Resonant extended states in driven quasiperiodic lattices: Aubry-Andre localization by design

We consider a quasiperiodic Aubry-Andre (AA) model and add a weak time-space-periodic perturbation. The undriven AA model is chosen to be well in the localized regime. The driving term controls the effective number of propagation channels. For a spatial resonance which reduces the reciprocal space dynamics to an effective one-dimensional two-leg ladder, the ac perturbation resonantly couples certain groups of localized eigenstates of the undriven AA model and turns them into extended ones. Slight detuning of the spatial and temporal frequencies off resonance returns these states into localized ones. We analyze the details of the resonant extended eigenstates using Floquet representations. In particular, we find that their size grows linearly with the system size. Initial wave packets overlap with resonant extended eigenstates and lead to ballistic spreading.

Figure 1

Energy spectrum $E$ vs $V_1$ of the undriven Aubry-Andre model $\left(V_2=0\right)$ with $\alpha =\sqrt{3}-1$. The shadowed region indicates the metallic regime for $V_1<2$. The arrow indicates the separation between edge and center bands for $V_1=2.5$ within the insulating regime.

Figure 2

Top: Density plot of highest participation number $HPN$ vs $\Omega$ and $\beta$. Dashed lines indicate the resonances $\Omega ={\omega }_0=2.9$ and $\Omega =2{\omega }_0=5.8$. Bottom: Same in a three-dimensional plot, with good visibility of the strong resonance at $\beta \approx 0.37$. Here $V_1=2.5,V_2=0.25,\alpha =\sqrt{3}-1$, and $N=200$.

Figure 3

Upper panel: Ordered values of $PN$ of the Floquet states vs Floquet state number $\nu$. Insets: The absolute value of the wave function vs lattice site $n$ for the state with smallest and largest $PN$. Lower panel: Same for the second moment $m_2$, with the sorting index from the upper panel. Inset: Enlargement of the region around the number 150 for a logarithmic scale. Here $V_2=V_1/10=0.25,\alpha =\sqrt{3}-1,\beta =0.37,\Omega =2.9$, and $N=200$.

Figure 4

Highest participation number $HPN$ as a function of the system size $N$ at resonance $\beta =0.37,\Omega =2.9$. Other parameters are $V_1=2.5,V_2=0.25,\alpha =\sqrt{3}-1$.

Figure 5

Density plot of the highest participation number $HPN$ vs $\Omega$ and $\beta$. Here $V_1=2.5,V_2=0.5,\alpha =\sqrt{3}-1$, and $N=200$.

Figure 6

Second moment $m_2$ of $|{\psi }_n(t={10}^3){|}^2$ as a function of $\delta$ for a single-site excitation at resonance $\beta =0.37,\Omega =2.9$. Other parameters are $V_1=2.5,V_2=0.25,\alpha =\sqrt{3}-1,N=2^8$.

Figure 7

${log}_{10}{m}_2$ vs ${log}_{10}t$. From top to bottom: $\Omega =2.9$ and $V_2=0.5$ (blue), $V_2=0.325$ (green), $V_2=0.275$ (magenta), $V_2=0.25$ (red), and $\Omega =2.6$ and $V_2=0.25$ (black, off resonance). The dash-dotted lines indicate normal diffusion $m_2\sim t$ and ballistic spreading $m_2\sim t^2$. In all cases, $\beta =0.37,V_1=2.5,\alpha =\sqrt{3}-1,\delta \approx 1.225,N=2^{15}$. Inset: Density distribution of the wave function at the final time of integration for $V_2=0.25$. Red extended curve: $\Omega =2.9$ (on resonance); black localized curve: $\Omega =2.6$ (off resonance). Other parameters as in the main figure.