General linearized RG theory

The above discussion illustrates the power of the linearized RG equations. We now generalize this approach to a general Hamiltonian with parameters . The RG equation

can be linearized about an unstable fixed point at according to

where

The matrix T need not be a symmetric matrix. Given this, we define a left eigenvalue equation for T according to

where the eigenvalues can be assumed to be real (although it cannot be proved). Finally, define a scaling variable, by

They are called scaling variables because they transform multiplicatively near a fixed point under the linearized RG flow:

Since scales with , it will increase if and will decrease if . Redefining the eigenvalues according to

we see that

By convention, the quantities are called the RG eigenvalues. These will soon be shown to determine the scaling relations among the critical exponents.

For the RG eigenvalues, three cases can be identified:

1.

. The scaling variable is called a relevant variable. Repeated RG transformations will drive it away from its fixed point value, .

2.

. The scaling variable is called an irrelevant variable. Repeated RG transformations will drive it toward 0.

3.

. The scaling variable is called a marginal variable. We cannot tell from the linearized RG equations if will iterate toward or away from the fixed point.

Typically, scaling variables are either relevant or irrelevant. Marginality is rare and will not be considered here. The number of relevant scaling variables corresponds to the number of experimentally `tunable' parameters or `knobs' (such as T and h in the magnetic system, or P and T in a fluid system; in the case of the former, the relevant variables are called the thermal and magnetic scaling variables, respectively).