Analyticity constraints bound the decay of the spectral form factor

Abstract

Quantum chaos cannot develop faster than $\lambda \le 2\pi /(\hslash \beta )$ 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 $\eta \le \pi /(2\hslash \beta )$. 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.

9+3 pages, 6+3 figures. Accepted for publication in Quantum

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• Quantum Dynamics in Krylov Space: Methods and Applications
• Exact universal bounds on quantum dynamics and fast scrambling
• Quantum Chaos and Coherence: Random Parametric Quantum Channels
• Universal Properties of the Spectral Form Factor in Open Quantum Systems
• Decomposing the Spectral Form Factor
• Family of Exact and Inexact Quantum Speed Limits for Completely Positive and Trace-Preserving Dynamics