🔵 Topological Insulator — Bulk-Edge Correspondence

The Su–Schrieffer–Heeger model describes electrons hopping along a one-dimensional chain with alternating amplitudes t₁ and t₂. It is the simplest system exhibiting a topological phase transition, making it the standard teaching example in condensed matter physics for bulk-edge correspondence.

Z counts how many times the vector h(k) = (t₁ + t₂cos k, t₂sin k) winds around the origin as k sweeps the Brillouin zone. Z = 0 gives the trivial phase (t₂ < t₁); Z = 1 gives the topological phase (t₂ > t₁), and the bulk gap closes exactly at t₁ = t₂.

t₁ and t₂ reshape the hopping pattern and move the system across the phase boundary; N sets how many sites are in the chain, which sharpens the spectrum; Disorder adds a random on-site potential so you can check whether the mid-gap edge states survive — they do, as long as chiral symmetry is not badly broken.

They are protected by chiral (sublattice) symmetry rather than by any specific value of the Hamiltonian. As long as that symmetry survives, local perturbations cannot mix the two edge states or push them away from zero energy, which is why they persist even when you add noise with the Disorder slider.

At t₂ = t₁ the bulk energy gap closes at the edge of the Brillouin zone, marking the topological phase transition; on one side no edge states exist, on the other two appear. The same bulk-edge principle extends to 2D quantum spin Hall insulators and 3D materials such as Bi₂Se₃, where protected states live on the surface instead of just at chain ends.