High-Activity Expansion for the Columnar Phase of the Hard Rectangle Gas

TL;DR

This paper develops a high-activity expansion to analyze the nematic-columnar phase transition in a system of hard rectangles on a lattice, providing bounds for critical parameters and insights into phase behavior.

Contribution

It introduces a novel high-activity expansion method for the columnar phase of hard rectangles with fixed orientation, deriving exact terms and bounds for critical points.

Findings

Derived exact first d+2 terms of the expansion

Provided lower bounds for critical density and activity

Bounds decrease with increasing aspect ratio k and decreasing m

Abstract

We study a system of monodispersed hard rectangles of size $m \times d$, where $d\geq m$ on a two dimensional square lattice. For large enough aspect ratio, the system is known to undergo three entropy driven phase transitions with increasing activity $z$: first from disordered to nematic, second from nematic to columnar and third from columnar to sublattice phases. We study the nematic-columnar transition by developing a high-activity expansion in integer powers of $z^{-1/d}$ for the columnar phase in a model where the rectangles are allowed to orient only in one direction. By deriving the exact expression for the first $d+2$ terms in the expansion, we obtain lower bounds for the critical density and activity. For $m$, $k\gg 1$, these bounds decrease with increasing $k$ and decreasing $m$.