Polarisation and transition by breaking of analyticity in a one-dimensional model for incommensurate structures in an electric field

The authors study the transition by breaking of analyticity on a variation of the Frenkel-Kontorova (FK) model with two sublattices, where a parameter E simulates the role of the electric field in certain incommensurate structures. For the incommensurate ground states, the polarisation response delta (E) at fixed incommensurability ratio zeta and as a function of the electric field E is found to be a smooth analytic curve when the parameter K which characterises the coupling to the lattice is small enough. It also becomes the sum of a smooth analytic curve and a devil's staircase for larger values of K. The change between these two regimes corresponds to the crossing of the critical line Kc(E, zeta ) of the transition by breaking of analyticity. It is shown that at fixed zeta irrational, this curve is not differentiable at an infinite set of cusps which form a Cantor set. These cusps are shown to be the terminating points of the lines corresponding to the edges of the plateaux of the devil's staircase delta (E) at fixed zeta .