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Energy magnetization and transport in systems with a non-zero Berry curvature in a magnetic field
by Archisman Panigrahi and Subroto Mukerjee
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Subroto Mukerjee · Archisman Panigrahi |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202204_00041v3 (pdf) |
| Date accepted: | 2023-05-22 |
| Date submitted: | 2023-05-12 09:09 |
| Submitted by: | Mukerjee, Subroto |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We demonstrate that the well-known expression for the charge magnetization of a sample with a non-zero Berry curvature can be obtained by demanding that the Einstein relation holds for the electric transport current. We extend this formalism to the transport energy current and show that the energy magnetization must satisfy a particular condition. We provide a physical interpretation of this condition, and relate the energy magnetization to circulating energy currents in Chern insulators due to chiral edge states. We further recover the expression for the energy magnetization with this alternative formalism. We also solve the Boltzmann Transport Equation for the non-equilibrium distribution function in 2D for systems with a non-zero Berry curvature in a magnetic field. This distribution function can be used to obtain the regular Hall response in time-reversal invariant samples with a non-zero Berry curvature, for which there is no anomalous Hall response.
Author comments upon resubmission
The referee says that our manuscript does not meet the stringent novelty and clarity criterion for publication in SciPost Physics. We hope that by making the changes suggested by the referee, we have addressed the questions about clarity. As for novelty, the referee has pointed us to a review paper, which they claim contains a summary of previous work on the perturbative solution of the Berry-Boltzmann equations to high orders in applied fields and which we have not cited. The referee says that the existence of this paper detracts from our claim of the novelty of our approach. We thank the referee for bringing this paper to our notice, which we had indeed missed earlier. However, as we explain below, there are questions about the correctness of the expressions in this paper. Moreover, the approach described in it does not account for temperature gradients, the inclusion of which is an important aspect of our calculations. We thus believe that our approach is indeed novel within the scope of general transport theory.
A further criticism of the referee is that we do not perform a calculation for a specific microscopic model and this fact disqualifies our paper from consideration for publication on SciPost Phys. We reiterate that our intention is to present a calculation of the energy magnetization in the most general terms. And, as we have pointed out in our previous response, there have been several papers in the recent past, which too have presented general treatments of different aspects of transport theory without performing calculations for specific models. While we once again, respectfully disagree with the referee that this should be a disqualification for publication on SciPost Phys., we are willing to accept the referee's recommendation of publication in Sci Post Phys. core instead.
Please find below a detailed point by point response to the referee's comments. We hope that this version of the manuscript will be considered acceptable for publication in Sci Post Phys. Core.
Best regards,
Archisman Panigrahi
Subroto Mukerjee
List of changes
We have made the following changes to the manuscript in the resubmission. They have been indicated in red in the manuscript wherever possible.
1. Added “in Chern insulators” to the abstract, as the referee suggested.
2. Removed the duplicate footnotes containing “the charge of electron is taken to be −e”, and added that to the main text.
3. Simplified the titles of the subsections of section III, as the referee suggested.
4. The acronym LHS (left-hand side) is defined where the first time it is used.
5. To denote the cyclotron frequency, the symbol ω is consistently used instead of ωc.
6. The Appendix titled “Derivation of the transport electric current density and derivation of the charge magne- tization” (which was Appendix C in revision 2) is now removed, and its contents are added to section III A of the main text.
7. The goals of all the appendices are mentioned at their beginning.
8. All the inline equations in the appendices are converted to numbered equations.
9. Some prior works related to the solution of Boltzmann Transport Equation (BTE) are cited (references 75, 76, 77, 78), and we have explained how our solution of the BTE addresses some problems which were not addressed previously.
10. The expression εk = ε0(k) − mk · B is now modified to εk = ε0(k) − mk · B − ms · B, (Eq. (4)) where ms is the spin magnetic moment.
11. All the factors of 2 arising from spin degeneracy are removed, and the corresponding quantities are written as sum over spins.
12. In Appendix A and B, the components of the tensor M are rewritten using Greek indices, Mμν.
13. What we mean by “regular Hall response” is now written in the second page of the main text, where it is
mentioned for the first time.
14. We have explained that the breaking of the inversion symmetry will not affect the spin degeneracy, as the spin-orbit coupling is assumed to be a negligible effect above Eqn. 12 on page 4.
15. We have replaced all the equations (Eqs. (7a), (7b), (9), (10), (11), (14a), (15), (16), (19)) with those for non-zero magnetic field, as the referee suggested.
16. We have obtained the formula for the charge and the energy magnetization at a non-zero B, and have demon- strated that they reproduce the previously known formula in the B → 0 limit.
17. Several footnotes were removed, and their contents were absorbed into the main text.
18. Appendix B is now invoked only within Appendix A, since the goal of Appendix B is to prove a certain result used in Appendix A.
19. We have reworded the opening paragraph of section III A, so that it does not anymore contain two similar sentences.
20. The “long inline equation immediately below figure 2” in the previous revision is now turned into a numbered equation (Eq. 17), as the referee suggested.
21. In revision 2, the 2nd column in page 5 had a large number of parenthetical remarks, which are removed in this revision.
22. Whenever we use the phrase “this term” or “this quantity”, where the sentence is not immediately after the corresponding “term” or “quantity”, the relevant equation has been mentioned to clarify which “term” or “quantity” we are referring to.
23. We have added a brief discussion in the first paragraph on page 8 of what happens to transport coefficients when the Fermi energy approaches a Dirac (or Weyl) point along with two references (79 and 80), to address one of the comments of the referee.
24. Several typos have been fixed.
Published as SciPost Phys. Core 6, 052 (2023)
Reports on this Submission
Report
I thank the authors for having made extensive changes to the manuscript in response to my comments and suggestions, and for the detailed and thoughtful responses to the report.
The revised manuscript has improved significantly in terms of clarity of presentation and overall polish. The novel aspect of the work, and its relation to the previous literature are also much clearer now.
I believe that in its present form the manuscript is a valuable addition to the literature, and I fully recommend its publication in SciPost Physics Core.
Here are a few typos to be corrected:
* Bottom of p. 2, 1st column: "Note that there is no fundamentally change the regular Hall respose." Please rephrase.
* Page 3, last sentence of Sec. II: "energy magnetization $m_k$" $\rightarrow$ "orbital magnetization $m_k$"?
* Below Eq. (15): "there are more than one bands" $\rightarrow$ "there is more than one band"