Centrosymmetric Double-Q Skyrmion Crystals Under Uniaxial Distortion and Bond-Dependent Anisotropy

Abstract

We theoretically investigate the stability of double-Q square skyrmion crystals under uniaxial distortion. Using an effective spin model with frustrated exchange interactions and bond-dependent anisotropy in momentum space, we construct the low-temperature magnetic phase diagram via simulated annealing. Our results reveal that uniaxial distortion drives a phase transition from the skyrmion crystal to a single-Q conical spiral state when the ratio of exchange interactions parallel and perpendicular to the uniaxial axis is reduced to about 95%. We further find that topologically trivial double-Q states, which emerge in the low- and high-field regimes, are more robust against uniaxial distortion than the skyrmion crystal appearing in the intermediate-field regime. Finally, we examine the role of bond-dependent anisotropy and demonstrate that a finite relative magnitude of this anisotropy is crucial for stabilizing the skyrmion crystal, even under uniaxial distortion. These findings highlight the delicate interplay between lattice distortions and bond-dependent interactions in determining the stability of multiple-Q magnetic textures, and they provide useful guidance for experimental efforts to manipulate skyrmion crystal phases in centrosymmetric magnets.

1. Introduction

Skyrmions are topologically stable objects characterized by a quantized topological number, which renders them robust against external perturbations and gives rise to emergent electromagnetic phenomena. Since their original introduction by Tony Skyrme [1,2], these topologically nontrivial entities have been explored in a wide variety of contexts, including crystalline liquids [3,4,5,6,7], quantum Hall magnets [8,9,10,11,12,13,14], and Bose–Einstein condensates [15,16,17,18,19,20]. In magnetic materials, where skyrmions manifest as nanoscale swirling spin textures, their first experimental discovery in noncentrosymmetric chiral magnets [21,22,23,24] triggered enormous interest in condensed matter physics and materials science [25,26,27,28,29]. They are now recognized not only as a novel realization of topological order but also as promising candidates for next-generation spintronic applications, including racetrack memories and neuromorphic computing devices [30,31,32,33,34,35,36].

In particular, skyrmion crystals (SkXs), where skyrmions form periodic lattices, provide an ideal platform for studying collective excitations [37,38,39,40,41,42], topological transport phenomena [43,44,45,46,47,48,49,50,51], and field/temperature-driven phase transitions [52,53,54,55]. While most early studies focused on hexagonal SkXs stabilized by the Dzyaloshinskii–Moriya (DM) interaction [56,57] in noncentrosymmetric systems [27,58], it has become increasingly clear that SkXs are not restricted to such environments. Recent theoretical and experimental works have shown that SkXs can also be stabilized in centrosymmetric magnets through the competition between frustrated exchange interactions [59,60,61,62,63] and magnetic anisotropies [64,65,66,67,68,69,70,71], thereby significantly broadening the scope of candidate materials [72].

Of particular interest are tetragonal or square-lattice magnets, where the interplay of symmetry-related ordering wave vectors can give rise to double-Q spin textures, including square SkXs. Recent studies have identified double-Q SkXs in tetragonal compounds such as ${\mathrm{Co}}_{10-x/2}$${\mathrm{Zn}}_{10-x/2}$${\mathrm{Mn}}_x$ [73,74,75,76,77] and ${\mathrm{Cu}}_2$OSeO₃ [78,79], GdRu₂Si₂ [80,81,82,83,84,85], and GdRu₂Ge₂ [86]. Understanding the stabilization mechanism and controllability of such square SkXs is crucial both for fundamental studies of nontrivial spin orders. So far, several mechanisms have been theoretically proposed, such as momentum–space frustration [87,88], bond-dependent interaction [89,90], dipolar interaction [91], and staggered DM interaction [92,93].

One of the central questions is how external perturbations affect the delicate stability of SkXs [94,95,96,97,98]. Among them, uniaxial distortion—arising from epitaxial strain, applied pressure, or lattice mismatch in thin films—is particularly important because it breaks the fourfold rotational symmetry of double-Q structures and lifts the degeneracy between orthogonal ordering wave vectors. An illustrative case is EuAl₄, where two types of the SkXs have been observed in a rhombic environment [99,100,101,102].

In this work, we investigate the stability of double-Q square SkXs on a square lattice under uniaxial distortion by employing an effective momentum–space spin model with frustrated exchange interactions and bond-dependent anisotropy. Using simulated annealing, we construct low-temperature phase diagrams and track the evolution of spin textures under applied magnetic fields. We find that moderate uniaxial distortion (interaction anisotropy of about 5%) destabilizes the SkX and drives a transition into a single-Q conical spiral state. By contrast, topologically trivial double-Q states that emerge in the low- and high-field regimes are more robust against distortion than the intermediate-field SkX. Furthermore, we demonstrate that a finite bond-dependent anisotropy is indispensable for stabilizing the SkX under distortion, underscoring the cooperative role of frustration and anisotropy. These results establish the essential stability conditions for square SkXs under uniaxial distortion and highlight the crucial role of bond-dependent anisotropy.

The remainder of this paper is organized as follows. Section 2 introduces the model and numerical method. Section 3 presents the phase diagrams and discusses the contrasting robustness of SkX and double-Q states under uniaxial distortion, together with the role of bond-dependent anisotropy. Section 4 summarizes the implications for materials exploration and strain-based control of multiple-Q textures.

2. Model and Method

We begin our analysis with an effective spin model defined on a two-dimensional square lattice with fourfold rotational symmetry as well as spatial inversion symmetry. We set the lattice constant of the square lattice to unity. The Hamiltonian is given by equations.

$$\mathcal{H}=-2\sum _{\nu }\sum _{\alpha \beta }{J}_{{\mathit{Q}}_{\nu }}{\mathsf{\Gamma }}_{{\mathit{Q}}_{\nu }}^{\alpha \beta }{S}_{{\mathit{Q}}_{\nu }}^{\alpha }{S}_{-{\mathit{Q}}_{\nu }}^{\beta }-H\sum _i{S}_i^z,$$

(1)

where ${\mathit{S}}_i=(S_i^x,S_i^y,S_i^z)$ denotes a classical localized spin at site i with a fixed length $|{\mathit{S}}_i|=1$. The Fourier component of the spin variable is written as $S_{{\mathit{Q}}_{\nu }}^{\alpha }=(1/\sqrt{N}){\sum }_i{\mathit{S}}_i{e}^{-i{\mathit{Q}}_{\nu }·{\mathit{r}}_i}$ for spin direction $\alpha =x,y,z$; N is the total number of spins and ${\mathit{r}}_i$ is the position vector.

The first term in Equation (1) describes bilinear exchange interactions resolved in momentum space; $J_{{\mathit{Q}}_{\nu }}$ is the coupling constant at wave vector ${\mathit{Q}}_{\nu }$. In principle, all wave vectors in the Brillouin zone contribute to the interaction, but in practice, only a few symmetry-related wave vectors dominate the low-energy magnetic instabilities. To capture the essential physics, we restrict our consideration to the ordering wave vectors

$$\begin{array}{c}\hfill {\mathit{Q}}_1=(Q,0),{\mathit{Q}}_2=(0,Q),{\mathit{Q}}_3=(Q,Q),{\mathit{Q}}_4=(-Q,Q),\end{array}$$

(2)

with $Q=\pi /4$. These vectors are related to each other: ${\mathit{Q}}_3={\mathit{Q}}_1+{\mathit{Q}}_2$ and ${\mathit{Q}}_4=-{\mathit{Q}}_1+{\mathit{Q}}_2$, so that ${\mathit{Q}}_3$ and ${\mathit{Q}}_4$ can be regarded as higher harmonics of ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$. We further assume that at zero field the magnetic instabilities at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ are stronger than those at ${\mathit{Q}}_3$ and ${\mathit{Q}}_4$ so as to satisfy $J_{{\mathit{Q}}_1}=J_{{\mathit{Q}}_2}>J_{{\mathit{Q}}_3}=J_{{\mathit{Q}}_4}$ under the tetragonal symmetry, while other higher harmonics beyond these wave vectors can be neglected. This approximation is well justified in localized spin systems, where ferromagnetic exchange interaction competes with further-neighbor antiferromagnetic exchange interactions [103,104,105,106,107], as well as in itinerant electron systems, where long-range interactions arise from Fermi surface nesting [108,109,110], known as the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction [111,112,113]. Consequently, the low-temperature phase diagram can be captured by considering only these few dominant interaction channels. In what follows, we set $J_{{\mathit{Q}}_1}=1$ as the unit of energy, which corresponds roughly to the ordering temperature scale. For other wave vector interactions, we take $J_{{\mathit{Q}}_2}={\gamma }_{{\mathit{Q}}_2}{J}_{{\mathit{Q}}_1}$ and $J_{{\mathit{Q}}_3}=J_{{\mathit{Q}}_4}={\gamma }_{{\mathit{Q}}_1+{\mathit{Q}}_2}{J}_{{\mathit{Q}}_1}$; we fix ${\gamma }_{{\mathit{Q}}_1+{\mathit{Q}}_2}=0.6$.

To describe the effect of magnetic anisotropy in spin space, we introduce the form factor ${\mathsf{\Gamma }}_{{\mathit{Q}}_{\nu }}^{\alpha \beta }$, which microscopically originates from the interplay of relativistic spin–orbit coupling, the crystalline electric field, and the underlying band structure [114]. For the tetragonal lattice symmetry, we adopt the following parameterization: ${\mathsf{\Gamma }}_{{\mathit{Q}}_1}^{yy}={\mathsf{\Gamma }}_{{\mathit{Q}}_2}^{xx}=1+I^{\mathrm{BA}}$, ${\mathsf{\Gamma }}_{{\mathit{Q}}_1}^{xx}={\mathsf{\Gamma }}_{{\mathit{Q}}_2}^{yy}=1-I^{\mathrm{BA}}$, ${\mathsf{\Gamma }}_{{\mathit{Q}}_1}^{zz}={\mathsf{\Gamma }}_{{\mathit{Q}}_2}^{zz}=1$, and ${\mathsf{\Gamma }}_{{\mathit{Q}}_3}^{xx}={\mathsf{\Gamma }}_{{\mathit{Q}}_3}^{yy}={\mathsf{\Gamma }}_{{\mathit{Q}}_3}^{zz}={\mathsf{\Gamma }}_{{\mathit{Q}}_4}^{xx}={\mathsf{\Gamma }}_{{\mathit{Q}}_4}^{yy}={\mathsf{\Gamma }}_{{\mathit{Q}}_4}^{zz}=1$, with all other components vanishing; $I^{\mathrm{BA}}$ denotes the bond-dependent anisotropy, which can become the microscopic origin of the SkX [115,116,117,118,119] and other topologically nontrivial spin textures [120,121,122,123]. By construction, ${\mathsf{\Gamma }}_{{\mathit{Q}}_{\nu }}^{\alpha \beta }$ is symmetric under the exchange of $\alpha$ and $\beta$, consistent with the inversion symmetry of the system. This bond-dependent anisotropy $I^{\mathrm{BA}}$ corresponds to a compass-type anisotropic exchange interaction, rather than a local single-ion anisotropy, in a real-space picture. As a consequence, no antisymmetric contribution of the form ${\mathsf{\Gamma }}_{{\mathit{Q}}_{\nu }}^{\alpha \beta }=-{\mathsf{\Gamma }}_{{\mathit{Q}}_{\nu }}^{\beta \alpha }$, which corresponds to the DM interaction, appears. The second term in Equation (1) corresponds to the Zeeman contribution, capturing the interaction with an external magnetic field H directed along the z axis.

Having formulated the model in the tetragonal limit, we next examine the impact of uniaxial distortion along the [100] axis. This distortion lowers the symmetry from tetragonal to orthorhombic and introduces inequivalence between the $\left[100\right]$ and $\left[010\right]$ directions. Consequently, the exchange constants satisfy $J_{{\mathit{Q}}_1}\ne J_{{\mathit{Q}}_2}$, while the relation $J_{{\mathit{Q}}_3}=J_{{\mathit{Q}}_4}$ is preserved. By systematically varying the ratio $J_{{\mathit{Q}}_2}/J_{{\mathit{Q}}_1}={\gamma }_{{\mathit{Q}}_2}$, we can map out how the SkX phases evolve under increasing distortion [124]. For simplicity, the ordering wave vectors themselves and all other microscopic parameters are fixed against distortion so that the effect of the anisotropy in $J_{{\mathit{Q}}_{\nu }}$ can be isolated.

To determine the phase diagram, we employ Monte Carlo simulations based on simulated annealing with a system size of $N={16}^2$ spins. Starting from random configurations at high temperatures $T/J_{{\mathit{Q}}_1}>1$, the temperature is gradually reduced according to $T_{n+1}=\tilde{\alpha }{T}_n$ with $\tilde{\alpha }=0.999995$–$0.999999$ until a final value $T=0.01$ is reached, such that thermal fluctuations are nearly negligible. At the lowest temperature, physical observables are obtained by averaging over ${10}^5$–${10}^6$ Monte Carlo sweeps after equilibration. This procedure allows us to reliably capture the low-temperature spin textures and phase transitions driven by uniaxial distortion.

To characterize the magnetic phases, we evaluate several order parameters and correlation functions that distinguish between single-Q, double-Q, and topologically nontrivial spin textures like the SkX. First, the uniform magnetization along the field direction is defined as

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

(3)

where N is the total number of lattice sites. In order to resolve the ordering wave vectors in momentum space, we compute the spin structure factor

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

(4)

with $\eta =x,y,z$. We also calculate the in-plane component of the spin structure factor as $S_s^{xy}\left(\mathit{q}\right)=S_s^x\left(\mathit{q}\right)+S_s^y\left(\mathit{q}\right)$. From the spin structure factor, the amplitude of the $\mathit{q}$ component of the spin moment is extracted as

$$\begin{array}{c}\hfill m_{\mathit{q}}^{\eta }=\sqrt{{\displaystyle \frac{S_s^{\eta }\left(\mathit{q}\right)}{N}}}.\end{array}$$

(5)

These quantities enable us to identify single-Q and double-Q orders.

To probe the topological nature of the spin textures, we evaluate the scalar spin chirality,

$$\begin{array}{c}\hfill {\chi }^{\mathrm{sc}}={\displaystyle \frac{1}{N}}\sum _i\sum _{\delta =\pm 1}{\mathit{S}}_i·\left({\mathit{S}}_{i+\delta \widehat{x}}\times {\mathit{S}}_{i+\delta \widehat{y}}\right),\end{array}$$

(6)

where $\widehat{x}$ ($\widehat{y}$) denotes a unit translation along the x (y) direction. A finite value of ${\chi }^{\mathrm{sc}}$ signals the emergence of the SkX or related noncoplanar textures.

3. Results

3.1. Effect of Uniaxial Distortion

Before discussing the influence of uniaxial distortion, we first summarize the sequence of magnetic phases obtained in the absence of distortion, i.e., for ${\gamma }_{{\mathit{Q}}_2}=1$ [125]. We fix the other model parameters as ${\gamma }_{{\mathit{Q}}_1+{\mathit{Q}}_2}=0.6$ and $I^{\mathrm{BA}}=0.1$. In this limit, the system retains fourfold rotational symmetry between the two ordering wave vectors ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$, and the double-Q states are stabilized in a more symmetric manner. When the magnetic field H is varied, three phases, denoted as double-Q I, SkX, and double-Q II states, are stabilized in addition to the fully polarized state, as shown by the phase diagram in Figure 1. Such a phase sequence among double-Q spin configurations has been identified in GdRu₂Si₂ [81]. At low fields, the ground state is the double-Q I state, in which a fourfold asymmetric spin texture emerges as a result of the inequivalent superposition of the proper-screw spiral wave at ${\mathit{Q}}_1$ and the in-plane sinusoidal wave at ${\mathit{Q}}_2$. The real-space spin texture shown in Figure 2a exhibits a periodic lattice of vortices and antivortices, while the spin structure factor shows sharp peaks at two ordering wave vectors, as shown in Figure 3a. This double-Q I state does not exhibit the net scalar spin chirality ${\chi }^{\mathrm{sc}}$, although it shows the density wave in terms of the scalar spin chirality. This means that the double-Q I state corresponds to a topologically trivial state. It should be noted that in the presence of fourfold rotational symmetry with ${\gamma }_{{\mathit{Q}}_2}=1$, two double-Q I domain states connected by a ${90}^{\circ }$ rotation are energetically degenerated. Such a degeneracy is lifted by considering the uniaxial distortion effect ${\gamma }_{{\mathit{Q}}_2}<1$, as detailed below.

As the magnetic field H is increased, the system undergoes transitions to the SkX, in which a superposition of the proper-screw spiral waves occurs at two ordering wave vectors ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$. In contrast to the double-Q I phase, showing partial twofold symmetry and displaying stripe-like vortex–antivortex arrangements in real space, the SkX is characterized by a square array of skyrmions, where the real-space spin texture exhibits a periodic lattice of vortices with quantized topological charge, as shown in Figure 2b. The spin structure factor in Figure 3b shows sharp peaks at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ owing to the fourfold rotational symmetry. A hallmark of the SkX is its nonzero scalar spin chirality ${\chi }^{\mathrm{sc}}$, directly signaling the topological nature of this phase. In the present model, the sign of the scalar spin chirality remains undetermined due to the energy degeneracy between SkX states with positive and negative scalar spin chirality; such a degeneracy is lifted by additionally considering the bond-dependent anisotropy at high-harmonic wave vectors ${\mathit{Q}}_3$ and ${\mathit{Q}}_4$ [126].

At higher fields, the system eventually reduces to the double-Q II state. This state is characterized by a superposition of two in-plane sinusoidal waves at the ordering wave vectors, ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$. In real space, the double-Q II state exhibits a fourfold symmetric pattern consisting of vortex–antivortex alignment, which is similar to the double-Q I state and SkX, as shown in Figure 2c. Meanwhile, the z-spin component is almost uniform, which indicates the cancellation of the scalar spin chirality between vortices and antivortices. In other words, the double-Q II state is also regarded as a topologically trivial state. The spin structure factor of the double-Q II state shows two sharp Bragg peaks of equal intensity at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ in the in-plane spin component, as shown in Figure 3c.

We now turn to the effect of uniaxial distortion along the [100] axis by introducing ${\gamma }_{{\mathit{Q}}_2}<1$. The introduction of distortion explicitly breaks the fourfold rotational symmetry and favors ordering wave vectors aligned with the distortion axis, thereby changing the stability of three magnetic phases realized at ${\gamma }_{{\mathit{Q}}_2}=1$. In the present model, ${\gamma }_{{\mathit{Q}}_2}<1$ corresponds to a reduction of the exchange interaction along the [010] direction, which can be regarded as the effect of a uniaxial tensile strain along that axis. Although the opposite case, ${\gamma }_{{\mathit{Q}}_2}>1$, corresponding to compressive strain, is not explicitly shown, the phase behavior is expected to be equivalent upon interchanging the $Q_1$ and $Q_2$ axes. Thus, the results for ${\gamma }_{{\mathit{Q}}_2}<1$ capture the general influence of uniaxial distortion on the stability of multiple-Q states, consistent with both tensile and compressive strain conditions observed in experiments.

Figure 2d–f display real-space snapshots at a moderate distortion ${\gamma }_{{\mathit{Q}}_2}=0.95$ for representative fields corresponding to the three phases that appear in the tetragonal limit. At low fields [Figure 2d], the system remains in the double-Q I state; for ${\gamma }_{{\mathit{Q}}_2}<1$, the tendency toward twofold anisotropy becomes more pronounced. This anisotropy stems from the imbalance between the two ordering wave vectors, with the component at ${\mathit{Q}}_1$ favored over ${\mathit{Q}}_2$. In reciprocal space, this is reflected as unequal Bragg intensities at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ in the in-plane spin structure factor, where the ratio $S_s^{xy}\left({\mathit{Q}}_2\right)/S_s^{xy}\left({\mathit{Q}}_1\right)$ is found to be suppressed at ${\gamma }_{{\mathit{Q}}_2}=0.95$ relative to ${\gamma }_{{\mathit{Q}}_2}=1$, as shown in Figure 3a,d.

At intermediate fields [Figure 2e], the SkX persists even under ${\gamma }_{{\mathit{Q}}_2}=0.95$, forming a square array of skyrmion cores, but each core becomes slightly elongated along the [010] direction imposed by the distortion. Correspondingly, the spin structure factor preserves two principal Bragg pairs at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$, yet their intensities become imbalanced both for $S_s^{xy}\left(\mathit{Q}\right)$ and $S_s^z\left(\mathit{Q}\right)$, as shown in Figure 3e, indicating a distortion-induced redistribution of weight among ordering wave vectors. The phase remains topological, as signaled by a finite scalar spin chirality. It is noteworthy that the stability of the SkX is more fragile than that of the double-Q I state, because the former spin texture preserves fourfold rotational symmetry, whereas the latter exhibits twofold symmetry at ${\gamma }_{{\mathit{Q}}_2}=1$. Accordingly, as ${\gamma }_{{\mathit{Q}}_2}$ decreases from 1, the phase boundary between these two phases shifts upward, indicating that the magnetostriction effect stabilizes the twofold-symmetric double-Q I state over the fourfold-symmetric SkX.

At higher fields, the double-Q II state is stabilized. In real space in Figure 2f, it forms a nearly checkerboard-like arrangement of vortex–antivortex textures, but the fourfold rotational symmetry is broken owing to the effect of distortion. In momentum space, the in-plane spin structure factor shows two sharp but unequal peaks at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$, whereas the out-of-plane component remains weak and featureless except for $\mathit{q}=\mathbf{0}$, as shown in Figure 3f.

Upon further increasing the distortion, the energetic splitting between ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ eventually suppresses multiple-Q superpositions altogether, and single-Q states take over the field windows that hosted double-Q and SkX phases in the tetragonal limit. Figure 4a–c illustrate typical real-space textures at ${\gamma }_{{\mathit{Q}}_2}=0.8$: a single-Q proper-screw spiral state at low fields, a single-Q conical spiral state at intermediate fields, and a single-Q fan state as the system approaches magnetic field saturation. Although the real-space spin configurations of the double-Q I state in Figure 2d and the single-Q proper-screw spiral state in Figure 4a appear similar, their difference is evident in momentum space. The double-Q I state shows two Bragg peaks at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ in the in-plane spin structure factor, corresponding to the coexistence of two spiral modulations, whereas the single-Q proper-screw spiral state exhibits only a single pair of peaks at ${\mathit{Q}}_1$, as shown in Figure 5a. This contrast in the spin structure factors distinguishes the two phases unambiguously. For the other single-Q phases, the single-Q conical spiral state shows a dominant in-plane peak at ${\mathit{Q}}_1$ with negligibly small $S_s^z\left({\mathit{Q}}_1\right)$, as shown in Figure 5b, and the single-Q fan state retains a single Bragg pair in the $xy$-spin component, while the z-component becomes nearly uniform, as shown in Figure 5c. These trends demonstrate that uniaxial distortion selects a unique helix axis and eliminates multiple-Q interference.

The ${\gamma }_{{\mathit{Q}}_2}$ dependence of squared magnetic moments, which correspond to order parameters, at fixed fields is quantified in Figure 6a–d. At low field $H=0.1$ in Figure 6a, the transition from the double-Q I state to the single-Q proper-screw spiral state occurs when ${\gamma }_{{\mathit{Q}}_2}$ is reduced below $\sim 0.94$: the in-plane moment at ${\mathit{Q}}_2$ collapses continuously while that at ${\mathit{Q}}_1$ remains robust, consistent with the real-space anisotropy seen in Figure 2d and Figure 4a. Thus, the transition between the double-Q I state and the single-Q proper-screw spiral state is of second order, although this transition does not show a distinct rotational symmetry breaking. At intermediate field $H=0.7$ in Figure 6b, a similar threshold separates the SkX from the single-Q conical spiral state; across this boundary, the out-of-plane modulation associated with the SkX (finite $m_{{\mathit{Q}}_1}^z$ and $m_{{\mathit{Q}}_2}^z$) disappears. The behavior of the squared scalar spin chirality corroborates this picture: ${\left({\chi }^{\mathrm{sc}}\right)}^2$ is essentially zero in the conical spiral state and becomes finite only inside the SkX regime for ${\gamma }_{{\mathit{Q}}_2}\gtrsim 0.95$, as shown in Figure 7. At higher fields $H=1.2$ in Figure 6c, the double-Q II state gives way to a single-Q conical spiral state upon increasing distortion, again via the progressive suppression of the ${\mathit{Q}}_2$ component. Finally, near saturation $H=1.8$ in Figure 6d, the system transitions from the double-Q II state to the single-Q fan state, in which the in-plane spin modulation persists only at ${\mathit{Q}}_1$. This transition between the double-Q II state and the single-Q fan state shows properties characteristic of a second-order phase transition.

Two robust messages emerge from these systematic datasets. First, among the multiple-Q textures, the SkX is the most sensitive to uniaxial distortion: a $\sim 5\%$ reduction of the ${\mathit{Q}}_2$ channel (i.e., ${\gamma }_{{\mathit{Q}}_2}\simeq 0.95$) already induces strong anisotropy in its spin structure factor, and a slightly stronger distortion drives a transition to a single-Q conical spiral state. Second, the topologically trivial double-Q states (I and II) are comparatively more resilient; they survive down to smaller ${\gamma }_{{\mathit{Q}}_2}$ before converting to single-Q spiral and/or fan states. Combined with the single-Q real- and momentum–space signatures in Figure 4 and Figure 5, these results establish an intuitive hierarchy under uniaxial distortion: SkX → double-Q→ single-Q, in the sense of decreasing robustness against anisotropic splitting of the two fundamental ordering wave vectors.

3.2. Effect of Bond-Dependent Anisotropy

We next examine the role of the bond-dependent anisotropy $I^{\mathrm{BA}}$, which couples spin components to lattice bonds in an anisotropic manner. In contrast to the isotropic Heisenberg exchange, $I^{\mathrm{BA}}$ originates from spin–orbit coupling and crystalline symmetry, thereby introducing an additional degree of control over the relative stability of competing magnetic textures. A central outcome of our analysis is that $I^{\mathrm{BA}}$ plays a decisive role in stabilizing the SkX even under uniaxial distortion, where otherwise the balance among competing magnetic orders is easily lost.

Figure 8 summarizes the phase diagram in the $I^{\mathrm{BA}}$–H plane at fixed ${\gamma }_{{\mathit{Q}}_2}=0.97$ and ${\gamma }_{{\mathit{Q}}_1+{\mathit{Q}}_2}=0.6$. Without $I^{\mathrm{BA}}$, the SkX phase is absent, and uniaxial distortion favors the single-Q conical spiral state with the ordering wave vector ${\mathit{Q}}_1$. Once $I^{\mathrm{BA}}$ is introduced, however, a finite SkX phase emerges in the intermediate-field regime, and it remains stable for $0.08\lesssim I^{\mathrm{BA}}\lesssim 0.25$. This clearly demonstrates that the bond-dependent anisotropy is indispensable for the robustness of SkX textures in the distorted tetragonal environment with bilinear exchange interactions.

The magnetization curves in Figure 9a–c provide further insight. For a small anisotropy $I^{\mathrm{BA}}=0.06$ in Figure 9a, the system exhibits a sequence of phase transitions: double-Q I → single-Q conical spiral → double-Q II → single-Q fan → fully polarized state. Here, the SkX is entirely absent, and the transitions are generally sharp except at the boundaries between the double-Q II and single-Q fan states and between the single-Q fan and fully polarized states. When $I^{\mathrm{BA}}$ is increased to $0.16$ in Figure 9b, the SkX phase is stabilized between the low-field double-Q I and high-field double-Q II states. In this case, the plateau-like behavior of the finite scalar spin chirality ${\chi }^{\mathrm{sc}}$ is found in the SkX region [Figure 10], marking the topological nature of the state. The abrupt onset and disappearance of ${\chi }^{\mathrm{sc}}$ at the phase boundaries signal the first-order nature of the SkX transitions, consistent with the sharp kinks in $M^z$. For a larger anisotropy $I^{\mathrm{BA}}=0.32$ in Figure 9c, the SkX region disappears, and the double-Q I and double-Q II states are smoothly connected against the magnetic field. This result confirms that the relative magnitude of $I^{\mathrm{BA}}$ plays an important role in realizing the SkX even in the presence of uniaxial distortion.

Finally, we note that the present model implicitly assumes a rigid and spatially uniform strain field, corresponding to the ideal case where the lattice deformation is homogeneously transmitted to the spin system. In experimental situations, strain is typically introduced through external stress, which may lead to partial relaxation depending on the sample geometry and boundary conditions. Nevertheless, the essential tuning parameter in the present model is the ratio between the exchange interactions associated with the two ordering wave vectors, $J_{{\mathit{Q}}_1}$ and $J_{{\mathit{Q}}_2}$. This ratio can be effectively modified by either rigid strain or externally applied stress. Thus, the qualitative behavior discussed here, including the strain-induced stabilization and suppression of multiple-Q states, is expected to be robust against the microscopic details of how the distortion is imposed.

4. Conclusions

We have investigated the stability of double-Q square SkXs in a centrosymmetric tetragonal host subject to uniaxial distortion and bond-dependent anisotropy. Using a momentum–space effective spin model with frustrated exchange and a symmetry-allowed anisotropic form factor, we mapped the low-temperature field-distortion phase behavior by simulated annealing and analyzed real- and momentum–space diagnostics. Our central findings are threefold.

First, in the tetragonal limit, the field evolution proceeds through a robust sequence of multiple-Q phases—double-Q I state at low fields, a square SkX at intermediate fields, and double-Q II state at higher fields—before reaching the fully polarized state. Introducing uniaxial distortion along the [100] direction splits the competing instabilities at ${\mathit{Q}}_1=(Q,0)$ and ${\mathit{Q}}_2=(0,Q)$ and promotes twofold anisotropy in both the real-space textures and the spin structure factors. Quantitatively, a modest ∼5% reduction in the ${\mathit{Q}}_2$ channel (${\gamma }_{{\mathit{Q}}_2}\approx 0.95$) already destabilizes the SkX and drives a transition into a single-Q conical spiral state, whereas the topologically trivial double-Q states remain comparatively more resilient against the same distortion. This establishes a clear robustness hierarchy under distortion: SkX → double-Q→ single-Q.

Second, we clarified the role of bond-dependent anisotropy $I^{\mathrm{BA}}$ as a key factor of the SkX under distortion. At fixed orthorhombicity as ${\gamma }_{{\mathit{Q}}_2}=0.97$, the SkX emerges only when $I^{\mathrm{BA}}$ exceeds a finite threshold and persists within an intermediate window of $I^{\mathrm{BA}}$, as further reflected by characteristic kinks in the magnization and a plateau-like onset/offset of the scalar spin chirality inside the SkX regime. Outside this window, the field evolution is dominated by topologically trivial double-Q or single-Q phases. These results demonstrate the importance of $I^{\mathrm{BA}}$ even under uniaxial distortion.

Third, our momentum-resolved analysis links qualitative changes in the phase diagram to distinctive experimental fingerprints. Distortion-induced anisotropy should manifest as unequal Bragg intensities at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ in the in-plane and out-of-plane spin structure factors, while topological phase boundaries can be tracked by sharp features in the magnetization and the abrupt appearance/disappearance of scalar spin chirality. These signatures are directly accessible via elastic magnetic scattering, such as small-angle neutron scattering and resonant x-ray measurements, bulk magnetometry, and probes of emergent electromagnetism associated with noncoplanar spin textures.

The present momentum–space spin Hamiltonian can be mapped onto a real-space model with competing exchange interactions and bond-dependent anisotropic exchanges, similar to those arising from RKKY-type or superexchange mechanisms [127]. The parameters $J_{{\mathit{Q}}_{\nu }}$ and $I_{\mathrm{BA}}$ can, in principle, be estimated by fitting theoretical spin structure factors and magnetization profiles to experimental data from neutron or resonant x-ray scattering and bulk magnetometry. Once such parameters are determined for a specific material, the model is expected to reproduce other regions of the phase diagram, reflecting its applicability to real systems that host competing multiple-Q magnetic instabilities.

From a broader perspective, our findings carry important implications for materials exploration and strain-based control of multiple-Q textures. The demonstrated sensitivity of square SkXs to small orthorhombic distortions suggests that epitaxial strain, uniaxial pressure, or chemical substitution could be used to selectively stabilize or suppress SkX phases in real tetragonal materials. Conversely, bond-dependent anisotropies, arising from spin–orbit–lattice entanglement, act as a tuning knob to restore SkX stability even under symmetry lowering. This interplay between strain and anisotropy suggests a potential route for tailoring spin textures in tetragonal magnets, particularly in thin films, pressure-tuned bulk crystals, and heterostructures where structural distortions can be controlled.