Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 275
(a) (b)
Figure 5.18: (a) The ground state of N = 5 Ising spins in an external magnetic field H. For
toroidal boundary conditions, the ground state energy is E0 = −5J −5H. (b) The flip of a single
spin of N = 5 Ising spins. The corresponding energy cost is 4J +2H.
(a) (b)
Figure 5.19: Microstates corresponding to two flipped spins of a system of N = 5 spins in one
dimension. In (a) the flipped spins are nearest neighbors and in (b) the flipped spins are not
nearest neighbors.
To understandthe natureofthis classofapproximationswe considerthe one-dimensionalIsing
model at low temperatures. We know that the ground state corresponds to all spins completely
aligned. When we raise the temperature slightly above T = 0, the system can raise its energy by
flipping oneormorespins. Atagiventemperaturewecanconsidertheexcited statescorresponding
to 1, 2,...,f flipped spins. These f spins may be connected or may consist of disconnected
groups. Each number of flipped spins corresponds to a term in the low temperature expansion of
the partition function.
As an example, consider a system of N = 5 spins with toroidal boundary conditions. The
ground state is shown in Figure 5.18(a). The energy cost of flipping a single spin is 4J +2H. A
typical microstate with one flipped spin is shown in Figure 5.18(b). (The energy of interaction of
the flipped spin with its two neighbors changes from −2J to +2J.) Because the flipped spin can
be at N = 5 different sites, we have
where E0 = −5(J +H).
Z = [1+5e −β(4J+2H) ]e −βE0 (f = 1), (5.146)
The next higher energy excitation consists of a pair of flipped spins with one type of contribution
arising from pairs that are nearest neighbors [see Figure 5.19(a)] and the other type arising
from pairs that are not nearest neighbors [see Figure 5.19(b)]. We leave it as an exercise (see
Problem 5.24) to determine the corresponding energies and the number of different ways that this
type of excitation occurs.
∗ Problem 5.24. Low temperature expansion for five spins
(a) Determine the contribution to the partition function corresponding to two flipped spins out of
N = 5. (Use toroidal boundary conditions.)
(b) Enumerate the 2 5 microstates of the N = 5 Ising model in one dimension and classify the
microstates corresponding to the energy of the microstate. Then use your results to find the