Analytic theory of the critical-polynomial method for Potts critical manifolds: finite-size scaling exponents, antiferromagnetic zero loci, and the limits of applicability

The method of critical polynomials estimates Potts-model critical points from the root of a graph polynomial $P_B(q,v)$ computed on a finite basis $B$ of the lattice, and it converges so fast that bond-percolation thresholds on the Archimedean lattices are known to 11-18 digits. Its key ingredients, however, are empirical: the finite-size correction exponents are fitted rather than derived (two conjectured integer sequences, $\Delta=6,7,8,\dots$ and $\Delta=4,6,8,\dots$, splitting the lattices into two unexplained classes); the antiferromagnetic zero loci - vertical rays at Beraha numbers, the extent of the Berker-Kadanoff phase, and a series of unexplained structures - have no analytic account beyond the square lattice; and no theory delimits when the method converges or fails. The problem, posed by Scullard and Jacobsen (2016) and sharpened by the follow-up literature, is to construct this analytic theory: derive the correction exponents and, if the conjectured two-class split holds, its mechanism, characterize the antiferromagnetic zero loci in the thermodynamic limit, and explain the method's fast convergence and applicability.

Background

The $q$-state Potts model generalizes the Ising model to $q$ possible spin states per site; in the Fortuin-Kasteleyn (FK) cluster representation its partition function on a graph $G=(V,E)$ is $Z=\sum_{A\subseteq E} v^{|A|} q^{k(A)}$, where $v=\mathrm{e}^K-1$ is the temperature variable, $|A|$ is the number of occupied edges, and $k(A)$ the number of connected components including isolated vertices. At $q=1$ the model reduces to bond percolation, in which edges are occupied independently with a probability related to $v$ and the critical point $p_{\mathrm{c}}$ is the percolation threshold. An Archimedean lattice is a vertex-transitive tiling of the plane by regular polygons; eleven exist, and for three of them (square, triangular, honeycomb) the critical manifolds are known exactly, while the other eight are unsolved.

The method of critical polynomials, developed by Scullard, Jacobsen and co-workers since 2008, produces by far the most precise thresholds for the unsolved cases. One chooses a basis $B$ - a finite subgraph of the lattice together with boundary identifications that tile the plane - and computes the FK weights of two connectivity events: $P_{2D}(q,v)$, the weight that tiling the plane with copies of $B$ (each carrying the same configuration) connects every copy, and $P_{0D}(q,v)$, the weight that no copy can be traversed. The critical polynomial

$$P_B(q,v) = P_{2D}(q,v) - q\,P_{0D}(q,v)$$

has a root that estimates the critical point; the estimate is exact on the solvable lattices for any $B$, and converges to the true critical point as $B$ grows - a conjecture essentially equivalent to the universality of crossing probabilities. What is striking is the convergence rate: bond-percolation thresholds on Archimedean lattices are now known to 11-13 digits (Scullard-Jacobsen 2016) and to 17-18 digits on a kagome hypergraph (Scullard-Jacobsen-Ziff 2021), far beyond Monte Carlo.

Jacobsen (2015) reformulated the method as an eigenvalue problem: $P_B=0$ is equivalent to equating the largest eigenvalues of two topologically distinct sectors (open and closed) of a row-by-row transfer matrix - built from the generators of the periodic Temperley-Lieb algebra, the algebra of non-crossing connectivity operators that assemble FK configurations - acting on a semi-infinite cylinder of circumference $n$. Conformal field theory (the continuum theory describing two-dimensional critical points, organized by scaling dimensions $x_i$) shows that both sectors are governed by the same scaling dimension, so their free-energy difference vanishes faster than $n^{-2}$ - suggestive of, but not implying, the observed corrections of order $n^{-4}$ and smaller.

Scullard and Jacobsen (2016) computed $P_B$ on $6\times 6$ bases for all Archimedean lattices and $8\times 8$ bases for percolation, and found the finite-size estimates obey $p_{\mathrm{c}}(n) = p_{\mathrm{c}} + A\,n^{-w}$ with $w$ apparently taking two values, $w_3 = 6.35\pm 0.05$ and $w_4 = 4.2\pm 0.1$, roughly correlated with the lattice's rotational symmetry. Scullard and Jacobsen (2020) extended the eigenvalue computations to $n=16$ and showed that the data instead split into two lattice classes with correction exponents $(\Delta_1,\Delta_2,\Delta_3)=(6,7,8)$ for the kagome, three-twelve and snub-hexagonal lattices and $(4,6,8)$ for the other five, conjectured to continue as all integers $\ge 6$ and all even integers $\ge 4$ respectively; they note that the split is not explained by rotational symmetry alone and that "it remains unclear how to derive them analytically". The structure is thus numerically sharp but theoretically unexplained.

The zero loci of $P_B(q,v)$ in the antiferromagnetic half-plane $v<0$ pose a second set of puzzles. On the square lattice, Saleur (1991) established the exact phase diagram: an extended-critical (Berker-Kadanoff) phase occupies part of the antiferromagnetic region, with special behaviour at the Beraha numbers $q = B_k = 4\cos^2(\pi/k)$. On this basis it was conjectured that on any lattice the thermodynamic-limit zero loci contain vertical rays - segments at constant $q$ - at $B_k$ for even $k$, extending as far as the Berker-Kadanoff phase reaches; the $n=6$ computations confirm and extend this, and the number and position of the rays is the best available probe of each lattice's Berker-Kadanoff phase. But beyond the square lattice nothing is explained: the computed diagrams show a persistent bump beside the $q=2$ ray on the kagome lattice, finger-like structures on the four-eight and cross lattices, an oval region straddling $B_8$ on the snub-square lattice, a flower-like structure near $(q,v)=(4,-2)$ on the snub-hexagonal lattice, and ever more rays at higher Beraha numbers as $n$ grows - none with an analytic explanation, and their thermodynamic-limit status conjectural. Exact results exist for the triangular lattice (the antiferromagnetic critical curve; exact exponents for $Q$-colourings) and for chromatic zeros under different boundary conditions, but no account for the unsolved lattices.

Third, the method's own applicability is not understood: it converges spectacularly on Archimedean (planar, periodic) lattices, yet for the non-Archimedean four-edge hypergraphs it does not beat Monte Carlo, and for nonplanar or continuum percolation it has so far been used only as a universality ansatz. The source paper poses "the broader question of why this method performs as well as it does".

Problem Statement

The problem is to replace the empirical ingredients of the critical-polynomial method for two-dimensional Potts models by an analytic theory. Let $P_B(q,v)=P_{2D}(q,v)-q\,P_{0D}(q,v)$ be the critical polynomial computed on a basis $B$ of $n\times n$ unit cells of a lattice (equivalently, in the eigenvalue formulation, on a semi-infinite cylinder of circumference $n$), where $P_{2D}$ and $P_{0D}$ are the Fortuin-Kasteleyn weights of the events that tiling the plane with copies of $B$ connects all copies, respectively leaves every copy isolated. The problem has three components, posed by Scullard and Jacobsen (2016) and sharpened by the follow-up literature; a complete solution must address all three.

(1) Finite-size scaling exponents. For bond percolation ($q=1$) on the eight Archimedean lattices whose critical manifolds are not known exactly - kagome $(3,6,3,6)$, three-twelve $(3,12^2)$, snub hexagonal $(3^4,6)$, four-eight $(4,8^2)$, frieze $(3^3,4^2)$, cross $(4,6,12)$, snub square $(3^2,4,3,4)$, ruby $(3,4,6,4)$ - derive analytically the correction exponents $\Delta_k$ in

$$p_{\mathrm{c}}(n)=p_{\mathrm{c}}+\sum_{k=1}^{\infty} A_k\,n^{-\Delta_k},$$

where $p_{\mathrm{c}}(n)$ is the critical-polynomial (or eigenvalue) estimate at basis size $n$. The derivation must establish or refute the conjecture of Scullard and Jacobsen (2020) that $\{\Delta_k\}=\{6,7,8,\dots\}$ for the kagome, three-twelve and snub-hexagonal lattices and $\{\Delta_k\}=\{4,6,8,\dots\}$ for the remaining five; if the conjecture holds it must identify the mechanism producing this two-class split, which is known not to follow from the lattices' rotational symmetries alone, whereas a refutation must exhibit the correct exponents.

(2) Antiferromagnetic zero loci. Characterize analytically the $n\to\infty$ limit of the zero locus of $P_B(q,v)$ in the antiferromagnetic half-plane $v<0$ on the same eight lattices. This requires determining which vertical rays (segments at constant $q$) at the Beraha numbers $q=B_k=4\cos^2(\pi/k)$, $k$ even, survive in the thermodynamic limit, as conjectured; determining the extent of the Berker-Kadanoff phase - the extended-critical antiferromagnetic phase established exactly on the square lattice by Saleur - on each lattice; and explaining, or proving to be finite-$n$ artifacts, the recurring unexplained structures: the bump to the left of the $q=2$ ray on the kagome lattice, the finger-like structures on the four-eight and cross lattices, the oval region straddling $B_8$ on the snub-square lattice, the flower-like structure near $(q,v)=(4,-2)$ on the snub-hexagonal lattice, and the proliferation of rays at ever higher Beraha numbers as $n$ grows. Any account must be consistent with the boundary-condition dependence of partition-function-zero accumulation loci (Jacobsen-Salas 2006, 2007).

(3) Convergence and applicability. Explain why the estimates converge as fast as they do - corrections of order $n^{-4}$ and smaller, faster than the generic conformal finite-size argument ($f_{\mathrm{open}}-f_{\mathrm{closed}}=o(n^{-2})$) accounts for - and address the broader question, posed by Scullard and Jacobsen (2016), of why this method performs as well as it does, including the observation that for non-Archimedean two-dimensional problems such as the four-edge hypergraphs it is not yet clear how to better Monte Carlo with critical polynomials, and that for nonplanar and continuum percolation $P_B$ has so far been validated only through universality-assisted Monte Carlo.

References

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