Liquid-solid transition and the fractional quantum-Hall effect

The critical Landau-level filling factor ${\nu }_c$ for the transition from Laughlin's liquid state to a Wigner crystal is determined by comparing the energies of these states. The Wigner-crystal energy is substantially improved over the Hartree-Fock result by using a variational wave function which includes particle correlations. The liquid-state energy is obtained from the Monte Carlo calculation of Levesque, Weis, and Mac-Donald. We find ${\nu }_c$ to be slightly larger than ? which is consistent with the experimental observation by Mendez and co-workers that the fractional quantum-Hall effect does not occur for $\nu <~\frac{1}{7}$. The improvement in the crystal energy by correlation is essential to this agreement since without correlations, ${\nu }_c\sim \frac{1}{10}$. In addition the crystal correlation energy explains the very low temperatures required to see the $\nu =\frac{1}{5}$ liquid state.