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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?

MMA13
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    "It is well known that for each symmetry in the system there must be a conserved quantum number" [citation needed] If you are referring to Noether's theorem, that only applies to continuous symmetries. – ACuriousMind Jul 04 '20 at 09:42
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    Noether's theorem works differently for discrete symmetries, but for these spacetime symmetries (e.g. parity, time-reversal), where the group is $\mathbb{Z}_2$, the charge is just $\pm 1$ (although it can be more complicated than this) – 4xion Jul 04 '20 at 16:35
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    here's a relevant phys.SE question: https://physics.stackexchange.com/q/8518/ – 4xion Jul 04 '20 at 16:36

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