In this blog post, I derive, step by step, the exact partition function for the ferromagnetic Ising model on the square lattice. The result is celebrated as “Onsager’s solution” of the 2-D Ising model. It was originally derived by Lars Onsager in 1942 and published in 1944 in Physical Review . That paper revolutionized the study of phase transitions and what we now call critical phenomena .
Somewhat ironically, I first heard about the Ising model when I was working in industry. I was 20 and held a summer job at what was then known as British Telecom Research Labs (BTRL), near Ipswich in the UK. This was before I had ever seen a cell phone or heard of the Internet (although I knew about BITNET and JANET). I worked there in the summer of 1990 and again for a month or so around April 1991. My job at BT involved writing C implementations of multilayer perceptrons and Hopfield neural nets. In those days, BT was interested in implementing hardware neural networks and my boss mentioned casually to me that certain kinds of neural nets are basically just special cases of the Ising model. (Indeed, the Hopfield network is closely related to the Ising spin glass.) Thus began my fascination with the Ising model. Later, in 1994 in Boston, I took a course given by Bill Klein at BU on statistical mechanics, where we went through the solution of the 1-D ferromagnetic Ising model. Still, I never had the chance to study properly the 2-D Ising model. As a PhD student, I would almost daily pass by a poster with a background photo of Lars Onsager (with a cigarette in his hand), hung near the office door of my advisor Gene Stanley, so I was regularly reminded of the 2-D Ising model. I kept telling myself that one day I would eventually learn how Onsager managed to do what seemed to me, at the time, an “impossible calculation.” That was 1994 and I am writing this in 2015!
In what follows, I solve the Ising model on the infinite square lattice, but I do not actually follow Onsager’s original argument. There are in fact several different ways of arriving at Onsager’s expression [3–9]. The method I use below is known as the combinatorial method and was developed by van der Waerden, Kac and Ward among others and relies essentially on counting certain kinds of closed graphs (see refs. [3,10–13]). I more or less follow Feynman  and I have also relied on the initial portions of ref. .
2. The 2-D Ising model
Consider a two dimensional lattice where at each point of the lattice is located a (somewhat idealized) spin- particle. Consider a finite subset of this lattice, of size and let . Let denote the set of pairs of integers such that spins and are nearest neighbors. In the ferromagnetic 2-D Ising model with nearest neighbor interactions, spins and interact if and only if . Each spin can assume only 2 values: .
Consider a system of spins. The Hamiltonian for a spin configuration is given by
The sum over the nearest neighbors should avoid double counting, so that and are not counted separately. Without loss of generality, we will assume for simplicity.
3. The canonical partition function
In the theory of equilibrium statistical mechanics, the canonical partition function contains all the information needed to recover the thermodynamic properties of a system with fixed number of particles, immersed in a heat bath, details of which can be found in any textbook on statistical mechanics [3-6,14].
I prefer to define the partition function as the two-sided Laplace transform of the degeneracy of the energy level . But traditionally, the partition function is defined as a sum or integral over all possible states of the system:
The two ways of thinking are equivalent. The Laplace transform variable is related to the thermodynamic temperature via , where is the Boltzmann constant. What follows is the exact calculation of .