1549380323-Statistical Mechanics Theory and Molecular Simulation

Linearized RG 643

ξ(K) =ξ 0 bn(K). (16.10.2)

As the starting pointKis chosen closer and closer toKc, the number of iterations
needed to reachK 0 increases. In the limit that the initial pointK→Kc, the number
of iterations is needed to reachK 0 approaches infinity. According to eqn. (16.10.2), as
KapproachesKc, the correlation length becomes infinite as expected in an ordered
phase. Thus, the new unstable fixed point must correspond to a critical point.
From this understanding of the correlation length behavior, we cananalyze the
exponentνusing the RG equation near the unstable fixed point. WhenK=Kc, the
fixed point condition requires thatKc=R(Kc). Near the fixed point, we can expand
the RG equation to give

K′≈R(Kc) + (K−Kc)R′(Kc) +···. (16.10.3)

Let us writeR′(Kc) asblnR
′(Kc)/lnb
and define an exponenty= lnR′(Kc)/lnb. Using
this exponent, eqn. (16.10.3) becomes

K′≈Kc+by(K−Kc). (16.10.4)

Near the critical point,ξdiverges according to

ξ∼|T−Tc|−ν∼

∣

∣

∣

∣

1

K

−

1

Kc

∣

∣

∣

∣

−ν

∼

∣

∣

∣

∣

K−Kc

K

∣

∣

∣

∣

−ν

∼

∣

∣

∣

∣

K−Kc

Kc

∣

∣

∣

∣

−ν

. (16.10.5)

Thus,ξ∼|K−Kc|−ν. However, sinceξ(K) =bξ(K′), it follows that

|K−Kc|−ν∼b|K′−Kc|−ν=b|by(K−Kc)|−ν, (16.10.6)

which is only possible if

ν=

1

y

. (16.10.7)

Eqn. (16.10.7) illustrates the general result that critical exponents are related to deriva-
tives of the RG transformation.

16.11General linearized RG theory

Our discussion in the previous section illustrates the power of the linearized RG
equations. We now generalize this approach to a Hamiltonian Θ 0 with parameters
K 1 ,K 2 ,K 3 ,...,≡K. Eqn. (16.9.6) for the RG transformation can be linearized about
an unstable fixed point atK∗according to

Ka′≈Ka∗+

∑

b

Tab(Kb−K∗b), (16.11.1)

where

Tab=

∂Ra

∂Kb

∣

∣

∣

∣

K=K∗

. (16.11.2)