Bäcker, Arnd and Haque, Masudul and Khaymovich, Ivan M. (2019) Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems. Working Paper. arXiv.

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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.

Item Type:	Monograph (Working Paper)
Additional Information:	Cite as: arXiv:1905.03099. This is the preprint version of the published article, which is available at 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 DOI:https://doi.org/10.1103/PhysRevE.100.032117
Keywords:	Multifractal dimensions; random matrices; chaotic quantum maps; many-body systems;
Academic Unit:	Faculty of Science and Engineering > Theoretical Physics
Item ID:	14836
Identification Number:	https://doi.org/10.1103/PhysRevE.100.032117
Depositing User:	Masud Haque
Date Deposited:	20 Sep 2021 16:49
Publisher:	arXiv
URI:	Publisher
Use Licence:	This item is available under a Creative Commons Attribution Non Commercial Share Alike Licence (CC BY-NC-SA). Details of this licence are available here

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