Floquet Bands Research Articles

Floquet band engineering with Bloch oscillations

This work provides a convenient and powerful means towards the engineering of Floquet bands via Bloch oscillations, by adding a tilted linear potential to periodically driven lattice systems. The added linear field not only restricts the spreading of a time-evolving wavepacket but also, depending on the ratio between the Bloch oscillation frequency and the modulation frequency of the periodic driving, dramatically modifies the band profile and topology. Specifically, we consider a driven Aubry-Andr\'e-Harper model as a working example, in the presence of a linear field. Almost flat Floquet bands or Floquet bands with large Chern numbers due to the interplay between the periodic driving and Bloch oscillations can be obtained, with the band structure and topology extensively tunable by adjusting the ratio of two competing frequencies. To confirm our finding, we further execute the Thouless pumping of one and two interacting bosons in such a lattice system and establish its connection with the topological properties of single- and two-particle Floquet bands.

All-Optically Controlled Topological Current Switcher Based on Silicene-Like Nanoribbons

Low-dimensional topological insulator materials, such as silicene-like 2-D materials, have wide applications in electronic, spintronic, and optoelectronic nanodevice, by utilizing their topologically protected edge states. In theoretical studies, Floquet theorem is usually employed in the nanosystems under light irradiation with high-frequency approximation, which may not be accessible in experimental research. In this work, we study the quasienergy spectra and edge-state transport of silicene-like nanoribbons under low-frequency light irradiation (in visible wavelengths), by extracting the zero-photon component from the nonperturbative Floquet bands. These silicene-like nanoribbons undergo topological phase transition from quantum spin Hall (QSH) to quantum anomalous Hall (QAH) insulator under circularly polarized light, accompanied by helical edge modes to chiral ones. The propagation direction of edge-mode current in QAH nanoribbons can be switched by left-/right-circularly polarized (LCP/RCP) light, and such a current-switching effect is then confirmed in heterojunction composed of QSH and QAH nanoribbons. Based on the all-optically controlled current-switching effect, we design a three-terminal current switcher, with one input terminal, two output ones, and central device irradiated by LCP/RCP light. Conductance and real-space local current distribution of the proposed nanosystem reveal that the input current can indeed be switched into specific output terminal by LCP/RCP light. The proposed current switcher changes the electron movement very quickly, and we believe that it would have potential applications in high-speed photoelectric nanodevices.

Topological invariants of Floquet topological phases under periodical driving

Owing to the replication of Floquet bands and the presence of additional gaps in the quasienergy dimension, the topological phases in a periodically driven system cannot be fully characterized by the conventional topological invariants used in static systems. In particular, an anomalous strong topological phase can be a host in driven systems, featured by nontrivial counterpropagating edge modes which cannot be characterized by the bulk band structure. In this paper, we propose a scheme to obtain a complete characterization of Floquet topological phases using only information about bulk dispersions under the condition that both the location and chirality of Floquet band-touching points are not changed by the periodical driving. A set of topological invariants associated with the band-touching points are formulated to establish a one-to-one correspondence to the number of edge modes. Finally, we discuss the experimental realization and detection scheme using cold atomic gases.

Floquet states and optical conductivity of an irradiated two-dimensional topological insulator

We study the topology of the Floquet states and time-averaged optical conductivity of the lattice model of a thin topological insulator subject to a circularly polarized light using the extended Kubo formalism. Two driving regimes, the off-resonant and on-resonant, and two models for the occupation of the Floquet states, the ideal and mean-energy occupation, are considered. In the ideal occupation the real part of DC optical Hall conductivity is shown to be quantized while it is not quantized for the mean-energy distribution. The optical transitions in the Floquet band structure depend strongly on the occupation and also the optical weight which consequently affect all components of optical conductivity. At high frequency regime we present an analytical calculation of the effective Hamiltonian and also its phase diagram which depends on the tunneling energy between two surfaces. The topology of the system shows rich phases when it is irradiated by a weak on-resonant drive giving rise to emergence of anomalous edge states.

Floquet engineering the band structure of materials with optimal control theory

We demonstrate that the electronic structure of a material can be deformed into Floquet pseudobands with arbitrarily tailored shapes. We achieve this goal with a combination of quantum optimal control theory and Floquet engineering. The power and versatility of this framework is demonstrated here by utilizing the independent-electron tight-binding description of the $\ensuremath{\pi}$ electronic system of graphene. We show several prototype examples focusing on the region around the $K$ (Dirac) point of the Brillouin zone: creation of a gap with opposing flat valence and conduction bands, creation of a gap with opposing concave symmetric valence and conduction bands (which would correspond to a material with an effective negative electron-hole mass), and closure of the gap when departing from a modified graphene model with a nonzero field-free gap. We employ time-periodic drives with several frequency components and polarizations, in contrast to the usual monochromatic fields, and use control theory to find the amplitudes of each component that optimize the shape of the bands as desired. In addition, we use quantum control methods to find realistic switch-on pulses that bring the material into the predefined stationary Floquet band structure, i.e., into a state in which the desired Floquet modes of the target bands are fully occupied, so that they should remain stroboscopically stationary, with long lifetimes, when the weak periodic drives are started. Finally, we note that although we have focused on solid state materials, the technique that we propose could be equally used for the Floquet engineering of ultracold atoms in optical lattices and for other nonequilibrium dynamical and correlated systems.

Topological Floquet bands in a circularly shaken dice lattice

The hoppings of noninteracting particles in the optical dice lattice result in the gapless dispersions in the band structure formed by the three lowest minibands. In our research, we find that once a periodic driving force is applied to this optical dice lattice, the original spectral characteristics could be changed, forming three gapped quasienergy bands in the quasienergy Brillouin zone. The topological phase diagram containing the Chern number of the lowest quasienergy band shows that when the hopping strengths of the nearest-neighbor hoppings are isotropic, the system persists in the topologically nontrivial phases with Chern number $C=2$ within a wide range of the driving strength. Accompanied by the anisotropic nearest-neighbor hopping strengths, a topological phase transition occurs, making the Chern number change from $C=2$ to $C=1$. This transition is further verified by our analytical method. Our theoretical work implies that it is feasible to realize the nontrivially topological characteristics of optical dice lattices by applying periodic shaking and that topological phase transition can be observed by independently tuning the strength of a type of nearest-neighbor hopping.

Light-induced topological phases in two-dimensional gapped Dirac materials

We present a comparative study of light-induced topological phases in two-dimensional massive Dirac models with and without quadratic corrections, which depicts different inversion symmetry-breaking Dirac materials. We find that a light has two effects on the Floquet band topology: it not only inverts the massive bands near the Dirac points but also introduces pseudospin-orbital coupling around the resonant points of light-matter interaction. Depending on the presence or absence of the quadratic corrections, the interplay of these two effects of light results in versatile topological phases with dissipationless valley polarized edge currents. In particular, without quadratic correction as exemplified by gapped graphene, a circularly polarized light induces a pair of in-gap edge states at each valley, which propagate on the same direction at a fixed boundary. A linearly polarized light also induces a pair of in-gap edge states, while propagating on the opposite direction at a fixed boundary. When including the quadratic correction as exemplified by monolayer transition-metal dichalcogenides, a circularly polarized light can induce two pairs of in-gap edge states located at a single valley, while a linearly polarized light does not induce any in-gap edge state. These edge currents as controlled by switching on/off the applied light and its polarization nature suggest an optical way to engineer valleytronic devices.

Time- and angle-resolved photoelectron spectroscopy of strong-field light-dressed solids: Prevalence of the adiabatic band picture

In recent years, strong-field physics in condensed-matter was pioneered as a novel approach for controlling material properties through laser-dressing, as well as for ultrafast spectroscopy via nonlinear light-matter interactions (e.g. harmonic generation). A potential controversy arising from these advancements is that it is sometimes vague which band-picture should be used to interpret strong-field experiments: the field-free bands, the adiabatic (instantaneous) field-dressed bands, Floquet bands, or some other intermediate picture. We here try to resolve this issue by performing 'theoretical experiments' of time- and angle-resolved photoelectron spectroscopy (Tr-ARPES) for a strong-field laser-pumped solid, which should give access to the actual observable bands of the irradiated material. To our surprise, we find that the adiabatic band-picture survives quite well, up to high field intensities (~10^12 W/cm^2), and in a wide frequency range (driving wavelengths of 4000 to 800nm, with Keldysh parameters ranging up to ~7). We conclude that to first order, the adiabatic instantaneous bands should be the standard blueprint for interpreting ultrafast electron dynamics in solids when the field is highly off-resonant with characteristic energy scales of the material. We then discuss weaker effects of modifications of the bands beyond this picture that are non-adiabatic, showing that by using bi-chromatic fields the deviations from the standard picture can be probed with enhanced sensitivity. Our work outlines a clear band picture for the physics of strong-field interactions in solids, which should be useful for designing and analyzing strong-field experimental observables and also to formulate simpler semi-empirical models.

Topological Pumping in a Floquet-Bloch Band.

Constructing new topological materials is of vital interest for the development of robust quantum applications. However, engineering such materials often causes technological overhead, such as large magnetic fields, spin-orbit coupling, or dynamical superlattice potentials. Simplifying the experimental requirements has been addressed on a conceptual level-by proposing to combine simple lattice structures with Floquet engineering-but there has been no experimental implementation. Here, we demonstrate topological pumping in a Floquet-Bloch band using a plain sinusoidal lattice potential and two-tone driving with frequencies ω and 2ω. We adiabatically prepare a near-insulating Floquet band of ultracold fermions via a frequency chirp, which avoids gap closings en route from trivial to topological bands. Subsequently, we induce topological pumping by slowly cycling the amplitude and the phase of the 2ω drive. Our system is well described by an effective Shockley model, establishing a novel paradigm to engineer topological matter from simple underlying lattice geometries. This approach could enable the application of quantized pumping in metrology, following recent experimental advances on two-frequency driving in real materials.

Hybrid skin-topological modes without asymmetric couplings

The non-Hermitian skin effect (NHSE) in non-Hermitian lattice systems, associated with a point gap on the complex energy plane, has attracted great theoretical and experimental interest. Much less is studied on the so-called second-order non-Hermitian skin effect, where the bulk does not support a point gap but localization at the corner still occurs. This work discovers a class of hybrid skin-topological modes as the second-order non-Hermitian skin effect without asymmetric couplings. Specifically, by only adding gain/loss to two-dimensional Chern insulators and so long as the gain/loss strength does not close the line gap, all the topological edge states are localized at one corner under the open boundary condition, with the bulk states extended. The resultant non-Hermitian Chern bands can be still topologically characterized by Chern numbers, whereas the hybrid skin-topological modes are understood via some auxiliary Hermitian systems that belong to either intrinsic or extrinsic second-order topological insulator phases. By proposing an innovative construction of auxiliary Hamiltonian, our generic route to hybrid skin-topological modes is further successfully extended to nonequilibrium topological systems with gain and loss, where the anomalous Floquet band topology is no longer captured by band Chern numbers. The extension thus leads to the intriguing finding of nonequilibrium hybrid skin-topological modes. In addition to offering a straightforward route to experimental realization of hybrid topological-skin effects, this study also opens up a promising perspective for the understanding of corner localization by revealing the synergy of three important concepts, namely, non-Hermitian topological insulator, second-order non-Hermitian skin effect, and second-order topological insulator.

Electron emission from plasmonically induced Floquet bands at metal surfaces

We explore the possibility of the existence of plasmonically generated electronic Floquet bands at metal surfaces by studying the gauge transformed electron-surface plasmon interaction in the prepumped plasmonic coherent state environment. These bands may promote non-Einsteinian electron emission from metal surfaces exposed to primary interactions with strong electromagnetic fields. Resonant behavior and scaling of emission yield with the parent electronic structure and plasmonic state parameters are estimated for Ag(111) surface. The obtained relative yield intensities from non-Einsteinian emission channels in photoelectron spectra offer the means to calibrate the mediating plasmonic fields and the therefrom ensuing surface Floquet bands.

Floquet electronic bands and transport in magic-angle bilayer graphene

We theoretically study Floquet band structures and transport properties of twisted bilayer graphene at the magic-angle under the irradiation of variously polarized light. The magic-angle bilayer graphene is depicted by the newly proposed ten-band tight-binding model and the iterative continued fraction method is adopted to facilitate the calculations of electronic properties in the low-frequency regime. The transitions between Floquet sidebands induce discontinuous electronic bands and energy gaps which further give rise to the antiresonances in longitudinal conductivity calculated by the Kubo formula. Furthermore, significant Hall conductivity is generated by circularly polarized light and its magnitude and sign are sensitive to light polarization as well as photoinduced bandgap-opening, offering a feasible way to tune Hall conductivity by manipulating light polarization. We finally take into account the interplay between light irradiation and short-range disorder, and reveal that disorder scattering remarkably enhances the photoinduced Hall conductivity and can be viewed as an extrinsic source to Hall conductivity.

Orbital magnetization of Floquet topological systems

A general expression for the orbital magnetization of a Floquet system is derived. The expression holds for a clean system and is valid for any driving protocol and arbitrary occupation of the bands. The orbital magnetization is shown to be large not only for Chern insulators, but also for anomalous phases where the Chern number does not fully account for the topology. In addition, the orbital magnetization is shown to take significant values both for a thermal equilibrium occupation of the Floquet bands and for occupations determined by a quantum quench from an initial state with zero orbital magnetization. For the latter case, the orbital magnetization is shown to be highly sensitive to van Hove singularities of the Floquet bands.

Prethermalization and entanglement dynamics in interacting topological pumps

We investigate the formation of quasisteady states in one-dimensional pumps of interacting fermions at non-integer filling fraction, in the regime where the driving frequency and interaction strength are small compared to the instantaneous single-particle band gap throughout the driving cycle. The system rapidly absorbs energy from the driving field, and approaches a quasisteady state that locally resembles a maximal entropy state subject to the constraint of fixed particle number in each of the system's single-particle Floquet bands. We explore the nature of this quasisteady state through one-body observables including the pumped current and natural orbital occupations, as well as the (many-body) entanglement spectrum and entropy. Potential disorder significantly reduces the amplitude of fluctuations of the quasisteady state current around its universal value, while the lifetime of the quasisteady state remains nearly unaffected for disorder strengths up to the scale of the single-particle band gap. Interestingly, the natural orbital occupations and entanglement entropy display patterns signifying the periodic entangling and disentangling of the system's degrees of freedom over each driving cycle. Moreover, prominent features in the system's time-dependent entanglement spectrum reveal the emergence of new long timescales associated with the equilibration of many-particle correlations.

High-temperature fractional quantum Hall state in the Floquet kagome flat band

A fractional quantum Hall effect (FQHE) has been predicted in a topological flat band (FB) by a single-particle band structure combined with phenomenological theory or solution of a many-body lattice Hamiltonian with fuzzy parameters. A long-standing roadblock toward the realization of a FB-FQHE is lacking the many-body solution of specific materials under realistic conditions. We demonstrate a combined study of single-particle Floquet band theory with exact diagonalization (ED) of a many-body Hamiltonian. We show that a time-periodic circularly polarized laser inverts the sign of second-nearest-neighbor hopping in a kagome lattice and enhances spin-orbit coupling in one spin channel to produce a Floquet FB with a high flatness ratio of bandwidth over band gap, as exemplified in monolayer ${\mathrm{Pt}}_{3}{\mathrm{C}}_{36}{\mathrm{S}}_{12}{\mathrm{H}}_{12}$. The ED of the resultant Floquet-kagome lattice Hamiltonian gives a one-third-filling ground state with a laser-dependent excitation gap of a FQH state, up to an estimated temperature above 70 K. Our findings pave the way for exploring the alluding high-temperature FB-FQHE.

Nonlinear signatures of Floquet band topology

We study how the nonlinear propagation dynamics of bulk states may be used to distinguish topological phases of slowly driven Floquet lattices. First, we show how instabilities of nonlinear Bloch waves may be used to populate Floquet bands and measure their Chern number via the emergence of nontrivial polarization textures in a similar manner to static (undriven) lattices. Second, we show how the nonlinear dynamics of nonstationary superposition states may be used to identify dynamical symmetry inversion points in the intracycle dynamics, thereby allowing anomalous Floquet phases to be distinguished from the trivial phase. The approaches may be readily implemented using light propagation in nonlinear waveguide arrays.

Dynamic bulk-boundary correspondence for anomalous Floquet topology

Periodically driven systems with internal and spatial symmetries can exhibit a variety of anomalous boundary behaviors at both the zero and $\ensuremath{\pi}$ quasienergies despite the trivial bulk Floquet bands. These phenomena are called anomalous Floquet topology (AFT) as they are unconnected from their static counterpart, emerging from the winding of the time-evolution unitary rather than the bulk Floquet bands at the end of the driving period. In this paper, we systematically derive the first and inversion-symmetric second-order AFT bulk-boundary correspondence for Altland-Zirnbauer (AZ) classes BDI, D, DIII, and AII. For each AZ class, we start a dimensional hierarchy with a parent dimension having $\mathbb{Z}$ classification, then use it as an interpolating map to classify the lower-dimensional descendants. From the Atiyah-Hirzebruch spectral sequence, we identify the subspace that contains topological information and faithfully derive the AFT bulk-boundary correspondence for both the parent and descendants. Our theory provides analytic tools for out-of-equilibrium topological phenomena.

Floquet band engineering and topological phase transitions in 1T’ transition metal dichalcogenides

Using ab initio tight-binding approaches, we investigate Floquet band engineering of the 1T’ phase of transition metal dichalcogenides (MX2, M = W, Mo and X = Te, Se, S) monolayers under the irradiation with circularly polarized light. Our first principles calculations demonstrate that light can induce important transitions in the topological phases of this emerging materials family. For example, upon irradiation, Te-based MX2 undergoes a phase transition from quantum spin Hall (QSH) semimetal to time-reversal symmetry broken QSH insulator with a nontrivial band gap of up to 92.5 meV. On the other hand, Se- and S-based MX2 undergoes the topological phase transition from the QSH effect to the quantum anomalous Hall effect and into trivial phases with increasing light intensity. From a general perspective, this theoretical work brings further insight into non-equilibrium topological systems.

Detecting light-induced Floquet band gaps of graphene via trARPES

We propose a realistic regime to detect the light-induced topological band gap in graphene via time-resolved angle-resolved photoelectron spectroscopy (trARPES), that can be achieved with current technology. The direct observation of Floquet-Bloch bands in graphene is limited by low-mobility, Fourier-broadening, laser-assisted photoemission (LAPE), probe-pulse energy-resolution bounds, space-charge effects and more. We characterize a regime of low driving frequency and high amplitude of the circularly polarized light that induces an effective band gap at the Dirac point that exceeds the Floquet zone. This circumvents limitations due to energy resolutions and band broadening. The electron distribution across the Floquet replica in this limit allow for distinguishing LAPE replica from Floquet replica. We derive our results from a dissipative master equation approach that gives access to two-point correlation functions and the electron distribution relevant for trARPES measurements.

Observing light-induced Floquet band gaps in the longitudinal conductivity of graphene

Floquet engineering presents a versatile method of dynamically controlling material properties. The light-induced Floquet-Bloch bands of graphene feature band gaps, which have not yet been observed directly. We propose optical longitudinal conductivity as a realistic observable to detect light-induced Floquet band gaps in graphene. These gaps manifest as resonant features in the conductivity, when resolved with respect to the probing frequency and the driving field strength. The electron distribution follows the light-induced Floquet-Bloch bands, resulting in a natural interpretation as occupations of these bands. Furthermore, we show that there are population inversions of the Floquet-Bloch bands at the band gaps for sufficiently strong driving field strengths. This strongly reduces the conductivity at the corresponding frequencies. Therefore our proposal puts forth not only an unambiguous demonstration of light-induced Floquet-Bloch bands, which advances the field of Floquet engineering in solids, but also points out the control of transport properties via light, that derives from the electron distribution on these bands.