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

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.

Figures (2)

• Figure 1: (a) Mapping a spin-1 state onto two Majorana stars, denoted as $\mathbf{P}_1$ and $\mathbf{P}_2$, on the Bloch sphere. The complete information of the spin-1 state is encoded in two key vectors: the average vector $\vec{\beta}$ (represented by the red arrow) and the relative vector $\vec{\gamma}$ (represented by the blue arrow). (b) The relative information can alternatively be expressed as a "vector" $\vec{\alpha}$. However, it's important to note that the $\vec{\alpha}$ vector transforms twice as fast as the actual relative vector $\vec{\gamma}$ under spin-rotation along $\vec{F}$. This $\vec{\alpha}$ vector effectively represents the expectation values of pseudo-spin operators with the second type subspace projection.
• Figure 2: (a) The trajectory of a magnetic soliton within a harmonic trap with a quadratic Zeeman energy of $q=0.19g_s n_b^{\text{peak}}$, where $n_b^{\text{peak}}$ is the peak density of ground state. The trajectory and depicted transition were originally presented in Ref. RelativePhaseDomainwall. The blue dashed lines indicate the transition points where the magnetic soliton changes between $0\pi$ and $2\pi$ type. We have selected three points- b1,b2 and b3- to highlight the geometric structure corresponding to the 0$\pi$ state, the transition point, and the $2\pi$ state, respectively.Their geometric configurations are detailed in figures (b1), (b2), and (b3). Since our simulations align the magnetic field along $x$-axis, we consider $\Gamma_y$, $\Gamma_z$and $S_x$ as the complete set of physical quantities. For the 0$\pi$ state shown in (b1), $\Gamma_z$ exhibits a small dip. In contrast, for the $2\pi$ state in (b3), $\Gamma_z$ forms a kink. At the transition point (b2), specifically at the soliton core, both $\Gamma_z$,$\Gamma_y$ become zero. This allows for a transformation where $\vec{\Gamma}$ on one side of the magnetic soliton effectively flips to $-\vec{\Gamma}$ on one sides of magnetic soliton (denoted by transparent lines), while maintaining the continuity of $\vec{\Gamma}$.This means that at the transition point, the dip and kink geometric configurations are equivalent under this transformation.