Anomalous isotope effect near a 2.5 Lifshitz transition in a multi-band multi-condensate superconductor made of a superlattice of stripes

The doping dependent isotope effect on the critical temperature (T_c) is calculated for multi-band multi-condensate superconductivity near a 2.5 Lifshitz transition. We consider a superlattice of quantum stripes with finite hopping between stripes near a 2.5 Lifshitz transition for the appearance of a new sub-band making a circular electron-like Fermi surface pocket. We describe a particular type of BEC (Bose-Einstein Condensate) to BCS (Bardeen-Cooper-Schrieffer condensate) crossover in multi-band/multi-condensate superconductivity at a metal-to-metal transition that is quite different from the standard BEC-BCS crossover at an insulator-to-metal transition. The results show that the isotope coefficient strongly deviates from the standard BCS value 0.5, when the chemical potential is tuned at the 2.5 Lifshitz transition for the metal-to-metal transition. The critical temperature T_c shows a minimum due to the Fano antiresonance in the superconducting gaps and the isotope coefficient diverges at the point where a BEC coexists with a BCS condensate. In contrast T_c reaches its maximum and the isotope coefficient vanishes at the crossover from a polaronic condensate to a BCS condensate in the newly appearing sub-band.