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

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 $M\to 0$ limits of $M$-layered O(n) models coupled each other via $\phi_{1,3}$ 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.