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I'm reading about the running coupling in QCD. I understand the vacuum polarization and its consequences. Also I've read that you can find the same phenomenon on the strong interaction, giving us the asymptotic freedom of quarks.

However, this "scale" things looks very, very related with phase transitions theory, which is important in condensed matter and also in complex systems/interdisciplinary physics. In Wikipedia looks like this is related to $\beta$-functions, so if $\mu$ is the energy scale and $g$ the coupling parameter,

$$\beta=\mu \dfrac{\partial g}{\partial \mu}$$

And they say that $\beta = 0$ if the theory is scale-invariant. The "scale-invariant" thing sounds to me a lot of complex systems -complex networks, fractals, this kind of stuff.

However, if I try to Google about running coupling I only find things about QED, QCD, great unification and all this things...

So, can you give me any examples of theories or models where the coupling parameters change with scale, as the fine structure constant do in QED? I'm interested also in interdisciplinary applications.

Thank you!

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    $\begingroup$ you might be interested in John Cardy's book "Scaling and renormalization in statistical physics" $\endgroup$
    – fqq
    Commented Jun 16, 2016 at 19:07

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The correct keyword for this is "renormalization". And, as you said yourself, high energy physics shares this field with condensed matter.

In condensed matter, you always encounter renormalization group of some kind if you are interested in critical phase transitions (which are scale invariant). Good references for this are Wikipedia and this book.

Condensed matter applications are naturally connected to graph/network-based problems like percolation

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  • $\begingroup$ Yes, I know about the existence of renormalization gruop, but I would be interested in a particular physics example. In Wikipedia it uses the example of "block of spins" (Ising model?) , and it says that the running coupling $J$ have only three values. Is this really running coupling? Can it be discrete instead of continuous? $\endgroup$
    – Victor Buendía
    Commented Jun 17, 2016 at 10:25
  • $\begingroup$ You confuse the fixed points of the renormalization transformation with the values of $J$. The idea that the properties of a critical system are universal and defined by its fixed points. Everything else is governed by renormalization flow. This is the main power of RG. McComb covers these topics in details, I really advise you to look into this book. $\endgroup$
    – Andrii Magalich
    Commented Jun 17, 2016 at 11:07
  • $\begingroup$ I read the book and I think I have understood it: there's a running coupling when I change the scale, and this flows towards the fixed points. Is this correct? $\endgroup$
    – Victor Buendía
    Commented Jun 20, 2016 at 16:05
  • $\begingroup$ Yes, more or less. Given the specific microscopic coupling to understand the behaviour on large scales, we need to see where the renormalization flow goes. $\endgroup$
    – Andrii Magalich
    Commented Jun 20, 2016 at 16:13
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Couplings that change (run) with scale occur in most quantum field theories! If you want a simple example, take a scalar theory with a quadratic and a quartic interaction (https://en.wikipedia.org/wiki/Quartic_interaction). In this theory, as you go to lower energies, the mass grows and the quartic coupling goes to zero. This theory is important both in particle physics (for describing the Higgs boson) and in condensed matter physics (phase transitions).

In fact, it's probably easier to talk about the QFTs who's couplings don't run! These are a very special subset of all QFTs, called scale invariant or conformal field theories (CFTs), where the $\beta$-function for all couplings is zero. These are very important in (among other areas) condensed matter physics, since they describe critical phenomena which occur in second order phase transitions (https://en.wikipedia.org/wiki/Critical_phenomena). A good reference might be Cardy's book (https://www.amazon.com/Scaling-Renormalization-Statistical-Physics-Cambridge/dp/0521499593).

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