R. G. Herb Condensed Matter Seminars |
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<Tc it is possible to have a coexistence of gapless Dirac fermions at the interface with a ferromagnetically ordered layer. For the case of an antiferromagnetic layer on a topological insulator substrate, we show that a second-order quantum phase transition occurs at the interface, and compute the corresponding critical exponents. In particular, we show that the electrons at the interface acquire an anomalous dimension at criticality. The critical behavior of the Néel order parameter is anisotropic and features large anomalous dimensions for both the longitudinal and transversal fluctuations.
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).