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