{"id":"35824680-7ebc-405e-8645-6d8d98ce3ee7","arxiv_id":"2510.14724","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"In 3D chiral d-wave superconducting altermagnets, altermagnetism deforms the usual surface flat bands into crossed, topologically protected zero-energy bands whose corner count follows the altermagnetic node structure.","lead":"A theory paper predicts that three-dimensional superconducting altermagnets host crossed zero-energy flat bands on their surfaces, protected by a mirror symmetry and detectable in tunneling conductance. The result turns 3D altermagnets into a candidate platform for higher-dimensional topological superconducting phases.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Self-contained counterexample: the SM's dz(x+y) 3D AM shows no crossed flat band, so the 'generic' 3D claim is too broad.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: the main text uses 2D AM terms embedded in a 3D superconductor, while the supplement's genuine 3D AM examples include a d-wave form that does not produce crossed flat bands. This is the most serious threat to the central claim because the claim is phrased generically for 3D superconducting altermagnets, not for a restricted subclass. The paper's internal numerics and symmetry analysis are otherwise consistent, and the g-wave 3D examples provide genuine positive evidence, so the issue is an overgeneralization that can be repaired by stating the required AM node structure. I therefore do not move the reader's CONDITIONAL verdict; I would keep the manuscript conditional pending either a qualified central claim or an explicit symmetry condition that includes the dz(x+y) counterexample.","tokens_in":56665,"tokens_out":6339,"duration_ms":59770,"concrete_test":"Compute the [001] surface spectral function and zero-bias conductance for a tight-binding model with M^d3_k = t_d3 sin kz (sin kx + sin ky) and the same chiral d-wave pairing, at t_d3 = Δ, μ = -4.5t, matching SM S3.2. If no zero-energy crossed flat band appears, as SM Fig. S5a indicates, then formulate and test an explicit node/projection condition—e.g., that the altermagnetic nodes must project to isolated lines in the surface Brillouin zone—and verify it for all claimed 3D AM forms. If the condition fails for any advertised model, the abstract and title must be narrowed from '3D superconducting altermagnets' to the particular AM symmetry classes that satisfy it.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that crossed flat bands appear in 3D superconducting altermagnets as a generic, symmetry-protected phenomenon—is contradicted by the paper's own supplementary calculation. The main-text model, Eq. (1), uses AM terms M^α_k depending only on kx,ky: dxy, dx2-y2, and gxy(x2-y2) forms. These are 2D altermagnetic order parameters embedded in a 3D chiral d-wave superconductor. SM S3.1 introduces genuine 3D AM forms from Ref. [65], including the d-wave form M^d3_k = t_d3 sin kz (sin kx + sin ky). SM S3.2 and Figs. S5-S6 report that for this dz(x+y) form no crossed flat bands appear and the projected DOS and conductance are essentially unchanged, whereas the g-wave forms gzx(x2-3y2) and gyz(3x2-y2) do produce crossed flat bands. Thus the phenomenon depends on the specific AM node structure and its surface projection, not merely on the coexistence of 3D chiral d-wave superconductivity and altermagnetism. Since dz(x+y) is itself a 3D d-wave altermagnet listed in the study, the headline assertion that 3D superconducting altermagnets host these flat bands as a generic effect is not supported. The topological winding-number discussion in SM S1.3 does not repair this gap: the chiral symmetries are established on mirror planes where the main-text AM terms vanish, but no nonzero winding number is explicitly computed, and the argument is not extended to the dz(x+y) case. This remains a real but fixable overgeneralization, not a fatal internal inconsistency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a 3D BdG model (Eq. 1) combining a spin-singlet chiral d-wave superconductor with 2D d- or g-wave altermagnetic terms M^α_k that depend only on kx, ky. The authors show analytically that zero-energy BFSs form when [M^α_k]^2 = ε^2(k)+|ψ(k)|^2 (Eq. 3 and surrounding text), and they argue that projection onto the [001] surface produces crossed zero-energy flat bands whose corners are set by the AM nodes, while the [100] surface hosts modified surface arcs. They further derive conductance formulas and present recursive Green's function numerics for junctions along z and x, reporting three coexisting power laws (2σ̄_N^0, σ̄_N^1, σ̄_N^2) in the momentum-resolved zero-bias conductance. The topological-protection claim is based on a pseudo-magnetic mirror symmetry and a chiral operator on mirror planes, with a winding number stated to be definable but not computed. The supplementary material extends the analysis to genuine 3D altermagnetic forms (dz(x+y), gzx(x2−3y2), gyz(3x2−y2)) and reports that crossed flat bands appear for the two g-wave forms but not for the dz(x+y) form.","tokens_in":57065,"tokens_out":2793,"duration_ms":27435,"significance":"If correctly scoped, the work is a valuable extension of flat-band and BFS physics from 2D to 3D superconducting altermagnets. The analytic BFS condition is clean and parameter-free within the model, and the conductance fingerprints (three distinct power laws) are concrete, falsifiable predictions that could guide tunneling experiments in Sr2RuO4-type materials. The numerical transport calculations are carefully described with all parameters disclosed, and the supplement makes a good-faith effort to test the 3D generalization. However, the paper's headline claim that crossed flat bands are a generic phenomenon in all 3D superconducting altermagnets is contradicted by the authors' own supplementary calculation for the dz(x+y) 3D d-wave altermagnet. The topological protection argument is also incomplete because no nonzero winding number is computed. These issues are load-bearing for the central 'generic, topologically protected 3D phenomenon' claim, but they are fixable by re-scoping the claim to the AM forms that actually exhibit the effect and by supplying the missing winding-number evaluation.","major_comments":[{"comment":"The main text and abstract state that crossed flat bands emerge in 'three-dimensional d- and g-wave altermagnets' as a 'generic topological phenomenon' (Eq. 1 effectively uses 2D AM terms, with a note that the findings 'remain in 3D AMs [65], see S3'). However, SM S3.1 introduces a genuine 3D d-wave altermagnet, dz(x+y) (Eq. S78), and SM S3.2 and Figs. S5–S6 report that this form does not produce crossed flat bands on the [001] surface and leaves the projected DOS and conductance essentially unchanged. Since dz(x+y) is itself a 3D d-wave altermagnet included in the study, the genericity claim is not supported. The phenomenon evidently depends on the specific AM node structure and its surface projection, not merely on the coexistence of 3D chiral d-wave superconductivity with altermagnetism. This is a load-bearing overgeneralization and should be corrected by restricting the claim (e.g.,","section":"SM S3.1, Eq. (S78), Figs. S5–S6; main text Abstract and Conclusions"},{"comment":"The paper claims the crossed flat bands and surface arcs are 'topologically protected'. The argument establishes a pseudo-magnetic mirror symmetry and a chiral operator Γ_k on mirror planes (SM S1.3, Eqs. S24–S29), and states that a winding number can be defined on symmetric lines. However, no winding number is actually computed for any of the models, and the text only says that a nonzero value 'implies' protection. Moreover, footnote [70] explicitly concedes that protection does not hold where BFSs exist. Thus the topological-protection claim is conditional and unsupported by an explicit topological invariant. I recommend either computing the winding number on the mirror planes (a finite calculation for the 2D AM forms) or softening the language to 'symmetry-protected in the regions where the BFS is absent'.","section":"Main text 'Topological origin' and SM S1.3"},{"comment":"The paper says the 'xy-plane nodal lines of chiral d-wave superconductivity ensure that the crossed flat bands appear at zero energy' and that 'the number of corners is determined by the altermagnetic nodes'. This is demonstrated for the 2D AM forms and for the g-wave 3D forms, but the dz(x+y) counterexample shows that the zero-energy condition is not sufficient: the AM term must also have the right node structure when projected onto the [001] surface. The statement should be qualified to avoid implying that any nodal superconducting AM will produce the effect. This is related to Major Comment 1 but deserves its own formulation because it concerns the physical mechanism claimed in the abstract.","section":"Main text, second paragraph and Fig. 1(c,d)"}],"minor_comments":[{"comment":"The phrase 'generic topological surface phenomenon' and 'number of corners determined by the crystal symmetry' is imprecise: the corner count is not determined by symmetry alone but by the specific nodal structure of the AM form. Rephrase to say 'determined by the altermagnetic nodes' as done later in the text.","section":"Abstract and Introduction"},{"comment":"The notation σ̄_N^0, σ̄_N^1, σ̄_N^2 is used in the main text and End Matter, but the definitions of σ̄_N and the superscripts are only implicit in the SM. Please define them explicitly at first use in the main text.","section":"Eq. (8) and SM S2.2"},{"comment":"The reference to the Supplemental Material is a placeholder ('See Supplemental Material for details.,.'). This should be completed with the actual citation details.","section":"Reference [66]"},{"comment":"In panels (f,g), the caption says 'crossed flat bands emerge (green) with their corners defined by the nodes of the 2D spin-polarized altermagnetic Fermi surfaces', but the main text says the corners are defined by the AM nodes. Clarify the relationship between the Fermi-surface nodes and the AM nodes, especially since the Fermi surfaces shown are for a 3D system projected onto the [001] surface.","section":"Fig. 1 caption"},{"comment":"The discussion of the gzx(x2−3y2) AM says 'point nodes mainly become BFS' but the figure shows a line node at kz = 0. Please reconcile the terminology (line node vs point node) for consistency.","section":"SM S3.3, Fig. S7"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a clear, analytically grounded core result for the specific AM forms studied in the main text, and the supplement's 3D extension is a useful stress test. The problem is that the supplement itself disproves the 'generic 3D' claim through the dz(x+y) case, while the topological-protection claim is left conditional. Both are fixable within the manuscript's scope by re-scoping the claims and performing the winding-number computation. I do not see this as a rejection-level internal inconsistency, but the central claim as currently written overreaches the evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: the crossed flat bands and the BFS-driven conductance power laws are real for the models actually solved, but the paper oversells them as a generic 3D superconducting-altermagnet phenomenon. The main text's Eq. (1) uses 2D altermagnetic terms M^α_k(kx,ky) embedded in a 3D chiral d-wave superconductor. That is an honest model, but it is not a genuine 3D altermagnet. The supplement then adds true 3D AM forms from Ezawa and finds crossed flat bands for gzx(x2−3y2) and gyz(3x2−y2) but not for dz(x+y). That is a self-contained counterexample to the abstract's sweeping claim. It does not kill the paper, but the authors should either narrow the claim or explain what symmetry condition separates the cases.\n\nCredit where due: the analytic band structure, the BFS condition E=0 for [M^α_k]^2 = ε^2+|ψ|^2, and the corner-count correspondence with AM nodes are clean and reproducible from the equations. The conductance numerics look internally consistent, all parameters are stated, and the supplement is unusually detailed about the recursive Green's function setup. The three distinct power laws (2σN, σN, σN^2) for flat band, BFS, and gapped regions are a nice, falsifiable fingerprint.\n\nSoft spots, in order: (1) The 3D overgeneralization is the main issue. The supplement's dz(x+y) result is not a minor caveat; it directly contradicts the phrase 'generic topological surface phenomenon' in the introduction. (2) The topological protection argument never actually computes a winding number. The chiral operator on mirror planes is constructed, but the paper says 'a nonzero winding number implies' without showing one. That is a gap, though probably fixable. (3) No code or data files are provided for the conductance plots, so the numerics are not independently checkable. That is standard for the subfield but worth noting.\n\nWho this is for: people working on altermagnet–superconductor hybrids and surface Andreev bound states. It will get more use after a revision that either proves the 3D cases with nonzero winding numbers or explicitly frames the result as symmetry-selective rather than generic.\n\nVerdict: worth sending to peer review. The core 2D results and the conductance signatures deserve referee time, and the 3D claim is a correctable overstatement rather than a fatal flaw.","headline":"A solid 2D-embedded model with clean conductance signatures, but the '3D generic' claim is overbroad and the paper's own supplement supplies the counterexample.","tokens_in":57550,"tokens_out":1247,"would_cite":false,"duration_ms":13012,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that in three-dimensional superconducting altermagnets, the nodal lines of chiral d-wave superconductivity force crossed zero-energy flat bands to appear on the [001] surface, with the number of corners fixed by the alterma","keywords":["altermagnetism","chiral d-wave superconductivity","crossed surface flat bands","Bogoliubov-Fermi surfaces","topological surface states","tunneling conductance","Sr2RuO4","nodal-line superconductor"],"falsifier":"A tunneling measurement on the [001] surface of a material whose altermagnetic order is of dz(x+y) type would be decisive: the paper's own supplement predicts no crossed flat bands for this symmetry, so observing them there would contradict the core claim. Conversely, for dxy or gxy(x2−y2) altermagnets, looking for a 4- or 8-corner crossed pattern in zero-bias conductance would test the predicted connection between altermagnetic nodes and corner count.","tokens_in":56574,"feed_emoji":"🧲","tokens_out":5466,"duration_ms":43880,"temperature":0.7,"pith_summary":"This work extends the study of superconducting altermagnets from two to three dimensions and argues that the combination of chiral d-wave superconductivity and altermagnetic spin splitting produces a new kind of topological surface state: crossed flat bands pinned at zero energy. The mechanism is geometric: the xy-plane nodal lines of the chiral d-wave order parameter guarantee zero-energy states on the [001] surface, while the nodes of the altermagnet decide how many corners the crossed bands have. The same bulk nodal lines become Bogoliubov-Fermi surfaces when altermagnetism is present, reshaping the surface arcs on the [100] face. If correct, these features would give tunneling experiments three distinct conductance power laws in one material, offering a concrete route to detect the topological phases.","feed_headline":"Crossed zero-energy flat bands predicted in 3D superconducting altermagnets","feed_subtitle":"Corners of the bands encode the altermagnet's symmetry, and tunneling conductance reveals them.","key_machinery":"The central objects are the altermagnetic order term M^α_k in the Bogoliubov–de Gennes Hamiltonian and the chiral d-wave pair potential ψ(k) = Δ sin k_z (sin k_x + i sin k_y). The nodal lines of ψ(k) in the kx–ky plane are what guarantee zero-energy flat bands on the [001] surface; the nodes of M^α_k (the spin-degenerate lines of the altermagnet) set the number of corners of the crossed flat bands. Topological protection comes from a pseudo-magnetic mirror symmetry (pMMS) — the product of pseudo-time-reversal and mirror symmetry — which survives despite broken time-reversal, and a winding number defined on the mirror-invariant lines. This is what distinguishes the crossed flat bands from mer","core_discovery":"The central claim is that a three-dimensional spin-singlet chiral d-wave superconductor with d- or g-wave altermagnetism hosts topologically protected crossed surface flat bands at zero energy on the [001] surface. These bands are the 3D analogue of zero-energy Andreev bound states; their zero-energy character is enforced by the nodal lines of the chiral pairing in the xy-plane, and their corners are fixed by the nodes of the altermagnetic order. The same nodal lines, once altermagnetism breaks the degeneracy, produce Bogoliubov-Fermi surfaces in the bulk, which modify the surface arcs on the [100] face. The protection is shown through a pseudo-magnetic mirror symmetry and a winding number d","pith_inferences":["The paper's main-text model embeds a strictly two-dimensional altermagnetic term in a three-dimensional superconductor; the supplement shows that a genuine 3D dz(x+y)-wave altermagnetism does not produce crossed flat bands. We infer that the effect is symmetry-specific: candidate materials must be screened for the correct altermagnetic node structure, not merely for the presence of altermagnetism.","The symmetry-based protection suggests that crossed flat bands should be robust to moderate disorder and surface roughness, making them a practical target for scanning tunneling microscopy; this robustness is not explicitly tested in the paper.","The demonstration that the barrier potential sharpens the conductance of the crossed flat bands implies that low-transparency junctions are the most favorable regime for experimental detection; this emphasis is our inference."],"forward_implications":["Crossed flat bands become a generic topological surface phenomenon in three-dimensional nodal superconductors with altermagnetism, not a fine-tuned feature of a single model.","The number of corners of the crossed flat bands reveals the crystal symmetry of the altermagnet: four corners for dxy or dx2−y2 altermagnetism and eight for gxy(x2−y2) altermagnetism.","Zero-bias tunneling conductance along the [001] direction shows a zero-bias peak originating from the crossed flat bands; its height is reduced but survives at strong altermagnetic strengths.","Along the [100] direction, conductance reflects surface arcs modified by Bogoliubov–Fermi surfaces, with direction-dependent transport and, for dx2−y2 altermagnetism, spin-split arcs producing two resonances.","The coexistence of three power laws in momentum-resolved conductance offers a direct way to detect flat bands and Bogoliubov–Fermi surfaces in tunneling spectroscopy, for example via Doppler-shift measurements."],"fun_headline_variants":["3D altermagnets host topological crossed flat bands","Crossed flat bands emerge in 3D superconducting altermagnets","Zero-energy flat bands in 3D altermagnets are shape-coded","Surface flat bands reveal altermagnet symmetry in 3D","Tunneling conductance maps altermagnet band corners"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The main-text model uses a two-dimensional altermagnetic term M^α(kx,ky) embedded in a three-dimensional superconductor, and the existence of crossed flat bands in a real three-dimensional altermagnet depends on that momentum dependence surviving with the right node structure (the supplement shows dz(x+y)-wave altermagnetism does not produce them).","fun_headline_variants_meta":{"raw":{"variants":["3D altermagnets host topological crossed flat bands","Crossed flat bands emerge in 3D superconducting altermagnets","Zero-energy flat bands in 3D altermagnets are shape-coded","Surface flat bands reveal altermagnet symmetry in 3D","Tunneling conductance maps altermagnet band corners"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000172,"raw_usage":{"total_tokens":1113,"prompt_tokens":745,"completion_tokens":368,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":293}},"tokens_in":489,"tokens_out":368,"duration_ms":3746,"temperature":1.0,"reasoning_tokens":293,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T09:30:55.674054+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A tunneling measurement on the [001] surface of a material whose altermagnetic order is of dz(x+y) type would be decisive: the paper's own supplement predicts no crossed flat bands for this symmetry, so observing them there would contradict the core claim. Conversely, for dxy or gxy(x2−y2) altermagnets, looking for a 4- or 8-corner crossed pattern in zero-bias conductance would test the predicted connection between altermagnetic nodes and corner count.","supporting_citations":[],"review_version":2}