Magnetic moment of electrons in systems with spin-orbit coupling

This paper investigates relativistic corrections to the electron magnetic moment operator in systems with spin-orbit coupling, introducing the concept of an "abnormal magnetic moment" to highlight ambiguities in conventional decompositions and challenges to standard orbital magnetization theories, while deriving a Kubo formula for the kinetic magnetoelectric effect rooted in the noncommutation of position and magnetic field derivative operators.

The Gist

The Big Picture: The Electron's "Magnetic Personality"

Imagine an electron not just as a tiny ball of charge, but as a tiny, spinning magnet. In physics, we call this its magnetic moment. It's like a microscopic compass needle.

For a long time, scientists have had a "rulebook" for calculating how strong this compass needle is. The rulebook says: "Take the electron's spin (how fast it's spinning) and its orbit (how it moves around the atom), add them up, and you get the magnetic moment."

The Problem: This rulebook works great for slow, lazy electrons. But in modern materials (like the chips in your phone or new quantum computers), electrons move fast and interact with the atomic structure in a way that creates Spin-Orbit Coupling (SOC). Think of SOC as a "dance" where the electron's spin and its movement get tangled up.

The authors of this paper discovered that the old rulebook is missing a few crucial pages. When electrons do this "SOC dance," the standard way of calculating their magnetic moment is wrong. It's like trying to calculate the weight of a suitcase by only looking at the handle, ignoring the heavy books inside.

The Core Discovery: The "Abnormal" Moment

The paper introduces a new concept called the "Abnormal Magnetic Moment."

The Analogy: The Tug-of-War
Imagine you are trying to measure how hard a magnet pulls on a piece of iron.

• The Old Way (Naive): You look at the magnet's label (the Hamiltonian, or energy equation) and assume the pull is exactly what the label says.
• The New Way (Real): The authors show that because the electron is "dancing" (SOC), the label changes as you try to measure it. The actual pull is different from what the label predicts.

This difference is the Abnormal Magnetic Moment. It's the "ghost" force that appears because the electron's environment is so complex that you can't just read the label; you have to account for the dance itself.

Key Concepts Explained Simply

1. The "Decoupling" Trick

To study these complex materials, scientists often try to separate the "good" electrons (conduction band) from the "bad" ones (valence band) to make the math easier. They use a mathematical "magic trick" (unitary transformation) to split them apart.

• The Catch: When you split them, the "good" electrons inherit some of the "bad" electrons' habits. They get a new, weird magnetic personality. The authors calculated exactly what this new personality looks like.

2. The "Spin" vs. "Orbit" Confusion

In the old days, we could clearly say: "This part of the magnetism comes from the spin, and that part comes from the orbit."

• The New Reality: With SOC, these two get mixed up like a smoothie. You can't separate the strawberry (spin) from the banana (orbit) anymore. The authors show that trying to label them separately is ambiguous and depends on how you look at them. It's like trying to separate the sound of a drum from the sound of a cymbal in a jazz solo—they blend into a single rhythm.

3. The "Non-Commuting" Mystery (The Hall Effect)

One of the most exciting parts of the paper is about how these electrons react to electric and magnetic fields.

• The Analogy: Imagine driving a car. Usually, if you turn the steering wheel (position), the car turns. But in these quantum systems, the "steering wheel" and the "gas pedal" don't work independently. If you try to turn and accelerate at the same time, the car does something unexpected because the order in which you do things matters.
• The Result: This "order matters" rule creates a new type of magnetic effect called the Kinetic Magnetoelectric Effect. The authors found a specific source for this: the fact that the electron's position and its sensitivity to magnetic fields don't play nicely together. This creates a current that flows sideways (like the Hall effect), generating a magnetic field out of thin air.

Why Should You Care?

You might think, "I don't care about abstract electron math." But this matters for the future of technology:

1. Spintronics: This is the next generation of electronics that uses electron spin instead of just charge. If we want to build faster, more efficient memory and processors, we need to know exactly how these electrons behave. If we use the old, wrong formulas, our devices might fail or be inefficient.
2. Quantum Computing: These materials are the playground for quantum computers. Understanding the "Abnormal Magnetic Moment" helps us control these qubits better.
3. Accuracy: The authors are essentially saying, "Stop guessing. Here is the precise math for how these electrons actually behave when they are dancing."

The Takeaway

This paper is a correction to the physics rulebook. It tells us that in the fast, quantum world of modern materials:

• The electron's magnetic pull is stronger and weirder than we thought.
• You can't easily separate "spin" from "orbit."
• There is a hidden "abnormal" force that we must include to get the math right.

By fixing these equations, the authors have provided a better map for navigating the future of magnetic and quantum technologies. They didn't just find a new particle; they found a new way to see the old ones.

Technical Summary

1. Problem Statement

The paper addresses a fundamental gap in the theoretical description of electron magnetic moments in systems with Spin-Orbit Coupling (SOC).

• Conventional Assumption: In standard treatments, the magnetic moment operator ($\mu$) is often assumed to be equivalent to the negative derivative of the Hamiltonian with respect to the magnetic field ($-\partial H/\partial B$). Similarly, the velocity operator is assumed to be $\partial H/\partial p$.
• The Issue: When relativistic corrections (specifically SOC) are included and bands are decoupled (e.g., via unitary transformations to isolate conduction bands), these assumptions break down. The operation $\partial/\partial B$ is not invariant under unitary transformations that depend on the magnetic field. Consequently, the projected magnetic moment operator in a specific band is not equal to $-\partial H_{\text{eff}}/\partial B$.
• Consequence: Neglecting this discrepancy leads to incorrect calculations of relativistic effects such as the Spin Edelstein Effect (SEE), Spin-Orbit Torques (SOT), Spin Hall Effect (SHE), and the kinetic magnetoelectric effect. Furthermore, the standard separation of magnetic moments into "spin" and "orbital" parts becomes ambiguous in the presence of these corrections.

2. Methodology

The authors employ a rigorous operator-based approach using unitary transformations to decouple electronic branches (e.g., separating electrons from positrons in vacuum, or conduction from valence bands in semiconductors).

• Unitary Decoupling: They apply a unitary transformation $U = e^{iW}$ to diagonalize the full Hamiltonian (Dirac or Kane), separating it into block-diagonal forms for different branches (e.g., $H^{(el)}$ and $H^{(p)}$).
• Operator Transformation: Instead of simply differentiating the transformed Hamiltonian, they transform the magnetic moment operator itself: $\mu^{(el)} = U^\dagger \mu^{(D)} U$.
• The $\partial/\partial B$ Operator: A key methodological innovation is treating the derivative with respect to the magnetic field, $\mathcal{B} = -\partial/\partial B$, as an operator. They derive the transformation rules for this operator under the unitary change of basis.
• Commutator Relations: They utilize the relation $\mu = [\mathcal{B}, H]$ to compute the projected magnetic moment, analogous to how velocity is computed via $v = [r, H]/i\hbar$.
• Linear Response Theory: They derive a general Kubo formula for the dc kinetic magnetoelectric effect in a two-branch system. They demonstrate how to "project" interband matrix elements of observables into expressions involving only intrabrand (branch-diagonal) operators by exploiting commutator identities.

3. Key Contributions and Results

A. Definition of "Abnormal Magnetic Moment"

The authors introduce the concept of the Abnormal Magnetic Moment ($\mu_{abn}$), defined as the difference between the true projected magnetic moment and the naive derivative of the Hamiltonian:
$$\mu_{abn} = \mu - \left( -\frac{\partial H}{\partial B} \right)$$

• Vacuum (Pauli Picture): They derive the explicit form of $\mu_{abn}$ to order $1/c^2$, showing it contains terms coupling spin and orbital degrees of freedom (SOC terms).
• Kane Model: They apply this to the 8-band Kane model for semiconductors (e.g., GaAs, InSb). They show that the conduction band magnetic moment acquires significant relativistic corrections that depend on the atomic orbital moment of $p$-functions, which were previously neglected in similar models.

B. Ambiguity of Spin vs. Orbital Decomposition

In the presence of SOC and relativistic corrections:

• The separation of the total magnetic moment into distinct "spin" and "orbital" components becomes basis-dependent and ambiguous.
• Terms that look like spin moments (proportional to $\sigma$) can originate from orbital transformations, and vice versa.
• Spin Commutation Violation: The projected spin operators in the decoupled bands no longer satisfy the standard $SU(2)$ commutation relations ($[S_i, S_j] \neq i\hbar \epsilon_{ijk} S_k$). The commutator acquires a correction term proportional to the SOC strength and momentum.

C. Linear Response and the Kinetic Magnetoelectric Effect

The authors derive a Kubo formula for the magnetoelectric tensor $\chi_{\zeta\eta}$ (relating induced magnetization $M$ to an electric field $E$) projected onto individual branches.

• Interband Contributions: They show that a proper linear response theory must account for interband matrix elements of the magnetic moment and velocity operators.
• Intrinsic Contribution ($M^{III}$): They identify a specific intrinsic contribution to the magnetoelectric effect arising from the non-commutation of the position operator ($r$) and the magnetic field derivative operator ($\mathcal{B}$) within a single branch.
  • Formula: $M^{III} \propto \langle [B_\zeta, r_\eta] \rangle$.
  • This contribution is analogous to the Berry curvature contribution to the Hall conductivity arising from non-commuting position operators.
• Physical Interpretation: For the Pauli picture and Kane model conduction band, this term corresponds to a Hall-like current proportional to $E \times M_\sigma$ (where $M_\sigma$ is spin magnetization).

D. Relation to Berry Curvature

The paper clarifies the connection between their commutator-based approach and the Berry curvature theory:

• The "abnormal" contributions and the $M^{III}$ term can be mapped to Berry curvature (and non-Abelian Berry curvature for degenerate bands) in periodic systems.
• However, the commutator approach is more general, as it does not require periodicity or homogeneity, making it applicable to systems with disorder, magnetic textures, or confinement potentials where Berry curvature definitions are difficult.

4. Significance

• Correcting Numerical Models: The work provides the necessary corrections for numerical simulations of spintronic devices. Ignoring the "abnormal magnetic moment" leads to quantitative errors in predicting SOC-driven phenomena like SOT and SEE.
• Theoretical Consistency: It resolves inconsistencies in the "modern theory of orbital magnetization," showing that the standard formulation fails when $v \neq \partial H/\partial p$. The authors propose that in the relativistic picture, the modern theory should be viewed as a theory of total magnetization.
• General Framework: The methodology for projecting linear response formulas onto decoupled branches using commutator identities offers a robust tool for analyzing relativistic effects in arbitrary two-branch systems, extending beyond the specific examples of vacuum and the Kane model.

In summary, this paper establishes that relativistic corrections fundamentally alter the definition of the electron magnetic moment operator in band-projected theories, necessitating the inclusion of "abnormal" terms and a re-evaluation of how spin and orbital moments are defined and calculated in spintronic materials.