Critical manifold of the kagome-lattice Potts model

Abstract

Any two-dimensional infinite regular lattice G can be produced by tiling the plane with a finite subgraph B⊆G; we call B a basis of G. We introduce a two-parameter graph polynomial PB(q, v) that depends on B and its embedding in G. The algebraic curve PB(q, v) = 0 is shown to provide an approximation to the critical manifold of the q-state Potts model, with coupling v = eK − 1, defined on G. This curve predicts the phase diagram not only in the physical ferromagnetic regime (v > 0), but also in the antiferromagnetic (v < 0) region, where analytical results are often difficult to obtain. For larger bases B the approximations become increasingly accurate, and we conjecture that PB(q, v) = 0 provides the exact critical manifold in the limit of infinite B. Furthermore, for some lattices G—or for the Ising model (q = 2) on any G—the polynomial PB(q, v) factorizes for any choice of B: the zero set of the recurrent factor then provides the exact critical manifold. In this sense, the computation of PB(q, v) can be used to detect exact solvability of the Potts model on G. We illustrate the method for two choices of G: the square lattice, where the Potts model has been exactly solved, and the kagome lattice, where it has not. For the square lattice we correctly reproduce the known phase diagram, including the antiferromagnetic transition and the singularities in the Berker–Kadanoff phase at certain Beraha numbers. For the kagome lattice, taking the smallest basis with six edges we recover a well-known (but now refuted) conjecture of F Y Wu. Larger bases provide successive improvements on this formula, giving a natural extension of Wu’s approach. We perform large-scale numerical computations for comparison and find excellent agreement with the polynomial predictions. For v > 0 the accuracy of the predicted critical coupling vc is of the order 10−4 or 10−5 for the six-edge basis, and improves to 10−6 or 10−7 for the largest basis studied (with 36 edges).This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.