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

Qidong He

TL;DR

The paper proves nematic order in Heilmann–Lieb’s 2D monomer-dimer model at low temperature under λ+a>0 and 3a>λ, establishing two extremal Gibbs measures that favor vertical or horizontal dimer alignment. It develops a mesoscopic framework of orientational order using sticks and mesoscopic rectangles, combined with a refined chessboard–reflection positivity technique and a disagreement-percolation method to show absence of translational order and decay of correlations. A key technical advance is extending infinite-volume chessboard estimates to finite products of periodic Gibbs measures, enabling control of interactions beyond nearest neighbors. The results substantially extend the parameter regime for nematic order and provide a pathway toward a complete understanding of orientational versus translational ordering in this classical lattice liquid-crystal model.

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.

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

TL;DR

The paper proves nematic order in Heilmann–Lieb’s 2D monomer-dimer model at low temperature under λ+a>0 and 3a>λ, establishing two extremal Gibbs measures that favor vertical or horizontal dimer alignment. It develops a mesoscopic framework of orientational order using sticks and mesoscopic rectangles, combined with a refined chessboard–reflection positivity technique and a disagreement-percolation method to show absence of translational order and decay of correlations. A key technical advance is extending infinite-volume chessboard estimates to finite products of periodic Gibbs measures, enabling control of interactions beyond nearest neighbors. The results substantially extend the parameter regime for nematic order and provide a pathway toward a complete understanding of orientational versus translational ordering in this classical lattice liquid-crystal model.

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 , interacting via a hard-core repulsion and an attractive interaction of strength 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 . In this paper, we prove the conjecture under the additional condition that , 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.

Paper Structure

This paper contains 42 sections, 48 theorems, 164 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model and main results
  3. Proof overview and comparison with Hadas--Peled
  4. The origin of the condition $3a>\lambda$
  5. Organization of the paper
  6. Preliminaries
  7. Basic definitions
  8. Graphical constructions
  9. Rectangles
  10. Dimer configurations
  11. The Hamiltonian and Gibbs measures
  12. Gibbs measures
  13. Comparison of boundary conditions
  14. Reflection positivity and the chessboard estimate
  15. Standard results
  16. ...and 27 more sections

Key Result

Theorem 1.1

Fix $\lambda+a>0,3a>\lambda$. There exist constants $\beta_0,C,c>0$ such that, for all $\beta\ge\beta_0$, the monomer-dimer model admits a unique Gibbs measure $\mu_\mathrm{ver}$ such that Let $\tau:\mathbb{R}^2\to\mathbb{R}^2$ be the reflection $(x,y)\mapsto(y,x)$. The push-forward of $\mu_\mathrm{ver}$ under $\tau$, denoted by $\mu_\mathrm{hor}$, is another Gibbs measure of the monomer-dimer mo

Figures (3)

  • Figure 1: The configuration graph associated to a dimer configuration in $\Omega^0_{\mathrm{R}_{8\times 8}}$. The correspondence between edge labels and edge styles is as follows: solid green edges correspond to the label $s$, dotted blue edges to $v$, dashed light green edges to $b$, and dash-dotted light green edges to $bv$ (color online)
  • Figure 2: Provided that a $1\times N\mathfrak{c}$ rectangle intersects only vertical dimers in both $\sigma$ and $\sigma'$, no disagreement $\ddag$-path in $\Delta_{\sigma,\sigma'}$ can cross a coincident vacancy or vertical dimer in that rectangle
  • Figure 3: Provided that a $1\times N\mathfrak{c}$ rectangle (a) intersects only vertical dimers in both $\sigma$ and $\sigma'$, (b) is not fully packed with dimers in either $\sigma$ or $\sigma'$, and (c) does not contain two adjacent vacancies in $\sigma$, the only way to rule out the existence of coincident vacancies and vertical dimers in the rectangle is to have a vacancy of $\sigma$ adjacent to a vacancy of $\sigma'$

Theorems & Definitions (99)

  • Theorem 1.1: Nematic order
  • Remark 2.1
  • Proposition 2.2: hadas2025columnar
  • Proposition 2.3: hadas2025columnar
  • Lemma 2.4: Reflection positivity
  • proof
  • Proposition 2.5: Chessboard estimate
  • Lemma 2.6: Recursive chessboard estimate, hadas2025columnar
  • proof
  • Proposition 2.7: hadas2025columnar
  • ...and 89 more