Convergence, exactness, and the antiferromagnetic phase diagrams of the critical-polynomial method for two-dimensional Potts models

The critical polynomial $P_B(q,v)=W^{(2D;B)}-q\,W^{(0D;B)}$, computed by transfer matrix on finite bases $B$, predicts critical manifolds of the $q$-state Potts model on two-dimensional lattices with an accuracy far exceeding Monte Carlo, yet its mathematical foundations remain incomplete. Jacobsen and Scullard (2013) established the method on the kagome, $(4,8^2)$ and $(3,12^2)$ lattices with bases of up to 96-243 edges and left open the questions that still stand: no proof (or disproof) that the roots of $P_B$ converge to the exact critical manifold as the basis grows; no accepted explanation of the basis-independent exactness observed at $q=2$, at special $q=0$ points, at $q\to\infty$, and at isolated antiferromagnetic points; and no analytical confirmation of the intricate antiferromagnetic structure the polynomials predict, namely vertical rays at Beraha numbers inside the Berker-Kadanoff phase, prongs, bubbles, and 'unexpected curves' emanating exactly from $(q,v)=(2,-1)$ on the kagome and $(3,12^2)$ lattices. Later work - an eigenvalue reformulation in the periodic Temperley-Lieb algebra corroborated by conformal field theory, computations on all Archimedean lattices, and proximity-to-exact-point explanations for two special cases - has sharpened the numerics and supplied partial rationales without settling any of these questions. The problem asks to establish the convergence and exactness of the method and to determine the thermodynamic-limit antiferromagnetic critical structure of the three lattices.

Background

The $q$-state Potts model assigns one of $q$ spins to each vertex of a lattice, with couplings on the edges. In the Fortuin-Kasteleyn (FK) cluster representation the physics is encoded by the edge variable $v=\mathrm{e}^{K}-1$, where $K$ is the dimensionless coupling, so the phase boundary of the two-dimensional model is a curve - the critical manifold - in the real $(q,v)$ plane. On the square lattice this manifold is known exactly from self-duality, but on the other Archimedean lattices (the vertex-transitive plane tilings, such as the kagome, $(4,8^2)$ and $(3,12^2)$ lattices) no exact solution is known and the critical manifolds must be approximated.

Jacobsen and Scullard (2012, 2013) introduced an algebraic approximation scheme. A finite subgraph $B$ of the lattice, called the basis, is tiled periodically, and a polynomial $P_B(q,v)$ is defined either through a deletion-contraction identity analogous to that of the Tutte polynomial, or probabilistically: partition the lattice into copies of $B$, let $W^{(2D;B)}$ be the weight of the event that an infinite two-dimensional cluster of open edges exists, and $W^{(0D;B)}$ the weight that no infinite cluster exists (the intermediate case is one-dimensional filaments). Then

$P_B(q,v)=W^{(2D;B)}-q\,W^{(0D;B)}.$

The real roots of $P_B$ approximate the critical manifold, and a transfer-matrix computation, exponential only in the cross-section of $B$ rather than in its edge count, gave access to bases of 96, 162 and 243 edges on the $(4,8^2)$, kagome and $(3,12^2)$ lattices respectively, producing ferromagnetic critical couplings of precision comparable to or better than the best simulations. Three empirical regularities define the open questions. First, the accuracy improves rapidly with basis size, but there is no proof that the roots converge to the exact manifold; the source paper calls the success of the condition 'quite mysterious' and states that it 'reveals some larger truth about critical Potts systems, the full mathematical implications of which are yet to be discovered'. Second, at special parameter values the roots are basis-independent and exact: the Ising case $q=2$ on every lattice studied (even in the antiferromagnetic region), exact passage through $(q,v)=(0,0)$ and through a second $q=0$ point ($v=-4$ for the square and $(4,8^2)$ lattices, $v=-3$ for kagome and $(3,12^2)$), the correct $q\to\infty$ first-order asymptotics, and isolated antiferromagnetic points, such as those on the kagome lattice established via a decorated-hexagonal transformation (a star-triangle mapping of the kagome lattice onto a decorated hexagonal lattice, which in turn maps onto the exactly solved hexagonal case). Third, in the antiferromagnetic region $v<0$ the polynomials predict a rich structure: a Berker-Kadanoff phase (the critical antiferromagnetic phase of the two-dimensional Potts model, described in the continuum by a conformal field theory) inside which vertical rays - transition lines of constant $q$ - appear at the Beraha numbers $B_k=4\cos^2(\pi/k)$, where a quantum-group symmetry produces cancellations allowing a reformulation as an RSOS (restricted solid-on-solid) height model; in addition there are prongs, bubbles and slivers whose thermodynamic-limit fate is uncertain, and 'unexpected curves' emanating exactly from $(q,v)=(2,-1)$ on the kagome and $(3,12^2)$ lattices, believed on the basis of several bases to be real effects.

Later work sharpened the method without settling these questions. Jacobsen (2015) reformulated it as an eigenvalue problem in the periodic Temperley-Lieb algebra (the algebra of bond-addition operators $e_i$ with $e_i^2=e_i$ and braid-type relations, which underlies the Potts transfer matrix): the critical condition equates the leading eigenvalues of two topologically distinct transfer-matrix sectors that determine the same critical exponent in the continuum limit, so conformal field theory corroborates the fast convergence; the same paper leaves the link between exact factorization and discrete holomorphicity (the lattice-exact conservation of non-local currents that underlies many known solvable critical models) as an open conjecture. Scullard and Jacobsen (2016) computed $n=6$ polynomials for all Archimedean lattices and found, for example, that the kagome lattice shows rays only at $B_4=2$ and $B_6=3$ plus an unexplained 'bump' to the left of the $q=2$ ray, while $(4,8^2)$ and $(3,12^2)$ develop rays at many more Beraha numbers; the paper states that only the square-lattice phase diagram is properly understood analytically. Scullard and Jacobsen (2020) produced reference-quality percolation thresholds and two lattice classes with distinct finite-size correction exponents ($\Delta=6,7,8$ versus $4,6,8$); Scullard, Jacobsen and Ziff (2021) traced the anomalously fast convergence on kagome and $(3,12^2)$ to their quantifiable proximity to an exactly solvable point on the same critical surface. Independent applications (Xu et al. 2021 on non-planar and continuum percolation; Bell et al. 2022 via Newman-Ziff sampling, a Monte Carlo scheme that sweeps through all bond configurations exactly once) confirm the phenomenon while leaving its origin unexplained.

Problem Statement

The problem is to establish the mathematical status and the thermodynamic-limit content of the critical-polynomial description of the $q$-state Potts model on the non-exactly-solved Archimedean lattices - kagome, $(4,8^2)$ and $(3,12^2)$ - as posed by Jacobsen and Scullard (2013). It has three linked parts.

(1) Convergence. Prove, or disprove by an explicit counterexample, that the real roots $v(q)$ of the critical polynomials $P_B(q,v)=W^{(2D;B)}-q\,W^{(0D;B)}$ converge, as the basis $B$ grows through the standard compatible families of square and hexagonal $n\times n$ bases (including the $m\to\infty$ eigenvalue formulation on semi-infinite cylinders), to the exact critical manifold of the Potts model on these lattices; and characterize the finite-size corrections, including the observed non-universal leading correction exponent (about 4 for some lattices and 6 for others) and its dependence on basis shape, twist, and lattice symmetry.

(2) Exactness. Explain mathematically why the method reproduces exact, basis-independent results at special points - the Ising case $q=2$ on every lattice studied, the exact passage through $(q,v)=(0,0)$ and the second $q=0$ points ($v=-4$ for the square and $(4,8^2)$ lattices, $v=-3$ for kagome and $(3,12^2)$), the correct $q\to\infty$ first-order asymptotics, and isolated antiferromagnetic points such as the kagome point found via the decorated-hexagonal transformation - by identifying the structural mechanism (candidate: the conjectured link with discrete holomorphicity or other manifestations of exact solvability) that determines for which lattices and parameter values the roots of $P_B$ are basis-independent.

(3) Antiferromagnetic phase diagram. Determine the thermodynamic-limit structure of the critical manifold in the antiferromagnetic region $v<0$ for the three lattices: the extent of the Berker-Kadanoff phase; the set of vertical rays at the Beraha numbers $B_k=4\cos^2(\pi/k)$ (in particular, whether the kagome lattice has rays only at $B_4=2$ and $B_6=3$ while $(4,8^2)$ and $(3,12^2)$ have rays at all $B_{2k}$); the fate of the prongs, bubbles and slivers observed at finite basis; and whether the 'unexpected curves' emanating exactly from $(q,v)=(2,-1)$ on the kagome and $(3,12^2)$ lattices, and the related bump to the left of the $q=2$ ray on the kagome lattice, are genuine thermodynamic-limit critical curves, and if so what transitions they represent.

These parts are linked by a single objective: to determine what the finite-basis observable $P_B$ actually computes about critical Potts systems. Part (2) concerns the special points where it is observed to be exact, part (3) the antiferromagnetic region where its predictions lack any analytical confirmation, and part (1) the convergence question that underlies both. A complete answer establishes all three; the computational scalability of the transfer-matrix algorithm itself is an engineering matter outside the problem.

References

1. Jesper Lykke Jacobsen and Christian R. Scullard, 'Transfer matrix computation of critical polynomials for two-dimensional Potts models', Journal of Physics A: Mathematical and Theoretical 46 (2013) 075001. DOI: 10.1088/1751-8113/46/7/075001; arXiv:1211.4335; https://arxiv.org/abs/1211.4335
2. Jesper Lykke Jacobsen and Christian R. Scullard, 'Critical manifold of the kagome-lattice Potts model', Journal of Physics A: Mathematical and Theoretical 45 (2012) 494003. DOI: 10.1088/1751-8113/45/49/494003; arXiv:1204.0622; https://arxiv.org/abs/1204.0622
3. Jesper Lykke Jacobsen, 'High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials', Journal of Physics A: Mathematical and Theoretical 47 (2014) 135001. DOI: 10.1088/1751-8113/47/13/135001; arXiv:1401.7847; https://arxiv.org/abs/1401.7847
4. Jesper Lykke 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 (2015) 454003. DOI: 10.1088/1751-8113/48/45/454003; arXiv:1507.03027; https://arxiv.org/abs/1507.03027
5. Christian R. Scullard and Jesper Lykke Jacobsen, 'Potts-model critical manifolds revisited', Journal of Physics A: Mathematical and Theoretical 49 (2016) 125003. DOI: 10.1088/1751-8113/49/12/125003; arXiv:1511.04374; https://arxiv.org/abs/1511.04374
6. Christian R. Scullard and Jesper Lykke Jacobsen, 'Bond percolation thresholds on Archimedean lattices from critical polynomial roots', Physical Review Research 2 (2020) 012050. DOI: 10.1103/PhysRevResearch.2.012050; arXiv:1910.12376; https://arxiv.org/abs/1910.12376
7. Christian R. Scullard, Jesper Lykke Jacobsen and Robert M. Ziff, 'Critical percolation on the kagome hypergraph', Journal of Physics A: Mathematical and Theoretical 54 (2021) 055006. DOI: 10.1088/1751-8121/abcddb; arXiv:2009.01913; https://arxiv.org/abs/2009.01913
8. Jesper Lykke Jacobsen, Jesus Salas and Christian R. Scullard, 'Phase diagram of the triangular-lattice Potts antiferromagnet', Journal of Physics A: Mathematical and Theoretical 50 (2017) 345002. DOI: 10.1088/1751-8121/aa778f; arXiv:1702.02006; https://arxiv.org/abs/1702.02006
9. Wenhui Xu, Junfeng Wang, Hao Hu and Youjin Deng, 'Critical polynomials in the nonplanar and continuum percolation models', Physical Review E 103 (2021) 022127. DOI: 10.1103/PhysRevE.103.022127; arXiv:2010.02887; https://arxiv.org/abs/2010.02887
10. Tolson H. Bell, Jerrell M. Cockerham, Clayton M. Mizgerd, Melita F. Wiles and Christian R. Scullard, 'Critical points of the random cluster model with Newman-Ziff sampling', Journal of Physics A: Mathematical and Theoretical 55 (2022) 044001. DOI: 10.1088/1751-8121/ac42ab; arXiv:2102.12716; https://arxiv.org/abs/2102.12716
11. Yi Yang and Shuigeng 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 (2024) 258001. DOI: 10.1088/1751-8121/ad4d2c; https://doi.org/10.1088/1751-8121/ad4d2c
12. Jesper Lykke Jacobsen, 'Reply to 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 (2024) 258002. DOI: 10.1088/1751-8121/ad4d33; https://doi.org/10.1088/1751-8121/ad4d33