Abstract
We theoretically analyse the equation of topological solitons in a chain of particles interacting via a repulsive power-law potential and confined by a periodic lattice. Starting from the discrete model, we perform a gradient expansion and obtain the kink equation in the continuum limit for a power-law exponent n ≥ 1 . The power-law interaction modifies the sine-Gordon equation, giving rise to a rescaling of the coefficient multiplying the second derivative (the kink width) and to an additional integral term. We argue that the integral term does not affect the local properties of the kink, but it governs the behaviour at the asymptotics. The kink behaviour at the center is dominated by a sine-Gordon equation and its width tends to increase with the power law exponent. When the interaction is the Coulomb repulsion, in particular, the kink width depends logarithmically on the chain size. We define an appropriate thermodynamic limit and compare our results with existing studies performed for infinite chains. Our formalism allows one to systematically take into account the finite-size effects and also slowly varying external potentials, such as for instance the curvature in an ion trap.
Highlights
The Frenkel–Kontorova model reproduces in one dimension the essential features of stick-slip motion between two surfaces [1,2,3]
In one dimension the elastic crystal is modelled by a periodic chain of classical particles with uniform equilibrium distance a, which interact with a sinusoidal potential with periodicity b [4,5]
When the interactions of the elastic crystal are nearest-neighbour, in the long-wavelength limit the dynamics of a single kink is governed by the integrable sine-Gordon equation [6,7,8]
Summary
The Frenkel–Kontorova model reproduces in one dimension the essential features of stick-slip motion between two surfaces [1,2,3]. In one dimension the elastic crystal is modelled by a periodic chain of classical particles with uniform equilibrium distance a, which interact with a sinusoidal potential with periodicity b [4,5]. The transition is characterized by proliferation of kinks, namely, of local distributions of excess particles (or holes) in the substrate potential. When the interactions of the elastic crystal are nearest-neighbour, in the long-wavelength limit the dynamics of a single kink is governed by the integrable sine-Gordon equation [6,7,8]
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