Universality classes of the non-Ising transitions in the six-vertex model on the kagome lattice

The six-vertex (ice-rule) model on the kagome lattice — the theoretical limit of the direct kagome artificial spin ice realized experimentally in 2024 — was systematically simulated in 2026 by Zhang et al. (Phys. Rev. B 113, 024407), revealing four vortex-lattice phases, BKT-type melting, and two unexplained anomalies: a vortex-lattice transition VL-T $\leftrightarrow$ VL-H with exponent $y_t=1.340(3)$ and no identifiable order parameter (possibly Landau-incompatible), and continuously varying ferromagnetic-to-paramagnetic thermal exponents $y_t=1.164(3)$ , $1.317(6)$ , $1.381(1)$ at vertex weights $a=4,5,6$ whose large- $a$ limit is unknown. This problem asks for the universality content underlying these transitions: the identification, or principled exclusion, of an order parameter and field-theoretic universality class for the VL-T $\leftrightarrow$ VL-H transition, and the convergence or nonconvergence of $y_t(a)$ as $a\to\infty$ . Both sub-questions serve one objective — establishing the critical behaviour of the kagome six-vertex model — and both remain open.

Background

The six-vertex (6V) model places arrows on lattice edges so that every vertex obeys the ice rule — two arrows point in and two point out — and assigns Boltzmann weights to the allowed vertex configurations. On the square lattice this model is exactly solvable, and its critical behavior includes a Berezinskii–Kosterlitz–Thouless (BKT) transition: an infinite-order transition driven by the unbinding of vortex–antivortex pairs, familiar from the two-dimensional XY model.

The kagome lattice — a tiling of corner-sharing triangles and hexagons in which every site has four neighbors — became an ice-rule platform in 2024, when "direct" kagome artificial spin ice was realized experimentally by placing nanomagnets on the edges of a kagome lattice, so that each kagome vertex touches four magnets (Yue et al., Nature Nanotechnology 19, 1101 (2024)). Because a degree-four vertex admits sixteen arrow configurations, the experimental system is naturally a 16-vertex model; the ice-rule (2-in-2-out) subset of six configurations defines the 6V model on the kagome lattice. Yue et al. observed two quasi-degenerate low-temperature toroidal phases — ferrotoroidicity and paratoroidicity, i.e. ordered patterns of toroidal moments that break both spatial-inversion and time-reversal symmetry — and a crossover to a non-toroidal paramagnetic phase.

Zhang et al. (Phys. Rev. B 113, 024407 (2026)) then simulated the theoretical 6V kagome model with directed-loop Monte Carlo, a Markov-chain method that updates arrows along closed directed loops and is efficient under ice-rule constraints, although loop closure in this method becomes increasingly rare at large vertex weight $a$ . The six allowed vertices carry pair weights $a$ , $b$ , $c$ , and the phase diagram was mapped in this weight space. Ordering the vertex "legs" (the four edge arrows at a vertex) distinguishes antiferromagnetic (AFM), ferromagnetic (FM), and paramagnetic (PM) leg order; separately, the circulation of arrows around each triangular or hexagonal face defines a vortex with integer winding number $k$ . Four vortex-lattice (VL) phases appear: VL-I (coexisting with AFM leg order), VL-II (coexisting with FM leg order), and — inside the leg-disordered PM regime — VL-T and VL-H, in which vortices crystallize on the triangular or hexagonal faces respectively, plus a vortex-disordered phase VL-D.

The central findings are: the FM $\to$ PM transition is Ising-like with thermal exponent $y_t=1$ at small weight $a$ , where $y_t=1/\nu$ is the inverse correlation-length exponent extracted from finite-size scaling of Binder ratios (dimensionless cumulants of magnetization); but along the same FM–PM boundary the exponent varies continuously at larger weights, $y_t=1.164(3)$ , $1.317(6)$ , $1.381(1)$ at $a=4,5,6$ ; the VL-T $\leftrightarrow$ VL-H transition occurs with the unusual exponent $y_t=1.340(3)$ , and no order parameter analogous to the Ising magnetization could be found, suggesting a possibly Landau-incompatible transition — a continuous transition between phases whose broken symmetries are not nested one inside the other, which Landau theory cannot describe (such transitions are known to occur classically in anisotropic two-dimensional XY models; Prakash and Jones, Phys. Rev. Lett. 134, 097103 (2025)); and VL-T and VL-H melt into VL-D by BKT-type transitions.

Problem Statement

Determine the critical behaviour of the six-vertex model on the kagome lattice at general vertex weights $(a,b,c)$ , as left open by Zhang et al. (Phys. Rev. B 113, 024407 (2026)). Two linked questions jointly define this objective:

The nature and universality class of the VL-T $\leftrightarrow$ VL-H vortex-lattice transition. Is there an order parameter — local, toroidal, or topological — distinguishing VL-T from VL-H, or is the transition Landau-incompatible? Independently confirm, refute, or refine the exponent $y_t=1.340(3)$ , and either identify the universality class field-theoretically or show why the transition falls outside the known classes.
The large- $a$ behaviour of the FM–PM thermal exponent. Along the ferromagnetic-to-paramagnetic boundary the measured exponent grows with the vertex weight $a$ ( $y_t=1$ at small $a$ ; $1.164(3)$ , $1.317(6)$ , $1.381(1)$ at $a=4,5,6$ ). Does $y_t(a)$ converge to a well-defined value as $a\to\infty$ , or does it vary without limit, indicating nonuniversal criticality?

These questions share one scientific objective: the universality content of the kagome six-vertex model's non-Ising transitions. The source paper also flags two adjacent future directions that concern different model classes and systems rather than this objective — extending the simulations to the 16-vertex and disordered kagome vertex models, and experimentally constraining direct-kagome artificial spin ice to the 6V subspace — which are not required for completion of this problem.

References

Wei Zhang, Wanzhou Zhang, Jie Zhang, Chengxiang Ding, and Youjin Deng, "Phase transitions and critical exponents in the six-vertex model on kagome lattices", Physical Review B 113, 024407 (2026). DOI: 10.1103/vb2v-gy4w; arXiv:2506.00804 (v1 posted under the title "New universality classes and BKT transition in the vortex lattice phases of Kagome Ice"). https://doi.org/10.1103/vb2v-gy4w
Wen-Cheng Yue, Zixiong Yuan, Peiyuan Huang, Yizhe Sun, Tan Gao, Yang-Yang Lyu, Xuecou Tu, Sining Dong, Liang He, Ying Dong, Xun Cao, Lin Kang, Huabing Wang, Peiheng Wu, Cristiano Nisoli, and Yong-Lei Wang, "Toroidic phase transitions in a direct-kagome artificial spin ice", Nature Nanotechnology 19, 1101–1107 (2024). DOI: 10.1038/s41565-024-01666-6; arXiv:2404.19377. https://doi.org/10.1038/s41565-024-01666-6
Abhishodh Prakash and Nick G. Jones, "Classical origins of Landau-incompatible transitions", Physical Review Letters 134, 097103 (2025). DOI: 10.1103/PhysRevLett.134.097103; arXiv:2404.19009. https://doi.org/10.1103/PhysRevLett.134.097103