\(C^*\)-Framework for Higher-Order Bulk-Boundary Correspondences

A typical crystal is a finite piece of a material which may be invariant under some point symmetry group. If it is a so-called intrinsic higher-order topological insulator or superconductor, then it displays boundary modes at hinges or corners protected by the crystalline symmetry and the bulk topology. We explain the mechanism behind such phenomena using operator K-theory. Specifically, we derive a groupoid \(C^*\)-algebra that (1) encodes the dynamics of the electrons in the infinite size limit of a crystal; (2) remembers the boundary conditions at the crystal’s boundaries, and (3) admits a natural action by the point symmetries of the atomic lattice. The filtrations of the groupoid’s unit space by closed subsets that are invariant under the groupoid and point group actions supply equivariant cofiltrations of the groupoid \(C^*\)-algebra. We show that specific derivations of the induced spectral sequences in twisted equivariant K-theories enumerate all non-trivial higher-order bulk-boundary correspondences.