Iterated projected position algorithm for constructing exponentially localized generalized Wannier functions for periodic and nonperiodic insulators in two dimensions and higher

Ultra-generalized Wannier bases: Are they relevant to topological transport?

“…This is exactly the simple but successful key idea in the paper by Prodan. We notice that more recently there has been another proposal by Stubbs, Lu and Watson to overcome such lack of compactness under further spectral assumptions on the operator P X j P (namely the uniformity of spectral gaps), see [34,33].…”

Localization of Generalized Wannier Bases Implies Chern Triviality in Non-periodic Insulators

“…The relation with the theory of almost-commuting operators has been explored in [ 37 ]. Some proposals to circumvent these difficulties appeared recently in [ 79 , 80 ], where the authors show that, under the additional crucial assumptions of uniform spectral gaps for the spectrum of the operator , it is possible to prove the existence of an exponentially decaying GWB for the projection P . Proving that satisfies such uniform spectral gaps hypothesis is still an open problem, but in [ 79 ] the authors present numerical simulations showing that in explicit tight-binding models the spectrum of has spectral gaps only when the projection P is Chern trivial, which is in accordance with Conjecture 3.2 .…”

“…These preliminary papers, together with the results in [ 62 ], resparked the interest of part of the community for the analysis of Wannier bases for non-periodic systems. Besides the aforementioned [ 79 , 80 ], we notice the preprint by Lu and Stubbs [ 52 ] (see also [ 51 ]) where they manage to show Theorem 3.1 with . Nevertheless, Theorem 3.1 with the optimal threshold is still an open problem.…”

Localised Module Frames and Wannier Bases from Groupoid Morita Equivalences

“…The regularity of Wannier bases changes drastically depending on the topological properties of the spectral band of the Schrödinger operator, where a delocalised Wannier basis can be used as an indicator that the system has a non-trival topological phase, see [15,16,24,29,32] for example. Wannier bases with exponential decay can be constructed for periodic and aperiodic Hamiltonians such that the compression of a position operator by the Fermi projection has uniform spectral gaps [40,41].…”