Laboratory 19 · One-dimensional band geometry · Self-directed exploration

Can a Berry phase diagnose the SSH chain's topology?

Compute the Zak phase by multiplying Bloch-eigenstate overlaps, test gauge invariance, and discover why the phase becomes undefined at a bulk-gap closing.

Complete independent study unitWilson-loop overlap productChiral winding and Zak phaseReproducible physics checks

Physical phenomenon

How does a complex hopping amplitude produce a quantized band phase?

In the SSH chain, a two-component Bloch state depends continuously on quasimomentum k. The Berry connection of the lower band integrates around the Brillouin zone into the Zak phase. Its phase angle is only defined modulo 2π and depends on conventions for the orbital unit cell. In the specified chiral-symmetric convention the result is restricted to 0 or π, and a jump between them requires a bulk gap closing. A discrete Wilson-loop product avoids arbitrary eigenvector phase choices.

Make a prediction. With positive t₁

Prerequisites: Bloch Hamiltonian, Berry connection, normalized eigenvectors, gauge changes and closed-loop phases.

Physics readout

Numerical and exact predictions

Bulk phase boundary

Energy gap and Zak phase under hopping changes

Sweep t₁ relative to fixed t₂. A smooth change in hopping can switch the topological region only through a closed bulk band gap.

The direct gap is 2|t₁−t₂|. The Zak phase is 0 when t₁>t₂ and π when t₂>t₁ for the specified cell; it is undefined at equality.

Gauge-invariant discrete holonomy

Wilson-loop discretization error

Refine the Brillouin-zone mesh and compare the computed Wilson phase against the exact lower-band Zak phase.

The product ∏〈u₋(kⱼ)|u₋(kⱼ₊₁)〉 is gauge invariant under arbitrary local eigenstate rephasings, with periodic closure. For this perfectly chiral two-band model the phase is exactly quantized away from gap closures for sufficiently sampled meshes.

Reconstruct the result

From complex hopping to a discrete Berry phase

Choose the textbook's Fourier convention q(k)=t₁+t₂e−ik and Bloch Hamiltonian h(k)=[[0,q(k)],[q*(k),0]]. The vector d=(t₁+t₂cosk,t₂sink) lies in a plane, and its oriented angle φ(k)=arg[q*(k)] winds once if t₂>t₁.

u₋(k)=2−1/2(−e−iφ(k),1)ᵀ,   A(k)=i〈u₋|∂k u₋〉=½∂kφ.

The lower-band Zak phase is γ=∫BZ A(k)dk=πν modulo 2π. On a finite mesh of N quasimomenta, use cyclic overlaps Z=∏ⱼ〈uⱼ|uⱼ₊₁〉 (including the final link back to u₀) and γN=−Arg Z. A gauge redefinition uⱼ→eiχⱼuⱼ cancels between incoming and outgoing overlaps, so a valid Wilson-loop phase is independent of phase choices at each k point.

At t₁=t₂ the loop touches zero at k=π and the bulk band gap collapses. An isolated lower band ceases to exist at this momentum, making the Zak phase undefined. For t₁ and t₂ of opposite sign or another unit-cell convention, phase orientations and edge interpretations require adjusting the definition.

An exactly soluble reference limit

When t₁≫t₂, d never winds around the origin, so γ=0. When t₂≫t₁, the unit circle encloses the origin once, giving γ=π. At t₁=t₂ and k=π the bands touch, so assigning a finite number to the Berry phase has no physical meaning. Integer gauge winding does not change e^{iγ}.

Assumptions and numerical reliability

Noninteracting, gapped, real positive SSH hoppings, exact sublattice/chiral symmetry Δ=0 and the unit-cell/Fourier convention stated above. The finite mesh is only a numerical representation of the Bloch eigenvectors; physical topology is defined in the continuous isolated-band limit.

Independent investigations

Three experiments to reason through

  1. Test the two winding regimes. Switch t₁ and t₂. Calculate the expected ν and the corresponding Zak phase, then compare the overlap phase and spectrum.
  2. Cross the critical point. Increase t₁ through t₂. Watch the gap vanish and explain the undefined invariant at the transition.
  3. Challenge the eigenvector gauge. Change m from −3 to +3 and the number of k points. Track whether the computed phase depends on the chosen eigenstate gauge while the spectral gap remains fixed.

Self-check: For real positive SSH hoppings t₁=0.6, t₂=1.4 and an isolated lower band, what is γZak mod 2π?

Extension: explicitly compute the finite-chain edge-state localization length ξ⁻¹=ln|t₂/t₁| and compare with the SSH boundary experiment.

Physical scope

What the experiment means

The Zak phase joins Bloch-band geometry to a topological invariant in a specific symmetry class and basis convention. Cyclic overlap products compute it without fixing a globally smooth eigenvector gauge. Its quantitative relation to boundaries requires consistent unit-cell termination and symmetry assumptions.

Compare with the manuscript: SSH winding and Zak phase · Boundary states.