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arXiv:2309.04363 · Physical review. B./Physical review. B

Strong relevance of zinc impurities in spin- 12 kagome quantum antiferromagnets: A variational study

Jianhua Yang, Tao Li

Published Mar 1, 2024• 2 citations
arXivHTMLPDF

Abstract

Copper hydroxyhalide materials herbertsmithite ${\mathrm{ZnCu}}_{3}{(\mathrm{OH})}_{6}{\mathrm{Cl}}_{2}$ and Zn-barlowite ${\mathrm{ZnCu}}_{3}{(\mathrm{OH})}_{6}\mathrm{FrBr}$ are thought to be the best realizations of the spin-1/2 kagome quantum antiferromagnetic Heisenberg model and are widely believed to host a spin liquid ground state. However, the exact nature of such a novel state of matter is still under strong debate, partly due to the complication related to the occupation disorder between the zinc and the copper ions in these systems. More specifically, some nonmagnetic ${\mathrm{Zn}}^{2+}$ ions in the system may occupy the ${\mathrm{Cu}}^{2+}$ site within the kagome plane and some spin-$1/2\phantom{\rule{4pt}{0ex}}{\mathrm{Cu}}^{2+}$ ions may occupy the ${\mathrm{Zn}}^{2+}$ site between the kagome planes. Both types of impurity can have a nontrivial effect on the dynamical behavior of the antiferromagnetically coupled ${\mathrm{Cu}}^{2+}$ spins on the kagome planes. Recent nuclear magnetic resonance measurements indicate that the magnetic response of the kagome plane is significantly spatial inhomogeneous, even though the content of the misplaced zinc or copper ions is very small. Here we use extensive variational optimization to show that the well-known U(1) Dirac spin liquid state is extremely sensitive to the introduction of the nonmagnetic ${\mathrm{Zn}}^{2+}$ impurity in the kagome plane. More specifically, we find that the zinc impurities can significantly reorganize the local spin correlation pattern around them and induce strong spatial oscillation in the magnetic response of the system. We argue that this is a general trend in highly frustrated quantum magnet systems, in which the nonmagnetic impurity may act as a strongly relevant perturbation on the emergent resonating valence bond structure in their spin liquid ground state. We also argue that the strong spatial oscillation in the magnetic response should be attributed to the free moment released by the doped ${\mathrm{Zn}}^{2+}$ ions and may serve as the smoking-gun evidence for the Dirac node in the U(1) Dirac spin liquid state on the kagome lattice.

DOI
10.1103/physrevb.109.115103

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Copper hydroxyhalide materials herbertsmithite ZnCu3(OH)6Cl2{\mathrm{ZnCu}}_{3}{(\mathrm{OH})}_{6}{\mathrm{Cl}}_{2}ZnCu3​(OH)6​Cl2​ and Zn-barlowite ZnCu3(OH)6FrBr{\mathrm{ZnCu}}_{3}{(\mathrm{OH})}_{6}\mathrm{FrBr}ZnCu3​(OH)6​FrBr are thought to be the best realizations of the spin-1/2 kagome quantum antiferromagnetic Heisenberg model and are widely believed to host a spin liquid ground state. However, the exact nature of such a novel state of matter is still under strong debate, partly due to the complication related to the occupation disorder between the zinc and the copper ions in these systems. More specifically, some nonmagnetic Zn2+{\mathrm{Zn}}^{2+}Zn2+ ions in the system may occupy the Cu2+{\mathrm{Cu}}^{2+}Cu2+ site within the kagome plane and some spin-1/2Cu2+1/2\phantom{\rule{4pt}{0ex}}{\mathrm{Cu}}^{2+}1/2Cu2+ ions may occupy the Zn2+{\mathrm{Zn}}^{2+}Zn2+ site between the kagome planes. Both types of impurity can have a nontrivial effect on the dynamical behavior of the antiferromagnetically coupled Cu2+{\mathrm{Cu}}^{2+}Cu2+ spins on the kagome planes. Recent nuclear magnetic resonance measurements indicate that the magnetic response of the kagome plane is significantly spatial inhomogeneous, even though the content of the misplaced zinc or copper ions is very small. Here we use extensive variational optimization to show that the well-known U(1) Dirac spin liquid state is extremely sensitive to the introduction of the nonmagnetic Zn2+{\mathrm{Zn}}^{2+}Zn2+ impurity in the kagome plane. More specifically, we find that the zinc impurities can significantly reorganize the local spin correlation pattern around them and induce strong spatial oscillation in the magnetic response of the system. We argue that this is a general trend in highly frustrated quantum magnet systems, in which the nonmagnetic impurity may act as a strongly relevant perturbation on the emergent resonating valence bond structure in their spin liquid ground state. We also argue that the strong spatial oscillation in the magnetic response should be attributed to the free moment released by the doped Zn2+{\mathrm{Zn}}^{2+}Zn2+ ions and may serve as the smoking-gun evidence for the Dirac node in the U(1) Dirac spin liquid state on the kagome lattice.
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.
The doping of quantum Heisenberg antiferromagnets on the kagome lattice by nonmagnetic impurities is investigated within the framework of a generalized quantum dimer model (QDM) describing (a) the valence-bond crystal (VBC), (b) the dimer liquid, and (c) the critical region on equal footing. Following the approach by Ralko et al. [Phys. Rev. Lett. 101, 117204 (2008)] for the square and triangular lattices, we introduce the (minimal) extension of the QDM on the kagome lattice to account for spontaneous creation of mobile S=1/2S=1/2S=1/2 spinons at finite magnetic field. Modulations of the dimer density (at zero or finite magnetic field) and of the local field-induced magnetization in the vicinity of impurities are computed using Lanczos exact diagonalization techniques on small clusters (48 and 75 sites). The VBC is clearly revealed from its pinning by impurities, while, in the dimer liquid, crystallization around impurities involves only two neighboring dimers. We also find that a next-nearest-neighbor ferromagnetic coupling favors VBC order. Unexpectedly, a small-size spinon-impurity bound state appears in some region of the dimer-liquid phase. In contrast, in the VBC phase the spinon delocalizes within a large region around the impurity, revealing the weakness of the VBC confining potential. Lastly, we observe that an impurity concentration as small as 4% enhances dimerization substantially. These results are confronted to the valence-bond glass scenario [R. R. P. Singh, Phys. Rev. Lett. 104, 177203 (2010)] and implications to the interpretation of the nuclear magnetic resonance spectra of the Herbertsmithite compound are outlined.
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.
Evidence for a Z2_22​ topological ordered quantum spin liquid in a kagome-lattice antiferromagnet
A quantum spin liquid with a Z2Z_2Z2​ topological order has long been thought to be important for the application of quantum computing and may be related to high-temperature superconductivity. While a two-dimensional kagome antiferromagnet may host such a state, strong experimental evidences are still lacking. Here we show that the spin excitations from the kagome planes in magnetically ordered Cu4_44​(OD)6_66​FBr and non-magnetically ordered Cu3_33​Zn(OD)6_66​FBr are similarly gapped although the content of inter-kagome-layer Cu2+^{2+}2+ ions changes dramatically. This suggests that the spin triplet gap and continuum of the intrinsic kagome antiferromagnet are robust against the interlayer magnetic impurities. Our results show that the ground state of Cu3_33​Zn(OD)6_66​FBr is a gapped quantum spin liquid with Z2_22​ topological order.
Abstract The kagome lattice of spin-1/2 copper atoms in herbertsmithite is conjectured to sustain a quantum spin liquid state with spinon quasiparticles. Ideally, the kagome crystal planes are each separated by a plane of spinless zinc atoms. However, in real crystals, some spin-1/2 copper atoms substitute randomly onto these inter-kagome zinc sites. Here we reconceptualize such ‘impurity’ atoms as quantum witness spins whose dynamics is designed to probe the spin liquid state. We then introduce spin noise spectroscopy to measure the frequency and temperature dependence of witness spin dynamics, demonstrating that their phenomenology is consistent with extensive interactions between witness spins mediated by propagation of spinons through a quantum spin liquid. Ultimately, a sharp transition occurs at around 260 mK, below which the properties of both spin noise and magnetic susceptibility suggest that the witness spins form a spin glass phase. Among the theoretical models considered, we demonstrate that our observations are only consistent with spinon-mediated interactions between witness spins by either a Z 2 or U(1) quantum spin liquid, with the former model more closely matching the data. Our work demonstrates that quantum mechanical witness spins may now conceivably be used as a widely applicable probe of quantum spin liquid physics.
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.