  Motivated by the fact that the idealized Kitaev chain toy model and the
experimental semiconductor-superconductor Majorana nanowire can both host the
Majorana zero modes, we theoretically investigate the question to what extent
the two models are equivalent or similar, using the perspective of the
corresponding dual spin models for both. We start with the duality between the
Kitaev chain and the transverse-field XY spin model through the Jordan-Wigner
transformation with the goal of establishing the connection between the Kitaev
chain and the nanowire. By applying the Jordan-Wigner transformation to the
nanowire, we find that the corresponding bosonic spin model is a generalized
spin cluster model, containing 3-spin and 4-spin terms, with staggered
couplings. By projecting out the upper band of the bare semiconductor with
higher energy, we obtain an effective low-energy spinless system from the
spinful nanowire system deep in the topological regime. Finally, we establish
the connection between the Kitaev chain and Majorana nanowire by showing that
the spinless Kitaev chain can be viewed as the first-order approximation of the
spinful Majorana nanowire deep in the topological regime.
