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Topological Insulator Magnetic Tunnel Junctions: Quantum Hall Effect and Fractional Charge via Folding

We provide a characterization of tunneling between coupled topological insulators in 2D and 3D under the influence of a ferromagnetic layer. We explore conditions for such systems to exhibit integer quantum Hall physics and localized fractional charge, also taking into account interaction effects for the 2D case. We show that the effects of tunneling are topologically equivalent to a certain deformation or folding of the sample geometry. Our key advance is the realization that the quantum Hall or fractional charge physics can appear in the presence of only a \emph{single} magnet unlike previous proposals which involve magnetic domain walls on the surface or edges of topological insulators respectively. We give illustrative topological folding arguments to prove our results and show that for the 2D case our results are robust even in the presence of interactions.