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Since the coupling constants J and h are dimensionless, the unit of length is set by the lattice spacing (which does not actually appear in the Hamiltonian). For example, the correlation function f(i, j) refers only to site numbers i and j, the "distance" between sites i and j being defined as the number of lattice spacings between the sites along a minimal path. Now divide the lattice into blocks of size b d , and define the block spin s' = 1 of a given block as some sort of average of the spins contained in the block. For example, we can use the "majority rule," s' = 1, if the majority of the spins in the block is . In case of a tie, we arbitrarily define S = Sl' where Sl is a particular spin, say the one at the upper left comer of the block. Denoting by S the sum of all spins in the block, we have

(S> 0) (S < 0) (S = 0)

After one block-spin transformation, the number of spins reduced by a factor b- d :

(18.4)

Note: To view the field, click the Home tab s Show/Hide button ( ).

(18.5)

The new lattice spacing is increased b-fold, which means that the unit of length is automatically increased by a factor b. For example, if the site spins are correlated over n lattice spacings, the block spins are correlated over nib lattice spacings. The process is illustrated in Fig. 18.1.

f----

\... - - (

FIGURE 9.14 When saucer patterns rst break out, they are not always as obvious as they are on a long-term chart.

- - - - ""1

'-- f - - - -

Fig. 18.1 Block-spin transformation: averaging the spins in a block,

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and then rescaling the lattice to the original size. In more than one dimension, the indirect interaction between Band C gives rise to next-to-nearest-neighbor interactions of the block spins.

Kadanoff assumes that sufficiently close to the critical point one can describe the system by a block-spin Hamiltonian, which again contains only nearestneighbor interactions:

5 A 6

E'{s'}

(i,j)

s:s; - h' L s:

FIGURE 9.15 This stock consolidated for almost two years before pushing higher, and it had already tripled in price by the time it broke out.

(18.6)

QN,(h', t') ==

e-N'g(h',t')

{s' }

e-E'{s'}

(18.7)

4 1 7

Note that the free energy g is the same function as in (18.3), because by assumption E' is the same function as E, except for the values of the coupling constants. Kadanoff now argues that the block-spin system is merely another way of describing the original system, and hence we must have QN' = QN' Using (18.5) one can write

has already proven itself with the pattern breakout. As it so often the case, it went on to much higher prices once the pattern completed. The old saying that history repeats itself is true for technical analysis (indeed, it s a signi cant part of the basis for technical analysis). This applies to patterns as well, particularly within the same stock. If you notice a particular stock is prone to a certain pattern (be it a saucer, head and shoulders, or what have you), it is likely you ll see more than one instance of that pattern for that stock. This can be helpful, since as you become acquainted with the price behavior of a particular stock, you can better anticipate its next move. The pattern isn t always the same length each time, as Figure 9.16 shows, but its shape and consequences tend to be similar. The longer the pattern, the more pronounced the consequences. Let s take a look at a few more saucers to complete the picture. As stated earlier, the length of the pattern has a lot to do with how much force comes afterward. Figure 9.17 has a couple of interesting features. First, the pattern is nearly six years long (with the breakout price being $5). Second, another bullish indication takes place during the formation of the pattern: a double bottom in price, where the price hit about $1.00 in the rst half of 1999 and again during the rst half of 2000. This refusal of the market to sell the stock at any lower prices gave buyers the con dence they needed to

g(h, t) = b-dg(h', t')

(18.8)

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