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arXiv:1002.2101 · Physical Review Letters

Phase Diagram of the Low-Density Two-Dimensional Homogeneous Electron Gas

N. D. Drummond, Richard J. Needs

Published Mar 24, 2009• 164 citations
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Abstract

We use quantum Monte Carlo methods to calculate the zero-temperature phase diagram of the two-dimensional homogeneous electron gas. We find a transition from a paramagnetic fluid to an antiferromagnetic triangular Wigner crystal at density parameter r(s)=31(1) a.u. and a transition to a ferromagnetic crystal at r(s)=38(5) a.u. The fully spin-polarized fluid is never stable. We search for, but do not find, the ferromagnetic "hybrid" phase proposed by H. Falakshahi and X. Waintal [Phys. Rev. Lett. 94, 046801 (2005)10.1103/PhysRevLett.94.046801].

DOI
10.1103/physrevlett.102.126402

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We use quantum Monte Carlo methods to calculate the zero-temperature phase diagram of the two-dimensional homogeneous electron gas. We find a transition from a paramagnetic fluid to an antiferromagnetic triangular Wigner crystal at density parameter r(s)=31(1) a.u. and a transition to a ferromagnetic crystal at r(s)=38(5) a.u. The fully spin-polarized fluid is never stable. We search for, but do not find, the ferromagnetic "hybrid" phase proposed by H. Falakshahi and X. Waintal [Phys. Rev. Lett. 94, 046801 (2005)10.1103/PhysRevLett.94.046801].
Variational and diffusion quantum Monte Carlo methods are employed to investigate the zero-temperature phase diagram of the three-dimensional homogeneous electron gas at very low density. Fermi fluid and body-centered cubic Wigner crystal ground-state energies are determined using Slater-Jastrow-backflow and Slater-Jastrow many-body wave functions at different densities and spin polarizations in finite simulation cells. Finite-size errors are removed using twist-averaged boundary conditions and extrapolation of the energy per particle to the thermodynamic limit of infinite system size. Unlike previous studies, our results show that the electron gas undergoes a first-order quantum phase transition directly from a paramagnetic fluid to a body-centered cubic crystal at density parameter rs=86.6(7){r}_{\text{s}}=86.6(7)rs​=86.6(7), with no region of stability for an itinerant ferromagnetic fluid.
Accurate diffusion Monte Carlo technique within the fixed-node approximation has been used to determine the equation of state of the two-dimensional electron gas in three phases: the normal paramagnetic liquid, the ferromagnetic (spin-polarized) liquid, and the Wigner crystal. Our backflow fixed-node calculation, which is a significant improvement over two previous conflicting Slater-Jastrow calculations, shows that there is only one clear phase transition from the paramagnetic liquid to the Wigner crystal as the density decreases. Unlike the three-dimensional case, we do not see the stability of the polarized liquid phase at intermediate densities before the crystallization.
Variational and fixed-node Green's-function Monte Carlo calculations have been performed to find the ground-state properties of the two-dimensional electron gas in the density range 1\ensuremath{\le}rs{r}_{s}rs​\ensuremath{\le}100. Our calculations predict a Wigner crystallization at the density rs{r}_{s}rs​\ensuremath{\simeq}37\ifmmode\pm\else\textpm\fi{}5. The electron system is found to be in the normal- (paramagnetic) fluid state below the transition density, but the fully polarized state is very close in energy. We have tabulated the values of pair distribution function g(r), the static structure factor S(k), and the momentum distribution n(k) at several densities of interest both in the normal and the polarized phases. An estimate of the spin susceptibility \ensuremath{\chi} is also given.
We calculate the ground-state phase diagram of the homogeneous electron gas in two dimensions within the Hartree-Fock approximation. At high density, we find stable solutions, where the electronic charge and spin density form an incommensurate crystal having more crystal sites than electrons, whereas the commensurate Wigner crystal is favored at lower densities, ${r}_{s}\ensuremath{\gtrsim}1.22$. Our explicit calculations demonstrate that the homogeneous Fermi-liquid state, though being an exact stationary solution of the Hartree-Fock equations, is never the Hartree-Fock ground state of the electron gas.
We determine numerically the ground state of the two-dimensional fully polarized electron gas within the Hartree-Fock approximation without imposing any particular symmetries on the solutions. At low electronic densities, the Wigner crystal solution is stable, but for higher densities (rs{r}_{s}rs​ less than $\ensuremath{\sim}2.7$) we obtain a ground state of different symmetry: the charge density forms a triangular lattice with about 11% more sites than electrons. We prove analytically that this conducting state with broken translational symmetry has lower energy than the uniform Fermi-gas state in the high-density region giving rise to a metal to insulator transition.
Motivated by potential realizations in cold-atom or cold-molecule systems, we have performed quantum Monte Carlo simulations of two-component gases of fermions in two dimensions with hard-core interactions. We have determined the gross features of the zero-temperature phase diagram by investigating the relative stabilities of paramagnetic and ferromagnetic fluids and crystals. We have also examined the effect of including a pairwise, long-range ${r}^{\ensuremath{-}3}$ potential between the particles. Our most important conclusion is that there is no region of stability for a ferromagnetic fluid phase, even if the long-range interaction is present. We also present results for the pair-correlation function, static structure factor, and momentum density of two-dimensional hard-core fluids.