Method of computation of energies in the fractional quantum Hall effect regime

Ammar Ammar,Bekhechi Bekhechi,Bentalha Bentalha

doi:10.5488/cmp.19.33702

Abstract

In a previous work, we reported exact results of energies of the ground state in the fractional quantum Hall effect (FQHE) regime for systems with up to $N_{\text{e}} = 6$ electrons at the filling factor $\nu = 1/3$ by using the method of complex polar coordinates. In this work, we display interesting computational details of the previous calculation and extend the calculation to $N_{\text{e}} = 7$ electrons at $\nu = 1/3$. Moreover, similar exact results are derived at the filling $\nu = 1/5$ for systems with up to $N_{\text{e}} = 6$ electrons. The results that we obtained by analytical calculation are in good agreement with their analogues ones derived by the method of Monte Carlo in a precedent work.

Highlights

The discovery of the fractional quantum Hall effect (FQHE) [1] was the beginning of a big revolution in the field of condensed matter
There are two world wide accepted theories in the field of FQHE, the theory of Laughlin [2] and the theory of Jain [3,4,5,6]. The former describes the ground state as an incompressible quantum fluid which successfully clarified the nature of states at the filling factors ν = 1/3, 1/5, 1/7, . . . . The latter is built upon the concept of composite fermions that are topological entities caricaturing the idea of electrons embracing a number of quantized vortices and gives satisfactory results regarding the ν = p/(2mp + 1) states, integer m and p
In Laughlin theory, the incompressible quantum fluid consists of strongly correlated electrons interacting with a strong magnetic field whereas in Jain theory it consists of weakly correlated composite fermions interacting with a reduced magnetic field

Summary

Introduction

The discovery of the fractional quantum Hall effect (FQHE) [1] was the beginning of a big revolution in the field of condensed matter. There are two world wide accepted theories in the field of FQHE, the theory of Laughlin [2] and the theory of Jain [3,4,5,6] The former describes the ground state as an incompressible quantum fluid which successfully clarified the nature of states at the filling factors ν = 1/3, 1/5, 1/7, . The most pronounced analytical method in this line of research was given by the author of [21], where ordinary polar coordinates (including ordinary Jacobi coordinates) are employed but technical calculational difficulties arise for systems with Ne > 4 electrons, and one can see an explicit dependence of the integrands upon the angles of the particles (see equation (21) in [21]).

Objectives

Results

Conclusion