Many-body quantum chaos: Analytic connection to random matrix theory

TL;DR

This paper establishes a theoretical connection between many-body quantum chaos and random matrix theory by explicitly computing spectral form factors in non-integrable spin systems, confirming RMT predictions in the ergodic phase.

Contribution

It provides the first analytical derivation of the spectral form factor for many-body quantum systems, demonstrating its agreement with RMT in the ergodic phase.

Findings

Spectral form factor $K(t)$ matches RMT predictions in non-integrable systems.

Explicit computation of $K(t)$ in leading orders shows agreement with RMT.

Analysis applies to simple many-body systems like kicked Ising spin chains.

Abstract

A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry (1985) within the so-called diagonal approximation of semiclassical periodic-orbit sums. Derivation of the full RMT spectral form factor $K(t)$ from semiclassics has been completed only much later in a tour de force by Mueller et al (2004). In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains. Such systems display two universal types of behaviour which are termed as `many-body localized phase' and `ergodic phase'. In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder. Here we provide the first theoretical explanation for these observations. We compute $K(t)$ explicitly in the leading two orders in $t$ and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.