Phase Transitions in a System of Hard Rectangles on Lattices

The study of rod like molecules with only excluded volume interaction has been of continuous theoretical and experimental interest. Although there are extensive studies in continuum, discrete versions are less understood. We study the system of monodispersed hard rectangles of size $m \times m k$ on two dimensional lattices and determine the phase diagram numerically for $m=2, 3$ and aspect ratio $k= 1,2,...., 7$. We show the existence of a disordered phase, a nematic phase with orientational order, a columnar phase with orientational and partial translational order, and a crystalline sublattice phase. The asymptotic behavior of the phase boundaries for large $k$ are determined using a combination of entropic arguments and a Bethe approximation. This allows us to generalize the phase diagram to larger $m$ and $k$, showing that for $k \geq 7 $, the system undergoes three entropy driven phase transitions with increasing density. The nature of the different phase transitions are established and the critical exponents for the continuous transitions are determined using finite size scaling.