The $K$-theoretic bulk-edge correspondence for topological insulators

Abstract

We study the application of Kasparov theory to topological insulator systems and the bulk-edge correspondence. We consider observable algebras as modelled by crossed products, where bulk and edge systems may be linked by a short exact sequence. We construct unbounded Kasparov modules encoding the dynamics of the crossed product. We then link bulk and edge Kasparov modules using the Kasparov product. Because of the anti-linear symmetries that occur in topological insulator models, real $C^{\ast }$-algebras and $KKO$-theory must be used.

Minor corrections, to appear in Annales Henri Poincaré

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• Finite volume calculation of $K$-theory invariants
• The K-Theoretic Bulk-Boundary Principle for Dynamically Patterned Resonators
• The Bulk-Edge Correspondence in Three Simple Cases
• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
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• The KO-valued spectral flow for skew-adjoint Fredholm operators
• The Cayley transform in complex, real and graded $K$-theory
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• Two-Dimensional Time-Reversal-Invariant Topological Insulators via Fredholm Theory
• Approximate symmetries and conservation laws in topological insulators and associated $\mathbb{Z}$-invariants
• Cyclic cohomology for graded $C^{\ast ,r}$-algebras and its pairings with van Daele $K$-theory
• Is the continuum SSH model topological?
• Crystallographic T-duality
• The bulk-dislocation correspondence for weak topological insulators on screw-dislocated lattices
• Application of semifinite index theory to weak topological phases
• Topological Lattice Defects by Groupoid Methods and Kasparov's KK-Theory
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• Classification of topological invariants related to corner states
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• Bulk-Boundary Correspondence for Topological Insulators with Quantized Magneto-Electric Effect
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