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[Naoki Kikugawa](https://orcid.org/0000-0003-3975-4478), [Shinya Uji](https://orcid.org/0000-0001-9351-6388), [Taichi Terashima](https://orcid.org/0000-0001-9239-0621)

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[Anomalous Hall effect in the magnetic Weyl semimetal NdAlGe with plateaus observed at low temperatures](https://mdr.nims.go.jp/datasets/2d0aa826-c5c1-4ad5-bf35-f7f2ab536758)

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Anomalous Hall effect in the magnetic Weyl semimetal NdAlGe with plateaus observed at low temperaturesPHYSICAL REVIEW B 109, 035143 (2024)Anomalous Hall effect in the magnetic Weyl semimetal NdAlGe withplateaus observed at low temperaturesNaoki Kikugawa ,1,* Shinya Uji ,2 and Taichi Terashima 21Center for Basic Research on Materials (CBRM), National Institute for Materials Science, 3-13 Sakura, Tsukuba, Ibaraki 305-0003, Japan2Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science,3-13 Sakura, Tsukuba, Ibaraki 305-0003, Japan(Received 29 November 2023; accepted 18 December 2023; published 18 January 2024)In the RAl(Si,Ge) (R: lanthanides) family, both spatial inversion and time-reversal symmetries are broken.This may offer opportunities to study Weyl-fermion physics in nontrivial spin structures emerging from a non-centrosymmetric crystal structure. In this study, we investigated the anomalous Hall effect (AHE) in NdAlGe viamagnetotransport, magnetization, and magnetic torque measurements down to 40 mK (0.4 K for magnetization).The single crystals grown by a laser-heated floating-zone method exhibit a single magnetic phase transition at TM= 13.5 K, where the TM is the transition temperature. With the magnetic field parallel to the easy [001] axis, theAHE gradually evolves as the temperature decreases below TM. The anomalous Hall conductivity (AHC) reaches∼320 �−1 cm−1 at 40 mK in the magnetically saturated state. Except in low-temperature low-field plateauphases, the AHC and magnetization are proportional, and their ratio agrees with the ratios for conventionalferromagnets, suggesting that the intrinsic AHE occurs by the Karplus-Luttinger mechanism. Below ∼0.6 K,the curves of Hall resistivity against the field exhibit plateaus at low fields below ∼0.5 T, correlating with theplateaus in the magnetization curve. For the first plateau, the magnetization is one order of magnitude smallerthan the magnetically saturated state, whereas the AHE is more than half that in the saturated state. This findingunder well below TM suggests that the AHE at the first plateau is not governed by the magnetization and may beinterpreted based on a multipole or spin chirality.DOI: 10.1103/PhysRevB.109.035143I. INTRODUCTIONTopologically nontrivial phases in condensed matter haveattracted much attention recently due to their novel physicalproperties [1–7]. Weyl semimetals are one such class of mate-rials. These semimetals have band crossings near the Fermilevel and host emergent relativistic quasiparticles, namely,Weyl fermions. The Weyl fermions can be realized wheneither spatial inversion or time-reversal symmetry is broken.Magnetic Weyl semimetals breaking time-reversal symme-try further offer opportunities to study the interplay betweenmagnetic interactions and topologically nontrivial electronicstructures; exhibiting novel phenomena such as the anomalousHall and Nernst effects with no (or negligibly small) mag-netization, optical Hall conductivity, and presence of axioninsulators, and chiral domain walls [8–12]. These phenom-ena may provide the basis for the next-generation spintronicsapplications [13–15].The RAl(Si,Ge) (R: lanthanides) family with the spacegroup I41md (No. 109) is a new class of magnetic Weylsemimetals where both the inversion and time-reversalsymmetries are broken [16,17]. Recent studies have sug-gested a topological magnetic order in SmAlSi [18] andNdAlSi [19,20], topological Hall effect in SmAlSi [18] andCeAl(Si,Ge) [18,21,22], anomalous Hall and Nernst effects*kikugawa.naoki@nims.go.jpin PrAl(Ge,Si) and NdAl(Si,Ge) [23–25], anomalous thermalconductivity [26], unusual quantum oscillations in NdAlSi[27,28], possible axial gauge fields in PrAlGe [25], domainwall chirality in CeAl(Si,Ge) [22,29], surface Fermi arcs andbulk Weyl fermion dispersion in PrAlGe [30] and NdAlSi[31], and reconstruction of the electronic structure across themagnetic transition in PrAlGe [32].We focused on NdAlGe in this study. The physical prop-erties of NdAlGe were investigated by several groups usingflux-grown crystals [33–36]. Yang et al. [34] reported thatNdAlGe undergoes two successive magnetic transitions: anincommensurate spin-density-wave transition at Tic = 6.8 K,where the spin structure is predominantly aligned in the (001)direction with small helical canting to the in-plane, and a com-mensurate ferrimagnetic transition at Tcom = 5.1 K, where thespin structure becomes a down-up-up structure propagatingin [110] or [11̄0] direction. The polarized up-up-up structurein high fields is formed through a metamagneticlike behaviorunder a magnetization of ∼3 T. Dhital et al. reported slightlydifferent transition temperatures Tic = 6.3 K and Tcom = 4.9 K[36]. The anomalous Hall effect (AHE) was observed in boththe down-up-up and polarized regions. This finding contrastswith the fact that the AHE was not detected in a sister material,NdAlSi [34]. Yang et al. argued that the AHE in the down-up-up and polarized regions was governed by an intrinsic andextrinsic origin, respectively, whereas Dhital et al. proposedan intrinsic origin of both regions [34,36]. Two other researchgroups also grew NdAlGe crystals by a flux method, but they2469-9950/2024/109(3)/035143(7) 035143-1 ©2024 American Physical Societyhttps://orcid.org/0000-0003-3975-4478https://orcid.org/0000-0001-9351-6388https://orcid.org/0000-0001-9239-0621https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.109.035143&domain=pdf&date_stamp=2024-01-18https://doi.org/10.1103/PhysRevB.109.035143KIKUGAWA, UJI, AND TERASHIMA PHYSICAL REVIEW B 109, 035143 (2024)observed a single magnetic transition at 5.2 K [33] or 6 K [35].Thus, we see that flux-grown crystals exhibit considerablesample dependence.Recently, we have succeeded in growing the single crystalsof NdAlGe by a laser-heated floating-zone technique [37].This technique can minimize accidental contamination byimpurities, allowing us to study the intrinsic properties ofNdAlGe. We describe magnetotransport and magnetic torquemeasurements down to 40 mK, and magnetization ones to0.4 K. We observed that AHE develops below a singlemagnetic transition temperature. We argue that, in all re-gions except a low-temperature low-field region, the observedAHE can be ascribed to the intrinsic Berry curvature and isconsistent with the Karplus-Luttinger theory developed forferromagnets [38]. At low temperatures and under low fields,the Hall conductivity versus magnetic field curves exhibitplateaus. A large AHE is observed at the first plateau, despitelow magnetization. We discuss the possible explanations forthis phenomenon.II. EXPERIMENTAL DETAILSSingle crystals of NdAlGe were grown by a laser-heatedfloating-zone method. The detailed growth procedure was de-scribed in Ref. [37]. Here, the grown crystals are deficientin aluminum due to the evaporation during growth. This isin sharp contrast to the aluminum-rich crystals that were ob-tained by flux methods [33–36].The crystals were cut and polished into rectangles witha typical size of 3.4 × 0.5 × 0.4 mm3 in [100], [010], and[001] directions, respectively. The electrical contacts werespot welded and supported with silver paste. The contactresistances were below 0.1 �. The magnetoresistivity andHall resistivity were measured simultaneously using a low-frequency (∼13 Hz) ac method. The measurements wereperformed in a top-loading 3He - 4He dilution refrigeratorat temperatures (T ) down to 40 mK with sweeping mag-netic field (H) between −17.5 and +17.5 T. The electricalcurrent and magnetic field were applied in the [100] and[001] directions, respectively, unless specified otherwise. Be-cause the misalignments of the contact electrodes can causemixing of the magnetoresistivity and Hall resistivity, themagnetoresistivity (ρxx) and Hall resistivity (ρyx) were ob-tained by symmetrizing ρxx = [(ρexpxx (H ) + ρexpxx (−H )]/2 andantisymmetrizing the experimental data ρyx = [(ρexpyx (H ) −ρexpyx (−H )]/2, respectively.The magnetic torque (τ ) was measured using a capac-itive cantilever method in a top-loading 3He - 4He dilutionrefrigerator. Because the magnetic torque vanishes for sym-metric directions, the measurements were performed underthe magnetic field applied 3◦ off from the exact [001] to [010]direction.The isothermal magnetization (M) measurements under H‖ [001] between −16 and +16 T were performed down to 2 Kusing the options of Physical Property Measurement System(PPMS, Quantum Design). The measurement at 0.4 K wasperformed using the Magnetic Property Measurement System(MPMS3, Quantum Design) with a 3He cooling option. Thespecific heat (CP) under zero field was measured down to0.4 K using the options of Physical Property MeasurementFIG. 1. (a) Temperature dependence of resistivity ρxx in NdAlGeunder zero magnetic field. The current flows in the [100] direction.The inset shows the closeup of the resistivity between 40 mK and2 K. (b) Hall resistivity ρyx of NdAlGe at 40 mK under the magneticfield between −17.5 and +17.5 T. The ordinary Hall coefficientR0 = +1.28 × 10−3cm3/C is evaluated from the slope of the ρyxabove 1 T.System (Quantum Design). The measurement was performedby a relaxation method.III. RESULTSFigure 1(a) shows the temperature (T ) dependence of re-sistivity ρxx under zero field. A clear kink observed at 13.5 Kcorresponds to the magnetic ordering temperature (TM). Theinset of Fig. 1(a) shows a closeup of ρxx below 2 K, whereno anomalies are seen down to 40 mK. This finding is consis-tent with the specific-heat measurement of our floating-zoneNdAlGe crystal down to 0.4 K (see Fig. S1 in the Supplemen-tal Material [39]), revealing a single sharp transition at TM.Notably, the transition width of the specific-heat jump at TM isas sharp as 0.4 K, suggesting that the floating-zone crystal inthis study is highly homogeneous. In comparison, two succes-sive transitions were observed at lower temperatures (∼5 K035143-2ANOMALOUS HALL EFFECT IN THE MAGNETIC WEYL … PHYSICAL REVIEW B 109, 035143 (2024)FIG. 2. Hall resistivity ρyx of NdAlGe at several temperatures (a) between 1.5 and 20 K across the magnetic ordering temperature TM =13.5 K, and (b) below 1 K. The magnetic fields are applied along the [001] direction. The anomalous Hall resistivity ρAHEyx at high temperaturesis defined by the extrapolations of the Hall resistivity from a high to zero magnetic field, as shown in (a). (c) Hall conductivity σxy = ρyx/(ρ2xx +ρ2yx ) of NdAlGe at 40 mK. The inset presents the magnetoconductivity σxx = ρxx/(ρ2xx + ρ2yx ).and ∼6 − 7 K) [34,36], or a single transition was observed at5 − 6 K in flux-grown NdAlGe crystals [33,35].Figure 1(b) shows the Hall resistivity ρyx at 40 mK ina wide magnetic field (H) range of −17.5 to +17.5 T.Hysteresis appears at low fields (< 0.5 T), which will bedetailed below. At |μ0H | > 1 T (μ0: magnetic permeabilityin vacuum), the ρyx exhibits a linear field dependence with apositive slope, suggesting that holes are the dominant carrier.The ordinary Hall coefficient (R0) deduced under high fieldgreater than 1 T is +1.28 × 10−3 cm3/C. A similar valueof +1.25 × 10−3 cm3/C was obtained for another sample(Fig. S2(a) [39]). The coefficients correspond to a carrierdensity of +4.88 to +4.99 × 1021 cm−3, assuming a singleband. In comparison, larger values of R0 ranging from +4 to+7 × 10−3 cm3/C, corresponding to smaller hole densities,were reported for flux-grown crystals [34–36]. The differenceof the values is likely related to the fact that while floating-zone crystals are aluminum deficient [37], flux-grown onesare aluminum rich [33–36]. No anomaly was found in ρyx forfields above 1 T [Fig. 1(b)]; this result is consistent with themagnetization (M) curve of our floating-zone crystal (Fig. S3[39]) and is in sharp contrast with the fact that ρyx and M influx-grown crystals exhibit an anomaly around 3 T [33–36].Figure 2(a) shows the Hall resistivity ρyx in a low-fieldregion at selected temperatures between 1.5 K and 20 K. Theanomalous Hall contribution is observed below TM = 13.5 K.The hysteresis between up and down field sweep is obviousbelow 10 K and is closely linked to the magnetization hys-teresis (Fig. S3(a) [39]). We notice the ρyx exhibits nonlinearbehavior in the hysteretic region at 1.5 and 5 K.Figure 2(b) shows the ρyx curves below 1.5 K. Clearplateaus are developed in the hysteretic region between −0.5and +0.5 T as the temperature decreases. As the field is sweptfrom the negative to positive values at the lowest temperatureof 40 mK, the ρyx jumps to the first plateau (ρyx = 1.1 μ� cm),second (ρyx = 1.8 μ� cm), and third (ρyx = 2.5 μ� cm)plateaus at 0.07 T, 0.22 T, and 0.36 T, respectively. Finally,ρyx reaches 2.7 μ� cm in the field-induced polarized stateabove 0.5 T. The plateau behavior in ρyx below 1 K wasreproduced in another sample, and the ρyx finally reached 2.7μ� cm above 0.5 T (Fig. S2(b) [39]). Such plateau behaviorhas not been reported in the flux-grown crystals. Figure 2(c)shows the Hall conductivity σxy at 40 mK, calculated us-ing the formula σxy = ρyx/(ρ2xx + ρ2yx ), and the inset showsthe behavior of magnetoconductivity σxx = ρxx/(ρ2xx + ρ2yx ).The Hall conductivity σxy reaches a large value of 320 �−1cm−1 in the high-field polarized state. For the polarized stateof flux-grown crystals, σxy is ∼1, 030 �−1 cm−1 [36] or0.5–1.5 × 103 �−1 cm−1 [34]. No anomaly was seen in themagnetoconductivity σxx in the same field region [inset ofFig. 2(c)].Figure 3 shows the magnetic field angle dependence of ρyxat a base temperature of 40 mK. Herein, the field angle (θ )was measured from [001] to [010] (inset of Fig. 3). The curvesmeasured at different angles collapse into a single curve whenplotted against μ0Hcosθ . This finding is consistent with theIsing character of the neodymium magnetism in NdAlGe[33–37].Figure 4 shows the plot of anomalous Hall resistivity(ρAHEyx ) against temperature. Herein, the ρAHEyx is defined byFIG. 3. Magnetic field angle dependence of the Hall resistivityρyx in NdAlGe at 40 mK. The field is tilted from the [001] to [010]axes as shown in the inset. The horizontal axis is μ0Hcosθ , wherethe θ is the tilting angle.035143-3KIKUGAWA, UJI, AND TERASHIMA PHYSICAL REVIEW B 109, 035143 (2024)FIG. 4. The anomalous Hall resistivity ρAHEyx plotted against tem-perature. The ρAHEyx gradually evolves below TM. The inset representsanomalous Hall conductivity σ AHExy against extrapolated zero-fieldmagnetization M0, obtained by extrapolating the high-field part ofa magnetization curve toward zero field (Fig. S3(a) [39]).the extrapolations of the Hall resistivity from the high to thezero magnetic field for high temperatures, as presented inFig. 2(a), and it is the residual Hall resistivity at μ0H = 0 Tfor low temperatures [Fig. 2(b)]. The ρAHEyx gradually evolvesbelow TM and reaches 2.7 μ� cm at the lowest temperature.The inset of Fig. 4 shows σ AHExy plotted against extrapolatedzero-field magnetization M0 for four temperatures (T = 2,10, 12, and 13 K), where M0 is obtained by extrapolatingthe high-field part of a magnetization curve toward zero fieldas presented in Fig. S3(a) [39]. A clear linear relation existsbetween σ AHExy and M0.Figures 5(a)–5(c) show a comparison of the magnetizationM, magnetic torque (τ ) divided by magnetic field τ/(μ0H ),and the Hall resistivity ρyx measured at ∼0.4 K, the lowestpossible temperature for magnetization measurements. Themagnetic torque is given by �τ = �M × �H . Considering theIsing nature of the neodymium moments [33–37], τ/(μ0H )can be roughly approximated to M. Similar to the behaviorof ρyx, as the field is increased from negative to positivevalues, both M and τ/(μ0H ) exhibit the first and secondplateaus before the field-polarized state is reached above 0.5 T.Although the M and τ/(μ0H ) values at the first plateau areclose to zero, the ρyx is more than half of the polarized state(∼1.5 μ� cm). The transition fields between the three regionsslightly differ among the three measurements. This differenceis mostly attributed to the sample difference and/or demagne-tization factor differences. In addition, only the magnetizationcurve shows a rather wide transitional (nonflat) region from∼0.18 to ∼0.28 T between the first and second plateaus. Thiswide transition region was possibly caused by a temperatureinstability or increased temperature during the magnetizationmeasurement; the samples were immersed in liquid 3He - 4Hemixture for the magnetic torque and ρyx measurements us-ing the top-loading dilution refrigerator, whereas a samplewas held in low-pressure 3He gas atmosphere for the mag-netization measurement, and thermal contact was providedFIG. 5. (a) Isothermal dc magnetization M, (b) magnetic torque(τ ) divided by field τ/μ0H , and (c) Hall resistivity ρyx of NdAlGe atT ∼ 0.4 K. The plateau structures in the Hall resistivity are closelyrelated to the magnetic properties.by copper wires attached to the sample. Consequently, thetemperature stability and accuracy may be worse in the mag-netization measurements.FIG. 6. Magnetic torque divided by the applied field [τ/(μ0H )]below 1 K.035143-4ANOMALOUS HALL EFFECT IN THE MAGNETIC WEYL … PHYSICAL REVIEW B 109, 035143 (2024)Figure 6 shows the τ/(μ0H ) curves measured at varioustemperatures below 1.1 K. Similar to the ρyx data (Figs. 2(b)and S2(b) [39]), the plateaus are sensitive to temperature: thefirst and second plateaus survive only up to 0.5 and 0.4 K,respectively. The heights of the plateau vary slightly withtemperature.IV. DISCUSSIONWe first argue that the observed anomalous Hall conduc-tivity (AHC) σ AHExy ∼ 320 �−1 cm−1 in the polarized state[Fig. 2(c)] is primarily attributed to the intrinsic Berry cur-vature. A theoretical scaling relation between σxy and σxx[40–42], experimentally supported, indicated three regimes.AHE in each regime is dominated by different mechanisms:in the high-conductivity regime (σxx > 106 �−1 cm−1), skewscattering dominates AHE and σ AHExy ∝ σxx; in the good-metal regime (σxx is 104 − 106 �−1 cm−1) where the intrinsicBerry-phase contribution dominates and σ AHExy is approxi-mately independent of σxx; in the bad-metal regime (σxx < 104�−1 cm−1) where σ AHExy ∝ σ 1.6xx . The AHC in the good-metalregime is of the order of 102 − 103 �−1 cm−1. Our results(σxx ∼ 1.1 × 104 �−1 cm−1 and σ AHExy ∼ 320 �−1 cm−1 atT = 40 mK) perfectly fit the good-metal regime. The intrinsicBerry curvature contribution to the AHE in the polarized stateof NdAlGe was theoretically calculated as σ AHExy ∼ 200 �−1cm−1 [34] or ∼ 270 �−1cm−1 [36]. These values are in goodagreement with our experimental values.Furthermore, according to the Karplus-Luttinger theory ofthe intrinsic AHE, ρAHEyx ∼ ρ2xxM [38]. Considering ρxx �|ρyx|, this relation indicates that σ AHExy (approximately equal toρAHEyx /ρ2xx) is proportional to M. The present data nicely satisfythis relation (inset of Fig. 4), further supporting the intrinsicmechanism. For various conventional ferromagnets exhibitingAHE, the ratio σ AHExy /M is in the range of a few tens to 1 ×103 �−1 cm−1/T (where M is measured in tesla) [10]. For thepolarized state of NdAlGe, where M = 0.45 T, the ratio is 7.1× 102 �−1 cm−1/T, lying in this empirical range.Our conclusion regarding the intrinsic AHE agrees withthe conclusion drawn from studies on flux-grown crystalsby Dhital et al. [36]. Conversely, Yang et al. [34] observedthat the temperature dependence of the AHC in the polarizedstate of flux-grown crystals differs with the sample and doesnot follow the magnetization; they argued that AHE in thepolarized state of NdAlGe has an extrinsic origin.Next, we focus on the first plateau in the Hall resistivity.Figure 5 indicates that τ/(μ0H ) and M of the first plateau areclose to zero and the magnetization is negative, respectively.Nevertheless, the ρyx has a substantial value of ρyx ∼ 1.5μ� cm, ∼55% of its value in the polarized state and has thesame positive sign. For the first plateau, where M = 0.043 T,the ratio σ AHExy /M is 4.3 × 103 �−1 cm−1/T, exceeding theaforementioned empirical range.Recent studies on anomalous Hall antiferromagnets indi-cated that the contribution of the Berry curvature to AHEis not necessarily controlled by magnetization [10,43,44].Mn3Sn, for example, is an antiferromagnet without net mag-netization (in the absence of spin-orbit coupling), but itexhibits a large AHE. In the case of such anomalous Hallantiferromagnets, breaking of the time-reversal symmetry thatis necessary for AHE and is attributed to the magnetic multi-poles composed of multiple atoms [43,44]. Another exampleis CoTa3S6 and CoNb3S6 [45,46]. Arguably, a fictitious mag-netic field associated with a scalar spin chirality induces agiant Hall response under the small net magnetization [46].For flux-grown NdAlGe samples, neutron diffractionmeasurements were performed, and the low-temperature zero-field spin structure was found to be composed of basicallyantiferromagnetic spin down-up-up chains along [110] direc-tions. Due to the Dzyaloshinskii-Moriya interaction, a part ofthe spins exhibits a slight canting from the [001] direction,resulting in a noncollinear spin structure [34,36]. The spinstructure at the first plateau of the floating-zone crystals maybe a variant with the same down-up-up motif. Determining theexact spin structure to identify the multipole or spin chiralityresponsible for the AHE of the first plateau is highly desirable.Finally, we discuss the differences between floating-zoneand flux-grown crystals. The floating-zone crystals are alu-minum deficient, whereas the flux-grown ones are aluminumrich. This difference likely explains the smaller ordinaryHall coefficient R0 (larger hole density) in the floating-zone crystals, and may also account for the smaller σ AHExyin the polarized state of these floating-zone crystals sincethe theoretical calculations indicate that the AHC causedby Berry curvature decreases with a decrease in the Fermilevel [34,36]. Further, because the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by itinerant carriers isreasonably assumed to be the primary exchange interactionbetween neodymium moments, the differences between themagnetic phase diagrams of the floating-zone and flux-growncrystals can be attributed to their different carrier densities.V. SUMMARYWe performed magnetotransport and magnetic torque mea-surements down to 40 mK and magnetization ones to 0.4 K onfloating-zone single crystals of NdAlGe. We observed onlyone magnetic phase transition at TM = 13.5 K, in contrastto the two transitions observed in some of the flux-growncrystals [34,36]. The AHE occurred below TM, and the AHCreached ∼ 320 �−1 cm−1 at 40 mK in the polarized state,comparable to the ab initio calculations of the intrinsic Berrycurvature contribution [34,36]. A comparison with the the-oretical scaling relation between σ AHExy and σxx supports theBerry curvature origin of the AHE. A linear relation existsbetween σxy and M0, and their ratio σ AHExy /M is in the typicalrange for ferromagnets. These results indicate that the AHE,except for the low-temperature low-field region, occurs withinthe framework of the Karplus-Luttinger theory. At low tem-peratures and fields below ∼0.6 K and ∼0.5 T, we observeplateaus in the curves of the Hall resistivity against the field.These plateaus in the Hall resistivity are correlated with theones in the magnetization curves. In the first plateau, althoughthe magnetization is one order of magnitude smaller than thatin the saturated state, we observe a large anomalous Hall resis-tivity, more than half the value observed in the magneticallysaturated state. This imbalance between the anomalous Halleffect and magnetization is analogous to anomalous Hall anti-ferromagnets such as Mn3Sn or Co(Nb, Ta)3S6. This finding035143-5KIKUGAWA, UJI, AND TERASHIMA PHYSICAL REVIEW B 109, 035143 (2024)indicates that a multipole or spin chirality governs the AHC inthe first plateau, and theoretical work on this topic is necessaryin the future.ACKNOWLEDGMENTSWe acknowledge Takanobu Hiroto for comments and sup-port, Masao Arai, and Jun-ichi Inoue for comments and dis-cussion, Ayumi Kawaguchi, Takashi Kato, Momoko Hayashi,Hitoshi Yamaguchi, Takeshi Shimada, Akira Kamimura,John McArthur, and Noritaka Kimura for support. 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