Stability of columnar order in assemblies of hard rectangles or squares

A system of $2\times d$ hard rectangles on square lattice is known to show four different phases for $d \geq 14$. As the covered area fraction $\rho$ 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 $\rho_1^*$ tends to $0$, and the critical density for the third transition $\rho_3^*$ tends to $1$. Interestingly, simulations have shown that the critical density for the second transition $\rho_2^*$ tends to a non-trivial finite value $\approx 0.73$, as $d \rightarrow \infty$, and $\rho_2^* \approx 0.93$ 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 $\rho_2^* = 0.746$, for $d\to \infty$, and $\rho_2^*=0.923$ for $d=2$. For all values of $d$, these estimates are in good agreement with Monte Carlo data.