Constraints on Beta Functions from Duality

TL;DR

This paper explores how duality symmetry constrains the beta function and phase transitions in spin systems, providing a method to determine the beta function and identify duality in models like the Potts model.

Contribution

It introduces a consistency condition linking duality, phase transitions, and correlation length, and offers a general procedure for finding duality symmetries in single-coupling models.

Findings

Duality invariance of correlation length in spin systems

Exact beta function derived for the 2D Potts model

Relation between self-dual points and phase transitions

Abstract

We analyze the way in which duality constrains the exact beta function and correlation length in single-coupling spin systems. A consistency condition we propose shows very concisely the relation between self-dual points and phase transitions, and implies that the correlation length must be duality invariant. These ideas are then tested on the 2-d Ising model, and used towards finding the exact beta function of the $q$-state Potts model. Finally, a generic procedure is given for identifying a duality symmetry in other single-coupling models with a continuous phase transition.