The Su-Schrieffer-Heeger (SSH) model is the simplest 1D topological insulator. Electrons hop between sites with alternating amplitudes t₁ (intracell) and t₂ (intercell). When t₂ > t₁, the chain enters a topological phase with a non-zero winding number Z = 1, and protected zero-energy states appear localised at both ends of the chain — even in the presence of weak disorder.
H = Σₙ (t₁ |A,n⟩⟨B,n| + t₂ |B,n⟩⟨A,n+1| + h.c.)
h(k) = t₁ + t₂ e^{ik} → E(k) = ±|h(k)|
Winding number: Z = (1/2πi) ∮ h*(k) dh(k)/dk dk / |h(k)|²
Gap = 2|t₁ - t₂| (closes at t₁ = t₂)
Topological edge states are robust to disorder because they are protected by symmetry, not by a specific Hamiltonian. This makes them candidates for fault-tolerant quantum computation. The SSH model was first proposed in 1979 to explain conductivity in polyacetylene — a plastic that can conduct electricity!
A topological insulator is a material that behaves as an insulator in its bulk but supports conducting states on its surface or edges. These edge states are topologically protected — they cannot be removed by smooth deformations of the Hamiltonian that preserve certain symmetries.
The SSH (Su-Schrieffer-Heeger) model describes electrons hopping on a 1D chain with alternating hopping amplitudes t1 (intracell) and t2 (intercell). It is the simplest model exhibiting a topological phase transition, making it a paradigmatic example in condensed matter physics.
The bulk-edge correspondence states that the number of topologically protected edge states at the boundary of a system is determined by a bulk topological invariant (like the winding number). When the winding number is 1, protected zero-energy edge states appear at both ends of an open chain.
The winding number Z counts how many times the vector (h_x(k), h_y(k)) winds around the origin as k traverses the Brillouin zone. For SSH: Z=0 when t2 < t1 (trivial phase), Z=1 when t2 > t1 (topological phase). At t1=t2 the gap closes and the phase transition occurs.
Edge states in the SSH model are protected by chiral symmetry (a sublattice symmetry). As long as this symmetry is preserved, perturbations cannot hybridize the two edge states and push them away from zero energy. This robustness is the hallmark of topological protection.
At t2/t1 = 1, the bulk energy gap closes at k = ±π/a. This gap closure signals a topological phase transition between the trivial (Z=0) and topological (Z=1) phases. Edge states only exist on one side of this critical point.
For an open chain of N sites, the Hamiltonian is an N×N tridiagonal real symmetric matrix. Odd off-diagonal entries are t1 and even off-diagonal entries are t2. Eigenvalues give the energy spectrum, and eigenvectors give the probability amplitudes |ψ|² on each site.
In the topological phase, edge state wavefunctions are exponentially localized at the chain ends. The localization length decreases as t2/t1 increases beyond 1. These states live exclusively on sublattice A at one end and sublattice B at the other end.
Yes. The SSH model is the 1D prototype. In 2D, the quantum spin Hall effect hosts helical edge states. In 3D, topological insulators like Bi₂Se₃ host Dirac-cone surface states. The full classification of topological phases in all symmetry classes and dimensions is given by the periodic table of topological insulators.
Topological insulators are promising for dissipationless electronics, spintronics, and quantum computing. Majorana fermions — potential building blocks for fault-tolerant qubits — can emerge at the ends of topological superconductors, which are closely related to the SSH model.