We investigate the electric-field driven power-law random banded matrix (PLRBM) model where a variation in the power-law exponent $\ensuremath{\alpha}$ yields a delocalization-to-localization phase transition. We examine the periodically driven PLRBM model with the help of the Floquet operator. The level spacing ratio and the generalized participation ratio of the Floquet Hamiltonian reveal a drive-induced weak multifractal (fractal) phase accompanied by diffusive (subdiffusive) transport on the delocalized side of the undriven PLRBM model. On the localized side, the time-periodic model remains localized---the average level-spacing ratio corresponds to Poisson statistics and logarithmic transport is observed in the dynamics. Extending our analysis to the aperiodic Thue-Morse driven system, we find that the aperiodically driven clean long-range hopping model (clean counterpart of the PLRBM model) exhibits the phenomenon of exact dynamical localization on tuning the drive parameters at special points. The disordered time-aperiodic system shows diffusive transport followed by relaxation to the infinite-temperature state on the delocalized side, and a prethermal plateau with subdiffusion on the localized side. Additionally, we compare this with a quasiperiodically driven Aubry-Andr\'e-Harper model that also undergoes a localization-delocalization transition. Unlike the disordered long-range model, it features a prolonged prethermal plateau followed by subdiffusion to the infinite temperature state, even on the delocalized side.
We investigate the electric-field driven power-law random banded matrix (PLRBM) model where a variation in the power-law exponent \ensuremathα yields a delocalization-to-localization phase transition. We examine the periodically driven PLRBM model with the help of the Floquet operator. The level spacing ratio and the generalized participation ratio of the Floquet Hamiltonian reveal a drive-induced weak multifractal (fractal) phase accompanied by diffusive (subdiffusive) transport on the delocalized side of the undriven PLRBM model. On the localized side, the time-periodic model remains localized---the average level-spacing ratio corresponds to Poisson statistics and logarithmic transport is observed in the dynamics. Extending our analysis to the aperiodic Thue-Morse driven system, we find that the aperiodically driven clean long-range hopping model (clean counterpart of the PLRBM model) exhibits the phenomenon of exact dynamical localization on tuning the drive parameters at special points. The disordered time-aperiodic system shows diffusive transport followed by relaxation to the infinite-temperature state on the delocalized side, and a prethermal plateau with subdiffusion on the localized side. Additionally, we compare this with a quasiperiodically driven Aubry-Andr'e-Harper model that also undergoes a localization-delocalization transition. Unlike the disordered long-range model, it features a prolonged prethermal plateau followed by subdiffusion to the infinite temperature state, even on the delocalized side.
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.
In this study, we examine whether periodic driving of a model with a quasiperiodic potential can generate interesting Floquet phases that have no counterparts in the static model. Specifically, we consider the Aubry-Andr\'e model, which is a one-dimensional time-independent model with an on-site quasiperiodic potential V0 and a nearest-neighbor hopping amplitude that is taken to have a staggered form. We add a uniform hopping amplitude that varies in time either sinusoidally or as a square pulse with a frequency $\ensuremath{\omega}$. Unlike the static Aubry-Andr\'e model, which has a simple phase diagram with only two phases (only extended or only localized states), we find that the driven model has four possible phases for the Floquet eigenstates: a phase with gapless quasienergy bands and only extended states, a phase with multiple mobility gaps separating different quasienergy bands, a mixed phase with coexisting extended, multifractal, and localized states, and a phase with only localized states. The multifractal states have generalized inverse participation ratios that scale with the system size with exponents that are different from the values for both extended and localized states. In addition, we observe intricate reentrant transitions between the different kinds of states when $\ensuremath{\omega}$ and V0 are varied. The appearance of such transitions is confirmed by the behavior of the Shannon entropy. Many of our numerical results can be understood from an analytic Floquet Hamiltonian derived using a Floquet perturbation theory that uses the inverse of the driving amplitude as the perturbation parameter. In the limit of high frequency and large driving amplitude, we find that the Floquet quasienergies match the energies of the undriven system, but the Floquet eigenstates are much more extended. We also study the spreading of a one-particle wave packet, and we find that it is always ballistic but the ballistic velocity varies significantly with the system parameters, sometimes showing a nonmonotonic dependence on V0 that does not occur in the static model. Finally, we compare the results for the driven model, which has a static staggered hopping amplitude, with a model that has a static uniform hopping amplitude, and we find some significant differences between the two cases. All of our results are found to be independent of the driving protocol, either sinusoidal or square pulse. We conclude that the interplay of the quasiperiodic potential and driving produces a rich phase diagram that does not appear in the static model.
We study the localization-delocalization transition of Floquet eigenstates in a driven fermionic chain with an incommensurate Aubry-Andr\'e potential and a hopping amplitude which is varied periodically in time. Our analysis shows the presence of a mobility edge separating single-particle delocalized states from localized and multifractal states in the Floquet spectrum. Such a mobility edge does not have any counterpart in the static Aubry-Andr\'e model and exists for a range of drive frequencies near the critical frequency at which the transition occurs. The presence of the mobility edge is shown to leave a distinct imprint on fermion transport in the driven chain; it also influences the Shannon entropy and the survival probability of the fermions at long times. In addition, we find the presence of CAT (linear superposition of localized single particle states) states in the Floquet spectrum with weights centered around a few nearby sites of the chain. This is shown to be tied to the flattening of Floquet bands over a range of quasienergies. We support our numerical studies with a semianalytic expression for the Floquet Hamiltonian (HF) computed within a Floquet perturbation theory. The eigenspectra of the perturbative HF so obtained exhibit qualitatively identical properties to the exact eigenstates of HF obtained numerically. Our results thus constitute an analytic expression of a HF whose spectrum supports multifractal and cat states. We suggest experiments which can test our theory.
We analyze the anisotropic Dicke model in the presence of a periodic drive and under a quasiperiodic drive. The study of drive-induced phenomena in this experimentally accessible model is important since, although it is simpler than full-fledged many-body quantum systems, it is still rich enough to exhibit many interesting features. We show that under a quasiperiodic Fibonacci (Thue-Morse) drive, the system features a prethermal plateau that increases as an exponential (stretched exponential) with the driving frequency before heating to an infinite-temperature state. In contrast, when the model is periodically driven, the dynamics reaches a plateau that is not followed by heating. In either case, the plateau value depends on the energy of the initial state and on the parameters of the undriven Hamiltonian. Surprisingly, this value does not always approach the infinite-temperature state monotonically as the frequency of the periodic drive decreases. We also show how the drive modifies the quantum critical point and discuss open questions associated with the analysis of level statistics at intermediate frequencies.
We study the many-body localization (MBL) transition of Floquet eigenstates in a driven, interacting fermionic chain with an incommensurate Aubry-Andr\'e potential and a time-periodic hopping amplitude as a function of the drive frequency ${\ensuremath{\omega}}_{D}$ using exact diagonalization (ED). We find that the nature of the Floquet eigenstates change from ergodic to Floquet-MBL with increasing frequency; moreover, for a significant range of intermediate ${\ensuremath{\omega}}_{D}$, the Floquet eigenstates exhibit nontrivial fractal dimensions. We find a possible transition from the ergodic to this multifractal phase followed by a gradual crossover to the MBL phase as the drive frequency is increased. We also study the fermion autocorrelation function, entanglement entropy, normalized participation ratio (NPR), fermion transport, and the inverse participation ratio (IPR) as a function of ${\ensuremath{\omega}}_{D}$. We show that the autocorrelation, fermion transport, and NPR display qualitatively different characteristics (compared to their behavior in the ergodic and MBL regions) for the range of ${\ensuremath{\omega}}_{D}$ which supports multifractal eigenstates. In contrast, the entanglement growth in this frequency range tend to have similar features as in the MBL regime; its rate of growth is controlled by ${\ensuremath{\omega}}_{D}$. Our analysis thus indicates that the multifractal nature of Floquet-MBL eigenstates can be detected by studying autocorrelation function and fermionic transport of these driven chains. We support our numerical results with a semianalytic expression of the Floquet Hamiltonian obtained using Floquet perturbation theory (FPT) and discuss possible experiments which can test our predictions.
Long-range hoppings in quantum disordered systems are known to yield quantum multifractality, the features of which can go beyond the characteristic properties associated with an Anderson transition. Indeed, critical dynamics of long-range quantum systems can exhibit anomalous dynamical behaviors distinct from those at the Anderson transition in finite dimensions. In this paper, we propose a phenomenological model of wave packet expansion in long-range hopping systems. We consider both their multifractal properties and the algebraic fat tails induced by the long-range hoppings. Using this model, we analytically derive the dynamics of moments and inverse participation ratios of the time-evolving wave packets, in connection with the multifractal dimension of the system. To validate our predictions, we perform numerical simulations of a Floquet model that is analogous to the power law random banded matrix ensemble. Unlike the Anderson transition in finite dimensions, the dynamics of such systems cannot be adequately described by a single parameter scaling law that solely depends on time. Instead, it becomes crucial to establish scaling laws involving both the finite size and the time. Explicit scaling laws for the observables under consideration are presented. Our findings are of considerable interest towards applications in the fields of many-body localization and Anderson localization on random graphs, where long-range effects arise due to the inherent topology of the Hilbert space.