Majorana bound states in encapsulated bilayer graphene

1. The authors propose a new way to engineer Majorana zero modes in graphene that do not require strong magnetic fields and electronic interactions.

2. The code to reproduce the results is provided.

3. Many aspects relevant to experiments are addressed.

4. The authors provide a comparison with other platforms.

1. Effects of disorder seem underestimated.

In this submission, the authors revisited many of the points suggested by the referees. I believe that the changes improved the manuscript. Before accepting for publication, I however still would like a better intuition on the interplay between lattice orientation and the existence of topological phases, as well as the effects of strong disorder due to the superconductor. As I comment below, I think that these points, although do not invalidate the theoretical model, have significant consequences of interest for upcoming experiments.

1. If I understand correctly, the authors argue that the difference between the armchair and zigzag orientations is just the dispersion. Namely, the main difference is that for zigzag orientation the Zeeman field opens a gap at zero energy, whereas for armchair edges the gap opens at finite energy. As a consequence, only for armchair interfaces there is an energy window with "spinless helical" edge states. Interestingly, the topological gap closing as a function of the crystal orientation occurs at $\sim 7^{\circ}$, which agrees with the transition from armchair to zigzag-like behavior of the boundary conditions in the continuum limit (https://doi.org/10.1103/PhysRevB.77.085423).

However, I point out that the nature of my question was not related to the effect of the proximitized $s$-wave gap in the helical edge states. Instead, I am curious on what are the symmetries that allow the existence of helical edge states in the first place.

In a similar setup, some of the authors have shown that intervalley scattering opens a trivial gap in the electronic structure that prevents the existence of topological superconductivity (https://doi.org/10.1103/PhysRevX.5.041042). Moreover, armchair boundary conditions break valley conservation (https://doi.org/10.1103/PhysRevB.77.085423) and are one of the origins of intervalley scattering in this previously proposed platform (https://doi.org/10.1103/PhysRevB.100.125411).

In this new setup, the system is gapped in the presence of Rashba spin-orbit coupling if the NS interface is armchair. So I wonder if the lack of valley conservation is somewhat related to this gap opening. Particularly, that would have important consequences under the presence of disorder, which I discuss in the next point.

2. Because the SOC gap and the superconducting gap have the same order of magnitude, the wavefunction should extend to regions with similar sizes on both sides. I, therefore, agree that disorder on any of the two sides should work.

However, I am not convinced that the form with which disorder was treated is compatible with experiments. I expect that the helical edge states will experience still a nearly ballistic edge, particularly because the envelope of the wavefunctions decay on a scale much larger than the region in which disorder is added (If I correctly interpreted the code in the GitHub repository, disorder is added as vacancies in the outermost row of atoms). Does this observation seem reasonable? I would imagine that an appropriate simulation should include disorder up to distances comparable with the wavefunctions envelope, as nearly half of the wavefunction should be inside a disordered superconductor. Am I missing something?

At this point, one could argue that the system is a quantum spin Hall insulator, and therefore, one would not expect disorder to have a significant effect. However, as I stated, I am still unsure which symmetries protect the quantum spin Hall state. And I think this point is important for the following reasons:

* One can model disorder in graphene edges as a random staggered potential $m({\mathbf{n}} \cdot \mathbf{r}) \sigma_z$, where $\mathbf{n}$ indicates the interface direction (https://doi.org/10.1103/PhysRevB.77.085423). In this case, both the symmetry operators necessary to preserve the band crossings no longer commute with the Hamiltonian ($[\sigma_z, \sigma_i]\neq 0$ for $i=x, y$). Thus I expect the crossings to no longer be protected. For this reason, I wonder how strong disorder will affect the topological phase. Particularly because I would expect $m({\mathbf{n}} \cdot \mathbf{r})$ to typically be orders of magnitude larger than the SOC, Zeeman, and superconducting gaps.

* (following the discussion in 1) If the presence of edge states depends on valley conservation, then disorder might have a significant effect. As we have recently shown (https://scipost.org/10.21468/SciPostPhysCore.5.3.045), MoRe side contacts with graphene likely introduce strong intervalley scattering. Thus, if related to intervalley scattering at all, the small gap opening at zero energy for armchair interfaces would be a lower bound. Realistic devices would have a much much larger scattering rate $\nu$. As a result, $h \nu \gtrsim \lambda_I$ would result in a trivial system.

Considering the two points above, I believe that disorder effects beyond what was simulated might be important to experimentalists. On the other hand, I expect that proximitized graphene, rather than side contacts, would suppress these problems both because of the interface smoothness and due to the screening of disorder by the superconductor (as recently explored in https://arxiv.org/abs/2207.02472). Therefore, I believe that the theory presented here might still be interesting for upcoming experiments. But the designed device might need some improvements that could be easily identified.

3. Since the authors investigated the effects of Fermi level mismatch, I suggest also providing the results on the GitHub repository.

1. Consider disorder over a larger region. For example, by adding a random onsite potential.

2. Show that time-reversal-invariant disorder can or cannot destroy the helical edge states.