Dynamical quantum phase transition from a critical quantum quench

We study the dynamical quantum phase transition of a 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 a 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 the 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.