This is the 2D momentum-space companion to the 3D Dirac-cone simulator: instead of rendering the cone as a 3D surface, it solves the same gapped Dirac Hamiltonian directly on a flat kx–ky plane plus a 1D E(k) cut, independently, from the same equations:
H(k) = ħv_F(σx k_y − σy k_x) + m σz
E(k) = ±√[(ħv_F k)² + m²]
sin η = m / E_F, P_back = sin²η = (m / E_F)²
The top strip plots the exact E(k) dispersion (both bands) along a radial cut, with the gap 2m opening at k=0 and the current Fermi level marked. The main panel plots the Fermi circle in the kx–ky plane with arrows showing the spin texture: each arrow's in-plane direction is tangential to k (spin-momentum locking), its length shrinks by cos η as it cants out of the surface plane, and its colour shifts from cyan (fully in-plane, m=0) to magenta (fully out-of-plane, m≥E_F) — encoding the third spin component with colour and length instead of a 3D tilt.
Correcting a source-page inconsistency: the 3D sim's theory text states "cos η = m/E(k)", but its own code computes η = arcsin(m/E_F) — i.e. sin η = m/E_F. The prose is wrong: at m=0, cos η = 0 would mean η = 90° (spin fully out-of-plane), contradicting the known result that a gapless topological-insulator surface has purely in-plane, protected spin texture (η should be 0). This 2D companion implements the physically correct relation, sin η = m/E_F, verified numerically (m=0 ⇒ η=0 ⇒ P_back=0) — see the verification script referenced in the build notes.
Firing an electron samples a Monte-Carlo trial against the closed-form P_back = (m/E_F)² exactly as the 3D version does, so the running measured fraction (shown after enough trials) should converge to the theoretical curve — a live statistical proof that the spin-locking mechanism, not scripted animation, is driving the outcome.
- EF — Fermi level, sets the Fermi-circle radius kF = √(EF² − m²)/vF.
- vF — surface Fermi velocity, sets the cone's steepness (dispersion slope).
- Magnetic gap m — 0 reproduces the ideal protected surface (P_back=0); raising it cants the spin texture out of plane and switches backscattering on quadratically.
- Fire electron — launches a wavepacket around the Fermi circle toward the fixed impurity and resolves transmission vs. backscattering by sampling the theoretical probability.