^{1}

^{*}

^{1}

For the
ground state of the homogeneous electron gas (jellium), it is shown how the
cumulant decomposition of the 2-matrix leads to the cumulant decomposition of
the structure factors S_{a,p}(q) for the antiparallel (a) and parallel
(p) spin pairs and how it simultaneously allows one to derive the momentum
distribution n(k), which is a one-body quantity [Phys. Rev. A 86, 012508 (2012)].
The small-q and large-q behavior of S_{a,p}(q), and
their normalizations are derived and compared with the results of P.
Gori-Giorgi et al. [Physica A 280,
199 (2000) and Phys. Rev. B 61, 7353 (2000)].

This paper deals with the ground state of an extended spin-unpolarized (paramagnetic) homogeneous electron gas (HEG), which is one of the most widely studied systems with correlated electrons [_{s} are the total energy per electron e, from which follows the interaction energy _{s} = 3.9 agrees with the results, obtained in Refs. [

Overhauser considered the pair densities PDs

A side product of this geminal analysis is the following. Usually the partitioning _{0} and the interaction-induced term

The paper is organized as follows. In Section 2, the 2-matrix

The functions mentioned above, i.e.

respectively. In Equation (2.1),

_{1}, and

Table 1. The behavior of the cumulant SFs vs. q with. Note that and. The index “a” means spin-antiparallel, “p” means spin-parallel, d means direct (2-line) diagram, and x means the corresponding exchange (1-line) diagram.. For the Löwdin parameter c see Equation (4.1), the other correlation parameters are, c_{1}, and. In addition to the normalization integrals of the last column, it holds that, see Equation (4.12).

CSF | |||
---|---|---|---|

0 |

Table 2. The behavior of the structure factors (SFs) vs. the momentum transfer q. Note that and . The spin-summed SF is, the magnetic SF is . In addition to the normalization integrals of the last column, it holds that, see Equation (4.12). Comment on: the small-q behavior of is not known so far, the assumption agrees with the Iwamoto expression for the total SF; the normalization of is due to the Pauli principle.

SF | normalization | ||
---|---|---|---|

of the ideal Fermi gas with

It is interesting to note that the plasmon sum rule [

The Feynman diagrams for the Coulomb repulsion in RPA with the partial summation (or screening replacement), the dashed line denotes the bare interaction, the closed loop denotes the particle-hole propagator of Equation (B.3), and the wavy line denotes the effectively screened interaction, where q is the momentum transfer, is the energy or frequency

where

In this paper, an alternative is presented, namely

Note that what is called

composition into the “Hartree-Fock” component_{s}. For example, according to the perturbation theory, _{s}. This is true for k values away from the Fermi edge (i.e.

The cumulant decomposition seems to be more complicated than its conventional counterpart but, on the oth- er hand, it may be considered a more natural partitioning. This is so because, within the many-body perturbation theory, the cumulant SF

As already mentioned above, the spin-summed SF is given by

where

The original definitions of the 1- and 2-body RDMs in terms of the anticommuting creation and annihiliation operators

The short hands

The hermiticity of

holds for the normalization. The corresponding contraction sum rule

describes how the 1-matrix

A way towards a size-extensive contraction is the cumulant decomposition. This means to define the cumulant 2-matrix

This matrix obviously has the spin structure

with the symmetric and antisymmetric 2-body spin functions

Equations (2.10) and (2.11) define

respectively. Here and in the following, the arguments

where

see also Appendix A. A consequence of Equation (2.14) and Equations (A.6)-(A.8) is that the short-range cor- relation determines the asymptotic large-q behavior of the SFs and their normalizations.

Where do the cumulant 2-matrices

following from Equation (2.12). So, the spin-averaged sum

Note the slightly different definitions of the spin-summed quantities

With

and

which can be written as

We move through the chain

with the abbreviations

When going from Equation (3.1) to Equation (3.2), the contour integration

2a denotes the lowest-order renormalized cumulant 2-matrix, 2b denotes the non-divergent cumulant PD or the cumulant SF (Ya- suhara) [46] , (Kimball) [38] , and 2c denotes the non-divergent interaction energy (Macke, Gell-Mann/Brueckner) [23] [24]

The RPA exchange terms corresponding to Figure 2 with 3a corresponding to (following from 2a corresponding to through the exchange replacement, 3b corresponding to or (non- divergent), and 3c corresponding to. As shown by Onsager et al. [54] , already does not diverge (needs no RPA renormalization). For the im- portance of “x” see [40]

qualitative behavior of

The Fourier transform of

[

with

also follows from

at the high-density limit [

The functions and

How does

in agreement with an ansatz of Iwamoto [

In the transitional region between the small and large values of

the ellipsis representing the terms beyond RPA. In the Cartesian space these jump discontinuities cause the Friedel oscillations in

How does

Are there perhaps corrections due to terms beyond RPA? Indeed, the electron-electron coalescence cusp theo- rem

with unknown correlation parameters

For the parallel-spin, SF it is appropriate to write it as

The quantity

The small- and large-q asymptotics of

respectively. The small

Together with

Next,

Its diagonal elements define the cumulant PD (within RPA)

and (after a tedious derivation) its Fourier transform turns out to be equal

Unfortunately this complicated integral is not known so far as an explicit function of

resulting in

with

This is the famous result of Onsager, Mittag, and Stephen [

Within the approximation

which agrees with Equation (3.5) in the corresponding approximation

Furthermore, the on-top curvature

Note, that

with

Next the small-q behavior of

where the first ellipsis stands for higher-order terms [of the order

What contributes to the second term as a function of

and thus (again for

in agreement with the plasmon sum rule and [

Next, the large-q behavior of

perhaps with an additional correlation parameter

which makes

The asymptotics of the SFs are summarized in

The focus of this paper is on the mathematics of the HEG, which on one hand is only a marginal point in the broad realm of correlated systems lacking rigorous solutions, but on the other constitutes an archetype of an extended many-body Fermi system. The small-q and large-q behaviors of the cumulant structure factors

It may be that in the present paper a deeply lying confrontation emerges, namely between 1) perturbation theory, which is linear in the Feynman diagrams, and 2) the cumulant decomposition, for which higher-order reduced density matrices in terms of products of lower-order ones are characteristic.

Going back to the starting point, which rests upon the belief that the geminals and the corresponding weights that diagonalize the cumulant 2-matrix, are the most “natural” 2-body quantities to describe an extended many- body Fermi system, establishes the direction of the future studies. If these quantities are known, then they de- termine the structure factors, the interaction energy

Note added in proof: Equations (4.7) and (4.16),

The authors thank P. Gori-Giorgi in particular for drawing their attention to the Kimball trick, and are grateful to C. Gutlé, and M. Holzmann for valuable discussions and to U. Saalmann for critically reading the manuscript. They acknowledge P. Fulde and the Max Planck Institute for the Physics of Complex Systems Dresden for supporting this work. One of the authors (J. C.) also acknowledges funding from NCN (Poland) under grant DEC-2012/07/B/ST4/00553.