Number conserving theory for topologically protected degeneracy in one-dimensional fermions

Semiconducting nanowires in proximity to superconductors are among promising candidates to search for Majorana fermions and topologically protected degeneracies, which may ultimately be used as building blocks for topological quantum computers. The prediction of neutral Majorana fermions in the proximity-induced superconducting systems ignores number conservation and thus leaves open the conceptual question of how a topological degeneracy that is robust to all local perturbations arises in a number-conserving system. In this work, we study how local attractive interactions generate a topological ground-state near-degeneracy in a quasi-one-dimensional superfluid using bosonization of the fermions. The local attractive interactions opens a topological quasiparticle gap in the odd channel wires (with more than one channel) with end Majorana modes associated with a topological near-degeneracy. We explicitly study the robustness of the topological degeneracy to local perturbations and find that such local perturbations result in quantum phase slips, which split of the topological degeneracy by an amount that does not decrease exponentially with the length of the wire, but still decreases rapidly if the number of channels is large. Therefore a bulk superconductor with a large number of channels is crucial for true topological degeneracy.

Figure 1

(a) Combination of Zeeman splitting, spin-orbit coupling, and weak attractive $s$-wave Feshbach resonance can induce end MFs in multichannel nanowire (shown as green rectangle). Spin-orbit coupling and Zeeman splitting ensures almost helical fermions at the fermi surface (shown as discs with arrows representing spin). Attractive $s$-wave interaction leads to the scattering of pairs of fermions with zero momentum (Cooper pairs) from one channel (represented by blue color) to another (represented in red). (b) Schematic band structure of multichannel nanowire with all degeneracies lifted by Zeeman splitting. Filling corresponding to an odd number of bands is in the topological phase.

Figure 2

(a) Tunneling conductance $G\left(V\right)$ into a Majorana site at the end of a long wire shows power-law peak (solid line) characteristic of tunneling into Luttinger liquid with attractive interactions. This differs qualitatively from the $\delta$-function peak (dashed line) expected from a proximity-coupled nanowire (Ref. 10). For the plot, we have chosen a Luttinger parameter $K=2$ and added a lead-induced Lorentzian broadening [${({\omega }^2+{\Gamma }^2)}^{-1}$] with width $\Gamma =0.02$. (b) Schematic figure to obtain a topological degeneracy using topological superfluid wires containing phase fluctuations and phase slips. Three topological wires (shown in black) each of which support MFs (discs at ends) are coupled by tunneling (shown as shaded regions) to obtain a topological degeneracy, which is robust to bulk scattering and respects number conservation.