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Moyu Kato, Yasuo Narumi, Katsuhiro Morita, [Yoshitaka Matsushita](https://orcid.org/0000-0002-4968-8905), Shuhei Fukuoka, Satoshi Yamashita, Yasuhiro Nakazawa, Migaku Oda, [Hiroaki Hayashi](https://orcid.org/0000-0001-7787-9082), [Kazunari Yamaura](https://orcid.org/0000-0003-0390-8244), Masayuki Hagiwara, Hiroyuki K. Yoshida

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[One-third magnetization plateau in Quantum Kagome antiferromagnet](https://mdr.nims.go.jp/datasets/c5d6cbe9-4f15-44e9-b7db-110af94480b9)

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One-third magnetization plateau in Quantum Kagome antiferromagnetcommunications physics Articlehttps://doi.org/10.1038/s42005-024-01922-0One-third magnetization plateau inQuantum Kagome antiferromagnetCheck for updatesMoyu Kato1, Yasuo Narumi 2, Katsuhiro Morita 3, Yoshitaka Matsushita 4, Shuhei Fukuoka 1,Satoshi Yamashita5, Yasuhiro Nakazawa 5, Migaku Oda1, Hiroaki Hayashi4,6, Kazunari Yamaura 4,6,Masayuki Hagiwara 2 & Hiroyuki K. Yoshida 1The emergence of nontrivial quantum states from competing interactions is a central issue in quantummagnetism. In particular, for the realization of the quantum spin-liquid state, extensive studies havebeen conducted on frustrated systems, such as kagome antiferromagnets and Kitaevmagnets. Novelquantum states in magnetic fields have remained elusive despite the prediction of rich physics. Thiscan be attributed to material scarcity and the difficulty of precise measurements under ultra-highmagnetic fields. Here, in this study, we develop theKapellasite-type compound InCu3(OH)6Cl3, whoseexchange interactions are in appropriate energy scale to comprehensively elucidate the magneticproperties of the frustratedS = 1/2 kagome antiferromagnet. The one-thirdmagnetization plateauwasclearly observed.Moreover, the large temperature-linear term in the heat capacitywas observed in themagnetic fields, indicating the excitation of gapless quasiparticles in the vicinity of the plateau. Theseresults shed light on the critical behaviors between quantum spin-liquid and -solid in kagomeantiferromagnets under high magnetic fields.The physics arising from kagome geometry provides understandings ofcondensed matter physics. Recently, not only a quantum spin liquid ininsulators but also the topological properties of the Dirac point and flatband, and exotic superconductivity have been discussed1–3. In particular,kagome frustratedmagnetism researchhas prioritized quantumspin liquidsfor a long time. Numerous theoretical models have been proposed for thequantum spin liquid, including Z2 topological and U(1) Dirac models4;however, microscopic properties such as the nature of magnetic excitationspectra have not yet been determined.On the other hand, the observation ofspin liquid has been proposed in S = 1/2 kagome antiferromagnet Her-bertsmithite (ZnCu3(OH)6Cl2) based on the continuum excitation spectraof inelastic neutron scattering experiments5.Furthermore, the formation of various quantum states in magneticfields theoretically predicted have been one of the intriguing interests. In theS = 1/2 kagome model considering a nearest-neighbor antiferromagneticinteraction and the Zeeman term in the Hamiltonian, the magnetizationjumps due to the collective excitation of resonant hexagonal magnons fromthe forced ferromagnetic state and (2n+ 1)/9 (n = 1, 2, and 3) plateaus ofhexagonal magnon crystallization inffiffiffi3p×ffiffiffi3pcells are expected6–8. Inaddition, a magnon supersolid state just below the 5/9 plateau is expectedwhen an intermagnon interaction is incorporated9. These are closely relatedto macroscopic quantum phenomena based on resonant hexagonal mag-non excitations and their bosonic statistics, as well as the localized/itinerantnature caused by interactions between excited quasiparticles. Moreover, the1/9magnetization plateau is also anticipated to be a field-induced quantumspin liquid, although the microscopic state is still controversial10,11. Eluci-dating these quantum many-body states is of great importance for under-standing kagome-frustrated magnetism.On the other hand, from an experimental perspective, actual quantumkagome antiferromagnets typically possess a large nearest-neighbor inter-action J1 which requires ultrahighmagnetic fields, often exceeding 100 T, toreach the saturation field12. Thus, it is generally difficult to search forquantum states despite the rich expectations of field-induced quantumstates. There are few examples except for a weak anomaly near the 1/3magnetization in Herbertsmithite around 190 T13, a metastable 1/3 plateauin Cu-titmb14, successive plateaus as candidates of the hexagonal magnoncrystallizations in CdCu3(OH)6(NO3)2·H2O (Cd-kapellasite)15, and a 1/9magnetization plateau in YCu3(OD)6+xBr3-x16.Here, we focused on Kapellasite-type compounds as quantumkagome magnets. Because the nonmagnetic ion is located at the center ofthe hexagon, a finite next-nearest-neighbor J2 and third-nearest-neighborJd across the diagonal of the hexagon are expected. These interactions and1Department of Physics, Faculty of Science,HokkaidoUniversity, Sapporo,Hokkaido, Japan. 2Center for AdvancedHighMagnetic FieldScience,GraduateSchoolof Science, Osaka University, Osaka, Japan. 3Department of Physics and Astronomy, Faculty of Science and Technology, Tokyo University of Science,Chiba, Japan. 4National Institute forMaterials Science (NIMS), Tsukuba, Ibaraki, Japan. 5Department of Chemistry, Graduate School of Science, Osaka University,Osaka, Japan. 6Graduate School of Chemical Sciences and Engineering, Hokkaido University, Sapporo, Hokkaido, Japan. e-mail: hyoshida@sci.hokudai.ac.jpCommunications Physics |           (2024) 7:424 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s42005-024-01922-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s42005-024-01922-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s42005-024-01922-0&domain=pdfhttp://orcid.org/0009-0000-4746-6535http://orcid.org/0009-0000-4746-6535http://orcid.org/0009-0000-4746-6535http://orcid.org/0009-0000-4746-6535http://orcid.org/0009-0000-4746-6535http://orcid.org/0009-0006-2087-5674http://orcid.org/0009-0006-2087-5674http://orcid.org/0009-0006-2087-5674http://orcid.org/0009-0006-2087-5674http://orcid.org/0009-0006-2087-5674http://orcid.org/0000-0002-4968-8905http://orcid.org/0000-0002-4968-8905http://orcid.org/0000-0002-4968-8905http://orcid.org/0000-0002-4968-8905http://orcid.org/0000-0002-4968-8905http://orcid.org/0000-0001-7291-7617http://orcid.org/0000-0001-7291-7617http://orcid.org/0000-0001-7291-7617http://orcid.org/0000-0001-7291-7617http://orcid.org/0000-0001-7291-7617http://orcid.org/0000-0002-6783-6430http://orcid.org/0000-0002-6783-6430http://orcid.org/0000-0002-6783-6430http://orcid.org/0000-0002-6783-6430http://orcid.org/0000-0002-6783-6430http://orcid.org/0000-0003-0390-8244http://orcid.org/0000-0003-0390-8244http://orcid.org/0000-0003-0390-8244http://orcid.org/0000-0003-0390-8244http://orcid.org/0000-0003-0390-8244http://orcid.org/0000-0002-1087-521Xhttp://orcid.org/0000-0002-1087-521Xhttp://orcid.org/0000-0002-1087-521Xhttp://orcid.org/0000-0002-1087-521Xhttp://orcid.org/0000-0002-1087-521Xhttp://orcid.org/0009-0004-7971-1311http://orcid.org/0009-0004-7971-1311http://orcid.org/0009-0004-7971-1311http://orcid.org/0009-0004-7971-1311http://orcid.org/0009-0004-7971-1311mailto:hyoshida@sci.hokudai.ac.jpwww.nature.com/commsphysthe nearest-neighbor interaction J1 compete, significantly affecting theground state of kagome magnets and leading to the formation of variousmagnetic orders, such as Cuboc1 and Cuboc2 structures, even in classicalsystems17,18. Moreover, in quantum systems, the formation of spin liquidsowing to quantum fluctuations around the phase boundaries isanticipated19,20. Therefore, the creation of quantum materials in thekapellasite system and the studies of their magnetism will promote thediscovery of quantummany-body states such as quantum spin liquids andmagnon crystal states, and will greatly facilitate our understanding of thephysics of frustrated kagome quantum spin systems. In fact, exoticmagnetic states have been observed, such as a spin liquid with Cuboc2correlations in Zn-kapellasite21,22, ferromagnetic order with spin fluctua-tion inMg-kapellasite23, the coexistence of negative vector chirality (NVC)order and spin liquid in Ca-kapellasite, and NVC order in Cd-kapellasiteand Y-kapellasite15,24–34. In the Y-kapellasite derivative compoundY3Cu9(OH)19Cl8, the dominant antiferromagnetic J1 has a complicatedspatial distribution. From a theoretical consideration of the magnitudeand spatial distribution of J1, phase diagrams that includemagnetic phasessuch as classical spin liquids have been proposed35,36.We synthesized a material InCu3(OH)6Cl3 as a member of theKapellasite-type kagome magnets and successfully found a 1/3magnetization plateau in high-magnetic fields. In this paper, wereport the results of structural analysis, magnetic susceptibility, heatcapacity, and high-field magnetization with theoretical calculation ofIn-kapellasite, which present insights into kagome frustratedmagnetism.Results and discussionCrystal structure, magnetic susceptibility and heat capacityFigure 1 shows the crystal structure of In-kapellasite. In-kapellasite crys-talizes in the space group P31m and the lattice constants area = 11.3235(6) Å and c = 6.0347(4) Å. The kagome layer is composed ofedge-shared CuO4Cl2 octahedra, in which the dx2-y2 orbital of the Cu2+ ionis occupied by an unpaired electron. There are threefold rotational axes onthe In sites, although each triangle of the Cu2+ ions is distorted into anisosceles triangle. Two isosceles triangles exist in the unit cell, one with twolong bonds and another with two short bonds. In the ab-plane, they alter-nately share vertices to form a kagome network with a threefold rotationalsymmetry. Because the threefold-axes are preserved, symmetry loweringsuch as the one-dimensionalization of the exchange interaction, which isoften observed in kagome systems with monoclinic distortion, is notexpected to occur12.Fig. 1 | Perspective view of the crystal structure ofIn-kapellasite. a, bWhole structure and in-planecrystal structure of In-kapellasite. The configurationof dx2-y2 orbitals occupied by the unpaired electronsof Cu2+ is depicted. c The four sets of Cu-Cu bonddistances and Cu-O-Cu bond angles, colored withred, blue, yellow, and green lines. The red bond hasthe longest Cu-Cu distance and largest Cu-O-Cubond angle, and the green bond has the shortestdistance and smallest angle.https://doi.org/10.1038/s42005-024-01922-0 ArticleCommunications Physics |           (2024) 7:424 2www.nature.com/commsphysThe temperature dependence of magnetic susceptibility measured forrandomlyoriented single crystals is shown inFig. 2. It obeys theCurie-Weisslaw above 40 K, and the Weiss temperature ΘW=−10.2 K and effectiveBohr magneton number peff = 1.89 were estimated from linear extrapola-tion. The magnetic susceptibility was reproduced well by high-temperatureseries expansion (HTSE) up to the 15th order, assuming a uniform anti-ferromagnetic J1 on a regular kagome network with S = 1/237. The J1 and g-values were estimated to be 11.5 K and 2.24, respectively. This interaction issmaller than those of Ca-, Cd-, and Y-kapellasites at 52, 45, and 99 K,respectively25,32,33. With decreasing temperature, a weak anomaly wasobserved at approximately 7 K, where the susceptibilities showed furtherincreases, deviating from the Curie-Weiss law. This anomaly was main-tained even at a magnetic field of 7 T, although the increase was slightlysuppressed upon increasing themagneticfield.No cusps or other anomalieswere observed at temperatures down to 2 K.The temperature dependence of the total specific heatmeasured for thesmall coaligned single crystals is shown in Fig. 3a. A broad peak wasobserved atTs = 7 Kwith zeromagnetic field, indicating the formation of anantiferromagnetic short-range order (SRO). A theoretical study predictedcharacteristic anomalies in heat capacity, where broad peaks and shoulder-like anomaly successively appeared at T/J ~ 0.6 and ~ 0.1, respectively37–39.The SRO temperatureTs /J1 ~ 0.6 is consistent with the broad peak positionof the theoretical expectation. Because Ts corresponds to the temperature atwhich the magnetic susceptibility increases further as illustrated in Fig. 2b,the enhancement in susceptibility is due to the formation of the SRO. Tsdecreases to 4.4 Kwhenapplyingmagneticfields of up to8 T, but in strongermagnetic fields the peak temperature increases up to 5 K at 14 T. With theapplication of magnetic fields, the peak becomes prominent, and moreentropy is released. Lower temperaturemeasurements in zeromagneticfieldrevealed a clear peak at TN = 1.8 K, indicating the development of a mag-netic long-range order (LRO). The peak shifts to lower temperatures withincreasing appliedmagnetic field and finally disappears at 8 T.Only a broadpeak exists at a low-temperature specific heat above 8 T.A temperature-linear (T-linear) term is observed under highmagneticfields. At low fields, it is difficult to estimate the T-linear term in the mea-sured temperature region because of the observed divergence in the specificheat associated with the LRO. However, above 10 T, the peak is suppressed,and an extrapolation of the C/T data has a finite value at T = 0, as is clearlyseen in the low-temperature region of the 14 T data for the C/T vs. T2 plot(Fig. 3b). The coefficient of the T-linear term was estimated to be 59.8 mJCu-mol−1 K−2 at 14 T. T-linear terms in insulators have been observed insome spin liquid candidates, such as κ-(BEDT-TTF)2Cu(CN)340,EtMe3Sb[Pd(dmit)2]241, andCa-kapellasite25, and their possible origins havebeen discussed in accordancewith the spinon Fermi surface and anisotropicweathervane excitation in the fluctuating zero-field ground state. In In-kapellasite, gaplessmagnetic quasiparticles are excited under highmagneticfields, which characterizes the peculiarity of this compound.To extract the magnetic entropy, we estimated the lattice contribu-tion by fitting the data above 40 K based on the Debye model as shown inFig. 2b42. We found that magnetic entropy of approximately 3 J Cu-mol−1 K−1 was released below 25 K under zero magnetic field. This sug-gests that approximately half of the magnetic entropy for S = 1/2 remainsbelow TN, which may appear as a T-linear term for In-kapellsite.Fig. 2 | Temperature dependence of bulk physical properties in magnetic fields.a Temperature dependence of magnetic susceptibility and its inverse measured atH = 1 T down to 2 K. The dashed black and solid yellow lines indicate the results ofCurie-Weiss and 15th HTSE fitting, respectively. b Low temperature magneticsusceptibility measured in several magnetic fields and heat capacity obtained atH = 0 T. The vertical dotted line indicates the short-range order temperature wherethe heat capacity shows the broad peak and the susceptibilities suddenly increase.The solid line is an approximated lattice contribution of heat capacity.Fig. 3 | Temperature dependence of the heat capacity of In-kapellsite.a Temperature dependence of heat capacity measured in several magnetic fieldsapplied along the c-axis of coaligned small single crystals. The data measured at eachfield is offset by 0.2. bC/T vs.T2 plot in variedmagneticfields. The solid line indicatesthe linear fitting.https://doi.org/10.1038/s42005-024-01922-0 ArticleCommunications Physics |           (2024) 7:424 3www.nature.com/commsphysHigh field magnetization and its theoretical analysesFigure 4 depicts the results of pulsed high-field magnetization measure-ments performed on randomly oriented single-crystal samples. The abso-lute value was corrected by themagnetization datameasured with a SQUIDmagnetometer, and the horizontal axis was normalized by J1 = 11.5 K. At1.3 K, the slope of magnetization, dM/dH, decreases at approximatelyH/J1 = 0.8 and increases again at approximately H/J1 = 1.6. Subsequently, apeak is observed atH/J1 = 3. Since dM/dH becomes small in the field regionfromH/J1 = 0.8 to 1.6 and the magnetization value in this region is close to1/3 of the saturation magnetization (Ms), this magnetization region isconsidered as a 1/3 magnetization plateau, though it does not show perfectflatness. Recently, the asymmetric plateau melting phenomenon at finitetemperatures has been theoretically discussed, in which kagome’s plateau isnot essentially flat under finite temperatures38,39,43. It may be related to thenon-flatness of 1/3 plateau in this study, however precise measurementsusing single crystals would be required to conclude. Similar behavior wasobserved at 4.2 K, but it was blunted.To better understand this magnetization behavior, we theoreticallyinvestigated the effects of finite temperature on the magnetizationprocess of a kagome antiferromagnet with uniform J1 on a 36-site clusterusing the orthogonalized finite-temperature Lanczosmethod (OFTLM).In Fig. 4, in addition to some finite-temperature calculation results, themagnetization obtained using the grand canonical density matrixrenormalization group (DMRG) method at T = 0 with a uniform J1 isrepresented by a black solid line10. We found that the magnetizationcurve measured at 1.3 K (T/J1 ~ 0.1) was in good agreement with thetheoretical curve at T/J1 = 0.1, although there was a difference nearsaturation. This indicates that the magnetic network of In-kapellasitecan be regarded as an ideal quantum kagome antiferromagnet consistingof uniform J1. The difference around the saturation field is due to theDzyaloshinskii-Moriya (DM) interaction, as observed for Cu benzoate.DM interaction creates a staggered field which significantly blunts themagnetization process near the saturation field; in particular, magneti-zation exhibits anisotropy depending on whether the external magneticfield is perpendicular or parallel to the staggered magnetic field44.Actually, anisotropic behavior was observed in our preliminary mag-netization measurements for In-kapellasite. Next, the theoretical mag-netization at T = 0, which includes multiple plateaus, was severelyblunted by finite-temperature effects, even at T/J1 = 0.1. To extractinformation from the magnetization process at finite temperature, wecompared the theoretical (T/J1 = 0.1) and experimental (T/J1 ~ 0.1(1.3 K)) dM/dH curves in Fig. 4; in practice, we need to compare thecharacteristic structure of the dM/dH around the 1/3 plateau and thesaturation, where the dM/dH change most significantly. Character-istically, theoretical dM/dH begins to decrease rapidly in the magneticfield where the plateau sets in, takes the smallest value at the end of theplateau, and then shows a peak at saturation. The experimental dM/dH,which decreases from H/J1 ~ 0.8 (7.0 T) and reaches a minimum at 1.6(13.7 T) and takes a peak at H/J1 ~ 3 (25.7 T), shows similar behavior tothe theoretical one. The experimental plateau width is comparable to thetheoretical width. Moreover, the value of magnetization between7.0 – 13.7 T was close to Ms/3. The agreement of the experimental andtheoretical results indicates that this behavior corresponds to the 1/3magnetization plateau, which should be a typical example of a 1/3 pla-teau in S = 1/2 kagome antiferromagnets12. It is generally difficult todetermine the plateau width from experiments at finite temperatures. Asdemonstrated here, a comparison of magnetization measurements andtheoretical calculations for In-kapellasite provides a good guide fordetermining the plateau width. In contrast, the 1/9, 5/9, and 7/9 plateauswith narrow widths were not observed in our experiments. This ismainly due to the finite temperature effect, which significantly obscuresthe fine structure of the magnetization curve; thus, further anomaliesmay be observed by performing magnetization measurements at lowertemperatures.Fig. 4 | Whole magnetization process of In-kapellasite.Magnetization curves normalized by thesaturation value (Ms) in high magnetic fields (upperpanel) and its differential dM/dH (lower panel). Themagnetization curve at 4.2 K is offset by 0.3. Theabsolute value of pulsed magnetization data wascorrected by the magnetization data obtained with aSQUID magnetometer at 4.2 K. The horizontal axisH/J1 is normalized by the nearest-neighbor anti-ferromagnetic interaction J1 = 11.5 K determined bythe HTSE fitting of magnetic susceptibility. Thedashed red line shows the calculated finite tem-perature magnetization curve assuming uniformnearest-neighbor interaction at T/J1 = 0.1 by theOFTLM method with N = 36. The dashed-dottedline shows the theoretical curve assuming threenearest-neighbor interactions with the ratio J1′(thick, solid, red) : J1 (thin, solid, blue) : J1″ (dotted,green) = 2 : 1 : 0.5 of which the spatial distribution isdepicted in the inset of the upper panel. The blackvertical arrows in the lower panel indicate thestarting and terminating magnetic fields of the 1/3plateau calculated with a DMRGmethod10. Blue andpurple vertical arrows show the experimental mag-netic fields of the 1/3 plateau corresponding to thecalculated one. Since the 4.2 K data is blunted, thelower end of the plateau field is determined by theintersection of extrapolated lines as shown in theupper panel. The saturation fields H/J1 = 3 for both1.3 K and 4.2 K are determined by the peak top ofdM/dH. The weak oscillation of theoretical dM/dHbetweenH/J = 1.5 – 2 in the lower panel is due to thefinite size effect.https://doi.org/10.1038/s42005-024-01922-0 ArticleCommunications Physics |           (2024) 7:424 4www.nature.com/commsphysPhase diagram and consideration of magnetic interactionsTheabove results are summarized in the temperature-fieldphasediagram inFig. 5. At zeromagnetic field, the antiferromagnetic SROdevelops belowTs;subsequently, the LRO is formed below TN = 1.8 K. This ordered phase issuppressed by magnetic fields of approximately 7 T. However, the SROpersists even at 8 T, although it shifts slightly to lower temperatures. The 1/3magnetization plateau phase appeared upon the application of magneticfields above 7 T, which protruded to higher-temperature regions comparedto the ordered phase. These results suggest that themagnetic states of the 1/3plateau and the LRO are not directly related.TheLROphase of In-kapellasite is considered to be inducedby theDMinteraction. The ratio of the DM (D) and nearest-neighbor interaction Jsignificantly affects the ground state of a quantumkagome antiferromagnet,and the theoretical critical point between the quantum spin liquid and themagnetic ordered state is aroundD/J ~ 0.145. The ratio ofD/J1 =Δg/g ~ 0.12was estimated using the g-value obtained from the HTSE fitting for In-kapellasite, which should disturb the spin liquid formation and stabilize theLRO. In fact, the Néel temperature 1.8 K roughly agrees with the DMinteraction D ~ 1.4 K, supporting the DM induced LRO scenario46. Thedecrease in TN may be explained by the effective suppression of the DMinteraction in the magnetic field, because the energy scale of the magneticfield of 7 T (~ 10 K) is larger than that of the DM interaction. Thus, themagnetismof the system is expected to be dominated by J1 and themagneticfield. The plateau phase appeared in such a magnetic field regime.In the case of a triangular lattice antiferromagnet, for comparison, the1/3 magnetization plateau with the up-up-down (UUD) arrangement isconsidered as the field-induced spin structure changed from the low-field120° state, as observed in CsCuCl347, Ba3CoSb2O948, RbFe(MoO4)249, andRb4Mn(MoO4)350. This indicates that the effects of the temperature as wellas magnetic field are very different from that for In-kapellasite. The 1/3plateau state with the UUD structure in these triangular lattice antiferro-magnets is in the LRO state. Thus, the temperature dependence of the heatcapacity in the magnetic fields of the plateau region exhibits a clear peak,which indicates the breaking of the translational symmetry of the lattice. Onthe other hand, in In-kapellasite, the temperature dependence of the heatcapacity in the magnetic fields of 10 and 14 T show a broad peak when theplateau state is stabilized. Although there is entropy release upon enteringthe plateau phase, it is not a typical λ-type peak of the second-ordertransition. If the 9-sites UUUUUUDDD structure or valence bond crystalstate7,10 is realized in the 1/3 plateau of In-kapellasite with a 9-site structuralunit cell, the translational symmetry of the lattice is not broken. We believethis corresponds to the absence of a sharp peak in our heat capacitymeasurements.Todate, the 1/3plateau state in quantumkagomeantiferromagnets hasnot been experimentally investigated. In particular, magnetic excitationsnear the plateau state are quite difficult to predict even with current theo-retical treatments. The observation of the 1/3 plateau with exotic excitationrepresented by the T-linear term at relatively low magnetic fields in In-kapellasite paves theway for the experimental verification of the anomalousproperties of quantum kagome antiferromagnets in high magnetic fieldssuch as magnon crystals and magnon BEC, etc.Finally, we consider the effects of structural distortion. In the Kapel-lasite series compounds without structural distortion, the Cu-O-Cu bondangle dependence of the nearest-neighbor superexchange interactionstrength was investigated using density functional theory (DFT)calculations19. As shown in Fig. 1, In-kapellasite has four Cu-O-Cu anglesbetween the nearest neighbor Cu2+ ions, and thus the four different inter-actions would be spatially distributed; based on DFT, 114.8°, 111.6°, 110.4°,and 108.6° bonds correspond to magnetic interactions of 60, 33, 27, and15 K, respectively. The validity of this situation was examined by theoreticalcalculations of the M-H curve. Because two of the four interactions withbond angles 111.6° and 110.4° are approximately equal to the averagemagnitude at 30 K, we consider for simplicity that there are three interac-tions J1, J1′, and J1″ with the ratio of J1′ : J1 : J1″ = 2 : 1 : 0.5. The spatialdistributions of these three interactions are shown in the inset of Fig. 4.There are two types of hexagons: hexagons with uniform interactions andhexagons with alternating large and small interactions. This is related to themagneticmodel of Y3Cu9(OH)19Cl8, although the spatial distribution of thehexagons differs35,36. However, when calculating the magnetization processaccording to this magnetic model, the 1/3 plateau was significantly stabi-lized, and the calculatedmagnetization did not reproduce the experimentalone, as depicted in Fig. 4.This result supports the idea that the nearest-neighbor interaction ofIn-kapellasite is essentially isotropic. Thismay be attributed to the degree oflocal orbital overlap caused by structural distortion. Owing to structuraldistortion, In-kapellasite possesses four nearest-neighbor bond lengthsbetween Cu2+ ions, which affects the degree of overlap of the dx2-y2 orbitalsof the Cu2+ ions. This factor also contributes to the magnitude of thesuperexchange interaction. Accordingly, the contribution of the four Cu-O-Cu bond angles to the superexchange interaction is compensated for by thefour bond lengths; for example, bonds with large Cu-O-Cu angles havesmall orbital overlaps due to large Cu-Cu distances, and vice versa. There-fore, the magnetic network of this system may be considered isotropic, asevidenced by the reproduction of the magnetic susceptibility and magne-tization curve by assuming only uniform J1.In this study, a temperature-field phase diagram is proposed for theS = 1/2 Kapellasite-type kagome antiferromagnet InCu3(OH)6Cl3 whichexhibits a 1/3 magnetization plateau. Importantly, the plateau state with anapparent 1/3 magnetization value is realized. Further microscopic andthermal investigations of thismaterial inmagnetic fields are required to leadthe frontier of high-fieldmagnetism of frustrated kagome antiferromagnets,such as the characterization of the dynamic properties of themagnetizationplateau as a quantum many-body state and the search for a hidden exoticmagnetic phase, such as a spin nematic state.MethodSample preparation and structural analysisA crystalline powder sample of InCu3(OH)6Cl3 was synthesized using ahydrothermal technique in a stainless-steel autoclave. Chemical reagents ofindium nitride and copper chloride with distilled water and lithiumhydroxide as a catalyst were put into the autoclave, and heated for 24 h at220 °C. Typical particle size of crystal is 0.1 mm in thickness and 0.2 mm indiagonal length.Fig. 5 | Temperature-field phase diagram of In-kapellasite.Magnetic phase dia-gram determined from the bulk magnetization and heat capacity measurements.Solid lines are eye-guides for each phase.https://doi.org/10.1038/s42005-024-01922-0 ArticleCommunications Physics |           (2024) 7:424 5www.nature.com/commsphysFor crystal structural analysis, a greenish-blue platelet crystal having0.088 × 0.075 × 0.112mmwas measured at 297 K on a Rigaku Saturn CCDdiffractometer with VariMax confocal optical system for Mo Kα radiation.Data were processed and corrected for absorption effects using the REQABalgorithm in the d*trek package of the CrystalClear software suite. Thestructure was determined using SHELXT51 and refined using SHELXL-201452 in the WinGX program suite53. A model with an In3+/Cu2+ mixturewas also examined; however, the R values did not improve. The Final Rvalues were Robs = 3.54% and wRall = 8.68%.Physical property measurementsThe temperature dependence of the magnetic susceptibility was measuredusing a SQUID magnetometer (Quantum Design, MPMS) in the tem-perature region 2 – 300 K in magnetic fields of up to 7 T. Heat capacitymeasurements were performed by the relaxationmethod using a QuantumDesign PPMS down to 2 K and up to 14 T, and with a self-developedcalorimeter below 2 K up to 14 T. Reproducibility of basic physical prop-erties was confirmed with some batches of samples. High-field magnetiza-tionmeasurements at 1.3 and4.2 Kwereperformedbyan inductionmethodin pulsedhighmagneticfields of up to 51 Tat theCenter forAdvancedHighMagnetic Field Science, Osaka University.Orthogonalized finite-temperature Lanczos methodThe Hamiltonian for the S = 1/2 kagome network with an In-kapellasite-type distortion in a magnetic field is defined asH ¼Xi;jh iJi;jSi � Sj � HXiSZi ; ð1Þwhere Si is the spin-half operator at the i-th site, SZi is the z component of Si,i; j� �runs over thenearest-neighbor spin pairs, Ji;j corresponds to J1, J1′, andJ1″ as shown inFig. 4, andH is themagnitudeof themagneticfield applied inthe z direction.Thefinite-temperatureLanczosmethod (FTLM) is useful for analyzingfrustratedquantum latticemodels54. TheOFTLMisamore accuratemethodthan the standard FTLM, particularly at low temperatures55,56. The partitionfunction using standard FTLM is as follows:Z T;Hð ÞFTL ¼XMsm¼�MsNðmÞstRXRr¼1XML�1j¼0e�βϵ rð Þj;m Hð Þ Vr;mjψrj;mD E��� ���2; ð2ÞwhereMs is the saturationmagnetization,Nst is thedimensionof theHilbertsubspace with SZtot ¼ m, R denotes the number of random samplings of theFTLM, ML denotes the dimension of the Krylov subspace, |Vr;mi is anormalized random initial vectorwith SZtot ¼ m, and |Vrj;mi ½ϵðrÞj;mðHÞ� are theeigenvectors (eigenvalues) in theML-th Krylov subspace with SZtot ¼ m. AsPi SZi is a conserved quantity, ϵ rð Þi;j ðHÞ can be expressed asϵ rð Þi;j Hð Þ ¼ ϵ rð Þi;j �mH. We define the order of fϵðrÞj;mg asϵ rð Þ0;m ≤ ϵ rð Þ1;m ≤ ϵ rð Þ2;m ≤ � � � ≤ ϵ rð ÞML�1 ;m. If ML is sufficiently large, ϵ rð Þ0;m becomesequal to the exact ground state energyE0;m. However, jhVr;m; j;ψrj;mij2 doesnot converge to the expected value, that is, dm=NðmÞst , where dm representsthe degeneracy of the ground state in the subspacewith SZtot ¼ m. Therefore,unless a sufficient number of random samples are considered, the accuracyof Z T;Hð ÞFTL will not improve at low temperatures.In the OFTLM, we first calculate several low-lying exact eigenvectorsjΨi;mi with NV levels.We define the order Ei;m� �as E0;m ≤ E1;m ≤ � � � ≤ ENV�1;m. We thencalculate the following modulated random vector:jV 0r;mi ¼ I �XNV�1i¼0jΨi;mihΨi;mjh ijVr;mi ð3Þwith normalizationjV 0r;mi )jV 0r;miffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihV 0r;mjV 0r;miq : ð4ÞThe partition function of the OFTLM is obtained as follows:Z T;Hð ÞOFTL ¼XMsm¼�MsN ðmÞst � NVRXRr¼1XML�1j¼0e�βϵ rð Þj;mðHÞ V 0r;mjψrj;mD E��� ���2"þXNV�1i¼0e�βEi;mðHÞi:ð5ÞSimilarly, in the OFTLM, the magnetizationM T;Hð ÞOFTL is obtainedas follows:M T;Hð ÞOFTL ¼1Z T;Hð ÞOFTLXMsm¼�MsN ðmÞst � NVR"×XRr¼1XML�1j¼0me�βϵ rð Þj;m Hð Þ V 0r;m; j;ψrj;mD E��� ���2þXNV�1i¼0e�βEi;mðHÞi;ð6ÞBecause the final terms in Eqs. (5) and (6) are exact values, they aremore accurate than those obtained using the standard FTLM, particularly atlow temperatures.We performed OFTLM calculations for a cluster of 36 sites underperiodic boundary conditions with R ¼ 10, NV ¼ 5, andML = 16057. Ourcalculations revealed that there were almost no finite-size effects on themagnetization for T/J > 0.157. Therefore, the analysis of the magnetizationcurve in this study was sufficiently accurate.Data availabilityThe data that support the findings of this study are available from thecorresponding author upon reasonable request.Code availabilityAll relevant code used in this study is available from the correspondingauthors upon reasonable request.Received: 27 January 2024; Accepted: 19 December 2024;References1. Ye, L. et al. Massive Dirac fermions in a ferromagnetic kagomemetal.Nature 555, 638–642 (2018).2. Kiesel, M. L., Platt, C. & Thomale, R. Unconventional Fermi surfaceinstabilities in the kagome Hubbard model. Phys. Rev. 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Isothermal and adiabatic magnetization pro- cesses of thespin-1/2 Heisenberg model on an anisotropic triangular lattice. Phys.Rev. B 105, 064428 (2022).57. Morita, K. Stability of the 1/3 magnetization plateau of the J1-J2kagome Heisenberg model. Phys. Rev. B 108, 184405 (2023).AcknowledgementsWe are grateful to Dr. Y. Ihara of Hokkaido University. This study wassupported by JSPS KAKENHI (grant numbers JP21H01035, JP19H01832,and JP23H04871). This work was partly conducted at the Center forAdvanced High Magnetic Field Science at Osaka University under theVisiting Researcher’s Program of the Institute for Solid-State Physics at theUniversity of Tokyo. The computations in this study were performed usingthe facilities at the Supercomputer Center, Institute for Solid State Physics,University of Tokyo.Author contributionsM.K. and H.K.Y. planned and designed the experiments. M.K., M.O., H.H.,K.Y., and H.K.Y. prepared the samples. The structural analysis wasperformed by Y.M. using the X-ray diffraction technique. M.K., S.F., S.Y.,H.K.Y., and Y.Nakazawa performed the heat capacity measurements undermagnetic fields. Magnetic susceptibility and pulsed high-magnetic-fieldmeasurements were performed by M.K., H.K.Y., Y.Narumi, and M.H. K.M.calculated the magnetization process using the orthogonalized finite-temperature Lanczos method. M.K. and H.K.Y. wrote the paper with con-siderable help from all authors. All the authors contributed to the discussionof the experimental results.Competing interestsThe authors declare no competing interests.Additional informationCorrespondence and requests for materials should be addressed toHiroyuki K. Yoshida.Peer review information Communications Physics thanks the anonymousreviewers for their contribution to the peer review of this work.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’snoteSpringerNature remainsneutralwith regard to jurisdictionalclaims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License,which permits any non-commercial use, sharing, distribution andreproduction in any medium or format, as long as you give appropriatecredit to the original author(s) and the source, provide a link to the CreativeCommons licence, and indicate if you modified the licensed material. Youdo not have permission under this licence to share adapted materialderived from this article or parts of it. 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Toview a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2024https://doi.org/10.1038/s42005-024-01922-0 ArticleCommunications Physics |           (2024) 7:424 8http://www.nature.com/reprintshttp://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/commsphys One-third magnetization plateau in Quantum Kagome antiferromagnet Results and discussion Crystal structure, magnetic susceptibility and heat capacity High field magnetization and its theoretical analyses Phase diagram and consideration of magnetic interactions Method Sample preparation and structural analysis Physical property measurements Orthogonalized finite-temperature Lanczos method Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information