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The two-dimensional Wigner crystals are studied with the variational quantumMonte Carlo method. The close relationship between the ground-statewavefunction and the collective excitations in the system is illustrated, andused to guide the construction of the ground-state wavefunction of the stronglycorrelated solid. Exchange, correlation, and magnetic field effects all giverise to distinct physical phenomena. In the absence of any external magneticfield, interesting spin-orderings are observed in the ground-state of theelectron crystal in various two-dimensional lattices. In particular,two-dimensional bipartite lattices are shown not to lead necessarily to anantiferromagnetic ground-state. In the quantum Hall effect regime, a strongmagnetic field introduces new energy and length scales. The magnetic fieldquenches the kinetic energy and poses constraints on how the electrons maycorrelate with each other. Care is taken to ensure the appropriatetranslational properties of the wavefunction when the system is in a uniformmagnetic field. We have examined the exchange, intra-Landau-level correlationas well as Landau-level-mixing effects with various variational wavefunctions.We also determine their dependences on the experimental parameters such as thecarrier effective mass at a modulation-doped semiconductor heterojunction. Ourresults, when combined with some recent calculations for the energy of thefractional quantum Hall liquid including Landau-level-mixing, showquantitatively that in going from $n$-doping to $p$-doping in $GaAS/AlGaAS$heterojunction systems, the crossover filling factor from the fractionalquantum Hall liquid to the Wigner crystal changes from filling factor $\nu \sim1/5$ to $\nu \sim 1/3$. This lends strong support to the claim that theComment: LaTex file, 14 figures available from zhu@physics.rutgers.ed

Variational quantum Monte Carlo study of two-dimensional Wigner crystals: Exchange, correlation, and magnetic-field effects