The High Density Phase of the $k$-NN Hard Core Lattice Gas Model

TL;DR

This paper investigates the high-density phases of the $k$-NN hard core lattice gas model on a square lattice, revealing that only a small subset of $k$ values exhibit columnar order, while most show sublattice order.

Contribution

It determines the nature of the full packing phase for a wide range of $k$ values, identifying which exhibit columnar versus sublattice order, extending understanding of phase behavior in this model.

Findings

Only 18 out of over 820,000 $k$ values show columnar order.

Most $k$ values exhibit solid-like sublattice order at full packing.

Columnar order occurs only for $k<4134$.

Abstract

The $k$-NN hard core lattice gas model on a square lattice, in which the first $k$ next nearest neighbor sites of a particle are excluded from being occupied by another particle, is the lattice version of the hard disc model in two dimensional continuum. It has been conjectured that the lattice model, like its continuum counterpart, will show multiple entropy-driven transitions with increasing density if the high density phase has columnar or striped order. Here, we determine the nature of the phase at full packing for $k$ up to $820302$. We show that there are only eighteen values of $k$, all less than $k=4134$, that show columnar order, while the others show solid-like sublattice order.