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arxiv: 2604.28088 · v1 · submitted 2026-04-30 · ❄️ cond-mat.mes-hall

Giant Spin Magnetization from Quantum Geometry in Altermagnets

Neelanjan Chakraborti , Sudeep Kumar Ghosh , Snehasish Nandy This is my paper

Pith reviewed 2026-05-07 05:28 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords altermagnetsquantum geometryspin magnetizationquantum metriccentrosymmetric systemsmagnetic field responsespintronicsBloch states
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The pith

Centrosymmetric altermagnets generate giant linear spin magnetization solely through the spin-rotation quantum metric.

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

Altermagnets host spin-split bands yet carry zero equilibrium spin magnetization due to symmetry. The paper constructs a generalized quantum geometric tensor that tracks both momentum translations and spin rotations of Bloch states, then decomposes the linear spin magnetization response into equilibrium, electric-field, and magnetic-field channels. Inversion symmetry rules out any linear electric-field contribution in centrosymmetric crystals, while C_n T symmetry eliminates the equilibrium term in altermagnets. This leaves only the magnetic-field-induced magnetization, which the authors prove is generated entirely by the spin-rotation quantum metric. Explicit calculations for FeSb2 and CrSb yield magnetization densities of order 10^{-2} μ_B nm^{-3} already at 10 mT, far larger than typical values in ordinary magnets. The framework therefore isolates a universal quantum-geometric mechanism for spin magnetization that operates cleanly in any centrosymmetric system.

Core claim

In centrosymmetric altermagnets the linear spin magnetization induced by an external magnetic field originates entirely from the spin-rotation quantum metric. A unified decomposition based on the generalized quantum geometric tensor shows that inversion symmetry forbids the electric-field-driven term while C_n T symmetry suppresses the equilibrium contribution, rendering the magnetic-field-driven term the sole symmetry-allowed linear response. In representative d-wave and g-wave compounds this metric directly sets the magnitude of the induced magnetization.

What carries the argument

The spin-rotation quantum metric, defined as the relevant component of the generalized quantum geometric tensor under spin rotations of Bloch states, which alone determines the magnetic-field-induced linear spin magnetization once inversion and C_n T symmetries are imposed.

If this is right

  • The magnetic-field-induced spin magnetization scales linearly with field strength and is fixed by the spin-rotation quantum metric in every centrosymmetric altermagnet.
  • No electric-field-driven linear spin magnetization is symmetry-allowed in any centrosymmetric system.
  • The equilibrium spin magnetization remains zero in altermagnets even when quantum geometry is taken into account.
  • The same spin-rotation metric mechanism operates universally in centrosymmetric materials, not only in altermagnets.
  • Low-field spintronic devices become feasible if the predicted magnetization density can be converted into detectable charge currents.

Where Pith is reading between the lines

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

  • The framework suggests that similar giant low-field responses could appear in other centrosymmetric systems once their spin-rotation quantum metric is computed.
  • Temperature dependence of the metric could be measured to test whether the effect survives to room temperature in candidate materials.
  • Orbital magnetization responses might be governed by an analogous orbital-rotation quantum metric under the same symmetry constraints.
  • Device designs that exploit this response would operate at millitesla fields rather than the tesla-scale fields required for conventional magnets.

Load-bearing premise

The generalized quantum geometric tensor that includes both momentum translations and spin rotations fully accounts for every linear contribution to spin magnetization, with no additional material-specific corrections or higher-order terms altering the symmetry-allowed channels in the compounds studied.

What would settle it

A direct measurement in FeSb2 or CrSb that finds either a detectable linear electric-field-induced spin magnetization or a magnetic-field-induced magnetization density orders of magnitude smaller than 10^{-2} μ_B nm^{-3} at 10 mT would falsify the claim that the spin-rotation quantum metric is the sole mechanism.

Figures

Figures reproduced from arXiv: 2604.28088 by Neelanjan Chakraborti, Snehasish Nandy, Sudeep Kumar Ghosh.

Figure 1
Figure 1. Figure 1: FIG. 1 view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 view at source ↗
Figure 3
Figure 3. Figure 3: (a) presents the spin-projected band structure of CrSb along the high-symmetry path −M′ (−0.5, 0, 0.25) → Γ ′ (0, 0, 0.25) → M′ (0.5, 0, 0.25), revealing pronounced momentum-dependent spin split￾ting along the −M′–Γ′–M′ directions [49]. The Fermi surface of CrSb shown in view at source ↗
read the original abstract

Altermagnets host spin-split band structures while exhibiting vanishing equilibrium spin magnetization, making field-induced responses a direct probe of their quantum geometry. A central question, in this regard, is which quantum-geometric mechanism can generate a linear spin magnetization in centrosymmetric systems. Here we develop a unified framework based on a generalized quantum geometric tensor that incorporates both momentum translations and spin rotations of Bloch states, and decompose spin magnetization into equilibrium, electric-field-driven, and magnetic-field-driven contributions. We show that inversion symmetry forbids the linear electric-field response in centrosymmetric systems, while $C_n T$ symmetry further suppresses the equilibrium contribution in altermagnets. Consequently, centrosymmetric altermagnets provide a particularly clean realization in which the magnetic-field-induced spin magnetization emerges as the only symmetry-allowed linear quantum-geometric response. We demonstrate that this contribution originates entirely from the spin-rotation quantum metric, establishing it as the sole linear quantum-geometric mechanism in such systems. Using representative centrosymmetric altermagnets, including the $d$-wave compound $\mathrm{FeSb}_2$ and the $g$-wave compound $\mathrm{CrSb}$, we show that the spin-rotation quantum metric directly controls this response. Crucially, we predict a giant linear spin magnetization of order $10^{-2}\mu_B\,\mathrm{nm}^{-3}$ at magnetic fields of $\sim 10\,\mathrm{mT}$, exceeding typical experimental values for conventional magnets by several orders of magnitude. Our results identify a universal quantum geometric mechanism of spin magnetization operative in centrosymmetric systems in general, and establish centrosymmetric altermagnets as an ideal platform for its experimental detection with potential applications in spintronics.

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

3 major / 2 minor

Summary. The manuscript develops a generalized quantum geometric tensor (QGT) framework that incorporates both momentum translations and spin rotations of Bloch states. It decomposes the linear response of spin magnetization into equilibrium, electric-field-driven, and magnetic-field-driven contributions. Symmetry arguments show that inversion symmetry forbids the linear electric response while C_n T symmetry suppresses the equilibrium contribution in altermagnets, leaving only the magnetic-field-induced term. This term is shown to originate entirely from the spin-rotation quantum metric. The framework is applied to centrosymmetric altermagnets FeSb2 (d-wave) and CrSb (g-wave), yielding a predicted giant linear spin magnetization of order 10^{-2} μ_B nm^{-3} at magnetic fields of ~10 mT.

Significance. If the central decomposition holds and the numerical predictions are robust, the work identifies a symmetry-protected, purely quantum-geometric mechanism for linear spin magnetization that is operative in any centrosymmetric system but particularly clean in altermagnets. The symmetry analysis cleanly isolates the spin-rotation metric contribution, and the predicted magnitude (orders of magnitude larger than conventional magnets) would provide a falsifiable signature for experimental detection and potential spintronic applications. The absence of free parameters in the symmetry constraints and the explicit link to a single geometric quantity are strengths.

major comments (3)
  1. [Generalized QGT decomposition] Generalized QGT decomposition (section deriving the magnetic-field response): The claim that the B-induced spin magnetization 'originates entirely from the spin-rotation quantum metric' and is the 'sole linear quantum-geometric mechanism' requires an explicit reduction of the Kubo formula for the spin operator response to a Zeeman field, showing that all other interband or intraband terms vanish or are absorbed into the metric at linear order in B. Without this step-by-step equivalence (including the precise definition of the generalized QGT components), it remains possible that material-specific interband matrix elements or Fermi-surface contributions reintroduce non-geometric terms, directly affecting the asserted magnitude.
  2. [Numerical evaluation for FeSb2 and CrSb] Numerical evaluation for FeSb2 and CrSb (results section on material calculations): The model Hamiltonian, tight-binding parameters, Brillouin-zone sampling, and Fermi-level placement used to compute the spin-rotation quantum metric are not specified in sufficient detail. Consequently, the reported value of order 10^{-2} μ_B nm^{-3} at 10 mT cannot be independently verified, and no error bars, convergence tests, or sensitivity analysis with respect to neglected many-body corrections or higher-order terms are provided. This undermines the quantitative 'giant' claim, which is load-bearing for the paper's central assertion.
  3. [Discussion of experimental implications] Comparison to conventional magnets (discussion of experimental implications): The statement that the predicted magnetization 'exceeds typical experimental values for conventional magnets by several orders of magnitude' is not supported by explicit citations to measured spin susceptibilities or linear response coefficients in standard ferromagnets or paramagnets. A quantitative benchmark (e.g., against known χ values) is needed to substantiate the 'giant' characterization.
minor comments (2)
  1. [Abstract and introduction] The abstract and introduction would benefit from a brief statement of the precise definition of the generalized QGT (including its decomposition into translation and rotation sectors) before the symmetry arguments are presented.
  2. [Figures] Figure captions for the band-structure or metric plots should explicitly state the magnetic-field strength, temperature, and Fermi level used in the calculations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their detailed and insightful comments on our manuscript. We have carefully considered each point and will revise the manuscript to address the concerns raised, thereby improving the clarity, rigor, and verifiability of our results. Our point-by-point responses are provided below.

read point-by-point responses

  1. Referee: Generalized QGT decomposition (section deriving the magnetic-field response): The claim that the B-induced spin magnetization 'originates entirely from the spin-rotation quantum metric' and is the 'sole linear quantum-geometric mechanism' requires an explicit reduction of the Kubo formula for the spin operator response to a Zeeman field, showing that all other interband or intraband terms vanish or are absorbed into the metric at linear order in B. Without this step-by-step equivalence (including the precise definition of the generalized QGT components), it remains possible that material-specific interband matrix elements or Fermi-surface contributions reintroduce non-geometric terms, directly affecting the asserted magnitude.

    Authors: We fully agree that an explicit derivation is necessary to rigorously establish the equivalence. In the revised manuscript, we will include a new appendix that provides a step-by-step reduction of the Kubo formula for the spin magnetization response to a Zeeman magnetic field. This derivation will show that at linear order in B, the response is precisely given by the spin-rotation component of the generalized quantum geometric tensor, with all other interband and intraband contributions either vanishing by symmetry (due to the altermagnetic properties) or being absorbed into the metric definition. We will also provide the precise definitions of the generalized QGT components used in our framework. This will eliminate any ambiguity regarding potential non-geometric terms and strengthen the central claim. revision: yes

  2. Referee: Numerical evaluation for FeSb2 and CrSb (results section on material calculations): The model Hamiltonian, tight-binding parameters, Brillouin-zone sampling, and Fermi-level placement used to compute the spin-rotation quantum metric are not specified in sufficient detail. Consequently, the reported value of order 10^{-2} μ_B nm^{-3} at 10 mT cannot be independently verified, and no error bars, convergence tests, or sensitivity analysis with respect to neglected many-body corrections or higher-order terms are provided. This undermines the quantitative 'giant' claim, which is load-bearing for the paper's central assertion.

    Authors: We acknowledge that the computational details in the original submission were not comprehensive enough for full reproducibility. In the revised manuscript, we will add a new subsection in the methods or results detailing the model Hamiltonians for FeSb2 and CrSb, including all tight-binding parameters, hopping integrals, and spin-orbit coupling strengths. We will specify the Brillouin zone sampling (e.g., a dense 200×200×200 k-grid), the exact Fermi level placement, and include convergence plots demonstrating that the spin magnetization value converges with increasing k-point density. Error bars will be provided based on variations in the Fermi energy within the insulating gap and small perturbations to the parameters. For many-body corrections, we will add a discussion noting that our calculations are within the single-particle approximation and that electron-electron interactions may renormalize the value, but the order of magnitude is expected to remain robust; we will suggest this as a direction for future ab initio studies. These additions will allow independent verification and support the 'giant' characterization. revision: yes

  3. Referee: Comparison to conventional magnets (discussion of experimental implications): The statement that the predicted magnetization 'exceeds typical experimental values for conventional magnets by several orders of magnitude' is not supported by explicit citations to measured spin susceptibilities or linear response coefficients in standard ferromagnets or paramagnets. A quantitative benchmark (e.g., against known χ values) is needed to substantiate the 'giant' characterization.

    Authors: We thank the referee for this important suggestion. In the revised discussion section, we will include explicit references to experimental measurements of spin susceptibilities in conventional materials. For instance, we will cite works reporting the magnetic susceptibility of ferromagnetic iron (χ ≈ 10^{-4} emu/g or equivalent in SI units) and paramagnetic metals or insulators. We will convert our predicted linear magnetization of 10^{-2} μ_B nm^{-3} at 10 mT into an effective susceptibility χ (in units of μ_B / T per volume) and provide a direct numerical comparison, demonstrating that it is indeed larger by 2-3 orders of magnitude than typical values in conventional magnets (e.g., χ ~ 10^{-5} - 10^{-6} in appropriate units). This quantitative benchmark will substantiate the 'giant' nature of the effect and its potential for experimental detection. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the quantum-geometric derivation for altermagnet spin magnetization

full rationale

The paper constructs a generalized quantum geometric tensor incorporating momentum translations and spin rotations of Bloch states, then decomposes the spin magnetization response into equilibrium, electric, and magnetic contributions. Symmetry constraints (inversion forbidding linear E-response; C_n T suppressing equilibrium magnetization) are applied to isolate the magnetic-field-induced term as originating from the spin-rotation quantum metric. This isolation follows directly from the explicit definitions and symmetry analysis within the single-particle Bloch framework rather than by redefinition or tautology. Numerical results for FeSb2 and CrSb are obtained by evaluating the derived expressions on model Hamiltonians, providing an independent computational check rather than a fitted or self-referential output. No load-bearing self-citations, ansatz smuggling, or renaming of known results are present; the central claim remains a direct consequence of the framework and symmetries without reducing to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard quantum-mechanical definitions of Bloch states and the quantum geometric tensor, plus domain assumptions about crystal symmetries in altermagnets. No new particles or forces are introduced. Material calculations for FeSb2 and CrSb presumably employ standard band-structure models, but specific free parameters are not listed in the abstract.

axioms (2)
  • standard math Bloch states in periodic crystals possess a well-defined quantum geometric tensor that can be generalized to include spin rotations
    Standard construction in condensed-matter theory for describing geometric properties of electronic bands.
  • domain assumption Inversion symmetry and C_n T symmetry impose strict constraints on allowed linear response tensors in centrosymmetric altermagnets
    Standard application of group theory to magnetic point groups in the altermagnet literature.

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