  In this paper, the effects of disorder on the dynamical quantum phase
transitions (DQPTs) in the transverse-field anisotropic XY chain are studied by
numerically calculating the Loschmidt echo after quench. We obtain the formula
for calculating the Loschmidt echo of the inhomogeneous system in real space.
By comparing the results with that of the homogeneous chain, we find that when
the quench crosses the Ising transition, the small disorder will cause a new
critical point. As the disorder increases, more critical points of the DQPTs
will occur, constituting a critical region. In the quench across the
anisotropic transition, the disorder will cause a critical region near the
critical point, and the width of the critical region increases by the
disordered strength. In the case of quench passing through two critical lines,
the small disorder leads to the system to have three additional critical
points. When the quench is in the ferromagnetic phase, the large disorder
causes the two critical points of the homogeneous case to become a critical
region. And for the quench in the paramagnetic phase, the DQPTs will disappear
for large disorder.
