Disordered O(n) Loop Model and Coupled Conformal Field Theories

Abstract

A family of models for fluctuating loops in a two dimensional random background is analyzed. The models are formulated as O(n) spin models with quenched inhomogeneous interactions. Using the replica method, the models are mapped to the M 0 limits of M-layered O(n) models coupled each other via φ1,3 primary fields. The renormalization group flow is calculated in the vicinity of the decoupled critical point, by an epsilon expansion around the Ising point (n=1), varying n as a continuous parameter. The one-loop beta function suggests the existence of a strongly coupled phase (0<n<n*) near the self-avoiding walk point (n=0) and a line of infrared fixed points (n*<n<1) near the Ising point. For the fixed points, the effective central charges are calculated. The scaling dimensions of the energy operator and the spin operator are obtained up to two-loop order. The relation to the random-bond q-state Potts model is briefly discussed.