Generalized Fermion Doubling Theorems: Classification of 2D Nodal Systems in Terms of Wallpaper Groups

Abstract

The Nielsen-Ninomiya Theorem has set up a ground rule for the minimal number of the topological points in a Brillouin zone. Notably, in the 2D Brillouin zone, chiral symmetry and space-time inversion symmetry can properly define topological invariants as charges characterizing the stability of the nodal points so that the non-zero charges protect these points. Due to the charge neutralization, the Nielsen-Ninomiya Theorem requires at least two stable topological points in the entire Brillouin zone. However, additional crystalline symmetries might duplicate the points. In this regard, for the wallpaper groups with crystalline symmetries, the minimal number of the nodal points in the Brillouin zone might be more than two. In this work, we determine the minimal numbers of the nodal points for the wallpaper groups in chiral-symmetric and space-time-inversion-symmetric systems separately and provide examples for new topological materials, such as topological nodal time-reversal-symmetric superconductors and Dirac semimetals. This generalized Nielsen-Ninomiya Theorem serves as a guide to search for 2D topological nodal materials and new platforms for twistronics. Furthermore, we show the Nielsen-Ninomiya Theorem can be extended to 2D non-Hermitian systems hosting topologically protected exceptional points and Fermi points for the 17 wallpaper groups and use the violation of the theorem on the surface to classify 3D Hermitian and non-Hermitian topological bulks.

arXiv commentSignificant extensions in the second version. 39 pages, 21 figures, 7 tables, any comments are welcome

References in corpus (116)

• The electronic properties of graphene
• Two-Dimensional Gas of Massless Dirac Fermions in Graphene
• Experimental Observation of Quantum Hall Effect and Berry's Phase in Graphene
• Magic-angle graphene superlattices: a new platform for unconventional superconductivity
• Quantum Spin Hall Effect in Graphene
• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Weyl and Dirac Semimetals in Three Dimensional Solids
• Discovery (theoretical prediction and experimental observation) of a large-gap topological-insulator class with spin-polarized single-Dirac-cone on the surface
• 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
• Two and one-dimensional honeycomb structures of silicon and germanium
• Topological Field Theory of Time-Reversal Invariant Insulators
• Discovery of a Weyl Fermion Semimetal and Topological Fermi Arcs
• Moire bands in twisted double-layer graphene
• Making Sense of Non-Hermitian Hamiltonians
• Classification of topological quantum matter with symmetries
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Discovery of a Three-dimensional Topological Dirac Semimetal, Na3Bi
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Edge states and topological invariants of non-Hermitian systems
• Periodic table for topological insulators and superconductors
• Complex Extension of Quantum Mechanics
• Experimental discovery of Weyl semimetal TaAs
• Localization transitions in non-Hermitian quantum mechanics
• Topological phases of non-Hermitian systems
• Symmetry and Topology in Non-Hermitian Physics
• Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems
• Topological Origin of Non-Hermitian Skin Effects
• Topological Band Theory for Non-Hermitian Hamiltonians
• Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase
• The (High Quality) Topological Materials In The World
• Experimental observation of Weyl points
• Anomalous edge state in a non-Hermitian lattice
• Non-Bloch Band Theory of Non-Hermitian Systems
• Correspondence between winding numbers and skin modes in non-hermitian systems
• Non-Hermitian Chern bands
• Catalogue of Topological Electronic Materials
• Origin of Magic Angles in Twisted Bilayer Graphene
• Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems
• Towards ideal topological materials: Comprehensive database searches using symmetry indicators
• Dirac Semimetals in Two Dimensions
• Topological Origin of Zero-Energy Edge States in Particle-Hole Symmetric Systems
• Observation of Bulk Fermi Arc and Polarization Half Charge from Paired Exceptional Points
• Topological classification of crystalline insulators through band structure combinatorics
• Semimetallic Two-Dimensional Boron Allotrope with Massless Dirac Fermions
• Non-Hermitian skin effect and chiral damping in open quantum systems
• Non-Hermitian bulk-boundary correspondence and auxiliary generalized Brillouin zone theory
• Untying the insulating and superconducting orders in magic-angle graphene
• Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas
• Topology of crystalline insulators and superconductors
• Hybrid higher-order skin-topological modes in non-reciprocal systems
• Second-Order Topological Phases in Non-Hermitian Systems
• Non-Hermitian Topological Invariants in Real Space
• Topological Transition in a Non-Hermitian Quantum Walk
• Topological phase transition in non-Hermitian quasicrystals
• The Rare Two-Dimensional Materials with Dirac Cones
• Cascade of Phase Transitions and Dirac Revivals in Magic Angle Graphene
• Fermion Path Integrals And Topological Phases
• Periodic Table for Topological Bands with Non-Hermitian Bernard-LeClair Symmetries
• Classification of Exceptional Points and Non-Hermitian Topological Semimetals
• Topological semi-metals with line nodes and drumhead surface states
• Fragile Topology and Wannier Obstructions
• 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
• Superconductivity without insulating states in twisted bilayer graphene stabilized by monolayer WSe${}_2$
• Random matrix model of QCD at finite density and the nature of the quenched limit
• Classification of topological insulators and superconductors in the presence of reflection symmetry
• New topological invariants in non-Hermitian systems
• Classification of reflection symmetry protected topological semimetals and nodal superconductors
• Non-Hermitian Many-Body Localization
• Higher-order topological corner states induced by gain and loss
• Magnetic Topological Quantum Chemistry
• Topological states of non-Hermitian systems
• Topological phases and surface flat bands in superconductors without inversion symmetry
• Quantum Oscillation from In-gap States and non-Hermitian Landau Level Problem
• A review of progress in the physics of open quantum systems: theory and experiment
• Optical realization of relativistic non-Hermitian quantum mechanics
• Protected boundary states in gapless topological phases
• Topological Correspondence between Hermitian and Non-Hermitian Systems: Anomalous Dynamics
• Non-Hermitian Hopf-Link Exceptional Line Semimetals
• Wallpaper Fermions and the Nonsymmorphic Dirac Insulator
• Exceptional Links and Twisted Fermi Ribbons in non-Hermitian Systems
• Effects of non-Hermitian perturbations on Weyl Hamiltonians with arbitrary topological charges
• Nodal Arc in Disordered Dirac Fermions: Connection to Non-Hermitian Band Theory
• Fermion doubling theorems in 2D non-Hermitian systems for Fermi points and exceptional points
• Local antiferromagnetic exchange and collaborative Fermi surface as key ingredients of high temperature superconductors
• Topological classification of non-Hermitian systems with reflection symmetry
• Two-Dimensional Dirac Fermions Protected by Space-Time Inversion Symmetry in Black Phosphorus
• Parity-Time Symmetry in Non-Hermitian Complex Optical Media
• Topologically protected flat zero-energy surface bands in non-centrosymmetric superconductors
• Topologically stable gapless phases of time-reversal invariant superconductors
• Exceptional Topological Insulators
• Refined symmetry indicators for topological superconductors in all space groups
• Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop
• Surface and 3D quantum Hall effects from engineering of exceptional points in nodal-line semimetals
• Two-dimensional Spin-Orbit Dirac Point in Monolayer HfGeTe
• Higher-Order Topological Superconductivity of Spin-Polarized Fermions
• Global anomalies on the surface of fermionic symmetry-protected topological phases in (3+1) dimensions
• Symmetry-based indicators for topological Bogoliubov-de Gennes Hamiltonians
• Fragile Phases As Affine Monoids: Classification and Material Examples
• Dirac cones beyond the honeycomb lattice: a symmetry based approach
• ${\mathbb{Z}}_2$-enriched symmetry indicators for topological superconductors in the 1651 magnetic space groups
• Exhaustive constructions of effective models in 1651 magnetic space groups
• Non-Hermitian Boundary State Engineering in Anomalous Floquet Topological Insulators
• Flat band in twisted bilayer Bravais lattices
• Nielsen-Ninomiya Theorem with Bulk Topology: Duality in Floquet and Non-Hermitian Systems
• Two-dimensional Dirac Semimetals without Inversion Symmetry
• Ni-based transition-metal trichalcogenide monolayer: a strongly correlated quadruple-layer graphene
• High-throughput Investigations of Topological and Nodal Superconductors
• Electronic structure of the candidate 2D Dirac semimetal SrMnSb2: a combined experimental and theoretical study
• Symmetry-based approach to nodal structures: Unification of compatibility relations and gapless point classifications
• Exact description of coalescing eigenstates in open quantum systems in terms of microscopic Hamiltonian dynamics
• Novel ${\mathbb{Z}}_2$-topological metals and semimetals
• Magic continuum in twisted bilayer square lattice with staggered flux
• $\eta$-pairing in correlated fermion models with spin-orbit coupling
• Topological superconductivity in Ni-based transition-metal trichalcogenide superconductors

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• Classification of time-reversal symmetric topological superconducting phases for conventional pairing symmetries
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry