Three `species' of Schrodinger cat states in an infinite-range spin model*

We explore a transverse-field Ising model that exhibits both spontaneous symmetry-breaking and eigenstate thermalization. Within its ferromagnetic phase, the exact eigenstates of the Hamiltonian of any large but finite-sized system are all Schrodinger cat states: coherent linear superpositions of states with `up' and `down' spontaneous magnetization. This model exhibits two dynamical phase transitions {\it within} the ferromagnetic phase between regimes where the motion of the order parameter between `up' and `down' is via quantum tunneling or not, and is always overdamped or not.