Pinning and phonon localization in Frenkel-Kontorova models on quasiperiodic substrates

Abstract

We show that the static and dynamic properties of the Frenkel-Kontorova (FK) model drastically change when an incommensurate harmonic is added to the periodic potential. Our model consists of a harmonic chain with spacing l on a quasiperiodic substrate potential of the form $V(x)=\ensuremath{\sigma}K/(2\ensuremath{\pi}{)}^{2}[1\ensuremath{-}\mathrm{cos}(2\ensuremath{\pi}x)]+(1\ensuremath{-}\ensuremath{\sigma})K/(2\ensuremath{\pi}\ensuremath{\tau}{)}^{2}[1\ensuremath{-}\mathrm{cos}(2\ensuremath{\pi}\ensuremath{\tau}x)],$ where the three relevant lengths l, 1, and ${\ensuremath{\tau}}^{\ensuremath{-}1}$ are chosen to be mutually incommensurate. Within this model we identify two classes of behavior. One presents a sliding mode up to an analyticity breaking, as in the FK model, and another is pinned for any strength of the additional harmonic. Besides, we show that in all cases if $\ensuremath{\sigma}\ensuremath{\ne}0$ or 1, localization of phonons exists beyond a critical value of the potential strength.