Visualizing near-coexistence of massless Dirac electrons and ultra-massive saddle point electrons

Dear Editors,

We thank the reviewers for their positive review and valuable suggestions for improving the manuscript. We have introduced into the text all the changes suggested by the reviewers. In the following we describe all the modification and provide a response to all the points raised by the reviewers.

Best regards,
Nurit Avraham
On behalf of all coauthors

Requested changes by reviewer 1:

1. What is the assumed energy as a function of momentum dependence of high-order van Hove singularity observed and shown in Fig.3. I would suggest Authors to expand discussion of symmetry properties of material that create and protect this van Hove singularity.

All the band structure calculations shown in Fig. 3 (d-g, and inset of a) are in fact ab-initio calculations. Regarding the symmetry properties that create and protect the van Hove singularity, as we explain in the text in the paragraph that starts with “Close inspection of the Bi(110) band structure around Γ…”, We believe that the enhanced power-low originates from the extended flattening of the surface band along the \gamma-X1 direction which continues almost across the whole Brillouin zone, and not from the immediate local surrounding of the C2 symmetric saddle points. As suggested by the reviewers, in the revised version we expand (page 3, last paragraph and page 6, second paragraph) on the symmetries properties of Bi in general and of Bi 110 surface in particular.

2. On page 18 the link to Overleaf appears. I would suggest Authors to use standardized ways to publish experimental data and codes for analysis such as Zenodo repositories.

This is now corrected

3. The DoS plots presented in Fig.1 and 3 contain a clear peaks related to the main (high-order) van Hove singularity and additional small peak structure that might be related to usual van Hove singularities. In the dispersion shown in Fig.3f, e, g Authors identify positions of van Hove singularities and the approximate shape of constant energy contours thus concluding about high-order type of saddle points. From these results it might be possible to identify which saddle points exactly produce high-order van Hove singularities and why divergence exponents are different on different sides of the peak in DoS. I would suggest Authors to extend the discussion of these features, as it seems that it is already partially present in the data shown.

The saddle points along the \gamma-x1 direction are regular saddle points that appear in the dft calculation at ~200 meV. These are identified with the small peak that we observe. The saddle points along the \Gamma-X2, above ~230meV are nearly C2 symmetric high order saddle points, however their tangential flat band extends across almost the whole Brillouin zone. The large peak that we observe is identified as the high order singularity that originates from the extended flat area associated with these saddle points. As suggested by the reviewer we clarify this further in the revised version (page 6, second paragraph).

4. What is the energy resolutions in DoS (dI/dV plot) presented in Fig.3

The energy resolution in Fig. 3 and throughout the paper is determined by or locking excitation, Vac, which is 10mV.

5. I would suggest Authors to correct typos in the text and in the references list (such as in Refs.15, 16, 34 etc)

We went over the text and corrected the typos.

Requested changes by reviewer 2:

(1) In the abstract, I suggest being clear about which quantity is diverging and exactly how the exponent is defined, for example DOS(ω)∝ω−0.7

As requested, we modified the line in question in the abstract to: “We show that the 110 surface of Bismuth exhibits high order VHS with an usually high density of states divergence ∼ (E)^−0.7”

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(2) In the abstract I would mention the role that theory played in reaching the conclusions about the band structure. For example, I would modify the existing sentence to the following: “Detailed mapping of the surface band structure using scanning tunneling microscopy and spectroscopy combined with first-principles calculations shows that this singularity occurs in close proximity to Dirac bands located at the center of the surface Brillouin zone. “

-We modified the sentence accordingly.

(3) For the opening sentence of the introduction I suggest “..ascertaining..” → “…determining..” since the DOS has a causal effect on the interacting aspects of the physics.

-We modified the text as suggested.

(4) Next line, “..upsurge..” →”…an upsurge..”. Following line, “..logarithmic…”
→“.. a logarithmic..”, etc. Please check use of articles throughout manuscript.

-We modified the text as suggested.

(5) In the section on “High-order van Hove singularity”, do the authors mean to quote a value of -0.25 (rather than 0.25) for magic angle graphene? (Also, angle is misspelled “angel” there.)

-Yes, we quote -0.25. This appears in Yuan, N. F. Q., Isobe, H. & Fu, L. Magic of high-order van Hove singularity. Nat. Commun.10, 5769 (2019), page 4, below equation 7.