Berry curvature Ω(k) and quantum metric g(k) in the Brillouin zone
Chern number: — | ∫g dk: —
The quantum geometric tensor Qnm(k) = ⟨∂ₙu|∂ₘu⟩ − ⟨∂ₙu|u⟩⟨u|∂ₘu⟩ unifies geometry and topology. Its real part is the quantum metric g(k) (measuring Wannier spread), imaginary part is Berry curvature Ω(k) (giving Hall conductance via TKNN). SSH model: H(k) = d(k)·σ; Chern number = ±1 in topological phase (t' > t). Left: band structure. Right: Berry curvature (blue/red) and metric (brightness).