Nambu identity and collective modes in superconductors and superfluid $^3$He

Collective modes manifest themselves in a variety of different physical systems ranging from superconductors to superfluid $^{3}$He. The collective modes are generated via the Higgs-Anderson mechanism that is based on the symmetry breaking double well potential. Recently collective modes were explored in superconducting NbN and InO in the presence of a strong terahertz laser field. In both cases a single collective mode that oscillates with twice the frequency of the superconducting energy gap $\Delta$ was discovered. Superfluid $^{3}$He is the host for a whole variety of collective modes. In particular, in the superfluid $^{3}$He B-phase, two massive collective modes were found with masses $\sqrt{{8}/{5}}\Delta$ and $\sqrt{{12}/{5}}\Delta$. We show that for both cases of the superconducting films and for the superfluid $^{3}$He B-phase, the collective modes satisfy the Nambu identity that relates the masses of different collective modes to the energy gap parameter $\Delta$.