Anomalous relaxation in the $\mathrm{XY}$ gauge glass

To study relaxation dynamics of the two-dimensional $\mathrm{XY}$ gauge glass, we integrate directly the equations of motion and investigate the energy function. As usual, it decays exponentially at high temperatures; at low but nonzero temperatures, it is found to exhibit an algebraic relaxation. We compute the relaxation time $\tau$ as a function of the temperature $T$ and find that the rapid increase of $\tau$ at low temperatures is well described by $\tau \sim (T-T_g{)}^{-b}$ with $T_g=0.22\mathrm{}\pm 0.02$ and $b=0.76\mathrm{}\pm 0.05$, which strongly suggests a finite-temperature glass transition. The decay of vorticity is also examined and explained in terms of a simple heuristic model, which attributes the fast relaxation at high temperatures to annihilation of unpinned vortices.