Classification of topological quantum matter with symmetries

2015/05/31 by Ching-Kai Chiu, Ching‐Kai Chiu, Jeffrey C. Y. Teo +3 · 1 voice · 2,927 citations
Physics and Astronomy · #Advanced Condensed Matter Physics #Homogeneous space #Physics #Quantum #Quantum many-body systems #Quantum mechanics #Theoretical physics #Topological Materials and Phenomena #Topology (electrical circuits) #cond-mat.mes-hall #cond-mat.str-el #cond-mat.supr-con

paper · pdf · doi:10.1103/revmodphys.88.035005

published in Reviews of Modern Physics 88(3) (American Physical Society) · 10 tables and 14 figures. Accepted for publication in Rev. Mod. Phys

arxiv created 2016/04/14 · openalex publication_date 2016/08/31 · arxiv updated 2016/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

In recent years an increasing amount of attention has been devoted to quantum materials with topological characteristics that are robust against disorder and other perturbations. In this context it was discovered that topological materials can be classified with respect to their dimension and symmetry properties. This review provides an overview of the classification schemes of both fully gapped and gapless topological materials and gives a pedagogical introduction into the field of topological band theory.

Citations

• Topological Insulator Materials
• Berry's Phase and Absence of Back Scattering in Carbon Nanotubes
• Solitons in conducting polymers
• Topological Stability of Majorana Zero Modes in Superconductor–Topological Insulator Systems
• Anyons in an exactly solved model and beyond
• Non-Abelian anyons and topological quantum computation
• Light-induced gauge fields for ultracold atoms

Cited by

• Topological phases of non-Hermitian SSH model with spin-orbit coupling
• Shiba duality and η-altermagnetism: Pairing and charge orders in bipartite attractive Hubbard models
• Counting Edge Modes with the Higher Berry Curvature: A Bulk Topological Order Parameter for Quantum Spin Chains
• Fermionic Genuine Multiparty Entanglement
• First-Order Topological FFLO Transition and Superconducting Diode Sign Reversal in Altermagnetic Nanowires
• Bulk spectra and the non-Hermitian skin effect in systems with long-range couplings
• Ultracold atomic lattice systems for simulating topological phases: A review
• Time-reversal symmetric topological superconductivity in Machida-Shibata lattices
• Topological constraints on the electronic band structure of hexagonal lattice in a magnetic field
• Higher-order exceptional ring semimetal with real hinge states in phononic crystals
• Electronic states at twist stacking faults in rhombohedral graphite
• Higher-Dimensional Information Lattice: Quantum State Characterization through Inclusion-Exclusion Local Information
• Quantum circuit algorithm for topological invariants of second order topological many-body quantum magnets
• Superconductivity in multi-Weyl semimetals: Conditions for the coexistence of topological and conventional phases
• Simulation of topological superconductors and their competing orders using photon-mediated interactions
• Field-free Josephson diode and tunable φ₀-junction in chiral kagome antiferromagnets
• Dynamical Scarring from Scrambling in Two Dimensional Topological Materials
• Current switching behaviour mediated via hinge modes in higher order topological phase using altermagnets
• Topological marker in three dimensions based on kernel polynomial method
• Many-body symmetry-protected zero boundary modes of synthetic photo-magnonic crystals
• Quantum cluster kink and ring frustration
• Topological superconductivity and superconducting diode effect mediated via unconventional magnet and Ising spin-orbit coupling
• Size optimization for observeing Majorana fermions
• Nonlinear Hall responses in tunable nodal Dirac semimetals
• Observation of topological phases without crystalline counterparts
• Real-space formulation of the Chern invariant and topological phases in a disordered Chern insulator
• Inhomogeneous SSH models and the doubling of orthogonal polynomials
• Topological Phases in Non-Hermitian Nonlinear-Eigenvalue Systems
• Hidden wallpaper fermion and third-order topological insulator
• Parametric resonance in a spin-1/2 chain: dynamical effects of nontrivial topology
• Ordinary lattice defects as probes of topology
• Superdiffusive transport protected by topology and symmetry in all dimensions
• Suppression of magnetism in Co₃Sn₂S₂ under external pressure
• Average Winding Number for Determinantal Curves associated with 2-Matrix Models in the Class AIII
• Measuring non-Abelian quantum geometry and topology in a multi-gap photonic lattice
• Proximity-induced superconductivity and emerging topological phases in altermagnet-based heterostructures
• Invariants for (2+1)D bosonic crystalline topological insulators for all 17 wallpaper groups
• Direct observation of the surface superconducting gap in the topological superconductor candidate β-PdBi2
• Topological Charges of Periodically Kicked Molecules
• Topological Magnons: A Review
• Complex nonlinear sigma model
• Phase diagram of amorphous quantum spin Hall insulators
• Impurity-induced topological decomposition
• Interband-Pairing-Boosted Supercurrent Diode Effect in Multiband Superconductors
• Discrete Shift and Polarization from Response to Symmetry Defects in Interacting Topological Phases
• Altermagnetism induced surface Chern insulator
• Spectral statistics and energy-gap scaling in k-local spin Hamiltonians
• Helical orbitals in electrical uni-directional molecular motors
• Coupled electric dipole model for a Su-Schrieffer-Heeger chain of optically resonant coreshell nanoparticles
• One-dimensional topological superconductors with nonsymmorphic symmetries
• Circular dichroism in high-order harmonic generation: Heralding topological phases and transitions in Chern insulators
• Hopf symmetry-protected topological phase at the intersection of magnetic and spin loop-current order
• Breakdown of Non-Bloch Bulk-Boundary Correspondence and Emergent Topology in Floquet Non-Hermitian Systems
• Theory of non-resonant Raman scattering from electrons in nodal and flat bands
• Spin Group Symmetry Criteria for Odd-parity Magnets
• Magnetic phase transitions protected by topological quantum geometry transitions: effects of electron-electron interactions in the Creutz ladder system
• Why is topology hard to learn?
• Non-Abelian interference of topological edge states
• Tunable quantum metric and band topology in bilayer Dirac models
• Quantum phases driven by strong correlations
• Selective bulk-boundary correspondence in higher-order topological insulators with anticommuting mirror and chiral symmetries
• Critical dynamics and superconducting state preparation in the quenched Kitaev chain with pairing imbalance
• Theory of quantum spin-Hall topological lasers
• Nontrivial topology in one- and two-dimensional asymmetric systems with chiral boundary states
• Breakdown of symmetry constraint in Floquet topological superconductor
• Universality of Energy-Space Entanglement in Quantum Impurity Models
• Non-Hermitian Skin Effect in a Non-Hermitian Electrical Circuit
• Angular momentum of vortex-core Majorana zero modes
• Intrinsic-perturbation induced anomalous higher-order boundary states in non-Hermitian systems
• Non-Fermi-liquid transport phenomena in bilayer nickelates: Impact of quasi-quantum metric
• Spin-Orbit Driven Topological Phases in Kagome Materials
• -Continuum limit of bipartite lattices -- The SSH model
• Continuum Landau surface states in a non-Hermitian Weyl semimetal
• Universal Boundary-Modes Localization from Quantum Metric Length
• Nonlinear Optical Spectroscopy of Nodal-Line Semimetals
• Signatures of emergent surface states across a displacive topological phase transition in Bi₄I₄
• Dissipation-Driven Topological Phase Transition in Quantum Open Systems Independent of system Hamiltonian
• A study of topological characterization and symmetries for a quantum simulated Kitaev chain
• Entanglement entropy as a probe of topological phase transitions
• Structures in higher-order quantum correlations due to non-spatial symmetries
• Subspace-Protected Topological Phases and Bulk-Boundary Correspondence
• Tunable multi-magnon Floquet topological edge states
• Linear Scaling Quantum Transport Methodologies
• Quasi-One-Dimensional Higher-Order Topological Insulators
• Interferences Measure Topology
• Dirac monopole magnets in non-Hermitian systems
• Emergence of non-trivial phases in interacting non-Hermitian quasiperiodic chains with power-law hopping
• Type-II Weyl Excitation in Vortex Arrays
• Dissipation-Induced Steady States in Topological Superconductors: Mechanisms and Design Principles
• A non-Hermitian Su-Schrieffer-Heeger model with the energy levels of free parafermions
• A Dynamical Bulk-Boundary Correspondence in Two Dimensional Topological Matter
• Extending fusion rules with finite subgroups: For a general understanding of quotient or gauging
• Fermion quantum criticality far from equilibrium
• Simulating Floquet non-Abelian topological insulator with photonic quantum walks
• Topological bound states in a lattice of rings with nearest-neighbour interactions
• Engineering Topological Materials
• Absence of dissipation-free topological edge states in quadratic open fermions
• Unconventional Floquet topological phases in the SSH lattice
• Topological band insulators without translational symmetry
• Observation of Embedded Topology in a Trivial Bulk via Projective Crystal Symmetry
• Quantum Spin Hall Effect with Extended Topologically Protected Features in Altermangetic Multilayers
• Intermediate-temperature topological Uhlmann phase on IBM quantum computers
• Explicit equivalence between the spectral localizer and local Chern and winding markers
• Projected branes as platforms for crystalline, superconducting, and higher-order topological phases
• Two-dimensional Disordered Projected Branes: Stability and Quantum Criticality via Dimensional Reduction
• Model Hamiltonian for Altermagnetic Topological Insulators
• Unconventional hybrid-order topological insulators
• Protected quantum gates using qubit doublons in dynamical optical lattices
• Identification and Properties of Topological States in the Bulk of Quasicrystals
• Theory of off-diagonal disorder in multilayer topological insulator
• Braiding Statistics of Vortices in 2+1d Topological Superconductors from Stacking
• Defect Engineering the Interacting Many-body SSH Model
• Quasicrystalline Altermagnetism
• Floquet composite Dirac semimetals
• Emergent topology by Landau level mixing in quantum Hall-superconductor nanostructures
• Topological phases and Edge states in an exactly solvable Gamma matrix model
• Large Chern numbers in a dissipative dice model
• Robustness of the flux-free sector of the Kitaev honeycomb against environment
• TopoMAS: Large Language Model Driven Topological Materials Multiagent System
• Ginzburg-Landau theory for unconventional surface superconductivity in PtBi₂
• Topological Braiding of Bloch Eigenmodes Protected by Non-Abelian Quaternion Invariants
• Defects Potentials for Two-Dimensional Topological Materials
• Quantum Hall Effect and Chern Phases in the 1/5-Depleted Square Lattice
• Identifying Anyonic Topological Order in Fractional Quantum Anomalous Hall Systems
• Metal-insulator phase transition and topology in a three-component system
• Observation of trapped fractional charge and topological states at disclination defects in higher-order topological insulators
• Spin-orbit interaction, band topology, and spin texture in BiInO3(001) surface
• Nonsymmorphic symmetry-enforced hourglass fermions and Rashba-Dresselhaus interaction in BiInO₃
• Topological Phase Transition under Infinite Randomness
• Topological crystals and soliton lattices in a Gross-Neveu model with Hilbert-space fragmentation
• Topological Phase Transitions and Edge-State Transfer in Time-Multiplexed Quantum Walks
• Two dimensional sub-wavelength topological dark state lattices
• Electric-field control of zero-dimensional topological states in ultranarrow germanene nanoribbons
• Higher-Order Topology in Monolayer FeSe
• Quantum metric and localization in a quasicrystal
• Emergent topology in thin films of nodal line semimetals
• Hierarchical Topological States without Dimension Reduction
• Dynamically stable topological edge states in an extended Su-Schrieffer-Heeger ladder with balanced perturbation
• Delicate Wannier insulators
• Quantum spin liquids in a square lattice subject to an Abelian flux
• Intra-unit-cell singlet pairing mediated by altermagnetic fluctuations
• Observation of average topological phase in disordered Rydberg atom array
• Multiple-order differential imaging based on two types of topological singularity in one dimensional photonic crystals
• Partons from stabilizer codes
• Robustness of topological edge states in alternating spin chains against environment
• Emergence of Diverse Topological States in Ge Doped MnBi2Te4
• Effective potential of scalar Lee-Wick pseudo-electrodynamics
• From Chern to Winding: Topological Invariant Correspondence in the Reduced Haldane Model
• A many-body marker for three-dimensional topological insulators with inversion symmetry
• Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology
• Majorana vortex phases in time-reversal invariant higher-order topological insulators and topologically trivial insulators
• Promoted current-induced spin polarization in inversion symmetry broken topological insulator thin films
• Composite quantum geometry of superconductors
• Uhlmann and scalar Wilczek-Zee phases of degenerate quantum systems
• Topological protection of Majorana polaritons in a cavity
• Bulk-Boundary Correspondence in Semi-Infinite Chains from Sublattice Zeros
• Experimental realization of Majorana hinge and corner modes in intrinsic organic topological superconductor without magnetic field at room temperature
• Hopf Bifurcation of Nonlinear Non-Hermitian Skin Effect
• Dimensional crossover of class D real-space topological invariants
• High Chern Number Quantum Anomalous Hall States in Haldane-Graphene Multilayers
• Quantum Hall Effect without Chern Bands
• Euler topology in classical spin liquids
• Superconductivity in Spin-Orbit coupled SU(8) Dirac Fermions on Honeycomb lattice
• Josephson current signature of Floquet Majorana and topological accidental zero modes in altermagnet heterostructures
• Physical Pictures for Quasisymmetry in Crystals
• Ferroelectrically Switchable Chirality in Topological Superconductivity
• Topological Time Crystals
• Topological characterization of chiral models through their long time dynamics
• Edge Solitons in a Nonlinear Mechanical Topological Insulator
• Nodal-line semimetals from Weyl superlattices
• Engineering symmetry-protected topological states in waveguide arrays
• Generic phase diagram for Weyl superconductivity in mirror-symmetric superconductors
• Spin-mixing-tunneling network model for Anderson transitions in two-dimensional disordered spinful electrons
• Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators
• Reflection-Symmetric Second-Order Topological Insulators and Superconductors
• Topological classification and edge states of magnons in honeycomb ferromagnets
• Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions
• Nonsymmorphic Topological Phases of Non-Hermitian Systems
• Emergence of Hermitian topology from non-Hermitian knots
• Shot noise in strongly correlated double quantum spin Hall edges
• Engineering Topological Phases Guided by Statistical and Machine Learning Methods
• Manipulating Anomalous Transport via Crystal Symmetry in 2D Altermagnets
• Time-reversal-invariant spin-orbit-coupled bilayer Bose-Einstein condensates
• Emergent fractals in dirty topological crystals
• Limits of thermal conductance quantization in chiral topological Josephson junctions
• Regularized universal topological markers for Dirac systems
• Quantum Floquet anomalous Hall states and quantized ratchet effect in one-dimensional dimer chain driven by two ac electric fields
• Intrinsic topological superconductivity with exactly flat surface bands in the quasi-one-dimensional A2Cr3As3 (A=Na, K, Rb, Cs) superconductors
• Interplay between short-range correlated disorder and Coulomb interaction in nodal-line semimetals
• Generalized Space Groups from Internal Configuration Spaces
• Controlling the direction of topological transport in a non-Hermitian time-reversal symmetric Floquet ladder
• Exponentially growing bulk Green functions as signature of nontrivial non-Hermitian winding number in one dimension
• Symmetry-enforced band nodes in 230 space groups
• Supersymmetric Free Fermions and Bosons: Locality, Symmetry and Topology
• Magnetic higher-order nodal lines
• Emergence of the Unconventional Type-II Nambu-Goldstone Modes with Topological Origin in Bose Superfluids
• Quantum Dynamical Characterization and Simulation of Topological Phases With High-Order Band Inversion Surfaces
• Piezoelectricity and topological quantum phase transitions in two-dimensional spin-orbit coupled crystals with time-reversal symmetry
• Exceptional surfaces in PT-symmetric non-Hermitian photonic systems
• Z2 topological quantum paramagnet on a honeycomb bilayer
• Dynamical fragile topology in Floquet crystals
• Higher-order band topology
• Scaling behavior in a multicritical one-dimensional topological insulator
• Controllable simulation of topological phases and edge states with quantum walk
• Unconventional Majorana fermions on the surface of topological superconductors protected by rotational symmetry
• Second-order topological modes in two-dimensional continuous media
• Complete random matrix classification of SYK models with N N = 0, 1 and 2 supersymmetry
• Skin effect and winding number in disordered non-Hermitian systems
• Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional Systems with Space-Time Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle
• Topological Anderson insulators induced by random binary disorders
• Weyl-loop half-metal in Li3(FeO3)2
• Mirror protected multiple nodal line semimetals and material realization
• Anatomy of Z2 fluxes in anyon Fermi liquids and Bose condensates
• Observation of Dislocation Non-Hermitian Skin Effect
• Non-Abelian Charged Nodal Links in a Dielectric Photonic Crystal
• Quantum Circuits Assisted by Local Operations and Classical Communication: Transformations and Phases of Matter
• Quantum Simulation for Three-Dimensional Chiral Topological Insulator
• Random-walk topological transition revealed via electron counting
• Chiral spin condensation in a one-dimensional optical lattice
• Unified bulk-boundary correspondence for band insulators
• Stability of Time-Reversal Symmetry Protected Topological Phases
• Extrinsic topology of Floquet anomalous boundary states in quantum walks
• Disorder-driven phase transition in the second-order non-Hermitian skin effect
• Fermiology of ZrTe with triply degenerate nodes and highly anisotropic magnetization
• Topological states in disordered arrays of dielectric nanoparticles
• Crystalline Weyl semimetal phase in Quantum Spin Hall systems under magnetic fields
• Non-Hermitian bulk-boundary correspondence in a periodically driven system
• Emergent Kitaev materials in synthetic Fermi-Hubbard bilayers
• Observation of topology of non-Hermitian systems without chiral symmetry
• Topological properties of the long-range Kitaev chain with Aubry-André-Harper modulation
• Klein-Gordon Representation of Acoustic Waves and Topological Origin of Surface Acoustic Modes
• Classification of Weyl points and nodal lines based on magnetic point groups for spin-(1)/(2) quasiparticles
• Chern-Simons fermionization approach to two-dimensional quantum magnets: Implications for antiferromagnetic magnons and unconventional quantum phase transitions
• The Lee-Wick-Chern-Simons pseudo-quantum electrodynamics
• Interaction-enhanced integer quantum Hall effect in disordered systems
• Higher-order non-Hermitian skin effect
• Non-Hermitian Hopf insulators
• Converting PT-Symmetric Topological Classes by Floquet Engineering
• Finite-size effects in non-Hermitian topological systems
• Real and imaginary edge states in stacked Floquet honeycomb lattices
• Mechanism for Unconventional Superconductivity in the Hole-Doped Rashba-Hubbard Model
• Three-dimensional topological insulators and bosonization
• Composite Dirac semimetals
• Symmetry-protected metallic and topological phases in penta-materials
• Topological Devil’s staircase in atomic two-leg ladders
• Topological classifications of quadratic bosonic excitations in closed and open systems with examples
• Floquet Second-Order Topological Insulators from Nonsymmorphic Space-Time Symmetries
• Phase Space Crystal Vibrations: Chiral Edge States with Preserved Time-reversal Symmetry
• Bogoliubov Fermi surfaces: General theory, magnetic order, and topology
• Shedding light on topological superconductors
• Quantum dynamics on a lossy non-Hermitian lattice*
• Disorder-enhanced topological protection and universal quantum criticality in a spin-32topological superconductor
• Weyl Nodal Point–Line Fermion in Ferromagnetic Eu5Bi3
• Topological exceptional surfaces in non-Hermitian systems with parity-time and parity-particle-hole symmetries
• Topological Phase Transition Driven by Infinitesimal Instability: Majorana Fermions in Non-Hermitian Spintronics
• Beyond universal behavior in the one-dimensional chain with random nearest-neighbor hopping
• Deep learning topological invariants of band insulators
• Point-gap topology with complete bulk-boundary correspondence and anomalous amplification in the Fock space of dissipative quantum systems
• Fidelity susceptibility near topological phase transitions in quantum walks
• Majorana modes in emergent-wire phases of helical and cycloidal magnet-superconductor hybrids
• Bulk-edge correspondence with generalized chiral symmetry
• Emergence of PT-symmetry breaking in open quantum systems
• Flat-band full localization and symmetry-protected topological phase on bilayer lattice systems
• Invariance of Topological Indices Under Hilbert Space Truncation
• Type-II Dirac line node in strained Na3N
• Choreographed entanglement dances: Topological states of quantum matter
• Theory of electron spin resonance in one-dimensional topological insulators with spin-orbit couplings: Detection of edge states
• Topological characterization of one-dimensional open fermionic systems
• Topological Entanglement-Spectrum Crossing in Quench Dynamics
• Topological critical materials of ternary compounds
• Exposing the quantum geometry of spin-orbit-coupled Fermi superfluids
• Anisotropic Magnetic Responses of Topological Crystalline Superconductors
• Quantized charge transport in chiral Majorana edge modes
• Quench-induced dynamical topology under dynamical noise
• Quantum phase transitions of a generalized compass chain with staggered Dzyaloshinskii–Moriya interaction
• Pseudo Dirac nodal sphere semimetal
• Emergence of a new symmetry class for Bogoliubov–de Gennes (BdG) Hamiltonians: expanding 10-fold symmetry classes
• Quantum Geometry of Finite XY Chains: A Comparison of Neveu-Schwarz and Ramond Sectors
• Novel phenomena in transition-metal oxide thin films and heterostructures with strong correlations and spin-orbit coupling
• Three-nodal surface phonons in solid-state materials: Theory and material realization
• Annular Majorana mode in a superconducting topological insulator
• Topological Charge-Transfer Excitons
• Topological edge solitons and their stability in a nonlinear Su-Schrieffer-Heeger model
• Temporal Berry Phase and the Emergence of Bose-Glass-Analog Phase in a Clean U(1) Superfluid
• Asymmetric real topology of conduction and valence bands
• Unsupervised learning of non-Abelian multi-gap topological phases
• Localization and Topology in Noncentrosymmetric Superconductors with Disorder
• Gross–Neveu–Wilson model and correlated symmetry-protected topological phases
• Topologically nontrivial phase in the hexagonal antiperovskites A3BiB (A=Ba,Sr; B=P,N)

Discussions

• In the Altland–Zirnbauer classification, single-particle quadratic fermion Hamiltonians are distinguished by certain algebraic conditions they satisfy. While these conditions are not symmetries in the [stackexchange, 1 points]

Related

• The space group classification of topological band insulators
• Topological Classification of Insulators: I. Non-interacting Spectrally-Gapped One-Dimensional Systems
• Symmetry Protected Topological phases of Quantum Matter
• Classification and Construction of Topological Phases of Quantum Matter
• Topological quantum materials: kagome, chiral, and square-net frameworks