Geometry of projected connections, Zak phase, and electric polarization

TL;DR

This paper offers a geometric interpretation of the Zak phase using projected connections, linking it to electric polarization and adiabatic currents in periodic systems, enhancing the understanding of polarization in quantum materials.

Contribution

It introduces a consistent geometric framework for the Zak phase via projected connections and clarifies its relation to electric polarization and Wannier functions.

Findings

Zak phase interpreted as a projected connection

Relation between Berry potential transformation and polarization currents

Wannier functions describe adiabatic current contributions

Abstract

The concept of the Zak phase lies at the core of the modern theory of electric polarization. It is defined using the components of the Bloch wave functions in a certain basis, which is not captured by the standard expression for Berry potential. We provide a consistent geometric interpretation of the Zak phase in terms of projected connections. In the context of Bloch states, we relate the transformation law of projected Berry potential with classical currents that contribute to the time derivative of the electric polarization. This gives a new argument for the Zak phase formula for the electronic contribution to the polarization. We demonstrate that the Wannier functions play a key role in the description of an adiabatic current in a periodic system.