Pinning transition of the discrete sine-Gordon equation

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Abstract
The ground state of the discrete sine-Gordon equation, used to model a one-dimensional solid in a periodic potential, is examined in the incommensurate region. The behavior of the system near the transition from an unpinned to a pinned phase (first discussed by Aubry) is investigated. A disorder parameter and a correlation length are defined and shown numerically to obey scaling relations on both sides of the transition. The system studied is equivalent to the "standard map" of dynamical systems theory, and this relationship is discussed. In particular, our results extend the scaling behavior found by Shenker and Kadanoff into the "chaotic" regime.