1

PROPAGATOR AND QUASI-PARTICLESLuca G Molinari

nov 2014, revised nov 2018

I. THE 1-PARTICLE PROPAGATOR

Given a Hamiltonian with ground state with N parti-cles |EN0 〉 and two single particle states |u〉 and |u′〉, thetime-ordered 1-particle propagator is

iG(u, t; v, t′) = 〈EN0 |Tψu(t)ψ†v(t′)|EN0 〉 (1)

=

{+ 〈EN0 |ψue−

i~ (H−E0)(t−t′)ψ†u′ |EN0 〉 t > t′

−〈EN0 |ψ†u′e−i~ (H−E0)(t′−t)ψu|EN0 〉 t′ > t

If t > t′, it is proportional to the amplitude for the propa-gation from time t′ to time t of a state formed by addinga particle |u′〉 to the ground state, to a state where aparticle |u〉 is added to the ground state24. If t′ > t theamplitude refers to a hole being created in |u〉 at time tand destroyed in |u′〉 at later time t′.Since it depends on t− t′, it has Fourier representation:

G(u, t;u′, t′) =

∫ +∞

−∞

dω

2πe−iω(t−t′)G(u, u′;ω) (2)

The discontinuity at t = t′ implies a slow decay of theFourier transform:

lim|ω|→∞

ωGu,u′(ω) = 〈u|u′〉 (3)

Proof: iG(u, t + η;u′, t) − iG(u, t − η;u′, t) = 〈u|u′〉 forη → 0+. After Fourier transform:

〈u|u′〉 =

∫ +∞

−∞

dω

πsin(ωη) G(u, u′;ω)

=

∫ +∞

−∞

dω

π

sinω

ω[ω′G(u, u′;ω′)]

where ω′ = ω/η. Since ω′ is as large as wanted, therelation is true for all |u〉 and |u′〉, if eq.(3) holds. �

If the system is confined in a box, the spectrum ofthe Hamiltonian is discrete, with eigenstates |ENa 〉. In-sertion of the identities 1 =

∑a |EN±1

a 〉〈EN±1a | in eq.(1)

makes the time-dependence explicit. In frequency spacethe propagator for position-spin states has the followingLehmann’s representation14

Gµµ′(x,x′;ω) =

∑a

〈EN0 |ψµ(x)|EN+1a 〉〈EN+1

a |ψ†µ′(x′)|EN0 〉ω − 1

~ (µ+ εN+1a ) + iη

+〈EN0 |ψ†µ′(x′)|EN−1

a 〉〈EN−1a |ψµ(x)|EN0 〉

ω − 1~ (µ− εN−1

a )− iη(4)

The values EN±1a − EN±1

0 = εN±1a ≥ 0 are the excita-

tion energies. In the differences EN±10 − EN0 = ±µ, the

chemical potential is insensitive to N , if N is large.

A. Homogeneous systems

For a homogeneous system confined in the box thepropagator has expansion

Gµµ′(x,x′, ω) =

1

V

∑k

eik·(x−x′)Gµµ′(k, ω)

Let us assume that Gµµ′ = δµµ′G. Since H and P com-mute, they share a basis of eigenvectors |ENa ,k〉; theground system has momentum equal to zero. By usingthe operator identity and its adjoint

ψµ(x) = exp(− i~

x ·P)ψµ(0) exp(i

~x ·P)

the Lehmann representation in k space is obtained:

G(k, ω) = 12V∑a,µ

|〈EN0 |ψµ(0)|EN+1a ,k〉|2

ω − 1~ (µ+ εN+1

a (k)) + iη

+|〈EN0 |ψ†µ(0)|EN−1

a ,−k〉|2ω − 1

~ (µ− εN−1a (−k))− iη

With the introduction of the spectral function

A(k, ω) =

V2

∑aµ

|〈EN0 |ψµ(0)|EN+1a ,k〉|2δ(ω − 1

~ (εN+1a (k) + µ))

+ |〈EN0 |ψ†µ(0)|EN−1a ,−k〉|2δ(ω + 1

~ (εN−1a (−k)− µ))

we get the remarkable expression:

G(k, ω) =

∫ ∞−∞

dω′A(k, ω′)

ω − ω′ + iη sign (~ω′ − µ)(5)

It pictures G as a superposition of independent-particlepropagators G0, weighted by the spectral function, whichis real positive and normalized for all k:∫ +∞

−∞dω A(k, ω) = 1 (6)

The independent-particle propagator

G0(k, ω) =1

ω − ω0k + iη sign(~ω0

k − µ0)

has a pole with Re ω = ω0k (the dispersion relation of

the particle; for free fermions ω0k = ~k2/2m) and in-

finitesimal imaginary part. The spectral function is adelta-function peaked on the pole: A0(k, ω) = δ(ω−ω0

k).As the two-particle interaction is turned on, the spectralfunction is expected to broaden and take structure, anda “quasi-particle” pole to survive.

The imaginary part of eq.(5) changes sign for all k atthe chemical potential:

ImG(k, ω) =

{πA(k, ω) > 0 if µ > ~ω

−πA(k, ω) < 0 if µ < ~ω(7)

2

This important property identifies the chemical poten-tial µ of the system with interacting particles: it is thefrequency value at which Im G changes sign for all k.Actually, Im G(k, µ/~) = 0.For the homogeneous electron gas (HEG) Luttinger7

showed that, near ω = µ/~, it is ImG(k, ω) ≈ C(k)(µ−~ω)2sign(µ− ~ω), with C(k) > 0.

The spectral function can be measured through thephotocurrent intensity of outgoing electrons in angle-resolved photoemission spectroscopy (ARPES)15. InHEG it is sharply peaked, with small equally spacedsatellites denoting plasmon excitations18.A high frequency expansion gives the spectral moments

G(k, ω) ≈ 1

ω+m1(k)

ω2+m2(k)

ω3+ . . .

m`(k) =∫∞−∞ ω`A(k, ω)dω. The first ones were evaluated

for HEG by Vogt et al.19. The variance m2−m21 measures

the width, that depends on k.

Exercise I.1 Show that, for a homogeneous system, andt > 0:

iG(k, t) =

∫ ∞µ/~

dω e−iωtA(k, ω) (8)

n(k) = 〈EN0 |a†kµakµ|EN0 〉 =

∫ µ/~

−∞dωA(k, ω) (9)

For independent fermions: iG0(k, t) = e−iω0ktθ(ω0

k − µ0

~ )

and n0(k) = θ(~µ0 − ω0k).

Exercise I.2 Show that for homogeneous (and, in gen-eral, interacting) systems, if the total energy per particleis a function of the density, E/N = ε(n), then the chem-ical potential and the pressure are (Hugenholtz)

µ(n) =∂E

∂N

∣∣∣V

=d

dn[nε(n)] (10)

p(n) = −∂E∂V

∣∣∣N

= n[µ(n)− ε(n)] (11)

Show that, for non-interacting fermions, it is µ0 = ~ωF .

Exercise I.3 Show that the retarded propagator for theideal electron gas G0ret(k, ω) = (ω − ω0

k + iη)−1, in realspace, coincides with the matrix element of the forwardtime-evolution operator of a single particle2:

iGret(xt,x′t′) = θ(t− t′)〈x| exp− i~p2

2m (t− t′)|x′〉

= θ(t− t′)[

m

2πi~(t− t′)

] 32

exp

[im

2~|x− x′|2t− t′

].

Example I.4 (Lorentzian approximation) A weakinteraction produces a broadening of the δ-shaped spec-tral line of the ideal gas. A useful approximation is aLorentzian of half-width Γk > 0,

A(k, ω) =1

π

Γk(ω − ωk)2 + Γ2

k

(12)

1 2 3 4 5

0.2

0.4

0.6

0.8

1

FIG. 1: The distribution nσ(k) for the Lorentzian spectralfunction. Units are such that µ = 1, ~Γ/µ = 0.1, 0.5, 0.01.

This distribution decays too slowly to have finite mo-menta. The propagator has the simple expression:

G(k, ω) =1

ω − ωk + iΓk sign(~ω − µ)(13)

The imaginary part has a finite (“pathological”) discon-tinuity. For Γk → 0 the propagator of the ideal Fermigas is reproduced, with the correct imaginary part of thepole. In the time domain (t > 0):

iG(k, t) = e−iωkt−Γkt − Γk

∫ µ/~

−∞

dω

π

e−iωt

(ω − ωk)2 + Γ2k

For k values ωk � µ/~+Γk the integral is negligible, andthe first term describes the propagation of a quasi-particlewith dispersion ω = ωk, that decays with lifetime Γ−1

k .The average occupation number in momentum space is

(see fig.1):

n(k) =1

2+

1

πarctg

µ− ~ωk~Γk

(14)

For infinitesimal broadness Γk we recover the step distri-bution of noninteracting fermions.

B. The Fermi surface

The change of sign of Im G reflects in another repre-sentation of the propagator:

G(k, ω) =1

ω − ω0k − Σ?(k, ω)

(15)

where ~ω0k is the single particle energy. It implies that

Im Σ?(k, ω) =

{> 0 if µ > ~ω< 0 if µ < ~ω

(16)

The frequency value where the imaginary part of the self-energy changes sign (vanishes), identifies the chemicalpotential µ of the interacting system of fermions. To thisspecial value there corresponds a surface in k space:

3

Definition I.5 (Fermi surface) The Fermi surface ofthe system is the set of k vectors such that

µ/~− ω0k − Σ?(k, µ/~) = 0 (17)

(the imaginary part of the self energy is zero for all k).

Theorem I.6 (Luttinger and Ward)1) The volume enclosed by the Fermi surface is indepen-dent of the interaction:

N

V=

∫dk

(2π)3θ(µ~− ω0

k − Σ?(k, µ/~))

(18)

=

∫dk

(2π)3θ

(µ0

~− ω0

k

)2) The occupation number nk is discontinuous across theFermi surface.

See the papers by Luttinger and Ward9, Luttinger8, thedissertation by Praz10, a lesson by Gurarie11 and, on asimilar line, the book by Giuliani and Vignale6.

For isotropic systems the Fermi surface is the surfaceof a sphere of radius kF fixed by the density.

C. Quasi-particles

In analogy with the propagator of independent parti-cles, a property that defines a quasi-particle is to be apole of the propagator:

0 = ω − ω0k − Σ?(k, ω) (19)

If the pole is ω1(k)+ iω2(k), the propagator has the formof a quasi-particle propagator plus a regular part:

G(k, ω) =Z(k)

ω − ω1(k)− iω2(k)+Greg(k, ω) (20)

The residue Z(k) is the “quasi-particle weight”, andthe sum rule for the spectral function implies that 0 ≤Z(k) ≤ 1. The Lehmann representation tells us thatω2 has the sign of µ − ~ω1. Accordingly, for t > 0 and~ω1(k) > µ it is

iG(k, t) = Z(k)e−iω1(k)t+ω2(k)t + iGreg(k, t) (21)

where ω2(k) < 0 describes the damping of the quasi-particle mode. For a quasi-particle weight close to unity,the quasi-particle term effectively describes the wholesystem in a region of k vectors. If the pole is near the realaxis, the lifetime is long and a frequency integral contain-ing the propagator is strongly enhanced in its vicinity,with a particle-like contribution weighted by the residue.

In conclusion, these are the properties that characte-rize a quasi-particle:i) it is a complex pole of the propagator,ii) the residue Z(k) is not negligible (order 1),

The momentum distribution of the homogeneous electron gas

Markus Holzmann1,2, Bernard Bernu2, Carlo Pierleoni3, Jeremy

McMinis4, David M. Ceperley4, Valerio Olevano5, and Luigi Delle Site6

1Univ. Grenoble 1/CNRS, LPMMC UMR 5493, Maison des Magisteres, 38042 Grenoble, France2 LPTMC, UMR 7600 of CNRS, UPMC, Jussieu, Paris, France

3Physics Department, University of L’Aquila, Via Vetoio, 67100 L’Aquila, Italy4Dept. of Physics and NCSA, U. of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

5Institut Neel, Grenoble, France and6Max-Planck-Institute for Polymer Research, Ackermannweg 10, D 55021 Mainz Germany

(Dated: May 19, 2011)

We calculate the o↵-diagonal density matrix of the homogeneous electron gas at zero tempera-ture using unbiased Reptation Monte Carlo for various densities and extrapolate the momentumdistribution, and the kinetic and potential energies to the thermodynamic limit. Our results on therenormalization factor allows us to validate approximate G0W0 calculations concerning quasiparticleproperties over a broad density region (1 rs . 10) and show that near the Fermi surface, vertexcorrections and self-consistency aspects almost cancel each other out.

PACS numbers: 05.30.Fk, 71.10.Ay, 71.10.Ca, 02.70.Ss

The uniform electron gas (jellium) is one of the mostfundamental models for understanding electronic proper-ties in simple metals and semiconductors. Knowledge ofits ground state properties, and, in particular, of modifi-cations due to electron correlation are at the heart of allapproximate approaches to the many-electron problem inrealistic models. Quantum Monte Carlo methods (QMC)[1] have provided the most precise estimates of the corre-lation energy, electron pair density and structure factorof jellium; basic quantities for constructing and parame-terizing the exchange-correlation energy used in densityfunctional theory (DFT) [2].

Correlations modify the momentum distribution, nk,of electrons, and introduce deviations from the idealFermi-Dirac step-function. The magnitude of the dis-continuity at the Fermi surface (kF ), the renormaliza-tion factor Z, quantifies the strength of a quasi-particleexcitation [3] and plays a fundamental role in Fermi liq-uid and many-body perturbation theory (GW) for spec-tral quantities. Whereas the momentum distribution (aswell as other spectral information) is inaccessible in cur-rent Kohn-Sham DFT formulations, the reduced single-particle density matrix – the Fourier transform of nk inhomogeneous systems – is the basic object in the so-calleddensity-matrix functional theory [4]; these theories relyon knowledge of nk of jellium. Inelastic x-ray scatteringmeasurement of the Compton profile of solid sodium [5]have determined nk, but experiments for elements withdi↵erent electronic densities are less conclusive.

In this paper, we calculate nk for the electron gas (jel-lium) by QMC in the density region 1 rs 10. Here,rs = (4⇡na3

B/3)�3 is the Wigner-Seitz density parame-ter, n is the density, and aB = ~2/me2 is the Bohr radius.In contrast to previous calculations [6], our calculationsare based on more precise backflow (BF) wave functions[7], and a careful extrapolation to the thermodynamic

0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

k/kF

nk

rs = 1

rs = 2

rs = 3.99

rs = 5

rs = 10

0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0

N ! 1N = 54rs=5

Z

FIG. 1: The momentum distribution (nk) of the unpolarizedelectron gas for various densities extrapolated to the thermo-dynamic limit. The inset shows the extrapolation of nk forrs = 5 from a system with N = 54 electrons to the thermo-dynamic limit, N ! 1,leading to a significant reduction ofthe renormalization factor Z.

limit [8, 9]. Similar to the worm algorithm in finite tem-perature path-integral and lattice Monte Carlo [10, 11],we have extended Reptation Monte Carlo (RMC) [12] toinclude the o↵-diagonal density matrix in order to ob-tain an unbiased estimator of the momentum distribu-tion [13, 14]. From our extrapolation scheme, we derivethe exact behavior of nk close to the Fermi surface. Bycomparing the renormalization factor, Z, with di↵erentapproximate GW theories, we can judge the importanceof self-consistency and vertex corrections within these ap-proaches. The excellent agreement of our QMC resultswith G0W0 over a broad density region indicate strongcancellations of vertex and self-consistency correctionsclose to the Fermi surface.

arX

iv:1

105.

2338

v2 [

cond

-mat

.str-

el]

17 M

ay 2

011

FIG. 2: The distribution n(k) for HEG, at various rs. Forsmall rs (large density) the gas is closer to ideal, with stepdistribution (Ceperley et al.1).

iii) |ω2(k)| is small (more precisely, |ω2| � |ω1 − µ/~|).

The presence of a quasi-particle reflects in a disconti-nuity of the occupation number

n(k) =

∫ ∞−∞

dω

2πiG(k, ω)eiωη (22)

across the Fermi surface (17), a fact first proven byMigdal13. The proof is straightforward. By evaluating(22) with the pole expansion (20), the residue theoremgives:

n(k) = nreg(k) +

{Z(k) if ~ω1(k) < µ

0 if ~ω1(k) > µ

The equation for the Fermi surface (17) is equivalent to~ω1(k) = µ. At a point kµ of the surface, the jump is :n(k<µ )− n(k>µ ) = Z(kµ).For metals, this step can be measured in Compton scat-tering experiments15. For HEG, an accurate Montecarloevaluation of the momentum distribution near kF wasdone by Holzmann et al.1 (Fig.2).

The concept of quasi-particle is useful if we require thatω2 is small. From eq.(19), we obtain

ω1(k)− ω0k − Re Σ?(k, ω1(k)) = 0 (23)

ω2(k) = Z(k) Im Σ?(k, ω1(k)) (24)

Z(k) =

[1− ∂

∂ωRe Σ?(k, ω)

]−1

ω=ω1(k)

(25)

The smallness of ω2 is ensured by ~ω1 close to to theFermi surface, where ImΣ?(k, ω) vanishes. The require-ment is then: |ω2| < |ω1 − µ/~|.

D. Effective mass

Long-living quasi-particles are found near the Fermisurface. If the residue is close to unity, this means that

4

the Fermi surface is sharp, i.e. the occupation numbershave finite (variable from point to point) discontinuouityacross it. It is then meaningful to expand the dispersionrelations near the Fermi surface.

For simplicity we consider the isotropic case. Theequation for the Fermi surface

µ− ω0k − ~Σ?(k,

µ

~) = 0 (26)

is solved by k equal to a constant, i.e. a spherical surfacewith the same radius kF for the free and interacting cases(Luttinger theorem). We then expand the dispersion lawsnear the surface:

~ω0k = µ0 +

~2kFm

(k − kF ) + . . .

~ω1(k) = µ+~2kFm∗

(k − kF ) + . . .

where m∗ is the effective mass of the quasi-particle.Linearization of eq.(23) near kF gives:

m

m∗= Z(kF )

[1 +

m

~kF∂

∂kRe Σ?(k,

µ

~)

]k=kF

(27)

The imaginary part of the pole defines the life-time ofthe quasi-particle, which is finite because of scatteringprocesses, and diverges near the Fermi surface:

~τ(k)

= −2Z(k) Im Σ?(k, ω1(k)) (28)

The mean free path is the length ` = ~kFm∗ τ(kF ).

The quasi-particle parameters m∗, Z(kF ) and τ(k) havebeen evaluated for the homogeneous electron gas in sev-eral approximations: R.P.A. (Quinn and Ferrell, 1958)20,G.W.A. (Krakovsky and Percus, 1996)17, G.W.A. withvertex corrections (Takada, 2001)18. We report thequasi-particle data in R.P.A.4

τ(k)−1 = 0.252√rs~(k − kF )2

2m+ . . . (29)

m

m∗= 1− 0.083 rs(log rs + 0.203) + . . . (30)

The interaction with phonons gives a much strongerrenormalization, about 40% in metals like Na or Al, and15% in Cs. An evaluation of self-energy with 1-phononexchange in made in Mahan’s book15? .

Exercise I.7 The self-energy diagram for HEG withscreened Coulomb potential in the Thomas-Fermi approx-imation is

Σ?(k) = −1

~

∫dq

(2π)3

4πe2

|k− q|2 + k2TF

θ(kF − q)

= − e2

2π~x

∫ 1

−1

dy y log[(x+ y)2 + a2]

= −~k2F

m

4

3π2r3/2s

{1 +

1 + a2

4xlog

(1 + x)2 + a2

(1− x)2 + a2

− (a+x

2a)

[arctan

x+ 1

a− arctan

x− 1

a

]}

where x = k/kF and a = kTF /kF = 3√

16/(3π2)√rs ≈

0.8145√rs. Evaluate the effective mass m?.

Exercise I.8 Consider the self-energy diagram for HEGwith 1-phonon exchange:

Σ?(k) =i

~

∫d4k′

(2π)4G0(k − k′)D(k′)

D(k) = g2 ω2k

(ω + iη)2 + ω2k

θ(ωD − ωk)

where ωD is Debye’s cutoff and ωk = ck (c is the veloc-ity of sound waves in the solid). Show that Im Σ?(k, ω)changes sign once, as a function of ω.

II. INHOMOGENEOUS SYSTEMS

In the general expression of Lehmann’s representation(4) it is useful to introduce the functions f ′aµ(x) andf ′′aµ(x), which carry the quantum numbers x, µ of a singleparticle, and the frequencies ω′a and ω′′a :

f ′aµ(x) = 〈EN0 |ψµ(x)|EN+1a 〉, ~ω′a = µ+ εN+1

a (31)

f ′′aµ(x) = 〈EN0 |ψ†µ(x)|EN−1a 〉, ~ω′′a = µ− εN−1

a (32)

It is ~ω′a > µ and ~ω′′a < µ. Eq.(4) becomes

Gµµ′(x,x′, ω)

=∑a

f ′aµ(x)f ′aµ′(x′)∗

ω − ω′a + iη+f ′′aµ′(x

′)f ′′aµ(x)∗

ω − ω′′a − iη

=∑a

faµ(x)faµ′(x′)∗

ω − ωa + iηsign(µ− ~ωa)(33)

where the sum is on the set {f ′a, f ′′a } corresponding to~ωa > µ and ~ωa < µ. Completeness of the eigenstates|EN±1aµ 〉 in the subspaces with N ± 1 particles implies∑

a

f ′aµ(x)f ′aµ′(x′)∗ = 〈EN0 |ψµ(x)ψ†µ′(x

′)|EN0 〉 (34)∑a

f ′′aµ(x)f ′′aµ′(x′)∗ = 〈EN0 |ψ†µ′(x′)ψµ(x)|EN0 〉 (35)

Their sum gives a completeness property in 1-particlespace: ∑

a

faµ(x)faµ′(x′)∗ = 〈xµ|x′µ′〉 (36)

In general, the functions are not orthogonal.In eq.(33) the propagator has the form of the propa-

gator of non-interacting particles or of the Hartree-Fockapproximation. Then, the functions fa would be eigen-states of a one-particle Hamiltonian, or solutions of theH.F. equations. For this analogy, the fa are called “quasi-particle” states.The functions solve a Schrodinger-like equation, which

5

was derived in 1952 by Julian Schwinger5? . Considerthe equation for the Green function:

(~ω − hx)Gµµ′(x,x′, ω)

= ~〈xµ|x′µ′〉+

∫dx′′Σ?µµ′′(x,x

′′, ω)Gµ′′µ′(x′′,x′, ω)

where h is the single particle operator, and the self-energy

acts as bi-local potential (the local term is included in h).Insert the representation (33) and take the limit ω → ωa:

(hfaµ)(x) +∑µ′′

∫dx′′~Σ?µµ′′(x,x

′′, ωa)faµ′′(x′′)

= ~ωafaµ(x) (37)

The ground-state average of a one-particle observable is

〈EN0 |O|EN0 〉 =∑µµ′

∫dxdx′〈xµ|o|x′µ′〉f ′′aµ(x)∗f ′′aµ(x′)

=∑a

〈fa|o|fa〉θ(µ− ~ωa) (38)

where the functions fa are treated as elements of theHilbert space of one particle with spin. In particular theground-state density of particles with spin µ is

〈EN0 |nµ(x)|EN0 〉 =∑a

|faµ(x)|2θ(µ− ~ωa) (39)

Integration and spin summation give a sum rule for thesquared norms in the one-particle Hilbert space:

N =∑aµ

‖fa‖2θ(µ− ~ωaµ) (40)

The total energy can be evaluated with the operatoridentity

∑µ

∫dxψ†µ(x)[ψµ(x), H] = H1 + 2H2, where

H = H1 +H2 is the Hamiltonian, and H1, H2 are 1 and2-particle operators. The average on the ground state|EN0 〉 gives

NEN0 −∑µ

∫dx 〈EN0 |ψ†µ(x)Hψµ(x)|EN0 〉 = 〈H1 + 2H2〉

A resolution of identity with states |EN−1a 〉 is inserted to

obtain an expression with quasi-particle states

〈H1 + 2H2〉 = NEN0 −∑a

EN−1a ‖fa‖2θ(µ− ~ωa)

=∑aµ

~ωa‖fa‖2θ(µ− ~ωa)

By adding the ground-state expectation value of H1 oneobtains the energy of the ground state:

EN0 = 12

∑a

〈fa|h+ ~ωa|fa〉θ(µ− ~ωa) (41)

A. Approximate quasi-particle evaluation

In applications, one may solve the quasi-particle equa-tion (37) perturbatively, by starting from approximatefunctions. In density functional theory (DFT) the inter-acting many body problem is replaced by a problem withindependent particles in a self-consistent Kohn-Sham po-tential. The particle density n(x), the chemical potentialµ and the ground state energy EN0 so obtained, are inprinciple the same as in the many-body problem.However, the accuracy of experimental data show thelimits of approximations to the unknown Kohn-Sham po-tential. For this reason, DFT is often used as the zeroorder for a many body calculation.Let us then start from the Kohn-Sham equation:

(hf0aµ)(x) + vKS(x)f0

aµ(x) = ~ω0af

0aµ(x) (42)

The functions f0a form an orthonormal system, and the

eigenvalues are real. If the potential vKS were exactlyknown, the solutions f0

a , though different from the quasi-particle functions fa, would provide the same density,chemical potential and total energy of the many-bodyproblem. Since this is not the case, let’s go back toSchwinger’s eq.(37) and take the scalar product with f0

b ,

and use the KS equation to eliminate h:∑µ,µ′

∫dxdyf0

bµ(x)∗Σ?µµ′(x,y, ωa)faµ′(y)

− 1~ 〈f0

b |vKS |fa〉 = (ωa − ω0b )〈f0

b |fa〉

The formula is rewritten as

(f0b |Σ?(ωa)|fa)− 1

~〈f0b |vKS |fa〉 = (ωa − ω0

b )〈f0b |fa〉.

At first order, the unknown amplitudes faµ(x) are re-placed with f0

aµ(x), and the self-energy is expanded

at the Kohn-Sham frequency: Σ?(ωa) ≈ Σ?(ω0a) +

∂ωΣ?(ω0a)(ωa−ω0

a). In the hypothesis that the functionsf0a are real, one obtains:

Reωa = ω0a + Za(f0

a |Re Σ?(ω0a)− 1

~vKSδ|f0a ) (43)

Imωa = Za (f0a | Im Σ?(ω0

a)|f0a ) (44)

Z−1a = 1− ∂

∂ω(f0a |Re Σ?(ω)|f0

a )∣∣∣ω=ω0

a

(45)

Za is the quasi-particle weight, δ means δ(x− y).

6

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