This question may have some overlaps with Can I integrate out the fermion field that is not gapped?
For a system which has isolated Fermi points, for example Weyl semimetal, what is the calculation procedure of integrating out the fermion fields and obtain the effective action for other fields, for example, gauge fields? I ask this because the way of "calculating the Gaussian integral and expanding the determinant" seems not safe when there are some zero modes (if putting them at Fermi points).
There are some papers doing such things. For example http://arxiv.org/pdf/1603.02674v1.pdf, which studies a Weyl semimetal model coupled to small lattice deformations $\vec{u}(R_{i})$ and calculates the Hall viscosity $\eta_{H}$. After integrating out the Weyl fermions, the effective action is a Chern-Simons like term of the deformation fields. But I do not understand how they get this result.
