Frenkel-Kontorova: Commensurate–Incommensurate Transition

Frenkel-Kontorova Model (1938)

The Hamiltonian is H = Σ [½(x_n+1−x_n−a₀)² + K(1−cos(2πx_n))/2], competing chain spacing a₀(1+δ) against substrate period 1.

• Commensurate (C): atoms lock to substrate minima; gap in phonon spectrum
• Incommensurate (IC): atoms cannot all sit in minima; solitons (kinks) form, each carrying one extra atom
• Aubry transition: at critical K*(δ), the ground state becomes analytically non-smooth (Cantorus)

Cyan atoms = near substrate minima (commensurate). Orange atoms = in soliton cores. The lower panel shows local deformation u_n = x_n − na₀.