Magnetic moment of electrons in systems with spin-orbit coupling

The work by Ado et al addresses one of the more convoluted issues in the semiclassical theory of electron motion in solid - the one of the orbital magnetic moment. It both addresses the microscopic definition of this quantity, and gives an example of a linear response calculation that involves it. I think that an attempt at "cleaning up" the situation in the literature is commendable, and should be seriously considered for publication.

Surprisingly, the paper does not include a careful comparison with the existing literature. For the first part of the paper, where the definition of the magnetic moment is considered, it would desirable to go over, say, the expressions given in a well-known review by Di Xiao and Qian Niu, Section IX of it, and compare the results of the present work to those given the review. In particular, the authors should comment on Section IX-F and IX-G. Given the extensive literature on the subject, one cannot simply add to it, there must be a critical comparison of the new results to the old ones.

Regarding the issue of the "spin" and "orbital" magnetic moment, one should point out that there is really no "spin" magnetic moment, it is all orbital, which has been elucidated by Huang in "On the Zitterbewegung of the Dirac Electron", Am. J. Phys. 20, 479–484 (1952).

Finally, it was unclear to me what the actual results of Section 4 of the paper were. The authors implied that the results of Refs 15 and 16 were incomplete, but never pointed out the errors or omissions in those works. In particular, Ref. 16 is close in spirit to the present work, so it should be critically evaluated.

I think the authors emphasized that single-band theories of KME based on the orbital magnetic moment of electrons in that particular band were incomplete. Unfortunately, I do not quite see why that is the case. The reason is the results of those one-band theories can be derived from the multi-band quantum kinetic equation, in which no reference to the magnetic moment is made at all. One just evaluates the interband coherences created by external fields, and evaluates the full microscopic current, which involves the interband matrix elements of the velocity operator. It so happens that the result can be expressed in a band-diagonal form, which involves the intrinsic orbital magnetic moment. If there are flaws in the outlined procedure, it would be great to know it - this will lead to a great deal of existing results being revised. But such serious claims need serious support.

I think the paper can be extremely useful for workers in the field if it makes detailed comparison of the obtained results, which often look quite convoluted, to the existing ones, with a critical analysis of errors made in the established literature. Without such an analysis the work will be another obscure contribution to a field full of misconceptions.

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The manuscript addresses an interesting point, the relativistic corrections to the magnetic moment of electrons. In a large number of works in the past 20 years these relativistic corrections have been ignored. The authors point out this for the kinetic magnetoelectric effect (or Edelstein effect) and the modern theory of orbital magnetization. It could thus be that additional relativistic contributions need to be considered. It is however not known how large these could be.
The authors arrive at several useful results, such as their expression for the abnormal magnetic moment . They point out the difficulty to separate in fact spin and orbital contributions, and they provide an analysis of the Kane 8 band model. Altogether, it will be interesting to see how large these relativistic contributions are in future studies of real materials.

One of the weaknesses of the manuscript is that there is a body of prior work related to the "proper" definition of the relativistic spin operator , the authors are possibly not aware of it (see their statement in 4th paragraph on p. 2). For a recent summary of previous achievements, see Mondal and Oppeneer, JPCM 32, 455802 (2020). Already Foldy and Wouthuysen and Pryce formulated relativistic forms of the spin operator; these spin operators are not identical but obey all the conditions for spin operators. The relativistic total angular momentum can then be shown to be conserved for a free Dirac particle. The corresponding orbital momentum operators are then also not identical, but the sum of spin and orbital momentum is zero.

The derivation of the authors, to go from the Dirac 4-vector description to the 2-component Pauli Hamiltonian in the presence of the electro-magnetic field is in principle good. It has been done previously by several authors, see for example Mondal et al, PRB 94, 144419 (2016). Are the results of the authors consistent with other previous derivations?

The derivation of the derived Hamiltonian with respect to the B field is used as definition of the electron magnetic moment. This can be done in this way, but then one has the total magnetic moment (spin + orbital parts). I am uncertain about the procedure to take the derivative with respect to B - the quantity $\pi$ that remains after the derivative still contains $A=[B \times r]/2$ so $B$ is still in the expression? This makes the defined "abnormal magnetic moment" (Eq. (18)) dependent on the external B field.

In the Kane model, the quantity $\lambda_2$ adopts the role of a spin-orbit parameter ($\sim 1/c^2$). Is $\lambda_2$ really small in this model? I didn't see a remark on this. In the non-relativistic limit $\lambda_2$ should be zero.

There is a procedure mentioned that could be dangerous. To define the spin magnetic moment by the derivative of the Hamiltonian with respect to the exchange field. This is correct in the non relativistic limit, but in the relativistic limit there are relativistic corrections to the exchange field, see e.g. Mondal et al, PRB 94, 144419 (2016); these are practically always ignored. The authors make a derivation without exchange field but one would need to consider it already in the Dirac Hamiltonian, to find the corresponding spin moment contribution later on, when takes the derivative.

The manuscript touches on topic of current interest (magnetoelectric effect, modern theory of orbital magnetism) and could be worthwhile for others working in these areas. However, the manuscript needs to be improved, see recommended changes below.

The authors are requested to consider previous work related to the definition of the relativistic spin operator and include this in their manuscript.
Address if their derivation provides the same result as previous derivations.
Check if their definition has become independent of B, or is the spin moment operator still dependent on B?
Check what happens with the derivation with respect to the exchange field, if there are not additional relativistic corrections to the exchange field that also might play a role. Possibly one can shift these corrections into the spin operator and then only have use the non relativistic exchange field.

Ask for minor revision