Unrestricted Hartree-Fock theory of Wigner crystals

Wigner first predicted that a system of electrons in a uniform potential would crystallize at low densities. Localizing electrons around lattice sites increases their kinetic energy, but at sufficiently low densities the reduction in interaction energy is always greater. The Wigner crystal remains a theoretical prediction in three dimensions ~3D!, but in 2D Wigner crystals have been created on a liquid helium surface and at the interface between two semiconductors. It has been suggested that electrons forming a Wigner crystal might eventually be used as quantum bits ~qubits! in a quantum computer. The widely studied model system of electrons in a uniform potential has yielded many insights into electronic many-body phenomena. The most accurate calculations performed to date for the zero-temperature ground-state phases of this system have used the diffusion quantum Monte Carlo ~DMC! method. A Wigner crystal may also be described as a vibrating lattice of electrons. When harmonic phonon vibrations and anharmonic terms are included the resulting energies are very similar to DMC ones. A recent Hartree-Fock study of small numbers of electrons confined by an external potential revealed a transition from a Fermi fluid to a Wigner molecule state. In this paper we also employ the HartreeFock approximation which gives a description of Wigner crystals in terms of Einstein oscillators, but including anharmonic and exchange effects. Within Hartree-Fock theory the paramagnetic ~unpolarized! fluid phase is unstable to the ferromagnetic ~fully polarized! fluid for values of the density parameter rs ~Ref. 11! greater than 2.01 in 2D and greater than 5.45 in 3D. HartreeFock theory also predicts that the paramagnetic fluid is unstable to the formation of a spin density wave. The introduction of electron correlation changes the picture dramatically, with the instability of the paramagnetic to the ferromagnetic fluid being shifted to rs.26 in 2D, while in 3D a second-order transition to a partially polarized fluid is predicted to occur at rs.50. The spin-density-wave instability may be entirely eliminated. DMC calculations also predict the occurrence of Wigner crystal phases for rs.35 ~Refs. 15 and 16! in 2D and rs.65–100 in 3D ~Refs. 7 and 17!. Within a mean-field theory of electron systems the interactions are replaced by a potential which acts on each electron orbital separately. The wave function is then simply a