I am confused about the quantum Hall state and topological insulating states. Following are the points (according to my naive understanding of this field) which confuse me:

  1. Topological insulator has insulating bulk and conducting edge states which are protected by time reversal symmetry.

  2. The quantum Hall state has conducting edge states and insulating bulk (Landau levels). And to achieve this Hall state we have to apply a strong magnetic field which breaks time-reversal symmetry.

Is not quantum Hall state a topological insulating state? it has insulating bulk and conducting edges. If yes, then how can we have topological insulator with broken time reversal symmetry? If no, why is it not a topological insulating state?

  • $\begingroup$ Quantum Hall effect is the same as Chern insulator, and you can find it in class A in the topological insulator/superconductor periodic table in class A. What you described in 1 is quantum spin hall, and it's in class AII $\endgroup$ – FangXie Jan 26 at 19:50

The first statement is incorrect for several reasons: first of all, there are many types of topological insulators, and models describing systems exhibiting the Quantum Hall Effect are just one particular type (class A using the Altland-Zirnbauer classification scheme). And for these, time-reversal symmetry is (and indeed, must be) broken.

However, the Quantum Spin Hall Effect is due to an odd time-reversal symmetry (a system of class AII), and here there is an edge spin current that is protected by time-reversal symmetry.


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