On superuniversality in the $q$-state Potts model with quenched disorder

TL;DR

This paper derives exact solutions for two-dimensional disordered Potts models, revealing superuniversality of some critical exponents and identifying fixed points with q-independent properties.

Contribution

It provides the first exact scale-invariant scattering solutions for disordered Potts models, highlighting superuniversality and characterizing multiple fixed points with q-independent features.

Findings

Existence of q-independent sectors indicating superuniversality.

Identification of fixed points corresponding to percolation and Nishimori-like phases.

Potential superuniversality of the magnetic exponent η.

Abstract

We obtain the exact scale invariant scattering solutions for two-dimensional field theories with replicated permutational symmetry $\mathbb{S}_q$. After sending to zero the number of replicas they correspond to the renormalization group fixed points of the $q$-state Potts model with quenched disorder. We find that all solutions with non-zero disorder possess $q$-independent sectors, pointing to superuniversality (i.e. symmetry independence) of some critical exponents. The solution corresponding to the random bond ferromagnet, for which disorder vanishes as $q\to 2$, allows for superuniversality of the correlation length exponent $\nu$ [PRL 118 (2017) 250601]. Of the two solutions which are strongly disordered for all values of $q$, one is completely $q$-independent and accounts for the zero-temperature percolation fixed point of the randomly bond diluted ferromagnet. The other is the main candidate to describe the Nishimori-like fixed point of the Potts model with $\pm J$ disorder, and leaves room for superuniversality of the magnetic exponent $\eta$, a possibility not yet excluded by available numerical data.