{"id":"99a524b0-f7da-48b0-bc99-c0f4b4c61d04","arxiv_id":"2603.00536","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"On a doped 3x3 frustrated Hubbard cluster, a hand-defined staggered spin-correlation measure grows with on-site repulsion and is claimed to indicate altermagnetism, but the diagnostic and abstract claims are incomplete.","lead":"Exact diagonalization of a 3x3 Hubbard cluster suggests that adding or removing one electron increases a d-wave-like spin-correlation measure that the authors call altermagnetism. The abstract claims additional results on other lattice geometries and a structure factor that do not appear in the full text, and the central diagnostic depends on an unspecified sign pattern.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central order parameter ⟨Δ_spin⟩ is underdefined: Eq. (2)'s η_ij is never specified, and with the reported A≈0 a simple d-wave NN sign choice would force ⟨Δ_spin⟩=0. The claimed effect may be an artifact of the projection.","rationale":"The reader's weakest_assumption identifies the same core issue: the central diagnostic is not well-defined because η_ij is never specified. I deliberately probe whether the nonzero ⟨Δ_spin⟩ can be interpreted as altermagnetic without a specified projection. The reported A≈0 in the doped sectors makes the simplest d-wave bond-sign interpretation inconsistent, since a C4-symmetric state would then give ⟨Δ_spin⟩=0. Therefore the positive values in Fig. 2 must come from an undocumented projection choice, making the result irreproducible and potentially an artifact. This is a correctness risk, not merely a presentational gap. The exact diagonalization method itself is sound, and the local-moment physics (¯m growing with U) is plausible, but the central order parameter is not falsifiable in the current form. The paper also has other issues the reader catalogued (missing promised results, no finite-size scaling), but the undefined η_ij is the most load-bearing because it undermines the primary evidence. Thus the reader's REJECT verdict remains appropriate; no verdict change is needed.","tokens_in":18025,"tokens_out":7463,"duration_ms":76015,"concrete_test":"Supply the exact η_ij pattern for the 3×3 cluster (e.g., as a 9×9 table) and recompute ⟨Δ_spin⟩ for two control choices: (i) a d-wave NN sign pattern, η_ij = +1 on x-bonds and −1 on y-bonds; (ii) a uniform η_ij ≡ +1. If the U-dependence of ⟨Δ_spin⟩ survives the uniform choice (ii) or vanishes in the d-wave choice (i), the 'altermagnetic enhancement' is not robust and the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's core evidence for correlation-driven altermagnetism is the growth of ⟨Δ_spin⟩ with U in the S=0 doped sectors (Fig. 2). This quantity is built from Ω_ij = ⟨S_i·S_j⟩ η_ij C_i (Eq. 2), with η_ij described only as 'a real-space staggered sign structure reflecting altermagnetic symmetry.' No explicit pattern or formula for η_ij is given anywhere in the text. This is not a cosmetic omission: ⟨Δ_spin⟩ is a linear projection of the spin-correlation matrix onto the chosen η_ij, so without knowing the projection, the numerical result cannot be reproduced or interpreted. The internal tension is sharp: the anisotropy parameter A (Eq. 6), which measures the x/y bond difference, is reported as essentially zero in exactly the doped sectors where ⟨Δ_spin⟩ is largest. If η_ij simply alternated sign between x- and y-directed NN bonds — the natural d-wave choice alluded to in the abstract — then a C4-symmetric state with A=0 would give Δ_i^spin=0 for every site. Nonzero ⟨Δ_spin⟩ with A=0 therefore requires a different, undocumented η_ij that is not odd under C4, and the physical connection to altermagnetism becomes a matter of arbitrary choice. The claim that the system exhibits 'a d-wave-like alternating sign structure' is thus asserted but not derivable from the reported data. At minimum, the manuscript must specify η_ij and demonstrate that the qualitative trend in Fig. 2 is robust under alternative sign conventions; without this, the central result could be an artifact of the projection.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports exact-diagonalization (ED) studies of the single-band and extended Hubbard model on a 3×3 periodic (toroidal) cluster at filling N=8, 9, 10, with on-site U up to 12 and nearest-neighbor V up to U/2. The authors define a site-resolved 'altermagnetic spin-response' Ω_ij = ⟨S_i·S_j⟩ η_ij C_i (Eq. 2), an averaged response ⟨Δ_spin⟩ (Eq. 4), the local moment m̄ (Eq. 5), and an anisotropy A (Eq. 6). The central claims are: (i) at half-filling the local moments are large but the altermagnetic response is small; (ii) in the S=0 doped sectors ⟨Δ_spin⟩ increases with U and with the degree of geometric frustration; (iii) a first-order-like transition occurs at V≈1.8 for N=10 at U=4; and (iv) these results establish a correlation-driven altermagnetic phase. The abstract additionally claims results for 2×3 and 2×4 cylinders, an altermagnetic structure factor S_alm(q), and a degeneracy–A relation, none of which appear in the body.","tokens_in":18444,"tokens_out":5303,"duration_ms":58941,"significance":"If substantiated, the claim that purely electronic correlations on a symmetric, frustrated lattice produce a compensated, anisotropic magnetic state without spin-orbit coupling or engineered hopping anisotropy would be a noteworthy contribution, especially because ED is an unbiased method for small clusters. The machine-checkable nature of ED and the explicit U- and V-dependence are strengths. However, the central diagnostic is not fully specified, the abstract promises results not contained in the text, and the ground-state sector is not verified. These issues currently prevent the reader from assessing whether the reported ⟨Δ_spin⟩ measures an emergent altermagnetic order or an artifact of the chosen projection.","major_comments":[{"comment":"The sign structure η_ij is never defined. The text only states that it is 'a real-space staggered sign structure reflecting altermagnetic symmetry.' Because ⟨Δ_spin⟩ is a linear projection of the spin-correlation matrix onto η_ij, the numerical result cannot be reproduced, and the physical interpretation is ambiguous. The manuscript must give an explicit formula (e.g., η_ij = +1 on x-directed bonds and −1 on y-directed bonds, or a translation-invariant pattern) and show that the U-dependence of ⟨Δ_spin⟩ is robust under alternative sign conventions. Without this, the central claim risks being an artifact of the projection.","section":"Eq. (2), Sec. II"},{"comment":"The abstract and introduction advertise results on the partially frustrated 2×3 cylinder and unfrustrated 2×4 cylinder, the altermagnetic structure factor S_alm(q), and the statement that A is nonzero only for degenerate ground states. None of these appear in the full text. In particular, the claim that geometric frustration is a 'fundamental prerequisite' relies on the 2×4 comparison, which is absent. The authors should either add the missing results or rewrite the abstract to match the body.","section":"Abstract vs. body"},{"comment":"The calculations are restricted to the lowest-total-spin sector (S=1/2 for N=9, S=0 for N=8,10) with the justification that these are magnetically compensated, but the paper never demonstrates that these sectors contain the actual ground state. If the ground state has a different total S, then claims about 'ground-state' correlations and phase transitions are unsupported. The authors should report the ground-state spin quantum number for each (U,V,N) and reconcile their sector choice with it.","section":"Sec. II, Sec. III.A"},{"comment":"There is an apparent tension between the reported A≈0 in the doped sectors and the claimed 'd-wave-like alternating sign structure.' If η_ij were the natural d-wave choice (opposite signs on x- and y-directed bonds), then a state with A=0 would require a specific relation between x- and y-bond correlations; the text does not explain how a nonzero ⟨Δ_spin⟩ coexists with A=0 under the adopted η. The paper must specify the transformation properties of η under C4 and show explicitly how the data satisfy the relation between A, ⟨Δ_spin⟩, and η. As written, the interpretation of the doped sectors as 'rotationally symmetric altermagnetic' is unclear.","section":"Eq. (6), Figs. 2 and 5, Sec. III.A"}],"minor_comments":[{"comment":"The manuscript contains numerous typos and stylistic issues ('Hubbar d model', 'career dopings', 'It therefore serves', 'identifies' vs 'identify' in the abstract). A careful proofreading pass is needed.","section":"General"},{"comment":"The heatmaps (Figs. 4, 8, 12) use colored matrices, but the color scales are sometimes unreadable and the sign of Ω_5j is not explained in relation to η_ij. The figure captions should state whether the plotted quantity is the raw correlation or the projected Ω_5j, and should give the explicit η pattern used.","section":"Figures"},{"comment":"The compensation factor C_i is defined as C_i=1−2|⟨n_i↑⟩−⟨n_i↓⟩|. For a singly occupied site with ⟨n_i↑⟩=1, ⟨n_i↓⟩=0, C_i=0, so the factor suppresses polarized sites. The text states C_i≈1 in the lowest spin sectors; this should be verified numerically or at least stated as an assumption.","section":"Sec. II, Eq. (2)"},{"comment":"The conclusion refers to 'spin-sector selectivity' as a central finding, but this is not demonstrated because only one spin sector per filling is studied. The paper would be stronger if a comparison with higher-spin sectors were shown.","section":"Conclusions"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to have been assembled from a larger project and the abstract overpromises. The core ED calculations are standard, but the central diagnostic is underdefined; if the authors cannot supply an explicit η_ij and demonstrate robustness to its choice, rejection would be warranted even after revision. I also note that several references are to the authors' own recent preprints, which is not itself a problem but should be checked for completeness."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core evidence is <Delta_spin>, built from Eq. (2), but eta_ij is never defined—no formula, no sign pattern. The stress-test note is right: if eta_ij were the natural d-wave NN alternation, a C4-symmetric state with A=0 would force <Delta_spin>=0. So the nonzero signal must come from some other, undocumented eta_ij. That makes the central result non-reproducible and the d-wave claim unsubstantiated.\n\nWhat the paper does well: the numerical work is straightforward exact diagonalization on a 3x3 Hubbard torus with a spin-adapted basis, an appropriate method for such clusters. The raw data—local moment growth with U, doping dependence of spin correlations, V-dependence—are likely honest and could be useful. The paper also tries to separate local moment formation from spatial order, which is the right question.\n\nSoft spots. Apart from the undefined diagnostic, the abstract promises results that are not in the body: 2x3/2x4 lattices, S_alm(q), and a degeneracy-anisotropy relation. That is a serious completeness problem. A is reported as essentially zero in the very sectors where <Delta_spin> is largest; the paper calls this 'rotationally symmetric altermagnetism,' but the connection between A and <Delta_spin> is never explained. The ground-state spin sector is assumed but not checked. With a 9-site cluster there is no finite-size control, so calling this a 'phase' overreaches. The self-citation list is heavy, but that is not the main issue.\n\nVerdict: not a desk reject—the question is relevant and the ED machinery is sound. But the central diagnostic must be specified and justified. If the authors can show the trend survives alternative sign conventions and supply the missing comparative data, the paper could be valuable. As written, I would not cite it.\n\nRecommendation: send it to a referee, but expect a major-revision or reject decision. It is a good reading-group case study in how a projection can define a result into existence.","headline":"The ED work is real but the central altermagnetic diagnostic is undefined (eta_ij is never specified), so the main claim cannot be checked; with A reported near zero, the interpretation is especially shaky.","tokens_in":18919,"tokens_out":2679,"would_cite":false,"duration_ms":28799,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding a single hole or electron to a 3×3 frustrated Hubbard torus creates altermagnetic spin order that grows with on-site repulsion, without spin–orbit coupling or engineered hopping anisotropy.","keywords":["altermagnetism","electronic correlations","geometric frustration","Hubbard model","exact diagonalization","doped Mott insulator","d-wave spin correlations","particle-hole asymmetry"],"falsifier":"Compute ⟨Δ_spin⟩ on the same 3×3 cluster with η_ij set to +1 on every bond (no staggered projection). If the doped S=0 sectors still show a U-dependent nonzero signal, the response is not an artifact of the η_ij choice; if it vanishes, the altermagnetic claim rests entirely on the hand-picked sign structure. Alternatively, check the ground-state spin by diagonalizing in all S sectors: if a higher-spin state is lower in energy for N=8 or N=10, the reported ⟨Δ_spin⟩ is computed in an excited sector.","tokens_in":17910,"feed_emoji":"🧲","tokens_out":7526,"duration_ms":70790,"temperature":0.7,"pith_summary":"This paper claims that altermagnetism—a collinear magnetic state with zero net magnetization but momentum-dependent spin splitting—can arise purely from electron–electron repulsion on a symmetric, geometrically frustrated lattice. Using exact diagonalization of the Hubbard model on a 3×3 torus, the authors find that local moment formation alone is not enough: at half-filling the average altermagnetic response ⟨Δ_spin⟩ is tiny, but doping by one hole or electron (in the S=0 sector) produces a nonzero ⟨Δ_spin⟩ that increases monotonically with U. The response has a d-wave-like sign pattern in real-space spin correlations, a broad momentum-space structure factor, and a strong electron–hole asymmetry favoring electron doping. The same correlations vanish on an unfrustrated 2×4 cylinder, establishing geometric frustration as a necessary condition, while nearest-neighbor repulsion V acts as a destabilizing control knob. If correct, this establishes a fluctuation-mediated, correlation-only route to altermagnetism on symmetric frustrated lattices, with carrier concentration and non-local Coulomb interactions as tunable parameters.","feed_headline":"Doping a frustrated 3x3 Hubbard torus creates altermagnetic order","feed_subtitle":"One added or removed electron produces d-wave-like spin order that grows with U — but only on frustrated lattices.","key_machinery":"The core diagnostic is the altermagnetic correlation matrix Ω_ij = ⟨S_i·S_j⟩ η_ij C_i (Eq. 2), where η_ij is a real-space staggered sign structure meant to encode altermagnetic (d-wave-like) symmetry and C_i is a local compensation factor that suppresses spin-polarized contributions. From Ω_ij the authors construct the site-resolved response Δ^spin_i, the global average ⟨Δ_spin⟩, and the anisotropy parameter A comparing x- and y-bond correlations. This projection, applied to exact equal-time spin correlators obtained in spin-adapted exact diagonalization, is the only order parameter used to claim altermagnetic order. The paper does not specify η_ij explicitly, making the projection the load-","core_discovery":"In exact diagonalization of the one-band Hubbard model on a fully frustrated 3×3 torus, the spin-0 doped sectors (N=8 holes and N=10 electrons) develop an altermagnetic spin texture whose average diagnostic ⟨Δ_spin⟩ grows monotonically with U and saturates at strong coupling, while the half-filled S=1/2 sector shows large local moments but negligible ⟨Δ_spin⟩. The doped states display a d-wave-like staggered sign pattern in the real-space correlation matrix Ω_5j, a broad distribution in the altermagnetic structure factor S_alm(q), and particle–hole asymmetry with stronger response for electron doping. The correlations vanish on the unfrustrated 2×4 cylinder, indicating geometric frustration","pith_inferences":["Because Eq. (2) projects spin correlations through a hand-chosen sign pattern η_ij, a cleaner test of genuine altermagnetism would be to extract a d-wave form factor directly from the raw spin–spin correlators without the η_ij weighting, or to compute the spin-resolved spectral function on the same cluster.","The nine-site result suggests that on larger frustrated clusters (e.g., 4×4 with flux, triangular lattices, or other odd-site tori) the single-carrier doped S=0 sector may also show enhanced altermagnetic correlations; larger-scale exact diagonalization or tensor-network studies could test whether the effect survives at thermodynamic sizes.","The observed particle–hole asymmetry may be a finite-size artifact of the 3×3 torus's Fermi surface; a comparison on a bipartite frustrated lattice with particle–hole symmetric dispersion could separate intrinsic asymmetry from cluster-geometry effects.","The report that anisotropy A is nonzero only for degenerate ground states implies that, on finite clusters, altermagnetic symmetry breaking is tied to degeneracy; in the thermodynamic limit this could mean the phase is stabilized only near a frustration-driven quantum critical point, possibly observable in specific heat or magnetic susceptibility measurements."],"forward_implications":["At half-filling on the 3×3 torus, local moment formation alone does not produce altermagnetic correlations; a mobile carrier must be present for ⟨Δ_spin⟩ to grow with U.","The altermagnetic response in the doped S=0 sectors is d-wave-like in real space and appears as a broad modulation in the momentum-space structure factor S_alm(q).","Electron doping yields a stronger altermagnetic response than hole doping, establishing a particle–hole asymmetry in this frustrated lattice.","On the unfrustrated 2×4 cylinder the altermagnetic correlations are zero, so geometric frustration is a necessary ingredient; on the partially frustrated 2×3 cylinder, V can promote altermagnetic order beyond a critical threshold.","Nearest-neighbor repulsion V monotonically weakens the 3×3 altermagnetic correlations; the electron-doped sector undergoes a first-order-like collapse at V≈1.8 for U=4 and at V≈4.9 for U=10, while the hole-doped sector remains more robust."],"fun_headline_variants":["Frustration plus one doped charge yields altermagnetic order","A single doped carrier creates altermagnetic order on a 3x3 torus","Electron or hole doping induces altermagnetism in frustrated lattices","Doping a frustrated Hubbard cluster induces d-wave-like spin order","One added carrier on a frustrated torus triggers altermagnetism"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The paper's results rise or fall on Eq. (2), where η_ij — the real-space sign pattern that encodes altermagnetic symmetry — is never specified; if η_ij is chosen to match a d-wave form factor, then a nonzero ⟨Δ_spin⟩ partly measures that choice rather than an emergent order, and the paper also assumes the lowest-total-spin sector is the ground-state sector for each filling.","fun_headline_variants_meta":{"raw":{"variants":["Frustration plus one doped charge yields altermagnetic order","A single doped carrier creates altermagnetic order on a 3x3 torus","Electron or hole doping induces altermagnetism in frustrated lattices","Doping a frustrated Hubbard cluster induces d-wave-like spin order","One added carrier on a frustrated torus triggers altermagnetism"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001121,"raw_usage":{"total_tokens":4554,"prompt_tokens":848,"completion_tokens":3706,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":3614}},"tokens_in":592,"tokens_out":3706,"duration_ms":24976,"temperature":1.0,"reasoning_tokens":3614,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T19:52:39.082843+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute ⟨Δ_spin⟩ on the same 3×3 cluster with η_ij set to +1 on every bond (no staggered projection). If the doped S=0 sectors still show a U-dependent nonzero signal, the response is not an artifact of the η_ij choice; if it vanishes, the altermagnetic claim rests entirely on the hand-picked sign structure. Alternatively, check the ground-state spin by diagonalizing in all S sectors: if a higher-spin state is lower in energy for N=8 or N=10, the reported ⟨Δ_spin⟩ is computed in an excited sector.","supporting_citations":[],"review_version":1}