# Fixed points of the Renormalization Group flow

Suppose we know ${\displaystyle R_{\ell }}$ (in Renormalization Group in coordinate space we will see explicitly how to construct them). If ${\displaystyle [K^{*}]}$ is a fixed point of the RG flow, by definition we have:

${\displaystyle R_{\ell }[K^{*}]=[K^{*}]}$
Then, in general from the Hamiltonian of a system we can determine the correlation length ${\displaystyle \xi }$, and if ${\displaystyle [K']=R_{\ell }[K]}$ we know that:
${\displaystyle \xi [K']=\xi [K]/\ell }$
Therefore for a fixed point we have:
${\displaystyle \xi [K^{*}]=\xi [K^{*}]/\ell }$
which implies that ${\displaystyle \xi [K^{*}]}$ is either zero or infinity. A fixed point with ${\displaystyle \xi =\infty }$ is called critical, while if ${\displaystyle \xi =0}$trivial. Clearly, every fixed point ${\displaystyle [K^{*}]}$ can have its own basin of attraction, i.e. a set of points that under the action of the RG flow tend to ${\displaystyle [K^{*}]}$.

An important result concerning the basin of attraction of critical fixed points is the following:

Theorem

The correlation length is infinite for every point in the basin of attraction of a critical fixed point of the RG flow.

Proof

Call ${\displaystyle [K]}$ the initial set of coupling constants, after ${\displaystyle n}$ iterations of the RG the correlation length of the system will be such that:

${\displaystyle \xi [K]=\ell \xi [K^{(1)}]=\cdots =\ell ^{n}\xi [K^{(n)}]\quad \Rightarrow \quad \xi [K]=\ell ^{n}\xi [K^{(n)}]}$
If we now take the limit ${\displaystyle n\to \infty }$ the right hand side diverges if ${\displaystyle K^{(n)}\to K^{*}}$, i.e. if ${\displaystyle [K]}$ belongs to the basin of attraction of ${\displaystyle [K^{*}]}$. Therefore, ${\displaystyle \xi [K]=\infty }$.

The basin of attraction of a critical fixed point is also called critical manifold. We can argue that the fact that all the points of a critical manifold flow towards the same fixed point (i.e. the same Hamiltonian) is the basic mechanism on which universality is based upon, but this is by no means a complete explanation, since universality involves the behaviour of systems near a critical point and we still have said nothing about that. We can however note the following fact: starting from any point in theory space, iterating the RG transformation and identifying the fixed point towards which the system flows, the phase of the original point in theory space (i.e. in the phase diagram) will be described by this fixed point. Therefore, every phase of the system is "represented" by a fixed point of the RG flow.

As we will later see (and will become clearer in Global properties of the Renormalization Group flow, Universality in the Renormalization Group and Renormalization Group in coordinate space), critical fixed points describe the singular critical behaviour while trivial fixed points are related to the bulk phases of the system: therefore, the knowledge of the location and nature of the fixed points of the RG flow can give us hints on the structure of the phase diagram of the system, and the behaviour of the flow near critical fixed points allows us to calculate the values of the critical exponents.