Session X58: Topological Phases in Floquet Systems

Phonon-assisted Floquet engineering of second-order topological phases

The co-existence of spatial and non-spatial symmetries together with appropriate commutation/anticommutation relations between them can give rise to static higher-order topological phases, which host gapless boundary modes of co-dimension higher than one. Alternative to spatial symmetries, space-time symmetries in a Floquet system can lead to anomalous Floquet boundary modes of higher co-dimensions, presumably with alterations in the commutation/anticommutation relations with respect to non-spatial symmetries. In this work, we discuss how a coherently excited phonon mode can be used to promote a spatial symmetry with which the static system is always trivial, to a space-time symmetry which supports non-trivial Floquet higher-order topological phase. We present two examples– one in class AIII and another in class D where a coherently excited phonon mode promotes the reflection symmetry to a time-glide symmetry such that the commutation/anticommutation relations between spatial and non-spatial symmetries are modified. These altered relations allow the previously trivial system to host gapless modes of co-dimension two at reflection-symmetric boundaries.   

Majorana states in a spin or qubit chain with a resonantly modulated coupling

We show that a spin or qubit chain with the qubit coupling modulated close to twice the qubit frequency maps on the Kitaev chain. This is an example of topologically nontrivial Floquet dynamics induced in a trivial system by a simple sinusoidal modulation. The nontrivial regime exists in the frequency range with a width determined by the strength of the qubit coupling. Preparation of the system in this regime involves going through a quantum phase transition where the excitation gap closes, in the limit of an infinite chain. The results are compared with a quantum phase transition induced by a resonant parametric modulation of a system of coupled nonlinear mesoscopic oscillators, which remains topologically trivial [1].
1. M. I. Dykman, Christoph Bruder, Niels Lorch, and Yaxing Zhang, Phys. Rev. B 98 , 195444 (2018)   

A periodic classification of local unitary operators

We present a complete topological classification of single particle local unitary operators, which are usually represented by local unitary matrices. The classification is performed in the ten symmetry classes and leads to a “periodic table”. Local unitary operators arise naturally in systems with an evolution generated by a quantum possibly time-varying (Hermitian) Hamiltonian. They also describe unitary operations in more general contexts, such as in quantum walks or models for information flow. The classification is complete in the sense that two local unitary operators have the same topological invariants if and only if they can be continuously deformed into each other. It allows one to distinguish “locally generated” unitary operators in a given symmetry class (generated by local Hamiltonians) from those that are not locally generated and provides a complete table of local operators which are not locally generated.   

Classification of Interacting Floquet Phases with U(1) Symmetry

Recently, it has become clear that periodically driven many-body systems can support new types of topological phases that have no counterpart in static systems. These anomalous Floquet topological phases have been studied for both interacting and non-interacting systems in different symmetry classes. In this talk, we consider interacting Floquet topological phases with U(1) charge conservation symmetry in two spatial dimensions. We propose a complete classification of Floquet phases of this kind, in the case where the Floquet eigenstates are short-range entangled, and we show that each phase can be labeled by a topological invariant which can be computed either at the edge of an open system or in the bulk. The latter formula provides a way to measure the invariant and also reveals a connection to the bulk magnetization density discussed in previous work.   

Emergent topological phases in synthetic dimensions with low-frequency laser pumping

In the Floquet-Bloch theory, when a physical system is driven by a strong external optical field, its dynamics can be conveniently represented in a higher dimensional Floquet lattice. Here, we consider the 2D Dirac systems under strong external optical driving. In the low-frequency limit, new topological states of matter emerge in the synthetic 2+1D, including Dirac nodal line, helix nodal lines, and Weyl fermions. In contrast to conventional Floquet topological states under the high-frequency limit, there is no anomalous Hall signal observed in this system although the time-reversal symmetry is broken and the Floquet energy gap opens at the crossing point. Furthermore, we use the quantum Liouville equation with phenomenological dissipation to simulate the evolution of 2D Dirac fermions under the strong optical field with low frequency, we confirm the existence of the Floquet states and find the simulated transport properties consistent with our prediction from Floquet theory in synthetic 2+1D.   

Floquet second-order topological insulator

Higher-order topological insulators (HOTIs) are a recent entrant to the field of topology in condensed matter physics. Usually, two-dimensional topological insulators host robust one-dimensional edge modes. These are related to the bulk properties via topological invariants such as the Chern number. However, in HOTIs, there are zero-dimensional corner modes, which, in case of a square-shaped sample, say, are confined to the four vertices of the square. In this work, a variant of the well-known Bernevig-Hughes-Zhang (BHZ) model is used to construct a two-dimensional HOTI. When this system is driven periodically by varying one of the model parameters in time, i.e., using a Floquet drive, multiple corner states may appear depending on the driving frequency and other parameters. The system can be characterized by topological invariants such as the Chern number and a diagonal winding number. The behavior of these invariants can be understood in terms of the value of the Floquet operator at some special points in momentum space.   

Out of equilibrium higher-order topological insulator: Floquet engineering and quench dynamics

Higher order topological (HOT) states, hosting topologically protected modes on lower-dimensional boundaries, such as hinges and corners, have recently extended the realm of the static topological phases. We here demonstrate the possibility of realizing a two-dimensional \emph{Floquet} second-order topological insulator, featuring corner localized zero quasienergy modes and characterized by quantized Floquet qudrupolar moment $Q^{\rm Flq}_{xy}=0.5$, by periodically kicking a quantum spin Hall insulator (QSHI) with a discrete four-fold ($C_4$) and time-reversal (${\mathcal T}$) symmetry breaking mass perturbation. We also analyze the dynamics of a corner mode after a sudden quench, when the $C_4$ and ${\mathcal T}$ symmetry breaking perturbation is switched off, and find that the corresponding survival probability displays periodic appearances of complete, partial and no revival for long time, encoding the signature of corner modes in a QSHI. Our protocol is sufficiently general to explore the territory of dynamical HOT phases in insulators (electrical and thermal) and gapless systems.   

Floquet topological flat bands in two-dimensional systems

Flat electronic bands in equilibrium condensed matter systems have been a common avenue 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 with nontrivial topology in 2D or quasi-2D systems subject to circularly-polarized light. By using a simple model that can interpolate between Schrödinger and Dirac electrons, we demonstrate their different band flattening behaviors within Floquet theory. The flat band condition, determined by the ratio between the time-periodic electric field strength and its frequency, can be qualitatively obtained from perturbation theory. Moreover, flat bands argued 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.   

Topological Correspondence between Hermitian and Non-Hermitian Systems: Anomalous Dynamics

The hallmark of symmetry-protected topological (SPT) phases is the existence of anomalous boundary states, which can only be realized with the corresponding bulk system. In this work, we show that for every Hermitian anomalous boundary mode of the ten Altland-Zirnbauer classes, a non-Hermitian counterpart can be constructed, whose long time dynamics provides a realization of the anomalous boundary state. We prove that the non-Hermitian counterpart is characterized by a point-gap topological invariant, and furthermore, that the invariant exactly matches that of the corresponding Hermitian anomalous boundary mode. We thus establish a correspondence between the topological classifications of $(d+1)$-dimensional gapped Hermitian systems and $d$-dimensional point-gapped non-Hermitian systems. We illustrate this general result with a number of examples in different dimensions. This work provides a new perspective on point-gap topological invariants in non-Hermitian systems.   

Bulk-edge correspondence and robustness of edge states in a non-unitary three-step quantum walk with PT symmetry

Topological phases in non-Hermitian systems have attracted great attention in recent years. It has been shown that a photonic quantum walk with effects of loss is an ideal platform to study topological phases in non-Hermitian systems with parity-time reversal symmetry (PT symmetry)[1]. In the present work, we study topological phases and the associated multiple edge states in non-unitary three-step quantum walks with PT symmetry in one dimension[2]. We show that the non-unitary quantum walk has large topological numbers and numerically confirm that multiple edge states appear as expected from the bulk-edge correspondence. We also study stability of the edge states and find extra stabilization mechanism of the edge states unique to non-unitary systems.

[1] L. Xiao, X. Zhan, Z.H. Bian, K.K. Wang, X.Zhang, X.P.Wang, J.Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B.C. Sanders, and P. Xue, Nature Physics, 13, 1117 (2017).
[2] M. Kawasaki, K. Mochizuki, N. Kawakami, H. Obuse, arXiv:1905.11098   

A bulk edge connection for Class AIII Floquet insulators

We propose edge and bulks invariants for Floquet systems in Class AIII in arbitrary dimensions. These indices are physically motivated and locally computable even in systems with disorder. Finally, we derive a bulk-edge correspondence which relates the nontrivial bulk behavior with the edge modes present on a boundary at the end of the evolution.   

Kagome lattice network model as a chiral Floquet topological insulator

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 there may be edge modes even when the Chern number for each band is zero. We describe a new topological invariant which is a modified version of the chiral Floquet invariant proposed by Rudner et al. and does not require the construction of an effective Hamiltonian, and which correctly counts the number of edge modes in each spectral gap. We apply this invariant to the Kagome lattice network model and show that it includes both Chern and chiral Floquet phases.   

Analysis of Topological States in a Floquet-driven Non-Hermitian System

Non-Hermitian Hamiltonians offer a good description of many open systems in which gain and loss are present; crucially, in contrast to their Hermtian counterparts, they may have a complex eigenspectrum. Interestingly, non-Hermitian Hamiltonians which possess PT symmetry [1] can be shown to admit an entirely real eigenspectrum within a certain range of their parameters. It has been shown [2] that certain PT-symmetric lattices can admit topologically non-trivial phases; however, this phase only coincides with the PT-symmetry broken phase, and the topological edge states correspond to imaginary eigenvalues. We examine Floquet driving of this system which, for high enough driving frequencies [3], has been shown to stabilize the edge states. By using a simple two-step, pulsed time dependence, we explore the entire range of driving frequencies to highlight new regions of stability, including those which are explicitly below the high-frequency regime.

[1] Bender, C. and Boettcher, S. Phys. Rev. Lett. 80, 5243 (1998)
[2] Rudner, M. S. and Levitov, L. S. Phys. Rev. Lett. 102, 065703 (2009)
[3] C. Yuce, Eur. Phys. J. D 69, 184 (2015)   

Light-induced Topological Insulator with Superlattice Structure

We study the superlattice structure created by light on 2D material. Specifically, we investigate the monolayer graphene irradiated by circularly-polarized (CP) beam where the beam amplitude has spatial periodicity. We numerically study the phase transition between the topological insulator and the normal insulator depending on the superlattice size. To examine the role of lattice shape in the topological property of the system, we investigate the topological phase transition happening in the sheared lattice. By superposing different kinds of beams, our scheme can create a more complicated lattice such as kagome lattice. This wide tunability of lattice can be used for further investigation of interesting dynamics of electrons.   

High-temperature Floquet fractional quantum Hall state in a Kagome lattice with photo-inverted hopping

High-temperature fractional quantum Hall (FQH) state is predicted to emerge from geometrically frustrated lattices with topological flat bands, but yet to be demonstrated in realistic systems due to very unusual lattice hopping requirements. Here we show that time-periodic circularly polarized laser (CPL) can effectively invert second-nearest-neighbor kinetic hopping in a Kagome lattice and simultaneously enhance spin-orbit coupling (SOC) in one spin channel, so as to produce an isolated flat Chern band exhibiting high-temperature Floquet FQH effect. CPL-driven monolayer HTT-Pt possesses an unprecedented high flatness ratio Δ/w ~ 40, which paves the way to observing the high-temperature FQH state in realistic 2D systems. Meanwhile, the CPL decreases the SOC strength in the other spin channel to form a chiral Kagome band with a high Chern number. Our approach may be generally applicable to engineering exotic topological quantum states in other crystal lattices.