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<div class=tx-hanmail-content-wrapper style="font-size: 10pt; font-family: 돋움; color: #333333; line-height: 1.5; background-color: transparent; browser: msie"><P>Dear Ivo and Jonathan</P>
<P> </P>
<P>Thank you for your great help.</P>
<P> </P>
<P>Kyu Won Lee</P>
<P> </P><SPAN style="FONT-FAMILY: arial, 돋움; MARGIN-LEFT: 0.8ex">--------- 원본 메일 ---------<BR>
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<DIV style="FONT-FAMILY: arial, 돋움; LINE-HEIGHT: 20px"><B>보낸사람</B>: Ivo Souza <ivo_souza@ehu.es><BR>
<B>받는사람</B> : Jonathan Yates <jonathan.yates@materials.ox.ac.uk><BR>
<B>참조</B> : "<wannier@quantum-espresso.org>" <wannier@quantum-espresso.org><BR>
<B>날짜</B>: 2014년 9월 17일 수요일, 02시 21분 45초 +0900<BR>
<B>제목</B>: Re: [Wannier] Integration over the half Brillouin zone<BR>
<DIV id=viewmail-textHtml class=viewmail-textHtml name="viewmail-textHtml"><!-- 내용 부분 Start-->Dear Kyu Won,<BR>
<BR>
Let me add to Jonathan's answer by saying something more about the<BR>
physics. It is useful to distinguish three cases:<BR>
<BR>
(i) Ferromagnetic metals<BR>
<BR>
(ii) Quantum anomalous Hall (Chern) insulators<BR>
<BR>
(iii) Quantum spin-Hall insulators<BR>
<BR>
What you currently have in wannier90 was really developed with case (i) in<BR>
mind, where you integrate the Berry curvature up to the Fermi energy to<BR>
get a non-quantized anomalous Hall conductivity.<BR>
<BR>
In principle the same approach can also be used for case (ii), but it will<BR>
not give you an exactly quantized Hall conductance except in the limit of<BR>
a very dense k-point mesh.<BR>
<BR>
A better approach for Chern insulators is to replace the Berry curvature<BR>
at each point k by a loop-Berry phase over a small square plaquette<BR>
surrounding the point, and eventually cover the entire 2D BZ with such<BR>
plaquettes. If you do things carefully, you are guaranteed to get an<BR>
integer Chern number, as explained in this paper:<BR>
<BR>
<A href="http://arxiv.org/abs/cond-mat/0503172" target=nlink>http://arxiv.org/abs/cond-mat/0503172</A><BR>
<BR>
This algorithm is quite straightforward to implement, and it will probably<BR>
make it into wannier90 eventually. But I don't think this will completely<BR>
solve your problem.<BR>
<BR>
You are interested in case (iii), which as I understand is a more subtle<BR>
numerical problem. I am aware of two algorithms:<BR>
<BR>
<A href="http://arxiv.org/abs/cond-mat/0611423" target=nlink>http://arxiv.org/abs/cond-mat/0611423</A><BR>
<BR>
and<BR>
<BR>
<A href="http://arxiv.org/abs/1102.5600" target=nlink>http://arxiv.org/abs/1102.5600</A><BR>
<BR>
I don't know if the first method is implemented in any easily available ab<BR>
intio code. But the authors of the second algorithm, Alexey Soluyanov and<BR>
David Vanderbilt, have made their code package available at<BR>
<BR>
<A href="http://www.phys.ethz.ch/~alexeys/z2.html" target=nlink>http://www.phys.ethz.ch/~alexeys/z2.html</A><BR>
<BR>
Hope this helps,<BR>
<BR>
Ivo<BR>
<BR>
<BR>
On Tue, 16 Sep 2014, Jonathan Yates wrote:<BR>
<BR>
><BR>
> On 16 Sep 2014, at 06:32, 고운 <<A href="http://mail2.daum.net/hanmail/mail/MailComposeFrame.daum?TO=rhdnsi@hanmail.net">rhdnsi@hanmail.net</A>> wrote:<BR>
><BR>
> > Dear experts<BR>
> ><BR>
> > I am trying to use Wannier90 for the quantum spin Hall (QSH) phase in<BR>
> > graphene and silicene. According to Kane and Melle, the Berry<BR>
> > curvature sould be integrated over the half Brillouin zone (but not<BR>
> > over the full Brillouin zone) for the spin Chern number or spin Hall<BR>
> > conductivity in the QSH phase.<BR>
> ><BR>
> > Wannier90 calculates the anomalous Hall conductivity (AHC) by<BR>
> > integrating the Berry curvature over the full Brillouin zone. Since<BR>
> > the quantized AHC is the Chern number in units of e^2/h, I expect that<BR>
> > the AHC calculated by integration of the Berry curvature over the half<BR>
> > Brillouin zone gives the spin Chern number in the QSH phase.<BR>
> ><BR>
> > The problem is how to integrate the Berry curvature over the half<BR>
> > Brillouin zone (but not over the full Brillouin zone) in Wannier90.<BR>
> > How can i overcome the problem? Is there a tricky way or a simple<BR>
> > modification of the code?<BR>
><BR>
> I think it is well worth your while reading the relevant sections of the<BR>
> code and seeing how they implement the physical equations. The Berry<BR>
> phase routines are largely the work of Ivo Souza, and they are very<BR>
> clearly presented and commented. In your case the key routine is in<BR>
> scr/postw90/berry.f90<BR>
><BR>
> In that file you will find a simple loop over points in the Brillouin<BR>
> Zone - I suspect it would not be hard to modify this to do what you<BR>
> want. You will also see from the code that there is an undocumented<BR>
> feature which allows you to read the points from a file (be aware that<BR>
> this will not work for all properties - you need to think carefully to<BR>
> see if this is appropriate for your problem).<BR>
><BR>
> Note this is a practical answer - I haven’t tried to think about the<BR>
> physics of your question.<BR>
><BR>
> Jonathan<BR>
><BR>
><BR>
> --<BR>
> Department of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH, UK<BR>
> tel: +44 (0)1865 612797 <A href="http://users.ox.ac.uk/~oums0549/" target=nlink>http://users.ox.ac.uk/~oums0549/</A><BR>
><BR>
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