  We consider a realistic nonequilibrium protocol, where a quantum system in
thermal equilibrium is suddenly subjected to an external force. Due to this
force, the system is driven out of equilibrium and the expectation values of
certain observables acquire a dependence on time. Eventually, upon switching
off the external force, the system unitarily evolves under its own Hamiltonian
and, as a consequence, the expectation values of observables equilibrate
towards specific constant long-time values. Summarizing our main results, we
show that, in systems which violate the eigenstate thermalization hypothesis
(ETH), this long-time value exhibits an intriguing dependence on the strength
of the external force. Specifically, for weak external forces, i.e., within the
linear response regime, we show that expectation values thermalize to their
original equilibrium values, despite the ETH being violated. In contrast, for
stronger perturbations beyond linear response, the quantum system relaxes to
some nonthermal value which depends on the previous nonequilibrium protocol.
While we present theoretical arguments which underpin these results, we also
numerically demonstrate our findings by studying the real-time dynamics of two
low-dimensional quantum spin models.
