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Proceedings of the Steklov Institute of Mathematics

Quasi-averages in Random Matrix Models

I. Ya. Aref’eva, Igor Vasil'evich Volovich

Published Sep 1, 2019• 1 citation
Publisher

Abstract

We use the Bogoliubov quasi-average approach to studying phase transitions in random matrix models related to a zero-dimensional version of the fermionic SYK model with replicas. We show that in the model with quartic interaction deformed by a quadratic term, there exist either two or four different phases with nonvanishing replica off-diagonal correlation functions.

DOI
10.1134/s0081543819050018

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We use the Bogoliubov quasi-average approach to studying phase transitions in random matrix models related to a zero-dimensional version of the fermionic SYK model with replicas. We show that in the model with quartic interaction deformed by a quadratic term, there exist either two or four different phases with nonvanishing replica off-diagonal correlation functions.
Abstract We investigate the nonperturbative effects in the Sadchev–Ye–Kitaev (SYK) model at large NNN . The replica-nondiagonal solutions are considered in the SYK model with an integer number of replicas M>1M > 1M>1 . An important property of such solutions is the fact that they realize spontaneous breaking of the time translation group on MMM replicas of the thermal circle. It is demonstrated in this study that this spontaneous symmetry breaking can be described in terms of the Bogolyubov quasi-averages, if we consider the system of nonlocally interacting replicas of the SYK model. The inclusion of the interaction of the replicas in this system transforms the subleading replica-nondiagonal saddle points from a pure SYK to thermodynamic phases, thereby generating a nontrivial phase structure. We investigate this phase structure exactly analytically in the SYK2 model and numerically in the SYK4 model.
Abstract We study the SYK model in the large N limit beyond the replica-diagonal approximation. First we show that there are exact replica-nondiagonal solutions of the saddle point equations for q = 2 for any finite replica number M.In the interacting q = 4 case we are able to construct the numerical solutions, which are in one-to-one correspondence to the analytic solutions of the quadratic model. These solutions are singular in the M → 0 limit in both quadratic and quartic interaction cases. The calculations of the on-shell action at finite integer M show that the nondiagonal replica-symmetric saddles are subleading in both quadratic and quartic cases. We also study replica-nondiagonal solutions of the SYK in the strong coupling limit. For arbitrary q we show that besides the usual solutions of the replica-diagonal saddle point equations in the conformal limit, there are also replica-nondiagonal solutions for any value of M (including zero). The specific configurations that we study, have factorized time and replica dependencies. The corresponding saddle point equations are separable at strong coupling, and can be solved using the Parisi ansatz from spin glass theory. We construct the solutions which correspond to the replica-symmetric case and to one-step replica symmetry breaking. We compute the regularizized free energy on these solutions in the limit of zero replicas. It is observed that there are nondiagonal solutions with the regularized free energy lower than that of the standard diagonal conformal solution.
We contrast some aspects of various SYK-like models with large-N melonic behavior. First, we note that ungauged tensor models can exhibit symmetry breaking, even though these are 0+1 dimensional theories. Related to this, we show that when gauged, some of them admit no singlets, and are anomalous. The uncolored Majorana tensor model with even N is a simple case where gauge singlets can exist in the spectrum. We outline a strategy for solving for the singlet spectrum, taking advantage of the results in arXiv: 1706.05364, and reproduce the singlet states expected in N = 2. In the second part of the paper, we contrast the random matrix aspects of some ungauged tensor models, the original SYK model, and a model due to Gross and Rosenhaus. The latter, even though disorder averaged, shows parallels with the Gurau-Witten model. In particular, the two models fall into identical Andreev ensembles as a function of N. In an appendix, we contrast the (expected) spectra of AdS 2 quantum gravity, SYK and SYK-like tensor models, and the zeros of the Riemann Zeta function.
We contrast some aspects of various SYK-like models with large-N melonic behavior. First, we note that ungauged tensor models can exhibit symmetry breaking, even though these are 0+1 dimensional theories. Related to this, we show that when gauged, some of them admit no singlets, and are anomalous. The uncolored Majorana tensor model with even N is a simple case where gauge singlets can exist in the spectrum. We outline a strategy for solving for the singlet spectrum, taking advantage of the results in arXiv:1706.05364, and reproduce the singlet states expected in N = 2. In the second part of the paper, we contrast the random matrix aspects of some ungauged tensor models, the original SYK model, and a model due to Gross and Rosenhaus. The latter, even though disorder averaged, shows parallels with the Gurau-Witten model. In particular, the two models fall into identical Andreev ensembles as a function of N . In an appendix, we contrast the (expected) spectra of AdS2 quantum gravity, SYK and SYK-like tensor models, and the zeros of the Riemann Zeta function.
We study the two-dimensional XY\mathrm{XY}XY model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to T=0T=0T=0 in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.