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

Majorana orthogonal transformation and Majorana zero modes in free fermionic systems