Crossover exponents, fractal dimensions and logarithms in Landau–Potts field theories

Abstract

We compute the crossover exponents of all quadratic and cubic deformations of critical field theories with permutation symmetry Sq in d= 6 - ϵ (Landau–Potts field theories) and d= 4 - ϵ (hypertetrahedral models) up to three loops. We use our results to determine the ϵ-expansion of the fractal dimension of critical clusters in the most interesting cases, which include spanning trees and forests (q→ 0), and bond percolations (q→ 1). We also explicitly verify several expected degeneracies in the spectrum of relevant operators for natural values of q upon analytic continuation, which are linked to logarithmic corrections of CFT correlators, and use the ϵ-expansion to determine the universal coefficients of such logarithms.