False signatures of non-ergodic behavior in disordered quantum many-body systems

Ergodic isolated quantum many-body systems satisfy the eigenstate thermalization hypothesis (ETH), i.e., the expectation values of local observables in the system's eigenstates approach the predictions of the microcanonical ensemble. However, the ETH does not specify what happens to expectation values of local observables within an energy window when the average over disorder realizations is taken. As a result, the expectation values of local observables can be distributed over a relatively wide interval and may exhibit nontrivial structure, as shown in [Phys. Rev. B \textbf{104}, 214201 (2021)] for a quasiperiodic disordered system for site-resolved magnetization. We argue that the non-Gaussian form of this distribution may \textit{falsely} suggest non-ergodicity and a breakdown of ETH. By considering various types of disorder, we find that the functional forms of the distributions of matrix elements of the site-resolved magnetization operator mirror the distribution of the onsite disorder. We argue that this distribution is a direct consequence of the local observable having a finite overlap with moments of the Hamiltonian. We then demonstrate how to adjust the energy window when analyzing expectation values of local observables in disordered quantum many-body systems to correctly assess the system's adherence to ETH, and provide a link between the distribution of expectation values in eigenstates and the outcomes of quench experiments.