# États de bord topologiques SSH

Modèle Su–Schrieffer–Heeger interactif : réglez la dimérisation t₁/t₂, observez la phase de Zak basculer et voyez apparaître des états de bord d'énergie nulle — la correspondance volume-bord en direct dans votre navigateur.

> Page canonique: https://elysiatools.com/fr/visualizations/ssh-topological-edge-states

- **Catégorie:** Physics

## Présentation

Interactive topological band theory — distinct from the existing bloch-sphere-gate-visualizer (single-qubit gate dynamics on the Bloch sphere), topological-quantum-walks (a quantum-walk algorithm), and skyrmion-racetrack-memory (magnetic spin textures); this is the only case treating electronic band topology and the bulk-boundary correspondence, the minimal textbook model of a topological insulator. The model is the full Su-Schrieffer-Heeger (1979) dimerized chain: N unit cells of (A,B) sites with intracell hop t1 and intercell hop t2, Bloch Hamiltonian H(k) = [[0, q(k)],[q*(k), 0]] with q(k) = t1 + t2·e^(ika), bands E±(k) = ±√(t1²+t2²+2t1t2·cos ka) and gap 2|t1−t2| closing only at the critical point t1 = t2 (k = π/a). Bulk topology is computed live: the winding number ν = (1/2π)∮d(arg q) of the q(k) circle around the origin, and the lower-band Zak phase γ via discretized parallel transport (γ = πν mod 2π, quantized by inversion + chiral symmetry). Open chains are diagonalized in the browser (Sturm-sequence bisection for eigenvalues, cyclic Jacobi rotations for orthonormal eigenvectors — verified against the exact uniform-chain cosine spectrum, the t1 = 0 dimer limit, Hψ = Eψ residuals < 1e-14, and chiral ±E spectral pairing to 1e-15). In the topological phase (t2 > t1) the gap hosts two edge states with the exact chiral profile ψ(Aₙ) ∝ (−t1/t2)ⁿ on sublattice A (left) and B (right), localization length ξ = 1/ln(t2/t1), single-mode IPR (1−r²)/(1+r²), and a finite-size splitting |E| ~ t1(t1/t2)^(N−1) verified exponentially in N. Disorder comes in two symmetry classes: bond disorder preserves chiral symmetry and the edge doublet stays pinned at zero (to exponential accuracy), while on-site disorder breaks it and the edge energy drifts linearly with W — the meaning of symmetry-protected topology, class BDI. Five visualization panels: (1) Bulk bands E±(k) with the gap shaded, dashed gap edges, the edge-doublet energies overlaid, and a hover crosshair reading out k, E±(k). (2) Open-chain eigenvalue spectrum as clickable ticks (cyan E<0, gold E>0, pulsing pink doublet nearest E = 0) with gap shading. (3) Edge-state wavefunction |ψ|² as per-site bars color-coded by sublattice A (cyan) / B (gold), with the dashed analytic zero-mode envelope when applicable, edge-region tints, site hover readout, and an animated Re(ψ·e^(−iEt)) overlay that visibly freezes at E = 0. (4) Complex q(k) winding plane: the traced circle with direction arrows, the t1 center marker, an origin dot that glows pink when enclosed (ν = 1) vs dim gold when avoided (ν = 0), and a Zak-phase gauge dial. (5) Phase diagram: |E| of the doublet vs δ = t1/t2 swept over 41 points live (topological side flat at ~0, trivial side rising into the band, clean-gap boundary and critical line overlaid) with the current operating point marked. Adjustable parameters: hopping ratio δ = t1/t2 (0–2, crossing the transition continuously), chain length N (10–50 cells), disorder strength W (0–1 × t2), disorder type (none / bonds / on-site), disorder seed re-roll, and an auto-sweep animation that ping-pongs δ through the transition. Five presets: trivial insulator, critical point, topological, topological + bond disorder (protection intact), topological + on-site disorder (protection broken). Real-time diagnostics: phase badge (topological/trivial/gap-closed), Zak phase γ, winding ν, bulk gap, doublet |E| (scientific notation for the exponential splitting), localization length ξ, selected-state IPR, and edge weight. Educational content covers the SSH Hamiltonian and its polyacetylene origin (2000 Chemistry Nobel), the q(k) winding and Zak/Berry phase quantization, the bulk-boundary correspondence and exact edge-state counting, and chiral symmetry protection vs its breaking (bond vs on-site disorder, Anderson localization outlook, BDI class). Multi-language support (zh, en, es, fr, de, ru, pt).

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