Geometric phases and Wannier functions of Bloch electrons in 1-dimension

We present a formal expression for Wannier functions of composite bands of 1-D 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 1-D model insulator and metal. Within first-principles density functional theory, we illustrate the formalism through the construction of Wannier functions of polyethylene and polyacetylene.