Tetratic Order in the Phase Behavior of a Hard-Rectangle System

TL;DR

This paper investigates the phase behavior of hard rectangles with an aspect ratio of two, revealing a tetratic, nonperiodic solid phase formed by random domino tilings, using Monte Carlo and molecular dynamics simulations.

Contribution

It demonstrates that simple convex shapes like rectangles can exhibit complex tetratic and nonperiodic solid phases, expanding understanding of two-dimensional phase behavior.

Findings

Identification of a tetratic, nonperiodic solid phase in hard-rectangle systems.

Observation of a two-stage KTHNY-like phase transition.

Use of novel molecular dynamics with rounded rectangles.

Abstract

Previous Monte Carlo investigations by Wojciechowski \emph{et al.} have found two unusual phases in two-dimensional systems of anisotropic hard particles: a tetratic phase of four-fold symmetry for hard squares [Comp. Methods in Science and Tech., 10: 235-255, 2004], and a nonperiodic degenerate solid phase for hard-disk dimers [Phys. Rev. Lett., 66: 3168-3171, 1991]. In this work, we study a system of hard rectangles of aspect ratio two, i.e., hard-square dimers (or dominos), and demonstrate that it exhibits a solid phase with both of these unusual properties. The solid shows tetratic, but not nematic, order, and it is nonperiodic having the structure of a random tiling of the square lattice with dominos. We obtain similar results with both a classical Monte Carlo method using true rectangles and a novel molecular dynamics algorithm employing rectangles with rounded corners. It is remarkable that such simple convex two-dimensional shapes can produce such rich phase behavior. Although we have not performed exact free-energy calculations, we expect that the random domino tiling is thermodynamically stabilized by its degeneracy entropy, well-known to be $1.79k_{B}$ per particle from previous studies of the dimer problem on the square lattice. Our observations are consistent with a KTHNY two-stage phase transition scenario with two continuous phase transitions, the first from isotropic to tetratic liquid, and the second from tetratic liquid to solid.