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

## TL;DR

This paper investigates the finite-size scaling of multifractal dimensions in complex quantum systems, comparing random matrices, quantum maps, and many-body systems, revealing system-specific deviations from ergodic behavior.

## Contribution

It provides analytical and numerical analysis of finite-size effects on multifractal dimensions across different quantum systems, highlighting deviations from random matrix theory.

## Key findings

- Random matrix ensembles show strong sample-to-sample variation in multifractal dimensions.
- Quantum maps closely follow random matrix predictions for multifractal dimensions.
- Many-body systems exhibit deviations indicating weak ergodicity.

## 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.