Fig. 3: Exotic magnetic states with hopfion rings.

a–h, Top, experimental Lorentz TEM images of magnetic states with different topological charges (Q = −11 (a), Q = −3 (b), Q = −2 (c), Q = −1 (d,e), Q = 0 (f), Q = 1 (g) and Q = 3 (h)) in different external magnetic fields in a 180-nm-thick sample recorded in the over-focus and under-focus regimes. The images in c, f, h were recorded at T = 95 K; the other images were recorded at T = 180 K. Bottom, theoretical Lorentz TEM images calculated for corresponding magnetic textures using micromagnetic simulations for identical external magnetic fields to the experimental values. All of the experimental and theoretical images have identical sizes of around 450 × 450 nm². The magnetization of the entire 1 μm × 1 μm domain and corresponding theoretical Lorentz TEM images are provided in Extended Data Fig. 7. The 3D magnetization vector fields of the magnetic states shown in a–d are provided in Fig. 4. See Extended Data Fig. 8 for the 3D magnetization vector fields of the magnetic states shown in e–h.