SciPost Submission Page
Conditions for fully gapped topological superconductivity in topological insulator nanowires
by Fernando de Juan, Jens H. Bardarson, Roni Ilan
This is not the current version.
|As Contributors:||Jens H Bardarson · Roni Ilan · Fernando de Juan|
|Submitted by:||de Juan, Fernando|
|Submitted to:||SciPost Physics|
|Subject area:||Condensed Matter Physics - Theory|
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.
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Submission & Refereeing History
- Report 2 submitted on 2019-05-06 14:41 by Anonymous
- Report 1 submitted on 2019-05-06 11:36 by Anonymous
Reports on this Submission
Anonymous Report 2 on 2018-12-3 Invited Report
- Cite as: Anonymous, Report on arXiv:1810.09576v2, delivered 2018-12-03, doi: 10.21468/SciPost.Report.693
1- It deals with a very timely and interesting problem: the topology of spin-orbit-coupled nanowires embedded in a SC shell and threaded by a magnetic flux
2- It includes some detailed derivations that could be useful for students and researchers entering this subfield
3- The quality of the presentation is decent, although somewhat verbose
4- Quite a few aspects of the problem are covered, including the role of lattice symmetries and supercurrent inhomogeneities
1- It has almost a 100% overlap with Ref. 48
2- It does not clarify whether any of the conclusions is different in a TI nanowire, respect the semiconducting nanowires of Ref. 48. Particularly, the most problematic result (gapless spectrum in clean systems) seems to be the same in both.
The study presented in this work has a very high overlap with Ref. 48, posted in the arXiv more than a month before this preprint. To accept this work for publication, it should highlight the differences with Ref 48 very clearly. In some aspects Ref. 48 does a considerably more thorough job in analysing the complicated physics of full-shell semiconducting nanowires, particularly as it probably had the benefit of knowing about the experimental results beforehand. This work should compensate for this in some way, or risk being ignored. Given my particular interest in seeing the SciPost initiative succeed (i.e. become a relatively high-impact journal), I would insist that the authors make this paper as competitive as possible with Ref. 48 before I can recommend publication. Essentially, it needs to be shown that the TI approach is different than the semiconducting nanowire approach, and is better in some significant sense. In more detail, I would suggest that the following issues are addressed.
It seems that the model employed here is essentially equivalent to that of Ref 48, except in the lack of a p^2 term in the Hamiltonian. If this is not so, please point out the differences very clearly. If that is indeed the only difference, does it lead different conclusions at all? In Ref. 48, it was already shown, for example, that the h/2e flux can only gap one mode, that with zero total angular mom,entum, while any higher angular momenta that are occupied will remain gapless. This seems also to be the case here, as eta=1/2 can only Cooper-pair one subband at zero energy. The gapless results in Sec. 3.1 seems to confirm this, unless rotation symmetry is broken (this possibility is also extensively discussed in Ref. 48). Unfortunately, the real reason for the gapless spectrum (gapless |l|>=1 modes) is not clearly given in this manuscript. As the occupation of different subbands of higher l is not readily tuneable in full shell nanowires due to electrostatic screening by the SC, and as rotation symmetry breaking is likely a small uncontrolled perturbation, it is not clear that the claim that a h/2e vortex can in general gap the system is actually justified.
Constraining the model of the TI nanowire to just the surface is very artificial. A thin nanowire will probably have considerable leakage of the surface wavefunction into the core. This scenario is nicely included in the lattice calculation of section 3, which shows that leakage doesn't seem to change the conclusions. It would be important to clarify under which conditions the topological transitions survive this leakage, and why, as it is not at all clear from the analytic description of the problem.
1 - There are two definitions of the flux quantum, the Dirac flux quantum h/e, relevant to the Aharonov-Bohm effect, and the superconducting flux quantum h/2e, relevant to superconducting vortices. The text should clarify that they are using the former definition when talking about "half a flux quantum".
2 - The manuscript says: "In a more thin film geometry, one may rather expect a roughly homogeneous order parameter which can be approximated by a constant ∆(x,θ)=∆, so nv=0". I don't fully understand what it is meant by this. The statement seems counterintuitive to me. A thin SC shell should more easily relax to the fluxoid of minimum energy, which will not be nv=0 as soon as the flux is greater than h/4e.
3 - The derivation of the continuum Hamiltonian in Sec. 5 seems much more complicated than is really necessary, and involves a number of spin rotations and change of variables. For comparison, the derivation in Ref. 48 is much simpler. Please try to present a more compact derivation of the Hamiltonian if possible.
4 - The topological invariant is defined in term of Pfaffians at high-symmetry points. Kitaev introduced this in a single-mode model. Is it justified in a generic multimode context with spin-orbit mode mixing? Does the topological gap inversion always take place at high-symmetry points in this case? Please give references.
1- Extend the paper with a comprehensive comparison to Ref. 48, highlighting relevant differences. If there are none that are relevant, I would not recommend publication.
2- Explain why bulk leakage doesn't change the results of the hollow core model.
3- Explain the origin of the gapless spectrum more clearly
4- Adresss "minor issues" 1 to 4 in the report.
Anonymous Report 1 on 2018-11-26 Invited Report
- Cite as: Anonymous, Report on arXiv:1810.09576v2, delivered 2018-11-26, doi: 10.21468/SciPost.Report.673
1. Importance of the subject among the current topics of research in the community
2. Timeliness with respect to experimental developments
3. Appropriate analytical and numerical techniques to approach the problem
4. Clarity of the presentation and of the message
5. Good introduction and description of the results in relationship to previous works
6. Novelty, in particular important differences from the conclusions of similar works addressing this subject in the past
1. The physical significance of the terms in Eq. 20 in not given, the coupling lambda's for example are not introduced, there is only a reference to a model in  which in my opinion is not enough.
2. The origin of the discrepancy with Ref.  is very vaguely explained, especially since the differences from Ref. seems to be one of the central and most important points of the paper. I believe a more thorough analysis is justified, especially in what concerns "the importance of choosing a suitable vector potential" and the "inhomogeneous supercurrent profile"; more generally the manuscript would benefit from a discussion of the relationship between the supercurrent profiles arising, their conservation and physical-ness in relationship also with the boundary conditions, and their dependence on choosing a gauge in finite-size discretized models.
This is a careful, clear, detailed and actual work concerning the formation of Majorana states in finite-width wires threaded by a magnetic flux. In particular the authors explore the topological phase diagram and the modification of the gap protecting the eventual topological states both in the presence and absence of a vortex in the superconducting providing the proximity effect necessary for the formation of Majorana states. Interestingly enough they find that when there is no vortex in the SC, such states cannot form in general due to a gaplessness of the system, unless the symmetry is either fully broken by geometry, disorder or inhomogeneities, but even so the induced gap is much smaller than when a vortex is present in the SC. The present results differ from a similar previous analysis which suggested that even in the absence of a vortex the Majorana states are protected by a topological gap.
I would recommend the publication of this manuscript in Scipost if the requested changes are made and the questions raised are answered, in particular if a careful analysis of the discrepancy with Ref.  and an analysis of the supercurrent profiles and of the gauge choices are performed.
1. Clarifications of the terms in Eq. 20.
2. Introduction of a thorough analysis of the significance of the choice of gauge, of the super current profiles, and of the origin of the discrepancy with the results of Ref. .