Semimetal-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 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.

Figure 1

Schematic comparison between two types of magnetization orientation on the surface of a TI. (a) In the case of out-of-plane magnetization the electronic spectrum at the surface is gapped. (b) For in-plane magnetization the spectrum is gapless.

Figure 2

Feynman diagram representing the vacuum polarization. The wiggly lines represent photons, and the internal lines represent fermionic propagators.