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

Abstract

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 noninteracting 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${}_{2}$Se${}_{3}$ heterostructures.