  Exact analytical calculations are performed to study the rotating
magnetoelectric effect in a ground state of a coupled spin-electron model on a
doubly decorated square lattice with and without presence of an external
magnetic field. Novel spatially anisotropic magnetic ground states emergent due
to a rotation in an external electric field are found at three physically
interesting electron concentrations ranging from a quarter up to a half
filling. In absence of the magnetic field existence of spatially anisotropic
structures requires a fractional electron concentration, where a significant
influence of spatial orientation of an electric field is observed. It turns out
that the investigated model exhibits a rotating magnetoelectric effect at all
three concentrations with one or two consecutive critical points in presence of
magnetic field. At the same time, the rotating electric field has a significant
effect on a critical value of an electrostatic potential, which can be enhanced
or lowered upon changing the electron hopping and the magnitude of an applied
magnetic field. Finally, we have found an intriguing interchange of magnetic
order between the horizontal and vertical directions driven by a rotation of
the electric field, which is however destabilized upon strengthening of the
magnetic field.
