Topological 0D defect states in 3D insulators

There has been intense interest in relating the electronic states bound to crystal defects to the bulk electronic structure of pristine crystals. By mapping the momentum-space Hamiltonians of Brillouin zone surfaces to the real-space surfaces between crystal defects, we develop a general formulation of topological (crystalline) defect states. We introduce topological invariants for these states using (nested) Wilson loops and Topological Quantum Chemistry. Our framework captures all previous results, including fractional charges bound to point defects in inversion-symmetric Chern insulators and helical modes bound to screw dislocations in weak topological insulators (TIs). However, we also discover new examples. In particular, we show that screw dislocations and edge disclinations in 3D higher-order TIs (HOTIs) can bind anomalous 0D higher-order "end states," which are equivalent to the fractionally charged corner modes of 2D "fragile" TIs and obstructed atomic limits, and persist under the relaxation of particle-hole symmetry, which is not present in real materials. Using density functional theory and tight-binding calculations, we demonstrate the presence of higher-order defect end states in the HOTI and topological crystalline insulator SnTe.