{"id":"77f11a1c-2703-4500-aa8a-fe30b0a49611","arxiv_id":"2506.19706","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Trilayer NbSe2 is predicted to host a layer-selective FFLO superconducting phase in which finite-momentum and zero-momentum Cooper pairs coexist.","lead":"This theory paper predicts a new superconducting state in three-layer NbSe2: Cooper pairs with nonzero momentum form on the outer layers while ordinary zero-momentum pairs form on the middle layer. The state, called layer-selective FFLO, has a clear tunneling signature and may explain recent experiments on layered NbSe2.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Layer-selective FFLO may hinge on the vertical-hopping approximation; a 2H-realistic interlayer bond would carry an orbital Peierls phase absent from H_perp.","rationale":"The paper's central claim is a qualitative prediction of a coexistence phase. The BdG calculation is internally consistent, the Chebyshev method is converged (Fig. S2), and the symmetry argument (Eq. (15)) distinguishes the two phases. The weakest point is the treatment of the orbital effect in the interlayer coupling. I examined whether the reader's concern actually lands: for a vertical interlayer bond, the Peierls phase is exactly zero in the Landau gauge A=Hz xhat, so the omission is not an internal inconsistency. The concern becomes material only because the model's momentum-conserving H_perp assumes vertical alignment, whereas 2H-NbSe2 has a lateral offset between adjacent layers. A realistic interlayer bond would have an in-plane displacement, making the Peierls phase nonzero and of the same order as the intralayer orbital phases. This would feed an orbital shift into the middle layer, precisely the ingredient whose absence makes the q=0 Ising pairing possible. The paper gives no estimate of this effect and no robustness check. I therefore do not change the reader's CONDITIONAL verdict: the high-field coexistence state is plausible and well computed within the model, but its material relevance rests on the vertical-hopping approximation. The proposed check (including the 2H Peierls phase in H_perp) would settle whether the phase survives.","tokens_in":17036,"tokens_out":21895,"duration_ms":230371,"concrete_test":"Extend the real-space BdG calculation to include interlayer hopping at the actual 2H offset: couple the top (bottom) layer to the middle layer with a lateral displacement equal to one of the triangular-lattice basis vectors, and attach the Peierls phase exp[-(i e/ℏ) ∫ A·dl] evaluated along the straight bond path, with A(z)=Hz xhat. Recompute the self-consistent gap at mu_B H_y/T_c0=1.6 and T/T_c0=0.4 on the same 400x100 lattice, and fit the middle-layer gap to Eq. (14). If the fitted Δ_Ising component remains nonzero and nodeless, the layer-selective FFLO is robust; if Δ_2(x) reverts to the cos(q0 x) nodal form, the coexistence is an artifact of the vertical-hopping approximation. As a control, scale the interlayer Peierls phase from zero to the full geometric value and track the size of the coexistence region.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the layer-selective FFLO phase at mu_B H_y/T_c0=1.6 and T/T_c0=0.4, where the middle layer develops a nodeless zero-momentum (Ising-type) gap while the outer layers carry q=+q0 and q=-q0 FFLO gaps. This coexistence requires the middle layer to experience essentially no orbital Fermi-surface shift. The model implements that by applying the layer-dependent vector potentials A^(1)=+Hd xhat, A^(2)=0, A^(3)=-Hd xhat only in the intralayer terms H_kin and H_Ising, while the interlayer hopping H_perp (Model section, Eq. (1) and following text) is a momentum-conserving t_perp with no Peierls phase. For a purely vertical bond this is gauge-consistent, because the line integral of A=Hz xhat along a vertical path vanishes. However, the momentum-conserving form of H_perp implicitly treats the Nb sites in adjacent layers as vertically aligned. In the actual 2H-NbSe2 stacking, adjacent layers are laterally offset by a triangular-lattice vector, so a realistic interlayer bond has an in-plane displacement. Under the same vector potential, the interlayer Peierls phase is nonzero and of the same dimensionless order as the intralayer orbital phases. Including this phase in H_perp would give the middle layer an effective orbital shift through interlayer hybridization, potentially suppressing the q=0 Ising component that is the defining feature of the layer-selective FFLO state. The paper neither justifies the vertical-hopping assumption nor tests the sensitivity of the coexistence phase to this neglected phase.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a minimal three-layer tight-binding model of NbSe2 with Ising spin-orbit coupling, including both the orbital and paramagnetic effects of an in-plane magnetic field. Using a linearized gap equation and real-space Bogoliubov-de Gennes calculations with the Chebyshev polynomial method, the authors map out the temperature-field phase diagram and identify two finite-momentum superconducting states: the orbital FFLO state, in which all layers carry a finite pair momentum, and a layer-selective FFLO state, in which the two outer layers carry momenta +q0 and -q0 while the middle layer develops a nodeless zero-momentum Ising-type gap. The paper claims these two states are separated by a second-order transition and can be distinguished spectroscopically by their local density of states.","tokens_in":17368,"tokens_out":9224,"duration_ms":105106,"significance":"If the layer-selective FFLO phase is robust, it is a new example of coexisting finite- and zero-momentum Cooper pairs in a single material, directly relevant to the active field of orbital FFLO and finite-momentum superconductivity in transition metal dichalcogenides. The paper gives a concrete, falsifiable prediction: the LDOS changes from residual zero-energy spectral weight in the orbital FFLO state to a V-shaped spectrum in the layer-selective FFLO state on the outer layers. The numerical work is careful in several respects: the Chebyshev implementation is documented, a convergence check is provided in Fig. S2, and the real-space solutions are consistent with the linearized gap equation. The principal risk is the model's treatment of interlayer hopping, which omits orbital Peierls phases and may affect the central coexistence phase.","major_comments":[{"comment":"The layer-selective FFLO phase relies on the middle layer being unaffected by the orbital field, which is implemented by setting A^(2)=0 and by omitting Peierls phases in H_perp. This is gauge-consistent only for strictly vertical interlayer bonds. In actual 2H-NbSe2 stacking, adjacent Nb layers are laterally offset, so a realistic interlayer bond has an in-plane displacement and, under A=Hz xhat, carries an orbital Peierls phase of the same order as the intralayer phases. Since t_perp is comparable to the intralayer hoppings (Table S1), including this phase would give the middle layer an effective orbital shift through interlayer hybridization and could weaken or destroy the zero-momentum Ising component that defines the layer-selective FFLO state. The authors should justify the vertical-bond approximation for their effective Nb-only model or test the sensitivity of the phase to a finite in-plane offset in the interlayer hopping.","section":"Model, Eq. (1)"},{"comment":"The conclusion that the orbital-FFLO to layer-selective-FFLO transition is second-order is based on the smooth temperature dependence of the order parameters Delta_FFLO and Delta_Ising extracted from a fit to Eq. (14). No free-energy comparison between the two BdG solutions is reported for this transition. Since BdG solutions are stationary points of the mean-field free energy, the thermodynamic stability of the layer-selective phase and the order of the transition should be verified by comparing the free energies of the two solutions as a function of temperature.","section":"Phase transition, Fig. 5"}],"minor_comments":[{"comment":"The order parameters Delta_FFLO, delta_Delta_FFLO, delta'_Delta_FFLO, Delta_Ising, and delta_Delta_Ising are extracted by fitting spatial profiles to Eq. (14), but the fitting procedure is not described. Please specify the fitting method (e.g., complex least squares), the number of free parameters, and show residuals or error estimates in Figs. 3-5.","section":"Eq. (14), Figs. 3-5"},{"comment":"The spectroscopic distinction is illustrated by LDOS at a single site in each phase. Because the layer-selective FFLO gap amplitude is spatially modulated, the LDOS varies with position, as shown in Figs. S5-S6. A statement on how representative the chosen sites are, or a spatially averaged LDOS, would make the prediction more robust.","section":"LDOS, Fig. 6"},{"comment":"The first-order boundary between the uniform Ising and layer-selective FFLO phases at mu_B H/T_c0 approximately 0.7 relies on q0 taking discrete values on the 400-site lattice, as the authors acknowledge. Please state the discretization step Delta_q and the resulting uncertainty in the boundary, or provide a finite-size scaling.","section":"Results, Fig. 2; SM Sec. III"},{"comment":"The convergence check in Fig. S2 verifies the uniform gap at mu_B H/T_c0 = 1.6 and T/T_c0 = 0.4. It would be helpful to state whether the same cutoff N_c = 10000 was tested for the inhomogeneous solutions at other parameter points.","section":"Method, Chebyshev implementation"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of cond-mat.supr-con and presents a potentially interesting new phase. The main risk is the vertical-hopping approximation in the orbital coupling; a sensitivity test with a laterally offset interlayer hopping would be feasible and would substantially strengthen the central claim. The free-energy comparison for the high-field transition is also important for the phase diagram. The low-field boundary is already acknowledged as limited by finite-size discretization, so that concern should not block publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper finds a genuinely new superconducting state—the layer-selective FFLO phase—in a trilayer Ising superconductor, where the middle layer keeps zero-momentum pairing while the outer layers carry opposite finite momenta. The claim is supported by internally consistent BdG calculations, and the numerics look careful. The soft spot is not the math; it is the model's handling of the interlayer hopping, which may matter more than the paper admits.\n\nThe genuinely useful part is the systematic treatment of the odd-layer case, which was missing from the orbital-FFLO literature. The mechanism is clear: the orbital field shifts the outer layers' Fermi surfaces in opposite directions, while the middle layer is unshifted in their gauge, so the middle layer's Ising pairing can survive at fields where the outer layers go FFLO. The real-space Chebyshev solver is documented with a convergence check (Fig. S2), and the gap profiles in Figs. 3 and 4 support the two-state picture. The LDOS difference between the orbital FFLO (residual zero-bias weight) and layer-selective FFLO (V-shaped) is a clean, testable fingerprint. The companion experiment by overlapping authors is a natural hook, not a circularity; the prediction is independent.\n\nWhere I would push back: the stress-test note is on target. In Eq. (1) and the text around it, the vector potential is coupled only to H_kin and H_Ising; H_perp has no Peierls phase. That is gauge-consistent for purely vertical bonds, but 2H-NbSe2 does not stack like that. Adjacent Nb layers are laterally offset, so real interlayer bonds have an in-plane component, and under A=Hz xhat those bonds acquire an orbital phase of the same dimensionless order as the intralayer one. That phase would hybridize the middle layer with shifted outer layers and give the middle layer an effective orbital shift, which could weaken or destroy the q=0 Ising component that defines the layer-selective phase. The paper never addresses this. It is not fatal—a minimal model is allowed to simplify—but the title says trilayer NbSe2, not 'toy trilayer,' so the authors should either justify the vertical-bond approximation or run a sensitivity check. The low-field transition line is a second, smaller caveat; the authors themselves flag the finite-size discretization there. And the lack of shipped code or data makes independent numerical confirmation harder, though that is a minor inconvenience more than a flaw.\n\nBottom line: the central phase is credible within the model, and the paper deserves a serious referee. I would send it out with a request for sensitivity analysis on the interlayer orbital phase and a cleaner low-field boundary. If the phase survives a realistic interlayer hopping, it is a strong and useful result; if it does not, the paper still contains a nice methodological framework for odd-layer systems.","headline":"A credible new odd-layer FFLO phase, but the model drops the interlayer orbital phase that real 2H stacking would introduce.","tokens_in":17922,"tokens_out":3069,"would_cite":true,"duration_ms":31968,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["74.20.-z","74.25.-q"],"model":"deepseek-v4-flash","headline":"The paper predicts that an in-plane magnetic field can drive trilayer NbSe2 into a superconducting phase in which finite-momentum and zero-momentum Cooper pairs coexist on different layers, visible as a V-shaped tunneling spectrum.","keywords":["finite-momentum superconductivity","orbital FFLO state","layer-selective FFLO phase","Ising superconductor","trilayer NbSe2","Bogoliubov-de Gennes equation","Chebyshev polynomial method","local density of states"],"falsifier":"Include the orbital Peierls phase in the interlayer hopping term of the model and re-solve the Bogoliubov-de Gennes equations at μ_B H_y/T_c0 = 1.6 and T/T_c0 = 0.4: if the middle-layer gap acquires nodes or vanishes, the layer-selective FFLO phase is an artifact of the gauge choice. Experimentally, scan the local density of states on the top layer of a clean trilayer NbSe2 device under a high in-plane field: a V-shaped spectrum with site-to-site modulation supports the phase, while a flat residual zero-bias spectrum in the high-field range contradicts it.","tokens_in":16792,"feed_emoji":"🧲","tokens_out":10480,"duration_ms":96115,"temperature":0.7,"pith_summary":"This paper asks what superconducting phases a three-layer Ising superconductor like NbSe2 can host under an in-plane magnetic field, when the orbital motion of electrons is included along with the usual Zeeman pair-breaking and spin-orbit locking. Solving the Bogoliubov-de Gennes equations on a large lattice, it finds two distinct finite-momentum states at high field. In the orbital FFLO phase, the outer layers carry Cooper pairs with momenta +q0 and -q0 while the middle layer is nearly normal. At lower temperature, the middle layer develops a nodeless order parameter that is a phase-wave superposition of zero-momentum Ising pairing and induced FFLO pairing: the authors call this the layer-selective FFLO phase, where finite- and zero-momentum Cooper pairs coexist. The two phases are separated by a second-order transition, and their local densities of states differ, so the coexistence phase should be visible in scanning-tunneling spectroscopy.","feed_headline":"Two Cooper-pair states can coexist in one superconductor","feed_subtitle":"Orbital-field theory predicts layer-selective FFLO in trilayer NbSe2, visible as V-shaped tunneling spectra.","key_machinery":"The load-bearing object is the layer-resolved orbital Fermi-surface shift created by the in-plane field: with vector potential A^(1) = +Hd x̂, A^(2) = 0, A^(3) = -Hd x̂, the outer layers' Fermi surfaces move by opposite amounts while the middle layer stays put, so the natural Cooper-pair momenta are +q0 on the top layer, 0 on the middle, and -q0 on the bottom, with |q0| ≃ 2eHd found numerically. The argument then decomposes each layer's gap into an FF component Δ_FFLO $e^{{±i q0 x}}$ and an Ising component Δ_Ising, plus small proximity corrections; the layer-selective FFLO phase is precisely the coexistence of these two order parameters, with the middle-layer gap fitted by Δ_Ising $e^{{iθ}}$ + δΔ_FFLO cos(q0(x - x̄)). Large-scale Bogoliubov-de Gennes solutions via the Chebyshev polynomial method supply the amplitude and phase profiles, the order-parameter temperature dependence, and the local density of states used to identify the phases.","core_discovery":"For the trilayer 2H-NbSe2 model, the paper claims that the orbital effect of an in-plane field shifts the top and bottom layers' Fermi surfaces by opposite momenta, ±eHd, while leaving the middle layer unshifted, and that the energetically favored Cooper-pair momenta are therefore layer-dependent. At μ_B H_y/T_c0 = 1.6 and T/T_c0 = 0.4, the Bogoliubov-de Gennes solution has outer-layer gaps close to Δ_FFLO $e^{{±i q0 x}}$ and a middle-layer gap close to Δ_Ising $e^{{iθ}}$ + δΔ_FFLO cos(q0(x - x̄)), with both Δ_FFLO and Δ_Ising nonzero: finite-momentum and zero-momentum Cooper pairs coexist. The paper calls this the layer-selective FFLO phase, argues that it is reached from the orbital FFLO phase by a second-order transition that breaks a discrete translational symmetry, and shows that the two phases have different local densities of states, the layer-selective one being V-shaped.","pith_inferences":["The near-π/2 relative phase between the Ising and FFLO components on the middle layer makes the effective order parameter complex-valued; this phase texture could host circulating supercurrents or a nonreciprocal diode-like response, though the paper does not compute transport.","Generalizing the odd-layer logic, a five-layer stack with layer-dependent shifts +, 0, -, 0, + would have two zero-momentum layers and could make the coexistence phase even more robust; this is a direct extension the paper leaves implicit.","Disorder is likely the main killer of this state, as the paper itself notes, so a clean trilayer flake with a long mean free path is the experimental target, and the predicted V-shaped local density of states gives a parameter-free check.","The structure of alternating finite-momentum outer layers and a zero-momentum middle layer forms a natural S-FFLO-S junction; Josephson-like interference between the layers might produce field oscillations in the critical current that could be tested in a trilayer device."],"forward_implications":["If the layer-selective FFLO phase is real, trilayer NbSe2 provides a tunable two-order-parameter superconductor whose middle-layer zero-momentum pairing and outer-layer finite-momentum pairing can be controlled independently by field and temperature.","The second-order transition from the orbital FFLO phase to the layer-selective FFLO phase breaks a discrete translational symmetry, so a Landau-type multicritical point should appear where the two transition lines meet.","Scanning tunneling spectroscopy can distinguish the phases without fitting the full gap structure: a V-shaped, spatially modulated local density of states signals the layer-selective phase, while residual zero-bias density of states signals the orbital FFLO phase.","Because the coexistence relies only on an odd-layer stack with an unshifted interior layer, the same phase should appear in other odd-layer transition-metal dichalcogenides, not only NbSe2."],"supporting_citations":[{"why":"Defines the finite-momentum Fulde-Ferrell pairing concept that the orbital FFLO state generalizes.","marker":"[1]"},{"why":"Defines the Larkin-Ovchinnikov modulated-gap structure used to describe the middle-layer gap.","marker":"[2]"},{"why":"Reports the orbital FFLO state in an Ising superconductor and motivates the high-field phase diagram.","marker":"[44]"},{"why":"Establishes the orbital-effect finite-momentum pairing mechanism in bilayer transition-metal dichalcogenides.","marker":"[52]"},{"why":"Develops the theory of unconventional pairing in bilayer dichalcogenides including the orbital effect.","marker":"[53]"},{"why":"Provides a theory of orbital FFLO pairing with a similar relation between momentum and orbital shift.","marker":"[56]"},{"why":"Gives the orbital FFLO state theory for Ising superconductors that this paper builds on.","marker":"[57]"},{"why":"Reports experiments on trilayer NbSe2 blocks suggesting multiple superconducting phases and a possible layer-selective FFLO state.","marker":"[64]"},{"why":"Supplies the tight-binding parameters for monolayer NbSe2 used to build the minimal trilayer model.","marker":"[65]"},{"why":"Provides the Chebyshev polynomial expansion method enabling the large-scale Bogoliubov-de Gennes calculation.","marker":"[67]"}],"fun_headline_variants":["Layer-selective FFLO: both zero- and finite-momentum pairs","Coexisting Cooper pairs with different momenta","Two Cooper-pair types coexist in NbSe2","Finite- and zero-momentum pairs share a superconducting phase"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The layer-selective FFLO phase rests on the assumption that the orbital magnetic field shifts only the outer layers' momenta, leaving the middle layer genuinely unshifted; the model enforces this by omitting the orbital Peierls phase from interlayer hopping, and if that phase matters the coexistence may be weakened or absent.","fun_headline_variants_meta":{"raw":{"variants":["Layer-selective FFLO: both zero- and finite-momentum pairs","Coexisting Cooper pairs with different momenta","Two Cooper-pair types coexist in NbSe2","Finite- and zero-momentum pairs share a superconducting phase"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000538,"raw_usage":{"total_tokens":2585,"prompt_tokens":949,"completion_tokens":1636,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":1567}},"tokens_in":565,"tokens_out":1636,"duration_ms":14359,"temperature":1.0,"reasoning_tokens":1567,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:27:40.459351+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Include the orbital Peierls phase in the interlayer hopping term of the model and re-solve the Bogoliubov-de Gennes equations at μ_B H_y/T_c0 = 1.6 and T/T_c0 = 0.4: if the middle-layer gap acquires nodes or vanishes, the layer-selective FFLO phase is an artifact of the gauge choice. Experimentally, scan the local density of states on the top layer of a clean trilayer NbSe2 device under a high in-plane field: a V-shaped spectrum with site-to-site modulation supports the phase, while a flat residual zero-bias spectrum in the high-field range contradicts it.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Larkin-Ovchinnikov modulated-gap structure used to describe the middle-layer gap."},{"cited_title":"Liu, Unconventional Superconductivity in Bilayer Transition Metal Dichalcogenides, Phys","cited_arxiv_id":null,"evidence_quote":"Develops the theory of unconventional pairing in bilayer dichalcogenides including the orbital effect."},{"cited_title":"Qiu and Y","cited_arxiv_id":null,"evidence_quote":"Provides a theory of orbital FFLO pairing with a similar relation between momentum and orbital shift."},{"cited_title":"Saito, Y","cited_arxiv_id":null,"evidence_quote":"Reports experiments on trilayer NbSe2 blocks suggesting multiple superconducting phases and a possible layer-selective FFLO state."},{"cited_title":"Misfit layered superconductor (PbSe)1.14(NbSe2)3 with possible layer-selective FFLO state","cited_arxiv_id":"2506.14106","evidence_quote":"Supplies the tight-binding parameters for monolayer NbSe2 used to build the minimal trilayer model."}],"review_version":2}