Quantum Effects in the High-magnetic-field Superconductivity of a Quasi-two-dimensional Multiple-band Superconductor*

We investigate the superconducting transition of a two-band layered system with deep and shallow bands in the presence of the out-of-plane magnetic field. As the shallow band’s Fermi energy and cyclotron frequency may be comparable, the Landau quantization has to be accounted for exactly. The interlayer tunneling lifts the Landau levels to dispersive bands along the out-of-plane momentum with two van Hove singularities at the zone center and the boundary. The pairing susceptibility has strong infrared divergence when either the same or opposite van Hove points simultaneously cross the Fermi level. The latter case yields the alternating Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state with sign change of the order parameter between the neighboring layers. This divergence may lead to the reentrant high-field superconductivity. In a general situation, when the Fermi level crosses the spin-up and spin-down bands, the FFLO states with a longer field-dependent modulating period are formed, which also may be reentrant.