$\mathbb{Z}_2$ 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_1$$v_2$$v_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_1$$v_2$$v_3$), would it be that I only take ($v_0$,$v_3$)? Or would that index be meaningless in a 2D system?



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