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Consider now a > 0. In this region m = is a minimum. The question is whether it is a lowest minimum. If a > and b > 0, then we see from (17.63) that m is complex. Hence the only minimum there is m = 0. Combined with the previous conclusion, we see that the positive b axis is a line of second-order transition, as shown in Fig. 17.3. If a > and b < 0, then there are three minima, i.e., m = and m = m +. Since t/J(O) = 0, the locus of points in the ab plane where the three minima are equal is determined by the condition t/J(m+) = 0, which leads to

2 3 4

(17.66)

-4/ea/3

(17.67)

FIGURE 9.5 The bene ts after a breakout can last decades. This shows 3M Corporation (symbol MMM) moving thousands of percent higher after a handsome breakout.

(17.68)

The conclusions we have reached are summarized in the phase diagram in the ab plane, in Fig. 17.3. The point a = b = marks the termination of a

Line of first-order transitions b = -4(ca/3) 1/2

(3alc) 1/4

Fig. 17.3 Phase diagram of the m model, in which a is the coefficient of m 2 , and b is the coefficient of m 4 in the Landau free energy. The various il1$ets show the forms of the Landau free energy in the particular regions.

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second-order phase transitlon, and the ordinary critical point of a first-order transition. Landau calls it a "critical point of the second-order transition." We adopt the common designation "tricritical point" suggested by Griffiths. * If we go up sufficiently far along the positive b axis, we should not be aware of the tricritical behavior and the system should undergo a second-order transition of the "classical" type. On the other hand, we expect the "classical" second-order behavior to be overshadowed by tricritical behavior near the origin.

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*R. B. Griffiths, Phys. Rev. Lett. 24, 715 (1970).

We therefore expect a "crossover" region in the quadrant a < 0, b > 0, which may be determined as follows: The root m is determined by the condition

In retrospect, of course, a person buying 3M in 1986 would indeed be getting it cheap. Because once the stock has moved up thousands of percent, whether you paid $1.25 a share or $1.10 a share doesn t make the least bit of difference. But at the time, few people want to pay what they see as top dollar for a stock and they prefer to wait until the stock gets cheap again. The fact is, sometimes that second opportunity will never come along. This, again, is where pullbacks represent a possible entry point for newcomers to a stock. There is absolutely no guarantee that a stock breaking above its base will provide such an opportunity. The momentum may simply carry it skyward, giving no one a second chance to get in cheaply. But if you have a lower tolerance for risk, you may want to acquire only during a pullback. Using 3M again as an example (Figure 9.6), the pullback took over a year to take place (and it was only because there was a major market crash to provide such a pullback), but the arrow illustrates where more cautious buyers could have acquired the stock at a relatively bargain price as weaker holders of the stock sold it off to prebreakout levels.

(17.69)

bm a,

(17.70)

(17.71)

(lal/c)1/2

(17.72)

This is indicated in Fig. 17.3 by a fuzzy line. The order parameter m as a function of a and b is shown in the threedimensional plot of Fig. 17.4. Note that since tf(m) is an even function of m, the solution - m is equally good (just as in the Ising model). We arbitrary choose ;n> o.

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./ ./

FIGURE 9.6 This pullback erased most of the gains enjoyed since the breakout, but it did not violate the pattern.

./ ./

Fig. 17.4 Equation of state of the m 6 model, showing the order parameter as a function of the parameters a and b.

To derive the critical exponents at the tricritical point, restore the external field H and approach the origin of the ab plane along b = 0, so that a becomes proportional to t:

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