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Hall vs Ohmic as the only proper generic partition of the nonlinear current
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/2106.06522v3 (pdf) 
Date submitted:  20211110 02:56 
Submitted by:  Tsirkin, Stepan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The symmetric and antisymmetric parts of the linear conductivity describe the dissipative (Ohmic) and nondissipative (Hall) parts of the current. 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. In this work, we extend that analysis beyond the linear regime. We consider all possible ways of partitioning the current at any order in the electric field without taking symmetry into account, and find that the Hall vs Ohmic decomposition is the only one that satisfies certain basic requirements. A simple prescription is given for achieving that decomposition.
Author comments upon resubmission
we would like to thank both you and the Referees for a careful
consideration of our work.
Referee 1 evaluated our work positively, and suggested just adding a
few more citations, which we did.
After reading the report of Referee 2, we realized that our main
message had not been sufficiently emphasized in the manuscript, and
that this lead to the following misunderstanding: the Referee took the
simple result for the partition of the nonlinear conductivity into a
fully symmetric part and a remainder to be our central finding, and on
that basis deemed the work not worthy of publication.
However, as elaborated in our reply to Referee 2, the scope of the
manuscript is significantly broader than what was implied by the
report. We not only arrive at a simple prescription for separating the
Hall and Ohmic reponses, but we give a rigorous proof that it leads to
the only proper way of partitioning the nonlinear current without
taking the specific symmetries of the system into account. In other
words, the Hall vs Ohmic partition is the only generic partition
possible.
To address these issues, we have made extensive revisions to the
manuscript.
We believe that the revised manuscript conveys the main ideas more
clearly, and that it does a better job at discussing the relevant
literature. We therefore hope that you Editorial Board will find
that in its present form the manuscipt is suitable for publication
in SciPost Physics.
With best regards,
Stepan Tsirkin and Ivo Souza.
List of changes
We mainly changed the writing of the Title and Abstract, Introduction and Conclusions, to emphasize the main nontrivial result of the manuscript.
In particular:
* Sec. 1 (Introduction) has been merged with the former Sec. 2 (Statement of the Problem)
* We now reveal the "trivial answer" mentioned by the Referee 2 much earlier (at the end of the extended Introduction, eq. 16), and the rest of the manuscript is devoted to the rigorous justification that it constitutes the only valid generic partition possible.
* We highlight the nontriviality of the uniquness of HallvsOhmic decomposition in Eq 17 and discussion around it.
*fixed typos in equations (24), (28) and (30) of Sec. 3 (Secondorder response)
* in the end of Sec.3 we make closer contact with the decomposition suggested in Refs. 15,20, and show that it may be corrected by a factor 4/3. (Eqs 3539 and discussion around them)
* Added more references to relevant literature
*Conclusions are revised accordingly
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 5) on 2022121 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2106.06522v3, delivered 20220121, doi: 10.21468/SciPost.Report.4212
Strengths
1 The present manuscript introduces a question of fundamental interest: How to properly define the ohmic and Hall part of nonlinear electric responses from basic principles.
Weaknesses
2 The manuscript mainly focuses on the generic definition of currents with respect to the permutation of electric fields. It is difficult (to me) to foresee practical consequences of the present manuscript in real experiments.
Report
After reading all the discussion between referees and authors, I find that all agree more or less in the basic content of the present manuscript, and with the principal result.
The question that, In my opinion, is not totally clear from the manuscript is what are the practical consequences of this.
The point is that the ohmic vs. Hall partition is unique without invoking crystalline symmetries or processes leading to the nonlinear conductivities. But it is not clear from the manuscript if different prescriptions for nonlinear conductivities have any experimental impact, beyond precise values of parameters. If I understand it well, the authors discuss that different prescriptions lead to different overall numerical prefactors, that can be accounted for in unknown parameters, like lifetimes or similar.
Taking the second order conductivity as an example, it is well known that inversion symmetry must be broken to get a nonzero response. According to the authors, this information is irrelevant for the partitioning they describe, but is of paramount importance in experiments, so I find this shift between what is important in the present manuscript, a little bit confusing.
To frame a little bit my concern, the question I would find quite useful to be answered in the present manuscript is ¿How the present discussion about partitioning might be relevant to understand experimental results? Discussing it for the second nonlinear conductivity would be enough (for me).
Requested changes
1 I strongly encourage the authors to enlarge the discussion section with the role of inversion symmetry and experimental consequences of their results in the case of second nonlinear conductivity, paying attention to any available physical example (the way shift , photogalvanic currents modify when performing the partioning procedure, etc).
Report #2 by Anonymous (Referee 4) on 20211228 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2106.06522v3, delivered 20211228, doi: 10.21468/SciPost.Report.4101
Report
It appears to me that the recent exchange of communications has not changed the opinions of anyone involved. I maintain that asking the question about "current partitioning" is just not a good idea. The fact that other groups made associated mistakes basically attests to this, they shouldn't have tried. Separating the conductivity tensor into totally symmetric part and the rest appears to me the only reasonable way to approach the problem. This separation is known to be unique. If I recall correctly, a tensor of rank higher than two can be split into totally symmetric part, totally antisymmetric part, and "the rest", and all of the three pieces are defined uniquely. I am not the type that remembers theorems, but this at least seems in accord with the common sense: once symmetrization is done, no further symmetrization can change the result. If the result is artificially altered by reapportioning the symmetric part, another symmetrization will recover the original partition. In summary, the authors spent a lot of effort solving a wellknown linear algebra problem. This is quite unfortunate, and I sympathize with their desire to get something out of this effort, but there is nothing one (or rather, I) can do about it.
Author: Stepan Tsirkin on 20211231 [id 2058]
(in reply to Report 2 on 20211228)
We thank the Referee and the Editor for maintaining the discussion during this holiday period. We would like to start by clarifying that our goal is not to "get something out of this effort". The goal of publishing a manuscript is not to increase one's publication list, but to share ideas, and to get feedback from a community which is broader than the one we could reach via email (as suggested by the Referee in his initial report).
One of the possible outcomes of our submission was that someone would point us to exactly which wheel we are reinventing. That was the reason why we chose SciPost Physics  a journal with a public peerreview process. However, we have not received so far any reference to a previous work that preempts ours. And naturally, we are unable to comment on the relation of our work to theorems that the Referee does not quite remember.
It is evident that a tensor may be separated into "something and the rest", and there are multiple ways to do so. Fully symmetric combination, fully antisymmetric one, and "the rest" are just particular cases of linear combinations of permutations of indices as defined by our Eq. 19, all of which transform as tensors. So, what is the reason to exclude all other possible combinations? What makes the symmetric combination unique, and in what sense is it "unique"? We have all been taught that to solve a mathematical problem means to find *all* the solutions and to prove that no other solutions may exist, and this is what we have achieved. Conversely, statements such as "it appears" or "it seems" are not accepted as valid solutions.
The Referee writes
"I maintain that asking the question about "current partitioning" is just not a good idea. The fact that other groups made associated mistakes basically attests to this, they shouldn't have tried."
We beg to differ:
the fact that we were able to formulate that question with mathematical rigor, and to obtain a rigorous answer for it, demonstrates that it is in fact a valid question to ask. If anything, it shows that other groups should have tried harder, rather than not trying at all. Moreover, the question we posed is a physics question about currents, not some linear algebra problem that the Referee claims is well known even though he/she cannot quite pin down or provide a reference for.
In closing, we wish to make the following comment. We are puzzled by the Referee's insistence that there is only "one reasonable way to approach the problem". Approaching a problem from different angles can only enrich our understanding of it. Our approach to the problem at hand offers a general solution which contains as special cases the one favored by the Referee, as well as a corrected version of an alternative solution discussed in the recent literature. This suggests that our approach has some merit to it, and that it constitutes a timely addition to the literature. Efforts to bring clarity to topics of current interest should be encouraged, not dismissed.
Report #1 by Anonymous (Referee 3) on 20211130 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2106.06522v3, delivered 20211130, doi: 10.21468/SciPost.Report.3976
Strengths
No change since the original report
Weaknesses
No change since the original report
Report
Dear Authors,
Thanks for your efforts updating the manuscript. I am still trying to understand the precise message of the paper, since you indicated that my original grasp of it was off. I would like to describe a point of view on this problem, which extends slightly the abstract of your work, and then ask you in what way you think it is incomplete. We seem to disagree on whether anything needs to be added to it. I think this is the shortest path to my understanding.
The Point of View: imagine a conductivity tensor has been calculated for certain medium. One can ask two questions related to the form of this tensor:
Q1: given an Efield, what is the total produced heat? The obvious answer to that is that one has to contract the Efield with the symmetric part of the tensor. There is no ambiguity here.
Q2: given a *measurement* of a current for a given Efield, can one split this current into "Ohmic" and "Hall" components? This is a bad question, and I think it is quite obvious. The current will in general have components along and perpendicular to the field. While the current along the field is definitely "Ohmic"/dissipative, the current perp to the field has contributions from both the Hall effect, and anisotropy of the symmetric part of the conductivity tensor (which we deemed "Ohmic"). It is impossible to separate the two using a single measurement. In fact, what's done in practice in linear transport is that one measures the current for two antiparallel orientation of the magnetic field, and subtracts the two to separate anisotropy and the Hall effect. Thus one is led to the conclusion that the difficulty of "Question 2" is not an abstract math one, it is a practical, physical matter.
Now my question is what the present work adds to this discussion? Why is it necessary to partition the current, and why is it a deeper/more meaningful procedure as compared to simply separating the totally symmetric part of the conductivity tensor? It seems to me that the games the authors play with the current partitioning, I mean those "new partitions" defined by fraction "x" of some current reshuffled between the two constituents, are not quite meaningful. Instead, separating the conductivity tensor into its symmetric part, and "the rest" is perfectly meaningful, and by construction obeys all the conditions the authors formulate for their "current partitioning", which now simply mean that the symmetric part of a tensor is unique, and is in fact a tensor from the point of view of its components transformation properties.
Author: Stepan Tsirkin on 20211224 [id 2054]
(in reply to Report 1 on 20211130)
Dear Referee, we thank you for the thoughtful remarks on our work, and
apologize for the late reply. We will do our best to maintain a more
lively conversation in the future.
We think that the source of our disagreement lies within the following
lines of your reply:
"One can ask two questions related to the form of this tensor:"
"[...] this is the sortest path to my understanding."
"We seem to disagree on whether anything needs to be added to it."
"Now my question is what the present work adds to this discussion?"
"[...] in what way you think [this point of view] is incomplete."
We do not think that Q1 and Q2 are the only valid and interesting
questions that one can ask regarding the form of conductivity
tensors. Apart from stating the obvious answer, our work adds nothing
to the discussion of Q1; it also does not add anything new to the
discussion of Q2. What our work does instead is to address a different
question which we believe is interesting in its own right, and whose
answer provides a fresh perspective on the Hall vs Ohmic partition of
the current.
The question we chose to address in our manuscript is the following:
*****************************************************
What are all the possible ways of partitioning the nonlinear current
into physically meaningful parts, without taking into account the
symmetries of the system?
*****************************************************
The answer we found is that the only nontrivial generic partition
possible is precisely the one that you discuss. (In itself, this
answer is not terribly surprising or exciting. But having been able to
rigoroulsy exclude any other possibilities was new to us, and rather
satisfying.) While the partial nonlinear currents are unique, the
partial nonlinear conductivities are  unlike their linear
counterparts  strongly nonunique. This is fully accounted for by our
answer to the question above, which takes the form of a set of
conditions that any meaningful partial conductivities must fulfill.
As the you noted, those conditions are satisfied by the "fully
symmetric" and "remainder" partial conductivities:
"separating the conductivity tensor into its symmetric part, and `the
rest' is perfectly meaningful"
One reason why we find this point of view incomplete is that those
conditions are also satisfied by other equally valid choices of Ohmic
and Hall nonlinear conductivities (which, of course, yield the same
Ohmic and Hall currents).
This brings us to a very practical reason why our work constitutes a
useful addition to the literature: the form of the quadratic Hall
conductivity tensor that has been frequently used in recent works
(e.g., Refs. 15 and 20) is different from the one you find to be the
most meaningful. This goes to show that one person's shortest path to
understanding may be different from that of another person, especially
when the question being asked affords multiple correct answers, as is
the case here. By taking into account their inherent nonuniqueness,
our work clarifies the most general definitions of nonlinear Hall and
Ohmic conductivities.
Finally, we feel that the "games that we are playing" around Eq. 17 do
serve a purpose: they demonstrate that without imposing the two
necessary conditions that we postulate, one can get literally any
value for the Hall current (this is, in fact, what happened in
Refs. 15 and 20). You are absolutely right that
"separating the conductivity tensor into its symmetric part, and 'the
rest' is perfectly meaningful,"
But what makes this procedure "unique", except for the fact that it is
formulated in an elegant way? Nature likes beauty, but aesthetics is
not a solid criteria for physics. After reshuffling via Eq. 17, both
the Hall and Ohmic parts remain "tensors from the point of view of
their components transformation" (because they are linear combination
of tensors). Our work strictly proves that the elegant partition (of
the current) that was obvious to you from the beginning is indeed the
only valid one. In addition, it provides a general recipe for all
possible valid separations between Hall and Ohmic conductivities.
We hope you find these comments helpful to understand our point of
view, and look forward to your further questions and comments.
Author: Stepan Tsirkin on 20220415 [id 2386]
(in reply to Report 3 on 20220121)We thank the Referee for a thoughtful report. Below we provide replies to the questions and concerns raised in the report, and describe the changes that have been made to the manuscript to address those concerns.
In the "Weaknesses" section, the Referee writes:

2 The manuscript mainly focuses on the generic definition of currents with respect to the permutation of electric fields. It is difficult (to me) to foresee practical consequences of the present manuscript in real experiments.

As a minor clarification, we note that the permutations we consider are not just with respect to the electric fields (represented by the last n indices of the nth order conductivity tensor), but with respect to all indices, including the 0th index representing the current: see Eqs .(19,20).
As for the practical consequences of our analysis, they are discussed below.
In the third paragraph of the Report, the Referee writes:

But it is not clear from the manuscript if different prescriptions for nonlinear conductivities have any experimental impact, beyond precise values of parameters. If I understand it well, the authors discuss that different prescriptions lead to different overall numerical prefactors, that can be accounted for in unknown parameters, like lifetimes or similar.

In the specific example discussed around Eqs.(613), different prescriptions that do not pass our sanity criteria give different overall numerical prefactors for the Hall current. More generally, incorrect Hall vs Ohmic partitions may also lead to Hall currents point along different directions. Moreover, Eq.(17) shows that even when the Hall current only changes by a numerical prefactor, the Ohmic current  which in general is not parallel to the applied field  may change direction. Finally, the numerical prefactors are important by themselves to allow the experimental benchmarking of parameterfree theories. In particular, the lifetimes may either be evaluated from first principles, or extracted from the measured linear Ohmic resistivity.
In the fourth paragraph of the Report, the Referee writes:

Taking the second order conductivity as an example, it is well known that inversion symmetry must be broken to get a nonzero response. According to the authors, this information is irrelevant for the partitioning they describe, but is of paramount importance in experiments, so I find this shift between what is important in the present manuscript, a little bit confusing.

As the Referee points out, if inversion symmetry is present the entire quadratic conductivity tensor vanishes; this is now mentioned explicitly below Eq. 4. In that case, our prescription gives the trivially correct result that the Hall and Ohmic parts of the quadratic current response vanish separately. Our analysis becomes nontrivial when the quadratic current is symmetry allowed, i.e., in acentric crystals, as elaborated in our answer to the comment below.
In the fifth and final paragraph of the Report and in the Requested Changes, the Referee writes:

To frame a little bit my concern, the question I would find quite useful to be answered in the present manuscript is How the present discussion about partitioning might be relevant to understand experimental results? Discussing it for the second nonlinear conductivity would be enough (for me).
1 I strongly encourage the authors to enlarge the discussion section with the role of inversion symmetry and experimental consequences of their results in the case of second nonlinear conductivity, paying attention to any available physical example (the way shift , photogalvanic currents modify when performing the partioning procedure, etc).

To address this concern and following the subsequent recommendation, we have made substantial revisions to the manuscript, namely:
We added a new Sec. 5 where we carry out a systematic symmetry analysis of the quadratic conductivity, broken down into four fundamental contributions: Hall vs Ohmic, and timereversal even vs timereversal odd. The results, summarized in Table 2, could guide the search for materials displaying only one type of quadratic response. We call attention to the timeeven Ohmic component, which is purely disorder mediated. We correct a result from the recent literature, and point out a possible oversight in recent papers on disordermediated quadratic responses, where the Ohmic part of those response may have been overlooked. These changes are reflected in part in the expanded abstract.
We revised the Discussion section, which now contains two paragraphs only. The second paragraph is new. In it, we comment on the relevance of our analysis for interpreting recent measurements of (i) a higherorder (cubic) nonlinear current, and (ii) spontaneous unidirectional magnetoresistance. In particular, we provide a sharp phenomenological definition of the latter effect, without invoking specific microscopic mechanisms.