# About the $Z_2$ topological invariant

In Kitaev 2001 it is shown that the topological invariant $Z_2$ in a topological superconductor (Class D or BDI, one dimensional) can be defined as $$(-1)^\nu={\rm sign\, Pf} [ A ]={\rm sign\, Pf}[ \tilde{A}(k=0)]\cdot {\rm sign\, Pf}[ \tilde{A}(k=\pi)]$$ where $A$ is the Hamiltonian of the system in the Majorana representation, and $\tilde{A}(k)$ is its Fourier transform, and $\rm Pf$ is the Pfaffian. Therefore one needs to calculate the Pfaffian only at the high-symmetry points $k=0$ and $k=\pi$.

How general is this identity? It is possible to generalize the identity to other classes of superconductors (e.g., classes C) and other dimensions (e.g., $d=2$ in class D and DIII)?

Note on class D: in this class the topological invariant is $Z$ for $d=2$ but the Pfaffian invariant $(-1)^\nu={\rm sign\, Pf} [ A ]$ still describes the "parity" of the invariant $Z$.

EDIT: Does one needs some special symmetries, such as, centrosymmetry or inversion parity, or this identity is really general? Regarding the dimensionality, does this identity generalize in higher dimensions simply to: $$(-1)^\nu={\rm sign\, Pf} [ A ]=\prod_{\bf k=-k}{\rm sign\, Pf}[ \tilde{A}(k)]$$ where the product is taken over all $\bf k$ in the $d$-dimensional Brillouin zone such that $\bf k=-k$?