Destruction of Bulk Ordering by Surface Randomness

We demonstrate that the arbitrarily weak quenched disorder on the surface of a system of continuous symmetry destroys long-range order in the bulk, and, instead, quasi-long-range order emerges. Correlation functions are calculated exactly for the two- and three-dimensional $XY$ models with surface randomness via the functional renormalization group. Even at strong quenched disorder the three-dimensional $XY$ model possesses topological order. We also determine roughness of a domain wall in the presence of surface disorder.