Asymptotic energy profile of a wavepacket in disordered chains

TL;DR

This paper studies the long-term energy distribution of wavepackets in disordered chains, revealing power-law decay and divergence of moments, with numerical evidence supporting theoretical predictions and exploring effects of anharmonicity.

Contribution

It provides a detailed analysis of the asymptotic energy profile in disordered harmonic chains and extends the investigation to anharmonic chains through numerical simulations.

Findings

Energy profile decays as a power law with specific exponents.

Second moment of energy diverges over time despite no wavepacket spreading.

Harmonic behavior influences energy tail in anharmonic chains at intermediate disorder.

Abstract

We investigate the long time behavior of a wavepacket initially localized at a single site $n_0$ in translationally invariant harmonic and anharmonic chains with random interactions. In the harmonic case, the energy profile $ \bar{< e_n(t)>}$ averaged on time and disorder decays for large $|n-n_0|$ as a power law $\bar{< e_n(t)>}\approx C|n-n_0|^{-\eta}$ where $\eta=5/2$ and 3/2 for initial displacement and momentum excitations, respectively. The prefactor $C$ depends on the probability distribution of the harmonic coupling constants and diverges in the limit of weak disorder. As a consequence, the moments $< m_{\nu}(t)>$ of the energy distribution averaged with respect to disorder diverge in time as $t^{\beta(\nu)}$ for $\nu \geq 2$, where $\beta=\nu+1-\eta$ for $\nu>\eta-1$. Molecular dynamics simulations yield good agreement with these theoretical predictions. Therefore, in this system, the second moment of the wavepacket diverges as a function of time despite the wavepacket is not spreading. Thus, this only criteria often considered earlier as proving the spreading of a wave packet, cannot be considered as sufficient in any model. The anharmonic case is investigated numerically. It is found for intermediate disorder, that the tail of the energy profile becomes very close to those of the harmonic case. For weak and strong disorder, our results suggest that the crossover to the harmonic behavior occurs at much larger $|n-n_0|$ and larger time.