Hopf Solitons in Helical and Conical Backgrounds of Chiral Magnetic Solids

Three-dimensional topological solitons attract a great deal of interest in fields ranging from particle physics to cosmology, but remain experimentally elusive in solid-state magnets. Here we numerically predict magnetic heliknotons, an embodiment of such nonzero-Hopf-index solitons localized in all spatial dimensions while embedded in a helical or conical background of chiral magnets. We describe conditions under which heliknotons emerge as metastable or ground-state localized nonsingular structures with fascinating knots of magnetization field in widely studied materials. We demonstrate magnetic control of three-dimensional spatial positions of such solitons, as well as show how they interact to form moleculelike clusters and possibly even crystalline phases comprising three-dimensional lattices of such solitons with both orientational and positional order. Finally, we discuss both fundamental importance and potential technological utility of magnetic heliknotons.

Figure 1

Magnetic heliknoton. (a) Helical (left) and conical fields (right). (b)–(d) Simulated cross sections of $\mathit{m}(\mathit{r})$ of a heliknoton in a helical background. $\mathit{m}(\mathit{r})$ is shown with arrows colored according to orientations on ${\mathbb{S}}^2$ (bottom-left insets). (e) Preimages in $\mathit{m}(\mathit{r})$ of a heliknoton colored according to their orientations shown as cones on ${\mathbb{S}}^2$ (bottom-right inset). The gray isosurface (bisected for clarity) shows the region with small deviation of $\mathit{q}(\mathit{r})$ from the background ${\mathit{q}}_0$. (f) Preimages of constant-polar-angle orientations as cones on ${\mathbb{S}}^2$ of heliknotons in a helical field (top, $\tilde{\mathit{H}}=0$) and in a conical field (bottom, $\tilde{\mathit{H}}=0.15\widehat{z}$). (g) Constant-polar-angle surfaces at angles shown on ${\mathbb{S}}^2$ of heliknotons in a helical field (left, $\tilde{\mathit{H}}=0$) and in a conical field (right, $\tilde{\mathit{H}}=0.15\widehat{z}$). Schematics of the nesting of tori surfaces are shown in the bottom-right insets. (h) Singular vortex lines in $\mathit{q}(\mathit{r})$ forming three mutually linked rings (schematic in bottom-right inset). (i) Visualization of ${\mathit{B}}_{\mathrm{em}}$ in a magnetic heliknoton by the isosurfaces colored by magnitude and streamlines with cones. (j)–(l) Simulated LTEM images of a heliknoton in (b)–(d) for different directions. Top-right insets in (k) and (l) show the images with the characteristic contrast of a heliknoton reduced in thick samples.

Figure 2

Screw motions of heliknotons induced by external magnetic fields. (a) Heliknotons, visualized by preimages of poles, with orientations ${\mathit{m}}_h$ parallel (left) or antiparallel (right) to the applied field being the metastable and stable state, respectively. (b) Screw motion of a heliknoton at $\tilde{\mathit{H}}=0.05\widehat{y}$ relaxing from metastable parallel to stable antiparallel orientation. (c) The energy, orientation, and vertical displacement of a heliknoton relaxing from the metastable parallel configuration to the stable antiparallel configuration upon a perturbing magnetic field; the energy is normalized by $\mathrm{\Delta }E=E_{\mathrm{parallel}}-E_{\mathrm{antiparallel}}$ and referenced by $E_{\mathrm{antiparallel}}$.

Figure 3

Heliknoton stability. (a)–(c) Energy surfaces and stability diagrams of heliknotons in a chiral magnetic with (a) uniaxial easy-plane anisotropy, (b) uniaxial easy-axis anisotropy, and (c) cubic anisotropy. The energy is presented as the difference between the heliknoton energy and that of the minimum of helical and conical states. Colored regions indicate stable (purple), metastable (green), and unstable (gray) heliknotons. Relative orientations of ${\mathit{q}}_0$, $\mathit{H}$, and $\widehat{k}$ are shown in the insets.

Figure 4

Deformed heliknotons. (a)–(c) A heliknoton at $\tilde{\mathit{H}}=0.4\widehat{y}$, shown using preimages of poles in ${\mathbb{S}}^2$ in (a) and $\mathit{m}(\mathit{r})$ colored according to orientations on ${\mathbb{S}}^2$ in midplane cross sections in (a) and (b), and singular vortex loops in $\mathit{q}(\mathit{r})$ in (c) shown by colored tubes. (d)–(f) A heliknoton at $\tilde{\mathit{H}}=0.45\widehat{y}$ and easy-axis uniaxial anisotropy ${\tilde{K}}_u=0.16$ along $\widehat{y}$, shown by preimages of poles in (d) and $\mathit{m}(\mathit{r})$ colored according to orientations on ${\mathbb{S}}^2$ in midplane cross sections in (d) and (e) and singular vortex lines in $\mathit{q}(\mathit{r})$ in (f).

Figure 5

Oligomeric assemblies of magnetic heliknotons. (a),(b) Isosurfaces of positive (red) and negative (green) perturbations in the background energy density by an individual heliknoton. (c) Energy per soliton of dimers, tetramer, and octamer, in units of $E_0$. Configurations of each oligomer are shown as insets. Tetramer (d) and octamer (e) of heliknotons shown by preimages at poles and isosurfaces of small deviation of $\mathit{q}(\mathit{r})$ from the background.