Micropolar continua as projective space of Skyrmions

Figures & data

Figure 1. The transformation of the inner structure of the microelement is illustrated with centroids positioned at P and p, in the reference configuration and the spatial configuration, respectively. This shows how the directors ${\mathbf{\Xi }}_K$ in the original body in the region ${\mathbb{R}}_0$ undergoes the microdeformation under ${\chi }_{kK}$ to become ${\mathit{\xi }}_k$ while the original body experiences displacement to become the deformed configuration in the region $\mathbb{R}$ under the macroscopic displacement $\mathit{u}$.

Figure 2. Two-dimensional vortex field configurations with various integers N using (34(34) ${\mathit{n}}_{\mathrm{v}}=\left(\cos \left(N\varphi \right(x,t\left)\right),\sin \left(N\varphi \right(x,t\left)\right)\right).$(34) ) and its corresponding ${\mathit{n}}_3$ fields on $S^2$ of (42(42) ${\mathit{n}}_3=\left(\sin \theta \cos N\varphi ,\sin \theta \sin N\varphi ,\cos \theta \right),$(42) ) are shown where ${\mathit{n}}_3(N=1)$ is essentially the isotropic distribution of the hedgehog configuration ${\mathit{n}}_{\mathrm{h}}$.

Figure 3. The correspondence between $S^3$ and its projection $\mathbb{R}{P}^3$ is shown with the asymptotic values of $U\in SU\left(2\right)$ acting on the point of $S^3$. In particular, the transition of a field configuration starting from ${\mathit{n}}_{\mathrm{\infty }}$ to ${\mathit{n}}_0$ is the phase rotation from zero to $2\pi$ on $S^3$. This is a transition that is projected on the $\mathbb{R}{P}^3$ plane by bringing a point from infinity to the origin.

Figure 4. Suppose we have started from two states with identical spin orientations on the same axis on $S^3$. As one spinor configuration undergoes a transition along the great circle, separating from the initial configuration which is kept in the initial state, the spin configuration changes gradually. When the phase reaches its $2\pi$ rotation, the spin configuration becomes complete opposite.