We study both analytically, using the renormalization group (RG) to two loop order, and numerically, using an exact polynomial algorithm, the disorder-induced glass phase of the two-dimensional $XY$ model with quenched random symmetry-breaking fields and without vortices. In the super-rough glassy phase, i.e., below the critical temperature $T_c$, the disorder and thermally averaged correlation function $B(r)$ of the phase field $\theta (\mathbf{x})$, $B(r)=\overline{⟨[\theta (\mathbf{x})-\theta (\mathbf{x}+\mathbf{r}){]}^2⟩}$ behaves, for $r\gg a$, as $B(r)\simeq A(\tau ){ln}^2(r/a)$ where $r=|\mathbf{r}|$ and $a$ is a microscopic length scale. We derive the RG equations up to cubic order in $\tau =(T_c-T)/T_c$ and predict the universal amplitude $A(\tau )=2{\tau }^2-2{\tau }^3+\mathcal{O}({\tau }^4)$. The universality of $A(\tau )$ results from nontrivial cancellations between nonuniversal constants of RG equations. Using an exact polynomial algorithm on an equivalent dimer version of the model we compute $A(\tau )$ numerically and obtain a remarkable agreement with our analytical prediction, up to $\tau \approx 0.5$.

• Received 27 April 2012

DOI:https://doi.org/10.1103/PhysRevLett.109.157205