Homogeneous Electron Liquid in Arbitrary Dimensions Exchange and Correlation Using the Singwi-Tosi-Land-Sj olander Approach L. V. Duc Pham1Pascal Sattler1Miguel A. L. Marques1and Carlos L. Benavides-Riveros2 3

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Homogeneous Electron Liquid in Arbitrary Dimensions:

Exchange and Correlation Using the Singwi-Tosi-Land-Sj¨olander Approach

L. V. Duc Pham,1Pascal Sattler,1Miguel A. L. Marques,1and Carlos L. Benavides-Riveros2, 3, ∗

1Institut f¨ur Physik, Martin-Luther-Universit¨at Halle-Wittenberg, 06120 Halle (Saale), Germany

2Pitaevskii BEC Center, CNR-INO and Dipartimento di Fisica, Universit`a di Trento, I-38123 Trento, Italy

3Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Str. 38, 01187, Dresden, Germany

(Dated: March 3, 2023)

The ground states of the homogeneous electron gas and the homogeneous electron liquid are

cornerstones in quantum physics and chemistry. They are archetypal systems in the regime of

slowly varying densities in which the exchange-correlation energy can be estimated with a myriad of

methods. For high densities, the behaviour the energy is well-known for 1, 2, and 3 dimensions. Here,

we extend this model to arbitrary integer dimensions, and compute its correlation energy beyond

the random phase approximation (RPA), using the celebrated approach developed by Singwi, Tosi,

Land, and Sj¨olander (STLS), which is known to be remarkably accurate in the description of the

full electronic density response for 2Dand 3D, both in the paramagnetic and ferromagnetic ground

states. For higher dimensions, we compare the results obtained for the correlation energy using the

STLS method with the values previously obtained using RPA. We found that at high dimensions

STLS tends to be more physical in the sense that the infamous sum rules are better satisﬁed by the

theory. We furthermore illustrate the importance of the plasmon contribution to STLS theory.

I. INTRODUCTION

The ground states of the homogeneous electron gas

(HEG) and the homogeneous electron liquid (HEL) have

played a prominent role in the modelling and under-

standing of a wide range of interacting electronic sys-

tems [1–12]. These systems are some of the most im-

portant models of choice to develop, improve and bench-

mark many approximate approaches to the full many-

electron problem, including some of the most popu-

lar exchange-correlation functionals of density functional

theory (DFT) [13–16].

An important question about these models is the de-

pendency of the correlation energy, and other physical

quantities, on D, the dimension of the physical space

in which the gas/liquid is embedded. Indeed, there is

a wealth of experimental electronic setups in which one

or two of the physical dimensions are much smaller than

the remaining ones. They can thus be modeled as one-

or two-dimensional quantum systems [17–21]. Further-

more, reduced dimensional systems often exhibit notable

physical properties, ranging from Luttinger physics [22]

to Moir´e superlattices [23]. More recently, progress in

the fabrication of artiﬁcial materials is paving the way

for the realization of non-integer dimensions, as fractal

substrates (e.g., Sierpi´nski carpets of bulk Cu) conﬁning

electron gases [24–26]. The possibility to circumvent the

von Neumann-Wigner theorem or to produce unconven-

tional topological phases by engineering (or mimicking)

additional synthetic dimensions [27–30] also highlights

the importance of realizing, studying and understanding

interacting fermionic and electronic systems embedded in

non-conventional dimensions.

∗cl.benavidesriveros@unitn.it

For the high-density spin-unpolarized HEG in D=

1,2, and 3, the energy per electron can be expanded in

terms of rs, the Wigner-Seitz radius, as follows [31]:

εD(rs→0) = aD

s−bD

+cDln rs+O(r0

s).(1)

The constants aD,bDand cDare independent of rs, and

their functional form in terms of the dimension Dis well

known [8]. Quite remarkably, the logarithm that appears

in Eq. (1) for the 3D case is due to the long range of the

Coulomb repulsion, and cannot be obtained from stan-

dard second-order perturbation theory [32]. In a recent

work [33], the HEG was extended to arbitrary integer

dimensions. It was found a very diﬀerent behavior for

D > 3: the leading term of the correlation energy does

not depend on the logarithm of rs[as in Eq. (1)], but

instead scales polynomially as cD/rγD

s, with the expo-

nent γD= (D−3)/(D−1). In the large-Dlimit, the

value of cDwas found to depend linearly with the di-

mension. This result was originally obtained within the

random-phase approximation (RPA) by summing all the

ring diagrams to inﬁnite order.

While RPA is known to be exact in the limit of the

dense gas, includes long-range interactions automati-

cally, and is applicable to systems where ﬁnite-order

many-body perturbation theories break down [34], it

has well-known deﬁciencies at the metallic (intermedi-

ate) and low densities of the typical HEL (1 ≤rs≤6).

Quantum Monte-Carlo (QMC) is an option for those

regimes [12, 35], but there are other high-quality ap-

proaches such as the celebrated Singwi, Tosi, Land, and

Sj¨olander (STLS) method that provides results compa-

rable to QMC [36]. This method attempts to tackle in

an approximate manner both the exchange and electronic

correlations through a local-ﬁeld correction. As such, this

scheme is often surprisingly accurate in the description

of the full electronic density response and is commonly

arXiv:2210.03024v3 [cond-mat.str-el] 1 Mar 2023

used to investigate quasi-one-dimensional [37], inhomo-

geneneuos [38–40] and warm dense electron liquids [41–

43]. It has also inspired the development of new func-

tional methods that explicitly retain the dynamical and

non-local nature of electronic correlations, while properly

accounting for the exchange contribution [44–48].

The purpose of this paper is to calculate the correla-

tion energy of the HEL for arbitrary integer dimensions

(in particular for D > 3) with the STLS method. As ex-

pected, the value of this portion of the energy improves

signiﬁcantly with respect to the RPA result (that tends

to over-correlate the liquid). The remainder of this paper

is organized as follows. In Section II we review the main

STLS equations and rewrite them explicitly in arbitrary

Ddimensions. As a result, we can compute the Lindhard

polarizability, the structure factor, and the so-called local

ﬁeld correction in the Hartree-Fock approximation, pro-

viding explicit formulae for some representative systems.

In Section III we explain how the correlation energy is

computed in this scheme, and discuss the fully polarized

case. In Sections V and VI we present and discuss our

numerical results for the exchange and correlation ener-

gies, the pair density, and its Fourier transform. We also

discuss the fulﬁllment of the compressibility sum rule.

Finally, in Section VII we present our main conclusions.

Two Appendices that give further technical details on

our calculations are presented at the end of the text.

II. STLS THEORY IN DDIMENSIONS

In this section we review the main STLS equations

and write them explicitly in arbitrary Ddimensions. We

follow the standard notation, namely, nis the D-dimen-

sional particle density, Ω is the volume occupied by the

electronic liquid, and qFis the usual Fermi wavelength.

The STLS theory departs from RPA (to include short-

range correlation between electrons) by writing the two-

particle density distribution f2(r,p,r0,p0, t) as follows:

f2(r,p,r0,p0, t) = f(r,p, t)f(r0,p0, t)g(r−r0),(2)

where f(r,p, t) is the one-particle phase-space density

and g(r) is the equilibrium, static pair distribution func-

tion. Eq. (2) can be seen as an ansatz that terminates

the hierarchy that otherwise would write the two-particle

distribution function in terms of the three-particle distri-

bution function, and so on. This leads to the following

density-density response function [36]:

χD(q, ω) = χ0

D(q, ω)

1−Φ(q)[1 −GD(q)]χ0

D(q, ω).(3)

Here χ0

D(q, ω) is the Lindhard polarizability, i.e., the in-

homogeneous non-interacting density response function

of an ideal Fermi gas in Ddimensions, GD(q) is the local

ﬁeld correction, and Φ(q) is the D-dimensional Fourier

transform of the Coulomb potential:

Φ(q) = (4π)D−1

2ΓD−1

2

qD−1,(4)

where Γ denotes the gamma function. The presence of

GD(q) in Eq. (3) is the key feature of the STLS equations

that gives the “beyond RPA” ﬂavor to the theory. This

local ﬁeld correction is a direct result of the short-range

correlation between the electrons. In arbitrary dimen-

sions, it is given by:

GD(q) = −1

nZq0·qqD−3

q0D−1[SD(q−q0)−1] dDq0

(2π)D,

(5)

with SD(q) being the structure factor. We can simplify

this integral to a two-dimensional one by substituting

q−q0=tand using the fact that SD(q) = SD(q) in

homogeneous systems. Afterwards, we rewrite the inte-

gral using D-dimensional spherical coordinates, where q

is parallel to the Dth-axis, and integrate over all angles

except the angle θbetween qand tto obtain:

GD(q) = −qD−3

(2π)Dn

2πD−1

ΓD−1

2Z∞

0Zπ

[SD(t)−1]×

[q2tD−1−qtDcos θ](sin θ)D−2

(q2+t2−2qt cos θ)D−1

dθdt . (6)

Note in passing that in the 2Dcase we must use po-

lar coordinates instead of the spherical coordinates and

obtain:

G2(q) = −1

(2π)2nZ∞

0Z2π

[S2(t)−1]×

qt −t2cos θ

(q2+t2−2qt cos θ)1/2dθdt . (7)

These equations, together with the equation for the

dielectric function, 1/D(q, ω) = 1 + Φ(q)χD(q, ω), lead

to an equation for the dielectric function within the STLS

theory, namely:

D(q, ω)=1−Φ(q)χ0

D(q, ω)

1 + GD(q)Φ(q)χ0

D(q, ω).(8)

Finally, the relation between the structure factor and

the dielectric function D(q, ω) (see, for instance, [49])

can be easily generalized for arbitrary dimensions:

SD(q) = −1

πnΦ(q)Z∞

Im 1

D(q, ω)dω . (9)

We can improve the readability of this equation by sep-

arating the contributions of the single-particle and plas-

mon excitations [50]:

SD(q) = −1

πnΦ(q)ZqvF+q2

Im 1

D(q, ω)dω

Φ(q)n∂ReD(q, ω)

∂ω −1

δ(ω−ωp(q)) ,(10)

where vFis the Fermi velocity and ωp(q) is the plasmon

dispersion [51]. Equation 10, Eq. 6 and Eq. 8 form the

core of the STLS set of equations. In this framework,

they can be evaluated self-consistently. Notice, indeed,

that GD(q) and SD(q) can be written symbolically as

GD=F1[SD] and SD=F2[GD], indicating the existence

of the mutual functional relations introduced above [47].

We should reiterate here that the crucial aspect of the

STLS method is the appearance of the density-density

pair distribution g(r−r0) in the decoupling of the equa-

tion of motion (2). Despite the crudeness of such a fac-

torization scheme, the formalism gives excellent correla-

tion energies for both 3D and 2D homogeneous electron

gases. Unfortunately, the factorization also results in a

number of well-known shortcomings, including negative

values of g(r−r0) for suﬃciently large values of rs, and

the failure to satisfy the compressibility sum rule (i.e., at

long wavelengths the exact screened density response is

determined by the isothermal compressibility). The ﬁrst

problem, the un-physical behavior of the pair distribu-

tion, is counter-balanced by the fact that the exchange-

correlation energy is in reality an integral over g(r): it

turns out that its value is still quite reasonable in the

metallic range as it beneﬁts from an error cancellation

[43]. As we will see below, this result, known for the

case of 2D and 3D, does also hold for larger dimen-

sions. To tackle the second problem, Vashista and Singwi

[52] provided a correction to the STLS method that

established the correct compressibility sum rule in the

metallic-density regime. This is however a partial solu-

tion as both the original STLS method and the Vashista-

Singwi extension under-estimate the exchange energy, as

pointed out by Sham [53].

A. Real part of the Lindhard function for D= 5,7

The expressions for χ0

2(q, ω) and χ0

3(q, ω) are widely

known since long ago (see for instance [31, 54]), but

higher dimensional expressions are missing in the litera-

ture. In Appendix A we detail their calculation for higher

dimensional settings by performing a linear perturbation

from equilibrium. We sketch here the main results.

The real part of χ0

D(q, ω) can be written explicitly as:

Reχ0

D(q, ω) = 2

(2π)DPZΘ(qF− |p−1

2q|)Θ(|p+1

2q| − qF)

ω−p·qdDp

−2

(2π)DPZΘ(|p−1

2q| − qF)Θ(qF− |p+1

2q|)

ω−p·qdDp,(11)

where Pdenotes the principal value. By evaluating these integrals, it is possible to obtain analytical expressions for

speciﬁc cases. For instance, for D= 5 and D= 7 one gets:

Reχ0

5(q, ω) = q3

8π3(1

96˜q5"3

2˜q2−2˜ω4+ 24˜q4−12˜q2(˜q2−2˜ω)2ln 

2˜q−˜q2+ 2˜ω

2˜q+ ˜q2−2˜ω

+3

2˜q2+ 2˜ω4+ 24˜q4−12˜q2(˜q2+ 2˜ω)2ln 

2˜q−˜q2−2˜ω

2˜q+ ˜q2+ 2˜ω

+ 12˜q7−16˜q5+ 144˜q3˜ω2#−2

(12)

and

Reχ0

7(q, ω) = q5

368640π4˜q5(60(16˜q4+ 3(˜q2+ 2˜ω)4−12(˜q3+ 2˜q˜ω)2)−15(˜q2+ 2˜ω)6

˜q2ln 

2˜q−˜q2−2˜ω

2˜q+ ˜q2+ 2˜ω

+60(16˜q4+ 3(˜q2−2˜ω)4−12(˜q3−2˜q˜ω)2)−15(˜q2−2˜ω)6

˜q2ln 

2˜q−˜q2+ 2˜ω

2˜q+ ˜q2−2˜ω

−4224˜q5+ 1280˜q3(˜q4+ 12˜ω2)−120˜q(˜q8+ 40˜q4˜ω2+ 80˜ω4)).(13)

A similar calculation gives a closed expression for the imaginary part:

Imχ0

D(q, ω) =











h(D)1

˜q1−ν2

−D−1

2−1−ν2

+D−1

2˜ω < ˜q−˜q2

2and ˜q < 2

0 ˜ω < ˜q−˜q2

2and ˜q > 2

h(D)1

˜q1−ν2

−D−1

2˜q−˜q2

2≤˜ω≤˜q+˜q2

0 ˜ω > ˜q+˜q2

(14)

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HomogeneousElectronLiquidinArbitraryDimensions:ExchangeandCorrelationUsingtheSingwi-Tosi-Land-SjolanderApproachL.V.DucPham,1PascalSattler,1MiguelA.L.Marques,1andCarlosL.Benavides-Riveros2,3,1InstitutfurPhysik,Martin-Luther-UniversitatHalle-Wittenberg,06120Halle(Saale),Germany2PitaevskiiBECCenter...

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Homogeneous Electron Liquid in Arbitrary Dimensions Exchange and Correlation Using the Singwi-Tosi-Land-Sj olander Approach L. V. Duc Pham1Pascal Sattler1Miguel A. L. Marques1and Carlos L. Benavides-Riveros2 3.pdf

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Homogeneous Electron Liquid in Arbitrary Dimensions Exchange and Correlation Using the Singwi-Tosi-Land-Sj olander Approach L. V. Duc Pham1Pascal Sattler1Miguel A. L. Marques1and Carlos L. Benavides-Riveros2 3

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