# Super-Rough Glassy Phase of the Random Field XY Model in Two Dimensions

### Anthony Perret 1, Zoran Ristivojevic 2, Pierre Le Doussal 2, Gregory Schehr 1, Kay J. Wiese 2

#### Physical Review Letters 109 (2012) 157205

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 $\theta(x)$, $B(r) = \bar{<[\theta(x) - \theta(x+ r) ]^2>}$ behaves, for $r \gg a$, as $B(r) \simeq A(\tau) \ln^2 (r/a)$ where $r = |r|$ 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 ${A}(\tau) = 2\tau^2-2\tau^3 + {\cal O}(\tau^4)$. 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 $\tau \approx 0.5$.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Laboratoire de Physique Théorique de l'ENS (LPTENS),
CNRS : UMR8549 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris