Abstract
Geometrical frustration and a high magnetic field are two key factors for realizing unconventional quantum states in magnetic materials. Specifically, conventional magnetic order can potentially be destroyed by competing interactions and may be replaced by an exotic state that is characterized in terms of quasiparticles called magnons, the density and chemical potential of which are controlled by the magnetic field. Here we show that a synthetic copper mineral, Cdkapellasite, which comprises a kagomé lattice consisting of cornersharing triangles of spin1/2 Cu^{2+} ions, exhibits an unprecedented series of fractional magnetization plateaus in ultrahigh magnetic fields of up to 160 T. We propose that these quantum states can be interpreted as crystallizations of emergent magnons localized on the hexagon of the kagomé lattice.
Introduction
Quantum manybody systems accommodate various exotic states and phenomena. One of the most notable examples is Bose–Einstein condensation (BEC), where a macroscopic number of bosonic particles occupy a single particle state as in a superfluid state of liquid ^{4}He and in cold atomic gases. With the aid of attractive force, fermions in pairs can also condensate as in a superconducting state of electrons. In most antiferromagnetic insulators, the elementary excitation is a bosonic excitation magnon, and this can form a BEC^{1,2,3}. Interestingly, interactions between magnons and couplings with the basal crystalline lattice lead to rich physics in quantum antiferromagnets, thereby distinguishing it from the canonical BEC.
The magnon picture has proven extremely fruitful for several antiferromagnets composed of spin1/2 pairs with a spinsinglet (S = 0) groundstate, and triplet (S = 1) excitations called triplons. The triplons are similar to conventional magnons excited in an ordered antiferromagnet because both carry the spin angular momentum of ħ, and thus the two terms are occasionally used interchangeably^{2,3}. At a critical applied magnetic field, the energy of one of the Zeemansplit triplet components intersects the groundstate singlet, thereby resulting in a longrange magnetic order. Specifically, the transition corresponds to a BEC of diluted triplons (magnons), and this is typically observed in TlCuCl_{3}^{4,5}. Above the critical field, the magnetization starts to increase linearly when the density of magnons increases with magnetic field. The magnetic field acts as a chemical potential for magnons, and thus controls the density of the magnons (which is proportional to the magnetization).
In simple spin systems, the magnetization increases smoothly with the magnetic field and eventually saturates. However, in certain quantum magnets, flat regions termed as magnetization plateaus appear at fractional magnetizations before saturation. There are two types of magnetization plateaus: a classical one that is described by a collinear arrangement of classical spins and a quantum state comprising entangled spins^{6}. Classical magnetization plateaus are observed in triangular magnets, such as Cs_{2}CuBr_{4}^{7} and Ba_{3}CoSb_{2}O_{9}^{8,9}, and the quantum plateaus in dimer magnets such as NH_{4}CuCl_{3}^{10} and SrCu_{2}(BO_{3})_{2}^{11}.
A transition to a quantum plateau as a function of magnetic field is considered to be a superfluidinsulator transition of hardcore bosons (magnons). Interacting magnons in a BEC state tend to localize due to the suppression of kinetic energy and eventually crystallize to become “insulating” such as Mott insulators in strongly correlated electron systems^{12}. The magnon crystal exhibits a fixed density of magnons, and thus the magnetization remains at a fractional value of the full magnetization in a field range^{6}. The fractional value of magnetization is attributed to the commensurability of the magnon crystal when there is no topological order. The number of magnons, Q_{mag}S(1 – m), in the magnetic unit cell should be an integer where Q_{mag}, S, m denote the number of spins in the magnetic unit cell, the spin quantum number, and the magnetization divided by the saturation magnetization, respectively^{13}. In SrCu_{2}(BO_{3})_{2}, which comprises pairs of Cu^{2+} ions arranged orthogonally to each other in the sheet to form a Shastry–Sutherland lattice^{11}, a series of magnetization plateaus appear at m = 1/8, 1/4, 1/3 (Q_{mag} = 16, 8, 12)^{14,15}; and nuclear magnetic resonance measurements directly confirmed spontaneous translational symmetry breaking in the magnon crystals^{16}.
In the spin1/2 kagomé antiferromagnet (KAFM)^{17,19,19}, the groundstate is a gapless or gapful spin liquid and the formation of nontrivial magnons is theoretically expected immediately below the saturation^{20}. When the magnetic field is set to infinitesimally smaller than the saturation field B_{s}, a magnon with total S^{z} = 2 in a hexagonal plaquette is generated in the fully polarized spin state, which is the vacuum of magnons as schematically depicted in Fig. 1. Each spin inside the hexagonal plaquette equally carries fractional magnetization, and thus, the ‘hexagonal magnon’’ corresponds to a highly quantum mechanical entity. Given the absence of energy cost for magnon generation, the density rapidly increases to 1/9 before the magnons overlap with each other to feel mutual repulsion. This results in an decrease in the magnetization from 1 to 7/9 at B_{s}^{20}. Subsequently, a crystalline phase with a superstructure of the \(\sqrt 3\) × \(\sqrt 3\) unit cell with Q_{mag} = 9 is formed in a range of fields, thereby yielding a 7/9 magnetization plateau. A large magnon is emergently generated on a hexagon of the kagomé lattice in the KAFM, which is significantly different from dimer magnets with singlet and triplet states that naturally occur on builtin pairs of Cu ions.
Here, we report the observation of a series of fractional magnetization plateaus in the kagomé antiferromagnet Cdkapellasite (CdK) and demonstrate the presence of emergent hexagonal magnons in the kagomé lattice. Some of the observed magnetization plateaus are reproduced by theoretical calculations for the simple KAFM model, while the others may be stabilized by lattice commensurability, additional longrange interactions, and potentially coupling to lattice.
Results
Theoretical predictions for multiple plateaus
Recent calculations by the densitymatrixrenormalizationgroup method, the exact diagonalization, and the tensor network method show that in addition to the wellestablished 7/9 plateau, three plateaus appear at m = 5/9, 1/3, 1/9^{21,23,24,24}. It is also suggested^{21,23,23} that the 5/9 and 1/3 plateaus are magnon crystals with Q_{mag} = 9, which are similar to that at m = 7/9 but with S^{z} = 1 and 0, respectively, as depicted in Fig. 1; the 1/9 plateau is supposed to be another state, possibly a spin liquid with topological order^{21}. However, one must be careful in concluding this because there are many competing phases in highly frustrated systems. In the present study, we calculated the complete magnetization process in the extended projected entangled pair states (PEPS) scheme, and found three plateaus at m = 1/3, 5/9, 7/9, as shown in Fig. 1, which supports the previous results as shown in Fig. 1. Additionally, we examined the magnetic structure at the 1/3 plateau and observed that a magnon crystal composed of hexagonal magnons with S^{z} = 0 in the \(\sqrt 3\) × \(\sqrt 3\) structure was associated with the lowest energy among other competing states such as the upupdown state (Supplementary Fig. 1). Thus, the hexagonal magnon must correspond to an entity that should be realized under the magnetic field for the KAFM, and this should be experimentally evidenced.
Experimental obstacles
Despite the intriguing predictions for the KAFM in magnetic fields, there is a paucity of experimental evidence due to the lack of ideal model compounds, and also the difficulties in experiments under high fields. Real materials always suffer from lattice distortion^{25} or disorder^{26}, and this tends to mask the intrinsic magnetism of the KAFM. The bestcharacterized S = 1/2 KAFM is herbertsmithite with a large interaction of J ~ 200 K^{27}, and this implies that ultrahigh magnetic fields not <B_{s} = 400 T are necessary to record the complete magnetization process. However, experimentally available static magnetic fields are only below 45 T and typical pulsed magnetic fields are limited to below 100 T^{28}. Furthermore, magnetization measurements by the conventional induction method are only available in pulsed magnetic fields below 100 T in the short time duration corresponding to several milliseconds;^{28} for a larger pulsed field, a much shorter time window of several microseconds is available, which causes a large electromagnetic noise preventing facile magnetization measurements. On the other hand, a stateoftheart measurement using a Faraday rotation technique made it possible to record the complete magnetization process of CdCr_{2}O_{4} up to 140 T, and this revealed a spinnematic phase just below the saturation field^{29}. In order to unveil the physics of the KAFM, both a suitable model compound with a relatively lowsaturation field around 100 T and an improvement in the Faraday rotation technique are necessary. It is noted that the required lowtemperature condition, T « J/k_{B}, is difficult to achieve for a compound with small J and B_{s} values.
Kagomé antiferromagnet CdK
To study the magnetism of S = 1/2 KAFM under magnetic fields, we selected Cdkapellasite (CdK), CdCu_{3}(OH)_{6}(NO_{3})_{2}·H_{2}O^{30}, which is isostructural to kapellasite, ZnCu_{3}(OH)_{6}Cl_{2}^{31}. The compound has a quasitwodimensional structure with an undistorted kagomé lattice of Cu^{2+} ions and a moderate antiferromagnetic interaction of J ~ 45 K (B_{s} ~ 100 T) (Fig. 2a)^{32}. The groundstate of CdK is not a spin liquid but a longrange order (LRO) with a q = 0 structure (a negative vector chirality order) below T_{N} ~ 4 K, which must be induced by a Dzyaloshinskii–Moriya (DM) interaction with a magnitude of approximately 10% of J. Other possible perturbations are the nextnearestneighbor interaction J_{2} and diagonal interaction J_{d} bridged via the Cd ion in the center of the hexagon as shown in Fig. 2b, which was observed as a low value (Supplementary Fig. 2).
Faraday rotation measurements up to 160 T
We measured small magnetization from a tiny hexagonal single crystal of CdK with a diameter of approximately 1 mm and a thickness of 150 µm (Fig. 3a) by a Faraday rotation technique optimized to a singleturn coil in magnetic fields of up to 160 T. The experimental setup and results are shown in Fig. 3b. The polarization angle of an incident light with λ = 532 nm (Supplementary Fig. 3) was rotated with a varying magnetic field in several microseconds via the Faraday effect from induced magnetization in the sample (Fig. 3c). Figure 3d shows magnetizations converted from the data in descending pulsed magnetic fields (Supplementary Fig. 4). The magnetization curve measured at 8 K smoothly increased and saturated at approximately 150 T. At 6 K, a faint wiggling was observed, which seemed to produce many anomalies at 5 K.
Figure 4a shows the magnetization process at 5 K and its field derivative, which is compared with the calculated magnetization process in the presence of DM interaction. Below 0.4 µ_{B}, the experimental magnetization is in good agreement with the calculation and deviates above that. At least seven anomalies are observed at magnetizations exceeding 0.4 µ_{B}. Above B_{s} = 160 T, M reaches 1.15 µ_{B}/Cu, and this was nearly equal to the saturation magnetization with fully polarized spins: gSµ_{B} with g_{c} ~ 2.3 from magnetic susceptibility (Supplementary Fig. 2).
Magnetization plateaus
We consider the anomalies as a series of blunt plateaus that were blurred or inclined due to the finite temperature effect or anisotropy such as DM interaction. In conventional ordered antiferromagnets a metamagnetic transition or a spinflop transition can occur in a magnetic field. However, weak anisotropy in Cu^{2+} spin systems makes it difficult for such transitions to occur. We consider blunting by temperature and define a critical field B_{k} and a magnetization M_{k} for each plateau at which the differential magnetization takes a local minimum; the B_{k} should correspond to the center of the field range for the plateau. The obtained values of [B_{k} (T), M_{k} (µ_{B}/Cu), m_{k}] (m_{k} = M_{k}/1.15) are (47.6, 0.42, 0.37), (72.6, 0.59, 0.51), (95.3, 0.78, 0.68), (113.8, 0.94, 0.81), (121.6, 0.98, 0.85), (129.1, 1.04, 0.9), and (139.7, 1.08, 0.94) for k = 1–7, respectively. It should be noted that the anomalies are distinct for k = 2, 3, 4, and 7 and weaker for k = 5 and 6. As mentioned above, it is expected that magnetization plateaus from magnon crystals with Q_{mag} = 9 appear only at m = 1/3, 5/9, and 7/9 for the simple KAFM. The values were near to m_{1}, m_{2}, and m_{4}, as shown in Fig. 4b, although they significantly deviated from them. The present observation of the increased number of plateaus in CdK significantly indicate the formation of other types of magnon crystals with unit cells exceeding Q_{mag} = 9.
Discussion
We considered possible magnetic structures realized at the magnetization plateaus in CdK. As in the case of the simple KAFM, we assumed that a type of localized hexagonal magnons is periodically aligned to reduce mutual repulsion while maintaining a sixfold rotational symmetry. Subsequently, such a series of magnon crystals as shown in Fig. 5 with unit cells of Q_{mag} = 9, 12, 21, 36 can appear wherein each involves three sequences with 1–3 magnons on the hexagon. All the experimentally observed m_{k} values are well reproduced. For example, m_{7} = 0.94 near the saturation is close to 34/36 ~ 0.944, and m_{6} = 0.90 is nearly equal to 19/21 ~ 0.905. Furthermore, m_{5} = 0.85, m_{4} = 0.82, m_{3} = 0.68, m_{2} = 0.51, and m_{1} = 0.37 are near 5/6 ~ 0.833, 17/21 ~ 0.810, 2/3 ~ 0.667, 1/2 = 0.5, and 1/3 ~ 0.333, respectively (Fig. 4b). Thus, one from the Q_{mag} = 9 series, three from Q_{mag} = 12, two from Q_{mag} = 21, and one from Q_{mag} = 36 were observed. The fact that all the three sequences appeared for Q_{mag} = 12 while only part appeared for the others prompted us to determine a rule for the magnon crystallization in CdK.
Emergence of larger unit cells in CdK must be attributed to additional farther–neighbor interactions such as J_{2} and J_{d}. Our calculations of magnetization without them and with 10% DM interaction does not exhibit any anomalies after an inclined 1/3 plateau with increasing fields (Fig. 4a). It is emphasized that the anomalies in CdK appear when the experimental magnetization curve deviates from the theoretical one, which suggests that effects excluded in the calculation play a crucial role. If J_{2} works as an effective repulsion among magnons, the Q_{mag} = 9 structure must become unstable (Supplementary Fig. 5). For the Q_{mag} = 12 structure, on the other hand, J_{2} does not work while J_{d} does. The experimental fact that all the three sequences are observed for the Q_{mag} = 12 series and only a part for the others (e.g., not 5/9 but 6/12 for m_{2}) indicates the relative stability of Q_{mag} = 12. This implies repulsive (ferromagnetic) J_{2} and attractive (antiferromagnetic) J_{d}. Larger unit cells, such as Q_{mag} = 21 and 36, can be stabilized by weak farther–neighbor interactions or the enlargement of the effective size of hexagonal magnons. Therefore, the appearance of multiple magnetization plateaus in CdK is reasonably explained by assuming a series of magnon crystals in a spin1/2 KAFM with additional interactions.
Finally, we focus on the reason for the appearance of magnetization plateaus in CdK after the deviation from the calculation based on the J + DM model; the smooth rise of the calculated curve indicates that spins are gradually forced to align along the magnetic field from the coplanar q = 0 structure as expected for a classical spin system. One possibility is the magnetostriction that induces a lattice distortion^{33}. For example, a hexagon where a localized magnon lives can deform to increase antiferromagnetic J, and thereby to stabilize a local magnon. When this type of a lattice distortion occurs periodically to form a superstructure, hexagonal magnons tend to crystallize at high fields with m ≥ 1/3. Therefore, a spinlattice coupling potentially makes a tilted q = 0 magnetic structure unfavorable and leads to a series of crystallizations of localized magnons with the aid of farther–neighbor interactions.
Methods
Sample preparation
Single crystals of Cdkapellasite, CdCu_{3}(OH)_{6}(NO_{3})_{2}·H_{2}O, were synthesized by a twostep hydrothermal method^{32}. A quartz ampoule with a length of 150 mm was filled with ingredients (Cu(OH)_{2}: 0.1 g, Cd(NO_{3})_{2}•4H_{2}O: 5 g, H_{2}O: 4 g) and sealed. A thick quartz tube with outer and inner diameters of 12 and 8 mm, respectively, was used to avoid bursting due to increased pressure inside the tube during heat treatment. The ampoule was placed horizontally in a twozone furnace and heated to 180 °C at the hot end and to 130 °C at the cold end for a week. Polycrystalline samples of CdK were quickly produced over the tube and subsequently slowly transported into the cold zone to condense into a bunch of single crystals 100 µm in diameter. After all polycrystalline samples were transported to the cold zone, the temperature gradient was reversed. The excool zone was set to 160 °C, and the exhot zone was set to 140 °C. This resulted in inverse transportation, thereby resulting in the growth of hexagonal platelike crystals as large as 1 mm in diameter and 150 µm in thickness as typically shown in Fig. 3a.
Magnetization measurements
Magnetization measurements in lowmagnetic fields below 7 T were conducted by using a single crystal of CdK with a weight of 2.38 mg in a MagneticPropertyMeasurementSystem 3 (MPMS3, Quantum Design). Magnetic susceptibilities along the a and c axes were measured at the constant field of 1 T. Magnetization at pulsed magnetic fields up to 60 T was measured at 4.2 K on stacked single crystals via the electromagnetic induction method. The absolute value of magnetization was calibrated by the data obtained from MPMS3.
Faraday rotation measurements in pulsed magnetic fields
A singleturn coil as shown in Fig. 3a was used to generate ultrahigh magnetic fields up to 160 T^{34}. A sample was not destroyed after the experiment because the coil explodes outwards along the direction of Maxwell stress. Faraday rotation was measured in a short time duration corresponding to 7 µs. The sample temperature was set to 5, 6, and 8 K and could slightly change at elevated fields due to the magnetocaloric effect in the adiabatic condition. A change in the polarization angle θ_{F} was proportional to magnetization M induced by the applied field in the sample: θ_{F} = αMd where α denotes the Verdet constant and d denotes the thickness of the sample. Specifically, θ_{F} is calculated from the intensities of the vertical and horizontal components, I_{p} and I_{s}, by the formula: θ_{F} = cos^{−1} {(I_{p} − I_{s})/(I_{p} + I_{s})}.
Tensor network calculations
Magnetization process of a S = 1/2 KAFM with the nearestneighbor interaction J is calculated by the infinite projected entangled pair state (iPEPS)^{35,37,37} method that expresses a wave function as an extended tensor product state termed as the projected simplex pair state (PESS)^{38,39}. Unit cells up to 18 sites are assumed. The wave function is optimized by the simple update method^{40,41} with bond dimensions D = 4–7. Physical quantities were typically calculated by the corner transfer matrix method^{42} with bond dimensions χ = D^{2}.
Data availability
References
 1.
Matsubara, T. & Matsuda, H. A lattice model of liquid helium, I. Prog. Theor. Phys. 16, 569–582 (1956).
 2.
Giamarchi, T., Rüegg, C. & Tchernyshyov, O. Bose–Einstein condensation in magnetic insulators. Nat. Phys. 4, 198–204 (2008).
 3.
Zapf, V., Jaime, M. & Batista, C. D. BoseEinstein condensation in quantum magnets. Rev. Mod. Phys. 86, 563–614 (2014).
 4.
Nikuni, T. et al. BoseEinstein condensation of dilute magnons in TlCuCl_{3}. Phys. Rev. Lett. 84, 5868–5871 (2000).
 5.
Rüegg, Ch. et al. Bose–Einstein condensation of the triplet states in the magnetic insulator TlCuCl_{3}. Nature 423, 62–65 (2003).
 6.
Takigawa, M. & Mila, F. in Introduction to Frustrated Magnetism (eds. Lacroix, C. et al.) 241–267 (Springer, Berlin Heidelberg, 2011).
 7.
Fortune, N. A. et al. Cascade of magneticfieldinduced quantum phase transitions in a spin1/2 triangularlattice antiferromagnet. Phys. Rev. Lett. 102, 257201 (2009).
 8.
Zhou, H. D. et al. Successive phase transitions and extended spinexcitation continuum in the S=1/2 triangularlattice antiferromagnet Ba_{3}CoSb_{2}O_{9}. Phys. Rev. Lett. 109, 267206 (2012).
 9.
Shirata, Y. et al. Experimental realization of a spin1/2 triangularlattice Heisenberg antiferromagnet. Phys. Rev. Lett. 108, 057205 (2012).
 10.
Shiramura, W. et al. Magnetization plateaus in NH_{4}CuCl_{3}. J. Phys. Soc. Jpn 67, 1548–1551 (1998).
 11.
Kageyama, H. et al. Exact dimer ground state and quantized magnetization plateaus in the twodimensional spin system SrCu_{2}(BO_{3})_{2}. Phys. Rev. Lett. 82, 3168–3171 (1999).
 12.
Imada, M., Atsushi, F. & Tokura, Y. Metalinsulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998).
 13.
Oshikawa, M., Yamanaka, M. & Affleck, I. Magnetization plateaus in spin chains: “Haldane gap” for halfinteger spins. Phys. Rev. Lett. 78, 1984–1987 (1997).
 14.
Matsuda, Y. H. et al. Magnetization of SrCu_{2}(BO_{3})_{2} in ultrahigh magnetic fields up to 118 T. Phys. Rev. Lett. 111, 137204 (2013).
 15.
Kodama, K. et al. Magnetic superstructure in the twodimensional quantum antiferromagnet SrCu_{2}(BO_{3})_{2}. Science 298, 395–399 (2002).
 16.
Yan, S., Huse, D. A. & White, S. R. Spinliquid ground state of the S = 1/2 kagome Heisenberg antiferromagnet. Science 332, 1173–1176 (2011).
 17.
He, Y. C. et al. Signatures of Dirac cones in a DMRG study of the kagome Heisenberg model. Phys. Rev. X 7, 031020 (2017).
 18.
Liao, H. J. et al. Gapless spinliquid ground state in the S = 1/2 kagome antiferromagnet. Phys. Rev. Lett. 118, 137202 (2017).
 19.
Schulenburg, J. et al. Macroscopic magnetization jumps due to independent magnons in frustrated quantum spin lattices. Phys. Rev. Lett. 88, 167207 (2002).
 20.
Nishimoto, S., Shibata, N. & Hotta, C. Controlling frustrated liquids and solids with an applied field in a kagome Heisenberg antiferromagnet. Nat. Commun. 4, 2287 (2012).
 21.
Capponi, S. et al. Numerical study of magnetization plateaus in the spin1/2 kagome Heisenberg antiferromagnet. Phys. Rev. B 88, 144416 (2013).
 22.
Chen, Xi et al. Thermodynamics of spin1/2 Kagomé Heisenberg antiferromagnet: algebraic paramagnetic liquid and finitetemperature phase diagram. Sci. Bull. 63, 1545–1550 (2018).
 23.
Picot, T. et al. SpinS kagome quantum antiferromagnets in a field with tensor networks. Phys. Rev. Lett. 93, 060407 (2016).
 24.
Hiroi, Z. et al. Spin1/2 kagomélike lattice in volborthite Cu_{3}V_{2}O_{7}(OH)_{2}· 2H_{2}O. J. Phys. Soc. Jpn 70, 3377–3384 (2001).
 25.
Kawamura, H., Watanabe, K. & Shimokawa, T. Quantum spinliquid behavior in the spin1/2 randombond Heisenberg antiferromagnet on the kagome lattice. J. Phys. Soc. Jpn 83, 103704 (2014).
 26.
Shores, M. P. et al. A structurally perfect S = 1/2 kagome antiferromagnet. J. Am. Chem. Soc. 127, 13462–13463 (2005).
 27.
N. Miura & F. Herlach. in Springer Topics in Applied Physics (ed. Herlach, F.) 247–350, Vol. 57 (Springer, Berlin, 1985).
 28.
Miyata, A., Takeyama, S. & Ueda, H. Magnetic superfluid state in the frustrated spinel oxide CdCr_{2}O_{4} revealed by ultrahigh magnetic fields. Phys. Rev. B 87, 214424 (2013).
 29.
Nytko, E. A. et al. CdCu_{3}(OH)_{6}(NO_{3})_{2}: An S = 1/2 Kagomé Antiferromagnet. Inorg. Chem. 48, 7782–7786 (2009).
 30.
Fåk, B. et al. Kapellasite: A kagome quantum spin liquid with competing interactions. Phys. Rev. Lett. 109, 037208 (2012).
 31.
Okuma, R. et al. Weak ferromagnetic order breaking the threefold rotational symmetry of the underlying kagome lattice in CdCu_{3}(OH)_{6}(NO_{3})_{2}·H_{2}O. Phys. Rev. B 95, 094427 (2017).
 32.
Becca, F. & Mila, F. Peierlslike transition induced by frustration in a twodimensional antiferromagnet. Phys. Rev. Lett. 89, 037204 (2002).
 33.
Miura, N., Osada, T. & Takeyama, S. Research in superhigh pulsed magnetic fields at the megagauss laboratory of the University of Tokyo. J. Low. Temp. Phys. 133, 139–158 (2003).
 34.
Verstraete, F. & Cirac, J. I. Valencebond states for quantum computation. Phys. Rev. A 70, 060302 (2004).
 35.
Verstraete, F. & Cirac, J. I. Renormalization algorithms for quantummany body systems in two and higher dimensions. Preprint at http://arXiv.org/abs/condmat/0407066 (2004).
 36.
Jordan, J., Orus, R., Vidal, G., Verstraete, F. & Cirac, J. I. Classical simulation of infinitesize quantum lattice systems in two spatial dimensions. Phys. Rev. Lett. 101, 250602 (2008).
 37.
Poilblanc, D. et al. Simplex Z_{2} spin liquids on the kagome lattice with projected entangled pair states: Spinon and vison coherence lengths, topological entropy, and gapless edge modes. Phys. Rev. B 87, 140407(R) (2012).
 38.
Xie, Z. Y. et al. Tensor renormalization of Quantum ManyBody Systems using projected entangled simplex states. Phys. Rev. X 4, 011025 (2014).
 39.
Jiang, H. C., Weng, Z. Y. & Xiang, T. Accurate determination of tensor network state of Quantum lattice models in two dimensions. Phys. Rev. Lett. 101, 090603 (2008).
 40.
Okubo, T. et al. Groundstate properties of Na_{2}IrO_{3} determined from an ab initio Hamiltonian and its extensions containing Kitaev and extended Heisenberg interactions. Phys. Rev. B 96, 054434 (2017).
 41.
Oru ́s, R. & Vidal, G. Simulation of twodimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction. Phys. Rev. B 80, 094403 (2009).
 42.
Nishino, T. & Okunishi, K. Corner transfer matrix renormalization group method. J. Phys. Soc. Jpn 65, 891–894 (1996).
Acknowledgements
We are grateful to T. Misawa, H. Tsunetsugu, and C. Hotta for helpful discussion. R.O. is supported by the Materials Education Program for the Future Leaders in Research, Industry, and Technology (MERIT) under the Ministry of Education, Culture, Sports, Science, and Technology of Japan (MEXT). The study was partially supported by KAKENHI (Grant No. 15K17701), the CoretoCore Program for Advanced Research Networks under the Japan Society for the Promotion of Science (JSPS), and MEXT of Japan as a social and scientific priority issue (Creation of new functional devices and highperformance materials to support nextgeneration industries; CDMSI) to be solved by using postK computer and “Exploratory Challenge on PostK computer” (Challenge of Basic Science–Exploring Extremes through MultiPhysics and MultiScale Simulations).
Author information
Affiliations
Contributions
R.O., D.N., S.T., and Z.H. conceived and designed the study. A. Miyake and A. Matsuo measured the magnetization curve up to 60 T under supervision of M.T.; K.K., R.O., D.N. measured the magnetization curve up to 180 T under supervision of S.T.; R.O., D.N., S.T., and Z.H. interpreted the experimental data. T.O. performed and interpreted PEPS calculations under supervision of N.K.; R.O. wrote the manuscript. All authors discussed and commented on the manuscript.
Corresponding author
Correspondence to R. Okuma.
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Journal peer review information: Nature Communications thanks Vivien Zapf and the other anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Okuma, R., Nakamura, D., Okubo, T. et al. A series of magnon crystals appearing under ultrahigh magnetic fields in a kagomé antiferromagnet. Nat Commun 10, 1229 (2019). https://doi.org/10.1038/s41467019090637
Received:
Accepted:
Published:
Further reading

Spin transport properties of anisotropic Heisenberg antiferromagnet on honeycomb lattice in the presence of magnetic field
The European Physical Journal B (2020)

Dynamical and static spin structure factors of Heisenberg antiferromagnet on honeycomb lattice in the presence of DzyaloshinskiiMoriya interaction
Physica E: Lowdimensional Systems and Nanostructures (2019)

Magnetic structure and highfield magnetization of the distorted kagome lattice antiferromagnet Cs2Cu3SnF12
Physical Review B (2019)

Magnon crystals and magnetic phases in a kagomestripe antiferromagnet
Physical Review B (2019)

Orbital Transitions and Frustrated Magnetism in the KagomeType Copper Mineral Volborthite
Inorganic Chemistry (2019)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.