Spinors in non-relativistic Chern-Simons electrodynamics

TL;DR

This paper derives a non-relativistic spinor equation from a higher-dimensional theory, demonstrates its Schrödinger symmetry, and constructs vortex solutions relevant to Chern-Simons gauge interactions.

Contribution

It introduces a lightlike reduction method to obtain a non-relativistic Dirac equation with Schrödinger symmetry and constructs explicit vortex solutions.

Findings

Derived non-relativistic Dirac equation via lightlike reduction.

Proved the system's Schrödinger symmetry.

Constructed static spinor vortex solutions.

Abstract

It is shown that the non-relativistic `Dirac' equation of L\'evy-Leblond, we used recently to describe a spin $1/2$ field interacting non-relativistically with a Chern-Simons gauge field, can be obtained by lightlike reduction from $3+1$ dimensions. This allows us to prove that the system is Schr\"odinger symmetric. A spinor representation of the Schr\"odinger group is presented. Static, self-dual solutions, describing spinor vortices are given and shown to be the non-relativistic limits of the fermionic vortices found by Cho et al. The construction is extended to external harmonic and uniform magnetic fields.