Nonequilibrium Floquet topological phases due to periodic driving are known to exhibit rich and interesting features with no static analogs. Various known topological invariants usually proposed to characterize static topological systems often fail to fully characterize Floquet topological phases. This fact has motivated extensive studies of Floquet topological phases to better understand nonequilibrium topological phases and to explore their possible applications. Here we present a theoretically simple Floquet topological insulating system that may possess an arbitrary number of counter-propagating chiral edge states. Further investigation into our system reveals another related feature by tuning the same set of system parameters, namely, the emergence of almost flat (dispersionless) edge modes. In particular, we employ two-terminal conductance and dynamical winding numbers to characterize counter-propagating chiral edge states. We further demonstrate the robustness of such edge states against symmetry preserving disorder. Finally, we identify an emergent chiral symmetry at certain sub-regimes of the Brillouin zone that can explain the presence of almost flat edge modes. Our results have exposed more interesting possibilities in Floquet topological matter.
The topological phases of periodically-driven, or Floquet systems, rely on a perfectly periodic modulation of system parameters in time. Even the smallest deviation from periodicity leads to decoherence, causing the boundary (end) states to leak into the system's bulk. Here, we show that in one dimension this decay of topologically protected end states depends fundamentally on the nature of the bulk states: a dispersive bulk results in an exponential decay, while a localized bulk slows the decay down to a diffusive process. The localization can be due to disorder, which remarkably counteracts …
Higher-order topological phases are characterized by protected states localized at the corners or hinges of the system. By applying time-periodic quenches to a two-dimensional lattice with balanced gain and loss, we obtain a rich variety of non-Hermitian Floquet second order topological insulating phases. Each of the phases is characterized by a pair of integer topological invariants, which predict the numbers of Floquet corner modes at zero and $π$ quasienergies. We establish the topological phase diagram of the model, and find a series of non-Hermiticity induced transitions between different…
Topological insulators represent unique phases of matter with insulating bulk and conducting edge or surface states, immune to small perturbations such as backscattering due to disorder. This stems from their peculiar band structure, which provides topological protections. While conventional tools (pressure, doping etc.) to modify the band structure are available, time periodic perturbations can provide tunability by adding time as an extra dimension enhanced to the problem. In this short review, we outline the recent research on topological insulators in non equilibrium situations. Firstly, w…
We present a non-Hermitian Floquet model with topological edge states in real and imaginary band gaps. The model utilizes two stacked honeycomb lattices which can be related via four different types of non-Hermitian time-reversal symmetry. Implementing the correct time-reversal symmetry provides us with either two counterpropagating edge states in a real gap, or a single edge state in an imaginary gap. The counterpropagating edge states allow for either helical or chiral transport along the lattice perimeter. In stark contrast, we find that the edge state in the imaginary gap does not propagat…
We theoretically investigate a periodically driven semimetal based on a square lattice. The possibility of engineering both Floquet Topological Insulator featuring Floquet edge states and Floquet higher order topological insulating phase, accommodating topological corner modes has been demonstrated starting from the semimetal phase, based on Floquet Hamiltonian picture. Topological phase transition takes place in the bulk quasi-energy spectrum with the variation of the drive amplitude where Chern number changes sign from $+1$ to $-1$. This can be attributed to broken time-reversal invariance (…
Periodically driven (Floquet) phases are attractive due to their ability to host unique physical phenomena with no static counterparts. We propose a general approach in nontrivially devising a square-root version of existing Floquet phases, applicable both in noninteracting and interacting setting. The resulting systems are found to yield richer physics that is otherwise absent in the original counterparts and is robust against parameter imperfection. These include the emergence of Floquet topological superconductors with arbitrarily many zero, $π$, and $π/2$ edge modes, as well as $4T$-period…
Interlayer Coupling and Floquet-Driven Topological Phases in Bilayer Haldane Lattices
Zhejiang Normal University（浙江师范大学）；Khazar University（哈扎尔大学）；Quaid-I Azam University Islamabad（巴基斯坦奎德-阿扎姆大学伊斯兰堡分校）；Institute for Research in Fundamental Sciences (IPM)（伊朗基础科学研究院）
We propose and model Floquet topological polaritons in semiconductor microcavities, using the interference of frequency detuned coherent fields to provide a time periodic potential. For arbitrarily weak field strength, where the Floquet frequency is larger than the relevant bandwidth of the system, a Chern insulator is obtained. As the field strength is increased, a topological phase transition is observed with an unpaired Dirac cone proclaiming the anomalous Floquet topological insulator. As the relevant bandwidth increases even further, an exotic Chern insulator with flat band is observed wi…
Recent experiments showed that the surface of a three dimensional topological insulator develops gaps in the Floquet-Bloch band spectrum when illuminated with a circularly polarized laser. These Floquet-Bloch bands are characterized by non-trivial Chern numbers which only depend on the helicity of the polarization of the radiation field. Here we propose a setup consisting of a pair of counter-rotating lasers, and show that one-dimensional chiral states emerge at the interface between the two lasers. These interface states turn out to be spin-polarized and may trigger interesting applications i…
Recent works have demonstrated that the Floquet-Bloch bands of periodically-driven systems feature a richer topological structure than their non-driven counterparts. The additional structure in the driven case arises from the periodicity of quasienergy, the energy-like quantity that defines the spectrum of a periodically-driven system. Here we develop a new paradigm for the topological classification of Floquet-Bloch bands, based on the time-dependent spectrum of the driven system's evolution operator throughout one driving period. Specifically, we show that this spectrum may host topologicall…
Fractional Chern insulators are the proposed phases of matter mimicking the physics of fractional quantum Hall states on a lattice without an overall magnetic field. The notion of Floquet fractional Chern insulators refers to the potential possibilities to generate the underlying topological bandstructure by means of Floquet engineering. In these schemes, a highly controllable and strongly interacting system is periodically driven by an external force at a frequency such that double tunneling events during one forcing period become important and contribute to shaping the required effective ene…
Periodic driving fields can induce topological phase transitions, resulting in Floquet topological phases with intriguing properties such as very large Chern numbers and unusual edge states. Whether such Floquet topological phases could generate robust edge state conductance much larger than their static counterparts is an interesting question. In this paper, working under the Keldysh formalism, we study two-lead transport via the edge states of irradiated quantum Hall insulators using the method of recursive Floquet-Green's functions. Focusing on a harmonically-driven Hofstadter model, we sho…
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, specific lattice geometries, strong spin-orbit coupling, synthetic dimensions, 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-…
Periodically driven non-Hermitian systems can exhibit rich topological band structure and non-Hermitian skin effect, without analogs in their static or Hermitian counterparts. In this work we investigate the exceptional band-touching points in the Floquet quasi-energy bands, the topological characterization of such exceptions points and the Floquet non-Hermitian skin effect (FNHSE). Specifically, we exploit the simplicity of periodically quenched two-band systems in one dimension or two dimensions to analytically obtain the Floquet effective Hamiltonian as well as locations of the many excepti…
We introduce and develop an approach to realizing a topological phase transition and non-Abelian statistics with dynamically induced Floquet Majorana Fermions (FMFs). When the periodic driving potential does not break fermion parity conservation, FMFs can encode quantum information. Quasi-energy analysis shows that a stable FMF zero mode and two other satellite modes exist in a wide parameter space with large quasi-energy gaps, which prevents transitions to other Floquet states under adiabatic driving. We also show that in the asymptotic limit FMFs preserve non-Abelian statistics and, thus, be…
While periodically-driven phases offer a unique insight into non-equilibrium topology that is richer than its static counterpart, their experimental realization is often hindered by ubiquitous decoherence effects. Recently, we have proposed a decoherence-free approach of realizing these Floquet phases. The central insight is that the reflection matrix, being unitary for a bulk insulator, plays the role of a Floquet time-evolution operator. We have shown that reflection processes off the boundaries of systems supporting higher-order topological phases (HOTPs) simulate non-trivial Floquet phases…
Recent theoretical work on time-periodically kicked Hofstadter model found robust counter-propagating edge modes. It remains unclear how ubiquitously such anomalous modes can appear, and what dictates their robustness against disorder. Here we shed further light on the nature of these modes by analyzing a simple type of periodic driving where the hopping along one spatial direction is modulated sinusoidally with time while the hopping along the other spatial direction is kept constant. We obtain the phase diagram for the quasienergy spectrum at flux 1/3 as the driving frequency $ω$ and the hop…
With the aim to reveal universal features of hadronic matter and correlated Dirac insulators in strong AC-electric fields, we study the $\mathcal{N}=2$ supersymmetric QCD with a finite quark mass driven by a rotating electric field $\mathcal{E}_x+i\mathcal{E}_y= E e^{iΩt}$. The analysis is done in the holographically dual D3/D7 system in the co-rotating frame, effectively. The nonequilibrium phase diagram is determined from the threshold electric field at which the insulator phase breaks down to a conductive phase due to the AC version of the Schwinger mechanism. The external field induces a r…
Floquet engineering provides a toolbox for the realization of novel quantum phases without static counterparts, while conventionally the realization may rely on the manipulation of complex temporal evolution. Here we propose a systematic and high-precision scheme to realize unconventional Floquet topological phases by engineering local band structures in particular momentum subspace called band inversion surfaces (BISs). This scheme is based on a new bulk-boundary correspondence that for a class of generic $d$-dimensional periodically driven systems, the local topological structure formed in e…
Floquet engineering is the concept of tailoring a system by a periodic drive. It has been very successful in opening new classes of Hamiltonians to the study with ultracold atoms in optical lattices, such as artificial gauge fields, topological band structures and density-dependent tunneling. Furthermore, driven systems provide new physics without static counterpart such as anomalous Floquet topological insulators. In this review article, we provide an overview of the exciting developments in the field and discuss the current challenges and perspectives.
Floquet states of periodically driven systems could exhibit rich topological properties. Many of them are absent in their static counterparts. One such example is the chiral edge states in anomalous Floquet topological insulators, whose description requires a new topological invariant and a novel type of bulk-edge correspondence. In this work, we propose a prototypical quenched lattice model, whose two Floquet bands could exchange their Chern numbers periodically and alternatively via touching at quasienergies 0 and $π$ under the change of a single system parameter. This process in principle a…
Confinement-Tunable Synthetic Gauge Fields and Floquet Topological Phenomena in a Driven Quantum Wire Qubit
University of Abomey-Calavi（阿博梅-卡利维大学）；University of Dschang（德尚大学）；Quantum Materials and Computing Group - QMaCG（量子材料与计算群 - QMaCG）；Instituteur principal de l’enseignement général, Délégation Régionale de l’Ouest, Délégation Départementale de la Menoua, Arrondissement de Nkong-Ni, École Publique de Batsing’la Chefferie, Bafou, Cameroun（喀麦隆巴富，班辛拉酋长国公立学校，孔克尼区，梅努瓦省教育厅，西部大区教育厅，普通教育首席教师）；University of Yaoundé I（雅温得第一大学）
Higher-order topological phases (HOTPs) are characterized by symmetry-protected bound states at the corners or hinges of the system. In this work, we reveal a momentum-space counterpart of HOTPs in time-periodic driven systems, which are demonstrated in a two-dimensional extension of the quantum double-kicked rotor. The found Floquet HOTPs are protected by chiral symmetry and characterized by a pair of topological invariants, which could take arbitrarily large integer values with the increase of kicking strengths. These topological numbers can also be measured from the chiral dynamics of wave …
We study the laser control of magnon topological phases induced by the Aharonov-Casher effect in insulating antiferromagnets (AFs). Since the laser electric field can be considered as a time-periodic perturbation, we apply the Floquet theory and perform the inverse frequency expansion by focusing on the high frequency region. Using the obtained effective Floquet Hamiltonian, we study nonequilibrium magnon dynamics away from the adiabatic limit and its effect on topological phenomena. We show that a linearly polarized laser can generate helical edge magnon states and induce the magnonic spin Ne…
In Hermitian topological systems, the bulk-boundary correspondence strictly constraints boundary transport to values determined by the topological properties of the bulk. We demonstrate that this constraint can be lifted in non-Hermitian Floquet insulators. Provided that the insulator supports an anomalous topological phase, non-Hermiticity allows us to modify the boundary states independently of the bulk, without sacrificing their topological nature. We explore the ensuing possibilities for a Floquet topological insulator with non-Hermitian time-reversal symmetry, where the helical transport …
We demonstrate the existence of a two-dimensional anomalous Floquet insulator (AFI) phase: an interacting (periodically-driven) non-equilibrium topological phase of matter with no counterpart in equilibrium. The AFI is characterized by a many-body localized bulk, exhibiting nontrivial micromotion within a driving period, and delocalized (thermalizing) chiral states at its boundaries. For a geometry without edges, we argue analytically that the bulk may be many-body localized in the presence of interactions, deriving conditions where stability is expected. We investigate the interplay between t…
Optical drives at terahertz and mid-infrared frequencies in quantum materials are increasingly used to reveal the nonlinear dynamics of collective modes in correlated many-body systems and their interplay with electromagnetic waves. Recent experiments demonstrated several surprising optical properties of transient states induced by driving, including the appearance of photo-induced edges in the reflectivity in cuprate superconductors, observed both below and above the equilibrium transition temperature. Furthermore, in other driven materials, reflection coefficients larger than unity have been…
The survival of the quantum spin Hall edge channels in presence of an external magnetic field has been a subject of experimental and theoretical research. The inversion of Landau levels that accommodates the quantum spin Hall effect is destroyed at a critical magnetic field, and a trivial insulating gap appears in the spectrum for stronger fields. In this work, we report the absence of this transport gap in disordered two dimensional topological insulators in perpendicular magnetic fields of up to 16 T. Instead, we observe that a topological edge channel (from band inversion) coexists with a c…
We show that H-phase transition metal dichalcogenides (TMDs) monolayers such as MoS$_2$ and WSe$_2$, are orbital Hall insulators. They present very large orbital Hall conductivity plateaus in their semiconducting gap, where the spin Hall conductivity vanishes. Our results open the possibility of using TMDs for orbital current injection and orbital torque transfers that surpass their spin-counterparts in spin-orbitronics devices. The orbital Hall effect (OHE) in TMD monolayers occurs even in the absence of spin-orbit coupling. It can be linked to exotic momentum-space Dresselhaus-like orbital t…
Universal driving protocol for symmetry-protected Floquet topological phasesWe propose a universal driving protocol for the realization of symmetry-protected topological phases in $2+1$ dimensional Floquet systems. Our proposal is based on the theoretical analysis of the possible symmetries of a square lattice model with pairwise nearest-neighbor coupling terms. Among the eight possible symmetry operators we identify the two relevant choices for topological phases with either time-reversal, chiral, or particle-hole symmetry. From the corresponding symmetry conditions on the protocol parameters…
Recently the creation of novel topological states of matter by a periodic driving field has attracted great attention. To motivate further experimental and theoretical studies, we investigate interesting aspects of Floquet bands and topological phase transitions in a continuously driven Harper model. In such a continuously driven system with an odd number of Floquet bands, the bands are found to have nonzero Chern numbers in general and topological phase transitions take place as we tune various system parameters, such as the amplitude or the period of the driving field. The nontrivial Floquet…
Due to studies in nonequilibrium (periodically-driven) topological matter, it is now understood that some topological invariants used to classify equilibrium states of matter do not suffice to describe their nonequilibrium counterparts. Indeed, in Floquet systems the additional gap arising from the periodicity of the quasienergy Brillouin zone often leads to unique topological phenomena without equilibrium analogues. In the context of Floquet Weyl semimetal, Weyl points may be induced at both quasienergy zero and $π/T$ ($T$ being the driving period) and these two types of Weyl points can be ve…
Recent experimental advances in Floquet engineering and controlling dissipation in open systems have brought about unprecedented flexibility in tailoring novel phenomena without any static and Hermitian analogues. It can be epitomized by the various Floquet and non-Hermitian topological phases. Topological classifications of either static/Floquet Hermitian or static non-Hermitian systems based on the underlying symmetries have been well established in the past several years. However, a coherent understanding and classification of Floquet non-Hermitian (FNH) topological phases have not been ach…
When a physical system is subjected to a strong external multi-frequency drive, its dynamics can be conveniently represented in the multi-dimensional Floquet lattice. The number of the Floquet lattice dimensions equals the number of {\em irrationally}-related drive frequencies, and the evolution occurs in response to a built-in effective "electric" field, whose components are proportional to the corresponding drive frequencies. The mapping allows to engineer and study temporal analogs of many real-space phenomena. Here we focus on the specific example of a two-level system under two-frequency …
Micromotion area as proxy for anomalous Floquet topological systems
Kyoto University（京都大学）；University of Hamburg（汉堡大学）；The Hamburg Centre for Ultrafast Imaging（汉堡超快成像中心）；Technische Universität Berlin（柏林工业大学）；TU Dortmund University（多特蒙德工业大学）
Motivated by recent experiments [P. Vaidya {\em et al.}, Science {\bf 368}, 160 (2020)] on spin pumping from sub-THz radiation-driven uniaxial antiferromagnetic insulator (AFI) MnF$_2$ into heavy metal (HM) Pt hosting strong spin-orbit (SO) coupling, we compute and compare pumped spin currents in Cu/MnF$_2$/Cu and Pt/MnF$_2$/Cu heterostructures. Recent theories of spin pumping by AFI have relied on simplistic Hamiltonians (such as tight-binding) and the scattering approach to quantum transport yielding the so-called interfacial spin mixing conductance (SMC), but the concept of SMC ceases to be…
A periodically driven quantum Hall system in a fixed magnetic field is found to exhibit a series of phases featuring anomalous edge modes with the "wrong" chirality. This leads to pairs of counter-propagating chiral edge modes at each edge, in sharp contrast to stationary quantum Hall systems. We show that the pair of Floquet edge modes are protected by the chiral (sublattice) symmetry, and that they are robust against static disorder. The existence of distinctive phases with the same Chern and winding numbers but very different edge state spectra points to the important role played by symmetr…
We analyze the dissipative conductance of the zero-plateau quantum Hall state appearing in undoped graphene in strong magnetic fields. Charge transport in this state is assumed to be carried by a magnetic domain wall, which forms by hybridization of two counter--propagating edge states of opposing spin due to interactions. The resulting non--chiral edge mode is a Luttinger liquid of parameter K, which enters a gapped, perfectly conducting state below a critical value K_c\approx 1/2. Backscattering in this system involves spin flip, so that interaction with localized magnetic moments generates …
Liouvillian topology and nonreciprocal dynamics in open Floquet chains
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Institute of Theoretical Physics, Technische Universität Dresden and Würzburg-Dresden Cluster of Excellence ctd.qmat（德累斯顿工业大学理论物理学院和维尔茨堡-德累斯顿卓越集群ctd.qmat）
