Skyrmion crystal with integer and fractional skyrmion numbers in a nonsymmorphic lattice structure with the screw axis

The emergence of a magnetic skyrmion crystal in a nonsymmorphic lattice system with the screw symmetry is numerically investigated. By performing the simulated annealing for a layered spin model with the isotropic exchange interaction and the antisymmetric Dzyaloshinskii-Moriya interaction, and then constructing a low-temperature phase diagram, we reveal that the skyrmion crystal is stabilized in both zero and nonzero fields even without the threefold rotational symmetry in the two-dimensional plane. We show that a competition between the ferromagnetic interlayer exchange interaction and the layer- and momentum-dependent anisotropic interactions is a source of the skyrmion crystals in the presence of the screw axis. Moreover, we find two types of layer-dependent skyrmion crystals, which are characterized as a coexisting state of the skyrmion crystal and the spiral state, in the narrow field region. Our result provides a reference in the further search for skyrmion crystals in nonsymmorphic lattice systems.

Figure 1

(a) Three layers (A, B, and C) connected by the threefold screw symmetry. (b) Schematics of the six relevant ordering wave vectors, $\pm {\mathit{Q}}_1, \pm {\mathit{Q}}_2$, and $\pm {\mathit{Q}}_3$, in the model in Eq. (5). The large (small) circles represent the dominant (subdominant) components in the momentum-resolved interactions.

Figure 2

Phase diagram of the model in Eq. (6) obtained by the simulated annealing at $\kappa =0.4$ while changing $J_{\parallel }$ and $H$. The contour shows the scalar chirality ${\chi }^{\mathrm{sc}}$.

Figure 3

$H$ dependences of (a) $M_{\eta }^z$, (b) ${\chi }_{\eta }^{\mathrm{sc}}$, (c) ${\left(m_{\eta {\mathit{Q}}_{\nu }}^{xy}\right)}^2$, and (d) ${\left(m_{\eta {\mathit{Q}}_{\nu }}^z\right)}^2$ for $J_{\parallel }=0.1$.

Figure 4

Real-space spin configurations of the $3Q$-I state on the layers A (left), B (middle), and C (right) at (a) $H=0.5$, (b) $H=1$, and (c) $H=1.6$ for $J_{\parallel }=0.1$. The color represents the $z$ component of the spin moment, and the arrows stand for the $xy$ components.

Figure 5

$H$ dependences of (a) $M_{\eta }^z$, (b) ${\chi }_{\eta }^{\mathrm{sc}}$, (c) ${\left(m_{\eta {\mathit{Q}}_{\nu }}^{xy}\right)}^2$, and (d) ${\left(m_{\eta {\mathit{Q}}_{\nu }}^z\right)}^2$ for $J_{\parallel }=0.4$. The vertical dashed lines represent the phase boundaries between the SkX and the other states.

Figure 6

Real-space spin configurations of (a) the $3Q$-I state at $H=0.2$, (b) the SkX at $H=1$, and (c) the $3Q$-I state at $H=1.3$ on the layers A (left), B (middle), and C (right) for $J_{\parallel }=0.4$. The color represents the $z$ component of the spin moment, and the arrows stand for the $xy$ components.

Figure 7

Real-space spin configurations of (a) and (b) the $3Q$-I state at $H=0.3$ and $H=0.35$, respectively, and (c) and (d) the SkX at $H=0.4$ and $H=0.5$, respectively, on layer A for $J_{\parallel }=0.4$. The color represents the $z$ component of the spin moment, and the arrows stand for the $xy$ components.

Figure 8

$H$ dependences of (a) $M_{\eta }^z$, (b) ${\chi }_{\eta }^{\mathrm{sc}}$, (c) ${\left(m_{\eta {\mathit{Q}}_{\nu }}^{xy}\right)}^2$, and (d) ${\left(m_{\eta {\mathit{Q}}_{\nu }}^z\right)}^2$ for $J_{\parallel }=1$. The vertical dashed lines represent the phase boundaries between the SkX and the other states.

Figure 9

Real-space spin configurations of (a) the $3Q$-II state at $H=0.1$, (b) the SkX at $H=0.7$, and (c) the $3Q$-I state at $H=1.3$ on the layers A (left), B (middle), and C (right) for $J_{\parallel }=1$. The color represents the $z$ component of the spin moment, and the arrows stand for the $xy$ components.

Figure 10

Phase diagram of the model in Eq. (6) obtained by the simulated annealing at $J_{\parallel }=0.2$ while changing $\kappa$ and $H$. The contour shows the scalar chirality ${\chi }^{\mathrm{sc}}$.

Figure 11

$H$ dependences of (a) $M_{\eta }^z$, (b) ${\chi }_{\eta }^{\mathrm{sc}}$, (c) ${\left(m_{\eta {\mathit{Q}}_{\nu }}^{xy}\right)}^2$, and (d) ${\left(m_{\eta {\mathit{Q}}_{\nu }}^z\right)}^2$ for $\kappa =0.76$. The vertical dashed lines represent the phase boundaries between the different phases.

Figure 12

Real-space spin configurations of (a) the f-SkX-I at $H=0.32$, (b) the f-SkX-II at $H=0.36$, and (c) the SkX at $H=0.4$ on the layers A (left), B (middle), and C (right) for $\kappa =0.76$. The color represents the $z$ component of the spin moment, and the arrows stand for the $xy$ components.