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Sandbox Physics

Topological Matter 02 · two-dimensional bulk–edge correspondence

Chern Insulator · Chiral Edge Transport

Integrate a two-component wave function on a finite QWZ Chern-insulator lattice. Sweep the mass through two gap closings to move from Chern plus one to zero to minus one, then drag a defect, open an inner boundary, and add disorder while probability current reroutes in real time.

Interactive modelChern Insulator · Chiral Edge Transport
Chern number CCC=+1C=+1
Bulk gap Δbulk\Delta_{\mathrm{bulk}}2.00 t2.00\,t
Boundary probability PedgeP_{\mathrm{edge}}96%96\%
Edge transportLeft-moving

Physics tutorial

Chern insulators: two-dimensional bulk topology becomes a one-way edge

BackgroundThe QWZ model uses a two-band lattice Hamiltonian whose Berry curvature integrates to an integer Chern number.

Why it mattersIt connects abstract bulk topology directly to observable edge propagation, Hall response, and suppressed backscattering.

Start with the essentials

Focus question
A local defect can bend the packet path, so why can it not reflect a one-way edge state back along the same edge?
One-sentence intuition
The Chern number fixes the net number of in-gap edge channels. With no reverse channel available, elastic local scattering can only reroute the packet.

Core mathematical model

QWZ Hamiltonian

H(k)=sin⁡kx σx+sin⁡ky σy+(m+cos⁡kx+cos⁡ky)σzH(\mathbf k)=\sin k_x\,\sigma_x+\sin k_y\,\sigma_y+\bigl(m+\cos k_x+\cos k_y\bigr)\sigma_z

The mass controls whether the Bloch-vector map wraps the sphere.

Chern number

C=12π∫BZΩxy(k) d2kC=\frac{1}{2\pi}\int_{\mathrm{BZ}}\Omega_{xy}(\mathbf k)\,\mathrm d^2k

The integral of Berry curvature over the Brillouin zone changes integer value only when the gap closes.

Continuity equation

∂ρ∂t+∇⋅j=0\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf j=0

The current overlay reveals how probability reroutes along outer and inner boundaries.

Common difficulties

Protection does not mean no scattering

Typical misconceptionA protected edge packet crosses every defect without any deformation.

Better mental modelDefects can delay and reshape it; what is suppressed is elastic backscattering that requires a reverse channel.

The Chern number belongs to a band

Typical misconceptionThe packet drawn on screen itself carries a Chern number.

Better mental modelThe invariant is defined for the complete Bloch band; the packet probes the edge spectrum forced by that bulk phase.

Run the experiment

  1. 01

    Reverse chirality

    Compare the C=+1C=+1 and C=−1C=-1 presets.

    What to observe: Propagation on the same edge reverses and the current arrows reverse with it.
  2. 02

    Drag the defect

    Drag the defect directly across the canvas and enlarge its inner-boundary radius.

    What to observe: The packet follows the new boundary around the hole while boundary probability remains high.
  3. 03

    Perform a topological quench

    Press “Quench across transition” and watch the mass pass continuously through zero.

    What to observe: Edge localization weakens at the bulk gap closing, then the channel rebuilds with opposite chirality.