Replica-nondiagonal solutions in the SYK model

Irina Aref’Eva,Mikhail Khramtsov,Maria Tikhanovskaya,Igor Volovich

doi:10.1007/jhep07(2019)113

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.

Highlights

The Sachdev-Ye-Kitaev model [1,2,3,4] is a quantum mechanical model of N Majorana fermions with disordered interactions that is solvable at large N in the strong coupling limit. It was proposed [2] as a solvable toy model for holographic description of quantum gravity in the AdS2 spacetime. This idea is justified by the fact that the SYK model displays emergent approximate conformal symmetry in the strong coupling regime [2, 3, 6] and that it exhibits maximal quantum chaos [2, 3, 7] at strong coupling
In the second part of the work, we focus on the strong coupling, or IR limit of the SYK model
We have found in the above section that the limit of zero replicas is singular in the replicanondiagonal solutions of the exact saddle point equations, and there are no replicanondiagonal solutions at M = 0 in the class which we studied

Summary

Introduction

The Sachdev-Ye-Kitaev model [1,2,3,4] is a quantum mechanical model of N Majorana fermions with disordered interactions that is solvable at large N in the strong coupling limit. It was proposed [2] as a solvable toy model for holographic description of quantum gravity in the AdS2 spacetime (see [5] for a review). While the precise bulk dual theory is still unknown, the SYK model has already allowed to obtain significant insight in the physics of black holes and wormholes [17, 18]

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Conclusion