On the C*-algebraic approach to topological phases for insulators

Abstract

The notion of a topological phase of an insulator is based on the concept of homotopy between Hamiltonians. It therefore depends on the choice of a topological space to which the Hamiltonians belong. We advocate that this space should be the $C^{\ast }$-algebra of observables. We relate the symmetries of insulators to graded real structures on the observable algebra and classify the topological phases using van Daele's formulation of $K$-theory. This is related but not identical to Thiang's recent approach to classify topological phases by $K$-groups in Karoubi's formulation.

Version 2 accidentally merged with version 1. Major generalisation of discussion of real structures. Version 4: Revision and errors corrected

References in corpus (3)

• Classification of topological insulators and superconductors in three spatial dimensions
• K-Theory for Real C*-algebras via Unitary Elements with Symmetries
• Classification of "Quaternionic" Bloch-bundles: Topological Quantum Systems of type AII

Cited by in corpus (7)

• Controlled topological phases and bulk-edge correspondence
• The $K$-theoretic bulk-edge correspondence for topological insulators
• The FKMM-invariant in low dimension
• Cyclic cohomology for graded $C^{\ast ,r}$-algebras and its pairings with van Daele $K$-theory
• Application of semifinite index theory to weak topological phases
• Notes on twisted equivariant $\mathrm{K}$-theory for ${\mathrm{C}}^{\ast }$-algebras
• Topological insulators from the perspective of non-commutative geometry and index theory