# how to determine the parity eigenvalues of time-reversal invariant momenta point from first principle calculation when we judge topological insulator?

This is a question of topological insulator.

Liang Fu and C. L. Kane proposed a method to judge whether an inversion symmetric insulator is a topological insulator or not in their article(L. Fu and C.L. Kane, Phys. Rev. B 76, 045302 (2007)). The method is just to determine the parity of the occupied band eigenstates at the eight or four(in two dimensions) time-reversal invariant momenta $\Gamma_i$in the Brillouin zone. The Z2 invariant is determined by quantity $${{\delta }_{i}}=\prod\limits_{m=1}^{N}{{{\xi }_{2m}}\left( {{\Gamma }_{i}} \right)}$$Where ${{\xi _{2m}}\left( {{\Gamma _i}} \right)}$is the parity eigenvalue of 2m band at $\Gamma_i$ point.

My question is how to determine the parity of band state at these points from first principle band calculation(like Wien2K band calculation)?

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You would normally write down the Bloch Hamiltonian $\mathcal{H}$ as a matrix in a convenient orbital and/or spin basis. Within this basis you know that $\mathcal{T}\left|\mathbf{k},\uparrow\right\rangle =\left|-\mathbf{k},\downarrow\right\rangle$. You can construct a matrix, in this basis, which satisfies that (and obviously $\left[\mathcal{H},\mathcal{T}\right]=0$). Then you can find the eigenvalues of the orbitals at the TRIMs. –  NanoPhys Sep 7 '13 at 16:23
@NanoPhys You mean something like slater-koster method to build the Hamiltonian directly? But, Is there a way to determine the parity from the first principle data(like Wien2K) directly? From this reference"Nature Physics 5(6): 438-442",according to their description I think they can directly determine the parity . –  user15964 Sep 8 '13 at 2:00
In the literature, that I have come across so far, people construct a model Hamiltonian $\mathcal{H}(\mathbf{k})$ which respects the $\mathcal{T}$-, rotational, inversion symmetries etc. with adjustable parameters. They diagonalize the Hamiltonian and fit the spectrum to first-principles calculations or experimental data to determine the values of these adjustable parameters; in your case the former. For example, you can look at the BHZ model in eq. (6) of arxiv.org/abs/0801.0901. The values of the parameters, on page 29, are computed from more rigorous calculations. –  NanoPhys Sep 8 '13 at 9:49