Interplay of Altermagnetic Order and Wilson Mass in the Dirac Equation: Helical Edge States without Time-Reversal Symmetry

Peng-Yi Liu; Qing-Feng Sun; Yu-Hao Wan

REVIEW 3 major objections 5 minor 81 references

Interplay of Altermagnetic Order and Wilson Mass in the Dirac Equation: Helical Edge States without Time-Reversal Symmetry

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Coupling a topological insulator thin film to altermagnetic order creates helical edge states even though the total Chern number stays zero.

desk verdict A clean model prediction of helical edge states in 3DTI/AM thin films, with quantized transport signatures; the protection symmetry is real but untested against realistic perturbations. read the letter →

arxiv 2509.03969 v1 pith:IU5TCEE2 submitted 2025-09-04 cond-mat.mes-hall

Yu-Hao Wan , Peng-Yi Liu , Qing-Feng Sun This is my paper · ORCID

classification cond-mat.mes-hall PACS 73.43.-f73.20.At

keywords altermagnetismtopologicalinsulatorthinfilmhelicaledgestatesWilsonmassChernnumberbandinversionsurfacenonlocaltransportquantumanomalousHall

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

The paper asks what happens when two quadratic mass terms of a lattice-regularized Dirac equation compete: the isotropic Wilson mass and an anisotropic d-wave altermagnetic mass. On a lattice, the altermagnetic mass bends the band-inversion surface into a hyperbolic shape, so the band topology is no longer fixed by the total Chern number alone; phases with identical Chern numbers are distinguished by which high-symmetry points the band-inversion surface encircles. The authors take this to a concrete platform, a three-dimensional topological insulator thin film with parallel altermagnetic Néel vectors on both surfaces. In a symmetric/antisymmetric surface basis, the Hamiltonian splits into two blocks carrying opposite Chern numbers at different high-symmetry points, giving counter-propagating edge modes while the total Chern number stays zero. These helical edge states are protected by crystalline and magnetic symmetries rather than time-reversal symmetry, and the paper's transport simulation predicts quantized nonlocal and local resistances that survive strong potential and magnetic disorder.

What carries the argument

The central object is the block-diagonal thin-film Hamiltonian H = h+ ⊕ h−, where each block takes the form h± = mkσz + vF(kyσx − kxσy) ± J(kx² − ky²)σz, with mk = m0 − Bk² the hybridization-induced mass. The sign of the altermagnetic mass is opposite in the two blocks, so each block develops a band-inversion surface at a different high-symmetry point, yielding opposite Chern numbers. This block structure, together with the high-symmetry-point Chern classification, carries the argument: it converts the interplay of Wilson and altermagnetic masses into a pair of counter-propagating edge channels with zero total Chern number.

What would settle it

In a six-terminal Hall bar made of a 3DTI/AM heterostructure, drive a current between leads 2 and 6 and measure R35,26 and R26,26. Quantized plateaus at 2h/3e² and 4h/3e² inside the bulk gap confirm the claim; applying an in-plane magnetic field, which couples the two blocks, and observing the plateaus break down would refute the block-decoupling mechanism.

Watch

Extended reading notes

Core claim

The paper's central claim is that altermagnetic order in a 3DTI thin film produces topological helical edge states even though the total Chern number remains zero. The mechanism works through the interplay of the Wilson mass, which comes from lattice regularization and produces a circular band-inversion surface, and the altermagnetic mass, which is anisotropic and produces a hyperbolic band-inversion surface. In the lattice model, the topology is refined by which high-symmetry points the band-inversion surface encircles, so phases with the same total Chern number can still be topologically distinct. In the thin-film model, surface hybridization and the altermagnetic exchange field drive a tr

Load-bearing premise

The load-bearing assumption is that the two surface sectors remain decoupled: the altermagnetic Néel vector is parallel on both surfaces and every perturbation, including disorder, preserves the block-diagonal form; if the two blocks mix, the helical edge gap can open and the quantized plateaus disappear.

Editorial extensions

If this is right

A 3DTI/AM heterostructure is predicted to show quantized nonlocal resistance plateaus at R35,26 = 2h/3e² and R26,26 = 4h/3e² inside the bulk gap, giving a direct experimental test.
Phases with equal Chern number but different band-inversion-surface positions cannot be adiabatically connected without closing the bulk gap or breaking inversion symmetry, so interfaces between them must host gapless modes.
The momentum location of an edge state, near kx = 0 or kx = π, encodes which high-symmetry point is encircled, allowing edge-state engineering by tuning the altermagnetic strength and the Dirac mass.
Unlike ferromagnetic order, which produces a single chiral quantum anomalous Hall edge mode, altermagnetic order in the same thin-film geometry produces a pair of helical modes.
Candidate materials such as Bi2Se3-family thin films coupled to MnTe, RuO2, or CrSb layers are proposed as feasible heterostructures for realizing the effect.

Reading between the lines

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

Editorial: The protection relies on the block-diagonal form of the Hamiltonian; any real perturbation that couples the two blocks—an in-plane magnetic moment, top/bottom surface asymmetry, or spin-orbit disorder—could open a gap at the crossing of the helical modes. The paper does not test such perturbations, so the practical stability of the plateaus in a real device remains an open question.
Editorial: The same high-symmetry-point classification argument should apply to other anisotropic, inversion-symmetric mass terms, so d-wave or p-wave magnetic orders in two-dimensional insulators could be engineered to select edge-state momentum in a similar way.
Editorial: The fractional quantized values 2h/3e² and 4h/3e² are tied to the six-terminal geometry; other lead configurations would yield different fractions but the same underlying quantization, so experiments should compare across geometries rather than against the 2e²/h value familiar from quantum spin Hall bars.
Editorial: A direct falsifying test is to measure the nonlocal resistance while continuously rotating the Néel vector away from the surface normal or applying a gate that makes the two surfaces inequivalent; loss of the plateaus would confirm the symmetry origin of the protection.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Probably you know the background: altermagnetism is hot, and people are asking what it does to TI surface states. This paper gives a concrete answer. The authors start from the modified Dirac equation with both Wilson mass and an altermagnetic mass, work out the high-symmetry-point-resolved Chern numbers, and then map it to a 3DTI thin film with AM order on both surfaces. The key claim is that even though the total Chern number is zero, opposite Chern numbers at X and Y produce a pair of helical edge states without time-reversal symmetry. The transport simulations then show quantized nonlocal resistance (2h/3e^2 and 4h/3e^2) that survives potential and out-of-plane Zeeman disorder. That's a nice, falsifiable experiment proposal.

What's good: the lattice model and phase diagram (Eq. 5) are internally consistent; the edge spectra in Figs. 5 and 6 match the topological expectation; the mapping from the ideal Dirac model to the thin film model is clearly explained; and they include an appendix on correlated magnetic disorder, which is more than most papers do. The paper is well-written and the figures are informative. The data is on Zenodo, which helps.

The soft spot is the robustness claim. The helical states exist because the Hamiltonian block-diagonalizes into h+ and h- sectors, which carry opposite Chern numbers. That block structure is protected by a specific symmetry—essentially top-bottom exchange times a spin rotation. The paper never names or tests this symmetry. The disorder they simulate (potential disorder and out-of-plane Zeeman) both commute with it, so those simulations don't probe the mechanism that keeps the counter-propagating modes from gapping each other. Any perturbation that couples the blocks—an in-plane magnetic field, a surface-asymmetric potential, or an AM layer on only one surface, which is the most natural experimental geometry—could open a gap in the helical edge spectrum. The authors actually say they assume the Néel vector is parallel on both surfaces, but they don't justify why that's realistic or check what happens if it isn't. That's a genuine gap in the argument, and it's the main thing I'd want fixed.

There are two smaller issues. The classification of phases with the same Chern number is taken from their own Ref. 57 without re-derivation; the paper's own figures make the distinction plausible, but the formal burden is on the prior work. And Eq. (2) is confusing because it drops the Wilson mass without a clear explanation.

Overall, the central idea is solid within the model, and the transport signature is interesting enough to be worth testing. The fragility issue is real but addressable with additional calculations. This paper deserves serious peer review, not a desk reject.

Referee Report

3 major / 5 minor

Summary. The paper studies a 2D modified Dirac model with Wilson and altermagnetic masses, proposes a high-symmetry-point-resolved classification of Chern numbers, and maps the model onto a 3D topological insulator thin film with altermagnetic proximity. The central claim is that the AM term produces two block-diagonal sectors with opposite Chern numbers at different high-symmetry points, so that the total Chern number is zero but helical edge states appear without time-reversal symmetry. The authors support this with nanoribbon band structures, Chern-number phase diagrams, and NEGF transport simulations showing quantized nonlocal resistance plateaus that are robust against certain potential and magnetic disorder. The paper closes with a proposal for 3DTI/altermagnet heterostructures as an experimental platform.

Significance. If the central claim is correct, the paper identifies a symmetry-enriched mechanism for helical edge transport that does not rely on TRS, which would be a useful conceptual addition to the altermagnet/TI literature. The lattice model, Chern-number calculations, phase boundaries in Eq. (5), edge spectra, and NEGF transport results are internally consistent, and the open-data statement is a positive feature. However, the robustness claim is conditioned on an unstated symmetry that keeps the two block-diagonal sectors decoupled, and the treatment of the Wilson mass is inconsistent between Eq. (2) and the later phase diagram. The paper is therefore valuable but needs additional analysis before the protection and material-platform claims can be accepted.

major comments (3)

[Sec. II, Eq. (2) and Eq. (5)] The lattice Hamiltonian in Eq. (2) has d_z = m + 2J(cos kx - cos ky), with no Wilson-mass term; the text says the Wilson mass is 'replaced by the altermagnetic term as appropriate.' Yet Fig. 2 and Eq. (5) subsequently use coexistence of the Wilson and altermagnetic masses, with factors m-4B±4J and m-8B that require a 2B(cos kx+cos ky-2) term. As written, a reader cannot reproduce the coexistence phase diagram from the displayed Hamiltonian. Please give the full lattice Hamiltonian used for Figs. 2-4 and clarify the role of B.
[Sec. III, Eq. (9), and Sec. IV] The block-diagonal form h+ ⊕ h- is the entire basis for the helical edge states, but the symmetry that protects this block structure is never stated. In the surface basis, the decomposition is preserved by operators such as τ_xσ_z, and the disorder terms used in Sec. IV (wa τ0⊗σ0 and wm τ0⊗σz) both commute with such an operator. The reported robustness therefore does not test the actual protection mechanism. The authors should identify the protecting symmetry explicitly and test representative symmetry-breaking perturbations, e.g., an in-plane Zeeman term wm τ0⊗σx, a surface-asymmetric potential Δτz⊗σ0, or AM proximity on only one surface. Without such tests, the claim that the helical states are protected by crystalline and magnetic symmetries is not established for the proposed realistic heterostructure.
[Sec. II and Sec. III, C_X/C_Y classification] The central distinction between phases with the same total Chern number is based on the high-symmetry-point-resolved invariants C_X and C_Y, imported from Ref. [57]. The paper does not define these invariants nor prove the statement that phases with identical Chern number but different BIS-encircled points cannot be adiabatically connected. Because this classification is load-bearing for the claim that a total-Chern-zero phase can host helical edge states, please provide a self-contained summary of the invariant and its physical content, or state the precise conditions under which the cited classification applies to the h± blocks.

minor comments (5)

[Sec. I and Sec. V] Typos and grammar: 'This findings' and 'These paves' should be corrected. Also 'Chen number' in Sec. II after Fig. 3 and 'resistences' in Appendix A.
[Sec. II, Eq. (1) and Eq. (7)] The altermagnetic term is written as J(k_y^2-k_x^2)σ_z in the text, but Eq. (7) uses J(k_x^2-k_y^2) diagonal entries. The sign convention should be made consistent, since the relative sign between J and the high-symmetry-point labels appears in the phase diagrams.
[Sec. IV, Eq. (12)] The Landauer-Büttiker formula would benefit from an explicit sign convention for the currents and voltages; the ideal transmission matrix used for the 6-terminal device is stated in the text but should be displayed as an equation for clarity.
[Appendix A] The correlated-disorder model is described as a Gaussian filter applied to random Zeeman fields. The physical interpretation of the correlation length ξ relative to the lattice constant and sample size should be stated, and the statement 'unphysical limit' should be justified.
[Sec. III, Fig. 5] In Fig. 5(f-h), the caption says 'without the magnetic order and SOC' but the figures are described in the text as the FM case; the labeling of panels with AM/FM could be made clearer.

Circularity Check

1 steps flagged · score 4.0 of 10

One load-bearing self-citation (Ref. 57) supplies the high-symmetry-point classification used to label the total-C=0 phase as a distinct symmetry-enriched topological phase; the edge-state and transport calculations are independent, but the central interpretation rests on the authors' own prior theorem.

uniqueness imported from authors [Sec. II, paragraphs after Fig. 1; also Sec. III and Conclusion]
"Importantly, two phases with the same Chern number but different skyrmion configurations cannot be adiabatically connected without closing the bulk gap or breaking inversion symmetry. This necessitates the appearance of gapless modes at their interface, highlighting the physical significance of this refined topological classification beyond the conventional Chern number [57]."

The refined classification (CX, CY, etc.) and the assertion that same-Chern-number phases are adiabatically distinct are taken verbatim from Ref. [57], which is by the same three authors (Wan, Liu, Sun). The paper's central physical conclusion—that the total-C=0 AM thin-film phase is a symmetry-enriched topological phase with protected helical edge states—depends on this classification to distinguish it from a trivial C=0 insulator. Since the theorem is cited rather than re-derived or independently checked, the load-bearing step reduces to a self-citation. The edge-state bands and NEGF transport are computed independently, so the circularity is partial, not total.

full rationale

The derivation chain from the lattice Dirac model to the 3DTI/AM thin film is self-contained: Eqs. (2), (6)-(11) are explicitly mapped, and the edge-state spectra (Figs. 1-3, 5-6) and NEGF transport (Fig. 6) are computed directly from the model. The helical edge states therefore do not reduce to a fit or to a definitional identity. The one significant circularity-adjacent element is the high-symmetry-point classification (C_X, C_Y, C_Gamma, C_M) and the adiabatic non-connectability of same-Chern-number phases, which are imported from Ref. [57], a paper by the same authors. This theorem is load-bearing for the paper's central interpretation that the total-C=0 phase is a distinct symmetry-enriched topological phase rather than a trivial insulator. The paper neither re-derives nor independently verifies this theorem; it cites it. However, because the edge states and quantized resistances are computed from the microscopic model, the central claim has independent content. In addition, the robustness claim is conditional: the disorder terms wa tau0⊗sigma0 and wm tau0⊗sigma_z both commute with the block-decoupling symmetry O=tau_x sigma_z (never stated), so O-breaking perturbations (in-plane magnetic disorder, surface asymmetry) could gap the helical states. This is a fragility/over-reach, not a circularity, and is reflected in the moderate score.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim of helical edge states requires only the sign structure of m, B, J, and the preservation of the block diagonal form. No experimental data are fitted. The main unstated input is the high-symmetry-point classification from the authors' own prior work, which is assumed rather than demonstrated.

free parameters (5)

m (Dirac mass) = 0, ±0.5
Hand-chosen model parameter; sign sets Chern number sign.
B (Wilson mass coefficient) = ±0.5
Hand-chosen model parameter; controls trivial-to-nontrivial crossover with J.
J (altermagnetic strength) = 0, ±0.5, ±1
Hand-chosen model parameter; the central tuning knob for the phase transitions studied.
m0 and B in thin film model = m0=0.5, B=-0.5
Chosen so the hybridization gap is topologically trivial; parameterizes film thickness and interface.
vF, A = 1
Energy and momentum scales set to unity; no effect on topology.

assumptions (6)

standard math Lattice regularization via Wilson term resolves fermion doubling and yields a well-defined Chern number on the torus.
Basis for treating the square-lattice model as a proper closed manifold; cited from Refs 19-26.
standard math Chern number computed by Berry curvature (Eqs 3-4) gives the bulk topological invariant and obeys bulk-boundary correspondence.
Standard topological band theory.
domain assumption The high-symmetry-point classification of Ref 57 (same authors) is correct and applicable here; phases with identical Chern numbers but different skyrmion locations are distinct and not adiabatically connected without closing the gap or breaking inversion symmetry.
The paper cites its own prior PRB for this classification and does not re-derive it.
domain assumption The 3DTI thin film can be modeled by the block-diagonal Hamiltonian (Eq 9) with the two blocks decoupled, and this decoupling is preserved by the considered perturbations (parallel AM order on both surfaces, out-of-plane Zeeman disorder).
The helical edge states require the two blocks to remain independent; in-plane disorder or surface asymmetry would couple them.
domain assumption The continuum-to-lattice mapping (replacing k^2 with cos terms and linear terms with sin terms) captures the low-energy topology; higher-order terms at BZ boundaries do not close the global bulk gap.
Acknowledged by the authors in Sec III but not verified beyond the chosen parameter set.
domain assumption Wide-band limit self-energies Sigma=-iI/2 for metallic leads give a valid NEGF transport description.
Standard simplification; ignores energy-dependent lead coupling.

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Cite this review

Pith. "Pith review of Interplay of Altermagnetic Order and Wilson Mass in the Dirac Equation: Helical Edge States without Time-Reversal Symmetry." pith.science (2026). https://pith.science/paper/IU5TCEE2

@misc{pith2026250903969,
  author       = {Pith},
  title        = {Pith review of: Interplay of Altermagnetic Order and Wilson Mass in the Dirac Equation: Helical Edge States without Time-Reversal Symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IU5TCEE2}},
  note         = {Machine review of arXiv:2509.03969}
}

read the original abstract

We investigate topological phases in three-dimensional topological insulator (3DTI) thin films interfaced with altermagnetic (AM) orders. Starting from a modified Dirac equation, we elucidate the interplay between the Wilson mass, arising from lattice regularization, and the altermagnetic mass, and show how this interplay fundamentally alters the band topology and boundary modes. In particular, we demonstrate that coupling a 3DTI thin film to AM order induces a topological phase transition: although the total Chern number remains zero across the transition, topological helical edge states emerge after the transition. These helical edge states arise from opposite Chern numbers at different high-symmetry points, and are distinct from both the chiral edge states of the quantum anomalous Hall phase and the helical edge states of the conventional quantum spin Hall states. The quantum transport simulations reveal robust, quantized nonlocal resistance plateaus associated with these helical edge states, which persist even under strong potential and magnetic disorder. Our results establish 3DTI/AM heterostructures as a feasible material platform for engineering and detecting helical topological edge transport without time-reversal symmetry, thus expanding the landscape of topological matter and providing new opportunities for quantum devices.

Figures

Figures reproduced from arXiv: 2509.03969 by the authors.

**Figure 2.** FIG. 2. (a) Chern number versus the altermagnetic parameter [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Chern number versus the altermagnetic parameter [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. The phase diagram of the Dirac equation with AM mass. (a), (c), and (d) are the cases where [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. (a,e) Schematics of 3DTI thin films (grey) with AM (dark red) and FM (dark blue) heterostructures, showing helical [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Robust nonlocal transport of helical edge states. (a-c) show the band structure of a nanoribbon of 3DTI/AM film with [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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