[Wannier] Integration over the half Brillouin zone

[Ivo Souza]

Dear Kyu Won,

  Let me add to Jonathan's answer by saying something more about the
physics. It is useful to distinguish three cases:

(i)   Ferromagnetic metals

(ii)  Quantum anomalous Hall (Chern) insulators

(iii) Quantum spin-Hall insulators

What you currently have in wannier90 was really developed with case (i) in
mind, where you integrate the Berry curvature up to the Fermi energy to
get a non-quantized anomalous Hall conductivity.

In principle the same approach can also be used for case (ii), but it will
not give you an exactly quantized Hall conductance except in the limit of
a very dense k-point mesh.

A better approach for Chern insulators is to replace the Berry curvature
at each point k by a loop-Berry phase over a small square plaquette
surrounding the point, and eventually cover the entire 2D BZ with such
plaquettes. If you do things carefully, you are guaranteed to get an
integer Chern number, as explained in this paper:

http://arxiv.org/abs/cond-mat/0503172

This algorithm is quite straightforward to implement, and it will probably
make it into wannier90 eventually. But I don't think this will completely
solve your problem.

You are interested in case (iii), which as I understand is a more subtle
numerical problem. I am aware of two algorithms:

http://arxiv.org/abs/cond-mat/0611423

and

http://arxiv.org/abs/1102.5600

I don't know if the first method is implemented in any easily available ab
intio code. But the authors of the second algorithm, Alexey Soluyanov and
David Vanderbilt, have made their code package available at

http://www.phys.ethz.ch/~alexeys/z2.html

  Hope this helps,

  Ivo


On Tue, 16 Sep 2014, Jonathan Yates wrote:

>
> On 16 Sep 2014, at 06:32, 고운 <rhdnsi at hanmail.net> wrote:
>
> > Dear experts
> >
> > I am trying to use Wannier90 for the quantum spin Hall (QSH) phase in
> > graphene and silicene. According to Kane and Melle, the Berry
> > curvature sould be integrated over the half Brillouin zone (but not
> > over the full Brillouin zone) for the spin Chern number or spin Hall
> > conductivity in the QSH phase.
> >
> > Wannier90 calculates the anomalous Hall conductivity (AHC) by
> > integrating the Berry curvature over the full Brillouin zone. Since
> > the quantized AHC is the Chern number in units of e^2/h, I expect that
> > the AHC calculated by integration of the Berry curvature over the half
> > Brillouin zone gives the spin Chern number in the QSH phase.
> >
> > The problem is how to integrate the Berry curvature over the half
> > Brillouin zone (but not over the full Brillouin zone) in Wannier90.
> > How can i overcome the problem? Is there a tricky way or a simple
> > modification of the code?
>
> I think it is well worth your while reading the relevant sections of the
> code and seeing how they implement the physical equations. The Berry
> phase routines are largely the work of Ivo Souza, and they are very
> clearly presented and commented. In your case the key routine is in
> scr/postw90/berry.f90
>
>  In that file you will find a simple loop over points in the Brillouin
> Zone - I suspect it would not be hard to modify this to do what you
> want. You will also see from the code that there is an undocumented
> feature which allows you to read the points from a file (be aware that
> this will not work for all properties - you need to think carefully to
> see if this is appropriate for your problem).
>
> Note this is a practical answer - I haven’t tried to think about the
> physics of your question.
>
>  Jonathan
>
>
> --
> Department of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH, UK
> tel: +44 (0)1865 612797                http://users.ox.ac.uk/~oums0549/
>
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