Families of Skyrmions in Two-Dimensional Spin-1/2 Systems

• Published in: IEEE journal of selected topics in quantum electronics 2020-11, Vol.26 (6), p.1-11
• Main Authors: Javed, Amaria, Al Sakkaf, Laila, Al Khawaja, Usama
• Format: Article
• Language: English
• Subjects: Biomedical optical imaging; Hypothetical particles; Manakov system; Nodes; Numerical analysis; Numerical methods; Numerical stability; Optical pulses; Optical waveguides; Particle theory; Power series; Skyrmions; Spin-1/2 system; Stability analysis; Two dimensional displays; Wave functions
• ISSN: 1077-260X; 1558-4542
• DOI: 10.1109/JSTQE.2020.2994555

We find Skyrmion-like topological excitations for a two-dimensional spin-1/2 system. Expressing the spinor wavefunction in terms of a rotation operator maps the spin-1/2 system to a Manakov system. We employ both analytical and numerical methods to solve the resulting Manakov system. Using a generalized similarity transformation, we reduce the two-dimensional Manakov system to the integrable one-dimensional Manakov system. Solutions obtained in this manner diverge at the origin. We employ a power series method to obtain an infinite family of localized and nondiverging solutions characterized by a finite number of nodes. A numerical method is then used to obtain a family of localized oscillatory solutions with an infinite number of nodes corresponding to a skyrmion composed of concentric rings with intensities alternating between the two components of the spinor. We investigate the stability of the skyrmion solutions found here by calculating their energy functional in terms of their effective size. It turns out that indeed the skyrmion is most stable when the phase difference between the concentric rings is \pi, i.e., alternating between spin up and spin down. Our results are also applicable to doubly polarized optical pulses.