Rhombic skyrmion lattice coupled with orthorhombic structural distortion in EuAl$_{4}$

The centrosymmetric tetragonal itinerant magnet EuAl$_{4}$ exhibits an intricate magnetic phase diagram including rhombic and square skyrmion-lattice (SkL) phases in the external magnetic field. Here, we report a dilatometric investigation on EuAl$_{4}$ for three principal axes by means of the fiber-Bragg-grating technique, revealing anisotropic magnetostriction and magnetovolume effect associated with successive phase transitions. The rhombic and square SkL phases are found to possess $\sim$0.10$\%$ and $\sim$0.03$\%$ orthorhombic structural distortion within the $ab$ plane, respectively. We propose that the coupling between the spin system and the lattice deformation should be essential for the structural instability in EuAl$_{4}$, yielding a rich variety of topological spin textures with spontaneous rotational-symmetry breaking as well as a potential controllability of the SkL phases by uniaxial stress or pressure.

Spontaneous rotational-symmetry breaking (SRSB) has long been a central issue in condensed-matter physics. A prominent recent topic is the electronic nematicity observed in heavy-fermion compounds [1], topological kagome metals [2], and copper/iron-based high-T c superconductors [3,4], where the contribution of electron-phonon coupling has often been controversial. The SRSB is more widely seen with the onset of magnetic long-range order. In the presence of magnetic frustration, the mutual coupling of spin and lattice degrees of freedom can induce a magnetostructural transition with crystal symmetry lowering [5], as observed in solid oxygen [6,7] and pyrochlore-based antiferromagnets [8,9].
Recently, the SRSB of SkL was discovered in a binary intermetallic EuAl 4 [30], which crystallizes in a centrosymmetric tetragonal structure (space group I4/mmm) with a square lattice of localized spin-7/2 Eu 2+ ions. Figure 1(a) shows the magnetic-field-versus-temperature (H-T ) phase diagram for H [001], where seven magnetic phases (I∼VII) appear as well as the paramagnetic (PM) and the forced ferromagnetic (FM) phases [30,40]. In zero magnetic field, four magnetic transitions from the PM phase to phases VII, VI, V, and I take place at T N1 = 15.4 K, T N2 = 13.2 K, T N3 = 12.2 K, and T N4 = 10.0 K, respectively [30,[40][41][42][43][44][45]. The former two are of the second order, whereas the latter two are of the first order. A single crystal x-ray diffraction study revealed a tetragonalto-orthorhombic structural transition with the B 1g -type distortion at T N3 [42]. When applying a magnetic field along [001], successive metamagnetic transitions from phase I to II, III, and IV take place below 5 K. Small-angle neutron and resonant x-ray scattering [30] unveiled the magnetic structures of all these phases; phases I and V are single-Q screw spiral with Q 1 = (0.19, 0, 0) and (0.17, 0, 0), respectively, II and III are double-Q SkL, IV and VII are double-Q vortex-antivortex lattice (VL), and VI is double-Q meron-antimeron lattice (ML). Two fundamental modulation vectors in phases III, VI, and VII are Q 1 = (q, q, 0) and Q 2 = (q, −q, 0), where q ∼ 0.085, so that the spin textures possess four-fold symmetry. In phases II and IV, on the other hand, Q 1 and Q 2 are tilted by θ q ∼ 5 • toward the [100] direction, resulting in the rhombic SkL and VL, respectively [ Fig. 1(b)]. In short, the SRSB is observed even in the double-Q states (II and IV) as well as in the single-Q states (I and V).
Theoretically, a rectangular SkL with four-fold symmetry breaking can appear on a square-lattice spin model by incorporating the competition among exchange interactions in momentum space J Q with Q 1 = (q, q, 0), Q 2 = (q, −q, 0), and   their higher harmonics, Q 1 + Q 2 and Q 1 − Q 2 [46,47], though this picture is incompatible with the rhombic SkL in EuAl 4 .
Recalling the structural transition at T N3 , it is natural to anticipate that the spin-lattice coupling is responsible for stabilizing the rhombic double-Q modulation in phases II and IV. To shed light on this, it is important to clarity the in-field crystalstructure changes, which cannot be sensitively detected by x-ray or neutron studies. Dilatometry using the capacitance method has sufficient sensitivity, and a previous study [40] succeeded in observing the relative change in the lattice constant a on a detwinned crystal of EuAl 4 while the information on the lattice constants b and c was lacking (we define a < b below T N3 ). Particularly, measurements of both a and b are essential to capture the magnitude of the B 1g -type distortion In this work, we investigate the lattice constant changes for three principal axes associated with the field-induced phase transitions in EuAl 4 using the fiber-Bragg-grating (FBG) technique, unveiling anisotropic magnetostriction and magnetovolume effect. Figure 1(c) shows a contour plot of ε mapped on the H-T phase diagram: e.g., ∼0.10% and ∼0.03% structural distortion is found in the rhombic and square SkL phases, respectively. Based on the microscopic consideration of a spin model, we argue that the spin-lattice coupling should act as the principal driving force for the multiple magnetostructural Ref. [42] Bragg grating [100] [001] transitions and the SRSB of SkL in EuAl 4 . The relative sample-length changes ∆L/L 0 along [100] and [001] were simultaneously measured, as illustrated in Fig. 2(a) (for experimental details, see Supplemental Material [48]). Throughout the paper, we define ∆k (k = L, a, b, c) as ∆k ≡ k − k 0 with the baseline value k 0 at 0 T and 18 K and assume a 0 = b 0 . The zero-field thermal expansion profiles measured along [100], (∆L/L 0 ) 1 and (∆L/L 0 ) 2 , which were taken under different sample settings, are shown by cyan lines in Fig. 2(b), along with the temperature evolution of ∆a/a 0 and ∆b/b 0 revealed by the previous x-ray diffraction study [42] (open circles). Opposite behaviors are observed between (∆L/L 0 ) 1 and (∆L/L 0 ) 2 below T N3 ; a positive thermal expansion for (∆L/L 0 ) 1 , while negative for (∆L/L 0 ) 2 . This discrepancy should be attributed to the difference in the magnetic domain patterns around the local area where the FBG was glued in each setting [ Fig. 2(c)]. Assuming (∆L/L 0 ) i = (1 − p i )(∆a/a 0 ) + p i (∆b/b 0 ), we determine p 1 = 0.49 for (∆L/L 0 ) 1 and p 2 = 0.74 for (∆L/L 0 ) 2 so that they match the x-ray data at 6.3 K. Importantly, p i is reproduced no matter how many times temperature or magnetic field is re-peatedly scanned in our experiments (Fig. S2 [48]). Thanks to this feature, we can decompose ∆a/a 0 and ∆b/b 0 from the two experimental data sets (∆L/L 0 ) 1 and (∆L/L 0 ) 2 in the whole measured H-T region using the following relations: ∆a/a 0 = [p 2 (∆L/L 0 ) 1 − p 1 (∆L/L 0 ) 2 ]/(p 2 − p 1 ) and The calculated ∆a/a 0 and ∆b/b 0 as a function of temperature are shown by orange and green lines in Fig. 2(b), respectively, which agree well with the x-ray data.
As shown in Fig. 2(d), the observed ∆L/L 0 along [001], which corresponds to ∆c/c 0 , exhibits negative thermal expansion below T N1 . The entire change in ∆c/c 0 across the phase transitions is much smaller than those of ∆a/a 0 and ∆b/b 0 . This trend is in contrast to GdRu 2 Si 2 [53] and suggests the weak out-of-plane spin-lattice coupling in EuAl 4 . We double check the reliability of our measurements by estimating thermodynamic quantities. Figure 2(e) shows the volume thermal expansion calculated as ∆V/V 0 = ∆a/a 0 + ∆b/b 0 + ∆c/c 0 and its temperature derivative α ≡ ∂(∆V/V 0 )/∂T . α jumps by ∆α ≈ −1.5 × 10 −4 K −1 at T N1 . Adopting this value in combination with the reported specific heat change ∆C p ≈ −6 J/(K·mol) at T N1 [40,41] and the volume V ≈ 107.1 Å 3 /f.u. at 20 K [54] to the Ehrenfest relation ∂T N /∂p = T N V(∆α/∆C p ), the pressure p dependence of T N1 is estimated to 2.5 K/GPa. This estimation agrees well with the previously obtained value ∂T N1 /∂p = 2.24 K/GPa [41].
Having confirmed the validity of our experimental and analytical methods from the thermal expansion data, we investigate the field-induced crystal-structure changes of EuAl 4 . Figures 3(a)-3(c) show magnetostriction curves for H [001] measured at various temperatures. We obtain the field evolution of ∆a/a 0 and ∆b/b 0 by the same procedure described above (Fig. S1 [48]). As the magnetic field increases at 2 K, ∆a/a 0 monotonically increases accompanied by jumps at firstorder transitions at H c1 , H c2 , and H c3 in accord with Ref. [40]. In contrast, ∆b/b 0 and ∆c/c 0 monotonically decrease, indicating that the structural distortion associated with the magnetic ordering is gradually suppressed toward higher fields. Furthermore, we performed similar sets of thermal expansion and magnetostriction measurements for ∆L/L 0 [100] on a crystal nearly detwinned by applying the thermal stress (Fig. S3  [48]); the observed ∆L/L 0 is closed to ∆b/b 0 and reasonably consistent with Figs. 2(b) and 3(a), while a uniaxial stress effect is seen in a shift of structural transition temperatures T N3 and T N4 by more than 1 K. The orthorhombic distortion ε calculated by ∆b/b 0 − ∆a/a 0 is visualized on the H-T phase diagram in Fig. 1(c). Note that ε may be slightly underestimated in high-field and high-temperature sides due to the missing incorporation of possible slight orthorhombic distortion at 0 T and 18 K [54] in our analysis (for details, see Supplemental Material [48]). The connection between the spin textures and orthorhombic structural distortion will be discussed later.
Our magnetostriction data provide insights on the pressure effect on the stability of the rhombic and square SkL phases. the demagnetization is corrected so as to accurately estimate the magnitudes of the magnetization jumps at H c1 , H c2 , and H c3 : ∆M 1 ≈ 1.6 µ B /f.u., ∆M 2 ≈ 0.3 µ B /f.u., and ∆M 3 ≈ 0.6 µ B /f.u., respectively. The corresponding volume jumps are (∆V/V 0 ) 1 ≈ 1.2×10 −4 , (∆V/V 0 ) 2 ≈ 0.7×10 −4 , and (∆V/V 0 ) 3 ≈ 0.5 × 10 −4 , respectively. According to Clausius-Clapeyron's equation ∂(µ 0 H c )/∂p = ∆V/∆M, the pressure dependence of each critical field is estimated to ∂(µ 0 H c1 )/∂p ≈ 0.8 T/GPa, ∂(µ 0 H c2 )/∂p ≈ 2.6 T/GPa, and ∂(µ 0 H c3 )/∂p ≈ 1.0 T/GPa. A relatively large value of ∂(µ 0 H c2 )/∂p is attributed to the substantial volume expansion at the transition to phase III in spite of the small magnetization jump. The predicted H-p phase diagram is shown in the inset of Fig. 3(e), suggesting that the square SkL can be annihilated by applying hydrostatic pressure lower than 0.1 GPa. The stability of each SkL phase may also be controllable by tuning the chemical pressure such as the isovalent Ga substitution for Al [55]. Indeed, the H-T phase diagrams of EuGa 4 and EuGa 2 Al 2 are simpler than that of EuAl 4 [56,57], presumably suggesting the absence of orthorhombic structural distortion and the rhombic SkL phase in these compounds. A systematic investigation on the phase diagram of Eu(Ga 1−x Al x ) 4 system would be intriguing.
To understand the magnetic instability in EuAl 4 , we start from the classical Heisenberg model on a simple square lattice: H = i, j k J k S i · S j + H ex , where J 1 < 0, J 2 > 0, and J 3 > 0 are the exchange interactions up to the third nearest neighbor (NN) [ Fig. 4(a)] and H ex includes extra terms. This model can host various kinds of single-Q and multiple-Q spin textures depending on H ex ; e.g., the double-Q ML is stabilized in the presence of the single-ion anisotropy and compass anisotropy [58], which would be relevant to phase VI in EuAl 4 [30]. If we neglect H ex , incommensurate ordering vectors Q = (±q, ±q, 0) with q = arccos(− J 1 2J 2 +4J 3 ) are selected for 2J 3 > J 2 whereas Q = (±q, 0, 0) or (0, ±q, 0) with q = arccos(− J 1 +2J 2 4J 3 ) for J 2 > 2J 3 [58]. The exchange parameter sets that agree with the Q vectors for the zero-field phases VI, V, and I [30,44] are plotted in a parameter space of J 2 /|J 1 | and J 3 /|J 1 | in Fig. 4(b). As phases VI, V, and I should compete with each other within a small energy scale, (J 2 /|J 1 |, J 3 /|J 1 |) ≈ (0.3, 0.15) is a reasonable parameter position for EuAl 4 . We note that the Q switching at T N3 and T N4 cannot occur within the frozen J 1 -J 2 -J 3 model.
We here propose the spin-lattice coupling as a driving force to modify the J's through a magnetostructural transition. This mechanism is reasonable to consider in EuAl 4 on the basis of the observed exceptionally large thermal expansion and magnetostriction associated with the magnetic transitions (∼10 −3 ) compared to those in other SkL-hosting Eu/Gd-based itinerant magnets (10 −4 ∼10 −5 ) [53,59,60]. This collective phenomenon is known as the spin Jahn-Teller effect, where the magnetic frustration is relieved by favoring one of the competing exchange interactions through the lattice distortion [5]. The B 1g -type distortion is selected for the transition from phase VI to V at T N3 [42], indicating that the FM J 1 along the a and b axes become inequivalent: J 1a ≈ J 1 [1 + (dJ 1 /dr)∆a] and J 1b ≈ J 1 [1 + (dJ 1 /dr)∆b], respectively [ Fig. 4(c)]. If dJ 1 /dr > 0, J 1b is favored whereas J 1a is disfavored, so that the Q vector should be oriented along the a axis [61]. This picture is compatible with the Q switching from (0.085, 0.085, 0) to (0.17, 0, 0) at T N3 because the total exchange energy can be reduced for the latter. The additional first-order transition from phase V to I at T N4 , which accompanies a modulationperiod change from q = 0.17 to 0.19 while conserving the orientation of the Q vector, suggests the competition of these two spiral states in the presence of the spin-lattice coupling; phase I is eventually stabilized as a ground state by enhancing the orthorhombic structural distortion ε ∼ 1.6 × 10 −3 and consequently increasing J 2 /|J 1a | (or J 2 /|J 1b |). We note that a uniaxial stress should facilitate the magnetostructural transitions at T N3 and T N4 because the system can save the elastic energy, which is indeed observed as mentioned above (Fig. S3  [48]).
The importance of the spin-lattice coupling is also corroborated from the strong correlation between the in-field spin textures and orthorhombic structural distortion. In the rhombic SkL phase (II), two fundamental Q vectors are tilted from Q 1 = (q, q, 0) and Q 2 = (q, −q, 0) (q ∼ 0.085) by θ q ∼ 5 • [30]. As can be seen from Fig. 1(c), a large structural distortion ε ∼ 1.0 × 10 −3 exists in phase II like in phase V, suggesting the importance of the spin-lattice coupling for the rhombic Q modulation. A discontinuous drop of θ q to zero at the SkL-SkL transition from phase II to III [30] is also a remarkable feature supporting this scenario. The fundamental modulations Q 1 = (0.083, 0.083, 0) and Q 2 = (0.083, −0.083, 0) in the square SkL phase (III) are almost identical with those in the zero-field phase VI [30,44]. However, a moderate structural distortion ε ∼ 3 × 10 −4 is found in phase III [ Fig. 1(c)], indicating that J 1a and J 1b remain inequivalent. Such a deviation from the tetragonal symmetry in phase III was not observed in the previous quantum beam experiments potentially due to insufficient resolution [30]. Interestingly, a reentrant symmetry breaking of the spin texture is seen in the highfield rhombic VL phase (IV) [30], though the orthorhombic structural distortion seems monotonically suppressed toward higher fields [ Fig. 1(c)]. The reason why the VL is more prone to deformation (in terms of spin textures) than the SkL is elusive at this stage. A theoretical framework incorporating local phonon modes or inequivalent J 1a and J 1b on the square lattice would be a promising approach to reproduce versatile magnetic phases in EuAl 4 , which is left for future work.
In summary, we have comprehensively revealed the crystalstructure changes of EuAl 4 associated with the field-induced phase transitions to address the microscopic origin of the SRSB of the SkL. The amplitudes of the orthorhombic structural distortion are quantitatively estimated for each magnetic phase. The appearance of two types of SkL phases should originate from the magnetic frustration in momentum space coupled with the lattice degrees of freedom. Furthermore, the orthorhombic structural distortion accompanies a pronounced magnetovolume effect. EuAl 4 would be an ideal playground to explore the tunability of the SkL phases by pressure as well as uniaxial stress.
The authors appreciate S. Hayami and H. Yoshimochi for fruitful discussions. The authors appreciate S. Shimomura for providing the data of the lattice constants of EuAl 4 in Ref. [42]. This work was financially supported by the JSPS KAKENHI Grants-In-Aid for Scientific Research (No.  Figure S1 shows the raw data of magnetostriction curves for H [001] measured along [100] at various temperatures: (a) (∆L/L 0 ) 1 for setting 1 and (b) (∆L/L 0 ) 2 for setting 2. As discussed in the main text, (∆L/L 0 ) 1 and (∆L/L 0 ) 2 were taken under different sample adhesion conditions. By using the relations ∆a/a 0 = [p 2 (∆L/L 0 ) 1 − p 1 (∆L/L 0 ) 2 ]/(p 2 − p 1 ) and ∆b/b 0 = [−(1 − p 2 )(∆L/L 0 ) 1 + (1 − p 1 )(∆L/L 0 ) 2 ]/(p 2 − p 1 ) with p 1 = 0.49 and p 2 = 0.74, the field evolutions of ∆a/a 0 and ∆b/b 0 at the corresponding temperatures are obtained as shown in Figs. 3(a) and 3(b) in the main text. III. ADDITIONAL EXPERIMENTAL DATA FOR SETTINGS 1 AND 2 Figure S2(a) shows thermal expansion curves measured along [100] at constant magnetic fields for setting 1. These data were obtained after a series of magnetostriction measurements shown in Fig. S1(a). By comparing Figs. S1(a) and S2(a), it can be seen that ∆L/L 0 has the same value under certain temperature and magnetic field regardless of the sweep process. This ensures that the volume ratio of the magnetic domains never changes with scanning temperature as well as magnetic field. Figure S2(b) shows magnetostriction curves measured along [100] at various temperatures for setting 2. These data were obtained on a different day (i.e., the probe was once taken out from the superconducting magnet and inserted again) without changing the sample setting in which the data shown in Fig. S2(b) were obtained, but they were surprisingly reproducible.