In fermionic systems with different types of quasi-particles, attractive interactions can give rise to exotic superconducting states, as pair density wave (PDW) superconductivity and breached pairing. In the last years the search for these new types
of ground states in cold atom and in metallic systems has been intense. In the case of metals the different quasi-particles may be the up and down spin bands in an external magnetic field or bands arising from distinct atomic orbitals that coexist at a common Fermi surface. These systems present a complex phase diagram as a function of the difference between the Fermi wave-vectors of the different bands. This can be controlled by external means, varying the density in the two-component cold atom system or, in a metal, by applying an external magnetic field or pressure. Here we study the zero temperature instability of the normal system as the Fermi wave-vectors mismatch of the quasi-particles (bands) is reduced and find a second order quantum phase transition to a PDW superconducting state. From the nature of the quantum critical fluctuations close to the superconducting quantum critical point (SQCP), we obtain its dynamic critical exponent. It turns out to be $z=2$ and this allows to fully characterize the SQCP for dimensions $d ge 2$.
In multi-band metals quasi-particles arising from different atomic orbitals coexist at a common Fermi surface. Superconductivity in these materials may appear due to interactions within a band (intra-band) or among the distinct metallic bands (inter-
band). Here we consider the suppression of superconductivity in the intra-band case due to hybridization. The fluctuations at the superconducting quantum critical point (SQCP) are obtained calculating the response of the system to a fictitious space and time dependent field, which couples to the superconducting order parameter. The appearance of superconductivity is related to the divergence of a generalized susceptibility. For a single band superconductor this coincides with the textit{Thouless criterion}. For fixed chemical potential and large hybridization, the superconducting state has many features in common with breached pair superconductivity with unpaired electrons at the Fermi surface. The T=0 phase transition from the superconductor to the normal state is in the universality class of the density-driven Bose-Einstein condensation. For fixed number of particles and in the strong coupling limit, the system still has an instability to the normal sate with increasing hybridization.
