Denjoy's anachronistic topological viewpoint on Aubry transition

TL;DR

This work casts the Aubry transition as a topological shift in the ground-state manifold, analyzed through Denjoy's circle-map theory. Ground states satisfy $x_{n+1}=F(x_n)$ where $F$ lifts a circle homeomorphism with irrational rotation number $\rho$, enabling a conjugacy to a rigid rotation in the sliding phase and a semiconjugacy to a Cantor function in the pinned phase. The transition arises from a change in the regularity of the Denjoy coordinate change $H$: injective $H$ yields a continuous hull function and dense $x_n$ mod $1$, while non-injective $H$ yields a Cantor-like structure with gaps and discontinuous hulls $x_{\pm}$. Numerical Frenkel-Kontorova results illustrate how $F$ becomes non-differentiable across the transition, consistent with Denjoy’s theory and hull-function discontinuities. The framework links analyticity breaking to topological conjugacy breaking and points to extensions to other incommensurate systems beyond the FK model.

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.

Figures (8)

• Figure 1: Periodic ground state configuration $\{x_n\}$ of the Frenkel-Kontorova model for $\rho=3/8$ and $K=1.5$ (top left). The order of the $x_n~$mod 1 variables (top right) is the same as that of the cyclic order of the rigid rotation with the same $\rho$ (bottom).
• Figure 2: The function $F$ constraining the ground states [see Eq. (\ref{['R']})] for the Frenkel-Kontorova model for $K<K_c$ (sliding phase). It converges to a single function $F$ when the sequence of rational approximants (only three are given here) converges to the irrational number $\rho=(3-\sqrt{5})/2$. It is a homeomorphism and seems to be differentiable.
• Figure 3: The same as Fig. \ref{['fig1a']} for $K>K_c$ (pinned phase). $F$ remains a homeomorphism but the broken line aspect indicates that it is no longer differentiable. It is obtained by constrained minimization.
• Figure 4: Change of variables $H$ and its representative diagram showing the equality (or semiconjugacy) between $H \circ F$ (upper path) and $R_{\rho} \circ H$ (lower path) applied to any point of $\mathbb{R}$. $F$ is a lift of a circle homeomorphism and $R_{\rho}$ an irrational translation by $\rho$, $H$ is the semiconjugacy that may not be invertible.
• Figure 5: (Top) Topological conjugacy function $H$ for an irrational $\rho$ and $K=0.5<K_c$. $H$ is a homeomorphism. (Bottom) Inverse function $H^{-1}$ corresponding to Aubry's hull function [Eq. (\ref{['sol']})] which is continuous for $K<K_c$.