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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
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Submission summary
Authors (as registered SciPost users):  Felix Flicker · Tobias Meng · Henrik Røising · Steve Simon 
Submission information  

Preprint Link:  https://arxiv.org/abs/1901.09933v2 (pdf) 
Date accepted:  20190418 
Date submitted:  20190410 02:00 
Submitted by:  Røising, Henrik 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We study Majorana fermions bound to vortex cores in a chiral $p$wave superconductor at temperatures nonnegligible compared to the superconducting gap. Thermal occupation of Caroli de GennesMatricon 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 zerotemperature result is exponentially recovered well below the minigap (lowestlying 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.
Published as SciPost Phys. 6, 055 (2019)
Author comments upon resubmission
Both referees requested that we emphasize the effect of the parity disparity on the intervortex force, leading to a protocol for readout 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 setups. 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 deemphasise 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 reintroduced a longer discussion of the specific setup 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 ingap 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 readout 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 overselling the likelihood of success with present setups.
(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 highertemperature effects is important, and the discussion combines naturally with the smallminigap 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 ingap 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 indepth study of dephasing effects is outside the scope of the current work, but plays a central role in a followup 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 hightemperature limit of the parity disparity with a continuum of vortex ingap 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 readout of Majorana qubits.
(5) References 9, 17, 20, 25, 27, 49, 53  59, and footnote 48 have been added.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 2019413 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1901.09933v2, delivered 20190413, doi: 10.21468/SciPost.Report.908
Report
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 nonequilibrium quasiparticles introduces a more complicated expression for the poisoning rate but this might be difficult to estimate at the current stage.
Author: Henrik Røising on 20190508 [id 508]
(in reply to Report 1 on 20190413)We thank the referee for their further helpful comments. We agree that, in a strictly twodimensional 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.