Aubry transition studied by direct evaluation of the modulation functions of infinite incommensurate systems

¹ Department of Chemical Engineering, Universiteit van Amsterdam Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands
² Institute for Theoretical Physics, NSRIM, University of Nijmegen Toernooiveld 1, 6525 ED Nijmegen, The Netherlands

Corresponding author: tsvanerp@science.uva.nl

Received: 14 January 2002
Accepted: 24 April 2002

Abstract

Incommensurate structures can be described by the Frenkel-Kontorova model. Aubry has shown that, at a critical value K_c of the coupling of the harmonic chain to an incommensurate periodic potential, the system displays the analyticity-breaking transition between a sliding and pinned state. The ground-state equations coincide with the standard map in non-linear dynamics, with smooth or chaotic orbits below and above K_c, respectively. For the standard map, Greene and MacKay have calculated the value . Conversely, evaluations based on the analyticity breaking of the modulation function have been performed for high commensurate approximants. Here we show how the modulation function of the infinite system can be calculated without using approximants but by Taylor expansions of increasing order. This approach leads to a value , implying the existence of a golden invariant circle up to .