SciPost Submission Page
Magnetic moment of electrons in systems with spin-orbit coupling
by Ivan A. Ado, Mikhail Titov, Rembert A. Duine, Arne Brataas
Submission summary
| Authors (as registered SciPost users): | Ivan Ado |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2503.10956v2 (pdf) |
| Date submitted: | April 11, 2025, 9:27 p.m. |
| Submitted by: | Ado, Ivan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
Magnetic effects originating from spin-orbit coupling (SOC) have been attracting major attention. However, SOC contributions to the electron magnetic moment operator are conventionally disregarded. In this work, we analyze relativistic contributions to the latter operator, including those of the SOC-type: in vacuum, for the semiconductor 8 band Kane model, and for an arbitrary system with two spectral branches. In this endeavor, we introduce a notion of relativistic corrections to the operation $\partial/\partial\boldsymbol B$, where $\boldsymbol B$ is an external magnetic field. We highlight the difference between the magnetic moment and $-\partial H/\partial\boldsymbol B$, where $H$ is the system Hamiltonian. We suggest to call this difference the abnormal magnetic moment. We demonstrate that the conventional splitting of the total magnetic moment into the spin and orbital parts becomes ambiguous when relativistic corrections are taken into account. The latter also jeopardize the ``modern theory of orbital magnetization'' in its standard formulation. We derive a linear response Kubo formula for the kinetic magnetoelectric effect projected to individual branches of a two branch system. This allows us, in particular, to identify a source of this effect that stems from noncommutation of the position and $\partial/\partial\boldsymbol B$ operators' components. This is an analog of the contribution to the Hall conductivity from noncommuting components of the position operator. We also report several additional observations related to the electron magnetic moment operator in systems with SOC and other relativistic corrections.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
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.
Weaknesses
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.
Report
Requested changes
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.
Recommendation
Ask for minor revision