Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos

TL;DR

This paper proves exactly that a periodically driven Ising model exhibits spectral correlations matching random matrix theory, providing a nonperturbative demonstration of quantum chaos in a system without a classical limit.

Contribution

It offers the first exact proof of RMT spectral form factor in a non-semiclassical many-body quantum system, using a novel duality approach.

Findings

Spectral form factor matches RMT for all integer times in the thermodynamic limit.

Self-dual cases serve as minimal models of many-body quantum chaos.

Disorder ensures ergodicity, ruling out many-body localization.

Abstract

The most general and versatile defining feature of quantum chaotic systems is that they possess an energy spectrum with correlations universally described by random matrix theory (RMT). This feature can be exhibited by systems with a well defined classical limit as well as by systems with no classical correspondence, such as locally interacting spins or fermions. Despite great phenomenological success, a general mechanism explaining the emergence of RMT without reference to semiclassical concepts is still missing. Here we provide the example of a quantum many-body system with no semiclassical limit (no large parameter) where the emergence of RMT spectral correlations is proven exactly. Specifically, we consider a periodically driven Ising model and write the Fourier transform of spectral density's two-point function, the spectral form factor, in terms of a partition function of a two-dimensional classical Ising model featuring a space-time duality. We show that the self-dual cases provide a minimal model of many-body quantum chaos, where the spectral form factor is demonstrated to match RMT for all values of the integer time variable $t$ in the thermodynamic limit. In particular, we rigorously prove RMT form factor for odd $t$, while we formulate a precise conjecture for even $t$. The results imply ergodicity for any finite amount of disorder in the longitudinal field, rigorously excluding the possibility of many-body localization. Our method provides a novel route for obtaining exact nonperturbative results in non-integrable systems.