  We study the interface physics of bipartite magnetic materials deposited on a
topological insulator. This comprises antiferromagnets as well as ferrimagnets
and ferromagnets with multiple magnetic moments per unit cell. If an energy gap
is induced in the Dirac states on the topological surface, a topological
magnetoelectric effect has been predicted. Here, we show that this effect can
act in opposite directions on the two components of the magnet in certain
parameter regions. Consequently, an electric field will mainly generate a
staggered field rather than a net magnetization in the plane. This is relevant
for the current attempts to detect the magnetoelectric effect experimentally,
as well as for possible applications. We take a field-theoretic approach that
includes the quantum fluctuations of both the Dirac fermions on the topological
surface as well as the fermions in the surface layer of the magnet in an
analytically solvable model. The effective Lagrangian and the Landau-Lifshitz
equation describing the interfacial magnetization dynamics are derived.
