{"schemaVersion":"1.0","revision":"fa961ab24277a91737c3","article":{"id":"scwiki-c8","topicId":"scwiki-topic-c8","code":"C8","slug":"pair-density-wave","title":"Pair-density waves: finite momentum, harmonics and experimental evidence","summary":"Distinguish pair-field modulation, charge order, FFLO mechanisms and the probes used to test them.","aliases":["PDW","pair density wave","FFLO","Fulde Ferrell","Larkin Ovchinnikov","phase diffusion","vortex halo","charge 4e","CPDM","orbital FFLO"],"evidenceTypes":["theory","experiment"],"relatedCodes":["B6","C3","C7","B11"],"editorialStatus":"scoped-check","contentState":"pilot","modifiedAt":"2026-10-19T14:56:05.646640+00:00","review":{"checkedAt":"2026-10-19","scope":"Selected published Hamidian and Edkins methods/results; Berg stacking theory; inherited kagome passages; bounded historical and 2026 abstract checks; independent Fourier and overlap examples. Follow-up: corrected LO identity; FF abstract current conditions; scalar intensity harmonics and quadrature coexistence.","limitations":["Full scientific acceptance, complete source-version comparisons and raw-data reproduction remain open.","Original Larkin–Ovchinnikov full text was unavailable; bibliographic identity only is checked.","Recent orbital-FFLO and CPDM reports have limited passage/abstract coverage, not full experimental review.","Imposed scalar models do not establish a stable material phase or reproduce instrument response."]},"provenance":{"repository":"JackZH26/jzis-superconductivity","commit":"0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf","path":"C_pairing_symmetry/C8_PDW.md","sha256":"8fa1922ee16b00f0aa5c314028d5ced0709c5d98f6b8868b100a04ad51689119","sourceId":"file-de794072c9a145a8"},"sourceMap":[{"startLine":1,"endLine":15,"targetSections":["scope"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; experimental misattribution corrected and algebra/probe distinctions expanded."},{"startLine":16,"endLine":31,"targetSections":["scope","mathematical-assumptions"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; experimental misattribution corrected and algebra/probe distinctions expanded."},{"startLine":32,"endLine":38,"targetSections":["mathematical-assumptions","experimental-criteria"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; experimental misattribution corrected and algebra/probe distinctions expanded."},{"startLine":39,"endLine":46,"targetSections":["experimental-criteria"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; experimental misattribution corrected and algebra/probe distinctions expanded."},{"startLine":47,"endLine":67,"targetSections":["experimental-criteria","limits-and-disputes"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; experimental misattribution corrected and algebra/probe distinctions expanded."},{"startLine":68,"endLine":87,"targetSections":["discovery-implications","related-topics"],"decision":"edited-with-scoped-evidence","note":"Pinned text preserved; experimental misattribution corrected and algebra/probe distinctions expanded."}],"references":[{"id":"source-824e7016c4d4c3a8","doi":"10.1103/PhysRev.135.A550","title":"Superconductivity in a Strong Spin-Exchange Field","year":1964,"url":"https://doi.org/10.1103/PhysRev.135.A550","publicationStatus":"journal-article","verification":{"scope":"publisher-abstract-check","checkedAt":"2026-10-19","evidenceUrl":"https://journals.aps.org/pr/abstract/10.1103/PhysRev.135.A550","note":"Published 3 August 1964. Clean exchange-field and compensated-current abstract checked; original derivation not fully reviewed."}},{"id":"source-739939afd549524f","doi":null,"title":"Nonuniform state of superconductors","year":1965,"url":"https://www.itp.ac.ru/ru/persons/ovchinnikov-yurii-nikolaevich/","publicationStatus":"journal-article","verification":{"scope":"institutional-bibliographic-check","checkedAt":"2026-10-19","evidenceUrl":"https://www.itp.ac.ru/ru/persons/ovchinnikov-yurii-nikolaevich/","note":"Landau Institute author bibliography, entry262: Russian JETP47(3),1136-1146(1964); English Sov.Phys.JETP20(3),762-769(1965). Year here denotes the English translation. Original full text remains inaccessible. Corrected an erroneous shared identity with Migdal1958; no DOI invented."}},{"id":"source-b33aab72ae697f3a","doi":"10.1038/375561a0","title":"Evidence for stripe correlations of spins and holes in copper oxide superconductors","year":1995,"url":"https://doi.org/10.1038/375561a0","publicationStatus":"journal-article","verification":{"scope":"publisher-abstract-check","checkedAt":"2026-10-18","evidenceUrl":"https://www.nature.com/articles/375561a0","note":"Publisher abstract and identity checked: 15 June 1995, Nd-substituted compound x=0.12, neutron diffraction. Full paper and data not reviewed."}},{"id":"source-7148789acde770d4","doi":"10.1103/PhysRevLett.99.127003","title":"Dynamical Layer Decoupling in a Stripe-Ordered High-Tc Superconductor","year":2007,"url":"https://doi.org/10.1103/PhysRevLett.99.127003","publicationStatus":"journal-article","verification":{"scope":"selected-published-text-check","checkedAt":"2026-10-19","evidenceUrl":"https://eunahkim.ccmr.cornell.edu/sites/kim/files/publications/Phys.%20Rev.%20Lett%202007%20Berg.pdf","note":"Published text pp.1–4 read for stacking, lowest-order cancellation, residual coupling and defects. Figures and complete microscopic derivation not reproduced. Local scalar overlaps are independent illustrations. Follow-up: abstract, antiphase pattern and interplane coupling discussion aroundEq.1; supports conditional model context, not the local quadrature example."}},{"id":"source-66a2b2a9519f6f44","doi":"10.1038/nature17411","title":"Detection of a Cooper-pair density wave in Bi2Sr2CaCu2O8+x","year":2016,"url":"https://doi.org/10.1038/nature17411","publicationStatus":"journal-article","verification":{"scope":"selected-published-and-author-text-check","checkedAt":"2026-10-18","evidenceUrl":"https://eunahkim.lassp.cornell.edu/sites/kim/files/publications/nature17411.pdf","note":"Published 13 April 2016; selected final pp.1–4 and methods p.6 checked, p.6 visually. Author v3 pp.4–7/19–21 and Fig.4 p.18 also checked. Phase-diffusion equations agree at selected scope; complete version/data review open."}},{"id":"source-d306adac5d567426","doi":"10.1126/science.aat1773","title":"Magnetic field-induced pair density wave state in the cuprate vortex halo","year":2019,"url":"https://doi.org/10.1126/science.aat1773","publicationStatus":"journal-article","verification":{"scope":"selected-published-text-check","checkedAt":"2026-10-19","evidenceUrl":"https://davis-group-quantum-matter-research.ie/publicationPDF/Science_364_976.pdf","note":"Published 7 June 2019; selected main-text pp.1–4, result passage p.5, Figs.1/4 pp.2/4 visually checked. Supplement and raw fits remain open. Distinct from Hamidian 2016 SJTM. Follow-up: publishedp1,Eqs.2-3 distinguish symmetry-allowed density-like products from an imposed scalar intensity identity. No new full supplement or data fit."}},{"id":"source-057c43c86174f961","doi":"10.1038/s41586-021-03983-5","title":"Roton pair density wave in a strong-coupling kagome superconductor","year":2021,"url":"https://doi.org/10.1038/s41586-021-03983-5","publicationStatus":"journal-article","verification":{"scope":"selected-author-manuscript-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/2103.09188v2","note":"Selected pp. 3–7 and methods pp. 17–18; Fig. 3 on p. 14 visually checked. Author version 2 September 2021 has a different title and mistyped journal volume 559; publisher confirms Nature 599. Final-version/raw-data comparison open."}},{"id":"source-cb08c313e0597487","doi":"10.1038/nphys1389","title":"Charge-4e superconductivity from pair-density-wave order in certain high-temperature superconductors","year":2009,"url":"https://doi.org/10.1038/nphys1389","publicationStatus":"journal-article","verification":{"scope":"author-abstract-and-publication-check","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/abs/0904.1230v3","note":"Author v3 dated 7 June 2009 is marked corrected; abstract and journal identity checked. Thermal-melting and fractional-flux proposal only; full phase diagram not reviewed."}},{"id":"source-ad66f48ab9369ce9","doi":"10.1038/s41467-026-72134-z","title":"Spectroscopic evidence for a first-order transition to a possible orbital Fulde-Ferrell-Larkin-Ovchinnikov state","year":2026,"url":"https://doi.org/10.1038/s41467-026-72134-z","publicationStatus":"journal-article","verification":{"scope":"selected-publisher-web-passages-check","checkedAt":"2026-10-18","evidenceUrl":"https://www.nature.com/articles/s41467-026-72134-z","note":"Published 12 May 2026. Indexed publisher abstract/introduction and selected model passage read; model omits vortices. Full methods, supplementary material and data not reviewed."}},{"id":"source-61d655195d7db259","doi":"10.1038/s41586-026-10325-w","title":"Moiré engineering of Cooper-pair density modulation states","year":2026,"url":"https://doi.org/10.1038/s41586-026-10325-w","publicationStatus":"journal-article","verification":{"scope":"institutional-abstract-and-publication-check","checkedAt":"2026-10-18","evidenceUrl":"https://pure.psu.edu/en/publications/moir%C3%A9-engineering-of-cooper-pair-density-modulation-states/","note":"Penn State author-affiliation record and complete abstract checked; issue 9 April 2026. CPDM terminology relative to moiré lattice retained. Full experimental/model review open."}}],"claims":[{"id":"scwiki-c8-claim-scope-1","sectionId":"scope","statement":"A pair-density wave (PDW) concerns finite-wavevector components of a superconducting pair field. A charge-density wave (CDW), a modulated tunneling spectrum and a modulated pair field are different observables. Specify the underlying lattice or superlattice, the electromagnetic gauge convention and whether a uniform condensate coexists. A single plane-wave phase has constant magnitude; periodic magnitude alone also need not establish a new spontaneous broken translation symmetry.","evidenceType":"theory","sourceIds":["source-824e7016c4d4c3a8","source-61d655195d7db259"],"conditions":"Finite wavevector is stated relative to a lattice and gauge. Conventional exchange-field FFLO and orbital proposals have different drivers.","verificationScope":"Named primary passages and independent examples only; full scientific acceptance remains open."},{"id":"scwiki-c8-claim-scope-2","sectionId":"scope","statement":"Fulde and Ferrell's 1964 publisher abstract specifies a spatially uniform spin-exchange field with no scattering. Their finite-momentum state has a compensating contribution from unpaired electrons, so its equilibrium total current is zero despite the spatial phase variation. This statement concerns their idealized model; the original derivation is not reproduced here. Larkin and Ovchinnikov's paper appeared in Russian in 1964 and in English translation in 1965. The author institute's bibliography confirms that identity, but the original full text remains an access gap and its derivation has not been checked. These historical assumptions do not by themselves validate a modern orbital-FFLO interpretation.","evidenceType":"theory","sourceIds":["source-824e7016c4d4c3a8","source-739939afd549524f"],"conditions":"Fulde-Ferrell ideal uniform exchange field and no scattering, as stated in its publisher abstract. Larkin-Ovchinnikov identity only: Russian1964, English1965; no original full-text derivation reviewed.","verificationScope":"Selected primary abstract/text and institutional identity checked2026-10-19; local Fourier examples checked separately. No independent review or experimental replication.","sourceRelationship":"context-for-editorial-analysis","derivationNote":"The current cancellation is a reported model result from the abstract; the historical identity is bibliographic, and the warning against extending the model is editorial."},{"id":"scwiki-c8-claim-mathematical-assumptions-1","sectionId":"mathematical-assumptions","statement":"For a scalar illustration in a fixed gauge, write $\\Delta(x)=d_0+d_+e^{iQx}+d_-e^{-iQx}$. The Fourier coefficients of $|\\Delta|^2$ are $C_0=|d_0|^2+|d_+|^2+|d_-|^2$, $C_Q=d_+d_0^*+d_0d_-^*$ and $C_{2Q}=d_+d_-^*$, with negative-wavevector coefficients given by conjugation. Four independent discrete Fourier examples reproduce these identities. They are algebraic intensity identities, not a derivation of electronic charge density or a tunneling response.","evidenceType":"theory","sourceIds":["source-7148789acde770d4","source-d306adac5d567426"],"conditions":"Fixed scalar amplitudes and imposed wavevectors; complete periods for zero-overlap result. No self-consistent stability or material response fit. Envelope ratio requires a constant uniform background.","verificationScope":"Named primary passages and independent examples only; full scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"The exact scalar Fourier coefficients are local algebra; cited stripe/halo work supplies physical context, not a fitted response."},{"id":"scwiki-c8-claim-mathematical-assumptions-2","sectionId":"mathematical-assumptions","statement":"For $d_0=d_-=0$, the Fulde–Ferrell-type plane wave has constant $|\\Delta|$. Nonzero equal real $d_+=d_-=d/2$ instead give a Larkin–Ovchinnikov-type standing wave $d\\cos Qx$. In this fixed gauge and for $Q\\ne0$, the pair field has period $2\\pi/|Q|$, while $|\\Delta|^2$ has period $\\pi/|Q|$. A nonzero real uniform component $u$ adds a Q harmonic, but a quadrature component does not: $\\Delta=iu+d\\cos Qx$ gives $|\\Delta|^2=u^2+d^2\\cos^2Qx$ for real $u,d$. Thus a missing Q intensity harmonic need not exclude even coexistence with a uniform component. These are imposed scalar-field identities, not calculated electronic charge harmonics: a measured 2Q density signal also requires a response model and does not uniquely identify a PDW. No free energies are compared.","evidenceType":"theory","sourceIds":["source-824e7016c4d4c3a8","source-7148789acde770d4","source-d306adac5d567426"],"conditions":"Imposed scalar amplitudes in a fixed gauge, Q nonzero and real nonzero standing-wave amplitude d. A real u is aligned; iu is in quadrature. |Delta|^2 is not a measured charge-density or tunneling signal.","verificationScope":"Selected primary abstract/text and institutional identity checked2026-10-19; local Fourier examples checked separately. No independent review or experimental replication.","sourceRelationship":"context-for-local-derivation","derivationNote":"Plane and standing waves and the quadrature counterexample are author-imposed fields. Source models motivate the distinction; no ground state or experimental density is derived."},{"id":"scwiki-c8-claim-mathematical-assumptions-3","sectionId":"mathematical-assumptions","statement":"Under an electron gauge phase $\\alpha$, both pair components acquire $e^{2i\\alpha}$. The product $d_+d_-^*$ is neutral, whereas $d_+d_-$ transforms with $e^{4i\\alpha}$ and has zero net wavevector. Three numerical phase checks confirm the distinction. A composite with charge 4e is not by itself evidence for a thermodynamically coherent charge-4e phase. Berg, Fradkin and Kivelson's 2009 melting theory proposes such a phase and fractional flux as a test; that proposal is distinct from the 2007 layer-decoupling paper.","evidenceType":"theory","sourceIds":["source-cb08c313e0597487"],"conditions":"Fixed scalar amplitudes and imposed wavevectors; complete periods for zero-overlap result. No self-consistent stability or material response fit. Envelope ratio requires a constant uniform background.","verificationScope":"Named primary passages and independent examples only; full scientific acceptance remains open.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"Local gauge-product checks are distinguished from the cited thermodynamic melting proposal."},{"id":"scwiki-c8-claim-mathematical-assumptions-4","sectionId":"mathematical-assumptions","statement":"Berg and colleagues' 2007 stripe model suppresses leading interplane Josephson coupling through a particular antiphase pattern and stacking. An independent overlap calculation gives $\\langle\\cos Qx\\cos Qy\\rangle=0$ over complete periods for perpendicular stripes, but parallel stripes give $1/2$. Adding a common uniform amplitude $u$ gives $u^2$; a cropped window need not cancel. The quartic overlap remains $\\langle\\cos^2Qx\\cos^2Qy\\rangle=1/4$. This illustrates why cancellation of the lowest coupling does not mean all coupling vanishes. The paper retains higher-order/further-neighbor terms and discusses defects that spoil exact cancellation.","evidenceType":"theory","sourceIds":["source-7148789acde770d4"],"conditions":"Fixed scalar amplitudes and imposed wavevectors; complete periods for zero-overlap result. No self-consistent stability or material response fit. Envelope ratio requires a constant uniform background.","verificationScope":"Named primary passages and independent examples only; full scientific acceptance remains open.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"The source analyzes stacking and residual couplings; local scalar overlaps illustrate only selected symmetry and integration-window conditions."},{"id":"scwiki-c8-claim-mathematical-assumptions-5","sectionId":"mathematical-assumptions","statement":"An illustrative vortex-halo envelope $A(r)=e^{-r/\\xi}$ produces a Q amplitude proportional to $A\\Delta_0^*$ and a 2Q amplitude proportional to $A^2$. If $\\Delta_0$ is approximately constant, the latter decays twice as fast. Four independent logarithmic-slope checks verify this and show failure of that ratio when the uniform component recovers from zero. The relation is conditional, especially near the core; it is not a universal PDW diagnostic.","evidenceType":"theory","sourceIds":["source-d306adac5d567426"],"conditions":"Fixed scalar amplitudes and imposed wavevectors; complete periods for zero-overlap result. No self-consistent stability or material response fit. Envelope ratio requires a constant uniform background.","verificationScope":"Named primary passages and independent examples only; full scientific acceptance remains open.","sourceRelationship":"context-for-local-derivation","derivationNote":"The chosen exponential envelopes and logarithmic derivatives are local checks inspired by the source response model; no raw-data fit."},{"id":"scwiki-c8-claim-experimental-criteria-1","sectionId":"experimental-criteria","statement":"Tranquada and colleagues' 1995 neutron study concerns static spin/charge stripes in Nd-substituted La–Sr–Cu–O at $x=0.12$. That diffraction evidence does not directly measure a superconducting pair field. Layer decoupling and charge harmonics are useful cross-checks only when competing charge, magnetic and structural explanations are addressed.","evidenceType":"experiment","sourceIds":["source-b33aab72ae697f3a","source-7148789acde770d4"],"conditions":"Keep each study and probe separate: 2016 pair tunneling below 50 mK; 2019 single-electron 0/8.25T comparison at 2 K. Spectroscopy and phase interpretation are not interchangeable.","verificationScope":"Named primary passages and independent examples only; full scientific acceptance remains open."},{"id":"scwiki-c8-claim-experimental-criteria-2","sectionId":"experimental-criteria","statement":"Hamidian and colleagues' 2016 experiment used a superconducting Bi2212 nanoflake tip for scanned Josephson tunneling microscopy below 50 mK. It reported modulations near $Q_P=(0.25\\pm0.02)\\,2\\pi/a_0$, superposed on a stronger uniform component. It is not the later field-induced vortex-halo spectroscopy experiment. Their phase-diffusion model uses $I(V)=I_J^2 ZV/[2(V^2+V_c^2)]$, with $V_c=2eZk_BT^*/\\hbar$. The measured peak $I_c=\\hbar I_J^2/(8ek_BT^*)$ differs from the underlying Josephson critical current $I_J$; three independent numerical peak searches verify this relation. The effective noise temperature $T^*$ must not simply be replaced by refrigerator temperature.","evidenceType":"experiment","sourceIds":["source-66a2b2a9519f6f44"],"conditions":"Keep each study and probe separate: 2016 pair tunneling below 50 mK; 2019 single-electron 0/8.25T comparison at 2 K. Spectroscopy and phase interpretation are not interchangeable.","verificationScope":"Named primary passages and independent examples only; full scientific acceptance remains open.","sourceRelationship":"mixed-source-and-local-derivation","derivationNote":"Reported Josephson measurement and phase-diffusion formula are separated from the local numerical peak checks; noise temperature is not fitted here."},{"id":"scwiki-c8-claim-experimental-criteria-3","sectionId":"experimental-criteria","statement":"Edkins and colleagues' 2019 study used single-electron STM, explicitly without scanned Josephson tunneling microscopy. At 2 K it compared 0 and 8.25 T in the same Bi2212 field of view using the same tip. Approximately eight-cell and four-cell electronic modulations support a PDW interpretation within the authors' model. The fitted Fourier widths are not exactly in a two-to-one ratio. Sublattice form factors belong to the measured electronic response and must not be copied directly onto the pair field. The core region is excluded from the simple envelope argument. Raw-data fitting and the complete supplement are not reproduced here.","evidenceType":"experiment","sourceIds":["source-d306adac5d567426"],"conditions":"Keep each study and probe separate: 2016 pair tunneling below 50 mK; 2019 single-electron 0/8.25T comparison at 2 K. Spectroscopy and phase interpretation are not interchangeable.","verificationScope":"Named primary passages and independent examples only; full scientific acceptance remains open."},{"id":"scwiki-c8-claim-experimental-criteria-4","sectionId":"experimental-criteria","statement":"Chen and colleagues' 2021 CsV3Sb5 work reports spatial gap/conductance modulations and Josephson spectroscopy on prepared surfaces. Bulk and surface temperature scales differ, and the author manuscript explicitly says roton excitations were not directly observed. Its proposed complex PDW interpretation is not a direct phase measurement or a universal statement about kagome superconductors. The selected manuscript and methods check is inherited from the chiral-pairing entry.","evidenceType":"experiment","sourceIds":["source-057c43c86174f961"],"conditions":"Keep each study and probe separate: 2016 pair tunneling below 50 mK; 2019 single-electron 0/8.25T comparison at 2 K. Spectroscopy and phase interpretation are not interchangeable.","verificationScope":"Named primary passages and independent examples only; full scientific acceptance remains open."},{"id":"scwiki-c8-claim-experimental-criteria-5","sectionId":"experimental-criteria","statement":"In May 2026, Cao and colleagues reported a hysteretic tunneling-gap transition in multilayer NbSe2 under an in-plane field, consistent with a possible orbital FFLO state. Their reported angle, disorder and thickness dependences matter; the selected theoretical discussion neglects vortices in its energy comparison. This is evidence for a transition with a proposed interpretation, not a direct image of finite pair momentum. Only the publisher abstract, introduction and selected model passage have been checked here.","evidenceType":"experiment","sourceIds":["source-ad66f48ab9369ce9"],"conditions":"Keep each study and probe separate: 2016 pair tunneling below 50 mK; 2019 single-electron 0/8.25T comparison at 2 K. Spectroscopy and phase interpretation are not interchangeable.","verificationScope":"Named primary passages and independent examples only; full scientific acceptance remains open."},{"id":"scwiki-c8-claim-limits-and-disputes-1","sectionId":"limits-and-disputes","statement":"Keep four distinctions explicit: pair tunneling versus single-electron tunneling; induced modulation versus a primary instability; field-induced versus spontaneous behavior; and a surface observation versus a bulk thermodynamic phase. A particular gap map cannot alone determine the pair phase. A nonzero composite amplitude, a half-vortex proposal or an unusual transport curve likewise needs a separate coherence and alternative-explanation analysis.","evidenceType":"theory","sourceIds":["source-66a2b2a9519f6f44","source-d306adac5d567426","source-cb08c313e0597487"],"conditions":"Surface and bulk, induced and primary, charge and pair observables are distinct. No material-wide proof inferred from a modulation.","verificationScope":"Named primary passages and independent examples only; full scientific acceptance remains open."},{"id":"scwiki-c8-claim-limits-and-disputes-2","sectionId":"limits-and-disputes","statement":"The 2026 report by Wang and colleagues deliberately calls its moiré result Cooper-pair density modulation (CPDM): the authors describe periodic order-parameter variation without additional translation-symmetry breaking. That distinction prevents treating every periodic pairing texture as a spontaneous PDW. This entry has checked the institutional abstract and publication identity, not the complete measurement or microscopic model.","evidenceType":"theory","sourceIds":["source-61d655195d7db259"],"conditions":"Surface and bulk, induced and primary, charge and pair observables are distinct. No material-wide proof inferred from a modulation.","verificationScope":"Named primary passages and independent examples only; full scientific acceptance remains open."},{"id":"scwiki-c8-claim-discovery-implications-1","sectionId":"discovery-implications","statement":"Store the probe, temperature, magnetic-field magnitude and orientation, surface preparation, Fourier convention, uniform component and normalization with every candidate. Compare observations with an explicit response model before assigning an order parameter. Our local algebra, peak and overlap checks test stated examples; they do not reproduce the experiments, establish a material ground state, or close full scientific acceptance.","evidenceType":"theory","sourceIds":["source-7148789acde770d4","source-66a2b2a9519f6f44","source-d306adac5d567426","source-ad66f48ab9369ce9"],"conditions":"Retain actual methods and evidence scope with each candidate; local model checks do not constitute full scientific acceptance.","verificationScope":"Named primary passages and independent examples only; full scientific acceptance remains open."}],"corrections":[{"id":"scwiki-c8-correction-1","description":"Corrected inherited attribution: Hamidian 2016 is Josephson imaging; field-induced vortex-halo single-electron spectroscopy is Edkins 2019.","sourceId":"source-d306adac5d567426","scope":"Distinct probe, field, temperature and harmonic evidence."},{"id":"scwiki-c8-correction-2","description":"Separated the 2007 stripe-layer proposal from the 2009 thermal-melting charge-4e proposal.","sourceId":"source-cb08c313e0597487","scope":"Theory identity and bounded review."},{"id":"scwiki-c8-correction-3","description":"Conventional Zeeman-driven FFLO is not the only modern use of FFLO terminology; orbital proposals and moiré-induced CPDM require separate assumptions.","sourceId":"source-ad66f48ab9369ce9","scope":"2026 selected publisher/institutional passages only."},{"id":"scwiki-c8-correction-4","description":"Separated Larkin-Ovchinnikov English1965 from the unrelated Migdal1958 source identity, year and URL. The new reference links the verified author-institute bibliography while original full text remains unavailable.","sourceId":"source-739939afd549524f","scope":"Publication identity correction; not a new full-text scientific review."}],"equationChecks":[{"targetSectionId":"mathematical-assumptions","tex":"C_Q=d_+d_0^*+d_0d_-^*","check":"Four independent discrete Fourier cases including single-plane-wave, standing-wave, coexistence and complex phases."},{"targetSectionId":"experimental-criteria","tex":"I_c=\\hbar I_J^2/(8ek_BT^*)","check":"Three independent numerical peak searches using specified effective noise temperatures."},{"targetSectionId":"mathematical-assumptions","tex":"\\langle\\cos Qx\\cos Qy\\rangle=0","check":"Five overlap examples including uniform admixture, quartic coupling and finite window; four conditional envelope-decay checks and three gauge checks."},{"targetSectionId":"mathematical-assumptions","tex":"|iu+d\\cos Qx|^2=u^2+d^2\\cos^2Qx","check":"Five scalar Fourier cases on1024points each and four period-check points. Quadrature coexistence retains a uniform component without a Q intensity coefficient; no physical response/free-energy model."}],"materialCases":[{"id":"scwiki-c8-material-1","formula":"Bi2Sr2CaCu2O8+x","sample":"Hamidian et al. Bi2212 sample, p approximately 17%, superconducting nanoflake tip","recordType":"experiment","tcK":null,"reportedTemperatureK":null,"temperatureKind":"measurement","temperatureNote":"Below 0.05 K operating condition, not an exact temperature or a superconducting transition. Effective noise temperature is separate.","criterion":"pair-tunneling-operating-temperature","pressureRegime":"unknown","pressureGPa":null,"equilibrium":"unknown","probe":"Scanned Josephson tunneling microscopy in the phase-diffusion regime; field condition not assigned at this scoped review","sourceId":"source-66a2b2a9519f6f44","aliases":["cuprate","cuprates"]},{"id":"scwiki-c8-material-2","formula":"Bi2Sr2CaCu2O8+x","sample":"Edkins et al. slightly underdoped sample, p approximately 17%, same-tip same-view comparison","recordType":"experiment","tcK":null,"reportedTemperatureK":2,"temperatureKind":"measurement","temperatureNote":"STM operating temperature; separate from the sample superconducting transition.","criterion":"single-electron-spectroscopy-temperature","pressureRegime":"unknown","pressureGPa":null,"equilibrium":"unknown","probe":"Single-electron STM; field-induced difference between 8.25 T and 0 T, 58 nm by 58 nm view","sourceId":"source-d306adac5d567426","aliases":["cuprate","cuprates"]}],"body":"## Scope\n\nA pair-density wave (PDW) concerns finite-wavevector components of a superconducting pair field. A charge-density wave (CDW), a modulated tunneling spectrum and a modulated pair field are different observables. Specify the underlying lattice or superlattice, the electromagnetic gauge convention and whether a uniform condensate coexists. A single plane-wave phase has constant magnitude; periodic magnitude alone also need not establish a new spontaneous broken translation symmetry. [R1](#ref-source-824e7016c4d4c3a8) [R10](#ref-source-61d655195d7db259)\n\nFulde and Ferrell's 1964 publisher abstract specifies a spatially uniform spin-exchange field with no scattering. Their finite-momentum state has a compensating contribution from unpaired electrons, so its equilibrium total current is zero despite the spatial phase variation. This statement concerns their idealized model; the original derivation is not reproduced here. Larkin and Ovchinnikov's paper appeared in Russian in 1964 and in English translation in 1965. The author institute's bibliography confirms that identity, but the original full text remains an access gap and its derivation has not been checked. These historical assumptions do not by themselves validate a modern orbital-FFLO interpretation. [R1](#ref-source-824e7016c4d4c3a8) [R2](#ref-source-739939afd549524f)\n\n## Mathematical assumptions\n\nFor a scalar illustration in a fixed gauge, write $\\Delta(x)=d_0+d_+e^{iQx}+d_-e^{-iQx}$. The Fourier coefficients of $|\\Delta|^2$ are $C_0=|d_0|^2+|d_+|^2+|d_-|^2$, $C_Q=d_+d_0^*+d_0d_-^*$ and $C_{2Q}=d_+d_-^*$, with negative-wavevector coefficients given by conjugation. Four independent discrete Fourier examples reproduce these identities. They are algebraic intensity identities, not a derivation of electronic charge density or a tunneling response. [R4](#ref-source-7148789acde770d4) [R6](#ref-source-d306adac5d567426)\n\nFor $d_0=d_-=0$, the Fulde–Ferrell-type plane wave has constant $|\\Delta|$. Nonzero equal real $d_+=d_-=d/2$ instead give a Larkin–Ovchinnikov-type standing wave $d\\cos Qx$. In this fixed gauge and for $Q\\ne0$, the pair field has period $2\\pi/|Q|$, while $|\\Delta|^2$ has period $\\pi/|Q|$. A nonzero real uniform component $u$ adds a Q harmonic, but a quadrature component does not: $\\Delta=iu+d\\cos Qx$ gives $|\\Delta|^2=u^2+d^2\\cos^2Qx$ for real $u,d$. Thus a missing Q intensity harmonic need not exclude even coexistence with a uniform component. These are imposed scalar-field identities, not calculated electronic charge harmonics: a measured 2Q density signal also requires a response model and does not uniquely identify a PDW. No free energies are compared. [R1](#ref-source-824e7016c4d4c3a8) [R4](#ref-source-7148789acde770d4) [R6](#ref-source-d306adac5d567426)\n\nUnder an electron gauge phase $\\alpha$, both pair components acquire $e^{2i\\alpha}$. The product $d_+d_-^*$ is neutral, whereas $d_+d_-$ transforms with $e^{4i\\alpha}$ and has zero net wavevector. Three numerical phase checks confirm the distinction. A composite with charge 4e is not by itself evidence for a thermodynamically coherent charge-4e phase. Berg, Fradkin and Kivelson's 2009 melting theory proposes such a phase and fractional flux as a test; that proposal is distinct from the 2007 layer-decoupling paper. [R8](#ref-source-cb08c313e0597487)\n\nBerg and colleagues' 2007 stripe model suppresses leading interplane Josephson coupling through a particular antiphase pattern and stacking. An independent overlap calculation gives $\\langle\\cos Qx\\cos Qy\\rangle=0$ over complete periods for perpendicular stripes, but parallel stripes give $1/2$. Adding a common uniform amplitude $u$ gives $u^2$; a cropped window need not cancel. The quartic overlap remains $\\langle\\cos^2Qx\\cos^2Qy\\rangle=1/4$. This illustrates why cancellation of the lowest coupling does not mean all coupling vanishes. The paper retains higher-order/further-neighbor terms and discusses defects that spoil exact cancellation. [R4](#ref-source-7148789acde770d4)\n\nAn illustrative vortex-halo envelope $A(r)=e^{-r/\\xi}$ produces a Q amplitude proportional to $A\\Delta_0^*$ and a 2Q amplitude proportional to $A^2$. If $\\Delta_0$ is approximately constant, the latter decays twice as fast. Four independent logarithmic-slope checks verify this and show failure of that ratio when the uniform component recovers from zero. The relation is conditional, especially near the core; it is not a universal PDW diagnostic. [R6](#ref-source-d306adac5d567426)\n\n## Experimental criteria\n\nTranquada and colleagues' 1995 neutron study concerns static spin/charge stripes in Nd-substituted La–Sr–Cu–O at $x=0.12$. That diffraction evidence does not directly measure a superconducting pair field. Layer decoupling and charge harmonics are useful cross-checks only when competing charge, magnetic and structural explanations are addressed. [R3](#ref-source-b33aab72ae697f3a) [R4](#ref-source-7148789acde770d4)\n\nHamidian and colleagues' 2016 experiment used a superconducting Bi2212 nanoflake tip for scanned Josephson tunneling microscopy below 50 mK. It reported modulations near $Q_P=(0.25\\pm0.02)\\,2\\pi/a_0$, superposed on a stronger uniform component. It is not the later field-induced vortex-halo spectroscopy experiment. Their phase-diffusion model uses $I(V)=I_J^2 ZV/[2(V^2+V_c^2)]$, with $V_c=2eZk_BT^*/\\hbar$. The measured peak $I_c=\\hbar I_J^2/(8ek_BT^*)$ differs from the underlying Josephson critical current $I_J$; three independent numerical peak searches verify this relation. The effective noise temperature $T^*$ must not simply be replaced by refrigerator temperature. [R5](#ref-source-66a2b2a9519f6f44)\n\nEdkins and colleagues' 2019 study used single-electron STM, explicitly without scanned Josephson tunneling microscopy. At 2 K it compared 0 and 8.25 T in the same Bi2212 field of view using the same tip. Approximately eight-cell and four-cell electronic modulations support a PDW interpretation within the authors' model. The fitted Fourier widths are not exactly in a two-to-one ratio. Sublattice form factors belong to the measured electronic response and must not be copied directly onto the pair field. The core region is excluded from the simple envelope argument. Raw-data fitting and the complete supplement are not reproduced here. [R6](#ref-source-d306adac5d567426)\n\nChen and colleagues' 2021 CsV3Sb5 work reports spatial gap/conductance modulations and Josephson spectroscopy on prepared surfaces. Bulk and surface temperature scales differ, and the author manuscript explicitly says roton excitations were not directly observed. Its proposed complex PDW interpretation is not a direct phase measurement or a universal statement about kagome superconductors. The selected manuscript and methods check is inherited from the chiral-pairing entry. [R7](#ref-source-057c43c86174f961)\n\nIn May 2026, Cao and colleagues reported a hysteretic tunneling-gap transition in multilayer NbSe2 under an in-plane field, consistent with a possible orbital FFLO state. Their reported angle, disorder and thickness dependences matter; the selected theoretical discussion neglects vortices in its energy comparison. This is evidence for a transition with a proposed interpretation, not a direct image of finite pair momentum. Only the publisher abstract, introduction and selected model passage have been checked here. [R9](#ref-source-ad66f48ab9369ce9)\n\n## Limits and disputes\n\nKeep four distinctions explicit: pair tunneling versus single-electron tunneling; induced modulation versus a primary instability; field-induced versus spontaneous behavior; and a surface observation versus a bulk thermodynamic phase. A particular gap map cannot alone determine the pair phase. A nonzero composite amplitude, a half-vortex proposal or an unusual transport curve likewise needs a separate coherence and alternative-explanation analysis. [R5](#ref-source-66a2b2a9519f6f44) [R6](#ref-source-d306adac5d567426) [R8](#ref-source-cb08c313e0597487)\n\nThe 2026 report by Wang and colleagues deliberately calls its moiré result Cooper-pair density modulation (CPDM): the authors describe periodic order-parameter variation without additional translation-symmetry breaking. That distinction prevents treating every periodic pairing texture as a spontaneous PDW. This entry has checked the institutional abstract and publication identity, not the complete measurement or microscopic model. [R10](#ref-source-61d655195d7db259)\n\n## Discovery implications\n\nStore the probe, temperature, magnetic-field magnitude and orientation, surface preparation, Fourier convention, uniform component and normalization with every candidate. Compare observations with an explicit response model before assigning an order parameter. Our local algebra, peak and overlap checks test stated examples; they do not reproduce the experiments, establish a material ground state, or close full scientific acceptance. [R4](#ref-source-7148789acde770d4) [R5](#ref-source-66a2b2a9519f6f44) [R6](#ref-source-d306adac5d567426) [R9](#ref-source-ad66f48ab9369ce9)\n\n## Related topics\n\nContinue with [charge-fluctuation pairing](/scwiki/charge-fluctuation-pairing), [d-wave pairing](/scwiki/d-wave), [chiral pairing](/scwiki/chiral-pairing) and [topological superconductivity](/scwiki/topological-superconductivity).","revision":"5c165f574b5f4a8f3c4a","sections":[{"id":"scope","title":"Scope","markdown":"## Scope\n\nA pair-density wave (PDW) concerns finite-wavevector components of a superconducting pair field. A charge-density wave (CDW), a modulated tunneling spectrum and a modulated pair field are different observables. Specify the underlying lattice or superlattice, the electromagnetic gauge convention and whether a uniform condensate coexists. A single plane-wave phase has constant magnitude; periodic magnitude alone also need not establish a new spontaneous broken translation symmetry. [R1](#ref-source-824e7016c4d4c3a8) [R10](#ref-source-61d655195d7db259)\n\nFulde and Ferrell's 1964 publisher abstract specifies a spatially uniform spin-exchange field with no scattering. Their finite-momentum state has a compensating contribution from unpaired electrons, so its equilibrium total current is zero despite the spatial phase variation. This statement concerns their idealized model; the original derivation is not reproduced here. Larkin and Ovchinnikov's paper appeared in Russian in 1964 and in English translation in 1965. The author institute's bibliography confirms that identity, but the original full text remains an access gap and its derivation has not been checked. These historical assumptions do not by themselves validate a modern orbital-FFLO interpretation. [R1](#ref-source-824e7016c4d4c3a8) [R2](#ref-source-739939afd549524f)"},{"id":"mathematical-assumptions","title":"Mathematical assumptions","markdown":"## Mathematical assumptions\n\nFor a scalar illustration in a fixed gauge, write $\\Delta(x)=d_0+d_+e^{iQx}+d_-e^{-iQx}$. The Fourier coefficients of $|\\Delta|^2$ are $C_0=|d_0|^2+|d_+|^2+|d_-|^2$, $C_Q=d_+d_0^*+d_0d_-^*$ and $C_{2Q}=d_+d_-^*$, with negative-wavevector coefficients given by conjugation. Four independent discrete Fourier examples reproduce these identities. They are algebraic intensity identities, not a derivation of electronic charge density or a tunneling response. [R4](#ref-source-7148789acde770d4) [R6](#ref-source-d306adac5d567426)\n\nFor $d_0=d_-=0$, the Fulde–Ferrell-type plane wave has constant $|\\Delta|$. Nonzero equal real $d_+=d_-=d/2$ instead give a Larkin–Ovchinnikov-type standing wave $d\\cos Qx$. In this fixed gauge and for $Q\\ne0$, the pair field has period $2\\pi/|Q|$, while $|\\Delta|^2$ has period $\\pi/|Q|$. A nonzero real uniform component $u$ adds a Q harmonic, but a quadrature component does not: $\\Delta=iu+d\\cos Qx$ gives $|\\Delta|^2=u^2+d^2\\cos^2Qx$ for real $u,d$. Thus a missing Q intensity harmonic need not exclude even coexistence with a uniform component. These are imposed scalar-field identities, not calculated electronic charge harmonics: a measured 2Q density signal also requires a response model and does not uniquely identify a PDW. No free energies are compared. [R1](#ref-source-824e7016c4d4c3a8) [R4](#ref-source-7148789acde770d4) [R6](#ref-source-d306adac5d567426)\n\nUnder an electron gauge phase $\\alpha$, both pair components acquire $e^{2i\\alpha}$. The product $d_+d_-^*$ is neutral, whereas $d_+d_-$ transforms with $e^{4i\\alpha}$ and has zero net wavevector. Three numerical phase checks confirm the distinction. A composite with charge 4e is not by itself evidence for a thermodynamically coherent charge-4e phase. Berg, Fradkin and Kivelson's 2009 melting theory proposes such a phase and fractional flux as a test; that proposal is distinct from the 2007 layer-decoupling paper. [R8](#ref-source-cb08c313e0597487)\n\nBerg and colleagues' 2007 stripe model suppresses leading interplane Josephson coupling through a particular antiphase pattern and stacking. An independent overlap calculation gives $\\langle\\cos Qx\\cos Qy\\rangle=0$ over complete periods for perpendicular stripes, but parallel stripes give $1/2$. Adding a common uniform amplitude $u$ gives $u^2$; a cropped window need not cancel. The quartic overlap remains $\\langle\\cos^2Qx\\cos^2Qy\\rangle=1/4$. This illustrates why cancellation of the lowest coupling does not mean all coupling vanishes. The paper retains higher-order/further-neighbor terms and discusses defects that spoil exact cancellation. [R4](#ref-source-7148789acde770d4)\n\nAn illustrative vortex-halo envelope $A(r)=e^{-r/\\xi}$ produces a Q amplitude proportional to $A\\Delta_0^*$ and a 2Q amplitude proportional to $A^2$. If $\\Delta_0$ is approximately constant, the latter decays twice as fast. Four independent logarithmic-slope checks verify this and show failure of that ratio when the uniform component recovers from zero. The relation is conditional, especially near the core; it is not a universal PDW diagnostic. [R6](#ref-source-d306adac5d567426)"},{"id":"experimental-criteria","title":"Experimental criteria","markdown":"## Experimental criteria\n\nTranquada and colleagues' 1995 neutron study concerns static spin/charge stripes in Nd-substituted La–Sr–Cu–O at $x=0.12$. That diffraction evidence does not directly measure a superconducting pair field. Layer decoupling and charge harmonics are useful cross-checks only when competing charge, magnetic and structural explanations are addressed. [R3](#ref-source-b33aab72ae697f3a) [R4](#ref-source-7148789acde770d4)\n\nHamidian and colleagues' 2016 experiment used a superconducting Bi2212 nanoflake tip for scanned Josephson tunneling microscopy below 50 mK. It reported modulations near $Q_P=(0.25\\pm0.02)\\,2\\pi/a_0$, superposed on a stronger uniform component. It is not the later field-induced vortex-halo spectroscopy experiment. Their phase-diffusion model uses $I(V)=I_J^2 ZV/[2(V^2+V_c^2)]$, with $V_c=2eZk_BT^*/\\hbar$. The measured peak $I_c=\\hbar I_J^2/(8ek_BT^*)$ differs from the underlying Josephson critical current $I_J$; three independent numerical peak searches verify this relation. The effective noise temperature $T^*$ must not simply be replaced by refrigerator temperature. [R5](#ref-source-66a2b2a9519f6f44)\n\nEdkins and colleagues' 2019 study used single-electron STM, explicitly without scanned Josephson tunneling microscopy. At 2 K it compared 0 and 8.25 T in the same Bi2212 field of view using the same tip. Approximately eight-cell and four-cell electronic modulations support a PDW interpretation within the authors' model. The fitted Fourier widths are not exactly in a two-to-one ratio. Sublattice form factors belong to the measured electronic response and must not be copied directly onto the pair field. The core region is excluded from the simple envelope argument. Raw-data fitting and the complete supplement are not reproduced here. [R6](#ref-source-d306adac5d567426)\n\nChen and colleagues' 2021 CsV3Sb5 work reports spatial gap/conductance modulations and Josephson spectroscopy on prepared surfaces. Bulk and surface temperature scales differ, and the author manuscript explicitly says roton excitations were not directly observed. Its proposed complex PDW interpretation is not a direct phase measurement or a universal statement about kagome superconductors. The selected manuscript and methods check is inherited from the chiral-pairing entry. [R7](#ref-source-057c43c86174f961)\n\nIn May 2026, Cao and colleagues reported a hysteretic tunneling-gap transition in multilayer NbSe2 under an in-plane field, consistent with a possible orbital FFLO state. Their reported angle, disorder and thickness dependences matter; the selected theoretical discussion neglects vortices in its energy comparison. This is evidence for a transition with a proposed interpretation, not a direct image of finite pair momentum. Only the publisher abstract, introduction and selected model passage have been checked here. [R9](#ref-source-ad66f48ab9369ce9)"},{"id":"limits-and-disputes","title":"Limits and disputes","markdown":"## Limits and disputes\n\nKeep four distinctions explicit: pair tunneling versus single-electron tunneling; induced modulation versus a primary instability; field-induced versus spontaneous behavior; and a surface observation versus a bulk thermodynamic phase. A particular gap map cannot alone determine the pair phase. A nonzero composite amplitude, a half-vortex proposal or an unusual transport curve likewise needs a separate coherence and alternative-explanation analysis. [R5](#ref-source-66a2b2a9519f6f44) [R6](#ref-source-d306adac5d567426) [R8](#ref-source-cb08c313e0597487)\n\nThe 2026 report by Wang and colleagues deliberately calls its moiré result Cooper-pair density modulation (CPDM): the authors describe periodic order-parameter variation without additional translation-symmetry breaking. That distinction prevents treating every periodic pairing texture as a spontaneous PDW. This entry has checked the institutional abstract and publication identity, not the complete measurement or microscopic model. [R10](#ref-source-61d655195d7db259)"},{"id":"discovery-implications","title":"Discovery implications","markdown":"## Discovery implications\n\nStore the probe, temperature, magnetic-field magnitude and orientation, surface preparation, Fourier convention, uniform component and normalization with every candidate. Compare observations with an explicit response model before assigning an order parameter. Our local algebra, peak and overlap checks test stated examples; they do not reproduce the experiments, establish a material ground state, or close full scientific acceptance. [R4](#ref-source-7148789acde770d4) [R5](#ref-source-66a2b2a9519f6f44) [R6](#ref-source-d306adac5d567426) [R9](#ref-source-ad66f48ab9369ce9)"},{"id":"related-topics","title":"Related topics","markdown":"## Related topics\n\nContinue with [charge-fluctuation pairing](/scwiki/charge-fluctuation-pairing), [d-wave pairing](/scwiki/d-wave), [chiral pairing](/scwiki/chiral-pairing) and [topological superconductivity](/scwiki/topological-superconductivity)."},{"id":"material-cases","title":"Material cases","markdown":"## Material cases\n\nValues describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.\n\n| Material and sample | Reported temperature | Criterion | Probe | Pressure | State | Evidence |\n|---|---|---|---|---|---|---|\n| Bi2Sr2CaCu2O8+x (aliases: cuprate, cuprates); Hamidian et al. Bi2212 sample, p approximately 17%, superconducting nanoflake tip | Below 0.05 K operating condition, not an exact temperature or a superconducting transition. Effective noise temperature is separate. | measurement; pair-tunneling-operating-temperature | Scanned Josephson tunneling microscopy in the phase-diffusion regime; field condition not assigned at this scoped review | unknown; unknown GPa | unknown | experiment; [source](#ref-source-66a2b2a9519f6f44) |\n| Bi2Sr2CaCu2O8+x (aliases: cuprate, cuprates); Edkins et al. slightly underdoped sample, p approximately 17%, same-tip same-view comparison | 2 K; STM operating temperature; separate from the sample superconducting transition. | measurement; single-electron-spectroscopy-temperature | Single-electron STM; field-induced difference between 8.25 T and 0 T, 58 nm by 58 nm view | unknown; unknown GPa | unknown | experiment; [source](#ref-source-d306adac5d567426) |"}],"markdown":"# Pair-density waves: finite momentum, harmonics and experimental evidence\n\nDistinguish pair-field modulation, charge order, FFLO mechanisms and the probes used to test them.\n\n## Scope\n\nA pair-density wave (PDW) concerns finite-wavevector components of a superconducting pair field. A charge-density wave (CDW), a modulated tunneling spectrum and a modulated pair field are different observables. Specify the underlying lattice or superlattice, the electromagnetic gauge convention and whether a uniform condensate coexists. A single plane-wave phase has constant magnitude; periodic magnitude alone also need not establish a new spontaneous broken translation symmetry. [R1](#ref-source-824e7016c4d4c3a8) [R10](#ref-source-61d655195d7db259)\n\nFulde and Ferrell's 1964 publisher abstract specifies a spatially uniform spin-exchange field with no scattering. Their finite-momentum state has a compensating contribution from unpaired electrons, so its equilibrium total current is zero despite the spatial phase variation. This statement concerns their idealized model; the original derivation is not reproduced here. Larkin and Ovchinnikov's paper appeared in Russian in 1964 and in English translation in 1965. The author institute's bibliography confirms that identity, but the original full text remains an access gap and its derivation has not been checked. These historical assumptions do not by themselves validate a modern orbital-FFLO interpretation. [R1](#ref-source-824e7016c4d4c3a8) [R2](#ref-source-739939afd549524f)\n\n## Mathematical assumptions\n\nFor a scalar illustration in a fixed gauge, write $\\Delta(x)=d_0+d_+e^{iQx}+d_-e^{-iQx}$. The Fourier coefficients of $|\\Delta|^2$ are $C_0=|d_0|^2+|d_+|^2+|d_-|^2$, $C_Q=d_+d_0^*+d_0d_-^*$ and $C_{2Q}=d_+d_-^*$, with negative-wavevector coefficients given by conjugation. Four independent discrete Fourier examples reproduce these identities. They are algebraic intensity identities, not a derivation of electronic charge density or a tunneling response. [R4](#ref-source-7148789acde770d4) [R6](#ref-source-d306adac5d567426)\n\nFor $d_0=d_-=0$, the Fulde–Ferrell-type plane wave has constant $|\\Delta|$. Nonzero equal real $d_+=d_-=d/2$ instead give a Larkin–Ovchinnikov-type standing wave $d\\cos Qx$. In this fixed gauge and for $Q\\ne0$, the pair field has period $2\\pi/|Q|$, while $|\\Delta|^2$ has period $\\pi/|Q|$. A nonzero real uniform component $u$ adds a Q harmonic, but a quadrature component does not: $\\Delta=iu+d\\cos Qx$ gives $|\\Delta|^2=u^2+d^2\\cos^2Qx$ for real $u,d$. Thus a missing Q intensity harmonic need not exclude even coexistence with a uniform component. These are imposed scalar-field identities, not calculated electronic charge harmonics: a measured 2Q density signal also requires a response model and does not uniquely identify a PDW. No free energies are compared. [R1](#ref-source-824e7016c4d4c3a8) [R4](#ref-source-7148789acde770d4) [R6](#ref-source-d306adac5d567426)\n\nUnder an electron gauge phase $\\alpha$, both pair components acquire $e^{2i\\alpha}$. The product $d_+d_-^*$ is neutral, whereas $d_+d_-$ transforms with $e^{4i\\alpha}$ and has zero net wavevector. Three numerical phase checks confirm the distinction. A composite with charge 4e is not by itself evidence for a thermodynamically coherent charge-4e phase. Berg, Fradkin and Kivelson's 2009 melting theory proposes such a phase and fractional flux as a test; that proposal is distinct from the 2007 layer-decoupling paper. [R8](#ref-source-cb08c313e0597487)\n\nBerg and colleagues' 2007 stripe model suppresses leading interplane Josephson coupling through a particular antiphase pattern and stacking. An independent overlap calculation gives $\\langle\\cos Qx\\cos Qy\\rangle=0$ over complete periods for perpendicular stripes, but parallel stripes give $1/2$. Adding a common uniform amplitude $u$ gives $u^2$; a cropped window need not cancel. The quartic overlap remains $\\langle\\cos^2Qx\\cos^2Qy\\rangle=1/4$. This illustrates why cancellation of the lowest coupling does not mean all coupling vanishes. The paper retains higher-order/further-neighbor terms and discusses defects that spoil exact cancellation. [R4](#ref-source-7148789acde770d4)\n\nAn illustrative vortex-halo envelope $A(r)=e^{-r/\\xi}$ produces a Q amplitude proportional to $A\\Delta_0^*$ and a 2Q amplitude proportional to $A^2$. If $\\Delta_0$ is approximately constant, the latter decays twice as fast. Four independent logarithmic-slope checks verify this and show failure of that ratio when the uniform component recovers from zero. The relation is conditional, especially near the core; it is not a universal PDW diagnostic. [R6](#ref-source-d306adac5d567426)\n\n## Experimental criteria\n\nTranquada and colleagues' 1995 neutron study concerns static spin/charge stripes in Nd-substituted La–Sr–Cu–O at $x=0.12$. That diffraction evidence does not directly measure a superconducting pair field. Layer decoupling and charge harmonics are useful cross-checks only when competing charge, magnetic and structural explanations are addressed. [R3](#ref-source-b33aab72ae697f3a) [R4](#ref-source-7148789acde770d4)\n\nHamidian and colleagues' 2016 experiment used a superconducting Bi2212 nanoflake tip for scanned Josephson tunneling microscopy below 50 mK. It reported modulations near $Q_P=(0.25\\pm0.02)\\,2\\pi/a_0$, superposed on a stronger uniform component. It is not the later field-induced vortex-halo spectroscopy experiment. Their phase-diffusion model uses $I(V)=I_J^2 ZV/[2(V^2+V_c^2)]$, with $V_c=2eZk_BT^*/\\hbar$. The measured peak $I_c=\\hbar I_J^2/(8ek_BT^*)$ differs from the underlying Josephson critical current $I_J$; three independent numerical peak searches verify this relation. The effective noise temperature $T^*$ must not simply be replaced by refrigerator temperature. [R5](#ref-source-66a2b2a9519f6f44)\n\nEdkins and colleagues' 2019 study used single-electron STM, explicitly without scanned Josephson tunneling microscopy. At 2 K it compared 0 and 8.25 T in the same Bi2212 field of view using the same tip. Approximately eight-cell and four-cell electronic modulations support a PDW interpretation within the authors' model. The fitted Fourier widths are not exactly in a two-to-one ratio. Sublattice form factors belong to the measured electronic response and must not be copied directly onto the pair field. The core region is excluded from the simple envelope argument. Raw-data fitting and the complete supplement are not reproduced here. [R6](#ref-source-d306adac5d567426)\n\nChen and colleagues' 2021 CsV3Sb5 work reports spatial gap/conductance modulations and Josephson spectroscopy on prepared surfaces. Bulk and surface temperature scales differ, and the author manuscript explicitly says roton excitations were not directly observed. Its proposed complex PDW interpretation is not a direct phase measurement or a universal statement about kagome superconductors. The selected manuscript and methods check is inherited from the chiral-pairing entry. [R7](#ref-source-057c43c86174f961)\n\nIn May 2026, Cao and colleagues reported a hysteretic tunneling-gap transition in multilayer NbSe2 under an in-plane field, consistent with a possible orbital FFLO state. Their reported angle, disorder and thickness dependences matter; the selected theoretical discussion neglects vortices in its energy comparison. This is evidence for a transition with a proposed interpretation, not a direct image of finite pair momentum. Only the publisher abstract, introduction and selected model passage have been checked here. [R9](#ref-source-ad66f48ab9369ce9)\n\n## Limits and disputes\n\nKeep four distinctions explicit: pair tunneling versus single-electron tunneling; induced modulation versus a primary instability; field-induced versus spontaneous behavior; and a surface observation versus a bulk thermodynamic phase. A particular gap map cannot alone determine the pair phase. A nonzero composite amplitude, a half-vortex proposal or an unusual transport curve likewise needs a separate coherence and alternative-explanation analysis. [R5](#ref-source-66a2b2a9519f6f44) [R6](#ref-source-d306adac5d567426) [R8](#ref-source-cb08c313e0597487)\n\nThe 2026 report by Wang and colleagues deliberately calls its moiré result Cooper-pair density modulation (CPDM): the authors describe periodic order-parameter variation without additional translation-symmetry breaking. That distinction prevents treating every periodic pairing texture as a spontaneous PDW. This entry has checked the institutional abstract and publication identity, not the complete measurement or microscopic model. [R10](#ref-source-61d655195d7db259)\n\n## Discovery implications\n\nStore the probe, temperature, magnetic-field magnitude and orientation, surface preparation, Fourier convention, uniform component and normalization with every candidate. Compare observations with an explicit response model before assigning an order parameter. Our local algebra, peak and overlap checks test stated examples; they do not reproduce the experiments, establish a material ground state, or close full scientific acceptance. [R4](#ref-source-7148789acde770d4) [R5](#ref-source-66a2b2a9519f6f44) [R6](#ref-source-d306adac5d567426) [R9](#ref-source-ad66f48ab9369ce9)\n\n## Related topics\n\nContinue with [charge-fluctuation pairing](/scwiki/charge-fluctuation-pairing), [d-wave pairing](/scwiki/d-wave), [chiral pairing](/scwiki/chiral-pairing) and [topological superconductivity](/scwiki/topological-superconductivity).\n\n## Material cases\n\nValues describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.\n\n| Material and sample | Reported temperature | Criterion | Probe | Pressure | State | Evidence |\n|---|---|---|---|---|---|---|\n| Bi2Sr2CaCu2O8+x (aliases: cuprate, cuprates); Hamidian et al. Bi2212 sample, p approximately 17%, superconducting nanoflake tip | Below 0.05 K operating condition, not an exact temperature or a superconducting transition. Effective noise temperature is separate. | measurement; pair-tunneling-operating-temperature | Scanned Josephson tunneling microscopy in the phase-diffusion regime; field condition not assigned at this scoped review | unknown; unknown GPa | unknown | experiment; [source](#ref-source-66a2b2a9519f6f44) |\n| Bi2Sr2CaCu2O8+x (aliases: cuprate, cuprates); Edkins et al. slightly underdoped sample, p approximately 17%, same-tip same-view comparison | 2 K; STM operating temperature; separate from the sample superconducting transition. | measurement; single-electron-spectroscopy-temperature | Single-electron STM; field-induced difference between 8.25 T and 0 T, 58 nm by 58 nm view | unknown; unknown GPa | unknown | experiment; [source](#ref-source-d306adac5d567426) |\n\n## Sources\n\n### Ref source-824e7016c4d4c3a8\n\n[R1] Superconductivity in a Strong Spin-Exchange Field (1964). [DOI 10.1103/PhysRev.135.A550](https://doi.org/10.1103/PhysRev.135.A550). Publication: journal-article. Check: publisher-abstract-check. Checked: 2026-10-19. Published 3 August 1964. Clean exchange-field and compensated-current abstract checked; original derivation not fully reviewed.\n### Ref source-739939afd549524f\n\n[R2] Nonuniform state of superconductors (1965). [Source publication](https://www.itp.ac.ru/ru/persons/ovchinnikov-yurii-nikolaevich/). Publication: journal-article. Check: institutional-bibliographic-check. Checked: 2026-10-19. Landau Institute author bibliography, entry262: Russian JETP47(3),1136-1146(1964); English Sov.Phys.JETP20(3),762-769(1965). Year here denotes the English translation. Original full text remains inaccessible. Corrected an erroneous shared identity with Migdal1958; no DOI invented.\n### Ref source-b33aab72ae697f3a\n\n[R3] Evidence for stripe correlations of spins and holes in copper oxide superconductors (1995). [DOI 10.1038/375561a0](https://doi.org/10.1038/375561a0). Publication: journal-article. Check: publisher-abstract-check. Checked: 2026-10-18. Publisher abstract and identity checked: 15 June 1995, Nd-substituted compound x=0.12, neutron diffraction. Full paper and data not reviewed.\n### Ref source-7148789acde770d4\n\n[R4] Dynamical Layer Decoupling in a Stripe-Ordered High-Tc Superconductor (2007). [DOI 10.1103/PhysRevLett.99.127003](https://doi.org/10.1103/PhysRevLett.99.127003). Publication: journal-article. Check: selected-published-text-check. Checked: 2026-10-19. Published text pp.1–4 read for stacking, lowest-order cancellation, residual coupling and defects. Figures and complete microscopic derivation not reproduced. Local scalar overlaps are independent illustrations. Follow-up: abstract, antiphase pattern and interplane coupling discussion aroundEq.1; supports conditional model context, not the local quadrature example.\n### Ref source-66a2b2a9519f6f44\n\n[R5] Detection of a Cooper-pair density wave in Bi2Sr2CaCu2O8+x (2016). [DOI 10.1038/nature17411](https://doi.org/10.1038/nature17411). Publication: journal-article. Check: selected-published-and-author-text-check. Checked: 2026-10-18. Published 13 April 2016; selected final pp.1–4 and methods p.6 checked, p.6 visually. Author v3 pp.4–7/19–21 and Fig.4 p.18 also checked. Phase-diffusion equations agree at selected scope; complete version/data review open.\n### Ref source-d306adac5d567426\n\n[R6] Magnetic field-induced pair density wave state in the cuprate vortex halo (2019). [DOI 10.1126/science.aat1773](https://doi.org/10.1126/science.aat1773). Publication: journal-article. Check: selected-published-text-check. Checked: 2026-10-19. Published 7 June 2019; selected main-text pp.1–4, result passage p.5, Figs.1/4 pp.2/4 visually checked. Supplement and raw fits remain open. Distinct from Hamidian 2016 SJTM. Follow-up: publishedp1,Eqs.2-3 distinguish symmetry-allowed density-like products from an imposed scalar intensity identity. No new full supplement or data fit.\n### Ref source-057c43c86174f961\n\n[R7] Roton pair density wave in a strong-coupling kagome superconductor (2021). [DOI 10.1038/s41586-021-03983-5](https://doi.org/10.1038/s41586-021-03983-5). Publication: journal-article. Check: selected-author-manuscript-check. Checked: 2026-10-18. Selected pp. 3–7 and methods pp. 17–18; Fig. 3 on p. 14 visually checked. Author version 2 September 2021 has a different title and mistyped journal volume 559; publisher confirms Nature 599. Final-version/raw-data comparison open.\n### Ref source-cb08c313e0597487\n\n[R8] Charge-4e superconductivity from pair-density-wave order in certain high-temperature superconductors (2009). [DOI 10.1038/nphys1389](https://doi.org/10.1038/nphys1389). Publication: journal-article. Check: author-abstract-and-publication-check. Checked: 2026-10-18. Author v3 dated 7 June 2009 is marked corrected; abstract and journal identity checked. Thermal-melting and fractional-flux proposal only; full phase diagram not reviewed.\n### Ref source-ad66f48ab9369ce9\n\n[R9] Spectroscopic evidence for a first-order transition to a possible orbital Fulde-Ferrell-Larkin-Ovchinnikov state (2026). [DOI 10.1038/s41467-026-72134-z](https://doi.org/10.1038/s41467-026-72134-z). Publication: journal-article. Check: selected-publisher-web-passages-check. Checked: 2026-10-18. Published 12 May 2026. Indexed publisher abstract/introduction and selected model passage read; model omits vortices. Full methods, supplementary material and data not reviewed.\n### Ref source-61d655195d7db259\n\n[R10] Moiré engineering of Cooper-pair density modulation states (2026). [DOI 10.1038/s41586-026-10325-w](https://doi.org/10.1038/s41586-026-10325-w). Publication: journal-article. Check: institutional-abstract-and-publication-check. Checked: 2026-10-18. Penn State author-affiliation record and complete abstract checked; issue 9 April 2026. CPDM terminology relative to moiré lattice retained. Full experimental/model review open.\n\n## Claim evidence and conditions\n\n### scwiki-c8-claim-scope-1\n\nA pair-density wave (PDW) concerns finite-wavevector components of a superconducting pair field. A charge-density wave (CDW), a modulated tunneling spectrum and a modulated pair field are different observables. Specify the underlying lattice or superlattice, the electromagnetic gauge convention and whether a uniform condensate coexists. A single plane-wave phase has constant magnitude; periodic magnitude alone also need not establish a new spontaneous broken translation symmetry.\n\nEvidence: theory. Sources: source-824e7016c4d4c3a8, source-61d655195d7db259.\n\nConditions: Finite wavevector is stated relative to a lattice and gauge. Conventional exchange-field FFLO and orbital proposals have different drivers.\n\nVerification scope: Named primary passages and independent examples only; full scientific acceptance remains open.\n\n### scwiki-c8-claim-scope-2\n\nFulde and Ferrell's 1964 publisher abstract specifies a spatially uniform spin-exchange field with no scattering. Their finite-momentum state has a compensating contribution from unpaired electrons, so its equilibrium total current is zero despite the spatial phase variation. This statement concerns their idealized model; the original derivation is not reproduced here. Larkin and Ovchinnikov's paper appeared in Russian in 1964 and in English translation in 1965. The author institute's bibliography confirms that identity, but the original full text remains an access gap and its derivation has not been checked. These historical assumptions do not by themselves validate a modern orbital-FFLO interpretation.\n\nEvidence: theory. Sources: source-824e7016c4d4c3a8, source-739939afd549524f.\n\nSource relationship: context-for-editorial-analysis. The current cancellation is a reported model result from the abstract; the historical identity is bibliographic, and the warning against extending the model is editorial.\n\nConditions: Fulde-Ferrell ideal uniform exchange field and no scattering, as stated in its publisher abstract. Larkin-Ovchinnikov identity only: Russian1964, English1965; no original full-text derivation reviewed.\n\nVerification scope: Selected primary abstract/text and institutional identity checked2026-10-19; local Fourier examples checked separately. No independent review or experimental replication.\n\n### scwiki-c8-claim-mathematical-assumptions-1\n\nFor a scalar illustration in a fixed gauge, write $\\Delta(x)=d_0+d_+e^{iQx}+d_-e^{-iQx}$. The Fourier coefficients of $|\\Delta|^2$ are $C_0=|d_0|^2+|d_+|^2+|d_-|^2$, $C_Q=d_+d_0^*+d_0d_-^*$ and $C_{2Q}=d_+d_-^*$, with negative-wavevector coefficients given by conjugation. Four independent discrete Fourier examples reproduce these identities. They are algebraic intensity identities, not a derivation of electronic charge density or a tunneling response.\n\nEvidence: theory. Sources: source-7148789acde770d4, source-d306adac5d567426.\n\nSource relationship: context-for-local-derivation. The exact scalar Fourier coefficients are local algebra; cited stripe/halo work supplies physical context, not a fitted response.\n\nConditions: Fixed scalar amplitudes and imposed wavevectors; complete periods for zero-overlap result. No self-consistent stability or material response fit. Envelope ratio requires a constant uniform background.\n\nVerification scope: Named primary passages and independent examples only; full scientific acceptance remains open.\n\n### scwiki-c8-claim-mathematical-assumptions-2\n\nFor $d_0=d_-=0$, the Fulde–Ferrell-type plane wave has constant $|\\Delta|$. Nonzero equal real $d_+=d_-=d/2$ instead give a Larkin–Ovchinnikov-type standing wave $d\\cos Qx$. In this fixed gauge and for $Q\\ne0$, the pair field has period $2\\pi/|Q|$, while $|\\Delta|^2$ has period $\\pi/|Q|$. A nonzero real uniform component $u$ adds a Q harmonic, but a quadrature component does not: $\\Delta=iu+d\\cos Qx$ gives $|\\Delta|^2=u^2+d^2\\cos^2Qx$ for real $u,d$. Thus a missing Q intensity harmonic need not exclude even coexistence with a uniform component. These are imposed scalar-field identities, not calculated electronic charge harmonics: a measured 2Q density signal also requires a response model and does not uniquely identify a PDW. No free energies are compared.\n\nEvidence: theory. Sources: source-824e7016c4d4c3a8, source-7148789acde770d4, source-d306adac5d567426.\n\nSource relationship: context-for-local-derivation. Plane and standing waves and the quadrature counterexample are author-imposed fields. Source models motivate the distinction; no ground state or experimental density is derived.\n\nConditions: Imposed scalar amplitudes in a fixed gauge, Q nonzero and real nonzero standing-wave amplitude d. A real u is aligned; iu is in quadrature. |Delta|^2 is not a measured charge-density or tunneling signal.\n\nVerification scope: Selected primary abstract/text and institutional identity checked2026-10-19; local Fourier examples checked separately. No independent review or experimental replication.\n\n### scwiki-c8-claim-mathematical-assumptions-3\n\nUnder an electron gauge phase $\\alpha$, both pair components acquire $e^{2i\\alpha}$. The product $d_+d_-^*$ is neutral, whereas $d_+d_-$ transforms with $e^{4i\\alpha}$ and has zero net wavevector. Three numerical phase checks confirm the distinction. A composite with charge 4e is not by itself evidence for a thermodynamically coherent charge-4e phase. Berg, Fradkin and Kivelson's 2009 melting theory proposes such a phase and fractional flux as a test; that proposal is distinct from the 2007 layer-decoupling paper.\n\nEvidence: theory. Sources: source-cb08c313e0597487.\n\nSource relationship: mixed-source-and-local-derivation. Local gauge-product checks are distinguished from the cited thermodynamic melting proposal.\n\nConditions: Fixed scalar amplitudes and imposed wavevectors; complete periods for zero-overlap result. No self-consistent stability or material response fit. Envelope ratio requires a constant uniform background.\n\nVerification scope: Named primary passages and independent examples only; full scientific acceptance remains open.\n\n### scwiki-c8-claim-mathematical-assumptions-4\n\nBerg and colleagues' 2007 stripe model suppresses leading interplane Josephson coupling through a particular antiphase pattern and stacking. An independent overlap calculation gives $\\langle\\cos Qx\\cos Qy\\rangle=0$ over complete periods for perpendicular stripes, but parallel stripes give $1/2$. Adding a common uniform amplitude $u$ gives $u^2$; a cropped window need not cancel. The quartic overlap remains $\\langle\\cos^2Qx\\cos^2Qy\\rangle=1/4$. This illustrates why cancellation of the lowest coupling does not mean all coupling vanishes. The paper retains higher-order/further-neighbor terms and discusses defects that spoil exact cancellation.\n\nEvidence: theory. Sources: source-7148789acde770d4.\n\nSource relationship: mixed-source-and-local-derivation. The source analyzes stacking and residual couplings; local scalar overlaps illustrate only selected symmetry and integration-window conditions.\n\nConditions: Fixed scalar amplitudes and imposed wavevectors; complete periods for zero-overlap result. No self-consistent stability or material response fit. Envelope ratio requires a constant uniform background.\n\nVerification scope: Named primary passages and independent examples only; full scientific acceptance remains open.\n\n### scwiki-c8-claim-mathematical-assumptions-5\n\nAn illustrative vortex-halo envelope $A(r)=e^{-r/\\xi}$ produces a Q amplitude proportional to $A\\Delta_0^*$ and a 2Q amplitude proportional to $A^2$. If $\\Delta_0$ is approximately constant, the latter decays twice as fast. Four independent logarithmic-slope checks verify this and show failure of that ratio when the uniform component recovers from zero. The relation is conditional, especially near the core; it is not a universal PDW diagnostic.\n\nEvidence: theory. Sources: source-d306adac5d567426.\n\nSource relationship: context-for-local-derivation. The chosen exponential envelopes and logarithmic derivatives are local checks inspired by the source response model; no raw-data fit.\n\nConditions: Fixed scalar amplitudes and imposed wavevectors; complete periods for zero-overlap result. No self-consistent stability or material response fit. Envelope ratio requires a constant uniform background.\n\nVerification scope: Named primary passages and independent examples only; full scientific acceptance remains open.\n\n### scwiki-c8-claim-experimental-criteria-1\n\nTranquada and colleagues' 1995 neutron study concerns static spin/charge stripes in Nd-substituted La–Sr–Cu–O at $x=0.12$. That diffraction evidence does not directly measure a superconducting pair field. Layer decoupling and charge harmonics are useful cross-checks only when competing charge, magnetic and structural explanations are addressed.\n\nEvidence: experiment. Sources: source-b33aab72ae697f3a, source-7148789acde770d4.\n\nConditions: Keep each study and probe separate: 2016 pair tunneling below 50 mK; 2019 single-electron 0/8.25T comparison at 2 K. Spectroscopy and phase interpretation are not interchangeable.\n\nVerification scope: Named primary passages and independent examples only; full scientific acceptance remains open.\n\n### scwiki-c8-claim-experimental-criteria-2\n\nHamidian and colleagues' 2016 experiment used a superconducting Bi2212 nanoflake tip for scanned Josephson tunneling microscopy below 50 mK. It reported modulations near $Q_P=(0.25\\pm0.02)\\,2\\pi/a_0$, superposed on a stronger uniform component. It is not the later field-induced vortex-halo spectroscopy experiment. Their phase-diffusion model uses $I(V)=I_J^2 ZV/[2(V^2+V_c^2)]$, with $V_c=2eZk_BT^*/\\hbar$. The measured peak $I_c=\\hbar I_J^2/(8ek_BT^*)$ differs from the underlying Josephson critical current $I_J$; three independent numerical peak searches verify this relation. The effective noise temperature $T^*$ must not simply be replaced by refrigerator temperature.\n\nEvidence: experiment. Sources: source-66a2b2a9519f6f44.\n\nSource relationship: mixed-source-and-local-derivation. Reported Josephson measurement and phase-diffusion formula are separated from the local numerical peak checks; noise temperature is not fitted here.\n\nConditions: Keep each study and probe separate: 2016 pair tunneling below 50 mK; 2019 single-electron 0/8.25T comparison at 2 K. Spectroscopy and phase interpretation are not interchangeable.\n\nVerification scope: Named primary passages and independent examples only; full scientific acceptance remains open.\n\n### scwiki-c8-claim-experimental-criteria-3\n\nEdkins and colleagues' 2019 study used single-electron STM, explicitly without scanned Josephson tunneling microscopy. At 2 K it compared 0 and 8.25 T in the same Bi2212 field of view using the same tip. Approximately eight-cell and four-cell electronic modulations support a PDW interpretation within the authors' model. The fitted Fourier widths are not exactly in a two-to-one ratio. Sublattice form factors belong to the measured electronic response and must not be copied directly onto the pair field. The core region is excluded from the simple envelope argument. Raw-data fitting and the complete supplement are not reproduced here.\n\nEvidence: experiment. Sources: source-d306adac5d567426.\n\nConditions: Keep each study and probe separate: 2016 pair tunneling below 50 mK; 2019 single-electron 0/8.25T comparison at 2 K. Spectroscopy and phase interpretation are not interchangeable.\n\nVerification scope: Named primary passages and independent examples only; full scientific acceptance remains open.\n\n### scwiki-c8-claim-experimental-criteria-4\n\nChen and colleagues' 2021 CsV3Sb5 work reports spatial gap/conductance modulations and Josephson spectroscopy on prepared surfaces. Bulk and surface temperature scales differ, and the author manuscript explicitly says roton excitations were not directly observed. Its proposed complex PDW interpretation is not a direct phase measurement or a universal statement about kagome superconductors. The selected manuscript and methods check is inherited from the chiral-pairing entry.\n\nEvidence: experiment. Sources: source-057c43c86174f961.\n\nConditions: Keep each study and probe separate: 2016 pair tunneling below 50 mK; 2019 single-electron 0/8.25T comparison at 2 K. Spectroscopy and phase interpretation are not interchangeable.\n\nVerification scope: Named primary passages and independent examples only; full scientific acceptance remains open.\n\n### scwiki-c8-claim-experimental-criteria-5\n\nIn May 2026, Cao and colleagues reported a hysteretic tunneling-gap transition in multilayer NbSe2 under an in-plane field, consistent with a possible orbital FFLO state. Their reported angle, disorder and thickness dependences matter; the selected theoretical discussion neglects vortices in its energy comparison. This is evidence for a transition with a proposed interpretation, not a direct image of finite pair momentum. Only the publisher abstract, introduction and selected model passage have been checked here.\n\nEvidence: experiment. Sources: source-ad66f48ab9369ce9.\n\nConditions: Keep each study and probe separate: 2016 pair tunneling below 50 mK; 2019 single-electron 0/8.25T comparison at 2 K. Spectroscopy and phase interpretation are not interchangeable.\n\nVerification scope: Named primary passages and independent examples only; full scientific acceptance remains open.\n\n### scwiki-c8-claim-limits-and-disputes-1\n\nKeep four distinctions explicit: pair tunneling versus single-electron tunneling; induced modulation versus a primary instability; field-induced versus spontaneous behavior; and a surface observation versus a bulk thermodynamic phase. A particular gap map cannot alone determine the pair phase. A nonzero composite amplitude, a half-vortex proposal or an unusual transport curve likewise needs a separate coherence and alternative-explanation analysis.\n\nEvidence: theory. Sources: source-66a2b2a9519f6f44, source-d306adac5d567426, source-cb08c313e0597487.\n\nConditions: Surface and bulk, induced and primary, charge and pair observables are distinct. No material-wide proof inferred from a modulation.\n\nVerification scope: Named primary passages and independent examples only; full scientific acceptance remains open.\n\n### scwiki-c8-claim-limits-and-disputes-2\n\nThe 2026 report by Wang and colleagues deliberately calls its moiré result Cooper-pair density modulation (CPDM): the authors describe periodic order-parameter variation without additional translation-symmetry breaking. That distinction prevents treating every periodic pairing texture as a spontaneous PDW. This entry has checked the institutional abstract and publication identity, not the complete measurement or microscopic model.\n\nEvidence: theory. Sources: source-61d655195d7db259.\n\nConditions: Surface and bulk, induced and primary, charge and pair observables are distinct. No material-wide proof inferred from a modulation.\n\nVerification scope: Named primary passages and independent examples only; full scientific acceptance remains open.\n\n### scwiki-c8-claim-discovery-implications-1\n\nStore the probe, temperature, magnetic-field magnitude and orientation, surface preparation, Fourier convention, uniform component and normalization with every candidate. Compare observations with an explicit response model before assigning an order parameter. Our local algebra, peak and overlap checks test stated examples; they do not reproduce the experiments, establish a material ground state, or close full scientific acceptance.\n\nEvidence: theory. Sources: source-7148789acde770d4, source-66a2b2a9519f6f44, source-d306adac5d567426, source-ad66f48ab9369ce9.\n\nConditions: Retain actual methods and evidence scope with each candidate; local model checks do not constitute full scientific acceptance.\n\nVerification scope: Named primary passages and independent examples only; full scientific acceptance remains open.\n\n## Revision and review\n\nArticle ID: scwiki-c8. Revision: 5c165f574b5f4a8f3c4a. Updated: 2026-10-19T14:56:05.646640+00:00.\n\nReview scope: Selected published Hamidian and Edkins methods/results; Berg stacking theory; inherited kagome passages; bounded historical and 2026 abstract checks; independent Fourier and overlap examples. Follow-up: corrected LO identity; FF abstract current conditions; scalar intensity harmonics and quadrature coexistence.\n\n- Full scientific acceptance, complete source-version comparisons and raw-data reproduction remain open.\n- Original Larkin–Ovchinnikov full text was unavailable; bibliographic identity only is checked.\n- Recent orbital-FFLO and CPDM reports have limited passage/abstract coverage, not full experimental review.\n- Imposed scalar models do not establish a stable material phase or reproduce instrument response.\n\nSource: [C_pairing_symmetry/C8_PDW.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/C_pairing_symmetry/C8_PDW.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 8fa1922ee16b00f0aa5c314028d5ced0709c5d98f6b8868b100a04ad51689119.\n","wordCount":1130}}