SciPost Submission Page
Fragility of the antichiral edge states under disorder
by Marwa Mannaï, Eduardo Filipe Vieira de Castro, Sonia Haddad
Submission summary
| Authors (as registered SciPost users): | Sonia Haddad |
| Submission information | |
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| Preprint Link: | scipost_202406_00054v5 (pdf) |
| Date submitted: | May 19, 2025, 9:27 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 insu- lator. 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. For energy conservation, the antichiral edge states are compensated by counter-propagating pseudo-bulk states. These counterin- tuitive edge modes are argued to be immune to backscattering and extremely robust against disorder. Here, we investigate the robustness of the antichiral edge states and their accompanying pseudo-bulk states in the presence of Anderson disorder. By com- puting different localization parameters, we show that, contrary to previous works, the antichiral edge modes are fragile against disorder since they exhibit a localization length comparable to that of their counter-propagating bulk states. They are found to be, as the bulk states, increasingly localized by increasing the disorder amplitude. We bench- mark this behavior with that of the chiral modes showing a strong robustness against localization regarding their substantially large localization length compared to the corre- sponding bulk states. We confirm the fragility of the anitchiral edge states by calculating their backscattering localization length of these edge modes. Our work provides insights to improve the transport efficiency in the burgeoning fields of antichiral topological pho- tonics 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
We are pleased to know that the first referee is “satisfied with our response” to his/her report and recommended the publication of our work in SciPost Physics.
We would like to thank the second referee for their valuable comments and recommendations pointed out in the second report.
We would like to resubmit to SciPost Physics our revised manuscript where we have carefully addressed the questions you raised. In the following, we give detailed responses to second referee’s report and a summary of the changes made in the text.
1. The second Referee mentioned that ”The newly added Fig. 8 does not perform the requested comparison between localization in a modified Haldane model and a regular Haldane model at a similar density of bulk states”
The figure was not intended to compare the localization lengths in the Haldane and the modified Haldane models, but to show the effect of the Dirac point energy shift on the localization of the pseudo-bulk states of the modified Haldane model. The latter are found to be particularly prone to localization regarding the decrease of the localization length with decreasing the Dirac energy offset, as the complex phase \Phi gets away from \Phi= \frac{\pi}2.
The localization of the pseudo-bulk states by tuning the shift of the Dirac point can be understood from the broken chiral symmetry induced by the mass term a_0= \sqrt{3} t_2 \cos{\Phi} emerging for \Phi_neq \frac{\pi}2. Since at 1D, the antichiral edge states are protected by the chiral symmetry, as argued by Colomes and Franz [23], breaking this symmetry fragilizes these modes and localizes the corresponding pseudo-bulk states as shown in Fig. 8.
To address the referee's suggestion and compare the localization in both models at a similar density of bulk states, we examined the localization length using the transfer matrix method. We considered macroscopically long cylinders with finite radii and equally long ribbons with finite widths. The transfer matrix method applied to the cylinders provides access to the localization length of bulk states at the specified energy. For ribbons, we anticipate an enhanced localization length whenever the system supports edge states at the chosen energy, as long as the bulk states exhibit a shorter localization length. The results are presented in figure 9.
In the left panel, we display the ratio between the localization length for the ribbon (with 'open' boundary conditions in the transverse direction) and the localization length for the cylinder (with 'closed' boundary conditions) as a function of the cylinder-diameter/ribbon-width (M). We selected an impurity strength of $U=3t$ and varying densities of n_I=0.3, 0.6, 1$. For the modified Haldane model (mHM), we plotted the localization length at zero energy for the three impurity densities. As shown in the right panel, the density of states (DOS) around zero energy is approximately the same for these densities (the DOS for the mHM is depicted in black). For the Haldane model (HM), we plotted the localization length at three different energies $E_1, E_2, E_3$, each associated with a given impurity density. These energies were chosen to ensure the model has a similar DOS to the mHM at zero energy. The three energies are indicated in the right panel.
From the left panel, we conclude that edge states in the mHM have a similar localization length to bulk states at the same energy, showing no special robustness of the antichiral edge states. This contrasts with the localization length of the chiral edge states in the HM at a similar density of bulk states: despite the small ribbon-width/cylinder-diameter sizes considered, which enable effective scattering between the two chiral edge states and bulk states due to disorder, the localization length of edge states is roughly an order of magnitude larger than that of bulk states.
We hope this addresses the referee's suggestion. These results have been added to the manuscript as subsection 3.2
2. The Referee pointed out that our “manuscript now concludes that the protection of the antichiral states is weaker than the protection of chiral states in the Haldane model” and that “this behavior matches the expectations of vast majority of researchers, and does not contradict the findings of Ref. 23”
In Ref. 23, the authors argued that the antichiral edge states are extremely robust and are immune against backscattering even at strong disorder. The authors have also found that the counterpropagating bulk states remain delocalized along the ribbon boundaries.
The authors did not discuss the mixing between the antichiral edge states and the corresponding pseudo-bulk states. Moreover, in Ref. [23] the authors do not discuss how the observed robustness translates into antichiral states which are expected to have a weaker protection than chiral states.
Contrary to the conclusions of Ref. 23, we showed that the antichiral edge states can be mixed with the pseudo-bulk states (Fig.5) and can be backscattered (Fig.9) at moderate disorder amplitude.
We also found that the pseudo-bulk states are localized by disorder or by tuning the energy offset between the Dirac points (Figs 7 and 8).
Furthermore, the new figure suggested by the referee helps clarify the underlying mechanism. The robustness reported in Ref. [23] can be understood as a consequence of the generally larger localization length of bulk states in the mHM compared to the HM, as evidenced by the comparison between Figs. 6 and 7. Since the antichiral edge states in the mHM exhibit localization lengths comparable to those of the bulk states, they appear robust at low disorder. However, this apparent robustness diminishes as disorder increases and the localization lengths of both edge and bulk states decrease. This behavior stands in stark contrast to the HM, where chiral edge states maintain significantly larger localization lengths than bulk states, even at comparable bulk state densities, highlighting their enhanced resilience to disorder.
Therefore, we believe our work provides, as stated by the first Referee, a “detailed discussion of disorder effect on antichiral edge states”.
3. The Referee pointed out that our “manuscript also puts forward as one of its findings in the abstract the fact that the antichiral states can be localized by coupling with the chiral states. This, however, was already clearly stated in Ref. 23.”
We did not mention in our work any possible coupling between antichiral states and chiral states. We assume the referee is referring to the coupling between antichiral edge states and counter-propagating bulk stats.
4. We addressed the minor changes required by the Referee. In particular, the data ranges and the font sizes in the plots.
We are looking forward to hearing from you.
Best regards,
The authors (Marwa Mannaï, Edurado V. Castro, and Sonia Haddad)
List of changes
List of changes (marked in red in the manuscript):
1. We changed the abstract to include the results dealing with the results presented in the added figure (Fig. 9).
2. We divided section 3 in two sub-sections to clearly distinguish the discussions concerning the localization of the bulk states and that of the edge modes.
3. We added three sentences in subsection 3.1 to discuss in more detail the comparison between the bulk states of the modified Haldane model and those of the Haldane model.
4. We added subsection 3.1 a paragraph to comment on the origin of the enhanced localization of the anti-chiral edge states depicted in Fig.8.
5. We added a new figure (Fig.9) and a paraph to comment this figure in the sub-section 3.2. This part has been suggested by the second Referee (the Editor in charge) to “clarify the new findings of the manuscript and their importance”.
6. We changed the conclusion to include the results of Fig.9.
Current status:
Reports on this Submission
Report #1 by Anton Akhmerov (Referee 2) on 2025-6-16 (Contributed Report)
Report
After reviewing the authors' response I have identified two factual inaccuracies and an open question that need to be addressed. Overall the manuscript improve, but unfortunately I cannot conclude that it contributes sufficiently positively to the literature in its current state.
In the updated version, the authors made their claims regarding the localization more precise. They have added Fig. 9, which compares localization in ribbons with periodic boundary conditions to ribbons and demonstrates the authors' claim that the localization length of antichiral states is comparable to that of bulk states. If correct, this claim would indeed be in a direct contradiction with the prior Ref. 23, which observed two different localization length scales: one for bulk states, one for the remaining edge and counter-propagating bulk states.
Factual inaccuracies
The authors' response and the manuscript makes several claims about the Ref. 23 and the relevant system that appear to be factually incorrect.
- Chiral symmetry
The authors write:
Since at 1D, the antichiral edge states are protected by the chiral symmetry, as argued by Colomes and Franz [23], breaking this symmetry fragilizes these modes and localizes the corresponding pseudo-bulk states as shown in Fig. 8.
The only mention of chiral symmetry in Ref. 23 is
In pristine graphene, the zigzag edge zero modes are protected by the chiral symmetry... which allows a topological winding number to be defined [15,16]. We review this topological protection in Supplemental Material [18] and show that it applies to the modified Haldane model as well.
The second sentence refers to the winding number, as evident from the appendix. Furthermore, chiral symmetry is broken already by the next-nearest neighbor hopping, and therefore it does not apply neither to the Haldane nor modified Haldane model.
- Discussion of localization and coupling to remaining bulk states.
The authors write:
In Ref. 23, the authors argued that the antichiral edge states are extremely robust and are immune against backscattering even at strong disorder. The authors have also found that the counterpropagating bulk states remain delocalized along the ribbon boundaries. The authors did not discuss the mixing between the antichiral edge states and the corresponding pseudo-bulk states. Moreover, in Ref. [23] the authors do not discuss how the observed robustness translates into antichiral states which are expected to have a weaker protection than chiral states.
Ref. [23] discusses the coupling between the antichiral states and the counter-propagating bulk states in the paragraph containing Eq. 5. That paragraph also explicitly states the expectation about the localization length of those states rather than claiming that they are immune against backscattering.
Open question
The estimate of the localization length of the antichiral states in the Eq. 11 of the manuscript is ∝W. On the other hand, the related Eq. 5 of Ref. 23 gives the localization length ∝W². This is a direct contradiction between two scaling estimates. Furthermore, the latter would imply that the protection of antichiral states grows faster with width than that of bulk states which is always ∝W. The newly added Fig. 9 seems to confirm the authors' claim and refute the observations of Ref. 23.
This evidence, however, is insufficient. Specifically, if the mean free path in the bulk becomes shorter than the spatial extent of the edge states, it is natural to expect that the distinction between the edge and bulk states vanishes. The data presented does not allow to determine whether the authors simulate a regime when the edge states are well defined. This could be established by reporting the bulk mean free path for the simulations in Fig. 9 and by comparing the localization lengths with periodic and open boundary conditions for a larger range of disorder strengths (smaller U or n_I).
An additional complication in analyzing the localization length is that it is likely that several length scales are present in the system, as shown in Fig. 3b of Ref. 23. Therefore the manuscript should explain in sufficient detail how the localization length was extracted, likely demonstrating the fits and demonstrating the data.
Furthermore, I believe it is appropriate for the authors to explain the difference between their estimate (Eq. 11) and the Eq. 5 of Ref. 23 since both are analytical computations.
If the authors convincingly confirm reliability of their analysis of localization length and explain the difference with the Ref. 23 analytical estimate, the manuscript would be suitable for a publication in SciPost Physics.
Requested changes
- Correct the discussion of the role of chiral symmetry: to the best of my understanding it does not apply.
- Correct the way Ref. 23 is discussed and do not attribute to it statements that are stronger than actually put forward in it.
- Present evidence that the localization length is extracted reliably, in particular in Fig. 9.
- Demonstrate that the comparison in Fig. 9 is performed in a regime where the antichiral edge states are not yet destroyed by disorder (compare with Fig. 3 of Ref. 23).
- Explain the difference in the estimate of the localization length (Eq. 11) with Eq. 5 of Ref. 23 and explain why Eq. 11 is correct rather than Ref. 23.
Recommendation
Ask for major revision