A space-adiabatic approach for bulk-defect correspondences in lattice models of topological insulators

TL;DR

This paper bridges space-adiabatic methods and operator K-theory to analyze topological insulators with defects, enabling the computation of topological invariants and bulk-defect correspondences in lattice models.

Contribution

It establishes a rigorous connection between space-adiabatic approximations and operator-theoretic approaches for topological insulators with defects, introducing a method to compute topological invariants via phase-space quantization.

Findings

Constructed lattice Hamiltonians with protected defect states

Computed topological invariants from symbol functions

Derived boundary maps in K-theory for bulk-defect relations

Abstract

In space-adiabatic approaches one can approximate Hamiltonians that are modulated slowly in space by phase-space functions that depend on position and momentum. In this paper, we establish a rigorous relation between this approach and the operator-theoretic approach for topological insulators with defects, which employs $C^*$-algebras and operator K-theory. Using such tools, we show that by quantizing phase-space functions one can construct lattice Hamiltonians which are gapped at certain spatial limits and carry protected states at defects such as boundaries, hinges, and corners. Moreover, we show that the topological invariants that protect the latter can be computed in terms of the symbol functions. This enables us to compute boundary maps in K-theory that are relevant for bulk-defect correspondences.