Extended regime of nematic order in an interacting monomer-dimer model of Heilmann and Lieb

Abstract

We revisit a two-dimensional model of liquid crystals introduced by Heilmann--Lieb (1979), which consists of dimers on the square lattice at chemical potential $\lambda$, interacting via a hard-core repulsion and an attractive interaction of strength $a>0$ between adjacent, colinear dimers. The model is conjectured to exhibit nematic order at low temperatures, in the sense of orientational symmetry breaking coupled with the absence of translational order, provided that $\lambda +a>0$. We prove this conjecture under the additional condition $3a>\lambda$, substantially extending the parameter regime in which nematic order is known. The extra condition corresponds exactly to the regime in which misaligned dimers remain rare on scales comparable to the correlation length of the one-dimensional reference system. Our proof begins with the microscopic characterization of orientational order used by Heilmann--Lieb, strengthens it into a quantitative characterization of the rarity of misaligned dimers under the strong-percolation formalism of Hadas--Peled (2025), and uses the improved estimate to implement a disagreement-percolation argument in the sense of van den Berg (1993). As a consequence, we establish uniqueness of Gibbs measure and anisotropic exponential decay of correlations within each orientational phase.

Proof strategy significantly simplified following suggestions by an anonymous referee. 33 pages, 2 figures