SSH Model — Topological Edge States

SSH (Su-Schrieffer-Heeger) model: A 1D tight-binding chain with alternating hoppings v (intra-cell) and w (inter-cell). The Bloch Hamiltonian H(k) = (v + w·cos k)σₓ + w·sin k·σᵧ traces a loop in the (d_x, d_y) plane as k traverses the Brillouin zone. When |w|>|v|, this loop winds around the origin — winding number ν=1, topological phase — and the finite chain hosts zero-energy edge states exponentially localized at both ends. When |v|>|w|, the loop misses the origin (ν=0, trivial). The gap closes at v=w — a topological phase transition.