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We establish nematic order in $d$-dimensional monomer-dimer models with lateral attraction for $d \ge 2$, under the condition $(4d-6)b > \mu$, which ensures misaligned dimers are sparse. The proof leverages techniques for orientational order from prior work on Model I, but crucially employs a custom cluster swapping method to demonstrate the absence of translational order. This leads to the uniqueness of the Gibbs measure and anisotropic exponential decay of correlations within each oriented phase.
Nematic order is proven for a broad class of $d$-dimensional liquid crystal models, resolving a long-standing conjecture and revealing a critical condition for crystalline behavior.
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 $\mu$, interacting via a hard-core repulsion and a lateral attraction between parallel dimers with coupling constant $b>0$. For $d\in\{2,3\}$, these models were conjectured to exhibit liquid-crystalline order at low temperatures provided that $\mu+2(d-1)b>0$. We establish nematic order for all $d\ge 2$ under the additional condition $(4d-6)b>\mu$, 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.