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

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##### Citation

Bäcker, A., Haque, M., & Khaymovich, I. M. (2019). Multifractal dimensions for
random matrices, chaotic quantum maps, and many-body systems.* Physical Review E,* *100*(3): 032117. doi:10.1103/PhysRevE.100.032117.

Cite as: https://hdl.handle.net/21.11116/0000-0009-A94C-4

##### 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.

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.