From skyrmions to ${\mathbb{Z}}_2$ vortices in distorted chiral antiferromagnets

Swirling topological spin configurations, known as magnetic skyrmions, are known to be stabilized by different mechanisms. In particular, in chiral ferromagnets with Dzyaloshinskii-Moriya interaction (DMI), they appear in a topological crystalline phase, termed the skyrmion crystal phase (SkX), for appropriate values of the external magnetic field. A similar phenomenon is present in the antiferromagnetic case for the Heisenberg triangular antiferromagnet (HTAF) with DMI. Here, the most striking feature is that the emergent topological phase consists of three SkX interpenetrated sublattices. On the other hand, the pure HTAF, being described by an SO(3) order parameter, can host ${\mathbb{Z}}_2$ vortices. This raises the fundamental question of whether both nontrivial structures are related. In this Rapid Communication we unravel a hidden connection between both topological entities by studying the HTAF with anisotropic DMI. To this end, we combine an effective field theory description, the Luttinger-Tisza approximation, and Monte Carlo simulations. We show that even a slight anisotropy in the DMI proves to be the key ingredient to deform the interpenetrated SkX structure and reveal a ${\mathbb{Z}}_2$-vortex crystal.

Figure 1

(a) Triangular lattice. The red arrows indicate the direction of the anisotropic DMI vectors: ${\mathbf{D}}^{\left(1\right)}=(D_x,0)$, ${\mathbf{D}}^{\left(2\right)}=(-D_x/2,\sqrt{3}{D}_y/2)$, and ${\mathbf{D}}^{\left(3\right)}=(-D_x/2,-\sqrt{3}{D}_y/2)$. The green solid arrows are unit vectors. Ordering vectors ${\mathbf{k}}_i^{*}$ for (b) $D_x=D_y=0.35J$, (c) $D_x=0.27J$ and $D_y=0.35J$, and (d) $D_x=0.35J$ and $D_y=0.27J$. The green squares correspond to the Bragg peaks of the in plane structure factor ${\mathbf{S}}_{\perp }\left(\mathbf{k}\right)$ calculated from 100 MC snapshots at $B/J=2.7$ and $T/J\approx 1\times {10}^{-2}$. The red circles indicate the set of minima of ${\lambda }^{-}\left(\mathbf{k}\right)$. The red hexagons delimit the first Brillouin zone of the triangular lattice. The black arrows represent qualitatively the displacement of the minima of ${\lambda }^{-}\left(\mathbf{k}\right)$ and the peaks of ${\mathbf{S}}_{\perp }\left(\mathbf{k}\right)$ with respect to the isotropic case.

Figure 2

Spin configuration for the full lattice (left) and for one sublattice (right) for $D_x/J=0.35$, $D_y/J=0.27$, and $B/J=2.7$ obtained from (a) MC simulations at $T/J\approx {10}^{-2}$ and (b) the discretization of the spin field from the ET results. The arrows represent the in-plane components of the spin; the color represents the $S_z$ component.

Figure 3

Vortex configuration for different values of $D_x$ and $D_y$ at $B/J=2.7$. On the left side, $D_y/D_x=1.3$ ($D_y/J=0.35$), in the central panel $D_x/D_y=1$ ($D_x/J=0.35$), and the right panel is for $D_y/D_x=0.77$ ($D_x/J=0.35$). The second rows correspond to a zoom of the region bounded by the inner black squares. In the zoomed regions the vorticity is $\mathcal{V}=-1$ along the green curves and $\mathcal{V}=1$ for the deep blue curves. (a) Results from MC simulations at $T/J\approx {10}^{-2}$. The green triangles represent the positions of the vortices. (b) Results from ET. The green points indicate the position of the vortices. In the complete graph the distance between the ticks is $8a$ ($a$ is the lattice parameter), and in the zoomed region it is $4a$. The arrows correspond to the in-plane component of the chirality vector, and the color represents its length.