Dynamically screened vertex correction to GW

Y. Pavlyukh∗Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle, Germany

G. StefanucciDipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy

R. van LeeuwenDepartment of Physics, Nanoscience Center, University of Jyväskylä, FI-40014 Jyväskylä, Finland

(Dated: April 14, 2020)Diagrammatic perturbation theory is a powerful tool for the investigation of interacting many-body systems,

the self-energy operator Σ encoding all the variety of scattering processes. In the simplest scenario of correlatedelectrons described by theGW approximation for the electron self-energy, a particle transfers a part of its energyto neutral excitations. Higher-order (in screened Coulomb interactionW ) self-energy diagrams lead to improvedelectron spectral functions (SF) by taking more complicated scattering channels into account and by adding cor-rections to lower order self-energy terms. However, they also may lead to unphysical negative spectral function.The resolution of this difficulty has been demonstrated in our previous works. The main idea is to represent theself-energy operator in a Fermi Golden rule form which leads to the manifestly positive definite SF and allowsfor a very efficient numerical algorithm. So far, the method has only been applied to 3D electron gas, whichis a paradigmatic system, but a rather simple one. Here, we systematically extend the method to 2D includingrealistic systems such as mono and bilayer graphene. We focus on one of the most important vertex functioneffects involving the exchange of two particles in the final state. We demonstrate that it should be evaluated withthe proper screening and discuss its influence on the quasiparticle properties.

I. INTRODUCTION

Numerous correlated electron calculations follow a canoni-cal scheme formulated by Hedin [1] in terms of dressed prop-agators. It is now well established that the lowest-order self-energy (SE) term, the so-called GW approximation is themajor source of electronic correlations. Much less is knownabout the next perturbative orders: there is no single standardway of evaluating them despite the fact that there is a singlesecond-order self-energy diagram (Fig. 1). There are multi-ple reasons for this. On one side, at the advent of many-bodyperturbation theory (MBPT) the computational power was in-sufficient to perform these demanding calculations, and onewas forced to use some drastic simplifications. On the otherside, there are several conceptual problems with the organi-zation of many-body perturbation theory (MBPT) for inter-acting electrons. For instance, it is known that higher-orderdiagrammatic approximations for the electron self-energy interms of the screened Coulomb interaction W leads to polesin the “wrong” part of the complex plane giving rise to nega-tive spectral densities. This observation has been made longtime ago byMinnhagen [2, 3], and in our recent works we pro-vided a general solution to this problem [4, 5] yielding positivedefinite (PSD) spectral functions. The idea was to write theself-energy in the Fermi Golden rule form well known fromthe scattering theory.

One interesting conclusion of our theory is that the second-order SE describes three distinct scattering processes that takeplace in a many-body system [6]: (I) A correction to the first-order scattering, involving the same final states as in GW .

∗ yaroslav.pavlyukh@gmail.com

This effect was numerically studied in Ref. [4], and has beenshown [6] to counteract the smearing out of spectral featuresin self-consistent calculations [7]. (II) Excitation of two plas-mons (pl), or two particle-hole pairs (p-ℎ), or a mixture ofthem in the final state. Especially the generation of two plas-mons is a prominent effect spectroscopically manifested as asecond satellite in the photoemission spectrum [8]. This effectcan be obtained from the cumulant expansion [9, 10], which,however, only works at the band bottom k = 0, or from theLangreth model [11, 12]. (III) A first-order scattering involv-ing the exchange of the two final state particles. This latterscattering process is the focus of the present work.Some manifestations of the mechanism (III) have already

been studied, albeit without realizing its deep connection withthe full Σ(2). First of all, for the two bare interaction lineswe get the so-called second-order exchange, which has beenshown to play an important role in correlated electronic calcu-lations for molecular systems as an ingredient of the second-Born approximation (2BA) [13–15]. Second, it yields a veryimportant total energy correction for the homogeneous elec-tron gas [16]. Third, the mechanism with screening hasbeen considered in the calculations of quasiparticle life-times.Reizer and Wilkins predicted that this diagram yields a 50%

Σ(2)= Γ (1)=

FIG. 1. A single second-order self-energy diagram and the associ-ated first-order vertex function in terms of the electron propagators(arrows) and the screened interactions (wavy lines).

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mailto:yaroslav.pavlyukh@gmail.com

2

reduction of the scattering rate in 2D electron gas calling it “anongolden-rule” contribution, whereas Qian and Vignale [17]correctly pointed out that it is “still described by the Fermigolden rule, provided one recognizes that the initial and fi-nal states are Slater determinants”, and that the coefficient isdifferent. Fourth, the mechanism is relevant for the scatter-ing theory [18]. With bare Coulomb interactions it representsthe so-called double photoemission (DPE) process, and if theinteraction is screened— the plasmon assisted DPE [19, 20].Finally, the considered mechanism has some features in com-mon with the second-order screened exchange (SOSEX) ap-proximation [21, 22]. However, there are also important dif-ferences in the constituent screened Coulomb interaction thatwill be explained below.

As can be seen from this list, the mechanism is an indis-pensable part of various physical processes. However, it hasnot been sufficiently emphasized that all of them can be de-rived from a single Σ(2) diagram. Moreover, there are no sys-tematic studies of its impact on the quasiparticle propertiesother than the life-times. These gaps are filled in here. Ourtheoretical derivations are illustrated by calculations for fourprominent systems: the homogeneous electron gas in two andthree dimensions and the mono- and bilayer graphene. Whilethe former two are very well studied model systems [23, 24],graphene is a real material, and while theGW calculations forit exists [25–27], MBPT has mostly been used in the renormal-ization group sense [28]. Little is known about the frequencydependence of higher-order self-energies.

Our approach consists of analytical and numerical parts. Forthe quasiparticle (qp) electron Green’s function (G0) and thescreened interaction (W0) in the random phase approximation(RPA), the frequency integration of a selected set of the elec-tron self-energy (Σ[G0,W0]) diagrams is performed in closedform using our symbolic algorithm implemented in MATHE-MATICA computer algebra system. The remaining momentumintegrals are performed numerically in line with our previousstudies using the Monte Carlo approach [4–6, 29] showing ex-cellent accuracy and scalability. First, we evaluate the scatter-ing rate function

Γ(k, !) = i[

Σ>c (k, !) − Σ<c (k, !)

]

, (1)and then the retarded self-energy via the Hilbert transform(Appendix A)

ΣR(k, !) = Σx(k) + ∫d!′

2�Γ(

k, !′)

! − !′ + i�, (2)

where Σx(k) is the frequency-independent exchange self-energy, and the meaning of greater> and lesser< componentsof the correlated self-energyΣc is explained in the next section.Via the Dyson equation (Appendix E), the retarded self-energydetermines correlated electronic structure.

Our work is structured as follows: we review our PSD ap-proach in Sec. II and illustrate it with a concrete set of diagramsin Sec. III. Next we discuss the building blocks of our diagram-matic perturbation theory and provide reference G0W0 calcu-lations for the four systems in Sec. IV. Efficient evaluation ofscreening is an important ingredient. In Sec. V we present

(a) (b) (c)

22

��

��

��

2

��

��

��

��

��

�� ����

FIG. 2. Half-diagrams for D(2), the constituent of the Σ ≡ Σ+−) can be treated analogously.The PSD property concerns the fact that the rate operator (1)

must be positive for all momentum k and frequency ! values.Σc with this property is diagrammatically constructed startingfrom any given set of diagrams as follows.

On the first step pluses and minuses are assigned to the di-agram vertices in all possible combinations. They carry in-formation about the contour times. The resulting decorateddiagrams are called partitions. Since we have shown that atzero temperature no isolated + or − islands can exist [4], the“cutting” procedure splits the diagrams for Σ

3

a meaning of the S-matrices describing various particle orhole scattering processes in a many-body system. The result-ing PSD self-energies have then the Fermi Golden rule form,which always leads to positive scattering rates. Topologicallydistinct S-matrices will be denoted as D diagrams. Diagramsthat can be obtained by the cutting procedure applied to Σc ofthe first- and second-order inW0 are depicted in Fig. 2.They are interpreted according to the standard diagram-matic rules. Consider for instance the half-diagrams withall the time-arguments on the + branch (such as depicted inFig. 2). In addition to the initial one-hole (1ℎ) state (with thecoordinate 2, where the composite position-spin and time vari-ables are abbreviated as i ≡ (xi, ti)), the final state is denotedby the two strings of numbers p = (p1,…pN ,pN+1) andq = (q1,…qN ) that specify composite coordinates of theoutgoing N + 1 holes and N particles, respectively. We fur-ther associate a single time-argument � with (p,q). � is thelatest time on the forward and the earliest time on the backwardcontour branches. With these notations, D(a) reads

D(a)p1,p2,q1 (2) = −(−1)1∫ d(4)W

++0 (4, 2)g

4

ig(q1�, 3), and that the lesser screened

interaction can be written in the formW

5

kk

W++(q2)W––(q1)

k1

k2

k3

k2

k1

k3

Σaa

=-<

ΣSOSEX

=v(q1)

W(q2)

(a)

(b)

FIG. 4. (a) Gluing two D(a) half-diagrams with one permutation ofthe two hole lines with momenta k1 = k − q1 and k2 = k − q2 and asingle hole line k3 = k−q1−q2 (dashed) yieldsΣ

6

0 1 2 3 4y

0

1

2�

(y,0)/

N0

0 2 4 6 8 10ξ

0

1

2

3

0 2 4 6 8 10ξ

0

1

2

3

4

-π/2Im

[1/ε 0R (

y,ξ)]ξ

-π/2Im

[1/ε 0R (

y,ξ)]ξ

MLGα=0.875

BLGrs=3.32

HEG 3D HEG 2D

BLG

MLGq=1.2kF

q=0.6 kF

q=1.8kF q=0.6kF

q=1.2kF

q=1.8kF

FIG. 6. Static irreducible polarization (q, 0) for the four studiedsystems normalized at the density of states at the Fermi levelN0 (a).N0 is given by ��(�F) for HEG and by ��,s(�F) for the two graphenesystems, y = q∕kF and � = !∕�F. Integrand of the f -sum rule forMLG (b) and BLG (c) demonstrating the divergence of the f -sum inthese systems.

the irreducible polarization (q, !) and to the density-densityresponse �(q, !),

�(q, !) = (q, !) + (q, !)v(q)�(q, !). (23)They determine the microscopic dielectric function and its in-verse, respectively,

"(q, !) = 1 − gv(q)(q, !), (24)"(q, !)−1 = 1 + gv(q)�(q, !). (25)

Here g is the degeneracy factor. For the homogeneous electrongas there is only spin degeneracy, g = gs = 2, whereas for themono- and bilayer graphene the valley degeneracy additionallyappears g = gsgv. For these systems gv = 2, but it can takelarger values for other systems [34, 35]. The density of statesat the Fermi energy N0 is a natural unit to measure and � ,because the static polarization (q, 0) for small values of q isexactly given by this quantity, Fig. 6(a).

The random phase approximation for the inverse dielec-tric function, Im "0(q, !)−1 is a very important ingredient ofthe subsequent correlated calculations because this gives (upto the Coulomb prefactor v(q)) the spectral function of thescreened interaction (Appendix B). A general overview of thisquantity is shown in Fig. 7. It is very fortunate that for all fourstudied systems it can be found in analytic form facilitatingnumerical calculations. Below, we collect all needed formu-las and additionally present the exchange self-energy, whichenters Eq. (2).

In the following we express the electron density n, whichis the central control parameter, in SI units in order to make aconnectionwith experiment. All other quantities are expressedin atomic units. Some simplification of formulas is possible toachieve by rescaling momenta and energies by the Fermi mo-mentum kF and energy �F, respectively. This will be explicitlyindicated.

A. 2D HEG

This is probably the best studied many-body system [24,36]. There is only one relevant parameter— the Wigner-Seitz

0 0.2 0.4 0.6 0.8 1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

6

7

HEG 3Drs 4.ωp 1.881

0 0.2 0.4 0.6 0.8 1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.00

2

4

6

8

10

HEG 2Drs 2.

0 0.2 0.4 0.6 0.8 1.0

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

2.0

2.5

MLG 2Dκ 2.5α 0.88

0 0.1 0.2 0.3 0.4 0.5

0.0 0.5 1.0 1.5 2.0 2.50

1

2

3

4

5

6

BLG 2Drs 3.32

ω/� F

-Imε-1(q,ω) -Imε-1(q,ω)

-Imε-1(q,ω)-Imε-1(q,ω)

q/kF q/kF

q/kFq/kF

ω/� F

ω/� F

ω/� F

FIG. 7. Imaginary part of the inverse dielectric function for the fourstudied systems. Red lines indicate the collective plasmonic mode.Blue lines separate different domains in the definition of polarizabil-ity such as shown in Fig. 5 and indicate removable singularities.

radius rs. It is given in terms of electronic density n as follows:

aBrs =( 1�n

)1∕2. (26)

In the case of systems with an effective electron mass m0 anda background dielectric constant � = 4�""0, one can redefinethe Bohr radius as

ãB =�ℏ2

m0e2(27)

and still have the same relation between the density and rs.The Coulomb potential v(q), the Fermi momentum kF and thedensity of states at the Fermi energy ��(�F) in atomic unitsread

v(q) = 2�q, kF =

1�2rs

, ��(�F) =12�, (28)

where we additionally defined the constant

�2 = 1∕√

2. (29)Introducing scaled variables

y = q∕kF, � = !∕�F, (30)

7

the dielectric function "R0 (k, !) ≡ "R0 (y, �) reads [37]

Re "R0 (y, �) = 1 +2�2rsy2

[

y + fr(�∕y − y) − fr(�∕y + y)]

,

(31)Im "R0 (y, �) =

2�2rsy2

[

fi(�∕y − y) − fi(�∕y + y)]

, (32)

withfr(z) = sign(z) �(1∕4z2 − 1)

√

1∕4z2 − 1,

fi(z) = �(1 − 1∕4z2)√

1 − 1∕4z2.

For small momentum values, the expression in brackets ofEq. (31) suffers from the precision loss. Therefore, in this limitthe approximate formula

Re "R0 (y, �) = 1 −qy�2, q = 4�2rs. (33)

should be used entailing the small-momenta plasmon disper-sion Ω(y) ≈ √yq. It can also be found analytically (seeEq. 5.54 of Ref. [38]):

Ω(y) =

√

y(2y + q)√

y4 + y3q + q2

q√

y + q. (34)

The critical wave-vector does not have a nice analytical ex-pression. However, one can show that kc ∼

√

q. Notice thateven though Ω(y) > y(y + 2) for y > kc there is no plas-mon above the critical vector because in reality the plasmonbecomes damped by entering the continuum, where the abovesolution is not valid.

The f -sum rule reads in rescaled units

−2� ∫

∞

0d� � Im

[

1"R(y, �)

]

= �2r3sy. (35)

The exchange part of the electron self-energy,

Σx(k) = −∫|q| 1.(38)

B. 3D HEG

This system also depends on a single parameter— theWigner-Seitz radius

aBrs =( 34�n

)1∕3. (39)

It has also been broadly studied [23]. The Coulomb potentialv(q), the Fermi momentum kF and the density of states at theFermi energy ��(�F) read in atomic units

v(q) = 4�q2, kF =

1�3rs

, ��(�F) =1

2�2�3rs, (40)

where the relevant constant is defined as

�3 =( 49�

)1∕3. (41)

The dielectric function is (the Lindhard result)Re "R0 (y, �) = 1 +

�3rs�y3

[

2y + fr(�∕y − y) − fr(�∕y + y)]

,

(42)Im "R0 (y, �) =

�3rsy3

[

fi(�∕y − y) − fi(�∕y + y)]

, (43)

withfr(z) = (1 − 1∕4z2) log[(z + 2)∕(z − 2)],

fi(z) = �(1∕4z2 − 1)(1 − 1∕4z2).

Notice a strong resemblance between the dielectric function in2D and 3D. This is due to the fact that upper and lower con-tinuum frequencies are the relevant parameters in both cases(Fig. 5). The shape of continuum is more complicated forMLG and BLG. However, we will see below that they like-wise enter expressions for "R0 (y, �).The f -sum rule is particularly simple in 3D systems. Thisis due to the form of the Coulomb interaction proportional toq−2 (40). Rescaling the frequency and momentum in the usualway (30) we get

−2� ∫

∞

0d� � Im

[

1"R(y, �)

]

= !2p, (44)

with the classical plasmon frequency (�F units)

!p = 4√

�3rs3�

. (45)

The exchange part of the electron self-energy reads

Σx(k) = −∫|q|

8

by the k ⋅ p equation ̂0F(r) = �F(r) [34, 40], where theHamiltonian readŝ0 = vF

(

0 p̂x − ip̂yp̂x + ip̂y 0

)

= vF(�xp̂x + �yp̂y), (49)

with p̂ = (p̂x, p̂y) being the momentum operator and vF theFermi velocity (can be expressed in terms of the hopping inte-gral and the lattice constant [41], the typically adopted valueis 106m∕s = 1∕2.188 a.u.). The wave-function is then

Fs,k(r) = |s,k⟩1Leik⋅r , |s,k⟩ = 1

√

2

(

e−i�ks

)

, (50)

where L2 is the area of the system, and kx = k cos �k, ky =k sin �k, k = |k|. The corresponding energy dispersion reads

�(k) = svFk, (51)which is different from previous cases in two important ways:i) the well-known linear momentum-dependence and ii) thepresence of two bands indicated by the band index s = ±1 and,as a consequence, the presence of additional matrix elementsin the Coulomb operator (Fig. 8)V̂ = 1

2L2∑

q,k1,k2

∑

s1,s2,s′1,s′2

⟨s′1,k1 + q|s1,k1⟩⟨s2,k2 − q|s′2,k2⟩

×∑

�,�′

2��qĉ†s′1,k1+q,�

ĉ†s2,k2−q,�′ ĉs′2,k2,�′ ĉs1,k1,� . (52)

Thus, basis functions are labeled by the momentum k, bandindex s and spin �. In view of the dispersion (51), the non-interacting GF is diagonal in s and �. We are interested inthe electron self-energy diagonal in the band indices. Further-more, the calculations are typically performed at finite doping(extrinsic graphene) and with dielectric function modified bythe presence of substrate. We will focus on the SiO2 substrate� = (1 + "SiO2 )∕2 = 2.45, consider the case of the electrondoping, i.e., that the Fermi level is above the Dirac point, andfollow the notations from the previous sections that

� = 4�""0, (53)with "0 being the vacuum electric permittivity. The Fermi mo-mentum and energy depend on the square root of the electrondensity

kF = aB√

4�ng

= 1⟨r⟩

, �F = vFkF, (54)

g = gsgv where gs = 2 is the spin and gv = 2 is the val-ley degeneracy, respectively. ⟨r⟩ is the averaged inter-electrondistance.

The parameter characterizing the level of correlations in thesystem is given by the ratio of the Coulomb EC and the ki-netic EK energies, as is therefore the counterpart of rs for thehomogeneous electron gas

� =ECEK

= 1� ⟨r⟩

1vFkF

= 1�vF

≃ 2.188�

, (55)

(a) (b)s1, k1, σ

s2ʹ, k2, σʹs2, k2–q, σʹ

s1, k1+q, σ

v(q)

s, k, σ

s,ʹ k+q, σ

FIG. 8. Diagrammatic representation of (a) the Coulomb interac-tion (52) and (b) the polarization bubble inMLG. The latter illustratesthe appearance of the spin gs and band gv degeneracy prefactors in thedielectric function, Eqs. (56,58). The overlap matrix elements (62)are represented by shaded vertices.

for instance, � = 2.188 for MLG in vacuum and � = 0.875 forthe SiO2 substrate. The dielectric function has been computedby Hwang and Das Sarma [42] and by Wunsch et al. [43]. Wewill use the latter form:

Re "R0 (y, �) = 1 +g�y+ g�f (y, �)Gr(y, �), (56)

Im "R0 (y, �) = g�f (y, �)Gi(y, �), (57)f (y, �) = 1

8y

√

|�2 − y2|, (58)

Functions Gr and Gi are defined in Appendix C. The densityof states at the Fermi level reads��,s = (2�vF)−1. (59)

The static polarizability normalized at this number is plotted inFig 6(a). Due to the presence of infinite sea of electrons belowthe Dirac point, the f -sum rule diverges as demonstrated byHwang, Throckmorton, and Das Sarma [44], the integrand ofthe f -sum is illustrated in Fig. 6(b).Due to this fact, a momentum cut-off kc needs to be intro-duced for the momentum integrals. In realistic system this is

not a problem because of the bands flattening due to lattice ef-fects [45]. For the idealistic model that we consider here, kcis an explicit parameter of the theory. We adopt

kc = yckF = 10kF. (60)The exchange self-energy can be written in the form

Σx,s(k) = −∑

s′=±1∫

d2q(2�)2

nF,s′ (k − q)2��qFs,s′ (k,k − q).

(61)In this equation, Fs1,s2 (k1,k2) takes into account the probabil-ity for an electron with momentum k1 in the band s1 to scatterinto the state with momentum k2 in the band s2. It depends onthe relative angle �12 between the two momenta,

⟨s2,k2|s1,k1⟩ =12 (1 + s1s2e

i�12 ), (62)Fs1,s2 (k1,k2) = |⟨s2,k2|s1,k1⟩|

2 = 12 (1 + s1s2 cos �12).(63)

9

Using the Fermi energy and momentum units, dividing intointrinsic (present in pristine graphene) and extrinsic (due tocarriers injection by doping or gating) contributions, shiftingby the constant so that the self-energy is zero at the Dirac point(Σx,±1(0) = 0) we obtain for Eq. (61)

Σ̄x,s(y) = Σ̄intx,s(y) + Σ̄extx,s(y) +

�2(1 + yc), (64)

withΣ̄intx,s(y) = −

�yc�

[

�2− sg

(

yyc

)]

, (65)

Σ̄extx,s(y) = −��

[

f2D(y) + sℎ(y)]

. (66)The function f2D(y) has already been defined for 2DHEG (38). The former intrinsic part results from the integra-tion over the s′ = −1 band from zero to the momentum cut-offkc = yckF. Functions ℎ(y) and g(y) have a representation (cor-recting g(y) in the original derivation by Hwang, Hu and DasSarma [46]):

ℎ(y) = y

⎧

⎪

⎪

⎨

⎪

⎪

⎩

�4 log

4ye1∕2 − ∫

y

0

dxx3

[

K(x) − E(x) − �x2

4

]

y ≤ 1,

∫

1∕y

0dx [K(x) − E(x)] y > 1;

(67)g(x) = 1

4 ∫

1

0dy∫

2�

0d�

x − y cos �√

x2 + y2 − 2xy cos �

= �2Re

{

3F2(

− 12 ,12 ,12 ; 1,

32 ;

1x2

)}

. (68)

D. 2D BLG

Consider now two parabolic energy bands�(k) = sk2∕(2m0). (69)

Unlike MLG, the dispersion is an idealization of several mate-rials with different number of valleys gv and with large flexi-bility in the properties control with the help of doping and thebackground dielectric constant. Here we focus on the gv = 2case pertinent to the bilayer graphene (a minimal two-bandmodel for the Bernal AB stacking [28]). We have the follow-ing relations determining the Fermi energy and momentum,and the Wigner-Seitz radius [47]kFaB

=

√

4�ngsgv

, �F =k2F2m0

, rs =gm0�kF

, ��,s(�F) =m02�.

(70)One may also define the Wigner-Seitz radius as the ratio oftwo energies

r̃s =ECEK

= 1�⟨r⟩

2m0k2F

=m0�

gsgvaB(4�n)1∕2

, aB⟨r⟩ =( 1�n

)1∕2.

Sensarma, Hwang and Das Sarma [47] derived the polariz-ability of this system separating the intra(inter)-band contribu-tions Π = Π1 + Π2. The former originates from the intraband(s = s′) and the latter from interband (s = −s′) transitions,Fig. 8(b),

Π1(y, �) =1� ∫

1

0dx∫

�

−�d� x

� + i� − 2xy cos(�) − y2

×[

1 −y2 sin2(�)

x2 + 2xy cos(�) + y2

]

, (71)

Π2(y, �) = −1� ∫

1

0dx∫

�

−�d� x

� + i� + 2x2 + 2xy cos(�) + y2

×y2 sin2(�)

x2 + 2xy cos(�) + y2. (72)

The retarded polarizability is given byRe(y, �) = ReΠ(y, �) + ReΠ(y,−�), (73)Im(y, �) = ImΠ(y, �) − ImΠ(y,−�). (74)

Π1 is fully defined in the paper [47]. There are, however, somemisprints in the extrinsic part that are corrected here in Ap-pendix D. The dielectric function is given by

"R0 (y, �) = 1 −rsy(y, �). (75)

Th static polarizability was derived by Hwang and DasSarma [48] and is plotted here for comparison with other sys-tems in Fig. 6(a). The f -sum rule diverges for this system forthe same reasons as for MLG. The integrand of the f -sum isillustrated in Fig. 6(c). The exchange self-energy is the sameas for MLG (66).

E. G0W0 calculations

Before presenting calculations with vertex functions, weoverview the electron self-energy in the simplest G0W0 ap-proximation, Eq. (14). The self-energy is depicted in Fig. 9,the respective electron spectral function

A(k, !) = −2 ImGR(k, !) (76)is shown in Fig. 10.a. HEG systems have a long history of studies:

Lundqvist [23], Hedin [1] and self-consistent GW calcu-lations by von Barth and Holm [7, 49] in 3D and Giulianiand Quinn [50], Santoro and Giuliani [24], Zhang and DasSarma [51], Lischner et al. [52] in 2D. Because of the wayhow 2D HEG is engineered (its properties can be tuned bydoping, the electron concentration), it is easy to go to stronglycorrelated regime and still have a homogeneous system.Therefore, correlations beyond G0W0 have been includedalmost from the beginning. Thus, Santoro and Giulianiincluded the many-body local fields G± in the calculation ofscreening and employed the plasmon-pole approximation.This yields the self-energy resembling the 3D case and resultsin a more pronounced plasmon peak as compared to theG0W0calculations.

10

-10 -5 0 5 10-1

0

1

2

-10 -5 0 5 10-2

0

2

4

6

-10 -5 0 5 10-1

0

1

2

-10 -5 0 5 10-1

0

1

2

(ω–μ)/εF (ω–μ)/εF

(ω–μ)/εF (ω–μ)/εF

ΣR(k

F,ω)/

ε FΣR

(kF,ω

)/ε F

HEG 2Drs=(πn)

–1/2/aB=2HEG 3Drs=(4/3πn)

–1/3/aBrs=4

MLG 2Dα=(κ vF/v0)

–1 α=0.875

BLG 2D

rs= =3.32κkFaB

gs gvm0_____

FIG. 9. Solution of the Dyson equation for the Fermi level k = kF.Intersection of straight line y = ! − �F − Δ� and y = ReΣR(kF, !)(thick black curve) yields the real part of quasiparticle energy. Thechemical potential shiftΔ� is selected (see Appendix E) as to have theimaginary part zero in accordance with the Fermi liquid assumption.Red curves stand for 1∕2Γ(kF, !) = − ImΣR(kF, !). In the case ofMLG and BLG thick/thin curves correspond to ΣRs=±1(kF, !).

b. Graphene There are two peculiarities in the case ofMLG and BLG systems. (i) The electron dispersion and self-energies additionally carry the valley index s resulting in thefollowing modification of Eq. (14):

Σ

11

-2 0 2 4 6 8-5

-4

-3

-2

-1

0

0.500.851.001.252.002.65

(ω-μ)/ωpl

½Γ(k,ω)

/ϵF

-5 0 5 10 15

-0.5

0.0

0.5

(ω-μ)/ωpl

½Γ(k,ω)

/ϵF

k=kF

=α2rs2( )3 2π2log(2) 3ζ(3)–Σ2xϵF

FIG. 11. Σ2x at different momentum values. The inset shows thereal self-energy part (solid line) obtained from the Hilbert transformof Γaā(kF, !). Its on-shell value (�2x(kF, �) = 0.204976) has only2.4% deviation from the indicated analytic result (0.210073).

is a product of the four wave-function overlaps,Fs0,s1s2s3 (k0;k1,k2,k3) = ⟨s0,k0|s1,k1⟩⟨s1,k1|s3,k3⟩

× ⟨s3,k3|s2,k2⟩⟨s2,k2|s0,k0⟩ =18

{

1 +∑

i

12

-1.5 -1 -0.5 0

½iΣ

< (k,ω)

/ϵF

(ω-μ)/ωpl

rs=4

k=0.85 kF

k=0.95 kF

k=1.05 kF

k=0.70 kF

forward scattering

backward scattering

q1 q2

k

k1=k-q1k2=k-k2

k3=k-q1-q2

q1 q2k

k1=k-q1k2=k-q2

k3=k-q1-q2

FIG. 13. Σ2x at different momentum values resolved with respectof forward (red) and backward (blue) scattering mechanisms. For ahole in the center of the Fermi sphere k ≪ kF, backward scatteringdominates, while for a hole close to the Fermi surface k ≈ kF forwardscattering with a small momentum transfer gives rise to a sharp peak.

guarantee that k1,2 ≤ kF and k3 > kF the interaction momentaq1,2 must be small and almost collinear with k. As a result wehave a “hot-spot” in themomentum space where all the permit-ted configurations contribute in a very narrow energy intervalgiving rise to a pronounced forward peak. If the initial stateis closer to the center of Fermi sphere, there are less restric-tions on the possible scattering angles. Therefore, the forwardpeak broadens, and for larger energy transfers the backwardscattering dominates. It is interesting to notice that the mixedmechanism, i. e., where one hole is in the forward and anotherin the backward direction, has a rather small contribution andtakes place at intermediate energies.

From our analysis follows that small momentum transfersq1,2 are important for the appearance of the forward peak. Inthis regime the Coulomb interaction is screened by plasmonssuggesting that the inclusion of screeningmay reduce the peak.Therefore, we performed three calculations for k = 0.85kFwith i) bare Coulomb lines, ii) with only plasmon screen-ing, and iii) using the fully screened RPA W0. They indeeddemonstrate that the plasmonic contribution to Σaā is essen-tial for compensating the singularity in the bare Coulomb term(Fig. 14). Notice that they both appear with the same sign be-cause the interactions enter quadratically in the expression forΣaā. As a result, a smooth frequency dependence free of anysingularities is obtained for the sum of all contributions.

C. Cancellations between Σaa (only p-ℎ excitations) and−iΣ>2x for large momentum values plotted on the logarithmic scale.The latter quantity is negative and therefore is multiplied with −1.A scheme on the right illustrates momentum configuration of p →p + p + ℎ scattering at large energy and momentum transfer, i. e. inthe asymptotic regime ! ≫ k2∕2≫ �F. In this limit k3 ≪ k and canbe integrated over.

would be evident. It is convenient to perform derivations usingscaled variables xi = ki∕kF, yi = qi∕kF, � = !∕�F.a. 3D HEG The G0W0 self-energy scales in the high-frequency limit [29] as

i2Σ

>aa(x, �) ≡

i2Σ

>aa(k∕kF, !∕�F)∕�F

�→∞⟶ c1

�23r2s

�3∕2, (80)

c1 =16√

23�

. (81)

Conversely, this determines the short-time behavior of theelectron GF. Unexpectedly, Vogt et al. [62] have demonstratedthat the second-order exchange Σ2x asymptotically scales inthe same way, but with an additional −1∕2 prefactor. This re-sult can be further generalized and derived as follows.The generalization concerns the fact that in the high fre-

quency limit, the screening is not important and the screenedinteractions in the expression for Σaā can be replaced with the

13

bare Coulomb, i. e.,W −−(q1, ! − �1) → v(q1),W ++(q2, ! −�2) → v(q2). This means that Σaā asymptitocally behaves asΣ2x. Using the momenta flow as in Fig. 3(a), the second-orderexchange reads

i2Σ

>2x(x, �) = −

�23r2s

�3 ∬d3y1y21

d3y2y22

n̄F(x1)n̄F(x2)nF(x3)

× �(� − �1 − �2 + �3). (82)In the asymptotic case � ≫ x2∕2≫ 1 we have x3 ≪ x1,2. Wechange the variables

y1,2 =12x ± z = x2,1

and integrate over x3 within the Fermi sphere (from thescheme in Fig. 15 it is evident that to a good approximationthe integrand is independent of x3) yielding the 4�∕3 prefac-tor to the following remaining integral

i2Σ

>2x(x, �) ≈ −

4�23r2s

3�2 ∫d3z

|

|

|

12x − z

|

|

|

2|

|

|

12x + z

|

|

|

2�(

� − 12x2 − 2z2

)

= −8�23r

2s

3�

tanh−1(

x√

2�−x2�

)

�x

= −8√

23�

�23r2s

�3∕2+ (x2). (83)

In the last step, we exploit the high-frequency assumption� ≫ x2∕2 and perform a series expansion over x to get theconjectured scaling iΣ>2x(x, �)

�→∞⟶ c2�23r

2s�−3∕2. The scaling

is verified numerically in Fig. 15 confirming that the constantc2 computed from Eq. (83) is momentum-independent. It isimportant, however, that c2 = − 12c1 ensuring the PSD prop-erty in the high-frequency limit.

b. 2D HEG The derivation follows the same line:i2Σ

>2x(x, �) ≈ −2∫

d2z|

|

|

12x − z

|

|

|

|

|

|

12x + z

|

|

|

�(

� − 12x2 − 2z2

)

= − 4|� − x2|

K

(

x√

x2 − 2�� − x2

)

= −2��+ (x2), (84)

whereK is a complete elliptic integral. As in the case of the 3DHEG, there is a universal (momentum-independent) asymp-totic scaling, see Fig. 16(a), and the ratio of the prefactors isthe same. This is yet another exact analytical statement aboutthe second-order self-energy.

c. Graphene systems The situation is much more com-plex in the case of graphene, Fig. 16(b,c). Besides the usualscattering processes considered above, Σaā contains processesin which particles change the band, Fig. 17. For instance, fork > kF our calculations indicate that Σ2x(k, !) is dominated

100 101 102

10-2

10-1

100

-60 -40 -20 0 20 40 60-0.5

0.0

0.5

1.0

-60 -40 -20 0 20 40 60-0.4

-0.3

-0.2

-0.1

0.0

0.1

3.03.54.04.55.01/ω

rs=2

Σaaph

Σ2xHEG 2D

MLG 2D BLG 2D2.03.04.05.0

(ω-μ)/ϵF

(ω-μ)/ϵF (ω-μ)/ϵF

2.03.04.05.0

α=0.875 rs=3.32

½Γ(k,ω)/ϵ

F½Γ(k,ω)/ϵ

F

(a)

(b) (c)Σ2x

Σ2x

FIG. 16. Top: asymptotic behavior of the rate functions iΣ>aa (onlyp-ℎ excitations) and −iΣ>2x for large momentum values plotted on thelogarithmic scale for 2D HEG. As in the 3D HEG, they posses thec1,2∕! scaling, respectively, with c2 = −1∕2c1. In panels (b,c) therate function of the two graphene systems is plotted for different mo-mentum values. There is no universal scaling.

-100 -50 0 50 100-0.4

-0.3

-0.2

-0.1

0.0

0.1

(ω-μ)/ϵF

½Γ(k,ω)

/ϵF

BLG 2Drs=3.32, k=3kF

Σ2x

ϵFk

k3

k2

k1

ϵFk

k3

k2k1

(a)

(b)

FIG. 17. Second-order exchange in the BLG system for k = 3kF.Besides common scattering channels, the ! → ∞ behavior is dom-inated by a processes (blue line) with one hole in the lower band,s3 = −1 (inset b), for ! → −∞ a mechanism with 2 holes in thelower band s1 = s2 = −1 is dominating (inset a).

by the process with s3 = −s1 = −s2 = 1 for ! < 0, and with−s3 = s1 = s2 = 1 for ! > 0, see Eq. (78). This leads to thescattering rates that do not tend to zero as !→ ±∞. They arecut-off dependent and should be treated as in the case of thefirst-order exchange.

14

-5 0 5 10 150.0

0.2

0.4

0.6

00

123456

-6 -4 -2 0 2 4 6 8 100.0

0.5

1.0

1.5

00

1246810

-3 -2 -1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

00

86421

-3 -2 -1 0 1 2 30.0

0.2

0.4

0.6

0.8

0

0.512358

(ω-μ)/ϵF (ω-μ)/ϵF

(ω-μ)/ϵF (ω-μ)/ϵF

A(k

F,ω)ϵ

FA

(kF,ω

)ϵF

HEG 2D HEG 3D

MLG 2D BLG 2D

rs rs

κ rs

FIG. 18. Electron spectral functions for quasiparticle states at theFermi surface (k = kF) for different interaction strengths. Dashedlines stand for theG0W0 results, full lines—additionally include Σaā.

D. Quasiparticle properties

All four considered systems possess very distinct spectralfunctions. In the vicinity of the quasiparticle peak they can berepresented in the Lorentzian form

A(k, !) = Zqp1∕�(k)

(! − �qp(k))2 + 1∕(2�(k))2, (85)

where the peak position �qp(k), the inverse life-time 1∕�(k),and the quasiparticle renormalization factor Zqp can be deter-mined by solving the Dyson equation with the given retardedself-energy operator (see Sec. 13.1 in Ref. [30]). In general,one has to take care whether the spectral density can really bewritten in this form with finite Zqp. For instance, this mightbe not the case in undoped graphene [63, 64], or 2D systemswith short-range repulsive interactions [65], but is the case forthe systems considered here. As can be seen from Fig. 18,the second-order self-energy has a rather small impact on theshape of quasiparticle peak and its satellites. Therefore, in or-der to quantify the effect we compute the quasiparticle peakstrength:

Zqp(k) =(

1 − ))!

ReΣR(k, !)|||!=�qp(k)

)−1. (86)

We further characterize the quasiparticle dispersion in termsof the effective mass:

1m∗

= 1kF

d�qp(k)dk

|

|

|

|

|k=kF, (87)

and the Fermi velocity (for MLG)

v∗F =d�qp(k)dk

|

|

|

|

|k=kF. (88)

0 2 4 6 8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Z qp

rs rs rs

G+G+ & G–VMC

GW †

scGW †

GW ‡

HEG 2D HEG 3D BLG 2D

G+G+ & G–SJ-VMCBF-RMCGWΓ

FIG. 19. Quasiparticle renormalization factor for k = kF and differ-ent interaction strengths: ΣG0W0—dashed, ΣG0W0 + Σaā—solid lines.For 2D HEG, the variational Monte-Carlo (VMC) results are takenfrom Ref. [66], and the VMC calculations using Slater-Jastrow (SJ)wave-functions and backflow (BF) reptation Monte-Carlo (RMC) for3D HEG from Ref. [67]. G+(q) and G+(q)&G−(q) denote methodsbased on the local structure factors from Ref. [68] (2D) and Ref. [69](3D). GW Γ is a scheme D of Ref. [71], it includes the first-ordervertex function in the finite temperature formalism. For BLG, †—Ref. [57], ‡— Ref. [72].

Finally, the inverse quasiparticle life-time is computed,�(k)−1 = Zqp(k)Γ(k, �qp(k)), (89)

12Γ(k, !) = − ImΣ

R(k, !). (90)We are mostly interested in the correlated regime rs ≫ 1.However, the asymptotic results rs → 0 are also shown whenavailable in order to demonstrate that they are valid in a rather

very narrow density interval. Our main comparison is withthree classes of theories. As benchmarks for the homogeneouselectron gas, the quantum Monte-Carlo results of Holzmannet al. [66] (2D) and [67] (3D) are used. The second classof methods has been advocated by Giuliani and co-workers:Ref. [68] (2D) and Ref. [69] (3D). They improve upon G0W0by using parameterized data from QMC calculations in termsof the charge and the spin static local fields factors, G+(q) andG−(q), respectively. In their method, the self-energy is in theGW form, however, the screened interaction is replaced bythe Kukkonen-Overhauser effective interaction [70]. Further-more, a diagrammatic approach based on the Bethe-Salpeterequation for the improved screened interaction byKutepov andKotliar [71] is also used for comparison. Unfortunately, noneof these theories are available for graphene systems.We start by compiling the data for homogeneous electron

gases in 2D and 3D and the bilayer graphene. In these systemsrs is the relevant parameter that can be controlled by doping orother means. In MLG, there is only an indirect possibility tocontrol � by changing the background dielectric constant and�. This system will be considered later.a. rs as a control parameter In Fig. 19 the quasiparti-cle renormalization factorZqp(kF) as a function of the densityparameter rs is shown. As expected, the agreement between

15

different methods deteriorates with increasing rs, moreoverthe quantum Monte-Carlo results are not available for BLG.Therefore, it is hard to say with absolute certainty what is the“right” value. On the positive side, we see a very nice conver-gence of all methods towards the linear asymptote

Zqp = 1 −(12+ 1�

)

�2rs, 2D (91)Zqp = 1 −

c�2�3rs, 3D. (92)

with c = − ∫ �∕20 log(1 − x cot x) dx ≈ 3.353. The 3D re-sult is by Daniel and Vosko [73], and the 2D asymptote to-gether with temperature corrections is due to Galitski and DasSarma [74]. For BLG, the effect of Σaā is negligible, and ourcalculations accurately reproduce the corrected results of Sen-sarma et al. [72], whereas the one-shot and the self-consistentGW calculations of Sabashvili et al. [57] deviate. For HEGin 2D and 3D, the inclusion of Σaā slightly increases the valueof Zqp. It is interesting to notice that the same trend is ob-served when the charge local field factor G+(q) is included.For HEG 2D, the additional inclusion of both local fields re-duces Zqp in agreement with variational MC calculations. Atvariance, for HEG 3D the effect of the spin local field is lesspronounced [69], and is nearly the same as in our calculationswithΣaā. The effect of the vertex function in Ref. [71] is rathersmall, therefore, it would be interesting if these calculationscould be extended towards larger rs, where even variationaland backflow reptation MC results are in disagreement.

Less accurate are the predictions of different theories forthe effective mass, Fig. 20. The situation gets complicateddue to different methods of its determination adopted in lit-erature [51, 68]. In order to avoid any ambiguities, the massesin our approach are obtained per definition, that is by solvingthe Dyson equation for �qp(k) and using Eq. (87), and not byusing the self-energy representation

1m∗

=Z−1qp

1 + 1kFdReΣR(k,�F)

dk|

|

|k=kF

. (93)

For weakly interacting systems, rs → 0 (high-density limit),we compare with asymptotic expansions. A general form [51]valid for 2D and 3D homogeneous electron gases reads

m∗ = 1 + ars(b + log rs). (94)The coefficients a and b in three dimensions can be inferredfrom the well-known result of Gell-Mann [76] for the specificheat. The correction in the linear temperature-dependent termdue to the electron-electron interaction is entirely attributed tothe mass renormalization [77], and therefore

m∗ = 1 + 12�3rs�

(

log�3rs�

+ 2)

. (95)In two dimensions, the original derivation is due to Janak [78],whereas the corrected formula 2 can be found in Saraga and

2 There has been some controversies in this derivation. For instance, some

0 2 4 6 8 10

1.0

1.2

1.4

1.6

1.8

0 2 4 6 8 100.9

1.0

1.1

1.2

0 2 4 6 8 100.7

0.8

0.9

1.0

m*/m

rs rs rs

VMCDMC

G+G+ & G– G+ & G–

GWΓ

G+

GW †

scGW †

GW ‡

HEG 2D HEG 3D BLG 2D

FIG. 20. Effective mass for different interaction strengths: ΣG0W0—dashed, ΣG0W0 + Σaā—solid lines. For 2D HEG, the variationalMonte-Carlo (VMC) results are taken from Ref. [66], and the DMCcalculations fromRef. [75]. G+(q) andG+(q)&G−(q) denotemethodsbased on the local structure factors from Ref. [68] (2D) and Ref. [69](3D). GW Γ is a scheme D of Ref. [71], it includes the first-ordervertex function in the finite temperature formalism. For BLG, †—Ref. [57], ‡— Ref. [72].

Loss [81]

m∗ = 1 +�2rs�

(

log�2rs2

+ 2)

. (96)

The asymptotic expressions are derived with the help of ad-ditional approximations (e. g. static screening), which quicklyinvalidates them as rs increases, see dotted lines in Fig. 20.Let us now inspect the influence of Σaā on the effectivemass, that is the difference between the full and the dashedlines. One impressive observation is that for 3D HEG, ourcalculations agree again very well with results of Simion andGiuliani [69], where both local field factors are taken into ac-count. The charge local field alone tends to underestimate theeffective mass for both systems. A rather poor performance ofthe Monte-Carlo methods is evident for the 2D HEG as well,further calculations of effective masses and extensive compar-isons can be found in Drummond and Needs [75]. However,the difficulties to extract excited state properties from thesemethods are understandable, and the work of Eich, Holzmannand Vignale [82] provides some justification.For BLG, the mass renormalization substantially deviate in

comparison with Sensarma et al. [72]. This might be due to adifferent procedure based on Eq. (93) adopted in this work.Large and negative mass renormalization indicates that forrs ≫ 1 the system goes into a correlated regime.b. � as a control parameter Let us recapitulate first that

we focus on the extrisicmonolayer graphene system here, thatis kF > 0. This is essentially a classical Fermi liquid in themarked contrast to the more complicated intrinsic graphene,

mistakes in the original result were pointed out Refs. [79, 80], but not ex-plicitly corrected; wrong coefficients a and b can be seen in Refs. [51, 74].

16

1 10

0.1

1

Λ=3Λ=10Λ=30Λ=23†

1 100.1

1MLG 2D

MLG 2D

κ κ

(1–Z

qp)/Z q

p

(v* F–

v F)/v F

1 +2

α

−α 5

3+ ln +

4ln

kckF

αα

FIG. 21. Renormalized Fermi velocity and the quasiparticle renor-malization factor of the monolayer graphene for different values ofthe momentum cut-offΛ = kc∕kF as a function of dielectric constant.The log-log plots are compared with the scalings from Ref. [83] (dot-ted lines). †—data from Ref. [56].

kF = 0. For the latter, we refer to a comprehensive summaryby Tang et al. [84]. While many conceptual problems do notarise in the extrinsic case, some important insight can be ob-tained from the intrinsic graphene. Consider for instance theexpression for the Fermi velocity renormalization derived in[83] and plotted in Fig. 21 (left) as dotted lines

v∗F − vFvF

= −��

(53+ log �

)

+ �4log

(

kckF

)

. (97)Here, the first part is extrinsic. It describes scattering pro-cesses in which the initial and the final state belong to the sameband s = s′ = 1 and contains no adjustable parameters. Thesecond part is intrinsic, it includes scattering processes chang-ing the band s = −s′ = 1 and therefore depends on the mo-mentum cut-off. In going to higher perturbative orders, suchas including Σaā, more and more processes involve interbandscatterings and the role of intraband scattering is diminishing(as explicitly demonstrated for BLG, Fig. 17).

Let us inspect the qualitative dependence of the renormal-ized velocity on the interaction parameter � ≃ 2.2∕� (55). Athigher �, the dependence deviates from linear. This is alreadyevident from the extrinsic part in Eq. (97). The intrinsic partshows a similar trend when computed beyond the leading or-der [84]. By consistently including other terms, one can im-prove the agreement of asymptotic theory with our numericalresults.

Generally, it is believed that G0W0 result are already veryaccurate [56]. However, higher-order diagrams have beentreated in Ref. [64], quantum Monte-Carlo calculations wereperformed in Ref. [84]. All of them concern with intrinsiccase, which is still relevant to some extent as stated above,however cannot be used for a direct comparison. They predicta slightly larger velocity renormalization, whereas, we observehere that the inclusion of Σaā leads to smaller values, Fig. 21.

c. Quasiparticle life-time 1∕�(k) is an essential ingredi-ent of the quasiparticle spectral function (85). We determine it

0 0.5 1 1.5 20

0.04

0.08

0.12

0.16

0 0.5 1 1.5 20

0.04

0.08

0.12

0.16

0 0.5 1 1.5 20

0.01

0.02

0.03

0.04

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

k/kF k/kF

1/ϵFτ

(k)

1/ϵFτ

(k)

HEG 2Drs=2

HEG 3Drs=4

MLG 2D α=0.875kc/kF=10

BLG 2Drs=3.32kc/kF=10

2 80

0.08

rs

1/ϵFτ(0)

2 100

0.05

rs

1/ϵFτ(0)

FIG. 22. Inverse quasiparticle life-time (89) as a function of mo-mentum. In the two insets, the values at the band bottom �(k = 0)−1is shown for different rs. Red lines additionally show the on-shellapproximation (98) without (dashed) and with (full) Σaā.

by solving the Dyson equation in the complex frequency plane.Also, it can be obtained from the Fermi Golden rule as advo-cated by Qian and Vignale [17]. From the discussion in Sec. IIwe know that the two are completely equivalent.In Fig. 22 we summarize our finding for �(k) computed for

0 < k < 2kF. In many studies, the “on-shell” imaginary self-energy is taken as a measure of the inverse life-time�(k)−1 = − ImΣR(k, �(k)), (98)

where �(k) is the bare dispersion relation. Apart from miss-ing the Zqp prefactor, this approach is reasonable for k ≈ kF,where the difference between the “true” �qp(k) and the bare�(k) spectrum is small. For �k = |�(k)−�F|∕�F ≫ 1 the differ-ence between Eq. (89) and the on-shell approximation (98) issubstantial, as can be seen by comparing black and red lines inFig. 22. We find, for instance for MLG, that the approximationincorrectly yields vanishing scattering rates at the Dirac point(k = 0). One consequence of this is the diverging inelasticmean free path at zero temperature predicted in Ref. [85]. Onthe other hand, Eq. (89) yields a finite value. It is worth not-ing that for the two HEG systems the correction upon the on-shell value is mostly associated with the quasiparticle strengthZqp renormalization, whereas, for graphene systems (MLGand BLG) the deviation of �qp(k) from �(k) also plays a role.Only for the two HEG systems the impact of Σaā is appre-ciable as depicted in the insets of Fig. 22 for �(0)−1 for differ-ent rs values. The dependence is not always monotonic. For�k ≪ 1, we can compare with the asymptotic expressions fromRef. [38]

1�(k)

= 14��2(rs)�2k log

(

4�k

)

, 2D; (99)1�(k)

= �8�3(rs)�2k, 3D. (100)

They have shown that the inclusion of exchange modifies thedensity-dependent prefactors �n without affecting the func-

17

0 0.5 1 1.5 2-0.6

-0.4

-0.2

0.0

0.2

0 0.5 1 1.5 2-0.6

-0.4

-0.2

0.0

0.2

0 2 4 6 8 10-0.5

-0.4

-0.3

-0.2

-0.1

0

0 2 4 6 8 10-0.5

-0.4

-0.3

-0.2

-0.1

0

1/ϵFτ

(k)

1/ϵFτ

(k)

τ(ex)

(kF)

–1/τ(d

) (kF)

–1

τ(ex)

(kF)

–1/τ(d

) (kF)

–1

rs rs

k/kF k/kF

τ(d)/τ(ex)τ(d)/τ(ex)HEG 3Drs=4

HEG 2Drs=2

Qian and Vignale (2005) Qian and Vignale (2005)

HEG 2D HEG 3D

FIG. 23. Top row: determination of the ratio � (ex)(k)−1∕� (d)(k)−1 forthe homogeneous electron gas in 2D and 3D illustrating numericaldifficulties of taking the k → kF limit. Bottom row: comparison withanalytical results of Ref. [17]: � (ex)n (rs)∕� (d)n (rs), see Eq. (101,102).

tional form. However, the fitting of small values with this formis not a trivial task because of (i) numerical issues and (ii) ourinsufficient knowledge of the subleading terms3. We followQian and Vignale [17], where the coefficients �n(rs) are de-rived:

�2(rs) =

[

1 + 12

�22r2s

(1 + �2rs)2

]

−[

14+ 12

�2rs1 + �2rs

]

, (101)

�3(rs) =12�

[

tan−1 � + �1 + �2

]

− 12�

[

1√

2 + �2cot−1 1

�√

2 + �2

]

, (102)

with � = �1∕2∕(�3rs)1∕2. Here, the first brackets originatefrom the direct (d) processes and the second— from the ex-change (ex). In Fig. 23, we determine the ratio of exchange todirect scattering rates using the on-shell approximation (98)with Σaā and ΣG0W0 , respectively. In 3D, the agreement withthe analytical results (99) is very good, whereas, in 2D wefind that the ratio is smaller. One possible explanation of thisdiscrepancy could be the absence of the plasmonic contribu-tions to the direct scattering in Ref. [17]. Besides ratios, wealso compared absolute values with analytical predictions andfound systematic underestimates. However, this should not bea surprise because the theory only yields the leading terms.

For MLG, the inverse life-time follows the same asymp-totic with famous logarithmic correction [50] as for HEG 2D,Eq. (99) [85]. Polini and Vignale provided a very pedagogicalderivation of this fact [88], however, exchange contributionswere not included. Our simulations show that they are indeed

3 In 2D, there are disagreements in the subleading terms [17, 86, 87].

1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4

0

1

2

3

4

5

ω/ϵF

ImΣ(1.6k

F,ω)/

ϵ F

κ=0.5

κ=1.0

κ=2.5

total

plasmon

p-h pair

FIG. 24. Imaginary self-energy part of MLG for different interac-tion strengths. The on-shell value is only weakly dependent on � inagreement with the analytical result of Polini and Vignale [88].

small for MLG and BLG systems, Fig. 22. One interestingconclusion of Ref. [88] is that the scatterings are dominated bythe “collinear scattering singularity”, that is, the momenta ofelectrons involved in the scattering are mostly parallel to eachother. We find it interesting because of its apparent similar-ities with the scattering processes shaping Σ2x, see our anal-ysis in Sec. VB. Another interesting conclusion of the ana-lytical formula is that the life-time is independent of the di-electric constant. While our numerics shows that its only ap-proximately true for full-fledged calculations using Eq. (89),an illustration on why this is the case for the on-shell �−1 isprovided in Fig. 24. There, we plot the imaginary self-energypart for k = 1.6kF and three different dielectric constants,� = 0.5, 1.0, 2.5 resolving plasmonic and p-ℎ contributions.Approaching the “on-shell” frequency marked as a verticaldashed line, the curves essentially fall on top of each other.The figure also illustrates the difficulties of the numerical de-termination of the asymptotic prefactors in Eq. (99) (for MLGthey also depend on the momentum cut-off).

VI. CONCLUSIONS

Hedin’s set of functional equations allows us to expand theelectron self-energy in terms of dressed G and W . The firstterm of such an expansion— the GW approximation—hasbeen successfully applied tomany systems. However, there arecases when this approximation is insufficient and higher-orderterms need to be taken into account. One such approximationhas been derived in our previous work starting from the first-and second-order SE [4]. Σaa+[Σcc+Σcc̄]+Σaā describes threedistinct scattering processes in many-body systems, comprisesall the first- and second-order terms and a subset of third andfourth order terms, and, crucially, has the PSD property [6].In this work focused on Σaā, relevant for small energy trans-fers, and evaluated it in the quasiparticle approximation forthe electron GF and RPA for the screened interaction. We

18

found that screening is important and must be determined con-sistently. For inconsistent screening, unphysical singularitieshave been observed in Σ2x, which is the bare Coulomb limitof Σaā. Nonetheless, Σ2x provides important corrections to thetotal energy in full agreement with analytic results of Onsageret al. [59].

We conducted a comprehensive investigation of the impactof Σaā on quasiparticle properties of the homogeneous elec-tron gas in 2D and 3D, and of the mono- and bilayer graphene.The quasiparticle renormalization factorZqp(kF), the effectivemass m∗, the Fermi velocity v∗F, and the quasiparticle life-time�(k) have been computed for a range of interaction strengthscontrolled by the density rs or the dielectric function �. Inthe weakly correlated limit (rs ≪ 1 or � ≫ 1), we comparedwith asymptotic expansions, and in the correlated regime withresults of other theories such as quantumMonte Carlo and per-turbative calculations including local field factors.

It is known that exchange processes encoded in Σaā re-duce the quasiparticle scattering rate. This has been shownin the asymptotic limit ! → ∞ by Vogt et al. [62] and in thevicinity of kF by Qian and Vignale [17]. Besides confirmingthis finding using a completely different methodology, we alsoobserved an appreciable effect of Σaā on the effective massm∗ and quasiparticle strength Zqp in 3D HEG. The effect issmaller in 2D, especially for graphene systems, as anticipated.Although we have focused on one particular scattering pro-

cess the PSD diagrammatic construction is versatile and appli-cable to realistic multiband systems. Furthermore, the PSD di-agrams remain PSD upon replacement of the zero-temperatureGreenâĂŹs function with the finite-temperature [89] or anyexcited-state GreenâĂŹs function. This latter possibilityopens the way toward the systematic inclusion of vertex cor-rections in the spectral function of systems in a (quasi) steady-state. Investigations in the field of, e.g., molecular transportand time-resolved (tr) and angle-resolved photoemission spec-troscopy (ARPES) are therefore foreseeable in the near future.The spectral function is indeed a key quantity to determinethe conductance of atomic-scale junctions, andMBPT calcula-tions have so far been limited to the GW [90–94] and second-Born [93, 94] approximation. Similarly, the tr-ARPES signalis related to the transient spectral function [95, 96] which insemiconductor or insulators can be evaluated using a steady-state approximation (provided that the carrier relaxation timeis much longer than the probe pulse). In this context [97] thePSD diagrammatic construction may provide a powerful toolin the field of light-induced exciton fluids, whose incoherentplasma phase [96, 98, 99] and coherent condensed phase [100–106] are currently under intense investigations.

ACKNOWLEDGMENTS

We thank A.-M.Uimonen for useful discussions. Thework has been performed under the Project HPC-EUROPA3(INFRAIA-2016-1-730897), with the support of the EC Re-search Innovation Action under the H2020 Programme; inparticular, Y.P. gratefully acknowledges the computer re-sources and technical support provided the CSC-IT Center

for Science (Espoo, Finland). Y.P. acknowledges support ofDeutsche Forschungsgemeinschaft (DFG), Collaborative Re-search Centre SFB/TRR 173 “Spin+X”. G.S. acknowledgesfunding from MIUR PRIN Grant No. 20173B72NB and fromINFN17_nemesys project. R.vL. likes to thank the Academyof Finland for support under grant no. 317139.

Appendix A: Hilbert transform and spectral functions

Hilbert transform is an important part of our numerical pro-cedure. We define

H[x](t) = 1� ∫

∞

−∞d�

x(�)t − �

. (A1)

It has the properties H[H[x]](t) = −x(t), H−1[x](t) =−H[x](t), and is computed using FFT. In particular we needthe relation between the real part of the correlation self-energyand the rate function Γ(k, !), Eq. (1), which in our approachis computed by the Monte-Carlo method,

ReΣRc (k, !) =12H[Γ(k)](!), (A2)

12Γ(k, !) = − ImΣ

Rc (k, !). (A3)

There are following possibilities to obtain positive spectralfunctions starting from the second-order self-energy:

∓iΣ≶aa(k, !) ≥ 0, (A4a)∓i

(

Σ≶cc + Σ≶cc̄

)

(k, !) ≥ 0, (A4b)∓i

(

Σ≶aa + Σ≶aā

)

(k, !) ≥ 0. (A4c)

Consequently, the sum of all contributions given by Eq. (8)is also PSD. By using the method from our earlier work [4],these results can also be generalized to any dressed GFs thatpossess a positive spectral function.

Appendix B: Equilibrium propagators

We define the bare electron propagators as averages ofthe field operators in the Heisenberg picture over the non-interacting stateg(1, 2) = −i⟨ ̂H (1) ̂†H (2)⟩0,

fulfilling the symmetry relationsig≶(1, 2) =

(

ig≶(2, 1))∗ . (B1)

Analogically, the density-density correlators are defined withrespect to the interacting ground state

�>(1, 2) = −i⟨Δn̂H (1)Δn̂H (2)⟩,�

19

with the density deviation Δn̂H (1) = n̂H (1) − ⟨n̂H (1)⟩. Theyfulfill the symmetriesi�≶(1, 2) =

(

i�≶(2, 1))∗ . (B2)

Because the screened interaction is directly related to �W (1, 2) = v(1, 2) +∬ d(3, 4) v(1, 3)�(3, 4)v(4, 3), (B3)

all the symmetry and analytic properties also hold forW .For homogeneous systems the momentum-energy represen-

tation is useful, which we formulate here in Fermi units (kF,�F). The Kubo-Martin-Schwinger conditions allow us to writethe lesser/greater propagators in terms of the retarded ones,

g(x, �) = −2i(nF(� ) − 1) Im gR(x, �); (B4b)W 0 (y, �) = +2i(nB(�) + 1) ImWR0 (y, �). (B4d)

nF/B are the Fermi/Bose distribution functions, which at zerotemperature reduce to simple step-functionsnF(� ) = �(1 − � ), n̄F(� ) = 1 − nF(� ); (B5a)nB(�) = −�(−�) = �(�) − 1, nB(�) + 1 = �(�). (B5b)

For the bare propagators we furthermore havegR(x, �) = 1

� − �(x) + i�, (B6)

andwe use a spectral representation of the screened interactionW R0 (y, �) = v(y) + ∫

∞

0d!

2!C(y, �)(� + i�)2 − !2

, (B7)where v(y) is the bare Coulomb interaction. It fulfills the sym-metry property

[

W R0 (y, �)]∗ = W R0 (y,−�). (B8)

Comparing it with the Hilbert transform of the inverse dielec-tric function

1"R(y, �)

= 1 − ∫

∞

0

d!�Im

[

1"R(y, !)

]

2!(� + i�)2 − !2

, (B9)we have for the spectral function of continuous spectrum

C(y, !) = v(y) Im[

−1�

1"R(y, !)

]

, (B10)and for plasmons C(y, !) = C(y)�(! − Ω(y)) with

C(y) = v(y)

[

) Re "R(y, �))�

|

|

|

|

|�=Ω(y)

]−1

. (B11)

The time-ordered (W T0 ≡ W −−0 ) and the anti-time-ordered(W T0 ≡ W ++0 ) screened interactions read

W T0 (y, �) = v(y) + ∫∞

0d!

2!C(y, !)�2 − (! − i�)2

, (B12)

W T0 (y, �) = −[

W T0 (y, �)]∗ , (B13)

withW T0 (y,−�) = W T0 (y, �).

Appendix C: Polarizability of MLG

The dynamical polarization of graphene at finite doping hasbeen computed by Hwang and Das Sarma [42] and byWunschet al. [43]. We present here for completeness the functionsGr,

Gr(y, �) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

0 1A� + G(z2) 2B−G>(z2) + G>(z1) 3B

, (C1)

and Gi,

Gi(y, �) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

G>(z2) − G>(−z1) 1AG>(z2) 2A0 3A0 1B� + G(z) = z

√

z2 − 1 − arccosh(z), (C4)G y as shown in Fig. 5.

Appendix D: Polarizability of BLG

We start by defining the four critical linesr1 = y2 + 2y + � + 2; r2 = y2 + �;

r3 = y2 − 2y + � + 2; r4 =y2

2+ �.

Furthermore, we introduce some auxiliary functions:

q1 =y2 − 2�4�

log 2 +r42�log |r4| −

r22�log |r2| �(y − 1)

+(

r24�log |1 + �| +

y2

4�log |y2|

)

sign(y − 1), (D1)

q2 =

[

|r2|4�

log|

|

|

|

|

|

(2 + �)√

r2r2 + �√

r1r3(2 + �)

√

r2r2 − �√

r1r3

|

|

|

|

|

|

−14log

|

|

|

|

|

|

√

r1r3 + 2 + �√

r1r3 − (2 + �)

|

|

|

|

|

|

]

(

�(−r1) − �(r3))

, (D2)

20

p1 =

[

r22�arctan

(

(2 + �)r2�√

−r1r3

)

+12arccos

(

2 + �y√

−2r4

)]

�(−r1r3), (D3)

p2 =�r24�sign(r2) �(−r1r3) +

�r4��(y − 2) �(−r4) �(r3)

−�y2

2��(−r1). (D4)

With the help of these definitionsReΠ2(y, �) = q1 + q2; ImΠ2(y, �) = p1 + p2. (D5)

Appendix E: Solution of the Dyson equation

Let us recapitulate possible approaches to the solution of theDyson equation

G(k, !) = g(k, !) + g(k, !)Σ(k, !)G(k, !) (E1)following Ref. [30]. In the preceding sections the self-energyis computed using bare propagators (B4), Σ = Σ[g,W0]. Thisapproach has an inherent problem that the k = kF state is nolonger a sharp quasiparticle state. We still can improve theone-shot calculations by applying some rigid shift Δ� to allpoles,

G0(k, !) =1

! − �0(k) − Δ� + i�= g(k, ! − Δ�). (E2)

The respective self-energy then readsΣ[G0,W0](k, !) = Σ[g,W0](k, ! − Δ�), (E3)

allowing to rewrite the quasiparticle approximation for theDyson’s equation

�(k) = �0(k) + Σ[g,W0](k, �(k) − Δ�), (E4)�̃(k) = �0(k) − Δ� + Σ[g,W0](k, �̃(k)), (E5)

Thus, the quasiparticle approximation for G reads

G(k, !) = 1! − �̃(k)

. (E6)

Now we demand that the solution of the Dyson equation atk = kF takes the form

G(kF, !) =1

! − � + i�= G0(kF, !). (E7)

and coincideswith the improved propagatorG0 (E2) represent-ing a sharp quasiparticle peak at the chemical potential. Theconsistency condition (E7) provides the interpretation of Δ�as the correlation shift of the chemical potential

Δ� = � − �F, (E8)and allows to determine it. To this end, we insert Eq. (E7) intoEq. (E4) leading to

� = �F + ReΣ[G0,W0](kF, �). (E9)This point and the connection of � to the total energy per elec-tron is explained in Ref. [107] (p. 82). Combining Eq. (E8)with Eq. (E9) we obtain

Δ� = ReΣ[g,W0](kF, �F). (E10)Thus, Δ� is expressed solely in terms of the self-energy fork = kF and ! = �F. In the case of MLG and BLG having twobands, one additionally sets the band index s consistent withthe doping (typically the chemical potential is above the Diracpoint implying s = +1).

[1] L. Hedin, Phys. Rev. 139, A796 (1965).[2] P. Minnhagen, J. Phys. C 7, 3013 (1974).[3] P. Minnhagen, J. Phys. C 8, 1535 (1975).[4] G. Stefanucci, Y. Pavlyukh, A.-M. Uimonen, and R. van

Leeuwen, Phys. Rev. B 90, 115134 (2014).[5] A.-M. Uimonen, G. Stefanucci, Y. Pavlyukh, and R. van

Leeuwen, Phys. Rev. B 91, 115104 (2015).[6] Y. Pavlyukh, A.-M. Uimonen, G. Stefanucci, and R. van

Leeuwen, Phys. Rev. Lett. 117, 206402 (2016).[7] B. Holm and U. von Barth, Phys. Rev. B 57, 2108 (1998).[8] J. M. Riley, F. Caruso, C. Verdi, L. B. Duffy, M. D. Wat-

son, L. Bawden, K. Volckaert, G. van der Laan, T. Hesjedal,M. Hoesch, F. Giustino, and P. D. C. King, Nat. Commun. 9,2305 (2018).

[9] B. Holm and F. Aryasetiawan, Phys. Rev. B 56, 12825 (1997).[10] M. Guzzo, J. J. Kas, L. Sponza, C. Giorgetti, F. Sottile,

D. Pierucci, M. G. Silly, F. Sirotti, J. J. Rehr, and L. Reining,Phys. Rev. B 89, 085425 (2014).

[11] D. C. Langreth, Phys. Rev. B 1, 471 (1970).[12] Y. Pavlyukh, Sci. Rep. 7, 504 (2017).[13] K. Balzer, S. Bauch, and M. Bonitz, Phys. Rev. A 82, 033427

(2010).[14] E. Perfetto, A.-M. Uimonen, R. van Leeuwen, and G. Ste-

fanucci, Phys. Rev. A 92, 033419 (2015).[15] M. Schüler and Y. Pavlyukh, Phys. Rev. B 97, 115164 (2018).[16] P. Ziesche, Ann. Phys. 16, 45 (2007).[17] Z. Qian and G. Vignale, Phys. Rev. B 71, 075112 (2005).[18] C.-O. Almbladh, J. Phys. Conf. Ser. 35, 127 (2006).[19] Y. Pavlyukh, M. Schüler, and J. Berakdar, Phys. Rev. B 91,

155116 (2015).[20] M. Schüler, Y. Pavlyukh, P. Bolognesi, L. Avaldi, and J. Be-

rakdar, Sci. Rep. 6, 24396 (2016).[21] A. Grüneis, M. Marsman, J. Harl, L. Schimka, and G. Kresse,

J. Chem. Phys. 131, 154115 (2009).[22] X. Ren, N. Marom, F. Caruso, M. Scheffler, and P. Rinke,

Phys. Rev. B 92, 081104(R) (2015).

21

[23] B. I. Lundqvist, Phys. Kondens. Mater. 7, 117 (1968).[24] G. E. Santoro and G. F. Giuliani, Phys. Rev. B 39, 12818

(1989).[25] E. H. Hwang and S. Das Sarma, Phys. Rev. B 77, 081412(R)

(2008).[26] M. Polini, R. Asgari, G. Borghi, Y. Barlas, T. Pereg-Barnea,

and A. H. MacDonald, Phys. Rev. B 77, 081411(R) (2008).[27] R. Sensarma, E. H. Hwang, and S. Das Sarma, Phys. Rev. B

84, 041408(R) (2011).[28] V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, and A. H.

Castro Neto, Rev. Mod. Phys. 84, 1067 (2012).[29] Y. Pavlyukh, A. Rubio, and J. Berakdar, Phys. Rev. B 87,

205124 (2013).[30] G. Stefanucci and R. van Leeuwen, Nonequilibrium Many-

Body Theory of Quantum Systems: A Modern Introduction(Cambridge University Press, Cambridge, 2013).

[31] G. Strinati, Riv. Nuovo Cimento 11, 1 (1988).[32] R. van Leeuwen and G. Stefanucci, Phys. Rev. B 85, 115119

(2012).[33] D. L. Freeman, Phys. Rev. B 15, 5512 (1977).[34] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev.Mod.

Phys. 83, 407 (2011).[35] D. N. Basov, M. M. Fogler, and F. J. Garcia de Abajo, Science

354, aag1992 (2016).[36] A. Czachor, A. Holas, S. R. Sharma, and K. S. Singwi, Phys.

Rev. B 25, 2144 (1982).[37] F. Stern, Phys. Rev. Lett. 18, 546 (1967).[38] G. Giuliani and G. Vignale, Quantum theory of the electron

liquid (Cambridge University Press, Cambridge, UK, 2005).[39] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series,

and products, 7th ed. (Elsevier, Amsterdam, 2007).[40] T. Ando, J. Phys. Soc. Jpn. 75, 074716 (2006).[41] D. N. Basov, M. M. Fogler, A. Lanzara, F. Wang, and

Y. Zhang, Rev. Mod. Phys. 86, 959 (2014).[42] E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418

(2007).[43] B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New J. Phys.

8, 318 (2006).[44] E. H. Hwang, R. E. Throckmorton, and S. Das Sarma, Phys.

Rev. B 98, 195140 (2018).[45] P. E. Trevisanutto, C. Giorgetti, L. Reining, M. Ladisa, and

V. Olevano, Phys. Rev. Lett. 101, 226405 (2008).[46] E. H. Hwang, B. Y.-K. Hu, and S. Das Sarma, Phys. Rev. Lett.

99, 226801 (2007).[47] R. Sensarma, E. H. Hwang, and S. Das Sarma, Phys. Rev. B

82, 195428 (2010).[48] E. H. Hwang and S. Das Sarma, Phys. Rev. Lett. 101, 156802

(2008).[49] U. von Barth and B. Holm, Phys. Rev. B 54, 8411 (1996).[50] G. F. Giuliani and J. J. Quinn, Phys. Rev. B 26, 4421 (1982).[51] Y. Zhang and S. Das Sarma, Phys. Rev. B 71, 045322 (2005).[52] J. Lischner, D. Vigil-Fowler, and S. G. Louie, Phys. Rev. B 89,

125430 (2014).[53] A. Bostwick, F. Speck, T. Seyller, K. Horn, M. Polini, R. As-

gari, A. H. MacDonald, and E. Rotenberg, Science 328, 999(2010).

[54] A. L. Walter, A. Bostwick, K.-J. Jeon, F. Speck, M. Ostler,T. Seyller, L. Moreschini, Y. J. Chang, M. Polini, R. Asgari,A. H. MacDonald, K. Horn, and E. Rotenberg, Phys. Rev. B84, 085410 (2011).

[55] J. P. Carbotte, J. P. F. LeBlanc, and E. J. Nicol, Phys. Rev. B85, 201411(R) (2012).

[56] S. Das Sarma and E. H. Hwang, Phys. Rev. B 87, 045425(2013).

[57] A. Sabashvili, S. Östlund, and M. Granath, Phys. Rev. B 88,085439 (2013).

[58] M. L. Glasser and G. Lamb, J. Phys. A 40, 1215 (2007).[59] L. Onsager, L. Mittag, and M. J. Stephen, Ann. Phys. 473, 71

(1966).[60] A. Isihara and L. Ioriatti, Phys. Rev. B 22, 214 (1980).[61] M. L. Glasser, in Many-body Approaches at Different Scales,

edited by G. Angilella and C. Amovilli (Springer InternationalPublishing, Cham, 2018) pp. 291–296.

[62] M. Vogt, R. Zimmermann, and R. J. Needs, Phys. Rev. B 69,045113 (2004).

[63] J. González, F. Guinea, and M. A. H. Vozmediano, Phys. Rev.B 59, R2474 (1999).

[64] E. Barnes, E. H. Hwang, R. E. Throckmorton, andS. Das Sarma, Phys. Rev. B 89, 235431 (2014).

[65] A. V. Chubukov, Phys. Rev. B 48, 1097 (1993).[66] M. Holzmann, B. Bernu, V. Olevano, R. M. Martin, and D. M.

Ceperley, Phys. Rev. B 79, 041308(R) (2009).[67] M. Holzmann, B. Bernu, C. Pierleoni, J. McMinis, D. M.

Ceperley, V. Olevano, and L. Delle Site, Phys. Rev. Lett. 107,110402 (2011).

[68] R. Asgari, B. Davoudi, M. Polini, G. F. Giuliani, M. P. Tosi,and G. Vignale, Phys. Rev. B 71, 045323 (2005).

[69] G. E. Simion and G. F. Giuliani, Phys. Rev. B 77, 035131(2008).

[70] C. A. Kukkonen and A. W. Overhauser, Phys. Rev. B 20, 550(1979).

[71] A. L. Kutepov and G. Kotliar, Phys. Rev. B 96, 035108 (2017).[72] R. Sensarma, E. H. Hwang, and S. Das Sarma, Phys. Rev. B

86, 079912(E) (2012).[73] E. Daniel and S. H. Vosko, Phys. Rev. 120, 2041 (1960).[74] V. M. Galitski and S. Das Sarma, Phys. Rev. B 70, 035111

(2004).[75] N. D. Drummond and R. J. Needs, Phys. Rev. B 80, 245104

(2009).[76] M. Gell-Mann, Phys. Rev. 106, 369 (1957).[77] G. Mahan, Many-particle physics, 3rd ed. (Kluwer Aca-

demic/Plenum Publishers, New York, 2000).[78] J. F. Janak, Phys. Rev. 178, 1416 (1969).[79] C. S. Ting, T. K. Lee, and J. J. Quinn, Phys. Rev. Lett. 34, 870

(1975).[80] T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437

(1982).[81] D. S. Saraga and D. Loss, Phys. Rev. B 72, 195319 (2005).[82] F. G. Eich, M. Holzmann, and G. Vignale, Phys. Rev. B 96,

035132 (2017).[83] S. Das Sarma, E. H. Hwang, and W.-K. Tse, Phys. Rev. B 75,

121406(R) (2007).[84] H.-K. Tang, J. N. Leaw, J. N. B. Rodrigues, I. F. Herbut, P. Sen-

gupta, F. F. Assaad, and S. Adam, Science 361, 570 (2018).[85] Q. Li and S. Das Sarma, Phys. Rev. B 87, 085406 (2013).[86] L. Zheng and S. Das Sarma, Phys. Rev. B 53, 9964 (1996).[87] M. Reizer and J. W. Wilkins, Phys. Rev. B 55, R7363 (1997).[88] M. Polini and G. Vignale, arXiv:1404.5728 [cond-mat]

(2014).[89] J. J. Kas and J. J. Rehr, Phys. Rev. Lett. 119, 176403 (2017).[90] K. S. Thygesen and A. Rubio, J. Chem. Phys. 126, 091101

(2007).[91] C. D. Spataru, M. S. Hybertsen, S. G. Louie, and A. J. Millis,

Phys. Rev. B 79, 155110 (2009).[92] J. B. Neaton, M. S. Hybertsen, and S. G. Louie, Phys. Rev.

Lett. 97, 216405 (2006).[93] P. Myöhänen, A. Stan, G. Stefanucci, and R. van Leeuwen,

Eurphys. Lett. 84, 67001 (2008).

22

[94] P. Myöhänen, A. Stan, G. Stefanucci, and R. van Leeuwen,Phys. Rev. B 80, 115107 (2009).

[95] J. K. Freericks, H. R. Krishnamurthy, and T. Pruschke, Phys.Rev. Lett. 102, 136401 (2009).

[96] E. Perfetto, D. Sangalli, A. Marini, and G. Stefanucci, Phys.Rev. B 94, 245303 (2016).

[97] M. Hyrkäs, D. Karlsson, and R. van Leeuwen, Phys. StatusSolidi B 256, 1800615 (2019).

[98] D. Semkat, F. Richter, D. Kremp, G. Manzke, W.-D. Kraeft,and K. Henneberger, Phys. Rev. B 80, 155201 (2009).

[99] A. Steinhoff, M. Florian, M. Rösner, G. Schönhoff, T. O.Wehling, and F. Jahnke, Nat. Commun. 8, 1166 (2017).

[100] A. Rustagi and A. F. Kemper, Phys. Rev. B 97, 235310 (2018).

[101] A. Rustagi and A. F. Kemper, Phys. Rev. B 99, 125303 (2019).[102] E. Perfetto, D. Sangalli, A. Marini, and G. Stefanucci, Phys.

Rev. Materials 3, 124601 (2019).[103] D. Christiansen, M. Selig, E. Malic, R. Ernstorfer, and

A. Knorr, Phys. Rev. B 100, 205401 (2019).[104] E. Perfetto, S. Bianchi, and G. Stefanucci, Phys. Rev. B 101,

041201(R) (2020).[105] R. Hanai, P. B. Littlewood, and Y. Ohashi, J. Low Temp. Phys.

183, 127 (2016).[106] K. W. Becker, H. Fehske, and V.-N. Phan, Phys. Rev. B 99,

035304 (2019).[107] L. Hedin and S. Lundqvist, in Solid State Physics, Vol. 23,

edited by D. T. a. H. E. Frederick Seitz (Academic Press, NewYork, 1970) pp. 1–181.

Dynamically screened vertex correction to GWAbstractI IntroductionII Summary of the PSD approachIII Self-energy approximations: physical meaning of diagramsA aa<B c< and cc<C a<

IV Systems and reference resultsA 2D HEGB 3D HEGC 2D MLGD 2D BLGE G0W0 calculations

V a: scattering accompanied by the generation of a ph-pair with exchange A Computation for MLG and BLG systemsB 2x results for 3D HEGC Cancellations between aa< and a< in the asymptotic regimeD Quasiparticle properties

VI Conclusions AcknowledgmentsA Hilbert transform and spectral functionsB Equilibrium propagatorsC Polarizability of MLGD Polarizability of BLGE Solution of the Dyson equation References