Geometric phases and Wannier functions of Bloch electrons in one dimension

We derive a formal expression for Wannier functions of composite bands of one-dimensional (1D) Bloch electrons in terms of parallel-transported Bloch functions and their non-Abelian geometric phases. Spatial decay properties of these Wannier functions are studied in the case of simple bands of 1D model insulators and metals. Within first-principles density functional theory, we illustrate the formalism through the construction of Wannier functions of polyethylene and polyacetylene. Maximal localization of these functions arises naturally from the properties of parallel-transported Bloch functions.