Why one can observe Quantum Hall Effect in 3D Topological Insulators in an external magnetic field when TRS is broken?

Question

In magnetotransport experiments scientists have observed the Quantum Hall effect in 3D topolgical insulators. Using a standard hall-bar geaometry in an external magnetic field they see plateaus in the horizontal and Shubnikov-de-Hass oscillations in the longitudinal conductivity. However, to my understanding, the Quantum Hall effect and 3D TIs belong to two different topological phases distinct by the validity of time reversal symmetry. In other words: The surface states in a 3D TI arise from the change of the Z2 invariant at the interface with a trivial insulator. This Z2 invariant can only be non-trivial in a system with unbroken TRS (whereas in a QHE state the Chern number is the topological invariant characterizing a TRS-broken system).

Doing magnetotransport on a 3D TI, the external magnetic field should brake the TRS (like in a QHE measurement) and the TI should behave like a trivial insulator (or semiconductor). How is it then possible to measure a QHE? I know that also the Quantum anomolous Hall effect has been observed. Though this also with magnetically doped TIs what again breaks TRS. The Quantum spin Hall effect has been measured and is only predicted in 2D TIs, to my knowledge...

So basically, why does it seem that one only observes interesting transport properties in 3D TIs when you break TRS what should make the system trivial? And why is it anyway that a QHE effect is observable in a 3D TI in external magnetic field?

Answer 1

If you only break time-reversal on the surface, not the bulk, then you can compare the relative Hall conductance between breaking the time-reversal one way on the surface and breaking it in the opposite way. This is still a signature of a time-reversal protected phase in the bulk.

If you consider a 2-D slab of a 3-D material, and you break time-reversal in one way on one surface, and the opposite way on the other surface, then the overall Hall conductance of the slab is the same as the relative Hall conductance I just mentioned. In some sense, applying a strong magnetic field is doing this, but of course it also breaks time-reversal in the bulk. But since it is claimed in the experimental paper you cited that, for the materials they studied, the Hall current is only flowing along the surface, it should be insensitive to the time-reversal breaking in the bulk. Of course, this depends on the slab being sufficiently thin. As the slab becomes thicker, the bulk electrons will surely occupy their own Landau levels and thereby contribute to the Hall conductance.