Figure 1

Schematic spin textures in the $xy$ plane for two different vortices in a spin-$\frac{1}{2}$ system which share the same discrete combined symmetry described by $e^{i\kappa 2\pi /3}{\mathcal{C}}_z(2\pi /3)$, where the total spin rotation about the rotating axis at $O$ is $2\pi$ for (a) and $-4\pi$ for (b).Reuse & Permissions

Figure 2

Numerically calculated ground-state spin textures for three different lattice phases showing triangular (a) and kagome [(e), (g)] structures for spin-orbit-coupled pseudo-spin-$\frac{1}{2}$ BECs with $v^{\prime }=15$, where (a) $\alpha =0.5\hslash {\omega }_{\perp }/N$ and $\beta =-0.1\alpha$, (e) $\alpha =0.3\hslash {\omega }_{\perp }/N$ and $\beta =0.1\alpha$, and (g) $\alpha =0.4\hslash {\omega }_{\perp }/N$ and $\beta =0.1\alpha$. The corresponding variationally calculated spin textures obtained by a superposition of several plane waves given in Eq. (15) are shown in (b), (f), and (h). Here the spin polarizations on the $x$-$y$ plane are indicated by arrows and its $z$ component $\langle {\sigma }_z\rangle$ is shown according to the color gauge on the bottom right. The phase distributions of the (c) spin-up and (d) spin-down components for the ground state of the triangular-lattice phase in (a) are also shown according to the color gauge on the bottom left. The size of each figure is $[-a_{\perp }/2,a_{\perp }/2]\times [-a_{\perp }/2,a_{\perp }/2]$. A unit cell of the order parameter is indicated by a solid hexagon. Singular vortices indicated by open circles form a triangular lattice for all cases. Two nonsingular vortices are indicated by filled circles with blue (${\text{A}}_3$) and red (${\text{A}}_4$) colors.Reuse & Permissions

Figure 3

Spin textures of the triangular-lattice phase [(a), (b)] and the square-lattice phase [(c)–(e)] of spin-orbit-coupled spin-1 [(a), (c)] and spin-2 [(b), (d), (e)] BECs. Each subfigure shows spatial variation of the corresponding order parameter in a unit cell visualized by plotting a surface with spherical coordinates $\left(\right|\Phi (\theta ,\phi )|,\theta ,\phi )$, where $\Phi (\theta ,\phi )={\sum }_m{\xi }_m{Y}_F^m(\theta ,\phi )$, $Y_F^{M_F}$ is the spherical harmonics of rank $F$, ${\xi }_m$ is the $m$th component of the spin wave function, and the phase $\arg \Phi$ is displayed according to the right color gauge. The profile of the spin states $\xi$ (from top to bottom and left to right: spin-1 ferromagnetic, polar, spin-2 ferromagnetic, ferromagnetic, uniaxial nematic, biaxial nematic, and cyclic phases) are shown in the spherical-harmonic representation in (f). For the triangular-lattice phase, there are vortices with cores at the vertices and the center of the triangles. For the square-lattice phase, vortex cores are located at the center of the squares. One representative vortex for each type of symmetry described by $e^{i\kappa 2\pi /3}{\mathcal{C}}_z(2\pi /3)$ [$e^{i\kappa \pi /2}{\mathcal{C}}_z(\pi /2)$] is enclosed in the circle “$\circ$” and denoted by $P_{\kappa }$ [$Q_{\kappa }$].Reuse & Permissions

Figure 4

Phase distributions for individual spin components ($M_F=1,0,-1$) [(a)–(c)] and the corresponding spin textures for two unit vectors using the Majorana representation [(d), (e)] of the numerically calculated ground state showing a triangular structure of a spin-orbit-coupled pseudo-spin-1 BEC with $v^{\prime }=15$, $\alpha =0.5\hslash {\omega }_{\perp }/N$, and $\beta =0.2\alpha$. A unit cell of the order parameter is indicated by a solid hexagon. Vortices denoted by $P_0$, $P_1$, and $P_2$ are the same as those in Fig. 3a.Reuse & Permissions

Figure 5

Two different Skyrmion crystals predicted from the symmetry classification for the spin-$\frac{1}{2}$ case [24], the density distributions of the spin-up and spin-down components, and the spin textures are shown from left to right. The corresponding order parameters are given by Eq. (15) with [(a)–(c)] $\mathbb{N}=4$ and ${\varphi }_j=j\pi /2$ ($j=0,\cdots ,3$) and [(d)–(f)] $\mathbb{N}=6$ and ${\varphi }_j=j2\pi /3$ ($j=0,\cdots ,5$). A unit cell is indicated by (c) a square and (f) a hexagon, both with $Q_{\mathrm{SkX}}=-2$.Reuse & Permissions