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

TL;DR

This paper develops a formalism for constructing Wannier functions of 1-D Bloch electrons using geometric phases, analyzing their decay properties and applying the method to real materials like polyethylene and polyacetylene.

Contribution

It introduces a new formal expression for Wannier functions in 1-D systems based on non-Abelian geometric phases, with practical applications to real materials.

Findings

Wannier functions can be expressed via parallel-transported Bloch functions and geometric phases.

Decay properties of Wannier functions are analyzed for simple 1-D insulators and metals.

Application to polyethylene and polyacetylene demonstrates the formalism's practical utility.

Abstract

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.