Devil's staircase and order without periodicity in classical condensed matter

Abstract

The existence of incommensurate structures proves that a crystalline ordering is not always the most stable one for non-quantum matter. Some properties of structures which are obtained by minimizing a free energy are investigated in the Frenkel-Kontorova and related models. It is shown that an incommensurate structure can be either quasi-sinusoidal with a phason mode or built out of a sequence of equidistant defects (discommensurations) which are locked to the lattice by the Peierls force. In that situation the variation of the commensurability ratio with physical parameters forms a « complete devil's staircase » with interesting physical consequences. Some general results for all structures which minimize a free energy are given. In addition to the known crystal and incommensurate structures, the existence of a new class of structures which have local order at all scale is predicted. Properties of the new class are described in physical terms and possible applications to certain amorphous or non-stoichiometric compounds are discussed