Algebraic localization of Wannier functions implies Chern triviality in non-periodic insulators

Abstract

For gapped periodic systems (insulators), it has been established that the insulator is topologically trivial (i.e., its Chern number is equal to $0$) if and only if its Fermi projector admits an orthogonal basis 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 paper, we extend one direction of this result to non-periodic gapped systems. In particular, we show that the existence of an orthogonal basis with slightly more decay ($\int |\mathbf{x}{|}^{2+\epsilon }|w(\mathbf{x}){|}^2\text{d}\mathbf{x}<\infty$ for any $\epsilon >0$) is a sufficient condition to conclude that the Chern marker, the natural generalization of the Chern number, vanishes.

arXiv comment12 pages, no figures. We found an error in the previous version of this paper. Because of this, the decay required for our main result is slightly worse ($2+\epsilon$ instead of finite second moment)

References in corpus (7)

• Maximally localized Wannier functions: Theory and applications
• Exponential localization of Wannier functions in insulators
• Optimal decay of Wannier functions in Chern and Quantum Hall insulators
• Beyond Diophantine Wannier diagrams: Gap labelling for Bloch-Landau Hamiltonians
• Existence and computation of generalized Wannier functions for non-periodic systems in two dimensions and higher
• The Haldane model and its localization dichotomy
• Algebraic localization implies exponential localization in non-periodic insulators

Cited by in corpus (1)

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