Algebraic localization implies exponential localization in non-periodic insulators

Abstract

Exponentially-localized Wannier functions are a basis of the Fermi projection of a Hamiltonian consisting of functions which decay exponentially fast in space. In two and three spatial dimensions, it is well understood for periodic insulators that exponentially-localized Wannier functions exist if and only if there exists an orthonormal basis for the Fermi projection with finite second moment (i.e. all basis elements satisfy $\int |\mathbf{x}{|}^2|w(\mathbf{x}){|}^2\text{d}\mathbf{x}<\infty$). In this work, we establish a similar result for non-periodic insulators in two spatial dimensions. In particular, we prove that if there exists an orthonormal basis for the Fermi projection which satisfies $\int |\mathbf{x}{|}^{5+\epsilon }|w(\mathbf{x}){|}^2\text{d}\mathbf{x}<\infty$ for some $\epsilon >0$ then there also exists an orthonormal basis for the Fermi projection which decays exponentially fast in space. This result lends support to the Localization Dichotomy Conjecture for non-periodic systems recently proposed by Marcelli, Monaco, Moscolari, and Panati

arXiv comment32 pages. Simplified and streamlined proofs using updated results from arxiv:2003.06676v5

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Cited by in corpus (4)

• Existence and computation of generalized Wannier functions for non-periodic systems in two dimensions and higher
• Localised module frames and Wannier bases from groupoid Morita equivalences
• Algebraic localization of Wannier functions implies Chern triviality in non-periodic insulators
• Ultra-generalized Wannier bases: Are they relevant to topological transport?