# Does every Renormalization Group fixed point correspond to a phase of the system?

(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?

• Somewhat similar: physics.stackexchange.com/q/649670 Nov 1, 2021 at 18:30
• My interpretation of the sentence in question and your prof's response is that the statement was not precise enough, causing some confusion: Goldenfeld is saying the stable fixed points ("bulk fixed points") represent phases of the system, whereas it sounds to me like your prof was interpreting your sentence to be specifically about the critical (fixed) points that form the critical manifold, which could be thought of as separating the phases. Nov 2, 2021 at 15:04
• Worth also pointing out that, similar to a point made in jesseylin's comment below, the stable fixed points aren't necessarily distinct phases: in the Ising model the fixed points at $h = \pm \infty$ and finite $T$ would both represent an ordered phase, for example (related by symmetry under inverting the spins and $h$). Nov 2, 2021 at 15:31

At the very least, unstable fixed points do not correspond to physical phases in the sense you describe (a la Goldenfeld's interpretation), because no point in the coupling constant space renormalizes to the unstable fixed point. For example, a $$\phi^4$$ theory in $$d < 4$$ is never in the "free field" phase corresponding to the Gaussian fixed point except when the $$\phi^4$$ interaction is turned off. To be more precise about fixed points and phases, perhaps one could discuss basins of attraction.
• Yes, by unstable I mean the RG flow takes the system away from the fixed point. I suppose depending on what you mean by phase you may interpret these as fixed points, but generally I would say determining the phases of the system requires more thought. For example, RG of the 1D Ising model in $(T,h)$ space yields two fixed points + a line of fixed points at $T=\infty$, but in general one would want to say that the 1D Ising model has one phase (paramagnetism). Here the unstable fixed point $T=0, h=0$ is not a critical point, and one would also hesitate to call it a phase. Nov 2, 2021 at 2:07