Algebraic Localization Implies Exponential Localization in Non-periodic Insulators

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 ∫ x ² w(x) ² \,dx < ∞ ∫ | 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 ∫ x 5 + ε w(x) ² \,dx < ∞ ∫ | 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].