{ "schema_version": "1.0.0", "artifact_id": "qft.artifact.many-body-quantum-matter.superfluidity-superconductivity-structure-and-dictionary", "title": "Paired-matter structure and dictionary", "classification": "original schematic dependency diagram", "reader_question": "Which new input is required when moving from a Cooper instability to a paired saddle, a Nambu or BdG spectrum, collective response, stiffness, defects, electromagnetic screening, fluxoid sectors, or Josephson relations?", "dominant_point": "A Cooper instability, paired saddle, Nambu spectrum, phase stiffness, defect response, and electromagnetic observable are distinct claim layers; each arrow adds information that the previous layer does not supply.", "status": { "schematic": true, "not_to_scale": true, "contains_experimental_data": false, "contains_simulation_data": false, "contains_material_specific_parameters": false }, "conventions": [ "The diagram uses the site's natural units hbar = c = k_B = 1.", "The Nambu kernel is written as G^{-1} = i omega tau_0 - xi tau_3 - Delta_1 tau_1 + Delta_2 tau_2, with Delta = Delta_1 + i Delta_2.", "Upsilon_ij denotes the helicity-modulus tensor for the pair phase; it is not a particle-number density or mass density.", "The gauge-covariant phase gradient is partial_i theta - q_* A_i, where q_* is the signed pair charge.", "Solid arrows add a mathematical or physical input needed for the downstream claim. Dashed arrows supply a convention, classification, or validity condition.", "Gray fill, white fill, solid outlines, dashed outlines, direct labels, and position redundantly encode every distinction; color carries no meaning." ], "nodes": [ { "id": "channel_classification", "kind": "condition", "label": "Antisymmetry, crystal irrep, normal-state cutoff", "meaning": "The Cooper kernel must be antisymmetrized, normalized, and classified in a declared normal-state shell before an eigenchannel is interpreted." }, { "id": "cooper_eigenchannel", "kind": "calculation", "label": "Cooper eigenchannel", "equation": "-Gamma^irr G G Delta = lambda Delta", "meaning": "An attractive eigenchannel and its growing susceptibility diagnose an instability of the normal state, not a completed ordered phase." }, { "id": "paired_saddle", "kind": "model result", "label": "Paired saddle and gap equation", "equation": "Delta_k = -sum_(k') V_(k k') Delta_(k') tanh(E_(k')/2T)/(2 E_(k'))", "meaning": "A self-consistent saddle conditionally supplies a gap function and quasiparticle kernel." }, { "id": "nambu_kernel", "kind": "representation", "label": "Nambu-Gor'kov kernel", "equation": "G^{-1} = i omega tau_0 - xi tau_3 - Delta_1 tau_1 + Delta_2 tau_2", "meaning": "Nambu space packages particle and hole propagation, with redundant plus and minus energy labels and convention-dependent off-diagonal signs." }, { "id": "gauge_and_vertex_condition", "kind": "condition", "label": "Gauge choice is not an observable; retain conserving vertices", "meaning": "Physical response must be gauge invariant and must use vertices consistent with the propagator and self-energy approximation." }, { "id": "spatial_bdg", "kind": "calculation branch", "label": "Spatial BdG problem and plus/minus E_n eigenpairs", "meaning": "Boundaries, vortices, disorder, and inhomogeneous pairing require a spatial eigenproblem with particle-hole-related eigenpairs." }, { "id": "fluctuation_kernel", "kind": "calculation branch", "label": "Phase, amplitude, and Leggett poles", "meaning": "Zeros of the analytically continued fluctuation kernel can define collective modes only after continua, damping, and conserving vertices are controlled." }, { "id": "phase_response", "kind": "response branch", "label": "Pair-phase helicity modulus", "equation": "Upsilon_ij (partial_i theta - q_* A_i)(partial_j theta - q_* A_j)", "meaning": "The free-energy curvature under a slow gauge-covariant twist defines phase rigidity in the stated normalization." }, { "id": "local_spectra", "kind": "conditional observable", "label": "Boundary, disorder, and vortex-core spectra", "meaning": "A local spectral claim additionally requires basis, geometry, boundary, and size convergence." }, { "id": "neutral_response", "kind": "conditional observable", "label": "Neutral sound, circulation, and vortices", "meaning": "A neutral phase with conserved U(1), nonzero stiffness, and no explicit locking supports a long-wavelength phase mode and quantized circulation sectors." }, { "id": "charged_response", "kind": "conditional observable", "label": "Meissner kernel and superfluid weight", "meaning": "Charged matter requires a gauge-consistent static transverse current kernel; the longitudinal phase-density mode and transverse screening are distinct response channels." }, { "id": "winding_and_weak_link", "kind": "conditional observable", "label": "Circulation, charged fluxoid sectors, and Josephson relations", "meaning": "Compact phase winding and a declared weak-link model yield neutral circulation, charged fluxoid sectors, and Josephson relations after screening and circuit assumptions are fixed." }, { "id": "limit_conditions", "kind": "condition", "label": "Dimension, thermodynamic and static limits, core and screening scales", "meaning": "These conditions determine which defect, fluxoid, screening, or weak-link conclusion is licensed." } ], "relations": [ { "from": "channel_classification", "to": "cooper_eigenchannel", "style": "dashed", "label": "classify", "meaning": "Fix antisymmetry, representation, cutoff, and normalization before interpreting the eigenvalue." }, { "from": "cooper_eigenchannel", "to": "paired_saddle", "style": "solid", "label": "leading lambda reaches unity", "meaning": "The normal-state instability motivates a paired saddle but does not itself solve it." }, { "from": "paired_saddle", "to": "nambu_kernel", "style": "solid", "label": "quadratic kernel", "meaning": "Expanding the saddle to quadratic fermion order supplies the Nambu kernel." }, { "from": "gauge_and_vertex_condition", "to": "nambu_kernel", "style": "dashed", "meaning": "Gauge and vertex conventions are required before matrix entries become physical response." }, { "from": "nambu_kernel", "to": "spatial_bdg", "style": "solid", "meaning": "Spatial dependence and boundary conditions turn the kernel into a BdG eigenproblem." }, { "from": "nambu_kernel", "to": "fluctuation_kernel", "style": "solid", "meaning": "Fluctuations about the saddle supply the collective kernel." }, { "from": "nambu_kernel", "to": "phase_response", "style": "solid", "meaning": "A source-consistent free-energy derivative supplies phase response; poles alone do not." }, { "from": "spatial_bdg", "to": "local_spectra", "style": "solid", "meaning": "Converged spatial eigenpairs conditionally determine local spectra." }, { "from": "fluctuation_kernel", "to": "neutral_response", "style": "solid", "meaning": "The neutral phase pole supplies sound only under its symmetry and damping hypotheses." }, { "from": "phase_response", "to": "neutral_response", "style": "solid", "meaning": "Nonzero helicity modulus licenses neutral phase rigidity and defect energetics." }, { "from": "phase_response", "to": "charged_response", "style": "solid", "meaning": "After charge coupling and Ward-consistent response, phase rigidity contributes to Meissner screening and superfluid weight." }, { "from": "neutral_response", "to": "winding_and_weak_link", "style": "solid", "meaning": "Compact neutral phase winding yields circulation sectors." }, { "from": "charged_response", "to": "winding_and_weak_link", "style": "solid", "meaning": "Charge, screening, and compact winding yield fluxoid and weak-link relations." }, { "from": "limit_conditions", "to": "winding_and_weak_link", "style": "dashed", "meaning": "Dimension, limit order, core scale, screening, and circuit dynamics bound the final inference." } ], "reading_order": [ "channel classification and Cooper eigenchannel", "paired saddle and Nambu-Gor'kov kernel", "three parallel branches: spatial BdG, collective fluctuations, and phase response", "local spectra, neutral response, and charged response", "compact winding and weak-link conclusions together with their limit conditions" ], "nonclaims": [ "The diagram does not identify a microscopic pairing mechanism from a Cooper eigenchannel or gap symmetry.", "A Nambu or BdG pole does not by itself establish nonzero helicity modulus, Meissner screening, or a topological phase.", "The charged branch does not assert that bare magnetic flux is always quantized; winding quantizes the fluxoid, and bare flux reaches the ideal quantum only in the appropriate current limit.", "A local zero-energy state is not by itself a bulk topological invariant or non-Abelian platform demonstration.", "Node sizes, spacings, gray values, and arrow lengths encode no numerical scale or probability." ], "scientific_checks": [ "The Nambu signs give E_k squared = xi_k squared + |Delta_k| squared.", "The gauge-covariant phase gradient is invariant under A -> A + grad chi and theta -> theta + q_* chi.", "Upsilon is consistently defined as the pair-phase helicity modulus rather than a density.", "The longitudinal charged collective mode is kept distinct from the static transverse Meissner kernel.", "Neutral circulation and charged fluxoid sectors are not conflated with bare magnetic flux.", "Every visual arrow has one matching relation in this record." ], "sources": [ { "citation": "Bardeen, Cooper, and Schrieffer, Theory of Superconductivity, Physical Review 108 (1957) 1175-1204", "doi": "10.1103/PhysRev.108.1175", "use": "paired saddle, gap equation, and quasiparticle spectrum" }, { "citation": "Nambu, Quasi-Particles and Gauge Invariance in the Theory of Superconductivity, Physical Review 117 (1960) 648-663", "doi": "10.1103/PhysRev.117.648", "use": "Nambu representation, gauge covariance, and conserving response" }, { "citation": "Anderson, Coherent Excited States in the Theory of Superconductivity, Physical Review 110 (1958) 827-835", "doi": "10.1103/PhysRev.110.827", "use": "phase response, gauge invariance, plasma mode, and Meissner effect" }, { "citation": "Byers and Yang, Theoretical Considerations Concerning Quantized Magnetic Flux in Superconducting Cylinders, Physical Review Letters 7 (1961) 46-49", "doi": "10.1103/PhysRevLett.7.46", "use": "gauge periodicity and charged winding sectors" }, { "citation": "Josephson, Possible New Effects in Superconductive Tunnelling, Physics Letters 1 (1962) 251-253", "doi": "10.1016/0031-9163(62)91369-0", "use": "weak-link current-phase and voltage-frequency relations" } ], "alt_text": "The Cooper eigenchannel leads to a paired saddle and Nambu propagator, which branch into BdG spectra, collective modes, and pair-phase stiffness before neutral vortices or charged Meissner, fluxoid, and Josephson responses are inferred.", "caption": "The paired-matter dictionary. A Cooper eigenvalue licenses an instability, the saddle licenses a conditional quasiparticle spectrum, and gauge-invariant stiffness licenses phase response. Defects, electromagnetic screening, fluxoid sectors, and boundary spectra require the additional dimensional, charge, and boundary data shown. Original schematic, not to scale.", "rights": { "basis": "Original QFT.org schematic derived from independently checked equations and relations; no third-party figure, geometry, data, or artwork was copied or restyled.", "creator": "OpenAI Codex, for QFT.org", "date": "2026-08-31" } }