In the quantum critical phenomena of condensed matter, the earlier work by Hertz, Moriya and Millis develope the the Hertz-Millis-Moriya (HMM) theory of quantum phase transition.
Naively, they integrate out Fermi surface quasiparticles and obtain an effective theory for the order parameter $\Phi$ alone.
This can be safe, if the Fermi surface quasiparticles are gapped, so we can integrate out the high energy modes. This can be dangerous, on one hand, if the Fermi surface quasiparticles are gapless. On the other hand, this may lead to non-local term in the $\Phi$ theory. Hertz focused on only the simplest such non-local term.
However, there are an infinite number of nonlocal terms at higher order, and these lead to a breakdown of the Hertz theory in higher dimensions.
What will be the criterion that Hertz-Millis-Moriya theory fails? Other than that the obvious case that Fermi surface quasiparticles are gapless?
Are the Pomeranchuk instability (i.e. the rotation symmetry of the Fermi surface broken spontaneously) and Fermi-liquid instabilities at magnetic quantum phase transitions some typical examples for the failure of Hertz-Millis-Moriya theory? What are some other examples?
How the dimensionality of the failure comes in for Hertz-Millis-Moriya theory? It seems that some people said that the 1+1d case the HMM works fine, but 2+1d and higher, there are many examples for HMM failure.