Spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions

TL;DR

This paper investigates a three-dimensional hyperbolic sigma model derived from supersymmetric methods in disordered electron systems, demonstrating spontaneous symmetry breaking and finite moments in the thermodynamic limit.

Contribution

It proves that the non-compact SU(1,1) symmetry in the hyperbolic sigma model is spontaneously broken in three dimensions, with moments remaining finite.

Findings

Moments of fields are finite in the thermodynamic limit.

The non-compact symmetry SU(1,1) is spontaneously broken.

Extended states are compatible with the observed symmetry breaking.

Abstract

Non-linear sigma models that arise from the supersymmetric approach to disordered electron systems contain a non-compact bosonic sector. We study the model with target space H^2, the two-hyperboloid with isometry group SU(1,1), and prove that in three dimensions moments of the fields are finite in the thermodynamic limit. Thus the non-compact symmetry SU(1,1) is spontaneously broken. The bound on moments is compatible with the presence of extended states.