Is RG fixed point always related to a second-order phase transition?

Question

In practice, usually one of the parameters is tuned (for example temperature in 3D Ising model, which is a relevant parameter) so that it coincides with the value of RG fixed point, then RG flow make sure the remaining irrelevant parameter flows to the fixed point.

Now suppose we have a theory where we have solved the fixed point exactly. We tune the relevant parameters and let the RG bring the system to the fixed point. Would that necessarily be a second-order phase transition point?

Related: Does every Renormalization Group fixed point correspond to a phase of the system? — Anyon, Mar 22 '24 at 13:39
I'm not an expert but I thought fixed points correspond to scale invariant theories. — lcv, Mar 24 '24 at 11:59
My hunch was correct and corresponds to this answer here https://physics.stackexchange.com/a/542238/1619 . In that sense I would say the answer is essentially yes — lcv, Mar 24 '24 at 12:02
Critical points correspond to scale invariant theories. There are fixed points that represent bulk phases (correlation length is not infinite), which are typically sinks in the RG flow. — bbrink, Mar 24 '24 at 14:14

score 0 · Answer 1 · answered Mar 24 '24 at 08:25

For starters, fixed points need not to be critical points, they can also correspond to phases. For example, in the Ising model case there are fixed points at $T=\infty$ and $T=0$, corresponding to the disordered and ordered phases respectively. You can also have a critical pahse like a BKT phase, where each temperature within the phase is a fixed point in RG, so here they don't correspond to a critical point, but a critical phase.

Is RG fixed point always related to a second-order phase transition?

1 Answers1