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 $etaleq pi/(2hbarbeta)$. 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.
Featured image: Illustration of the proof of the bound (upper), renormalization of the strip for extensive inflection exponent (lower left) and spectral form factor along with its exponential decay (lower right)
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Popular summary
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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