Random Bond Ising Model and Massless Phase of the Gross - Neveu Model

The O(n) Gross-Neveu model for n < 2 presents a massless phase that can be characterized by right-left mover scattering processes. The limit n → 0 describes the on-shell properties of the random bond Ising model. Permanent address: International School for Advanced Studies and Istituto Nazionale di Fisica Nucleare, Trieste, Italy. 1. Aim of this paper is to discuss the S-matrix formulation of the O(n) Gross-Neveu (GN) model for n < 2, and in particular the scattering theory associated to the limit n → 0. There is a well-defined physical problem related to this limit, which is the analysis of the influence of random impurities, induced by thermal fluctuations, on the critical behaviour of pure homogeneous systems. It is therefore useful to briefly remind the relationship between the GN model and the random systems. For “random temperature” kinds of impurities, i.e. defects or dislocations in the material induced by thermal fluctuations, there is a simple criterion [1] for extimating the relevance of weak disorder on a critical system. According to this criterion, the effect of the disorder depends on the sign of the specific heat critical index α of the pure material. For α > 0 the impurities are expected to completely suppress the long range fluctuations of the pure system, canceling all singularities in the thermodynamical quantities. On the contrary, for α < 0, the impurities may produce a shift of the critical temperature but they do not affect the critical behaviour, i.e. the critical exponents are the same as in the pure system. The marginal case α = 0 is special and must be separately analysed. This situation occurs in the two-dimensional Ising model with random bond distribution, and it has been initially considered by Dotsenko and Dotsenko [2]. In particular, these authors have shown that near the critical temperature, the class of universality of the random bond Ising model is described by the O(n) Gross-Neveu model in the limit n → 0 (see also [3, 4, 5]). The mapping is realized as follows. The two-dimensional homogeneous Ising model can be described in the continuum limit by a massive Majorana fermion Ψ, where the mass m is a linear measurament of the deviation of the temperature T from the critical value Tc. The partition function for the pure system is given by Z[m] = ∫