Algebraic Localization Implies Exponential Localization in Non-periodic Insulators
- 1Department of Mathematics, Department of Physics, and Department of Chemistry, Duke University, Box 90320, Durham, NC, 27708, USA
- 2Department of Mathematics, University of California - Berkeley, 970 Evans Hall, Berkeley, CA, 94720-3840, USA
- 3Department of Mathematics, University of Minnesota - Twin Cities, 127 Vincent Hall, Minneapolis, MN, 55455, USA
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 (that is all basis elements satisfy ∫ | varvecx| 2 | w(varvecx)| 2 dvarvecx < ∞ ∫ | x | 2 | w ( x ) | 2 d x < ∞ ). 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 ∫ | varvecx| 5 + ε | w(varvecx)| 2 dvarvecx < ∞ ∫ | x | 5 + ε | w ( x ) | 2 d x < ∞ for some ε > 0 ε > 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 in [1, 2].
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BibTeX
@article{lu2026algebraic,
title = {{Algebraic Localization Implies Exponential Localization in Non-periodic Insulators}},
author = {Jianfeng Lu and Kevin D. Stubbs},
journal = {Archive for Rational Mechanics and Analysis},
year = {2026},
month = sep,
doi = {10.1007/s00205-026-02242-z},
url = {https://doi.org/10.1007/s00205-026-02242-z},
note = {CC BY},
}
RIS
TY - JOUR
TI - Algebraic Localization Implies Exponential Localization in Non-periodic Insulators
AU - Jianfeng Lu
AU - Kevin D. Stubbs
PY - 2026
DA - 2026/09/20
JO - Archive for Rational Mechanics and Analysis
DO - 10.1007/s00205-026-02242-z
UR - https://doi.org/10.1007/s00205-026-02242-z
AB - 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 (that is all basis elements satisfy ∫ | varvecx| ^2 | w(varvecx)| ^2 dvarvecx < ∞ ∫ | x | 2 | w ( x ) | 2 d x < ∞ ). 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 ∫ | varvecx| ^5 + ε | w(varvecx)| ^2 dvarvecx < ∞ ∫ | x | 5 + ε | w ( x ) | 2 d x < ∞ for some ε > 0 ε > 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 in [1, 2].
SN - 0003-9527
SN - 1432-0673
ER -