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arXiv:cond-mat/0509177 · Physical Review B

On-top pair-correlation function in the homogeneous electron liquid

Zhixin Qian

Published Jan 9, 2006• 10 citations
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Abstract

The ladder theory, in which the Bethe-Goldstone equation for the effective potential between two scattering particles plays a central role, is well known for its satisfactory description of the short-range correlations in the homogeneous electron liquid. By solving exactly the Bethe-Goldstone equation in the limit of large transfer momentum between two scattering particles, we obtain accurate results for the on-top pair-correlation function $g(0)$, in both three dimensions and two dimensions. Furthermore, we prove, in general, that the ladder theory satisfies the cusp condition for the pair-correlation function $g(r)$ at zero distance $r=0$.

DOI
10.1103/physrevb.73.035106

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The ladder theory, in which the Bethe-Goldstone equation for the effective potential between two scattering particles plays a central role, is well known for its satisfactory description of the short-range correlations in the homogeneous electron liquid. By solving exactly the Bethe-Goldstone equation in the limit of large transfer momentum between two scattering particles, we obtain accurate results for the on-top pair-correlation function g(0)g(0)g(0), in both three dimensions and two dimensions. Furthermore, we prove, in general, that the ladder theory satisfies the cusp condition for the pair-correlation function g(r)g(r)g(r) at zero distance r=0r=0r=0.
We have evaluated the density-density response of the two-dimensional electron gas at zero temperature by solving the Dyson equation for the particle-hole Green's function, including exchange Coulomb matrix elements and short-range contributions in the ladder approximation. We study the effect of these correlations on the total energy, compressibility per particle, local field factor G ( q ), static structure factor and pair-correlation function. Results are compared with the normal random-phase approximation, local field theories and quantum Monte Carlo calculations.
The effects of short-range correlations on the properties of the electron gas are studied using an approach phrased in the polarization potential language of Pines and Nozi\`eres. It was argued by Lowy and Brown that two-body terms make up the dominant contribution to the effective interaction for large momentum transfers. In the region of small-momentum transfers it is known that the random-phase approximation provides an adequate picture of the electron gas. With these two regions of large- and small-momentum transfers in mind, a polarization propagator is constructed to interpolate between them. A number of sum rules, among them the fff-sum rule, involving the imaginary part of the polarization propagator, were found to be satisfied. The failure of our theory to satisfy the compressibility sum rule is discussed in detail. We present here the results for the local ttt-matrix, the dynamic and static form factors, and the pair correlation function. That our pair-correlation function remains positive and is essentially the same as in the paper of Lowy and Brown indicates that the physics and not the formalism is responsible for the results we obtain.
In this paper, we study correlations in quasi-one-dimensional electron systems. To account for the short-range correlations we have used a ladder approximation (LA), and to account for the long-range correlations we have used the Singwi, Tosi, S\"olander, and Land (STLS) scheme. After solving the integral equation appearing in the LA and solving the set of self-consistent equations of the STLS scheme, we join the resulting static structure factor in the large- and small-qqq limit smoothly. We then calculate the pair-correlation functions, the ground-state energy, the screened potential, and the plasmon dispersion relation. Our pair-correlation function remains positive everywhere (for rs< 6.5{r}_{s}<~6.5rs​< 6.5) and its value at the origin, g(0),g(0),g(0), is compared with the results of other calculational schemes. The screened potential is repulsive for rs< 2{r}_{s}<~2rs​< 2 and shows a weak attractive part for rs>2.{r}_{s}>2.rs​>2. We have argued that in these systems the sss-wave intrinsic superconductivity is absent. Finally, the quantities calculated in this paper are compared with the results of other calculational schemes.
In this second work in a series of papers the coupled-cluster or exp(S)\mathrm{exp}(S)exp(S) formalism is applied to the problem of ground-state correlations in the one-component Fermi plasma. The main approximation employed is the so-called SUB2 approximation for the two-body subsystem correlation operator S2{S}_{2}S2​ which provides a measure of the two-particle---two-hole component in the true ground-state wave function, and in which the coupling to larger subsystems is neglected. The SUB2 equations for Fermi systems are brought into tractable form by a "state-averaging" procedure which we develop by analogy with the comparable Bose equations, and which is shown by comparison with earlier exact results to be accurate in the metallic density regime at about the 1% level. Our final results for metallic densities include the completely integrated and self-consistent effects of the terms which by themselves generate (i) the random-phase approximation (RPA) and its associated exact long-range screening effects, (ii) the extra random-phase approximation exchange terms necessary to keep the RPA explicitly antisymmetric, (iii) the self-consistent particle-particle ladders (LAD) that describe two-particle scattering within the many-body medium and which describe the exact short-range behavior, (iv) a class of particle-hole ladder terms, and (v) the self-consistent hole-potential terms. Particular attention is paid to the important effects caused by the interference at intermediate separations of the long-range RPA and the short-range LAD effects. By comparison with recent and essentially exact stochastic simulations of the many-body Schr\"odinger equation, our results are seen to be accurate to about 1% for metallic densities, and hence to provide what is probably the currently best microscopic description available for this system. We show also that even in the low-density limit the SUB2 approximation provides a good "translationally-invariant-solid" description of both charged Fermi and Bose systems in this exact Wigner crystal regime.
The theory of an electron gas at high density is generalized so as to include the exchange scattering of a particle and hole in the singlet as well as in the triplet spin state. This approximation is the maximum possible linearization of a quantum-mechanical many-body problem, and corresponds to the theory of small-amplitude oscillations of a classical many-body system. The linearized Hamiltonian is that of Wentzel's meson pair theory, in which now two kinds of mesons are involved, one with spin zero corresponding to the singlet state of a particle and hole and the other with spin one corresponding to the triplet state. The correlation energy is shown to be the sum of the change in zero-point energies of the two-meson fields with correction only in the second-order term. The theory is then formulated in terms of Green's function, and it is proved explicitly that the linearized theory (or the random-phase approximation with exchange effects taken into account) is equivalent to the "ladder approximation" for the particle-hole scattering. The particle-particle scattering by means of the screened interaction is also discussed. The case of the $\ensuremath{\delta}$-function potential is explicitly solved, and it is shown that the correlation energy becomes complex in the region where the instability of the paramagnetic state occurs.