Dynamics of correlation spreading after a quantum quench in low-dimensional transverse-field Ising models

Motivated by recent development in quantum simulation of quantum Ising models with Rydberg atoms, we investigate dynamics of spatial correlations after a quantum quench starting from a magnetically disordered state in the transverse-field Ising model at one (1D) and two spatial dimensions (2D). For theoretical analyses, we utilize several methods, including the exact analytical method for 1D, the tensor-network method based on the projected-entangled pair states (PEPS) for 2D, and the linear spin-wave approximation (LSWA) for both 1D and 2D, in order to calculate the longitudinal and transverse spin-spin correlation functions at equal time. In 1D, the comparison between the exact results and the LSWA results indicates that the LSWA can quantitatively capture the exact group velocity when the transverse field after the quench is sufficiently large. However, LSWA fails to capture the detailed time dependence of the correlation functions. In 2D, we estimate the propagation velocity to be $Ja/(2hbar)$ at the strong-field limit by using LSWA, where $J$ is the Ising interaction and $a$ is the lattice spacing. We also utilize PEPS in order to provide quantitatively accurate time-evolution of the correlation functions for a relatively short time. Our findings will be useful as benchmark for quantum simulation experiments of correlation spreading and theoretical refinement of the Lieb-Robinson bound.