It is well known that for each symmetry in the system there must be a conserved quantum number, I would like to know what are the conserved quantum numbers associated with inversion and time-reversal symmetries. For example, in the continuum approximation AB-stacked bilayer graphene has the following Hamiltonian:
$H=\left(
\begin{array}{cccc}
0 & v_{F}\pi^{+} & 0 & 0 \\
v_{F}\pi^{-} &0 & \gamma_{1} & 0\\
0 & \gamma_{1} & 0& v_{F}\pi^{+} \\
0 & 0& v_{F}\pi^{-} &0 \\
\end{array}%
\right)$, with $\pi^{\pm}=p_{x}+ip_{y}$, $p_{x,y}=-i\hbar \partial_{x,y}$, and $\gamma_1$ is the interlayer coupling (constant).
This Hamiltonian preserves both symmetries, so what are the associated quantum numbers?
