Wannier Decay and the Thouless Conjecture

Abstract Nontrivial Chern classes pose an obstruction to the existence of exponentially decaying Wannier functions that provide natural bases for spectral subspaces. For nontrivial Bloch bundles, we obtain decay rates of Wannier functions in dimensions $d=2,3$. For $d=2$, we construct Wannier functions with full asymptotics and optimal decay rate $\mathcal{O}(|x|^{-2})$ as conjectured by Thouless; for $d=3$, we construct Wannier functions with the uniform decay rate $\mathcal{O}(|x|^{-7/3})$.