{"id":"41269753-0e0a-4369-8080-13fc2b97e959","arxiv_id":"2601.14526","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Magnetic quantum oscillations in CrSb reveal a g-wave (Y_4^{-3}) spin-splitting order parameter with nodal planes every 60 degrees, confirming CrSb as a g-wave metallic altermagnet.","lead":"Quantum oscillation measurements on the metallic antiferromagnet CrSb map a spin-split Fermi surface whose splitting follows a g-wave pattern, establishing CrSb as a prototypical g-wave altermagnet. The work shows that bulk-sensitive quantum oscillations can directly reveal the symmetry of a magnetic order parameter.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"DFT topology is patched by ad hoc band shifts; this weakens the DFT confirmation but not the symmetry-enforced g-wave nodes, so the central claim is only partly affected.","rationale":"The reader's weakest_assumption correctly identifies the ad hoc DFT topology correction as a real weakness, but its impact on the central claim is more limited than stated. The g-wave symmetry assignment rests primarily on the experimentally observed nodes—where the spin-split QO branches merge—and these nodes are symmetry-enforced by the magnetic space group, not by the specific Fermi-surface topology. Thus even if the dogbone pocket assignment were wrong, the same nodal structure would appear for any Fermi-sheet in the altermagnetic state. The topology issue does, however, undermine the quantitative DFT comparison and the use of the simulated branches to confirm the spin-split-daughter interpretation. A first-principles check (e.g., DFT+U) could settle whether the rigid shift is a legitimate band-structure correction or a symptom of a deeper discrepancy. Given that the central symmetry evidence remains intact, the reader's CONDITIONAL verdict seems appropriate but for slightly different reasons; the concern is real but not fatal, so no change to the verdict is needed.","tokens_in":15747,"tokens_out":19798,"duration_ms":197689,"concrete_test":"Recompute the CrSb band structure including a Hubbard U on Cr 3d (or a hybrid functional) tuned to reproduce the measured Neel temperature and magnetic moment, without any rigid band shift. If the dogbone sheets close naturally and the QO frequencies match, the 0.11 eV shift is a legitimate correction; if they remain open, the DFT Fermi surface is genuinely inconsistent and the simulated branch assignment in Figs. 2–3 requires re-evaluation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper admits (Methods, 'DFT calculations') that converged DFT yields open cylindrical 'dogbone' Fermi-surface sheets, inconsistent with the observed quantum oscillations, and then applies rigid band shifts—dogbone down by 0.11 eV, web up by 0.015 eV—to force closure and match measured frequencies. This is an explicit two-parameter, ad hoc correction. Consequently, the DFT-vs-experiment agreement in Figs. 2 and 3 is not an independent validation; the simulated frequency branches carry the imprint of the fitted shifts. If the true topology were open or differently connected, the assignment of the high-frequency QO branches to a single closed dogbone pocket could be wrong, which would affect the quantitative mapping of Δ(θ,φ). However, the central g-wave symmetry evidence—single QO peaks in the nodal planes and split peaks in antinodal planes—is a direct experimental observation, independent of the DFT shifts. Since those nodes are symmetry-enforced and are reproduced at multiple orientations, the topology correction does not destroy the symmetry claim, but it does limit confidence in the detailed comparison and in the identification of the split branches as spin-split daughters of the same pocket.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15953,"tokens_out":9273,"duration_ms":110351,"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":[{"comment":"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.","section":"Methods, 'DFT calculations'"},{"comment":"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.","section":"Section 'Unconventional magnetic g-wave order parameter symmetry in CrSb' (p. 16)"},{"comment":"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.","section":"Fig. 4 and Eq. (4)"}],"minor_comments":[{"comment":"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.","section":"Abstract / Extended Data Fig. 1"},{"comment":"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.","section":"Page 8, 'Symmetry-enforced Kramers spin-degenerate nodal planes'"},{"comment":"Typo: 'altermagetic' should be 'altermagnetic'.","section":"Abstract"},{"comment":"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.","section":"Fig. 2i"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"This is a high-profile claim with a strong experimental core, but the current manuscript overstates the uniqueness of the g-wave assignment and the DFT validation is compromised by the manual band shifts. The symmetry-enforced nodal observations are solid and should survive revision. The main risk is that the 'g-wave' label is presented as quantitatively established when only the nodal structure is directly measured. I recommend major revision with emphasis on either a quantitative angular fit of the splitting or appropriately softened language, and a transparent treatment of the DFT fitting procedure. The relationship to Ref. 36 should also be clarified for the editor."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the take: the experiment is the real result, and the g-wave symmetry assignment stands on direct observation of symmetry-enforced nodal planes, not on the DFT. The ad hoc band shifts in the Methods are a legitimate weakness, but they do not sink the central claim.\n\nWhat is actually new: the explicit experimental assignment of the Y_4^{-3} g-wave symmetry to the altermagnetic spin-splitting in CrSb, based on the collapse of the split QO branches at phi = 0, 60, 120 degrees and theta = 90 degrees. The single-frequency-to-double-frequency pattern going from nodal to antinodal orientations is direct and convincing, and the mass ratio check (m1*/m2* ~ sqrt(f1/f2)) is a clean, parameter-free corroboration that the two branches are spin-split daughters of the same pocket. The paper also does a good job with the RuO2 history and makes the case for QOs as a bulk probe. It is not the first 3D QO mapping of CrSb—Yang et al. did that—but the explicit g-wave identification and the tilted-plane rotation that connects the nodal planes are genuine increments.\n\nSoft spots, in proportion: the DFT simulation is patched. The Methods admit that the converged DFT gives open cylindrical 'dogbone' sheets, inconsistent with quantum oscillations, and then rigidly shift the dogbone bands down by 0.11 eV and the web bands up by 0.015 eV to close the sheets and match frequencies. That is a two-parameter ad hoc fix, and it means the DFT-vs-experiment agreement in Figs. 2 and 3 is not independent validation. If the true Fermi surface topology is open or differently connected, the assignment of the high-frequency branches to a single closed dogbone pocket would need re-evaluation, which would compromise the quantitative Delta(theta,phi) mapping. But the central symmetry evidence—single QO peaks in the nodal planes and split peaks away from them—does not depend on the DFT shifts, because the nodes are symmetry-enforced and are reproduced at multiple orientations. So the g-wave assignment is on solid experimental ground; the DFT patch limits confidence in the detailed band-structure comparison, not in the symmetry claim.\n\nMinor: the splitting profiles have no error bars, the mapping is limited to a single pocket (the dogbone), and the paper does not sharply distinguish its result from Yang et al. beyond the explicit Y_4^{-3} labeling. Those are fixable.\n\nWho this is for: the altermagnetism and quantum-oscillations communities. The data are high quality, the paper is readable, and it deserves a serious referee. The revision should address the ad hoc shifts transparently—ideally with a sensitivity analysis or a more self-consistent calculation—and spell out the incremental claim relative to Yang et al.\n\nRecommendation: send to peer review with the expectation of a solid, revisable paper.","headline":"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.","tokens_in":16557,"tokens_out":2620,"would_cite":true,"duration_ms":27153,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Quantum oscillations map the g-wave magnetic order parameter of CrSb in three dimensions.","keywords":["altermagnetism","CrSb","quantum oscillations","de Haas-van Alphen effect","g-wave order parameter","spin splitting","Fermi surface","spherical harmonics"],"falsifier":"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.","tokens_in":15557,"feed_emoji":"🧲","tokens_out":5365,"duration_ms":53976,"temperature":0.7,"pith_summary":"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.","feed_headline":"Quantum oscillations reveal g-wave order in CrSb","feed_subtitle":"Rotating the field through nodal planes shows spin splitting that follows the Y4-3 spherical harmonic, pinning down a long-sought phase.","key_machinery":"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","core_discovery":"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)","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["g-wave altermagnetic order mapped in CrSb","CrSb's spin splitting follows a g-wave pattern","3D mapping reveals g-wave order in CrSb","CrSb shows bulk g-wave altermagnetic order","Spin splitting in CrSb obeys g-wave harmonic"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["g-wave altermagnetic order mapped in CrSb","CrSb's spin splitting follows a g-wave pattern","3D mapping reveals g-wave order in CrSb","CrSb shows bulk g-wave altermagnetic order","Spin splitting in CrSb obeys g-wave harmonic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001068,"raw_usage":{"total_tokens":4373,"prompt_tokens":867,"completion_tokens":3506,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":611,"completion_tokens_details":{"reasoning_tokens":3444}},"tokens_in":611,"tokens_out":3506,"duration_ms":30715,"temperature":1.0,"reasoning_tokens":3444,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T09:08:45.295570+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}