Hidden orders and (anti-)Magnetoelectric Effects in Cr$_2$O$_3$ and $\alpha$-Fe$_2$O$_3$

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.