Tag Archives: Ising model

Fermionization of the 2-D Ising model: The method of Schultz, Mattis and Lieb

F. A da Costa, R. T. G. de Oliveira, G. M. Viswanathan

This blog post was written in co-authorship with my physics department colleague Professor Francisco “Xico” Alexandre da Costa and Professor Roberto Teodoro Gurgel de Oliveira, of the UFRN mathematics department. Xico obtained his doctorate under Professor Sílvio Salinas at the University of São Paulo. Roberto was a student of Xico many years ago, but left physics to study mathematics at IMPA in Rio de Janeiro in 2010. During 2006–2007, Roberto and Xico had written up a short text in Portuguese that included the exact solution of the Ising model on the infinite square lattice using the method of fermion operators developed by Schultz, Mattis and Lieb. With the aim of learning this method, I adapted their text and expanded many of the calculations for my own convenience. I decided to post it on this blog since others might also find it interesting. I have previously written an introduction to the 2-D Ising model here, where I review a combinatorial method of solution.

1. Introduction

The spins in the Ising model can only take on two values, {\pm 1}. This behavior is not unlike how the occupation number {n} for some single particle state for fermions can only take on two values, {n=0,1}. It thus makes sense to try to solve the Ising model via fermionization. This is what Schultz, Mattis and Lieb accomplished in their well-known paper of 1964. In turn, their method of solution is a simplified version of Bruria Kaufman’s spinor analysis method, which is in turn a simplification of Onsager’s original method.

We will proceed as follows. First we will set up the transfer matrix. Next we will reformulate it in terms of Pauli’s spin matrices for spin-{\tfrac 1 2} particles. Recall that in quantum field theory boson creation and annihilation operators satisfy the well-known commutation relations of the quantum harmonic oscillator, whereas fermion operators satisfy analogous anticommutation relations. The spin annihilation and creation operators {\sigma_j^\pm } do not anticommute at distinct sites {j} but instead commute, whereas fermion operators must anticommute at different sites. This problem of mixed commutation and anticommutation relations can be solved using a method known as the Jordan-Wigner transformation. This step completes the fermionic reformulation of the 2-D Ising model. To obtain the partition function in the thermodynamic limit, which is the largest eigenvalue of the transfer matrix, one diagonalizes the fermionized transfer matrix using appropriate canonical transformations.

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Onsager’s solution of the 2-D Ising model: The combinatorial method

1. Introduction

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 [1]. That paper revolutionized the study of phase transitions and what we now call critical phenomena [2].

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 [3] and I have also relied on the initial portions of ref. [13].

2. The 2-D Ising model

Consider a two dimensional lattice {\Bbb Z^2} where at each point of the lattice is located a (somewhat idealized) spin-{\tfrac{1}{2}} particle. Consider a finite subset of this lattice, of size {L\times L} and let {i,j=1,2,\ldots L^2}. Let {\cal N} denote the set of pairs of integers {(i,j)} such that spins {\sigma_i} and {\sigma_j} are nearest neighbors. In the ferromagnetic 2-D Ising model with nearest neighbor interactions, spins {i} and {j} interact if and only if {(i,j)\in \cal N}. Each spin {\sigma_i} can assume only 2 values: {\sigma=\pm 1}.

Consider a system of {N=L^2} spins. The Hamiltonian for a spin configuration {\sigma=(\sigma_1,\sigma_2,\ldots \sigma_N)} is given by

\displaystyle H(\sigma)= -J \displaystyle\sum_{i,j\in \cal N} \sigma_i \sigma_j ~. \ \ \ \ \ (1)

The sum over the nearest neighbors should avoid double counting, so that {(i,j)} and {(j,i)} are not counted separately. Without loss of generality, we will assume {J=1} 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 {\Omega(E)} of the energy level {E}. But traditionally, the partition function is defined as a sum or integral over all possible states of the system:

\displaystyle Z(\beta) = \displaystyle\sum _{\sigma} e^{-\beta H(\sigma)} ~. \ \ \ \ \ (2)

The two ways of thinking are equivalent. The Laplace transform variable {\beta} is related to the thermodynamic temperature {T} via {\beta =1/k_B T}, where {k_B} is the Boltzmann constant. What follows is the exact calculation of {Z(\beta)}.

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