  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.
