Good Wannier bases in Hilbert modules associated to topological insulators

Abstract

For a large class of physically relevant operators on a manifold with discrete group action, we prove general results on the (non-)existence of a basis of smooth well-localised Wannier functions for their spectral subspaces. This turns out to be equivalent to the freeness of a certain Hilbert module over the group $C^{\ast }$-algebra canonically associated to the spectral subspace. This brings into play $K$-theoretic methods and justifies their importance as invariants of topological insulators in physics.

arXiv comment35 pages, 5 Figures, 1 Table. Revised version for publication in J. Math. Phys

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Cited by in corpus (7)

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• Large-scale geometry obstructs localization
• Twisted crystallograpic T-duality via the Baum--Connes isomorphism
• Localised module frames and Wannier bases from groupoid Morita equivalences
• Ultra-generalized Wannier bases: Are they relevant to topological transport?
• On topological obstructions to the existence of non-periodic Wannier bases