Fig. 4: Micromagnetic simulations of magnetic textures with hopfion rings.

a–d, Each row of images corresponds to the magnetic textures shown in Fig. 3a–d, with Q = −11 (a), Q = −3 (b), Q = −2 (c) and Q = −1 (d). The first column shows the magnetization field in the middle plane of the sample. The isosurfaces m_z = 0 shown in the second and third columns are identical and differ only in their representations. In particular, the second column shows the isosurface of the hopfion ring alone, to highlight the linkage of the pre-images. The third column shows complete isosurfaces of hopfion rings and skyrmions. Analogous images for the magnetic textures depicted in Fig. 3e–h are provided in Extended Data Fig. 8a–d. e–h, Images showing hopfion rings in a bulk system. As well as the isosurfaces m_z = 0, we show the magnetization at the edges of the simulated box to visualize the conical modulations along the z axis (see also Extended Data Fig. 8e-h for other hopfion rings in the bulk). A pair of indices (Q, H) defines the skyrmion–hopfion topological charge of the corresponding texture (see Methods). e and f show topologically identical states of different morphology. g, The pair of hopfion rings on a single skyrmion string. h, The pair of hopfion rings on three skyrmion strings. Note that, for the states shown in b,c, the Hopf charge is uncertain.