Session X46: Floquet Topological Insulators

Live

Periodically-driven or Floquet systems can realize anomalous topological phenomena that do not exist in any equilibrium states of matter, whose classification and characterization require new theoretical ideas that are beyond the well-established paradigm of static topological phases. In this work, we provide a general framework to understand anomalous Floquet higher-order topological insulators (AFHOTIs), the classification of which has remained a challenging open question. In two dimensions (2D), such AFHOTIs are defined by their robust, symmetry-protected corner modes pinned at special quasienergies, even though all their Floquet bands feature trivial band topology. The corner-mode physics of an AFHOTI is found to be generically indicated by 3D Dirac/Weyl-like topological singularities living in the phase spectrum of the bulk time-evolution operator. Physically, such a phase-band singularity is essentially a "footprint" of the topological quantum criticality, which separates an AFHOTI from a trivial phase adiabatically connected to a static limit. We establish the above higher-order bulk-boundary correspondence through a dimensional reduction technique, which also allows for a systematic classification of 2D AFHOTIs protected by point group symmetries.   

Live

In twisted bilayer graphene (TBG), the choice of the twist angle allows for tailored engineering of the low energy absorption spectrum. This tunability makes TBG a perfect playground for Floquet engineering. Motivated by the measurement of ultrafast light-induced Hall currents in monolayer graphene [1], we investigate the topological properties of twisted bilayer graphene for an intermediate twisting angle in and out of equilibrium on the basis a full Moiré-unit-cell tight-binding model [2]. By breaking time-reversal symmetry with a circularly polarized light field, we induce a transition to a topologically non-trivial Floquet band structure with a Berry curvature analogous to a Chern insulator, which can be controlled via inversion-symmetry-breaking back-gate potentials. Additionally, I will discuss preliminary results of my ongoing work on light-matter couplings in magic-angle twisted bilayer graphene (MATBG).

[1] J. M. McIver et al., Nature Physics 16, 38-41 (2020)
[2] G. E. Topp et al. PRR 1, 023031 (2019)   

Live

We study the effects of circularly polarized light propagating in free space and confined in a waveguide on the band structure and topological properties of twisted double bilayer graphene (TDBG) samples. These two Floquet protocols allow us to selectively tune different parameters of the system by varying the intensity and light frequency. For the drive protocol in free space, we find that in TDBG with AB/BA stacking, we can selectively close the zone-center quasienergy gaps around one valley while increasing the gaps near the opposite valley by tuning the parameters of the drive. In TDBG with AB/AB stacking, a similar effect can be obtained upon the application of a perpendicular static electric field. We study the topological properties of the driven system in different settings, provide accurate effective Floquet Hamiltonians, and show that relatively strong drives can generate flat bands. Longitudinal light confined in a waveguide couples to the components of the interlayer hopping that are perpendicular to the TDBG sheet, allowing for selective engineering of the bandwidth of Floquet zone-center quasienergy bands.   

Live

Anderson localization in two-dimensional topological insulators takes place via the so-called levitation and pair annihilation process. As disorder is increased, extended bulk states carrying opposite topological invariants move towards each other in energy, reducing the size of the topological gap, eventually meeting and localizing. This results in a topologically trivial Anderson insulator. Here, we introduce the anomalous levitation and pair annihilation, a process unique to periodically driven, or Floquet, systems. Due to the periodicity of the quasienergy spectrum, we find it is possible for the topological gap to increase as a function of disorder strength. Thus, after all bulk states have localized, the system remains topologically nontrivial, forming an anomalous Floquet-Anderson insulator (AFAI) phase.   

Live

In many examples of topological phases, there exists a bulk-boundary correspondence that relates topological properties of the edge to topological properties of the bulk. In this talk, we present a bulk-boundary correspondence for non-interacting and interacting Floquet MBL systems in two spatial dimensions. Using this correspondence, we derive bulk invariants for several classes of Floquet systems, including interacting systems without symmetry and with U(1) symmetry, and non-interacting systems. The bulk invariants do not require translation symmetry or flux threading. In systems with $U(1)$ symmetry, the bulk invariant can be related to both a magnetization density and a conserved edge current.   

Live

Quantum walk, which is a kind of Floquet systems where time evolves in a discrete manner, can possess nontrivial topological phases. Recently, quantum walks with nonlinear effects have been proposed theoretically. Taking these features into account, we study the stability of topologically protected edge states in nonlinear quantum walks. In contrast to the previous work [1] which ignores the discrete-time nature, we analyze the stability taking the discreteness of time into account [2]. As a result, we find a new bifurcation where edge states change from stable attractors to unstable repellers. The bifurcation is unique to Floquet systems since it originates from the discreteness of time. Furthermore, because of the simpleness of the quantum walk, we analytically derive the bifurcation thresholds, which are generally difficult to obtain in a wide range of nonlinear systems.

[1] Y. Gerasimenko, B. Tarasinski, and C.W. J. Beenakker, Phys. Rev. A 93, 022329 (2016).
[2] K. Mochizuki, N. Kawakami, and H. Obuse, Journal of Physics A 53, 085702 (2020).   

Live

Flat electronic bands and Landau levels in equilibrium condensed matter systems have been common avenues to nontrivial correlation effects, with the twisted bilayer graphene being a most recent prominent example. It is expected that flat quasienergy bands can also enhance interaction effects in time-periodic Floquet systems and may lead to novel interaction driven metastable phases. Here, we propose a general approach to realizing Floquet flat bands and Landau levels with nontrivial topology in 2D or quasi-2D systems subject to circularly-polarized light. The Floquet flat bands and Landau levels in this work can be realized without the need of fine tuning in contrast to twisted bilayer graphene. Our proposal may pave the way to novel interaction-driven phases in nonequilibrium systems.   

Live

The magnetic proximity effect can open a gap in the spectrum of Dirac electrons at the surface of a topological insulator; on the other hand, topological defects in the magnetization can host topologically protected localized (isolated skyrmion) and propagating (domain walls) states. Here we argue that the electronic structure of Dirac electrons coupled to the skyrmion lattice phase in an insulating magnet (for example, Cu₂OSeO₃) can be described by the Chalker-Coddington network model (CCN) with the Kagome geometry. We study this model relying on a recent insight that CCN should be thought of as a chiral Floquet topological insulator. While in static systems the number of edge modes is completely determined by calculation of the Chern number for each energy band, in Floquet systems a chiral Floquet phase may exist in which there are edge modes even when the Chern number for each band is zero. We describe a band reconstruction procedure which allows us to show that the Dirac electrons in a Kagome geometry end up forming a Chern band.   

Live

We investigate Floquet topology in a tight-binding model on a finite-sized, two-dimensional, square lattice. Hoppings between neighboring sites vary in time according to a periodic protocol. An appropriate driving protocol makes the bulk of the system insulating while topologically protected edge currents flow around its borders.

A single particle in our two-dimensional system is equivalent to two particles in one dimension. If these two particles interact, they can form doublons, i.e., quasiparticles with both particles localized on the same lattice sites. To model interaction in the two-dimensional system, we modify the lattice sites on the diagonal by adding on-site potentials. This leads to energetically separated, and therefore robust, doublon states which allow the particle to travel along the diagonal.

We combine a driving protocol for the hoppings with a diagonal on-site potential. Small parameter changes flip between particle propagation along the edge and the diagonal.

A possible experimental realization of this Floquet topological system is in a photonic lattice. By flipping between the two propagation modes, light can be steered along the edges or the diagonal of the waveguide array.   

Live

Over the past decades, intriguing topological states have been extensively studied in electronic, photonic, ultracold atomic, and other systems. How topological edge states behave in the presence of inter-particle interactions and nonlinearity is an important and open question in this field. Here we demonstrate unidirectional soliton-like nonlinear states on the edge of photonic Floquet topological insulators formed by modulated optical waveguides. A well-controlled modulation of waveguide paths gives rise to a topologically nontrivial photonic band, characterized by an integer-valued topological invariant (winding number). Optical Kerr nonlinear interaction is introduced by using intense laser pulses. The observed non-diffracting, soliton-like wavepackets slowly radiate power because of the intrinsic gaplessness of the system. The rich and distinct localization characteristics measured as a function of nonlinearity confirms the existence of these soliton-like edge states. Our results are universal to other interacting bosonic systems, described by the focusing nonlinear Schrödinger equation or the attractive Gross Pitaevskii equation.


Ref. S. Mukherjee, M. C. Rechtsman, arXiv:2010.11359 [physics.optics] (2020).   

Live

Network models for equilibrium integer quantum Hall (IQH) transitions are described by unitary scattering matrices that can also be viewed as representing nonequilibrium Floquet systems. The resulting Floquet bands have zero Chern number, and are instead characterized by a chiral Floquet winding number. This begs the question, how can a model without Chern number describe IQH systems? We resolve this puzzle by showing that nonzero Chern number is recovered from the network model via the energy dependence of network model scattering parameters. This relationship shows that, despite their topologically distinct origins, IQH and chiral Floquet topology-changing transitions share identical universal scaling properties.   

Not Participating

The spectrum of periodically driven (i.e. “Floquet”) crystalline systems is characterized by energy bands in momentum space. Due to the discrete nature of their time-translation symmetry, Floquet systems conserve energy only modulo the angular frequency of the drive, resulting in the well-known compactification of the real energy axis into a looping “quasi-energy”. The resulting quasi-energy bands can exhibit topologically non-trivial gaps, robust band singularities, and anomalous boundary modes -- including ones that are impossible in static systems.

In this work [1], we derive a classification of singularities of quasi-energy bands. Besides the “band node” singularity known from the static case, we find that Floquet systems in the adiabatic limit further exhibit a novel type of singularity, which we dub “band screw”. We topologically characterize both species of singularities using homotopy theory. This task requires a generalization of the methods used by Ref. [2] in the static case, in particular by considering relative homotopy groups and multi-gap classifying spaces [3]. In this talk, our classification results would be presented.

[1] (in preparation, 2021)
[2] T. Bzdušek and M. Sigrist, PRB 96, 155105 (2017)
[3] A. Bouhon, T. Bzdušek, and R.-J. Slager, PRB 102, 115135 (2020)   

Live

Symmetry plays an important role in topological states. In a band insulator, usually the band topology is a global property of Bloch wavfunctions over the whole Brillouin zone. In the presence of crystalline symmetries, however, the band topology can sometimes be inferred just from knowledge of Bloch wavefunctions at high symmetry momenta in k space, known as symmetry indicators. In this work, we generalize the concept of symmetry indicators from spatial crystalline symmetries to spacetime symmetry in periodically driven Floquet systems. In the presence of spacetime screw symmetry, a combination of time translation and spatial rotation, we derive the symmetry indicators for Floquet topological insulators in 2+1-D. This provides a convenient tool to identify and engineer nontrivial Floquet topological phases in periodically driven materials.   

Live

Gauge pumps are spatially-resolved probes that can reveal discrete symmetries due to nontrivial topology. We introduce the Floquet gauge pump whereby a dynamically engineered Floquet Hamiltonian is employed to reveal the inherent topology of the ground state in interacting systems. We demonstrate this concept in a 1D XY model with periodically driven couplings and a transverse field. In the high-frequency limit, we obtain a Floquet Hamiltonian consisting of the static XY and dynamically generated Dzyaloshinsky-Moriya interactions (DMI) terms. We show that anisotropy in the couplings facilitates a magnetization current across a dynamically imprinted junction. In fermionic language, this corresponds to an unconventional Josephson junction with both hopping and pairing tunneling terms. The magnetization current depends on the phases of complex coupling terms, with the XY interaction as the real and DMI as the imaginary part. It shows 4π periodicity revealing the topological nature of the ground state manifold in the ordered phase, in contrast to the trivial topology in the disordered phase. We discuss the requirements to realize the Floquet gauge pump with interacting trapped ions.   

Live

Graphene antidot lattices are a convenient platform for exploring the properties of quantum-confined Dirac fermions in 2D. Changing the confinement strength by varying the geometric parameters (hole diameter and spacing) results in a widely tunable band gap. In similarity to pristine graphene [1], periodic driving with time-reversal symmetry breaking circularly polarized light has the potential to induce non-trivial topology in the quasienergy spectra. Based on the Floquet-Kubo formalism, the conductivity is calculated as the photon energy and electric field amplitude are varied across the topological transition, and constitutes a starting point for bridging between theory and future experiments. Furthermore, time-dependent density functional theory calculations are used to validate the simple Dirac Hamiltonian and Peierls substitution approaches. Floquet graphene antidot lattices may find application in solar cells, solid state spin qubit arrays, and parallel DNA sequencers.

[1] M. Schüler et al., Phys. Rev. X, 10, 041013 (2020)