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Topological magneto-electric effects in thin films of topological insulators

We propose that the topological magneto-electric (ME) effect, a hallmark of topological insulators (TIs), can be realized in thin films of TIs in the $ν=0$ quantum Hall state under magnetic field or by doping two magnetic ions with opposite signs of exchange coupling. These setups have the advantage compared to previously proposed setups that a uniform configuration of magnetic field or magnetization is sufficient for the realization of the topological ME effect. To verify our proposal, we numerically calculate ME response of TI thin films in the cylinder geometry and that of effective 2D models of surface Dirac fermions. The ME response is shown to converge to the quantized value corresponding to the axion angle $θ=\pm π$ in the limit of the large top and bottom surface area of TI films, where non-topological contributions from the bulk and the side surface are negligible.