Test of Guttmann and Enting’s conjecture in the eight-vertex model

We investigate the analyticity property of the partially resummed series expansion(PRSE) of the partition function for the eight-vertex model. Developing a graphical technique, we have obtained a first few terms of the PRSE and found that these terms have a pole only at one point in the complex plane of the coupling constant. This result supports the conjecture proposed by Guttmann and Enting concerning the “solvability” in statistical mechanical lattice models. Recently Guttmann and Enting introduced a new point of view into the study of the solvability of lattice models in the statistical mechanics, including models for combinatorial problems [1]. Their approach is based on a study of the connection between the solvability of lattice models by use of the inversion relation and the analyticity properties of their associated series expansion, which is called the partially resummed series expansion(PRSE). For exactly solvable models, the inversion relation for the partition function was derived using the star-triangle relation [2,3]. Subsequently, it was shown to hold true for the partition function of (so far) unsolved models and other physical quantities as well [4]. Thus the inversion relation can be used for clarifying the difference between exactly solved quantities and unsolved ones in statistical lattice models. The importance of the analyticity properties of the PRSE was first pointed out by Baxter [5]. He considered the PRSE for the reduced free energy lnΛ(t1, t2) of the 1 zero-field anisotropic Ising model on the square lattice, lnΛ(t1, t2) = ∑ Rn(t 2 1)t 2n 2 , and showed that the functions Rn(t 2 1) can be determined recursively from the inversion relation, ln Λ(t1, t2)+ lnΛ(t −1 1 ,−t2) = ln(1− t 2 2), and the symmetry relation Λ(t1, t2) = Λ(t2, t1), provided Rn(t 2 1) have singularities only at a single point in the t 2 1-complex plane. This realization led to a series of investigations on the partial sums Rn(t 2 1) for unsolved quantities such as the partition function of the 2D Ising model in a field [5], non-critical 2D Potts model [4], zero-field 3D Ising model [6] or the susceptibility of the 2D Ising model [1]. The results for these quantities indicate that infinitely many poles appear in partial sums Rn in the limit n → ∞ in clear contrast with the solvable case. The new approach of Guttmann and Enting to the solvability of lattice models is as follows. They defined a given quantity to be solvable if its solution can be expressed with a D-finite function [7] as a function of the coupling constant. D-finite functions are defined as functions which satisfy a linear ordinary differential equation of finite order with polynomial coefficients. The definition of the solvability was extended so that solvable quantities do not necessarily have a closed form solution. However, it is restrictive in the sense that those quantities which are expressible with the solution of a nonlinear differential equation are classified as “unsolvable” ones. Guttmann and Enting further put forward a criterion which determines whether a given quantity is solvable or not in the sense of D-finiteness. First, generalize the model such that it possesses two (or more) anisotropic coupling constants t1 and t2. Let the wanted quantity be f(t1, t2). One can calculate exact coefficients of the series expansion of f , f(t1, t2) =