{"id":"7d2a4f3d-e821-46ec-8725-7d1169069de0","arxiv_id":"2608.07099","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Stacking altermagnetic layers in an SSH chain leaves a compensated antiferromagnet in bulk and spin-split topologically protected states at the surface.","lead":"This paper proposes stacking two-dimensional magnetic layers in an alternating Su-Schrieffer-Heeger pattern to create surface altermagnetic states at the boundaries. The design could let spintronics use ordinary antiferromagnets as platforms for altermagnetism, with an electric field as a probe.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The topological-protection claim is not established at generic in-plane momenta: the altermagnetic term breaks the chiral symmetry used for the Zak-phase argument, so the shifted edge modes are exact but not topologically protected away from kx=±ky.","rationale":"The model calculation is internally consistent: the exact ±2J(cos kx−cos ky) energy shift follows from the sublattice-pure zero modes of the SSH chain, and the numerical figures are plausible as a demonstration of the toy model. My concern is not that the surface spin texture disappears, but that the claim of topological protection goes beyond what the model can support. The Zak-phase argument in the 'Topological nature' section applies only to the chiral-symmetric lines kx=±ky; at generic momenta the altermagnetic term breaks chiral symmetry, so no quantized invariant protects the shifted edge modes. The phrase 'adiabatically connected' is weaker than 'topologically protected', and the abstract and conclusion use the stronger language. The proposed numerical test with a small same-sublattice hopping would settle whether the states survive a generic chiral-symmetry-breaking perturbation. If they do not survive, the correct formulation is that a fine-tuned nearest-layer model hosts surface altermagnetic states, not that topological boundaries generically protect them. This supports the reader's CONDITIONAL verdict rather than changing it, so I recommend UNCHANGED.","tokens_in":9194,"tokens_out":18812,"duration_ms":197153,"concrete_test":"Numerically diagonalize a finite slab of Eq. (1) at a generic k∥ with kx ≠ ky (e.g., (0.4π, 0.3π)) after adding a small same-sublattice interlayer hopping t'Σ_i(c†_{A,i}c_{A,i+1} + c†_{B,i}c_{B,i+1} + h.c.), with equal t' on the A-A and B-B channels so that PT remains a symmetry, and with |t'| ≪ |t2−t1| so the bulk gap at that k∥ stays open. Track the two in-gap edge states. If their energies remain exactly at ε±2J(cos kx−cos ky) and the d-wave spin pattern of Fig. 1(e) is unchanged, the strong protection claim survives; if the states move off those energies, hybridize with the continuum, or lose spin polarization, then the surface altermagnetism away from kx=±ky is an exact but fine-tuned property of the nearest-layer model rather than a topologically protected state.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central assertion that the surface altermagnet is 'topologically protected' rests on the Zak-phase/chiral-symmetry argument in the 'Topological nature' section. That argument is valid only on the nodal lines kx=±ky, where the ±2J(cos kx - cos ky)sz term in h(′) vanishes. At a generic k∥, after removing the momentum-dependent energy shift, the 1D fixed-k∥ Hamiltonian in each spin sector has the form H_σ(kz) = m(k∥)σz + (t1 + t2 e^{-ikz})σ+ + h.c., with m = ±2J(cos kx - cos ky). This operator has no chiral symmetry because all three Pauli components are present; hence no quantized Zak phase protects its boundary modes. The edge states at E = ε ± m exist as exact eigenstates only because the zero modes of the m = 0 SSH chain are sublattice-pure and the altermagnetic term is diagonal in that sublattice basis. A generic same-sublattice interlayer hopping or an edge perturbation that is not diagonal on the edge-state support can mix these states and move them off ε ± m without closing the bulk gap. The paper's statement that the finite-energy states are 'adiabatically connected' to protected zero modes does not establish protection: adiabatic continuity without an invariant does not imply topological robustness. This is not merely semantics, because the abstract, introduction, and conclusion advertise topological protection as the mechanism that makes the surface altermagnet robust beyond the dimerized limit.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a tight-binding construction in which two-dimensional altermagnetic layers are stacked in a Su-Schrieffer-Heeger pattern. The two layers per unit cell are PT partners with opposite altermagnetic spin splittings, so the bulk is a compensated, spin-degenerate antiferromagnet. At an open boundary the local PT symmetry is broken, and the authors show numerically and analytically that edge modes appear inside the bulk gap with energies shifted by ±2J(cos kx - cos ky); the surface spin-resolved LDOS, the layer-resolved spin splitter angle, and a proposed perpendicular-electric-field detection scheme are presented. The mechanism is generalized to p-wave and f-wave altermagnetic layers. The central advertised claim is that these surface altermagnetic states are topologically protected.","tokens_in":9656,"tokens_out":13074,"duration_ms":136987,"significance":"If fully established, the paper would offer a simple design principle for surface altermagnetism in artificial multilayers and van der Waals stacks, and it connects the recent surface-altermagnetism literature with textbook SSH topology. The strengths are that the exact energy shift of the boundary modes follows from the sublattice polarization of the SSH edge states, and the numerical spectra and LDOS in Figs. 1-3 support the existence of strongly spin-polarized boundary states within the model. The paper is not fitted to any material and the topological-protection assertion is the main point of concern.","major_comments":[{"comment":"The central claim that the surface altermagnet is topologically protected is not supported by the presented argument. The Zak-phase argument is valid only on the nodal lines kx=±ky, where the ±2J(cos kx - cos ky)sz term vanishes. At a generic in-plane momentum, after removing the scalar energy shift, the 1D Hamiltonian per spin sector has the form m(k∥)σz + [t1 + t2 e^{-ikz}]σ+ + h.c., which has no chiral symmetry; hence no quantized Zak phase protects its boundary modes. The exact edge states at E=ε±m exist because the SSH zero modes are sublattice-pure, not because of a topological invariant. Adiabatic connection to the nodal-line zero modes does not imply robustness: a perturbation that mixes the two sublattices at the surface can move these states off ε±m without closing the bulk gap. I recommend either deriving a genuine invariant for the full Brillouin zone, for example by exploiting the exact pseudo-chiral relation C[H(k)-ε(k∥)I]C^{-1}=-[H(k)-ε(k∥)I], or substantially revising the title, abstract, and conclusion so that they claim exact but not generically topologically protected surface states.","section":"Topological nature"},{"comment":"The construction in Eq. (1) assumes that the two layers in each unit cell are exact PT partners with opposite altermagnetic splittings, and that the surface termination leaves an uncompensated layer in the dimerized limit or a sublattice-pure boundary mode for t1≠0. If a realistic material has spin-dependent interlayer hybridization, interlayer exchange, or surface reconstruction, the sublattice purity is lost and the exact ±2J(cos kx - cos ky) shift ceases to be an eigenstate property. This is a structural limitation of the model; it should be stated explicitly, and the robustness claim should be tested against a perturbation that directly couples the two sublattices at the boundary.","section":"Model and dimerized limit"},{"comment":"The total spin splitter angle is defined as α(θ)=Σ_z α(z,θ), where each α(z,θ) is 2 arctan[(σ↑-σ↓)/(σ↑+σ↓)] for a single layer. Summing per-layer arctangents is not generally equivalent to the spin splitter angle computed from the total spin and charge conductivities, which would be 2 arctan[Σ_z(σ↑-σ↓)/Σ_z(σ↑+σ↓)]. Since the electric-field detection prediction in Fig. 2 relies on this total quantity, the summation formula should be justified or replaced by the conductivity-weighted total response.","section":"Layer-resolved spin splitter effect"}],"minor_comments":[{"comment":"The text says 'p-wave and g-wave altermagnetism' but the model and Fig. 3 are for p-wave and f-wave; please correct this inconsistency.","section":"Generalization to more scenarios"},{"comment":"The proof of the exact ±2J(cos kx - cos ky) shift is relegated to the Supplementary Material; since this is the main analytic result, a sketch of the sublattice-polarization argument should appear in the main text.","section":"Supplementary Material"},{"comment":"The layer-dependent potential is written as V_z = V(2z - Nz -1)/(Nz -1), but the parentheses and typesetting are ambiguous; the formula should be clarified.","section":"Layer-resolved spin splitter effect"},{"comment":"In the definition of α(z,θ), the text refers to the 'j-th layer' although the variable is z; this should be corrected.","section":"Layer-resolved spin splitter effect"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a toy-model proposal with no material parameters or ab initio input; its value is conceptual. The main risk is the overclaim of topological protection away from the nodal lines. If the authors can either provide a valid full-Brillouin-zone invariant or rewrite the paper as a model with exact but model-specific surface states, the paper could be acceptable. The relationship to the existing surface-altermagnetism literature (Refs. [32-35]) is properly cited, but the phrase 'topologically protected' should be reserved for what is actually proven."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the SSH-stacked model is a clean, concrete realization of surface altermagnetism, and the exact edge-state calculation is right. The soft spot is the topological-protection claim: away from the nodal lines kx=±ky, the system has no chiral symmetry and no quantized invariant, so \"topologically protected\" is stronger than what the argument shows.\n\nThe paper does several things well. The tight-binding model is transparent, and the analytic argument that the SSH edge modes shift by ±2J(cos kx - cos ky) is correct because the edge states are sublattice-polarized and the altermagnetic term is diagonal on that support. The numerical spectra and LDOS confirm the picture, and the generalizations to p-wave and f-wave altermagnets are a nice touch. The electric-field detection proposal is practical; citing the 10^-3 sensitivity of current spin-splitter measurements is a reasonable anchor.\n\nThe main problem is the claim that the surface altermagnet is topologically protected. The Zak-phase argument is valid only on the nodal lines, where the ±2J(cos kx - cos ky)sz term vanishes. At generic k∥, the effective 1D Hamiltonian has all three Pauli components, so no chiral symmetry and no quantized Zak phase. \"Adiabatic continuity\" to protected zero modes does not imply robustness against perturbations that mix the two sublattice polarizations. So the statements in the abstract and conclusion that the surface states are \"topologically protected\" and \"robust beyond the dimerized limit\" are not established by the paper's own math. A more accurate phrasing is that the boundary modes are exact and survive as long as perturbations do not couple the sublattice-polarized edge-state subspace.\n\nTwo secondary issues: the construction assumes the two layers in each unit cell are exact PT partners with opposite altermagnetic splitting, and that a clean surface leaves one uncompensated layer. If real interlayer coupling mixes the orders, or surface reconstruction removes that layer, the bulk is not a compensated antiferromagnet and the surface altermagnetism is not separated. The detection signal is plausible but not fully quantified; the odd-V dependence is a clear qualitative prediction.\n\nCitation pattern is fine; the paper properly cites ref. 32 for the prior concept of topological surface altermagnetism. The new element is the concrete SSH construction and the field-driven detection.\n\nThis deserves a serious referee. The model is worth publishing, but the claims need to be narrowed or supported. I would send it out and ask the referees to focus on the robustness argument: either prove an invariant that survives at generic momenta, or explicitly state the perturbation class under which the edge modes remain exact. With that revision, it would be a solid contribution.","headline":"A clean SSH-stacking model for surface altermagnetism with an overstated topological-protection claim; the exact edge-state calculation is right, but the robustness story needs to be narrowed.","tokens_in":10039,"tokens_out":3158,"would_cite":true,"duration_ms":27760,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A stack of alternating magnetic layers ordered like an SSH chain produces a topologically protected surface altermagnet, with the bulk remaining a spin-degenerate antiferromagnet.","keywords":["topological surface altermagnetism","Su-Schrieffer-Heeger model","antiferromagnet","spin splitter effect","d-wave spin splitting","surface states","electric-field detection","hidden altermagnet"],"falsifier":"A slab calculation for a real material candidate (for instance a van der Waals stack with alternating interlayer spacings) that finds the bulk bands spin-split, or finds the surface states not following the ±2J(cos kx − cos ky) pattern, would show that the exact PT-pairing assumption fails; equally, measuring the surface spin-splitter angle and seeing no signal in the nontrivial regime t1 < t2 would directly falsify the claimed topological surface altermagnetism.","tokens_in":9001,"feed_emoji":"🧲","tokens_out":9199,"duration_ms":77094,"temperature":0.7,"pith_summary":"The paper proposes a scheme to create surface altermagnetism in a conventional antiferromagnet by stacking two-dimensional magnetic layers in a Su-Schrieffer-Heeger (SSH) sequence of alternating interlayer bonds. In the bulk, adjacent layers are parity-times-time-reversal partners with opposite altermagnetic spin splittings, so the bands remain spin-degenerate and the bulk looks like an ordinary compensated antiferromagnet. Cutting the stack breaks this local symmetry and leaves an uncompensated altermagnetic layer at the boundary, whose surface states are topologically protected inside the gap. The authors show that at each in-plane momentum the model is an SSH chain, and the boundary modes shift by exactly ±2J(cos kx − cos ky), producing a d-wave surface altermagnet that persists for finite intra-layer hopping; the same stacking idea works for p- and f-wave altermagnets. They further propose detecting the surface states through a layer-resolved spin-splitter angle and a perpendicular electric field that unbalances the two surfaces.","feed_headline":"Alternating-bond stacks turn antiferromagnet edges into altermagnets","feed_subtitle":"Spin-split surface states hide in the bulk gap and emerge under a perpendicular electric field for easy detection.","key_machinery":"The load-bearing object is the SSH-stacked Hamiltonian H(k) = ((h(k∥), t1 + t2 $e^{{−ikz}}$); (t1 + t2 $e^{{ikz}}$, h′(k∥))) with h(′)(k∥) = 2t(cos kx + cos ky)s0 ± 2J(cos kx − cos ky)sz. At fixed in-plane momentum the two layers of each unit cell form the two sublattices of an SSH chain, and the bulk PT symmetry pairs them so that the bands are spin-degenerate. The topological content is captured by the Zak phase along the nodal lines where the altermagnetic term vanishes, which is quantized to 0 or π depending on t1/t2; moving away from the nodes, the chiral-symmetry-breaking term ±2J(cos kx − cos ky) shifts the boundary modes by exactly that amount. The surface altermagnet is thus the continuous, momentum-resolved collection of SSH zero modes dressed by the in-plane altermagnetic splitting.","core_discovery":"The central claim is that geometric truncation of an SSH-stacked magnetic multilayer converts an ordinary, spin-degenerate antiferromagnet into a topological surface altermagnet. The bulk Hamiltonian contains two layer blocks that are PT partners carrying opposite altermagnetic splittings ±2J(cos kx − cos ky)sz, so the combined system is PT-symmetric and spin-degenerate. At an open boundary the local PT symmetry is lost, leaving an uncompensated layer whose altermagnetic character shows up as spin-split surface states. Because in-plane momentum is conserved, the stack decouples into independent SSH chains, and when the interlayer hoppings satisfy t1 < t2 these chains are topologically nontrivial: their zero-energy boundary modes, protected by chiral symmetry along the nodal lines kx = ±ky, are shifted by the exact amplitude ±2J(cos kx − cos ky) away from the nodes. The continuous family of these shifted boundary modes across the Brillouin zone is the d-wave surface altermagnet, and it remains well-defined within the direct bulk gap even when a uniform in-plane hopping adds dispersion. The same mechanism is shown for p- and f-wave altermagnetic layers, establishing topological boundaries as a general platform for surface altermagnetism.","pith_inferences":["Because the boundary-mode shift equals the single-layer term ±2J(cos kx − cos ky), the surface spin splitting is essentially set by the layer's intrinsic exchange coupling J, independent of the topological gap size; this suggests the surface altermagnetism could be tuned by choosing or straining the magnetic layer.","Since the topological protection is per-k∥ SSH chains, weak disorder that scatters between momenta could in principle degrade the surface spin polarization; an interesting test would be to add finite in-plane impurity scattering and check whether the d-wave spin texture survives.","The dependence of the surface state on the ratio t1/t2 implies that the same stacking geometry could serve as a topological switch: changing the interlayer bond order (e.g., by pressure or by inserting a spacer) would turn surface altermagnetism on or off without changing the magnetic order."],"forward_implications":["A surface altermagnet can be engineered from any conventional antiferromagnet by arranging its layers in an alternating-bond SSH stack, without needing bulk altermagnetic order.","The surface spin-split states remain inside the direct bulk gap and localized at the boundary for finite intra-layer hopping, and they disappear entirely in the trivial regime t1 > t2.","Applying a perpendicular electric field destroys the exact cancellation between top and bottom surfaces and produces a net, angle-dependent spin-splitter signal, giving a measurable transport signature.","The same stacking design with p- or f-wave altermagnetic layers yields correspondingly different surface spin textures, showing that the topological mechanism is not restricted to d-wave order."],"supporting_citations":[{"why":"Supplies the Su-Schrieffer-Heeger chain model whose dimerized limit gives the topological boundary modes at each in-plane momentum.","marker":"[36]"},{"why":"Defines d-wave altermagnetism and the layer Hamiltonian with ±2J(cos kx − cos ky) spin splitting used in the model.","marker":"[1]"},{"why":"Prior proposal of topologically protected surface altermagnetism on antiferromagnets, the concept this paper builds on and extends.","marker":"[32]"},{"why":"Textbook SSH topology used to identify the Zak phase quantization and the robustness of the boundary modes.","marker":"[47]"},{"why":"Layer Hall effect experiment that establishes the perpendicular-electric-field detection approach the paper adopts for spin splitter signals.","marker":"[52]"},{"why":"Reported experimental sensitivity to spin-splitter signals of order 10^-3, used to argue the predicted macroscopic signal is detectable.","marker":"[54]"},{"why":"Introduces the hidden altermagnet concept justifying that PT-paired altermagnetic layers mask their splitting in the bulk.","marker":"[37]"}],"fun_headline_variants":["Topological edges turn antiferromagnet stacks into surface altermagnets","SSH-stacked layers: bulk antiferromagnet, surface altermagnet","Truncated SSH stack generates topological surface altermagnetism","Spin-split surface states emerge from antiferromagnet with SSH stacking","Electric field reveals topological surface altermagnet in magnetic stack"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The scheme assumes each unit cell contains two layers that are exact PT partners with opposite altermagnetic spin splitting, and that a clean surface termination leaves one uncompensated layer at the boundary; if real interlayer coupling mixes the two orders or surface reconstruction removes the uncompensated layer, the bulk is not a clean compensated antiferromagnet and the surface state loses its altermagnetic character.","fun_headline_variants_meta":{"raw":{"variants":["Topological edges turn antiferromagnet stacks into surface altermagnets","SSH-stacked layers: bulk antiferromagnet, surface altermagnet","Truncated SSH stack generates topological surface altermagnetism","Spin-split surface states emerge from antiferromagnet with SSH stacking","Electric field reveals topological surface altermagnet in magnetic stack"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000363,"raw_usage":{"total_tokens":1966,"prompt_tokens":960,"completion_tokens":1006,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":576,"completion_tokens_details":{"reasoning_tokens":913}},"tokens_in":576,"tokens_out":1006,"duration_ms":9179,"temperature":1.0,"reasoning_tokens":913,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:54:39.092027+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A slab calculation for a real material candidate (for instance a van der Waals stack with alternating interlayer spacings) that finds the bulk bands spin-split, or finds the surface states not following the ±2J(cos kx − cos ky) pattern, would show that the exact PT-pairing assumption fails; equally, measuring the surface spin-splitter angle and seeing no signal in the nontrivial regime t1 < t2 would directly falsify the claimed topological surface altermagnetism.","supporting_citations":[{"cited_title":"Solitons in polyacetylene","cited_arxiv_id":null,"evidence_quote":"Supplies the Su-Schrieffer-Heeger chain model whose dimerized limit gives the topological boundary modes at each in-plane momentum."},{"cited_title":"Beyond con- ventional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation sym- 5 metry","cited_arxiv_id":null,"evidence_quote":"Defines d-wave altermagnetism and the layer Hamiltonian with ±2J(cos kx − cos ky) spin splitting used in the model."},{"cited_title":"The su- schrieffer-heeger (ssh) model","cited_arxiv_id":null,"evidence_quote":"Textbook SSH topology used to identify the Zak phase quantization and the robustness of the boundary modes."},{"cited_title":"Layer Hall effect in a 2D topological axion antiferromagnet","cited_arxiv_id":null,"evidence_quote":"Layer Hall effect experiment that establishes the perpendicular-electric-field detection approach the paper adopts for spin splitter signals."},{"cited_title":"Spin- to charge-current conversion in altermagnetic candidate ruo 2 probed by terahertz emission spectroscopy","cited_arxiv_id":null,"evidence_quote":"Reported experimental sensitivity to spin-splitter signals of order 10^-3, used to argue the predicted macroscopic signal is detectable."},{"cited_title":"Hidden altermagnetism","cited_arxiv_id":null,"evidence_quote":"Introduces the hidden altermagnet concept justifying that PT-paired altermagnetic layers mask their splitting in the bulk."}],"review_version":1}