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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.