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