  Improper ferroelectricity (trimerization) in the hexagonal manganites
RMnO$_3$ leads to a network of coupled structural and magnetic vortices that
induce domain wall magnetoelectricity and magnetization neither of which,
however, occurs in the bulk. Here we combined first-principles calculations,
group-theoretic techniques, and microscopic spin models to show how the
trimerization not only induces a polarization but also a bulk magnetization and
bulk magnetoelectric (ME) effect. This results in the existence of a bulk
linear ME vortex structure or a bulk ME coupling such that if direction of
polarization reverses so does magnetization. To measure the predicted ME
vortex, we suggest RMnO$_3$ under large magnetic field. We suggest a family of
materials, the hexagonal RFeO$_3$ ferrites, also display the predicted
phenomena in their ground state.
