Symmetry-prohibited thermalization after a quantum quench

The observable long-time behavior of an isolated many-body system after a quantum quench is considered, i.e., an eigenstate (or an equilibrium ensemble) of some pre-quench Hamiltonian $H$ serves as initial condition which then evolves in time according to some post-quench Hamiltonian $H_p$. Absence of thermalization is analytically demonstrated for a large class of quite common pre- and post-quench spin Hamiltonians. The main requirement is that the pre-quench Hamiltonian must exhibit a $Z_2$ (spin-flip) symmetry, which would be spontaneously broken in the thermodynamic limit, though we actually focus on finite (but large) systems. On the other hand, the post-quench Hamiltonian must violate the $Z_2$ symmetry, but for the rest may be non-integrable and may obey the eigenstate thermalization hypothesis for (sums of) few-body observables.