Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

Arnd Bäcker, Masudul Haque, and Ivan M. Khaymovich
Phys. Rev. E 100, 032117 – Published 11 September 2019

Abstract

Multifractal dimensions allow for characterizing the localization properties of states in complex quantum systems. For ergodic states the finite-size versions of fractal dimensions converge to unity in the limit of large system size. However, the approach to the limiting behavior is remarkably slow. Thus, an understanding of the scaling and finite-size properties of fractal dimensions is essential. We present such a study for random matrix ensembles, and compare with two chaotic quantum systems—the kicked rotor and a spin chain. For random matrix ensembles we analytically obtain the finite-size dependence of the mean behavior of the multifractal dimensions, which provides a lower bound to the typical (logarithmic) averages. We show that finite statistics has remarkably strong effects, so that even random matrix computations deviate from analytic results (and show strong sample-to-sample variation), such that restoring agreement requires exponentially large sample sizes. For the quantized standard map (kicked rotor) the multifractal dimensions are found to follow the random matrix predictions closely, with the same finite statistics effects. For a XXZ spin-chain we find significant deviations from the random matrix prediction—the large-size scaling follows a system-specific path towards unity. This suggests that local many-body Hamiltonians are “weakly ergodic,” in the sense that their eigenfunction statistics deviate from random matrix theory.

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  • Received 8 May 2019

DOI:https://doi.org/10.1103/PhysRevE.100.032117

©2019 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied PhysicsNonlinear Dynamics

Authors & Affiliations

Arnd Bäcker1,2, Masudul Haque2,3, and Ivan M. Khaymovich2

  • 1Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
  • 2Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
  • 3Department of Theoretical Physics, Maynooth University, Co. Kildare, Ireland

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Issue

Vol. 100, Iss. 3 — September 2019

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