1549380323-Statistical Mechanics Theory and Molecular Simulation

644 Critical phenomena

Note that the matrix T is not required to be symmetric. Consequently, we define a
left eigenvalue equation for T according to

∑

a

φiaTab=λiφib (16.11.3)

and ascaling variableuias

ui=

∑

a

φia(Ka−Ka∗). (16.11.4)

The term “scaling variable” arises from the fact thatuitransforms multiplicatively
near a fixed point under the linearized RG flow:

u′i=

∑

a

φia(K′a−Ka∗)

=

∑

a

∑

b

φiaTab(Kb−Kb∗)

=

∑

b

λiφib(Kb−K∗b)

=λiui. (16.11.5)

Suppose the eigenvaluesλiare real. Sinceu′i=λiui,uiwill increase ifλi>1 and
decrease ifλi<1. Redefining the eigenvaluesλias

λi=byi, (16.11.6)

we see that
u′i=byiui. (16.11.7)

By convention, the quantities{yi}are referred to as the RG eigenvalues.
From the discussion in the preceding paragraph, three cases can be identified for
the RG eigenvalues:

1. Ifyi>0, the scaling variableuiis calledrelevantbecause repeated iteration of
   the RG transformation drives it away from its fixed point value atui= 0.


2. Ifyi<0, the scaling variableuiis calledirrelevantbecause repeated iteration of
   the RG transformation drives it toward 0.

3.yi= 0. The scaling variableuiis referred to as marginalbecause we cannot

determine from the linearized RG equations whetheruiwill iterate towards or

away from the fixed point.

Typically, scaling variables are either relevant or irrelevant; marginality is rare. The
number of relevant scaling variables corresponds to the number ofexperimentally
tunable parameters such asPandTin a fluid system orTandhin a magnetic system.
For the former, the relevant variables are called thethermalandmagneticscaling
variables, respectively. The thermal and magnetic scaling variableshave corresponding