The Global Minimum of Energy Is Not Always a Sum of Local Minima - a Note on Frustration

TL;DR

This paper constructs a classical lattice gas model with finite-range competing interactions that lacks periodic ground states, yet has a unique translation-invariant ground state measure, highlighting complexities in energy minimization.

Contribution

It introduces a novel lattice gas model demonstrating that the global energy minimum can be nonperiodic and not decomposable into local minima, challenging traditional assumptions.

Findings

No periodic ground states exist for the model.

The model's ground state is a unique translation-invariant measure.

The construction relies on nonperiodic ground state configurations.

Abstract

A classical lattice gas model with translation-invariant finite range competing interactions, for which there does not exist an equivalent translation-invariant finite range nonfrustrated potential, is constructed. The construction uses the structure of nonperiodic ground state configurations of the model. In fact, the model does not have any periodic ground state configurations. However, its ground state - a translation-invariant probability measure supported by ground state configurations - is unique.