On thermalization in random free fermions: Statistical origins and energy-dependent ETH structure

Abstract

We study the thermalization in the random free fermion model by a detailed random-matrix analysis. By computing the ensemble average and fluctuations of $\operatorname{Tr}(A\rho(t))$ for a single-particle observable $A$, we derive the thermalization function $g^2(t/\tau_\lambda)$ with $\tau_\lambda = \hbar/(2\eta\sqrt{N})$, prove its $|t|^{-3}$ asymptotic decay, and show that the variance of the fluctuations vanishes as $O(1/N)$ in the thermodynamic limit.Particle number conservation is also incorporated into the model, by which we further study the energy-shell eigenstate statistics and prove a factorization theorem of the eigenstate statistics. We further show that under a controlled eigenvector-eigenvalue correlation deformation, the diagonal energy-resolved slope and the off-diagonal eigenstate thermalization hypothesis spectral function acquire an energy dependence.Finally, we compute the fluctuations of correlation functions and compare the fluctuation scales with those of a fully chaotic system, revealing quantitative differences rooted in the Gaussian nature of the eigenstates.Our work establishes the random free fermion model as an analytically solvable realization of weak eigenstate thermalization hypothesis type self-averaging, and as a controllable setting in which eigenbasis chaos can be separated from spectral chaos.