Monopole-based formalism for the diagonal magnetoelectric response

We develop the formalism of the macroscopic monopolization-that is, the magnetoelectric monopole moment per unit volume-in periodic solids, and discuss its relationship to the magnetoelectric effect. For the series of lithium transition metal phosphate compounds, we use first-principles density functional theory to calculate the contributions to the macroscopic monopolization from the global distribution of magnetic moments within the unit cell, as well as from the distribution of magnetization around the atomic sites. We find one example within the series (LiMnPO4) that shows a macroscopic monopolization corresponding to a ferromonopolar ordering consistent with its diagonal magnetoelectric response. The other members of the series (LiMPO4, with M = Co, Fe, and Ni) have zero net monopolization but have antiferromonopolar orderings that should lead to q-dependent diagonal magnetoelectric effects.