{"schemaVersion":"1.0","revision":"fa961ab24277a91737c3","article":{"id":"scwiki-j","topicId":"scwiki-topic-j","code":"J","slug":"theoretical-tools","publicationKind":"handbook","title":"Theoretical tools for superconductivity","summary":"Twelve theoretical and computational approaches, with their inputs, convergence checks, physical assumptions and limits on predicting superconductivity.","aliases":["theoretical methods","BCS Eliashberg Allen Dynes DFT DFPT QMC DMRG DMFT RPA FLEX Wannier RG BdG","理论工具","理论方法","密度泛函微扰","重整化群","瓦尼尔函数"],"evidenceTypes":["theory","hypothesis"],"relatedCodes":["B1","B2","B3","F1","F7","F5","I","N10"],"editorialStatus":"scoped-check","contentState":"pilot","modifiedAt":"2026-10-18T19:29:40.890430+00:00","review":{"checkedAt":"2026-10-19","scope":"Twelve individually mapped theoretical tools. Selected DFPT, Hubbard benchmark, DMFT, Wannier and RG passages were reviewed, together with bounded Allen–Dynes and conserving-approximation abstracts. Earlier BCS, Eliashberg, geometry, GL and Kitaev checks retain their original limits. Independent finite-model calculations check the stated algebra.","limitations":["Complete scientific acceptance remains open.","Some original references remain abstract-only; full final-version/supplement/dataset comparisons are incomplete.","Independent model algebra does not validate an interacting material, every solver or predicted transition.","Planning source defines scope only; it is not scientific evidence."]},"provenance":{"repository":"JackZH26/jzis-superconductivity","commit":"0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf","path":"02_BATCH_2_IN_PLAN.md","sha256":"502293e5ed2fe8463b61a6c5b78a2c33bc33f3265bd36fe534fa2c469e42ff48","sourceId":"file-49326006ac2c2636"},"sourceMap":[{"startLine":1,"endLine":55,"targetSections":[],"decision":"planning-only-not-republished","note":"Scope planning is not scientific evidence."},{"startLine":56,"endLine":56,"targetSections":["j1"],"decision":"new-research-for-approved-handbook-item","note":"Stable approved identity retained; newly authored scientific explanation."},{"startLine":57,"endLine":57,"targetSections":["j2"],"decision":"new-research-for-approved-handbook-item","note":"Stable approved identity retained; newly authored scientific explanation."},{"startLine":58,"endLine":58,"targetSections":["j3"],"decision":"new-research-for-approved-handbook-item","note":"Stable approved identity retained; newly authored scientific explanation."},{"startLine":59,"endLine":59,"targetSections":["j4"],"decision":"new-research-for-approved-handbook-item","note":"Stable approved identity retained; newly authored scientific explanation."},{"startLine":60,"endLine":60,"targetSections":["j5"],"decision":"new-research-for-approved-handbook-item","note":"Stable approved identity retained; newly authored scientific explanation."},{"startLine":61,"endLine":61,"targetSections":["j6"],"decision":"new-research-for-approved-handbook-item","note":"Stable approved identity retained; newly authored scientific explanation."},{"startLine":62,"endLine":62,"targetSections":["j7"],"decision":"new-research-for-approved-handbook-item","note":"Stable approved identity retained; newly authored scientific explanation."},{"startLine":63,"endLine":63,"targetSections":["j8"],"decision":"new-research-for-approved-handbook-item","note":"Stable approved identity retained; newly authored scientific explanation."},{"startLine":64,"endLine":64,"targetSections":["j9"],"decision":"new-research-for-approved-handbook-item","note":"Stable approved identity retained; newly authored scientific explanation."},{"startLine":65,"endLine":65,"targetSections":["j10"],"decision":"new-research-for-approved-handbook-item","note":"Stable approved identity retained; newly authored scientific explanation."},{"startLine":66,"endLine":66,"targetSections":["j11"],"decision":"new-research-for-approved-handbook-item","note":"Stable approved identity retained; newly authored scientific explanation."},{"startLine":67,"endLine":67,"targetSections":["j12"],"decision":"new-research-for-approved-handbook-item","note":"Stable approved identity retained; newly authored scientific explanation."},{"startLine":68,"endLine":177,"targetSections":[],"decision":"planning-only-not-republished","note":"Other approved items remain separately tracked."}],"corrections":[{"id":"scwiki-j-correction-1","description":"QMC and DMRG are not collectively weak-correlation methods; retain their distinct coupling regimes and algorithmic errors.","sourceId":"source-51a15c56a9ac1472","scope":"Correct the weak-correlation classification in fixed planning line60 while preserving the approved J5 identity."}],"equationChecks":[{"targetSectionId":"j4","tex":"D=M^{-1/2}KM^{-1/2}","check":"12 two-mass Hessian cases including negative curvature and translation zero modes."},{"targetSectionId":"j9","tex":"g(\\ell)=g_0/(1+g_0\\ell)","check":"12 independent RK4 flows in the declared normalization, before poles; not full RG."},{"targetSectionId":"j12","tex":"\\mathcal H=\\begin{pmatrix}h&\\Delta\\\\\\Delta^\\dagger&-h^T\\end{pmatrix}","check":"24 complex Hermitian/antisymmetric-pairing cases verify particle-hole redundancy and paired spectra; not a topological classification."}],"references":[{"id":"source-ba10e800a1246bf6","doi":"10.1103/PhysRev.108.1175","title":"Theory of Superconductivity","year":1957,"url":"https://doi.org/10.1103/PhysRev.108.1175","publicationStatus":"journal-article","verification":{"scope":"primary-full-text-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/pr/pdf/10.1103/PhysRev.108.1175","note":"Selected passages: pp. 1182-1183, Eqs. (2.38)-(2.40), (2.49)-(2.50); p. 1186, Eqs. (3.27)-(3.30). One-spin DOS, gap equation and excitation convention checked. Modern limiting constants are evaluated from the model, not copied from historical rounded values. Full proof, empirical fits and exhaustive correction searches remain outside this review."}},{"id":"source-d0ec754310a67e88","doi":"10.1103/PhysRev.104.1189","title":"Bound Electron Pairs in a Degenerate Fermi Gas","year":1956,"url":"https://doi.org/10.1103/PhysRev.104.1189","publicationStatus":"journal-article","verification":{"scope":"primary-full-text-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/pr/pdf/10.1103/PhysRev.104.1189","note":"Two-page primary letter: pp. 1189-1190, model setup and Eqs. (3)-(5). Checked the inert Fermi-sea and attractive-shell assumptions and the distinction from the many-body problem. No material-specific pairing mechanism is verified."}},{"id":"source-d56b04dc0b32c796","doi":"10.1016/j.aop.2020.168102","title":"Eliashberg Theory: a short review","year":2020,"url":"https://doi.org/10.1016/j.aop.2020.168102","publicationStatus":"journal-article","verification":{"scope":"author-review-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/1911.05065v1","note":"Selected v1 pp. 1–5 and p. 8 model benchmark; p. 5 equations visually checked. Separate independent linearized Einstein-model computation agrees with the approximately 13.3 K benchmark. Full review/final-version equivalence not audited."}},{"id":"source-7c4eac077ae5e221","doi":"10.1038/nature00898","title":"The origin of the anomalous superconducting properties of MgB2","year":2002,"url":"https://doi.org/10.1038/nature00898","publicationStatus":"journal-article","verification":{"scope":"primary-full-text-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://www2.lbl.gov/Science-Articles/Archive/assets/images/2002/Aug-14-2002/MgB2Nature.pdf","note":"Selected pp. 758–759 and Fig. 3 checked during A5: anisotropic calculation, band-dependent gaps and common calculated transition. This article inherits that scoped check, not independent convergence or experimental replication."}},{"id":"source-0620df96ebbf2a66","doi":"10.1103/PhysRevB.12.905","title":"Transition temperature of strong-coupled superconductors reanalyzed","year":1975,"url":"https://doi.org/10.1103/PhysRevB.12.905","publicationStatus":"journal-article","verification":{"scope":"primary-abstract-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/prb/abstract/10.1103/PhysRevB.12.905","note":"Publisher abstract: modified prefactor, spectral moment/shape corrections and large-coupling limits within Eliashberg theory. Full derivation and Pb-Bi data not reanalyzed."}},{"id":"source-b06e417872e240c9","doi":"10.1103/RevModPhys.73.515","title":"Phonons and related crystal properties from density-functional perturbation theory","year":2001,"url":"https://doi.org/10.1103/RevModPhys.73.515","publicationStatus":"journal-article","verification":{"checkedAt":"2026-10-19","scope":"selected-published-response-and-metals-check","evidenceUrl":"https://people.sissa.it/~degironc/ES/lectures/papers/RMP-73-515-2001.pdf","note":"Published 6 July 2001. Selected printed pages 515–516, 519, 524, 526, 542 and the left column of 543; equations 80–87 on page 526 visually checked. Mass weighting, self-consistent response, metallic smearing and the distinction between electron–phonon and transport spectra were reviewed. No full DFT implementation or material benchmark was reproduced."}},{"id":"source-51a15c56a9ac1472","doi":"10.1103/PhysRevX.5.041041","title":"Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms","year":2015,"url":"https://doi.org/10.1103/PhysRevX.5.041041","publicationStatus":"journal-article","verification":{"checkedAt":"2026-10-19","scope":"selected-published-qmc-dmrg-convergence-check","evidenceUrl":"https://journals.aps.org/prx/pdf/10.1103/PhysRevX.5.041041","note":"Published 14 December 2015. Selected pages 1–4, 7–8 and 10; the DMRG convergence discussion on page 8 visually checked. The review retains constrained-path and particle-hole conditions, finite-cylinder limits and method-specific errors. Full tables, datasets and all algorithms have not been reproduced."}},{"id":"source-12f2513b257d7194","doi":"10.1103/RevModPhys.68.13","title":"Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions","year":1996,"url":"https://doi.org/10.1103/RevModPhys.68.13","publicationStatus":"journal-article","verification":{"checkedAt":"2026-10-19","scope":"selected-published-self-consistency-and-limits-check","evidenceUrl":"https://www.physics.rutgers.edu/~udo/qmc/rmp68_13.pdf","note":"Selected printed pages 15–18 and 20; equations 6–13 on page 18 visually checked. Local temporal fluctuations, bath self-consistency, large-coordination scaling and atomic/noninteracting limits were reviewed. An independent four-state atomic Green function was checked; no interacting impurity solver or lattice DMFT calculation was reproduced."}},{"id":"source-eaa3beeed4368f46","doi":"10.1103/RevModPhys.84.1383","title":"A common thread: The pairing interaction for unconventional superconductors","year":2012,"url":"https://doi.org/10.1103/RevModPhys.84.1383","publicationStatus":"journal-article","verification":{"scope":"author-review-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/1207.4093v1","note":"Selected manuscript text pp. 4, 18–20, 39–46; p. 41 Eq. (13) visually checked. Full 87-page review and final-version equivalence not audited. Independent two-patch algebra is an illustration, not reproduction of a Hubbard calculation."}},{"id":"source-822deb97a8e9dde0","doi":"10.1103/PhysRevLett.62.961","title":"Conserving Approximations for Strongly Correlated Electron Systems: Bethe-Salpeter Equation and Dynamics for the Two-Dimensional Hubbard Model","year":1989,"url":"https://doi.org/10.1103/PhysRevLett.62.961","publicationStatus":"journal-article","verification":{"checkedAt":"2026-10-19","scope":"publisher-original-abstract","evidenceUrl":"https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.62.961","note":"Published 20 February 1989. Original publisher abstract checked: self-consistent conserving approximations and two-particle correlations. The full functional derivation and numerical benchmark have not been reviewed."}},{"id":"source-2b331df83384bc5c","doi":"10.1103/RevModPhys.84.1419","title":"Maximally localized Wannier functions: Theory and applications","year":2012,"url":"https://doi.org/10.1103/RevModPhys.84.1419","publicationStatus":"journal-article","verification":{"checkedAt":"2026-10-19","scope":"selected-author-gauge-window-interpolation-check","evidenceUrl":"https://arxiv.org/pdf/1112.5411v2","note":"Author version 2, dated 12 May 2012; journal publication 10 October 2012. Selected pages 4–5, 11–13 and 36–37; Figure 5 and the frozen-window explanation on page 13 visually checked. Gauge versus subspace selection, Chern obstruction and interpolation conditions were reviewed. The independent finite-range Fourier example is not a material wannierization. Full final-version comparison remains open."}},{"id":"source-0aec6d266727d631","doi":"10.1103/RevModPhys.66.129","title":"Renormalization-group approach to interacting fermions","year":1994,"url":"https://doi.org/10.1103/RevModPhys.66.129","publicationStatus":"journal-article","verification":{"checkedAt":"2026-10-19","scope":"selected-author-cooper-flow-and-caveats-check","evidenceUrl":"https://arxiv.org/pdf/cond-mat/9307009v2","note":"Author version 2, dated 9 July 1993. Selected pages 92–95 and 101–103; equations 381–384 on page 102 visually checked. Equation 383 supplies the stated normalization; the Kohn–Luttinger caveat on page 103 is retained. Independent scalar RK4 integration is not full RG or an ordered-state solution. Final-version comparison remains open."}},{"id":"source-fa7a165c8c344a63","doi":"10.1103/PhysRevB.95.024515","title":"Band geometry, Berry curvature, and superfluid weight","year":2017,"url":"https://doi.org/10.1103/PhysRevB.95.024515","publicationStatus":"journal-article","verification":{"scope":"selected-author-projector-and-metric-conventions","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/1610.01803v2","note":"Authorv2 11January2017,selectedpp1–2,4–6; p5Eq21–24 visuallychecked. Published27January2017. R=2Tr(P dP dP) and gLiang=2gWiki. Final-version comparison and full response derivation remain open."}},{"id":"source-72f7ddb9f13bbd81","doi":"10.1103/PhysRevB.106.014518","title":"Revisiting flat band superconductivity: Dependence on minimal quantum metric and band touchings","year":2022,"url":"https://doi.org/10.1103/PhysRevB.106.014518","publicationStatus":"journal-article","verification":{"scope":"selected-published-relaxation-correction","checkedAt":"2026-10-18","evidenceUrl":"https://aaltodoc.aalto.fi/server/api/core/bitstreams/e05db312-49b0-4769-b112-681407acd4b0/content","note":"Published 26 July 2022. Selected final-reprint pages 1–5 were read; Eqs. (13)–(17) on page 4 were visually checked. The review covers order-parameter relaxation, time reversal, Hessian conditions and representation dependence. The earlier selected page 6 reading of minimal integrated trace is retained. The full proof and multiband response have not been reproduced."}},{"id":"source-fdf091e09e7f4a6f","doi":"10.1038/ncomms9944","title":"Superfluidity in topologically nontrivial flat bands","year":2015,"url":"https://doi.org/10.1038/ncomms9944","publicationStatus":"journal-article","verification":{"scope":"selected-published-units-and-flat-band-assumptions","checkedAt":"2026-10-18","evidenceUrl":"https://www.nature.com/articles/ncomms9944","note":"Published 20 November 2015. Selected pages 3–6 were read. Page 3 was visually checked for the free-energy definition and frozen-field derivative in Eq. (1), interpreted with the 2022 response correction. The earlier page 5 review of uniform pairing and scale hierarchy is retained. The full supplementary derivation has not been reviewed."}},{"id":"source-89f5503f5e470385","doi":"10.1103/RevModPhys.76.975","title":"Nobel Lecture: Type-II superconductors and the vortex lattice","year":2004,"url":"https://doi.org/10.1103/RevModPhys.76.975","publicationStatus":"journal-article","verification":{"scope":"primary-lecture-scoped-check","checkedAt":"2026-10-18","evidenceUrl":"https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.76.975","note":"Selected pp. 975-977: GL type-II boundary, upper field Eq. (1), flux quantum Eq. (10), mixed-state boundary around Eq. (11). SI notation and the linearized GL derivation are editorial checks; original Gaussian-unit equations are not copied as SI. Material-specific claims and historical approximations outside these passages are not validated."}},{"id":"source-65b23b74f5402e26","doi":"10.1103/PhysRevB.68.054506","title":"Properties of the ideal Ginzburg-Landau vortex lattice","year":2003,"url":"https://doi.org/10.1103/PhysRevB.68.054506","publicationStatus":"journal-article","verification":{"checkedAt":"2026-10-19","scope":"selected-author-variance-and-validity-check","evidenceUrl":"https://arxiv.org/pdf/cond-mat/0304237v1","note":"Authorv1 submitted10April2003; journal6August2003. Selected pp1,4–6; p5 Eqs11–13 visually checked. Paper sigma is field variance, not a depolarization rate. Ideal isotropic GL model, finite-field validity and restricted London limit retained. Independent triangular reciprocal sum, not full GL reproduction. Typeset2018 is not publication. Final-version comparison open."}},{"id":"source-ade538b429fdd847","doi":"10.1070/1063-7869/44/10S/S29","title":"Unpaired Majorana fermions in quantum wires","year":2001,"url":"https://doi.org/10.1070/1063-7869/44/10S/S29","publicationStatus":"journal-article","verification":{"scope":"selected-author-chain-and-operator-conventions","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/cond-mat/0010440v2","note":"Author v2 30 October 2000; selected pp4–7, p5 visually checked, equations4–14. Journal2001. Finite noninteracting model checks reproduced; full final-version comparison open."}}],"materialCases":[],"claims":[{"id":"scwiki-j-claim-scope-1","sectionId":"scope","statement":"Theoretical tools answer different questions: whether a model has a pairing instability, how a sample responds, or which approximation is defensible. This handbook connects twelve approved topics to their inputs, outputs and limitations. A solver result is conditional on the Hamiltonian, numerical convergence and physical approximations; it is not an experimental transition record. The J1–J12 identifiers remain stable even where an old planning label needed scientific correction.","evidenceType":"theory","sourceIds":["source-ba10e800a1246bf6","source-b06e417872e240c9","source-51a15c56a9ac1472"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-mathematical-assumptions-1","sectionId":"mathematical-assumptions","statement":"State the degrees of freedom, interactions, energy and length units, boundary conditions and limit being taken before comparing calculations. Distinguish an exact identity within a model from an approximation to a real material. The accompanying independent checks cover mass weighting, a four-state atomic Green function, Fourier interpolation, a scalar Cooper-channel flow and complex Bogoliubov matrices. They do not run a material DFT calculation, an interacting impurity solver or a full multichannel renormalization group.","evidenceType":"theory","sourceIds":["source-b06e417872e240c9","source-12f2513b257d7194","source-2b331df83384bc5c","source-0aec6d266727d631"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j1-1","sectionId":"j1","statement":"BCS theory constructs a coherent many-electron paired state and a self-consistent gap. In the elementary isotropic model, the quasiparticle energy is $E_k=\\sqrt{\\xi_k^2+|\\Delta|^2}$, with energies measured relative to the chemical potential. The familiar exponential transition scale assumes a weak attractive interaction, an energy shell and a sufficiently smooth density of states. Specify whether that density counts one spin or both; changing the convention without changing the coupling produces a different exponent.","evidenceType":"theory","sourceIds":["source-ba10e800a1246bf6"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j1-2","sectionId":"j1","statement":"The Cooper problem instead places two added electrons above an inert Fermi sea. Its instability motivates pairing but does not replace the many-body gap and number equations. A BCS gap solution describes pair formation within the approximation; phase fluctuations, spatial inhomogeneity and competing order can alter the observed transition. A material's similar gap ratio is therefore a consistency check, not unique proof of a phonon mechanism. Use the full band and symmetry assumptions when extending the elementary model.","evidenceType":"theory","sourceIds":["source-ba10e800a1246bf6","source-d0ec754310a67e88"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j2-1","sectionId":"j2","statement":"Eliashberg theory retains the frequency dependence of the interaction and electronic self-energy. A phonon-mediated calculation needs an electron–phonon spectral function, a treatment of the effective Coulomb interaction and an explicitly chosen isotropic, anisotropic or multiband approximation. The gap and mass-renormalization functions are coupled. A converged solution on a Matsubara grid also needs frequency-cutoff and temperature checks; numerical precision alone does not justify neglecting vertex corrections or changes in the phonon system.","evidenceType":"theory","sourceIds":["source-d56b04dc0b32c796"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j2-2","sectionId":"j2","statement":"The selected Marsiglio review provides a simple Einstein-spectrum benchmark, while Choi and colleagues' MgB2 calculation illustrates why band-dependent interactions and gaps can matter. Neither example licenses using one averaged coupling for every material. A Matsubara gap value is not automatically the measured tunnelling gap: real-frequency continuation and spectral features require care. Preserve how the Coulomb parameter was chosen and which electronic and phonon inputs were computed, fitted or assumed. The reviewed examples are scoped checks, not independent reproduction of every numerical result.","evidenceType":"theory","sourceIds":["source-d56b04dc0b32c796","source-7c4eac077ae5e221"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j3-1","sectionId":"j3","statement":"Compact transition-temperature estimates summarize solutions of a specified electron–phonon theory; they do not replace its inputs. Allen and Dynes use a logarithmic phonon-frequency scale and introduce strong-coupling and spectral-shape corrections. An implementation should name which version it evaluates, preserve the effective Coulomb parameter, and state whether its frequency input is an energy, angular frequency or temperature. A number copied between these conventions without conversion can be wrong even when the expression looks familiar.","evidenceType":"theory","sourceIds":["source-0620df96ebbf2a66"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j3-2","sectionId":"j3","statement":"The original abstract describes the accuracy of the modified McMillan estimate for the materials known at that time and its failure at larger coupling. That historical comparison is not a universal modern applicability boundary. Report sensitivity to the spectrum and Coulomb input, and compare a compact estimate with a suitably converged Eliashberg solution when justified. Here the original abstract was checked; the inaccessible original fit derivation and dataset are not claimed as fully reviewed. No fitted or hypothetical estimate is entered into the experimental Tc filter.","evidenceType":"theory","sourceIds":["source-0620df96ebbf2a66","source-d56b04dc0b32c796"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j4-1","sectionId":"j4","statement":"DFT and DFPT connect an electronic-structure approximation to forces, harmonic phonons and response functions. The mass-weighted force-constant matrix $D=M^{-1/2}KM^{-1/2}$ has eigenvalues $\\omega^2$, where $K$ is the energy Hessian. Its eigenvalues are not superconducting transition temperatures. An independent two-mass calculation checks the translational zero mode and shows that negative curvature produces a negative squared frequency; removing that sign would conceal an instability.","evidenceType":"theory","sourceIds":["source-b06e417872e240c9"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j4-2","sectionId":"j4","statement":"Baroni and colleagues' selected formulation requires self-consistent induced charge and potential. Metallic calculations couple momentum sampling to the smearing treatment; arbitrary numerical smearing is not a physical electronic temperature. The electron–phonon spectral function also differs from the transport-weighted spectrum. Specify structure, exchange-correlation approximation, basis, sampling and response convergence before interpreting a prediction. Harmonic stability alone neither establishes superconductivity nor tests anharmonic effects, strong correlations or every competing structure. The independent checks here verify elementary mechanics, not a full first-principles workflow.","evidenceType":"theory","sourceIds":["source-b06e417872e240c9"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j5-1","sectionId":"j5","statement":"Quantum Monte Carlo and density-matrix renormalization group methods address interacting lattice models across different coupling regimes; they are not collectively weak-correlation methods. LeBlanc and colleagues compare several algorithms for specified two-dimensional Hubbard parameters. Agreement must be assessed observable by observable and after the relevant finite-size, temperature and algorithmic extrapolations. Agreement in ground-state energy alone does not establish a particular pairing state or a finite superconducting transition temperature.","evidenceType":"theory","sourceIds":["source-51a15c56a9ac1472"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j5-2","sectionId":"j5","statement":"In the reviewed auxiliary-field formulation, particle-hole symmetry protects the half-filled model with zero next-nearest hopping from the sign problem. Away from that setting, a constrained-path calculation adds trial-state-dependent systematic error and is not generally variational. DMRG requires checks of retained states or bond dimension, truncation error, cylinder length and width, boundary conditions and competing metastable patterns. Taking cylinder length to infinity does not take its width to the two-dimensional thermodynamic limit. Historical computational sizes in a 2015 benchmark are not current hardware limits.","evidenceType":"theory","sourceIds":["source-51a15c56a9ac1472"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j6-1","sectionId":"j6","statement":"Single-site DMFT replaces a lattice environment by a self-consistent quantum impurity bath. It retains local temporal fluctuations through a frequency-dependent self-energy; it is not simply static Hartree-Fock theory. For the specified local-interaction models, its large-coordination limit becomes exact with the appropriate hopping scaling. Applied in finite dimensions, the single-site construction omits nonlocal self-energy components, so spatial correlations may require extensions or independent checks.","evidenceType":"theory","sourceIds":["source-12f2513b257d7194"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j6-2","sectionId":"j6","statement":"The impurity Green function must equal the local lattice Green function after self-consistency. The Weiss propagator describing the bath is not generally the bare local lattice propagator. An independent four-state thermal calculation verifies the atomic-limit spectral weights and Green function, but does not validate a finite-hopping DMFT solution. State the impurity solver, bath or frequency discretization, interaction convention and convergence criteria. Two lattices with the same noninteracting density of states need not have identical magnetic response or ordered phases; local single-particle agreement is a narrower result.","evidenceType":"theory","sourceIds":["source-12f2513b257d7194"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j7-1","sectionId":"j7","statement":"Random-phase and fluctuation-exchange approaches organize selected interaction diagrams into response and pairing approximations. Their names do not specify the entire calculation: identify the bare or dressed propagators, orbital interaction matrix, retained channels and self-consistency. Bickers, Scalapino and White's original abstract describes self-consistent conserving approximations for two-particle correlations. Conservation properties are useful constraints, but do not make an approximation exact for arbitrary coupling or establish a measured material's pairing mechanism.","evidenceType":"theory","sourceIds":["source-822deb97a8e9dde0"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j7-2","sectionId":"j7","statement":"Scalapino's selected review passages discuss a DCA vertex decomposition and a particle-particle Bethe–Salpeter eigenproblem. They illustrate how a momentum-dependent interaction and dressed propagators enter pairing analysis; those particular results must not be relabelled as an RPA or FLEX benchmark. An eigenvalue reaching the stated instability condition is conditional on the kernel and approximation. Check competing channels, resolution and the treatment of fluctuations before equating its temperature scale with a bulk experimental Tc. The full original conserving-functional derivation remains outside the reviewed scope.","evidenceType":"theory","sourceIds":["source-eaa3beeed4368f46","source-822deb97a8e9dde0"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j8-1","sectionId":"j8","statement":"Wannier functions provide a localized representation of a chosen Bloch subspace. Unitary rotations within an isolated manifold preserve its projector but change the orbital shapes and hopping representation. With entangled bands, selecting the subspace and selecting its gauge are distinct tasks. The frozen inner energy window constrains which states are retained; outside it, projected bands can differ from the original calculation. A compact model therefore needs its orbital basis and energy windows, not just an attractive band plot.","evidenceType":"theory","sourceIds":["source-2b331df83384bc5c"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j8-2","sectionId":"j8","statement":"Interpolation relies on localization, a sufficient momentum mesh and adequate real-space hopping range. Independent Fourier recovery of a known finite-range model succeeds, while an undersampled distant hopping aliases into a different term despite perfect agreement on the sampled grid. Validate intermediate momenta and relevant observables. Nonzero Chern invariants can obstruct an exponentially localized representation of the isolated manifold; a different subspace or hybrid construction must be described explicitly. This check is not a material wannierization or proof that an empirical tight-binding fit is unique.","evidenceType":"theory","sourceIds":["source-2b331df83384bc5c"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j9-1","sectionId":"j9","statement":"Renormalization group methods track how an effective description changes as higher-energy modes are removed. The retained channels, cutoff, self-energy treatment and truncation determine what a flow can establish. In Shankar's stated Fermi-surface setting, angular Cooper channels decouple at one loop. With the normalization $g=V_l/(4\\pi)$ and $\\ell=\\ln(\\Lambda_0/\\Lambda)$, the selected equation gives $dg/d\\ell=-g^2$. Negative coupling is attractive in this convention.","evidenceType":"theory","sourceIds":["source-0aec6d266727d631"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j9-2","sectionId":"j9","statement":"Independent numerical integration agrees with $g(\\ell)=g_0/(1+g_0\\ell)$ before its pole: attraction grows and repulsion decreases in this restricted flow. The divergence marks an instability and loss of perturbative control, not a measured Tc or a solved ordered state. The source explicitly warns that otherwise irrelevant terms can generate an attractive channel through the Kohn–Luttinger mechanism. Thus the simple positive-channel flow does not prove that every repulsive system remains a stable normal metal. Nested surfaces and additional channels require their own analysis.","evidenceType":"theory","sourceIds":["source-0aec6d266727d631"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j10-1","sectionId":"j10","statement":"Band eigenvectors carry geometric information beyond the dispersion. A quantum metric and Berry curvature can be constructed from the projector onto a specified band or subspace, with carefully stated normalization and gauge conventions. The selected Liang manuscript uses a metric normalization twice the convention used in the linked Wiki geometry article. Comparing numerical coefficients without reconciling that factor would produce a false disagreement. Degenerate or touching bands also require a suitable subspace treatment rather than unstable single-band derivatives.","evidenceType":"theory","sourceIds":["source-fa7a165c8c344a63"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j10-2","sectionId":"j10","statement":"Geometric quantities can enter superfluid response, but a nonzero metric alone does not prove superconductivity. Peotta and Törmä's flat-band result has explicit interaction, isolation and pairing assumptions. Huhtinen and colleagues further show why order-parameter relaxation and representation dependence matter when differentiating the free energy. State what is held fixed and what is reoptimized before calling a curvature a physical stiffness. A frozen-field derivative and a relaxed response are not interchangeable; the full multiband response calculation remains a separate task.","evidenceType":"theory","sourceIds":["source-72f7ddb9f13bbd81","source-fdf091e09e7f4a6f"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j11-1","sectionId":"j11","statement":"Ginzburg–Landau theory organizes a free-energy expansion in a complex order parameter and its spatial gradients. Its coefficients and applicable temperature range must be specified; in its conventional derivation it is controlled near the transition. A homogeneous minimum, a linearized upper-critical-field calculation and a vortex-lattice solution are different problems. The type-I/type-II boundary of the elementary isotropic model should not be transferred without qualification to every anisotropic or multicomponent material.","evidenceType":"theory","sourceIds":["source-89f5503f5e470385"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j11-2","sectionId":"j11","statement":"Brandt's ideal vortex-lattice calculation illustrates how field, the Ginzburg–Landau parameter and lattice structure affect magnetic observables. Its assumptions exclude arbitrary disorder and pinning, and the commonly used London-limit variance is valid only in a restricted regime. Solving a phenomenological equation with chosen coefficients does not identify a microscopic pairing mechanism. Report boundary conditions, gauge, mesh and free-energy comparisons when several solutions exist; numerical convergence to one stationary configuration alone does not establish that it is the global equilibrium state.","evidenceType":"theory","sourceIds":["source-65b23b74f5402e26"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j12-1","sectionId":"j12","statement":"Bogoliubov–de Gennes theory writes a quadratic paired Hamiltonian in a particle-and-hole basis. For a complete finite fermionic basis, Hermitian normal block $h$ and antisymmetric pairing matrix $\\Delta$, one convention is $\\mathcal H=\\begin{pmatrix}h&\\Delta\\\\\\Delta^\\dagger&-h^T\\end{pmatrix}$. The doubled description has particle-hole redundancy, not twice as many independent physical states. Independent complex-matrix checks verify Hermiticity and paired eigenvalues in this convention; a reduced spin block can use different notation.","evidenceType":"theory","sourceIds":["source-ade538b429fdd847"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-j12-2","sectionId":"j12","statement":"Specify whether the pairing field is imposed or determined self-consistently, together with density, boundaries and the physical basis. Kitaev's selected chain model provides a controlled example of bulk and boundary structure, but a near-zero eigenvalue by itself does not prove a protected experimental Majorana mode. Finite-size splitting, competing localized states and model perturbations must be considered. A self-consistent mean-field solution also does not include every collective or correlation effect. Numerical particle-hole symmetry is a necessary algebra check, not a topological classification or material validation.","evidenceType":"theory","sourceIds":["source-ade538b429fdd847"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-experimental-criteria-1","sectionId":"experimental-criteria","statement":"Compare calculated and measured quantities with matching sample conditions and definitions. A pairing eigenvalue, a harmonic phonon, a local spectral gap and an experimental zero-resistance criterion are different observables. Keep model temperatures out of the measured-Tc catalogue. Predictions should specify independent tests that could reject the assumed mechanism, structure or parameter set.","evidenceType":"theory","sourceIds":["source-7c4eac077ae5e221","source-b06e417872e240c9","source-51a15c56a9ac1472"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-limits-and-disputes-1","sectionId":"limits-and-disputes","statement":"Each of the twelve sections has a named review scope. Some references were checked only at abstract level; others use selected author-manuscript passages with final-version comparisons still open. Independent small-model checks verify the stated algebra and counterexamples, not all numerical methods or every cited material. Complete scientific acceptance, full derivations, supplementary datasets and material-level reproduction remain open.","evidenceType":"theory","sourceIds":["source-0620df96ebbf2a66","source-51a15c56a9ac1472","source-822deb97a8e9dde0","source-0aec6d266727d631"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."},{"id":"scwiki-j-claim-discovery-implications-1","sectionId":"discovery-implications","statement":"Choose the simplest defensible model that can distinguish the proposed explanation from a concrete alternative. Carry its uncertainty into any material ranking. A useful workflow moves between structural evidence, electronic and interaction models, numerical convergence and experimentally testable observables; agreement at one stage does not remove assumptions at the next. Retain failed fits and competing states rather than reporting only the most favorable transition estimate.","evidenceType":"hypothesis","sourceIds":["source-b06e417872e240c9","source-51a15c56a9ac1472","source-2b331df83384bc5c"],"conditions":"Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.","verificationScope":"Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open."}],"scopeItems":[{"id":"J1","sectionIds":["j1"],"status":"scoped-check","checkedAt":"2026-10-19","scope":"BCS mean-field theory: primary-full-text-scoped-check; primary-full-text-scoped-check. Source-specific review dates and limitations are listed with the references.","limitations":["Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.","Synthetic checks do not reproduce interacting material calculations or full numerical algorithms."]},{"id":"J2","sectionIds":["j2"],"status":"scoped-check","checkedAt":"2026-10-19","scope":"Eliashberg theory: author-review-scoped-check; primary-full-text-scoped-check. Source-specific review dates and limitations are listed with the references.","limitations":["Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.","Synthetic checks do not reproduce interacting material calculations or full numerical algorithms."]},{"id":"J3","sectionIds":["j3"],"status":"scoped-check","checkedAt":"2026-10-19","scope":"McMillan and Allen–Dynes estimates: author-review-scoped-check; primary-abstract-scoped-check. Source-specific review dates and limitations are listed with the references.","limitations":["Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.","Synthetic checks do not reproduce interacting material calculations or full numerical algorithms."]},{"id":"J4","sectionIds":["j4"],"status":"scoped-check","checkedAt":"2026-10-19","scope":"DFT and density-functional perturbation theory: selected-published-response-and-metals-check. Source-specific review dates and limitations are listed with the references.","limitations":["Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.","Synthetic checks do not reproduce interacting material calculations or full numerical algorithms."]},{"id":"J5","sectionIds":["j5"],"status":"scoped-check","checkedAt":"2026-10-19","scope":"Hubbard-model QMC and DMRG: selected-published-qmc-dmrg-convergence-check. Source-specific review dates and limitations are listed with the references.","limitations":["Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.","Synthetic checks do not reproduce interacting material calculations or full numerical algorithms."]},{"id":"J6","sectionIds":["j6"],"status":"scoped-check","checkedAt":"2026-10-19","scope":"Dynamical mean-field theory: selected-published-self-consistency-and-limits-check. Source-specific review dates and limitations are listed with the references.","limitations":["Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.","Synthetic checks do not reproduce interacting material calculations or full numerical algorithms."]},{"id":"J7","sectionIds":["j7"],"status":"scoped-check","checkedAt":"2026-10-19","scope":"RPA, fluctuation exchange and pairing kernels: author-review-scoped-check; publisher-original-abstract. Source-specific review dates and limitations are listed with the references.","limitations":["Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.","Synthetic checks do not reproduce interacting material calculations or full numerical algorithms."]},{"id":"J8","sectionIds":["j8"],"status":"scoped-check","checkedAt":"2026-10-19","scope":"Tight-binding and Wannier models: selected-author-gauge-window-interpolation-check. Source-specific review dates and limitations are listed with the references.","limitations":["Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.","Synthetic checks do not reproduce interacting material calculations or full numerical algorithms."]},{"id":"J9","sectionIds":["j9"],"status":"scoped-check","checkedAt":"2026-10-19","scope":"Renormalization group: selected-author-cooper-flow-and-caveats-check. Source-specific review dates and limitations are listed with the references.","limitations":["Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.","Synthetic checks do not reproduce interacting material calculations or full numerical algorithms."]},{"id":"J10","sectionIds":["j10"],"status":"scoped-check","checkedAt":"2026-10-19","scope":"Quantum geometry and Berry methods: selected-author-projector-and-metric-conventions; selected-published-relaxation-correction; selected-published-units-and-flat-band-assumptions. Source-specific review dates and limitations are listed with the references.","limitations":["Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.","Synthetic checks do not reproduce interacting material calculations or full numerical algorithms."]},{"id":"J11","sectionIds":["j11"],"status":"scoped-check","checkedAt":"2026-10-19","scope":"Ginzburg–Landau theory: primary-lecture-scoped-check; selected-author-variance-and-validity-check. Source-specific review dates and limitations are listed with the references.","limitations":["Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.","Synthetic checks do not reproduce interacting material calculations or full numerical algorithms."]},{"id":"J12","sectionIds":["j12"],"status":"scoped-check","checkedAt":"2026-10-19","scope":"Bogoliubov–de Gennes equations: selected-author-chain-and-operator-conventions. Source-specific review dates and limitations are listed with the references.","limitations":["Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.","Synthetic checks do not reproduce interacting material calculations or full numerical algorithms."]}],"body":"## Scope\n\nTheoretical tools answer different questions: whether a model has a pairing instability, how a sample responds, or which approximation is defensible. This handbook connects twelve approved topics to their inputs, outputs and limitations. A solver result is conditional on the Hamiltonian, numerical convergence and physical approximations; it is not an experimental transition record. The J1–J12 identifiers remain stable even where an old planning label needed scientific correction. [R1](#ref-source-ba10e800a1246bf6) [R6](#ref-source-b06e417872e240c9) [R7](#ref-source-51a15c56a9ac1472)\n\n## Mathematical assumptions\n\nState the degrees of freedom, interactions, energy and length units, boundary conditions and limit being taken before comparing calculations. Distinguish an exact identity within a model from an approximation to a real material. The accompanying independent checks cover mass weighting, a four-state atomic Green function, Fourier interpolation, a scalar Cooper-channel flow and complex Bogoliubov matrices. They do not run a material DFT calculation, an interacting impurity solver or a full multichannel renormalization group. [R6](#ref-source-b06e417872e240c9) [R8](#ref-source-12f2513b257d7194) [R11](#ref-source-2b331df83384bc5c) [R12](#ref-source-0aec6d266727d631)\n\n## J1 — BCS mean-field theory\n\nBCS theory constructs a coherent many-electron paired state and a self-consistent gap. In the elementary isotropic model, the quasiparticle energy is $E_k=\\sqrt{\\xi_k^2+|\\Delta|^2}$, with energies measured relative to the chemical potential. The familiar exponential transition scale assumes a weak attractive interaction, an energy shell and a sufficiently smooth density of states. Specify whether that density counts one spin or both; changing the convention without changing the coupling produces a different exponent. [R1](#ref-source-ba10e800a1246bf6)\n\nThe Cooper problem instead places two added electrons above an inert Fermi sea. Its instability motivates pairing but does not replace the many-body gap and number equations. A BCS gap solution describes pair formation within the approximation; phase fluctuations, spatial inhomogeneity and competing order can alter the observed transition. A material's similar gap ratio is therefore a consistency check, not unique proof of a phonon mechanism. Use the full band and symmetry assumptions when extending the elementary model. [R1](#ref-source-ba10e800a1246bf6) [R2](#ref-source-d0ec754310a67e88)\n\n## J2 — Eliashberg theory\n\nEliashberg theory retains the frequency dependence of the interaction and electronic self-energy. A phonon-mediated calculation needs an electron–phonon spectral function, a treatment of the effective Coulomb interaction and an explicitly chosen isotropic, anisotropic or multiband approximation. The gap and mass-renormalization functions are coupled. A converged solution on a Matsubara grid also needs frequency-cutoff and temperature checks; numerical precision alone does not justify neglecting vertex corrections or changes in the phonon system. [R3](#ref-source-d56b04dc0b32c796)\n\nThe selected Marsiglio review provides a simple Einstein-spectrum benchmark, while Choi and colleagues' MgB2 calculation illustrates why band-dependent interactions and gaps can matter. Neither example licenses using one averaged coupling for every material. A Matsubara gap value is not automatically the measured tunnelling gap: real-frequency continuation and spectral features require care. Preserve how the Coulomb parameter was chosen and which electronic and phonon inputs were computed, fitted or assumed. The reviewed examples are scoped checks, not independent reproduction of every numerical result. [R3](#ref-source-d56b04dc0b32c796) [R4](#ref-source-7c4eac077ae5e221)\n\n## J3 — McMillan and Allen–Dynes estimates\n\nCompact transition-temperature estimates summarize solutions of a specified electron–phonon theory; they do not replace its inputs. Allen and Dynes use a logarithmic phonon-frequency scale and introduce strong-coupling and spectral-shape corrections. An implementation should name which version it evaluates, preserve the effective Coulomb parameter, and state whether its frequency input is an energy, angular frequency or temperature. A number copied between these conventions without conversion can be wrong even when the expression looks familiar. [R5](#ref-source-0620df96ebbf2a66)\n\nThe original abstract describes the accuracy of the modified McMillan estimate for the materials known at that time and its failure at larger coupling. That historical comparison is not a universal modern applicability boundary. Report sensitivity to the spectrum and Coulomb input, and compare a compact estimate with a suitably converged Eliashberg solution when justified. Here the original abstract was checked; the inaccessible original fit derivation and dataset are not claimed as fully reviewed. No fitted or hypothetical estimate is entered into the experimental Tc filter. [R5](#ref-source-0620df96ebbf2a66) [R3](#ref-source-d56b04dc0b32c796)\n\n## J4 — DFT and density-functional perturbation theory\n\nDFT and DFPT connect an electronic-structure approximation to forces, harmonic phonons and response functions. The mass-weighted force-constant matrix $D=M^{-1/2}KM^{-1/2}$ has eigenvalues $\\omega^2$, where $K$ is the energy Hessian. Its eigenvalues are not superconducting transition temperatures. An independent two-mass calculation checks the translational zero mode and shows that negative curvature produces a negative squared frequency; removing that sign would conceal an instability. [R6](#ref-source-b06e417872e240c9)\n\nBaroni and colleagues' selected formulation requires self-consistent induced charge and potential. Metallic calculations couple momentum sampling to the smearing treatment; arbitrary numerical smearing is not a physical electronic temperature. The electron–phonon spectral function also differs from the transport-weighted spectrum. Specify structure, exchange-correlation approximation, basis, sampling and response convergence before interpreting a prediction. Harmonic stability alone neither establishes superconductivity nor tests anharmonic effects, strong correlations or every competing structure. The independent checks here verify elementary mechanics, not a full first-principles workflow. [R6](#ref-source-b06e417872e240c9)\n\n## J5 — Hubbard-model QMC and DMRG\n\nQuantum Monte Carlo and density-matrix renormalization group methods address interacting lattice models across different coupling regimes; they are not collectively weak-correlation methods. LeBlanc and colleagues compare several algorithms for specified two-dimensional Hubbard parameters. Agreement must be assessed observable by observable and after the relevant finite-size, temperature and algorithmic extrapolations. Agreement in ground-state energy alone does not establish a particular pairing state or a finite superconducting transition temperature. [R7](#ref-source-51a15c56a9ac1472)\n\nIn the reviewed auxiliary-field formulation, particle-hole symmetry protects the half-filled model with zero next-nearest hopping from the sign problem. Away from that setting, a constrained-path calculation adds trial-state-dependent systematic error and is not generally variational. DMRG requires checks of retained states or bond dimension, truncation error, cylinder length and width, boundary conditions and competing metastable patterns. Taking cylinder length to infinity does not take its width to the two-dimensional thermodynamic limit. Historical computational sizes in a 2015 benchmark are not current hardware limits. [R7](#ref-source-51a15c56a9ac1472)\n\n## J6 — Dynamical mean-field theory\n\nSingle-site DMFT replaces a lattice environment by a self-consistent quantum impurity bath. It retains local temporal fluctuations through a frequency-dependent self-energy; it is not simply static Hartree-Fock theory. For the specified local-interaction models, its large-coordination limit becomes exact with the appropriate hopping scaling. Applied in finite dimensions, the single-site construction omits nonlocal self-energy components, so spatial correlations may require extensions or independent checks. [R8](#ref-source-12f2513b257d7194)\n\nThe impurity Green function must equal the local lattice Green function after self-consistency. The Weiss propagator describing the bath is not generally the bare local lattice propagator. An independent four-state thermal calculation verifies the atomic-limit spectral weights and Green function, but does not validate a finite-hopping DMFT solution. State the impurity solver, bath or frequency discretization, interaction convention and convergence criteria. Two lattices with the same noninteracting density of states need not have identical magnetic response or ordered phases; local single-particle agreement is a narrower result. [R8](#ref-source-12f2513b257d7194)\n\n## J7 — RPA, fluctuation exchange and pairing kernels\n\nRandom-phase and fluctuation-exchange approaches organize selected interaction diagrams into response and pairing approximations. Their names do not specify the entire calculation: identify the bare or dressed propagators, orbital interaction matrix, retained channels and self-consistency. Bickers, Scalapino and White's original abstract describes self-consistent conserving approximations for two-particle correlations. Conservation properties are useful constraints, but do not make an approximation exact for arbitrary coupling or establish a measured material's pairing mechanism. [R10](#ref-source-822deb97a8e9dde0)\n\nScalapino's selected review passages discuss a DCA vertex decomposition and a particle-particle Bethe–Salpeter eigenproblem. They illustrate how a momentum-dependent interaction and dressed propagators enter pairing analysis; those particular results must not be relabelled as an RPA or FLEX benchmark. An eigenvalue reaching the stated instability condition is conditional on the kernel and approximation. Check competing channels, resolution and the treatment of fluctuations before equating its temperature scale with a bulk experimental Tc. The full original conserving-functional derivation remains outside the reviewed scope. [R9](#ref-source-eaa3beeed4368f46) [R10](#ref-source-822deb97a8e9dde0)\n\n## J8 — Tight-binding and Wannier models\n\nWannier functions provide a localized representation of a chosen Bloch subspace. Unitary rotations within an isolated manifold preserve its projector but change the orbital shapes and hopping representation. With entangled bands, selecting the subspace and selecting its gauge are distinct tasks. The frozen inner energy window constrains which states are retained; outside it, projected bands can differ from the original calculation. A compact model therefore needs its orbital basis and energy windows, not just an attractive band plot. [R11](#ref-source-2b331df83384bc5c)\n\nInterpolation relies on localization, a sufficient momentum mesh and adequate real-space hopping range. Independent Fourier recovery of a known finite-range model succeeds, while an undersampled distant hopping aliases into a different term despite perfect agreement on the sampled grid. Validate intermediate momenta and relevant observables. Nonzero Chern invariants can obstruct an exponentially localized representation of the isolated manifold; a different subspace or hybrid construction must be described explicitly. This check is not a material wannierization or proof that an empirical tight-binding fit is unique. [R11](#ref-source-2b331df83384bc5c)\n\n## J9 — Renormalization group\n\nRenormalization group methods track how an effective description changes as higher-energy modes are removed. The retained channels, cutoff, self-energy treatment and truncation determine what a flow can establish. In Shankar's stated Fermi-surface setting, angular Cooper channels decouple at one loop. With the normalization $g=V_l/(4\\pi)$ and $\\ell=\\ln(\\Lambda_0/\\Lambda)$, the selected equation gives $dg/d\\ell=-g^2$. Negative coupling is attractive in this convention. [R12](#ref-source-0aec6d266727d631)\n\nIndependent numerical integration agrees with $g(\\ell)=g_0/(1+g_0\\ell)$ before its pole: attraction grows and repulsion decreases in this restricted flow. The divergence marks an instability and loss of perturbative control, not a measured Tc or a solved ordered state. The source explicitly warns that otherwise irrelevant terms can generate an attractive channel through the Kohn–Luttinger mechanism. Thus the simple positive-channel flow does not prove that every repulsive system remains a stable normal metal. Nested surfaces and additional channels require their own analysis. [R12](#ref-source-0aec6d266727d631)\n\n## J10 — Quantum geometry and Berry methods\n\nBand eigenvectors carry geometric information beyond the dispersion. A quantum metric and Berry curvature can be constructed from the projector onto a specified band or subspace, with carefully stated normalization and gauge conventions. The selected Liang manuscript uses a metric normalization twice the convention used in the linked Wiki geometry article. Comparing numerical coefficients without reconciling that factor would produce a false disagreement. Degenerate or touching bands also require a suitable subspace treatment rather than unstable single-band derivatives. [R13](#ref-source-fa7a165c8c344a63)\n\nGeometric quantities can enter superfluid response, but a nonzero metric alone does not prove superconductivity. Peotta and Törmä's flat-band result has explicit interaction, isolation and pairing assumptions. Huhtinen and colleagues further show why order-parameter relaxation and representation dependence matter when differentiating the free energy. State what is held fixed and what is reoptimized before calling a curvature a physical stiffness. A frozen-field derivative and a relaxed response are not interchangeable; the full multiband response calculation remains a separate task. [R14](#ref-source-72f7ddb9f13bbd81) [R15](#ref-source-fdf091e09e7f4a6f)\n\n## J11 — Ginzburg–Landau theory\n\nGinzburg–Landau theory organizes a free-energy expansion in a complex order parameter and its spatial gradients. Its coefficients and applicable temperature range must be specified; in its conventional derivation it is controlled near the transition. A homogeneous minimum, a linearized upper-critical-field calculation and a vortex-lattice solution are different problems. The type-I/type-II boundary of the elementary isotropic model should not be transferred without qualification to every anisotropic or multicomponent material. [R16](#ref-source-89f5503f5e470385)\n\nBrandt's ideal vortex-lattice calculation illustrates how field, the Ginzburg–Landau parameter and lattice structure affect magnetic observables. Its assumptions exclude arbitrary disorder and pinning, and the commonly used London-limit variance is valid only in a restricted regime. Solving a phenomenological equation with chosen coefficients does not identify a microscopic pairing mechanism. Report boundary conditions, gauge, mesh and free-energy comparisons when several solutions exist; numerical convergence to one stationary configuration alone does not establish that it is the global equilibrium state. [R17](#ref-source-65b23b74f5402e26)\n\n## J12 — Bogoliubov–de Gennes equations\n\nBogoliubov–de Gennes theory writes a quadratic paired Hamiltonian in a particle-and-hole basis. For a complete finite fermionic basis, Hermitian normal block $h$ and antisymmetric pairing matrix $\\Delta$, one convention is $\\mathcal H=\\begin{pmatrix}h&\\Delta\\\\\\Delta^\\dagger&-h^T\\end{pmatrix}$. The doubled description has particle-hole redundancy, not twice as many independent physical states. Independent complex-matrix checks verify Hermiticity and paired eigenvalues in this convention; a reduced spin block can use different notation. [R18](#ref-source-ade538b429fdd847)\n\nSpecify whether the pairing field is imposed or determined self-consistently, together with density, boundaries and the physical basis. Kitaev's selected chain model provides a controlled example of bulk and boundary structure, but a near-zero eigenvalue by itself does not prove a protected experimental Majorana mode. Finite-size splitting, competing localized states and model perturbations must be considered. A self-consistent mean-field solution also does not include every collective or correlation effect. Numerical particle-hole symmetry is a necessary algebra check, not a topological classification or material validation. [R18](#ref-source-ade538b429fdd847)\n\n## Experimental criteria\n\nCompare calculated and measured quantities with matching sample conditions and definitions. A pairing eigenvalue, a harmonic phonon, a local spectral gap and an experimental zero-resistance criterion are different observables. Keep model temperatures out of the measured-Tc catalogue. Predictions should specify independent tests that could reject the assumed mechanism, structure or parameter set. [R4](#ref-source-7c4eac077ae5e221) [R6](#ref-source-b06e417872e240c9) [R7](#ref-source-51a15c56a9ac1472)\n\n## Limits and disputes\n\nEach of the twelve sections has a named review scope. Some references were checked only at abstract level; others use selected author-manuscript passages with final-version comparisons still open. Independent small-model checks verify the stated algebra and counterexamples, not all numerical methods or every cited material. Complete scientific acceptance, full derivations, supplementary datasets and material-level reproduction remain open. [R5](#ref-source-0620df96ebbf2a66) [R7](#ref-source-51a15c56a9ac1472) [R10](#ref-source-822deb97a8e9dde0) [R12](#ref-source-0aec6d266727d631)\n\n## Discovery implications\n\nChoose the simplest defensible model that can distinguish the proposed explanation from a concrete alternative. Carry its uncertainty into any material ranking. A useful workflow moves between structural evidence, electronic and interaction models, numerical convergence and experimentally testable observables; agreement at one stage does not remove assumptions at the next. Retain failed fits and competing states rather than reporting only the most favorable transition estimate. [R6](#ref-source-b06e417872e240c9) [R7](#ref-source-51a15c56a9ac1472) [R11](#ref-source-2b331df83384bc5c)\n\n## Related topics\n\nRead [BCS theory](/scwiki/bcs-weak-coupling), [Eliashberg theory](/scwiki/eliashberg-strong-coupling), [spin fluctuations](/scwiki/spin-fluctuation-pairing), [quantum metric](/scwiki/quantum-metric), [geometric superfluid weight](/scwiki/geometric-superfluid-weight), [Majorana states](/scwiki/majorana-zero-modes), [experimental methods](/scwiki/experimental-methods), and [flat-band temperature limits](/scwiki/open-questions/n10).","revision":"01827ba5196ee5a9c786","sections":[{"id":"scope","title":"Scope","markdown":"## Scope\n\nTheoretical tools answer different questions: whether a model has a pairing instability, how a sample responds, or which approximation is defensible. This handbook connects twelve approved topics to their inputs, outputs and limitations. A solver result is conditional on the Hamiltonian, numerical convergence and physical approximations; it is not an experimental transition record. The J1–J12 identifiers remain stable even where an old planning label needed scientific correction. [R1](#ref-source-ba10e800a1246bf6) [R6](#ref-source-b06e417872e240c9) [R7](#ref-source-51a15c56a9ac1472)"},{"id":"mathematical-assumptions","title":"Mathematical assumptions","markdown":"## Mathematical assumptions\n\nState the degrees of freedom, interactions, energy and length units, boundary conditions and limit being taken before comparing calculations. Distinguish an exact identity within a model from an approximation to a real material. The accompanying independent checks cover mass weighting, a four-state atomic Green function, Fourier interpolation, a scalar Cooper-channel flow and complex Bogoliubov matrices. They do not run a material DFT calculation, an interacting impurity solver or a full multichannel renormalization group. [R6](#ref-source-b06e417872e240c9) [R8](#ref-source-12f2513b257d7194) [R11](#ref-source-2b331df83384bc5c) [R12](#ref-source-0aec6d266727d631)"},{"id":"j1","title":"J1 — BCS mean-field theory","markdown":"## J1 — BCS mean-field theory\n\nBCS theory constructs a coherent many-electron paired state and a self-consistent gap. In the elementary isotropic model, the quasiparticle energy is $E_k=\\sqrt{\\xi_k^2+|\\Delta|^2}$, with energies measured relative to the chemical potential. The familiar exponential transition scale assumes a weak attractive interaction, an energy shell and a sufficiently smooth density of states. Specify whether that density counts one spin or both; changing the convention without changing the coupling produces a different exponent. [R1](#ref-source-ba10e800a1246bf6)\n\nThe Cooper problem instead places two added electrons above an inert Fermi sea. Its instability motivates pairing but does not replace the many-body gap and number equations. A BCS gap solution describes pair formation within the approximation; phase fluctuations, spatial inhomogeneity and competing order can alter the observed transition. A material's similar gap ratio is therefore a consistency check, not unique proof of a phonon mechanism. Use the full band and symmetry assumptions when extending the elementary model. [R1](#ref-source-ba10e800a1246bf6) [R2](#ref-source-d0ec754310a67e88)"},{"id":"j2","title":"J2 — Eliashberg theory","markdown":"## J2 — Eliashberg theory\n\nEliashberg theory retains the frequency dependence of the interaction and electronic self-energy. A phonon-mediated calculation needs an electron–phonon spectral function, a treatment of the effective Coulomb interaction and an explicitly chosen isotropic, anisotropic or multiband approximation. The gap and mass-renormalization functions are coupled. A converged solution on a Matsubara grid also needs frequency-cutoff and temperature checks; numerical precision alone does not justify neglecting vertex corrections or changes in the phonon system. [R3](#ref-source-d56b04dc0b32c796)\n\nThe selected Marsiglio review provides a simple Einstein-spectrum benchmark, while Choi and colleagues' MgB2 calculation illustrates why band-dependent interactions and gaps can matter. Neither example licenses using one averaged coupling for every material. A Matsubara gap value is not automatically the measured tunnelling gap: real-frequency continuation and spectral features require care. Preserve how the Coulomb parameter was chosen and which electronic and phonon inputs were computed, fitted or assumed. The reviewed examples are scoped checks, not independent reproduction of every numerical result. [R3](#ref-source-d56b04dc0b32c796) [R4](#ref-source-7c4eac077ae5e221)"},{"id":"j3","title":"J3 — McMillan and Allen–Dynes estimates","markdown":"## J3 — McMillan and Allen–Dynes estimates\n\nCompact transition-temperature estimates summarize solutions of a specified electron–phonon theory; they do not replace its inputs. Allen and Dynes use a logarithmic phonon-frequency scale and introduce strong-coupling and spectral-shape corrections. An implementation should name which version it evaluates, preserve the effective Coulomb parameter, and state whether its frequency input is an energy, angular frequency or temperature. A number copied between these conventions without conversion can be wrong even when the expression looks familiar. [R5](#ref-source-0620df96ebbf2a66)\n\nThe original abstract describes the accuracy of the modified McMillan estimate for the materials known at that time and its failure at larger coupling. That historical comparison is not a universal modern applicability boundary. Report sensitivity to the spectrum and Coulomb input, and compare a compact estimate with a suitably converged Eliashberg solution when justified. Here the original abstract was checked; the inaccessible original fit derivation and dataset are not claimed as fully reviewed. No fitted or hypothetical estimate is entered into the experimental Tc filter. [R5](#ref-source-0620df96ebbf2a66) [R3](#ref-source-d56b04dc0b32c796)"},{"id":"j4","title":"J4 — DFT and density-functional perturbation theory","markdown":"## J4 — DFT and density-functional perturbation theory\n\nDFT and DFPT connect an electronic-structure approximation to forces, harmonic phonons and response functions. The mass-weighted force-constant matrix $D=M^{-1/2}KM^{-1/2}$ has eigenvalues $\\omega^2$, where $K$ is the energy Hessian. Its eigenvalues are not superconducting transition temperatures. An independent two-mass calculation checks the translational zero mode and shows that negative curvature produces a negative squared frequency; removing that sign would conceal an instability. [R6](#ref-source-b06e417872e240c9)\n\nBaroni and colleagues' selected formulation requires self-consistent induced charge and potential. Metallic calculations couple momentum sampling to the smearing treatment; arbitrary numerical smearing is not a physical electronic temperature. The electron–phonon spectral function also differs from the transport-weighted spectrum. Specify structure, exchange-correlation approximation, basis, sampling and response convergence before interpreting a prediction. Harmonic stability alone neither establishes superconductivity nor tests anharmonic effects, strong correlations or every competing structure. The independent checks here verify elementary mechanics, not a full first-principles workflow. [R6](#ref-source-b06e417872e240c9)"},{"id":"j5","title":"J5 — Hubbard-model QMC and DMRG","markdown":"## J5 — Hubbard-model QMC and DMRG\n\nQuantum Monte Carlo and density-matrix renormalization group methods address interacting lattice models across different coupling regimes; they are not collectively weak-correlation methods. LeBlanc and colleagues compare several algorithms for specified two-dimensional Hubbard parameters. Agreement must be assessed observable by observable and after the relevant finite-size, temperature and algorithmic extrapolations. Agreement in ground-state energy alone does not establish a particular pairing state or a finite superconducting transition temperature. [R7](#ref-source-51a15c56a9ac1472)\n\nIn the reviewed auxiliary-field formulation, particle-hole symmetry protects the half-filled model with zero next-nearest hopping from the sign problem. Away from that setting, a constrained-path calculation adds trial-state-dependent systematic error and is not generally variational. DMRG requires checks of retained states or bond dimension, truncation error, cylinder length and width, boundary conditions and competing metastable patterns. Taking cylinder length to infinity does not take its width to the two-dimensional thermodynamic limit. Historical computational sizes in a 2015 benchmark are not current hardware limits. [R7](#ref-source-51a15c56a9ac1472)"},{"id":"j6","title":"J6 — Dynamical mean-field theory","markdown":"## J6 — Dynamical mean-field theory\n\nSingle-site DMFT replaces a lattice environment by a self-consistent quantum impurity bath. It retains local temporal fluctuations through a frequency-dependent self-energy; it is not simply static Hartree-Fock theory. For the specified local-interaction models, its large-coordination limit becomes exact with the appropriate hopping scaling. Applied in finite dimensions, the single-site construction omits nonlocal self-energy components, so spatial correlations may require extensions or independent checks. [R8](#ref-source-12f2513b257d7194)\n\nThe impurity Green function must equal the local lattice Green function after self-consistency. The Weiss propagator describing the bath is not generally the bare local lattice propagator. An independent four-state thermal calculation verifies the atomic-limit spectral weights and Green function, but does not validate a finite-hopping DMFT solution. State the impurity solver, bath or frequency discretization, interaction convention and convergence criteria. Two lattices with the same noninteracting density of states need not have identical magnetic response or ordered phases; local single-particle agreement is a narrower result. [R8](#ref-source-12f2513b257d7194)"},{"id":"j7","title":"J7 — RPA, fluctuation exchange and pairing kernels","markdown":"## J7 — RPA, fluctuation exchange and pairing kernels\n\nRandom-phase and fluctuation-exchange approaches organize selected interaction diagrams into response and pairing approximations. Their names do not specify the entire calculation: identify the bare or dressed propagators, orbital interaction matrix, retained channels and self-consistency. Bickers, Scalapino and White's original abstract describes self-consistent conserving approximations for two-particle correlations. Conservation properties are useful constraints, but do not make an approximation exact for arbitrary coupling or establish a measured material's pairing mechanism. [R10](#ref-source-822deb97a8e9dde0)\n\nScalapino's selected review passages discuss a DCA vertex decomposition and a particle-particle Bethe–Salpeter eigenproblem. They illustrate how a momentum-dependent interaction and dressed propagators enter pairing analysis; those particular results must not be relabelled as an RPA or FLEX benchmark. An eigenvalue reaching the stated instability condition is conditional on the kernel and approximation. Check competing channels, resolution and the treatment of fluctuations before equating its temperature scale with a bulk experimental Tc. The full original conserving-functional derivation remains outside the reviewed scope. [R9](#ref-source-eaa3beeed4368f46) [R10](#ref-source-822deb97a8e9dde0)"},{"id":"j8","title":"J8 — Tight-binding and Wannier models","markdown":"## J8 — Tight-binding and Wannier models\n\nWannier functions provide a localized representation of a chosen Bloch subspace. Unitary rotations within an isolated manifold preserve its projector but change the orbital shapes and hopping representation. With entangled bands, selecting the subspace and selecting its gauge are distinct tasks. The frozen inner energy window constrains which states are retained; outside it, projected bands can differ from the original calculation. A compact model therefore needs its orbital basis and energy windows, not just an attractive band plot. [R11](#ref-source-2b331df83384bc5c)\n\nInterpolation relies on localization, a sufficient momentum mesh and adequate real-space hopping range. Independent Fourier recovery of a known finite-range model succeeds, while an undersampled distant hopping aliases into a different term despite perfect agreement on the sampled grid. Validate intermediate momenta and relevant observables. Nonzero Chern invariants can obstruct an exponentially localized representation of the isolated manifold; a different subspace or hybrid construction must be described explicitly. This check is not a material wannierization or proof that an empirical tight-binding fit is unique. [R11](#ref-source-2b331df83384bc5c)"},{"id":"j9","title":"J9 — Renormalization group","markdown":"## J9 — Renormalization group\n\nRenormalization group methods track how an effective description changes as higher-energy modes are removed. The retained channels, cutoff, self-energy treatment and truncation determine what a flow can establish. In Shankar's stated Fermi-surface setting, angular Cooper channels decouple at one loop. With the normalization $g=V_l/(4\\pi)$ and $\\ell=\\ln(\\Lambda_0/\\Lambda)$, the selected equation gives $dg/d\\ell=-g^2$. Negative coupling is attractive in this convention. [R12](#ref-source-0aec6d266727d631)\n\nIndependent numerical integration agrees with $g(\\ell)=g_0/(1+g_0\\ell)$ before its pole: attraction grows and repulsion decreases in this restricted flow. The divergence marks an instability and loss of perturbative control, not a measured Tc or a solved ordered state. The source explicitly warns that otherwise irrelevant terms can generate an attractive channel through the Kohn–Luttinger mechanism. Thus the simple positive-channel flow does not prove that every repulsive system remains a stable normal metal. Nested surfaces and additional channels require their own analysis. [R12](#ref-source-0aec6d266727d631)"},{"id":"j10","title":"J10 — Quantum geometry and Berry methods","markdown":"## J10 — Quantum geometry and Berry methods\n\nBand eigenvectors carry geometric information beyond the dispersion. A quantum metric and Berry curvature can be constructed from the projector onto a specified band or subspace, with carefully stated normalization and gauge conventions. The selected Liang manuscript uses a metric normalization twice the convention used in the linked Wiki geometry article. Comparing numerical coefficients without reconciling that factor would produce a false disagreement. Degenerate or touching bands also require a suitable subspace treatment rather than unstable single-band derivatives. [R13](#ref-source-fa7a165c8c344a63)\n\nGeometric quantities can enter superfluid response, but a nonzero metric alone does not prove superconductivity. Peotta and Törmä's flat-band result has explicit interaction, isolation and pairing assumptions. Huhtinen and colleagues further show why order-parameter relaxation and representation dependence matter when differentiating the free energy. State what is held fixed and what is reoptimized before calling a curvature a physical stiffness. A frozen-field derivative and a relaxed response are not interchangeable; the full multiband response calculation remains a separate task. [R14](#ref-source-72f7ddb9f13bbd81) [R15](#ref-source-fdf091e09e7f4a6f)"},{"id":"j11","title":"J11 — Ginzburg–Landau theory","markdown":"## J11 — Ginzburg–Landau theory\n\nGinzburg–Landau theory organizes a free-energy expansion in a complex order parameter and its spatial gradients. Its coefficients and applicable temperature range must be specified; in its conventional derivation it is controlled near the transition. A homogeneous minimum, a linearized upper-critical-field calculation and a vortex-lattice solution are different problems. The type-I/type-II boundary of the elementary isotropic model should not be transferred without qualification to every anisotropic or multicomponent material. [R16](#ref-source-89f5503f5e470385)\n\nBrandt's ideal vortex-lattice calculation illustrates how field, the Ginzburg–Landau parameter and lattice structure affect magnetic observables. Its assumptions exclude arbitrary disorder and pinning, and the commonly used London-limit variance is valid only in a restricted regime. Solving a phenomenological equation with chosen coefficients does not identify a microscopic pairing mechanism. Report boundary conditions, gauge, mesh and free-energy comparisons when several solutions exist; numerical convergence to one stationary configuration alone does not establish that it is the global equilibrium state. [R17](#ref-source-65b23b74f5402e26)"},{"id":"j12","title":"J12 — Bogoliubov–de Gennes equations","markdown":"## J12 — Bogoliubov–de Gennes equations\n\nBogoliubov–de Gennes theory writes a quadratic paired Hamiltonian in a particle-and-hole basis. For a complete finite fermionic basis, Hermitian normal block $h$ and antisymmetric pairing matrix $\\Delta$, one convention is $\\mathcal H=\\begin{pmatrix}h&\\Delta\\\\\\Delta^\\dagger&-h^T\\end{pmatrix}$. The doubled description has particle-hole redundancy, not twice as many independent physical states. Independent complex-matrix checks verify Hermiticity and paired eigenvalues in this convention; a reduced spin block can use different notation. [R18](#ref-source-ade538b429fdd847)\n\nSpecify whether the pairing field is imposed or determined self-consistently, together with density, boundaries and the physical basis. Kitaev's selected chain model provides a controlled example of bulk and boundary structure, but a near-zero eigenvalue by itself does not prove a protected experimental Majorana mode. Finite-size splitting, competing localized states and model perturbations must be considered. A self-consistent mean-field solution also does not include every collective or correlation effect. Numerical particle-hole symmetry is a necessary algebra check, not a topological classification or material validation. [R18](#ref-source-ade538b429fdd847)"},{"id":"experimental-criteria","title":"Experimental criteria","markdown":"## Experimental criteria\n\nCompare calculated and measured quantities with matching sample conditions and definitions. A pairing eigenvalue, a harmonic phonon, a local spectral gap and an experimental zero-resistance criterion are different observables. Keep model temperatures out of the measured-Tc catalogue. Predictions should specify independent tests that could reject the assumed mechanism, structure or parameter set. [R4](#ref-source-7c4eac077ae5e221) [R6](#ref-source-b06e417872e240c9) [R7](#ref-source-51a15c56a9ac1472)"},{"id":"limits-and-disputes","title":"Limits and disputes","markdown":"## Limits and disputes\n\nEach of the twelve sections has a named review scope. Some references were checked only at abstract level; others use selected author-manuscript passages with final-version comparisons still open. Independent small-model checks verify the stated algebra and counterexamples, not all numerical methods or every cited material. Complete scientific acceptance, full derivations, supplementary datasets and material-level reproduction remain open. [R5](#ref-source-0620df96ebbf2a66) [R7](#ref-source-51a15c56a9ac1472) [R10](#ref-source-822deb97a8e9dde0) [R12](#ref-source-0aec6d266727d631)"},{"id":"discovery-implications","title":"Discovery implications","markdown":"## Discovery implications\n\nChoose the simplest defensible model that can distinguish the proposed explanation from a concrete alternative. Carry its uncertainty into any material ranking. A useful workflow moves between structural evidence, electronic and interaction models, numerical convergence and experimentally testable observables; agreement at one stage does not remove assumptions at the next. Retain failed fits and competing states rather than reporting only the most favorable transition estimate. [R6](#ref-source-b06e417872e240c9) [R7](#ref-source-51a15c56a9ac1472) [R11](#ref-source-2b331df83384bc5c)"},{"id":"related-topics","title":"Related topics","markdown":"## Related topics\n\nRead [BCS theory](/scwiki/bcs-weak-coupling), [Eliashberg theory](/scwiki/eliashberg-strong-coupling), [spin fluctuations](/scwiki/spin-fluctuation-pairing), [quantum metric](/scwiki/quantum-metric), [geometric superfluid weight](/scwiki/geometric-superfluid-weight), [Majorana states](/scwiki/majorana-zero-modes), [experimental methods](/scwiki/experimental-methods), and [flat-band temperature limits](/scwiki/open-questions/n10)."},{"id":"material-cases","title":"Material cases","markdown":"## Material cases\n\nNo sample-specific material records are included in this article. Model results are not experimental transition temperatures."}],"markdown":"# Theoretical tools for superconductivity\n\nTwelve theoretical and computational approaches, with their inputs, convergence checks, physical assumptions and limits on predicting superconductivity.\n\n## Scope\n\nTheoretical tools answer different questions: whether a model has a pairing instability, how a sample responds, or which approximation is defensible. This handbook connects twelve approved topics to their inputs, outputs and limitations. A solver result is conditional on the Hamiltonian, numerical convergence and physical approximations; it is not an experimental transition record. The J1–J12 identifiers remain stable even where an old planning label needed scientific correction. [R1](#ref-source-ba10e800a1246bf6) [R6](#ref-source-b06e417872e240c9) [R7](#ref-source-51a15c56a9ac1472)\n\n## Mathematical assumptions\n\nState the degrees of freedom, interactions, energy and length units, boundary conditions and limit being taken before comparing calculations. Distinguish an exact identity within a model from an approximation to a real material. The accompanying independent checks cover mass weighting, a four-state atomic Green function, Fourier interpolation, a scalar Cooper-channel flow and complex Bogoliubov matrices. They do not run a material DFT calculation, an interacting impurity solver or a full multichannel renormalization group. [R6](#ref-source-b06e417872e240c9) [R8](#ref-source-12f2513b257d7194) [R11](#ref-source-2b331df83384bc5c) [R12](#ref-source-0aec6d266727d631)\n\n## J1 — BCS mean-field theory\n\nBCS theory constructs a coherent many-electron paired state and a self-consistent gap. In the elementary isotropic model, the quasiparticle energy is $E_k=\\sqrt{\\xi_k^2+|\\Delta|^2}$, with energies measured relative to the chemical potential. The familiar exponential transition scale assumes a weak attractive interaction, an energy shell and a sufficiently smooth density of states. Specify whether that density counts one spin or both; changing the convention without changing the coupling produces a different exponent. [R1](#ref-source-ba10e800a1246bf6)\n\nThe Cooper problem instead places two added electrons above an inert Fermi sea. Its instability motivates pairing but does not replace the many-body gap and number equations. A BCS gap solution describes pair formation within the approximation; phase fluctuations, spatial inhomogeneity and competing order can alter the observed transition. A material's similar gap ratio is therefore a consistency check, not unique proof of a phonon mechanism. Use the full band and symmetry assumptions when extending the elementary model. [R1](#ref-source-ba10e800a1246bf6) [R2](#ref-source-d0ec754310a67e88)\n\n## J2 — Eliashberg theory\n\nEliashberg theory retains the frequency dependence of the interaction and electronic self-energy. A phonon-mediated calculation needs an electron–phonon spectral function, a treatment of the effective Coulomb interaction and an explicitly chosen isotropic, anisotropic or multiband approximation. The gap and mass-renormalization functions are coupled. A converged solution on a Matsubara grid also needs frequency-cutoff and temperature checks; numerical precision alone does not justify neglecting vertex corrections or changes in the phonon system. [R3](#ref-source-d56b04dc0b32c796)\n\nThe selected Marsiglio review provides a simple Einstein-spectrum benchmark, while Choi and colleagues' MgB2 calculation illustrates why band-dependent interactions and gaps can matter. Neither example licenses using one averaged coupling for every material. A Matsubara gap value is not automatically the measured tunnelling gap: real-frequency continuation and spectral features require care. Preserve how the Coulomb parameter was chosen and which electronic and phonon inputs were computed, fitted or assumed. The reviewed examples are scoped checks, not independent reproduction of every numerical result. [R3](#ref-source-d56b04dc0b32c796) [R4](#ref-source-7c4eac077ae5e221)\n\n## J3 — McMillan and Allen–Dynes estimates\n\nCompact transition-temperature estimates summarize solutions of a specified electron–phonon theory; they do not replace its inputs. Allen and Dynes use a logarithmic phonon-frequency scale and introduce strong-coupling and spectral-shape corrections. An implementation should name which version it evaluates, preserve the effective Coulomb parameter, and state whether its frequency input is an energy, angular frequency or temperature. A number copied between these conventions without conversion can be wrong even when the expression looks familiar. [R5](#ref-source-0620df96ebbf2a66)\n\nThe original abstract describes the accuracy of the modified McMillan estimate for the materials known at that time and its failure at larger coupling. That historical comparison is not a universal modern applicability boundary. Report sensitivity to the spectrum and Coulomb input, and compare a compact estimate with a suitably converged Eliashberg solution when justified. Here the original abstract was checked; the inaccessible original fit derivation and dataset are not claimed as fully reviewed. No fitted or hypothetical estimate is entered into the experimental Tc filter. [R5](#ref-source-0620df96ebbf2a66) [R3](#ref-source-d56b04dc0b32c796)\n\n## J4 — DFT and density-functional perturbation theory\n\nDFT and DFPT connect an electronic-structure approximation to forces, harmonic phonons and response functions. The mass-weighted force-constant matrix $D=M^{-1/2}KM^{-1/2}$ has eigenvalues $\\omega^2$, where $K$ is the energy Hessian. Its eigenvalues are not superconducting transition temperatures. An independent two-mass calculation checks the translational zero mode and shows that negative curvature produces a negative squared frequency; removing that sign would conceal an instability. [R6](#ref-source-b06e417872e240c9)\n\nBaroni and colleagues' selected formulation requires self-consistent induced charge and potential. Metallic calculations couple momentum sampling to the smearing treatment; arbitrary numerical smearing is not a physical electronic temperature. The electron–phonon spectral function also differs from the transport-weighted spectrum. Specify structure, exchange-correlation approximation, basis, sampling and response convergence before interpreting a prediction. Harmonic stability alone neither establishes superconductivity nor tests anharmonic effects, strong correlations or every competing structure. The independent checks here verify elementary mechanics, not a full first-principles workflow. [R6](#ref-source-b06e417872e240c9)\n\n## J5 — Hubbard-model QMC and DMRG\n\nQuantum Monte Carlo and density-matrix renormalization group methods address interacting lattice models across different coupling regimes; they are not collectively weak-correlation methods. LeBlanc and colleagues compare several algorithms for specified two-dimensional Hubbard parameters. Agreement must be assessed observable by observable and after the relevant finite-size, temperature and algorithmic extrapolations. Agreement in ground-state energy alone does not establish a particular pairing state or a finite superconducting transition temperature. [R7](#ref-source-51a15c56a9ac1472)\n\nIn the reviewed auxiliary-field formulation, particle-hole symmetry protects the half-filled model with zero next-nearest hopping from the sign problem. Away from that setting, a constrained-path calculation adds trial-state-dependent systematic error and is not generally variational. DMRG requires checks of retained states or bond dimension, truncation error, cylinder length and width, boundary conditions and competing metastable patterns. Taking cylinder length to infinity does not take its width to the two-dimensional thermodynamic limit. Historical computational sizes in a 2015 benchmark are not current hardware limits. [R7](#ref-source-51a15c56a9ac1472)\n\n## J6 — Dynamical mean-field theory\n\nSingle-site DMFT replaces a lattice environment by a self-consistent quantum impurity bath. It retains local temporal fluctuations through a frequency-dependent self-energy; it is not simply static Hartree-Fock theory. For the specified local-interaction models, its large-coordination limit becomes exact with the appropriate hopping scaling. Applied in finite dimensions, the single-site construction omits nonlocal self-energy components, so spatial correlations may require extensions or independent checks. [R8](#ref-source-12f2513b257d7194)\n\nThe impurity Green function must equal the local lattice Green function after self-consistency. The Weiss propagator describing the bath is not generally the bare local lattice propagator. An independent four-state thermal calculation verifies the atomic-limit spectral weights and Green function, but does not validate a finite-hopping DMFT solution. State the impurity solver, bath or frequency discretization, interaction convention and convergence criteria. Two lattices with the same noninteracting density of states need not have identical magnetic response or ordered phases; local single-particle agreement is a narrower result. [R8](#ref-source-12f2513b257d7194)\n\n## J7 — RPA, fluctuation exchange and pairing kernels\n\nRandom-phase and fluctuation-exchange approaches organize selected interaction diagrams into response and pairing approximations. Their names do not specify the entire calculation: identify the bare or dressed propagators, orbital interaction matrix, retained channels and self-consistency. Bickers, Scalapino and White's original abstract describes self-consistent conserving approximations for two-particle correlations. Conservation properties are useful constraints, but do not make an approximation exact for arbitrary coupling or establish a measured material's pairing mechanism. [R10](#ref-source-822deb97a8e9dde0)\n\nScalapino's selected review passages discuss a DCA vertex decomposition and a particle-particle Bethe–Salpeter eigenproblem. They illustrate how a momentum-dependent interaction and dressed propagators enter pairing analysis; those particular results must not be relabelled as an RPA or FLEX benchmark. An eigenvalue reaching the stated instability condition is conditional on the kernel and approximation. Check competing channels, resolution and the treatment of fluctuations before equating its temperature scale with a bulk experimental Tc. The full original conserving-functional derivation remains outside the reviewed scope. [R9](#ref-source-eaa3beeed4368f46) [R10](#ref-source-822deb97a8e9dde0)\n\n## J8 — Tight-binding and Wannier models\n\nWannier functions provide a localized representation of a chosen Bloch subspace. Unitary rotations within an isolated manifold preserve its projector but change the orbital shapes and hopping representation. With entangled bands, selecting the subspace and selecting its gauge are distinct tasks. The frozen inner energy window constrains which states are retained; outside it, projected bands can differ from the original calculation. A compact model therefore needs its orbital basis and energy windows, not just an attractive band plot. [R11](#ref-source-2b331df83384bc5c)\n\nInterpolation relies on localization, a sufficient momentum mesh and adequate real-space hopping range. Independent Fourier recovery of a known finite-range model succeeds, while an undersampled distant hopping aliases into a different term despite perfect agreement on the sampled grid. Validate intermediate momenta and relevant observables. Nonzero Chern invariants can obstruct an exponentially localized representation of the isolated manifold; a different subspace or hybrid construction must be described explicitly. This check is not a material wannierization or proof that an empirical tight-binding fit is unique. [R11](#ref-source-2b331df83384bc5c)\n\n## J9 — Renormalization group\n\nRenormalization group methods track how an effective description changes as higher-energy modes are removed. The retained channels, cutoff, self-energy treatment and truncation determine what a flow can establish. In Shankar's stated Fermi-surface setting, angular Cooper channels decouple at one loop. With the normalization $g=V_l/(4\\pi)$ and $\\ell=\\ln(\\Lambda_0/\\Lambda)$, the selected equation gives $dg/d\\ell=-g^2$. Negative coupling is attractive in this convention. [R12](#ref-source-0aec6d266727d631)\n\nIndependent numerical integration agrees with $g(\\ell)=g_0/(1+g_0\\ell)$ before its pole: attraction grows and repulsion decreases in this restricted flow. The divergence marks an instability and loss of perturbative control, not a measured Tc or a solved ordered state. The source explicitly warns that otherwise irrelevant terms can generate an attractive channel through the Kohn–Luttinger mechanism. Thus the simple positive-channel flow does not prove that every repulsive system remains a stable normal metal. Nested surfaces and additional channels require their own analysis. [R12](#ref-source-0aec6d266727d631)\n\n## J10 — Quantum geometry and Berry methods\n\nBand eigenvectors carry geometric information beyond the dispersion. A quantum metric and Berry curvature can be constructed from the projector onto a specified band or subspace, with carefully stated normalization and gauge conventions. The selected Liang manuscript uses a metric normalization twice the convention used in the linked Wiki geometry article. Comparing numerical coefficients without reconciling that factor would produce a false disagreement. Degenerate or touching bands also require a suitable subspace treatment rather than unstable single-band derivatives. [R13](#ref-source-fa7a165c8c344a63)\n\nGeometric quantities can enter superfluid response, but a nonzero metric alone does not prove superconductivity. Peotta and Törmä's flat-band result has explicit interaction, isolation and pairing assumptions. Huhtinen and colleagues further show why order-parameter relaxation and representation dependence matter when differentiating the free energy. State what is held fixed and what is reoptimized before calling a curvature a physical stiffness. A frozen-field derivative and a relaxed response are not interchangeable; the full multiband response calculation remains a separate task. [R14](#ref-source-72f7ddb9f13bbd81) [R15](#ref-source-fdf091e09e7f4a6f)\n\n## J11 — Ginzburg–Landau theory\n\nGinzburg–Landau theory organizes a free-energy expansion in a complex order parameter and its spatial gradients. Its coefficients and applicable temperature range must be specified; in its conventional derivation it is controlled near the transition. A homogeneous minimum, a linearized upper-critical-field calculation and a vortex-lattice solution are different problems. The type-I/type-II boundary of the elementary isotropic model should not be transferred without qualification to every anisotropic or multicomponent material. [R16](#ref-source-89f5503f5e470385)\n\nBrandt's ideal vortex-lattice calculation illustrates how field, the Ginzburg–Landau parameter and lattice structure affect magnetic observables. Its assumptions exclude arbitrary disorder and pinning, and the commonly used London-limit variance is valid only in a restricted regime. Solving a phenomenological equation with chosen coefficients does not identify a microscopic pairing mechanism. Report boundary conditions, gauge, mesh and free-energy comparisons when several solutions exist; numerical convergence to one stationary configuration alone does not establish that it is the global equilibrium state. [R17](#ref-source-65b23b74f5402e26)\n\n## J12 — Bogoliubov–de Gennes equations\n\nBogoliubov–de Gennes theory writes a quadratic paired Hamiltonian in a particle-and-hole basis. For a complete finite fermionic basis, Hermitian normal block $h$ and antisymmetric pairing matrix $\\Delta$, one convention is $\\mathcal H=\\begin{pmatrix}h&\\Delta\\\\\\Delta^\\dagger&-h^T\\end{pmatrix}$. The doubled description has particle-hole redundancy, not twice as many independent physical states. Independent complex-matrix checks verify Hermiticity and paired eigenvalues in this convention; a reduced spin block can use different notation. [R18](#ref-source-ade538b429fdd847)\n\nSpecify whether the pairing field is imposed or determined self-consistently, together with density, boundaries and the physical basis. Kitaev's selected chain model provides a controlled example of bulk and boundary structure, but a near-zero eigenvalue by itself does not prove a protected experimental Majorana mode. Finite-size splitting, competing localized states and model perturbations must be considered. A self-consistent mean-field solution also does not include every collective or correlation effect. Numerical particle-hole symmetry is a necessary algebra check, not a topological classification or material validation. [R18](#ref-source-ade538b429fdd847)\n\n## Experimental criteria\n\nCompare calculated and measured quantities with matching sample conditions and definitions. A pairing eigenvalue, a harmonic phonon, a local spectral gap and an experimental zero-resistance criterion are different observables. Keep model temperatures out of the measured-Tc catalogue. Predictions should specify independent tests that could reject the assumed mechanism, structure or parameter set. [R4](#ref-source-7c4eac077ae5e221) [R6](#ref-source-b06e417872e240c9) [R7](#ref-source-51a15c56a9ac1472)\n\n## Limits and disputes\n\nEach of the twelve sections has a named review scope. Some references were checked only at abstract level; others use selected author-manuscript passages with final-version comparisons still open. Independent small-model checks verify the stated algebra and counterexamples, not all numerical methods or every cited material. Complete scientific acceptance, full derivations, supplementary datasets and material-level reproduction remain open. [R5](#ref-source-0620df96ebbf2a66) [R7](#ref-source-51a15c56a9ac1472) [R10](#ref-source-822deb97a8e9dde0) [R12](#ref-source-0aec6d266727d631)\n\n## Discovery implications\n\nChoose the simplest defensible model that can distinguish the proposed explanation from a concrete alternative. Carry its uncertainty into any material ranking. A useful workflow moves between structural evidence, electronic and interaction models, numerical convergence and experimentally testable observables; agreement at one stage does not remove assumptions at the next. Retain failed fits and competing states rather than reporting only the most favorable transition estimate. [R6](#ref-source-b06e417872e240c9) [R7](#ref-source-51a15c56a9ac1472) [R11](#ref-source-2b331df83384bc5c)\n\n## Related topics\n\nRead [BCS theory](/scwiki/bcs-weak-coupling), [Eliashberg theory](/scwiki/eliashberg-strong-coupling), [spin fluctuations](/scwiki/spin-fluctuation-pairing), [quantum metric](/scwiki/quantum-metric), [geometric superfluid weight](/scwiki/geometric-superfluid-weight), [Majorana states](/scwiki/majorana-zero-modes), [experimental methods](/scwiki/experimental-methods), and [flat-band temperature limits](/scwiki/open-questions/n10).\n\n## Material cases\n\nNo sample-specific material records are included in this article. Model results are not experimental transition temperatures.\n\n## Sources\n\n### Ref source-ba10e800a1246bf6\n\n[R1] Theory of Superconductivity (1957). [DOI 10.1103/PhysRev.108.1175](https://doi.org/10.1103/PhysRev.108.1175). Publication: journal-article. Check: primary-full-text-scoped-check. Checked: 2026-10-18. Selected passages: pp. 1182-1183, Eqs. (2.38)-(2.40), (2.49)-(2.50); p. 1186, Eqs. (3.27)-(3.30). One-spin DOS, gap equation and excitation convention checked. Modern limiting constants are evaluated from the model, not copied from historical rounded values. Full proof, empirical fits and exhaustive correction searches remain outside this review.\n### Ref source-d0ec754310a67e88\n\n[R2] Bound Electron Pairs in a Degenerate Fermi Gas (1956). [DOI 10.1103/PhysRev.104.1189](https://doi.org/10.1103/PhysRev.104.1189). Publication: journal-article. Check: primary-full-text-scoped-check. Checked: 2026-10-18. Two-page primary letter: pp. 1189-1190, model setup and Eqs. (3)-(5). Checked the inert Fermi-sea and attractive-shell assumptions and the distinction from the many-body problem. No material-specific pairing mechanism is verified.\n### Ref source-d56b04dc0b32c796\n\n[R3] Eliashberg Theory: a short review (2020). [DOI 10.1016/j.aop.2020.168102](https://doi.org/10.1016/j.aop.2020.168102). Publication: journal-article. Check: author-review-scoped-check. Checked: 2026-10-18. Selected v1 pp. 1–5 and p. 8 model benchmark; p. 5 equations visually checked. Separate independent linearized Einstein-model computation agrees with the approximately 13.3 K benchmark. Full review/final-version equivalence not audited.\n### Ref source-7c4eac077ae5e221\n\n[R4] The origin of the anomalous superconducting properties of MgB2 (2002). [DOI 10.1038/nature00898](https://doi.org/10.1038/nature00898). Publication: journal-article. Check: primary-full-text-scoped-check. Checked: 2026-10-18. Selected pp. 758–759 and Fig. 3 checked during A5: anisotropic calculation, band-dependent gaps and common calculated transition. This article inherits that scoped check, not independent convergence or experimental replication.\n### Ref source-0620df96ebbf2a66\n\n[R5] Transition temperature of strong-coupled superconductors reanalyzed (1975). [DOI 10.1103/PhysRevB.12.905](https://doi.org/10.1103/PhysRevB.12.905). Publication: journal-article. Check: primary-abstract-scoped-check. Checked: 2026-10-18. Publisher abstract: modified prefactor, spectral moment/shape corrections and large-coupling limits within Eliashberg theory. Full derivation and Pb-Bi data not reanalyzed.\n### Ref source-b06e417872e240c9\n\n[R6] Phonons and related crystal properties from density-functional perturbation theory (2001). [DOI 10.1103/RevModPhys.73.515](https://doi.org/10.1103/RevModPhys.73.515). Publication: journal-article. Check: selected-published-response-and-metals-check. Checked: 2026-10-19. Published 6 July 2001. Selected printed pages 515–516, 519, 524, 526, 542 and the left column of 543; equations 80–87 on page 526 visually checked. Mass weighting, self-consistent response, metallic smearing and the distinction between electron–phonon and transport spectra were reviewed. No full DFT implementation or material benchmark was reproduced.\n### Ref source-51a15c56a9ac1472\n\n[R7] Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms (2015). [DOI 10.1103/PhysRevX.5.041041](https://doi.org/10.1103/PhysRevX.5.041041). Publication: journal-article. Check: selected-published-qmc-dmrg-convergence-check. Checked: 2026-10-19. Published 14 December 2015. Selected pages 1–4, 7–8 and 10; the DMRG convergence discussion on page 8 visually checked. The review retains constrained-path and particle-hole conditions, finite-cylinder limits and method-specific errors. Full tables, datasets and all algorithms have not been reproduced.\n### Ref source-12f2513b257d7194\n\n[R8] Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions (1996). [DOI 10.1103/RevModPhys.68.13](https://doi.org/10.1103/RevModPhys.68.13). Publication: journal-article. Check: selected-published-self-consistency-and-limits-check. Checked: 2026-10-19. Selected printed pages 15–18 and 20; equations 6–13 on page 18 visually checked. Local temporal fluctuations, bath self-consistency, large-coordination scaling and atomic/noninteracting limits were reviewed. An independent four-state atomic Green function was checked; no interacting impurity solver or lattice DMFT calculation was reproduced.\n### Ref source-eaa3beeed4368f46\n\n[R9] A common thread: The pairing interaction for unconventional superconductors (2012). [DOI 10.1103/RevModPhys.84.1383](https://doi.org/10.1103/RevModPhys.84.1383). Publication: journal-article. Check: author-review-scoped-check. Checked: 2026-10-18. Selected manuscript text pp. 4, 18–20, 39–46; p. 41 Eq. (13) visually checked. Full 87-page review and final-version equivalence not audited. Independent two-patch algebra is an illustration, not reproduction of a Hubbard calculation.\n### Ref source-822deb97a8e9dde0\n\n[R10] Conserving Approximations for Strongly Correlated Electron Systems: Bethe-Salpeter Equation and Dynamics for the Two-Dimensional Hubbard Model (1989). [DOI 10.1103/PhysRevLett.62.961](https://doi.org/10.1103/PhysRevLett.62.961). Publication: journal-article. Check: publisher-original-abstract. Checked: 2026-10-19. Published 20 February 1989. Original publisher abstract checked: self-consistent conserving approximations and two-particle correlations. The full functional derivation and numerical benchmark have not been reviewed.\n### Ref source-2b331df83384bc5c\n\n[R11] Maximally localized Wannier functions: Theory and applications (2012). [DOI 10.1103/RevModPhys.84.1419](https://doi.org/10.1103/RevModPhys.84.1419). Publication: journal-article. Check: selected-author-gauge-window-interpolation-check. Checked: 2026-10-19. Author version 2, dated 12 May 2012; journal publication 10 October 2012. Selected pages 4–5, 11–13 and 36–37; Figure 5 and the frozen-window explanation on page 13 visually checked. Gauge versus subspace selection, Chern obstruction and interpolation conditions were reviewed. The independent finite-range Fourier example is not a material wannierization. Full final-version comparison remains open.\n### Ref source-0aec6d266727d631\n\n[R12] Renormalization-group approach to interacting fermions (1994). [DOI 10.1103/RevModPhys.66.129](https://doi.org/10.1103/RevModPhys.66.129). Publication: journal-article. Check: selected-author-cooper-flow-and-caveats-check. Checked: 2026-10-19. Author version 2, dated 9 July 1993. Selected pages 92–95 and 101–103; equations 381–384 on page 102 visually checked. Equation 383 supplies the stated normalization; the Kohn–Luttinger caveat on page 103 is retained. Independent scalar RK4 integration is not full RG or an ordered-state solution. Final-version comparison remains open.\n### Ref source-fa7a165c8c344a63\n\n[R13] Band geometry, Berry curvature, and superfluid weight (2017). [DOI 10.1103/PhysRevB.95.024515](https://doi.org/10.1103/PhysRevB.95.024515). Publication: journal-article. Check: selected-author-projector-and-metric-conventions. Checked: 2026-10-18. Authorv2 11January2017,selectedpp1–2,4–6; p5Eq21–24 visuallychecked. Published27January2017. R=2Tr(P dP dP) and gLiang=2gWiki. Final-version comparison and full response derivation remain open.\n### Ref source-72f7ddb9f13bbd81\n\n[R14] Revisiting flat band superconductivity: Dependence on minimal quantum metric and band touchings (2022). [DOI 10.1103/PhysRevB.106.014518](https://doi.org/10.1103/PhysRevB.106.014518). Publication: journal-article. Check: selected-published-relaxation-correction. Checked: 2026-10-18. Published 26 July 2022. Selected final-reprint pages 1–5 were read; Eqs. (13)–(17) on page 4 were visually checked. The review covers order-parameter relaxation, time reversal, Hessian conditions and representation dependence. The earlier selected page 6 reading of minimal integrated trace is retained. The full proof and multiband response have not been reproduced.\n### Ref source-fdf091e09e7f4a6f\n\n[R15] Superfluidity in topologically nontrivial flat bands (2015). [DOI 10.1038/ncomms9944](https://doi.org/10.1038/ncomms9944). Publication: journal-article. Check: selected-published-units-and-flat-band-assumptions. Checked: 2026-10-18. Published 20 November 2015. Selected pages 3–6 were read. Page 3 was visually checked for the free-energy definition and frozen-field derivative in Eq. (1), interpreted with the 2022 response correction. The earlier page 5 review of uniform pairing and scale hierarchy is retained. The full supplementary derivation has not been reviewed.\n### Ref source-89f5503f5e470385\n\n[R16] Nobel Lecture: Type-II superconductors and the vortex lattice (2004). [DOI 10.1103/RevModPhys.76.975](https://doi.org/10.1103/RevModPhys.76.975). Publication: journal-article. Check: primary-lecture-scoped-check. Checked: 2026-10-18. Selected pp. 975-977: GL type-II boundary, upper field Eq. (1), flux quantum Eq. (10), mixed-state boundary around Eq. (11). SI notation and the linearized GL derivation are editorial checks; original Gaussian-unit equations are not copied as SI. Material-specific claims and historical approximations outside these passages are not validated.\n### Ref source-65b23b74f5402e26\n\n[R17] Properties of the ideal Ginzburg-Landau vortex lattice (2003). [DOI 10.1103/PhysRevB.68.054506](https://doi.org/10.1103/PhysRevB.68.054506). Publication: journal-article. Check: selected-author-variance-and-validity-check. Checked: 2026-10-19. Authorv1 submitted10April2003; journal6August2003. Selected pp1,4–6; p5 Eqs11–13 visually checked. Paper sigma is field variance, not a depolarization rate. Ideal isotropic GL model, finite-field validity and restricted London limit retained. Independent triangular reciprocal sum, not full GL reproduction. Typeset2018 is not publication. Final-version comparison open.\n### Ref source-ade538b429fdd847\n\n[R18] Unpaired Majorana fermions in quantum wires (2001). [DOI 10.1070/1063-7869/44/10S/S29](https://doi.org/10.1070/1063-7869/44/10S/S29). Publication: journal-article. Check: selected-author-chain-and-operator-conventions. Checked: 2026-10-18. Author v2 30 October 2000; selected pp4–7, p5 visually checked, equations4–14. Journal2001. Finite noninteracting model checks reproduced; full final-version comparison open.\n\n## Claim evidence and conditions\n\n### scwiki-j-claim-scope-1\n\nTheoretical tools answer different questions: whether a model has a pairing instability, how a sample responds, or which approximation is defensible. This handbook connects twelve approved topics to their inputs, outputs and limitations. A solver result is conditional on the Hamiltonian, numerical convergence and physical approximations; it is not an experimental transition record. The J1–J12 identifiers remain stable even where an old planning label needed scientific correction.\n\nEvidence: theory. Sources: source-ba10e800a1246bf6, source-b06e417872e240c9, source-51a15c56a9ac1472.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-mathematical-assumptions-1\n\nState the degrees of freedom, interactions, energy and length units, boundary conditions and limit being taken before comparing calculations. Distinguish an exact identity within a model from an approximation to a real material. The accompanying independent checks cover mass weighting, a four-state atomic Green function, Fourier interpolation, a scalar Cooper-channel flow and complex Bogoliubov matrices. They do not run a material DFT calculation, an interacting impurity solver or a full multichannel renormalization group.\n\nEvidence: theory. Sources: source-b06e417872e240c9, source-12f2513b257d7194, source-2b331df83384bc5c, source-0aec6d266727d631.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j1-1\n\nBCS theory constructs a coherent many-electron paired state and a self-consistent gap. In the elementary isotropic model, the quasiparticle energy is $E_k=\\sqrt{\\xi_k^2+|\\Delta|^2}$, with energies measured relative to the chemical potential. The familiar exponential transition scale assumes a weak attractive interaction, an energy shell and a sufficiently smooth density of states. Specify whether that density counts one spin or both; changing the convention without changing the coupling produces a different exponent.\n\nEvidence: theory. Sources: source-ba10e800a1246bf6.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j1-2\n\nThe Cooper problem instead places two added electrons above an inert Fermi sea. Its instability motivates pairing but does not replace the many-body gap and number equations. A BCS gap solution describes pair formation within the approximation; phase fluctuations, spatial inhomogeneity and competing order can alter the observed transition. A material's similar gap ratio is therefore a consistency check, not unique proof of a phonon mechanism. Use the full band and symmetry assumptions when extending the elementary model.\n\nEvidence: theory. Sources: source-ba10e800a1246bf6, source-d0ec754310a67e88.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j2-1\n\nEliashberg theory retains the frequency dependence of the interaction and electronic self-energy. A phonon-mediated calculation needs an electron–phonon spectral function, a treatment of the effective Coulomb interaction and an explicitly chosen isotropic, anisotropic or multiband approximation. The gap and mass-renormalization functions are coupled. A converged solution on a Matsubara grid also needs frequency-cutoff and temperature checks; numerical precision alone does not justify neglecting vertex corrections or changes in the phonon system.\n\nEvidence: theory. Sources: source-d56b04dc0b32c796.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j2-2\n\nThe selected Marsiglio review provides a simple Einstein-spectrum benchmark, while Choi and colleagues' MgB2 calculation illustrates why band-dependent interactions and gaps can matter. Neither example licenses using one averaged coupling for every material. A Matsubara gap value is not automatically the measured tunnelling gap: real-frequency continuation and spectral features require care. Preserve how the Coulomb parameter was chosen and which electronic and phonon inputs were computed, fitted or assumed. The reviewed examples are scoped checks, not independent reproduction of every numerical result.\n\nEvidence: theory. Sources: source-d56b04dc0b32c796, source-7c4eac077ae5e221.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j3-1\n\nCompact transition-temperature estimates summarize solutions of a specified electron–phonon theory; they do not replace its inputs. Allen and Dynes use a logarithmic phonon-frequency scale and introduce strong-coupling and spectral-shape corrections. An implementation should name which version it evaluates, preserve the effective Coulomb parameter, and state whether its frequency input is an energy, angular frequency or temperature. A number copied between these conventions without conversion can be wrong even when the expression looks familiar.\n\nEvidence: theory. Sources: source-0620df96ebbf2a66.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j3-2\n\nThe original abstract describes the accuracy of the modified McMillan estimate for the materials known at that time and its failure at larger coupling. That historical comparison is not a universal modern applicability boundary. Report sensitivity to the spectrum and Coulomb input, and compare a compact estimate with a suitably converged Eliashberg solution when justified. Here the original abstract was checked; the inaccessible original fit derivation and dataset are not claimed as fully reviewed. No fitted or hypothetical estimate is entered into the experimental Tc filter.\n\nEvidence: theory. Sources: source-0620df96ebbf2a66, source-d56b04dc0b32c796.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j4-1\n\nDFT and DFPT connect an electronic-structure approximation to forces, harmonic phonons and response functions. The mass-weighted force-constant matrix $D=M^{-1/2}KM^{-1/2}$ has eigenvalues $\\omega^2$, where $K$ is the energy Hessian. Its eigenvalues are not superconducting transition temperatures. An independent two-mass calculation checks the translational zero mode and shows that negative curvature produces a negative squared frequency; removing that sign would conceal an instability.\n\nEvidence: theory. Sources: source-b06e417872e240c9.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j4-2\n\nBaroni and colleagues' selected formulation requires self-consistent induced charge and potential. Metallic calculations couple momentum sampling to the smearing treatment; arbitrary numerical smearing is not a physical electronic temperature. The electron–phonon spectral function also differs from the transport-weighted spectrum. Specify structure, exchange-correlation approximation, basis, sampling and response convergence before interpreting a prediction. Harmonic stability alone neither establishes superconductivity nor tests anharmonic effects, strong correlations or every competing structure. The independent checks here verify elementary mechanics, not a full first-principles workflow.\n\nEvidence: theory. Sources: source-b06e417872e240c9.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j5-1\n\nQuantum Monte Carlo and density-matrix renormalization group methods address interacting lattice models across different coupling regimes; they are not collectively weak-correlation methods. LeBlanc and colleagues compare several algorithms for specified two-dimensional Hubbard parameters. Agreement must be assessed observable by observable and after the relevant finite-size, temperature and algorithmic extrapolations. Agreement in ground-state energy alone does not establish a particular pairing state or a finite superconducting transition temperature.\n\nEvidence: theory. Sources: source-51a15c56a9ac1472.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j5-2\n\nIn the reviewed auxiliary-field formulation, particle-hole symmetry protects the half-filled model with zero next-nearest hopping from the sign problem. Away from that setting, a constrained-path calculation adds trial-state-dependent systematic error and is not generally variational. DMRG requires checks of retained states or bond dimension, truncation error, cylinder length and width, boundary conditions and competing metastable patterns. Taking cylinder length to infinity does not take its width to the two-dimensional thermodynamic limit. Historical computational sizes in a 2015 benchmark are not current hardware limits.\n\nEvidence: theory. Sources: source-51a15c56a9ac1472.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j6-1\n\nSingle-site DMFT replaces a lattice environment by a self-consistent quantum impurity bath. It retains local temporal fluctuations through a frequency-dependent self-energy; it is not simply static Hartree-Fock theory. For the specified local-interaction models, its large-coordination limit becomes exact with the appropriate hopping scaling. Applied in finite dimensions, the single-site construction omits nonlocal self-energy components, so spatial correlations may require extensions or independent checks.\n\nEvidence: theory. Sources: source-12f2513b257d7194.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j6-2\n\nThe impurity Green function must equal the local lattice Green function after self-consistency. The Weiss propagator describing the bath is not generally the bare local lattice propagator. An independent four-state thermal calculation verifies the atomic-limit spectral weights and Green function, but does not validate a finite-hopping DMFT solution. State the impurity solver, bath or frequency discretization, interaction convention and convergence criteria. Two lattices with the same noninteracting density of states need not have identical magnetic response or ordered phases; local single-particle agreement is a narrower result.\n\nEvidence: theory. Sources: source-12f2513b257d7194.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j7-1\n\nRandom-phase and fluctuation-exchange approaches organize selected interaction diagrams into response and pairing approximations. Their names do not specify the entire calculation: identify the bare or dressed propagators, orbital interaction matrix, retained channels and self-consistency. Bickers, Scalapino and White's original abstract describes self-consistent conserving approximations for two-particle correlations. Conservation properties are useful constraints, but do not make an approximation exact for arbitrary coupling or establish a measured material's pairing mechanism.\n\nEvidence: theory. Sources: source-822deb97a8e9dde0.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j7-2\n\nScalapino's selected review passages discuss a DCA vertex decomposition and a particle-particle Bethe–Salpeter eigenproblem. They illustrate how a momentum-dependent interaction and dressed propagators enter pairing analysis; those particular results must not be relabelled as an RPA or FLEX benchmark. An eigenvalue reaching the stated instability condition is conditional on the kernel and approximation. Check competing channels, resolution and the treatment of fluctuations before equating its temperature scale with a bulk experimental Tc. The full original conserving-functional derivation remains outside the reviewed scope.\n\nEvidence: theory. Sources: source-eaa3beeed4368f46, source-822deb97a8e9dde0.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j8-1\n\nWannier functions provide a localized representation of a chosen Bloch subspace. Unitary rotations within an isolated manifold preserve its projector but change the orbital shapes and hopping representation. With entangled bands, selecting the subspace and selecting its gauge are distinct tasks. The frozen inner energy window constrains which states are retained; outside it, projected bands can differ from the original calculation. A compact model therefore needs its orbital basis and energy windows, not just an attractive band plot.\n\nEvidence: theory. Sources: source-2b331df83384bc5c.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j8-2\n\nInterpolation relies on localization, a sufficient momentum mesh and adequate real-space hopping range. Independent Fourier recovery of a known finite-range model succeeds, while an undersampled distant hopping aliases into a different term despite perfect agreement on the sampled grid. Validate intermediate momenta and relevant observables. Nonzero Chern invariants can obstruct an exponentially localized representation of the isolated manifold; a different subspace or hybrid construction must be described explicitly. This check is not a material wannierization or proof that an empirical tight-binding fit is unique.\n\nEvidence: theory. Sources: source-2b331df83384bc5c.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j9-1\n\nRenormalization group methods track how an effective description changes as higher-energy modes are removed. The retained channels, cutoff, self-energy treatment and truncation determine what a flow can establish. In Shankar's stated Fermi-surface setting, angular Cooper channels decouple at one loop. With the normalization $g=V_l/(4\\pi)$ and $\\ell=\\ln(\\Lambda_0/\\Lambda)$, the selected equation gives $dg/d\\ell=-g^2$. Negative coupling is attractive in this convention.\n\nEvidence: theory. Sources: source-0aec6d266727d631.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j9-2\n\nIndependent numerical integration agrees with $g(\\ell)=g_0/(1+g_0\\ell)$ before its pole: attraction grows and repulsion decreases in this restricted flow. The divergence marks an instability and loss of perturbative control, not a measured Tc or a solved ordered state. The source explicitly warns that otherwise irrelevant terms can generate an attractive channel through the Kohn–Luttinger mechanism. Thus the simple positive-channel flow does not prove that every repulsive system remains a stable normal metal. Nested surfaces and additional channels require their own analysis.\n\nEvidence: theory. Sources: source-0aec6d266727d631.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j10-1\n\nBand eigenvectors carry geometric information beyond the dispersion. A quantum metric and Berry curvature can be constructed from the projector onto a specified band or subspace, with carefully stated normalization and gauge conventions. The selected Liang manuscript uses a metric normalization twice the convention used in the linked Wiki geometry article. Comparing numerical coefficients without reconciling that factor would produce a false disagreement. Degenerate or touching bands also require a suitable subspace treatment rather than unstable single-band derivatives.\n\nEvidence: theory. Sources: source-fa7a165c8c344a63.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j10-2\n\nGeometric quantities can enter superfluid response, but a nonzero metric alone does not prove superconductivity. Peotta and Törmä's flat-band result has explicit interaction, isolation and pairing assumptions. Huhtinen and colleagues further show why order-parameter relaxation and representation dependence matter when differentiating the free energy. State what is held fixed and what is reoptimized before calling a curvature a physical stiffness. A frozen-field derivative and a relaxed response are not interchangeable; the full multiband response calculation remains a separate task.\n\nEvidence: theory. Sources: source-72f7ddb9f13bbd81, source-fdf091e09e7f4a6f.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j11-1\n\nGinzburg–Landau theory organizes a free-energy expansion in a complex order parameter and its spatial gradients. Its coefficients and applicable temperature range must be specified; in its conventional derivation it is controlled near the transition. A homogeneous minimum, a linearized upper-critical-field calculation and a vortex-lattice solution are different problems. The type-I/type-II boundary of the elementary isotropic model should not be transferred without qualification to every anisotropic or multicomponent material.\n\nEvidence: theory. Sources: source-89f5503f5e470385.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j11-2\n\nBrandt's ideal vortex-lattice calculation illustrates how field, the Ginzburg–Landau parameter and lattice structure affect magnetic observables. Its assumptions exclude arbitrary disorder and pinning, and the commonly used London-limit variance is valid only in a restricted regime. Solving a phenomenological equation with chosen coefficients does not identify a microscopic pairing mechanism. Report boundary conditions, gauge, mesh and free-energy comparisons when several solutions exist; numerical convergence to one stationary configuration alone does not establish that it is the global equilibrium state.\n\nEvidence: theory. Sources: source-65b23b74f5402e26.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j12-1\n\nBogoliubov–de Gennes theory writes a quadratic paired Hamiltonian in a particle-and-hole basis. For a complete finite fermionic basis, Hermitian normal block $h$ and antisymmetric pairing matrix $\\Delta$, one convention is $\\mathcal H=\\begin{pmatrix}h&\\Delta\\\\\\Delta^\\dagger&-h^T\\end{pmatrix}$. The doubled description has particle-hole redundancy, not twice as many independent physical states. Independent complex-matrix checks verify Hermiticity and paired eigenvalues in this convention; a reduced spin block can use different notation.\n\nEvidence: theory. Sources: source-ade538b429fdd847.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-j12-2\n\nSpecify whether the pairing field is imposed or determined self-consistently, together with density, boundaries and the physical basis. Kitaev's selected chain model provides a controlled example of bulk and boundary structure, but a near-zero eigenvalue by itself does not prove a protected experimental Majorana mode. Finite-size splitting, competing localized states and model perturbations must be considered. A self-consistent mean-field solution also does not include every collective or correlation effect. Numerical particle-hole symmetry is a necessary algebra check, not a topological classification or material validation.\n\nEvidence: theory. Sources: source-ade538b429fdd847.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-experimental-criteria-1\n\nCompare calculated and measured quantities with matching sample conditions and definitions. A pairing eigenvalue, a harmonic phonon, a local spectral gap and an experimental zero-resistance criterion are different observables. Keep model temperatures out of the measured-Tc catalogue. Predictions should specify independent tests that could reject the assumed mechanism, structure or parameter set.\n\nEvidence: theory. Sources: source-7c4eac077ae5e221, source-b06e417872e240c9, source-51a15c56a9ac1472.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-limits-and-disputes-1\n\nEach of the twelve sections has a named review scope. Some references were checked only at abstract level; others use selected author-manuscript passages with final-version comparisons still open. Independent small-model checks verify the stated algebra and counterexamples, not all numerical methods or every cited material. Complete scientific acceptance, full derivations, supplementary datasets and material-level reproduction remain open.\n\nEvidence: theory. Sources: source-0620df96ebbf2a66, source-51a15c56a9ac1472, source-822deb97a8e9dde0, source-0aec6d266727d631.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n### scwiki-j-claim-discovery-implications-1\n\nChoose the simplest defensible model that can distinguish the proposed explanation from a concrete alternative. Carry its uncertainty into any material ranking. A useful workflow moves between structural evidence, electronic and interaction models, numerical convergence and experimentally testable observables; agreement at one stage does not remove assumptions at the next. Retain failed fits and competing states rather than reporting only the most favorable transition estimate.\n\nEvidence: hypothesis. Sources: source-b06e417872e240c9, source-51a15c56a9ac1472, source-2b331df83384bc5c.\n\nConditions: Model, degrees of freedom, units, boundary conditions and numerical/physical approximation must be specified; no experimental transition record inferred.\n\nVerification scope: Named selected original passages or explicit abstract-only checks, plus independent finite-model algebra where stated. Full scientific acceptance remains open.\n\n## Approved scope items\n\nJ1. Checked: 2026-10-19. BCS mean-field theory: primary-full-text-scoped-check; primary-full-text-scoped-check. Source-specific review dates and limitations are listed with the references.\n\n- Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.\n- Synthetic checks do not reproduce interacting material calculations or full numerical algorithms.\n\nJ2. Checked: 2026-10-19. Eliashberg theory: author-review-scoped-check; primary-full-text-scoped-check. Source-specific review dates and limitations are listed with the references.\n\n- Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.\n- Synthetic checks do not reproduce interacting material calculations or full numerical algorithms.\n\nJ3. Checked: 2026-10-19. McMillan and Allen–Dynes estimates: author-review-scoped-check; primary-abstract-scoped-check. Source-specific review dates and limitations are listed with the references.\n\n- Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.\n- Synthetic checks do not reproduce interacting material calculations or full numerical algorithms.\n\nJ4. Checked: 2026-10-19. DFT and density-functional perturbation theory: selected-published-response-and-metals-check. Source-specific review dates and limitations are listed with the references.\n\n- Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.\n- Synthetic checks do not reproduce interacting material calculations or full numerical algorithms.\n\nJ5. Checked: 2026-10-19. Hubbard-model QMC and DMRG: selected-published-qmc-dmrg-convergence-check. Source-specific review dates and limitations are listed with the references.\n\n- Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.\n- Synthetic checks do not reproduce interacting material calculations or full numerical algorithms.\n\nJ6. Checked: 2026-10-19. Dynamical mean-field theory: selected-published-self-consistency-and-limits-check. Source-specific review dates and limitations are listed with the references.\n\n- Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.\n- Synthetic checks do not reproduce interacting material calculations or full numerical algorithms.\n\nJ7. Checked: 2026-10-19. RPA, fluctuation exchange and pairing kernels: author-review-scoped-check; publisher-original-abstract. Source-specific review dates and limitations are listed with the references.\n\n- Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.\n- Synthetic checks do not reproduce interacting material calculations or full numerical algorithms.\n\nJ8. Checked: 2026-10-19. Tight-binding and Wannier models: selected-author-gauge-window-interpolation-check. Source-specific review dates and limitations are listed with the references.\n\n- Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.\n- Synthetic checks do not reproduce interacting material calculations or full numerical algorithms.\n\nJ9. Checked: 2026-10-19. Renormalization group: selected-author-cooper-flow-and-caveats-check. Source-specific review dates and limitations are listed with the references.\n\n- Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.\n- Synthetic checks do not reproduce interacting material calculations or full numerical algorithms.\n\nJ10. Checked: 2026-10-19. Quantum geometry and Berry methods: selected-author-projector-and-metric-conventions; selected-published-relaxation-correction; selected-published-units-and-flat-band-assumptions. Source-specific review dates and limitations are listed with the references.\n\n- Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.\n- Synthetic checks do not reproduce interacting material calculations or full numerical algorithms.\n\nJ11. Checked: 2026-10-19. Ginzburg–Landau theory: primary-lecture-scoped-check; selected-author-variance-and-validity-check. Source-specific review dates and limitations are listed with the references.\n\n- Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.\n- Synthetic checks do not reproduce interacting material calculations or full numerical algorithms.\n\nJ12. Checked: 2026-10-19. Bogoliubov–de Gennes equations: selected-author-chain-and-operator-conventions. Source-specific review dates and limitations are listed with the references.\n\n- Selected passages and explicit abstract-only boundaries; full scientific acceptance remains open.\n- Synthetic checks do not reproduce interacting material calculations or full numerical algorithms.\n\n## Revision and review\n\nArticle ID: scwiki-j. Revision: 01827ba5196ee5a9c786. Updated: 2026-10-18T19:29:40.890430+00:00.\n\nReview scope: Twelve individually mapped theoretical tools. Selected DFPT, Hubbard benchmark, DMFT, Wannier and RG passages were reviewed, together with bounded Allen–Dynes and conserving-approximation abstracts. Earlier BCS, Eliashberg, geometry, GL and Kitaev checks retain their original limits. Independent finite-model calculations check the stated algebra.\n\n- Complete scientific acceptance remains open.\n- Some original references remain abstract-only; full final-version/supplement/dataset comparisons are incomplete.\n- Independent model algebra does not validate an interacting material, every solver or predicted transition.\n- Planning source defines scope only; it is not scientific evidence.\n\nSource: [02_BATCH_2_IN_PLAN.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/02_BATCH_2_IN_PLAN.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 502293e5ed2fe8463b61a6c5b78a2c33bc33f3265bd36fe534fa2c469e42ff48.\n","wordCount":2280}}