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Advanced Solid State Theory

Thomas Pruschke

Chapter 1

Elements of many-particle physics

1.1 Green’s functions in condensed matter physicsAll interesting phenomena in solid state physics are due to interactions: super-conductivity, magnetism, metal–insulator transitions, thermal expansion and struc-tural transitons, resistivity, optical properties and so on cannot be understood withouttaking Coulomb interactions between electrons and nuclei into account. This can bedone by attempting a full numerical solution of Schrödinger’s equation, for example,with big and fast computers.However, even then the size of the systemwe can treat is atmost ten to twenty particles. Given that in a true solid the actual number is 1023, it isevident that such a calculation cannot even come close to the desired solution.

Another approach tries to develop approximate solutions to Schrödinger’sequation, usually based on some kind of perturbation theory. It allows the treatmentof the solid in the thermodynamic limit, but is of course not exact and we quite oftenneed what we call ‘physical intuition’ to find the ‘right’ approximation.

The language used in both cases is based on the concepts of correlation functions,respectively Green’s functions.

1.1.1 Scattering and correlation functions

In inelastic scattering experiments we measure the differential cross-sectionðd2σ=dΩdħωÞ. In the lowest order, the first Born approximation, we obtain thisquantity by applying Fermi’s Golden Rule (a derivation can be found for example inMarshall and Lovesey (Theory of Thermal Neutron Scattering (Clarendon Press,Oxford) 1971)) as

d2σ

dΩdħω¼ α

k 0

k

Xi, f

pijhk0, f jHintjk, iij2δðħω� Ef þ EiÞ ð1:1Þ

The Hamiltonian H int describes the interaction between the scattered objectsand the target, ħk, ħk0 are the momenta of the incident and scattered particles,respectively (see figure 1.1), i and f denote the initial and final states and pi ¼ e�βEi=Z

doi:10.1088/978-1-627-05328-0ch1 1-1 ª Morgan & Claypool Publishers 2014

is the probability for a given initial state of the target. The δ-function finally takescare of energy conservation, and the prefactor α collects details of the initial andfinal states.

The orbital part of the wave functions of the scattered particles can be describedby plane waves eikr, i.e. the transition matrix elements become (the spin will beneglected for simplicity here)

hk0f jH intjk, ii ¼ hf jH intð�qÞjii ð1:2Þwhere

H intðqÞ ¼Z

d3r e�iqrH intðrÞ ð1:3Þ

and q ¼ k � k0 is the scattering vector (see figure 1.1).We can furthermore get rid of the explicit energies of target states by writing the

delta function as

δðħωÞ ¼ 1

ħδðωÞ ¼ 1

2πħ

Z þN

�Ndt eiωt ð1:4Þ

and introduce the Heisenberg representation

H intðq, tÞ ¼ expi

ħHt

� �H intðqÞ exp � i

ħH t

� �ð1:5Þ

where H denotes the target Hamiltonian with H jiðf Þi ¼ Eiðf Þjiðf Þi. Finally, we mayrewrite X

i

pi? ¼ h?i

as the thermal expectation values in the canonical ensemble.With these manipulations we obtain

d2σ

dΩdħω¼ α

k 0

k

1

2πħ

Z þN

�Ndt eiωthHy

intð�q, tÞH intð�q, 0Þi ð1:6Þ

for the differential cross-section.

Counter

Target

→k'

→k

→q

Figure 1.1. Scattering geometry.

Advanced Solid State Theory

1-2

The further evaluation depends on the details of the interaction between targetand scattered objects. For scattering of light or neutrons from nuclei in a solid, wecan represent this interaction as the sum of independent potentials centred at thepositions of the nuclei, i.e.

H intðrÞ ¼Xj

Vjðr� RjÞ: ð1:7Þ

The interaction between nuclei and neutrons or light is very short-range, i.e.approximately

V ðr� RjÞ ¼ V0δðr� RjÞ: ð1:8Þ

The cross-section then becomes

d2σ

dΩdħω¼ αV 2

0

k 0

k

1

2πħ

Z þN

�Ndt eiωt

Xjl

he�iqRjðtÞeiqRlð0Þi

¼ αV 20

ħk 0

kSðq, ωÞ ð1:9Þ

where we introduced the dynamical structure factor

Sðq, ωÞ ¼ 1

2π

Z þN

�Ndteiωt

Xjl

he�iqRjðtÞeiqRlð0Þi: ð1:10Þ

Let us introduce the nuclear density and its Fourier transform

nðrÞ ¼Xl

δðr� RlÞ , nðqÞ ¼Z

dr3e�iqr nðrÞ ¼Xl

e�iqRl ð1:11Þ

to express the dynamical structure factor as

Sðq, ωÞ ¼ 1

2π

Z þN

�Ndteiωt hnðq, tÞnyðq, 0Þi ð1:12aÞ

¼ 1

2π

Z þN

�Ndteiωt

Zd3r

Zd3r0 e�iqðr�r0Þhnðr, tÞnðr0, 0Þi: ð1:12bÞ

In this last equation we have expressed the structure factor through the quantity

Cnnðr, r0, tÞ :¼ hnðr, tÞnðr0, 0Þi ð1:13Þ

the density–density correlation function. Correlation functions of this type frequentlyoccur in the description of spectroscopic probes and are thus of fundamental interest.

1.1.2 Linear response and generalized susceptibilities

A frequent experimental setup is to probe the solid with a spatial and temporalvarying field. The measurement then collects data on a certain observable A

Advanced Solid State Theory

1-3

(magnetization, particle density, current density, . . . ) as a function of time. We willdenote the Hamiltonian of the solid with H0 and the action of the external field byH sðtÞ. In most practical cases the latter will be of linear form

H sðtÞ ¼ �Z

d3rBðrÞ U f ðr, tÞ

where BðrÞ is another observable of the system, and f(r, t) a real field. Frequentexamples will be electromagnetic fields E(r, t) and B(r, t), respectively, and the cor-responding operators in the solids are electronic or nuclear charge densities e nðr)where e denotes themagnitude of the elementary charge, and currents jðrÞ or spins sðrÞ.

To keep the notation simple, we will ignore the spatial dependence in thefollowing, i.e. use

H sðtÞ ¼ �B U f ðtÞ:

We further assume that at a certain time t0 the field will be switched on, and that fortimes t < t0 the solid is in thermal equilibrium, i.e. can be described by a statisticaloperator ρ ¼ e�βH0=Z. The Hamiltonian H0 has a set of eigenvalues En witheigenvectors 9ni. We seek the time dependence of the thermal expectation value ofanother observable A due to the influence of the external field, hAiðtÞ.

For simplicity, we set t0 ¼ 0 in the following. For t < t0 ¼ 0 we then have

hAi ¼Xn

e�βEn

ZhnjAjni

as the system is in thermal equilibrium. For t > 0 we first have to choose whichpicture we want to study the time dependence in. We here choose the Heisenbergpicture, i.e. the operators are time-dependent. Note that the statistical operator istime-independent in this picture. With this convention we obtain

hAiðtÞ ¼Xn

e�βEn

ZhnjAðtÞjni

for t > 0. For the time-dependence of the operator A we obtain with H t ¼ H0 þ H sðtÞ

dA

dt¼ i

ħ[H tðtÞ, AðtÞ]

which we solve by the ansatz

AðtÞ ¼ UyðtÞeiH t=ħAe�iH t=ħ UðtÞ ¼ U

yðtÞ AHðtÞ UðtÞ:

Taking the derivative yields

dAðtÞdt

¼ dUyðtÞdt

AHðtÞUðtÞ þ UyðtÞ dAHðtÞ

dtUðtÞ þ U

yðtÞAHðtÞdUðtÞdt

¼! i

ħ½H tðtÞ, AðtÞ�:

Advanced Solid State Theory

1-4

With the standard result

dAHðtÞdt

¼ i

ħ[H , AHðtÞ]

we obtain as differential equation for the unitary operator UðtÞdUðtÞdt

¼ 1

iħeiH t=ħHsðtÞ e�iHt=ħUðtÞ ¼ 1

iħH s,HðtÞUðtÞ

where we used HHðtÞ ¼ H and UyðtÞ H þ H s,HðtÞ� �

UðtÞ ¼ H tðtÞ. The initial con-dition is Uð0Þ ¼ 1.

Thus, we have to evaluate

hAiðtÞ ¼ hU yðtÞ AHðtÞ UðtÞi:

Up to now everything was exact. Of course we cannot calculate the operatorUðtÞ exactly. The simplest approximation is to assume that the external field is‘weak’ and expand UðtÞ to lowest order in the field, i.e.

UðtÞ � 1þ 1

iħ

Z t

0

dt0 H s,Hðt0Þ ¼ 1� 1

iħ

Z t

0

dt0 f ðt0Þ U BHðt0Þ:

Furthermore, we have assumed that B is an observable of the system, i.e. B ¼ By,

and f(t) is a real field. Under these conditions we obtain

hAiðtÞ ¼ hAi þ i

ħ

Z t

0

dt0 hAHðtÞBHðt0Þ � BHðt0ÞAHðtÞi U f ðt0Þ: ð1:14Þ

This result describes the response of a system to an external field to linear order inthe field. This approximation is therefore called a linear response. We can further-more define the quantity1

χABðt � t0Þ :¼ i

ħΘðt � t0Þh[AðtÞ, Bðt0Þ]i; ð1:15Þ

which we call generalized susceptibility, and write formula (1.14) in the compactand intuitive form

hAiðtÞ ¼ hAi þZ N

�Ndt0 χABðt � t0Þ U f ðt0Þ ð1:16Þ

which is also called the Kubo formula.Some remarks:• The susceptibility depends only on the properties of the solid (through H0),but not on the applied field. Such a quantity is called the material property. Ingeneral the external field will modify the properties of the solid, which thenresults in so-called non-equilibrium or non-linear effects. These can actually be

1From now on the index ‘H’ denoting the time evolution in the Heisenberg picture with respect to H0 will bedropped.

Advanced Solid State Theory

1-5

quite important; for example, in optics it leads to effects like mode doubling,non-linear refraction, parametric amplification and so on.

• The susceptibility depends in thermal equilibrium only on the time differencet � t0. This property reflects the time-translational invariance of a solid inthermal equilibrium.

• As f(t) ¼ 0 for t < t0, the lower integration bound is actually t0.• Due to the Heaviside function, the upper integration bound is t, which meansthat in the response only properties of the system for times in the interval [t0, t]contribute. This property reflects causality in the physical process.

One particularly important time-dependence for an external perturbation is theharmonic one, e.g.

f ðtÞ ¼ f0 U cosðωtÞΘðt � t0Þ

with some frequency ω. The Heaviside function realizes the switching at t ¼ t0. Sucha ‘hard’ switching is, however, not really desirable. From a theoretical point ofview, we would rather like to have a very slow switching process (at least ontimescales relevant to the excitations in the solid), such that at each time we canthink of a system in quasi-equilibrium. Such a switching is called adiabatic. For-mally, we can realize it by replacing the Heaviside function by a term eδt, whereδ > 0 denotes an infinitesimal positive number, and take the limit t0 ! �N. Theexternal field then acquires the form2

f ðtÞ ¼ f0 eδt=ħ cosðωtÞ:

In the integral (1.16), this modification only affects the lower integration bound,which is now in fact �N. Due to the exponential factor, the integral is well definedfor t0 < 0. For t0 > 0, the exponential growth may be disturbing at first sight.However, the integral is still cut off at t0 ¼ t, i.e. here, too, the integral is well-definedand the limit δ ! 0 can be taken.

The peculiar time-dependence for the adiabatic switching can be most con-veniently re-expressed in complex notation as

~f zðtÞ ¼ f0 e�izt=ħ , z ¼ ħωþ iδ

which then leads to

hδAiðtÞ ¼Z N

�Ndt0 χABðt � t0Þ f0 e�izt0=ħ

¼ f0 e�izt=ħ

Z t

�Ndt0 χABðt � t0Þeizðt�t0Þ=ħ

¼ ~f zðtÞZ N

0

dt0 χABðt0Þeizt0=ħ

¼ ~f zðtÞχABðzÞ

2 The factor ħ�1 is convenience only.

Advanced Solid State Theory

1-6

where we introduced the Laplace transform

χABðzÞ ¼Z N

0

dt0 χABðt0Þeizt0=ħ ð1:17Þ

of the susceptibility, where hδAiðtÞ ¼ hAiðtÞ � hAi.We may now ask how the complex notation is related to the (real) measurement.

This is done, as usual, by using

f ðtÞ ¼ f0

2e�izt=ħ þ eiz*t=ħh i

and hence

hδAiðtÞ ¼ 1

2~f zðtÞχABðzÞ þ ~f z*ðtÞχABð�z*Þ� �

:

Since by assumption both A and B are observables, i.e. hermitian operators, the niceproperties

χABðzÞ* ¼ χBAðz*Þ ð1:18aÞχABð�zÞ ¼ χBAðzÞ ð1:18bÞ

follow, which leads to

hδAiðtÞ ¼ f0R e�iωt χABðħωþ iδÞ� �

:

Note, that we may set δ ¼ 0 in the term e�iωtþδt=ħ, but should always keep thisadiabatic factor in the argument of the susceptibility, as it determines the analyticalproperties and hence the result of taking the real part. Omitting it carelessly will leadto wrong results!

1.1.3 Susceptibilities and energy dissipation

To gain some feeling of what the imaginary part of the susceptibility means, let usinvestigate the rate of change of energy

PðtÞ ¼ d

dthH tðtÞi

which describes the momentary power absorbed or emitted by the solid. Using theHeisenberg equation of motion we have quite generally

d

dthAiðtÞ ¼ i

ħh[H tðtÞ, AðtÞ]i þ

@AðtÞ@t

����exp

* +

and in particular with A ¼ H t the simple result

PðtÞ ¼ �@t f ðtÞhBiðtÞ:

Advanced Solid State Theory

1-7

With our result (1.16) we can write

PðtÞ ¼ � df

dt

Z N

�Ndt0χBBðt � t0Þ f ðt0Þ:

Using again a harmonic time dependence for the field, i.e.

f ðtÞ ¼ f0

2eδt=ħ e�iωt þ eiωt

� �we obtain as before

hBiðtÞ ¼ f0

2e�iωtχBBðħωþ iδÞ þ eiωtχBBð�ħωþ iδÞ� �

:

Furthermore,

df

dt¼ f0

2�iωe�iωt þ iωeiωt� �

:

Multiplying the two expressions and taking the temporal average over one periodT ¼ 2π/ω, we obtain

B ¼ � f 204

iωχBBðħωþ iδÞ � iωχBBð�ħωþ iδÞ½ �:

Employing again the nice properties (1.18a) and (1.18b) for this special case A ¼ Bwe find

χBBð�ħωþ iδÞ ¼ χBBðħω� iδÞ ¼ χBBðħωþ iδÞ*

and particularly

R χBBð�ħωþ iδÞ ¼ R χBBðħωþ iδÞI χBBð�ħωþ iδÞ ¼ �I χBBðħωþ iδÞ

With these relations, we finally obtain

P ¼ f 202ωI χBBðħωþ iδÞ: ð1:19Þ

In words: the energy absorbed, respectively the power dissipated, by the solidfrom the experimental apparatus is controlled by the imaginary part of the corre-sponding susceptibility.

Advanced Solid State Theory

1-8

Example 1.1

Let us discuss a simple but important example, namely a particle in a harmonicpotential subject to an oscillatory field, i.e.

H t ¼p2

2mþ mω2

0

2x2 � x U f ðtÞ

and we want to calculate hxiðtÞ in linear response. For that we need

χxxðtÞ ¼i

ħΘðtÞh[xðtÞ, x 0ð Þ]i:

In Quantum Mechanics you surely have solved the Heisenberg equations for theposition operator (if not I urge you to do it) with the result

xðtÞ ¼ xcosðω0tÞ þp

mω0sinðω0tÞ

and therefore

[xðtÞ, x] ¼ �iħ1

mω0sinðω0tÞ:

The susceptibility then becomes

χxxðtÞ ¼ ΘðtÞ sinðω0tÞmω0

:

Note that ħ does not appear in the result any more. This is a peculiarity of theharmonic oscillator, telling us that the response to an external field will be identical inquantum and classical statistics.

Finally, the Laplace transform can be calculated straightforwardly as

χxxðzÞ ¼1

zþ ω0� 1

z� ω0:

In particular,

I χxxðωþ iδÞ ¼ πδðω� ω0Þ � πδðωþ ω0Þi.e. the poles of the susceptibility denote the fundamental oscillation frequencies of themodel.

This feature of the susceptibility is true in general: if we find, for a certainobservable of a system, that χA,A(q, z) develops a simple pole at some frequency ω(q),we can infer that this frequency belongs to a fundamental eigenmode or elementaryexcitation of the system, and the observable A is the generalized coordinate associatedwith this mode. In many cases we actually do not find a true pole, but rather a Lor-entian with a certain width Γ. If Γ� characteristic energy scales of the system, we thenstill speak of a fundamental mode or excitation of the system. Bearing in mind that thelife-time τ ¼ ħ=Γ� typical decay times in the system, such an excitation corre-spondingly has a life-time that is much larger than other typical times and in this sensecan be viewed as elementary excitation.

Advanced Solid State Theory

1-9

1.1.4 Isothermal versus adiabatic susceptibilities

In the previous sections we studied the response of a system to a perturbation, whichwas switched on slowly, i.e. assuming an adiabatic development of the system underthe perturbation. The resulting susceptibilities are correspondingly referred to asadiabatic susceptibilities. A particularly interesting limit is ω ! 0, which in the spiritof the Laplace transform corresponds to t ! N. In this limit the so-called transientresponse of the system to the switching should have been damped out, and theresponse should be that of the solid in the presence of the static external field.Therefore,

limω!0

χABðħωþ iδÞ

is called adiabatic static susceptibility.This concept does not work if either observable A or B commutes with the

Hamiltonian of the solid, because then h[AHðtÞ, B]i ¼ 0, respectively χABðtÞ ¼ 0, forall times t. Let us consider as a specific example a collection of non-interacting spinsin a homogeneous magnetic field h in the z direction, i.e.

H sðtÞ ¼ �hXi

Sz,iΘðtÞ:

Statistical physics tells us that the static susceptibility of free spins in a magneticfield is given by (Curie law)

χzzðTÞ ¼ � @2F

@h2¼ 1

kBT

SðS þ 1Þ3

while the result from our linear response theory would give χzzðTÞ ¼ 0.The reason for this apparent contradiction lies in the fact that Sz is a constant of

motion. Therefore, by slowly switching on the external field as in the case of theadiabatic susceptibility, the solid always stays in thermal equilibrium and wenecessarily must have hSziðtÞ ¼ hSzið0Þ ¼ 0.

To avoid this problem we must include the perturbation exactly into our calcu-lation. This is usually not possible with the full time dependence, but in many casesat least for a static field. We thus now study a solid described by a Hamiltonian H inthe presence of a static external field Hs ¼ �h U B. Expectation values can be cal-culated in the usual way as

hAih ¼ Tr ρh A

where

ρh ¼1

Zhe�βðH�hUBÞ

Zh ¼ Tr e�βðH�hUBÞ:

Advanced Solid State Theory

1-10

We want to calculate the static susceptibility

χABðTÞ :¼ limh!0

@hAih@h

also called isothermal static susceptibility.The calculation becomes trivial if [H0, B] ¼ 0. In the general case the derivative of

the statistical operator with respect to h is more subtle. To perform it we use arepresentation similar to the interaction picture in the time evolution, i.e.

e�βðH�hUBÞ ¼ e�βH Tτe�R β

0HsðτÞdτ

where

H sðτÞ :¼ eτH H se�τH

and Tτ a ‘time-ordering operator’, which ensures that the times in the multipleintegrals in the expansion of the exponential are always ordered decreasing from leftto right.

We can now again expand the exponential to lowest order and obtain

e�βðH�hUBÞ � e�βH 1þ h

Z β

0

dτ BðτÞ� �

:

For the expectation value this means

hAih �Z0

ZhhAi0 þ h

Z β

0

dτ hBðτÞAi0� �

and

Zh � Z0 1þ h

Z β

0

dτhBðτÞi0� �

¼ Z0 1þ βhhBi0� �

:

The latter relation can be used to calculate

Z0

Zh� 1

1þ βhhBi0�h!0

1� βhhBi0

and finally in order h

χABðTÞ ¼Z β

0

dτ hBðτÞAi � hBi U hAi: ð1:20Þ

The expectation values have to be calculated with respect to the Hamiltonian H ofthe solid without the external field.

Advanced Solid State Theory

1-11

For our example with H ¼ 0 and A ¼ B ¼ Sz the integral and expectation valuescan be performed trivially: we have hSzi ¼ 0, hSzðτÞSzi ¼ hS 2

z i and therefore

χzzðTÞ ¼ βhS 2

z i

Furthermore, the Hamiltonian H ¼ 0 preserves rotational invariance, i.e.hS 2

z i ¼ hS 2

x i ¼ hS 2

y i ¼ hS 2i=3 ¼ SðS þ 1Þ=3, which leads to the desired result.

1.2 Formal properties of Green’s functions1.2.1 Definition and equation of motion

To keep the notation simple, from now on we will use the convention ħ ¼ 1. Thediscussion in the previous section suggests that the following quantity is an inter-esting object:

GABðtÞ ¼ �iΘðt � t0Þ h AðtÞ, B� �

si ð1:21Þ

where A, B� �

s:¼ A U Bþ sB U A, i.e. for s ¼ �1 we have the usual commutator, while

for s ¼ þ1 we obtain an anticommutator. We adopt the convention that we alwayshave to take s ¼ �1 if at least one of the two operators is of bosonic type (i.e. anobservable or a creation/annihilation operator for boson). Only if both operators areodd in fermionic annihilation or creation operators is the anticommutator to be used.Confused? A simple example will make the convention clear in a few moments.

With this convention, our adiabatic susceptibilities are given by χABðtÞ ¼ �GABðtÞ.How the correlation functions and isothermal susceptibilites fall into place is thecentral theme of this section.

Why do we call these objects ‘Green’s functions’? To motivate this nomenclature,let us study the equation of motion for the quantity GABðtÞ. With the Heisenbergequation we find

d

dtGABðtÞ ¼ �iΘðtÞ

h_AðtÞ, B

is

D E� iδðtÞh A, B

� �si

¼ �iG[A,H ], BðtÞ � iδðtÞh A, B� �

si:

This equation has the structure D[G] ¼ δðtÞ, where D is some differential operator,i.e. the usual definition for a Green’s function in the theory of differential equations.We can again use the Laplace transform3 (1.17) to formally convert the differentialequation into an algebraic one. The result is

zGABðzÞ ¼ h A, B� �

si þ G[A,H ], BðzÞ: ð1:22Þ

3Due to the Heaviside function, the Fourier transform is of little use here.

Advanced Solid State Theory

1-12

This relation (1.22) is called the equation of motion. It is an exact relation and, astypical with such relations, is only of limited value for actual calculations. However,for simple systems or formal considerations it can often be quite useful.

1.2.2 Spectral representation and dissipation–fluctuation theorem

For the application of the formalism a very important aspect is the concept of thespectral representation. To this end let us consider the two correlation functions

C >ABðtÞ :¼ hAðtÞBi

C <ABðtÞ :¼ hBAðtÞi:

Example 1.2

Let us consider a simple example, namely non-interacting electrons. The Hamiltonianin second quantization is

H ¼Xkσ

εk cykσ ckσ ð1:23Þ

where εk is the dispersion of the band. We can now try to calculate all kinds of Green’sfunctions for this system. One, which is at first sight rather funny, is the so-called single-particle Green’s function

GkσðzÞ � Gckσ , c

ykσðzÞ

For the equation of motion we need the anticommutator [ckσ , cykσ ]þ ¼ 1 and the

commutator [ckσ , H ] ¼ εk ckσ. Note that the use of the commutator [ckσ , cykσ]� ¼

1� 2cykσ ckσ would be possible, too. However, the anticommutator directly ensures thatthe algebra of the fermionic operators is properly reflected by the Green’s function.This feature is very important for the consistency of the theory.

Inserting these two commutators into the equation of motion (1.22), we find

zGkσðzÞ ¼ 1þ εkGkσðzÞrespectively

GkσðzÞ ¼1

z� εk: ð1:24Þ

Note that as in the case of the harmonic oscillator in example 1.1, here, too, the poleof the Green’s function is located at the corresponding single-particle energy.

Performing the back transformation yields

GkσðtÞ ¼ �iΘðtÞe�iεk t:

Of course, this result could have been obtained without the detour through thecomplex plane!

Advanced Solid State Theory

1-13

Quite obviously,

GABðtÞ ¼ �iΘðtÞ C >ABðtÞ þ s UC <

ABðtÞ� �

:

These functions can be Fourier-transformed,

CαABðωÞ ¼

Z N

�Ndt eiωtCα

ABðtÞ

and with the definition of the Laplace transform (1.17) we can write

GABðzÞ ¼Z N

0

dteizt GABðtÞ

¼ �i

Z N

0

dt eiztZ N

�N

dω

2πe�iωt C >

ABðωÞ þ s UC <ABðωÞ

� �¼I z> 0�i

Z N

�N

dω

2π

�1

iðz� ωÞ C >ABðωÞ þ s UC <

ABðωÞ� �

:

Let us define the spectral function

ρABðωÞ :¼1

2πC >ABðωÞ þ s UC <

ABðωÞ� �

ð1:25Þ

to obtain the spectral representation

GABðzÞ ¼Z N

�Ndω

ρABðωÞz� ω

: ð1:26Þ

This relation has been obtained by assuming I z> 0. However, a similar argumentcan be made for a definition of ~GABðtÞ involving a factor Θð�tÞ4, leading to exactlythe same result, but this time with I z< 0. We can thus read off the formula (1.26)the analytical properties of GABðzÞ:

GABðzÞ is analytical in the upper, respectively lower, half plane. The spectralfunction describes the singularity on the real axis, because

ρABðωÞ ¼ � 1

2πiGABðωþ iδÞ � GABðω� iδÞ½ � ¼: � 1

πG00

ABðωÞ:

4One frequently denotes the Green’s function (1.21) as the retarded Green’s function, and its partner withΘð�tÞ as the advanced Green’s function.

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1-14

Often (but not always!) we have the situation that G00ABðωÞAℝ. Quite generally, we

can prove that

GABðzÞ* ¼ GBy,Ay ðz*Þ ð1:27aÞ

GABð�zÞ ¼ �s UGBAðzÞ: ð1:27bÞ

The proofs for these relations are left as exercises.With these relations and under the condition that B ¼ A

yit follows

GA,Ay ðzÞ* ¼ GA,Ayðz*Þ ð1:28aÞG00

A,AyðωÞ ¼ IGA,Ay ðωþ iδÞ: ð1:28bÞ

Moreover, we can prove (exercise!) that

C <ABðωÞ ¼ e�βωC >

ABðωÞ

and thus

ρABðωÞ ¼1

2πC >ABðωÞ 1þ se�βω

¼ � 1

πG00

ABðωÞ

With the above formula we can obtain

hAðtÞBi ¼ �Z N

�N

dω

π

G00ABðωÞe�iωt

1þ s U e�βωð1:29Þ

which is the promised connection between the Green’s function and the correlationfunction.

With equation (1.29), the special choice B ¼ Ay

and replacing GA, Ay ðzÞ ¼�χA, Ay ðzÞ we obtain for the special case t ¼ 0

hAAyi ¼Z N

�N

dω

π

χ 00A,Ay ðωÞ

1þ s U e�βωð1:30aÞ

¼Z N

�N

dω

π

I χA,Ay ðωþ iδÞ1þ s U e�βω

: ð1:30bÞ

The relation (1.30a) is called the dissipation–fluctuation theorem, because it relatesthe dissipation of energy due to the quantity A with its fluctuation. The latter aspectbecomes particularly apparent when A is an observable, i.e. A

y ¼ A holds. In this

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1-15

case we necessarily must have a ‘bosonic’ operator and s ¼ �1. The fluctuation–dissipation theorem then becomes

hA2i ¼Z N

�N

dω

π

I χAAðωþ iδÞ1� e�βω

Finally, if we write GABðzÞ ¼ G0ABðzÞ þ iG00

ABðzÞ, we can prove the Kramers–Kronigrelations5

G0ðzÞ ¼ � 1

πPZ N

�Ndζ

G00ðζÞω� ζ

ð1:31aÞ

G00ðzÞ ¼ þ 1

πPZ N

�Ndζ

G0ðζÞω� ζ

ð1:31bÞ

where PR: : : denotes the Cauchy principal value integral.

1.2.3 Thermodynamic Green’s functions

The last missing link is the one to the isothermal susceptibilities. To this end weintroduce a third variant6 of Green’s functions, defined as

GðτÞ :¼ �hTτ A[τ]Bi ¼�hA[τ]Bi for τ> 0

s U hBA[τ]i for τ< 0

(ð1:32Þ

where we introduce the notation7,8 A[τ] ¼ eτH Ae�τH and Tτ orders the times from leftto right in decreasing order. The factor s, as before, is þ1 if both A and B are offermionic type, �1 otherwise.

An important property of the Green’s function (1.32) follows from the invarianceof the trace under cyclic permutations:

hA[τ þ β]Bi ¼ hBA[τ]i:

With this relation, we have for �β � τ< 0

Gðτ þ βÞ ¼ �sGðτÞ

5 For simplicity the operators are omitted here.6We will, however, see in a moment that it is actually nothing new.7 To distinguish this unconventional time evolution from the actual Heisenberg time dependence, which has ani in the exponent.8 Formally, one can write A[τ] ¼ Að�iħτÞ. This ‘rotation’ t ! �iħτ in the complex plane is calledWick rotation.

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i.e. GðτÞ is a periodic function for bosonic operators, and an antiperiodic one forfermions. This periodicity can be used to perform a discrete Fourier transform

ωn :¼π

βn GðnÞ ¼ 1

2

Z β

�βdτeiωnτGðτÞ:

With the above (anti-) symmetry we can rewrite this relation as

GðnÞAB ¼

Z β

0

dτeiωnτGABðτÞ, ωn ¼

2πn

βfor s ¼ �1

ð2nþ 1Þπβ

for s ¼ þ1

8>>>><>>>>:

ð1:33Þ

GABðτÞ ¼1

β

XNn¼�N

e�iωnτGðnÞAB : ð1:34Þ

The Fourier frequencies ωn are calledMatsubara frequencies. In particular, for B andA observables, i.e. bosonic operators, the Fourier coefficient for the frequencyω0 ¼ 0 is

Gð0ÞAB ¼

Z β

0

dτ GABðτÞ ¼ �Z β

0

dτhA[τ]Bi

which is, apart from the subtraction of the product of expectation values, nothingbut the definition of the isothermal static susceptibility (1.20).

The important aspect of these thermodynamic Green’s functions is that they arebasically the same quantities as the adiabatic Green’s functions, but are much easierto use for both formal considerations and for applications. Using the spectralrepresentation, one can show that (try it as exercise or study [6, 7])

GðnÞAB ¼ GABðz ¼ iωnÞ: ð1:35Þ

We will therefore drop the distinction between G and G and use GðzÞ with zAℂinstead.

If we know Gðiωn) analytically for all nAℤ, it is by virtue of relation (1.35)trivial to obtain G(z) and from it the spectral function ρ(ω). However, withnumerical methods we can usually only calculate a finite set of Matsubara coeffi-cients, which are often also plagued with numerical or statistical errors. Then it turnsout that the process GðiωnÞ ! Gðωþ iδÞ is a highly ill-defined task, called analyticalcontinuation.

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1.2.4 Poisson’s summation formula

In working with thermodynamic Green’s functions a frequent task to perform is thecalculation of sums over the Matsubara frequencies, which are of the form

1

β

XNn¼�N

FðiωnÞ:

Here, F(z) is some given function of a complex variable. The trick is to find ameromorphic function g(z), which has isolated, simple poles at zn ¼ iωn with residue�β�1. Then

1

β

XNn¼�N

FðiωnÞ ¼ � 1

2πi

IC

gðzÞ UFðzÞdz: ð1:36Þ

The closed curve C encircles all poles of the function g(z), and there should be nosingularities of F(z) within C. This formula is called Poisson’s summation formula.

Example 1.3

We again study the non-interacting electron system defined by the Hamiltonian (1.23).For τ > 0 the thermodynamic single-particle Green’s function is

GkσðτÞ ¼ �hckσ[τ]cykσi:

With the Baker–Campbell–Hausdorff formula it is easy to show that ckσ [τ] ¼e�εkτ ckσ (try it as exercise) and thus

GðnÞkσ ¼ �

Z β

0

dτeiωnτ e�ɛkτ hckσ cykσi

¼ � eiωnβ e�ɛkτ � 1

iωn � ɛkhckσ c

ykσi:

Since we deal with fermionic operators, we have ωn ¼ ð2nþ 1Þπ=β, henceeiωnβ ¼ �1. Finally, from statistical physics we know that

hckσ cykσi ¼

1

1þ e�εkτ

so that the final result is

GðnÞkσ ¼ 1

iωn � εk:

This is, as predicted, just Gkσðz ¼ iωnÞ. Note that the use of the anticommutator inthe definition (1.21) of the retarded Green’s function becomes essential, as otherwisethis property would not hold.

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A possible realization is shown by the full line in figure 1.2. Note that the pole forωn ¼ 0 in the case of bosonic operators must not appear here and has to be treatedseparately, because the Green’s function is usually not analytic on the real axis!

What kinds of functions are suitable? The answer is given in the following table:

bosons fermions

g1ðzÞ � 1

eβz � 1

1

eβz þ 1

g2ðzÞ1

e�βz � 1� 1

e�βz þ 1

g3ðzÞ � 1

2coth

βz

2� 1

2tanh

βz

2

These are either the Fermi or the Bose function, or can be expressed by them.Which of the three we have to choose depends on the asymptotic behaviour of thefunction F(z). In order to understand how this choice has to be made, let us considera specific example:

a)

iωn iωn

b)

z

z

Figure 1.2. Integration paths and their possible deformations in the application of Poisson’s summationformula.

Example 1.4

The task is to calculate the expectation value hcykσ ckσi. This expectation value can beexpressed via the thermodynamic Green’s function as

Gkσð�δÞ ¼ �hTτckσ[�δ]cykσi ¼ hcykσ ckσ[δ]i

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1-19

This example shows how to choose the proper function. It also illustrates the wayPoisson’s formula works in simple cases with poles only. How do we proceed ifwe do not know the poles of GkσðzÞ?

where δ > 0 is infinitesimal. On the other hand,

Gkσð�δÞ ¼ 1

β

XNn¼�N

e�iωnð�δÞGkσðiωnÞ ¼1

β

XNn¼�N

eiωnδ

iωn � εk:

With Poisson’s summation formula, we obtain

hcykσ ckσi ¼ � 1

2πi

IC

eδz

z� εkgðzÞdz

where the curve C is the full line in figure 1.2(a). Note that it does not touch the realaxis, and that all Matsubara frequencies ωn 6¼ 0. The only singularity of the integrand,except for the ones at the Matsubara frequencies, is at z ¼ εk, i.e. on the real axis.Provided the integrand decays fast enough for 9z9 ! N, we may deform the contourinto the dashed one in figure 1.2(a), which is denoted with C 0, leaving only the pole onthe real axis in its interior. For R z ! �N, the term eδz ensures convergence of theintegral. To ensure fast enough decay for R z ! þN, we need a g(z) from thefermionic sector, that goes to zero faster as eδz diverges. The one that does the job isg1ðzÞ, i.e. we obtain (we can now safely drop the ‘convergence term’ eδz)

hcykσ ckσi ¼1

2πi

IC0

1

z� εk

1

eβz þ 1dz ¼ 1

1þ eβεk:

i.e. Fermi’s function, as expected.

Example 1.5

Well, we still know that the Green’s function is analytical in the upper and lower halfplanes; all its singularities are restricted to the real axis, and we still need to ensure theconvergence for 9z9 ! N, i.e. we must choose g1(z) in Poisson’s summation formula.It now reads

hcykσ ckσi ¼ � 1

2πi

IC

GkσðzÞ1

eβz þ 1dz:

Since we do not know either the location or the type of the singularities of Gkσ(z),we cannot use the deformation in figure 1.2(a), but we can still deform C according tofigure 1.2(b), i.e. have integration paths parallel to the real axis with an infinitesimal offsetinto either the upper or lower half plane. With this choice, the integral becomes

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1-20

1.3 Perturbation theory and Feynman diagrams

Up to now we have been concerned with the formal properties of the creatures calledGreen’s functions. Of course, we are interested in doing physics, i.e. how can wecalculate them for systems of interest? One possibility would be the equation ofmotion (1.22). As already mentioned, this usually does not lead very far, only in verysimple cases do we obtain a closed set of equations. Usually, we find an infinitehierarchy of equations, and nothing is gained.

Another way is to treat the problematic part of the Hamiltonian as perturbation,and hope that the corresponding series is simple enough to be handled. Althoughslightly more formal, we will pursue this way and learn about the modern way ofdoing calculations graphically, namely via Feynman diagrams.

Example 1.6

Finally, you may wonder what the result for a function

FðiωnÞ ¼1

iωn � a1

1

iωn � a2

might be. Of course, a1 and a2 have to be such that they do not introduce poles at theMatsubara frequencies ωn, but can otherwise be arbitrary complex numbers. The firstthing to note is that F(z) goes to zero like 1/9z92 for 9z9 ! N, i.e. the choice of g(z) heredoes not matter. The result is (prove as an exercise)

1

β

XNn¼�N

FðiωnÞ ¼gða1Þ � gða2Þ

a1 � a2ð1:38Þ

Again: the special pole at ωn ¼ 0 for bosons is not included here and must be treatedseparately!

hcykσ ckσi ¼ � 1

2πi

Z N

�NGkσðωþ iδÞ � Gkσðω� iδÞ½ � 1

eβω þ 1dω

¼Z N

�N

1

eβω þ 1� 1

π

!I Gkσðωþ iδÞdω

¼Z N

�N

ρkσðωÞeβω þ 1

dω: ð1:37Þ

This result is exact and can be used to calculate the occupation number for Fer-mions, provided we know the spectral function ρkσðωÞ.

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1-21

In the following, we will study the Hamiltonian of electrons in a solid. We willassume that the Born–Oppenheimer approximation has been carried through, i.e.the lattice leads to a static potential UðrÞ for the electrons. In second quantizationthe Hamiltonian then reads

H ¼ H0 þ H s ð1:39Þ

H0 ¼Xσ

Zd3r Ψ

yσðrÞ � r

2mþ UðrÞ

� �ΨσðrÞ ð1:40Þ

H s ¼1

2

Xσ1;σ2

Zd3r1d

3r2Ψyσ1ðr1ÞΨ

yσ2ðr2ÞV ðr1 � r2ÞΨσ2

ðr2ÞΨσ1ðr1Þ ð1:41Þ

where the field operators obey the anti-commutation relations

[ΨyσðrÞ, Ψσ0 ðr0Þ]þ ¼ δσ,σ0 δðr� r0Þ

and V(r) is the Coulomb interaction between the electrons. As this notation requiresa large number of indices, respectively arguments, to be cared for, a short-handnotation is appropriate. We will abbreviate the set (r1, σ1, τ1) ! 1, i.e. writeΨσ1

ðr1, τ1Þ ! Ψð1Þ,V(r1 � r2)δ(τ1 � τ)δ(τ2 � τ) ! V(1, 2; τ) andP

σ1

Rd3r1

Rdτ1 !R

d1 and so on in the following.We want to calculate the single-particle Green’s function

Gð1, 2Þ ¼ �hT τΨð1ÞΨyð2Þi:

To deal with the (imaginary) time dependence of the operators, we use the inter-action picture, where

e�τH ¼ e�τH0 S½τ; 0�

S[τ; 0� ¼ T τ exp �Z τ

0

dλHs[λ]� �

, H s[λ] ¼ eλH0H se�λH0

In this picture, we can write the Green’s function as

Gð1; 2Þ ¼ � 1

ZTr e�βH0 S[β, τ1]Ψð1ÞS[τ1, τ2]Ψ

yð2ÞS[τ2; 0]n o

where now the time dependence of all operators is with respect to H0. Due to thepresence of the time ordering operators in the evolution operators S[τ, τ0], we canformally (!) rearrange the operators at will, and obtain

G 1, 2ð Þ ¼ � 1

ZTr e�βH0 S[β; 0]Ψð1ÞΨyð2Þn o

:

Expanding the evolution operator in its Taylor series this leads to

Gð1, 2Þ ¼ � Z0

Z

XNn¼0

ð�1Þn

n!

Z β

0

dτ1?dτnhT Hs[τ1]?H s[τn]Ψð1ÞΨyð2Þi0

Z

Z0¼XNn¼0

ð�1Þn

n!

Z β

0

dτ1?dτnhT H s[τ1]?H s[τn]i0

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1-22

where h?i0 means expectation value with respect to H0.The general structure of the Green’s function is

G 1, 2ð Þ ¼ � numerator

denominator

and both numerator and denominator are given as a series

numerator ¼ numeratorð0Þ þ numeratorð1Þ þ?

denominator ¼ denominatorð0Þ þ denominatorð1Þ þ?:

For the particular Hamiltonian (1.39), these different contributions have theexplicit form

numeratorð0Þ ¼ hT τΨð1ÞΨyð2Þi0

numeratorð1Þ ¼ ð�1Þ1

1! U 2

Z β

0

dτ

Zd3

Zd4V ð3, 4; τÞ

× hT τΨyð3ÞΨyð4ÞΨð4ÞΨð3ÞΨð1ÞΨyð2Þi0

denominatorð0Þ ¼ 1

denominatorð1Þ ¼ ð�1Þ1

1! U 2

Z β

0

dτ

Zd3

Zd4V ð3, 4; τÞ× hT τΨ

yð3ÞΨyð4ÞΨð4ÞΨð3Þi0

and so on. The quantities we must evaluate are thus of the type

hT τΨð1ÞΨð2Þ?ΨðnÞΨyðn0Þ?Ψyð20ÞΨyð10Þi0:

This looks rather hopeless at first sight. However, the special form of H0, moreprecisely that it is bilinear in the field operators, allows for a considerable simplifi-cation of this expression. The underlying theorem is as follows.

Theorem 1.1 (Wick’s theorem). Let H0 be bilinear in Bose or Fermi field operators andαi an arbitrary linear combination of these field operators. Furthermore, letαi[τ] ¼ eτH0 αie�τH0 . Then

Tτ Ln

k¼1

αk [τk ]

* +0

¼Xni¼2

ð�sÞihTτα1[τ1]αi[τi]i0 Tτ Li�1

k¼2

αk [τk ] Ln

k¼iþ1

αk [τk ]

* +0

:

The statistical factor s ¼ þ1 if the field operators are fermions, s ¼ �1 otherwise.

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1-23

Sounds complicated? Well, it is not. Take as an example hc1c2cy20cy10i0, where

cðyÞi annihilates (creates) a fermion with quantum numbers i. Wick’s theorem thenstates with s ¼ 1

hT τc1c2cy20cy10i0 ¼ hT τc1c2i0 hT τc

y20cy10i0 þ hT τc1c

y10i0hT τc2c

y20i0

� hT τc1cy20i0hT τc2c

y10i0:

The first term hTτc1c2i0 violates particle number conservation and thus usuallyvanishes. A notable example is superconductivity, where precisely these expectationvalues become finite.

Wick’s theorem can be applied recursively. When, after building the first con-traction, the second term is still a product of more than two operators, we simplyapply the theorem again, until finally only a sum over products of expectation valuesof pairs of operators remains. The general recipe is thus fairly simple. From thegiven set of operators, pick a partitioning into pairs, build the expectation values forthese individual pairs, multiply them and sum over all possible partitioning. If fer-mions are being dealt with, count the number of commutations needed to bring thepairs together. If this number is even, nothing happens; if it is odd, the correspondingterm acquires a minus sign.

Let us apply this scheme to the expectation value in the term numerator(1). It reads

hT τΨyð3ÞΨyð4ÞΨð4ÞΨð3ÞΨð1ÞΨyð2Þi

¼Wick’s theorem � hT τΨyð3ÞΨð4ÞihT τΨ

yð4ÞΨð3ÞihT τΨð1ÞΨyð2Þi

þ hT τΨyð3ÞΨð4ÞihT τΨ

yð4ÞΨð1ÞihT τΨð3ÞΨyð2Þi

þ hT τΨyð3ÞΨð3ÞihT τΨ

yð4ÞΨð4ÞihT τΨð1ÞΨyð2Þi

� hT τΨyð3ÞΨð3ÞihT τΨ

yð4ÞΨð1ÞihT τΨð4ÞΨyð2Þi

� hT τΨyð3ÞΨð1ÞihT τΨ

yð4ÞΨð4ÞihT τΨð3ÞΨyð2Þi

þ hT τΨyð3ÞΨð1ÞihT τΨ

yð4ÞΨð3ÞihT τΨð4ÞΨyð2Þi

¼ Gð0Þð4, 3ÞGð0Þð3, 4ÞGð0Þð1, 2Þ � Gð0Þð4, 3ÞGð0Þð1, 4ÞGð0Þð3, 2Þ� Gð0Þð3, 3ÞGð0Þð4, 4ÞGð0Þð1, 2Þ þ Gð0Þð3, 3ÞGð0Þð1, 4ÞGð0Þð4, 2Þþ Gð0Þð1, 3ÞGð0Þð4, 4ÞGð0Þð3, 2Þ � Gð0Þð1, 3ÞGð0Þð3, 4ÞGð0Þð4, 2Þ:

The superscript (0) attached to the Green’s functions indicates that these are theGreen’s functions for the non-interacting system described by H0 only. Careful: inconverting the expectation values into Green’s functions, we have to remember thesigns in the definition, for example

hT τΨyð3ÞΨð4Þi ¼ �hT τΨð4ÞΨ

yð3Þi ¼ Gð4, 3Þ:

Well, this expression does not really much look simpler, does it? The important ideaof Richard Feynman was to now convert these expressions into a graphic language.

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1-24

For this purpose let us introduce the pictogram 1 2 ¼ G(0)(1, 2) for the non-interacting Green’s function and 1 2 ¼ V(1, 2) for the interaction. The point

, where Green’s function lines and interaction lines meet, we call a vertex.With this convention, the various terms in the above expression, after multipli-

cation with the proper interaction terms, give

1 2

3 4

1 24 3

1 2

3 4

1 24

3

1 23

4

1 2 .3 4

These diagrams consist of two distinct subclasses. The first and the third are calleddisconnected diagrams, and after integration over the variables 3 and 4 are of theform Γ Gð0Þ(1, 2). If we write down the diagrams for the partition function Z/Z0,we will observe that the parts Γ multiplying the expansion of G(1, 2) are preciselythe terms appearing there. In fact, one can prove that summing up all disconnecteddiagrams with a fixed lower part simply gives a factor Z/Z0, i.e. the factor denom-inator is cancelled exactly in the series expansion for the Green’s function.

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1-25

The rather cumbersome proof can be found in standard text books on many-bodyphysics, e.g. [7]. This is the first important result:

We only need to perform a perturbation expansion for the numerator.

The remaining four diagrams are called connected diagrams. Here, too, somegeneral features can be identified: diagrams 2 and 6, respectively 4 and 5, areidentical, except for the position of the vertices with the internal variables 3 and 4.Since these variables are integrated over, the contributions are the same, and cancelthe factor 1/2 from the interaction. If we perform this analysis for a general nthorder diagram, we will find that for the 2n internal variables there exist 2 n!arrangements, which give identical results after integration over the internalvariables, i.e. the factor (2 n!)�1 in front of the nth order term will be cancelledexactly (see e.g. [7]).

Finally, for the actual calculation we can make use of the temporal and spatialtranslational invariance of the Green’s functions. As usual, I will confine the solid toa cube of volume Ω ¼ L3 and impose periodic boundary conditions. Thus theallowed k vectors will be discrete with values k ¼ 2π

LZ. We can now perform Fourier

transformations with respect to space and imaginary time and replaceZd3r

Zdτ ! 1

β

Xk

Xωn

:

After these preliminaries we can now formulate the rules for Feynman diagrams.

1. Draw all topologically distinct connected diagrams.2. Assign a momentum, Matsubara frequency and spin to each Green’s function line,

and a momentum to each interaction line. At each vertex, the sum of incoming andoutgoing momenta, spins and frequencies must add up to zero (momentum, spinand energy conservation at each vertex)9.

3. Sum over all internal variables. For a closed loop beginning and ending at the samevertex, add a factor eiωnδ, where ωn denotes the internal Matsubara frequency of thisloop.

4. Each diagram carries a sign (�1)nþF, where n is the order in the interaction V and Fthe number of closed fermionic loops.

Let us write down all diagrams up to order n ¼ 2 in the interaction, where a labelk1 ¼ ðk1, ωn1 , σ1Þ now collects momentum, Matsubara frequency and spin quantumnumber.

9 Each Greens’s function line is replaced by a factor Gkσðiωn) and each interaction line by the Fouriertransform of the Coulomb potential ωq ¼ 4πe2/q2.

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We have introduced as a pictogram for the full Green’s function a double linewith an arrow.

Again, we can identify certain structures in the diagrams: the second in the thirdline is the product of the two in the second line. Similarly, the first in the fourth line isthe product of the second from the second line with itself. Diagrams of this type arecalled single-particle reducible, meaning that one can divide them into separatepieces by cutting a single Green’s function line. The four diagrams starting fromthe second in the fourth line each have a diagram of the type in the second line asinsertion in one internal line. Only the first in the third line and the last diagramconstitute a new class, they can neither be divided into smaller substructures bycutting only one Green’s function line, nor are they insertions of such simplersubstructures in Green’s function lines. Such diagrams are called irreducible.

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Let us look at the reducible diagrams more closely. The schematic structure is

where the diagram collects all possible irreducible diagrams. It is now obvious

that with the help of this irreducible single-particle self-energy or proper single-particle self-energy one can rewrite the perturbation series of the single-particleGreen’s function as

If we denote the self-energy with the symbol ΣkσðiωnÞ, then the above diagrammaticseries can be written as

GkσðiωnÞ ¼ Gð0Þkσ ðiωnÞ

XNn¼0

ΣkσðiωnÞ UGð0Þkσ ðiωnÞ

h in¼ G

ð0Þkσ ðiωnÞ

1� Gð0Þkσ ðiωnÞΣkσðiωnÞ

:

Noting that Gð0Þkσ ðiωnÞ ¼ iωn � εk½ ��1, we obtain as the exact expression for the

single-particle Green’s function

GkσðiωnÞ ¼1

iωn � εk � ΣkσðiωnÞ: ð1:42Þ

As usual with such exact relations, they are not really useful, because to calculateΣkσðiωnÞ is exactly as complicated as the calculation of GkσðiωnÞ. However, at leastthe perturbation series contains only irreducible diagrams.

Another way of presenting the series is the so-called Dyson equation

.

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Finally, we can write down the first few diagrams in the series expansion of the self-energy as

ð1:43Þ

Note that in the Feynman series for ΣðkÞ we have ‘amputated’ the external ‘legs’,representing the factor Gð0ÞðkÞ, but must fix the external quantum numbers kto one of these external legs. This is denoted by the labels k appearing at theoutermost vertices. The quantum numbers we have to sum over are only theinternal ones.

There are, in (1.43), a few irreducible second order diagrams missing, namely

+ + +

These are, from right to left, two diagrams similar to the first in (1.43), but withthe same self-energy parts inserted into the internal Green’s function line. The leftmostdiagrams are similar to the second in (1.43), again with the first two inserted into theinternal Green’s function line. Obviously, with a fixed external line configuration,we can always add to the internal line a complete series of self-energy diagrams, thustransforming each internal Gð0ÞðkÞ ! GðkÞ. With this observation in mind, we canwrite (1.43) also in the form

ð1:44Þ

The self-energy diagrams with all internal Green’s function lines replaced by thefull interacting Green’s function are called skeleton diagrams, and the series corre-spondingly skeleton series.

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Let us apply our new knowledge to calculate the self-energy in the lowest order inthe interaction, i.e. retain the first two terms only. This is nothing but the Hartree–Fock approximation:

The first diagram, the tadpole, is also referred to as Hartree diagram, the secondconsequently as Fock or exchange diagram.

The signs of the two diagrams are according to the diagram rules. Note thatin evaluating the second contribtution, we also need to introduce a factor eiωn0δ,because the Green’s function alone is not sufficient to ensure convergence. In sucha case, we use the convention that the Fermi, respectively Bose, function is taken inthe integral10.

If we use the skeleton series for the self-energy instead, and keep only the first twodiagrams, we arrive at the self-consistent Hartree–Fock approximation

where Ek :¼ εk þPHF

kσ , i.e. this equation is a self-consistency condition forPHF

kσ .The Hartree-Fock approximation to the self-energy is independent of the

Matsubara frequency, and in particular IPHF

kσ¼ 0. We only obtain a renormali-zation of the band energies, Ek :¼ εk þ

PHFkσ . Lifetime effects, i.e. damping of these

single-particle excitations, occur in the next order, for example through the thirddiagram in (1.43).

10 The actual reason is that the frequency independence of the interaction enforces a δ(τ) in the time domain,i.e. both operators have to be evaluated at the same time. In this case the ordering cykσ ckσ is appropriate, whichis enforced by the choice of the convergence factor.

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