Majorana Fermions in a Box

TL;DR

This paper investigates the boundary conditions and spectral properties of Majorana fermions confined to finite regions, revealing restrictions on boundary parameters and exploring their quantum mechanical interpretation.

Contribution

It provides a detailed analysis of self-adjoint boundary conditions for Majorana fermions in various dimensions, highlighting differences from Dirac fermions and addressing interpretational issues.

Findings

Majorana boundary parameters are restricted to b1 i in 1D.

The spectrum of confined Majorana fermions is computed.

The study clarifies the quantum interpretation of the Majorana equation.

Abstract

Majorana fermion dynamics may arise at the edge of Kitaev wires or superconductors. Alternatively, it can be engineered by using trapped ions or ultracold atoms in an optical lattice as quantum simulators. This motivates the theoretical study of Majorana fermions confined to a finite volume, whose boundary conditions are characterized by self-adjoint extension parameters. While the boundary conditions for Dirac fermions in $(1+1)$-d are characterized by a 1-parameter family, $\lambda = - \lambda^*$, of self-adjoint extensions, for Majorana fermions $\lambda$ is restricted to $\pm i$. Based on this result, we compute the frequency spectrum of Majorana fermions confined to a 1-d interval. The boundary conditions for Dirac fermions confined to a 3-d region of space are characterized by a 4-parameter family of self-adjoint extensions, which is reduced to two distinct 1-parameter families for Majorana fermions. We also consider the problems related to the quantum mechanical interpretation of the Majorana equation as a single-particle equation. Furthermore, the equation is related to a relativistic Schr\"odinger equation that does not suffer from these problems.