Stability of the renormalization group in the 2D random Ising and Baxter models with respect to replica symmetry breaking

Abstract

We study the critical properties of the weakly disordered two-dimensional Ising and Baxter models in terms of the renormalization group (RG) theory generalized to take into account the replica symmetry breaking (RSB) effects. Recently it has been shown that the traditional replica-symmetric RG flows in the dimensions D = 4−ǫ are unstable with respect to the RSB potentials and a new spin-glass type critical phenomena has been discovered [11],[12]. In contrast, here it is demonstrated that in the considered two-dimensional systems the renormalization-group flows are stable with respect to the RSB modes. It is shown that the solution of the renormalization group equations with arbitrary starting RSB coupling matrix exhibits asymptotic approach to the traditional replica-symmetric ones. Thus, in the leading order the non-perturbative RSB degrees of freedom does not effect the critical phenomena in the two-dimensional weakly disordered Ising and Baxter models studied earlier.The effects produced by weak quenched disorder on the critical phenomena near the phase transition point have been studied since many years ago [1]- [6]. According to the Harris criterion [1], the disorder effects the critical behaviour only if α, the specific heat exponent of the pure system, is positive. In this case a new universal critical behavior, with new critical exponents, is established sufficiently close to the phase transition point, where u ≪ 1 is the parameter which describes the strength of the disorder. In contrast, when α < 0, the disorder appears to be irrelevant for the critical behavior.Originally the modified critical behaviour has been derived for the classical φ 4 model near four dimensions [2], and later it has been studied for the two-dimensional Ising [3], Baxter [4] and Potts [5] models by various renormalization group (RG) techniques, and by numerical simulations [6].In dealing with the quenched disorder the traditional approach is the replica method, and in terms of replicas all the results obtained for the systems listed above correspond to the so-called replica-symmetric (RS) solutions. Physically it means that olny unique ground state is assumed to be relevant for the observable thermodynamics. The problem, however, is that in the presence of the quenched disorder there exist numerous local minimum configurations separated by finite barriers, and in this case the direct application of the traditional replica-symmetric RG scheme may be questioned.On the other hand, it is the Parisi Replica Symmetry Breaking (RSB) scheme which has been developed specifically for dealing with disordered systems which exhibit numerous local minima states (see e.g. [7]). Recent studies show that besides mean-field theory of spin-glasses the RSB approach can also be generalized for situations where one has to deal with fluctuations as well [8],[9],[10]. In the paper [11] qualitative arguments were presented demonstrating the mechanism how the summation over multiple local minima configurations could provide additional non-trivial RSB inte...

Emergent universal critical behavior of the 2D N-color Ashkin-Teller model in the presence of correlated disorder

“…The article is organized as follows. In section 2, we consider the formulation of the problem in terms of Dirac (complex) fermions following [25,40,41] and restrict our consideration to the case of the same disorder for all fermion flavors [i.e., the disorder potentials seen by different Ising components in the models (1.1) and (1.3) are completely correlated]. In section 2, we introduce a fermion representation for both models.…”