Controlled topological phases and bulk-edge correspondence

TL;DR

This paper develops a new framework for topological phases that incorporates metric structures and extends existing invariants, providing a rigorous mathematical proof of bulk-edge correspondence.

Contribution

It introduces a metric-based topological phase framework and defines new bulk and edge invariants using twisted equivariant K-groups, generalizing previous invariants.

Findings

Unified framework for periodic, non-periodic, and crystalline insulators

New proof of bulk-edge correspondence using coarse Mayer-Vietoris sequence

Extension of topological invariants to Roe algebra K-groups

Abstract

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.