Monopole and Berry Phase in Momentum Space in Noncommutative Quantum Mechanics

TL;DR

This paper explores the connection between noncommutative quantum mechanics, monopoles, and Berry phase in momentum space, proposing a dual Dirac monopole interpretation linked to the anomalous Hall effect.

Contribution

It introduces the necessity of extending the noncommutative parameter to a momentum-dependent field, identified as a dual Dirac monopole, connecting noncommutative geometry with Berry curvature.

Findings

Noncommutative parameter extended to a momentum-dependent field.

Identification of this field as a dual Dirac monopole.

Proposed link between noncommutative field and Berry curvature in momentum space.

Abstract

To build genuine generators of the rotations group in noncommutative quantum mechanics, we show that it is necessary to extend the noncommutative parameter $\theta $ to a field operator, which one proves to be only momentum dependent. We find consequently that this field must be obligatorily a dual Dirac monopole in momentum space. Recent experiments in the context of the anomalous Hall effect provide for a monopole in the crystal momentum space. We suggest a connection between the noncommutative field and the Berry curvature in momentum space which is at the origine of the anomalous Hall effect.