# Simulation Study of Dynamic Rotation and Deformation for Plasmonic Electric Field-Skyrmions
## 3 Results and Discussions
### Dynamic Deformation for Skyrmions  
We have discovered that while controlling the rotation direction of the skyrmion, we can independently deform the skyrmion. As shown in Figure 4, we can deform the skyrmion under different rotations by changing the phase difference of the incident light source. This deformation can be explained from the perspective of the skyrmions' number: skyrmions gradually become standard circles with skyrmions no longer equal to 1 or, more precisely, real numbers less than 1. The detailed results and discussions on the three-dimensional characteristics of the electric field vector of standard skyrmions and the skyrmion number \(S\) under the initialization conditions of the simulation experiment are presented in the Appendix A.  
Figure 3: The value of the squared electric field in the X–Y plane. Based on the morphology mentioned above, the rotational characteristics of the skyrmions are labeled specifically in (**a–d**). The blue dashed lines in (**a–d**) contain the generated skyrmions and their topological domain walls. The yellow dashed arrow indicates the long axis direction of the skyrmion under the deformation, and the green dashed arrow indicates the direction of the positive X-axis. The corresponding positions of regulated slits are shown in (**e–h**). Specifically, the blue dashed arrow indicates the direction of the position vector of the regulated slits, which pass through the center of the arc-shaped slit structure. Therefore, we clearly indicate in Table 1 the different conditions of the phase value that need to be set to obtain the results.  
Figure 4: _Cont._  
It is precisely because of the independence of deformation and rotation that we combined these two methods in Figure 4b and discovered the unique topological structure. At this point, the skyrmion is simultaneously affected by both horizontal and vertical rotations, exhibiting a composite rotation state of inclinations. At the same time, by gradually increasing the phase difference between the two adjusted positions, we found that the skyrmions in the synthesized rotational state were still deformed. To further analyze the rotations in Table 2, especially the acute-angle tilted state, corresponding to Figure 4b, we added the detailed instructions in the Appendix B.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Rotation State** & **Deformations** & **Adjusted Values of Phase** & **Positions of Slits** \\ \hline \multirow{4}{*}{Horizontal state \(\theta_{r}=0 rad\)} & Standard initial-state skyrmions & \(\varphi_{n}=0,(n=1,\dots,12)\) & / \\ \cline{2-4}  & Obviously deformed to the right & \(\varphi_{1}=\pi_{r},\varphi_{n}=0\), \((n=1,\dots,12,n\
eq 1)\) &  
+++ ==WARNING: Truncated because of repetitions==
4 \\ \hline \multirow{4}{*}{Acute angle tilted state \(\theta_{r}=\  
++  
The detailed simulation parameters of the results, as shown in Figure 4, are shown in Table 2. As the phase difference increases, the degree of deformation in Figure 4a,b increases (the topological domain walls gradually open, and breaks appear). By comparing Figure 4a,c, we assign precise rotational states to the skyrmion while completing its deformation. It can be seen that the skyrmion transforms from Figure 4a (the left figure) to Figure 4a (the right figure), and then rotates to Figure 4c. However, this process utilizes the principle of coordinated regulation of deformation and rotation, and we only need one step.  
With the gradual changes of phase difference, there were continuous deformations of skyrmions. Assuming that the value of \(\theta_{p}\) is \((2/3)\pi\) rad, more detailed deformations are presented in the Appendix C.