A cluster expansion approach to the Heilmann-Lieb liquid crystal model

TL;DR

This paper investigates a monomer-dimer lattice model with short-range attractions, demonstrating the absence of long-range translational order among horizontal dimers, supporting the Heilmann-Lieb conjecture.

Contribution

It introduces a cluster expansion method to analyze the Heilmann-Lieb liquid crystal model, providing rigorous evidence for the conjecture.

Findings

Horizontal dimers lack long-range order

Model supports Heilmann-Lieb conjecture

Cluster expansion effectively analyzes the system

Abstract

A monomer-dimer model with a short-range attractive interaction favoring colinear dimers is considered on the lattice $\mathbb{Z}^2$. Although our choice of the chemical potentials results in more horizontal than vertical dimers, the horizontal dimers have no long-range traslational order - in agreement with the Heilmann-Lieb conjecture.