Lattice representations in solids

A Bravais-lattice operator is defined in one band of a solid. Its eigenstates are covariantly defined Wannier functions and its eigenvalues are all the points of the Bravais lattice. This operator establishes a convenient phase convention for Bloch functions. The newly defined Bravais-lattice operator is conjugate to the quasimomentum and together they form a complete set of operators by means of which any one-band operator can be expressed. The Wannier functions for different bands and sites are shown to be eigenfunctions of a band index and the Bravais-lattice operators. It is shown that the one-band position operator has a discrete spectrum with the structure of a Stark ladder in solids. A $\mathrm{kq}$ representation is defined for one band which leads to symmetric coordinates for superlattices. The conjugate operators to these symmetric coordinates form the superlattice representation of McIrvine and Overhauser.