Statistical Mechanics - Physics at Oregon State University

9.7. RENORMALIZATION GROUP THEORY. 219
Z(T, N, h) =
{σ1,···,σN }
N
N
βJ
e i=1 σiσi+1+βh i=1 σi (9.144)
The transformation we have in mind is a decrease of the magnification by
a factor of two. Hence we plan to combine spins 2j and 2j + 1 into a single
average spin with value σ ′ j = σ2j. We also combine βJ into one symbol ˜ J and
similarly write ˜ h = βh. For simplicity we assume that N is even. Of course, in
the end we need to take the limit N → ∞, and the fact that N is even or odd
does not play a role. The parameters in the partition function are therefore ˜ J,
˜h, and N. Our goal is to write the partition function in the form
Z(N, ˜ J, ˜ h) =
σ ′ 1 ,···,σ′ N 2
e E′ (σ ′
1 ,···,σ′ N )
2
(9.145)
This can be accomplished very easily by separating out the odd and even values
of i in the original formula of the partition function. Using the periodic boundary
conditions to give the original Hamiltonian a symmetrical form we find
Z(N, ˜ J, ˜ h) =
σ2,σ4,··· σ1,σ3,···
N
N
βJ
e i=1 σiσi+1+βh i=1 σi (9.146)
We can think about this in terms bonds between spins. If we consider the spins
with even indices to be the ”master” spin, the spins with odd indices represent
bonds between these master spins. Summation over all possible odd spin states
is equivalent to a summation over all possible bonds between master spins.
Next, we rewrite the summation in the exponent as a summation over odd
spin indices only. That is easy, and we have
N
˜J σiσi+1 + ˜ N
h σi =
i=1
which gives
i=1
Z(N, ˜ J, ˜ h) =
i odd
σ2,σ4,··· σ1,σ3,··· i odd
˜Jσi(σi−1 + σi+1) + ˜ h(σi + 1
2 (σi−1
+ σi+1))
e ˜ Jσi(σi−1+σi+1)+ ˜ h(σi+ 1
2 (σi−1+σi+1))
(9.147)
(9.148)
Now we perform the summation over the variables with odd indices. Each such
sum occurs only in one factor of the product of exponents, and hence these sums
can be done term by term. We arrive at
or
Z(N, ˜ J, ˜ h) =
σ2,σ4,··· i odd σ
e ˜ Jσ(σi−1+σi+1)+ ˜ h(σ+ 1
2 (σi−1+σi+1))
(9.149)