Phase diagram
C → K → F → U
C — Compression
Electron density n falling. Phase space of trajectories compresses. Positional correlations grow.
Drive parameter r_s
rs = potential/kinetic 12.0
Temperature T (normalised) 1.00
B field (kinetic-energy suppression → favours crystal) 0.0
Contact normal form · Chapter W
| Parameter | Symbol | Value |
|---|---|---|
| Transverse Lyapunov | μmax | −0.31 |
| Limit cycle freq. | ω | 0.19 |
| Decay constant | β | 1.7 |
| Critical threshold | rs* | ≈ 30–40 |
| Lattice geometry | — | Triangular / moiré |
Theorem W.1
The Wigner crystallization is G = U∘F∘K∘C acting on the 2D electron gas manifold X. At rs = rs* ≈ 30–40, the Jacobian loses rank by 1 (translational mode softens). The triangular lattice is the stable fixed point of the dm³ flow.
Sources
rs* ≈ 30–40 brackets both standing QMC estimates: Tanatar &
Ceperley (PRB 39:5005, 1989), rs ≈ 37 ± 5; Drummond & Needs
(PRL 102:126402, 2009), rs = 31(1). Triangular ground state: Bonsall &
Maradudin (PRB 15:1959, 1977). The B-field control raises effective rs —
Landau quantisation suppresses kinetic energy and so favours the crystal.
Two modelling choices, not derived facts: the order of the 2D fluid-to-crystal transition is open in the literature, so the fold picture picks a side; and breaking continuous translation in 2D gives two acoustic phonon branches, so “loses rank by 1” is the house corank-1 convention. μmax, ω and β above are asserted, with no derivation here or in Lean.
Checks:
Two modelling choices, not derived facts: the order of the 2D fluid-to-crystal transition is open in the literature, so the fold picture picks a side; and breaking continuous translation in 2D gives two acoustic phonon branches, so “loses rank by 1” is the house corank-1 convention. μmax, ω and β above are asserted, with no derivation here or in Lean.
Checks:
Orthogenesis/Architecture/WignerChecks.lean §5.