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 $M\to 0$ limits of $M$-layered O(n) models coupled each other via ${\varphi }_{1,3}$ 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 ($0<n<n_{\ast }$) near the self-avoiding walk point ($n=0$) and a line of infrared fixed points ($n_{\ast }<n<1$) 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.

48 pages, 12 figures, uses iopart

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Cited by in corpus (8)

• Fractal dimensions of self-avoiding walks and Ising high-temperature graphs in 3D conformal bootstrap
• The Random-Bond Ising Model in 2.01 and 3 Dimensions
• Disorder in Large-N Theories
• Particles, conformal invariance and criticality in pure and disordered systems
• Critical lines in the pure and disordered $O(N)$ model
• Exact results for the $O(N)$ model with quenched disorder
• Phase diagram and the strong-coupling fixed point in the disordered O(n) loop model
• Coulomb Gas Partition Function of a Layered Loop Model