Effect of TRS potential on Topological insulator (QSH)

[mohsen2002]

Hi every body,

I faced a paradox. The topological insulator is robust against a potential that does not breaks the TRS.
But in the original work of Kane-Mele (PRL 95, 146802), the "staggered sublattice potential" that does not breaks the TRS,, makes zigzag ribbon trivial insulator (figure 1 in the PRL).
Is there any explanation?

 

[radium]

The staggered sublattice potential can open a gap. Think of boron nitride. Because the A and B sublattice are in equivalent so it is gapped. If the gap becomes large enough in the QSHE we have a phase transition, the gap closes and are now not inverted so we have a trivial insulator. So there can be a staggered sublattice potential but at a critical value it will close the gap due to spin orbit and the system becomes a trivial insulator.

Also think of the Haldane model. The Kane mele model is just two copies of the Haldane model, one for each spin. The sigma z term in the Haldane model is proportional to a Haldane mass, and there is a region where you can have a chiral edge state also due to the gap closing and reopening.

 

[mohsen2002]

Thank you for your reply. I can understand the gap, but is it match with our expectation of robustness of TI phase against interactions?

 

[radium]

A correction to the above, the Haldane mass is actually proportional to sigma z Sz. In the QSHE this mass would correspond to sigma z tau z s z. The staggered sublattice potential is just proportional to sigma z and produces the gap of the same size at K and K'

The Haldane mass is a different kind of mass term than from the sublattice potential as it introduces gaps of opposite signs at the points K and K' (location of the Dirac points in graphene). If the sublattice potential is too large, the bands will not invert since the gaps at K and K' will have the same sign even with the Haldane mass. The transition comes when the gap closes. So basically they are competing.

Usually when you think of interactions in the QSHE, you could think of something like an impurity somewhere or something that could cause backscattering. If such a thing does not break time reversal, backscattering cannot happen since the states are chiral.

 

[mohsen2002]

Thank you so much for helpful discussion and your time.