Bounding multifractality by observables

Tuomas I. Vanhala; Niklas Järvelin; Teemu Ojanen

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arXiv · 2501.17649

Bounding multifractality by observables

Tuomas I. Vanhala·Niklas Järvelin·Teemu Ojanen

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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.

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Tuomas I. Vanhala, Niklas Järvelin, Teemu Ojanen. 2025-01-29. Bounding multifractality by observables. https://arxiv.org/abs/2501.17649

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