{"id":"2f26600e-9d6b-41f7-8b77-ca8ff877f974","arxiv_id":"2602.21135","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"Lowering CrSb to two-fold rotational symmetry replaces diagonal nodal planes with band-specific fragmented nodal curves and permits anomalous Hall conductivity for in-plane and out-of-plane Néel vectors.","lead":"A computational symmetry analysis of the altermagnet CrSb claims that lowering its six-fold rotational symmetry to two-fold creates fragmented nodal curves in momentum space and can turn on an anomalous Hall effect. If correct, this gives spintronics researchers a strain- or doping-based switch for Hall transport in antiferromagnets.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (10) in Section V yields only opposite-momentum spin relations; it does not force same-k degeneracy, so the claimed symmetry-enforced FNCs are unsupported.","rationale":"The reader's REJECT verdict is well founded. The paper's central claim—that lowering symmetry to C2z produces fragmented nodal curves that are symmetry-enforced—rests on Section V's derivation. That derivation, summarized in Eq. (10), only relates opposite-spin energies at C2z- or Mz-related momenta. It does not constrain energies at the same momentum except on the invariant sets (kx=ky=0 for C2z, kz=0 for Mz). Thus the plotted FNCs are not explained by the stated symmetry operations. The manuscript's internal statement that MS-V lacks an inversion center is also inconsistent with assigning it to P21/m, a centrosymmetric space group, which compounds the concern. Even if one assumes [E||TR,L] provides ε(k,σ)=ε(-k,σ), the non sequitur from opposite-momentum equality to same-momentum crossing remains. The AHC section is on firmer ground, since allowed tensor components follow from the magnetic space group, but it does not rescue the advertised FNC mechanism. A revision that proves an additional degeneracy-forcing symmetry or explicitly reclassifies the curves as accidental crossings would be needed to support the central claim. Therefore no adjustment to the reader's verdict is needed.","tokens_in":16010,"tokens_out":16657,"duration_ms":159475,"concrete_test":"Build a two-band model satisfying exactly the symmetries of Section V: set E↑(k)=f(k), E↓(k)=f(-kx,-ky,kz) (and also E↓(k)=f(kx,ky,-kz)) for a smooth generic f (e.g., f=cos kx+cos ky+cos kz). If the degeneracy condition E↑(k)=E↓(k) is then satisfied only on {kx=ky=0}∪{kz=0}, this demonstrates that the stated symmetries do not force FNCs at generic k. As a numerical cross-check, recompute the MS-V DFT band structure under a small symmetry-preserving distortion (e.g., 1–2% shift of Sb along an allowed mode) and test whether the reported FNCs persist; if they gap out or move discontinuously, they are accidental rather than symmetry-enforced.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section V, the derivation of FNCs rests on Eq. (10): ε(kx,ky,kz,σ)=ε(-kx,-ky,kz,-σ)=ε(kx,ky,-kz,-σ). These identities relate opposite-spin eigenvalues at different momenta. A same-k degeneracy ε(k,↑)=ε(k,↓) follows only when the spatial operation leaves k invariant—i.e., on the C2z axis (kx=ky=0) or the Mz plane (kz=0). At a generic point of the kx-ky plane, no equality is forced. The paper's claim that 'two spin-opposite bands have to intersect each other at a given point in the kx-ky plane' is therefore a non sequitur. The computed FNCs may be real accidental band crossings, but they are not symmetry-enforced by the operations listed. This is load-bearing because the headline claim is the symmetry-driven formation of FNCs. The additional use of [E||TR,L] as a symmetry of non-centrosymmetric MS-V/strained CrSb is also not justified, but it is not the only obstacle: even granting ε(k,σ)=ε(-k,σ), the logical gap between Eq. (10) and same-k degeneracy remains. A revision would need to prove an additional degeneracy-forcing symmetry or explicitly reframe the FNCs as accidental.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies altermagnetic spin splitting in CrSb and in a set of hypothetical model structures built by vacancy/interstitial engineering, with the aim of understanding how lowering the crystal symmetry from sixfold to twofold affects the momentum-space spin polarization. The authors report that when the symmetry is reduced to a C2z rotation (together with Mz), the three diagonal nodal planes of pristine CrSb disappear and are replaced by what they call fragmented nodal curves (FNCs) — band-specific curves in the Brillouin zone along which opposite-spin sublattice bands become degenerate. They support this observation with DFT constant-energy surfaces for a model structure (MS-V), for uniaxially strained CrSb, and for RbMnPO4, and they show that the same symmetry lowering allows finite anomalous Hall conductivity for both in-plane and out-of-plane Néel-vector orientations.","tokens_in":16277,"tokens_out":11408,"duration_ms":107200,"significance":"If the symmetry analysis were correct, FNCs would be a genuinely new type of nodal feature in altermagnets, distinct from nodal planes and nodal axes, and the proposal that such curves can be engineered by strain or doping would be of practical interest. The paper also contains an independent, non-fitted symmetry analysis (spin-space-group based), a separate validation compound (RbMnPO4), and standard AHC calculations; these are positive features. However, the central derivation of the FNCs is not sound as written, and the paper's main conceptual novelty therefore rests on a claim that is not established. The AHC results may be correct regardless of the FNC interpretation, but the title and abstract hinge on the FNC mechanism.","major_comments":[{"comment":"Equation (10) states ε(kx,ky,kz,σ)=ε(-kx,-ky,kz,-σ)=ε(kx,ky,-kz,-σ). This is a relation between opposite-spin eigenvalues at different momenta, not a same-k degeneracy. For a generic (kx,ky) with kz≠0, the operation C2z maps k to (-kx,-ky,kz), so it does not leave the momentum invariant. The text then claims that 'the former implies that two spin-opposite bands have to intersect each other at a given point (nodal point) in the kx-ky plane.' This implication does not follow from Eq. (10) alone. To force a crossing at the same k, one would need an additional symmetry that leaves k invariant or a separate topological/band-representation argument. As written, the proof does not establish that the plotted FNCs are symmetry-enforced; they may be accidental crossings. This is load-bearing for the paper's central claim.","section":"Section V, Eq. (10) and following paragraph"},{"comment":"The FNCs are identified visually from constant-energy surfaces (e.g., Fig. 5 lower panel, Fig. 6, Fig. 7(f,g)). The text states that as kz varies continuously, nodal points form a nodal curve, but no direct calculation of the spin-resolved band gap or a definition of the degeneracy locus is provided. A constant-energy contour intersection at selected energies is not by itself a proof of a continuous 1D nodal curve in the full 3D Brillouin zone. To make the claim quantitative, the authors should compute the direct gap between the opposite-spin pairs, identify the zero-gap locus in 3D, and confirm its dimension and band-specificity. Without this, the 'fragmented nodal curves' remain a visual feature of constant-energy plots.","section":"Section V and Figures 5-7"},{"comment":"If, after the above is fixed, no symmetry-enforcing mechanism can be found, the paper should explicitly reframe the FNCs as accidental crossings that are compatible with, but not required by, the C2z and [E||TR,L] symmetries. The current abstract and Section V assert a 'discovery' of symmetry-driven FNC formation. A reframing would substantially lower the novelty claim but would make the paper internally consistent. The AHC results and the comparative study with RbMnPO4 can stand independently of the symmetry-enforcement issue.","section":"Section V and abstract"}],"minor_comments":[{"comment":"The text says 'Eqs. (6) - (8)' when referring to Eqs. (7)-(9); please correct the equation numbering in the sentence above Eq. (10).","section":"Section V"},{"comment":"The notation [C2||M2z] is inconsistent with the earlier notation Mz used throughout the paper (e.g., in Section II and Fig. 1). Define whether M2z means Mz or a different mirror operation.","section":"Section V, Eq. (7)"},{"comment":"The phrase 'when in an altermagnetic material when the symmetry is restricted' is grammatically awkward and should be revised.","section":"Abstract and Section V"},{"comment":"The term 'random FNCs' is imprecise; the curves appear to be band-specific but not random. Clarify what is meant by 'random'.","section":"Section V"},{"comment":"For strained CrSb, the spin-space group and magnetic space group are not explicitly listed, although the text states that C2z and Mz are retained. Providing the exact SSG/MSG for the strained structure would help the reader connect the AHC analysis in Table II to the strained system.","section":"Section V.A"}],"recommendation":"major_revision","confidential_remarks":"The reader's recommendation of reject is understandable because the symmetry derivation in Section V is flawed. However, the flaw appears fixable within the manuscript's scope: the authors can either find a correct symmetry argument for the degeneracy loci or explicitly reclassify the FNCs as accidental crossings. The AHC calculations and the RbMnPO4 test are independent and appear sound. I therefore recommend major revision rather than outright rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front. The paper is a serious computational study of CrSb under symmetry lowering, and the strain-dependent AHC result is the part worth keeping. The headline claim—that lowering to two-fold rotation creates symmetry-enforced 'fragmented nodal curves'—does not survive contact with the paper's own equations.\n\nSection V's derivation is a non sequitur. Eq. (10) states ε(kx,ky,kz,σ)=ε(-kx,-ky,kz,-σ)=ε(kx,ky,-kz,-σ). That is a relation between a state at k and opposite-spin states at −k or at (kx,ky,−kz). A same-k degeneracy ε(k,↑)=ε(k,↓) is forced only where the spatial operation leaves k invariant—on the C2z axis or on the kz=0 plane. For a generic point of the kx–ky plane, nothing forces the two spin bands to touch. The sentence 'two spin-opposite bands have to intersect each other at a given point in the kx–ky plane' does not follow from Eq. (10). The plotted curves may be genuine accidental crossings, but the paper provides no analysis showing they are anything more.\n\nA second, related problem is the use of [E||TR,L] as if it were an inversion symmetry for MS-V, when the text explicitly says MS-V lacks inversion. Even if we grant that operation, the logical gap above remains.\n\nWhat is genuinely good: the AHC section. The symmetry-allowed tensor components in Table II follow from the MSG, and the computed finite out-of-plane AHC in strained CrSb is a concrete, useful prediction for altermagnet spintronics. The RbMnPO4 calculation is an independent cross-check, though it inherits the same derivation gap. Circularity is low—no parameters are fitted to the FNCs, and the SSG formalism is taken from the literature rather than invented.\n\nSo: the computational results are worth a referee's time, but the central claim needs major rework. If a referee can push the authors to either prove a genuine same-k degeneracy under C2z (unlikely from these symmetries) or explicitly reframe the curves as accidental crossings with supporting convergence checks, the paper could become publishable. As is, the advertised discovery is unsupported.","headline":"Useful AHC prediction under strain, but the symmetry-enforced FNC claim collapses: Eq. (10) relates opposite momenta, not same-k degeneracies.","tokens_in":16848,"tokens_out":6838,"would_cite":false,"duration_ms":66477,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In altermagnetic CrSb, reducing six-fold rotational symmetry to two-fold replaces the three diagonal nodal planes with band-specific fragmented nodal curves, and this enables anomalous Hall conductivity for both in-plane and out-of-plane Né","keywords":["altermagnetism","CrSb","fragmented nodal curves","momentum-space spin polarization","anomalous Hall effect","spin space group","symmetry lowering","strain engineering"],"falsifier":"Check whether [E||TR,L] belongs to the spin space group of the relaxed MS-V or 5%-strained CrSb crystal by querying the actual magnetic space group (the paper does not list it for strained CrSb, and for MS-V it lists P21/m without inversion in the chemical space group). If it is absent, then compute the band eigenvalues along one of the plotted FNCs: if the opposite-spin bands are not exactly degenerate, the FNC is not a symmetry-required crossing. A more direct test is to add small spin-orbit coupling, which breaks [E||TR,L]; if the degeneracy lifts, the curve is a non-relativistic symmetry a","tokens_in":15765,"feed_emoji":"🧲","tokens_out":7545,"duration_ms":69445,"temperature":0.7,"pith_summary":"This paper sets out to show what happens to altermagnetism in CrSb when the crystal symmetry is lowered from six-fold to two-fold rotation. It claims that the three diagonal nodal planes of the pristine material disappear and are replaced by fragmented nodal curves (FNCs): band-specific lines in the Brillouin zone along which pairs of opposite-spin bands are exactly degenerate, with non-degenerate splitting elsewhere. The authors argue this follows from the combined action of a two-fold spin-space rotation and the non-relativistic time-reversal/inversion operation, and they support the claim with first-principles calculations on intentionally modified model structures, 5% uniaxially strained CrSb, and the compound RbMnPO4. A key consequence they stress is that the symmetry lowering allows a finite anomalous Hall conductivity for both in-plane and out-of-plane Néel vector orientations, which pristine CrSb forbids for the out-of-plane orientation.","feed_headline":"Two-fold symmetry turns CrSb nodal planes into curves","feed_subtitle":"The curves replace CrSb's nodal planes and enable anomalous Hall in both Néel directions.","key_machinery":"The central object is the fragmented nodal curve (FNC): a band-specific curve in the Brillouin zone along which two opposite-spin sublattice bands are degenerate, replacing the rigid nodal planes of the high-symmetry altermagnet. The argument is carried by the spin-space-group operations [C2||C2z] and [C2||Mz] together with the non-relativistic dual operation [E||TR,L] (spin-preserving reversal of all momenta, equivalent to inversion in the non-relativistic limit). The paper uses these operations to derive an eigenvalue relation that, in its reading, forces the spin-opposite pair bands to cross at points in the kx-ky plane for each kz; as k varies, these points trace out the FNC. The work al","core_discovery":"The central discovery is the formation of fragmented nodal curves (FNCs) in an altermagnet when the spin-space symmetry is restricted to a two-fold rotation. In pristine CrSb, the high-symmetry operations [C2||C6z] and [C2||Mz] generate four nodal planes (three diagonal and one basal). Once the symmetry is lowered to C2z, the paper shows that the three diagonal planes are no longer symmetry-forced; instead, for each pair of spin-opposite sublattice bands there appear curves in reciprocal space along which the two bands are degenerate. The paper argues from spin-space-group operations that these curves must exist, and it verifies their presence in density-functional band structures of model s","pith_inferences":["If the FNC degeneracies are real, one could engineer the anomalous Hall response of CrSb by choosing strain magnitude and direction; the paper shows AHC around the Fermi level is already higher under 5% strain but does not map how it grows or changes sign with further strain.","The band-specific nature of FNCs implies that shifting the Fermi level by doping or gating will switch between different subsets of curves, making the AHC sharply tunable with carrier density — a prediction the paper does not directly test.","The same symmetry-lowering logic used here for CrSb could be applied to other hexagonal altermagnets; verifying FNCs in a second family would test whether the mechanism is generic.","A useful next step would be spin-resolved ARPES or a direct experimental probe of the constant-energy surfaces under strain; the predicted FNC pattern would show as spin-degenerate lines surrounded by opposite-spin splitting, which is measurable in principle."],"forward_implications":["A 5% uniaxial strain along the basal axis of CrSb reduces the symmetry to C2 and produces FNCs in the calculated constant-energy surfaces, offering a concrete experimental route to create them.","In the symmetry-lowered systems, anomalous Hall conductivity becomes finite for both out-of-plane and in-plane Néel vector orientations, not just in-plane as in the pristine case.","Because each band pair has its own FNC, the momentum-space spin polarization becomes selectable by energy and band index rather than fixed by symmetry planes.","The appearance of FNCs in RbMnPO4, which has only a two-fold screw symmetry, indicates the phenomenon is not restricted to CrSb but should occur in any altermagnet with C2 spin-space symmetry.","Within the non-relativistic picture, proper six-fold rotation and six-fold roto-inversion produce the same nodal planes, so improper rotations can sustain altermagnetism just as well."],"fun_headline_variants":["Symmetry reduction in CrSb turns nodal planes into curves","Two-fold symmetry spawns fragmented nodal curves in CrSb","Fragmented nodal curves emerge in CrSb with two-fold symmetry","CrSb's two-fold symmetry enables anomalous Hall via nodal curves","Lowered symmetry in CrSb creates fragmented nodal curves"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument's load-bearing step is the assumption that the combined operation [E||TR,L] (spin-preserving time reversal plus inversion) is a symmetry of the C2-symmetric, non-centrosymmetric structures used for MS-V and strained CrSb; if that operation is not actually present, the eigenvalue relation derived in Section V does not force opposite-spin bands to be degenerate at the same momentum, and the fragmented nodal curves would not be symmetry-guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Symmetry reduction in CrSb turns nodal planes into curves","Two-fold symmetry spawns fragmented nodal curves in CrSb","Fragmented nodal curves emerge in CrSb with two-fold symmetry","CrSb's two-fold symmetry enables anomalous Hall via nodal curves","Lowered symmetry in CrSb creates fragmented nodal curves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000214,"raw_usage":{"total_tokens":1302,"prompt_tokens":823,"completion_tokens":479,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":396}},"tokens_in":567,"tokens_out":479,"duration_ms":4717,"temperature":1.0,"reasoning_tokens":396,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T21:08:11.441045+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether [E||TR,L] belongs to the spin space group of the relaxed MS-V or 5%-strained CrSb crystal by querying the actual magnetic space group (the paper does not list it for strained CrSb, and for MS-V it lists P21/m without inversion in the chemical space group). If it is absent, then compute the band eigenvalues along one of the plotted FNCs: if the opposite-spin bands are not exactly degenerate, the FNC is not a symmetry-required crossing. A more direct test is to add small spin-orbit coupling, which breaks [E||TR,L]; if the degeneracy lifts, the curve is a non-relativistic symmetry a","supporting_citations":[],"review_version":1}