Abstract
The magnetization of a quantum magnet can be pinned at a fraction of its saturated value by collective effects. One example of such a plateau phase is found in spin-1/2 triangular-lattice antiferromagnets. They feature strong geometrical frustration and the plateau phase therein is often interpreted as arising from an order-by-disorder mechanism driven by quantum fluctuations. Here we observe a one-third magnetization plateau under an applied magnetic field in the spin-1 antiferromagnet Na3Ni2BiO6 with a honeycomb lattice, which, with conventional magnetic interactions, would not be geometrically frustrated. Based on our elastic neutron scattering measurements, we propose the spin structure of the plateau phase to be an unusual partial spin-flop ferrimagnetic order. Our theoretical calculations indicate that bond-anisotropic Kitaev interactions are the source of frustration that produces the plateau. These results suggest that Kitaev interactions provide a different route to frustration and phases driven by quantum fluctuations in high-spin magnets.
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Data availability
Data supporting the findings of this study are available from the corresponding author J.W. upon reasonable request. Source data are provided with this paper.
Change history
10 October 2023
A Correction to this paper has been published: https://doi.org/10.1038/s41567-023-02280-4
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Acknowledgements
The work was supported by National Key Projects for Research and Development of China with grant no. 2021YFA1400400 (J.W. and J.-X.L.), National Natural Science Foundation of China with grant nos. 12225407 (J.W.), 12074174 (J.W.), 92165205 (J.-X.L.), 12074175 (S.-L.Y.), 11904170 (Z.-Y.D.), 12047503 (W.L.), 11974036 (W.L.), 12222412 (W.L.), 12004191 (W.W.) and 12204160 (Z.M.), Natural Science Foundation of Jiangsu province with grant nos. BK20190436 (Z.-Y.D.) and BK20200738 (W.W.), China Postdoctoral Science Foundation with grant nos. 2022M711569 (S.B.) and 2022T150315 (S.B.), Jiangsu Province Excellent Postdoctoral Programme with grant no. 20220ZB5 (S.B.), Hubei Provincial Natural Science Foundation of China with grant no. 2021CFB238 (Z.M.), CAS Project for Young Scientists in Basic Research with grant no. (YSBR-003) (W.L.) and Fundamental Research Funds for the Central Universities. We acknowledge the neutron beam time from ANSTO with proposal no. P9334 and the support from G. Davidson in setting up and operation of the 7 T superconducting magnet and the beam time from the Japan Proton Accelerator Research Complex with proposal no. 2022A0039. We thank Y. Han at High Magnetic Field Laboratory of the Chinese Academy of Sciences for assisting us in measuring the magnetization under high magnetic fields.
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J.W. conceived the project. Y.S. prepared the samples. Y.S. carried out the magnetization and specific heat measurements with assistance from Z.M., Z.H., J.L., X.Z., B.Z., S.C., H.X., S.Z. and F.S. Y.S., S.B., D.Y., R.A.M., N.M., S.O.-K., L.H. and J.H. performed the neutron scattering experiments. Y.S., S.B. and J.W. analysed the experimental data. Z.-Y.D., N.X., Y.-P.G., W.W., Z.Q., Q.-B.Y., W.L., S.-L.Y. and J.-X.L. performed the theoretical calculations and analyses. J.W., Y.S., W.L. and J.-X.L. wrote the paper with inputs from all co-authors.
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Extended data
Extended Data Fig. 1 Additional magnetometry data for Na3Ni2BiO6 single crystals and results of tensor-network calculations.
a, Magnetisation as a function of magnetic field applied parallel to the a − b plane for a Na3Ni2BiO6 single crystal. b, Derivative of the magnetization in a. Data are offset vertically for clarity. c, Ground state magnetisation curve of the spin-1 J1-J3-K-D model obtained by iPEPS. The magnetic field H is applied parallel to the a − b plane. The magnetisation increases linearly before it saturates, consistent with the 2-K data shown in a. As the field ramps up, the zigzag order parameter gradually decreases and falls to zero at the field where the magnetisation saturates. d, Derivative of the product χT (magnetic susceptibility times temperature) as a function of temperature under different magnetic fields applied perpendicular to the a − b plane, obtained from Fig. 1e in the main text. Black arrows denote the transition temperatures which form the phase boundary in the magnetic phase diagram in Fig. 4 of the main text.
Extended Data Fig. 2 Additional elastic neutron scattering results.
a and b, Contour maps of elastic scattering in the (H, K, 0) plane measured on PELICAN with Ei = 3.70 meV at 1.5 K (below the TN) under magnetic field of μ0H⊥ = 0 and 6.6 T, respectively. c, Contour map of elastic scattering in the (H, K, 0) plane measured at 80 K (well above the TN) under zero field. It is used as the background data for subtraction.
Extended Data Fig. 3 Refinement of the magnetic structure in zero field.
a, The Gaussian distribution simulation of the two characteristic peaks M (0, 1, 0) and \({M}^{{\prime} }\) (1, 2, 0). The results are obtained from AMATERAS. Because of the finer resolution, the two peaks are much sharper than those obtained from PELICAN, which allows us to fit them properly. The intensity of elastic neutron scattering is proportional to \(I(k)\propto | F(k){| }^{2}{\sum }_{\alpha \beta }\,({\delta }_{\alpha \beta }-{\hat{k}}_{\alpha }{\hat{k}}_{\beta }){M}^{\alpha }(k){M}^{\beta }(k)\), where F(k) is the magnetic form factor and Mα(k) is one of the three components of the Fourier transform of the spin distribution. For the zigzag order, there are two characteristic peaks M and \({M}^{{\prime} }\), as marked in Fig. 3a of the main text. The dip angle θ of the magnetic order defined by the angle off from c* can be refined by a simulation of the two peaks, by \(I(M)\propto F{(M)}^{2}({M}^{x}{(M)}^{2}+{M}^{x}\,{(M)}^{2}),I({M}^{{\prime} })\propto \,F{({M}^{{\prime} })}^{2}\) \(({M}^{x}({M}^{{\prime} }),{M}^{y}({M}^{{\prime} }),{M}^{z}({M}^{{\prime} }))\), where \(\left({M}^{x}({M}^{{\prime} }),{M}^{y}({M}^{{\prime} }),{M}^{z}({M}^{{\prime} })\right.=(\sin \theta ,0,\cos \theta )| M(k)|\). The relationship between the ratio of the intensities of the two peaks and the dip angle in the Gaussian simulation is, \(\frac{I({M}^{{\prime} })F\,{\left(M\right)}^{2}}{I(M)F\,{\left({M}^{{\prime} }\right)}^{2}}=(0.7727+0.1814\cos (2\theta )-0.1373\sin (2\theta ))\) \(=\frac{{A}^{{\prime} }{\sigma }^{{\prime} }}{A\sigma }\). Here, A and σ are the amplitude and variance of the Gaussian function, respectively. From the fit, we obtain \(A=37.3\pm 0.4,\sigma =0.0144\pm 0.0002,\) \({A}^{{\prime} }=16.8\pm 0.2,{\sigma }^{{\prime} }=0.0264\pm 0.0003\). So the dip angle is about θ = 19.7 ± 2. 6°. b, DFT calculations on the angle dependence of the energy taking the spin-orbit coupling into account. Zero degree is the direction perpendicular to the a − b plane (c*). The dashed line represents the c axis.
Extended Data Fig. 4 Possible magnetic configurations for the 1/3 plateau phase based on the zigzag ground state.
a and b, Configurations of the honeycomb-lattice clusters with 24 and 6 sites, respectively. c-e, Magnetic structures with a 6-site chain with partial spin flip. These three 6-site chains differ in the position of spins where the spin-flip takes place and c, d, and e can be symbolized as ↓↑↑↑↓↑, ↑↓↑↑↓↑, and ↓↓↑↑↑↑, respectively. f-i Magnetic structures with a 6-site chain with partial spin flop. These four 6-site chains differ in the position of spins where the spin-flop take place, and they can be symbolized as ↑\(\circ\)\(\circ\)↓↑↑, ↑\(\circ\)↓\(\circ\)↑↑, \(\circ\)↑↓\(\circ\)↑↑, and \(\circ\)↑\(\circ\)↓↑↑, respectively. The up and down spins are aligned along c*, parallel to the magnetic field. The dotted ellipse and circle indicate that the spin at this position has been flipped and flopped, respectively.
Extended Data Fig. 5 Calculations of the magnetic structure factors corresponding to the spin configurations in Extended Data Fig. 4.
a and b, Magnetic structure factors for the honeycomb clusters with 24 and 6 sites, respectively; c-e, for 6-site chain structures with partial spin flip; f and i, for 6-site chain structures with partial spin flop. As shown in a and b, there are major Bragg peaks at the M and K, respectively, which are not present in the experimental results. For c, the strongest peak is located at the \({\Gamma }^{{\prime} }\) point, which contradicts with the experiment. In d, the six peaks around the triangle centres in the second Brillouin zone exhibit comparable intensities, which is quite different from the dramatically different intensities of the two groups of peaks rotated by 180° in the experimental results. In e-g, the stronger peaks in the first Brillouin zone are significantly more intense than those in the second Brillouin zone, which conflicts with the experimental results. The patterns in h and i both appear to closely resemble the experiment. We pick i which corresponds to the spin configuration symbolised as \(\circ\)↑\(\circ\)↓↑↑ depicted in Extended Data Fig. 4i to be the magnetic structure in the plateau phase, according to the least-square criterion as discussed in Extended Data Table 2. The white dotted box contains four distinctive peaks along the [0, K, 0] direction, namely M1, M0, M2 and M3, based on which we are going to determine the magnetic structure.
Extended Data Fig. 6 Tensor-network method used in calculating the spin-1 J1-J3-K-D model.
a, The projected entangled pair states (PEPS) setup on the honeycomb lattice. T and S tensors are placed on the honeycomb sites, and each tensor consists of a physical spin indices i and three auxiliary ones α, β, γ. b, The unit cell setup in the PEPS calculations performed in an infinitely large system, and we adopt a 24-site unit cell as surrounded in the red parallelogram. The zigzag order pattern can be contained in such a unit cell, with S1 to S12 tensors indicated explicitly. c, Flowchart for the simple update scheme for the PEPS using the higher-order singular value decomposition (HOSVD). In practical calculations, the entanglement environment of local tensor cluster consisted of an S tensor and three T tensors around it are approximated by diagonal matrices Λ. We update the tensor cluster through imaginary time evolution, and conduct the decomposition and truncation via HOSVD, optimizing the tensors \(\tilde{S}\) and \({\tilde{T}}_{a,b,c}\) to simulate the ground state of the system.
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Shangguan, Y., Bao, S., Dong, ZY. et al. A one-third magnetization plateau phase as evidence for the Kitaev interaction in a honeycomb-lattice antiferromagnet. Nat. Phys. 19, 1883–1889 (2023). https://doi.org/10.1038/s41567-023-02212-2
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DOI: https://doi.org/10.1038/s41567-023-02212-2
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