Stability of columnar order in assemblies of hard rectangles or squares

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A system of 2× d hard rectangles on square lattice is known to show four different phases for d ≥ 14. As the covered area fraction ρ is increased from 0 to 1, the system goes from low-density disordered phase, to orientationally-ordered nematic phase, to a columnar phase with orientational order and also broken translational invariance, to a high density phase in which orientational order is lost. For large d, the threshold density for the first transition ρ₁* tends to 0, and the critical density for the third transition ρ₃* tends to 1. Interestingly, simulations have shown that the critical density for the second transition ρ₂* tends to a non-trivial finite value ≈ 0.73, as d → ∞, and ρ₂* ≈ 0.93 for d=2. We provide a theoretical explanation of this interesting result. We develop an approximation scheme to calculate the surface tension between two differently ordered columnar phases. The density at which the surface tension vanishes gives an estimate ρ₂* = 0.746, for d→ ∞, and ρ₂*=0.923 for d=2. For all values of d, these estimates are in good agreement with Monte Carlo data.