Phase transitions between helices, vortices, and hedgehogs driven by spatial anisotropy in chiral magnets

Superpositions of spin helices can yield topological spin textures, such as two-dimensional vortices and skyrmions and three-dimensional hedgehogs. Their topological nature and spatial dimensionality depend on the number and relative directions of the constituent helices. This allows mutual transformation between the topological spin textures by controlling the spatial anisotropy. Here, we theoretically study the effect of anisotropy of the magnetic interactions in an effective spin model for chiral magnetic metals. By variational calculations for both triple and quadruple superpositions, we find that the hedgehog lattices, which are stable in the isotropic case, are deformed by the anisotropy and eventually changed into other spin textures with reduced dimension, such as helices and vortices. We also clarify the changes in topological properties by tracing the real-space positions of emergent magnetic monopoles and antimonopoles as well as the emergent magnetic field generated by the noncoplanar spin textures. Our results suggest possible control of the topological spin textures, e.g., by uniaxial pressure and chemical substitution in chiral materials.

Figure 1

Real-space spin configurations of (a) a helical state ($1Q$-H), (b) a vortex crystal composed of two helices ($2Q$-VC), (c) a skyrmion lattice composed of three helices ($3Q$-SkL), (d) a hedgehog lattice composed of three helices ($3Q$-HL) and (e) another hedgehog lattice composed of four helices ($4Q$-HL). The arrows represent the spins and their colors display the $z$ component. Each inset shows the wave vectors of the constituent helices.

Figure 2

Phase diagrams of the model in Eq. (1) obtained by the variational calculations for (a)–(c) the $3Q$ case and (d)–(f) the $4Q$ case: (a) and (d) $J^{\prime }/J=0.8$ (hard axis), (b) and (e) $J^{\prime }/J=1.0$ (isotropic), and (c) and (f) $J/J^{\prime }=0.8$ (easy axis). $1Q$-H, $2Q$-VC, $2Q$-CS, $3Q\left(4Q\right)$-HL, $3Q\left(4Q\right)$, and $3Q\left(4Q\right)$-NC represent the $1Q$ helical, the $2Q$ vortex crystal, the $2Q$ chiral stripe, the $3Q\left(4Q\right)$ hedgehog lattice, the topologically trivial $3Q\left(4Q\right)$ states, and the $3Q(4Q$) nonchiral states, respectively. The light and dark colors for $2Q$-VC, $2Q$-CS, and $1Q$-H denote the differences in the constituent wave vectors. We take $J\left(J^{\prime }\right)$ as the energy unit for $J^{\prime }/J<1(>1)$.

Figure 3

Anisotropy dependences of the variational energy per site, $E$, the amplitude of the helix in the $J^{\prime }$ direction, $a_3$ or $a_4$, the magnetic moment with the wave vector ${\mathbf{Q}}_{\eta }$, $m_{{\mathbf{Q}}_{\eta }}$ [Eq. (5)], and the number of hedgehogs and antihedgehogs, $N_{\mathrm{m}}$ [Eq. (9)], for the (a) $3Q$ and (b) $4Q$ cases. The dashed lines represent the topological transitions with the changes in $N_{\mathrm{m}}$. The results are calculated for $D=0.3$ and $K=0.8$ ($D^{\prime }=0.3$ and $K^{\prime }=0.8$) for $J^{\prime }/J<1(>1)$ in Eq. (1). We take $J$ ($J^{\prime }$) as the energy unit for $J^{\prime }/J<1$ ($>1$).

Figure 4

Real-space configurations of the scalar spin chirality ${\mathit{\chi }}_{{\mathbf{r}}_l}$ (arrows) and the monopoles and antimonopoles (magenta and cyan balls, respectively) in a magnetic unit cell obtained from the variational calculations for the $3Q$ case: (a) $J^{\prime }/J=0.875$, (b) $J^{\prime }/J=0.9$, (c) $J^{\prime }/J=1.0$, (d) $J/J^{\prime }=0.9$, and (e) $J/J^{\prime }=0.875$. ${\mathit{\chi }}_{{\mathbf{r}}_l}$ are plotted when their lengths are longer than a certain threshold for better visibility. The color of the arrows represents the $z$ component of ${\mathit{\chi }}_{{\mathbf{r}}_l}$; pale color is used for the arrows in the neighboring magnetic unit cells.

Figure 5

Plots similar to Fig. 4, but for the $4Q$ case: (a) $J^{\prime }/J=0.75$, (b) $J^{\prime }/J=0.775$, (c) $J^{\prime }/J=1.0$, (d) $J/J^{\prime }=0.775$, and (e) $J/J^{\prime }=0.75$. The notations are common to those in Fig. 4. Note that the magnetic unit cell is one-fourth of the whole cube in (a). The green dashed cubes are guides for the eyes; see the text for details.