Physical question
Does the bulk winding guarantee an edge mode for every possible cut?
A bulk one-dimensional winding number is defined only after specifying its Bloch basis, unit cell and protecting symmetry. Edge states require a compatible physical termination: which bonds are cut matters. A finite chain also contains two ends, and modes localized near opposite boundaries may overlap and split away from exact zero. The presence or absence of a near-zero mode, and its robustness under a staggered potential, provide a concrete route to understanding both the power and the limits of bulk–boundary correspondence.
Prerequisites: The SSH Bloch Hamiltonian, alternating hopping terms, sublattice/chiral symmetry, and finite-dimensional matrix diagonalization.
Calculated quantum predictions
Live readout and numerical experiments
Finite-chain levels
Exact ordered energies of the open chain
The near-middle levels make the edge-state splitting visible while all other eigenstates remain part of the complete finite-dimensional spectrum.
Each point corresponds to an eigenvalue of a real 2N×2N tight-binding Hamiltonian. Orange highlights the eigenstate closest to zero. With Δ=0 levels pair at ±E by chiral symmetry.
Boundary localization
Where is the closest-to-zero eigenvector?
A boundary mode has concentrated probability near the chain's edges; an ordinary bulk state extends through many sites.
A/B sites are ordered from left to right. Total probability is one. A near-zero energy without boundary localization would not, by itself, demonstrate an edge state.
Finite-size physics
How does the middle spectral gap depend on chain length?
Edge-mode overlap typically decreases exponentially when opposite ends are separated in a gapped topological regime.
The plotted splitting is E(N)−E(N−1), the gap between the two middle energy-ordered levels of a 2N-site open chain, evaluated by exact diagonalization. Its decay depends on termination, hopping ratio and onsite term.
Mathematical reconstruction
Termination and symmetry are part of bulk–boundary correspondence
With sublattice orbitals Aⱼ and Bⱼ, the original real-space SSH Hamiltonian has alternating hoppings t₁A†ⱼBⱼ and t₂A†ⱼ₊₁Bⱼ, plus Hermitian conjugates. An open chain containing complete cells A₁,B₁,…,Aₙ,Bₙ ends with intracell t₁ bonds. The alternative cut shifts the 2N-site chain by one lattice site, so its end bonds are t₂ while its interior alternation is reversed.
For the weak-end termination and t₂>t₁, the semi-infinite zero-energy recurrence on the A sublattice gives t₁Aⱼ+t₂Aⱼ₊₁=0. Hence successive edge amplitudes are multiplied by −t₁/t₂ and the localization length (in cells) is ξ=1/ln(t₂/t₁). The edge state is normalizable if |t₁/t₂|<1.
On a finite chain two boundaries support overlapping exponentially localized solutions. Exact eigenenergies generally split into ±ε rather than remaining exactly zero. Switching the termination changes which weak and strong links are cut, while the original bulk Bloch convention stays fixed. Adding ±Δ onsite energies on A and B breaks the chiral anticommutation: its edge-related energies can move away from zero without forcing a bulk-band transition.
Exactly checkable limits
For t₁=0 and t₂>0, the intercell-cut chain has two exactly disconnected end sites and two exact zero modes; the shifted termination forms N disconnected t₂ dimers and has no zero mode. The UI uses t₁≥0.1, but a numerical limit approaches this behavior. At t₁=t₂, the bulk gap closes and a winding number ceases to be defined.
Assumptions and error sources
Noninteracting single-particle SSH tight-binding chain with two alternating positive real hoppings, two open-boundary terminations and possible sublattice-staggered onsite energy. The term 'topological' references the fixed original unit cell and Δ=0; the shifted cut and symmetry breaking modify which boundary statements apply.
Investigate independently
Follow three scientific questions
- Hold the bulk, change the cut. Use t₁=.5,t₂=1.5 and N=10. Compare the exact spectrum and closest-to-zero eigenvector for two endings. Explain which termination supports weakly paired boundary orbitals.
- Explore finite-size splitting. Choose the intercell cut, Δ=0 and t₂>t₁. Change N and inspect the middle levels. Relate the shrinking overlap to powers of |t₁/t₂|.
- Remove symmetry protection. Add a staggered Δ. Check how the midgap states respond and why the two-band gap need not close when exact zero-energy pinning is lost.
Self-check: For t₂>t₁>0 and Δ=0, which termination has exponentially localized SSH end modes?
Extension: compare eigenstate localization as t₁/t₂ approaches unity, and distinguish edge-state hybridization from its disappearance at the bulk gap closure.
Interpretation and further reading
What the result teaches
The physically relevant boundary must match the bulk Bloch and unit-cell convention. Topological winding predicts robust features under its stated symmetry and termination assumptions; finite-size hybridization and staggered onsite terms demonstrate what the integer alone cannot determine.
Mathematical source in the lecture notes: SSH edge-state construction · Bulk winding conventions.