Controlled topological phases and bulk-edge correspondence

In this paper, we introduce a variation of the notion of topological phase reflecting metric structure of the position space. This framework contains not only periodic and non-periodic systems with symmetries in Kitaev’s periodic table but also topological crystalline insulators. We also define the bulk and edge indices as invariants taking values in the twisted equivariant $\mathrm{K}$-groups of Roe algebras as generalizations of existing invariants such as the Hall conductance or the Kane–Mele ${\mathbb{Z}}_2$-invariant. As a consequence, we obtain a new mathematical proof of the bulk-edge correspondence by using the coarse Mayer-Vietoris exact sequence.