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Is Fermi liquid topologically protected?

The book by Volovik [1] contains the argument, which can be considered as the topological proof of the Luttinger theorem. The Green function of the ideal Fermi gas has a pole in the (E,|p|) plane (where E and p are energy and momentum). This pole is considered to be analogous to a vortex in liquid helium. Since a vortex is topologically stable against restricted perturbations, one can include interaction adiabatically (as a succession of small perturbations), and observe transformation of the Fermi gas pole to the Fermi liquid pole. In this argument, the topological stability arises already on the level of the ideal Fermi gas, which is in conflict with its Cooper instability. We discuss the origin of this controversy.