Influence of weak disorder on the dynamical quantum phase transitions in the anisotropic XY chain

We study the effects of disorder on the dynamical quantum phase transitions (DQPTs) in the transverse-field anisotropic XY chain by numerically calculating the Loschmidt echo after quench. 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, since there are already two dynamical phase transitions in the homogeneous chain, 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 larger disorder will cause the DQPTs to disappear.