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Phonon modes in the Frenkel-Kontorova chain: exponential localization and the number theory properties of frequency bands

O. V. Zhirov·G. Casati·D. L. Shepelyansky
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Abstract

We study numerically phonon modes of the classical one-dimensional Frenkel-Kontorova chain, in the regime of pinned phase characterized by the phonon gap and devil's staircase, as well as by a large number of states (configurational excitations), which energy splitting from the ground state is exponentially small. We demonstrate, these states behave like disorder media: their phonon modes are {\it exponentially} localized, in contrast to the phonon modes in the ground state, where phonons are {\it prelocalized} only. We demonstrate also, the phonon frequency spectrum of the ground state has an hierarchical structure, a direct manifestation of hierarchical spatial structure, found for the ground state of the FK chain in our recent work.

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O. V. Zhirov, G. Casati, D. L. Shepelyansky. 2005-01-10. Phonon modes in the Frenkel-Kontorova chain: exponential localization and the number theory properties of frequency bands. https://arxiv.org/abs/cond-mat/0501188

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