Microscopic analytical theory of a correlated, two-dimensional N-electron gas in a magnetic field

We present a microscopic, analytical theory describing a confined N-electron gas in two dimensions subject to an external magnetic field. The number of electrons N and strength of the electron-electron interaction can be arbitrarily large, and all Landau levels are included implicitly. For any value of the magnetic field B, the correlated N-electron states are determined by the solution to a universal effective problem which resembles that of a fictitious particle moving in a multi-dimensional space, without a magnetic field, occupied by potential minima corresponding to the classical N-electron equilibrium configurations. Introducing the requirement of total wavefunction antisymmetry selects out the allowed minimum-energy N-electron states. It is shown that low-energy minima can exist at filling factors v = p/(2n + 1) where p and n are any positive integers. These filling factors correspond to the experimentally observed fractional (FQHE) and integer (IQHE) quantum Hall effects. The energy gaps calculated analytically at v = p/3 aie found to be consistent with experimental data as a function of magnetic field, over a range of samples.