Majorana Fermions and Orthogonal Complex Structures

TL;DR

This paper presents a new approach to analyzing Majorana fermions using orthogonal complex structures, eliminating the need for Hilbert space doubling and clarifying the connection to topological invariants.

Contribution

It introduces a method that removes Hilbert space redundancy in fermionic quadratic Hamiltonians by leveraging orthogonal complex structures, enhancing physical and mathematical understanding.

Findings

Reduces Hilbert space redundancy in fermionic systems

Establishes a clear link between orthogonal complex structures and topological invariants

Provides a transparent framework for Majorana fermions analysis

Abstract

Ground states of quadratic Hamiltonians for fermionic systems can be characterized in terms of orthogonal complex structures. The standard way in which such Hamiltonians are diagonalized makes use of a certain "doubling" of the Hilbert space. In this work we show that this redundancy in the Hilbert space can be completely lifted if the relevant orthogonal structure is taken into account. Such an approach allows for a treatment of Majorana fermions which is both physically and mathematically transparent. Furthermore, an explicit connection between orthogonal complex structures and the topological $\mathbb Z_2$-invariant is given.