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REVIEW 3 major objections 4 minor 79 references

Investigation of non-Hermitian and Hermitian models of Altermagnets

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2509.03320 v1 pith:UEHTWHHI submitted 2025-09-03 cond-mat.mes-hall cond-mat.mtrl-sci

Partha Goswami This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-sci PACS 73.43.-f
keywords altermagnetismDzyaloshinskii-Moriyainteractionquantumgeometrictensord-waveandg-waveorderingChernnumberanomalousHalleffectnon-HermitianHamiltonianPTsymmetry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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 $\gamma = 0.23$.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Appendix A, Eq. (23) and Fig. 5/6 captions] 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.
  2. [Section 2, Eq. (3) and Appendix A] 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.
  3. [Section 3, Eq. (28) and Fig. 6] 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.
minor comments (4)
  1. [Fig. 2 caption] 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).
  2. [Section 2, paragraph after Eq. (7)] 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.
  3. [Section 2, around Eq. (33)] 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.
  4. [Section 5 heading] The heading reads 'Concluing remarks'; it should read 'Concluding remarks'.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the Chern number is computed directly from the model eigenstates, and the only self-citations are non-load-bearing references to standard Berry-curvature expressions.

full rationale

The central claim, C=0.999994≈+1, is obtained by evaluating the RR Berry curvature of the occupied bands of H_insul(k) (Eqs. 7 and 28) from the explicitly constructed eigenvectors in Appendix A. No parameter is fitted to data and no output quantity is used to define an input: the g-wave form, DMI terms, and imaginary potential are ansatz inputs, and the Chern number is an independent integral of a curvature derived from those inputs. Refs. [88,89] are self-citations but are used only for the standard expression of the anomalous Hall conductance and the RR-BC formula; Eq. (28) follows from the QGT definitions in Eqs. (19)-(20), so the self-citation is not load-bearing. The skeptical objection that the PT-symmetry inequality in Appendix A may be violated at Γ for the Fig. 5/6 parameters is a correctness and parameter-consistency concern rather than a circularity: even if real, it would make the Chern-number computation ill defined, not make the result equivalent to its inputs by construction. No step in the derivation reduces to a fit, a renamed known result, or a uniqueness theorem imported from the authors' prior work.

Assumptions & free parameters 12 free parameters · 5 assumptions · 0 invented entities

The model depends on a large set of hand-chosen parameters and several unproved modeling assumptions. The central Chern number result is a self-contained calculation for one parameter set, but its connection to real altermagnets is assumed rather than demonstrated.

free parameters (12)
  • t = 1 (energy unit)
    Nearest-neighbor hopping amplitude, chosen as the unit of energy.
  • epsilon_A, epsilon_B = 0.41, 0.32
    Sublattice on-site energies, chosen to open a band gap.
  • M_s = 0.95
    Semenoff mass, chosen to place the insulating model in a nontrivial Chern phase.
  • psi = pi/3
    Magnetic flux phase for nearest-neighbor hopping, chosen by hand.
  • gamma = 0.50, 0.23, 0.032/0.04
    Imaginary potential for gain/loss in various models; values chosen arbitrarily.
  • Delta_g0 / Delta_d0 = 0.50 / 0.14 and 0.50
    g-wave and d-wave gap amplitudes, chosen to produce spectral gaps at the high-symmetry points.
  • lambda_1, lambda_2 = 0.01, 0.001
    Next-nearest-neighbor hoppings, chosen small and positive.
  • J = 0.80
    Exchange interaction strength in the insulating altermagnet Hamiltonian.
  • lambda_SOC / lambda_R = 0.01 / 0.62-0.63
    Spin-orbit coupling and Rashba strengths, chosen to produce spin splitting.
  • D_1, D_2 = 0.62, 0.53
    Dzyaloshinskii-Moriya coefficients, chosen so that the PT-symmetry inequality is satisfied.
  • M = 0.125
    Perpendicular magnetization in the metallic altermagnet model.
  • Gamma = 0.7 or 0.8
    Self-energy broadening from coupling to a ferromagnetic lead in the metallic model.
assumptions (5)
  • standard math Pauli matrix algebra, Berry curvature and Chern number quantization formulas.
    Used throughout Section 3 and Appendix A to construct the QGT and compute topological invariants.
  • domain assumption The 2D tight-binding models capture the low-energy physics of insulating and metallic altermagnets such as MnTe and RuO2.
    The paper provides a qualitative 2D approximation argument but no quantitative comparison with ab initio or experimental results.
  • domain assumption The complex potential iγ models dissipation/gain, and the PT-symmetry condition yields real eigenvalues in the parameter regime used.
    This underpins the non-Hermitian analysis and the spectral plots in Figures 2 and 3.
  • ad hoc to paper Imagined p-orbitals lower the local site symmetry from C4 to C2, producing the altermagnetic order.
    Introduced in Section 1 and Figure 1(a); no microscopic derivation is provided.
  • ad hoc to paper The g-wave gap function Delta_g(k) = Delta_g0 cos(4 arctan(ak_y/ak_x)) represents l=4 pairing.
    The paper's symmetry justification is incorrect, yet this form is central to the insulating altermagnet model.

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Pith. "Pith review of Investigation of non-Hermitian and Hermitian models of Altermagnets." pith.science (2026). https://pith.science/paper/UEHTWHHI

@misc{pith2026250903320,
  author       = {Pith},
  title        = {Pith review of: Investigation of non-Hermitian and Hermitian models of Altermagnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UEHTWHHI}},
  note         = {Machine review of arXiv:2509.03320}
}
read the original abstract

Insulating altermagnets like MnTe exhibit spin configurations where opposing spins are not only aligned antiparallel but also rotated relative to each other. This is an arrangement reminiscent of antiferromagnetism with a twist of spin canting. This study investigates a model Hamiltonian that captures the essential physics of such systems, incorporating key interactions including Dzyaloshinskii-Moriya and conventional exchange terms, relativistic spin-orbit coupling, and d-wave and g-wave orderings. Non-Hermitian dynamics are introduced through complex potentials that simulate energy dissipation and amplification. The paper delves into the behavior of the quantum geometric tensor and the emergence of the quantum anomalous Hall effect within the topologically insulating regime. It also broadens the scope to encompass non-Hermitian metallic altermagnets, focusing on phases characterized by symmetry-breaking d-wave and g-wave order parameters.

Figures

Figures reproduced from arXiv: 2509.03320 by the authors.

Figure 1
Figure 1. (a) (and (b)) The prototypical insulating (metallic) AM model to be investigated is schematically represented in Figure (a) ((b)). In Figure (a), a corresponding unit cell is clearly shown by four sites forming a square. In fact, it shows two types of sites, A and B, each attributed with an imagined p-orbital. The NN (t) and NNN (λ1 and λ2 ) hopping amplitudes are indicated in the figure a. The p orbitals on site A … view at source ↗
Figure 2
Figure 2. (a) and (b) The 2D plots of the eigenvalues 𝐸(𝒌) = ௛ಲ(𝒌)ା௛ಳ(𝒌) ଶ ± ඥ𝐽଴(𝑘௫, 𝑘௬ ) are presented in [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. (a), (b), (c) and (d)The 2D plots of the single-particle excitation spectra 𝐸ଵ , 𝐸ଶ = (𝐴(𝑘௫ , 𝑘௬) ± ඥ𝐽ଵ (𝑘௫ , 𝑘௬ )) of the d-wave ( (a), and (b) ) and the g-wave ( (c), and (d) ) ordered systems as a function of the wavenumber component 𝑎𝑘௫ ( The wavevector component 𝑎𝑘௬ = 0 and 𝜋 in Figures (a, c) and Figures (b, d), respectively. ).The numerical values of the parameters used in the plot are 𝑡 = 1, 𝑀 = 0.125, 𝜆ோௌை஼… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: (a) and (b) The 2D plots of the single-particle excitation spectrum 𝐸ௗ,௡ (𝑛 = 1,2,3,4) given by Eq. (A.5), obtained from the d-wave ordered model of AM, as a function of the wavenumber component 𝑎𝑘௫ ( The wavevector component 𝑎𝑘௬ = 0 and 𝜋 in Figure (a) and Figure (b),…
Figure 5
Figure 5. Figure 5: (a) The contour plot of 𝑔௡,ఓఔ ோோ ൫𝑘௫ , 𝑘௬൯ = 𝑅𝑒𝐺௡,ఓఔ ோோ ൫𝑘௫ , 𝑘௬ ൯ for the lowest band over [− π,π]×[− π,π], assuming half-filling, using the parameters values 𝑡 = 1, 𝜀஺ = 0.41, 𝑀௦ = 0.95, 𝜓 = గ ଷ , 𝛾 = 0.23, 𝐽 = 0.80, Δ୥଴ = 0.50, 𝜆ଵ = 0.01, 𝜆ଶ = 0.001, 𝜆௥௘௟ = 0.01, 𝐷ଵ…
Figure 6
Figure 6. Figure 6: (a) (b)The contour plot of the Berry curvature of the occupied bands in Fig.2 using the Hamiltonian matrix for IAM. The numerical values of the parameters to be used in the plot are 𝑡 = 1, 𝜀஺ = 0.41, 𝑀௦ = 0.95, 𝜓 = గ ଷ , 𝛾 = 0.23, 𝐽 = 0.80, μ = 0.50, Δ୥଴ = 0.50, 𝜆ଵ = 0…

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