Dynamics of correlation spreading in low-dimensional transverse-field Ising models

We investigate the dynamical spreading 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). We analyze specifically the longitudinal and transverse spin-spin correlation functions at equal time with use of several methods. From the comparison of the results in 1D obtained by the linear spin-wave approximation (LSWA) and those obtained by the rigorous analytical approach, we show that the LSWA can asymptotically reproduce the exact group velocity in the limit of strong transverse fields while it fails to capture the detailed time dependence of the correlation functions. By applying the LSWA to the 2D case, in which the rigorous analytical approach is unavailable, we estimate the propagation velocity to be $Ja/(2\hbar)$ at the strong-field limit, where $J$ is the Ising interaction and $a$ is the lattice spacing. We also utilize the tensor-network method based on the projected-entangled pair states for 2D and quantitatively compute the time evolution of the correlation functions for a relatively short time. Our findings provide useful benchmarks for quantum simulation experiments of correlation spreading and theoretical refinement of the Lieb-Robinson bound in the future.