Phonon Localization in One-Dimensional Quasiperiodic Chains

TL;DR

This study investigates phonon localization in 1D quasiperiodic chains, revealing that localization can occur independently of spectral changes and depends on the specific model and excitation mode.

Contribution

It provides a numerical analysis of phonon localization in two incommensurate 1D chains, highlighting differences in excitation behavior across the analyticity-breaking transition.

Findings

Phonon localization occurs in Lennard-Jones chains on both sides of the transition.

Eigenfunctions in FK chains satisfy quasi-Bloch theorem below transition but not above.

Spectrum changes from continuous to Cantor-set-like when crossing the transition.

Abstract

Quasiperiodic long range order is intermediate between spatial periodicity and disorder, and the excitations in 1D quasiperiodic systems are believed to be transitional between extended and localized. These ideas are tested with a numerical analysis of two incommensurate 1D elastic chains: Frenkel-Kontorova (FK) and Lennard-Jones (LJ). The ground state configurations and the eigenfrequencies and eigenfunctions for harmonic excitations are determined. Aubry's "transition by breaking the analyticity" is observed in the ground state of each model, but the behavior of the excitations is qualitatively different. Phonon localization is observed for some modes in the LJ chain on both sides of the transition. The localization phenomenon apparently is decoupled from the distribution of eigenfrequencies since the spectrum changes from continuous to Cantor-set-like when the interaction parameters are varied to cross the analyticity--breaking transition. The eigenfunctions of the FK chain satisfy the "quasi-Bloch" theorem below the transition, but not above it, while only a subset of the eigenfunctions of the LJ chain satisfy the theorem.