topological invariants calculation on 2D system

[Yifan GAO]

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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