Abstract
Determining ground states of correlated electron systems is fundamental to understanding unusual phenomena in condensed-matter physics. A difficulty, however, arises in a geometrically frustrated system in which the incompatibility between the global topology of an underlying lattice and local spin interactions gives rise to macroscopically degenerate ground states1, potentially prompting the emergence of quantum spin states, such as resonating valence bond2,3,4,5,6,7 and valence-bond solid8,9,10,11 (VBS). Although theoretically proposed to exist in a kagome lattice—one of the most highly frustrated lattices in two dimensions being comprised of corner-sharing triangles—such quantum-fluctuation-induced states have not been observed experimentally. Here we report the first realization of the ‘pinwheel’ VBS ground state in the S=1/2 deformed kagome lattice antiferromagnet Rb2Cu3SnF12 (refs 12, 13). In this system, a lattice distortion breaks the translational symmetry of the ideal kagome lattice and stabilizes the VBS state.
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Theoretically, significant progress has been made towards understanding the ground state of the quantum (S=1/2) kagome lattice antiferromagnet using various approaches. Diverse classes of new states have been proposed; those include a gapless U(1)-Dirac-spin-liquid state2,5, a gapped-spin-liquid state3,4,6,7 and a VBS state8,9,10,11. One leading approach is based on a quantum dimer model, which was first introduced to describe spin dynamics of singlet pairs (dimers) in high-Tc superconductors14,15. On the basis of this model, two kinds of quantum state promptly emerge, namely the VBS and resonating-valence-bond states6,7,9,10,11. The VBS state has long-range dimer–dimer correlations that break the translational symmetry of the ideal kagome lattice with spin-1 singlet-to-triplet excitations, whereas the gapped resonating-valence-bond state, in which different dimer coverings resonate among themselves, has short-range dimer–dimer correlations and topological order with continuum spin-1/2 excitations (spinons). As these quantum states are non-magnetic and lack static order, one needs to look for their distinct features in magnetic excitations by means of inelastic neutron scattering to distinguish them in a real system.
Experimentally, realizations of these states have been extensively studied in quasi-one dimensional16,17,18 and Shastry–Sutherland two-dimensional systems19. In contrast, the rarity of model systems has hitherto precluded experimental attempts to observe these states in the kagome lattice. In recent years, the discoveries of the kagome lattice in herbertsmithite20,21,22 and distorted kagome lattice in volborthite23,24 have been generating a lot of excitement and debate over their plausible ground states. However, the lack of single crystals and Zn–Cu intersite disorder in herbertsmithite hinder the study of these systems. In this work, we present a single-crystal study of a stoichiometrically pure S=1/2 deformed kagome lattice antiferromagnet Rb2Cu3SnF12 with full occupancy of spin-1/2 at the Cu sites. Combined neutron scattering measurements on a large single crystal and advanced numerical analysis provide the first direct evidence for the pinwheel VBS state.
The magnetic S=1/2 Cu2+ ions of Rb2Cu3SnF12 reside at the corners of the triangles of the deformed single-layer kagome lattice, and are inside distorted octahedral cages formed by F− ions. At room temperature, the system crystallizes in the hexagonal space group with the lattice constants a=13.917(2) Å and c=20.356(3) Å (ref. 12; Fig. 1a). The slightly distorted kagome lattice yields four antiferromagnetic exchange interactions (Fig. 1b), which are characterized by different Cu2+–F−–Cu2+ bond angles ranging from 124° for the weakest bond to 138° for the strongest bond. The system does not exhibit magnetic order down to 1.3 K (refs 12, 13), which is confirmed by our elastic neutron scattering on a powder sample. At low temperatures, magnetic susceptibility rapidly decreases towards zero, a signature of a non-magnetic, spin-singlet (S=0) ground state. Furthermore, low-temperature magnetization sharply rises when the applied field H exceeds a critical field Hc≈20 T, indicative of singlet-pair breaking and closing of a singlet-to-triplet gap. Finite magnetic susceptibility observed at the base temperature of 1.8 K when the field is applied perpendicular to the crystallographic c axis is attributed to the mixing of the singlet and triplet states through the antisymmetric Dzyaloshinskii–Moriya interaction12. Therefore, to a first approximation, the spin Hamiltonian is given by:
where Ji j are the nearest-neighbour interactions for four different bonds J1>J2>J3>J4 as labelled in Fig. 1b, and Di j=D1, D2, D3 and D4 are the Dzyaloshinskii–Moriya vectors. The in-plane components of these vectors point towards the centre of a triangle and the out-of-plane components alternately point into and out of the kagome plane13. The spatial distribution of the non-uniform exchange interactions suggests the stabilization of the pinwheel VBS state25 in which singlet dimers are formed between two spins bonded by J1∼20 meV (ref. 12). This pinwheel VBS state is different from the VBS state theoretically proposed for the ideal kagome lattice, because it is stabilized by the lattice distortion and the long-range dimer–dimer correlations do not break the translational symmetry of the lattice. A theoretical study of the J4-depleted kagome lattice26, which is defined by J1=J2=J3≡J and J4≡α J, shows that the most energetically favoured VBS state proposed for the ideal kagome lattice, which corresponds to α=1, is destabilized against this pinwheel state even at α∼0.97, highlighting the importance of geometrical frustration, which leads to the diversity and closeness of exotic phases in the quantum kagome lattice.
To verify the pinwheel VBS state, we carried out inelastic neutron scattering on a single-crystal sample (see the Methods section). The scattering intensity can be described by the following equation:
where (γ r0/2)2=72.65×10−3 b/μB2, f(Q) is a magnetic form factor for Cu2+, S(ℏω,q;Qm) is a dynamical structure factor and Q=Qm+q. An energy scan at the Brillouin zone centre Qm=(0,2,0) (Fig. 2d) clearly shows two singlet-to-triplet gaps. The convoluted fit with the resolution function yields the gap energies Δ1=2.35(7) meV and Δ2=7.3(3) meV. The thermal renormalization of the spectral weight and damping of the spin excitations (Fig. 3) are typical of triplet modes in an interacting dimer system27. In addition, energy scans in an applied magnetic field (Fig. 4) reveal the Zeeman splitting of the lower gap and no splitting of the upper gap. These results demonstrate that, whereas the upper gap is a result of the excitations from the singlet to Stotz=0 triplet states, the lower gap, which is twofold degenerate at zero field, is due to the excitations from the singlet to Stotz=±1 triplet states (Stotz represents a triplet spin state of a single dimer). The Zeeman splitting of the Stotz=±1 triplet gap can be described by the following relation Δ1,±(H)=Δ1−g±μBStotzH, where μB is the Bohr magneton. For Stotz=1 and −1, we obtained g+=2.58(30) and g−=2.34(20), respectively. Extrapolating the Stotz=1 line to intercept the H axis, that is, Δ1,+(H=Hc)=0, yields the critical field Hc≈21 T, consistent with the magnetization data12. The magnitude of the Dzyaloshinskii–Moriya vector can be estimated from |D|/Ja v∼Δg/g∼0.2 () (ref. 28). Both the Landé g factor and are in good agreement with the values obtained from fitting the susceptibility data to the exact diagonalization for the 12-site kagome cluster12,13.
For Rb2Cu3SnF12, six dimers in the unit cell give rise to six triplet modes. In the limit J2−J40, these modes can be characterized by the irreducible representations of the cyclic group C3 as four E and two A localized (flat) modes (see Fig. 6 in ref. 26 and Supplementary Fig. S3). The observed triplet excitations belong to the lowest-energy E mode. The partial lifting of this E-mode triplet degeneracy can be attributed to the Dzyaloshinskii–Moriya interaction. For a single dimer, one can easily show that the Dzyaloshinskii–Moriya interaction raises the energy of the Stotz=0 state, while retaining the degeneracy of the two Stotz=±1 states.To explain the observed large bandwidth and small Stotz=±1 triplet gap, we carried out the dimer series expansion on the pinwheel VBS ground state using the model Hamiltonian given in equation (1). We assume that the strength of the Dzyaloshinskii–Moriya vectors scales with the respective exchange interactions, that is, dz=Diz/Ji (dp=Dip/Ji), where i=1,2,3 and 4 for the out-of-plane (in-plane) component. In our first-order dimer expansion (see Supplementary Fig. S3), the out-of-plane component dz of the Dzyaloshinskii–Moriya interaction lowers the Stotz=±1 branch at the Γ-point, which strongly suggests that the observed two branches are the lowest-energy E-mode in the Stotz=±1 and 0 sectors. On the other hand, the in-plane component dp determines only detailed structure of further energy splitting, and thus is omitted in the higher-order calculations. The linked cluster expansion algorithm29 was used to obtain a longer series. The lowest-energy triplet excitation spectra were then calculated using the Dlog-Padé approximation up to eighth order in the interdimer exchange and Dzyaloshinskii–Moriya interactions. Under the constraints J1>J2>J3>J4 and a fixed Ja v, we obtain the best fit to the excitation spectra (Fig. 2a–c) for J1=18.6 meV, J2=0.95J1, J3=0.85J1, J4=0.55J1 and dz=0.18. The agreement between the measured and calculated triplet dispersions verifies the pinwheel VBS state. The integrated dynamic structure factor (Fig. 2e) was calculated to fifth order, assuming equal contributions from two domains, one domain with pinwheel arms in the clockwise direction and the other with pinwheel arms in the anticlockwise direction. These two pinwheel motifs correspond to two equivalent structural domains, which are a mirror image of each other (see Supplementary Fig. S5). The intensity pattern with the periodicity four times the reciprocal lattice vector correlates with the intra-dimer distance, which is one fourth of the in-plane lattice constant. The maximum scattering intensity around (2, 0, 0), (0, 2, 0) and (2, 2, 0) also supports the pinwheel VBS state in Rb2Cu3SnF12 (see Supplementary Information).
In contrast to triplet excitations in other dimer systems, the highly dispersive triplet bands and the small triplet gap (Δ1/J1∼0.13) in Rb2Cu3SnF12 are due to the strong interdimer interactions, the Dzyaloshinskii–Moriya interaction and the unique pinwheel arrangement of dimers. The bandwidth of the Stotz=0 branch is denominated by J2–J4; therefore, the rather small J4≈J2/2 is essential to explain its relatively large bandwidth. It is also worth noting that the Stotz=0 branch is virtually unaffected by dz and has a gap minimum at the K-point, as predicted by the bond operator mean field theory without the Dzyaloshinskii–Moriya interaction26. On the other hand, dz changes the gap minimum of the Stotz=±1 branch from the K-point to the Γ-point, and significantly lowers the gap energy. If the strength of dz exceeded 0.25, the gap at the Γ-point would be closed and magnetic order would instead be observed.
The macroscopically degenerate ground states in the kagome lattice are particularly sensitive to small perturbations, and thereby could inherently give rise to a number of different states in a real system. In Rb2Cu3SnF12, our results reveal the pinwheel VBS state that has been suggested to be in close proximity to other phases11,26. Recently, we have studied other related compounds, in which a different ground state was observed13. In particular, Cs2Cu3SnF12 exhibits magnetic order but possesses the low-temperature phase that shares the crystal structure with Rb2Cu3SnF12, which suggests similar magnetic interactions. The magnetic order in Cs2Cu3SnF12 could, therefore, be ascribed to a small difference in exchange coupling and/or a large Dzyaloshinskii–Moriya interaction30. In this case, the former could play a more prominent role in determining the ground state because the Dzyaloshinskii–Moriya interactions in these systems are essentially comparable13. In addition, other phases could be obtained by tuning external parameters such as magnetic field and pressure. Further experimental and theoretical studies are desirable to determine precisely the driving mechanism that leads to these different ground states. A more comprehensive understanding of this mechanism could shed some light on many remaining questions regarding the quantum kagome lattice, particularly in relation to the sought-after quantum spin liquid.
Methods
Experiments.
Inelastic neutron scattering experiments were carried out on the HER and GPTAS triple-axis spectrometers operated by the Institute for Solid State Physics, University of Tokyo. Vertically focusing pyrolytic graphite crystals were used to monochromate the incident neutron beam. For the measurements on the HER spectrometer, the scattered neutrons with a fixed final energy of 5 meV were analysed by the central three blades of a seven-blade doubly focused pyrolytic graphite analyser. A cooled Be or oriented pyrolytic graphite crystal filter was placed in the incident beam, and a room-temperature Be filter in the scattered beam to remove higher-order contamination. For the measurements on the GPTAS spectrometer, vertically focused (horizontally flat) pyrolytic graphite crystals were used to analyse scattered neutrons with fixed final energies of 14.7 and 13.7 meV. The horizontal collimation sequence of 40′–80′–sample–80′–80′ (LR) or 40′–40′–sample–40′–80′ (HR) was employed with a pyrolytic graphite filter placed in the scattered beam. A single crystal of mass 0.94 g was mounted in the h k0 and h0l zones. The single-crystal sample used in this study was synthesized using the method described in ref. 23. The sample was sealed in an aluminium container in a He gas environment for heat exchange, and cooled by a closed-cycle 4He cryostat. For the measurements in a magnetic field, the sample was aligned in the h k0 zone, and the field was applied along the crystallographic c axis, perpendicular to the kagome plane, using a vertical magnet with a maximum field of 5 T. The sample was then cooled by a liquid 4He cryostat equipped with the magnet.
Data analysis.
In Fig. 2a, the measurements were done around (2, 0, 0) for energies between 2 and 4.5 meV, and around (2, 2, 0) between 4.5 and 9 meV. In Fig. 2b, the measurements were done around (2, 2, 0) along two equivalent high-symmetry directions; for energies between 2 and 4 meV, the scans were taken along (H, H, 0), and between 4 and 9 meV along (H, 6-2 H, 0). In Fig. 2a, the triplet branch around (1.5, 0, 0) could be a result of higher-order corrections to the exchange interactions owing to small atomic displacements as evidenced by weak superlattice reflections; we have ruled out the contribution of other triplet modes (see Supplementary Information). All scans were analysed using equation (2) and an empirical dispersion relation, , where α=1 or 2 for two non-degenerate branches, and q=(kx,ky) is a wave vector away from a Brillouin zone centre Qm. Within the resolution of the instrument of ∼1 meV (GPTAS-LR), the out-of-plane dispersion is flat, attesting to the two-dimensionality of the system, and hence we set vα,z=0. The observed peaks were fitted with narrow, resolution-limited Lorenztians, convoluted with the resolution function taking into account the empirical dispersion relation. The line shapes are governed by the convolution with the resolution function. The peak positions in units of Å−1 were obtained by fitting vα and an offset Δq (see Supplementary Fig. S2).
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Acknowledgements
We thank M. Nishi, C. Broholm, K. Iida and M. Kofu for useful discussions. The work at Tokyo Institute of Technology was supported by a Grant-in-Aid for Scientific Research from JSPS and a Global COE Program funded by MEXT Japan.
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K. Matan, T.O. and T.J.S. carried out neutron scattering measurements. J.Y. carried out X-ray diffraction measurements. Y.F. carried out numerical calculations. T.O., M.Y., K. Morita and H.T. contributed to sample synthesis and characterization.
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Matan, K., Ono, T., Fukumoto, Y. et al. Pinwheel valence-bond solid and triplet excitations in the two-dimensional deformed kagome lattice. Nature Phys 6, 865–869 (2010). https://doi.org/10.1038/nphys1761
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DOI: https://doi.org/10.1038/nphys1761
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