Asymptotic energy profile of a wave packet in disordered chains

Abstract

We investigate the long-time behavior of a wave packet 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 $\overline{⟨{e}_{n}(t)⟩}$ averaged on time and disorder decays for large $|n\ensuremath{-}{n}_{0}|$ as a power law $\overline{⟨{e}_{n}(t)⟩}\ensuremath{\approx}C{|n\ensuremath{-}{n}_{0}|}^{\ensuremath{-}\ensuremath{\eta}}$, where $\ensuremath{\eta}=5/2$ 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 $⟨{m}_{\ensuremath{\nu}}(t)⟩$ of the energy distribution averaged with respect to disorder diverge in time as ${t}^{\ensuremath{\beta}(\ensuremath{\nu})}$ for $\ensuremath{\nu}\ensuremath{\ge}2$, where $\ensuremath{\beta}=\ensuremath{\nu}+1\ensuremath{-}\ensuremath{\eta}$ for $\ensuremath{\nu}>\ensuremath{\eta}\ensuremath{-}1$. Molecular-dynamics simulations yield good agreement with these theoretical predictions. Therefore, in this system, the second moment of the wave packet diverges as a function of time despite the wave packet is not spreading. Thus, this only criterion, 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 disorders, our results suggest that the crossover to the harmonic behavior occurs at much larger $|n\ensuremath{-}{n}_{0}|$ and larger time.