Exponentially-localized Wannier functions are a basis of the Fermi projection of a Hamiltonian consisting of functions which decay exponentially quickly 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 |x| 2 |w(x)| 2 dx < ∞). 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+ǫ |w(x)| 2 dx < ∞ for some ǫ > 0 then there also exists an orthonormal basis for the Fermi projection which decays exponentially quickly in space. This result lends support to the Localization Dichotomy Conjecture for non-periodic systems recently proposed by Marcelli, Monaco, Moscolari, and Panati in [10,11].
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