We present an effective field theory approach to the topological response of Floquet systems with symmetry group $G$. This is achieved by introducing a background $G$ gauge field in the Schwinger-Keldysh formalism, which is suitable for far from equilibrium systems. We carry out this program for chiral topological Floquet systems (anomalous Floquet-Anderson insulators) in two spatial dimensions, and the group cohomology models of topological Floquet unitaries. These response actions serve as many-body topological invariants for topological Floquet unitaries. The effective action approach also leads us to propose novel topological response functions.
Periodically driven (Floquet) systems have been under active theoretical and experimental investigations. This paper aims at a systematic study in the following aspects of Floquet systems: (i) A systematic formulation of topological invariants of Floquet systems based on the cooperation of topology and symmetries. Topological invariants are constructed for the ten symmetry classes in all spatial dimensions, for both homogeneous Floquet systems (Floquet topological insulators and superconductors) and Floquet topological defects. Meanwhile, useful representative Dirac Hamiltonians for all the sy…
We study Floquet topological phases in periodically driven systems that are protected by "time glide symmetry", a combination of reflection and half time period translation. Time glide symmetry is an analog of glide symmetry with partial time translation replacing the partial space translation, and hence, is an intrinsically dynamical symmetry which may be engineered in periodically driven systems by exploiting the controllability of driving. We present lattice models of time glide symmetric Floquet topological insulators in two and three dimensions. The topological numbers characterizing thos…
Backscattering-immune chiral modes arise along certain line defects in three-dimensional materials. In this paper, we study Floquet chiral modes along Floquet topological defects, namely, the defects come entirely from spatial modulations of periodic driving. We define a precise topological invariant that counts the number of Floquet chiral modes, which is expressed as an integral on a five-dimensional torus parameterized by $(k_x,k_y,k_z,θ,t)$. This work demonstrates the possibility of creating chiral modes in three-dimensional bulk materials by modulated driving. We hope that it will stimula…
We show that Floquet chiral topological superconductivity arises naturally in Josephson junctions made of magnetic topological insulator-superconductor sandwich structures. The Josephson phase modulation associated with an applied bias voltage across the junction drives the system into the anomalous Floquet chiral topological superconductor hosting chiral Majorana edge modes in the quasienergy spectrum, with the bulk Floquet bands carrying zero Chern numbers. The bias voltage acts as a tuning parameter enabling novel dynamical topological quantum phase transitions driving the system into a myr…
Spatially uniform excitations can induce Floquet topological bandstructures within insulators which have equal characteristics to those of topological insulators. Going beyond we demonstrate in this article the evolution of Floquet topological quantum states for electromagnetically driven semiconductor bulk matter. We show the direct physical impact of the mathematical precision of the Floquet-Keldysh theory when we solve the driven system of a generalized Hubbard model with our framework of dynamical mean field theory (DMFT) in the non-equilibrium. We explain the physical consequences of the …
Floquet topological insulators are topological phases of matter generated by the application of time-periodic perturbations on otherwise conventional insulators. We demonstrate that spatial variations in the time-periodic potential lead to localized quasi-stationary states in two-dimensional systems. These states include one-dimensional interface modes at the nodes of the external potential, and fractionalized excitations at vortices of the external potential. We also propose a setup by which light can induce currents in these systems. We explain these results by showing a close analogy to px+…
Dynamical phases with novel topological properties are known to arise in driven systems of free fermions. In this paper, we obtain a `periodic table' to describe the phases of such time-dependent systems, generalizing the periodic table for static topological insulators. Using K-theory, we systematically classify Floquet topological insulators from the ten Altland-Zirnbauer symmetry classes across all dimensions. We find that the static classification scheme described by a group $G$ becomes $G\times G$ in the time-dependent case, and interpret the two factors as arising from the bipartite deco…
Topological semimetals with nodal line are a novel class of topological matter extending the concept of topological matter beyond topological insulators and Weyl/Dirac semimetals. Here, we show that a Floquet topological semimetal with nodal helix can be generated by irradiating graphene or the surface of a topological insulator with circularly polarized light. Nodal helix is a form of nodal line running across the Brillouin zone with helical winding. Specifically, it is shown that the dynamics of irradiated graphene is described by the time Stark Hamiltonian, which can host a Floquet topologi…
Electronic bands featuring nontrivial bulk topological invariant manifest through robust gapless modes at the boundaries, e.g., edges and surfaces. As such this bulk-boundary correspondence is also operative in driven quantum materials. For example, a suitable periodic drive can convert a trivial insulator into a Floquet topological insulator (FTI) that accommodates nondissipative dynamic gapless modes at the interfaces with vacuum. Here we theoretically demonstrate that dislocations, ubiquitous lattice defects in crystals, can probe FTIs as well as unconventional $π$-trivial insulator in the …
We predict the existence of a novel Floquet topological insulator in three-dimensional two-band systems, the Floquet Hopf insulator, which possesses two distinct topological invariants. One is the Hopf $\mathbb{Z}$ invariant, a linking number characterizing the (non-driven) Hopf topological insulator. The second invariant is an intrinsically Floquet $\mathbb{Z}_2$ invariant, and represents a condensed matter realization of the topology underlying the Witten anomaly in particle physics. Both invariants arise from topological defects in the system's time-evolution, subject to a process in which …
Network models for equilibrium integer quantum Hall (IQH) transitions are described by unitary scattering matrices, that can also be viewed as representing non-equilibrium Floquet systems. The resulting Floquet bands have zero Chern number, and are instead characterized by a chiral Floquet (CF) winding number. This begs the question: How can a model without Chern number describe IQH systems? We resolve this apparent paradox by showing that non-zero Chern number is recovered from the network model via the energy dependence of network model scattering parameters. This relationship shows that, de…
Spatially uniform optical excitations can induce Floquet topological band structures within insulators which can develop similar or equal characteristics as are known from three-dimensional topological insulators. We derive in this article theoretically the development of Floquet topological quantum states for electromagnetically driven semiconductor bulk matter and we present results for the lifetime of these states and their occupation in the non-equilibrium. The direct physical impact of the mathematical precision of the Floquet-Keldysh theory is evident when we solve the driven system of a…
Symmetry protected topological order in one dimension leads to protected degeneracies between symmetry blocks of the reduced density matrix. In the presence of periodic driving, topological Floquet phases can be identified in terms of a cycling of these symmetry blocks between different charge quantum numbers. We discuss an example of this phenomenon with an Ising $\mathbb{Z}_2$ symmetry, using both analytic methods and real quantum computers. By adiabatically moving along the phase diagram, we demonstrate that the cycling periodicity is broken in Floquet topological phase transitions. An equi…
We introduce the concept of Floquet topological magnons --- a mechanism by which a synthetic tunable Dzyaloshinskii-Moriya interaction (DMI) can be generated in quantum magnets using circularly polarized electric (laser) field. The resulting effect is that Dirac magnons and nodal magnons in two-dimensional (2D) and three-dimensional (3D) quantum magnets can be tuned to magnon Chern insulators and Weyl magnons respectively under circularly polarized laser field. The Floquet formalism also yields a tunable intrinsic DMI in insulating quantum magnets without an inversion center. We demonstrate th…
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…
We investigate a mechanism to transiently stabilize topological phenomena in long-lived quasi-steady states of isolated quantum many-body systems driven at low frequencies. We obtain an analytical bound for the lifetime of the quasi-steady states which is exponentially large in the inverse driving frequency. Within this lifetime, the quasi-steady state is characterized by maximum entropy subject to the constraint of fixed number of particles in the system's Floquet-Bloch bands. In such a state, all the non-universal properties of these bands are washed out, hence only the topological propertie…
A single Weyl fermion, which is prohibited in static lattice systems by the Nielsen-Ninomiya theorem, is shown to be realized in a periodically driven three-dimensional lattice system with a topologically nontrivial Floquet unitary operator, manifesting the chiral magnetic effect. We give a topological classification of Floquet unitary operators in the Altland-Zirnbauer symmetry classes for all dimensions, and use it to predict that all gapless surface states of topological insulators and superconductors can emerge in bulk quasienergy spectra in Floquet systems.
Floquet-Weyl states at one-photon resonances in three-dimensional topological insulators
The University of Tokyo（东京大学）；Tokyo University of Science（东京理科大学）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所 物质创造科学研究中心）；Hiroshima University（广岛大学）
We propose a bulk topological invariant for one-dimensional Floquet systems with chiral symmetry which quantifies the particle transport on each sublattice during the evolution. This chiral flow is physically motivated, locally computable, and improves on existing topological invariants by being applicable to systems with disorder. We derive a bulk-edge correspondence which relates the chiral flow to the number of protected dynamical edge modes present on a boundary at the end of the evolution. In the process, we introduce two real-space edge invariants which classify the dynamical topological…
The periodic driving of a quantum system can enable new topological phases without analogs in static systems. This provides a route towards preparing non-equilibrium quantum phases rooted into the non-equilibrium nature by periodic driving engineering. Motivated by the ongoing considerable interest in topological semimetals, we are interested in the novel topological phases in the periodically driven topological semimetals without a static counterpart. We propose to design non-equilibrium topological semimetals in the regime of weakly driving field where the spectrum width of shares the same m…
The Floquet Hamiltonian has often been used to describe a time-periodic system. Nevertheless, because the Floquet Hamiltonian depends on a micro-motion parameter, the Floquet Hamiltonian with a fixed micro-motion parameter cannot faithfully represent a driven system, which manifests as the anomalous edge states. Here we show that an accurate description of a Floquet system requires a set of Hamiltonian exhausting all values of the micro-motion parameter, and this micro-motion parameter can be viewed as an extra synthetic dimension of the system. Therefore, we show that a $d$-dimensional Floque…
Higher order topological insulators (HOTI) have emerged as a new class of phases, whose robust in-gap "corner" modes arise from the bulk higher-order multipoles beyond the dipoles in conventional topological insulators. Here, we incorporate Floquet driving into HOTI's, and report for the first time a dynamical polarization theory with anomalous non-equilibrium multipoles. Further, a proposal to detect not only corner states but also their dynamical origin in cold atoms is demonstrated, with the latter one never achieved before. Experimental determination of anomalous Floquet corner modes are a…
We investigate Floquet Topological Insulators in the presence of spatially-modulated light. We extend on previous work to show that light can be used to generate and control localized modes in the bulk of these systems. We provide examples of bulk modes generated through modulation of different properties of the light, such as its phase, polarization and frequency. We show that these effects may be realized in a variety of systems, including a zincblende model and graphene, and also provide a generalization of these results to three dimensional systems.
We study periodically driven pure Kitaev model and ferromagnetic phase of the Kitaev-Heisenberg model on the honeycomb lattice by off-resonant linearly and circularly-polarized lights at zero magnetic field. Using a combination of linear spin wave and Floquet theories, we show that the effective time-independent Hamiltonians in the off-resonant regime map onto the corresponding anisotropic static spin model, plus a tunable photoinduced magnetic field along the $[111]$ direction, which precipitates Floquet topological magnons and chiral magnon edge modes. They are tunable by the light amplitude…
We theoretically investigate possible photoinduced topological phase transitions in the organic salt $α$-(BEDT-TTF)$_2$I$_3$, which possesses a pair of inclined massless Dirac-cone bands between the conduction and valence bands under uniaxial pressure. The Floquet analyses of a driven tight-binding model for this material reveal rich photoinduced variations of band structures, Chern numbers, and Hall conductivities under irradiation with elliptically polarized light. The obtained phase diagrams contain a variety of nonequilibrium steady phases, e.g., the Floquet Chern insulator, Floquet semime…
We uncover an infinite family of time-reversal symmetric 3d interacting topological insulators of bosons or spins, in time-periodically driven systems, which we term Floquet topological paramagnets (FTPMs). These FTPM phases exhibit intrinsically dynamical properties that could not occur in thermal equilibrium, and are governed by an infinite set of $Z_2$-valued topological invariants, one for each prime number. The topological invariants are physically characterized by surface magnetic domain walls that act as unidirectional quantum channels, transferring quantized packets of information duri…
