Definitions

# Wannier function

The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier.

The Wannier functions for different lattice sites in a crystal are orthogonal, allowing a convenient basis for the expansion of electron states in certain regimes. Wannier functions have found widespread use, for example, in the analysis of binding forces acting on electrons; they have proven to be in general localized, at least for insulators, in 2006. Specifically, these functions are also used in the analysis of excitons and condensed Rydberg matter.

## Simplest definition

Although Wannier functions can be chosen in many different ways, the original, simplest, and most common definition in solid-state physics is as follows. Choose a single band in a perfect crystal, and denote its Bloch states by

$psi_\left\{mathbf\left\{k\right\}\right\}\left(mathbf\left\{r\right\}\right) = e^\left\{imathbf\left\{k\right\}cdotmathbf\left\{r\right\}\right\}u_\left\{mathbf\left\{k\right\}\right\}\left(mathbf\left\{r\right\}\right)$
where $, u_\left\{mathbf\left\{k\right\}\right\}\left(mathbf\left\{r\right\}\right)$ has the same periodicity as the crystal. Then the Wannier functions are defined by
$phi_\left\{mathbf\left\{R\right\}\right\}\left(mathbf\left\{r\right\}\right) = frac\left\{1\right\}\left\{sqrt\left\{N\right\}\right\} sum_\left\{mathbf\left\{k\right\}\right\} e^\left\{-imathbf\left\{k\right\}cdotmathbf\left\{R\right\}\right\} psi_\left\{mathbf\left\{k\right\}\right\}\left(mathbf\left\{r\right\}\right)$,
where

$sum_\left\{mathbf\left\{k\right\}\right\} longrightarrow frac\left\{N\right\}\left\{Omega\right\} int_\left\{BZ\right\} d^3mathbf\left\{k\right\}$
where "BZ" denotes the Brillouin zone, which has volume Ω.

### Properties

On the basis of this definition, the following properties can be proven to hold:

• For any lattice vector R' ,

$phi_\left\{mathbf\left\{R\right\}\right\}\left(mathbf\left\{r\right\}\right) = phi_\left\{mathbf\left\{R\right\}+mathbf\left\{R\right\}\text{'}\right\}\left(mathbf\left\{r\right\}+mathbf\left\{R\right\}\text{'}\right)$
In other words, a Wannier function only depends on the quantity (r-R). As a result, these functions are often written in the alternative notation
$phi\left(mathbf\left\{r\right\}-mathbf\left\{R\right\}\right) := phi_\left\{mathbf\left\{R\right\}\right\}\left(mathbf\left\{r\right\}\right)$

• The Bloch functions can be written in terms of Wannier functions as follows:

$psi_\left\{mathbf\left\{k\right\}\right\}\left(mathbf\left\{r\right\}\right) = frac\left\{1\right\}\left\{sqrt\left\{N\right\}\right\} sum_\left\{mathbf\left\{R\right\}\right\} e^\left\{imathbf\left\{k\right\}cdotmathbf\left\{R\right\}\right\} phi_\left\{mathbf\left\{R\right\}\right\}\left(mathbf\left\{r\right\}\right)$,
where the sum is over each lattice vector R in the crystal.

• The set of wavefunctions $phi_\left\{mathbf\left\{R\right\}\right\}$ is an orthonormal basis for the band in question.

It is generally assumed that the function $phi_\left\{mathbf\left\{R\right\}\right\}$ is localized around the point R, and rapidly goes to zero away from that point. However, quantifying and proving this assertion can be difficult, and is the subject of ongoing research.