A Fermi Surface Descriptor Quantifying the Correlations between Anomalous Hall Effect and Fermi Surface Geometry

The authors' response letter is quoted below:

Below I attach the point-by-point referee reply.

Referee 1

i) My main criticism is that the comparison between what can be achieved by this approached and what was previously accomplished based on calculating the Berry spectrum does not look entirely fair. The upper panel of Figure 2c is a plot of the experimentally measured AHC vs the dimensionless hyperpolic factor. The lower panel is a plot is a plot of the experimentally measured AHC vs the theoretically predicted AHC. The authors argue that the correlation seen in the upper panel is stronger. This is convincing. But the completion is not fair. The upper plot does not make any prediction on the absolute value. In order to make their case more crystal clear, the authors should make a statement the slope of the upper panel in addition to the difference in the standard deviation in the linear fits. Moreover, what is the physical meaning of the finite intercept in the upper panel?

• We thank the referee for this comment and addressing the issues raised above we added some clarifications to the text: “Even though as defined H_F does not give any predicted value of AHC, the found slope value of m = 1573 of the correlation can used as an empirical normalizing factor for the use of H_F as a predictive descriptor of AHC so that σ_AHC = wH_F , w = 1573(Ωcm)^−1.” The finite intersect is marked in figure as “limit for one EBR”.

ii) For the same reason, I recommend that they include a table listing the experimental quantities and the theoretically expected values in the two competing pictures. iii) As for experimental data, I strongly recommend to the authors to add more references and try to make an exhaustive list in order to dissipate any suspicion of cherry picking. It looks like that Dresden has been favored as a source of experimental data. Since in most cases, there is an experimental consensus, adding more references should strengthen and not weaken their case. Moreover, data on crystals are more reliable than data on films. I recommend to use the latter only when the former are absent, which is not the case of, for example Co2MnGa.

• We thank the referee for these fair and insightful comments. It may indeed appear that Dresden is favored as a source of experimental data, but this is simply due to our reliance on publicly available data from the work E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao, S.-Y. Yang, D. Liu, A. Liang, Q. Xu, J. Kroder, V. Süß et al., Giant anomalous Hall effect in a ferromagnetic kagome-lattice semimetal, Nature Physics 14(11), 1125 (2018), doi:10.1038/s41567-018-0234-5, where the requested by the referee table is present in the supplementary materials. There one can find the list of the experimentally measured values of AHC with the reference to the literature, where it can be found additionally with the corresponding calculated prediction. The work has been published in a reputable journal and authors are trusted experts, so we have no reason to suspect any manipulation of the data. Rather than duplicating the table, we chose to give proper credit by referencing their work. To address the referee's concerns, we have now explicitly referenced the corresponding table in the text.: “We compared those experimental AHC values (mainly from [21], supplementary information table S3) ...”

iv) I also think that the authors should discuss a number of specific cases. For example, what insight does this new approach provide in the specific case of nickel? What about KV3Sb5? The non-linear Hall response there arises in absence of magnetic order. Is it really an AHC or is it a feature of the geometry of the Fermi surface?

• We thank the referee for this comment and agree that it would be an interesting discussion to include. However, since our study is presented at an empirical level without a rigorous justification for the introduced metrics, such a discussion would not only feel a little speculative but also fail to address some of the concerns raised by the other referee.

v) Finally, there are a number of minor issues:

Page 2: “The accuracy issues appeared e.g. with simple compounds like Ni [4], where prediction gives a significant offset from experimentally observed AHC.” It may be a good idea to be more quantitative and specify that the agreement between theory and experiment is excellent for Co (477 vs. 480), not bad for iron (750 vs. 1032), and very bad for Ni (-2275 vs. -646).

• We thank the referee for this helpful suggestion and have incorporated it into the text: “The accuracy issues appeared even with simple compounds: while for Co and Fe Berry curvature based predictions give reasonable errors within 30% compared to the experiment, the N i predictions have about 250% mismatch [4].“

Page 2, introduction. It is worth to inform early the the uninitiated reader that EBR is “an elementary band representation”.

• We thank the referee for the comment and corrected to: “There are various reasons for that, but one could be that the current computational method fundamentally does not take into account the possible coexistence of multiple unconnected sets of bands, having one elementary band representation (EBR) at the Fermi level, as was recently presented in topological quantum chemistry as multi-EBR bandstructures “

Page 3 introduction. “Our research represents a significant advancement in the field of topological materials, offering a valuable tool for both theoretical investigations and practical applications.” I agree, but wouldn’t be wiser to avoid self-congratulation and let other investigators make such an acknowledgement?

• We thank the referee for the comment and corrected to: “Our phenomenological research not only provides a valuable tool for practical applications in the prediction of AHC, but also underscores limitations in conventional transport theories and suggests possible directions for further theoretical exploration. In the following sections, we outline the development and application of the description HF , providing evidence of its predictive capabilities and highlighting its potential impact on the field of topological materials research and its technological applications.”

Referee 2

(1) A general tone I sensed from the manuscript is that it claims their approach is superior to the more conventional Berry curvature approach. However, after carefully reading the text, it will only be fair for the authors to state more clearly that the method in the manuscript is empirical. The derivation in section 2 does not establish that these two approaches are equivalent or directly related.

• We thank the referee for this comment. It was not our intention to imply that our method is superior to others, pointing out its computational simplicity, which is fair to say compare to the Berry curvature approach. We just attempted to present H_F metric as an alternative to the Berry curvature, coming from a different theoretical foundation, which needs to be explored more (including its connection to the Berry curvature), motivated by this empirical study. In the revised manuscript, we have adjusted the tone throughout the text.
• We have made as well a significant extension of Section 2 to provide more theoretical motivations and clarify the challenges involved in developing a formal theory with rigorous predictive power, explaining why, at this stage, we are presenting only an empirical study. The requested explicit statement regarding the empirical nature of our work has been highlighted at the end of Section 2, and we hope it is now sufficiently clear: “For now, we aim to conduct a preliminary qualitative numerical study to provide compelling evidence that justifies a more rigorous investigation into the geometric description of transport theories.”
• Also some edits with regards to the connection with topological transport theory were made. Firstly in the introduction: “Nevertheless, an indirect link between the Fermi surface shape and the topological properties of the eigenstates can be assumed. Since the corresponding eigenvalues represent the action of the Hamiltonian on the eigenstates, i.e. ϵn (k) = H(ψnk ), when H behaves, for instance, as a homeomorphism, it preserves the topological properties [6], i.e. topological connectivity remains for {ϵn (k)}n,k and thus it should be reflected in the shape of the FS. However, a detailed analysis of the exact expressions for the equivalents of the Berry connection and curvature on the space {ϵn (k)}n,k falls outside the scope of this work. Instead, we focus on a preliminary qualitative exploration of the FS shape in relation to anomalous quantum transport phenomena, utilizing the standard Riemannian metric on {ϵn (k)}k .”
• In section 3.1: “Assuming that the shape of the FS reflects topological connectivity, the summation is performed over all connected bands (i.e., one EBR bands) that form the FS. This approach implicitly accounts for inter-band exchange by positing that the dynamics governed by the semiclassical equation 3 occur across different bands within one EBR”.
• “Such an extraordinary correlation with Berry curvature-based calculations would be highly unlikely if HF was entirely unrelated to the topological quantities. This numerical evidence suggests a potential for expressing transport dominated by the topology of eigenstates in terms of the band’s geometry, supporting the hypothesis of a deeper connection between the two. Exploring their relationship more rigorously could be a promising direction for future research.”
• In the conclusion: “Important future work includes exploring the full theoretical motivation of the method and its relation to the topological transport theory. The graphs in Figure 2b and Figure 3b suggest that hyperbolicity may result from the topological connectivity of the eigenstates. However, a direct equivalence with the Berry curvature cannot be concluded. For instance, as seen in Figure 3b, HF follows the general trend of Berry curvature-based values but smooths and averages its peaks. This may imply that HF reflects not only the Berry curvature but also a more general metric that encompasses it, such as e.g. the Fubini-Study metric [33].”

(2) In line with the above point, it is important that the authors clarify the limitations of the hyperbolic Fermi surface approach. For instance, the HF index does not describe the anomalous Hall and spin Hall conductivities of quantum anomalous hall insulators and quantum spin Hall insulators (as there are no Fermi surfaces in these cases). The manuscript should explicitly address these limitations.

• We thank the referee for this comment and added discussion on this matter in Section 2: “We emphasize that the same reasoning can be applied not only to the FS, but also to the energy band and the dynamics governed by d x / d t = 1 /ħ ∇ϵn (k). Consequently, non-trivial transport in insulating and semiconducting materials can also be linked to the underlying geometry and hyperbolic dynamics. In this case, however, since the band is a 3-dimensional manifold, there are not only more geometric types, but the dynamics on such manifolds is an active area of study. For example, the entropy formula for certain dynamical systems on them was discovered only recently, which famously led to the proof of the Poincaré conjecture [20].”
• And in Results section: “Quantum anomalous Hall insulators and quantum spin Hall insulators, however, cannot be included in our consideration as they don’t have a Fermi surface and the influence of geometry needs to be considered differently.”

(3) In the manuscript AHC is expressed in the format number*\hbar/e (Ohm cm)-1 a few times. The inclusion of \hbar/e is redundant and not correct.

• We apologize for this typo and corrected the units so they are consistent with figure 2.

(4) The discussion in section 2 on the mixing of orbits in a hyperbolic Fermi surface is not clear. More clarification and literature references are needed. What is the direction of magnetic field in Fig.1 for the hyperbolic Fermi surface?

• We thank the referee for their comment and understand the source of confusion. We face the challenge of making the manuscript both accessible and not overly burdened with theoretical details, while still maintaining the necessary level of mathematical rigour. We have added the requested explanations and references in Section 2, pages 4-5.

Other more minor points:

(1) Introduction Paragraph 1: “The are various contributions to AHE…” should be “There are…”

• We thank the referee and corrected the typo.

(2) Introduction Paragraph 4: “which computation does not include…” should be “whose…”

• We thank the referee and corrected the typo.

(3) Results Paragraph 4: what is “W3W”? The same typo appears in Fig. 2B.

• We thank the referee for this comment, but we didn’t identify the mentioned typo. If it was related to the spelling, then in our file-viewer everywhere it is shown as W3W with 3 as subscript, which perhaps may be different in a different pdf viewing system. If the referee meant conceptual confusion in this notation, we added clarification in the introduction “We also find that the HF matches predictions of the spin Hall conductivity (SHC) for Pt, Beta-W (W3W), and TaGa3.”

(4) Co, Fe, Mn3Ge, CuCr2Se4 are strictly speaking not 2D materials. The statement that the approach used in the manuscript is limited to the cases of 2D materials is not accurate.

• We thank the referee for the comment. Indeed, strictly speaking most of the materials from figure 2 are not pure 2D. We corrected the sentence in the conclusion to “… and also is limited to the cases of quasi 2D materials”
• To clarify restrictions in dimensionality we also added: “Since the HF parameter is currently defined for 2D conductance, where transport effects in the third dimension are negligibly small, we compared the calculated HF with measured values for thin films or layered crystal structures (indicated as (l) in Figure 2), where the layers have enough separation, so the contribution of the third dimension to the overall effect is relatively weak. In these cases, the AHE can reasonably be considered quasi-2D and compared to the HF.”

(5) No supplementary materials are available albeit referenced a few times in the manuscript

• We apologize for missing the supplementary information and provide it with the revision.