The Ising model and counting graphs

The last example we will consider is the Ising model. Unlike the previous three examples in which animal and polyomino enumeration arise quite directly, one must venture a little way beyond the definition of the Ising model before the animals can be found lurking.

The Ising model is perhaps the most famous and widely studied model in statistical mechanics. It models the effect of temperature and external magnetic fields on the properties of a magnet. When a piece of iron is placed in a magnetic field it becomes magnetised. If the magnet is then heated, the strength of the iron's magnetic field weakens until it disappears -- the temperature at which this happens is called the Curie temperature. This change in behaviour is much like the transition water undergoes when it evaporates, and is known as a phase transition. Phase transitions are also observed in combinatorial models and exhibit themselves as changes in their asymptotic behaviour.

The Ising model was solved in one dimension by Ising [24], but was shown not to undergo any phase transition. Onsager [34] solved²⁹ the two dimensional Ising model (in the case of no external magnetic field), and it was shown to undergo a phase transition like those exhibited by real magnets. The three dimensional Ising model remains unsolved.

Let us demonstrate how animals arise in this model. Consider a finite portion of the square lattice with magnetic ``spins'' placed at the vertices. These spins could be vectors, scalars or even quantum mechanical spin operators. The Ising model considers only the simplest case; each spin is either spin up ( $\sigma_{_P}= +1$) or spin down ( $\sigma_{_P}= -1$) (see figure 22).

Figure 22: Ising model

The interaction energy of a configuration $\{\sigma \}$ is defined to be

$$E\{\sigma\} = -J \sum_{P,Q}^* \sigma_{_P}\sigma_{_Q}- H\sum_P \sigma_{_P},$$ (10)

where $\sum^*_{P,Q}$ denotes the sum over all pairs of nearest neighbour spins, $P$ and $Q$³⁰, $J$ is the ``coupling constant'' that defines the strength of the interaction between spins, and $H$ represents the interaction between the spins and the external magnetic field (if there is one).

The key to any statistical mechanical problem is the computation of the partition function which is given by

$$Z_N = \sum_{\{\sigma\}} \exp\left(-\beta E(\sigma)\right),$$ (11)

where the sum is over all possible configurations, $\sigma$, of the system, $E(\sigma)$ is the energy of a given configuration, $N$ is the number of sites in the lattice, $\beta = 1/kT$, $k$ is Boltzmann's constant and $T$ is the absolute temperature. A knowledge of the partition function of a system is sufficient to find many other relevant thermodynamic quantities, such as the internal energy and entropy.

Onsager's solution of the two dimensional Ising model in zero external field ($H=0$) is far from trivial, and we will not discuss it in this thesis (except for a brief discussion of transfer matrices in chapter 8). Van der Waerden [41] showed how the evaluation of the partition function can be translated into a bond animal enumeration problem. We give an outline of this approach.

When $H=0$ the partition function becomes

$$Z_N = \sum_{\{\sigma\}} \prod^*_{P,Q} \exp(\nu \sigma_{_P}\sigma_{_Q}),$$ (12)

where $\nu = J/kT$ and $\prod^*_{P,Q}$ denotes the product over all pairs of nearest-neighbour spins, $P$ and $Q$. It is not difficult to verify that

$$\exp(\nu \sigma_{_P}\sigma_{_Q}) = % latex2html id marker 4021 \left\{ \begin{array}{cl} e^{-\n... ... \qquad \mbox{if $\sigma_{_P}= \sigma_{_Q}$} \end{array}\right.$$

$$= (\cosh \nu)\Big(1 + \sigma_{_P}\sigma_{_Q}\tanh \nu \Big).$$ (13)

Using this we can rewrite the partition function as

$$Z_N = ( \cosh \nu)^B \sum_{\{\sigma\}} \prod^*_{P,Q} \left(1 + \omega \sigma_{_P}\sigma_{_Q}\right),$$ (14)

where $\omega = \tanh \nu$, and $B$ is the number of bonds in the lattice. From here it is not hard to rewrite this as a sum over graphs on the square grid.

Figure 23: Typical graphs that contribute to the partition function -- some of these graphs are self-avoiding polygons. It is possible for the graphs to be disconnected.

Let us consider each bond on the grid to be either occupied or vacant, and let ${\cal G}$ be the set of all possible combinations of occupied bonds (each combination forms a graph with vertex set equal to the $N$ sites in the lattice). Note that though these graphs are similar to bond animals, they are not bond animals, since they can be disconnected and they are not translationally invariant. The partition function can be rewritten as

$$Z_N = (\cosh \nu)^B \left( \sum_{G \... ...vert G\vert} \prod_P \sum_{\sigma_{_P}= \pm 1} \sigma_{_P}^{\deg_G(P)} \right),$$ (15)

where $\prod_{P}$ is a product over all $N$ sites, $\vert G\vert$ is the number of occupied bonds in $G$ and $\deg_G(P)$ is the number of occupied bonds incident on the vertex $P$ in the graph $G$. Now, for any spin

$$\sum_{\sigma_{_P}=\pm1} \sigma_{_P}^... ...l} 0 & \mbox{ if $n$\ is odd}\\ 2 & \mbox{ if $n$\ is even} \end{array} \right.$$ (16)

and so when we sum over the possible spin configurations in equation (15), any graph, $G$, that has a vertex of odd degree will contribute zero while all other graphs will contribute $2^N$. Let us define ${\cal G} '$ to be the subset of ${\cal G}$ such that for every $G \in {\cal G} '$ every vertex in $G$ has even degree. If we define $n(r)$ to be the number of graphs in ${\cal G} '$ that contain $r$ bonds, then the partition function is

$$Z_N = 2^N (\cosh \nu)^B \sum_{r=0}^{\infty} n(r) \omega^r,$$ (17)

where $n(0)=1$. So the problem has been reduced to finding the generating function of a set of graphs, which are related to bond animals (though they are not restricted to be connected, nor are they defined up to translation). These graphs (see figure ) look like self-avoiding polygons, or groups of overlapping self-avoiding polygons -- in fact, self-avoiding polygons were introduced by Temperley [40] as a special case of these graphs.

Figure 24: Typical graphs that contribute to the correlation function -- some may be self-avoiding walks. The vertices of odd degree are highlighted.

Another quantity of interest is the pair correlation function, $\langle \sigma_{_k}\sigma_{_l} \rangle$, which is defined by

$$\langle \sigma_{_k}\sigma_{_l} \rangle = Z_N^{-1} \sum_{\{\sigma\}} \sigma_{_k}\sigma_{_l} \prod^*_{P,Q} \exp(\nu\sigma_{_P}\sigma_{_Q}).$$ (18)

This function is a measure of the degree of order within the state of the system -- if there is long range order in the system, then separated spins will tend to be in the same state and the pair correlation function will take a value bounded away from 0. Following the same argument used for the partition function, we can rephrase the pair correlation function as the generating function of a new set of graphs in which every vertex is of even degree excepting $k$ and $l$, which must be of odd degree (see figure 24). In the same way that self-avoiding polygons arose in the partition function, we find that self-avoiding walks arise in the graphs contributing to the pair correlation function.

The $N$-vector model discussed above is a generalisation of the Ising model, in which the spins can take one of $N$ states. These $N$ states are equally spaced unit vectors in $\mathbb{R}^N$, and the interaction energy of a configuration $\{\sigma \}$ is defined to be

$$E\{\sigma\} = -J \sum_{P,Q}^* \vec{\sigma_{_P}} \cdot \vec{\sigma_{_Q}},$$ (19)

and so when $N=2$ we arrive back at the Ising model. In the limit $N \rightarrow 0$ the $N$-vector model is equivalent (in some sense) to self-avoiding walks [11].