Eigenstate thermalization hypothesis through the lens of autocorrelation functions

Matrix elements of observables in eigenstates of generic Hamiltonians are described by the Srednicki ansatz within the eigenstate thermalization hypothesis (ETH). We study a quantum chaotic spin-fermion model in a one-dimensional lattice, which consists of a spin-1/2 XX chain coupled to a single itinerant fermion. In our study, we focus on translationally invariant observables including the charge and energy current, thereby also connecting the ETH with transport properties. Considering observables with a Hilbert-Schmidt norm of one, we first perform a comprehensive analysis of ETH in the model taking into account latest developments. A particular emphasis is on the analysis of the structure of the offdiagonal matrix elements $|\langle \alpha | \hat O | \beta \rangle|^2$ in the limit of small eigenstate energy differences $\omega = E_\beta - E_\alpha$. Removing the dominant exponential suppression of $|\langle \alpha | \hat O | \beta \rangle|^2$, we find that: (i) the current matrix elements exhibit a system-size dependence that is different from other observables under investigation, (ii) matrix elements of several other observables exhibit a Drude-like structure with a Lorentzian frequency dependence. We then show how this information can be extracted from the autocorrelation functions as well. Finally, our study is complemented by a numerical analysis of the fluctuation-dissipation relation for eigenstates in the bulk of the spectrum. We identify the regime of $\omega$ in which the well-known fluctuation-dissipation relation is valid with high accuracy for finite systems.