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.

72 pages, 52 figures

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• Mode-Shell correspondence, a unifying phase space theory in topological physics -- Part I: Chiral number of zero-modes
• Majorana corner modes and flat-band Majorana edge modes in superconductor/topological-insulator/superconductor junctions
• Realizing Corner States in Artificial Crystals Based on Topological Spin Textures
• Higher-order topology protected by latent crystalline symmetries
• Probing Chiral-Symmetric Higher-Order Topological Insulators with Multipole Winding Number
• AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states
• Topological Wannier cycles for the bulk and edges
• Topologically localized insulators
• Multi higher-order Dirac and nodal line semimetals
• Multiorder topological superfluid phase transitions in a two-dimensional optical superlattice
• Crystalline-electromagnetic responses of higher order topological semimetals
• Higher-order topological corner states induced solely by onsite potentials with mirror symmetry
• Controllable Weyl Nodes and Fermi Arcs from Floquet Engineering Triple Fermions
• Gapless higher-order topology and corner states in Floquet systems
• Fractional boundary charges with quantized slopes in interacting one- and two-dimensional systems
• Higher-order topological insulators in a crisscross antiferromagnetic model
• Second-order topological insulator under strong magnetic field: Landau levels, Zeeman effect, and magnetotransport
• Edge and corner superconductivity in a 2D topological model
• Controlling Majorana modes by $p$-wave pairing in two-dimensional $p+id$ topological superconductors
• Fractal Surface States in Three-Dimensional Topological Quasicrystals
• Disorder-Induced Phase Transitions in Three-Dimensional Chiral Second-Order Topological Insulator
• Gapless Fermionic Systems as Phase-space Topological Insulators: Non-perturbative Results from Anomalies
• Direct demonstration of bulk-boundary correspondence in higher-order topological superconductors with chiral symmetry
• Corner modes in non-Hermitian next-nearest-neighbor hopping model
• Corner and Edge States in Topological Sierpinski Carpet Systems
• Topological Corner Modes by Composite Wannier States in Glide-Symmetric Photonic Crystal
• Fractional second-order topological insulator from a three-dimensional coupled-wires construction
• Competing topological phases in a non-Hermitian time-reversal symmetry-broken Bernevig-Hughes-Zhang model
• Unveiling Higher-Order Topology via Polarized Topological Charges
• Remote control of spin polarization of topological corner states
• Intrinsic-perturbation induced anomalous higher-order boundary states in non-Hermitian systems
• Electric polarization and its quantization in one-dimensional non-Hermitian chains
• Local and energy-resolved topological invariants for Floquet systems
• Extrinsic topology of Floquet anomalous boundary states in quantum walks
• Topological Classical Systems with Generalized Chiral Symmetry
• Microwave hinge states in a simple-cubic-lattice photonic crystal insulator
• Berezinskii-Kosterlitz-Thouless localization-localization transitions in disordered two-dimensional quantized quadrupole insulators
• Many-body multipole index and bulk-boundary correspondence
• Velocity Scanning Tomography for Room-Temperature Quantum Simulation
• A mono-atomic orbital-based 1D topological crystalline insulator
• Synthetic topology and Floquet dynamic quantum phase transition in a periodically driven Raman lattice
• Two-Dimensional Altermagnetic Iron Oxyhalides: Real Chern Topology and Valley-Spin-Lattice Coupling
• Spin-Dependent High-Order Topological Insulator and Two Types of Distinct Corner Modes in Monolayer FeSe/GdClO Heterostructure
• Thermodynamics of a Higher-Order Topological Insulator
• Theory of Anomalous Floquet Higher-Order Topology: Classification, Characterization, and Bulk-Boundary Correspondence
• Hierarchical Topological States in Thermal Diffusive Networks
• Chern dartboard insulator: sub-Brillouin zone topology and skyrmion multipoles
• Fractional corner magnetization of collinear antiferromagnets
• Fractional corner charges in threefold-symmetric two-dimensional materials with fragile topology
• Orbital Edelstein effect of electronic itinerant orbital motion at edges
• Floquet chiral hinge modes and their interplay with Weyl physics in a three-dimensional lattice
• Spin-orbit-coupled superconductivity with spin-singlet non-unitary pairing
• Higher-order topology and corner triplon excitations in two-dimensional quantum spin-dimer models
• Higher-order topology in twisted multilayer systems: a review
• Anisotropic-scaling localization in higher-dimensional non-Hermitian systems
• Generic Approach to Intrinsic Magnetic Second-order Topological Insulators via Inverted $p-d$ Orbitals
• Magnonics along the wall in Bimeron Chain Domain Walls
• Engineering rich two-dimensional higher-order topological phases by flux and periodic driving
• Topological metals constructed by sliding quantum wire arrays
• Interaction-Driven Topological Switch in a $P$-Band Honeycomb Lattice
• Second-order topological insulator in a coinless discrete-time quantum walk
• Emergent metallicity at the grain boundaries of higher-order topological insulators
• XFe4Ge2 (X = Y, Lu) and Mn3Pt: Filling-enforced magnetic topological metals
• Fermionic tensor networks for higher order topological insulators from charge pumping
• Second-order topology and supersymmetry in two-dimensional 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
• Chiral Hinge Transport in Disordered Non-Hermitian Second-Order Topological Insulators
• Higher-order topological superconductor phases in a multilayer system
• Tailoring higher-order topological phases via orbital hybridization
• 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
• Zero-Energy State Localized near an Arbitrary Edge in Quadrupole Topological Insulators
• Single-layer tensor network approach for three-dimensional quantum systems
• 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
• A topological current divider
• Two-dimensional orbital-obstructed insulators with higher-order band topology
• Magnetic impurities along the edge of a quantum spin Hall insulator: Realizing a one-dimensional AIII insulator
• Unconventional hybrid-order topological insulators
• Observation of nonlinear higher-order topological insulators with unconventional boundary truncations
• Kitaev Building-block Construction for Inversion-Protected Higher-order Topological Superconductors
• Generalization of the Nested Wilson Loop Formalism in Topological Dirac Semimetals with Higher-order Fermi Arcs
• Topology in quasicrystals
• Second-order topological insulator in periodically driven lattice
• Topological characteristics of gap closing points in nonlinear Weyl semimetals
• Corner states in second-order mechanical topological insulator
• General corner charge formulas in various tetrahedral and cubic space groups
• Model non-Hermitian topological operators without skin effect: A general principle of construction
• Optical Berry Curvature of Hinge State and Its Detection
• Protecting Quantum Superposition and Entanglement with Photonic Higher-Order Topological Crystalline Insulator
• Second-order topological insulator in Bilayer borophene
• Majorana corner states in square and Kagome quantum spin liquids
• Scattering theory of higher order topological phases
• Higher-order topological superconductors characterized by Fermi level crossings
• Observation of trapped fractional charge and topological states at disclination defects in higher-order topological insulators
• Adiabatic charge transport in extended SSH models
• Recipe for higher-order topology on the triangular lattice
• Hybrid-Order Topological Phase And Transition in 1H Transition Metal Compounds
• Exceptional second-order topological insulators
• Theoretical analysis of glide-Z_2 magnetic topological photonic crystals
• Fermion-fermion interaction driven phase transitions in rhombohedral trilayer graphene
• Topological states constructed by two different trivial quantum wires
• Mirror-symmetry protected higher-order topological zero-frequency boundary and corner modes in Maxwell lattices
• Homotopy invariant in time-reversal and twofold rotation symmetric systems
• Extrinsic higher-order topological corner states in AB-stacked transition-metal dichalcogenides
• Higher order topological state induced by $d$-wave competing orders in high-T${}_c$ superconductor based heterostructure
• Realization of a three-dimensional quantum Hall effect in a Zeeman-induced second order topological insulator on a torus
• Fragmented exceptional points and their bulk and edge realizations in lattice models
• Unconventional Topological Insulators from Extended Topological Band Degeneracies
• Cavity-induced topological edge and corner states
• Finite-temperature topological invariant for higher-order topological insulators
• Hinge Majorana Flat Band in Type-II Dirac Semimetals
• Tunable symmetry-protected higher-order topological states with fermionic atoms in bilayer optical lattices
• Magnetic flux induced higher-order topological superconductivity
• Theory of fractional corner charges in cylindrical crystal shapes
• Anisotropic Penetration Depths of Corner States in a Higher-Order Topological Insulator
• Takagi Topological Insulator on the Honeycomb Lattice
• Topological parafermion corner states in clock-symmetric non-Hermitian second-order topological insulator
• The destiny of obstructed atomic insulator under correlation
• Mixed-parity octupolar pairing and corner Majorana modes in three dimensions
• Fragile topology in nodal-line semimetal superconductors
• Tunable corner states in topological insulators with long-range hoppings and diverse shapes
• Unconventional phonon spectra and obstructed edge phonon modes
• Altermagnet-Superconductor Heterostructure: a Scalable Platform for Braiding of Majorana Modes
• Persistent Corner Spin Mode at the Quantum Critical Point of a Plaquette Heisenberg Model
• Nonlinear quadrupole topological insulators
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• $Z_2\times Z_2$ symmetry and $Z_4$ Berry phase of bosonic ladder
• Many-Body Quadrupolar Sum Rule for Higher-Order Topological Insulator
• RKKY interaction in helical higher-order topological insulators
• Chiral Majorana Edge Modes and Vortex Majorana Zero Modes in Superconducting Antiferromagnetic Topological Insulator
• Topological Higgs Amplitude Modes in Strongly Interacting Superfluids
• Coexistence of gapless and gapped vortex modes with Majorana corner states in a 2D second-order topological superconductor
• Topology reconstruction for asymmetric systems by isomorphic mapping or perturbation approximation
• Observation of Embedded Topology in a Trivial Bulk via Projective Crystal Symmetry
• Third-order Orbital Corner State and its Realization in Acoustic Crystals
• Topological insulators and geometry of vector bundles
• Exceptionally deficient topological square-root insulators
• Surface theory of a second-order topological insulator beyond the Dirac approximation
• Correlated second-order Dirac semimetals with Coulomb interactions
• Quasiperiodic Quadrupole Insulators
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Engineering Majorana corner modes from two-dimensional hexagonal crystals
• Floquet-Anderson localization in the Thouless pump and how to avoid it
• Boundary topological insulators and superconductors of Altland-Zirnbauer tenfold classes
• Effects of the Vertices on the Topological Bound States in a Quasicrystalline Topological Insulator
• Arboreal Obstructed Atomic Insulating and Metallic Phases of Fermions
• Multiple topological corner states in the continuum of extended kagome lattice
• Topological Aspects of Dirac Fermions in a Kagomé Lattice
• Perfectly localized Majorana corner modes in fermionic lattices
• Fractional chiral second-order topological insulator from a three-dimensional array of coupled wires
• Phase diagram of the J1-J2 Heisenberg second-order topological quantum magnet
• Zeeman field induced corner states in the Kane-Mele-Hubbard model
• Interaction-Induced Higher-Order Topological Insulator via Floquet Engineering
• Extended Haldane model -- a modern gateway to topological insulators
• Disorder-induced delocalization and reentrance in a Chern-Hopf insulator
• Nonlinearity-induced corner states in a kagome lattice
• Network model for magnetic higher-order topological phases
• Quantum-vacuum-protected topological edge polaritons
• 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
• Symmetry-protected gates of Majorana qubits in a high-$T_c$ superconductor platform
• Unified Multipole Bott Indices for Non-Hermitian Skin Effect in Different Orders
• Fractional ac Josephson effect as evidence of topological hinge states in a Dirac semimetal NiTe2
• Wilson Loop and Topological Properties in 3D Woodpile Photonic Crystal
• Theory of Topological Superconductivity in Doped IV-VI Semiconductors
• Hybrid Higher-Order Skin Topological Modes in the Two-Dimensional Su-Schrieffer-Heeger Model with Nonreciprocal Hoppings
• Topological states in chiral electronic chains
• Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry
• Higher-Order Topology in Monolayer FeSe
• Hidden wallpaper fermion and third-order topological insulator
• Gapless States Localized along a Staircase Edge in Second-Order Topological Insulators
• Obstructed Atomic Insulators and Superfluids of Fermions Coupled to ${\mathbb{Z}}_2$ Gauge Fields
• Photonic Higher-Order Topological States Induced by Long Range Interactions
• Higher order topology in a Creutz ladder
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry
• Non-Hermitian Higher-Order Weyl Semimetals
• Dynamical Symmetry Indicators for Floquet Crystals
• Enormous Berry-Curvature-Driven Anomalous Hall Effect in Topological Insulator (Bi,Sb)2Te3 on Ferrimagnetic Europium Iron Garnet beyond 400 K
• Corner Flat Bands Induced by $d$-Density Wave and Partial Corner State Modification Due to Competition with $d$-Wave Superconductivity
• Quantized edge magnetizations and their symmetry protection in one-dimensional quantum spin systems
• Pseudospin-valley-coupled phononic topological insulator with edge and corner states
• Topological Photonics on a Small Scale
• Magnetic-Field Tunable Möbius and Higher-Order Topological Insulators in Three-Dimensional Layered Octagonal Quasicrystals
• Direct visualization of topological transitions and higher-order topological states in photonic metasurfaces
• Fourth-Order Topological Insulator via Dimensional Reduction
• Boundary Condition Analysis of First and Second Order Topological Insulators
• Electronic structure and hinge states of strained half-Heusler compounds LiSbZn and LiBiZn
• Second-order topological insulator induced by compensated altermagnetism without bulk spin splitting
• Hinge modes of three-dimensional Euler insulators
• An all-dielectric photonic crystal with unconventional higher-order topology
• Emergent One-Dimensional Helical Channel in Higher-Order Topological Insulators with Step Edges
• Application of Jordan Decomposition to Non-Hermitian Lattice Models with Spectrally-Isolated Lower Dimensional States
• Layer-number parity induced topological phase transition
• Proposal for realizing and probing topological crystalline insulators in optical lattices
• Wilson loop invariants and bulk-boundary correspondence in higher-order topological insulators with two anticommuting mirror symmetries
• Graphyne as a second-order and real Chern topological insulator in two dimensions
• When Wannier centers jump: Critical points between atomic insulating phases
• Topological energy pumping in a quasi-periodically driven four-level system
• 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
• Gapless hinge states from adiabatic pumping of axion coupling
• Majorana vortex phases in time-reversal invariant higher-order topological insulators and topologically trivial insulators
• Entanglement as a topological marker in harmonically confined attractive Fermi-Hubbard chains
• Non-Hermitian physics and topological phenomena in convective thermal metamaterials
• Topological defects in a double-mirror quadrupole insulator displace diverging charge
• Topological Classification of Gapped/Gap-preserving Rational Space-Time Crystal Systems
• Higher-order topological phases protected by noninvertible and subsystem symmetries
• Topological phases of monolayer and bilayer depleted Lieb lattices
• Supermetal-insulator transition in a non-Hermitian network model
• Higher-order topological quantum paramagnets
• Non-Hermitian Higher-Order Dirac Semimetals