Even and odd normalized zero modes in random interacting Majorana models respecting the Parity $P$ and the Time-Reversal-Symmetry $T$

For random interacting Majorana models where the only symmetries are the Parity $P$ and the Time-Reversal-Symmetry $T$, various approaches are compared to construct exact even and odd normalized zero modes $\Gamma$ in finite size, i.e. hermitian operators that commute with the Hamiltonian, that square to the Identity, and that commute (even) or anticommute (odd) with the Parity $P$. Even Normalized Zero-Modes $\Gamma^{even}$ are well known under the name of 'pseudo-spins' $\tau^z_n$ in the field of Many-Body-Localization or more precisely 'Local Integrals of Motion' (LIOMs) in the Many-Body-Localized-Phase where the pseudo-spins happens to be spatially localized. Odd Normalized Zero-Modes $\Gamma^{odd}$ are popular under the name of 'Majorana Zero Modes' or 'Strong Zero Modes'. Explicit examples for small systems are described in detail. Applications to real-space renormalization procedures based on blocks containing an odd number of Majorana fermions are also discussed.