1549380323-Statistical Mechanics Theory and Molecular Simulation

Two-dimensional Ising model 627

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1 2 3 n n + 1

1

2

3

n

n + 1

n 1

n 1

Fig. 16.9Two-dimensional Ising system subject to periodic boundaryconditions.

∆(N,h,T) =

∑

σ 1 , 1 =± 1

···

∑

σn, 1 =± 1

∑

σ 1 , 2 =± 1

···

∑

σn, 2 =± 1

···

∑

σ 1 ,n=± 1

···

∑

σn,n=± 1

exp







βJ

∑n

i,j=1

[σi,jσi+1,j+σi,jσi,j+1] +βh

∑n

i,j=1

σi,j







. (16.7.2)

As there areN=n^2 spin sums, each having two terms, the total number of terms
represented by the spin sums is 2N= 2n

2
.
The form of the Hamiltonian and partition function in eqns. (16.7.1) and (16.7.2)
suggests that a matrix multiplication analogous to eqn. (16.6.7) involves entire columns
of spins and that the elements of the transfer matrix should be determined by the
columns rather than by the single spins of the one-dimensional case. (Note that we
could also have used rows of spins and written eqn. (16.7.2) “row-wise” rather than
“column-wise.”) Let us we define a full column of spins by a variableμj

μj={σ 1 ,j,σ 2 ,j,...,σn,j}. (16.7.3)

The Hamiltonian can then be conveniently represented in terms of interactions between
full columns of spins. We first introduce the two functions