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 LaAlO${}_3$/SrTiO${}_3$ interfaces.

Figure 1

A semiconductor nanowire in contact with a bulk 3D superconductor.

Figure 2

A semiconductor nanowire in contact with a 1D superconducting wire. The superconducting wire could be a coating that (a) completely covers the semiconductor or (b) only covers part of it.

Figure 3

The moduli space of semiclassical vacua is the torus, which is here depicted as a cylinder with the top and bottom edges identified. For any fixed ${\theta }_{\rho }$, there are two different semiclassical ground states, depicted by the intersection points between the dotted vertical line and the line of fixed $\sqrt{2}{\theta }_{\rho }-2\theta$, which winds twice around the cylinder. If ${\theta }_{\rho }$ is, indeed, fixed, as in Sec. 2, then there is a tunneling barrier between these two ground states. However, if the total charge mode $\frac{1}{\sqrt{2}}{\theta }_{\rho }+\theta$ can fluctuate, as in Sec. 3, then the entire line of fixed $\sqrt{2}{\theta }_{\rho }-2\theta$ is the same quantum ground state, and there is no degeneracy.

Figure 4

A schematic depiction of the different processes that split the two states of the qubit. Process (a): A vortex can tunnel between a semiconducting wire and the superconducting wire. This causes a splitting that is exponentially small in $\ell$. Process (b): An electron can tunnel from one semiconducting wire to another through the superconducting wire. This causes a splitting that is exponentially small in $L-2\ell$. Process (c): A vortex can tunnel through the superconducting wire between the semiconducting wires as a result of an electron backscattering process. This causes a splitting that decays as a power of $L$. This model applies also to a situation in which there is a single semiconducting nanowire that extends from $-L/2$ to $L/2$ and is in a topological phase for $L/2-\ell <\left|x\right|<L$ (similar to our two semiconducting nanowires) and is in a nontopological phase for $\left|x\right|<L/2-\ell$ (similar to our superconducting wire).