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

TL;DR

This paper introduces a discrete geometric framework for analyzing the Hopf index in magnetic textures, revealing the existence of fractional and mixed topologies, and providing a physical basis for non-integer topological invariants.

Contribution

It presents a novel discrete geometric definition of the Hopf index for magnetic textures, enabling the identification of fractional and mixed topological states.

Findings

Fractional Hopfions can exist in magnetic systems.

The Hopf index can be non-integer or fractional in certain textures.

A physical foundation for mixed topologies in magnetic textures.

Abstract

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.