Farey Tree and the Frenkel-Kontorova Model

Abstract

We solved the Frenkel-Kontorova model with the potential $V(u)=-\frac{1}{2}|\lambda |(u-\mathrm{Int}[u]-\frac{1}{2}{)}^2$ exactly. For given $|\lambda |$, there exists a positive integer $q_c$ such that for almost all values of the tensile force $\sigma$, the winding number $\omega$ of the ground state configuration is a rational number in the $q_c$-th level Farey tree. For fixed $\omega =p/q$, there is a critical ${\lambda }_c$ when a first order phase transition occurs. This phase transition can be understood as the dissociation of a large molecule into two smaller ones in a manner dictated by the Farey tree. A kind of ``commensurate-incommensurate'' transition occurs at critical values of $\sigma$ when two sizes of molecules co-exist. ``Soliton'' in the usual sense does not exist but induces a transformation of one size of molecules into the other.

9 pages, uses Revtex.

Cited by in corpus (1)

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