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Asymptotic Behavior of the Isotropic-Nematic and Nematic-Columnar Phase Boundaries for the System of Hard Rectangles on a Square lattice

A system of hard rectangles of size $m\times mk$ on a square lattice undergoes three entropy driven phase transitions with increasing density for large enough aspect ratio $k$: first from a low density isotropic to an intermediate density nematic phase, second from the nematic to a columnar phase, and third from the columnar to a high density sublattice phase. In this paper we show, from extensive Monte Carlo simulations of systems with $m=1,2$ and $3$, that the transition density for the isotropic-nematic transition is $\approx A_1/k$ when $k \gg 1$, where $A_1$ is independent of $m$. We estimate $A_1=4.80\pm 0.05$. Within a Bethe approximation, we obtain $A_1=2$ and the virial expansion truncated at second virial coefficient gives $A_1=1$. The critical density for the nematic-columnar transition when $m=2$ is numerically shown to tend to a value less than the full packing density as $k^{-1}$ when $k\to \infty$. We find that the critical Binder cumulant for this transition is non-universal and decreases as $k^{-1}$ for $k \gg 1$. However, the transition is shown to be in the Ising universality class.