# Is my interpretation of the fixed points of the renormalization group correct?

I would like to know whether or not I understood the meaning of the fixed points of the RG concerning the phase diagram of a system. This is how I understand it:

Since a RG transformation leaves the partition function unchanged, such a transformation does not change the thermodynamic behaviour of the system. Then, since all points in coupling-space approach a fixed point under repeated RG transformations, we only need to know the thermodynamic behaviour at the fixed points to know all possible "behaviours". Therefore, every fixed point corresponds to a possible phase-transition point of the system (I have read that fixed points with correlation length $$\xi = 0$$ correspond to bulk phases, and those with $$\xi = \infty$$ correspond to a unique phase at a critical point).

The phenomenon you mention of all points in coupling space approaching a fixed point is a good rule of thumb but there are exceptions because RG flows can have limit cycles. Assuming we are not in this exotic situation, which fixed point we reach can be determined by considering the unstable directions which are associated with the so called relevant operators. The typical example, present in most statistical systems, is the reduced temperature. If $$T > T_c$$, RG steps make the effective temperature rise to a larger value. Experimentally, this corresponds to "losing sight" of domains and their typical size as you zoom out leading to a $$\xi = 0$$ phase as you mention. However, when $$T = T_c$$, RG steps will leave you there. The diminishing effect of irrelevant operators means that the system's behaviour will stabilize onto something non-trivially scale invariant as you zoom out. This has $$\xi = \infty$$ meaning there are clusters of all sizes and this is what we call a critical point.