Exact critical manifold of the q-state Potts model on the kagome lattice and the convergence of its critical polynomials

The $q$-state Potts model on the kagome lattice is a canonical unsolved problem of two-dimensional lattice statistics: unlike the square, triangular and hexagonal lattices, its critical manifold in the real $(q,v)$ plane is not known exactly. Jacobsen and Scullard (2012) introduced the critical (graph) polynomial $P_B(q,v)$, defined on a finite basis $B$ of the lattice, whose zero set gives approximations to the exact critical manifold that improve with basis size, and posed the central open questions: whether $P_B(q,v)=0$ converges to the exact critical manifold when both basis dimensions tend to infinity at fixed finite aspect ratio; whether the limiting manifold is algebraic or non-algebraic (they conjectured non-algebraic); whether systematic factorization of $P_B$ provably signals exact solvability; and which exact finite-basis features persist in the thermodynamic limit. Follow-up work by the same authors (2012-2016) turned the scheme into the most precise numerical tool available for two-dimensional percolation and Potts thresholds (bases of up to 882 edges, thresholds to $10^{-13}$-$10^{-15}$, and one new exact critical point found and then proven by a star-triangle transformation), but every one of the analytical questions remains open.

Background

The $q$-state Potts model assigns to each vertex of a lattice a spin taking one of $q$ values, with Boltzmann weights favouring equal spins on adjacent sites. In the Fortuin-Kasteleyn (FK) cluster representation its partition function on a graph $G=(V,E)$ reads $Z=\sum_{A\subseteq E} v^{|A|} q^{k(A)}$, where $v=e^K-1$ is the temperature variable ($K$ being the dimensionless coupling), the sum runs over all edge subsets $A$, and $k(A)$ counts connected components including isolated vertices. This representation makes sense for any real $q$ and $v$, so the model can be studied on the whole real $(q,v)$ plane: $v>0$ is the physical ferromagnet, $v<0$ the antiferromagnet, $q=1$ is bond percolation, $q=0$ is related to spanning trees and forests (connected acyclic subgraphs, and arbitrary acyclic subgraphs, respectively), and integer $q\ge 2$ are the spin systems. The critical manifold is the set of points in the $(q,v)$ plane at which the infinite-lattice model stands at a phase transition.

On the square, triangular and hexagonal lattices the Potts model is exactly solved, and the critical manifolds are simple algebraic curves,

$$(v^2-q)(v^2+4v+q)=0 \quad \text{(square)}, \qquad v^3+3v^2-q=0 \quad \text{(triangular)}, \qquad v^3-3qv-q^2=0 \quad \text{(hexagonal)}.$$

The kagome lattice — the Archimedean tiling $(3,6,3,6)$, where the notation lists in cyclic order around each vertex the sizes of the polygons meeting there (here triangle, hexagon, triangle, hexagon), ubiquitous in frustrated magnetism — is the canonical case that resists solution: apart from the Ising case $q=2$ (solved by Kano and Naya in 1953), the $q=0$ spanning-tree and spanning-forest problems, and isolated special points, no exact result for its Potts critical manifold is known. A sixth-order algebraic conjecture for this manifold published by F. Y. Wu in 1979 stood for thirty years before being refuted numerically, although it is extraordinarily accurate, with deviations of order $10^{-4}$ to $10^{-5}$.

Jacobsen and Scullard (2012) introduced a systematic approach to such unsolved manifolds. Any infinite two-dimensional lattice $G$ can be tiled by a finite subgraph $B$, called a basis, whose boundary vertices are identified according to an embedding rule. They defined a two-parameter graph polynomial $P_B(q,v)$ — closely related to the Tutte polynomial, the two-variable graph invariant that encodes the Potts partition function, and computable recursively by the contraction-deletion identity (the rule that computes the polynomial by summing the terms obtained by contracting or deleting an edge) — such that the algebraic curve $P_B(q,v)=0$ approximates the critical manifold of the Potts model on $G$, with accuracy increasing with the size of $B$. For the square lattice the polynomial factorizes for every choice of $B$, always shedding the same small factor $(v^2-q)(v^2+4v+q)$ whose zero set is the exact critical manifold; this suggested that factorization of $P_B$ acts as a detector of exact solvability. On the kagome lattice, bases of up to 36 edges gave approximations deviating from high-precision transfer-matrix numerics (the transfer matrix being the finite matrix whose iterated product builds the partition function strip by strip) by only $10^{-6}$ to $10^{-7}$ in the ferromagnetic regime, revealed almost-vertical rays at the Beraha numbers $q=B_4=2$ and $q=B_6=3$ (the numbers $B_n=4\cos^2(\pi/n)$, at which the square-lattice antiferromagnet has singularities within its Berker-Kadanoff phase — the critical, algebraically ordered low-temperature antiferromagnetic regime), rays that are conjectured to become exactly vertical in the thermodynamic limit, and led, since even the 32nd-order curve failed to reproduce the numerics perfectly, to the conjecture that the exact kagome critical manifold is non-algebraic.

The follow-up literature transformed this scheme into the most precise numerical tool available for two-dimensional percolation and Potts thresholds: a probabilistic redefinition of $P_B$, introduced for percolation in 2012 and extended to the Potts model in 2013, enabled transfer-matrix computation on bases of hundreds of edges; a formulation within the periodic Temperley-Lieb algebra — the diagrammatic algebra generated by the join and detach operators of the FK loop representation — reached 882 edges and $10^{-13}$ precision (2014); an eigenvalue method on semi-infinite cylinders, which determines criticality by equating the largest eigenvalues of two topologically distinct sectors of the transfer matrix, doubled the accessible circumference and fixed the kagome bond-percolation threshold to fifteen digits, $p_c=0.524404999167439(4)$ (2015); and a parallel implementation computed exact polynomials on $6\times 6$ bases (2016). Yet the analytical questions that motivated the method are untouched: the convergence of $P_B(q,v)=0$ to the exact critical manifold has no proof (it is recognized to be comparable in difficulty to proving universality of crossing probabilities, that is, a rigorous conformal-invariance theorem), the algebraic or transcendental nature of the limiting kagome manifold is undecided, the factorization heuristic has no proof or criterion, and only one further exact critical point has been found — $q=3.476950573042399\ldots$, $v=-2.392646781702640\ldots$, established by a star-triangle transformation (the exact local replacement of a triangle of pairwise-coupled spins by an equivalent three-legged star) mapping to a decorated hexagonal lattice (2013) — underscoring how sparse exact knowledge remains.

Problem Statement

Determine the exact critical manifold of the $q$-state Potts model on the kagome lattice in the real $(q,v)$ plane, and settle the mathematical status of the critical-polynomial scheme that approximates it. The common objective is to convert the sequence of finite-basis algebraic approximants $P_B(q,v)=0$ into exact, provable knowledge of the kagome critical manifold; four linked questions jointly support it.

First, convergence: prove or refute that the zero sets of the critical polynomials $P_B(q,v)$ converge to the exact critical manifold of the kagome-lattice Potts model as the basis $B$ grows with both linear dimensions $N,M\to\infty$ at fixed finite aspect ratio $\eta=N/M$, the limit being expected to be independent of $\eta$; this includes establishing the observed fast convergence — empirically described along sequences of $n\times n$ bases by the scaling law $p_c(n)-p_c \sim A\,n^{-w}$ for the bond-percolation threshold (with the same exponent for $v_c(n)$ under the smooth change of variables $p=v/(1+v)$), with non-universal exponent $w\approx 4$ to $6$ — and, ideally, determining the correction-to-scaling exponent $w$ governing it.

Second, the nature of the limit: decide whether the limiting critical manifold is an algebraic curve — and if so, exhibit that curve — or non-algebraic, as conjectured by Jacobsen and Scullard; in the latter case, provide a constructive characterization (for example, a transcendental or implicit representation from which critical points can be computed to arbitrary precision) together with a proof that it is the exact critical manifold.

Third, factorization and exact solvability: prove or refute that systematic factorization of $P_B(q,v)$ — the same factor shed for every choice of basis $B$ — is sufficient for the zero set of that factor to be an exact part of the critical manifold of the Potts model on a two-dimensional lattice $G$, and characterize the lattices and values of $q$ for which such factorization occurs, including its relation to integrability (Yang-Baxter solvability, the existence of a parametrized family of local Boltzmann weights satisfying the Yang-Baxter consistency equation that underlies all exactly solved two-dimensional lattice models, and discrete holomorphicity, the exact lattice conservation of certain holomorphic observables that underlies rigorous conformal-invariance proofs). Necessity is known to fail — the exactly solvable point $(q,v)=(3,-1)$ of the kagome model does not arise from factorization — so the open question concerns sufficiency and the characterization, not necessity.

Fourth, persistence of exact features: establish analytically which exact features of the finite-basis curves survive in the thermodynamic limit — the passage of every basis through $(q,v)=(0,-3)$ and $(2,-1)$, the conjectured exactly vertical rays at the Beraha numbers $q=2$ and $q=3$ bounding the Berker-Kadanoff phase, the passage of the antiferromagnetic branch through $(3,-1)$, and the extent of the Berker-Kadanoff phase in the kagome phase diagram.

A complete answer determines the exact kagome critical manifold with proof (questions two and four) together with the foundations of the approximation scheme (questions one and three); a rigorous resolution of any one of the four questions, such as a proof or refutation of convergence alone, constitutes meaningful partial progress and is admissible on its own terms.

References

1. J. L. Jacobsen and C. R. Scullard, 'Critical manifold of the kagome-lattice Potts model', Journal of Physics A: Mathematical and Theoretical 45, 494003 (2012), DOI: 10.1088/1751-8113/45/49/494003, arXiv:1204.0622, https://arxiv.org/abs/1204.0622
2. C. R. Scullard and J. L. Jacobsen, 'Transfer matrix computation of generalized critical polynomials in percolation', Journal of Physics A: Mathematical and Theoretical 45, 494004 (2012), DOI: 10.1088/1751-8113/45/49/494004, arXiv:1209.1451, https://arxiv.org/abs/1209.1451
3. J. L. Jacobsen and C. R. Scullard, 'Transfer matrix computation of critical polynomials for two-dimensional Potts models', Journal of Physics A: Mathematical and Theoretical 46, 075001 (2013), DOI: 10.1088/1751-8113/46/7/075001, arXiv:1211.4335, https://arxiv.org/abs/1211.4335
4. J. L. Jacobsen, 'High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials', Journal of Physics A: Mathematical and Theoretical 47, 135001 (2014), DOI: 10.1088/1751-8113/47/13/135001, arXiv:1401.7847, https://arxiv.org/abs/1401.7847
5. J. L. Jacobsen, 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley-Lieb algebras', Journal of Physics A: Mathematical and Theoretical 48, 454003 (2015), DOI: 10.1088/1751-8113/48/45/454003, arXiv:1507.03027, https://arxiv.org/abs/1507.03027
6. C. R. Scullard and J. L. Jacobsen, 'Potts-model critical manifolds revisited', Journal of Physics A: Mathematical and Theoretical 49, 125003 (2016), DOI: 10.1088/1751-8113/49/12/125003, arXiv:1511.04374, https://arxiv.org/abs/1511.04374
7. F. Y. Wu, 'Critical point of planar Potts models', Journal of Physics C: Solid State Physics 12, L645-L651 (1979), DOI: 10.1088/0022-3719/12/17/002, https://doi.org/10.1088/0022-3719/12/17/002
8. K. Kano and S. Naya, 'Antiferromagnetism. The Kagome Ising Net', Progress of Theoretical Physics 10, 158-172 (1953), DOI: 10.1143/ptp/10.2.158, https://doi.org/10.1143/ptp/10.2.158
9. Y. Yang and S. Zhou, 'Comment on Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley-Lieb algebras', Journal of Physics A: Mathematical and Theoretical 57, 258001 (2024), DOI: 10.1088/1751-8121/ad4d2c, https://doi.org/10.1088/1751-8121/ad4d2c