Hopf characterization of two-dimensional Floquet topological insulators

Abstract

We present a topological characterization of time-periodically driven two-band models in 2+1 dimensions as Hopf insulators. The intrinsic periodicity of the Floquet system with respect to both time and the underlying two-dimensional momentum space constitutes a map from a three dimensional torus to the Bloch sphere. As a result, we find that the driven system can be understood by appealing to a Hopf map that is directly constructed from the micromotion of the drive. Previously found winding numbers are shown to correspond to Hopf invariants, which are associated with linking numbers describing the topology of knots in three dimensions. Moreover, after being cast as a Hopf insulator, not only the Chern numbers, but also the winding numbers of the Floquet topological insulator become accessible in experiments as linking numbers. We exploit this description to propose a feasible scheme for measuring the complete set of their Floquet topological invariants in optical lattices.

arXiv comment6 pages, 3 figures + 2 pages, 1 figure supplementary

References in corpus (45)

• Topological Insulators
• Topological insulators and superconductors
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Experimental realisation of the topological Haldane model
• Topological Crystalline Insulators
• Periodic table for topological insulators and superconductors
• Topological quantum chemistry
• Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices
• Topological characterization of periodically-driven quantum systems
• Atomic quantum gases in periodically driven optical lattices
• Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices
• Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• The space group classification of topological band insulators
• Topological classification of crystalline insulators through band structure combinatorics
• Observation of photonic anomalous Floquet Topological Insulators
• Experimental reconstruction of the Berry curvature in a topological Bloch band
• Experimental observation of anomalous topological edge modes in a slowly-driven photonic lattice
• Observation of a dynamical topological phase transition
• Periodic Table for Floquet Topological Insulators
• Bulk-boundary correspondence of topological insulators from their Green's functions
• Impurity Bound States and Greens Function Zeroes as Local Signatures of Topology
• Measuring topology by dynamics: Chern number from linking number
• Topological surface states in three-dimensional magnetic insulators
• Bloch state tomography using Wilson lines
• Uncover Topology by Quantum Quench Dynamics
• Measuring Topological Number of a Chern-Insulator from Quench Dynamics
• Universal probes of two-dimensional topological insulators: Dislocation and pi-flux
• Dynamical classification of topological quantum phases
• Tomography of band insulators from quench dynamics
• Measurement of a topological edge invariant in a microwave network
• Unified bulk-boundary correspondence for band insulators
• Hopf Insulators and Their Topologically Protected Surface States
• The translational side of topological band insulators
• Topological Bloch oscillations
• Symmetry Protected Topological Hopf Insulator and its Generalizations
• Topological Hopf-Chern Insulators and the Hopf Superconductor
• Driving protocol for a Floquet topological phase without static counterpart
• State-recycling and time-resolved imaging in topological photonic lattices
• How to Directly Measure Floquet Topological Invariants in Optical Lattices
• Self-organized pseudo-graphene on grain boundaries in topological band insulators
• Observation of topological links associated with Hopf insulators in a solid-state quantum simulator
• Phase vortices of the quenched Haldane Model
• Inner topological structure of Hopf invariant
• Three dimensional two-band Floquet topological insulator with $Z_2$ index

Cited by in corpus (56)

• Tailoring quantum gases by Floquet engineering
• Topological Euler class as a dynamical observable in optical lattices
• Geometric approach to fragile topology beyond symmetry indicators
• Topological correspondence between magnetic space group representations
• Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates
• Bimorphic Floquet Topological Insulators
• Topological Invariants for Quantum Quench Dynamics from Unitary Evolution
• Realization of a Hopf insulator in circuit systems
• Unified theory to characterize Floquet topological phases by quench dynamics
• Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase
• Nonadiabatic Topological Energy Pumps with Quasiperiodic Driving
• Multicellularity of delicate topological insulators
• Multi-gap topology and non-Abelian braiding of phonons from first principles
• Subdimensional topologies, indicators and higher order phases
• Teleportation of Berry curvature on the surface of a Hopf insulator
• Realizing Hopf Insulators in Dipolar Spin Systems
• Generating quantum multi-criticality in topological insulators by periodic driving
• $N$-band Hopf insulator
• Floquet engineering topological Dirac bands
• Phase transitions of non-Abelian charged nodal links in a spring-mass system
• Non-Abelian Hopf-Euler insulators
• Signatures of topology in quantum quench dynamics and their interrelation
• Massless multifold Hopf semimetals
• Nested-sphere description of the N-level Chern number and the generalized Bloch hypersphere
• Linking invariant for the quench dynamics of a two-dimensional two-band Chern insulator
• Interferometry of non-Abelian band singularities and Euler class topology
• Extrinsic and Intrinsic Nonlinear Hall Effects across Berry-Dipole Transitions
• Floquet and Anomalous Floquet Weyl Semimetals
• Three-dimensional $\mathcal{P}\mathcal{T}$-symmetric topological phases with Pontryagin index
• Real Hopf Insulator
• Wave packet dynamics and edge transport in anomalous Floquet topological phases
• Quench dynamics of Hopf insulators
• Dipolar Weyl semimetals
• Topological Micromotion of Floquet Quantum Systems
• Non-Hermitian three-dimensional two-band Hopf insulator
• Quantum quenches in a pseudo-Hermitian Chern insulator
• Estimating heating times in periodically driven quantum many-body systems via avoided crossing spectroscopy
• Floquet topological insulators with hybrid edges
• Second Euler number in four dimensional synthetic matter
• Flat Bands in Three-dimensional Lattice Models with Non-trivial Hopf Index
• Characterizing Floquet topological phases by quench dynamics: A multiple-subsystem approach
• Representation-protected topology of spin-singlet $s$-wave superconductors
• Non-Hermitian Hopf insulators
• Anomalous multi-gap topological phases in periodically driven quantum rotors
• PT-symmetric Non-Hermitian Hopf Metal
• Steering edge currents through a Floquet topological insulator
• Hopf Semimetals
• Emergent topological phases and coexistence of gapless and spectral-localized Floquet quantum spin Hall states via electron-phonon interaction
• Topological Phase Diagram of Optimally Shaken Honeycomb Lattices: A Dual Perspective from Stroboscopic and Non-Stroboscopic Floquet Hamiltonians
• Tuning the topology of $p$-wave superconductivity in an analytically solvable two-band model
• Disorder-induced delocalization and reentrance in a Chern-Hopf insulator
• Breakdown of Non-Bloch Bulk-Boundary Correspondence and Emergent Topology in Floquet Non-Hermitian Systems
• Phase transitions from Heating to non-heating in SU(1, 1) quantum dynamics: applications to Bose-Einstein condensates and periodically driven coupled oscillators
• Emergence of Nodal-Knot Transitions by Disorder
• Correlated Hopf insulators
• From orthogonal link to phase vortex in generalized dynamical Hopf insulators