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.

27 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 (24)

• Topological Insulators with Inversion Symmetry
• 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
• Reflection symmetric second-order topological insulators and superconductors
• Higher-order topological insulators and semimetals on the breathing Kagome and pyrochlore lattices
• Topological Node-Line Semimetal in Three Dimensional Graphene Networks
• Topological nodal line semimetals
• Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals
• Computing topological invariants without inversion symmetry
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials
• Double crystallographic groups and their representations on the Bilbao Crystallographic Server
• Higher-Order Weyl Semimetals
• Topological Euler class as a dynamical observable in optical lattices
• $PT$ Symmetric Real Dirac Fermions and Semimetals
• Geometric approach to fragile topology beyond symmetry indicators
• Graph Theory Data for Topological Quantum Chemistry
• Boundary criticality of $PT$-invariant topology and second-order nodal-line semimetals
• Topological acoustic triple point
• Filling anomaly for general 2D and 3D $C_4$ symmetric lattices
• Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond
• Topology of Triple-Point Metals
• Generalized Bloch theorem and topological characterization
• Triple nodal points characterized by their nodal-line structure in all magnetic space groups

Cited by in corpus (9)

• 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