SSH Model — Topological Band Theory

Bulk topological invariant (Zak phase/winding number), bulk-boundary correspondence, edge states

Band Structure E(k)

Intra-cell hopping v0.8

Inter-cell hopping w1.4

SSH Hamiltonian: H(k) = (v+w cos k)σₓ + w sin k σᵧ
Energy: E(k) = ±√(v²+w²+2vw cos k)
Gap closes at k=π when v=w (topological phase transition).

Winding Number & Zak Phase

v/w ratio0.57

Winding number: ν = (1/2π) ∮ (∂θ/∂k)dk where θ(k) = arg(v+we^(ik)).
v<w: ν=1 (topological), Zak phase = π. v>w: ν=0 (trivial), Zak phase = 0.

Finite Chain — Edge States

Chain length N14

v hopping0.6

w hopping1.5

Bulk-boundary correspondence: topological phase (v<w) → 2 zero-energy edge modes exponentially localized at chain ends. Edge state energy: E_edge → 0 exactly in the thermodynamic limit.

Adiabatic Pumping Cycle

Pump phase φ0.00

Thouless pump: adiabatically cycle parameters around the topological phase transition → quantized charge transport ΔQ = ν per cycle (in units of e). Direct consequence of Zak phase = Berry phase of band.