A thin film of ferromagnetically ordered material proximate to the surface of a three-dimensional topological insulator explicitly breaks the time-reversal symmetry of the surface states. For an out-of-plane ferromagnetic order parameter on the surface\, parity is also broken\, since the Dirac fermions become massive. This leads in turn to the generation of a Chern-Simons term by quantum fluctuations. On the other hand\, for an in-plane magnetization the surface states remain metallic.

We consider a theory for a two-dimensional interacting conduction electron system with strong spin-orbit coupling on the interface between a topological insulator and the magnetic (ferromagnetic or antiferromagnetic) layer. For the ferromagnetic case we derive the Landau-Lifshitz equation\, which features a contribution proportional to a fluctuation-induced electric field obtained by computing the topological (Chern-Simons) contribution from the vacuum polarization. We also show that fermionic quantum fluctuations reduce the critical temperature T*c at the interface relative to the critical temperature Tc of the bulk\, so that in the interval T*c ≤T

In addition we study the possibility of spontaneous breaking of parity due to a dynamical gap generation on the surface. We find that in the absence of interaction between the fermions there is no spontaneous gap generation. In the presence of a local\, Hubbard-like\, interaction of strength g\, a gap and a Chern-Simons term are generated for g larger than some critical value provided the number of Dirac fermions\, N is odd. For an even number of Dirac fermions the masses are generated in pairs having opposite signs\, and no Chern-Simons term is generated. Our result offers a possible explanation to recent experiments showing a gap opening even when the topological insulator is proximate to a planar ferromagnet.

[1] F.S. Noguiera and Ilya Eremin\, Phys. Rev. Lett. 109\, 237203 (2012)

[2] F.S. Noguiera and Ilya Eremin\, arXiv:1304.2933 (unpublished).

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