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 $-\beta f=\ln (2\cosh 2K)-{\kappa }^2{}_4{F}_3[1,1,\frac{3}{2},\frac{3}{2};2,2,2;16{\kappa }^2]$, where ${}_p{F}_q$ is the generalized hypergeometric function, $K=\beta J$, and $2\kappa =\tanh 2K\mathrm{sech}2K$.

Final version as published in JSTAT

References in corpus (1)

• Fuchs versus Painlevé

Cited by in corpus (5)

• Asymptotic expansions of the hypergeometric function with two large parameters $-$ application to the partition function of a lattice gas in a field of traps
• The double hypergeometric series for the partition function of the 2D anisotropic Ising model
• Correspondence between spanning trees and the Ising model on a square lattice
• Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?
• Low-temperature series expansion of square lattice Ising model: A study based on Fisher zeros