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Finite temperature effects on Majorana bound states in chiral $p$-wave superconductors

by Henrik Schou Røising, Roni Ilan, Tobias Meng, Steven H. Simon, Felix Flicker

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Felix Flicker · Tobias Meng · Henrik Røising · Steve Simon
Submission information
Preprint Link:  (pdf)
Date accepted: 2019-04-18
Date submitted: 2019-04-10 02:00
Submitted by: Røising, Henrik
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approaches: Theoretical, Computational


We study Majorana fermions bound to vortex cores in a chiral $p$-wave superconductor at temperatures non-negligible compared to the superconducting gap. Thermal occupation of Caroli de Gennes-Matricon states, below the full gap, causes the free energy difference between the two fermionic parity sectors to decay algebraically with increasing temperature. The power law acquires an additional factor of $T^{-1}$ for each bound state thermally excited. The zero-temperature result is exponentially recovered well below the minigap (lowest-lying CdGM level). Our results suggest that temperatures larger than the minigap may not be disastrous for topological quantum computation. We discuss the prospect of precision measurements of pinning forces on vortices as a readout scheme for Majorana qubits.

Author comments upon resubmission

We kindly thank the editor for taking the time to consider our manuscript, and the referees for their useful reports and recommendations. We will address all points raised by the referees. We believe the manuscript is now suitable for publication in SciPost.

Both referees requested that we emphasize the effect of the parity disparity on the inter-vortex force, leading to a protocol for read-out of the qubit state. This is a helpful suggestion, and indeed earlier versions of the draft contained a significantly greater discussion of this point, including detailed calculations for specific experimental set-ups. We found that the required sensitivity of measurement, and precision of control in manipulating the vortices, place force measurements slightly beyond the reach of current techniques, and so we opted to de-emphasise the extent to which we proposed force measurements as a useful protocol. That said, since both referees request further discussion on this point, we have re-introduced a longer discussion of the specific set-up we found to be the best candidate given existing reported data.

Response to referee 1:

We thank Prof. Sau for his positive assessment of our work, and for his insightful and helpful comments and recommendations. We address each of the requested changes in turn.

(1) We have added a new paragraph at the end of Sec. III.C in which we calculate the parity disparity with a continuum of in-gap states. Our result, derived in Eq. (15), matches the expression suggested by Prof. Sau. We emphasize, in the Introduction and Conclusions, the precise manner in which a small level spacing would be detrimental to the read-out of the Majorana qubit parity, providing specific examples.

(2) In order to emphasise possible force measurements we have made Sec. IV a more central part of the paper, extending it with estimates of the required force sensitivity and a discussion of braiding timescales. We now also make this point in the abstract. Several experimental references have also been added to make contact with present technologies. We feel these additions give a fair representation of the plausibility of this approach without over-selling the likelihood of success with present set-ups.

(3) As part of the added subsection IV.A we discuss the consequences of thermal vortex motion and the required properties of the vortex pinning potential to avoid a substantial suppression of the parity disparity.

Response to referee 2:

We again thank the referee for their positive assessment of our work, and their helpful suggestions.

(1) Further to our general comments on force measurements above, we have included specific estimates concerning the compound in Ref. 28 (previously Ref. 22) in the two last paragraphs before subsection IV.A. The suggestion that we discuss the effect of pinning potentials is also helpful. We have added a discussion of pinning potentials in the second paragraph of subsection IV.A.

(2) We agree with the referee that a greater discussion of higher-temperature effects is important, and the discussion combines naturally with the small-minigap effects suggested by the first referee. We have refined the consequences of Eq. (14) by adding a sentence below that equation along the lines suggested by the referee. We have also added another paragraph to Sec. III.C on the consequences of having a continuum of in-gap states. We emphasize these points in the Introduction and Conclusions.

(3) We again thank the referee for this helpful suggestion. We now discuss the poisoning time, leading to dephasing, in subsection IV.A, with associated references added. We agree that quantifying these effects is vital to any realistic proposal for implementing and measuring Majorana braiding; an in-depth study of dephasing effects is outside the scope of the current work, but plays a central role in a follow-up project.

List of changes

(1) A paragraph (including Eq. (15)) and a sentence have been added to the end of Sec. III.C. The new paragraph addresses the high-temperature limit of the parity disparity with a continuum of vortex in-gap states.

(2) Section IV is extended to include (at the end) two paragraphs on the force measurement consideration in an iron based compound and a new subsection, IV.A, on quasiparticle poisoning and thermal vortex motion in pinning potentials.

(3) Three sentences at the end of the Introduction, and one sentence at the end of the first paragraph of the Conclusions, have been added to emphasize the validity of the main result.

(4) A sentence has been added to the end of the abstract to advertise that we consider applications to read-out of Majorana qubits.

(5) References 9, 17, 20, 25, 27, 49, 53 - 59, and footnote 48 have been added.

Published as SciPost Phys. 6, 055 (2019)

Reports on this Submission

Anonymous Report 1 on 2019-4-13 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1901.09933v2, delivered 2019-04-13, doi: 10.21468/SciPost.Report.908


With the resubmission I find that the manuscript improved and the authors addressed all my previous comments/questions. However, it seems that the newly introduced discussion of quasiparticle poisoning is actually too pessimistic. The authors estimate the poisoning rate by the thermal activation above the minigap. If Eq. (17) would indeed be the limiting factor then the regime of temperatures of the order of the minigap that is discussed throughout the paper would lead to unpractically short poisoning rates. In fact, Eq. (17) would reintroduce the condition that temperatures much smaller than the minigap are required. Fortunately this is not necessary since the poisoning rate of interest is that of external quasiparticles entering the vortex. Ideally, the latter rate is exponentially suppressed with the full gap of the bulk superconductor. In practice the possible presence of non-equilibrium quasiparticles introduces a more complicated expression for the poisoning rate but this might be difficult to estimate at the current stage.

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

Author:  Henrik Røising  on 2019-05-08  [id 508]

(in reply to Report 1 on 2019-04-13)

We thank the referee for their further helpful comments. We agree that, in a strictly two-dimensional geometry, it is the full gap rather than the minigap which should appear in Eq. (17). We have corrected this in the published version of the manuscript.

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