# How to calculate the parity of a band at a particular point in Brillouin zone

Some references mentioned that the calculation of $$Z_2$$ topological invariant of a crystal can be greatly simplified if the crystal contains inversion symmetry. But it involves the calculation of the parity of the occupied band at a specific TRIM point. I have a very basic question about this:

If I have a tight binding Hamiltonian obtained by fitting the data from a first principles calculation, how can we calculate the parity of the occupied band at particular TRIM point?

• +1 But I've made some edits which I think you should look over so that no one has to do that next time. Also the question of how "parity of a band" is defined ought to be a separate question, but you can add a link to that second question here if you wish. Mar 11, 2021 at 19:24
• @NikeDattani Thank you for your help. I will open a new post to ask the second question. Mar 12, 2021 at 2:47

Consider the energy eigenstate $$|\psi_{n\mathbf{k}}\rangle$$, where $$n$$ is the band you are interested in and $$\mathbf{k}$$ is wave vector of the TRIM point. Then, if the system has inversion symmetry, these energy eigenstates are also eigenstates of the parity operator $$\hat{\pi}$$. This means that to determine the parity of that state, you can calculate the expectation value of the parity operator with respect to the energy eigenstate:
$$\langle\psi_{n\mathbf{k}}|\hat{\pi}|\psi_{n\mathbf{k}}\rangle$$
This can only be $$\pm1$$, and the parity is "even" for $$+1$$ and "odd" for $$-1$$.
• The parity operator can be defined by its action on position eigenkets: $\hat{\pi}|\mathbf{r}\rangle=|-\mathbf{r}\rangle$. So I imagine that a way forward would be to expand the energy eigenstates in the position basis, and then apply the operator? Mar 12, 2021 at 15:53