Z2 invariant for 2D material (or a semi-infinite slab)

[Dominik Gresch]

Dear Koushik,

Two-dimensional materials are characterized by a single Z2 invariant, 
since there is only a single plane in k-space. This is exactly the case 
described in the Kane-Mele paper: 
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.146802

For Z2Pack, there are examples showing the BHZ model: 
https://github.com/Z2PackDev/Z2Pack/blob/dev/current/examples/hm/bhz_model/bhz.py

and a simple 2D tight-binding model: 
http://z2pack.ethz.ch/doc/2.1/examples/tb.html

For the cases where your model is actually three-dimensional, you can 
just choose a fixed kz because your model should not depend on kz if it 
is effectively two-dimensional.

Best regards,

Dominik


On 24.11.2017 05:30, koushik pal wrote:
> Hello All,
>
> I was wondering if it's possible to calculate Z2 invariant
> for a 2D material (or a semi-infinite slab) using the Z2Pack code.
>
> Let's say I have  a slab model with (001) surface orientation
> (i.e. vacuum is along the z-direction of the slab). To determine
> the topological invariant of  this (001) slab,  shall I calculate the
> Z_2 invariant for kz=0 and kz=0.5 planes, and take their difference
> modulo 2?
>
> The problem here is the following.  As this is a slab, kz=0.5 plane
> does not have any physical meaning. So I don't know if Z2 invariant
> on kz=0.5 plane has some significance.
> Please correct me if I am missing something here, and suggest me
> ways to calculate Z2 invariant for such a slab.
>
> Thanks in advance.
>
> Best regards,
> Koushik
>
> Graduate student
> JNCASR
> Bangalore, India