[Wannier] Integration over the half Brillouin zone

[David Vanderbilt]

Kyu Won,

Let me add just a couple of comments, because I think your email
indicates some misunderstandings.

You talked about three different quantities, and they are not
all the same.  Are you clear about which one you really want to
calculate?

One is the spin Hall conductivity.  This is NOT quantized even
in the QSH phase, and I don't think any of the methods that have
been discussed in this thread yield this quantity.

The second is the Z2 index.  Kane and co-workers found a way
of expressing the Z2 index in terms of an integral of the Berry
curvature over a half BZ, but if I remember correctly, it also
involved a Berry phase.  My memory is a bit hazy at the moment.
But I think their observation does not lead to any very practical
scheme for computing the Z2 index.  I think the two methods that
Ivo mentioned are the more-or-less now-standard ones for computing
the Z2 index.

The third is the spin Chern number.  This is a somewhat tricky
beast.  In my view it is not a true topological index; for example,
it is possible to construct a TR-invariant insulator for which
it is not defined.  When it is defined, it is an integer, but
again I don't think any of the methods that have been discussed
in this thread yield this quantity.

David Vanderbilt


On Wed, 17 Sep 2014, 고운 wrote:

> 
> Dear Ivo and Jonathan
> 
>  
> 
> Thank you for your great help.
> 
>  
> 
> Kyu Won Lee