Two-dimensional Ising model 631Now consider just the first few terms of the product (i,j= 1,2), which contribute
the following factors to the partition function:
(1 +vσ 1 , 1 σ 2 , 1 )(1 +vσ 1 , 2 σ 2 , 2 )(1 +vσ 1 , 1 σ 1 , 2 )(1 +vσ 2 , 1 σ 2 , 2 ).Multiplying this expression out gives
1 +v[σ 1 , 1 σ 2 , 1 +σ 1 , 2 σ 2 , 2 +σ 1 , 1 σ 1 , 2 +σ 2 , 1 σ 2 , 2 ]+v^2 [σ 1 , 1 σ 2 , 1 σ 1 , 2 σ 2 , 2 +σ 1 , 1 σ 2 , 1 σ 1 , 1 σ 1 , 2 +σ 1 , 1 σ 2 , 1 σ 2 , 1 σ 2 , 2
+σ 1 , 2 σ 2 , 2 σ 1 , 1 σ 1 , 2 +σ 1 , 2 σ 2 , 2 σ 2 , 1 σ 2 , 2 +σ 1 , 1 σ 1 , 2 σ 2 , 1 σ 2 , 2 ]+v^3 [σ 1 , 1 σ 2 , 1 σ 1 , 2 σ 2 , 2 σ 1 , 1 σ 1 , 2 +σ 1 , 1 σ 2 , 1 σ 2 , 1 σ 2 , 2 σ 2 , 2 σ 1 , 2
+σ 1 , 1 σ 2 , 1 σ 2 , 1 σ 2 , 2 σ 1 , 1 σ 1 , 2 +σ 1 , 1 σ 1 , 2 σ 1 , 2 σ 2 , 2 σ 2 , 2 σ 2 , 1 ]+v^4 [σ 1 , 1 σ 2 , 1 σ 2 , 1 σ 2 , 2 σ 2 , 2 σ 1 , 2 σ 1 , 2 σ 1 , 1 ]. (16.7.17)As the power ofvincreases, the number of spin factors increases. Thus, in order to
keep track of the “bookkeeping,” we introduce a graphical notation. Each index pair
(i,j) on the spin variablesσi,jcorresponds to a point on the lattice. We identify these
points as the vertices of one or more graphs that can be drawn on the lattice. We
also identify a spin productσi,jσi′,j′with an edge joining the vertices (i,j) and (i′,j′).
Thus, in eqn. (16.7.17), there are four vertices, (1,1), (1,2), (2,1), (2,2), corresponding
to four points on the lattice, and at thenth power ofv,n= 0,...,4, there arenedges
joining the vertices. Fig. 16.11 shows the complete set of graphs corresponding to the
terms in eqn. (16.7.17).
We next ask what each graph contributes to the overall partition function. The
first graph, which contains no edges, obviously contributes exactly 1, and when this is
summed overNspins, each of which can take on two values, we obtain a contribution
of 2N. The graphs that arise from thev^1 term contain a single edge. Consider, for
example, the productσ 1 , 1 σ 2 , 1 , which must be summed overσ 1 , 1 andσ 2 , 1. The spin sum
produces four terms corresponding to (σ 1 , 1 ,σ 2 , 1 ) = (1,1),(− 1 ,1),(1,−1),(− 1 ,−1).
When the spin products are taken, two of these terms will be 1 and the other two−1,
and the sum yields 0 overall. The same is true for each of the remainingspin products
in thev^1 term. After some reflection, we see that all of the terms proportional tov^2
andv^3 also sum to 0. However, thev^4 term, represented by a closed graph in which
each vertex is included in two edges, does not vanish, since each spinvariable appears
twice. The contribution from this single graph (seev^4 term in Fig. 16.11), when the
sum over allNspins is carried out, isv^42 N. Now, on a lattice ofNspins, it is possible
to drawN−1 such graphs containing four vertices. Thus, the total contribution from
graphs containing four vertices is (N−1)v^42 N.
The analysis in the preceding paragraph suggests that the problemof evaluating
the partition function becomes one of counting the numbern(r) of closed graphs that
can be drawn on the lattice containingredges and then summing the result overr,
wherer= 0, 4 , 6 , 8 ,.... Fig. 16.12 shows some examples of graphs that occur when