Variations of polarisation in external electrostatic fields

TL;DR

This paper revisits the concept of polarization variations in condensed matter, clarifying their relation to Berry phase and providing a more general derivation applicable to external electric fields, enhancing computational methods.

Contribution

It offers a generalized derivation of polarization differences that accommodates external electric fields and flexible boundary conditions, improving theoretical understanding and computational implementation.

Findings

Demonstrates polarization differences can be defined with external fields.

Provides equations suitable for modern first-principles codes.

Clarifies the relation between polarization and Berry phase.

Abstract

It is nowadays a quite diffuse idea that variations of polarisation in condensed matter theory are related to a "Berry phase". The derivation of the latter geometric phase is correct $\it{only if}$ the restrictive periodic gauge\cite{KS-V} is used for its derivation, aside giving rise to a "quantum of polarisation" that is of difficult theoretical interpretation. Its derivation has not been demonstrated in the general case of an external homogeneous electric field interacting with the electronic field. In the present paper we give a brief derivation of the polarisation differences in a general manner than in\cite{KS-V} allowing for a general boundary conditions freedom showing that is possible to define it even in the presence of an external electric field interacting with the electronic field and writing equations in a way directly implementable in modern first principles codes.