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

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.