# Renormalization Group flow near a fixed point

In order to study the behaviour of the RG flow near a fixed point ${\displaystyle {\vec {K}}^{*}}$, let us take a slight perturbation from it, namely we set:

${\displaystyle {\vec {K}}={\vec {K}}^{*}+\delta {\vec {K}}}$
where ${\displaystyle \delta {\vec {K}}}$ is a small displacement. Applying the RG flow, in components we will have:
${\displaystyle K_{j}'=R_{\ell }({\vec {K}}^{*}+\delta {\vec {K}})_{j}=K_{j}^{*}+\sum _{i}{\frac {\partial K_{j}'}{\partial K_{i}}}_{|K^{*}}\delta K_{i}+O(\delta K^{2})}$
Neglecting all the terms beyond the linear ones, we can write the action of the linearised RG transformation in terms of the displacements ${\displaystyle \delta {\vec {K}}}$ and ${\displaystyle \delta {\vec {K}}'}$ as:
${\displaystyle \delta {\vec {K}}'={\boldsymbol {T}}\delta {\vec {K}}\qquad \qquad {\text{where}}\;{\boldsymbol {T}}_{ij}={\frac {\partial K_{j}'}{\partial K_{i}}}_{|K^{*}}}$
Of course ${\displaystyle {\boldsymbol {T}}}$ 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 ${\displaystyle {\boldsymbol {T}}}$ to be symmetric (which, as before, is almost always the case) so that it can be diagonalized. If we call ${\displaystyle \lambda _{\ell }{}^{(\sigma )}}$ and ${\displaystyle {\vec {e}}{\text{ }}^{(\sigma )}}$ the ${\displaystyle \sigma }$-th eigenvalue and relative eigenvector of ${\displaystyle {\boldsymbol {T}}{}^{(\ell )}}$ (where we are explicitly writing the length scale of the decimation), in components the action ${\displaystyle {\boldsymbol {T}}^{(\ell )}}$ will be:
${\displaystyle {\boldsymbol {T}}_{ij}^{(\ell )}e_{j}^{(\sigma )}=\lambda _{\ell }^{(\sigma )}e_{i}^{(\sigma )}}$
From the semigroup property Basic ideas of the Renormalization Group of the RG transformation we have:
${\displaystyle {\boldsymbol {T}}^{(\ell )}{\boldsymbol {T}}^{(\ell ')}={\boldsymbol {T}}^{(\ell \ell ')}}$
and so from  :
${\displaystyle \lambda _{\ell }^{(\sigma )}\lambda _{\ell '}^{(\sigma )}=\lambda _{\ell \ell '}^{(\sigma )}}$
This is a functional equation which can be solved in the following way: if we write the eigenvalues explicitly as functions of ${\displaystyle \ell }$, namely ${\displaystyle \lambda _{\ell }{}^{(\sigma )}=\lambda (\ell ){}^{(\sigma )}}$, then differentiating with respect to ${\displaystyle \ell '}$:
${\displaystyle \lambda ^{(\sigma )}(\ell )\lambda '^{(\sigma )}(\ell ')=\ell \lambda '^{(\sigma )}(\ell \ell ')}$
where with ${\displaystyle \lambda '}$ we mean that ${\displaystyle \lambda }$ has been differentiated with respect to its argument. Setting now ${\displaystyle \ell '=1}$ and defining ${\displaystyle \lambda '{}^{(\sigma )}(1)=y_{\sigma }^{-1}}$ we get:
${\displaystyle {\frac {\lambda '^{(\sigma )}(\ell )}{\lambda (\ell )}}=\ell y_{\sigma }}$
which is easily solved to give:
${\displaystyle \lambda _{\ell }^{(\sigma )}=\ell ^{y_{\sigma }}}$
where, as we have defined it, ${\displaystyle y_{\sigma }}$ is a number (to be determined) independent of ${\displaystyle \ell }$. To see how ${\displaystyle \delta {\vec {K}}}$ changes under the action of ${\displaystyle {\boldsymbol {T}}}$ let us find out how its components along the directions determined by the eigenvectors ${\displaystyle {\vec {e}}{\text{ }}^{(\sigma )}}$ change[1]. In other words, we write:
${\displaystyle \delta {\vec {K}}=\sum _{\sigma }a^{(\sigma )}{\vec {e}}{\text{ }}^{(\sigma )}\quad \qquad a^{(\sigma )}={\vec {e}}{\text{ }}^{(\sigma )}\cdot \delta {\vec {K}}}$
and applying ${\displaystyle {\boldsymbol {T}}{}^{(\ell )}}$:
${\displaystyle \delta {\vec {K}}'={\boldsymbol {T}}\delta {\vec {K}}={\boldsymbol {T}}\sum _{\sigma }a^{(\sigma )}{\vec {e}}{\text{ }}^{(\sigma )}=\sum _{\sigma }a^{(\sigma )}\lambda _{\ell }^{(\sigma )}{\vec {e}}{\text{ }}^{(\sigma )}:=\sum _{\sigma }a'^{(\sigma )}{\vec {e}}{\text{ }}^{(\sigma )}}$
where in the last step we have defined the components ${\textstyle a'^{(\sigma )}}$ of ${\textstyle \delta {\vec {K}}'}$ along ${\textstyle {\vec {e}}{\text{ }}^{(\sigma )}}$. We therefore see that the behaviour of ${\displaystyle \delta {\vec {K}}}$ along the eigenvectors ${\displaystyle {\vec {e}}{\text{ }}^{(\sigma )}}$ depends on the magnitudes of the eigenvalues ${\displaystyle \lambda _{\ell }{}^{(\sigma )}}$. In particular, we can distinguish three cases:

• ${\displaystyle {\boldsymbol {|\lambda _{\ell }^{(\sigma )}|>1}}}$: this implies that ${\displaystyle a'^{(\sigma )}}$ grows
• ${\displaystyle {\boldsymbol {|\lambda _{\ell }^{(\sigma )}|=1}}}$: this implies that ${\displaystyle a'^{(\sigma )}}$ doesn't change (its behaviour can depend on the higher orders in the expansion that we have neglected)
• ${\displaystyle {\boldsymbol {|\lambda _{\ell }^{(\sigma )}|<1}}}$: this implies that ${\displaystyle a'^{(\sigma )}}$ 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 ${\displaystyle {\boldsymbol {T}}}$, 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.

1. Note: since ${\displaystyle {\boldsymbol {T}}{}^{(\ell )}}$ is diagonalizable its eigenvectors are orthonormal.