Algebraic Localization Implies Exponential Localization in Non-periodic Insulators — UC Berkeley

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 $$\int \vert \varvec{x}\vert ^2 \vert w(\varvec{x})\vert ^2 \,\text {d}{\varvec{x}} < \infty $$ ∫ | 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 $$\int \vert \varvec{x}\vert ^{5 + \varepsilon } \vert w(\varvec{x})\vert ^2 \,\text {d}{\varvec{x}} < \infty $$ ∫ | x | 5 + ε | w ( x ) | 2 d x < ∞ for some $$\varepsilon > 0$$ 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].