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We investigate the onset of superconductivity in magnetic field for a cleantwo-dimensional multiple-band superconductor in the vicinity of the Lifshitztransition when one of the bands is very shallow. Due to small number ofcarriers in this band, the quasiclassical Werthamer-Helfand approximationbreaks down and Landau quantization has to be taken into account. We found thatthe transition temperature TC2(H) has giant oscillations and is resonantlyenhanced at the magnetic fields corresponding to full occupancy of the Landaulevels in the shallow band. This enhancement is especially pronounced for thelowest Landau level. As a consequence, the reentrant superconducting regions inthe temperature-field phase diagram emerge at low temperatures near themagnetic fields at which the chemical potential matches the Landau levels. Thespecific behavior depends on the relative strength of the intraband andinterband pairing interactions and the reentrance is most pronounced in thepurely interband coupling scenario. The reentrant behavior is suppressed by theZeeman spin splitting in the shallow band, the separated regions disappearalready for very small spin-splitting factors. On the other hand, thereentrance is restored in the resonance cases when the spin-splitting energyexactly matches the separation between the Landau levels. The predictedbehavior may realize in the gate-tuned FeSe monolayer.

Strong Landau-quantization effects in high-magnetic-field superconductivity of a two-dimensional multiple-band metal near the Lifshitz transition