Can somebody explain to me what the fixed points of a renormalization group mean? What is their physical significance in the sense that why do we study them and what do we get to know from them?

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    $\begingroup$ If you have a graduate student level in physics, then the review by Shankar is the best starting point. Here is arXiv version (it was published in Rev. Mod. Phys.) arxiv.org/abs/cond-mat/9307009 $\endgroup$ – Vadim Apr 7 '20 at 16:01

The RG describes how a QFT changes under an overall change of scale -- loosely speaking, as we "zoom in" or "zoom out". Fixed points of the RG are scale-invariant QFTs: they look the same at all scales. If you start with some generic QFT and follow the RG flow to its low- or high-resolution limit, the usual expectation is that it approaches a scale-invariant QFT (if the limit is defined at all).

Many different QFTs may flow to the same fixed point, so we can think of the scale-invariant QFTs as unifying landmarks in the landscape of all QFTs. This is what universality is about. (This is also closely related to why renormalizable QFTs are often sufficient in applications. Roughly speaking, at low enough resolution, any QFT looks the same as a renormalizable one, even if we don't go all the way to the fixed point.) Knowing what scale-invariant QFTs lie at the low- and high-resolution endpoints of a given RG flow can tell us a lot about the QFTs that lie along intermediate points on that flow. Conversely, one way to approach the study of QFT is to start with a scale-invariant QFT and perturb it in different ways, which is like taking little steps out into the various RG flows that go to/from that particular fixed point in various directions.

The study of critical phenomena in statistical mechanics, like the critical point that terminates the liquid-vapor phase transition of a typical fluid, uses the same ideas. (This was probably the inspiration for some of the key ideas.)

Interestingly, scale-invariant QFTs tend to be invariant under a larger set of transformations, the conformal transformations. This sometimes allows additional non-perturbative results to be derived, especially in 2-d spacetime.

  • $\begingroup$ Dear @ChialAnomaly Sorry, it took me quite a long time to appreciate your answer. I am a little confused. If I understood it right, then you are saying that at the fixed points (which could be at a high scale or at a low scale) a QFT is scale-invariant. However, when you say that different QFTs flow into the same fixed point, what do you mean? What do you mean by different here? $\endgroup$ – mithusengupta123 May 13 '20 at 17:41
  • $\begingroup$ As far as I could understand from your post, universality has to do with many different QFTs behaving as scale-invariant theories at the fixed point and in the context of critical phenomena, this 'fixed point' refers to the critical point. When you say that many different QFTs flow to the same fixed point, do you mean different systems such as Fe and Ni (in the other answer) described by the same Heisenberg model but with different parameters, flows into a scale-invariant theory? I am not sure. $\endgroup$ – mithusengupta123 May 13 '20 at 18:23
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    $\begingroup$ @mithusengupta123 By "different" QFTs, I simply mean QFTs whose physical predictions differ. Saying that two different QFT's can flow to the same IR fixed point (for example) means that if we only consider processes with sufficiently low momentum, then the two theories' predictions for those processes are indistinguishable. If we take the limit all the way to the IR, so that the distinctions between the original QFTs are lost, we get the "fixed point" QFT, which is scale-invariant. Not only to the both flow to a scale-invariant theory, they both flow to the same scale-invariant theory. $\endgroup$ – Chiral Anomaly May 13 '20 at 18:46

Chiral anomaly gave an excellent answer in QFT language but I'll provide a more physical way of thinking about fixed points.

Consider two distinct physical systems (for example Ni or Fe magnet) that flow to the same fixed point under RG, then near the critical point, they have similar behaviors in heat capacity or magnetic susceptibility measurements. More precisely, the critical exponents, which characterize how these quantities diverge, are the same. That is to say they correspond to the same universality class, since they flow to the same fixed point.

But you can say the fixed points don't exactly describe the physical systems we are studying so why bother? Because studying Fe/Ni exactly is too complicated. We are content enough to correctly get the exponents that describe a non-generic measurement.


To be more explicit, the example of ferromagnetism I gave can be described by the Landau-Ginzburg hamiltonian $$\beta H = \int d^dx \, [\frac{t}{2}m^2 + \frac{K}{2} (\nabla m)^2+..] + u \int d^dx \,m^4$$

where $t$ is the reduced temperature and $u$ the interaction term. The RG flow diagram is given by Kardar chapter 5 which I copy below, enter image description here

for $d<4$, the non-zero fixed point is called Wilson-Fisher fixed point, which describe the paramagnet/ferromagnet phase transition. You can see there are two relatively straight lines flow into the same WF fixed point. Real materials like Ni/Fe will have some curve in this t-u parameter plane that people measure and at some point both will cross the straight line and flow into WF fixed point.

  • $\begingroup$ When you say, that Fe and Ni flow to the same fixed point under RG, what microscopic/field theory do you have in mind for them? $\endgroup$ – mithusengupta123 May 13 '20 at 17:58
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    $\begingroup$ I'm referring to ferromagnetic Fe/Ni, which are described by Landau-Ginzburg hamiltonian or O(N) model (see Kardar Chapter 5). I'll edit my answer a bit $\endgroup$ – Histoscienology May 15 '20 at 14:26
  • $\begingroup$ Thank you for your edits. Unfortunately, I cannot accept more than one answer. $\endgroup$ – mithusengupta123 May 16 '20 at 3:06

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