Universal higher-order bulk-boundary correspondence of triple nodal points

Abstract

Triple nodal points are degeneracies of energy bands in momentum space at which three Hamiltonian eigenstates coalesce at a single eigenenergy. For spinless particles, the stability of a triple nodal point requires two ingredients: rotation symmetry of order three, four or six; combined with mirror or space-time-inversion symmetry. However, despite ample studies of their classification, robust boundary signatures of triple nodal points have until now remained elusive. In this work, we first show that pairs of triple nodal points in semimetals and metals can be characterized by Stiefel-Whitney and Euler monopole invariants, of which the first one is known to facilitate higher-order topology. Motivated by this observation, we then combine symmetry indicators for corner charges and for the Stiefel-Whitney invariant in two dimensions with the classification of triple nodal points for spinless systems in three dimensions. The result is a complete higher-order bulk-boundary correspondence, where pairs of triple nodal points are characterized by fractional jumps of the hinge charge. We present minimal models of the various species of triple nodal points carrying higher-order topology, and illustrate the derived correspondence on Sc${}_3$AlC which becomes a higher-order triple-point metal in applied strain. The generalization to spinful systems, in particular to the WC-type triple-point material class, is briefly outlined.

arXiv comment27 pages (20 figures, 3 tables) + 16 pages of Appendices (10 figures, 8 tables) + references + 10 pages of supplement; (v2) extension to non-symmorphic space groups, addition of supplement; (v3) updated with accepted version

References in corpus (100)

• Electronic Structure of Pyrochlore Iridates: From Topological Dirac Metal to Mott Insulator
• Topological Insulators with Inversion Symmetry
• Maximally localized Wannier functions: Theory and applications
• Quantized Electric Multipole Insulators
• Wannier90 as a community code: new features and applications
• Dirac semimetal in three dimensions
• Topological nodal semimetals
• Higher-Order Topological Insulators
• Topological quantum chemistry
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Topolectrical circuit realization of topological corner modes
• Reflection symmetric second-order topological insulators and superconductors
• Observation of a phononic quadrupole topological insulator
• Topological nodal line semimetals with and without spin-orbital coupling
• Higher-order topological insulators and semimetals on the breathing Kagome and pyrochlore lattices
• Higher-Order Topology in Bismuth
• Dirac Line Nodes in Inversion Symmetric Crystals
• Demonstration of a quantized microwave quadrupole insulator with topologically protected corner states
• Topological Nodal Line Semimetal and Dirac Semimetal State in Antiperovskite Cu${}_3$PdN
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Quantization of fractional corner charge in $C_n$-symmetric higher-order topological crystalline insulators
• Topological Node-Line Semimetal in Three Dimensional Graphene Networks
• Topological nodal line semimetals
• Topological classification of crystalline insulators through band structure combinatorics
• Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals
• Computing topological invariants without inversion symmetry
• Topological protection of photonic mid-gap cavity modes
• Inversion Symmetric Topological Insulators
• Nodal-chain metals
• Photonic quadrupole topological phases
• Higher-order topological insulators and superconductors protected by inversion symmetry
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Corner states of light in photonic waveguides
• Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials
• High-throughput calculations of magnetic topological materials
• Second-order topological insulators and superconductors with an order-two crystalline symmetry
• 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
• Irvsp: to obtain irreducible representations of electronic states in the VASP
• Experimental Observation of Three-Component 'New Fermions' in Topological Semimetal MoP
• Higher-Order Topology, Monopole Nodal Lines, and the Origin of Large Fermi Arcs in Transition Metal Dichalcogenides XTe${}_2$ (X=Mo,W)
• Triple Point Topological Metals
• Topological Semimetals with Triply Degenerate Nodal Points in θ-phase Tantalum Nitride
• Higher-order bulk-boundary correspondence for topological crystalline phases
• Double crystallographic groups and their representations on the Bilbao Crystallographic Server
• Magnetic Topological Quantum Chemistry
• Higher-Order Topological Insulator in Twisted Bilayer Graphene
• Band Topology and Linking Structure of Nodal Line Semimetals with Z2 Monopole Charges
• Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
• Non-Abelian band topology in noninteracting metals
• Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems
• Higher Order Topological Phases: A General Principle of Construction
• Strong and Fragile Topological Dirac Semimetals with Higher-Order Fermi Arcs
• Observation of Quadrupole Transitions and Edge Mode Topology in an LC network
• Higher-order topological insulators protected by inversion and rotoinversion symmetries
• Non-Abelian reciprocal braiding of Weyl points and its manifestation in $\text{ZrTe}$
• Higher-Order Weyl Semimetals
• Topological Quadrupolar Semimetals
• Two-dimensional higher-order topology in monolayer graphdiyne
• Higher-order topological semimetal in acoustic crystals
• Topological Euler class as a dynamical observable in optical lattices
• Higher-order Weyl Semimetals
• Hourglass Dirac Chain Metal in Rhenium Dioxide
• Coexistence of four-band nodal rings and triply-degenerate nodal points in centrosymmetric metal diborides
• Lattice models with exactly solvable topological hinge and corner states
• $PT$ Symmetric Real Dirac Fermions and Semimetals
• Geometric approach to fragile topology beyond symmetry indicators
• New fermions on the line in topological symmorphic metals
• Three-component fermions with surface Fermi arcs in topological semimetal tungsten carbide
• Graph Theory Data for Topological Quantum Chemistry
• Magnetic second-order topological insulators and semimetals
• Nexus and Dirac lines in topological materials
• Diagnosis for topological semimetals in the absence of spin-orbital coupling
• Boundary criticality of $PT$-invariant topology and second-order nodal-line semimetals
• Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop
• Non-abelian topology of nodal-line rings in $\mathcal{P}\mathcal{T}$-symmetric systems
• Higher-order Dirac sonic crystals
• Braiding Majorana corner modes in a second-order topological superconductor
• General corner charge formula in two-dimensional C_n-symmetric higher-order topological insulators
• Momentum-space structure of surface states in a topological semimetal with a nexus point of Dirac lines
• The centrosymmetric Li2NaN material: an ideal topological semimetal with critical-type triply degenerate nodal points
• Crystallographic splitting theorem for band representations and fragile topological photonic crystals
• Topological acoustic triple point
• Simple model for second-order topological insulators and loop-nodal semimetals in Transition Metal Dichalcogenides XTe${}_2$ (X=Mo,W)
• Conversion rules for Weyl points and nodal lines in topological media
• Qsymm: Algorithmic symmetry finding and symmetric Hamiltonian generation
• Filling anomaly for general 2D and 3D $C_4$ symmetric lattices
• Corner charge and bulk multipole moment in periodic systems
• Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond
• From triple-point materials to multiband nodal links
• Parallel Transport and Band Theory in Crystals
• Topology of Triple-Point Metals
• Nearly triple nodal point topological phase in half-metallic GdN
• Fractional Corner Charge of Sodium Chloride
• Quadrupole moments, edge polarizations, and corner charges in the Wannier representation
• Hopf-chain networks evolved from triple points
• Generalized Bloch theorem and topological characterization
• Nexus networks in carbon honeycombs
• Triple nodal points characterized by their nodal-line structure in all magnetic space groups
• On the topological character of three-dimensional Nexus triple point degeneracies

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• Four-band non-Abelian topological insulator and its experimental realization
• Euler Obstructed Cooper Pairing: Nodal Superconductivity and Hinge Majorana Zero Modes
• Experimental demonstration of non-adjacent band topology connecting multiple nodal links
• Triple nodal points characterized by their nodal-line structure in all magnetic space groups
• Bulk-edge correspondence of Stiefel-Whitney and Euler insulators through the entanglement spectrum and cutting procedure
• Demonstration of non-Abelian frame charge flow in photonic crystals
• Family of third-order topological insulators from Su-Schrieffer-Heeger stacking
• Punctured-Chern topological invariants for semi-metallic bandstructures
• Perfectly localized Majorana corner modes in fermionic lattices