Role of Single-Ion Anisotropy in Stabilizing Higher-Order Skyrmion Crystals in D_3d-Symmetric Magnets

Abstract

We investigate the role of single-ion anisotropy in stabilizing higher-order skyrmion crystal phases in centrosymmetric magnets under D_3d symmetry. Using a classical spin model that incorporates both a local single-ion anisotropy arising from the two-dimensional crystal symmetry and a D_3d-type magnetic anisotropy originating from the D_3d point-group symmetry, we perform simulated annealing calculations to explore the ground-state spin configurations. We find that a skyrmion crystal with a skyrmion number of two is stabilized over a wide range of parameters of single-ion anisotropy and D_3d-type anisotropy. We also show that the skyrmion core position shifts from an interstitial site to an on-site location as the magnitude of the easy-axis single-ion anisotropy increases. Furthermore, we demonstrate that the magnetic field drives a variety of topological phase transitions depending on the sign and magnitude of the single-ion and D_3d-type anisotropies. These results provide a possible microscopic understanding of how complex topological spin textures can be stabilized in centrosymmetric D_3d magnets, suggesting that multiple phases with topological spin textures could emerge even in the absence of the Dzyaloshinskii–Moriya interaction.

1. Introduction

Magnetic skyrmions, particle-like topological spin textures characterized by a quantized winding number [1,2,3,4], have been extensively studied owing to their fundamental importance in topological magnetism and their potential applications in spintronic devices [5,6,7,8,9,10,11,12,13]. The nontrivial topology of skyrmion spin textures gives rise to emergent electromagnetism, such as the topological Hall effect [14,15,16,17,18,19,20], and endows them with remarkable stability against perturbations [13,21,22,23,24,25]. In many noncentrosymmetric systems, skyrmion crystals (SkX), which correspond to a periodic alignment of skyrmions, are stabilized by the Dzyaloshinskii–Moriya (DM) interaction [26,27], which favors spin twisting [3,4,28,29,30,31,32,33]. This mechanism has been realized in various chiral and polar magnets, including metallic B20 compounds like MnSi [17,34,35,36,37,38], Fe_1−xCo_xSi [39,40,41,42], and FeGe [43,44], and multiferroic insulators like Cu₂OSeO₃ [45,46,47] and VOSe₂O₅ [48,49], where the interplay between ferromagnetic exchange interaction and the DM interaction leads to the formation of hexagonal and tetragonal SkXs under external magnetic fields.

In contrast, the emergence of SkXs in centrosymmetric magnets has attracted increasing attention in recent years, as it demonstrates that the DM interaction is not a prerequisite for the formation of SkX spin textures [50,51,52]. In such systems, competing symmetric exchange interactions and magnetic anisotropies can induce multiple-Q states, in which several spin density waves with distinct propagation wave vectors coexist [53,54,55,56,57]. The superposition of these spin density waves naturally gives rise to noncoplanar spin configurations with finite scalar spin chirality, thereby mimicking the topological properties of skyrmions even in the absence of DM interaction [58]. This mechanism provides an alternative and broader route toward realizing SkXs in a wide class of frustrated magnets, including both itinerant and localized spin systems on triangular and square lattices. Experimental realizations have also been reported in centrosymmetric magnets such as the rare earth intermetallics Gd₂PdSi₃ [59,60,61,62,63], Gd₃Ru₄Al₁₂ [64,65,66,67], and GdRu₂Si₂ [68,69,70,71].

Among the various types of SkXs, higher-order SkXs, which possess a skyrmion number greater than one, have recently been recognized as intriguing topological spin textures [72,73,74,75]. These textures host multiple vortex cores within a single magnetic unit cell and exhibit intricate internal structures distinct from conventional SkX with the skyrmion number of one. The growing interest in higher-order SkXs stems from their potential to generate topological Hall and nonreciprocal transport responses even without external magnetic fields, net magnetization, or relativistic spin–orbit coupling [76,77,78]. Theoretically, such higher-order SkXs can emerge from multi-spin interactions [73,79,80,81] and bond-dependent anisotropy [82,83], whereas experimental observations of higher-order SkX in centrosymmetric materials have not yet been reported. Nevertheless, the microscopic mechanisms that determine the stability range, internal structure, and response of these higher-order SkXs remain incompletely understood. For the experimental realization of such higher-order SkXs, a further theoretical understanding of their stabilization mechanisms is crucial.

The D_3d point-group symmetry offers an ideal platform for exploring higher-order SkXs with the skyrmion number of two, as its symmetry allows characteristic anisotropic spin interactions that naturally bring about the formation of such phases [84]. In this work, we investigate a classical spin model on a two-dimensional triangular lattice possessing D_3d symmetry by focusing on the role of single-ion anisotropy on the stability of the higher-order SkX. By performing simulated annealing calculations, we construct the low-temperature zero-field phase diagram as functions of both the single-ion anisotropy and the D_3d-type magnetic anisotropy. Our results reveal a broad region in parameter space where the higher-order SkX becomes energetically favorable. Moreover, our analysis reveals that the application of a magnetic field induces a series of topological phase transitions, whose nature depends sensitively on both the sign and the strength of the single-ion and D_3d-type anisotropies. These findings shed light on the possibility of the higher-order SkXs in centrosymmetric D_3d magnets even without the DM interaction.

The remainder of this paper is organized as follows. In Section 2, we introduce an effective spin model on the triangular lattice with D_3d symmetry and present the relevant magnetic anisotropy terms, including the single-ion and D_3d-type anisotropies. The numerical method based on simulated annealing is also described there. In Section 3, we present the ground-state phase diagram at zero magnetic field and show the stability region of the higher-order SkXs by the cooperative effects of the two anisotropies. We then discuss in detail the real-space spin textures, scalar spin chirality distributions, and spin structure factors appearing in the phase diagram. We also examine the effects of an external magnetic field and demonstrate a variety of field-induced phase transitions for representative parameter sets. Finally, Section 4 summarizes our main findings and discusses their implications for the realization of higher-order SkXs in centrosymmetric magnets.

2. Model and Method

To clarify the influence of single-ion anisotropy on the stability of higher-order SkX under centrosymmetric D_3d point-group symmetry, we consider a classical spin model with magnetic anisotropic interactions on a two-dimensional triangular lattice. The Hamiltonian is written as

$$\begin{array}{c}\hfill \mathcal{H}=-2J\sum _{\nu }\sum _{\alpha \beta }{I}_{{\mathit{Q}}_{\nu }}^{\alpha \beta }{S}_{{\mathit{Q}}_{\nu }}^{\alpha }{S}_{-{\mathit{Q}}_{\nu }}^{\beta }-A\sum _i{\left(S_i^z\right)}^2-H\sum _i{S}_i^z,\end{array}$$

(1)

where S_i denotes a classical spin vector with unit length (|S_i| = 1) at site i. The spin components in momentum space at wave vector Q_ν are defined as ${\mathit{S}}_{{\mathit{Q}}_{\nu }}=(1/\sqrt{N}){\sum }_i{\mathit{S}}_i{e}^{-i{\mathit{Q}}_{\nu }·{\mathit{r}}_i}$, with N and r_i being the total number of spins and the position vector at site i, respectively. The exchange constant J sets the energy scale and is fixed to J = 1 throughout this study. The second term in Equation (1) represents the single-ion anisotropy A, originating from spin–orbit coupling under the local crystalline electric field. Its magnitude is not solely determined by crystal-field parameters, but also depends on the electronic configuration of the magnetic ion and hybridization effects. It controls the preferred spin orientation relative to the z axis: A > 0 corresponds to an easy-axis anisotropy, whereas A < 0 favors an easy-plane alignment. For rare earth magnets, ions such as Gd³⁺, with nearly vanishing orbital angular momentum, are expected to exhibit very weak single-ion anisotropy, whereas ions such as Dy³⁺, possessing large orbital angular momentum and strong spin–orbit coupling, typically show much stronger easy-axis anisotropy. In this study, A is treated as a tunable parameter to clarify the qualitative effects of weak and strong single-ion anisotropy on the stability of higher-order SkXs. Such an effect of the single-ion anisotropy has been extensively studied in discussing the stability of the SkXs in various systems, including noncentrosymmetric systems [38,53,85,86,87,88,89]. The third term in Equation (1) stands for the Zeeman coupling in the presence of the out-of-plane magnetic field H.

We restrict the interaction to three symmetry-equivalent ordering wave vectors

$$\begin{array}{c}\hfill {\mathit{Q}}_1=(\pi /3,0),{\mathit{Q}}_2=(-\pi /6,\sqrt{3}\pi /6),{\mathit{Q}}_3=(-\pi /6,-\sqrt{3}\pi /6),\end{array}$$

(2)

by supposing that the dominant magnetic instabilities on the triangular lattice occur at these ordering wave vectors. Such a situation arises from competing exchange interactions in frustrated insulating magnets [90,91,92,93] and from the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction [94,95,96] in itinerant metallic magnets [97] so that the instability toward the single-Q spiral state is induced [98,99,100]. All other q components are neglected under the assumption that the magnetic susceptibility exhibits sharp peaks at these Q_ν. Such a reduced description effectively captures the essential physics of multiple-Q states in a numerically efficient way and has been successfully applied to a variety of frustrated systems, including SkX and other multiple-Q states [101,102,103,104,105,106,107,108], such as Y₃Co₈Sn₄ [109], EuPtSi [110], and EuNiGe₃ [111].

The matrix $I_{{\mathit{Q}}_{\nu }}^{\alpha \beta }$ in Equation (1) encodes the symmetry-allowed anisotropic exchange interactions associated with the trigonal D_3d point-group symmetry. Its explicit tensor form is given by

$$\begin{array}{cc}\hfill I^{\mathrm{D}3\mathrm{d}}\equiv & I_{{\mathit{Q}}_1}^{yz}=I_{{\mathit{Q}}_1}^{zy}=-{\displaystyle \frac{2}{\sqrt{3}}}{I}_{{\mathit{Q}}_2}^{xz}=-{\displaystyle \frac{2}{\sqrt{3}}}{I}_{{\mathit{Q}}_2}^{zx}=-2I_{{\mathit{Q}}_2}^{yz}=-2I_{{\mathit{Q}}_2}^{zy}\hfill \\ \hfill & ={\displaystyle \frac{2}{\sqrt{3}}}{I}_{{\mathit{Q}}_3}^{xz}={\displaystyle \frac{2}{\sqrt{3}}}{I}_{{\mathit{Q}}_3}^{zx}=-2I_{{\mathit{Q}}_3}^{yz}=-2I_{{\mathit{Q}}_3}^{zy}.\hfill \end{array}$$

(3)

Here, I^D3d parameterizes the D_3d-type anisotropy, which favors sinusoidal spin modulations perpendicular to the propagation vectors Q_ν. In contrast to the compass- and Kitaev-type bond-dependent anisotropies [112,113,114,115,116,117], which often favor purely in-plane sinusoidal spin modulations [82], the present D_3d-type anisotropy gives rise to spin modulations whose oscillation plane involves both in-plane and out-of-plane components [58]. This anisotropy, enabled under trigonal crystal symmetry, introduces a momentum-resolved anisotropic exchange interaction that can promote noncoplanar multiple-Q spin textures like the higher-order SkX [84].

To determine the equilibrium spin configurations at low temperatures, we perform simulated annealing based on the Metropolis algorithm on 24 × 24 lattices with periodic boundary conditions. Starting from a high temperature T₀ = 1, the system is gradually cooled according to $T_{n+1}=\tilde{\alpha }{T}_n$ with $\tilde{\alpha }$ between 0.999995 and 0.999999, until the final temperature reaches T = 0.0001. Each annealing sequence consists of 10⁵–10⁶ Monte Carlo sweeps for both equilibration and measurements. Independent simulations are conducted from different random initial configurations to ensure reproducibility and to identify the ground-state phases.

The magnetic structures are characterized through both momentum-space and real-space observables. The spin structure factor for the spin component α = x, y, z is defined as

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

(4)

where q is the wave vector in the Brillouin zone. The in-plane contribution is given by $S_s^{xy}\left(\mathit{q}\right)=S_s^x\left(\mathit{q}\right)+S_s^y\left(\mathit{q}\right)$. The corresponding Fourier amplitudes at Q_ν are obtained from

$$\begin{array}{c}\hfill m_{{\mathit{Q}}_{\nu }}^{\alpha }=\sqrt{{\displaystyle \frac{S_s^{\alpha }\left({\mathit{Q}}_{\nu }\right)}{N}}}.\end{array}$$

(5)

The averaged magnetization along the field direction is given by

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

(6)

The topological nature of each spin configuration is evaluated via the scalar spin chirality

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

(7)

where μ = (u, d) distinguishes upward and downward triangles, and (i, j, k) denote the three sites forming a triangle centered at R in counterclockwise order. This quantity is related to a discrete measure of the skyrmion number [118] and serves to distinguish topologically trivial and nontrivial magnetic phases in the low-temperature phase diagram.

In the following, we vary the parameters single-ion anisotropy A, D_3d-type anisotropic interaction I^D3, and magnetic field H. We summarize model parameters used in Section 3 in

Table 1. Summary of model parameters and numerical settings used in the simulated annealing calculations. In all cases, the triangular lattice with the system size 24 × 24 under the D_3d symmetry is adopted. ✓ represents the presence of the SkX phases. χ^sc(H) represents the H dependence of the scalar spin chirality when the magnetic field increases in the SkX phase.

Case	$\mathit{A}/\mathit{J}$	${\mathit{I}}^{\mathbf{D}3\mathbf{d}}/\mathit{J}$	SkX	${\mathit{\chi }}^{\mathbf{sc}}\left(\mathit{H}\right)$
Zero-field phase diagram	−0.5 to 0.5	0 to 0.5	✓	–
Weak D3d, easy-axis	0.3	0.1	✓	jump
Weak D3d, easy-plane	−0.3	0.1	–	–
Strong D3d, easy-axis	0.3	0.3	✓	jump
Strong D3d, easy-plane	−0.3	0.3	✓	continuous

.

3. Results

We show the stability of higher-order SkX by focusing on the interplay between the D_3d-type anisotropic interaction I^D3d and single-ion anisotropy A on the triangular lattice. First, we discuss the results at zero magnetic field in Section 3.1, where the higher-order SkX with the skyrmion number of two is relatively robust against the easy-axis single-ion anisotropy compared to the easy-plane one. Then, we discuss the field-induced phase transitions by introducing the magnetic field in Section 3.2.

3.1. Zero Magnetic Field

Figure 1 presents the low-temperature phase diagram obtained by the simulated annealing at the zero magnetic field (H = 0), which reveals how the interplay between the single-ion anisotropy A and the D_3d-type anisotropy I^D3d governs the emergence of distinct magnetic phases. A prominent feature is the stabilization of the higher-order SkX with the skyrmion number of two (SkX-2) in a broad region at intermediate and large I^D3d [58]. In contrast, smaller I^D3d values favor a double-Q (2Q) phase, and for weak I^D3d combined with strong |A|, the system stabilizes a single-Q (1Q) spiral phase. These results demonstrate that the D_3d-type anisotropy plays a key role in stabilizing noncoplanar multiple-Q states, while the single-ion anisotropy modulates the degree of spin canting and the spatial structure in magnetic phases.

In the absence of the single-ion anisotropy (A = 0), the SkX-2 state emerges as a robust ground state for I^D3d ≳ 0.1. The real-space spin configuration in the case of I^D3d = 0.3 in Figure 2a exhibits a periodic triangular array of swirling textures, each magnetic unit cell containing two inequivalent vortex cores with different topological winding senses. The color scale represents $S_i^z$, where red and blue correspond to spins pointing up and down, respectively, while the arrows indicate the in-plane components forming a swirling vortex pattern. This configuration comprises two types of vortices: vortices centered at $S_i^z=+1$ with the vorticity +1, and antivortices centered at $S_i^z=-1$ with the vorticity −2. The asymmetry between their populations results in a net topological charge of n_sk = 2 per magnetic unit cell. Such a composite topological texture arises from the phase-locked sinusoidal superposition of three spin density waves with propagation ordering wave vectors Q₁, Q₂, and Q₃, which are mutually separated by 120° [84]. The scalar spin chirality map in Figure 3a shows a uniform hexagonal array of positive ${\chi }_{\mathit{R}}^{\mathrm{sc}}$, corresponding to the vortex and antivortex centers. The overall positive average of ${\chi }_{\mathit{R}}^{\mathrm{sc}}$ indicates a nontrivial topological state; it is noted that the state with the opposite sign of the scalar spin chirality has the same energy. The spin structure factor in Figure 4a displays sharp Bragg peaks at all three Q_ν positions in both the in-plane and out-of-plane components, confirming that this higher-order SkX arises from the coherent superposition of triple-Q modulations. This spin configuration spontaneously shows a nonzero averaged magnetization, $M^z\ne 0$, which arises from a triple-Q sinusoidal superposition, even at zero magnetic field.

When the single-ion anisotropy A is introduced, the SkX-2 state continuously evolves without changing its topological character over a wide range of parameters. For the easy-axis case (A > 0), as shown in Figure 2b, the spins acquire stronger out-of-plane polarization, leading to a more pronounced z-spin modulation. According to the out-of-plane polarization enhancement, the local scalar spin chirality becomes smaller, as shown in Figure 3b. The spin structure factor correspondingly displays an increased amplitude in the out-of-plane components at the Q_ν points, while the in-plane components are slightly suppressed, as shown in Figure 4b. In contrast, for the easy-plane case (A < 0), the out-of-plane spin component is reduced, while the in-plane spin component is enhanced, as shown by both the real-space and momentum-space plots in Figure 2c and Figure 4c. Such a change has less influence on the local scalar spin chirality, as shown in Figure 3c.

The systematic evolution of these quantities is summarized in Figure 5. As shown in Figure 5a, the total magnetization M_z tends to increase gradually with A; there is a kink structure around A ≃ 0.2. The scalar spin chirality χ^sc in Figure 5b exhibits a dome-like variation, which is maximized around A ≃ −0.2. For the easy-axis anisotropy, χ^sc is rapidly suppressed owing to the collinear tendency of the internal spin configuration. The Fourier amplitudes of in-plane and out-of-plane components in Figure 5c,d show opposite trends with respect to A, which is understood from the role of the single-ion anisotrpy; A > 0 tends to favor the z-directional oscillations, while A < 0 tends to favor the xy-plane oscillations. These results clearly demonstrate that the degree of the easy-axis and easy-plane single-ion anisotropy continuously tunes the spin canting angle and the noncoplanarity in the SkX-2 phase while keeping the skyrmion number.

A particularly remarkable effect of the single-ion anisotropy appears in the spatial position of the skyrmion cores [119]. In the easy-plane regime (A < 0), the skyrmion cores are located at interstitial sites of the triangular lattice, i.e., at the centers of the triangle plaquettes, as shown in Figure 2c. With increasing A, the skyrmion cores shift toward the lattice sites, and in the easy-axis regime (A > 0), they become centered directly on the atomic positions, as shown in Figure 2b. This crossover from interstitial to on-site positions is discontinuous, taking place around A ≃ 0.2, where the kink structure appears in the magnetization shown in Figure 5a, indicating the modification of the underlying multiple-Q interference pattern.

For weaker D_3d-type anisotropy, the SkX-2 phase becomes unstable and gives way to simpler magnetic structures. At I^D3d = 0.08 with small single-ion anisotropy, a double-Q state emerges, as indicated in Figure 2d, Figure 3d and Figure 4d. In this phase, two symmetry-related ordering wave vectors dominate the spin structure, producing a noncoplanar but spatially anisotropic pattern with density waves in terms of the scalar spin chirality. In contrast to the SkX-2 phase, this phase does not show the averaged scalar spin chirality in the whole system. Upon increasing |A|, single-Q spiral phases become stabilized. When the anisotropy is easy-axis, the spiral is of the vertical type, characterized by spin rotations mainly along the perpendicular plane to the xy plane, whose real-space and momentum-space spin quantities are shown in Figure 2e and Figure 4e, respectively. Meanwhile, for the easy-plane case, a conical spiral wave develops, in which spins mainly lies in an in-plane, although the z-spin modulation is weak but finite owing to the presence of the D_3d-type anisotropy, as shown in Figure 2f and Figure 4f. These results indicate that the D_3d-type anisotropy is essential for stabilizing triple-Q SkX at zero magnetic field, while the single-ion anisotropy continuously tunes the balance between in-plane and out-of-plane spin modulations, controlling the noncoplanarity and the position of the skyrmion cores.

3.2. Nonzero Magnetic Fields

Next, we investigate the influence of an external magnetic field on the ground-state spin configurations discussed in the previous section and demonstrate a variety of field-induced phase transitions across four representative parameter sets.

3.2.1. Weak $D_{3\mathrm{d}}$-Type and Easy-Axis Anisotropies

We consider the case of weak D_3d-type anisotropy under an easy-axis single-ion anisotropy, corresponding to I^D3d = 0.1 and A = 0.3, when the out-of-plane magnetic field H is applied. At zero magnetic field, the system forms the single-Q state without the net scalar spin chirality. By increasing the magnetic field, the single-Q state turns into the SkX-2 with the jumps of the magnetization [Figure 6a] and the scalar spin chirality [Figure 6b]. Although the Fourier amplitudes of the magnetic moments at Q₁–Q₃ change in this state for 0.15 ≲ H ≲ 0.65, as shown in Figure 6c,d, the real-space spin configurations are almost the same, as shown in Figure 7a–d. The slight difference found in the position of the skyrmion core, which is located at the lattice (interstitial) site for small (large) H. This result indicates that the SkX-2 can appear even for small D_3d-type anisotropy when the external magnetic field is applied.

As the magnetic field is further increased to H ≳ 0.65, the system undergoes a topological transition from the SkX with the skyrmion number of two to another SkX with the skyrmion number of one, followed by a topologically trivial multiple-Q state. The real-space snapshots in Figure 7e,f show that the core position of $S_i^z<0$ changes from the vortex with the vorticity 2 to that with the vorticity 1. This results in the reduction of the skyrmion number from two to one in the magnetic unit cell. Indeed, this transition is accompanied by a sudden drop in χ^sc, as shown in Figure 6b. Finally, the SkX with the skyrmion number of one changes into the topologically trivial triple-Q state without the skyrmion number. In this way, the topological transitions in terms of the skyrmion number occur as 0 → 2 → 1 → 0 in this parameter region.

3.2.2. Weak $D_{3\mathrm{d}}$-Type and Easy-Plane Anisotropies

Next, we discuss the magnetic-field evolution for the case of weak D_3d-type anisotropy under an easy-plane single-ion anisotropy, corresponding to I^D3d = 0.1 and A = −0.3, where a coplanar single-Q spiral state is the ground state at zero magnetic field. Upon applying a small out-of-plane magnetic field, the magnetization is gradually developed, as shown in Figure 8a, while the scalar spin chirality remains zero in the whole H region, as shown in Figure 8b. This indicates that the topologically nontrivial spin configuration, i.e., SkX, does not appear in the case of the easy-plane single-ion anisotropy for weak D_3d-type anisotropy.

Meanwhile, one finds that the single-Q spiral state changes into the triple-Q state close to the saturation magnetic field, as shown in Figure 8c,d. The spin configuration of this triple-Q state is characterized by the periodic alignment of vortex and antivortex without the z-spin oscillations, whose real-space spin configuration is presented in Figure 9; no net scalar spin chirality emerges in this state. A similar triple-Q state has also been found in other spin models, where the frustrated competing interactions play an important role [54,120].

3.2.3. Strong $D_{3\mathrm{d}}$-Type and Easy-Axis Anisotropies

Next, we discuss the situation with strong D_3d-type anisotropy, I^D3d = 0.3. In the case of the easy-axis single-ion anisotropy at A = 0.3, the zero-field state corresponds to the SkX-2. The SkX-2 phase is robust against the external magnetic field up to H ≃ 1.3, which is in contrast to the weak D_3d-type anisotropy with I^D3d = 0.1. The spin configuration of the SkX-2 smoothly changes by increasing the magnetic field, as found by the behaviors of the magnetization in Figure 10a and the scalar spin chirality in Figure 10b. For the small magnetic field around H ≃ 0.2, the intensities of magnetic moments at Q₁–Q₃ are inequivalent, as shown in Figure 10c,d, although their real-space spin configurations are almost the same, as shown in Figure 11a,b, indicating the same topological property irrespective of H.

By further increasing the magnetic field, the SkX-2 turns into the SkX with the skyrmion number of one at H ≃ 1.3, similar to the case of I^D3d = 0.1. The real-space spin configuration in Figure 11c resembles that in Figure 7e. The SkX with the skyrmion number of one finally turns into the fully polarized state with a clear jump of all the physical quantities in Figure 10. Thus, the strong D_3d-type anisotropy tends to favor the SkX-2 even under the magnetic field, and the topological transition between the SkXs with different skyrmion numbers is commonly found in the case of the easy-axis single-ion anisotropy.

3.2.4. Strong $D_{3\mathrm{d}}$-Type and Easy-Plane Anisotropies

Finally, we discuss the case of strong D_3d-type anisotropy under an easy-plane single-ion anisotropy, corresponding to I^D3d = 0.3 and A = −0.3, when the out-of-plane magnetic field H is applied. At zero magnetic field, the SkX-2 is stabilized, where the finite scalar spin chirality is present. As the magnetic field increases, the spins tend to align the magnetic-field direction in a gradual way while keeping the triple-Q spin configuration with equal intensities as shown in Figure 12c,d. Such a gradual change of the spin configuration is found in the continuous change of the magnetization in Figure 12a and the scalar spin chirality in Figure 12b. The real-space spin configurations for different values of H are shown in Figure 13a–c, where the sign of $S_i^z$ in the skyrmion core around $S_i^z=-1$ at zero field is reversed while increasing the magnetic field. Thus, the topological phase transition occurs in the intermediate field, although the spin and scalar spin chirality quantities show a continuous change. Such a field-induced change of the spin configuration is clearly different from that for the easy-axis single-ion anisotropy, the latter of which shows a clear phase transition from the SkX-2 to the SkX with the skyrmion number of one.

4. Conclusions

We have theoretically investigated the emergence and stability of higher-order SkX in centrosymmetric magnets with the trigonal D_3d symmetry by systematically analyzing the effects of the single-ion anisotropy and the anisotropic exchange interaction that arises from the D_3d symmetry. Using real-space simulated annealing calculations of a classical spin model on the two-dimensional triangular lattice, we constructed comprehensive phase diagrams at both zero and finite magnetic fields and clarified the microscopic mechanisms responsible for stabilizing nontrivial multiple-Q topological spin textures.

At zero magnetic field, we demonstrated that the cooperative interplay between the D_3d-type magnetic anisotropy and the single-ion anisotropy gives rise to a higher-order SkX with the skyrmion number of two. This SkX phase emerges as a robust ground state in a wide parameter range of both parameters. We also showed that the position of the skyrmion core undergoes a crossover from on-site to interstitial with changing the degree of single-ion anisotropy, indicating the sensitivity of the real-space spin texture to local anisotropic terms. Under finite magnetic fields, a series of field-induced topological transitions was observed. For both weak and strong D_3d-type anisotropies, the system exhibits a sequential evolution of spin textures as 2 → 1 → 0 in terms of the skyrmion number under the easy-axis single-ion anisotropy. The SkX with the skyrmion number of two remains remarkably stable over a wide field range, particularly in the presence of strong D_3d-type anisotropy. In contrast, under easy-plane anisotropy, the higher-order SkX only appears for the strong D_3d-type anisotropy, which turns into the topologically trivial triple-Q state without the skyrmion number. These results indicate that the D_3d-type anisotropy is essential for the formation and stability of higher-order SkXs, while the single-ion anisotropy affects the topological phase transitions driven by the magnetic field. Our findings provide microscopic insight into the stabilization mechanism of higher-order SkXs in centrosymmetric trigonal magnets, suggesting that the interplay between local and momentum-resolved anisotropies can realize complex noncoplanar spin textures even in the absence of the DM interaction.

Finally, we remark on the relation between the present phase diagrams and real materials. Rare earth intermetallic compounds such as Gd₂PdSi₃ and GdRu₂Si₂ were mentioned as representative examples of centrosymmetric skyrmion-hosting systems, rather than as direct realizations of the D_3d-symmetric model considered here. From a symmetry viewpoint, materials such as NiI₂ are more closely related [74]; however, realistic modeling of such systems typically involves a larger number of microscopic interaction parameters. The purpose of the present study is therefore to construct a minimal model and a qualitative phase diagram that clarifies generic stabilization mechanisms of higher-order SkXs in centrosymmetric D_3d magnets.

We also comment on the system-size dependence and the physical relevance of the model parameters. Although topological phase transitions, especially those involving changes in the skyrmion number such as the 2 → 1 → 0 transitions, can be sensitive to finite-size effects, the present simulations on 24 × 24 lattices are intended to capture qualitative and robust features of the phase diagram rather than precise critical boundaries. From a materials perspective, the higher-order SkX is stabilized for intermediate values of the D_3d-type anisotropy I^D3d/J combined with weak to moderate single-ion anisotropy A/J. Such a parameter regime is expected to be physically realistic for centrosymmetric D_3d magnets with appreciable spin–orbit coupling and trigonal crystal fields, and the present results therefore provide qualitative guidance for experimental searches of higher-order SkX phases.