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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 The values of the parameters, on page 29, are computed from more rigorous calculations. –  NanoPhys Sep 8 '13 at 9:49

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