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In the famous seminal paper by K G Wilson and J Kogut in Physics Reports (Aug 1974) on The renormalization group and the ε expansion, they achieve a pinnacle of uniting the Ginzburg-Landau theory and the renormalization group approach to study the phase transitions and its critical theory. Together with the concept of Ginzburg-Landau symmetry-breaking theory using the local order parameters to distinguish phases, their theory has been called the Ginzburg-Landau-Wilson theory.

It seems that current theoretical research theme is to search new physics and to go beyond the Ginzburg-Landau-Wilson theory.

One obvious thing to go beyond the Ginzburg-Landau-Wilson theory is through the studying of topological order and topological quantum field theory (TQFT). The reason is that topological order and TQFT cannot be detected by local order parameters. This is not within the framework of Ginzburg-Landau-Wilson theory, so some new tools are required.

Question: However, if we exclude topological order and TQFT, what have we learned that had gone beyond the Ginzburg-Landau-Wilson theory?

Some understandings that may be new beyond themes:

1) One story is emergent global symmetries or emergent conformal symmetries for the critical theory/phase transition critical point. Or the emergent gauge fields. The emergent gauge fields can be related to the fractionalized anyons or the underlying topological orders.

2) Non-Fermi liquid: Another related story is the broken down of the Landau Fermi liquid theory, for example, the non-Fermi liquid theory. In some case, it is due to the emergent gauge fields or fractionalized excitations, through the physics of 1).

3) Fermi surface: A third story is that the renormalization group treatment for the system with a Fermi surface may be more subtle. Furthermore together with the gauge fields coupling to Fermi surface, it can be a difficult challenge.

4) Conformal bootstrap: A fourth story is utilizing the conformal symmetry to do the conformal bootstrap. This bootstrap program may be overlooked in the past by Wilson. But Wilson surely mentions that the Migdal—Polyakov bootstrap and the conformal invariance are interesting. Wilson cited their work in his 1974 paper.

So, apart from the topological order and TQFT require new concepts beyond the Ginzburg-Landau-Wilson theory, do we really have some conceptual breakthroughs that go beyond Ginzburg-Landau-Wilson theory? Or are all the issues in non-Fermi liquid/Fermi surface renormalization group and conformal bootstrap are part of development and extension of Ginzburg-Landau-Wilson theory?

For example, one may say that the interacting conformal field theories may have no good quasi-particle descriptions --- but isn't that the same common part of the story also encountered in the past of the critical theory of Ginzburg-Landau-Wilson? Do they require some new ideas beyond Ginzburg-Landau-Wilson?

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  • $\begingroup$ Perhaps one may consider the Ginzburg-Landau-Wilson theory (GLW) as a condensed matter statement of effective field theory. In this sense it is a general paradigm. Now if one insists that the ingredients of GLW must necessarily be those originally envisioned, i.e. bosonic variables/geometric order parameters, then a simple way to go beyond is to introduce itinerant fermions (a la model of quantum critical metals). If you look at deconfined criticality its GLW theory of partons, although technically one is going beyond simple geometric interpretation of the bosonic variables. $\endgroup$
    – vik
    Commented Nov 22, 2017 at 7:26

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A lot of those things aren't really ''beyond Landau Ginsburg".

Landau Ginsburg describes phase transitions as due to a loss of symmetry: there is some order parameter and when it's nonzero some symmetry is broken, and when it is zero the symmetry is there. The conformal bootstrap, to take your example, is basically a tool to study conformal field theory -- there's nothing about it that is inconsistent with LG. Indeed the conformal bootstrap has been very useful in analysing the critical points of phase transitions within the LG paradigm -- it has been used to calculate the critical exponents of the 3d Ising model incredibly precisely for example.

One idea that's outside of LG but needn't involve topological order is that of ''deconfined quantum criticality", see https://arxiv.org/pdf/cond-mat/0404718.pdf, which is closely related to fractionalisation as you mention.

The basic point is that you can have a phase transition between two states of completely different broken symmetries. For instance, the Neel to VBS transition is between two ground states of a spin system, one of which breaks spin rotational symmetry and the other breaks lattice symmetry. This is forbidden in LG because you'd need two order parameters to describe each broken symmetry, and to get a phase transition between two such states you'd need to tune both of those parameters so that they break/unbreak at the same point in the phase diagram. Deconfined quantum criticality is basically the idea that there are monopole events in the theory that carry quantum numbers of both symmetries, and so if these proliferate at the critical point you can break/unbreak both symmetries at the same critical point.

That's a bit of a terse explanation, but you can read the link if you want to learn more.

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    $\begingroup$ I don't think the deconfined quantum critical point is really beyond Landau theory, since it is still described by symmetry breaking, except now in the presence of an 't Hooft anomaly. We explained this in arxiv.org/abs/1707.07686 for instance. $\endgroup$
    – Ryan Thorngren
    Commented Nov 25, 2018 at 7:08
  • $\begingroup$ Hey this looks great (and I'll definitely read it in full) but is there a TLDR on how the anomaly gets you past the fact you naively need to tune two parameters so can't have a second order transition? $\endgroup$
    – user213887
    Commented Nov 25, 2018 at 15:35
  • $\begingroup$ Well the anomaly does not guarantee a second order transition, but it does guarantee an order-to-order transition (which might be discontinuous). The point is that once the anomaly is there, no point in the phase diagram can have a nondegenerate gapped ground state. All points must be ordered unless you break the symmetry explicitly. $\endgroup$
    – Ryan Thorngren
    Commented Nov 25, 2018 at 16:12
  • $\begingroup$ I see, I'll have a closer read. From the sounds of it, DQC is still beyond Landau theory though, since Landau theory would forbid a second order transition here. $\endgroup$
    – user213887
    Commented Nov 25, 2018 at 21:53
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    $\begingroup$ We don't really know yet. It still seems to me that something like an emergent deconfined SU(2) gauge field cannot be described by a spontaneously broken global symmetry in the ordinary sense, since the gauge field itself (which would be the Goldstone mode) is not gauge invariant. Likewise it's hard to describe nonabelian TQFTs this way, although it seems possible. You should really call them higher symmetries by the way, it's more precise. $\endgroup$
    – Ryan Thorngren
    Commented Nov 26, 2018 at 9:16

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