The fractional quantum Hall effect and the spherical shell model

When a magnetic field B is applied perpendicular (z-direction) to a two dimensional surface with a current in the x-direction, jx, an electric field is induced in the y-direction, Ey. Generally the predicted linear dependence on magnetic field of the Hall transverse resistivity, ρyx = (Ey/jx), is observed. However, at low temperatures there is a plateau in the Hall resistivity when the magnetic field is the inverse of an integer times the electron density. Furthermore the longitudinal resistivity, ρxx = Ex/jx, goes through a minimum at these values of the magnetic field. These phenomena can be explained by taking into account the quantum motion of the electrons in quantized degenerate single particle levels called Landau levels, and they occur when these levels are filled. However these phenomena can occur also for certain fractional fillings of the Landau levels. This fractional quantum Hall effect (FQHE) is thought to be the result of correlations between electrons induced by their mutual Coulomb interactions. Following Haldane, the author maps the problem onto the surface of a sphere, so that the number of states in each Landau level is finite. The author diagonalizes the Coulomb interaction in this spherical shell model and compares the exact wave function, energies, and two-electron density with the Laughlin/Haldane ground state ansatz for fractional fillings 1/m, m an odd integer. While this ansatz accounts for many of the J=0 ground states found in numerical diagonalizations, it does not explain all such states. From numerical and analytical studies, the author suggests a generalization of this ansatz to more general observed fractional fillings