Does quasi-long-range order in the two-dimensional XY model really survive weak random phase fluctuations?

Effective theories for random critical points are usually non-unitary, and thus may contain relevant operators with negative scaling dimensions. To study the consequences of the existence of negative dimensional operators, we consider the random-bond XY model. It has been argued that the XY model on a square lattice, when weakly perturbed by random phases, has a quasi-long-range ordered phase (the random spin wave phase) at sufficiently low temperatures. We show that infinitely many relevant perturbations to the proposed critical action for the random spin wave phase were omitted in all previous treatments. The physical origin of these perturbations is intimately related to the existence of broadly distributed correlation functions. We find that those relevant perturbations do enter the Renormalization Group equations, and affect critical behavior. This raises the possibility that the random XY model has no quasi-long-range ordered phase and no Kosterlitz-Thouless (KT) phase transition.