Multiple-Q States in Bilayer Triangular-Lattice Systems with Bond-Dependent Anisotropic Interaction

Abstract

We investigate magnetic instabilities toward multiple-Q states in centrosymmetric bilayer triangular-lattice systems. By focusing on the interplay between the layer-dependent Dzyaloshinskii–Moriya interaction and layer-independent bond-dependent anisotropic interaction, both of which originate from the relativistic spin-orbit coupling, we construct a low-temperature phase diagram based on an effective spin model that also includes frustrated isotropic exchange interactions. Employing simulated annealing, we reveal the stabilization of three distinct double-Q phases in the absence of an external magnetic field, each characterized by noncoplanar spin textures with spatially modulated local scalar spin chirality. Under applied magnetic fields, we identify field-induced phase transitions among single-Q, double-Q, and triple-Q states, some of which exhibit a finite net scalar spin chirality indicative of topologically nontrivial order. These findings highlight centrosymmetric systems with sublattice-dependent Dzyaloshinskii–Moriya interactions as promising platforms for realizing a variety of multiple-Q spin textures.

1. Introduction

Frustrated magnetic systems have long served as fertile ground for the emergence of exotic spin textures. Among them, multiple-Q magnetic states, which are characterized by the superposition of spin modulations with distinct wave vectors, have attracted considerable attention owing to their rich topological properties and potential applications in spintronics [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. These states typically arise from a delicate interplay of competing exchange interactions, magnetic anisotropies, and external magnetic fields, and encompass diverse configurations such as skyrmion crystals, vortex lattices, and noncoplanar chiral spin states [15,16,17,18,19].

One of the key ingredients behind the stabilization of such nontrivial spin textures is the Dzyaloshinskii–Moriya (DM) interaction, which originates from relativistic spin–orbit coupling in systems lacking spatial inversion symmetry [20,21]. In such noncentrosymmetric crystals, the competition between the ferromagnetic exchange interaction and the DM interaction induces a preferred spin rotation between neighboring sites, often giving rise to spiral spin textures. Under an applied external magnetic field, these spiral states may evolve into topologically nontrivial magnetic configurations, such as skyrmion crystals [22,23,24,25,26,27,28,29,30], as observed in chiral magnets like MnSi [31,32,33,34,35,36,37,38,39,40], ${\mathrm{Fe}}_{1-x}$${\mathrm{Co}}_x$Si [41,42,43,44], FeGe [45,46], ${\mathrm{Cu}}_2$${\mathrm{OSeO}}_3$ [47,48,49,50,51,52], and EuPtSi [53,54,55,56,57,58], polar magnets like ${\mathrm{GaV}}_4$${\mathrm{S}}_8$ [59,60,61], ${\mathrm{GaV}}_4$${\mathrm{Se}}_8$ [62,63,64,65], and ${\mathrm{VOSe}}_2$${\mathrm{O}}_5$ [66,67], and other noncentrosymmetric magnets like ${\mathrm{Mn}}_{1.4}$${\mathrm{Pt}}_{0.9}$${\mathrm{Pd}}_{0.1}$Sn [68,69,70] and ${\mathrm{Fe}}_{1.9}$${\mathrm{Ni}}_{0.9}$${\mathrm{Pd}}_{0.2}$P [71]. The specific type of skyrmion crystals is determined by the form of the Lifshitz invariants ${\mathcal{L}}_{ij}^{\left(k\right)}=m_i{\partial }_k{m}_j-m_j{\partial }_k{m}_i$, where $m_i$ denotes the magnetic moment component for $i=x,y,z$ [22,23,72,73].

More recently, centrosymmetric materials that retain spatial inversion symmetry have also emerged as promising hosts of multiple-Q states. In such systems, even in the absence of the DM interaction, yet alternative mechanisms such as the Fermi surface nesting [10], frustrated exchange interaction [74,75], dipolar interaction [76,77], and crystalline-dependent magnetic anisotropy [78,79,80,81,82] have been shown to stabilize skyrmion and other complex magnetic states. Notable examples include Gd-based compounds, such as ${\mathrm{Gd}}_2$${\mathrm{PdSi}}_3$ [83,84,85], ${\mathrm{Gd}}_3$${\mathrm{Ru}}_4$${\mathrm{Al}}_{12}$ [86,87,88], and ${\mathrm{GdRu}}_2$${\mathrm{Si}}_2$ [89,90,91,92].

Meanwhile, even within centrosymmetric crystal structures, the DM interaction can still arise in a sublattice-dependent manner, preserving global spatial inversion symmetry [93,94,95,96]. This so-called hidden or staggered DM interaction opens a promising path towards realizing topological magnetic states in artificial multilayers, bilayers, and other engineered heterostructures [97,98].

In the present study, we explore the emergence of multiple-Q states in centrosymmetric bilayer triangular-lattice systems. We construct an effective spin model that incorporates frustrated isotropic Heisenberg interactions, a layer-dependent DM interaction, and a bond-dependent anisotropic interaction by taking into account the effect of the relativistic spin–orbit coupling. By employing simulated annealing, we map out the low-temperature phase diagram and analyze the magnetic textures stabilized in both zero and finite external magnetic fields. Our results reveal a variety of noncoplanar double-Q and triple-Q magnetic phases, some of which exhibit nonzero scalar spin chirality. In addition, these findings suggest the crucial role of the competition between frustration and spin–orbit coupling in locally noncentrosymmetric systems, which can be a source of rich multiple-Q states.

The rest of this paper is organized as follows. In Section 2, we introduce the effective spin model on the bilayer triangular lattice. We consider four types of interactions: intralayer Heisenberg exchange interaction, interlayer Heisenberg exchange interaction, layer-dependent DM interaction, and bond-dependent anisotropic interaction. We also introduce the effect of the external magnetic field through the Zeeman coupling. Then, we briefly describe simulated annealing used in obtaining optimal spin configurations at each model parameter. In Section 3, we discuss numerical results. First, we present the low-temperature phase diagram in the absence of the magnetic field. We show that double-Q instabilities occur when the DM interaction or bond-dependent anisotropic interaction increases. Next, we discuss field-induced multiple-Q states in the presence of the magnetic field. In both cases, we describe spin and scalar spin chirality quantities in each magnetic phase. Finally, our conclusions are summarized in Section 4.

2. Model and Method

To investigate multiple-Q instabilities in bilayer triangular-lattice systems, we consider the following model Hamiltonian, which is given by

$$\begin{array}{cc}\hfill \mathcal{H}=& {\mathcal{H}}^{\perp }+{\mathcal{H}}^{\Vert }+{\mathcal{H}}^{\mathrm{Z}},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\mathcal{H}}^{\perp }=& \sum _{\eta }\sum _{ij}\left[J_{ij}\left({\mathit{S}}_i·{\mathit{S}}_j+\sum _{\alpha \beta }{I}_{ij}^{\alpha \beta }{S}_i^{\alpha }{S}_j^{\beta }\right)-{\mathit{D}}_{ij}^{\left(\eta \right)}·({\mathit{S}}_i\times {\mathit{S}}_j)\right],\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\mathcal{H}}^{\Vert }=& J_{\Vert }\sum _i{\mathit{S}}_i·{\mathit{S}}_{i+\widehat{z}},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\mathcal{H}}^{\mathrm{Z}}=& -H\sum _{\eta }\sum _i{S}_i^z,\hfill \end{array}$$

(4)

where ${\mathit{S}}_i$ represents a classical spin vector of unit length $|{\mathit{S}}_i|=1$ at site i. The bilayer triangular lattice consists of two triangular-lattice planes stacked along the z axis with identical $xy$ coordinates, resulting in a total of $2\times L^2$ spins. We refer to the lower and upper layers as layer A and layer B, respectively.

The first term in Equation (1), ${\mathcal{H}}^{\perp }$, describes the intralayer interactions. It includes a layer-independent symmetric exchange interaction with $J_{ij}$ and a layer-dependent staggered DM interaction with ${\mathit{D}}_{ij}^{\left(\mathrm{A}\right)}=-{\mathit{D}}_{ij}^{\left(\mathrm{B}\right)}$. We also incorporate a bond-dependent anisotropic exchange interaction, whose form factor is given by $I_{ij}^{\alpha \beta }={\tilde{I}}^{\mathrm{BA}}{\widehat{r}}_{ij}^{\alpha }{\widehat{r}}_{ij}^{\beta }$, where ${\widehat{r}}_{ij}^{\alpha }$ is the $\alpha =x,y,z$ component of the unit vector connecting sites i and j. This form arises naturally from discrete rotational symmetry considerations [99].

The layer-dependent DM interaction originates from the lack of local inversion symmetry at each layer. Owing to opposing local crystalline electric fields, the resulting polar DM vector lies perpendicular to both the in-plane bond and the z directions. Such staggered DM interactions are known to stabilize skyrmion crystals in centrosymmetric systems [97,98], and are closely related to odd-parity multipole and topological phenomena, as demonstrated in other locally noncentrosymmetric lattice structures, such as the zigzag-chain [100,101,102,103,104,105,106], honeycomb [107,108,109,110,111], and diamond structures [112,113,114,115,116,117].

The second term in Equation (1), ${\mathcal{H}}^{\Vert }$, describes the interlayer exchange interactions. We consider the antiferromagnetic isotropic exchange interaction $J_{\Vert }>0$. The third term in Equation (1), ${\mathcal{H}}^{\mathrm{Z}}$, represents the Zeeman coupling due to an external magnetic field along the z direction.

To efficiently explore the lowest-energy spin configurations over a wide range of model parameters, we reduce the complexity of the intralayer Hamiltonian by retaining only the dominant contributions at low temperatures. This is achieved by expressing the intralayer Hamiltonian in momentum space:

$$\begin{array}{cc}\hfill {\tilde{\mathcal{H}}}^{\perp }=& \sum _{\nu }\sum _{\eta }\left[-J({\mathit{S}}_{{\mathit{Q}}_{\nu }}^{\left(\eta \right)}·{\mathit{S}}_{-{\mathit{Q}}_{\nu }}^{\left(\eta \right)}+I_{{\mathit{Q}}_{\nu }}^{\alpha \beta }{S}_{{\mathit{Q}}_{\nu }}^{\alpha \left(\eta \right)}{S}_{-{\mathit{Q}}_{\nu }}^{\beta \left(\eta \right)})-i{\mathit{D}}_{{\mathit{Q}}_{\nu }}^{\left(\eta \right)}·({\mathit{S}}_{{\mathit{Q}}_{\nu }}^{\left(\eta \right)}\times {\mathit{S}}_{-{\mathit{Q}}_{\nu }}^{\left(\eta \right)})\right],\hfill \end{array}$$

(5)

where $S_{{\mathit{Q}}_{\nu }}^{\alpha \left(\eta \right)}$ is the Fourier component of the spins along the direction $\alpha =x,y,z$ and wave vector ${\mathit{Q}}_{\nu }$ on layer $\eta$. The six dominant wave vectors are chosen as ${\mathit{Q}}_1=(\pi /3,0)$, ${\mathit{Q}}_2=(-\pi /6,\sqrt{3}\pi /6)$, ${\mathit{Q}}_3=(-\pi /6,-\sqrt{3}\pi /6)$, ${\mathit{Q}}_4=-{\mathit{Q}}_1$, ${\mathit{Q}}_5=-{\mathit{Q}}_2$, and ${\mathit{Q}}_6=-{\mathit{Q}}_3$, preserving the six-fold rotational symmetry of the bilayer triangular-lattice system. This momentum–space truncation is justified by the sharp peak structure in the momentum-resolved exchange interaction at ${\mathit{Q}}_{\nu }$, which captures nearly the full ground-state energy when high-harmonic contributions are negligible [75,118]. The interaction parameters at the dominant ordering wave vectors are given by J and $D\equiv |{\mathit{D}}_{{\mathit{Q}}_{\nu }}^{\left(\eta \right)}|$. The bond-dependent anisotropic form factor in momentum space, $I_{{\mathit{Q}}_{\nu }}^{\alpha \beta }$, is given by $-I_{{\mathit{Q}}_1}^{xx}=I_{{\mathit{Q}}_1}^{yy}=2I_{{\mathit{Q}}_2}^{xx}=-2I_{{\mathit{Q}}_2}^{yy}=2I_{{\mathit{Q}}_2}^{xy}/\sqrt{3}=2I_{{\mathit{Q}}_2}^{yx}/\sqrt{3}=2I_{{\mathit{Q}}_3}^{xx}=-2I_{{\mathit{Q}}_3}^{yy}=-2I_{{\mathit{Q}}_3}^{xy}/\sqrt{3}=-2{\tilde{I}}_{{\mathit{Q}}_3}^{yx}/\sqrt{3}\equiv I^{\mathrm{BA}}$ (the others are zero) [118]. Positive (negative) $I_{{\mathit{Q}}_{\nu }}^{\alpha \beta }$ tends to favor the spiral plane perpendicular (parallel) to the ordering wave vector.

To investigate the ground-state spin configuration rather than the finite-temperature one, we use simulated annealing to minimize the energy of the total Hamiltonian $\mathcal{H}={\tilde{\mathcal{H}}}^{\perp }+{\mathcal{H}}^{\Vert }+{\mathcal{H}}^{\mathrm{Z}}$. The energy unit is fixed as $J=1$, and the interlayer exchange interaction is set to $J_{\Vert }=2$ or $J_{\Vert }=1.5$, representing a regime of dominant interlayer coupling; the results for the weak interlayer exchange interaction ($J_{\Vert }<J$) have been shown in ref. [97], where the skyrmion crystal is stabilized in the external magnetic field. The other model parameters $(I^{\mathrm{BA}},D,H)$ are treated as variables in investigating the low-temperature phase diagram. Simulations begin at a high temperature $T_0/J=1$ and progressively cool the system using a temperature reduction schedule defined by $T_{n+1}=0.999999T_n$, where $T_n$ denotes the temperature at the nth step. This process continues until the final temperature reaches $T=0.001$. At each temperature step, all the spins are locally updated in real space using the standard Metropolis algorithm. Once the system reaches the final temperature, we carry out an additional ${10}^5$–${10}^6$ Monte Carlo sweeps to ensure thermalization and to perform measurements. Simulations are conducted for $L=24$ under periodic boundary conditions. The above procedure is independently applied for each parameter set of $(I^{\mathrm{BA}},D,H)$. Near phase boundaries, simulations are additionally initialized from previously obtained low-temperature spin configurations, and we adopt the lowest-energy state among them.

To distinguish between different magnetic phases, which are characterized by distinct spin textures and scalar spin chiralities, we evaluate these quantities based on the spin configurations obtained at the lowest temperature. For the spin sector, we calculate the spin structure factor for each spin component $\alpha =x,y,z$ and layer $\eta =\mathrm{A},\mathrm{B}$, defined as

$$\begin{array}{c}\hfill S_{\eta }^{\alpha }\left(\mathit{q}\right)={\displaystyle \frac{1}{L^2}}\sum _{j,l\in \eta }{S}_j^{\alpha }{S}_l^{\alpha }{e}^{i\mathit{q}·({\mathit{r}}_j-{\mathit{r}}_l)},\end{array}$$

(6)

where j and l are site indices restricted to layer $\eta$. The in-plane spin structure factor is obtained as $S_{\eta }^{xy}\left(\mathit{q}\right)=S_{\eta }^x\left(\mathit{q}\right)+S_{\eta }^y\left(\mathit{q}\right)$. The magnetization along the z axis for each layer is given by $M_{\eta }^z=(1/L^2){\sum }_{i\in \eta }{S}_i^z$.

For the scalar spin chirality, we compute its layer-resolved form as

$$\begin{array}{c}\hfill {\chi }_{\eta }^{\mathrm{sc}}={\displaystyle \frac{1}{L^2}}\sum _{\mathit{R}\in \eta }{\mathit{S}}_j·({\mathit{S}}_k\times {\mathit{S}}_l),\end{array}$$

(7)

where $\mathit{R}$ denotes the position of the triangle center, and the three sites $(j,k,l)$ form a triangle at $\mathit{R}$ in counterclockwise order. The local scalar spin chirality on triangle $\mathit{R}$ is given by ${\chi }_{\mathit{R}}^{\mathrm{sc}}={\mathit{S}}_j·({\mathit{S}}_k\times {\mathit{S}}_l)$. The total scalar chirality ${\chi }^{\mathrm{sc}}={\chi }_{\mathrm{A}}^{\mathrm{sc}}+{\chi }_{\mathrm{B}}^{\mathrm{sc}}$ serves as a hallmark of the topologically nontrivial magnetic order, which exhibits the topological Hall effect [119,120,121].

3. Results

First, we discuss the low-temperature magnetic phases in the absence of the external magnetic field, i.e., $H=0$. Figure 1a,b present the magnetic phase diagram as a function of the layer-dependent DM interaction D and bond-dependent anisotropic form factor $I^{\mathrm{BA}}$ for $J_{\Vert }=2$ and $J_{\Vert }=1.5$, respectively. In both cases, the phase diagram includes three double-Q phases denoted as 2Q I, 2Q II, and 2Q III in addition to the single-Q phase denoted as 1Q.

The 1Q phase tends to be stabilized in the regime of large D and small $I^{\mathrm{BA}}$. The real-space spin configurations for layers A and B are presented in the upper and lower panels of Figure 2a, respectively, where the spin moments in layer A are antiparallel to those in layer B due to the strong antiferromagnetic interlayer exchange interaction. The spiral plane orientation is dictated by the competition between the DM interaction and bond-dependent anisotropic interaction: The former tends to favor the cycloidal spiral configuration with opposite helicities in the two layers, while the latter tends to favor the proper-screw spiral configuration. Because the DM interaction imposes opposite helicities for each layer, it incurs an energy cost in the presence of antiferromagnetic interlayer coupling, which favors antiparallel alignment. Consequently, the spiral plane aligns nearly perpendicularly to the wave vector, dominated by the bond-dependent anisotropy. The resulting spin structure is a nearly pure proper-screw spiral configuration, with a slight tilt due to finite D, as shown in Figure 2a. The corresponding spin structure factor of the 1Q state for layer A is shown in Figure 3a; the spin structure factor for layer B is the same.

As $I^{\mathrm{BA}}$ increases, the system transitions to the 2Q I state, as shown by the phase diagram in Figure 1. In contrast to the 1Q state, the in-plane spin component is characterized by the double-Q modulations, as shown by the spin structure factor for layer A in the upper panel of Figure 3b. Meanwhile, the out-of-plane spin component remains the single-Q modulation, similarly to the 1Q state, as shown in the lower panel of Figure 3b. The spin structure factor for layer B is the same as that for layer A. Reflecting double-Q modulations for the in-plane spin component, the meron-type vortices and antivortices are found in the real-space spin configuration, as shown in Figure 2b; the spin moments for layers A and B are antiparallel to each other in order to gain the antiferromagnetic interlayer exchange energy. These vortices with noncoplanar spin textures accompany a local scalar spin chirality, as shown in Figure 4a. The local scalar spin chirality is spatially modulated, meaning the emergence of chirality density wave [122]. The direction of the chirality density wave is characterized by the direction of the sub-dominant ordering wave vectors; in the case of Figure 4a, the chirality density wave occurs along the ${\mathit{Q}}_1$ direction, where the intensity of the spin structure factor at ${\mathit{Q}}_1$ is smaller than that at ${\mathit{Q}}_3$ in Figure 3b. There is no uniform component of the scalar spin chirality.

Upon further increasing $I^{\mathrm{BA}}$, the system enters the 2Q II phase. Here, both in-plane and out-of-plane spin components exhibit nearly isotropic double-Q modulations with equal intensities, as shown in Figure 3c in the case of $I^{\mathrm{BA}}=0.42$, leading to a checkerboard-like pattern in the z component of the spin texture, as shown in Figure 2c. Similarly to the 2Q I state, the spin moments for layer A are antiparallel to those for layer B. The scalar spin chirality becomes nonzero in the region where the vortices with the finite z-spin component appear in real-space, resulting in a checkerboard-like distribution in both layers. On the other hand, the sign of scalar spin chirality is opposite for layers A and B due to antiferromagnetic interlayer alignment, as shown in Figure 4b; there is no net scalar spin chirality in the whole system. This state resembles the meron-antimeron crystal reported in various systems [123,124,125,126,127,128,129,130,131].

Since $I^{\mathrm{BA}}$ tends to favor the in-plane spin configuration energetically, the intensities of the in-plane spin structure factor are much larger than those of the out-of-plane spin structure factor, as shown in Figure 3c. Such a tendency becomes more pronounced as $I^{\mathrm{BA}}$ increases. We show the real-space spin configuration for larger $I^{\mathrm{BA}}=0.5$ in Figure 2d, where the z-spin modulation is barely discernible. In addition, the intensities of the out-of-plane spin structure factor are almost negligible, as shown in the lower panel of Figure 3d. Accordingly, the local scalar spin chirality is significantly suppressed as $I^{\mathrm{BA}}$ increases. Meanwhile, the intensities of the in-plane spin structure factor are almost the same as those at $I^{\mathrm{BA}}=0.42$, as shown in the upper panel of Figure 3d.

Finally, the 2Q III phase emerges in the region where D is ranged from 0.42 to 0.5 (0.37 to 0.5) at $I^{\mathrm{BA}}=0$, as shown by the phase diagram in Figure 1a,b. The spin configuration of this phase consists of anisotropic double-Q superpositions in both $xy$ and z spin components, as shown in Figure 3e. In contrast to the other double-Q states, the peak structures of the spin structure factor for layers A and B are different from each other, as shown in Figure 3e,f; the dominant peak appears at ${\mathit{Q}}_1$ (${\mathit{Q}}_3$) for layer A (B), while the subdominant peak appears at ${\mathit{Q}}_3$ (${\mathit{Q}}_1$) in both $xy$ and z components. Accordingly, the real-space spin configurations for layers A and B are different from each other, as shown in Figure 2e, where the spiral plane directions are different. In addition, the local scalar spin chiralities for layers A and B are characterized by the different spatial distributions, as shown in Figure 4c. Such a difference is understood from the fact that the 2Q III state is stabilized for large D but without $I^{\mathrm{BA}}$. In such a situation, the system tends to gain energy from the DM interaction as well as the antiferromagnetic interlayer exchange interaction. Such a tendency was also found in a similar model with a smaller interlayer exchange interaction so that the effect of the DM interaction is relatively large [97].

Next, we discuss the magnetic phase evolution under a finite external magnetic field, i.e., $H\ne 0$, with a particular focus on the stabilization of field-induced multiple-Q states for $J_{\Vert }=2$. We show the H dependence of the magnetization and the scalar spin chirality for three representative values of the bond-dependent anisotropic form factor: $I^{\mathrm{BA}}=0.1$ in Figure 5a, $I^{\mathrm{BA}}=0.2$ in Figure 5b, and $I^{\mathrm{BA}}=0.4$ in Figure 5c. All the results are obtained at a fixed DM interaction $D=0.2$, where the zero-field states correspond to the 1Q, 2Q I, and 2Q II states, respectively.

For $I^{\mathrm{BA}}=0.1$, the 1Q state is stabilized in the system at zero field. As the magnetic field is introduced, a second-order phase transition occurs at $H\simeq 0.45$ to a 3Q state. Although the magnetization curve shows no clear anomaly, the scalar spin chirality for both layers A and B becomes finite, indicating the emergence of a noncoplanar spin texture, as shown in Figure 5d. The scalar spin chirality is not perfectly canceled between layers, resulting in a net component over the whole system. Figure 6a displays the real-space spin configuration in this 3Q state, while Figure 7a shows the corresponding scalar spin chirality distribution. While visually similar to the 2Q I state in Figure 2b and Figure 4a, the 3Q state is distinguished by the presence of small but finite intensities at all three $\mathit{Q}$ ordering wave vectors in both in-plane and out-of-plane spin components, as shown in Figure 8a.

Upon further increasing the magnetic field, the system undergoes a first-order phase transition to the 1Q state at $H\simeq 1.3$, as shown in Figure 5a. This 1Q state corresponds to the zero-field 1Q state in Figure 2a, although the present 1Q state exhibits a net magnetization due to the presence of the external magnetic field. Accordingly, the spiral plane lies in almost the $xy$ plane in order to gain the Zeeman energy, as shown in Figure 6b. This is also evidenced by the spin structure factor in Figure 8b, where the intensity of the in-plane component is larger than that of the out-of-plane component. Thus, this spin configuration is almost characterized by the conical spiral wave. Accordingly, the scalar spin chirality exhibits a sign change between upward and downward triangles, while maintaining the same sign between layers A and B due to the parallel alignment of the z-spin component, as shown in Figure 7b.

The 1Q state is replaced by the 2Q IV state with a jumps of scalar spin chirality at $H\simeq 2.7$, as shown in Figure 5a. The 2Q IV state exhibits nonzero uniform scalar spin chirality for both layers A and B, as shown in Figure 5d. The $xy$ component of the real-space spin configuration in the 2Q IV state resembles that in the 2Q II state, as shown in Figure 2b and Figure 6c, although the behaviors of the z-spin component are different from each other. In the 2Q IV state, the intensities of the z component of the spin structure factor at ${\mathit{Q}}_1$–${\mathit{Q}}_3$ are almost negligible, as shown in the lower panel of Figure 8c. Consequently, the z component of the real-space spin is characterized by the $\mathit{q}=\mathbf{0}$ uniform component. Such an almost uniform z-spin component combined with the double-Q-modulated $xy$-spin component leads to the checkerboard-type distribution of the scalar spin chirality, as shown in Figure 7c, although the positive and negative contributions are not perfectly canceled with each other. The 2Q IV state smoothly evolves into the fully polarized state with increasing H.

For larger bond-dependent anisotropy, i.e., $I^{\mathrm{BA}}=0.2$, the system starts in the 2Q I state at zero field. With an increasing H, the 2Q I state shows a similar behavior to the 3Q state in terms of the magnetization [Figure 5b] and the scalar spin chirality [Figure 5e]. Meanwhile, the double-Q structure is maintained against the magnetic field, indicating that there is no additional intensity at the third ordering wave vector, as confirmed by the absence of additional peaks in the spin structure factor. The 2Q I state undergoes a transformation into the 2Q IV state with a jump of the scalar spin chirality when H increases.

For stronger bond-dependent anisotropy, i.e., $I^{\mathrm{BA}}=0.4$, the system initially resides in the 2Q II state. Upon increasing the magnetic field, the 2Q II state exhibits a similar behavior to the 3Q and 2Q I states in the magnetic field, as shown in Figure 5c,f. This state also turns into the 2Q IV state, where both magnetization and scalar spin chirality look continuous in the transition. Thus, the 2Q IV state always appears in the high-field region irrespective of the magnitude of the bond-dependent anisotropic interaction.

4. Conclusions

In this study, we have investigated the emergence and stability of multiple-Q magnetic phases in centrosymmetric bilayer triangular-lattice systems, focusing on the interplay between layer-dependent DM interactions and bond-dependent anisotropic interactions, both originating from the relativistic spin–orbit coupling. Using simulated annealing techniques, we constructed the low-temperature phase diagrams in both zero and finite external magnetic fields. Our results revealed the existence of three distinct double-Q phases, denoted as 2Q I, 2Q II, and 2Q III, in the absence of an external magnetic field, each exhibiting the different spatial distributions of local scalar spin chirality, without the uniform component, due to their noncoplanar spin textures. Under applied magnetic fields, we further identified transitions between single-Q, double-Q, and triple-Q states, including the appearance of topologically nontrivial configurations such as the 3Q and 2Q IV phases. Especially, the field-induced multiple-Q phases show a net scalar spin chirality. Such a feature to exhibit nonzero scalar spin chirality persists across a wide range of parameter values, indicating its robust nature. These findings underscore the crucial role of frustration and spin–orbit coupling in stabilizing exotic multiple-Q spin textures in systems without local spatial inversion symmetry. Our results demonstrate that centrosymmetric bilayer systems with sublattice-dependent DM interactions serve as a promising platform for realizing topologically rich magnetic phases, including topological chiral magnetic states and chirality density waves, and may guide future explorations in spintronic applications.