{"id":"ebce3824-dd20-45a6-9952-f8c6d60fd8ea","arxiv_id":"2608.08843","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A bilayer antiferromagnetic s-wave superconductor with nonsymmorphic stacking can host three gate-tunable phases in which chiral Majorana edge states and vortex zero modes are locked to one layer.","lead":"This paper shows a way to make special electronic edge states live on just one layer of a two-layer superconductor, and to flip the chosen layer with a gate voltage. The mechanism works even when the layers are strongly coupled, and could be used in future topological quantum devices.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Universal claim not secured: nonsymmorphic symmetry forces the nearest-neighbor interlayer form factor in Eq.","rationale":"Both the reader and this pass identify the same weakest link: the symmetry argument exempts only one interlayer hopping channel. The abstract's language of a universal mechanism operating regardless of interlayer coupling strength makes the absence of other interlayer channels load-bearing. Because layer-resolved Chern numbers are not quantized by topology, the near values in Fig. 2 are not stable consequences of a bulk invariant; they depend on the transition occurring where H_inter = 0. That condition is not invariant under adding symmetry-preserving longer-range interlayer terms, so the numerical results demonstrate the model rather than the proposed universal principle. The suggested test is decisive: if the layer-resolved Chern numbers and edge-state localization persist with a symmetry-allowed second-neighbor term, then the mechanism is more robust than the nearest-neighbor form suggests; if not, the paper's scope must be narrowed. I therefore keep the reader's CONDITIONAL verdict unchanged.","tokens_in":11707,"tokens_out":9290,"duration_ms":106531,"concrete_test":"Recompute Fig. 2(a)-(d) after adding to Eq. (1) the smallest symmetry-allowed second-neighbor interlayer hopping that is nonzero at M, with amplitude η2 = 0.1η and then η2 = 0.5η, for the same parameters t=1, μ=3.8, η=0.5, λso=0.5, Mz=0.5, Δ=0.3, at V_z = -0.4, 0, and 0.4. If |C_t - 1| or |C_t + C_b| moves beyond the paper's near-quantization tolerance (about 0.05), or the edge-state layer-weight ratio drops below roughly 5:1, the cancellation is not symmetry-protected and the universality claim fails. If C_t and C_b remain within tolerance for all η2 up to η, the concern does not land.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (1) contains only the nearest-neighbor interlayer hopping 4η cos(kx/2)cos(ky/2), and the mechanism relies on its vanishing at X, Y, and M so that the phase transitions in Fig. 2 occur at points where the layers decouple. The nonsymmorphic glide and screw symmetries cited in the text relate the phases of symmetry-equivalent interlayer bonds; they do not by themselves set the total interlayer Hamiltonian to zero. A real bilayer with the same space group generically has additional interlayer overlaps, such as second-neighbor hops or substrate-mediated tunneling, whose Bloch sums typically do not vanish at the zone boundary. Once such a term is added at M, the gap-closing condition changes and the topological transition is no longer pinned to the decoupling line, so the layer-resolved Berry curvature need not be concentrated at zero interlayer coupling. The paper itself notes after Eq. (2) that C_z is not a genuine topological invariant; its near-quantized values (0.981-0.999) are therefore a property of the truncated Hamiltonian, not of the symmetry alone. Without proof that all symmetry-allowed interlayer channels vanish at the transition momentum, the claimed universal mechanism and the FeSe suggestion are not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a symmetry-based mechanism for realizing layer-locked topological phases in clean bilayers: when two adjacent layers are mutually mapped by nonsymmorphic symmetries, the interlayer hopping form factor cos(kx/2)cos(ky/2) vanishes on the Brillouin-zone boundary, so topological phase transitions occurring there see effectively decoupled layers. The authors implement this idea in a bilayer antiferromagnetic s-wave Bogoliubov-de Gennes model [Eq. (1)] with spin-orbit coupling, a magnetic exchange field, and a layer-staggered gate potential V_z. They compute the total Chern number C and layer-resolved Chern numbers C_t, C_b, and identify three gate-tunable phases: a bilayer-locking phase with C=0 and C_t=-C_b approximately -1, and two single-layer-locking phases with C=+/-1 where the nonzero Chern number is confined almost entirely to one layer. They corroborate these bulk quantities with ribbon-edge spectra showing layer-locked chiral Majorana edge states, vortex simulations showing layer-locked Majorana zero modes, and finite-temperature layer-resolved thermal Hall calculations. The paper closes with a proposal that the mechanism could apply to FeSe and related nonsymmorphic layered superconductors.","tokens_in":11985,"tokens_out":12947,"duration_ms":134595,"significance":"If the universal mechanism holds, this work would provide a disorder-free route to the quantum anomalous layer Hall effect and its superconducting analogue, with electrically switchable layer-locked Majorana edge states, vortex Majorana zero modes, and thermal Hall responses. The strengths of the paper are that the layer-resolved Chern numbers are computed directly from the Hamiltonian and checked against independent finite-system simulations (edge spectra, vortex profiles, thermal Hall), and the model is explicit enough to be reproduced. The central limitation is that C_z is not a genuine topological invariant, as the authors acknowledge after Eq. (2), and the near-quantized values reported (0.981-0.999) are demonstrated for a single parameter set and for a Hamiltonian containing only nearest-neighbor interlayer hopping. The universality and robustness claims therefore require additional support before the mechanism can be regarded as established.","major_comments":[{"comment":"The central claim that the nonsymmorphic symmetries 'dictate the interlayer hopping possesses a form factor cos(kx/2)cos(ky/2)' is established only for the nearest-neighbor interlayer hopping included in Eq. (1). The three nonsymmorphic operations listed constrain the phases of symmetry-equivalent interlayer bonds, but they do not, as presented, eliminate all symmetry-allowed interlayer channels; a next-nearest-neighbor or substrate-mediated interlayer hopping generically has a Bloch sum that does not vanish at X, Y, and M. Because the paper itself states after Eq. (2) that C_z is not a genuine topological invariant, the near-quantized values are properties of the truncated Hamiltonian, not of the symmetry alone. Without a proof that every symmetry-allowed interlayer channel vanishes at the transition momentum, or a numerical demonstration that adding such a term leaves C_t and C_b nearly quantized, the claimed universality of the mechanism and the FeSe suggestion are not supported.","section":"Model Hamiltonian, Eq. (1) and following paragraph"},{"comment":"The text states that the mechanism operates 'regardless of interlayer coupling strength', but all numerical results use a single value of the interlayer hopping, eta=0.5 (Figs. 2-4 and the common-parameter lists). No sweep over eta is shown, nor over lambda_so, M_z, Delta, or mu beyond the one listed parameter set. Since the phase boundaries in Fig. 2(a) are pinned to the gap-closing condition at M only for the specific parameters chosen, the claim that the near-quantization of C_z is robust to interlayer coupling strength is not demonstrated. At minimum, the authors should show C_t and C_b versus eta, including large eta, and verify that the topological transitions remain at the decoupling line.","section":"Layer-locked chiral topological superconductivity, Fig. 2(a)"},{"comment":"In the bilayer-locking phase the total Chern number is zero, so the two vortex-core Majorana modes are not protected by the standard bulk-vortex correspondence. The paper asserts that the nearly quantized layer-resolved Chern numbers 'prevent their hybridization', but this does not obviously follow from the bulk C_z values: the vortex wave packet is not a Bloch eigenstate, and the momentum-dependent interlayer coupling need not vanish for a localized core state. The finite-size eigenvalues are stated to be nonzero, yet no energy splitting or system-size scaling is reported, and no symmetry argument for the degeneracy is given. To support the claim of two coexisting robust MZMs, the authors should provide the finite-size scaling of the splitting and a test with a symmetry-allowed interlayer perturbation.","section":"Layer-locked vortex-core MZMs, Fig. 3(a)"}],"minor_comments":[{"comment":"There are typographical issues such as 'withs-wave' in the abstract, 'p±ip' without a space, and inconsistent hyphenation of 's-wave'; these should be corrected.","section":"Abstract / Introduction"},{"comment":"The text says near-perfect quantization is effectively exact, but the bilayer-locking phase has C_t=-0.981 rather than -1. The difference between 'approximately -1' in Fig. 2(a) and 'holds exactly' in the thermal Hall section should be stated consistently so the reader knows the numerical precision.","section":"After Eq. (2) and Fig. 2(a)"},{"comment":"The degeneracy of the two edge-state branches at ky=pi is attributed to the lattice symmetry; the authors should identify which of the glide or screw symmetries protects this crossing and why the two modes do not hybridize for finite eta.","section":"Fig. 2(b) discussion"},{"comment":"Because sigma_z(E) in Eq. (5) is defined through the layer-resolved Berry curvature, the zero-temperature relation kappa_xy_z=C_z*kappa_0 follows directly from the definition when C_z is the full integral; the numerical agreement is a useful consistency check but should not be described as an independent verification.","section":"Layer-resolved thermal Hall effects, Eq. (4)"},{"comment":"The FeSe proposal should be qualified: FeSe's actual interlayer hoppings are not shown to be restricted to the nearest-neighbor channel, and replacing the antiferromagnetic exchange field by an external Zeeman field changes the model assumptions.","section":"Discussions and conclusions"}],"recommendation":"major_revision","confidential_remarks":"The manuscript overlaps with the authors' previous works on the same nonsymmorphic Dirac-semimetal model (Refs. [19,66-69]), and the novel element is the layer-resolved C_z interpretation and the gate-tunable phase diagram. I recommend that the editor require the authors to clearly delineate the new claims and to provide the robustness checks described in the major comments before considering publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has a crisp, concrete idea: in a nonsymmorphic bilayer, the nearest-neighbor interlayer hopping form factor vanishes at zone-boundary momenta, so the layers effectively decouple at the gap-closing points. That makes the layer-resolved Chern number nearly quantized even for strong interlayer coupling. The specific application to a superconducting BdG model with three gate-tunable phases — bilayer-locking (C=0), top-layer-locking (C=-1), bottom-layer-locking (C=+1) — is new as far as I know. The numerical evidence is multi-pronged and mutually consistent: edge-state spectra, layer-resolved density profiles, vortex-core Majorana modes, and thermal Hall coefficients all agree with the computed layer Chern numbers. It is also honest that C_z is not a genuine topological invariant, and it does not try to hide that the values are 0.98–0.999, not exactly integer.\n\nWhere it is soft: the word \"universal\" is doing too much work. The glide/screw symmetries enforce the form factor only for the nearest-neighbor interlayer hopping that is actually in Eq. (1). Any additional interlayer term — second-neighbor hops, substrate-mediated tunneling — generically does not vanish at the same momenta, and the near quantization would degrade. So \"regardless of interlayer coupling strength\" really means \"regardless of the value of η\" within the truncated model, not a statement about all symmetry-allowed interlayer channels. The stress-test note is correct on this point. One parameter set is shown, and there is no perturbative argument that the effect survives moderate extra interlayer hopping. The FeSe suggestion is brief and speculative: FeSe in its superconducting state is not the antiferromagnet assumed here, and replacing the staggered exchange by a uniform Zeeman field is a nontrivial modification.\n\nThe thermal Hall relation κ_xy^z = C_z κ_0 follows directly from the definitions of the layer-resolved coefficients, so it is a consistency check rather than an independent prediction. That said, the circularity burden is low: the layer Chern numbers are computed from the Hamiltonian and then compared with independently simulated edge and vortex states, so the main claim is not fitted into existence.\n\nWho gets value: researchers in layer Hall physics and chiral topological superconductivity will find the model instructive as a proof of principle. The paper deserves peer review; a good referee will push the authors to temper the universality claim and ideally show how the mechanism degrades when additional interlayer terms are added. I would send it out with the expectation of revision.","headline":"Clean minimal model, but the 'universal' overstates what is actually shown — the mechanism works within the truncated Hamiltonian, not as a general symmetry principle.","tokens_in":12472,"tokens_out":2760,"would_cite":false,"duration_ms":31929,"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":"Nonsymmorphic lattice symmetry can lock chiral Majorana edge states, vortex zero modes, and thermal Hall transport to one layer, with a gate voltage switching the locking layer.","keywords":["layer-locked topological superconductivity","layer-resolved Chern number","chiral Majorana edge states","Majorana zero modes","nonsymmorphic symmetry","quantum anomalous layer Hall effect","thermal Hall effect","gate-tunable topological phases"],"falsifier":"Add a next-nearest-neighbor or substrate-mediated interlayer hopping term to the tight-binding Hamiltonian and recompute the layer-resolved Chern numbers at the same parameters; if the near-integer values shift substantially, the symmetry-enforced decoupling is not robust. A layer-resolved thermal Hall measurement that fails to show a gate-voltage sign switch with near-quantized magnitude would also refute the proposal.","tokens_in":11505,"feed_emoji":"🔀","tokens_out":5880,"duration_ms":58197,"temperature":0.7,"pith_summary":"The paper proposes a symmetry-based mechanism for layer-locked topological superconductivity: when two layers are related by nonsymmorphic lattice symmetries, interlayer hopping vanishes along high-symmetry lines in the Brillouin zone. In a bilayer antiferromagnetic s-wave superconductor, this makes layer-resolved Chern numbers nearly quantized, so chiral Majorana edge states, vortex-core zero modes, and the thermal Hall response lock to one layer. A gate voltage can switch between three phases: a bilayer-locked phase with zero total Chern number and two single-layer-locked phases with Chern number $\\pm 1$. The mechanism works at any interlayer coupling strength and without disorder, unlike earlier proposals.","feed_headline":"Symmetry locks chiral Majorana modes to a single layer","feed_subtitle":"A gate voltage flips which layer carries the edge current, giving three switchable superconducting phases.","key_machinery":"The load-bearing object is the interlayer hopping form factor $\\cos(k_x/2)\\cos(k_y/2)$ in the Bogoliubov-de Gennes Hamiltonian, which vanishes at the Brillouin-zone boundary because each site couples to four nearest neighbors in the other layer and the paths interfere destructively. This zero, enforced by the glide and screw symmetries relating the layers, decouples the layers exactly at the momenta where the gap closes. Together with the layer projection operator $P_z$, it makes the layer-resolved Berry curvature concentrate near those momenta, so the layer-resolved Chern numbers become nearly quantized even for strong interlayer coupling.","core_discovery":"The central claim is that a universal, disorder-free mechanism produces layer-locked topological phases: if adjacent layers can be mapped to each other by nonsymmorphic symmetries, interlayer hopping is symmetry-enforced to vanish along specific high-symmetry lines, so topological transitions occurring at those momenta make the layer-resolved Berry curvature nearly independent between layers. Applied to a $PT$-symmetric bilayer antiferromagnetic metal with s-wave pairing, the paper finds three gate-tunable phases with almost integer layer-resolved Chern numbers. In the bilayer-locking phase $C=0$ with $C_t=-C_b\\simeq -1$; in the top-layer-locking phase $C=-1$ with $C_t\\simeq -1$ and $C_b\\simeq 0$; and in the bottom-layer-locking phase $C=1$ with $C_t\\simeq 0$ and $C_b\\simeq 1$. The corresponding chiral Majorana edge states, vortex-core Majorana zero modes, and layer-resolved thermal Hall coefficients follow the layer-resolved Chern numbers, giving fully electric-field-tunable chiral topological superconductivity.","pith_inferences":["The mechanism should transfer to insulating platforms: any topological phase transition pinned to the zone-boundary momenta in a nonsymmorphic bilayer should produce nearly quantized layer-resolved Chern numbers and layer-locked chiral edge states there too.","Real materials will have subleading interlayer hoppings, so the vanishing is only approximate; estimating the tolerable size of next-nearest-neighbor interlayer hopping would be a natural extension of the calculation.","A clean experimental discriminant is the gate-voltage sign switch of the thermal Hall response combined with local imaging of the edge mode's layer, which would separate this symmetry mechanism from disorder-induced localization.","The bilayer-locking phase's two spatially separated Majorana zero modes on one vortex suggest a route to doubled Majorana storage; whether braiding operations survive finite-size hybridization is left open by the paper."],"forward_implications":["If the mechanism holds, layer-locked chiral topological superconductivity provides a disorder-free route to the superconducting analogue of the quantum anomalous layer Hall effect.","Reversing the gate voltage reverses the chirality and switches the hosting layer of the chiral Majorana edge state, making the topological response fully electric-field tunable.","At low temperature the layer-resolved thermal Hall coefficients are predicted to be nearly quantized integer multiples of $\\kappa_0$, with the two layers contributing opposite signs in the bilayer-locking phase.","The same symmetry-based argument should extend to multilayer stacks, yielding larger Chern numbers and designed layer-resolved edge-state patterns, and to helical and second-order topological phases."],"supporting_citations":[{"why":"Supplies the nonsymmorphic bilayer lattice model in which the interlayer hopping form factor vanishes at the zone boundary, providing the platform for the topological phases.","marker":"[62]"},{"why":"Reports the experimental observation of the layer Hall effect in $PT$-symmetric even-layer MnBi2Te4, the phenomenon whose quantized superconducting analogue this paper constructs.","marker":"[42]"},{"why":"Proposes the earlier disorder-based mechanism for the quantum anomalous layer Hall effect, the baseline this paper's symmetry mechanism improves upon by working in clean systems.","marker":"[44]"},{"why":"Documents the coexisting twofold Majorana zero modes in a related phase and invokes symmetry protection, a result the paper connects to its layer-resolved picture.","marker":"[68]"},{"why":"Shows the same nonsymmorphic band structure supports chiral Majorana edge states and Bogoliubov Fermi surfaces, grounding the present model in a known topological phase diagram.","marker":"[19]"},{"why":"Supplies the bulk-vortex correspondence $n_M = C \\bmod 2$ for chiral topological superconductors, which the layer-resolved Majorana zero-mode analysis uses.","marker":"[10]"},{"why":"Identifies FeSe's lattice symmetries as matching the bilayer setup, giving a concrete candidate material for experimental realization.","marker":"[77]"}],"fun_headline_variants":["Gate switches Majorana edge current between layers","Three gate-tunable phases for layer-locked Majorana modes","Nonsymmorphic bilayer yields switchable chiral Majorana modes","Layer-locked superconductivity with gate-controlled Majorana states","Electric field selects which layer hosts chiral Majorana modes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes nearest-neighbor interlayer hopping is the only interlayer tunneling; any additional interlayer hopping that does not vanish at the same zone-boundary momenta would spoil the near-quantization.","fun_headline_variants_meta":{"raw":{"variants":["Gate switches Majorana edge current between layers","Three gate-tunable phases for layer-locked Majorana modes","Nonsymmorphic bilayer yields switchable chiral Majorana modes","Layer-locked superconductivity with gate-controlled Majorana states","Electric field selects which layer hosts chiral Majorana modes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000239,"raw_usage":{"total_tokens":1467,"prompt_tokens":848,"completion_tokens":619,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":464,"completion_tokens_details":{"reasoning_tokens":538}},"tokens_in":464,"tokens_out":619,"duration_ms":6998,"temperature":1.0,"reasoning_tokens":538,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:22:54.564315+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Add a next-nearest-neighbor or substrate-mediated interlayer hopping term to the tight-binding Hamiltonian and recompute the layer-resolved Chern numbers at the same parameters; if the near-integer values shift substantially, the symmetry-enforced decoupling is not robust. A layer-resolved thermal Hall measurement that fails to show a gate-voltage sign switch with near-quantized magnitude would also refute the proposal.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the nonsymmorphic bilayer lattice model in which the interlayer hopping form factor vanishes at the zone boundary, providing the platform for the topological phases."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports the experimental observation of the layer Hall effect in $PT$-symmetric even-layer MnBi2Te4, the phenomenon whose quantized superconducting analogue this paper constructs."},{"cited_title":"Chen, H.-P","cited_arxiv_id":null,"evidence_quote":"Proposes the earlier disorder-based mechanism for the quantum anomalous layer Hall effect, the baseline this paper's symmetry mechanism improves upon by working in clean systems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents the coexisting twofold Majorana zero modes in a related phase and invokes symmetry protection, a result the paper connects to its layer-resolved picture."},{"cited_title":"Shimizu, A","cited_arxiv_id":null,"evidence_quote":"Shows the same nonsymmorphic band structure supports chiral Majorana edge states and Bogoliubov Fermi surfaces, grounding the present model in a known topological phase diagram."},{"cited_title":"Kallin and J","cited_arxiv_id":null,"evidence_quote":"Supplies the bulk-vortex correspondence $n_M = C \\bmod 2$ for chiral topological superconductors, which the layer-resolved Majorana zero-mode analysis uses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies FeSe's lattice symmetries as matching the bilayer setup, giving a concrete candidate material for experimental realization."}],"review_version":1}