Polarization operator in the 2+1 dimensional quantum electrodynamics with a nonzero fermion density in a constant uniform magnetic field

V R Khalilov,I V Mamsurov

doi:10.1140/epjc/s10052-015-3389-6

Abstract

The polarization operator (tensor) for planar charged fermions in a constant uniform magnetic field is calculated in the one-loop approximation of $$2+1$$ -dimensional quantum electrodynamics (QED $$_{2+1}$$ ) with a nonzero fermion density. We construct the Green function of the Dirac equation with a constant uniform external magnetic field in QED $$_{2+1}$$ at a finite chemical potential, find the imaginary part of this Green function, and then obtain the polarization tensor related to the combined contribution from real particles occupying the finite number of energy levels and magnetic field. We expect that some physical effects under consideration seem likely to be revealed in a monolayer graphene sample in the presence of an external constant uniform magnetic field $$B$$ perpendicular to it.

Highlights

Planar charged fermions governed by the Dirac equation with external electromagnetic fields attract considerable interest in connection with problems of the quantum Hall effect [1], high-temperature superconductivity [2] as well as graphene
We have shown that the one-loop polarization tensor induces physical effects, which seem to be likely to be revealed in a monolayer graphene sample in a strong constant uniform magnetic field aligned perpendicularly to the sample
The polarization tensor contains the contributions from virtual and real charged particles occupying a finite number of the Landau levels

Summary

Introduction

Planar charged fermions governed by the Dirac equation with external electromagnetic fields attract considerable interest in connection with problems of the quantum Hall effect [1], high-temperature superconductivity [2] as well as graphene (see, e.g., [3,4,5,6]). Polarization effects in the massive QED2+1 with a constant uniform magnetic field and with a nonzero fermion density were studied in [20,21]. The contribution of the induced Chern–Simons term to the polarization tensor and the effective Lagrangian with the electron density corresponding to the occupation of n Landau levels in an uniform magnetic field were calculated in [21].

Results

Conclusion