Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets

Abstract

Magnetic hopfions are three-dimensional topological solitons embedded into a homogeneously magnetized background. The internal structure of hopfions is distinguished by the linked preimages—closed loops with a single orientation of the magnetization on the target space S2—and is thus characterized by the integer Hopf index QH. Alternatively, hopfions can be visualized as a result of the swirling of two-dimensional bimerons around the direction of an applied magnetic field. Since the bimeron consists of a circular core and an anti-skyrmion crescent, two hopfion varieties can be achieved with either bimeron constituent facing the hopfion interior. In bulk cubic helimagnets, however, the applied magnetic field leads to a spontaneous collapse of hopfions, i.e., the eigen-energy of hopfions has the minimum for zero hopfion radius R. Anti-hopfions with QH=−1, in this case, pass through the intermediate toron state with two-point defects. Here, we demonstrate that the competing cubic and exchange anisotropies inherent in cubic non-centrosymmetric magnets (e.g., in the Mott insulator Cu2OSeO3) as a third level of the hierarchy of energy scales following the exchange and Dzyaloshinskii–Moriya interactions, may shift the energy minimum into the region of finite hopfion radii.