(Sorry if this has been asked before but I can't find a similar question on here)
My bachelors thesis had some connections to renormalization group theory. I covered some theory briefly and used the sentence "Each fixed point corresponds to one possible phase the system can be in". My professor told me that this was wrong and said something like "fixed points are more related to phase transitions and not phases per se" (I was too shy to ask further questions and don't remember what he said exactly).
I wrote the sentence because I read the following in "Lectures on phase transitions and the renormalization group" (ch. 9.3.3) by Nigel Goldenfeld:
"The global behaviour of RG flows [...] determines the phase diagram of the system. The basic idea is simple: starting from any point in coupling constant space (i.e. in the phase diagram), iterate the RG transformation and identify the fixed point to which the system flows. The state of the system described by this fixed point represents the phase at the original point in the phase diagram."
I interpret this in the way that corresponds to my sentence: Since a RG transformation does not change the large scale behaviour of the system (i.e. if it was in an ordered state before, it will be ordered afterwards and vice versa) and every state flows to some fixed point after repeated RG transformations, each fixed point corresponds to one phase of the system. So why is my sentence wrong afterall?