C*-framework for higher-order bulk-boundary correspondences

TL;DR

This paper develops a C*-algebraic framework using operator K-theory to understand higher-order bulk-boundary correspondences in crystalline topological insulators and superconductors, capturing boundary modes protected by symmetry.

Contribution

It introduces a groupoid C*-algebra model that encodes bulk dynamics, boundary conditions, and symmetry actions, providing a new mathematical approach to higher-order topological phenomena.

Findings

Derivation of spectral sequences in twisted equivariant K-theory

Enumeration of all non-trivial higher-order bulk-boundary correspondences

Framework applicable to crystals with point symmetries

Abstract

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.