Frenkel-Kontorova Model

Frenkel-Kontorova Model: N atoms connected by springs (stiffness 1) sit in a sinusoidal substrate potential of strength K. The natural atom spacing is δ × a, where a=2π is the substrate period.

Aubry Transition: Below a critical coupling K_c (which depends on δ), the ground state is incommensurate — atoms slide freely (sliding phase, hull function is smooth). Above K_c, the system is commensurate — atoms are locked to substrate wells and kinks (solitons) are pinned with a finite Peierls-Nabarro barrier.

Kinks: In the incommensurate phase, topological solitons (kinks) carry excess density and move freely. Watch them propagate along the chain.