Dynamical relaxation behavior of extended XY chain with gapless phase following a quantum quench

We investigate the dynamical relaxation behavior of the two-point correlation in extended XY models with a gapless phase after quenches from various initial states. Specifically, we study the XY chain with gapless phase induced by the additional interactions: Dzyaloshinskii-Moriya interaction and XZY-YZX type of three-site interaction. When quenching from the gapped phase, we observe that the additional interactions have no effect on the relaxation behavior. The relaxation behavior is $\delta C_{mn}(t)\sim t^{-3\mathrm{/}2}$ and $\sim t^{-1\mathrm{/}2}$ for the quench to the commensurate phase and the incommensurate phase, respectively. However, when quenching from the gapless phase, we demonstrate that the scaling behavior of $\delta C_{mn}(t)$ is changed to $\sim t^{-1}$ for the quench to the commensurate phase, and the decay of $\delta C_{mn}(t)$ follows $\sim t^{-1}$ or $\sim t^{-1\mathrm{/}2}$ for the quench to the incommensurate phase depending on the parameters of pre-quench Hamiltonian. We also establish the dynamical phase diagrams based on the dynamical relaxation behavior of $\delta C_{mn}(t)$ in the extended XY models.