Experiment 03 · Bulk topology and boundaries

Where do the SSH chain's edge states come from?

Change the two hopping amplitudes, observe the Bloch bands, trace the complex winding loop, and diagonalize a finite open chain. Test what happens when the gap closes, the ends connect, or chiral symmetry is broken.

15–35 min core experimentExact finite-chain diagonalizationWinding and bulk gapsBoundary-state localization

Start from a question

Can a property of bulk Bloch states predict boundary behavior?

The Su–Schrieffer–Heeger chain alternates an intracell coupling t₁ and an intercell coupling t₂. It has only two sublattice orbitals, A and B, per unit cell. Nevertheless, its infinite periodic system can carry a nontrivial winding number, and a finite open chain with the corresponding termination develops states concentrated near its ends. Both statements require care about gap, symmetry, unit-cell choice and boundary termination.

Predict first: for positive t₁ and t₂, what happens as t₂ crosses t₁? Watch the bulk gap close and reopen. Does the open finite chain acquire states at precisely zero energy, or can finite-size hybridization split them?

Real-space lattice

Alternating bonds and boundary conditions

Blue A and orange B sublattice sites; bond thickness reflects |t₁| and |t₂|. A periodic boundary adds the last B-to-first A intercell hopping.

Infinite periodic system

Bulk bands E₊(k), E₋(k)

Exact Bloch dispersion including staggered on-site potential. Band gaps belong to the infinite periodic model and remain well defined even when the finite open-chain edge levels split.

Complex plane geometry

The winding loop q*(k) = t₁ + t₂eⁱᵏ

We display q*(k), matching the book's positive winding convention. A complete circuit around the origin gives ν=+1 when t₂>t₁, provided the bulk is gapped and chiral symmetry is present.

The finite system

Exact open- or periodic-chain spectrum

Each dot is one numerically diagonalized energy eigenvalue, sorted from low to high. Near-zero eigenvalues in the open topological chain can hybridize and split at finite N.

Selected eigenvector

Site-resolved probability |ψⱼ|²

Bars list alternating A and B sites from the left to right boundary. A boundary-localized mode has large weight near the ends, whereas bulk eigenstates extend through the chain. Total probability equals one.

From hopping to topology

Deriving the bulk winding

With the note's Fourier convention an=N−1/2∑keiknak, the Bloch Hamiltonian is

h(k) = [[Δ, q(k)], [q*(k), −Δ]],   q(k)=t₁+t₂e−ik.

For Δ=0, Γ=σz anticommutes with h(k), and the Bloch vector lies in a plane. Its loop, seen as q*(k)=t₁+t₂eik, winds around zero whenever the circle radius t₂ exceeds its center displacement t₁.

E±(k)=±√[t₁²+t₂²+2t₁t₂ cos k+Δ²],   gap=2√[(t₁−t₂)²+Δ²].
Why does the winding become undefined at t₁=t₂?

At k=π, q(k)=t₁−t₂. Equality makes the complex loop pass through its reference origin. Its phase is then undefined at that momentum. At Δ=0 the bulk gap closes at the same point. When Δ≠0, the energy gap can remain open, but chiral symmetry and the resulting protected winding classification have already been lost.

Where do the boundary amplitudes come from?

For a semi-infinite chiral chain, the E=0 sublattice equations imply t₁ Aj+t₂ Aj+1=0, hence Aj+1=−(t₁/t₂)Aj. A normalizable left-localized mode requires |t₁/t₂|<1 for this termination. On a finite chain, left and right edge wavefunctions overlap and generally create two eigenenergies ±ε instead of exact zero. Compare the splitting as N increases.

What does the Zak phase measure?

The lower Bloch band's Berry connection is A(k)=i〈u₋|∂ku₋〉. With the specified Bloch basis and unit cell, γZak=πν modulo 2π when the band remains gapped and chiral symmetry protects the quantization. An orbital-position or unit-cell convention change requires tracking its effect on Berry phases; observe the present q*(k) loop only under the convention stated above.

Controlled investigations

Experiments that reveal the assumptions

  1. Topological transition. Set t₂=1, sweep t₁ through 1. Check the gap and whether ν changes. The exact winding value is undefined at the transition.
  2. Boundary versus bulk. In the t₂>t₁ regime, compare N=4 to N=16. Select the state nearest E=0 and observe localization and finite-size energy splitting. Then change to periodic boundaries: the physical ends disappear.
  3. Break chiral symmetry. Turn on Δ. The periodic band gap can stay open while the protected winding label disappears. Compare the finite-chain energies and the q loop, now identified as a reference curve only.

Self-check: for Δ=0, t₁=0.6 and t₂=1.3, what is the bulk winding ν in the chapter's convention?

Further study: distinguish the chiral 1D SSH classification from the 2D Berry-curvature/Chern-number calculation later in the chapter.

Scientific scope and manuscript links

The finite Hamiltonian uses a real symmetric 2N×2N matrix, exactly diagonalized for the displayed parameter choices. The model describes a single-particle tight-binding band; filling and many-body interactions are outside this experiment. The selected termination is explicit. A nonzero staggered Δ is an illustrative symmetry-breaking deformation and the two-band q loop is not a protected topological invariant in that case. At t₁=t₂, Δ=0, the undefined winding is shown explicitly.

Follow the derivations: SSH bulk Hamiltonian and winding and boundary-state recurrence and localization. For adiabatic geometry, see Berry phase.