Fig. 1. Ground state of the rotating ferromagnetic BEC of ⁸⁷Rb for the different spin-orbit coupling strengths under the toroidal trap. The first, second and third columns show the particle number densities. The fourth, fifth and sixth columns show phase distributions. Changing the strength of spin-orbit coupling can control the number of half-skyrmion in the system and the symmetry of half-skyrmion with circular distribution. The parameters are set as follows: (a) ; (b) ; (c) ; (d) . And the rest of parameters are , , , , and ω = 2π × 250 Hz.

Fig. 2. Effects of the different rotation frequency on ground state. With the increase of rotation frequency, the system transforms from plane wave phase to half-skyrmion chain phase with circular symmetry arrangement. The first, second and third columns are the particle number densities. The fourth, the fifth and the sixth columns are the corresponding phase distributions. The parameters are set as follows: (a) ; (b) ; (c) ; (d) . And the other parameters are , , , , and ω = 2π × 250 Hz.

Fig. 3. Effects of the different spin-independent and spin-dependent interactions on ground state. Increasing the strength of spin-independent interaction can induce the transition of half-skyrmion distribution from a circular monolayer arrangement to a circular bilayer arrangement. The influence of different spin-dependent interaction on the number of half-skyrmion is weak, which only makes the ring arrangement of half-skyrmion more regular. The parameters are set as follows: (a1) , ; (a2) 4600, ; (b1) , ; (b2) , . And the other parameters are , , , and ω = 2π × 250 Hz.

Fig. 4. Effects of the width and the central height of the toroidal potential on ground state. The ring distribution of half-skyrmion chain can be controlled by changing the width and height of potential well. The particle number densities of ground state are shown in the first, second and third columns. The corresponding phase distributions are shown in the fourth, fifth and sixth columns. The parameters are set as follows: (a1) , ; (a2) , ; (b1) , ; (b2) , . And the other parameters are , , , and ω = 2π × 250 Hz.

Fig. 5. The spin texture of the ground state: (a) Spin texture corresponding to the Fig. 2(a); (b) spin texture corresponding to the Fig. 1(b); (c) spin texture corresponding to the Fig. 2(c); (d) spin texture corresponding to the Fig. 2(d). The circle region in the graph represents a half-skyrmion structure. Values of the spin density are from –1 (blue) to 1 (red).