How periodic driving stabilizes and destabilizes Anderson localization on random trees

Motivated by the link between Anderson localization on high-dimensional graphs and many-body localization, we study the effect of periodic driving on Anderson localization on random trees. The time dependence is eliminated in favor of an extra dimension, resulting in an extended graph wherein the disorder is correlated along the new dimension. The extra dimension increases the number of paths between any two sites and allows for interference between their amplitudes. We study the localization problem within the forward scattering approximation (FSA), which we adapt to this extended graph. At low frequency, this favors delocalization as the availability of a large number of extra paths dominates. By contrast, at high frequency, it stabilizes localization compared to the static system. These lead to a regime of re-entrant localization in the phase diagram. Analyzing the statistics of path amplitudes within the FSA, we provide a detailed theoretical picture of the physical mechanisms governing the phase diagram.

Figure 1

(a) The Shirley picture for a driven tight-binding model on a tree. The different copies of the tree are indexed by $n$. The blue line shows the static path between the root site and another, say ${\alpha }_0$ and ${\alpha }_d$, whereas the different colored lines show some of the driving-induced paths between the root site and the same site's copy, $|{\alpha }_d,n\rangle$, on other Shirley layers. (b) Schematic of the FSA amplitude on sites $|{\alpha }_d,n\rangle$ starting from $|{\alpha }_0,0\rangle$. The amplitudes on the former can be both positive and negative so they can interfere to give the total amplitude ${\psi }_{{\alpha }_d}\propto {\sum }_n{\psi }_{{\alpha }_d,n}$. (c) Summary of the localization phase diagram in the $(\mathrm{\Gamma },\omega )$ plane for the model in (a). At high frequencies, driving favors localiation and enhances the critical hopping compared to the static case, ${\mathrm{\Gamma }}_c^{\mathrm{s}}$, whereas at low frequencies, it favors delocalization and ${\mathrm{\Gamma }}_c$ decreases with decreasing $\omega$.

Figure 2

Results for the localization phase diagram obtained from the FSA in the space of dimensionless parameters $\mathrm{\Gamma }/W$ and $\omega /W$. We set $W=1$ and the disordered potential is drawn from a standard Normal distribution ${\epsilon }_{\alpha }\sim \mathcal{N}(0,W=1)$. The cumulative distribution of the rescaled FSA amplitudes, $C\left({\mathrm{\Lambda }}_d\right)$ [with $|{\psi }_d{|}^2$ obtained via Eq. (7)] for values of $d=8,16,...,64$ and four different values of $\omega$ labeled by the four color maps. The crossing points of the curves indicate the $ln{(1/\mathrm{\Gamma })}_c$ for the corresponding $\omega$. The inset shows the distributions $P\left({\mathrm{\Lambda }}_d\right)$ for $\omega =0.5$ and $\omega =2$ which get sharper with increasing $d$. Results are for $K=1.05$. (b) The critical ${\mathrm{\Gamma }}_c$ as a function of $\omega$ for different values of $K$ obtained from the crossing of $C\left({\mathrm{\Lambda }}_d\right)$ exemplified in (a) and obtained using Eq. (8). Note the nonmonotonicity in ${\mathrm{\Gamma }}_c$ as a function of $\omega$. Statistics for all data were obtained over $5\times {10}^6$ realizations.

Figure 3

Schematic picture for the Shirley resonances in the low-frequency regime, $\omega \ll W$. In the plots above ${\omega }_1>{\omega }_2$. The resonances denoted by the red dots, where the effective site energies $|{\mathrm{\Delta }}_i+n_{{\alpha }_i}^{*}\omega |$ are within $\omega$ typically stay within a strip (shaded in orange) in the Shirley direction of width $W/\omega$. The larger $\omega /W$, the greater the likelihood of a resonance occurring per unit length. A path $q$ is resonant if the $n_{\alpha }^{\left(p\right)}$ passes within a distance 1 of these resonances, as indicated by the blue-shaded “swept area” of each path.