Relation between Spontaneous Symmetry Breaking and Renormalization Group

Question

I have two different pictures in my head of how a phase transition occurs, but I am not sure of the relation between these two pictures.

1. SSB: Our theory has a global symmetry and when the parameters of theory change, our order parameter acquires a VEV.

2. RG: we can change the parameters of the theory, and when we move to a different basin of attraction, we can flow to different theories (in the IR)—these different fixed points represent the phases.

First of all, is this an accurate description? Second of all, if this is the case, then I do not see the role of symmetry in the RG description. In other words, why should flowing to a different fixed point necessarily come with a broken symmetry?

Answer 1

From the perspective of RG, there are different IR fixed points representing different IR phases of the theory, including the critical point. It is perhaps useful to start from a fixed point that represents the transition itself. There are different directions in the space of couplings, and imagine adding various kinds of perturbations, which takes one out of the fixed point (but still in its neighborhood) in different directions. If it's moving into a direction that represents a relevant coupling, then the theory is going to flow under the RG to a new fixed point. Depending on some details (e.g. actual value of the coupling), this new fixed point may or may not break some symmetry.

So short answer: flowing to a new fixed point does not necessarily mean SSB. Symmetry still constrains what sort of couplings are allowed in the theory, so we are not looking at the parameter space with completely arbitrary couplings, but only those that preserve the symmetry. But the actual dynamics of the theory (i.e. what exactly these new fixed points are) needs to be analyzed on a case-by-case basis, and is generally a difficult question.

Answer 2

The second perspective is more general than the first. There are phases of matter that are characterized by a symmetry-breaking condensate. But there are also phases that do not have anything to do with symmetry breaking, at least not in any straightforward way. The latter define the so-called "beyond Landau" paradigm, and they are a very active area of research. Amusingly, Meng has several brilliant papers on this subject, so do check them out!

So, all in all, the most general perspective is the one you call "RG". The "SSB" perspective only describes a subset of all possible phases of matter.

Answer 3

I'm a bit more used to these concepts in the context of high energy physics, so I might use terminology that is not so familiar, but the concepts are pretty much the same.

Firstly, let me address your descriptions of SSB and RG.

SSB

While the parameters might change, they need not. The point of spontaneous symmetry breaking is "the Lagrangian has a symmetry which the vacuum does not". One way of that happening is with the parameters changing some way, such as when the temperature changes and eventually leads to the parameters assuming values leading to SSB. However, this isn't the only possibility. In the Standard Model at zero temperature, the parameters are fixed in a potential with SSB.

RG

We don't really change the parameters by hand. The point of the renormalization group is to analyze how the theory changes when we zoom out. We start with a description using a lattice with spacing $a$, for example. We may then wonder what would happen if we used a lattice with spacing $2a$. The renormalization group allows us to compute how we must change the parameters of the theory to properly describe the behaviour in this new lattice with larger spacing. Notice that this is essentially the idea of "zooming out": we are losing detail as we look at larger scales. The parameters changing are a consequence, not a cause.

We can then analyze how these parameters change with scale, and eventually find out that in the IR they are attracted to some values, repelled from others, and so on.

Does the RG require a broken symmetry?

Not really. We can perfectly talk about the renormalization group flow of theories which do not possess SSB. For example, we can talk about the renormalization flow of Quantum Electrodynamics, which has an unbroken $U(1)$ gauge symmetry. It still has a non-trivial RG flow, even though it does not undergo SSB.