{ "schema_version": "1.0.0", "artifact_id": "qft.artifact.many-body-quantum-matter.fermi-surface-frg-coupled-channel-validation", "title": "Coupled functional-RG channels, projection tests, and the claim ceiling", "scope": { "system": "weak-coupling normal-state interacting fermions with a resolved Fermi surface", "formalism": "regulated one-particle-irreducible effective-action functional renormalization group", "hierarchy": "the exact Wetterich hierarchy is closed by retaining the self-energy and four-point vertex while neglecting or approximating higher vertices", "external_leg_convention": "legs 1 and 2 are incoming and legs 3 and 4 are outgoing; P = k1 + k2, direct particle-hole Q = k3 - k1, and crossed particle-hole Q' = k4 - k1", "model_dependence": "no particular dispersion, interaction, regulator, patch mesh, form-factor basis, or frequency grid is fixed", "units": "hbar = k_B = 1" }, "scale_status": { "overall": "schematic and nonquantitative", "panel_a": "Boxes encode the logical coupling of one-loop transfer structures using the stated incoming-first P, Q, and Q' names, not graph multiplicities, signs, combinatorial factors, or a unique channel-allocation convention.", "panel_b": "The branch order is a validation workflow; box position and arrow length do not encode eigenvalues, susceptibilities, uncertainties, or physical scales." }, "flow_architecture": { "exact_parent": "The scale-dependent one-particle-irreducible effective action obeys the exact Wetterich equation before truncation.", "displayed_closure": "The displayed vertex flow is the retained self-energy plus four-point-vertex closure; six-point and higher vertices are omitted or represented only through a chosen completion.", "shared_running_inputs": "Every retained loop uses the same running full four-point vertex and the same regulated propagators G_Lambda[Sigma_Lambda,R_Lambda], so self-energy feedback affects all channel contributions.", "channel_contributions": [ { "id": "particle-particle", "symbol": "T_pp", "distinguished_transfer": "pair total momentum P", "common_projection_targets": "pairing eigenmodes" }, { "id": "crossed-particle-hole", "symbol": "T_ph,cr", "distinguished_transfer": "crossed particle-hole transfer Q' = k4 - k1", "common_projection_targets": "spin, charge, and other finite-wave-vector particle-hole modes" }, { "id": "direct-particle-hole", "symbol": "T_ph,d", "distinguished_transfer": "direct particle-hole transfer Q = k3 - k1", "common_projection_targets": "forward-scattering and Pomeranchuk modes" } ], "reconstruction": "Within the displayed closure, the three transfer contributions update one reconstructed antisymmetrized four-point vertex rather than three independent vertices. The updated full vertex returns as the shared input to all three loop transfers at the next RG step.", "finite_truncation_qualification": "The allocation of a contribution among channel functions depends on projection and parametrization conventions at finite truncation; reconstructed vertices, observable susceptibilities, and symmetry-resolved eigenmodes are the comparison objects." }, "projection_and_modes": { "projection_axes": [ "Fermi-surface patch or momentum resolution", "form-factor basis", "frequency dependence" ], "mode_families": [ "pairing modes", "spin and charge modes at finite ordering vector Q", "forward-scattering and Pomeranchuk modes near q = 0" ], "comparison_rule": "Compare symmetry-resolved eigenmodes and susceptibilities after reconstructing the full vertex; do not infer competition from isolated channel amplitudes alone." }, "validation_gates": [ { "id": "regulator-family", "test": "Repeat with admissible regulator families and compare reconstructed observables and the leading-mode hierarchy.", "failure_meaning": "Material regulator drift signals unresolved truncation error." }, { "id": "basis-and-frequency-resolution", "test": "Refine the patch or form-factor resolution and the retained frequency dependence.", "failure_meaning": "A changing hierarchy is not a robust leading tendency at the stated resolution." }, { "id": "feedback-order", "test": "Compare self-energy feedback and, where applicable, Katanin substitution and increasing multiloop order or loop convergence.", "failure_meaning": "Sensitivity to feedback order is part of the truncation uncertainty, not evidence for a phase." }, { "id": "identities-and-crossing", "test": "Monitor crossing symmetry and relevant Ward-identity residuals.", "failure_meaning": "Large residuals invalidate a symmetry-resolved hierarchy until the approximation is improved." }, { "id": "quasiparticle-integrity", "test": "Track quasiparticle residue, damping, and pole quality over the modes and momenta used by the projection.", "failure_meaning": "If the pole premise is lost, stop interpreting the flow as a weak-coupling Fermi-surface instability analysis." } ], "outcomes_and_claim_ceiling": { "stable_hierarchy": "A hierarchy stable under the declared refinements at the declared strong-coupling threshold licenses a leading symmetry-resolved tendency together with the normal-state stopping scale Lambda_star.", "unstable_hierarchy": "If the hierarchy drifts under admissible refinements, report no robust leading channel at the stated resolution.", "normal_state_threshold": "Crossing the declared strong-coupling threshold ends the symmetric normal-state integration and defines Lambda_star; the hierarchy at that stop must still pass the declared refinements.", "failure_exit": "Loss of the quasiparticle pole premise invalidates the weak-coupling Fermi-surface instability interpretation and ends this analysis without licensing a leading-channel claim.", "claim_ceiling": "Lambda_star is a regulator- and truncation-qualified stopping scale, not by itself a gap, transition temperature, ordered phase, or proof of long-range order.", "optional_continuation": "A gap, T_c, or phase claim requires a separate broken-symmetry continuation or another controlled low-energy calculation." }, "visual_encoding": { "solid_arrows": "required flow, projection, reconstruction, explicit next-step iteration, or decision path", "dashed_arrow_and_box": "optional broken-symmetry continuation beyond the normal-state stopping analysis", "dotted_arrow_and_box": "failure exit from the controlled symmetric normal-state interpretation", "gray_boxes": "retained computation, projection, or validation operations", "heavy_white_boxes": "reconstructed comparison objects or licensed outcomes", "canvas": "explicit white background with black and gray marks for light, dark, monochrome, and print use" }, "exclusions": [ "No numerical channel-versus-scale curves or universal ordering among pairing, density-wave, and Pomeranchuk modes are asserted.", "No exact combinatorial factors, regulator, or channel-allocation convention beyond the stated incoming-first transfer naming is fixed.", "No claim is made that Katanin substitution or a finite multiloop order removes every truncation error.", "No ordered-state spectrum, gap amplitude, transition temperature, or thermodynamic phase boundary is computed.", "The workflow does not cover flows whose low-energy degrees of freedom are no longer fermionic quasiparticles on a resolved Fermi surface." ], "sources": [ { "citation": "Walter Metzner, Manfred Salmhofer, Carsten Honerkamp, Volker Meden, and Kurt Schoenhammer, Functional Renormalization Group Approach to Correlated Fermion Systems, Reviews of Modern Physics 84 (2012) 299-352", "url": "https://arxiv.org/abs/1105.5289", "use": "exact effective-action hierarchy, self-energy and four-point truncations, competing pairing and particle-hole channels, and stopping-scale interpretation" }, { "citation": "Carsten Husemann and Manfred Salmhofer, Efficient Parametrization of the Vertex Function, Omega Scheme, and the (t,t') Hubbard Model at Van Hove Filling, Physical Review B 79 (2009) 195125", "url": "https://arxiv.org/abs/0812.3824", "use": "three singular transfer channels, exchange-boson or form-factor parametrization, and finite-parametrization channel allocation" }, { "citation": "Fabian B. Kugler and Jan von Delft, Multiloop Functional Renormalization Group That Sums Up All Parquet Diagrams, Physical Review Letters 120 (2018) 057403", "url": "https://arxiv.org/abs/1703.06505", "use": "multiloop completion of the vertex flow and the relation to parquet diagrams" }, { "citation": "Fabian B. Kugler and Jan von Delft, Multiloop Functional Renormalization Group for General Models, Physical Review B 97 (2018) 035162", "url": "https://arxiv.org/abs/1707.04536", "use": "vertex and self-energy corrections, loop convergence, and regulator dependence within the parquet approximation" } ], "semantic_equivalent": { "status": "realized", "page_route": "/many-body-quantum-matter/fermi-liquids-beyond/functional-rg-competing-instabilities/", "figure_anchor": "fermi-surface-frg-coupled-channel-validation", "adjacent_table": { "columns": [ "Flow object", "Resolved transfer or projection", "Data that must be retained", "Convergence question", "Invalid inference" ], "rows": [ { "flow_object": "Particle-particle loop", "resolved_transfer_or_projection": "Pair total P", "data_that_must_be_retained": "Relative momenta, frequencies, spin parity", "convergence_question": "Does the pairing eigenfunction stabilize?", "invalid_inference": "One attractive entry proves superconductivity" }, { "flow_object": "Crossed particle-hole loop", "resolved_transfer_or_projection": "Transfer Q'", "data_that_must_be_retained": "Spin/charge tensor and finite-Q structure", "convergence_question": "Do magnetic or density peaks survive refinement?", "invalid_inference": "A large vertex component is a susceptibility" }, { "flow_object": "Direct particle-hole loop", "resolved_transfer_or_projection": "Transfer Q", "data_that_must_be_retained": "Forward and exchange limits, crossing partners", "convergence_question": "Are uniform and finite-Q limits separated?", "invalid_inference": "Every forward enhancement is nematic order" }, { "flow_object": "Reconstructed Gamma_Lambda^(4)", "resolved_transfer_or_projection": "All three transfers", "data_that_must_be_retained": "Smooth remainder and every retained basis coefficient", "convergence_question": "Are crossing and antisymmetry residuals small?", "invalid_inference": "Channel pieces may be compared without reconstruction" }, { "flow_object": "Physical response", "resolved_transfer_or_projection": "Source vertex or Bethe-Salpeter kernel", "data_that_must_be_retained": "Bubble weights, operator normalization, and wave vector", "convergence_question": "Is the response hierarchy stable across admissible schemes?", "invalid_inference": "Lambda_star is a transition temperature" } ] }, "additional_equivalents": [ "The preceding prose defines P, direct Q, and crossed Q' and states that the updated full vertex feeds every channel at the next RG step.", "The later practical-validation and warranted-claim tables expand the figure's refinement gates, failure exit, stopping-scale ceiling, and broken-symmetry continuation." ], "note": "The adjacent five-row table is the primary reflowing text equivalent for the transfer, reconstruction, convergence, and invalid-inference relations shown in the SVG." } }