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Fragility of the antichiral edge states under disorder
by Marwa Mannaï, Eduardo Filipe Vieira de Castro, Sonia Haddad
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Sonia Haddad |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202406_00054v3 (pdf) |
| Date submitted: | Nov. 28, 2024, 7:57 p.m. |
| Submitted by: | Haddad, Sonia |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Chiral edge states are the fingerprint of the bulk-edge correspondence in a Chern insulator. Co-propagating edge modes, known as antichiral edge states, have been predicted to occur in the so-called modified Haldane model describing a two-dimensional semi-metal with broken time reversal symmetry. These counterintuitive edge modes are argued to be immune to backscattering and extremely robust against disorder. Here, we investigate the robustness of the antichiral edge states in the presence of Anderson disorder. By computing different localization parameters, we show that these edge states are relatively fragile against disorder compared to the chiral modes. We confirm this fragility by calculating the backscattering localization length of the antichiral edge states. Our work provides insights to improve the transport efficiency in the burgeoning fields of antichiral topological photonics and acoustics.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Thank you very much for accepting to examinate our manuscript scipost_202406_00054v2 entitled “Fragility of the antichiral edge states under disorder” and for raising valuable comments which we have carefully addressed
We would like to resubmit our revised manuscript to SciPost Physics. In the following, we give responses to your comments.
We prefer to answer your feedback before you open a review round.
In your feedback, you mentioned that the antichiral edge modes may only localize by first scattering into the bulk modes.
We agree that the 1D description of the antichiral edge states, presented in the previous version of the manuscript, can be confusing regarding their connection to the pseudo-bulk states.
We then focused on the analytical derivation of the localization length l_edge of the antichiral edge modes due to their backscattering into the pseudo-bulk states.
We assumed that the latter are not fully extended over the width of the ribbon but are rather localized along the boundaries along a lengthscale \xi^{\prime}.
This derivation is already presented in Ref. Colomès and Franz (Ref.23 in the new version) but the authors assumed a fully extended bulkstates, which as mentioned by the authors is nt consistent with the numerical results of the conductance suggesting that the pseudo-bulk states are concentrated around the ribbon boundaries.
Using the Fermi golden rule, we found that l_edge is proportional to the ratio \xi^{\prime}/\xi, where \xi is the decay length of the wavefunction of the antichiral edge state. This ratio is inversely proportional to the corresponding IPR.
We then computed the disorder dependence of this ratio as a function of disorder. The results are presented in Fig.8 showing a drastic collapse of the ratio, which reflects a strong suppression of l_edge indicating the fragility of the antichiral edge states.
Our result agrees with a recent work (Guan et al, PRB 110, 165303 (2024)) where the autors computed the conductance of ribbons described by the Haldane and the modified Haldane model as a function of disorder. They found that the conductance of the modified Haldane model collapses while that of the Haldane model maintains its plateau under relatively strong disorder.
We are looking forward to hearing from you.
Best regards,
The authors (Marwa Mannaï, Edurado V. Castro, and Sonia Haddad)
List of changes
Current status:
Reports on this Submission
Report #2 by Anton Akhmerov (Referee 2) on 2025-3-3 (Contributed Report)
Strengths
Weaknesses
Report
An earlier work (Ref. 23 of the manuscript) reported that edge states in graphene nanoribbons with valley-dependent shift of Dirac points enjoy protection from localization. The closest analog of these systems are Fermi arcs in Weyl semimetals, and antichiral graphene nanoribbons is an interesting 2D simplification of the phenomenon.
This manuscript further investigates the claimed robustness and clarifies its context, in particular claiming in its current form that the protection is limited. I find this analysis relevant to the question whether antichiral graphene nanoribbons are a relevant research topic, and because of the earlier claim, it is a worthy addition to the literature both if the conclusion is positive or negative.
The manuscript puts forward a Fermi golden rule estimate for the scattering rate between the edge states and the bulk states. This one is width-independent. It should be compared to the 1D localization length of the bulk states in a nanoribbon, which is proportional to its width. That aspect of the analysis is somewhat implicit in the manuscript, but evident from the authors plotting λM/M (localization length normalized to system width).
The theme of analysis being somewhat implicit is my main feedback on the manuscript, see the requested changes below.
Requested changes
- The main numerical evidence of scaling of localization length is in Fig. 7. The figure shows localization length for the complete bandwidth, while only E=0 is relevant to the manuscript. Seeing the finite energy data makes it harder for the reader to figure out what happens at E=0, and also limits the amount of information provided about E=0.
- The Haldane model provided for comparison in Fig. 6 is somewhat confusing: there the localization length seems to be the smallest in the gap, whereas in this regime the chiral edge states should have an exponentially large localization length. Furthermore, because the Haldane model is only a validity check, this plot unlikely deserves an amount of attention comparable to Fig. 7.
- I believe a clear demonstration of the (lack of) robustness of antichiral edge states would be to compare the localization length with chemical potential corresponding to the Dirac point shift in the modified Haldane model. The manuscript would be strengthened by such a comparison.
- Comparing the numerical results of Fig. 7 with the analytical estimate of the localization length Eq. 11 would provide readers with a lot more confidence in the validity of the estimate.
In addition to the above qualitative improvements of the presentation, I would like to raise a minor concern regarding the plot quality:
- The font size in some plots is less than half of the text font size, which makes the plots only readable with high zoom.
- The data ranges sometimes make the plots contain relatively little information. For example, most of the plot area in Fig. 5 is occupied by the continuous spectrum of the bulk states.
Recommendation
Ask for minor revision
Strengths
- Detailed discussion of disorder effect on antichiral edge states
- Detailed computations of disorder effect on antichiral edge states
Weaknesses
- The conclusions are a bit overstated
Report
Requested changes
I feel that conclusions are a bit overstated. In the intro the authors state that "AC edge modes are not robust and can be *easily* localized
by defects ...", while in conclusions "topological invariant is *drastically suppressed* even at small disorder amplitude ...".
In reality, a clear distinction between mHM and HM is evident only by comparing panels (i) of Figures 6 and 7; these show that all states are getting localized in mHM while some states remain extended in HM. But, by any reasonable standard, this parameter regime corresponds to the limit of extremely strong disorder. Consider: at n_I=1 there is an impurity at every site of the lattice and its average strength is comparable to the bandwidth.
Given the above observation I would like to request that the authors soften their language somewhat and acknowledge (in the intro and conclusions) that very strong disorder is actually required to localize the antichiral edge states.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Author: Sonia Haddad on 2025-04-20 [id 5387]
(in reply to Report 1 on 2025-02-09)
We are pleased to know that the Referee recommended the publication of our work in SciPost Physics after amending our conclusions which, according to the Referee, are “ a bit overstated”.
The Referee mentioned that “ the conclusions are a bit overstated. In the intro the authors state that "AC edge modes are not robust and can be *easily* localized
by defects ...", while in conclusions "topological invariant is *drastically suppressed* even at small disorder amplitude ...".
The Referee stressed that “a clear distinction between mHM and HM is evident only by comparing panels (i) of Figures 6 and 7”.
We agree with the Referee that the behaviors of the bulk states associated to, respectively, the chiral and the antichiral edge states, are distinguishable only at high impurity concentrations, as shown in panels (i) of figures 6 and 7. These figures suggest that the pseudo-bulk states of the modified Haldane model (mHM) are robust against disorder and localize in the limit of high impurity concentration.
However, the robustness of the pseudo-bulk states is not a guarantee that the antichiral edge states will remain extended under disorder. Figure 8 shows that the localization length of a single antichiral channel collapses, at a moderate disorder amplitude, expressing the fragility of the antichiral edge states against disorder compared to the chiral states, for which the plateau at the Chern number C = 1 persists up to a strong disorder amplitude (U ∼ 4t ) (see reference 62).
We changed the sentence in the conclusion and the introduction as follows:
Introduction
By computing different localization parameters, we show that the AC edge modes are not
as robust as the chiral states, and can be localized by defects which mix them with their counterbalancing bulk modes. We confirm this fragility by calculating the backscattering
localization length of the antichiral edge states.
Conclusion:
Our numerical results show that, the quantization of the winding number associated to the antichiral edge states is strongly suppressed by disorder, contrary to the chiral states showing a persistent plateau at a Chern number C=1 up to a moderate disorder regime [62].
On the other hand, the results on the disorder dependence of the localization lengths revealed a rather robust character of the counter-propagating bulk states accompanying the antichiral edge states. The robustness of the former does not guarantee the immunity of the latter against disorder. Our analytical calculation showed that, at moderate disorder amplitude, the localization length of the antichiral edge states is reduced, by two orders of magnitude, compared to the pristine case.
We are looking forward to hearing from you.
Best regards,
The authors
Anonymous on 2025-04-28 [id 5421]
(in reply to Sonia Haddad on 2025-04-20 [id 5387])I am satisfied with the authors' response to my comments and recommend this version for publication.
Author: Sonia Haddad on 2025-04-20 [id 5386]
(in reply to Report 2 by Anton Akhmerov on 2025-03-03)We extend our gratitude to the referee for their insightful report. The referee raised several pertinent points, which we have addressed in the revised manuscript. Specifically, the referee noted that our work "provides a careful analysis of localization properties of the antichiral states in nanoribbons" but also mentioned that "the numerical data is sometimes hard to follow, and the comparison with the analytical estimate is missing."
We have thoroughly considered all recommendations and suggestions. Below, we provide detailed responses to each point.
1. The Referee pointed out that considering the complete bandwidth in figure 7 is somehow confusing for the reader since “the relevant energy window is around E=0”.
We changed the different panels in figure 7 as suggested by the Referee to clearly show the scaling behavior of the localization length.
2. The Referee pointed out that “’the Haldane model provided for comparison in Fig. 6 is somewhat confusing: there the localization length seems to be the smallest in the gap, whereas in this regime the chiral edge states should have an exponentially large localization length. Furthermore, because the Haldane model is only a validity check, this plot unlikely deserves an amount of attention comparable to Fig. 7.
Figure 6 shows the behavior of the bulk states of the Haldane model and not that of the chiral edge states (we use periodic boundary conditions in the transverse direction of the ribbon). Both localized and extended bulk states are seen as a function of energy. The later, which are the characteristic of disordered Chern insulators [67–73] appear far from the chemical potential and carry the Chern number. We have kept Figure 6 in the revised manuscript to benchmark the robustness of the pseudo-bulk states of the modified Haldane model to these extended bulk states.
3. The Referee proposed us to study the behavior of the localization length of the modified Haldane model as a function of the energy shift of the Dirac points.
We added a figure (Fig. 8) showing the localization length Lambda_M/M, computed by TMM for a nanoribbon width M=32, as a function of the complex phase Phi which governs the Dirac point energy offset. The latter is given, around the Dirac points, by a=\xi \sqrt{3} t_2 \sin{\Phi}.
The results show that the pseudo-bulk states of the modified Haldane model get less extended as the energy shift decreases towards zero, where the model reduces to a zigzag graphene ribbon with a flat E=0 edge state. The localization of the pseudo-bulk states by tuning the Dirac point energy offset, is a signature of the fragility of the antichiral edge states which strongly depend on their counterpropagating bulk states, as seen through the Fermi’s golden rule analysis.
4. The Referee suggested us to compare “the numerical results of Fig. 7 with the analytical estimate of the localization length Eq. 11” to “provide readers with a lot more confidence in the validity of the estimate.”
Figure 7 shows the localization lengths of the pseudo-bulk states while Eq. 11 provides an estimation of the localization length of the antichiral edge states. Therefore, the two results are meant to be complementary, and should not be compared.
To improve the plot quality, we have increased the font size and rearranged the data ranges as proposed by the Referee.
Best regards
The authors