Spectral Form Factor of a Quantum Spin Glass

TL;DR

This paper analyzes the spectral form factor of quantum spin glasses, revealing that their level statistics resemble a sum of independent random matrices linked to metastable states, thus probing ergodicity-breaking.

Contribution

It introduces an analytic approach to connect level spacing statistics with metastable configurations in quantum spin glasses, extending the concept of complexity beyond semi-classical models.

Findings

Level statistics match a sum of independent random matrices.

Number of matrices equals the exponential of the spin glass complexity.

Poissonian distribution observed for the number of metastable states.

Abstract

It is widely expected that systems which fully thermalize are chaotic in the sense of exhibiting random-matrix statistics of their energy level spacings, whereas integrable systems exhibit Poissonian statistics. In this paper, we investigate a third class: spin glasses. These systems are partially chaotic but do not achieve full thermalization due to large free energy barriers. We examine the level spacing statistics of a canonical infinite-range quantum spin glass, the quantum $p$-spherical model, using an analytic path integral approach. We find statistics consistent with a direct sum of independent random matrices, and show that the number of such matrices is equal to the number of distinct metastable configurations -- the exponential of the spin glass "complexity" as obtained from the quantum Thouless-Anderson-Palmer equations. We also consider the statistical properties of the complexity itself and identify a set of contributions to the path integral which suggest a Poissonian distribution for the number of metastable configurations. Our results show that level spacing statistics can probe the ergodicity-breaking in quantum spin glasses and provide a way to generalize the notion of spin glass complexity beyond models with a semi-classical limit.