High-harmonic generation in Weyl semimetal β-WP2 crystals

As a quantum material, Weyl semimetal has a series of electronic-band-structure features, including Weyl points with left and right chirality and corresponding Berry curvature, which have been observed in experiments. These band-structure features also lead to some unique nonlinear properties, especially high-order harmonic generation (HHG) due to the dynamic process of electrons under strong laser excitation, which has remained unexplored previously. Herein, we obtain effective HHG in type-II Weyl semimetal β-WP2 crystals, where both odd and even orders are observed, with spectra extending into the vacuum ultraviolet region (190 nm, 10th order), even under fairly low femtosecond laser intensity. In-depth studies have interpreted that odd-order harmonics come from the Bloch electron oscillation, while even orders are attributed to Bloch oscillations under the “spike-like” Berry curvature at Weyl points. With crystallographic orientation-dependent HHG spectra, we further quantitatively retrieved the electronic band structure and Berry curvature of β-WP2. These findings may open the door for exploiting metallic/semimetallic states as solid platforms for deep ultraviolet radiation and offer an all-optical and pragmatic solution to characterize the complicated multiband electronic structure and Berry curvature of quantum topological materials.

2 But, before I finally suggest accepting this draft, I wish authors clarify/correct four important issues described as follows.
(1) In the simulation of the experimental θ-dependent HHG in β-WP2 crystals, why the authors just considered the two hopping coefficients (the nearest and next-nearest neighboring hopping). I wish authors make more discussion about this issue.
(2) Recently Nature Physics published a paper [Bai, Y. et al. Nat. Phys. (2020). https://doi.org/10.1038/s41567-020-01052-8] that reported the observation of HHG in topological insulator BiSbTeSe2. Authors should compare their results with this Nature Physics paper, and discuss the similarities and differences between these two works.
(3) There are some theoretical and experimental works reporting the odd-order HHG in (2020).). Authors should add them in the introduction to respect these works of HHG in topological materials. Both Dirac and Weyl semimetals may be good candidates to generate ultraviolet laser beam.
(4) English in this draft should be polished carefully. There are some obvious typo errors in the current version.

Reviewer #2 (Remarks to the Author):
Lv et al. report an interesting study of high-order harmonic generation (HHG) in semimetal β-WP2 crystals. There are two important results: First, they observe polarization-dependent HHG in a type-II Weyl semimetal. Second, they demonstrate how the polarization dependence of odd-and even-order harmonics can be used to extract the electronic band structure and Berry curvature around the Fermi-energy of the crystal. These are two major results, both of which are interesting and intriguing, however, a number of conceptual and minor issues should be addressed as discussed below.
1. I highly encourage the authors to check the English. Some writing is not correct and even confusing. I recommend a full manuscript language editing by a native English speaker, checking for typos, grammar etc.. that it HHG in a Weyl semimetal hasn't been reported before, but the manuscript would strongly benefit from a more scientific motivation.
3. While I find the overall study exciting, the manuscript needs improving for consistency. In particular, I would recommend introducing the experimental data before presenting the theoretical analysis of the even-and odd-order harmonics (which is hard to follow without reading the entire SI). The authors could achieve this by moving the theory paragraph (line 114-138) below the experimental paragraph. My recommendation is to have the experimental data stand for itself, independent of the specific interpretation, where the odd-order harmonics are attributed to Bloch oscillations, and the even-harmonics are interpreted in terms of Berry curvature effects. Figure 2e), you plot the HHG intensity versus pump intensity. Can you include uncertainties for the extracted fitting exponents? 5. For the discussion and theory results, I would have appreciated some more literature citations reflecting the open questions and controversial discussions regarding the physical mechanisms underlying HHG in solids. This deficiency makes it difficult to evaluate the validity of the model, seemingly matching the experimental data on a quantitative level. In addition, could you further elaborate why both odd-and evenorder harmonics show a quadratic intensity-dependence despite the seemingly different underlying physical mechanisms? Could this hint to a common thermal origin? 6. In Figure 4, the red lines retrieved through fitting the experimental data show jumps at the gamma points. Can you discuss this feature? Is it a fitting artifact? 7. In the supplement, the x-axis in Figure S3 d) is incorrectly labeled as B(T), instead of T(K). Overall, I believe this is a solid and interesting study that both established unambiguously the possibility of HHG in Weyl semimetals, and also uncovers interesting puzzles for future work concerning the nature of HHG in topological materials. Assuming that the minor issues and questions raised can be addressed, I recommend this manuscript for publication.

Reviewer #3 (Remarks to the Author):
The manuscript reports on a study of near-infrared/optical high-harmonic generation in the Weyl semimetal WP2. Even-and odd-order harmonic radiation up to the 10th orders is observed by exciting the material by 1900 nm laser pulses. While the observed odd harmonics are ascribed to Bloch oscillations, the even harmonics are interpreted as a result of Berry curvatures which is charateristic for a Weyl semimetal. Nonlinear optical effects are in general a very interesting topic, thus the present study fits the scope of 5 this journal. However, the interpretation of the experimental results has clear flaws.
1. For many solid-state materials the observation of only odd-order harmonics (see Table S1) is guaranteed by the existence of centrosymmetry in the crystal structure.
Here, WP2 crystalizes in a non-centrosymmetric structure, thus it is natural to observe the even harmonics, even without invoking the effects of Berry curvatures.
2. The band structure exhibits several bands crossing the Fermi surface, but only selected regions are compared to the experimental results as shown in Fig.4. 3. Since the energy of the laser pulses is quite high and allows interband transitions, even more bands can be involved in the dynamical processes, which however is not discussed in the manuscript.
With these considerations I cannot recommend the present manuscript for publication in Nature Communications.

Reviewer #1 (Remarks to the Author):
Lv, et al. reported the observation of both even-and odd-order the high harmonic generation in Weyl semimetal β-WP2 crystals. The highest order of harmonic generation can be as large as ten. Through analysis of polarization dependent intensity of harmonic generation, they concluded that the even-order harmonic generations are attributed to significant Berry curvature; while the odd ones come from the Bloch oscillations.
The discovery of Weyl semimetals is one of most important progresses in condensed matter physics within past 10 years. Up to now, the electronic band structures of Weyl points, as well as Fermi arc of electronic surface state, have been verified experimentally. Currently, this field faces an important question: can we find any useful functionality of these Weyl semimetals, except these remarkably electronic features?
Lv et al. reported the efficient the high harmonic generation (up to ultra-violet regime) in Weyl semimetal β-WP2 crystals. According to my knowledge, this draft reports the first experimental observation of the high-harmonic-generations (HHG) in Weyl semimetal metals.
Considering this background, I think this draft has two bright spots. Firstly, for HHG, this work dramatically expanded the materials candidate besides the normal semiconductors. The Weyl semimetals were proposed by the authors to generate uniform HHG due to the superior advantages including extremely high carrier mobility and "spike-like" Berry curvatures. Secondly, they can retrieve the electronic band structure and Berry curvature of multi-bands Weyl semimetals β-WP2 crystals through HHG experiments. This work demonstrates the HHG as a complementary method for ARPES to study the electronic band structure of topological materials. Therefore, I believe that this work can attract broad interests in the physics, optics, and materials science communities. Therefore, in my opinion, this paper is timely and systematic, and deserves the publication in Nature Communications. important issues described as follows.
Answer: Firstly, we'd like to thank you for your review time and your constructive comments on our draft. We carefully read your comments and re-analyze the original data in accordance with your suggestions. The responses to your comments are outlined as follows. The corresponding revises at the main text are labeled by red color.
(1) In the simulation of the experimental θ-dependent HHG in β-WP2 crystals, why the authors just considered the two hopping coefficients (the nearest and next-nearest neighboring hopping). I wish authors make more discussion about this issue.
Answer: Thanks for the reviewer's comments. In the simulation, we just considered the two hopping coefficients. There are two reasons: (1) the HHG is predominantly contributed by the nearest and next-nearest neighboring hopping; (2) considering more than two coefficients would lead to the uncertainty of the fitting results due to the more curve-fitting parameters.
(2) Recently Nature Physics published a paper [Bai, Y. et al. Nat. Phys. (2020). https://doi.org/10.1038/s41567-020-01052-8] that reported the observation of HHG in topological insulator BiSbTeSe2. Authors should compare their results with this Nature Physics paper, and discuss the similarities and differences between these two works.
Answer: Thanks for the reviewer's good suggestion. Bai, Y. et al. reported the observation of HHG in topological insulator BiSbTeSe2. They claimed that even-and odd-order harmonics have different origins, and the even-order harmonics that are produced from a freshly cleaved BiSbTeSe2 surface and the odd-order harmonics origin in the bulk. The even-order HHG can be the fingerprints of strong-field-driven helical Dirac fermions in the topological surface states. In our paper, we discovered effective HHG in type-II Weyl semimetal β-WP2 crystals under fairly low femtosecond laser intensity, where the odd-order harmonics come from the Bloch electron oscillation, while the even orders were attributed to "spike-like" Berry curvature at Weyl points.
Although the two papers all reported the observation of HHG in topological materials, they have many differences (especially, the one is topological insulator, and the other Weyl semimetal). Now the paper of Bai, Y. et al. is cited and discussed in our new draft.
(3) There are some theoretical and experimental works reporting the odd-order HHG in (4) English in this draft should be polished carefully. There are some obvious typo errors in the current version.

Answer:
We have carefully read the whole draft and corrected some typos. The corrected languages parts have been labeled by red color.
We hope that our replies could answer your doubts. Thank you again for your careful review on our work.

Reviewer #2 (Remarks to the Author):
Lv et al. report an interesting study of high-order harmonic generation (HHG) in semimetal β-WP2 crystals. There are two important results: First, they observe polarization-dependent HHG in a type-II Weyl semimetal. Second, they demonstrate how the polarization dependence of odd-and even-order harmonics can be used to extract the electronic band structure and Berry curvature around the Fermi-energy of the crystal. These are two major results, both of which are interesting and intriguing, however, a number of conceptual and minor issues should be addressed as discussed below.
Answer: We'd like to thank you for your review time and your constructive comments on how to make our draft more solid and clear. Your comments are addressed one-byone as follows.
1. I highly encourage the authors to check the English. Some writing is not correct and even confusing. I recommend a full manuscript language editing by a native English speaker, checking for typos, grammar etc..

Answer:
We are sorry for the problem in our old manuscript. We have revised the whole manuscript and carefully proof-read the manuscript to minimize typographical, grammatical, and bibliographical errors. In addition, we have invited a native English speaker to check the language. We believe that the language is now acceptable for the review process.
2. The introduction suffers from exaggerating language, e.g. ('Discovery of Weyl semimetals is one of the most important progress in physics … ,'The most remarkable features of Weyl semimetals are existences of Weyl points … , … we believe that HHG is an ideal tool …'. Keep in mind that the introduction is not just a sales pitch. The reader may not want to know what you consider as most remarkable, or what you believe in. In addition, I would recommend adding one or two sentences stating a clear research question at the end of the introduction paragraph. What is the goal? It is clear that it HHG in a Weyl semimetal hasn't been reported before, but the manuscript would strongly benefit from a more scientific motivation.
Answer: Thanks for reviewer's good suggestions. We sincerely accepted your suggestions. In the new draft, we have modified the introduction and given a more scientific motivation to our present study. For your convenience, we copied new introduction as follows.
2. While I find the overall study exciting, the manuscript needs improving for consistency. In particular, I would recommend introducing the experimental data before presenting the theoretical analysis of the even-and odd-order harmonics (which is hard to follow without reading the entire SI). The authors could achieve this by moving the theory paragraph (line 114-138) below the experimental paragraph. My recommendation is to have the experimental data stand for itself, independent of the specific interpretation, where the odd-order harmonics are attributed to Bloch oscillations, and the even-harmonics are interpreted in terms of Berry curvature effects.
Answer: Thanks for reviewer's good suggestions. We are sorry for the problem in our old manuscript. According to the suggestions, we have moved the theory paragraph below the experimental paragraph. And the corresponding figures were recombined in the new draft. These changes will not influence the content of the paper.
3. In Figure   In what follows, we try to understand the physical mechanism of the abovementioned HHG in β-WP2. Currently, the physical mechanisms of HHG in solids, especially even-order HHG, are still unknown. For example, interband transitions and successive Bloch oscillations, sole Bloch oscillations of intraband electrons, and interband resonant high-harmonic generation have been proposed to explain HHG in solids [24,[28][29][30][39][40][41]. In this work, we tentatively propose that even-order HHG in β-WP2 comes from the Berry curvature mechanism, while odd-order HHG is attributed to intraband Bloch oscillations. Some qualitative discussions that rule out interband transition/resonance mechanisms and perturbative nonlinear optics mechanisms can be found in the SI.
The third reviewer also raised a similar comment, especially on the mechanism of even-order HHG in β-WP2. At what follows, we copied the answer here for your reference.
According to our literature review, the generation of even-order harmonics is proposed by two mechanisms: 1) the interband transition and successive the Bloch In our discussion, we took the Berry curvature mechanism. There are two reasons: 1) It has been established that there are Weyls points in β-WP2 crystals and corresponding Berry curvature [Kumar, N. et al. Nat. Commun. 8, 1642(2017; Zhang, K. X. et al. arXiv:2008K. X. et al. arXiv: .13553 (2020.]. In this condition, the kinetic equations of Bloch electrons naturally have the term of Berry curvature. 2) In the previous report, it has been proposed that the interband transition leads to the generation of even-order HHG (e.g. in semiconductor GaSe). But the intensity of even-order HHG due to interband transition is much weaker than odd-order HHG [Kaneshima, K. et al. Phys. Rev. Lett. 120, 243903 (2018).]. And in our experiment, we can see that the intensity of evenorder (for example second-order) is quite comparable to odd-order (third-harmonic generation). Therefore, interband mechanism generating the even-order plays minor role in our experiment, compared to Berry curvature mechanism. Similar discussions have been reported previously [Liu, H. Z. et al. Nat. Phys. 13, 262-265 (2017); Luu, T.
As to the power-law observed in the experiment, we double-checked fitting and the experimental data. The results indicate that the power law is not exact 2, but ranged from 2.1 to 2.5. We also found that the power-law of intensity-dependence high In addition, both odd-and even-order harmonics are originated from the Bloch oscillations, although internal magnetic field (Berry curvature) is involved in evenorder HHG. Thus, odd-order and even-order harmonic generation should follow the power-law with a similar power index. Here, we'd like to add a semi-quantitative discussion about the same power-law observed in even-and odd-order harmonic generation. The following is a brief proof.
The intensity of odd-order HHG ( ) In the above equation, we have used one asymptotic relationship The intensity of even-order HHG  6. In the supplement, the x-axis in Figure S3 d) is incorrectly labeled as B(T), instead of T(K).

Answer:
We are very sorry for this negligence in the old manuscript. Now the Figure   S3(d) is improved in our new draft (see Fig. A3).

Fig. A3
The dependence of charge carrier densities ne and nh, as well as carrier mobility μe and μh of electrons and holes on temperature, respectively.
Overall, I believe this is a solid and interesting study that both established unambiguously the possibility of HHG in Weyl semimetals, and also uncovers interesting puzzles for future work concerning the nature of HHG in topological materials. Assuming that the minor issues and questions raised can be addressed, I recommend this manuscript for publication.
In conclusion, we did carefully solid/elucidate our point in the new draft. We'd like to thank you again for your very constructive comments on how to refine our work, as well as how to present the work scientifically.

Reviewer #3 (Remarks to the Author):
The manuscript reports on a study of near-infrared/optical high-harmonic generation in the Weyl semimetal WP2. Even-and odd-order harmonic radiation up to the 10th orders is observed by exciting the material by 1900 nm laser pulses. While the observed odd harmonics are ascribed to Bloch oscillations, the even harmonics are interpreted as a result of Berry curvatures which is charateristic for a Weyl semimetal. Nonlinear optical effects are in general a very interesting topic, thus the present study fits the scope of this journal. However, the interpretation of the experimental results has clear flaws.
Answer: Firstly, we'd like to thank you for your review time. Though the comments are quite critical but really constructive to make us re-consider/re-examine this work.
As follows, we'd like to make our points more clearly and solid.
1. For many solid-state materials the observation of only odd-order harmonics (see Table S1) is guaranteed by the existence of centrosymmetry in the crystal structure.
Here, WP2 crystalizes in a non-centrosymmetric structure, thus it is natural to observe the even harmonics, even without invoking the effects of Berry curvatures.

Answer:
We'd admit that we have not clearly discussed this issue carefully in the old draft. According to our literature review, there are about three mechanisms (nonlinear optical process, interband transition and Berry curvature mechanisms) for even-order harmonic generation in the materials with the non-centrosymmetric structure.
(1) In the viewpoint of nonlinear optics, if the crystals have non-centrosymmetric structure, the nonlinear electrical polarization P can be written as: We do can observe the high-order harmonic generation (HHG), especially even-order is not zero if there is no inversion symmetry. And we can see that intensity of second-harmonic generation is still proportional to the square of incident intensity. But if the order is higher than two, the intensity of corresponding n-order harmonic generation will proportional to n 0 I . For example, the intensity of fourth-harmonic generation 4 ∝ 0 4 . In our experiment, we found that the intensity of n-order-harmonic generation in β-WP2 crystals is approximatively proportional to 0 , power index x ranged from 2.1 to 2.5. Obviously, it is different from 2N-power in perturbative nonlinear optics. Therefore, we firstly rule out the possibility of the conventional nonlinear optical process in our even-order HHG experiment.
(2) In some papers, it has been proposed that the inter-band transition and successive the Bloch oscillation in conduction band leads to the generation of evenorder HHG (e.g. in semiconductor GaSe). But there are three evidences violating the inter-band transition mechanism: 1) In the answer of comment 3 described at follows, we experimentally substantiated that interband transition in WP2 is nearly negligible.
Accordingly, even-order HHG coming from interband transition contributes little to experimental one. 2) the intensity of even-order HHG due to inter-band transition is much weaker than odd-order HHG [Kaneshima, K. et al. Phys. Rev. Lett. 120, 243903 (2018);Liu, H. Z. et al. Nat. Phys. 13, 262-265 (2017).]. Differently, we can see that the intensity of even-order (for example second-order) is quite comparable to odd-order (third harmonic generation) in our experiment. 3) According to symmetry, the electrical polarization along c-axis is finite while that of a-axis is zero (a-and c-axis are two inplane directions of the surface of our β-WP2 sample). If the even-order HHG is generated by non-inversion symmetry, in theory, even-order HHG should show a twoleaf pattern, but in our experimental, experimental even-order HHG has four-leaf shape (see Fig. 4 at main text). Therefore, even-order HHG only considering symmetry is not (3) In some papers, the even harmonics generation is attributed to nonvanishing Berry curvature in the electronic bands of the materials with the non-centrosymmetric structure. In Table S1 at the supplementary information, we found 6 papers that reported the observation of even-harmonic-generation in semiconductors, 3 papers (α-quartz, Based on the above discussions, we think that the mechanism of even-order harmonic generation attributed to Berry curvature is the most natural explanation for our experimental observation. The relevant discussions have been added in the supplementary information of the new draft.
2. The band structure exhibits several bands crossing the Fermi surface, but only selected regions are compared to the experimental results as shown in Fig.4.

Answer:
We are sorry that we have not discussed this problem carefully in the old draft.
In our experiments, the infrared laser is perpendicularly incident to the surface of β-WP2 crystals, and then the electrical field of the laser beam is on the ac-plane of β-WP2 crystals, Accordingly, the A-vector is also along the ac-plane (in this work we use the   We'd like to mention that in each either conduction-or valence-band, there are two bands of quite similar shapes, which come from splitting of two-fold degenerated bands by spin-orbit-interaction. The parameters in the tight-bonding model of these twobranch conduction-/valence-bands are quite close. Numerically, two sets of tightbonding models cannot be distinguished after numerical optimization even initially we set two conduction-/valence-bands with different tight-binding-model parameters. So, we used only one band model to simulate these two-branch conduction-/valence-bands.
3. Since the energy of the laser pulses is quite high and allows interband transitions, even more bands can be involved in the dynamical processes, which however is not discussed in the manuscript.

Answer:
We did not discuss this important issue at the old draft. Actually, there are several reasons that we did not discuss the contributions of interband transitions in the manuscript. spectroscopy is a precise technique to investigate the interband dynamic process of materials. We performed TA measurement here to identify the probability of interband transition in β-WP2 system using a homemade femtosecond pump-probe system. The laser source was a commercial Ti:sapphire mode-locked laser centered at 800 nm with 120-fs duration and 80-MHz repetition rate. The output laser was split into two beams, with one frequency-doubled through BBO as the pump, and the other to be the probe beam. The sample was subjected to the two beams in the refection geometry.
As shown in Fig. A4 For the 1900-nm excited harmonics generation on β-WP2 in our work, two-photon absorption mechanism is responsible for the interband transition since 1900-nm photon energy (0.65 eV) is lower than the bandgap of ~0.8 eV. As we know, two-photon absorption is a third-order process, whose absorption cross section is typically several  With these considerations I cannot recommend the present manuscript for publication in Nature Communications.
In summary, we'd like to thank you for your constructive comments on our draft. We wish our explanations can elucidate your comments. Really thank you.

Reviewer #1 (Remarks to the Author):
The authors have addressed my concerns. I would now recommend the publication of this manuscript in Nature Communications.

Reviewer #3 (Remarks to the Author):
The authors have indeed spent efforts to answer my questions in their response letter, and also made modifications in the revised manuscript and SI. However, my major concerns have not been convincingly addressed.
1. Symmetry consideration is independent on the perturbative or nonperturbative nature of the dynamic processes. Indeed the experiment revealed a nonperturbative regime for the observed HHG, but solely based this one cannot claim that the even-order harmonics are not primarily due to the breaking of inversion symmetry. On the phenomenological level, for example, there is no a priori reason to omit the sine-function terms in the equation in Line 165. It is neither obvious why the cosine terms should be absent in Line 184.
2. It is unclear how the 400nm pump 800nm probe measurement can rule out an interband transition excited by the 1900nm pulse. For example, the 1900nm photon is sufficient to excite an interband transition close to the Z point along the T-Z direction.
Along the \gamma-Y direction the energy is also sufficient to excite an electron to higher conduction bands.
3. In the response various references were mentioned to support the claims of the authors, but most of them are not necessarily relevant to the specific Weyl system treated here. For example, the authors argued that " the intensity of even-order (for example second-order) is quite comparable to odd-order (third harmonic generation) in our experiment. This feature is also used to support the even-order HHG coming from Berry curvature mechanism in some papers [Liu, H. Z. et al. Nat. Phys. 13, 262-265 (2017); Luu, T. T. & Wörner, H. J. Nat. Commun. 9, 916 (2018).]". However, Liu et al dealed with a monolayer sample which has centrosymmetry in bulk, which is very different to the situation here. Moreover, isn't it more natural to ascribe the comparable intensity of the even-and odd-harmonics to the same mechanism?
To summarize, the flaws of the interpretation remains. It is not convincing to assign the observed even-and odd-harmonics to completely different mechanisms. Without addressing the raised points convincingly in the manuscript, I cannot recommend publication in Nature Communications.

Reviewer #1 (Remarks to the Author):
The authors have addressed my concerns. I would now recommend the publication of this manuscript in Nature Communications.

Answer:
We thank the reviewer for accepting our manuscript.

Reviewer #3 (Remarks to the Author):
The authors have indeed spent efforts to answer my questions in their response letter, and also made modifications in the revised manuscript and SI. However, my major concerns have not been convincingly addressed.
Answer: we'd like to thank you for your review time. These comments are critical and quite useful for us to re-consider our conclusion more deeply and present our data more clearly. Accordingly, we did further experimental work and clarifications to address these comments. All revises in the main text have been highlighted by red color. The replies to these questions are outlines as follows.
Before the detailed answer the comments, we'd like to point out the two points that are not clearly in the old draft and last reply.
(1) We have not clearly presented the crucial role of breaking of inversion-symmetry in even-order HHG. Actually, breaking of inversion symmetry gives rise to the Weyl semimetal state in β-WP2 crystals. Accordingly, there is "spike-like" Berry curvature in β-WP2 crystals. In turn, there is significant even-order HHG here.
(2) Both odd-and even-order HHG in β-WP2 come from the Bloch oscillation. Under laser acceleration, Bloch electrons are accelerated, this motion generates the normal odd-order HHG; simultaneously, if there is finite Berry curvature, under effect of Berry curvature (equivalent to an internally spontaneous magnetic field), electrons will have an additional anomalous velocity term (cyclotron movement). This anomalous velocity term leads to even-order HHG, which still belongs to Bloch oscillation. The schematic of above two-process is shown in Fig. 1.   Fig. 1. The schematic showing the electron's Bloch oscillation under both external electric field E of laser beam and Berry curvature Ωy(k). ∥ represents the velocity parallel to electric field E, while ⊥ does the anomalous velocity term coming from Berry curvature Ωy(k). ∥ and ⊥ give rise to odd-order and even-order HHG, respectively. Corresponding mathematic formulas can be found in equation (1)-(4) at the main text and section 10 of supplementary information.
1. Symmetry consideration is independent on the perturbative or nonperturbative nature of the dynamic processes. Indeed the experiment revealed a nonperturbative regime for the observed HHG, but solely based this one cannot claim that the even-order harmonics are not primarily due to the breaking of inversion symmetry. On the phenomenological level, for example, there is no a priori reason to omit the sine-function terms in the equation in Line 165. It is neither obvious why the cosine terms should be absent in Line 184.

Answer:
We do agree with referee that symmetry consideration is independent on the perturbative or non-perturbative nature of dynamic process. And we find that our presentation about the role of breaking-inversion symmetry does have problem at the old draft and last reply to the reviewers' comments.
Actually, breaking inversion-symmetry has close relationship to significant Berry curvature in β-WP2. Here breaking inversion-symmetry gives rise to the Weyl semimetal state in β-WP2, and accordingly there is a "spike-like" Berry curvature in β-WP2 crystals. In other words, here even-order HHG can be attributed to "spike-like" Berry curvature in Weyl semimetal β-WP2 that is resulted from breaking of inversionsymmetry. Therefore, we emphasized the role of breaking of inversion symmetry in the revised draft. The revised parts have been highlighted by red color in the main text.  Along the \gamma-Y direction the energy is also sufficient to excite an electron to higher conduction bands.

Answer:
We do admit that it is too hasty for us to rule out the interband transition excited by the 1900 nm pulse. And we do agree with you that there is possible electron transitions close Z point along the T-Z direction and along gamma-Y direction, as seen from electronic band structure.
To make the work more rigorous, we directly measured the dynamic process of interband transient absorption (TA) probed at 1900 nm (~100 fs pulse duration, 1 kHz repetition rate) with pumped at 800 nm (800-nm photons can excite all transition whose energy is lower than 1.55 eV) in a femtosecond pump-probe system. If there is obvious interband transition at the points along the T-Z direction and along gamma-Y direction under 1900 nm photons, clear decay in TA response on 1900-nm probe pulse should be detected. However, as shown in Fig. 4, we did not detect any discernible decay response at 1900 nm, even under the pump strength of 6.2 GW/cm 2 . For comparison, we also carried out the TA at several shorter probe wavelengths under the same 800-nm pump intensity, as shown in Fig. 4. One can see the distinct absorption response at 1000 nm, which means that there is stronger absorption than 1900 nm case. These results confirm the weak interband transition at 1900 nm.
To cross-check above experimental data, we calculated the imaging part εi(E) of dielectric constant ε(E) (E is the energy, and can be converted to wavelength of electromagnetic-wave by timing 1240 nm/eV) by first-principles LDA method. The data is presented in Fig. 5. One can see that the εI at 0.65 eV (1900 nm) is as small as 5.2, in contrast, there are a giant Drude peak at zero-energy and εI being as large as 20.0 at 2.1 eV. We also can see that theoretical εI is in line with TA experiment shown in Fig.   4, absorption at 1000 nm is larger than that at 1900 nm.
Combining above two data, we do believe that the inter-band transition of β-WP2 is quite weak and its role in observed HHG is immaterial. can see that the small ∆absorption at 1900 nm. The 800-nm pump intensity is 6.2 GW/cm 2 , and the probe intensities at different wavelengths are fixed at 0.6 GW/cm 2 .
One can see that at this energy range, dielectric absorptions have two peaks, one is the Drude peak at 0-energy (εimag=30); the other is the peak at 2.1 eV (εimag=20.0). εimag at 0.65 eV (corresponding electromagnetic-wavelength of 1900 nm used in our HHG experiment) is around 5.2.
3. In the response various references were mentioned to support the claims of the authors, but most of them are not necessarily relevant to the specific Weyl system treated here. For example, the authors argued that " the intensity of even-order (for example second-order) is quite comparable to odd-order (third harmonic generation) in our experiment. This feature is also used to support the even-order HHG coming from Answer: We admit that we have not discussed this problem clearly in the last reply, which may be due to our poor language ability.
Actually, the aim in citing these papers is to express that authors in these works used the Berry curvature to explain the observed the HHG. The similarity between our work and cited works is that there is finite Berry curvature in these systems, rather than samples used in cited paper are Weyl semimetals. The only difference in our work is that the Berry curvature shows "spike-like" feature in our case (β-WP2) that is a Weyl semimetal, but Berry curvature is a much smooth function in the samples in cited papers. The comparison of Berry curvature in our and cited works is shown in Fig. 6. Evidently, the Berry curvature in cited works is smoother/smaller than that in our case. Citing "Liu et al work of monolayer MoS2" is also because in mono-layer MoS2, there are broken-spatial-inversion-symmetry and finite Berry curvature, these features are quite similar to our β-WP2. We do not want to compare our β-WP2 to bulk MoS2.
One more thing we'd like to mention is that: in the samples in the cited papers, the crystal structures of these samples all have broken-spatial-inversion-symmetry.
This feature is the same as our sample (β-WP2).
As to "Moreover, isn't it more natural to ascribe the comparable intensity of the even-and odd-harmonics to the same mechanism?" comment, based on equations (1)-(4) at main text of this manuscript, even-order harmonic generation can be explained as  To summarize, the flaws of the interpretation remains. It is not convincing to assign the observed even-and odd-harmonics to completely different mechanisms. Without addressing the raised points convincingly in the manuscript, I cannot recommend publication in Nature Communications.
Answer: In summary, in the new draft we do have emphasize the role of broken inversion-symmetry in even-order HHG. It is the broken inversion-symmetry generating Weyl state in β-WP2 crystal; and even-order harmonic generation can be explained as Bloch oscillation under "spike-like" Berry curvature, while odd-one is attributed to Bloch oscillation. The even-and odd-order HHG in β-WP2 can be explained within the same quasi-classical dynamics of Bloch electrons.
We wish our new experiments and clarifications address your comments.

Reviewer #3 (Remarks to the Author):
If one plots the dispersion relation along a specific a-b-a direction of high symmetry (such as \gamma-z-\gamma in Fig.2 of the "Response to Referees Letter"), one should be surprised not to get an even function. In reality the band structure is of three dimension. Is anywhere parallel this high-symmetry direction a perfect even function?
I do not insist on further revisions, but leave this to the authors' decision. I am fine if this work is published together with the response letter.
Answer: Thanks for reviewer's comments. We are very sorry for the un-clear description for this problem at the last Response letter. This problem comes from our un-clear description/definition of high-symmetry points. Actually, we crossed the two adjacent Brillouin zones to choose k and -k (k is the Bloch wavevector) points to check whether electronic band structures E (k) are even-function of k. Here we have corrected the figure 5 of the main text (see Fig. 5 in the revised manuscript) and the figure 2 of the last "Response to Referees Letter" (see Fig. 1 here). Point Z, as well as X and X1, is the boundary of the Brillouin zone. Point Γ represents the center of the Brillouin zone, and point Γ' is the center of another adjacent Brillouin zone. As labeled in Figs. 1cd and Fig. 2, Γ-Z and Z-Γ' are two different paths with center inversion symmetry in the k-space. One can see clearly that eigenvalue energy E(k)=E(-k) (k is the Bloch wavevector) along the high-symmetry lines, therefore we can expand the E(k) as Fourier series only consisting of cosine terms.
As to the question "Is anywhere parallel this high-symmetry direction a perfect even function?", generally it is not. If chosen k-line does not cross the Γ-point (k=0), though it is parallel to the high-symmetry direction, we cannot find k and -k points in this specific line. Therefore, we cannot determine whether it is a perfect even function of k. But if chosen k-line cross the Γ-point (k=0), E(k) is definitely an even function of k, which is guaranteed by time-reversal-symmetry in β-WP2 and numerically confirmed by the first-principles calculations.
And more, we do agree to publish our manuscript together with the whole response letter of the reviewer. Wish the above-description answer this question! Really thank you for your comments on how to solid and refine our draft.