Pinwheel valence-bond solid and triplet excitations in the two-dimensional deformed kagome lattice

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 bond^2,3,4,5,6,7 and valence-bond solid^8,9,10,11 (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.

Main

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 state^2,5, a gapped-spin-liquid state^3,4,6,7 and a VBS state^8,9,10,11. One leading approach is based on a quantum dimer model, which was first introduced to describe spin dynamics of singlet pairs (dimers) in high-T_c superconductors^14,15. On the basis of this model, two kinds of quantum state promptly emerge, namely the VBS and resonating-valence-bond states^6,7,9,10,11. 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 dimensional^16,17,18 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 herbertsmithite^20,21,22 and distorted kagome lattice in volborthite^23,24 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 H_c≈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 J_{i j} are the nearest-neighbour interactions for four different bonds J₁>J₂>J₃>J₄ as labelled in Fig. 1b, and D_{i 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.

Figure 1: Crystal structure of Rb₂Cu₃SnF₁₂.

a, A crystal structure in the a b plane shows the connectivity of the Cu²⁺ ions (red) forming a deformed kagome lattice. b, Dimers (blue bonds) form the pinwheel VBS state. The exchange interactions are labelled as J₁>J₂>J₃>J₄.

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/μ_B², f(Q) is a magnetic form factor for Cu²⁺, S(ℏω,q;Q_m) is a dynamical structure factor and Q=Q_m+q. An energy scan at the Brillouin zone centre Q_m=(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 S_tot^z=0 triplet states, the lower gap, which is twofold degenerate at zero field, is due to the excitations from the singlet to S_tot^z=±1 triplet states (S_tot^z represents a triplet spin state of a single dimer). The Zeeman splitting of the S_tot^z=±1 triplet gap can be described by the following relation Δ_1,±(H)=Δ₁−g_±μ_BS_tot^zH, where μ_B is the Bohr magneton. For S_tot^z=1 and −1, we obtained g₊=2.58(30) and g₋=2.34(20), respectively. Extrapolating the S_tot^z=1 line to intercept the H axis, that is, Δ_1,+(H=H_c)=0, yields the critical field H_c≈21 T, consistent with the magnetization data¹². The magnitude of the Dzyaloshinskii–Moriya vector can be estimated from |D|/J_{a 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 cluster^12,13.

Figure 2: Energy–momentum contour maps.

a,b, Intensity contour maps were measured at T=3 K around (2, 0, 0) and (2, 2, 0) along two high-symmetry directions, along k_x (a) and along k_y (b). c, Triplet dispersions were measured along the path shown in the inset. The error bars of the experimental data correspond to three times the statistical error obtained from the fitting (see the Methods section). The solid lines represent the best fits to the dimer series expansion described in the text. The error bars of the calculated curves reflect the differences of various Padé approximations. d, An energy scan at Q_m= (0, 2, 0) (Γ-point) and T=5 K shows two spin gaps at 2.35(7) and 7.3(3) meV. The error bars correspond to the counting-statistical error. e, The calculated dynamic structure factor integrated over triplet excitations shows a high scattering intensity around (2, 0, 0), (0, 2, 0) and (2, 2, 0), consistent with the experimental data.

Figure 3: Representative energy scans and temperature dependence.

a–c, Energy scans of the lower gap at three temperatures. The error bars correspond to the counting-statistical error. The solid lines denote the fits to the dispersion relation described in the Methods section, convoluted with the instrumental resolution function. Peak widths are resolution limited at the base temperature and become broader with increasing temperatures. d, Temperature dependence of the two triplet gaps; the error bars correspond to three times the statistical error obtained from the fitting.

Figure 4: Triplet excitations in a magnetic field.

a–c, Energy scans measured at T=5 K show the splitting of the lower gap at three different fields. The error bars correspond to the counting-statistical error. The solid lines denote the fits to the dispersion relation described in the Methods section, convoluted with the instrumental resolution function. The yellow and green shading denotes the S_tot^z=1 and S_tot^z=−1 triplet excitations at the Γ-point, respectively. d, Field dependence of the triplet gaps; the error bars correspond to three times the statistical error obtained from the fitting.

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 S_tot^z=0 state, while retaining the degeneracy of the two S_tot^z=±1 states.To explain the observed large bandwidth and small S_tot^z=±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, d_z=D_i^z/J_i (d_p=D_i^p/J_i), 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 d_z of the Dzyaloshinskii–Moriya interaction lowers the S_tot^z=±1 branch at the Γ-point, which strongly suggests that the observed two branches are the lowest-energy E-mode in the S_tot^z=±1 and 0 sectors. On the other hand, the in-plane component d_p 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 J_{a 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 d_z=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 S_tot^z=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 S_tot^z=0 branch is virtually unaffected by d_z 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, d_z changes the gap minimum of the S_tot^z=±1 branch from the K-point to the Γ-point, and significantly lowers the gap energy. If the strength of d_z 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 phases^11,26. 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=(k_x,k_y) is a wave vector away from a Brillouin zone centre Q_m. 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 v_α,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 v_α and an offset Δq (see Supplementary Fig. S2).