On Long-Range Order in Low-Dimensional Lattice-Gas Models of Nematic Liquid Crystals

TL;DR

This paper investigates the conditions under which low-dimensional lattice-gas models of nematic liquid crystals exhibit long-range orientational order, highlighting the roles of interaction range and temperature.

Contribution

It demonstrates that long-range isotropic interactions can induce orientational order in low-dimensional lattice models, which is not possible with short-range interactions.

Findings

Long-range isotropic interactions lead to orientational order at high packing and low temperature.

Short-range interactions do not support long-range order in low dimensions.

Existence of an orientational ordering transition is proven for certain long-range interaction conditions.

Abstract

The problem of the orientational ordering transition for lattice-gas models of liquid crystals is discussed in the low-dimensional case $d=1,2$. For isotropic short-range interactions, orientational long-range order at finite temperature is excluded for any packing of molecules on the lattice $Z^d$; on the other hand, for reflection-positive long-range isotropic interactions, we prove existence of an orientational ordering transition for high packing ($\mu > \mu_0$) and low temperatures ($\beta > \beta_c(\mu)$).