# Topological Quantum Walks — Chiral Edge States

A QWZ (Qi-Wu-Zhang) Chern insulator on a 2D lattice. Sweep the mass parameter M and the Chern number C jumps between 0, −1 and +1. Whenever C ≠ 0, the sample's edge carries one-way, defect-immune states — the bulk-boundary correspondence behind topological insulators and the quantum Hall effect.

> Canonical page: https://elysiatools.com/en/visualizations/topological-quantum-walks

- **Category:** Physics

## Overview

Interactive QWZ (Qi-Wu-Zhang 2006) Chern-insulator visualization demonstrating the bulk-boundary correspondence — the deep theorem that whenever a 2D lattice's bulk carries a nonzero topological invariant (the Chern number C), its boundary must host chiral, defect-immune one-way edge states. Distinct from existing single-body quantum cases (quantum-tunneling, quantum-harmonic-oscillator, hydrogen-wave-function) and quantum-wave-collapse: this is the only case showing topologically protected transport. The model realizes the two-band Bloch Hamiltonian H(k) = sin kₓ σₓ + sin k_y σ_y + (M + cos kₓ + cos k_y) σ_z on an L×L lattice with open boundaries (each site a 2-component spinor, dimension 2L²). The Bloch vector d̂(k) maps the Brillouin zone onto S² and the Chern number is computed exactly as its winding C = (1/4π)∫ d²k d̂·(∂k_y d̂ × ∂k_x d̂), quantized to an integer. Sweeping the mass M = 2.6·cos φ closes the bulk gap at M = −2, 0, +2 and the Chern number jumps between three phases: C=0 (trivial, |M|>2), C=−1 (−2<M<0, counter-clockwise edge), C=+1 (0<M<2, clockwise edge). Real-space evolution uses a Crank-Nicolson unitary propagator ψ(t+dt) = (1−iHdt/2)/(1+iHdt/2)ψ solved by complex Gaussian elimination, exactly norm-preserving. Three visualization panels: (1) Main lattice heatmap showing |ψ|² probability density evolving under the Hamiltonian — an edge-started wave packet hugs the boundary and runs one way around the perimeter in a topological phase (C≠0), but diffracts into the bulk in the trivial phase (C=0); center-of-mass marker, chirality-colored border ring, and an optional red defect block (strong on-site potential) that a topological packet skirts around without reflection. (2) Band Structure E±(k)=±|d(k)| along ky=0 showing the bulk gap (2·min|d|) pinch shut at the transitions. (3) M–C Phase Diagram showing the Chern-number step function vs M with phase-region shading and a live marker dot at the current M. Adjustable parameters: phase φ (0–2π, driving M=2.6cosφ), mass nudge (−2 to +4), lattice size L (6–24), and an insert-defect toggle. Six phase presets: Trivial (C=0), Chiral C=−1, Chiral C=+1, Deep C=−1, C=−1+Defect (demonstrating immunity), Mid-gap C=+1. Real-time statistics: mass M, Chern number C, bulk gap, phase classification, edge-fraction (probability in the boundary ring — high in topological phases, low in trivial), and chirality direction. Educational content covers the bulk-boundary correspondence, the Chern number as the winding of the Bloch vector / lattice analogue of Hall conductivity σxy = C·e²/h, and applications (1980 quantum Hall effect measured to 1 part in 10⁹, Haldane's 1988 Chern insulator with no net field, Kane-Mele and Bernevig-Hughes-Zhang 2005 Z₂ topological insulators in HgTe wells, chip-scale topological photonic waveguides, Majorana zero modes for topological quantum computing). Multi-language support (zh, en, es, fr, de, ru, pt).

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