{"id":"32b59e74-edeb-4fd1-85fe-b115ab4c2419","arxiv_id":"2506.21482","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A d_xy+is superconductor formed in a d/s bilayer hosts frustrated, topologically trivial edge currents whose apparent charge non-conservation is resolved by bulk superflow and detectable as magnetic flux patterns.","lead":"This paper predicts that stacking an s-wave superconductor on top of a d-wave superconductor can generate persistent electric currents along the edges of the stack, even though the combined superconductor has no topological protection. The currents appear to flow into and out of corners, but the paper shows with self-consistent calculations how charge conservation is restored by currents through the bulk, and predicts magnetic fields strong enough for SQUID detection.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central prediction presupposes that the d±is phase is the equilibrium state; the paper assumes C>0 in Eq. (1) but never derives C from the BdG model or compares the free energies of competing relative phases.","rationale":"The paper's central claim is that a d/s bilayer spontaneously forms a d_xy±is superconductor with large, frustrated edge currents despite trivial topology. Two calculations support the current phenomenology: the GL analysis of the mixed-gradient term and fully self-consistent BdG solutions showing localized currents at s-island edges. Both calculations, however, presuppose that the d±is phase is the thermodynamically stable state. The GL free energy stabilizes this phase only if the coefficient C in Eq. (1) is positive; the paper never derives C from the microscopic BdG Hamiltonian. The BdG solutions reported in Sec. III are self-consistent stationary points, but no comparison is shown with the free energy of the real relative-phase states φ=0 or π. If the real-phase state has lower free energy for the chosen parameters, the system would not spontaneously break TRS and the predicted edge currents would vanish. The reader's identified weakest assumption, the uncomputed mixed-gradient coefficient γ_v, is related but less fundamental: γ_v controls the magnitude and direction of currents within the d±is phase, whereas the existence of the phase itself is controlled by C. I therefore recommend CONDITIONAL acceptance: the theoretical analysis of the d±is state is coherent and the BdG currents are credible once that state is granted, but the central claim as stated in the abstract requires demonstrating that d±is is the equilibrium phase, not merely a converged self-consistent solution.","tokens_in":17982,"tokens_out":38352,"duration_ms":456668,"concrete_test":"For the BdG parameters used in Fig. 2 (long strip) and one island geometry of Fig. 4, initialize the gap equations with relative phases φ=0, π, +π/2, and -π/2, iterate each to convergence, and evaluate the mean-field grand potential of each converged solution at the same μ_s, μ_d, and T. If the φ=±π/2 states do not have the lowest free energy, the d±is phase is not the equilibrium phase and the central edge-current claim does not apply as stated. Equivalently, compute C by expanding the microscopic free energy in powers of |s| and |d| to verify that C>0 in the parameter regime.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The abstract claims the d/s bilayer 'spontaneously breaks time reversal symmetry' and generates edge currents. In the GL treatment this requires the quartic Josephson coefficient C in Eq. (1) to be positive so that f(φ)=E0+2C|s|^2|d|^2 cos 2φ is minimized at φ=±π/2 (Eq. (2)). The paper assumes C>0 implicitly but gives no microscopic derivation of C and no explicit free-energy comparison among the φ=0, π, and ±π/2 solutions of the BdG model in Sec. III. A converged self-consistent d+is solution is a stationary point; it is not automatically the global minimum. If for the parameters of Figs. 2 and 4 the real-phase solution has lower free energy, TRS is not broken in equilibrium and the predicted frustrated edge currents would not occur. The γ_v assumption flagged by the reader is secondary: it affects the magnitude and sign of currents within the d±is phase, whereas the existence of the phase itself is decided by C. The BdG currents are legitimate evidence once the d±is phase is granted, but the paper's central claim includes the spontaneous stabilization of that phase.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies heterostructures formed by an s-wave superconductor island on a d-wave superconducting substrate, focusing on the resulting d±is state. The authors argue from a Ginzburg-Landau free energy that this state spontaneously breaks time-reversal symmetry and, despite being topologically trivial, supports large edge currents for d_xy+is order along certain edge orientations, while d_x2−y2+is does not. They support this with fully self-consistent BdG lattice calculations in strip and island geometries, showing that edge currents are frustrated at corners and resolve through bulk superflow into vortex-like patterns, and they estimate that the resulting magnetic fields are detectable by scanning SQUID for BSCCO/iron-based superconductor parameters.","tokens_in":18275,"tokens_out":4964,"duration_ms":67588,"significance":"If the central claim holds, the paper establishes a conceptually important result: time-reversal-symmetry-breaking superconductors can host sizable edge currents without any nontrivial bulk topology, and the corner frustration provides a concrete experimental fingerprint. The BdG calculations are a strength: they are fully self-consistent, explicitly enforce current conservation, and the contrast between d_xy+is and d_x2−y2+is is supported by a spectral-function argument. The experimental estimates for SQUID-visible magnetic fields (0.1–0.6 μT at 30 nm height) make the prediction falsifiable. The main gaps are that the stabilization of the d±is phase itself is assumed rather than demonstrated from the microscopic model, and the mixed-gradient coupling γ_v that drives the GL edge-current mechanism is not derived from the BdG theory.","major_comments":[{"comment":"The paper assumes C>0 in the Josephson free energy f(φ)=E0+2C|s|^2|d|^2 cos 2φ, which selects the time-reversal-symmetry-breaking minima at φ=±π/2. However, C is never computed from the microscopic BdG model, and the self-consistent dxy+is solutions in Figs. 2 and 4 are stationary points, not necessarily global minima. If C<0 for those parameters, the equilibrium state would be a real relative phase (φ=0 or π) with no spontaneous TRS breaking, and the predicted edge currents would not occur. Please compute the BdG free energies of the φ=0, π, and ±π/2 solutions (or equivalently derive C from the microscopic parameters g, V_s, V_d, etc.) and demonstrate that the complex-phase solution is the global minimum for the parameters used in Figs. 2, 4, and 7.","section":"Sec. I, Eq. (1)–(2); Sec. III, Eq. (11); Figs. 2 and 4"},{"comment":"The GL edge-current expressions in Eq. (4) and the GL vortex patterns in Fig. 6 depend on the mixed-gradient coefficient γ_v, which is assumed positive but never derived from the BdG model. The BdG simulations independently show edge currents, so the central phenomenon does not collapse without γ_v, but the quantitative GL estimates and the interpretation that 'these supercurrents originate from mixed gradient terms' (Fig. 1 caption) are contingent on the sign and magnitude of a coupling that is not computed. Please estimate γ_v from the microscopic model or otherwise establish its sign and magnitude for the parameters of Sec. V, or clearly state that the GL analysis is schematic while the microscopic calculation provides the quantitative prediction.","section":"Sec. II, Eq. (4); Sec. IV A; Sec. V"},{"comment":"The projection of the GL current onto a solenoidal field, j=∇×h with h obtained from −∇^2 h=∇×j, is introduced to enforce current conservation after discretization. While this is a reasonable numerical procedure, the paper does not quantify the divergence of the raw current before projection or verify that the projected current agrees with the raw current away from discontinuities. If the projection substantially modifies the current pattern, the 'vortex-like' structures in Fig. 6 could be artifacts of the projection rather than physical. Please provide a comparison of raw and projected currents, or an estimate of ∇·j before projection, to validate the method.","section":"Sec. IV C, Eqs. (27)–(28)"}],"minor_comments":[{"comment":"There are several typographical errors: 'spontea-neously' in the Introduction, 'Threfore' after Eq. (4), 'ans−wave' in the Fig. 1 caption, and 'µd = µd' in the Fig. 7 caption (presumably µ_s = µ_d = −1.2 t_d).","section":"Throughout"},{"comment":"The spectral-function argument for the absence of edge currents in d_x2−y2+is is qualitative: the statement that occupied modes appear symmetrically for positive and negative k is not quantified. A short quantitative statement (e.g., the integrated contribution to Eq. (16) for the two order parameters) would make the explanation more convincing.","section":"Sec. III B, Fig. 3"},{"comment":"The net supercurrent I_net is defined as a sum over half the strip width, but the strip contains two step edges. Please specify whether this quantity includes only one edge and whether the two edges carry equal and opposite currents, as implied by current conservation.","section":"Sec. V"},{"comment":"The estimate treats the edge current as an infinitely thin line current and ignores the magnetic back-action on the superconducting order parameters. For the small fields predicted this is likely negligible, but a sentence explicitly justifying this approximation would be useful.","section":"Sec. V, Eq. (29)"}],"recommendation":"major_revision","confidential_remarks":"The paper is well-written and the BdG calculations are careful, but the phase-stability issue is the main obstacle: the central claim that the d±is state is spontaneously stabilized requires either a microscopic derivation of C>0 or a direct free-energy comparison among relative phases. If the authors can supply this (it should be straightforward within their existing BdG framework), the paper would be suitable for publication. The γ_v derivation and the validation of the GL current projection are secondary but should also be addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this paper. It predicts something genuinely new: a d_xy+is superconductor realized in a d/s bilayer supports large edge currents that flow in opposite directions on adjacent edges, look frustrated at corners, and are resolved by bulk vorticity. All of that happens despite the system being topologically trivial. The contrast with d_x2-y2+is, which shows no such currents, is explained cleanly by symmetry and by the spectral-function argument. The authors back the claim with two independent methods: GL theory with a mixed-gradient coupling, and fully self-consistent BdG on lattice clusters with explicit current conservation. The BdG and GL results agree qualitatively, and the SQUID estimates (0.1–0.6 µT at 30 nm) are in a detectable range. This is careful, physical work.\n\nMy main reservation is about the stability of the d±is phase. The GL analysis assumes C>0 in the quartic Josephson term, and nowhere do the authors derive C from the microscopic model or compare the free energies of the φ=0, π, ±π/2 solutions. The BdG self-consistent solutions converge to d+is, but a converged stationary point is not obviously the global minimum. If for the parameters used in Figs. 2 and 4 the real-phase solution has lower free energy, the edge currents would not occur in equilibrium. The prior literature (Ref. [15]) argues d+is is a robust phase in d/s bilayers, so my prior is that the phase is fine, but the authors should have checked. This is a genuine, referee-addressable gap, and it is the only load-bearing one.\n\nThe assumed value of γ_v (the mixed gradient coefficient) is a secondary concern. The BdG currents do not rely on it, so the existence of the phenomenon is secure; only the quantitative GL estimates and the interpretation via Eq. (4) depend on it. It would be good to compute γ_v from the BdG model, but I wouldn't call it fatal.\n\nOther minor points: the periodic boundary conditions for the d-wave layer avoid pair-breaking edge states, and real samples may have additional extended s-wave order, but the authors explicitly state this and design the s-wave island to isolate the effect. That is acceptable.\n\nOverall, the central claim holds up, with a missing free-energy comparison that should be requested rather than treated as a dealbreaker. The paper deserves a serious referee; I'd send it out.","headline":"Solid, interesting prediction of non-chiral edge currents in a topologically trivial d+is bilayer; needs a free-energy check to confirm the d±is phase is the ground state.","tokens_in":18805,"tokens_out":2288,"would_cite":true,"duration_ms":25858,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82D55"],"pacs":["74.20.-z","74.25.-q","74.25.Ha"],"model":"deepseek-v4-flash","headline":"A d/s bilayer can carry large non-chiral edge currents without any bulk topology, with corner frustration resolved by bulk superflow.","keywords":["d-wave superconductor","s-wave superconductor","time-reversal symmetry breaking","edge currents","Ginzburg-Landau theory","Bogoliubov-de Gennes","scanning SQUID","d+is superconductor"],"falsifier":"Map the stray field of a single square s-wave island on a d_xy substrate with a scanning SQUID: the paper predicts a quadrupolar field pattern with opposite currents on adjacent edges and bulk return flow, at peak fields of 0.1–0.6 µT for sensor heights near 30 nm. A null result at that sensitivity, or an equally strong signal when the same island is placed on a d_x2-y2 substrate, where the paper predicts no edge currents in this geometry, would falsify the central claim.","tokens_in":17717,"feed_emoji":"🧲","tokens_out":9147,"duration_ms":90856,"temperature":0.7,"pith_summary":"This paper argues that placing an s-wave superconducting island on a d-wave superconducting substrate can create a d_xy + i s state whose two superconducting components are locked at a quarter-cycle phase difference, spontaneously breaking time-reversal symmetry. Although this state has no bulk topology—no Chern number and no protected chiral edge modes—it supports large persistent supercurrents along edges aligned with the nodal directions of the d-wave order. Adjacent perpendicular edges carry currents in opposite directions, so the pattern appears frustrated at corners, as if charge were being created and destroyed. Using fully self-consistent Bogoliubov–de Gennes and Ginzburg–Landau calculations, the paper shows that the true ground state conserves current by sending superflow through the bulk, producing vortex-like magnetic flux patterns. For a Bi-2212 substrate with an iron-based s-wave island, the estimated stray fields are strong enough for state-of-the-art scanning SQUID magnetometry.","feed_headline":"Edge currents appear in a topologically trivial bilayer","feed_subtitle":"An s-wave island on a d-wave flake drives corner-frustrated supercurrents visible to SQUID magnetometry.","key_machinery":"The central object is the mixed gradient coupling $\\gamma_v$ in the Ginzburg-Landau free energy, the term $\\gamma_v[(\\Pi_x s)^*(\\Pi_y d)+(\\Pi_y s)^*(\\Pi_x d)+\\mathrm{c.c.}]$. This term makes currents flow perpendicular to gradients of the other order parameter, so it converts the local suppression of the s order at an island edge into tangential supercurrent. On the microscopic side the equivalent machinery is the self-consistent BdG gap equation together with the bond-current operator; in a translationally invariant strip the current can be expressed through the spectral function $A_k(y,\\omega)$ of the BdG Hamiltonian, and the sign of the $k$-integral is controlled by whether the occupied spectrum is asymmetric in $k$. In $d_{xy}+is$ the occupied states contribute predominantly at one sign of $k$, producing a net current; in $d_{x^2-y^2}+is$ the occupied states are symmetric, and the odd $\\sin k$ factor in the current expression cancels the integral.","core_discovery":"The paper's central claim is that a d_xy ± i s superconductor—realized when an s-wave island sits on a d_xy substrate—produces large edge currents along (100) and (010) directions despite being topologically trivial, and that those currents are not chiral but frustrated. The mechanism is a mixed gradient term in the Ginzburg-Landau free energy that couples spatial gradients of the s- and d-order parameters crosswise: an x-gradient of one order drives a y-directed current in the other, and vice versa. Near an edge, where the s-order parameter is suppressed, this crosswise coupling yields opposite-directed supercurrents on adjacent perpendicular edges, so the currents seem to emanate from two corners and sink into the other two. Fully self-consistent BdG and GL solutions show that charge conservation is restored by bulk superflow, with the return currents forming geometry-dependent patterns: quadrupolar flux for a square island, dipolar for a triangle, unipolar for a rectangle, and a vortex-antivortex-like texture for an array of islands. The microscopic calculation also explains the orientation dependence: in a strip geometry, d_xy + i s has an asymmetric occupied spectral function in momentum, so the current integral does not cancel, whereas d_x2-y2 + i s has a symmetric spectrum and produces no edge current in this setup.","pith_inferences":["A natural test the paper does not run is to pattern two adjacent s-wave islands of different orientations on the same d-wave substrate and use SQUID maps of the bulk return flow to infer the sign of the mixed-gradient coupling; the current direction around each corner encodes whether the phase is d + i s or d − i s.","If a microscopic estimate of the mixed-gradient coupling were computed, the Ginzburg-Landau current magnitudes could be made fully quantitative; until then the microscopic BdG results, not the GL estimate, are the firmer quantitative anchor for the edge current amplitude.","The island-array idea points toward a tunable platform for frustration engineering: by choosing island shapes and separations, one could deliberately design unquantized vortex-antivortex-like current patterns that mimic classical or quantum spin models.","Stacking an s-wave island with topological surface states, as in some iron-based superconductors, would superpose these bulk-superflow currents with Majorana physics; whether the two effects interfere or simply coexist is left open by the paper."],"forward_implications":["If the central claim is right, a topologically trivial superconductor can produce edge currents of comparable character to chiral topological ones, so edge currents alone cannot be read as evidence of bulk topology in d/s heterostructures.","For a BSCCO substrate with an iron-based s-wave island, the predicted edge currents produce peak magnetic fields of order 0.1–0.6 µT at a 30 nm sensor height, within reach of scanning SQUID microscopes; a null or much weaker signal would rule out the d_xy + i s phase in that geometry.","The apparent corner frustration is resolved by bulk superflow, so magnetometry images of finite islands should show vortex-like flux patterns rather than purely edge-localized fields, with the pattern's character encoding the island's shape.","Because the current reverses with edge orientation, the same bilayer geometry can serve as a directional diagnostic of the underlying d-wave component: d_xy + i s edges carry currents in this setup, while d_x2-y2 + i s edges do not."],"supporting_citations":[{"why":"Supplies the spectral-function expression for edge currents and the prior context of edge currents as a topology probe that this paper contrasts with.","marker":"[12]"},{"why":"Provides the method for iteratively solving multi-component Ginzburg-Landau equations on a lattice and the form of the mixed-gradient free energy.","marker":"[20,21]"},{"why":"The antecedent twisted double-layer cuprate proposal that motivates time-reversal-breaking d+id superconductivity and the broader bilayer framework used here.","marker":"[7]"},{"why":"Proposes the d/s bilayer as a superconducting qubit platform and underlines the bulk d+is order parameter to which the edge currents are tied.","marker":"[15]"},{"why":"Provides the material parameters, gap values, and critical temperatures for iron-based superconductors used in the experimental estimates.","marker":"[22]"},{"why":"Supplies the line-current magnetic field model and scanning SQUID sensitivity framework used to estimate detectability.","marker":"[25]"},{"why":"Establishes the state-of-the-art SQUID sensor heights and sensitivities used in the detectability conclusions.","marker":"[30]"}],"fun_headline_variants":["Frustrated edge currents appear in trivial s-wave/d-wave bilayers","Edge currents emerge despite trivial topology in d+is superconductor","Corner-oriented supercurrents in a topologically trivial bilayer","s-wave island on d-wave flake yields frustrated edge supercurrents"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The quantitative Ginzburg-Landau current formula assumes a nonzero mixed-gradient coupling between the two superconducting order parameters, and the paper takes it positive without computing its value from the microscopic model; the BdG calculation independently produces the edge currents, so the phenomenon does not rest on that constant, but the GL estimates do.","fun_headline_variants_meta":{"raw":{"variants":["Frustrated edge currents appear in trivial s-wave/d-wave bilayers","Edge currents emerge despite trivial topology in d+is superconductor","Corner-oriented supercurrents in a topologically trivial bilayer","s-wave island on d-wave flake yields frustrated edge supercurrents"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000708,"raw_usage":{"total_tokens":3220,"prompt_tokens":1006,"completion_tokens":2214,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":622,"completion_tokens_details":{"reasoning_tokens":2138}},"tokens_in":622,"tokens_out":2214,"duration_ms":20560,"temperature":1.0,"reasoning_tokens":2138,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:25:39.848088+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Map the stray field of a single square s-wave island on a d_xy substrate with a scanning SQUID: the paper predicts a quadrupolar field pattern with opposite currents on adjacent edges and bulk return flow, at peak fields of 0.1–0.6 µT for sensor heights near 30 nm. A null result at that sensitivity, or an equally strong signal when the same island is placed on a d_x2-y2 substrate, where the paper predicts no edge currents in this geometry, would falsify the central claim.","supporting_citations":[{"cited_title":"Pathak, O","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral-function expression for edge currents and the prior context of edge currents as a topology probe that this paper contrasts with."},{"cited_title":"Patel, V","cited_arxiv_id":null,"evidence_quote":"Proposes the d/s bilayer as a superconducting qubit platform and underlines the bulk d+is order parameter to which the edge currents are tied."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the line-current magnetic field model and scanning SQUID sensitivity framework used to estimate detectability."},{"cited_title":"Persky, I","cited_arxiv_id":null,"evidence_quote":"Establishes the state-of-the-art SQUID sensor heights and sensitivities used in the detectability conclusions."}],"review_version":1}