where p = d - 2 + 1/. In the presence of an external field H the above generalized to

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(16.40)

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where y is some constant. This may be justified physically as follows: The magnetic moments in the system are strongly correlated within a correlation length, which tends to infinity as t ~ o. Thus, there is a natural tendency for very large blocks of magnetic moments to line up. The effect of an external field is thereby magnified, with a magnification factor proportional to some power of the correlation length. Hence we expect H to occur only in the combination He. Making the replacement

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SPECIAL TOPICS IN STATISTICAL MECHANICS

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This is called a "scaling form" of the correlation function. The quantity ~ is sometimes called the "gap exponent," because (16.41) gains a factor t a when integrated with respect to H. It is instructive to derive the scaling laws again from the scaling forms. By the fluctuation-dissipation theorem and (16.39), we have

By changing the variable of integration to

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which yields y = v(2 - 1/), Fisher's scaling law. In a similar manner, when H = 0, we use the fluctuation-dissipation theorem together with (16.41) to deduce

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Integrating this with respect to H gives the scaling form of the equation of state:

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which, for H

0, says M - Itl a - y. Therefore the gap exponent is given by

~={3+y

(16.45)

Integrating (16.44) with respect to H yields the scaling form of the Gibbs free energy:

G - Itl2t.-y~ (H/ltla)

(16.46)

At H

0, we have G - ItI 2a -r, and hence C - ItI 2a - y- 2 - ItI 2/1+y-2

(16.47)

which gives a + 2{3 + y = 2, Rushbrooke's law. At t = and H =1= 0, the order parameter is assumed to be finite. Therefore the function .A in (16.44) must have the following behavior:

.A (H/ltla)_IW+a-2 = Itl-/1

1--+0

(16.48)

We have used the relation 2 - a - ~ = {3, obtainable from (16.45), and the Rushbrooke scaling law. Using (16.44), we now find

M_l t l/1(H/l t l a)/1/ a = H/1/a

1--+0

(16.49)

CRITICAL PHENOMENA

which says [) = 11/f3. Use of (16.45) then leads to the Widom scaling law y=f3([)-l). Finally, to derive the Josephson scaling law from the scaling form, we have the to make an extra assumption known as "hyperscaling," that at H = amount of free energy residing in a spatial volume of linear size ~ is of the order of kT. This is consistent with the idea that ~ is the only length scale, so that there can be no fluctuations with wavelengths shorter than ~. Accordingly the total free energy is of the order of kTV/~d. As t ~ we have

(16.50) Comparison with (16.46) at H = gives vd = vd = 2 - lX, the Josephson scaling law.

- "I, which by (16.45) leads to

Wldom's Scaling Form

If there are conjugate-variable pairs in the theory other than M and H, say ep; and J;, then the scaling form of the free energy may be generalized to the following:

- It I

2-arg

ItI118'~' Itla2'

(16.51)

which is known as Widom's scaling form. The fields H, J1 , J2 , . are called scaling fields. The associated exponents 11,11 1 ,11 2 , , called "crossover exponents," control the relative importance of the fields near t = 0. For example, if 11; < 0, then the dependence on 1. drops out near t = 0, and the field is said to be "irrelevant"; if 11; > 0, the field .l; is "relevant"; while if 11; = 0, we would have a "marginal" case. We should bear in mind that the scaling form above specifically refers to the neighborhood of a particular critical point. A system may have more than one critical point, and a form like (16.51) is supposed to hold near each of them, with different sets of crossover exponents.

16.5 SCALE INVARIANCE

The scaling hypothesis states that ~ is the only characteristic length of a system in the neighborhood of t = O. When combined with the experimental observation that ~ diverges at t = 0, it leads to the conclusion that the system has no characteristic length, and is therefore invariant under scale transformations.