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where Sd(k) is the surface area of a d-sphere of radius k. The arbitrary constant A is called the renormalization point. Changing it only leads to a finite additive constant to the free energy density, without affecting the infrared singularities of the integral in the limit t ---> O. This explains why all dimensions are canonical in the Gaussian model. In more complicated models A could be entangled with the infrared singularity at t = 0 and lead to anomalous dimensions. We can of course obtain the critical exponents by direct calculation of the thermodynamic functions, as was done in Section 17.7, except that there are no divergent constants anywhere.
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We now discuss m 4 theory, the simplest nontrivial case in Landau theory*:
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(18.78) Renormallzatlon-Group Equations
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We shall integrate out an infinitesimally thin shell of k values in momentum space, effectively lowering the cutoff to Alb (b "'" 1). In a large but finite volume V, the number of states in the shell is
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N = ~VSAA) ~k
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where SAA) is the surface area of a d-sphere of radius A. We write m(x) = m(x) + 8m(x), where 8m(x) contains only Fourier components to be integrated out, and m(x) is the average order parameter, a little bit more coarse-grained than m(x). Instead of Fourier analyzing cSm(x) we expand it in a set of N real wave packets <Pl(X), ... , <PN(X), which are formed by superposing plane waves whose k values lie in the thin shell, and their centers are so chosen that they cover the whole space of the system. They are rather extended objects with dimension ~x - l/Ak, containing waves of extremely short wavelengths (-I/A), and have the following properties:
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f (dx ) [ <Pi (x)]
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j ( dx ) [ <Pi ( X)]
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Only approximate orthogonality is required, to avoid having long tails in the wave packets. In view of the qualitative nature of subsequent calculations, there is no point in specifying these states in greater detail. Thus, in summary,
+ 8m(x)
8m(x) =
L Ci<Pi(X)
To carry out step 1 in the RG recipe, substitute (18.81) into (18.78) and integrate out the terms containing 8 m( x). We use (18.80) plus the following
See K. G. Wilson and J. Kogut, Phys. Rep. C 12,75 (1974), for detailed review and original literature. The qualitative treatment given here follows K. G. Wilson, Rev. Mod. Phys. 55, 583 (1983).
( a)
m(x) is assumed to be constant over a wave packet.
(b) j(dx)1 'Vcf>;1 2 == A 2
( c) (8 m ) 4 terms are neglected. Some estimates are given below:
j(dxHI'V [m(x) + 8m(x)]
== j(dx)I'Vm(x) 1 + N
L c;
(18.82) (18.83)
j(dx)[m(x) + 8m(x)]2 == j(dx)m 2(x) + j(dx)[m(x) + 8m(x)]4 == j(dx)m 4(x) +
L c;
6L c;m 2(xJ
where m(x;) denotes m(x) at the center of the ith wave packet. Thus
E[m] +
L [HA2+ '0) + 6u om2(x;)]c;
The RG-transformed Hamiltonian E' is defined by
eE'[m l
TI j
dc; e-E[m l
= e-E[mlTI /7T/2
[N +'0 + 12u om2 (xJl- 1/ 2
which gives
E[m] +
~ ;~1 IOg{(2A f 2 7T
)[1 +
'02 + 12~Om2(xJ]}
Note that in this approximation the coefficient of I 'V m12 is not changed by coarse-graining. This immediately tells us that the rescaling of m (x) will be dictated by naive dimensional analysis, as in the Gaussian model. The same considerations as we went through in the Gaussian model will yield a canonical value for the dimension of the external field: d+2 D h = -2(18.87) Returning to (18.86), we can replace the sum there by an integral, by noting (18.79):
SAA) b.k j(dx)
CdAd(log b) j(dx)
Expanding the logarithm in (18.86), keeping only terms up to second order in ro and u o, and dropping an irrelevant constant, we obtain
Uo rouo ) 36u6 ] E'[m]=E[m] + Cd Ad log b (dx) 6 N- A4 m 2 ( x ) - 7 m4 (x)
(18.89) Using (18.78), we rewrite this in the form
f [(