Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1811.04831 · Journal of High Energy Physics

Replica-nondiagonal solutions in the SYK model

I. Ya. Aref’eva, Mikhail Alexandrovich Khramtsov, Maria Tikhanovskaya, Igor Volovich

Published Jul 1, 2019• 41 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/jhep07(2019)113

Related papers

On replica-nondiagonal large N saddles in the SYK model

I. Ya. Aref’eva, Mikhail Alexandrovich Khramtsov, Maria Tikhanovskaya +1 more

We study the saddle points of the SYK model, formulated in terms of the replica bilocal fields, beyond the replica-diagonal assumption. We find a family of replica-nondiagonal saddle points in the IR limit, where the saddle point equations are separable. We use the Parisi ansatz to find the replicanondiagonal solutions and take the replica limit. The free energy on these solutions is computed, and we find that some of the replica-nondiagonal saddles have lower free energy than the replica-diagonal saddle point.

EPJ Web of Conferences•Jan 1, 2018•10 citations

Revealing Nonperturbative Effects in the SYK Model

I. Ya. Aref’eva, I. V. Volovich, M. A. Khramtsov

In the large-N limit, we study saddle points of two SYK chains coupled by an interaction that is nonlocal in Euclidean time. We study the free model with the order of the fermionic interaction q = 2 analytically and also investigate the model with interaction in the case q = 4 numerically. We show that in both cases, there is a nontrivial phase structure with an infinite number of phases. Each phase corresponds to a saddle point in the noninteracting two-replica SYK. The nontrivial saddle points have a nonzero value of the replica-nondiagonal correlator in the sense of quasiaveraging if the coupling between replicas is turned off. The nonlocal interaction between replicas thus provides a protocol for turning the nonperturbatively subleading effects in SYK into nonequilibrium configurations that dominate at large N. For comparison, we also study two SYK chains with local interaction for q = 2 and q = 4 . We show that the q= 2 model has a similar phase structure, while the phase structure differs in the q = 4 model, dual to the traversable wormhole.

arXiv:1905.04203•Theoretical and Mathematical Physics•Nov 1, 2019•6 citations

Spontaneous Symmetry Breaking in the Sachdev–Ye–Kitaev Model

Mikhail Alexandrovich Khramtsov

Abstract We investigate the nonperturbative effects in the Sadchev–Ye–Kitaev (SYK) model at large $$N$$ . The replica-nondiagonal solutions are considered in the SYK model with an integer number of replicas $$M > 1$$ . An important property of such solutions is the fact that they realize spontaneous breaking of the time translation group on $$M$$ 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.

Physics of Particles and Nuclei•Jul 1, 2020•0 citations

Spectral form factor in the double-scaled SYK model

Mikhail Alexandrovich Khramtsov, Elena Nikolaevna Lanina

Abstract In this note we study the spectral form factor in the SYK model in large q limit at infinite temperature. We construct analytic solutions for the saddle point equations that describe the slope and the ramp regions of the spectral form factor time dependence. These saddle points are obtained by taking different approaches to the large q limit: the slope region is described by a replica-diagonal solution and the ramp region is described by a replica-nondiagonal solution. We find that the onset of the ramp behavior happens at the Thouless time of order q log q. We also evaluate the one-loop corrections to the slope and ramp solutions for late times, and study the transition from the slope to the ramp. We show this transition is accompanied by the breakdown of the perturbative 1/q expansion, and that the Thouless time is defined by the consistency of extrapolation of this expansion to late times.

arXiv:2011.01906•Journal of High Energy Physics•Mar 1, 2021•16 citations

SYK Meets Non-Hermiticity I: Emergent Replica Conformal Symmetry

Pengfei Zhang, Shao-Kai Jian, Chunxiao Liu +1 more

Recently, the steady states of non-unitary free fermion dynamics are found to exhibit novel critical phases with power-law squared correlations and a logarithmic subsystem entanglement. In this work, we theoretically understand the underlying physics by constructing solvable static/Brownian quadratic Sachdev-Ye-Kitaev chains with non-Hermitian dynamics. We find the action of the replicated system generally shows (one or infinite copies of) $O(2)\times O(2)$ symmetries, which is broken to $O(2)$ by the saddle-point solution. This leads to an emergent conformal field theory of the Goldstone modes. We derive their effective action and obtain the universal critical behaviors of squared correlators. Furthermore, the entanglement entropy of a subsystem $A$ with length $L_A$ corresponds to the energy of the half-vortex pair $S\sim \rho_s \log L_A$, where $\rho_s$ is the stiffness of the Goldstone mode. We also discuss special limits where more than one Goldstone mode exists and comment on interaction effects.

•Apr 8, 2021•3 citations

A derivation of the soft modes effective action for the SYK spectral form factor

Matteo A. Cardella

The SYK model spectral form factor exhibits absence of information loss in the form of a ramp and a plateau that are typical of random matrix theory. In a large N collective fields description, the ramp was reproduced by Saad, Shenker and Stanford by a connected component of the analytically continued thermal partition function two point correlator. The existence of an \emph{off shell} connected component for the real time partition function is a consequence of disorder averaging over a statistical ensemble of two replica systems. In particular, the ramp arises from a family of vanishing action replica-non-diagonal conformal saddles. We provide a derivation of the soft modes effective action for fluctuations around the ramp conformal saddles, by adapting to the two replica system a method used by Kitaev and Suh in regular SYK.

•Jul 22, 2019•0 citations
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 study the saddle points of the SYK model, formulated in terms of the replica bilocal fields, beyond the replica-diagonal assumption. We find a family of replica-nondiagonal saddle points in the IR limit, where the saddle point equations are separable. We use the Parisi ansatz to find the replicanondiagonal solutions and take the replica limit. The free energy on these solutions is computed, and we find that some of the replica-nondiagonal saddles have lower free energy than the replica-diagonal saddle point.
In the large-N limit, we study saddle points of two SYK chains coupled by an interaction that is nonlocal in Euclidean time. We study the free model with the order of the fermionic interaction q = 2 analytically and also investigate the model with interaction in the case q = 4 numerically. We show that in both cases, there is a nontrivial phase structure with an infinite number of phases. Each phase corresponds to a saddle point in the noninteracting two-replica SYK. The nontrivial saddle points have a nonzero value of the replica-nondiagonal correlator in the sense of quasiaveraging if the coupling between replicas is turned off. The nonlocal interaction between replicas thus provides a protocol for turning the nonperturbatively subleading effects in SYK into nonequilibrium configurations that dominate at large N. For comparison, we also study two SYK chains with local interaction for q = 2 and q = 4 . We show that the q= 2 model has a similar phase structure, while the phase structure differs in the q = 4 model, dual to the traversable wormhole.
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 In this note we study the spectral form factor in the SYK model in large q limit at infinite temperature. We construct analytic solutions for the saddle point equations that describe the slope and the ramp regions of the spectral form factor time dependence. These saddle points are obtained by taking different approaches to the large q limit: the slope region is described by a replica-diagonal solution and the ramp region is described by a replica-nondiagonal solution. We find that the onset of the ramp behavior happens at the Thouless time of order q log q. We also evaluate the one-loop corrections to the slope and ramp solutions for late times, and study the transition from the slope to the ramp. We show this transition is accompanied by the breakdown of the perturbative 1/q expansion, and that the Thouless time is defined by the consistency of extrapolation of this expansion to late times.
Recently, the steady states of non-unitary free fermion dynamics are found to exhibit novel critical phases with power-law squared correlations and a logarithmic subsystem entanglement. In this work, we theoretically understand the underlying physics by constructing solvable static/Brownian quadratic Sachdev-Ye-Kitaev chains with non-Hermitian dynamics. We find the action of the replicated system generally shows (one or infinite copies of) O(2)×O(2)O(2)\times O(2)O(2)×O(2) symmetries, which is broken to O(2)O(2)O(2) by the saddle-point solution. This leads to an emergent conformal field theory of the Goldstone modes. We derive their effective action and obtain the universal critical behaviors of squared correlators. Furthermore, the entanglement entropy of a subsystem AAA with length LAL_ALA​ corresponds to the energy of the half-vortex pair S∼ρslog⁡LAS\sim \rho_s \log L_AS∼ρs​logLA​, where ρs\rho_sρs​ is the stiffness of the Goldstone mode. We also discuss special limits where more than one Goldstone mode exists and comment on interaction effects.
The SYK model spectral form factor exhibits absence of information loss in the form of a ramp and a plateau that are typical of random matrix theory. In a large N collective fields description, the ramp was reproduced by Saad, Shenker and Stanford by a connected component of the analytically continued thermal partition function two point correlator. The existence of an \emph{off shell} connected component for the real time partition function is a consequence of disorder averaging over a statistical ensemble of two replica systems. In particular, the ramp arises from a family of vanishing action replica-non-diagonal conformal saddles. We provide a derivation of the soft modes effective action for fluctuations around the ramp conformal saddles, by adapting to the two replica system a method used by Kitaev and Suh in regular SYK.