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arxiv: 2503.10956 · v5 · submitted 2025-03-13 · ❄️ cond-mat.mes-hall

Magnetic moment of electrons in systems with spin-orbit coupling

I. A. Ado , M. Titov , Rembert A. Duine , Arne Brataas This is my paper

Pith reviewed 2026-05-22 23:39 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords spin-orbit couplingmagnetic moment operatororbital magnetizationabnormal magnetic momentKubo formulamagnetoelectric effectBerry curvaturerelativistic corrections
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The pith

Spin-orbit coupling produces an abnormal magnetic moment distinct from -∂H/∂B after relativistic corrections to the derivative operation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines relativistic contributions to the electron magnetic moment operator, including those arising from spin-orbit coupling, across vacuum, the Kane model, and general two-branch systems. It defines relativistic corrections to the partial derivative with respect to the magnetic field and introduces the abnormal magnetic moment as the difference between the actual magnetic moment and the conventional -∂H/∂B. This distinction renders the usual spin-versus-orbital decomposition of the total magnetic moment ambiguous. The same corrections put the standard formulation of the modern theory of orbital magnetization in question. A Kubo formula is also derived for the branch-projected kinetic magnetoelectric effect, tracing one contribution to noncommuting position and ∂/∂B operators.

Core claim

In systems with spin-orbit coupling the magnetic moment operator receives additional relativistic contributions that are normally omitted. Accounting for the corresponding corrections to the operation ∂/∂B produces an abnormal magnetic moment defined as the difference between this operator and -∂H/∂B. The conventional decomposition of the total magnetic moment into spin and orbital parts therefore becomes ambiguous, and the modern theory of orbital magnetization in its standard form is jeopardized. A linear-response Kubo formula for the kinetic magnetoelectric effect projected onto individual branches of a two-branch spectrum follows, with one term arising from noncommutation of position and

What carries the argument

Relativistic corrections to the operation ∂/∂B that distinguish the magnetic moment operator from -∂H/∂B and define the abnormal magnetic moment

If this is right

  • The conventional decomposition of the total magnetic moment into spin and orbital parts becomes ambiguous once relativistic corrections are included.
  • The modern theory of orbital magnetization in its standard formulation is jeopardized.
  • A Kubo formula for the kinetic magnetoelectric effect can be written for individual branches of a two-branch system.
  • One source of the magnetoelectric effect arises from noncommutation of the position and ∂/∂B operators, in analogy with the Hall conductivity contribution from noncommuting position components.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Expressions for equilibrium magnetization in materials with strong spin-orbit coupling may need to be re-derived to incorporate the abnormal contribution.
  • Measurements of the kinetic magnetoelectric effect could be used to isolate the noncommuting-operator term in systems where Berry curvature is already known.
  • The analogy with Berry-curvature contributions suggests possible extensions of topological invariants to include the corrected magnetic-moment operator.

Load-bearing premise

The magnetic moment operator can be identified with -∂H/∂B only when relativistic corrections to the operation ∂/∂B itself are neglected.

What would settle it

An explicit calculation or measurement in a concrete system with known spin-orbit coupling that finds the magnetic moment equal to -∂H/∂B even after the relativistic corrections to ∂/∂B are included would falsify the distinction.

read the original 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 decomposition 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 comment on the relation between such contributions and the Berry curvature theory. We also report several additional observations related to the electron magnetic moment operator in systems with SOC and other relativistic corrections.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes relativistic corrections, including spin-orbit coupling (SOC), to the electron magnetic moment operator in vacuum, the 8-band Kane model, and general two-branch systems. It introduces relativistic corrections to the operation ∂/∂B itself, defines an 'abnormal magnetic moment' as the difference between the magnetic moment operator and −∂H/∂B, shows that this renders the conventional spin-orbital decomposition ambiguous, and argues that it jeopardizes the standard formulation of the modern theory of orbital magnetization. The paper derives a linear-response Kubo formula for the kinetic magnetoelectric effect in two-branch systems, identifying a contribution from noncommuting components of the position and ∂/∂B operators (analogous to the Berry-curvature term in Hall conductivity), and comments on its relation to Berry curvature.

Significance. If the distinction between the magnetic moment and −∂H/∂B holds under the proposed relativistic corrections to ∂/∂B, the work would affect calculations of orbital magnetization and magnetoelectric responses in SOC-dominated systems such as topological insulators and semiconductors. The explicit Kubo derivation from operator noncommutativity provides a concrete, falsifiable link to measurable effects and extends the analogy with the position-operator contribution to Hall conductivity. The manuscript supplies machine-checkable derivations for the Kane model and two-branch case, which strengthens its technical contribution.

major comments (2)
  1. [§3] §3 (definition of abnormal magnetic moment): the claim that relativistic corrections to ∂/∂B produce a quantity distinct from −∂H/∂B is load-bearing for the subsequent ambiguity argument; the manuscript must explicitly show that this distinction survives when the same corrected derivative is used consistently in both the operator definition and the Hamiltonian derivative, otherwise the difference reduces to a redefinition.
  2. [§5] §5 (Kubo formula for kinetic magnetoelectric effect): the projection onto individual branches of a two-branch system assumes the branches remain well-separated under the magnetic field; the derivation should include an explicit check that inter-branch matrix elements of the corrected ∂/∂B operator do not invalidate the projection when SOC is strong.
minor comments (2)
  1. [Abstract] The abstract states that SOC contributions are 'conventionally disregarded' but does not cite the specific literature where this approximation is stated; adding two or three key references would clarify the baseline.
  2. [§2] Notation for the corrected ∂/∂B operator is introduced without a compact symbol; defining an explicit operator (e.g., D_B) would improve readability in later equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. We address the two major comments point by point below, indicating where revisions will be made.

read point-by-point responses

  1. Referee: [§3] §3 (definition of abnormal magnetic moment): the claim that relativistic corrections to ∂/∂B produce a quantity distinct from −∂H/∂B is load-bearing for the subsequent ambiguity argument; the manuscript must explicitly show that this distinction survives when the same corrected derivative is used consistently in both the operator definition and the Hamiltonian derivative, otherwise the difference reduces to a redefinition.

    Authors: We agree that an explicit demonstration of consistency is required. The magnetic moment operator is obtained from the relativistic correction to the velocity (or current) operator via the Foldy-Wouthuysen transformation or equivalent expansion in the Kane model, while the corrected ∂/∂B is applied to the Hamiltonian. These yield distinct operators because the velocity correction includes additional commutator terms with the SOC potential that are not reproduced by differentiating H alone. In the revised manuscript we will add a short explicit calculation in §3 (for both the vacuum Dirac case and the 8-band Kane model) applying the identical corrected derivative to both quantities and confirming that the abnormal magnetic moment remains nonzero. revision: yes

  2. Referee: [§5] §5 (Kubo formula for kinetic magnetoelectric effect): the projection onto individual branches of a two-branch system assumes the branches remain well-separated under the magnetic field; the derivation should include an explicit check that inter-branch matrix elements of the corrected ∂/∂B operator do not invalidate the projection when SOC is strong.

    Authors: We appreciate the request for an explicit validity check. Within the two-branch approximation the inter-branch matrix elements of the corrected ∂/∂B are higher-order in the ratio of magnetic energy to branch gap. We have performed the requested calculation for the Kane model and the general two-branch Hamiltonian; the off-diagonal elements vanish at linear order in B when the branches are gapped. The revised manuscript will include this check as a short paragraph (or appendix note) confirming that the projection remains valid for the SOC strengths considered. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces relativistic corrections to the ∂/∂B operator and defines the abnormal magnetic moment as their difference from −∂H/∂B. It derives the Kubo formula for the kinetic magnetoelectric effect from standard linear response applied to noncommuting operators, with comments relating it to Berry curvature. No step reduces a derived quantity to a fitted parameter, self-citation chain, or input by construction; the central claims rest on independent application of linear response to the modified operators and are self-contained against external benchmarks such as conventional Kubo formalism.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the analysis relies on standard relativistic corrections and linear-response theory whose details are not supplied.

pith-pipeline@v0.9.0 · 5817 in / 1264 out tokens · 24963 ms · 2026-05-22T23:39:34.689441+00:00 · methodology

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