Monte Carlo simulations of two-dimensional hard core lattice gases

Abstract

Monte Carlo simulations are used to study lattice gases of particles with extended hard cores on a two dimensional square lattice. Exclusions of one and up to five nearest neighbors (NN) are considered. These can be mapped onto hard squares of varying side length, λ (in lattice units), tilted by some angle with respect to the original lattice. In agreement with earlier studies, the 1NN exclusion undergoes a continuous order-disorder transition in the Ising universality class. Surprisingly, we find that the lattice gas with exclusions of up to second nearest neighbors (2NN) also undergoes a continuous phase transition in the Ising universality class, while the Landau-Lifshitz theory predicts that this transition should be in the universality class of the XY model with cubic anisotropy. The lattice gas of 3NN exclusions is found to undergo a discontinuous order-disorder transition, in agreement with the earlier transfer matrix calculations and the Landau-Lifshitz theory. On the other hand, the gas of 4NN exclusions once again exhibits a continuous phase transition in the Ising universality class -contradicting the predictions of the Landau-Lifshitz theory. Finally, the lattice gas of 5NN exclusions is found to undergo a discontinuous phase transition.

Discontinuous phase transition in a dimer lattice gas

“…This model also suffers a discontinuous phase transition [5], as would be expected from its similarity to the q = 10 Potts model. It seems natural to conjecture, then, that a lattice gas having exactly q equivalent maximum-density states (or minimum-energy states) will exhibit a discontinuous phase transition if q is sufficiently large, i.e., q > 4 in two dimensions.…”

“…The converse to this statement does not, however, appear to hold. Thus, the number of maximum-density configurations of the lattice gas with exclusion out to fifth neighbors is infinite, but the model nevertheless exhibits a discontinuous transition [5]. Finally, I note that the dimer model may also be mapped to a particle model, i.e., to a lattice in which each bond of the original square lattice corresponds to a site on the new lattice, which is also square.…”

“…The lattice gas with exclusion out to fourth neighbors has a continuous fluid-solid transition, while exclusion out to fifth neighbors leads to a discontinuous transition [5] …”

“…Thus the model with nearest-neighbor exclusion on bipartite lattices belongs to the Ising universality class, while the corresponding model on the triangular lattice -the hard-hexagon model -has a transition in the q = 3 Potts universality class [6]. Lattice gases with extended exclusion regions on the square lattice were recently studied by Fernandes et al [5]. For exclusion of up to n-th neighbors, these authors found, for n = 1 to 5, that the fluid-solid transition is discontinuous for n = 3 and 5, and continuous in the other cases.…”

“…Athermal models, characterized by purely excluded volume interactions, are of particular interest in this context for their simplicity, given that (off-lattice) systems of hard spheres and hard disks exhibit fluid-solid transitions [5]. Analyzing athermal lattice models of this kind is relevant not only to the equilibrium melting transition, but to modeling diffusion in dense fluids, glassy dynamics, disordered solids, and granular materials.…”