Topological classification with additional symmetries from Clifford algebras

Abstract

We classify topological insulators and superconductors in the presence of additional symmetries such as reflection or mirror symmetries. For each member of the 10 Altland-Zirnbauer symmetry classes, we have a Clifford algebra defined by operators of the generic (time-reversal, particle-hole, or chiral) symmetries and additional symmetries, together with gamma matrices in Dirac Hamiltonians representing topological insulators and superconductors. Following Kitaev's approach, we classify gapped phases of non-interacting fermions under additional symmetries by examining all possible distinct Dirac mass terms which can be added to the set of generators of the Clifford algebra. We find that imposing additional symmetries in effect changes symmetry classes and causes shifts in the periodic table of topological insulators and superconductors. Our results are in agreement with the classification under reflection symmetry recently reported by Chiu et al. Several examples are discussed including a topological crystalline insulator with mirror Chern numbers and mirror superconductors.

18 pages

References in corpus (10)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Classification of topological insulators and superconductors in three spatial dimensions
• 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
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• The space group classification of topological band insulators

Cited by in corpus (217)

• Classification of topological quantum matter with symmetries
• Topological superconductors: a review
• Topological phases of non-Hermitian systems
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Symmetry and Topology in Non-Hermitian Physics
• Reflection symmetric second-order topological insulators and superconductors
• Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials
• Topological classification of crystalline insulators through band structure combinatorics
• Topology of crystalline insulators and superconductors
• Periodic Table for Topological Bands with Non-Hermitian Bernard-LeClair Symmetries
• Second-order topological insulators and superconductors with an order-two crystalline symmetry
• Topological semi-metals with line nodes and drumhead surface states
• Majorana Fermions and Topology in Superconductors
• Spin-filtered Edge States with an Electrically Tunable Gap in a Two-Dimensional Topological Crystalline Insulator
• Classification of reflection symmetry protected topological semimetals and nodal superconductors
• Higher-order bulk-boundary correspondence for topological crystalline phases
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
• New classes of topological crystalline insulators with unpinned surface Dirac cones
• Gauging spatial symmetries and the classification of topological crystalline phases
• Topological superconductivity in Dirac semimetals
• Z2-topology in nonsymmorphic crystalline insulators: Mobius twist in surface states
• Are the double Fermi arcs of Dirac semimetals topologically protected?
• Topology of nonsymmorphic crystalline insulators and superconductors
• Unified theory of PT and CP invariant topological metals and nodal superconductors
• Lattice models with exactly solvable topological hinge and corner states
• Geometric approach to fragile topology beyond symmetry indicators
• Topological Phase Transition driven by Infinitesimal Instability: Majorana Fermions in Non-Hermitian Spintronics
• Dynamical classification of topological quantum phases
• Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups -
• Symmetry Protected Topological Superfluids and Superconductors --- From the Basics to ${}^3$He ---
• Topological Insulators from Group Cohomology
• Nonsymmorphic symmetry-required band crossings in topological semimetals
• Topological classification of non-Hermitian systems with reflection symmetry
• Non-Hermitian second-order skin and topological modes
• Exceptional rings protected by emergent symmetry for mechanical systems
• Symmetry Protected Topological States of Interacting Fermions and Bosons
• Topological semimetals protected by off-centered symmetries in nonsymmorphic crystals
• Topological Blount's theorem of odd-parity superconductors
• Theory of interacting topological crystalline insulators
• Floquet topological phases protected by time glide symmetry
• Breakdown of the topological classification Z for gapped phases of noninteracting fermions by quartic interactions
• Topological crystalline superconductivity and second-order topological superconductivity in nodal-loop materials
• Many-body topological invariants for fermionic symmetry-protected topological phases
• Weyl and Dirac semimetals with Z_2 topological charge
• Topological classification of defects in non-Hermitian systems
• Controlled topological phases and bulk-edge correspondence
• Braiding Majorana corner modes in a second-order topological superconductor
• New class of topological superconductors protected by magnetic group symmetries
• Torsion, Parity-odd Response and Anomalies in Topological States
• Crystalline topological phases as defect networks
• Phase diagram of disordered higher-order topological insulator: A machine learning study
• Higher-Order topological insulator in a dodecagonal quasicrystal
• Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem
• Global anomalies on the surface of fermionic symmetry-protected topological phases in (3+1) dimensions
• Topological charges of three-dimensional Dirac semimetals with rotation symmetry
• Shift insulators: rotation-protected two-dimensional topological crystalline insulators
• Symmetry-based indicators for topological Bogoliubov-de Gennes Hamiltonians
• Topological zero modes and Dirac points protected by spatial symmetry and chiral symmetry
• Scattering matrix invariants of Floquet topological insulators
• Mirror winding number and helical edge modes in honeycomb lattice with hopping-energy texture
• Quantum simulation of exotic PT-invariant topological nodal loop bands with ultracold atoms in an optical lattice
• Patterns of Electro-magnetic Response in Topological Semi-metals
• Interacting Topological Insulator and Emergent Grand Unified Theory
• Topological Bloch oscillations
• ${\mathbb{Z}}_2$ Topological Invariant for Magnon Spin Hall Systems
• Higher-order topological band structures
• Symmetry-protected Topological Phases, Generalized Laughlin Argument and Orientifolds
• Classification of Crystalline Topological Insulators and Superconductors with Point Group Symmetries
• Topological crystalline superconductivity in locally noncentrosymmetric CeRh${}_2$As${}_2$
• Classification of topological crystalline insulators based on representation theory
• Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena
• Dimensional transmutation from non-Hermiticity
• Bulk invariants and topological response in insulators and superconductors with nonsymmorphic symmetries
• Topological Crystalline Superconductivity in Locally Non-centrosymmetric Multilayer Superconductors
• Topologically stable gapless phases in nonsymmorphic superconductors
• Non-Hermitian band topology with generalized inversion symmetry
• Topological classification under nonmagnetic and magnetic point group symmetry: Application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology
• Non-Hermiticity and topological invariants of magnon Bogoliubov-de Gennes systems
• Finite-size-effect-induced topological phase transition in a topological crystalline insulator
• Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups
• Symmetry indicators for topological superconductors
• Exceptional band touching for strongly correlated systems in equilibrium
• Anderson localization and the topology of classifying spaces
• Type-II Dirac surface states in topological crystalline insulators
• Topological Phases Protected By Reflection Symmetry and Cross-cap States
• Topological Crystalline Materials of $J=3/2$ Electrons: Antiperovskites, Dirac points, and High Winding Topological Superconductivity
• Symmetry Protected Topological Superfluid ${}^3$He-B
• CPT theorem and classification of topological insulators and superconductors
• First and Second Order Topological Phases on Ferromagnetic Breathing Kagome Lattice
• Möbius topological superconductivity in UPt${}_3$
• Non-Abelian Bloch oscillations in higher-order topological insulators
• Symmetry and topological classification of Floquet non-Hermitian systems
• $Z_4$ Topological Superconductivity in UCoGe
• Coupled-layer description of topological crystalline insulators
• Characterizing topological phases by quantum quenches: A general theory
• Crystalline Kitaev spin liquids
• $Z_N$ Berry Phases in Symmetry Protected Topological Phases
• Majorana flat bands, chiral Majorana edge states, and unidirectional Majorana edge states in noncentrosymmetric superconductors
• Tenfold Topology of Crystals: Unified classification of crystalline topological insulators and superconductors
• Axion coupling in the hybrid Wannier representation
• qeirreps: an open-source program for Quantum ESPRESSO to compute irreducible representations of Bloch wavefunctions
• Symmetry-protected line nodes and Majorana flat bands in nodal crystalline superconductors
• Paramagnetically induced gapful topological superconductors
• Surface electronic state of superconducting topological crystalline insulator
• Disorder-Induced Quantum Phase Transitions in Three-Dimensional Second-Order Topological Insulators
• Second-Order Bulk-Boundary Correspondence in Rotationally Symmetric Topological Superconductors from Stacked Dirac Hamiltonians
• Unification of bosonic and fermionic theories of spin liquids on the kagome lattice
• Symmetry indicator in non-Hermitian systems
• Interaction effects on the classification of crystalline topological insulators and superconductors
• The classification of surface states of topological insulators and superconductors with magnetic point group symmetry
• Topology and entanglement in quench dynamics
• Spin-directed network model for the surface states of weak three-dimensional ${\mathbb{Z}}_2^{}$ topological insulators
• Layered Topological Crystalline Insulators
• Amorphous topological phases protected by continuous rotation symmetry
• Stability of surface states of weak ${\mathbb{Z}}_2$ topological insulators and superconductors
• Extended topological group structure due to average reflection symmetry
• Quantum dynamical characterization and simulation of topological phases with high-order band inversion surfaces
• Topological Superconductivity Induced by Magnetic Texture Crystals
• Zeeman-field-induced nontrivial topological phases in a one-dimensional spin-orbit-coupled dimerized lattice
• Higher-order topological insulators in a magnetic field
• Bott periodicity for the topological classification of gapped states of matter with reflection symmetry
• Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery
• Correlation effects on topological crystalline insulators
• Anisotropic magnetic responses of topological crystalline superconductors
• Microscopic Theory of Surface Topological Order for Topological Crystalline Superconductors
• Interaction Enabled Topological Crystalline Phases
• Topology invisible to eigenvalues in obstructed atomic insulators
• Topological invariants to characterize universality of boundary charge in one-dimensionalinsulators beyond symmetry constraints
• One-dimensional topological metal
• No-go theorem for topological insulators and sure-fire recipe for Chern insulators
• Symmetries as the guiding principle for flattening bands of Dirac fermions
• Generation of higher-order topological insulators using periodic driving
• Multiple band-crossings and Fermi surface topology: role of double nonsymmorphic symmetries in MnP-type crystal structures
• Glide-symmetric magnetic topological crystalline insulators with inversion symmetry
• Symmetry-protected entangling boundary zero modes in crystalline topological insulators
• Exceptional points for chiral Majorana fermions in arbitrary dimensions
• Surface charge theorem and topological constraints for edge states: An analytical study of one-dimensional nearest-neighbor tight-binding models
• Unified characterization for higher-order topological phase transitions
• Mirror-symmetry-protected dynamical quantum phase transitions in topological crystalline insulators
• Space group constraints on weak indices in topological insulators
• Topology of the planar phase of superfluid ${}^3$He and bulk-boundary correspondence for three dimensional topological superconductors
• Universal Relation among Many-Body Chern Number, Rotation Symmetry, and Filling
• Possible Realization of Topological Crystalline Superconductivity with Time-Reversal Symmetry in UTe2
• Topological D+p-wave superconductivity in Rashba systems
• Classification of crystalline topological semimetals with an application to Na${}_3$Bi
• Characterizing the weak topological properties: Berry phase point of view
• Robustness of Helical Hinge States of Weak Second-Order Topological Insulators
• Symmetry protected vortex bound state in superfluid ${}^3$He B-phase
• Boundary edge networks induced by bulk topology
• New types of topological superconductors under local magnetic symmetries
• Quasiperiodicity and 2D Topology in 1D Charge Ordered Materials
• Catalog of noninteracting tight-binding models with two energy bands in one dimension
• Dislocation Majorana zero modes in perovskite oxide 2DEG
• Majorana multipole response: General theory and application to wallpaper groups
• Dynamically characterizing topological phases by high-order topological charges
• Anomalous Floquet topological crystalline insulators
• Unconventional Floquet topological phases from quantum engineering of band inversion surfaces
• Real Hopf Insulator
• Correlation-induced Fermi surface evolution and topological crystalline superconductivity in CeRh2As2
• Topological crystalline insulators from stacked graphene layers
• Topology in time-reversal symmetric crystals
• Glide-symmetric topological crystalline insulator phase in a nonprimitive lattice
• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
• Topological d-wave superconductivity in two dimensions
• Rational boundary charge in one-dimensional systems with interaction and disorder
• Magnetic Response of Majorana Kramers Pairs Protected by Z2 Invariants
• Magnetic response of Majorana Kramers pairs with an order-two symmetry
• Non-Hermitian Origin of Detachable Boundary States in Topological Insulators
• Engineering of Chern insulators and circuits of topological edge states
• Phase space methods for Majorana fermions
• Symmetry-protected topological invariant and Majorana impurity states in time-reversal-invariant superconductors
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Generic phase diagram for Weyl superconductivity in mirror-symmetric superconductors
• Approximate symmetries and conservation laws in topological insulators and associated $\mathbb{Z}$-invariants
• Disorder-induced topology in quench dynamics
• Revealing the spatial nature of sublattice symmetry
• Twisted crystallograpic T-duality via the Baum--Connes isomorphism
• Electric Multipoles of Double Majorana Kramers Pairs
• Electromagnetic Response of Three-dimensional Topological Crystalline Insulators
• Extrinsic topology of Floquet anomalous boundary states in quantum walks
• BerryEasy: A GPU enabled python package for diagnosis of nth-order and spin-resolved topology in the presence of fields and effects
• Fractional topological insulator precursors in spin-orbit fermion ladders
• Rigidity of topological invariants to symmetry breakings
• Unveiling Higher-Order Topology via Polarized Topological Charges
• Spin-Orbit Coupling and Topological States in $F=\frac{3}{2}$ Cold Fermi Gas
• Nodal lines and mapping to mirror Chern numbers in Ca${}_2$As family
• Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras
• Symmetry-enriched Bose-Einstein condensates in spin-orbit coupled bilayer system
• Entanglement Chern Number of the Kane-Mele Model with Ferromagnetism
• Unconventional hybrid-order topological insulators
• Manipulating quantum channels in weak topological insulator nanoarchitectures
• Second Euler number in four dimensional synthetic matter
• Robust semi-Dirac points and unconventional topological phase transitions in doped superconducting Sr2IrO4 tunnel coupled to t2g electron systems
• Interacting Chern Insulator in Infinite Spatial Dimensions
• Stability against contact interactions of a topological superconductor in two-dimensional space protected by time-reversal and reflection symmetries
• Discriminant indicator with generalized rotational symmetry
• Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: $E_2$ pages for 1651 magnetic space groups
• Layer Construction of Three-Dimensional Z2 Monopole Charge Nodal Line Semimetals and prediction of the abundant candidate materials
• Majorana multipole response with magnetic point group symmetry
• Non-Hermitian ${\mathbb{Z}}_4$ skin effect protected by glide symmetry
• Index of Dirac operators and classification of topological insulators
• Topological Crystalline Metal in Orthorhombic Perovskite Iridates
• Helical Topological Superconducting Pairing at Finite Excitation Energies
• Effective field theories for interacting boundaries of 3D topological crystalline insulators through bosonisation
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Three-dimensional $\mathbb{Z}$ topological insulators without reflection symmetry
• Interaction-driven phase transition in one dimensional mirror-symmetry protected topological insulator
• Lieb-Schultz-Mattis Theorem and the Filling Constraint
• Boundary topological insulators and superconductors of Altland-Zirnbauer tenfold classes
• Modulating Hamiltonian Approach to Quantum Many-Body Systems and Crystalline Topological Phases Protected by Generalized Magnetic Translations
• Tunable plasmon modes and topological transitions in single- and bilayer semi-Dirac materials
• Stability of the X-Y-phase of the two-dimensional C4 point group insulator
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry
• Fermi-surface diagnosis for topological superconductivity with $s$-wave-like pairing symmetries
• Fragile topological insulators protected by rotation symmetry without spin-orbit coupling
• Glide-resolved photoemission spectroscopy: Measuring topological invariants in nonsymmorphic space groups