  Transport properties of the classical antiferromagnetic XXZ model on the
square lattice have been theoretically investigated, putting emphasis on how
the occurrence of a phase transition is reflected in spin and thermal
transports. As is well known, the anisotropy of the exchange interaction
$\Delta\equiv J_z/J_x$ plays a role to control the universality class of the
transition of the model, i.e., either a second-order transition at $T_N$ into a
magnetically ordered state or the Kosterlitz-Thouless (KT) transition at
$T_{KT}$, which respectively occur for the Ising-type ($\Delta >1$) and
$XY$-type ($\Delta <1$) anisotropies, while for the isotropic Heisenberg case
of $\Delta=1$, a phase transition does not occur at any finite temperature. It
is found by means of the hybrid Monte-Carlo and spin-dynamics simulations that
the spin current probes the difference in the ordering properties, while the
thermal current does not. For the $XY$-type anisotropy, the longitudinal
spin-current conductivity $\sigma^s_{xx}$ ($=\sigma^s_{yy}$) exhibits a
divergence at $T_{KT}$ of the exponential form, $\sigma^s_{xx} \propto
\exp\big[ B/\sqrt{T/T_{KT}-1 }\, \big]$ with $B={\cal O}(1)$, while for the
Ising-type anisotropy, the temperature dependence of $\sigma^s_{xx}$ is almost
monotonic without showing a clear anomaly at $T_{N}$ and such a monotonic
behavior is also the case in the Heisenberg-type spin system. The significant
enhancement of $\sigma^s_{xx}$ at $T_{KT}$ is found to be due to the
exponential rapid growth of the spin-current-relaxation time toward $T_{KT}$,
which can be understood as a manifestation of the topological nature of a
vortex whose lifetime is expected to get longer toward $T_{KT}$. Possible
experimental platforms for the spin-transport phenomena associated with the KT
topological transition are discussed.
