I try to understand the concept of topological invariants in condensed matter specially in the case of the $\mathbb{Z}_{2}$ invariant and graphene.

From Fradkins book I know that the $\mathbb{Z}_{2}$ invariant can be written as

$$ \left(-1\right)^{\nu} = \prod_{i}\delta_{i} = \prod_{i}\frac{\sqrt{\det\left(w\left(\vec{Q}_{i}\right)\right)}}{\text{pf}\left(w\left(\vec{Q}_{i}\right)\right)} $$

where, $w_{mn}\left(\vec{k}\right) = \langle u_{m}\left(-\vec{k}\right)|\Theta|u_{n}\left(\vec{k}\right)\rangle$, $\Theta$ is time-reversal operator, $u_{n}\left(\vec{k}\right)$ are the Bloch eigenfunctions and $\vec{Q}_{i}$ are the time-reversal invariant points in the Brillouin zone $\vec{Q}_{i} = -\vec{Q}_{i} + \vec{G}$, usually $\vec{Q}_{i}\in\{\left(0,0\right),\left(\pi,0\right),\left(0,\pi\right),\left(\pi,\pi\right)\}$.

My question is now what happens in the case of graphene (i.e. the Kane and Mele model) with the definition of $\vec{Q}_{i}$? In the book by Bernevig the Pfaffian is calculated at the Dirac points $K$ and $K^{\prime}$ (as long as the $C_{3}$ is unbroken). Therefore, in the $\mathbb{Z}_{2}$ invariant should have the following form for graphene:

$$ \left(-1\right)^{\nu} = \frac{\sqrt{\det\left(w\left(K\right)\right)}}{\text{pf}\left(w\left(K\right)\right)}\cdot\frac{\sqrt{\det\left(w\left(K^{\prime}\right)\right)}}{\text{pf}\left(w\left(K^{\prime}\right)\right)}\text{.} $$

Intuitively, it is clear to me why, because at these points the energy gap is closing. However, I also want to understand from a mathematical point of view why we can write the topological invariant as a Pfaffian at specific points in the Brillouin zone.


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