Graphene Moiré Magic-Angle Phase Map (2D)

Twisted Bilayer

Twist angle θ 1.10°

Interlayer coupling w 110 meV

band renormalization: —

Live Readouts

Moiré period L— nm

Coupling ratio α = w/(ħv_Fk_θ)—

Renorm. velocity v*/v_F—

How it works

This is the 2D companion to the 3D moiré-landscape sim, computed independently rather than viewed from a different camera angle. Instead of rendering a single real-space density landscape, the left panel plots the entire parameter space: every pixel is one (twist angle θ, interlayer coupling w) pair, colored by the Bistritzer–MacDonald renormalized Dirac velocity

L = a / (2 sin(θ/2)),  kθ = 2K sin(θ/2)
α = w / (ħ v_F kθ)
v* / v_F = (1 − 3α²) / (1 + 6α²)

The bright gold contour traces the true magic-angle locus — every (θ,w) pair solving α(θ,w) = 1/√3, i.e. v*/v_F = 0 exactly — computed analytically, not interpolated from the color grid. The crosshair marks your current slider position live, so you can see directly how far you sit from flat-band condition and how raising the coupling shifts the whole magic-angle curve to larger twist angles.

The right panel is a from-scratch 2D real-space density grid ρ(r) = Σᵢ cos(Gᵢ·r) over the three moiré reciprocal vectors at your current (θ,w), raised to the same sharpening power the 3D sim uses — recomputed pixel by pixel on a flat grid, with no 3D geometry or camera behind it, so the AA-stacking puddles you see concentrating are a direct 2D field calculation.

Twist angle / coupling sliders — move the crosshair through parameter space and redraw the real-space grid at that point.
Snap to magic angle — jumps θ onto the current contour (solves α(θ) = 1/√3 for the current w).
Magic-angle contour toggle — hides/shows the analytic locus overlay so you can read the raw color field underneath.

Flat band / magic angle (v*≈0) Dispersive (v*/v_F large) Band-inverted (v*/v_F < 0)