Ground States of a Generalized Frenkel-Kontorova Model^*

We studied the modulated structures and commensurate-incommensurate transitions in a generalized Frenkel–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 $\hbar$. For the case of $\mu_A=\mu_B=1$ and $\mu_C=2$, with the increasing of $\hbar$, the critical point $K_c$ first decreases, then increases. The minimum value of $K_c$ occurs at nearly $\hbar$ = 0.035. When $K\lt K_c$, the phonon gap $\Omega_G=0$ and the Lyapunov exponent $\gamma=0$, the system is in the sliding state. When $K\gt K_c$, the phonon gap and the Lyapunov exponent show great difference from those of the standard FK model.