Topological Band Inversion & Berry Phase

Band inversion occurs when a parameter like mass m passes through zero, swapping the orbital character of the valence and conduction bands. The Berry phase γ = i∮⟨u(k)|∂_k|u(k)⟩dk is quantized (0 or π) by inversion/time-reversal symmetry — this is the Z₂ topological invariant. When γ=π (topological phase), the bulk-boundary correspondence guarantees protected edge states crossing the gap. The SSH model H(k) = (t₁+t₂cos k)σ_x + t₂sin k σ_y + mσ_z demonstrates this: |m|<|t₁| gives topological, |m|>|t₁| gives trivial.