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Conditions for fully gapped topological superconductivity in topological insulator nanowires
by Fernando de Juan, Jens H. Bardarson, Roni Ilan
Among the different platforms to engineer Majorana fermions in
one-dimensional topological superconductors, topological insulator nanowires
remain a promising option. Threading an odd number of flux quanta through these
wires induces an odd number of surface channels, which can then be gapped with
proximity induced pairing. Because of the flux and depending on energetics, the
phase of this surface pairing may or may not wind around the wire in the form
of a vortex. Here we show that for wires with discrete rotational symmetry,
this vortex is necessary to produce a fully gapped topological superconductor
with localized Majorana end states. Without a vortex the proximitized wire
remains gapless, and it is only if the symmetry is broken by disorder that a
gap develops, which is much smaller than the one obtained with a vortex. These
results are explained with the help of a continuum model and validated
numerically with a tight binding model, and highlight the benefit of a vortex
for reliable use of Majorana fermions in this platform.
Author comments upon resubmission
We have considered all issues and criticisms raised by the reviewers, and we have modified our manuscript to address them. Please the referee reports for detailed answered.
List of changes
- We have stated explicitly that we consider the Aharonov-Bohm flux quantum h/e, and that half a flux quantum refers to h/2e.
- At the beginning of section 2 we have explained why the surface model is appropriate and under what conditions.
- We have clarified what we meant by the "thin film limit" of the bulk superconductor in the text.
- We have done a major rewriting of section 2.1 simplifying the equations and addressing the general case of arbitrary $\eta$, $\l$ and $n_v$. We have included the operators for time-reversal and inversion symmetries.
- We have also rewritten section 2.2 to discuss the implications of time-reversal symmetry in bulk vs surface models. We have merged into this discussion what used to be appendix 5.3.
- We have emphasized throughout the text that when $\eta = n_v/2$ we get a topological state for an arbitrary value of the chemical potential.
- For the lattice model, we have defined the operators of inversion and mirror symmetries.
- We have introduced the significance of every parameter in the lattice model after Eq. 20. Note also we have relabeled the orbital degree of freedom from $\sigma$ to $\rho$, and we have made consistent the notation for the transformation to the Majorana basis.
- We have significantly extended the discussion on the discrepancy with Ref. 24 in Sec. 3.1, elaborating on the choice of gauge and inhomogeneous supercurrents.
- We have significantly extended the discussion of the results of the lattice model calculations with difference discrete rotation symmetries, and explained the origin of the gapped and gapless regions.
- In the discussion section, we have included a paragraph about possible mechanisms that can break the effective time-reversal symmetry at $\eta = n_v/2$.
- We have removed that added note about Refs. 47 and 48, and instead included another paragraph in the discussion section commenting on the main differences between our work and Ref. 48. (note in the current manuscript Ref numbers 47 and 48 are interchanged compared to the original manuscript). In our response to referees we have used the numbering corresponding to the original manuscript.
- We have also fixed several minor typos through the text.