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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_00004v2  (pdf)
Date submitted: 2023-07-20 14:42
Submitted by: Bespalov, Anton
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
Approach: Theoretical

Abstract

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


Author comments upon resubmission

Dear Editor,

Thank you for processing my submission. I am also grateful to the Referee for their comments, especially for pointing out important references that I missed. Below I provide a list of changes in the text and the replies to the Referee’s comments.

Best regards,
Anton Bespalov

List of changes

REPLY TO THE REFEREES AND LIST OF CHANGES IN THE MANUSCRIPT

----------------------------------------------------------------------
Report of the First Referee (Anonymous Report 1 on 2023-6-13)
----------------------------------------------------------------------

The Referee wrote:

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 α,β>2, since the leading order momentum terms of the dispersion relation change across this boundary.

Reply:

My statement concerning the asymptotic behavior of s_l in the first version of the manuscript was, probably, not very clear. In the revised version of the manuscript, Eq. (28) [formerly Eq.(33)] unambiguously indicates what the leading term looks like. For alpha>1, beta>1 and alpha≠beta the asymptotic behavior is always a power law. Including further terms in Eq. (A10) may be labor consuming, and it does not cancel the leading term. A brief discussion of this can be found after Eq. (A10).
A reference to the published version of [arXiv:2301.12514] (Ref. [13]) has been added in the introduction and in the beginning of Sec. IIIA. In the published version, the authors admit that the asymptotic behavior of s_l follows a power law even for alpha>3 and beta>2, however, their approach for some reason fails to capture this. In my opinion, this approach is not mathematically justified. It relies on the assumption that the BdG equations for an infinite chain have solutions proportional to a growing/decaying exponent. However, if we substitute such coefficients u_l and v_l into Eq. (2), we obtain divergent series for the power-law model. Thus, the applicability of the scattering approach to this model is questionable. A discussion of this has been added in a footnote - currently, Ref. [29].

The Referee wrote:

The author should also acknowledge Ref. [arXiv:2211.15690] which deals on similar matters.

Reply:

This is a very important reference. This paper (currently published, Ref. [20]) has an overlap with my work in the general technique and in the part where the power-law model with alpha>1 and beta>1 is considered. I give full credit to the authors, as they obtained their results a bit earlier than me. To minimize the overlap and to enhance the novelty of my work, I made major changes to the manuscript. They are as follows:
- the abstract has been rewritten;
- a part of the introduction has been rewritten after the words "However, it turns out that for a wide class of systems an exact analytical solution of our Wiener-Hopf problem can be obtained...";
- Ref. [20] is now cited where appropriate;
-the end of Sec. II around Eq. (20) has been extended;
-the section devoted to finite-range models (formerly Sec III) has been removed, as it is covered by Ref. [20];
- results related to the power law model are now contained in Section III;
- in the end of Sec. IIIC., new results concerning the numerical solution of the BdG equations in finite open chains with alpha<1 have been added. These results illustrate and complement the analytical considerations. A new figure (Figure 1) has been added.
- the analysis of the model with exponential falloff of hopping amplitudes is now contained in Section IV.
- in the end of Sec. V, new numerical results concerning the E_0 vs. chain length dependence in the case alpha<1 have been added. A new figure (Figure 2) has been added.
- the conclusion has been reworked.

The Referee wrote:

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.

Reply:

In the revised version, the unusual scaling of the entanglement entropy is mentioned in the introduction and in the beginning of Sec. IIIC. Also, a new reference (Ref. [27]) has been added. It seems to me that the Wiener-Hopf technique and the edge states are not directly related to these interesting features. Indeed, the logarithmic entanglement scaling takes place not only in the topological phase, but also in the trivial phase (in particular, for alpha<1 or beta<1), where edge states are absent. A short remark about this has been added in the beginning of Sec. IIIC. In addition, I mention there that the power-law model with alpha<1 and beta<1 might be relevant for the description of magnetic atom chains on two-dimensional superconductors.

----------------------------------------------------------------------
Report of the Second Referee (Anonymous Report 2 on 2023-7-4)
----------------------------------------------------------------------

The Referee wrote:

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).

Reply:

The papers [Phys. Rev. Lett. 130, 246601 (2022)] and [arXiv:2301.12514] (currently published) are now appropriately cited, and the manuscript has been reworked accordingly - see reply to First Referee. The paper by Jäger, Dell’Anna and Morigi (currently, Ref. [19]) gives me some concerns. The analytical derivation of the Majorana wave function is contradictory: the projection operator {\cal Q} does not act according to its definition. In addition, it does not seem right that the Majorana wave function [Eqs. (34)-(36)] depends on characteristics of an arbitrary chosen Hamiltonian H_0. However, in many cases an incorrect formula for the wave function may produce correct asymptotic behavior. These issues are discussed in a footnote (currently, Ref. [29], which has been extended compared to the previous version). The numerical results from Ref. [19] look reasonable, but they are related to the case alpha>1 and beta>1, while the present paper focuses on the case alpha<1.

The Referee wrote:

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.

Reply:

Calculations of correlation functions and of the entanglement entropy require the knowledge of the whole spectrum of the BdG equations. The Majorana edge mode does not seem to have a decisive influence on the entanglement entropy: indeed, the unusual logarithmic scaling occurs both in the topological and trivial phases (when edge states are absent). To calculate the mentioned quantities, a different technique should be used, as the Wiener-Hopf technique is insufficient. A remark about this has been added in the beginning of Sec. IIIC.

The Referee wrote:

The results are interesting and I would suggest publication, after requested changes are taken into account.

Reply:

I thank the Referee for their positive evaluation of my work and useful comments.


Reports on this Submission

Anonymous Report 2 on 2023-8-10 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202302_00004v2, delivered 2023-08-10, doi: 10.21468/SciPost.Report.7626

Report

I read the revised manuscript from the author and I acknowledge that he made a serious effort to address mine and the other referee concerns. Unfortunately, part of these modifications are bound to generate confusion rather than clarify the open issues. In my previous report I requested the author to clarify the issue of the decay of the Majorana edge states for power-law interactions with \alpha,\beta>2 due to the mismatch between the result of the author (and of Ref. [20]) with the ones of Ref. [13]. The author, in agreement with Ref.[20], obtains the power-law decay in Eq.(28)

s_{l}\propto l^{-\delta} with \delta=min(\alpha,\beta).

Ref.[13] finds that the above result is only applicable for \alpha<3 or \beta<2, while at larger decays the approximation of Ref.[13] yields that long-range interactions are in-influential to the edge modes decay.

In the revised version of the manuscript the author claims that the result in Eq.(28) is the leading one and that all additional terms are subleading. In order to justify the discrepancy with the finding of Ref.[13] the author also makes some general claims in a footnote (Ref. [29]), where he states that Ref.[13] is "most likely erroneous". In order to sustain this latter claim the author states that assuming $u_{l},v_{l}\propto exp(-\lambda l)$ with $Re(\lambda)>0$ yields a diverging result in Eq.(2) .

Actually, it is the claim of the author to be erroneous. In fact, if one assumes the exponential decay of the BdG amplitudes with positive decay exponent the summation on the l.h.s. of Eq.(2) is convergent and, accordingly, cannot serve as an argument against the scattering approach. I advice the author to remove his fallacious statement and amend the footnote accordingly. Also, I would advise the author to be more cautious when implicating possible mistakes in other published works.

More in general, it should be noted that all approaches of Ref. [13], of Ref.[20] and of the current manuscript, albeit using different methods, produce similar equations. The main discrepancy between the approaches of Ref. [13] and Ref.[20] depends on the way the polylogarithm singularities in the complex plane are treated. Indeed, Ref.[20] always keeps the non-analytic term in the expansion of the PolyLog. On the contrary, Ref.[13] only focuses on the leading order contribution, which becomes analytic for $\alpha>3$ and $\beta>2$.

The author's result is specular to the one of Ref.[20], but the author states that Eq. (28) can only be applied to the case of non-integer $\alpha,\beta$. This is in agreement with the aforementioned picture in which the non-analyticity of the polylog function plays a crucial role in the Wiener-Hopf method. However, this result leaves open two alternative scenarios:

a) For integer power-laws e.g. $\alpha=6$ and $\beta=4$, the decay of the modes is still power-law.

b) The power-law behaviour only occurs at non-integer values and the integer case displays exponential behaviour.

The author stated that his analysis unambiguously identifies Eq. (28) as the leading behaviour for the decay of the edge mode, but what scenario is to be expected? In the case (a) the author shall clarify how the power-law behaviour is recovered in absence of the polylog branch cut. In the case (b) the result will be quite unphysical since all integer power-law decays will behave differently from the non-analytic case. These question is not physically irrelevant since most of the naturally occurring power-law potentials display integer power-laws (gravity, coulomb interaction, dipolar, van-der-Wals,...).

In the revised version of the manuscript the author also includes some finite-size numerical study of the edge mode decay. This is certainly commendable. Nonetheless, the discussion of these results is incomplete and it generates several further issues rather than clarifying the aforementioned questions.

Issue 1: The author states: "When $\alpha< 1$ the slopes exhibit
significant variations with site number $l$, however, the
variations become less pronounced with increasing chain
length (compare the blue dotted and solid curves, corresponding to L = 10000 and L = 20000, respectively)". The poor convergence of the numerical data does not come as a surprise to me, since strong long-range couplings $\alpha,\beta<1$ are known to generate large finite size corrections. Yet, the numerical curves in Fig.1 clearly do not converge to a constant slope and the author only presents the study of a single size (apart from 1 set of values where both $L=10^{4}$ and $L=2*10^{4}$). This is far from being enough to argue convergence of the numerics to the analytics. I suggest the author to present some form of finite size scaling for various (exponentially increasing) system sizes 2^10, 2^{12}, 2^{14}, and so on, to show that the central slope is converging to the expected value of Eq. (28).

Issue 2: While I can live with some finite size corrections at $\alpha,\beta<1$, the black curve in Fig.1 with $\alpha=1.5$ and $\beta=3$ also does not reproduce the result in Eq. (28). Worst of all, the curve seem to converge to a fixed slope, but the slope does not coincide with the theoretical prediction. On this matter the author only comments: "The curve corresponding to $\alpha = 1.5$ exhibts a relatively stable slope, which is somewhat smaller than the one predicted analytically [Eq. (28)]", but this is far from being sufficient. Eq. (28) was supposed to be exact (at least at $1<\alpha,\beta<2$) how can the numerics not reproduce it? Why is the result smaller than the prediction? Are those very weak finite size effects? I urge the author to present a finite size scaling for the mode decay also in this region and present convincing evidence that Eq. (28) holds in the thermodynamic limit.

Issue 3: Since the author is performing numerical analysis, I would reccommend doing so also in the region $\alpha>3$ and $\beta>2$ where the aforementioned controversy between Eq. (28) and the exponential decay took place. The author may also try some large integer power laws, e.g. \alpha=6 and \beta=4 to investigate what one may expect in the limit of the applicability of Eq. (28).

Unfortunately, in the present form the paper is not suitable to be published, but a more extensive numerical study and a softening of the current harsh claims may make it a valuable addition to the literature.

Requested changes

-) Amend/soften incorrect/harsh statements in the footnote Ref.[29].

-) Comment on the destiny of the edge mode power-law tails for large integer $\alpha$ and $\beta$.

-) Numerical finite size scaling of the edge mode decay at $\alpha,\beta<1$ to verify its convergence to Eq. (32).

-) Numerical finite size scaling of the edge mode decay at $\alpha,\beta>1$ to verify its convergence to Eq. (28).

-) Numerical finite size scaling of the edge mode decay at $\alpha=6,\beta=4$ or large values to verify the possible limitations of Eq. (28).

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

Anonymous Report 1 on 2023-8-3 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202302_00004v2, delivered 2023-08-03, doi: 10.21468/SciPost.Report.7601

Strengths

1. The analytical results are strong
2. Good presentation and comparison with previous results

Weaknesses

1. Numerical checks are not properly explained and interpreted

Report

The authors has carefully considered the Referees' suggestions and changed the paper accordingly.
In particular, now recent literature is correctly mentioned and acknowledged.
I think the theoretical and analytical calculations are now complete.

However I have some questions regarding the added part on numerical evaluations of s_l and E_0(l).
Finite size effects can be very strong (especially when alpha<1) so that it is necessary to consider very long chains and values of l far from the (right) boundary of the finite chain considered in numerics. Also the theoretical predictions holds for l large. i.e. far from the left boundary l=0. This can be clearly seen in Fig 1(a), from which one can reasonably argue that the correct range to perform any fit lies between l=40 and l=5000, at least for L=20000.
This is indeed the range considered for the evaluation of the energy E_0(l) (Fig.2), which exhibits results agreeing with the theoretical predictions. Instead, the author has not used the same procedure for s_l and I wonder why a similar fit (in the same range) is not reported for curves such as the one reported in Fig 1(a), for different values of alpha.
Also, it would be appropriate to have an estimate of the error of the fit.
Finally, I have doubts on the relevance of the behaviour of d(ln s_l^2)/d(ln l) as function of l itself -shown in Fig 1(b). This quantity is calculated locally from a quantity such as s_l that on a chain of length L is for sure affected strongly by both the finiteness of the chain itself and of the lattice spacing.

Requested changes

Analysis of the validity of the numerics

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

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