Majorana orthogonal transformation and Majorana zero modes in free fermionic systems

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.