632 Critical phenomena
(1,1) (1,2)(2,1) (2,2)- v
(1,1) (1,2)(2,1) (2,2)(
(1,1) (1,2)(2,1) (2,2)(1,1) (1,2)(2,1) (2,2)(1,1) (1,2)(2,1) (2,2)(
- v^2
(
(1,1) (1,2)(2,1) (2,2)(1,1) (1,2)(2,1) (2,2)(1,1) (1,2)(2,1) (2,2)(1,1) (1,2)(2,1) (2,2)(1,1) (1,2)(2,1) (2,2)(1,1) (1,2)(2,1) (2,2)(
v^3(
(1,1) (1,2)(2,1) (2,2)(1,1) (1,2)(2,1) (2,2)(1,1) (1,2)(2,1) (2,2)(1,1) (1,2)(2,1) (2,2)(
- v
4(
(1,1) (1,2)(2,1) (2,2)(
Fig. 16.11 Graphical representation of eqn. (16.7.17).r= 6 and whenr= 8. Oncen(r) is known, the partition function can be shown to
take the following form:
Q(N,T) = 2N[cosh(K)]ν∑
rn(r)vr. (16.7.18)We conclude this brief discussion by illustrating how eqn. (16.7.18) canbe applied
to the one-dimensional Ising model on a periodic lattice. The partition function was
given by eqn. (16.6.10), which forh= 0 becomes
Q(N,T) = 2N
[
coshN(K) + sinhN(K)]
. (16.7.19)
We now show that the graph-theoretic approach can be used to derive eqn. (16.7.19).
First, note that on a one-dimensional periodic lattice, only two graphs contribute: the
graph withr= 0 and the graph withr=N, which is the only closed graph that can
be drawn forr >0,i.e., one that involves each vertex in two edges. Thus,n(N) = 1,