## Introduction

Recently, the concept of topology is attracting attention as a source for various exotic phenomena in solids. One typical example is magnetic skyrmion, i.e., a noncoplanar swirling spin structure with topologically stable particle-like nature (Fig. 1c). It is characterized by non-zero integer skyrmion number Nsk defined as:

$${N}_{{{{{{{\mathrm{sk}}}}}}}}=\frac{1}{4\pi }\int {{{{{\bf{n}}}}}}\cdot \left(\frac{\partial {{{{{\bf{n}}}}}}}{\partial x}\times \frac{\partial {{{{{\bf{n}}}}}}}{\partial y}\right){dxdy}$$
(1)

with n(r) = m(r)/|m(r)| being the unit vector along the direction of local magnetic moment m(r), which represents how many times n(r) wrap a sphere1. In metallic compounds, skyrmion spin textures induce fictitious magnetic fluxes acting on the conduction electrons through quantum-mechanical Berry phase, which leads to various exotic transport phenomena, such as the topological Hall effect2,3,4. Conversely, an applied electric current can efficiently drive the skyrmion motion through a spin transfer torque, whose threshold current density is often five orders of magnitude smaller than for conventional ferromagnetic domain walls3,4. These features highlight skyrmions as potential novel information carriers with high-energy efficiency and information density.

Previously, magnetic skyrmions were mostly observed in noncentrosymmetric systems, where a competition between the ferromagnetic exchange and Dzyaloshinskii-Moriya (DM) interactions stabilizes a triangular skyrmion lattice (SkL) state1,5,6,7,8,9,10,11,12,13,14,15. On the other hand, a SkL has recently been discovered for centrosymmetric rare-earth-based compounds. In the latter systems, the symmetry of the SkL usually reflects that of the underlying crystal lattice, such as triangular SkL in hexagonal Gd2PdSi3 and Gd3Ru4Al12 and square SkL in tetragonal GdRu2Si216,17,18. Notably, these centrosymmetric materials commonly host extremely small skyrmions of diameter less than 5 nm, which is almost one order of magnitude smaller than the typical skyrmion diameter in conventional DM-based bulk magnets. For such systems, a distinct skyrmion formation mechanism mediated by itinerant electrons has been proposed19,20,21,22, and this theoretical framework successfully explains the magnetic phase diagram and the associated spin-charge coupling in GdRu2Si217,23. To date, the examples of skyrmion-hosting centrosymmetric magnets are limited to a few Gd-based intermetallics, and further search of simpler model compounds and better understanding of their intricate magnetic phase diagram are highly demanded.

In this work, we focus on a simple centrosymmetric binary compound EuAl4, and investigate its magnetic structure by means of neutron and X-ray scattering experiments. Unlike previously reported systems, EuAl4 turns out to host two distinctive topological magnetic phases, i.e., a square and a rhombic lattice state of skyrmions with a diameter of 3.5 nm, which is accompanied by the multiple-step reorientation of fundamental magnetic modulation vector as a function of magnetic field. The present results highlight EuAl4 as the simple model system to embody the unique itinerant-electron-mediated skyrmion formation mechanism, and suggest that a similar competition among multiple topological magnetic phases can be widely expected in other rare-earth-based intermetallic systems.

## Results

Our target material EuAl4 has the centrosymmetric tetragonal crystal structure, which belongs to the space group I4/mmm (No. 139) as shown in Fig. 1a24. This compound is reported to form a charge density wave characterized by the incommensurate ordering vector along the [001] axis below 145 K25 (See Supplementary Note X for the detailed discussion on the charge density wave and crystal symmetry). The magnetism is governed by the Eu2+ (S = 7/2, L = 0) ions with Heisenberg magnetic moment, and the square lattice layers of Eu2+ and nonmagnetic Al layers are alternately stacked along the [001] axis. According to the previous neutron scattering experiment under zero magnetic field, this compound hosts incommensurate magnetic orders below TN = 15.4 K26. It is characterized by the weak easy-axis magnetic anisotropy, and shows multiple-step metamagnetic transitions as sweeping the external magnetic field H || [001]24. Very recently, the non-monotonous H-dependence of Hall resistivity was reported, and its possible relevance to topological Hall effect was discussed27. At this stage, the magnetic structure for each phase in magnetic fields has not been identified experimentally.

First, we investigated the magnetic phase diagram of EuAl4 for H || [001]. Figure 1e–g shows the H-dependence of magnetization M, resistivity ρxx, and Hall resistivity ρyx measured at 4 K for H || [001] and I || [100], respectively. The magnetization curves show multiple-step metamagnetic transitions before reaching the forced ferromagnetic (FM) state, which is accompanied by non-monotonous changes in ρxx and ρyx. These behaviors are in agreement with previous reports24,27 (See Supplementary Note V for the detailed interpretation of ρyx profile). By performing similar measurements at selected temperatures, the HT (temperature) magnetic phase diagram is summarized in Fig. 1b. Below 8 K, four distinctive magnetic phases (i.e., phases I, II, III, and IV) are identified. The H-values for the anomalies observed in the M, ρxx and ρyx profiles coincide with each other, which suggests a strong coupling between the magnetism and the electrical transport properties.

Next, in order to identify the magnetic modulation vector in each phase, a series of small-angle neutron scattering (SANS) experiments for the (001) plane at 5.0 K in H || [001] of various strengths were performed. Here, both the direction of the incident neutron beam and that of the external magnetic field are parallel to the [001] direction, as shown in Fig. 1d. When the Fourier transform of magnetic structure contains the modulated spin component ($$\hat{{{{{{\bf{m}}}}}}}({{{{{\bf{Q}}}}}}){\exp }$$[iQ∙r]+c.c.) with $$\hat{{{{{{\bf{m}}}}}}}({{{{{\bf{Q}}}}}})$$ being a complex vector and c.c. representing complex conjugate, SANS is generally sensitive to the component of $$\hat{{{{{{\bf{m}}}}}}}({{{{{\bf{Q}}}}}})$$ normal to the magnetic modulation vector Q28. Figure 2a–d shows the typical SANS patterns obtained in phases I, II, III and IV, respectively. Here, we define θQ as the angle between the Q-vector and the [110] axis, and Q1 as the direction of fundamental magnetic modulation vector. In phase I at H = 0 (Fig. 2a), we observed magnetic reflections at Q = Q1 along the 〈100〉 directions (i.e. θQ ~ 45°). By increasing the magnetic field, the Q1-direction switches to θQ ~ ±5° upon entering phase II (Fig. 2b), and is further aligned along the 〈110〉 direction in phase III (i.e. θQ ~ 0°) (Fig. 2c). At higher fields, the Q1-direction is tilted back to θQ ~ ±5° in phase IV (Fig. 2d). Figure 3e, f exhibits the magnetic-field variations of the wavenumber |Q| and the azimuth angle θQ for the fundamental magnetic reflection Q = Q1, respectively. θQ takes distinctive values in each phase, demonstrating the multiple-step reorientation of the fundamental modulation vector in this compound. In phase I, |Q| is almost constant at around 0.28 Å-1, which corresponds to a modulation period of 2.2 nm. In phases II–IV, |Q| is around 0.18 Å-1 giving a modulation period of 3.5 nm. Note that the observed fundamental modulation vector Q1 is always incommensurate, i.e. Q1 ~ (0, 0.19, 0), (0.073, 0.097, 0), (0.083, 0.083, 0), and (0.070, 0.092, 0) in phases I, II, III, and IV, respectively. Here, reflecting the tetragonal symmetry of the underlying crystal lattice, multiple equivalent magnetic reflections are observed in the SANS pattern for each phase. In phase III, for instance, four fundamental magnetic reflections corresponding to Q1 = (q, q, 0), Q2 = (q, -q, 0) as well as -Q1 and -Q2 are identified (Fig. 2c). Such scattering patterns can originate from a multiple-Q spin order hosting multiple number of modulation vectors, or the spatial coexistence of symmetrically equivalent magnetic domains that are related by symmetry elements lost during the phase transition. One straightforward method to distinguish between these two possibilities is the detection of intensity at the Q1 + Q2 position, which should appear only in the multiple-Q state17,29,30. Remarkably, we have identified the clear Q1 + Q2 reflection in phases II and III (Fig. 2b, c), demonstrating that phases II and III are double-Q states. Figure 3g, h indicates the magnetic-field dependence of the wavenumber |Q| and the azimuth angle θQ for the Q1 + Q2 position. The experimentally observed |Q| and θQ values for the Q1 + Q2 reflections are consistent with the ones expected from the fundamental reflections (solid lines in Fig. 3g, h). As detailed in Supplementary Notes I, II, and III, similar Q1 + Q2 (as well as the associated Q1 – Q2) reflections have been identified in the resonant X-ray scattering experiments characterized by better momentum-space resolution. Note that in phase II, the Q1 and Q2 directions are not orthogonal to each other, and therefore two double-Q states related by the lost four-fold symmetry can coexist as detailed in Supplementary Note II. This explains the observed appearance of eight fundamental reflection spots in phase II (Fig. 2b). By considering a similar magnetic domain formation, the scattering patterns in phases I and IV can be also reasonably explained.

Figure 3c, d exhibits the magnetic-field variations of the SANS integrated intensities measured for Area 1 and Area 2, i.e., the boxed regions along the 〈100〉 and 〈110〉 directions as indicated in Fig. 3a, b, respectively. In phase I with Q1 || 〈100〉, the SANS intensity is observed only in Area 1. In phases II, III and IV, the SANS intensity emerges in Area 2, reflecting the reorientation of fundamental magnetic modulation vector Q1. In phases II and III, finite SANS intensity is also observed in Area 1, which corresponds to the Q1 + Q2 reflections originating from the double-Q nature of these phases.

To investigate the detailed spin orientations in each phase, we have further performed polarized SANS measurements with the experimental setup shown in Fig. 4a. Here, the neutron spin orientation Sn is aligned parallel or antiparallel to the incident neutron beam (|| [001]), and the intensities of spin-flip and non-spin-flip scattering (ISF and INSF, respectively) are measured separately (Fig. 4a). In this configuration, ISF reflects the component of $$\hat{{{{{{\bf{m}}}}}}}({{{{{\bf{Q}}}}}})$$ normal to both Sn || [001] and Q, and INSF reflects the component of $$\hat{{{{{{\bf{m}}}}}}}({{{{{\bf{Q}}}}}})$$ parallel to Sn || [001] (Fig. 4b)28. To access phases II, III and IV, temperature was swept with μ0H = 1 T applied along the [001] direction. In Fig. 4f–i, the line-scan profiles of ISF and INSF are plotted for the fundamental magnetic reflections in phases II and III. The corresponding line-scan direction in the reciprocal space is presented in Fig. 4d, e. Magnetic reflections are observed for both ISF and INSF channels, which proves that $$\hat{{{{{{\bf{m}}}}}}}({{{{{\bf{Q}}}}}})$$ possesses both in-plane and out-of-plane components normal to Q. This result suggests that $$\hat{{{{{{\bf{m}}}}}}}({{{{{\bf{Q}}}}}})$$ in phases II and III are characterized by the screw-type spin modulation (Fig. 4b), where their neighboring spins rotate within a plane normal to the magnetic modulation vector Q. By considering the double-Q nature of phases II and III, their spin textures m(r)= s(r)/|s(r)| should be approximately described as

$${{{{{\bf{s}}}}}}({{{{{\bf{r}}}}}})={{{{{{\bf{s}}}}}}}_{0}^{z}+\mathop{\sum}\limits_{i=1,2}\Big({{{{{{\bf{s}}}}}}}_{{{{{{{\bf{Q}}}}}}}_{i}}^{{xy}}{{\cos }}\left({{{{{{\bf{Q}}}}}}}_{i}\cdot {{{{{\bf{r}}}}}}\right)+{{{{{{\bf{s}}}}}}}_{{{{{{{\bf{Q}}}}}}}_{i}}^{z}{{\sin }}\left({{{{{{\bf{Q}}}}}}}_{i}\cdot {{{{{\bf{r}}}}}}\right)\Big)$$
(2)

which represents the superposition of screw spin helices characterized by two obliquely or orthogonally arranged magnetic modulation vectors Q1 and Q2, respectively. $${{{{{{\bf{s}}}}}}}_{0}^{z}$$ is the H-induced uniform magnetization component along the [001] axis, and $${{{{{{\bf{s}}}}}}}_{{{{{{{\bf{Q}}}}}}}_{i}}^{{xy}}$$ ($${{{{{{\bf{s}}}}}}}_{{{{{{{\bf{Q}}}}}}}_{i}}^{z}$$) represents the modulated spin component normal to both magnetic modulation vector Qi and the [001] axis (parallel to the [001] axis). The resultant spin textures for phases II and III are illustrated in Fig. 2f, g, which can be considered as the rhombic and square lattice of magnetic skyrmions, respectively, according to Eq. (1). In Fig. 4c, the temperature dependence of ISF and INSF measured at 1 T is summarized. In phase IV above 11 K, the magnetic scattering appears only in the ISF channel but not in the INSF channel. It suggests that $$\hat{{{{{{\bf{m}}}}}}}({{{{{\bf{Q}}}}}})$$ in phase IV consists only of the in-plane component normal to Q. A similar conclusion is also obtained from the resonant elastic X-ray scattering results for phase IV, as discussed in Supplementary Note I. Note that additional neutron scattering data in Supplementary Fig. 8 indicate the existence of very weak but discernible Q1 + Q2 reflection in phase IV, which suggests the formation of double-Q vortex-lattice spin state described by the superposition of two sinusoidally-modulated in-plane spin components (Fig. 2h). For the detailed discussion on the spin texture for phase IV, see Supplementary Note VII and VIII.

## Discussion

In the following, we discuss the microscopic origin for the appearance of square and rhombic SkL states in EuAl4. Since the crystal structure of EuAl4 is centrosymmetric, the contribution of DM interaction should be absent and some different mechanism must be considered. According to the recent theoretical studies19,20,21,22, the SkL formation in centrosymmetric rare-earth compounds are approximately explained by the combination of Ruderman-Kiettel-Kasuya-Yosida (RKKY) interaction $$\hat{{{{J}}}}\left({{{{{\bf{Q}}}}}}\right)[\hat{{{{{{\bf{m}}}}}}}\left({{{{{\bf{Q}}}}}}\right)\cdot \hat{{{{{{\bf{m}}}}}}}\left({{{{{\boldsymbol{-}}}}}}{{{{{\bf{Q}}}}}}\right)]$$ and four-spin interaction $$\hat{K}\left({{{{{\bf{Q}}}}}}\right){[\hat{{{{{{\bf{m}}}}}}}\left({{{{{\bf{Q}}}}}}\right)\cdot \hat{{{{{{\bf{m}}}}}}}(-{{{{{\bf{Q}}}}}})]}^{2}$$, which are derived from the Kondo lattice model consisting of localized magnetic moments and itinerant electrons. Here, the former RKKY term stabilizes the magnetic modulation with a specific wave vector, and the latter four-spin interaction term enhances the stability of the multiple-Q orders. This model well explains the recently reported square SkL formation in a centrosymmetric tetragonal magnet GdRu2Si217,23, and similar mechanism would be also relevant for EuAl4 having the same tetragonal crystal structure.

Compared to the known skyrmion-hosting materials, the present EuAl4 is characterized by several unique features. First, EuAl4 shows multiple-step reorientation of the fundamental magnetic modulation vector as a function of external magnetic field, which leads to the appearance of not only square, but also rhombic form of SkL states. These behaviors are distinctive from the previously reported centrosymmetric Gd-compounds16,17,18, where the magnetic modulation vector is always fixed along the one specific direction and the associated SkL reflects the symmetry of underlying crystal lattice. (For example, only the square SkL appears for tetragonal GdRu2Si2 with Q ||〈100〉17). In general, the orientation and length of stable magnetic modulation vector are determined by the peak position in $$\hat{J}\left({{{{{\bf{Q}}}}}}\right)$$ reflecting the character of the associated Fermi surfaces or electronic structure. The presently observed multiple-step Q-reorientation in EuAl4 implies that almost comparable amplitude of $$\hat{J}\left({{{{{\bf{Q}}}}}}\right)$$ peaks exist for several Q-positions. In Supplementary Note IX and Supplementary Fig. 10, we performed theoretical simulation based on this assumption, which turned out to well reproduce the observed H-induced transition between rhombic and square SkL states. This suggests that the symmetry of SkL sensitively depends on the electronic structure20 and potentially can be controlled by chemical substitution or external stimuli, which would be a unique feature of itinerant-electron-mediated mechanism compared with the traditional DM-based mechanism.

Second, EuAl4 is the first example of centrosymmetric binary compound to host skyrmion spin texture, which just consists of the alternate stacking of magnetic Eu layer and nonmagnetic Al layer (Fig. 1a). The interactions among the Eu2+ localized magnetic moments are mediated by itinerant electrons, whose Fermi surfaces are mostly governed by Al according to the recent ARPES (angle-resolved photoemission spectroscopy) measurements31. The above features highlight EuAl4 as the simple model system to embody the unique skyrmion formation mechanisms mediated by itinerant electrons. It is remarkable that even such a minimum centrosymmetric binary compound can host a variety of distinctive skyrmion orders with intricate multiple-step Q-reorientation. The present experimental results demonstrate that rare-earth intermetallics with delicate balance of magnetic interactions19,20,21,22,32,33,34,35,36 can be a promising platform to realize/control the competition of multiple topological magnetic phases in a single compound, which would be a good guideline for further search of exotic materials with emergent functional responses.

## Methods

### Sample preparation and characterization

Single crystals of EuAl4 were grown by the Al self-flux method. The samples were characterized by powder X-ray diffraction, which confirmed the purity of the grown crystals. Crystal orientations were determined by Laue X-ray diffraction, and then the samples were cut out from the crystal with a wire saw and carefully mechanically polished.

### Magnetization and resistivity measurements

Magnetization was measured using a SQUID magnetometer (Magnetic Property Measurement System, Quantum Design). The longitudinal (ρxx) and Hall (ρyx) resistivities were measured using the ac-transport option in a physical property measurement system (PPMS, Quantum Design). Measurements of magnetic and electrical transport properties were performed on the same crystal with a dimension of 1 mm by 0.9 mm by 0.5 mm.

### SANS measurements

SANS measurements were carried out at the time-of-flight type small-and-wide-angle neutron scattering instrument TAIKAN (BL15) at Materials and Life Science Experimental Facility (MLF) in Japan Proton Accelerator Research Complex (J-PARC)37. A thin crystal with a thickness of 0.2 mm having the widest surface of 5 mm by 2.5 mm parallel to the (001) plane was installed in a cryostat equipped with a horizontal-field superconducting magnet. An incident neutron beam with a wavelength ranging from 0.7 to 7.7 Å was exposed on the sample. The incident beam was always directed along the [001] axis within the accuracy of 2 degrees. The typical measuring time for a SANS pattern was 200 s. The SANS patterns (Fig. 2) were measured with rotating and tilting the sample up to 2 degrees. Background data were obtained at 15.5 K above Tc in a magnetic field of 0.05 T, and subtracted from the data obtained at low T to leave only the magnetic signal. As for the magnetic-field scans, SANS signals were collected with continuously increasing magnetic field at a sweeping rate of 0.01 T/min.

In the SANS experiments with longitudinal polarization analysis, a polarized incident neutron beam was obtained by a supermirror polarizer. The neutron spin polarization at the sample position was controlled to be parallel to the [001] axis of the sample by guide fields and the horizontal-field superconducting magnet, which applied a magnetic field of 1.0 T. A supermirror spin analyzer was set between the sample and the detectors, and was used to separate the spin-flip (SF) and non-spin-flip (NSF) scattering signals. Total beam polarization measured using a direct beam was 0.85 for the neutrons with the wavelengths longer than 3.5 Å. The incident and scattering angles for the fundamental magnetic Bragg reflections near the (H,H,0) line were tuned so that the wavelengths for the Bragg reflections were approximately 3.5 Å. The mixing of the SF and NSF signals was corrected taking into account of the total beam polarization.

### X-ray measurements

Resonant elastic X-ray scattering experiment was carried out using circularly polarized X-rays of an incoming photon energy tuned approximately to the Eu L2 absorption edge (7617 eV) in Laue-transmission geometry on beamline P09 at the PETRA-III synchrotron, DESY. A (001)-oriented thin plate of EuAl4 with a thickness of 10 μm and an area of 600 μm by 300 μm was attached to a Si3N4 substrate (almost X-ray transparent). The substrate was attached to a sample holder mounted at the bottom of a long probe and inserted into the variable temperature insert (VTI). The X-ray beam was focused to a spot size of ~70 × 200 μm2. Diffraction patterns were recorded by a two-dimensional detector PILATUS300K by rocking the sample. A magnetic field was applied parallel to the incident X-ray beam and perpendicular to the thin plate (|| [001]) by a cryogen-free vector magnet. Background data were measured at 14.5 K where no magnetic scattering was observed.