Wannier decay and the Thouless conjecture

Abstract

Non-trivial Chern classes pose an obstruction to the existence of exponentially decaying Wannier functions which provide natural bases for spectral subspaces. For non-trivial 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 $\mathcal{O}(|x{|}^{-7/3})$.

Outsourced from a previous paper, which included a discussion of fragile topology arXiv:2502.03442v2. arXiv admin note: substantial text overlap with arXiv:2502.03442