Magnetic hopfions in solids

APL Materials, vol. 10

Abstract

Hopfions are an intriguing class of string-like solitons, named according to a classical topological concept classifying three-dimensional direction fields. The search for hopfions in real physical systems has been ongoing for nearly half a century, starting with the seminal work of Faddeev. However, so far, realizations in bulk solids are missing. Here, we show that hopfions appear as emergent particles of the classical Heisenberg model with competing exchange interactions. This requires going beyond the model approach used in prior work and deriving a general micromagnetic energy functional directly from a spin-lattice Hamiltonian. We present a definite parameter space in which the existence of hopfions is possible. This opens a concrete vista to combine computational approaches such as density functional theory with material informatics to find magnetic crystals that can host hopfions. As proof of principle, we show how zero-field hopfions can be visualized by the means of off-axis electron holography in a transmission electron microscope.

Authors 6

1. Filipp N. Rybakov corresponding

   KTH Royal Institute of Technology · Uppsala University

   Affiliation as printed

   Department of Physics and Astronomy, Uppsala University 1 , SE-75120 Uppsala, Sweden

   Department of Physics, KTH Royal Institute of Technology 2 , SE-10691 Stockholm, Sweden

2. Nikolai S. Kiselev

   Forschungszentrum Jülich · Jülich Aachen Research Alliance

   Affiliation as printed

   Peter Grünberg Institut and Institute for Advanced Simulation 3 , Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany

3. Aleksandr Borisovich Borisov
   Affiliation as printed

   Independent Researcher 4 , 68b Amundsen Street, Ekaterinburg 620146, Russia

4. Lukas Döring Aachen Department of Mathematics and JARA FIT

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics and JARA FIT, RWTH Aachen University 5 , 52056 Aachen, Germany

5. Christof Melcher Aachen Department of Mathematics and JARA FIT

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics and JARA FIT, RWTH Aachen University 5 , 52056 Aachen, Germany

6. Stefan Blügel

   Forschungszentrum Jülich · Jülich Aachen Research Alliance

   Affiliation as printed

   Peter Grünberg Institut and Institute for Advanced Simulation 3 , Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany