Documentation
Model overview, valid parameter ranges, and topological protection concepts.
TopoChain Designer operates on a 1D synthetic topological hopping chain — a finite tight-binding Hamiltonian with alternating bond strengths arranged in a periodic pattern.
Each site is connected to its nearest neighbors by hopping amplitudes. Every third bond carries a tunable amplitude Δ (or derived tA values for edge and interface control), while the remaining bonds are normalized to unity. This produces a gapped spectrum whose width W and in-gap edge/interface states can be targeted through inverse design.
Given target quantities — band gap width W_target, optional left-edge energy E_left_target, and optional interface energy E_iface_target — the private solver computes the hopping parameters (Δ, tA_left, tA_iface) that realize those targets, then optionally verifies them by diagonalizing finite N-site Hamiltonians.
| Parameter | Range | Notes |
|---|---|---|
| W_target | 0.1 – 2.5 | Target band gap width |
| E_left_target | Within band gap | Must lie between z₋ and Δ for the designed chain |
| E_iface_target | 0.74 – 1.79 | Interface-state control range |
| N (chain size) | 150, 300, or 600 | Finite chain length for verification (default 300) |
| Δ (Delta) | Derived | Primary hopping amplitude, determined by W_target |
The Zak phase is a topological invariant characterizing the bulk band structure of the 1D chain. When Δ > 1, the system is in a topological phase with Zak phase π, supporting protected edge states within the band gap.
When Δ ≤ 1, the chain is in a trivial phase (Zak phase 0) with no topologically protected edge states. Designs in this regime may still have in-gap states, but they lack topological protection against disorder or perturbations.
The designer displays the Zak phase badge after each run. Warnings are shown when E_iface is outside [0.74, 1.79], when E_left exceeds Δ/2 (near the continuum edge), or when Δ ≤ 1 (trivial phase).
| Action | Credits |
|---|---|
| Design (basic) | 1 |
| Design + verification | 3 |
| HTML report download | 10 |