Principles of higher-order topology

The higher-order topological insulators are crystalline-symmetry-protected phases of matter, which can support zero-energy states, localized at the sharp corners and hinges of a sample. Despite rapidly developing inter-disciplinary research on diverse aspects of higher-order topology, the fundamental organizing principles remain obscured by the absence of bulk topological invariants. In this work, we develop a unified theory of different orders of topological insulators, based on first and second homotopy classifications of non-Abelian, Berry's connections. The bulk invariants are computed from the windings of gauge-invariant eigenvalues of Wilson lines and planar Wilson loops. We show the various orders of topological insulators can be distinguished by the number of high-symmetry axes that can support non-trivial windings of Wannier centers. These high-symmetry directions determine the orientations of surfaces, which can host massless, Dirac fermions as gapless, surface states.