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arXiv:1707.09666 · Physical review. B./Physical review. B

Microscopic theory of the nearest-neighbor valence bond sector of the spin- 12 kagome antiferromagnet

Arnaud Ralko, Frédéric Mila, Ioannis Rousochatzakis

Published Mar 1, 2018• 24 citations
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Abstract

The spin-$\frac{1}{2}$ Heisenberg model on the kagome lattice, which is closely realized in layered Mott insulators such as ${\mathrm{ZnCu}}_{3}{(\mathrm{OH})}_{6}{\mathrm{Cl}}_{2}$, is one of the oldest and most enigmatic spin-$\frac{1}{2}$ 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 ${Z}_{2}$ 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-$\frac{1}{2}$ 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 ${Z}_{2}$ 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 $d$, there is a transition between the ${Z}_{2}$ spin liquid at small $d$ and the diamond valence bond crystal at large $d$, the prediction of the present microscopic description for the two-dimensional lattice. These results show that, if the ground state of the spin-$\frac{1}{2}$ 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.

DOI
10.1103/physrevb.97.104401

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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.
We revisit the description of the low-energy singlet sector of the spin-12\frac{1}{2}21​ Heisenberg antiferromagnet on kagome in terms of an effective quantum dimer model. With the help of exact diagonalizations of appropriate finite-size clusters, we show that the embedding of a given process in its kagome environment leads to dramatic modifications of the amplitudes of the elementary loop processes, an effect not accessible to the standard approach based on the truncation of the Hamiltonian to the nearest-neighbor valence-bond basis. The resulting parameters are consistent with a Z2{Z}_{2}Z2​ spin liquid rather than with a valence-bond crystal, in agreement with the last density matrix renormalization group results.
The nature of quantum spin liquids is studied for the spin-1/21/21/2 antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites, J1J_1J1​, and those between next-nearest-neighbor sites, J2J_2J2​. We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region 0.4<J2/J1≤0.60.4< J_2/J_1\le 0.60.4<J2​/J1​≤0.6 sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of 0.4<J2/J1≤0.50.4 < J_2/J_1 \le 0.50.4<J2​/J1​≤0.5, while the gapped valence-bond-crystal (VBC) phase is stabilized for 0.5<J2/J1≤0.60.5 < J_2/J_1 \le 0.60.5<J2​/J1​≤0.6. The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the C4vC_{4v}C4v​ symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation ⟨S(0)S(r)⟩∝1/rz+η\langle S(0)S(r)\rangle \propto 1/r^{z+η}⟨S(0)S(r)⟩∝1/rz+η at a long distance rrr appears to increase from z+η∼1z+η\sim 1z+η∼1 at J2/J1∼0.4J_2/J_1\sim0.4J2​/J1​∼0.4 toward the continuous transition to the VBC phase at J1/J1∼0.5J_1/J_1\sim0.5J1​/J1​∼0.5. Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.
We study the ground-state phase diagram of the quantum spin-1/2\text{spin-}1/2spin-1/2 Heisenberg model on the kagome lattice with first- (J1<0)({J}_{1}<0)(J1​<0), second- (J2<0)({J}_{2}<0)(J2​<0), and third-neighbor interactions (Jd>0)({J}_{d}>0)(Jd​>0) by means of analytical low-energy field theory and numerical density-matrix renormalization group (DMRG) studies. The results offer a consistent picture of the Jd-dominant{J}_{d}\text{-dominant}Jd​-dominant regime in terms of three sets of spin chains weakly coupled by the ferromagnetic interchain interactions J1,2{J}_{1,2}J1,2​. When either J1{J}_{1}J1​ or J2{J}_{2}J2​ is much stronger than the other one, the model is found to support one of two cuboctohedral phases, cuboc1, and cuboc2. These cuboc states host noncoplanar long-ranged magnetic order and possess finite scalar spin chirality. However, in the compensated regime ${J}_{1}\ensuremath{\simeq}{J}_{2}$, a valence bond crystal phase emerges between the two cuboc phases. We find excellent agreement between an analytical theory based on coupled spin chains and unbiased DMRG calculations, including at a very detailed level of comparison of the structure of the valence bond crystal state. To our knowledge, this is the first such comprehensive understanding of a highly frustrated two-dimensional quantum antiferromagnet. We find no evidence of either the one-dimensional gapless spin liquid or the chiral spin liquids, which were previously suggested by parton mean-field theories.
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 present numerical evidence for the existence of an extended valence bond solid (VBS) phase at T=0T=0T=0 in the kagome S=1/2S=1/2S=1/2 Heisenberg antiferromagnet with ferromagnetic further-neighbor interactions. The VBS is located at the boundary between two magnetically ordered regions and extends close to the nearest-neighbor Heisenberg point. It exhibits a diamondlike singlet covering pattern with a 12-site unit cell. Our results suggest the possibility of a direct, possibly continuous, quantum phase transition from the neighboring q=0\mathbf{q}=0q=0 coplanar magnetically ordered phase into the VBS phase. Moreover, a second phase which breaks lattice symmetries, and is of likely spin-nematic type, is found close to the transition to the ferromagnetic phase. The results have been obtained using large-scale numerical exact diagonalization. We discuss implications of our results on the nature of the nearest-neighbor Heisenberg antiferromagnet.
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