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

TL;DR

This paper investigates the super-rough glassy phase of the 2D XY model with quenched disorder, deriving universal predictions for correlation functions via RG and confirming them numerically with high accuracy.

Contribution

It provides the first analytical derivation of the universal amplitude for the super-rough phase and validates it through precise numerical simulations.

Findings

Analytical RG predicts A(τ) = 2τ^2 - 2τ^3 + ...

Numerical simulations confirm the analytical prediction up to τ ≈ 0.5

Correlation function exhibits a ln^2(r) behavior in the super-rough phase.

Abstract

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 $\tau \approx 0.5$.