Majorana polarization in disordered quasi-one-dimensional hybrid nanowires

Recent studies have proposed Majorana polarization as a diagnostic tool for identifying topological Majorana bound states (MBS) in engineered $p$-wave heterostructures. In this work, we analyze the behavior of Majorana polarization in two systems: a one-dimensional semiconducting nanowire and a quasi-one-dimensional semiconducting system, both subject to Rashba spin-orbit coupling, proximity-induced superconductivity, and disorder. Our analysis shows that Majorana polarization provides valuable information on the spatial structure of low-energy states, but by itself does not fully differentiate true topological MBS from partially separated Andreev bound states (psABS) or quasi-Majorana modes in realistic disordered systems. While earlier literature proposes that true topological MBS must satisfy the condition ${\mathcal{P}}_{\mathrm{left}}^{*}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathcal{P}}_{\mathrm{right}}\ensuremath{\approx}\ensuremath{-}1$, where ${\mathcal{P}}_{\mathrm{left}}$ and ${\mathcal{P}}_{\mathrm{right}}$ denote the Majorana polarizations in the left and right halves of the wire, it is well-established that two additional criteria must also be met for the identification of topological MBS and their applicability in topological quantum computation: the presence of a topological band gap and the localization of wave functions at the edges. We identify scenarios where the condition ${\mathcal{P}}_{\mathrm{left}}^{*}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathcal{P}}_{\mathrm{right}}\ensuremath{\approx}\ensuremath{-}1$ is satisfied without fulfilling the two additional conditions mentioned above. Our conclusions remain robust irrespective of the chosen definition of Majorana polarization, whether based on the chiral or the particle-hole framework. Taken together, our results place Majorana polarization within a broader set of complementary diagnostics and provide clear guidelines for reliably distinguishing true MBS from trivial modes in disordered systems, with direct relevance for topological quantum computation.