Recursion relations for the partition function of the two-dimensional Ising model

TL;DR

This paper derives recursion relations for the partition function of the 2D Ising model on a square lattice, revealing a complex structure and enabling more efficient computer algebra calculations.

Contribution

It introduces new recursion relations for the 2D Ising model's partition function expressed as polynomials with integer coefficients.

Findings

Recursion relations for system sizes $2^m\times 2^n$ are established.

Polynomials with integer coefficients are derived from low-temperature expansions.

Recursions improve the efficiency of computer algebra calculations.

Abstract

The partition function of the two-dimensional Ising model on a square lattice with nearest-neighbour interactions and periodic boundary conditions is investigated. Kaufman [Phys. Rev. 76, 1232--1243 (1949)] gave a solution for this function consisting of four summands. The summands are rewritten as functions of a low-temperature expansion variable, resulting in polynomials with integer coefficients. Considering these polynomials for system sizes $2^m\times 2^n$ ($m,n\in\N$), a variety of recursion relations in $m,n$ are found. The recursions reveal a rich structure of the partition function and can be employed to render the computer algebra calculation of the microcanonical partition function more efficient.