630 Critical phenomena
an unsolved problem. Should an exact solution emerge, the exponents for thed= 3,
n= 1 universality class would be known.
Graph-theoretic approach: The second approach is a combinatorial one that leads
directly to the partition of the two-dimensional Ising model in the zero-field limit. We
begin by introducing the shorthand notationK=βJ, and again we assume periodic
boundary conditions. The partition function in the zero-field limit canbe written as
Q(N,T) =∑
σ 1 , 1 =± 1∑
σ 2 , 1 =± 1···
∑
σn, 1 =± 1∑
σ 1 , 2 =± 1∑
σ 2 , 2 =± 1···
∑
σn, 2 =± 1··· (16.7.14)
×
∑
σ 1 ,n=± 1∑
σ 2 ,n=± 1···
∑
σn,n=± 1exp
K
∑ni,j=1[σi,jσi+1,j+σi,jσi,j+1]
.
The combinatorial approach starts with an identity derived from the fact that the
product of spinsσi,jσi′,j′ =±1 so that exp[Kσi,jσi′,j′] = exp[±K] = cosh(K)±
sinh(K). From these relations, it follows that
exp [Kσi,jσi′,j′] = cosh(K) +σi,jσi′,j′sinh(K)= cosh(K) [1 +σi,jσi′,j′tanh(K)]= cosh(K) [1 +vσi,jσi′,j′], (16.7.15)wherev= tanh(K). Converting exponentiated sums into products, the partition func-
tion can be written as
Q(N,T) =
∑
{σ}=± 1exp
K
∑
i,j[σi,jσi+1,j+σi,jσi,j+1]
=
∑
{σ}=± 1exp
K
∑
i,jσi,jσi+1,j
exp
K
∑
i,jσi,jσi,j+1
=
∑
{σ}=± 1
∏
i,jeKσi,jσi+1,j
∏
i,jeKσi,jσi,j+1
=
∑
{σ}=± 1
∏
i,jcosh(K) (1 +vσi,jσi+1,j)
∏
i,jcosh(K) (1 +vσi,jσi,j+1)
= [cosh(K)]ν∑
{σ}=± 1∏
i,j[1 +vσi,jσi+1,j] [1 +vσi,jσi,j+1], (16.7.16)whereνis the total number of nearest neighbors on the two-dimensional lattice and
the notation
∑
{σ}=± 1 indicates that all spins are summed over.