Anisotropic sub-band splitting mechanisms in strained HgTe: a first principles study

1- Relevance to experiments. The work provides an explanation for subtle splittings observed in ARPES and could help guide interpretation of future measurements.

2- Impact potential. The connection to Berry curvature dipoles and superconducting diode effects enhances the broader interest of the study.

1. Weak treatment of symmetries.

2. The discussion about the different terms in the Hamiltonian goes mostly via numerical evaluation, making the conceptual understanding of how they affect the band structure a bit hidden (particularly the competition by the C4 term and the BIA).

1) Regarding the Weyl-node positions, I have some concerns. The manuscript remarks that in the k·p model the Weyl points lie in the ky = 0 plane, while in DFT+Wannier interpolation they are shifted away from ky = 0. On the other hand, the total number of Weyl nodes related by symmetry depends on their orbit of the k point upon action of the different symmetries.  If the ky=0 plane is a high-symmetry plane, then the total number of k-points is different if a Weyl node exists at ky=0 or at ky!=0. There can well be topological constraints for a Weyl node to leave from or to exist in  a high-symmetry plane, depending on what are the symmetries of the system.

Having a look at the small numbers associated with the coordinates in Table 2 (the ky coordinate given is -0.000270), one possibility is that this is just numerical uncertainty. One way to solve the issue lies in computing the associated Chern number with spheres centered at the Weyl nodes and smoothly increasing the radii.  The uncertainty may be estimated as the size of the smaller radii with which one gets nearly 1 for the Chern number. If this radii is of the order of ky, then it is simply not true that the Weyl node is away from the ky=0 plane, it is just the uncertainty. The authors may come out with a different way of approaching the issue. The bottom line is that some discussion and perhaps a calculation that convincingly explain/shows that the number of Weyl nodes in both DFT and model are the same, some sort of error bar associated with the Weyl node positions and, more importantly, what shall one expect from  the symmetries in the real material, are in order.

2) Related to the point 1, given that symmetry is at the heart of both strain responses and Weyl-node multiplicities, the absence of a systematic symmetry analysis weakens the conceptual foundation of the work. One would also like to know which symmetries are obeyed by the Hamiltonian Eq. 5 (at least, what of the symmetries in the actual compound are obeyed by the Hamiltonian).

3) A further clarification I would request concerns the relative role of the C4 strain terms and the BIA terms. As it stands, if I am not missing something, the manuscript shows numerically that the C4 terms dominate along some crystallographic directions (e.g. Γ–X) while the BIA terms dominate along others (e.g. Γ–K or Γ–L). However, this outcome may appear counterintuitive since BIA is an intrinsic property of HgTe, present even without strain, and one might expect it to always provide the leading contribution. The explanation is presumably that the symmetry of the BIA Hamiltonian forces its contribution to vanish along certain high-symmetry lines, leaving the strain-induced C4 term as the only effective source of splitting in those directions. I suggest that the authors make this reasoning explicit, rather than leaving it to numerical examples, so that the reader can more directly see why and where each term becomes relevant.

4) Small detail: Please check the font sizes in the figures. Some variations in the size of the fonts one can understand, but, e.g. in Figure 5 one can count like five different font sizes between labels, axis, etc.

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