Quantizing Majorana Fermions in a Superconductor

TL;DR

This paper demonstrates that Majorana fermions in superconductors are characterized by real quantum fields, a property that extends beyond zero-energy modes and is rooted in the fundamental symmetries of the system.

Contribution

It shows that the Majorana feature is inherent to the full quantum field in superconductors and provides a generic quantization procedure based on Hamiltonian imaginary transformations.

Findings

Full quantum fields in these systems are real, not just zero modes.

The Hamiltonian can be transformed to an imaginary form, simplifying the equations of motion.

A natural two-dimensional fermion parity-preserving Fock space representation is identified.

Abstract

A Dirac-type matrix equation governs surface excitations in a topological insulator in contact with an s-wave superconductor. The order parameter can be homogenous or vortex valued. In the homogenous case a winding number can be defined whose non-vanishing value signals topological effects. A vortex leads to a static, isolated, zero energy solution. Its mode function is real, and has been called "Majorana." Here we demonstrate that the reality/Majorana feature is not confined to the zero energy mode, but characterizes the full quantum field. In a four-component description a change of basis for the relevant matrices renders the Hamiltonian imaginary and the full, space-time dependent field is real, as is the case for the relativistic Majorana equation in the Majorana matrix representation. More broadly, we show that the Majorana quantization procedure is generic to superconductors, with or without the Dirac structure, and follows from the constraints of fermionic statistics on the symmetries of Bogoliubov-de Gennes Hamiltonians. The Hamiltonian can always be brought to an imaginary form, leading to equations of motion that are real with quantized real field solutions. Also we examine the Fock space realization of the zero mode algebra for the Dirac-type systems. We show that a two-dimensional representation is natural, in which fermion parity is preserved.