Gapless Fermi Surfaces in anisotropic multiband superconductors in magnetic field.*

We propose that a new state with a fully gapless Fermi surface appears in quasi-2D multiband superconductors in magnetic field applied parallel to the plane. It is characterized by a paramagnetic moment caused by a finite density of states on the open Fermi surface. We calculate thermodynamic and magnetic properties of the gapless state for both s-wave and d-wave cases, and discuss the details of the 1-st order metamagnetic phase transition that accompanies the appearance of the new phase in s-wave superconductors. We suggest possible experiments to detect this state both in the s-wave (2-H NbSe$_2$) and d-wave (CeCoIn$_5$) superconductors.