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For random interacting Majorana models where the only symmetries are theParity $P$ and the Time-Reversal-Symmetry $T$, various approaches are comparedto construct exact even and odd normalized zero modes $\Gamma$ in finite size,i.e. hermitian operators that commute with the Hamiltonian, that square to theIdentity, 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 moreprecisely 'Local Integrals of Motion' (LIOMs) in the Many-Body-Localized-Phasewhere the pseudo-spins happens to be spatially localized. Odd NormalizedZero-Modes $\Gamma^{odd}$ are popular under the name of 'Majorana Zero Modes'or 'Strong Zero Modes'. Explicit examples for small systems are described indetail. Applications to real-space renormalization procedures based on blockscontaining an odd number of Majorana fermions are also discussed.

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