Higher-order topological band structures

Abstract

The interplay of topology and symmetry in a material's band structure may result in various patterns of topological states of different dimensionality on the boundary of a crystal. The protection of these "higher-order" boundary states comes from topology, with constraints imposed by symmetry. We review the bulk-boundary correspondence of topological crystalline band structures, which relates the topology of the bulk band structure to the pattern of the boundary states. Furthermore, recent advances in the K-theoretic classification of topological crystalline band structures are discussed.

arXiv commentExpanded version of the published review article in physica status solidi (b). v2: discussion of experimental realizations of higher-order topological band structures

References in corpus (61)

• Topological Insulators
• Topological insulators and superconductors
• Unpaired Majorana fermions in quantum wires
• Classification of topological quantum matter with symmetries
• Topological Crystalline Insulators
• Quantized Electric Multipole Insulators
• Periodic table for topological insulators and superconductors
• Higher-Order Topological Insulators
• Topological quantum chemistry
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• The (High Quality) Topological Materials In The World
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Topolectrical circuit realization of topological corner modes
• Reflection symmetric second-order topological insulators and superconductors
• Observation of a phononic quadrupole topological insulator
• Catalogue of Topological Electronic Materials
• Higher-Order Topology in Bismuth
• Demonstration of a quantized microwave quadrupole insulator with topologically protected corner states
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Quantization of fractional corner charge in $C_n$-symmetric higher-order topological crystalline insulators
• Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials
• Higher-order topological insulators and superconductors protected by inversion symmetry
• Topology of crystalline insulators and superconductors
• Second-order topological insulators and superconductors with an order-two crystalline symmetry
• Classification of topological insulators and superconductors in the presence of reflection symmetry
• Classification of Topological Insulators and Superconductors
• Quantized Response and Topology of Insulators with Inversion Symmetry
• Topological classification with additional symmetries from Clifford algebras
• Surface State Magnetization and Chiral Edge States on Topological Insulators
• Robust zero-energy modes in an electronic higher-order topological insulator: the dimerized Kagome lattice
• Observation of Quadrupole Transitions and Edge Mode Topology in an LC network
• Scattering formula for the topological quantum number of a disordered multi-mode wire
• Higher-order topological insulators protected by inversion and rotoinversion symmetries
• Classification of Two Dimensional Topological Crystalline Superconductors and Majorana Bound States at Disclinations
• Building crystalline topological phases from lower-dimensional states
• Majorana Fermions and Disclinations in Topological Crystalline Superconductors
• Real Space Invariants: Twisted Bulk-Boundary Correspondence of Fragile Topology
• Rotation Anomaly and Topological Crystalline Insulators
• Topological insulators in magnetic fields: Quantum Hall effect and edge channels with non-quantized θ-term
• Ballistic edge states in Bismuth nanowires revealed by SQUID interferometry
• Theory of interacting topological crystalline insulators
• Refined symmetry indicators for topological superconductors in all space groups
• Resolving the Topological Classification of Bismuth with Topological Defects
• Stiefel-Whitney classes and topological phases in band theory
• Unified understanding of symmetry indicators for all internal symmetry classes
• Symmetry-based indicators for topological Bogoliubov-de Gennes Hamiltonians
• Difficulties in operator-based formulation of the bulk quadrupole moment
• Topological superfluid 3He-B in magnetic field and Ising variable
• Classification and construction of higher-order symmetry protected topological phases of interacting bosons
• Topological Mirror Insulators in One Dimension
• Classification of Crystalline Topological Insulators and Superconductors with Point Group Symmetries
• Atomic limit and inversion-symmetry indicators for topological superconductors
• Topological classification under nonmagnetic and magnetic point group symmetry: Application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology
• Symmetry indicators for topological superconductors
• Coupled-layer description of topological crystalline insulators
• Pairing Obstructions in Topological Superconductors
• Second-Order Bulk-Boundary Correspondence in Rotationally Symmetric Topological Superconductors from Stacked Dirac Hamiltonians
• Classification of crystalline insulators without symmetry indicators: atomic and fragile topological phases in twofold rotation symmetric systems
• Bott periodicity for the topological classification of gapped states of matter with reflection symmetry
• Embedded Topological Insulators

Cited by in corpus (45)

• Topological Non-Hermitian skin effect
• Boundary-obstructed topological phases
• Higher-Order Topological Dirac Superconductors
• Effective Model for Fractional Topological Corner Modes in Quasicrystals
• Hierarchy of higher-order topological superconductors in three dimensions
• Non-Hermitian Floquet second order topological insulators in periodically quenched lattices
• Higher-order topological phases in crystalline and non-crystalline systems: a review
• Intrinsic first and higher-order topological superconductivity in a doped topological insulator
• $N$-band Hopf insulator
• Bulk-boundary-defect correspondence at disclinations in rotation-symmetric topological insulators and superconductors
• Bulk-and-edge to corner correspondence
• Quadrupole spin polarization as signature of second-order topological superconductors
• Generation of higher-order topological insulators using periodic driving
• Unified characterization for higher-order topological phase transitions
• Constructing higher-order topological states in higher dimension
• Corner states, hinge states and Majorana modes in SnTe nanowires
• Dynamical bulk boundary correspondence and dynamical quantum phase transitions in higher order topological insulators
• Exact Solutions Disentangle Higher-Order Topology in 2D Non-Hermitian Lattices
• Hinge Spin Polarization in Magnetic Topological Insulators Revealed by Resistance Switch
• Floquet second-order topological Anderson insulator hosting corner localized modes
• Higher-order topological semimetals and nodal superconductors with an order-two crystalline symmetry
• Dirac cones and higher-order topology in quasi-continuous media
• Topological phases of an interacting Majorana Benalcazar-Bernevig-Hughes model
• 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
• Green's Function Approach to Interacting Higher-order Topological Insulators
• Floquet higher-order topological phases in momentum space
• Gapless higher-order topology and corner states in Floquet systems
• Mode-Shell correspondence, a unifying phase space theory in topological physics -- Part I: Chiral number of zero-modes
• Unveiling Higher-Order Topology via Polarized Topological Charges
• 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
• Boundary obstructed topological superconductor in buckled honeycomb lattice under perpendicular electric field
• Unconventional hybrid-order topological insulators
• Scattering theory of higher order topological phases
• Hinge States of Second-Order Topological Insulators as a Mach-Zehnder Interferometer
• Selective branching, quenching, and converting of topological modes
• Higher-order topological phases for time-reversal-symmetry breaking superconductivity in UTe${}_2$
• Multiple topological corner states in the continuum of extended kagome lattice
• Perfectly localized Majorana corner modes in fermionic lattices
• Network model for magnetic higher-order topological phases
• Proposal for realizing and probing topological crystalline insulators in optical lattices
• A Dynamical Bulk-Boundary Correspondence in Two Dimensional Topological Matter
• Topological energy pumping in a quasi-periodically driven four-level system