2$k_F$ instability and chiral spin density wave at the 1/9 magnetization plateau in the kagome antiferromagnets

Kagome lattice antiferromagnets exhibit plethora of intriguing phases of matter. Particularly interesting state appears at the magnetic field-induced $1/9$ magnetization plateau observed in several recent experimental studies. The nature and exotic physical properties of the plateau however remain controversial due to an exceptional complexity of the state generated by geometrical frustration. Among candidate states recent studies found a $Z_3$ quantum spin liquid state, a valence bond crystal exhibiting an hourglass pattern and a valence bond crystal state with a $3\times 3$ periodicity and a windmill-shaped motif. Recent torque magnetometry measurements on YCOB single-crystal samples however indicate presence of Dirac-like spinons at $1/9$ magnetization plateau. We study properties of the plateau state using novel machine learning technique that combines variational Monte Carlo, symmetry enhanced neural network quantum states and flux insertion method. Our machine learning study reveals that the ground state at the $1/9$ plateau is a gapless $1\times 1$ chiral spin density wave caused by 2$k_F$ instability of the underlying composite Fermi liquid. The spin wave chirality results from the correlated spin order that reflects its nontrivial topology.

💡 Research Summary

The paper investigates the long‑standing puzzle of the 1/9 magnetization plateau observed in kagome‑lattice antiferromagnets. Earlier theoretical work proposed several competing ground‑state candidates: a Z₃ quantum spin liquid (QSL), an hour‑glass valence‑bond crystal (VBC), and a 3×3 “wind‑mill’’ VBC. Recent torque‑magnetometry experiments on YCOB single crystals, however, reported signatures of Dirac‑like spinons, suggesting a gapless state.

To resolve this controversy the authors employ a state‑of‑the‑art machine‑learning framework that combines variational Monte‑Carlo (VMC) with symmetry‑enhanced neural‑network quantum states (GCNNs) and a flux‑insertion technique. The GCNN ansatz respects all space‑group symmetries of the kagome lattice (translations, rotations, reflections) and is optimized using stochastic reconfiguration (natural‑gradient descent) to minimize a free‑energy loss that includes an entropy‑regularization term. Flux insertion is used to probe the response of the many‑body wavefunction to twisted boundary conditions, thereby detecting emergent gauge flux.

The numerical results show that the lowest‑energy state in the 1/9 plateau sector carries a finite chiral order parameter, indicating broken time‑reversal symmetry while preserving lattice translations (a 1×1 pattern). The spin‑structure factor displays sharp peaks at wave vectors corresponding to 2k_F scattering of an underlying composite Fermi surface of spinons bound to Z₃ flux (a composite fermion liquid). These peaks are interpreted as the hallmark of a 2k_F density‑wave instability, which drives the formation of a chiral spin‑density wave (CSDW). The CSDW order modulates the spin component parallel to the field with a uniform 1×1 periodicity, analogous to charge‑density‑wave order in kagome metals, but with a non‑trivial topology encoded in the chirality.

The authors argue that pairing instabilities are suppressed by the emergent gauge boson, leaving the 2k_F instability as the dominant channel. The “hot spots’’ on the spinon Fermi surface, where the nesting condition is optimal, give rise to non‑Fermi‑liquid behavior: linear‑in‑temperature magnetic susceptibility, vanishing quasiparticle weight, and anomalous power‑law correlations, consistent with the experimental torque data. The study also reconciles the apparent contradiction between earlier DMRG/VMC (favoring a Z₃ QSL) and iPEPS (favoring a VBC) by showing that those methods missed the subtle chiral order that only emerges when the full symmetry‑enhanced GCNN space and flux insertion are employed.

In summary, the paper presents a comprehensive computational analysis that identifies the 1/9 plateau of kagome antiferromagnets as a gapless, translation‑invariant, chiral spin‑density‑wave state arising from a 2k_F instability of a composite fermion spinon liquid with Z₃ flux. This resolves experimental observations of Dirac‑like excitations and predicts characteristic signatures—sharp 2k_F peaks in neutron‑scattering, finite scalar chirality, and linear‑T susceptibility—that can be tested in future experiments.