Wannier Function - Definition

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

where has the same periodicity as the crystal. Then the Wannier functions are defined by

,

where

• R is any lattice vector (i.e., there is one Wannier function for each Bravais lattice vector);
• N is the number of primitive cells in the crystal;
• The sum on k includes all the values of k in the Brillouin zone (or any other primitive cell of the reciprocal lattice) that are consistent with periodic boundary conditions on the crystal. This includes N different values of k, spread out uniformly through the Brillouin zone. Since N is usually very large, the sum can be written as an integral according to the replacement rule:

where "BZ" denotes the Brillouin zone, which has volume Ω.