The paper analyzes a lattice model of liquid crystals made of dimers — pairs of occupied nearest-neighbor sites — and shows that these dimers line up in a preferred direction (nematic order) while not forming a full crystal. The authors prove this behavior for any dimension d ≥ 2, under a specific inequality that relates two physical parameters: the chemical potential μ and the lateral attraction strength b. In plain terms, when the attraction between parallel dimers is strong enough relative to μ, the system prefers a common orientation but does not lock into a positional lattice.

The model studied is a d-dimensional generalization of two Heilmann–Lieb models from 1979 (called Models II and V for d = 2 and d = 3). Dimers live on the edges of a hypercubic lattice. They cannot overlap (hard-core repulsion) and parallel, adjacent dimers attract each other with strength b > 0. The chemical potential μ controls how many dimers are present. Earlier work had shown that orientations can align in low-temperature regimes, but it left open whether that alignment comes with unwanted positional order. The present paper addresses that gap.

The main result is a rigorous proof of nematic order for all dimensions d ≥ 2, provided (4d − 6) b > μ. This extra condition means misaligned dimers are rare on the length scale set by a certain one-dimensional reference system. The authors combine several tools. They analyze the one-dimensional reference model by transfer-matrix methods to get precise estimates. To rule out translational order, they adapt a technique called cluster swapping (introduced by Sheffield) to this setting. Cluster swapping lets them control large connected regions of disagreement in a flexible way. They also use a percolation picture with “sealed boxes” that confine problematic clusters to quasi-one-dimensional columns where they can be handled.