The double hypergeometric series for the partition function of the 2D anisotropic Ising model

TL;DR

This paper expresses the partition function of the 2D anisotropic Ising model using a double hypergeometric function, extending previous formulas limited to the isotropic case and enabling advanced calculations.

Contribution

It introduces a novel formulation of the anisotropic Ising model's partition function using Kampé de Fériet functions, a significant generalization over prior special function representations.

Findings

Reformulation of the anisotropic partition function in terms of double hypergeometric functions.

Derivation of hypergeometric formulas for generating functions of multipolygons.

Facilitation of arbitrary order calculations for the isotropic series.

Abstract

In 1944 Lars Onsager published the exact partition function of the ferromagnetic Ising model on the infinite square lattice in terms of a definite integral. Only in the literature of the last decade, however, has it been recast in terms of special functions. Until now all known formulas for the partition function in terms of special functions have been restricted to the important special case of the isotropic Ising model with symmetric couplings. Indeed, the anisotropic model is more challenging because there are two couplings and hence two reduced temperatures, one for each of the two axes of the square lattice. Hence, standard special functions of one variable are inadequate to the task. Here, we reformulate the partition function of the anisotropic Ising model in terms of the Kamp\'e de F\'eriet function, which is a double hypergeometric function in two variables that is more general than the Appell hypergeometric functions. Finally, we present hypergeometric formulas for the generating function of multipolygons of given length on the infinite square lattice, for isotropic as well as anisotropic edge weights. For the isotropic case, the results allow easy calculation, to arbitrary order, of the celebrated series found by Cyril Domb.