### 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 language | English |
---|---|

Pages (from-to) | 239-252 |

Number of pages | 14 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 104 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 1 Jan 1997 |

MoE publication type | Not Eligible |

### Fingerprint

### Keywords

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

### Cite this

*Physica D: Nonlinear Phenomena*,

*104*(3-4), 239-252. https://doi.org/10.1016/S0167-2789(97)00005-5

}

*Physica D: Nonlinear Phenomena*, vol. 104, no. 3-4, pp. 239-252. https://doi.org/10.1016/S0167-2789(97)00005-5

**"Critical" phonons of the supercritical Frenkel-Kontorova model : Renormalization bifurcation diagrams.** / Ketoja, Jukka A.; Satija, Indubala I.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - "Critical" phonons of the supercritical Frenkel-Kontorova model

T2 - Renormalization bifurcation diagrams

AU - Ketoja, Jukka A.

AU - Satija, Indubala I.

PY - 1997/1/1

Y1 - 1997/1/1

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.

KW - Cantorus

KW - Frenkel-Kontorova model

KW - Phonon

KW - Quasiperiodicity

KW - Renormalization

KW - Standard map

UR - http://www.scopus.com/inward/record.url?scp=4243927660&partnerID=8YFLogxK

U2 - 10.1016/S0167-2789(97)00005-5

DO - 10.1016/S0167-2789(97)00005-5

M3 - Article

VL - 104

SP - 239

EP - 252

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 3-4

ER -