereas the treewidth and branchwidth of any graph are closely related, its cliquewidth can be very different from both of them. For example, the treewidth of the complete graph on $n$ vertices is equal to $n-1$, while its cliquewidth is equal to $2$. However, the cliquewidth of a graph is always bounded by a function of its treewidth [@CourcelleO00]. This makes cliquewidth an interesting graph parameter to consider on graphs where the treewidth or branchwidth is too high for obtaining efficient algorithms.
Concerning the first step of the general two-step approach, we note that, like the other two parameters, computing the cliquewidth of general graphs is [$\mathcal{NP}$]{}-hard [@FellowsRRS09]. However, graphs of cliquewidth 1, 2, or 3 can be recognised in polynomial time [@CorneilHLRR00]. For $k \ge 4$, there is a fixed-parameter-tractable algorithm that, given a graph of cliquewidth $k$, outputs a $2^{k+1}$ expression.
Concerning the second step, the first singly-exponential-time algorithm for [Dominating Set]{} on clique decompositions of width $k$ is an ${\ensuremath{\mathcal{O}^*}}(16^k)$-time algorithm by Kobler and Rotics [@KoblerR03]. The previously fastest algorithm for this problem has a running time of ${\ensuremath{\mathcal{O}^*}}(8^k)$, obtained by transforming the problem to a problem on boolean decompositions [@Bui-XuanTV09]. In this paper, we present a direct algorithm that runs in ${\ensuremath{\mathcal{O}^*}}(4^k)$ time. We also show that one can count the number of dominating sets of each given size at the cost of an extra polynomial factor in the running time. Furthermore, we show that one can solve [Independent Dominating Set]{} in ${\ensuremath{\mathcal{O}^*}}(4^k)$ and [Total Dominating Set]{} in ${\ensuremath{\mathcal{O}^*}}(4^k)$ time.
We note that there are other width parameters of graphs that potentially have lower values than cliquewidth, for example rankwidth (see [@OumS06]) and booleanwidth (see [@Bui-XuanTV09]). These width parameters are related since a problem is fixed-parameter tractable parameterised by cliquewidth if and only if it is fixed-parameter tractable parameterised by rankwidth or booleanwidth [@Bui-XuanTV09; @OumS06]. However, for many problems the best known running times for these problems are often much better as a function of the cliquewidth than as a function of the rankwidth or booleanwidth. For example, dominating set can be solved on rank decompositions of width $k$ in ${\ensuremath{\mathcal{O}^*}}(2^{\frac{3}{4}k^2+\frac{23}{4}k})$ time [@Bui-XuanTV10; @GanianH10] and on boolean decompositions of width $k$ in ${\ensuremath{\mathcal{O}}}(8^k)$ time [@Bui-XuanTV09].
#### Optimality, Polynomial Factors.
We note that our results attain, or are very close to, intuitive natural lower bounds for the problems considered, namely a polynomial times the amount of space used by any dynamic programming algorithm for these problems on graph decompositions. Similarly, it makes sense to think about the number of states necessary to represent partial solutions as the best possible base of the exponent in the running time: this equals the space requirements. Currently, this is ${\ensuremath{\mathcal{O}^*}}(3^k)$ space for [Dominating Set]{} on tree decompositions and branch decompositions and ${\ensuremath{\mathcal{O}^*}}(4^k)$ space on clique decompositions.
Very recently, this intuition has been strengthened by a result of Lokshtanov et al. [@LokshtanovMS10]. They prove that it is impossible to improve the exponential part of the running time for a number of tree-decomposition-based algorithms that we present in this paper, unless the *Strong Exponential-Time Hypothesis* fails. That is, unless there exist an algorithm for the general [Satisfiability]{} problem running in ${\ensuremath{\mathcal{O}}}((2-\epsilon)^n)$ time, for any $\epsilon > 0$. In particular, this holds for our tree decomposition based algorithms for [Dominating Set]{} and [Partition Into Triangles]{}.
We see that our algorithms on tree decompositions and clique decompositions all attain this intuitive lower bound. On branch decompositions, we are very close. When the number of states that one would naturally use to represent partial solutions equals $s$, then our algorithms run in ${\ensuremath{\mathcal{O}^*}}(s^{\frac{\omega}{2}k})$ time, where $\omega/2 < 1.188$. Under the hypothesis that $\omega = 2$, which could be the true value of $\omega$, our algorithms do attains this space bound.
Because of these seemingly-optimal exponential factors in the running times of our algorithms, we spend quite some effort to make the polynomial factors involved as small as possible. In order to improve these polynomial factors, we need to distinguish between different problems based on a technical property for each type of graph decomposition that we call the *de Fluiter property*. This property is related to the concept of *finite integer index* [@BodlaenderA01; @Fluiter97].
Considering the polynomial factors of the running times of our algorithms sometimes leads to seemingly strange situations when the matrix multiplication constant is involved. To see this, notice that $\omega$ is defined as the smallest constant such that two $n \times n$ matrices can be multiplied in ${\ensuremath{\mathcal{O}}}(n^\omega)$ time. Consequently, any polylogarithmic factor in the running time of the corresponding matrix-multiplication algorithm disappears in an infinitesimal increase of $\omega$. These polylogarithmic factors are additional polynomial factors in the running times of our algorithms on branch decompositions. In our analyses, we pay no extra attention to this, and we only explicitly give the polynomial factors involved that are not related to the time required to multiply matrices.
Also, because many of our algorithms use numbers which require more than a constant amount of bits to represent (often $n$-bit numbers are involved), the time and space required to represent these numbers and perform arithmetic operations on these numbers affects the polynomial factors in the running times of our algorithms. We will always include these factors and highlight them using a special notation ($i_+(n)$ and $i_\times(n)$).
#### Model of Computation.
In this paper, we use the *Random Access Machine* (RAM) model with ${\ensuremath{\mathcal{O}}}(k)$-bit word size [@FredmanW93] for the analysis of our algorithms. In this model, memory access can be performed in constant time for memory of size ${\ensuremath{\mathcal{O}}}(c^k)$, for any constant $c$. We consider addition and multiplication operations on ${\ensuremath{\mathcal{O}}}(k)$-bit numbers to be unit-time operations. For an overview of this model, see for example [@Hagerup98].
We use this computational model because we do not want the table look-up operations to influence the polynomial factors of the running time. Since the tables have size ${\ensuremath{\mathcal{O}^*}}(s^k)$, for a problem specific integer $s \geq 2$, these operations are constant-time operations in this model.
#### Paper Organisation
This paper is organised as follows. We start with some preliminaries in Section \[sec:prelim\]. In Section \[sec:treewidth\], we present our results on tree decompositions. This is followed by our results on branch decompositions in Section \[sec:branchwidth\] and clique decompositions in Section \[sec:cliquewidth\]. To conclude, we briefly discuss the relation between the de Fluiter properties and finite integer index in Section \[sec:fluiterprop\]. Finally, we give some concluding remarks in Section \[sec:conclusion\].
Preliminaries {#sec:prelim}
=============
Let $G=(V,E)$ be an $n$-vertex graph with $m$ edges. We denote the open neighbourhood of a vertex $v$ by $N(v)$ and the closed neighbourhood of a $v$ by $N[v]$, i.e., $N(v) = \{ u \in V \;|\; \{u,v\} \in E\}$ and $N[v] = N(v) \cup \{v\} $. For a vertex subset $U \subseteq V$, we denote by $G[U]$ the subgraph induced by $U$, i.e., $G[U] = (U, E \cap (U \times U))$. We denote the powerset of a set $S$ by $2^S$.
For a decomposition tree $T$, we often identify $T$ with the set of nodes in $T$, and we write $E(T)$ for the edges of $T$. We often associate a table with each node or each edge in a decomposition tree $T$. Such a table $A$ can be seen as a function, while we write $|A|$ for the size of $A$, that is, the total space required to store all entries of $A$.
We denote the time required to add and multiply $n$-bit numbers by $i_+(n)$ and $i_\times(n)$, respectively. Currently, $i_\times(n) = n \log(n)2^{{\ensuremath{\mathcal{O}}}(\log^*(n))}$ due to Fürer’s algorithm [@Furer09], and $i_+(n) = {\ensuremath{\mathcal{O}}}(n)$. Note that $i_\times(k) = i_+(k) = {\ensuremath{\mathcal{O}}}(k)$ due to the chosen model of computation.
Combinatorial Problems Studied {#sec:problems}
------------------------------
A *dominating set* in a graph $G$ is a set of vertices $D \subseteq V$ such that for every vertex $v \in V \setminus D$ there exists a vertex $d \in D$ with $\{v,d\}\in E$, i.e, $\bigcup_{v \in D} N[v] = V$. A dominating set $D$ in $G$ is a *minimum dominating set* if it is of minimum cardinality among all dominating sets in $G$. The classical [$\mathcal{NP}$]{}-hard problem [Dominating Set]{} asks to find the size of a minimum dominating set in $G$. Given a (partial) solution $D$ of [Dominating Set]{}, we say that a vertex $d \in D$ *dominates* a vertex $v$ if $v \in N[d]$, and that a vertex $v$ is *undominated* if $N[v] \cap D = \emptyset$. Besides the standard minimisation version of the problem, we also consider counting the number of minimum dominating sets, and counting the number of dominating sets of each given size.
A *matching* in $G$ is a set of edges $M \subseteq E$ such that no two edges are incident to the same vertex. A vertex that is an endpoint of an edge in $M$ is said to be *matched* to the other endpoint of this edge. A *perfect matching* is a matchings in which every vertex $v \in V$ is matched. Counting the number of perfect matchings in a graph is a classical [$\mathcal{\#P}$]{}-complete problem [@Valiant79].
A *$[\rho, \sigma]$-dominating set* is a generalisation of a dominating set introduced by Telle in [@Telle94; @TelleP97].
Let $\rho, \sigma \subseteq {\ensuremath{\mathbb{N}}}$. A $[\rho, \sigma]$-dominating set in a graph $G$ is a subset $D \subseteq V$ such that:
- for every $v \in V \setminus D$: $|N(v) \cap D| \in \rho$;
- for every $v \in D$: $|N(v) \cap D| \in \sigma$.
The $[\rho, \sigma]$-domination problems are the computational problems of finding $[\rho, \sigma]$-dominating sets; see Table \[tab:rhosigma\]. Of these problems, we consider several variants: the *$[\rho, \sigma]$-existence problems* ask whether a $[\rho, \sigma]$-dominating set exists in a graph $G$; the *$[\rho, \sigma]$-minimisation* and *$[\rho, \sigma]$-maximisation problems* ask for the minimum or maximum cardinality of a $[\rho, \sigma]$-dominating set in a graph $G$; and the *$[\rho, \sigma]$-counting problems* ask for the number of $[\rho, \sigma]$-dominating sets in a graph $G$. In a $[\rho, \sigma]$-counting problem, we sometimes restrict ourselves to counting $[\rho, \sigma]$-dominating sets of minimum size, maximum size, or of each given size.
Throughout this paper, we assume that $\rho$ and $\sigma$ are either finite or cofinite.
$\rho$ $\sigma$ Standard problem description
-------------------- -------------------- ---------------------------------------
$\{0,1,\ldots\}$ $\{0\}$ Independent Set
$\{1,2,\ldots\}$ $\{0,1,\ldots\}$ Dominating Set
$\{0,1\}$ $\{0\}$ Strong Stable Set/2-Packing/
Distance-2 Independent Set
$\{1\}$ $\{0\}$ Perfect Code/Efficient Dominating Set
$\{1,2,\ldots\}$ $\{0\}$ Independent Dominating Set
$\{1\}$ $\{0,1,\ldots\}$ Perfect Dominating Set
$\{1,2,\ldots\}$ $\{1,2,\ldots\}$ Total Dominating Set
$\{1\}$ $\{1\}$ Total Perfect Dominating Set
$\{0,1\}$ $\{0,1,\ldots\}$ Nearly Perfect Set
$\{0,1\}$ $\{0,1\}$ Total Nearly Perfect Set
$\{1\}$ $\{0,1\}$ Weakly Perfect Dominating Set
$\{0,1,\ldots\}$ $\{0,1,\ldots,p\}$ Induced Bounded Degree Subgraph
$\{p,p+1,\ldots\}$ $\{0,1,\ldots\}$ $p$-Dominating Set
$\{0,1,\ldots\}$ $\{p\}$ Induced $p$-Regular Subgraph
: $[\rho,\sigma]$-domination problems (taken from [@Telle94; @TelleP97]).[]{data-label="tab:rhosigma"}
Another type of problems we consider are clique covering, packing, and partitioning problems. Because we want to give general results applying to many different problems, we will define a class of problems of our own: we define the notion of *$\gamma$-clique covering*, *$\gamma$-clique packing*, and *$\gamma$-clique partitioning problems*.
We start by defining the $\gamma$-clique problems and note that their definitions somewhat resemble the definition of $[\rho,\sigma]$-domination problems.
\[def:cliqueproblems\] Let $\gamma \subseteq {\ensuremath{\mathbb{N}}}\setminus\{0\}$, let $G$ be a graph, and let ${\ensuremath{\mathcal{C}}}$ be a collection of cliques from $G$ such that the size of every clique in ${\ensuremath{\mathcal{C}}}$ is contained in $\gamma$. We define the following notions:
- ${\ensuremath{\mathcal{C}}}$ is a $\gamma$-clique cover of $G$ if ${\ensuremath{\mathcal{C}}}$ covers the vertices of $G$, i.e, $\bigcup_{C \in {\ensuremath{\mathcal{C}}}} C = V$.
- ${\ensuremath{\mathcal{C}}}$ is a $\gamma$-clique packing of $G$ if the cliques are disjoint, i.e, for any two $C_1, C_2 \in {\ensuremath{\mathcal{C}}}$: $C_1 \cap C_2 = \emptyset$.
- ${\ensuremath{\mathcal{C}}}$ is a $\gamma$-clique partitioning of $G$ if it is both a $\gamma$-clique cover and a $\gamma$-clique packing.
The corresponding computational problems are defined in the following way. The $\gamma$-*clique covering* problems asks for the cardinality of the smallest $\gamma$-clique cover. The $\gamma$-*clique packing* problems asks for the cardinality of the largest $\gamma$-clique packing. The $\gamma$-*clique partitioning* problems asks whether a $\gamma$-clique partitioning exists. For these problems, we also consider their minimisation, maximisation, and counting variants. See Table \[tab:cliqueproblems\] for some concrete example problems. We note that clique covering problems in the literature often ask to cover all the edges of a graph: here we cover only the vertices.
Throughout this paper, we assume that $\gamma$ is decidable in polynomial time, that is, for every $j \in {\ensuremath{\mathbb{N}}}$ we can decide in time polynomial in $j$ whether $j \in \gamma$.
$\gamma$ problem type Standard problem description
---------------------- ---------------------------- ------------------------------------
$\{1,2,\ldots\}$ partitioning, minimisation Minimum clique partition
$\{2\}$ partitioning, counting Count perfect matchings
$\{3\}$ covering Minimum triangle cover of vertices
$\{3\}$ packing Maximum triangle packing
$\{3\}$ partitioning Partition into triangles
$\{p\}$ partitioning Partition into $p$-cliques
$\{1,3,5,7,\ldots\}$ covering Minimum cover by odd-cliques
: $\gamma$-clique covering, packing and partitioning problems.[]{data-label="tab:cliqueproblems"}
Graph Decompositions
--------------------
We consider dynamic programming algorithms on three different kinds of graph decompositions, namely tree decompositions, branch decompositions, and clique decompositions.
### Tree Decompositions
The notions of a tree decomposition and treewidth were introduced by Robertson and Seymour [@RobertsonS86] and measure the tree-likeness of a graph.
A *tree decomposition* of a graph $G$ consists of a tree $T$ in which each node $x \in T$ has an associated set of vertices $X_x \subseteq V$ (called a *bag*) such that $\bigcup_{x \in T} X_x = V$ and the following properties hold:
1. for each $\{u,v\} \in E$, there exists a node $x \in T$ such that $\{u,v\} \in X_x$.
2. if $v \in X_x$ and $v \in X_y$, then $v \in X_z$ for all nodes $z$ on the path from node $x$ to node $y$ in $T$.
The *width* $tw(T)$ of a tree decomposition $T$ is the size of the largest bag of $T$ minus one. The treewidth of a graph $G$ is the minimum treewidth over all possible tree decompositions of $G$. We note that the minus one in the definition exists to set the treewidth of trees to one. In this paper, we will always assume that tree decompositions of the appropriate width are given.
Dynamic programming algorithms on tree decompositions are often presented on nice tree decompositions, which were introduced by Kloks [@Kloks94]. We give a slightly different definition of a nice tree decomposition.
\[def:nicetreedecomp\] A *nice tree decomposition* is a tree decomposition with one special node $z$ called the *root* with $X_z = \emptyset$ and in which each node is one of the following types:
1. *Leaf node*: a leaf $x$ of $T$ with $X_x = \{v\}$ for some vertex $v \in V$.
2. *Introduce node*: an internal node $x$ of $T$ with one child node $y$; this type of node has $X_x = X_y \cup \{v\}$, for some $v \notin X_y$. The node is said to *introduce* the vertex $v$.
3. *Forget node*: an internal node $x$ of $T$ with on child node $y$; this type of node has $X_x = X_y \setminus \{v\}$, for some $v \in X_y$. The node is said to *forget* the vertex $v$.
4. *Join node*: an internal node $x$ with two child nodes $l$ and $r$; this type of node has $X_x = X_r = X_l$.
We note that this definition is slightly different from the usual. In our definition, we have the extra requirements that a bag $X_x$ associated with a leaf $x$ of $T$ consists of a single vertex $v$ ($X_x = \{v\}$), and that the bag $X_z$ associated with the root node $Z$ is empty ($X_z = \emptyset$).
Given a tree decomposition consisting of ${\ensuremath{\mathcal{O}}}(n)$ nodes, a nice tree decomposition of equal width and also consisting of ${\ensuremath{\mathcal{O}}}(n)$ nodes can be found in ${\ensuremath{\mathcal{O}}}(n)$ time [@Kloks94]. By adding a series of forget nodes to the old root, and by adding a series of introduce nodes below an old leaf node if its associated bag contains more than one vertex, we can easily modify any nice tree decomposition to have our extra requirements within the same running time.
By fixing the root of $T$, we associate with each node $x$ in a tree decomposition $T$ a vertex set $V_x \subseteq V$: a vertex $v$ belongs to $V_x$ if and only if there exists a bag $y$ with $v \in X_y$ such that either $y=x$ or $y$ is a descendant of $x$ in $T$. Furthermore, we associate with each node $x$ of $T$ the induced subgraph $G_x = G[V_x]$ of $G$. I.e., $G_x$ is the following graph: $$G_x = G\!\!\left[\bigcup \{ X_y \;|\; \textrm{$y = x$ or $y$ is a descendant of $x$} \} \right]$$
For an overview of tree decompositions and dynamic programming on tree decompositions, see [@BodlaenderK08; @HicksKK05].
### Branch Decompositions {#sec:defbw}
Branch decompositions are related to tree decompositions and also originate from the series of papers on graph minors by Robertson and Seymour [@RobertsonS91].
A *branch decomposition* of a graph $G$ is a tree $T$ in which each internal node has degree three and in which each leaf $x$ of $T$ has an assigned edge $e_x \in E$ such that this assignment is a bijection between the leaves of $T$ and the edges $E$ of $G$.
If we would remove any edge $e$ from a branch decomposition $T$ of $G$, then this cuts $T$ into two subtrees $T_1$ and $T_2$. In this way, the edge $e \in E(T)$ partitions the edges of $G$ into two sets $E_1$, $E_2$, where $E_i$ contains exactly those edges in the leaves of subtree $T_i$. The *middle set* $X_e$ associated to the edge $e \in E(T)$ is defined to be the set of vertices $X_e \subseteq V$ that are both an endpoint of an edge in the edge partition $E_1$ and an endpoint of an edge in the edge partition $E_2$, where $E_1$ and $E_2$ are associated with $e$. That is, if $V_i = \bigcup E_i$, then $X_e = V_1 \cap V_2$.
The *width* $bw(T)$ of a branch decomposition $T$ is the size of the largest middle set associated with the edges of $T$. The branchwidth $bw(G)$ of a graph $G$ is the minimum width over all possible branch decompositions of $G$. In this paper, we always assume that a branch decomposition of the appropriate width is given.
Observe that vertices $v$ of degree one in $G$ are not in any middle set of a branch decomposition $T$ of $G$. Let $u$ be the neighbour of such a vertex $v$. We include the vertex $v$ in the middle set of the edge $e$ of $T$ incident to the leaf of $T$ that contains $\{u,v\}$. This raises the branchwidth to $\max\{2,bw(G)\}$. Throughout this paper, we ignore this technicality.
The treewidth $tw(G)$ and branchwidth $bw(G)$ of any graph are related in the following way:
\[prop:1.5\] For any graph $G$ with branchwidth $bw(G) \geq 2$: $$bw(G) \leq tw(G) + 1 \leq \left\lfloor \frac{3}{2} bw(G) \right\rfloor$$
To perform dynamic programming on a branch decomposition $T$, we need $T$ to be rooted. To create a root, we choose any edge $e \in E(T)$ and subdivide it, creating edges $e_1$ and $e_2$ and a new node $y$. Next, we create another new node $z$, which will be our root, and add it together with the new edge $\{y,z\}$ to $T$. The middle sets associated with the edges created by the subdivision are set to $X_e$, i.e., $X_{e_1} = X_{e_2} = X_e$. Furthermore, the middle set of the new edge $\{y,z\}$ is the empty set: $X_{\{y,z\}} = \emptyset$.
We use the following terminology on the edges in a branch decomposition $T$ giving similar names to edges as we would usually do to vertices. We call any edge of $T$ that is incident to a leaf but not the root a *leaf edge*. Any other edge is called an *internal edge*. Let $x$ be the lower endpoint of an internal edge $e$ of $T$ and let $l$, $r$ be the other two edges incident to $x$. We call the edges $l$ and $r$ the *child edges* of $e$.
\[def:middlesets\] For a branch decomposition $T$, let $e \in E(T)$ be an edge not incident to a leaf with left child $l \in E(T)$ and right child $r \in E(T)$. We define the following partitioning of $X_e \cup X_l \cup X_r$:
1. The *intersection vertices*: $I = X_e \cap X_l \cap X_r$.
2. The *forget vertices*: $F = (X_l \cap X_r) \setminus I$.
3. The *vertices passed from the left*: $L = (X_e \cap X_l) \setminus I$.
4. The *vertices passed from the right*: $R = (X_e \cap X_r) \setminus I$.
Notice that this is a partitioning because any vertex in at least one of the sets $X_e$, $X_l$, $X_r$ must be in at least two of them by definition of a middle set.
Because each bag has size at most $k$, the partitioning satisfies the properties: $$|I| + |L| + |R| \leq k \qquad\qquad |I| + |L| + |F| \leq k \qquad\qquad |I| + |R| + |F| \leq k$$
We associate with each edge $e \in E(T)$ of a branch decomposition $T$ the induced subgraph $G_e = G[V_e]$ of $G$. A vertex $v \in V$ belongs to $V_e$ in this definition if and only if there is a middle set $f$ with $f=e$ or $f$ below $e$ in $T$ with $v \in X_f$. That is, $v$ is in $V_e$ if and only if $v$ is an endpoint of an edge associated with a leaf of $T$ that is below $e$ in $T$, i.e.: $$G_e = G\!\!\left[\bigcup \{ X_f \;|\; \textrm{$f = e$ or $f$ is below $e$ in $T$} \} \right]$$
For an overview of branch decomposition based techniques, see [@HicksKK05].
### $k$-Expressions and Cliquewidth {#sec:defcw}
Another notion related to the decomposition of graphs is cliquewidth, introduced by Courcelle et al. [@CourcelleER93].
A *$k$-expression* is an expression combining any number of the following four operations on labelled graphs with labels $\{1,2,\ldots,k\}$:
1. *create a new graph*: create a new graph with one vertex having any label,
2. *relabel*: relabel all vertices with label $i$ to $j$ ($i \not= j$),
3. *add edges*: connect all vertices with label $i$ to all vertices with label $j$ ($i \not= j$),
4. *join graphs*: take the disjoint union of two labelled graphs.
The *cliquewidth* $cw(G)$ of a graph $G$ is defined to be the minimum $k$ for which there exists a $k$-expression that evaluates to a graph isomorphic to $G$.
The definition of a $k$-expression can also be turned into a rooted decomposition tree. In this decomposition tree $T$, leafs of the tree $T$ correspond to the operations that create new graphs, effectively creating the vertices of $G$, and internal vertices of $T$ correspond to one of the other three above operations described above. We call this tree a *clique decomposition* of width $k$. In this paper, we always assume that a given decomposition of the appropriate width is given.
In this paper, we also ssume that any $k$-expression does not contain superfluous operations, e.g., a $k$-expression does apply the operation to add edges between vertices with labels $i$ and $j$ twice in a row without first changing the sets of vertices with the labels $i$ and $j$, and it does not relabel vertices with a given label or add edges between vertices with a given label if the set of vertices with such a label is empty. Under these conditions, it is not hard to see that any $k$-expressions consists of at most ${\ensuremath{\mathcal{O}}}(n)$ join operations and ${\ensuremath{\mathcal{O}}}(nk^2)$ other operations.
More information on solving problems on graphs of bounded cliquewidth can be found in [@CourcelleMR00].
Fast Algorithms to Speed Up Dynamic Programming {#sec:matrixmultiplic}
-----------------------------------------------
\[sec:fastsubsetconv\] In this paper, we will use fast algorithms for two standard problems as subroutines to speed up dynamic programming. These are fast multiplication of matrices, and fast subset convolution.
#### Fast Matrix Multiplication.
In this paper, we let $\omega$ be the smallest constant such that two $n \times n$ matrices can be multiplied in ${\ensuremath{\mathcal{O}}}(n^\omega)$ time; that is, $\omega$ is the matrix multiplication constant. Currently, $\omega < 2.376$ due to the algorithm by Coppersmith and Winograd [@CoppersmithW90].
For multiplying an $(n \times p)$ matrix $A$ and a $(p \times n)$ matrix $B$, we differentiate between $p \leq n$ and $p > n$. Under the assumption that $\omega = 2.376$, an ${\ensuremath{\mathcal{O}}}(n^{1.85}p^{0.54})$ time algorithm is known if $p \leq n$ [@CoppersmithW90]. Otherwise, the matrices can be multiplied in ${\ensuremath{\mathcal{O}}}(\frac{p}{n}n^\omega) = {\ensuremath{\mathcal{O}}}(pn^{\omega-1})$ time by matrix splitting: split the matrices $A$ and $B$ into $\frac{p}{n}$ many $n \times n$ matrices $A_1,\ldots A_\frac{p}{n}$ and $B_1,\ldots B_\frac{p}{n}$, multiply each of the $\frac{p}{n}$ pairs $A_i \times B_i$, and sum up the results.
#### Fast Subset Convolution.
Given a set $U$ and two functions $f,g: 2^{U} \rightarrow {\ensuremath{\mathbb{Z}}}$, their *subset convolution* $(f * g)$ is defined as follows: $$(f * g)(S) = \sum_{X \subseteq S} f(X) g(S \setminus X)$$ The fast subset convolution algorithm by Björklund et al. can compute this convolution using ${\ensuremath{\mathcal{O}}}(k^2 2^k)$ arithmetic operations [@BjorklundHKK07].
Similarly, Björklund et al. define the *covering product* $(f *_c g)$ and the *packing product* $(f *_p g)$ of $f$ and $g$ in the following way: $$(f *_c g)(S) = \mathop{\sum_{X, Y \subseteq S}}_{X \cup Y = S} f(X) g(Y) \qquad \qquad
(f *_p g)(S) = \mathop{\sum_{X, Y \subseteq S}}_{X \cap Y = \emptyset} f(X) g(Y)$$ These products can be computed using ${\ensuremath{\mathcal{O}}}(k 2^k)$ arithmetic operations [@BjorklundHKK07].
In this paper, we will not directly use the algorithms of Björklund et al. as subroutines. Instead, we present their algorithms based on what we will call state changes. The result is exactly the same as using the algorithms by Björklund et al. as subroutines. We choose to present our results in this way because it allows us to easily generalise the fast subset convolution algorithm to a more complex setting than functions with domain $2^U$ for some set $U$.
Dynamic Programming on Tree Decompositions {#sec:treewidth}
==========================================
Algorithms solving [$\mathcal{NP}$]{}-hard problems in polynomial time on graphs of bounded treewidth are often dynamic programming algorithms of the following form. The tree decomposition $T$ is traversed in a bottom-up manner. For each node $x \in T$ visited, the algorithm constructs a table with partial solutions on the subgraph $G_x$, that is, the induced subgraph on all vertices that are in a bag $X_y$ where $y = x$ or $y$ is a descendant of $x$ in $T$. Let an *extension* of such a partial solution be a solution on $G$ that contains the partial solution on $G_x$, and let two such partial solutions $P_1$, $P_2$ have the same *characteristic* if any extension of $P_1$ also is an extension of $P_2$ and vice versa. The table for a node $x \in T$ does not store all possible partial solutions on $G_x$: it stores a set of solutions such that it contains exactly one partial solution for each possible characteristic. While traversing the tree $T$, the table for a node $x \in T$ is computed using the tables that had been constructed for the children of $x$ in $T$.
This type of algorithm typically has a running time of the form ${\ensuremath{\mathcal{O}}}(f(k)poly(n))$ or even ${\ensuremath{\mathcal{O}}}(f(k)n)$, for an some function $f$ that grows at least exponentially. This is because the size of the computed tables often is (at least) exponential in the treewidth $k$, but polynomial (or even constant) in the size of the graph $G$. See Proposition \[prop:simpletwdsalg\] for an example algorithm.
In this section, we improve the exponential part of running time for many dynamic programming algorithms on tree decompositions for a large class of problems. When the number of partial solutions of different characteristics stored in the table is ${\ensuremath{\mathcal{O}^*}}(s^k)$, previous algorithms typically run in time ${\ensuremath{\mathcal{O}^*}}(r^k)$ for some $r > s$. This is because it is hard for these algorithms to compute a new table for a node in $T$ with multiple children. In this case, the algorithm often needs to inspect exponentially many combinations of partial solutions from it children per entry of the new table. We will show that algorithms with a running time of ${\ensuremath{\mathcal{O}^*}}(s^k)$ exist for many problems.
This section is organised as follows. We start by setting up the framework that we use for dynamic programming on tree decompositions by giving a simple algorithm in Section \[sec:twintro\]. Here, we also define the de Fluiter property for treewidth. Then, we give our results on [Dominating Set]{} in Section \[sec:dstw\], our results on counting perfect matchings in Section \[sec:countpmtwalg\], our results on $[\rho,\sigma]$-domination problems in Section \[sec:rhosigmatw\], and finally our results on the $\gamma$-clique covering, packing, and partitioning problems in Section \[sec:cliquetwalg\].
General Framework on Tree Decompositions {#sec:twintro}
----------------------------------------
We will now give a simple dynamic programming algorithm for the [Dominating Set]{} problem. This algorithm follows from standard techniques for treewidth-based algorithms, and we will give faster algorithms later.
\[prop:simpletwdsalg\] There is an algorithm that, given a tree decomposition of a graph $G$ of width $k$, computes the size of a minimum dominating set in $G$ in ${\ensuremath{\mathcal{O}}}(n 5^k i_+(\log(n)))$ time.
state meaning
------- ---------------------------------------------------------------------------
$1$ this vertex is in the dominating set.
$0_1$ this vertex is not in the dominating set and has already been dominated.
$0_0$ this vertex is not in the dominating set and has not yet been dominated.
$0_?$ this vertex is not in the dominating set and may or may not be dominated.
: Vertex states for the [Dominating Set]{} problem.[]{data-label="tab:dsstates"}
First, we construct a nice tree decomposition $T$ of $G$ of width $k$ from the given tree decomposition in ${\ensuremath{\mathcal{O}}}(n)$ time.
Similar to Telle and Proskurowski [@TelleP97], we introduce vertex states $1$, $0_1$, and $0_0$ that characterise the ‘state’ of a vertex with respect to a vertex set $D$ that is a partial solution of the [Dominating Set]{} problem: $v$ has state $1$ if $v \in D$; $v$ has state $0_1$ if $v \not\in D$ but $v$ is dominated by $D$, i.e., there is a $d \in D$ with $\{v,d\} \in E$; and, $v$ has state $0_0$ if $v \not\in D$ and $v$ is not dominated by $D$; see also Table \[tab:dsstates\].
For each node $x$ in the nice tree decomposition $T$, we consider partial solutions $D \subseteq V_x$, such that all vertices in $V_x \setminus X_x$ are dominated by $D$. We characterise these sets $D$ by the states of the vertices in $X_x$ and the size of $D$. More precisely, we will compute a table $A_x$ with an entry $A_x(c) \in \{0,1,\ldots,n\} \cup \{\infty\}$ for each $c \in \{1,0_1,0_0\}^{|X_x|}$. We call $c \in \{1,0_1,0_0\}^{|X_x|}$ a *colouring* of the vertices in $X_x$. A table entry $A_x(c)$ represents the size of the partial solution $D$ of [Dominating Set]{} in the induced subgraph $G_x$ associated with the node $x$ of $T$ that satisfies the requirements defined by the states in the colouring $c$, or infinity if no such set exists. That is, the table entry gives the size of the smallest partial solution $D$ in $G_x$ that contains all vertices in $X_x$ with state $1$ in $c$ and that dominates all vertices in $G_x$ except those in $X_x$ with state $0_0$ in $c$, or infinity if no such set exists. Notice that these $3^{|X_x|}$ colourings correspond to $3^{|X_x|}$ partial solutions with different characteristics, and that it contains a partial solution for each possible characteristic.
We now show how to compute the table $A_x$ for the next node $x \in T$ while traversing the nice tree decomposition $T$ in a bottom-up manner. Depending on the type of the node $x$ (see Definition \[def:nicetreedecomp\]), we do the following:
[*Leaf node*]{}: Let $x$ be a leaf node in $T$. The table consists of three entries, one for each possible colouring $c \in \{1,0_1,0_0\}$ of the single vertex $v$ in $X_x$. $$A_x(\{1\}) = 1 \qquad\qquad A_x(\{0_1\}) = \infty \qquad\qquad A_x(\{0_0\}) = 0$$ Here, $A_x(c)$ corresponds to the size of the smallest partial solution satisfying the requirements defined by the colouring $c$ on the single vertex $v$.
[*Introduce node*]{}: Let $x$ be an introduce node in $T$ with child node $y$. We assume that when the $l$-th coordinate of a colouring of $X_x$ represents a vertex $u$, then the same coordinate of a colouring of $X_y$ also represents $u$, and that the last coordinate of a colouring of $X_x$ represents the newly introduced vertex $v$. Now, for any colouring $c \in \{1,0_1,0_0\}^{|X_y|}$: $$\begin{aligned}
A_x(c \times \{0_1\}) & = & \left\{ \begin{array}{ll} A_y(c) & \textrm{if $v$ has a neighbour with state $1$ in $c$} \\ \infty & \textrm{otherwise} \end{array} \right. \\
A_x(c \times \{0_0\}) & = & \left\{ \begin{array}{ll} A_y(c) & \textrm{if $v$ has no neighbour with state $1$ in $c$} \\ \infty & \textrm{otherwise} \end{array} \right.\end{aligned}$$ For colourings with state $1$ for the introduced vertex, we say that a colouring $c_x$ of $X_x$ *matches* a colouring $c_y$ of $X_y$ if:
- For all $u \in X_y \setminus N(v)$: $c_x(u) = c_y(u)$.
- For all $u \in X_y \cap N(v)$: either $c_x(u)\!=\!c_y(u)\!=\!1$, or $c_x(u) \!=\! 0_1$ and $c_y(u) \!\in\! \{0_1,0_0\}$.
Here, $c(u)$ is the state of the vertex $u$ in the colouring $c$. We compute $A_x(c)$ by the following formula: $$\begin{aligned}
A_x(c \times \{1\}) & \!=\! & \left\{ \begin{array}{ll} \infty & \textrm{if $c(u)=0_0$ for some $u \in N(v)$} \\
1 + \min \{ A_y(c') \;|\; \textrm{$c'$ matches $c$} \} & \textrm{otherwise} \end{array} \right.\end{aligned}$$ It is not hard to see that $A_x(c)$ now corresponds to the size of the partial solution satisfying the requirements imposed on $X_x$ by the colouring $c$..
[*Forget node*]{}: Let $x$ be a forget node in $T$ with child node $y$. Again, we assume that when the $l$-th coordinate of a colouring of $X_x$ represents a vertex $u$, then the same coordinate of a colouring of $X_y$ also represents $u$, and that the last coordinate of a colouring of $X_y$ represents vertex $v$ that we are forgetting. $$A_x(c) = \min\{ A_y(c \times \{1\}), A_y(c \times \{0_1\}) \}$$ Now, $A_x(c)$ corresponds to the size of the smallest partial solution satisfying the requirements imposed on $X_x$ by the colouring $c$ as we consider only partial solutions that dominate the forgotten vertex.
[*Join node*]{}: Let $x$ be a join node in $T$ and let $l$ and $r$ be its child nodes. As $X_x = X_l = X_r$, we can assume that the same coordinates represent the same vertices in a colouring of each of the three bags.
Let $c_x(v)$ be the state that represents the vertex $v$ in colouring $c_x$ of $X_x$. We say that three colourings $c_x$, $c_l$, and $c_r$ of $X_x$, $X_l,$ and $X_r$, respectively, *match* if for each vertex $v \in X_x$:
- either $c_x(v) = c_l(v) = c_r(v) = 1$,
- or $c_x(v) = c_l(v) = c_r(v) = 0_0$,
- or $c_x(v) = 0_1$ while $c_l(v)$ and $c_r(v)$ are $0_1$ or $0_0$, but not both $0_0$.
Notice that three colourings $c_x$, $c_l$, and $c_r$ match if for each vertex $v$ the requirements imposed by the states are correctly combined from the states in the colourings on both child bags $c_l$ and $c_r$ to the states in the colourings of the parent bag $c_x$. That is, if a vertex is required by $c_x$ to be in the vertex set of a partial solution, then it is also required to be so in $c_l$ and $c_r$; if a vertex is required to be undominated in $c_x$, then it is also required to be undominated in $c_l$ and $c_r$; and, if a vertex is required to be not in the partially constructed dominating set but it is required to be dominated in $c_x$, then it is required not to be in the vertex sets of the partial solutions in both $c_l$ and $c_r$, but it must be dominated in one of both partial solutions.
The new table $A_x$ can be computed by the following formula: $$A_x(c_x) = \min_{c_x, c_l, c_r \,\textrm{\scriptsize match}} A_l(c_l) + A_r(c_r) - \#_1(c_x)$$ Here, $\#_1(c)$ stands for the number of $1$-states in the colouring $c$. This number needs to be subtracted from the total size of the partial solution because the corresponding vertices are counted in each entry of $A_l(c_l)$ as well as in each entry of $A_r(c_r)$. One can easily check that this gives a correct computation of $A_x$.
After traversing the nice tree decomposition $T$, we end up in the root node $z \in T$. As $X_z = \emptyset$ and thus $G_z = G$, we find the size of the minimum dominating set in $G$ in the single entry of $A_z$.
It is not hard to see that the algorithm stores the size of the smallest partial solution of [Dominating Set]{} in $A_x$ for each possible characteristic on $X_x$ for every node $x \in T$. Hence, the algorithm is correct.
For the running time, observe that, for a leaf or forget node, ${\ensuremath{\mathcal{O}}}(3^{|X_x|}i_+(\log(n)))$ time is required since we work with $\log(n)$-bit numbers. In an introduce node, we need more time as we need to inspect multiple entries from $A_y$ to compute an entry of $A_x$. For a vertex $u$ outside $N(v)$, we have three possible combinations of states, and for a vertex $u \in N(v)$ we have four possible combinations we need to inspect: the table entry with $c_x(u) = c_y(u) = 0_0$, colourings with $c_x(u) = c_y(u) = 1$, and colourings with $c_x(u) = 0_1$ while $c_y(u) = 0_0$ or $c_y(u) = 0_1$. This leads to a total time of ${\ensuremath{\mathcal{O}}}(4^{|X_x|}i_+(\log(n)))$ for an introduce node. In a join node, five combinations of states need to be inspected per vertex requiring ${\ensuremath{\mathcal{O}}}(5^{|X_x|}i_+(\log(n)))$ time in total. As the largest bag has size at most $k+1$ and the tree decomposition $T$ has ${\ensuremath{\mathcal{O}}}(n)$ nodes, the running time is ${\ensuremath{\mathcal{O}}}(n 5^k i_+(\log(n)))$.
Many of the details of the algorithm described in the proof of Proposition \[prop:simpletwdsalg\] also apply to other algorithms described in this section. We will not repeat these details: for the other algorithms we will only specify how to compute the tables $A_x$ for all four kinds of nodes.
We notice that the above algorithm computes only the size of a minimum dominating set in $G$, not the dominating set itself. To construct a minimum dominating set $D$, the tree decomposition $T$ can be traversed in top-down order (reverse order compared to the algorithm of Proposition \[prop:simpletwdsalg\]). We start by selecting the single entry in the table of the root node, and then, for each child node $y$ of the current node $x$, we select an the entry in $A_y$ which was used to compute the selected entry of $A_x$. More specifically, we select the entry that was either used to copy into the selected entry of $A_x$, or we select one, or in a join node two, entries that lead to the minimum that was computed for $A_x$. In this way, we trace back the computation path that computed the size of $D$. During this process, we construct $D$ by adding each vertex that is not yet in $D$ and that has state $1$ in $c$ to $D$. As we only use colourings that lead to a minimum dominating set, this process gives us a minimum dominating set in $G$.
Before we give a series of new, fast dynamic programming algorithms for a broad range of problems, we need the following definition. We use it to improve the polynomial factors involved in the running times of the algorithms in this section.
\[def:fluiterproptw\] Given a graph optimisation problem $\Pi$, consider a method to represent the different characteristics of partial solutions used in an algorithm that performs dynamic programming on tree decomposition to solve $\Pi$. Such a representation of partial solutions has the *de Fluiter property for treewidth* if the difference between the objective values of any two partial solutions of $\Pi$ that are associated with a different characteristic and can both still be extended to an optimal solution is at most $f(k)$, for some non-negative function $f$ that depends only on the treewidth $k$.
This property is named after Babette van Antwerpen-de Fluiter, as this property implicitly play an important role in her work reported in [@BodlaenderA01; @Fluiter97]. Note that although we use the value $\infty$ in our dynamic programming tables, we do not consider such entries since they can never be extended to an optimal solution. Hence, these entries do not influence the de Fluiter property. Furthermore, we say that a problem has the *linear de Fluiter property for treewidth* if $f$ is a linear function in $k$.
Consider the representation used in Proposition \[prop:simpletwdsalg\] for the [Dominating Set]{} problem. This representation has the de Fluiter property for treewidth with $f(k) = k + 1$ because any table entry that is more than $k+1$ larger than the smallest value stored in the table cannot lead to an optimal solution. This holds because any partial solution of [Dominating Set]{} $D$ that is more than $k+1$ larger than the smallest value stored in the table cannot be part of a minimum dominating set. Namely, we can obtain a partial solution that is smaller than $D$ and that dominates the same vertices or more by taking the partial solution corresponding to the smallest value stored in the table and adding all vertices in $X_x$ to it.
For a discussion of the de Fluiter properties and their relation to the related property *finite integer index*, see Section \[sec:fluiterprop\].
Minimum Dominating Set {#sec:dstw}
----------------------
Alber et al. showed that one can improve the straightforward result of Proposition \[prop:simpletwdsalg\] by choosing a different set of states to represent characteristics of partial solutions [@AlberBFKN02; @AlberN02]: they obtained an ${\ensuremath{\mathcal{O}^*}}(4^k)$-time algorithm using the set of states $\{1,0_1,0_?\}$ (see Table \[tab:dsstates\]). We obtain an ${\ensuremath{\mathcal{O}^*}}(3^k)$-time algorithm by using yet another set of states, namely $\{1,0_0,0_?\}$.
Note that $0_?$ represents a vertex $v$ that is not in the vertex set $D$ of a partial solution of [Dominating Set]{}, while we do not specify whether $v$ is dominated; i.e., given $D$, vertices with state $0_1$ and with state $0_0$ could also have state $0_?$. In particular, there is no longer a unique colouring of $X_x$ with states for a specific partial solution: a partial solution can correspond to several such colourings. Below, we discuss in detail how we can handle this situation and how it can lead to faster algorithms.
Since the state $0_0$ represents an undominated vertex and the state $0_?$ represents a vertex that may or may not be dominated, one may think that it is impossible to guarantee that a vertex is dominated using these states. We circumvent this problem by not just computing the *size* of a minimum dominating set, but by computing the *number* of dominating sets of each fixed size $\kappa$ with $0 \leq \kappa \leq n$. This approach does not store (the size of) a solution per characteristic of the partial solutions, but counts the number of partial solutions of each possible size per characteristic. We note that the algorithm of Proposition \[prop:simpletwdsalg\] can straightforwardly be modified to also count the number of (minimum) dominating sets.
For our next algorithm, we use dynamic programming tables in which an entry $A_x(c,\kappa)$ represents the number of partial solutions of [Dominating Sets]{} on $G_x$ of size exactly $\kappa$ that satisfy the requirements defined by the states in the colouring $c$. That is, the table entries give the number of partial solutions in $G_x$ of size $\kappa$ that dominate all vertices in $V_x \setminus X_x$ and all vertices in $X_x$ with state $0_1$, and that do not dominate the vertices in $X_x$ with state $0_0$. This approach leads to the following result.
\[thrm:countingtwdsalg\] There is an algorithm that, given a tree decomposition of a graph $G$ of width $k$, computes the number of dominating sets in $G$ of each size $\kappa$, $0 \leq \kappa \leq n$, in ${\ensuremath{\mathcal{O}}}(n^3 3^k i_\times(n))$ time.
We will show how to compute the table $A_x$ for each type of node $x$ in a nice tree decomposition $T$. Recall that an entry $A_x(c,\kappa)$ counts the number of partial solution of [Dominating Set]{} of size exactly $\kappa$ in $G_x$ satisfying the requirements defined by the states in the colouring $c$.
[*Leaf node*]{}: Let $x$ be a leaf node in $T$ with $X_x = \{v\}$. We compute $A_x$ in the following way: $$\begin{aligned}
A_x(\{1\},\kappa) & = & \left\{ \begin{array}{ll} 1 & \textrm{if $\kappa = 1$} \\ 0 & \textrm{otherwise} \end{array} \right. \\
A_x(\{0_0\},\kappa) & = & \left\{ \begin{array}{ll} 1 & \textrm{if $\kappa = 0$} \\ 0 & \textrm{otherwise} \end{array} \right. \\
A_x(\{0_?\},\kappa) & = & \left\{ \begin{array}{ll} 1 & \textrm{if $\kappa = 0$} \\ 0 & \textrm{otherwise} \end{array} \right.\end{aligned}$$ Notice that this is correct since there is exactly one partial solution of size one that contains $v$, namely $\{v\}$, and exactly one partial solution of size zero that does not contain $v$, namely $\emptyset$.
[*Introduce node*]{}: Let $x$ be an introduce node in $T$ with child node $y$ that introduces the vertex $v$, and let $c \in \{1,0_1,0_0\}^{|X_y|}$. We compute $A_x$ in the following way: $$\begin{aligned}
A_x(c \times \{1\}, \kappa) & = & \left\{ \begin{array}{ll} 0 & \textrm{if $v$ has a neighbour with state $0_0$ in $c$} \\
0 & \textrm{if $\kappa = 0$} \\
A_y(c, \kappa-1) & \textrm{otherwise} \end{array} \right. \\
A_x(c \times \{0_0\}, \kappa) & = & \left\{ \begin{array}{ll} 0 & \textrm{if $v$ has a neighbour with state $1$ in $c$} \\
A_y(c, \kappa) & \textrm{otherwise} \end{array} \right. \\
A_x(c \times \{0_?\}, \kappa) & = & A_y(c, \kappa) \end{aligned}$$ As the state $0_?$ is indifferent about domination, we can copy the appropriate value from $A_y$. With the other two states, we have to set $A_x(c,\kappa)$ to zero if a vertex with state $0_0$ can be dominated by a vertex with state $1$. Moreover, we have to update the size of the set if $v$ gets state $1$.
[*Forget node*]{}: Let $x$ be a forget node in $T$ with child node $y$ that forgets the vertex $v$. We compute $A_x$ in the following way: $$A_x(c, \kappa) = A_y(c \times \{1\}, \kappa) + A_y(c \times \{0_?\}, \kappa) - A_y(c \times \{0_0\}, \kappa)$$ The number of partial solutions of size $\kappa$ in $G_x$ satisfying the requirements defined by $c$ equals the number of partial solutions of size $\kappa$ that contain $v$ plus the number of partial solutions of size $\kappa$ that do not contain $v$ but where $v$ is dominated. This last number can be computed by subtracting the number of such solutions in which $v$ is not dominated (state $0_0$) from the total number of partial solutions in which $v$ may be dominated or not (state $0_?$). This shows the correctness of the above formula.
The computation in the forget node is a simple illustration of the principle of inclusion/exclusion and the related Möbius transform; see for example [@BjorklundHK09].
[*Join node*]{}: Let $x$ be a join node in $T$ and let $l$ and $r$ be its child nodes. Recall that $X_x = X_l = X_r$.
If we are using the set of states $\{1,0_0,0_?\}$, then we do not have the consider colourings with matching states in order to compute the join. Namely, we can compute $A_x$ using the following formula: $$A_x(c,\kappa) = \sum_{\kappa_l + \kappa_r = \kappa + \#_1(c)} A_l(c,\kappa_l) \cdot A_r(c,\kappa_r)$$ The fact that this formula does not need to consider multiple matching colourings per colouring $c$ (see Proposition \[prop:simpletwdsalg\]) is the main reason why the algorithm of this theorem is faster than previous results.
To see that the formula is correct, recall that any partial solution of [Dominating Set]{} on $G_x$ counted in the table $A_x$ can be constructed from combining partial solutions $G_l$ and $G_r$ that are counted in $A_l$ and $A_r$, respectively. Because an entry in $A_x$ where a vertex $v$ that has state $1$ in a colouring of $X_x$ counts partial solutions with $v$ in the vertex set of the partial solution, this entry must count combinations of partial solutions in $A_l$ and $A_r$ where this vertex is also in the vertex set of these partial solutions and thus also has state $1$. Similarly, if a vertex $v$ has state $0_0$, we count partial solutions in which $v$ is undominated; hence $v$ must be undominated in both partial solutions we combine and also have state $0_0$. And, if a vertex $v$ has state $0_?$, we count partial solutions in which $v$ is not in the vertex set of the partial solution and we are indifferent about domination; hence, we can get all combinations of partial solutions from $G_l$ and $G_r$ if we also are indifferent about domination in $A_l$ and $A_r$ which is represented by the state $0_?$. All in all, if we fix the sizes of the solutions from $G_l$ and $G_r$ that we use, then we only need to multiply the number of solutions from $A_r$ and $A_l$ of this size which have the same colouring on $X_x$. The formula is correct as it combines all possible combinations by summing over all possible sizes of solutions on $G_l$ and $G_r$ that lead to a solution on $G_x$ of size $\kappa$. Notice that the term $\#_1(c)$ under the summation sign corrects the double counting of the vertices with state $1$ in $c$.
After the execution of this algorithm, the number of dominating sets of $G$ of size $\kappa$ can be found in the table entry $A_z(\emptyset,\kappa)$, where $z$ is the root of $T$.
For the running time, we observe that in a leaf, introduce, or forget node $x$, the time required to compute $A_x$ is linear in the size of the table $A_x$. The computations involve $n$-bit numbers because there can be up to $2^n$ dominating sets in $G$. Since $c \in \{1,0_0,0_?\}^{|X_x|}$ and $0 \leq \kappa \leq n$, we can compute each table $A_x$ in ${\ensuremath{\mathcal{O}}}(n3^k i_+(n))$ time. In a join node $x$, we have to perform ${\ensuremath{\mathcal{O}}}(n)$ multiplications to compute an entry of $A_x$. This gives a total of ${\ensuremath{\mathcal{O}}}(n^2 3^k i_\times(n))$ time per join node. As the nice tree decomposition has ${\ensuremath{\mathcal{O}}}(n)$ nodes, the total running time is ${\ensuremath{\mathcal{O}}}(n^3 3^k i_\times(n))$.
The algorithm of Theorem \[thrm:countingtwdsalg\] is exponentially faster in the treewidth $k$ compared to the previous fastest algorithm of Alber et al. [@AlberBFKN02; @AlberN02]. Also, no exponentially faster algorithm exists unless the Strong Exponential-Time Hypothesis fails [@LokshtanovMS10]. The exponential speed-up comes from the fact that we use a different set of states to represent the characteristics of partial solutions: a set of states that allows us to perform the computations in a join node much faster. We note that although the algorithm of Theorem \[thrm:countingtwdsalg\] uses underlying ideas of the covering product of [@BjorklundHKK07], no transformations associated with such an algorithm are used directly.
To represent the characteristics of the partial solutions of the [Dominating Set]{} problem, we can use any of the following three sets of states: $\{1,0_1,0_0\}$, $\{1,0_1,0_?\}$, $\{1,0_0,0_?\}$. Depending on which set we choose, the number of combinations that we need to inspect in a join node differ. We give an overview of this in Figure \[fig:jointablesds\]: each table represents a join using a different set of states, and each state in an entry of such a table represents a combination of the states in the left and right child nodes that need to be inspected to the create this new state. The number of non-empty entries now shows how many combinations have to be considered per vertex in a bag of a join node. Therefore, one can easily see that a table in a join node can be computed in ${\ensuremath{\mathcal{O}^*}}(5^k)$, ${\ensuremath{\mathcal{O}^*}}(4^k)$, and ${\ensuremath{\mathcal{O}^*}}(3^k)$ time, respectively, depending on the set of states used. These tables correspond to the algorithm of Proposition \[prop:simpletwdsalg\], the algorithm of Alber et al. [@AlberBFKN02; @AlberN02], and the algorithm of Theorem \[thrm:countingtwdsalg\], respectively.
The way in which we obtain the third table in Figure \[fig:jointablesds\] from the first one reminds us of Strassen’s algorithm for matrix multiplication [@Strassen69]: the speed-up in this algorithm comes from the fact that one multiplication can be omitted by using a series of extra additions and subtractions. Here, we do something similar by adding up all entries with a $0_1$-state or $0_0$-state together in the $0_?$-state and computing the whole block of four combinations at once. We then reconstruct the values we need by subtracting to combinations with two $0_0$-states.
[3]{}
$\times$ $1$ $0_1$ $0_0$
---------- ----- ------- -------
$1$ $1$
$0_1$ $0_1$ $0_1$
$0_0$ $0_1$ $0_0$
$\times$ $1$ $0_1$ $0_?$
---------- ----- ------- -------
$1$ $1$
$0_1$ $0_1$
$0_?$ $0_1$ $0_?$
$\times$ $1$ $0_?$ $0_0$
---------- ----- ------- -------
$1$ $1$
$0_?$ $0_?$
$0_0$ $0_0$
The exponential speed-up obtained by the algorithm of Theorem \[thrm:countingtwdsalg\] comes at the cost of extra polynomial factors in the running time. This is $n^2$ times the factor due to the fact that we work with $n$-bit numbers. Since we compute the number of dominating sets of each size $\kappa$, $0 \leq \kappa \leq n$, instead of computing a minimum dominating set, some extra polynomial factors in $n$ seem unavoidable. However, the ideas of Theorem \[thrm:countingtwdsalg\] can also be used to count only *minimum* dominating sets Using that [Dominating Set]{} has the de Fluiter property for treewidth, this leads to the following result, where the factor $n^2$ is replaced by the much smaller factor $k^2$.
\[cor:countmdstwalg\] There is an algorithm that, given a tree decomposition of a graph $G$ of width $k$, computes the number of minimum dominating sets in $G$ in ${\ensuremath{\mathcal{O}}}(n k^2 3^k i_\times(n))$ time.
We notice that the representation of the different characteristics of partial solutions used in Theorem \[thrm:countingtwdsalg\] has the linear de Fluiter property when used to count the number of minimum dominating sets. More explicitly, when counting the number of minimum dominating sets, we need to store only the number of partial solutions of each different characteristic that are at most $k+1$ larger in size than the smallest partial solution with a non-zero entry. This holds, as larger partial solutions can never lead to a minimum dominating set since taking any set corresponding to this smallest non-zero entry and adding all vertices in $X_x$ leads to a smaller partial solution that dominates at least the same vertices.
In this way, we can modify the algorithm of Theorem \[thrm:countingtwdsalg\] such that, in each node $x \in T$, we store a number $\xi_x$ representing the size of the smallest partial solution and a table $A_x$ with the number of partial solutions $A_x(c,\kappa)$ with $\xi_x \leq \kappa \leq \xi_x+k+1$.
In a leaf node $x$, we simply set $\xi_x = 0$. In an introduce or forget node $x$ with child node $y$, we first compute the entries $A_x(c,\kappa)$ for $\xi_y \leq \kappa \leq \xi_y + k +1$ and then set $\xi_x$ to the value of $\kappa$ corresponding to the smallest non-zero entry of $A_x$. While computing $A_x$, the algorithm uses $A_y(c,\kappa) = 0$ for any entry $A_y(c,\kappa)$ that falls outside the given range of $\kappa$. Finally, in a join node $x$ with child nodes $r$ and $l$, we do the same as in Theorem \[thrm:countingtwdsalg\], but we compute only the entries with $\kappa$ in the range $\xi_l + \xi_r - (k+1) \leq \kappa \leq \xi_l + \xi_r + (k+1)$. Furthermore, as all terms of the sum with $\kappa_l$ or $\kappa_r$ outside the range of $A_l$ and $A_r$ evaluate to zero, we now have to evaluate only ${\ensuremath{\mathcal{O}}}(k)$ terms of the sum. It is not hard to see that all relevant combinations of partial solutions from the two child nodes $l$ and $r$ fall in this range of $\kappa$.
The modified algorithm computes ${\ensuremath{\mathcal{O}}}(n)$ tables of size ${\ensuremath{\mathcal{O}}}(k3^k)$, and the computation of each entry requires at most ${\ensuremath{\mathcal{O}}}(k)$ multiplications of $n$-bit numbers. Therefore, the running time is ${\ensuremath{\mathcal{O}}}(n k^2 3^k i_\times(n))$.
A disadvantage of the direct use of the algorithm of Corollary \[cor:countmdstwalg\] compared to Proposition \[prop:simpletwdsalg\] is that we cannot reconstruct a minimum dominating set in $G$ by directly tracing back the computation path that gave us the size of a minimum domination set. However, as we show below, we can transform the tables computed by Theorem \[thrm:countingtwdsalg\] and Corollary \[cor:countmdstwalg\] that use the states $\{1,0_0,0_?\}$ in ${\ensuremath{\mathcal{O}^*}}(3^k)$ time into tables using any of the other sets of states. These transformations have two applications. First of all, they allow us to easily construct a minimum dominating set in $G$ from the computation of Corollary \[cor:countmdstwalg\] by transforming the computed tables into different tables as used in Proposition \[prop:simpletwdsalg\] and thereafter traverse the tree in a top-down fashion as we have discussed earlier. Secondly, they can be used to switch from using $n$-bit numbers to ${\ensuremath{\mathcal{O}}}(k)$-bit numbers, further improving the polynomial factors of the running time if we are interested only in solving the [Dominating Set]{} problem.
\[lem:dsstates\] Let $x$ be a node of a tree decomposition $T$ and let $A_x$ be a table with entries $A_x(c,\kappa)$ representing the number of partial solutions of [Dominating Set]{} of $G_x$ of each size $\kappa$, for some range of $\kappa$, corresponding to each colouring $c$ of the bag $X_x$ with states from one of the following sets: $$\{1,0_1,0_0\} \qquad \qquad \{1,0_1,0_?\} \qquad \qquad \{1,0_0,0_?\} \qquad \qquad \textrm{(see Table~\ref{tab:dsstates})}$$ The information represented in the table $A_x$ does not depend on the choice of the set of states from the options given above. Moreover, there exist transformations between tables using representations with different sets of states using ${\ensuremath{\mathcal{O}}}(|X_x||A_x|)$ arithmetic operations.
We will transform $A_x$ such that it represents the same information using a different set of states. The transformation will be given for fixed $\kappa$ and can be repeated for each $\kappa$ in the given range.
The transformations work in $|X_x|$ steps. In step $i$, we assume that the first $i-1$ coordinates of the colouring $c$ in our table $A_x$ use the initial set of states, and the last $|X_x|-i$ coordinates use the set of states to which we want to transform. Using this as an invariant, we change the set of states used for the $i$-th coordinate at step $i$.
Transforming from $\{1,0_1,0_0\}$ to $\{1,0_0,0_?\}$ can be done using the following formula in which $A_x(c,\kappa)$ represents our table for colouring $c$, $c_1$ is a subcolouring of size $i-1$ using states $\{1,0_1,0_0\}$, and $c_2$ is a subcolouring of size $|X_x|-i$ using states $\{1,0_0,0_?\}$. $$A_x(c_1 \times \{0_?\} \times c_2, \kappa) = A_x(c_1 \times \{0_1\} \times c_2, \kappa) + A_x(c_1 \times \{0_0\} \times c_2, \kappa)$$ We keep entries with states $1$ and $0_0$ on the $i$-th vertex the same, and we remove entries with state $0_1$ on the $i$-th vertex after computing the new value. In words, the above formula counts the number partial solutions that do not containing the $i$-th vertex $v$ in their vertex sets by adding the number of partial solutions that do not contain $v$ in their vertex sets and dominate it to the number of partial solutions that do not contain $v$ in the vertex sets and do not dominate it. This completes the description of the transformation.
To see that the new table contains the same information, we can apply the reverse transformation from the set of states $\{1,0_0,0_?\}$ to the set $\{1,0_1,0_0\}$ by using the same transformation with a different formula to introduce the new state: $$A_x(c_1 \times \{0_1\} \times c_2, \kappa) = A_x(c_1 \times \{0_?\} \times c_2, \kappa) - A_x(c_1 \times \{0_0\} \times c_2, \kappa)$$ A similar argument applies here: the number of partial solutions that dominate but do not contain the $i$-th vertex $v$ in their vertex sets equals the total number of partial solutions that do not contain $v$ in their vertex sets minus the number of partial solutions in which $v$ is undominated.
The other four transformations work similarly. Each transformation keeps the entries of one of the three states $0_1$, $0_0$, and $0_?$ intact, computes the entries for the new state by a coordinate-wise addition or subtraction of the other two states, and removes the entries using the third state from the table. To compute an entry with the new state, either the above two formula can be used if the new state is $0_1$ or $0_?$, or the following formula can be used if the new state is $0_0$: $$A_x(c_1 \times \{0_0\} \times c_2, \kappa) = A_x(c_1 \times \{0_?\} \times c_2, \kappa) - A_x(c_1 \times \{0_1\} \times c_2, \kappa)$$
For the above transformations, we need $|X_x|$ additions or subtractions for each of the $|A_x|$ table entries. Hence, a transformation requires ${\ensuremath{\mathcal{O}}}(|X_x||A_x|)$ arithmetic operations.
We are now ready to give our final improvement for [Dominating Set]{}.
\[cor:solvedstwalg\] There is an algorithm that, given a tree decomposition of a graph $G$ of width $k$, computes the size of a minimum dominating set in $G$ in ${\ensuremath{\mathcal{O}}}(n k^2 3^k)$ time.
We could give a slightly shorter proof than the one given below. This proof would directly combine the algorithm of Proposition \[prop:simpletwdsalg\] with the ideas of Theorem \[thrm:countingtwdsalg\] using the transformations from Lemma \[lem:dsstates\]. However, combining our ideas with the computations in the introduce and forget nodes in the algorithm of Alber et al. [@AlberBFKN02; @AlberN02] gives a more elegant solution, which we prefer to present.
On leaf, introduce, and forget nodes, our algorithm is exactly the same as the algorithm of Alber et al. [@AlberBFKN02; @AlberN02], while on a join node it is similar to Corollary \[cor:countmdstwalg\]. We give the full algorithm for completeness.
For each node $x \in T$, we compute a table $A_x$ with entries $A_x(c)$ containing the size of a smallest partial solution of [Dominating Set]{} that satisfies the requirements defined by the colouring $c$ using the set of states $\{1,0_1,0_?\}$.
[*Leaf node*]{}: Let $x$ be a leaf node in $T$. We compute $A_x$ in the following way: $$A_x(\{1\}) = 1 \qquad \qquad A_x(\{0_1\}) = \infty \qquad \qquad A_x(\{0_?\}) = 0$$
[*Introduce node*]{}: Let $x$ be an introduce node in $T$ with child node $y$ introducing the vertex $v$. We compute $A_x$ in the following way: $$\begin{aligned}
A_x(c \times \{0_1\}) & = & \left\{ \begin{array}{ll} A_y(c) & \textrm{if $v$ has a neighbour with state 1 in $c$} \\ \infty & \textrm{otherwise} \end{array} \right. \\
A_x(c \times \{0_?\}) & = & A_y(c) \\
A_x(c \times \{1\}) & = & 1 + A_y(\phi_{N(v):0_1 \rightarrow 0_?}(c))\end{aligned}$$ Here, $\phi_{N(v):0_1 \rightarrow 0_?}(c)$ is the colouring $c$ with every occurrence of the state $0_1$ on a vertex in $N(v)$ replaced by the state $0_?$.
[*Forget node*]{}: Let $x$ be a forget node in $T$ with child node $y$ forgetting the vertex $v$. We compute $A_x$ in the following way: $$A_x(c) = \min\{ A_y(c \times \{1\}), A_y(c \times \{0_1\}) \}$$ Correctness of the operations on a leaf, introduce, and forget node are easy to verify and follow from [@AlberBFKN02; @AlberN02].
[*Join node*]{}: Let $x$ be a join node in $T$ and let $l$ and $r$ be its child nodes. We first create two tables $A'_l$ and $A'_r$. For $y \in \{l,r\}$, we let $\xi_y = \min\left\{ A_y(c') \;|\; c' \in \{1,0_1,0_?\}^{|X_y|} \right\}$ and let $A'_y$ have entries $A'_y(c,\kappa)$ for all $c \in \{1,0_1,0_?\}^{|X_y|}$ and $\kappa$ with $\xi_y \leq \kappa \leq \xi_y + k + 1$: $$A'_y(c,\kappa) = \left\{ \begin{array}{ll} 1 & \textrm{if $A_y(c)=\kappa$} \\ 0 & \textrm{otherwise} \end{array} \right.$$ After creating the tables $A'_l$ and $A'_r$, we use Lemma \[lem:dsstates\] to transform the tables $A'_l$ and $A'_r$ such that they use colourings $c$ with states from the set $\{1,0_0,0_?\}$. The initial tables $A'_y$ do not contain the actual number of partial solutions; they contain a $1$-entry if a corresponding partial solution exists. In this case, the tables obtained after the transformation count the number $1$-entries in the tables before the transformation. In the table $A'_x$ computed for the join node $x$, we now count the number of combinations of these $1$-entries. This suffices since any smallest partial solution in $G_x$ that is obtained by joining partial solutions from both child nodes must consist of minimum solutions in $G_l$ and $G_r$.
We can compute $A'_x$ by evaluating the formula for the join node in Theorem \[thrm:countingtwdsalg\] for all $\kappa$ with $\xi_l + \xi_r - (k+1) \leq \kappa \leq \xi_l + \xi_r + (k+1)$ using the tables $A'_l$ and $A'_r$. If we do this in the same way as in Corollary \[cor:countmdstwalg\], then we consider only the ${\ensuremath{\mathcal{O}}}(k)$ terms of the formula where $\kappa_l$ and $\kappa_r$ fall in the specified ranges for $A_l$ and $A_r$, respectively, as other terms evaluate to zero. In this way, we obtain the table $A'_x$ in which entries are marked by colourings with states from the set $\{1,0_0,0_?\}$. Finally, we use Lemma \[lem:dsstates\] to transform the table $A'_x$ such that it again uses colourings with states from the set $\{1,0_1,0_?\}$. This final table gives the number of combinations of $1$-entries in $A_l$ and $A_r$ that lead to partial solutions of each size that satisfy the associated colourings. Since we are interested only in the size of the smallest partial solution of [Dominating Set]{} of each characteristic, we can extract these values in the following way: $$A_x(c) = \min\{ \kappa \;|\; A'_x(c,\kappa) \geq 1; \; \xi_l + \xi_r - (k+1) \leq \kappa \leq \xi_l + \xi_r + (k+1) \}$$
For the running time, we first consider the computations in a join node. Here, each state transformation requires ${\ensuremath{\mathcal{O}}}(k^23^k)$ operations by Lemma \[lem:dsstates\] since the tables have size ${\ensuremath{\mathcal{O}}}(k3^k)$. These operations involve ${\ensuremath{\mathcal{O}}}(k)$-bit numbers since the number of $1$-entries in $A_l$ and $A_r$ is at most $3^{k+1}$. Evaluating the formula that computes $A'_x$ from the tables $A'_l$ and $A'_r$ costs ${\ensuremath{\mathcal{O}}}(k^23^k)$ multiplications. If we do not store a $\log(n)$-bit number for each entry in the tables $A_x$ in any of the four kinds of nodes of $T$, but store only the smallest entry using a $\log(n)$-bit number and let $A'_x$ contain the difference to this smallest entry, then all entries in any of the $A'_x$ can also be represented using ${\ensuremath{\mathcal{O}}}(k)$-bit numbers. Since there are ${\ensuremath{\mathcal{O}}}(n)$ nodes in $T$, this gives a running time of ${\ensuremath{\mathcal{O}}}(n k^2 3^k)$. Note that the time required to multiply the ${\ensuremath{\mathcal{O}}}(k)$-bit numbers disappears in the computational model with ${\ensuremath{\mathcal{O}}}(k)$-bit word size that we use.
Corollary \[cor:solvedstwalg\] gives the currently fastest algorithm for [Dominating Set]{} on graphs given with a tree decomposition of width $k$. Essentially, what the algorithm does is fixing the $1$-states and applying the covering product of Björklund et al. [@BjorklundHKK07] on the $0_1$-states and $0_?$-states, where the $0_1$-states need to be covered by the same states from both child nodes. We chose to present our algorithm in a way that does not use the covering product directly, because reasoning with states allows us to generalise our results in Section \[sec:rhosigmatw\].
We conclude by stating that we can directly obtain similar results for similar problems using exactly the same techniques:
For each of the following problems, there is an algorithm that solves them, given a tree decomposition of a graph $G$ of width $k$, using the following running times:
- [Independent Dominating Set]{} in ${\ensuremath{\mathcal{O}}}(n^3 3^k)$ time,
- [Total Dominating Set]{} in ${\ensuremath{\mathcal{O}}}(n k^2 4^k)$ time,
- [Red-Blue Dominating Set]{} in ${\ensuremath{\mathcal{O}}}(n k^2 2^k)$ time,
- [Partition Into Two Total Dominating Sets]{} in ${\ensuremath{\mathcal{O}}}(n 6^k)$ time.
Use the same techniques as in the rest of this subsection. We emphasise only the following details.
With [Independent Dominating Set]{}, the factor $n^3$ comes from the fact that this (minimisation) problem does not have the de Fluiter property for treewidth. However, we can still use ${\ensuremath{\mathcal{O}}}(k)$-bit numbers. This holds because, even though the expanded tables $A'_l$ and $A'_r$ have size at most $n3^k$, they still contain the value one only once for each of the $3^k$ characteristics before applying the state changes. Therefore, the total sum of the values in the table, and thus also the maximum value of an entry in these tables after the state transformations is $3^k$; these can be represented by ${\ensuremath{\mathcal{O}}}(k)$-bit numbers.
With [Total Dominating Set]{}, the running time is linear in $n$ while the extra polynomial factor is $k^2$. This is because this problem does have the linear de Fluiter property for treewidth.
With [Red-Blue Dominating Set]{}, an exponential factor of $2^k$ suffices as we can use two states for the red vertices (in the red-blue dominating set or not) and two different states for the blue vertices (dominated or not).
With [Partition Into Two Total Dominating Sets]{}, we note that we can restrict ourselves to using six states when we modify the tree decomposition such that every vertex always has at least one neighbour and hence is always dominated by at least one of the two partitions. Furthermore, the polynomial factors are smaller because this is not an optimisation problem and we do not care about the sizes of both partitions.
Counting the Number of Perfect Matchings {#sec:countpmtwalg}
----------------------------------------
The next problem we consider is the problem of computing the number of perfect matchings in a graph. We give an ${\ensuremath{\mathcal{O}^*}}(2^k)$-time algorithm for this problem. This requires a slightly more complicated approach than the approach of the previous section. The main difference is that here every vertex needs to be matched *exactly* once, while previously we needed to dominate every vertex *at least* once. After introducing state transformations similar to Lemma \[lem:dsstates\], we will introduce some extra counting techniques to overcome this problem.
The obvious tree-decomposition-based dynamic programming algorithm uses the set of states $\{0,1\}$, where $1$ means this vertex is matched and $0$ means that it is not. It then computes, for every node $x \in T$, a table $A_x$ with entries $A_x(c)$ containing the number of matchings in $G_x$ with the property that the only vertices that are not matched are exactly the vertices in the current bag $X_x$ with state $0$ in $c$. This algorithm will run in ${\ensuremath{\mathcal{O}^*}}(3^k)$ time; this running time can be derived from the join table in Figure \[fig:jointablepm\]. Similar to Lemma \[lem:dsstates\] in the previous section, we will prove that the table $A_x$ contains exactly the same information independent of whether we use the set of states $\{0,1\}$ or $\{0,?\}$, where $?$ represents a vertex for which we do not specify whether it is matched or not. I.e., for a colouring $c$, we count the number of matchings in $G_x$, where all vertices in $V_x \setminus X_x$ and all vertices in $X_x$ with state $1$ in $c$ are matched, all vertices in $X_x$ with state $0$ in $c$ are unmatched, and all vertices in $X_x$ with state $?$ can either be matched or not.
[2]{}
$\times$ $0$ $1$
---------- ----- -----
$0$ $0$ $1$
$1$ $1$
$\times$ $0$ $?$
---------- ----- -----------
$0$ $0$
$?$ $\not\,?$
\[lem:pmstates\] Let $x$ be a node of a tree decomposition $T$ and let $A_x$ be a table with entries $A_x(c)$ representing the number of matchings in $G_x$ matching all vertices in $V_x \setminus X_x$ and corresponding to each colouring $c$ of the bag $X_x$ with states from one of the following sets: $$\{1,0\} \qquad \qquad \qquad \{1,?\} \qquad \qquad \qquad \{0,?\}$$ The information represented in the table $A_x$ does not depend on the choice of the set of states from the options given above. Moreover, there exist transformations between tables using representations with different sets of states using ${\ensuremath{\mathcal{O}}}(|X_x||A_x|)$ arithmetic operations.
If one defines a vertex with state $1$ or $?$ to be in a set $S$, and a vertex with state $0$ not to be in $S$, then the state changes essentially are Möbius transforms and inversions, see [@BjorklundHKK07]. The transformations in the proof below essentially are the fast evaluation algorithms from [@BjorklundHKK07].
The transformations work almost identical to those in the proof of Lemma \[lem:dsstates\]. In step $1 \leq i \leq |X_x|$, we assume that the first $i-1$ coordinates of the colouring $c$ in our table use one set of states, and the last $|X_x|-i$ coordinates use the other set of states. Using this as an invariant, we change the set of states used for the $i$-th coordinate at step $i$.
Transforming from $\{0,1\}$ to $\{0,?\}$ or $\{1,?\}$ can be done using the following formula. In this formula, $A_x(c)$ represents our table for colouring $c$, $c_1$ is a subcolouring of size $i-1$ using states $\{0,1\}$, and $c_2$ is a subcolouring of size $|X_x|-i$ using states $\{0,?\}$: $$A_x(c_1 \times \{?\} \times c_2) = A_x(c_1 \times \{0\} \times c_2) + A_x(c_1 \times \{1\} \times c_2)$$ In words, the number of matchings that may contain some vertex $v$ equals the sum of the number of matchings that do and the number of matchings that do not contain $v$.
The following two similar formulas can be used for the other four transformations: $$A_x(c_1 \times \{1\} \times c_2) = A_x(c_1 \times \{?\} \times c_2) - A_x(c_1 \times \{0\} \times c_2)$$ $$A_x(c_1 \times \{0\} \times c_2) = A_x(c_1 \times \{?\} \times c_2) - A_x(c_1 \times \{1\} \times c_2)$$
In these transformations, we need $|X_x|$ additions or subtractions for each of the $|A_x|$ table entries. Hence, a transformation requires ${\ensuremath{\mathcal{O}}}(|X_x||A_x|)$ arithmetic operations.
Although we can transform our dynamic programming tables such that they use different sets of states, this does not directly help us in obtaining a faster algorithm for counting the number of perfect matchings. Namely, if we would combine two partial solutions in which a vertex $v$ has the $?$-state in a join node, then it is possible that $v$ is matched twice in the combined solution: once in each child node. This would lead to incorrect answers, and this is why we put a $\not\,?$ instead of a $?$ in the join table in Figure \[fig:jointablepm\]. We overcome this problem by using some additional counting tricks that can be found in the proof below.
\[thrm:countingpmtwalg\] There is an algorithm that, given a tree decomposition of a graph $G$ of width $k$, computes the number of perfect matchings in $G$ in ${\ensuremath{\mathcal{O}}}(n k^2 2^k i_\times(k\log(n)))$ time.
For each node $x \in T$, we compute a table $A_x$ with entries $A_x(c)$ containing the number of matchings that match all vertices in $V_x \setminus X_x$ and that satisfy the requirements defined by the colouring $c$ using states $\{1,0\}$. We use the extra invariant that vertices with state $1$ are matched only with vertices outside the bag, i.e., vertices that have already been forgotten by the algorithm. This prevents vertices being matched within the bag and greatly simplifies the presentation of the algorithm.
[*Leaf node*]{}: Let $x$ be a leaf node in $T$. We compute $A_x$ in the following way: $$A_x(\{1\}) = 0 \qquad \qquad A_x(\{0\}) = 1$$ The only matching in the single vertex graph is the empty matching.
[*Introduce node*]{}: Let $x$ be an introduce node in $T$ with child node $y$ introducing the vertex $v$. The invariant on vertices with state $1$ makes the introduce operation trivial: $$A_x(c \times \{1\}) = 0 \qquad \qquad A_x(c \times \{0\}) = A_y(c)$$
[*Forget node*]{}: Let $x$ be a forget node in $T$ with child node $y$ forgetting the vertex $v$. If the vertex $v$ is not matched already, then it must be matched to an available neighbour at this point: $$A_x(c) = A_y(c \times \{1\}) + \sum_{u \in N(v), c(u)=1} A_y(\phi_{u:1 \rightarrow 0}(c) \times \{0\})$$ Here, $c(u)$ is the state of $u$ in $c$ and $\phi_{u:1 \rightarrow 0}(c)$ is the colouring $c$ where the state of $u$ is changed from $1$ to $0$. This formula computes the number of matchings corresponding to $c$, by adding the number of matchings in which $v$ is matched already to the number of matchings of all possibly ways of matching $v$ to one of its neighbours. We note that, because of our extra invariant, we have to consider only neighbours in the current bag $X_x$. Namely, if we would match $v$ to an already forgotten vertex $u$, then we could have matched $v$ to $u$ in the node where $u$ was forgotten.
[*Join node*]{}: Let $x$ be a join node in $T$ and let $l$ and $r$ be its child nodes.
The join is the most interesting operation. As discussed before, we cannot simply change the set of states to $\{0,?\}$ and perform the join similar to [Dominating Set]{} as suggested by Table \[fig:jointablepm\]. We use the following method: we expand the tables and index them by the number of matched vertices in $X_l$ or $X_r$, i.e., the number of vertices with state $1$. Let $y \in \{l,r\}$, then we compute tables $A'_l$ and $A'_r$ as follows: $$A'_y(c,i) = \left\{ \begin{array}{ll} A_y(c) & \textrm{if $\#_1(c) = i$}\\ 0 & \textrm{otherwise} \end{array} \right.$$ Next, we change the state representation in both tables $A'_y$ to $\{0,\!?\}$ using Lemma \[lem:pmstates\]. These tables do not use state $1$, but are still indexed by the number of $1$-states used in the previous representation. Then, we join the tables by combining all possibilities that arise from $i$ $1$-states in the previous representation using states $\{0,1\}$ (stored in the index $i$) using the following formula: $$A'_x(c,i) = \sum_{i_l + i_r = i} A'_l(c,i_l) \cdot A'_r(c,i_r)$$ As a result, the entries $A'_x(c,i)$ give us the total number of ways to combine partial solutions from $G_l$ and $G_r$ such that the vertices with state $0$ in $c$ are unmatched, the vertices with state $?$ in $c$ can be matched in zero, one, or both partial solutions used, and the total number of times the vertices with state $?$ are matched is $i$.
Next, we change the states in the table $A'_x$ back to $\{0,1\}$ using Lemma \[lem:pmstates\]. It is important to note that the $1$-state can now represent a vertex that is matched twice because the $?$-state used before this second transformation represented vertices that could be matched twice as well. However, we can find those entries in which no vertex is matched twice by applying the following observation: the total number of $1$-states in $c$ should equal the sum of those in its child tables, and this sum is stored in the index $i$. Therefore, we can extract the number of perfect matchings for each colouring $c$ using the following formula: $$A_x(c) = A'_x(c,\#_1(c))$$ In this way, the algorithm correctly computes the tables $A_x$ for a join node $x \in T$. This completes the description of the algorithm.
The computations in the join nodes again dominate the running time. In a join node, the transformations of the states in the tables cost ${\ensuremath{\mathcal{O}}}(k^22^k)$ arithmetic operations each, and the computations of $A'_x$ from $A'_l$ and $A'_r$ also costs ${\ensuremath{\mathcal{O}}}(k^22^k)$ arithmetic operations. We will now show that these arithmetic operations can be implemented using ${\ensuremath{\mathcal{O}}}(k\log(n))$-bit numbers. For every vertex, we can say that the vertex is matched to another vertex at the time when it is forgotten in $T$, or when its matching neighbour is forgotten. When it is matched at the time that it is forgotten, then it is matched to one of its at most $k+1$ neighbours. This leads to at most $k+2$ choices per vertex. As a result, there are at most ${\ensuremath{\mathcal{O}}}(k^n)$ perfect matchings in $G$, and the described operations can be implemented using ${\ensuremath{\mathcal{O}}}(k\log(n))$-bit numbers.
Because a nice tree decomposition has ${\ensuremath{\mathcal{O}}}(n)$ nodes, the running time of the algorithm is ${\ensuremath{\mathcal{O}}}(n k^2 2^k i_\times(k\log(n)))$.
The above theorem gives the currently fastest algorithm for counting the number of perfect matchings in graphs with a given tree decompositions of width $k$. The algorithm uses ideas from the fast subset convolution algorithm of Björklund et al. [@BjorklundHKK07] to perform the computations in the join node.
$[\rho,\sigma]$-Domination Problems {#sec:rhosigmatw}
-----------------------------------
We have shown how to solve two elementary problems in ${\ensuremath{\mathcal{O}^*}}(s^k)$ time on graphs of treewidth $k$, where $s$ is the number of states per vertex used in representations of partial solutions. In this section, we generalise our result for [Dominating Set]{} to the $[\rho,\sigma]$-domination problems. We show that we can solve all $[\rho,\sigma]$-domination problems with finite or cofinite $\rho$ and $\sigma$ in ${\ensuremath{\mathcal{O}^*}}(s^k)$ time. This includes the existence (decision), minimisation, maximisation, and counting variants of these problems.
For the $[\rho,\sigma]$-domination problems, one can also use colourings with states to represent the different characteristics of partial solutions. Let $D$ be the vertex set of a partial solution of a $[\rho,\sigma]$-domination problem. One set of states that we use involves the states $\rho_j$ and $\sigma_j$, where $\rho_j$ and $\sigma_j$ represent vertices not in $D$, or in $D$, that have $j$ neighbours in $D$, respectively. For finite $\rho$, $\sigma$, we let $p = \max \{\rho\}$ and $q = \max \{\sigma\}$. In this case, we have the following set of states: $\{\rho_0,\rho_1,\ldots,\rho_p,\sigma_0,\sigma_1,\linebreak[0]\ldots,\sigma_q\}$. If $\rho$ or $\sigma$ are cofinite, we let $p = 1 + \max\{ {\ensuremath{\mathbb{N}}}\setminus \rho \}$ and $q = 1 + \max\{ {\ensuremath{\mathbb{N}}}\setminus \sigma \}$. In this case, we replace the last state in the given sets by $\rho_{\geq p}$ or $\rho_{\geq q}$, respectively. This state represents a vertex in the vertex set $D$ of the partial solution of the $[\rho,\sigma]$-domination problem that has at least $p$ neighbours in $D$, or a vertex not in $D$ with at least $q$ neighbours in $D$, respectively. Let $s=p+q+2$ be the number of states involved.
Dynamic programming tables for the $[\rho,\sigma]$-domination problems can also be represented using different sets of states that contain the same information. In this section, we will use three different sets of states. These sets are defined as follows.
\[def:rsstates\] Let State Set I, II, and III be the following sets of states:
- State Set I: $\{\rho_0,\rho_1,\rho_2,\ldots,\rho_{p-1}, \rho_p / \rho_{\geq p}, \sigma_0,\sigma_1,\sigma_2,\ldots,\sigma_{q-1},\sigma_q / \sigma_{\geq q} \}$.
- StateSet II: $\{\rho_0,\rho_{\leq1},\rho_{\leq2},\ldots,\rho_{\leq p-1},\rho_{\leq p} / \rho_{{\ensuremath{\mathbb{N}}}}, \sigma_0,\sigma_{\leq 1},\sigma_{\leq 2},\ldots,\sigma_{\leq q-1},\sigma_{\leq q} / \sigma_{{\ensuremath{\mathbb{N}}}} \}$.
- State Set III: $\{\rho_0,\rho_1,\rho_2,\ldots \rho_{p-1}, \rho_p / \rho_{\geq p-1}, \sigma_0,\sigma_1,\sigma_2,\ldots,\sigma_{q-1},\sigma_q / \sigma_{\geq q-1} \}$.
The meaning of all the states is self-explanatory: $\rho_{condition}$ and $\sigma_{condition}$ consider the number of partial solutions of the $[\rho,\sigma]$-domination problem that do not contain ($\rho$-state) or do contain ($\sigma$-state) this vertex with a number of neighbours in the corresponding vertex sets satisfying the $condition$. The subscript ${\ensuremath{\mathbb{N}}}$ stands for no condition at all, i.e., $\rho_{{\ensuremath{\mathbb{N}}}} = \rho_{\geq 0}$: all possible number of neighbours in ${\ensuremath{\mathbb{N}}}$. We note that the notation $\rho_p / \rho_{\geq p}$ in Definition \[def:rsstates\] is used to indicate that this set uses the state $\rho_p$ if $\rho$ is finite and $\rho_{\geq p}$ if $\rho$ is cofinite.
\[lem:rsstates\] Let $x$ be a node of a tree decomposition $T$ and let $A_x$ be a table with entries $A_x(c,\kappa)$ representing the number of partial solutions of size $\kappa$ to the $[\rho,\sigma]$-domination problem in $G_x$ corresponding to each colouring $c$ of the bag $X_x$ with states from any of the three sets from Definition \[def:rsstates\]. The information represented in the table $A_x$ does not depend on the choice of the set of states from the options given in Definition \[def:rsstates\]. Moreover, there exist transformations between tables using representations with different sets of states using ${\ensuremath{\mathcal{O}}}(s|X_x||A_x|)$ arithmetic operations.
We apply transformations that work in $|X_x|$ steps and are similar to those in the proofs of Lemmas \[lem:dsstates\] and \[lem:pmstates\]. In the $i$-th step, we replace the states at the $i$-th coordinate of $c$. We use the following formulas to create entries with a new state.
We will give only the formulas for the $\rho$-states. The formulas for the $\sigma$-states are identical, but with $\rho$ replaced by $\sigma$ and $p$ replaced by $q$. We note that we slightly abuse notation below since we use that $\rho_{\leq 0} = \rho_0$.
To obtain states from State Set I not present in State Set II or III, we can use: $$\begin{aligned}
A_x(c_1 \times \{\rho_j \} \times c_2,\kappa) & = & A_x(c_1 \times \{\rho_{\leq j}\} \times c_2, \kappa) - A_x(c_1 \times \{\rho_{\leq j-1}\} \times c_2, \kappa) \\
A_x(c_1 \times \{\rho_{\geq p} \} \times c_2,\kappa) & = & A_x(c_1 \times \{\rho_{{\ensuremath{\mathbb{N}}}}\} \times c_2, \kappa) - A_x(c_1 \times \{\rho_{\leq p-1}\} \times c_2, \kappa) \\
A_x(c_1 \times \{\rho_{\geq p} \} \times c_2,\kappa) & = & A_x(c_1 \times \{\rho_{\geq p-1}\} \times c_2, \kappa) - A_x(c_1 \times \{\rho_{p-1}\} \times c_2, \kappa)\end{aligned}$$
To obtain states from State Set II not present in State Set I or III, we can use: $$\begin{aligned}
A_x(c_1 \times \{\rho_{\leq j} \} \times c_2,\kappa) & = & \sum_{l=0}^j A_x(c_1 \times \{\rho_l\} \times c_2, \kappa) \\
A_x(c_1 \times \{\rho_{{\ensuremath{\mathbb{N}}}} \} \times c_2,\kappa) & = & A_x(c_1 \times \{\rho_{\geq p}\} \times c_2, \kappa) + \sum_{l=0}^{p-1} A_x(c_1 \times \{\rho_l\} \times c_2, \kappa) \\
A_x(c_1 \times \{\rho_{{\ensuremath{\mathbb{N}}}} \} \times c_2,\kappa) & = & A_x(c_1 \times \{\rho_{\geq p-1}\} \times c_2, \kappa) + \sum_{l=0}^{p-2} A_x(c_1 \times \{\rho_l\} \times c_2, \kappa)\end{aligned}$$
To obtain states from State Set III not present in State Set I or II, we can use the same formulas used to obtain states from State Set I in combination with the following formulas: $$\begin{aligned}
A_x(c_1 \times \{\rho_{\geq p-1} \} \times c_2,\kappa) & = & A_x(c_1 \times \{\rho_{\geq p}\} \times c_2, \kappa) + A_x(c_1 \times \{\rho_{p-1}\} \times c_2, \kappa) \\
A_x(c_1 \times \{\rho_{\geq p-1} \} \times c_2,\kappa) & = & A_x(c_1 \times \{\rho_{{\ensuremath{\mathbb{N}}}}\} \times c_2, \kappa) - A_x(c_1 \times \{\rho_{\leq p-2}\} \times c_2, \kappa)\end{aligned}$$
As the transformations use $|X_x|$ steps in which each entry is computed by evaluating a sum of less than $s$ terms, the transformations require ${\ensuremath{\mathcal{O}}}(|X_x||A_x|)$ arithmetic operations.
We note that similar transformations can also be used to transform a table into a new table that uses different sets of states on different vertices in a bag $X_x$. For example, we can use State Set I on the first two vertices (assuming some ordering) and State Set III on the other $|X_x|-2$ vertices. We will use a transformation of this type in the proof of Theorem \[thrm:rstwalg\].
To prove our main result for the $[\rho,\sigma]$-domination problems, we will also need more involved state transformations than those given above. We need to generalise the ideas of the proof of Theorem \[thrm:countingpmtwalg\]. In this proof, we expanded the tables $A_l$ and $A_r$ of the two child nodes $l$ and $r$ such that they contain entries $A_l(c,i)$ and $A_r(c,i)$, where $i$ was an index indicating the number of $1$-states used to create the $?$-states in $c$. We will generalise this to the states used for the $[\rho,\sigma]$-domination problems.
Below, we often say that a colouring $c$ of a bag $X_x$ using State Set I from Definition \[def:rsstates\] is *counted* in a colouring $c'$ of $X_x$ using State Set II. We let this be the case when, all partial solutions counted in the entry with colouring $c$ in a table using State Set I are also counted in the entry with colouring $c'$ in the same table when transformed such that it uses State Set II. I.e., when, for each vertex $v \in X_x$, $c(v)$ and $c'(v)$ are both $\sigma$-states or both $\rho$-states, and if $c(v) = \rho_i$ or $c(v) = \sigma_i$, then $c'(v) = \rho_{\leq j}$ or $c'(v) = \sigma_{\leq j}$ for some $j \geq i$.
Consider the case where $\rho$ and $\sigma$ are finite. We introduce an *index vector* $\vec{i} = (i_{\rho1}, i_{\rho2}, \ldots, i_{\rho p},i_{\sigma1},i_{\sigma2},\ldots,i_{\sigma q})$ that is used in combination with states from State Set II from Definition \[def:rsstates\]. In this index vector, $i_{\rho j}$ and $i_{\sigma j}$ represent the sum over all vertices with state $\rho_{\leq j}$ and $\sigma_{\leq j}$ of the number of neighbours of the vertex in $D$, respectively. We say that a solution corresponding to a colouring $c$ using State Set I from Definition \[def:rsstates\] *satisfies* a combination of a colouring $c'$ using State Set II and an index vector $\vec{i}$ if: $c$ is counted in $c'$, and for each $i_{\rho j}$ or $i_{\sigma j}$, the sum over all vertices with state $\rho_{\leq j}$ and $\sigma_{\leq j}$ in $c'$ of the number of neighbours of the vertex in $D$ equals $i_{\rho j}$ or $i_{\sigma j}$, respectively.
We clarify this with an example. Suppose that we have a bag of size three and a dynamic programming table indexed by colourings using the set of states $\{\rho_0,\rho_1,\rho_2,\sigma_0\}$ (State Set I) that we want to transform to one using the set states $\{\rho_0,\rho_{\leq1},\rho_{\leq2},\sigma_0\}$ (State Set II): thus $\vec{i} = (i_{\rho1},i_{\rho2})$. Notice that a partial solution corresponding to the colouring $c=(\rho_0,\rho_1,\rho_2)$ will be counted in both $c'_1 = (\rho_{0},\rho_{\leq 2},\rho_{\leq 2})$ and $c'_2 = (\rho_{\leq 1},\rho_{\leq 1},\rho_{\leq 2})$. In this case, $c$ satisfies the combination $(c'_1, \vec{i} = (0,3))$ since the sum of the subscripts of the states in $c$ of the vertices with state $\rho_{\leq 1}$ in $c'_1$ equals zero and this sum for the vertices with state $\rho_{\leq 2}$ in $c'_1$ equals three. Also, $c$ satisfies no combination of $c'_1$ with an other index vector. Similarly, $c$ satisfies the combination $(c'_2, \vec{i}=(1,2))$ and no other combination involving $c'_2$.
In the case where $\rho$ or $\sigma$ are cofinite, the index vectors are one shorter: we do not count the sum of the number of neighbours in $D$ of the vertices with state $\rho_{{\ensuremath{\mathbb{N}}}}$ and $\sigma_{{\ensuremath{\mathbb{N}}}}$.
What we will need is a table containing, for each possible combination of a colouring using State Set II with an index vector, the number of partial solutions that satisfy these. We can construct such a table using the following lemma.
\[lem:rsstates2\] Let $x$ be a node of a tree decomposition $T$ of width $k$. There exists an algorithm that, given a table $A_x$ with entries $A_x(c,\kappa)$ containing the number of partial solutions of size $\kappa$ to the $[\rho,\sigma]$-domination problem corresponding to the colouring $c$ on the bag $X_x$ using State Set I from Definition \[def:rsstates\], computes in ${\ensuremath{\mathcal{O}}}(n(sk)^{s-1} s^{k+1} i_+(n))$ time a table $A'_x$ with entries $A'_x(c,\kappa,\vec{i})$ containing the number partial solutions of size $\kappa$ to the $[\rho,\sigma]$-domination problem satisfying the combination of a colouring using State Set II and the index vector $\vec{i}$.
We start with the following table $A'_x$ using State Set I: $$A'_x(c,\kappa,\vec{i}) = \left\{ \begin{array}{ll} A_x(c,\kappa) & \textrm{if $\vec{i}$ is the all-0 vector} \\ 0 & \textrm{otherwise} \end{array} \right.$$ Since there are no colourings with states $\rho_{\leq j}$ and $\sigma_{\leq j}$ yet, the sum of the number of neighbours in the vertex set $D$ of the partial solutions of vertices with these states is zero.
Next, we change the states of the $j$-th coordinate at step $j$ similar to Lemma \[lem:rsstates\], but now we also updates the index vector $\vec{i}$: $$\begin{aligned}
A'_x(c_1 \times \{\rho_{\leq j} \} \times c_2,\kappa,\vec{i} ) & = & \sum_{l=0}^j A'_x(c_1 \times \{\rho_l\} \times c_2, \kappa, \vec{i}_{i_{\rho j} \rightarrow (i_{\rho j}-l)}) \\
A'_x(c_1 \times \{\sigma_{\leq j} \} \times c_2,\kappa,\vec{i} ) & = & \sum_{l=0}^j A'_x(c_1 \times \{\sigma_l\} \times c_2, \kappa, \vec{i}_{i_{\sigma j} \rightarrow (i_{\sigma j}-l)})\end{aligned}$$ Here, $\vec{i}_{i_{\rho j} \rightarrow (i_{\rho j}-l)}$ denotes the index vector $\vec{i}$ with the value of $i_{\rho j}$ set to $i_{\rho j}-l$.
If $\rho$ or $\sigma$ are cofinite, we simply use the formula in Lemma \[lem:rsstates\] for every fixed index vector $\vec{i}$ for the $\rho_{{\ensuremath{\mathbb{N}}}}$-states and $\sigma_{{\ensuremath{\mathbb{N}}}}$-states. We do so because we do not need to keep track of any index vectors for these states.
For the running time, note that each index $i_{\rho j}$, $i_{\sigma j}$ can have only values between zero and $sk$ because there can be at most $k$ vertices in $X_x$ that each have at most $s$ neighbours in $D$ when considered for a state of the form $\rho_{\leq j}$ or $\sigma_{\leq j}$, as $j < p$ or $j < q$, respectively. The new table has ${\ensuremath{\mathcal{O}}}(n (sk)^{s-2} s^{k+1})$ entries since we have $s^{k+1}$ colourings, $n + 1$ sizes $\kappa$, and $s-2$ indices that range over $sk$ values. Since the algorithm uses at most $k+1$ steps in which it computes a sum with less than $s$ terms for each entry using $n$-bit numbers, this gives a running time of ${\ensuremath{\mathcal{O}}}(n(sk)^{s-1} s^{k+1} i_+(n))$.
We are now ready to prove our main result of this section.
\[thrm:rstwalg\] Let $\rho, \sigma \subseteq {\ensuremath{\mathbb{N}}}$ be finite or cofinite, and let $p$, $q$ and $s$ be the values associated with the corresponding $[\rho,\sigma]$-domination problem. There is an algorithm that, given a tree decomposition of a graph $G$ of width $k$, computes the number of $[\rho,\sigma]$-dominating sets in $G$ of each size $\kappa$, $0 \leq \kappa \leq n$, in ${\ensuremath{\mathcal{O}}}(n^3 (sk)^{2(s-2)} s^{k+1} i_\times(n))$ time.
Notice that, for any given $[\rho,\sigma]$-domination problem, $s$ is a fixed constant. Hence, Theorem \[thrm:rstwalg\] gives us ${\ensuremath{\mathcal{O}^*}}(s^k)$-time algorithms for these problems.
Before we give the computations involved for each type of node in a nice tree decomposition $T$, we slightly change the meaning of the subscript of the states $\rho_{condition}$ and $\sigma_{condition}$. In our algorithm, we let the subscripts of these states count only the number of neighbours in the vertex sets $D$ of the partial solution of the $[\rho,\sigma]$-domination problem that have already been forgotten by the algorithm. This prevents us from having to keep track of any adjacencies within a bag during a join operation. We will update these subscripts in the forget nodes. This modification is similar to the approach for counting perfect matchings in the proof of Theorem \[thrm:countingpmtwalg\], where we matched vertices in a forget node to make sure that we did not have to deal with vertices that are matched within a bag when computing the table for a join node.
We will now give the computations for each type of node in a nice tree decomposition $T$. For each node $x \in T$, we will compute a table $A_x(c,\kappa)$ containing the number of partial solutions of size $\kappa$ in $G_x$ corresponding to the colouring $c$ on $X_x$ for all colourings $c$ using State Set I from Definition \[def:rsstates\] and all $0 \leq \kappa \leq n$. During this computation, we will transform to different sets of states using Lemmas \[lem:rsstates\] and \[lem:rsstates2\] when necessary.
[*Leaf node*]{}: Let $x$ be a leaf node in $T$.
Because the subscripts of the states count only neighbours in the vertex set of the partial solutions that have already been forgotten, we use only the states $\rho_0$ and $\sigma_0$ on a leaf. Furthermore, the number of $\sigma$-states must equal $\kappa$. As a result, we can compute $A_x$ in the following way: $$A_x(c,\kappa) = \left\{ \begin{array}{ll} 1 & \textrm{if $c = \{\rho_0\}$ and $\kappa = 0$} \\ 1 & \textrm{if $c = \{\sigma_0\}$ and $\kappa = 1$} \\ 0 & \textrm{otherwise} \end{array} \right.$$
[*Introduce node*]{}: Let $x$ be an introduce node in $T$ with child node $y$ introducing the vertex $v$.
Again, the entries where $v$ has the states $\rho_j$ or $\sigma_j$, for $j \geq 1$, will be zero due to the definition of the (subscripts of) the states. Also, we must again keep track of the size $\kappa$. Let $\varsigma$ be the state of the introduced vertex. We compute $A_x$ in the following way: $$A_x(c \times \{\varsigma\},\kappa) = \left\{ \begin{array}{ll} A_y(c,\kappa) & \textrm{if $\varsigma=\rho_0$} \\ A_y(c,\kappa-1) & \textrm{if $\varsigma=\sigma_0$ and $\kappa \geq 1$} \\ 0 & \textrm{otherwise} \end{array} \right.$$
[*Forget node*]{}: Let $x$ be a forget node in $T$ with child node $y$ forgetting the vertex $v$.
The operations performed in the forget node are quite complicated. Here, we must update the states such that they are correct after forgetting the vertex $v$, and we must select those solutions that satisfy the constraints imposed on $v$ by the specific $[\rho,\sigma]$-domination problem. We will do this in three steps: we compute intermediate tables $A_1$, $A_2$ in the first two steps and finally $A_x$ in step three. Let $c(N(v))$ be the subcolouring of $c$ restricted to vertices in $N(v)$.
*Step 1*: We update the states used on the vertex $v$. We do so to include the neighbours in $D$ that the vertex $v$ has inside the bag $X_x$ in the states used to represent the different characteristics. Notice that after including these neighbours, the subscripts of the states on $v$ represent the total number of neighbours that $v$ has in $D$. The result will be the table $A_1$, which we compute using the following formulas where $\#_\sigma(c)$ stands for the number of $\sigma$-states in the colouring $c$: $$\begin{aligned}
A_1(c \times \{ \rho_j \}, \kappa) & = & \left\{ \begin{array}{ll} A_y(c \times \{ \rho_{j - \#_\sigma(c(N(v)))} \} , \kappa) & \textrm{if $j \geq \#_\sigma(c(N(v)))$} \\ 0 & \textrm{otherwise} \end{array} \right. \\
A_1(c \times \{ \sigma_j \}, \kappa) & = & \left\{ \begin{array}{ll} A_y(c \times \{ \sigma_{j - \#_\sigma(c(N(v)))} \} , \kappa) & \textrm{if $j \geq \#_\sigma(c(N(v)))$} \\ 0 & \textrm{otherwise} \end{array} \right.\end{aligned}$$ If $\rho$ or $\sigma$ are cofinite, we also need the following formulas: $$\begin{aligned}
A_1(c \times \{ \rho_{\geq p} \}, \kappa) & = & A_y(c \times \{ \rho_{\geq p} \}, \kappa) + \sum_{i = p - \#_\sigma(c(N(v)))}^{p-1} A_y(c \times \{ \rho_i \},\kappa) \\
A_1(c \times \{ \sigma_{\geq q} \}, \kappa) & = & A_y(c \times \{ \sigma_{\geq q} \}, \kappa) + \sum_{i = q - \#_\sigma(c(N(v)))}^{q-1} A_y(c \times \{ \sigma_i \},\kappa)\end{aligned}$$ Correctness of these formulas is easy to verify.
*Step 2*: We update the states representing the neighbours of $v$ such that they are according to their definitions after forgetting $v$. All the required information to do this can again be read from the colouring $c$.
We apply Lemma \[lem:rsstates\] and change the state representation for the vertices in $N(v)$ to State Set III (Definition \[def:rsstates\]) obtaining the table $A_1'(c,\kappa)$; we do not change the representation of other vertices in the bag. That is, if $\rho$ or $\sigma$ are cofinite, we replace the last state $\rho_{\geq p}$ or $\sigma_{\geq q}$ by $\rho_{\geq p-1}$ or $\sigma_{\geq q-1}$, respectively, on vertices in $X_y \cap N(v)$. We can do so as discussed below the proof of \[lem:rsstates\].
This state change allows us to extract the required values for the table $A_2$, as we will show next. We introduce the function $\phi$ that will send a colouring using State Set I to a colouring that uses State Set I on the vertices in $X_y \setminus N(v)$ and State Set III on the vertices in $X_y \cap N(v)$. This function updates the states used on $N(v)$ assuming that we would put $v$ in the vertex set $D$ of the partial solution. We define $\phi$ in the following way: it maps a colouring $c$ to a new colouring with the same states on vertices in $X_y \setminus N(v)$ while it applies the following replacement rules on the states on vertices in $X_y \cap N(v)$: $\rho_1 \mapsto \rho_0$, $\rho_2 \mapsto \rho_1$, …, $\rho_p \mapsto \rho_{p-1}$, $\rho_{\geq p} \mapsto \rho_{\geq p-1}$, $\sigma_1 \mapsto \sigma_0$, $\sigma_2 \mapsto \sigma_1$, …, $\sigma_q \mapsto \sigma_{q-1}$, $\sigma_{\geq q} \mapsto \sigma_{\geq q-1}$. Thus, $\phi$ lowers the counters in the conditions that index the states by one for states representing vertices in $N(v)$. We note that $\phi(c)$ is defined only if $\rho_0, \sigma_0 \not\in c$.
Using this function, we can easily update our states as required: $$\begin{aligned}
A_2( c \times \{ \sigma_j \}, \kappa) & = & \left\{ \begin{array}{ll} A'_1(\phi(c) \times \{\sigma_j\},\kappa) & \textrm{if $\rho_0,\sigma_0 \not\in c(N(v))$} \\ 0 & \textrm{otherwise} \end{array} \right. \\
A_2( c \times \{ \rho_j \}, \kappa) & = & A'_1(c \times \{ \rho_j \}, \kappa)\end{aligned}$$ In words, for partial solutions on which the vertex $v$ that we will forget has a $\sigma$-state, we update the states for vertices in $X_y \cap N(v)$ such that the vertex $v$ is counted in the subscript of the states. Entries in $A_2$ are set to 0 if the states count no neighbours in $D$ while $v$ has a $\sigma$-state in $c$ and thus a neighbour in $D$ in this partial solution.
Notice that after updating the states using the above formula the colourings $c$ in $A_2$ again uses State Set I from Definition \[def:rsstates\].
*Step 3*: We select the solutions that satisfy the constraints of the specific $[\rho,\sigma]$-domination problem on $v$ and forget $v$. $$A_x(c, \kappa) = \left( \sum_{i \in \rho} A_2(c \times \{ \rho_i \}, \kappa) \right) + \left( \sum_{i \in \sigma} A_2(c \times \{ \sigma_i \}, \kappa) \right)$$ We slightly abuse our notation here when $\rho$ or $\sigma$ are cofinite. Following the discussion of the construction of the table $A_x$, we conclude that this correctly computes the required values.
[*Join node*]{}: Let $x$ be a join node in $T$ and let $l$ and $r$ be its child nodes. Computing the table $A_x$ for the join node $x$ is the most interesting operation.
First, we transform the tables $A_l$ and $A_r$ of the child nodes such that they use State Set II (Definition \[def:rsstates\]) and are indexed by index vectors using Lemma \[lem:rsstates2\]. As a result, we obtain tables $A'_l$ and $A'_r$ with entries $A'_l(c,\kappa,\vec{g})$ and $A'_r(c,\kappa,\vec{h})$. These entries count the number of partial solutions of size $\kappa$ corresponding to the colouring $c$ such that the sum of the number of neighbours in $D$ of the set of vertices with each state equals the value that the index vectors $\vec{g}$ and $\vec{h}$ indicate. Here, $D$ is again the vertex set of the partial solution involved. See the example above the statement of Lemma \[lem:rsstates2\].
Then, we compute the table $A_x(c,\kappa,\vec{i})$ by combining identical states from $A'_l$ and $A'_r$ using the formula below. In this formula, we sum over all ways of obtaining a partial solution of size $\kappa$ by combining the sizes in the tables of the child nodes and all ways of obtaining index vector $\vec{i}$ from $\vec{i} = \vec{g} + \vec{h}$. $$A'_x(c,\kappa,\vec{i}) = \sum_{\kappa_l + \kappa_r = \kappa + \#_\sigma(c)} \left( \sum_{i_{\rho1}=g_{\rho1}+h_{\rho1}} \!\!\cdots\!\! \sum_{i_{\sigma q}=g_{\sigma q}+h_{\sigma q}} A'_l(c,\kappa_l,\vec{g}) \cdot A'_r(c,\kappa_r,\vec{h}) \right)$$
We observe the following: a partial solution $D$ in $A'_x$ that is a combination of partial solutions from $A'_l$ and $A'_r$ is counted in an entry in $A'_x(c,\kappa,\vec{i})$ if and only if it satisfies the following three conditions.
1. The sum over all vertices with state $\rho_{\leq j}$ and $\sigma_{\leq j}$ of the number of neighbours of the vertex in $D$ of this combined partial solution equals $i_{\rho j}$ or $i_{\sigma j}$, respectively.
2. The number of neighbours in $D$ of each vertex with state $\rho_{\leq j}$ or $\sigma_{\leq j}$ of both partial solutions used to create this combined solution is at most $j$.
3. The total number of vertices in $D$ in this joined solution is $\kappa$.
Let $\Sigma_\rho^l(c)$, $\Sigma_\sigma^l(c)$ be the weighted sums of the number of $\rho_j$-states and $\sigma_j$-states with $0 \leq j \leq l$ in $c$, respectively, defined by: $$\Sigma_\rho^l(c) = \sum_{j=1}^l j \cdot \#_{\rho_j}(c) \qquad \qquad \Sigma_\sigma^l(c) = \sum_{j=1}^l j \cdot \#_{\sigma_j}(c)$$ We note that $\Sigma_\rho^1(c) = \#_{\rho_1}(c)$ and $\Sigma_\sigma^1(c) = \#_{\sigma_1}(c)$.
Now, using Lemma \[lem:rsstates\], we change the states used in the table $A'_x$ back to State Set I. If $\rho$ and $\sigma$ are finite, we extract the values computed for the final table $A_x$ in the following way: $$A_x(c,\kappa) = A'_x \left( c, \; \kappa, \; (\Sigma_\rho^1(c),\Sigma_\rho^2(c),\ldots,\Sigma_\rho^p(c),\Sigma_\sigma^1(c),\Sigma_\sigma^2(c),\ldots,\Sigma_\sigma^q(c)) \; \right)$$ If $\rho$ or $\sigma$ are cofinite, we use the same formula but omit the components $\Sigma_\rho^p(c)$ or $\Sigma_\sigma^q(c)$ from the index vector of the extracted entries, respectively.
Below, we will prove that the entries in $A_x$ are exactly the values that we want to compute. We first give some intuition. In essence, the proof is a generalisation of how we performed the join operation for counting the number of perfect matchings in the proof of Theorem \[thrm:countingpmtwalg\]. State Set II has the role of the $?$-states in the proof of Theorem \[thrm:countingpmtwalg\]. These states are used to count possible combinations of partial solutions from $A_l$ and $A_r$. These combinations include incorrect combinations in the sense that a vertex can have more neighbours in $D$ than it should have; this is analogous to counting the number of perfect matchings, where combinations were incorrect if a vertex is matched twice. The values $\Sigma_\rho^l(c)$ and $\Sigma_\sigma^l(c)$ represent the total number of neighbours in $D$ of the vertices with a $\rho_j$-states or $\sigma_j$-states with $0 \leq j \leq l$ in $c$, respectively. The above formula uses these $\Sigma_\rho^l(c)$ and $\Sigma_\sigma^l(c)$ to extract exactly those values from the table $A'_x$ that correspond to correct combinations. That is, in this case, correct combinations for which the number of neighbours of a vertex in $D$ is also correctly represented by the new states.
We will now prove that the computation of the entries in $A_x$ gives the correct values. An entry in $A_x(c,\kappa)$ with $c \in \{\rho_0,\sigma_0\}^k$ is correct: these states are unaffected by the state changes and the index vector is not used. The values of these entries follow from combinations of partial solutions from both child nodes corresponding to the same states on the vertices.
Now consider an entry in $A_x(c,\kappa)$ with $c \in \{\rho_0,\rho_1,\sigma_0\}^k$. Each $\rho_1$-state comes from a $\rho_{\leq 1}$-state in $A_x'(c,\kappa,\vec{i})$ and is a combination of partial solutions from $A_l$ and $A_r$ with the following combinations of states on this vertex: $(\rho_0,\rho_0)$, $(\rho_0,\rho_1)$, $(\rho_1,\rho_0)$, $(\rho_1,\rho_1)$. Because we have changed states back to State Set I, each $(\rho_0,\rho_0)$ combination is counted in the $\rho_0$-state on this vertex, and thus subtracted from the combinations used to form state $\rho_1$: the other three combinations remain counted in the $\rho_1$-state. Since we consider only those solutions with index vector $i_{\rho_1} = \Sigma_\rho^1(c)$, the total number of $\rho_1$-states used to form this joined solution equals $\Sigma_\rho^1(c) = \#_{\rho_1}(c)$. Therefore, no $(\rho_1,\rho_1)$ combination could have been used, and each partial solution counted in $A(c,\kappa)$ has exactly one neighbour in $D$ on each of the $\rho_1$-states, as required.
We can now inductively repeat this argument for the other states. For $c \in \{\rho_0,\rho_1,\rho_2,\sigma_0\}^k$, we know that the entries with only $\rho_0$-states and $\rho_1$-states are correct. Thus, when a $\rho_2$-state is formed from a $\rho_{\leq 2}$-state during the state transformation of Lemma \[lem:rsstates\], all nine possibilities of getting the state $\rho_{\leq 2}$ from the states $\rho_0$, $\rho_1$, and $\rho_2$ in the child bags are counted, and from this number all three combinations that should lead to a $\rho_0$ and $\rho_1$ in the join are subtracted. What remains are the combinations $(\rho_0,\rho_2)$, $(\rho_1,\rho_2)$, $(\rho_2,\rho_2)$, $(\rho_1,\rho_1)$, $(\rho_1,\rho_2)$, $(\rho_2,\rho_0)$. Because of the index vector of the specific the entry we extracted from $A'_x$, the total sum of the number of neighbours in $D$ of these vertices equals $\Sigma_\rho^2$, and hence only the combinations $(\rho_0,\rho_2)$, $(\rho_1,\rho_1)$, and $(\rho_2,\rho_0)$ could have been used. Any other combination would raise the component $i_{\rho2}$ of $\vec{i}$ to a number larger than $\Sigma_\rho^2$.
If we repeat this argument for all states involved, we conclude that the above computation correctly computes $A_x$ if $\rho$ and $\sigma$ are finite. If $\rho$ or $\sigma$ are cofinite, then the argument can also be used with one small difference. Namely, the index vectors are one component shorter and keep no index for the states $\rho_{{\ensuremath{\mathbb{N}}}}$ and $\sigma_{{\ensuremath{\mathbb{N}}}}$. That is, at the point in the algorithm where we introduce these index vectors and transform to State Set II using Lemma \[lem:rsstates2\], we have no index corresponding to the sum of the number of neighbours in the vertex set $D$ of the partial solution of the vertices with states $\rho_{{\ensuremath{\mathbb{N}}}}$ and $\sigma_{{\ensuremath{\mathbb{N}}}}$. However, we do not need to select entries corresponding to having $p$ or $q$ neighbours in $D$ for the states $\rho_{\geq p}$ and $\sigma_{\geq q}$ since these correspond to all possibilities of getting at least $p$ or $q$ neighbours in $D$. When we transform the states back to State Set I just before extracting the values for $A_x$ from $A'_x$, entries that have the state $\rho_{\geq p}$ or $\sigma_{\geq q}$ after the transformation count all possible combinations of partial solutions except those counted in any of the other states. This is exactly what we need since all combinations with less than $p$ (or $q$) neighbours are present in the other states.
After traversing the whole decomposition tree $T$, one can find the number of $[\rho,\sigma]$-dominating sets of size $\kappa$ in the table computed for the root node $z$ of $T$ in $A_z(\emptyset,\kappa)$.
We conclude with an analysis of the running time. The most time-consuming computations are again those involved in computing the table $A_x$ for a join node $x$. Here, we need ${\ensuremath{\mathcal{O}}}(n(sk)^{s-1} s^{k+1} i_+(n))$ time for the transformations of Lemma \[lem:rsstates2\] that introduce the index vectors since $\max\{ |X_x| \;|\; x \in T\} = k+1$. However, this is still dominated by the time required to compute the table $A'_x$: this table contains at most $s^{k+1} n (sk)^{s-2}$ entries $A'_x(c,\kappa,\vec{i})$, each of which is computed by an $n (sk)^{s-2}$-term sum. This gives a total time of ${\ensuremath{\mathcal{O}}}(n^2 (sk)^{2(s-2)} s^{k+1} i_\times(n))$ since we use $n$-bit numbers. Because the nice tree decomposition has ${\ensuremath{\mathcal{O}}}(n)$ nodes, we conclude that the algorithm runs in ${\ensuremath{\mathcal{O}}}(n^3 (sk)^{2(s-2)} s^{k+1} i_\times(n))$ time in total.
This proof generalises ideas from the fast subset convolution algorithm [@BjorklundHKK07]. While convolutions use ranked Möbius transforms [@BjorklundHKK07], we use transformations with multiple states and multiple ranks in our index vectors.
The polynomial factors in the proof of Theorem \[thrm:rstwalg\] can be improved in several ways. Some improvements we give are for $[\rho,\sigma]$-domination problems in general, and others apply only to specific problems. Similar to $s = p + q + 2$, we define the value $r$ associated with a $[\rho, \sigma]$-domination problems as follows: $$r = \left\{ \begin{array}{ll} \max\{p-1,q-1\} & \textrm{if $\rho$ and $\sigma$ are cofinite} \\ \max\{p,q-1\} & \textrm{if $\rho$ is finite and $\sigma$ is cofinite} \\ \max\{p-1,q\} & \textrm{if $\rho$ is confinite and $\sigma$ is finite} \\ \max\{p,q\} & \textrm{if $\rho$ and $\sigma$ are finite} \end{array} \right.$$
\[cor:generalrstwalg\] Let $\rho, \sigma \subseteq {\ensuremath{\mathbb{N}}}$ be finite or cofinite, and let $p$, $q$, $r$, and $s$ be the values associated with the corresponding $[\rho,\sigma]$-domination problem. There is an algorithm that, given a tree decomposition of a graph $G$ of width $k$, computes the number of $[\rho,\sigma]$-dominating sets in $G$ of each size $\kappa$, $0 \leq \kappa \leq n$, in ${\ensuremath{\mathcal{O}}}(n^3 (rk)^{2r} s^{k+1} i_\times(n))$ time. Moreover, there is an algorithm that decides whether there exist a $[\rho,\sigma]$-dominating set of size $\kappa$, for each individual value of $\kappa$, $0 \leq \kappa \leq n$, in ${\ensuremath{\mathcal{O}}}(n^3 (rk)^{2r} s^{k+1} i_\times(log(n)+k\log(r)))$ time.
We improve the polynomial factor $(sk)^{2(s-2)}$ to $(rk)^{2r}$ by making the following observation. We never combine partial solutions corresponding to a $\rho$-state in one child node with a partial solution corresponding to a $\sigma$-state on the same vertex in the other child node. Therefore, we can combine the components of the index vector related to the states $\rho_j$ and $\sigma_j$ for each fixed $j$ in a single index. For example consider the $\rho_1$-states and $\sigma_1$-states. For these states, this means the following: if we index the number of vertices used to create a $\rho_1$-state and $\sigma_1$-state in $i_1$ and we have $i_1$ vertices on which a partial solution is formed by considering the combinations $(\rho_0,\rho_1)$, $(\rho_1,\rho_0)$, $(\rho_1,\rho_1)$, $(\sigma_0,\sigma_1)$, $(\sigma_1,\sigma_0)$, or $(\sigma_1,\sigma_1)$, then non of the combinations $(\rho_1,\rho_1)$ and $(\sigma_1,\sigma_1)$ could have been used. Since the new components of the index vector range between $0$ and $rk$, this proves the first running time in the statement of the corollary.
The second running time follows from reasoning similar to that in Corollary \[cor:solvedstwalg\]. In this case, we can stop counting the number of partial solutions of each size and instead keep track of the existence of a partial solution of each size. The state transformations then count the number of $1$-entries in the initial tables instead of the number of solutions. After computing the table for a join node, we have to reset all entries $e$ of $A_x$ to $\min\{1,e\}$. For these computations, we can use ${\ensuremath{\mathcal{O}}}(\log(n)+k\log(r))$-bit numbers. This is because of the following reasoning. For a fixed colouring $c$ using State Set II, each of the at most $r^{k+1}$ colourings using State Set I that can be counted in $c$ occur with at most one index vector in the tables $A'_l$ and $A'_r$. Note that these are $r^{k+1}$ colourings, not $s^{k+1}$ colourings, since $\rho$-states are never counted in a colouring $c$ where the vertex has a $\sigma$-state and vice versa. Therefore, the result of the large summation over all index vectors $\vec{g}$ and $\vec{h}$ with $\vec{i}=\vec{g}+\vec{h}$ can be bounded from above by $(r^k)^2$. Since we sum over $n$ possible combinations of sizes, the maximum is $nr^{2k}$ allowing us to use ${\ensuremath{\mathcal{O}}}(\log(n)+k\log(r))$-bit numbers.
As a result, we can, for example, compute the size of a minimum-cardinality perfect code in ${\ensuremath{\mathcal{O}}}(n^3k^23^k i_\times(\log(n)))$ time. Note that the time bound follows because the problem is fixed and we use a computational model with ${\ensuremath{\mathcal{O}}}(k)$-bit word size.
\[cor:defluiterrstwalg\] Let $\rho, \sigma \subseteq {\ensuremath{\mathbb{N}}}$ be finite or cofinite, and let $p$, $q$, $r$, and $s$ be the values associated with the corresponding $[\rho,\sigma]$-domination problem. If the standard representation using State Set I of the minimisation (or maximisation) variant of this $[\rho,\sigma]$-domination problem has the de Fluiter property for treewidth with function $f$, then there is an algorithm that, given a tree decomposition of a graph $G$ of width $k$, computes the number of minimum (or maximum) $[\rho,\sigma]$-dominating sets in $G$ in ${\ensuremath{\mathcal{O}}}(n (f(k))^2 (rk)^{2r} s^{k+1} i_\times(n))$ time. Moreover, there is an algorithm that computes the minimum (or maximum) size of such a $[\rho,\sigma]$-dominating set in ${\ensuremath{\mathcal{O}}}(n (f(k))^2 (rk)^{2r} s^{k+1} i_\times(log(n)+k\log(r)))$ time.
The difference with the proof of Corollary \[cor:generalrstwalg\] is that, similar to the proof of Corollary \[cor:countmdstwalg\], we can keep track of the minimum or maximum size of a partial solution in each node of the tree decomposition and consider only other partial solutions whose size differs at most $f(k)$ of this minimum or maximum size. As a result, both factors $n$ (the factor $n$ due to the size of the tables, and the factor $n$ due to the summation over the sizes of partial solutions) are replaced by a factor $f(k)$.
As an application of Corollary \[cor:defluiterrstwalg\], it follows for example that [2-Dominating Set]{} can be solved in ${\ensuremath{\mathcal{O}}}(n k^6 4^k i_\times(\log(n)))$ time.
\[cor:decisionrstwalg\] Let $\rho, \sigma \subseteq {\ensuremath{\mathbb{N}}}$ be finite or cofinite, and let $p$, $q$, $r$, and $s$ be the values associated with the corresponding $[\rho,\sigma]$-domination problem. There is an algorithm that, given a tree decomposition of a graph $G$ of width $k$, counts the number of $[\rho,\sigma]$-dominating sets in $G$ in ${\ensuremath{\mathcal{O}}}(n (rk)^{2r} s^{k+1} i_\times(n))$ time. Moreover, there is an algorithm that decides whether there exists a $[\rho,\!\sigma]$-dominating set in ${\ensuremath{\mathcal{O}}}(n(rk)^{2r} s^{k+1} i_\times(log(n)+k\log(r)))$ time.
This result follows similarly as Corollary \[cor:defluiterrstwalg\]. In this case, we can omit the size parameter from our tables, and we can remove the sum over the sizes in the computation of entries of $A'_x$ completely.
As an application of Corollary \[cor:decisionrstwalg\], it follows for example that we can compute the number of strong stable sets (distance-2 independent sets) in ${\ensuremath{\mathcal{O}}}(nk^23^k i_\times(n))$ time.
Clique Covering, Packing and Partitioning Problems {#sec:cliquetwalg}
--------------------------------------------------
The final class of problems that we consider for our tree decomposition-based-algorithms are the clique covering, packing, and partitioning problems. To give a general result, we defined the $\gamma$-clique covering, $\gamma$-clique packing, and $\gamma$-clique partitioning problems in Section \[sec:problems\]; see Definition \[def:cliqueproblems\]. For these $\gamma$-clique problems, we obtain ${\ensuremath{\mathcal{O}^*}}(2^k)$ algorithms.
Although any natural problem seems to satisfy this restriction, we remind the reader that we restrict ourselves to polynomial-time decidable $\gamma$, that is, given an integer $j$, we can decide in time polynomial in $j$ whether $j \in \gamma$ or not. This allows us to precompute $\gamma \cap \{1,2,\ldots,k+1\}$ in time polynomial in $k$, after which we can decide in constant time whether a clique of size $l$ is allowed to be used in an associated covering, packing, or partitioning.
We start by giving algorithms for the $\gamma$-clique packing and partitioning problems.
\[thrm:cliqueparttwalg\] Let $\gamma \subseteq {\ensuremath{\mathbb{N}}}\setminus \{0\}$ be polynomial-time decidable. There is an algorithm that, given a tree decomposition of a graph $G$ of width $k$, computes the number of $\gamma$-clique packings or $\gamma$-clique partitionings of $G$ using $\kappa$, $0 \leq \kappa \leq n$, cliques in ${\ensuremath{\mathcal{O}}}(n^3 k^2 2^k i_\times(nk + n\log(n)))$ time.
Before we start dynamic programming on the tree decomposition $T$, we first compute the set $\gamma \cap \{1,2,\dots,k+1\}$.
We use states $0$ and $1$ for the colourings $c$, where $1$ means that a vertex is already in a clique in the partial solution, and $0$ means that the vertex is not in a clique in the partial solution. For each node $x \in T$, we compute a table $A_x$ with entries $A_x(c,\kappa)$ containing the number of $\gamma$-clique packings or partitionings of $G_x$ consisting of exactly $\kappa$ cliques that satisfy the requirements defined by the colouring $c \in \{1,0\}^{|X_x|}$, for all $0 \leq \kappa \leq n$.
The algorithm uses the well-known property of tree decompositions that for every clique $C$ in the graph $G$, there exists a node $x \in T$ such that $C$ is contained in the bag $X_x$ (a nice proof of this property can be found in [@BodlaenderM93]). As every vertex in $G$ is forgotten in exactly one forget node in $T$, we can implicitly assign a unique forget node $x_C$ to every clique $C$, namely the first forget node that forgets a vertex from $C$. In this forget node $x_C$, we will update the dynamic programming tables such that they take the choice of whether to pick $C$ in a solution into account.
[*Leaf node*]{}: Let $x$ be a leaf node in $T$. We compute $A_x$ in the following way: $$A_x(\{0\},\kappa) = \left\{ \begin{array}{ll} 1 & \textrm{if $\kappa = 0$} \\ 0 & \textrm{otherwise} \end{array} \right. \qquad \qquad A_x(\{1\},\kappa) = 0$$ Since we decide to take cliques in a partial solution only in the forget nodes, the only partial solution we count in $A_x$ is the empty solution.
[*Introduce node*]{}: Let $x$ be an introduce node in $T$ with child node $y$ introducing the vertex $v$. Deciding whether to take a clique in a solution in the corresponding forget nodes makes the introduce operation trivial since the introduced vertex must have state $0$: $$A_x(c \times \{1\},\kappa) = 0 \qquad \qquad A_x(c \times \{0\},\kappa) = A_y(c,\kappa)$$
[*Join node*]{}: In contrast to previous algorithms, we will first present the computations in the join nodes. We do so because we will use this operation as a subroutine in the forget nodes.
[2]{}
$\times$ $0$ $1$
---------- ----- -----
$0$ $0$ $1$
$1$ $1$
\
$\times$ $0$ $1$
---------- ----- -----
$0$ $0$ $1$
$1$ $1$ $1$
Let $x$ be a join node in $T$ and let $l$ and $r$ be its child nodes. For the $\gamma$-clique partitioning and packing problems, the join is very similar to the join in the algorithm for counting the number of perfect matchings (Theorem \[thrm:countingpmtwalg\]). This can be seen from the corresponding join table; see Figure \[fig:jointableclique\]. The only difference is that we now also have the size parameter $\kappa$. Hence, for $y \in \{l,r\}$, we first create the tables $A'_y$ with entries $A'_y(c,\kappa,i)$, where $i$ indexes the number of $1$-states in $c$. Then, we transform the set of states used for these tables $A'_y$ from $\{1,0\}$ to $\{0,?\}$ using Lemma \[lem:pmstates\], and compute the table $A'_x$, now with the extra size parameter $\kappa$, using the following formula: $$A'_x(c,\kappa,i) = \sum_{\kappa_l + \kappa_r = \kappa} \sum_{i = i_l + i_r} A'_l(c,\kappa_l,i_l) \cdot A'_r(c,\kappa_r,i_r)$$ Finally, the states in $A'_x$ are transformed back to the set $\{0,1\}$, after which the entries of $A_x$ can be extracted that correspond to the correct number of 1-states in $c$. Because the approach described above is a simple extension of the join operation in the proof of Theorem \[thrm:countingpmtwalg\] which was also used in the proof of Theorem \[thrm:rstwalg\], we omit further details.
[*Forget node*]{}: Let $x$ be a forget node in $T$ with child node $y$ forgetting the vertex $v$. Here, we first update the table $A_y$ such that it takes into account the choice of taking any clique in $X_y$ that contains $y$ in a solution or not.
Let $M$ be a table with all the (non-empty) cliques $C$ in $X_y$ that contain the vertex $v$ and such that $|C| \in \gamma$, i.e., $M$ contains all the cliques that we need to consider before forgetting the vertex $v$. We notice that the operation of updating $A_y$ such that it takes into account all possible ways of choosing the cliques in $M$ is identical to letting the new $A_y$ be the result of the join operation on $A_y$ and the following table $A_M$: $$\begin{aligned}
A_M(c \!\times\! \{1\},\kappa) & \!\!\!=\!\!\! & \left\{ \begin{array}{ll} 1 & \textrm{if all the 1-states in $c$ form a clique with $v$ in $M$ and $\kappa \!=\! 1$} \\ 1 & \textrm{if $c$ is the colouring with only 0-states and $\kappa \!=\! 0$} \\ 0 & \textrm{otherwise} \end{array} \right. \\
A_M(c \!\times\! \{0\},\kappa) & \!\!\!=\!\!\! & 0\end{aligned}$$ It is not hard to see that this updates $A_y$ as required since $A_M(c,\kappa)$ is non-zero only when a clique in $M$ is used with size $\kappa = 1$, or if no clique is used and $\kappa = 0$.
If we consider a partitioning problem, then $A_x(c,\kappa) = A_y(c \times \{1\},\kappa)$ since $v$ must be contained in a clique. If we consider a packing problem, then $A_x(c,\kappa) = A_y(c \times \{1\},\kappa) + A_y(c \times \{0\},\kappa)$ since $v$ can but does not need to be in a clique. Clearly, this correctly computes $A_y$.
After computing $A_z$ for the root node $z$ of $T$, the number of $\gamma$-clique packings or partitionings of each size $\kappa$ can be found in $A_z(\emptyset,\kappa)$.
For the running time, we first observe that there are at most ${\ensuremath{\mathcal{O}}}(n2^k)$ cliques in $G$ since $T$ has ${\ensuremath{\mathcal{O}}}(n)$ nodes that each contain at most $k+1$ vertices. Hence, there are at most ${\ensuremath{\mathcal{O}}}((n2^k)^n)$ ways to pick at most $n$ cliques, and we can work with ${\ensuremath{\mathcal{O}}}(nk + n\log(n))$-bit numbers. As a join and a forget operation require ${\ensuremath{\mathcal{O}}}(n^2 k^2 2^k)$ arithmetical operations, the running time is ${\ensuremath{\mathcal{O}}}(n^3 k^2 2^k i_\times(nk + n\log(n)))$.
For the $\gamma$-clique covering problems, the situation is different. We cannot count the number of $\gamma$-clique covers of all possible sizes, as the size of such a cover can be arbitrarily large. Even if we restrict ourselves to counting covers that contain each clique at most once, then we need numbers with an exponential number of bits. To see this, notice that the number of cliques in a graph of treewidth $k$ is at most ${\ensuremath{\mathcal{O}^*}}(2^k)$ since there are at most ${\ensuremath{\mathcal{O}}}(2^k)$ different cliques in each bag. Hence, there are at most $2^{{\ensuremath{\mathcal{O}^*}}(2^k)}$ different clique covers, and these can be counted using only ${\ensuremath{\mathcal{O}^*}}(2^k)$-bit numbers. Therefore, we will restrict ourselves to counting covers of size at most $n$ because minimum covers will never be larger than $n$.
A second difference is that, in a forget node, we now need to consider covering the forgotten vertex multiple times. This requires a slightly different approach.
\[thrm:cliquecovertwalg\] Let $\gamma \subseteq {\ensuremath{\mathbb{N}}}\setminus \{0\}$ be polynomial-time decidable. There is an algorithm that, given a tree decomposition of a graph $G$ of width $k$, computes the size and number of minimum $\gamma$-clique covers of $G$ in ${\ensuremath{\mathcal{O}}}(n^3 \log(k) 2^k i_\times(nk + n\log(n)))$ time.
The dynamic programming algorithm for counting the number of minimum $\gamma$-clique covers is similar to the algorithm of Theorem \[thrm:cliqueparttwalg\]. It uses the same tables $A_x$ for every $x \in T$ with entries $A_x(c,\kappa)$ for all $c \in \{0,1\}^{|X_x|}$ and $0 \leq \kappa \leq n$. And, the computations of these tables in a leaf or introduce node of $T$ are the same.
[*Join node*]{}: Let $x$ be a join node in $T$ and let $l$ and $r$ be its child nodes. The join operation is different from the join operation in the algorithm of Theorem \[thrm:cliqueparttwalg\] as can be seen from Figure \[fig:jointableclique\]. Here, the join operation is similar to our method of handling the $0_1$-states and $0_0$-states for the [Dominating Set]{} problem in the algorithm of Theorem \[thrm:countingtwdsalg\]. We simply transform the states in $A_l$ and $A_r$ to $\{0,?\}$ and compute $A_x$ using these same states by summing over identical entries with different size parameters: $$A_x(c,\kappa) = \sum_{\kappa_l + \kappa_r = \kappa} A_l(c,\kappa_l) \cdot A_r(c,\kappa_r)$$ Then, we obtain the required result by transforming $A_x$ back to using the set of states $\{1,0\}$. We omit further details because this works analogously to Theorem \[thrm:countingtwdsalg\]. The only difference with before is that we use the value zero for any $A_l(c,\kappa_l)$ or $A_r(c,\kappa_r)$ with $\kappa_l, \kappa_r < 0$ or $\kappa_l, \kappa_r > n$ as these never contribute to minimum clique covers.
[*Forget node*]{}: Let $x$ be a forget node in $T$ with child node $y$ forgetting the vertex $v$. In contrast to Theorem \[thrm:cliqueparttwalg\], we now have to consider covering $v$ with multiple cliques. In a minimum cover, $v$ can be covered at most $k$ times because there are no vertices in $X_x$ left to cover after using $k$ cliques from $X_x$.
Let $A_M$ be as in Theorem \[thrm:cliqueparttwalg\]. What we need is a table that contains more than just all cliques that can be used to cover $v$: it needs to count all combinations of cliques that we can pick to cover $v$ at most $k$ times indexed by the number of cliques used. To create this new table, we let $A_M^0 = A_M$, and let $A_M^j$ be the result of the join operation applied to the table $A_M^{j-1}$ with itself. Then, the table $A_M^j$ counts all ways of picking a series of $2^j$ sets $C_1,C_2,\ldots,C_{2^j}$, where each set is either the empty set or a clique from $M$. To see that this holds, compare the definition of the join operation for this problem to the result of executing these operations repeatedly. The algorithm computes $A_M^{\lceil\log(k)\rceil}$. Because we want to know the number of clique covers that we can choose, and not the number of series of $2^{\lceil\log(k)\rceil}$ sets $C_1,C_2,\ldots,C_{2^{\lceil\log(k)\rceil}}$, we have to compensate for the fact that most covers are counted more than once. Clearly, each cover consisting of $\kappa$ cliques corresponds to a series in which $2^{\lceil\log(k)\rceil} - \kappa$ empty sets are picked: there are $\binom{2^{\lceil\log(k)\rceil}}{\kappa}$ possibilities of picking the empty sets and $\kappa!$ permutations of picking each of the $\kappa$ cliques in any order. Hence, we divide each entry $A_M(c,\kappa)$ by $\kappa! \, \binom{2^{\lceil\log(k)\rceil}}{\kappa}$. Now, $A_M^{\lceil\log(k)\rceil}$ contains the numbers we need for a join with $A_y$.
After performing the join operation with $A_y$ and $A_M^{\lceil\log(k)\rceil}$ obtaining a new table $A_y$, we select the entries of $A_y$ that cover $v$: $A_x(c,\kappa) = A_y(c \times \{1\},\kappa)$.
If we have computed $A_z$ for the root node $z$ of $T$, the size of the minimum $\gamma$-clique cover equals the smallest $\kappa$ for which $A_z(\emptyset,\kappa)$ is non-zero, and this entry contains the number of such sets.
For the running time, we find that in order to compute $A_M^{\lceil\log(k)\rceil}$, we need ${\ensuremath{\mathcal{O}}}(\log(k))$ join operations. The running time then follows from the same analysis as in Theorem \[thrm:cliqueparttwalg\].
Similar to previous results, we can improve the polynomial factors involved.
\[cor:cliqueparttwalg\] Let $\gamma \subseteq {\ensuremath{\mathbb{N}}}\setminus \{0\}$ be polynomial-time decidable. There are algorithms that, given a tree decomposition of a graph $G$ of width $k$:
1. decide whether there exists a $\gamma$-clique partition of $G$ in ${\ensuremath{\mathcal{O}}}(n k^2 2^k)$ time.
2. count the number of $\gamma$-clique packings in $G$ or the number of $\gamma$-clique partitionings in $G$ in ${\ensuremath{\mathcal{O}}}(n k^2 2^k i_\times(nk + n\log(n)))$ time.
3. compute the size of a maximum $\gamma$-clique packing in $G$, maximum $\gamma$-clique partitioning in $G$, or minimum $\gamma$-clique partitioning in $G$ of a problem with the de Fluiter property for treewidth in ${\ensuremath{\mathcal{O}}}(n k^4 2^k)$ time.
4. compute the size of a minimum $\gamma$-clique cover in $G$ of a problem with the de Fluiter property for treewidth in ${\ensuremath{\mathcal{O}}}(n k^2 \log(k) 2^k)$ time, or in in ${\ensuremath{\mathcal{O}}}(n k^2 2^k)$ time if $|\gamma|$ is a constant.
5. compute the number of maximum $\gamma$-clique packings in $G$, maximum $\gamma$-clique partitionings in $G$, or minimum $\gamma$-clique partitionings in $G$ of a problem with the de Fluiter property for treewidth in ${\ensuremath{\mathcal{O}}}(n k^4 2^k i_\times(nk + n\log(n)))$ time.
6. compute the number of minimum $\gamma$-clique covers in $G$ of a problem with the de Fluiter property for treewidth in ${\ensuremath{\mathcal{O}}}(n k^2 \log(k) 2^k i_\times(nk + n\log(n)))$ time, or in ${\ensuremath{\mathcal{O}}}(n k^2 2^k i_\times(nk + n\log(n)))$ time if $|\gamma|$ is a constant.
Similar to before. Either use the de Fluiter property to replace a factor $n^2$ by $k^2$, or omit the size parameter to completely remove this factor $n^2$ if possible. Moreover, we can use ${\ensuremath{\mathcal{O}}}(k)$-bit numbers instead of ${\ensuremath{\mathcal{O}}}(nk + n\log(n))$-bit numbers if we are not counting the number of solutions. In this case, we omit the time required for the arithmetic operations because of the computational model that we use with ${\ensuremath{\mathcal{O}}}(k)$-bit word size. For the $\gamma$-clique cover problems where $|\gamma|$ is a constant, we note that we can use $A_M^p$ for some constant $p$ because, in a forget node, we need only a constant number of repetitions of the join operation on $A_M^0$ instead of $\log(k)$ repetitions.
By this result, [Partition Into Triangles]{} can be solved in ${\ensuremath{\mathcal{O}}}(n k^2 2^k)$ time. For this problem, Lokshtanov et al. proved that the given exponential factor in the running time is optimal, unless the Strong Exponential-Time Hypothesis fails [@LokshtanovMS10].
We note that in Corollary \[cor:cliqueparttwalg\] the problem of deciding whether there exists a $\gamma$-clique cover is omitted. This is because this problem can easily be solved without dynamic programming on the tree decomposition by considering each vertex and testing whether it is contained in a clique whose size is a member of $\gamma$. This requires ${\ensuremath{\mathcal{O}}}(nk2^k)$ time in general, and polynomial time if $|\gamma|$ is a constant.
Dynamic Programming on Branch Decompositions {#sec:branchwidth}
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Dynamic programming algorithms on branch decompositions work similar to those on tree decompositions. The tree is traversed in a bottom-up manner while computing tables $A_e$ with partial solutions on $G_e$ for every *edge* $e$ of $T$ (see the definitions in Section \[sec:defbw\]). Again, the table $A_e$ contains partial solutions of each possible *characteristic*, where two partial solutions $P_1$ and $P_2$ have the same characteristic if any extension of $P_1$ to a solution on $G$ also is an extension of $P_2$ to a solution on $G$. After computing a table for every edge $e \in E(T)$, we find a solution for the problem on $G$ in the single entry of the table $A_{\{y,z\}}$, where $z$ is the root of $T$ and $y$ is its only child node. Because the size of the tables is often (at least) exponential in the branchwidth $k$, such an algorithm typically runs in ${\ensuremath{\mathcal{O}}}(f(k)poly(n))$ time, for some function $f$ that grows at least exponentially. See Proposition \[prop:simplebwdsalg\] for an example algorithm.
In this section, we improve the exponential part of the running time for many dynamic programming algorithms on branch decompositions. A difference to our results on tree decompositions is that when the number of partial solutions stored in a table is ${\ensuremath{\mathcal{O}^*}}(s^k)$, then our algorithms will run in ${\ensuremath{\mathcal{O}^*}}(s^{\frac{\omega}{2}k})$ time. This difference in the running time is due to the fact that the structure of a branch decomposition is different to the structure of a tree decomposition. A tree decomposition can be transformed into a nice tree decomposition, such that every join node $x$ with children $l$, $r$ has $X_x = X_r = X_l$. But a branch decomposition does not have such a property: here we need to consider combining partial solutions from both tables of the child edges while forgetting and introducing new vertices at the same time.
This section is organised as follows. We start by setting up the framework that we use for dynamic programming on branch decompositions by giving a simple algorithm in Section \[sec:bwframework\]. Hereafter, we give our results on [Dominating Set]{} in Section \[sec:bwds\], our results on counting perfect matchings in Section \[sec:bwcpm\], and our results on the $[\rho,\sigma]$-domination problems in Section \[sec:bwrhosigma\].
General Framework on Branch Decompositions {#sec:bwframework}
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We will first give a simple dynamic programming algorithm for the [Dominating Set]{} problem. This algorithm follows from standard techniques on branchwidth-based algorithms and will be improved later.
\[prop:simplebwdsalg\] There is an algorithm that, given a branch decomposition of a graph $G$ of width $k$, counts the number of dominating sets in $G$ of each size $\kappa$, $0 \leq \kappa \leq n$, in ${\ensuremath{\mathcal{O}}}(m n^2 6^k i_\times(n))$ time.
Let $T$ be a branch decomposition of $G$ rooted at a vertex $z$. For each edge $e \in E(T)$, we will compute a table $A_e$ with entries $A_e(c,\kappa)$ for all $c \in \{1,0_1,0_0\}^{X_e}$ and all $0 \leq \kappa \leq n$. Here, $c$ is a colouring with states $1$, $0_1$, and $0_0$ that have the same meaning as in the tree-decomposition-based algorithms: see Table \[tab:dsstates\]. In the table $A_e$, an entry $A_e(c,\kappa)$ equals the number of partial solutions of [Dominating Set]{} of size $\kappa$ in $G_e$ that satisfy the requirements defined by the colouring $c$ on the vertices in $X_e$. That is, the number of vertex sets $D \subseteq V_e$ of size $\kappa$ that dominate all vertices in $V_e$ except for those with state $0_0$ in colouring $c$ of $X_e$, and that contain all vertices in $X_e$ with state $1$ in $c$.
The described tables $A_e$ are computed by traversing the decomposition tree $T$ in a bottom-up manner. A branch decompositions has only two kinds of edges for which we need to compute such a table: leaf edges, and internal edges which have two child edges.
[*Leaf edges*]{}: Let $e$ be an edge of $T$ incident to a leaf of $T$ that is not the root. Then, $G_e = G[X_e]$ is a two-vertex graph with $X_e = \{u,v\}$. Note that $\{u,v\} \in E$.
We compute $A_e$ in the following way: $$A_e(c,\kappa) = \left\{ \begin{array}{ll} 1 & \textrm{if $\kappa = 2$ and $c=(1,1)$} \\ 1 & \textrm{if $\kappa = 1$ and either $c = (1,0_1)$ or $c = (0_1,1)$} \\ 1 & \textrm{if $\kappa = 0$ and $c=(0_0,0_0)$} \\ 0 & \textrm{otherwise} \end{array} \right.$$ The entries in this table are zero unless the colouring $c$ represents one of the four possible partial solutions of [Dominating Set]{} on $G_e$ and the size of this solution is $\kappa$. In these non-zero entries, the single partial solution represented by $c$ is counted.
[*Internal edges*]{}: Let $e$ be an internal edge of $T$ with child edges $l$ and $r$. Recall the definition of the sets $I$, $L$, $R$, $F$ induced by $X_e$, $X_l$, and $X_r$ (Definition \[def:middlesets\]).
Given a colouring $c$, let $c(I)$ denote the colouring of the vertices of $I$ induced by $c$. We define $c(L)$, $c(R)$, and $c(F)$ in the same way. Given a colouring $c_e$ of $X_e$, a colouring $c_l$ of $X_l$, and a colouring $c_r$ of $X_r$, we say that these colourings *match* if they correspond to a correct combination of two partial solutions with the colourings $c_l$ and $c_r$ on $X_l$ and $X_r$ which result is a partial solution that corresponds to the colouring $c_e$ on $X_e$. For a vertex in each of the four partitions $I$, $L$, $R$, and $F$ of $X_e \cup X_l \cup X_r$, this means something different:
- For any $v \in I$: either $c_e(v) = c_l(v) = c_r(v) \in \{1,0_0\}$, or $c_e(v) = 0_1$ while $c_l(v), c_r(v) \in \{0_0,0_1\}$ and not $c_l(v) = c_r(v) = 0_0$. (5 possibilities)
- For any $v \in F$: either $c_l(v) = c_r(v) = 1$, or $c_l(v), c_r(v) \in \{0_0,0_1\}$ while not $c_l(v) = c_r(v) = 0_0$. (4 possibilities)
- For any $v \in L$: $c_e(v) = c_l(v) \in \{1,0_1,0_0\}$. (3 possibilities)
- For any $v \in R$: $c_e(v) = c_r(v) \in \{1,0_1,0_0\}$. (3 possibilities)
That is, for vertices in $L$ or $R$, the properties defined by the colourings are copied from $A_l$ and $A_r$ to $A_e$. For vertices in $I$, the properties defined by the colouring $c_e$ is a combination of the properties defined by $c_l$ and $c_r$ in the same way as it is for tree decompositions (as in Proposition \[prop:simpletwdsalg\]). For vertices in $F$, the properties defined by the colourings are such that they form correct combinations in which the vertices may be forgotten, i.e., such a vertex is in the vertex set of both partial solutions, or it is not in the vertex set of both partial solutions while it is dominated.
Let $\kappa_{\#_1} = \#_{1}(c_r(I \cup F))$ be the number of vertices that are assigned state $1$ on $I \cup F$ in any matching triple $c_e$, $c_l$, $c_r$. We can count the number of partial solutions on $G_e$ satisfying the requirements defined by each colouring $c_e$ on $X_e$ using the following formula: $$A_e(c_e,\kappa) = \sum_{c_e, c_l, c_r \,\textrm{\scriptsize match}} \; \sum_{\kappa_l + \kappa_r = \kappa + \kappa_{\#_1}} A_l(c_l,\kappa_l) \cdot A_r(c_r,\kappa_r)$$ Notice that this formula correctly counts all possible partial solutions on $G_e$ per corresponding colouring $c_e$ on $X_e$ by counting all valid combinations of partial solutions on $G_l$ corresponding to a colouring $c_l$ on $X_l$ and partial solutions $G_r$ corresponding to a colouring $c_r$ on $X_r$.
Let $\{y,z\}$ be the edge incident to the root $z$ of $T$. From the definition of $A_{\{y,z\}}$, $A_{\{y,z\}}(\emptyset,\kappa)$ contains the number of dominating sets of size $\kappa$ in $G_{\{y,z\}} = G$.
For the running time, we observe that we can compute $A_{e}$ in ${\ensuremath{\mathcal{O}}}(n)$ time for all leaf edges $e$ of $T$. For the internal edges, we have to compute the ${\ensuremath{\mathcal{O}}}(n 3^k)$ values of $A_e$, each of which requires ${\ensuremath{\mathcal{O}}}(n)$ terms of the above sum per set of matchings states. Since each vertex in $I$ has 5 possible matching states, each vertex in $F$ has 4 possible matching states, and each vertex in $L$ or $R$ has 3 possible matching states, we compute each $A_e$ in ${\ensuremath{\mathcal{O}}}(n^25^{|I|}4^{|F|}3^{|L|+|R|} i_\times(n))$ time.
Under the constraint that $|I|+|L|+|R|, |I|+|L|+|F|, |I|+|R|+|F| \leq k$, the running time is maximal if $|I| = 0$, $|L|=|R|=|F|=\frac{1}{2}k$. As $T$ has ${\ensuremath{\mathcal{O}}}(m)$ edges and we work with $n$-bit numbers, this leads to a running time of ${\ensuremath{\mathcal{O}}}(m n^2 4^{\frac{1}{2}k} 3^k i_\times(n)) = {\ensuremath{\mathcal{O}}}(m n^2 6^k i_\times(n))$.
The above algorithm gives the framework that we use in all of our dynamic programming algorithms on branch decompositions. In later algorithms, we will specify only how to compute the tables $A_e$ for both kinds of edges.
#### De Fluiter Propery for Branchwidth.
We conclude this subsection with a discussion on a de Fluiter property for branchwidth. We will see below that such a property for branchwidth is identical to the de Fluiter property for treewidth.
One could define a de Fluiter property for branchwidth by replacing the words treewidth and tree decomposition in Definition \[def:fluiterproptw\] by branchwidth and branch decomposition. However, the result would be a property equivalent to the de Fluiter property for treewidth. This is not hard to see, namely, consider any edge $e$ of a branch decomposition with middle set $X_e$. A representation of the different characteristics on $X_e$ of partial solutions on $G_e$ used on branch decompositions can also be used as a representation of the different characteristics on $X_x$ of partial solutions on $G_x$ on tree decompositions, if $X_e = X_x$ and $G_e = G_x$. Clearly, an extension of a partial solution on $G_e$ with some characteristic is equivalent to an extension of the same partial solution on $G_x = G_e$, and hence the representations can be used on both decompositions. This equivalence of both de Fluiter properties follows directly.
As a result, we will use the de Fluiter property for treewidth in this section. In Section \[sec:cliquewidth\], we will also define a de Fluiter property for cliquewidth; this property will be different from the other two.
Minimum Dominating Set {#sec:bwds}
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We start by improving the above algorithm of Proposition \[prop:simplebwdsalg\]. This improvement will be presented in two steps. First, we use state changes similar to what we did in Section \[sec:dstw\]. Thereafter, we will further improve the result by using fast matrix multiplication in the same way as proposed by Dorn in [@Dorn06]. As a result, we will obtain an ${\ensuremath{\mathcal{O}^*}}(3^{\frac{\omega}{2}k})$ algorithm for [Dominating Set]{} for graphs given with *any* branch decomposition of width $k$.
Similar to tree decompositions, it is more efficient to transform the problem to one using states $1$, $0_0$, and $0_?$ if we want to combine partial solutions from different dynamic programming tables. However, there is a big difference between dynamic programming on tree decompositions and on branch decompositions. On tree decompositions, we can deal with forget vertices separately, while this is not possible on branch decompositions. This makes the situation more complicated. On branch decompositions, vertices in $F$ must be dealt with simultaneously with the computation of $A_e$ from the two tables $A_l$ and $A_r$ for the child edges $l$ and $r$ of $e$. We will overcome this problem by using different sets of states simultaneously. The set of states used depends on whether a vertex is in $L$, $R$, $I$ or $F$. Moreover, we do this asymmetrically as different states can be used on the same vertices in a different table $A_e$, $A_l$, $A_r$. This use of *asymmetrical vertex states* will later allow us to easily combine the use of state changes with fast matrix multiplication and obtain significant improvements in the running time.
We state this use of different states on different vertices formally. We note that this construction has already been used in the proof of Theorem \[thrm:rstwalg\].
\[lem:asymdsstates\] Let $e$ be an edge of a branch decomposition $T$ with corresponding middle set $|X_e|$, and let $A_e$ be a table with entries $A_e(c,\kappa)$ representing the number of partial solutions of [Dominating Set]{} in $G_e$ of each size $\kappa$, for some range of $\kappa$, corresponding to all colourings of the middle set $X_e$ with states such that for every individual vertex in $X_e$ one of the following fixed sets of states is used: $$\{1,0_1,0_0\} \qquad \{1,0_1,0_?\} \qquad \{1,0_0,0_?\} \qquad \textrm{(see Table~\ref{tab:dsstates})}$$ The information represented in the table $A_e$ does not depend on the choice of the set of states from the options given above. Moreover, there exist transformations between tables using representations with different sets of states on each vertex using ${\ensuremath{\mathcal{O}}}(|X_x||A_x|)$ arithmetic operations.
Use the same $|X_e|$-step transformation as in the proof of Lemma \[lem:dsstates\] with the difference that we can choose a different formula to change the states at each coordinate of the colouring $c$ of $X_e$. At coordinate $i$ of the colouring $c$, we use the formula that corresponds to the set of states that we want to use on the corresponding vertex in $X_e$.
We are now ready to give the first improvement of Proposition \[prop:simplebwdsalg\].
\[prop:secondbwdsalg\] There is an algorithm that, given a branch decomposition of a graph $G$ of width $k$, counts the number of dominating sets in $G$ of each size $\kappa$, $0 \leq \kappa \leq n$, in ${\ensuremath{\mathcal{O}}}(m n^2 3^{\frac{3}{2}k} i_\times(n))$ time.
The algorithm is similar to the algorithm of Proposition \[prop:simplebwdsalg\], only we employ a different method to compute $A_e$ for an internal edge $e$ of $T$.
[*Internal edges*]{}: Let $e$ be an internal edge of $T$ with child edges $l$ and $r$.
We start by applying Lemma \[lem:asymdsstates\] to $A_l$ and $A_r$ and change the sets of states used for each individual vertex in the following way. We let $A_l$ use the set of states $\{1,0_?,0_0\}$ on vertices in $I$ and the set of states $\{1,0_1,0_0\}$ on vertices in $L$ and $F$. We let $A_r$ use the set of states $\{1,0_?,0_0\}$ on vertices in $I$, the set of states $\{1,0_1,0_0\}$ on vertices in $R$, and the set of states $\{1,0_1,0_?\}$ on vertices in $F$. Finally, we let $A_e$ use the set of states $\{1,0_?,0_0\}$ on vertices in $I$ and the set of states $\{1,0_1,0_0\}$ on vertices in $L$ and $R$. Notice that different colourings use the same sets of states on the same vertices with the exception of the set of states used for vertices in $F$; here, $A_l$ and $A_r$ use different sets of states.
Now, three colourings $c_e$, $c_l$ and $c_r$ *match* if:
- For any $v \in I$: $c_e(v) = c_l(v) = c_r(v) \in \{1,0_?,0_0\}$. (3 possibilities)
- For any $v \in F$: either $c_l(v) = c_r(v) = 1$, or $c_l(v) = 0_0$ and $c_r(v) = 0_1$, or $c_l(v) = 0_1$ and $c_r(v) = 0_?$. (3 possibilities)
- For any $v \in L$: $c_e(v) = c_l(v) \in \{1,0_1,0_0\}$. (3 possibilities)
- For any $v \in R$: $c_e(v) = c_r(v) \in \{1,0_1,0_0\}$. (3 possibilities)
For the vertices on $I$, these matching combinations are the same as used on tree decompositions in Theorem \[thrm:countingtwdsalg\], namely the combinations with states from the set $\{1,0_?,0_0\}$ where all states are the same. For the vertices on $L$ and $R$, we do exactly the same as in the proof of Proposition \[prop:simplebwdsalg\].
For the vertices in $F$, a more complicated method has to be used. Here, we can use only combinations that make sure that these vertices will be dominated: combinations with vertices that are in vertex set of the partial solution, or combinations in which the vertices are not in this vertex set, but in which they will be dominated. Moreover, by using different states for $A_l$ and $A_r$, every combination of partial solutions is counted exactly once. To see this, consider each of the three combinations on $F$ used in Proposition \[prop:simplebwdsalg\] with the set of states $\{1,0_1,0_0\}$. The combination with $c_l(v) = 0_0$ and $c_r(v) = 0_1$ is counted using the same combination, while the other two combinations ($c_l(v) = 0_1$ and $c_r(v) = 0_0$ or $c_r(v) = 0_1$) are counted when combining $0_1$ with $0_?$.
In this way, we can compute the entries in the table $A_e$ using the following formula: $$A_e(c_e, \kappa) = \sum_{c_e, c_l, c_r \,\textrm{\scriptsize match}} \; \sum_{\kappa_l + \kappa_r = \kappa + \kappa_{\#_1}} A_l(c_l,\kappa_l) \cdot A_r(c_r, \kappa_r )$$ Here, $\kappa_{\#_1} = \#_{1}(c_r(I \cup F))$ again is the number of vertices that are assigned state $1$ on $I \cup F$ in any matching triple $c_e$, $c_l$, $c_r$.
After having obtained $A_e$ in this way, we can transform the set of states used back to $\{1,0_1,0_0\}$ using Lemma \[lem:asymdsstates\].
For the running time, we observe that the combination of the different sets of states that we are using allows us to evaluate the above formula in ${\ensuremath{\mathcal{O}}}(n^2 3^{|I|+|L|+|R|+|F|} i_\times(n))$ time. As each state transformation requires ${\ensuremath{\mathcal{O}}}(n 3^k i_+(n))$ time, the improved algorithm has a running time of ${\ensuremath{\mathcal{O}}}(m n^2 3^{|I|+|L|+|R|+|F|} i_\times(n))$. Under the constraint that $|I|+|L|+|R|, |I|+|L|+|F|, |I|+|R|+|F| \leq k$, the running time is maximal if $|I| = 0$, $|L|=|R|=|F|=\frac{1}{2}k$. This gives a total running time of ${\ensuremath{\mathcal{O}}}(m n^2 3^{\frac{3}{2}k} i_{\times}(n))$.
We will now give our faster algorithm for counting the number of dominating sets of each size $\kappa$, $0 \leq \kappa \leq n$ on branch decompositions. This algorithm uses fast matrix multiplication to speed up the algorithm of Proposition \[prop:secondbwdsalg\]. This use of fast matrix multiplication in dynamic programming algorithms on branch decompositions was first proposed by Dorn in [@Dorn06].
\[thrm:countdsbwalg\] There is an algorithm that, given a branch decomposition of a graph $G$ of width $k$, counts the number of dominating sets in $G$ of each size $\kappa$, $0 \leq \kappa \leq n$, in ${\ensuremath{\mathcal{O}}}(m n^2 3^{\frac{\omega}{2}k} i_\times(n))$ time.
Consider the algorithm of Proposition \[prop:secondbwdsalg\]. We will make one modification to this algorithm. Namely, when computing the table $A_e$ for an internal edge $e \in E(T)$, we will show how to evaluate the formula for $A_e(c,\kappa)$ for a number of colourings $c$ simultaneously using fast matrix multiplication. We give the details below. Here, we assume that the states in the tables $A_l$ and $A_r$ are transformed such that the given formula for $A_e(c,\kappa)$ in Proposition \[prop:secondbwdsalg\] applies.
We do the following. First, we fix the two numbers $\kappa$ and $\kappa_l$, and we fix a colouring of $I$. Note that this is well-defined because all three tables use the same set of states for colours on $I$. Second, we construct a $3^{|L|} \times 3^{|F|}$ matrix $M_l$ where each row corresponds to a colouring of $L$ and each column corresponds to a colouring of $F$ where both colourings use the states used by the corresponding vertices in $A_l$. We let the entries of $M_l$ be the values of $A_l(c_l,\kappa_l)$ for the $c_l$ corresponding to the colourings of $L$ and $F$ of the given row and column of $M_l$, and corresponding to the fixed colouring on $I$ and the fixed number $\kappa_l$. We also construct a similar $3^{|F|}\times 3^{|R|}$ matrix $M_r$ with entries from $A_r$ such that its rows correspond to different colourings of $F$ and its columns correspond to different colourings of $R$ where both colourings use the states used by the corresponding vertices in $A_r$. The entries of $M_r$ are the values of $A_r(c_r,\kappa - \kappa_l - \kappa_{\#_1})$ where $c_r$ corresponds to the colouring of $R$ and $F$ of the given row and column of $M_r$, and corresponding to the fixed colouring on $I$ and the fixed numbers $\kappa$ and $\kappa_l$. Here, the value of $\kappa_{\#_1} = \#_{1}(c_r(I \cup F))$ depends on the colouring $c_r$ in the same way as in Proposition \[prop:secondbwdsalg\]. Third, we permute the rows of $M_r$ such that column $i$ of $M_l$ and row $i$ of $M_r$ correspond to matching colourings on $F$.
Now, we can evaluate the formula for $A_e$ for all entries corresponding to the fixed colouring on $I$ and the fixed values of $\kappa$ and $\kappa_l$ simultaneously by computing $M_e = M_l \cdot M_r$. Clearly, $M_e$ is a $3^{|L|} \times 3^{|R|}$ matrix where each row corresponds to a colouring of $L$ and each column corresponds to a colouring of $R$. If one works out the matrix product $M_l \cdot M_r$, one can see that each entry of $M_e$ contains the sum of the terms of the formula for $A_e(c_e,\kappa)$ such that the colouring $c_e$ corresponds to the given row and column of $M_e$ and the given fixed colouring on $I$ and such that $\kappa_l + \kappa_r = \kappa + \kappa_{\#_1}$ corresponding to the fixed $\kappa$ and $\kappa_l$. That is, each entry in $M_e$ equals the sum over all possible allowed matching combinations of the colouring on $F$ for the fixed values of $\kappa$ and $\kappa_l$, where the $\kappa_r$ involved are adjusted such that the number of $1$-states used on $F$ is taken into account.
In this way, we can compute the function $A_e$ by repeating the above matrix-multiplication-based process for every colouring on $I$ and every value of $\kappa$ and $\kappa_l$ in the range from $0$ to $n$. As a result, we can compute the function $A_e$ by a series of $n^2\,3^{|I|}$ matrix multiplications.
The time required to compute $A_e$ in this way depends on $|I|$, $|L|$, $|R|$ and $|F|$. Under the constraint that $|I| + |L| + |F|, |I|+|R|+|F|, |I|+|L|+|R| \leq k$ and using the matrix-multiplication algorithms for square and non-square matrices as described in Section \[sec:matrixmultiplic\], the worst case arises when $|I|=0$ and $|L|=|R|=|F| = \frac{k}{2}$. In this case, we compute each table $A_e$ in ${\ensuremath{\mathcal{O}}}(n^2 (3^{\frac{k}{2}})^\omega i_{\times}(n))$ time. This gives a total running time of ${\ensuremath{\mathcal{O}}}(m n^2 3^{\frac{\omega}{2}k} i_\times(n))$.
Using the fact that [Dominating Set]{} has the de Fluiter property for treewidth, and using the same tricks as in Corollaries \[cor:countmdstwalg\] and \[cor:solvedstwalg\], we also obtain the following results.
\[cor:countmdsbwalg\] There is an algorithm that, given a branch decomposition of a graph $G$ of width $k$, counts the number of minimum dominating sets in $G$ in ${\ensuremath{\mathcal{O}}}(m k^2 3^{\frac{\omega}{2}k} i_\times(n))$ time.
\[cor:solvedsbwalg\] There is an algorithm that, given a branch decomposition of a graph $G$ of width $k$, computes the size of a minimum dominating set in $G$ in ${\ensuremath{\mathcal{O}}}(m k^2 3^{\frac{\omega}{2}k} i_\times(k))$ time.
Counting the Number of Perfect Matchings {#sec:bwcpm}
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The next problem we consider is the problem of counting the number of perfect matchings in a graph. To give a fast algorithm for this problem, we use both the ideas introduced in Theorem \[thrm:countingpmtwalg\] to count the number of perfect matchings on graphs given with a tree decomposition and the idea of using fast matrix multiplication of Dorn [@Dorn06] found in Theorem \[thrm:countdsbwalg\]. We note that [@Dorn06; @Dorn07] did not consider counting perfect matchings. The result will be an ${\ensuremath{\mathcal{O}^*}}(2^{\frac{\omega}{2}k})$ algorithm.
From the algorithm to count the number of dominating sets of each given size in a graph of bounded branchwidth, it is clear that vertices in $I$ and $F$ need special attention when developing a dynamic programming algorithm over branch decompositions. This is no different when we consider counting the number of perfect matchings. For the vertices in $I$, we will use state changes and an index similar to Theorem \[thrm:countingpmtwalg\], but for the vertices in $F$ we will require only that all these vertices are matched. In contrast to the approach on tree decompositions, we will not take into account the fact that we can pick edges in the matching at the point in the algorithm where we forget the first endpoint of the edge. We represent this choice directly in the tables $A_e$ of the leaf edges $e$: this is possible because every edge of $G$ is uniquely assigned to a leaf of $T$.
Our algorithm will again be based on state changes, where we will again use different sets of states on vertices with different roles in the computation.
\[lem:asympmstates\] Let $e$ be an edge of a branch decomposition $T$ with corresponding middle set $|X_e|$, and let $A_e$ be a table with entries $A_e(c,\kappa)$ representing the number of matchings in $H_e$ matching all vertices in $V_e \setminus X_e$ and corresponding to all colourings of the middle set $X_e$ with states such that for every individual vertex in $X_e$ one of the following fixed sets of states is used: $$\{1,0\} \qquad \{1,?\} \qquad \{0,?\} \qquad \textrm{(meaning of the states as in Lemma~\ref{lem:pmstates})}$$ The information represented in the table $A_e$ does not depend on the choice of the set of states from the options given above. Moreover, there exist transformations between tables using representations with different sets of states on each vertex using ${\ensuremath{\mathcal{O}}}(|X_x||A_x|)$ arithmetic operations.
The proof is identical to that of Lemma \[lem:asymdsstates\] while using the formulas from the proof of Lemma \[lem:pmstates\].
Now, we are ready to proof the result for counting perfect matchings.
\[thrm:countpmbwalg\] There is an algorithm that, given a branch decomposition of a graph $G$ of width $k$, counts the number of perfect matchings in $G$ in ${\ensuremath{\mathcal{O}}}(m k^2 2^{\frac{\omega}{2}k} i_{\times}(k\log(n)))$ time.
Let $T$ be a branch decomposition of $G$ of branchwidth $k$ rooted at a vertex $z$.
For each edge $e \in E(T)$, we will compute a table $A_e$ with entries $A_e(c)$ for all $c \in \{1,0\}^{X_e}$ where the states have the same meaning as in Theorem \[thrm:countingpmtwalg\]. In this table, an entry $A_e(c)$ equals the number of matchings in the graph $H_e$ matching all vertices in $V_e \setminus X_e$ and satisfying the requirements defined by the colouring $c$ on the vertices in $X_e$. These entries do not count matchings in $G_e$ but in its subgraph $H_e$ that has the same vertices as $G_e$ but contains only the edges of $G_e$ that are in the leaves below $e$ in $T$.
[*Leaf edges*]{}: Let $e$ be an edge of $T$ incident to a leaf of $T$ that is not the root. Now, $H_e = G_e = G[X_e]$ is a two vertex graph with $X_e = \{u,v\}$ and with an edge between $u$ and $v$.
We compute $A_e$ in the following way: $$A_e(c) = \left\{ \begin{array}{ll} 1 & \textrm{if $c=(1,1)$ or $c=(0,0)$} \\ 0 & \textrm{otherwise} \end{array} \right.$$ The only non-zero entries are the empty matching and the matching consisting of the unique edge in $H_e$. This is clearly correct.
[*Internal edges*]{}: Let $e$ be an internal edge of $T$ with child edges $l$ and $r$.
Similar to the proof of Theorem \[thrm:countingpmtwalg\], we start by indexing the tables $A_l$ and $A_r$ by the number of 1-states used for later use. However, we now count only the number of 1-states used on vertices in $I$ in the index. We compute indexed tables $A'_l$ and $A'_r$ with entries $A'_l(c_l,i_l)$ and $A'_r(c_r,i_r)$ using the following formula with $y \in \{l,r\}$: $$A'_y(c_y,i_y) = \left\{ \begin{array}{ll} A_y(c_y) & \textrm{if $\#_{1}(c_y(I))) = i_y$} \\ 0 & \textrm{otherwise} \end{array} \right.$$ Here, $\#_{1}(c_y(I))$ is the number of $1$-entries in the colouring $c_y$ on the vertices in $I$.
Next, we apply state changes by using Lemma \[lem:asympmstates\]. In this case, we change the states used for the colourings in $A'_r$ and $A'_l$ such that they use the set of states $\{0,1\}$ on $L$, $R$, and $F$, and the set of states $\{0,?\}$ on $I$. Notice that the number of $1$-states used to create the $?$-states is now stored in the index $i_l$ of $A'_l(c,i_l)$ and $i_r$ of $A'_r(c,i_r)$.
We say that three colourings $c_e$ of $X_e$, $c_l$ of $X_l$ and $c_r$ of $X_r$ using these sets of states on the different partitions of $X_e \cup X_l \cup X_r$ *match* if:
- For any $v \in I$: $c_e(v) = c_l(v) = c_r(v) \in \{0,?\}$. (2 possibilities)
- For any $v \in F$: either $c_l(v) = 0$ and $c_r(v) = 1$, or $c_l(v) = 1$ and $c_r(v) = 0$. (2 possibilities)
- For any $v \in L$: $c_e(v) = c_l(v) \in \{1,0\}$. (2 possibilities)
- For any $v \in R$: $c_e(v) = c_r(v) \in \{1,0\}$. (2 possibilities)
Now, we can compute the indexed table $A'_e$ for the edge $e$ of $T$ using the following formula: $$A'_e(c_e, i_e) = \sum_{c_e, c_l, c_r \,\textrm{\scriptsize match}} \; \sum_{i_l + i_r = i_e} A'_l(c_l,i_l) \cdot A'_r(c_r, i_r )$$ Notice that we can compute $A'_e$ efficiently by using a series of matrix multiplications in the same way as done in the proof of Theorem \[thrm:countdsbwalg\]. However, the index $i$ should be treated slightly differently from the parameter $\kappa$ in the proof of Theorem \[thrm:countdsbwalg\]. After fixing a colouring on $I$ and the two values of $i_e$ and $i_l$, we still create the two matrices $M_l$ and $M_r$. In $M_l$ each row corresponds to a colouring of $L$ and each column corresponds to a colouring of $F$, and in $M_r$ each rows again corresponds to a colourings of $F$ and each column corresponds to a colouring of $R$. The difference is that we fill $M_l$ with the corresponding entries $A'_l(c_l,i_l)$ and $M_r$ with the corresponding entries $A'_r(c_r,i_r)$. That is, we do not adjust the value of $i_r$ for the selected $A'_r(c_r,i_r)$ depending on the states used on $F$. This is not necessary here since the index counts the total number of $1$-states hidden in the $?$-states and no double counting can take place. This in contrast to the parameter $\kappa$ in the proof of Theorem \[thrm:countdsbwalg\]; this parameter counted the number of vertices in a solution, which we had to correct to avoid double counting of the vertices in $F$.
After computing $A'_e$ in this way, we again change the states such that the set of states $\{1,0\}$ is used on all vertices in the colourings used in $A'_e$. We then extract the values of $A'_e$ in which no two $1$-states hidden in a $?$-state are combined to a new $1$-state on a vertex in $I$. We do so using the indices in the same way as in Theorem \[thrm:countingpmtwalg\] but with the counting restricted to $I$: $$A_e(c) = A'_e(c,\#_1(c(I)))$$
After computing the $A_e$ for all $e \in E(T)$, we can find the number of perfect matchings in $G = G_{\{y,z\}}$ in the single entry in $A_{\{y,z\}}$ where $z$ is the root of $T$ and $y$ is its only child.
Because the treewidth and branchwidth of a graph differ by at most a factor $\frac{3}{2}$ (see Proposition \[prop:1.5\]), we can conclude that the computations can be done using ${\ensuremath{\mathcal{O}}}(k\log(n))$-bit numbers using the same reasoning as in the proof of Theorem \[thrm:countingpmtwalg\]. For the running time, we observe that we can compute each $A_e$ using a series of $k^22^{|I|}$ matrix multiplications. The worst case arises when $|I|=0$ and $|L|=|R|=|F|=\frac{k}{2}$. Then the matrix multiplications require ${\ensuremath{\mathcal{O}}}(k^2 2^{\frac{\omega}{2}k})$ time. Since $T$ has ${\ensuremath{\mathcal{O}}}(m)$ edges, this gives a running time of ${\ensuremath{\mathcal{O}}}(m k^2 2^{\frac{\omega}{2}k} i_{\times}(k\log(n)))$ time.
$[\rho,\sigma]$-Domination Problems {#sec:bwrhosigma}
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We have shown how to solve two fundamental graph problems in ${\ensuremath{\mathcal{O}^*}}(s^{\frac{\omega}{2}k})$ time on branch decompositions of width $k$, where $s$ is the natural number of states involved in a dynamic programming algorithm on branch decompositions for these problem. Similar to the results on tree decompositions, we generalize this and show that one can solve all $[\rho,\sigma]$-domination problems with finite or cofinite $\rho$ and $\sigma$ in ${\ensuremath{\mathcal{O}^*}}(s^{\frac{\omega}{2}k})$ time.
For the $[\rho,\sigma]$-domination problems, we use states $\rho_j$ and $\sigma_j$, where $\rho_j$ and $\sigma_j$ represent that a vertex is not in or in the vertex set $D$ of the partial solution of the $[\sigma,\rho]$-domination problem, respectively, and has $j$ neighbours in $D$. This is similar to Section \[sec:rhosigmatw\]. Note that the number of states used equals $s = p + q + 2$.
On branch decompositions, we have to use a different approach than on tree decompositions, since we have to deal with vertices in $L$, $R$, $I$, and $F$ simultaneously. It is, however, possible to reuse part of the algorithm of Theorem \[thrm:rstwalg\]. Observe that joining two children in a tree decomposition is similar to joining two children in a branch decomposition if $L=R=F=\emptyset$. Since we have demonstrated in the algorithms earlier in this section that one can have distinct states and perform different computations on $I$, $L$, $R$, and $F$, we can essentially use the approach of Theorem \[thrm:rstwalg\] for the vertices in $I$.
\[thrm:rsbwalg\] Let $\rho,\sigma \subseteq {\ensuremath{\mathbb{N}}}$ be finite or cofinite. There is an algorithm that, given a branch decomposition of a graph $G$ of width $k$, counts the number of $[\rho,\sigma]$-dominating sets of $G$ of each size $\kappa$, $0 \leq \kappa \leq n$, of a fixed $[\rho,\sigma]$-domination problem involving $s$ states in ${\ensuremath{\mathcal{O}}}(m n^2 (sk)^{2(s-2)} s^{\frac{\omega}{2} k} i_\times(n))$ time.
Let $T$ be the branch decomposition of $G$ of width $k$ rooted at the vertex $z$.
Recall the definitions of State Sets I and II defined in Definition \[def:rsstates\]. Similar to the proof of Theorem \[thrm:rstwalg\], we will use different sets of states to prove this theorem. In this proof, we mostly use State Set I while we let the subscripts of the states count only neighbours in $D$ outside the current middle set. That is, we use states $\rho_j$ and $\sigma_j$ for our tables $A_e$, $A_f$, and $A_g$ such that the subscripts $j$ represent the number of neighbours in the vertex set $D$ of each partial solution of the $[\sigma,\rho]$-domination problem outside of the vertex sets $X_e$, $X_f$ and $X_g$, respectively. Using these states for colourings $c$, we compute the table $A_e$ for each edge $e \in E(T)$ such that the entry $A_e(c,\kappa)$ contains the number of partial solutions of the $[\rho,\sigma]$-domination problem on $G_e$ consisting of $\kappa$ vertices that satisfy the requirements defined by $c$.
[*Leaf edges*]{}: Let $e$ be an edge of $T$ incident to a leaf of $T$ that is not the root. Now, $G_e = G[X_e]$ is a two vertex graph.
We compute $A_e$ in the following way: $$A_e(c, \kappa) = \left\{ \begin{array}{ll} 1 & \textrm{if $c=(\rho_0,\rho_0)$ and $\kappa = 0$} \\ 1 & \textrm{if $c=(\rho_0,\sigma_0)$ or $c=(\sigma_0,\rho_0)$, and $\kappa = 1$} \\ 1 & \textrm{if $c=(\sigma_0,\sigma_0)$ and $\kappa = 2$} \\ 0 & \textrm{otherwise} \end{array} \right.$$ Since the subscripts of the states count only vertices in the vertex set of a partial solutions of the $[\rho,\sigma]$-domination problem on $G_e$ that are outside the middle set $X_e$, we only count partial solutions in which the subscripts are zero. Moreover, the size parameter $\kappa$ must equal the number of $\sigma$-states since these represent vertices in the vertex set of the partial solutions.
[*Internal edges*]{}: Let $e$ be an internal edge of $T$ with child edges $l$ and $r$.
The process of computing the table $A_e$ by combining the information in the two tables $A_l$ and $A_r$ is quite technical. This is mainly due to the fact that we need to do different things on the different vertex sets $I$, $L$, $R$, and $F$. We will give a three-step proof.
[*Step 1*]{}: As a form of preprocessing, we will update the entries in $A_l$ and $A_r$ such that the subscripts will not count only the vertices in vertex sets of the partial solutions outside of $X_l$ and $X_r$, but also some specific vertices in the vertex sets of the partial solutions in the middle sets. Later, we will combine the information from $A_l$ and $A_r$ to create the table $A_e$ according to the following general rule: combining $\rho_i$ and $\rho_j$ gives $\rho_{i+j}$, and $\sigma_i$ and $\sigma_j$ gives $\sigma_{i+j}$. In this context, the preprocessing makes sure that the subscripts of the states in the result in $A_e$ correctly count the number of vertices in the vertex sets of the partial solutions of the $[\rho,\sigma]$-domination problem.
Recall that for an edge $e$ of the branch decomposition $T$ the vertex set $V_e$ is defined to be the vertex set of the graph $G_e$, that is, the union of the middle set of $e$ and all middle sets below $e$ in $T$. We update the tables $A_l$ and $A_r$ such that the subscripts of the states $\rho_j$ and $\sigma_j$ count the number of neighbours in the vertex sets of the partial solutions with the following properties:
- States used in $A_l$ on vertices in $L$ or $I$ count neighbours in $(V_l \setminus X_l) \cup F$.
- States used in $A_l$ on vertices in $F$ count neighbours in $(V_l \setminus X_l) \cup L \cup I \cup F$.
- States used in $A_r$ on vertices in $I$ count neighbours in $(V_r \setminus X_r)$ (nothing changes here).
- States used in $A_r$ on vertices in $R$ count neighbours in $(V_r \setminus X_r) \cup F$.
- States used in $A_r$ on vertices in $F$ count neighbours in $(V_r \setminus X_r) \cup R$.
If we now combine partial solutions with state $\rho_i$ in $A_l$ and state $\rho_j$ in $A_r$ for a vertex in $I$, then the state $\rho_{i+j}$ corresponding to the combined solution in $A_e$ correctly counts the number of neighbours in the partial solution in $V_e \setminus X_e$. Also, states for vertices in $L$ and $R$ in $A_e$ count their neighbours in the partial solution in $V_e \setminus X_e$. And, if we combine solutions with a state $\rho_i$ in $A_l$ and a state $\rho_j$ in $A_r$ for a vertex in $F$, then this vertex will have exactly $i+j$ neighbours in the combined partial solution.
Although one must be careful which vertices to count and which not to count, the actual updating of the tables $A_l$ and $A_r$ is simple because one can see which of the counted vertices are in the vertex set of a partial solution ($\sigma$-state) and which are not ($\rho$-state).
Let $A^*_y$ be the table before the updating process with $y \in \{l,r\}$. We compute the updated table $A_y$ in the following way: $$A_y(c,\kappa) = \left\{ \begin{array}{ll} 0 & \textrm{if $\phi(c)$ is not a correct colouring of $X_y$} \\ A^*_y(\phi(c),\kappa) & \textrm{otherwise} \end{array} \right.$$ Here, $\phi$ is the inverse of the function that updates the subscripts of the states, e.g., if $y = l$ and we consider a vertex in $I$ with exactly one neighbour with a $\sigma$-state on a vertex in $F$ in $c$, then it changes $\rho_2$ into $\rho_1$. The result of this updating is not a correct colouring of $X_y$ if the inverse does not exist, i.e., if the strict application of subtracting the right number of neighbours results in a negative number. For example, this happens if $c$ contains a $\rho_0$- or $\sigma_0$-state while it has neighbours that should be counted in the subscripts.
[*Step 2*]{}: Next, we will change the states used for the tables $A_l$ and $A_r$, and we will add index vectors to these tables that allow us to use the ideas of Theorem \[thrm:rstwalg\] on the vertices in $I$.
We will not change the states for vertices in $L$ in the table $A_l$, nor for vertices in $R$ in the table $A_r$. But, we will change the states for the vertices in $I$ in both $A_l$ and $A_r$ and on the vertices in $F$ in $A_r$. On $F$, simple state changes suffice, while, for vertices on $I$, we need to change the states and introduce index vectors at the same time.
We will start by changing the states for the vertices in $F$. On the vertices in $F$, we will not change the states in $A_l$, but introduce a new set of states to use for $A_r$. We define the states $\bar{\rho_j}$ and $\bar{\sigma_j}$. A table entry with state $\bar{\rho_j}$ on a vertex $v$ requires that the vertex has an allowed number of neighbours in the vertex set of a partial solution when combined with a partial solution from $A_l$ with state $\rho_j$. That is, a partial solution that corresponds to the state $\rho_i$ on $v$ is counted in the entry with state $\bar{\rho_j}$ on $v$ if $i + j \in \rho$. The definition of $\bar{\sigma_j}$ is similar.
Let $A^*_r$ be the result of the table for the right child $r$ of $e$ obtained by Step 1. We can obtain the table $A_r$ with the states on $F$ transformed as described by a coordinate-wise application of the following formula on the vertices in $F$. The details are identical to the state changes in the proofs of Lemmas \[lem:asymdsstates\] and \[lem:asympmstates\]. $$\begin{aligned}
A_r(c_1 \times \{\bar{\rho_j}\} \times c_2) & = & \sum_{i+j \in \rho} A^*_r(c_1 \times \{\rho_i\} \times c_2) \\
A_r(c_1 \times \{\bar{\sigma_j}\} \times c_2) & = & \sum_{i+j \in \sigma} A^*_r(c_1 \times \{\sigma_i\} \times c_2)\end{aligned}$$ Notice that if we combine an entry with state $\rho_j$ in $A_l$ with an entry with state $\bar{\rho_j}$ from $A_r$, then we can count all valid combinations in which this vertex is not in the vertex set of a partial solution of the $[\rho,\sigma]$-domination problem. The same is true for a combination with state $\sigma_j$ in $A_l$ with state $\bar{\sigma_j}$ in $A_r$ for vertices in the vertex set of the partial solutions.
As a final part of Step 2, we now change the states in $A_l$ and $A_r$ on the vertices in $I$ and introduce the index vectors $\vec{i} = (i_{\rho1}, i_{\rho2}, \ldots, i_{\rho p},i_{\sigma1},i_{\sigma2},\ldots,i_{\sigma q})$, where $i_{\rho j}$ and $i_{\sigma j}$ index the sum of the number of neighbours in the vertex set of a partial solution of the $[\rho,\sigma]$-domination problem over the vertices with state $\rho_{\leq j}$ and $\sigma_{\leq j}$, respectively. That is, we change the states used in $A_l$ and $A_r$ on vertices in $I$ to State Set II of Definition \[def:rsstates\] and introduce index vectors in exactly the same way as in the proof of Lemma \[lem:rsstates2\], but only on the coordinates of the vertices in $I$, similar to what we did in the proofs of Lemmas \[lem:asymdsstates\] and \[lem:asympmstates\]. Because the states $\rho_{\leq j}$ and $\sigma_{\leq j}$ are used only on $I$, we note that that the component $i_{\rho_j}$ of the index vector $\vec{i}$ count the total number of neighbours in the vertex sets of the partial solutions of the $[\rho,\sigma]$-domination problem of vertices with state $\rho_{\leq j}$ on $I$. As a result, we obtain tables $A'_l$ and $A'_r$ with entries $A'_l(c_l,\kappa_l,\vec{g})$ and $A'_r(c_r,\kappa_r,\vec{h})$ with index vectors $\vec{g}$ and $\vec{h}$, where these entries have the same meaning as in Theorem \[thrm:rstwalg\]. We note that the components $i_{\rho p}$ and $i_{\sigma q}$ of the index vector are omitted if $\rho$ or $\sigma$ is cofinite, respectively.
We have now performed all relevant preprocessing and are ready for the final step.
[*Step 3*]{}: Now, we construct the table $A_e$ by computing the number of valid combinations from $A'_l$ and $A'_r$ using fast matrix multiplication.
We first define when three colourings $c_e$, $c_l$, and $c_r$ match. They *match* if:
- For any $v \in I$: $c_e(v) = c_l(v) = c_r(v)$ with State Set II. ($s$ possibilities)
- For any $v \in F$: either $c_l(v) = \rho_j$ and $c_r(v) = \bar{\rho_j}$, or $c_l(v) = \sigma_j$ and $c_r(v) = \bar{\sigma_j}$, with State Set I used for $c_l$ and the new states used for $c_r$. ($s$ possibilities)
- For any $v \in L$: $c_e(v) = c_l(v)$ with State Set I. ($s$ possibilities)
- For any $v \in R$: $c_e(v) = c_r(v)$ with State Set I. ($s$ possibilities)
State Set I and State Set II are as defined in Definition \[def:rsstates\]. That is, colourings match if they forget valid combinations on $F$, and have identical states on $I$, $L$, and $R$.
Using this definition, the following formula computes the table $A'_e$. The function of this table is identical to the same table in the proof of Theorem \[thrm:rstwalg\]: the table gives all valid combinations of entries corresponding to the colouring $c$ that lead to a partial solution of size $\kappa$ with the given values of the index vector $\vec{i}$. The index vectors allow us to extract the values we need afterwards. $$A'_e(c_e,\kappa,\vec{i}) = \!\!\!
\mathop{\sum_{c_e, c_l, c_r}}_{\textrm{\scriptsize match}} \;\;
\sum_{\kappa_l + \kappa_r = \kappa + \#_\sigma(c)} \!\!
\left( \sum_{i_{\rho1}=g_{\rho1}+h_{\rho1}} \!\!\cdots\!\! \sum_{i_{\sigma q}=g_{\sigma q}+h_{\sigma q}} \!\!\!\!
A'_l(c_l,\kappa_l,\vec{g}) \cdot A'_r(c_r,\kappa_r,\vec{h})
\right)$$ Here, $\#_\sigma = \#_\sigma(c_r(I \cup F))$ is the number of vertices that are assigned a $\sigma$-state on $I \cup F$ in any matching triple $c_e$, $c_l$, $c_r$.
We will now argue what kind of entries the table $A'_e$ contains by giving a series of observations.
\[obs:1\] For a combination of a partial solutions on $G_l$ counted in $A'_l$ and a partial solution on $G_r$ counted in $A'_r$ to be counted in the summation for $A'_e(c,\kappa,\vec{i})$, it is required that both partial solutions contains the same vertices on $X_l \cap X_r$ ($= I \cap F$).
This holds because sets of matching colourings have a $\sigma$-state on a vertex if and only if the other colourings in which the same vertex is included also have a $\sigma$-state on this vertex.
\[obs:2\] For a combination of a partial solutions on $G_l$ counted in $A'_l$ and a partial solution on $G_r$ counted in $A'_r$ to be counted in the summation for $A'_e(c,\kappa,\vec{i})$, it is required that the total number of vertices that are part of the combined partial solution is $\kappa$.
This holds because we demand that $\kappa$ equals the sum of the sizes of the partial solutions on $G_l$ and $G_r$ used for the combination minus the number of vertices in these partial solutions that are counted in both sides, namely, the vertices with a $\sigma$-state on $I$ or $F$.
\[obs:3\] For a combination of a partial solutions on $G_l$ counted in $A'_l$ and a partial solution on $G_r$ counted in $A'_r$ to be counted in the summation for $A'_e(c,\kappa,\vec{i})$, it is required that the subscripts $j$ of the states $\rho_j$ and $\sigma_j$ used in $c$ on vertices in $L$ and $R$ correctly count the number of neighbours of this vertex in $V_e \setminus X_e$ in the combined partial solution.
This holds because of the preprocessing we performed in Step 1.
\[obs:4\] For a combination of a partial solutions on $G_l$ counted in $A'_l$ and a partial solution on $G_r$ counted in $A'_r$ to be counted in the summation for $A'_e(c,\kappa,\vec{i})$, it is required that the forgotten vertices in a combined partial solution satisfy the requirements imposed by the specific $[\rho,\sigma]$-domination problem. I.e., if such a vertex is not in the vertex set $D$ of the combined partial solutions, then it has a number of neighbours in $D$ that is a member of $\rho$, and if such a vertex is in the vertex set $D$ of the combined partial solution, then it has a number of neighbours in $D$ that is a member of $\sigma$. Moreover, all such combinations are considered.
This holds because we combine only entries with the states $\rho_j$ and $\bar{\rho_j}$ or with the states $\sigma_j$ and $\bar{\sigma_j}$ for vertices in $F$. These are all required combinations by definition of the states $\bar{\rho_j}$ and $\bar{\sigma_j}$.
\[obs:5\] For a combination of a partial solutions on $G_l$ counted in $A'_l$ and a partial solution on $G_r$ counted in $A'_r$ to be counted in the summation for $A'_e(c,\kappa,\vec{i})$, it is required that the total sum of the number of neighbours outside $X_e$ of the vertices with state $\rho_{\leq j}$ or $\sigma_{\leq j}$ in a combined partial solution equals $i_{\rho j}$ or $i_{\sigma j}$, respectively.
This holds because of the following. First the subscripts of the states are updated such that every relevant vertex is counted exactly once in Step 1. Then, these numbers are stored in the index vectors at Step 2. Finally, the entries of $A'_e$ corresponding to a given index vector combine only partial solutions which index vectors sum to the given index vector $\vec{i}$.
\[obs:6\] Let $D_l$ and $D_r$ be the vertex set of a partial solution counted in $A_l$ and $A_r$ that are used to create a combined partial solution with vertex set $D$, respectively. After the preprocessing of Step 1, the vertices with state $\rho_{\leq j}$ or $\sigma_{\leq j}$ have at most $j$ neighbours that we count in the vertex sets $D_l$ and $D_r$, respectively. And, if a vertex in the partial solution from $A_l$ has $i$ such counted neighbours in $D_l$, and the same vertex in the partial solution from $A_r$ has $j$ such counted neighbours in $D_r$, then the combined partial solution has a total of $i+j$ neighbours in $D$ outside of $X_e$.
The last statement holds because we count each relevant neighbour of a vertex either in the states used in $A_l$ or in the states used in $A_r$ by the preprocessing of Step 1. The first part of the statement follows from the definition of the states $\rho_{\leq j}$ or $\sigma_{\leq j}$: here, only partial solutions that previously had a state $\rho_i$ and $\sigma_i$ with $i \leq j$ are counted.
We will now use Observations \[obs:1\]-\[obs:6\] to show that we can compute the required values for $A_e$ in the following way. This works very similar to Theorem \[thrm:rstwalg\]. First, we change the states in the table $A'_e$ back to State Set I (as defined in Definition \[def:rsstates\]). We can do so similar as in Lemma \[lem:rsstates\] and Lemmas \[lem:asymdsstates\] and \[lem:asympmstates\]. Then, we extract the entries required for the table $A_e$ using the following formula: $$A_e(c,\kappa) = A'_e \left( c, \; \kappa, \; (\Sigma_\rho^1(c),\Sigma_\rho^2(c),\ldots,\Sigma_\rho^p(c),\Sigma_\sigma^1(c),\Sigma_\sigma^2(c),\ldots,\Sigma_\sigma^q(c)) \; \right)$$ Here, $\Sigma_\rho^l(c)$ and $\Sigma_\sigma^l(c)$ are defined as in the proof of Theorem \[thrm:rstwalg\]: the weighted sums of the number of $\rho_j$- and $\sigma_j$-states with $0 \leq j \leq l$, respectively.
If $\rho$ or $\sigma$ is cofinite, we use the same formula but omit the components $\Sigma_\rho^p(c)$ or $\Sigma_\sigma^q(c)$ from the index vector of the extracted entries, respectively.
That the values of these entries equal the values we want to compute follows from the following reasoning. First of all, any combination considered leads to a new partial solution since it uses the same vertices (Observation \[obs:1\]) and forgets vertices that satisfy the constraints of the fixed $[\rho,\sigma]$-domination problem (Observation \[obs:4\]). Secondly, the combinations lead to combined partial solutions of the correct size (Observation \[obs:2\]). Thirdly, the subscripts of the states used in $A_e$ correctly count the number of neighbours of these vertices in the vertex set of the partial solution in $V_e \setminus X_e$. For vertices in $L$ and $R$, this directly follows from Observation \[obs:3\] and the fact that for any three matching colourings the states used on each vertex in $L$ and $R$ are the same. For vertices in $I$, this follows from exactly the same arguments as in the last part of the proof of Theorem \[thrm:rstwalg\] using Observation \[obs:5\] and Observation \[obs:6\]. This is the argument where we first argue that any entry which colouring uses only the states $\rho_0$ and $\sigma_0$ is correct, and thereafter inductively proceed to $\rho_j$ and $\sigma_j$ for $j > 0$ by using correctness for $\rho_{j-1}$ and $\sigma_{j-1}$ and fact that we use the entries corresponding to the chosen values of the index vectors.
All in all, we see that this procedure correctly computes the required table $A_e$.
After computing $A_e$ in the above way for all $e \in E(T)$, we can find the number of $[\rho,\sigma]$-dominating sets of each size in the table $A_{\{y,z\}}$, where $z$ is the root of $T$ and $y$ its only child because $G = G_{\{y,z\}}$ and $X_{\{y,z\}} = \emptyset$.
For the running time, we note that we have to compute the tables $A_e$ for the ${\ensuremath{\mathcal{O}}}(m)$ edges $e \in E(T)$. For each table $A_e$, the running time is dominated by evaluating the formula for the intermediate table $A'_e$ with entries $A'_e(c,\kappa,\vec{i})$. We can evaluate each summand of the formula for $A'_e$ for all combinations of matching states by $s^{|I|}$ matrix multiplications as in Theorem \[thrm:countdsbwalg\]. This requires ${\ensuremath{\mathcal{O}}}(n^2 (sk)^{2(s-2)} s^{|I|})$ multiplications of an $s^{|L|} \times s^{|F|}$ matrix and an $s^{|F|} \times s^{|R|}$ matrix. The running time is maximal if $|I| = 0$ and $|L|=|R|=|F| = \frac{k}{2}$. In this case, the total running time equals ${\ensuremath{\mathcal{O}}}(m n^2 (sk)^{2(s-2)} s^{\frac{\omega}{2}k} i_\times(n) )$ since we can do the computations using $n$-bit numbers.
Similar to our results on the $[\rho,\sigma]$-domination problem on tree decompositions, we can improve the polynomial factors of the above algorithm in several ways. The techniques involved are identical to those of Corollaries \[cor:generalrstwalg\], \[cor:defluiterrstwalg\], and \[cor:decisionrstwalg\]. Similar to Section \[sec:rhosigmatw\], we define the value $r$ associated with a $[\rho,\sigma]$-domination problems as follows: $$r = \left\{ \begin{array}{ll} \max\{p-1,q-1\} & \textrm{if $\rho$ and $\sigma$ are cofinite} \\ \max\{p,q-1\} & \textrm{if $\rho$ is finite and $\sigma$ is cofinite} \\ \max\{p-1,q\} & \textrm{if $\rho$ is confinite and $\sigma$ is finite} \\ \max\{p,q\} & \textrm{if $\rho$ and $\sigma$ are finite} \end{array} \right.$$
\[cor:generalrsbwalg\] Let $\rho, \sigma \subseteq {\ensuremath{\mathbb{N}}}$ be finite or cofinite, and let $p$, $q$, $r$ and $s$ be the values associated with the corresponding $[\rho,\sigma]$-domination problem. There is an algorithm that, given a branch decomposition of a graph $G$ of width $k$, computes the number of $[\rho,\sigma]$-dominating sets in $G$ of each size $\kappa$, $0 \leq \kappa \leq n$, in ${\ensuremath{\mathcal{O}}}(m n^2 (rk)^{2r} s^{\frac{\omega}{2}k} i_\times(n))$ time. Moreover, there is an algorithm that decides whether there exist a $[\rho,\sigma]$-dominating set of size $\kappa$, for each individual value of $\kappa$, $0 \leq \kappa \leq n$, in ${\ensuremath{\mathcal{O}}}(m n^2 (rk)^{2r} s^{\frac{\omega}{2}k} i_\times(log(n)+k\log(r)))$ time.
Apply the modifications to the algorithm of Theorem \[thrm:rstwalg\] that we have used in the proof of Corollary \[cor:generalrstwalg\] for $[\rho,\sigma]$-domination problems on tree decompositions to the algorithm of Theorem \[thrm:rsbwalg\] for the same problems on branch decompositions.
\[cor:defluiterrsbwalg\] Let $\rho, \sigma \subseteq {\ensuremath{\mathbb{N}}}$ be finite or cofinite, and let $p$, $q$, $r$ and $s$ be the values associated with the corresponding $[\rho,\sigma]$-domination problem. If the standard representation using State Set I of the minimisation (or maximisation) variant of this $[\rho,\sigma]$-domination problem has the de Fluiter property for treewidth with function $f$, then there is an algorithm that, given a branch decomposition of a graph $G$ of width $k$, computes the number of minimum (or maximum) $[\rho,\sigma]$-dominating sets in $G$ in ${\ensuremath{\mathcal{O}}}(m [f(k)]^2 (rk)^{2r} s^{\frac{\omega}{2}k} i_\times(n))$ time. Moreover, there is an algorithm that computes the minimum (or maximum) size of such a $[\rho,\sigma]$-dominating set in ${\ensuremath{\mathcal{O}}}(m [f(k)]^2 (rk)^{2r} s^{\frac{\omega}{2}k} i_\times(log(n)+k\log(r)))$ time.
Improve the result of Corollary \[cor:generalrsbwalg\] in the same way as Corollary \[cor:defluiterrstwalg\] improves Corollary \[cor:generalrstwalg\] on tree decompositions.
Let $\rho, \sigma \subseteq {\ensuremath{\mathbb{N}}}$ be finite or cofinite, and let $p$, $q$, $r$ and $s$ be the values associated with the corresponding $[\rho,\sigma]$-domination problem. There is an algorithm that, given a branch decomposition of a graph $G$ of width $k$, counts the number of $[\rho,\sigma]$-dominating sets in $G$ of a fixed $[\rho,\sigma]$-domination problem in ${\ensuremath{\mathcal{O}}}(m (rk)^{2r} s^{\frac{\omega}{2}k} i_\times(n))$ time. Moreover, there is an algorithm that decides whether there exists a $[\rho,\!\sigma]$-dominating set in ${\ensuremath{\mathcal{O}}}(m (rk)^{2r} s^{\frac{\omega}{2}k} i_\times(log(n)+k\log(r)))$ time.
Improve the result of Corollary \[cor:defluiterrsbwalg\] in the same way as Corollary \[cor:decisionrstwalg\] improves upon Corollary \[cor:defluiterrstwalg\] on tree decompositions.
Dynamic Programming on Clique Decompositions {#sec:cliquewidth}
============================================
On graphs of bounded cliquewidth, we mainly consider the [Dominating Set]{} problem. We show how to improve the complex ${\ensuremath{\mathcal{O}^*}}(8^{k})$-time algorithm, which computes a boolean decomposition of a graph of cliquewidth at most $k$ to solve the [Dominating Set]{} problem [@Bui-XuanTV09], to an ${\ensuremath{\mathcal{O}^*}}(4^{k})$-time algorithm. Similar results for [Independent Dominating Set]{} and [Total Dominating Set]{} follow from the same approach.
\[thrm:dscwalg\] There is an algorithm that, given a $k$-expression for a graph $G$, computes the number of dominating sets in $G$ of each size $0 \leq \kappa \leq n$ in ${\ensuremath{\mathcal{O}}}(n^{3} (k^{2} + i_{\times}(n))\, 4^{k})$ time.
An operation in a $k$-expression applies a procedure on zero, one, or two labelled graphs with labels $\{1,2,\ldots,k\}$ that transforms these labelled graphs into a new labelled graph with the same set of labels. If $H$ is such a labelled graph with vertex set $V$, then we use $V(i)$ to denote the vertices of $H$ with label $i$.
For each labelled graph $H$ obtained by applying an operation in a $k$-expression, we will compute a table $A$ with entries $A(c,\kappa)$ that store the number of partial solutions of [Dominating Set]{} of size $\kappa$ that satisfy the constraints defined by the colouring $c$. In contrast to the algorithms on tree and branch decompositions, we do not use colourings that assign a state to each individual vertex, but colourings that assign states to the sets $V(1), V(2), \ldots, V(k)$.
The states that we use are similar to the ones used for [Dominating Set]{} on tree decompositions and branch decomposition. The states that we use are tuples representing two attributes: inclusion and domination. The first attribute determines whether at least one vertex in $V(i)$ is included in a partial solution. We use states $1$, $0$, and $?$ to indicate whether this is true, false, or any of both, respectively. The second attribute determines whether all vertices of $V(i)$ are dominated in a partial solution. Here, we also use states $1$, $0$, and $?$ to indicate whether this is true, false, or any of both, respectively. Thus, we get tuples of the form $(s,t)$, where the first components is related to inclusion and the second to domination, e.g., $(1,?)$ for vertex set $V(i)$ represents that the vertex set contains a vertex in the dominating set while we are indifferent about whether all vertices in $V(i)$ are dominated.
We will now show how to compute the table $A$ for a $k$-expression obtained by using any of the four operations on smaller $k$-expressions that are given with similar tables for these smaller $k$-expressions. This table $A$ contains an entry for every colouring $c$ of the series of vertex sets $\{V(1),V(2),\ldots,V(k)\}$ using the four states $(1,1)$, $(1,0)$, $(0,1)$, and $(0,0)$. We note that the other states will be used to perform intermediate computations. By a recursive evaluation, we can compute $A$ for the $k$-expression that evaluates to $G$.
[*Create a new graph*]{}: In this operation, we create a new graph $H$ with one vertex $v$ with any label $j \in \{1,2,\ldots,k\}$. We assume, without loss of generality by permuting the labels, that $j = k$. We compute $A$ by using the following formula where $c$ is a colouring of the first $k-1$ vertex sets $V(i)$ and $c_k$ is the state of $V(k)$: $$A( c \times \{c_k\} ,\kappa) = \left\{ \begin{array}{ll} 1 & \textrm{if $c_k=(1,1)$, $\kappa = 1$, and $c = \{(0,1)\}^{k-1}$} \\ 1 & \textrm{if $c_k=(0,0)$, $\kappa = 0$, and $c = \{(0,1)\}^{k-1}$} \\ 0 & \textrm{otherwise} \end{array} \right.$$ Since $H$ has only one vertex and this vertex has label $k$, the vertex sets for the other labels cannot have a dominating vertex, therefore the first attribute of their state must be $0$. Also, all vertices in these sets are dominated, hence the second attribute of their state must be $1$. The above formula counts the only two possibilities: either taking the vertex in the partial solution or not.
[*Relabel*]{}: In this operation, all vertices with some label $i \in \{1,2,\ldots,k\}$ are relabelled such that they obtain the label $j \in \{1,2,\ldots,k\}$, $j \not= i$. We assume, without loss of generality by permuting the labels, that $i = k$ and $j=k-1$.
Let $A'$ be the table belonging to the $k$-expression before the relabelling and let $A$ be the table we need to compute. We compute $A$ using the following formulas: $$\begin{aligned}
A( c \!\times\! \{(0,1)\} \!\times\! \{(i,d)\},\kappa) & \!\!=\!\! & \mathop{\sum_{i_1, i_2 \in \{0,1\}}}_{\max\{i_1, i_2\} = i} \; \mathop{\sum_{d_1, d_2 \in \{0,1\}}}_{\min\{d_1,d_2\} = d} \!\! A'(c \!\times\! \{(i_1,d_1)\} \!\times\! \{(i_2,d_2)\}, \kappa) \\
A( c \!\times\! \{(i^*,d^*)\} \!\times\! \{(i,d)\},\kappa) & \!\!=\!\! & 0 \qquad \qquad \textrm{if $(i^*,d^*) \not= (0,1)$}\end{aligned}$$ These formula correctly compute the table $A$ because of the following observations. For $V(i)$, the first attribute must be $0$ and the second attribute must be $1$ in any valid partial solution because $V(i) = \emptyset$ after the operations; this is similar to this requirement in the ‘create new graph’ operation. If $V(j)$ must have a vertex in the dominating set, then this vertex must be in $V(i)$ or $V(j)$ originally. And, if all vertices in $V(j)$ must be dominated, then all vertices in $V(i)$ and $V(j)$ must be dominated. Note that the minimum and maximum under the summations correspond to ‘and’ and ‘or’ operations, respectively.
[*Add edges*]{}: In this operation, all vertices with some label $i \in \{1,2,\ldots,k\}$ are connected to all vertices with another label $j \in \{1,2,\ldots,k\}$, $j \not= i$. We again assume, without loss of generality by permuting the labels, that $i = k-1$ and $j=k$.
Let $A'$ be the table belonging to the $k$-expression before adding the edges and let $A$ be the table we need to compute. We compute $A$ using the following formula: $$A(c \times \{(i_1,d_1)\} \times \{(i_2,d_2)\}, \kappa) = \mathop{\sum_{d'_1 \in \{0,1\}}}_{\max\{d'_1,i_2\} = d_1} \; \mathop{\sum_{d'_2 \in \{0,1\}}}_{\max\{d'_2,i_1\} = d_2} \!\! A(c \times (i_1, d'_1) \times (i_2, d'_2), \kappa)$$ This formula is correct as a vertex sets $V(i)$ and $V(j)$ contain a dominating vertex if and only if they contained such a vertex before adding the edges. For the property of domination, correctness follows because the vertex sets $V(i)$ and $V(j)$ are dominated if and only if they were either dominated before adding the edges, or if they become dominated by a vertex from the other vertex set because of the adding of the edges.
[*Join graphs*]{}: This operation joins two labelled graphs $H_1$ and $H_2$ with tables $A_1$ and $A_2$ to a labelled graph $H$ with table $A$. To do this efficiently, we first apply state changes similar to those used in Sections \[sec:treewidth\] and \[sec:branchwidth\]. We use states $0$ and $?$ for the first attribute (inclusion) and states $1$ and $?$ for the second attribute (domination).
Changing $A_1$ and $A_2$ to tables $A^*_1$ and $A^*_2$ that use this set of states can be done in a similar manner as in Lemmas \[lem:dsstates\] and \[lem:asymdsstates\]. We first copy $A_y$ into $A^*_y$, for $y \in \{1,2\}$ and then iteratively use the following formulas in a coordinate-wise manner: $$\begin{aligned}
A^*_y(c_1 \times (0,1) \times c_2,\kappa) & = & A^*_y(c_1 \times (0,1) \times c_2, \kappa) \\
A^*_y(c_1 \times (?,1) \times c_2,\kappa) & = & A^*_y(c_1 \times (1,1) \times c_2, \kappa) + A^*_y(c_1 \times (0,1) \times c_2, \kappa) \\
A^*_y(c_{1} \times (0,?) \times c_2,\kappa) & = & A^*_y(c_1 \times (0,1) \times c_2, \kappa) + A^*_y(c_1 \times (0,0) \times c_2, \kappa) \\
A^*_y(c_1 \times (?,?) \times c_2,\kappa) & = & A^*_y(c_1 \times (1,1) \times c_2, \kappa) + A^*_y(c_1 \times (1,0) \times c_2, \kappa) +\\
&& A^*_y(c_1 \times (0,1) \times c_2, \kappa) + A^*_y(c_1 \times (0,0) \times c_2, \kappa)\end{aligned}$$
We have already seen many state changes similar to these in Sections \[sec:treewidth\] and \[sec:branchwidth\]. Therefore, it is not surprising that we can now compute the table $A^*$ in the following way, where the table $A^*$ is the equivalent of the table $A$ we want to compute only using the different set of states: $$A^*(c, \kappa) = \sum_{\kappa_1 + \kappa_2 = \kappa} A^{*}_{1}(c,\kappa_1) \cdot A^{*}_{2}(c,\kappa_2)$$
Next, we apply state changes to obtain $A$ from $A^*$. These state changes are the inverse of those given above. Again, first copy $A^*$ into $A$ and then iteratively transform the states in a coordinate-wise manner using the following formulas: $$\begin{aligned}
A(c_1 \times (0,1) \times c_2, \kappa) & = & A(c_1 \times (0,1) \times c_2, \kappa) \\
A(c_1 \times (1,1) \times c_2, \kappa) & = & A(c_1 \times (?,1) \times c_2, \kappa) - A(c_1 \times (0,1) \times c_2, \kappa) \\
A(c_1 \times (0,0) \times c_2, \kappa) & = & A(c_1 \times (0,?) \times c_2, \kappa) - A(c_1 \times (0,1) \times c_2, \kappa) \\
A(c_1 \times (1,0) \times c_2, \kappa) & = & A(c_1 \times (?,?) \times c_2, \kappa) - A(c_1 \times (0,?) \times c_2, \kappa) \\
&&- A(c_1 \times (?,1) \times c_2, \kappa) + A(c_1 \times (0,1) \times c_2, \kappa)\end{aligned}$$ Correctness of the computed table $A$ follows by exactly the same reasoning as used in Theorem \[thrm:countingtwdsalg\] and in Proposition \[prop:secondbwdsalg\]. We note that the last of the above formulas is a nice example of an application of the principle of inclusion/exclusion: to find the number of sets corresponding to the $(1,0)$ -state, we take the number of sets corresponding to the $(?,?)$-state; then, we subtract what we counted to much, but because we subtract some sets twice, we need to add some number of sets again to obtain the required value.
The number of dominating sets in $G$ of size $\kappa$ can be computed from the table $A$ related to the final operation of the $k$-expression for $G$. In this table, we consider only the entries in which the second property is $1$, i.e., the entries corresponding to partial solutions in which all vertices in $G$ are dominated. Now, the number of dominating sets in $G$ of size $\kappa$ equals the sum over all entries $A(c,\kappa)$ with $c \in \{(0,1),(1,1)\}$.
For the running time, we observe that each of the ${\ensuremath{\mathcal{O}}}(n)$ join operations take ${\ensuremath{\mathcal{O}}}(n^{2} 4^{k} i_{\times}(n))$ time because we are multiplying $n$-bit numbers. Each of the ${\ensuremath{\mathcal{O}}}(nk^{2})$ other operations take ${\ensuremath{\mathcal{O}}}(n^{2} 4^{k})$ time since we need ${\ensuremath{\mathcal{O}}}(n4^k)$ series of a constant number of additions using $n$-bit numbers, and $i_+(n)={\ensuremath{\mathcal{O}}}(n)$. The running time of ${\ensuremath{\mathcal{O}}}(n^{3} (k^{2} + i_{\times}(n)) \, 4^{k})$ follows.
Similar to the algorithms for [Dominating Set]{} on tree decompositions and branch decompositions in Section \[sec:dstw\] and \[sec:bwds\], we can improve the polynomial factors in the running time if we are interested only in the size of a minimum dominating set, or the number of these sets. To this end, we will introduce a notion of a de Fluiter property for cliquewidth. This notion is defined similarly to the de Fluiter property for treewidth; see Definition \[def:fluiterproptw\].
\[def:fluiterpropcw\] Consider a method to represent the different partial solutions used in an algorithm that performs dynamic programming on clique decompositions ($k$-expressions) for an optimisation problem $\Pi$. Such a representation has the *de Fluiter property for cliquewidth* if the difference between the objective values of any two partial solutions of $\Pi$ that are stored for a partially evaluated $k$-expression and that can both still lead to an optimal solution is at most $f(k)$, for some function $f$. Here, the function $f$ depends only on the cliquewidth $k$.
The definition of the de Fluiter property for cliquewidth is very similar to the same notion for treewidth. However, the structure of a $k$-expression is different from tree decompositions and branch decompositions in such a way that the de Fluiter property for cliquewidth does not appear to be equivalent to the other two. This in contrast to the same notion for branchwidth that is equivalent to this notion for treewidth; see Section \[sec:bwframework\]. The main difference is that $k$-expressions deal with sets of equivalent vertices instead of the vertices themselves.
The representation used in the algorithm for the [Dominating Set]{} problem above also has the de Fluiter property for cliquewidth.
The representation of partial solutions for the [Dominating Set]{} problem used in Theorem \[thrm:dscwalg\] has the de Fluiter property for cliquewidth with $f(k) = 2k$.
Consider any partially constructed graph $H$ from a partial bottom-up evaluation of the $k$-expression for a graph $G$, and let $S$ be the set of vertices of the smallest remaining partial solution stored in the table for the subgraph $H$. We prove the lemma by showing that by adding at most $2k$ vertices to $S$, we can dominate all future neighbours of the vertices in $H$ and all vertices in $H$ that will receive future neighbours. We can restrict ourselves to adding vertices to $S$ that dominate these vertices and not vertices in $H$ that do not receive future neighbours, because Definition \[def:fluiterpropcw\] considers only partial solutions on $H$ that can still lead to an optimal solution on $G$. Namely, a vertex set $V(i)$ that contains undominated vertices that will not receive future neighbours when continuing the evaluation of the $k$-expression will not lead to an optimal solution on $G$. This is because the selection of the vertices that will be in a dominating set happens only in the ‘create a new graph’ operations.
We now show that by adding at most $k$ vertices to $S$, we can dominate all vertices in $H$, and by adding another set of at most $k$ vertices to $S$, we can dominated all future neighbours of the vertices in $H$. To dominate all future neighbours of the vertices in $H$, we can pick one vertex from each set $V(i)$. Next, consider dominating the vertices in each of the vertex sets $V(i)$ and are not yet dominated and that will receive future neighbours. Since the ‘add edges’ operations of a $k$-expression can only add edges between future neighbours and all vertices with the label $i$, and since the ‘relabel’ operation can only merges the sets $V(i)$ and not split them, we can add a single vertex to $S$ that is a future neighbour of a vertex in $V(i)$ to dominate all vertices in $V(i)$.
Using this property, we can easily improve the result of Theorem \[thrm:dscwalg\] for the case where we want to count only the number of minimum dominating sets. This goes in a way similar to Corollaries \[cor:countmdstwalg\], \[cor:solvedstwalg\], \[cor:countmdsbwalg\], and \[cor:solvedsbwalg\].
\[cor:countmdscwalg\] There is an algorithm that, given a $k$-expression for a graph $G$, computes the number of minimum dominating sets in $G$ in ${\ensuremath{\mathcal{O}}}(n k^2 4^k i_\times(n))$ time.
For each colouring $c$, we maintain the size $B(c)$ of any minimum partial dominating set inducing $c$, and the number $A(c)$ of such sets. This can also be seen as a table $D(c)$ of tuples. Define a new function $\oplus$ such that $$(A(c), B(c)) \oplus (A(c'), B(c')) \ = \ \left\{\begin{array}{ll}
(A(c) + A(c'), B(c)) & \mbox{if}\ B(c) = B(c')\\
(A(c^{*}), B(c^{*})) & \mbox{otherwise}
\end{array}\right.$$ where $c^{*} = \arg\min\{B(c),B(c')\}$. We will use this function to ensure that we count only dominating sets of minimum size.
We now modify the algorithm of Theorem \[thrm:dscwalg\] to use the tables $D$. For the first three operations, simply omit the size parameter $\kappa$ from the formula and replace any $+$ by $\oplus$. For instance, the computation for the third operation that adds new edges connecting all vertices with label $V(i)$ to all vertices with label $V(j)$, becomes: $$D(c \times \{(i_1,d_1)\} \times \{(i_2,d_2)\}) = \mathop{\bigoplus_{d'_1 \in \{0,1\}}}_{\max\{d'_1,i_2\} = d_1} \; \mathop{\bigoplus_{d'_2 \in \{0,1\}}}_{\min\{d'_2, i_1\} = d_2} D(c \times (i_1, d'_1) \times (i_2, d'_2))$$
For the fourth operation, where we take the union of two labelled graphs, we need to be more careful. Here, we use that the given representation of partial solutions has the de Fluiter property for cliquewidth. We first discard solutions that contain vertices that are undominated and will not receive new neighbours in the future, that is, we set the corresponding table entries to $D(c) = (\infty,0)$. Then, we also discard any remaining solutions that are at least $2k$ larger than the minimum remaining solution.
Let $D_1(c) = (A_1(c),B_1(c))$ and $D_2(c) = (A_2(c),B_2(c))$ be the two resulting tables for the two labelled graphs $H_1$ and $H_2$ we need to join. To perform the join operation, we construct tables $A_1(c,\kappa)$ and $A_2(c,\kappa)$ as follows, with $y \in \{1,2\}$: $$A_y(c, \kappa) = \left\{ \begin{array}{ll} A_y(c) & \mbox{if}\ B_y(c) = \kappa \\ 0 & \mbox{otherwise} \end{array} \right.$$ In these tables, $\kappa$ has a range of size $2k$ and thus this table has size ${\ensuremath{\mathcal{O}}}(k\, 4^{k})$.
Now, we can apply the same algorithm for the join operations as described in Theorem \[thrm:dscwalg\]. Afterwards, we retrieve the value of $D(c)$ by setting $A(c) = A(c,\kappa')$ and $B(c) = \kappa'$, where $\kappa'$ is the smallest value of $\kappa$ for which $A(c,\kappa)$ is non-zero.
For the running time, we observe that each of the ${\ensuremath{\mathcal{O}}}(k^2n)$ operations that create a new graph, relabel vertex sets, or add edges to the graph compute ${\ensuremath{\mathcal{O}}}(4^k)$ tuples that cost ${\ensuremath{\mathcal{O}}}(i_+(n))$ time each since we use a constant number of additions and comparisons of an $\log(n)$-bits number and an $n$-bits number. Each of the ${\ensuremath{\mathcal{O}}}(n)$ join operations cost ${\ensuremath{\mathcal{O}}}(k^2 4^k i_\times(n))$ time because of the reduced table size. In total, this gives a running time of ${\ensuremath{\mathcal{O}}}(n k^2 4^k i_\times(n))$.
Finally, we show that one can use ${\ensuremath{\mathcal{O}}}(k)$-bit numbers when considering the decision version of this minimisation problem instead of the counting variant.
\[cor:solvedscwalg\] There is an algorithm that, given a $k$-expression for a graph $G$, computes the size of a minimum dominating sets in $G$ in ${\ensuremath{\mathcal{O}}}(n k^2 4^k)$ time.
Maintain only the size $B(c)$ of any partial solution satisfying the requirements of the colouring $c$ in the computations involved in any of the first three operations. Store this table by maintaining the size $\xi$ of the smallest solution in $B$ that has no undominated vertices that will not get future neighbours, and let $B$ contain ${\ensuremath{\mathcal{O}}}(\log(k))$-bit numbers giving the difference in size between the size of the partial solutions and the number $\xi$; this is similar to, for example, Corollary \[cor:solvedstwalg\].
For the fourth operation, follow the same algorithm as in Corollary \[cor:countmdscwalg\], using $A(c,\kappa) = 1$ if $B(c)=\kappa$ and $A(c,\kappa) = 0$ otherwise. Since the total sum of all entries in this table is $4^k$, the computations for the join operation can now be implemented using ${\ensuremath{\mathcal{O}}}(k)$-bit numbers. See also, Corollaries \[cor:solvedstwalg\] and \[cor:solvedsbwalg\]. In the computational model with ${\ensuremath{\mathcal{O}}}(k)$-bit word size that we use, the term in the running time for the arithmetic operations disappears since $i_\times(k) = {\ensuremath{\mathcal{O}}}(1)$.
We conclude by noticing that ${\ensuremath{\mathcal{O}^*}}(4^k)$ algorithms for [Independent Dominating Set]{} and [Total Dominating Set]{} follow from the same approach. For [Total Dominating Set]{}, we have to change only the fact that a vertex does not dominate itself at the ’create new graph’ operations. For [Independent Dominating Set]{}, we have to incorporate a check such that no two vertices in the solution set become neighbours in the ‘add edges’ operation.
Relations Between the de Fluiter Properties and Finite Integer Index {#sec:fluiterprop}
====================================================================
In the previous sections, we have defined *de Fluiter properties* for all three types of graph decompositions. This property is highly related to the concept *finite integer index* as defined in [@BodlaenderA01]. Finite integer index is a property used in reduction algorithms for optimisation problems on graphs of small treewidth [@BodlaenderA01] and is also used in meta results in the theory of kernelisation [@BodlaenderFLPST09]. We will conclude by explaining the relation between the de Fluiter properties and finite integer index.
We start with a series of definitions. Let a *terminal graph* be a graph $G$ together with an ordered set of distinct vertices $X = \{x_1,x_2,\ldots,x_l\}$ with each $x_i \in V$. The vertices $x_i \in X$ are called the *terminals* of $G$. For two terminal graphs $G_1$ and $G_2$ with the same number of terminals, the addition operation $G_1 + G_2$ is defined to be the operation that takes the disjoint union of both graphs, then identifies each pair of terminals with the same number $1,2,\ldots,t$, and finally removes any double edges created.
For a graph optimisation problem $\Pi$, Bodlaender and van Antwerpen-de Fluiter define an equivalence relation $\sim_{\Pi,l}$ on terminal graphs with $l$ terminals [@BodlaenderA01]: $G_1 \sim_{\Pi,l} G_2$ if and only if there exists an $i \in {\ensuremath{\mathbb{Z}}}$ such that for all terminal graphs $H$ with $l$ terminals: $$\pi(G_1 + H) = \pi(G_2 + H) + i$$ Here, the function $\pi(G)$ assigns the objective value of an optimal solution of the optimisation problem $\Pi$ to the input graph $G$.
An optimisation problem $\Pi$ is of *finite integer index* if $\sim_{\Pi,l}$ has a finite number of equivalence classes for each fixed $l$.
When one proves that a problem has finite integer index, one often gives a representation of partial solutions that has the de Fluiter property for treewidth; see for example [@Fluiter97]. That one can prove that a problem has finite integer index in this way can see from the following proposition.
\[prop:fii\] If a problem $\Pi$ has a representation of its partial solutions of different characteristics that can be used in an algorithm that performs dynamic programming on tree decompositions and that has the de Fluiter property for treewidth, then $\Pi$ is of finite integer index.
Let $l$ be fixed, and consider an $l$-terminal graph $G$. Construct a tree decomposition $T$ of $G$ such that bag associated the root of $T$ equals the set of terminals $X$ of $G$. Note that this is always possible since we have not specified a bound on the treewidth of $T$. For an $l$-terminal graph $H$, one can construct a tree decomposition of $G + H$ by making a similar tree decomposition of $H$ and identifying the roots, which both have the same vertex set $X$.
Let $G_1$, $G_2$ be two $l$-terminal graphs to which we both add another $l$-terminal graph $H$ through addition, i.e., $G_i + H$, and let $T_1$, $T_2$ be tree decompositions of these graphs obtained in the above way. For both graph, consider the dynamic programming table constructed for the node $x_X$ associated with the vertex set $X$ by a dynamic programming algorithm for $\Pi$ that has the de Fluiter property for treewidth. For these tables, we assume that the induced subgraph associated with $x_X$ of the decompositions equals $G_i$, that is, the bags of the nodes below $x_X$ contain all vertices in $G_i$, and vertices in $H$ only occur in bags associated with nodes that are not descendants of $x_X$ in $T_i$.
Clearly, $\pi(G_1 + H) = \pi(G_2 + H)$ if both dynamic programming tables are the same and $G_1[X] = G_2[X]$, that is, if the tables are equal and both graphs have the same edges between their terminals. Let us now consider a more general case where we first normalise the dynamic programming tables such that the smallest valued entry equals zero, and all other entries contain the difference in value to this smallest entry. In this case, it is not hard to see that if both normalised dynamic programming tables are equal and $G_1[X] = G_2[X]$, then there must exists an $i \in {\ensuremath{\mathbb{Z}}}$ such that $\pi(G_1 + H) = \pi(G_2 + H) + i$.
The dynamic programming algorithm for the problem $\Pi$ can compute only finite size tables. Moreover, as the representation used by the algorithm has the de Fluiter property for treewidth, the normalised tables can only have values in the range $0,1,\ldots,f(k)$. Therefore, there are only a finite number of different normalised tables and a finite number of possible induced subgraphs on $l$ vertices (terminals). We conclude that the relation $\sim_{\Pi,l}$ has a finite number of equivalence classes.
By the same Proposition it follows that problems that are not of finite integer index (e.g., [Independent Dominating Set]{}) do not have a representation of partial solutions that has the de Fluiter property for treewidth. We note that the converse of Proposition \[prop:fii\] is not necessarily true.
While we focused on the relation between finite integer index and the de Fluiter property for treewidth (or branchwidth), we do not know the relation between these concepts and the de Fluiter property for cliquewidth. This property seems to be very different to the other two. Finding the details on the relations between all these properties is beyond the scope of this paper as we only used the de Fluiter properties to improve the polynomial factors in the running times of the presented algorithms.
Conclusion {#sec:conclusion}
==========
We have presented faster algorithms for a broad range of problems on three different types of graph decompositions. These algorithms were obtained by using generalisations of the fast subset convolution algorithm, sometimes combined with using fast multiplication of matrices. On tree decompositions and clique decompositions the exponential factor in the running times equal the space requirement for such algorithms. On branch decompositions, the running times of our algorithms come very close to this space bound. Additionally, a further improvement of the exponential factor in the running time for some problems on tree decompositions would contradict the Strong Exponential-Time Hypothesis.
We like to mention that, very recently, ${\ensuremath{\mathcal{O}^*}}(c^k)$-time algorithms for various problems on tree decompositions have been obtained, for some constant $c \geq 2$, for which previously only ${\ensuremath{\mathcal{O}^*}}(k^k)$-time algorithms existed [@CyganNPPRW11]. This includes problems like [Hamilton Cycle]{}, [Feedback Vertex Set]{}, [Steiner Tree]{}, [Connected Dominating Set]{}. Our techniques play an important role in this paper to make sure that the constants $c$ in these algorithms are small and equal the space requirement. Here, $c$ is often also small enough such that no faster algorithms exist under the Strong Exponential-Time Hypothesis [@CyganNPPRW11]. It would be interesting to find a general result stating which properties a problem (or join operation on nice tree decompositions) must have to admit ${\ensuremath{\mathcal{O}}}(c^k)$-time algorithms, where $c$ is the space requirement of the dynamic programming algorithm.
To conclude, we note that, for some problems like counting perfect matchings, the running times of our algorithms on branch decompositions come close to the running times of the currently-fastest exact exponential-time algorithms for these problems. For this we use that the branchwidth of any graph is at most $\frac{2}{3}n$, e.g., see [@Hicks05]. In this way, we directly obtain an ${\ensuremath{\mathcal{O}}}(2^{\frac{\omega}{2} \cdot \frac{2}{3} n}) = {\ensuremath{\mathcal{O}}}(1.7315^n)$-time algorithm for counting the number of perfect matchings. This running time is identical to the fast-matrix-multiplication-based algorithm for this problem by Björklund et al. [@BjorklundH08]. We note that this result has recently been improved to ${\ensuremath{\mathcal{O}}}(1.6181^n)$ by Koivisto [@Koivisto09]. Our algorithm improves this result on graphs for which we can compute a branch decomposition of width at most $0.5844n$ in polynomial time; this is a very large family of graphs since this bound is not much smaller that the given upper bound of $\frac{2}{3}n$.
### Acknowledgements {#acknowledgements .unnumbered}
The first author is grateful to Jan Arne Telle for introducing him to several problems solved in this paper at Dagstuhl seminar 08431, and for the opportunity to visit Bergen to work with Martin Vatshelle. The first author also thanks Jesper Nederlof for several useful discussions.
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[^1]: Department of Information and Computing Sciences, Utrecht University, P. O. Box 80.089, NL-3508 TB Utrecht, The Netherlands, `[email protected], [email protected]`
[^2]: Department of Informatics, University of Bergen, P. O. Box 7803, NO-5020 Bergen, Norway, `[email protected], [email protected]`
[^3]: Department of Computer Science, RWTH Aachen University, DE-52056 Aachen, Germany, `[email protected]`
[^4]: Preliminary parts of this paper have appeared under the title ‘Dynamic Programming on Tree Decompositions Using Generalised Fast Subset Convolution’ on the 17th Annual European Symposium on Algorithms (ESA 2009), Lecture Notes in Computer Science 5757, pages 566-577, and under the title ‘Faster Algorithms on Branch and Clique Decompositions’ on the 35th International Symposium Mathematical Foundations of Computer Science (MFCS 2010), Lecture Notes in Computer Science 6281, pages 174-185.
|
---
abstract: 'It is tacitly accepted that, for practical basis sets consisting of N functions, solution of the two-electron Coulomb problem in quantum mechanics requires storage of O(N$^4$) integrals in the small N limit. For localized functions, in the large N limit, or for planewaves, due to closure, the storage can be reduced to O(N$^2$) integrals. Here, it is shown that the storage can be further reduced to O($N^{2/3}$) for separable basis functions. A practical algorithm, that uses standard one-dimensional Gaussian-quadrature sums, is demonstrated. The resulting algorithm allows for the simultaneous storage, or fast reconstruction, of any two-electron coulomb integral required for a many-electron calculation, on each and every processor of massively parallel computers even if such processors have very limited memory and disk space. For example, for calculations involving a basis of 9171 planewaves, the memory required to effectively store all coulomb integrals decreases from 2.8Gbytes to less than 2.4 Mbytes.'
author:
- 'Mark R. Pederson'
title: 'Practical and Rigorous Reduction of the Many-Electron Quantum Mechanical Coulomb Problem to O(N$^{2/3}$) Storage'
---
Introduction
============
In this communication a workable algorithm is derived and presented that allows each processor to store all information required to quickly look up any two-electron integral, involving four basis functions, needed for either density-functional or multiconfigurational wavefunction methods. The method is demonstrated by applications of a uniform electron gas, confined to a cubic box, for electrons with wavevectors that are enclosed in a Fermi sphere.
Strategies for rapid calculation or efficient storage of two-electron integrals, for density-functional [@HK; @KS] calculations, or multiconfigurational active space methods [@molcas; @LG] continue to evolve as different mathematical techniques and different types of computing platforms arise and as different types of basis functions are implemented for use in electronic structure calculations. A recent comprehensive review of these efforts by Reine [*et al*]{} [@reine] includes discussions of least-square variational fitting methods [@koster; @dunlap] and Rys polynomials [@king]. Other methods such as direct methods [@almlof], analytic algebraic decompositions [@nrlmol], tensor hypercontraction [@parrish] and multipole methods [@Lambrecht] are also widely used. Many of these methods support the hypothesis that the space of two-electron integrals is smaller than naively expected. This paper seeks to formally prove, for separable functions used in electronic structure calculations, that the set of information on which the N$^4$ Coulomb integrals truly depends is much smaller than expected from a permutational analysis. Further a practical approach is developed and applied to the uniform electron gas. The algorithm is based upon a three-dimensional Fourier transform, a one-dimensional Laplace transform, an additional one-dimensional integral transform, and the use of Gaussian quadrature. The storage requirements needed to calculate matrix elements associated with the coulomb operator is reduced to O($N^{2/3}$) for either planewaves or Gaussians.
Another motivation for this work is that the development of massively parallel methods requires one to break a problem up into many independent subtasks that can then be performed simultaneously by a large number of computer processors [@nrlmol]. To achieve high efficiency on massively parallel architectures, it is necessary to ensure that the amount of information exchanged between processors is small and that the rate of information exchange is intrinsically faster than the computing time used by any processor. For future low-power computing platforms it is desireable, if not expected, for each processor to have a very limited amount of computer memory. Thus, in reference to many-electron quantum mechanics or density functional theory [@HK; @PW92; @PBE; @MHG; @DT], it is appropriate to reconsider whether there are other means for reconstructing matrix elements that might be more efficient on modern massively parallel architectures. For such systems it would be ideal to allow each processor to quickly reconstruct any possible coulomb integral needed for a quantum-mechanical simulation without information transfer to or from other processors.
Derivation
==========
There is one important aspect of this derivation that appears to be universally correct for many, possibly all, choices of separable basis functions and that is definitely correct for planewave and Gaussian basis functions. Therefore some general considerations are discussed before moving the focus of this paper to applications within planewave basis sets. Given a set of infinitely differentiable and continuous one-dimensional functions, labeled as f$_l(x)$, it is possible to create three-dimensional basis functions $g_{\bf I}({\bf r})$ according to: $$g_{\bf I}({\bf r}) = f_l(x)f_m(y)f_n(z)=\prod_x f_{I_x}(x),$$ with ${\bf I}=(l,m,n)$. Common examples of such basis functions include planewaves inside a box or unit cell or products of one-dimension Gaussian functions which generally also have separable polynomial prefactors. In the former case one generally uses all possible products subject to the constraint that $\frac{2\pi}{L}|{\bf I}|<k_c$ and then seeks convergence by performing the calculation as a function of the cutoff wavenumber ($k_c$). Assuming one chooses a total of N three-dimensional basis functions, it is then clear that there are approximately $N^{1/3}$ one-dimensional basis functions for each cartesian coordinate. For simplicity, but not actually required for this observation, the assumption is that the same one-dimensional basis sets are used for each cartesian component. So, even though there are $N^{2}$ pairs of three dimensional basis functions, there are only N$^{2/3}$ one dimensional products of basis functions. For planewaves, the complexity is further reduced to $2N^{1/3}$ since the product of a planewave is a plane wave. For Gaussians this number becomes $\eta N^{1/3}$, with $\eta$ a characteristic number of neighbors, since the product of two well separated Gaussians is identically zero. The matrix elements that are needed to solve the Coulomb problem in density functional theory or to determine matrix elements required for either Hartree-Fock or Multi-Configurational calculations are given by
$$C_{\bf IJKL}=<g_{\bf I} g_{\bf J}|\frac{1}{|{\bf r-r'}|}| g_{\bf K} g_{\bf L}>=\int \int d^3r d^3r' \frac{1}{|\bf {r-r'}|}
g_{\bf I}({\bf r}) g_{\bf J}({\bf r})
g_{\bf K}({\bf r}) g_{\bf L}({\bf r}).$$
However, by using a continuous Fourier transform of $\frac{1}{|{\bf r}-{\bf r'}|}$, followed by a Laplace transform of $\frac{1}{p^2}$, the above equation can be written in quasi-separable form according to: $$\begin{aligned}
C_{\bf IJKL}=&&<g_{\bf I} g_{\bf J}|\frac{1}{|{\bf r-r'}|}| g_{\bf K} g_{\bf L}> \\
&&=4 \pi \int d^3p \int d^3r \int d^3r' \frac{e^{i{\bf p (r-r')}}}{p^2}
g_{\bf I}({\bf r}) g_{\bf J}({\bf r})
g_{\bf K}({\bf r'}) g_{\bf L}({\bf r'}).\\
%C_{\bf IJKL}
%&&=<g_{\bf I} g_{\bf J}|\frac{1}{|{\bf r-r'}|}| g_{\bf K} g_{\bf L}> \\
&&=4 \pi \int_0^\infty d \alpha \int d^3p \int d^3r \int d^3r' e^{i{\bf p (r-r')}}e^{-\alpha p^2}
g_{\bf I}({\bf r}) g_{\bf J}({\bf r})
g_{\bf K}({\bf r'}) g_{\bf L}({\bf r'}) \\
%&&=<g_{\bf I} g_{\bf J}|\frac{1}{|{\bf r-r'}|}| g_{\bf K} g_{\bf L}>
&&=4\pi \int_0^\infty d\alpha
\prod_x A_x(\alpha,I_x,J_x,K_x,L_x) \\
&&=4\pi \int_0^{\alpha_c} d\alpha
\prod_x A_x(\alpha,I_x,J_x,K_x,L_x)
+4\pi \int_{\alpha_c}^\infty d\alpha
\prod_x A_x(\alpha,I_x,J_x,K_x,L_x).\end{aligned}$$ Eq. 4 follows from Eq. 3 by a continuous Fourier transform of $1/|{\bf r- r'|}$. Eq. 5 follows from Eq. 4 by a continuous Laplace transform of $1/p^2$. Eq. 6 follows from Eq. 5 since all functions are separable. In the above equation, the nine-dimensional integral is reduced to a triple product. Each one of these products are composed of three dimensional integrals that is defined according to: $$A_x(\alpha,I_x,J_x,K_x,L_x)= \int dx \int dx' \int dp_x e^{-\alpha p_x^2} e^{i p_x(x-x')}
f_{I_x}(x)f_{J_x}(x')
f_{K_x}(x)f_{L_x}(x').$$
For either one-dimensional planewaves or Gaussians, the above three-dimensional integral can be determined, as a function of $\alpha$, without significant difficulty. It is possible that for other separable functions these integrals would be difficult to calculate. However, since in the worst case there are only $N^{4/3}$ of these integrals, one can imagine calculating them only once and storing them forever. This means that one only needs to find an efficient numerical method for performing the Laplace integral in Eq. 6. From this standpoint, an observation that is absolutely key to capitalizing on this quasi-separable form is that by integrating the above expression (Eq. 8) over $p_x$, the $\alpha$-dependent part of the, now, two dimensional integral, can in principal, be reduced to products of quantities with the following form: $$\frac{exp(-\frac{(x-x')^2}{4 \alpha})}{\sqrt{\alpha}} = \Sigma_{n=0}^{\infty}
a_n \frac{(x-x')^{2n}}{\alpha^{n+\frac{1}{2}}},$$ with $a_n=(-1)^n/n!$. Therefore, for a large enough value of $\alpha_c$, it follows that Eq. (7) may be rewritten, to any desired precision, according to: $$C_{\bf IJKL}=
=4\pi \int_0^{\alpha_c} d\alpha
\prod_x A_x(\alpha,I_x,J_x,K_x,L_x)
+4\pi \Sigma_{n=0}^{\infty} \Gamma_n({\bf I,J,K,L}) \int_{\alpha_c}^\infty d\alpha
\frac{1}{\alpha^{n+\frac{3}{2}}}.$$ In the above equation the $\Gamma_n$ are hard-to-determine constants that depend upon the functional form of separable basis sets, the Taylor expansion coefficients, $a_n$, in Eq. (9), a lot of really complicated algebra, triple products of two-dimensional integrals associated with Eq. (8), and the collection of common coefficients of $1/\alpha^{n+3/2}$ arising from the occurrence of triple summations associated with each cartesian coordinate. It would be algebraically difficult and computationally inefficient but not impossible to calculate these numbers. [**However, for the purpose here it is only necessary to know that the value of $\Gamma_n$ could, in principle, be found and to accept that knowledge about the asymptotic power law associated with the Laplace integrand provides very important information about how to numerically evaluate the integral which extends to infinity.**]{} To make further progress, the second term in the Eq. 10 is temporarily rewritten by making the substitution $t=\frac{1}{\sqrt{\alpha}}$, and $dt=-\frac{1}{2}\frac{d\alpha}{\alpha^{3/2}}$. This leads to: $$C_{\bf IJKL}=
4\pi \int_0^{\alpha_c} d\alpha
\prod_x A_x(\alpha,I_x,J_x,K_x,L_x)
+2\pi \Sigma_{n=0}^{\infty} \Gamma_n({\bf I,J,K,L})
\int^{\frac{1}{\sqrt{\alpha_c}}}_{0} t^{2n} dt.$$ Now, since both definite integrals are to be evaluated over a finite interval, these integrals can be evaluated using Gaussian-quadrature or other one-dimensional numerical integration meshes according to: $$\begin{aligned}
C_{\bf IJKL}&&=
4\pi \Sigma_{i=1}^Q w_{1i}
\prod_x A_x(\alpha_i,I_x,J_x,K_x,L_x) \\ \nonumber
+&&4\pi
\Sigma_{i=1}^Q \frac{w_{2i} }{2}
\Sigma_{n=0}^{\infty} \Gamma_n({\bf I,J,K,L})
\frac{\alpha_i^{3/2}}{\alpha_i^{3/2}} t_i^{2n}.\end{aligned}$$ In the above expressions the two sets of Gaussian-quadrature weights and points, ${w_{1i},\alpha_i}$ and ${w_{2i},t_i}$ depend only on the choice of $\alpha_c$ and methods and codes for choosing these points are widely available and well known [@vmesh; @recipes]. A back transformation of the right-hand sum, obtained by setting $\frac{1}{\alpha_i}=t_i^2$, and defining $\Omega_i=\frac{1}{2}w_{2i} \alpha_i^{3/2}t_{i}^{2n}$, the integral collapses to the original recognizable form: $$\begin{aligned}
C_{\bf IJKL}&&=
=4\pi \Sigma_{i=1}^Q w_i[0,\alpha_c]
\prod_x A_x(\alpha_i,I_x,J_x,K_x,L_x) \\ \nonumber
+&&4\pi
\Sigma_{i=1}^Q \Omega_i
\Sigma_{n=0}^{\infty} \Gamma_n({\bf I,J,K,L})
\frac{1}{\alpha_i^{n+\frac{3}{2}}}.\end{aligned}$$ With a suitable redefinition of notation for the volume elements and the recognition that the second term includes a summation which is exactly equal to $\prod_x A_x(\alpha_i,I_x,J_x,K_x,L_x)$, the Laplace integral is reduced to quadratures over products of three one-dimensional integrals (Eq. 8). Here, it is emphasized, that Eq. (7) could have been immediately written in terms of numerical integrals. However the analysis followed allows one to determine how the asymptotic form of the integrand scales so that the particular case of Gaussian quadrature methods, that are amenable to numerical evaluation of polynomials over finite intervals, may be used for performing the integrations. As written, it has been demonstrated that one needs to store at most N$^{4/3}$ one dimensional integrals to reconstruct any of the N$^{4}$ integrals. Based on past usage of quadrature methods, it is reasonable to expect that one can perform multiscale numerical one-dimensional integration, such as the Laplace transformation here, with approximately 30-100 sampling points [@vmesh]. $$\begin{aligned}
C_{\bf IJKL}&&
=4\pi \Sigma_{i=1}^{2Q} \Omega_i
\prod_x A_x(\alpha_i,I_x,J_x,K_x,L_x) \end{aligned}$$ While the results discussed here are a factor of 2-4 away from this goal, it is likely that the number of sampling points can be significantly decreased by determining the value of $\alpha_c$ which allows for the most efficient numerical integration, by breaking the $\alpha$ integral (Laplace transformation) into more than two intervals, and/or by using techniques similar to the variational one-dimensional exponential quadrature methods of Ref [@vmesh]. For example, a quadrature mesh constructed to integrate polynomials of $x^2$, rather than x, would be twice as efficient as the standard Gaussian quadratures meshes. Except for the clear need to exploit the $t=1/\sqrt{\alpha}$ transformation for the final interval that extends to $\infty$, finding the best quadrature sums are expected to depend on the form of the separable functions being employed. Here, for simplicity and reproducibility by others, only standard Gaussian-quadrature methods, with $\alpha_c \equiv 1$, are used.
Reduction of Storage to $4N^{2/3}$ for Plane Waves: Exact Exchange for the Uniform Electron Gas
===============================================================================================
For planewaves, the product of the one-dimensional functions $f_{I_x}f_{j_x}$ reduce to a product of two one-dimensional planes waves which is itself a planewave. If one starts with $N^{1/3}$ one dimensional planewaves (e.g. $f_I=exp(i2I\pi/L)$, the products will only provide $2N^{1/3}$ plane waves. Therefore the number of one-dimensional integrals that are required is reduced to $4N^{2/3}$. As a simple application, the M-dependence of the exact exchange energy of an unpolarized gas of 2M electrons in a box with finite volume (V=LxLxL) is determined in this section. As M gets very large, the exchange energy will converge to the Kohn-Sham value of $E_{KS}=-(3/4)(6/\pi)^{1/3} M^{4/3}/L$. It is also easy to verify based on scaling arguments that for any number of planewaves placed inside such a box, the exact exchange energy will scale a $\beta(M,\{q_{\bf k}\})$/L with $\beta$ depending on the occupations $\{q_{\bf k}\}$ as a function of wavevector and the number of electrons M placed in the box. Here to validate the numerics, the standard choice of occupation numbers are taken to be unity for all planewaves enclosed in a Fermi sphere of various radii. The radii, or Fermi wavevector, are chosen so that there are shell closings in reciprocal space.
For a finite system, it is possible to fully occupy a Fermi sphere for a well defined cutoff wavevector if one chooses M= 7, 33, 123, 257, 515, 925,1419, 2109, 3071, 4169, 5575, 7153, or 9171 electrons of each spin. In Fig. 1, the ratio of the exact exchange energy to the Kohn-Sham energy is presented as a function of $1/M^{1/3}$. In the large M limit, it is evident that this ratio converges linearly to 1. This indicates that all the integrals are being performed accurately. In Fig. 2, the time required per electron, as a function of the total number of electrons, is shown. For cases where each KS-orbital is identically equal to a planewave the time required for the calculation of the exchange (or coulomb) interaction scales as the square of the number of electrons. For 9171 electrons, the Hartree-Fock exchange energy can be calculated in four seconds on a MacBook Air. In Table I, the convergence of the Hartree-Fock energy for M=9171 parallel spin electrons is shown as a function of Gaussian-quadrature mesh. For purposes of reproducibility, the first mesh is determined by Q quadrature points on the interval between 0 and 1. These points, designated by $\{\alpha_i,w_{1i}\}$ in Eq. (12) are then transformed as described above to reduce the calculation of each exchange integral to the form shown in Eq. 14 (e.g. a total of 2Q mesh points for the two intervals). The results show that with standard quadrature methods, and an overly simple tesselation into only two sub-intervals, it is difficult to efficiently converge the energy due to sharp structure near $\alpha=0$. However, as shown in the right-most columns, if one further breaks the first interval into sub-intervals defined by $[0,1/5^7],[1/5^7,1/5^6],...,[1/5,1]$ and then uses 5-, 10-, and 15-point quadrature meshes in each of these sub intervals, convergence of the energy for M=9171 electrons is achieved.
Mesh 1 (Q) Interval 1 Total Mesh 2 (Q) Interval 1 Total
------------ ------------ ---------- ------------ ------------ ----------
90 0.936629 0.965018 8x5 0.936961 0.965350
105 0.936884 0.965272 8x10 0.937043 0.965431
120 0.936953 0.965341 8x15 0.937043 0.965431
150 0.937001 0.965389
180 9.937019 0.965408
: Ratio of exchange energy to Kohn-Sham exchange energy for a cube containing 2M=18342 electrons as a function of the number of quadrature points used in Eq. 14. Mesh 1 uses Q quadrature points on an interval between 0 and 1. Mesh 2, which breaks interval 1 into eight sub-intervals with geometrically varying length scales is numerically more efficient and allows for at least six-place precision. This suggests that the variational exponential quadrature methods, used for radial integrations in Ref. [@vmesh] may be more efficient
.
To summarize, this paper provides a practical and systematically improvable algorithm that reduces the storage required for the coulomb integrals to $O(N^{2/3})$ for the special cases of basis sets that are commonly used in electronic structure calculations. For the case of planewave calculations, it is only recently that researchers have begun to entertain the possibility of performing multiconfigurational corrections using such basis sets. The results of this paper significantly lower the storage requirements needed for either DFT, Hartree-Fock, or multi-configurational methods based upon planewaves. Future improvements of this method, with initial applications of the self-interaction correction [@perzun; @mrp1; @mrp2] to the uniform electron gas calculations are in progress [@jianwei]. As compared to structurally simpler plane-wave methods, conversion of this algorithm for use withing Gaussian-based-orbital methodologies, will require a large investment of programming time but are fully expected to provide the same reduction of memory/disk requirements for reconstruction of the two-electron integrals.
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