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arXiv:2610.24065

Two quantum spin liquid states coexisting in the kagome antiferromagnet herbertsmithite

Arthur C. Campello, Shengtao Jiang, Wei He, Jiajia Wen, Daniel M. Pajerowski, Steven Allan Kivelson, Hong-Chen Jiang, Young Sun Lee

Published Oct 16, 2026• 0 citations
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Abstract

Herbertsmithite (ZnCu$_3$(OH)$_6$Cl$_2$) is one of the most extensively studied quantum spin liquid (QSL) candidates, yet the precise nature of its ground state remains unresolved. Here, we compare the results of high-resolution inelastic neutron scattering on single crystals of herbertsmithite (Zn$_\mathrm{x}$Cu$_\mathrm{4-x}$(OD)$_6$Cl$_2$, $x\simeq 0.85$) with large-scale density-matrix renormalization group (DMRG) and dynamical DMRG simulations to benchmark the spin Hamiltonian and uncover fundamental properties of the ground state. The measured spin correlations determine an energy density for the Heisenberg model that closely matches simulations, an essential verification of the Hamiltonian. In parallel, DMRG simulations reveal two nearly degenerate QSL states. Both preserve translational symmetry, but one retains the lattice point-group symmetries whereas the other develops anisotropic spin correlations that break point-group symmetry. These two QSLs have distinct spin structure factors, and, remarkably, close agreement with the neutron spectra is achieved with an admixture of the two states. Our results establish a quantitative correspondence between neutron scattering and simulations, and suggest that herbertsmithite realizes a regime in which two proximate gapped QSL states coexist.

DOI
10.48550/arxiv.2610.24065

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Herbertsmithite (ZnCu3_33​(OH)6_66​Cl2_22​) is one of the most extensively studied quantum spin liquid (QSL) candidates, yet the precise nature of its ground state remains unresolved. Here, we compare the results of high-resolution inelastic neutron scattering on single crystals of herbertsmithite (Znx_\mathrm{x}x​Cu4−x_\mathrm{4-x}4−x​(OD)6_66​Cl2_22​, x≃0.85x\simeq 0.85x≃0.85) with large-scale density-matrix renormalization group (DMRG) and dynamical DMRG simulations to benchmark the spin Hamiltonian and uncover fundamental properties of the ground state. The measured spin correlations determine an energy density for the Heisenberg model that closely matches simulations, an essential verification of the Hamiltonian. In parallel, DMRG simulations reveal two nearly degenerate QSL states. Both preserve translational symmetry, but one retains the lattice point-group symmetries whereas the other develops anisotropic spin correlations that break point-group symmetry. These two QSLs have distinct spin structure factors, and, remarkably, close agreement with the neutron spectra is achieved with an admixture of the two states. Our results establish a quantitative correspondence between neutron scattering and simulations, and suggest that herbertsmithite realizes a regime in which two proximate gapped QSL states coexist.
Abstract Spin-1/2 kagome antiferromagnets are leading candidates for realizing quantum spin liquid (QSL) ground states. While QSL ground states are predicted for the pure Heisenberg model, understanding the robustness of the QSL to additional interactions that may be present in real materials is a forefront question in the field. Here we employ large-scale density-matrix renormalization group simulations to investigate the effects of next-nearest neighbor exchange couplings J 2 and Dzyaloshinskii-Moriya interactions D , which are relevant to understanding the prototypical kagome materials herbertsmithite and Zn-barlowite. By utilizing clusters as large as XC12 and extrapolating the results to the thermodynamic limit, we precisely delineate the scope of the QSL phase, which remains robust across an expanded parameter range of J 2 and D . Direct comparison of the simulated static and dynamic spin structure factors with inelastic neutron scattering reveals the parameter space of the Hamiltonians for herbertsmithite and Zn-barlowite, and, importantly, provides compelling evidence that both materials exist within the QSL phase. These results establish a powerful convergence of theory and experiment in this most elusive state of matter.
A quantum spin liquid (QSL) is an exotic quantum state of matter characterized by fluctuating spins which may exhibit long-range entanglement. Among the possible host candidates for a QSL ground state, the SSS=1/2 kagome lattice antiferromagnet is particularly promising. Using high resolution inelastic neutron scattering measurements on Zn-barlowite (Znx_\mathrm{x}x​Cu4−x_\mathrm{4-x}4−x​(OD)6_\mathrm{6}6​FBr, x≃0.80x\simeq 0.80x≃0.80), we measure a spin excitation spectrum consistent with a QSL ground state. Continuum scattering above $\sim1meVmatchesthatofherbertsmithite(Zn1 meV matches that of herbertsmithite (Zn1meVmatchesthatofherbertsmithite(Zn_\mathrm{x}CuCuCu_\mathrm{4-x}(OD)(OD)(OD)_6ClClCl_2$, x≃0.85x\simeq 0.85x≃0.85), another prominent kagome QSL material, indicating universal spinon excitations. A detailed analysis of the spin-spin correlations, compared with density matrix renormalization group calculations, further indicate a QSL ground state for the physically relevant Hamiltonian parameters. The measured spectra in Zn-barlowite are consistent with gapped behavior with a gap size Δ=1.1(2)Δ= 1.1(2)Δ=1.1(2) meV. Comparison with a simple pair correlation model allows us to clearly distinguish intrinsic kagome correlations from impurity-induced correlations. Our results clarify the behavior that is universal within this important family of QSL candidate materials.
It has been suggested that the nearest-neighbor antiferromagnetic Heisenberg model on the Kagome lattice may be a good starting point for understanding the spin-liquid behavior discovered in herbertsmithite. In this work, we investigate possible quantum spin liquid phases in the presence of spin-rotation symmetry-breaking perturbations such as Dzyaloshinskii-Moriya and Ising interactions, as well as second-neighbor antiferromagnetic Heisenberg interactions. Experiments suggest that such perturbations are likely to be present in herbertsmithite. We use the projective symmetry group analysis within the framework of the slave-fermion construction of quantum spin liquid phases and systematically classify possible spin liquid phases in the presence of perturbations mentioned above. The dynamical spin-structure factor for relevant spin liquid phases is computed and the effect of those perturbations are studied. Our calculations reveal dispersive features in the spin structure factor embedded in a generally diffuse background due to the existence of fractionalized spin-1/21/21/2 excitations called spinons. For two of the previously proposed Z2{Z}_{2}Z2​ states, the dispersive features are almost absent, and diffuse scattering dominates over a large energy window throughout the Brillouin zone. This resembles the structure factor observed in recent inelastic neutron-scattering experiments on singlet crystals of herbertsmithite. Furthermore, one of the Z2{Z}_{2}Z2​ states with the spin structure factor with mostly diffuse scattering is gapped, and it may be adiabatically connected to the gapped spin liquid state observed in recent density-matrix renormalization group calculations for the nearest-neighbor antiferromagnetic Heisenberg model. The perturbations mentioned above are found to enhance the diffuse nature of the spin structure factor and reduce the momentum dependencies of the spin gap. We also calculate the electron spin resonance (ESR) absorption spectra that further characterize the role of spin-rotation symmetry-breaking perturbations in these states. We suggest that the measurement of ESR spectra can shed more light into the nature of the ground state in herbertsmithite.
We find that the best resonating valence bond (RVB) state of the spin-12\frac{1}{2}21​ kagome antiferromagnetic Heisenberg model (spin-12\frac{1}{2}21​ KAFH) is described by a Z2{Z}_{2}Z2​ mean field ansatz belonging to the ${Z}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ class, which hosts a mean field spinon dispersion very different from that of the widely studied U(1)U(1)U(1) Dirac spin liquid state. However, we find that the physical spin fluctuation spectrum calculated from the Gutzwiller projected random phase approximation (RPA) (GRPA) theory above such an RVB state is actually gapless and is almost identical to that above the U(1)U(1)U(1) Dirac spin liquid state. We find that such a peculiar behavior can be attributed to the unique flat band physics on the kagome lattice, which makes the mapping between the mean field ansatz and the RVB state noninjective. We find that the spin fluctuation spectrum of the spin-12\frac{1}{2}21​ KAFH is not at all featureless, but is characterized by a prominent spectral peak at about 0.25J0.25J0.25J around the M\mathbf{M}M point, which is immersed in a broad continuum extending to 2.7J2.7J2.7J. Based on these results, we argue that the spectral peak below 2 meV in the inelastic neutron scattering spectrum of hebertsmithite ZnCu3(OH)6Cl2{\mathrm{ZnCu}}_{3}{(\mathrm{OH})}_{6}{\mathrm{Cl}}_{2}ZnCu3​(OH)6​Cl2​, which has been attributed to the contribution of Cu2+{\mathrm{Cu}}^{2+}Cu2+ impurity spins occupying the Zn2+{\mathrm{Zn}}^{2+}Zn2+ site, should rather be understood as the intrinsic contribution from the kagome layer. We propose to verify such a picture by measuring the Knight shift on the Cu site, rather than the O site, which is almost blind to the spin fluctuation at the M\mathbf{M}M point as a result of the strong antiferromagnetic correlation between nearest neighboring spins on the kagome lattice.
We study the ground-state properties of the quantum spin liquid (QSL) phases of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the triangular lattice with nearest- (J1)({J}_{1})(J1​), next-nearest- (J2)({J}_{2})(J2​), and third-neighbor (J3)({J}_{3})(J3​) interactions by using the density-matrix renormalization group (DMRG) method. By combining parallel DMRG with SU(2)\mathrm{SU}(2)SU(2) spin rotational symmetry, we are able to obtain accurate results on large cylinders with length up to Lx=48{L}_{x}=48Lx​=48 and circumference Ly=6−−12{L}_{y}=6--12Ly​=6−−12. Our results suggest that the QSL phase of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model is gapped, which is characterized by the absence of a gapless mode and by short-range spin-spin and dimer-dimer correlations. In the presence of J3{J}_{3}J3​ interaction, we find that a critical QSL with a single gapless mode emerges. While both spin-spin and scalar chiral-chiral correlations are short ranged, dimer-dimer correlations are quasi long ranged and decay as a power law at long distances.