  We study the dynamical quantum phase transition of the critical quantum
quench, in which the prequenched Hamiltonian, or the postquenched Hamiltonian,
or both of them are set to be the critical points of equilibrium quantum phase
transitions, we find half-quantized or unquantized dynamical topological order
parameter and dynamical Chern number; these results and also the existence of
dynamical quantum phase transition are all closely related to the singularity
of the Bogoliubov angle at the gap-closing momentum. The effects of the
singularity may also be canceled out if both the prequenched and postquenched
Hamiltonians are critical, then the dynamical topological order parameter and
dynamical Chern number restore to integer ones. Our findings show that the
widely accepted definitions of dynamical topological order parameter and
dynamical Chern number are problematic for the critical quenches in the
perspective of topology, which call for new definitions of them.
