The arboreal gas and the supersphere sigma model

TL;DR

This paper explores the connections between the Potts model, arboreal gas, and supersphere sigma models, revealing critical points and symmetries that involve non-compact degrees of freedom and continuous spectra.

Contribution

It identifies the Potts antiferromagnetic critical point with a sigma model critical point and analyzes the role of topological properties in these transitions.

Findings

Critical points of Potts model and arboreal gas coincide

The sigma model exhibits non-linearly realized OSP(2/2) symmetry

Spectrum of critical exponents is continuous

Abstract

We discuss the relationship between the phase diagram of the Q=0 state Potts model, the arboreal gas model, and the supersphere sigma model S^{0,2} = OSP(1/2) / OSP(0/2). We identify the Potts antiferromagnetic critical point with the critical point of the arboreal gas (at negative tree fugacity), and with a critical point of the sigma model. We show that the corresponding conformal theory on the square lattice has a non-linearly realized OSP(2/2) = SL(1/2) symmetry, and involves non-compact degrees of freedom, with a continuous spectrum of critical exponents. The role of global topological properties in the sigma model transition is discussed in terms of a generalized arboreal gas model.