Existence and Computation of Generalized Wannier Functions for Non-Periodic Systems 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.…”

“…Moreover, in [ 26 ] it has been shown that the construction of [ 71 ] is optimal, in the sense that the obtained GWB has the same exponential decay of the associated Fermi projection. An alternative strategy to construct a GWB for a generic two-dimensional non-periodic system has been recently suggested and numerically validated [ 79 ]. We shortly review the whole topic of GWB in Sect.…”

Purely linear response of the quantum Hall current to space-adiabatic perturbations

“…In a more general setting, where periodicity is broken, one can still define a generalized notion of Wannier basis for an isolated spectral island, but the relation of its existence with the Chern marker remains to be fully understood. In fact, while the existence of a generalized Wannier basis (with suitable localization) has been shown to imply the vanishing of the Chern marker (see [28,33]), the converse implication [29] remains a challenging and interesting line of research.…”