634 Critical phenomena
correlation function should depend only onr. Heuristically,G(r) is assumed to decay
exponentially according to
G(r)≡〈σiσj〉−〈σi〉〈σj〉∼
e−r/ξ
rd−2+η(16.8.2)
forT > Tc(Ma, 1976). The quantityξis called thecorrelation length. As a critical
point is approached from above, long-range order sets in, and we expectξto diverge
asT→Tc+. This divergence is characterized by an exponentνsuch that
ξ∼|T−Tc|−ν. (16.8.3)AsT→Tc+,ξ→ ∞, and the exponential numerator inG(r) becomes 1. In this
case,G(r) decays in a manner characteristic of a system with long-range order, i.e.,
as a small inverse power ofr. The exponentηappearing in the expression forG(r)
characterizes this decay atT=Tc.
The exponentsνandηcannot be determined from mean-field theory, as the mean-
field approximation neglects all spatial correlations. In order to calculate these expo-
nents, fluctuations must be restored at some level. One method that treats correlations
explicitly is a field-theoretic approach known as the Landau–Ginzberg theory (Huang,
1963; Ma, 1976). This theory uses a continuous spin field to define afree energy func-
tional and provides a prescription for deriving the spatial correlation functions from
the external field dependence of the partition function via functional differentiation.
Owing to its mathematical complexity, a detailed discussion of this theory is beyond
the scope of this book; instead, in the next section, we will focus onan elegant ap-
proach that is motivated by a few simple physical considerations derived from the
long-range behavior spin-spin correlations.
16.9 Introduction to the renormalization group
Therenormalization group(RG) theory is based on ideas first introduced by L. P.
Kadanoff (1966) and K. G. Wilson (1971) and posits that near a critical point, where
long-range correlations dominate, the system possesses self-similarity at any scale. It
then proposes a series of coarse-graining operations that leave the system invariant,
from which the ordered phases can be correctly identified.^2 The RG framework also
offers an explanation of universality, provides a framework for calculating the critical
exponents (Wilson and Fisher, 1972; Bonanno and Zappal`a, 2001), and through a
hypothesis known as thescaling hypothesis, generates sets of relations calledscaling
relations(Widom, 1965; Cardy, 1996) among the critical exponents. Although we will
only explore here how the RG approach applies to the study of magnetic systems,
the technique is very general and has been employed in problems ranging from fluid
dynamics to quantum chemistry (see, for example, Baer and Head-Gordon (1998)).
(^2) The term “renormalization group” has little to do with grouptheory in the usual mathematical
sense. Although the RG does employ a series of transformations based on the physics of a system
near its critical point, the RG transformations are not unique and do not form a mathematical group.
Hence, references to “the” renormalization group are also misleading, but as this usage has become
the common parlance, we will continue this usage here.