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.
This simulation visualises the Su–Schrieffer–Heeger (SSH) model, the simplest one-dimensional topological insulator. A chain of sites is coupled by alternating hopping amplitudes t₁ (intracell) and t₂ (intercell); diagonalising the resulting tridiagonal Hamiltonian reveals whether the chain sits in a trivial or topological phase, characterised by a bulk winding number Z that predicts protected zero-energy edge states at the chain ends.
The energy spectrum of an open chain of N sites, built by diagonalising a tridiagonal Hamiltonian with intracell hopping t₁ and intercell hopping t₂. When t₂ > t₁, two mid-gap eigenvalues appear near E = 0; their eigenvectors are exponentially localised at opposite ends of the chain — the hallmark of topologically protected edge states.
Drag the t₁ and t₂ sliders (0.1–3.0) to change hopping strengths and cross the t₂/t₁ = 1 phase boundary. The N slider (8–80) sets chain length and Disorder (0–1.5) adds random on-site noise to test how robust the edge states are. Toggle View swaps the spectrum/edge-density panels for the periodic band structure E(k) and the winding-number path h(k); Pause and Reset control the animation.
The SSH model was proposed in 1979 to explain electrical conduction in polyacetylene, a conjugated polymer whose alternating single and double bonds are a real chemical example of the same dimerisation pattern that drives this topological phase transition.
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.