  We present ab initio calculations of hidden magnetoelectric multipolar order
in Cr$_2$O$_3$ and its iron-based analogue, $\alpha$-Fe$_2$O$_3$. First, we
discuss the connection between the order of such hidden multipoles and the
linear magnetoelectric effect. Next, we show the presence of hidden
antiferroically-ordered magnetoelectric multipoles in both the prototypical
magnetoelectric material Cr$_2$O$_3$, and centrosymmetric $\alpha$-Fe$_2$O$_3$,
which has the same crystal structure as Cr$_2$O$_3$, but a different magnetic
dipolar ordering. In turn, we predict anti-magnetoelectric effects, in which
local magnetic dipole moments are induced in opposite directions under the
application of an external electric field, to create an additional
antiferromagnetic ordering. We confirm the predicted induced moments using
first-principles calculations. Our results demonstrate the existence of hidden
magnetoelectric multipoles leading to local linear magnetoelectric responses
even in centrosymmetric materials, where a net bulk linear magnetoelectric
effect is forbidden by symmetry.
