Unified Geometric Perspective for Spin-1 Systems: Bridging Nematic Director and Majorana Stars

TL;DR

This paper introduces a unified geometric framework for spin-1 systems that links various representations and enables new insights into magnetic soliton dynamics and transitions.

Contribution

It provides a novel geometric approach connecting different spin-1 representations and maps soliton dynamics onto a kink model, enhancing understanding of their behavior.

Findings

Unified geometric representation for spin-1 systems

Mapping of magnetic solitons to a kink model

Discovery of geometric transitions in soliton dynamics

Abstract

We present a unified geometric approach for spin-1 systems that connects seemingly distinct geometric representations such as the nematic director, the Cartesian representation and the Majorana stellar representation. Starting from a product state of two distinguishable spin-1/2 particles, we provide a direct way to capture crucial geometric information. This perspective reveals the fundamental interplay between subspace projection and geometric constraints. This approach effectively maps magnetic solitons onto a kink model, allowing us to derive their equations of motion, a task not readily achieved with traditional methods. This simplified dynamical description reveals that the novel transition of these solitons in a harmonic trap corresponds to a fundamental transformation between kink and dip structures in the underlying geometry.