Stability of columnar order in assemblies of hard rectangles or squares

Abstract

A system of $2\times d$ hard rectangles on square lattice is known to show four different phases for $d\ge 14$. As the covered area fraction $\rho$ is increased from $0$ to $1$, the system goes from low-density disordered phase, to orientationally-ordered nematic phase, to a columnar phase with orientational order and also broken translational invariance, to a high density phase in which orientational order is lost. For large d, the threshold density for the first transition ${\rho }_1^{\ast }$ tends to $0$, and the critical density for the third transition ${\rho }_3^{\ast }$ tends to $1$. Interestingly, simulations have shown that the critical density for the second transition ${\rho }_2^{\ast }$ tends to a non-trivial finite value $\approx 0.73$, as $d\to \infty$, and ${\rho }_2^{\ast }\approx 0.93$ for $d=2$. We provide a theoretical explanation of this interesting result. We develop an approximation scheme to calculate the surface tension between two differently ordered columnar phases. The density at which the surface tension vanishes gives an estimate ${\rho }_2^{\ast }=0.746$, for $d\to \infty$, and ${\rho }_2^{\ast }=0.923$ for $d=2$. For all values of $d$, these estimates are in good agreement with Monte Carlo data.

6 pages, 3 figures

References in corpus (10)

• Classical dimers with aligning interactions on the square lattice
• Monte Carlo simulations of 2d hard core lattice gases
• Quantum criticality, lines of fixed points, and phase separation in doped two-dimensional quantum dimer models
• Random trimer tilings
• Asymptotic Behavior of the Isotropic-Nematic and Nematic-Columnar Phase Boundaries for the System of Hard Rectangles on a Square lattice
• Stability of columnar order in assemblies of hard rectangles or squares
• Evidence of columnar order in the fully frustrated transverse field Ising model on the square lattice
• Equation of state for hard square lattice gases
• Phase transitions in systems of hard rectangles with non-integer aspect ratio
• Exact entropy of dimer coverings for a class of lattices in three or more dimensions

Cited by in corpus (16)

• Stability of columnar order in assemblies of hard rectangles or squares
• Entropy of fully packed hard rigid rods on $d$-dimensional hyper-cubic lattices
• The High Density Phase of the $k$-NN Hard Core Lattice Gas Model
• Different phases of a system of hard rods on three dimensional cubic lattice
• Ordering of hard rectangles in strong confinement
• Phase transitions in a system of hard $Y$-shaped particles on the triangular lattice
• Phase diagram of a system of hard cubes on the cubic lattice
• Estimating the Critical Parameters of the Hard Square Lattice Gas Model
• Husimi lattice solutions and the coherent-anomaly-method analysis for hard-square lattice gases
• The columnar-disorder phase boundary in a mixture of hard squares and dimers
• The freezing phase transition in hard core lattice gases on triangular lattice with exclusion up to seventh next-nearest neighbor
• Hard core lattice gas with third next-nearest neighbor exclusion on triangular lattice: one or two phase transitions?
• Positional ordering of hard adsorbate particles in tubular nanopores
• Phases of the hard-plate lattice gas on a three-dimensional cubic lattice
• Rejection-free cluster Wang-Landau algorithm for hard-core lattice gases
• Upper bounds for the connective constant of weighted self-avoiding walks