Dynamical quantum phase transition in XY chains with the Dzyaloshinskii-Moriya and XZY-YZX three-site interactions

We study the dynamical quantum phase transitions (DQPTs) in the XY chains with the Dzyaloshinskii-Moriya interaction and the XZY-YZX type of three-site interaction after a sudden quench. Both the models can be mapped to the spinless free fermion models by the Jordan-Wigner and Bogoliubov transformations with the form $H=\sum_{k}\varepsilon_{k}(\eta^†_{k}\eta_{k}-\frac{1}{2})$, where the quasiparticle excitation spectra $\varepsilon_{k}$ may be smaller than 0 for some $k$ and are asymmetrical ($\varepsilon_{k}\neq\varepsilon_{-k}$). It's found that the factors of Loschmidt echo equal 1 for some $k$ corresponding to the quasiparticle excitation spectra of the pre-quench Hamiltonian satisfying $\varepsilon_{k}\cdot\varepsilon_{-k}<0$, when the quench is from the gapless phase. By considering the quench from different ground states, we obtain the conditions for the occurrence of DQPTs for the general XY chains with gapless phase, and find that the DQPTs may not occur in the quench across the quantum phase transitions regardless of whether the quench is from the gapless phase to gapped phase or from the gapped phase to gapless phase. This is different from the DQPTs in the case of quench from the gapped phase to gapped phase, in which the DQPTs will always appear. Besides, we also analyze the different reasons for the absence of DQPTs in the quench from the gapless phase and the gapped phase. The conclusion can also be extended to the general quantum spin chains.