  We show that there exist dynamical phase transitions (DPTs), as defined in
[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 found in [Phys. Rev. B 88 184303 (2013)]. Studying
the dynamics of these special states under the evolution of the TFIM
Hamiltonian, we find temporal non-analtyicities 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.
