Construction of Hopfion Crystals

Abstract

Hopfions, three-dimensional topological solitons characterized by nontrivial Hopf indices, represent a fundamental class of field configurations that emerge across diverse areas of physics. Despite extensive studies of isolated hopfions, a framework for constructing spatially ordered arrays of hopfions, i.e., hopfion crystals, has been lacking. Here, we present a systematic approach for generating hopfion crystals with cubic symmetry by combining the Hopf map with rational mapping techniques. By superposing helical waves in $\mathbb{R}^4$, we construct hopfion crystals with tunable Hopf indices and controllable topology. We demonstrate simple cubic, facecentered cubic, and body-centered cubic hopfion crystals, and extend our framework to create crystals of more complex topological structures, including axially symmetric tori, torus links, and torus knots with higher Hopf indices. Our results provide a foundation for searching hopfions in real materials and studying their collective phenomena.

Figures (6)

• Figure 1: Two mapping pathways from $\mathbb{R}^3$ to $\mathbb{S}^2$. The lower one is the standard construction for a single hopfion, and the upper one is the path for hopfion crystals. $sp$ stands for sterographic projection, and $rp$ stands for rational map.
• Figure 2: $Q_{H}=1$ simple cubic hopfion crystal. The left figure is the simple cubic $Q_{H}=1$ hopfion crystal with $c=2$ and $x,y,z$ are from $-2\pi$ to $2\pi$. The right top figure indicates the the relationships between colors and the values of $\arctan(n_{y}/n_{x})$ and $n_z$. The right bottom figure is the spin texture of $z=\pi$ plane.
• Figure 3: Hopfion crystal structures with differnet $c$ values. (a) $c=-0.5$ hopfion crystal structures with the spin textures at $z=\pi$ and $z=2\pi$ planes. (b) $c=0$ hopfion crystal structures with the spin textures at $z=\pi$ and $z=2\pi$ planes. (c) $c=-0.5$ hopfion crystal structures with the spin textures at $z=\pi$ and $z=2\pi$ planes.
• Figure 4: (a) $Q_{H}=1$ face-centered cubic hopfion crystal. (b) $Q_{H}=1$ body-centered cubic hopfion crystal. We make $c=2$ for both of them.
• Figure 5: The simple cubic structures of $(m,n)$-tori. (a) $(m,n)=(1,2)$. (b) $(m,n)=(2,1)$. (c) $(m,n)=(3,2)$. The left figure of each one is the simple cubic structure of $(m,n)-$torus. The right top figure is the equi-spin contour lines of one torus in the simple cubic structure and the right bottom figure indicates how the spin fields wind in the corresponding structure.