A theory for the multiferroic compound LiCu$_{2}$O$_{2}$

We investigate the possible coupling between ferroelectricity and magnetic structure in the zig-zag spin chain compound LiCu$_{2}$O$_{2}$. Based on a group theory analysis, we construct a multi-order parameter phenomenological model and show that a coupling involving the inter-chain magnetic structures and ferroelectricity is necessary in order to understand the experimental results of Park \emph{et. al}. The model is able to account for the electric polarization flip through $\pi /2$ and explain the occurrence of an electric polarization parallel to an applied external magnetic field.