Results for Particular Cases of $\varvec{\upsilon =1/2}$ States of ${\varvec{FQHE}}$ in Disk Geometry

Abstract

In this article, we report on analytic studies in the particular case $\upsilon$ =1/2 of the fractional quantum Hall effect(FQHE). We studied the properties of a 2D electronic gas subjected to a strong magnetic field and cooled at a low temperature. According to composite fermions (CF) theory for a stable system, particular values of the filling of the electronic states lead to more stable ground states with smaller energy. For strongly correlated system, is it possible to see the CF at 1/2. The results that we obtained by analytical calculations, where the representation for certain integrals of products of Meijer G-functions is obtained. We considering two special wave functions of the form of: (a) Bose-Laughlin wave function in Ref. [22] for half-filling and (b) Slater determinant functions in Ref. [23](International Journal of Modern Physics B Vol. 24, No. 18 (2010) 3489-3499), and we extend this results for systems with up to N=10 electrons. We note that none of these two wave functions is related to the real Fermi wave function at filling 1/2 which is the Rezayi-Read wave function in Eq.(2). We compare our analytical results with the method of Monte Carlo of Jain [8], Park [21] as well as of Ciftja [22,23,24] for some electrons and concluded that the different values of energy in good agreement with each other. In the thermodynamic limit, the ground state energy per particle is obtained by an extrapolation of the three energies, the energy of the interaction between electron–electron(e–e), electron-background(e–b), and background-background(b–b).