{"id":"5d03d030-4fae-41a4-b9d5-5423e1e5f972","arxiv_id":"2509.03969","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Altermagnetic order in 3D topological insulator thin films produces helical edge states with zero total Chern number, protected by crystalline and magnetic symmetry, with measurable quantized nonlocal resistance plateaus.","lead":"This paper shows that coupling a thin film of a three-dimensional topological insulator to altermagnetic order can create pairs of counter-propagating (helical) edge states even though the total Chern number stays zero. A generalist might care because it identifies a concrete material platform where topological edge transport could be engineered and detected without relying on time-reversal symmetry.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Helical edge states are protected only by a top-bottom mirror/spin symmetry (τ_x σ_z) that generic one-sided AM/TI interfaces, surface asymmetry, or in-plane magnetic disorder break; the paper never tests such perturbations, so the robustness claim is conditional.","rationale":"The paper's headline phenomenon is a pair of helical edge states with zero total Chern number. In a purely two-dimensional insulator with no symmetry, counterpropagating edge modes are not topologically protected: they can hybridize and open a gap. Protection therefore requires an exact symmetry that separates the two counterpropagating channels. Here that symmetry is the block label behind Eqs. (8)-(9), namely O = τ_x σ_z. The paper's transport simulations only include disorder terms that commute with this symmetry, so they do not test the most dangerous perturbations. Moreover, the proposed experimental platform—a thin film of a 3DTI in proximity to an altermagnet—will in general have the AM layer on one side only, with the Néel vector not necessarily identical on both surfaces, and may have inequivalent top/bottom interfaces. These conditions break O and can destroy the helical edge states. This is exactly the fragility the reader identified. Because the reader already rendered a CONDITIONAL verdict and this concern supports rather than overturns that verdict, no adjustment is needed. If the proposed concrete test shows the plateaus persist under O-breaking perturbations, the central claim would be strengthened; if not, the paper should be revised to state the symmetry requirement explicitly and soften the robustness claim.","tokens_in":16969,"tokens_out":17134,"duration_ms":178863,"concrete_test":"Add the smallest symmetry-breaking perturbation δH = Δ τ_z⊗σ0 (top/bottom surface asymmetry) or δH = w_in τ0⊗σx (in-plane magnetic disorder) to the lattice version of Sec. III, with Δ and w_in small compared with the bulk gap, and recompute the nanoribbon spectrum and the NEGF resistances R35,26 and R26,26. A decisive variant is to replace HAM in Eq. (7) with the one-sided form diag(Jd, -Jd, 0, 0), modeling AM proximity from only the top surface. If a gap appears in the edge spectrum or the resistance plateaus leave 2h/3e2 and 4h/3e2 as Δ,w_in → 0, the helical modes are not robust in a realistic heterostructure; if the plateaus persist, the symmetry protection is stronger than the block-diagonal limit and the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the h± blocks in Eq. (9) remain decoupled when assigning opposite Chern numbers (CX=+1, CY=-1) to the two sectors. This decoupling is not automatic: it is enforced by a symmetry operator O = τ_x σ_z (top-bottom exchange combined with a π spin rotation about z), which commutes with both Hsf and the AM term in Eq. (7). The paper never states or tests O. In Sec. IV, the disorder terms wa τ0⊗σ0 and wm τ0⊗σz both commute with O, so the reported robustness does not probe the mechanism that keeps the two counterpropagating edge modes from gapping each other. Any O-breaking perturbation—an in-plane magnetic disorder term wm τ0⊗σx, a surface-asymmetric potential Δ τz⊗σ0, or an AM layer on only one surface—couples h+ and h- and can open a gap in the helical edge spectrum. Since the proposed 3DTI/AM heterostructure is naturally one-sided, the assumption that the Néel vector is identical on both surfaces (Sec. III) and that the slab is mirror-symmetric is not self-evidently realizable. Thus the claim that the helical edge states are 'protected by the crystalline and magnetic symmetries' is established only within the artificially symmetric model, not for the realistic platform.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17337,"tokens_out":7118,"duration_ms":78007,"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":[{"comment":"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.","section":"Sec. II, Eq. (2) and Eq. (5)"},{"comment":"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.","section":"Sec. III, Eq. (9), and Sec. IV"},{"comment":"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.","section":"Sec. II and Sec. III, C_X/C_Y classification"}],"minor_comments":[{"comment":"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.","section":"Sec. I and Sec. V"},{"comment":"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.","section":"Sec. II, Eq. (1) and Eq. (7)"},{"comment":"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.","section":"Sec. IV, Eq. (12)"},{"comment":"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.","section":"Appendix A"},{"comment":"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.","section":"Sec. III, Fig. 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically sound within its chosen model, but the robustness and material-platform claims are broader than what the calculations demonstrate. The missing symmetry analysis is the main stumbling block; if the authors can identify the protecting symmetry and show that at least one realistic symmetry-breaking perturbation does not destroy the quantization, the paper would be suitable. I would also encourage them to de-emphasize reliance on Ref. [57] by summarizing the invariant."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nThe 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.\n\nThere 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.\n\nOverall, 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.","headline":"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.","tokens_in":17791,"tokens_out":2967,"would_cite":true,"duration_ms":27554,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["73.43.-f","73.20.At"],"model":"deepseek-v4-flash","headline":"Coupling a topological insulator thin film to altermagnetic order creates helical edge states even though the total Chern number stays zero.","keywords":["altermagnetism","topological insulator thin film","helical edge states","Wilson mass","Chern number","band inversion surface","nonlocal transport","quantum anomalous Hall"],"falsifier":"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.","tokens_in":16890,"feed_emoji":"🧲","tokens_out":6196,"duration_ms":57822,"temperature":0.7,"pith_summary":"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.","feed_headline":"Altermagnetic order yields helical edge states at zero Chern number","feed_subtitle":"Six-terminal Hall bars should show quantized nonlocal resistances—a clear fingerprint of altermagnetic edge transport.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the modified Dirac equation, Wilson mass, and Chern number formula used throughout the paper.","marker":"[19]"},{"why":"Defines d-wave altermagnetism and its spin-symmetry structure, the source of the altermagnetic mass term.","marker":"[29]"},{"why":"Provides the ferromagnetic 3DTI thin-film model and its block-diagonalization, which the paper adapts to altermagnetic order.","marker":"[56]"},{"why":"Gives the high-symmetry-point classification of Chern numbers used to distinguish phases with the same total Chern number.","marker":"[57]"},{"why":"Establishes nonlocal transport as the experimental probe for helical edge states and supplies the multi-terminal measurement scheme.","marker":"[63]"},{"why":"Provides the Landauer-Büttiker and non-equilibrium Green's function transport formalism used for the resistance simulations.","marker":"[65]"},{"why":"Identifies altermagnetic materials that can form heterostructures with 3DTI thin films, supporting the experimental feasibility claim.","marker":"[75]"}],"fun_headline_variants":["Zero Chern number, yet helical edge states emerge","Altermagnets create helical edges without time-reversal","Interplay of masses yields topological helical edge states","Quantized nonlocal resistance signals zero-Chern helical edges"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Zero Chern number, yet helical edge states emerge","Altermagnets create helical edges without time-reversal","Interplay of masses yields topological helical edge states","Quantized nonlocal resistance signals zero-Chern helical edges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000555,"raw_usage":{"total_tokens":2476,"prompt_tokens":739,"completion_tokens":1737,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":1674}},"tokens_in":483,"tokens_out":1737,"duration_ms":17506,"temperature":1.0,"reasoning_tokens":1674,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T10:30:35.509218+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Shen, Topological Insulators: Dirac Equation in Condensed Matters, Springer Series in Solid-State Sci- ences, Vol","cited_arxiv_id":null,"evidence_quote":"Supplies the modified Dirac equation, Wilson mass, and Chern number formula used throughout the paper."},{"cited_title":"Wan, P.-Y","cited_arxiv_id":null,"evidence_quote":"Gives the high-symmetry-point classification of Chern numbers used to distinguish phases with the same total Chern number."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes nonlocal transport as the experimental probe for helical edge states and supplies the multi-terminal measurement scheme."},{"cited_title":"and ˇSmejkal, L","cited_arxiv_id":null,"evidence_quote":"Identifies altermagnetic materials that can form heterostructures with 3DTI thin films, supporting the experimental feasibility claim."}],"review_version":1}