{"schemaVersion":"1.0","revision":"fa961ab24277a91737c3","article":{"id":"scwiki-f1","topicId":"scwiki-topic-f1","code":"F1","slug":"quantum-metric","title":"Quantum metric: state distance, conventions and response limits","summary":"Define Bloch-state geometry with explicit normalization, distinguish metric from topology, and keep geometric response models separate from measured total stiffness.","aliases":["quantum metric","Fubini Study","Fubini-Study","quantum geometric tensor","fidelity","minimal quantum metric"],"evidenceTypes":["theory","experiment"],"relatedCodes":["F7","B13","E9","E8"],"editorialStatus":"scoped-check","contentState":"pilot","modifiedAt":"2026-10-18T13:49:34.227083+00:00","review":{"checkedAt":"2026-10-18","scope":"Selected Liang author equations,Peotta published assumptions,Huhtinen published response correction,Tanaka author methods and independent bounded state-geometry models.","limitations":["Complete scientific acceptance, full supplementary and raw-data reviews remain open.","Provost original is publisher-abstract scope; Liang and Tanaka final-version comparisons incomplete.","No material quantum metric, geometric stiffness fraction or transition temperature reconstructed.","Huhtinen full minimal-metric proof,band-touching response and complete self-consistent calculation not reproduced."]},"provenance":{"repository":"JackZH26/jzis-superconductivity","commit":"0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf","path":"F_quantum_geometry/F1_quantum_metric.md","sha256":"f4f76f73cf21723199f60f45a93acb2d4ace612337ebbf851629966f88dad668","sourceId":"file-4e31089d4d599b5b"},"references":[{"id":"source-fbbc7a50ad47a3f8","doi":"10.1007/BF02193559","title":"Riemannian structure on manifolds of quantum states","year":1980,"url":"https://doi.org/10.1007/BF02193559","publicationStatus":"journal-article","verification":{"scope":"publisher-abstract-and-bibliography","checkedAt":"2026-10-18","evidenceUrl":"https://link.springer.com/article/10.1007/BF02193559","note":"Original publisher abstract and September1980 citation checked; full original not retrieved. General state-manifold background only."}},{"id":"source-fdf091e09e7f4a6f","doi":"10.1038/ncomms9944","title":"Superfluidity in topologically nontrivial flat bands","year":2015,"url":"https://doi.org/10.1038/ncomms9944","publicationStatus":"journal-article","verification":{"scope":"selected-published-flat-band-assumptions","checkedAt":"2026-10-18","evidenceUrl":"https://www.nature.com/articles/ncomms9944","note":"Published20November2015. Existing local final PDF selectedpp4–6, especially uniform pairing and bandwidth/interaction/bandgap hierarchy onp5; prior B13p5visual reused. Full supplementary derivation not reproduced."}},{"id":"source-fa7a165c8c344a63","doi":"10.1103/PhysRevB.95.024515","title":"Band geometry, Berry curvature, and superfluid weight","year":2017,"url":"https://doi.org/10.1103/PhysRevB.95.024515","publicationStatus":"journal-article","verification":{"scope":"selected-author-projector-and-metric-conventions","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/1610.01803v2","note":"Authorv2 11January2017,selectedpp1–2,4–6; p5Eq21–24 visuallychecked. Published27January2017. R=2Tr(P dP dP) and gLiang=2gWiki. Final-version comparison and full response derivation remain open."}},{"id":"source-be7282623ca96056","doi":"10.1038/s41586-024-08494-7","title":"Superfluid stiffness of magic-angle twisted bilayer graphene","year":2025,"url":"https://doi.org/10.1038/s41586-024-08494-7","publicationStatus":"journal-article","verification":{"scope":"selected-author-circuit-and-temperature-definitions","checkedAt":"2026-10-18","evidenceUrl":"https://arxiv.org/pdf/2406.13740v2","note":"Tanaka et al, published5February2025. Authorv2selectedpp5–7 and prior B13/E9 Supplement3p31 circuitreview.20mKbase/40mKdevice,half-resistance near1K,active aspect ratio and parallel edge model; fullfinal/rawdata comparison open."}},{"id":"source-72f7ddb9f13bbd81","doi":"10.1103/PhysRevB.106.014518","title":"Revisiting flat band superconductivity: Dependence on minimal quantum metric and band touchings","year":2022,"url":"https://doi.org/10.1103/PhysRevB.106.014518","publicationStatus":"journal-article","verification":{"scope":"selected-published-response-correction","checkedAt":"2026-10-18","evidenceUrl":"https://aaltodoc.aalto.fi/server/api/core/bitstreams/e05db312-49b0-4769-b112-681407acd4b0/content","note":"Published26July2022. Final reprint introduction and selectedpp1–5 derivative/orbital-position discussion plusp6minimal-integrated-trace statement read as extractedtext. Full minimal-metric proof,band-touching calculation and complete response model not reproduced."}}],"claims":[{"id":"scwiki-f1-claim-scope-1","sectionId":"scope","statement":"The quantum metric measures how a normalized quantum state changes with parameters after removing its unobservable overall phase. For Bloch bands, those parameters can be crystal momentum. It describes wavefunctions rather than only energy dispersion. A flat band can therefore have nonzero metric, while a large metric alone neither supplies an attractive interaction nor establishes superconductivity. Provost and Vallée developed a general state-manifold construction; later multiband superconductivity work connected band geometry to current response under specified assumptions.","evidenceType":"theory","sourceIds":["source-fbbc7a50ad47a3f8","source-fdf091e09e7f4a6f","source-fa7a165c8c344a63"],"conditions":"Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.","verificationScope":"Selected primary passages and bounded model checks; complete scientific acceptance open."},{"id":"scwiki-f1-claim-scope-2","sectionId":"scope","statement":"This page uses a single isolated band or a specified smooth band subspace. It does not identify a single-particle Bloch metric with every many-body fidelity susceptibility. Keep the Hilbert space, parameter coordinates, orbital basis and selected subspace explicit. A geometric quantity can vary continuously without becoming a quantized topological invariant.","evidenceType":"theory","sourceIds":["source-fbbc7a50ad47a3f8","source-fa7a165c8c344a63"],"conditions":"Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.","verificationScope":"Selected primary passages and bounded model checks; complete scientific acceptance open."},{"id":"scwiki-f1-claim-mathematical-assumptions-1","sectionId":"mathematical-assumptions","statement":"For a normalized state $|u\\rangle$, let $P=|u\\rangle\\langle u|$ and $Q_{ij}=\\langle\\partial_i u|(1-P)|\\partial_j u\\rangle$. We define $g_{ij}=\\mathrm{Re}\\,Q_{ij}$ and $\\Omega_{ij}=-2\\mathrm{Im}\\,Q_{ij}$, consistent with $A_i=i\\langle u|\\partial_i u\\rangle$. Thus $Q_{ij}=g_{ij}-i\\Omega_{ij}/2$. The nearby-state fidelity obeys $1-|\\langle u(\\mathbf k)|u(\\mathbf k+d\\mathbf k)\\rangle|^2=g_{ij}dk_i dk_j+O(|d\\mathbf k|^3)$. With physical wavevectors, the metric components have units of length squared; dimensionless lattice coordinates change those units.","evidenceType":"theory","sourceIds":["source-fbbc7a50ad47a3f8","source-fa7a165c8c344a63"],"conditions":"Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.","verificationScope":"Selected primary passages and bounded model checks; complete scientific acceptance open."},{"id":"scwiki-f1-claim-mathematical-assumptions-2","sectionId":"mathematical-assumptions","statement":"For a rank-one projector, $Q_{ij}=\\mathrm{Tr}[P(\\partial_iP)(\\partial_jP)]$ and $g_{ij}=\\tfrac12\\mathrm{Tr}[(\\partial_iP)(\\partial_jP)]$. The projector is unchanged by $|u\\rangle\\rightarrow e^{i\\phi(\\mathbf k)}|u\\rangle$, so this construction removes phase-gauge artifacts. For an isolated group of touching bands, use the smooth subspace projector and state the trace convention. A chosen individual eigenvector can become singular or ambiguous at a degeneracy. Phase-gauge invariance within a fixed orbital representation is not permission to ignore a momentum-dependent change of the underlying basis.","evidenceType":"theory","sourceIds":["source-fa7a165c8c344a63","source-72f7ddb9f13bbd81"],"conditions":"Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.","verificationScope":"Selected primary passages and bounded model checks; complete scientific acceptance open."},{"id":"scwiki-f1-claim-mathematical-assumptions-3","sectionId":"mathematical-assumptions","statement":"Positivity of the two-coordinate matrix $Q$ gives $\\det g\\geq\\Omega_{xy}^2/4$ and $g_{xx}+g_{yy}\\geq|\\Omega_{xy}|$. The trace statement refers to the selected common coordinate convention. For the spinor $u=(\\cos(\\theta/2),e^{i\\varphi}\\sin(\\theta/2))$, our convention gives $g_{\\theta\\theta}=1/4$, $g_{\\varphi\\varphi}=\\sin^2\\theta/4$ and $\\Omega_{\\theta\\varphi}=-\\sin\\theta/2$. This two-level example saturates the determinant inequality; a three-level family can satisfy it strictly. These are model checks, not measurements of a material.","evidenceType":"theory","sourceIds":["source-fa7a165c8c344a63"],"conditions":"Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.","verificationScope":"Selected primary passages and bounded model checks; complete scientific acceptance open."},{"id":"scwiki-f1-claim-mathematical-assumptions-4","sectionId":"mathematical-assumptions","statement":"Normalization must be translated before comparing formulas. Liang's checked manuscript defines $\\mathcal R_{ij}=2\\mathrm{Tr}[P(\\partial_iP)(\\partial_jP)]$ and calls its real part the metric. Consequently $g^{\\rm Liang}=2g$ in this page's convention, and that paper writes the fidelity distance with an extra factor of one half. Its curvature sign must likewise be translated consistently. This is a convention difference, not a contradiction or a factor-of-two enhancement of a physical stiffness.","evidenceType":"theory","sourceIds":["source-fa7a165c8c344a63"],"conditions":"Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.","verificationScope":"Selected primary passages and bounded model checks; complete scientific acceptance open."},{"id":"scwiki-f1-claim-mathematical-assumptions-5","sectionId":"mathematical-assumptions","statement":"As a counterexample to equating metric with topology, take $u=(\\cos(k_x/2),\\sin(k_x/2))$, independent of $k_y$. The projector is periodic on a $2\\pi$ torus, $g_{xx}=1/4$, and the Berry curvature vanishes. The Hamiltonian $H=E_0(2P-I)$ has flat eigenvalues $\\pm E_0$ for positive constant $E_0$. It has nonzero metric and zero Chern number without introducing pairing. Our finite-difference checks compare projector derivatives, phase-transformed states and symmetric fidelity curvature; they do not calculate a superconducting response.","evidenceType":"theory","sourceIds":["source-fa7a165c8c344a63"],"conditions":"Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.","verificationScope":"Selected primary passages and bounded model checks; complete scientific acceptance open."},{"id":"scwiki-f1-claim-experimental-criteria-1","sectionId":"experimental-criteria","statement":"An experimental response may constrain band geometry through a model and matrix elements; it need not uniquely reconstruct every metric component. For superconductivity, distinguish the measured total stiffness from a decomposition into conventional and geometric terms. Report carrier filling, band window, temperature, units and the response-to-metric conversion. A successful fit does not by itself identify a unique microscopic pairing mechanism.","evidenceType":"experiment","sourceIds":["source-fa7a165c8c344a63","source-be7282623ca96056"],"conditions":"Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.","verificationScope":"Selected primary passages and bounded model checks; complete scientific acceptance open."},{"id":"scwiki-f1-claim-experimental-criteria-2","sectionId":"experimental-criteria","statement":"Tanaka and colleagues' 2025 MATBG study combines microwave kinetic inductance and transport. The checked author version includes a parallel edge-proximity circuit and an assumed active-region aspect ratio. It distinguishes a 20 mK refrigerator base from an approximately 40 mK wired-device temperature, and a half-normal-resistance transition near 1 K across much of the gate range. These are different quantities. Its comparison supports a connection to quantum geometry without directly isolating a model-independent geometric fraction. This page stores the 40 mK value only as a measurement condition.","evidenceType":"experiment","sourceIds":["source-be7282623ca96056"],"conditions":"Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.","verificationScope":"Selected primary passages and bounded model checks; complete scientific acceptance open."},{"id":"scwiki-f1-claim-limits-and-disputes-1","sectionId":"limits-and-disputes","statement":"Peotta–Törmä's flat-band result assumes an interaction window between the bandwidth and separating band gaps, together with an appropriate uniform-pairing construction. Liang extends the response analysis, but the isolated-band expression retains time-reversal, orbital-uniform pairing and band-separation assumptions. Huhtinen and colleagues subsequently identify missing order-parameter-response terms in earlier formulas and connect isolated-flat-band stiffness to a minimal quantum metric. Therefore an arbitrary raw metric integral should not be presented as an automatically complete quantitative stiffness prediction.","evidenceType":"theory","sourceIds":["source-fdf091e09e7f4a6f","source-fa7a165c8c344a63","source-72f7ddb9f13bbd81"],"conditions":"Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.","verificationScope":"Selected primary passages and bounded model checks; complete scientific acceptance open."},{"id":"scwiki-f1-claim-limits-and-disputes-2","sectionId":"limits-and-disputes","statement":"The review covers selected Peotta published passages, Liang author-version equations and Tanaka author methods; it does not complete their raw-data, full supplementary or final-version audits. Provost's original is checked at publisher-abstract scope. Huhtinen's published introduction and selected response derivation, pages 1–4, were checked; its full minimal-metric proof and band-touching analysis were not reproduced. The inherited F1 bibliography misattributes the MATBG paper to Randeria; the published authors are Tanaka and colleagues. Complete scientific acceptance remains open.","evidenceType":"theory","sourceIds":["source-fbbc7a50ad47a3f8","source-fdf091e09e7f4a6f","source-fa7a165c8c344a63","source-be7282623ca96056","source-72f7ddb9f13bbd81"],"conditions":"Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.","verificationScope":"Selected primary passages and bounded model checks; complete scientific acceptance open."},{"id":"scwiki-f1-claim-discovery-implications-1","sectionId":"discovery-implications","statement":"Calculate geometry with a reproducible orbital convention and subspace, check mesh convergence and preserve normalization factors. Test pairing and total phase stiffness separately. A topological lower bound can be useful under its assumptions, but zero Chern number does not exclude useful geometry, and nonzero metric does not guarantee superconductivity. Geometry is one constraint in a material-specific calculation, alongside interactions, filling, disorder and phase coherence.","evidenceType":"theory","sourceIds":["source-fdf091e09e7f4a6f","source-fa7a165c8c344a63","source-72f7ddb9f13bbd81"],"conditions":"Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.","verificationScope":"Selected primary passages and bounded model checks; complete scientific acceptance open."}],"corrections":[{"id":"scwiki-f1-correction-1","description":"The inherited F1 bibliography attributes the2025 MATBG paper to Randeria; publisher identifies Tanaka and colleagues. Fixed source bytes retained.","sourceId":"source-be7282623ca96056","scope":"Published author identity; final paper not fully reviewed.","targetSectionId":"limits-and-disputes"}],"equationChecks":[{"targetSectionId":"mathematical-assumptions","tex":"Q_{ij}=\\mathrm{Tr}[P(\\partial_iP)(\\partial_jP)]","check":"12 bounded spinor checks compare finite differences, projectors, U(1) phase changes and symmetric fidelity curvature."},{"targetSectionId":"mathematical-assumptions","tex":"\\det g\\geq\\Omega_{xy}^2/4","check":"12 two-level saturating and3 three-level strict examples;4 flat real-projector examples have nonzero metric and zero curvature. No material response calculated."}],"materialCases":[{"id":"scwiki-f1-material-1","formula":"C","sample":"Tanaka wired MATBG device","recordType":"experiment","reportedTemperatureK":0.04,"tcK":null,"temperatureKind":"measurement","temperatureNote":"Approximately40mK effective device temperature, distinct from20mKbase andnear1Khalf-resistance scale; no Tc assigned here.","criterion":"effective-device-temperature","pressureRegime":"ambient","pressureGPa":null,"equilibrium":"equilibrium","probe":"Microwave kinetic inductance and transport","sourceId":"source-be7282623ca96056","aliases":["MATBG","Tanaka device","twisted bilayer graphene"]}],"sourceMap":[{"startLine":1,"endLine":73,"targetSections":["scope","mathematical-assumptions","experimental-criteria","limits-and-disputes","discovery-implications","related-topics"],"decision":"edited-with-scoped-evidence","note":"Pinned source preserved; corrected author attribution, translated metric normalization and added subsequent response-correction limits."}],"body":"## Scope\n\nThe quantum metric measures how a normalized quantum state changes with parameters after removing its unobservable overall phase. For Bloch bands, those parameters can be crystal momentum. It describes wavefunctions rather than only energy dispersion. A flat band can therefore have nonzero metric, while a large metric alone neither supplies an attractive interaction nor establishes superconductivity. Provost and Vallée developed a general state-manifold construction; later multiband superconductivity work connected band geometry to current response under specified assumptions. [R1](#ref-source-fbbc7a50ad47a3f8) [R2](#ref-source-fdf091e09e7f4a6f) [R3](#ref-source-fa7a165c8c344a63)\n\nThis page uses a single isolated band or a specified smooth band subspace. It does not identify a single-particle Bloch metric with every many-body fidelity susceptibility. Keep the Hilbert space, parameter coordinates, orbital basis and selected subspace explicit. A geometric quantity can vary continuously without becoming a quantized topological invariant. [R1](#ref-source-fbbc7a50ad47a3f8) [R3](#ref-source-fa7a165c8c344a63)\n\n## Mathematical assumptions\n\nFor a normalized state $|u\\rangle$, let $P=|u\\rangle\\langle u|$ and $Q_{ij}=\\langle\\partial_i u|(1-P)|\\partial_j u\\rangle$. We define $g_{ij}=\\mathrm{Re}\\,Q_{ij}$ and $\\Omega_{ij}=-2\\mathrm{Im}\\,Q_{ij}$, consistent with $A_i=i\\langle u|\\partial_i u\\rangle$. Thus $Q_{ij}=g_{ij}-i\\Omega_{ij}/2$. The nearby-state fidelity obeys $1-|\\langle u(\\mathbf k)|u(\\mathbf k+d\\mathbf k)\\rangle|^2=g_{ij}dk_i dk_j+O(|d\\mathbf k|^3)$. With physical wavevectors, the metric components have units of length squared; dimensionless lattice coordinates change those units. [R1](#ref-source-fbbc7a50ad47a3f8) [R3](#ref-source-fa7a165c8c344a63)\n\nFor a rank-one projector, $Q_{ij}=\\mathrm{Tr}[P(\\partial_iP)(\\partial_jP)]$ and $g_{ij}=\\tfrac12\\mathrm{Tr}[(\\partial_iP)(\\partial_jP)]$. The projector is unchanged by $|u\\rangle\\rightarrow e^{i\\phi(\\mathbf k)}|u\\rangle$, so this construction removes phase-gauge artifacts. For an isolated group of touching bands, use the smooth subspace projector and state the trace convention. A chosen individual eigenvector can become singular or ambiguous at a degeneracy. Phase-gauge invariance within a fixed orbital representation is not permission to ignore a momentum-dependent change of the underlying basis. [R3](#ref-source-fa7a165c8c344a63) [R5](#ref-source-72f7ddb9f13bbd81)\n\nPositivity of the two-coordinate matrix $Q$ gives $\\det g\\geq\\Omega_{xy}^2/4$ and $g_{xx}+g_{yy}\\geq|\\Omega_{xy}|$. The trace statement refers to the selected common coordinate convention. For the spinor $u=(\\cos(\\theta/2),e^{i\\varphi}\\sin(\\theta/2))$, our convention gives $g_{\\theta\\theta}=1/4$, $g_{\\varphi\\varphi}=\\sin^2\\theta/4$ and $\\Omega_{\\theta\\varphi}=-\\sin\\theta/2$. This two-level example saturates the determinant inequality; a three-level family can satisfy it strictly. These are model checks, not measurements of a material. [R3](#ref-source-fa7a165c8c344a63)\n\nNormalization must be translated before comparing formulas. Liang's checked manuscript defines $\\mathcal R_{ij}=2\\mathrm{Tr}[P(\\partial_iP)(\\partial_jP)]$ and calls its real part the metric. Consequently $g^{\\rm Liang}=2g$ in this page's convention, and that paper writes the fidelity distance with an extra factor of one half. Its curvature sign must likewise be translated consistently. This is a convention difference, not a contradiction or a factor-of-two enhancement of a physical stiffness. [R3](#ref-source-fa7a165c8c344a63)\n\nAs a counterexample to equating metric with topology, take $u=(\\cos(k_x/2),\\sin(k_x/2))$, independent of $k_y$. The projector is periodic on a $2\\pi$ torus, $g_{xx}=1/4$, and the Berry curvature vanishes. The Hamiltonian $H=E_0(2P-I)$ has flat eigenvalues $\\pm E_0$ for positive constant $E_0$. It has nonzero metric and zero Chern number without introducing pairing. Our finite-difference checks compare projector derivatives, phase-transformed states and symmetric fidelity curvature; they do not calculate a superconducting response. [R3](#ref-source-fa7a165c8c344a63)\n\n## Experimental criteria\n\nAn experimental response may constrain band geometry through a model and matrix elements; it need not uniquely reconstruct every metric component. For superconductivity, distinguish the measured total stiffness from a decomposition into conventional and geometric terms. Report carrier filling, band window, temperature, units and the response-to-metric conversion. A successful fit does not by itself identify a unique microscopic pairing mechanism. [R3](#ref-source-fa7a165c8c344a63) [R4](#ref-source-be7282623ca96056)\n\nTanaka and colleagues' 2025 MATBG study combines microwave kinetic inductance and transport. The checked author version includes a parallel edge-proximity circuit and an assumed active-region aspect ratio. It distinguishes a 20 mK refrigerator base from an approximately 40 mK wired-device temperature, and a half-normal-resistance transition near 1 K across much of the gate range. These are different quantities. Its comparison supports a connection to quantum geometry without directly isolating a model-independent geometric fraction. This page stores the 40 mK value only as a measurement condition. [R4](#ref-source-be7282623ca96056)\n\n## Limits and disputes\n\nPeotta–Törmä's flat-band result assumes an interaction window between the bandwidth and separating band gaps, together with an appropriate uniform-pairing construction. Liang extends the response analysis, but the isolated-band expression retains time-reversal, orbital-uniform pairing and band-separation assumptions. Huhtinen and colleagues subsequently identify missing order-parameter-response terms in earlier formulas and connect isolated-flat-band stiffness to a minimal quantum metric. Therefore an arbitrary raw metric integral should not be presented as an automatically complete quantitative stiffness prediction. [R2](#ref-source-fdf091e09e7f4a6f) [R3](#ref-source-fa7a165c8c344a63) [R5](#ref-source-72f7ddb9f13bbd81)\n\nThe review covers selected Peotta published passages, Liang author-version equations and Tanaka author methods; it does not complete their raw-data, full supplementary or final-version audits. Provost's original is checked at publisher-abstract scope. Huhtinen's published introduction and selected response derivation, pages 1–4, were checked; its full minimal-metric proof and band-touching analysis were not reproduced. The inherited F1 bibliography misattributes the MATBG paper to Randeria; the published authors are Tanaka and colleagues. Complete scientific acceptance remains open. [R1](#ref-source-fbbc7a50ad47a3f8) [R2](#ref-source-fdf091e09e7f4a6f) [R3](#ref-source-fa7a165c8c344a63) [R4](#ref-source-be7282623ca96056) [R5](#ref-source-72f7ddb9f13bbd81)\n\n## Discovery implications\n\nCalculate geometry with a reproducible orbital convention and subspace, check mesh convergence and preserve normalization factors. Test pairing and total phase stiffness separately. A topological lower bound can be useful under its assumptions, but zero Chern number does not exclude useful geometry, and nonzero metric does not guarantee superconductivity. Geometry is one constraint in a material-specific calculation, alongside interactions, filling, disorder and phase coherence. [R2](#ref-source-fdf091e09e7f4a6f) [R3](#ref-source-fa7a165c8c344a63) [R5](#ref-source-72f7ddb9f13bbd81)\n\n## Related topics\n\nCompare [geometric superfluid weight](/scwiki/geometric-superfluid-weight), [flat-band pairing](/scwiki/flat-band-pairing), [twist angle](/scwiki/twist-angle) and [dimensionality](/scwiki/dimensionality).","revision":"f2d7e183c1f7e7ab67c8","sections":[{"id":"scope","title":"Scope","markdown":"## Scope\n\nThe quantum metric measures how a normalized quantum state changes with parameters after removing its unobservable overall phase. For Bloch bands, those parameters can be crystal momentum. It describes wavefunctions rather than only energy dispersion. A flat band can therefore have nonzero metric, while a large metric alone neither supplies an attractive interaction nor establishes superconductivity. Provost and Vallée developed a general state-manifold construction; later multiband superconductivity work connected band geometry to current response under specified assumptions. [R1](#ref-source-fbbc7a50ad47a3f8) [R2](#ref-source-fdf091e09e7f4a6f) [R3](#ref-source-fa7a165c8c344a63)\n\nThis page uses a single isolated band or a specified smooth band subspace. It does not identify a single-particle Bloch metric with every many-body fidelity susceptibility. Keep the Hilbert space, parameter coordinates, orbital basis and selected subspace explicit. A geometric quantity can vary continuously without becoming a quantized topological invariant. [R1](#ref-source-fbbc7a50ad47a3f8) [R3](#ref-source-fa7a165c8c344a63)"},{"id":"mathematical-assumptions","title":"Mathematical assumptions","markdown":"## Mathematical assumptions\n\nFor a normalized state $|u\\rangle$, let $P=|u\\rangle\\langle u|$ and $Q_{ij}=\\langle\\partial_i u|(1-P)|\\partial_j u\\rangle$. We define $g_{ij}=\\mathrm{Re}\\,Q_{ij}$ and $\\Omega_{ij}=-2\\mathrm{Im}\\,Q_{ij}$, consistent with $A_i=i\\langle u|\\partial_i u\\rangle$. Thus $Q_{ij}=g_{ij}-i\\Omega_{ij}/2$. The nearby-state fidelity obeys $1-|\\langle u(\\mathbf k)|u(\\mathbf k+d\\mathbf k)\\rangle|^2=g_{ij}dk_i dk_j+O(|d\\mathbf k|^3)$. With physical wavevectors, the metric components have units of length squared; dimensionless lattice coordinates change those units. [R1](#ref-source-fbbc7a50ad47a3f8) [R3](#ref-source-fa7a165c8c344a63)\n\nFor a rank-one projector, $Q_{ij}=\\mathrm{Tr}[P(\\partial_iP)(\\partial_jP)]$ and $g_{ij}=\\tfrac12\\mathrm{Tr}[(\\partial_iP)(\\partial_jP)]$. The projector is unchanged by $|u\\rangle\\rightarrow e^{i\\phi(\\mathbf k)}|u\\rangle$, so this construction removes phase-gauge artifacts. For an isolated group of touching bands, use the smooth subspace projector and state the trace convention. A chosen individual eigenvector can become singular or ambiguous at a degeneracy. Phase-gauge invariance within a fixed orbital representation is not permission to ignore a momentum-dependent change of the underlying basis. [R3](#ref-source-fa7a165c8c344a63) [R5](#ref-source-72f7ddb9f13bbd81)\n\nPositivity of the two-coordinate matrix $Q$ gives $\\det g\\geq\\Omega_{xy}^2/4$ and $g_{xx}+g_{yy}\\geq|\\Omega_{xy}|$. The trace statement refers to the selected common coordinate convention. For the spinor $u=(\\cos(\\theta/2),e^{i\\varphi}\\sin(\\theta/2))$, our convention gives $g_{\\theta\\theta}=1/4$, $g_{\\varphi\\varphi}=\\sin^2\\theta/4$ and $\\Omega_{\\theta\\varphi}=-\\sin\\theta/2$. This two-level example saturates the determinant inequality; a three-level family can satisfy it strictly. These are model checks, not measurements of a material. [R3](#ref-source-fa7a165c8c344a63)\n\nNormalization must be translated before comparing formulas. Liang's checked manuscript defines $\\mathcal R_{ij}=2\\mathrm{Tr}[P(\\partial_iP)(\\partial_jP)]$ and calls its real part the metric. Consequently $g^{\\rm Liang}=2g$ in this page's convention, and that paper writes the fidelity distance with an extra factor of one half. Its curvature sign must likewise be translated consistently. This is a convention difference, not a contradiction or a factor-of-two enhancement of a physical stiffness. [R3](#ref-source-fa7a165c8c344a63)\n\nAs a counterexample to equating metric with topology, take $u=(\\cos(k_x/2),\\sin(k_x/2))$, independent of $k_y$. The projector is periodic on a $2\\pi$ torus, $g_{xx}=1/4$, and the Berry curvature vanishes. The Hamiltonian $H=E_0(2P-I)$ has flat eigenvalues $\\pm E_0$ for positive constant $E_0$. It has nonzero metric and zero Chern number without introducing pairing. Our finite-difference checks compare projector derivatives, phase-transformed states and symmetric fidelity curvature; they do not calculate a superconducting response. [R3](#ref-source-fa7a165c8c344a63)"},{"id":"experimental-criteria","title":"Experimental criteria","markdown":"## Experimental criteria\n\nAn experimental response may constrain band geometry through a model and matrix elements; it need not uniquely reconstruct every metric component. For superconductivity, distinguish the measured total stiffness from a decomposition into conventional and geometric terms. Report carrier filling, band window, temperature, units and the response-to-metric conversion. A successful fit does not by itself identify a unique microscopic pairing mechanism. [R3](#ref-source-fa7a165c8c344a63) [R4](#ref-source-be7282623ca96056)\n\nTanaka and colleagues' 2025 MATBG study combines microwave kinetic inductance and transport. The checked author version includes a parallel edge-proximity circuit and an assumed active-region aspect ratio. It distinguishes a 20 mK refrigerator base from an approximately 40 mK wired-device temperature, and a half-normal-resistance transition near 1 K across much of the gate range. These are different quantities. Its comparison supports a connection to quantum geometry without directly isolating a model-independent geometric fraction. This page stores the 40 mK value only as a measurement condition. [R4](#ref-source-be7282623ca96056)"},{"id":"limits-and-disputes","title":"Limits and disputes","markdown":"## Limits and disputes\n\nPeotta–Törmä's flat-band result assumes an interaction window between the bandwidth and separating band gaps, together with an appropriate uniform-pairing construction. Liang extends the response analysis, but the isolated-band expression retains time-reversal, orbital-uniform pairing and band-separation assumptions. Huhtinen and colleagues subsequently identify missing order-parameter-response terms in earlier formulas and connect isolated-flat-band stiffness to a minimal quantum metric. Therefore an arbitrary raw metric integral should not be presented as an automatically complete quantitative stiffness prediction. [R2](#ref-source-fdf091e09e7f4a6f) [R3](#ref-source-fa7a165c8c344a63) [R5](#ref-source-72f7ddb9f13bbd81)\n\nThe review covers selected Peotta published passages, Liang author-version equations and Tanaka author methods; it does not complete their raw-data, full supplementary or final-version audits. Provost's original is checked at publisher-abstract scope. Huhtinen's published introduction and selected response derivation, pages 1–4, were checked; its full minimal-metric proof and band-touching analysis were not reproduced. The inherited F1 bibliography misattributes the MATBG paper to Randeria; the published authors are Tanaka and colleagues. Complete scientific acceptance remains open. [R1](#ref-source-fbbc7a50ad47a3f8) [R2](#ref-source-fdf091e09e7f4a6f) [R3](#ref-source-fa7a165c8c344a63) [R4](#ref-source-be7282623ca96056) [R5](#ref-source-72f7ddb9f13bbd81)"},{"id":"discovery-implications","title":"Discovery implications","markdown":"## Discovery implications\n\nCalculate geometry with a reproducible orbital convention and subspace, check mesh convergence and preserve normalization factors. Test pairing and total phase stiffness separately. A topological lower bound can be useful under its assumptions, but zero Chern number does not exclude useful geometry, and nonzero metric does not guarantee superconductivity. Geometry is one constraint in a material-specific calculation, alongside interactions, filling, disorder and phase coherence. [R2](#ref-source-fdf091e09e7f4a6f) [R3](#ref-source-fa7a165c8c344a63) [R5](#ref-source-72f7ddb9f13bbd81)"},{"id":"related-topics","title":"Related topics","markdown":"## Related topics\n\nCompare [geometric superfluid weight](/scwiki/geometric-superfluid-weight), [flat-band pairing](/scwiki/flat-band-pairing), [twist angle](/scwiki/twist-angle) and [dimensionality](/scwiki/dimensionality)."},{"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| C (aliases: MATBG, Tanaka device, twisted bilayer graphene); Tanaka wired MATBG device | 0.04 K; Approximately40mK effective device temperature, distinct from20mKbase andnear1Khalf-resistance scale; no Tc assigned here. | measurement; effective-device-temperature | Microwave kinetic inductance and transport | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-be7282623ca96056) |"}],"markdown":"# Quantum metric: state distance, conventions and response limits\n\nDefine Bloch-state geometry with explicit normalization, distinguish metric from topology, and keep geometric response models separate from measured total stiffness.\n\n## Scope\n\nThe quantum metric measures how a normalized quantum state changes with parameters after removing its unobservable overall phase. For Bloch bands, those parameters can be crystal momentum. It describes wavefunctions rather than only energy dispersion. A flat band can therefore have nonzero metric, while a large metric alone neither supplies an attractive interaction nor establishes superconductivity. Provost and Vallée developed a general state-manifold construction; later multiband superconductivity work connected band geometry to current response under specified assumptions. [R1](#ref-source-fbbc7a50ad47a3f8) [R2](#ref-source-fdf091e09e7f4a6f) [R3](#ref-source-fa7a165c8c344a63)\n\nThis page uses a single isolated band or a specified smooth band subspace. It does not identify a single-particle Bloch metric with every many-body fidelity susceptibility. Keep the Hilbert space, parameter coordinates, orbital basis and selected subspace explicit. A geometric quantity can vary continuously without becoming a quantized topological invariant. [R1](#ref-source-fbbc7a50ad47a3f8) [R3](#ref-source-fa7a165c8c344a63)\n\n## Mathematical assumptions\n\nFor a normalized state $|u\\rangle$, let $P=|u\\rangle\\langle u|$ and $Q_{ij}=\\langle\\partial_i u|(1-P)|\\partial_j u\\rangle$. We define $g_{ij}=\\mathrm{Re}\\,Q_{ij}$ and $\\Omega_{ij}=-2\\mathrm{Im}\\,Q_{ij}$, consistent with $A_i=i\\langle u|\\partial_i u\\rangle$. Thus $Q_{ij}=g_{ij}-i\\Omega_{ij}/2$. The nearby-state fidelity obeys $1-|\\langle u(\\mathbf k)|u(\\mathbf k+d\\mathbf k)\\rangle|^2=g_{ij}dk_i dk_j+O(|d\\mathbf k|^3)$. With physical wavevectors, the metric components have units of length squared; dimensionless lattice coordinates change those units. [R1](#ref-source-fbbc7a50ad47a3f8) [R3](#ref-source-fa7a165c8c344a63)\n\nFor a rank-one projector, $Q_{ij}=\\mathrm{Tr}[P(\\partial_iP)(\\partial_jP)]$ and $g_{ij}=\\tfrac12\\mathrm{Tr}[(\\partial_iP)(\\partial_jP)]$. The projector is unchanged by $|u\\rangle\\rightarrow e^{i\\phi(\\mathbf k)}|u\\rangle$, so this construction removes phase-gauge artifacts. For an isolated group of touching bands, use the smooth subspace projector and state the trace convention. A chosen individual eigenvector can become singular or ambiguous at a degeneracy. Phase-gauge invariance within a fixed orbital representation is not permission to ignore a momentum-dependent change of the underlying basis. [R3](#ref-source-fa7a165c8c344a63) [R5](#ref-source-72f7ddb9f13bbd81)\n\nPositivity of the two-coordinate matrix $Q$ gives $\\det g\\geq\\Omega_{xy}^2/4$ and $g_{xx}+g_{yy}\\geq|\\Omega_{xy}|$. The trace statement refers to the selected common coordinate convention. For the spinor $u=(\\cos(\\theta/2),e^{i\\varphi}\\sin(\\theta/2))$, our convention gives $g_{\\theta\\theta}=1/4$, $g_{\\varphi\\varphi}=\\sin^2\\theta/4$ and $\\Omega_{\\theta\\varphi}=-\\sin\\theta/2$. This two-level example saturates the determinant inequality; a three-level family can satisfy it strictly. These are model checks, not measurements of a material. [R3](#ref-source-fa7a165c8c344a63)\n\nNormalization must be translated before comparing formulas. Liang's checked manuscript defines $\\mathcal R_{ij}=2\\mathrm{Tr}[P(\\partial_iP)(\\partial_jP)]$ and calls its real part the metric. Consequently $g^{\\rm Liang}=2g$ in this page's convention, and that paper writes the fidelity distance with an extra factor of one half. Its curvature sign must likewise be translated consistently. This is a convention difference, not a contradiction or a factor-of-two enhancement of a physical stiffness. [R3](#ref-source-fa7a165c8c344a63)\n\nAs a counterexample to equating metric with topology, take $u=(\\cos(k_x/2),\\sin(k_x/2))$, independent of $k_y$. The projector is periodic on a $2\\pi$ torus, $g_{xx}=1/4$, and the Berry curvature vanishes. The Hamiltonian $H=E_0(2P-I)$ has flat eigenvalues $\\pm E_0$ for positive constant $E_0$. It has nonzero metric and zero Chern number without introducing pairing. Our finite-difference checks compare projector derivatives, phase-transformed states and symmetric fidelity curvature; they do not calculate a superconducting response. [R3](#ref-source-fa7a165c8c344a63)\n\n## Experimental criteria\n\nAn experimental response may constrain band geometry through a model and matrix elements; it need not uniquely reconstruct every metric component. For superconductivity, distinguish the measured total stiffness from a decomposition into conventional and geometric terms. Report carrier filling, band window, temperature, units and the response-to-metric conversion. A successful fit does not by itself identify a unique microscopic pairing mechanism. [R3](#ref-source-fa7a165c8c344a63) [R4](#ref-source-be7282623ca96056)\n\nTanaka and colleagues' 2025 MATBG study combines microwave kinetic inductance and transport. The checked author version includes a parallel edge-proximity circuit and an assumed active-region aspect ratio. It distinguishes a 20 mK refrigerator base from an approximately 40 mK wired-device temperature, and a half-normal-resistance transition near 1 K across much of the gate range. These are different quantities. Its comparison supports a connection to quantum geometry without directly isolating a model-independent geometric fraction. This page stores the 40 mK value only as a measurement condition. [R4](#ref-source-be7282623ca96056)\n\n## Limits and disputes\n\nPeotta–Törmä's flat-band result assumes an interaction window between the bandwidth and separating band gaps, together with an appropriate uniform-pairing construction. Liang extends the response analysis, but the isolated-band expression retains time-reversal, orbital-uniform pairing and band-separation assumptions. Huhtinen and colleagues subsequently identify missing order-parameter-response terms in earlier formulas and connect isolated-flat-band stiffness to a minimal quantum metric. Therefore an arbitrary raw metric integral should not be presented as an automatically complete quantitative stiffness prediction. [R2](#ref-source-fdf091e09e7f4a6f) [R3](#ref-source-fa7a165c8c344a63) [R5](#ref-source-72f7ddb9f13bbd81)\n\nThe review covers selected Peotta published passages, Liang author-version equations and Tanaka author methods; it does not complete their raw-data, full supplementary or final-version audits. Provost's original is checked at publisher-abstract scope. Huhtinen's published introduction and selected response derivation, pages 1–4, were checked; its full minimal-metric proof and band-touching analysis were not reproduced. The inherited F1 bibliography misattributes the MATBG paper to Randeria; the published authors are Tanaka and colleagues. Complete scientific acceptance remains open. [R1](#ref-source-fbbc7a50ad47a3f8) [R2](#ref-source-fdf091e09e7f4a6f) [R3](#ref-source-fa7a165c8c344a63) [R4](#ref-source-be7282623ca96056) [R5](#ref-source-72f7ddb9f13bbd81)\n\n## Discovery implications\n\nCalculate geometry with a reproducible orbital convention and subspace, check mesh convergence and preserve normalization factors. Test pairing and total phase stiffness separately. A topological lower bound can be useful under its assumptions, but zero Chern number does not exclude useful geometry, and nonzero metric does not guarantee superconductivity. Geometry is one constraint in a material-specific calculation, alongside interactions, filling, disorder and phase coherence. [R2](#ref-source-fdf091e09e7f4a6f) [R3](#ref-source-fa7a165c8c344a63) [R5](#ref-source-72f7ddb9f13bbd81)\n\n## Related topics\n\nCompare [geometric superfluid weight](/scwiki/geometric-superfluid-weight), [flat-band pairing](/scwiki/flat-band-pairing), [twist angle](/scwiki/twist-angle) and [dimensionality](/scwiki/dimensionality).\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| C (aliases: MATBG, Tanaka device, twisted bilayer graphene); Tanaka wired MATBG device | 0.04 K; Approximately40mK effective device temperature, distinct from20mKbase andnear1Khalf-resistance scale; no Tc assigned here. | measurement; effective-device-temperature | Microwave kinetic inductance and transport | ambient; unknown GPa | equilibrium | experiment; [source](#ref-source-be7282623ca96056) |\n\n## Sources\n\n### Ref source-fbbc7a50ad47a3f8\n\n[R1] Riemannian structure on manifolds of quantum states (1980). [DOI 10.1007/BF02193559](https://doi.org/10.1007/BF02193559). Publication: journal-article. Check: publisher-abstract-and-bibliography. Checked: 2026-10-18. Original publisher abstract and September1980 citation checked; full original not retrieved. General state-manifold background only.\n### Ref source-fdf091e09e7f4a6f\n\n[R2] Superfluidity in topologically nontrivial flat bands (2015). [DOI 10.1038/ncomms9944](https://doi.org/10.1038/ncomms9944). Publication: journal-article. Check: selected-published-flat-band-assumptions. Checked: 2026-10-18. Published20November2015. Existing local final PDF selectedpp4–6, especially uniform pairing and bandwidth/interaction/bandgap hierarchy onp5; prior B13p5visual reused. Full supplementary derivation not reproduced.\n### Ref source-fa7a165c8c344a63\n\n[R3] Band geometry, Berry curvature, and superfluid weight (2017). [DOI 10.1103/PhysRevB.95.024515](https://doi.org/10.1103/PhysRevB.95.024515). Publication: journal-article. Check: selected-author-projector-and-metric-conventions. Checked: 2026-10-18. Authorv2 11January2017,selectedpp1–2,4–6; p5Eq21–24 visuallychecked. Published27January2017. R=2Tr(P dP dP) and gLiang=2gWiki. Final-version comparison and full response derivation remain open.\n### Ref source-be7282623ca96056\n\n[R4] Superfluid stiffness of magic-angle twisted bilayer graphene (2025). [DOI 10.1038/s41586-024-08494-7](https://doi.org/10.1038/s41586-024-08494-7). Publication: journal-article. Check: selected-author-circuit-and-temperature-definitions. Checked: 2026-10-18. Tanaka et al, published5February2025. Authorv2selectedpp5–7 and prior B13/E9 Supplement3p31 circuitreview.20mKbase/40mKdevice,half-resistance near1K,active aspect ratio and parallel edge model; fullfinal/rawdata comparison open.\n### Ref source-72f7ddb9f13bbd81\n\n[R5] Revisiting flat band superconductivity: Dependence on minimal quantum metric and band touchings (2022). [DOI 10.1103/PhysRevB.106.014518](https://doi.org/10.1103/PhysRevB.106.014518). Publication: journal-article. Check: selected-published-response-correction. Checked: 2026-10-18. Published26July2022. Final reprint introduction and selectedpp1–5 derivative/orbital-position discussion plusp6minimal-integrated-trace statement read as extractedtext. Full minimal-metric proof,band-touching calculation and complete response model not reproduced.\n\n## Claim evidence and conditions\n\n### scwiki-f1-claim-scope-1\n\nThe quantum metric measures how a normalized quantum state changes with parameters after removing its unobservable overall phase. For Bloch bands, those parameters can be crystal momentum. It describes wavefunctions rather than only energy dispersion. A flat band can therefore have nonzero metric, while a large metric alone neither supplies an attractive interaction nor establishes superconductivity. Provost and Vallée developed a general state-manifold construction; later multiband superconductivity work connected band geometry to current response under specified assumptions.\n\nEvidence: theory. Sources: source-fbbc7a50ad47a3f8, source-fdf091e09e7f4a6f, source-fa7a165c8c344a63.\n\nConditions: Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.\n\nVerification scope: Selected primary passages and bounded model checks; complete scientific acceptance open.\n\n### scwiki-f1-claim-scope-2\n\nThis page uses a single isolated band or a specified smooth band subspace. It does not identify a single-particle Bloch metric with every many-body fidelity susceptibility. Keep the Hilbert space, parameter coordinates, orbital basis and selected subspace explicit. A geometric quantity can vary continuously without becoming a quantized topological invariant.\n\nEvidence: theory. Sources: source-fbbc7a50ad47a3f8, source-fa7a165c8c344a63.\n\nConditions: Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.\n\nVerification scope: Selected primary passages and bounded model checks; complete scientific acceptance open.\n\n### scwiki-f1-claim-mathematical-assumptions-1\n\nFor a normalized state $|u\\rangle$, let $P=|u\\rangle\\langle u|$ and $Q_{ij}=\\langle\\partial_i u|(1-P)|\\partial_j u\\rangle$. We define $g_{ij}=\\mathrm{Re}\\,Q_{ij}$ and $\\Omega_{ij}=-2\\mathrm{Im}\\,Q_{ij}$, consistent with $A_i=i\\langle u|\\partial_i u\\rangle$. Thus $Q_{ij}=g_{ij}-i\\Omega_{ij}/2$. The nearby-state fidelity obeys $1-|\\langle u(\\mathbf k)|u(\\mathbf k+d\\mathbf k)\\rangle|^2=g_{ij}dk_i dk_j+O(|d\\mathbf k|^3)$. With physical wavevectors, the metric components have units of length squared; dimensionless lattice coordinates change those units.\n\nEvidence: theory. Sources: source-fbbc7a50ad47a3f8, source-fa7a165c8c344a63.\n\nConditions: Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.\n\nVerification scope: Selected primary passages and bounded model checks; complete scientific acceptance open.\n\n### scwiki-f1-claim-mathematical-assumptions-2\n\nFor a rank-one projector, $Q_{ij}=\\mathrm{Tr}[P(\\partial_iP)(\\partial_jP)]$ and $g_{ij}=\\tfrac12\\mathrm{Tr}[(\\partial_iP)(\\partial_jP)]$. The projector is unchanged by $|u\\rangle\\rightarrow e^{i\\phi(\\mathbf k)}|u\\rangle$, so this construction removes phase-gauge artifacts. For an isolated group of touching bands, use the smooth subspace projector and state the trace convention. A chosen individual eigenvector can become singular or ambiguous at a degeneracy. Phase-gauge invariance within a fixed orbital representation is not permission to ignore a momentum-dependent change of the underlying basis.\n\nEvidence: theory. Sources: source-fa7a165c8c344a63, source-72f7ddb9f13bbd81.\n\nConditions: Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.\n\nVerification scope: Selected primary passages and bounded model checks; complete scientific acceptance open.\n\n### scwiki-f1-claim-mathematical-assumptions-3\n\nPositivity of the two-coordinate matrix $Q$ gives $\\det g\\geq\\Omega_{xy}^2/4$ and $g_{xx}+g_{yy}\\geq|\\Omega_{xy}|$. The trace statement refers to the selected common coordinate convention. For the spinor $u=(\\cos(\\theta/2),e^{i\\varphi}\\sin(\\theta/2))$, our convention gives $g_{\\theta\\theta}=1/4$, $g_{\\varphi\\varphi}=\\sin^2\\theta/4$ and $\\Omega_{\\theta\\varphi}=-\\sin\\theta/2$. This two-level example saturates the determinant inequality; a three-level family can satisfy it strictly. These are model checks, not measurements of a material.\n\nEvidence: theory. Sources: source-fa7a165c8c344a63.\n\nConditions: Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.\n\nVerification scope: Selected primary passages and bounded model checks; complete scientific acceptance open.\n\n### scwiki-f1-claim-mathematical-assumptions-4\n\nNormalization must be translated before comparing formulas. Liang's checked manuscript defines $\\mathcal R_{ij}=2\\mathrm{Tr}[P(\\partial_iP)(\\partial_jP)]$ and calls its real part the metric. Consequently $g^{\\rm Liang}=2g$ in this page's convention, and that paper writes the fidelity distance with an extra factor of one half. Its curvature sign must likewise be translated consistently. This is a convention difference, not a contradiction or a factor-of-two enhancement of a physical stiffness.\n\nEvidence: theory. Sources: source-fa7a165c8c344a63.\n\nConditions: Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.\n\nVerification scope: Selected primary passages and bounded model checks; complete scientific acceptance open.\n\n### scwiki-f1-claim-mathematical-assumptions-5\n\nAs a counterexample to equating metric with topology, take $u=(\\cos(k_x/2),\\sin(k_x/2))$, independent of $k_y$. The projector is periodic on a $2\\pi$ torus, $g_{xx}=1/4$, and the Berry curvature vanishes. The Hamiltonian $H=E_0(2P-I)$ has flat eigenvalues $\\pm E_0$ for positive constant $E_0$. It has nonzero metric and zero Chern number without introducing pairing. Our finite-difference checks compare projector derivatives, phase-transformed states and symmetric fidelity curvature; they do not calculate a superconducting response.\n\nEvidence: theory. Sources: source-fa7a165c8c344a63.\n\nConditions: Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.\n\nVerification scope: Selected primary passages and bounded model checks; complete scientific acceptance open.\n\n### scwiki-f1-claim-experimental-criteria-1\n\nAn experimental response may constrain band geometry through a model and matrix elements; it need not uniquely reconstruct every metric component. For superconductivity, distinguish the measured total stiffness from a decomposition into conventional and geometric terms. Report carrier filling, band window, temperature, units and the response-to-metric conversion. A successful fit does not by itself identify a unique microscopic pairing mechanism.\n\nEvidence: experiment. Sources: source-fa7a165c8c344a63, source-be7282623ca96056.\n\nConditions: Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.\n\nVerification scope: Selected primary passages and bounded model checks; complete scientific acceptance open.\n\n### scwiki-f1-claim-experimental-criteria-2\n\nTanaka and colleagues' 2025 MATBG study combines microwave kinetic inductance and transport. The checked author version includes a parallel edge-proximity circuit and an assumed active-region aspect ratio. It distinguishes a 20 mK refrigerator base from an approximately 40 mK wired-device temperature, and a half-normal-resistance transition near 1 K across much of the gate range. These are different quantities. Its comparison supports a connection to quantum geometry without directly isolating a model-independent geometric fraction. This page stores the 40 mK value only as a measurement condition.\n\nEvidence: experiment. Sources: source-be7282623ca96056.\n\nConditions: Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.\n\nVerification scope: Selected primary passages and bounded model checks; complete scientific acceptance open.\n\n### scwiki-f1-claim-limits-and-disputes-1\n\nPeotta–Törmä's flat-band result assumes an interaction window between the bandwidth and separating band gaps, together with an appropriate uniform-pairing construction. Liang extends the response analysis, but the isolated-band expression retains time-reversal, orbital-uniform pairing and band-separation assumptions. Huhtinen and colleagues subsequently identify missing order-parameter-response terms in earlier formulas and connect isolated-flat-band stiffness to a minimal quantum metric. Therefore an arbitrary raw metric integral should not be presented as an automatically complete quantitative stiffness prediction.\n\nEvidence: theory. Sources: source-fdf091e09e7f4a6f, source-fa7a165c8c344a63, source-72f7ddb9f13bbd81.\n\nConditions: Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.\n\nVerification scope: Selected primary passages and bounded model checks; complete scientific acceptance open.\n\n### scwiki-f1-claim-limits-and-disputes-2\n\nThe review covers selected Peotta published passages, Liang author-version equations and Tanaka author methods; it does not complete their raw-data, full supplementary or final-version audits. Provost's original is checked at publisher-abstract scope. Huhtinen's published introduction and selected response derivation, pages 1–4, were checked; its full minimal-metric proof and band-touching analysis were not reproduced. The inherited F1 bibliography misattributes the MATBG paper to Randeria; the published authors are Tanaka and colleagues. Complete scientific acceptance remains open.\n\nEvidence: theory. Sources: source-fbbc7a50ad47a3f8, source-fdf091e09e7f4a6f, source-fa7a165c8c344a63, source-be7282623ca96056, source-72f7ddb9f13bbd81.\n\nConditions: Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.\n\nVerification scope: Selected primary passages and bounded model checks; complete scientific acceptance open.\n\n### scwiki-f1-claim-discovery-implications-1\n\nCalculate geometry with a reproducible orbital convention and subspace, check mesh convergence and preserve normalization factors. Test pairing and total phase stiffness separately. A topological lower bound can be useful under its assumptions, but zero Chern number does not exclude useful geometry, and nonzero metric does not guarantee superconductivity. Geometry is one constraint in a material-specific calculation, alongside interactions, filling, disorder and phase coherence.\n\nEvidence: theory. Sources: source-fdf091e09e7f4a6f, source-fa7a165c8c344a63, source-72f7ddb9f13bbd81.\n\nConditions: Fixed orbital representation, selected smooth subspace, explicit metric/curvature normalization and units; models separate from material measurements.\n\nVerification scope: Selected primary passages and bounded model checks; complete scientific acceptance open.\n\n## Revision and review\n\nArticle ID: scwiki-f1. Revision: f2d7e183c1f7e7ab67c8. Updated: 2026-10-18T13:49:34.227083+00:00.\n\nReview scope: Selected Liang author equations,Peotta published assumptions,Huhtinen published response correction,Tanaka author methods and independent bounded state-geometry models.\n\n- Complete scientific acceptance, full supplementary and raw-data reviews remain open.\n- Provost original is publisher-abstract scope; Liang and Tanaka final-version comparisons incomplete.\n- No material quantum metric, geometric stiffness fraction or transition temperature reconstructed.\n- Huhtinen full minimal-metric proof,band-touching response and complete self-consistent calculation not reproduced.\n\nSource: [F_quantum_geometry/F1_quantum_metric.md](https://github.com/JackZH26/jzis-superconductivity/blob/0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf/F_quantum_geometry/F1_quantum_metric.md), commit 0a7c9d6696804ec26aa12f8eed68b55c9b3c0fcf, SHA-256 f4f76f73cf21723199f60f45a93acb2d4ace612337ebbf851629966f88dad668.\n","wordCount":848}}