In-Plane Magnetic Field-Induced Multiple-Q Magnetic Phases in Frustrated Magnets with Easy-Plane and Bond-Dependent Anisotropy

Abstract

We numerically investigate instabilities toward bimeron crystals and multiple-Q magnetic states induced by an in-plane external magnetic field in centrosymmetric magnets with magnetic anisotropy. By focusing on the interplay between easy-plane single-ion anisotropy and bond-dependent anisotropy originating from relativistic spin–orbit coupling in crystalline environments, we construct the magnetic phase diagram of an effective spin model with competing momentum-resolved interactions using simulated annealing. Our analysis reveals that the bimeron crystal is stabilized within the regime of weak bond-dependent anisotropy, independent of its sign, whereas increasing the strength of bond-dependent anisotropy drives the system into a topologically trivial triple-Q magnetic state. The obtained bimeron crystal is characterized by finite scalar spin chirality and triple-Q modulations in both the in-plane and out-of-plane spin components. These findings demonstrate that centrosymmetric easy-plane magnets provide a fertile platform for realizing nontrivial topological spin textures without relying on Dzyaloshinskii–Moriya interactions, thereby opening new avenues for inducing emergent topological transport phenomena in centrosymmetric materials.

1. Introduction

Topological spin textures, such as skyrmions and merons, have attracted considerable attention in condensed matter physics owing to their emergent electrodynamics, robustness against perturbations, and potential applications in spintronic devices [1,2,3,4,5]. Among them, the bimeron, which corresponds to a counterpart of the skyrmion in systems with easy-plane anisotropy, has recently emerged as a promising spin texture hosting nontrivial topology while preserving compatibility with planar spin configurations [6,7,8,9,10]. Bimerons and related multiple-Q magnetic states are anticipated to play a central role in broadening the variety of topological magnetic phases and in providing alternative pathways to manipulate spin textures beyond the conventional skyrmion paradigm [11,12]. The topological aspects of bimerons and their associated transport properties continue to attract active investigation, especially with regard to their promise for spintronic applications [13,14,15,16,17].

Most theoretical and experimental studies on topological spin textures have focused on noncentrosymmetric magnets, where the Dzyaloshinskii–Moriya (DM) interaction [18,19] plays a crucial role in stabilizing skyrmion crystal and bimeron crystal (BmX) phases [20,21,22,23,24,25,26,27,28,29,30]. Typical realizations include chiral magnets such as MnSi [23,31,32,33,34,35,36,37,38,39], ${\mathrm{Fe}}_{1-x}$${\mathrm{Co}}_x$Si [25,27,28,40], FeGe [29,41], ${\mathrm{Cu}}_2$OSe${\mathrm{O}}_3$ [42,43,44,45,46,47], and EuPtSi [48,49,50,51,52,53,54,55], polar magnets like Ga${\mathrm{V}}_4$${\mathrm{S}}_8$ [56,57,58], Ga${\mathrm{V}}_4$${\mathrm{Se}}_8$ [59,60,61,62], and VO${\mathrm{Se}}_2$${\mathrm{O}}_5$ [63,64], and other noncentrosymmetric magnets like ${\mathrm{Mn}}_{1.4}$${\mathrm{Pt}}_{0.9}$${\mathrm{Pd}}_{0.1}$Sn [65,66,67] and ${\mathrm{Fe}}_{1.9}$${\mathrm{Ni}}_{0.9}$${\mathrm{Pd}}_{0.2}$P [68], where noncentrosymmetric crystal structures give rise to the DM interaction, resulting in swirling spin textures in an external magnetic field.

In contrast, centrosymmetric magnets lack intrinsic DM interactions, and topological spin textures are instead stabilized by competing exchange interactions [69,70,71,72], long-range dipolar interactions [73,74], or crystal-dependent anisotropies [75,76,77,78,79,80]. Representative examples include hexagonal intermetallic compounds such as ${\mathrm{Gd}}_2$${\mathrm{PdSi}}_3$ [81,82,83,84,85,86] and ${\mathrm{Gd}}_3$${\mathrm{Ru}}_4$${\mathrm{Al}}_{12}$ [87,88,89,90], and tetragonal ones such as Gd${\mathrm{Ru}}_2$${\mathrm{Si}}_2$ [91,92,93,94,95,96] and Gd${\mathrm{Ru}}_2$${\mathrm{Ge}}_2$ [97]. Despite the absence of the DM interaction, these centrosymmetric materials exhibit skyrmion crystals, BmXs, and other multiple-Q magnetic states. Interestingly, it has also been pointed out that breaking of the local inversion symmetry can reintroduce sublattice-dependent DM interactions even in globally centrosymmetric systems, further enriching the possible stabilization mechanisms [98,99].

This contrast highlights an important open problem: while the DM-driven mechanisms in noncentrosymmetric magnets are well-established, the microscopic routes by which centrosymmetric magnets, especially those with easy-plane and bond-dependent anisotropy, stabilize topological spin textures remain far less explored. In particular, bond-dependent anisotropy, originating from relativistic spin–orbit coupling in crystalline environments, can strongly influence the stability of multiple-Q magnetic states [75,77,100,101], yet its interplay with easy-plane anisotropy under in-plane external magnetic fields has not been systematically addressed.

In the present study, we investigate the emergence of BmXs and multiple-Q magnetic states in centrosymmetric systems, focusing on the competition between easy-plane single-ion anisotropy and bond-dependent anisotropy in the presence of in-plane external magnetic fields. By performing simulated annealing to construct the phase diagram of an effective spin model with momentum-resolved interactions and easy-plane single-ion anisotropy, we demonstrate that BmXs appear robustly in the intermediate magnetic-field region whenever the bond-dependent anisotropy is weak, irrespective of its sign. As the bond-dependent anisotropy is strengthened, the system undergoes a transition into a topologically trivial triple-Q magnetic state without out-of-plane spin modulations. Our results indicate that centrosymmetric easy-plane magnets can accommodate nontrivial topological spin textures under realistic conditions, thereby adding to the variety of known topological magnetic states. Beyond the fundamental interest, these findings may also offer potential for spintronic applications where emergent responses are induced by external fields.

The remainder of this paper is organized as follows: Section 2 introduces the effective spin model employed in our analysis. This model includes easy-plane single-ion anisotropy and bond-dependent anisotropic interactions in momentum space. The section also outlines the simulated annealing procedure used to determine the low-temperature phase diagram. Section 3 presents the stability regions of the BmX and various multiple-Q magnetic states when the magnitude and the sign of the bond-dependent anisotropic interaction are varied. Section 4 summarizes the main outcomes in the present paper.

2. Model and Method

We study a classical spin model with momentum-resolved interactions on a two-dimensional triangular lattice [102], defined as

$$\begin{array}{cc}\hfill \mathcal{H}& =-2\sum _{\nu }{J}_{{\mathit{Q}}_{\nu }}\left({\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{\nu }}+\sum _{\alpha \beta }{I}_{{\mathit{Q}}_{\nu }}^{\alpha \beta }{S}_{{\mathit{Q}}_{\nu }}^{\alpha }{S}_{-{\mathit{Q}}_{\nu }}^{\beta }\right)-A\sum _i{\left(S_i^z\right)}^2-H_x\sum _i{S}_i^x.\hfill \end{array}$$

(1)

Here, ${\mathit{S}}_i=(S_i^x,S_i^y,S_i^z)$ denotes a classical spin of unit length at site i ($|{\mathit{S}}_i|=1$), and ${\mathit{S}}_{{\mathit{Q}}_{\nu }}=(S_{{\mathit{Q}}_{\nu }}^x,S_{{\mathit{Q}}_{\nu }}^y,S_{{\mathit{Q}}_{\nu }}^z)$ represents the Fourier component of the spin moment at wave vector ${\mathit{Q}}_{\nu }$ in the first Brillouin zone. The first term in Equation (1) describes momentum–space bilinear interactions parameterized by the coupling constant $J_{{\mathit{Q}}_{\nu }}$. It consists of an isotropic two-spin exchange interaction together with a bond-dependent anisotropic contribution. This anisotropy is encoded in the ${\mathit{Q}}_{\nu }$-dependent form factor $I_{{\mathit{Q}}_{\nu }}^{\alpha \beta }$ for $\alpha ,\beta =x,y,z$, which originates from the cooperative effects of spin–orbit coupling and the crystalline electric field on the triangular lattice [103,104]; $I_{\mathit{Q}\nu }^{\alpha \beta }$ has different forms depending on ${\mathit{Q}}_{\nu }$, which reflects the bond-dependent interactions in real space. This momentum-resolved formulation can be equivalently expressed in real space through inverse Fourier transformation, highlighting the direct correspondence between exchange interactions in momentum and real space.

The second term represents the single-ion anisotropy, known to affect the stability of skyrmion crystals and BmXs [70,105,106]. In this study, we focus on the easy-plane regime, $A<0$, in which planar spin configurations and BmX textures are naturally favored [8]. The third term is the Zeeman coupling to an in-plane external magnetic field along the x direction $\left(H_x\right)$, which tunes the spin alignment and drives field-induced phase transitions.

In Equation (1), we further reduce the complexity of the problem by retaining only the momentum-resolved interactions at a small set of ordering wave vectors ${\mathit{Q}}_1$–${\mathit{Q}}_3$ that dominate the ground-state energy. Such finite-Q magnetic states, including a single-Q spiral state, arise naturally when the Fourier transform of the exchange interaction develops maxima at characteristic ${\mathit{Q}}_{\nu }$ vectors [107]. This situation is realized, for example, by frustrated competition between the nearest-neighbor ferromagnetic and further-neighbor antiferromagnetic exchange interactions, or by long-range interactions mediated by conduction electrons, such as the Ruderman–Kittel–Kasuya–Yosida interaction [108,109,110]. Representative material examples include frustrated triangular-lattice magnets like Ni${\mathrm{Ga}}_2$${\mathrm{S}}_4$ [111,112], as well as itinerant magnets like ${\mathrm{Gd}}_2$Pd${\mathrm{Si}}_3$ [113,114]; although quantum effects are expected to play a role in these materials [115,116], our analysis focuses on the magnetic instabilities in the classical limit, providing a qualitative framework that can later be extended to incorporate quantum fluctuations. Concretely, we set ${\mathit{Q}}_1=(Q,0)$, ${\mathit{Q}}_2=(-Q/2,\sqrt{3}Q/2)$, and ${\mathit{Q}}_3=(-Q/2,-\sqrt{3}Q/2)$ with $Q=\pi /3$, and the triangular lattice constant taken as unity. These three vectors are related by the threefold rotational symmetry intrinsic to the triangular lattice, and thus, we set the common coupling constant at these wave vectors to $J_{{\mathit{Q}}_{\nu }}\equiv J$. Our results remain qualitatively insensitive to the magnitude of Q, as long as the ordering vectors lie inside the Brillouin zone. This effective model, owing to its efficiency and low computational cost, has been widely used to capture the essential microscopic origins of topological spin phases in a variety of lattice structures and interaction settings [102]. Remarkably, it reproduces experimental phase diagrams, including the skyrmion crystals and other various multiple-Q magnetic states, such as Gd${\mathrm{Ru}}_2$${\mathrm{Si}}_2$ [92], Gd${\mathrm{Ru}}_2$${\mathrm{Ge}}_2$ [97], EuPtSi [54], ${\mathrm{Y}}_3$${\mathrm{Co}}_8$${\mathrm{Sn}}_4$ [117], and EuNi${\mathrm{Ge}}_3$ [118].

For these ordering wave vectors, the anisotropic interaction matrix $I_{{\mathit{Q}}_{\nu }}^{\alpha \beta }$ is chosen to respect the sixfold rotational symmetry of the triangular lattice. Concretely, the matrix elements satisfy the relations [104]

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

(2)

In this notation, $I^{\mathrm{BA}}>0$ ($I^{\mathrm{BA}}<0$) favors the spin oscillations perpendicular (parallel) to ${\mathit{Q}}_{\nu }$. The resulting form factor $I^{\mathrm{A}}$ represents a symmetric, bond-dependent anisotropy reminiscent of compass- and Kitaev-type interactions [119,120,121,122,123], which are known as microscopic origins of exotic topological spin textures [100,101,124,125] and nonreciprocal magnon excitations [123,126]. On the triangular lattice, this type of bond-dependent anisotropy can stabilize a skyrmion crystal, even in the absence of higher-order multi-spin interactions or easy-axis single-ion anisotropy when the magnetic field is applied along the out-of-plane direction [77,89,127]. We here examine the interplay between bond-dependent anisotropy and in-plane magnetic field in centrosymmetric magnets with easy-plane single-ion anisotropy.

To map out magnetic phase diagrams, we fix $J=1$ and $A=-0.25$ (taking J as the energy unit) and vary $I^{\mathrm{BA}}$ and $H_x$. Simulated annealing is performed using the Metropolis algorithm with local spin updates on a triangular lattice of $N={24}^2$ sites under periodic boundary conditions. Starting from a random initial spin configuration, the temperature is gradually decreased from a high initial value ($T_0=1$–5) down to $T={10}^{-4}$ according to $T_{n+1}=0.999999T_n$, where $T_n$ is the temperature at the nth step. At each temperature, we update spins locally in real space. At the final temperature $T={10}^{-4}$, we further perform between ${10}^5$ and ${10}^6$ Monte Carlo sweeps to optimize the spin configurations and ensure convergence. The procedure is repeated independently for each set of $(I^{\mathrm{BA}},H^x)$ to construct the phase diagrams in an unbiased way. In addition, to avoid trapping in metastable states near phase boundaries, simulations are also initialized from low-temperature spin configurations obtained in neighboring parameter regions.

To characterize the resulting phases, we evaluate the spin structure factor, corresponding to the magnetic order parameter. The expression is given by

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

(3)

where $\eta =x,y,z$ denotes the spin component and ${\mathit{r}}_i$ the position of site i. The uniform magnetization is given by the real-space average,

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

(4)

In addition, we compute the scalar spin chirality, which serves as an indicator of topological character:

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

(5)

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

(6)

where $\mu =(u,d)$ labels upward and downward triangles of the triangular lattice in the lattice structure, and $(i,j,k)$ are the three sites of a given triangle $\mathit{R}$ taken in counterclockwise order. A finite ${\chi }^{\mathrm{sc}}$ signifies a noncoplanar spin texture and topologically nontrivial magnetic order, since it produces an emergent gauge flux that couples to itinerant electrons via the spin Berry phase mechanism [128,129,130,131,132].

3. Results

We discuss the magnetic phase diagrams of the model in Equation (1). Section 3.1 presents the results for positive bond-dependent anisotropy ($I^{\mathrm{BA}}>0$), while Section 3.2 analyzes the case of negative anisotropy ($I^{\mathrm{BA}}<0$), where distinct selections of ordering wave vectors occur although the phase taxonomy remains the same.

3.1. Positive Bond-Dependent Anisotropy

Figure 1 displays the magnetic phase diagram versus $I^{\mathrm{BA}}>0$ and $H_x$ at low temperatures. Apart from the fully polarized phase, which is stabilized in the high-field region, six magnetic phases appear. The results at $I^{\mathrm{BA}}=0$ are consistent with those reported in the previous study [8]. By introducing $I^{\mathrm{BA}}>0$, two additional phases, denoted as the triple-Q I and triple-Q II phases, are obtained.

We first examine the sequence of magnetic phases as a function of the applied magnetic field for the case of weak bond anisotropy, $I^{\mathrm{BA}}=0.02$. At low fields, a single-Q vertical spiral state is stabilized, whose real-space spin pattern is plotted in the leftmost column of Figure 2a. The spin configuration is characterized by the single-Q spiral wave lying on the $xy$ plane parallel to the magnetic field, where the spiral plane is elliptical owing to the presence of the magnetic field and bond-dependent anisotropy. Among the three ordering wave vectors, ${\mathit{Q}}_1$, which is aligned with the magnetic field, is selected so as to maximize the exchange interaction energy for $I^{\mathrm{BA}}>0$. Since $I^{\mathrm{BA}}>0$ favors the spin oscillation perpendicular to the ordering wave vector ${\mathit{Q}}_{\nu }$, the spin structure factor $S_s^{yy}\left({\mathit{Q}}_1\right)$ is larger than $S_s^{xx}\left({\mathit{Q}}_1\right)$, as shown in the middle-left and middle-right panels of Figure 2a. Meanwhile, there are no intensities in the z-spin component owing to the easy-plane single-ion anisotropy, as shown in the rightmost column of Figure 2a. Reflecting the coplanar spin texture, there is no net scalar spin chirality, as shown in Figure 3a. In this state, the magnetization increases linearly with the applied magnetic field.

When the magnetic field increases, the single-Q vertical spiral state turns into the triple-Q I state with a jump of the magnetization $M^x$, as shown in Figure 3a. Similarly to the single-Q vertical spiral state, the triple-Q I state exhibits the in-plane spin configuration with a negligibly small z-spin modulation, as shown in the rightmost column of Figure 2b. On the other hand, the in-plane spin configuration is expressed as a superposition of triple-Q spin density waves with different intensities, as shown in the middle-left and middle-right panels of Figure 2b. Accordingly, the vortex-type spin configuration can be seen in a real-space plot, as shown in the leftmost panel of Figure 2b. This triple-Q I state has been reported in the previous model even without $I^{\mathrm{BA}}$ by increasing the magnitude of A [8]. This result indicates that the bond-dependent anisotropy $I^{\mathrm{BA}}$ enhances the effect of the easy-axis single-ion anisotropy. It is noted that a small but nonzero $S_s^{zz}\left({\mathit{Q}}_{\nu }\right)$ in this state leads to small scalar spin chirality ${\chi }^{\mathrm{sc}}$, as shown in Figure 3a. Since a finite ${\chi }^{\mathrm{sc}}$, indicative of topological spin textures, also appears in the BmX as will be described below, we distinguish between the two phases by setting a cutoff criterion; states with ${\left({\chi }^{\mathrm{sc}}\right)}^2<0.005$ are classified as the triple-Q I state, whereas those with ${\left({\chi }^{\mathrm{sc}}\right)}^2>0.005$ are identified as BmX.

The triple-Q I state is replaced by the BmX with increasing the magnetic field. The first-order phase transition between them occurs by looking at ${\chi }^{\mathrm{sc}}$, although the magnetization seems to be almost continuous, as shown in Figure 3a. The BmX exhibits a similar in-plane spin configuration to the triple-Q I state, as shown in the middle-left and middle-right panels of Figure 2c. The distinct difference appears in the z-spin component; in the BmX, the triple-Q intensities at ${\mathit{Q}}_1$–${\mathit{Q}}_3$ are clearly found, as shown in the rightmost panel of Figure 2c. Thus, the spin configuration of the BmX is noncoplanar, as illustrated in the leftmost panel of Figure 2c; the positive z-spin configuration appears around the vortex with the positive vorticity, whereas the negative one appears around the antivortex with the negative vorticity. The periodic alignment of the vortices and antivortices gives rise to a finite scalar spin chirality in the whole system, as shown in Figure 3a. It is noted that the BmX also shows a net magnetization along the out-of-plane direction shown in the $\mathit{q}=\mathbf{0}$ component of the spin structure factor in the rightmost column of Figure 2c, whose appearance is attributed to the fact that the scalar spin chirality and z-spin magnetization are the same symmetry. Thus, it is expected that the BmX shows the topological and anomalous Hall effects even without the out-of-plane magnetic field.

The increase in the magnetic field leads to the phase transition from the BmX to the single-Q conical spiral state, as shown in Figure 1. Both the magnetization and scalar spin chirality exhibit discontinuous change, as shown in Figure 3a. The main contribution to the spin configuration is the y- and z-spin components at ${\mathit{Q}}_1$, i.e., $S_s^{yy}\left({\mathit{Q}}_1\right)$ and $S_s^{zz}\left({\mathit{Q}}_1\right)$, as shown in the middle-right and rightmost panels of Figure 2d, where the intensity of the y-spin component is much larger than that of the z-spin component owing to the easy-plane single-ion anisotropy. The real-space spin configuration is shown in the leftmost panel of Figure 2d. Although the spin configuration is noncoplanar, there is no net scalar spin chirality.

When the magnetic field is further increased, the magnitude of the z-spin component gradually decreases and vanishes. This corresponds to the appearance of the single-Q fan state. The in-plane spin configuration of the single-Q fan state is similar to that of the single-Q conical spiral state, as shown in the left three panels of Figure 2e. Meanwhile, there is no structure in the z-spin component, as shown in the rightmost panel of Figure 2e. The single-Q fan state finally turns into the fully polarized state by further increasing the magnetic field.

At $I^{\mathrm{BA}}=0.02$, the stability regions of the triple-Q I and single-Q conical spiral phases in Figure 3a appear extremely sharp. This reflects the first-order nature of these transitions at low temperatures: in the simulated annealing process, such changes occur within a single field increment. While in real crystals thermal and quantum fluctuations would smooth the boundary to some extent, the sharp character of the transition should still be observable at sufficiently low temperatures.

For the strong bond-dependent anisotropy, the BmX is perfectly replaced by the triple-Q I state, where the typical critical value of $I^{\mathrm{BA}}$ is 0.03. In the low-field region, the single-Q vertical spiral state is also replaced by another triple-Q state, which is referred to as the triple-Q II state, as shown in Figure 1. The typical real-space spin configuration of the triple-Q II state is illustrated in the leftmost panel of Figure 2f. Similarly to the triple-Q I state and BmX, the periodic alignment of the vortices and antivortices appears. The difference between them is found in the relative intensities among the triple-Q spin density waves. As shown in the middle-left and middle-right panels of Figure 2f, in the triple-Q II state, the intensities at ${\mathit{Q}}_2$ and ${\mathit{Q}}_3$ are different from each other, while they are the same in the triple-Q I state and BmX. This state does not show the scalar spin chirality owing to the absence of the z-spin component, as shown in the rightmost panel of Figure 2f. In addition, the triple-Q II state is distinguished from the triple-Q I state by the magnetization jump, as shown in Figure 3b. It is noted that this state reduces to the double-Q state in the limit of the absence of the magnetic field $H_x\to 0$ [77].

3.2. Negative Bond-Dependent Anisotropy

For negative bond-dependent anisotropy ($I^{\mathrm{BA}}<0$), spin oscillations parallel to the ordering wave vectors are energetically favored. This distinction modifies the stability tendency of the phase diagram when the in-plane magnetic field is applied. The resulting low-temperature magnetic phase diagram as functions of $I^{\mathrm{BA}}<0$ and $H_x$ is presented in Figure 4. When compared with the phase diagram for $I^{\mathrm{BA}}>0$ shown in Figure 1, the set of magnetic phases appearing in the phase diagram in Figure 5 remains the same. Moreover, the spin configurations of these phases are qualitatively similar to those obtained for $I^{\mathrm{BA}}>0$. The key differences between the two cases are summarized below.

Because a magnetic field applied along the x direction favors spin orientations perpendicular to the field, the selection of ordering wave vectors in each phase depends sensitively on the sign of $I^{\mathrm{BA}}$. For magnetic phases characterized by single-Q spin density waves, such as the single-Q vertical spiral, single-Q conical spiral, and single-Q fan states, the ordering wave vector ${\mathit{Q}}_2$ or ${\mathit{Q}}_3$ is selcted rather than ${\mathit{Q}}_1$, as shown in Figure 5a, Figure 5d, and Figure 5e, respectively. In a similar manner, for multiple-Q states, including triple-Q I, BmX, and triple-Q II, the intensity plots of the spin structure factor for $I^{\mathrm{BA}}<0$ are different from those for $I^{\mathrm{BA}}>0$, as shown in Figure 5b, Figure 5c, and Figure 5f, respectively. This difference reflects how to gain the energy by $I^{\mathrm{BA}}$ and $H_x$. The $H_x$ dependence of the magnetization and scalar spin chirality at $I^{\mathrm{BA}}=-0.02$ and $I^{\mathrm{BA}}=-0.2$ is shown in Figure 6a and Figure 6b, respectively.

Furthermore, a key quantitative difference between the phase diagrams for $I^{\mathrm{BA}}<0$ and $I^{\mathrm{BA}}>0$ lies in the stability of the BmX against bond-dependent anisotropy; the BmX remains stable up to $|I^{\mathrm{BA}}|\simeq 0.06$ for $I^{\mathrm{BA}}<0$, whereas it remains stable up to $|I^{\mathrm{BA}}|\simeq 0.03$ for $I^{\mathrm{BA}}>0$. This demonstrates that negative bond-dependent anisotropy significantly enhances the robustness of the BmX, extending its stability range to nearly twice the critical anisotropy strength required for a transition to topologically trivial multiple-Q states.

4. Conclusions

We have constructed low-temperature phase diagrams under in-plane magnetic fields for a centrosymmetric triangular-lattice spin model that incorporates easy-plane single-ion anisotropy together with a bond-dependent symmetric exchange interaction formulated in momentum space. By means of simulated annealing, we demonstrated that the BmX is robustly stabilized at intermediate fields when the bond-dependent anisotropy is weak, regardless of its sign, whereas stronger anisotropy drives the system into topologically trivial triple-Q phases.

The BmX is characterized by pronounced triple-Q intensity in both the in-plane and out-of-plane spin components, a finite scalar spin chirality, and a concomitant $\mathit{q}=\mathbf{0}$ weight in the z-spin component, which corresponds to distinctive hallmarks that point to emergent-field phenomena such as the topological Hall effects. Upon reversing the sign of the bond-dependent anisotropy, the sequence of phases remains unchanged, but the selected ordering vectors rotate in momentum space, consistent with the tendency of an in-plane magnetic field to favor spin modulations perpendicular to its direction. Notably, negative bond-dependent anisotropy substantially enhances the stability of the BmX: the critical anisotropy strength required to destabilize the BmX is approximately doubled compared to the positive case. This sign asymmetry offers a concrete principle for extending the stability window of the BmX in real materials.

Overall, our findings establish centrosymmetric easy-plane magnets as promising hosts for topological spin textures without the DM interaction, and they provide practical insights for guiding the search for novel platforms to realize BmXs and related emergent electrodynamic phenomena.