Magnetic hopfions in solids (1904.00250v1)

Abstract: Hopfions are an intriguing class of string-like solitons, named according to a classical topological concept classifying three-dimensional direction fields. The search of hopfions in real physical systems is going on for nearly half a century, starting with the seminal work of Faddeev. But so far realizations in solids are missing. Here, we present a theory that identifies magnetic materials featuring hopfions as stable states without the assistance of confinement or external fields. Our results are based on an advanced micromagnetic energy functional derived from a spin-lattice Hamiltonian. Hopfions appear as emergent particles of the classical Heisenberg model. Magnetic hopfions represent three-dimensional particle-like objects of nanometre-size dimensions opening the gate to a new generation of spintronic devices in the framework of a truly three-dimensional architecture. Our approach goes beyond the conventional phenomenological models. We derive material-realistic parameters that serve as concrete guidance in the search of magnetic hopfions bridging computational physics with materials science.

Overview of "Magnetic Hopfions in Solids"

The paper "Magnetic Hopfions in Solids" presents a theoretical framework for the existence of magnetic hopfions as stable solitonic structures in solid-state systems, using an advanced micromagnetic energy functional derived from a classical spin-lattice Hamiltonian. This paper fills a gap in the condensed matter physics domain by proposing hopfions as stable entities in specific magnetic materials, without the need for external stabilizing factors like confinement or applied fields.

Key Findings and Contributions

The authors focus on identifying conditions under which hopfions—a type of three-dimensional topological soliton traditionally elusive in condensed matter systems—could exist and remain stable. The framework builds on a sophisticated micromagnetic model that captures the essence of competitive exchange interactions on a lattice. These interactions give rise to energy configurations that acknowledge the requirement for higher order stabilization terms analogous to those in the Skyrme model, which are absent in conventional micromagnetic functionals.

• Energy Functional and Stability: The paper proposes an energy functional that encompasses higher-order gradient terms, derived from a microscopic spin-lattice Hamiltonian. This formulation allows for stability of hopfions without external fields, contrasting with traditional models which don't typically account for such stabilizer terms.
• Topological Nature: Hopfions are characterized by a topological invariant known as the Hopf index. The paper demonstrates the fractional power law governing the minimum energy scaling with the Hopf index, $E \sim |H|^{3/4}$. This sublinear scaling implies that minimal energy states exist across multiple homotopy classes, suggesting the presence of stable solutions with varied indices.
• Numerical Insights: Numerical simulations validate the existence of stable hopfions for specific exchange parameters, offering a spectrum of such solitons with different morphological characteristics in three-dimensional lattices. These findings support the hypothesized stability criteria based on the effective micromagnetic parameters.

Implications and Future Directions

This research suggests a paradigm where magnetic hopfions may serve as foundational components in developing new types of spintronic devices. Given their distinct topological properties and three-dimensional characteristics, hopfions could enable the creation of advanced data storage and processing devices with true three-dimensional architectures.

• Spintronics and Information Technology: Hopfions, due to their emergent electromagnetic fields, offer potential as information carriers, with applications in next-generation memory and neuromorphic computing. Their unique responses to spin currents further underscore the possibility of controlling hopfion motions through electrical manipulation.
• Materials Design and Discovery: The paper advocates a computational material science approach, supported by DFT, to guide the experimental search for materials hosting stable hopfions. This approach bridges quantum mechanical models with macroscopic materials science, aiding in the tailored design of candidate materials.

Conclusion

The theoretical and numerical findings reported in this paper provide a robust foundation for the potential realization of magnetic hopfions in condensed matter systems. As experimental methodologies such as X-ray magnetic tomography evolve, the practical detection and application of hopfions could revolutionize the design and function of technology platforms reliant on magnetic information carriers. The pursuit of hopfions in materials science thus represents an intriguing and promising frontier in the advancement of spintronic technologies.