Classification of Topological Insulators and Superconductors

Abstract

An exhaustive classification scheme of topological insulators and superconductors is presented. The key property of topological insulators (superconductors) is the appearance of gapless degrees of freedom at the interface/boundary between a topologically trivial and a topologically non-trivial state. Our approach consists in reducing the problem of classifying topological insulators (superconductors) in d spatial dimensions to the problem of Anderson localization at a (d-1) dimensional boundary of the system. We find that in each spatial dimension there are precisely five distinct classes of topological insulators (superconductors). The different topological sectors within a given topological insulator (superconductor) can be labeled by an integer winding number or a Z_2 quantity. One of the five topological insulators is the 'quantum spin Hall' (or: Z_2 topological) insulator in d=2, and its generalization in d=3 dimensions. For each dimension d, the five topological insulators correspond to a certain subset of five of the ten generic symmetry classes of Hamiltonians introduced more than a decade ago by Altland and Zirnbauer in the context of disordered systems (which generalizes the three well known "Wigner-Dyson'' symmetry classes).

12 pages, 5 figures; Proceedings of the L.D. Landau Memorial Conference, 2008

Cited by in corpus (232)

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• Topological Photonics
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• Topological Defects and Gapless Modes in Insulators and Superconductors
• Topological protection of photonic mid-gap cavity modes
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• Wannier representation of Z_2 topological insulators
• Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials
• Second-order topological insulators and superconductors with an order-two crystalline symmetry
• New topological invariants in non-Hermitian systems
• Classification of reflection symmetry protected topological semimetals and nodal superconductors
• Topological Zero-Energy Modes in Gapless Commensurate Aubry-André-Harper Models
• Edge states and the bulk-boundary correspondence in Dirac Hamiltonians
• Topological invariants for spin-orbit coupled superconductor nanowires
• Topological classification with additional symmetries from Clifford algebras
• Topological Superconductivity in a Planar Josephson Junction
• Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems
• Protected boundary states in gapless topological phases
• Scattering theory of topological insulators and superconductors
• Topological Classification and Stability of Fermi Surfaces
• Topological surface states in nodal superconductors
• Charged Skyrmions and Topological Origin of Superconductivity in Magic Angle Graphene
• Altermagnetic Routes to Majorana Modes in Zero Net Magnetization
• Majorana Fermions and Non-Abelian Statistics in Three Dimensions
• Types of topological surface states in nodal noncentrosymmetric superconductors
• Interaction-induced criticality in Z_2 topological insulators
• Symmetry Protected Topological Superfluids and Superconductors --- From the Basics to ${}^3$He ---
• Network Models of Photonic Floquet Topological Insulators
• Topological Insulators from Group Cohomology
• Multipartite entanglement in topological quantum phases
• Superconductivity in the doped topological insulator Cu${}_x$Bi${}_2$Se${}_3$ under high pressure
• Symmetry Protected Topological States of Interacting Fermions and Bosons
• Majorana nanowires for topological quantum computation
• Aperiodic Weak Topological Superconductors
• Smooth gauge for topological insulators
• Complex classes of periodically driven topological lattice systems
• Topologically stable gapless phases of time-reversal invariant superconductors
• Emergent topological phenomena in thin films of pyrochlore iridates
• Fermion zero modes at the boundary of superfluid 3He-B
• Berry-phase description of Topological Crystalline Insulators
• Majorana Fermions in Chiral Topological Ferromagnetic Nanowires
• Topological classification of defects in non-Hermitian systems
• Inflated nodes and surface states in superconducting half-Heusler compounds
• Emergent Weyl spinors in multi-fermion systems
• From the adiabatic theorem of quantum mechanics to topological states of matter
• Topological invariant for superfluid 3He-B and quantum phase transitions
• Long-range topological insulators and weakened bulk-boundary correspondence
• Higher-Order topological insulator in a dodecagonal quasicrystal
• Topological networks for quantum communication between distant qubits
• Lattice model of three-dimensional topological singlet superconductor with time-reversal symmetry
• Exotic Quantum Phase Transitions of $(2+1)d$ Dirac fermions
• Majorana fermions in ferromagnetic chains on the surface of bulk spin-orbit coupled $s$-wave superconductors
• Shift insulators: rotation-protected two-dimensional topological crystalline insulators
• A study on correlation effects in two dimensional topological insulators
• Hidden-Symmetry-Protected Topological Semimetals on a Square Lattice
• Almost Commuting Matrices, Localized Wannier Functions, and the Quantum Hall Effect
• Symmetries, Dimensions, and Topological Insulators: the mechanism behind the face of the Bott clock
• Particle-hole symmetry, many-body localization, and topological edge modes
• Statistical translation invariance protects a topological insulator from interactions
• The topological classification of one-dimensional symmetric quantum walks
• Twisted Injectivity in PEPS and the Classification of Quantum Phases
• Topological superfluid 3He-B in magnetic field and Ising variable
• Interacting Topological Insulator and Emergent Grand Unified Theory
• Topological Bloch oscillations
• Higher-order topological band structures
• Spherical topological insulator
• Classification of Crystalline Topological Insulators and Superconductors with Point Group Symmetries
• Topological minigap in quasi-one-dimensional spin-orbit-coupled semiconductor Majorana wires
• Metal-insulator transition in 2D random fermion systems of chiral symmetry classes
• Topology Classification from Chiral Symmetry: Chiral Phase Index and Spin Correlations in Graphene Nanoribbons
• Magnetic field induced localization in 2D topological insulators
• Symmetry Protected Topological Hopf Insulator and its Generalizations
• From Standard Model of particle physics to room-temperature superconductivity
• Topological invariants for Standard Model: from semi-metal to topological insulator
• Quantum impurity spin in Majorana edge fermions
• Revealing a topological connection between stabilities of Fermi surfaces and topological insulators/superconductors
• Non-Abelian topological spin liquids from arrays of quantum wires or spin chains
• Finite-size energy gap in weak and strong topological insulators
• Magnetic Field Response and Chiral Symmetry of Time Reversal Invariant Topological Superconductors
• Weyl and Dirac Loop Superconductors
• Multicellularity of delicate topological insulators
• Anderson localization and the topology of classifying spaces
• Topological Properties of Time Reversal Symmetric Kitaev Chain and Applications to Organic Superconductors
• Time reversal invariant topological superconductivity in correlated non-centrosymmetric systems
• Dissipation-induced topological insulators: A no-go theorem and a recipe
• Wire constructions of Abelian topological phases in three or more dimensions
• Scattering theory of chiral Majorana fermion interferometry
• Transport through a quantum spin Hall quantum dot
• $Z_4$ Topological Superconductivity in UCoGe
• Disorder-induced Majorana zero modes in a dimerized Kitaev superconductor chain
• Higher-order topological phases in crystalline and non-crystalline systems: a review
• Magnetic and superconducting phase diagram of the half-Heusler topological semimetal HoPdBi
• Majorana flat bands, chiral Majorana edge states, and unidirectional Majorana edge states in noncentrosymmetric superconductors
• Criticality of the metal-topological insulator transition driven by disorder
• Topological Fermi arcs in superfluid ${}^3$He
• Notes on topological insulators
• Construction and properties of a topological index for periodically driven time-reversal invariant 2D crystals
• Spectral flows associated to flux tubes
• Models of three-dimensional fractional topological insulators
• Density of states in a two-dimensional chiral metal with vacancies
• Andreev-Majorana bound states in superfluids
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• Topological invariants for interacting topological insulators: II. Breakdown of the Green's function formalism
• Winding number and Zak phase in multi-band SSH models
• Current transport properties and phase diagram of a Kitaev chain with long-range pairing
• Interaction effects on the classification of crystalline topological insulators and superconductors
• Bridging Fermionic and Bosonic Short Range Entangled States
• Robustness of Majorana bound states in the short junction limit
• Quantum Monte Carlo study of strange correlator in interacting topological insulators
• Fermion masses without symmetry breaking in two spacetime dimensions
• The structure of spinful quantum Hall states: a squeezing perspective
• Topological Insulators and Superconductors from String Theory
• Time-reversal symmetric Kitaev model and topological superconductor in two dimensions
• Four-dimensional semimetals with tensor monopoles: From surface states to topological responses
• Majorana/Andreev crossover and the fate of the topological phase transition in inhomogeneous nanowires
• Three Dimensional Z2 Topological Phases enriched by Time-Reversal Symmetry
• On gapped phases with a continuous symmetry and boundary operators
• Disordered Weyl semimetals and their topological family
• Index theorem and Majorana zero modes along a non-Abelian vortex in a color superconductor
• Complete homotopy invariants for translation invariant symmetric quantum walks on a chain
• Topological number and Fermion Green's function of Strongly Interacting Topological Superconductors
• Topological Insulators and Superconductors from D-branes
• Extended topological group structure due to average reflection symmetry
• Evolutions of helical edge states in disordered HgTe/CdTe quantum wells
• Long-range Kitaev Chains via Planar Josephson Junctions
• Bulk-boundary-defect correspondence at disclinations in rotation-symmetric topological insulators and superconductors
• Classifying Quantum Phases With The Torus Trick
• Experimental observation of exceptional bound states in a classical circuit network
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• Holographic fractional topological insulators in 2+1 and 1+1 dimensions
• Dimensional hierarchy of fermionic interacting topological phases
• Edge insulating topological phases in a two-dimensional long-range superconductor
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• Interacting topological quantum chemistry of Mott atomic limits
• Bulk-and-edge to corner correspondence
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• Non-Abelian statistics of vortices with multiple Majorana fermions
• Unwinding Short-Range Entanglement
• Novel ${\mathbb{Z}}_2$-topological metals and semimetals
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• Bulk-boundary correspondence and singularity-filling in long-range free-fermion chains
• (3+1)-dimensional topological quantum field theory from a tight-binding model of interacting spinless fermions
• Topology of the planar phase of superfluid ${}^3$He and bulk-boundary correspondence for three dimensional topological superconductors
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• Homotopical Foundations of Parametrized Quantum Spin Systems
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• Unwinding Fermionic SPT Phases: Supersymmetry Extension
• Majorana zero modes bound to a vortex line in a topological superconductor
• Interference of Majorana fermions in NS junctions
• Dissipation driven dynamical topological phase transitions in two-dimensional superconductors
• Helical spin texture of surface states in topological superconductors
• Topological Edge and Interface states at Bulk disorder-to-order Quantum Critical Points
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• Hall conductivity as bulk signature of topological transitions in superconductors
• Quantum distance and the Euler number index of the Bloch band in a 1D spin model
• Visualizing a Bosonic Symmetry Protected Topological Phase in an Interacting Fermion Model
• Quantum walks: Schur functions meet symmetry protected topological phases
• Novel topological insulators from crystalline symmetries
• Helical Majorana surface states of strongly disordered topological superconductors with time-reversal symmetry
• Topological linear response of hyperbolic Chern insulators
• Helical Majorana edge mode in a superconducting antiferromagnetic quantum spin Hall insulator
• Bosonic Short Range Entangled states Beyond Group Cohomology classification
• Symmetry protected weak topological phases in a superlattice
• Entanglement entropy and entanglement spectrum of triplet topological superconductors
• Majorana Doublets, Flat Bands, and Dirac Nodes in s-Wave Superfluids
• Interplay between singlet and triplet pairings in multi-band two-dimensional oxide superconductors
• Spherical representations of Lie supergroups
• Type-I and type-II topological nodal superconductors with $s$-wave interaction
• Strange correlators for topological quantum systems from bulk-boundary correspondence
• Evolution of edge states in topological superfluids during the quantum phase transition
• Topological thermoelectric effects in spin-orbit coupled electron and hole doped semiconductors
• Interacting Topological Insulators with Synthetic Dimensions
• Topological invariants beyond symmetry indicators: Boundary diagnostics for twofold rotationally symmetric superconductors
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Non-Abelian statistics of vortices with non-Abelian Dirac fermions
• Mutual information for fermionic systems
• Interacting Valley Chern Insulator and its Topological Imprint on Moiré Superconductors
• Out of equilibrium chiral higher order topological insulator on a $\pi$-flux square lattice
• Simulating Floquet topological phases in static systems
• The Harish-Chandra isomorphism for reductive symmetric superpairs
• Dispersive 1D Majorana modes with emergent supersymmetry in 1D proximitized superconductors via spatially-modulated potentials and magnetic fields
• Reproducible synthetic method for the topological superconductor CuxBi2Se3
• Topologically localized insulators
• Topological mechanical metamaterial with nonrectilinear constraints
• Majorana Zero-modes and Topological Phases of Multi-flavored Jackiw-Rebbi model
• Topological Orders with Global Gauge Anomalies
• Beyond the standard model of topological Josephson junctions: From crystalline anisotropy to finite-size and diode effects
• Symmetry analysis of transport properties in helical superconductor junctions
• Holographic Wilson Loops, Dielectric Interfaces, and Topological Insulators
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• Zero-dimensional topologically nontrivial state in a superconducting quantum dot
• Majorana multipole response with magnetic point group symmetry
• Interacting Majorana modes at surfaces of noncentrosymmetric superconductors
• Theory of surface spectroscopy for noncentrosymmetric superconductors
• Majorana bound state of a Bogoliubov-de Gennes-Dirac Hamiltonian in arbitrary dimensions
• C-P-T Fractionalization
• 4$\pi$ and 8$\pi$ dual Josephson effects induced by symmetry defects
• Dynamics of Anderson localization in disordered wires
• Representation-protected topology of spin-singlet $s$-wave superconductors
• Instanton analysis for the spin quantum Hall symmetry class: Non-perturbative corrections to physical observables and generalized multifractal spectrum
• Interacting Dirac fields in an expanding universe: dynamical condensates and particle production
• Effects of domain walls in bilayer graphene in an external magnetic field
• Tunable Flux Vortices in 2D Dirac Superconductors
• Free-Fermion Dynamics with Measurements: Topological Classification and Adaptive Preparation of Topological States
• Phase diagrams of superconducting topological surface states
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Neutrino zeromodes on electroweak strings in light of topological insulators
• Microtubules as electron-based topological insulators
• Enriched classification of parafermionic gapped phases with time reversal symmetry
• Topological zero modes and bounded modes at smooth domain walls: Exact solutions and dualities
• Nodal lines and boundary modes in topological Dirac semimetals with magnetism
• Synthesis of Majorana mass terms in low-energy quantum systems
• Refined Characterization of Lattice Chern Insulators by Bulk Entanglement Spectrum
• Quantum Fractality on the Surface of Topological Insulators
• Fermi arc criterion for surface Majorana modes in superconducting time-reversal symmetric Weyl semimetals
• Euler band topology in superfluids and superconductors
• Antichiral edge states and Bogoliubov Fermi surfaces in a two-dimensional proximity-induced superconductor
• Manipulation of Majorana wave packets at surfaces of nodal noncentrosymmetric superconductors
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry
• Network model for magnetic higher-order topological phases
• Glide-resolved photoemission spectroscopy: Measuring topological invariants in nonsymmorphic space groups
• Lattice-induced wavefunction effects on trapped superfluids
• Symmetry Reduction and Boundary Modes for Fe-Chains on an s-wave Superconductor
• Combined spontaneous symmetry-breaking and symmetry-protected topological order from cluster charge interaction
• Persistent Haldane Phase in Carbon Tetris Chains
• Strictly one dimensional behavior emerging from dispersive two dimensional system: implications on metallic nanowires on semiconducting substrates
• Zero energy modes with Gaussian, exponential, or polynomial decay: Exact solutions in hermitian and nonhermitian regimes