Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop

We study the bulk and boundary properties of fragile topological insulators (TIs) protected by inversion symmetry, mostly focusing on the class A insulator. First, we propose an efficient method for diagnosing fragile band topology by using the symmetry data in momentum space. Second, we study the bulk-boundary correspondence of fragile TIs protected by inversion symmetry. In particular, we show that a minimal fragile TI with the filling anomaly corresponds to the dD (d+1)th-order TI. Although dD (d+1)th-order TIs have no in-gap states, the boundary mass terms carry an odd winding number along the boundary, which induces localized charges on the boundary at the positions where the boundary mass terms change abruptly. Also by studying the (nested) Wilson loop spectra, we show that all the spectral windings of the (nested) Wilson loop should be unwound to resolve the Wannier obstruction of fragile TIs. By counting the minimal number of bands required to unwind the spectral winding of the Wilson loop and nested Wilson loop, we determine the minimal number of bands to resolve the Wannier obstruction, which is consistent with our diagnosis method of fragile topology.