Fig. 4: Geometric transformation of Hopf solitons driven by electric field.
From: Geometric transformation and three-dimensional hopping of Hopf solitons
a Schematic illustration of the experimental setup and the switching of topologically trivial backgrounds; \({\theta }_{c}\) at midplane decreases from 90° to 0° as the applied voltage increases. b–d Vertical midplane cross-sections through the Hopf soliton going through intertransformation from a heliknoton to a hopfion. b–d correspond to simulated \({{{{{\boldsymbol{n}}}}}}({{{{{\boldsymbol{r}}}}}})\) at 0.00%, 0.19%, 6.07% of the total simulation time in f–g. The 100% simulation time corresponds to the total inter-transformation time (48.06 s) in experiments. e Experimental POM snapshots of Hopf soliton switching from a heliknoton (0 s) at U = 0 V to a hopfion (11.73 s) at U = 3.85 V, and back to a heliknoton (48.06 s) at U = 0 V. \(d/{p}_{0}=3\), \(d=7\) µm and scale bars are 5 µm. f–g Simulated transformation shown by preimages of two antiparallel orientations (f; white: \(+\hat{z}\), black: \(-\hat{z}\)) and simulated POM micrographs (g). The vertical red line going through the center of the volume in f is a guide to the eye. The progress of inter-transformation in numerical simulations shown as percentage of the total simulation time is labeled in each panel. In simulations, \({K}_{33}/{K}_{22}=1\) and \(d/{p}_{0}=3\).
