The Energy Density in the Maxwell-Chern-Simons Theory

TL;DR

This paper analyzes the energy density correction in a 2D fermion system coupled to gauge fields, relevant for quantum Hall effect and anyon superconductivity, using RPA and numerical evaluations.

Contribution

It provides a detailed calculation of polarization tensors and energy density corrections in the Maxwell-Chern-Simons theory, highlighting their dependence on particle density and cutoff parameters.

Findings

Energy density correction varies significantly with particle density.

Minimum energy density correction occurs at specific hole concentration.

RPA correction diverges logarithmically as cutoff is removed.

Abstract

A two-dimensional nonrelativistic fermion system coupled to both electromagnetic gauge fields and Chern-Simons gauge fields is analysed. Polarization tensors relevant in the quantum Hall effect and anyon superconductivity are obtained as simple closed integrals and are evaluated numerically for all momenta and frequencies. The correction to the energy density is evaluated in the random phase approximation (RPA), by summing an infinite series of ring diagrams. It is found that the correction has significant dependence on the particle number density. In the context of anyon superconductivity, the energy density relative to the mean field value is minimized at a hole concentration per lattice plaquette (0.05 \sim 0.06) (p_c a/\hbar)^2 where p_c and a are the momentum cutoff and lattice constant, respectively. At the minimum the correction is about -5 % \sim -25 %, depending on the ratio (2m \omega_c)/(p_c^2) where \omega_c is the frequency cutoff. In the Jain-Fradkin-Lopez picture of the fractional quantum Hall effect the RPA correction to the energy density is very large. It diverges logarithmically as the cutoff is removed, implying that corrections beyond RPA become important at large momentum and frequency.