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.


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


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. enter image description here

  • $\begingroup$ 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 $P_3$ $\endgroup$
    – user10001
    Aug 28, 2012 at 17:40
  • $\begingroup$ 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 $P_3$. $\endgroup$
    – Heidar
    Aug 28, 2012 at 18:33
  • $\begingroup$ 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. $\endgroup$
    – Machine
    Aug 29, 2012 at 3:40

2 Answers 2


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.

  • $\begingroup$ Thank you for your answer! But it may not be too bad with a finite magnetic field. $\endgroup$
    – Machine
    Sep 11, 2012 at 14:31
  • $\begingroup$ Hi Prof. Wen, thanks for your answer. If topological insulator cannot be defined in magnetic field, why Qi's paper "topological field theory of time-reversal insulators" still studies the Maxwell equations in topological insulators? $\endgroup$
    – Timothy
    Oct 12, 2012 at 4:50
  • $\begingroup$ " topological insulator cannot be defined in magnetic field" means that in magnetic field, topological insulator and trivial band insulator become the same phase. This does not prevent people to study Maxwell equation in such phase. $\endgroup$ Oct 13, 2012 at 0:26

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.

  • $\begingroup$ You mentioned that 'in the TI case, "topological" means universal, independent of material properties'. But TI is not very universal and not very independent of material properties, since time-reversal-symmetry-breaking perturbation destroy TI. So the essence of TI is not topology, but symmetry. $\endgroup$ Sep 12, 2012 at 1:16
  • $\begingroup$ I see, in this sense, it is true. $\endgroup$
    – Machine
    Sep 12, 2012 at 9:11

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