topological invariants calculation on 2D system

Dominik Gresch greschd at
Thu Sep 27 20:22:31 CEST 2018

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,


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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