Nematic-columnar phase transition in oriented hard rectangles

We consider an assembly of monodispersed hard rectangles of size $2 \times d$ on a square lattice with only hard core interactions amongst them. The long axes of the rectangles can be oriented along the horizontal or vertical directions. For large enough aspect ratio, it is known that this system undergoes three phase transitions as the density ($\rho$) of rectangles is increased: first an isotropic-nematic transition (at $\rho^*_1$), second a nematic-columnar transition (at $\rho^*_2$), and third a columnar-sublattice transition (at $\rho^*_3$). In the nematic phase, only the orientational symmetry is broken. The columnar and sublattice phases correspond to additional broken translational symmetries along one (perpendicular to the nematic orientation) and both directions respectively. Interestingly, the critical value $\rho^*_2$ remains finite, approximately $0.73$, even as $d \to \infty$. We develop a systematic high density expansion for the surface tension between two differently-ordered columnar phases. Keeping only the first order perturbative correction term and setting this surface tension to zero, we get an estimate of $\rho^*_2$ in excellent agreement with estimates from Monte Carlo simulations, for all $d\geq2$.