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[Tomomi Furuhashi](https://orcid.org/0009-0008-8017-5375), Keita Nagasawa, [Yuichi Yamasaki](https://orcid.org/0000-0002-8560-3462), [Hironori Nakao](https://orcid.org/0000-0003-4020-537X), [Yusuke Kozuka](https://orcid.org/0000-0001-7674-600X), [Yoshihiro Tsujimoto](https://orcid.org/0000-0003-2140-3362), [Kazunari Yamaura](https://orcid.org/0000-0003-0390-8244), [Jun Fujioka](https://orcid.org/0000-0003-1340-0268)

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©2025 American Physical Society[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Spin fluctuation induced anomalous Hall effect in the distorted kagome antiferromagnet DyAgGe](https://mdr.nims.go.jp/datasets/819ff760-c2c3-4827-9b46-c28930f08de9)

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Spin-Fluctuation Induced Anomalous Hall Effect in DistortedKagome Antiferromagnet DyAgGeTomomi Furuhashi,1 Keita Nagasawa,2, 3 Yuichi Yamasaki,4, 5 Hironori Nakao,6Yusuke Kozuka,7 Yoshihiro Tsujimoto,7 Kazunari Yamaura,7 and Jun Fujioka8, 91Graduate School of Science and Technology,University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan2College of Engineering Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan3Institute for Solid State Physics, The University of Tokyo,Kashiwanoha 5-1-5, Kashiwa 277-8581, Japan4Center for Basic Research on Materials,National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan5International Center for Synchrotron Radiation Innovation Smart,Tohoku University, Sendai 980-8577, Japan6Institute of Materials Structure Science,High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan7Research Center for Materials Nanoarchitechtonics (MANA),National Institute for Materials Science (NIMS), Namiki, Tsukuba 305-0044, Japan8Department of Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan9Research Center for Organic-Inorganic Quantum Spin Science and Technology (OIQSST),University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan(Dated: October 30, 2025)1AbstractWe have investigated the magnetic property and charge-transport property for the kagome frus-trated antiferromagnet DyAgGe with the ZrNiAl-type crystal structure by measurements of mag-netization, resistivity and resonant X-ray scattering. The magnetization curve shows the 1/9-, 1/3-,5/9- and 7/9-plateaus of saturated magnetization under the magnetic field along the c-axis beforeentering the forced ferromagnetic phase at 7.3 T and 2 K. Magnetic superlattice X-ray reflectionswith the modulation vector (1/3, 1/3, 0), resonantly enhanced at the Dy L3 edge, are observedin these plateau phases. All of these phases, except for the 1/9-plateau phase, evolve into theparamagnetic phase around 12-15 K with strong competition among them in the field range of 4-6T. Furthermore, we found the large negative magnetoresistivity and anomalous Hall resistivity notproportional to magnetic field or magnetization near and above the magnetic transition tempera-ture around 3-7 T. For comparison, we also investigated the charge-transport for the isostructuralmagnet DyPtIn, which does not exhibit the field-induced multiple phase transitions. In contrastwith the behavior of DyAgGe, DyPtIn does not show remarkable magnetoresistivity and anoma-lous Hall resistivity near and above the magnetic transition temperature. The anomalous Halleffect persisting above the magnetic transition temperature is likely caused by spin fluctuationsenhanced by the competing multiple magnetic phases through the scalar spin chirality mechanismin DyAgGe.2I. INTRODUCTIONElectromagnetic response arising from nontrivial spin textures in magnetic metals hasreceived of great interest in modern condensed matter physics. The anomalous Hall effect(AHE) or topological Hall effect (THE) is a typical example, where the Hall effect is inducedby the magnetism in metals and semiconductors [1]. This is well exemplified by skyrmionmagnets, which typically emerge in chiral and geometrically frustrated magnetic systems[2, 3]. In skyrmion magnets, itinerant electrons moving on noncoplanar spin texture acquirethe spin Berry phase proportional to the scalar spin chirality (SSC), which is defined by thesolid angle subtended by three spins as χ = S1 · (S2 × S3) [4–9]. The spin Berry phase actsas the fictitious magnetic field for itinerant electrons, thereby giving rise to the topologicalHall effect. In particular, the SSC is scaled to the Skyrmion density in real space and/ormomentum space, and its modulation often results in the giant THE, offering potential forspintronic applications.In addition to the case of long-range ordered noncoplanar spin textures, the Hall effectcan also arise from fluctuating or spatially inhomogeneous spin textures that lack long-rangemagnetic ordering. Conventionally, it is known that electron scattering off localized mag-netic defects can give rise to the extrinsic AHE via skew scattering or side-jump mechanisms[1]. The extrinsic AHE is generally small in its magnitude, but recent studies demonstratethat it can be remarkably enhanced by spin fluctuation or magnetic domain walls in topo-logical semimetals [10, 11]. On the other hand, it has been also known that the transientnoncoplanar spin textures generated by thermal spin fluctuations can induce AHE throughthe SSC. Indeed, in double-exchange ferromagnets, the AHE emerging near the transitiontemperature has been attributed to thermally induced topological spin texture [12, 13]. Morerecently, it has been demonstrated that spin fluctuations can trigger the large AHE with thetangent of anomalous Hall angle (tan θH) more than 0.01 [14–20]. Notably, the large AHEdue to the thermally induced SSC has been observed even in collinear antiferromagnets. Forexample, a recent study argues that thermally induced solitonic spin fluctuations can causelarge AHE that persists far above the magnetic transition temperature in collinear antiferro-magnets exhibiting a magnetic devil’s staircase [18]. Nevertheless, it remains unclear whatkind of magnetic ordering or phase transition induces the large AHE persisting above thetransition temperature, and thus its observation has been limited to a few magnetic metals.3In this context, geometrically frustrated magnets with unconventional spin excitation ormagnetic transition offer a platform to search the remarkable spin-fluctuation induced AHE.The RAgGe (R =Tb-Lu) is one of typical frustrated magnetic metals, which crystallize in theZrNiAl-type structure [21]. As shown in Figs. 1(a) and (b), the R-site forms the distortedkagome lattice in the ab-plane, which stacks along the c-axis. The magnetic interaction ofR-4f moment is mediated by conduction electrons, and the competing intersite interactionor geometrically frustrated interaction induce a variety of unconventional magnetic statesfor R=Tb, Dy and Ho [21–23]. For example, HoAgGe exhibits several magnetic phases withnoncollinear magnetic structure including the spin ice state under the magnetic field alongthe b-axis [23–25]. In particular, the recent studies demonstrate that the AHE persists inwide temperature regime up to about four times of the magnetic transition temperature,which is attributed to the skew scattering from fluctuating spins [26].Field-induced multiple phase transitions are also observed in DyAgGe, which exhibitscollinear antiferromagnetic orderings. In this material, the antiferromagnetic ordering occursat 15 K (= TN1), followed by a second magnetic transition at 12 K (= TN2) under zeromagnetic field [22, 23]. The wave vector of the antiferromagnetic ordering is characterizedby (1/3, 1/3, 0), and the Dy magnetic moment lies in the bc-plane, forming a tilt angleof about 50◦ with respect to the c-axis, as shown in Figs. 1(a) and (b). Such multiplemagnetic phase transitions on the kagome lattice may induce peculiar spin fluctuations andmagneto-transport properties in DyAgGe, but these phenomena have not been elucidated sofar. In this study, we investigated the magnetic property and magneto-transport propertyfor DyAgGe by measurements of magnetization, Hall resistivity and the resonant magneticX-ray scattering.II. EXPERIMENTAL METHODSSingle crystalline samples of DyAgGe were grown by using the Ag-Ge rich self-flux tech-nique [23]. The starting materials Dy, Ag, and Ge were placed in an aluminum cruciblein a molar ratio of 1 : 6.8 : 2.3, and sealed in an evacuated quartz ampoule. The quartzampoule was heated up to 1100 ◦C, held at that temperature for 24 h, and then cooledto 850 ◦C over 135 h. Following slow cooling, the ampoule was quenched to room tem-perature, and the remaining solution was separated by centrifugation. The single crystal4was ground into powder form, which was examined by X-ray diffraction using a commercialdiffractometer (Rigaku MiniFlex) with Cu-Kα1 and -Kα2 radiation sources [see also Fig. S1][27]. Resistivity measurements were performed using the four-probe method in a PhysicalProperty Measurement System (Quantum Design) over the temperature range of 2–300 Kand in magnetic field up to 14 T. The magnetization measurements were performed usingthe Dynacool System equipped with the VSM option from 2 K to 300 K under the magneticfield up to 14 T. The resonant magnetic X-ray scattering was performed under the magneticfield at the BL-3A, Photon Factory of KEK, Japan. The magnetic field was applied alongthe c-axis up to 6 T by using a commercial superconducting magnet.III. RESULTS AND DISCUSSIONFigure 1(c) shows the temperature dependence of magnetization for DyAgGe under themagnetic field (B) along the c-axis. For B = 0.1 T and 0.5 T, the magnetization shows twoanomalies at TN1 and TN2, consistent with previous studies [22, 23]. Above 2 T, the anomalyat TN1 remains, whereas that at TN2 is not clearly observed except for the case of B = 5 T.Figure 1(d) shows the field dependence of magnetization at various temperatures. At 2 K,the magnetization exhibits plateaus at 1/9, 1/3, 5/9 and 7/9 of the saturated magnetization[see also the inset to Fig. 1(d)] and is nearly constant above 7.6 T. Here, the 5/9-plateauphase is clearly visible only during the process of increasing the magnetic field [see inset toFig. 1(d)]. All data shown below under finite magnetic fields were measured during theprocess of increasing the magnetic field. We note that the saturated magnetization is about6.4 µB/f.u., which is consistent with the expected value for the Dy-4f moment (9.9 µB)tilted by approximately 50◦ from the c-axis. With increasing temperatures, these plateausare gradually smeared out and vanish around 14 K.Based on the magnetization results, we constructed the magnetic phase diagram in thetemperature–field plane as shown in Fig. 2 (a). The 1/9-plateau phase is confined to thelow-field, low-temperature region below approximately 1.3 T and TN2, while the 1/3-plateauphase (7/9-plateau phase) extends over a relatively wide field–temperature region of 1–4.5T (5.4–7.1 T). Notably, the 5/9-plateau phase emerges within a narrow region betweenthe 1/3- and 7/9-plateau phases. According to the previous study, at 1.5 K, the Dy-4fmoments show the collinear antiferromagnetic structure with a wave vector of (1/3, 1/3, 0),5forming the√3 ×√3 magnetic unit cell at zero magnetic field [22]. To gain insight intothe magnetic state under the magnetic field, we investigated the magnetic wave vector bymeans of resonant X-ray scattering.Figure 2(b) shows the scattering spectrum of (H, 8/3, 0) magnetic superlattice reflectionmeasured at 4.5 K [see also Fig. S2] [27]. The incident X-ray is nearly π-polarized, but thescattered X-ray includes both σ′- and π′-polarized components. Here, σ and π denote thepolarization component perpendicular and parallel to the scattering plane, respectively. At 0T, a clear peak is observed around H = 0.67, which is consistent with the antiferromagneticstructure with the wave vector (1/3, 1/3, 0). The peak becomes more pronounced at 2 T butdiminishes above 4.8 T. We plot the integrated intensity and full width at half maximumof peak as a function of magnetic field in Fig. 2(c). With increasing the magnetic field,the peak intensity slightly increases in the 1/3-plateau phase, decreases sharply near Bc2(= 4.7 T) and becomes nearly constant above 5 T. This is in contrast with the behaviorof peak width, which does not significantly change under the magnetic field. These resultssuggest that the antiferromagnetic structures with√3×√3 magnetic unit cell is maintainedin the 1/3-, 5/9-, and 7/9-plateau phases, while the detail of magnetic structure could notbe determined at the present stage [see also Fig. S3] [27].With the magnetic phase diagram established, we proceed to discuss the charge transportproperties. Figure 3(a) shows the temperature dependence of resistivity (ρxx). At 0 T, theresistivity shows a sharp peak at TN1, attributed to the carrier scattering by spin fluctuation,while only a slight change in gradient is observed at TN2. Figure 4(a) shows the magnetore-sistivity measured for B||c at various temperatures. At 2 K, four anomalies due to thefield-induced phase transition are clearly observed in agreement with the results of magne-tization. With increasing temperature, the anomalies gradually diminish, and only a singlepeak remains at 14 K. Above 16 K, the peak structure is no longer observed, but the mag-netoresistivity exhibits a downturn around 3-7 T. The negative magnetoresistivity is mostpronounced near TN1, but remains observable even at high temperatures. To quantify thetemperature dependence of negative magnetoresistivity, we plotted the magnetoresistivity-ratio defined as ∆ρxx/ρxx(0 T) with ∆ρxx = [ρxx(14 T) − ρxx(0 T)] in Fig. 3(b). Thesignificant negative magnetoresistivity is observed not only near TN1 but also up to around60 K. This suggests that the spin fluctuation occurs over wide temperature range above TN1,resulting in the remarkable electrons scattering.6Figure 4(b) shows the Hall resistivity (ρyx) at various temperatures. At 2 K, ρyx in-creases almost linearly with the magnetic field up to 5 T and above 7.6 T, primarily due tothe ordinary Hall effect, and displays several anomalies due to the magnetic transitions inintermediate field region. As the temperature increases, the anomalies gradually diminishand evolve into a hump-like structure around 6 T at 12 K. Interestingly, the remnant of thehump-like structure persists even above TN1, leading to a B-nonlinear behavior of ρyx up toaround 40 K. Finally, the B-linear dependence is recovered at around 100 K.In general, the Hall effect in magnetic metals originates from the anomalous or topologicalHall effect in addition to the ordinary Hall effect. To qualitatively evaluate each contribution,we derived the Hall conductivity σxy = ρyx/(ρ2xx + ρ2yx), and analyzed it using the followingthree-component-model,σxy = σOxy + σMxy + σnotMxy (1)The first, second and third terms represent the ordinary Hall component [σOxy = ROB/(ρ2xx+ρ2yx)], the anomalous Hall component in proportion to M (σMxy = SAM) and the residualcomponent (σnotMxy ), which is proportional neither to B nor to M , respectively. Here, weassumed the intrinsic mechanism in the coherent transport regime for σMxy [1, 28]. Figure 5displays the analyzed results as well as the magnetization curve. At 2 K, σxy shows severalsteps or dips at field-induced magnetic transitions. Except these anomalies, σxy is well re-produced by the sum of σOxy and σMxy (σOxy + σMxy) as exemplified in Fig. 5(a2). Consequently,σnotMxy is vanishingly small over the entire magnetic field range [see Fig. 5(a3)]. At 14 K,σOxy + σMxy is slightly deviated from σxy around 6 T, resulting in a peak structure of σnotMxy .The deviation is much enhanced at 30 K, and consequently, σnotMxy exhibits a pronouncedpeak around 8 T. At 50 K, a similar behavior is observed, but σnotMxy appears to be slightlysuppressed. At 100 K, σxy is again well reproduced by σOxy + σMxy, resulting in the negligibleσnotMxy .To visualize the field-temperature variation of σnotMxy , we show the contour plot of σnotMxyon the temperature-field plane in Fig. 6(a). σnotMxy is small in the magnetically orderedphases but is remarkably enhanced near and above TN1 around 4-10 T, below which the1/3-, 5/9-, 7/9-plateaus and enforced ferromagnetic phases are keenly competing with eachother. We also show the temperature dependence of σMxy and σnotMxy at 8.5 T in Fig. 3(c).7σnotMxy is typically less than 50 Ω−1cm−1 below TN1, but remarkably increases above TN1,reaching about 150 Ω−1cm−1 around 30 K. At higher temperatures, it gradually decreasesand becomes substantially small at 80 K. On the other hand, σMxy is nearly temperatureindependent below TN2 but is discontinuously enhanced at TN2. In particular, the tangentof anomalous Hall angle (tan θAH) reaches more than 0.02 at 12 K. With increasing tem-perature, σMxy monotonically decreases but remains to be substantial at 80 K. We note thatσnotMxy reaches about one-half of σMxy around 30 K, whereas the former is smaller than thelatter in all temperature regions.To clarify the relationship between the magnetic ordering and anomalous Hall response,it is instructive to compare with the results for DyPtIn, which also crystallizes in the ZrNiAl-type structure but exhibits ferromagnetic ordering with canted spin structure at 27 K [29, 30].Figures 7 (a)-(d) show the resistivity, magnetization, σxy and σnotMxy , respectively. As shownin Fig. 7(a), the resistivity shows a typical metallic behavior with a small anomaly around 22K, which is slightly lower than the transition temperature in the previous report [see also theinset to Fig. 7(a)]. This is in contrast with the remarkable anomaly at TN1 for DyAgGe. Asshown in Fig. 7(b), the magnetization curve shows a simple ferromagnetic-like behavior withthe saturated magnetization about 6 µB/f.u.. This is consistent with the previous report,which demonstrates that the easy axis is tilted by about 37◦ from c-axis, similar to the caseof DyAgGe [30]. As shown in Fig. 7(c), σxy shows a step-like structure below 1 T and nearlylinearly increases with the magnetic field at 2 K. The former and latter are attributed tothe anomalous Hall component and ordinary Hall component, respectively. As temperatureincreases, the step-like anomaly becomes progressively smeared out, leading to a broad diparound 4 T at 40 K. Notably, σxy is well fitted by σOxy + σMxy over the entire temperaturerange, resulting in the small σnotMxy . These results suggest that the enhanced σnotMxy above TN1in DyAgGe originates from the unconventional spin fluctuation associated with the multiplefield-induced phase transitions, or equivalently, competing magnetic phases.On the basis of these results, we consider the mechanism of AHE persisting above themagnetic transition temperature. At first, we consider the mechanisms of σMxy. One possiblemechanism is conventional skew scattering from spin fluctuation. However, it is recognizedthat the magnitude of skew scattering is not significant unless the charge transport is in thehighly conductive regime ρxx < 10−5 Ωcm. In the present material, the resistivity is morethan about 5×10−4 Ωcm [see Fig. 3(a)]. Therefore, the relatively large tan θAH (> 0.02) may8not be explained by the conventional skew scattering mechanism. More plausible scenario isthe intrinsic mechanism (Berry curvature mechanism) in momentum space. Indeed, recentstudy argues that the Weyl nodes emerge near the Fermi energy in HoAgGe [26]. Thesignificant Berry curvature due to Weyl nodes may also induced σMxy with relatively largeanomalous Hall angle in the present material.On the other hand, σnotMxy gradually increases above TN1 and reaches the maximum around30 K, indicating that its origin is different from that of σMxy. The enhancement of σnotMxy abovemagnetic transition temperature is unconventional, but recent studies suggest that the SSCinduced by thermal spin fluctuation can cause the anomalous Hall effect or topological Halleffect even above the magnetic transition temperature [16, 17, 19, 31, 32]. It is well knownthat the sum of the thermally induced SSC among three neighbouring sites on the kagomelattice over the entire crystal is finite, which manifests as the finite Hall signal [5–7]. InFig. 6 (b), we plot σnotMxy as a function of the magnetization normalized by the saturationmagnetization (M/Ms), where Ms is defined as the magnetization at 2 K and 14 T. Near andabove TN1, σnotMxy at various temperatures commonly exhibits a maximum around M/Ms =0.6−0.8, indicating that σnotMxy is enhanced in partially spin polarized state. Similar AHE orTHE are often observed in ferromagnets or frustrated magnets with complex spin texture,which are attributed to the SSC-induced Berry phase or multiple skew scattering [13, 17–19,33–35]. In this context, it is likely that σnotMxy originates from the SSC-induced Berry phaseor multiple skew scattering mechanism in the present material. Recent theoretical studiesbased on J1-J2-J3 Ising model on the kagome lattice propose the emergence of classicalspin liquid phase with 1/9-, 1/3-, 5/9- and 17/27-magnetization plateaus, which can beviewed as the self-organization of unconventional domain-wall structure or spin clusters[36, 37]. Although the magnetic structure of each plateau phase is not clear in DyAgGe,the fluctuation of similar domain-wall structure or cluster spin excitation promoted by themultiple phase competition may cause the SSC and resultant AHE persisting in the widetemperature-field region.IV. CONCLUSIONIn this study, we have investigated the magnetic property and charge-transport for thekagome antiferromagnet DyAgGe, which crystallizes in the ZrNiAl-type structure, with a9focus on the anomalous Hall effect due to the spin fluctuation. The magnetization curveshows the 1/9-, 1/3-, 5/9- and 7/9-plateau under the magnetic field before reaching theforced ferromagnetic phase at 2 K. The 1/9-plateau (5/9-plateau) phase is observed in nar-row region below 1 T (around 5 T), below 10-11 K, respectively. On the contrary, the 1/3-and 7/9-plateau phases appear over relatively wide field ranges and turn into the param-agnetic state around 13-15 K. The results of resonant magnetic X-ray scattering show thatthe magnetic superlattice reflection with the wave vector (1/3, 1/3, 0) commonly appearsin these plateau phases, indicating the antiferromagnetic or ferrimagnetic ordering with the√3×√3 unit cell. The negative magnetoresistivity becomes remarkable around 3-7 T and15 K, below which the 1/3-, 5/9- and 7/9-plateau phases are strongly competing with eachother. Moreover, the anomalous Hall conductivity not proportional to magnetic field or mag-netization grows up above 15 K with increasing temperature. In particular, the anomalousHall conductivity exhibits a maximum around 30 K and persists up to approximately 60 K.As a comparison, we also examined the magnetic property and magneto-transport propertyfor the isostructural DyPtIn without complex magnetic transitions. In this material, thepronounced anomalous Hall resistivity not proportional to the magnetic field or magneti-zation is not observed both below and above the magnetic transition temperature. Basedon the analysis of Hall conductivity, it is likely that the thermal spin fluctuation, enhancedby the competing multiple magnetic phases, gives rise to the scalar-spin-chirality-relatedanomalous Hall effect persisting well above the magnetic transition temperature.ACKNOWLEDGEMENTSThis work was partly supported by Grant-In-Aid for Science Research (Nos. 21K18813,22H01177, 24K03205, 24H01685, 25K01657) from the MEXT, by Iketani Foundation forMaterials Science and Engineering, Japan, by JST FOREST Program (Grant Number:JPMJFR203D), and by JST-CREST (Grant Number: JPMJCR2435). MANA is supportedby World Premier International Research Center Initiative (WPI), MEXT, Japan. 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(c) The temperatureof magnetization of DyAgGe for B||c . Open and closed triangles denote TN1 and TN2, respectively.(d) The field dependence of magnetization of DyAgGe for B||c. The dashed line denotes the 1/9,1/3, 5/9 and 7/9-plateau. The inset shows the magnified view of magnetization for 5/9-plateau.140.50.40.30.20.10.06543210Magnetic Field (T)1.20.80.40.0T=4.5 K(2/3, 8/3, 0)Bc1=1.2 T Bc2=4.7 TBc3=5.1 TB || cFig. 2: phase diagram60402000.690.680.670.660.650.64HT=4.5 K(H, 8/3, 0) 0 T 2.0 T 4.0 T 4.4 T 4.8 T 4.9 T 5.1 T 5.5 T 6.0 TB || c(b)(c)(a)14121086420Magnetic field (T)2520151050Temperature (K)Bc1Bc2Bc3Bc4Tc1Tc2B || cFM7/95/91/31/9FIG. 2. (a) The magnetic phase diagram for DyAgGe. Closed (open) symbols are determined fromthe field (temperature) dependence of magnetization. (b) The resonant magnetic X-ray scatteringaround (H, 8/3, 0). The incident beam is π-polarized, while the scattered X-ray contains bothσ′- and π′-polarization. (c) The closed circles and open squares denote the field dependence ofscattering intensity and full width at half maximum (FWHM) of (2/3, 8/3, 0) reflection. The thickline is the guide to eyes.15-0.5-0.4-0.3-0.2-0.10.0B || c4003002001000806040200Temperature (K)8.5 TTN1TN2B || cσxyσxyMnot M806040200ρ xx (μΩcm)TN1=15 KTN2=12 KI || abFig. 3(b)(c)(a)(b)を14 Tに変更(c)データラベルを修正FIG. 3. (a) The temperature dependence of resistivity in the ab-plane under the zero magneticfield. (b) The temperature dependence of magnetoresistivity-ratio defined as ∆ρxx/ρxx(0 T) with∆ρxx = [ρxx(14 T)−ρxx(0 T)]. (c) The anomalous Hall conductivity in proportion to magnetizationσMxy (open triangles) and that not proportional to magnetic field and magnetization σnotMxy (closedcircles) at 8.5 T.16353025201510501050Magnetic field (T)2 K8 K12 K14 K16 K20 K30 K60 K100 K40 K80 KB || cρ xx (μΩcm)Fig. 4(a) (b)磁場上げ過程のデータに変更FIG. 4. (a) The magnetoresistivity (b) the Hall resistivity for DyAgGe. ρxx (ρyx) below 100 K isoffset by 20µΩcm (2µΩcm) for clarity. The magnetoresistivity and Hall resistivity were measuredas the magnetic field was increased. The closed circle, closed triangle, open triangle and closedsquare denote the magnetic transitions at Bc1, Bc2, Bc3 and Bc4, respectively. The open circle in(b) shows the hump-like structure of ρyx (see also the main text).17200150100500-5012840B (T)3.02.01.00.0Fig. 5: Electrical conductivity and Hall conductivity864202 K12840B (T)12840B (T)12840B (T)12840B (T)(a1) (b1)(a2)(a3)(b2)(b3)(c1)(c2)(c3)(e1)(e2)(e3)(d1)(d2)(d3)FIG. 5. The magnetization M , Hall conductivity σxy and Hall conductivity not proportional tothe magnetic field and magnetization σnotMxy for DyAgGe. The dashed curves in (a2)-(e2) denotethe sum of the ordinary Hall conductivity and anomalous Hall conductivity proportional to M(σOxy + σMxy). The vertical dashed lines denote the magnetic transitions at Bc1, Bc2, Bc3 and Bc4.18200150100500σ xy     (Ω-1cm-1)1.00.80.60.40.20.0M / Ms 14 K 16 K 20 K 24 K 30 K 34 K 40 KnotMFig. 6(a) (b)FIG. 6. (a) The contour plot of anomalous Hall conductivity not proportional to M (σnotMxy ) forDyAgGe. The closed circle, closed triangle, open triangle and closed square denote the magnetictransitions at Bc1, Bc2, Bc3 and Bc4, respectively. (b) σnotMxy as a function of normalized magneti-zation (M/Ms) with Ms being the magnetization at 2 K and 14 T.19σ xy    (Ω-1cm-1)notMFig. 786420Magnetization (μ B/f. u. )12840Magnetic field (T)2 K18 K30 K40 KB || cDyPtIn-80-4004080 2 K 18 K 30 K 40 KB || cDyPtInρ xx (μΩcm)(a)(b)(c)(d)118.0117.0116.0115.03020100Temperature (K)FIG. 7. (a) The temperature dependence of resistivity for DyPtIn under zero magnetic field. (b)Magnetization curve of DyPtIn for B||c. (c) The anomalous Hall conductivity proportional to M(σMxy). The dashed curves denote the sum of the ordinary Hall conductivity and anomalous Hallconductivity proportional to M (σOxy + σMxy). (d) Anomalous Hall conductivity not proportional toM (σnotMxy ).20