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arXiv:2306.09563 · Physical review. B./Physical review. B

Nature of the 1/9-magnetization plateau in the spin- 1 2 kagome Heisenberg antiferromagnet

Da-zhi Fang, Ning Xi, Shi-Ju Ran, Gang Su

Published Jun 13, 2023• 23 citations
arXivHTMLPDF

Abstract

The nature of the 1/9-magnetization plateau of the spin-$\frac{1}{2}$ kagome Heisenberg antiferromagnet remains controversial due to the exotic physical properties and high complexity induced by the geometrical frustration. Instead of a ${\mathbb{Z}}_{3}$ 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 $c=1$ conformal field theory.

DOI
10.1103/physrevb.107.l220401

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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 3\ifmmode×\else\texttimes\fi3\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}3​\ifmmode×\else\texttimes\fi3​ spatial symmetry, demonstrated by magnetizations, bond energies, and three-body correlators. The spatial inversion symmetry in the 3\ifmmode×\else\texttimes\fi3\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}3​\ifmmode×\else\texttimes\fi3​ 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.
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Gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids have been proposed as candidates for the ground state of the S=1/2S=1/2S=1/2 quantum antiferromagnet on the kagome lattice. We extend the use of projected entangled pair states to construct (on the cylinder) resonating valence bond (RVB) states including both nearest-neighbor and next-nearest-neighbor singlet bonds. Our ansatz---dubbed ``simplex spin liquid''---allows for an asymmetry between the two types of triangles (of order 22%--3%2 in the energy density after optimization) leading to the breaking of inversion symmetry. We show that the topological Z2{\mathbb{Z}}_{2}Z2​ structure is still preserved and, by considering the presence or the absence of spinon and vison lines along an infinite cylinder, we explicitly construct four orthogonal RVB minimally entangled states. The spinon and vison coherence lengths are extracted from a finite size scaling with regard to the cylinder perimeter of the energy splittings of the four sectors and are found to be of the order of the lattice spacing. The entanglement spectrum of a partitioned (infinite) cylinder is found to be gapless, suggesting the occurrence, on a cylinder with real open boundaries, of gapless edge modes formally similar to Luttinger liquid (nonchiral) spin and charge modes. When inversion symmetry is spontaneously broken, the RVB spin liquid exhibits an extra Ising degeneracy, which might have been observed in recent exact diagonalization studies.
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