Topological insulators with time-reversal symmetry: How to derive Z2 invariant from the zeros of the pfaffian?

5 months ago by
Following the book ''Topological Insulators and Topological Superconductors'' by B. Bernevig (esp. chapter 10.1), I want to understand how to derive a Z2 invariant starting from the zeros of the off-diagonal element \(P(k)\) of the matrix

\[<u_i(k)|T|u_j(k)> = -<u_j(k)|T|u_i(k)> = \epsilon_{ij}P(k)\]

which gives the overlap between a state \(|u_i(k)>\) and it's time reversed partner \(T|u_j(k)>\) with \(i=1,2\) the index of one of the two filled bands (minimal required number of filled bands due to Kramer's theorem) and \(T\) with \(T^2 = -1\) the time-reversal operator for spin-1/2 particles. \(P(k)\) is in this case equal to the pfaffianof the matrix.

According to the book, two important subspaces of the Hilbertspace are the so called even and odd subspaces. In an even subspace, the space spanned by \(\{|u_i(k)>\}\) and \(\{T|u_i(k)>\}\) are the same, while in an odd subspace they are orthogonal. Since \(T\) reverses momentum, the eigenfunctions \(|u_i(k^*)>\) at time-reversal-invariant momenta \(k^* = -k^* + G\), with \(G\) a reciprocal lattice vector, will span an even subspace, while for all other \(k\) the above matrix is zero, the spaces spanned by \(|u_i(k)>\) and \(T|u_i(k)>\) therefore orthogonal (they have different momenta) and we have an odd subspace. This is at least what I think.

In the book they talk about special points where \(P(k)\), the off-diagonal matrix-element, is zero. One can then assign a vorticity to these points and derive the Z2 invariant (number of vortices in half of the BZ) from there, since an even number of vortices can always annihilate.

My problem is that as I understand it there should not be some special points where \(P(k) =0\), but \(P(k)\) should be zero everywhere except for TR-invariant points. Can you spot my error? Help is very appreciated!

The mentioned chapter of the book follows the article Z2 Topological Order and the Quantum Spin Hall Effect by Kane and Mele, 2005, by the way
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