1

NOTES ON QUASIPARTICLES

Luca G Molinari

The many-body time-ordered propagator

iGσσ′ (xt,x′t′) = 〈Tψσ(xt)ψ†σ′ (x

′t′)〉 (1)

is the amplitude for the propagation of a state where aparticle is added in x′ at time t′ < t to a ground stateof identical particles, propagated from time t′ up to timet, and destroyed in x. If t′ > t the amplitude refers to ahole being created in (x, t) and destroyed later in (x′, t′).

For systems denoted Fermi liquids, the evolution canbe identified, in a certain energy range, with that of asingle quasiparticle, the effect of the other particles be-ing a renormalization of the bare parameters, such asmass and dispersion law. Normal metals are Fermi liq-uids, and to some extent they can be described in termsof non-interacting quasiparticles (as in Sommerfeld’s de-scription of metals). The concept of quasiparticle was in-troduced phenomenologically in 1957 by Landau3,4 andapplied with success to the study of 3He. A microscopicjustification in many body theory was provided later byGalitskii, who evaluated self-energy corrections by re-summing ladder diagrams for two-particle scattering5,14.

As the interaction is turned on, the one particle ex-citations of the ideal Fermi gas evolve into quasiparticleexcitations of a Fermi liquid. The identification is notsharp and involves the notion of lifetime and weight ofthe quasiparticle. The origin of this simple picture isthat in Fermi liquids the interaction preserves a Fermisurface, wherein most degrees of freedom are frozen atlow energy.

In other instances, quasiparticle appear as a result ofcanonical transformations, which bring a Hamiltonian Hto the form H ≈ Egs +

∑

a ~ωac†aca, up to small terms,

with a canonical set of operators ca and c†a. The opera-tors ca and c†a destroy and create a quasiparticle, whichappears as an excitation of a normal mode of the model.As a consequence, the fully dressed propagator of theoriginal theory takes the semblance, in the new basis, ofa propagator for independent dressed particles.

We begin by considering a system of fermions which ishomogeneous and isotropic.

I. HOMOGENEOUS SYSTEMS

Homogeneous systems are simple in (k, ω) space.We also assume a spin-independent and rotation in-variant two-particle interaction, so that Gσσ′ (k, ω) =δσσ′G(k, ω). The ground-state average of any one-particle observable is

〈O〉 =

∫

d3k

(2π)3

∑

σ

〈kσ|o|kσ〉nσ(k) (2)

where nσ(k) is the one-particle density in k space perspin component:

nσ(k) = 〈a†σ(k)aσ(k)〉 = −i∫

dω

2πG(k, ω)eiωη (3)

In our hypothesis the density depends on k and not onσ. If the Green function had poles in ω, close the realaxis, the frequency integral would be dominated by them.Such poles identify the quasiparticles. Let us first reviewparticles.

A. Free fermions

The ground state of a system of fermions with isotropicdispersion (energy- momentum relation) ǫ = ~ω0(k) andspin s is a sphere in momentum space filled up to a radiuskF determined by the particle density: k3

F = 32 (2s +

1)π2(N/V ). The chemical potential µ0 coincides withthe Fermi energy ~ω0(kF ). The N−particle time-orderedpropagator is

G(k, ω) =1

ω − ω0k + iηsign(k − kF )

(4)

Its pole ω = ω0(k) specifies the dispersion relation andis endowed with an infinitesimal imaginary part that isessential to evaluate frequency integrals. This imaginarypart selects the contribution of the filled Fermi sphere forthe evaluation of the density:

n0σ(k) = θ(kF − k). (5)

The retarded propagator is insensitive to kF , and henceto Pauli principle. In real space it coincides with thematrix element of the forward time-evolution operator ofone particle, and has the simple analytic expression1

iGR(xt,x′t′) = θ(t− t′)〈x|U (t− t′)|x′〉

= θ(t− t′)

[

m

2πi~(t− t′)

]3/2

eim

2~

|x−x′|2

t−t′ (6)

B. Interacting fermions

In momentum space, the time-ordered propagator hasthe Lehmann’s representation14

G(k, ω) =

∫ ∞

−∞

dω′ A(k, ω′)

ω − ω′ + iη sign (~ω′ − µ)(7)

It can be read as a superposition of independent-particlepropagators, weighted by a spectral function A(k, ω),which is positive and normalized

∫ +∞

−∞

dω A(k, ω) = 1 (8)

2

For independent fermions, the spectral function ispeaked around the dispersion relation of a free particle:A0(k, ω) = δ(ω − ω0

k), with ω0k = ~k2/2m.

The spectral function A can be measured through thephotocurrent intensity of outgoing electrons in angle-resolved photoemission spectroscopy (ARPES)15. In theuniform electron gas it is sharply peaked, with smallsatellites denoting plasmon excitations18. If the spectralrepresentation (7) is used in eq.(9), the following expres-sion for the occupation of momentum states is obtained:

nσ(k) =

∫ µ/~

−∞

dωA(k, ω) (9)

Integration in k gives the density of one-particle excita-tions:

ρ(ω) = 4π

∫ ∞

0

dkk2A(k, ω) (10)

By extracting the imaginary part of eq.(7) one gets

ImG(k, ω) = πA(k, ω) sign (µ− ~ω) (11)

It follows that the imaginary part of the propagator hasthe same sign as µ−~ω, independently of k. Luttinger11

has shown that, for ω close to µ/~, it is

ImG(k, ω) ≈ C(k)(µ − ~ω)2sign(µ− ~ω) (12)

with C(k) > 0. The property reflects in the alternativeexact representation of the propagator:

G(k, ω) =1

ω − ω0k − Σ(k, ω)

(13)

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

ImΣ(k, ω) has the same sign as µ − ~ω, independentlyof k. The frequency value where the self-energy changessign (and vanishes, according to Luttinger’s proposition),identifies the chemical potential of the interacting systemof fermions.

In the time domain, the expression of the propagatoris

iG(k, t) =

∫ ∞

−∞

dωe−iωtA(k, ω) × (14)

{θ(t)θ(~ω − µ) − θ(−t)θ(µ − ~ω)}

If A is a smooth function of the real variable ω, one canmake general statements on the Fourier integral. If thesupport is well inside the half-line ω > µ/~, the spec-tral function is unaffected by the step function. Thenthe propagator decays at least exponentially in time andit is zero for t < 0. The situation is reversed if thesupport is contained in the other half-line. If A is non-zero in the proximity of the value ω = µ/~, the productA(k, ω)θ(~ω − µ) is discontinuous on the real line, andtherefore its Fourier integral decays slowly, as t−1.

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 and0.01.

C. An example

It is plausible that the turning on of a weak interactionproduces a broadening of the δ-shaped spectral line A0

of the ideal gas. A useful approximation is a Lorentzianshape of half-width Γk > 0,

A(k, ω) =1

π

Γk

(ω − ωk)2 + Γ2k

(15)

This form of spectral function leads to a simple expres-sion for the propagator in momentum space:

G(k, ω) =1

ω − ωk + iΓk sign(~ω − µ)(16)

For Γk → 0 the propagator of the ideal Fermi gas isreproduced, with the correct sign of the small imaginarypart of the pole. However, the lorentzian distributiondecays too slowly to have finite momenta, and low orderones have been evaluated for the exact spectral functionof the jellium model19.

In the time domain:

G(k, t) =

∫ ∞

−∞

dω

2πG(k, ω)e−iωt

= iθ(t)e−iωkt−Γkt +

∫ µ/~

−∞

dω

2πe−iωt 2iΓk

(ω − ωk)2 + Γ2k

If ωk ≫ µ/~ + Γk the integral is negligible, and the firstterm describes the propagation of a quasiparticle withdispersion ω = ωk, that decays in time, with lifetimemeasured by Γ−1

k .

For t = 0− we obtain the average occupation numberin momentum space, eq.(9) and Fig.1:

nσ(k) =1

2+

1

πarctg

µ− ~ωk

~Γk(17)

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

3

D. Quasiparticles

A first requirement that defines a quasiparticle is to bea pole of the propagator. So we study the equation

0 = ω − ω0k − Σ(k, ω) (18)

The equation is solved by a function ω1(k)+ iω2(k). Thepropagator gains the form of a quasiparticle propagatorplus a regular part:

G(k, ω) =Z(k)

ω − ω1(k) − iω2(k)+Greg(k, ω), (19)

When evaluating one-particle averages, this pole pro-duces the result as if the particles were independent,weighted by the residue Z(k), the quasiparticle weight.Its value at the Fermi momentum measures the jump inthe occupation number in spin-momentum space at theFermi value13:

nσ(kF − ǫ) − nσ(kF + ǫ) = Z(kF )

The proof is straightforward:

nσ(k) =

∫

dω

2πiG(k, ω)eiωη = Z(k)θ(µ−~ω1(k))+n

regσ (k)

Necessarily, it is Z(kF ) ≤ 1. For metals, this step canbe measured in Compton scattering experiments15. IfZ(kF ) is close to unity, the Fermi surface is meaningful,with radius kF and chemical potential µ = ω1(kF ).

The concept of quasiparticle is useful if we require thatω2 ≪ ω1. From eq.(18), we obtain

ω1(k) − ω0k − ReΣ(k, ω1(k)) = 0 (20)

ω2(k) = Z(k)ImΣ(k, ω1(k)) (21)

Z(k) =

(

1 − ∂

∂ωReΣ(k, ω)

)−1

ω=ω1(k)

(22)

The smallness of ω2 is ensured by ~ω1(k) close to µ, whichis the point where ImΣ(k, ω1) changes sign (vanishes).With the pole close to the real axis, the quasiparticleterm dominates in the evaluation ground-state averagesof single-particle observables.

The Fermi surface is a sphere, and the Fermi energy isµ = ω1(kF ) or, equivalently, it solves the equation

µ− µ0 − ~ReΣ(kF , µ/~) = 0 (23)

Close to the Fermi energy, single-particle energies havethe expansion

~ω0k = µ0 +

~2kF

m(k − kF ) + . . . (24)

For a quasiparticle we write the same expansion, withthe linear term defining the effective mass:

~ω1(k) = µ+~

2kF

m∗(k − kF ) + . . . (25)

By evaluating the derivative in k of eq.(20) in kF , oneevaluates

m

m∗= Z(kF )

[

1 +m

~kF

∂

∂kReΣ(k, µ/~)

]

k=kF

(26)

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

~

τ(k)= −2 ImΣ(k, ω1(k)) (27)

The mean free path is the length ℓ = ~kF

m∗ τ(kF ). Thequasiparticle parameters m∗, Z(kF ) and τ(k) have beenevaluated for the homogeneous electron gas in severalapproximations: R.P.A. (Quinn and Ferrell, 1958)21,G.W.A. (Krakovsky and Percus, 1996)17, G.W.A. withvertex corrections (Takada, 2001)18. The Hartree-Fockapproximation gives the wrong result m∗ = 0. We reportthe quasiparticle data in R.P.A.6

τ(k)−1 = 0.252√rs

~(k − kF )2

2m+ . . . (28)

m

m∗= 1 − 0.083rs(log rs + 0.203) + . . . (29)

The interaction with phonons gives a much strongerrenormalization, of about 40% in metals like Na or Al,and 15% in Cs. An evaluation of self energy with onephonon exchange in made in Mahan’s book15,22.

II. NON-HOMOGENEOUS SYSTEMS

We consider a system of N interacting fermions in abox, in the ground state |EN

0 〉. The spectrum of theHamiltonian is discrete, with exact eigenstates |EN

a 〉.The chemical potential µ is EN+1

0 − EN0 , and it is in-

sensitive to N , if N is large. The time-ordered propaga-tor has the following general Lehmann’s representationin frequency space14

Gαβ(x,x′, ω) = (30)

∑

a

〈EN0 |ψα(x)|EN+1

a 〉〈EN+1a |ψ†

β(x′)|EN0 〉

ω − 1~(µ+ ǫN+1

a ) + iη

+〈EN

0 |ψ†β(x′)|EN−1

a 〉〈EN−1a |ψα(x)|EN

0 〉ω − 1

~(µ− ǫN−1

a ) − iη

The values ǫN±1a = EN±1

a −EN±10 are the excitation ener-

gies, and are positive. It is useful to introduce two sets offunctions f ′

a and f ′′a and corresponding frequency values

ω′a and ω′′

a , as follows:

f ′a(x, α) = 〈EN

0 |ψα(x)|EN+1a 〉, ~ω′

a = µ+ ǫN+1a (31)

f ′′a (x, α) = 〈EN−1

a |ψα(x)|EN0 〉, ~ω′′

a = µ− ǫN−1a (32)

4

It is ~ω′a > µ and ~ω′′

a < µ. It is customary to removeapices and denote both functions and frequencies as fa

and ωa. With these functions the propagator gains thesimple form

Gαβ(x,x′, ω) =∑

a

fa(x, α)fa(x′, β)∗

ω − ωa + iη sign(~ωa − µ)(33)

It is the same form of the propagator of independent par-ticles or in Hartree-Fock approximation, where the func-tions fa are the eigenstates of the one-particle Hamilto-nian or solutions of the Hartree-Fock equations. For thisanalogy, the fa are called “quasiparticle” states.

A. Quasiparticle states

By inserting Lehmann’s representation (33) intoDyson’s equation for the propagator, and by taking thelimit ω → ωa, one obtains a Schrodinger-like equation forthe amplitudes, which was first derived in 1952 by JulianSchwinger7,8:

(hfa)(x, α) + ~

∑

β

∫

d3yΣαβ(x,y, ωa)fa(y, β)

= ~ωafa(x, α) (34)

h is the one-particle operator in the many-body hamil-tonian, and the self-energy acts as a non-local potential.The eigenvalues ωa in general are not real. In the caseof space-translation (and rotation) invariance, the equa-tion is solved by the eigenstates of momentum and yieldseq.(18) for the quasiparticle frequency ω(k).

Completeness of the eigenstates |EN±1n 〉 in spaces of

N ± 1 particles imply

∑

a

f ′a(x, α)f ′

a(x′, β)∗ = 〈EN0 |ψα(x)ψ†

β(x′)|EN0 〉 (35)

∑

a

f ′′a (x, α)f ′′

a (x′, β)∗ = 〈EN0 |ψ†

β(x′)ψα(x)|EN0 〉 (36)

When summed together, the resulting canonical anticom-mutation rule of field operators gives the completeness

relation:∑

a

fa(x, α)fa(x′, β)∗ = δαβδ3(x − x′) (37)

There is no relation of orthogonality for these functions.The ground-state expectation value of a one-particle

observable O =∑

o(i) is

〈EN0 |O|EN

0 〉

=∑

αβ

∫

d3xd3x′〈xα|o|x′β〉f ′′a (x, α)∗f ′′

a (x′, β)

=∑

a

〈fa|o|fa〉θ(µ − ~ωa) (38)

where the functions fa are treated formally as elementsof the one-particle Hilbert space, where the one-particleoperator o acts. In particular the ground-state density ofparticles with spin σ is

〈EN0 |nσ(x)|EN

0 〉 =∑

a

|fa(x, σ)|2θ(µ− ~ωa) (39)

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

N = (2s+ 1)∑

a

‖fa‖2θ(µ− ~ωa) (40)

The total energy can be evaluated from the operatoridentity

∑

α

∫

d3xψ†α(x)[ψα(x), H ] = H1 + 2H2 (41)

where H = H1 + H2 is the total Hamiltonian, H1 andH2 are its single-particle and two-particle (interaction)parts. The expectation value on the N -particle groundstate |EN

0 〉 is taken

NEN0 −

∑

α

∫

d3x〈ψ†α(x)Hψα(x)〉 = 〈H1 + 2H2〉

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

to obtain an expression with quasiparticle states

∑

a

~ωa‖fa‖2θ(µ− ~ωa) = 〈H1 + 2H2〉

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

1

2

∑

a

〈fa|h+ ~ωa|fa〉θ(µ− ~ωa) = EN0 (42)

B. Approximate quasiparticle evaluation

In applications, one usually solves the quasi-particleequation (34) perturbatively, by starting from approxi-mate functions.

Density functional theory (DFT) represents the inter-acting many body problem as one with independent par-ticles in a self-consistent appropriate potential. As Kohnputs it, DFT is the exactification of mean field (Hartree)theory. The particle density n(x), chemical potential µand ground state energy EN

0 do in principle coincide withthe results of the many-body calculations. Thus DFT isa one-particle description that accounts for correlationsexactly, for the purpose of evaluating n, µ and EN

0 .The equality of the densities leads to the Sham-

Schluter equation. If G, GKS and GH are respectivelythe exact, K.S. and Hartree Green functions, then G =GH + GHΣG and GKS = GH + GHvKSGKS . Subtrac-tion gives G − GKS = GH [ΣG − vKSGKS ]. For x = x′

5

and t′ = t+, i.e. x′ = x+, the left hand side vanishes be-cause both Green functions yield the same density. Whatremains is the Sham-Scluter equation25:

0 =

∫

d4x′′d4x′′′GH(x, x′′)Σ(x′′, x′′′)G(x′′′, x+)

−∫

d4x′′GH(x, x′′)vKS(x′′)GKS(x′′, x+) (43)

In practice the identification with the many-body sys-tem is often pushed beyond this controlled boundary, andDFT has proven a simple and cheap alternative to many-body calculations. With the improvement of experimen-tal data, techniques, and computing power, DFT is thezero level for many body GW calculations or, even better,calculations with vertex corrections. Quasiparticle prop-erties are presently a main topic in in condensed matter.Let us then start from the Kohn-Sham equation

(hf0a)(xα) + vKS(x)f0

a (x, α) = ~ω0af

0a (x, α) (44)

Note that the functions f0a do form an orthonormal sys-

tem, with real eigenvalues. In principle, if the exact po-tential vKS were used, the resulting solutions f0

a , whichare different from the quasiparticle functions fa, shouldprovide the same density, chemical potential and totalenergy of the manybody problem.

By taking the scalar product of eq.(34) with f0b one

gets:

∑

α,β

∫

d3xd3yf0b (x, α)∗Σαβ(x,y, ωa)fa(y, β)

−1

~〈f0

b |vKS |fa〉 = (ωa − ω0b )〈f0

b |fa〉

The formula is rewritten as

(f0b |Σ(ωa)|fa) − 1

~〈f0

b |vKS |fa〉 = (ωa − ω0b )〈f0

b |fa〉.

Next one replaces the unknown amplitudes fa with f0a

and expands the self-energy in the Kohn-Sham frequency,Σ(ωa) ≈ Σ(ω0

a)+∂ωΣ(ω0a)(ωa−ω0

a) Under the hypothesisthat the functions f0

a are real, one obtains:

Reωa = ω0a +

Za

~(f0

a |ReΣ(ω0a) − vKSδ|f0

a ) (45)

Imωa =Za

~(f0

a |ImΣ(ω0a)|f0

a ) (46)

Z−1a = 1 − ∂

∂ω(f0

a |ReΣ(ω)|f0a )ω=ω0

a(47)

Za is the quasiparticle weight.

1 L. Schiff, Quantum Mechanics, (McGraw Hill, 1968)2 Lieb3 L.D.Landau, The theory of a Fermi Liquid, Soviet Physics

JETP 3, 920-925 (1957).4 P.Nozieres and D.Pines, The theory of quantum liquids,

(1966, Perseus Books Group).5 V.M.Galitskii, The energy spectrum of a non-ideal Fermi

gas, Soviet Physics JETP 34, 104-112 (1958).6 R.D.Mattuck, A guide to Feynman Diagrams in the many-

body problem, (1974, Dover).7 Schwinger8 E.K.U.Gross, E.Runge and O.Heinonen, Many Particle

Theory (Adam Hilger, 1991 Bristol).26 Giuliani and Vignale, Quantum Theory of the Electron

Liquid, (2005, Cambridge).10 V.M.Galitskii and A.B.Migdal, Application of quantum

field theory methods to the many body problem, SovietPhysics JETP 34, 96-104 (1958).

11 J.M.Luttinger, Analytic properties of single-particle prop-

agators for many-fermion systems, Phys. Rev. 121, 942-949 (1961).

12 J.M.Luttinger, Fermi surface and some simple equilibrium

properties of a system of interacting fermions, Phys. Rev.119, 1153-1163 (1960).

13 A.B.Migdal, The momentum distribution of interacting

Fermi particles , Soviet Physics JETP 5, 333-334 (1957).14 A.L.Fetter and J.D.Walecka, Quantum Theory of Many

Particle Systems (1971 MacGraw Hill, New York).

15 G.D.Mahan, Many-Particle Physics, 3rd edition, (2000Kluwer Academic/Plenum Publishers).

16 Z.Quian and Giovanni Vignale, Lifetime of a quasiparticle

in an electron liquid, Phys. Rev. B 71, 075112 (2005).17 A.Krakovsky and J.K.Percus, Quasiparticle effective mass

for the two- and three-dimensional electron gas, Phys. Rev.B 53, 7352 (1996).

18 Y.Takada, Inclusion of vertex corrections in the self-

consistent calculation of quasiparticles in metals, Phys.Rev. Lett. 22, 226402 (2001).

19 M.Vogt, R.Zimmermann and R.J.Needs, Spectral mo-

ments in the homogeneous electron gas, Phys. Rev. B 69,045113 (2004).

20 Chulkov, Echenique...21 J.J.Quinn and R.A.Ferrell, Electron self-energy approach

to correlation in a degenerate electron gas,Phys. Rev. 112,812 (1958).

22 S.Engelsberg and J.R.Schrieffer, Coupled Electron-Phonon

System, Phys. Rev 131, 993 (1963).23 A.Fleszar and W.Hanke, Spectral properties of a quasipar-

ticle in a semiconductor, Phys. Rev. B 56 (1997) 10228.24 A.A.Kordyuk, S.V.Borisenko, A.Koitzsch, J.Fink,

M.Knupfer and H.Berger, Bare electron dispersion

from experiment: Self-consistent self-energy analysis of

photoemission data, Phys. Rev. B 71 (2005) 214513..25 Sham and Schluter, Phys. Rev. Lett. 51 (1983) 1888.26 Vignale and Giuliani, Cambridge University Press.