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Enter a Symbol entry box. This is where you can enter any valid symbol that you want. It can be anything chartable an index, a stock, a mutual fund, a commodity, an option just type in the symbol and click Add Symbol, and it will be added to the Displayed Items list. Displayed Items. These are the items which will be compared to the original security charted. There may be anywhere from zero items (normal chart, no comparisons) to multiple items in this box. You can erase everything to start over by clicking Delete All, or if you want to get rid of just one item, click on it and click Delete. Comparison Type. The radio buttons, Percentage Comparison and Price Comparison, control what kind of comparison will be done. Percentage is the default, since that is usually more useful. Figure 6.26 shows that the S&P 100 has been added, as well as the symbol IWM. The list of symbols in the lower portion of the dialog box will appear in alphabetical order, and as you add symbols, you will see the resulting comparison chart instantly in the background. When you are done, you can click the Close button.

(Helmholtz free energy)

+ [Y( 12

+ h zN]

To change your view of tasks in the Tasks list, you can click the Change View button and choose a view option from the menu that appears.

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The energy (14.1) of the Ising model in the absence of external field is invariant when all the spins change sign simultaneously: Sj -> - Sj' For H = 0, therefore. the magnetization as defined by (14.6) is zero, because the contribution of ever:spin configuration {Sj} is canceled by that of { - Sj}' As we have discussed in Section 12.5, however, the correct way to calculate the spontaneous magnetization, which is a case of spontaneous symmetry breaking, is to take the thermodynamic limit of (14.1) in the presence of a arbitrarily small H, and then let H -> 0 Figure 14.4 illustrates the typical behavior of the magnetization for a system exhibiting spontaneous magnetization. The magnetization per unit volume M j I' is plotted as a function of temperature T for a range of values of the external field H. As H -> , MjV approaches one or the other branch of the H = (I curve. It would be physically incorrect to average the two branches (thus getting 0), for reasons discussed in Section 12.5.

Magnetization per unit volume as a function of temperature for various external fields H, typical of a ferromagnet. Below 1;, for H = 0 there is spontaneous symmetry breaking. The mathematical limit that corresponds to physical observations consists of first taking the thermodynamic limit, and then the limit H ---> 0 . For T < 1; this gives either the upper or the lower branch of the curve, but not the average of the two.

We now show that the existence of spontaneous magnetization in the Ising model depends on the dimensionality of the model.

When all spins are aligned, as shown in Fig. 14.5a, the energy is at the absolute minimum, but the entropy is zero. Now create a "domain wall" by flipping all the spins to the right of some site, as shown in Fig. 14.5b. The energy increases by 2f, but the entropy increases by k log (N - 1), since we have N - 1 choices to place the wall. Thus the free energy associated with the creation of one domain wall is

2f - kT log (N - 1)

FIGURE 6.25 When using the Comparison Chart function, choose from a list of Indexes or enter your own symbols for the comparison.

(14.24)

For T> 0 and N ~ 00, the creation of a domain wall lowers the free energy. Hence more will be created until the spins are randomized. Therefore there can be no spontaneous magnetization at T> 0 in one-dimensional models. In an antiferromagnetic model, where f < 0, the analog of spontaneous magnetization is a staggered magnetization, in which the neighboring spins are alternately 1. Again, there can be no staggered magnetization in one-dimensional antiferromagnetic models at T> O. We can see this by mentally separating the antiferromagnetic lattice into two interpenetrating sublattices, one consisting of every other spin, the other consisting of the remaining spins. Existence of staggered magnetization means that each sublattice is spontaneously magnetized. This is not possible for T > 0 because the proof we gave for the ferromagnetic case applies to each sublattice separately. It should be emphasized that the discussion above is valid only for T > O. Phase transitions in one-dimensional systems can occur at absolute zero, and can actually be seen in the laboratory. *

Peierls t has shown that in two dimensions there will be spontaneous magnetization below a finite temperature. We follow the proof given by Griffiths.:j:

'For a more detailed discussion, see R. Birgeneau and G. Shirane, Phys. Today (Dec. 1978). Peierls, Proc. Camb. Phi/os. Soc. 32, 477 (1936). *R. B. Griffiths, Phys. Rev. A, 136, 437 (1964).

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