EFFECTIVE DISCOMMENSURATIONS IN THE INCOMMENSURATE GROUND-STATES OF THE EXTENDED FRENKEL-KONTOROWA MODELS

For the Frenkel-Kontorowa model and its extensions in several dimensions, with several neighbor interactions, etc., it is proven that the hull function of an incommensurate ground state is purely discrete when the phonon spectrum exhibits a non-zero gap. The same result also holds when the Lyapunov coefficient of the corresponding set of trajectories in the associated twist map (when it is definable) is strictly positive. When this theorem applies, the Fourier coefficients of the incommensurate modulation can be described by an analytic hull function. This is in some sense a dual result to the Kolmogorov-Arnol'd-Moser theorem, which proves under different hypotheses that incommensurate modulation in real space (instead of reciprocal space) can be described by an analytic hull function. The physical implication of this theorem is that the incommensurate ground states can be decomposed into a linear superposition of localized effective discommensurations. The shape of these discommensurations depends on the model parameters and on their density. Approaching the transition by breaking of analyticity (TBA) from above, the size of these discommensurations diverges. Below the TBA, these discommensurations cannot be described anymore as localized objects in the analytic phase. These results confirm that TBA in non-linear models extends the concept of localization of eigenstates, which up to now was only meaningful for linear operators.