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

Abstract

In 1944 Onsager published the formula for the partition function of the Ising model for the infinite square lattice. He was able to express the internal energy in terms of a special function, but he left the free energy as a definite integral. Seven decades later, the partition function and free energy have yet to be written in closed form, even with the aid of special functions. Here we evaluate the definite integral explicitly, using hypergeometric series. Let $\beta$ denote the reciprocal temperature, $J$ the coupling and $f$ the free energy per spin. We prove that $-\beta f = \ln(2 \cosh 2K) - \kappa^2\, {}_4F_3 [1,1,\tfrac{3}{2},\tfrac{3}{2};\ 2,2,2 ;\ 16 \kappa^2 ] $, where $_p F_q$ is the generalized hypergeometric function, $K=\beta J$, and $2\kappa= {\rm tanh} 2K {\rm sech} 2K$.