{"id":"5e433ab8-d7d6-4079-92cb-238360fedcdb","arxiv_id":"2505.17349","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"For s-wave superconductors, the diamagnetic superfluid weight equals a momentum integral of the quasihole quantum metric, and can be mapped to a site-resolved marker for disorder studies.","lead":"This paper rewrites the diamagnetic superfluid weight of conventional s-wave superconductors as an integral of a quantum metric of the superconducting quasihole states, giving a geometric description of the Meissner effect. It then introduces a real-space marker that maps this weight to individual lattice sites, showing how nonmagnetic impurities suppress the local diamagnetic current in 2D and 3D.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Diamagnetic-only marker understates the disorder problem: the omitted paramagnetic kernel is exactly what grows in dirty superconductors, so the Pippard comparison is not yet established.","rationale":"The reader's weakest assumption is exactly the point that is load-bearing: the paper identifies the Meissner response with the diamagnetic superfluid weight D^d and neglects D^p, despite itself flagging that D^p becomes significant in highly disordered superconductors. My read agrees. Eq. (14) is derived consistently in Appendix A and is a legitimate geometric rewriting of D^d; I have no objection to the clean-limit statement. The marker construction in Eqs. (16)-(24) is a plausible real-space generalization, and the self-consistent BdG treatment of disorder is a genuine computational step forward. But the disorder-oriented headline claims, especially the claimed consistency with Pippard's experiment, rest on the assumption D^d >> D^p, which is precisely the regime that disorder destroys. The paper's own caveat in the last paragraph of Sec. III C is an admitted limitation, not a repair. Because this gap is acknowledged and the central Eq. (14) stands, the appropriate outcome is unchanged from the reader's CONDITIONAL verdict: the geometric identity can be accepted, while the Pippard interpretation should be conditional on a full calculation of the total superfluid weight in the disordered lattices. The proposed check is a direct, numerically feasible computation that would settle whether the omitted paramagnetic term changes the disorder trend or only its magnitude.","tokens_in":13187,"tokens_out":11702,"duration_ms":98091,"concrete_test":"On the same 81x9x9 and 175x15 disordered BdG lattices used in Figs. 2 and 4, compute the full zero-temperature Meissner kernel including the paramagnetic contribution via the standard Kubo/linear-response formula D^s_munu = D^d_munu + lim_{q->0} Pi_munu(q, omega=0), where Pi_munu involves current-current matrix elements between positive- and negative-energy BdG eigenstates, and then form lambda_L^full from the trace of D^s. Compare this with lambda_L from Eq. (21), which uses D^d only, for n_imp=16% and U_imp in [0,10]. If lambda_L^full differs from the diamagnetic-only estimate by more than about 20%, the Pippard consistency claim is not supported by the marker calculation alone; if the two track within numerical noise, the neglect of D^p is benign.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central identity Eq. (14) is a correct rewriting of the second-order vector-potential term, and in a clean T=0 s-wave superconductor the paramagnetic kernel vanishes, so the geometric statement is meaningful there. The load-bearing problem is the disordered-system application. Eq. (15) sets 1/(mu0 lambda_L^2) equal to D^d_mumu by assuming D^d >> D^p, and Eqs. (20)-(21) turn the marker into a local London penetration depth in exactly the impurity regimes where this inequality is least secure. In a dirty s-wave superconductor the full response is D^d + D^p; D^p is negative and grows as disorder reduces the superfluid stiffness. The authors concede this in the final paragraph of Sec. III C, stating that in highly disordered SCs the paramagnetic current 'may contribute significantly and diminish the Meissner effect,' and Appendix A says D^p contains a gamma_z matrix element that prevents a local-marker representation. Consequently Figs. 1-4 and the Pippard comparison report only the diamagnetic component. A reduction in D^d alone does not establish the measured increase of lambda_L, because the measured kernel is the sum; D^d and D^p can change by comparable amounts, leaving lambda_L different from the marker estimate. The abstract's claim of consistency with Pippard therefore goes beyond what is actually computed.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper derives an identity for the diamagnetic superfluid weight D^d_μν of a multiband s-wave superconductor in terms of the quantum metric of the Bogoliubov quasihole states (Eq. 14). It introduces a 'fidelity number' and a quasihole Wannier spread, then constructs a real-space marker D^d_μν(r) using projectors onto positive- and negative-energy BdG eigenstates (Eqs. 16–24). The marker is applied to 2D square and 3D cubic lattices with self-consistently solved BdG equations and nonmagnetic impurities; the authors report that strong impurities suppress and 'turbulently' redirect the diamagnetic current, and that the resulting increase of London penetration depth is consistent with Pippard.","tokens_in":13454,"tokens_out":12580,"duration_ms":120438,"significance":"The central identity is a rigorous analytic rewriting of the second-order vector-potential term; the derivation in Appendix A checks out for momentum-independent s-wave pairing. This provides a clean geometric interpretation of D^d in the clean limit, and the real-space marker is a potentially useful numerical tool for studying local diamagnetic responses. The paper is self-contained analytically and the numerics are presented transparently. However, the experimental claim about penetration-depth increase is based only on D^d; the full Meissner kernel also contains the paramagnetic term D^p, which is not computed. The disordered-system conclusions are therefore not yet established.","major_comments":[{"comment":"The replacement of 1/(μ_0 λ_L²) by D^d_μμ assumes D^d ≫ D^p, but the paper's own final paragraph of Sec. III C concedes that in highly disordered superconductors the paramagnetic current 'may contribute significantly and diminish the Meissner effect,' and Appendix A states that D^p contains a γ_z matrix element that prevents a local-marker representation. Since the measured superfluid weight is D^d + D^p and D^p is negative and grows with disorder in dirty s-wave superconductors, a reduction in D^d alone does not establish the Pippard-type increase of λ_L. The abstract and conclusion should either compute the full kernel (e.g., by evaluating D^p via linear response for the same disorder realizations) or explicitly restrict the claim to the diamagnetic component and remove the Pippard consistency statement.","section":"Sec. III C, Eq. (21), Figs. 1–4"},{"comment":"The marker is constructed with the position operator diag(1,2,3,...)⊗γ, which is an ad hoc lattice replacement for the true position operator. The total D^d is gauge-invariant, but the local values D^d(r) and the flow patterns in Figs. 1–4 depend on this chosen operator. The authors acknowledge boundary inaccuracy but do not test the robustness of the impurity-induced turbulence to other position-operator conventions. Please add a benchmark using a different, gauge-equivalent position operator (or a shifted unit-cell convention) and discuss the resulting ambiguity of the local marker, or clearly state that the marker is a non-unique local decomposition.","section":"Sec. II D, Eq. (19)"}],"minor_comments":[{"comment":"There are several typos: 'detemrined' after Eq. (31), 'latttice' in Sec. II D, 'suppoort' in the Acknowledgments, and 'the the superfluid weight' in Sec. III B.","section":"Sec. III B and Acknowledgments"},{"comment":"The sentence stating that the type of SC is 'directly determined by (D^d_μμ)^{-1/2}' should be qualified, because the Ginzburg-Landau parameter is controlled by the total superfluid kernel, not by the diamagnetic part alone.","section":"Sec. II C, after Eq. (15)"},{"comment":"The lattice constant a is introduced as a regularization of a parabolic-band integral; the resulting Ω_I depends on a and therefore is a model-dependent estimate rather than an intrinsic material property. This should be stated explicitly where the analytical formulas are presented.","section":"Eqs. (10) and (27)"},{"comment":"The term 'superfluid weight' is used interchangeably with the diamagnetic component D^d. Since the paper does not compute the full superfluid weight, please consistently reserve 'superfluid weight' for D^d + D^p, and call D^d the 'diamagnetic superfluid weight' throughout, or define the shorthand at first use.","section":"Abstract and Sec. II C"}],"recommendation":"major_revision","confidential_remarks":"The core derivation is sound and the paper is a reasonable contribution to the quantum-geometry literature on superconductivity. My main reservation is the overclaim in the disorder application: the diamagnetic-only marker cannot, by itself, establish the Pippard-type penetration-depth increase. If the authors either compute the paramagnetic contribution for the same disorder configurations or explicitly restrict the conclusions to the diamagnetic component, I would support publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The takeaway: the core identity in Eq. (14) is a solid, correctly derived result, but the paper oversells the disorder comparison with Pippard by computing only the diamagnetic part of the response. The central mathematics is fine; the application needs a caveat.\n\nWhat's new: this generalizes the flat-band quantum metric superfluid weight to arbitrary multiband s-wave superconductors, showing the diamagnetic weight is a momentum integral of the quasihole quantum metric weighted by quasiparticle energy differences. That's a genuine extension of the authors' earlier work, and the derivation in Appendix A is clean. The superfluid weight marker is also new—a real-space decomposition of the diamagnetic response that can be computed from BdG eigenstates. Used on 2D and 3D lattice models, it produces plausible pictures of how impurities suppress and divert the diamagnetic current locally. That is a useful tool.\n\nThe soft spot is the disorder application. Eq. (15) identifies 1/lambda_L^2 with the diamagnetic superfluid weight D^d alone, assuming D^d >> D^p. In dirty superconductors, the paramagnetic term is not negligible; it grows as disorder suppresses stiffness, and the authors admit this in the last paragraph of Sec. III C. They also note in Appendix A that the paramagnetic kernel contains a gamma_z that makes a local-marker representation non-obvious. So Figs. 1-4 and the comparison with Pippard's Sn data report only the diamagnetic component. A reduction in D^d does not by itself establish the measured increase of lambda_L; the measured kernel is the sum D^d + D^p, and both can change. So the abstract's phrase \"consistent with experiments\" is too strong for what is computed.\n\nThat said, this is not a fatal flaw for the central geometric statement. In a clean s-wave superconductor at T=0, the paramagnetic term vanishes, so Eq. (14) is a correct geometric interpretation there. The marker remains a promising local diagnostic for inhomogeneous systems, with the caveat that it only captures the diamagnetic response.\n\nWho should read this: people working on quantum geometry in superconductors and on real-space markers for disordered systems. It deserves a serious referee; the mathematics is sound and the marker is a reasonable new contribution. I'd advise the editor to send it out, and recommend that the authors either compute the paramagnetic response in the disordered regime or explicitly frame the results as the diamagnetic contribution only.","headline":"A correct and useful geometric identity for the diamagnetic superfluid weight, but the disorder comparison with Pippard overreaches because the paramagnetic response is omitted.","tokens_in":14001,"tokens_out":2100,"would_cite":true,"duration_ms":14817,"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":"The Meissner effect in s-wave superconductors is shown to be governed by the quantum metric of quasihole states, with the diamagnetic superfluid weight equal to the energy-weighted momentum integral of that metric.","keywords":["quantum geometry","Meissner effect","superfluid weight","quantum metric","Bogoliubov-de Gennes","London penetration depth","disorder","s-wave superconductivity"],"falsifier":"Calculate the full Meissner kernel of a single-band s-wave lattice model by linear response, including the paramagnetic term, and compare it with Eq. (14) evaluated from the same Bogoliubov–de Gennes eigenstates: if the two disagree as disorder is increased, the diamagnetic-only assumption fails. A more direct geometric test would measure the quasihole quantum metric and the superfluid weight independently (for example through a superconducting-state optical or dielectric sum rule, if one is established) and check the energy-weighted integration identity at several gap sizes.","tokens_in":12931,"feed_emoji":"🧲","tokens_out":12835,"duration_ms":95161,"temperature":0.7,"pith_summary":"The paper argues that the Meissner effect in conventional s-wave superconductors is a quantum-geometric phenomenon. Its central result is an exact identity, Eq. (14): at zero temperature the diamagnetic superfluid weight that controls the London penetration depth equals $-2 e^2/V_{\\mathrm{cell}}$ times the Brillouin-zone integral of the quasihole quantum metric weighted by quasiparticle energy differences. This places the superconducting-state quantum metric, not the normal-state band structure, at the origin of the diamagnetic response, and it applies to any multiband s-wave superconductor with arbitrary intraband and interband pairing. The same geometric object yields the gauge-invariant spread of quasihole Wannier functions, proposed as a measure of superconducting-state stability. The paper also converts the superfluid weight into a site-resolved marker computable from self-consistent Bogoliubov–de Gennes equations, and uses it to show that nonmagnetic impurities suppress and divert the local diamagnetic current while increasing the London penetration depth, in agreement with the classic disorder experiments.","feed_headline":"Meissner effect traced to quantum geometry of quasihole states","feed_subtitle":"A new derivation ties the superfluid weight of any s-wave superconductor to the quantum metric of its quasihole states.","key_machinery":"The central object is the quasihole quantum metric, defined by the overlap of fully antisymmetric filled quasihole states at neighboring momenta: $|\\langle u_h(\\mathbf{k})|u_h(\\mathbf{k}+\\delta\\mathbf{k})\\rangle| = 1 - \\frac12 g_{\\mu\\nu}(\\mathbf{k})\\,\\delta k_\\mu \\delta k_\\nu$, with $g_{\\mu\\nu}(\\mathbf{k}) = \\sum_{nm} \\frac12[\\langle\\partial_\\mu n|m\\rangle\\langle m|\\partial_\\nu n\\rangle + (\\mu\\leftrightarrow\\nu)]$. Its role is to convert the second-derivative expectation value in the diamagnetic superfluid weight, Eq. (13), into a product of energy differences and metric elements, Eq. (14). The momentum integral of the metric defines the fidelity number $G_{\\mu\\nu}$, whose trace gives the gauge-invariant spread $\\Omega_I$ of quasihole Wannier functions. The real-space marker is carried by the projectors $\\hat{P} = \\sum_{E_n<0}|E_n\\rangle\\langle E_n|$, $\\hat{P}_E = \\sum_{E_n<0} E_n|E_n\\rangle\\langle E_n|$, and the analogous $\\hat{Q}$, $\\hat{Q}_E$ for positive-energy states, together with the position operator, which replace the momentum-space integral by a trace over lattice eigenstates.","core_discovery":"On the paper's own terms, the discovery is that the diamagnetic superfluid weight $D^d_{\\mu\\nu}$ of a conventional superconductor is not a kinetic, band-dispersion quantity but a geometric one: it is the energy-weighted Brillouin-zone integral of the quasihole quantum metric $g^{nm}_{\\mu\\nu}$, where $n$ runs over filled quasihole states and $m$ over empty quasiparticle states. Because this quasihole quantum metric is built from the overlap of fully antisymmetric quasihole Bloch states at neighboring momenta, it is present even when the normal-state bands are geometrically trivial, making the Meissner effect a generic quantum-geometric property of all s-wave superconductors. The paper further shows that the momentum integral of the metric, the fidelity number, equals the gauge-invariant part of the spread of quasihole Wannier functions; in parabolic-band models this spread is inversely proportional to the superconducting gap, so a larger gap localizes the quasihole wave function and, in the paper's proposal, stabilizes the superconducting state. Finally, by rewriting the momentum-space formula in terms of lattice projectors and the position operator, the paper obtains a real-space superfluid weight marker $D^d_{\\mu\\nu}(\\mathbf{r})$ that gives the local diamagnetic current at each lattice site, allowing disorder to be included by self-consistently solving the Bogoliubov–de Gennes equations.","pith_inferences":["Beyond the paper: if the quasihole quantum metric controls the superfluid weight at $T=0$, it should also regulate phase fluctuations at finite temperature, so a metric-based upper bound on the superconducting transition temperature may exist in low-dimensional s-wave systems, analogous to the flat-band bounds but arising from the superconducting-state geometry.","Beyond the paper: the paper leaves the paramagnetic current without a real-space marker; constructing such a marker would complete the local Meissner kernel and would allow strong-disorder cases to be checked against the $D^d\\gg D^p$ assumption directly.","Beyond the paper: the predicted local suppression and circumvention of the diamagnetic current around a single nonmagnetic impurity could be probed with atomic-scale magnetic imaging on a clean 2D superconductor, providing a direct spatial test of the marker.","Beyond the paper: because the quasihole Wannier spread is proposed as a stability measure, one can test it by computing the response of the order parameter to local perturbations (e.g., a magnetic impurity or a phase slip) in lattice Bogoliubov–de Gennes simulations and correlating it with $\\Omega_I$."],"forward_implications":["If Eq. (14) is correct, the London penetration depth of every s-wave superconductor, including ordinary single-band ones, is set by the quasihole quantum metric, so quantum-geometric superfluid weight is not limited to flat-band materials.","Because $\\lambda_L$ enters the Ginzburg-Landau parameter $\\kappa=\\lambda_L/\\xi_{GL}$, the type-I/type-II boundary is partly controlled by the average magnitude of the energy-weighted quasihole quantum metric: larger geometric weight shortens $\\lambda_L$ and pushes the superconductor toward type-II behavior.","The gauge-invariant quasihole Wannier spread, inversely proportional to the pairing gap in the parabolic-band estimate, gives a geometric criterion for superconducting-state stability: larger gap, more localized quasihole wave function, more stable condensate.","Disorder, treated self-consistently through the Bogoliubov–de Gennes equations, suppresses the local diamagnetic superfluid weight, makes the local diamagnetic current turbulent and circumventing impurities, and increases the London penetration depth, matching the classic experimental trend.","The superfluid weight marker provides a site-resolved route to compute the Meissner response in inhomogeneous and disordered superconductors, going beyond the homogeneous clean-limit formulas."],"supporting_citations":[{"why":"Defines the single-band quasihole quantum metric and supplies the parabolic-band fidelity-number results that the multiband generalization and the marker build on.","marker":"[12]"},{"why":"Establishes the vector-potential expansion of the Hamiltonian and the paramagnetic-current expression against which the diamagnetic formula is defined.","marker":"[5]"},{"why":"Provides the Ginzburg-Landau parameter criterion for vortex formation, used to connect the geometric superfluid weight to the type-I/type-II boundary.","marker":"[13]"},{"why":"Supplies the Wannier-spread formalism whose gauge-invariant part is identified with the momentum-integrated quantum metric.","marker":"[14–16]"},{"why":"Provides the real-space topological-marker approach adapted to turn the superfluid weight into a local marker.","marker":"[17–20]"},{"why":"Reports the experimental increase of London penetration depth with disorder that the marker calculations are designed to reproduce.","marker":"[21]"},{"why":"Introduces the fidelity-marker projector formalism whose structure the superfluid-weight marker modifies.","marker":"[25,35]"}],"fun_headline_variants":["Meissner effect traced to quantum metric of quasiholes","Superfluid weight is a quantum geometric marker","Quantum geometry sets superfluid weight","Superfluidity from quasihole band geometry","Meissner effect rooted in quantum geometry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation identifies the London penetration depth with the diamagnetic superfluid weight alone, assuming the paramagnetic current is negligible ($D^d \\gg \\langle D^p\\rangle$); the paper itself notes that in highly disordered superconductors the paramagnetic current can become significant and diminish the Meissner effect, so the disorder conclusions depend on this clean-limit dominance holding.","fun_headline_variants_meta":{"raw":{"variants":["Meissner effect traced to quantum metric of quasiholes","Superfluid weight is a quantum geometric marker","Quantum geometry sets superfluid weight","Superfluidity from quasihole band geometry","Meissner effect rooted in quantum geometry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001024,"raw_usage":{"total_tokens":4364,"prompt_tokens":1038,"completion_tokens":3326,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":654,"completion_tokens_details":{"reasoning_tokens":3257}},"tokens_in":654,"tokens_out":3326,"duration_ms":20927,"temperature":1.0,"reasoning_tokens":3257,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:49:36.621738+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Calculate the full Meissner kernel of a single-band s-wave lattice model by linear response, including the paramagnetic term, and compare it with Eq. (14) evaluated from the same Bogoliubov–de Gennes eigenstates: if the two disagree as disorder is increased, the diamagnetic-only assumption fails. A more direct geometric test would measure the quasihole quantum metric and the superfluid weight independently (for example through a superconducting-state optical or dielectric sum rule, if one is established) and check the energy-weighted integration identity at several gap sizes.","supporting_citations":[{"cited_title":"Porlles \\ and\\ author W","cited_arxiv_id":null,"evidence_quote":"Defines the single-band quasihole quantum metric and supplies the parabolic-band fidelity-number results that the multiband generalization and the marker build on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports the experimental increase of London penetration depth with disorder that the marker calculations are designed to reproduce."}],"review_version":1}