Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators

Abstract

We extend the theory of dipole moments in crystalline insulators to higher multipole moments. In this paper, we expand in great detail the theory presented in Ref. 1, and extend it to cover associated topological pumping phenomena, and a novel class of 3D insulator with chiral hinge states. In quantum-mechanical crystalline insulators, higher multipole bulk moments manifest themselves by the presence of boundary-localized moments of lower dimension, in exact correspondence with the electromagnetic theory of classical continuous dielectrics. In the presence of certain symmetries, these moments are quantized, and their boundary signatures are fractionalized. These multipole moments then correspond to new SPT phases. The topological structure of these phases is described by "nested" Wilson loops, which reflect the bulk-boundary correspondence in a way that makes evident a hierarchical classification of the multipole moments. Just as a varying dipole generates charge pumping, a varying quadrupole generates dipole pumping, and a varying octupole generates quadrupole pumping. For non-trivial adiabatic cycles, the transport of these moments is quantized. An analysis of these interconnected phenomena leads to the conclusion that a new kind of Chern-type insulator exists, which has chiral, hinge-localized modes in 3D. We provide the minimal models for the quantized multipole moments, the non-trivial pumping processes and the hinge Chern insulator, and describe the topological invariants that protect them.

arXiv comment72 pages, 52 figures

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• Hinged Quantum Spin-Hall Effect in Antiferromagnetic Topological Insulators
• Hidden Breathing Kagome Topology in Hexagonal Transition Metal Dichalcogenides
• Theory of inversion-${\mathbb{Z}}_4$ protected topological chiral hinge states and its applications to layered antiferromagnets
• Topological Mechanics from Supersymmetry
• Topological excitonic corner states and nodal phase in bilayer quantum spin Hall insulators
• Existence of Corner Modes in Elastic Twisted Kagome Lattices
• Symmetry-enforced Band Nodes in 230 Space Groups
• Appearance of hinge states in second-order topological insulators via the cutting procedure
• Interaction Enabled Fractonic Higher-Order Topological Phases
• Higher-order topological spin Hall effect of sound
• Bulk-and-edge to corner correspondence
• Quadrupole moments, edge polarizations, and corner charges in the Wannier representation
• Quadrupole spin polarization as signature of second-order topological superconductors
• Topological classification of Higher-order topological phases with nested band inversion surfaces
• Fractional Chiral Hinge Insulator
• Boundary-obstructed topological phases of a massive Dirac fermion in a magnetic field
• Fragile phonon topology on the honeycomb lattice with time-reversal symmetry
• Generation of higher-order topological insulators using periodic driving
• Higher-Order Topological Crystalline Insulating Phase and Quantized Hinge Charge in Topological Electride Apatite
• Embedded Topological Insulators
• Corner- and sublattice-sensitive Majorana zero modes on the kagome lattice
• Magnon corner states in twisted bilayer honeycomb magnets
• Second-order topological corner states in zigzag graphene nanoflake with different types of edge magnetic configurations
• Breaking and resurgence of symmetry in the non-Hermitian Su-Schrieffer-Heeger model in photonic waveguides
• Engineering second-order topological insulators via coupling two first-order topological insulators
• Many-Body Invariants for Chern and Chiral Hinge Insulators
• Klein-bottle quadrupole insulators and Dirac semimetals
• Second-order and real Chern topological insulator in twisted bilayer $\alpha$-graphyne
• Observation of backscattering induced by magnetism in a topological edge state
• Thermodynamic approach to electric quadrupole moments
• Glide-symmetric magnetic topological crystalline insulators with inversion symmetry
• Non-Compact Atomic Insulators
• Multidimensional sound propagation in 3D high-order topological sonic insulator
• Unified characterization for higher-order topological phase transitions
• Sublattice-enriched tunability of bound states in second-order topological insulators and superconductors
• Floquet quadrupole photonic crystals protected by space-time symmetry
• Strain engineered higher order topological phases for spin-3/2 Luttinger fermions
• Quasicrystalline second-order topological semimetals
• Vortex solitons in topological disclination lattices
• $Z_4$ parafermion $\pm \pi /2$ modes in interacting periodically driven superconducting chain
• Nodal higher-order topological superconductivity from a ${\mathcal{C}}_4$-symmetric Dirac semimetal
• Interacting Topological Quantum Chemistry in 2D: Many-body Real Space Invariants
• Higher-Order Topological Insulators via Momentum-Space Nonsymmorphic Symmetries
• Higher-order Klein bottle topological insulator in three-dimensional acoustic crystals
• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
• Mirror-symmetry-protected dynamical quantum phase transitions in topological crystalline insulators
• Universal higher-order bulk-boundary correspondence of triple nodal points
• Topological Crystalline Insulators with $C_2$ Rotation Anomaly
• Topological near-field heat flow in a honeycomb lattice
• Identifying Higher Order Topology and Fractional Corner Charge Using Entanglement Spectra
• Fractional corner charges in a 2D super-lattice Bose-Hubbard model
• Density-driven higher-order topological phase transitions in amorphous solids
• The Bulk-Hinge Correspondence and Three-Dimensional Quantum Anomalous Hall Effect in Second Order Topological Insulators
• Surface Bogoliubov-Dirac cones and helical Majorana hinge modes in superconducting Dirac semimetals
• Dynamical bulk boundary correspondence and dynamical quantum phase transitions in higher order topological insulators
• Topological Gaps in Quasi-Periodic Spin Chains: A Numerical and K-Theoretic Analysis
• Interacting second-order topological insulators in one-dimensional fermions with correlated hopping
• Chiral Majorana Hinge Modes in Superconducting Dirac Materials
• Constructing higher-order topological states in higher dimension
• Lattice distortion induced first and second order topological phase transition in rectangular high-T${}_{\mathrm{c}}$ superconducting monolayer
• Crystalline Responses for Rotation-Invariant Higher-Order Topological Insulators
• A Dirac fermion model associated with second order topological insulator
• Impurity bound states as detectors of topological band structures revisited
• Large shift current, $\pi$ Zak phase and unconventional nature in Se and Te
• ${\mathbb{Z}}_2$ Spin Hopf Insulator: Helical Hinge States and Returning Thouless Pump
• Higher-Order Topological Mott Insulator on the Pyrochlore Lattice
• Topological properties of two-dimensional photonic square lattice without $C_4$ and $M_{x(y)}$ symmetries
• Higher-Order Topological Insulator on a Martini Lattice and Its Square Root Descendant
• Higher-order topological superconductors from Weyl semimetals
• Higher-order Topological Phases of Magnons in van der Waals Honeycomb Ferromagnets
• Topological States in Two-Dimensional Su-Schrieffer-Heeger Models
• Boundary edge networks induced by bulk topology
• Interplay between boundary conditions and Wilson's mass in Dirac-like Hamiltonians
• Quantized surface magnetism and higher-order topology: Application to the Hopf insulator
• Hinge mode dynamics of periodically driven higher-order Weyl semimetals
• Second-Order Topological Insulator in van der Waals Heterostructures of CoBr${}_2$/Pt${}_2$HgSe${}_3$/CoBr${}_2$
• Second-Order Topological Insulator in Two-Dimensional C2N and Its Derivatives
• Engineering non-Hermitian Second Order Topological Insulator in Quasicrystals
• Topological states in a dimerized system with staggered magnetic fluxes
• Second-order topological modes in two-dimensional continuous media
• Hinge states in a system of coupled Rashba layers
• Higher-order topological heat conduction on a lattice for detection of corner states
• Topological Transitions with an Imaginary Aubry-Andre-Harper Potential
• Generalization of Benalcazar-Bernevig-Hughes model to arbitrary dimensions
• Topological corner modes in a brick lattice with nonsymmorphic symmetry
• Detecting Bulk Topology of Quadrupolar Phase from Quench Dynamics
• Entanglement in a second order topological insulator on a square lattice
• Engineering topologically protected zero-dimensional interface end states in antiferromagnetic heterojunction graphene nanoflakes
• Role of the edges in a quasicrystalline Haldane model
• Theory of Dipole Insulators
• Dynamically characterizing topological phases by high-order topological charges
• Floquet second-order topological Anderson insulator hosting corner localized modes
• Topological phases of the dimerized Hofstadter butterfly
• Engineering topological states and quantum-inspired information processing using classical circuits
• Topological invariants for interacting systems: from twisted boundary condition to center-of-mass momentum
• High-temperature Majorana corner modes in a $d+i{d}^{\prime }$ superconductor heterostructure: Application to twisted bilayer cuprate superconductors
• Correlation Effects in Quadrupole Insulators: a Quantum Monte Carlo Study
• Theory of edge states based on the hermiticity of tight-binding Hamiltonian operators
• Dynamic bulk-boundary correspondence for anomalous Floquet topology
• The General Principle behind Magnetization-induced Second-Order Topological Corner States in the Kane-Mele Model
• Mirror real Chern insulator in two and three dimensions
• Competition of First-Order and Second-Order Topology on the Honeycomb Lattice
• High-Order Topological Phase Diagram Revealed by Anomalous Nernst Effect in Janus ScClI Monolayer
• Dephasing enhanced strong Majorana zero modes in 2D and 3D higher-order topological superconductors
• Dirac cones and higher-order topology in quasi-continuous media
• Topological nature of higher-order hinge states revealed by spin transport
• Non-Hermitian higher-Order Weyl semimetal with surface diabolic points
• Chirality polarizations and spectral bulk-boundary correspondence
• Magnetic Flux Periodicity in Second Order Topological Superconductors
• Higher-order Topological Insulators and Semimetals in Three Dimensions without Crystalline Counterparts
• Engineering second-order nodal-line semimetals by breaking $\mathcal{P}\mathcal{T}$ symmetry and periodic driving
• Higher-order topological semimetals and nodal superconductors with an order-two crystalline symmetry
• Anomalous topological edge modes in a periodically-driven trimer lattice
• Classification and construction of interacting fractonic higher-order topological phases
• Effects of Disorder On Thouless Pumping In Higher-Order Topological Insulators
• Symmetry indicators in commensurate magnetic flux
• Quantized charge-pumping in higher-order topological insulators
• From magnetoelectric response to optical activity
• Higher order topological matter and fractional chiral states
• Persistence of Monoclinic Crystal Structure in Three-Dimensional Second-Order Topological Insulator Candidate 1T'-MoTe2 Thin Flake without Structural Phase transition
• Effective action approach to the filling anomaly in crystalline topological matter
• Topological Dipole Conserving Insulators and Multipolar Responses
• Transport Induced Dimer State from Topological Corner States
• Observation of emergent Dirac physics at the surfaces of acoustic higher-order topological insulators
• Orbital shift-induced boundary obstructed topological materials with a large energy gap
• Third-order topological insulator induced by disorder
• Interlayer Coupling Induced Topological Phase Transition to Higher Order
• Observation of Dual-band Topological Corner Modes in Acoustic Kagome Lattice with Long-range Interactions
• Observation of gapless corner modes in synthetic translation dimensions
• High-order topological pumping on a superconducting quantum processor
• Tailoring Quadrupole Topological Insulators with Periodic Driving and Disorder
• Quasidisorder Induced Topology
• Topological phonon analysis of the 2D buckled honeycomb lattice: an application to real materials
• Inducing a many-body topological state of matter through Coulomb-engineered local interactions
• Fractional hinge and corner charges in various crystal shapes with cubic symmetry
• A General Method to Design Acoustic Higher-Order Topological Insulators
• Real-space Obstruction in Quantum Spin Hall Insulators
• Topology with memory in nonlinear driven-dissipative photonic lattices
• Topologically distinct atomic insulators
• Topological insulators and higher-order topological insulators from gauge-invariant 1D lines
• Topological quantum transition driven by charge-phonon coupling in higher-order topological insulators
• Towards a theory of surface orbital magnetization
• Multiple higher-order topological phases with even and odd pairs of zero-energy corner modes in a $C_3$ symmetry broken model
• Aharonov-Bohm Effect in Three-dimensional Higher-order Topological Insulator
• Controlling the Deformation of Metamaterials: Corner Modes via Topology
• Topological and dynamical features of periodically driven spin ladders
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Generating many Majorana corner modes and multiple phase transitions in Floquet second-order topological superconductors
• Time evolution of Majorana corner modes in Floquet second-order topological superconductor
• Thermodynamic electric quadrupole moments of nematic phases from first-principles calculation
• Network model for higher-order topological phases
• Rotational symmetry breaking and partial Majorana corner states in a high-Tc superconductor based heterostructure
• Two-Dimensional Higher-Order Topological Metals
• Topological invariants beyond symmetry indicators: Boundary diagnostics for twofold rotationally symmetric superconductors
• Glide Symmetry Protected Higher-Order Topological Insulators from Semimetals with butterfly-like Nodal Lines
• Realizing and detecting Stiefel-Whitney insulators in an optical Raman lattice
• Abundance of second order topology in $C_3$ symmetric two-dimensional insulators
• Hinge Modes and Surface States in Second-Order Topological Three-Dimensional Quantum Hall Systems induced by Charge Density Modulation
• Connection topology of step edge state bands at the surface of a three dimensional topological insulator
• Out of equilibrium chiral higher order topological insulator on a $\pi$-flux square lattice
• Wannier Topology and Quadrupole Moments for a generalized Benalcazar-Bernevig-Hughes Model
• Nonlinear Corner States in Topologically Nontrivial Kagome Lattice
• Topologically Protected Edge States in Triangular Lattices
• Revisiting (higher order and crystalline) topology in old models of lead telluride
• 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
• Third-order topological insulator in three-dimensional lattice of magnetic vortices
• Simulating Floquet topological phases in static systems
• Interlayer electric multipoles induced by in-plane field from quantum geometric origins
• A semiclassical treatment of spinor topological effects in driven, inhomogeneous insulators under external electromagnetic fields
• Wannier band transitions in disordered $\pi$-flux ladders
• Observation of Bulk Polarization Transitions and Higher-Order Embedded Topological Eigenstates for Sound
• Quantum Monte Carlo study of topological phases on a spin analogue of Benalcazar-Bernevig-Hughes model
• Multi-Dimensional Quantum Walks: a Playground of Dirac and Schrödinger Particles
• Magnetic wallpaper Dirac fermions and topological magnetic Dirac insulators
• Electric Polarization in Inhomogeneous Crystals
• Green's Function Approach to Interacting Higher-order Topological Insulators
• Protected Gapless Edge States In Trivial Topology
• Floquet higher-order topological phases in momentum space
• Anomalous levitation and annihilation in Floquet topological insulators
• Energy density as a probe of band representations in photonic crystals
• Rotoinversion-symmetric bulk-hinge correspondence and its applications to higher-order Weyl semimetals
• Controllable Weyl Nodes and Fermi Arcs from Floquet Engineering Triple Fermions
• Gapless higher-order topology and corner states in Floquet systems
• Holstein polaron in a pseudospin-$1$ quantum spin Hall system: first and second order topological phase transitions
• Multi higher-order Dirac and nodal line semimetals
• Higher-order topology protected by latent crystalline symmetries
• Probing Chiral-Symmetric Higher-Order Topological Insulators with Multipole Winding Number
• Mode-Shell correspondence, a unifying phase space theory in topological physics -- Part I: Chiral number of zero-modes
• Crystalline-electromagnetic responses of higher order topological semimetals
• Majorana corner modes and flat-band Majorana edge modes in superconductor/topological-insulator/superconductor junctions
• Higher-order topological corner states induced solely by onsite potentials with mirror symmetry
• Topologically localized insulators
• Topological Wannier cycles for the bulk and edges
• Fractional boundary charges with quantized slopes in interacting one- and two-dimensional systems
• AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states
• Multiorder topological superfluid phase transitions in a two-dimensional optical superlattice
• Realizing Corner States in Artificial Crystals Based on Topological Spin Textures
• Higher-order topological insulators in a crisscross antiferromagnetic model
• A Bipartite Kronig-Penney Model with Dirac Potential Scatterers
• Geometry of projected connections, Zak phase, and electric polarization
• Berezinskii-Kosterlitz-Thouless localization-localization transitions in disordered two-dimensional quantized quadrupole insulators
• Two-Dimensional Altermagnetic Iron Oxyhalides: Real Chern Topology and Valley-Spin-Lattice Coupling
• Controlling Majorana modes by $p$-wave pairing in two-dimensional $p+id$ topological superconductors
• Local and energy-resolved topological invariants for Floquet systems
• Velocity Scanning Tomography for Room-Temperature Quantum Simulation
• Second-order topological insulator under strong magnetic field: Landau levels, Zeeman effect, and magnetotransport
• Unveiling Higher-Order Topology via Polarized Topological Charges
• Competing topological phases in a non-Hermitian time-reversal symmetry-broken Bernevig-Hughes-Zhang model
• Corner and Edge States in Topological Sierpinski Carpet Systems
• Corner modes in non-Hermitian next-nearest-neighbor hopping model
• Remote control of spin polarization of topological corner states
• Direct demonstration of bulk-boundary correspondence in higher-order topological superconductors with chiral symmetry
• Gapless Fermionic Systems as Phase-space Topological Insulators: Non-perturbative Results from Anomalies
• Electric polarization and its quantization in one-dimensional non-Hermitian chains
• Fractal Surface States in Three-Dimensional Topological Quasicrystals
• A mono-atomic orbital-based 1D topological crystalline insulator
• Disorder-Induced Phase Transitions in Three-Dimensional Chiral Second-Order Topological Insulator
• Fractional second-order topological insulator from a three-dimensional coupled-wires construction
• Theory of Anomalous Floquet Higher-Order Topology: Classification, Characterization, and Bulk-Boundary Correspondence
• Microwave hinge states in a simple-cubic-lattice photonic crystal insulator
• Edge and corner superconductivity in a 2D topological model
• Spin-Dependent High-Order Topological Insulator and Two Types of Distinct Corner Modes in Monolayer FeSe/GdClO Heterostructure
• Many-body multipole index and bulk-boundary correspondence
• Synthetic topology and Floquet dynamic quantum phase transition in a periodically driven Raman lattice
• Extrinsic topology of Floquet anomalous boundary states in quantum walks
• Thermodynamics of a Higher-Order Topological Insulator
• Topological Classical Systems with Generalized Chiral Symmetry
• Topological Corner Modes by Composite Wannier States in Glide-Symmetric Photonic Crystal
• Intrinsic-perturbation induced anomalous higher-order boundary states in non-Hermitian systems
• Emergent metallicity at the grain boundaries of higher-order topological insulators
• Higher-order topology in twisted multilayer systems: a review
• Spin-orbit-coupled superconductivity with spin-singlet non-unitary pairing
• Fermionic tensor networks for higher order topological insulators from charge pumping
• Chern dartboard insulator: sub-Brillouin zone topology and skyrmion multipoles
• Second-order topological insulator in a coinless discrete-time quantum walk
• Engineering rich two-dimensional higher-order topological phases by flux and periodic driving
• Fractional corner magnetization of collinear antiferromagnets
• XFe4Ge2 (X = Y, Lu) and Mn3Pt: Filling-enforced magnetic topological metals
• Generic Approach to Intrinsic Magnetic Second-order Topological Insulators via Inverted $p-d$ Orbitals
• Floquet chiral hinge modes and their interplay with Weyl physics in a three-dimensional lattice
• Topological metals constructed by sliding quantum wire arrays
• Hierarchical Topological States in Thermal Diffusive Networks
• Orbital Edelstein effect of electronic itinerant orbital motion at edges
• Higher-order topology and corner triplon excitations in two-dimensional quantum spin-dimer models
• Interaction-Driven Topological Switch in a $P$-Band Honeycomb Lattice
• Fractional corner charges in threefold-symmetric two-dimensional materials with fragile topology
• Anisotropic-scaling localization in higher-dimensional non-Hermitian systems
• Magnonics along the wall in Bimeron Chain Domain Walls
• Second-order topology and supersymmetry in two-dimensional topological insulators
• Magnetic impurities along the edge of a quantum spin Hall insulator: Realizing a one-dimensional AIII insulator
• A topological current divider
• Tailoring higher-order topological phases via orbital hybridization
• Chiral Hinge Transport in Disordered Non-Hermitian Second-Order Topological Insulators
• Zero-Energy State Localized near an Arbitrary Edge in Quadrupole Topological Insulators
• Topological Field Theories of Three-Dimensional Rotation Symmetric Insulators: Coupling Curvature and Electromagnetism
• Effects of first- and second-order topological phases on equilibrium crystal shapes
• Light-Matter interactions in Hofstadter lattice with the next-nearest neighbor couplings
• Corner states of two-dimensional second-order topological insulators with a chiral symmetry and broken time reversal and charge conjugation
• Higher-order topological superconductor phases in a multilayer system
• Orbital Hall effect and topology on a two-dimensional triangular lattice: from bulk to edge
• Mixed higher-order topology and nodal and nodeless flat band topological phases in a superconducting multiorbital model
• Single-layer tensor network approach for three-dimensional quantum systems
• Distinguishing trivial and topological quadrupolar insulators by Wannier-Stark ladders
• Topological Corner States in Bilayer and Trilayer Systems with Vertically Stacked Topological Heterostructures
• Characterization of higher-order topological superconductors using Bott indices
• Family of third-order topological insulators from Su-Schrieffer-Heeger stacking
• Two-dimensional orbital-obstructed insulators with higher-order band topology
• Unconventional hybrid-order topological insulators
• Observation of nonlinear higher-order topological insulators with unconventional boundary truncations
• Generalization of the Nested Wilson Loop Formalism in Topological Dirac Semimetals with Higher-order Fermi Arcs
• Topology in quasicrystals
• Kitaev Building-block Construction for Inversion-Protected Higher-order Topological Superconductors
• Majorana corner states in square and Kagome quantum spin liquids
• Topological characteristics of gap closing points in nonlinear Weyl semimetals
• Recipe for higher-order topology on the triangular lattice
• Model non-Hermitian topological operators without skin effect: A general principle of construction
• Second-order topological insulator in periodically driven lattice
• Coexisting surface states in the weak and crystalline topological insulator Bi_2TeI
• Topological states constructed by two different trivial quantum wires
• Observation of trapped fractional charge and topological states at disclination defects in higher-order topological insulators
• Fermion-fermion interaction driven phase transitions in rhombohedral trilayer graphene
• Protecting Quantum Superposition and Entanglement with Photonic Higher-Order Topological Crystalline Insulator
• Higher-order topological superconductors characterized by Fermi level crossings
• Corner Modes and Ground-State Degeneracy in Models with Gauge-Like Subsystem Symmetries
• Second-order topological insulator in Bilayer borophene
• Hybrid-Order Topological Phase And Transition in 1H Transition Metal Compounds
• Corner states in second-order mechanical topological insulator
• Optical Berry Curvature of Hinge State and Its Detection
• Exceptional second-order topological insulators
• General corner charge formulas in various tetrahedral and cubic space groups
• Scattering theory of higher order topological phases
• Adiabatic charge transport in extended SSH models
• Theoretical analysis of glide-Z_2 magnetic topological photonic crystals
• Unconventional Topological Insulators from Extended Topological Band Degeneracies
• Hinge Majorana Flat Band in Type-II Dirac Semimetals
• Persistent Corner Spin Mode at the Quantum Critical Point of a Plaquette Heisenberg Model
• Cavity-induced topological edge and corner states
• Realization of a three-dimensional quantum Hall effect in a Zeeman-induced second order topological insulator on a torus
• Extrinsic higher-order topological corner states in AB-stacked transition-metal dichalcogenides
• The destiny of obstructed atomic insulator under correlation
• Unconventional phonon spectra and obstructed edge phonon modes
• Mirror-symmetry protected higher-order topological zero-frequency boundary and corner modes in Maxwell lattices
• Theory of fractional corner charges in cylindrical crystal shapes
• Finite-temperature topological invariant for higher-order topological insulators
• Photonic quadrupole topological phases in zero-dimensional cavity with synthetic dimensions
• Takagi Topological Insulator on the Honeycomb Lattice
• Anisotropic Penetration Depths of Corner States in a Higher-Order Topological Insulator
• Homotopy invariant in time-reversal and twofold rotation symmetric systems
• Higher order topological state induced by $d$-wave competing orders in high-T${}_c$ superconductor based heterostructure
• Tunable corner states in topological insulators with long-range hoppings and diverse shapes
• Altermagnet-Superconductor Heterostructure: a Scalable Platform for Braiding of Majorana Modes
• Magnetic flux induced higher-order topological superconductivity
• Mixed-parity octupolar pairing and corner Majorana modes in three dimensions
• Fragmented exceptional points and their bulk and edge realizations in lattice models
• Topological parafermion corner states in clock-symmetric non-Hermitian second-order topological insulator
• Tunable symmetry-protected higher-order topological states with fermionic atoms in bilayer optical lattices
• Fragile topology in nodal-line semimetal superconductors
• Floquet-Anderson localization in the Thouless pump and how to avoid it
• Nonlinear quadrupole topological insulators
• Topology reconstruction for asymmetric systems by isomorphic mapping or perturbation approximation
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Boundary topological insulators and superconductors of Altland-Zirnbauer tenfold classes
• Topological insulators and geometry of vector bundles
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• Topological Aspects of Dirac Fermions in a Kagomé Lattice
• Third-order Orbital Corner State and its Realization in Acoustic Crystals
• Coexistence of gapless and gapped vortex modes with Majorana corner states in a 2D second-order topological superconductor
• Chiral Majorana Edge Modes and Vortex Majorana Zero Modes in Superconducting Antiferromagnetic Topological Insulator
• Many-Body Quadrupolar Sum Rule for Higher-Order Topological Insulator
• Topological Higgs Amplitude Modes in Strongly Interacting Superfluids
• Perfectly localized Majorana corner modes in fermionic lattices
• Engineering Majorana corner modes from two-dimensional hexagonal crystals
• Quasiperiodic Quadrupole Insulators
• Correlated second-order Dirac semimetals with Coulomb interactions
• RKKY interaction in helical higher-order topological insulators
• Exceptionally deficient topological square-root insulators
• Arboreal Obstructed Atomic Insulating and Metallic Phases of Fermions
• Observation of Embedded Topology in a Trivial Bulk via Projective Crystal Symmetry
• Surface theory of a second-order topological insulator beyond the Dirac approximation
• $Z_2\times Z_2$ symmetry and $Z_4$ Berry phase of bosonic ladder
• Multiple topological corner states in the continuum of extended kagome lattice
• Effects of the Vertices on the Topological Bound States in a Quasicrystalline Topological Insulator
• Hidden wallpaper fermion and third-order topological insulator
• Hybrid Higher-Order Skin Topological Modes in the Two-Dimensional Su-Schrieffer-Heeger Model with Nonreciprocal Hoppings
• Photonic Higher-Order Topological States Induced by Long Range Interactions
• Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry
• Fractional ac Josephson effect as evidence of topological hinge states in a Dirac semimetal NiTe2
• Quantum-vacuum-protected topological edge polaritons
• Network model for magnetic higher-order topological phases
• Theory of Topological Superconductivity in Doped IV-VI Semiconductors
• Disorder-induced delocalization and reentrance in a Chern-Hopf insulator
• Zeeman field induced corner states in the Kane-Mele-Hubbard model
• Phase diagram of the J1-J2 Heisenberg second-order topological quantum magnet
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry
• Interaction-Induced Higher-Order Topological Insulator via Floquet Engineering
• Extended Haldane model -- a modern gateway to topological insulators
• Fractional chiral second-order topological insulator from a three-dimensional array of coupled wires
• Nonlinearity-induced corner states in a kagome lattice
• Local expression of fractional corner charges in obstructed atomic insulators and relationship with the fractional disclination charges
• Current switching behavior mediated via hinge modes in higher-order topological phases using altermagnets
• Tunable cornerlike states in topological type-II hyperbolic lattices
• Recent progress on disorder-induced topological phases
• Engineering Two-Dimensional Hybrid-Order Topological Insulators via Trilayer Coupling
• Magnetic doping-induced second-order and first-order topological phase transition inthe photonic alloy
• Higher-order topological insulators in two-dimensional antiferromagnetic and altermagnetic chromium-based group-IV chalcogenides
• Magnetic-flux tunable electronic transport through domain walls in a three-dimensional second-order topological insulator
• Unified Multipole Bott Indices for Non-Hermitian Skin Effect in Different Orders
• Wilson Loop and Topological Properties in 3D Woodpile Photonic Crystal
• Symmetry-protected gates of Majorana qubits in a high-$T_c$ superconductor platform
• Topological states in chiral electronic chains
• Higher-Order Topology in Monolayer FeSe
• Obstructed Atomic Insulators and Superfluids of Fermions Coupled to ${\mathbb{Z}}_2$ Gauge Fields
• Gapless States Localized along a Staircase Edge in Second-Order Topological Insulators
• Higher order topology in a Creutz ladder
• Classification of topological insulators and superconductors with multiple order-two point group symmetries
• Projected branes as platforms for crystalline, superconducting, and higher-order topological phases
• An all-dielectric photonic crystal with unconventional higher-order topology
• Dynamical Symmetry Indicators for Floquet Crystals
• Boundary Condition Analysis of First and Second Order Topological Insulators
• Enormous Berry-Curvature-Driven Anomalous Hall Effect in Topological Insulator (Bi,Sb)2Te3 on Ferrimagnetic Europium Iron Garnet beyond 400 K
• Application of Jordan Decomposition to Non-Hermitian Lattice Models with Spectrally-Isolated Lower Dimensional States
• Fourth-Order Topological Insulator via Dimensional Reduction
• Topological phases of monolayer and bilayer depleted Lieb lattices
• Pseudospin-valley-coupled phononic topological insulator with edge and corner states
• Topological Classification of Gapped/Gap-preserving Rational Space-Time Crystal Systems
• Majorana vortex phases in time-reversal invariant higher-order topological insulators and topologically trivial insulators
• Higher-order topological phases protected by noninvertible and subsystem symmetries
• Electronic structure and hinge states of strained half-Heusler compounds LiSbZn and LiBiZn
• Supermetal-insulator transition in a non-Hermitian network model
• Non-Hermitian Higher-Order Dirac Semimetals
• Non-Hermitian Higher-Order Weyl Semimetals
• Altermagnetism-induced higher-order topological Dirac semimetals in two-dimensional nonsymmorphic systems
• Direct visualization of topological transitions and higher-order topological states in photonic metasurfaces
• Non-Hermitian physics and topological phenomena in convective thermal metamaterials
• Quantized edge magnetizations and their symmetry protection in one-dimensional quantum spin systems
• Graphyne as a second-order and real Chern topological insulator in two dimensions
• Emergent One-Dimensional Helical Channel in Higher-Order Topological Insulators with Step Edges
• Wilson loop invariants and bulk-boundary correspondence in higher-order topological insulators with two anticommuting mirror symmetries
• Entanglement as a topological marker in harmonically confined attractive Fermi-Hubbard chains
• Corner Flat Bands Induced by $d$-Density Wave and Partial Corner State Modification Due to Competition with $d$-Wave Superconductivity
• Higher-order topological quantum paramagnets
• Gapless hinge states from adiabatic pumping of axion coupling
• Layer-number parity induced topological phase transition
• When Wannier centers jump: Critical points between atomic insulating phases
• Hinge modes of three-dimensional Euler insulators
• Second-order topological insulator induced by compensated altermagnetism without bulk spin splitting
• Proposal for realizing and probing topological crystalline insulators in optical lattices
• Valley-Hall-like second-order photonic topological insulators in Kagome lattice
• Topological defects in a double-mirror quadrupole insulator displace diverging charge
• Topological Photonics on a Small Scale
• Floquet first- and higher-order topological superconductors: A review
• Magnetic-Field Tunable Möbius and Higher-Order Topological Insulators in Three-Dimensional Layered Octagonal Quasicrystals
• Topological energy pumping in a quasi-periodically driven four-level system