Majorana orthogonal transformation and Majorana zero modes in free fermionic systems

TL;DR

This paper explores Majorana zero modes in free fermionic systems using Majorana orthogonal transformations, providing new models, analytical methods, and insights into topological superconductors and lattice systems.

Contribution

It introduces a systematic approach to construct models with Majorana zero modes via Majorana orthogonal transformations, including explicit lattice models and examples in 1D and 2D.

Findings

Majorana orthogonal transformation links different fermionic models.

Explicit lattice models for p+ip superconductor with Majorana zero modes.

Systematic construction of models hosting Majorana zero modes from independent Majorana hopping models.

Abstract

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.