Quantum metric: state distance, conventions and response limits

Scope

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. R1 R2 R3

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. R1 R3

Mathematical assumptions

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. R1 R3

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. R3 R5

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. R3

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. R3

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. R3

Experimental criteria

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. R3 R4

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. R4

Limits and disputes

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. R2 R3 R5

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. R1 R2 R3 R4 R5

Discovery implications

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. R2 R3 R5

Related topics

Compare geometric superfluid weight, flat-band pairing, twist angle and dimensionality.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.