Emergent Nonperturbative Universal Floquet Localization

TL;DR

The paper addresses whether periodic driving can induce robust, global localization in quasiperiodic lattices irrespective of their static properties. It combines exact Floquet calculations with Floquet perturbation theory and a superasymptotic Van-Vleck analysis to identify a finely tuned amplitude-to-frequency regime where all Floquet states localize, even in the presence of dense resonances. The nonperturbative localization emerges from the interplay of drive-induced long-range modulated hopping and onsite potentials, with VVPT capturing the non-resonant sector up to an optimal truncation while rare resonances drive a breakdown of the perturbative series. The findings imply a universal Floquet localization plateau that persists across drive protocols and model parameters, offering a route to engineer localization in driven quantum systems and enabling experimental tests in ultracold atoms and photonic lattices.

Abstract

We show that a robust, nonperturbative localization plateau emerges in periodically driven quasiperiodic lattices, independent of the static localization properties and drive protocol. Using exact Floquet dynamics, Floquet perturbation theory, and optimal-order van Vleck analysis, we identify a fine-tuned amplitude-to-frequency ratio where all Floquet states become localized despite dense resonances. The van Vleck expansion achieves superasymptotic accuracy up to an optimal orde; it ultimately breaks down due to resonant hybridization at a weak quasiperiodic potential, revealing that the observed localization is nonperturbative.

Figures (8)

• Figure 1: a)$\langle IPR\rangle_g$(in color code) is plotted for all Floquet eigenstates $\chi^m$ at $\lambda=1, 2, 3$ of AA model and $(\lambda,\beta)=(-0.8,0.5)$ of GAA model for both sinusoidal and square wave drive protocols, for $L=512, N=200$. A universal localization regime is found near $(a,\omega)=(1,2)$. b) The density plot of the local probability $\langle p_{nT}(j) \rangle_g$ for same parameter values, models, and drive protocols. c)$\mathcal{O}(1)$ IPR scaling is observed at $(a,\omega)=(1,2)$ for AA at $\lambda=1,2,3$, d) The same $\mathcal{O}(1)$ scaling is seen across different driving protocols (square/sine pulses) and models (AA:$\lambda=1$; GAA: $(\lambda,\beta)=(-0.8,0.5)$). e),f) For AA: $\lambda=1$, IPR $\sim1/L$ for points lying outside the localized patch marked by yellow $\star$ in phase diagram 1(a) (Ballistic transport behavior of those regions shown in Fig.3 of supplementary section). Here, $L=500, 1000, 1500$ and $2000$, the trotter steps $N=500$, and sinusoidal drive is used for scaling plots. In IPR scaling plots, states are organized in ascending order of IPR magnitude.
• Figure 2: a) Using $2^{\rm nd}$ order VV, some eigenstates are found to be localized at $\omega=1.3$ for $a=1$, $\lambda=1.5$, while the revival of high frequency delocalized static limit is seen at $\omega=10$. b) Equivalent localized behavior can be depicted using FPT at $a=2.15,\omega=2.45$; while at low frequencies $\omega=1$ the FPT suggests retrieved delocalized phase. c),d) Heuristic extrapolation of VVPT up to an optimal order reveals enhanced localization achieved at $a=1,\omega=1.5$ universal localization region compared to $a=5,\omega=7.1$ distant regions. Here, L=1500, 2000, 3000, 4000 are used for (c)-(d) IPR scaling and sine drive is considered throughout this analysis. In all plots, states are organized in ascending order of IPR magnitude.
• Figure 3: a)-c) Rapid wave packet spreading observed for the points lying outside universal localization $(a,\omega)=(0.1,2.45),(3.5,2.45)$ and $(2.15,3.5)$ at $\lambda=1$, indicating ballistic transport with scaling exponent $\eta\approx1$ as quantified in d). Here, $L=1024$, $N=1000$ and sine drive used for the analysis. e) IPR scaling plotted for $\lambda=0.1$, demonstrating drive-induced universal localization for weak disorder. Plots f)-g) illustrate localized scaling in the form of IPR and slow wave packet evolution in the form of $\sigma(n)$ and wave packet profile using geometric mean of square and sine drive, proving the existence of universal localization. $L=500,1000,1500,2000$ is used for scaling and $L=1024$ for transport.
• Figure 4: a),b) The combined effect of $H_{F1}^{(1)}$ and $H_{F2}^{(1)}$ decreases the diagonal weight relative to off-diagonals at $(a,\omega)=(2.15,1)$, resulting in delocalization. Whereas, c),d)$(a,\omega)=(2.15,{2.45})$ shows enhanced diagonal weight leading to localization. Here, $L=500$ and sine drive is implemented.
• Figure 5: a) The count of resonance $\mathcal{R_r}$ is high and grows modearately with increasing order $2k$ at intermediate $\lambda=1,1.5$. b),c) At higher $\lambda$, resonances are fewer but stronger, reflected in the form of average and rms $\mathcal{R_r}$ strength, while $\lambda=1$ contains a consistent high $\mathcal{R_r}$ average and rms in initial orders proving the breakdown of perturbation at this size $L=4000$. Here, $a=1$ and $\omega=1.5$; sine drive is used for calculation.