The bulk-edge correspondence for the quantum Hall effect in Kasparov theory

Abstract

We prove the bulk-edge correspondence in $K$-theory for the quantum Hall effect by constructing an unbounded Kasparov module from a short exact sequence that links the bulk and boundary algebras. This approach allows us to represent bulk topological invariants explicitly as a Kasparov product of boundary invariants with the extension class linking the algebras. This paper focuses on the example of the discrete integer quantum Hall effect, though our general method potentially has much wider applications.

arXiv comment16 pages. Minor corrections and introduction expanded. To appear in Letters in Mathematical Physics

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• Topological invariants and corner states for Hamiltonians on a three-dimensional lattice
• Notes on topological insulators
• Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases
• T-duality simplifies bulk-boundary correspondence
• Generalized Connes-Chern characters in KK-theory with an application to weak invariants of topological insulators
• T-duality simplifies bulk-boundary correspondence: some higher dimensional cases
• Bulk-edge correspondence and the cobordism invariance of the index
• Good Wannier bases in Hilbert modules associated to topological insulators
• T-duality and the bulk-boundary correspondence
• Topological Lattice Defects by Groupoid Methods and Kasparov's KK-Theory
• Classification of topological invariants related to corner states
• C*-framework for higher-order bulk-boundary correspondences