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arXiv:1108.5257 · Physical Review B

Spin cluster operator theory for the kagome lattice antiferromagnet

Kyusung Hwang, Yong Baek Kim, Jaejun Yu, Kwon Park

Published Nov 21, 2011• 22 citations
arXivHTMLPDF

Abstract

The spin-1/2 quantum antiferromagnet on the kagome lattice provides a quintessential example in strongly correlated electron physics in which the effects of both geometric frustration and quantum fluctuation are pushed to their limit. Among possible nonmagnetic ground states, the valence bond solid (VBS) with a 36-site unit cell is one of the most promising candidates. A natural theoretical framework for the analysis of such VBS order is to consider quantum states on a bond connecting the nearest-neighboring sites as fundamental quantum modes of the system and treat them as effectively independent ``bond particles.'' While correctly describing the VBS order in the ground state, this approach, known as the bond operator theory, significantly overestimates the lowest spin excitation energy. To overcome this problem, we take a next logical step in this paper to improve the bond operator theory and consider extended spin clusters as fundamental building blocks of the system. Depending on two possible configurations of the VBS order, various spin clusters are considered: (i) in the VBS order with staggered hexagonal resonance, we consider one spin cluster for a star of David shape and two spin clusters with each composed of a perfect hexagon and three attached dimers, and (ii) in the VBS order with uniform hexagonal resonance, we consider one spin cluster composed of a star of David and three attached dimers. It is shown that the majority of low-energy spin excitations are nearly or perfectly flat in energy. With most of its weight coming from the star of David, the lowest spin excitation has a gap much lower than the previous value obtained by the bond operator theory, narrowing the difference against exact diagonalization results. Despite the better agreement, however, the fact that the lowest spin excitation mostly comes from the star-of-David structure is at odds with the bond operator theory as well as previous series expansion studies. Expecting that the true low-energy spin excitations have contributions from both the star of David and its surroundings, we propose using two superclusters of spin in future work, treating spin excitations from both contributions on a more equal footing.

DOI
10.1103/physrevb.84.205133

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The spin-1/2 quantum antiferromagnet on the kagome lattice provides a quintessential example in strongly correlated electron physics in which the effects of both geometric frustration and quantum fluctuation are pushed to their limit. Among possible nonmagnetic ground states, the valence bond solid (VBS) with a 36-site unit cell is one of the most promising candidates. A natural theoretical framework for the analysis of such VBS order is to consider quantum states on a bond connecting the nearest-neighboring sites as fundamental quantum modes of the system and treat them as effectively independent ``bond particles.'' While correctly describing the VBS order in the ground state, this approach, known as the bond operator theory, significantly overestimates the lowest spin excitation energy. To overcome this problem, we take a next logical step in this paper to improve the bond operator theory and consider extended spin clusters as fundamental building blocks of the system. Depending on two possible configurations of the VBS order, various spin clusters are considered: (i) in the VBS order with staggered hexagonal resonance, we consider one spin cluster for a star of David shape and two spin clusters with each composed of a perfect hexagon and three attached dimers, and (ii) in the VBS order with uniform hexagonal resonance, we consider one spin cluster composed of a star of David and three attached dimers. It is shown that the majority of low-energy spin excitations are nearly or perfectly flat in energy. With most of its weight coming from the star of David, the lowest spin excitation has a gap much lower than the previous value obtained by the bond operator theory, narrowing the difference against exact diagonalization results. Despite the better agreement, however, the fact that the lowest spin excitation mostly comes from the star-of-David structure is at odds with the bond operator theory as well as previous series expansion studies. Expecting that the true low-energy spin excitations have contributions from both the star of David and its surroundings, we propose using two superclusters of spin in future work, treating spin excitations from both contributions on a more equal footing.
One of the most promising candidate ground states for the quantum antiferromagnetic Heisenberg model on the kagome lattice is the valence bond solid (VBS) with a 36-site unit cell. We present a theory of triplet excitation spectra about this ground state using bond operator formalism. In particular we obtain dispersions of all 18 triplet modes in the reduced Brillouin zone. In the bond operator mean-field theory, it is found that a large number of triplet modes are nondispersive. In particular, the lowest triplet excitation is nondispersive and degenerate with a dispersive mode at the zone center. Away from the zone center, the lowest triplet is separated from two other flat modes by a small energy gap. Quantum fluctuations are considered by taking into account scattering processes of two triplets and their bound-state formation, which leads to a downward renormalization of the lowest spin triplet gap. The dispersion of the lowest triplet excitation in the VBS state is compared with the dispersive lower bound of the triplet continuum expected in competing spin liquid phases. Implications to future neutron-scattering experiments are discussed.
Using series expansions around the dimer limit, we find that the ground state of the spin-1/2 Heisenberg antiferromagnet on the kagome lattice appears to be a valence bond crystal (VBC) with a 36 site unit cell, and ground-state energy per site $E=\ensuremath{-}0.433\ifmmode\pm\else\textpm\fi{}0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}$. It consists of a honeycomb lattice of ``perfect hexagons.'' The energy difference between the ground state and other ordered states with the maximum number of perfect hexagons, such as a stripe-ordered state, is of order 0.001J0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}0.001J. The expansion is also done for the 36 site system with periodic boundary conditions; its energy per site is $0.005\ifmmode\pm\else\textpm\fi{}0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}$ lower than the infinite system, consistent with exact diagonalization results. Every unit cell of the VBC has two singlet states whose degeneracy is not lifted to sixth order in the expansion. We estimate this energy difference to be less than 0.001J0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}0.001J. The dimerization order parameter is found to be robust. Two leading orders of perturbation theory give lowest triplet excitations to be dispersionless and confined to the perfect hexagons.
Motivated by a recent experiment on Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​, where spin-1/2 Cu2+{\text{Cu}}^{2+}Cu2+ moments reside on the layers of kagome-like lattices, we investigate quantum ground states of the antiferromagnetic Heisenberg model on a series of deformed kagome lattices. The deformation is characterized by a weaker exchange coupling $\ensuremath{\alpha}J$ on certain lattice links appropriate for Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​ with $\ensuremath{\alpha}=1$ corresponding to the ideal kagome lattice. In particular, we study possible valence bond solid phases using the perturbation theory around isolated dimer limits, dimer series expansion, and self-consistent bond operator mean-field theory. It is shown that the valence bond solid phase with a 36-site unit cell of the ideal kagome lattice is quite sensitive to a small lattice distortion as the kind discovered in Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​. As a result, we find that a more likely quantum ground state in Rb2Cu3SnF12{\text{Rb}}_{2}{\text{Cu}}_{3}{\text{SnF}}_{12}Rb2​Cu3​SnF12​ is the valence bond solid phase with a 12-site unit cell, where six dimers form a pinwheel structure, leading to strong modification of the elementary triplet and singlet excitation spectra in the deformed kagome lattices.
A proposed ground state for the kagome lattice Heisenberg model consists of a valence bond crystal (VBC) with a 36-site unit cell. We calculate the low-lying triplet and singlet excitations in the VBC phase for the infinite-lattice model and for the 36-site cluster. For the infinite lattice, the lowest triplet excitation is found to have a spin gap of approximately $0.08\ifmmode\pm\else\textpm\fi{}0.02J$ and a bandwidth of only about 0.01J0.01J0.01J. For the 36-site cluster, which consists of a single unit cell with periodic boundary conditions, there are substantial finite-size effects: the spin gap there is estimated to be approximately 0.2J0.2J0.2J, which is close to the exact diagonalization result of 0.164J0.164J0.164J. The triplet excitations attract one another and form many bound states in the spin-singlet channel. We find a large number of such bound states for the 36-site cluster, many of which appear to lie below the spin gap, which is again in agreement with the results from exact diagonalization.
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.
The singlet dynamics, which plays a major role in the physics of the spin-1/2 quantum Heisenberg antiferromagnet (QHAF) on the kagome lattice, can be approximately described by projecting onto the nearest-neighbor valence bond (NNVB) singlet subspace. We revisit here the effective quantum dimer model, which originates from the latter NNVB-projected Heisenberg model via a nonperturbative Rokhsar-Kivelson-like scheme. By using Lanczos exact diagonalization on a 108-site cluster supplemented by a careful symmetry analysis, it is shown that a previously found 36-site valence bond crystal (VBC) in fact competes with a new type of 12-site ``resonating-columnar'' VBC. Interestingly, these two VBCs ``emerge'' in different topological sectors. The exceptionally large degeneracy of the ground-state multiplets (144 on our 108-site cluster) and the proximity of a Z2{\mathbb{Z}}_{2}Z2​ dimer liquid have implications for the interpretation of numerical results on the QHAF, which are outlined. The possibility of a chiral VBC (i.e., spontaneously breaking time-reversal symmetry) is also discussed.