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)

• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Topological Insulators with Inversion Symmetry
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• 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
• Non-Abelian Topological Order in S-Wave Superfluids of Ultracold Fermionic Atoms
• 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 (55)

• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• 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
• 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
• Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem
• Topologically stable gapless phases in nonsymmorphic superconductors
• Topological Phases Protected By Reflection Symmetry and Cross-cap States
• Symmetry Protected Topological Superfluid ${}^3$He-B
• CPT theorem and classification of topological insulators and superconductors
• Excess charges as a probe of one-dimensional topological crystalline insulating phases
• Möbius topological superconductivity in UPt${}_3$
• Topological space-time crystal
• Superconducting doped topological materials
• Crystalline Kitaev spin liquids
• Valley-dependent Corner States in Honeycomb Photonic Crystal without Inversion Symmetry
• Disorder-Induced Quantum Phase Transitions in Three-Dimensional Second-Order Topological Insulators
• Bott periodicity for the topological classification of gapped states of matter with reflection symmetry
• Anisotropic magnetic responses of topological crystalline superconductors
• Microscopic Theory of Surface Topological Order for Topological Crystalline Superconductors
• Topology and symmetry of surface Majorana arcs in cyclic superconductors
• Magnetic order on a topological insulator surface with warping and proximity-induced superconductivity
• Possible Realization of Topological Crystalline Superconductivity with Time-Reversal Symmetry in UTe2
• Topological insulators in the NaCaBi family with large SOC gaps
• Classification of crystalline topological semimetals with an application to Na${}_3$Bi
• Symmetry protected vortex bound state in superfluid ${}^3$He B-phase
• Topological properties of a generalized spin-orbit-coupled Su-Schrieffer-Heeger model
• Boundary edge networks induced by bulk topology
• A gap-protected zero-Hall effect state in the quantum limit of the nonsymmorphic metal KHgSb
• Odd-frequency pairing and Ising spin susceptibility in time-reversal invariant superfluids and superconductors
• Topological Crystalline Magnets: Symmetry-Protected Topological Phases of Fermions
• Catalog of noninteracting tight-binding models with two energy bands in one dimension
• Majorana vortex-bound states in three-dimensional nodal noncentrosymmetric superconductors
• Prediction of Topological Crystalline Insulator and Topological Phase Transitions in Two-dimensional PbTe Films
• Self-localized topological states in three dimensions
• Revealing the spatial nature of sublattice symmetry
• Larkin-Ovchinnikov state of superconducting Weyl metals: Fundamental differences between pairings restricted and extended in the $\text{k}$-space
• Symmetry classes of dissipative topological insulators with edge dark state
• Interacting topological phases in thin films of topological mirror Kondo insulators
• Non-Hermitian ${\mathbb{Z}}_4$ skin effect protected by glide symmetry
• Observation of trapped fractional charge and topological states at disclination defects in higher-order topological insulators
• From orthosymplectic structure to super topological matter
• Projective Spacetime Symmetry of Spacetime Crystals
• Artificial oxide heterostructures with non-trivial topology
• Phase diagrams of superconducting topological surface states
• Stability of the X-Y-phase of the two-dimensional C4 point group insulator
• Theory of Hofstadter Superconductors
• Possible topological phases in quantum anomalous Hall insulator/unconventional superconductor hybrid systems
• Proximity-Induced Superconductivity in a Topological Crystalline Insulator