Peierls Instability in Hexagonal Lattices

TL;DR

This paper analyzes how lattice distortions in graphene influence its stability, proving that small elastic forces favor Kekulé patterns, while large forces favor uniform structures, revealing phase transitions in lattice configurations.

Contribution

It rigorously demonstrates the conditions under which Kekulé distortions or uniform lattices are energetically favored in a tight-binding model of graphene.

Findings

Kekulé O-type symmetry is optimal for small elasticity.

Non-translation-invariant configurations are favored at low elasticity.

Translation-invariant lattice is optimal at high elasticity.

Abstract

We investigate a conventional tight-binding model for graphene, where distortion of the honeycomb lattice is allowed, but penalized by a quadratic energy. We prove that the optimal 3-periodic lattice configuration has Kekul\'e O-type symmetry, and that for a sufficiently small elasticity parameter, the minimizer is not translation-invariant. Conversely, we prove that for a large elasticity parameter the translation-invariant configuration is the unique minimizer.