Universality and linearized RG 645RG eigenvaluesytandyh. An analysis of the scaling properties of the singular part
̃g(h,T) of the Gibbs free energyg(h,T), which obeys ̃g(h,T) =b−dg ̃(byhh,byt,T),
leads to the following relations for the primary critical exponents:
α= 2−d
yt, β=d−yh
yt, γ=2 yh−d
yt, δ=yh
d−yh. (16.11.8)
These relations are obtained by differentiating ̃g(h,T) to obtain the heat capacity,
magnetization, and magnetic susceptibility. From eqns. (16.11.8), the relationsα+
2 β+γ = 2 andα+β(1 +δ) = 2, which are examples of scaling relations, can be
easily derived. Two other scaling relations can be derived from the scaling behavior of
the spin-spin correlation functionG ̃(r) =b−2(d−yh)G(r/b,bytt), wheret= (T−Tc)/T.
These areα= 2−dνandγ=ν(2−η). Because of such scaling relations, we do not
need to determine all of the critical exponents individually. For the Ising model, we see
that there are four such scaling relations, indicating that only two of the exponents,
νandη, of the six total are independent. Because a subset of the critical exponents
still needs to be determined by some method, numerical simulations play an important
role in the implementation of the RG, and techniques such as the Wang-Landau and
M(RT)^2 schemes carried out on a lattice are useful approaches that can be employed
(see Problems 16.8 and 16.9).
16.12Understanding universality from the linearized RG theory
In the linearized RG theory, at a fixed point, all scaling variables are zero, regardless
of whether they are relevant, irrelevant, or marginal. Let us assume for the present
discussion that there are no marginal scaling variables. From the definitions of relevant
and irrelevant scaling variables, we can propose a formal procedure for locating fixed
points. Begin with the space spanned by the full set of eigenvectors of T, and project
out the relevant subspace by setting all the relevant scaling variables to zero by hand.
The remaining subspace is spanned by the irrelevant eigenvectors of T, which defines
a hypersurface in the full coupling constant space. This surface iscalled thecritical
hypersurface. Any point on the critical hypersurface belongs to the irrelevant subspace
and iterates to zero under successive RG transformations. This procedure defines a
trajectory on the hypersurface that leads to a fixed point, as illustrated in Fig. 16.18.
This fixed point, called thecritical fixed point, is stable with respect to irrelevant
scaling variables and unstable with respect to relevant scaling variables.
In order to understand the importance of the critical fixed point,consider a simple
model in which there is one relevant and one irrelevant scaling variable. Let these
be denoted asu 1 andu 2 , respectively, and let these variables have corresponding
couplingsK 1 andK 2. In an Ising model,K 1 might represent the reduced nearest-
neighbor coupling, andK 2 might represent a next-nearest-neighbor coupling. Relevant
variables also include experimentally tunable parameters such as temperature and
magnetic field. The reasonu 1 is relevant andu 2 is irrelevant is that there must be
a nearest-neighbor coupling for the existence of a critical point and ordered phase at
h= 0, but magnetization can occur even if there is no next-nearest-neighbor coupling.
According to the procedure of the preceding paragraph, the condition
u 1 (K 1 ,K 2 ) = 0