标题： 具有混合拓扑的三维磁性纹理：解锁可调节的霍普夫指标 显示英文标题

标题： 3D Magnetic Textures with Mixed Topology: Unlocking the Tunable Hopf Index

摘要： 绳结和链环在理解拓扑性和自然中的离散性中起着关键作用。 在磁系统中，扭曲、打结和编织的涡旋管表现为Skyrmions、Hopfions或螺旋位错。 这些复杂的纹理由拓扑非平凡的量来表征，例如Skyrmion数、Hopf指标$H$、Burgers矢量（由整数$\nu$量化）和链接数。 在本工作中，我们为周期性磁纹理引入了$H$的离散几何定义，该定义可以分离出自旋链接和互旋链接的贡献。 我们表明，分数Hopfions或具有非整数值的$H$的纹理自然出现，并可解释为“混合拓扑”状态，这些状态可以连续变换到多个可能的拓扑区域中的一个。 我们的发现证明了Hopf指标可以根据系统的底层拓扑取整数、非整数或特定分数值的坚实物理基础。 显示英文摘要

摘要： Knots and links play a crucial role in understanding topology and discreteness in nature. In magnetic systems, twisted, knotted and braided vortex tubes manifest as Skyrmions, Hopfions, or screw dislocations. These complex textures are characterized by topologically non-trivial quantities, such as a Skyrmion number, a Hopf index $H$, a Burgers vector (quantified by an integer $\nu$), and linking numbers. In this work, we introduce a discrete geometric definition of $H$ for periodic magnetic textures, which can be separated into contributions from the self-linking and inter-linking of flux tubes. We show that fractional Hopfions or textures with non-integer values of $H$ naturally arise and can be interpreted as states of ``mixed topology" that are continuously transformable to one of the multiple possible topological sectors. Our findings demonstrate a solid physical foundation for the Hopf index to take integer, non-integer, or specific fractional values, depending on the underlying topology of the system.