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Sandbox Physics

Topological Matter 01 · one-dimensional bulk–edge correspondence

SSH Topological Chain · Edge State & Winding

Evolve a two-sublattice tight-binding chain in real time. Exchange intracell and intercell hopping, watch the gap close and the winding number flip, then add bond disorder or a domain wall to test whether localization follows geometry or bulk topology.

Interactive modelSSH Topological Chain · Edge State & Winding
Winding number ν\nu11
Bulk gap Δ\Delta1.20 t1.20\,t
Edge weight PedgeP_{\mathrm{edge}}98%98\%
Localization length ξ\xi1.44 a1.44\,a

Physics tutorial

The SSH chain: how bulk winding determines a boundary

BackgroundThe SSH model divides a one-dimensional lattice into A and B sublattices with alternating intracell and intercell hopping.

Why it mattersIt is the minimal tight-binding model for learning topological invariants, gap closings, and bulk–edge correspondence.

Start with the essentials

Focus question
Why can exchanging the strengths of two bonds create a zero-energy state at the end of a chain?
One-sentence intuition
When the Bloch vector winds around the origin, the bulk winding is one and an open boundary must host an E=0E=0 mode that cannot be removed continuously.

Core mathematical model

Bloch Hamiltonian

H(k)=(t1+t2cos⁡k)σx+t2sin⁡k σyH(k)=\bigl(t_1+t_2\cos k\bigr)\sigma_x+t_2\sin k\,\sigma_y

(dx,dy)(d_x,d_y) traces a circle with momentum; whether it encloses the origin is the geometric winding test.

Bands and gap

E±(k)=±t12+t22+2t1t2cos⁡k,Δ=2∣t1−t2∣E_\pm(k)=\pm\sqrt{t_1^2+t_2^2+2t_1t_2\cos k},\qquad \Delta=2|t_1-t_2|

Topology can change only through t1=t2t_1=t_2, where the bulk gap closes.

Edge localization length

ξ−1=∣ln⁡∣t1/t2∣∣\xi^{-1}=\left|\ln\left|t_1/t_2\right|\right|

Near the transition the two hoppings approach one another, so the edge state spreads into the bulk.

Common difficulties

A weak bond is not automatically topological

Typical misconceptionAny chain containing weak bonds must have an edge state.

Better mental modelThe unit-cell convention, termination, and winding of the complete Bloch vector matter together.

Robust does not mean immune to every perturbation

Typical misconceptionA topological edge state stays exactly at zero energy under arbitrary disorder.

Better mental modelBond disorder that preserves chiral symmetry cannot easily remove it; symmetry-breaking onsite potentials can shift it away from zero.

Run the experiment

  1. 01

    Cross the gap closing

    Sweep δ\delta slowly from negative to positive.

    What to observe: Δ\Delta closes and reopens while winding and edge weight change.
  2. 02

    Compare edge and bulk packets

    Switch between the topological-edge and trivial-bulk presets.

    What to observe: Edge probability stays localized while the bulk packet disperses across the chain.
  3. 03

    Build a topological domain wall

    Choose the domain-wall preset and move the wall cell.

    What to observe: The localized state follows the interface between different windings rather than a geometric outer end.