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Properties of a finite fully spin-polarized free homogeneous one-dimensional electron gas

Orion Ciftja

Published Jan 1, 2015• 8 citations
Publisher

Abstract

The homogeneous electron gas model has been quite successful to predict the bulk properties of systems of electrons at various densities. In many occasions, a simplified free homogeneous electron gas model represents a powerful first approximation to a real system. Despite our considerable knowledge on the bulk properties of a homogeneous electron gas, advances in nanoscience and nanotechnology call for a greater effort to understand the opposite limit of small finite systems of electrons with size-dependent properties. In this work, we provide a detailed description of the properties of a finite fully spin-polarized (spinless) free homogeneous one-dimensional electron gas, the simplest of the free homogeneous electron gases. We derive exact analytical results for various quantities such as the one-particle density function, two-particle density function, one-particle density matrix, pair correlation function and energy of finite systems with an arbitrary number of electrons. The results obtained provide a detailed view on how various quantities corresponding to a finite system approach their bulk (thermodynamic limit) value.

DOI
10.1063/1.4907104

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The homogeneous electron gas model has been quite successful to predict the bulk properties of systems of electrons at various densities. In many occasions, a simplified free homogeneous electron gas model represents a powerful first approximation to a real system. Despite our considerable knowledge on the bulk properties of a homogeneous electron gas, advances in nanoscience and nanotechnology call for a greater effort to understand the opposite limit of small finite systems of electrons with size-dependent properties. In this work, we provide a detailed description of the properties of a finite fully spin-polarized (spinless) free homogeneous one-dimensional electron gas, the simplest of the free homogeneous electron gases. We derive exact analytical results for various quantities such as the one-particle density function, two-particle density function, one-particle density matrix, pair correlation function and energy of finite systems with an arbitrary number of electrons. The results obtained provide a detailed view on how various quantities corresponding to a finite system approach their bulk (thermodynamic limit) value.
The selfconsistent high-density expansion for thermodynamic quantities of the quantum spin-polarized fermion lattice gas, derived in the preceding paper, is extended to higher orders. The (1/z)2 terms in the free energy are evaluated by summing two-loop diagrams and the corresponding equation for critical temperature in the classical limit is discussed. Ground state phase diagram of the system for arbitrary electron density is derived in the first and second order of the expansion. Properties of the half-filled, one-dimensional spinless fermion model at T = 0, obtained here in the first three orders in 1/z, are compared with the results of exact calculations and of other approximate methods. This provides a test for the unrenormalized high-density expansion method.
The momentum distributions of a homogeneous electron gas, arising from a simple one-matrix functional for the electron-electron repulsion energy that generalizes the Hartree-Fock and Goedecker-Umrigar approximations, are analyzed in detail. Their properties are found to depend strongly on the exponent \ensuremath{\beta} that enters the functional. Smooth momentum distributions that, together with the corresponding energies per particle, exhibit simple scaling behavior with respect to the electron density \ensuremath{\rho}, are obtained only for $\frac{4}{5}<\ensuremath{\beta}<\frac{4}{3}$ and $\ensuremath{\rho}<~{\ensuremath{\rho}}_{\mathrm{crit}}(\ensuremath{\beta}).$
Scaling arguments show that the exchange-correlation energy density of a spin-unpolarized homogeneous electron gas equals $A{\ensuremath{\omega}}^{4∕3}$, where $\ensuremath{\omega}$ is given by a simple functional of the momentum distribution n(k)n(k)n(k) and AAA is a constant. This functional possesses the property of yielding the true n(k)n(k)n(k) (including the discontinuity at the Fermi momentum, which is not reproduced by any of the presently known density matrix functionals) upon minimization of the energy density eee constrained by the electron density $\ensuremath{\rho}$. At the limit of $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\omega}$ coincides with $\ensuremath{\rho}$ and the dependence of $\ensuremath{\omega}$ on n(k)n(k)n(k), which can be readily recovered from the actual values of eee and n(k)n(k)n(k), is given by an explicit analytical formula. Application of this formalism to general Coulombic systems yields a local version of the density matrix theory, in which the exchange-correlation energy is given by a spatial integral of an expression that depends on the amplitudes of natural orbitals at a given point and the respective occupation numbers.
Accurate treatment of the electronic correlation in inhomogeneous electronic systems, combined with the ability to capture the correlation energy of the homogeneous electron gas, allows to reach high predictive power in the application of density-functional theory. For two-dimensional systems we can achieve this goal by generalizing our previous approximation [Phys. Rev. B 79, 085316 (2009)] to a parameter-free form, which reproduces the correlation energy of the homogeneous gas while preserving the ability to deal with inhomogeneous systems. The resulting functional is shown to be very accurate for finite systems with an arbitrary number of electrons with respect to numerically exact reference data.
Thermodynamic properties of the interacting homogeneous electron gas are calculated using a finite-temperature cumulant Green's function approach over a broad range of densities and temperatures up to the warm dense matter regime $T\ensuremath{\sim}{T}_{F}$, where TF{T}_{F}TF​ is the Fermi degeneracy temperature. These properties can be separated into independent particle and exchange-correlation contributions, and our focus here is on the latter. Our approach is based on the Galitskii-Migdal-Koltun and electron number sum rules from the finite temperature many-body Green's function formalism, together with an extension of the cumulant Green's function to finite temperature. Previously this approach yielded exchange-correlation energies and potentials in good agreement with quantum Monte-Carlo calculations. Here the method is extended for various thermodynamic quantities including the chemical potential, total energy, Helmholtz free-energy, electronic equation of state, specific heat, and isothermal compressibility, which optionally include spin dependence. We find that the exchange-correlation contributions are weakly varying at low temperature but exhibit significant temperature dependence in the WDM regime, as well as a crossover from exchange- to correlation-dominated behavior. In contrast to the T=0+T={0}^{+}T=0+ limit, we also find that renormalization effects are largely but not completely suppressed at finite temperature. Comparisons with other approaches at various levels of approximation are also discussed.
The two-dimensional (2D) homogeneous electron gas (HEG) is a fundamental model in quantum many-body physics. It is important to theoretical and computational studies, where exchange-correlation energies computed in it serve as the foundation for density-functional calculations. It is also of direct relevance to a variety of experimental settings, especially with the rapid recent growth in 2D materials and moir\'e systems. In these experiments, metallic gates are often present, which screen the Coulomb interaction between electrons. The effect of the screening can qualitatively change the behavior of the 2D HEG, and requires accurate many-body computations to capture. In this paper, we perform state-of-the-art diffusion Monte Carlo (DMC) calculations in the 2D HEG subjected to symmetric dual-gate screening. We systematically compute the correlation energy across a range of densities and gate separations for both spin unpolarized and fully polarized systems. A global fit is obtained for the correlation energy, using these data and imposing various limiting behaviors obtained from perturbation analysis. The functional will allow density-functional calculations to be performed for a variety of realistic experimental setups, which can accurately account for the presence of gates. We also investigate how the gate screening affects the bulk modulus, pair correlation function, and the structure factor of the 2D HEG, which can potentially be probed in experiments.