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Majorana edge states in Kitaev chains of the BDI symmetry class

by Anton Bespalov

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Submission summary

Authors (as registered SciPost users): Anton Bespalov
Submission information
Preprint Link: scipost_202302_00004v1  (pdf)
Date submitted: 2023-02-02 15:23
Submitted by: Bespalov, Anton
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


Majorana edge states in Kitaev chains possessing an effective time reversal symmetry with one fermionic site per unit cell are studied. It is found that for a semi-infinite chain the equations for the wave functions of Majorana zero modes can be reduced to a single Wiener-Hopf equation, for which an exact analytical solution exists. The obtained solution can be used to analyze the wave functions of Majorana modes in Kitaev chains with finite-range and infinite-range hopping and pairing on common footing. We determine the asymptotic behaviors of the wave functions at large distances from the edge of the chain for several infinite-range models described in the literature. For these models we also determine the asymptotic behavior of the energy of the fermionic state composed of two Majorana modes in the limit of long (finite) Kitaev chains.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2023-7-4 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202302_00004v1, delivered 2023-07-04, doi: 10.21468/SciPost.Report.7444


1. The paper presents a novel approach to the problem that makes use of robust and rigorous analytical techniques.
2. The results are presented in a very clear and exhaustive way.


The paper studies the structure of edge states, but does not address other questions about the description of the different quantum phases, especially in the controversial regime of alpha,beta<1.


The paper deals with a class of interesting models (Kitaev chain with long-range interactions) that, despite their apparent simplicity, displays unusual properties and a rich physics that is yet to be fully understood. The proposed rigorous analytical method is crucial to obtain important results about edge states in this model, which are carefully explained in the manuscript.
The only concern is about a more precise comparison with previous results, such as the one cited in the bibliography (in particular ref. [18]) as well as more recent ones (e.g. Phys. Rev. Lett. 130, 246601 (2022), arXiv:2301.12514). This also implies to enlarge the analysis to other quantities, such as entanglement entropy and/or correlation functions that might shed light on the universal properties of the models. I understand this might be a lot of work, but at least some comments are at order.
The results are interesting and I would suggest publication, after requested changes are taken into account.

Requested changes

Comments about:
1. a stricter comparison with previous results and
2. about the possibility to calculate other physical quantities (e.g. entanglement entropy and/or correlation functions).

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

Anonymous Report 1 on 2023-6-13 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202302_00004v1, delivered 2023-06-13, doi: 10.21468/SciPost.Report.7343


1) Clarity and exhaustiveness of the presentation.

2) Novel and relevant technical developments.


3) Lacks further discussion of crucial aspect of the problem.


I have read the manuscript by the author and I was positively impressed by the in depth investigation of the edge state of the linear chain with different couplings shape. The outlined analytic method appears to be very powerful and i have no reason to doubt the correctness of the calculations. I am inclined to recommend the paper for publication provided that the author is willing to consider a few revision.

The major question mark concerning the paper appears in below Eq. (33) where the author reproduces the result of Ref. [18], but the author expresses some doubts about whether this is the leading behaviour. The author shall make some major comment on what is expected to occur to Eq. (33) upon including further terms in the expansion in Eq. (A10). Indeed, the recent work [arXiv:2301.12514 ] showed how the result in Eq. (33) is not recovered by the scattering approach for $\alpha,\beta>2$, since the leading order momentum terms of the dispersion relation change across this boundary. The author should also acknowledge Ref. [arXiv:2211.15690] which deals on similar matters.

It would be also interesting to comment on how the edge states influences other quantities aside the energy, such as the entanglement entropy of the open hand portion of the chain with respect to the infinite system. In this perspective, please notice that enhanced entanglement scaling is one of the major features of long-range Kitaev chains


Requested changes

1) Comment on the long distance scaling of edge states for \alpha,\beta>2

2) Comment on the possibility to evaluate the entanglement entropy of the system with the method described in the paper.

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

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