Floquet mobility edges and transport in a periodically driven generalized Aubry-André model

Abstract: We investigate the effect of a periodic electric field drive on the generalized Aubry-André model, also known as the Ganeshan-Pixley-Das Sarma (GPD) model, which is well known as a host of mobility edges. Our study of the Floquet spectrum of the driven GPD model uncovers the emergence of two distinct Floquet mobility edges, a delocalized--localized (DL) edge in the bounded regime, and a multifractal--localized (ML) edge in the unbounded regime. Using analytical results derived from Avila's global theory applied to the high frequency effective Hamiltonian, together with numerical diagnostics such as the fractal dimension and inverse participation ratio, we demonstrate that these mobility edges can be effectively controlled by the amplitude and frequency of the electric field drive. We also identify drive-induced localization at specific values of the driving parameters, corresponding to dynamical localization points in the absence of quasiperiodic potential. Furthermore, the dynamical study of the periodically driven GPD model demonstrates superdiffusive to almost ballistic transport in the bounded regime corresponding to the DL edges, whereas subdiffusive transport is observed in the unbounded regime associated with the ML edges. We also analyze deviations from the high-frequency effective description by explicitly examining the low-frequency driving regime, where significant and counterintuitive deviations in both spectral properties and transport behavior are observed. Our study highlights the interplay of a quasiperiodic potential and a periodically varying electric field drive as a powerful mechanism to engineer mobility edges and control transport in systems with rich spectral features.

• The paper introduces analytically determined Floquet mobility edges in a driven generalized Aubry-André model using Avila’s global theory.
• It applies high-frequency Floquet-Magnus expansion and numerical diagnostics, such as fractal dimensions and IPR, to map localization and multifractality.
• The study reveals that external drive parameters can toggle transport between superdiffusive, subdiffusive, and localized regimes.

Floquet Mobility Edges and Quantum Transport in a Periodically Driven Generalized Aubry-André Model

Introduction

This work analyzes localization and transport in the periodically driven Ganeshan-Pixley-Das Sarma (GPD) model—a generalization of the Aubry-André-Harper (AAH) Hamiltonian that supports cooperatively extended, localized, and multifractal phases. Specifically, the study applies high-frequency Floquet theory and Avila’s global theory of quasiperiodic Schrödinger operators to reveal the conditions for Floquet mobility edges in the presence of a time-periodic electric field. The investigation develops a comprehensive phase diagram for the driven GPD model, quantifies the spectral and dynamical signatures of mobility edges, and characterizes the rich interplay between periodic driving, quasiperiodic potential, and multifractality.

Theoretical Framework: Driven GPD Model and Floquet Analysis

The GPD model’s Hamiltonian is defined on a one-dimensional chain with incommensurate onsite potential parameterized by $\beta$, and subject to a monochromatic field of amplitude $K$ and frequency $\omega$. Without driving ($K=0$) and for $|\beta|<1$, the model possesses a delocalized–localized (DL) mobility edge; for $|\beta|\geq 1$, a multifractal–localized (ML) edge. The inclusion of periodic driving is treated via a gauge transformation to the rotating frame, with the Floquet-Magnus expansion applied in the high-frequency limit. This leads to a renormalized effective hopping $J_{\rm eff}=J J_0(\mathcal{K})$ ($\mathcal{K}=K/\omega$), where $J_0$ is a Bessel function. The analytical condition for the Floquet mobility edge is then derived, and is fully tunable via $K$ and $K$0.

Static and Floquet Mobility Edge Structure

The spectral landscape of the undriven GPD model demonstrates coexistence of extended and localized states separated by sharp mobility edges, distinguished by the fractal dimension $K$1.

Figure 1: $K$2 for eigenstates of the undriven GPD model as a function of energy $K$3 and $K$4, contrasting DL ($K$5) and ML ($K$6) edges.

Periodic driving preserves the essential dual mobility edge properties (DL/ML) but introduces continuous tunability through $K$7. At the zeros of $K$8, $K$9 vanishes and the spectrum becomes fully localized. The analytical mobility edge conditions obtained from the high-frequency effective Hamiltonian show quantitative agreement with numerics, and the location of mobility edges oscillates periodically with $\omega$0.

Figure 2: Floquet eigenstate $\omega$1 for $\omega$2 as a function of quasienergy and $\omega$3, showing tunable DL and ML edge evolution and drive-induced localization at Bessel zeros.

Numerical Diagnostics: Multifractality and Finite-Size Effects

The fractal dimension $\omega$4, inverse participation ratio $\omega$5, and the spatial standard deviation $\omega$6 are computed for the Floquet eigenstates. For $\omega$7 (DL edge), clear separation exists between extended and localized regimes, and all states are localized at the Bessel zeros.

Figure 3: $\omega$8 for $\omega$9 under various $K=0$0 and system sizes, highlighting finite-size scaling and localization at Bessel zeros.

In the multifractal regime ($K=0$1), $K=0$2 shows states with $K=0$3, and a significant portion of eigenstates within the multifractal band show slow decay with system size—demonstrating the non-ergodic extended character.

Figure 4: $K=0$4 for $K=0$5 as a function of system size and $K=0$6, indicating multifractality and enhanced localization at the Bessel zeros.

The standard deviation $K=0$7 of eigenstates further isolates extended, localized, and multifractal states, correlating with the analytically predicted edge positions.

Figure 5: $K=0$8 for $K=0$9; extended states dominate except for pronounced localization at Bessel zeros and spectral edges.

Figure 6: $|\beta|<1$0 for $|\beta|<1$1; strong state-to-state variance characteristic of multifractal eigenstates.

Impact of Drive Frequency and Higher-Order Corrections

Deviations from high-frequency Floquet behavior, introduced by lower $|\beta|<1$2, are treated via the second-order corrections in the Floquet-Magnus expansion. The next-nearest-neighbor hopping and renormalized onsite potential generated at this order act in competition: NNN hopping enhances delocalization, while onsite potential corrections increase localization—especially within the multifractal regime at low frequencies. This interplay leads to the counterintuitive result that decreasing drive frequency can enhance localization within multifractal bands, contrary to expectations from generic Floquet heating phenomena.

Quantum Transport and Entanglement Dynamics

Transport is probed through the root mean squared displacement (RMSD) $|\beta|<1$3, entanglement entropy $|\beta|<1$4, and return probability $|\beta|<1$5. For $|\beta|<1$6 (DL), transport is superdiffusive to near-ballistic except at the drive-induced localization points, where spreading collapses.

Figure 7: RMSD $|\beta|<1$7 for varying $|\beta|<1$8, $|\beta|<1$9, and $|\beta|\geq 1$0; drive-tunable transitions between ballistic, subdiffusive, and localized regimes.

For $|\beta|\geq 1$1 (ML), transport is predominantly subdiffusive, with suppression further pronounced at lower drive frequencies and at Bessel zeros. Entanglement entropy exhibits analogous scaling and suppression patterns.

Figure 8: Entanglement entropy $|\beta|\geq 1$2 for various parameter sets, illustrating correspondence between transport and entanglement growth rates.

Phase Diagram and Parameter Dependence

Systematic analysis of the $|\beta|\geq 1$3 plane reveals that increasing potential strength or proximity to Bessel zeros favors localization, as measured by the average fractal dimension $|\beta|\geq 1$4.

Figure 9: $|\beta|\geq 1$5 maps for (a) DL and (b) ML regimes as functions of $|\beta|\geq 1$6 and $|\beta|\geq 1$7, showing tunable transitions by the drive.

Return probabilities corroborate the spectral and dynamical localization observed.

Figure 10: Return probability $|\beta|\geq 1$8 dynamics, demonstrating long-term memory retention in localized regimes.

Conclusions

This paper rigorously demonstrates that the periodically driven GPD model supports robust, tunable Floquet mobility edges, with delocalized–localized or multifractal–localized separation precisely manipulated via external drive amplitude and frequency. The authors provide analytic expressions for Floquet mobility edges using Avila’s global theory, validate them numerically across spectrally and dynamically distinct regimes, and reveal that the conventional expectation of Floquet-induced delocalization can break down in multifractal bands at low frequencies. The results establish the driven GPD model as a versatile setting for quantum control of localization, multifractality, and anomalous transport.

Experimental realization is feasible in optical lattice or photonic quasiperiodic systems where drive amplitude and frequency are accessible knobs. Theoretical implications extend to questions on the stability of mobility edges under interactions and the thermalization dynamics of driven many-body systems, opening avenues for further exploration in Floquet engineering of strongly correlated quantum matter.