The Frenkel-Kontorova Model with Nonconvex Interparticle Interactions

Abstract

We present an analytical and numerical study of a chain of atoms moving in a periodic potential with nonlinear, nonconvex interparticle interactions, described by the Hamiltonian

$$ H = \sum\limits_n {\frac{1}{2}u_n^2 + A(u_{n + 1} - u_n )^4 - B(u_{n + 1} - u_n )^2 - cos(u_n ).} $$

The ground state is shown to be homogeneous for B < 1/8 and dimerized for B > 1/8. The nonconvexity is shown to also play an important role when excitations are considered. In the dimerized phase, we define the staggered order parameter v_n = (−l)ⁿ u_n and map the model to the on-site Φ⁴ problem. In particular we find a localized kink solution, v = tgh(x/d), with a width d varying from infinity at B = 1/8 to zero at B = 3/16, where the interparticle interactions in the ground state cross over from the nonconvex region to the convex one. We also show that at this point the kinks are pinned to the lattice. These results are verified by a direct numerical simulation of the discrete model. Finite temperature effects are discussed in terms of a displacive phase transition at B = 1/8 becoming order disorder transition at B = 3/16. When coupling between the springs is introduced by adding a strain gradient term G(u_n−1 – 2u_n + u_n+l)² to the Hamiltonian, we observe a crossover from an infinite kink width at B = 1/8 to a finite width for B > 3/16, determined by the competition between the effective double well and the interspring coupling strengths.