Wannier function
The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier. Wannier functions are the counterpart of localized molecular orbitals for crystalline systems.
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; the existence of exponentially localized Wannier functions in insulators has been proved in 2006. Specifically, these functions are also used in the analysis of excitons and condensed Rydberg matter.
Although, like localized molecular orbitals, 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 uk(r) has the same periodicity as the crystal. Then the Wannier functions are defined by
where
where "BZ" denotes the Brillouin zone, which has volume Ω.
On the basis of this definition, the following properties can be proven to hold:
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
where the sum is over each lattice vector R in the crystal.
Wannier functions have been extended to nearly periodic potentials as well.
The Bloch states ψk(r) are defined as the eigenfunctions of a particular Hamiltonian, and are therefore defined only up to an overall phase. By applying a phase transformation eiθ(k) to the functions ψk(r), for any (real) function θ(k), one arrives at an equally valid choice. While the change has no consequences for the properties of the Bloch states, the corresponding Wannier functions are significantly changed by this transformation.
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