Nematic order in monomer-dimer models of Heilmann and Lieb with lateral attraction in dimensions $d\ge 2$

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.