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

Abstract

Abstract We compute the crossover exponents of all quadratic and cubic deformations of critical field theories with permutation symmetry $$S_q$$ S q in $$d=6-\epsilon $$ d = 6 - ϵ (Landau–Potts field theories) and $$d=4-\epsilon $$ d = 4 - ϵ (hypertetrahedral models) up to three loops. We use our results to determine the $$\epsilon $$ ϵ -expansion of the fractal dimension of critical clusters in the most interesting cases, which include spanning trees and forests ( $$q\rightarrow 0$$ q → 0 ), and bond percolations ( $$q\rightarrow 1$$ 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 $$\epsilon $$ ϵ -expansion to determine the universal coefficients of such logarithms.