The question of integrating out gapless fermions in an interacting theory of bosons and fermions usually arises when one studies
- the critical point of a 2nd order quantum phase transition from a metal to some ordered state, such as an anti-ferromagnet.
- a critical phase, such a gauge field interacting with fermions.
Gapless fermionic systems can be of two types based on the dispersion relation: energy and momentum of the low energy fermion excitations
A) vanish at the same point in the momentum space, as in the case of electrons in QED, electrons in graphene with suitably tuned chemical potential, "band touching" situations, etc. These systems are said to possess Fermi points in the momentum space.
B) vanish at different points in the momentum space, as is the case of fermions at finite density such as traditional metals that host extended Fermi surfaces and QCD at finite density.
In general, integrating out gapless modes is dangerous because the procedure may generate non-local relevant or marginal (in RG sense) terms in the effective action which cannot be renormalized within local RG schemes. However, it appears to be the case that if there is a single zero energy mode, as is the case for gapless bosons and cases 1A and 2A above, the integrating out procedure is "safe" and no problematic non-local terms are generated.
However, cases 1B and 2B are more subtle. In the field of quantum critical phenomena, works by Hertz, Moriya and Millis that resulted in the Hertz-Millis-Moriya (HMM) theory of quantum phase transition, relies on integrating out fermion thereby constructing an effective Landau-Ginzburg theory for the bosonic order parameter. This was considered "safe enough" for decades, until it was realized that the "safety" related to integrating out fermions around the Fermi surface crucially depends on the dimension and type of Fermi surface (extended vs. single point). For example, in (2+1)D integrating out fermions with momenta around an extended Fermi surface generates non-local marginal vertices in the effective bosonic theory, leading to the failure of the HMM paradigm. (Further reading: look here and here).
With our current understanding the moral of the story appears to be that integrating out fermions
- with momenta around an extended Fermi surface is unsafe, especially in (2+1)D, although it seems to be safe in (3+1)D. The physical reason is that the extended Fermi surface provides the system with an infinite number of zero modes which contributes substantially to the low energy properties of the system. Also in lower dimensions increase in quantum fluctuations makes the Fermi surface contribution even more significant, such that integrating out fermions generates stronger IR singular vertices for the effective bosonic theory.
- with momenta around one or several Fermi points is generally safe in any dimensions.