"Critical" phonons of the supercritical Frenkel-Kontorova model

Renormalization bifurcation diagrams

Jukka A. Ketoja, Indubala I. Satija

Research output: Contribution to journalArticleScientificpeer-review

6 Citations (Scopus)

Abstract

The phonon modes of the Frenkel-Kontorova model are studied both at the pinning transition as well as in the pinned (cantorus) phase. We focus on the minimal frequency of the phonon spectrum and the corresponding generalized eigenfunction. Using an exact decimation scheme, the eigenfunctions are shown to have non-trivial scaling properties not only at the pinning transition point but also in the cantorus regime. Therefore the phonons defy localization and remain critical even where the associated area-preserving map has a positive Lyapunov exponent. In this region, the critical scaling properties vary continuously and are described by a line of renormalization limit cycles. Interesting renormalization bifurcation diagrams are obtained by monitoring the cycles as the parameters of the system are varied from an integrable case to the anti-integrable limit. Both of these limits are described by a trivial decimation fixed point. Very surprisingly we find additional special parameter values in the cantorus regime where the renormalization limit cycle degenerates into the above trivial fixed point. At these "degeneracy points" the phonon hull is represented by an infinite series of step functions. This novel behavior persists in the extended version of the model containing two harmonics. Additional richnesses of this extended model are the one to two-hole transition line, characterized by a divergence in the renormalization cycles, non-exponentially localized phonons, and the preservation of critical behavior all the way upto the anti-integrable limit.

Original languageEnglish
Pages (from-to)239-252
Number of pages14
JournalPhysica D: Nonlinear Phenomena
Volume104
Issue number3-4
DOIs
Publication statusPublished - 1 Jan 1997
MoE publication typeNot Eligible

Fingerprint

Frenkel-Kontorova Model
Phonons
Bifurcation Diagram
Renormalization
phonons
Phonon
diagrams
Eigenvalues and eigenfunctions
Decimation
cycles
Limit Cycle
Eigenfunctions
Trivial
Fixed point
Scaling
eigenvectors
Cycle
Step function
Line
Infinite series

Keywords

  • Cantorus
  • Frenkel-Kontorova model
  • Phonon
  • Quasiperiodicity
  • Renormalization
  • Standard map

Cite this

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title = "{"}Critical{"} phonons of the supercritical Frenkel-Kontorova model: Renormalization bifurcation diagrams",
abstract = "The phonon modes of the Frenkel-Kontorova model are studied both at the pinning transition as well as in the pinned (cantorus) phase. We focus on the minimal frequency of the phonon spectrum and the corresponding generalized eigenfunction. Using an exact decimation scheme, the eigenfunctions are shown to have non-trivial scaling properties not only at the pinning transition point but also in the cantorus regime. Therefore the phonons defy localization and remain critical even where the associated area-preserving map has a positive Lyapunov exponent. In this region, the critical scaling properties vary continuously and are described by a line of renormalization limit cycles. Interesting renormalization bifurcation diagrams are obtained by monitoring the cycles as the parameters of the system are varied from an integrable case to the anti-integrable limit. Both of these limits are described by a trivial decimation fixed point. Very surprisingly we find additional special parameter values in the cantorus regime where the renormalization limit cycle degenerates into the above trivial fixed point. At these {"}degeneracy points{"} the phonon hull is represented by an infinite series of step functions. This novel behavior persists in the extended version of the model containing two harmonics. Additional richnesses of this extended model are the one to two-hole transition line, characterized by a divergence in the renormalization cycles, non-exponentially localized phonons, and the preservation of critical behavior all the way upto the anti-integrable limit.",
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"Critical" phonons of the supercritical Frenkel-Kontorova model : Renormalization bifurcation diagrams. / Ketoja, Jukka A.; Satija, Indubala I.

In: Physica D: Nonlinear Phenomena, Vol. 104, No. 3-4, 01.01.1997, p. 239-252.

Research output: Contribution to journalArticleScientificpeer-review

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N2 - The phonon modes of the Frenkel-Kontorova model are studied both at the pinning transition as well as in the pinned (cantorus) phase. We focus on the minimal frequency of the phonon spectrum and the corresponding generalized eigenfunction. Using an exact decimation scheme, the eigenfunctions are shown to have non-trivial scaling properties not only at the pinning transition point but also in the cantorus regime. Therefore the phonons defy localization and remain critical even where the associated area-preserving map has a positive Lyapunov exponent. In this region, the critical scaling properties vary continuously and are described by a line of renormalization limit cycles. Interesting renormalization bifurcation diagrams are obtained by monitoring the cycles as the parameters of the system are varied from an integrable case to the anti-integrable limit. Both of these limits are described by a trivial decimation fixed point. Very surprisingly we find additional special parameter values in the cantorus regime where the renormalization limit cycle degenerates into the above trivial fixed point. At these "degeneracy points" the phonon hull is represented by an infinite series of step functions. This novel behavior persists in the extended version of the model containing two harmonics. Additional richnesses of this extended model are the one to two-hole transition line, characterized by a divergence in the renormalization cycles, non-exponentially localized phonons, and the preservation of critical behavior all the way upto the anti-integrable limit.

AB - The phonon modes of the Frenkel-Kontorova model are studied both at the pinning transition as well as in the pinned (cantorus) phase. We focus on the minimal frequency of the phonon spectrum and the corresponding generalized eigenfunction. Using an exact decimation scheme, the eigenfunctions are shown to have non-trivial scaling properties not only at the pinning transition point but also in the cantorus regime. Therefore the phonons defy localization and remain critical even where the associated area-preserving map has a positive Lyapunov exponent. In this region, the critical scaling properties vary continuously and are described by a line of renormalization limit cycles. Interesting renormalization bifurcation diagrams are obtained by monitoring the cycles as the parameters of the system are varied from an integrable case to the anti-integrable limit. Both of these limits are described by a trivial decimation fixed point. Very surprisingly we find additional special parameter values in the cantorus regime where the renormalization limit cycle degenerates into the above trivial fixed point. At these "degeneracy points" the phonon hull is represented by an infinite series of step functions. This novel behavior persists in the extended version of the model containing two harmonics. Additional richnesses of this extended model are the one to two-hole transition line, characterized by a divergence in the renormalization cycles, non-exponentially localized phonons, and the preservation of critical behavior all the way upto the anti-integrable limit.

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