{"id":"2ad8e5a7-950e-4808-a971-23c5e72eed22","arxiv_id":"2607.29620","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"In a semi-infinite superfluid, the local pairing amplitude rises from zero at a hard wall with 2k_F Friedel oscillations in the BCS regime that are progressively washed out toward the BEC regime.","lead":"This paper calculates how the local pairing amplitude of attractively interacting fermions varies near a hard wall across the BEC-BCS crossover, finding Friedel oscillations in the weak-coupling regime that disappear at stronger coupling. The result matters for interpreting boundary effects in box-trapped ultracold Fermi gases and at superconducting edges.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central no-surface-state and below-T_c claims rest on explicitly unproven completeness of the continuum kernel eigenfunctions; Appendix C's bound allows bound states below the continuum, so the Sec. VI basis expansion could miss a localized surface-pairing component.","rationale":"The reader's weakest assumption matches the main load-bearing gap: completeness of the continuum eigenfunctions. The paper is otherwise careful: the renormalization procedure is plausible, the eigenfunction cutoff-independence argument is explicit, and the numerical diagonalization is described in detail. However, the authors themselves flag the missing completeness proof, and the Appendix C bound is too weak to exclude a surface bound state. Such a bound state would be a qualitative change, not a small correction, so a conditional verdict is appropriate. The arbitrary β in the cubic nonlinearity is secondary because it cancels in normalized profiles, but the model dependence of the below-T_c result reinforces the need for an independent check.","tokens_in":24777,"tokens_out":10056,"duration_ms":119522,"concrete_test":"Numerically verify the spectral closure Eq. (39): from the diagonalized eigenfunctions, form the truncated reconstruction K_rec(x,x') = Σ_{i≤N} E_Λ(q_i)\\barΔ_{q_i}(x)\\barΔ_{q_i}(x') and compare pointwise against the direct kernel Eq. (8) for x,x' within a few k_F^{-1} of the wall; a missing bound state produces a systematic discrepancy that does not shrink as N and L grow. Independently, solve the homogeneous real-space integral equation (7) for square-integrable eigenfunctions with a high-resolution collocation (L≥10^4 k_F^{-1}, N≥10^6, including points very near x=0) and scan eigenvalues in [2EΛ(0), EΛ(0)); if a convergent eigenpair below EΛ(0) appears, the no-surface-state claim fails. As a complementary check, run a self-consistent BdG calculation in the half-space and compare Δ(x)/Δ_bulk with Figs. 7-8 to test whether the cubic model preserves the oscillations.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim has two parts: (i) the edge profile is universal with 2k_F Friedel oscillations below T_c, and (ii) there is no surface-bound-state pairing instability. Both are built on the generalized eigenfunctions \\barΔ_q(x) forming a complete basis, Eq. (39), which is used in Eq. (48) to expand the nonlinear solution and in Eq. (40) to construct the renormalized kernel. The authors explicitly state in Sec. V that completeness is not proven ('we have not directly proven completeness') and that bound states were only checked numerically. Appendix C proves E_bound ≥ 2EΛ(0), not E_bound ≥ EΛ(0). Because EΛ(0)<0, this leaves the entire interval [2EΛ(0), EΛ(0)) open for a square-normalizable surface state below the continuum. Such a state would change T_c, invalidate the 'no surface pairing instability' claim, and make the expansion Eq. (48) incomplete, so the predicted edge profile could acquire a localized component not captured by the continuum basis. The below-T_c equation (43) is also a phenomenological local cubic model with arbitrary β; while normalized profiles are β-independent via rescaling, the persistence of Friedel oscillations is only demonstrated for this model, not for self-consistent BCS/BdG. The completeness gap is the more fundamental issue because it affects both the spectral conclusion and the below-T_c solution.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the local pairing amplitude of attractively interacting fermions in a three-dimensional semi-infinite half-slab with a hard wall, across the BEC-BCS crossover. At T_c, the authors analyze the linearized pairing kernel in a cosine basis, use Weyl's theorem and an analytic ansatz to show that the essential spectrum is the bulk spectrum, and construct numerically the corresponding continuum eigenfunctions. They normalize these eigenfunctions and build a renormalized, cutoff-independent pairing kernel. Below T_c, they add a Ginzburg-Landau-type cubic nonlinearity and solve the resulting equation in the eigenfunction basis. The central claims are that (i) the edge pairing profile rises from zero at the wall with 2k_F Friedel oscillations in the weak-coupling BCS regime, which are suppressed toward the BEC regime, and (ii) within this model there is no surface-bound-state (surface-superconductivity) instability.","tokens_in":25182,"tokens_out":11122,"duration_ms":114859,"significance":"If the claims hold, the paper gives a concrete, falsifiable prediction for box-trapped cold-atom experiments and provides an interesting contrast to surface-superconductivity instabilities found in one-dimensional and two-dimensional settings. The linearized T_c analysis is careful: the analytic ansatz and numerical diagonalization agree, the cutoff-independence argument based on convergent differences is convincing, and the numerical methods are documented in enough detail to be reproduced. The strengths of the paper are the explicit regularization procedure and the demonstration that the eigenfunction shapes are universal. However, the no-surface-state conclusion and the below-T_c Friedel-oscillation persistence both rest on an unproven completeness assumption and on a phenomenological nonlinear model, so the significance is conditional on closing those gaps.","major_comments":[{"comment":"The completeness of the continuum eigenfunctions \\barΔ_q is assumed, not established. The authors state in Sec. V that they have not directly proven completeness. Appendix C proves only E ≥ 2EΛ(0); since EΛ(0)<0, the interval [2EΛ(0), EΛ(0)) remains open for a square-normalizable bound state below the continuum. The numerical diagonalization in Appendix D is finite-dimensional and cannot exclude such a state. If a bound state exists, Eq. (39) is false, the spectral representation Eq. (40) is incomplete, and the expansion Eq. (48) misses a localized surface-pairing component. This affects both the 'no surface pairing instability' conclusion and the below-T_c edge profile. Please either prove the required completeness (or a bound excluding bound states) or explicitly reformulate the central results as conditional on the numerically supported absence of bound states.","section":"§V, Eqs. (39)-(40); §VI, Eq. (48); Appendix C"},{"comment":"The below-T_c equation is a Ginzburg-Landau-inspired cubic with an arbitrary coefficient β, not a self-consistent BdG or microscopic BCS equation. The cancellation of β in Δ(x)/Δ_bulk normalizes the overall scale only; it does not establish that the local cubic model reproduces the correct nonlocal self-consistent pairing. The persistence of 2k_F Friedel oscillations below T_c is therefore demonstrated only within this phenomenological model. The paper acknowledges this in Sec. VII, but the abstract and Fig. 1 present the result more generally. A derivation of the nonlinear term from the microscopic action or a BdG calculation in the half-slab geometry (at least in the BCS regime) is needed to make the main claim fully load-bearing.","section":"§VI, Eq. (43)"}],"minor_comments":[{"comment":"The blue solid curve in Fig. 1 corresponds to βc μ=5, i.e., k_B T_c/μ≃0.2, but the text says 'blue solid (k_B T_c/μ≃0.1 or k_F a_s≃−1.39)', which duplicates the red dashed value. Please correct this inconsistency.","section":"Fig. 1 and Sec. I text"},{"comment":"The nonlinear term is written Δ^3 in Eq. (43), |Δ|^3 in Eq. (D18), and β|Δ|²Δ in Appendix B. Since the order parameter can be complex, use β|Δ|²Δ consistently, or explicitly state the reality/positivity convention.","section":"Eq. (43) and Appendix D"},{"comment":"β denotes inverse temperature in the figures (βc μ) and in Appendix D, while it is the GL nonlinear coefficient in Eq. (43) and Appendix B. Rename one of them to avoid ambiguity.","section":"Notation"},{"comment":"Ref. [38] (the 2D case) is cited as unpublished; please provide a preprint identifier or publication status if available.","section":"References"},{"comment":"The abstract uses 'BEC-BCS crossover' while the title and body use 'BCS-BEC crossover' or 'BEC-BCS crossover' inconsistently; unify the terminology.","section":"Abstract and Title"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the unproven completeness assumption. If the authors can prove the absence of bound states or derive the below-T_c equation microscopically, the paper would be suitable for publication. Without that, the strong claims in the abstract and figures outrun the demonstrated results. I would not reject outright, but the revision must either close the gap or explicitly reframe the results as conditional on numerical evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a careful, honest extension of BCS boundary theory to the BEC-BCS crossover. The genuinely new piece is a cutoff-independent renormalized pairing kernel for a half-slab, built from continuum eigenfunctions that are shown to be universal even though the eigenvalues are cutoff dependent. The numerics are extensive and the semi-analytic and full numerical solutions agree well. The 2k_F Friedel oscillations in the BCS regime were already in Stojković and Valls, but the coupling-dependent suppression across the crossover and the clean construction of the renormalized kernel are new.\n\nThe soft spots are real but the authors mostly acknowledge them. The strongest claim—no surface-bound-state pairing instability—rests on numerical search plus an unproven completeness of the continuum eigenfunctions. Appendix C proves only E_bound >= 2E_Lambda(0), not E_bound >= E_Lambda(0), and since E_Lambda(0)<0 the interval below the continuum is open. The stress-test note is right about that. The authors say explicitly they have not proven completeness, so this is an honest limitation, not a hidden one. It limits the claim to \"no numerical evidence,\" which is what they actually state.\n\nThe below-T_c analysis is a GL-inspired cubic equation with an arbitrary beta_GL. Normalized profiles cancel beta, so the shape prediction is robust within that model, but the persistence of Friedel oscillations below T_c is only demonstrated for that model, not for self-consistent BdG. The paper itself flags that as future work. So the headline result about below-T_c behavior is conditional.\n\nThe citation pattern looks fair: they cite the prior boundary-oscillation work and the 1D surface-state papers. No code or data is shipped, but the numerical method in Appendix D is detailed enough to reproduce.\n\nOverall this is a solid subfield paper, not a breakthrough. It deserves a serious referee. I would like the revision to (1) soften or prove the no-surface-state claim, and (2) either do a BdG check for at least one coupling or clearly frame the below-T_c result as model-dependent. The completeness gap is the more fundamental issue but it is an open problem, not a fatal error.","headline":"Careful, honest extension of BCS boundary theory to the crossover; the no-surface-state claim is numerically supported but not proven, and the below-T_c analysis is a GL model rather than full BdG.","tokens_in":25571,"tokens_out":2970,"would_cite":true,"duration_ms":32306,"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":"In a half-space Fermi superfluid, the local pairing amplitude rises from zero at the hard wall and shows Friedel oscillations at wavevector 2k_F that are suppressed as coupling moves from the BCS to the BEC regime.","keywords":["BEC-BCS crossover","local pairing amplitude","hard-wall boundary","Friedel oscillations","surface superconductivity","box trap","pairing kernel","renormalization"],"falsifier":"A direct numerical search for square-normalizable eigenfunctions of the pairing kernel with eigenvalues below the continuum bottom E_Lambda(0), using a fine grid and a large box, would settle the surface-state question; the authors' numerics find none. Experimentally, measuring the local pairing profile near the wall of a box-trapped Fermi gas across the crossover—e.g., via momentum-resolved photoemission or rf spectroscopy—would reveal the 2k_F oscillations in the BCS regime and their absence at unitarity.","tokens_in":24699,"feed_emoji":"⚛️","tokens_out":6161,"duration_ms":56839,"temperature":0.7,"pith_summary":"This paper asks how superfluidity is modified by a hard boundary, as encountered in box-shaped traps for ultracold Fermi gases or at the edge of a superconductor. Treating a semi-infinite system with a wall at x=0, the authors show that the local pairing amplitude is not uniform up to the wall: it drops to zero at the boundary and, in the weak-coupling BCS regime, displays Friedel-like oscillations at wavevector 2k_F. As the interaction is strengthened toward the BEC side of the crossover, these oscillations are suppressed and the pairing rises smoothly to its bulk value. The authors further argue that the edge profile is universal, independent of the ultraviolet cutoff, and that there is no surface-bound-state pairing instability in three dimensions with contact interactions.","feed_headline":"Edge pairing in a Fermi superfluid oscillates at 2k_F","feed_subtitle":"The oscillations fade as interactions strengthen, offering a direct probe of the BEC-BCS crossover at the boundary.","key_machinery":"The central object is the pairing kernel K_Lambda(x,x') for the linearized T_c gap equation in the half-slab geometry, expressed in a cosine basis as a singular diagonal piece plus a compact off-diagonal piece that imposes the hard-wall condition. Its continuum eigenfunctions, normalized with a residue R(q) that emerges from principal-value products, are universal (cutoff-independent) even though the eigenvalues are not; the Lippmann-Schwinger renormalization replaces the eigenvalues by E_R(q) while leaving the eigenfunctions untouched. This basis then carries the expansion of the nonlinear pairing problem below T_c.","core_discovery":"The authors solve the linearized BCS gap equation for a semi-infinite superfluid with a hard-wall boundary, treating the pairing instability as an eigenvalue problem for an integral kernel K_Lambda. They find that the kernel's generalized eigenfunctions form a continuum labeled by wavevector q, with eigenvalues equal to the bulk values; the off-diagonal part that enforces the boundary condition does not alter the essential spectrum, and no bound states (which would signal surface superconductivity) appear numerically. These eigenfunctions are cutoff-independent, so the kernel can be renormalized by the same Lippmann-Schwinger procedure used for the bulk. Below T_c, adding a Ginzburg-Landau n","pith_inferences":["If the predicted 2k_F oscillations are real, they should be visible in a box-trapped Fermi gas as a spatial modulation of the pairing gap near the walls, measurable with local probes such as momentum-resolved photoemission or rf spectroscopy.","The contrast between 1D models (which show surface bound states) and 2D/3D contact-interaction models (which do not) suggests a dimensionality threshold: surface pairing enhancement may require quasi-1D confinement or a longer-ranged interaction kernel.","A rigorous proof or disproof of completeness of the continuum eigenfunctions would settle the surface-state question; the weak bound derived in the paper (E ≥ 2E(0)) does not rule out a bound state below the continuum but above 2E(0).","The universal edge profile implies that the boundary of a box trap is a clean place to measure the BEC-BCS crossover parameter, since the shape of Δ(x) near the wall encodes coupling strength in a cutoff-independent way."],"forward_implications":["In the weak-coupling BCS regime, the local pairing amplitude near a hard wall follows approximately Δ(x) = Δ_bulk[1 − sin(2k_F x)/(2k_F x)], with oscillations at wavevector 2k_F and a rise on a length scale ~(2k_F)^{-1}.","As coupling increases toward the BEC regime, Friedel oscillations are suppressed and the edge profile approaches the smooth Ginzburg-Landau solution, with a quadratic rise near the wall rather than a linear one.","The edge pairing profile is universal: after renormalization it is independent of the ultraviolet cutoff.","No surface-bound-state pairing instability (surface superconductivity) exists for 3D contact-interaction superfluids; the half-slab transition temperature equals the bulk T_c.","The same kernel-eigenfunction approach can be extended to a self-consistent Bogoliubov-de Gennes treatment below T_c or to a finite box geometry, where the box T_c may differ from the bulk T_c."],"fun_headline_variants":["Friedel oscillations in edge pairing fade towards BEC","Boundary pairing oscillates in BCS, suppressed in BEC","Edge pairing in Fermi gas: Friedel-like oscillations near BCS","Hard-wall trap reveals pairing oscillations that die with coupling","Box boundary pairing: BCS oscillations, BEC suppression"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The central result depends on the unproven completeness of the continuum eigenfunctions of the pairing kernel: the authors state they have not directly proven completeness and find no numerical evidence for bound states, but if a square-normalizable bound state existed below the continuum, the edge pairing could acquire a localized surface component.","fun_headline_variants_meta":{"raw":{"variants":["Friedel oscillations in edge pairing fade towards BEC","Boundary pairing oscillates in BCS, suppressed in BEC","Edge pairing in Fermi gas: Friedel-like oscillations near BCS","Hard-wall trap reveals pairing oscillations that die with coupling","Box boundary pairing: BCS oscillations, BEC suppression"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000193,"raw_usage":{"total_tokens":1136,"prompt_tokens":645,"completion_tokens":491,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":389,"completion_tokens_details":{"reasoning_tokens":408}},"tokens_in":389,"tokens_out":491,"duration_ms":5413,"temperature":1.0,"reasoning_tokens":408,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T03:20:19.136281+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct numerical search for square-normalizable eigenfunctions of the pairing kernel with eigenvalues below the continuum bottom E_Lambda(0), using a fine grid and a large box, would settle the surface-state question; the authors' numerics find none. Experimentally, measuring the local pairing profile near the wall of a box-trapped Fermi gas across the crossover—e.g., via momentum-resolved photoemission or rf spectroscopy—would reveal the 2k_F oscillations in the BCS regime and their absence at unitarity.","supporting_citations":[],"review_version":1}