Chiral symmetry and bulk--boundary correspondence in periodically driven one-dimensional systems

Abstract

Over the past few years, topological insulators have taken center stage in solid state physics. The desire to tune the topological invariants of the bulk and thus control the number of edge states has steered theorists and experimentalists towards periodically driving parameters of these systems. In such periodically driven setups, by varying the drive sequence the effective (Floquet) Hamiltonian can be engineered to be topological: then, the principle of bulk--boundary correspondence guarantees the existence of robust edge states. It has also been realized, however, that periodically driven systems can host edge states not predicted by the Floquet Hamiltonian. The exploration of such edge states, and the corresponding topological phases unique to periodically driven systems, has only recently begun. We contribute to this goal by identifying the bulk topological invariants of periodically driven one-dimensional lattice Hamiltonians with chiral symmetry. We find simple closed expressions for these invariants, as winding numbers of blocks of the unitary operator corresponding to a part of the time evolution, and ways to tune these invariants using sublattice shifts. We illustrate our ideas on the periodically driven Su-Schrieffer-Heeger model, which we map to a discrete time quantum walk, allowing theoretical results about either of these systems to be applied to the other. Our work helps interpret the results of recent simulations where a large number of Floquet Majorana fermions in periodically driven superconductors have been found, and of recent experiments on discrete time quantum walks.

References in corpus (21)

• Photonic Floquet Topological Insulators
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Floquet Topological Insulator in Semiconductor Quantum Wells
• Observation of Floquet-Bloch states on the surface of a topological insulator
• Topological characterization of periodically-driven quantum systems
• Topological States and Adiabatic Pumping in Quasicrystals
• Transport properties of non-equilibrium systems under the application of light: Photo-induced quantum Hall insulators without Landau levels
• Majorana Fermions in Equilibrium and Driven Cold Atom Quantum Wires
• Observation of topologically protected bound states in a one dimensional photonic system
• Exploring Topological Phases With Quantum Walks
• Floquet-Bloch theory and topology in periodically driven lattices
• Non-Abelian gauge fields and topological insulators in shaken optical lattices
• Topological Criticality in the Chiral-Symmetric AIII Class at Strong Disorder
• Floquet generation of Majorana end modes and topological invariants
• Bulk--Boundary Correspondence for Chiral Symmetric Quantum Walks
• Transport signatures of Floquet Majorana fermions in driven topological superconductors
• Symmetries, Topological Phases and Bound States in the One-Dimensional Quantum Walk
• Optically engineering the topological properties of a spin Hall insulator
• Generating Many Majorana Modes via Periodic Driving: A Superconductor Model
• Floquet Edge States with Ultracold Atoms
• Scattering theory of topological phases in discrete-time quantum walks

Cited by in corpus (177)

• Topological Bands for Ultracold Atoms
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• Absolute Stability and Spatiotemporal Long-Range Order in Floquet systems
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• Classification of Interacting Topological Floquet Phases in One Dimension
• Classification of topological phases in periodically driven interacting systems
• Topology and Broken Symmetry in Floquet Systems
• Phase Structure of 1d Interacting Floquet Systems I: Abelian SPTs
• Topological invariants of Floquet systems: General formulation, special properties, and Floquet topological defects
• Floquet topological transitions in a driven one-dimensional topological insulator
• Observation of anomalous π modes in photonic Floquet engineering
• Topological characterization of chiral models through their long time dynamics
• Anomalous Topological Phases and Unpaired Dirac Cones in Photonic Floquet Topological Insulators
• Higher-order Floquet topological phases with corner and bulk bound states
• Floquet Topological Phases of Non-Hermitian Systems
• Abelian Floquet symmetry-protected topological phases in one dimension
• Floquet topological phases protected by time glide symmetry
• Complex classes of periodically driven topological lattice systems
• Floquet engineering of long-range p-wave superconductivity
• Floquet Topological Order in Interacting Systems of Bosons and Fermions
• Topological Effects in Chiral Symmetric Driven Systems
• Generating a second-order topological insulator with multiple corner states by periodic driving
• Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk
• Nonlinearity induced topological physics in momentum space and real space
• Anomalous bulk-edge correspondence in continuous media
• Topological classification of quasi-periodically driven quantum systems
• Scattering matrix invariants of Floquet topological insulators
• Space-time crystal and space-time group
• Dynamically manipulating topological physics and edge modes in a single degenerate optical cavity
• Measuring a Dynamical Topological Order Parameter in Quantum Walks
• Quantum computation via Floquet topological edge modes
• Dynamically enriched topological orders in driven two-dimensional systems
• Floquet higher order topological insulator in a periodically driven bipartite lattice
• Robustness of topologically protected edge states in quantum walk experiments with neutral atoms
• Quantized magnetization density in periodically driven systems
• Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk
• Defining Time Crystals via Representation Theory
• Measuring the winding number in a large-scale chiral quantum walk
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• Observation of pi/2 modes in an acoustic Floquet system
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• Digital Simulation of Topological Matter on Programmable Quantum Processors
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• Observation of $\pi$ solitons in oscillating waveguide arrays
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• Gauge-induced Floquet topological states in photonic waveguides
• Edge and bulk localization of Floquet topological superconductors
• Topological states and quantized current in helical molecules
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• Line nodes and surface Majorana flat bands in static and kicked p-wave superconducting Harper model
• Square-root Floquet topological phases and time crystals
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• Floquet higher-order Weyl and nexus semimetals
• Edge states in dynamical superlattices
• Central Charge of Periodically Driven Critical Kitaev Chains
• Topology and localization of a periodically driven Kitaev model
• Dynamical relaxation of correlators in periodically driven integrable quantum systems
• Disordered Chern insulator with a two step Floquet drive
• Chiral 1D Floquet topological insulators beyond rotating wave approximation
• Persistence of Topological Phases in Non-Hermitian Quantum Walks
• Revivals in Quantum Walks with quasi-periodically time-dependent coin
• Edge mode dynamics of quenched topological wires
• Attractor-repeller pair of topological zero-modes in a nonlinear quantum walk
• Bulk-edge correspondence in nonunitary Floquet systems with chiral symmetry
• Topological boundary invariants for Floquet systems and quantum walks
• High-fidelity and long-distance entangled-state transfer with Floquet topological edge modes
• Critical phase boundaries of static and periodically kicked long-range Kitaev chain
• Topological quantum walks: theory and experiments
• Counter-propagating edge states in Floquet topological insulating phases
• Edge mode manipulation through commensurate multifrequency driving
• Generating end modes in a superconducting wire by periodic driving of the hopping
• Quantum Control in Open and Periodically Driven Systems
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• Topological Signatures in Quantum Transport in Anomalous Floquet-Anderson Insulators
• Hybrid-order topological odd-parity superconductors via Floquet engineering
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• Topology of critical chiral phases: multiband insulators and superconductors
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• Effective vacua for Floquet topological phases: A numerical perspective on switch-function formalism
• Strong zero modes in integrable quantum circuits
• Chiral Flow in One-dimensional Floquet Topological Insulators
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• Robust edge states induced by electron-phonon interaction in graphene nanoribbons
• Floquet second-order topological Anderson insulator hosting corner localized modes
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• Floquet edge solitons in modulated trimer waveguide arrays
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• Floquet Supersymmetry
• Quasiperiodic disorder induced critical phases in a periodically driven dimerized $p$-wave Kitaev chain
• Floquet topological insulators: from band structure engineering to novel non-equilibrium quantum phenomena
• Transport signatures in topological systems coupled to AC fields
• A Universal Model of Floquet Operator Krylov Space
• Generating many Majorana corner modes and multiple phase transitions in Floquet second-order topological superconductors
• Interacting Floquet topological phases in three dimensions
• Stability of topologically protected edge states in nonlinear quantum walks: Additional bifurcations unique to Floquet systems
• Combating quasiparticle poisoning with multiple Majorana fermions in a periodically-driven quantum wire
• Topological properties of a periodically driven Creutz ladder
• Fate of Topological Edge States in Disordered Periodically-driven Nonlinear Systems
• Simulating Floquet topological phases in static systems
• Topological Phases of Tight-Binding Trimer Lattice in the BDI Symmetry Class
• Transport signatures of symmetry protection in 1D Floquet topological insulators
• The Streda Formula for Floquet Systems: Topological Invariants and Quantized Anomalies from Cesaro Summation
• Unsupervised identification of Floquet topological phase boundaries
• Disconnected entanglement entropy as a marker of edge modes in a periodically driven Kitaev chain
• Non-Floquet engineering in periodically driven non-Hermitian systems
• Mode-Shell correspondence, a unifying phase space theory in topological physics -- Part I: Chiral number of zero-modes
• Anomalous Floquet Phases. A resonance phenomena
• Odd-frequency superconducting pairing due to multiple Majorana edge modes in driven topological superconductors
• Symmetry Protected Topological Phases and Majorana Mode in One-dimensional Quantum Walk with Boundary
• Local and energy-resolved topological invariants for Floquet systems
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• Light driven magnetic transitions in transition metal dichalcogenide heterobilayers
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• Topological reflection matrix
• Single and multi-frequency driving protocols in a Rashba nanowire proximitized to an s-wave superconductor
• Driven Topological Systems in the Classical Limit
• Disorder induced topological phase transition in a driven Majorana chain
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• Floquet valley Hall edge solitons
• Revivals in One-dimensional Quantum Walks with a Time and Spin-dependent Phase Shift
• Dynamical Chiral Symmetry and Symmetry-Class Conversion in Floquet Topological Insulators
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• Resonant edge-site pumping of polaritonic Su-Schrieffer-Heeger lattices
• Non-Abelian Anyons in Periodically Driven Abelian Spin Liquids
• Designing nontrivial one-dimensional Floquet topological phases using a spin-1/2 double-kicked rotor
• Moment method and continued fraction expansion in Floquet Operator Krylov Space
• Floquet defect solitons
• Investigating the quench dynamics of the bound states in a spin-orbital coupling system using a trapped ion
• Quantum Walks in Weak Stochastic Gauge Fields
• Topology reconstruction for asymmetric systems by isomorphic mapping or perturbation approximation
• Generalized Majorana edge modes in a number-conserving periodically driven $p$-wave superconductor
• Emergent topological phases and coexistence of gapless and spectral-localized Floquet quantum spin Hall states via electron-phonon interaction
• Steering edge currents through a Floquet topological insulator
• Proposal for realizing anomalous Floquet insulators via Chern band annihilation
• Quantum-vacuum-protected topological edge polaritons
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• Floquet Non-Bloch Formalism for a Non-Hermitian Ladder: From Theoretical Framework to Topolectrical Circuits
• Strong and almost strong modes of Floquet spin chains in Krylov subspaces
• Floquet states on disclinations
• Low-frequency and Moiré Floquet engineering: a review
• Symmetry-protected topological corner modes in a periodically driven interacting spin lattice
• Green Function Invariants for Floquet Topological Superconductivity Induced by Proximity Effects
• Dynamical Signatures of Floquet Topology in Wave Packet Dynamics
• Wilson loop invariants and bulk-boundary correspondence in higher-order topological insulators with two anticommuting mirror symmetries
• Noise resilience of two-dimensional Floquet topological phases
• Breakdown of the symmetry constraint in a Floquet topological insulator
• Digital Quantum Simulation of Floquet Topological Phases with a Solid-State Quantum Simulator