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Multipole theory of optical spatial dispersion in crystals
by Óscar Pozo Ocaña and Ivo Souza
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Submission summary
| Authors (as registered SciPost users): | Óscar Pozo Ocaña · Ivo Souza |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202212_00003v1 (pdf) |
| Date submitted: | 2022-12-01 16:21 |
| Submitted by: | Pozo Ocaña, Óscar |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Natural optical activity is the paradigmatic example of an effect originating in the weak spatial inhomogeneity of the electromagnetic field on the atomic scale. In molecules, such effects are well described by the multipole theory of electromagnetism, where the coupling to light is treated semiclassically beyond the electric-dipole approximation. That theory has two shortcomings: it is limited to bounded systems, and its building blocks - the multipole transition moments - are origin dependent. In this work, we recast the multipole theory in a translationally-invariant form that remains valid for periodic crystals. Working in the independent-particle approximation, we introduce "intrinsic'' multipole transition moments that are origin independent and transform covariantly under gauge transformations of the Bloch eigenstates. Electric-dipole transitions are given by the interband Berry connection, while magnetic-dipole and electric-quadrupole transitions are described by matrix generalizations of the intrinsic magnetic moment and quantum metric. In addition to multipole-like terms, the response of crystals at first order in the wavevector of light contains band-dispersion terms that have no counterpart in molecular theories. The rotatory-strength sum rule for crystals is found to be equivalent to the topological constraint for a vanishing chiral magnetic effect in equilibrium, and the formalism is validated by numerical tight-binding calculations.
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 3) on 2023-2-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202212_00003v1, delivered 2023-02-07, doi: 10.21468/SciPost.Report.6700
Strengths
1. Detailed and concise formulation.
2. The formulation is numerical friendly.
Weaknesses
Lengthy
Report
This work provides systematic and very detailed derivations of the multipole theory of spatial dispersion in crystals. It started with the expansion of the interaction Hamiltonian in the velocity form, and the multipole moments were obtained in the length form. The formulation of the individual gauge invariant components is not only numerical friendly, but also can provide much physical insights. In addition, the band dispersion term was clearly formulated which was usually neglected in the quantum chemistry community. Finally, the formulation was reduced to the molecular theory but with the origin problem eliminated, which cannot be easily solved for the quantum chemistry formulation. It is expected that the work may stimulate further ab-initio works due to the clear and easy-to-implement formulation and open the way towards light-matter phenomena of solids beyond dipole approximation. In summary, the manuscript is already in well-conditioned form. I have no comments or suggestions on any change.
Requested changes
N/A
Report #2 by Anonymous (Referee 4) on 2023-2-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202212_00003v1, delivered 2023-02-07, doi: 10.21468/SciPost.Report.6698
Report
This feels like a solid theoretical work, and a timely contribution to a research topic that is attracting increasing interest. The somewhat heavy algebra presented here reflects the difficulties that one has to face in dealing with spatial dispersion effects in the framework of first-principles theory; the authors do a good job at discussing the related formal subtleties in an orderly and clear way. Extensive and careful referencing to the relevant literature is provided in order to place the material in its context. Overall, I believe that this is a quality work that is likely to become a milestone in its field.
I am leaning towards recommending publication in the present form, but first I'd encourage the authors to consider the following (optional) questions.
As far as I understand, the origin dependence of the traditional quadrupolar and magnetic contributions to the gyration tensor can be traced back to the gauge freedom of electromagnetism: the magnetic moment of a current-density field J, for example, is well defined only if J(r) is purely solenoidal. Here the authors demonstrate that these contributions can be hacked in a way that they become separately origin independent.
1) Does this operation have an obvious physical interpretation (e.g., in terms of fixing the EM gauge) or should it be mostly regarded as a convenient trick to improve the numerics (preventing basis set truncation errors etc.)?
2) Related question: are the two "intrinsic" multipolar contributions that the authors propose here separately meaningful (and possibly measurable), as one would expect from gauge-invariant quantities? How about the five terms in Eq.(28)?
Author: Óscar Pozo Ocaña on 2023-02-23 [id 3398]
(in reply to Report 2 on 2023-02-07)
We thanks the Referee for the thoughtful report. In the following, we answer the questions that were raised. Whenever specific equations or references are mentioned, they refer to the resubmitted version of the manuscript.
1) We agree that the origin dependence of the traditional quadrupolar and magnetic contributions to the gyration tensor can be traced back to the gauge freedom of electromagnetism; this is clearly discussed in Sec. 3.3.1 of Ref. 9.
On the other hand, the way in which we have "hacked" those contributions was by enforcing gauge invariance in a different sense: invariance under k-dependent phase changes of the Bloch eigenstates.
We do not have a clear idea on how to relate our "hack" to electromagnetic gauge transformations, but we feel that it would be worthwhile to investigate this matter further. At present, we regard it as a convenient formulation that remains well-defined for crystals, and which may lead to improved numerics for molecules.
2) This is another excellent question, and in this case we believe we can provide a more definite answer.
It is indeed possible, on the basis of purely phenomenological considerations, to break down the optical conductivity at first order in q into separately meaningful parts, namely, into magnetoelectric and quadrupolar contributions. This was recognized in the seminal work of Ref. 3, and Ref. 4 discusses how a purely quadrupolar optical response can be isolated for some crystal classes.
More generally, magnetoelectric and quadrupolar responses can be in principle measured separately in the static limit where electric and magnetic fields become decoupled. For example, the fifth term in Eq. (26) [the old Eq. (28) mentioned by the Referee] describes, in the static limit, a "current-induced magnetization" (or "kinetic magnetoelectric effect", also known as "Edelstein effect") in gyrotropic conducting crystals.
To address this important point, and also to help readers make sense of the long equations derived in Sec. 3, we have added at the end of that section a new subsection where these matters are briefly discussed. Equation (35), which reveals the physical meaning of each separate term, is the key result of that subsection.
Report #1 by Anonymous (Referee 5) on 2023-2-2 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202212_00003v1, delivered 2023-02-02, doi: 10.21468/SciPost.Report.6663
Strengths
1-A very detailed rigorous derivation of natural optical activity and of the key ingredients, the multipole transition moments.
2-Excellent, thorough overview of the existing literature and the issues. In particular, for any term, it is referred whether it appeared in previous derivation.
3-Beside the derivation, sum rules, the physical meaning of each term and its molecular limit are given.
4-The conclusions provide concrete examples of application of the formulae beyond the work. In fact, a concurrent work has implemented the formulae within an ab-initio context
Weaknesses
1- It is a long-read and not a light one, even though the authors detailed a lot, the reader needs to work a bit in order to follow the details. This is 'intrinsic' in this type of research work.
Report
This work can be used as a reference by specialists working in the field. Non-specialists which can be interest (e.g. user of codes that employ the derived expressions) can still refer to the well-written introduction and conclusions for the background, main results and significance of the work. I do not have to propose particular changes to the work, which I believe should be published on Scipost as it met these criteria:
1. It "connects the dots" on a previously-identified and long-standing research stumbling block
2. It is written clearly and precisely
3. As reported in the strengths, the introduction provides the context, the work is well detailed, the existing literature correctly referred, results are objectively summarised, conclusions are clear, linking with the rest of the work and offering perspectives of future work.
Requested changes
No changes to suggest. A minor point, I believe that the '-' at the end of the first line of (16d) should be a '+'.
Author: Óscar Pozo Ocaña on 2023-02-23 [id 3397]
(in reply to Report 1 on 2023-02-02)We thank the referee for the thoughtful report. We have implemented the suggested change, by removing the '-' at the end of the first line of (16d) [now (14d)].
Author: Óscar Pozo Ocaña on 2023-02-23 [id 3399]
(in reply to Report 3 on 2023-02-07)We thank the Referee for the thoughtful report.