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We present a comprehensive study of two of the most experimentally relevantextensions of Kitaev's spinless model of a 1D p-wave superconductor: thoseinvolving (i) longer range hopping and superconductivity and (ii) inhomogeneouspotentials. We commence with a pedagogical review of the spinless model and, asa means of characterizing topological phases exhibited by the systems studiedhere, we introduce bulk topological invariants as well as those derived from anexplicit consideration of boundary modes. In time-reversal invariant systems,we find that the longer range hopping leads to topological phases characterizedby multiple Majorana modes. In particular, we investigate a spin model, whichrespects a duality and maps to a fermionic model with multiple Majorana modes;we highlight the connection between these topological phases and the brokensymmetry phases in the original spin model. In the presence of time-reversalsymmetry breaking terms, we show that the topological phase diagram ischaracterized by an extended gapless regime. For the case of inhomogeneouspotentials, we explore phase diagrams of periodic, quasiperiodic, anddisordered systems. We present a detailed mapping between normal statelocalization properties of such systems and the topological phases of thecorresponding superconducting systems. This powerful tool allows us to leveragethe analyses of Hofstadter's butterfly and the vast literature on Andersonlocalization to the question of Majorana modes in superconducting quasiperiodicand disordered systems, respectively. We briefly touch upon the synergisticeffects that can be expected in cases where long-range hopping and disorder areboth present.

Majorana Fermions in superconducting wires: effects of long-range hopping, broken time-reversal symmetry and potential landscapes