Effects of Uniaxial Distortion on the Stability of Square Skyrmion Crystals in Noncentrosymmetric Magnets

1. Introduction

Magnetic skyrmions are topologically protected spin textures characterized by a quantized topological number, which endows them with robustness against perturbations and gives rise to a variety of emergent electromagnetic phenomena. Since their theoretical conception by Skyrme [1,2], skyrmion structures have been studied in diverse physical systems, such as liquid crystals [3,4,5,6,7], quantum Hall magnets [8,9,10,11,12,13,14], Bose-Einstein condensates [15,16,17,18,19,20], and ferroelectrics [21,22,23,24,25]. In magnetic materials, skyrmions appear as nanoscale swirling arrangements of localized spins [26,27], and their periodic arrangement, the skyrmion crystal (SkX), was first discovered in noncentrosymmetric chiral magnets such as MnSi [28,29,30], Fe_1−xCo_xSi [31,32], FeGe [33,34], and Cu₂OSeO₃ [35,36,37]. These discoveries sparked enormous research interest, motivated by both the fundamental topological nature of skyrmions and their promise for spintronic applications such as racetrack memory and neuromorphic computing [38,39,40,41,42,43,44].

In noncentrosymmetric chiral magnets, the Dzyaloshinskii-Moriya (DM) interaction [45,46] plays a decisive role in stabilizing a single-Q spiral state and SkXs [47,48,49,50]. These SkXs provide an ideal platform to explore a wide range of emergent phenomena, including the topological Hall effect [30,51,52,53,54,55,56], collective excitations [57,58,59,60,61,62], and field- or temperature-induced phase transitions [63,64,65,66]. Meanwhile, extensive theoretical efforts have demonstrated that similar topological spin textures can also emerge in centrosymmetric magnets, where SkXs arise from competing exchange interactions [67,68,69,70,71], magnetic anisotropies [72,73,74,75,76,77,78,79], dipolar interactions [80,81], and sublattice-dependent staggered DM interactions [82,83]. These findings have expanded the scope of materials hosting SkXs beyond conventional chiral systems.

One of the intriguing research directions has been paid to square-lattice magnets, since such a system provides a simpler situation compared to triangular-lattice magnets owing to the absence of geometrically frustrated structures. There, multiple symmetry-related ordering wave vectors naturally allow for double-Q spin textures, such as square SkXs. Such SkXs characterized by the double-Q spin configuration have been observed in both noncentrosymmetric and centrosymmetric systems, including Co-Zn-Mn alloys [84,85,86,87,88], Cu₂OSeO₃ [89,90], GdRu₂Si₂ [91,92,93,94,95], and EuNiGe₃ [96]. Theoretically, their stability has been attributed to the interplay between multiple interactions and magnetic anisotropies, which collectively govern the relative phases of helical modulations. The typical mechanisms include the DM interaction [63], momentum-space frustration [97,98], bond-dependent interaction [99,100], dipolar interaction [80], and multi-orbital effects [101]. Understanding how these microscopic ingredients stabilize or destabilize the square SkX is thus a central question in the study of topological spin textures [102].

Among various external perturbations, uniaxial distortion, which originates from epitaxial strain and applied pressure, has emerged as a key factor that can significantly modify the stability of SkXs [103,104,105,106,107]. By lowering the lattice symmetry and breaking the equivalence of orthogonal ordering wave vectors, distortion introduces a natural anisotropy in the competing double-Q spin configurations [108]. Experimentally, such effects have been reported in the centrosymmetric tetragonal compound EuAl₄, where multiple SkX phases occur under rhombic distortion [109,110,111,112,113]. However, a comprehensive theoretical understanding of how uniaxial distortion affects the stability of square SkXs, especially in noncentrosymmetric systems with the DM interaction, remains limited.

In this study, we investigate the stability and evolution of square SkXs under uniaxial distortion in noncentrosymmetric tetragonal magnets. We employ an effective spin model on a two-dimensional square lattice that includes momentum-resolved frustrated exchange interactions and DM interaction. Using simulated annealing, we construct low-temperature magnetic phase diagrams and clarify how uniaxial anisotropy drives topological phase transitions among the double-Q SkX, single-Q tilted conical spiral state, and the single-Q vertical spiral state. We also demonstrate that stronger DM interaction enhances the robustness of the SkX against distortion by widening its stability region in the phase diagram. The results provide microscopic insight into the cooperative role of uniaxial strain and DM interactions in realizing topological spin textures.

The remainder of this paper is organized as follows. Section 2 introduces the model Hamiltonian and numerical methods. Section 3 presents the calculated magnetic phase diagrams, magnetic-field and uniaxial distortion dependence of order parameters, and real-space spin structures in several representative model parameters. Section 4 summarizes the main findings and discusses their implications for strain-engineered control of SkXs in noncentrosymmetric magnets.

2. Model and Method

We study a classical spin system on a two-dimensional square lattice that possesses fourfold rotational symmetry but lacks spatial inversion symmetry, corresponding to the polar point group $C_{4\mathrm{v}}$. The lattice constant is set to unity throughout this work. The magnetic interactions are formulated in momentum space so that the dominant instabilities at specific wave vectors can be treated explicitly. The effective spin Hamiltonian is written as

$$\begin{array}{cc}\hfill \mathcal{H}=& -2\sum _{\nu }\left[J_{{\mathit{Q}}_{\nu }}{\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{\nu }}+i{\mathit{D}}_{{\mathit{Q}}_{\nu }}·({\mathit{S}}_{{\mathit{Q}}_{\nu }}\times {\mathit{S}}_{-{\mathit{Q}}_{\nu }})\right]-\sum _i\mathit{H}·{\mathit{S}}_i,\hfill \end{array}$$

(1)

where ${\mathit{S}}_{{\mathit{Q}}_{\nu }}=(S_{{\mathit{Q}}_{\nu }}^x,S_{{\mathit{Q}}_{\nu }}^y,S_{{\mathit{Q}}_{\nu }}^z)$ is the Fourier component of the classical spin ${\mathit{S}}_i=(S_i^x,S_i^y,S_i^z)$ with fixed length $|{\mathit{S}}_i|=1$. The summation runs over two symmetry-related ordering wave vectors, ${\mathit{Q}}_1=(\pi /4,0)$ and ${\mathit{Q}}_2=(0,\pi /4)$, by the fourfold rotational operation. The prefactor of two in Eq. (1) accounts for the equivalent contributions from $\pm {\mathit{Q}}_{\nu }$.

The first term describes the momentum-resolved isotropic exchange interaction, characterized by the coupling constants $J_{{\mathit{Q}}_{\nu }}$. In the tetragonal structure, the fourfold rotational symmetry ensures $J_{{\mathit{Q}}_1}=J_{{\mathit{Q}}_2}$. To capture anisotropic effects, we define the ratio ${\gamma }_{{\mathit{Q}}_2}=J_{{\mathit{Q}}_2}/J_{{\mathit{Q}}_1}$ and set $J_{{\mathit{Q}}_1}=1$ as the energy unit. The form of this interaction originates from frustrated short-range exchange interactions [114,115,116,117,118] or long-range Ruderman-Kittel-Kasuya-Yosida-type interactions [119,120,121] arising from Fermi-surface nesting [122,123,124], both of which can produce multiple wave-vector instabilities in momentum space. We restrict the model to the ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ channels, assuming that these wave vectors dominate the magnetic instability, while higher harmonics are neglected for simplicity. Such a simplification allows an efficient exploration of the emergence and stability of multiple-Q states, offering microscopic insight into experimentally observed behaviors in materials such as Y₃Co₈Sn₄ [125], EuNiGe₃ [96], and EuPtSi [126].

The second term in Eq. (1) represents the antisymmetric DM interaction allowed by the lack of spatial inversion symmetry. For the $C_{4\mathrm{v}}$ symmetry, the DM vectors satisfy $D_{{\mathit{Q}}_1}^y=-D_{{\mathit{Q}}_2}^x\equiv D$, and all other components vanish. This term introduces a fixed helicity to the spin modulation, favoring a spiral structure determined by the sign of D. Other anisotropic terms that are symmetry-allowed but secondary in strength for small spin–orbit coupling, such as bond-dependent symmetric exchange interaction [127,128,129,130,131,132,133], are omitted here for clarity [134]. The final term denotes the Zeeman coupling to an external magnetic field $\mathit{H}=(0,0,H)$ applied along the out-of-plane direction.

To investigate the effect of symmetry lowering, we introduce a uniaxial distortion along the $\left[100\right]$ direction. This perturbation reduces the crystal symmetry from tetragonal to orthorhombic and removes the equivalence between the two ordering wave vectors. In our model, this distortion is incorporated solely through the anisotropy of the coupling constants, so that $J_{{\mathit{Q}}_1}\ne J_{{\mathit{Q}}_2}$ (or equivalently ${\gamma }_{{\mathit{Q}}_2}\ne 1$). By varying ${\gamma }_{{\mathit{Q}}_2}$ systematically, we can trace how the distortion modifies the balance between single-Q, double-Q, and SkX phases, while keeping all other parameters fixed.

The phase diagram is determined by large-scale Monte Carlo simulations based on simulated annealing. We use a lattice of $N={16}^2$ spins with periodic boundary conditions and employ standard Metropolis updates in real space. The simulation begins from random spin configurations at high temperature ($T/J_{{\mathit{Q}}_1}>1$) and cools according to $T_{n+1}=\tilde{\alpha }{T}_n$, where $\tilde{\alpha }=0.999995$–$0.999999$, until a final temperature $T=0.01$ is reached, where thermal fluctuations can be negligible. Physical quantities are averaged over ${10}^5$–${10}^6$ Monte Carlo sweeps after equilibration, ensuring well-converged low-temperature states and reliable identification of phase boundaries.

The magnetic order is characterized by several quantities. The uniform magnetization along the field direction is defined as

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

(2)

The spin correlations are analyzed via the spin structure factor,

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

(3)

and the in-plane component is given by $S_s^{xy}\left(\mathit{q}\right)=S_s^x\left(\mathit{q}\right)+S_s^y\left(\mathit{q}\right)$. Here, ${\mathit{r}}_i$ is the position vector at site i and $\mathit{q}$ is the wave vector in the Brillouin zone. The amplitude of the magnetic Fourier mode is then

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

(4)

which distinguishes single-Q state from multiple-Q states. To capture the topological aspect of the spin textures, we evaluate the scalar spin chirality,

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

(5)

where $\widehat{x}$ and $\widehat{y}$ denote unit translations along the x and y directions, respectively. A nonzero ${\chi }^{\mathrm{sc}}$ identifies noncoplanar spin configurations such as SkXs, while its sign distinguishes the chirality of the underlying spin structure.

3. Results

We begin by mapping the low-temperature magnetic phase diagram in the $({\gamma }_{{\mathit{Q}}_2},H)$ plane, as shown in Figure 1. Three primary phases are stabilized: a single-Q vertical spiral state denoted as 1Q VS at low fields, a double-Q SkX at intermediate fields, and a single-Q tilted conical spiral state denoted as 1Q TCS at higher fields approaching saturation. Another double-Q state also appears in a narrow region of the phase diagram at ${\gamma }_{{\mathit{Q}}_2}=1$. As the uniaxial distortion increases (corresponding to smaller ${\gamma }_{{\mathit{Q}}_2}$), the SkX window narrows while single-Q phases expand, reflecting enhanced exchange anisotropy between the two orthogonal modulation channels ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$.

Representative real-space spin textures at ${\gamma }_{{\mathit{Q}}_2}=0.95$ are presented in Figure 2. In the single-Q vertical spiral state shown in Figure 2(a), the spins rotate within a plane containing the field axis to gain energy by the DM interaction, forming stripes with alternating $S_i^z$ components and negligible in-plane vorticity. At intermediate fields, the system reorganizes into a square array of vortices and antivortices, constituting the SkX phase characterized by a finite scalar spin chirality ${\chi }^{\mathrm{sc}}$, as shown in Figure 2(b). At higher fields, the spin texture transforms into the single-Q tilted conical spiral texture, where a single helical modulation occurs in the $xy$ plane, as shown in Figure 2(c). It is noted that the spiral plane of this state is slightly tilted from the $xy$ plane to the $zx$ plane owing to the presence of the DM interaction.

The corresponding spin structure factors in Figure 3 provide a clear reciprocal-space fingerprint of each magnetic phase. For both the single-Q vertical spiral and the tilted conical spiral states, the spectrum is dominated by a single pair of Bragg peaks at $\pm {\mathit{Q}}_1$, reflecting a unidirectional helical modulation. In the single-Q vertical spiral phase, comparable intensities are found in the in-plane ($S_s^{xy}$) and out-of-plane ($S_s^z$) components, as shown in Figure 3(a), consistent with spins rotating within a plane that includes the magnetic-field direction. By contrast, in the single-Q tilted conical spiral state, the spectral weight in $S_s^z\left({\mathit{Q}}_1\right)$ is strongly suppressed while $S_s^{xy}\left({\mathit{Q}}_1\right)$ remains pronounced, as shown in Figure 3(c), indicating that the cone axis is slightly canted from the z direction.

In striking contrast, the SkX exhibits four principal Bragg peaks located at $\pm {\mathit{Q}}_1$ and $\pm {\mathit{Q}}_2$, which form a characteristic square pattern in reciprocal space that mirrors the real-space spin texture, as shown in Figure 3(b). Both $S_s^{xy}\left(\mathit{q}\right)$ and $S_s^z\left(\mathit{q}\right)$ display finite and comparable intensities at these ordering wave vectors, signifying that the spin texture contains substantial modulations in all three spin components. This spectral distribution evidences the superposition of two helical modes with orthogonal propagation vectors, giving rise to a noncoplanar double-Q spin configuration responsible for the finite scalar spin chirality and topological character of the SkX phase.

The evolution of these magnetic phases as a function of the magnetic field is summarized in Figure 4, Figure 5, and Figure 6 for different values of ${\gamma }_{{\mathit{Q}}_2}$. Each panel displays the field dependence of the uniform magnetization $M^z$ in (a), the squared Fourier amplitudes of the spin moments ${\left(m_{{\mathit{Q}}_{\nu }}^{\eta }\right)}^2$ for $\eta =x,y,z$ in (b), and the squared scalar spin chirality ${\left({\chi }^{\mathrm{sc}}\right)}^2$ in (c). These quantities provide complementary insights into how the sequence of magnetic phases evolves with uniaxial distortion.

In the tetragonal limit, i.e.,${\gamma }_{{\mathit{Q}}_2}=1$, shown in Figure 4, five distinct regions appear as the magnetic field H increases. Figure 4(a) indicates that $M^z$ grows in a stepwise manner with increasing field, reflecting successive phase transitions among the single-Q vertical spiral state, the SkX, the single-Q tilted conical spiral state, the double-Q state, and finally the fully polarized state. In the low-field region, the magnetization remains small, consistent with the nearly coplanar nature of the single-Q vertical spiral state. Upon entering the SkX phase, $M^z$ exhibits a sudden enhancement, accompanied by partial alignment of the spins with the field due to the formation of vortex cores in the SkX phase. A further gradual increase in $M^z$ occurs as the system transitions to the single-Q tilted conical spiral and double-Q states, where the conical and sinusoidal modulations are gradually suppressed by the Zeeman energy. Figure 4(b) shows the corresponding evolution of the order parameters: in the single-Q vertical spiral phase, the Fourier weights in the in-plane $m_{{\mathit{Q}}_1}^{x,y}$ and out-of-plane $m_{{\mathit{Q}}_1}^z$ components are almost the same. The magnitude of $m_{{\mathit{Q}}_1}^z$ becomes smaller while increasing H, since the spins tend to align along the perpendicular direction of the magnetic field to gain the energy by $J_{{\mathit{Q}}_1}$. Entering the SkX region, finite components at both ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ emerge in all spin directions, confirming the superposition of two orthogonal modulations that produce a noncoplanar spin texture. At higher fields, the in-plane components $m_{{\mathit{Q}}_{\nu }}^{x,y}$ dominate. Figure 4(c) highlights the field dependence of the squared scalar spin chirality ${\left({\chi }^{\mathrm{sc}}\right)}^2$: it is zero in the single-Q vertical spiral phase, exhibits a pronounced plateau-like behavior in the SkX regime, signaling robust topological order, and vanishes again in the single-Q tilted conical spiral, the double-Q, and fully polarized states. These combined trends establish that the SkX appears only in a limited field window surrounded by topologically trivial spiral states.

Figure 5 presents the corresponding results for moderate uniaxial distortion at ${\gamma }_{{\mathit{Q}}_2}=0.95$. Similarly to the case of ${\gamma }_{{\mathit{Q}}_2}=1$, the magnetic field evolution in Figure 5(a) shows stepwise changes in $M^z$, but the range of the SkX phase becomes narrower than in the tetragonal limit. An additional single-Q tilted conical spiral phase intervenes between the single-Q vertical spiral and SkX regions, suggesting that the anisotropy between ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ channels partially suppresses the double-Q interference essential for stabilizing the SkX. In Figure 5(b), the reduction of the ${\mathit{Q}}_2$ component is evident: $m_{{\mathit{Q}}_2}^{x,y}$ and $m_{{\mathit{Q}}_2}^z$ acquire smaller amplitudes than their ${\mathit{Q}}_1$ counterparts, producing an asymmetric intensity distribution in momentum space that mirrors the twofold distortion in real space. Nevertheless, within a narrow field window, the SkX retains finite contributions from both ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$, maintaining its topological character. Figure 5(c) further confirms this behavior: ${\left({\chi }^{\mathrm{sc}}\right)}^2$ develops a distinct but narrower plateau compared with Figure 4(c), indicating that the SkX remains stable only within a limited field range before being replaced by a topologically trivial single-Q tilted conical spiral state at higher fields. This result demonstrates that a modest ($\sim 5\%$) anisotropy already leads to substantial deformation of the SkX and reduction of its stability window.

For stronger uniaxial distortion at ${\gamma }_{{\mathit{Q}}_2}=0.9$, the results in Figure 6 show that the SkX phase is completely suppressed. Figure 6(a) reveals a smoother and more monotonic increase in magnetization without distinct jumps, consistent with continuous evolution from the single-Q vertical spiral state to the single-Q tilted conical spiral state, and then to the fully polarized state. Figure 6(b) exhibits a single dominant Fourier component at ${\mathit{Q}}_1$ throughout the field range; the secondary ${\mathit{Q}}_2$ mode is completely absent, signifying the collapse of double-Q superposition. Both $m_{{\mathit{Q}}_1}^z$ and $m_{{\mathit{Q}}_1}^{x,y}$ gradually decrease. The small decrease of $m_{{\mathit{Q}}_1}^{x,y}$ compared to $m_{{\mathit{Q}}_1}^z$ reflects the tendency of the spin flop in terms of the field direction. ${\left({\chi }^{\mathrm{sc}}\right)}^2$ remains vanishingly small over the entire field range (not shown), which means the topologically trivial nature of these states. These results collectively establish that increasing distortion systematically suppresses the noncoplanar SkX and double-Q phases, favoring simpler single-Q spiral textures whose ordering wave vector aligns with the distortion axis.

Figure 7 presents the ${\gamma }_{{\mathit{Q}}_2}$ dependence of the squared magnetic Fourier components ${\left(m_{{\mathit{Q}}_{\nu }}^{\eta }\right)}^2$ at three representative magnetic fields, $H=1$, $1.3$, and $1.6$, which correspond respectively to the single-Q vertical spiral state, the SkX, and the single-Q tilted conical spiral state at ${\gamma }_{{\mathit{Q}}_2}=1$. These results reveal in detail how uniaxial distortion, represented by the deviation of ${\gamma }_{{\mathit{Q}}_2}$ from unity, continuously modifies the relative intensities of the two ordering wave vectors ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ and thereby controls the stability of each spiral structure.

In the case of $H=1$ in Figure 7(a), the system resides in the single-Q vertical spiral phase irrespective of the magnitude of distortion. The dominant Fourier components ${\left(m_{{\mathit{Q}}_1}^x\right)}^2$ and ${\left(m_{{\mathit{Q}}_1}^z\right)}^2$ exhibit almost no dependence on ${\gamma }_{{\mathit{Q}}_2}$. This is because the change of ${\gamma }_{{\mathit{Q}}_2}$ does not affect the internal energy of the single-Q vertical spiral phase. Similarly, the single-Q tilted conical spiral state remains stable against distortion, as shown in Figure 7(c).

Meanwhile, in the region where the SkX phase is stabilized in the absence of distortion at $H=1.3$ in Figure 7(b), there is a clear topological phase transition driven by the variation of ${\gamma }_{{\mathit{Q}}_2}$. Upon decreasing ${\gamma }_{{\mathit{Q}}_2}$, the Fourier amplitudes associated with ${\mathit{Q}}_2$ systematically decrease, while those at ${\mathit{Q}}_1$ are gradually enhanced. The imbalance between the two modulation wave vectors grows continuously, signaling that the uniaxial distortion suppresses the interference between the orthogonal spiral components. At a critical value of ${\gamma }_{{\mathit{Q}}_2}\simeq 0.93$, the ${\mathit{Q}}_2$ contribution vanishes, and the system transitions into the single-Q tilted conical spiral state, as indicated by the vertical dashed line in Figure 7(b). This behavior demonstrates that even a modest uniaxial anisotropy is sufficient to destabilize the SkX by lifting the degeneracy between ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$. The corresponding evolution of the topological property is captured in Figure 8, which displays the ${\gamma }_{{\mathit{Q}}_2}$ dependence of the squared scalar spin chirality ${\left({\chi }^{\mathrm{sc}}\right)}^2$ at the same field of $H=1.3$. As ${\gamma }_{{\mathit{Q}}_2}$ decreases, ${\left({\chi }^{\mathrm{sc}}\right)}^2$ sharply drops to zero according to the transition to the single-Q tilted conical spiral state.

Next, we discuss the stability of the SkX under distortion for different values of the DM interaction. Figure 9(a) and Figure 9(b) show the low-temperature magnetic phase diagrams in the ${\gamma }_{{\mathit{Q}}_2}$–H plane for $D=0.1$ and $D=0.5$, respectively. For the weaker DM interaction $D=0.1$, three phases are stabilized: the single-Q vertical spiral state at low magnetic fields, the single-Q tilted conical spiral state at intermediate fields, and the fully polarized state at high fields, as shown in Figure 9(a); the SkX does not appear in the phase diagram. Since both the single-Q vertical spiral and tilted conical spiral states are characterized by the single-Q peak structure at ${\mathit{Q}}_1$, the change of ${\gamma }_{{\mathit{Q}}_2}$ does not alter their stability. The phase boundaries do not depend on ${\gamma }_{{\mathit{Q}}_2}$.

In contrast, for the stronger DM interaction $D=0.5$, the SkX region dramatically expands both in ${\gamma }_{{\mathit{Q}}_2}$ and H, forming a robust pocket bounded by the single-Q vertical spiral and single-Q tilted conical spiral phases, as shown in Figure 9(b). This comparison clearly demonstrates that increasing D enhances the energetic stability of the SkX and compensates for the anisotropy introduced by the uniaxial distortion, thereby broadening the region where the double-Q spin configuration is stabilized.

The real-space spin configurations corresponding to the SkX at different degrees of distortion are illustrated in Figure 10 for $H=1.7$. In the absence of distortion (${\gamma }_{{\mathit{Q}}_2}=1$) in Figure 10(a), the SkX forms a perfectly symmetric square array of skyrmions, with nearly circular cores and uniform periodicity. At ${\gamma }_{{\mathit{Q}}_2}=0.95$ shown in Figure 10(b), the SkX remains topologically intact but exhibits a weak elongation along the distortion axis, and the skyrmion cores deform into slightly elliptical shapes. With further distortion at ${\gamma }_{{\mathit{Q}}_2}=0.88$ in Figure 10(c) and ${\gamma }_{{\mathit{Q}}_2}=0.81$ in Figure 10(d), the anisotropy becomes more pronounced, and the lattice transforms into a rectangular arrangement with compressed cores along the ${\mathit{Q}}_2$ direction.

Figure 11 further quantifies this distortion-driven evolution at $H=1.7$ by showing the ${\gamma }_{{\mathit{Q}}_2}$ dependence of the squared Fourier amplitudes ${\left(m_{{\mathit{Q}}_{\nu }}^{\eta }\right)}^2$ and the squared scalar spin chirality ${\left({\chi }^{\mathrm{sc}}\right)}^2$. In Figure 11(a), the ${\mathit{Q}}_2$ components gradually decrease as ${\gamma }_{{\mathit{Q}}_2}$ is reduced, while the ${\mathit{Q}}_1$ components are slightly enhanced, illustrating the imbalance between the two modulation directions. The vertical dashed line denotes the critical distortion where the ${\mathit{Q}}_2$ amplitude nearly vanishes, signaling the transition from the SkX to the single-Q tilted conical spiral phase. In Figure 11(b), ${\left({\chi }^{\mathrm{sc}}\right)}^2$ exhibits finite values, reflecting the stable topological SkX, but rapidly drops to zero around ${\gamma }_{{\mathit{Q}}_2}\approx 0.8$. The concurrent disappearance of the ${\mathit{Q}}_2$ mode and ${\left({\chi }^{\mathrm{sc}}\right)}^2$ demonstrates that uniaxial distortion induces a topological phase transition from the SkX to the single-Q tilted conical spiral state. These results show that the cooperative effect of DM interaction and exchange anisotropy on the formation and suppression of the double-Q SkX.

4. Conclusions

We have theoretically investigated the effects of uniaxial distortion on the stability of the square SkX in noncentrosymmetric tetragonal magnets by analyzing a classical spin model that incorporates momentum-resolved frustrated interactions and DM interactions. Using simulated-annealing calculations, we clarified how the interplay between exchange anisotropy and DM interaction governs the formation, deformation, and collapse of topological spin textures.

The results revealed that in the absence of distortion, the double-Q SkX phase emerges between the single-Q vertical spiral state and the single-Q tilted conical spiral state in the intermediate magnetic-field region. As uniaxial distortion is introduced, the equivalence between the two ordering wave vectors ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ is lifted, suppressing the interference between the orthogonal spiral components. This effect continuously weakens the noncoplanar modulation characteristic of the SkX, and beyond a critical anisotropy ratio ${\gamma }_{{\mathit{Q}}_2}\sim 0.93$, the ${\mathit{Q}}_2$ contribution vanishes, driving a topological phase transition from the SkX to the single-Q tilted conical spiral phase. The concomitant disappearance of the scalar spin chirality confirms that this transition involves a change from a topologically nontrivial to a trivial spin texture.

Furthermore, by systematically varying the magnitude of the DM interaction, we demonstrated that the stability region of the SkX strongly depends on D. For weak DM interaction, only single-Q spiral states are stabilized. Meanwhile, when the DM interaction is further increased, the SkX region expands substantially in both magnetic field and anisotropy range, highlighting the cooperative role of the DM interaction in compensating the exchange anisotropy induced by distortion. The real-space spin textures and reciprocal-space spin structure factors consistently reflect these trends, showing the deformation and eventual collapse of the SkX as the uniaxial distortion strengthens. Our study establishes a unified understanding of how strain and DM interactions interplay to control the stability of square SkXs in noncentrosymmetric magnets.