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Reproducing topological properties with quasiMajorana 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:  https://arxiv.org/abs/1806.02801v2 (pdf) 
Date submitted:  20190331 01:00 
Submitted by:  Vuik, Adriaan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Andreev bound states in hybrid superconductorsemiconductor devices can have nearzero energy in the topologically trivial regime as long as the confinement potential is sufficiently smooth. These quasiMajorana states show zerobias 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 quasiMajorana states, also the coupling of these states across a tunnel barrier to the outside is exponentially different. As a consequence, quasiMajorana states mimic most of the proposed Majorana signatures: quantized zerobias 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 quasiMajorana states for braiding. Therefore, while the appearance of quasiMajorana states complicates the observation of topological Majorana states, it opens an alternative route towards braiding of nonAbelian anyons and topological quantum computation.
Current status:
Reports on this Submission
Anonymous Report 1 on 201954 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1806.02801v2, delivered 20190504, doi: 10.21468/SciPost.Report.933
Strengths
1 discusses braiding operations with quasiMajorana 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 lowenergy conductance
Weaknesses
1 the main idea is not new
2 some claims (e.g., regarding the smoothness of the effective potential, the spin of the quasiMajorana states, the spatial properties on the quasiMajorana wave functions, the conductance in the open regime) may require amendment/clarification.
Report
This manuscript studies the properties of lowenergy states emerging in Majorana superconductorsemiconductor devices in the topologicallytrivial 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 nontopological, quasiMajorana states that mimic the (local) phenomenology of topologicallyprotected Majorana zero modes. The main idea (i.e. that quasiMajorana 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 quasiMajorana 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 lowenergy conductance. Given the potential experimental relevance of the quasiMajorana states, I believe that the manuscript contains enough new elements to warrant its publication. There are, however, a few points that need clarification.
1Is “smoothness” a strong requirement for the effective potential? For example one can imagine a potential with a “sharp” steplike variation in the vicinity of the chemical potential. Would it be inconsistent with the presence of quasiMajorana states? In this context, it would be helpful to state explicitly that all quasiMajorana 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 quasiMajorana state).
2The authors emphasize the importance of the two quasiMajoranas having (nearly) opposite spins. How is the “spin” of the Majorana mode defined? Which of the statements regarding the spin of the quasiMajorana states will hold (or not hold) in the presence of strong transverse spinorbit coupling?
3It is pointed out that the quasiMajorana 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 quasiMajoranas should be amended. Also, for completeness, the lowenergy spectra corresponding to wave functions shown in Fig. 4 should be provided (as function of E_Z).
4Regarding 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 lowenergy “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.
5Finally, a few minor observations. The couplings defined below Eq. (13) do not have the appropriate dimension (some density of states is missing). The quasiMajorana 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 “quasiMajorana 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 confinementinduced subband), not in general.
Requested changes
Clarifications/changes that address points 15 of the report.
Author: Anton Akhmerov on 20190516 [id 515]
(in reply to Report 1 on 20190504)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 quasiMajorana 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 quasiMajorana 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:
In our view, our main result is to make the observation of the exponentially different couplings of quasiMajoranas, and through that to conclude that quasiMajoranas 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 20190524 [id 527]
(in reply to Anton Akhmerov on 20190516 [id 515])I think that the authors have correctly identified the main aspects of the Majorana versus quasiMajorana problem. The cited works are representative for the relevant developments in this area (and I do not believe that an exhaustive list is necessary).