Variational study of the magnetization plateaus in the spin-1/2 kagome Heisenberg antiferromagnet: an approach from vision transformer neural quantum states

TL;DR

This work investigates magnetization plateaus in the spin-1/2 kagome Heisenberg antiferromagnet under a magnetic field using Vision Transformer-based neural quantum states (ViT-NQS). The authors confirm robust plateaus at $m=\frac{1}{9},\frac{1}{3},\frac{5}{9},\frac{7}{9}$ and show that the high-field plateaus host $\sqrt{3}\times\sqrt{3}$ valence-bond crystals, while the $m=\frac{1}{9}$ plateau exhibits competing $3\times3$ valence-bond patterns with distinct symmetry content. The ViT-NQS framework, with patch-translation invariance and symmetry-projected analyses, provides variational energies competitive with or lower than previous results and yields detailed irrep and local-observable characterizations of the plateau states. The findings offer experimentally testable predictions for local magnetization modulations and demonstrate the potential of ViT-based neural quantum states for studying frustrated quantum magnets and plateau phenomena.

Abstract

We analyze the magnetization curve of the spin-1/2 kagome Heisenberg model in a magnetic field. Using state-of-the-art variational wavefunctions based on neural networks, we confirm the presence of robust magnetization plateaus at $m=1/3$, $5/9$ and $7/9$ of the saturation value, stabilized by a spontaneous symmetry breaking of lattice translations with a $\sqrt{3}\times \sqrt{3}$ unit cell. Regarding the more challenging $m=1/9$ 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 $3\times 3$ unit cell. Such quantum states with local modulations of the magnetization average values could be observed experimentally in the near future.

Figures (9)

• Figure 1: Kagome lattice and ViT patch geometry. Blue (red) arrows indicate the lattice (patch-superlattice) primitive vectors and the corresponding primitive unit cell (superlattice unit cell, i.e. the $3\times 3$ ViT patch). The choice of patch induces folding of the original Brillouin zone (blue) into the reduced superlattice zone (red). The nine marked points are the momenta in the $\mathcal{K}_\Gamma$ set (see Eq. \ref{['eq:vit_momenta']}) representable by the ViT for this patch geometry.
• Figure 2: Magnetization curve for the spin-1/2 kagome Heisenberg antiferromagnet with $L=6,9$ lattice unit cells per linear dimension. The dotted lines correspond to the $m=1/9,1/3,5/9$ and $7/9$ plateaus.
• Figure 3: Magnetization per site and bond energy (see Eq. \ref{['eq:mag_bond_energy']}) for the VBC states at the three high-field plateaus $m= 1/3$, $5/9$, $7/9$. Disk diameter and bond width encode the magnitude of the observables as estimated directly from $2^{16}$ Monte Carlo samples, while red (blue) indicates positive (negative) sign. Spatially averaged expectation values and standard deviations are reported in Tables \ref{['tab:magnetization']} and \ref{['tab:bond_energies']}.
• Figure 4: Rescaled space group irrep weights $\tilde{N}_{\alpha}$ (see Eq. (\ref{['eq:rescaled_weights']})) for the optimized NQS plateau states at $m=1/3$, $5/9$, $7/9$, for several system sizes. In all cases the weight is concentrated in the $\Gamma$ and $K$ momentum sectors, consistent with the threefold-degenerate $\sqrt{3} \times \sqrt{3}$ VBC picture. The size-dependent $\Gamma A1 - \Gamma B1$ irrep flipping for the $m=1/3, 7/9$ states can be attributed to the internal symmetry of the corresponding VBCs, see Sec. \ref{['sec:high_field_plateaus']}. The small residual leakage into other irreps at $m=1/3$ is a result of weak spatial fluctuations of the converged NQS (see Tables \ref{['tab:magnetization']} and \ref{['tab:bond_energies']}). Irrep sectors not shown carry zero weight up to machine precision.
• Figure 5: Magnetization per site and bond energy (see Eq. \ref{['eq:mag_bond_energy']}) for the VBC A and VBC B states obtained with unguided optimization for $L=6$ and $L=9$, respectively, at the $m=1/9$ plateau. Disk diameter and bond width encode the magnitude of the observables, while red (blue) indicates positive (negative) sign (see Appendix \ref{['sec:App-local']} for numerical values). The unshaded region indicates the $3\times 3$ extended unit cell of the two VBCs.