Representation of Berry phase by the trajectories of Majorana stars

TL;DR

This paper introduces a geometric approach to understanding Berry phase in high-dimensional quantum systems using Majorana star trajectories, providing a new formula and exploring entanglement relations.

Contribution

It presents a novel formula linking Berry phase to Majorana star loops and extends the approach to arbitrary-dimensional states.

Findings

Berry phase expressed via solid angles of Majorana star loops

Applicable to states with arbitrary dimensions

Relation between star correlations and quantum entanglement

Abstract

The Majorana's stellar representation, which represents the evolution of a quantum state with the trajectories of the Majorana stars on a Bloch sphere, provides an intuitive way to study a physical system with high dimensional projective Hilbert space. In this Letter, we study the Berry phase by these stars and their loops on the Bloch sphere. It is shown that the Berry phase of a general spin state can be expressed by an elegant formula with the solid angles of Majorana star loops. Furthermore, these results can be naturally used to a general state with arbitrary dimensions. To demonstrate our theory, we study a two mode interacting boson system. Finally, the relation between stars' correlations and quantum entanglement is discussed.