where W is the matrix whose elements are W

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JRa(K) ---a/3JK

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(18.40)

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Substituting (18.39) into (18.38), using (18.37), we obtain

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K(n+l) - K*

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*In principle behavior.

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K(n)

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W(K(n) - K*)

(18.41)

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may approach a fixed point, go into a limit cycle, or exhibit ergodic

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Now choose K * as the origin in the coupling-constant space, and introduce new coordinate axes along the directions defined by the left eigenvectors of the matrix W: (18.42) There are of course many different eigenvectors. We suppressed their labeling for simplicity. The vector K(n) can be represented by the coordinates

urn)

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L<P,,(K(n) - K*)"

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(18.43)

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which are called "scaling fields." Their usefulness lies in the fact that they do not mix with one another under the RG transformation, as the following calculation shows:

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(18.44)

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Under the RG transformation, v is said to "scale" with a factor A. It increases if A > 1, and decreases if A < 1. Since the RG transformation increases the unit of length by a factor b, we expect A to have the form

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(18.45)

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where y is the dimension of u. In the neighborhood of a fixed point, it is convenient to use the scaling fields {v 1 ' v2 ' } as independent variables, replacing the coupling constants { K l' K 2' ... }. The fixed point corresponds to all v = O. We can rewrite (18.35) in the form

(18.46)

There being no reason for I.t to be singular at the fixed point, we shall assume it is regular. We identify the fixed point with a critical point, and identify the second term in (18.46) as the singular part of the free energy. It satisfies the homogeneity rule

(18.47)

A scaling field is called "irrelevant" if A < 1, because it tends to 0 under repeated coarse-graining. In the neighborhood of a critical point the system behaves as if it had never existed. It is called "relevant" if A > 1, for any nonzero initial value will be magnified under coarse-graining. To be at the critical point, we have to specially set it to zero. The case A = 1 is called "marginal," which we shall not consider, for it depends on the details of the system. In the coupling-constant space (K space), the fixed point lies on a hypersurface, called the "critical surface," defined by Vi = 0 for all the relevant scaling fields Vi. A point on the critical surface will approach the fixed point under successive RG transformations, while a point not on the surface will eventually veer away from the fixed point, as illustrated schematically in Fig. 18.4. Since

RENORMALIZATION GROUP

- - - - - - - K2

Trajectories under RG transformation

The critical surface for a particular fixed point. It is a hypersurface in couplingconstant space obtained by setting all relevant variables to zero. Points on this surface correspond to systems in the same universality class, with the same critical exponents.

Fig. 18.4

each point in K space represents a physical system, the critical surface contains different systems belonging to a universality class, sharing the same critical properties. To illustrate how the critical exponents can be obtained from the eigenvalues A, let us specialize to a familiar case by assuming that there are two relevant fields, VI and v2 , identified respectively with field h and temperature t:

= h,

= b Dh =

v2 = t,

(18.48)

The behavior of the correlation length ~ at h = 0 can be deduced as follows. Under an RG transformation the unit of length increases by a factor b. Hence ~' = ~/b. By definition t' = fjD,t. Hence ~D,t is invariant under the RG, i.e., ~D,t - 1. Therefore 1 . p=(18.49) .. Dr a result we had assumed earlier in (16.56). The argument above can be rephrased more physically. Under successive block-spin transformations, there will be come a point when the correlation length is equal to the size of a block, so that nearest-neighbor blocks are uncorrelated. This point must correspond to a definite to, which cannot depend