$Z_2$ invariants in quantum spin hall (QSH)

Question

In a recent literature survey, I learned that $Z_2$ topological invariant is defined as zeros of Pfaffians in half a Brillouin Zone, where Pfaffians are defined as $P(k)=Pf[<u_i(k)|T|u_j(k)>]$.

However, since time reversal operator $T$ takes $k$ to $-k$, if $k\ne -k$, $Pf[<u_i(k)|T|u_j(k)>]$ should always be zero because $T|u_j(k)>$ and $|u_i(k)>$ is eigenstates of different eigenvalues($-k$ and $k$) of translate operator.

The above argument leads to a conclusion that there are infinite zeros of Pfaffians in half a Brillouin Zone, which is obviously not true.

Can anyone point out what's wrong with my argument?

(Ref: http://link.aps.org/doi/10.1103/PhysRevLett.95.146802)

Answer 1

It is not true that if $\mathbf{k}\neq -\mathbf{k}$ the matrix element $\langle u_i(\mathbf{k}|T|u_j(\mathbf{k})\rangle$ vanishes. Remember that $u(\mathbf{k})$ are Bloch wavefunctions, which are eigenvectors of the momentum space Hamiltonian $H(\mathbf{k})$ (e.g. $H(\mathbf{k})$ in Kane-Mele model is just a $4\times 4$ matrix at a given $\mathbf{k}$). Essentially they are just some column vectors. You are probably thinking in terms of real-space wavefunctions obtained by putting back the plane wave factor $e^{i\mathbf{k}\cdot\mathbf{r}}$, which makes wavefunctions at different $\mathbf{k}$ orthogonal. But for the Pfaffian invariant you need to use the Bloch wavefunctions.