# Fixed points of the Renormalization Group flow

Suppose we know (in Renormalization Group in coordinate space we will see explicitly how to construct them). If is a fixed point of the RG flow, by definition we have:

*critical*, while if

*trivial*. Clearly, every fixed point can have its own

*basin of attraction*, i.e. a set of points that under the action of the RG flow tend to .

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 the initial set of coupling constants, after iterations of the RG the correlation length of the system will be such that:

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.