Majorana zero modes in one-dimensional quantum wires without long-ranged superconducting order

We show that long-ranged superconducting order is not necessary to guarantee the existence of Majorana fermion zero modes at the ends of a quantum wire. We formulate a concrete model, which applies, for instance, to a semiconducting quantum wire with strong spin-orbit coupling and Zeeman splitting coupled to a wire with algebraically decaying superconducting fluctuations. We solve this model by bosonization and show that it supports Majorana fermion zero modes. We show that electron backscattering in the superconducting wire, which is caused by potential variations at the Fermi wave vector, generates quantum phase slips that cause a splitting of the topological degeneracy, which decays as a power law of the length of the superconducting wire. The power is proportional to the number of channels in the superconducting wire. Other perturbations give contributions to the splitting that decay exponentially with the length of either the superconducting or semiconducting wires. We argue that our results are generic and apply to a large class of models. We discuss the implications for experiments on spin-orbit coupled nanowires coated with superconducting film and for LaAlO3/SrTiO3 interfaces.