Figure 1

Unit cell for the minimal-energy crystalline spin texture. Axes are in units of the lattice constants $a_{\parallel }=90$ $\mu$m, $a_{\perp }=42$ $\mu$m. Green vertical lines indicating glide reflection lines, purple horizontal lines indicating mirror lines, and white spheres indicating rotation centers describe symmetry operations. See Fig. 2 for a depiction of symmetry operations. Red background (light gray, center of figure) indicates positive skyrmion density $q$, while blue background (dark gray, left and right edges) indicates negative $q$. Black two-dimensional (2D) arrows indicate the superfluid velocity $\mathbf{v}$, and shaded 3D arrows the magnetization $\mathrm{n̂}$. The magnetic field $\mathrm{B̂}=\mathrm{x̂}$ inducing Larmor precession lies along the horizontal axis in the plane in real space. In spin space, white (black) 3D arrows point along $+\mathrm{B̂}$ ($-\mathrm{B̂}$).Reuse & Permissions

Figure 2

(a)–(d) illustrate spin group operations that combine nontrivial spin-space and real-space actions. One element of the spin group acts as a real-space reflection about the thick purple horizontal mirror line combined with spin-space reflection ${\mathrm{n̂}}_{\parallel }\to -{\mathrm{n̂}}_{\parallel }$ of the component along $\mathrm{B̂}$. Recall that 3D arrows indicate the magnetization $\mathrm{n̂}$ with white (black) 3D arrows pointing along $+\mathrm{B̂}$ ($-\mathrm{B̂}$). This action of this operation on the back panel is shown in the front panel of (a). Notice that spins pointing along $-\mathrm{B̂}$ below the thick purple line map to those along $+\mathrm{B̂}$ above. Those on the thick horizontal purple line are mapped to themselves and are perpendicular to $\mathrm{B̂}$ in spin space. (b) shows a vertical translation followed by reflection along a glide mirror line (thick green vertical line) combined with ${\mathrm{n̂}}_{\perp ,1}\to -{\mathrm{n̂}}_{\perp ,1}$, ${\mathrm{n̂}}_{\parallel }\to -{\mathrm{n̂}}_{\parallel }$. (c) $\pi$ rotation about a rotation center (large white sphere) combined with ${\mathrm{n̂}}_{\perp ,1}\to -{\mathrm{n̂}}_{\perp ,1}$. (d) Horizontal translation combined with ${\mathrm{n̂}}_{\perp ,1}\to -{\mathrm{n̂}}_{\perp ,2}$ where ${\mathrm{n̂}}_{\perp ,1}$,${\mathrm{n̂}}_{\perp ,2}$ are the two components perpendicular to $\mathrm{B̂}$.Reuse & Permissions

Figure 3

Momentum-space structure factor $\mathrm{n̂}(k)$ for the minimal-energy crystalline spin texture of Fig. 1. The magnetic field $\mathrm{B̂}$ is along the horizontal axis while the axes are in units of the inverse wave vector with lattice constants $a_{\parallel }=90$ $\mu$m, $a_{\perp }=42$ $\mu$m. The area of dark gray (blue) [light gray (red)] disks is proportional to the magnitude of components parallel (perpendicular) to $\mathrm{B̂}$.Reuse & Permissions

Figure 4

Brillouin zone for a space group SG with rectangular Bravais lattice $T_{\mathrm{rect}}$ and $\mathrm{SG}/T_{\mathrm{rect}}=D_2$ point group. Each wave vector $k$ in the Brillouin zone describes an inequivalent and irreducible representation ${\varphi }_k$ of the translation group $T_{\mathrm{rect}}$. The little group SG${}_k$ consists of elements of SG that leave ${\varphi }_k$ fixed under conjugation. The wave vector point group ${\mathrm{SG}}_k/T_{\mathrm{rect}}$ is the trivial group $C_1$ for a generic wave vector in the Brillouin zone, $D_1$ on high-symmetry lines, and $D_2$ on high-symmetry points.Reuse & Permissions