Multiple-$q$ States and the Skyrmion Lattice of the Triangular-Lattice Heisenberg Antiferromagnet under Magnetic Fields

Ordering of the frustrated classical Heisenberg model on the triangular lattice with an incommensurate spiral structure is studied under magnetic fields by means of a mean-field analysis and a Monte Carlo simulation. Several types of multiple-$q$ states including the Skyrmion-lattice state is observed in addition to the standard single-$q$ state. In contrast to the Dzyaloshinskii-Moriya interaction driven system, the present model allows both Skyrmions and anti-Skyrmions, together with a new thermodynamic phase where Skyrmion and anti-Skyrmion lattices form a domain state.

Figure 1

Phase diagram of the $J_1\mathrm{\text{−}}{J}_3$ model with $J_1/J_3=-1/3$ in the temperature-field plane, obtained by a Monte Carlo simulation. Transition temperatures between the paramagnetic and the ordered phases are determined from the specific-heat-peak position, while those between the triple-$q$ and the single-$q$ (or the double-$q$) phases are determined by the mixed-phase method [29].

Figure 2

The intensity plot of the spin structure factor for $J_1/J_3=-1/3$, in the single-$q$ phase [(a),(b)], the double-$q$ phase [(c),(d)], and the triple-$q$ phase [(e),(f)]. The lattice size is $L=72$. The upper and lower figures represent the spin structure factors of the $xy$ component $S_{\perp }(\mathit{q})=\frac{1}{N}\sum _{\mu =x,y}⟨|\sum _i{S}_{i,\mu }{e}^{-i\mathit{q}·{\mathit{r}}_i}{|}^2⟩$ and of the $z$ component $S_{\parallel }(\mathit{q})=\frac{1}{N}⟨|\sum _i{S}_{i,z}{e}^{-i\mathit{q}·{\mathit{r}}_i}{|}^2⟩$. The gray (color) scale represents $\sqrt{S_{\perp ,\parallel }(\mathit{q})}$.

Figure 3

Typical real-space spin configurations (left figures) and the intensity plots of the Skyrmion density (right figures), in the triple-$q$ phase (a)–(d) and in the $Z$ phase (e),(f). Upper (middle) figures represent the Skyrmion (anti-Skyrmion) lattice. The $xy$ component of spins is represented by the arrow, while the $z$ component is given by the gray (color) scale. Short-time average over 10 MCS is made to reduce the thermal noise. The lattice size is $L=72$.

Figure 4

The temperature dependence of the specific heat per spin (a) and of the total scalar chirality per plaquette (b) for $H/|J_3|=2.0$. The spin structure factor of the $xy$ component (c) and of the $z$ component (d) in the $Z$ phase. $J_1/J_3=-1/3$ in all cases.