Semi-metal-insulator transition on the surface of a topological insulator with in-plane magnetization

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, the parity is also broken, since the Dirac fermions become massive. This leads in turn to the generation of a topological Chern-Simons term by quantum fluctuations. On the other hand, for an in-plane magnetization the surface states remain gapless for the non-interacting Dirac fermions. In this work we study the possibility of spontaneous breaking of parity due to a dynamical gap generation on the surface in the presence of a local, Hubbard-like, interaction of strength $g$ between the Dirac fermions. A gap and a Chern-Simons term are generated for $g$ larger than some critical value, g_c, 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. We discuss our results in the context of recent experiments in EuS/Bi_2Se_3 heterostructures.