Eigenstate thermalization hypothesis through the lens of autocorrelation functions

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