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arXiv:2610.34439

Theory of the $1/9$ Magnetization Plateau of the Spin-$1/2$ Kagome Heisenberg Antiferromagnet

Pratyay Ghosh, Frédéric Mila

Published Oct 25, 2026• 0 citations
arXivHTMLPDF

Abstract

We investigate the $1/9$ magnetization plateau of the spin-$1/2$ kagome Heisenberg antiferromagnet using infinite projected entangled-pair states and identify a spin-density-wave state with a $27$-site unit cell and a lower variational energy than previous approaches. In contrast to higher-magnetization plateaus, which are well described by localized magnons on nonoverlapping hexagons and preserve a sixfold rotational symmetry, the $1/9$ plateau only keeps a threefold rotational symmmetry and consists of a tiling by three types of $9$-site clusters related by $\pm2π/3$ rotations. These conclusions are supported by a complementary cluster mean-field theory which correctly preserves the symmetry of the higher plateaus but leads to the same spontaneous symmetry breaking for the $1/9$ plateau with a qualitatively similar pattern of nine different local magnetizations.

DOI
10.48550/arxiv.2610.34439

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Theory of the 1/91/91/9 Magnetization Plateau of the Spin-1/21/21/2 Kagome Heisenberg Antiferromagnet
We investigate the 1/91/91/9 magnetization plateau of the spin-1/21/21/2 kagome Heisenberg antiferromagnet using infinite projected entangled-pair states and identify a spin-density-wave state with a 272727-site unit cell and a lower variational energy than previous approaches. In contrast to higher-magnetization plateaus, which are well described by localized magnons on nonoverlapping hexagons and preserve a sixfold rotational symmetry, the 1/91/91/9 plateau only keeps a threefold rotational symmmetry and consists of a tiling by three types of 999-site clusters related by ±2π/3\pm2π/3±2π/3 rotations. These conclusions are supported by a complementary cluster mean-field theory which correctly preserves the symmetry of the higher plateaus but leads to the same spontaneous symmetry breaking for the 1/91/91/9 plateau with a qualitatively similar pattern of nine different local magnetizations.
The spin-12\frac{1}{2}21​ Heisenberg model on a kagome lattice is a typical frustrated quantum spin system. The basic structure of a kagome lattice is also present in the kagome-strip lattice in one dimension, where a similar type of frustration is expected. We thus study the magnetization plateaus of the spin-12\frac{1}{2}21​ Heisenberg model on a kagome-strip chain with three-independent antiferromagnetic exchange interactions using the density-matrix renormalization-group method. In a certain range of exchange parameters, we find twelve kinds of magnetization plateaus, nine of which have magnetic structures breaking translational and/or reflection symmetry spontaneously. The structures are classified by an array of five-site unit cells with specific bond-spin correlations. In a case with a nontrivial plateau, namely a 3/10 plateau, we find long-period magnetic structure with a period of four unit cells.
We clarify the existence of several magnetization plateaus for the kagome S=12S=\frac{1}{2}S=21​ antiferromagnetic Heisenberg model in a magnetic field. Using approximate or exact localized magnon eigenstates, we are able to describe in a similar manner the plateau states that occur for magnetization per site m=13m=\frac{1}{3}m=31​, 59\frac{5}{9}95​, and 79\frac{7}{9}97​ of the saturation value. These results are confirmed using large-scale exact diagonalization on lattices up to 63 sites.
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.
We analyze the magnetization curve of the spin-1/2 kagome Heisenberg model in a magnetic field. Using state-of-the-art variational wave functions based on neural networks, we confirm the presence of robust magnetization plateaus at m=1/3m=1/3m=1/3, 5/95/95/9 and 7/97/97/9 of the saturation value, stabilized by a spontaneous symmetry breaking of lattice translations with a 3×3\sqrt{3}\times \sqrt{3}3​×3​ unit cell. Regarding the more challenging m=1/9m=1/9m=1/9 plateau, we find two competing valence bond crystals depending on the system size, both breaking translation as well as point group symmetries and with a larger 3×33\times 33×3 unit cell. Such quantum states with local modulations of the magnetization average values could be observed experimentally in the near future.
We investigate the anisotropic effect of magnetization plateaus in the antiferromagnetic Heisenberg model on a kagome strip chain. The kagome strip chain Heisenberg model, composed of a hexagonal net of triangles forming five-site unit cells, exhibits four magnetization plateaus in the presence of an applied magnetic field. Using numerical density matrix renormalization group method, we find that the magnetization plateaus are stable against anisotropic interactions in the same direction of the applied magnetic field, but the plateaus become much smaller with anisotropic interactions in other directions. We further show the anisotropic effect of the magnon excitations of the 0.6 plateau state using linear spin wave theory. The magnon bandwidth remains small when tuning the anisotropic interactions along the field where the magnetization plateau is stable, while the band becomes more dispersive with anisotropic interactions perpendicular to the field. In addition, upon tuning down the interaction strength for the two lower legs below a critical value, the Hamiltonian of the kagome strip chain is dominated by two separate spin chains. This can be used to determine the effective lattice structure in materials with strong distortions. Our results enhance the theoretical understanding of the anisotropic effect and the nature of magnetization plateaus in kagome strip chain materials, which can contribute to the design and manipulation of kagome materials with tailored properties.
Magnetization plateaus in quantum spin systems emerges in two-dimensional frustrated systems such as kagome lattice. The spin-1/2 antiferromagnetic Heisenberg model on square-kagome lattice is also appropriate for the study of magnetization plateau. Motivated by recent experimental finding of such a square-kagome lattice with nonequivalent three bonds, we investigate the phase diagrams and magnetization plateaus of the lattice by the exact diagonalization method. In addition to previously reported the 1/3 and 2/3 plateaus in the model with two equivalent bonds, we find a new 2/3 plateau whose magnetic structure is characterized by spontaneously broken four-fold rotational symmetry. The plateau appears only in the case of three nonequivalent bonds. We propose the possibility of finding plateaus including the new one.