Spin anisotropy and crossover in the Potts model

A group-theoretical analysis is presented of the perturbation's in the $n+1$-state Potts model, which arise when the equivalence between the states is broken. In the continuum version of this model, described by n real fields, all possible quadratic perturbations transform according to two irreducible representations of the underlying symmetry group, so there are two associated cross-over exponents. One of these exponents is equal to the order-parameter exponent β for any n. In the limit n→0 the other exponent φ describes crossover between percolation and random Ising behavior and is also related to an exponent describing the conductivity of a random resistor network close to the percolation threshold. We show that φ=1, in this limit, to all orders in perturbation theory. Consequently, there are no new exponents associated with crossover to anisotropic behavior for n→0.