Strong Pumping of Fragile Topology in Higher-Order Topological Insulators and Semimetals*

To gain insight on their response and surface states, topological insulators (TIs) and semimetals are frequently re-expressed as periodic tuning cycles of lower-dimensional insulators with nontrivial topology or multipole moments. The recent discovery of 3D higher-order TIs (HOTIs) and semimetals, which exhibit gapless 1D hinges, poses a clear question: “what are the possible 2D insulators that may be pumped to realize higher-order topology?” We answer this question by demonstrating that 2D “fragile” TIs, which are only topologically nontrivial for small numbers of valence bands, exhibit the same anomalous 0D corner modes as 2D obstructed atomic limits. We reveal that the most general HOTIs, including the well-studied “axion insulator,” are equivalent to pumping cycles of fragile-phase corner charge, and introduce nested Wilson loop invariants to diagnose corner charges in 2D and strong pumping in 3D. We then extend discussion to more exotic examples. Finally, we show that the fragile-phase pumping formulation directly implies the crystal defect and flux responses of 3D HOTIs and weak fragile TIs.