Three-dimensional topological superstructure of magnetic hopfions threaded by meron strings in easy-plane magnets

Topological spin textures exhibit a hierarchical nature. For instance, magnetic skyrmions, which possess a particlelike 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 type of independent particle. 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 $H=\ensuremath{-}1$ 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 ${N}_{\mathrm{sk}}=2$ per magnetic unit cell on any two-dimensional cut parallel to the easy plane. We show that this hopfion superstructure is metastable against angular noise, supported by an energy barrier comparable to the dominant exchange interaction, and remains robust 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.