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 and Lieb (1979), which consists of a system of dimers on the square lattice at chemical potential $λ$, 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 $λ+a>0$. In this paper, we prove the conjecture under the additional condition that $3a>λ$, which corresponds physically to the regime where vacancies, as opposed to misaligned dimers, are the dominant mechanism for decorrelation, significantly extending the parameter regime under which the conjecture is known to hold. Our proof adapts the strategy of Hadas and Peled (2025) for proving the existence of a columnar phase in the hard-square model, combining a mesoscopic characterization of orientational order with the disagreement percolation method of van den Berg (1993) to prove the absence of translational order. To deal with the non-nearest neighbor interactions in the model, we also introduce an extension of the chessboard estimate applicable to finite products of periodic Gibbs measures.