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

Article Physics, Multidisciplinary

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

Giovanna Marcelli et al.

Summary: This study investigates the relation between the localization of generalized Wannier bases and the topological properties of two-dimensional gapped quantum systems in a disordered background, including magnetic fields. It is proven that the existence of well-localized generalized Wannier bases implies the vanishing of the Chern character.

ANNALES HENRI POINCARE (2023)

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Article Mathematics, Applied

Existence and Computation of Generalized Wannier Functions for Non-Periodic Systems in Two Dimensions and Higher

Jianfeng Lu et al.

Summary: This study aims to explore the construction method of ELWFs in insulating materials without necessarily having a crystalline structure. While the existence of ELWFs has been proven in one-dimensional case, we successfully extend this result to two-dimensional and higher dimensions under a certain assumption.

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS (2022)

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Article Physics, Mathematical

Large-scale geometry obstructs localization

Matthias Ludewig et al.

Summary: We explain the coarse geometric origin of the fact that certain spectral subspaces of topological insulator Hamiltonians cannot admit an orthonormal basis of localized wavefunctions.

JOURNAL OF MATHEMATICAL PHYSICS (2022)

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Article Mathematics, Applied

Localised Module Frames and Wannier Bases from Groupoid Morita Equivalences

Chris Bourne et al.

Summary: By utilizing the operator algebraic approach, this paper constructs frames of translates for the Hilbert space localisation of the Morita equivalence bimodule arising from a groupoid equivalence, and investigates the conditions under which these frames are orthonormal bases if the module is free. The study is then applied to the localised Wannier bases of spectral subspaces of Schrodinger operators with atomic potentials supported on Delone sets, with noncommutative Chern numbers serving as a topological obstruction to fast-decaying Wannier bases. Additionally, the stability of this result under deformations of the underlying Delone set is demonstrated.

JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS (2021)

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Article Materials Science, Multidisciplinary

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

Kevin D. Stubbs et al.

Summary: In this study, the problem of calculating exponentially localized generalized Wannier functions in various systems is addressed and the Iterated Projected Position (IPP) algorithm is introduced as a solution. Numerical experiments demonstrate the efficacy of the IPP algorithm in computing exponentially localized bases for different models.

PHYSICAL REVIEW B (2021)

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Article Mathematics, Applied