Wannier functions

A Wannier function is \[ w_n(\mathbf{r}-\mathbf{R}) = \frac{1}{(2\pi)^d}\int_{BZ}d\mathbf{k}e^{-i\mathbf{k}\cdot\mathbf{R}}u_{n\mathbf{k}}(\mathbf{r}), \] this means \(u_{nk}\)

Thouless in 1984 showed that the localization of Wannier function led to the bands having a zero Chern number. If the bands feature a non-zero Chern number, it is impossible to construct exponentially localized Wannier functions! He conjectured that there exist Wannier functions with decay \(O(|x|^{-2})\). However, in 3D this decay rate can be inhomogeneous (see [https://arxiv.org/pdf/2505.01999]).

Mathematically, Wannier functions establish a natural base for spectral subspace.