ARXIV:1901.10485 • BM CONTINUUM • MATTG JJ-LAB

MoiréTwist 3D — Magic-Angle Hierarchy & Superconductivity

Coherent Flat-Band Superconductor (BKT Phase)
STACK / SCALING λk(n) n=3, k=1 • √2 (1.4142)
TWIST θ / MAGIC θ* 1.527° / 1.527°
MOIRÉ LM & ATOMS 9.23 nm • 16,896 C
vF* / vF & WIDTH W (analytic) 0.008 • 2.81 meV
TBKT & STIFFNESS Js 2.64 K • Js 3.68 K
PAULI Bc∥/BP & MAG MIVC 2.70× • M=0.31
STM / Raycast Probe: click or hover 3D superlattice — (x=0.0 nm, y=0.0 nm) • AA weight=1.00 • Local ΔSC=0.88 • Spin MIVC=0.12
Alternating-Twist Stack (±θ) λ₁⁽³⁾ = 2 cos(π/4) = √2
Magic Hierarchy Snap & Sweep
Camera & Overlays Keys 1–6 • Click to probe
Real-Space Moiré Map
Inhomogeneity & Magnetic Field JJ array & IVC order
Load experiment —
Band Filling & Chiral Relaxation Pauli violation & D-field
BKT dome TBKT(θ), stiffness Js & competing order MIVC Quantum geometric weight: 92%
Next Magic Number Formula Explorer
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BM numerical audit: computing…
Analytic vF*(θ)/vF & W(θ) (closed-form proxy) See tab 4 for BM numerics
Hierarchy table (n=2..8) vs. experiment Click a row to load
n, kExact λk(n)m=1 θ*m=2 θ*Exp. θ*ΔDirac
Moiré bands E(k) along Km→Γm→Mm→K'm + DOS exact sector decomposition
BM numerics: computing…
Numerical magic angles: vF*/vF vs αeff = σk wAB/(ħvFkθ) scanning…
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Solver settings —
Sector mode uses the exact Khalaf et al. decoupling (U = 0). Odd n adds an uncoupled Dirac cone (rose).
Rendering & theme —
Moiré clarity visual, not physical
Per-layer visibility & focus alternating ±θ/2
Capture & export zero external dependencies
Ready to export high-resolution 3D render, band data or hierarchy tables.
Scientific grounding verified equations
  • Khalaf, Kruchkov, Tarnopolsky & Vishwanath, PRB 100, 085109 (arXiv:1901.10485): alternating-twist n-layers map exactly onto ⌊n/2⌋ bilayers with λk = 2 cos(πk/(n+1)) (√2, φ, √3, …) plus a Dirac cone for odd n.
  • Tarnopolsky, Kruchkov & Vishwanath, PRL 122, 106405: chiral magic couplings αm = 0.586, 2.221, 3.751, …, Δα → 3/2.
  • Bistritzer & MacDonald, PNAS 108, 12233: continuum model solved here by plane-wave diagonalisation (tab 4), reproducing the exact chiral flat band at α₁.
  • MATTG experiments: Park et al. Nature 590 (2021); Cao et al. Nature 595 (2021) — Pauli-limit violation 2–3×; twist-disordered Josephson networks with competing in-plane magnetic order.
Cloudflare Workers & Pages GitHub auto-deploy ready

Includes wrangler.toml + worker.js (Cloudflare Workers Static Assets) and _headers (Cloudflare Pages), plus bundled vendor/three.module.min.js & Web Worker band-worker.js.

  1. Upload repo to GitHub & connect in Cloudflare Workers & Pages
  2. Workers build command: npx wrangler deploy • Pages output dir: /
  3. 📦 Open Verified macOS Package Downloader (.zip & .tar.gz)