Relaxational dynamics study of the classical Heisenberg spin $\mathrm{XY}$ model in spherical coordinate representation

The two- and three-dimensional classical Heisenberg spin $\mathrm{XY}$ $\left(\mathrm{CHS}XY\right)$ models, with the spherical coordinates of spins taken as dynamic variables, are numerically investigated. We allow the polar θ 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 $\mathrm{XY}$ model with two-component spins, as well as the $\mathrm{CHS}\mathrm{XY}$ 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 $z\approx d/2\mathrm{}$ $(d=$ spatial dimension) found previously with spin dynamics simulation of the conventional $\mathrm{CHS}\mathrm{XY}$ model. Comparisons with the usual two-component classical $\mathrm{XY}$ model are also made.