Renormalization Group Solution for the Two-dimensional Random Bond Potts Model with Broken Replica Symmetry

We find a new solution of the renormalization group for the Potts model with ferromagnetic random valued coupling constants. The solution exhibits universality and broken replica symmetry. It is argued that the model reaches this universality class if the replica symmetry is broken initially. Otherwise the model stays with the replica symmetric renormalization group flow and reaches the fixed point which has been considered before. Also at the Landau Institute for Theoretical Physics, Moscow Laboratoire associé No. 280 au CNRS The problem of new critical behavior induced by randomness in spin systems has a considerable history. Starting with a classical φ problem, the modified critical behavior has later been studied for the two-dimensional Ising and Potts models by various renormalization group techniques, and by numerical simulations. Incomplete list of references is provided in [1-12]. Replicas has been used generally to deal with the quenched disorder and replica symmetric solutions have generally been looked for. The first example of replica symmetry broken solutions of the renormalization group has been suggested in [13], in the context of the φ model. In this letter we report on the replica symmetry broken solution for the two-dimensional Potts model with random bonds. The model reaches this solution if the replica symmetry is broken initially. In contrast, the two-dimensional Ising model turns out to be stable with respect to replica symmetry breaking [4]. It reaches always the replica symmetric critical behavior which has been studied earlier [5-12]. For theoretical study one uses models with a weak disorder, e.g. models with spin couplings having small fluctuations around a mean ferromagnetic value. This gives a possibility to study the model in continuum, because one reaches the critical point sufficiently close before the randomness becomes important. For the two-dimensional Potts model in particular this allows to use eventually the renormalization group based on the conformal theory of the unperturbed model. In this approach the effective theory could be described by the Hamiltonian H = H0 + ∫ dxm(x)ε(x) (1) where H0 represents, symbolically, the conformal theory of the unperturbed model, while the second term with a spatially random mass m(x) coupled to the energy operator represents the effective randomness due to spatially inhomogeneous coupling constants of spins. Replicating the model and taking the average of the partition function over m(x) one gets the effective homogeneous theory with the Hamiltonian: H = n