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?