The fractional quantum Hall effect: What is the ground state of a two-dimensional electron system at the Landau level filling factor $\nu=1/3$?

The fractional quantum Hall effect was experimentally discovered in 1982: It was observed that the Hall conductivity of a two-dimensional electron system is quantized, $\sigma_{yx}=e^2/3h$, at the Landau level filling factor $\nu=1/3$. In 1983, Laughlin proposed a many-body variational wave function, which he claimed described a ``new state of matter'' -- a homogeneous incompressible liquid with fractionally charged quasiparticles. Here we calculate the exact ground state energies and the ground state wave functions of $N$-particle systems of Coulomb interacting two-dimensional electrons, with $N$ up to $N= 7$, on a compensating positive background having the form of a disk. We also calculate the energy and other physical properties of the Laughlin state for a system of $N\le 8$ electrons. We show that, while the properties of the exact and Laughlin states are quite close at $N\lesssim 4$ electrons, at larger $N$ they strongly deviate from each other. In particular, at $N\gg 1$ the density of electrons in the Laughlin state takes the form of a ring, with a vanishingly small density in the center and with a very high density near the edge of the positively charged disk. The projection of the Laughlin state on the exact ground state wave function varies from 95\% and 91\% at $N=3$ and 4 respectively, down to $\sim 38$\% at $N=7$. The true ground state has a substantially lower energy than that of the Laughlin state and has the form of a sliding Wigner crystal.Comment: 26 pages, 11 figure