The laminations of a crystal near an anti-continuum limit

TL;DR

This paper studies the structure of solutions called laminations in a lattice model near an anti-continuum limit, showing their topological organization relates to the local minima of the potential.

Contribution

It generalizes previous results by characterizing the space of laminations as homeomorphic to a simplex near the anti-continuum limit.

Findings

Laminations form a space homeomorphic to an (N-1)-dimensional simplex.

The structure of solutions persists under small coupling.

Generalizes results from twist maps to lattice models.

Abstract

The anti-continuum limit of a monotone variational recurrence relation consists of a lattice of uncoupled particles in a periodic background. This limit supports many trivial equilibrium states that persist as solutions of the model with small coupling. We investigate when a persisting solution generates a so-called lamination and prove that near the anti-continuum limit the collection of laminations of solutions is homeomorphic to the (N-1)-dimensional simplex, with N the number of distinct local minima of the background potential. This generalizes a result by Baesens and MacKay on twist maps near an anti-integrable limit.