Ground States of a Generalized Frenkel-Kontorova Model

We studied the modulated structures and commensurate-incommensurate transitions in a generalized Ftenkel-Kontorova (FK) model where the chain is composed of three kinds of atoms. For the classical ground states, the phase diagram and the phonon spectrum show the multi-atomic effects. For the quantum ground states, the multi-atomic effects are reflected by the distribution of atoms in external potential, the phase diagram, the correspondences between the ground states and the orbits of the area-preserving maps, as well as the phonon spectrum and coherence length. For the most incommensurate structure, the critical point K_c at which the transition by breaking of analyticity occurs does not vary monotonically with A. For the case of μ_A= μ_B = 1 and μ_c = 2, with the increasing of A, the critical point K_c first decreases, then increases. The minimum value of K_c occurs at nearly h = 0.035. When K < K_c, the phonon gap Ω_G = 0 and the Lyapunov exponent γ = 0, the system is in the sliding state. When K > K_c, the phonon gap and the Lyapunov exponent show great difference from those of the standard FK model.