Replica-nondiagonal solutions in the SYK 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 replicasymmetric 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.arXiv:1811.04831v4 [hep-th]

Revealing Nonperturbative Effects in the SYK Model

“…Comparing to the results of section 3.1 in [32] in the decoupled case ν = 0, we see that in the limit ν → 0 solutions 1 and 2 turn into the replica-diagonal solutions, whereas the solutions 3 and 4 turn into the replica-nondiagonal solutions. It is also important to note that for solutions 1 and 2 the subleading powers in ν do not change the leading UV asymptotic ω n → ∞.…”

“…This form of equations is valid for both zero and finite temperature cases for q = 2. In the case ν = 0 one recovers the equations for the decoupled replicas for M = 2, which were solved in [32]. Before we proceed to discussion of the solutions, let us note that the peculiar feature of the q = 2 case is that the source η in the form (2.6) respects the conformal symmetry in the IR limit and transforms with fixed conformal dimension ∆ = 1 2 .…”

“…where G 0 and G 1 are both assumed to be odd functions in the frequency and time representations, and to be antiperiodic in time. In [32] the constructed solutions for non-interacting replica case had the same assumptions applied.…”

“…In particular, it was shown that the replica-nondiagonal saddle points of the replica field path integral are related to the ramp behavior of the spectral form-factor [14]. Replica-nondiagonal structures of saddle points of SYK-like models were also studied in [24,[29][30][31][32][33][34], in particular in relation to the issue of spin glasses and replica symmetry breaking. From the gravity point of view, the nonperturbative effects arising from the subleading saddle points are related to the topologies of higher genus [15,18,35], and they probe discrete spectrum of black hole microstates and are important for the recovery of lost information [13][14][15].…”

“…In the previous work [32] we have obtained a family of exact replica-nondiagonal saddle points in the SYK model with M replicas, where M is a positive integer number. We have shown that these saddle points give nonperturbative contributions to the path integrals, that are suppressed at large N as e −N δI compared to the replica-diagonal saddle point.…”