Microscopic many-particle wave functions for the fractional quantum Hall effect in periodic geometry

A two-dimensional system of electrons with repulsive two-particle interaction is considered in a homogeneous magnetic field. Assuming the limit of high magnetic field the motion of the electrons is restricted to the lowest Landau level. We present quantum-mechanical wave functions for the many-particle system of N spin-polarized fermions on a parallelogram with periodic boundary conditions and with ${\mathit{N}}_{\mathrm{\Phi }}$ quanta of magnetic flux flowing through its area. These wave functions are generalizations of the Laughlin wave function to more general rational values of the occupation fraction ν=N/${\mathit{N}}_{\mathrm{\Phi }}$. We divide the N electrons into n groups and assume Laughlin-Jastrow-type correlations for electrons within the same group, while electrons in different groups are correlated with a second Jastrow function. The corresponding wave functions exhibit a special structure that depends on ν but is not affected by the interaction between the particles. Furthermore, we consider excited states with quasielectrons and quasiholes. Our theory leads to a unified view of the composite fermion theory of Jain and the hierarchy theory of Haldane, Laughlin, and Halperin. We present a qualitative argument why the fractions of Jain are observed preferably in the experiments.