Dynamical phase transitions, time-integrated observables, and geometry of states

We show that there exist dynamical phase transitions (DPTs), as defined by Heyl et al. Phys. Rev. Lett. 110, 135704 (2013), in the transverse-field Ising model (TFIM) away from the static quantum critical points. We study a class of special states associated with singularities in the generating functions of time-integrated observables as found by Hickey et al. Phys. Rev. B 87, 184303 (2013). Studying the dynamics of these special states under the evolution of the TFIM Hamiltonian, we find temporal nonanalyticities in the initial-state return probability associated with dynamical phase transitions. By calculating the Berry phase and Chern number we show the set of special states have interesting geometric features similar to those associated with static quantum critical points.