Fig. 6: 3D Hopping of Hopf solitons.

a–b Translational and orientational displacement of a Hopf soliton by repeated voltage switching shown at its initial (a) and final (b) position. The 2D trajectory is color-coded by time and the long-axis orientations of the Hopf soliton in helical background at intermediate positions are shown by double arrows (plotted for every two switching cycles). c Distance and accumulated change in orientation in each transformation cycle (\(\Delta \phi\)) of a hopping Hopf soliton shown in a–b. d Hopf solitons in different long-axis orientations in a helical background with perpendicular confinement. e Displacement in \(z\) as a function of initial position \({z}_{i}\) of a hopfion in the uniform background. f Displacement in \(z\) as a function of initial position \({z}_{i}\) of a heliknoton in the helical background. g–h Free energy (g) and orientation (h) dependence of a heliknoton on equilibrium position \({z}_{f}\). In f–h, heliknotons with \(d=3{p}_{0}\) and \(d=4{p}_{0}\) are shown in blue and orange, respectively. The insets in e, f show the initial and final equilibrated Hopf solitons of the corresponding data points by polar preimages. Insets in h show the simulated POM images (viewed along \(\hat{z}\)) and the preimages (viewed along \(\hat{y}\)) of the solitons with \(z\) positions 0, 0.42\({p}_{0}\), and 1\({p}_{0}\) relative to the midplane (left to right) for \(d=3{p}_{0}\). The line in h shows \(\phi =2\pi ({z}_{f}/{p}_{0})\) for bulk heliknotons is a guide to the eye. Free energy is in units of \(K{p}_{0}\) and \({k}_{B}T\), where \(K=6.47\) pN is the average elastic constant of 5CB and \({p}_{0}=2.33\) µm, \({k}_{B}\) is the Boltzmann constant, and temperature \(T=300\) K. \(d=3{p}_{0}\) in experiments in a–d and scale bars are 10 µm.