MIT OpenCourseWare http://ocw.mit.edu 8.334 Statistical Mechanics II: Statistical Physics of Fields Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Z III.D The Renormalization Group (Conceptual) Success of the scaling theory in correctly predicting various exponent identities strongly supports the assumption that close to the critical point the correlation length ξ, is the only important length scale, and that microscopic length scales are irrelevant. The critical behavior is dominated by fluctuations that are self–similar up to the scale ξ. The self–similarity is of course only statistical, in that a ma gnetization configuration is generated with a weight W [~ m(x)]}.m(x)] ∝ exp{−βH[ ~ Kadanoff suggested taking advan-tage of the self–similarity o f the fluctuations to gradually eliminate the correlated degrees of freedom at length scales x ≪ ξ, until one is left with the relatively simple, uncorrelated degrees o f freedom at scale ξ. This is achieved through a procedure called the renor-malization group (RG), whose conceptual foundation is the three steps outl ined in this section. (1) Coarse Grain: There is an implicit short distance length scale a, for allowed variations of ~m(x) in the system. This is the l attice spacing for a model of spins, or t he coarse graining scale that underlies the Landau–Ginzburg Hamiltonian. In a digital picture of the system, a corresponds to the pixel size. The first step of the RG is to decrease the resolution by changing t his minimum scale to ba (b > 1). The coarse–grained magnetization is then given by Z mi(x) =1 dd x ′ mi(x ′ ). (III.27) bd Cell centered at x (2) Rescale: Due to the change in resolution, the coarse grained ‘picture’ is grainier than the original. The original resolution of a can be restored by decreasing all length scales by a factor o f b, i.e. by setting xold xnew = . (III.28) b (3) Renormalize: The variations of fluctuations in the rescaled magnetization profile is in general different from the original, i.e. there is a difference in contrast between the pictures. This can be remedied by introducing a change of contrast by a factor ζ, through defining a renormalized magnetization ~ ) =1 dd x ′ ~′ ).mnew (xnewm(x (III.29) ζbd Cell centered at bxnew By following these steps, for each configuration ~mold(x), we generate a renormalized configuration ~ (x). Eq.(III.29) can be regarded as a mapping from one set of random mnew vari ables to another, and can be used to construct the probability distribution, or weight 41( ( Wb[~ (x)] mnew (x)]}. Kadanoff’s insight was that since on length scales mnew ≡ exp{−βHb[~less than ξ, the renormalized configurations are statistically similar to the original ones, they may be distributed by a Hamiltonian βHb that is also ‘close’ to the original. In particular, the original Hamiltonian becomes critical by tuning the two parameter t and h to zero, at which point the dominant configurations are similar to those of the rescaled system; the critical Hami ltonian is thus i nvariant under such rescaling. In the original problem, one moves away from criticality for finite t and h. Kadanoff’s assumption is that the corresponding new Hamiltonian is also described by non-zero tnew or hnew. The assumption that t he vicinity of the original and renormalized Hamiltonians to criticality is described by the two parameters t and h greatly simplifies t he analysis. The effect of the RG transformation on the probability of configurations is now described by the two parameter mappings tnew ≡ tb(told, hold) and hnew ≡ hb(told, hold). The next assumption is that since the transformation only involves changes at the shortest length scales, it cannot cause any singularities. The renormalized parameters must be analytic functions of the original ones, and hence expandable as tb(t, h) = A(b)t + B(b)h + · · · . (III.30) hb(t, h) = C(b)t + D(b)h + · · · Note that there are no constant terms in the above Taylor expansions. This expresses the condition that if βH is at its critical p o int (t = h = 0), then βHb is also at criticality, and tnew = hnew = 0. Furthermore, due to rotat ional symmetry, under the combined transfor-mation (m(x) 7→ −m(x), h 7→ −h, t 7→ t) the weight of a configuration is unchanged. As this symmetry i s preserved by the RG, the coefficients B and C in the above expression must be zero, leading to the further simplifications tb(t, h) = A(b)t + · · · . (III.31) hb(t, h) = D(b)h + · · · The remaining coefficients A(b) and D(b) depend on the (arbitrary) rescaling factor b, and trivially A(1) = D(1) = 1 for b = 1. Since the above transformations can be carried out in sequence, and the net effect of rescalings of b1 and b2 is a change of scale by b1b2, the RG procedure is sometimes referred to as a semi-group. The term applies to the action of RG on the space of configurations: each magnetization profile is mapped uniquely to one at larger scale, but the inverse process is non-unique as some short scale information is lost in the coarse graining. (There is in fact no problem with inverting the 42Z Z transformation in the space of the parameters of the Hamiltonian.) The dependence of A and D in eqs.(III.31) on b can be deduced from this g ro up property. Since at b = 1, A = D = 1, and t(b1b2) ≈ A(b1)A(b2)t ≈ A(b1b2)t; we must have A(b) = byt , and similarly D(b) = byh , yielding ( ′ t ≡ tb = byt t + · · · . (III.32) h ′ ≡ hb = byh h + · · · If βHold is slightly away from criticality, it is described by a large but finite correlation length ξold. After the RG transformation, due to the rescaling in eq.(III.28), the new