Stability of Higher-Order Skyrmion Crystals Under Competing Magnetic Anisotropies in D_3d Systems

Abstract

We investigate the stability of higher-order skyrmion crystals with large topological charges in the presence of crystal-dependent magnetic anisotropies. Focusing on the competition between two types of bond-dependent anisotropy allowed by $D_{3\mathrm{d}}$ crystalline symmetry on a two-dimensional triangular lattice, we systematically construct a low-temperature magnetic phase diagram using simulated annealing. Our analysis reveals that the stability of the higher-order skyrmion crystal with skyrmion number of two is strongly controlled by the relative sign of the bond-dependent anisotropy to the $D_{3\mathrm{d}}$-type anisotropy: a positive anisotropy, which favors spin oscillations perpendicular to the ordering wave vector, enhances its stability, whereas a negative anisotropy, favoring oscillations parallel to the ordering wave vector, suppresses it and instead stabilizes a topologically trivial double-Q state. We further examine the field evolution of these phases under an out-of-plane magnetic field and show that distinct types of skyrmion crystals with the skyrmion number of one emerge in the intermediate-field regime. These results highlight that the competition between different magnetic anisotropies in crystalline systems is a key factor governing the stability of both zero-field and field-induced skyrmion crystals.

1. Introduction

Magnetic skyrmions—particle-like, topologically nontrivial spin textures—were first conceived in field theory as solitonic configurations [1,2] and later realized in the field of magnetism as nanoscale swirls of local magnetization [3,4,5]. The unique emergent electromagnetics of skyrmions—manifesting in a topological Hall response [6,7,8,9,10,11,12,13,14], emergent inductance [15,16,17,18,19,20,21], resilience to moderate disorder [22,23,24,25,26,27], and efficient current-driven mobility [28,29,30,31,32,33,34,35]—have established them as a central theme in contemporary spintronics and magnonics [36,37]. In crystalline magnets, skyrmion crystals (SkXs) are most commonly stabilized in noncentrosymmetric hosts [11,38,39,40], where spin–orbit coupling and broken inversion symmetry produce Dzyaloshinskii–Moriya (DM) interactions that favor helical twists [41,42] and, under magnetic fields, triangular SkXs [5,43,44,45,46]. Beyond such chiral settings, a rapidly growing body of work has uncovered a variety of stabilization routes that do not rely on the DM interaction, broadening both the materials base and the controllability of SkX phases [47,48,49,50,51].

A unifying lesson from these developments is that magnetic anisotropy is a decisive control knob for skyrmion stability. Single-ion anisotropy selects easy-axis/easy-plane manifolds and tunes the balance among conical, helicoidal, and multiple-Q states [48,52,53,54,55,56,57]. Exchange anisotropy—particularly bond-dependent forms locked to crystallographic directions—selects the orientation of spin oscillations relative to the ordering wave vectors $\mathit{Q}$, lifts degeneracies among Bloch, Néel, and antiskyrmion textures [58,59,60], and can fix skyrmion helicity or vorticity [61,62,63,64,65]. From a momentum-space viewpoint, such anisotropies act as angle-selective filters in $\mathit{Q}$-space, biasing the superposition of multiple helices that underpins many SkX phases [66]. In itinerant magnets, additional itinerant frustration arising from Fermi-surface geometry enhances the tendency of noncoplanar multiple-Q orders and high-topological-charge textures, while anisotropy determines which superpositions ultimately prevail [63,67,68].

Particularly intriguing are centrosymmetric magnets with spatial inversion symmetry, where SkXs form without bulk DM interaction. Here, competition among isotropic exchanges (including frustrated exchange interactions in insulators [69,70,71,72] and Ruderman–Kittel–Kasuya–Yosida-type (RKKY) interactions in metals [73,74,75]), higher-order spin interactions, dipolar or easy-axis anisotropy, and bond-dependent interactions can stabilize SkXs at zero or finite fields [63,76,77,78]. Theory and experiments have reported triangular and square SkXs, helicity-locked textures, and distinct lattice symmetries in centrosymmetric hosts, including tetragonal and hexagonal systems, such as Gd₂PdSi₃ [79,80,81,82,83,84,85,86,87,88], Gd₃Ru₄Al₁₂ [89,90,91,92], and GdRu₂Si₂ [93,94,95,96].

In the present study, we investigate how competing bond-dependent anisotropies permitted by $D_{3\mathrm{d}}$ symmetry, relevant for trigonal-symmetric triangular-lattice layers, govern the stability of higher-order SkXs [62,97]. Our model incorporates two symmetry-distinct anisotropic exchange interactions: a bond-dependent in-plane anisotropy, which selects spin oscillations parallel versus perpendicular to the ordering wave vector, and a $D_{3\mathrm{d}}$-symmetric anisotropy that imposes trigonal constraints on inter-bond correlations. Using simulated annealing at low temperatures, we map out the competition between topologically trivial double-Q states and SkXs with quantized skyrmion numbers. The central finding is that the relative sign of the bond-dependent anisotropy to the $D_{3\mathrm{d}}$-symmetric anisotropy decisively selects the zero-field phase: positive anisotropy, favoring oscillations perpendicular to the ordering wave vector, stabilizes the higher-order SkX with the skyrmion number of two, whereas negative anisotropy, which favors oscillations parallel to the ordering wave vector, suppresses the SkX in favor of a topologically trivial double-Q state. This sharp selection is evident in the low-temperature phase diagram and confirmed by real-space spin and scalar spin chirality textures. We also clarify how an out-of-plane magnetic field reorganizes these phases in centrosymmetric $D_{3\mathrm{d}}$ systems. Intermediate-field windows host distinct SkXs even when the zero-field ground state is a higher-order SkX or double-Q state. These results highlight that competing anisotropies not only stabilize zero-field SkX but also reshape the hierarchy of field-induced SkXs, enabling multiple and switchable skyrmion species within a single centrosymmetric host [97].

The remainder of the paper is organized as follows: Section 2 introduces the model and simulation approach; Section 3 presents phase diagrams, characterizations of SkX with the skyrmion number of two and other double-Q states, and field evolution; Section 4 concludes with implications for stabilizing higher-order zero-field SkXs and field-induced SkXs in inversion-symmetric systems from the viewpoint of two competing magnetic anisotropies.

2. Model and Method

We investigate an effective spin model on a two-dimensional triangular lattice under $D_{3\mathrm{d}}$ symmetry in trigonal environments. We consider the effective spin model with the momentum-resolved exchange interactions with symmetry-allowed magnetic anisotropies [66]. The model is given by

$$\begin{array}{cc}\hfill \mathcal{H}& =-2J\sum _{\nu }\sum _{\alpha \beta }{I}_{{\mathit{Q}}_{\nu }}^{\alpha \beta }{S}_{{\mathit{Q}}_{\nu }}^{\alpha }{S}_{-{\mathit{Q}}_{\nu }}^{\beta }-H\sum _i{S}_i^z,\hfill \end{array}$$

(1)

where ${\mathit{S}}_{{\mathit{Q}}_{\nu }}$ is the momentum-space spin at wave vector ${\mathit{Q}}_{\nu }$ ($\nu$ is the index of wave vector in the Brillouin zone), which is related to the real-space spin at site i, ${\mathit{S}}_i$, via the Fourier transformation as ${\mathit{S}}_{{\mathit{Q}}_{\nu }}=(1/\sqrt{N}){\sum }_i{\mathit{S}}_i{e}^{-i{\mathit{Q}}_{\nu }·{\mathit{r}}_i}$; N is the total number of spins and ${\mathit{r}}_i$ is the position vector. We suppose that ${\mathit{S}}_i$ is a classical spin with a fixed length $|{\mathit{S}}_i|=1$. The first term stands for the bilinear exchange interaction, where J represents bilinear couplings, which is treated as the energy unit of the model, i.e., $J=1$. The second term denotes the Zeeman coupling with field strength H, applied along the out-of-plane (z) direction.

In the first term, for simplicity, we focus on the three symmetry-related wave vectors ${\mathit{Q}}_1=(\pi /3,0)$, ${\mathit{Q}}_2=(-\pi /6,\sqrt{3}\pi /6)$, and ${\mathit{Q}}_3=(-\pi /6,-\sqrt{3}\pi /6)$, which are assumed to dominate the magnetic susceptibility of the triangular lattice. Other $\mathit{q}$ components, including $\mathit{q}=\mathbf{0}$, are neglected under the assumption of sharp susceptibility peaks at ${\mathit{Q}}_{\nu }$. Such a simplification is justified when the energy of the ground-state spin configurations is almost determined by the interaction at ${\mathit{Q}}_{\nu }$. Although the interaction at the $\mathit{q}=\mathbf{0}$ component affects the critical and saturation magnetic fields associated with the phase transitions, neglecting such an effect does not qualitatively alter the phase sequence or the nature of the multiple-Q states. Indeed, a similar effective spin model excluding the $\mathit{q}=\mathbf{0}$ component succeeded in reproducing the experimental magnetic-field–temperature phase diagram in GdRu₂Si₂ [98].

The tensor $I_{{\mathit{Q}}_{\nu }}^{\alpha \beta }$ describes the magnetic anisotropy that originates from the crystal symmetry. We consider two types of magnetic anisotropies. One is the bond-dependent anisotropy, which is expressed as

$$\begin{array}{c}\hfill I^{\mathrm{BA}}\equiv -I_{{\mathit{Q}}_1}^{xx}=I_{{\mathit{Q}}_1}^{yy}=2I_{{\mathit{Q}}_{2,3}}^{xx}=-2I_{{\mathit{Q}}_{2,3}}^{yy}={\displaystyle \frac{2}{\sqrt{3}}}{I}_{{\mathit{Q}}_2}^{xy}={\displaystyle \frac{2}{\sqrt{3}}}{I}_{{\mathit{Q}}_2}^{yx}=-{\displaystyle \frac{2}{\sqrt{3}}}{I}_{{\mathit{Q}}_3}^{xy}=-{\displaystyle \frac{2}{\sqrt{3}}}{I}_{{\mathit{Q}}_3}^{yx}.\end{array}$$

(2)

This anisotropy originates from the discrete sixfold rotational symmetry, which is present even in hexagonal systems under $D_{6\mathrm{h}}$ symmetry. Microscopically, such an anisotropy arises from the interplay of spin–orbit coupling, crystal-field effects, and the underlying electronic structure [66], which tends to favor spin oscillations either perpendicular or parallel to the ordering wave vector ${\mathit{Q}}_{\nu }$. A positive $I_{\mathrm{BA}}$ corresponds to the former case, stabilizing a proper-screw spiral spin configuration, whereas a negative $I_{\mathrm{BA}}$ promotes cycloidal spiral modulations. In other words, the sign of $I_{\mathrm{BA}}$ effectively determines whether the spin rotation plane is transverse or longitudinal to ${\mathbf{Q}}_{\nu }$. The anisotropy form factor $I^{\mathrm{BA}}$ captures a symmetric, bond-directional exchange that is conceptually analogous to the compass and Kitaev interactions [99,100,101,102,103]. Such bond-dependent couplings, rooted in spin–orbit entanglement, are widely recognized as microscopic mechanisms giving rise to unconventional magnetic phenomena, including the stabilization of topological spin textures [104,105,106,107,108] and the emergence of nonreciprocal magnon excitations [103,109].

The other originates from distinct forms of bond-dependent anisotropy inherent to the $D_{3\mathrm{d}}$ symmetry. The form factor 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)

The role of such an anisotropy has been studied in both localized spin and itinerant electron models, where the SkX with the skyrmion number of two is stabilized at zero magnetic field [62,97]. Throughout this paper, we take $I^{\mathrm{D}3\mathrm{d}}>0$, which favors the spin oscillations perpendicular to ${\mathit{Q}}_{\nu }$.

To clarify the microscopic nature of the momentum-space model in Equation (1), we note that its first term can be equivalently expressed in the real-space representation as

$${\mathcal{H}}^{\mathrm{real}-\mathrm{space}}=-\sum _{i,j}\sum _{\alpha ,\beta }{J}_{ij}^{\alpha \beta }{S}_i^{\alpha }{S}_j^{\beta },$$

(4)

where $J_{ij}^{\alpha \beta }$ is the Fourier transform of the momentum-resolved exchange coupling $2J{I}_{{\mathbf{Q}}_{\nu }}^{\alpha \beta }$. Because the interaction is sharply peaked at the symmetry-related wave vectors ${\mathbf{Q}}_{\nu }$, the real-space coupling $J_{ij}^{\alpha \beta }$ is of long-range character and oscillatory form, as found in itinerant magnets with the RKKY interaction or in localized spin systems with competing exchange interactions. This framework has been widely adopted to reproduce complex multiple-Q ordering tendencies in magnetic materials such as Y₃Co₈Sn₄ [110], EuNiGe₃ [111], and EuPtSi [112]. From a microscopic viewpoint, the bond-dependent and $D_{3d}$-type anisotropies included in our model can originate from anisotropic RKKY couplings or orbital-dependent superexchange processes, which selectively enhance or suppress the spin components parallel or perpendicular to the ordering wave vectors.

To explore the phase diagram, we perform simulated annealing using the Metropolis algorithm on lattices of size $N={24}^2$ under the periodic boundary conditions. Starting from high temperature around $T_0=1$, the temperature is decreased as $T_{n+1}=\tilde{\alpha }{T}_n$ with $\tilde{\alpha }=0.999995$–$0.999999$, until $T=0.0001$. Each run involves ${10}^5$–${10}^6$ Monte Carlo sweeps for both annealing and measurements, ensuring proper thermalization. The simulations are repeated with different initial states for a different parameter set to determine phase boundaries in an unbiased way.

Magnetic phases are identified using both momentum-space and real-space observables. The magnetic spin structure factor for the component $\alpha =x,y,z$ is defined as

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

(5)

and the corresponding Fourier amplitudes at ${\mathit{Q}}_{\nu }$ are given by

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

(6)

Topological aspects are analyzed through the scalar spin chirality,

$$\begin{array}{cc}\hfill {\chi }^{\mathrm{sc}}& ={\displaystyle \frac{1}{N}}\sum _{\mu }\sum _{\mathit{R}\in \mu }{\chi }_{\mathit{R}}^{\mathrm{sc}},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\chi }_{\mathit{R}}^{\mathrm{sc}}& ={\mathit{S}}_i·({\mathit{S}}_j\times {\mathit{S}}_k).\hfill \end{array}$$

(8)

Here, $\mu =(u,d)$ distinguishes the upward and downward triangles of the lattice, while $(i,j,k)$ denotes the three spin sites forming a triangle centered at position $\mathit{R}$, listed in counterclockwise sequence. The scalar spin chirality is related to the discretized skyrmion number $n_{\mathrm{sk}}$ [113]. This quantity allows us to distinguish between topologically trivial and nontrivial phases.

3. Results

3.1. Zero Magnetic Field

At zero magnetic field ($H=0$), the triangular-lattice system with $D_{3\mathrm{d}}$ symmetry hosts a rich variety of multiple-Q spin textures that arise from the interplay between the bond-dependent anisotropies $I^{\mathrm{BA}}$ and $I^{\mathrm{D}3\mathrm{d}}$. The $D_{3\mathrm{d}}$-type anisotropy $I^{\mathrm{D}3\mathrm{d}}$ serves as a phase-locking mechanism that couples noncoplanar spin modulations along the three symmetry-equivalent directions, thereby favoring triple-Q configurations as its magnitude increases [97]. In contrast, the relative sign of the bond-dependent anisotropy $I^{\mathrm{BA}}$ determines whether this coupling stabilizes or frustrates the triple-Q order: a positive $I^{\mathrm{BA}}$ cooperates with $I^{\mathrm{D}3\mathrm{d}}$ to favor a noncoplanar SkX with skyrmion number $n_{\mathrm{sk}}=2$ (SkX-2), whereas a negative $I^{\mathrm{BA}}$ promotes cycloidal modulations that compete with the phase locking and instead stabilize topologically trivial double-Q states.

The resulting zero-field phase diagram, shown in Figure 1, comprises three ordered phases: the SkX-2, and two double-Q phases (2Q I and 2Q II) that differ in the relative orientation and amplitude of their spin modulations. For $I^{\mathrm{BA}}>0$, the cooperative interplay between the proper-screw-type bond-dependent anisotropy and $I^{\mathrm{D}3\mathrm{d}}$ expands the stability region of SkX-2 over a wide range of parameters. Conversely, for $I^{\mathrm{BA}}<0$, the cycloidal tendency weakens the triple-Q coupling, reducing the noncoplanarity and giving rise to the double-Q phases that dominate the negative-$I^{\mathrm{BA}}$ side of the diagram.

First, we discuss the case for $I^{\mathrm{BA}}>0$. The calculated spin textures for representative parameter sets are presented in Figure 2. The double-Q I phase exhibits a noncoplanar spin arrangement characterized by two dominant ordering wave vectors. Although the overall modulation retains a stripe-like appearance, the spins are not confined to a single plane: the in-plane components ($S_i^x$, $S_i^y$) form a periodic alignment of vortices and antivortices, while the out-of-plane component $S_i^z$ displays stripe-like modulations that generate density waves in terms of the scalar spin chirality, as shown by the real-space plots of spin in Figure 2a and scalar spin chirality in Figure 3a. Although local noncoplanar spin arrangements exist, the scalar spin chirality averages to zero when summed over the whole system; there is no uniform component. This noncoplanarity arises from a superposition of single-Q proper-screw spiral and single-Q sinusoidal modulations at the different ordering wave vectors. The spin structure factor in Figure 4a reveals that the intensities at the two ordering wave vectors differ significantly, indicating an inequivalent contribution of each modulation component; the out-of-plane spin component exhibits only weak amplitude variation compared to the in-plane spin component. When $I^{\mathrm{D}3\mathrm{d}}$ increases, the out-of-plane spin modulations become more prominent so as to gain energy by $I^{\mathrm{D}3\mathrm{d}}$.

By increasing $I^{\mathrm{D}3\mathrm{d}}$, the double-Q I state is replaced by the SkX-2. In sharp contrast, the SkX-2 phase displays a fully developed noncoplanar triple-Q configuration, forming a triangular lattice of skyrmions. The out-of-plane spin component $S_i^z$ alternates periodically between up- and down-spin cores surrounded by in-plane swirling spins, as shown in Figure 2b. This structure corresponds to an equal-weight superposition of three sinusoidal waves with mutual phase locking among ${\mathit{Q}}_1$, ${\mathit{Q}}_2$, and ${\mathit{Q}}_3$, whose spin ansats in real space are given by

$$\begin{array}{c}\hfill {\mathit{S}}_i\propto {\left(\begin{array}{c}{\displaystyle \frac{\sqrt{3}}{2}}(cos{\mathit{Q}}_2·{\mathit{r}}_i-cos{\mathit{Q}}_3·{\mathit{r}}_i)\\ cos{\mathit{Q}}_1·{\mathit{r}}_i-{\displaystyle \frac{1}{2}}(cos{\mathit{Q}}_2·{\mathit{r}}_i+cos{\mathit{Q}}_3·{\mathit{r}}_i)\\ {\displaystyle \frac{1}{\sqrt{2}}}(cos{\mathit{Q}}_1·{\mathit{r}}_i+cos{\mathit{Q}}_2·{\mathit{r}}_i+cos{\mathit{Q}}_3·{\mathit{r}}_i)\end{array}\right)}^{\mathrm{T}}.\end{array}$$

(9)

The equal-weight, phase-locked triple-Q superposition in Equation (9) implies that the three spin modulations with wave vectors ${\mathit{Q}}_1$, ${\mathit{Q}}_2$, and ${\mathit{Q}}_3$ are coherently coupled with relative phases fixed by the $D_{3\mathrm{d}}$-type anisotropy. As a result, the local spin direction on each lattice site undergoes two full rotations on the unit sphere when the position traverses a magnetic unit cell, producing a mapping that wraps the sphere twice and yielding the quantized skyrmion number $n_{\mathrm{sk}}=2$. In real space, this texture can be viewed as a vortex lattice: the superposition of three sinusoidal modulations creates vortices with winding number $+1$ and antivortices with $-2$, whose population imbalance gives rise to a net skyrmion number of two. According to the quantized skyrmion number, the scalar spin chirality is distributed uniformly in real space, as shown in Figure 3b. It is noted that the sign of the scalar spin chirality (skyrmion number) is not determined within the present model. The spin structure factor also exhibits the triple-Q peak structures at ${\mathit{Q}}_1$–${\mathit{Q}}_3$ for both in-plane and out-of-plane spin components, as shown in Figure 4b. A similar SkX-2 spin configuration has been reported in the Kondo lattice model without the magnetic anisotropy, where multi-spin interactions arising from the nesting of the Fermi surfaces play an important role [114].

For larger $I^{\mathrm{BA}}$ and smaller $I^{\mathrm{D}3\mathrm{d}}$, the double-Q II phase appears instead of the double-Q I phase. The double-Q II phase also consists of two ordering wave vectors but is distinguished from the double-Q I phase by the relative amplitude of spin density waves in each channel. In this case, the intensities of the spin structure factor at double-Q ordering wave vectors are equivalent for both in-plane and out-of-plane components, while those in the double-Q I state are inequivalent, as shown in Figure 4a,c. Accordingly, the out-of-plane spin component exhibits a checkerboard pattern rather than a stripe pattern, as shown in Figure 2c. Based on this real-space spin configuration, the texture can be interpreted as a meron-antimeron crystal structure, where topological entities carrying opposite signs of half the skyrmion number are periodically aligned [115,116]. Meanwhile, this state exhibits only a negligible local scalar spin chirality, as shown in Figure 3c. This is attributed to the phase difference between in-plane and out-of-plane spin components; there is no z-spin component around the core center of the vortices and antivortices, as shown in Figure 2c. Thus, the double-Q II state is a topologically trivial state. It is noted that the small out-of-plane spin modulation is almost suppressed for larger $I^{\mathrm{BA}}$, resulting in a double-Q configuration that becomes essentially coplanar [63].

Next, we discuss the case for $I^{\mathrm{BA}}<0$. As shown in Figure 1, the system for negative $I^{\mathrm{BA}}$ predominantly stabilizes double-Q spin textures, in sharp contrast to the $I^{\mathrm{BA}}>0$ case where the SkX-2 phase is widely stabilized. This difference originates from the cycloidal nature favored by negative $I^{\mathrm{BA}}$, which competes with the different-directional phase-locking tendency induced by $I^{\mathrm{D}3\mathrm{d}}$ and thereby suppresses the triple-Q sinusoidal coupling required for the SkX-2 formation. Consequently, the SkX-2 phase appears only in a narrow region, while the topologically trivial double-Q states occupy most of the phase diagram.

At small $I^{\mathrm{D}3\mathrm{d}}$ and $|I^{\mathrm{BA}}|$, the system exhibits the double-Q I phase. Similarly to the case for $I^{\mathrm{BA}}>0$, this state is characterized by a superposition of the spiral wave and sinusoidal wave at different ordering wave vectors, as shown in Figure 2d. Accordingly, a similar real-space distribution in terms of the scalar spin chirality is observed, as shown in Figure 3d. The combination of these modulations results in a spatially varying scalar spin chirality that alternates in sign but averages to zero over the entire lattice. The spin structure factor in Figure 4d exhibits two inequivalent intensity peaks at ${\mathit{Q}}_2$ and ${\mathit{Q}}_3$, as found in the case of $I^{\mathrm{BA}}>0$.

By increasing $I^{\mathrm{D}3\mathrm{d}}$, the system stabilizes the SkX-2 phase in Figure 2e, but the stability region is much narrower than that for $I^{\mathrm{BA}}>0$. The real-space and momentum-space spin quantities are similar to those for $I^{\mathrm{BA}}>0$, as shown in Figure 2e and Figure 4e. Accordingly, the scalar spin chirality distribution in Figure 3e is similar to that in Figure 3b.

For large $|I^{\mathrm{BA}}|$, the double-Q II phase emerges, characterized by two spin density waves of comparable amplitude and a checkerboard-like $S_i^z$ modulation, as shown in Figure 4f. In contrast to the case of $I^{\mathrm{BA}}>0$, the local maxima and minima of $S_i^z$ are located at the center of the vortices and antivortices, as found in the meron-antimeron crystal [54,116,117,118]. As a consequence of this spin arrangement, the local scalar spin chirality develops a checkerboard pattern, as shown in Figure 3f, yet its uniform component vanishes owing to the alternating sign distribution.

Finally, the dependence of the spatially averaged scalar spin chirality ${\chi }^{\mathrm{sc}}$ on the $D_{3\mathrm{d}}$ anisotropy $I^{\mathrm{D}3\mathrm{d}}$, shown in Figure 5a, reveals the cooperative and competitive roles of the two anisotropies. For $I^{\mathrm{BA}}=+0.1$, ${\chi }^{\mathrm{sc}}$ remains nearly zero at small $I^{\mathrm{D}3\mathrm{d}}$, corresponding to the double-Q I phase, but sharply increases beyond a critical threshold, indicating the transition to the SkX-2 phase, where it saturates at a large positive value. In contrast, for $I^{\mathrm{BA}}=-0.1$, a finite ${\chi }^{\mathrm{sc}}$ appears only at substantially larger $I^{\mathrm{D}3\mathrm{d}}$, and the stable region of SkX-2 is significantly reduced, demonstrating the suppressive influence of the negative bond-dependent anisotropy.

These results collectively indicate that the formation of the $n_{\mathrm{sk}}=2$ SkX at zero magnetic field arises from the cooperative effect between the $D_{3\mathrm{d}}$ anisotropy, which locks the relative phases of the three $\mathit{Q}$ modes, and the bond-dependent anisotropy $I^{\mathrm{BA}}$, whose relative sign determines whether these phase lockings stabilize or frustrates the triple-Q configuration. Thus, even in the absence of an external magnetic field, a noncoplanar spin texture with a finite emergent magnetic field can be realized solely through bond-dependent anisotropic interactions, providing a microscopic mechanism for the emergence of high-topological-number SkXs in centrosymmetric magnets.

Let us comment on the experimental identification of the $n_{\mathrm{sk}}=2$ SkX by experiments based on elastic neutron or resonant X-ray magnetic scattering. The $n_{\mathrm{sk}}=2$ SkX can be experimentally distinguished from the double-Q and triple-Q vortex phases as follows. The SkX shows three equivalent Bragg peaks at ${\mathit{Q}}_1$, ${\mathit{Q}}_2$, and ${\mathit{Q}}_3$ with nearly equal intensity for both in-plane and out-of-plane spin components, reflecting the fully developed noncoplanar triple-Q order. In contrast, the double-Q I/II phases display only two dominant peaks with markedly anisotropic intensity ratios, and the triple-Q vortex phase shows reduced out-of-plane intensity due to its suppressed z-spin modulation, as discussed in the next section. These characteristic differences in the magnetic scattering profiles provide clear experimental fingerprints for identifying the $n_{\mathrm{sk}}=2$ SkX phase in real materials.

3.2. Effect of Magnetic Field

We now examine the evolution under a magnetic field applied along the z axis and summarize the response of magnetization, scalar spin chirality, and wave-vector-resolved magnetic amplitudes together with real-space snapshots. We first discuss the case of $I^{\mathrm{BA}}>0$, where the zero-field state corresponds to the SkX-2. The magnetic field dependence at $I^{\mathrm{BA}}=0.3$ and $I^{\mathrm{D}3\mathrm{d}}=0.3$ are compiled in Figure 6: (a) the uniform magnetization $M^z$ increases monotonically with H and approaches saturation at high fields; (b) the spatially averaged scalar spin chirality ${\chi }^{\mathrm{sc}}$ decreases nearly monotonically from a large value at low fields to almost zero as the system becomes a fully polarized state; (c) the in-plane Fourier amplitudes ${\left(m_{{\mathit{Q}}_{\nu }}^{xy}\right)}^2$ for $\nu =1$–3, which are nearly equal in the triple-Q regime, are gradually suppressed with H; and (d) the out-of-plane components ${\left(m_{{\mathit{Q}}_{\nu }}^z\right)}^2$ are likewise diminished when H increases. These observables together delineate the continuous weakening and eventual destruction of the noncoplanar triple-Q texture by the out-of-plane magnetic field.

At $H=0.5$, as shown in Figure 7a, the system maintains a well-developed SkX closely resembling the configuration at zero field. The color scale indicates alternating $S_i^z$ cores corresponding to vortex and antivortex centers, while the arrows show robust in-plane swirling spin arrangements forming a threefold-symmetric pattern. This texture consists of two distinct types of vortices: those with a winding number of one centered around $S_i^z=-1$ and antivortices with the winding number of minus two centered around $S_i^z=+1$. The former vortices appear with twice the number of the latter, resulting in a net topological charge of $n_{\mathrm{sk}}=2$ per magnetic unit cell. Such an asymmetry between vortex and antivortex populations originates from the phase-locked triple-Q configuration favored by the $D_{3\mathrm{d}}$ anisotropy. The corresponding scalar spin chirality distribution in Figure 8a exhibits a uniform pattern on a triangular lattice, in close correspondence with the $H=0$ configuration shown in Figure 3b. These results demonstrate that the application of a small magnetic field does not disrupt the $n_{\mathrm{sk}}=2$ SkX but instead enhances its uniform magnetization while preserving its topological structure.

At $H=1.2$, as shown in Figure 7b, the spin texture undergoes a qualitative transformation. One of the vortices with a winding number of one shows a reversed sign of the z-spin component around its core, indicating that one sublattice of vortices has switched polarity under the Zeeman effect. This change effectively reduces the total skyrmion number in the magnetic unit cell from two to one [63]. The resulting structure can be viewed as a topological transition from an $n_{\mathrm{sk}}=2$ SkX to a conventional $n_{\mathrm{sk}}=1$ SkX. In the corresponding real-space scalar spin chirality map [Figure 8b], one of the vortex cores exhibits a positive chirality contribution consistent with the sign reversal of $S_i^z$ around the core, while the other maintains the original sign. This partial reversal produces an imbalance between positive and negative chirality regions, signifying the breakdown of the uniform pattern observed at lower fields. The magnetic field thus selectively alters one component of the triple-Q superposition, driving a topological reconstruction of the SkX.

Upon further increasing the magnetic field to $H=2$, as shown in Figure 7c, the remaining vortices also reverse the sign of the z-spin component around their cores. As a result, the skyrmion number per magnetic unit cell becomes zero, although the local scalar spin chirality remains finite, as confirmed by Figure 6b. In this intermediate regime, the system no longer supports a net topological charge but retains local noncoplanar spin textures that contribute to a finite scalar spin chirality. The real-space distribution of ${\chi }_{\mathit{R}}^{\mathrm{sc}}$, shown in Figure 8c, reveals an arrangement of alternating chiral domains. This behavior indicates that the magnetic field induces partial unwinding of the SkX into a non-topological, but still noncoplanar, triple-Q state. The persistence of local scalar spin chirality even after the loss of the net skyrmion number highlights the robustness of triple-Q interference effects against moderate fields, as the Zeeman coupling cannot completely remove the transverse spin modulations.

Finally, at $H=2.5$, the spins become almost fully aligned along the magnetic field direction, as illustrated in Figure 7d. The system approaches a fully polarized state, where only minor transverse spin components survive as small fluctuations around the positive z-spin configuration. In this high-field regime, the scalar spin chirality map shown in Figure 8d becomes nearly featureless, signifying the complete suppression of the noncoplanar components and the vanishing of ${\chi }^{\mathrm{sc}}$. Similar triple-Q spin textures without the skyrmion number have been reported in frustrated triangular-lattice systems [119,120].

The destruction of the SkX is thus continuous, proceeding through sequential reversals of $S_i^z$ around the vortex cores, which successively reduce the skyrmion number from $n_{\mathrm{sk}}=2$ to $n_{\mathrm{sk}}=1$, and finally to $n_{\mathrm{sk}}=0$. Such field-induced topological transitions are a hallmark of multiple-Q spin systems where competing anisotropic interactions allow different superpositions of ordering wave vectors [63]. Overall, the results demonstrate that the applied out-of-plane magnetic field plays a dual role: it aligns the spins along the field direction while simultaneously modulating the phase relations among the triple-Q components, thereby governing the sequence of topological transitions in the SkX.

Next, we discuss the case of $I^{\mathrm{BA}}<0$, for which the zero-field ground state corresponds to the double-Q I phase rather than the SkX-2 phase. Figure 9, Figure 10, Figure 11 summarize the evolution of the spin textures, magnetic responses, and real-space distributions of the spin and scalar spin chirality under an external magnetic field applied along the z axis. In contrast to the $I^{\mathrm{BA}}>0$ case, the field response is qualitatively different: the system exhibits a transition from an anisotropic multiple-Q state to an isotropic one, as evidenced by the evolution of the Fourier amplitudes in Figure 9c,d. Concomitantly, the spatially averaged scalar spin chirality ${\chi }^{\mathrm{sc}}$ is markedly enhanced in the isotropic triple-Q region, as shown in Figure 9b. Furthermore, no discrete topological transitions in the skyrmion number from $n_{\mathrm{sk}}=2$ to 1 observed in the SkX-2 phase for $I^{\mathrm{BA}}>0$ are found for $I^{\mathrm{BA}}<0$. On the other hand, the topological transition from $n_{\mathrm{sk}}=1$ to 0 occurs in both cases.

The field dependence of the relevant magnetic quantities at $(I^{\mathrm{BA}},I^{\mathrm{D}3\mathrm{d}})=(-0.3, 0.3)$ is displayed in Figure 9. The uniform magnetization $M^z$ [Figure 9a] increases almost linearly at low fields, reflecting the gradual canting of spins toward the field direction while almost preserving the underlying multiple-Q modulations. Interestingly, the double-Q configuration at zero field transforms into a triple-Q state even for infinitesimally small H, as indicated by the emergence of three Fourier components at ${\mathit{Q}}_1$–${\mathit{Q}}_3$ in Figure 9c,d. The anisotropic superposition of triple-Q spin density waves might be attributed to the energy balance between the bond-dependent anisotropy, favoring the cycloidal oscillation, and the Zeeman coupling. This anisotropic tendency is suppressed by increasing H, where the SkX appears. At higher fields, $M^z$ approaches saturation as the spins align with the external field, eventually leading to a nearly fully polarized spin configuration. The averaged scalar spin chirality ${\chi }^{\mathrm{sc}}$, plotted in Figure 9b, exhibits a pronounced enhancement in the intermediate-field range, reflecting the stabilization of a more isotropic triple-Q noncoplanar state induced by the competition between the Zeeman effect and the bond-dependent anisotropy.

The real-space spin textures for representative magnetic fields are displayed in Figure 10a–d. At $H=0.5$ in Figure 10a, the system exhibits an anisotropic triple-Q spin configuration, which is similar to the double-Q I spin configuration exhibiting a periodic arrangement of vortices and antivortices in the in-plane spin components. Unlike the proper-screw-type helices found for $I^{\mathrm{BA}}>0$, the present pattern reflects the cycloidal character induced by the negative bond-dependent anisotropy, where spins rotate mainly within a plane perpendicular to one of the ordering wave vectors. This does not affect the alignment of local scalar spin chirality that alternates in sign across the lattice. The scalar spin chirality map in Figure 11a exhibits a weak checkerboard-like distribution of alternating scalar spin chirality, whose spatial average remains nearly zero, confirming the topologically trivial nature of this phase.

At $H=1.2$, as shown in Figure 10b, the Zeeman coupling enhances the out-of-plane spin component $S_i^z$. Although the system retains its multiple-Q character, the previously anisotropic triple-Q spin texture at lower fields becomes more isotropic at this field, signaling the emergence of the SkX. The in-plane vortex structure reorganizes into a more symmetric pattern, with the vortex and antivortex cores forming a nearly triangular-lattice arrangement. This reorganization is evidenced by the equalization of the three Fourier components in Figure 9c,d. In the real-space scalar spin chirality map shown in Figure 11b, the magnitude of ${\chi }_{\mathbf{R}}^{\mathrm{sc}}$ is enhanced and distributed more uniformly, reflecting the development of noncoplanar spin arrangements associated with the SkX. This behavior indicates that the magnetic field stabilizes a topologically nontrivial triple-Q configuration through the cooperative effect of Zeeman coupling, bond-dependent anisotropy, and the $D_{3\mathrm{d}}$-type anisotropy. Unlike the $I^{\mathrm{BA}}>0$ case, where the SkX appears at low fields and is gradually destabilized with increasing H, here the SkX emerges only after the anisotropic triple-Q state becomes isotropic, representing a field-induced realization of the topological transition.

When the magnetic field is further increased to $H=2$, as shown in Figure 10c, the isotropic SkX observed at $H=1.2$ gradually transforms into a distinct triple-Q vortex state. In this regime, the amplitudes of the three $\mathit{Q}$ components remain comparable, preserving the overall triple-Q character, but the relative intensities of the z-spin modulations to those of the $xy$-spin ones at ${\mathit{Q}}_{\nu }$ change, leading to the disappearance of the SkX. The uniform component becomes dominant in the z-spin structure, while the in-plane spins form a swirling pattern around the vortex centers, which leads to the different spatial distribution of the scalar spin chirality in Figure 11c from the SkX in Figure 11b; the emergent magnetic flux almost cancels globally but remains finite locally. As a consequence, the skyrmion number in the magnetic unit cell vanishes, but the local noncoplanarity persists owing to the surviving triple-Q noncoplanar superposition. This field-induced transition from the isotropic SkX to the triple-Q vortex phase represents a reorganization of the phase-locked triple-Q structure in terms of the z-spin component. It highlights that the Zeeman coupling modulates the relative out-of-plane modulations among the three ordering wave vectors, thereby converting the topologically nontrivial SkX into a non-topological yet noncoplanar multiple-Q vortex texture.

At $H=2.5$, as shown in Figure 10d, the spins are further aligned along the magnetic field direction, and the triple-Q modulation is almost suppressed. The color map of $S_i^z$ becomes nearly uniform, and the in-plane components are strongly diminished, signifying the approach to the fully polarized state. The corresponding scalar spin chirality map in Figure 11d is almost featureless, confirming that ${\chi }^{\mathrm{sc}}$ vanishes as the residual noncoplanarity is eliminated. Thus, the evolution for $I^{\mathrm{BA}}<0$ proceeds through a sequence of transitions from the anisotropic triple-Q state at low fields, to the isotropic SkX at intermediate fields, and finally to the triple-Q vortex and fully polarized states at high fields.

The results presented in Figure 9, Figure 10 and Figure 11 emphasize the qualitative difference between positive and negative bond-dependent anisotropies under an applied magnetic field. For $I^{\mathrm{BA}}>0$, the magnetic field drives a sequence of quantized topological transitions ($n_{\mathrm{sk}}=2\to 1\to 0$) through successive reversals of $S_i^z$ around the vortex cores. In contrast, for $I^{\mathrm{BA}}<0$, the evolution is discontinuous from an anisotropic triple-Q texture to an isotropic SkX, and then it continuously changes into a triple-Q vortex configuration with a discrete change in the skyrmion number ($n_{\mathrm{sk}}=0\to 1\to 0$). The negative bond-dependent anisotropy favors cycloidal modulations that weaken the proper-screw coupling among the three ordering wave vectors by $I^{\mathrm{D}3\mathrm{d}}$, thereby suppressing the formation of a stable SkX. Consequently, while both signs of $I^{\mathrm{BA}}$ enable multiple-Q order at zero field, only the positive $I^{\mathrm{BA}}$ case supports robust SkX phases over a broader field range. These findings demonstrate that the interplay between bond-dependent anisotropy and Zeeman coupling determines not only the stability but also the sequence of noncoplanar magnetic states, with the relative sign of $I^{\mathrm{BA}}$ to $I^{\mathrm{D}3\mathrm{d}}$ playing a crucial role in governing whether the magnetic field stabilizes or destabilizes topological magnetic textures in centrosymmetric magnets.

4. Conclusions

We have elucidated how two symmetry-allowed bond-dependent anisotropies permitted by the $D_{3\mathrm{d}}$ crystal environment cooperate and compete to stabilize multiple-Q spin textures, including a higher-order SkX with the skyrmion number of two, on a triangular lattice. A central outcome is that the relative sign of the sixfold symmetric bond-dependent anisotropy to the $D_{3\mathrm{d}}$-type anisotropy decisively selects the zero-field ground state: for positive bond-dependent anisotropy, the proper-screw tendency cooperates with the $D_{3\mathrm{d}}$-type anisotropy to stabilize the SkX with the skyrmion number of two over a broad parameter window; for negative bond-dependent anisotropy, the cycloidal tendency competes with the $D_{3\mathrm{d}}$-type anisotropy, and topologically trivial double-Q states prevail. These trends are encoded in the low-temperature phase diagram.

At zero magnetic field, the SkX with the skyrmion number of two emerges as an equal-weight, phase-locked triple-Q superposition of three sinusoidal modes, forming a hexagonal lattice of alternating vortex cores and exhibiting triple-Q peaks in the spin structure factor together with a uniform scalar spin chirality. In contrast, the two double-Q phases differ in the relative weights and phase relations of their constituent modulations with vanishing net scalar spin chirality: the double-Q I phase shows inequivalent intensities at the two ordering wave vectors and a stripe-like z-spin modulation, while the double-Q II phase exhibits nearly equal double-Q peaks and a checkerboard z-spin pattern similar to a meron-antimeron crystal. Under an out-of-plane magnetic field, the system with positive bond-dependent anisotropy exhibits a continuous weakening of the SkX with the skyrmion number of two: the skyrmion number changes sequentially from 2 to 1 and then to 0. In contrast, for negative bond-dependent anisotropy, a weak magnetic field induces a transition from a double-Q I to an anisotropic triple-Q state, followed by the emergence of a SkX with enhanced scalar spin chirality and its subsequent evolution into a topologically trivial triple-Q vortex state at higher fields.

These findings show that bond-directional anisotropies determine the phase locking of spin components at each ordering wave vector, thereby controlling the stabilization of higher-order and conventional SkXs in centrosymmetric magnets across zero and finite fields. By tuning the sign and magnitude of bond-dependent anisotropy relative to $D_{3\mathrm{d}}$-type anisotropy as well as the out-of-plane external magnetic field, one can selectively realize distinct multiple-Q spin textures—including 2Q stripe, $n_{\mathrm{sk}}=2$ SkX, field-induced $n_{\mathrm{sk}}=1$ SkX, and triple-Q vortex states. This anisotropy-based viewpoint helps rationalize the distinct phase stabilities and field-dependent evolutions identified in this work and may guide future efforts to control topological spin textures in centrosymmetric magnets.

Although no experimental realization of the $n_{\mathrm{sk}}=2$ SkX in a compound with exact $D_{3\mathrm{d}}$ symmetry has been reported so far, a promising candidate material is NiI₂ [62], where the emergence of the $n_{\mathrm{sk}}=2$ SkX has been theoretically suggested. Since the sign and magnitude of the anisotropy are sensitive to electronic band structures, it could be tuned by chemical substitution or by applying uniaxial pressure. These considerations highlight NiI₂ as an appealing platform for future experimental exploration of higher-order SkXs in centrosymmetric magnets.