Atypicality of most few-body observables

Understanding how isolated quantum systems thermalize has recently gathered renewed interest among theorists, thanks to the experimental realizations of such systems. The eigenstate thermalization hypothesis (ETH) is particularly investigated as a sufficient condition for the approach to thermal equilibrium. It states that diagonal matrix elements of an observable for the energy eigenstates are almost the same within a small energy shell. The ETH is justified for an observable and a Hamiltonian whose respective eigenbases are typically oriented to each other; i.e., for almost all unitary transformations of these two eigenbases with respect to the uniform Haar measure.

In this seminar, we consider a Hamiltonian with few-body interactions and random observables without assuming the uniform Haar measure. These observables are chosen in an operational manner as random linear combinations of the operator basis of spins. We show that most few-body observables have atypical matrix elements when the energy width is not exponentially small with the system size. Namely, the maximum fluctuation for diagonal matrix elements is larger than that predicted by the uniform Haar measure.