We discuss an alternative accurate Monte Carlo method to calculate the ground-state energy and related quantities for Laughlin states of the fractional quantum Hall effect in a disk geometry. This alternative approach allows us to obtain accurate bulk regime ͑thermodynamic limit͒ values for various quantities from Monte Carlo simulations with a small number of particles ͑much smaller than that needed with standard Monte Carlo approaches͒.

We obtain analytical expressions for the total magnetic moment and the static spin-correlation functions of the classical Heisenberg model for ultrasmall systems of spins ͑unit vectors͒, that interact via isotropic, nearest-neighbor ͑n-n͒ exchange and that are subject to a uniform dc magnetic field of arbitrary strength. Explicit results are presented for the dimer, equilateral triangle, square, and regular tetrahedron arrays of spins. These systems provide a useful theoretical framework for calculating the magnetic properties of several recently synthesized molecular magnets. The tetrahedron as well as the equilateral triangle systems, each considered for n-n antiferromagnetic exchange, are of particular interest since they exhibit frustrated spin ordering for sufficiently low temperatures and weak magnetic fields. ͓S0163-1829͑99͒01538-6͔

We investigate broken rotational symmetry ͑BRS͒ states at half-filling of the valence Landau level ͑LL͒. We generalize Rezayi and Read's ͑RR͒ trial wave function, a special case of Jain's composite fermion ͑CF͒ wave functions, to include anisotropic coupling of the flux quanta to electrons, thus generating a nematic order in the underlying CF liquid. Using the Fermi hypernetted-chain method, which readily gives results in the thermodynamic limit, we determine the properties of these states in detail. By using the anisotropic pair distribution and static structure functions we determine the correlation energy and find that, as expected, RR's state is stable in the lowest LL, whereas BRS states may occur at half-filling of higher LL's, with a possible connection to the recently discovered quantum Hall liquid crystals.

We study the ground-state properties of two-dimensional quantum-dot helium in zero external magnetic field (a system of two interacting electrons in a two-dimensional parabolic confinement potential) by using perturbation and variational theory. We introduce a family of ground-state trial wave functions with one, two, and three variational parameters. We compare the perturbation and variational energies with numerically exact diagonalization results and earlier unrestricted Hartree-Fock studies. We find that the three-parameter variational wave function is an excellent representation of the true ground state and argue on how to generalize such a wave function for larger quantum dots with arbitrary numbers of electrons.