Coarse geometry and topological phases

Abstract

We propose the Roe C*-algebra from coarse geometry as a model for topological phases of disordered materials. We explain the robustness of this C*-algebra and formulate the bulk-edge correspondence in this framework. We describe the map from the K-theory of the group C*-algebra of Z^d to the K-theory of the Roe C*-algebra, both for real and complex K-theory.

References in corpus (12)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Topological Insulators in Three Dimensions
• Periodic table for topological insulators and superconductors
• Model Hamiltonian for Topological Insulators
• The Non-Commutative Geometry of the Quantum Hall Effect
• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• The strong side of weak topological insulators
• Controlled topological phases and bulk-edge correspondence
• Bott periodicity for $Z_2$ symmetric ground states of gapped free-fermion systems
• Classification of "Real" Bloch-bundles: Topological quantum systems of type AI
• Property A and the operator norm localization property for discrete metric spaces
• Warped cones and property A

Cited by in corpus (15)

• Complete homotopy invariants for translation invariant symmetric quantum walks on a chain
• Gaplessness of Landau Hamiltonians on hyperbolic half-planes via coarse geometry
• Rigidity of ${\ell }^p$ Roe-type algebras
• Remarks on anomalous symmetries of C*-algebras
• Good Wannier bases in Hilbert modules associated to topological insulators
• Large-scale geometry obstructs localization
• Locally equivalent quasifree states and index theory
• Corona Rigidity
• Geometric construction of classes in van Daele's K-theory
• C*-rigidity of bounded geometry metric spaces
• On topological obstructions to the existence of non-periodic Wannier bases
• Coarse geometric approach to topological phases: Invariants from real-space representations
• A vanishing theorem in $K$-theory for spectral projections of a non-periodic magnetic Schrödinger operator
• A quantization of coarse spaces and uniform Roe algebras
• Robustness of topological phases on aperiodic lattices