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REVIEW 2 major objections 7 minor 46 references

Local spin magnetization in itinerant non-collinear magnets: The local spin Berry curvature

T0 review · 2 major / 7 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Spin magnetization redefined: Berry curvature term reshapes local magnetic maps

desk verdict Solid formal result with an unverified but reasonable physical premise; deserves serious peer review. read the letter →

arxiv 2607.06253 v1 pith:3MPUXM62 submitted 2026-07-07 cond-mat.mes-hall

Doru Sticlet , Frédéric Piéchon This is my paper
classification cond-mat.mes-hall PACS 75.30.-m75.50.-y75.70.Ak
keywords spinlocalmagnetizationmagnetsthermodynamicapproachitinerantnon-collinear
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the standard way to compute local spin magnetization in itinerant magnets — taking the equilibrium spin density at each site — misses a physically distinct contribution that becomes relevant whenever the magnetic order is non-collinear. The authors propose an alternative thermodynamic definition: differentiate the local grand potential with respect to an external magnetic field at zero field. This procedure yields two terms. The first, a diagonal term, reproduces the conventional result and corresponds to the Zeeman shift of energy levels. The second, an off-diagonal term the authors call the local spin Berry curvature, arises from the field-induced deformation of energy eigenstates and has no counterpart in the conventional formula. The two definitions agree for collinear magnets but diverge qualitatively and quantitatively for non-collinear ones, producing different site-resolved magnetization distributions and, most strikingly, allowing local magnetization to vary linearly with chemical potential inside spectral gaps — a feature the conventional theory forbids. The discrepancy is invisible to bulk probes because the off-diagonal corrections cancel when integrated over a full unit cell, which the authors suggest explains why the effect has gone unnoticed.

What carries the argument

Thermodynamic definition of local magnetization via grand-potential derivative; decomposition into diagonal Zeeman-shift term and off-diagonal Berry-curvature term; exact analytical results for a four-band non-collinear ferrimagnet on a square lattice; numerical comparison on a honeycomb lattice with skyrmion textures; supplementary results for a magnetic dimer, a dice lattice, and a hexagonal lattice with Rashba coupling and skyrmion field.

What would settle it

If spin-polarized STM or equivalent site-resolved probes measure the equilibrium spin density rather than the thermodynamic response, the Berry-curvature contribution and its linear-in-gap signature would be experimentally invisible.

Watch

Extended reading notes

Core claim

The central object is the local spin Berry curvature, defined as a sum over off-diagonal spin matrix elements between energy eigenstates weighted by inverse energy denominators. When the local spin magnetization is defined thermodynamically as the zero-field derivative of the local grand potential with respect to an external magnetic field, this Berry-curvature term enters weighted by the integrated density of states rather than the Fermi function, producing predictions that differ from the conventional equilibrium spin density. The term vanishes for collinear magnets, where eigenstates are also spin eigenstates, but is generically finite for non-collinear textures where spin coherences are非

Load-bearing premise

The paper assumes that the thermodynamic definition — differentiating the local grand potential with respect to an external field — is the physically correct definition of local spin magnetization for site-resolved probes, but does not demonstrate experimentally or by microscopic derivation that existing probes like SP-STM actually measure this thermodynamic quantity rather than the conventional equilibrium spin density.

Editorial extensions

If this is right

  • Site-resolved probes such as spin-polarized STM could distinguish the two theories by measuring whether local magnetization varies linearly with chemical potential inside spectral gaps, a signature unique to the thermodynamic definition.
  • The thermodynamic identity relating magnetization derivative with respect to chemical potential to density derivative with respect to field suggests a possible measurement protocol using non-magnetic STM to probe field-dependent local density of states.
  • Any first-principles calculation of site-resolved magnetization in non-collinear magnets — skyrmions, spin spirals, frustrated magnets — that uses only the equilibrium spin density may be systematically misassigning magnetization among sublattices.
  • The formal unification of local spin and orbital magnetization through a common Berry-curvature structure suggests that computational packages for orbital magnetization could be extended to spin magnetization with minimal modification.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. The manuscript proposes a thermodynamic definition of local spin magnetization in itinerant non-collinear magnets, defined as the zero-field derivative of a local grand potential with respect to an external magnetic field. Starting from the standard perturbative expansion of the Green's function in a Zeeman field, the authors derive Eqs. (2)–(4), which contain a diagonal term identical to the conventional equilibrium spin density and a novel off-diagonal Berry-curvature-like term Ω_n(r). The two definitions coincide for collinear magnets but differ for non-collinear textures. The authors illustrate the differences analytically in a four-band ferrimagnet (Eq. 16) and a magnetic dimer (SM Sec. II), and numerically in a skyrmion texture on a honeycomb lattice (Fig. 2) and a dice lattice (SM Sec. III). The most striking qualitative prediction is a linear variation of local spin magnetization inside spectral gaps, absent in the conventional theory.

Significance. The derivation from the Green's function identity G(ε,B) = G(ε) + μ_B G(ε)(B·σ)G(ε) to Eqs. (2)–(4) is mathematically clean and self-contained, with no free parameters fitted to the target result. The formal analogy to the local orbital magnetization theory (Ref. [42]) is a genuine strength: substituting the spin magnetic moment by the orbital magnetic moment recovers the established local orbital magnetization expression, providing a unified framework. The analytical results for the four-band ferrimagnet (Eq. 16) and the magnetic dimer (SM Sec. II) are worked out explicitly and are internally consistent. The prediction of linear-in-μ magnetization in gaps is falsifiable and qualitatively distinct from the conventional theory. The Maxwell relation ∂M(r)/∂μ = ∂n(r)/∂B provides a concrete experimental route via non-magnetic STM. These are substantive contributions. The central open question is whether the thermodynamic definition corresponds to what local probes actually measure, which the paper acknowledges but does not resolve.

major comments (2)
  1. The central physical claim — that the thermodynamic definition M̃(r) = −∂_B Ξ(r)|_{B=0} yields the physically relevant local spin magnetization for site-resolved probes — rests on the premise that this quantity is what SP-STM or related techniques measure, rather than the conventional equilibrium spin density (Eq. 1). The paper offers two supporting arguments: (i) the analogy with orbital magnetization, where the thermodynamic approach is well-established, and (ii) the Maxwell relation ∂M(r)/∂μ = ∂n(r)/∂B. However, as the authors themselves note, for orbital magnetization there is no competing equilibrium local density, whereas for spin the conventional spin density is well-defined, widely used in DFT and SP-STM theory, and already validated. The Maxwell relation (argument ii) is a consistency identity that follows from commutativity of partial derivatives on Ξ(r); it confirms internal数学
  2. The definition of the local grand potential Ξ(r,μ,B) = −∫dε F(ε)ρ(r,ε,B) involves the local density of states ρ(r,ε,B) = −Im/π ⟨r|G(ε,B)|r⟩, which is a site-diagonal quantity. The thermodynamic magnetization M̃(r) = −∂_B Ξ(r)|_{B=0} is therefore defined without explicit reference to how the magnetic field couples to the spin operator at position r. The derivation uses the global Zeeman coupling B·σ in the Green's function expansion, and the locality emerges from the projection ⟨r|...|r⟩. The authors should clarify whether this prescription is unique, or whether alternative local decompositions of the grand potential (e.g., involving the full spin-resolved local density of states rather than the spin-averaged one) would yield different local magnetizations. This is load-bearing because the claim of a unique thermodynamic local spin magnetization depends on the uniqueness of the localgrand
minor comments (7)
  1. In Eq. (4), the sum over m≠n excludes the degenerate case. The SM (Sec. II.B) handles the antiferromagnetic degenerate limit carefully, but the main text does not mention this subtlety. A brief comment in the main text pointing to the SM discussion would help readers.
  2. The notation in Eq. (12) uses ρ(r,ε,B) on the left-hand side but the right-hand side involves band-resolved quantities with delta functions. It would be clearer to state that this is an effective low-field expansion of the LDOS, not an exact identity.
  3. In the skyrmion example (Fig. 2), the parameters are M=t and T=0.01t, but the lattice radius R=27a is only mentioned in the caption. The main text should state R explicitly for reproducibility, as was done for the ferrimagnet example.
  4. The paper states that the differences between the two theories 'integrate to zero over a complete unit cell' (page 4). A brief derivation or reference for this cancellation would strengthen this important claim, which explains why the effect is invisible to bulk probes.
  5. Reference [42] is cited as a 2025 arXiv preprint (arXiv:2512.12343). If this work is not yet published, the authors should note its status. If it has been published since submission, the reference should be updated accordingly.
  6. In SM Eq. (9), the expression for Ω_nk involves Δ_k in the denominator. Near points where Δ_k → 0 (e.g., g_k = 0 and h_+ · h_- = 0), this expression may diverge. The authors should comment on the behavior near such points and whether the Brillouin-zone integral remains well-defined.
  7. The abstract states that the thermodynamic approach has 'several distinct observable consequences.' Given that the experimental measurability is the main open question (see major comment 1), softening this to 'several distinct predictions' or 'potential observable consequences' would be more precise.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee raises two major comments, both concerning the physical interpretation and uniqueness of our thermodynamic definition of local spin magnetization. We address each below and propose targeted revisions to clarify the scope and limitations of our definition.

read point-by-point responses

  1. Referee: The central physical claim — that the thermodynamic definition M̃(r) = −∂_B Ξ(r)|_{B=0} yields the physically relevant local spin magnetization for site-resolved probes — rests on the premise that this quantity is what SP-STM or related techniques measure, rather than the conventional equilibrium spin density (Eq. 1). The paper offers two supporting arguments: (i) the analogy with orbital magnetization, where the thermodynamic approach is well-established, and (ii) the Maxwell relation ∂M(r)/∂μ = ∂n(r)/∂B. However, as the authors themselves note, for orbital magnetization there is no competing equilibrium local density, whereas for spin the conventional spin density is well-defined, widely used in DFT and SP-STM theory, and already validated. The Maxwell relation (argument ii) is a consistency identity that follows from commutativity of partial derivatives on Ξ(r); it confirms internal [

    Authors: We agree with the referee that the question of which definition corresponds to what local probes actually measure is not resolved by our manuscript. We wish to clarify our position on three points. First, we do not claim that the thermodynamic definition supersedes the conventional spin density in all contexts. The conventional equilibrium spin density (Eq. 1) is indeed well-established and validated in DFT and SP-STM theory, particularly for collinear magnets where the two definitions coincide. Our claim is more specific: for non-collinear magnets, the two definitions are inequivalent, and the thermodynamic definition carries distinct physical content — specifically, the field-induced deformation of eigenstates captured by the local spin Berry curvature. Second, regarding the orbital magnetization analogy: the referee correctly notes that the analogy is imperfect because, for spin, there exists a competing local density. We acknowledge this asymmetry explicitly. The role of the analogy is not to prove that the thermodynamic definition must be what SP-STM measures, but rather to show that the mathematical structure (diagonal Zeeman-shift term plus off-diagonal Berry-curvature term) is shared between spin and orbital local magnetization, providing a unified framework. Third, regarding the Maxwell relation: we agree that it is a consistency identity and does not by itself establish that the thermodynamic definition corresponds to what a given probe measures. Its role is more modest — it provides a concrete experimental route (via non-magnetic STM measuring ∂n(r)/∂B) that is specific to the thermodynamic definition and has no counterpart in the conventional theory. This is a falsifiable prediction: if the linear-in-gap behavior of ∂M̃(r)/∂μ is observed via the field-depend revision: no

  2. Referee: The definition of the local grand potential Ξ(r,μ,B) = −∫dε F(ε)ρ(r,ε,B) involves the local density of states ρ(r,ε,B) = −Im/π ⟨r|G(ε,B)|r⟩, which is a site-diagonal quantity. The thermodynamic magnetization M̃(r) = −∂_B Ξ(r)|_{B=0} is therefore defined without explicit reference to how the magnetic field couples to the spin operator at position r. The derivation uses the global Zeeman coupling B·σ in the Green's function expansion, and the locality emerges from the projection ⟨r|...|r⟩. The authors should clarify whether this prescription is unique, or whether alternative local decompositions of the grand potential (e.g., involving the full spin-resolved local density of states rather than the spin-averaged one) would yield different local magnetizations. This is load-bearing because the claim of a unique thermodynamic local spin magnetization depends on the uniqueness of the local [

    Authors: This is a well-taken point. We will add a clarifying discussion to the revised manuscript. The referee is correct that the local grand potential Ξ(r,μ,B) = −∫dε F(ε)ρ(r,ε,B) is built from the spin-averaged local density of states, and that the locality emerges from the projection ⟨r|...|r⟩ applied to the global Zeeman coupling. We will clarify the following: (1) The choice of spin-averaged LDOS is the natural one for defining a local grand potential from which a spin magnetization is extracted by differentiation with respect to B, because the Zeeman coupling B·σ is the perturbation, and the derivative −∂_B Ξ(r)|_{B=0} projects out the spin response. Using the spin-resolved LDOS would amount to a different (and redundant) decomposition, since the spin information is already encoded in the B-dependence of ρ(r,ε,B). (2) However, we acknowledge that the question of uniqueness of the local decomposition is deeper than this. The local grand potential Ξ(r) is itself a choice — one could imagine alternative local decompositions of the total grand potential (e.g., based on different real-space partitioning schemes or Wannier representations). Our definition follows the same prescription as the local orbital magnetization theory of Ref. [42], where the local density of states at position r is the fundamental object. The uniqueness of the resulting local magnetization is therefore tied to the uniqueness of the LDOS as a local observable, which is well-established in the orbital case. (3) We will add a sentence acknowledging that alternative local decompositions are in principle possible and that the uniqueness of our prescription is inherited from the LDOS-based approach, which is the standard choice in the orbital magnetization literature. We believe this clarification addresses revision: no

  3. Referee: The central physical claim — that the thermodynamic definition M̃(r) = −∂_B Ξ(r)|_{B=0} yields the physically relevant local spin magnetization for site-resolved probes — rests on the premise that this quantity is what SP-STM or related techniques measure, rather than the conventional equilibrium spin density (Eq. 1). The paper offers two supporting arguments: (i) the analogy with orbital magnetization, where the thermodynamic approach is well-established, and (ii) the Maxwell relation ∂M(r)/∂μ = ∂n(r)/∂B. However, as the authors themselves note, for orbital magnetization there is no competing equilibrium local density, whereas for spin the conventional spin density is well-defined, widely used in DFT and SP-STM theory, and already validated. The Maxwell relation (argument ii) is a consistency identity that follows from commutativity of partial derivatives on Ξ(r); it confirms internal [

    Authors: We agree with the referee that the question of which definition corresponds to what local probes actually measure is not resolved by our manuscript. We wish to clarify our position on three points. First, we do not claim that the thermodynamic definition supersedes the conventional spin density in all contexts. The conventional equilibrium spin density (Eq. 1) is indeed well-established and validated in DFT and SP-STM theory, particularly for collinear magnets where the two definitions coincide. Our claim is more specific: for non-collinear magnets, the two definitions are inequivalent, and the thermodynamic definition carries distinct physical content — specifically, the field-induced deformation of eigenstates captured by the local spin Berry curvature. Second, regarding the orbital magnetization analogy: the referee correctly notes that the analogy is imperfect because, for spin, there exists a competing local density. We acknowledge this asymmetry explicitly. The role of the analogy is not to prove that the thermodynamic definition must be what SP-STM measures, but rather to show that the mathematical structure (diagonal Zeeman-shift term plus off-diagonal Berry-curvature term) is shared between spin and orbital local magnetization, providing a unified framework. Third, regarding the Maxwell relation: we agree that it is a consistency identity and does not by itself establish that the thermodynamic definition corresponds to what a given probe measures. Its role is more modest — it provides a concrete experimental route (via non-magnetic STM measuring ∂n(r)/∂B) that is specific to the thermodynamic definition and has no counterpart in the conventional theory. This is a falsifiable prediction: if the linear-in-gap behavior of ∂M̃(r)/∂μ is observed via the field-depend revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation is self-contained from standard perturbation theory, with no fitted parameters and no load-bearing self-citation chain.

full rationale

The paper's central derivation chain is self-contained. It starts from the standard perturbative expansion of the Green's function in a Zeeman field, G(ε,B) = G(ε) + μ_B G(ε)(B·σ)G(ε) + O(B²), and arrives at Eqs. (2)–(4) by inserting the spectral decomposition G(ε) = Σ_n |n⟩⟨n|/(ε−ε_n) into Eq. (9). No parameters are fitted to the target result. The off-diagonal Berry-curvature-like term Ω_n(r) in Eq. (4) emerges directly from this algebraic manipulation — it is not defined in terms of the quantity it purports to predict. The analogy to local orbital magnetization (Ref. [42], co-authored by one of the present authors) is noted as a formal parallel, but the spin result is derived independently from first principles within the paper. The eigenprojector formalism of Ref. [44] (also co-authored) is a calculational tool for evaluating matrix elements in specific models, not a load-bearing premise for the existence or form of Ω_n(r). The Maxwell relation ∂M(r)/∂μ = ∂n(r)/∂B is presented as a consistency check, not as the derivation of the main result. The physical question of whether local probes measure the thermodynamic vs. conventional spin density is a correctness/physical-relevance concern, not a circularity issue — the paper is transparent that it is proposing a definition and exploring its consequences, not claiming experimental validation. No step in the derivation reduces to its own inputs by construction, no prediction is a renamed fit, and no self-citation is load-bearing for the mathematical content of the central result. The paper is self-contained against external benchmarks (the four-band ferrimagnet, dimer, dice lattice, and skyrmion models are all solved from the Hamiltonian without fitting to the target formula). This is a normal, honest theoretical paper with no significant circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 1 invented entities

The paper introduces one new conceptual entity (local spin Berry curvature) that is derived, not postulated. No free parameters are fitted to produce the central claims; model parameters are chosen for illustration. The main axiom — that the thermodynamic definition is the physically relevant one for local probes — is a domain assumption whose validity remains unconfirmed experimentally.

free parameters (4)
  • h_A, h_B (exchange fields) = h_A=(4,0,0)t, h_B=(0,2,0)t in Fig. 1; various in other models
    Model parameters chosen to illustrate the theory in specific regimes; not fitted to experimental data but selected to demonstrate qualitative differences.
  • t (hopping amplitude) = used as energy unit
    Standard tight-binding parameter set to unity; not a fitting parameter.
  • M (skyrmion field amplitude) = M=t
    Chosen for the skyrmion lattice example; not fitted to data.
  • T (temperature) = T=0.01t
    Low temperature chosen to approximate zero-temperature limit; not fitted.
assumptions (4)
  • domain assumption The local spin magnetization is defined as M(r) = −∂_B Ξ(r,µ,B)|_{B=0}, i.e., the derivative of the local grand potential with respect to external magnetic field at zero field.
    Stated in the section 'Thermodynamic theory of local spin magnetization.' This is the foundational definition that distinguishes the thermodynamic from the conventional approach. Its physical validity for local probes is assumed, not proven.
  • domain assumption Only Zeeman coupling to the magnetic field is considered; orbital coupling is neglected.
    Stated: 'we consider only the Zeeman coupling to the magnetic field.' This simplifies the derivation but excludes effects from orbital coupling to B, which could matter in real materials.
  • domain assumption The system is described by non-interacting quasiparticles (mean-field level).
    The Hamiltonian is a single-particle tight-binding model with exchange fields treated as static. Electron-electron interactions beyond mean field are not considered.
  • standard math The perturbative expansion of the Green's function to linear order in B is valid.
    Standard Dyson equation expansion G(ε,B) = G(ε) + µ_B G(ε)(B·σ)G(ε) + O(B²). This is a well-established perturbation theory result.
invented entities (1)
  • Local spin Berry curvature Ω_n(r) independent evidence
    purpose: Quantifies the field-induced deformation of eigenstates contributing to local spin magnetization, analogous to the Berry curvature in local orbital magnetization.
    The term is derived from standard perturbation theory (Eq. 4) and makes a falsifiable prediction: linear variation of local spin magnetization inside spectral gaps, detectable by SP-STM. The prediction is not used to fit any parameter. However, experimental confirmation is not yet provided.

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Cite this review

Pith. "Pith review of Local spin magnetization in itinerant non-collinear magnets: The local spin Berry curvature." pith.science (2026). https://pith.science/paper/3MPUXM62

@misc{pith2026260706253,
  author       = {Pith},
  title        = {Pith review of: Local spin magnetization in itinerant non-collinear magnets: The local spin Berry curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3MPUXM62}},
  note         = {Machine review of arXiv:2607.06253}
}
read the original abstract

Conventionally, the local spin magnetization in itinerant magnets is determined from the equilibrium local spin density. Here, we propose a thermodynamic approach in which the local spin magnetization is defined from the response of the system to an infinitesimal external magnetic field. The predictions of the two theories are identical for collinear magnets, but differ qualitatively and quantitatively for non-collinear magnets. In the present thermodynamic approach, the spin coherences determine an alternative distribution of local spin magnetization due to the field-induced deformation of the energy eigenstates. This effect is captured by a Berry-curvature-like contribution reminiscent of orbital magnetization and has several distinct observable consequences. We explore the differences between the conventional and thermodynamic approaches in several test cases.

Figures

Figures reproduced from arXiv: 2607.06253 by the authors.

Figure 1
Figure 1. FIG. 1. Off-diagonal (interband) local spin magnetization [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) The sites of the honeycomb lattice circumscribed [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 1
Figure 1. FIG. 1. Comparison between the total local spin magnetization in the conventional theory [ [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figures from the paper (4 more)
Figure 2
Figure 2. Figure 2: FIG. 2. Effect of an alternating potential on (a) the local density of states [PITH_FULL_IMAGE:figures/full_fig_p010_2.png]
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Sketch of the magnetic dimer ( [PITH_FULL_IMAGE:figures/full_fig_p011_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Schematic representation of the dice lattice with primitive vectors [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Sketch of the hexagonal lattice circumscribed by a circle of radius [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

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Reviewed July 8, 2026 · model on record in the stance chip above.