{"id":"ab7406f6-afe7-46c2-bbe6-fd7080b4fce4","arxiv_id":"2509.03320","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":12,"one_line_summary":"A model of an insulating altermagnet with g-wave order and non-Hermitian terms is claimed to have Chern number +1, implying a quantum anomalous Hall insulator.","lead":"This paper builds toy models of altermagnets, magnetic materials with zero net magnetization but momentum-dependent spin splitting, and adds non-Hermitian gain and loss terms. It reports a Chern number of about +1 for an insulating altermagnet model, which would imply a quantum anomalous Hall phase.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The PT-symmetry inequality stated in Appendix A is violated at Γ for the parameter set used in the Chern-number calculation, so the claimed C≈+1 is computed in a regime where the occupied bands are not real and the Chern number is not well defined.","rationale":"The reader's weakest assumption already flagged the PT-symmetry inequality, and my check sharpens it into a concrete internal failure: with the exact parameters used in the Chern-number calculation, the inequality is violated at Γ. This is more decisive than a complaint about material realism, because the paper explicitly says it 'prefer[s] to work with PT-symmetric case for simplicity' and then selects occupied bands by E_n < μ. If the spectrum is complex at Γ, the notion of an occupied band below the chemical potential is ambiguous, and the RR Berry curvature integral cannot be interpreted as the Chern number of a gapped real-band insulator. The paper does not offer an alternative non-Hermitian occupancy rule, so the reported C≈+1 is unsupported. I also note the g-wave symmetry argument in Sect. 2 contains a genuine error (cos(4φ) has period π/2, i.e., C4 symmetry, not 8-fold), but that error does not by itself destroy the topological claim; the PT-violation does. Because the reader's REJECT verdict is already based on high correctness risk and unreproducible numerics, my concern does not change the verdict, but it supplies a sharper, checkable reason why the central claim fails on the paper's own terms.","tokens_in":31543,"tokens_out":6355,"duration_ms":54738,"concrete_test":"Evaluate the PT inequality at Γ using the exact Fig. 6 parameter values: compute LHS=|2(D1(0)+D2(0))|=4.60 and RHS=sqrt(M_s^2+(t(0)-Δ(0))^2+t'(0)^2) with Δ(0)=Δ_g0 (and also Δ(0)=-Δ_g0); if LHS>RHS in either case, recompute the Chern number of H_insul by the Fukui-Hatsugai-Suzuki method on a 200×200 mesh both for the reported parameters and for a modified set (e.g., D1=0.3, D2=0.2) that satisfies the inequality at every k. A robust integer C=1 should survive both checks; a non-integer or parameter-dependent result indicates the reported 0.999994 is not a well-defined topological invariant.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is C=0.999994≈+1 for the IAM Hamiltonian H_insul(k) (Eq. 7) with the parameters in the Fig. 5/6 captions. The calculation presumes a real spectrum: the paper states that H_insul is PT-symmetric with real eigenvalues when |2(D1(k)+D2(k))| < sqrt(M_s^2 + (t(k)∓Δ(k))^2 + t'^2(k)) for all k (Appendix A), and then selects 'occupied bands' by E_n < μ=0.50. With the listed parameters (t=1, M_s=0.95, ψ=π/3, γ=0.23, Δ_g0=0.50, D1=0.62, D2=0.53, ℏ=1), at Γ=(0,0) we have D1(k)+D2(k)=2D1+2D2, so |2(D1+D2)|=4.60, while t(Γ)=-2, t'(Γ)=-2√3+0.23≈-3.23, and Δ(Γ)=±0.5, giving RHS≈4.2 (or 3.7). Thus 4.60 > RHS: the stated PT inequality fails at Γ. Consequently e∓ = sqrt(M_s^2+(t∓Δ)^2+t'^2-b^2) is imaginary at Γ, the spectrum is complex, and the Fermi-level 'occupied bands' below μ are not well-defined real bands. The RR-BC formula (Eq. 28), integrated over a complex 'occupied' manifold, can produce a number near 1 without representing a gapped Chern insulator. This is an internal inconsistency, not merely a question of material realism: the parameter window used for the headline result is outside the PT-symmetric regime the paper says it works in.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes non-Hermitian tight-binding models for insulating and metallic altermagnets, incorporating Dzyaloshinskii-Moriya interaction, relativistic spin-orbit coupling, d-wave and g-wave orderings, and imaginary potentials. The central result is a claim that for the insulating model with Semenoff mass M_s=0.95, the occupied-band Chern number is C=0.999994≈+1, implying a Chern insulator with a quantum anomalous Hall effect. The paper also computes the quantum geometric tensor, discusses phase rigidity and exceptional points in the metallic models, and outlines possible applications of altermagnets. The appendices provide the explicit 4x4 Hamiltonian, eigenvalues, and eigenvector formulas used in the topological calculation.","tokens_in":32088,"tokens_out":5633,"duration_ms":48403,"significance":"If the topological claim were correct, the paper would present a non-Hermitian altermagnet realization of a Chern insulator, which is a timely and interesting contribution to both altermagnetism and non-Hermitian topology. The manuscript contains explicit model Hamiltonians and analytic eigenvector expressions that could serve as a useful starting point for further study. However, the central result is not reliable as presented: the PT-symmetry condition stated by the authors is violated at the Γ point for the parameter set used in the Chern number calculation, the reported g-wave symmetry is mischaracterized, and the numerical computation of the Chern number is not reproducible from the information given. These issues undermine the paper's main claim.","major_comments":[{"comment":"The PT-symmetry inequality stated in Appendix A is violated at Γ=(0,0) for the parameter set used in the Chern number calculation (t=1, M_s=0.95, ψ=π/3, γ=0.23, Δ_g0=0.50, D1=0.62, D2=0.53). With ℏ=1, one obtains b(Γ)=2(D1(Γ)+D2(Γ))=4.60, while the right-hand side sqrt(M_s^2+(t(Γ)∓Δ(Γ))^2+t'^2(Γ)) evaluates to approximately 4.2 (or 4.0) for Δ(Γ)=±0.5. Therefore e∓ in Eq. (23) is imaginary at Γ, the spectrum is complex, and the 'occupied bands' below μ=0.50 are not well-defined real bands. The RR-BC formula (Eq. 28) integrated over this complex manifold does not define a Chern number of a gapped insulator, so the headline claim C=0.999994≈+1 is not supported.","section":"Appendix A, Eq. (23) and Fig. 5/6 captions"},{"comment":"The g-wave gap Δ_g(k)=Δ_g0 cos(4 arctan(ak_y/ak_x)) is described as having 8-fold rotational symmetry because 4φ has period π/2. This is incorrect: cos(4φ) has period π/2 and changes sign under a rotation by π/4, so it is invariant under 4-fold rotations only. The identification with an l=4 g-wave order is consistent with the factor 4 in the argument, but the stated 8-fold symmetry is wrong and indicates a misunderstanding of the angular dependence.","section":"Section 2, Eq. (3) and Appendix A"},{"comment":"The Chern number is computed using direct derivatives of right eigenstates without the gauge-invariant Fukui-Hatsugai-Suzuki method, and the manuscript provides no code, data, mesh size, or convergence analysis. The statement that the result converges to |C|<10^{-5} is not verifiable, and the manuscript does not demonstrate that the value 0.999994 is robust to gauge choices and numerical discretization. Given that this number is the central claim of the paper, the calculation must be made reproducible and properly benchmarked.","section":"Section 3, Eq. (28) and Fig. 6"}],"minor_comments":[{"comment":"The caption lists D1=0,62 with a comma instead of a decimal point, and refers to 'in (a) and (b)' for the IAM spectra that are actually shown in panels (c) and (d).","section":"Fig. 2 caption"},{"comment":"The sentence 'Despite being non-Hermitian, the Hamiltonian H_insul(k) becomes PT symmetric yielding real eigenvalues under certain conditions' appears twice verbatim in the same paragraph.","section":"Section 2, paragraph after Eq. (7)"},{"comment":"The basis for the d-wave Hamiltonian H_d(k) is given as (c_{A↑}, c_{B↓}, c_{A↓}, c_{B↓}, c_{B↑})^T, which contains five components for a 4x4 matrix; this appears to be a typographical error that should be corrected.","section":"Section 2, around Eq. (33)"},{"comment":"The heading reads 'Concluing remarks'; it should read 'Concluding remarks'.","section":"Section 5 heading"}],"recommendation":"reject","confidential_remarks":"The manuscript would require a major reworking to make its central claim viable: a parameter set that satisfies the PT condition, a reproducible Chern number calculation with a standard gauge-invariant method, and correction of the symmetry statements. As it stands, the PT violation at Γ undermines the headline result, and the lack of numerical reproducibility prevents verification. The topic is timely, but this version does not meet the standard for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the headline: the paper's central result, C≈+1 for the insulating altermagnet model, fails under the paper's own rules. At Γ, with the parameters listed for the Chern-number figures (t=1, M_s=0.95, ψ=π/3, γ=0.23, D1=0.62, D2=0.53), the PT-symmetry inequality stated in Appendix A gives |2(D1+D2)|=4.60, while the right-hand side is at most about 4.2 (or 3.7 for the other sign). The inequality is violated, the eigenvalues e∓ become imaginary, and the \"occupied bands\" below μ are not real bands. The Chern number computed from the right-right Berry curvature on a complex manifold can produce a number near 1 without representing a gapped Chern insulator. This is an internal inconsistency, not just a question of material realism.\n\nWhat is new and worth credit: the specific combination of non-Hermitian terms, g-wave order, Rashba SOC, and quantum geometric tensor analysis in an altermagnet model is not in the cited literature. The author writes down the 4x4 Hamiltonian and explicit eigenvectors, which makes the calculation checkable in principle. The QGT discussion for non-Hermitian systems is standard, but applying it to this model is a legitimate exercise.\n\nThe soft spots are substantial. The g-wave form cos(4φ) is claimed to have 8-fold symmetry; it actually has 4-fold symmetry (period π/2). The Chern number is computed without the standard FHS gauge-invariant method, with a looser tolerance (10^-5 vs. the usual 10^-8), and no code or data are provided. The parameters are hand-picked with no material-specific justification—for MnTe this is a toy model. The manuscript is also riddled with editing errors: duplicated text, garbled equations, and inconsistent parameter listings, which further undercut confidence in the numerics.\n\nRecommendation: reject. The central claim is not supported because the calculation runs in a PT-broken regime where the Chern number is not well-defined. If the author corrects the PT-regime issue, fixes the symmetry claim, and provides reproducible numerics, a revised version might be worth a second look. As is, I would not send it to peer review; the internal contradiction is decisive.","headline":"The paper's headline Chern number C≈+1 is computed in a regime where its own PT-symmetry condition fails, so the central topological claim is not supported.","tokens_in":32515,"tokens_out":3744,"would_cite":false,"duration_ms":31600,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["73.43.-f"],"model":"deepseek-v4-flash","headline":"The paper claims that a g-wave model of the insulating altermagnet MnTe has occupied-band Chern number +1, making it a Chern insulator with a quantized anomalous Hall effect.","keywords":["altermagnetism","Dzyaloshinskii-Moriya interaction","quantum geometric tensor","d-wave and g-wave ordering","Chern number","quantum anomalous Hall effect","non-Hermitian Hamiltonian","PT symmetry"],"falsifier":"Recompute the occupied-band Chern number on the same parameter set with the standard gauge-invariant lattice method that the paper cites as reference [83] and states it did not use, demanding convergence below $10^{-8}$; if the integral does not converge to an integer, the claimed $C \\approx +1$ and the resulting topological phase do not follow. A second, independent check is to evaluate numerically whether the PT-symmetry inequality $|2(D_1(\\mathbf{k}) + D_2(\\mathbf{k}))| < \\sqrt{M_s^2 + (t(\\mathbf{k}) \\mp \\Delta(\\mathbf{k}))^2 + t'^2(\\mathbf{k})}$ is satisfied at every $\\mathbf{k}$ in the first Brillouin zone for $M_s = 0.95$, $D_1 = 0.62$, $D_2 = 0.53$, and $\\gamma = 0.23$.","tokens_in":31363,"feed_emoji":"🧲","tokens_out":24512,"duration_ms":185656,"temperature":0.7,"pith_summary":"Altermagnets are a recently identified third class of magnetic order: neighboring spins are antiparallel with zero net magnetization, yet the electronic bands are spin-split because opposite-spin sublattices are related by lattice rotations rather than by inversion or translation. The paper builds minimal two-dimensional model Hamiltonians for an insulating altermagnet (MnTe) and for metallic altermagnets, adding Dzyaloshinskii-Moriya terms, relativistic spin-orbit coupling, d-wave or g-wave order parameters, and an imaginary potential that makes the Hamiltonians non-Hermitian. The central claim is that the g-wave ordered insulating model, for a specific parameter window, has occupied-band Chern number $C = 0.999994 \\approx +1$; if correct, this describes an insulating altermagnet that is a Chern insulator with a quantized anomalous Hall effect despite having no net magnetization. The paper also computes the quantum geometric tensor (quantum metric and Berry curvature) for this model, shows where Kramers degeneracies occur for d-wave versus g-wave order, and locates exceptional points in the non-Hermitian metallic models through the vanishing of phase rigidity.","feed_headline":"Insulating altermagnet model hosts Chern number +1","feed_subtitle":"If the g-wave MnTe model is right, this is a topological Chern insulator with zero net magnetization.","key_machinery":"The argument is carried by three objects. First, the momentum-space Hamiltonian $H_{\\rm insul}(\\mathbf{k})$ is a 4x4 matrix in the basis of two sublattices and two spin states; its entries combine the hopping $t(\\mathbf{k})$, the g-wave gap $\\Delta_g(\\mathbf{k}) = \\Delta_{g0}\\cos(4\\arctan(ak_y/ak_x))$ with its eightfold-symmetric angular dependence, Dzyaloshinskii-Moriya terms entering through $b(\\mathbf{k}) = 2(D_1(\\mathbf{k}) + D_2(\\mathbf{k}))$, and the imaginary potential $\\gamma$ that makes the system non-Hermitian. Second, a PT-symmetry condition keeps the spectrum real: the inequality $|2(D_1(\\mathbf{k}) + D_2(\\mathbf{k}))| < \\sqrt{M_s^2 + (t(\\mathbf{k}) \\mp \\Delta(\\mathbf{k}))^2 + t'^2(\\mathbf{k})}$ must hold for every $\\mathbf{k}$ in the first Brillouin zone, and its failure would produce complex eigenvalues and exceptional points. Third, the topological invariant is the right-right Berry curvature $\\Omega_{xy}(\\mathbf{k}) = -2\\sum_{\\text{occupied}} \\operatorname{Im}\\langle \\partial_{k_x}u_n | \\partial_{k_y}u_n \\rangle$, integrated over the Brillouin zone to give the Chern number; the eigenvectors are the explicit analytic expressions of Appendix A, normalized through the biorthonormality condition $\\langle v^{(m)} | u^{(n)} \\rangle = \\delta_{mn}$ appropriate to a non-Hermitian system.","core_discovery":"The paper's central claim is that the 4x4 Hamiltonian $H_{\\rm insul}(\\mathbf{k})$ for the insulating altermagnet MnTe — built on two sublattices and two spin states, with g-wave pairing $\\Delta_g(\\mathbf{k}) = \\Delta_{g0}\\cos(4\\arctan(ak_y/ak_x))$, momentum-dependent Dzyaloshinskii-Moriya terms, relativistic spin-orbit coupling, and an imaginary potential $i\\gamma$ — possesses a topologically nontrivial insulating phase. For the parameter set $t = 1$, $\\varepsilon_A = 0.41$, $M_s = 0.95$, $\\psi = \\pi/3$, $\\gamma = 0.23$, $J = 0.80$, $\\mu = 0.50$, $\\Delta_{g0} = 0.50$, $\\lambda_1 = 0.01$, $\\lambda_2 = 0.001$, $\\lambda_{\\rm SOC} = 0.01$, $D_1 = 0.62$, $D_2 = 0.53$, the Berry curvature of the occupied bands integrates to $C = 0.999994 \\approx +1$. The paper reads this as a Chern insulator: topological edge modes and a quantized anomalous Hall conductance $\\sigma_{xy} = Ce^2/h$ realized in a compensated magnet with no net magnetization. It further reports that increasing the Semenoff mass $M_s$ closes and reopens the band gap, redistributes the Berry curvature, and drives the Chern number to zero, marking a topological phase transition from Chern insulator to trivial insulator.","pith_inferences":["If the integer value +1 survives a gauge-invariant recomputation, the natural next step the paper does not take is to map the full phase diagram in the Semenoff mass $M_s$, the loss rate $\\gamma$, and the Dzyaloshinskii-Moriya strengths, marking where the $C = \\pm 1$ plateau ends at the PT-symmetry boundary.","The imaginary potential $\\gamma$ could then act as a dissipation-based control knob: pushing the system toward the PT-symmetry boundary should soften the gap and eventually destroy the quantized plateau, a signature a substrate-engineered non-Hermitian altermagnet might show.","Applying the same machinery to the metallic d-wave and g-wave models, one could map where exceptional points cross the Fermi surface and test whether they produce transport anomalies beyond the Hall response — a question the paper's phase-rigidity plots raise but leave open.","The quantum-metric contour plots suggest a check the paper does not perform: whether the non-Hermitian quantum geometric tensor still satisfies the positive-semidefiniteness bound $g_{xx}g_{yy} \\ge \\Omega_{xy}^2/4$; a violation would signal geometry that is genuinely non-Hermitian."],"forward_implications":["If the Chern number calculation is right, the g-wave ordered insulating altermagnet has a quantized anomalous Hall conductance $\\sigma_{xy} = Ce^2/h$ with $C = +1$, and it hosts topological edge modes within the parameter window.","The topological phase is confined to a window: increasing the Semenoff mass $M_s$ closes and reopens the band gap, redistributes the Berry curvature, and drives the Chern number to zero, a transition from Chern insulator to trivial insulator.","Because the anomalous Nernst effect is set by Berry curvature near the Fermi level while the quantized Hall conductance sums the curvature of all occupied bands, the two responses can differ in size; the paper uses this distinction to motivate future Nernst calculations.","In the metallic altermagnet models, exceptional points appear where the phase rigidity $P_j = |\\langle v^{(j)} | u^{(j)} \\rangle|/|\\langle u^{(j)} | u^{(j)} \\rangle|$ falls to zero, giving a concrete non-Hermitian signature in the d-wave and g-wave ordered phases.","Kramers degeneracies sit at different high-symmetry points for d-wave order (X, Y, M) than for g-wave order (Gamma, X), so the two orderings are distinguishable by where their band crossings survive."],"supporting_citations":[{"why":"Identifies MnTe as the insulating altermagnet whose low-energy physics the paper's 4x4 Hamiltonian is meant to capture.","marker":"[73]"},{"why":"Provides the Haldane-model framework, extended to a square lattice, on which the spinless insulating model is built.","marker":"[67,68]"},{"why":"Reports the spontaneous anomalous Hall effect in a compensated magnetic phase and its quantization for insulating altermagnets, motivating the Chern-number computation.","marker":"[24]"},{"why":"Supplies the standard gauge-invariant lattice method for Chern numbers that the paper explicitly positions its own computation against.","marker":"[83]"},{"why":"Origin of the Dzyaloshinskii-Moriya interaction, whose momentum-dependent terms enter the insulating Hamiltonian and feed the PT-symmetry inequality.","marker":"[70-72]"},{"why":"Provide the anomalous Hall conductance formula that connects the computed Berry curvature to the quantized transport response.","marker":"[88,89]"},{"why":"Define altermagnetism's distinguishing symmetry of zero net magnetization with spin-split bands that the model Hamiltonians are constructed to realize.","marker":"[4,5]"},{"why":"Gives the Semenoff mass term its meaning as a staggered on-site potential that opens the gap; the value 0.95 for this mass is the key parameter of the claimed topological window.","marker":"[63]"},{"why":"Supplies the PT-symmetry formalism used to keep the non-Hermitian spectrum real through the stated inequality.","marker":"[20]"}],"fun_headline_variants":["Non-Hermitian altermagnet hosts Chern +1 phase","Altermagnet model yields topological Chern insulator","Chern number +1 in a non-Hermitian altermagnet","g-wave altermagnet: Chern insulator with zero moment","Spin-canted antiferromagnet becomes Chern insulator"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the hand-written four-by-four Hamiltonian in Appendix A faithfully represents the low-energy physics of the insulating altermagnet MnTe, and that the PT-symmetry inequality stated there holds for every wavevector in the first Brillouin zone at the parameter values used for the Chern number calculation; if either fails, the claimed topological phase with $C \\approx +1$ does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Non-Hermitian altermagnet hosts Chern +1 phase","Altermagnet model yields topological Chern insulator","Chern number +1 in a non-Hermitian altermagnet","g-wave altermagnet: Chern insulator with zero moment","Spin-canted antiferromagnet becomes Chern insulator"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1436,"prompt_tokens":1032,"completion_tokens":404,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":648,"completion_tokens_details":{"reasoning_tokens":323}},"tokens_in":648,"tokens_out":404,"duration_ms":3790,"temperature":1.0,"reasoning_tokens":323,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:30:46.846013+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the occupied-band Chern number on the same parameter set with the standard gauge-invariant lattice method that the paper cites as reference [83] and states it did not use, demanding convergence below $10^{-8}$; if the integral does not converge to an integer, the claimed $C \\approx +1$ and the resulting topological phase do not follow. A second, independent check is to evaluate numerically whether the PT-symmetry inequality $|2(D_1(\\mathbf{k}) + D_2(\\mathbf{k}))| < \\sqrt{M_s^2 + (t(\\mathbf{k}) \\mp \\Delta(\\mathbf{k}))^2 + t'^2(\\mathbf{k})}$ is satisfied at every $\\mathbf{k}$ in the first Brillouin zone for $M_s = 0.95$, $D_1 = 0.62$, $D_2 = 0.53$, and $\\gamma = 0.23$.","supporting_citations":[{"cited_title":"Semenoff, Phys","cited_arxiv_id":null,"evidence_quote":"Gives the Semenoff mass term its meaning as a staggered on-site potential that opens the gap; the value 0.95 for this mass is the key parameter of the claimed topological window."}],"review_version":2}