  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.
