In order to study the behaviour of the RG flow near a fixed point , let us take a slight perturbation from it, namely we set:
is a small displacement. Applying the RG flow, in components we will have:
Neglecting all the terms beyond the linear ones, we can write the action of the linearised RG transformation in terms of the displacements
is a square matrix but in general it is not
symmetric, so it is not diagonalizable and its eigenvalues can be complex (and we also must distinguish between left and right eigenvectors).
However we suppose
to be symmetric (which, as before, is almost always the case) so that it can be diagonalized.
If we call
-th eigenvalue and relative eigenvector of
(where we are explicitly writing the length scale of the decimation), in components the action
From the semigroup property Basic ideas of the Renormalization Group
of the RG transformation we have:
and so from :
This is a functional equation which can be solved in the following way: if we write the eigenvalues explicitly as functions of
, then differentiating with respect to
we mean that
has been differentiated with respect to its argument. Setting now
which is easily solved to give:
where, as we have defined it,
is a number (to be determined) independent of
To see how
changes under the action of
let us find out how its components along the directions determined by the eigenvectors
. In other words, we write:
where in the last step we have defined the components
. We therefore see that the behaviour of
along the eigenvectors
depends on the magnitudes of the eigenvalues
. In particular, we can distinguish three cases:
- : this implies that grows
- : this implies that doesn't change (its behaviour can depend on the higher orders in the expansion that we have neglected)
- : this implies that shrinks
These three cases are given, respectively, the following terminology:
- relevant eigenvalues/directions/eigenvectors
- marginal eigenvalues/directions/eigenvectors
- irrelevant eigenvalues/directions/eigenvectors
The number of irrelevant directions of a fixed point is equal to the dimension of its critical manifold, while the number of relevant directions is equal to its codimension.
Let us note that the eigenvalues and their possible relevance depend on the matrix , which in turn depends on the fixed point considered: this means that the terms "relevant", "irrelevant" or "marginal" must always be specified with respect to the particular fixed point considered.
- ↑ Note: since is diagonalizable its eigenvectors are orthonormal.