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

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.

The $P_3$ appeared in my question comes from Qi's paper topological field theory of time-reversal insulators. • Chen it would be beneficial for everyone (and would make your question more clear) if you can please include definition of what you mean by gapped surface, and $P_3$ Aug 28, 2012 at 17:40
• Well, the bulk is only robust against time-reversal invariant perturbations not those that break it. If you want a dynamic "axion" field on the boundary, you have to make sure that the bulk preserves time-reversal invariance while the boundary break it. As dushya said, it would be nice if you could define $P_3$. Aug 28, 2012 at 18:33
• I think the right answer for my second question would be when T-symmetry is broken, there are no topological insulators at all since they are not well-defined. Aug 29, 2012 at 3:40

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.