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Reproducing topological properties with quasi-Majorana states

by A. Vuik, B. Nijholt, A. R. Akhmerov, M. Wimmer

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Submission summary

Authors (as registered SciPost users): Anton Akhmerov · Adriaan Vuik · Michael Wimmer
Submission information
Preprint Link:  (pdf)
Date submitted: 2019-03-31 01:00
Submitted by: Vuik, Adriaan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approaches: Theoretical, Computational


Andreev bound states in hybrid superconductor-semiconductor devices can have near-zero energy in the topologically trivial regime as long as the confinement potential is sufficiently smooth. These quasi-Majorana states show zero-bias conductance features in a topologically trivial phase, thereby mimicking spatially separated topological Majorana states. We show that in addition to the suppressed coupling between the quasi-Majorana states, also the coupling of these states across a tunnel barrier to the outside is exponentially different. As a consequence, quasi-Majorana states mimic most of the proposed Majorana signatures: quantized zero-bias peaks, the $4\pi$ Josephson effect, and the tunneling spectrum in presence of a normal quantum dot. We identify a quantized conductance dip instead of a peak in the open regime as a distinguishing feature of true Majorana states in addition to having a bulk topological transition. Because braiding schemes rely only on the ability to couple to individual Majorana states, the exponential control over coupling strengths allows to also use quasi-Majorana states for braiding. Therefore, while the appearance of quasi-Majorana states complicates the observation of topological Majorana states, it opens an alternative route towards braiding of non-Abelian anyons and topological quantum computation.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2019-5-4 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1806.02801v2, delivered 2019-05-04, doi: 10.21468/SciPost.Report.933


1- discusses braiding operations with quasi-Majorana states
2- proposes of a new distinctive signature of a topological phase (based on conductance measurements in the open regime)
3- derives of an explicit expression for the low-energy conductance


1- the main idea is not new
2- some claims (e.g., regarding the smoothness of the effective potential, the spin of the quasi-Majorana states, the spatial properties on the quasi-Majorana wave functions, the conductance in the open regime) may require amendment/clarification.


This manuscript studies the properties of low-energy states emerging in Majorana superconductor-semiconductor devices in the topologically-trivial regime due to the presence of a soft effective potential. The work is part of a series of studies (starting with Ref. 15 and including several recent works) dedicated to non-topological, quasi-Majorana states that mimic the (local) phenomenology of topologically-protected Majorana zero modes. The main idea (i.e. that quasi-Majorana states generated in the presence of a smooth potential can reproduce all the local properties of topological Majorana states) was discussed extensively in other works. In my opinion, the new contributions of this study include the discussion of braiding operations with quasi-Majorana states, the proposal of a new distinctive signature of a topological phase (based on conductance measurements in the open regime), and the derivation of an explicit expression for the low-energy conductance. Given the potential experimental relevance of the quasi-Majorana states, I believe that the manuscript contains enough new elements to warrant its publication. There are, however, a few points that need clarification.

1-Is “smoothness” a strong requirement for the effective potential? For example one can imagine a potential with a “sharp” step-like variation in the vicinity of the chemical potential. Would it be inconsistent with the presence of quasi-Majorana states? In this context, it would be helpful to state explicitly that all quasi-Majorana states discussed in this work are generated by a smooth potential (i.e., including the case of a proximitized wire coupled to a quantum dot; here, the dot plays no role in the emergence of the quasi-Majorana state).

2-The authors emphasize the importance of the two quasi-Majoranas having (nearly) opposite spins. How is the “spin” of the Majorana mode defined? Which of the statements regarding the spin of the quasi-Majorana states will hold (or not hold) in the presence of strong transverse spin-orbit coupling?

3-It is pointed out that the quasi-Majorana states can be either partially separated or spatially fully overlapping. However, the examples provided in Fig. 4 appear qualitatively the same. In both panel (a) and panel (b) the wave functions have a substantial overlap, with the leftmost peaks being slightly displaced by something on the order of 1/k_F. The difference between the two situations is quantitative, k_F being much larger in (b). Without additional evidence, the statement regarding the spatial properties of the quasi-Majoranas should be amended. Also, for completeness, the low-energy spectra corresponding to wave functions shown in Fig. 4 should be provided (as function of E_Z).

4-Regarding the proposed distinctive signature of the topological phase, I could imagine some potential issues. First, the authors assume that the effective potential at the end of the wire can be made perfectly flat. It is not obvious that this is the case in experiment. In other words, it may be possible that the system enters the “open regime” while a significant potential inhomogeneity persists near the end of the wire. This inhomogeneity may still support low-energy “trivial” states that produce a conductance peak lower than 4e^2/h. On the other hand, I am not sure whether or not the contributions to the differential conductance from states above the induced gap where properly accounted for. In principle, these contributions could “flood” the gap and mask the signatures discussed by the authors.

5-Finally, a few minor observations. The couplings defined below Eq. (13) do not have the appropriate dimension (some density of states is missing). The quasi-Majorana states are not topologically protected; one should be careful when stating that they open “an alternative route towards (…) topological quantum computation.” It is stated that “quasi-Majorana states emerge for smaller magnetic fields (…) resulting in smaller energy splittings.” This is true for a given chemical potential (away from the bottom of a confinement-induced sub-band), not in general.

Requested changes

Clarifications/changes that address points 1-5 of the report.

  • validity: high
  • significance: high
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Author:  Anton Akhmerov  on 2019-05-16  [id 515]

(in reply to Report 1 on 2019-05-04)

Dear referee,

We thank you for your careful analysis of our work. In the resubmitted version we would like to do our best in addressing your feedback, and for that we would like to ask for a clarification.

In your report you write:

"The main idea (i.e. that quasi-Majorana states generated in the presence of a smooth potential can reproduce all the local properties of topological Majorana states) was discussed extensively in other works."

We recognize that quasi-Majorana states are an active topic, and therefore it is important to give proper credit to the prior works. We aimed to do that in the manuscript.

To the best of our understanding the following statements are true:

  1. Before our work it was known that quasi-Majoranas may reproduce some local signatures of Majorana bound states (Refs. 15–23 of the manuscript).
  2. There were also proposals to distinguish quasi-Majoranas from Majoranas relying on a tunneling measurement that our manuscript invalidates (Refs. 26–28 of the manuscript).
  3. That the tunnel couplings of quasi-Majorana states are exponentially different in the barrier smoothness was not demonstrated.
  4. It was therefore not known that quasi-Majorana states mimic all local tunneling signatures of Majoranas.

In our view, our main result is to make the observation of the exponentially different couplings of quasi-Majoranas, and through that to conclude that quasi-Majoranas reproduce all local tunneling Majorana signatures.

If we have overlooked something, we would be happy to amend the discussion of the prior literature following specific suggestions.

Anonymous on 2019-05-24  [id 527]

(in reply to Anton Akhmerov on 2019-05-16 [id 515])
answer to question

I think that the authors have correctly identified the main aspects of the Majorana versus quasi-Majorana problem. The cited works are representative for the relevant developments in this area (and I do not believe that an exhaustive list is necessary).

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