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 $S_q$ in $d=6-\epsilon$ (Landau-Potts field theories) and $d=4-\epsilon$ (hypertetrahedral models) up to three loops.We use our results to determine the $\epsilon$-expansion of the fractal dimension of critical clusters in the most interesting cases, which include spanning trees and forests ($q\to 0$), and bond percolations ($q\to 1$). 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 $\epsilon$-expansion to determine the universal coefficients of such logarithms.

arXiv comment15 pages, 3 tables, 3 figures; v2: improved version with estimates relevant for the cubic model, to appear in EPJC

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• Conjecture on the lower bound of the length-scale critical exponent $\nu$ at continuous phase transitions