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On the separation of Hall and Ohmic nonlinear responses
by Stepan S. Tsirkin, Ivo Souza
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Submission summary
Authors (as registered SciPost users):  Ivo Souza · Stepan Tsirkin 
Submission information  

Preprint Link:  https://arxiv.org/abs/2106.06522v2 (pdf) 
Date submitted:  20210618 13:07 
Submitted by:  Tsirkin, Stepan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
As is well known, the symmetric and antisymmetric parts of the linear conductivity tensor describe the dissipative (Ohmic) and nondissipative (Hall) parts of the current response, respectively. The Hall current is always transverse to the applied electric field regardless of its orientation; the Ohmic current is purely longitudinal in cubic crystals, but in lowersymmetry crystals it has a transverse component whenever the field is not aligned with a principal axis. Beyond the linear regime, the partition of the current into Hall and Ohmic parts becomes more subtle. One reason is that nonlinear conductivities are inherently nonunique: different conductivity tensors can be used to describe the same current response, which amounts to a gauge freedom. In the present article, we discuss how to extract the Hall and Ohmic parts of the nonlinear current starting from a conductivity tensor in an arbitrary gauge. The procedure holds at all orders in the electric field, and it yields a unique partition of the current. The nonlinear Ohmic and Hall conductivities are instead strongly nonunique, but the former can always be chosen as the fully symmetrized nonlinear conductivity tensor, and the latter as the remainder.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2021812 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2106.06522v2, delivered 20210812, doi: 10.21468/SciPost.Report.3381
Strengths
1. The paper addresses an apparent difficulty with identifying the physical nature of nonlinear currents in recent literature.
Weaknesses
1. The scope of the paper is rather limited. It would be appropriate for a journal like "Americal Journal of Physics", which often focuses on works important from the education point of view.
Report
I can attest that the paper is correct, because its result can be derived in two lines. The Joule heating is given by j\cdotE=\sigma_{abc}E_aE_bE_c (quadratic response for simplicity of typing only). This means that only the part of \sigma symmetric in all pairs of indices contributes to it. This part is given by 1/(# of indices)!*(sum of all permutations of indices of \sigma). Subtract this part from the original tensor, the resultant tensor defines the current that does not contribute to Joule heating, which the authors call the "Hall current". That's it. I would just communicate this information in the form of an email to those who made mistakes in their papers to make sure they do not do it again.
Report #1 by Anonymous (Referee 1) on 202189 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2106.06522v2, delivered 20210809, doi: 10.21468/SciPost.Report.3366
Strengths
1Deals with a topic of current interest.
2Highlights and explains how to resolve an ambiguity of nonlinear Hall conductivity tensors.
3Useful to people working in the field.
Report
This paper reports on some technical clarification on the ambiguities related to extracting nonlinear Hall conductivities at arbitrary order of electric fields. The discussion and clarifications could be useful to people working in this field.
Requested changes
Other papers which have discussed the symmetrization of the conductivity tensors in order to extract nonlinear Hall conductivities (and that might suffer from related ambiguities to those pointed out by the current paper) include:
 S. Nandy and I. Sodemann, Phys. Rev. B 100, 195117 (2019).
 ChengPing Zhang, XueJian Gao, YingMing Xie, Hoi Chun Po, K. T. Law.
arXiv:2012.15628.
The current paper seems to have missed these studies.
Author: Stepan Tsirkin on 20211110 [id 1930]
(in reply to Report 1 on 20210809)
We thank the Referee for the positive evaluation of our work.
When quoting a statement from the review by C. Ortix in our original
submission, we failed to notice that it actually came from the above
work by Nandy and Sodemann. In the resubmitted manuscript, the two
papers are cited together. More importantly, we have added at the end
of Sec. 3 a critical discussion of the prescription given in those two
works for separating the Hall and Ohmic quadratic currents, and for
repackaging the quadratic Hall conductivity as a lowerrank tensor.
In fact, that discussion serves as an excellent application of our
general formalism, and illustrates its usefulness.
As for the preprint by Zhang et al, it was actually cited in our
original manuscript, around Eq. (7) [Eq.(10) in the resubmitted
manuscript]. Elsewhere that work is less relevant to ours, as it
considers the transformation properties of the "Berry curvature
multipoles" under certain crystal symmetries, while we do the
opposite: we search for a procedure that is independent of any crystal
symmetries, which leads us to only consider how the nonlinear
conductivities transform under permutations of indices.
Author: Stepan Tsirkin on 20211110 [id 1931]
(in reply to Report 2 on 20210812)Referee 2 writes that "the paper is correct, because its result can be derived in two lines", and goes on to describe in his/her own words the discussion around Eqs.(39,40) of the original manuscript, where we show that the result can indeed be written in a simple form: the Joule heating can be attributed to the conductivity tensor symmetrized over all indices, and the remainder is dissipationless (Halllike).
If that was all there was to it, we would not have bothered to write a manuscript and submit it for publication in any journal. The reason we went ahead is that in the process of answering the original question that motivated the work, namely,
we realized that we could answer a much more general question, namely,
This is the main question that our manuscript is devoted to, and the answer we found is that there are only two valid generic partitions of the nonlinear current: a trivial "all or nothing" partition, and the Hall vs Ohmic partition. (Note that the formulation of Question 2 says nothing about Hall vs Ohmic, which only appears in the answer.)
We believe that this is a nontrivial result of broad interest, which therefore deserves to be published rather than burried in a private email communication with a few colleagues.
However, the fact that neither of the two Referees commented on Question 2 made us realize that we had failed to emphasize it properly in the original manuscript. Indeed, the Abstract and Introduction were too narrowly focussed on the original question that motivated the work, and not on the more general question that we ended up answering.
To address this issue, we have made extensive revisions to the manuscript. The changes are mainly in the Title and Abstract, and in Sec. 1 (Introduction), which has been merged with the former Sec. 2 (Statement of Problem), in Sec. 3 (Secondorder response), and in Sec. 5 (Conclusions). We now reveal the "trivial answer" mentioned by the referee much earlier (at the end of the extended Introduction), and the rest of the manuscript is devoted to the rigorous justification that it constitutes the only valid generic partition possible. We believe that the revised manuscript conveys the main ideas more clearly, and that it does a better job at discussing the relevant literature (see our reply to Referee 1).
We would also like to point out that already Question 1 is not as trivial as it may seem at first. While the fully symmetric form of the Ohmic conductivity "feels right", it is not evident a priori that there are no other valid partitions of the current. For example, if we separate the current into Hall (j^H) and Ohmic (j^O) parts in that way, we can construct another partition j^H' = (1x)j^H and j^O' = j^O + xj^H, where where 'x' is any real number. Again, j^H' is dissipationless, and all dissipation is contained in j^O'. The reason why this alternative partition is not valid is subtle, and our manuscript provides rigorous criteria for discarding it. To emphasize this point, we added Eq.(17) and some discussion around it.
Thus, already from the narrower perspective of Question 1, our manuscript is a valuable contribution to the literature. The number of papers devoted to nonlinear currents in solids keeps increasing, and yet in most of them the fundamental definition of Hall and Ohmic nonlinear conductivities is given, if at all, by naive analogy with the linear case (symmetric vs antisymmetric parts). A clear and wellargued discussion of this question would therefore constitute a welcome addition to the literature.
Finally, we would also like to point out that our manuscript has already been cited twice (arXiv:2106.12695, arxiv:2109.08775) by authors who are not our collaborators, which suggests that the work is timely and of interest to the community.