A noncommutative framework for topological insulators

Abstract

We study topological insulators, regarded as physical systems giving rise to topological invariants determined by symmetries both linear and anti-linear. Our perspective is that of noncommutative index theory of operator algebras. In particular we formulate the index problems using Kasparov theory, both complex and real. We show that the periodic table of topological insulators and superconductors can be realised as a real or complex index pairing of a Kasparov module capturing internal symmetries of the Hamiltonian with a spectral triple encoding the geometry of the sample's (possibly noncommutative) Brillouin zone.

32 pages, final version

References in corpus (10)

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• Quantum Spin Hall Insulator State in HgTe Quantum Wells
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Cited by in corpus (26)

• Controlled topological phases and bulk-edge correspondence
• The $K$-theoretic bulk-edge correspondence for topological insulators
• The Noncommutative Index Theorem and the Periodic Table for Disordered Topological Insulators and Superconductors
• Bulk-boundary correspondence for disordered free-fermion topological phases
• Wannier functions and Z_2 invariants in time-reversal symmetric topological insulators
• Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases
• Topological and conventional phases of a three dimensional electron glass
• Generalized Connes-Chern characters in KK-theory with an application to weak invariants of topological insulators
• Index theory and topological phases of aperiodic lattices
• Fredholm Homotopies for Strongly-Disordered 2D Insulators
• The KO-valued spectral flow for skew-adjoint Fredholm operators
• The FKMM-invariant in low dimension
• Bulk-edge correspondence and the cobordism invariance of the index
• The non-commutative topology of two-dimensional dirty superconductors
• A groupoid approach to interacting fermions
• Application of semifinite index theory to weak topological phases
• Locally equivalent quasifree states and index theory
• Spin geometry of the rational noncommutative torus
• Symmetric Fermi projections and Kitaev's table: topological phases of matter in low dimensions
• Cwikel Estimates and Negative Eigenvalues of Schroedinger Operators on Noncommutative Tori
• The cohomology invariant for class DIII topological insulators
• Majorana Fermions and Orthogonal Complex Structures
• Dynamic noncommutative BTZ black holes
• A ${\mathbb{Z}}_2$-Topological Index for Quasi-Free Fermions
• Robustness of topological phases on aperiodic lattices
• Topological junctions for one-dimensional systems