An Ultimate Frustration in Classical-Lattice Gas Models

TL;DR

This paper constructs a broad class of classical lattice-gas models with unique ground states that are not representable as tiling systems, revealing a larger diversity of ground states than previously known.

Contribution

It introduces an uncountable family of models with unique ground states that defy representation as tilings, expanding understanding of ground-state measure structures.

Findings

Ground-state measures are not always tiling measures.

The family of models with unique ground states exceeds tiling-based models.

Ground states resemble 2D analogs of 1D convex interaction configurations.

Abstract

We constructed an uncountable family of classical lattice-gas models with unique ground-state measures which are not uniquely ergodic measures of any tiling system, or more generally, of any system of finite type. Therefore, we have shown that the family of structures which are unique ground states of some translation-invariant, finite-range interactions is larger than the family of tilings which form single isomorphism classes. Such ground-state measures cannot be ground-state measures of any translation-invariant, finite-range, nonfrustrated potential. Our ground-state configurations are two-dimensional analogs of one-dimensional, most homogeneous ground-state configurations of infinite-range, convex, repulsive interactions in models with devil's staircases.