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

TL;DR

This study investigates phase transitions in a system of hard rectangles on a square lattice, revealing how transition densities scale with aspect ratio and identifying the universality class of the nematic-columnar transition through extensive simulations and theoretical approximations.

Contribution

The paper provides extensive Monte Carlo simulation data and theoretical estimates for phase transition densities in hard rectangle systems, highlighting the scaling behavior and universality class of the nematic-columnar transition.

Findings

Transition density for isotropic-nematic transition scales as 1/k with coefficient ~4.80

Nematic-columnar transition critical density approaches a value less than full packing density as k increases

Transition is in the Ising universality class despite non-universal Binder cumulant behavior

Abstract

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.