Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 277
where q is the number of nearest neighbor sites of a given site; q = 2 for one dimension and q = 4
for a square lattice.
To make this procedure explicit, consider an Ising chain with toroidal boundary conditions
for N = 3. For this case p = 3(2)/2 = 3, and there are three factors in the product in (5.151):
(1+vs1s2)(1+vs2s3)(1+vs3s1). We can expand this product in powers of v to obtain the 2 p = 8
terms in the partition function:
ZN=3 = (coshβJ) 3
1
1
1
s1=−1 s2=−1 s3=−1
1+v(s1s2 +s2s3 +s3s1)
+v 2 (s1s2s2s3 +s1s2s3s1 +s2s3s3s1)+v 3 s1s2s2s3s3s1
. (5.153)
It is convenient to introduce a one-to-one correspondence between each of the eight terms in
the brackets in (5.153) and a diagram. The set of eight diagrams is shown in Figure 5.20. Because
v enters into the product in (5.153) as vsisj, a diagram of order v n has n v-bonds. We can use the
topology of the diagrams to help us to keep track of the terms in (5.153). The term of order v 0 is
2 N=3 = 8. Because
si=±1 si = 0, each of the terms of order v vanishes. Similarly, each of the
three terms of order v 2 contains at least one of the spin variables raised to an odd power so that
these terms also vanish. For example, s1s2s2s3 = s1s3, and both s1 and s3 enter to first-order. In
general, we have
1
si=−1
si n =
2 n even,
0 n odd.
From (5.154) we see that only terms of order v 0 and v 3 contribute so that
(5.154)
ZN=3 = cosh 3 βJ[8+8v 3 ] = 2 3 (cosh 3 βJ +sinh 3 βJ). (5.155)
We can now generalize our analysis to arbitrary N. We have observed that the diagrams that
correspond to nonvanishing terms in Z are those that have an even number of bonds from each
vertex; these diagrams are called closed. A bond from site i corresponds to a product of the form
sisj. An even number of bonds from site i implies that si raised to an even power enters into
the sum in (5.151). Hence, only diagrams with an even number of bonds from each vertex yield a
nonzero contribution to ZN.
For the Ising chain only two bonds can come from a given site. Hence, although there are
2 N diagrams for a Ising chain of N spins with toroidal boundary conditions, only the diagrams of
order v 0 (with no bonds) and of order v N contribute to ZN. We conclude that
ZN = (coshβJ) N [2 N +2 N v N ]. (5.156)
Problem 5.26. The form of ZN in (5.156) is not the same as the form of ZN given in (5.39). Use
the fact that v < 1 and take the thermodynamic limit N → ∞ to show the equivalence of the two
results for ZN.
Problem 5.27. High temperature expansion for four spins
Draw the diagrams that correspond to the nonvanishing terms in the high temperature expansion
of the partition function for the N = 4 Ising chain.