High-density hard-core model on $\mathbb{Z}^2$ and norm equations in ring $\mathbb{Z} [{\sqrt[6]{-1}}]$

Abstract

We study the Gibbs statistics of high-density hard-core configurations on a unit square lattice $\mathbb{Z}^2$, for a general Euclidean exclusion distance $D$. As a by-product, we solve the disk-packing problem on $\mathbb{Z}^2$ for disks of diameter $D$. The key point is an analysis of solutions to norm equations in $\mathbb{Z}[{\sqrt[6]{-1}}]$. We describe the ground states in terms of M-triangles, i.e., non-obtuse $\mathbb{Z}^2$-triangles of a minimal area with the side-lengths $\geq D$. There is a finite class (Class S) formed by values $D^2$ generating sliding, a phenomenon leading to countable families of periodic ground states. We identify all $D^2$ with sliding. Each of the remaining classes is proven to be infinite; they are characterized by uniqueness or non-uniqueness of a minimal triangle for a given $D^2$, up to $\mathbb{Z}^2$-congruencies. For values of $D^2$ with uniqueness (Class A) we describe the periodic ground states as admissible sub-lattices in $\mathbb{Z}^2$ of maximum density. By using the Pirogov-Sinai theory, it allows us to identify the extreme Gibbs measures (pure phases) for large values of fugacity and describe symmetries between them. Next, we analyze the values $D^2$ with non-uniqueness. For some $D^2$ all M-triangles are ${\mathbb{R}}^2$-congruent but not $\mathbb{Z}^2$-congruent (Class B0). For other values of $D^2$ there exist non-${\mathbb{R}}^2$-congruent M-triangles, with different collections of side-lengths (Class B1). Moreover, there are values $D^2$ for which both cases occur (Class B2). The large-fugacity phase diagram for Classes B0, B1, B2 is determined by dominant ground states. Classes A, B0-B2 are described in terms of cosets in $\mathbb{Z}[{\sqrt[6]{-1}}]$ by the group of units.