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Ginzburg-Landau theory of the zig-zag transition in quasi-one-dimensional classical Wigner crystals

We present a mean-field description of the zig-zag phase transition of a quasi-one-dimensional system of strongly interacting particles, with interaction potential $r^{-n}e^{-r/λ}$, that are confined by a power-law potential ($y^α$). The parameters of the resulting one-dimensional Ginzburg-Landau theory are determined analytically for different values of $α$ and $n$. Close to the transition point for the zig-zag phase transition, the scaling behavior of the order parameter is determined. For $α=2$ the zig-zag transition from a single to a double chain is of second order, while for $α>2$ the one chain configuration is always unstable and for $α<2$ the one chain ordered state becomes unstable at a certain critical density resulting in jumps of single particles out of the chain.