XU AiGu0, WANG GuangRui, CHEN ShiGang, YANG ZhanRu
理论物理通讯. 2000, 34(3): 419-432.
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 Kc at which the transition by breaking of analyticity occurs does not vary monotonically with A. For the case of μA= μB = 1 and μc = 2, with the increasing of A, the critical point Kc first decreases, then increases. The minimum value of Kc occurs at nearly h = 0.035. When K < Kc, the phonon gap ΩG = 0 and the Lyapunov exponent γ = 0, the system is in the sliding state. When K > Kc, the phonon gap and the Lyapunov exponent show great difference from those of the standard FK model.
and 
are introduced via their acting on the q-number states. The even and odd number eigenstates of twophoton operators 
and 
are constructed.
beyond vacuum saturation approximation by including the contribution of r and u mesons. The numerical results show that there is a sizeable correction of the four-quark condensate in comparison with its factorized value using the vacu urn saturation approximation.
