Spin Polarization of Majorana Zero Modes and Topological Quantum Phase Transition in Semiconductor Majorana Nanowires

A number of recent works have discussed the issue of spin polarization of a Majorana zero mode in condensed matter systems. Here we show that the spin polarization density of a Majorana zero mode, computed as an average of the spin operator over its wave function, is identically zero. A single non-degenerate Majorana zero mode, therefore, does not couple to an applied magnetic field, except via hybridization with higher energy excited states (if present), which may perturb its wave function. If spin is defined by considering only the particle components of the wave function, as has been done in some recent works, Majorana zero modes do have a non-zero spatial profile of this quantity, which is measurable in scanning tunneling microscopy (STM) experiments. However, if such a quantity is measured in spin-resolved tunneling experiments (without spatial resolution), we show that it cannot be used as a unique signature of Majorana zero modes in the topologically non-trivial phase. As a byproduct, we show that in spatially inhomogeneous systems (specifically, in systems with a soft boundary), accidental zero energy modes (which for all practical purposes behave as Majorana zero modes) can appear with increasing magnetic field even in the absence of a topological quantum phase transition (TQPT). But only after gap closing and the associated TQPT, the modes are localized near the system edges, resulting in the maximum topological protection. In light of these considerations, demonstrating the nonlocal character of the topologically protected Majorana pair and its emergence after the systems undergo a TQPT, become critical tasks for the ongoing experimental search for Majorana bound states in condensed matter systems.