Hopfion in magnetic solids -- Theory, Stability, Material, Experiment

Topological magnetization texturonics, a spintronics that makes use of two (2D)- and three-dimensional (3D) topological magnetisation textures on the nanometer scale and localized in space with particle like properties, such as skyrmions is a very exciting, timely and multidisciplinary field, where mathematicians, theoretical and computational physicists meet with experts on film and multilayer growth, on high-resolution magnetization and electronic structure characterization, transport and dynamics, and on device concepts and devices in order to integrate these particles into the field of spintronics. 3D magnetization textures in bulk solids such as twisted skrymion strings [1], hybrid particles like bobbers [2-4] or hopfions [5, 6] open a novel dimension. It is very fascinating to observe that simple spin models, which can be determined for real materials by density functional theory calculations, lead to micromagnetic equations whose solutions are localized topological textures in an homogeneous solid can now be studied by experimental methods, e.g. by means of quantitative off-axis electron holography [1, 4, 6]. I will give an elementary overview about this field, but then focus on hopfions in bulk solids, introduce the primary interactions that are responsible for its existence. In this presentation I report on our advanced micromagnetic theory [7] in which we derived analytical conditions of Heisenberg exchange parameters under which isolated Hopfions can be expected. We combine atomistic spin-dynamics with harmonic transition state theory [8] to investigate the lifetime of hopfions and related decay mechanisms. I report on our recent experimental observation of hopfion rings on skyrmions strings [4] and comment on its impact on the extension of the classification from topological groups to product groups.