Denjoy's anachronistic topological viewpoint on Aubry transition

TL;DR

This paper reinterprets the Aubry transition using Denjoy's topological framework, linking it to circle homeomorphisms and cyclic order, and illustrates this with numerical analysis of the Frenkel-Kontorova model.

Contribution

It introduces a topological perspective on the Aubry transition based on Denjoy's theory, offering a novel conceptual understanding of the transition.

Findings

Identifies two types of cyclic order related to the Aubry transition.

Rephrases the breaking of analyticity as a breaking of topological conjugacy.

Numerical calculations support the topological interpretation.

Abstract

The Aubry transition is a phase transition between two types of incommensurate states, originally described as a transition by ``breaking of analyticity''. Here we present Denjoy's (anachronistic) viewpoint, who almost hundred years ago described certain mathematical properties of circle homeomorphisms with irrational rotation numbers. The connection between the two lies in the existence of a change of variables from the incommensurate ground state variables to new simple phase variables that rotate by a constant irrational angle. This confers a cyclic order, an essential property of models with the Aubry transition. Denjoy's description indicates that there are two types of cyclic order, distinguished by the regular or singular nature of the change of variables or, in mathematical terms, by the distinction between topological conjugacy versus semiconjugacy. This allows rephrasing the breaking of analyticity as a breaking of topological conjugacy. We illustrate this description with numerical calculations on the Frenkel-Kontorova model.