What happens when a bare 3d topological insulator is subject to a magnetic field?

Question

Effective field theory of 3d topological insulators (TI) predict some novel electromagnetic effects. Unfortunately it require a gapped surface which is hard to achieve experimentally. Then I have two questions.

1.

Is $\nabla P_3=0$ for a bare TI (without magnetic coating), and hence it behaves like a trivial insulator?

2.

Why not put the bare TI inside a uniform magnetic field which, of course, opens a gap on the surface? Can I still use the topological field theory without worrying about the bulk?

I am afraid that the bulk is no longer "topological" since time-reversal symmetry is broken. But the bulk band gap is robust under perturbation. As long as the gap is not closed, it remains a TI.

Take a look at the article topological quantization in unit of $\alpha$, in which a $B$ field is applied instead of magnetic coating. I got more confused after reading this paper.

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The $P_3$ appeared in my question comes from Qi's paper topological field theory of time-reversal insulators.

Answer 1

Topological insulator, by definition, cannot exist in magnetic field. This is because the topological insulator is NOT topological. A topological insulator is a material with time reversal symmetry and particle number conservation. Without time-reversal symmetry, topological insulators cannot exist, since they become the same as trivial band insulators. So a magnetic field destroys the topological insulator. True topological phases (ie phases with non-trivial topologically orders) are robust against any perturbations, including magnetic field.

Answer 2

Thank you for you answer. But there might be different interpretations of the word "topology." In the TI case, "topological" means universal, independent of material properties while when it comes to the concept of "topological orders," "topology" implies robustness. After all, it is merely a name whose meaning varies from person to person. Yet, we can still uniquely specify them with their "physical names" -- the classification through symmetry group or tensor category.

However, as the golden rule of science, "principle of uniformity" says, though magnetic field breaks $\cal T$, as long as it is small, the effects would be small (linear in $B$). In other words, we can still do the measurement in lab and extrapolate useful results in the limit $B\rightarrow 0^+$. Although TIs are not topological, it doesn't matter at least to experimentalists.