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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 accepted: 2019-10-28
Date submitted: 2019-10-14 02:00
Submitted by: Vuik, Adriaan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
  • 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, 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 for increasing magnetic field. 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 protected quantum computation.

List of changes

We thank the referee for the favorable assessment of our manuscript. We have addressed
the requests for minor clarifications raised by the referee:

1.) Indeed, the smoothness requirement is not absolute. We now refer explicitly to
previous works that have found near-zero energy states with potentials involving
kinks or abrupt changes. We have also clarified that it is indeed the smooth potential
slope of the tunnel barrier that is responsible for the appearance of quasi-Majorana
states in all of our systems, and that the quantum dot in system Fig. 2(b) serves
only as a probe.

2.) We have now clarified that with "spin" we mean the expectation value of
Furthermore, note that quasi-Majorana states are actually favored by
weak spin-orbit interaction, as shown in Fig. 2(c) and (d). For this reason,
we focus on the case of weak spin-orbit interaction, where the transverse
part of the spin-orbit coupling is of negligible influence. In fact, in the
3D model of Sec. VI we do include the transverse part explicitly. For the weak
spin-orbit strengths considered there, quasi-Majorana states appear. We have
extended the discussion in this regard in the paper.

3.) We have now clarified that we compare the separation of the quasi-Majorana
states to the coherence length \xi. Furthermore, we have added to Fig. 4 the spectrum
corresponding to the wave functions as suggested by the referee.

4.) In relation to the questions raised by the referee, we would like to point out that
Fig. 8 was computed using our microscopic model, i.e. the contributions of above-gap
states are explicitly included. We have clarified this aspect in the text.
Furthermore, we have now added at the end of Section V an extended discussion on how
robust the different features of the conductance shown in Fig. 8 are.

5.) We have clarified that the coupling elements t in Eq. (12) and (14) are in units of sqrt(energy).
With respect to quasi-Majoranas allowing for topological operations, we would like to
point out that in Sec. VIII we do discuss the fact that quasi-Majoranas themselves are
not topologically protected. To avoid confusion, we have renamed topological quantum
computation with quasi-Majorana states to protected quantum computation.
With respect to the occurrence of quasi-Majoranas at smaller fields, we have now
added the clarification that this indeed holds for a given chemical potential.

Published as SciPost Phys. 7, 061 (2019)

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