标题： 易平面磁体中由梅隆弦编织的三维拓扑超结构的磁荷子结构 显示英文标题

标题： Three-dimensional Topological Superstructure of Magnetic Hopfions Threaded by Meron Strings in Easy-plane Magnets

摘要： 拓扑自旋纹理表现出层次性。 例如，具有类似粒子性质的磁skyrmions可以聚集成如skyrmion弦和skyrmion晶格等超结构。 磁Hopfions也被视为由扭曲的skyrmion弦闭合环构成的超结构，它们作为另一种独立的粒子行为。 然而，这种磁Hopfions是否也能聚集成更高级的超结构仍然不明确。 在这里，我们报告了一种稳定的超结构，在具有易平面各向异性的受挫自旋模型中，磁Hopfions具有三维周期排列。 通过全面检查两个Hopfions之间的有效相互作用，我们通过将一维Hopfion链以Hopf数$H=+1$和$H=-1$的交错排列方式构建Hopfion超结构，这些链垂直于易平面排列。 每个Hopfion链被磁meron弦穿过，在任何平行于易平面的二维切片上，每个磁单位晶胞都具有非平凡的拓扑纹理，其skyrmion数为$N_{\rm sk}=2$。 我们表明，Hopfion超结构在Hopfion密度的一定范围内保持稳定，作为亚稳态。 此外，我们证明具有更高Hopf数的超结构也可以被稳定。 我们的发现扩展了现有拓扑磁体的层次结构，并为探索新的量子现象和自旋动力学铺平了道路。 显示英文摘要

摘要： Topological spin textures exhibit a hierarchical nature. For instance, magnetic skyrmions, which possess a particle-like nature, can aggregate to form superstructures such as skyrmion strings and skyrmion lattices. Magnetic hopfions are also regarded as superstructures constructed from closed loops of twisted skyrmion strings, which behave as another independent particles. However, it remains elusive whether such magnetic hopfions can also aggregate to form higher-level superstructures. Here, we report a stable superstructure with three-dimensional periodic arrangement of magnetic hopfions in a frustrated spin model with easy-plane anisotropy. By comprehensively examining effective interactions between two hopfions, we construct the hopfion superstructure by a staggered arrangement of one-dimensional hopfion chains with Hopf number $H=+1$ and $H=-1$ running perpendicular to the easy plane. Each hopfion chain is threaded by a magnetic meron string, resulting in a nontrivial topological texture with skyrmion number $N_{\rm sk}=2$ per magnetic unit cell on any two-dimensional cut parallel to the easy plane. We show that the hopfion superstructure remains robust as a metastable state across a range of the hopfion density. Furthermore, we demonstrate that superstructures with higher Hopf number can also be stabilized. Our findings extend the existing hierarchy of topological magnets and pave the way for exploring new quantum phenomena and spin dynamics.