Analyticity constraints bound the decay of the spectral form factor

TL;DR

This paper establishes a universal bound on the early-time decay of the spectral form factor in quantum systems, extending the concept of chaos bounds to spectral correlations and applicable beyond chaotic regimes.

Contribution

It introduces the inflection exponent to bound spectral form factor decay and demonstrates its universality across various quantum systems.

Findings

The inflection exponent is bounded by cb8/(2cb2b1).

The bound applies to regular, chaotic, and tunable quantum systems.

The relation to quantum speed limits is discussed.

Abstract

Quantum chaos cannot develop faster than $\lambda \leq 2 \pi/(\hbar \beta)$ for systems in thermal equilibrium [Maldacena, Shenker & Stanford, JHEP (2016)]. This `MSS bound' on the Lyapunov exponent $\lambda$ is set by the width of the strip on which the regularized out-of-time-order correlator is analytic. We show that similar constraints also bound the decay of the spectral form factor (SFF), that measures spectral correlation and is defined from the Fourier transform of the two-level correlation function. Specifically, the inflection exponent $\eta$, that we introduce to characterize the early-time decay of the SFF, is bounded as $\eta\leq \pi/(2\hbar\beta)$. This bound is universal and exists outside of the chaotic regime. The results are illustrated in systems with regular, chaotic, and tunable dynamics, namely the single-particle harmonic oscillator, the many-particle Calogero-Sutherland model, an ensemble from random matrix theory, and the quantum kicked top. The relation of the derived bound with other known bounds, including quantum speed limits, is discussed.