Quench dynamics in randomly generated extended quantum models

We analyze the thermalization properties and the validity of the eigenstate thermalization hypothesis in a generic class of quantum Hamiltonians where the quench parameter explicitly breaks a Z(2) symmetry. Natural realizations of such systems are given by random matrices expressed in a block form where the terms responsible for the quench dynamics are the off-diagonal blocks. Our analysis examines both dense and sparse random matrix realizations of the Hamiltonians and the observables. Sparse random matrices may be associated with local quantum Hamiltonians and they show a different spread of the observables on the energy eigenstates with respect to the dense ones. In particular, the numerical data seem to support the existence of rare states, i.e., states where the observables take expectation values that are different compared to the typical ones sampled by the microcanonical distribution. In the case of sparse random matrices, we also extract the finite-size behavior of two different time scales associated with the thermalization process.