Fragility of the antichiral edge states under disorder

Dear Editor,

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 (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.