{"id":"30ea3f27-8ff0-4383-82a8-6b3aca676a1d","arxiv_id":"2601.19105","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Quantum oscillation measurements on CrSb map spin-split Fermi surfaces that confirm altermagnetic g-wave band splitting in the bulk.","lead":"Quantum oscillations in the antiferromagnet CrSb reveal spin-split Fermi surfaces consistent with altermagnetic band structure. The result gives bulk evidence for a hotly debated class of magnetic materials that could enable new spintronic devices.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Band-3 Fermi-surface topology in the DFT+U calculation is explicitly sensitive to the Hubbard U; the assignment of β and δ to small reconstructed pockets may be an artifact of the single chosen U=0.01 Ry.","rationale":"I read the paper as a careful, multi-technique quantum-oscillation study that aims to confirm the altermagnetic spin-split Fermi surface of CrSb predicted by DFT+U. The reader's weakest assumption was the rigid energy shifts used to match the α frequency, and the presence of unobserved calculated orbits. I agree with that as a general concern, but I find a more specific and load-bearing soft spot: the Hubbard U parameter is not just a small correction but qualitatively alters the band-3 Fermi surface topology. The authors themselves state that introducing U=0.01 Ry changes the connectivity of band-3 and creates small pockets, to which they then assign the β and δ branches. Since U is an adjustable input with no sensitivity analysis, the assignment of those branches could be an artifact of parameter choice. This does not invalidate the paper—the β and δ data are real, and the dominant α frequency and band-1/2 assignments are less sensitive—but it does mean the 'definitive' claim, especially regarding the comprehensive Fermi-surface picture including band-3, is conditional on the U value. A systematic U scan would resolve this. The reader's verdict of CONDITIONAL remains appropriate; my concern reinforces it rather than changing it. I therefore recommend UNCHANGED. I partially agree with the reader because they identified the related issue of parameter dependence but focused on the rigid shifts rather than the more critical U sensitivity.","tokens_in":15474,"tokens_out":6686,"duration_ms":77979,"concrete_test":"Recompute the DFT+U band structure and quantum-oscillation frequencies for several Hubbard U values (e.g., U = 0, 0.005, 0.01, 0.02 Ry), re-optimizing the rigid Fermi-level shifts to reproduce the α frequency in each case. Check whether the β and δ branches and their angular dependences (orbits 3-gp1 and 3-gp2) still match the same experimental branches. If the band-3 reconstruction and the associated orbits appear only in a narrow U window, the central claim is weakened; if the assignments are robust across a reasonable U range, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the measured quantum-oscillation frequencies are consistently explained by the DFT+U spin-split Fermi surface, providing definitive bulk evidence for altermagnetic spin splitting. However, this conclusion depends on the calculated Fermi-surface topology, and the Supplemental Material states that 'the band-3 Fermi surface undergoes a significant reconstruction upon the introduction of a small Hubbard U. The inclusion of U = 0.01 Ry enlarges the volumes of the band-1 and -2 Fermi surfaces and breaks the complex connectivity of the band-3 Fermi surface, generating several small pockets.' The observed β and δ branches are assigned to the general-point orbits 3-gp1 and 3-gp2 on these reconstructed pockets. Because U is an adjustable parameter and no systematic scan over U is presented, it is possible that the band-3 topology—and hence the β/δ assignments—is a spurious consequence of choosing U = 0.01 Ry. The rigid Fermi-level shifts (bands 1-2 up 0.046 eV, bands 3-4 down 0.007 eV) are less worrisome because they conserve the band-structure shape, but the U value changes the connectivity itself. Additionally, the paper does not compare the data to a spin-degenerate calculation, so the QO data alone would not prove spin splitting if the known magnetic order were not already established. The combination of U sensitivity and the emphasis on 'definitive' evidence makes this the most load-bearing assumption.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports a quantum-oscillation (SdH and dHvA) study of the altermagnetic metal CrSb, combining high-field transport and torque measurements with DFT+U calculations including spin-orbit coupling. Multiple frequency branches (α, β, δ, ε, ζ) are tracked over wide angular ranges and assigned to extremal orbits on the calculated four-band Fermi surface. The authors conclude that their results provide definitive bulk-sensitive evidence for the altermagnetic spin-split Fermi surface and that bands 1 and 2 consist of tubular sheets plus A-centered pockets, with band-3 and band-4 orbits explaining the remaining branches. They also argue that the Berry phase of the α oscillation cannot be reliably inferred because α is a sum of two near-degenerate orbits.","tokens_in":15864,"tokens_out":4352,"duration_ms":50871,"significance":"If the assignments are correct, the paper would be an important bulk-sensitive confirmation of altermagnetic spin splitting in CrSb, complementing surface-sensitive ARPES studies. The experimental work is substantial: high-quality crystals, multiple techniques, wide angular tracking, and careful treatment of the Lifshitz–Kosevich analysis. The paper is also honest about the ambiguity in the Berry-phase determination. However, the central claim of definitive evidence is weakened by the explicit use of adjustable rigid Fermi-level shifts to match the dominant α frequency and by the sensitivity of the band-3 Fermi-surface topology to the chosen Hubbard U. Those points need to be addressed before the conclusion can be accepted at face value.","major_comments":[{"comment":"The α frequency is reproduced by rigidly shifting the Fermi level of bands 1 and 2 by +0.046 eV and of bands 3 and 4 by −0.007 eV. Thus the agreement of the calculated 1-A-o orbit with the experimental α branch is not an independent prediction but a least-squares-style fit. The paper should state the unshifted DFT+U frequency for this orbit, quantify the discrepancy, and justify the shift independently (e.g., by comparison with ARPES or with a self-consistent renormalization). Without this, the agreement for α is a tuning result and cannot be cited as evidence for the altermagnetic calculation.","section":"Supplemental, 'Band structure calculations'"},{"comment":"The band-3 Fermi surface is explicitly reconstructed by the introduction of U = 0.01 Ry, which 'breaks the complex connectivity ... generating several small pockets.' The β and δ (|θ| ≥ 40°) branches are assigned to the 3-gp1 and 3-gp2 orbits on these reconstructed pockets. Because no systematic scan over U or over double-counting schemes is presented, it is possible that the existence and shape of these pockets are artifacts of a single chosen value of U. This is load-bearing: β and δ are the only branches assigned to band-3, and the paper uses them to claim comprehensive Fermi-surface coverage. A U-dependence study or an independent check of the pocket sizes (e.g., via carrier densities or ARPES) is needed.","section":"Supplemental, 'Band structure calculations'"},{"comment":"The abstract states that 'bands 1 and 2 form closed pockets centered at the A point, rather than the tubular c-axis-open sheets ... proposed in previous studies.' This directly contradicts the main text, which says each band-1 and band-2 Fermi surface 'consists of an outer tubular sheet along the ΓA line and an inner closed pocket at the A point.' The abstract must be corrected; as written, it misstates the paper's own calculated and argued Fermi-surface topology and could mislead readers about the central result.","section":"Abstract vs. main text, p. 2"},{"comment":"The paper compares the measured frequencies only to a spin-polarized DFT+U calculation that contains the known altermagnetic order. It does not provide a comparison with a spin-degenerate (nonmagnetic or artificially paramagnetic) calculation. Since the magnetic order is already established by neutron diffraction and the DFT+U calculation builds it in, the observation of multiple frequencies does not by itself demonstrate altermagnetic spin splitting. A direct comparison with a nonmagnetic calculation would clarify which frequency branches are specifically attributable to the spin splitting, rather than to the underlying band structure. This comparison is needed to support the 'definitive evidence' claim.","section":"General comparison with a spin-degenerate calculation"}],"minor_comments":[{"comment":"Typo: 'quantum os cillation' should be 'quantum oscillation'.","section":"Title"},{"comment":"The sentence 'the lowest frequency peaks ... correspond to oscillations with periods of 1.5 or lower within the measured field range' is unclear. Presumably the frequencies are below about 1.5 T, corresponding to long periods; please rephrase to avoid confusion between frequency and period.","section":"Fig. 1, caption and text"},{"comment":"The assumption A1 = A2 and ϕB,1 = −ϕB,2 for the two α components is introduced without comment. Since this assumption underlies the explanation of the near-zero intercept, it should be explicitly justified or at least flagged as an assumption that cannot be independently verified from the present data.","section":"Supplemental, Eq. (S2) and following"},{"comment":"The experimental branches are shown with different symbols for different samples/field windows, but no legend appears in Fig. 2(a). Adding a legend or a clear table matching symbols to conditions would improve readability.","section":"Fig. 2 and Table SI"},{"comment":"References [33] and [34] are parallel studies; the paper compares frequencies with them. It would be helpful to state explicitly which of their samples/methods are being compared, and whether the differences in field windows could affect the frequency resolution.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is a careful experimental work with honest discussion of limitations, but the 'definitive evidence' claim is currently supported by a calculation that has been adjusted to match the dominant frequency and whose band-3 topology is sensitive to an ad hoc Hubbard U. I believe the technical issues are fixable within the scope of a revision (U scan, null-model comparison, abstract correction), so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The field has been sorting out the Fermi surface of CrSb, and this paper is the most complete quantum-oscillation measurement so far. The important news is that the main Fermi surface picture — tubular sheets along Γ-A plus A-centered pockets for bands 1 and 2 — is likely right. That part is anchored by ARPES and by the angle-dependent data, and it disagrees with the dogbone topology in ref [34] in a way that seems correct. The band-3 and band-4 assignments are more provisional, especially β and δ on band-3.\n\nWhat the paper does well: it tracks five frequency branches over wide rotations in two planes, uses both SdH and torque, and checks consistency across flux-grown and CVT-grown crystals. The analysis is careful about the α frequency being a sum of two near-degenerate orbits, and it honestly says the Berry phase cannot be pinned down. The effective masses and Dingle times are reported and are consistent with the assignments.\n\nThe soft spots. The rigid Fermi-level shifts are the obvious one: α is matched by construction, so it isn't a prediction. The paper should show the unshifted frequencies and the resulting mismatch. More concerning is the band-3 reconstruction with Hubbard U. The Supplemental states that U = 0.01 Ry breaks the connectivity of the band-3 surface and creates the small pockets that give β and δ. Without a U scan, those two assignments are conditional. I don't think this destroys the central claim — the band-1/2 spin splitting is supported by the near-degenerate α pair and by ARPES — but it means the 'definitive' language in the abstract is too strong for the band-3 part.\n\nA note on the stress-test concern: it says the QO data alone wouldn't prove spin splitting without the known magnetic order. That's true but minor, since the magnetic order is well established; the paper is using QO to confirm the bulk Fermi surface, not to discover magnetism.\n\nVerdict: the paper deserves a serious referee. The data are valuable, the analysis is honest, and the disagreements with the parallel studies need to be sorted out. The referee should ask for the unshifted predictions, a U scan or an explicit statement of the band-3 uncertainty, and a tone-down of 'definitive.' After that, it would be a solid PRB.","headline":"The most complete quantum-oscillation study of CrSb so far, with a solid band-1/2 Fermi surface picture but band-3 assignments that ride on a U-sensitive reconstruction.","tokens_in":16358,"tokens_out":4411,"would_cite":true,"duration_ms":49708,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["71.18.+y","75.50.Ee","71.20.-b"],"model":"deepseek-v4-flash","headline":"All measured quantum-oscillation frequencies in CrSb match the spin-split Fermi surface of an altermagnetic band calculation, providing bulk-sensitive evidence for altermagnetic spin splitting.","keywords":["altermagnetism","CrSb","quantum oscillations","Fermi surface","spin splitting","Shubnikov-de Haas effect","de Haas-van Alphen effect","DFT+U"],"falsifier":"Treat the rigid band shifts and the Hubbard U as free parameters and simultaneously fit all five measured branches across both rotation planes; if no single set of shifts reproduces the observed angular dependences of α, β, δ, ε, and ζ, the altermagnetic Fermi-surface assignment fails. Alternatively, angle-resolved photoemission at the wave vectors of the 3-gp orbits (around kz ≈ 0.21·2π/c) should directly image the band-3 pockets assigned to the β branch; their absence would falsify that assignment.","tokens_in":15347,"feed_emoji":"🧲","tokens_out":11935,"duration_ms":114177,"temperature":0.7,"pith_summary":"The paper sets out to verify, in the bulk, the altermagnetic band structure of CrSb, a collinear antiferromagnet whose up- and down-spin sublattices are related by rotation and whose bands should be spin-split with g-wave symmetry. Using Shubnikov–de Haas and magnetic-torque measurements in fields up to 41.5 T, combined with DFT+U calculations including spin-orbit coupling, it identifies five quantum-oscillation frequency branches and assigns each to extremal orbits on four spin-non-degenerate bands. The central result is that the measured frequencies are consistently explained by the calculated altermagnetic Fermi surface, with bands 1 and 2 forming outer tubular sheets along ΓA plus new closed pockets at A, band 3 contributing ring-like and general-point orbits, and band 4 ellipsoidal pockets at M. This matters because it gives definitive bulk measurement, as opposed to surface-sensitive photoemission, of altermagnetic spin splitting, and it settles the Fermi-surface topology of a prototypical altermagnet.","feed_headline":"Confirm altermagnetic spin splitting in CrSb by quantum oscillations","feed_subtitle":"All five measured frequency branches match a spin-split four-band Fermi surface, settling the bulk topology of a prototypical altermagnet.","key_machinery":"The key machinery is the Onsager relation F=(ħ/2πe)A, connecting each measured frequency to an extremal Fermi-surface cross-section, and the Lifshitz–Kosevich formula for fitting amplitudes. The calculated Fermi surface comes from fully relativistic DFT+U (U=0.01 Ry on Cr d orbitals) with spin-orbit coupling, giving four spin-split bands. To match the dominant α frequency, the Fermi levels of bands 1–2 are shifted up by 0.046 eV and bands 3–4 down by 0.007 eV; the comparison of these shifted calculations with the measured angle-dependent branches is the load-bearing evidence.","core_discovery":"The measured quantum-oscillation frequencies in CrSb—α, β, δ, ε, and ζ—are each assigned to extremal orbits of a DFT+U Fermi surface in which four bands cross the Fermi level and are spin-split by altermagnetic order. Bands 1 and 2 each contribute an outer tubular sheet along Γ-A and an inner closed pocket at A, which explains the δ branch near B∥c and corrects earlier open-tube or dogbone proposals. Band 3 provides the general-point orbits behind the β and high-angle δ branches and the T-line orbit behind ζ; band 4 gives ellipsoidal pockets at M that explain ε. Effective masses are only moderately enhanced over band masses, and no Zeeman-driven frequency splitting is seen; the paper present","pith_inferences":["The same measurement strategy could be applied to other metallic altermagnet candidates: predicted momentum-dependent spin splitting should appear as a characteristic multiplicity of frequency branches, and its absence would test the altermagnetic assignment.","Because the α branch is a sum of two nearly degenerate orbits (1-A-o and 2-A-o), a careful beat analysis at higher fields could separate the two and directly reveal the spin-orbit-induced splitting between them—a finer check than the frequency assignment itself.","If the inner A pockets are real, measurements beyond 41.5 T should reveal additional low-frequency branches; detecting or failing to detect them is a concrete extension of this work.","The predicted Zeeman-driven merging and re-splitting of altermagnetic frequency branches is a natural next experiment at 60 T or higher: confirming the predicted behavior would further validate the altermagnetic band structure."],"forward_implications":["CrSb's Fermi-surface topology is now pinned down by a bulk probe: bands 1 and 2 carry inner closed pockets at A plus outer tubular sheets, ruling out the earlier open-tube and dogbone proposals.","Quantum oscillations are shown to be a workable bulk tool for detecting altermagnetic spin splitting in metals, complementing surface-sensitive photoemission.","The assignment of the β branch and the high-angle δ branch to band-3 general-point orbits rounds out the experimental picture of the higher bands, which previous studies had not addressed.","Effective masses only moderately exceed band masses (m*/m_band ≈ 1.3–1.8), indicating that electron correlations are not strong in CrSb and that the DFT+U description is a credible starting point.","Derived mobilities (roughly 400–700 cm²/Vs) are consistent with the high-mobility multicarrier transport reported for CrSb, tying the Fermi-surface picture to bulk transport behavior."],"fun_headline_variants":["Quantum oscillations settle CrSb Fermi surface topology","Altermagnet CrSb spin-split bands revealed by quantum oscillations","CrSb Fermi surface: closed pockets at A, not tubes or dogbones","Five oscillation branches trace spin-split Fermi surface in CrSb"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The comparison between experiment and theory relies on rigidly shifting the calculated Fermi energies—up 0.046 eV for bands 1–2 and down 0.007 eV for bands 3–4—to fit the dominant α frequency; if those shifts, or the Hubbard U=0.01 Ry, misrepresent the true band structure, the orbit assignments, especially for the band-3 general-point orbits and the band-1/2 A pockets, could be wrong.","fun_headline_variants_meta":{"raw":{"variants":["Quantum oscillations settle CrSb Fermi surface topology","Altermagnet CrSb spin-split bands revealed by quantum oscillations","CrSb Fermi surface: closed pockets at A, not tubes or dogbones","Five oscillation branches trace spin-split Fermi surface in CrSb"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00069,"raw_usage":{"total_tokens":2927,"prompt_tokens":673,"completion_tokens":2254,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":417,"completion_tokens_details":{"reasoning_tokens":2183}},"tokens_in":417,"tokens_out":2254,"duration_ms":18002,"temperature":1.0,"reasoning_tokens":2183,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T07:43:32.855775+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Treat the rigid band shifts and the Hubbard U as free parameters and simultaneously fit all five measured branches across both rotation planes; if no single set of shifts reproduces the observed angular dependences of α, β, δ, ε, and ζ, the altermagnetic Fermi-surface assignment fails. Alternatively, angle-resolved photoemission at the wave vectors of the 3-gp orbits (around kz ≈ 0.21·2π/c) should directly image the band-3 pockets assigned to the β branch; their absence would falsify that assignment.","supporting_citations":[],"review_version":1}