Concurrence of dynamical phase transitions at finite temperature in the fully connected transverse-field Ising model

We construct the finite-temperature dynamical phase diagram of the fully connected transverse-field Ising model from the vantage point of two disparate concepts of dynamical criticality. An analytical derivation of the classical dynamics and exact diagonalization simulations are used to study the dynamics after a quantum quench in the system prepared in a thermal equilibrium state. The different dynamical phases characterized by the type of non-analyticities that emerge in an appropriately defined Loschmidt-echo return rate directly correspond to the dynamical phases determined by the spontaneous breaking of $\mathbb{Z}_2$ symmetry in the long-time steady state. The dynamical phase diagram is qualitatively different depending on whether the initial thermal state is ferromagnetic or paramagnetic. Whereas the former leads to a dynamical phase diagram that can be directly related to its equilibrium counterpart, the latter gives rise to a divergent dynamical critical temperature at vanishing final transverse-field strength.