Visualizing Majorana bound states in one and two dimensions using the generalized Majorana polarization

We study the solutions of generic Hamiltonians exhibiting particle-hole mixing. We show that there exists a universal quantity that can describe locally the Majorana nature of a given state. This pseudo-spin-like two-component quantity is in fact a generalization of the Majorana polarization (MP) measure introduced by Sticlet, Bena, and Simon [Phys. Rev. Lett. 108, 096802 (2012)], which was applicable only for some models with specific spin and symmetry properties. We apply this to an open two-dimensional Kitaev system, as well as to a one-dimensional topological wire. We show that the MP characterization is a necessary and sufficient criterion to test whether a state is a Majorana or not, and use it to numerically determine the topological phase diagram.