Microcanonical Operator Spectrum, BPS Chaos and Free Probability Theory
Abstract
Matrix elements of simple operators are central to thermalization in chaotic quantum many-body systems. Their moments are often described statistically through the Eigenstate Thermalization Hypothesis. In this paper, we study a more refined characterization of these matrix elements: the eigenspectrum of simple operators once projected to microcanonical energy windows. The spectrum encodes all moments of the matrix elements at once, and is particularly relevant when the spectrum of the Hamiltonian is degenerate. Assuming that the eigenbasis of the simple operator is Haar-randomly oriented with respect to that of the Hamiltonian, we prove that the spectrum of an operator projected to a small microcanonical window obeys Gaussian random matrix statistics. We do this directly by studying the probability distribution inherited from the random orientation, as well as using tools from Free Probability Theory. We discuss the implications for BPS chaos and more generally for eigenstate thermalization.
Related
- Discrete Geometric Physics. Volume 1: Computational Topodynamics of Media. Volume 2: From Chaos to Infinity. — Complete Theory in a Single Document
- The Pauli Probability Spectrum Carries the Pure-State Quantum Fisher Metric
- Theory of the Edge-of-Chaos Advantage in Quantum Reservoir Computing
- Conditional probability density functional theory for solids
- Microcanonical Hamiltonian Monte Carlo and the Helmholtz Theorem
BibTeX
@misc{belin2026microcanonical,
title = {{Microcanonical Operator Spectrum, BPS Chaos and Free Probability Theory}},
author = {Alexandre Belin and Yichao Fu and Federico La Rocca},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.22422},
eprint = {2609.22422},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22422},
}
RIS
TY - UNPB
TI - Microcanonical Operator Spectrum, BPS Chaos and Free Probability Theory
AU - Alexandre Belin
AU - Yichao Fu
AU - Federico La Rocca
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.22422
UR - https://arxiv.org/abs/2609.22422
AB - Matrix elements of simple operators are central to thermalization in chaotic quantum many-body systems. Their moments are often described statistically through the Eigenstate Thermalization Hypothesis. In this paper, we study a more refined characterization of these matrix elements: the eigenspectrum of simple operators once projected to microcanonical energy windows. The spectrum encodes all moments of the matrix elements at once, and is particularly relevant when the spectrum of the Hamiltonian is degenerate. Assuming that the eigenbasis of the simple operator is Haar-randomly oriented with respect to that of the Hamiltonian, we prove that the spectrum of an operator projected to a small microcanonical window obeys Gaussian random matrix statistics. We do this directly by studying the probability distribution inherited from the random orientation, as well as using tools from Free Probability Theory. We discuss the implications for BPS chaos and more generally for eigenstate thermalization.
ER -