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Universal level statistics of the out-of-time-ordered operator

Efim B. Rozenbaum, Sriram Ganeshan, and Victor Galitski
Phys. Rev. B 100, 035112 – Published 15 July 2019

Abstract

The out-of-time-ordered correlator has been proposed as an indicator of chaos in quantum systems due to its simple interpretation in the semiclassical limit. In particular, its rate of possible exponential growth at 0 is closely related to the classical Lyapunov exponent. Here we explore how this approach to quantum chaos relates to the random-matrix theoretical description. To do so, we introduce and study the level statistics of the logarithm of the out-of-time-ordered operator, Λ̂(t)=lnx̂(t),p̂x(0)2/(2t), that we dub the “Lyapunovian” or “Lyapunov operator” for brevity. The Lyapunovian's level statistics is calculated explicitly for the quantum stadium billiard. It is shown that in the bulk of the filtered spectrum, this statistics perfectly aligns with the Wigner-Dyson distribution. One of the advantages of looking at the spectral statistics of this operator is that it has a well-defined semiclassical limit where it reduces to the matrix of uncorrelated classical finite-time Lyapunov exponents in a partitioned phase space. We provide a heuristic picture interpolating these two limits using Moyal quantum mechanics. Our results show that the Lyapunov operator may serve as a useful tool to characterize quantum chaos and in particular quantum-to-classical correspondence in chaotic systems by connecting the semiclassical Lyapunov growth at early times, when the quantum effects are weak, to universal level repulsion that hinges on strong quantum interference effects.

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  • Received 22 February 2018
  • Revised 25 June 2019

DOI:https://doi.org/10.1103/PhysRevB.100.035112

©2019 American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied PhysicsNonlinear Dynamics

Authors & Affiliations

Efim B. Rozenbaum1,2,*, Sriram Ganeshan3,4, and Victor Galitski1,2

  • 1Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA
  • 2Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA
  • 3Simons Center for Geometry and Physics, Stony Brook, New York 11794, USA
  • 4Department of Physics, City College, City University of New York, New York, New York 10031, USA

  • *efimroz@umd.edu

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Issue

Vol. 100, Iss. 3 — 15 July 2019

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