Quantized Electric Multipole Insulators

Abstract

In this article we extend the celebrated Berry-phase formulation of electric polarization in crystals to higher electric multipole moments. We determine the necessary conditions under which, and minimal models in which, the quadrupole and octupole moments are topologically quantized electromagnetic observables. Such systems exhibit gapped boundaries that are themselves lower-dimensional topological phases. Furthermore, they manifest topologically protected corner states carrying fractional charge, i.e., fractionalization at the boundary of the boundary. To characterize these new insulating phases of matter, we introduce a new paradigm whereby `nested' Wilson loops give rise to a large number of new topological invariants that have been previously overlooked. We propose three realistic experimental implementations of this new topological behavior that can be immediately tested.

Main text: 9 pages, 6 figures. Supplementary Material: 37 pages, 15 figures. Submitted on Jul 25, 2016

References in corpus (4)

• Topological Field Theory of Time-Reversal Invariant Insulators
• Observation of the topological soliton state in the Su-Schrieffer-Heeger model
• Topological Insulators from Group Cohomology
• Electric polarization in a Chern insulator

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• General corner states in 2D SSH model with intracelluar next-nearest-neighbour hopping
• Buckled honeycomb group-$V$--$S_6$ symmetric $(d-2)$ higher order topological insulators
• Second-order topological insulator in breathing square lattice of magnetic vortices
• Rotation Symmetry-Protected Topological Phases of Fermions
• Application of the induction procedure and the Smith Decomposition in the calculation and topological classification of electronic band structures in the 230 space groups
• Matryoshka approach to Sine-Cosine topological models
• Thouless Pumps and Bulk-Boundary Correspondence in Higher-Order Symmetry-Protected Topological Phases
• Classification of Dirac points with higher-order Fermi arcs
• Anti-chiral edge states and hinge states based on the Haldane model
• Non-Abelian Generalizations of the Hofstadter model: Spin-orbit-coupled Butterfly Pairs
• Classification of crystalline insulators without symmetry indicators: atomic and fragile topological phases in twofold rotation symmetric systems
• Transition from metal to higher-order topological insulator driven by random flux
• Piezoelectricity and Topological Quantum Phase Transitions in Two-Dimensional Spin-Orbit Coupled Crystals with Time-Reversal Symmetry
• Aspects of topological superconductivity in 2D systems: noncollinear magnetism, skyrmions, and higher-order topology
• Direct dynamical characterization of higher-order topological insulators with nested band inversion surfaces
• Observation of higher-order topological states on a quantum computer
• $N$-band Hopf insulator
• Observation of Large-Number Corner Modes in $\mathbb{Z}$-class Higher-Order Topolectrical Circuits
• Quantum dynamical characterization and simulation of topological phases with high-order band inversion surfaces
• Higher-order topological insulators in a magnetic field
• Extended Bloch theorem for topological lattice models with open boundaries
• Bott periodicity for the topological classification of gapped states of matter with reflection symmetry
• Higher-Order Entanglement and Many-Body Invariants for Higher-Order Topological Phases
• Microwave signature of robust topological Andreev-level crossings in superconducting proximitized crystalline Bi nanowires
• Higher-order topological phases emerging from the Su-Schrieffer-Heeger stacking
• Higher-order valley vortices enabled by synchronized rotation in a photonic crystal
• Topological Crystalline Phases in a Disordered Inversion-Symmetric Chain
• Floquet higher-order Weyl and nexus semimetals
• Dynamical construction of Quadrupolar and Octupolar topological superconductors
• Higher-form Gauge Symmetries in Multipole Topological Phases
• Euler Obstructed Cooper Pairing: Nodal Superconductivity and Hinge Majorana Zero Modes
• Hinge solitons in three-dimensional second-order topological insulators
• Non-Hermitian spatial symmetries and their stabilized normal and exceptional topological semimetals
• Interaction driven surface Chern insulator in nodal line semimetals
• Comb tensor networks
• Edge states, corner states, and flat bands in a two-dimensional $\mathcal{P}\mathcal{T}$ symmetric system
• Quantized Topological Anderson-Thouless Pump
• Bulk-boundary-defect correspondence at disclinations in rotation-symmetric topological insulators and superconductors
• Flat bands and higher-order topology in polymerized triptycene: Tight-binding analysis on decorated star lattices
• Engineering exotic second-order topological semimetals by periodic driving
• Bulk-boundary correspondence in disordered higher-order topological insulators
• Wannier-type photonic higher-order topological corner states induced solely by gain and loss
• Dirty higher-order Dirac semimetal: Quantum criticality and bulk-boundary correspondence
• Fractional Corner Charge of Sodium Chloride
• Ferroelectric higher-order topological insulator in two dimensions
• Boundaries of boundaries: a systematic approach to lattice models with solvable boundary states of arbitrary codimension
• Appearance of hinge states in second-order topological insulators via the cutting procedure
• Hinged Quantum Spin-Hall Effect in Antiferromagnetic Topological Insulators
• Interaction Enabled Fractonic Higher-Order Topological Phases
• Topogivity: A Machine-Learned Chemical Rule for Discovering Topological Materials
• Current-phase relation of a WTe2 Josephson junction
• Topology invisible to eigenvalues in obstructed atomic insulators
• Existence of Corner Modes in Elastic Twisted Kagome Lattices
• Hidden Breathing Kagome Topology in Hexagonal Transition Metal Dichalcogenides
• Topological Mechanics from Supersymmetry
• Theory of inversion-${\mathbb{Z}}_4$ protected topological chiral hinge states and its applications to layered antiferromagnets
• Topological excitonic corner states and nodal phase in bilayer quantum spin Hall insulators
• Higher-order topological spin Hall effect of sound
• Quadrupole moments, edge polarizations, and corner charges in the Wannier representation
• Shift vector as the geometric origin of beam shifts
• The hybrid-order topology of weak topological insulators
• Higher-Order Topological Crystalline Insulating Phase and Quantized Hinge Charge in Topological Electride Apatite
• Simulation of higher-order topological phases in 2D spin-phononic crystal networks
• Fractional Chiral Hinge Insulator
• Quadrupole spin polarization as signature of second-order topological superconductors
• Corner- and sublattice-sensitive Majorana zero modes on the kagome lattice
• Spin-Polarized Fractional Corner Charges and Their Photonic Realization
• Embedded Topological Insulators
• Time-reversal-invariant $C_2$-symmetric higher-order topological superconductors
• Fragile phonon topology on the honeycomb lattice with time-reversal symmetry
• Superconductors with anomalous Floquet higher-order topology
• Generation of higher-order topological insulators using periodic driving
• Magnon corner states in twisted bilayer honeycomb magnets
• Topological classification of Higher-order topological phases with nested band inversion surfaces
• Boundary-obstructed topological phases of a massive Dirac fermion in a magnetic field
• Edge states and corner modes in second-order topological phononic crystal plates
• Second-order topological corner states in zigzag graphene nanoflake with different types of edge magnetic configurations
• Engineering second-order topological insulators via coupling two first-order topological insulators
• Second-order and real Chern topological insulator in twisted bilayer $\alpha$-graphyne
• Many-Body Invariants for Chern and Chiral Hinge Insulators
• Glide-symmetric magnetic topological crystalline insulators with inversion symmetry
• Absence of topological protection of the interface states in ${\mathbb{Z}}_2$ photonic crystals
• Thermodynamic approach to electric quadrupole moments
• Breaking and resurgence of symmetry in the non-Hermitian Su-Schrieffer-Heeger model in photonic waveguides
• Optimization and robustness of topological corner state in second-order topological photonic crystal
• Observation of backscattering induced by magnetism in a topological edge state
• Klein-bottle quadrupole insulators and Dirac semimetals
• Quasicrystalline second-order topological semimetals
• Non-Compact Atomic Insulators
• Floquet quadrupole photonic crystals protected by space-time symmetry
• Unified characterization for higher-order topological phase transitions
• Average Symmetry Protected Higher-order Topological Amorphous Insulators
• Vortex solitons in topological disclination lattices
• Multidimensional sound propagation in 3D high-order topological sonic insulator
• Strain engineered higher order topological phases for spin-3/2 Luttinger fermions
• Sublattice-enriched tunability of bound states in second-order topological insulators and superconductors
• Higher-Order Topological Insulators via Momentum-Space Nonsymmorphic Symmetries
• $Z_4$ parafermion $\pm \pi /2$ modes in interacting periodically driven superconducting chain
• Out-of-Time-Order Correlation as a Witness for Topological Phase Transitions
• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
• Higher-order Klein bottle topological insulator in three-dimensional acoustic crystals
• Interacting Topological Quantum Chemistry in 2D: Many-body Real Space Invariants
• Nodal higher-order topological superconductivity from a ${\mathcal{C}}_4$-symmetric Dirac semimetal
• Mirror-symmetry-protected dynamical quantum phase transitions in topological crystalline insulators
• Topological edge and corner states and fractional corner charges in blue phosphorene
• Dynamical bulk boundary correspondence and dynamical quantum phase transitions in higher order topological insulators
• Dual Topological Insulator Device with Disorder Robustness
• Identifying Higher Order Topology and Fractional Corner Charge Using Entanglement Spectra
• Density-driven higher-order topological phase transitions in amorphous solids
• Universal signatures of Dirac fermions in entanglement and charge fluctuations
• Surface Bogoliubov-Dirac cones and helical Majorana hinge modes in superconducting Dirac semimetals
• Topological phases and Anderson localization in off-diagonal mosaic lattices
• The Bulk-Hinge Correspondence and Three-Dimensional Quantum Anomalous Hall Effect in Second Order Topological Insulators
• Topological Superconductivity from Unconventional Band Degeneracy with Conventional Pairing
• Fractional corner charges in a 2D super-lattice Bose-Hubbard model
• Chiral Majorana Hinge Modes in Superconducting Dirac Materials
• Constructing higher-order topological states in higher dimension
• Universal higher-order bulk-boundary correspondence of triple nodal points
• Topological Crystalline Insulators with $C_2$ Rotation Anomaly
• Higher-order Weyl superconductors with anisotropic Weyl-point connectivity
• ${\mathbb{Z}}_2$ Spin Hopf Insulator: Helical Hinge States and Returning Thouless Pump
• Crystalline Responses for Rotation-Invariant Higher-Order Topological Insulators
• Evidencing non-Bloch dynamics in temporal topolectrical circuits
• Large shift current, $\pi$ Zak phase and unconventional nature in Se and Te
• A Dirac fermion model associated with second order topological insulator
• Measuring Zak phase in room-temperature atoms
• Robustness of Helical Hinge States of Weak Second-Order Topological Insulators
• Topological hinge modes in Dirac semimetals
• Weyl-like points from band inversions of spin-polarised surface states in NbGeSb
• Impurity bound states as detectors of topological band structures revisited
• Higher-order Topological Phases of Magnons in van der Waals Honeycomb Ferromagnets
• Higher-Order Topological Insulator on a Martini Lattice and Its Square Root Descendant
• Competition of non-Hermitian skin effect and topological localization of corner states observed in circuits
• Higher-order topological superconductors from Weyl semimetals
• Engineering topological phases with a three-dimensional nodal-loop semimetal
• Phononic Stiefel-Whitney topology with corner vibrational modes in two-dimensional Xenes and ligand-functionalized derivatives
• Escape from black holes in materials: Type II Weyl semimetals and generic edge states
• Topological States in Two-Dimensional Su-Schrieffer-Heeger Models
• Boundary edge networks induced by bulk topology
• Single-mode lasing based on PT-breaking of two-dimensional photonic higher-order topological insulator
• Discovering two-dimensional magnetic topological insulators by machine learning
• Higher-Order Topological Mott Insulator on the Pyrochlore Lattice
• Robustness of chiral edge modes in fractal-like-lattices below two dimensions: A case study
• Lecture Notes on Topological Crystalline Insulators
• Multiplicative topological phases
• Topological properties of two-dimensional photonic square lattice without $C_4$ and $M_{x(y)}$ symmetries
• Engineering non-Hermitian Second Order Topological Insulator in Quasicrystals
• Hybrid-order topological odd-parity superconductors via Floquet engineering
• 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
• Hinge mode dynamics of periodically driven higher-order Weyl semimetals
• Topological states in a dimerized system with staggered magnetic fluxes
• Two-dimensional weak-type topological insulators in inversion symmetric crystals
• Second-order topological modes in two-dimensional continuous media
• Quantum anomalous Hall phase in synthetic bilayers via twistless twistronics
• Topological Edge and Interface states at Bulk disorder-to-order Quantum Critical Points
• Interplay between boundary conditions and Wilson's mass in Dirac-like Hamiltonians
• Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases
• Topological transitions and Anderson localization of light in disordered atomic arrays
• Tunable two-dimensional laser arrays with zero-phase locking
• Hinge states in a system of coupled Rashba layers
• Symmetry-protected localized states at defects in non-Hermitian systems
• Generalization of Benalcazar-Bernevig-Hughes model to arbitrary dimensions
• Topological corner modes in a brick lattice with nonsymmorphic symmetry
• Engineering topologically protected zero-dimensional interface end states in antiferromagnetic heterojunction graphene nanoflakes
• ${}^3$He Universe 2020
• Toeplitz operators on concave corners and topologically protected corner states
• Role of the edges in a quasicrystalline Haldane model
• Higher-order topological heat conduction on a lattice for detection of corner states
• Light bullets in Su-Schrieffer-Heeger photonic topological insulators
• Topological Transitions with an Imaginary Aubry-Andre-Harper Potential
• Entanglement in a second order topological insulator on a square lattice
• Theory of Dipole Insulators
• Majorana corner states on the dice lattice
• Sublattice-sensitive Majorana Modes
• Detecting Bulk Topology of Quadrupolar Phase from Quench Dynamics
• Hinge Magnons from Non-collinear Magnetic Order in Honeycomb Antiferromagnet
• Theory of edge states based on the hermiticity of tight-binding Hamiltonian operators
• Dynamically characterizing topological phases by high-order topological charges
• Correlation Effects in Quadrupole Insulators: a Quantum Monte Carlo Study
• Dephasing enhanced strong Majorana zero modes in 2D and 3D higher-order topological superconductors
• Floquet second-order topological Anderson insulator hosting corner localized modes
• Robust zero modes in non-Hermitian systems without global symmetries
• Interaction-induced quantum spin Hall insulator in the organic Dirac electron system $\alpha$-(BEDT-TSeF)${}_2$I${}_3$
• Competition of First-Order and Second-Order Topology on the Honeycomb Lattice
• 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
• Eight-color chiral spin liquid in the $S=1$ bilinear-biquadratic model with Kitaev interactions
• Multi-frequency optical lattice for dynamic lattice-geometry control
• High-temperature Majorana corner modes in a $d+i{d}^{\prime }$ superconductor heterostructure: Application to twisted bilayer cuprate superconductors
• Engineering topological states and quantum-inspired information processing using classical circuits
• Topological phases of the dimerized Hofstadter butterfly
• High-Order Topological Phase Diagram Revealed by Anomalous Nernst Effect in Janus ScClI Monolayer
• Photonic nodal lines with quadrupole Berry curvature distribution
• Magnetic Flux Periodicity in Second Order Topological Superconductors
• Lieb Schultz Mattis-Type Theorems and Other Non-perturbative Results for Strongly Correlated Systems with Conserved Dipole Moments
• Classification and construction of interacting fractonic higher-order topological phases
• Effective field theories of topological crystalline insulators and topological crystals
• Effects of Disorder On Thouless Pumping In Higher-Order Topological Insulators
• Group theory study of the vibrational modes and magnetic order in the topological antiferromagnet MnBi${}_2$Te${}_4$
• Symmetry indicators in commensurate magnetic flux
• From magnetoelectric response to optical activity
• Higher-order Topological Insulators and Semimetals in Three Dimensions without Crystalline Counterparts
• Quantized charge-pumping in higher-order topological insulators
• Non-Hermitian higher-Order Weyl semimetal with surface diabolic points
• Topological nature of higher-order hinge states revealed by spin transport
• Chirality polarizations and spectral bulk-boundary correspondence
• Higher order topological matter and fractional chiral states
• Anomalous topological edge modes in a periodically-driven trimer lattice
• Transparent Josephson Junctions in Higher-Order Topological Insulator WTe2 via Pd Diffusion
• Dirac cones and higher-order topology in quasi-continuous media
• Inducing a many-body topological state of matter through Coulomb-engineered local interactions
• Interlayer Coupling Induced Topological Phase Transition to Higher Order
• Tailoring Quadrupole Topological Insulators with Periodic Driving and Disorder
• Persistence of Monoclinic Crystal Structure in Three-Dimensional Second-Order Topological Insulator Candidate 1T'-MoTe2 Thin Flake without Structural Phase transition
• Topological Edge and Corner States in Biphenylene Network
• Observation of topologically distinct corner states in "bearded" photonic Kagome lattices
• Effective action approach to the filling anomaly in crystalline topological matter
• Topological phonon analysis of the 2D buckled honeycomb lattice: an application to real materials
• Topological Dipole Conserving Insulators and Multipolar Responses
• A General Method to Design Acoustic Higher-Order Topological Insulators
• Orbital shift-induced boundary obstructed topological materials with a large energy gap
• High-order topological pumping on a superconducting quantum processor
• Observation of gapless corner modes in synthetic translation dimensions
• Topological d-wave superconductivity in two dimensions
• Third-order topological insulator induced by disorder
• Dynamical Detection of Topological Spectral Density
• Symmetric non-Hermitian skin effect with emergent nonlocal correspondence
• Observation of Dual-band Topological Corner Modes in Acoustic Kagome Lattice with Long-range Interactions
• Transport Induced Dimer State from Topological Corner States
• Observation of emergent Dirac physics at the surfaces of acoustic higher-order topological insulators
• Controlling the Deformation of Metamaterials: Corner Modes via Topology
• Granular Graphene: direct observation of zigzag and armchair edge waves
• Fully-Polarized Topological Isostatic Metamaterials in Three Dimensions
• Topological quantum transition driven by charge-phonon coupling in higher-order topological insulators
• Aharonov-Bohm Effect in Three-dimensional Higher-order Topological Insulator
• Towards a theory of surface orbital magnetization
• Topologically distinct atomic insulators
• Real-space Obstruction in Quantum Spin Hall Insulators
• Spin-polarized antichiral exciton-polariton edge states
• Single branch of chiral Majorana modes from doubly degenerate Fermi surfaces
• Higher-order Skin Effect through a Hermitian-non-Hermitian Correspondence and Its Observation in an Acoustic Kagome Lattice
• Multiple higher-order topological phases with even and odd pairs of zero-energy corner modes in a $C_3$ symmetry broken model
• Polarization-independent second-order photonic topological corner states
• Topology with memory in nonlinear driven-dissipative photonic lattices
• Zeros of Green Functions in Topological Insulators
• Interacting topological mirror excitonic insulator in one dimension
• Topological and dynamical features of periodically driven spin ladders
• Intertwined Weyl phases emergent from higher-order topology and unconventional Weyl fermions via crystalline symmetry
• Topological Corner States in Graphene by Bulk and Edge Engineering
• Topological insulators and higher-order topological insulators from gauge-invariant 1D lines
• Analytical results for Green's functions of lattice fermions
• Time evolution of Majorana corner modes in Floquet second-order topological superconductor
• Generation of twisted magnons via spin-to-orbital angular momentum conversion
• Two-Dimensional Higher-Order Topological Metals
• Glide Symmetry Protected Higher-Order Topological Insulators from Semimetals with butterfly-like Nodal Lines
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Network model for higher-order topological phases
• Zak Phase Induced Topological Nonreciprocity
• Abundance of second order topology in $C_3$ symmetric two-dimensional insulators
• Generating many Majorana corner modes and multiple phase transitions in Floquet second-order topological superconductors
• Rotational symmetry breaking and partial Majorana corner states in a high-Tc superconductor based heterostructure
• Hinge Modes and Surface States in Second-Order Topological Three-Dimensional Quantum Hall Systems induced by Charge Density Modulation
• Qubit-photon corner states in all dimensions
• Topological invariants beyond symmetry indicators: Boundary diagnostics for twofold rotationally symmetric superconductors
• Topological phases of an interacting Majorana Benalcazar-Bernevig-Hughes model
• Realizing and detecting Stiefel-Whitney insulators in an optical Raman lattice
• Universal Higher-Order Topology from 5D Weyl semimetal: Edge topology, edge Hamiltonian and nested Wilson loop
• Thermodynamic electric quadrupole moments of nematic phases from first-principles calculation
• Wannier Topology and Quadrupole Moments for a generalized Benalcazar-Bernevig-Hughes Model
• Nonlinear Corner States in Topologically Nontrivial Kagome Lattice
• Interlayer electric multipoles induced by in-plane field from quantum geometric origins
• Hybrid-order topology in unconventional magnets of Eu-based Zintl compounds with surface-dependent quantum geometry
• Observation of nonlinear topological corner states originating from different spectral charges
• Mobile Kink Solitons in a Van der Waals Charge-Density-Wave Layer
• Third-order topological insulator in three-dimensional lattice of magnetic vortices
• Spontaneous magnetization in unitary superconductors with time reversal symmetry breaking
• Out of equilibrium chiral higher order topological insulator on a $\pi$-flux square lattice
• On the relation of the entanglement spectrum to the bulk polarization
• Revisiting (higher order and crystalline) topology in old models of lead telluride
• Connection topology of step edge state bands at the surface of a three dimensional topological insulator
• Simulating Floquet topological phases in static systems
• Magnetic field filtering of the hinge supercurrent in unconventional metal NiTe${}_2$-based Josephson junctions
• A Concise Review of Recently Synthesized 2D Carbon Allotropes: Amorphous Carbon, Graphynes, Biphenylene and Fullerene Networks
• Hermitian chiral boundary states in non-Hermitian topological insulators
• Quasiparticle excitations in a one-dimensional interacting topological insulator: Application for dopant-based quantum simulation
• Pseudo topological insulators
• Floquet higher-order topological phases in momentum space
• Magnetic wallpaper Dirac fermions and topological magnetic Dirac insulators
• Quantum Monte Carlo study of topological phases on a spin analogue of Benalcazar-Bernevig-Hughes model
• Electric Polarization in Inhomogeneous Crystals
• Anderson Critical Metal Phase in Trivial States Protected by Average Magnetic Crystalline Symmetry
• Wannier band transitions in disordered $\pi$-flux ladders
• Observation of Multiple Topological Corner States in Thermal Diffusion
• Higher-order topological phase of interacting photon pairs
• Rotoinversion-symmetric bulk-hinge correspondence and its applications to higher-order Weyl semimetals
• Topological obstructed atomic limit by annihilating Dirac fermions
• Chiral orbital order of interacting bosons without higher bands
• Versatile Control of Nonlinear Topological States in Non-Hermitian Systems
• Long-Time Magnetic Relaxation in Antiferromagnetic Topological Material EuCd${}_2$As${}_2$
• Anomalous levitation and annihilation in Floquet topological insulators
• Energy density as a probe of band representations in photonic crystals
• A semiclassical treatment of spinor topological effects in driven, inhomogeneous insulators under external electromagnetic fields
• Higher-order topological insulator in a modified Haldane-Hubbard model
• Entanglement Signature of Hinge Arcs, Fermi Arcs, and Crystalline Symmetry Protection in Higher-Order Weyl Semimetals
• Multi-Dimensional Quantum Walks: a Playground of Dirac and Schrödinger Particles
• Green's Function Approach to Interacting Higher-order Topological Insulators
• Protected Gapless Edge States In Trivial Topology
• Supersymmetry on the lattice: Geometry, Topology, and Flat Bands
• Multi higher-order Dirac and nodal line semimetals
• Controllable Weyl Nodes and Fermi Arcs from Floquet Engineering Triple Fermions
• Mode-Shell correspondence, a unifying phase space theory in topological physics -- Part I: Chiral number of zero-modes
• Engineering Topological Models with a General-Purpose Symmetry-to-Hamiltonian Approach
• Majorana corner modes and flat-band Majorana edge modes in superconductor/topological-insulator/superconductor junctions
• Higher-order topological insulators in a crisscross antiferromagnetic model
• Effect of local Coulomb interaction on Majorana corner modes: weak and strong correlation limits
• Holstein polaron in a pseudospin-$1$ quantum spin Hall system: first and second order topological phase transitions
• Realizing Corner States in Artificial Crystals Based on Topological Spin Textures
• Transport features of topological corner states in honeycomb lattice with multihollow structure
• Multiorder topological superfluid phase transitions in a two-dimensional optical superlattice
• Higher-order topological corner states induced solely by onsite potentials with mirror symmetry
• AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states
• Tunable boson-assisted finite-range interaction and engineering Majorana corner modes in optical lattices
• Probing Chiral-Symmetric Higher-Order Topological Insulators with Multipole Winding Number
• Rotational Symmetry Protected Edge and Corner States in Abelian topological phases
• Gapless higher-order topology and corner states in Floquet systems
• Higher-order topological superconductivity in monolayer WTe${}_2$ from repulsive interactions
• Topological edge and corner states in coupled wave lattices in nonlinear polariton condensates
• Robust Bulk States
• Fractional boundary charges with quantized slopes in interacting one- and two-dimensional systems
• Crystalline-electromagnetic responses of higher order topological semimetals
• Second-order topological insulator under strong magnetic field: Landau levels, Zeeman effect, and magnetotransport
• Controlling Majorana modes by $p$-wave pairing in two-dimensional $p+id$ topological superconductors
• Topological Insulators with Hybrid-order Boundary States
• Dirac Solitons and Topological Edge States in the $\beta$-Fermi-Pasta-Ulam-Tsingou dimer lattice
• Two-Dimensional Altermagnetic Iron Oxyhalides: Real Chern Topology and Valley-Spin-Lattice Coupling
• Direct demonstration of bulk-boundary correspondence in higher-order topological superconductors with chiral symmetry
• Many-body multipole index and bulk-boundary correspondence
• Topological Corner Modes by Composite Wannier States in Glide-Symmetric Photonic Crystal
• Corner modes in non-Hermitian next-nearest-neighbor hopping model
• Simulating Higher-Order Topological Insulators in Density Wave Insulators
• Velocity Scanning Tomography for Room-Temperature Quantum Simulation
• Berezinskii-Kosterlitz-Thouless localization-localization transitions in disordered two-dimensional quantized quadrupole insulators
• Corner and Edge States in Topological Sierpinski Carpet Systems
• Topologically protected dynamics in three-dimensional nonlinear antisymmetric Lotka-Volterra systems
• Edge and corner superconductivity in a 2D topological model
• Bulk-boundary-transport correspondence of the second-order topological insulators
• Spin-Dependent High-Order Topological Insulator and Two Types of Distinct Corner Modes in Monolayer FeSe/GdClO Heterostructure
• Gapless Fermionic Systems as Phase-space Topological Insulators: Non-perturbative Results from Anomalies
• BerryEasy: A GPU enabled python package for diagnosis of nth-order and spin-resolved topology in the presence of fields and effects
• Thermodynamics of a Higher-Order Topological Insulator
• Remote control of spin polarization of topological corner states
• Electric polarization and its quantization in one-dimensional non-Hermitian chains
• Topological Classical Systems with Generalized Chiral Symmetry
• Extrinsic topology of Floquet anomalous boundary states in quantum walks
• Intrinsic-perturbation induced anomalous higher-order boundary states in non-Hermitian systems
• Local and energy-resolved topological invariants for Floquet systems
• Unveiling Higher-Order Topology via Polarized Topological Charges
• Competing topological phases in a non-Hermitian time-reversal symmetry-broken Bernevig-Hughes-Zhang model
• Fractional second-order topological insulator from a three-dimensional coupled-wires construction
• A mono-atomic orbital-based 1D topological crystalline insulator
• Chern dartboard insulator: sub-Brillouin zone topology and skyrmion multipoles
• Emergent metallicity at the grain boundaries of higher-order topological insulators
• Topological metals constructed by sliding quantum wire arrays
• Structural, electronic and topological properties of 3D TmBi compound
• Anisotropic-scaling localization in higher-dimensional non-Hermitian systems
• Hierarchical Topological States in Thermal Diffusive Networks
• Higher-order topology and corner triplon excitations in two-dimensional quantum spin-dimer models
• Nontrivial gapless electronic states at the stacking faults of weak topological insulators
• Engineering rich two-dimensional higher-order topological phases by flux and periodic driving
• Second-order topological insulator in a coinless discrete-time quantum walk
• Discovery of topological metamaterials by symmetry relaxation and smooth topological indicators
• Interaction-Driven Topological Switch in a $P$-Band Honeycomb Lattice
• Higher-order topology in twisted multilayer systems: a review
• Fractional corner magnetization of collinear antiferromagnets
• Orbital Edelstein effect of electronic itinerant orbital motion at edges
• Generic Approach to Intrinsic Magnetic Second-order Topological Insulators via Inverted $p-d$ Orbitals
• Takagi topological insulator with odd $\mathcal{P}\mathcal{T}$ pairs of corner states
• Fermionic tensor networks for higher order topological insulators from charge pumping
• Floquet chiral hinge modes and their interplay with Weyl physics in a three-dimensional lattice
• Zero-Energy State Localized near an Arbitrary Edge in Quadrupole Topological Insulators
• Fragile topological phase on the triangular kagome lattice and its bulk-boundary correspondence
• Topologically compatible non-Hermitian skin effect
• Additive rule of real and reciprocal space topologies at disclinations
• Effects of first- and second-order topological phases on equilibrium crystal shapes
• Tailoring higher-order topological phases via orbital hybridization
• Corner states of two-dimensional second-order topological insulators with a chiral symmetry and broken time reversal and charge conjugation
• A compact and stable incidence-plane-rotating second harmonics detector
• Generalization of the Nested Wilson Loop Formalism in Topological Dirac Semimetals with Higher-order Fermi Arcs
• Majorana corner pairs in a two-dimensional $s$-wave cold atomic superfluid
• Chiral Hinge Transport in Disordered Non-Hermitian Second-Order Topological Insulators
• Family of third-order topological insulators from Su-Schrieffer-Heeger stacking
• Mixed higher-order topology and nodal and nodeless flat band topological phases in a superconducting multiorbital model
• Magnetic impurities along the edge of a quantum spin Hall insulator: Realizing a one-dimensional AIII insulator
• Topological reflection matrix
• Second Euler number in four dimensional synthetic matter
• Unconventional hybrid-order topological insulators
• Topological electronic states in holey graphyne
• Topological optical Raman superlattices
• A topological current divider
• Restoration of non-Hermitian bulk-boundary correspondence by counterbalancing skin effect
• Second-order topology and supersymmetry in two-dimensional topological insulators
• Characterization of higher-order topological superconductors using Bott indices
• Higher-order topological superconductor phases in a multilayer system
• On the role of geometric phase in the dynamics of elastic waveguides
• Classification of topological invariants related to corner states
• Topological Corner States in Bilayer and Trilayer Systems with Vertically Stacked Topological Heterostructures
• Distinguishing trivial and topological quadrupolar insulators by Wannier-Stark ladders
• Topological Field Theories of Three-Dimensional Rotation Symmetric Insulators: Coupling Curvature and Electromagnetism
• Observation of nonlinear higher-order topological insulators with unconventional boundary truncations
• Kitaev Building-block Construction for Inversion-Protected Higher-order Topological Superconductors
• PHOENIX -- Paderborn highly optimized and energy efficient solver for two-dimensional nonlinear Schrödinger equations with integrated extensions
• Two-dimensional orbital-obstructed insulators with higher-order band topology
• Higher-order topological superconductors characterized by Fermi level crossings
• Pumping with Symmetry
• A new class of higher-order topological insulators that localize energy at arbitrary multiple sites
• Theory of Josephson Current on a Magnetically Doped Topological Insulator
• Topological states constructed by two different trivial quantum wires
• Second-order topological insulator in periodically driven lattice
• Anomalous in-gap edge states in two-dimensional pseudospin-1 Dirac insulators
• Majorana modes in trapped-ion system and their Floquet engineering
• Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators
• Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: $E_2$ pages for 1651 magnetic space groups
• Higher-order topology in honeycomb lattice with Y-Kekulé distortions
• An index theorem for quarter-plane Toeplitz operators via extended symbols and gapped Invariants related to corner states
• Inexorable Edge Kondo Breakdown in Topological Kondo Insulators
• Corner Modes and Ground-State Degeneracy in Models with Gauge-Like Subsystem Symmetries
• Exceptional second-order topological insulators
• Model non-Hermitian topological operators without skin effect: A general principle of construction
• Scattering theory of higher order topological phases
• Dimensional hierarchy of topological bound states in the continuum
• Majorana corner states in square and Kagome quantum spin liquids
• Hinge States of Second-Order Topological Insulators as a Mach-Zehnder Interferometer
• Theoretical analysis of glide-Z_2 magnetic topological photonic crystals
• Boundary and defect criticality in topological insulators and superconductors
• Topological characteristics of gap closing points in nonlinear Weyl semimetals
• Second-order topological insulator in Bilayer borophene
• Spectral Form Factors of Topological Phases
• Optical Berry Curvature of Hinge State and Its Detection
• Recipe for higher-order topology on the triangular lattice
• Extrinsic higher-order topological corner states in AB-stacked transition-metal dichalcogenides
• Topological phases in discrete stochastic systems
• Floquet topological phases with time-reversal and space inversion symmetries and dynamical detection of topological charges
• Topological parafermion corner states in clock-symmetric non-Hermitian second-order topological insulator
• Mixed-parity octupolar pairing and corner Majorana modes in three dimensions
• Hermitian and non-Hermitian topology in active matter
• Finite-temperature topological invariant for higher-order topological insulators
• Selective branching, quenching, and converting of topological modes
• Engineering Hierarchical Symmetries
• Edge States with Hidden Topology in Spinner Lattices
• Persistent Corner Spin Mode at the Quantum Critical Point of a Plaquette Heisenberg Model
• Mode-Shell correspondence, a unifying phase space theory in topological physics -- part II: Higher-dimensional spectral invariants
• Topological semimetal with coexisting nodal points and nodal lines
• Revealing Higher-Order Topological Bulk-boundary Correspondence in Bismuth Crystal with Spin-helical Hinge State Loop and Proximity Superconductivity
• Quantized crystalline-electromagnetic responses in insulators
• Tunable corner states in topological insulators with long-range hoppings and diverse shapes
• Fragmented exceptional points and their bulk and edge realizations in lattice models
• Higher-Order Topological Systems and Their Sub-Symmetry-Protected Topology
• Altermagnetism and its induced higher-order topology on the Lieb lattice
• Tunable symmetry-protected higher-order topological states with fermionic atoms in bilayer optical lattices
• Magnetic flux induced higher-order topological superconductivity
• Topologically trivial gap-filling in superconducting Fe(Se,Te) by one dimensional defects
• Dynamical characterization of topological phases beyond the minimal models
• Realization of a three-dimensional quantum Hall effect in a Zeeman-induced second order topological insulator on a torus
• Higher order topological state induced by $d$-wave competing orders in high-T${}_c$ superconductor based heterostructure
• The destiny of obstructed atomic insulator under correlation
• Cavity-induced topological edge and corner states
• Mirror-symmetry protected higher-order topological zero-frequency boundary and corner modes in Maxwell lattices
• Hinge Majorana Flat Band in Type-II Dirac Semimetals
• Anisotropic Penetration Depths of Corner States in a Higher-Order Topological Insulator
• Unconventional phonon spectra and obstructed edge phonon modes
• Hybrid symmetry class topological insulators
• Unconventional Topological Insulators from Extended Topological Band Degeneracies
• Effects of the Vertices on the Topological Bound States in a Quasicrystalline Topological Insulator
• Arboreal Obstructed Atomic Insulating and Metallic Phases of Fermions
• Quasiperiodic Quadrupole Insulators
• Chiral Majorana Edge Modes and Vortex Majorana Zero Modes in Superconducting Antiferromagnetic Topological Insulator
• Cubic Dirac Semimetals: General Theory and Application to Rare-Earth Magnets
• Surface theory of a second-order topological insulator beyond the Dirac approximation
• Third-order Orbital Corner State and its Realization in Acoustic Crystals
• Multiple topological corner states in the continuum of extended kagome lattice
• Spin separation and filtering assisted by topological corner states in the Kekulé lattice
• Engineering Majorana corner modes from two-dimensional hexagonal crystals
• Floquet-Anderson localization in the Thouless pump and how to avoid it
• Many-Body Quadrupolar Sum Rule for Higher-Order Topological Insulator
• Observation of Embedded Topology in a Trivial Bulk via Projective Crystal Symmetry
• Topological defect-mediated corner states and higher-order bulk topology in a two-dimensional crystalline insulator
• Perfectly localized Majorana corner modes in fermionic lattices
• Floquet Möbius topological insulators
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• Boundary topological insulators and superconductors of Altland-Zirnbauer tenfold classes
• Topological insulators and geometry of vector bundles
• Topological Higgs Amplitude Modes in Strongly Interacting Superfluids
• Correlated second-order Dirac semimetals with Coulomb interactions
• Inducing ${\mathbb{Z}}_2$ Topology in Twisted Nodal Superconductors
• Topology reconstruction for asymmetric systems by isomorphic mapping or perturbation approximation
• Breakdown of boundary criticality and exotic topological semimetals in $\mathcal{P}\mathcal{T}$-invariant systems
• Thresholdless corner vortex solitons in fractal Sierpiński topological insulators
• Visualizing dissipative charge carrier dynamics at the nanoscale with superconducting charge qubit microscopy
• Exceptionally deficient topological square-root insulators
• Topological Aspects of Dirac Fermions in a Kagomé Lattice
• RKKY interaction in helical higher-order topological insulators
• Floquet-Weyl semimetals generated by an optically resonant interband-transition
• C*-framework for higher-order bulk-boundary correspondences
• Nonlinear quadrupole topological insulators
• Dots and Boxes Algorithm for Peierls Substitution: Application to Multidomain Topological Insulators
• Coexistence of gapless and gapped vortex modes with Majorana corner states in a 2D second-order topological superconductor
• Network model for magnetic higher-order topological phases
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry
• Engineering Two-Dimensional Hybrid-Order Topological Insulators via Trilayer Coupling
• Gapless States Localized along a Staircase Edge in Second-Order Topological Insulators
• 3D Topologically Polarized Elastic Metamaterials Enable Asymmetric Energy Isolation at Low Frequencies
• Crystalline finite-size topology
• Non-Hermitian topological filters
• Fate of chiral order and impurity self-pinning in flat bands with local symmetry
• Quantum-vacuum-protected topological edge polaritons
• Obstructed Atomic Insulators and Superfluids of Fermions Coupled to ${\mathbb{Z}}_2$ Gauge Fields
• Phase diagram of the J1-J2 Heisenberg second-order topological quantum magnet
• Zeeman field induced corner states in the Kane-Mele-Hubbard model
• Disorder-induced delocalization and reentrance in a Chern-Hopf insulator
• Theory of Topological Superconductivity in Doped IV-VI Semiconductors
• Nonlinearity-induced corner states in a kagome lattice
• The Quadrupole Moment of Higher-Order Topological Insulator at Finite temperature
• Fractional chiral second-order topological insulator from a three-dimensional array of coupled wires
• Recent progress on disorder-induced topological phases
• Unified Multipole Bott Indices for Non-Hermitian Skin Effect in Different Orders
• Intrinsic higher-order topological states in 2D honeycomb Z_2 quantum spin Hall insulators
• Extended Haldane model -- a modern gateway to topological insulators
• Magnetic-flux tunable electronic transport through domain walls in a three-dimensional second-order topological insulator
• Magnetic doping-induced second-order and first-order topological phase transition inthe photonic alloy
• Tunable cornerlike states in topological type-II hyperbolic lattices
• Topological edge states of continuous Hamiltonians
• Interaction-Induced Higher-Order Topological Insulator via Floquet Engineering
• Hybrid Higher-Order Skin Topological Modes in the Two-Dimensional Su-Schrieffer-Heeger Model with Nonreciprocal Hoppings
• Current switching behavior mediated via hinge modes in higher-order topological phases using altermagnets
• Topological defects in a double-mirror quadrupole insulator displace diverging charge
• Valley- and Orbital-Controlled 2D Chern Insulators Without Spin-orbit Interaction
• Second-order topological insulator induced by compensated altermagnetism without bulk spin splitting
• Floquet second-order topological insulator in strained graphene
• Hinge modes of three-dimensional Euler insulators
• Bulk-boundary correspondence in topological two-dimensional non-Hermitian systems: Toeplitz operators and singular values
• Layer-number parity induced topological phase transition
• Theoretical description of interface states in a tetragonal lattice of bianisotropic resonators
• Non-Hermitian second-order topological insulator with point gap
• Coupled-wire construction of non-Abelian higher-order topological phases
• 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
• Topologically nontrivial flat bands and quantum Hall crossovers in square-octagon lattice materials
• Orbital Frontiers: Harnessing Higher Modes in Photonic Simulators
• Non-Hermitian physics and topological phenomena in convective thermal metamaterials
• When Wannier centers jump: Critical points between atomic insulating phases
• Nanoimprinted topological laser in the visible
• Boundary Condition Analysis of First and Second Order Topological Insulators
• Topological energy pumping in a quasi-periodically driven four-level system
• Classification of topological insulators and superconductors with multiple order-two point group symmetries
• Proposal for realizing and probing topological crystalline insulators in optical lattices
• Single $\pi$-flux hosting topological defect modes in bilayer acoustic metamaterials
• Projected branes as platforms for crystalline, superconducting, and higher-order topological phases
• Code CFTs and Topological Matter
• 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
• Symmetry-protected topological wire in a topological vacuum
• Electronic structure and hinge states of strained half-Heusler compounds LiSbZn and LiBiZn
• Supermetal-insulator transition in a non-Hermitian network model
• Anti-higher-order Weyl semimetal
• Graphyne as a second-order and real Chern topological insulator in two dimensions
• Magnetically tuned topological phase in graphene nanoribbon heterojunctions
• Non-Hermitian Higher-Order Dirac Semimetals
• Non-Hermitian Higher-Order Weyl Semimetals
• Quantized edge magnetizations and their symmetry protection in one-dimensional quantum spin systems
• Topological squashed entanglement: nonlocal order parameter for one-dimensional topological superconductors
• Topological Photonics on a Small Scale
• Higher-order topological insulator in cubic semiconductor quantum wells
• Application of Jordan Decomposition to Non-Hermitian Lattice Models with Spectrally-Isolated Lower Dimensional States
• Corner Flat Bands Induced by $d$-Density Wave and Partial Corner State Modification Due to Competition with $d$-Wave Superconductivity
• Vortex bound states in dimerized $\pi$-flux optical lattices: characterization, state preparation and current measurement
• Dynamical Symmetry Indicators for Floquet Crystals
• Three-dimensional higher-order topological insulator protected by cubic symmetry
• Weak ${\mathbb{Z}}_2$ Supertopology
• Dynamics of 2D topological quadrupole insulator and Chern insulator induced by real-space topological changes
• Higher-order topological quantum paramagnets
• Glide-resolved photoemission spectroscopy: Measuring topological invariants in nonsymmorphic space groups
• Hofstadter Butterflies in Topological Insulators
• Synchronized in-gap edge states and robust copropagation in topological insulators without magnetic flux
• Semimetals without correspondence between the topological charge of nodal line/surface and Fermi arc
• Magnetic-Field Tunable Möbius and Higher-Order Topological Insulators in Three-Dimensional Layered Octagonal Quasicrystals
• Topological phases of monolayer and bilayer depleted Lieb lattices