topological invariants calculation on 2D system

[Dominik Gresch]

Dear Patric,


For the Z2 calculation, the surface should go only across half the 
Brillouin zone. So, the surface should probably be lambda s, t: [s / 2, 
t, 0].


The reason for this is that the second half is a mirror image of the 
first. When you go across the full BZ, the result for the Z2 invariant 
will always be zero.


Best regards,

Dominik


On 28.09.2018 12:04, Yifan GAO wrote:
>
> Dear Dominik,
>
>
> Thanks for your reply! I'm really a rookie in this area.
>
>
> Now I'm testing silicene(10.1103/PhysRevLett.107.076802) with Z2Pack. 
> I successfully get the edge states by wannier90 calculation. The Z2 
> invariant should be 1 according to the reference. I set 
> /num_band=num_wann=8/ to integrate the bands under the fermi level and 
> chose /lambda s, t: (s, t, 0)/ and to calculate lines along ky.
>
>
> The result given by Z2Pack shows Z2 = 0.  I don't know if there's 
> anything wrong with my calculation?
>
>
> All the parameters are similar to the Bi example. I changed 
> /search_shell/ in range of 24~96 in order to finish the calculation.
>
>
> Best regards,
>
> Patric
>
>
> ------------------------------------------------------------------------
> *From:* Dominik Gresch <greschd at phys.ethz.ch>
> *Sent:* Thursday, September 27, 2018 11:22:31 AM
> *To:* Yifan GAO; z2pack at lists.phys.ethz.ch
> *Subject:* Re: topological invariants calculation on 2D system
>
> Dear Patric,
>
>
> When calculating 2D invariants, the only Chern and Z2 invariants that 
> can be calculated are those on the 2D Brillouin zone itself. Since you 
> are using VASP, the 2D system is represented as a 3D system, where one 
> dimension is very long in real space, and correspondingly very narrow 
> in reciprocal space. As a result, the bands will be almost flat in 
> that direction. So, the surface you need to choose for calculating the 
> 2D invariant should be extended in the remaining two dimensions.
>
>
> As a simple examle, if your unit cell is
>
>
> a_1 = (1, 0, 0)
>
> a_2 = (0, 1, 0)
>
> a_3 = (0, 0, 200)
>
>
> then you want the surface to span k_x and k_y, e.g. lambda s, t: (s, 
> t, 0) for a Chern number. The k_z value can be chosen to be zero, but 
> it could really be anything since the system should not depend on k_z.
>
>
> Best regards,
>
> Dominik
>
>
> On 27.09.2018 14:38, Yifan GAO wrote:
>>
>> Hello,
>>
>>
>> I don't quite understand how can I get the topological invariants, 
>> like Z2 and Chern, within a 2D system? I'm feeling puzzled when 
>> building the surface.
>>
>> BTW I'm using vasp.
>>
>>
>> Thank you,
>>
>> Patric
>>
>>
>

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