Bounding multifractality by observables

TL;DR

This paper explores how the structure of observables in quantum many-body systems can be used to bound fractal dimensions, providing insights into ergodicity, localization, and the energy spectrum's arc-shape.

Contribution

It introduces a method to bound fractal dimensions using observable structures, linking eigenstate properties to many-body localization and ergodicity.

Findings

Upper bounds on fractal dimensions explain their arc-shape across the spectrum.

Bounds serve as proxies for fractal dimensions during the many-body localization transition.

Connects single-particle and Fock space perspectives in quantum chaos.

Abstract

Fractal dimensions have been used as a quantitative measure for structure of eigenstates of quantum many-body systems, useful for comparison to random matrix theory predictions or to distinguish many-body localized systems from chaotic ones. For chaotic systems at midspectrum the states are expected to be ``ergodic'', infinite temperature states with all fractal dimensions approaching 1 in the thermodynamic limit. However, when moving away from midspectrum, the states develop structure, as they are expected to follow the eigenstate thermalization hypothesis, with few-body observables predicted by a finite-temperature ensemble. We discuss how this structure of the observables can be used to bound the fractal dimensions from above, thus explaining their typical arc-shape over the energy spectrum. We then consider how such upper bounds act as a proxy for the fractal dimension over the many-body localization transition, thus formally connecting the single-particle and Fock space pictures discussed in the literature.