Frenkel-Kontorova Model

Atoms in a periodic potential — commensurate/incommensurate phases and Aubry transition




COMMENSURATE

Atom Positions on Substrate Potential

Displacement Field u(n)

Hull Function (Aubry)

Frenkel-Kontorova (1938): H = Σ [½(xₙ₊₁-xₙ-a)² + K/4π² · (1-cos(2πxₙ))] where a is the natural atomic spacing and the substrate has period 1. When a is a rational p/q (commensurate), atoms lock into a periodic structure — the Peierls-Nabarro barrier is nonzero. For irrational a (incommensurate), the Aubry transition occurs at a critical K*: below K*, atoms slide freely (KAM tori survive, hull function continuous); above K*, the hull function becomes discontinuous (cantorus) and the chain is pinned. Kinks/solitons (discommensurations) carry topological charge and mediate transport.