Critical exponents for an incommensurate structure with several length scales

Abstract

We study the critical behavior of an extended Frenkel-Kontorova model (discrete static double–sine-Gordon model), representing an incommensurate structure with three length scales. The critical line separating the sliding and pinned phases of the ground state is constructed in (${\lambda }_1$,${\lambda }_2$) space, ${\lambda }_1$ and ${\lambda }_2$ being the amplitudes of the first and second harmonics of the external potential. Using nonlinear-dynamics techniques to study the pinning transition, we numerically determine the critical exponents for the correlation length, the Peierls-Nabarro barrier, and the gap in the excitation spectrum. When ${\lambda }_2$ becomes sufficiently large a scaling anomaly occurs with the critical exponents varying with the external potential. We conjecture that this behavior signals a crossover to a new universality class.