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 $\operatorname{Tr}(Aρ(t))$: we derive the relaxation envelope $g^2(t/τ_λ)$ with $τ_λ= \hbar/(2η\sqrt{N})$ 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 $H(\varepsilon)=H_0+\varepsilon H_1$ then generates, at first order in $\varepsilon$, 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.