Provable first-order transitions for liquid crystal and lattice gauge models with continuous symmetries

TL;DR

This paper rigorously proves the existence of first-order phase transitions in certain nonlinear sigma and lattice gauge models with continuous symmetries, clarifying previous debates and providing new insights into their phase behavior.

Contribution

It offers the first rigorous proof of first-order transitions in models with continuous gauge symmetry and clarifies the nature of such transitions in nematic liquid crystal and lattice gauge models.

Findings

First-order transitions in RP^{N-1} models in dimensions ≥2.

First-order transitions in lattice gauge models in dimensions ≥3.

Resolution of a controversy regarding phase transitions in 2D models.

Abstract

We consider various sufficiently nonlinear sigma models for nematic liquid crystal ordering of RP^{N-1} type and of lattice gauge type with continous symmetries. We rigorously show that they exhibit a first-order transition in the temperature. The result holds in dimension 2 or more for the RP^{N-1} models and in dimension 3 or more for the lattice gauge models. In the two-dimensional case our results clarify and solve a recent controversy about the possibility of such transitions. For lattice gauge models our methods provide the first proof of a first-order transition in a model with a continuous gauge symmetry.