The Wannier Function is the Fourier transform of the bloch functions. Since bloch function have a gauge freedom, that one can choose a phase function that is periodic in K space and won’t change the bloch function’s functionality, so the wanneir function can be formed in varies ways.

Usually the phase function is chosed to be smooth enough, so that its fourier transform - Wannier Function, will be at most localized in real space.

The workflow looks like this:

First, we isolate the fitting band that we want to represent using wannier functions.

Second, the projection matrix of each k is constructed. We know the projection matrix is invariant when represented using different bloch function.

Third, the WF is constructed from the bloch wave function of the selected band region, one WF for each band.

Wannier By Projection:

One way to find a phase function that make bloch function smooth in k space is constructing a set of projection from well localized orbitals in real space. E.G. LCAO, then it is projected by the kspace projection operator constructed by selecting the index of bands we want, like:

After the projection, we can get a bloch orbital in k space. (why? what garanttee this? Try to derive this out from the projection formula of some localized basis.)

Then, the smooth bloch orbital can be transform to WFs using Fourier transform.