Analytic Properties of Bloch Waves and Wannier Functions

TLDR

The study examines one‑dimensional Schrödinger equations with periodic symmetric potentials, focusing on Bloch waves and energy bands as analytic, periodic functions of complex wave‑vector k that possess branch points off the real axis, which influence the properties of Wannier functions. The authors aim to provide a simple procedure for locating the branch points of these analytic functions. They apply this procedure to the power‑series and Fourier‑series expansions of the Bloch waves and energy bands. They show that each band admits a unique Wannier function that is real and either symmetric or antisymmetric under reflection, decaying exponentially with distance, and that the decay rate is set by the distance of the branch points from the real axis.

Abstract

The one-dimensional Schr\"odinger equation with a periodic and symmetric potential is considered, under the assumption that the energy bands do not intersect. The Bloch waves, ${\ensuremath{\phi}}_{n,k}$, and energy bands, ${E}_{n,k}$, are studied as functions of the complex variable, $k$. In the complex plane, they are branches of multivalued analytic and periodic functions, ${\ensuremath{\phi}}_{k}$, and ${E}_{k}$, with branch points, ${k}^{\ensuremath{'}}$, off the real axis. A simple procedure is described for locating the branch points. Application is made to the power series and Fourier series developments of these functions. The analyticity and periodicity of ${\ensuremath{\phi}}_{n,k}$ has some consequences for the form of the Wannier functions. In particular, it is shown that for each band there exists one and only one Wannier function which is real, symmetric or antisymmetric under an appropriate reflection, and falling off exponentially with distance. The rate of falloff is determined by the distance of the branch points ${k}^{\ensuremath{'}}$ from the real axis.