Wannier decay and the Thouless conjecture

TL;DR

This paper investigates the decay properties of Wannier functions in two and three dimensions, confirming Thouless's conjecture in 2D and providing explicit decay rates for 3D.

Contribution

It establishes decay rates for Wannier functions in non-trivial Bloch bundles in 2D and 3D, including full asymptotics in 2D and optimal decay in 3D.

Findings

In 2D, Wannier functions decay as |x|^{-2} with full asymptotics.

In 3D, Wannier functions decay as |x|^{-7/3}.

Decay rates match conjectures for non-trivial Chern classes.

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 $\mathcal{O}(|x|^{-7/3})$.