  We study free fermionic models that host Majorana zero modes using the
Majorana orthogonal transformation, which is a type of transformation between
different fermionic models under Majorana representation of complex fermions.
Using Majorana orthogonal transformation, a U(1) topological gauge theory for
the doubled $p_{x}+ip_{y}$ topological superconductor is obtained; the vortex
Majorana zero modes and the degeneracy splitting of multiple vortices are
studied using field theoretical method. For lattice Majorana hopping models, we
perform real-space analysis on the Majorana zero modes. In one dimension, the
decoupled Su-Schrieffer-Heeger model and the Kitaev chain are discussed as
examples and building blocks for composite models. In two dimensions a simple
lattice model realizing the $p_{x}+ip_{y}$ superconductor is introduced, and
its defect Majorana zero mode is written down explicitly. We introduce a
systematic way to obtain models hosting Majorana zero modes in which composite
models are constructed from two independent Majorana hopping models by Majorana
orthogonal transformations. Three one-dimensional models are proposed and
discussed as examples.
