{"id":"0e1e02c6-67e8-4480-a811-e8f44f5b1002","arxiv_id":"2504.21138","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Torque magnetometry and quantum oscillations show RuO2 single crystals are paramagnetic, with no evidence for altermagnetic order.","lead":"High-quality RuO2 crystals show no detectable magnetic order in high-field torque and magnetization experiments, indicating they are paramagnetic rather than altermagnetic. This matters because RuO2 is the most studied testbed for the proposed altermagnet class in spintronics.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Weak-moment sensitivity gap: the χ∥→0 criterion and spherical Fermi surface do not quantitatively rule out the reported 0.05 μB order.","rationale":"The reader identified the same load-bearing assumption: the vanishing parallel susceptibility criterion is used to distinguish paramagnetic Case I from c-axis altermagnetic Case II. My stress-test agrees and widens the concern to the Fermi-surface comparison, because the altermagnetic DFT ratios quoted in Table I correspond to a large ordered moment; a 0.05 μB moment would produce far weaker spin splitting and a nearly spherical Γ pocket, so the measured pocket shape does not provide an independent null test unless the weak-moment case is computed. The data are internally consistent and strongly favor 'no detectable long-range order', so a rejection is not warranted. However, the stronger conclusion that RuO2 is 'not altermagnetic' depends on an unquantified sensitivity assumption: the paper never demonstrates that its torque, magnetization, and quantum-oscillation measurements would have detected the specific 0.05 μB state reported in the neutron and X-ray literature. A single constrained-moment DFT calculation, including orbital susceptibility, would settle whether that state is excluded or lies below the experimental detection floor. Because that calibration is missing and the stronger inference rests on it, I would adjust the verdict from ACCEPT to CONDITIONAL rather than leave it unchanged; the authors should either add the sensitivity calculation or explicitly restrict their conclusion to a bound on the detectable ordered moment.","tokens_in":9291,"tokens_out":13786,"duration_ms":165713,"concrete_test":"Perform constrained DFT (e.g., VASP or WIEN2k) on RuO2 with the Ru moment fixed to 0.05 μB along c, and compute (i) the full static susceptibility tensor including orbital (Van Vleck) contributions and (ii) the Γ-pocket Fermi-surface radii along ΓM, ΓZ, and ΓX. Compare χc − χa and ΓM/ΓZ against the experimental values (≈2.4×10−5 and 1.13) and against the torque and quantum-oscillation noise floors. If the 0.05 μB state still yields χa > χc and ΓM/ΓZ well above 1.13, the reported state is truly excluded by the data; if it yields χc > χa and ΓM/ΓZ ≈ 1.1, the central exclusion fails and the conclusion must be softened to a quantitative detection limit.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central inference excluding c-axis altermagnetism (Case II) is made in the paragraph beginning 'It is well-established that systems with local or itinerant collinear magnetic order exhibit a smaller magnetic susceptibility parallel to the Néel vector that approaches zero at 0 K...' This textbook result is a local-moment, Heisenberg-type statement, but the relevant regime for RuO2 is a weak itinerant antiferromagnet with reported ordered moments of only ~0.05 μB (refs. 26, 39). In an itinerant spin-density-wave picture, the uniform susceptibility parallel to the staggered moment need not vanish; it can remain of order the Pauli susceptibility. Thus finite χc at low temperature and even the observed sign χc > χa do not by themselves exclude Case II. The measured susceptibility is also a sum of spin and orbital (Van Vleck/Landau) contributions, and the orbital part can be anisotropic in a 4d oxide; without isolating the spin-only anisotropy, the sign of χc − χa is not a robust order-parameter diagnostic. The quantum-oscillation comparison to altermagnetic DFT (Fig. 4, Table I) has the same sensitivity gap: those calculations assume a large ordered moment, whereas a 0.05 μB moment would produce a much smaller spin splitting and a nearly spherical Γ pocket, matching paramagnetic DFT even if weak order is present. The data convincingly establish 'no detectable order', but they do not, as written, rule out the reported small-moment altermagnetic state.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports torque magnetometry, magnetization, and magnetic quantum oscillation measurements on RuO2 single crystals at fields up to 31 T and temperatures down to 0.5 K. The authors show that the torque in the (001) plane is zero, that the torque responses in the (0-10) and (1-10) planes are identical with H^2 scaling, and that the magnetization anisotropy satisfies χc > χa with no tendency of χ toward zero at low temperature. Quantum oscillations reveal a nearly spherical Fermi surface pocket at the Brillouin zone center. From these observations they argue that the magnetic susceptibility tensor is uniaxial, that the commonly assumed c-axis collinear (altermagnetic) state is inconsistent with experiment, and that the Fermi surface matches paramagnetic DFT calculations. They conclude that high-quality RuO2 single crystals are itinerant paramagnets with no detectable long-range magnetic order and, by extension, are not altermagnets.","tokens_in":9565,"tokens_out":9484,"duration_ms":110231,"significance":"If the conclusions hold, this is an important resolution of a long-running controversy about a canonical altermagnetic candidate, with direct consequences for the interpretation of thin-film work and for the altermagnetism literature more broadly. The experimental program is strong: the torque and magnetization data are internally consistent, the anisotropies obtained by the two techniques agree quantitatively, the results are reproduced on multiple samples and instruments, and the Fermi-surface comparison uses independent DFT benchmarks rather than fitted parameters. The near-spherical Γ pocket is a particularly clean and falsifiable diagnostic. The main weakness is that the quantitative exclusion of a weak-moment (≈0.05 μB) antiferromagnetic state is not fully established, because the sensitivity of the measurements to the small anisotropy produced by such a moment is not demonstrated.","major_comments":[{"comment":"The exclusion of the c-axis altermagnetic Case II rests on the premise that χ parallel to the Néel vector approaches zero at 0 K. This premise is a local-moment Heisenberg result and is not established for the weak itinerant antiferromagnetic regime relevant to the reported ~0.05 μB ordered moments in RuO2 (refs. 26, 39). In an itinerant spin-density-wave picture the uniform susceptibility along the staggered moment can remain Pauli-like, and the measured total susceptibility also contains anisotropic orbital (Van Vleck/Landau) contributions, so the observed χc > χa with finite low-temperature χc does not by itself rule out a small-moment c-axis antiferromagnetic state. The authors should either provide a quantitative estimate of the susceptibility anisotropy expected for a 0.05 μB moment and demonstrate that the torque and magnetization noise floors are below it, or restrict the claim to \"no detectable order at the achieved sensitivity.\"","section":"Case II exclusion (paragraph beginning 'It is well-established...')"},{"comment":"The Fermi-surface comparison also has a small-moment sensitivity gap. The altermagnetic DFT calculations cited in Table I assume ordered moments large enough to produce substantial d-wave spin splitting, whereas a 0.05 μB moment would produce a much smaller splitting and a nearly spherical Γ pocket. The measured ratios ΓM/ΓZ = 1.13(1) and ΓX/ΓM = 1.06(1) are therefore consistent not only with paramagnetism but also with a weak-moment altermagnetic state. The authors should either compute the Γ-pocket anisotropy for a small-moment magnetic state or explicitly state, in the discussion of Table I, that the quantum-oscillation data cannot exclude small-moment altermagnetism.","section":"Table I and Fig. 4"}],"minor_comments":[{"comment":"There is a typo in \"non-relativisitic\" and the accents in \"Néel\" are used inconsistently.","section":"Abstract"},{"comment":"The plane labels in panels (g)-(i) omit the overbar in one \"(110)\" label, which is inconsistent with the text's \"(1¯10)\" notation; also, \"Data for the (d) (001) configuration are not periodic\" should be rephrased to indicate that no periodic torque signal is observed.","section":"Fig. 2 caption"},{"comment":"The space group should be typeset as P42/mnm with appropriate subscripts.","section":"Introduction"},{"comment":"The phrase \"smaller, non-spherical pocket(s)\" should be checked for singular/plural agreement, and the assignment of the low frequencies could be clarified by naming the DFT bands or pockets they correspond to.","section":"After Fig. 4"},{"comment":"Reference [49] is a placeholder URL; it should be replaced with the actual Supplemental Material link in the published version.","section":"Supplemental Material reference"}],"recommendation":"major_revision","confidential_remarks":"The manuscript presents a strong multi-technique dataset and addresses a timely controversy, so it is well within the journal's scope. The main concern is that the headline conclusion that RuO2 is not altermagnetic is currently phrased more strongly than the sensitivity analysis supports. If the authors add a quantitative upper bound on the torque/magnetization sensitivity, discuss the applicability of the χ∥→0 criterion to weak itinerant systems, and soften the Case II exclusion accordingly, the paper would be acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this one. First, the torque magnetometry up to 31 T in three crystallographic planes is a solid piece of work: zero torque in the (001) plane, identical sin(2θ) torque in (0-10) and (1-10) planes, clean H^2 scaling, reproduced on multiple samples. That symmetry analysis genuinely narrows the allowed magnetic ground states and rules out in-plane Néel vectors and field-induced reorientations. Second, the paper's quantitative argument against c-axis altermagnetism (Case II) rests on the textbook χ∥→0 criterion plus a comparison of the Γ-pocket shape to altermagnetic DFT. Both of those have a weak-moment sensitivity problem, and the stress-test note you passed me is right on target.\n\nWhat's new: the high-field torque symmetry, the quantitative Fermi-surface anisotropy ratios (ΓM/ΓZ = 1.13(1), ΓX/ΓM = 1.06(1)), and the explicit comparison to both paramagnetic and altermagnetic DFT. The paper also aggregates the existing muSR, ARPES, heat capacity, and transport results into a coherent case. No fitted parameters; the DFT benchmarks are external. That is reproducible, checkable evidence.\n\nThe soft spot: the claim that the data are 'inconsistent with collinear magnetic order possessing a Néel vector along the c-axis' is stronger than what the measurements can actually deliver. For a weak itinerant antiferromagnet with an ordered moment of ~0.05 μB, the uniform susceptibility parallel to the staggered moment need not vanish at T→0; the orbital (Van Vleck/Landau) part of the measured susceptibility is also anisotropic and not separated out. And the altermagnetic DFT Fermi surfaces in Table I are computed for a large moment; a 0.05 μB moment would leave the Γ pocket nearly spherical, so the quantum-oscillation comparison cannot distinguish a weak-moment altermagnet from a paramagnet. The paper's own conclusion is careful about this — 'no detectable long-range magnetic order' — but the abstract and one or two sentences go further. The fix is to soften those passages and add a sentence acknowledging the sensitivity floor.\n\nOverall: this is a serious experimental contribution to a high-profile dispute, and the main conclusion is well supported by convergent evidence. It deserves peer review. I'd send it out with a request to address the weak-moment caveat, not because the central claim is wrong — I think it's probably right — but because the paper currently asks the quantitative arguments to carry more weight than they can bear.","headline":"Solid torque and quantum-oscillation evidence that bulk RuO2 is paramagnetic, but the paper overclaims how definitively it rules out the 0.05 μB altermagnetic state.","tokens_in":10125,"tokens_out":3430,"would_cite":true,"duration_ms":34873,"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":"The authors show that high-quality RuO2 single crystals are itinerant paramagnets with no long-range magnetic order, and therefore not altermagnets.","keywords":["altermagnetism","RuO2","torque magnetometry","magnetic susceptibility","quantum oscillations","itinerant paramagnetism","Néel vector","Fermi surface"],"falsifier":"A decisive test would be a direct zero-field structure determination on the same high-quality crystals, for example neutron or resonant X-ray diffraction with sensitivity to ordered moments below $0.05\\ \\mu_B$, looking for a Bragg peak at the altermagnetic propagation vector. A second test is to extend torque and magnetization measurements to lower temperatures and check whether $\\chi_c$ extrapolates to zero as $T \\to 0$, which the collinear-order scenario requires and the present data show it does not. Null results in both would settle the paramagnetic ground state; positive results would overturn the paper's conclusion.","tokens_in":9097,"feed_emoji":"🧲","tokens_out":10026,"duration_ms":93805,"temperature":0.7,"pith_summary":"This paper tackles a foundational question in the young field of altermagnetism: is the most widely cited candidate, $\\mathrm{RuO_2}$, actually magnetically ordered? Drawing on torque magnetometry, longitudinal magnetization, and magnetic quantum oscillations, the authors argue that high-quality single crystals of $\\mathrm{RuO_2}$ are itinerant paramagnets with no detectable long-range magnetic order. If correct, the result removes $\\mathrm{RuO_2}$ from the role of archetypal altermagnet and redirects attention to extrinsic routes, such as epitaxial strain or defects, for realizing altermagnetic behavior. The question matters because altermagnets promise momentum-dependent spin-splitting without net magnetization, a property that would be absent in a genuinely paramagnetic material.","feed_headline":"RuO2 single crystals show no magnetic order and are not altermagnets","feed_subtitle":"Torque, magnetization, and Fermi-surface oscillations agree: the archetypal altermagnet is an itinerant paramagnet.","key_machinery":"The argument is carried by the magnetic susceptibility tensor as constrained jointly by crystal symmetry and possible magnetic point groups, and measured directly with torque magnetometry. Torque detects the angular derivative of magnetization; the observed $\\sin(2\\theta)$ response with identical amplitude and periodicity in the $(0\\bar{1}0)$ and $(1\\bar{1}0)$ planes, together with zero torque in the $(001)$ plane, fixes uniaxial anisotropy with $\\chi_a = \\chi_b \\neq \\chi_c$. The second load-bearing element is the textbook result that in a collinear antiferromagnet the susceptibility parallel to the Néel vector (the axis of antiparallel spin alignment) approaches zero at $T \\to 0$, while the perpendicular susceptibility remains finite; the authors use the observed $\\chi_c > \\chi_a$ with $\\chi_c$ not tending to zero to eliminate $c$-axis Néel order. The third element is the Fermi-surface shape: paramagnetic and altermagnetic density-functional calculations predict measurably different forms for the Brillouin-zone-center pocket, nearly spherical versus d-wave-distorted, and the quantum-oscillation radius ratios match the paramagnetic predictions.","core_discovery":"The paper's central claim is that bulk $\\mathrm{RuO_2}$ single crystals do not exhibit the collinear antiferromagnetic order required for altermagnetism. Symmetry-based torque measurements fix the magnetic susceptibility tensor as uniaxial with isotropic response within the $ab$-plane and a larger response along the $c$-axis, a form compatible with either a paramagnet or a $c$-axis Néel vector but incompatible with any in-plane Néel vector, with magnetic domains, or with field-induced Néel-vector reorientation. The measured sign of the torque and the magnetization data show $\\chi_c > \\chi_a$ with no tendency for $\\chi_c$ to vanish as $T \\to 0$, which the authors use to exclude the remaining $c$-axis altermagnetic case on the grounds that a collinear antiferromagnet's susceptibility parallel to its Néel vector should approach zero at low temperature. Quantum oscillations reveal a nearly spherical Fermi-surface pocket at the Brillouin-zone center with radius ratios $\\Gamma M/\\Gamma Z \\approx 1.13$ and $\\Gamma X/\\Gamma M \\approx 1.06$, quantitatively matching paramagnetic band-structure calculations rather than the distorted four-fold pocket predicted for altermagnetic order. Taken together, the authors conclude that high-quality $\\mathrm{RuO_2}$ single crystals are itinerant paramagnets without long-range magnetic order and, by extension, are not altermagnets.","pith_inferences":["The symmetry-based exclusion of in-plane Néel vectors holds regardless of moment size, whereas the exclusion of a weak $c$-axis moment depends on the susceptibility rule; a cautious reading is 'no order above roughly the torque noise floor.'","The Fermi-surface radius ratios could be used as a quick predictive screen for other candidate altermagnets: a measured $\\Gamma M/\\Gamma Z$ near unity indicates a paramagnetic-like pocket, while a large ratio would flag a distorted pocket worth pursuing.","If bulk $\\mathrm{RuO_2}$ is truly paramagnetic, then reported altermagnetic transport and spin-split band features in $\\mathrm{RuO_2}$ films most plausibly originate from epitaxial strain, interfacial effects, or defects rather than from the intrinsic bulk electronic structure—an interpretation the authors gesture toward but do not test.","The absence of any phase transition up to 31 T and 0.5 K opens the possibility of probing field-induced or quantum-critical magnetic order at even lower temperatures, where a small ordered moment might finally appear."],"forward_implications":["Bulk $\\mathrm{RuO_2}$ should no longer be treated as a confirmed altermagnet; theoretical altermagnetic predictions for this material would apply at most to thin films or defect-engineered specimens, not to the intrinsic bulk ground state.","The reported neutron and X-ray signatures of roughly $0.05\\ \\mu_B$ order in $\\mathrm{RuO_2}$ need re-examination, since the thermodynamic and Fermi-surface data presented here are inconsistent with robust long-range order in high-quality crystals.","Combining torque-determined susceptibility symmetry with quantum-oscillation Fermi-surface shapes offers a practical two-step fingerprint for vetting other candidate altermagnets before pursuing spintronic applications.","The sharp contrast between bulk crystals and thin films shifts the search for altermagnetism in rutile oxides toward strain, thickness, and point defects as the controlling variables."],"supporting_citations":[{"why":"Reported neutron-diffraction evidence of ~0.05 μB antiferromagnetic order in RuO2, the claim the present measurements are designed to test and contradict.","marker":"[26]"},{"why":"Introduced the altermagnetism concept and designated RuO2 as the archetypal material, defining the theoretical backdrop for the study.","marker":"[4, 5]"},{"why":"Reported altermagnetic signatures and spin reorientation in RuO2 thin films, supplying the contrasting thin-film dataset that the bulk measurements are compared against.","marker":"[20]"},{"why":"Prior single-crystal magnetization study whose linear, weakly temperature-dependent susceptibility the present data quantitatively reproduce.","marker":"[29]"},{"why":"Recent independent single-crystal measurements that agree with the observed susceptibility magnitude and anisotropy, corroborating the bulk behavior.","marker":"[54]"},{"why":"Earlier quantum-oscillation measurements on RuO2 single crystals, providing the Fermi-surface frequencies that the present data extend and compare.","marker":"[55]"},{"why":"Density-functional calculations of paramagnetic and altermagnetic Fermi surfaces used to quantify the expected anisotropy of the Γ-point pocket.","marker":"[58]"},{"why":"Additional density-functional Fermi-surface calculation supplying the paramagnetic and altermagnetic radius ratios listed in Table I.","marker":"[57]"},{"why":"Textbook result that collinear antiferromagnets have susceptibility parallel to the Néel vector approaching zero at 0 K, underpinning the exclusion of c-axis order.","marker":"[52, 53]"}],"fun_headline_variants":["RuO2 single crystals show no magnetic order, not altermagnets","Torque and magnetization data rule out altermagnetism in RuO2","RuO2 is an itinerant paramagnet, not an altermagnet","Isotropic susceptibility in RuO2 kills altermagnetic claims","Quantum oscillations confirm RuO2 has no magnetic order"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the textbook premise that a collinear antiferromagnet's susceptibility parallel to its ordered moment approaches zero at absolute zero, so the observed $\\chi_c > \\chi_a$ with $\\chi_c$ not vanishing is taken as proof against $c$-axis order; that premise may fail for very small itinerant moments, and the paper gives no independent calibration showing the torque sensitivity is below the anisotropy that a $0.05\\ \\mu_B$ moment would produce.","fun_headline_variants_meta":{"raw":{"variants":["RuO2 single crystals show no magnetic order, not altermagnets","Torque and magnetization data rule out altermagnetism in RuO2","RuO2 is an itinerant paramagnet, not an altermagnet","Isotropic susceptibility in RuO2 kills altermagnetic claims","Quantum oscillations confirm RuO2 has no magnetic order"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000608,"raw_usage":{"total_tokens":2888,"prompt_tokens":1054,"completion_tokens":1834,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":670,"completion_tokens_details":{"reasoning_tokens":1742}},"tokens_in":670,"tokens_out":1834,"duration_ms":11580,"temperature":1.0,"reasoning_tokens":1742,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:11:53.347848+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive test would be a direct zero-field structure determination on the same high-quality crystals, for example neutron or resonant X-ray diffraction with sensitivity to ordered moments below $0.05\\ \\mu_B$, looking for a Bragg peak at the altermagnetic propagation vector. A second test is to extend torque and magnetization measurements to lower temperatures and check whether $\\chi_c$ extrapolates to zero as $T \\to 0$, which the collinear-order scenario requires and the present data show it does not. Null results in both would settle the paramagnetic ground state; positive results would overturn the paper's conclusion.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reported altermagnetic signatures and spin reorientation in RuO2 thin films, supplying the contrasting thin-film dataset that the bulk measurements are compared against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Prior single-crystal magnetization study whose linear, weakly temperature-dependent susceptibility the present data quantitatively reproduce."},{"cited_title":"Lidiard, Proceedings of the Royal Society A 224, 161 (1953)","cited_arxiv_id":null,"evidence_quote":"Recent independent single-crystal measurements that agree with the observed susceptibility magnitude and anisotropy, corroborating the bulk behavior."},{"cited_title":"Kiefer, F","cited_arxiv_id":null,"evidence_quote":"Earlier quantum-oscillation measurements on RuO2 single crystals, providing the Fermi-surface frequencies that the present data extend and compare."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Density-functional calculations of paramagnetic and altermagnetic Fermi surfaces used to quantify the expected anisotropy of the Γ-point pocket."},{"cited_title":"Shoenberg, Magnetic Oscillations in Metals (Cam- bridge University Press, 1984)","cited_arxiv_id":null,"evidence_quote":"Additional density-functional Fermi-surface calculation supplying the paramagnetic and altermagnetic radius ratios listed in Table I."}],"review_version":1}