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

TL;DR

This paper calculates crossover exponents and fractal dimensions in Landau-Potts field theories near critical dimensions, revealing logarithmic corrections and operator degeneracies relevant to percolation and spanning structures.

Contribution

It provides three-loop epsilon-expansions of crossover exponents and fractal dimensions for Landau-Potts models, including analysis of logarithmic corrections and operator degeneracies.

Findings

Computed crossover exponents up to three loops in $6-psilon$ and $4-psilon$ dimensions.

Determined epsilon-expansion of fractal dimensions for percolation-related models.

Identified operator degeneracies linked to logarithmic corrections in conformal 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 ($q\to0$), and bond percolations ($q\to1$). 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.