Schematic pictures of the SkXs and the crystal structure. (a-c) SkXs characterized by three spiral and sinusoidal waves along the Q 1 , Q 2 , and Q 3 directions: (a) the n sk = 1 SkX for ψ η = 0 , (b) the n sk = 2 SkX for ψ η = π/2 , and (c) the n sk = 1 T-SkX for ψ η = π/6 in Eq. (1). (d) Centrosymmetric trigonal structure without the horizontal mirror plane. The blue spheres represent magnetic sites, while the gray spheres shifted by +c ( −c ) from the center of the downward (upward) triangles on the magnetic layer represent nonmagnetic sites on a layer A (B).

Schematic pictures of the SkXs and the crystal structure. (a-c) SkXs characterized by three spiral and sinusoidal waves along the Q 1 , Q 2 , and Q 3 directions: (a) the n sk = 1 SkX for ψ η = 0 , (b) the n sk = 2 SkX for ψ η = π/2 , and (c) the n sk = 1 T-SkX for ψ η = π/6 in Eq. (1). (d) Centrosymmetric trigonal structure without the horizontal mirror plane. The blue spheres represent magnetic sites, while the gray spheres shifted by +c ( −c ) from the center of the downward (upward) triangles on the magnetic layer represent nonmagnetic sites on a layer A (B).

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We theoretically investigate a new stabilization mechanism of a skyrmion crystal (SkX) in centrosymmetric itinerant magnets with magnetic anisotropy. By considering a trigonal crystal system without the horizontal mirror plane, we derive an effective spin model with an anisotropic Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction for a multi-band pe...

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... of each spin density wave. A variety of the SkXs are described by Eq. (1); a superposition of spiral waves for e η � e z × Q η ( e η Q η ) and ψ 1 = ψ 2 = ψ 3 = 0 or π represents the Bloch-type (Néel-type) SkX, while that for ψ 1 = ψ 2 = 0 and ψ 3 = π represents the anti-type SkX. The realspace spin texture for the Bloch-type SkX is shown in Fig. 1a. All the SkXs have the skyrmion number of one, n sk ≡ |N sk | = 1 , in the magnetic unit cell and breaks the spatial inversion symmetry irrespective of e η and ψ η 8 . We call them the n sk = 1 SkXs. The n sk = 1 SkXs are stabilized by the Dzyaloshinskii-Moriya (DM) interaction 17,18 in chiral/polar magnets 4,19 or the competing ...
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... the spiral density waves are not necessarily for the formation of the SkX. By considering the superposition of the sinusoidal waves characterized by a different ψ η , another type of the SkX can emerge, as shown in Fig. 1b 23,24 . In contrast to the n sk = 1 SkX, this spin texture exhibits the skyrmion number of two in a magnetic unit cell ( n sk = 2 SkX), whose spatial inversion and/or sixfold rotational symmetries are broken depending on φ η on a discrete lattice. For example, the n sk = 2 SkX with φ η = π shown in Fig. 1b has the inversion symmetry, ...
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... type of the SkX can emerge, as shown in Fig. 1b 23,24 . In contrast to the n sk = 1 SkX, this spin texture exhibits the skyrmion number of two in a magnetic unit cell ( n sk = 2 SkX), whose spatial inversion and/or sixfold rotational symmetries are broken depending on φ η on a discrete lattice. For example, the n sk = 2 SkX with φ η = π shown in Fig. 1b has the inversion symmetry, but the n sk = 2 SkX with φ 1 = 4π/3 , φ 2 = 2π/3 , and φ 3 = π shows the inversion symmetry breaking. Although the n sk = 2 SkX seems to be rare compared to the n sk = 1 one, it is stabilized by a ...
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... from the absence of the mirror symmetry in the crystal structure. By constructing a microscopic effective spin model and performing simulated annealing for triangular itinerant magnets, we show that an anisotropic Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [28][29][30] arising from the absence of the mirror symmetry on a magnetic layer [ Fig. 1d] induces the SkXs with n sk = 1 and n sk = 2 . The anisotropic RKKY interaction stabilizes the SkXs even without the DM, competing exchange, and multi-spin interactions 25,[31][32][33][34][35] . The obtained SkXs exhibit different symmetry breaking compared to that found in previous studies 4,19 . The spin texture in the SkX with n sk ...
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... the SkXs even without the DM, competing exchange, and multi-spin interactions 25,[31][32][33][34][35] . The obtained SkXs exhibit different symmetry breaking compared to that found in previous studies 4,19 . The spin texture in the SkX with n sk = 1 does not have the sixfold rotational symmetry in addition to the inversion symmetry, as shown in Fig. 1c, which is different from that in chiral and frustrated magnets in Fig. 1a. We here call this state the n sk = 1 threefold-rotational-symmetric SkX (T-SkX). Meanwhile, the n sk = 2 SkX shows the inversion symmetry breaking. Furthermore, we elucidate that topological phase transitions between the n sk = 1 T-SkX, the n sk = 2 SkX, and ...
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... 25,[31][32][33][34][35] . The obtained SkXs exhibit different symmetry breaking compared to that found in previous studies 4,19 . The spin texture in the SkX with n sk = 1 does not have the sixfold rotational symmetry in addition to the inversion symmetry, as shown in Fig. 1c, which is different from that in chiral and frustrated magnets in Fig. 1a. We here call this state the n sk = 1 threefold-rotational-symmetric SkX (T-SkX). Meanwhile, the n sk = 2 SkX shows the inversion symmetry breaking. Furthermore, we elucidate that topological phase transitions between the n sk = 1 T-SkX, the n sk = 2 SkX, and another non-topological triple-Q state are caused by a change with respect to ...
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... localized electron spin S i at site i ( |S i | = 1 ), and the coefficient 2 arises from the −Q η contribution. , originate from the atomic spin-orbit coupling [42][43][44] . The number of Q η and nonzero components of the interactions are determined by the lattice symmetry. For the above effective spin model, we consider the lattice structure in Fig. 1d consisting of a magnetic layer sandwiched by two nonmagnetic layers. The nonmagnetic ions at z = c ( z = −c ) are located above (below) the downward (upward) triangles on the magnetic layer at z = 0 , which breaks the horizontal mirror symmetry at z = 0 while keeping the inversion symmetry. The lattice symmetry is compatible with the D ...

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Citations

... Another distinguishing feature of SkXs is the phase degree of freedom [70,71]. The relative phases among the constituent spin density waves influence spin configurations, resulting in structures such as the meron-antimeron crystals [45,[72][73][74][75], tetra-axial vortex crystals [70], and SkXs with high skyrmion numbers [76,77], thereby altering their topological properties. Furthermore, the number of constituent waves also influences the SkX structure. ...
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We conduct a numerical investigation into the stability of a quadruple-Q skyrmion crystal, a structure generated by the superposition of four spin density waves traveling in distinct directions within three-dimensional space, hosted on a centrosymmetric body-centered tetragonal lattice. Using simulated annealing applied to an effective spin model that includes momentum-resolved bilinear and biquadratic interactions, we construct a magnetic phase diagram spanning a broad range of model parameters. Our study finds that a quadruple-Q skyrmion crystal does not emerge within the phase diagram when varying the biquadratic interaction and external magnetic field. Instead, three distinct quadruple-Q states with topologically trivial spin textures are stabilized. However, we demonstrate that the quadruple-Q skyrmion crystal can become the ground state when an additional high-harmonic wave–vector interaction is considered. Depending on the magnitude of this interaction, we obtain two types of quadruple-Q skyrmion crystals exhibiting the skyrmion numbers of one and two. These findings highlight the emergence of diverse three-dimensional multiple-Q spin states in centrosymmetric body-centered tetragonal magnets.
... 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 S i = N i [c 1 cos(Q · r i ), c 2 sin(Q · r i ), 0] (c 1 ̸ = 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. ...
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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.
... The distribution of the scalar spin chirality is shown in the right panel of Figure 3b, where the uniform component exists. Similar SkX with high skyrmion numbers has also been found when the anisotropic exchange interaction [88][89][90], higher-harmonic-wave vector interaction [91], and dynamical electric field [92] are considered, although the present SkX-II does not require such factors. Similar to the SkX-I, the SkX-II has a degeneracy in terms of the sign of the skyrmion number. ...
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... Its low-computational cost enables us to perform a number of simulations while changing the lattice structures, magnetic interactions, anisotropy, and so on. The effective spin model has so far uncovered new stabilization mechanisms of the SkXs: biquadratic interaction [159], symmetric anisotropic interaction [160,161,162], staggered DM interaction [163], high-harmonic wave-vector interaction [164,165], and so on. Along the line, the realization of the SkXs has given a deep understanding of the experimental phase diagrams in SkX-hosting materials, such as Gd 3 Ru 4 Al 12 [109], GdRu 2 Si 2 [112], EuAl 4 [121], EuPtSi [166], EuNiGe 3 [99], and GdRu 2 Ge 2 [127]. ...
... We summarize the correspondence between the point groups and (D Q , E Q , F Q ) for Q = (Q, 0, 0) in Table 3. By using the effective spin model incorporating the anisotropic spin interactions in Eq. (13) and external magnetic field, various multiple-Q instabilities have been revealed, such as the S-SkX under D 4h [160,115], the S-bubble crystal under D 4h [170], the hybrid S-SkX under C 4v [188] and C 4h [199], the T-SkX under D 6h [161], the T-MAX under C 6v [168], the n sk = 2 SkX under D 3d [162], the 6Q SkX under O [186], the hedgehog crystal under T [185] and T h [177], and the distorted T-SkX under D 2h [193]. It also accounts for the important ingredients to reproduce the experimental phase diagrams in SkX-hosting materials, such as Gd 3 Ru 4 Al 12 [109], GdRu 2 Si 2 [112], EuPtSi [166], and EuNiGe 3 [99]. ...
... The qualitatively same phase diagram is obtained for E z Q 1 . In Fig. 15(c), E x Q 1 stabilizes two types of T-SkXs: T-SkX with the skyrmion number of one and the n sk = 2 T-SkX with the skyrmion number of two depending on the magnetic field [162]. The real-space spin configurations of two T-SkXs are shown in the middle and right panels of Fig. 15(c). ...
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... In EuAl 4 , the consequences also include a field-induced skyrmion lattice [17,21,22]. Topological spin textures in centrosymmetric crystals are of great interest precisely because of the highly tunable equilibrium of correlations [23,24], which replaces the traditional scenario of inversion-symmetrybroken skyrmion hosts with Dzyaloshinskii-Moriya exchange [25][26][27][28][29][30][31]. The magnetism of EuAl 4 has been thoroughly characterized by transport measurements [19,32], the anomalous Hall effect (AHE) [14,33], neutron diffraction [16], and resonant elastic x-ray scattering [21,22]. ...
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... However, recent experimental studies have shown that skyrmions can be realized even in centrosymmetric materials such as triangular lattices [4][5][6]. This is because of the frustration or oscillation of the magnetic exchange coupling constant ij , which is also known as the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [4][5][6][7]. The presence of a skyrmion in a centrosymmetric material offers several advantages; for example, the helicity and polarity can be controlled as additional degrees of freedom, which can be useful when developing future computing hardware such as skyrmion qubits in quantum computing or neuromorphic computing [7][8][9][10][11]. ...
... This is because of the frustration or oscillation of the magnetic exchange coupling constant ij , which is also known as the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [4][5][6][7]. The presence of a skyrmion in a centrosymmetric material offers several advantages; for example, the helicity and polarity can be controlled as additional degrees of freedom, which can be useful when developing future computing hardware such as skyrmion qubits in quantum computing or neuromorphic computing [7][8][9][10][11]. Thus, exploring other possible mechanisms to realize skyrmions in a new class of centrosymmetric materials is an important task for both theoretical and experimental frameworks. ...
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