Spin polarization of Majorana zero modes and topological quantum phase\n transition in semiconductor Majorana nanowires

A number of recent works have discussed the issue of spin polarization of a\nMajorana zero mode in condensed matter systems. Here we show that the spin\npolarization density of a Majorana zero mode, computed as an average of the\nspin operator over its wave function, vanishes everywhere. A single\nnon-degenerate Majorana zero mode, therefore, does not couple to an applied\nmagnetic field, except via hybridization with higher energy excited states (if\npresent), which may perturb its wave function. If `spin' is defined by\nconsidering only the particle components of the wave function, as has been done\nin some recent works, Majorana zero modes do have a non-zero spatial profile of\nthis quantity, measurable in scanning tunneling microscopy (STM) experiments.\nHowever, if such a quantity is measured in spin-resolved tunneling experiments\n(without spatial resolution), we show that it cannot be used as a unique\nsignature of Majorana zero modes in the topologically non-trivial\nsuperconducting phase. As a byproduct, we show that in spatially inhomogeneous\nsystems, accidental zero energy modes, which for all practical purposes behave\nas Majorana zero modes (including giving rise to a zero bias conductance peak\nof height 2e^2/h), can appear with increasing magnetic field even in the\nabsence of a topological quantum phase transition (TQPT). But only after gap\nclosing and the associated TQPT, the modes are localized near the system edges,\nresulting in the maximum topological protection. In the light of these\nconsiderations, demonstrating the nonlocal character of the\ntopologically-protected Majorana pair and its emergence {\\em after} the systems\nundergo a TQPT, become critical tasks for the ongoing experimental search for\nMajorana bound states in condensed matter systems.\n