Symmetry properties of finite Frenkel-Kontorova chains

We investigate the symmetry properties of finite Frenkel-Kontorova chains with free-end boundary conditions. A symmetry-breaking transition is found as one varies η, the relative strength of the harmonic and substrate interactions. This transition, from a symmetric configuration to a configuration with broken symmetry at η=${\mathrm{\eta }}_{\mathit{c}}$, is characterized by a gap in the phonon spectrum and by disorder and reflection-symmetry parameters, all of which display the scaling behavior (${\mathrm{\eta }}_{\mathit{c}}$-η${)}^{0.5}$. The chains are shown to be pinned both below and above the symmetry-breaking transition.