Relaxational dynamics study of the classical Heisenberg spin XY model in spherical coordinate representation

The two- and three-dimensional classical Heisenberg spin XY (CHSXY) models, with the spherical coordinates of spins taken as dynamic variables, are numerically investigated. We allow the polar $\theta$ and azimuthal $\phi$ angles to have uniform values in $[0,\pi)$ and $[-\pi, \pi)$, respectively, and the static universality class is shown to be identical to the classical XY model with two-component spins, as well as the CHSXY model with a different choice of dynamic variables, conventionally used in the literature. The relaxational dynamic simulation reveals that the dynamic critical exponent $z$ is found to have the value $z \approx 2.0$ for both two and three dimensions, in contrast to $z \approx d/2$ ($d=$ spatial dimension) found previously with spin dynamics simulation of the conventional CHSXY model. Comparisons with the usual two-component classical XY model are also made.