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arXiv:2008.07829 · Physical review. B./Physical review. B

Variational study of the ground state and spin dynamics of the spin- 1 2 kagome antiferromagnetic Heisenberg model and its implication for herbertsmithite ZnCu 3 ( OH ) 6 Cl 2

Chun Zhang, Tao Li

Published Nov 4, 2020• 27 citations
arXivHTMLPDF

Abstract

We find that the best resonating valence bond (RVB) state of the spin-$\frac{1}{2}$ kagome antiferromagnetic Heisenberg model (spin-$\frac{1}{2}$ KAFH) is described by a ${Z}_{2}$ 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 $U(1)$ 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 $U(1)$ 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-$\frac{1}{2}$ KAFH is not at all featureless, but is characterized by a prominent spectral peak at about $0.25J$ around the $\mathbf{M}$ point, which is immersed in a broad continuum extending to $2.7J$. Based on these results, we argue that the spectral peak below 2 meV in the inelastic neutron scattering spectrum of hebertsmithite ${\mathrm{ZnCu}}_{3}{(\mathrm{OH})}_{6}{\mathrm{Cl}}_{2}$, which has been attributed to the contribution of ${\mathrm{Cu}}^{2+}$ impurity spins occupying the ${\mathrm{Zn}}^{2+}$ 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 $\mathbf{M}$ point as a result of the strong antiferromagnetic correlation between nearest neighboring spins on the kagome lattice.

DOI
10.1103/physrevb.102.195106

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

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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.

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Identifying Universal Spin Excitations in Spin-1/2 Kagome Quantum Spin Liquid Materials

Aaron Breidenbach, Arthur C. Campello, Jiajia Wen +4 more

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 $S$=1/2 kagome lattice antiferromagnet is particularly promising. Using high resolution inelastic neutron scattering measurements on Zn-barlowite (Zn$_\mathrm{x}$Cu$_\mathrm{4-x}$(OD)$_\mathrm{6}$FBr, $x\simeq 0.80$), we measure a spin excitation spectrum consistent with a QSL ground state. Continuum scattering above $\sim$1 meV matches that of herbertsmithite (Zn$_\mathrm{x}$Cu$_\mathrm{4-x}$(OD)$_6$Cl$_2$, $x\simeq 0.85$), 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 $Δ= 1.1(2)$ 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.

arXiv:2504.06491•Apr 8, 2025•0 citations
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 Z2[0,\ensuremathπ]\ensuremathβ{Z}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}Z2​[0,\ensuremathπ]\ensuremathβ 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.
Z2Z_{2}Z2​ spin liquid ground state of the spin-12\frac{1}{2}21​ Kagome antiferromagnetic Heisenberg model
We show that the best variational ground state of the spin-12\frac{1}{2}21​ Kagome antiferromagnetic Heisenberg model with nearest-neighboring exchange coupling(NN-KAFH) is a Z2Z_{2}Z2​ spin liquid state, rather than the widely believed U(1)U(1)U(1) Dirac spin liquid state. The spinon excitation in the Z2Z_{2}Z2​ spin liquid state has a small gap of about 1/401/401/40 of the spinon band width. We find that while the Z2Z_{2}Z2​ and the U(1)U(1)U(1) spin liquid state have a large overlap on finite clusters and are thus very close in energy, they host totally different spinon excitation spectrum. The strength of the RVB theory becomes particularly clear in such a situation, as it provides not only a variational understanding of the ground state structure, but also a comprehensive picture for the excitation spectrum of the system. Our result indicates that the spin-12\frac{1}{2}21​ NN-KAFH should be better understood as a nearly critical system, rather than a prototypical gapped Z2Z_{2}Z2​ spin liquid system.
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.
Z2_{2}2​ spin liquid phase on the kagome lattice: a new saddle point
We have performed large scale variational search for the best RVB ansatz for the spin-12\frac{1}{2}21​ kagome antiferromagnet with both nearest-neighboring(NN) and next-nearest-neighboring(NNN) exchanges assuming only translational symmetry. We find the best RVB state is always fully symmetric and has an mean field ansatz that is gauge equivalent to a previous proposed Z2_{2}2​ spin liquid ansatz. The Z2_{2}2​ state is found to be slightly more stable than the extensively studied U(1) gapless Dirac spin liquid state in both the J2≠0J_{2}\neq0J2​=0 and the J2=0J_{2}=0J2​=0 case and to possess a small spinon gap for J2<0.2J_{2}<0.2J2​<0.2. The breaking of the U(1) gauge symmetry in the Z2_{2}2​ state is found to increase with J2J_{2}J2​ and to be quite substantial around J2=0.15J_{2}=0.15J2​=0.15. However, we find the Z2_{2}2​ state is always very close to the gapless U(1) Dirac spin liquid state, although they have very different RVB parameters. We argue the kagome antiferromagnet should be better understood as a near critical system, rather than a system deep inside a gapped spin liquid phase with well established Z2_{2}2​ topological order.
In this paper, we do a complete classification of valence-bond crystals (VBCs) on the kagomé lattice based on general arguments of symmetry only and thus identify many new VBCs for different unit cell sizes. For the spin-1/2 Heisenberg antiferromagnet, we study the relative energetics of competing gapless spin liquids (SLs) and VBC phases within the class of Gutzwiller-projected fermionic wave functions using variational Monte Carlo techniques, hence implementing exactly the constraint of one fermion per site. By using a state-of-the-art optimization method, we conclusively show that the U(1) Dirac SL is remarkably stable towards dimerizing into all 6-, 12- and 36-site unit cell VBCs. This stability is also preserved on addition of a next-nearest-neighbor super-exchange coupling of both antiferromagnetic and ferromagnetic (FM) type. However, we find that a 36-site unit cell VBC is stabilized on addition of a very small next-nearest-neighbor FM super-exchange coupling, i.e. | J 2 | ≈ 0.045, and this VBC is the same in terms of space-group symmetry as that obtained in an effective quantum dimer model study. It breaks reflection symmetry, has a nontrivial flux pattern and is a strong dimerization of the uniform RVB SL.
The spin-12\frac{1}{2}21​ Heisenberg model on the kagome lattice, which is closely realized in layered Mott insulators such as ZnCu3(OH)6Cl2{\mathrm{ZnCu}}_{3}{(\mathrm{OH})}_{6}{\mathrm{Cl}}_{2}ZnCu3​(OH)6​Cl2​, is one of the oldest and most enigmatic spin-12\frac{1}{2}21​ lattice models. While the numerical evidence has accumulated in favor of a quantum spin liquid, the debate is still open as to whether it is a Z2{Z}_{2}Z2​ spin liquid with very short-range correlations (some kind of resonating valence bond spin liquid), or an algebraic spin liquid with power-law correlations. To address this issue, we have pushed the program started by Rokhsar and Kivelson in their derivation of the effective quantum dimer model description of Heisenberg models to unprecedented accuracy for the spin-12\frac{1}{2}21​ kagome, by including all the most important virtual singlet contributions on top of the orthogonalization of the nearest-neighbor valence bond singlet basis. Quite remarkably, the resulting picture is a competition between a Z2{Z}_{2}Z2​ spin liquid and a diamond valence bond crystal with a 12-site unit cell, as in the density-matrix renormalization group simulations of Yan et al. Furthermore, we found that, on cylinders of finite diameter ddd, there is a transition between the Z2{Z}_{2}Z2​ spin liquid at small ddd and the diamond valence bond crystal at large ddd, the prediction of the present microscopic description for the two-dimensional lattice. These results show that, if the ground state of the spin-12\frac{1}{2}21​ kagome antiferromagnet can be described by nearest-neighbor singlet dimers, it is a diamond valence bond crystal, and, a contrario, that, if the system is a quantum spin liquid, it has to involve long-range singlets, consistent with the algebraic spin liquid scenario.
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.