Wigner Crystal Formation

Electrons crystallize at low density — rs > 37 (2D) marks the quantum phase transition
Phase: Fermi Liquid
Structural order ψ₆:
Step: 0
Wigner crystal: Eugene Wigner (1934) predicted that electrons, despite their kinetic energy, will form a crystalline lattice when the Coulomb repulsion dominates. This happens at low density, quantified by rs = a/aB where a is the average inter-electron spacing and aB is the Bohr radius. In 2D, the transition occurs near rs ≈ 37 (quantum Monte Carlo, Tanatar & Ceperley 1989). The equilibrium structure is a triangular (hexagonal) lattice, minimizing Madelung energy. This simulation uses Monte Carlo relaxation with Coulomb energy — watch hexagonal order emerge as rs increases. The order parameter ψ₆ = |⟨e^{6iθ}⟩| measures hexatic/crystalline order (0 = disordered, 1 = perfect crystal).