{"id":"c87d87b7-8c0e-4764-8062-e676cf812305","arxiv_id":"2507.09906","paper_version":5,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Odd-parity altermagnetism in collinear magnets requires broken nonmagnetic time-reversal symmetry plus a C2 rotation combined with inversion or mirror symmetry connecting opposite-spin sublattices, and the Haldane-Hubbard model realizes it.","lead":"This paper derives symmetry conditions under which collinear antiferromagnets can show odd-parity spin splitting, a form of magnetism called odd-parity altermagnetism. It then shows that the Haldane-Hubbard model with opposite sublattice currents satisfies these conditions, giving a concrete platform for the effect.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Criterion (iii) is not sufficient as stated: [C2||M] alone enforces only E_{k,sigma}=E_{Mk,-sigma}, not the defining odd-parity relation E_{k,sigma}=E_{-k,-sigma}.","rationale":"The reader's no-spin-orbit-coupling scoping concern is legitimate but is a clarity issue rather than an internal gap. The more load-bearing problem is that criterion (iii) in the abstract lists [C2||M] as sufficient for odd-parity ALM, while the paper's own defining relation for odd-parity ALM requires a global inversion-type operation. A mirror alone does not map generic k to -k, so [C2||M] cannot by itself enforce E_{k,sigma}=E_{-k,-sigma}. The square-lattice example works because it additionally has [C2||\\bar{E}_{BL}tau]; the hexagon example works because of [C2||\\bar{E}_{BL}tau]. This is a concrete, testable gap in the central sufficiency claim, and it reinforces the need for the CONDITIONAL verdict: the criteria need to be restated with the additional requirement that enforces full odd-parity momentum reversal, or the definition of p-wave odd-parity ALM must be clarified. The proposed tight-binding test would settle whether the [C2||M] disjunct is genuinely sufficient in isolation.","tokens_in":13188,"tokens_out":26070,"duration_ms":288111,"concrete_test":"Build a minimal 2D two-sublattice tight-binding model on a rectangular lattice whose spin-group operations include [C2||M_yz] (M_yz: k_x -> -k_x) plus spin-only rotations, with collinear opposite-spin sublattices and no real-space inversion or C2z symmetry. Compute the band structure with a sublattice-staggered Zeeman term. If E(k_x,k_y,up) = E(-k_x,-k_y,down) fails for generic momentum, the [C2||M] disjunct of criterion (iii) is not sufficient as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central sufficiency claim in the abstract (criterion iii) is overbroad as written. The paper defines odd-parity ALM by E_{k,sigma}=E_{-k,-sigma} (paragraph after Eq. 1), but a mirror reflection M maps k to Mk, not to -k. A symmetry [C2||M] therefore enforces only E_{k,sigma}=E_{Mk,-sigma}, which reduces to E_{k,sigma}=E_{-k,-sigma} on the mirror-invariant locus or when a second operation relating the remaining momentum component is present. In the square-lattice example, the full odd-parity relation is obtained from [C2||\\bar{E}_{BL}tau], not from [C2||M] alone (text near Fig. 2(a)); in the hexagon case, similarly from [C2||\\bar{E}_{BL}tau]. Thus the abstract's unconditional statement that [C2||M] is sufficient for p-wave odd-parity ALM is not established: a collinear compensated magnet with [C2||M] but no [C2||\\bar{E}] (or an equivalent second mirror/rotation) need not satisfy E_{k,sigma}=E_{-k,-sigma}.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript uses spin-group theory to propose sufficient conditions for odd-parity altermagnetism (ALM) in collinear compensated magnets: (i) broken nonmagnetic time-reversal symmetry, (ii) long-range collinear compensated magnetism, and (iii) a symmetry of the type [C2||Ebar] or [C2||M] connecting opposite-spin sublattices. The authors argue that these criteria produce the odd-parity spin-splitting relation E_{k,sigma}=E_{-k,-sigma}, and they illustrate the proposal with the Haldane-Hubbard model, where opposite sublattice currents break nonmagnetic TRS. Using the cluster slave-spin method, they present constant-energy contours, a phase diagram in the (lambda, U) plane, and the momentum dependence of spin-resolved gaps, identifying an odd-parity ALM Chern insulator and an odd-parity ALM insulator.","tokens_in":13386,"tokens_out":2871,"duration_ms":32310,"significance":"If the proposed criteria are correct and stated with the appropriate qualifications, the paper would provide a compact symmetry-based route to odd-parity spin splitting in collinear magnets without spin-orbit coupling, a topic of current interest. The formal derivation is a genuine strength: it is built from the cited spin-group formalism with no fitted parameters, and it shows clearly how breaking nonmagnetic TRS removes [Cbar2||T] while retaining the collinear spin-only group. The Haldane-Hubbard illustration is also useful as a concrete model, although it is an example chosen to satisfy the criteria rather than a test that could falsify them. The main weakness is that one of the central criteria, as stated in the abstract, is overbroad: [C2||M] alone does not enforce the odd-parity relation E_{k,sigma}=E_{-k,-sigma} except on special momentum loci or when combined with another operation. In addition, the no-spin-orbit-coupling scope is not explicitly stated, and the numerical phase diagram lacks error estimates and sufficient reproducibility details.","major_comments":[{"comment":"The abstract's criterion (iii) is not sufficient as stated for the odd-parity relation E_{k,sigma}=E_{-k,-sigma} defined after Eq. (1). A symmetry [C2||M] with M a mirror plane enforces only E_{k,sigma}=E_{Mk,-sigma}; it yields E_{k,sigma}=E_{-k,-sigma} only on the mirror-invariant locus or when an additional operation, such as [C2||Ebar_{BL}tau], relates the remaining momentum components. The text itself indicates this: near Fig. 2(a) the p-wave dalternagnet is described by [C2||Ebar_{BL}tau], not by [C2||M] alone. The criterion should therefore be reformulated as '[C2||Ebar], or [C2||M] together with an additional operation that connects -k to Mk', or the definition of odd-parity ALM should be generalized to include spin splitting antisymmetric under a mirror operation rather than only under inversion. This is a load-bearing point because the central claim in the abstract and conclusion is built on this criterion.","section":"Abstract and Symmetry analysis (paragraphs around Eq. (1) and the discussion of [C2||M])"},{"comment":"The spin-group decomposition into a direct product of a spin-only group and a nontrivial spin group assumes that spin and real-space degrees of freedom are decoupled, i.e., that spin-orbit coupling is absent. This no-SOC scope is not stated in the abstract or conclusion, and the criteria are presented as general conditions for odd-parity ALM. Since in real materials with SOC the operation [C2||M] acting independently on spin and lattice is not a symmetry, the sufficient conditions may fail or need modification. The manuscript should explicitly state that all derivations and criteria apply in the nonrelativistic limit (negligible spin-orbit coupling) and discuss the implications when this limit is violated.","section":"Symmetry analysis, paragraph after Eq. (2)"},{"comment":"The numerical demonstration of odd-parity ALM relies on the cluster slave-spin approximation, but the paper gives no error bars, no cluster-size convergence analysis, and no explicit statement of the cluster size used for the phase diagram. The phase diagram in Fig. 3(c) is therefore not reproducible from the text alone, particularly because the text states that U_AFM ≈ 3.1 in the cluster slave-spin method while the large-scale QMC value is ≈ 3.8, yet no systematic extrapolation is shown. Since the phase boundaries are a supporting illustration rather than the core symmetry claim, this issue does not invalidate the central derivation, but it should be addressed by providing the missing numerical details and a discussion of the approximation's reliability.","section":"Haldane-Hubbard model, Eq. (8) and Fig. 3(c)"},{"comment":"The claim that the square-lattice system exhibits a time-reversal-symmetric energy band, E_{k,sigma}=E_{-k,-sigma}, is attributed to the symmetry [C2||Ebar_{BL}tau], but the action of the translation tau in this symmetry is not explained. The text states that tau is the minimal vector connecting opposite-spin sublattices, but it does not specify how the symmetry acts on the Hamiltonian, which is essential to verify that the combination actually produces the odd-parity relation. This needs a concise clarification in the main text or a reference to a specific equation in the supplemental material.","section":"Haldane-Hubbard model, discussion of the square-lattice system"}],"minor_comments":[{"comment":"There are several typographical errors, including 'Bellow' for 'Below' in the Haldane-Hubbard section and 'in spit of' for 'in spite of' in the symmetry analysis.","section":"Throughout"},{"comment":"The notation E_{k,sigma}=E_{-k+(-)sigma} is ambiguous; it should be written clearly as E_{k,sigma}=E_{-k,-sigma} for odd parity and E_{k,sigma}=E_{-k,sigma} for even parity.","section":"Eq. (1) and surrounding text"},{"comment":"The phrase 'nonmagnetic time reversal symmetry' and its identification with 'real-space TRS' could be defined more explicitly. The connection between the spin-only symmetry [Cbar2||E] and the absence of [Cbar2||T] is central to the argument, so it would help to state precisely that the nonmagnetic TRS operation acts only on real space and not on spin.","section":"Symmetry analysis, paragraph defining nonmagnetic TRS"},{"comment":"The phase diagram is described in the text as having regions of different colors, but the figure is not reproduced in color in the text, and the legend is not described in sufficient detail. Please ensure the figure is legible in grayscale or add a clear description of the phase boundaries.","section":"Fig. 3(c)"},{"comment":"The author list of Ref. [34] appears garbled ('S.-W. Gedik, Nuh Cheong' and 'B. Ilyas, E. Erge¸cen' in particular); please check and correct the reference.","section":"Reference list, Ref. [34]"},{"comment":"The phrase 'standard odd-parity ALM' is not defined; it would be clearer to say 'the usual odd-parity spin splitting E_{k,sigma}=E_{-k,-sigma}' or to define the term at first use.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The core symmetry derivation is sound in its formal structure, but the overbroad statement of criterion (iii) in the abstract and conclusion will mislead readers and must be corrected by reformulating the criterion or the definition of odd-parity ALM. The no-SOC scope should also be made explicit. The numerical part is illustrative and needs more transparency, but it does not undermine the symmetry argument. The manuscript would also benefit from a clearer comparison with the related work in Ref. [38], which already discusses sublattice-current-induced odd-parity ALM in the Haldane-Hubbard model. These issues are fixable within the scope of the paper, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper is a useful spin-group systematization of odd-parity altermagnetism in collinear magnets, but the abstract overstates the sufficiency of the mirror criterion. I checked the stress-test concern about [C2||M] and it holds up. [C2||M] alone enforces E_{kσ}=E_{Mk,-σ}, not the defining E_{kσ}=E_{-k,-σ}. The paper's own square-lattice example gets the odd-parity relation from [C2||\\bar{E}_{BL}τ], not from [C2||M] alone. So criterion (iii) as written is not sufficient; it needs to be limited to [C2||\\bar{E}] or qualified with the additional operations that complete momentum inversion.\n\nWhat is genuinely new: the identification of the spin-group type R^IV_s for collinear magnets with broken nonmagnetic TRS, and the unification of p-wave and higher-harmonic cases under one framework. That is a real step beyond the existing even-parity classification. The Haldane-Hubbard illustration is reasonable, with opposite sublattice currents breaking nonmagnetic TRS and the band structure showing the expected splitting.\n\nSoft spots beyond the main issue: the numerics are cluster slave-spin with no error bars, no shipped code or data, and the phase diagram is not reproducible from the text alone. The no-SOC assumption is also implicit. These are moderate; the symmetry issue is the important one.\n\nThe reader's CONDITIONAL verdict is about right. The central claim needs rewording, but the framework is sound and likely fixable. This deserves a serious referee: I'd send it to review and ask for a revision that corrects criterion (iii) and makes the numerical basis more transparent.\n\nFor the reading group, yes—if only to argue about whether [C2||M] ever suffices without [C2||\\bar{E}].\n\nBest,","headline":"A useful spin-group framework for odd-parity altermagnetism, but the abstract's [C2||M] criterion is not sufficient as stated and needs amending.","tokens_in":13972,"tokens_out":6013,"would_cite":true,"duration_ms":56057,"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":"Collinear antiferromagnets can produce odd-parity spin splitting without spin-orbit coupling when three symmetry conditions are met.","keywords":["altermagnetism","odd-parity spin splitting","spin group","collinear antiferromagnet","Haldane-Hubbard model","nonmagnetic time-reversal symmetry","sublattice currents","p-wave magnet"],"falsifier":"Measure spin-resolved photoemission on a collinear compensated antiferromagnet with sublattice currents and $[C_2||\\bar{E}]$ symmetry: if the bands satisfy $E_{k\\sigma}=E_{-k\\sigma}$ rather than $E_{k\\sigma}=E_{-k,-\\sigma}$, the predicted odd-parity splitting is absent.","tokens_in":12926,"feed_emoji":"🧲","tokens_out":5692,"duration_ms":56740,"temperature":0.7,"pith_summary":"This paper tries to establish exactly when a collinear antiferromagnet can show odd-parity spin splitting—band splitting that changes sign under inversion in momentum space—without any spin-orbit coupling. Using spin-group analysis, the authors derive three sufficient conditions: time-reversal symmetry in real space must be broken, the magnetic order must be long-range, collinear, and compensated, and opposite-spin sublattices must be connected by a 180-degree spin rotation combined with either inversion or a mirror reflection. These conditions produce what they call odd-parity altermagnetism, with high-harmonic ($l\\ge3$) or $p$-wave ($l=1$) spin splitting. They then show that the Haldane-Hubbard model with opposite sublattice currents satisfies the conditions and exhibits the predicted splitting in its collinear altermagnetic ground state. If correct, this gives a route to Rashba-like spin-momentum locking in materials without heavy elements or spin-orbit coupling.","feed_headline":"Three symmetry conditions generate odd-parity altermagnets","feed_subtitle":"Spin-group analysis shows collinear antiferromagnets can split bands like Rashba systems without spin-orbit coupling.","key_machinery":"The carrying object is the spin group, a symmetry group whose elements $[R_i||R_j v]$ act independently on spin space ($R_i$) and real space ($R_j$ followed by translation $v$). For collinear magnets the spin-only subgroup is $\\{C_\\infty, \\bar{C}_2\\}$. The paper classifies nontrivial spin groups by how the coset decomposition of the crystallographic Laue group $G = H + AH$ is paired with the spin rotation $C_2$; in even-parity altermagnets $A$ cannot be inversion, but in odd-parity altermagnets $A$ is inversion $\\bar{E}$ or mirror $M$. The Haldane-Hubbard model supplies the microscopic realization: opposite sublattice currents from the complex next-nearest-neighbor Haldane hopping break nonmagnetic TRS, and the symmetry $[C_2||\\bar{E}]$ connects opposite-spin sublattices. The cluster slave-spin computation then yields the staggered magnetization and the four-regime phase diagram.","core_discovery":"The central claim is that odd-parity altermagnetism is a distinct class of collinear compensated magnetism, not a variant of conventional antiferromagnetism. The paper derives that a collinear antiferromagnet exhibits odd-parity spin splitting when (i) nonmagnetic time-reversal symmetry—time reversal acting only on real space—is broken; (ii) long-range collinear compensated magnetic order is present; and (iii) the symmetry operation $[C_2||\\bar{E}]$ (a 180-degree spin rotation around the axis perpendicular to the spins combined with spatial inversion) or $[C_2||M]$ (the same spin rotation combined with a mirror reflection) connects opposite-spin sublattices. The first criterion removes the combined symmetry $[\\bar{C}_2||T]$ that forces even-parity bands, while the third supplies the odd-parity connection. The inversion-type operation yields high-order harmonics ($l\\ge3$) and the mirror-type yields $p$-wave ($l=1$) spin splitting. The Haldane-Hubbard model with opposite sublattice currents from Haldane hopping is identified as a concrete realization: the currents break nonmagnetic TRS, and the reversed currents on opposite-spin sublattices enforce $[C_2||\\bar{E}]$, producing the predicted odd-parity splitting in the collinear altermagnetic ground state.","pith_inferences":["If the criteria are sufficient, they should transfer to other bipartite lattices with sublattice currents, so one can systematically scan two-dimensional magnets for odd-parity ALM by checking the three symmetries rather than by brute-force band calculation.","The analysis assumes negligible spin-orbit coupling; in real compounds with weak but finite SOC, the odd-parity splitting should acquire corrections and the exact symmetry labels may soften—a testable prediction for first-principles calculations.","The same spin-group logic could classify odd-parity spin splitting in three dimensions, where the rotation $C_{2z}$ is not automatically inversion, possibly yielding new harmonic orders beyond $l=1$ and $l=3$.","The phase diagram's ALM Chern insulator with $C=2$ suggests that odd-parity altermagnets could host chiral edge states, which the authors do not explicitly analyze."],"forward_implications":["Odd-parity spin splitting can occur in collinear compensated magnets without spin-orbit coupling, so Rashba-like spin-momentum locking is not restricted to noncollinear or coplanar magnets.","The three criteria give a concrete materials search: look for collinear antiferromagnets with broken nonmagnetic TRS (via sublattice currents, light, or orbital order) and $[C_2||\\bar{E}]$ or $[C_2||M]$ symmetry.","In the Haldane-Hubbard model, the odd-parity altermagnetic phase coexists with a Chern insulator ($C=2$) at intermediate interaction, so the phase diagram contains four regimes: AFMI, CI, odd-parity ALM CI, and odd-parity ALMI.","Quasiparticle scattering interference can identify the phase experimentally, because up- and down-spin constant-energy contours are centered on the two inequivalent Dirac points and do not intersect."],"supporting_citations":[{"why":"Defines even-parity altermagnetism and the spin-group language that this paper extends to odd parity.","marker":"[25]"},{"why":"Introduces spin-group theory as the symmetry framework used throughout the derivation.","marker":"[27]"},{"why":"Formalizes spin point groups, giving the coset construction the paper relies on.","marker":"[28]"},{"why":"Proposes odd-parity altermagnetism from sublattice currents, the starting point of the Haldane-Hubbard analysis.","marker":"[38]"},{"why":"Supplies the Haldane model whose opposite sublattice currents break nonmagnetic time reversal.","marker":"[48]"},{"why":"Supplies the cluster slave-spin method used to compute the phase diagram and staggered magnetization.","marker":"[45-47]"},{"why":"Provides an earlier topological phase diagram of the Haldane-Hubbard model that the paper's four-phase diagram builds on.","marker":"[54]"},{"why":"Gives a large-scale quantum Monte Carlo value for the antiferromagnetic transition that the paper compares with its slave-spin result.","marker":"[62]"}],"fun_headline_variants":["Odd-parity altermagnetism arises from three symmetry conditions","Spin-group study reveals odd-parity magnetic spin splitting","No spin-orbit needed: odd-parity altermagnetic splitting","Haldane-Hubbard model realizes odd-parity altermagnetism"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that spin and real-space degrees of freedom decouple completely—spin-orbit coupling is absent—so the spin group factors into a direct product and the symmetry $[C_2||\\bar{E}]$ can act independently on spins and lattice; if SOC is significant, the derived conditions are no longer sufficient.","fun_headline_variants_meta":{"raw":{"variants":["Odd-parity altermagnetism arises from three symmetry conditions","Spin-group study reveals odd-parity magnetic spin splitting","No spin-orbit needed: odd-parity altermagnetic splitting","Haldane-Hubbard model realizes odd-parity altermagnetism"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000259,"raw_usage":{"total_tokens":1676,"prompt_tokens":1125,"completion_tokens":551,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":741,"completion_tokens_details":{"reasoning_tokens":480}},"tokens_in":741,"tokens_out":551,"duration_ms":5793,"temperature":1.0,"reasoning_tokens":480,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:44:49.060989+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure spin-resolved photoemission on a collinear compensated antiferromagnet with sublattice currents and $[C_2||\\bar{E}]$ symmetry: if the bands satisfy $E_{k\\sigma}=E_{-k\\sigma}$ rather than $E_{k\\sigma}=E_{-k,-\\sigma}$, the predicted odd-parity splitting is absent.","supporting_citations":[{"cited_title":"Sorella, Y","cited_arxiv_id":null,"evidence_quote":"Gives a large-scale quantum Monte Carlo value for the antiferromagnetic transition that the paper compares with its slave-spin result."}],"review_version":1}