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 (i.e. all basis elements satisfy integral squared w squared, d < infinity). 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 integral 5 + ε w squared, d < infinity for some 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