  Dynamical phase transitions (DPTs) are signaled by the non-analytical time
evolution of the dynamical free energy after quenching some global parameters
in quantum systems. The dynamical free energy is calculated from the overlap
between the initial and the time evolved states (Loschmidt amplitude). In a
recent study it was suggested that DPTs are related to the equilibrium phase
transitions (EPTs) (M. Heyl et al., Phys. Rev. Lett. \textbf{110}, 135704
(2013)). We here study an exactly solvable model, the extended $XY$ model, the
Loschmidt amplitude of which provides a counterexample. We show analytically
that the connection between the DPTs and the EPTs does not hold generally.
Analysing also the general compass model as a second example, assists us to
propound the physical condition under which the DPT occurs without crossing the
equilibrium critical point, and also no DPT by crossing the equilibrium
critical point.
