Spin and thermal transport and critical phenomena in three-dimensional antiferromagnets

We investigate spin and thermal transport near the N\'{e}el transition temperature $T_N$ in three dimensions, by numerically analyzing the classical antiferromagnetic $XXZ$ model on the cubic lattice, where in the model, the anisotropy of the exchange interaction $\Delta=J_z/J_x$ plays a role to control the universality class of the transition. It is found by means of the hybrid Monte-Carlo and spin-dynamics simulations that in the $XY$ and Heisenberg cases of $\Delta \leq 1$, the longitudinal spin conductivity $\sigma^s_{\mu\mu}$ exhibits a divergent enhancement on cooling toward $T_N$, while not in the Ising case of $\Delta>1$. In all the three cases, the temperature dependence of the thermal conductivity $\kappa_{\mu\mu}$ is featureless at $T_N$, being consistent with experimental results. The divergent enhancement of $\sigma^s_{\mu\mu}$ toward $T_N$ is attributed to the spin-current relaxation time which gets longer toward $T_N$, showing a power-law divergence characteristic of critical phenomena. It is also found that in contrast to the $XY$ case where the divergence in $\sigma^s_{\mu\mu}$ is rapidly suppressed below $T_N$, $\sigma^s_{\mu\mu}$ likely remains divergent even below $T_N$ in the Heisenberg case, which might experimentally be observed in the ideally isotropic antiferromagnet RbMnF$_3$.