Topological magneto-electric response in non-magnetic and anti-ferromagnetic topological insulators

The theory of magneto-electric response in topological insulators is confusing because bulk calculations yield a topologically protected non-zero result that is forbidden by time-reversal symmetry. The theoretical bulk response is expressed as an integral over three-dimensional momentum space that is quantized. This conundrum can be resolved by recognizing the importance of time-reversal symmetry breaking at the surface. Evidently in this instance the bulk theory does not fully explain the magneto-electric phenomenology of real devices, and this has created some confusion in the literature. For instance, in thin-films for which time-reversal symmetry is not broken either in the bulk or at the surface, topological insulators have no magneto-electric response. Here we consider the magneto-electric response of non-magnetic and anti-ferromagnetic multilayer crystals. We develop tight-binding Hamiltonians for both cases, which are special in the sense that the bulk magneto-electric response coefficient can be evaluated analytically in a Hamiltonian gauge. Motivated by the modern theory of magnetization, and by numerical thin-film calculations for the same model which are presented in a related talk by Lei, Mahon, and MacDonald, we compare the structures of the bulk momentum-space integrands between the two cases and discuss bulk/surface decompositions of the total response.