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Physical review. B, Condensed matter

Quantum Monte Carlo calculation of the Fermi-liquid parameters in the two-dimensional electron gas

Yongkyung Kwon, David M. Ceperley, Richard M. Martin

Published Jul 15, 1994• 131 citations
Publisher

Abstract

Excitations of the two-dimensional electron gas, including many-body effects, are calculated with a variational Monte Carlo method. Correlated sampling is introduced to calculate small energy differences between different excitations. The usual pair-product (Slater-Jastrow) trial wave function is found to lack certain correlations entirely so that backflow correlation is crucial. From the excitation energies calculated here, we determine Fermi-liquid parameters and related physical quantities such as the effective mass and the Land\'e g factor of the system. Our results for the effective mass are compared with previous analytic calculations.

DOI
10.1103/physrevb.50.1684

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Excitations of the two-dimensional electron gas, including many-body effects, are calculated with a variational Monte Carlo method. Correlated sampling is introduced to calculate small energy differences between different excitations. The usual pair-product (Slater-Jastrow) trial wave function is found to lack certain correlations entirely so that backflow correlation is crucial. From the excitation energies calculated here, we determine Fermi-liquid parameters and related physical quantities such as the effective mass and the Land'e g factor of the system. Our results for the effective mass are compared with previous analytic calculations.
We have used the variational and diffusion quantum Monte Carlo methods to calculate the energy, pair-correlation function, static structure factor, and momentum density of the ground state of the two-dimensional (2D) homogeneous electron gas. We have used highly accurate Slater-Jastrow-backflow trial wave functions and twist averaging to reduce finite-size effects where applicable. We compare our results with others in the literature and construct a local-density-approximation exchange-correlation functional for 2D systems.
We calculate the ground-state energy and the effective spin Land\'e g factor (g∗{\mathit{g}}^{\mathrm{*}}g∗) of a two-dimensional electron gas in strong magnetic fields using a variational Monte Carlo method and many-body trial wave functions composed of the Jastrow correlation factor and the Laughlin-type functions. We find that g∗{\mathit{g}}^{\mathrm{*}}g∗ increases with decreasing of electron density and also shows increasing behavior as the filling factor \ensuremath{\nu} decreases in the region of 12. At \ensuremath{\nu}=1.0, g∗{\mathit{g}}^{\mathrm{*}}g∗ is estimated to be about 11 for a dimensionless parameter rs{\mathit{r}}_{\mathit{s}}rs​=1.33, as compared to the measured value of 7.3, while considering impurity and phonon scatterings gives a better agreement of g∗{\mathit{g}}^{\mathrm{*}}g∗=9. \textcopyright{} 1996 The American Physical Society.
Following previous work in which a variational method for the many-body problem was proposed wherein an effective potential $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ was introduced as a variational parameter to construct a trial wave function, we apply the method to the two-dimensional electron gas. In the method, every physical quantity is expanded in powers of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ instead of the bare potential as in the usual perturbation-theoretic approach. Because of this choice of expansion parameter, the expansion series converges rapidly even in the strong-coupling region. The result for the correlation energy in the two-dimensional electron gas agrees very well with that given by the variational Monte Carlo method even in the lowest-order calculation (i.e., to second order in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$). The difference is within several percent for 1<rs<1001<{r}_{s}<1001<rs​<100.
We present a study of spin-unpolarized and spin-polarized two-dimensional uniform electron liquids using variational and diffusion quantum Monte Carlo (VMC and DMC) methods with Slater-Jastrow-backflow trial wave functions. Ground-state VMC and DMC energies are obtained in the density range 1 ≤ r s ≤ 40 . Single-particle and many-body finite-size errors are corrected using canonical-ensemble twist-averaged boundary conditions and extrapolation of twist-averaged energies to the thermodynamic limit of infinite system size. System-size-dependent errors in Slater-Jastrow-backflow DMC energies caused by partially converged VMC energy minimization calculations are discussed. We find that, for 1 ≤ r s ≤ 5 , optimizing the backflow function at each twist lowers the twist-averaged DMC energy at finite system size. However, nonsystematic system-size-dependent effects remain in the DMC energies, which can be partially removed by extrapolation from multiple finite system sizes to infinite system size. The DMC energies in the thermodynamic limit are used to parametrize a local spin density approximation correlation functional for inhomogeneous electron systems. Our zero-temperature phase diagram shows a single transition from a paramagnetic fluid to a hexagonal Wigner crystal at r s = 35 ( 1 ) , with no region of stability for a ferromagnetic fluid. Published by the American Physical Society 2024
The correlation energy of the homogeneous three-dimensional interacting electron gas is calculated using the variational and fixed-node diffusion Monte Carlo methods, with trial functions that include backflow and three-body correlations. In the high-density regime (rs< 5){(r}_{s}<~5)(rs​< 5) the effects of backflow dominate over those due to three-body correlations, but the relative importance of the latter increases as the density decreases. Since the backflow correlations vary the nodes of the trial function, this leads to improved energies in the fixed-node diffusion Monte Carlo calculations. The effects are comparable to those found for the two-dimensional electron gas, leading to much improved variational energies and fixed-node diffusion energies similar to the released-node energies of Ceperley and Alder.
Fermi liquid theory is the basic paradigm within which we understand the normal behavior of interacting electron systems, but quantitative values for the parameters that occur in this theory are currently unknown in many important cases. One such case is the two-dimensional homogeneous electron gas (2D HEG), which is realized in a wide variety of semiconductor devices. We have used quantum Monte Carlo (QMC) methods to calculate the Landau interaction functions between pairs of quasiparticles. We use these to study the Fermi liquid parameters, finding that finite-size effects represent a serious obstacle to the direct determination of Fermi liquid parameters in QMC calculations. We have used QMC data in the literature for other properties of the 2D HEG to assemble a set of ``best available'' values for the Fermi liquid parameters.