Quantum ergodicity in the SYK model

TL;DR

This paper develops a replica path integral approach to analyze quantum ergodicity in the SYK model, revealing non-ergodic modes that relax over time and lead to an ergodic regime described by random matrix theory.

Contribution

It introduces a novel replica path integral framework to understand the transition from non-ergodic to ergodic behavior in the SYK model, identifying collective modes related to the Clifford algebra.

Findings

Identification of non-ergodic collective modes in the SYK model.

Explanation of the system's approach to ergodicity via mode competition.

Agreement of spectral correlation functions with numerical data.

Abstract

We present a replica path integral approach describing the quantum chaotic dynamics of the SYK model at large time scales. The theory leads to the identification of non-ergodic collective modes which relax and eventually give way to an ergodic long time regime (describable by random matrix theory). These modes, which play a role conceptually similar to the diffusion modes of dirty metals, carry quantum numbers which we identify as the generators of the Clifford algebra: each of the $2^N$ different products that can be formed from $N$ Majorana operators defines one effective mode. The competition between a decay rate quickly growing in the order of the product and a density of modes exponentially growing in the same parameter explains the characteristics of the system's approach to the ergodic long time regime. We probe this dynamics through various spectral correlation functions and obtain favorable agreement with existing numerical data.