Geometric currents in piezoelectricity

TL;DR

This paper models piezoelectricity using a gas of non-interacting electrons in a periodic potential, demonstrating that the macroscopic current in the adiabatic limit is governed by the geometry of the Bloch bundle, and deriving the King-Smith and Vanderbilt formula with high precision.

Contribution

It establishes a geometric framework for piezoelectricity and rigorously derives the King-Smith and Vanderbilt formula in the adiabatic limit.

Findings

Macroscopic current determined by Bloch bundle geometry

Derivation of King-Smith and Vanderbilt formula with negligible errors

Validation of geometric approach in piezoelectricity modeling

Abstract

As a simple model for piezoelectricity we consider a gas of infinitely many non-interacting electrons subject to a slowly time-dependent periodic potential. We show that in the adiabatic limit the macroscopic current is determined by the geometry of the Bloch bundle. As a consequence we obtain the King-Smith and Vanderbilt formula up to errors smaller than any power of the adiabatic parameter.