Failure of the linearized phonon approximation for a one-dimensional crystal

A partly soluble model is presented. The model describes a one-dimensional system of mass points interacting via two-body nearest-neighbor and three-body next-nearest-neighbor potentials. The interactions are chosen to obtain a simple form for the ground state. Certain long-wavelength properties of the system are compared with the same properties in the harmonic approximation for the same Hamiltonian. It is shown that certain correlations are of a much longer range in the harmonic approximation and that the spectrum of the true Hamiltonian is much softer than that of its harmonic approximant. If the coordinates in the Hamiltonian are cyclic the spectrum seems to have a gap while the correlations remain very short ranged.