Critical behaviour at the transition by breaking of analyticity in the discrete Frenkel-Kontorova model

The authors study numerically the transition by breaking of analyticity which occurs in the incommensurate ground state of the Frenkel-Kontorova model (1938) when the amplitude lambda of its periodic-potential V(u) is increased beyond a critical value lambda _c. A brief review of the properties of this transition and its connection with the standard map is given. They consider four quantities which are critical when lambda goes to lambda _c from upper values: the gap in the phonon spectrum, the coherence length of the ground state, the Peierls-Nabarro barrier and the depinning force. The numerical method is discussed and the authors show in particular that the mapping method is unpracticable for lambda > lambda _c. They observe the transition by breaking of analyticity and show that in the stochastic region ( lambda > lambda _c) the ground state is never chaotic. Nevertheless in this region metastable chaotic states can be obtained. The numerical calculations, performed with a ratio of the atomic mean distance to the period of the potential V(u), l/2a=(3- square root 5)/2 equivalent to the golden mean, show that critical exponents can be defined for the four critical quantities studied. Their values reveal two scaling laws which are empirically explained.