# Volovik's argument and superconductivity

In Volovik's book he describes the Fermi surface as a vortex in energy+momentum space. Due to a winding number the Fermi surface is topologically protected.

I don't understand how the above topological protection is compatible with superconductivity, which destroys the Fermi surface even for small attractive interactions. If it is a topological phase transition, there should be some type of gap closing, e.g. Fermi surface shrinking to points, which is seemingly not the case. Or is it that the pole in the Green's function still exists in a superconductor, although then I am wondering what Volovik's argument really says about the Fermi surface.

I am familiar with the terminology of Chern numbers, topological insulators etc. I would be very grateful if someone could explain this to me using that language, if possible.

The picture is compatible with superconductivity. Because in the presence of the pairing term, we rewrite the Hamiltonian in the Nambu basis, where both particle and hole Fermi surfaces should be considered. If the particle Fermi surface corresponds to a vortex, then the hole Fermi surface corresponds to an antivortex. They coincide exactly in the energy-momentum space. The pairing term mixes the particle and hole together, allowing the vortex and anti-vortex to annihilate with each other, which destroys the Fermi surface. However, if we impose the $\mathrm{U}(1)$ symmetry, the vortex-antivortex annihilation will be forbidden and the Fermi surface becomes stable. So Volovik's argument works only in the presence of the $\mathrm{U}(1)$ symmetry.