Peierls instability for the Holstein model with rational density

We consider the static Holstein model, describing a chain of fermions interacting with a classical phonon field, when the interaction is weak and the density is a rational number p = P/Q, with P, Q relative prime integers. We show that the energy of the system, as a function of the phonon field, has one (if Q is even) or two (if Q is odd) stationary points, defined up to a lattice translation, which are local minima in the space of fields periodic with period equal to the inverse of the density.