A Frenkel-Kontorova Model with Two Spring Constants: New Phases and   Phase Transitions
  A discrete Frenkel-Kontorova model with two alternate spring constants $k_{1}$ and $k_{2}$ is studied. It is found that this model has many surprising behaviours. The continuum version of this model is different from the sine-Gordon equation, which is the continuum version of the standard FK model. More interestingly, it has an unpinned commensurate phase which is translationally invariant. A phase transition takes place on a lattice point in the parameter space. We have also predicted another interesting phase transition in which an unlocked commensurate phase breaking translational invariance will take place as the strength of the substrate potential is increased. 
Bambi Hu and Jian Zhou