Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 279
s 1
Figure 5.21: The simplest cluster on the square lattice used in the Bethe approximation. The
interaction of the central spin with its q = 4 nearest neighbors is treated exactly.
For a square lattice q = 4. Note that the fluctuating field acting on the nearest neighbor spins
s1,...,sq has been replaced by the effective field Heff.
The cluster partition function Zc is given by
Zc =
s 4
s 0
s 2
s0=±1,sj=±1
We first do the sum over s0 = ±1 using (5.157b) and write
βH
Zc = e
sj=±1
s 3
e β(J+Heff)( q j=1 sj)
−βH
+e
e −βHc . (5.158)
sj=±1
e β(−J+Heff)( q
j=1 sj) . (5.159)
For simplicity, we will evaluate the partition function of the cluster for the one-dimensional Ising
model for which q = 2. Because the two neighboring cluster spins can take the values ↑↑, ↑↓, ↓↑,
and ↓↓, the sums in (5.159) yield
Zc = e βH e 2β(J+Heff) +2+e −2β(J+Heff)
+e −βH e 2β(−J+Heff) +2+e −2β(−J+Heff)
(5.160a)
= 4 e βH cosh 2 β(J +Heff)+e −βH cosh 2 β(J −Heff) . (5.160b)
The expectation value of the central spin is given by
〈s0〉 = 1 ∂lnZc 4 βH 2
= e cosh β(J +Heff)−e
β ∂H Zc
−βH cosh 2 β(J −Heff) . (5.161)
In the following we will set H = 0 to find the critical temperature.
We also want to calculate the expectation value of the spin of the nearest neighbors 〈sj〉 for
j = 1,...,q. Because the system is translationally invariant, we require that 〈s0〉 = 〈sj〉 and find
the effective field Heff by requiring that this condition be satisfied. From (5.159) we see that
〈sj〉 = 1
q
∂lnZc
. (5.162)
∂(βHeff)