Singularities in the Fermi liquid description of a partially filled Landau level and the energy gaps of fractional quantum Hall states

TL;DR

This paper investigates the low energy excitations of a two-dimensional electron system near even denominator Landau level fillings, revealing a divergence in effective mass and a delta-function interaction, which impact the energy gaps of fractional quantum Hall states.

Contribution

It extends Landau's Fermi liquid theory to describe fractional quantum Hall states, showing a logarithmic divergence in effective mass and a delta-function interaction at even denominator fillings.

Findings

Effective mass diverges logarithmically at the Fermi level.

Quasi-particle interaction function approaches a delta function.

Diverging effective mass cancels in linear response functions.

Abstract

We consider a two dimensional electron system in an external magnetic field at and near an even denominator Landau level filling fraction. Using a fermionic Chern--Simons approach we study the description of the system's low energy excitations within an extension of Landau's Fermi liquid theory. We calculate perturbatively the effective mass and the quasi--particle interaction function characterizing this description. We find that at an even denominator filling fraction the fermion's effective mass diverges logarithmically at the Fermi level, and argue that this divergence allows for an {\it exact} calculation of the energy gaps of the fractional quantized Hall states asymptotically approaching these filling fractions. We find that the quasi--particle interaction function approaches a delta function. This singular behavior leads to a cancelation of the diverging effective mass from the long wavelength low frequency linear response functions at even denominator filling fractions.