标题： Hopfion 晶体的构造 显示英文标题

标题： Construction of Hopfion Crystals

摘要： Hopf子（Hopfions），即由非平凡Hopf指数表征的三维拓扑孤立子，代表了物理学多个领域中出现的一类基本场构型。尽管孤立Hopf子的研究已经很广泛，但构建空间有序阵列的Hopf子（即Hopf子晶体）的框架一直缺失。在这里，我们提出了一种系统的方法，通过结合Hopf映射与有理映射技术，生成具有立方对称性的Hopf子晶体。通过叠加$\mathbb{R}^4$中的螺旋波，我们可以构造具有可调Hopf指数和可控拓扑的Hopf子晶体。我们展示了简单的立方、面心立方和体心立方的Hopf子晶体，并将我们的框架扩展到创建更复杂拓扑结构的晶体，包括具有更高Hopf指数的轴对称环面、环面链和环面纽结。我们的研究为在真实材料中寻找Hopf子并研究它们的集体现象奠定了基础。 显示英文摘要

摘要： 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.