Disentangling dynamical phase transitions from equilibrium phase transitions

Dynamical phase transitions (DPT) occur after quenching some global parameters in quantum systems and are signalled by the non-analytical time evolution of the dynamical free energy, which is calculated from the Loschmidt overlap between the initial and time evolved states. In a recent letter (M. Heyl et al., Phys. Rev. Lett. \textbf{110}, 135704 (2013)), it was suggested that DPTs are closely related to equilibrium phase transitions (EPT) for the transverse field Ising model. By studying a minimal model, the XY chain in transverse magnetic field, we show analytically that this connection does not hold generally. We present examples where DPT occurs without crossing any equilibrium critical lines by the quench, and a nontrivial example with no DPT but crossing a critical line by the quench. Albeit the non-analyticities of the dynamical free energy on the real time axis do not indicate the presence or absence of an EPT, the structure of Fisher-lines for complex times reveal a qualitative difference.