Energy magnetization and transport in systems with a non-zero Berry curvature in a magnetic field

We thank the referee for their very careful reading of our manuscript and for highlighting the strengths and weaknesses in it along with providing several suggestions for its improvements. Please find below our response to the referee’s comments, suggestions and criticisms.

The referee says :
Strengths: The strength of the manuscript lies on the detailed derivation of the charge and energy magnetization of Bloch electrons using an alternative approach based on Einstein relation.

Our response: We thank the referee for highlighting the strengths of our manuscript.

The referee says: Weaknesses

Our response: We thank the referee for also pointing out the weaknesses in our manuscript. However, as we explain below, we respectfully disagree with the referee that these are necessarily weaknesses.

The referee says: 1. Although the authors provide detailed derivation of charge and energy magnetization, they have not applied their formalism to any particular system.

Our response: As we have also mentioned in our response to a similar comment by referee 1, our intention here is to provide a treatment of transport and diamagnetic currents in systems with both B ̸= 0 and Ω ̸= 0 in the most general terms that would be applicable to any system rather than focus on specific systems. In this regard, it is very similar in scope to previous papers on topological systems such as Refs. 58 and 62, which too did not perform calculations on model systems. We thus believe our paper will also be a valuable contribution to the broad field like those other papers.

The referee says: 2. They do not provide the explicit expression of Hall conductivity (which is one of their main results) in the main text.

Our response: This point too has been brought up by referee 1. As we have clearly mentioned in the manuscript, we have derived an expression for the non-equilibrium distribution function in the presence of both a non-zero Berry curvature and non-zero magnetic field. This expression can be used to obtain all charge and heat transport coefficients, in linear response in temperature and potential gradients and not just the Hall response. The terms involving the scalar product of the field and Berry curvature, i.e. B.Ω provide contributions to these coefficients (including the Hall conductivity) in addition to those already derived in the literature in the presence of either B or Ω alone. The scope of the current work is limited to deriving only the expression for the non-equilibrium distribution function and not providing detailed expressions for all the transport coef-ficients. Nevertheless, the reason we singled out the Hall response for mention is that it is the most widely studied of the transport coefficients in topological systems and so the existence of the additional contributions to transport in general is perhaps best highlighted using the Hall response as an example.

The referee says: Report The authors provide an alternative approach (based on the Einstein relation) to derive charge and energy magnetization. They use the approach only to recover old results. There- fore, I feel there are not many new results in the manuscript. Another point is that the authors did not compare their approach with the old one (why one should use their approach to derive charge and energy magnetization?).

Our response: We thank the referee for their comments. As we have already emphasized, our approach is based on general considerations related to the Einstein relations rather than based on specifics of the microscopics of model systems. The energy magnetization is a rather opaque quantity compared to its charge counterpart. We feel that our derivation based on the above general considerations simplifies its understanding compared to those that appeal to microscopics, which currently exist in the literature. We have added a line to this effect in the modified manuscript.

The referee says: The authors claim that the non-equilibrium distribution function obtained in this work can generate a new type of regular Hall response in time-reversal invariant samples with a non-zero Berry curvature. Although the authors have presented detailed derivation for the rest of the calculations, however, they did not provide the expression of Hall conductivity anywhere. Regarding that I have few questions

Our response: Please find below the responses to the questions brought up by the referee.

The referee says: 1. Please write down the explicit expression of the different components of the Hall conductivity.

Our response: As we have mentioned earlier, the Hall response is not the focus of this work. Rather, it is the expression for the non-equilibrium distribution function in the presence of both Ω ̸= 0 and B ̸= 0 from which, in principle, one can derive all the heat and charge trans- port coefficients by integrating with appropriate kernels. The Hall response is non-linear in the magnetic field (as also pointed out in our response to referee 1) and thus does not have a simple closed form expression. Nevertheless, we have added a discussion after Eqn. 24 on page 7 explaining the effect of each of the terms to Hall response.

The referee says: 2. How one can distinguish different Hall components from each other in experiment?

Our response: The different components of the Hall response contribute with different functional dependences on the magnetic field. Since the field in the non-equilibrium distribution functions appears only through the combination e Ω(k).B, with the momentum k be- ing integrated over, the functional dependence on B will be determined by the precise dependence of Ω(k) on k. There is thus no simple way to parse the contributions experimentally without doing an explicit calculation for a specific microscopic system. However, a general feature of the field dependence coming from the presence of the term e Ω(k).B is that it is non-linear in B as opposed to the Hall response in a regular metal 􏰀 or even Chern insulator, in which it is linear. The deviation from linearity in the field could be an experimental signature of the contribution from the non-equilibrium distribution function that we have derived.

The referee says: 3. What is τ scaling for the new regular Hall effect? What are symmetry constraints for this Hall conductivity? Our response: In the discussion added after Eqn. 24, we have commented on the τ dependence of the different terms in the expression for the Hall conductivity. We are not sure what the referee means by “symmetry constraints” for this Hall conductivity but it obeys the usual Onsager symmetry relations like the Hall conductivity in any other situation.

The referee says: 4. The authors should calculate the new regular Hall conductivity for bilayer graphene system and then compare with other coexisting Hall components.

Our response: As we have emphasized before, the Hall conductivity is not the focus of this work. It is the calculation of the non-equilibrium distribution function from which all charge and energy transport coefficients can, in principle, be obtained. The Hall conductivity is just one of these, even if it is the most commonly studied one in the literature. A calculation of only the Hall conductivity detracts from the significance of the distribution function to the calculation of all transport coefficients. At the same time, a calculation of all the transport coefficients is too involved to be within the scope of what we are trying to achieve here. Also, as mentioned before, our work develops a general formalism without focusing on any specific microscopic system along the lines of other works such as Refs. 58 and 62. We mention bilayer graphene since it is an example of a system in which Ω ̸= 0 due to broken inversion and not time reversal symmetry and thus is of the type that our formalism might be useful to study. However, there could be other such systems and so we do not think that studying this specific system as an example is necessary. This is more so because the calculation for Dirac materials can be technically quite involved due to the singular behavior of the Berry curvature, as we have pointed out in the discussion added after Eqn. 24. It should thus be the subject of a self-contained investigation of its own well beyond the scope of our general treatment here.

The referee says: 5. The authors consider relaxation time approximation. Please discuss the regime of validity of this approximation.

Our response: The relaxation time approximation employed here has the same regime of validity as in all other such studies based on a non-interacting description of topological systems. The main assumption is that transport can be described in terms of non-interacting electrons or quasiparticles with momentum relaxation in the bulk due to interactions with other degrees of freedom like phonons and impurities. The approximation breaks down in the so-called “ hydrodynamic regime” in which the electrons collectively behave as a fluid with bulk momentum conservation. This regime is usually accessible in a rather restricted region of electron density and temperature in which the time scale obtained from the shear viscosity of the electron fluid is shorter than that from interactions between the electrons and other degrees of freedom. The relaxation time approximation thus applies in a much broader regime of parameters.

The referee says: Requested changes Major changes 1. The authors should write down the explicit expression of different Hall components using their formalism and explain how they can be distinguished from each other in experiment. 2. They should calculate the new regular Hall response for the bilayer graphene system.

Our response: We respectfully disagree with the referee that either of these changes is required since, as we have emphasized above, our work is about the derivation of the non-equilibrium distribution function from which all transport coefficients can be derived and not just the Hall response. Further, our formalism is also meant to be general in scope and not focused on any particular system. We have added a short discussion after Eqn. 24 on why the calculation of the Hall conductivity for specific Dirac systems can be technically involved. We thus think it should be the subject of a self-contained investigation of its own and it is not very reasonable to expect us to fit it within the scope of our general treatment here.

The referee says: Minor changes 1. Appendix A and Appendix J can be included in the main text. 2. Appendix B is very well known. So the authors can remove it from the manuscript. 3. There are several typos and grammatical errors in the manuscript which authors should fix in the revised version. 4. Some of the equations in the main text do not align properly. The authors should consider them to align properly within the margin.

Our response: We thank the referee for suggesting these minor changes, which we have now incorporated in the manuscript.

Response to referee 1:

We thank the referee for their very careful reading of our manuscript and for highlighting the strengths and weaknesses in it along with providing several suggestions for its improve- ments. Please find below our response to the referee’s comments, suggestions and criticisms.

The referee says: Strengths 1. Proposes an alternative way of deriving the expressions for the charge and energy mag- netization of Bloch electrons [Eqs. (8) and (14) in the manuscript]. 2. A useful and detailed Introduction, with abundant references to previous works. 3. Detailed derivations are given in the Appendices. See however point 4 in the Weaknesses section.

Our response: We thank the referee for highlighting the strengths of our manuscript.

The referee says: Weaknesses

Our response: We thank the referee for also pointing out the weaknesses in our manuscript. However, as we explain below, we respectfully disagree with the referee that these are necessarily weaknesses.

The referee says: 1. A significant part of the manuscript is devoted to rederiving in a new way results that were previously obtained in the literature, namely Eqs. (8) and (14).

Our response: It is indeed true that the expression for the energy magnetization has been derived before in the literature, a fact that we did acknowledge in our manuscript. The utility of our work is that it provides a derivation that is perhaps more straightforward to understand and adapt to other situations, based as it is on general considerations of the Einstein relation rather than specific microscopics.

The referee says: 2. The authors claim to have identified a new type of regular Hall response in time reversal invariant systems coming from the Berry curvature and from the nonequilibrium part of the distribution function given by Eq.(18). However, an actual expression for that Hall response is never written down. As detailed in the Report, I believe that such a contribution to the regular Hall response cannot exist.

Our response: As we have clearly mentioned in the manuscript, we have derived an expression for the non-equilibrium distribution function in the presence of both a non-zero Berry curvature and non-zero magnetic field. This expression can be used to obtain all charge and heat transport coefficients, in linear response in temperature and potential gradients and not just the Hall response. The terms involving the scalar product of the field and Berry curvature, i.e. B.Ω provide contributions to these coefficients (including the Hall conductivity) in addition to those already derived in the literature in the presence of either B or Ω alone. The scope of the current work is limited to deriving only the expression for the non-equilibrium distribution function and not providing detailed expressions for all the transport coefficients. Nevertheless, the reason we singled out the Hall response for mention is that it is the most widely studied of the transport coefficients in topological systems. Thus, the additional contributions to transport that we obtain in our work are perhaps best highlighted using the Hall response as an example. The referee’s contention that the additional contribution to the regular Hall response cannot exist is perhaps due to the expectation that the term “Hall response” should only be used to describe an effect that is linear in the magnetic field. If that is the case, we agree. However, we have not claimed anywhere that the additional terms arising from our calculations are linear in the magnetic field (since they are not) and have thus also assiduously avoided any mention of the Hall resistance, as opposed to the Hall response since the former term is usually employed in situations where the response is linear in the field.

The referee says: 3. The formal results obtained in the manuscript are not illustrated by any explicit calculations on model systems.

Our response: Our intention here is to provide a treatment of transport and diamagnetic currents in systems with both B ̸= 0 and Ω ̸= 0 in the most general terms that would be applicable to any system rather than focus on specific systems. In this regard, it is very similar in scope to previous papers on topological systems such as Refs. 58 and 62, which too did not perform calculations on model systems. We thus believe our work will also be a valuable contribution to the broad field like those papers.

The referee says: 4. The main text is not sufficiently self contained. I had to constantly flip back and forth between the main text and the numerous appendices (as well as the numerous footnotes) to keep up with the manuscript.

Our response: We have followed the referee’s very useful suggestions for improvement of the presenta tion of the manuscript and made modifications accordingly.

The referee says: Report The authors have shown that requiring the transport charge and energy currents to obey the Einstein relation provides an alternative route for deriving the known expressions for the charge and energy magnetization of Bloch electrons at finite temperature [Eqs.(8) and (14) in the manuscript]. This approach complements the standard approach [e.g., Ref. 58] where one starts from those expressions for the magnetizations, subtracts their curl from the expressions for the local (or total) current densities to obtain the transport current densities, and then verifies a posteriori that the resulting transport coefficients satisfy the Einstein relation. The new approach in this manuscript provides a fresh perspective. But that fact that it is only used to recover previously known results limits its impact somewhat.

Our response: We are happy to note that the referee feels that our approach provides a fresh perspective. As we have also mentioned above, we do not claim that the result obtained is entirely new. Rather, it is the approach based on a general application of the Einstein relation as opposed to an appeal to microscopics for specific models that is the focus of our work.

The referee says: In addition to the new derivation of Eqs.(8) and (14), the other main result of the manuscript is the expression in Eq.(18), valid for 2D systems, of the correction to the distribu- tion function to linear order in the electric field, chemical potential gradient, and temperature gradient. In previous works, the emphasis had been on the intrinsic part of the response in ferromagnetic systems, associated with the equilibrium distribution function. The authors claim that plugging Eq.(18) into the expressions in Eqs.(15a,15b) for the transport currents yields a regular Hall response in systems like bilayer graphene, which possess a nonzero Berry curvature due to broken inversion symmetry, but display no anomalous Hall effect due to time reversal invariance.

Our response: We thank the referee for this summary of the significance of our work. As they say, previous calculations have indeed focused on systems with a non-zero Ω due to broken time reversal but no magnetic field. Our calculation applies to general situations with non-zero Ω and non-zero B. In particular, as the referee mentions, it is of interest for systems with non-zero Ω due to broken inversion and not time reversal which require a magnetic field to produce off-diagonal transport responses.

The referee says: However, the expression for that Hall response is never written down explicitly. From which of the various terms in Eq.(18) does it arise? Is it from the first term which is linear in tau, or from the second or third term, both of which are quadratic in tau? I don’t quite see how such a Hall response will come about. For time reversal invariant systems, it follows from the Onsager relation that the Hall response is odd in B, see https://doi.org/10.1016/0038-1098(65)90178-X On the other hand, for time reversal invariant systems one can easily show that terms in the magnetoconductivity with odd powers of B have even powers of the relaxation time tau. See, e.g., below Eq.(9) in https://doi.org/10.1103/PhysRevB.105.045421 This seems to rule out the first term in Eq.(18) as a candidate for a Hall response in time reversal invariant systems, since it it is linear (odd) in tau. The other two terms in Eq.(18) are quadratic in tau, but at linear order in B they are independent of the Berry curvature: the Berry curvature only contributes at secondorder in Bv iathe energydenominators 1+(e/hbar)B⃗.Ω⃗,since there is already a factor of B in the numerator of each of those terms. The above considerations seem to rule out any regular (linear in B) Hall response in time reversal invariant systems coming from the Berry-curvature and from the nonequilibrium distribution function.

Our response: We thank the referee for taking the time to carefully analyze the expression we have obtained term by term to determine its contribution to the Hall response. As we have mentioned earlier, we are not using the term “Hall response” to mean a response that is linear in B, which is the sense in which the referee is presumably using it. Hence, their conclusion that our expression produces no Hall response. We have clearly indicated in the new version of the manuscript that the term “Hall response” does not imply linear in B response. We have also commented on the contribution of each of the different terms that appears in the expression for the non-equilibrium distribution function to the Hall response.

The referee says: Note that there is actually a regular Hall response in time reversal invariant systems coming from the product of the Berry curvature with the orbital moment. It is however associated with the equilibrium part of the distribution function, not with the nonequilibrium part. That type of response was first identified in http://dx.doi.org/10.1103/PhysRevLett.112.166601 [see the second term in Eq.(12) therein], and is also discussed in https://doi.org/10.1103/PhysRevB.103.125432 That response is zeroth order in tau and linear in B, and so it complies with the above- mentioned constraints for Hall responses in time reversal invariant systems. The authors appear to have missed this contribution in their analysis.

Our response: We thank the referee for bringing up this point. The intrinsic equilibrium contribution the Hall response is present in our analysis through the second term of Eqn. 17a. We had not highlighted this fact earlier since the intrinsic equilibrium Hall response has been a subject of detailed investigation already as the referee points out and we did not have anything to add to what is known. Our focus, as mentioned earlier, is on the contribution to transport arising from the non-equilibrium distribution function. However, in the interest of clarity, we have now explicitly mentioned exactly how our calculation also includes the equilibrium response after En. 19.

The referee says: Overall, I feel that this submission does not quite meet the strict acceptance criteria of SciPost Physics. However, I would recommend publication in SciPost Physics Core once the manuscript has been revised taking this report into account.

Our response: We respectfully disagree that our work does not meet the criteria for publication in Sci- Post Physics. We hope we have managed to convey from our responses above that both of the important results of our work are of significance to the field of transport in topologi- cal systems. And that our results are correct and produce the already studied equilibrium Hall response in addition to the hitherto unexplored non-equilibrium response for Ω ̸= 0 and B ̸= 0. To recap the significance of our work: The energy magnetization is a rather opaque quantity compared to its charge counterpart. We feel that our derivation based on general considerations related to the Einstein relation simplifies its understanding more than those that appeal to microscopics and currently exist in the literature. Further, our expression for the non-equilibrium distribution function in systems with Ω ̸= 0 and B ̸= 0 is a new result, which can be used to obtain any charge or heat transport coefficient. We have clarified that this distribution function does indeed provide a Hall response, just not one that is linear in B, and our calculation also contains the intrinsic equilibrium Hall response. These seem to have been the referee’s main concerns about the validity of our results and we believe that we have addressed them.

The referee says: Requested Changes Concerning the physics content of the manuscript, my main request would be to revise the discussion of the types of responses (Hall vs Ohmic) that arise from the equilibrium and non-equilibrium parts of the distribution function, taking into account the above comments. Concerning the presentation, I wonder if it would make more sense to put the paragraph containing Eq.(14) immediately after Eq.(11). The paragraphs containing Eqs.(12,13) and Fig. 2 could then be placed in a separate subsection III.C, where the physical interpretation for Chern insulators is discussed. Minor suggestions • The wording of the sentence ”We further obtain [...]” in the abstract suggests that the expression obtained for the energy magnetization is new. In view of the references given below Eq.(14), that sentence should probably be revised. The same comment applies to the corresponding sentences in the Conclusions section. • The inline equation below Eq.(F2) is the same as Eq.(8), no need to write it explicitly.

Our response: We thank the referee for these suggestions and have now incorporated them in the manuscript.

The referee says: Suggestions on the formatting of the manuscript: 1. The authors should consider numbering all equations that are not in-line. 2. The manuscript contains a large number of footnotes, which are not readily identified by the reader as such since they are placed in the references. It would be helpful if the footnotes were placed on the page where they are called from, and if their overall number was reduced. 3. A related issue is that several of the Appendices are only mentioned in footnotes. It would be better if the Appendices were called from the main text. 4. The number of Appendices is quite large. For example, I wonder if the two very short appendices A and H could be disposed of, and their contents inserted at the appropriate places in the main text. 5. Some Appendices have a high concentration of in-line equations, sometime with several of them in the same paragraph: see top of p. 12, beginning of Appendix G, Appendix I.1, and below Eq.(I1). This is a matter of personal taste, but I find this format hard to read. 6. In Appendix D, the notation with ”all space” or ”cell” on both sides of the inner products is somewhat cumbersome. Maybe place them on the right-hand side only?

Our response: We thank the referee for these formatting suggestions and have now incorporated them in the manuscript. Please see the list of changes for details.

The referee says: Typos 1. In the last paragraph of the 1st column of p. 2 , it is written ”the the validity” . 2. In the first two lines below Eq. (3), remove the band indices from the cell-periodic Bloch states. Same for Eq. (C1) and the in-line equations below it. 3. At the beginning of Sec. III.B, ”The expression in Eq.(6)” should read ”The expression in Eq.(6b)”. 4. In footnote 79, one reference appears (twice) as a question mark. 7

Our response: We thank the referee for pointing out these typos and have now corrected them and a few others that we found upon rereading the manuscript.