Fragile topology protected by inversion symmetry: Diagnosis, bulk-boundary correspondence, and Wilson loop

Abstract

We study the bulk and boundary properties of fragile topological insulators (TIs) protected by inversion symmetry, mostly focusing on the class A of the Altland-Zirnbauer classification. First, we propose an efficient method for diagnosing fragile band topology by using the symmetry data in momentum space. Using this method, we show that among all the possible parity configurations of inversion-symmetric insulators, at least $17%$ of them have fragile topology in two dimensions while fragile TIs are less than $3%$ in three dimensions. Second, we study the bulk-boundary correspondence of fragile TIs protected by inversion symmetry. In particular, we generalize the notion of $d$-dimensional ($d\mathrm{D}$) $k\mathrm{th}\text{\ensuremath{-}}\mathrm{order}$ TIs, which is normally defined for $0<k\ensuremath{\le}d$, to the cases with $k>d$, and show that they all have fragile topology. In terms of the Dirac Hamiltonian, a $d\mathrm{D}\phantom{\rule{4pt}{0ex}}k\mathrm{th}\text{\ensuremath{-}}\mathrm{order}$ TI has $(k\ensuremath{-}1)$ boundary mass terms. We show that a minimal fragile TI with the filling anomaly can be considered as the $d\mathrm{D}\phantom{\rule{4pt}{0ex}}(d+1)\mathrm{th}\text{\ensuremath{-}}\mathrm{order}$ TI, and all the other $d\mathrm{D}\phantom{\rule{4pt}{0ex}}k\mathrm{th}$-order TIs with $k>(d+1)$ can be constructed by stacking $d\mathrm{D}\phantom{\rule{4pt}{0ex}}(d+1)\mathrm{th}\text{\ensuremath{-}}\mathrm{order}$ TIs. Although $d\mathrm{D}\phantom{\rule{4pt}{0ex}}(d+1)\mathrm{th}\text{\ensuremath{-}}\mathrm{order}$ 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. In the cases with $k>(d+1)$, we show that the net parity of the system with boundaries can distinguish topological insulators and trivial insulators. Also, by studying the Wilson loop and nested Wilson loop spectra, we show that all the spectral windings of the Wilson loop and 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 the prediction from our diagnosis method of fragile topology. Finally, we show that a $(d+1)\mathrm{D}\phantom{\rule{4pt}{0ex}}(k\ensuremath{-}1)\mathrm{th}\text{\ensuremath{-}}\mathrm{order}$ TI can be obtained by an adiabatic pumping of $d\mathrm{D}\phantom{\rule{4pt}{0ex}}k\mathrm{th}\text{\ensuremath{-}}\mathrm{order}$ TI, which generalizes the previous study of the $2\mathrm{D}\phantom{\rule{4pt}{0ex}}\mathrm{third}\text{\ensuremath{-}}\mathrm{order}$ TI.