SSH Model: Topological Phase Transition
The Su-Schrieffer-Heeger (SSH) model describes electrons on a 1D chain with alternating hopping amplitudes t₁ and t₂. When t₂/t₁ > 1, the system is in a topological phase with protected zero-energy edge states. The topological invariant is the winding number W ∈ {0, 1}.
Trivial Phase (W=0)
Wavefunction |ψ|² for selected states
Chain visualization: thick bonds = t₁, thin bonds = t₂ (or vice versa); orange dots = edge-state weight
SSH Hamiltonian:
H = Σ_n [t₁ c†_{n,A} c_{n,B} + t₂ c†_{n+1,A} c_{n,B} + h.c.]
In k-space: H(k) = d(k)·σ where d(k) = (t₁+t₂cos k, −t₂sin k, 0)
Topological invariant (winding number):
W = (1/2π) ∮ dφ(k) where φ(k) = arg(d_x + i·d_y)
W=0 (trivial): t₁ > t₂. W=1 (topological): t₂ > t₁ → zero-energy edge modes protected by chiral symmetry (sublattice symmetry). Bulk-boundary correspondence: W = number of edge states per edge.