Double-Q Instability in Noncentrosymmetric Tetragonal Magnets with Bond-Dependent Magnetic Anisotropy Under an In-Plane Magnetic Field

Abstract

We investigate the instability toward a double-Q state, which consists of a superposition of two spin density waves at different wave vectors, on a two-dimensional noncentrosymmetric square lattice in an in-plane external magnetic field. By performing the simulated annealing for the spin model with the Dzyaloshinskii–Moriya interaction and bond-dependent anisotropic interaction, we obtain four types of double-Q states depending on the sign of the bond-dependent anisotropic interaction. On the other hand, only the single-Q spiral state appears in the absence of the bond-dependent anisotropic interaction. The present results suggest that the interplay between the Dzyaloshinskii–Moriya interaction and bond-dependent anisotropic interaction can give rise to multiple-Q states for both zero and nonzero in-plane magnetic fields.

1. Introduction

A multiple-Q state, which is formed by superposing multiple spin density waves in crystals, has long been studied in magnetism, since it often leads to noncollinear and noncoplanar spin textures [1,2,3,4,5,6,7,8,9,10,11]. A variety of multiple-Q states has been identified under various lattice structures, such as triple-Q noncoplanar states under triangular [12,13,14,15,16,17,18,19], kagome [20,21,22,23,24,25], square-kagome [26], honeycomb [27,28,29,30,31], maple-leaf [32], face-centered-cubic [33,34,35], and pyrochlore [7,36,37,38] structures and double-Q noncoplanar states on square [39,40,41,42,43], checkerboard [44], and Shastry–Sutherland [45,46] structures. These multiple-Q states include topologically nontrivial spin textures, such as magnetic skyrmion [47,48,49,50,51,52,53,54,55,56] and hedgehog crystals [57,58,59,60,61,62,63,64], where the unconventional anomalous Hall effect, i.e., the topological Hall effect, is intrinsically invoked [65,66,67,68,69,70,71]. Since the emergence of multiple-Q states is rare compared to that of collinear antiferromagnetic states and single-Q spiral states, it is important to elucidate when and how multiple-Q states are stabilized.

From an energetic viewpoint, multiple-Q states compete with the single-Q spiral state. When the ordering of classical spins with a fixed spin length at each lattice site is supposed, the Heisenberg-type isotropic exchange interaction favors the single-Q spiral state rather than multiple-Q states, since the spin configuration of the single-Q spiral state, which is given by ${\mathit{S}}_i=[cos(\mathit{Q}·{\mathit{r}}_i),sin(\mathit{Q}·{\mathit{r}}_i),0]$ ($\mathit{Q}$ represents the ordering wave vector and ${\mathit{r}}_i$ represents the position vector at site i), satisfies the local constraint in terms of the spin length. Meanwhile, multiple-Q states do not satisfy such a local constraint in most cases owing to nonzero spin intensities at higher-harmonic wave vectors arising from a multiple-Q superposition, which leads to an energy loss compared to the single-Q state. In order to stabilize multiple-Q states, there are mainly two principles [72]. One is the introduction of higher-order spin interactions beyond two-spin interactions, which often leads to energy gain (loss) for the multiple-Q (single-Q) state [12,73,74,75,76,77,78,79,80]. The other is the introduction of anisotropic spin interactions and/or external fields that deform the spiral plane from a circular one to an elliptical one [81,82,83,84]. This is because an elliptical spiral spin configuration, which is characterized by ${\mathit{S}}_i=N_i[c_{1}cos(\mathit{Q}·{\mathit{r}}_i),c_{2}sin(\mathit{Q}·{\mathit{r}}_i),0]$ ($c_1\ne c_2$ are numerical coefficients and $N_i$ is the normalized constant), induces additional spin intensities at high-harmonic wave vectors in order to satisfy the local constraint in terms of the spin length. Accordingly, the energy of the single-Q state can be higher than that of the multiple-Q state. In this way, magnetic anisotropy and/or multi-spin interactions are significant in realizing the multiple-Q states.

In the present study, we further investigate the instability toward the multiple-Q states by focusing on the role of bond-dependent magnetic anisotropy, which arises from the relativistic spin–orbit coupling under the discrete rotational symmetry [82,85,86,87,88,89,90,91,92,93,94,95,96]. We especially search for multiple-Q states in an in-plane external magnetic field on a two-dimensional noncentrosymmetric square lattice. Through the numerical analyses based on the simulated annealing for the spin model with the bond-dependent magnetic anisotropy and Dzyaloshinskii–Moriya (DM) interaction, we find that four types of double-Q states appear in zero and nonzero in-plane magnetic fields. The emergence of four double-Q states is accounted for by the competition among the bond-dependent magnetic anisotropy, DM interaction, and in-plane magnetic field. The present results indicate a rich possibility of multiple-Q states by considering multiple anisotropic interactions.

The rest of this paper is organized as follows. In Section 2, we show an effective spin model with the DM interaction and bond-dependent magnetic anisotropy in order to investigate the instability toward the multiple-Q states under an in-plane external magnetic field. We also present a numerical method based on the simulated annealing. Then, we show the magnetic phase diagram and discuss the obtained multiple-Q states in Section 3. Section 4 is devoted to the conclusion of this paper.

2. Model and Method

We consider a localized spin model on a two-dimensional square lattice belonging to the chiral point group $D_4$, since square-lattice magnets provide rich multiple-Q physics in a variety of materials, such as GdRu₂Si₂ [97,98,99,100,101], GdRu₂Ge₂ [102], and EuNiGe₃ [103,104]. We set the lattice constant to be unity. The Hamiltonian is given by

$$\begin{array}{c}\hfill \mathcal{H}=\sum _{i,j,\alpha ,\beta }{\Gamma }_{ij}^{\alpha \beta }{S}_i^{\alpha }{S}_j^{\beta }-H^x\sum _i{S}_i^x,\end{array}$$

(1)

where $S_i^{\alpha }$ for $\alpha =x,y,z$ is the classical localized spin at site i, whose magnitude is given by $|{\mathit{S}}_i|=1$. The first term represents the exchange interaction between the spins at sites i and j. The coupling constant ${\Gamma }_{ij}^{\alpha \beta }$ is the $3\times 3$ matrix, which is taken for the x-directional spin pair as

$$\begin{array}{c}\hfill {\left({\Gamma }_{ij}\right)}_x=\left(\begin{array}{ccc}{J}_{ij}-I_{ij}^{\mathrm{BA}}& 0& 0\\ 0& J_{ij}+I_{ij}^{\mathrm{BA}}& D_{ij}\\ 0& -D_{ij}& J_{ij}\end{array}\right),\end{array}$$

(2)

and for the y-directional spin pair

$$\begin{array}{c}\hfill {\left({\Gamma }_{ij}\right)}_y=\left(\begin{array}{ccc}{J}_{ij}+I_{ij}^{\mathrm{BA}}& 0& -D_{ij}\\ 0& J_{ij}-I_{ij}^{\mathrm{BA}}& 0\\ D_{ij}& 0& J_{ij}\end{array}\right),\end{array}$$

(3)

where $J_{ij}$, $I_{ij}^{\mathrm{BA}}$, and $D_{ij}$ stand for the Heisenberg exchange interaction, bond-dependent anisotropic interaction, and DM interaction, respectively; the bond-dependent anisotropic interaction and DM interaction arise from the relativistic spin–orbit coupling [85,105,106,107] and the anisotropy of hopping integrals between different orbitals, both of which lead to the instability toward the skyrmion crystal when the magnetic field is applied along the z direction [50,54,108,109,110]. We ignore the effect of surface anisotropy [111,112,113,114]. The second term denotes the Zeeman coupling to take into account the effect of the in-plane external magnetic field. We do not consider the effect of random disorder in the presence of the impurity by supposing the clean sample [115].

For the exchange interaction ${\Gamma }_{ij}^{\alpha \beta }$, we choose $J_{ij}$, $I_{ij}^{\mathrm{BA}}$, and $D_{ij}$ so that the Fourier transform of ${\Gamma }_{ij}^{\alpha \beta }$, i.e., ${\Gamma }^{\alpha \beta }\left(\mathit{q}\right)$, gives the lowest energy at ${\mathit{Q}}_1=(2\pi /5,0)$ and ${\mathit{Q}}_2=(0,2\pi /5)$; we ignore the possibility of the triple-Q state to have the ordering wave vectors at low-symmetric positions, as found in EuNiGe₃ [103]. When the interactions at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ are dominant, the ground-state spin configuration is described by a linear combination of the spin density waves at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$. In such a situation, we drop off the contributions of the interactions at the other wave vectors. Then, the spin model with the real-space interaction in Equation (1) turns into the effective momentum-resolved spin model, which is given by [116]

$$\begin{array}{c}\hfill \mathcal{H}=-2\sum _{\nu =1,2}{\Gamma }^{\alpha \beta }\left({\mathit{Q}}_{\nu }\right)S_{{\mathit{Q}}_{\nu }}^{\alpha }{S}_{-{\mathit{Q}}_{\nu }}^{\beta }-H^x\sum _i{S}_i^x,\end{array}$$

(4)

where $S_{{\mathit{Q}}_{\nu }}^{\alpha }$ is the Fourier transform of $S_i^{\alpha }$, and $\nu$ is the label of the dominant ordering wave vectors. The prefactor 2 in the first term is owing to the contributions from $-{\mathit{Q}}_{\nu }$. The interaction matrices at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ are explicitly given by

$$\begin{array}{cc}\hfill \Gamma \left({\mathit{Q}}_1\right)& =\left(\begin{array}{ccc}J-I^{\mathrm{BA}}& 0& 0\\ 0& J+I^{\mathrm{BA}}& iD\\ 0& -iD& J\end{array}\right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \Gamma \left({\mathit{Q}}_2\right)& =\left(\begin{array}{ccc}J+I^{\mathrm{BA}}& 0& -iD\\ 0& J-I^{\mathrm{BA}}& 0\\ iD& 0& J\end{array}\right).\hfill \end{array}$$

(6)

This momentum-resolved interaction satisfies the symmetry of the chiral point group $D_4$ [116]. Hereafter, we set $J=1$ as the energy unit of the model in Equation (4) and $D=0.2$. The magnetic phase diagram of the model in Equation (4) under the out-of-plane magnetic field direction was investigated [117], where both Néel-type and Bloch-type SkXs are stabilized. Meanwhile, the investigation of the model under the in-plane magnetic field has yet to be performed.

For the momentum-resolved spin model in Equation (4), we perform the simulated annealing in order to obtain the lowest-energy spin configuration at low temperatures. We set the system size as $N={10}^2$ under the periodic boundary conditions. In the simulations, we start from a random spin configuration at high temperatures $T_0=1$–10. Then, we gradually reduce the temperature as $T_{n+1}=\alpha T_n$ with the annealing rate $\alpha =0.99999$–$0.999999$ in each Monte Carlo sweep; $T_n$ is the nth temperature. In each temperature, we update all the spins one by one in real space based on the Metropolis algorithm [118]. Once the temperature reaches the final temperature $T={10}^{-4}$, we further perform ${10}^5$–${10}^6$ Monte Carlo sweeps for measurements. The above process is independently performed in each $I^{\mathrm{BA}}$ and $H^x$.

We compute the spin structure factor in order to identify magnetic phases, which is defined by

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

(7)

where $\mathit{q}$ is the wave vector in the first Brillouin zone and ${\mathit{r}}_i$ is the position vector at site i. The net magnetization along the $\mu$ direction is given by $M^{\mu }=(1/N){\sum }_i{S}_i^{\mu }$.

3. Results

We show the magnetic phase diagram in the plane of $I^{\mathrm{BA}}$ and $H^x$ at $D=0.2$ in this section. We obtain rich magnetic phases including single-Q and double-Q spin configurations depending on the sign of $I^{\mathrm{BA}}$, although no SkX appears in the in-plane magnetic field in contrast to the easy-plane magnets [119]. We discuss the appearance of magnetic phases in the cases of $I^{\mathrm{BA}}=0$ in Section 3.1, $I^{\mathrm{BA}}>0$ in Section 3.2, and $I^{\mathrm{BA}}<0$ in Section 3.3.

3.1. Case of $I^{\mathrm{BA}}=0$

First, we show the result for $I^{\mathrm{BA}}=0$. The zero-field state corresponds to the single-Q proper-screw spiral (1Q PS) state; the spiral plane is perpendicular to ${\mathit{Q}}_1$ or ${\mathit{Q}}_2$ in order to gain the energy by the DM interaction. For $I^{\mathrm{BA}}=0$, the 1Q PS state with the ${\mathit{Q}}_1$ modulation has the same energy as that with the ${\mathit{Q}}_2$ modulation. Meanwhile, the energy for the former state becomes lower when the magnetic field is applied in the x direction, since the spiral plane is perpendicular to the field direction for the ${\mathit{Q}}_1$ modulation, which leads to the energy gain for both J and $H^x$. We refer to the 1Q PS state with the ${\mathit{Q}}_1$ modulation as the 1$Q_1$ PS state. When the magnetic field increases, the spin configuration of the 1$Q_1$ PS state smoothly changes into that of the fully polarized state.

3.2. Case of $I^{\mathrm{BA}}>0$

We consider the effect of $I^{\mathrm{BA}}$. The positive value of $I^{\mathrm{BA}}$ tends to stabilize the proper-screw spiral state, since $I^{\mathrm{BA}}$ favors the spin oscillations perpendicular to the ${\mathit{Q}}_{\nu }$ and z directions, as shown in the bottom-right panel of Figure 1. In other words, $I^{\mathrm{BA}}$ enhances the stability of the 1$Q_1$ PS state in collaboration with the DM interaction. The real-space spin configuration of the 1$Q_1$ PS state at $I^{\mathrm{BA}}=0.08$ and $H^x=0.5$ is shown in the leftmost panel of Figure 2a. In the presence of the in-plane magnetic field, the 1$Q_1$ PS state exhibits the peak structures at $\mathit{q}=\mathbf{0}$ in the x-spin component and $\mathit{q}={\mathit{Q}}_1$ in the y- and z-spin components, as shown in the right three panels of Figure 2a. For almost all the regions for $I^{\mathrm{BA}}>0$ in the phase diagram in Figure 1, the 1$Q_1$ PS state is stabilized.

Meanwhile, three phases appear for a large $I^{\mathrm{BA}}$ and a small $H^x$ in the phase diagram. This is because a large $I^{\mathrm{BA}}$ deforms the spiral plane from circular to elliptic, which results in the energy loss of the single-Q state compared to the multiple-Q state. For $0.22\lesssim I^{\mathrm{BA}}\lesssim 0.34$, the 1$Q_2$ PS state, which is characterized by the proper-screw spiral state with the ${\mathit{Q}}_2$ modulation, is realized; the real-space spin configuration and spin structure factor are shown in Figure 2b. This result indicates that the bond-dependent anisotropy $I^{\mathrm{BA}}$ favors the spin modulations parallel to the field direction rather than those perpendicular to the field for a small magnetic field. When the magnetic field increases, there is a spin-flop transition from the ${\mathit{Q}}_2$ spiral state (1$Q_2$ PS state) to the ${\mathit{Q}}_1$ one (1$Q_1$ PS state) in order to gain the Zeeman energy.

For a larger $I^{\mathrm{BA}}\gtrsim 0.34$, the 2Q I state is stabilized in the small $H^x$ region. The 2Q I state mainly consists of the superposition of the proper-screw spiral wave with ${\mathit{Q}}_2$ and the y-directional sinusoidal wave with ${\mathit{Q}}_1$, as shown in the right three panels of Figure 2c. Owing to the double-Q superposition, the spin configuration is noncoplanar, which is regarded as the vortex–antivortex crystal, as shown in the leftmost panel of Figure 2c. The 2Q I state shows the density wave in terms of the scalar spin chirality along the sinusoidal-wave (${\mathit{Q}}_1$) direction. With the increase in $H^x$ within the 2Q I state, another double-Q state appears, which is referred to as the 2Q II state. In contrast to the 2Q I state, the 2Q II state consists of the superposition of the proper-screw spiral wave with ${\mathit{Q}}_1$ and the x-directional sinusoidal wave with ${\mathit{Q}}_2$, as shown in the right three panels of Figure 2d. In other words, the dominant spiral wave modulation changes from the ${\mathit{Q}}_2$ to ${\mathit{Q}}_1$ component. Such a difference is found in the real-space spin configuration in the leftmost panel of Figure 2d, where the z-spin oscillation in the 2Q II state is different from that in the 2Q I state. Accordingly, the 2Q II state exhibits the density wave in terms of the scalar spin chirality along the ${\mathit{Q}}_2$ direction. When the magnetic field further increases, the sinusoidal modulation with ${\mathit{Q}}_2$ becomes smaller, and the 2Q II state is replaced by the 1$Q_1$ PS state.

We briefly comment on the finite-temperature effect. When the temperature increases, the magnitude of the ${\mathit{Q}}_{\nu }$ component of spins in each magnetic phase tends to decrease irrespective of the sign of $I^{\mathrm{BA}}$. Since the magnitude of magnetic moments at the sub-dominant ordering wave vector decreases faster than that at the dominant ordering wave vector, the 2Q I and 2Q II states are expected to show the phase transition to the single-Q state [120].

3.3. Case of $I^{\mathrm{BA}}<0$

Next, we investigate the effect of a negative $I^{\mathrm{BA}}$. In contrast to the situation for $I^{\mathrm{BA}}>0$, $I^{\mathrm{BA}}<0$ tends to favor the cycloidal spiral wave, whose spiral plane is parallel to the ${\mathit{Q}}_{\nu }$ and z direction, as shown in the bottom-left panel of Figure 1. Thus, $I^{\mathrm{BA}}$ and D tend to stabilize the different types of spiral waves, which results in frustration. Indeed, there are five additional phases for $I^{\mathrm{BA}}<0$, as shown in Figure 1, which results from the competition between $I^{\mathrm{BA}}$ and D.

For $-0.14\lesssim I^{\mathrm{BA}}\lesssim -0.1$, the spiral plane is tilted from the plane perpendicular to ${\mathit{Q}}_1$; we refer to this state as the single-Q tilted spiral (1$Q_1$ TS) state. As shown in the real-space spin configuration in the leftmost panel of Figure 3a, the spin moments at each site are tilted from the y to x direction. Accordingly, all the spin components show the peak structures at ${\mathit{Q}}_1$, as shown in the right three panels of Figure 3a. The emergence of the 1$Q_1$ TS state is owing to the synergy between $I^{\mathrm{BA}}$ and D.

When the magnitude of $I^{\mathrm{BA}}$ increases, the ordering wave vector changes from ${\mathit{Q}}_1$ to ${\mathit{Q}}_2$, which indicates the appearance of the 1$Q_2$ TS state. The spin textures of the 1$Q_2$ TS state in real and momentum spaces are shown in Figure 3b. The change of the ordering wave vector from ${\mathit{Q}}_1$ to ${\mathit{Q}}_2$ is understood from the fact that the cycloidal spiral state, which is favored by $I^{\mathrm{BA}}<0$, gains the Zeeman energy more when the spiral plane perpendicular to the magnetic field direction. When the magnetic field is close to the saturation field, the 1$Q_2$ TS state turns into the 1$Q_2$ S state, where the sinusoidal spin modulation occurs in the $yz$ plane, as shown by the real-space spin configuration and the spin structure factor in Figure 3c.

For $-0.4\lesssim I^{\mathrm{BA}}\lesssim -0.22$, the instability toward two double-Q states occurs. In the small $H^x$ region, the 2Q III state appears next to the 1$Q_2$ TS state. The spin configuration of the 2Q III state is mainly characterized by the tilted spiral modulation with ${\mathit{Q}}_1$ and the sinusoidal modulation with ${\mathit{Q}}_2$, as shown by the spin structure factor in the right three panels of Figure 3d. Similarly to the 2Q I state in Figure 2c, this state also shows a noncoplanar spin texture to have vortex and antivortex, as found in the real-space spin configuration in the leftmost panel of Figure 3d. It is noted that there is no uniform scalar spin chirality, although this state shows the density wave in terms of the scalar spin chirality along the sinusoidal-wave (${\mathit{Q}}_2$) direction. For $I^{\mathrm{BA}}=-0.4$, the 2Q IV state appears when the magnetic field increases in the 2Q III state. In the 2Q IV state, the z-spin modulation is suppressed, as shown in the rightmost panel of Figure 3e; this spin configuration is almost characterized by the double-Q fan state. Since the spin structure factors in the x- and y-spin components are similar between the 2Q III and 2Q IV states shown in the leftmost panel of Figure 3d and Figure 3e, respectively, the 2Q IV state is also characterized by the vortex–antivortex crystal.

Finally, we discuss the effect of the DM interaction in the phase diagram. Since the DM interaction tends to favor the single-Q state, the large (small) value of D tends to increase (decrease) the critical value of $|I^{\mathrm{BA}}|$ in the phase boundaries between the single-Q and double-Q states. Thus, the smaller DM interaction is favored for the realization of the double-Q states.

4. Conclusions

We have investigated the instability toward multiple-Q states on the noncentrosymmetric square lattice as a consequence of the competition between magnetic anisotropy and in-plane magnetic field. We have shown that the spin model with the DM interaction and the bond-dependent anisotropic interaction gives rise to rich double-Q states depending on their relative sign by performing the simulated annealing. Our results indicate that competing anisotropic interactions provide an intriguing playground for various types of multiple-Q states, which would stimulate a further search for new types of exotic spin states and magnetic excitations [121].