The nematic phase of a system of long hard rods

TL;DR

This paper rigorously proves the existence of a nematic phase in a two-dimensional lattice model of long hard rods, demonstrating orientational order without positional order at intermediate densities.

Contribution

It introduces a rigorous proof of nematic phase emergence in a lattice model of long rods using a two-scales cluster expansion and Pirogov-Sinai methods.

Findings

Existence of a nematic phase with orientational order

No positional order at intermediate densities

Application of cluster expansion and contour models

Abstract

We consider a two-dimensional lattice model for liquid crystals consisting of long rods interacting via purely hard core interactions, with two allowed orientations defined by the underlying lattice. We rigorously prove the existence of a nematic phase, i.e., we show that at intermediate densities the system exhibits orientational order, either horizontal or vertical, but no positional order. The proof is based on a two-scales cluster expansion: we first coarse grain the system on a scale comparable with the rods' length; then we express the resulting effective theory as a contour's model, which can be treated by Pirogov-Sinai methods.