Nematic order in monomer-dimer models of Heilmann and Lieb with lateral attraction in dimensions d≥ 2
Abstract
We study the d-dimensional generalization of Models II (d=2) and V (d=3) for liquid crystals introduced by Heilmann--Lieb (1979), which consists of dimers on the hypercubic lattice at chemical potential μ, interacting via a hard-core repulsion and a lateral attraction between parallel dimers with coupling constant b>0. For d∈{2,3}, these models were conjectured to exhibit liquid-crystalline order at low temperatures provided that μ+2(d−1)b>0. We establish nematic order for all d≥ 2 under the additional condition (4d−6)b>μ, which corresponds to the physical regime in which misaligned dimers are rare on scales comparable to the correlation length of the one-dimensional reference system. While the treatment of orientational order here follows the approach of our recent work on Model I, the proof of the absence of translational order relies on a custom implementation of cluster swapping in the sense of Sheffield (2005). Consequently, we deduce uniqueness of Gibbs measure and anisotropic exponential decay of correlations within each oriented phase.
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BibTeX
@misc{he2026nematic,
title = {{Nematic order in monomer-dimer models of Heilmann and Lieb with lateral attraction in dimensions $d\ge 2$}},
author = {Qidong He},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08600},
eprint = {2610.08600},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08600},
}
RIS
TY - UNPB
TI - Nematic order in monomer-dimer models of Heilmann and Lieb with lateral attraction in dimensions d≥ 2
AU - Qidong He
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.08600
UR - https://arxiv.org/abs/2610.08600
AB - We study the d-dimensional generalization of Models II (d=2) and V (d=3) for liquid crystals introduced by Heilmann--Lieb (1979), which consists of dimers on the hypercubic lattice at chemical potential μ, interacting via a hard-core repulsion and a lateral attraction between parallel dimers with coupling constant b>0. For d∈{2,3}, these models were conjectured to exhibit liquid-crystalline order at low temperatures provided that μ+2(d−1)b>0. We establish nematic order for all d≥ 2 under the additional condition (4d−6)b>μ, which corresponds to the physical regime in which misaligned dimers are rare on scales comparable to the correlation length of the one-dimensional reference system. While the treatment of orientational order here follows the approach of our recent work on Model I, the proof of the absence of translational order relies on a custom implementation of cluster swapping in the sense of Sheffield (2005). Consequently, we deduce uniqueness of Gibbs measure and anisotropic exponential decay of correlations within each oriented phase.
ER -