Classical higher-order topological insulators

Abstract

Topological states nurtures the emergence of devices with unprecedented functions in photonics, plasmonics, acoustics and phononics. As one of the recently discovered members, higher-order topological insulators (HOTIs) have been increasingly explored, featuring lower-dimensional topological boundary states, leading to rich mechanisms for topological manipulation, guiding and trapping of classical waves. Here, we provide an overview of current developments of HOTIs in classical waves including basic principles, unique physical properties, various experimental realizations, novel phenomena and potential applications. Based on these discussions, we remark on the trends and challenges in this field and the impacts of higher-order topology on other research fields.

arXiv comment21 pages, 4 figures

References in corpus (169)

• Topological Insulators
• Topological insulators and superconductors
• Weyl and Dirac Semimetals in Three Dimensional Solids
• Topological Photonics
• Topological Photonics
• Photonic Floquet Topological Insulators
• Two-Dimensional Material Nanophotonics
• Classification of topological quantum matter with symmetries
• Topological Crystalline Insulators
• Quantized Electric Multipole Insulators
• Non-Hermitian Physics
• Scheme to Achieve Silicon Topological Photonics
• Higher-Order Topological Insulators
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Topological Acoustics
• Acoustic topological insulator and robust one-way sound transport
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Topolectrical circuit realization of topological corner modes
• Reflection symmetric second-order topological insulators and superconductors
• Observation of a phononic quadrupole topological insulator
• Non-Hermitian Chern bands
• Higher-order topological insulators and semimetals on the breathing Kagome and pyrochlore lattices
• Acoustic higher-order topological insulator on a Kagome lattice
• Higher-Order Topology in Bismuth
• Demonstration of a quantized microwave quadrupole insulator with topologically protected corner states
• A topological quantum optics interface
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Quantization of fractional corner charge in $C_n$-symmetric higher-order topological crystalline insulators
• A Thouless Quantum Pump with Ultracold Bosonic Atoms in an Optical Superlattice
• Topological Thouless Pumping of Ultracold Fermions
• Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quantum Computation with Majorana Zero Modes
• Observation of topologically protected bound states in a one dimensional photonic system
• Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals
• Nonlinear topological photonics
• Direct observation of corner states in second-order topological photonic crystal slabs
• Topological protection of photonic mid-gap cavity modes
• Photonic quadrupole topological phases
• Higher-order topological insulators and superconductors protected by inversion symmetry
• Second-order photonic topological insulator with corner states
• Visualization of unidirectional optical waveguide using topological photonic crystals made of dielectric material
• Floquet topological insulators
• Hybrid higher-order skin-topological modes in non-reciprocal systems
• Second-Order Topological Phases in Non-Hermitian Systems
• Novel Topological Phase with Zero Berry Curvature
• Observation of second-order topological insulators in sonic crystals
• Corner states of light in photonic waveguides
• Experimental observation of anomalous topological edge modes in a slowly-driven photonic lattice
• Second-order topological insulators and superconductors with an order-two crystalline symmetry
• Topological quantum matter in synthetic dimensions
• Interacting topological insulators: a review
• Elastic higher-order topological insulator with topologically protected corner states
• Photonic crystal nanocavity based on a topological corner state
• Realization of a three-dimensional photonic topological insulator
• Higher-order Topology of Axion Insulator EuIn${}_2$As${}_2$
• Higher-Order Topology, Monopole Nodal Lines, and the Origin of Large Fermi Arcs in Transition Metal Dichalcogenides XTe${}_2$ (X=Mo,W)
• Two-dimensional second-order topological insulator in graphdiyne
• Experimental discovery of bulk-disclination correspondence
• Acoustic realization of quadrupole topological insulators
• Higher-order topological corner states induced by gain and loss
• Low-threshold topological nanolasers based on second-order corner state
• Surface State Magnetization and Chiral Edge States on Topological Insulators
• Higher-Order Topological Insulator in Twisted Bilayer Graphene
• High-Temperature Majorana Corner States
• Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence
• Realization of an acoustic third-order topological insulator
• Higher-Order Topological Insulators in Quasicrystals
• Controlling the Spontaneous Emission Rate of Monolayer MoS${}_2$ in a Photonic Crystal Nanocavity
• Lasing from multipole topological corner states
• A fractional corner anomaly reveals higher-order topology
• Impurity Bound States and Greens Function Zeroes as Local Signatures of Topology
• Efficient Topological Materials Discovery Using Symmetry Indicators
• Helical Topological Edge States in a Quadrupole Phase
• Majorana Kramers pairs in higher-order topological insulators
• Higher-order topological insulators protected by inversion and rotoinversion symmetries
• Higher-Order Weyl Semimetals
• Non-linear second-order topological insulators
• Observation of an acoustic octupole topological insulator
• Minimal model for higher-order topological insulators and phosphorene
• Engineering Corner States from Two-Dimensional Topological Insulators
• Demonstration of a 3rd order hierarchy of higher order topological states in a three-dimensional acoustic metamaterial
• Demonstration of a quantized acoustic octupole topological insulator
• Higher-order topological semimetal in acoustic crystals
• Tunable Majorana corner states in a two-dimensional second-order topological superconductor induced by magnetic fields
• Surfaces of Axion Insulators
• Non-Hermitian route to higher-order topology in an acoustic crystal
• Majorana corner states in a two-dimensional magnetic topological insulator on a high-temperature superconductor
• Higher-order Weyl Semimetals
• Bound states in the continuum of higher-order topological insulators
• Dimensional hierarchy of higher-order topology in three-dimensional sonic crystals
• Higher Order Topology and Nodal Topological Superconductivity in Fe(Se,Te) Heterostructures
• Higher-order topological electric circuits and topological corner resonance on the breathing Kagome and pyrochlore lattices
• Observation of a phononic higher-order Weyl semimetal
• Non-Hermitian higher-order topological states in nonreciprocal and reciprocal systems with their electric-circuit realization
• Rotation Anomaly and Topological Crystalline Insulators
• Helical Hinge Majoranas in Iron-Based Superconductors
• Topolectrical-circuit octupole insulator with topologically protected corner states
• Symmetry Enforced Chiral Hinge States and Surface Quantum Anomalous Hall Effect in Magnetic Axion Insulator ${\text{Bi}}_{2-x}{\text{Sm}}_x{\text{Se}}_3$
• Lattice models with exactly solvable topological hinge and corner states
• Topological phases without crystalline counterparts
• Floquet second-order topological insulators from nonsymmorphic space-time symmetries
• Many-Body Electric Multipole Operators in Extended Systems
• Higher-order topological insulators in amorphous solids
• Quadrupole Topological Photonic Crystals
• Many-Body Order Parameters for Multipoles in Solids
• Higher-order topological odd-parity superconductors
• Topological switch between second-order topological insulators and topological crystalline insulators
• Dynamical Singularities of Floquet Higher-Order Topological Insulators
• Topological insulators in magnetic fields: Quantum Hall effect and edge channels with non-quantized θ-term
• Higher-order Floquet topological phases with corner and bulk bound states
• Coupled-wire construction of static and Floquet second-order topological insulators
• Evidence of Higher Order Topology in Multilayer WTe${}_2$ from Josephson Coupling through Anisotropic Hinge States
• Higher-order topological insulators in synthetic dimensions
• Floquet Higher-Order Topological Insulators with Anomalous Dynamical Polarization
• Symmetry-protected hierarchy of anomalous multipole topological band gaps in nonsymmorphic metacrystals
• Magnetic second-order topological insulators and semimetals
• Inversion-protected higher order topological superconductivity in monolayer WTe${}_2$
• Higher-order photonic topological states in surface-wave photonic crystals
• Higher order symmetry-protected topological states for interacting bosons and fermions
• Connecting higher-order topological insulators to lower-dimensional topological insulators
• Higher-order topological phases in tunable $C_3$-symmetric photonic crystals
• Braiding photonic topological zero modes
• Majorana-like zero modes in Kekulé distorted sonic lattices
• Cavity Quantum Electrodynamics with Second-Order Topological Corner State
• Higher-order Topological Mott Insulators
• Generating a second-order topological insulator with multiple corner states by periodic driving
• Braiding of Majorana corner states in electric circuits and its non-Hermitian generalization
• Non-Abelian braiding of light
• Coupling between Exciton-Polariton Corner Modes through Edge States
• Gapped Topological Kink States and Topological Corner States in Graphene
• Braiding Majorana corner modes in a second-order topological superconductor
• Phase diagram of disordered higher-order topological insulator: A machine learning study
• Higher-Order topological insulator in a dodecagonal quasicrystal
• Higher-order topological phases in a spring-mass model on a breathing kagome lattice
• Measurement-only quantum computation with Majorana corner modes
• Higher Order Topological Insulator via Periodic Driving
• Twisted quadrupole topological photonic crystals
• ${\mathbb{Z}}_Q$ Berry Phase for Higher-Order Symmetry-Protected Topological Phases
• Symmetry indicators of band topology
• Classification and construction of higher-order symmetry protected topological phases of interacting bosons
• Topological Bloch oscillations
• Evidence for Helical Hinge Zero Modes in an Fe-Based Superconductor
• Observation of symmetry protected zero modes in topolectrical circuits
• Floquet higher-order topological insulators and superconductors with space-time symmetries
• Vortex and Surface Phase Transitions in Superconducting Higher-order Topological Insulators
• Quantization in chiral higher order topological insulators: circular dichroism and local Chern marker
• Splitting the hinge mode of higher-order topological insulators
• Topological classification under nonmagnetic and magnetic point group symmetry: Application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology
• Higher order topological superconductors as generators of quantum codes
• Plasmon-polaritonic quadrupole topological insulators
• Spin-orbit-parity coupled superconductivity in topological monolayer WTe${}_2$
• Inversion-symmetry protected chiral hinge states in stacks of doped quantum Hall layers
• Corner states and topological transitions in two-dimensional higher-order topological sonic crystals with inversion symmetry
• `Unhinging' the surfaces of higher-order topological insulators and superconductors
• High-order topological insulators from high-dimensional Chern insulators
• Second-order topological corner states with ultracold atoms carrying orbital angular momentum in optical lattices
• Interaction-induced lattices for bound states: Designing flat bands, quantized pumps and higher-order topological insulators for doublons
• Kane-Mele with a twist: Quasicrystalline higher-order topological insulators with fractional mass kinks
• Three-dimensional higher-order topological acoustic system with multidimensional topological states
• Type-II quadrupole topological insulators
• Non-Abelian Bloch oscillations in higher-order topological insulators
• Anomalous Quadrupole Topological Insulators in 2D Nonsymmorphic Sonic Crystals
• Nonlinear higher-order polariton topological insulator
• Wannier-type photonic higher-order topological corner states induced solely by gain and loss
• Higher-order topological spin Hall effect of sound
• Fractional Chiral Hinge Insulator
• Multidimensional sound propagation in 3D high-order topological sonic insulator
• Hinge Magnons from Non-collinear Magnetic Order in Honeycomb Antiferromagnet
• Second-order topological insulator in a coinless discrete-time quantum walk
• Protecting Quantum Superposition and Entanglement with Photonic Higher-Order Topological Crystalline Insulator

Cited by in corpus (137)

• Topological acoustics
• Applications of Bound States in the Continuum in Photonics
• Tailoring Structure-borne Sound Through Bandgap Engineering in Phononic Crystals and Metamaterials: A Comprehensive Review
• Controlling mass and energy diffusion with metamaterials
• Experimental realization of single-plaquette gauge flux insertion and topological Wannier cycles
• Acoustic Möbius insulators from projective symmetry
• Photonic quadrupole topological insulator using orbital-induced synthetic flux
• A Universal Mirror-stacking Approach for Constructing Topological Bound States in the Continuum
• Second Order Topological Insulator State in Hexagonal Lattices and its Abundant Material Candidates
• Kekule Lattice in Graphdiyne: Coexistence of Phononic and Electronic Higher-Order Band Topology
• Field-Tunable One-Sided Higher-Order Topological Hinge States in Dirac Semimetals
• Observation of nonlinear disclination states
• Topological thermal transport
• Realization of a Z classified chiral-symmetric higher-order topological insulator in a coupling-inversion acoustic crystal
• Higher-order topological states mediated by long-range coupling in $D_4$-symmetric lattices
• Large-area quantum-spin-Hall waveguide states in a three-layer topological photonic crystal heterostructure
• Non-Hermitian higher-order topological superconductors in two-dimension: statics and dynamics
• Higher-order topological phases in crystalline and non-crystalline systems: a review
• Multifrequency and multimode topological waveguides in a Stampfli-triangle photonic crystal with large valley Chern numbers
• Floquet Weyl semimetal phases in light-irradiated higher-order topological Dirac semimetals
• Acoustic corner states in topological insulators with built-in Zeeman-like fields
• Realization of Surface-Obstructed Topological Insulators
• Ultra-sensitive integrated circuit sensors based on high-order nonHermitian topological physics
• An operator-based approach to topological photonics
• Observation of Topological Corner State Arrays in Photonic Quasicrystals
• Observation of $\pi$ solitons in oscillating waveguide arrays
• Topological photonic crystal fibers based on second-order corner modes
• Systematic generation of the cascade of anomalous dynamical first and higher-order modes in Floquet topological insulators
• Chiral edge modes in evolutionary game theory: a kagome network of rock-paper-scissors
• Higher-order topological Anderson insulators in quasicrystals
• Observation of nonlinear fractal higher-order topological insulator
• Local markers for crystalline topology
• Thouless Pumps and Bulk-Boundary Correspondence in Higher-Order Symmetry-Protected Topological Phases
• Observation of Large-Number Corner Modes in $\mathbb{Z}$-class Higher-Order Topolectrical Circuits
• Generalized bulk-boundary correspondence in periodically driven non-Hermitian systems
• Dynamical construction of Quadrupolar and Octupolar topological superconductors
• Nonlinear photonic disclination states
• Generation of higher-order topological insulators using periodic driving
• A robust and tunable Luttinger liquid in correlated edge of transition-metal second-order topological insulator Ta${}_2$Pd${}_3$Te${}_5$
• Unified characterization for higher-order topological phase transitions
• Sublattice-enriched tunability of bound states in second-order topological insulators and superconductors
• Wannier Function Methods for Topological Modes in 1D Photonic Crystals
• Customizing pseudospin unidirectional states of acoustic and electromagnetic waves in two-dimensional phoxonic topological insulators via multi-objective strategies
• Vortex solitons in topological disclination lattices
• Non-Hermitian Second-Order Topological Phases and Bipolar Skin Effect in Photonic Kagome Crystals
• Dynamical bulk boundary correspondence and dynamical quantum phase transitions in higher order topological insulators
• Density-driven higher-order topological phase transitions in amorphous solids
• Observation of nonlinearity-controlled switching of topological edge states
• Topological properties of two-dimensional photonic square lattice without $C_4$ and $M_{x(y)}$ symmetries
• Exact Solutions Disentangle Higher-Order Topology in 2D Non-Hermitian Lattices
• Engineering non-Hermitian Second Order Topological Insulator in Quasicrystals
• Observation of squeezed Chern insulator in an acoustic fractal lattice
• Non-Hermitian second-order topological superconductors
• Experimental quantum simulation of non-Hermitian dynamical topological states using stochastic Schrödinger equation
• Floquet second-order topological Anderson insulator hosting corner localized modes
• Efficient Design of Helical Higher-Order Topological Insulators in 3D Elastic Medium
• Mirror real Chern insulator in two and three dimensions
• Dirac cones and higher-order topology in quasi-continuous media
• Dynamical Detection of Topological Spectral Density
• Topological solitonic macromolecules
• Observation of gapless corner modes in synthetic translation dimensions
• Observation of Dual-band Topological Corner Modes in Acoustic Kagome Lattice with Long-range Interactions
• Tailoring Quadrupole Topological Insulators with Periodic Driving and Disorder
• Multiple higher-order topological phases with even and odd pairs of zero-energy corner modes in a $C_3$ symmetry broken model
• Realizing and detecting Stiefel-Whitney insulators in an optical Raman lattice
• Time evolution of Majorana corner modes in Floquet second-order topological superconductor
• Generating many Majorana corner modes and multiple phase transitions in Floquet second-order topological superconductors
• Planar Hall Plateau in Magnetic Weyl Semimetals
• Nonlinear Corner States in Topologically Nontrivial Kagome Lattice
• Observation of nonlinear topological corner states originating from different spectral charges
• Magnetic field filtering of the hinge supercurrent in unconventional metal NiTe${}_2$-based Josephson junctions
• Topological gap solitons in Rabi Su-Schrieffer-Heeger lattices
• Topological Wannier cycles for the bulk and edges
• Topological edge states in photonic Floquet insulator with unpaired Dirac cones
• Probing Chiral-Symmetric Higher-Order Topological Insulators with Multipole Winding Number
• Gapless higher-order topology and corner states in Floquet systems
• Transport features of topological corner states in honeycomb lattice with multihollow structure
• Remote control of spin polarization of topological corner states
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• Competing topological phases in a non-Hermitian time-reversal symmetry-broken Bernevig-Hughes-Zhang model
• Two-Dimensional Altermagnetic Iron Oxyhalides: Real Chern Topology and Valley-Spin-Lattice Coupling
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• Microwave hinge states in a simple-cubic-lattice photonic crystal insulator
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• Orbital Edelstein effect of electronic itinerant orbital motion at edges
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• On the role of geometric phase in the dynamics of elastic waveguides
• Observation of nonlinear higher-order topological insulators with unconventional boundary truncations
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• Hybrid-Order Topological Phase And Transition in 1H Transition Metal Compounds
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