Analyticity constraints bound the decay of the spectral form factor

Pablo Martinez-Azcona and Aurélia Chenu

Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg

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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 $\textit{inflection exponent}$ $\eta$, 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.

Classical chaos is quantified using the Lyapunov exponent, which measures the distance between trajectories with slightly different initial conditions. A quantum analog of this exponent has been defined from a 4-point Out of Time Order Correlator, and it is known to be bounded by the temperature of the system: the hotter a quantum system is, the more chaotic it can be.

Using tools from complex analysis, we find a similar bound on the initial decay of a quantity called the Spectral Form Factor (SFF), which is defined from the system partition function at complex temperatures. The hotter the system, the faster the early-time decay of the SFF can be. This bound is universal and not restricted to chaotic dynamics. We illustrate the results in systems that are conceptually very different and discuss the connections between other known bounds, such as quantum speed limits.

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