Effective equilibrium picture in the $xy$ model with exponentially correlated noise

We study the effect of exponentially correlated noise on the $xy$ model in the limit of small correlation time, discussing the order-disorder transition in the mean field and the topological transition in two dimensions. We map the steady states of the nonequilibrium dynamics into an effective equilibrium theory. In the mean field, the critical temperature increases with the noise correlation time $\tau$, indicating that memory effects promote ordering. This finding is confirmed by numerical simulations. The topological transition temperature in two dimensions remains untouched. However, finite-size effects induce a crossover in the vortices proliferation that is confirmed by numerical simulations.

Figure 1

Phase diagram. Mean-field phase diagram obtained minimizing the free energy (11). The red curve is the analytical computation of the critical line given by (15), black symbols are numerical simulations, and the white dashed curve is the small $\tau$ expansion of (15).

Figure 2

Two-dimensional simulations. (a) Comparison between theory (22) and numerical simulations. Triangles, circles, and squares are $\tau =0.01,0.05,0.1$, respectively. At small temperatures $T<0.5$, data collapse on the same curve. (b) Monotonic shift in the temperature of the topological transition as a function of $\tau$. Blue symbols are simulations, and the red line is the linear fit $T_{\mathrm{BKT}}\left(\tau \right)=a+b\tau$, with $a=0.93\left(1\right)$ and $b=0.67\left(3\right)$.