A unique one-body position operator for periodic systems

TL;DR

This paper proves the uniqueness of a recently proposed one-body position operator for periodic systems, ensuring it satisfies key physical constraints and is compatible with periodic boundary conditions.

Contribution

It provides a rigorous proof that the proposed position operator is unique up to a phase factor and additive constant, under general physical constraints.

Findings

The operator is unique modulo a phase and constant.

The proof confirms compatibility with periodic boundary conditions.

The approach uses fundamental physical constraints.

Abstract

In this work we proof that the one-body position operator for periodic systems that we have recently proposed [Phys. Rev. B 99, 205144] is unique modulo a phase factor and an additive constant. The proof uses several general physical constraints that a periodic one-body position operator should satisfy. We show that these constraints are sufficient to uniquely define a position operator that is compatible with periodic boundary conditions.