fig1

Figure 1. 3D topological spin textures of hopfions and torons. The set of preimages with s_z = 0 for hopfions at 0 T (A) and toron at 0.2 T (B) in a square domain with side length L = 150 nm and thickness t = 90 nm. The preimage refers to a mapping of the 3D space that contains spins with the same orientations. The left and right insets show the series of cross-sectional spin textures for the hopfion and the toron in the xy plane, respectively. The color sphere denotes the spin orientation in three dimensions. The color wheel denotes the spin orientation in the xy plane. (C) The xz midplane cross-section of the hopfion. (D) The dependence of the 2D topological charge (N_s) on the perpendicular position z for the hopfion and the toron. (E) The xz midplane cross-section of the toron.