Effect of High-Harmonic Wave-Vector Interactions on the Single-Q Spiral State

Abstract

We investigate the role of high-harmonic wave-vector interactions, which affect the stability of the single-Q spiral state and often result in the formation of multiple-Q states. By performing simulated annealing for an effective spin model on a two-dimensional square lattice, we examine the modulation of the single-Q spiral spin configuration by the high-harmonic wave-vector interaction. As a result, we find that the interactions at particular high-harmonic wave vectors affect the stability of the single-Q spiral state. In particular, the incorporation of interactions at high-harmonic wave vectors formed by the sum of two mutually perpendicular ordering wave vectors can lead to the emergence of three double-Q states and a square skyrmion crystal. The present study unveils the importance of high-harmonic wave-vector interactions in order to realize complicated noncoplanar spin textures.

1. Introduction

Noncollinear magnetic textures have attracted considerable attention in recent years owing to their rich physical properties like the anomalous Hall effect [1,2,3,4,5,6,7] and potential applications in spintronic devices [8,9,10,11,12]. Among them, a spiral (helical) magnetic state, in which spin moments rotate periodically in real space, represents a prototypical system that exhibits various spin-order-driven physical phenomena. For example, reflecting the breaking of spatial inversion symmetry by the spiral spin texture, the system exhibits an emergent electric polarization [13,14,15,16,17,18] and antisymmetric spin-split band structures without relying on the relativistic spin-orbit coupling [19,20,21], which results in the Edelstein effect [22,23,24,25,26], the spin Hall effect [27,28,29,30,31], and unconventional spin-orbital coupling [32,33,34,35].

From the viewpoint of theoretical modeling, there are three main stabilization mechanisms for the spiral state. The first is the competition between the ferromagnetic exchange interaction and the Dzyaloshinskii–Moriya interaction [36,37], the latter of which arises from the relativistic spin-orbit coupling under the breaking of the spatial inversion symmetry. The second is the competition between the ferromagnetic exchange interaction and further-neighbor antiferromagnetic exchange interactions [38,39,40,41,42]. The third is the long-range interaction in itinerant magnets [43,44,45], which is the so-called Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction [46,47,48]. Although the first mechanism relies on a noncentrosymmetric lattice structure, the other two are effective irrespective of the lattice symmetry.

Since the spin configuration of the spiral state is characterized by a single dominant ordering wave vector Q in momentum space, this can be referred to as a single-Q state. In contrast, the superposition of the single-Q spin configuration at different ordering wave vectors gives rise to more complex spin configurations, which is denoted as a multiple-Q state; we refer to the spin state as the double-Q (triple-Q) state when a double-Q (triple-Q) superposition of the single-Q spiral state occurs [49,50,51,52]. Such multiple-Q states are of great interest due to their potential to host topologically nontrivial magnetic textures, including skyrmion crystals (SkXs) [53,54,55,56,57,58,59,60,61]. For example, the superposition of two orthogonal spiral waves leads to the square skyrmion crystal (SkX), and the superposition of three spiral waves leads to the triangular SkX, both of which accompany nonzero topological charges. Depending on the relative phases among the constituent waves, one can construct topologically distinct multiple-Q states, such as meron–antimeron crystals consisting of merons with the topological number of $-1/2$ ($+1/2$) and antimerons with the topological number of $+1/2$ ($-1/2$), leading to a cancellation of the net topological charge [62,63,64,65,66], which differs from another topologically nontrivial multiple-Q state, bimeron crystal with a nonzero topological number [67,68,69,70,71]. In order to stabilize such multiple-Q states, including SkXs, one needs to take into account other effects in addition to the interactions inducing the single-Q spiral state [72], such as thermal fluctuations [73,74,75], the bond-dependent symmetric anisotropic exchange interaction [76,77,78,79,80,81,82,83,84,85,86], multiple-spin interactions [87,88,89,90,91,92], and nonmagnetic impurities [93] in addition to an external magnetic field [55,60,94,95].

In the present study, we investigate the stability of single-Q and multiple-Q states by focusing on a mechanism that induces multiple-Q instabilities: interactions at high-harmonic wave vectors in momentum space. While the primary contributions to the stabilization of single-Q and multiple-Q states are typically governed by interactions at the fundamental ordering wave vectors, recent studies have demonstrated that interactions at higher-harmonic wave vectors can also play an important role in stabilizing multiple-Q states [96,97]. For example, the high-harmonic wave-vector interaction at wave vectors given by the sum of two ordering wave vectors serves as the microscopic origin of multiple SkX phases observed in GdRu₂Ge₂ [98]. Here, we examine the effect of high-harmonic wave-vector interactions on the stability of the single-Q spiral state, identifying the specific interaction channels that can trigger multiple-Q instabilities. To this end, we analyze an effective spin model with momentum-resolved interactions on a two-dimensional square lattice by employing simulated annealing. As a result, we find that interactions at second-harmonic wave vectors affect the stability of the single-Q spiral state, whereas those at third-harmonic wave vectors have negligible influence, although neither interaction leads to the emergence of multiple-Q states. Furthermore, we show that four distinct multiple-Q states, including the SkX, are stabilized when interactions are introduced at specific high-harmonic wave vectors formed by linear combinations of two ordering wave vectors. These findings provide another guideline for engineering multiple-Q states and SkXs through the design of microscopic interaction models.

The rest of this paper is organized as follows: In Section 2, we introduce an effective spin model with momentum-resolved interactions on the square lattice. We also outline the numerical method based on simulated annealing. In Section 3, we show the results obtained by the simulated annealing. After presenting the result without high-harmonic wave-vector interactions, where only the single-Q spiral state appears, we discuss the effect of five-type high-harmonic wave-vector interactions on the single-Q spiral state. We show that only one of the high-harmonic wave-vector interactions results in the instabilities toward double-Q states. Finally, we conclude our results in Section 4.

2. Model and Method

As a starting point, we consider a localized spin model on a centrosymmetric square lattice in two dimensions; we set the lattice constant of the square lattice as unity. The classical spin model is given by

$$\begin{array}{c}\hfill \mathcal{H}=\sum _{i,j}[J_{ij}(S_i^x{S}_j^x+S_i^y{S}_j^y)+J_{ij}^z{S}_i^z{S}_j^z]-H\sum _i{S}_i^z,\end{array}$$

(1)

where ${\mathit{S}}_i=(S_i^x,S_i^y,S_i^z)$ denotes the classical localized spin at site i, with its magnitude set to unity. The Hamiltonian consists of the exchange interactions in the first term and the Zeeman coupling in the second term. In the first term, we consider the $xxz$-type exchange interaction between ith and jth spins, since such an Ising-type interaction often leads to multiple-Q and SkX instabilities [94,95,99,100,101,102,103]. We consider competing exchange interactions by taking the ferromagnetic exchange interaction between the nearest-neighbor spins and the antiferromagnetic exchange interactions between the further-neighbor spins so that the Fourier transform of $J_{ij}$ and $J_{ij}^z$ shows the energy minima at ${\mathit{Q}}_1=(\pi /5,0)$ and ${\mathit{Q}}_2=(0,\pi /5)$ for the isotropic case, i.e., $J_{ij}=J_{ij}^z$; the interactions at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ are equivalent owing to the fourfold rotational symmetry of the square lattice. As an example, such peak structures at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ can be realized when the parameters are chosen as $(J_1,J_2,J_3)\simeq (-1,0.4,0.0618034)$, $(J_1,J_2,J_3)\simeq (-1,0.3,0.12361)$, and so on, for the first-, second-, and third-neighbor interactions, $J_1$, $J_2$, and $J_3$.

Since the internal energy in the ground-state spin configuration is largely determined by the dominant ordering wave vectors, we simplify the interaction term accordingly:

$$\begin{array}{cc}\hfill \sum _{\langle ij\rangle }[J_{ij}(S_i^x{S}_j^x+S_i^y{S}_j^y)+J_{ij}^z{S}_i^z{S}_j^z]\sim & 2J\sum _{\nu }[S_{{\mathit{Q}}_{\nu }}^x{S}_{-{\mathit{Q}}_{\nu }}^x+S_{{\mathit{Q}}_{\nu }}^y{S}_{-{\mathit{Q}}_{\nu }}^y+(1+I^z)S_{{\mathit{Q}}_{\nu }}^z{S}_{-{\mathit{Q}}_{\nu }}^z]\hfill \\ \hfill & +2J^{\prime }\sum _{{\nu }^{\prime }}({\mathit{S}}_{{\mathit{Q}}_{{\nu }^{\prime }}}·{\mathit{S}}_{-{\mathit{Q}}_{{\nu }^{\prime }}}),\hfill \end{array}$$

(2)

where ${\mathit{S}}_{{\mathit{Q}}_{\nu }}=(S_{{\mathit{Q}}_{\nu }}^x,S_{{\mathit{Q}}_{\nu }}^y,S_{{\mathit{Q}}_{\nu }}^z)$ is the ${\mathit{Q}}_{\nu }$ component of localized spins ${\mathit{S}}_i=(S_i^x,S_i^y,S_i^z)$, which is derived by the Fourier transformation. The first term represents the contribution from the dominant ordering wave vectors, where $\nu =1, 2$ is the label of the dominant ordering wave vectors: ${\mathit{Q}}_1=(\pi /5, 0)$ and ${\mathit{Q}}_2=(0,\pi /5)$. The prefactor 2 means the contributions at $+{\mathit{Q}}_{\nu }$ and $-{\mathit{Q}}_{\nu }$. The second term represents the contribution from the other wave vectors, where ${\nu }^{\prime }$ is the label of the corresponding wave vectors. In order to clarify the role of the high-harmonic wave-vector interactions, we consider the following five situations:

(i)

${\mathit{Q}}_{1^{\prime }}=2{\mathit{Q}}_1$ and ${\mathit{Q}}_{2^{\prime }}=2{\mathit{Q}}_2$

(ii)

${\mathit{Q}}_{1^{\prime }}=3{\mathit{Q}}_1$ and ${\mathit{Q}}_{2^{\prime }}=3{\mathit{Q}}_2$

(iii)

${\mathit{Q}}_{1^{\prime }}={\mathit{Q}}_1+{\mathit{Q}}_2$ and ${\mathit{Q}}_{2^{\prime }}={\mathit{Q}}_1-{\mathit{Q}}_2$

(iv)

${\mathit{Q}}_{1^{\prime }}=2{\mathit{Q}}_1+2{\mathit{Q}}_2$ and ${\mathit{Q}}_{2^{\prime }}=2{\mathit{Q}}_1-2{\mathit{Q}}_2$

(v)

${\mathit{Q}}_{1^{\prime }}={\mathit{Q}}_1+2{\mathit{Q}}_2$, ${\mathit{Q}}_{2^{\prime }}={\mathit{Q}}_1-2{\mathit{Q}}_2$, ${\mathit{Q}}_{3^{\prime }}=2{\mathit{Q}}_1+{\mathit{Q}}_2$, and ${\mathit{Q}}_{4^{\prime }}=2{\mathit{Q}}_1-{\mathit{Q}}_2$.

We set the magnitude of the high-harmonic wave-vector interactions as $J^{\prime }\equiv J{\alpha }_{2{\mathit{Q}}_1}$ for case (i), $J^{\prime }\equiv J{\alpha }_{3{\mathit{Q}}_1}$ for case (ii), $J^{\prime }\equiv J{\alpha }_{{\mathit{Q}}_1+{\mathit{Q}}_2}$ for case (iii), $J^{\prime }\equiv J{\alpha }_{2{\mathit{Q}}_1+2{\mathit{Q}}_2}$ for case (iv), and $J^{\prime }\equiv J{\alpha }_{{\mathit{Q}}_1+2{\mathit{Q}}_2}$ for case (v). Among (i)–(v), the contributions at (i) and (ii) play an important role when the effect of $I^z$ is considered, whereas the contributions at (iii), (iv), and (v) are important in discussing multiple-Q instabilities consisting of ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$. We ignore the $xxz$-type anisotropic effect for the high-harmonic wave-vector interaction for simplicity. We also neglect the interactions for other wave vectors, since they have a negligible effect on the ground-state energy when the spin configuration is characterized by ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$. In other words, a superposition of wave vectors at symmetry-unrelated positions usually leads to energy loss compared to the single-Q spiral state [104]. Meanwhile, the above high-harmonic wave vectors of ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ sometimes assist the formation of multiple-Q states. In the following, we set $J=1$ as the energy unit of the model in Equation (1), and we fix $I^z=0.2$, which favors the easy-axis spin configuration; we confirm that the following results are not altered qualitatively for $I^z=0.1$. The second term in Equation (1) denotes the effect of the out-of-plane magnetic field with the strength H.

To explore the ground-state spin configurations as functions of ${\alpha }_{{\mathit{Q}}_{{\nu }^{\prime }}}$ and H, we employ the simulated annealing combined with the Metropolis algorithm for updating localized spins in real space. Each simulation begins with a randomly generated spin configuration at a high initial temperature $T_0\in [1,10]$. The system is then gradually cooled down to a final temperature $T={10}^{-4}$ using an exponential annealing schedule defined by $T_{n+1}=\alpha T_n$ in each Monte Carlo step, where the annealing rate $\alpha$ ranges from $0.99999$ to $0.999999$. After reaching T, we perform ${10}^5$–${10}^6$ Monte Carlo sweeps to calculate thermal averages. This procedure is repeated independently for each set of model parameters to ensure proper sampling. The system size is taken at $N={10}^2$, chosen to be commensurate with the ordering wave vectors ${\mathit{Q}}_1$; the system size is also taken at $N={40}^2$, where we confirmed that the obtained magnetic phases are the same.

To characterize the spin textures obtained in the simulations, we evaluate the magnetic moment resolved by wave vector $\mathit{q}$ and spin component $\mu =x,y,z$, denoted by $m_{\mathit{q}}^{\mu }$. It is computed via the spin structure factor $S_s^{\mu }\left(\mathit{q}\right)$ as

$$\begin{array}{cc}\hfill m_{\mathit{q}}^{\mu }& =\sqrt{{\displaystyle \frac{S_s^{\mu }\left(\mathit{q}\right)}{N}}},\hfill \end{array}$$

(3)

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

(4)

where $\langle \cdots \rangle$ denotes the Monte Carlo average over thermalized spin configurations, and N is the total number of lattice sites; ${\mathit{r}}_i$ represents the position vector at site i. From the rotational symmetry around the z-axis in spin space, the $xy$ components of the magnetic moments ${\left(m_{\mathit{q}}^{xy}\right)}^2={\left(m_{\mathit{q}}^x\right)}^2+{\left(m_{\mathit{q}}^y\right)}^2$ and spin structure factor $S_s^{xy}\left(\mathit{q}\right)=S_s^x\left(\mathit{q}\right)+S_s^y\left(\mathit{q}\right)$ are calculated. In addition, the uniform magnetization along the z-axis is calculated as $M^z=(1/N){\sum }_i{S}_i^z$.

3. Result

In this section, we show the results obtained by the simulated annealing. First, we discuss the magnetic phases appearing in the absence of high-harmonic wave-vector interactions in Section 3.1. Then, we discuss the effect of high-harmonic wave-vector interactions, which can be a source of multiple-Q states, in Section 3.2.

3.1. Without High-Harmonic Wave-Vector Interaction

Figure 1a shows the H dependence of the uniform magnetization $M^z$ for the model in Equation (1), where the effect of high-harmonic wave-vector interactions is neglected, i.e., $J^{\prime }=0$. There are three magnetic phases, which are distinguished by the behavior of the $M^z$-H curve: the single-Q vertical spiral (1Q VS) state, the single-Q conical spiral (1Q CS) state, and the fully polarized (FP) state. The 1Q VS state stabilized in the low-field regions has the single-Q spiral spin configuration at ${\mathit{Q}}_1$ or ${\mathit{Q}}_2$, as shown in Figure 1b. Owing to the presence of the $xxz$-type easy-axis anisotropy $I^z$, the z component of magnetic moments, ${\left(m_{{\mathit{Q}}_1}^z\right)}^2$ is larger than the $xy$ component, ${\left(m_{{\mathit{Q}}_1}^{xy}\right)}^2$. When the magnetic field H increases, ${\left(m_{{\mathit{Q}}_1}^z\right)}^2$ is slightly suppressed so as to gain the Zeeman energy. The real-space spin configuration of the 1Q VS state is given by ${\mathit{S}}_i=N_i(acos{\mathit{Q}}_1·{\mathit{r}}_i,0,m^z+bsin{\mathit{Q}}_1·{\mathit{r}}_i)$, where a, b, and $m^z$ stand for the variational parameters depending on $H^z$, and $N_i$ represents the normalization factor; the spiral plane lies on the $zx$ plane and its shape is elliptical. It is noted that the 1Q VS state with the spiral plane on the $zy$ plane has the same energy owing to the presence of the spin rotational symmetry around the z-axis.

For $H\simeq 0.9$, the 1Q VS state shows a phase transition to the 1Q CS state with a jump in the magnetization, as shown in Figure 1a. Accordingly, the magnetic structure of the 1Q CS state is modulated so as to vanish ${\left(m_{{\mathit{Q}}_1}^z\right)}^2$, as shown in Figure 1b. Thus, this transition is regarded as the spin-flop transition, where the spiral plane changes from the $zx$ plane to the $xy$ plane. Indeed, the real-space spin configuration is characterized by the circular spiral spin configuration on the $xy$ plane expressed as ${\mathit{S}}_i=N_i(acos{\mathit{Q}}_1·{\mathit{r}}_i,asin{\mathit{Q}}_1·{\mathit{r}}_i,m^z)$. The 1Q VS state continuously turns into the FP state with ${\mathit{S}}_i=(0,0,1)$.

Owing to the local spin length constraint, there is a finite intensity of the spin moment at high-harmonic wave vectors. Figure 1c,d show the H dependence of the magnetic moments at the second-harmonic wave vector $2{\mathit{Q}}_1$ and the third-harmonic wave vector $3{\mathit{Q}}_1$, respectively. The data show that both ${\left(m_{2{\mathit{Q}}_1}^{xy,z}\right)}^2$ and ${\left(m_{3{\mathit{Q}}_1}^{xy,z}\right)}^2$ become nonzero in the 1Q VS state, while they vanish in the 1Q CS state. The appearance of the intensities at high-harmonic wave vectors is attributed to the elliptical spiral plane in the 1Q VS state under the local spin length constraint. The behaviors of ${\left(m_{2{\mathit{Q}}_1}^{xy,z}\right)}^2$ and ${\left(m_{3{\mathit{Q}}_1}^{xy,z}\right)}^2$ against H are different from each other; ${\left(m_{2{\mathit{Q}}_1}^{xy,z}\right)}^2$ tend to be enhanced for larger H, while ${\left(m_{3{\mathit{Q}}_1}^{xy,z}\right)}^2$ tend to be suppressed for larger H. In particular, the intensity at the second-harmonic wave vector is due to the presence of the magnetic field [105,106]. Such a different behavior affects the stability of the 1Q VS state, as will be discussed in the subsequent section.

3.2. With High-Harmonic Wave-Vector Interaction

We discuss the effect of high-harmonic wave-vector interactions on the stability of the single-Q spiral state. We show the results under the five types of high-harmonic wave-vector interactions introduced in Section 2 one by one below.

3.2.1. Case (i)

We consider the case of the interaction at the second-harmonic wave vectors, i.e., ${\mathit{Q}}_{1^{\prime }}=2{\mathit{Q}}_1$ and ${\mathit{Q}}_{2^{\prime }}=2{\mathit{Q}}_2$. Figure 2a shows the phase diagram in the plane of ${\alpha }_{2{\mathit{Q}}_1}$ and H constructed by performing the simulated annealing. Although no additional phases appear in the phase diagram for ${\alpha }_{2{\mathit{Q}}_1}\le 0.8$, the phase boundary between the 1Q VS and 1Q CS states shifts upward as ${\alpha }_{2{\mathit{Q}}_1}$ increases. This shift in the phase boundary is due to the second-harmonic wave-vector interaction, which is understood from the results in Figure 1c. In other words, nonzero intensity at $2{\mathit{Q}}_1$ can contribute to the internal energy when the corresponding interaction is considered, which enlarges the region of the 1Q VS state with the intensity at $2{\mathit{Q}}_1$ in the spin structure factor. Thus, the second-harmonic wave-vector interaction tends to favor the 1Q VS state rather than the 1Q CS state.

3.2.2. Case (ii)

We show the phase diagram in Figure 2b when the third-harmonic wave-vector interaction with ${\mathit{Q}}_{1^{\prime }}=3{\mathit{Q}}_1$ and ${\mathit{Q}}_{2^{\prime }}=3{\mathit{Q}}_2$ is included. In contrast to the second-harmonic wave-vector interaction in Figure 2a, there is no change in the phase boundary, which means that the third-harmonic wave-vector interaction has little effect on the stability of the single-Q spiral state. Such a behavior is also understood from the H dependence of ${\left(m_{3{\mathit{Q}}_1}^{xy,z}\right)}^2$ in Figure 1d. Both ${\left(m_{3{\mathit{Q}}_1}^{xy}\right)}^2$ and ${\left(m_{3{\mathit{Q}}_1}^z\right)}^2$ become smaller in the vicinity of the phase boundary, which indicates that there is almost no energy gain by $J^{\prime }$. Thus, we can neglect the contribution of the third-harmonic wave-vector interaction in the effective spin model with the momentum-resolved interaction.

3.2.3. Case (iii)

We consider the effect of the high-harmonic wave-vector interaction of ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$, i.e., ${\mathit{Q}}_{1^{\prime }}={\mathit{Q}}_1+{\mathit{Q}}_2$ and ${\mathit{Q}}_{2^{\prime }}={\mathit{Q}}_1-{\mathit{Q}}_2$ [107]. As shown in the phase diagram in Figure 3, four additional phases appear in the phase diagram in the intermediate-field regions, which are denoted as SkX, 2Q I, 2Q II, and 2Q III. Among them, the most notable phase is the SkX, which consists of the periodic alignment of the skyrmions, as shown by the real-space spin configuration in Figure 4a. The SkX is characterized by a superposition of two vertical spiral waves at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$. As shown in the spin structure factor in Figure 4e,i, both $xy$ and z components show the fourfold symmetric peak structures at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ with the same intensity. Owing to the vortex-type spin configuration, this state exhibits a nonzero quantized skyrmion number $\pm 1$ calculated by the solid angle among three neighboring spins [108]; it is noted that the spin configuration with the skyrmion number of $+1$ [Figure 4a] has the same energy as that with the skyrmion number of $-1$ owing to the spin rotational symmetry around the z-axis. The appearance of the SkX with the skyrmion number of $+1$ in Figure 4a is due to initial spin configurations; we also obtain the spin configuration with the skyrmion number of $-1$ in the simulations for different initial spin configurations. Such a degeneracy can be lifted by considering the bond-dependent magnetic anisotropy arising from the discrete fourfold rotational symmetry of the square lattice [109]. The SkX is stabilized for large ${\alpha }_{{\mathit{Q}}_1+{\mathit{Q}}_2}$, which indicates that the high-harmonic wave-vector interaction at ${\mathit{Q}}_1+{\mathit{Q}}_2$ becomes the essence to induce the SkX [107]. Since the present SkX in centrosymmetric magnets shares the same real-space topological properties as the SkX stabilized by the Dzyaloshinskii–Moriya interaction, the spin waves in the present SkX are supposed to be topological [110,111]. In contrast to the previous study, we do not obtain the instability toward the rectangular SkX characterized by different intensities of magnetic moments at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ [107].

We describe the spin configurations for the other three double-Q states stabilized for nonzero ${\alpha }_{{\mathit{Q}}_1+{\mathit{Q}}_2}$, all of which are topologically trivial. The 2Q I state emerges when the magnetic field increases in the 1Q VS region. As shown by the real-space spin configuration in Figure 4b, the single-Q spiral modulation at ${\mathit{Q}}_2$ seems to be dominant. Indeed, the data in terms of the spin structure factor show that the 2Q I state is characterized by the dominant peak at ${\mathit{Q}}_2$ and the subdominant peak at ${\mathit{Q}}_1$, as shown in Figure 4f,j. Since there is no intensity of $S_s^z\left({\mathit{Q}}_1\right)$, this state is described by the superposition of the vertical spiral wave at ${\mathit{Q}}_2$ and the sinusoidal wave at ${\mathit{Q}}_1$, where the sinusoidal-wave direction is perpendicular to the spiral plane.

The 2Q II state is stabilized in the region adjacent to the 1Q CS state upon decreasing H. This spin configuration resembles the 1Q CS state, as shown by the real-space spin configuration in Figure 4c. By closely looking at the spin configuration, one finds that there is an oscillation in the z-spin component, as shown by the ${\mathit{Q}}_2$ component of the z-spin structure factor in Figure 4k. On the other hand, the $xy$ spin component is almost characterized by the ${\mathit{Q}}_1$ component, as shown in Figure 4g, which indicates the presence of the conical spiral configuration at ${\mathit{Q}}_1$. Thus, the spin configuration of the 2Q II state is represented by a superposition of the conical spiral wave at ${\mathit{Q}}_1$ and the sinusoidal wave at ${\mathit{Q}}_2$. Additionally in this case, the sinusoidal-wave direction is perpendicular to the spiral plane. The phase sequence at ${\alpha }_{{\mathit{Q}}_1+{\mathit{Q}}_2}=0.8$ is shown by the H–$M^z$ plot in Figure 5.

The 2Q III state appears in the narrow region sandwiched by the 2Q I and 2Q II states, as shown in the phase diagram in Figure 3. The real-space spin configuration of the 2Q III state is shown in Figure 4d; the $xy$-spin configuration resembles the conical spiral state, while the z-spin configuration resembles the SkX. In the spin structure factor, both $xy$ and z components show the anisotropic peak structures at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$, as shown in Figure 4h and Figure 4l, respectively.

Common to all the double-Q states and SkX, the double-Q superposition occurs so as to have the intensity at ${\mathit{Q}}_1+{\mathit{Q}}_2$ and ${\mathit{Q}}_1-{\mathit{Q}}_2$ in Figure 4e–l. This is due to the energy gain by the high-harmonic wave-vector interactions. In other words, the high-harmonic wave-vector interactions at ${\mathit{Q}}_1+{\mathit{Q}}_2$ lead to double-Q instabilities, which play an essential role in stabilizing the SkX.

3.2.4. Case (iv)

We consider a different type of high-harmonic wave-vector interaction at ${\mathit{Q}}_{1^{\prime }}=2{\mathit{Q}}_1+2{\mathit{Q}}_2$ and ${\mathit{Q}}_{2^{\prime }}=2{\mathit{Q}}_1-2{\mathit{Q}}_2$. Figure 6a shows the magnetic phase diagram against ${\alpha }_{2{\mathit{Q}}_1+2{\mathit{Q}}_2}$ and H. The results show no change in the phase boundaries while ${\alpha }_{2{\mathit{Q}}_1+2{\mathit{Q}}_2}$ is varied, which indicates that the high-harmonic wave-vector interaction at $2{\mathit{Q}}_1+2{\mathit{Q}}_2$ plays a less significant role in inducing multiple-Q states.

3.2.5. Case (v)

The last case is the inclusion of the high-harmonic wave-vector interaction at ${\mathit{Q}}_{1^{\prime }}={\mathit{Q}}_1+2{\mathit{Q}}_2$, ${\mathit{Q}}_{2^{\prime }}={\mathit{Q}}_1-2{\mathit{Q}}_2$, ${\mathit{Q}}_{3^{\prime }}=2{\mathit{Q}}_1+{\mathit{Q}}_2$, and ${\mathit{Q}}_{4^{\prime }}=2{\mathit{Q}}_1-{\mathit{Q}}_2$. Similarly to the result in Figure 6a, the phase boundary in Figure 6b shows no dependence of ${\alpha }_{{\mathit{Q}}_1+2{\mathit{Q}}_2}$. Thus, the high-harmonic wave-vector interaction at ${\mathit{Q}}_1+2{\mathit{Q}}_2$ also plays a less significant role in inducing multiple-Q states.

4. Conclusions

In conclusion, we have examined the role of the high-harmonic wave-vector interaction on the stability of single-Q and multiple-Q states in a tetragonal crystalline system. Using simulated annealing applied to an effective spin model on the two-dimensional centrosymmetric square lattice, we have investigated how these interactions modulate the single-Q spiral spin configurations. Our findings revealed that specific high-harmonic wave-vector components play a stabilizing role in maintaining the single-Q vertical spiral state. More importantly, when the interactions involve wave vectors corresponding to the sum of two orthogonal ordering wave vectors, the system exhibits a rich variety of multiple-Q phases, including three distinct double-Q states and a square SkX. These results highlight the essential role of high-harmonic wave-vector interactions in generating complex noncollinear spin textures, which provide a new microscopic pathway toward engineering exotic magnetic phases in centrosymmetric systems. One of the important consequence is that the interactions at ${\mathit{Q}}_1+{\mathit{Q}}_2$ and ${\mathit{Q}}_1-{\mathit{Q}}_2$ can be a source of various multiple-Q phases rather than those at other high-harmonic wave vectors. Therefore, when applying the model to real materials, it is desirable to evaluate the effective interactions not only at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$, but also at ${\mathit{Q}}_1+{\mathit{Q}}_2$ and ${\mathit{Q}}_1-{\mathit{Q}}_2$, based on first-principles calculations. Furthermore, recent studies have demonstrated that topological spin textures such as hedgehogs and hopfions can be represented by superpositions of spin density waves [112]. Given that such superpositions give rise to intensities in the high-harmonic wave-vector components of the spin structure factor, the interaction explored in this work may play a role in stabilizing these exotic spin phases.