Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 241
Problem 5.6. Thermodynamics of the Ising chain
(a) What is the ground state of the Ising chain?
(b) What is the entropy S in the limits T → 0 and T → ∞? The answers can be found without
doing an explicit calculation.
(c) Use (5.48) for the free energy F to verify the following results for the entropy S, the mean
energy E, and the heat capacity C of the Ising chain:
S = Nk ln(e 2βJ 2βJ
+1)−
1+e −2βJ
, (5.49)
E = −NJ tanhβJ, (5.50)
C = Nk(βJ) 2 (sechβJ) 2 . (5.51)
Verify that the results in (5.49)–(5.51) reduce to the appropriate behavior for low and high
temperatures.
(d) A plot of the T dependence of the heat capacity in the absence of a magnetic field is given in
Figure 5.4. Explain why it has a maximum.
5.5.2 Spin-spin correlation function
We can gain further insight into the properties of the Ising model by calculating the spin-spin
correlation function G(r) defined as
G(r) = sksk+r −sk sk+r. (5.52)
Because the average of sk is independent of the choice of the site k (for toroidal boundary conditions)
and equals m = M/N, G(r) can be written as
G(r) = sksk+r −m 2 . (5.53)
The average is over all microstates. Because all lattice sites are equivalent, G(r) is independent of
thechoiceofk anddependsonlyontheseparationr (foragivenT andH), wherer istheseparation
between the two spins in units of the lattice constant. Note that G(r = 0) = m 2 −m 2 ∝ χ [see
(5.17)].
The spin-spin correlation function G(r) is a measure of the degree to which a spin at one
site is correlated with a spin at another site. If the spins are not correlated, then G(r) = 0. At
high temperatures the interaction between spins is unimportant, and hence the spins are randomly
oriented in the absence of an external magnetic field. Thus in the limit kT ≫ J, we expect that
G(r) → 0 for any r. For fixed T and H, we expect that, if spin k is up, then the two adjacent
spins will have a greater probability of being up than down. For spins further away from spin k,
we expect that the probability that spin k +r is up or correlated will decrease. Hence, we expect
that G(r) → 0 as r → ∞.