Topology of crystalline insulators and superconductors

Abstract

We complete a classification of topological phases and their topological defects in crystalline insulators and superconductors. We consider topological phases and defects described by non-interacting Bloch and Bogoliubov de Gennes Hamiltonians that support additional order-two spatial symmetry, besides any of ten classes of symmetries defined by time-reversal symmetry and particle-hole symmetry. The additional order-two spatial symmetry we consider is general and it includes $Z_2$ global symmetry, mirror reflection, two-fold rotation, inversion, and their magnetic point group symmetries. We find that the topological periodic table shows a novel periodicity in the number of flipped coordinates under the order-two spatial symmetry, in addition to the Bott-periodicity in the space dimensions. Various symmetry protected topological phases and gapless modes will be identified and discussed in a unified framework. We also present topological classification of symmetry protected Fermi points. The bulk classification and the surface Fermi point classification provide a novel realization of the bulk-boundary correspondence in terms of the K-theory.

58 pages, final version

References in corpus (34)

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• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Topological Crystalline Insulators
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
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• Antiferromagnetic topological insulators
• The space group classification of topological band insulators
• Topological Phases of Noncentrosymmetric Superconductors: Edge States, Majorana Fermions, and the Non-Abelian Statistics
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• On the $Z_2$ classification of Quantum Spin Hall Models
• Non-Abelian Topological Orders and Majorana Fermions in Spin-Singlet Superconductors
• Topological spin-current in non-centrosymmetric superconductors
• Topological properties of spin-triplet superconductors and the Fermi surface topology in the normal state
• Anomalous Andreev bound state in non-centrosymmetric superconductors
• Time-reversal invariant topological superconductivity in doped Weyl semimetals
• Surface density of states and topological edge states in non-centrosymmetric superconductors
• Majorana Fermions in Nodal Superconductors: Gapless Topological Phase
• Topological field theory and thermal responses of interacting topological superconductors
• Majorana bound states and non-local spin correlations in a quantum wire on an unconventional superconductor
• Nodal Structure of Superconductors with Time-Reversal Invariance and Z2 Topological Number
• Axion topological field theory of topological superconductors
• Symmetry Protected Topological Order and Spin Susceptibility in Superfluid 3He-B
• Surface spectral function in the superconducting state of a topological insulator
• From the adiabatic theorem of quantum mechanics to topological states of matter
• Topological protection, disorder, and interactions: Survival at the surface of 3D topological superconductors
• Quantum impurity spin in Majorana edge fermions
• Electromagnetic and thermal responses of Z topological insulators and superconductors in odd spatial dimensions
• Index theorem for topological heterostructure systems

Cited by in corpus (188)

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• Reflection symmetric second-order topological insulators and superconductors
• Novel Topological Phase with Zero Berry Curvature
• Periodic Table for Topological Bands with Non-Hermitian Bernard-LeClair Symmetries
• Non-Hermitian topological phenomena: A review
• Classification of reflection symmetry protected topological semimetals and nodal superconductors
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• New classes of topological crystalline insulators with unpinned surface Dirac cones
• Topological surface states in nodal superconductors
• Z2-topology in nonsymmorphic crystalline insulators: Mobius twist in surface states
• Geometric approach to fragile topology beyond symmetry indicators
• Graph Theory Data for Topological Quantum Chemistry
• Topological correspondence between magnetic space group representations
• Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates
• Superconductivity and Local Inversion-Symmetry Breaking
• Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem
• Photonic ${\mathbb{Z}}_2$ topological Anderson insulators
• Topological crystalline superconductivity in locally noncentrosymmetric CeRh${}_2$As${}_2$
• Topological classification and diagnosis in magnetically ordered electronic materials
• Topologically stable gapless phases in nonsymmorphic superconductors
• Non-Hermitian band topology with generalized inversion symmetry
• Two-dimensional Shiba lattices as possible platform for crystalline topological superconductivity
• Multi-gap topology and non-Abelian braiding of phonons from first principles
• Topological Phases Protected By Reflection Symmetry and Cross-cap States
• Exact Higher-order Bulk-boundary Correspondence of Corner-localized States
• Excess charges as a probe of one-dimensional topological crystalline insulating phases
• CPT theorem and classification of topological insulators and superconductors
• Symmetry Protected Topological Superfluid ${}^3$He-B
• Möbius topological superconductivity in UPt${}_3$
• Topological space-time crystal
• Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond
• Observation of topological Euler insulators with a trapped-ion quantum simulator
• Superconducting doped topological materials
• Higher-order topological phases in crystalline and non-crystalline systems: a review
• Crystalline Kitaev spin liquids
• Valley-dependent Corner States in Honeycomb Photonic Crystal without Inversion Symmetry
• Optical manifestations and bounds of topological Euler class
• Disorder-Induced Quantum Phase Transitions in Three-Dimensional Second-Order Topological Insulators
• Topological properties of subsystem-symmetry-protected edge states in an extended quasi-one-dimensional dimerized lattice
• Symmetry indicator in non-Hermitian systems
• Faithful derivation of symmetry indicators: A case study for topological superconductors with time-reversal and inversion symmetries
• Amorphous topological phases protected by continuous rotation symmetry
• $N$-band Hopf insulator
• General Theory of Momentum-Space Nonsymmorphic Symmetry
• Topological Superconductivity Induced by Magnetic Texture Crystals
• Topological continuum charges of acoustic phonons in 2D
• Bott periodicity for the topological classification of gapped states of matter with reflection symmetry
• Anisotropic magnetic responses of topological crystalline superconductors
• Noncrystalline topological superconductors
• Microscopic Theory of Surface Topological Order for Topological Crystalline Superconductors
• Topology and symmetry of surface Majorana arcs in cyclic superconductors
• Topology invisible to eigenvalues in obstructed atomic insulators
• Phase transitions of non-Abelian charged nodal links in a spring-mass system
• Magnetic order on a topological insulator surface with warping and proximity-induced superconductivity
• Landau Levels of the Euler Class Topology
• Generation of higher-order topological insulators using periodic driving
• Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D
• Mirror-symmetry-protected dynamical quantum phase transitions in topological crystalline insulators
• Projected Topological Branes
• Dirac surface states in magnonic analogs of topological crystalline insulators
• Theory of polarization textures in crystal supercells
• Possible Realization of Topological Crystalline Superconductivity with Time-Reversal Symmetry in UTe2
• Topological insulators in the NaCaBi family with large SOC gaps
• Dynamical bulk boundary correspondence and dynamical quantum phase transitions in higher order topological insulators
• Quantum anomalous Hall insulator in ionic Rashba lattice of correlated electrons
• Classification of crystalline topological semimetals with an application to Na${}_3$Bi
• Topological properties of a generalized spin-orbit-coupled Su-Schrieffer-Heeger model
• Boundary edge networks induced by bulk topology
• Symmetry protected vortex bound state in superfluid ${}^3$He B-phase
• Odd-frequency pairing and Ising spin susceptibility in time-reversal invariant superfluids and superconductors
• A gap-protected zero-Hall effect state in the quantum limit of the nonsymmorphic metal KHgSb
• Three-dimensional $\mathcal{P}\mathcal{T}$-symmetric topological phases with Pontryagin index
• Topological Crystalline Magnets: Symmetry-Protected Topological Phases of Fermions
• Catalog of noninteracting tight-binding models with two energy bands in one dimension
• Manipulation of Majorana Kramers Qubit and its Tolerance in Time-Reversal-Invariant Topological Superconductor
• Majorana vortex-bound states in three-dimensional nodal noncentrosymmetric superconductors
• Topological Mott insulator in the odd-integer filled Anderson lattice model with Hatsugai-Kohmoto interactions
• Correlation-induced Fermi surface evolution and topological crystalline superconductivity in CeRh2As2
• Embedded Topological Semimetals
• Optical $N$-insulators: topological obstructions in the atomistic susceptibility tensor
• Majorana multipole response: General theory and application to wallpaper groups
• Higher-order topological semimetals and nodal superconductors with an order-two crystalline symmetry
• Effective field theories of topological crystalline insulators and topological crystals
• Prediction of Topological Crystalline Insulator and Topological Phase Transitions in Two-dimensional PbTe Films
• Non-Hermitian Origin of Detachable Boundary States in Topological Insulators
• Topological classification of intrinsic 3D superconductors using anomalous surface construction
• Non-Abelian Half-Quantum Vortices in ${}^3{P}_2$ Topological Superfluids
• Topological d-wave superconductivity in two dimensions
• Generalized Fermion Doubling Theorems: Classification of 2D Nodal Systems in Terms of Wallpaper Groups
• Topologically distinct atomic insulators
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Topological invariants beyond symmetry indicators: Boundary diagnostics for twofold rotationally symmetric superconductors
• Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors
• Exceptional points and non-Hermitian skin effects under nonlinearity of eigenvalues
• Unitary Symmetry-Protected Non-Abelian Statistics of Majorana Modes
• Anomalous geometric transport signatures of topological Euler class
• Self-localized topological states in three dimensions
• Exact projected entangled pair ground states with topological Euler invariant
• Larkin-Ovchinnikov state of superconducting Weyl metals: Fundamental differences between pairings restricted and extended in the $\text{k}$-space
• Higher-order topological superconductivity in a topological metal 1T${}^{\prime }$-MoTe${}_2$
• Protected Gapless Edge States In Trivial Topology
• Disorder-induced topology in quench dynamics
• Revealing the spatial nature of sublattice symmetry
• Topology of chalcogen chains
• Higher-order topological corner states induced solely by onsite potentials with mirror symmetry
• Electric Multipoles of Double Majorana Kramers Pairs
• Classification of time-reversal symmetric topological superconducting phases for conventional pairing symmetries
• Anomalous Crystal Shapes of Topological Crystalline Insulators
• Anisotropic paramagnetic response of topological Majorana surface states in the superconductor ${\text{UTe}}_2$
• Symmetry classes of dissipative topological insulators with edge dark state
• Surface-dependent Majorana vortex phases in topological crystalline insulators
• Distinct Floquet topological classifications from color-decorated frequency lattices with space-time symmetries
• Nontrivial gapless electronic states at the stacking faults of weak topological insulators
• Optical signatures of Euler superconductors
• Crystal-symmetry-protected gapless vortex-line phases in superconducting Dirac semimetals
• Interacting topological phases in thin films of topological mirror Kondo insulators
• Unconventional hybrid-order topological insulators
• Dual topological insulator with mirror symmetry protected helical edge states
• Dynamical Majorana Ising spin response in a topological superconductor-magnet hybrid by microwave irradiation
• Higher-order topological superconductor phases in a multilayer system
• Effects of first- and second-order topological phases on equilibrium crystal shapes
• Second Euler number in four dimensional synthetic matter
• Second-order topology and supersymmetry in two-dimensional topological insulators
• Additive rule of real and reciprocal space topologies at disclinations
• Quantized thermal and spin transports of dirty planar topological superconductors
• Stability against contact interactions of a topological superconductor in two-dimensional space protected by time-reversal and reflection symmetries
• Model non-Hermitian topological operators without skin effect: A general principle of construction
• Non-Hermitian ${\mathbb{Z}}_4$ skin effect protected by glide symmetry
• Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: $E_2$ pages for 1651 magnetic space groups
• General corner charge formulas in various tetrahedral and cubic space groups
• Bulk-entanglement spectrum correspondence in $PT$- and $PC$-symmetric topological insulators and superconductors
• Observation of trapped fractional charge and topological states at disclination defects in higher-order topological insulators
• Renormalization approach to the analysis and design of Hermitian and non-Hermitian interfaces
• Odd-frequency pairs and anomalous proximity effect in nematic and chiral states of superconducting topological insulators
• Flow of unitary matrices: Real-space winding numbers in one and three dimensions
• Majorana multipole response with magnetic point group symmetry
• Topological insulators on fractal lattices: A general principle of construction
• From orthosymplectic structure to super topological matter
• Quantized electrical, thermal, and spin transports of non-Hermitian clean and dirty two-dimensional topological insulators and superconductors
• Topological nonsymmorphic insulator versus Dirac semimetal in KZnBi
• Topological Micro-Electro-Mechanical Systems
• Theory of fractional corner charges in cylindrical crystal shapes
• Mirror-protected Majorana zero modes in $f$-wave multilayer graphene superconductors
• One-dimensional ${\mathbb{Z}}_4$ topological superconductor
• Two-dimensional flat band on the (011) surface of UTe${}_2$: Implication for STM measurements with a superconducting tip
• Higher-order topological phases for time-reversal-symmetry breaking superconductivity in UTe${}_2$
• Homotopy invariant in time-reversal and twofold rotation symmetric systems
• Observation of dislocation-induced topological modes in a three-dimensional acoustic topological insulator
• Controlling topology through targeted composite symmetry manipulation in magnetic systems
• Dynamic melting and condensation of topological dislocation modes
• Hybrid symmetry class topological insulators
• Takagi Topological Insulator on the Honeycomb Lattice
• Projective Spacetime Symmetry of Spacetime Crystals
• Artificial oxide heterostructures with non-trivial topology
• Topological Spectral Bands with Frieze Groups
• Tunable zero-energy Dirac and Luttinger nodes in a two-dimensional topological superconductor
• Generic reduction theory for Fermi sea topology in metallic systems
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Interaction-driven phase transition in one dimensional mirror-symmetry protected topological insulator
• Modulating Hamiltonian Approach to Quantum Many-Body Systems and Crystalline Topological Phases Protected by Generalized Magnetic Translations
• Anomalous multi-gap topological phases in periodically driven quantum rotors
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• Magnon-mediated superconductivity on ferromagnetic wallpaper fermions
• Stability of the X-Y-phase of the two-dimensional C4 point group insulator
• Artificial topological insulator realized in a two-terminal Josephson junction with Rashba spin-orbit interaction
• Phase diagrams of superconducting topological surface states
• Engineering Helical Superconductors with Multiple Majorana Kramers Pairs via Higher-Order Rashba Spin-Orbit Coupling
• Fermi-surface diagnosis for topological superconductivity with $s$-wave-like pairing symmetries
• Nuclear spin relaxation rate of nonunitary Dirac and Weyl superconductors
• Local expression of fractional corner charges in obstructed atomic insulators and relationship with the fractional disclination charges
• Symmetry-protected gates of Majorana qubits in a high-$T_c$ superconductor platform
• Isotropic 3D topological phases with broken time reversal symmetry
• Fragile topological insulators protected by rotation symmetry without spin-orbit coupling
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry
• Intrinsic non-Hermitian topological phases
• Nodal Line Topological Superfluid and Multiply Protected Majorana Fermi Arc in a Three-Dimensional Time-Reversal-Invariant Superfluid Model
• Possible topological phases in quantum anomalous Hall insulator/unconventional superconductor hybrid systems
• Gapped spinful phases obtained via Gutzwiller projections of Euler states
• Projected branes as platforms for crystalline, superconducting, and higher-order topological phases
• Symmetry-protected topological wire in a topological vacuum
• One-dimensional topology and topolectrics of nonsymmorphic Kramers degenerate systems
• Orbital magnetization reveals multiband topology
• Dynamical Symmetry Indicators for Floquet Crystals
• Bulk-boundary correspondence in topological two-dimensional non-Hermitian systems: Toeplitz operators and singular values
• Theory of Hofstadter Superconductors
• Three-dimensional spinless Euler insulators with rotational symmetry
• Classification of topological insulators and superconductors with multiple order-two point group symmetries
• Proximity-Induced Superconductivity in a Topological Crystalline Insulator