Wavepacket dynamics of the nonlinear Harper model

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.

Figure 1

(Color online) The variance ${⟨m_2\left(t\right)⟩}_{\varphi }$ for various $\chi$ values of the NLHM. The $\chi ={10}^{-4}$ case is shifted downward in order to distinguish it from the $\chi =0$ case. Dashed lines have slopes as indicated in the figure and are drawn to guide the eyes. Inset: the decay of $P\left(t\right)\sim 1∕t^{D_2^{\mu }}$ for the linear Harper model $(\chi =0)$.

Figure 2

The fitting values (stars) of the power-law exponent $\alpha$ vs $V$ for the Fibonacci model. Circles are the fitting exponent $2\beta$ of the linear model for the spreading $⟨m_2\left(t\right)⟩\sim t^{2\beta }$, while diamonds are the extracted exponents $2\beta -D_2^{\mu }$, where $D_2^{\mu }$ was obtained from the decay of $P\left(t\right)\sim 1∕t^{D_2^{\mu }}$ of the linear model.

Figure 3

(Color online) The variance vs scaled time. $t^{*}$ is extracted by scaling the time axis so that the variance curves overlap in the time region where Eq. (1) is valid. Inset: the scaling of $t^{*}$ vs $\chi$. The dashed line indicates the least-squares fit with slope 3.96 [the theoretical prediction, Eq. (2), estimates it to be $3.45\pm 0.35$].

Figure 4

(Color online) The scaled probability distributions $P_s\left(x\right)$ vs $x$ of the NLHM at different times $t$ (including $t>t^{*}$) and for various $\chi >{\chi }^{*}$. The nice overlap of the curves at the core $\mid x\mid ⩽8$ confirms the validity of the scaling law, Eq. (3). The scaling is lost for $\mid x\mid >8$. The lower inset shows the behavior near the origin. The solid line is the fitting curve, Eq. (16). For comparison, we also report in the upper inset the probability distribution $P_s\left(x\right)$ of the linear Harper model.

Figure 5

The integrated survival probability ${⟨P\left(T\right)⟩}_T$ vs $\chi$. For $\chi ⩾5.5\times {10}^{-4}$, we observe an increase in ${⟨P\left(T\right)⟩}_T$. The first point in the curve corresponds to the case $\chi =0$ and is included in the log-log plot as a reference point. Upper inset shows the scaling of the $P_2$ and $-S$ vs the system size $q_i$. The horizontal dashed line is $S=-2.2\times {10}^{-4}$.