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[Hiroaki Hayashi](https://orcid.org/0000-0001-7787-9082), Moyu Kato, [Taichi Terashima](https://orcid.org/0000-0001-9239-0621), [Naoki Kikugawa](https://orcid.org/0000-0003-3975-4478), [Hiroya Sakurai](https://orcid.org/0000-0003-1964-6023), Hiroyuki K. Yoshida, [Kazunari Yamaura](https://orcid.org/0000-0003-0390-8244)

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[Exploring the Magnetic Phase Diagram and Hall Resistivity Suppression in Centrosymmetric GdOs<sub>2</sub>Si<sub>2</sub> Single Crystal](https://mdr.nims.go.jp/datasets/50732d10-5a9f-4071-8ad1-39f5cebfbb71)

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1 Exploring the Magnetic Phase Diagram and Hall Resistivity Suppression in Centrosymmetric GdOs2Si2 Single Crystal  Hiroaki Hayashi,1,2,* Moyu Kato,3 Taichi Terashima,1 Naoki Kikugawa,4 Hiroya Sakurai,1  Hiroyuki K. Yoshida,3 Kazunari Yamaura 1,2  1 Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan 2 Graduate School of Chemical Sciences and Engineering, Hokkaido University, North 10 West 8, Kita-ku, Sapporo, Hokkaido 060-0810, Japan 3 Department of Physics, Hokkaido University, North 10 West 8, Kita-ku, Sapporo, Hokkaido 060-0810, Japan 4 Research Center for Energy and Environmental Materials, National Institute for Materials Science, 3-13 Sakura, Tsukuba, Ibaraki 305-0003, Japan   Magnetic skyrmions were initially identified in non-centrosymmetric materials. However, recent theoretical advancements have suggested their potential emergence in centrosymmetric compounds, which has been experimentally observed in cases such as GdRu2Si2. In this study, we comprehensively explored the magnetic properties of GdOs2Si2, a centrosymmetric counterpart to GdRu2Si2, with a focused investigation into the crystal direction-dependent behavior of key parameters. By combining magnetic, thermal, and transport measurements, we constructed a detailed magnetic phase diagram along two crystal directions, revealing intriguing similarities and differences compared to GdRu2Si2. Notably, we observed a distinct suppression of the Hall resistivity in GdOs2Si2—a phenomenon absent in its counterpart compound.   1. Introduction   Initially, magnetic skyrmions were observed only in materials without crystallographic inversion centers. Examples include MnSi, EuPtSi, and FeGe, where the Dzyaloshinski-Moriya interaction plays a pivotal role in magnetism [1–5]. However, theoretical advancements have proposed the possibility of skyrmion formation even in compounds with centrosymmetry [6–11]. For instance, in J1-J2(J3) triangular-lattice antiferromagnets, geometrical frustration was predicted to lead to the emergence of skyrmion lattices, despite these systems being insulating, 2 unlike their predecessors [12]. Additionally, skyrmions observed in the Gd3+ triangular lattices in Gd2PdSi3 [13–16] and the breathing kagome networks in Gd3Ru4Al12 [17–19] further support the influence of this magnetic frustration.  More recently, it has been suggested that the skyrmion lattice may also manifest in non-frustrated lattices with centrosymmetry and metallic attributes. The interplay between the Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction and multiple spin interaction is considered crucial in this context. This possibility was demonstrated in studies of the tetragonal compound GdRu2Si2 [20–25]. These findings suggest the ubiquitous presence of skyrmions across a variety of material systems.  Another important aspect of the skyrmion lattice is its characteristic transport properties. In particular, the topological Hall effect manifests itself in connection with the skyrmion lattice [1,26–28] due to the scalar spin chirality Si‧(Sj×Sk), which acts like a fictitious magnetic field. However, only a few metallic compounds with spatial inversion symmetry currently studied show the existence of a skyrmion lattice [16,19–21,29]. Among them, GdRu2Si2 and EuAl4 have attracted great attention due to their multi-Q-skirmion lattice and complex magnetic phase diagram [29,30].  In this study, we shift our focus to GdOs2Si2 [31,32], an osmium-substituted variant of GdRu2Si2. While the magnetic properties of GdOs2Si2 remain unresolved, it shares the centrosymmetry and layered structure stacking along the c-axis with GdRu2Si2. Given the slight variations in lattice constants and energy levels in the 4d and 5d electron bands between GdOs2Si2 and GdRu2Si2, one can reasonably anticipate differences in the nature of the Gd-spin interaction in these two compounds. Consequently, the magnetic phase diagram of GdOs2Si2 may also exhibit some alterations. We have meticulously summarized our measurements of GdOs2Si2 single crystals, including electrical resistivity, magnetization, and specific heat capacity. These results were compiled into a temperature-field phase diagram and juxtaposed with the phase diagram obtained for GdRu2Si2. The results show that there is a new magnetic phase in GdOs2Si2, which has not been observed before, and that the Hall resistivity is significantly suppressed in this phase.   2. Experimental  The crystal growth process of GdOs2Si2 was meticulously documented in a conference proceeding [31]. For the purposes of our study, we selected two crystals from the aforementioned batch, one of which is visually presented in Fig. S1 [33]. To definitively 3 establish the crystal’s identity as GdOs2Si2, characterized by a ThCr2Si2-type structure with body-centered tetragonal symmetry (I4/mmm, #139) [31], an in-depth Laue X-ray diffraction analysis was conducted. The determination of crystal axes was carried out employing the same Laue method. Subsequently, the resultant lattice constants were determined to be a = 4.1549(2) Å and c = 9.8117(8) Å. These values exhibit a moderate decrease in a and a substantial increase in c when compared to the dimensions observed in GdRu2Si2 [a = 4.164(2) Å, c = 9.616(5) Å] [34].    We conducted measurements of magnetic, thermal, and transport properties across a temperature range spanning from 2 K to 300 K, as well as under varying magnetic fields (H), utilizing a single crystal sample. Initially, a slender plate-shaped crystal with approximate dimensions of 1.1 mm × 0.50 mm × 0.23 mm was chosen for the study. Additionally, to explore sample dependence, another crystal underwent similar investigations. A SQUID magnetometer, specifically the MPMS3 model from Quantum Design Inc., was utilized to collect magnetic data. The temperature-dependent magnetization (M) was recorded under field cooling conditions, and the magnetic susceptibility was calculated as χ = M/H. To determine the specific heat capacity (Cp), the heat relaxation method was employed, with a magnetic field applied parallel to the c-axis. This measurement setup employed a commercially available system, specifically the PPMS model from Quantum Design Inc.   For the assessment of longitudinal resistivity (ρxx) and Hall resistivity (ρyx) (equal to -ρxy), a magnetic field parallel to the c-axis was applied using the transport option of the PPMS system. To secure the gold wires onto the crystal, silver paste was utilized, with the distance between the voltage terminals of the crystal set at 0.35 mm for ρxx and 0.38 mm for ρyx for the first crystal. Throughout these measurements, an excitation current of 20 mA was passed along the crystal’s [100] direction.   3. Results and Discussion  Figure 1(a) shows the temperature dependence of χ curves (left) and their differentials dχ/dT curves (right) measured at various magnetic fields below 40 K. For the case of H//[001], distinct features labeled as T1 (arrow), T2 (triangle), and T3 (diamond) were observed at different temperatures, corresponding to the peak positions in the dχ/dT curves. While the change at T1 seems relatively smaller compared to the more pronounced peak-like changes at T2 and T3 in the differential curves, it becomes evident from the data at μ0H = 2.4 T and 3 T that T1 and T2 are indeed clearly separated at elevated magnetic fields. T2 displays a more pronounced 4 decrease than T1 as the magnetic field increases from 25 K (at μ0H = 1 T) to 6 K (at 3 T). Moreover, an anomaly emerges around 13 K at μ0H = 2 T (T3), which fades as the magnetic field is further increased.   Regarding the situation where H//[100], at least two distinct changes were identified, as portrayed in Fig. 1(b). Although the overall profile of alterations appears to display a more gradual and uniform trend, qualitatively similar associations with temperature and magnetic field can be observed. For the sake of convenience, we label the temperatures corresponding to these distinctive changes as T4 and T5.   The upper portion of Fig. 2(a) shows the isothermal magnetization curve and its derivative curve obtained at 1.8 K while applying a magnetic field perpendicular to the crystal’s c-plane (H//[001]). Within this depiction, the magnetization process demonstrates swift alterations at specific magnetic field strengths: μ0H1 = 2.5 T, μ0H2 = 2.8 T, and μ0H3 = 3.2 T, along with a gradual kink manifesting at μ0H4 = 6.5 T. These variations manifest more prominently within the differential curves. Notably, the transformations at H1, H2, and H3 exhibit hysteresis in response to the magnetic field, underscoring their connection with first-order phase transitions. Conversely, the absence of noticeable hysteresis is evident at H4, where the magnetization of Gd3+ ions achieves saturation at a value of 7 μB.  The outcomes of the measurements capturing magnetic transitions driven by varying temperatures and magnetic fields are presented in the lower section of Fig. 2(a). Notably, all transition fields—designated as H1, H2, H3, and H4—shift towards lower magnetic field strengths as the temperature increases. Specifically, H3 diminishes around 10 K, while H1 either merges with H2 or vanishes entirely at approximately 20 K. As for H4, it becomes progressively less distinguishable at elevated temperatures and is indiscernible beyond 20 K. This behavior can be attributed to the fact that, as elucidated later, the phase boundaries above 20 K align nearly perpendicular to the temperature axis.  At the upper section of Fig. 2(b), we present the M-H and dM/dH-H curves at 1.8 K, considering an external magnetic field aligned with H//[100]. When examining the H//[100] configuration, distinct anomalies are observed in the dM/dH curve at specific magnetic fields: μ0H5 = 1.1 T, μ0H6 = 2.5 T, μ0H7 = 5.7 T, and μ0H8 = 6.8 T. Notably, the anomaly at H5 exhibits hysteresis. Moreover, a broader peak anomaly is noticeable in the dM/dH curve at H6. In contrast, a well-defined peak manifests at H7, positioned just slightly below the saturation field H8. The prominence of this peak increases with rising temperature, as illustrated in the lower portion of Fig. 2(b). With increasing temperature, these critical fields shift towards lower 5 magnetic field values.   The phase diagram for H//[001] (left) and H//[100] (right) was established based on χ-T and M-H measurements, as illustrated in Fig. 3. The diagram features notable points such as T1, T2, T3, H1, H2, H3, and H4, along with T4, T5, H5, H6, H7, and H8. In the H//[001] phase diagram, it is evident that four distinct magnetic phases (I, II, II’, and III) exist before reaching saturation, contingent upon temperature and magnetic field conditions. The overall structure of this phase diagram closely resembles that of GdRu2Si2 [30], except for the possible inclusion of the phase II’.   To better understand the magnetic structure, longitudinal resistivity, ρxx, and Hall resistivity, ρyx, measurements were performed, and the results are shown in Fig. 4. Clear changes are observed at H1, H2, H3, and H4 for ρxx and ρyx, indicating a robust interaction between the local moment of Gd3+ and the conduction electrons. The magnetic field dependence is similar to that of GdRu2Si2 [21], except for phase II’. In particular, the prominent anomaly of ρxx and anomalous Hall effect of ρyx and σyx (see Fig. S2) [35] observed in phase II closely resembles the topological Hall effect observed in the double-Q square skyrmion lattice state of GdRu2Si2. Although skyrmion formation is theoretically predicted in GdOs2Si2 [36], our analysis did not successfully extract the topological Hall component by subtracting normal and anomalous Hall terms using ρyx = ρyxN  + ρyxA  + ρyxT  (see Fig. S3) [37]. This limitation may stem from the crucial roles played by both real-space and momentum-space Berry curvature in the transverse motion of conduction electrons atop topological magnetic textures with relatively short periods, as discussed in previous studies [38,39]. The analysis method, which assumes scattering by a single skyrmion, may therefore be inadequate for application to GdOs2Si2, expected to involve a crossover between real-space and momentum-space descriptions.  On the other hand, the anomalous Hall effect in ρyx is suddenly suppressed, while ρxx sharply drops at H2, with the minimum value of ρxx observed in Phase II'. Such a minimum value of ρxx and a drastic change of ρyx have not been identified in GdRu2Si2. This change is attributed to a rearrangement of the band structure near the Fermi surface due to the substitution of Os for Ru, resulting in a modification of the RKKY interactions between Gd ions. Consequently, the characteristic aspect of GdOs2Si2 is the II’ phase, representing a magnetic state with an undisclosed spin configuration. Recent theoretical studies have suggested the existence of a topologically trivial double-Q state in the immediate vicinity of the double-Q square skyrmion lattice state [40]. The emergence of a topologically trivial spin texture, the meron-antimeron pair state, has been reported in GdRu2Ge2 [41]. The phase II’ observed here 6 may reflect such a state, but further microscopic techniques are needed to accurately evaluate the exact magnetic structure.   In the study for H//[100], the existence of four magnetically ordered phases is suggested, namely phases I, III’, IV, and V, which are qualitatively similar to those reported for GdRu2Si2 [30]. In GdRu2Si2, these phases are associated with anisotropic double-Q states (screw + sinusoidal), anisotropic double-Q states associated with a double-Q square vortex lattice, single Q-screw states, and single Q-fan states [30]. While it remains to be confirmed whether these states directly apply to GdOs2Si2, it is suggested that phase I of GdOs2Si2 extends closer to the boundary of the paramagnetic region and is potentially more stable than phase I of GdRu2Si2 [30]. In contrast, phase V only exists in the high-field part of the phase diagram and ceases well below the zero-field antiferromagnetic transition temperature. This persists even when the magnetic field is no longer small, continuing almost up to the zero-field transition temperature. The differences in these quantitative changes are likely attributed to variations in the strength of the magnetic interaction between the respective Gd3+ moments in GdRu2Si2 and GdOs2Si2.   The temperature dependence of Cp at zero magnetic field is shown in Fig. 5 (inset), where a distinct λ-type anomaly is observed at T1. Since the isostructural nonmagnetic compounds usually used to separate the lattice contribution from the Cp data were not available, we attempted to analyze the data using the Einstein and Debye models: Cp = γT + CE + CD. In this equation, the first term represents the conduction electron contribution and γ is the Sommerfeld coefficient. The second and third terms correspond to the phonon contribution according to the Einstein and Debye models and are given by:     CE = 3nER xex(ex-1)2  and    CD = 9nDR( TθD)3∫ y4ey(ey-1)2θD/T0  ,  where x = θE/T, y = θD/T, θE and θD are the Einstein and Debye temperatures, respectively. The scale factors nE and nD correspond to the number of vibrational modes per formula unit for the Einstein and Debye models, respectively, so we applied the constraint nE + nD = 5. Fitting these equations yielded γ = 24.4(9) mJ mol⁻¹, nE = 2.44(5), nD = 2.55(5), θE = 142(2) K, and θD = 609(9) K.   Furthermore, the magnetic specific heat (Cmag) of GdOs2Si2 was obtained by 7 subtracting the contributions of conduction electrons and lattice vibrations from the Cp data, as shown in Fig. 5. The behavior of Cmag decreases rapidly as the temperature passes T1. A broad hump then appears, and eventually, Cmag approaches zero in the low-temperature limit. Similar humps have been observed in other Gd compounds in the temperature range below the magnetic transition temperature [42]. It has been argued that a peak like this appears at approximately 1/4 of magnetic transition temperatures, based on a mean-field model, originating from the splitting of the Gd3+ ground-state multiplet due to an internal field [43–45].  It is important to note that in the ρxx vs T measurements, negative magnetoresistance, indicative of magnetic precursor effects, was observed above T1 (Fig. S4) [46]. However, the broad peak of Cmag at high temperatures, characteristic of the magnetic precursor effect [47], was not clearly observed. The calculated magnetic entropy Smag = 15.7 J mol-1 K-1 is slightly smaller than the theoretical value Rln(2J+1) = 17.3 J mol-1 K-1 (J = 7/2). This discrepancy may have been due to the limited temperature range in the analysis, which may have affected the fitting process.   Magnetic transitions at T1, T2, and T3 are discernible in the Cmag curves when a magnetic field is aligned parallel to [001], as depicted in Fig. S5 [48]. Notably, Cmag exhibits sharp increases at both T1 and T2, signifying distinct phase transitions. However, while the magnetization notably decreases at T3, there is no significant change observed in the Cmag curves. This outcome indicates the relatively minor entropy change associated with the phase transition occurring at T3.   In addition, we conducted similar measurements on another crystal of GdOs2Si2 to explore the crystal-dependent aspects of the magnetic phase diagram. In this supplementary investigation, we observed slight variations in the magnetic transition temperatures. These subtle differences could be attributed to undetected disorder related to the distribution of elements between crystallographic sites within the ThCr2Si2-type structure [49], which can be challenging to assess using conventional X-ray diffraction methods. Furthermore, we examined the magnetic field dependence and noted some distinctions. However, these variations did not induce intrinsic and substantial changes in the overall structure of the magnetic phase diagram (detailed results can be found in Fig. S6) [50]. This suggests that the II and II’ phases, which are expected to be associated with the appearance of the skyrmion lattice, represent stable intrinsic phases of the GdOs2Si2 crystal and are not easily disrupted by minor crystallographic disturbances.   8 4. Conclusion   This study presents the results obtained from investigating GdOs2Si2 using a single crystal, leading to the construction of a magnetic phase diagram along H//[100] and H//[001], revealing notable similarities with the phase diagram of GdRu2Si2. Of particular interest is the distinct Hall resistivity anomaly observed in phase II, reminiscent of the topological Hall effect seen in the skyrmion state of GdRu2Si2. This intriguing parallel suggests the potential occurrence of skyrmion formation in phase II of GdOs2Si2. Additionally, our investigation unveils an unrecognized phase (phase II’) absent in GdRu2Si2. The appearance of this phase coincides with a distinct recovery pattern in the Hall resistivity, implying a significant shift in the spin structure. An in-depth understanding of the complex magnetic configurations in GdOs2Si2 could be facilitated by future endeavors involving advanced microscopic techniques like resonant X-ray diffraction measurements and the direct observation of possible skyrmions through electron microscopy.    Acknowledgment  We gratefully acknowledge the support of the World Premier International Research Center Initiative (WPI) of MEXT, Japan, which has enabled the contributions of MANA to this research. We extend our appreciation to H. Nakao (KEK, Japan) and Y. Tsujimoto (NIMS, Japan) for their invaluable discussions. This research has been partially funded by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (No. JP22H04601), as well as the Kazuchika Okura Memorial Foundation (No. 2022-11) and the Casio Science Promotion Foundation (39-10).   *E-mail: hhayashi@issp.u-tokyo.ac.jp  [1] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Niklowitz, and P. Böni, Phys. Rev. Lett. 102, 186602 (2009). [2] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Nat. Mater. 10, 106 (2011). 9 [3] N. Kanazawa, J.-H. Kim, D. S. Inosov, J. S. White, N. Egetenmeyer, J. L. Gavilano, S. Ishiwata, Y. Onose, T. Arima, B. Keimer, and Y. Tokura, Phys. Rev. B 86, 134425 (2012). [4] A. Bogdanov and A. Hubert, J. Magn. Magn. Mater. 138, 255 (1994). [5] M. Kakihana, D. Aoki, A. Nakamura, F. Honda, M. Nakashima, Y. Amako, T. Takeuchi, H. Harima, M. Hedo, T. Nakama, and Y. Ōnuki, J. Phys. Soc. Jpn. 88, 094705 (2019). [6] S. Z. 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T2 T3 1 T2 T  2.4 T3 T  4 T2.4 TT4 T5 1 T2 T  3 T  4 T13  Fig. 2. (Color online) Isothermal magnetization curve (left axis) and its derivative dM/dH (right axis) measured at 1.8 K, with increasing and decreasing magnetic fields up to 7 T applied parallel to [001] (a) and parallel to [100] (b). The symbols represent critical fields: H1, H2, H3, and H4 for H//[001] and H5, H6, H7, and H8 for H//[100]. The thermal evolution of dM/dH is shown at the bottom. The data have been shifted for clarity.    14   Fig. 3. (Color online) The magnetic field-temperature phase diagram for GdOs2Si2, with the magnetic field applied parallel to [001] (left) and [100] (right), based on magnetic measurements. The inset in the left figure depicts the crystal structure of GdOs2Si2, with red, black, and gray spheres representing Gd, Os, and Si atoms, respectively.       Phase IPhase IVPhase VPhase III’   [010][001][100]GdOsSi15   Fig. 4. (Color online) Magnetic field dependence of ρyx at 5 K for H//[001] and I//[100]. The black and gray curves represent ρyx with increasing and decreasing magnetic fields, respectively. The regions bordered by H1 and H2 (gray), H2 and H3 (yellow) correspond to the phases II and II’, respectively.   16   Fig. 5. (Color online) Temperature dependence of magnetic specific heat (Cmag) and magnetic entropy at μ0H = 0 T. The inset displays the Cp-T data and the fitting curve (solid blue light line) obtained using the Einstein and Debye formula.