Global universality in the Frenkel-Kontorova model

The Frenkel-Kontorova model of a chain of atoms in an external periodic potential is studied. The phase diagram of this model contains infinitely many tongues of commensurate phases separated by gaps of incommensurate structures. The period of these structures is described by a devil’s staircase (DS) function when the parameters which define the model are varied. The model exhibits a critical line above which the phason mode of the incommensurate phases is pinned and the DS is complete, while below the line the phason mode is unpinned and the DS is incomplete. We evaluate the critical line numerically and show that it has a fractal nature. The Hausdorff dimension $D_0$ and the spectrum of singularities f(α) of the gaps along the critical line are calculated. The analysis is performed for several forms of the periodic potential. The resulting $D_0$ and f(α) seem to be independent of the details of the potential with $D_0$=0.87±0.02. It is interesting to note that $D_0$ is equal, within the numerical uncertainty, to the Hausdorff dimension corresponding to the critical line of dissipative systems, although the f(α) of the two cases are found to be different.