Wannier function phase factor

The Wannier functions (5.5) are not defined uniquely because each Bloch function (r) can be multiplied by an arbitrarily chosen phase factor of absolute value unity. 

The mathematics of Wannier functions involves a number of subtleties which will not be covered here. The choice of phase factors in (15), the case of overlapping or degenerate bands, and the possibility of more general transformations than (15) are some of these. Some comments on past and potential utility of the use of Wannier functions follow. 

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