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

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.

• The paper uses vision-transformer neural quantum states with symmetry-resolved analysis to reproduce plateaus at m=1/3, 5/9, and 7/9 with √3×√3 valence bond crystals and state-of-the-art variational energies.
• The paper finds that the contentious m=1/9 plateau favors competing 3×3 valence bond crystals with energies of −0.42069 and −0.42122 per site, substantially below proposed Z₃ spin-liquid states.
• The paper demonstrates a promising method for frustrated quantum magnets while leaving the thermodynamic-limit order at m=1/9 unresolved because larger clusters fail to converge and conflicting neural-network results remain unexplained.

Overview and motivation

The spin-1/2 Heisenberg antiferromagnet on the kagome lattice in a magnetic field is a canonical frustrated quantum magnet whose zero-field ground state remains contested, while its finite-field behavior exhibits a series of magnetization plateaus at rational fractions of saturation. Raikos, Capponi, and Alet address this problem using neural quantum states (NQS) based on vision transformers (ViT) with factored attention (2602.12998). Their central contributions are: (i) confirmation of robust plateaus at $m=1/3$, $5/9$, and $7/9$ hosting $\sqrt{3}\times\sqrt{3}$ valence bond crystals (VBCs), consistent with most prior work; and (ii) evidence that the more contentious $m=1/9$ plateau hosts competing VBCs with large $3\times 3$ unit cells, rather than the topological $\mathbb{Z}_3$ quantum spin liquid proposed by some earlier studies.

The plateau physics is framed via the standard hardcore-boson mapping, where commensurate fillings favor insulating (VBC) states over superfluid ones. A constraint from Oshikawa–Yamanaka–Affleck/Hastings-type arguments implies that plateaus at $m=1/9$, $5/9$, $7/9$ necessarily carry ground-state degeneracy or gapless excitations, whereas $5/9$0 permits a featureless gapped state — which does not occur here.

Methodology: ViT NQS and symmetry analysis

The variational ansatz maps spin configurations to amplitudes through a deep real-valued ViT encoder followed by a shallow complex fully-connected output layer. The key architectural ingredient is factored attention, where query-key dot products are replaced by learnable weights depending only on relative patch displacement. Combined with sum-pooling over patches, this enforces exact patch-translation invariance, so the wavefunction lives at the $5/9$1 point of the Brillouin zone folded onto the patch superlattice. The authors use a 27-site ($5/9$2) patch, which accommodates both translation-invariant phases and all candidate symmetry-broken states considered. Models have roughly 1.1 million parameters ($5/9$3, $5/9$4, $5/9$5) and are optimized with SPRING natural-gradient descent within fixed-$5/9$6 sectors on $5/9$7 ($5/9$8) and $5/9$9 ($7/9$0) rhombic-torus clusters.

A notable methodological element is the symmetry-resolved characterization of plateau states. The authors decompose optimized wavefunctions into irreps of the full space group $7/9$1 via Monte Carlo estimation of irrep projectors, and compare against group-theoretic predictions from the character-stabilizer formula applied to idealized VBC patterns. This dual approach cleanly distinguishes candidate broken-symmetry orders. They also note a caveat: the character-stabilizer counting assumes a single idealized representative state and can miss finite-size internal structure, an effect they explicitly diagnose for the magnon crystal and VBC A states.

Magnetization curve and high-field plateaus

Optimizing within every fixed-magnetization sector and taking the lower convex hull of $7/9$2 yields magnetization curves for both cluster sizes displaying four plateaus at $7/9$3. Plateau widths are nearly size-independent; e.g., the $7/9$4 plateau spans $7/9$5 for $7/9$6 and $7/9$7 for $7/9$8. The exact metamagnetic jump above $7/9$9 to saturation is well reproduced. Asymmetric jumps above versus below each plateau are interpreted as possible signatures of neighboring supersolid phases, though the authors caution that simulations outside the plateaus were less thoroughly converged.

For the three high-field plateaus, the results are unambiguous:

• $\sqrt{3}\times\sqrt{3}$0: the NQS converges to the exact magnon crystal of Schulenburg et al., with vertex sites fully polarized ($\sqrt{3}\times\sqrt{3}$1) and hexagon sites carrying one delocalized magnon each ($\sqrt{3}\times\sqrt{3}$2). The energy per site, $\sqrt{3}\times\sqrt{3}$3, matches the exact value $\sqrt{3}\times\sqrt{3}$4. The even–odd switching between $\sqrt{3}\times\sqrt{3}$5 and $\sqrt{3}\times\sqrt{3}$6 irreps follows from the sign structure of hexagon-localized $\sqrt{3}\times\sqrt{3}$7 magnons.
• $\sqrt{3}\times\sqrt{3}$8: the same hexagram VBC persists with reduced local moments; vertices are not fully polarized. Energies per site reach $\sqrt{3}\times\sqrt{3}$9 ($m=1/9$0) and $m=1/9$1 ($m=1/9$2). The state retains full point-group symmetry, contradicting iPEPS reports of additional rotational breaking.
• $m=1/9$3: again the $m=1/9$4 VBC, with vertex magnetization approximately $m=1/9$5. The energies $m=1/9$6 ($m=1/9$7) and $m=1/9$8 ($m=1/9$9) are claimed to be the lowest reported so far. These results directly contradict a recent fermionic-parton VMC study reporting negative (field-opposed) local magnetizations outside hexagons; the ViT results instead agree closely with the RVB-based study of Cheng and Li.

Across all three plateaus, the irrep decomposition shows weight concentrated in $3\times 3$0 (one third) and $3\times 3$1 (two thirds) sectors, exactly as expected for threefold-degenerate $3\times 3$2 VBCs. Since these energies match or improve upon prior studies, the authors use them to validate their methodology before tackling the harder low-field case.

The $3\times 3$3 plateau: two competing valence bond crystals

The $3\times 3$4 plateau is the paper's main focus, given the spread of prior proposals: a gapped topological $3\times 3$5 QSL from DMRG and parton VMC, an 18-fold degenerate or gapless $3\times 3$6 VBC from iPEPS, and a windmill-shaped $3\times 3$7 VBC from recent RVB variational work. Experimentally, the plateau was observed in Y-based kagome materials, with specific heat and magnetic oscillation measurements suggesting charge-neutral Dirac fermionic excitations nearby.

Unguided optimization yields two distinct VBCs depending on system size:

State	Cluster	Energy/site	Degeneracy	Point-group content
VBC A ("windmill")	$3\times 3$8	$3\times 3$9	36	$\mathbb{Z}_3$0 only
VBC B	$\mathbb{Z}_3$1	$\mathbb{Z}_3$2	18	$\mathbb{Z}_3$3 plus three mirrors

Both share a common motif on the central hexagram — staggered hexagon magnetization, uniform negative hexagon bonds, positively magnetized vertices — but differ in the ordering of strong outward bonds at the vertices: clockwise-monotonic for VBC A (breaking mirror symmetry), alternating for VBC B (preserving three mirror axes). Both break translations down to a 27-site cell. Imprinting experiments, in which each pattern is seeded on the alternate cluster size, converge to higher energies while retaining the imprinted symmetries, confirming genuine size-dependent energetic competition between metastable states separated by a narrow energy window.

Both VBCs achieve substantially lower variational energies than the $\mathbb{Z}_3$4 QSL candidate ($\mathbb{Z}_3$5) and the gapless $\mathbb{Z}_3$6 VBC of Fang et al. ($\mathbb{Z}_3$7), and slightly below the windmill VBC of Cheng and Li ($\mathbb{Z}_3$8). VBC B has not been previously reported. The irrep analysis confirms that the two orders are cleanly distinguishable by their space-group content, with finite-size effects on VBC A (nonzero overlap under $\mathbb{Z}_3$9 rotations, decaying with system size) explained quantitatively via modified stabilizer counting.

Optimization at $m=1/9$0 proved markedly harder than at high field, with the optimizer frequently trapped in local minima at $m=1/9$1. Notably, these metastable states always broke translation down to the $m=1/9$2 cell, never converging to smaller unit cells — indirect evidence against the $m=1/9$3 candidates.

Limitations and open questions

Several limitations are stated plainly. As a variational method, NQS carries no guarantee of reaching the true ground state, though the high-field benchmarks support the approach. The 27-site patch forbids representing states with larger unit cells, and enlarging it is computationally expensive. Most importantly, the discrepancy between VBC A ($m=1/9$4) and VBC B ($m=1/9$5) prevents any conclusion about the thermodynamic-limit ground state; attempts at $m=1/9$6 ($m=1/9$7) failed to converge due to initialization-dependent trapping in local minima. The authors also flag a significant unresolved conflict: a concurrent GCNN-NQS preprint reports a much lower energy ($m=1/9$8) interpreted as a gapless chiral spin density wave. The authors argue this value is physically implausible — it lies below their derived rigorous lower bound $m=1/9$9 and below best known non-magnetic energies — but they concede they have no simple explanation for the discrepancy between otherwise similar methods. Finally, whether the experimentally suggested fermionic excitations near the plateau are compatible with a VBC ground state remains open.

Conclusion

This work establishes ViT-based NQS with factored attention as a competitive tool for frustrated magnets in a field, reproducing the established $5/9$0 VBC physics at $5/9$1, $5/9$2, and $5/9$3 with state-of-the-art variational energies, and providing the strongest variational evidence to date that the $5/9$4 plateau is a valence bond crystal with at least a 27-site unit cell rather than a topological spin liquid. The identification of two nearly degenerate, size-dependent VBC patterns — one novel — underscores a crowded low-energy landscape at $5/9$5 whose thermodynamic resolution, and reconciliation with conflicting concurrent NQS results, remain open problems. The predicted local magnetization modulations are directly testable via NMR on Y-based kagome materials.