In this section we examine in detail the simplest form of the FK model. A detailed version of this derivation can be found in the literature. The model describes a one-dimensional chain of atoms with a harmonic nearest neighbor interaction and subject to a sinusoidal potential. Transverse motion of the atoms is ignored, i.e. the atoms can only move along the chain. The Hamiltonian for this situation is given by 1, where we specify the interaction potential to be