{ "schema_version": "1.0.0", "artifact_id": "qft.artifact.many-body-quantum-matter.phase-amplitude-collective-modes-pole-continuum-map", "title": "Phase, amplitude, and Leggett poles relative to pair-breaking continua", "classification": "original qualitative schematic; coordinates are layout fixtures, not physical data", "reader_question": "Which paired-matter fluctuations give poles, which are reorganized by Coulomb interaction, and when do fermionic continua damp them?", "dominant_point": "A collective pole is established by an isolated simple zero of the analytically continued kernel on a sheet where it is analytic; attributing a feature in a chosen probe additionally requires nonzero residue and comparison with every coupled continuum.", "scope": { "state": "clean, fully gapped paired matter at zero temperature", "approximation": "Gaussian fluctuations about a self-consistent saddle, with conserving vertices and a declared electromagnetic environment", "momentum_regime": "long wavelength", "leggett_threshold_scope": "At q = 0 in the simple band-diagonal fully gapped model, each fermionic threshold is 2 min_k E_i(k); the labels 2 Delta_< and 2 Delta_> assume isotropic gaps whose minima lie on the relevant Fermi surfaces.", "scale_status": "not to scale; the ordering of the three-dimensional plasma frequency and pair-breaking energy is material-dependent" }, "panels": [ { "id": "common_phase_and_amplitude", "axes": { "horizontal": "wave-number magnitude q", "vertical": "real frequency omega" }, "relations": [ { "id": "neutral_phase_sound", "formula": "omega = c_s q", "limit": "omega tends to zero as q tends to zero", "style": "solid black curve" }, { "id": "charged_common_phase_3d", "formula": "omega squared = omega_p squared + c_s squared q squared", "limit": "omega tends to omega_p as q tends to zero", "style": "dashed black curve", "caveat": "This is the local three-dimensional long-wavelength form; it is not the two-dimensional square-root plasmon law." }, { "id": "two_quasiparticle_continuum", "formula": "omega_th(q) = min_k [E_(k+q/2) + E_(k-q/2)], with omega_th(0) = 2 Delta_0 in the stated ideal model", "style": "dashed gray boundary with gray region above" }, { "id": "amplitude_threshold_response", "relation": "The weak-coupling amplitude response meets the two-quasiparticle edge at q = 0.", "classification": "threshold feature; the drawing does not assert an isolated simple pole", "style": "thick gray segment" } ] }, { "id": "relative_phase_leggett_damping", "axis": "real frequency omega", "thresholds": [ { "id": "smaller_gap", "value_label": "2 Delta_<" }, { "id": "larger_gap", "value_label": "2 Delta_>" } ], "ordered_regions": [ { "range": "omega < 2 Delta_<", "fermionic_channels": "closed in the ideal model", "example": "sharp pole omega_L" }, { "range": "2 Delta_< < omega < 2 Delta_>", "fermionic_channels": "smaller-gap band open", "example": "complex pole Omega_L - i Gamma_L" }, { "range": "omega > 2 Delta_>", "fermionic_channels": "both band continua open", "example": "no undamped pole is asserted" } ] } ], "analytic_continuation": { "retarded_boundary_value": "i Omega_m maps to omega + i 0^+", "damped_pole": "omega_star = Omega_star - i Gamma_star with Gamma_star nonnegative, on the relevant continuation of the retarded response", "pole_test": "A real-axis maximum or branch point is not a pole. An isolated simple determinant zero away from branch points establishes a collective pole, while the analytically continued probe vertices establish whether that pole is visible in a chosen response." }, "layout_fixture": { "common_sector": { "neutral_phase": { "start": [ 0, 0 ], "end": [ 5.55, 4.56 ], "style": "solid" }, "charged_phase_3d": { "start": [ 0, 2.12 ], "end": [ 6.45, 3.28 ], "style": "dashed" }, "continuum": { "start": [ 0, 4.38 ], "control_1": [ 2, 4.43 ], "control_2": [ 4.4, 4.65 ], "end": [ 7.15, 5.1 ], "style": "gray boundary with shaded region above" }, "amplitude_response": { "start": [ 0.42, 4.34 ], "end": [ 2.18, 4.4 ], "style": "thick gray" } }, "relative_phase": { "smaller_gap_threshold_y": 2.55, "larger_gap_threshold_y": 4.62, "sharp_pole_y": 1.42, "complex_pole_y": 3.25 } }, "nonclaims": [ "The curves are not material-specific dispersions and their vertical separations are not quantitative.", "The three-dimensional plasma branch does not represent a two-dimensional charged layer.", "The amplitude mark does not assert a Lorentz-invariant Higgs particle or an isolated pole.", "Being below every coupled fermionic threshold closes those decay channels but does not guarantee visibility; a pole above a threshold can remain visible as a finite-width resonance, and an open channel with zero coupling need not broaden it.", "Nodes, disorder, finite temperature, other collective modes, and nonconserving approximations can open additional damping channels or invalidate the ideal thresholds." ], "accessibility": { "color_independence": "Branches use direct labels, solid or dashed lines, thickness, shaded regions, and threshold labels; color is not required.", "reading_order": "Read the common phase and amplitude panel first, then compare the Leggett pole with the two ordered band thresholds.", "structured_equivalent": "This record preserves every branch, threshold, region, caveat, line-style distinction, and scale limitation encoded by the figure." }, "sources": [ { "citation": "P. W. Anderson, Random-Phase Approximation in the Theory of Superconductivity, Physical Review 112 (1958) 1900-1916", "doi": "10.1103/PhysRev.112.1900", "use": "neutral common-phase sound and the three-dimensional charged plasma reorganization" }, { "citation": "A. J. Leggett, Number-Phase Fluctuations in Two-Band Superconductors, Progress of Theoretical Physics 36 (1966) 901-930", "doi": "10.1143/PTP.36.901", "use": "two-band relative-phase coordinate and nonzero Leggett frequency" }, { "citation": "P. B. Littlewood and C. M. Varma, Amplitude Collective Modes in Superconductors and Their Coupling to Charge-Density Waves, Physical Review B 26 (1982) 4883-4893", "doi": "10.1103/PhysRevB.26.4883", "use": "amplitude response and probe dependence" }, { "citation": "T. Cea, C. Castellani, G. Seibold, and L. Benfatto, Nonrelativistic Dynamics of the Amplitude (Higgs) Mode in Superconductors, Physical Review Letters 115 (2015) 157002", "doi": "10.1103/PhysRevLett.115.157002", "use": "pair-breaking-edge versus isolated amplitude-pole distinction" } ], "alt_text": "Neutral phase sound rises from zero below a two-quasiparticle continuum, a dashed three-dimensional charged common-phase branch starts at a plasma frequency, and the amplitude response meets the continuum edge. A second panel places a sharp Leggett pole below the smaller pair-breaking threshold and a complex damped pole between the smaller and larger pair-breaking thresholds.", "caption": "Pole and continuum structure for clean, fully gapped paired matter at zero temperature. The common-phase panel is schematic: the ordering of omega_p and 2 Delta_0 is material-dependent. The relative-phase panel gives only the robust kinematic test: a Leggett pole below 2 Delta_< cannot decay into either band's quasiparticles in the ideal model, while a root above that threshold can acquire a width when the relevant matrix element is nonzero. Nodes, disorder, finite temperature, and other modes can open lower-energy decay channels.", "rights": { "basis": "Original QFT.org schematic generated from the declared equations and qualitative threshold relations; no third-party figure, geometry, data, or artwork was copied or restyled.", "creator": "OpenAI Codex, for QFT.org", "date": "2026-08-31" } }