  The Kane-Mele model is known to show a quantized spin Hall conductivity at
zero temperature. Including Hubbard interactions at each site leads to a
quantum phase transition to an XY antiferromagnet at sufficiently high
interaction strength. Here, we use the two-particle self-consistent approach
(TPSC), which we extend to include spin-orbit coupling, to investigate the
Kane-Mele-Hubbard model at finite temperature and half-filling. TPSC is a weak
to intermediate coupling approach capable of calculating a frequency- and
momentum-dependent self-energy from spin and charge fluctuations. We present
results for the spin Hall conductivity and correlation lengths for
antiferromagnetic spin fluctuations for different values of temperature,
spin-orbit coupling and Hubbard interaction. The vertex corrections, which here
are analogues of Maki-Thompson contributions, show a strong momentum dependence
and give a large contribution in the vicinity of the phase transition at all
temperatures. Their inclusion is necessary to observe the quantization of the
spin Hall conductivity for the interacting system in the zero temperature
limit. At finite temperature, increasing the Hubbard interaction leads to a
decrease of the spin Hall conductivity. This decrease can be explained by
band-gap renormalization from scattering of electrons on antiferromagnetic spin
fluctuations.
