A Mathematical Aspect of A Tunnel-Junction for Spintronic Qubit

TL;DR

This paper explores the mathematical classification of boundary conditions for a Dirac particle in a 1D junction, revealing explicit formulas and parameter reductions relevant for spintronic qubits.

Contribution

It provides explicit formulas linking self-adjoint extensions of the Dirac operator to boundary conditions, classifies these conditions into two types, and refines a four-parameter family into a three-parameter characterization.

Findings

Explicit correspondence between self-adjoint extensions and boundary conditions.

Classification of boundary conditions into two parameter types.

Reduction of Benvegnu and Dabrowski's four-parameter family to three parameters.

Abstract

We consider the Dirac particle living in the 1-dimensional configuration space with a junction for a spintronic qubit. We give concrete formulae explicitly showing the one-to-one correspondence between every self-adjoint extension of the minimal Dirac operator and the boundary condition of the wave functions of the Dirac particle. We then show that the boundary conditions are classified into two types: one of them is characterized by two parameters and the other is by three parameters. Then, we show that Benvegnu and Dabrowski's four-parameter family can actually be characterized by three parameters, concerned with the reflection, penetration, and phase factor.