{ "$schema": "https://qft.org/data/schemas/qft-many-body-comparison-1.0.0.schema.json", "schema_version": "1.0.0", "artifact_id": "qft.artifact.many-body-quantum-matter.quantum-phase-transitions-critical-metals-claim-validity-ledger", "table_id": "quantum-phase-transitions-critical-metals-claim-validity-ledger", "canonical_source": { "route": "/many-body-quantum-matter/quantum-phase-transitions-critical-metals/", "heading": "Mechanism and evidence table", "anchor": "mechanism-and-evidence-table" }, "caption": "Low-energy mechanisms, control, and evidence for quantum phase transitions and critical metals.", "common_scope": { "markdown": "", "math": [], "links": [] }, "qualifications": { "markdown": "", "math": [], "links": [] }, "columns": [ { "key": "mechanism", "label": "Mechanism" }, { "key": "retained_low_energy_fields", "label": "Retained low-energy fields" }, { "key": "dynamics", "label": "Dynamics" }, { "key": "controlled_limit", "label": "Controlled limit" }, { "key": "dangerous_variable_or_hyperscaling_issue", "label": "Dangerous variable or hyperscaling issue" }, { "key": "transport_assumption", "label": "Transport assumption" }, { "key": "finite_temperature_scope", "label": "Finite-temperature scope" }, { "key": "evidence_ceiling", "label": "Supported conclusion" } ], "rows": [ { "key": "relativistic-lgw", "row_id": "quantum-phase-transitions-critical-metals-claim-validity-ledger--relativistic-lgw", "cells": { "mechanism": { "markdown": "Relativistic LGW", "math": [], "links": [] }, "retained_low_energy_fields": { "markdown": "local $N$-component order parameter", "math": [ { "kind": "inlineMath", "tex": "N" } ], "links": [] }, "dynamics": { "markdown": "$z=1$ in the displayed kernel", "math": [ { "kind": "inlineMath", "tex": "z=1" } ], "links": [] }, "controlled_limit": { "markdown": "$4-(d+z)$ expansion, large $N$, or known fixed point", "math": [ { "kind": "inlineMath", "tex": "4-(d+z)" }, { "kind": "inlineMath", "tex": "N" } ], "links": [] }, "dangerous_variable_or_hyperscaling_issue": { "markdown": "quartic $u$ dangerous above $d+z=4$", "math": [ { "kind": "inlineMath", "tex": "u" }, { "kind": "inlineMath", "tex": "d+z=4" } ], "links": [] }, "transport_assumption": { "markdown": "transport requires a separately defined current", "math": [], "links": [] }, "finite_temperature_scope": { "markdown": "fan below UV and above other gaps", "math": [], "links": [] }, "evidence_ceiling": { "markdown": "identifies an order-parameter universality class", "math": [], "links": [] } } }, { "key": "antiferromagnetic-hertz-millis", "row_id": "quantum-phase-transitions-critical-metals-claim-validity-ledger--antiferromagnetic-hertz-millis", "cells": { "mechanism": { "markdown": "Antiferromagnetic Hertz–Millis", "math": [], "links": [] }, "retained_low_energy_fields": { "markdown": "damped order parameter after fermions are removed", "math": [], "links": [] }, "dynamics": { "markdown": "$q^2+\\gamma\\lvert\\omega\\rvert$, Gaussian $z=2$", "math": [ { "kind": "inlineMath", "tex": "q^2+\\gamma\\lvert\\omega\\rvert" }, { "kind": "inlineMath", "tex": "z=2" } ], "links": [] }, "controlled_limit": { "markdown": "regular generated vertices; often above upper critical dimension", "math": [], "links": [] }, "dangerous_variable_or_hyperscaling_issue": { "markdown": "$u$ controls thermal mass and $\\psi$", "math": [ { "kind": "inlineMath", "tex": "u" }, { "kind": "inlineMath", "tex": "\\psi" } ], "links": [] }, "transport_assumption": { "markdown": "momentum relaxation not supplied by the boson action alone", "math": [], "links": [] }, "finite_temperature_scope": { "markdown": "can be preempted by pairing", "math": [], "links": [] }, "evidence_ceiling": { "markdown": "establishes damping kinematics, not harmless fermion removal", "math": [], "links": [] } } }, { "key": "clean-ferromagnetic-hertz-theory", "row_id": "quantum-phase-transitions-critical-metals-claim-validity-ledger--clean-ferromagnetic-hertz-theory", "cells": { "mechanism": { "markdown": "Clean ferromagnetic Hertz theory", "math": [], "links": [] }, "retained_low_energy_fields": { "markdown": "small-$q$ order parameter", "math": [ { "kind": "inlineMath", "tex": "q" } ], "links": [] }, "dynamics": { "markdown": "$q^2+\\gamma\\lvert\\omega\\rvert/q$, Gaussian $z=3$", "math": [ { "kind": "inlineMath", "tex": "q^2+\\gamma\\lvert\\omega\\rvert/q" }, { "kind": "inlineMath", "tex": "z=3" } ], "links": [] }, "controlled_limit": { "markdown": "weak-coupling reduction before nonanalytic feedback", "math": [], "links": [] }, "dangerous_variable_or_hyperscaling_issue": { "markdown": "$u$ dangerous; soft fermion modes add nonlocality", "math": [ { "kind": "inlineMath", "tex": "u" } ], "links": [] }, "transport_assumption": { "markdown": "small-angle scattering weakly relaxes current", "math": [], "links": [] }, "finite_temperature_scope": { "markdown": "first-order or modulated order may intervene", "math": [], "links": [] }, "evidence_ceiling": { "markdown": "a $z=3$ fit does not prove a continuous local action", "math": [ { "kind": "inlineMath", "tex": "z=3" } ], "links": [] } } }, { "key": "spin-fermion-hot-spots", "row_id": "quantum-phase-transitions-critical-metals-claim-validity-ledger--spin-fermion-hot-spots", "cells": { "mechanism": { "markdown": "Spin-fermion hot spots", "math": [], "links": [] }, "retained_low_energy_fields": { "markdown": "order parameter plus discrete hot fermions", "math": [], "links": [] }, "dynamics": { "markdown": "coupled, generally beyond fixed $z=2$ one-loop picture", "math": [ { "kind": "inlineMath", "tex": "z=2" } ], "links": [] }, "controlled_limit": { "markdown": "$\\epsilon$, emergent velocity ratio, or specified model", "math": [ { "kind": "inlineMath", "tex": "\\epsilon" } ], "links": [] }, "dangerous_variable_or_hyperscaling_issue": { "markdown": "pairing and composite channels introduce lower scales", "math": [], "links": [] }, "transport_assumption": { "markdown": "cold regions and momentum sinks must be included", "math": [], "links": [] }, "finite_temperature_scope": { "markdown": "above superconducting and curvature crossovers", "math": [], "links": [] }, "evidence_ceiling": { "markdown": "identifies hot-region breakdown within the stated control", "math": [], "links": [] } } }, { "key": "gauge-or-nematic-patches", "row_id": "quantum-phase-transitions-critical-metals-claim-validity-ledger--gauge-or-nematic-patches", "cells": { "mechanism": { "markdown": "Gauge or nematic patches", "math": [], "links": [] }, "retained_low_energy_fields": { "markdown": "extended Fermi surface plus low-$q$ boson", "math": [ { "kind": "inlineMath", "tex": "q" } ], "links": [] }, "dynamics": { "markdown": "anisotropic $q_y:q_x:\\omega=1:2:3$", "math": [ { "kind": "inlineMath", "tex": "q_y:q_x:\\omega=1:2:3" } ], "links": [] }, "controlled_limit": { "markdown": "controlled deformation; naive large $N$ is nonuniform", "math": [ { "kind": "inlineMath", "tex": "N" } ], "links": [] }, "dangerous_variable_or_hyperscaling_issue": { "markdown": "patch count produces hyperscaling issues", "math": [], "links": [] }, "transport_assumption": { "markdown": "vertex corrections and antipodal coupling sign matter", "math": [], "links": [] }, "finite_temperature_scope": { "markdown": "often hidden by pairing", "math": [], "links": [] }, "evidence_ceiling": { "markdown": "supports a critical Fermi surface, not a unique global metal", "math": [], "links": [] } } }, { "key": "syk-dot-or-array", "row_id": "quantum-phase-transitions-critical-metals-claim-validity-ledger--syk-dot-or-array", "cells": { "mechanism": { "markdown": "SYK dot or array", "math": [], "links": [] }, "retained_low_energy_fields": { "markdown": "all-to-all large-$N$ fermions; spatial couplings added for arrays", "math": [ { "kind": "inlineMath", "tex": "N" } ], "links": [] }, "dynamics": { "markdown": "local conformal time, no dot momentum", "math": [], "links": [] }, "controlled_limit": { "markdown": "$N\\to\\infty$ before deep infrared", "math": [ { "kind": "inlineMath", "tex": "N\\to\\infty" } ], "links": [] }, "dangerous_variable_or_hyperscaling_issue": { "markdown": "residual entropy and finite-$N$ scale depend on limits", "math": [ { "kind": "inlineMath", "tex": "N" } ], "links": [] }, "transport_assumption": { "markdown": "dot has no conductivity; arrays require explicit current", "math": [], "links": [] }, "finite_temperature_scope": { "markdown": "$J/N\\ll T\\ll J$ for conformal dot window", "math": [ { "kind": "inlineMath", "tex": "J/N\\ll T\\ll J" } ], "links": [] }, "evidence_ceiling": { "markdown": "proves solvable local criticality for the specified model", "math": [], "links": [] } } }, { "key": "planckian-transport-fit", "row_id": "quantum-phase-transitions-critical-metals-claim-validity-ledger--planckian-transport-fit", "cells": { "mechanism": { "markdown": "Planckian transport fit", "math": [], "links": [] }, "retained_low_energy_fields": { "markdown": "current response and fitted weight", "math": [], "links": [] }, "dynamics": { "markdown": "$\\alpha=\\hbar\\Gamma_{\\mathrm{tr}}/(k_BT)$", "math": [ { "kind": "inlineMath", "tex": "\\alpha=\\hbar\\Gamma_{\\mathrm{tr}}/(k_BT)" } ], "links": [] }, "controlled_limit": { "markdown": "optical/DC consistency and covariance, not an RG limit", "math": [], "links": [] }, "dangerous_variable_or_hyperscaling_issue": { "markdown": "changing Drude weight can mimic a rate", "math": [], "links": [] }, "transport_assumption": { "markdown": "explicit umklapp, disorder, phonons, or other momentum sink", "math": [], "links": [] }, "finite_temperature_scope": { "markdown": "declared linear window", "math": [], "links": [] }, "evidence_ceiling": { "markdown": "establishes an order-one fitted rate, not a universal bound", "math": [], "links": [] } } }, { "key": "evidence-comparison", "row_id": "quantum-phase-transitions-critical-metals-claim-validity-ledger--evidence-comparison", "cells": { "mechanism": { "markdown": "Evidence comparison", "math": [], "links": [] }, "retained_low_energy_fields": { "markdown": "predictions from several mechanisms", "math": [], "links": [] }, "dynamics": { "markdown": "model-specific", "math": [], "links": [] }, "controlled_limit": { "markdown": "common covariance and held-out tests", "math": [], "links": [] }, "dangerous_variable_or_hyperscaling_issue": { "markdown": "corrections and analytic backgrounds retained", "math": [], "links": [] }, "transport_assumption": { "markdown": "compare DC, optical, Hall, thermal, and one-particle channels", "math": [], "links": [] }, "finite_temperature_scope": { "markdown": "searched windows reported", "math": [], "links": [] }, "evidence_ceiling": { "markdown": "mechanism only when alternatives make and fail distinct predictions", "math": [], "links": [] } } } ], "references": [ { "id": "ref-8656c6a62127696e", "markdown": "Hertz, J. A. “Quantum Critical Phenomena.” *Physical Review B* 14 (1976): 1165–1184. [DOI](https://doi.org/10.1103/PhysRevB.14.1165).", "urls": [ "https://doi.org/10.1103/PhysRevB.14.1165" ] } ], "reference_scope": "The reference catalog reproduces the canonical chapter bibliography. Cell links are associated with references only when their resolved URL matches; uncited chapter references are not asserted as claim-specific support." }