Exact critical coupling of the q-state Potts model on the kagome lattice

The ferromagnetic q-state Potts model on the kagome lattice - the Archimedean lattice of corner-sharing triangles - has resisted exact solution for more than forty-five years; Wu's 1982 review called the kagome lattice "the simplest lattice on which an exact solution has not yet been found." The critical coupling is known exactly on the square, triangular, and honeycomb lattices and for the Ising case $q=2$ ($v_c=\sqrt{3+2\sqrt{3}}-1$), but for $q\ne 2$ on the kagome lattice only conjectures and extraordinarily precise numerics exist. Wu's 1979 sixth-order polynomial conjecture was refuted at 95-205 standard deviations, Tsallis' 1982 alternative was rigorously disproved in its percolation case, and the 2011 internal-energy strip-crossing conjecture of Baek, Mäkelä, Minnhagen, and Kim was disproved by the authors' own $L=3$ transfer-matrix calculation, which showed that finite-size corrections are nonzero for $q\ne 2$. The surviving open core is an exact closed-form expression for the critical coupling $v_c(q)$ as a function of $q$ - including the $q\to 1$ limit, the kagome bond-percolation threshold, known numerically to seventeen digits ($p_c=0.52440499916744820(1)$) but with no closed form. The problem remains open.

Background

The q-state Potts model is a lattice spin model in which each site carries a variable $\sigma_i$ taking one of $q$ discrete values and each nearest-neighbor bond contributes an energy $-J\,\delta(\sigma_i,\sigma_j)$ that favors alignment, with $J>0$ the ferromagnetic coupling. Its critical behavior is governed by the dimensionless coupling $K=\beta J$ (with $\beta$ the inverse temperature), or equivalently by $v=e^K-1$, which grows from 0 at infinite temperature to infinity at zero temperature. In the Fortuin-Kasteleyn (FK) representation the model is rewritten as a bond-percolation ensemble on the same lattice, in which each bond is independently occupied with probability $p=1-e^{-K}$ and every connected cluster carries a combinatorial weight $q$; setting $q=1$ recovers ordinary bond percolation, so the percolation threshold is the $q\to 1$ limit of the Potts critical point, with $p=v/(1+v)$.

The kagome lattice is the Archimedean (vertex-transitive) tiling of the plane by alternating triangles and hexagons, denoted $(3,6,3,6)$, in which every site has four nearest neighbors (coordination number 4); it is the lattice of corner-sharing triangles and the medial lattice of the honeycomb lattice. Its planar dual (obtained by placing a site in each face and joining sites of adjacent faces) is the dice lattice, and the critical couplings of the two lattices are related by the exact duality relation $v_c\,\tilde v_c=q$. On the square lattice the ferromagnetic Potts transition is known to be continuous for $q\le 4$ and first order for $q>4$, and this is believed to hold on any two-dimensional lattice.

Exact Potts critical points are known only on a restricted class of planar lattices: the square lattice (self-dual), and the triangular and honeycomb lattices, whose solution combines duality with the star-triangle transformation - an exact local equivalence between a triangle of three couplings and a three-legged star, which is the algebraic mechanism underlying solvability. For the kagome lattice the Ising case $q=2$ was solved exactly by Kano and Naya in 1953, giving $v_c(2)=\sqrt{3+2\sqrt{3}}-1\approx 1.542459756$; the Ising model is special, being solvable by duality and star-triangle arguments on essentially any planar lattice. For $q\ne 2$ no exact solution is known on the kagome lattice. Wu's 1982 review of the Potts model describes the kagome lattice as "the simplest lattice on which an exact solution has not yet been found" - a characterization still current in the recent literature - and two closed-form conjectures long stood: Wu (1979) proposed that the kagome critical manifold is the zero locus of the sixth-order algebraic curve

$$v^6+6v^5+9v^4-2qv^3-12qv^2-6q^2v-q^3=0,$$

derived from exact equivalences plus a non-rigorous homogeneity assumption, and Tsallis (1982) proposed an alternative form. Both are now refuted: Jacobsen and Scullard (2012) showed that numerical critical values rule out Wu's curve at a confidence level of 95 (respectively 205) standard deviations for $q=3$ (respectively $q=4$), identifying it as exactly the smallest (6-edge) approximation in a convergent sequence with no exact status, and Wierman, Yu, and Huang (2015) rigorously disproved Tsallis' conjecture in its bond-percolation case.

Baek, Mäkelä, Minnhagen, and Kim (2011) attacked the problem with a transfer-matrix method. A transfer matrix propagates the statistical weight of a spin configuration from one slice of a semi-infinite strip of width $L$ to the next; its largest eigenvalue yields the strip free energy, and derivatives with respect to $\beta$ yield observables such as the internal energy per spin $u_L$. From exact $L=1$ and $L=2$ calculations performed in arbitrary-precision arithmetic, they observed that curves of $u_L$ versus $K$ for different widths appear to cross at a width-independent point, and combined this crossing with the kagome-dice duality relation to conjecture the exact critical couplings $K_c(q=3)=1.0565094269290$ and $K_c(q=4)=1.1493605872292$. They also searched for a sixth-order polynomial with small integer coefficients having their value $v_c(q=3)=1.876313463895$ as a root, finding only a near miss (a polynomial whose root is $1.876313463898$) and concluding that no such polynomial exists in the tested coefficient ranges, and they posed the exact $q$-dependence of $K_c(q)$, encompassing the $q=1$ percolation case, as the open problem.

The width-invariance premise was disproved within two years. Baek and Mäkelä (2013) computed the $L=3$ transfer matrix exactly in factorized form and found $u_{L=3}=-1.62949\ldots$ at the conjectured critical point, deviating from the conjectured $-1.6295437063996$: finite-size corrections are nonzero for $q\ne 2$. Jacobsen and Scullard (2012) reached the same conclusion independently, noting that exact internal-energy crossings hold only for the square-lattice Potts model (by duality) and for the Ising model on general lattices, and that the 2011 values are incompatible with their own numerics. The 2011 "exact" values are therefore refuted; the refined analysis of Baek and Mäkelä yields the bounds $1.0560<K_c(q=3)<1.0565615$.

The state of the art is the critical-polynomial (graph-polynomial) method of Jacobsen and Scullard. For a finite basis $B$ - a chosen cluster of edges of the lattice - one computes, by the contraction-deletion recursion satisfied by the FK weight, a bivariate polynomial $P_B(q,v)$ whose zero locus approximates the critical manifold; Wu's conjecture is precisely the 6-edge basis result. Larger bases give systematically improving approximations that converge to the numerically observed critical points, and a reformulation in terms of eigenvalue identities in the periodic Temperley-Lieb algebra - the algebra of non-crossing loop operators describing the FK clusters - makes bases of up to $n=16$ unit cells tractable. The resulting values are extraordinarily precise but still not exact: the extrapolations $v_c(q=3)=1.876459(2)$ and $v_c(q=4)=2.156252(2)$ (Jacobsen-Scullard 2012), and, for $q=1$ bond percolation, the kagome threshold $p_c=0.52440499916744820(1)$ (Scullard-Jacobsen 2020) with rigorous bounds $0.522551<p_c<0.526490$. Scullard and Jacobsen emphasize that exact solutions remain available only on a restricted class of lattices, the kagome lattice being among the unsolved Archimedean lattices. One further hint of possible exact structure: Anglès d'Auriac (1999) presented random-matrix-theory evidence that the critical point of the three-state kagome Potts model is integrable, but no algebraic solution has been derived from it since.

Problem Statement

Determine the exact critical coupling of the ferromagnetic q-state Potts model on the infinite kagome lattice. In the variable $v=e^K-1$ (with $K=\beta J$), the goal is an exact closed-form expression for the critical value $v_c(q)$ as a function of the state count $q$ - for instance an explicit algebraic curve $P(q,v)=0$ whose physical branch gives $v_c(q)$ - together with a proof that the expression is exactly the critical point of the model on the infinite lattice. The primary targets are $q=3$ and $q=4$, for which sharp numerical anchors exist but no closed form is known: $v_c(3)=1.876459(2)$ and $v_c(4)=2.156252(2)$. A complete answer also fixes the general integer-$q$ behavior and the $q\to 1$ limit, which is the bond-percolation threshold of the kagome lattice, $p_c=v_c(1)/(1+v_c(1))$, known numerically to $p_c=0.52440499916744820(1)$ but not exactly; at minimum, an acceptable answer must settle $q=3$ and $q=4$ exactly and state precisely what it establishes for general integer $q$ and for the $q\to 1$ limit. The Ising case $q=2$ is already solved exactly, $v_c(2)=\sqrt{3+2\sqrt{3}}-1$ (Kano-Naya), and is not part of the open problem. An answer may alternatively consist of a rigorous exact determination of $v_c(q)$ by other means - for example, proving the infinite-basis limit of the kagome critical polynomials $P_B(q,v)$, or exhibiting a new exact solvability (integrable structure) of the kagome Potts model. This is the problem behind Wu's 1982 characterization of the kagome lattice as "the simplest lattice on which an exact solution has not yet been found".

References

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