Wavepacket dynamics of the nonlinear Harper model

TL;DR

This paper investigates how weak nonlinearity affects wavepacket spreading in the critical Harper model, revealing subdiffusive behavior and a recovery of anomalous diffusion at longer times, with implications for waveguides and Bose-Einstein condensates.

Contribution

It demonstrates the impact of nonlinearity on the multifractal properties and diffusion dynamics of the Harper model at criticality, providing new insights into wave propagation in nonlinear systems.

Findings

Subdiffusive growth of the second moment with exponent α

Existence of a crossover time t* depending on nonlinearity strength

Recovery of anomalous diffusion law beyond t* with altered profile

Abstract

The destruction of anomalous diffusion of the Harper model at criticality, due to weak nonlinearity $\chi$, is analyzed. It is shown that the second moment grows subdiffusively as $<m_2> \sim t^{\alpha}$ up to time $t^*\sim \chi^{\gamma}$. The exponents $\alpha$ and $\gamma$ reflect the multifractal properties of the spectra and the eigenfunctions of the linear model. For $t>t^*$, the anomalous diffusion law is recovered, although the evolving profile has a different shape than in the linear case. These results are applicable in wave propagation through nonlinear waveguide arrays and transport of Bose-Einstein condensates in optical lattices.