  Multiferroic materials combine multiple ferroic orders that may enable
cross-functionalities, e.g. control of one ferroic order by the field conjugate
to the other one. BaCoSiO$_4$, a recently proposed multiferroic combines
multiple unit cell-tripling modes, structural chirality and ferroelectric
polarization, whose interactions result in a peculiar ground state. We derive
the Landau free energy for interacting trimerization, electric polarization and
structural chiral modes and find the corresponding coefficients from
first-principles calculations. This results in a quantitative model for the
description of domain walls and vortices, observed in recent experiments.
