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arXiv:2410.21757 · Journal of the Physical Society of Japan

Valence Bond Crystal Ground State of the 1/9 Magnetization Plateau in the Spin-1/2 Kagome Lattice

Katsuhiro Morita

Published Nov 27, 2024• 8 citations
arXivHTMLPDF

Abstract

We investigate the ground state of a spin-1/2 kagome antiferromagnet at the 1/9 magnetization plateau, focusing primarily on six types of valence bond crystal (VBC) distortions. Among six types of VBC distortions, type 1 consistently exhibits the lowest ground-state energy. Analysis of the second derivative of the energy with respect to the distortion strength $J_{\rm d}$ using exact diagonalization reveals that, as the system size approaches the thermodynamic limit, the type 1 VBC state remains stable down to $J_{\rm d} = 1$, corresponding to the undistorted kagome lattice. Perturbation theory further supports the stability of the type 1 VBC state, providing energy values that agree with exact diagonalization results within 3%. We also explore the possibility that other types of VBC distortions could become the ground state, but type 1 VBC still exhibited the lowest energy. These findings suggest that the ground state of the 1/9 plateau in the kagome lattice is the type 1 VBC state.

DOI
10.7566/jpsj.93.123706

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We investigate the ground state of a spin-1/2 kagome antiferromagnet at the 1/9 magnetization plateau, focusing primarily on six types of valence bond crystal (VBC) distortions. Among six types of VBC distortions, type 1 consistently exhibits the lowest ground-state energy. Analysis of the second derivative of the energy with respect to the distortion strength JdJ_{\rm d}Jd​ using exact diagonalization reveals that, as the system size approaches the thermodynamic limit, the type 1 VBC state remains stable down to Jd=1J_{\rm d} = 1Jd​=1, corresponding to the undistorted kagome lattice. Perturbation theory further supports the stability of the type 1 VBC state, providing energy values that agree with exact diagonalization results within 3%. We also explore the possibility that other types of VBC distortions could become the ground state, but type 1 VBC still exhibited the lowest energy. These findings suggest that the ground state of the 1/9 plateau in the kagome lattice is the type 1 VBC state.
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.
The nature of the 1/9-magnetization plateau of the spin-12\frac{1}{2}21​ kagome Heisenberg antiferromagnet remains controversial due to the exotic physical properties and high complexity induced by the geometrical frustration. Instead of a Z3{\mathbb{Z}}_{3}Z3​ quantum spin liquid revealed on a cylinder, we show on an infinite-size lattice that the 1/9 plateau can be described by a valence bond crystal (VBC) that breaks spatial translational invariance. Consistent results are achieved by two accurate tensor network methods, namely the full-update infinite projected-entangled pair states and the projected-entangled simplex states. The VBC exhibits an hourglass pattern with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ spatial symmetry, demonstrated by magnetizations, bond energies, and three-body correlators. The spatial inversion symmetry in the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC is instantly broken with the presence of the difference between the coupling strengths in the up and down triangles, suggesting the existence of gapless excitations. The gapless nature of the 1/9 plateau is further indicated by the scaling behaviors of the entanglement entropy and the correlation length, which indicate a c=1c=1c=1 conformal field theory.
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.
We investigate the spin-1 kagome-lattice Heisenberg antiferromagnet using large-scale Lanczos diagonalization and the finite-temperature Lanczos method. The zero-temperature magnetization process exhibits plateaus at m = 0 , 1 / 3 , 7 / 9 , and 8 / 9 , where m is the normalized magnetization. The m = 0 plateau is identified as a trimer valence-bond-crystal state, while the high-field plateaus at m = 7 / 9 and 8 / 9 are identified as magnon crystals. In particular, the m = 8 / 9 plateau corresponds to the exact localized-magnon crystal state. A smoothed zero-temperature magnetization curve constructed using the Gaussian-kernel smoothing method indicates magnetization jumps at the lower-field edge of the m = 1 / 3 plateau and at the upper-field edges of the m = 7 / 9 and 8 / 9 plateaus. At finite temperatures, the specific heat exhibits a double-peak structure with peaks around T / J ≃ 0.1 and T / J ≃ 1.1 , and the low-temperature peak may be related to trimer valence-bond-crystal ordering. The finite-temperature magnetization curves show that the m = 1 / 3 plateau remains visible at low temperatures, whereas the high-field plateaus are rapidly smeared out by thermal effects. These results provide benchmark data for thermodynamic and high-field magnetization measurements in candidate spin-1 kagome-lattice materials.
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 study the ground-state phase diagram of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the square-kagome lattice using infinite projected entangled-pair states (iPEPS). By systematically varying the ratio of exchange couplings on triangular and square plaquettes, we establish a complete quantum phase diagram in the thermodynamic limit. In the intermediate-coupling regime, we identify four distinct nonmagnetic phases that are unambiguously characterized as valence-bond crystals (VBCs) by their symmetry-inequivalent patterns of strong and weak spin-spin correlations. These include a plaquette crossed-dimer VBC, a loop-six VBC stabilized around the isotropic point, a generalized pinwheel VBC with reduced rotational symmetry, and a decorated loop-six VBC proximate to ferrimagnetic order. We determine the phase boundaries using a combination of bond-resolved correlation functions, entanglement entropy, and magnetization. For transitions not accompanied by sharp entanglement signatures, we extract the spin gap from finite-field simulations, allowing us to distinguish gapped and gapless VBC phases. At larger coupling ratios, the system undergoes transitions into imperfect and perfect ferrimagnetic states. Our results resolve long-standing ambiguities in the square-kagome Heisenberg model and provide a quantitatively reliable reference for future theoretical and experimental studies of frustrated quantum magnets.