Frequency-independent Optical Spin Injection in Weyl Semimetals

We would like to thank the referee for many helpful questions and comments. Below we give a point-by-point reply to referee’s comments.

“In Fig. 1(b), shouldn't the green (spin) arrows in hc all be pointing outwards, and those in hv all be pointing inwards ("hedgehog" pattern)?”
[REPLY] The hedgehog pattern is of course one example of the helicity. But generally speaking, a nontrivial helicity does not need to be tied to the hedgehog pattern. Here the pattern shown in Fig.1(b) actually corresponds to the model above Eq. 8, i.e., from a lattice model of the Weyl Hamiltonian. In our revised manuscript, we make this point clear.

“The caption of Fig. 1 refers to the "chirality" of light. Shouldn't it be "helicity", since it is referenced to the propagation direction?”
[REPLY] Both chirality and helicity are used for circular light in literatures. To avoid confusion, we use “circular polarization of light” in the caption instead in the revised manuscript.

“Figures 2 and 3 are almost identical, the only difference being that Fig. 3 contains one additional curve in panels (b,c,d). Would it make sense to simply replace Fig. 2 with Fig. 3, and change the text accordingly? (Possibly expand the last paragraph of Sec. 4 to include some of the material in the last paragraph of Appendix C)”
[REPLY] We thank the referee for this suggestion. In our revised manuscript, we replace Fig. 2 with Fig. 3 and expand the discussions.

“In both Figs. 2 and 3, the frequency range in panel (c) goes exactly from 0 to 3, whereas in panels (b,c) it goes slightly beyond 3, so that panels (b,d) are slightly "misaligned". Once they are aligned the tick labels above panel (b) could be removed, making the figure less busy.”
[REPLY] We thank the referee for this suggestion. We have modified the figure accordingly.

“Equation (10) is the same as (4), with an extra equality in the middle. Maybe replace Eq. (4) with (10), reducing by one the total number of equations?”
[REPLY] We have made this replacement.
“In Appendix C, τ1 and τ2 are dummy time-integration variables, while τ0 is the relaxation time. Would it be more clear to rename τ1 and τ2 as t1 and t2?”
[REPLY] We have made this replacement of labels.

“The relaxation time τ0 appears out of nowhere in Eq. (41). The way it is usually introduced is in the adiabatic switching on of the coupling to light. But do I understand correctly that then it should always appear in the combination ω+i/τ0? That does not seem to be the case in the denominator of Eq. (41). Also, in the definition of Gln below Eq. (41) is it really just ω→−ω, or should the sign change affect i/τ0 as well?”
[REPLY] As stated by the referee, the relaxation time appears due to the adiabatic switching. Mathematically, it appears as the imaginary part of the frequency as pointed out the referee. It always appear in the combination of frequencies. The reason is that, for positive and negative frequency, their imaginary part has the same sign due to the causality.
In the revised manuscript, we have added discussions about the imaginary part of the frequency.

“Slightly inconsistent notation throughout the text concerning the traced quantity.”
[REPLY] We have made the notation consistent in the revised manuscript.

“The title refers to "spin injection", but the abstract talks about "injection spin", and both forms are used in the main text. Use consistently the first form?”
[REPLY] We use the term “spin injection” in the revised manuscript.

“In the abstract, "multiband and lattice effect" should be replaced for clarity with "multiband corrections and lattice-regularization effects".”
[REPLY] We thank the referee for this suggestion. We have made the replacement in the revised manuscript.

“In the 5th line of the caption of Fig. 1, replace "velocity" with "velocity v", so that the symbol v has been defined before it appears again three lines below. Similarly, need to define (Δv)i below Eq. (2).”
[REPLY] We have made the replacement in the revised manuscript.

“Is the "static photoinduced spin magnetization" the same as the inverse Faraday effect? Is it correct that it does not require absorption/dissipation, while spin injection does?”
[REPLY] The spin injection is one mechanism of the inverse Faraday effect and as stated by the referee, it involves the absorption. It can generalize a static photoinduced spin magnetization after reaching the steady state. There is a similar phenomenon: the circular photogalvanic effect. It generates an injecting current which is static at the steady state.

“Miscellaneous typos: Second line of Sec. 2: "injectin". Below Eq. (2), after ωln=εl−εn there should be a comma, not a period.
Replace the arXiv preprint in Ref. [26] by the published article.
Fix capitalization in the article titles in the bibliography. For example, "taas" vs "TaAs" in Ref. [9], and many other similar issues. In bibtex, this can be achieved by placing the title entry inside double curly brackets.
The text could benefit from one more round of polishing revisions. Some examples: "the three plots shows" (p. 3); "the changing rate" instead of "the rate of change" (p. 3), "can be heuristically described [as] in Fig. 1(a)" (p. 3); the title of Sec. 3 would read better as "Spin injection and the helicity of a Weyl node"; "but the expression of βinj ii" ("of" → "for") (p. 5); "located at the kz axis" → "located on the kz axis" (p. 5), etc.”
[REPLY] We thank the referee for many helpful suggestions. We have made the revision accordingly.