Dynamical phase transitions, eigenstate thermalization, and Schrodinger cats within the ferromagnetic phase of an infinite-range quantum Ising model

An isolated quantum many-body system may be a reservoir that thermalizes its constituents. I will explore an example of the interplay of this thermalization and spontaneous symmetry-breaking, in the ferromagnetic phase of an infinite-range quantum Ising model. For a system of N spins, the barrier that must be crossed to flip the magnetization is of order N. The unitary quantum dynamics of this system can cross this barrier either by thermally activating itself over the barrier, or by quantum tunneling through the barrier. In the limit of large N, there is a dynamical phase transition, with the thermal process happening at temperatures above the phase transition. There is another phase transition at a lower temperature between phases where a "Schrodinger cat" state can and cannot be made to coherently oscillate between the two opposite magnetizations.
Although these things can be calculated "exactly", there is an assumption of thermalization. There are few examples of quantum many-body systems where thermalization can be proven; could this become one?