Skyrmion Crystal Induced by Four-Spin Interactions in Itinerant Triangular Magnets

Abstract

We investigate the emergence of magnetic skyrmion crystals with swirling topological spin textures in itinerant magnets with an emphasis on momentum-resolved multi-spin interactions. By performing the simulated annealing for the effective spin model with the two-spin and four-spin interactions on a two-dimensional triangular lattice, we show that various types of four-spin interactions become the microscopic origin of the magnetic skyrmion crystal with the skyrmion numbers of one and two. We find that the four-spin interactions between the different wave vectors lead to the skyrmion crystal with the skyrmion number of one, whereas those at the same wave vectors lead to the skyrmion crystals with the skyrmion number of one and two. Our results indicate that the multi-spin interactions arising from the itinerant nature of electrons provide rich topological spin textures in magnetic metals.

1. Introduction

Noncoplnar magnetic states have attracted much interest in condensed matter physics, as they give rise to fascinating physical phenomena induced by an emergent magnetic field through the spin Berry phase mechanism [1,2,3,4,5,6]. One of the examples is the topological Hall effect with neither an external magnetic field nor uniform magnetization [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. Microscopically, the uniform alignment of the scalar spin chirality, which is defined by the triple scalar product as ${\mathit{S}}_i·({\mathit{S}}_j\times {\mathit{S}}_k)$ (${\mathit{S}}_i$ represents the spin at site i), becomes an indicator, since it has the same symmetry as the ferromagnetic spin state. Another example is the nonlinear nonreciprocal transport when noncoplanar magnetic textures break the spatial inversion symmetry in addition to the time-reversal symmetry [22,23]. In this case, the uniform component of the scalar spin chirality is not necessary [24,25]. Moreover, since these phenomena are purely driven by the noncoplanar magnetic orderings, the relativistic spin–orbit coupling plays a less important role, which indicates that candidate materials are not limited to those with heavy elements.

In contrast to rich physical phenomena, it is usually difficult to stabilize the noncoplanar magnetic states compared to the collinear and noncollinear ones. Multiple-Q states, which are expressed as a superposition of multiple spin density waves, naturally give rise to noncoplanar spin textures [26,27,28,29,30]. They include the magnetic skyrmion crystal (SkX) as a superposition of the double-Q (triple-Q) spiral waves on a two-dimensional square (triangular) lattice [31,32,33,34,35,36,37,38,39,40,41,42,43] and magnetic hedgehog crystal as a superposition of the triple-Q or quadruple-Q spiral waves on a three-dimensional cubic lattice [44,45,46,47,48,49,50,51,52,53,54]. Such multiple-Q states have often been found in itinerant magnets, where long-range exchange interactions with the sign change are present. Indeed, a plethora of multiple-Q states with noncoplanar spin textures have been theoretically identified in itinerant triangular-lattice systems [12,55,56,57,58,59,60,61,62,63,64,65] and itinerant tetragonal-lattice systems [66,67,68,69,70,71]. Meanwhile, extracting the essentially important interactions in itinerant magnets is difficult owing to the competition of the interactions including higher-order multi-spin interactions. The approach based on perturbative analyses [72,73,74] and machine learning analyses [75] has mainly been used. However, systematic investigations to clarify the necessary multi-spin interactions to cause the instability toward noncoplanar magnetic states have not been fully performed.

In the present study, we explore a new stabilization mechanism of the SkX by focusing on four-spin interactions that arise from the interplay between charge and spin degrees of freedom in itinerant magnets. We investigate the instability toward the SkX for five-type four-spin interactions. By performing the simulated annealing for the effective spin model incorporating the four-spin interactions on a two-dimensional triangular lattice, we find that four out of five four-spin interactions lead to the SkX as the ground state in an external magnetic field. Among them, the four-spin interactions consisting of the single wave-vector channels give rise to two types of SkXs with a skyrmion number of one and two, while those consisting of multiple wave-vector channels give rise to the SkX with a skyrmion number of one. The present result provides a possibility of stabilizing the SkX by multi-spin interactions in itinerant magnets, which might be relevant to why many of the SkX-hosting materials are metallic.

2. Model and Method

We start from the Kondo lattice model with classical spins ${\mathit{S}}_i$ ($|{\mathit{S}}_i|=1$) on a two-dimensional triangular lattice, which is one of the typical models for itinerant magnets. The Kondo lattice model consists of itinerant electrons and localized spins, and it is given by

$$\begin{array}{c}\hfill {\mathcal{H}}_{\mathrm{KLM}}=-t\sum _{i,j,\sigma }{c}_{i\sigma }^{†}{c}_{j\sigma }+J_{\mathrm{K}}\sum _{i,\sigma ,{\sigma }^{\prime }}{c}_{i\sigma }^{†}{\mathit{\sigma }}_{\sigma {\sigma }^{\prime }}·{\mathit{S}}_i{c}_{i{\sigma }^{\prime }},\end{array}$$

(1)

where $c_{i\sigma }^{†}$ ($c_{i\sigma }$) is a creation (annihilation) operator of an itinerant electron at site i and spin $\sigma$. The first term represents the hopping of itinerant electrons on the triangular lattice and the second term represents the exchange coupling between itinerant electron spins and localized spins ${\mathit{S}}_i$ with the coupling constant $J_{\mathrm{K}}$; $\mathit{\sigma }=({\sigma }^x,{\sigma }^y,{\sigma }^z)$ is the vector of Pauli matrices. Owing to the centrosymmetric lattice structure, no spin-dependent hopping, which is referred to as the antisymmetric spin–orbit interaction, appears in the Hamiltonian.

When $J_{\mathrm{K}}$ is much smaller than the bandwidth of itinerant electrons, the Kondo lattice model reduces to a spin model with effective spin interactions between localized spins after tracing out the itinerant electron degree of freedom [72,73,74]. Up to the four-spin interaction, the effective spin model is generally given by

$$\begin{array}{cc}\hfill \mathcal{H}=& -\sum _{\mathit{q}}{J}_{\mathit{q}}{\mathit{S}}_{\mathit{q}}·{\mathit{S}}_{-\mathit{q}}+{\displaystyle \frac{1}{N}}\sum _{{\mathit{q}}_1,{\mathit{q}}_2,{\mathit{q}}_3,{\mathit{q}}_4}{K}_{{\mathit{q}}_1,{\mathit{q}}_2,{\mathit{q}}_3,{\mathit{q}}_4}{\delta }_{{\mathit{q}}_1+{\mathit{q}}_2+{\mathit{q}}_3+{\mathit{q}}_4,l\mathit{G}}\hfill \\ \hfill & \times \left[({\mathit{S}}_{{\mathit{q}}_1}·{\mathit{S}}_{{\mathit{q}}_2})({\mathit{S}}_{{\mathit{q}}_3}·{\mathit{S}}_{{\mathit{q}}_4})+({\mathit{S}}_{{\mathit{q}}_1}·{\mathit{S}}_{{\mathit{q}}_4})({\mathit{S}}_{{\mathit{q}}_2}·{\mathit{S}}_{{\mathit{q}}_3})-({\mathit{S}}_{{\mathit{q}}_1}·{\mathit{S}}_{{\mathit{q}}_3})({\mathit{S}}_{{\mathit{q}}_2}·{\mathit{S}}_{{\mathit{q}}_4})\right],\hfill \end{array}$$

(2)

where N is the total number of sites, $\delta$ is the Kronecker delta, and $\mathit{G}$ is the reciprocal lattice vector (l is an interger). ${\mathit{S}}_{\mathit{q}}$ is the Fourier transform of localized spins ${\mathit{S}}_i$. The first term represents the bilinear interaction with the coupling constant $J_{\mathit{q}}$ proportional to $J_{\mathrm{K}}^2$, whereas the second term represents the four-spin interaction with the coupling constant $K_{{\mathit{q}}_1,{\mathit{q}}_2,{\mathit{q}}_3,{\mathit{q}}_4}$ proportional to $J_{\mathrm{K}}^4$. The former bilinear interaction is referred to as the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction [76,77,78]. Since the magnitude of $K_{{\mathit{q}}_1,{\mathit{q}}_2,{\mathit{q}}_3,{\mathit{q}}_4}$ is determined by the products of Green’s functions of the itinerant electrons as well as the factor $J_{\mathrm{K}}^4$ in a rigorous way, it can be large depending on the electronic state [72]. It is noted that no Dzyaloshinskii–Moriya interaction [79,80] appears owing to the absence of the antisymmetric spin–orbit interaction in the Kondo lattice model [81]; the isotropic form of the spin interaction is caused by the spin rotational symmetry shown in Equation (1).

In order to investigate the instability toward the SkX in the model in Equation (2), we consider the situation where the interaction at $\pm {\mathit{Q}}_1=\pm (\pi /3,0)$ corresponds to the dominant interaction by supposing that the nesting of the Fermi surfaces occurs at ${\mathit{Q}}_1$. In other words, we consider the Fermi surfaces, which invokes the instability toward the magnetic orderings with the magnetic modulations by ${\mathit{Q}}_1$. In such a situation, the interactions at $\pm {\mathit{Q}}_2=\pm (-\pi /6,\sqrt{3}\pi /6)$ and $\pm {\mathit{Q}}_3=\pm (-\pi /6,-\sqrt{3}\pi /6)$ also give the dominant contributions owing to the threefold rotational symmetry of the triangular lattice Figure 1. By considering the contributions from the $\pm {\mathit{Q}}_1$, $\pm {\mathit{Q}}_2$, and $\pm {\mathit{Q}}_3$ channels, the model in Equation (2) is simplified as follows [74]:

$$\begin{array}{cc}\hfill {\mathcal{H}}_{\mathrm{eff}}=& -2J\sum _{\nu }{\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{\nu }}+{\displaystyle \frac{2K_1}{N}}\sum _{\nu }{({\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{\nu }})}^2+{\displaystyle \frac{2K_2}{N}}\sum _{\nu }({\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{{\mathit{Q}}_{\nu }})({\mathit{S}}_{-{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{\nu }})\hfill \\ \hfill & +{\displaystyle \frac{K_3}{N}}\sum _{\nu \ne {\nu }^{\prime }}({\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{\nu }})({\mathit{S}}_{{\mathit{Q}}_{{\nu }^{\prime }}}·{\mathit{S}}_{-{\mathit{Q}}_{{\nu }^{\prime }}})+{\displaystyle \frac{K_4}{N}}\sum _{\nu \ne {\nu }^{\prime }}({\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{{\mathit{Q}}_{{\nu }^{\prime }}})({\mathit{S}}_{-{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{{\nu }^{\prime }}})\hfill \\ \hfill & +{\displaystyle \frac{K_5}{N}}\sum _{\nu \ne {\nu }^{\prime }}({\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{{\nu }^{\prime }}})({\mathit{S}}_{-{\mathit{Q}}_{\nu }}·{\mathit{S}}_{{\mathit{Q}}_{{\nu }^{\prime }}}),\hfill \end{array}$$

(3)

where $\nu =1$–3 is the index for the ordering wave vectors ${\mathit{Q}}_{\nu }$ and $J\equiv J_{{\mathit{Q}}_1}=J_{{\mathit{Q}}_2}=J_{{\mathit{Q}}_3}$. The first term represents the RKKY interaction, while the second to sixth terms represent the four-spin interactions with the coupling constants $K_1$–$K_5$. In the following section, we investigate the effects of the four-spin interactions on the stabilization of the SkX. We set $J=1$ as the energy unit of the model. In addition, we also consider the effect of the external magnetic field H in the form of the Zeeman coupling as

$$\begin{array}{c}\hfill {\mathcal{H}}^Z=-H\sum _i{S}_i^z.\end{array}$$

(4)

In the following section, we analyze the total Hamiltonian as

$$\begin{array}{c}\hfill \mathcal{H}={\mathcal{H}}_{\mathrm{eff}}+{\mathcal{H}}^Z.\end{array}$$

(5)

It is noted that the ground-state spin configuration is given as the single-Q spiral state irrespective of H when all the four-spin interactions are zero.

To obtain the stable spin configurations of the model in Equation (5), we perform the simulated annealing combined with Monte Carlo simulations based on the standard single-spin-flip Metropolis algorithm. Starting from a high temperature $T_0/J=1$, we gradually reduce the temperature at the ratio of $\alpha =0.999999$ until the final temperature $T/J=0.01$ is reached. At each temperature, we perform Monte Carlo sweeps, and we further perform ${10}^5$–${10}^6$ Monte Carlo sweeps at the final temperature for measurements. We also start the simulations from the spin patterns obtained at low temperatures in the vicinity of the phase boundaries.

For the obtained spin configurations, we calculate the spin structure factor given by

$$\begin{array}{c}\hfill S_s^{\eta \eta }\left(\mathit{q}\right)={\displaystyle \frac{1}{N}}\sum _{i,j}{S}_i^{\eta }{S}_j^{\eta }{e}^{i\mathit{q}·({\mathit{r}}_i-{\mathit{r}}_j)},\end{array}$$

(6)

where $\eta =x,y,z$. ${\mathit{r}}_i$ is the position vector at site i, and $\mathit{q}$ is the wave vector. The total spin structure factor is given by $S_s\left(\mathit{q}\right)=S_s^{xx}\left(\mathit{q}\right)+S_s^{yy}\left(\mathit{q}\right)+S_s^{zz}\left(\mathit{q}\right)$, and its in-plane spin component is given by $S_s^{\perp }\left(\mathit{q}\right)=S_s^{xx}\left(\mathit{q}\right)+S_s^{yy}\left(\mathit{q}\right)$. The magnetization along the magnetic field direction given by

$$\begin{array}{c}\hfill M={\displaystyle \frac{1}{N}}\sum _i{S}_i^z,\end{array}$$

(7)

is also calculated. Meanwhile, the scalar spin chirality is used to identify whether the spin configurations are topologically nontrivial or not, whose expression is given by

$$\begin{array}{c}\hfill {\chi }^{\mathrm{sc}}={\displaystyle \frac{1}{N}}\sum _{\mathit{R}}{\mathit{S}}_j·({\mathit{S}}_k\times {\mathit{S}}_l),\end{array}$$

(8)

where $\mathit{R}$ represents the position vector at the centers of triangles; the sites j, k, and l form the triangle at $\mathit{R}$ in the counterclockwise order. The magnetic ordering with nonzero ${\chi }^{\mathrm{sc}}$ exhibits the topological Hall effect.

3. Results

In this section, we discuss the stability of the SkX in the presence of the five-type four-spin interactions in Equation (5). In order to understand the role of each four-spin interaction, we investigate its effect one by one below. We discuss the effect of $K_1$ in Section 3.1, that of $K_2$ in Section 3.2, that of $K_3$ in Section 3.3, that of $K_4$ in Section 3.4, and that of $K_5$ in Section 3.5. We especially focus on the emergence of the SkX in each case. In the following section, the result for $H<0$ is the same as that for $H>0$ when the sign of $S_i^z$ is reversed [82].

3.1. Effect of $K_1$

First, we discuss the effect of $K_1$, which has the form of ${({\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{\nu }})}^2$, by setting $K_2=K_3=K_4=K_5=0$. This type of interaction corresponds to the biquadratic interaction. Since this interaction plays an important role when the Fermi surface is strongly nested at ${\mathit{Q}}_{\nu }$ [72,73,74], its effect on the stability of the SkX has been investigated in the case of the triangular lattice [83] and square lattice [84]. Thus, we briefly discuss the role of $K_1$ in the following. Since the SkX does not appear for $K_1<0$, we focus on the situation for $K_1>0$.

Figure 2a shows the H dependence of the magnetization M and the scalar spin chirality ${\chi }^{\mathrm{sc}}$. For $0.5\lesssim H\lesssim 1.0$, the magnetic state with nonzero ${\chi }^{\mathrm{sc}}$ indicates the emergence of the SkX. We plot the real-space spin and scalar spin chirality configurations in this state in Figure 3a. The skyrmion cores located at $S_i^z=-1$ form the triangular lattice, as shown in the left panel of Figure 3a; the vorticity in terms of the $xy$-spin component around the skyrmion core is given by $-1$, which means that the SkX phase exhibits the skyrmion number of $+1$ [43]; we denote this as the SkX-I. The nonzero skyrmion number is also found in the distribution of the scalar spin chirality in the right panel of Figure 3a, where the region with the positive scalar spin chirality is much larger than that with the negative scalar spin chirality. It is noted that the SkX with the skyrmion number of $+1$ is energetically degenerated with that with the skyrmion number of $-1$ owing to the spin rotational symmetry in the model. Such a degeneracy can be lifted when considering the symmetric anisotropic exchange interaction that arises from the relativistic spin–orbit coupling [85,86]. Another characteristic feature in the SkX-I is the triple-Q peaks in both $xy$ and z components of the spin structure factor with the equal intensity at ${\mathit{Q}}_1$, ${\mathit{Q}}_2$, and ${\mathit{Q}}_3$, i.e., $S_s^{\perp }\left({\mathit{Q}}_1\right)=S_s^{\perp }\left({\mathit{Q}}_2\right)=S_s^{\perp }\left({\mathit{Q}}_3\right)$ and $S_s^{zz}\left({\mathit{Q}}_1\right)=S_s^{zz}\left({\mathit{Q}}_2\right)=S_s^{zz}\left({\mathit{Q}}_3\right)$.

When the magnitude of $K_1$ becomes larger, another SkX phase appears in the low magnetic field region, as shown in the case of $K_1=0.3$ in Figure 3b. This SkX is also characterized by the isotropic triple-Q peaks in the spin structure factor with $S_s\left({\mathit{Q}}_1\right)=S_s\left({\mathit{Q}}_2\right)=S_s\left({\mathit{Q}}_3\right)$. The scalar spin chirality in this low-field SkX phase is larger than that in the SkX-I. By looking into the real-space spin configuration in this state in the left panel of Figure 3b, one finds that the vorticity around the skyrmion core with positive $S_i^z$ is given by $-2$. Thus, this spin configuration has a skyrmion number of two, which is denoted as the SkX-II [62,87]. The distribution of the scalar spin chirality is shown in the right panel of Figure 3b, where the uniform component exists. Similar SkX with high skyrmion numbers has also been found when the anisotropic exchange interaction [88,89,90], higher-harmonic-wave vector interaction [91], and dynamical electric field [92] are considered, although the present SkX-II does not require such factors. Similar to the SkX-I, the SkX-II has a degeneracy in terms of the sign of the skyrmion number.

3.2. Effect of $K_2$

Next, we consider the effect of $K_2$ in the form of $({\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{{\mathit{Q}}_{\nu }})({\mathit{S}}_{-{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{\nu }})$; we set $K_1=K_3=K_4=K_5=0$. In contrast to $K_1$, the effect of $K_2$ has not been investigated so far. In the case of $K_2$, the SkX appears only for $K_2<0$, as shown below.

For $K_2=-0.2$, there is no SkX phase, as shown in Figure 4a; no scalar spin chirality is induced in the whole magnetic field region. Only the topological trivial triple-Q state, whose spin configuration is characterized by a superposition of the dominant in-plane cycloidal spiral wave at ${\mathit{Q}}_1$ and subdominant spin density waves at ${\mathit{Q}}_2$ and ${\mathit{Q}}_3$ appears instead of the single-Q spiral state. With the increase in $|K_2|$, both SkX-I and SkX-II appear against H, as shown in Figure 4b. The real-space spin and scalar spin chirality configurations of the SkX-I and SkX-II are shown in Figure 5a and Figure 5b, respectively. For the SkX-I, the spin configuration in Figure 5a is similar to that in Figure 3a, except for the vorticity. Since there is a degeneracy in terms of the vorticity in the presence of $K_2$, these states are identical to each other. Meanwhile, the spin configuration of the SkX-II in Figure 5b looks different from that in Figure 3b. Nevertheless, the spin configuration in Figure 5b also exhibits the skyrmion number of two [93]. According to the integer skyrmion number, we refer to this state as the SkX-II. Indeed, the distribution of the scalar spin chirality in Figure 5b is similar to that in Figure 3, which indicates the similarity of the topological properties; a similar triple-Q peak structure is also found in the spin structure factor.

While further increasing $|K_2|$, the SkX-II remains stable whereas the SkX-I vanishes, as shown by $K_2=-0.5$ in Figure 4c. In contrast to the situation in the presence of $K_1$, the data indicate the appearance of another magnetic state with nonzero scalar spin chirality for $0.25\lesssim H\lesssim 1.15$, although this state does not exhibit nonzero skyrmion numbers. The real-space spin and scalar spin chirality configurations are shown in the left and right panels of Figure 5c, respectively. As shown in Figure 5c, the region with the positive scalar spin chirality is almost the same as that with the negative scalar spin chirality, although they are not perfectly canceled out with each other. This is why this state shows a small value of ${\chi }_{\mathrm{sc}}$ compared to the SkX-I and SkX-II, as shown in Figure 5c. We refer to this state as the chiral state in order to distinguish it from other topologically trivial states. The spin structure factor in this state shows the dominant double-Q peaks in the $xy$-spin component at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$, as well as the subdominant double-Q peaks in the z-spin component at $2{\mathit{Q}}_1$ and $2{\mathit{Q}}_2$, where the intensities at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ are slightly different from each other. The SkX-II vanishes for further large $K_2=-0.6$, as shown in Figure 4d; the stability of the chiral state is extended to the zero-field region.

3.3. Effect of $K_3$

We consider the effect of $K_3$ in the form of $({\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{\nu }})({\mathit{S}}_{{\mathit{Q}}_{{\nu }^{\prime }}}·{\mathit{S}}_{-{\mathit{Q}}_{{\nu }^{\prime }}})$ for $\nu \ne {\nu }^{\prime }$ by setting $K_1=K_2=K_4=K_5=0$. When $K_3$ is negative (positive), this interaction corresponds to the attractive (repulsive) interaction between the different ordering wave vectors. Thus, the multiple-Q instability is expected only for $K_3<0$; we confirmed that no SkX appears for $K_3>0$.

Figure 6 shows the H dependence of M and ${\chi }^{\mathrm{sc}}$ for $K_3=-0.1$. Similarly to the case of $K_1=0.1$ in Figure 2a, only the SkX-I appears in the intermediate-field region. The real-space spin and scalar spin chirality configurations shown in Figure 7 are similar to those in Figure 3a. Thus, the $K_3$-type four-spin interaction also gives rise to the SkX-I. On the other hand, the SkX-II does not appear when the magnitude of $K_3$ increases; only the SkX-I appears, at least, up to $K_3=-0.8$.

3.4. Effect of $K_4$

In the case of $K_4$, which is characterized by the different types of four-spin interactions in the form of $({\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{{\mathit{Q}}_{{\nu }^{\prime }}})({\mathit{S}}_{-{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{{\nu }^{\prime }}})$ for $\nu \ne {\nu }^{\prime }$ with $K_1=K_2=K_3=K_5=0$, similar SkX instability occurs. We show the H dependence of M and ${\chi }^{\mathrm{sc}}$ at $K_4=-0.2$ in Figure 8; only the SkX-I appears against H. The spin and scalar spin chirality configurations are shown in the left and right panels of Figure 9, respectively. Also, in this case, the SkX-II is not stabilized, at least, for $K_4\ge -0.8$. Thus, $K_4$ plays a similar role in stabilizing the SkX-I to $K_3$.

3.5. Effect of $K_5$

Finally, let us discuss the effect of $K_5$, which has the form of $({\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{{\nu }^{\prime }}})({\mathit{S}}_{-{\mathit{Q}}_{\nu }}·{\mathit{S}}_{{\mathit{Q}}_{{\nu }^{\prime }}})$, where $K_1=K_2=K_3=K_4=0$. In contrast to the situations with $K_1$–$K_4$, any SkXs do not appear for both positive and negative $K_5$; see the H dependence of ${\chi }^{\mathrm{sc}}$ at $K_5=-0.8$ in Figure 10. Instead of the SkX, the topologically trivial triple-Q state appears even for a large $|K_5|=0.8$ value, whose spin configuration is shown in Figure 11a; the spin structure factor in this state is characterized by a superposition of the dominant in-plane cycloidal spiral at ${\mathit{Q}}_1$ and the subdominant spin density waves at ${\mathit{Q}}_2$ and ${\mathit{Q}}_3$, as found in the case of $K_2=-0.2$. Thus, this interaction just leads to the topologically trivial triple-Q structure for $K_5<0$.

When $|K_5|$ is larger than J, where the perturbation analysis is no longer valid, a triple-Q magnetic vortex appears in the high-field region. We show the obtained spin configuration at $K_5=-2.5$ and $H=1.3$ in Figure 11b, which consists of the periodic alignment of vortex and antivortex, as found in frustrated magnets [94,95,96]. In this state, there are isotropic triple-Q peaks at ${\mathit{Q}}_1$, ${\mathit{Q}}_2$, and ${\mathit{Q}}_3$ in the $xy$ component of the spin structure factor satisfying $S_s^{\perp }\left({\mathit{Q}}_1\right)=S_s^{\perp }\left({\mathit{Q}}_2\right)=S_s^{\perp }\left({\mathit{Q}}_3\right)$. These results indicate that $K_5$ does not play an important role in inducing the SkX.

4. Conclusions

We have investigated the instability toward the SkX driven by four-spin interactions that originate from the interplay between spin and charge degrees of freedom in itinerant magnets. By analyzing the optimal spin configurations for the effective spin model with five-type four-spin interactions on the triangular lattice, we found that four out of five cases lead to the SkX with the skyrmion number of one in the external magnetic field. In addition, we also found that two-type four-spin interactions which correspond to the interactions at the same wave-vector channel give rise to the emergence of the higher-order SkX with the skyrmion number of two even without the external magnetic field. The present results indicate the possibility of further topological spin textures induced by multi-spin interactions in itinerant magnets. Since the SkXs have been found in materials with the interplay between itinerant electrons and localized spins, such as Gd₂PdSi₃ [18,97,98,99], the materials with similar chemical compositions might be promising for the realization of SkXs based on the four-spin interaction mechanism.