Fubini–Study quantum metric
Quantum states have a natural notion of distance that measures how quickly distinguishable states change.
$$g_{ij}=\mathrm{Re}\big[\langle\partial_i u|(1-|u\rangle\langle u|)|\partial_j u\rangle\big]$$
What this means
Quantum states have a natural notion of distance that measures how quickly distinguishable states change.
Deep dive: mathematical and physical mechanism
The quantum geometric tensor is $Q_{ij}=\langle\partial_i u|(1-|u\rangle\langle u|)|\partial_j u\rangle$. Its real part is the quantum metric, while its imaginary antisymmetric part is related to Berry curvature. The metric affects localization, flat-band superfluidity, and response.
How it appears in quantum phenomena
Quantum metric contributes to the superfluid weight in some flat-band systems.
Assumptions, caveats and limitations
Metric and topology are related but not interchangeable; a large metric does not imply nonzero Chern number.
Connect the ideas
For context, see the Geometric Quantum Mechanics topic guide and the reading references.