Drive-induced delocalization in the Aubry-André model

Motivated by the recent experiment by Bordia et al. [Nat. Phys. 13, 460 (2017)], we study the single particle delocalization phenomena of the Aubry-André (AA) model subjected to periodic drives. In two distinct cases we construct an equivalent classical description to illustrate that the drive-induced delocalization phenomena stems from an instability and the onset of chaos in the underlying dynamics. In the first case we analyze the delocalization and the thermalization in a time modulated AA potential with respect to driving frequency and demonstrate that there exists a threshold value of the amplitude of the drive. In the next example, we show that the periodic modulation of the phase of the hopping amplitude induced by a gauge field leads to an unusual effect on delocalization with a nonmonotonic dependence on the driving frequency. Within a window of such a driving frequency a delocalized Floquet band with a mobility edge appears, exhibiting multifractality in the spectrum as well as in the Floquet eigenfunctions. Finally, we explore the effect of interaction and discuss how the results of the present analysis can be tested experimentally.

Figure 1

Stroboscopic phase-space portraits with (a)–(c) increasing values of $\epsilon =0,0.033,0.33$ for $T=10$ and (d)–(f) for increasing values of $T=1,10,50$ for $\epsilon =0.1$. Here and in all other figures we set $\lambda =3$. The different colors indicate different initial conditions.

Figure 2

(a) $\langle r\rangle$ as a function of $T$ for different values of the driving strength $\epsilon$. (b) A phase diagram on the $T\text{−}\epsilon$ plane where the color bar indicates $S_{\mathrm{en}}$. The delocalization phase boundary asymptotically converges to ${\epsilon }_c=1/3$.

Figure 3

(a)–(c) Stroboscopic phase-space portraits for increasing values of $T=0.1,1.0,10$. Here and in the rest of the plots we set $\mathrm{\Delta }=1$.

Figure 4

(a) Stability of $\overline{R}$ on the $T\text{−}(\lambda /{\lambda }_c)$ plane, and the red line corresponds to $\lambda =3$. The color plots of (b) IPR and (c) SPEE for different Floquet states. (d)–(f) IPR and SPEE of all eigenmodes for different values of $T$.

Figure 5

(a) Scaling of $\mathrm{\Delta }N$ with $\mathrm{\Delta }\varphi$ with different exponents for band center ($\varphi \sim 0$) and band edge ($\varphi \sim 0.35$). (b) ${\tau }_q$ vs $q$ for different eigenphases at $T=3.4$.

Figure 6

(a) Entanglement entropy calculated from the many body eigenstates of the Floquet operator as a function of $T$. (b) $\langle r\rangle$ is plotted as a function of $T$. The spacing distributions of the eigenphases are shown for different values of $T$ (indicated by the arrowhead) in the inset. The other parameters are $\lambda =3,V=0.1$, and $\mathrm{\Delta }=1$ for system size $L=14$.