Dynamical universality classes of the superconducting phase transition

We present a finite temperature Monte Carlo study of the $\mathrm{XY}$ model in the vortex representation and study its dynamical critical behavior in two limits. The first neglects magnetic-field fluctuations, corresponding to the absence of screening, which should be a good approximation in high-$T_c$ superconductors $\left(\overrightarrow{\kappa }\infty \right)$ except extremely close to the critical point. Here, from finite-size scaling of the linear resistivity we find the dynamical critical exponent of the vortex motion to be $z\approx 1.5$. The second limit includes magnetic-field fluctuations in the strong screening limit $\left(\overrightarrow{\kappa }0\right)$ corresponding to the true asymptotic inverted $\mathrm{XY}$ critical regime, where we find the unexpectedly large value $z\approx \mathrm{2.7.}$ We compare these results, obtained from dissipative dynamics in the vortex representation, with the universality class of the corresponding model in the phase representation with propagating (spin-wave) modes. We also discuss the effect of disorder and the relevance of our results for experiments.