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3D bulk-resolved $g$-wave altermagnetic order parameter in CrSb

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Quantum oscillations map the g-wave magnetic order parameter of CrSb in three dimensions.

desk verdict The g-wave claim survives the DFT patch because the experiment carries the symmetry assignment, but the paper needs revision to clean up the ad hoc band shifts. read the letter →

arxiv 2601.14526 v2 pith:GR25G3AD submitted 2026-01-20 cond-mat.mtrl-sci cond-mat.mes-hallcond-mat.str-el

Mengmeng Long , Theodore I. Weinberger , Zheyu Wu , Mads F. Hansen , Ran Tao , Mridul Shrestha , Dave Graf , Yurii Skourski
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F. Malte Grosche Alexander G. Eaton
This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.mes-hallcond-mat.str-el
keywords altermagnetismCrSbquantumoscillationsdeHaas-vanAlpheneffectg-waveorderparameterspinsplittingFermisurfacesphericalharmonics
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 claims that the altermagnet CrSb carries a g-wave magnetic order parameter: the momentum-space splitting between up- and down-spin Fermi surfaces follows the angular profile of the real spherical harmonic Y4^-3 = zy(3x²-y²), vanishing on four nodal planes at theta=0°, 90° and phi=0°, 60°, 120°. It establishes this by rotating a magnetic field through high- and low-symmetry planes and using quantum oscillations to resolve whether a given Fermi pocket produces one frequency (spin-degenerate, at nodal orientations) or two split frequencies (away from nodes). The frequency branches are verified as spin-split daughters of the same mother sheet by matching the ratio of cyclotron masses to the square root of the frequency ratio. If right, this demonstrates that quantum oscillations can provide a bulk-sensitive, 3D mapping of an unconventional magnetic order parameter, establishing CrSb as a prototypical g-wave metallic altermagnet.

What carries the argument

The central object is the real spherical harmonic Y4^-3(theta,phi) = zy(3x²-y²), expressed in Cartesian coordinates, which describes the angular anisotropy of the altermagnetic spin-splitting Δ(theta,phi). Its role is to encode the symmetry-enforced nodal planes: the function vanishes when theta = 0°, 90° and when phi = 0°, 60°, 120°, exactly the orientations at which the measured quantum-oscillation frequencies remain single and spin-degenerate. The carrying mechanism is magnetic quantum oscillation (de Haas-van Alphen) torque magnetometry: because a field in a nodal plane yields orbits of equal area for up and down spins (single frequency), while a field away from a nodal plane yields diff

What would settle it

Re-measure the Fermi surface with a probe that distinguishes open vs closed orbit topology (e.g., angle-dependent magnetoresistance oscillations) or perform high-resolution ARPES near the A point to test whether the dogbone sheet is truly closed; if the sheet is open, the observed single-frequency assignments at the a- and ab-axes would have to be reinterpreted. Alternatively, measure QO with the field along the c-axis: the g-wave model predicts a single degenerate frequency at theta = 0°, so resolving two well-separated frequencies there would falsify the nodal-plane structure.

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Extended reading notes

Core claim

By tracking the de Haas-van Alphen frequencies of the primary 'dogbone' Fermi surface as the field is rotated, the authors find that the exchange splitting between majority- and minority-spin sheets obeys the symmetries of the real spherical harmonic Y4^-3(theta,phi) ∝ P4^3(cos theta) sin 3phi = zy(3x²-y²)/r^4. They identify four nodal planes—three at phi = 0°, 60°, 120° for all theta, and one at theta = 90° for all phi—where Kramers degeneracy is symmetry-enforced and a single QO frequency is observed; in the antinodal c-ab plane, a single frequency at theta = 90° splits into two branches separated by ~1 kT within 4° of rotation. The mass ratio of the split branches (m1*/m2* = 0.94 ± 0.04)

Load-bearing premise

The identification of the two measured frequency branches as spin-split daughters of a single closed dogbone pocket depends on an ad hoc 0.11 eV downward shift of the dogbone bands and a 0.015 eV upward shift of the web bands in the DFT calculation; if the true Fermi surface is open or differently connected, the symmetry mapping extracted from these frequencies could change.

Editorial extensions

If this is right

  • CrSb is established as a prototypical g-wave metallic altermagnet with a room-temperature (T_N ~ 740 K) compensated magnetic ground state and low residual resistivity (~2-1 µΩcm), making it a platform for spintronic applications.
  • Quantum oscillation spectroscopy becomes a bulk-sensitive, high-resolution tool for determining the symmetry of unconventional magnetic order parameters—an approach that is notoriously difficult in superconductors.
  • The splitting of a single QO peak into two branches in antinodal planes, collapsing to one in nodal planes, provides a distinctive experimental signature of altermagnetic ordering.
  • The spin-splitting magnitude at the Fermi level, ~25 meV, is directly quantified, enabling comparison with band-structure calculations and models of altermagnetic exchange.
  • The same measurement protocol could, in principle, classify higher-order even-parity symmetries such as i-wave in other altermagnets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A quantitative test would compare the full angular dependence of the splitting magnitude, not just the nodal structure, against the Y4^-3 profile; the paper maps nodes and splitting patterns but does not yet demonstrate that the amplitude of the splitting follows sin³θ cosθ sin3φ across a continuous arc.
  • If the ad hoc DFT band shifts are disputed, the g-wave assignment could also be checked by other bulk probes sensitive to spin-texture, such as spin-resolved ARPES or neutron scattering, though surface sensitivity would need to be ruled out.
  • Because the nodal planes are at fixed crystallographic orientations, this symmetry implies an anisotropic directional response in spin-transport devices—spin currents could be generated or detected preferentially along antinodal directions.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This manuscript reports torque magnetometry de Haas-van Alphen measurements on single-crystal CrSb and presents quantum-oscillation frequencies in three rotation planes: the nodal c–a plane, the antinodal c–ab plane, and a tilted low-symmetry plane. The key observation is that a single QO frequency is seen when the field lies in symmetry-enforced nodal orientations, while two well-resolved frequency branches appear away from those orientations, with a beat structure that is highly angle-sensitive. The authors assign the two branches to spin-split daughters of the same dogbone Fermi-surface sheet, supported by a Lifshitz–Kosevich mass analysis giving m1*/m2* = sqrt(f1/f2). They interpret the angular dependence as showing that the altermagnetic exchange splitting follows the real spherical harmonic Y_4^{-3} = zy(3x^2 - y^2), i.e. a g-wave order parameter. DFT simulations of the QO spectra are used as supporting evidence, after applying rigid band-edge shifts to close the DFT Fermi-surface topology.

Significance. The paper presents a conceptually important idea: quantum oscillations can serve as a bulk-sensitive, angle-resolved probe of the order-parameter symmetry of an altermagnet. The experimental design is strong, and the central dichotomy — single frequency in nodal planes, split frequencies in antinodal planes — is a direct symmetry-based observation that does not depend on the DFT calculation. The mass-ratio test is a useful parameter-free consistency check, and the crystal quality is high (RRR 28). If the g-wave assignment can be justified quantitatively rather than inferred from nodal topology alone, this would be a substantial advance and would establish CrSb as a canonical g-wave metallic altermagnet. However, the manual DFT band shifts and the uniqueness of the Y_4^{-3} assignment are concerns that need to be addressed before the strongest claims can be accepted.

major comments (3)
  1. [Methods, 'DFT calculations'] The converged DFT calculation yields open cylindrical 'dogbone' sheets, inconsistent with the observed QO frequencies. The manuscript then applies rigid shifts of −0.11 eV to the dogbone bands and +0.015 eV to the web bands to close the pockets and match the measured frequencies. These are two ad hoc fitting parameters. As a result, the simulated frequency branches in Figs. 2b,f and 3b are not an independent validation; they carry the imprint of the shifts. The symmetry-enforced nodal observations are independent of this patch, but the quantitative mapping of Δ(θ,φ), the closed-pocket assignment, and the claim of 'excellent correspondence' (Fig. 3) are not. Please either provide a DFT treatment that yields closed dogbone sheets without manual shifts, or explicitly present the simulation as a schematic fitted to the data and correspondingly temper the supporting claims.
  2. [Section 'Unconventional magnetic g-wave order parameter symmetry in CrSb' (p. 16)] The identification of Δ(θ,φ) with the specific harmonic Y_4^{-3} is based on the observed nodal loci (θ = 0°, 90°; φ = 0°, 60°, 120°). These nodes are symmetry-enforced and independently confirmed, but they do not by themselves select l = 4. Other functions with the same B1g symmetry and the same nodal structure would also fit the data as presented. No quantitative fit of the measured splitting amplitude away from the nodes to |P_4^3(cosθ) sin(3φ)| is provided; Fig. 4 gives only a single orientation. The title claim of a 'g-wave' order parameter is therefore stronger than the evidence unless higher-order harmonics are shown to be negligible or an angular-profile fit is added. I recommend either adding such a test or rephrasing to 'symmetry-compatible with Y_4^{-3}' / B1g symmetry.
  3. [Fig. 4 and Eq. (4)] The mass-ratio argument m1*/m2* ≈ sqrt(f1/f2) is a useful consistency check, but it assumes that the two orbits are related by the same local Fermi velocity, i.e. that the spin splitting is essentially a rigid shift of the mother sheet. In an altermagnet the momentum-dependent splitting can vary around the orbit, so this assumption is not automatic. The angular continuity of the split branches in Fig. 3 helps support the same-sheet assignment, but the paper should state the rigidity assumption explicitly and discuss how a momentum-dependent Δ(k) would affect the mass-ratio test. This does not invalidate the data, but it is relevant to the quantitative estimate of Δ ≈ 25 meV from the same framework.
minor comments (5)
  1. [Abstract / Extended Data Fig. 1] The abstract states residual resistivities 'down to ~1 µΩcm', while Extended Data Fig. 1 gives a value of 2.08(1) µΩcm and the main text says 'as low as 2 µΩcm'. Please harmonize these numbers.
  2. [Page 8, 'Symmetry-enforced Kramers spin-degenerate nodal planes'] The sentence 'g-wave splitting should result in nodal planes every θ = 60° for rotations in the a−ab plane, and every φ = 90° when rotating in the c−ab plane' appears to be a typo. Based on the stated nodes of Y_4^{-3}, rotations in the a−ab plane should encounter nodes every φ = 60°, and rotations in the c−ab plane should encounter the node at θ = 90°. Please correct.
  3. [Abstract] Typo: 'altermagetic' should be 'altermagnetic'.
  4. [Fig. 2i] The caption says 'high-pass filtered the raw data' but does not give the filter parameters; the main text refers only to the Methods generally. Please specify the Butterworth/LOESS settings used for this panel, as done for other figures.
  5. [References] Ref. 36 (Yang et al., Nat. Commun. 16, 1442 (2025)) has a closely related title on three-dimensional mapping of altermagnetic spin splitting in CrSb. The present manuscript should explicitly differentiate its new contribution — namely the bulk QO determination of the g-wave symmetry — from that earlier work.

Circularity Check

1 steps flagged · score 2.0 of 10

Central g-wave claim is experimentally self-contained; only the secondary DFT comparison involves fitted band shifts, making it a partial fitted-input rather than a true independent prediction.

  1. fitted input called prediction [Methods, 'DFT calculations'; Fig. 3 section]
    "To 'close' the open Fermi surface sheets of our DFT calculations, we shift our band edges relative to the Fermi energy. The dogbone-like sheets were shifted down by 0.11 eV so that the calculated frequencies along the a, ab, and c-directions are in good agreement with the quantum oscillation data."

    The two rigid band shifts are chosen to reproduce the QO frequencies that the DFT simulation is later presented against as 'theoretical prediction' (Fig. 3: 'correspondence between experimental observation... and theoretical prediction... is excellent'). Absolute frequency agreement is therefore partly constructed rather than predicted. However, the fitted shifts do not set the angular locations of the nodal planes; those are observed directly and are symmetry-enforced, so the step is peripheral to the g-wave claim.

full rationale

The central claim—that CrSb's altermagnetic spin-splitting Δ(θ,φ) has the nodal structure of Y_4^-3, with nodes at θ = 0°, 90° and φ = 0°, 60°, 120°—is established experimentally from direct dHvA torque measurements: single QO peaks are resolved in nodal planes and split peaks in antinodal planes (Figs. 1g–j, 2a,e, 3a,d). The comparison to Y_4^-3 is a symmetry classification of these measured degeneracy loci, not a fit to the measured splitting magnitude; the harmonic's nodal surfaces are not used as inputs to select the data. The effective-mass ratio test m1*/m2* = sqrt(f1/f2) is an independent consistency relation. The only place where an input is fed back into a 'prediction' is the DFT simulation: converged DFT gives open cylindrical dogbone sheets, and the authors then apply ad hoc rigid shifts (dogbone −0.11 eV, web +0.015 eV) to force closed topology and match the QO frequencies. The resulting DFT frequency profiles in Figs. 2–3 are thus calibrated, so calling them 'theoretical prediction' and citing the agreement is partly circular. But this does not propagate to the g-wave conclusion: the nodal angles are symmetry-enforced by the magnetic space group and are reproduced at multiple independent orientations without using the fitted shifts. No load-bearing self-citation chain is present; prior work on CrSb's altermagnetism is external to this author list, and the only self-citations are methodological. Score 2 reflects a minor fitted-input issue in a secondary comparison, not circularity of the central derivation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on two fitted band-edge shifts in the DFT calculation, on the standard symmetry assumptions of altermagnetism, and on the implicit assumption that Y_4^{-3} is the only relevant harmonic. No new physical entities are introduced, but the Fermi-surface closure correction is a significant ad hoc element.

free parameters (2)
  • Dogbone band-edge shift = -0.11 eV
    Shifted down to close the open dogbone Fermi surface sheets in DFT and to match the calculated QO frequencies along a, ab, and c with experiment. This is an ad hoc adjustment to fix a topology inconsistency.
  • Web band-edge shift = +0.015 eV
    Shifted up to keep the total carrier number constant after the dogbone shift; also fitted to maintain charge neutrality, not independently determined.
assumptions (4)
  • domain assumption The magnetic structure of CrSb consists of alternating Cr moments along the c-axis, connected by a 6_3 screw plus time-reversal operation.
    Taken from prior neutron diffraction (ref 35) and used as input for DFT and the symmetry analysis; the paper does not redetermine the magnetic structure.
  • domain assumption The altermagnetic spin-splitting obeys Δ(g k) = −Δ(k) and thus has symmetry-enforced nodal planes where spin degeneracy is restored.
    Standard altermagnetism framework from refs 9–13; the interpretation of the single/double QO frequency peaks as nodal/antinodal relies on this symmetry rule.
  • domain assumption The two observed QO frequencies at a given orientation are spin-split daughters of a single closed dogbone Fermi surface pocket, not distinct pockets or magnetic breakdown orbits.
    Supported by the mass ratio m1*/m2* = sqrt(f1/f2), but the uniqueness of this assignment depends on the Fermi surface topology, which the DFT had to be artificially closed.
  • ad hoc to paper The angular dependence of the spin-splitting is fully captured by the lowest-order real spherical harmonic Y_4^{-3}; higher-order harmonics with the same nodal structure are negligible.
    The paper compares the observed nodal planes to Y_4^{-3} but does not test or exclude higher-l harmonics (e.g., Y_6^{-3}) that share the same sin(3φ) and θ=0,90 nodes.

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Pith. "Pith review of 3D bulk-resolved $g$-wave altermagnetic order parameter in CrSb." pith.science (2026). https://pith.science/paper/GR25G3AD

@misc{pith2026260114526,
  author       = {Pith},
  title        = {Pith review of: 3D bulk-resolved $g$-wave altermagnetic order parameter in CrSb},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GR25G3AD}},
  note         = {Machine review of arXiv:2601.14526}
}
abstract

Electronic phases of matter, such as magnetism and superconductivity, are defined and distinguished by their order parameters quantifying the spontaneous symmetry breaking underlying each phase. Simple cases include the uniform magnetisation of ferromagnets and isotropic gap function of conventional superconductors. Unconventional superconductors often have a nodal gap function, where the gap changes sign at nodes on the Fermi surface. This concept of unconventional/nodal order parameter symmetry has recently been extended to numerous magnetic systems, including altermagnets, in which up- and down-spin species are non-degenerate around the Fermi surface. Here we demonstrate that magnetic quantum oscillation measurements can provide a high resolution, bulk-sensitive, 3D mapping of the order parameter in an unconventional magnet. By rotating a magnetic field through high- and low-symmetry directions of the CrSb Brillouin zone, we show that this material's altermagnetic band structure leads to a reduction of symmetry for each spin-split Fermi sheet away from nodal orientations. In momentum space, the exchange splitting between up and down spins follows the profile of the $\mathcal{Y}_{4}^{-3}=zy(3x^2-y^2)$ real spherical harmonic - analogous to a $g$-orbital of the hydrogen atom. While notoriously difficult to resolve in unconventional superconductors, our work demonstrates that the order parameter symmetry of unconventional magnets can be precisely mapped via quantum-oscillatory quasiparticle spectroscopy, establishing CrSb as a prototypical $g$-wave metallic altermagnet.

Figures

Figures reproduced from arXiv: 2601.14526 by the authors.

Figure 1
Figure 1. Nodal planes bisect 𝑔-wave-symmetric spin-split Fermi surface sheets in CrSb. a, The crystal structure of CrSb, with alternating magnetic moments on the Cr sites oriented along the 𝑐-axis. Red indicates spin-up, with blue correspond￾ing to spin-down. b, The trigonal arrangement of the antimony ions means that the mapping of a red chromium site to a blue one cannot be performed by trans￾lation and time-reversal opera… view at source ↗
Figure 1
Figure 1. (cont.) altermagnetic spin-splitting in CrSb. g, Quantum oscillations in the background-subtracted magnetic torque Δ𝜏, rescaled to the same maximal amplitude for ease of presentation (see Methods for experimental and analytical details) collected at magnetic field H orientations as indicated, and h, their fast Fourier transform (FFT) frequency spectra. A singular FFT peak is observed in the three nodal orientations,… view at source ↗
Figure 2
Figure 2. Mapping the altermagnetically spin-split Fermi surface of CrSb. a, [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figures from the paper (3 more)
Figure 3
Figure 3. Figure 3: Confirmation of 𝑔-wave altermagnetic spin texture in CrSb. a, Heatmap of quantum oscillation frequencies versus rotation angle and b, simulated spin-up (red) and spin-down (blue) angular frequency profile computed by DFT. Here we rotate by an angle 𝛼 through a plane of…
Figure 4
Figure 4. Figure 4: Spin-split effective mass study. a, Quantum oscillations at incremental temperatures for 𝜃 = 84.8 ◦ , 𝜑 = 21.4 ◦ and b, their corresponding frequency spectra, with temperatures indicated. Δ𝜏 was high-pass filtered to focus on two frequency components, at 𝑓1 = 3.41 kT a…
Figure 4
Figure 4. Figure 4: (cont.) The raw torque for 𝜃 = 𝜑 = 90◦ has been scaled by a factor of 5 so that its magnitude is comparable to that of the torque at 𝜃 = 6 ◦ , 𝜑 = 90◦ . f, Second derivative of torque with respect to field, 𝜏 ′′, for rotations in the nodal 𝑐 − 𝑎 plane and g, in the ant…

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Altermagnetic spin-split Fermi surfaces in CrSb revealed by quantum oscillation measurements

    cond-mat.mtrl-sci 2026-01 conditional novelty 6.0 of 10

    Quantum oscillation measurements on CrSb map spin-split Fermi surfaces that confirm altermagnetic g-wave band splitting in the bulk.

  2. Band splitting in the altermagnet CrSb

    cond-mat.str-el 2026-02 unverdicted novelty 5.0 of 10

    The proposed spin–orbit band splitting for CrSb (Eq. 10) is not invariant under the paper's own magnetic group (Eq. 9), and the abstract's toroid-altermagnet result is absent from the text.

Reference graph

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