On thermalization in random free fermions: Statistical origins and energy-dependent ETH structure
Abstract
We analyze thermalization and eigenstate thermalization in the random free fermion model, a quadratic Hamiltonian with a Gaussian Orthogonal Ensemble hopping matrix, by an exact random-matrix calculation. For a single-particle observable A we compute the ensemble average and fluctuations of Tr(Aρ(t)): we derive the relaxation envelope g2(t/τλ) with τλ= ℏ/(2η√N) and its oscillatory |t|−3 tail, both properties of the semicircular limiting density, and we show that the variance of the fluctuations is O(1/N), so that the relaxation profile self-averages at fixed scaled time. Incorporating particle-number conservation, we prove a factorization theorem for the energy-shell statistics of the many-body eigenstates: the energy-independent ETH structure follows from the statistical independence of the eigenvector matrix and the spectrum, and at fixed realization the energy constraint enters only through a covariance subtraction of relative order 1/N. A controlled eigenvector--eigenvalue correlation deformation H(ε)=H0+ε H1 then generates, at first order in ε, the energy dependence of the diagonal energy-resolved slope and of the off-diagonal spectral function, a single-particle diagnostic of the ETH structure in the allowed particle-hole sector. Correlation-function fluctuations and the single-particle spectral form factor finally identify the subleading O(1/N) late-time fingerprints of two-level spectral correlations. The model thus separates eigenbasis chaos from spectral chaos and identifies the statistical origin of energy-independent versus energy-dependent ETH.
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BibTeX
@misc{guo2026thermalization,
title = {{On thermalization in random free fermions: Statistical origins and energy-dependent ETH structure}},
author = {Xiao-Kan Guo and Zhiqiang Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01414},
eprint = {2610.01414},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01414},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On thermalization in random free fermions: Statistical origins and energy-dependent ETH structure
AU - Xiao-Kan Guo
AU - Zhiqiang Huang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.01414
UR - https://arxiv.org/abs/2610.01414
AB - We analyze thermalization and eigenstate thermalization in the random free fermion model, a quadratic Hamiltonian with a Gaussian Orthogonal Ensemble hopping matrix, by an exact random-matrix calculation. For a single-particle observable A we compute the ensemble average and fluctuations of Tr(Aρ(t)): we derive the relaxation envelope g^2(t/τ_λ) with τ_λ= ℏ/(2η√N) and its oscillatory |t|^−3 tail, both properties of the semicircular limiting density, and we show that the variance of the fluctuations is O(1/N), so that the relaxation profile self-averages at fixed scaled time. Incorporating particle-number conservation, we prove a factorization theorem for the energy-shell statistics of the many-body eigenstates: the energy-independent ETH structure follows from the statistical independence of the eigenvector matrix and the spectrum, and at fixed realization the energy constraint enters only through a covariance subtraction of relative order 1/N. A controlled eigenvector--eigenvalue correlation deformation H(ε)=H_0+ε H_1 then generates, at first order in ε, the energy dependence of the diagonal energy-resolved slope and of the off-diagonal spectral function, a single-particle diagnostic of the ETH structure in the allowed particle-hole sector. Correlation-function fluctuations and the single-particle spectral form factor finally identify the subleading O(1/N) late-time fingerprints of two-level spectral correlations. The model thus separates eigenbasis chaos from spectral chaos and identifies the statistical origin of energy-independent versus energy-dependent ETH.
ER -