Quantum Metric Tensor & Berry Curvature

Quantum geometric tensor Q_μν = ⟨∂_μu|∂_νu⟩ − ⟨∂_μu|u⟩⟨u|∂_νu⟩. Real part = quantum metric g_μν (measures distinguishability of states). Imaginary part = Berry curvature Ω_μν/2. Berry curvature Ω = Im⟨∂_kx u|∂_ky u⟩ integrated over BZ gives Chern number C = (1/2π)∫Ω d²k ∈ ℤ. At |m| < 2: C = ±1 (topological). At |m| > 2: C = 0 (trivial). Gap closes at |m| = 2 — topological transition.