{"id":"19fe246d-e84c-4491-af36-1beb50aa2276","arxiv_id":"2608.09751","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Nonreciprocal conductance between adjacent terminals maps the angular structure of unconventional pairing, offering a symmetry-selective probe for hidden s+id and s+p states.","lead":"This paper predicts that a measurable difference in conductance when source and detector terminals are swapped in a multi-terminal superconductor device directly reveals the symmetry of the underlying Cooper pairing, even when the exotic pairing component is hidden beneath a dominant conventional gap. The result offers a transport-based route to identify pairing symmetries that are invisible to standard spectroscopic probes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The exact π/(2l) angular mapping in Eqs. (8)–(10) relies on SO(2)-symmetric leads assumed in SM Eq. (S10); faceted leads or anisotropic band structure can distort the pattern, so the one-to-one encoding is not model-independent.","rationale":"The paper's strongest claim is a quantitative, model-independent correspondence between the angular dependence of nonreciprocal conductance and the momentum-space harmonic of hidden unconventional pairing. The derivation in SM Eq. (S10) obtains this correspondence by rotating the leads through an SO(2) transformation, which requires the lead broadening matrices to be rotationally symmetric. This assumption is explicitly stated in the SM ('we assume that the leads are rotationally symmetric'), and it is the load-bearing link between the measurable interface angle φ and the momentum-space angle φ_k. The robustness checks in SM §S5 are real evidence: they show that varying the lead-coupling strength γ and adding random interface roughness up to W/µ<1 leave the angular pattern intact, and the AFe2Se2 calculation with elliptical Fermi surfaces provides partial support for anisotropic band structure. However, those checks do not isolate the SO(2) lead-shape assumption, which is the precise condition under which Eqs. (8)–(10) are derived. If a wide, flat lead with a fixed orientation is used instead of an SO(2)-symmetric lead, the pre-factor D_{σ,σ'} in SM Eq. (S14) can acquire a φ-dependence, injecting a shape-dependent envelope into ∆G(φ) and shifting the zero crossings. That would weaken the claim from 'directly encodes' to 'qualitatively selects the pairing symmetry'. The reader's verdict of CONDITIONAL is appropriately calibrated: the internal derivation is consistent and the numerics support the ideal case, but the model-independence is not yet demonstrated. My proposed test would either confirm the universal rotation for anisotropic dispersions with realistic leads or reveal the effective-angle correction needed, thereby settling whether the conditional verdict should be upgraded or maintained.","tokens_in":24147,"tokens_out":10743,"duration_ms":97341,"concrete_test":"Recompute the minimal-model result of Fig. 1(e) (s+id, ψ_un=icos(2φ_k)) on a lattice with an elliptical dispersion ϵ(k)=B_x k_x^2 + B_y k_y^2 (B_x≠B_y) using a wide, flat lead attached along a straight edge (as in Fig. S3), and extract the angular positions of the zero crossings of ∆G_c(φ) for several eccentricities. If the zero crossings lie at φ = π/4 + nπ/2 (i.e., the zeros of Im[ψ_un(φ+π/4)]) to within the numerical resolution for all eccentricities, the SO(2) concern is resolved; if they shift with B_x/B_y, the claimed universal π/(2l) rotation is not model-independent and Eq. (8) requires an effective-angle correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim—∆G_c(φ) ∝ Im[ψ_un(φ+π/(2l))] and ∆G_s_n(φ) ∝ d_n(φ+π/(2l)]—is proven only under the assumption, stated in SM §S2 after Eq. (S10), that the leads are rotationally symmetric, so that the broadening matrices for a lead at normal angle φ are obtained by an SO(2) rotation of a reference lead. Real multiterminal devices employ leads with finite width and faceted interfaces; the lead self-energy is then not SO(2)-invariant, and Γ_j(φ) ≠ U_R(φ) Γ_1 U_R^†(φ). The same issue arises for anisotropic Fermi surfaces, where rotating the crystal changes both the pairing orientation and the normal-state Green's function, so the momentum-space angle φ_k and the real-space interface angle φ are not simply related by a rotation. The robustness checks in SM §S5 vary only the coupling strength γ and a random edge roughness W; they do not remove the SO(2) assumption on the lead geometry, and for W/µ=5 the correspondence visibly degrades. Since Eqs. (8)–(10) are the basis for claiming that the angular pattern 'directly encodes' the pairing harmonic, a failure of rotational symmetry would reduce the method to a qualitative symmetry probe rather than a quantitative fingerprint.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a symmetry-resolved transport probe for unconventional superconductivity. In a multiterminal superconductor device, the difference in charge or spin conductance obtained upon exchanging source and detector terminals (the nonreciprocal conductance) is studied as a function of the interface orientation angle φ. The authors derive, using a Landauer–Büttiker formalism and first-order perturbation in the unconventional pairing strength, that the nonreciprocal charge conductance for time-reversal-breaking singlet pairing obeys ∆G_c(φ) ∝ Im[ψ_un(φ + π/(2l))] (Eq. 8), while the nonreciprocal spin conductance for triplet pairing obeys ∆G_s_n(φ) ∝ d_n(φ + π/(2l)) (Eq. 10). The conventional s-wave background does not contribute because it preserves the relevant symmetries. The correspondence is verified numerically in minimal models and in models for AFe2Se2 and noncentrosymmetric superconductors, and experimental protocols using multiterminal devices and multi-tip STM are outlined. The central claim is that the angular dependence of the nonreciprocal conductance directly encodes the momentum-space structure of the hidden unconventional pairing component, even when that component is obscured by a dominant s-wave gap.","tokens_in":24407,"tokens_out":7508,"duration_ms":67503,"significance":"If the correspondence holds, the proposal provides a qualitatively new probe for identifying pairing symmetries that are masked by an s-wave background, with the crucial advantage of being sensitive to symmetry rather than gap magnitude. The paper contains a detailed derivation (SM S2), numerical confirmation in both minimal and material-specific models, a symmetry-based table linking pairing channels to transport channels, and explicit experimental protocols. It also carefully distinguishes pairing-induced nonreciprocity from edge-state and vortex contributions. The main caveats are that the exact angular mapping is derived under a restrictive rotational-symmetry assumption for the leads and under a perturbative expansion in the unconventional pairing strength, while the material-model simulations are self-consistent checks within the same Hamiltonian framework.","major_comments":[{"comment":"The derivation of the central angular correspondence, Eqs. (8)–(10) of the main text, explicitly assumes in SM S2 that the leads are rotationally symmetric, so that the broadening matrix for a lead at normal angle φ is obtained from a reference lead by an SO(2) rotation. This assumption is violated for finite-width or faceted leads and for anisotropic Fermi surfaces, where the interface angle φ and the Fermi-surface momentum angle φ_k are not related by a rigid rotation. The robustness checks in SM §S5 vary the lead coupling γ and a random edge roughness W, but they do not relax the rotational symmetry of the lead self-energy, and for W/µ=5 the authors state that the angular correspondence becomes increasingly indistinct. Because the exact π/(2l) rotation is the quantitative fingerprint claimed in the abstract and conclusion, the manuscript should either prove stability of the mapping under non-SO(2) lead geometries and anisotropic dispersions (e.g., by showing the response is a convolution with a narrow kernel) or explicitly weaken the claim from exact angular encoding to qualitative symmetry-selective behavior.","section":"SM S2, Eq. (S10)"},{"comment":"The formulas in Eqs. (8)–(10) are derived in SM S2 by a first-order expansion in the unconventional pairing amplitude λ. The paper nevertheless claims in SM S6 that for a topologically nontrivial s+ spinful chiral p-wave state with λ=1 (which exceeds the s-wave gap ∆_s=0.2), the angular correspondence “remains valid,” even though the perturbative expansion is not controlled in that regime. No nonperturbative derivation is provided. The authors should either supply a nonperturbative symmetry argument that yields the same angular mapping, or explicitly restrict the validity claims of Eqs. (8)–(10) to the regime where first-order perturbation theory is justified.","section":"Main text Eqs. (8)–(10); SM §S6"}],"minor_comments":[{"comment":"The phrase “dominants-wave component” should read “dominant s-wave component”.","section":"Abstract"},{"comment":"In the caption, “s+id-wave (d-e) consider” appears to be a typo; the intended reference to panels (e)–(g) should be clarified.","section":"Fig. 1 caption"},{"comment":"References [24] and [47] are duplicate entries for Wakatsuki and Nagaosa, Phys. Rev. Lett. 121, 026601 (2018); one duplicate should be removed.","section":"References"},{"comment":"The sentence “Even if this symmetry is moderately broken, it would only introduce minor errors in the angular resolution” is an assertion without supporting analysis; a quantitative estimate of the error in the angular pattern for a concrete non-SO(2) lead model would strengthen this statement.","section":"SM S2, after Eq. (S10)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript makes a potentially important proposal, and the derivation in the SM is careful within its stated assumptions. The main issue is whether the headline claim of exact angular encoding survives when the SO(2) lead-symmetry assumption is relaxed; this is a load-bearing point that the robustness section does not fully address. The paper would be suitable for publication after a revision that either generalizes the derivation or tempers the claim to qualitative symmetry selectivity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth a serious referee. The core idea is new and clean: in a multiterminal device, the difference between reciprocal conductances is symmetry-forbidden for pure s-wave and first-order in the unconventional pairing, so the angular pattern of ΔG directly tracks the harmonic structure of the pairing. That is a nice way to isolate hidden d- or p-wave components under a dominant s-wave gap.\n\nWhat the paper does well: the SM derivation is transparent, the numerics on the minimal model and on AFe2Se2 and the Rashba NCS model confirm the predicted angular patterns, and the spin-resolved channel decomposition (spin-flip vs equal-spin) is physically clear and backed by Table I. The robustness checks against lead coupling and moderate edge roughness are a plus, and the comparison to edge-state nonreciprocity is sensible.\n\nThe main soft spot is that the quantitative mapping ΔG(φ) ∝ pairing harmonic at φ + π/(2l) is derived under the assumption that the leads are rotationally symmetric (SM S2), so the lead broadening matrices are related by SO(2) rotations. That is an idealization. Real faceted interfaces or strongly anisotropic Fermi surfaces can break the simple relation between real-space angle and momentum angle. The paper's own Fig. S5 shows the pattern degrades for roughness W/µ = 5. The material-model numerics with elliptical Fermi surfaces are encouraging, but they don't systematically test how much distortion a non-SO(2) lead geometry or strongly warped FS causes. The claim that the correspondence is 'independent of model details' is therefore stronger than what is demonstrated; for a quantitative fingerprint, the paper should either prove or numerically establish that the mapping survives generic lead shapes and FS anisotropy.\n\nNo code or data is provided, which for a theory paper is not fatal but would help reproducibility. The self-consistency of the numerics (same Hamiltonian for pairing and transport) is fine here because the point is to show the mapping works, not to fit experimental data.\n\nRecommendation: send to peer review. The method is promising and the derivation is careful, but the authors should be pushed to test the angular mapping under non-rotationally-symmetric leads and to temper the model-independence claim. It deserves referee time.","headline":"A genuinely new symmetry-based transport probe for hidden pairing components, with a careful derivation, but the precise angular fingerprint is proven only under idealized lead symmetry and should be tempered to a robustness-backed qualitative probe.","tokens_in":24947,"tokens_out":2475,"would_cite":true,"duration_ms":22582,"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":"Nonreciprocal conductance in a multiterminal superconductor maps the momentum-space angular structure of unconventional pairing components that are hidden beneath a dominant s-wave gap.","keywords":["nonreciprocal transport","unconventional superconductivity","pairing symmetry","multiterminal conductance","time-reversal symmetry breaking","spin-triplet pairing","s+id pairing","Landauer-Büttiker formalism"],"falsifier":"Numerically simulate the same multiterminal device with faceted or otherwise non-rotationally-symmetric lead interfaces—for example, with a lead broadening matrix that is not a rotated copy of a reference lead—and compare the computed $\\Delta G_c(\\varphi)$ and $\\Delta G_{s_n}(\\varphi)$ to Eqs. (8) and (10); if the angular pattern deviates beyond small angular-resolution errors, the central correspondence is false.","tokens_in":23942,"feed_emoji":"🔄","tokens_out":8866,"duration_ms":77724,"temperature":0.7,"pith_summary":"This paper claims that a measurable transport asymmetry—the difference in conductance between adjacent terminals of a multiterminal superconductor when the source and detector are exchanged—directly encodes the momentum-space angular structure of unconventional pairing components that are usually hidden under a dominant s-wave gap. The central statement is a pair of proportionalities: for time-reversal-breaking singlet pairing with orbital harmonic l, the nonreciprocal charge conductance obeys $\\Delta G_c(\\varphi)\\propto \\mathrm{Im}[\\psi_{\\mathrm{un}}(\\varphi+\\pi/(2l))]$, and for spin-triplet pairing the nonreciprocal spin conductance obeys $\\Delta G_{s_n}(\\varphi)\\propto d_n(\\varphi+\\pi/(2l))$. Because a pure s-wave component is perfectly reciprocal, it contributes no intrinsic nonreciprocal signal, so the measurement acts as a symmetry filter that removes the s-wave background. The authors verify the correspondence in minimal BCS models, in two-orbital models of iron-based superconductors with $s+id$ pairing, and in Rashba noncentrosymmetric superconductors with $s+$ helical $p$ pairing, and they propose multiterminal and multi-tip STM protocols for measuring it.","feed_headline":"Swapping probes reveals a superconductor's hidden pairing","feed_subtitle":"A measured angle-dependent asymmetry encodes the momentum structure of the unconventional gap, even beneath a dominant s-wave.","key_machinery":"The load-bearing object is the spin-resolved nonreciprocal conductance between neighboring terminals, $\\Delta G_{\\sigma,\\sigma'}(\\varphi)=G_{j+1\\sigma,j\\sigma'}-G_{j\\sigma,j+1\\sigma'}$, computed from normal and Andreev transmission coefficients in a multiterminal Landauer–Büttiker formalism. The argument runs on two symmetry links: time-reversal symmetry $T$ enforces reciprocity in the spin-flip scattering channel, and the combined spin-time symmetry $s_{n_\\perp}T$ enforces reciprocity in the equal-spin channel; a pure $s$-wave superconductor preserves both, making all $\\Delta G_{\\sigma,\\sigma'}$ vanish. An unconventional pairing potential $V_{\\mathrm{un}}$ that depends only on the Fermi-surface angle $\\varphi_k$ breaks one or both of these symmetries, and because the leads are assumed rotationally symmetric, the $l$-th angular harmonic of $V_{\\mathrm{un}}$ enters the conductance as $V_{\\mathrm{un}}^l(\\varphi+\\pi/(2l))$, producing the rotated proportionalities in Eqs. (8)–(10). In other words, the orbital harmonic $l$ of the hidden pairing component is transferred to a measurable $l$-fold angular pattern of the nonreciprocal conductance, with the rotation offset $\\pi/(2l)$ fixed by the harmonic order.","core_discovery":"At the center of the paper is the nonreciprocal conductance $\\Delta G(\\varphi)$ between two adjacent terminals, defined by $\\Delta G=G_{j+1,j}-G_{j,j+1}$, obtained within a Landauer–Büttiker treatment that includes both normal and Andreev transmission. The central discovery is a symmetry-enforced correspondence: unconventional pairing components selectively break either time-reversal symmetry $T$ or spin-rotation symmetry, and the resulting nonreciprocity carries the angular profile of the pairing amplitude. For time-reversal-breaking spin-singlet pairing with even orbital angular momentum $l$, rotational symmetry constrains $\\Delta G_c(\\varphi)\\propto \\mathrm{Im}[\\psi_{\\mathrm{un}}(\\varphi+\\pi/(2l))]$; for spin-triplet pairing with odd $l$, the spin-resolved channels obey $\\Delta G_{\\uparrow_n,\\uparrow_n}(\\varphi)\\propto \\mathrm{Re}[d_n(\\varphi+\\pi/(2l))]$ and $\\Delta G_{\\downarrow_n,\\uparrow_n}(\\varphi)\\propto \\mathrm{Im}[d_n(\\varphi+\\pi/(2l))]$, so the total spin conductance satisfies $\\Delta G_{s_n}(\\varphi)\\propto d_n(\\varphi+\\pi/(2l))$. The paper further decomposes these signals into spin-flip and equal-spin scattering channels, showing that $T$ governs the former and the combined spin-time symmetry $s_{n_\\perp}T$ governs the latter, and that a pure $s$-wave state preserves both symmetries and therefore produces no nonreciprocity. Numerical simulations on AFe$_2$Se$_2$ and on noncentrosymmetric superconductors show that the predicted angular patterns persist in the $s$-wave-dominated regime, where the unconventional gap structure is invisible in the quasiparticle spectrum.","pith_inferences":["The same symmetry-filter logic could be turned into a materials-screening tool: scanning the angle-resolved nonreciprocal conductance over many devices would allow one to classify superconductors by the harmonic content of their pairing without prior knowledge of the gap magnitude.","Because the derivation only needs the pairing potential's angular harmonic, the protocol should extend to higher harmonics and to mixed-$l$ states by Fourier-transforming $\\Delta G(\\varphi)$, a step the paper leaves mostly implicit.","A controlled validation experiment could be built in a hybrid system where a normal multiterminal device is proximity-coupled to a superconductor with artificially engineered pairing, allowing the predicted $\\pi/(2l)$ rotation to be tested before applying the method to unknown materials."],"forward_implications":["A dominant $s$-wave component no longer masks unconventional pairing: because the $s$-wave channel is reciprocal, the measured nonreciprocal angular pattern is generated only by the symmetry-breaking component, and its orbital harmonic $l$ can be read off from the periodicity.","Charge nonreciprocity in a multiterminal device provides a direct probe of time-reversal-breaking singlet pairing, while spin nonreciprocity probes triplet pairing; when both are present, the two signals separate the channels.","For spin-triplet pairing, the equal-spin and spin-flip contributions carry $\\mathrm{Re}(d_n)$ and $\\mathrm{Im}(d_n)$ separately, so the measurement is phase-sensitive and can distinguish, for instance, a helical from a chiral $p$-wave state.","Materials in which the unconventional component is too small to alter the density of states—iron-based, kagome, moiré, and noncentrosymmetric superconductors—become accessible to pairing-symmetry identification through transport alone."],"supporting_citations":[{"why":"Supplies the generalized reciprocity principle whose violation defines nonreciprocal transport in the paper.","marker":"[40]"},{"why":"Provides the multiterminal conductance framework and terminal-exchange notation used to define $\\Delta G$.","marker":"[43]"},{"why":"Basis of the Landauer transmission formalism through which normal and Andreev conductances are computed.","marker":"[73]"},{"why":"Extends the Landauer formula to interacting systems, underlying the spin-resolved current expression used here.","marker":"[74]"},{"why":"Gives the two-orbital iron-based model with competing $s$- and $d$-wave channels that produces the $s+id$ state used for numerical demonstration.","marker":"[29]"},{"why":"Provides the noncentrosymmetric superconductor model in which Rashba spin-orbit coupling locks the $d$-vector to a helical $p$-wave pattern.","marker":"[20]"},{"why":"Reports nonreciprocal transport in a superconducting device, serving as the experimental reference point for the proposed protocol.","marker":"[25]"}],"fun_headline_variants":["Nonreciprocal flow reveals hidden pairing symmetry","Angle-dependent asymmetry exposes unconventional gap","Swapping terminals unmask superconductor's pairing","A new transport probe for obscured pairing order"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The exact angular correspondence assumes that the leads are rotationally symmetric, so that a lead attached at angle $\\varphi$ behaves exactly like a rotated copy of a reference lead; a real device with faceted interfaces or strongly angle-dependent lead coupling can distort the angular pattern even if the qualitative symmetry-selective behavior survives.","fun_headline_variants_meta":{"raw":{"variants":["Nonreciprocal flow reveals hidden pairing symmetry","Angle-dependent asymmetry exposes unconventional gap","Swapping terminals unmask superconductor's pairing","A new transport probe for obscured pairing order"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000265,"raw_usage":{"total_tokens":1690,"prompt_tokens":1112,"completion_tokens":578,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":728,"completion_tokens_details":{"reasoning_tokens":532}},"tokens_in":728,"tokens_out":578,"duration_ms":10837,"temperature":1.0,"reasoning_tokens":532,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:28:32.476331+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically simulate the same multiterminal device with faceted or otherwise non-rotationally-symmetric lead interfaces—for example, with a lead broadening matrix that is not a rotated copy of a reference lead—and compare the computed $\\Delta G_c(\\varphi)$ and $\\Delta G_{s_n}(\\varphi)$ to Eqs. (8) and (10); if the angular pattern deviates beyond small angular-resolution errors, the central correspondence is false.","supporting_citations":[{"cited_title":"B ¨uttiker, Phys","cited_arxiv_id":null,"evidence_quote":"Provides the multiterminal conductance framework and terminal-exchange notation used to define $\\Delta G$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Basis of the Landauer transmission formalism through which normal and Andreev conductances are computed."},{"cited_title":"Meir and N","cited_arxiv_id":null,"evidence_quote":"Extends the Landauer formula to interacting systems, underlying the spin-resolved current expression used here."},{"cited_title":"Khodas and A","cited_arxiv_id":null,"evidence_quote":"Gives the two-orbital iron-based model with competing $s$- and $d$-wave channels that produces the $s+id$ state used for numerical demonstration."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports nonreciprocal transport in a superconducting device, serving as the experimental reference point for the proposed protocol."}],"review_version":1}