# Topological Insulator Edge States

Interactive 2D Chern insulator (Qi–Wu–Zhang model): Berry curvature in momentum space, the quantized Chern number, chiral edge channels from bulk–boundary correspondence, and why they cannot backscatter.

> Canonical page: https://elysiatools.com/en/visualizations/topological-insulator-edge-states

- **Category:** Physics

## Overview

Interactive 2D Chern insulator — the full Qi–Wu–Zhang two-band model H(k) = sin kx σx + sin ky σy + (u + cos kx + cos ky) σz, distinct from the existing ssh-topological-edge-states (the 1D Z2/chiral prototype; this case is the 2D Chern descendant with Berry curvature over a Brillouin zone), bloch-sphere-gate-visualizer (qubit gates), and skyrmion-racetrack-memory (magnetic textures). Bulk physics: bands E± = ±|d| with gap 2·min(|u+2|, |u|, |u−2|) closing at u = −2 (M), 0 (X/Y), +2 (Γ); Berry curvature computed from the exact reduction d·(∂kx d × ∂ky d) = cos kx + cos ky + u·cos kx·cos ky (verified against finite differences of d̂) and rendered as a diverging Brillouin-zone heatmap where the peaks migrate and annihilate at the transitions; the first Chern number computed live with the gauge-exact Fukui–Hatsugai–Suzuki link-variable algorithm (C = (1/2π)Σ Im ln[UxUyUx*Uy*]) and cross-checked against the analytic curvature integral (agreement < 0.02; C = −1 for −2<u<0, +1 for 0<u<2, 0 outside). Ribbon physics (open y, periodic x — real-symmetric in the σ eigenbasis, block-tridiagonal, solved by cyclic Jacobi with residuals < 1e-14 and the kx = 0 spectrum verified particle-hole symmetric): strip spectrum E(kx) where |C| chiral branches per edge cross the bulk gap (bulk–boundary correspondence — the strip is gapless iff C ≠ 0, with the exponentially small e^(−Ny/ξ) avoided crossing between counter-propagating edges made visible), edge-localization classification per state, per-edge channel velocities showing opposite propagation directions that flip with the sign of C, and layer disorder (x-translation-invariant random potentials) through which the chiral channel survives with direction intact. Wave-packet dynamics: a kx-diagonal Gaussian packet built from branch-continuity-tracked edge eigenstates races along the chosen edge with the branch group velocity (top edge one way, bottom the other; trivial phase — no packet, no channel), animated as a strip heatmap; launching on the opposite edge or flipping u reverses the motion. Backscattering test: two projected 1D channels hit a disorder wall — the chiral channel i∂tψ = −iv∂xψ + V(x)ψ passes with exactly zero reflection (measured ~0%) while a trivial ± pair partially bounces (65% at strong disorder), both with unitary norm conservation to 1e-13. Five visualization panels: (1) Berry-curvature heatmap with colorbar; (2) bulk bands along Γ–X–M with gap shading plus the u-axis phase diagram (C = 0/−1/+1/0 regions, critical lines, live marker); (3) strip spectrum with bulk cloud, red top-edge and blue bottom-edge states; (4) animated strip packet heatmap with direction arrow and the C = 0 no-channel notice; (5) the chiral-vs-trivial backscattering playback with wall shading and reflection statistics. Controls: mass u (−4…4, sweeping through all three transitions), strip width Ny (12–24), layer disorder, scattering-wall disorder, packet edge selection + relaunch, disorder seed, and five presets (C = −1, C = +1, critical, trivial, topological + disorder). Real-time diagnostics: phase badge, Chern number, bulk gap, strip gap, both edge-channel velocities, packet velocity, chiral and trivial reflection percentages. Educational content covers the QWZ model and its role of the mass term, Berry curvature and the TKNN/Chern invariant with the FHS algorithm, bulk–boundary correspondence and chiral channels (σxy = Ce²/h), and why chirality forbids backscattering — the heart of quantized Hall transport. Multi-language support (zh, en, es, fr, de, ru, pt).

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