Fig. 1. Spin configurations of hopfions in the lowest-energy state.

(A and B) Isosurfaces n_z = 0 of (A) an H-1 hopfion and (B) an H-2 hopfion. The direction of the Neel field is represented by colors in the hue, saturation, and lightness (HSL) color space. The pale green planes are the xy and yz planes. The top left insets are the yz slices of the hopfions, and the bottom right insets are the xy slices of the hopfions. It can be seen from the axis orientation that the H-1 hopfion is oriented toward the $[1\overline{1}1]$ direction, while the H-2 hopfion is oriented toward the [001] direction. (C and D) Preimages of the Neel vectors with xy angles of 0°, 90°, 180°, and 270° for (C) the H-1 hopfion and (D) the H-2 hopfion. The white torus represents the isosurface n_z = 0.