Three-sublattice skyrmion crystal in the antiferromagnetic triangular lattice

The frustrated classical antiferromagnetic Heisenberg model with Dzyaloshinskii-Moriya (DM) interactions on the triangular lattice is studied under a magnetic field by means of semiclassical calculations and large-scale Monte Carlo simulations. We show that even a small DM interaction induces the formation of an antiferromagnetic skyrmion crystal (AF-SkX) state. Unlike what is observed in ferromagnetic materials, we show that the AF-SkX state consists of three interpenetrating skyrmion crystals (one by sublattice), and most importantly, the AF-SkX state seems to survive in the limit of zero temperature. To characterize the phase diagram we compute the average of the topological order parameter which can be associated with the number of topological charges or skyrmions. As the magnetic field increases this parameter presents a clear jump, indicating a discontinuous transition from a spiral phase into the AF-SkX phase, where multiple Bragg peaks coexist in the spin structure factor. For higher fields, a second (probably continuous) transition occurs into a featureless paramagnetic phase.

Figure 1

(a) Triangular lattice. The green arrows indicate the primitive translation vectors of the direct lattice ${\vec{e}}_1=(1,0),{\vec{e}}_2=(1/2,\sqrt{3}/2)$, red arrows the Dzyaloshinskii-Moriya vectors. The sites with labels ${\mathbf{r}}_i,{\mathbf{r}}_1,\cdots ,{\mathbf{r}}_4$ indicate the sites involved in the calculation of the local chirality. (b) Area of the triangle to compute for the discretized skyrmion number $A_{\mathbf{r}}^{ab}$ [Eq. (8)].

Figure 2

Minimum eigenvalue $w^{\text{min}}\left(\mathbf{k}\right)$ in the spherical approximation for $D/J=1/2$. For any value of $D/J>0$ we find that the ground state corresponds to a triple-$q$ state by sublattice.

Figure 3

Snapshots by sublattice for $L=48,D/J=1/2,T/J=9\times {10}^{-3}$, and $h/J=1\left(\mathrm{i}\right),2.4\left(\mathrm{ii}\right)$, and 6.2 (iii). The green box indicates which sublattice is plotted (A, B, or C). In panel (iv) we show all three sublattices together.

Figure 4

Top: Magnetization $M$ (blue) and magnetic susceptibility vs magnetic field $h$ for $L=48,D/J=0.5$, and $T/J=9\times {10}^{-3}$. Center: Discretized skyrmion number ${\chi }_Q$ and total chirality ${\chi }_L$, both quantities per sublattice, vs $h/J$ for $L=48,D/J=1/2$. Red triangles, green rhombus, and purple triangles indicate the discretized skyrmion numbers $A,B$, and $C$. Blue dots show the total chirality. Bottom: Complete h-T phase diagram. The limit of the phases is obtained by the peaks in the specific heat and from the changes in the chirality order parameter.

Figure 5

Intensity plot of the static spin structure factor for $L=96$ for the spiral phase (top), AF-SkX phase (center), and disordered phase (bottom).

Figure 6

Specific heat vs temperature for $h/J=1.4$, 3, and 6. The positions of the peaks are used to locate the lines separating the different phases.

Figure 7

Dependence of the free energy density given by Eq. (16) as a function of the skyrmion radius $R$ for two values of the skyrmion core: $R_0=0.5,1$ $\left(\kappa =1\right)$.

Figure 8

The Monte Carlo time evolution of the magnetization, discretized skyrmion number ${\chi }_Q$, and total chirality ${\chi }_L$ defined by Eqs. (8) and (9), calculated at $T=9\times {10}^{-3}J$ for sizes of $L=24,36,48,60$. The initial state corresponds to the AF-SkX phase (Fig. 3). Each curve corresponds to an average of 500 independent time evolutions.