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

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) = <(<[theta(x) - theta(x + r)](2)>)over bar> behaves, for r >> a, as B(r) similar or equal to A(tau)ln(2)(r/a) where r = vertical bar r vertical bar 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(2) + O(tau(4)). The universality of A(tau) results from non-trivial 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 approximate to 0.5.