Level statistics in the fractal phase of generalized Rosenzweig--Porter models
Abstract
The Rosenzweig--Porter (RP) random matrix ensemble has emerged as a minimal model for the integrability-to-chaos crossover in quantum many-body systems. Its phase diagram features a region with fractal eigenstates, exhibiting intermediate spectral and localization properties between the fully localized and fully delocalized regimes. In this work, we explore several generalizations of the RP model and determine their level statistics at the scale of the Thouless energy ET, which characterizes the crossover. Using tools from free probability theory and the replica method, we compute the full counting statistics in the limit of large system size, and show that it takes a simple, universal scaling form around ET, shared across all variations of the model. We validate our analytical predictions using exact numerical diagonalization of large samples, and large-deviation algorithms that resolve the full counting statistics down to probabilities as low as 10−40. We also contrast our predictions with measurements on the quantum random energy model, which is the simplest model displaying many-body localization.
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BibTeX
@misc{delapalme2026level,
title = {{Level statistics in the fractal phase of generalized Rosenzweig--Porter models}},
author = {Victor Delapalme and Leticia F. Cugliandolo and Alexander K. Hartmann and Marco Tarzia and Davide Venturelli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.09444},
eprint = {2607.09444},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.09444},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Level statistics in the fractal phase of generalized Rosenzweig--Porter models
AU - Victor Delapalme
AU - Leticia F. Cugliandolo
AU - Alexander K. Hartmann
AU - Marco Tarzia
AU - Davide Venturelli
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2607.09444
UR - https://arxiv.org/abs/2607.09444
AB - The Rosenzweig--Porter (RP) random matrix ensemble has emerged as a minimal model for the integrability-to-chaos crossover in quantum many-body systems. Its phase diagram features a region with fractal eigenstates, exhibiting intermediate spectral and localization properties between the fully localized and fully delocalized regimes. In this work, we explore several generalizations of the RP model and determine their level statistics at the scale of the Thouless energy E_T, which characterizes the crossover. Using tools from free probability theory and the replica method, we compute the full counting statistics in the limit of large system size, and show that it takes a simple, universal scaling form around E_T, shared across all variations of the model. We validate our analytical predictions using exact numerical diagonalization of large samples, and large-deviation algorithms that resolve the full counting statistics down to probabilities as low as 10^−40. We also contrast our predictions with measurements on the quantum random energy model, which is the simplest model displaying many-body localization.
ER -