Dynamical relaxation behavior of an extended XY chain with a 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 following additional interactions: Dzyaloshinskii-Moriya interaction and the 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 $\delta C_{mn}\left(t\right)\sim t^{-3/2}$ and $\sim t^{-1/2}$ 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 $\delta C_{mn}\left(t\right)$ is changed to $\sim t^{-1}$ for the quench to the commensurate phase, and the decay of $\delta C_{mn}\left(t\right)$ follows $\sim t^{-1}$ or $\sim t^{-1/2}$ for the quench to the incommensurate phase depending on the parameters of prequench Hamiltonian. We also establish the dynamical phase diagrams based on the dynamical relaxation behavior of $\delta C_{mn}\left(t\right)$ in the extended XY models.