# Z2 Topological Invariant for 2D Systems

$$\mathbb{Z}_2$$ topological invariant is typically calculated based on the parity of occupied bands at time-reversal invariant momenta (TRIM) in the Brillouin zone. The WannierTools docs states that we can get $$\mathbb{Z}_2$$ topological Index ($$v_0$$, $$v_1v_2v_3$$) from the $$\mathbb{Z}_2$$ calculations of six time reversal invariant planes, i.e. (a) $$k_1=0.0$$; (b) $$k_1=0.5$$; (c) $$k_2=0.0$$; (d) $$k_2=0.5$$; (e) $$k_3=0.0$$; (f) $$k_3=0.5$$

In the same documentation, $$\mathbb{Z}_2$$ index for 3D system is given as:

$$v_0= (\mathbb{Z}_2(k_i=0)+\mathbb{Z}_2(k_i=0.5)) mod(2)$$

$$v_i= \mathbb{Z}_2(k_i=0.5)$$

After obtaitning the values from those six planes, we can get the index based on the previous equations, and check whether we have a strong or weak topological insulator. The documentaion also states: for 2D system, if you set the Z axis as the stack axis, please only take the $$\mathbb{Z}_2$$ number at $$k_3=0$$ plane.

In the case of 2D system, how should I treat the index ($$v_0$$, $$v_1v_2v_3$$), would it be that I only take ($$v_0$$,$$v_3$$)? Or would that index be meaningless in a 2D system?