Many-particle physics

Lecture notes for P654, Cornell University, spring 2003.c©Piet Brouwer, 2004

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1 Preliminaries

1.1 Thermal average

In the literature, two types of theories are constructed to describe a systemof many particles: Theories of the many-particle ground state, and theoriesthat address a thermal average at temperature T . In the book by Doniachand Sondheimer, both approaches are described. Here, we’ll limit ourselvesto the finite-temperature theory. In all cases of interest to us, results atzero temperature can be obtained as the zero temperature limit of the finite-temperature theory.

The thermal average of an operator A is defined as

〈A〉 = trAρ, (1.1)

where ρ is the density matrix. It describes the thermal distribution over thedifferent eigenstates of the system. The symbol tr denotes the “trace” of anoperator, the summation over the expectation values of the operator over anorthonormal basis set,

trAρ =∑

n

〈n|Aρ|n〉. (1.2)

The basis set |n〉 can be the collection of many-particle eigenstates of theHamiltonian H , or any other orthonormal basis set.

In the canonical ensemble of statistical thermodynamics, the trace is takenover all states with N particles and one has

ρ =1

Ze−H/T , Z = tr e−H/T . (1.3)

Using the basis of N -particle eigenstates |n〉 of the Hamiltonian H , the ther-mal average 〈A〉 can then be written as

〈A〉 =

∑

n e−En/T 〈n|A|n〉∑

n e−En/T

. (1.4)

In the grand-canonical ensemble, the trace is taken over all states, irre-spective of particle number, and one has

ρ =1

Ze−(H−µN)/T , Z = tr e−(H−µN)/T . (1.5)

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1.2 Schrodinger, Heisenberg, and interaction picture

There exist formally different but mathematically equivalent ways to for-mulate the quantum theoretical dynamics. These different formulations arecalled “pictures”. The idea behind the different pictures is that the physi-cal content of the quantum theoretical formalism is contained in expectationvalues only. For an operator A, the expectation value in the quantum state|ψ〉 is

A = 〈ψ|A|ψ〉. (1.6)

The time-dependence of the expectation value of A can be encoded throughthe time-dependence of the quantum state ψ, or through the time-dependenceof the operator A, or through both.1

In the Schrodinger picture, all time-dependence is encoded in the quantumstate |ψ(t)〉; operators are time independent, except for a possible explicittime dependence.2 Probably, this is the picture of quantum mechanics thatyou are best used to. The time-evolution of the quantum state |ψ(t)〉 isgoverned by a unitary evolution operator U(t, t0) that relates the quantumstate at time t to the quantum state at a reference time t0,

|ψ(t)〉 = U(t, t0)|ψ(t0)〉. (1.7)

Quantum states at two arbitrary times t and t′ are related by the evolutionoperator U(t, t′) = U(t, t0)U

†(t′, t0). The evolution operator U satisfies thegroup properties U(t, t) = 1 and U(t, t′) = U(t, t′′)U(t′′, t′) for any three timest, t′, and t′′, and the unitarity condition

U(t, t′) = U †(t′, t). (1.8)

The time-dependence of the evolution operator U(t, t0) is given by the Schro-dinger equation

i~∂U(t, t0)

∂t= HU(t, t0). (1.9)

1You can find a detailed discussion of the pictures in a textbook on quantum mechanics,e.g., chapter 8 of Quantum Mechanics, by A. Messiah, North-Holland (1961).

2An example of an operator with an explicit time dependence is the velocity operatorin the presence of a time-dependent vector potential,

~v =1

m

(

−i~~∇− e ~A(t))

.

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For a time-independent Hamiltonian H , the evolution operator U(t, t0) is thewell-known exponential function of H ,

U(t, t′) = exp[−iH(t− t′)/~]. (1.10)

If the Hamiltonian H is time dependent, there is no such simple result as Eq.(1.10) above, and one has to solve the Schrodinger equation (1.9).

In the Heisenberg picture, the time-dependence is encoded in operatorsA(t), whereas the quantum state |ψ〉 is time independent. The Heisenbergpicture operator A(t) is related to the Schrodinger picture operator A as

A(t) = U(t0, t)AU(t, t0), (1.11)

where t0 is a reference time. The Heisenberg picture quantum state |ψ〉 hasno dynamics and is equal to the Schrodinger picture quantum state |ψ(t0)〉at the reference time t0. The time evolution of A(t) then follows from Eq.(1.9) above,

dA(t)

dt=i

~[H(t), A(t)]− +

∂A(t)

∂t(1.12)

where H(t) = U(t0, t)H(t0)U(t, t0) is the Heisenberg representation of theHamiltonian and the last term is the Heisenberg picture representation of aneventual explicit time dependence of the observable A. The square brackets[·, ·]− denote a commutator. The Heisenberg picture Hamiltonian H(t) alsosatisfies the evolution equation (1.12),

dH(t)

dt=i

~[H(t), H(t)]− +

∂H

∂t=∂H(t)

∂t.

For a time-independent Hamiltonian, the solution of Eq. (1.12) takes thefamiliar form

A(t) = eiH(t−t0)/~Ae−iH(t−t0)/~. (1.13)

You can easily verify that the Heisenberg and Schrodinger pictures areequivalent. Indeed, in both pictures, the expectation values (A)t of the ob-servable A at time t is related to the operators A(t0) and quantum state|ψ(t0)〉 at the reference time t0 by the same equation,

(A)t = 〈ψ(t)|A|ψ(t)〉= 〈ψ(t0)|U(t0, t)AU(t, t0)|ψ(t0)〉= 〈ψ|A(t)|ψ〉. (1.14)

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The interaction picture is a mixture of the Heisenberg and Schrodingerpictures: both the quantum state |ψ(t)〉 and the operator A(t) are timedependent. The interaction picture is usually used if the Hamiltonian isseparated into a time-independent unperturbed part H0 and a possibly time-dependent perturbation H1,

H(t) = H0 +H1. (1.15)

In the interaction picture, the operators evolve according to the unperturbedHamiltonian H0,

A(t) = eiH0(t−t0)/~A(t0)e−iH0(t−t0)/~, (1.16)

whereas the quantum state |ψ〉 evolves according to a modified evolutionoperator UI(t, t0),

UI(t, t0) = eiH0(t−t0)/~U(t, t0). (1.17)

The evolution operator that relates interaction picture quantum states attwo arbitrary times t and t′ is

UI(t, t′) = eiH0(t−t0)/~U(t, t′)e−iH0(t′−t0)/~. (1.18)

The interaction picture evolution operator also satisfies the group and uni-tarity properties.

You easily verify that this assignment leads to the same time-dependentexpectation value (1.14) as the Schrodinger and Heisenberg pictures.

A differential equation for the time dependence of the operator A(t) isreadily obtained from the definition (1.16),

dA(t)

dt=i

~[H0, A(t)]− +

∂A(t)

∂t. (1.19)

The time dependence of the interaction picture evolution operator is governedby the interaction-picture representation of the perturbation H1 only,

i~∂UI(t, t0)

∂t= H1(t)UI(t, t0), (1.20)

whereH1(t) = eiH0(t−t0)/~H1e

−iH0(t−t0). (1.21)

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Note that, in the interaction picture, the perturbation H1 acquires an ad-ditional time dependence from the evolution according to the unperturbedHamiltonian H0.

In these notes we’ll use all three pictures. Mostly, the context providessufficient information to find out which picture is used. As a general rule,operators without time argument are operators in the Schrodinger picture,whereas operators with time argument are in the Heisenberg and interactionpictures. We’ll use this convention even if the Schrodinger picture operatorshave an explicit time dependence, cf. Eq. (1.15) above.

1.3 imaginary time

When performing a calculation that involves a thermal average, it is oftenconvenient to introduce operators that are a function of the imaginary time

τ .Imaginary times have no direct physical meaning. Real times do, e.g., for

time-dependent correlation functions. We use imaginary time for technicalconvenience only. Later in this course, when we develop the diagrammaticperturbation theory, it will become clear why. At this moment we can saythe following: A thermal average brings about a negative exponents e−H/T ,whereas the real time dependence brings about imaginary exponents eiHt. Forcalculations, it would have been much easier if all exponents were either realor imaginary. To treat both real and imaginary exponents at the same time israther awkward. Therefore, we’ll opt to consider operators that depend on animaginary time argument, so that the imaginary exponent eiHt is replaced byeHτ . If only imaginary times are used, only real exponents occur. This leadsto much simpler calculations, as we’ll see soon. The drawback of the changefrom real times to imaginary times is that, at the end of the calculation,one has to perform an analytical continuation from imaginary times to realtimes, which requires considerable mathematical care.

In the Schrodinger picture, the imaginary time variable appears in theevolution operator U(τ, τ ′), which now reads

U(τ, τ ′) = e−H(τ−τ ′)/~, −~∂

∂τU(τ, τ ′)HU(τ, τ ′). (1.22)

Since a thermal average necessarily involves a time-independent Hamiltonian,we can give an explicit expression for the evolution operator. Note that

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t

τ

0

1/T

Figure 1: Relation between real time t and imaginary time τ .

the evolution operator (1.22) satisfies the group properties U(τ, τ) = 1 andU(τ, τ ′)U(τ ′, τ ′′) = U(τ, τ ′′), but that it is no longer unitary.

In the Heisenberg picture, the imaginary time variable appears throughthe relation

A(τ) = eHτ/~Ae−Hτ/~,∂A(τ)

∂τ=

1

~[H,A]−, (1.23)

where A is the corresponding real-time operator. Again, H is assumed to betime independent.

In the interaction picture, the time dependence of the operators is givenby

A(τ) = eH0τ/~Ae−H0τ/~,∂A(τ)

∂τ=

1

~[H0, A]−, (1.24)

whereas the time dependence of the quantum state |ψ(τ)〉 is given by theinteraction picture evolution operator

UI(τ, τ′) = eH0τe−H(τ−τ ′)e−H0τ ′

, −~∂

∂τUI(τ, τ

′) = H1(τ)U(τ, τ ′). (1.25)

Here H1(τ) = eH0τH1e−H0τ is the perturbation in the interaction picture.

The relation between the real time t and the imaginary time τ is illus-trated in Fig. 1.

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1.4 Exercises

Exercise 1.1: Evolution equation for expectation value

Show that the time dependence of the expectation value of an observable Asatisfies the differential equation

d

dt

(

A)

t=i

~

(

[H,A]−

)

t,

if the operator A has no explicit time dependence.

Exercise 1.2: Free particle, Schrodinger picture.

For a free particle with mass m, express the expectation values (xα)t and(pα)t of the components of the position and momentum at time t in termsof the corresponding expectation values at time 0 (α = x, y, z). Use theSchrodinger picture.

Exercise 1.3: Free particle, Heisenberg picture

Derive the Heisenberg picture evolution equations for the position and mo-mentum operators of a free particle. Use your answer to express the expec-tation values (xα)t and (pα)t of the components of the position and momen-tum at time t in terms of the corresponding expectation values at time 0(α = x, y, z).

Exercise 1.4: Harmonic oscillator

Derive and solve the Heisenberg picture evolution equations for the positionand momentum operators x and p of a one-dimensional harmonic oscillatorwith mass m and frequency ω0. Do the same for the corresponding creationand annihilation operators

a† = (ω0m/2~)1/2x− i(2~ω0m)−1/2p

a = (ω0m/2~)1/2x+ i(2~ω0m)−1/2p. (1.26)

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Verify that the Hamiltonian is time-independent.

Exercise 1.5: Imaginary time

Derive and solve the imaginary-time Heisenberg picture evolution equationsfor the position and momentum operators of a one-dimensional harmonicoscillator with mass m and frequency ω0. Do the same for the correspondingcreation and annihilation operators, see Eq. (1.26).

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2 General formalism; equilibrium theory

Throughout this course we’ll make use of “Green functions”. There aremany ways one can define Green functions, and there are important relationsbetween the different definitions. Quite often, a certain application calls forone Green function, whereas it is easier to calculate a different Green functionfor the same system.

We now present the various definitions of Green functions in the generalcase and the general relations between the Green functions. In the laterchapters we discuss various physical applications that call for the use ofGreen functions.

We consider Green functions (or “correlation functions”) defined for twooperators A and B, which do not need to be hermitian. In the applications,we’ll encounter cases where the operators A and B are fermion or bosoncreation or annihilation operators, or displacements of atoms in a lattice,or current or charge densities. We say that the operators A and B satisfycommutation relations if they describe bosons or if they describe fermionsand they are even functions of fermion creation and annihilation operators.We’ll refer to this case as the “boson” case, and use a − sign in the formulasbelow. We say that the operators A and B satisfy anticommutation relationsif they describe fermions and they are odd functions of in fermion creationand annihilation operators. We’ll call this case the “fermion case”, and usea + sign in the formulas of the next sections.

2.1 Temperature Green functions

The temperature Green function GA;B(τ1; τ2) for the operators A and B isdefined as3

GA;B(τ1; τ2) = −〈Tτ [A(τ1)B(τ2)]〉. (2.1)

Here, the brackets denote a thermal average and the symbol Tτ denotes time-ordering,

Tτ [A(τ1)B(τ2)] = θ(τ1 − τ2)A(τ1)B(τ2) − (±1)θ(τ2 − τ1)B(τ2)A(τ1). (2.2)

As discussed previously, the + sign applies if the operators A and B satisfyfermion anticommutation relations, i.e., if they are of odd degree in fermion

3DS denote the phonon temperature Green function as G. Their definition differs aminus-sign with respect to that of Eq. (2.1).

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creation/annihilation operators. The − sign applies if A and B are bosonoperators or if they are of even degree in fermion creation/annihilation op-erators. The temperature Green function is defined on the interval −~/T <τ1, τ2 < ~/T . In order to extend the definition to the entire imaginary timeaxis −∞ < τ1, τ2 < ∞ one makes use of the facts that (i) G depends onthe difference τ = τ1 − τ2 only and (ii) G is periodic with period ~/T ,G(τ) = G(τ + ~/T ) for bosons, and antiperiodic with the same period,G(τ) = −G(τ +~/T ) for fermions. With these properties, G can be extendedto the entire imaginary time axis.

If the operators A and B and the Hamiltonian H are all symmetric,the temperature Green function is symmetric or antisymmetric in the timeargument, for the boson and fermion cases, respectively,

GA;B(τ) = −(±1)GA;B(−τ) if A, B, H symmetric. (2.3)

Using the periodicity of G, one may write G as a Fourier series,4

GA;B(τ) = T∑

n

e−iωnτGA;B(iωn), (2.4)

with Matsubara frequencies ωn = 2πn/T , n integer, for bosons and ωn =(2n+ 1)π/T for fermions. The inverse relation is

Giα;jβ(iωn) =

∫

~/T

0

dτeiωnτGiα;jβ(τ). (2.5)

2.2 Green functions with real time arguments

In addition to the temperature Green functions, Green functions with realtime arguments are used. The retarded, advanced, and time-ordered Greenfunctions are defined as

GRA;B(t; t′) = −iθ(t− t′)〈[A(t), B(t′)]±〉, (2.6)

GAA;B(t; t′) = iθ(t′ − t)〈[A(t), B(t′)]±〉, (2.7)

GA;B(t; t′) = −i〈Tt[A(t)B(t′)]〉= −iθ(t− t′)〈A(t)B(t′)〉 ± iθ(t′ − t)〈B(t′)A(t)〉, (2.8)

4DS define the Fourier transform without the prefactor T .

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respectively. Here, [·, ·]+ is the anticommutator, [A,B]+ = AB + BA. Oneverifies that

GAB†;A†(t

′; t) = GRA;B(t; t′)∗. (2.9)

By definition, GR is zero for t− t′ < 0 and GA is zero for t− t′ > 0. Further,by virtue of their definitions, one has the relation

GRA;B(t; t′) = −(±1)GA

B;A(t′; t), (2.10)

where the + sign applies to fermions and the − sign to bosons.Since the averages 〈. . .〉 are taken in thermal equilibrium, the Green func-

tions depend on the time difference t− t′ only. Then, Fourier transforms ofthese Green functions are defined as

GRA;B(ω) =

∫ ∞

0

dteiωtGRA;B(t), (2.11)

GAA;B(ω) =

∫ 0

−∞

dteiωtGAA;B(t), (2.12)

GA;B(ω) =

∫ ∞

−∞

dteiωtGA;B(t). (2.13)

The retarded Green function GR(ω) is analytic for Imω > 0, while the ad-vanced Green function GA(ω) is analytic for Imω < 0.

The inverse Fourier transforms are

GRA;B(t) =

1

2π

∫ ∞

−∞

dωe−iωtGRA;B(ω), (2.14)

GAA;B(t) =

1

2π

∫ ∞

−∞

dωe−iωtGAA;B(ω), (2.15)

GA;B(t) =1

2π

∫ ∞

−∞

dωe−iωtGA;B(ω). (2.16)

Note that the fact that GR(ω) is analytic for Imω > 0 implies that GR(t) = 0for t < 0, and, similarly, that the fact that GA(ω) is analytic for Imω < 0implies that GR(t) = 0 for t > 0.

For some applications it is important to consider Green functions withouttime ordering. These are the “greater” and “lesser” Green functions

G>A;B(t, t′) = −i〈A(t)B(t′)〉, (2.17)

G<A;B(t, t′) = ±i〈B(t′)A(t)〉, (2.18)

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where the + sign is for fermions and the − sign for bosons. The retarded,advanced, and time-ordered Green functions can be written in terms of thesetwo functions as

GRA;B(t, t′) = θ(t− t′)[G>

A;B(t, t′) −G<A;B(t, t′)],

GAA;B(t, t′) = θ(t′ − t)[G<

A;B(t, t′) −G>A;B(t, t′)], (2.19)

GA;B(t, t′) = θ(t− t′)G>A;B(t, t′) + θ(t′ − t)G<

A;B(t, t′).

Inversely, the greater and lesser Green functions can be expressed as

G>A;B(t, t′) = GA;B(t, t′) −GA

A;B(t, t′)

G<A;B(t, t′) = GA;B(t, t′) −GR

A;B(t, t′). (2.20)

Since averages are taken in thermal equilibrium, the Green functions G> andG< depend on the time difference t − t′ only. The Fourier transform of thegreater and lesser Green functions is defined as

G>A;B(ω) =

∫ ∞

−∞

eiωtG>A;B(t), (2.21)

G<A;B(ω) =

∫ ∞

−∞

eiωtG<A;B(t). (2.22)

2.3 Relations between Green functions

In fact, all six Green functions defined in the previous two sections are related.Before we explain those relations, it is helpful to define a “real part” and“imaginary part” of a Green function.

In the time representation, we define the “real part” ℜG of the retarded,advanced, and time-ordered Green functions as

ℜGRA;B(t; t′) =

1

2

(

GRA;B(t; t′) +GR

B†;A†(t′; t)∗

)

, (2.23)

ℜGAA;B(t; t′) =

1

2

(

GAA;B(t; t′) +GA

B†;A†(t′; t)∗

)

, (2.24)

ℜGA;B(t; t′) =1

2

(

GA;B(t; t′) +GB†;A†(t′; t)∗)

. (2.25)

Similarly, we define the “imaginary part” ℑG as

ℑGRA;B(t; t′) =

1

2i

(

GRA;B(t; t′) −GR

B†;A†(t′; t)∗

)

, (2.26)

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ℑGAA;B(t; t′) =

1

2i

(

GAA;B(t; t′) −GA

B†;A†(t′; t)∗

)

, (2.27)

ℑGA;B(t; t′) =1

2i

(

GA;B(t; t′) −GB†;A†(t′; t)∗)

. (2.28)

These “real” and “imaginary” parts of Green functions are different from thestandard real and imaginary parts ReG and ImG. In particular, ℜG andℑG are not necessarily real numbers. The quantities ℜG and ℑG are definedwith respect to a “hermitian conjugation” that consists of ordinary complexconjugation, interchange and hermitian conjugation of the operators A andB, and of the time arguments. The assignments ℜ and ℑ are invariant underthis hermitian conjugation and are preserved under Fourier transform of thetime variable t,

ℜGA;B(ω) =1

2

(

GA;B(ω) +GB†;A†(ω)∗)

, (2.29)

ℑGA;B(ω) =1

2i

(

GA;B(ω) −GB†;A†(ω)∗)

. (2.30)

with similar expressions for the retarded and advanced Green functions.By virtue of Eq. (2.9), the retarded and advanced Green functions are

hermitian conjugates, hence

ℜGR(ω) = ℜGA(ω), (2.31)

ℑGR(ω) = −ℑGA(ω). (2.32)

Similarly, by employing their definitions and with repeated use of the relation

〈A(t)B(t′)〉∗ = 〈B†(t′)A†(t)〉, (2.33)

one finds a relation between the real parts of the time-ordered and the re-tarded or advanced Green functions,

ℜGR(ω) = ℜG(ω), (2.34)

ℜGA(ω) = ℜG(ω). (2.35)

Shifting the t-integration from the real axis to the line t − i/T , one derivesthe general relation

∫

dteiωt〈A(t)B(0)〉 = e~ω/T

∫

dteiωt〈B(0)A(t)〉. (2.36)

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Using Eq. (2.36), together with Eq. (2.33), one finds a relation for the imag-inary parts of the time-ordered and retarded or advanced Green functions.This relation follows from the fact that the imaginary parts of the retarded,advanced, and time-ordered Green functions do not involve theta functions.Indeed, for the imaginary part of the retarded and advanced functions onefinds

ℑGRA;B(ω) = −ℑGA

A;B(ω)

= −1

2

∫ ∞

−∞

dteiωt〈A(t)B(0) ± B(0)A(t)〉

= −1

2

(

1 ± e−~ω/T)

∫ ∞

−∞

dteiωt〈A(t)B(0)〉, (2.37)

whereas for the time-ordered Green function one has

ℑGA;B(ω) = −1

2

∫ ∞

−∞

dteiωt〈A(t)B(0) − (±1)B(0)A(t)〉

= −1

2

(

1 − (±1)e−~ω/T)

∫ ∞

−∞

dteiωt〈A(t)B(0)〉. (2.38)

Hence, we find

ℑGR(ω) =1 + (±1)e−~ω/T

1 − (±1)e−~ω/TℑG(ω), (2.39)

ℑGA(ω) = −1 + (±1)e−~ω/T

1 − (±1)e−~ω/TℑG(ω), (2.40)

where the + sign is for fermion operators and the − sign for boson operators.With the help of the Fourier representation of the step function,

θ(t) =1

2πi

∫ ∞

−∞

dω′ eiω′t

ω′ − iη, (2.41)

where η is a positive infinitesimal, and of the second line of Eq. (2.37), oneshows that

GRAB(ω) = − 1

2π

∫

dω′ 1

ω′ − ω − iη

×∫ ∞

−∞

dteiω′t〈A(t)B(0) ±B(0)A(t)〉

=1

π

∫ ∞

−∞

dω′ℑGR

A;B(ω′)

ω′ − ω − iη. (2.42)

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Similarly, one finds

GAA;B(ω) = −1

π

∫ ∞

−∞

dω′ℑGA

A;B(ω′)

ω′ − ω + iη. (2.43)

The numerator of the fractions in Eqs. (2.42) and (2.43) is known as the“spectral density”,

AA;B(ω) = −2ℑGR(ω) = 2ℑGA(ω). (2.44)

Other definitions of the spectral density can be found in the literature,5 butthey all have in common that the spectral density is the imaginary part of aGreen function. The spectral density satisfies the normalization condition

1

2π

∫ ∞

−∞

dωAA;B(ω) = 〈[A,B]±〉. (2.45)

Using Eq. (2.20) and the results derived above, we can write the Fouriertransforms G>(ω) and G<(ω) of the greater and lesser Green functions as

G>A;B(ω) = − iAA;B(ω)

1 ± e−~ω/T, (2.46)

G<A;B(ω) =

iAA;B(ω)

1 ± e~ω/T. (2.47)

We conclude that the greater and lesser Green functions are purely imaginary.Finally, by shifting integration contours as shown in Fig. 2, one can write

the Fourier transform of the temperature Green function for ωn > 0 as

GA;B(iωn) = −∫

~/T

0

dτeiωnτ 〈A(τ)B(0)〉

= −i∫ ∞

0

dt〈A(t)B(0)〉e−ωnt

+ i

∫ ∞

0

dt〈A(t− i/T )B(0)〉e−ωnt+iωn/T

5DS define two spectral densities J1(ω) and J2(ω) for the boson case, with

J1(ω) = iG>A;B(ω), J2(ω) = −iG<

A;B(ω),

see Eq. (2.46) below.

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τ0 1/T

Figure 2: Integration contours for derivation of Eq. (2.48).

= −i∫ ∞

0

dt〈A(t)B(0)〉e−ωnt

+ i

∫ ∞

0

dt〈B(0)A(t)〉e−ωnt+iωn/T

= GRA;B(iωn). (2.48)

Here, we made use of the fact that exp(iωn/T ) = −(±1). In order to findthe temperature Green function for negative Matsubara frequencies, one canmake use of the relation

GA;B(iωn) = GB†;A†(−iωn)∗. (2.49)

Writing the retarded Green function in terms of the spectral density usingEqs. (2.42) and (2.44), one thus obtains the general expression

GA;B(iωn) = − 1

2π

∫

dω′AA;B(ω′)

ω′ − iωn

, (2.50)

which is valid for positive and for negative Matsubara frequencies.In many applications is it difficult, if not impossible, to calculate Green

functions exactly. Instead, we have to rely on perturbation theory. Whereasthe retarded, advanced, greater, and lesser Green functions are the onesthat appear in most physical applications, good diagrammatic methods tocalculate Green functions in perturbation theory exist for temperature andtime-ordered Green functions only. The above relations allows one to relateretarded, advanced, and temperature Green functions to the temperatureand time-ordered Green functions. Hence, these relations are a crucial linkbetween what can be calculated easily and what is desired to be calculated.

16

2.4 Exercises

Exercise 2.1: Lehmann representation

Explicit representations for Green functions can be obtained using the set ofmany-particle eigenstates |n〉 of the Hamiltonian H , as a basis set. Forexample, the greater Green function in the canonical ensemble can be writtenas

G>A;B(t, t′) = − i

ZtrA(t)B(t′)e−H/T

= − i

Z

∑

n

e−En/T 〈n|A(t)B(t′)|n〉, (2.51)

where Z =∑

n e−En/T is the canonical partition function. (In the grand

canonical ensemble, similar expressions are obtained.) Inserting a completebasis set and using

∑

n′

|n′〉〈n′| = 1,

we write Eq. (2.51) as

G>A;B(t, t′) = − i

Z

∑

n,n′

e−En/T+i(t−t′)(En−En′)〈n|A|n′〉〈n′|B|n〉. (2.52)

Performing the Fourier transform to time, one finds

G>A;B(ω) = −2πi

Z

∑

n,n′

e−En/T δ(En − En′ + ω)〈n|A|n′〉〈n′|B|n〉. (2.53)

With the Lehmann representation, the general relations we derived in Sec.2.3 can be verified explicitly.

(a) Verify that G>A;B is purely imaginary (with respect to hermitian conju-

gation, as defined in Sec. 2.3).

(b) Derive Lehmann representations of the retarded, advanced, lesser, time-ordered, and temperature Green functions.

(c) Derive the Lehmann representation of the spectral density AA;B(ω).

17

(d) Verify the relations between the different types of Green functions thatwere derived in Sec. 2.3 using the Lehmann representations derived in(b) and (c).

Exercise 2.2: Spectral density

In this exercise we consider the spectral density AA;B(ω) for the case A = c,B = c† that the operators A and B are fermion or boson annihilation andcreation operators, respectively. In this case, the spectral density Ac;c†(ω)can be viewed as the energy resolution of a particle created by the creationoperator c†. To show that this is a plausible idea, you are asked to provesome general relations for the spectral density Ac;c†(ω). Consider the casesof fermions and bosons separately.

(a) The spectral density Ac;c†(ω) is normalized,

1

2π

∫ ∞

−∞

dωAc;c†(ω) = 1. (2.54)

(b) For fermions, the spectral density Ac;c†(ω) ≥ 0. For bosons, Ac;c†(ω) ≥0 if ω > 0 and Ac;c†(ω) ≤ 0 if ω < 0. (Hint: use the Lehmannrepresentation of the spectral density.)

(c) The average occupation n = 〈c†c〉 is

n =1

2π

∫ ∞

−∞

dωAc;c†(ω)1

1 ± e~ω/T. (2.55)

(d) Now consider a Hamiltonian that is quadratic in creation and annihi-lation operators,

H =∑

n

εnc†ncn.

For this Hamiltonian, calculate the spectral density A(n, ω) ≡ Acn;c†n(ω).

18

Exercise 2.3: Harmonic oscillator

For a one-dimensional harmonic oscillator with mass m and frequency ω0,calculate the retarded, advanced, greater, lesser, time-ordered, and temper-ature Green functions for the following choices of the operators A and B:

(a) Both A and B are equal to the position operator x.

(b) The operator A is equal to the position operator x, while B is equal tothe momentum operator p.

(c) The operator A is equal to the annihilation operator a, while B is equalto the creation operator a†, see Eq. (1.26).

19

20

3 Introduction: phonons

3.1 Normal modes

We consider the dynamics of the atoms of a lattice. The atoms interact via apotential V(~xi) that depends on the location ~xi of all atoms i = 1, . . . , N .Each atom has an equilibrium position ~ri. For small displacements ~ui = ~xi−~ri

from equilibrium, the Hamiltonian for the lattice atoms can be expanded in aseries in the displacements ~ui. The first order term in the expansion vanishes,since the potential V is a minimum at the equilibrium positions of the atoms.If we truncate the expansion at the second order, we have

H =N∑

i=1

~p2i

2M+

1

2

∑

i,j

∑

αβ

uiαujβVα;β(~ri, ~rj), (3.1)

where the numbers Vα;β(~ri, ~rj) are derivatives of the potential V at the equi-librium positions ~ri, ~rj ,

Vα;β(~ri, ~rj) = ∂xiα∂xjβ

V(~ri). (3.2)

The displacement and momentum operators ~ui and ~pj have canonical com-mutation relations,

[piα, pjβ]− = [uiα, ujβ]− = 0 (3.3)

[piα, ujβ]− = −i~δαβδij. (3.4)

For an infinite lattice or for a lattice with periodic boundary conditions, thereal numbers Vα;β(~ri, ~rj)) depend on the difference ~ri − ~rj only. Inversionsymmetry implies Vα;β(~ri − ~rj) = Vα;β(~rj − ~ri), and, since derivatives can betaken in any order, Vα;β(~ri − ~rj) = Vβ;α(~ri − ~rj).

The approximation (3.1) in which the Hamiltonian of the lattice atomsis truncated after the quadratic term in the displacements is known as the“harmonic approximation”.

Doniach and Sondheimer study the case that V is a sum of pair potentials,

V(~xi) =∑

i6=j

U(~xi − ~xj). (3.5)

In that case, one has the expressions

Viα;jβ = −∂α∂βU(~ri − ~rj) if i 6= j, (3.6)

Viα;iβ =∑

j 6=i

∂α∂βU(~ri − ~rj). (3.7)

21

The Hamiltonian (3.1) can be brought to diagonal form by a suitablebasis transformation. We construct this basis transformation in two steps.First, we perform a Fourier transform with respect to the lattice coordinate~ri and define

~p~k =1√N

∑

j

ei~k·~rj~pj (3.8)

~u~k =1√N

∑

j

e−i~k·~rj~uj. (3.9)

Here and below, the wavevector ~k is in the first Brillouin zone of the reciprocallattice. After Fourier transform, the operators ~p~k and ~u~k still obey canonicalcommutation relations,

[

p~kα, p~k′β

]

−=

[

u~kα, u~k′β

]

−= 0 (3.10)

[

p~kα, u~k′β

]

−= −i~δαβδ~k~k′. (3.11)

The momenta and displacements corresponding to opposite wavevectors ~kand −~k are related by hermitian conjugation, ~p~k = ~p†

−~kand ~u~k = ~u†

−~k. Note

that the total number of degrees of freedom has not changed upon Fouriertransform.

In terms of these new variables, the Hamiltonian reads

H =∑

~k

∑

α,β

(

1

2Mp~kαpβ,−~k +

1

2Vαβ,~ku~kαuβ,−~k

)

, (3.12)

whereVαβ,~k =

∑

~r

ei~k·~rVα;β(~r). (3.13)

The matrix 3 × 3 matrix V~k is symmetric because Vα;β(~r) is symmetric and

real because Vα;β(~r) = Vα;β(−~r). Moreover, Vαβ,~k = Vα;β(−~k). Since thematrix V~k is real and symmetric, it is diagonalized by three real and orthog-onal eigenvectors. We denote the eigenvectors as vλ

~kα, λ = 1, 2, 3, and the

corresponding eigenvalues as Mω2~kλ

. Then we can introduce normal modes

pλ~k

=∑

α

vλαp~kα, (3.14)

uλ~k

=∑

α

vλαu~kα, (3.15)

22

that bring H into diagonal form,

H =∑

~k,λ

(

1

2Mpλ

~kpλ−~k

+1

2Mω2

~kλuλ

~kuλ−~k

)

. (3.16)

One verifies that the normal mode operators pλ~k

and uλ~k

still obey canonicalcommutation relations.

Finally, we write the Hamiltonian in terms of phonon creation and anni-hilation operators. These are defined as

bλ~k = uλ~k

√

Mω~kλ

2~+ ipλ

−~k

√

1

2Mω~kλ~, (3.17)

bλ†~k= uλ

−~k

√

Mω~kλ

2~− ipλ

~k

√

1

2Mω~kλ~, (3.18)

and obey the usual commutation rules for creation and annihilation opera-tors,

[

bλ~k , bλ′

~k′

]

−=

[

bλ†~k, bλ

′†~k′

]

−

= 0,[

bλ~k , bλ′†~k′

]

−= δ~k~k′δλλ′ . (3.19)

In terms of the new operators bλ†~kand bλ~k , the Hamiltonian reads

H =∑

~kλ

~ω~kλ

(

bλ†~kbλ~k +

1

2

)

. (3.20)

Since the operators bλ†~kand bλ~k satisfy creation and annihilation operators

for bosons, the Hamiltonian (3.20) is interpreted as describing “phonons”,quantized lattice vibrations with boson statistics. With or without suchinterpretation, what is important for us is that the Hamiltonian (3.20) isdiagonal, so that any average of displacements and momenta of lattice atomscan be calculated straightforwardly once it is expressed in terms of the op-erators bλ†~k

and bλ~k . For this, it is important to have an expression for the

23

inverse of the normal-mode transformation we just derived,

ujα =1√N

∑

~kλ

ei~k·~rjvλα

√

~

2Mω~kλ

(

bλ~k + bλ†−~k

)

, (3.21)

pjα =1√N

∑

~kλ

e−i~k·~rjvλα

√

M~ω~kλ

2

(

ibλ†~k− ibλ

−~k

)

. (3.22)

3.2 Phonon Green functions

For phonons we define Green functions with respect to the displacementoperators uiα. Following the formalism outlined in Sec. 2, we introduce thephonon temperature Green function6

Diα;jβ(τ1; τ2) = −〈Tτ [uiα(τ1)ujβ(τ2)]〉, (3.23)

where the brackets denote a thermal average and the symbol Tτ denotestime-ordering,

Tτ [uiα(τ1)ujβ(τ2)] = θ(τ1−τ2)uiα(τ1)ujβ(τ2)+θ(τ2−τ1)ujβ(τ2)uiα(τ1). (3.24)

Since the displacement operators u obey commutation relations (phonons arebosons), we choose the boson variant of the expressions in Sec. 2.

For phonons, D is real, since u and H are real. According to Eq. (2.3),this implies

Diα;jβ(τ) = Diα;jβ(−τ). (3.25)

Using the periodicity of D, one may write D as a Fourier series,7

Diα;jβ(τ) = T∑

n

e−iωnτDiα;jβ(iωn), (3.26)

where the ωn = 2πn/T , n integer, are the boson Matsubara frequencies.The retarded, advanced, and time-ordered phonon Green functions are

defined as

DRiα;jβ(t− t′) = −iθ(t − t′)〈[uiα(t), ujβ(t

′)]−〉. (3.27)

6DS denote the phonon temperature Green function as G. Their definition differs aminus-sign with respect to that of Eq. (3.23).

7DS define the Fourier transform without the prefactor T .

24

DAiα;jβ(t− t′) = iθ(t′ − t)〈[uiα(t), ujβ(t

′)]−〉. (3.28)

Diα;jβ(t− t′) = −i〈Tt[uiα(t)ujβ(t′)]〉

= −iθ(t − t′)〈uiα(t)ujβ(t′)〉− iθ(t′ − t)〈ujβ(t

′)uiα(t)〉, (3.29)

and their Fourier transforms are as given in Sec. 2. Finally, the phonon lesserand greater Green functions are defined as

D>iα;jβ(t− t′) = −i〈uiα(t)ujβ(t

′)〉, (3.30)

D<iα;jβ(t− t′) = i〈ujβ(t

′)uiα(t)〉. (3.31)

3.3 Calculation of phonon Green functions

3.3.1 Calculation using diagonal form of the Hamiltonian

The simplest method to calculate the phonon Green functions is to use thediagonal form (3.16) of the phonon Hamiltonian. The time-dependence ofthe phonon annihilation and creation operators is

bλ~k(t) = e−iω~kλtbλ~k(0), bλ†~k

(t) = eiω~kλtbλ†~k

(0). (3.32)

In thermal equilibrium, averages involving the phonon annihilation and cre-ation operators are particularly simple:

〈bλ~k(t)bλ′

~k′(0)〉 = 0, (3.33)

〈bλ†~k(t)bλ

′†~k′

(0)〉 = 0, (3.34)

〈bλ†~k(t)bλ

′

~k′(0)〉 = δ~k~k′δλλ′eiω~kλt〈n~kλ〉, (3.35)

〈bλ~k(t)bλ′†~k′

(0)〉 = δ~k~k′δλλ′e−iω~kλt〈n~kλ + 1〉, (3.36)

where

〈n~kλ〉 =1

e~ω~kλ/T − 1

(3.37)

is the phonon distribution function. We can use these averages, togetherwith the variable transformation (3.21) and find for the time-ordered Greenfunction for t > 0

Diα;jβ(t > 0) = −i〈uiα(t)ujβ(0)〉

= −i∑

~kλ

(

~

2MNω~kλ

)

vλαv

λβe

i~k·(~ri−~rj)

× (〈bλ~k(t)bλ†~k

(0) + bλ†−~k

(t)bλ−~k

(0)〉), (3.38)

25

where we omitted terms that give zero after thermal averaging. Performingthe thermal average in Eq. (3.38), and repeating the same analysis for t < 0,we find

Diα;jβ(t) = −i∑

~kλ

(

~

2MNω~kλ

)

vλαv

λβe

i~k·(~ri−~rj)

×(

〈n~kλ〉eiω~kλ|t| + 〈n~kλ + 1〉e−iω~kλ

|t|)

. (3.39)

Similarly, for the retarded and advanced Green functions, one finds

DRiα;jβ(t) = −θ(t)

∑

~kλ

(

~

MNω~kλ

)

vλαv

λβe

i~k·(~ri−~rj) sin(ω~kλt), (3.40)

DAiα;jβ(t) = θ(−t)

∑

~kλ

(

~

MNω~kλ

)

vλαv

λβe

i~k·(~ri−~rj) sin(ω~kλt). (3.41)

Performing a Fourier transform to time, we have

DRiα;jβ(ω) =

∑

~kλ

(

~

2MNω~kλ

)

vλαv

λβe

i~k·(~ri−~rj)

×(

1

ω − ω~kλ + iη− 1

ω + ω~kλ + iη

)

. (3.42)

The spectral density can now be found by taking the imaginary part ofDR(ω),

Aiα;jβ(ω) = 2π∑

~kλ

(

~

2MNω~kλ

)

vλαv

λβe

i~k·(~ri−~rj)

×(

δ(ω − ω~kλ) − δ(ω + ω~kλ))

. (3.43)

The temperature Green function can be found using Eq. (2.50) or bydirect calculation using the imaginary-time dependence of the creation andannihilation operators. Here we use the latter method. Starting point is thegeneral expression for the dependence of the phonon creation and annihilationoperators on the imaginary time τ . For 0 ≤ τ ≤ ~/T one has

〈bλ~k(τ)bλ′

~k′(0)〉 = 0, (3.44)

〈bλ†~k(τ)bλ

′†~k′

(0)〉 = 0, (3.45)

〈bλ†~k(τ)bλ

′

~k′(0)〉 = δ~k~k′δλλ′eω~kλτ 〈n~kλ〉, (3.46)

〈bλ~k(τ)bλ′†~k′

(0)〉 = δ~k~k′δλλ′e−ω~kλτ 〈n~kλ + 1〉, (3.47)

26

Calculation of the temperature Green function then yields, for 0 ≤ τ ≤ ~/T ,

Diα;jβ(τ) = −〈uiα(τ)ujβ(0)〉

= −∑

~kλ

(

~

2MNω~kλ

)

vλαv

λβe

i~k·(~ri−~rj)

×(

eω~kλτ

e~ω~kλ/T − 1

+e−ω~kλ

τ

1 − e−~ω~kλ/T

)

. (3.48)

It is easily verified that Diα;jβ(0) = Diα;jβ(~/T ), so that this result can beextended periodically to the entire imaginary time axis. Fourier transformto the imaginary time τ yields

Diα;jβ(iΩn) = −∑

~kλ

(

~

MN

)

vλαv

λβe

i~k·(~ri−~rj)1

ω2~kλ

+ Ω2n

. (3.49)

3.3.2 Calculation using the equation of motion

A different method to calculate the Green function is to use the “equationof motion”. As an illustration of this method, we repeat the calculation ofthe phonon temperature Green function below.

For the Heisenberg operators uiα(τ) and piα(τ) the equations of motionare

∂τuiα(τ) =1

~[H, uiα]− = − i

Mpiα, (3.50)

∂τpiα(τ) =1

~[H, piα]− = i

∑

jβ

Vα;β(~ri − ~rj)ujβ. (3.51)

Then, using the definition of the temperature Green function

Diα;jβ(τ) = −θ(τ)〈uiα(τ)ujβ(0)〉 − θ(−τ)〈ujβ(0)uiα(τ)〉, (3.52)

we find

∂τDiα;jβ(τ) = −δ(τ)〈uiα(τ)ujβ(0) − ujβ(0)uiα(τ)〉+

i

Mθ(τ)〈piα(τ)ujβ(0)〉

+i

Mθ(−τ)〈ujβ(0)piα(τ)〉. (3.53)

27

The first term on the r.h.s. of Eq. (3.53), multiplying the delta function δ(τ),is zero. Taking one more derivative of the remaining two terms, we find

∂2τDiα;jβ(τ) =

i

Mδ(τ)〈piα(τ)ujβ(0) − ujβ(0)piα(τ)〉

+i

Mθ(τ)〈∂τpiα(τ)ujβ(0)〉

+i

Mθ(−τ)〈ujβ(0)∂τpiα(τ)〉

=~

Mδ(τ)δijδαβ +

1

M

∑

lγ

Vα;γ(~ri − ~rl)Dlγ;jβ(τ). (3.54)

Rearranging terms, we find that D(τ) satisfies an inhomogeneous differentialequation8

∑

lγ

(

∂2τ δilδαγ −

1

MVα;γ(~ri − ~rl)

)

Dlγ;jβ(τ) =~

Mδ(τ)δijδαβ. (3.55)

This equation can be solved by Fourier transform to the imaginary time τ ,

−∑

lγ

(

Ω2nδliδγα +

1

MVα;γ(~ri − ~rl)

)

Dlγ;jβ(iΩn) =~

Mδijδαβ . (3.56)

The remaining problem of dealing with the lattice indices and the coordi-nate directions can be tackled by a further Fourier transform. Based on ourprevious analysis of the normal modes, we can make use of the equation

Ω2n +

1

MVα;γ(~ri − ~rl) =

1

N

∑

~kλ

vλαv

λγe

i~k·(~ri−~rl)(Ω2n + ω2

~kλ) (3.57)

and the completeness relation

1

N

∑

lγ

vλγv

λ′

γ ei~rl·(~k−~k′) = δλλ′

δ~k~k′, (3.58)

to find

Diα;jβ(iΩn) = − ~

MN

∑

~kλ

vλ~kαvλ

β~kei~k·(~ri−~rj)

1

Ω2n + ω2

~kλ

, (3.59)

in agreement with Eq. (3.49) above.

8Note that DS have a minus sign on the right hand side, because definition of thetemperature Green function differs a minus sign from ours.

28

3.4 Neutron scattering cross-section

As an application of the use of Green functions, Doniach and Sondheimerconsider inelastic neutron scattering. An important quantity to calculate ina scattering experiment is the differential cross section. It tells us the amountof scattered neutrons that emerge from the sample at a given energy ~ω andsolid angle Ω.

In the first Born approximation, the differential scattering cross sectionper unit solid angle and per unit energy is9

d2σ

dΩdω=∑

f

k′

k

(

M

2π

)2⟨

|〈~k′, f |H ′|~k, i〉|2⟩

δ(ω + Ei − Ef), (3.60)

where ~k is the wavevector of the incoming neutron, ~k′ is the wavevector ofthe outgoing neutron, M is the reduced mass of the neutron, i and f referto initial state and final state, respectively, H ′ is the interaction betweenthe neutron and the target, and the outer brackets 〈. . .〉 indicate a thermalaverage over the initial states i of the target.

Doniach and Sondheimer describe how Eq. (3.60) can be rewritten as

d2σ

dΩdω=

k′

2πk

(

M

2π

)2

|V~q|2∫ ∞

−∞

dteiωtF (~q, t), (3.61)

where ~q = ~k − ~k′ and the correlation function F (~q, t) is defined as

F (~q, t) =∑

j,l

ei~q·(~rl−~rj)⟨

e−i~q·~uj(t)ei~q·~ul(0)⟩

. (3.62)

In Eq. (3.62), the summation is over the lattice sites j and l and the opera-tors ~uj(t), ~ul(0) are the corresponding displacements of the lattice atoms inthe Heisenberg picture. The equilibrium positions of the lattice atoms aredenoted ~rj and ~rl. In Eq. (3.61), V~q is the Fourier transform of the potentialthat describes how a neutron scatters off a single atom. The combined effectof many atoms in a vibrating lattice is described by the correlation functionF (~q, t).

9This result is derived in standard texts on quantum mechanics, e.g., Quantum Me-

chanics, by A. Messiah, North-Holland (1961).

29

For small ~q, we now write

F (~q, t) =∑

j,l

ei~q·(~rl−~rj)⟨

e−i~q·(~uj(t)−~ul(0))+12[~q·~uj(t),~q·~ul(0)]−

⟩

, (3.63)

where the expression in the exponent is correct up to corrections of order q3.In order to perform the thermal average, we use the cumulant expansion,10

and obtain

F (~q, t) =∑

j,l

ei~q·(~rl−~rj)−12〈(~q·~uj(t))

2〉−12〈(~q·~ul(0))

2〉+〈(~q·~uj(t))(~q·~ul(0))〉, (3.64)

again up to corrections of order q3 in the exponent.11

The second and third terms in the exponent do not depend on j, l, or t;they form an overall prefactor known as the Debije-Waller factor. Writingthe Debije-Waller factor as e−2W , we may calculate W from the time-orderedphonon Green function

W =i

2

∑

αβ

qαqβ limt↓0

Djα;jβ(t). (3.65)

Substituting the exact result (3.39) for the time-ordered Green function, weobtain

W =∑

~kλ

∑

αβ

(

~

4MNω~kλ

)

qαqβvλαv

λβ coth(~ω~kλ/2T ). (3.66)

We further expand F in the fourth term of the exponent, and separate Finto an elastic and inelastic contribution,

F (~q, t) = F el(~q, t) + F inel(~q, t) (3.67)

10According to the cumulant expansion, any average of the form 〈exp(A)〉 can be cal-culated as

〈eA〉 = exp(

〈A〉 + 12varA+ . . .

)

,

where varA = 〈A2〉−〈A〉2 is the variance (i.e., the second cumulant) and the dots indicatehigher cumulants. If the probability distribution of A is Gaussian, the first two terms inthe cumulant expansion are sufficient.

11In fact, Eqs. (3.63) and (3.64) are exact because the Hamiltonian is quadratic in thedisplacements ~ui, i = 1, . . . , N .

30

with

F el(~q, t) = e−2W∑

j,l

ei~q·(~rl−~rj) (3.68)

F inel(~q, t) = e−2W∑

j,l

ei~q·(~rl−~rj)∑

αβ

qαqβ〈ujα(t)ulβ(0)〉. (3.69)

Fourier transform of F el gives a term proportional to δ(ω) in the scatteringcross section, corresponding to elastic scattering from the lattice without theexcitation of phonons. Fourier transform of F inel gives

∫ ∞

−∞

dteiωtF inel(~q, t) = ie−2W∑

j,l

ei~q·(~rl−~rj)∑

αβ

qαqβD>jα;lβ(ω), (3.70)

where D>jα;lβ(ω) is the phonon greater Green function, see Eq. (3.31) above.

Using Eqs. (2.46) together with the exact result (3.43) to calculate D>jα;lβ(ω),

we find∫ ∞

−∞

dteiωtF inel(~q, t) = Ne−2W∑

~kλ

∑

αβ

qαqβvλαv

λβ

π~eω/2T

2Mω sinh(ω/2T )

×(

δ(ω − ω~kλ) − δ(ω + ω~kλ))

. (3.71)

3.5 Calculation of the phonon Green function using

diagrammatic perturbation theory

In the harmonic approximation, the phonon problem is exactly solvable.However, most cases of interest are not exactly solvable. In order to demon-strate the techniques used for a more general calculation, Doniach and Sond-heimer return to the calculation of the phonon Green function.

In this section, we put ~ = 1, restoring ~ in the final formulas only.Following Doniach and Sondheimer, we ignore the vector nature of the

displacement ~ui, i = 1, . . . , N , and consider a scalar displacement ui instead,with the Hamiltonian

H =∑

i

p2i

2M+

1

2

∑

i,j

uiujV (~ri − ~rj). (3.72)

We separate the potential energy in diagonal and off-diagonal parts,

H = H0 +H1 (3.73)

31

with

H0 =∑

i

(

p2i

2M+

1

2Mω2

0u2i

)

, (3.74)

H1 =1

2

∑

ij

uiujV′(~ri − ~rj), (3.75)

where we abbreviated Mω20 = V (0) and

V ′(~r) =

V (~r) if ~r 6= 0,0 if ~r = 0.

(3.76)

Below, we’ll consider H1 as a “perturbation”, although there is no param-eter that guarantees that this term is small. In the absence of the “perturba-tion”, the calculation of the temperature Green function is straightforward,

D0ij(τ) = − ~δij

2Mω0

(

eω0τ

e~ω0/T − 1+

e−ω0τ

1 − e−~ω0/T

)

. (3.77)

The purpose of the calculation is to develop a systematic way to includethe perturbation H1, without resorting to the exact solution of the previoussections.

In order to incorporate the “perturbation” H1 we switch to an imaginary-time version of the so-called “interaction picture” of quantum mechanics.The standard “interaction picture” is a mixture of the Heisenberg and Schro-dinger pictures: the time-dependence of the operators is given by the unper-turbed Hamiltonian H0, while the remaining time dependence is encoded inthe wavefunction. For imaginary times, this procedure is implemented withthe introduction of an imaginary time “evolution operator” for the quantumstate,

U(τ, τ ′) = eτH0e−(τ−τ ′)(H0+H1)e−τ ′H0. (3.78)

The middle factor in Eq. (3.78) represents the exact time evolution with thetotal Hamiltonian H = H0 +H1, whereas the right and left factors “offset”for the time-evolution with the unperturbed Hamiltonian H0 which will beencoded in the operators.

The operator U satisfies the evolution equation

∂U

∂τ= −H1(τ)U(τ, τ ′), (3.79)

32

where the time-dependence of H1 is defined with respect to the unperturbedHamiltonian H0 only,

H1(τ) = eH0τH1e−H0τ . (3.80)

The evolution operator U(τ, τ ′) has the group property

U(τ, τ ′)U(τ ′, τ ′′) = U(τ, τ ′′) (3.81)

and U(τ, τ) = 1. However, note that U(τ, τ ′) is not a unitary operator, unlikethe evolution operator in a real-time formalism.

The solution of the evolution equation (3.79) can be written as

U(τ, τ ′) = 1 −∫ τ

τ ′

dτ1H1(τ1)

+

∫ τ

τ ′

dτ1

∫ τ1

τ ′

dτ2H1(τ1)H1(τ2) + . . . . (3.82)

It is important to note that in Eq. (3.82) the time arguments of the pertur-bation H1 always appear in descending order if τ > τ ′. Hence, formally, Eq.(3.82) may be written as

U(τ, τ ′) = Tτ exp

(

−∫ τ

τ ′

H1(τ′′)dτ ′′

)

, (3.83)

where Tτ denotes τ -ordering.With the help of the evolution operator U(τ, τ ′) a thermal average with

respect to the full Hamiltonian H = H0 + H1 can be expressed in termsof a thermal average with respect to the unperturbed Hamiltonian H0 only.That is a useful transformation if the Hamiltonian H0 is simple, whereasthe full Hamiltonian H0 + H1 is complicated. For example, for τ > τ ′ thetemperature Green function Dij(τ) can be written as

Dij(τ, τ′) =

tr e−(H0+H1)/T e(H0+H1)τuie−(H0+H1)τe(H0+H1)τ ′

uje−(H0+H1)τ ′

tr e−(H0+H1)/T

=tr e−H0/TU(1/T, τ)ui(τ)U(τ, τ ′)uj(τ

′)U(τ ′, 0)

tr e−H0/TU(1/T, 0), (3.84)

where the operator ui(τ) is defined as

ui(τ) = eH0τuie−H0τ , (3.85)

33

in analogy to Eq. (3.80). Similarly, for τ < τ ′ we find

Dij(τ, τ′) =

tr e−H0/TU(1/T, τ ′)uj(τ′)U(τ ′, τ)ui(τ)U(τ, 0)

tr e−H0/TU(1/T, 0). (3.86)

We can summarize Eqs. (3.84) and (3.86) with the concise expression

Dij(τ, τ′) =

〈Tτ [U(1/T, 0)ui(τ)uj(τ′)]〉0

〈TτU(1/T, 0)〉0, (3.87)

where Tτ [. . .] denotes τ -ordering and the brackets 〈. . .〉0 denote a thermalaverage with respect to the unperturbed Hamiltonian H0.

The reason why it is favorable to express the temperature Green functionin terms of an thermal average with respect to the unperturbed HamiltonianH0 is that such an average can be done with the help of Wick’s theorem.Wick’s theorem applies to a time-ordered average with respect to a Hamilto-nian that is quadratic in creation/annihilation operators. According to theWick theorem, the average of any product of creation/annihilation operatorscan be found by forming all pairs of operators and replacing these by theiraverage. By definition, the average of a pair of operators is the unperturbedtemperature Green function D0. Hence, for the case of the phonon Greenfunction, Wick’s theorem states that

〈Tτui1(τ1) . . . ui2n(τ2n)〉0 =

(

−1

2

)n∑

P

D0iP (1),iP (2)

(τP (1), τP (2)) × . . .

×D0iP (2n−1),iP (2n)

(τP (2n−1), τP (2n)), (3.88)

where the summation is over all permutations P of the numbers 1, 2, . . . , 2n.The factor (−1)n accounts for the minus sign in the definition of Di,j; thefactor 2−n accounts for the possible exchanges 1 ↔ 2, 3 ↔ 4, . . . , (2n−1) ↔2n. For a proof of the Wick theorem, see the book Methods of quantum field

theory in statistical physics, by A. A. Abrikosov, L. P. Gorkov, and I. E.Dzyaloshinski, Dover (1963).

For the first terms in a series expansion of the numerator in Eq. (3.87)one thus finds

−〈Tτ [U(1/T, 0)ui(τ)uj(τ′)]〉0 =

= D0ij(τ, τ

′) +1

2

∑

kl

V ′(~rl − ~rk)

∫ 1/T

0

dτ1(

D0ij(τ, τ

′)D0kl(τ1, τ1)

+ D0ik(τ, τ1)D0

lj(τ1, τ′) + D0

il(τ, τ1)D0kj(τ1, τ

′))

+ . . . . (3.89)

34

τ’j τ i

V’kl

τ’j τ i

τ1l

τ1k

τ’j τ i

τ1k

τ1lV’kl

V’kl

τ’j τ i

τ1l

τ1k

τ1k

τ1l

V’kl

0th order

1st order + +

numerator:

denominator:

1st order

Figure 3: Diagrammatic representation of Eqs. (3.89) and (3.90).

Similarly, the denominator gives

〈TτU(1/T, 0)〉0 = 1 +1

2

∑

kl

V ′(~rl − ~rk)

∫ 1/T

0

dτ1D0kl(τ1, τ1) + . . . .

(3.90)

It is possible to represent these terms diagrammatically, see Fig. 3. With thehelp of such a diagrammatic representation, higher-order terms can be easilyclassified.

For the terms contributing to the numerator to second order, we dis-criminate two types of contributions: disconnected diagrams (the left-mostone) and connected diagrams (the middle and right diagrams in Fig. 3). Infact, one can easily convince oneself that the disconnected diagrams in theperturbative expansion of the numerator cancel against the diagrammaticexpansion of the numerator. Thus, for the calculation of D we can ignorethe denominator if we keep connected diagrams only.

For this calculation, the two connected diagrams that contribute to firstorder are equal, because Vij = Vji. Moreover, the unperturbed Green func-tion is diagonal in the site indices i and j, cf. Eq. (3.77) above. With thisin mind, and with a slight change of diagrammatic notation, we write thediagrammatic expansion of the phonon Green function as in Fig. 4. Notethat for the higher order diagrams (second order and higher), the combina-torial factors arising from the ways in which one can assign the indices inthe diagram cancel the factorials in the Taylor expansion of the exponent

35

τ τ1 τ’

V’ij

j i+ +

τ’ τ

j=i

τ τ1 τ’

kj

j k

τ2

i

V’ik

+ ...V’

Figure 4: Diagrammatic representation the diagrams that contribute to thephonon Green function.

and the prefactor 1/2 in the definition of the perturbation H1, cf. Eq. (3.75).Hence, every order has unit weight.

Summing over all orders, we then find

Dij(τ, τ′) = D0

ii(τ, τ′) δij +

∞∑

n=1

∑

l1,...,ln−1

∫ 1/T

0

dτ1 . . .

∫ 1/T

0

dτnD0ii(τ, τ1)

× V ′(~ri − ~rl1)D0l1l1

(τ1, τ2) . . . V′(~rln−1 − ~rj)D0

jj(τn, τ′).

(3.91)

We note that Eq. (3.91) contains a convolution of unperturbed Green func-tions. Taking the Fourier transforms with respect to imaginary time andposition,

D(iΩn, ~k) =1

N

∫ 1/T

0

dτeiΩnτ∑

i,j

e−i~k·(~ri−~rj)Dij(τ), (3.92)

all convolutions transform into simple products,

D(iΩn, ~k) =

∞∑

n=0

D0(iΩn, ~k)n+1(V ′

~k)n

=D0(iΩn, ~k)

1 −D0(iΩn, ~k)V′

~k

, (3.93)

where V ′~k

is the Fourier transform of the perturbation,

V ′~k

=∑

~r 6=0

ei~k·~rV (~r)

= V~k −Mω20

= Mω2~k−Mω2

0. (3.94)

36

Substitution of

D0(iΩn, ~k) = − 1

M(Ω2n + ω2

0), (3.95)

yields

D(iΩn, ~k) = − 1

M(Ω2n + ω2

0) + V ′~k

= − 1

M(Ω2n + ω2

~k). (3.96)

This is the same result as that obtained in Eq. (3.49).The calculation we just performed is a good example of the use of dia-

grammatic perturbation theory. We collected all parts of the Hamiltonianthat we knew how to deal with in the “unperturbed Hamiltonian” H0 andtreated the remaining terms in the Hamiltonian as a perturbation. We thenexpanded the Green function in the perturbation using the diagrammaticrepresentation as a tool to organize the expansion. This problem was spe-cial, because we could sum the perturbation theory to all orders. Quite often,that is not the case, and we can only include a subset of diagrams.

3.6 Diagrammatic calculation of the phonon Free en-

ergy

As another application of the diagrammatic technique, we calculate the freeenergy of the lattice vibrations. Of course, this problem can be solved exactlyusing the normal mode representation. Here, we’ll show how one can obtainthe same answer using the perturbation methods of the previous section.

As before, we ignore the vector nature of the displacement and momen-tum operators and separate the Hamiltonian into an unperturbed part anda perturbation, see Eq. (3.73). Calculation of the free energy for the unper-turbed part is simple. The corresponding partition function is that for Nharmonic oscillators of frequency ω0,

Z0 =

(

1

2 sinh(~ω0/2T )

)N

, (3.97)

so that the corresponding Free energy reads

F0 = −T lnZ0 = NT ln[2 sinh(~ω0/2T )]. (3.98)

Using the definition (3.78) of the imaginary-time evolution operator, thepartition function Z for the total Hamiltonian H = H0+H1 can be expressed

37

τ’ τ’+1 + +2!2

1!

V’ij V’

j

i

+ ...

j=i

V’ii

2!1

k=l

V’kk

j=i

V’ii

τ ττ

ij

Figure 5: Diagrammatic representation the diagrams that contribute to thephonon partition function.

asZ

Z0=

tr e−H0/TU(1/T, 0)

tr e−H0/T= 〈U(1/T, 0)〉0, (3.99)

where the brackets 〈. . .〉 denote a thermal average with respect to the unper-turbed Hamiltonian H0.

We now expand the evolution operator U(1/T, 0) in powers of the per-turbation H1 in order to calculate the free energy difference

∆F = F − F0 = −T ln(Z/Z0) = −T ln〈U(1/T, 0)〉0. (3.100)

A diagrammatic expansion of 〈U(1/T, 0)〉0 can be obtained using the rulesderived in the previous section. In the first few orders, we find the diagramsshown in Fig. 5. As before, we distinguish two types of diagrams: connecteddiagrams (all but the third diagram in Fig. 5) and disconnected diagrams(the third diagram in Fig. 5). Upon taking the logarithm, the disconnecteddiagrams cancel, and one is left with the connected diagrams only.

The combinatorial factor for the connected diagrams is slightly more com-plicated than for the calculation of the phonon Green function. You can verifythat, for the nth order diagram, there are 2n−1(n− 1)! connected diagrams.Combining this combinatorial factor with the factor (−1)n/n! from the Tay-lor expansion of the exponent and the prefactor 1/2 in the perturbation H1,we find that the free energy is

∆F = −T∞∑

n=1

1

2n

∑

i1,...,in

∫ 1/T

0

dτ1 . . . dτnD0i1,i1(τ1 − τ2)V

′(~ri1 − ~ri2)

×D0i2,i2(τ2 − τ3) . . .D0

in,in(τn − τ1)V′(~rin − ~ri1). (3.101)

In order to make progress, we change to Matsubara frequencies, and performa Fourier transform with respect to the lattice coordinate ~ri. We then find

∆F = −T∑

~k

∑

m

∞∑

n=1

1

2n

(

V ′~kD0

~k(iωm)

)n

38

ω ωk−ω−ω

k

C1 C1

C2

00

C2

Figure 6: Integration contours for the evaluation of the Matsubara sum of Eq.(3.102).

=T

2

∑

~k

∑

m

ln

(

ω2m + ω2

~k

ω2m + ω2

0

)

. (3.102)

At zero temperature, one can replace the summation over Matsubarafrequencies by an integration. Since the zero temperature free energy equalsthe ground state energy EG, we find the result

∆EG =~

4π

∫

dω∑

~k

ln

(

ω2 + ω2~k

ω2 + ω20

)

=~

2

∑

~k

(ω~k − ω0). (3.103)

For finite temperatures one has to perform the summation over Matsubarafrequencies. Hereto we represent the summation as an integration in thecomplex plane, as shown in Fig. 6,

∆F =~

8πi

∑

~k

∫

C1

dz coth(z~/2T ) ln

(

−z2 + ω2~k

−z2 + ω20

)

. (3.104)

To see that Eq. (3.104) is correct, note that the function coth z has poles atthe points z = 2πmiT/~, with m integer. Performing the integration alongthe contour C1 picks out precisely these poles. The integrand is analyticeverywhere else in the complex plane, except for branch cuts on the real axis,for ω2

0 < z2 < ω2~k, see Fig. 6. Hence we can deform the integration contour

C1 into a contour C2 that is around the branch cuts instead of around the

39

Matsubara poles. The real part of the logarithm is continuous at the branchcuts, so that it does not contribute to the integral. The imaginary part jumpsby 2π at the branch cuts. Adding contributions from both branch cuts, wefind

∆F =~

2

∑

~k

∫ ω~k

ω0

dω coth(~ω/2T )

= T∑

~k

ln

(

sinh(~ω~k/2T )

sinh(~ω0/2T )

)

. (3.105)

Combining the result (3.105) for ∆F with that for the unperturbed freeenergy F0, Eq. (3.98), we find

F = T∑

~k

ln(

sinh(~ω~k/2T ))

. (3.106)

This is the same answer as one would have obtained from the normal-modeanalysis.

40

3.7 Exercises

Exercise 3.1: Jellium model for phonons

Sometimes, one wants a simplified description of phonons without referenceto the lattice. This may be the case, e.g., if one is interested in long-wavelength phenomena only, or if one only needs an order-of-magnitude es-timate of the phonon contribution to a certain physical observable. Such“structureless” models that do not refer to the underlying crystal lattice arecalled “jellium models”.

We start with a description of lattice ions by means of their charge den-sity ρion(~r), which is taken to be a continuous function of the coordinate~r. We first treat the electron gas as a negatively charged static backgroundthat neutralizes the positive charge of the ions. Then any deviation fromequilibrium δρion = ρion − ρ0

ion is associated with an electric field

~∇ · E =1

ǫ0δρion, (3.107)

where ǫ0 is the electric permittivity of vacuum. The electric field, in turnexerts a force field on the ions,

~f = ρ0ion~E. (3.108)

(a) Show that the ion charge density satisfies the equation

M∂2

∂t2δρion +

Zeρ0ion

ǫ0δρion = 0, (3.109)

where M is the ionic mass and Ze is the charge on one ion.

(b) Describe the possible excitations of the ion “jellium”. These are notphonons. Why not?

The physical cause that the jellium excitations we found are not phononsis the long range of the Coulomb interaction. In fact, those excitations areknown as “plasmon” excitations. They do not exist for a real ion lattice, butthey are relevant for electrons.

In a real material, the long range Coulomb interaction between the ions isscreened by the electrons. Since electrons are much lighter than the phonons,

41

they react quickly to any change in the ion density. In fact, to a good ap-proximation we can assume that the electrons follow the ion density instanta-neously, maintaining charge neutrality throughout the material at all times.

As a consequence, the energy price for a perturbation δρion of the iondensity is not the Coulomb energy, but the kinetic energy cost of increasingor decreasing the electron density: If the ionic charge density is changed byan amount δρion, the electronic charge density needs to be changed by −δρion

in order to maintain charge neutrality. The energy cost associated with thechange of electron density leads to a restoring force for the ions.

(b) The easiest way to estimate that restoring force, is to view it as an“electron pressure”. Show that, for a free electron gas, the groundstate energy for N electrons confined to a volume V is

Ee =~

2 (3π2N)5/3

10π2mV 2/3. (3.110)

(c) Use your answer to (b) to argue that the electronic pressure equals

pe =~

2

15π2m

(

3π2ρe

)5/3. (3.111)

(d) The ionic motion is driven by the gradient of pe. Show that the changeδρion obeys the wave equation

M∂2

∂t2δρion −

2

3ZεF

~∇2δρion = 0. (3.112)

(e) From here, derive the dispersion relation for phonons in the jelliummodel. What is the phonon velocity? This expression for the phonondisperson is known as the “Bohm-Staver law”.

(f) In the jellium model, you find only one phonon mode per wavevector.For a real lattice, there are three phonon modes per wavevector: onelongitudinal mode and two transverse modes. Is the single mode youfind in the jellium model a longitudinal mode or a transverse mode?What happened to the other two modes of the lattice model?

42

Exercise 3.2: Free energy

In this exercise we consider an alternative method to calculate the phononfree energy using diagrammatic perturbation theory. This alternative methoduses the diagrammatic rules for Green functions, and, thus, avoid the ap-pearance of the nontrivial combinatorial factors that appeared in the directcalculation of the free energy in Sec. 3.6.

In this calculation we consider a Hamiltonian H(λ) in which the pertur-bation H1 is multiplied by a coupling constant λ,

H(λ) = H0 + λH1. (3.113)

Hence, H(0) corresponds to the unperturbed Hamiltonian H0 whereas H(1)corresponds to the full Hamiltonian H0 +H1. For this Hamiltonian, the Freeenergy becomes a function of λ

F (λ) = −T lnZ(λ), Z(λ) = tr e−(H0+λH1)/T . (3.114)

Taking a derivative to λ, we find

∂F (λ)

∂λ=

trH1e−(H0+λH1)/T

tr e−(H0+λH1)/T=

1

λ〈λH1〉λ (3.115)

where the brackets 〈. . .〉λ denote a thermal average with respect to the Hamil-tonian H(λ).

Integrating Eq. (3.115) from λ = 0 to λ = 1 we find the free energydifference ∆F between the full Hamiltonian H0 + H1 and the unperturbedHamiltonian H0,

∆F =

∫ 1

0

dλ1

λ〈λH1〉λ. (3.116)

Hence, if we know the average 〈λH1〉λ for all λ, we can find the change inthe free energy.

The average 〈λH1〉λ can be expressed in terms of the phonon temperatureGreen function for the Hamiltonian H(λ),

〈λH1〉λ = −λ2

∑

ij

V ′(~ri − ~rj) limτ↓0

D(λ)ij (τ), (3.117)

whereD(λ)

ij (τ) = −〈Tτui(τ)uj(0)〉λ. (3.118)

43

(a) Verify Eq. (3.117).

(b) Calculate ∆F using diagrammatic perturbation theory and Eq. (3.116).

(c) Verify that your result is the same as that of the direct perturbationexpansion of ∆F , order by order in the perturbation theory in H1. Howare the combinatorial factors taken into account in the method you usehere?

The trick with the integration over the coupling constant can be appliedto other systems as well.

44

4 Scattering from a single impurity

In this section, we consider non-interacting electrons in the presence of animpurity. This is another problem for which an exact solution is possible.In the next two subsections we first describe this exact solution. In thelast subsection we then show how the same results can be obtained usingthe many-electron Green function formalism and diagrammatic perturbationtheory.

In first quantization notation, the Hamiltonian we consider is

H = − 1

2m∂2

~r + U(~r), (4.1)

where U(~r) is the impurity potential. In second quantization notation, theHamiltonian is written in terms of creation and annihilation operators ψ†

σ(~r)and ψσ(~r) of an electron at position ~r and with spin σ,

H = − 1

2m

∑

σ

∫

d~rψ†σ(~r)∂2

~rψσ(~r) +∑

σ

∫

d~rψ†σ(~r)U(~r)ψσ(~r). (4.2)

In the Heisenberg picture, the operators ψ† and ψ are time dependent, andtheir time-dependence is given by

∂tψσ(~r, t) =i

~[H, ψσ(~r, t)]−. (4.3)

Using the anticommutation relations for the creation and annihilation oper-ators ψ†

σ(~r) and ψσ(~r), the time-evolution can be rewritten in terms of thefirst-quantized Hamiltonian in a form that is quite similar to the Schrodingerequation,

∂tψσ(~r, t) = − i

~Hψσ(~r, t). (4.4)

4.1 Scattering theory and Friedel Sum Rule

4.1.1 General formalism

We first consider the problem of one electron in the presence of an impuritywith scattering potential U(~r). There are no other electrons present. Wedrop reference to the spin index σ.

45

For the one-electron problem, the expectation values 〈. . .〉 in the Greenfunction definitions are taken with respect to the vacuum |0〉. The temper-ature Green function has no meaning in that case. In the absence of otherelectrons, the retarded and time-ordered Green functions coincide,

GR(~r, ~r′; t) = −iθ(t)〈0|[ψ(~r, t), ψ†(~r′, 0)]−|0〉= −iθ(t)〈0|Tt[ψ(~r, t)ψ†(~r′, 0)]|0〉= G(~r, ~r′; t). (4.5)

Using the equation of motion for the operator ψ(~r, t), one can derive anequation of motion for the retarded Green function,

∂tGR(~r, ~r′; t) = −iδ(t)〈0|ψ(~r, t)ψ†(~r′, 0)|0〉 − iθ(t)〈0|∂tψ(~r, t)ψ†(~r′, 0)|0〉

= −iδ(t)δ(~r − ~r′) − i

~HGR(~r, ~r′; t). (4.6)

where H is the first-quantization form of the single-particle Hamiltonianoperating on the first argument of the Green function, see Eq. (4.4) above.Performing a Fourier transform, one finds

(ω − ~−1H)GR(~r, ~r′;ω) = δ(~r − ~r′). (4.7)

A differential equation like Eq. (4.7) requires the specification of boundaryconditions as |~r−~r′| → ∞. Since the retarded Green function is zero for t < 0,we require outgoing wave boundary conditions as |~r − ~r′| → ∞. Similarly,the advanced Green function GA the solution of Eq. (4.7) with ingoing waveboundary conditions.

The advanced and retarded Green functions can be expressed in terms ofthe eigenvalues εµ and eigenfunctions φµ(~r) of H,

GR(~r, ~r′;ω) =∑

µ

φµ(~r)φ∗µ(~r

′)

ω − εµ + iη, (4.8)

GA(~r, ~r′;ω) =∑

µ

φµ(~r)φ∗µ(~r′)

ω − εµ − iη, (4.9)

where η is a positive infinitesimal. One verifies that both the retarded and theadvanced Green functions are analytical in the upper and lower half planesof the complex plane, respectively, and that they satisfy the relation

trG(ε) = − ∂

∂εln detG(ε). (4.10)

46

In the absence of an impurity potential, the Hamiltonian is

H0 = − ~2

2m∂2

~r . (4.11)

In the basis of plane wave states |~k〉, the retarded and advanced Green func-tions are

G0R~k,~k′(ω) =

δ~k,~k′

ω + iη − ε~k, G0A

~k,~k′(ω) =δ~k,~k′

ω − iη − ε~k. (4.12)

In coordinate representation one has

G0(~r, ~r′;ω) =1

V

∑

~k′

ei~k′·(~r−~r′)

ω ± iη − ε~k′

, (4.13)

where η is a positive infinitesimal and the + (−) sign applies to the retarded(advanced) Green function. In order to further simplify this result, we definek through εk = ω and linearize the spectrum,

ε~k′ = ω + v(|~k′| − k), (4.14)

where v is the velocity. Then, performing the Fourier transform, one finds

G0(~r, ~r′;ω) =1

(2π)3

∫ 2π

0

dφ

∫ 1

−1

d cos θ

∫ ∞

0

k′2dk′eik′|~r−~r′| cos θ

±iη − v(k′ − k)

=1

(2π)2i|~r − ~r′|

∫ ∞

0

k′dk′eik′|~r−~r′| − e−ik′|~r−~r′|

±iη − v(k′ − k)

≈ k

4π2i|~r − ~r′|v

∫ ∞

−∞

dxei(k+x/v)|~r−~r′| − e−i(k+x/v)|~r−~r′|

±iη − x

= − m

2π~2

e±ik|~r−~r′|

|~r − ~r′| . (4.15)

This result can be derived without approximations for a quadratic disper-sion relation ε~k = ~

2|~k|2/2m. The derivation given here shows that theresult (4.15) is not restricted to a parabolic dispersion relation. For non-infinitesimal η, the final result needs to be multiplied by e−η|~r−~r′|/v.

Let us now add an impurity to the system, which is described by a thepotential U(~r). We want to write a solution |ψ~k〉 of the total Hamiltonian

47

H = H0 +U with incoming boundary condition corresponding to the unper-turbed state |~k〉. This solution can be written as12

|ψ~k〉 = |k〉 +G0R(ε~k)U |ψ~k〉= |k〉 +G0R(ε~k)U |~k〉 +G0R(ε~k)UG

0R(ε~k)U |~k〉 + . . . ,

where the symbol U should be interpreted as the quantum operator cor-responding to the potential U(~r). The series (4.16) is known as the Bornseries. Truncating the series after the nth order in U yields the nth Bornapproximation. Defining the T -matrix as

T (ω+) = U + UG0R(ω)U + . . .

= U [1 −G0R(ω)U ]−1

= U + UG0R(ω)T (ω+), (4.16)

where ω+ = ω + iη, η being a positive infinitesimal, we can rewrite this as

|ψ~k〉 = |k〉 +∑

~k′

T~k′~k(ε~k + iη)

ε~k − ε~k′ + iη|k′〉, (4.17)

with T~k′~k(ω) = 〈~k′|T (ω)|~k〉. The T -matrix is related to the retarded Greenfunction GR of the total Hamiltonian H0 + U as

GR(ω) = G0R(ω) +G0R(ω)T (ω+)G0R(ω). (4.18)

Note that in this notation, where we have a summation over discrete set ofwavevectors, the T -matrix is of order 1/V . Using Eq. (4.15) one then findsthat the asymptotic form of the wavefunction of the scattered wave (4.17) is

ei~k~r + f(~k′, ~k)ei~k′·~r

r, f(~k,~k′) = − mV

2π~2T~k~k′(ε~k + iη), (4.19)

where the wavevector ~k′ is defined as kr, r being the unit vector in thedirection of ~r.

With this result we can now write expressions for two useful quantitiesfrom collision theory: the differential cross section dσ/dΩ, where dΩ denotesan element of solid angle, which is

dσ

dΩ= |f(~k′, ~k)|2 =

m2V 2

4π2~4|T~k′,~k(ε~k + iη)|2, (4.20)

12You can find a good discussion of stationary scattering theory in section 37 of Quantum

Mechanics, by L. I. Schiff, McGraw-Hill (1968).

48

and the transition rate

W~k′~k =2π

~|T~k′~k(ε~k + iη)|2δ(ε~k′ − ε~k). (4.21)

The transition rate gives the probability per unit time that a particle isscattered out of plane wave state ~k into plane wave state ~k′. To lowest orderin the impurity potential U , Eq. (4.21) is nothing but the Fermi golden rule.

The total rate for scattering out of the state |~k〉 is found by summation ofEq. (4.21) over all outgoing momenta,

1

τ~k=

2π

~

∑

~k′

|T~k′~k(ε~k + iη)|2δ(ε~k′ − ε~k). (4.22)

Taking the hermitian conjugate of Eq. (4.16) and substituting U = U † =T † − T †G0AU in the second term of Eq. (4.16), one finds

T = U + T †G0RT − T †G0AUG0RT . (4.23)

The first and third term on the right hand side are hermitian, so that

ℑT = T †ℑG0RT . (4.24)

Using Eq. (4.12), this result can be written as

1

2i

(

T~k~k′(ω+) − T~k′~k(ω

+)∗)

= −π∑

~k1

T ∗~k1

~k(ω+)T~k1

~k′(ω+)δ(ω − ε~k1

). (4.25)

This result is known as the “generalized optical theorem”. The “standard”optical theorem is obtained setting ~k′ = ~k,

ImT~k~k(ω+) = −π

∑

~k1

|T~k1~k(ω

+)|2δ(ω − ε~k1). (4.26)

The density of levels dN(ω)/dω is expressed in terms of the Green func-tion as

dN(ω)

dω=∑

µ

δ(ω − εµ) = −1

πIm trGR(ω). (4.27)

The impurity density of states (dN/dω)imp is defined as the difference betweenthe density of states with and without impurity. Using Eq. (4.10) above, onecan show that

(

dN

dω

)

imp

=1

π

∂δ(ω)

∂ω, (4.28)

49

where δ(ω) = arg det T (ω) is the so-called “total scattering phase shift”.Zero eigenvalues of T , which correspond to states that do not scatter at all,should be left out from the calculation of the total scattering phase shift.Equation (4.28) is known as the “Friedel sum rule”. At zero temperature,as a result of the introduction of the impurity, the number of electrons in amany-electron system is changed by the amount

Nimp =

∫ εF

−∞

(

dN

dω

)

imp

dω =δ(εF )

π, (4.29)

where we took δ(−∞) = 0 and εF is the Fermi energy.

4.1.2 Example of an s-wave scatterer

Following Doniach and Sondheimer, we study the case U(~r) = uδ(~r) of adelta-function potential at the origin in detail. Fourier transforming to thebasis of plane-wave states and using second-quantization notation, the po-tential U is written as

U =u

V

∑

~k,~q

ψ†~k+~q

ψ~k. (4.30)

Then the equation for the T -matrix becomes

T~k,~k′(ω + iη) =u

V

1

1 − uG0R(ω), (4.31)

where we abbreviated

G0R(ω) =1

V

∑

~k

G0R~k~k

(ω) =1

V

∑

~k

1

ω + iη − ε~k′

. (4.32)

(A similar definition can be made for G0A(ω).) Note that the T -matrix isisotropic: For a delta-function scatterer, scattering is equally probable in alldirections. Since T is isotropic, only one of the eigenvalues of T is nonzero.The simplest way to calculate the corresponding phase shift δ is to note thatT (ω + iη) = T (ω − iη)∗, so that

e2iδ(ω) =T (ω + iη)

T (ω − iη)=

1 − uG0(ω − iη)

1 − uG0(ω + iη). (4.33)

50

The real part of G0(ω + iη) depends weakly on ω. Formally, for a parabolicpotential and a delta-function scatterer, ReG0(ω + iη) is ultraviolet diver-gent. The high-energy part of the energy integration can be cut off by in-troducing a finite width of the conduction band, or by noting that for highenergies the approximation of a delta function scatterer breaks down. Theimaginary part of G0(ω + iη) is related to the density of states ν0 (per unitvolume) at the energy ω in the absence of the impurity,

ImG0(ω + iη) = −πν0(ω) = − k2

2π~v, (4.34)

Here v = ~−1∂ε/∂k is the velocity at wavevector ~k.

With the help of Eq. (4.34), we can express T in terms of δ and ν0,

T~k~k′(ω ± iη) = − 1

πν0Ve±iδ(ω) sin δ(ω)

= −2π~v

k2Ve±iδ(ω) sin δ(ω). (4.35)

Hence, in this case we find for the differential cross section

dσ

dΩ=m2v2

k4~2sin2 δ(ω), (4.36)

and for the total scattering cross section (the integral of dσ/dΩ over the fullsolid angle)

σ =4πm2v2

k4~2sin2 δ(ω). (4.37)

4.2 Alternative derivation of the Friedel Sum Rule

The Friedel sum rule relates the change ∆N in the total number of particlesas the result of the presence of an impurity to the phase shifts that particlesexperience while scattering off the impurity. In terms of the T -matrix, theFriedel sum-rule reads

∆N =δtot(εF )

π, (4.38)

where the total phase shift δtot is defined as13

δtot = arg detT (ε+ i0).

13Strictly speaking, Eq. (4.38) and the derivation given below apply to potential scat-tering only. One may, however, extend the Friedel sum rule to the case of scattering froma dynamical impurity. See, e.g., chapter 5 of The Kondo problem to heavy fermions, byA. C. Hewson.

51

You have seen a formal derivation of the Friedel sum rule in the previoussection. Here, we will consider a more elementary derivation of the Friedelsum rule for the case of a spherically symmetric scatterer.

If a flux of noninteracting particles is elastically scattered on a sphericallysymmetric potential at ~r = 0, the wave function at large values of r will beof the form

Ψ ≈ eikz + f(θ)eikr

r, (4.39)

where z is the direction of the incoming flux. The first term represents theincoming flux (with wavenumber k), the second one the scattered wave. TheT -matrix is related to the function f(θ) as f(θ) = −(m/2π~

2)T~k~k′, where

|~k| = |~k′| = k and θ is the angle between ~k and ~k′, see Eq. (4.19) above.Since the scattered wave in Eq. (4.39) has axial symmetry about the

z-axis, it may be expanded in Legendre polynomials,

Ψ =

∞∑

ℓ=0

aℓPℓ(cos θ)Rkℓ(r) (4.40)

The constants aℓ are just expansion coefficients, and the radial functions Rkℓ

satisfy the “radial Schrodinger equation”

r−2 ∂

∂rr2∂Rkℓ

∂r+

(

k2 − ℓ(ℓ+ 1)

r2− 2m

~2U(r)

)

Rkℓ = 0,

where the energy eigenvalues are written as ~2k2/2m, the energy of a free

particle, and U(r) is the scattering potential.At distances sufficiently large that U(r) ≈ 0, the radial functions Rkℓ will

approach the asymptotic expansion

Rkℓ ≈2

rsin(kr − ℓπ/2 + δℓ) (4.41)

Both the factor of 2 and the factor ℓπ/2 are introduced for convenience, sothat the asymptotic behavior of Rkℓ reduces to the solutions for Rkℓ whenU(r) vanishes identically, by putting δℓ = 0. The δℓ are the so-called scat-tering phases. They depend on the scattering potential U(r) and approachzero when U(r) vanishes.

The change in the number of states near the Fermi surface as a resultof the scattering potential generally depends on the δℓ’s. To see this, we

52

first have to connect the general δℓ’s in the expression (4.41) for the radialfunctions in the expansion of the total wavefunction (4.40) to the functionf(θ) defined in (4.39). Apart from an overall normalization, the coefficientsaℓ in (4.40) are fixed by the requirement that the function Ψ − eikz is of theform f(θ)eikr/r, i.e. has only terms of the form of an outgoing wave, eikr/r.This requirement leads to the identification

f(θ) =1

2ik

∞∑

ℓ=0

(2ℓ+ 1)[

e2iδℓ − 1]

Pℓ(cos θ). (4.42)

You may derive Eq. (4.42) using the fact that for large r the plane wave eikz

can be expanded in terms of Legendre polynomials as

eikz = eikr cos θ =1

2ikr

∞∑

ℓ=0

(2ℓ+ 1)Pℓ(cos θ)[

(−1)ℓ+1e−ikr + eikr]

.

Let us now consider the change in the number of states in the presence ofthe impurity. Hereto we imagine a large sphere of radius L about the origin,and impose that the wavefunction vanishes on this sphere. In the absence ofthe potential (δℓ = 0), Eq. (4.41) then yields

kL− ℓπ

2= nπ =⇒ knℓ =

ℓπ

2L+nπ

L(4.43)

In the presence of the scattering potential, the wavenumbers k′nℓ satisfy k′nℓ =ℓπ/(2L) + nπ/L− δℓ/L, so we get for the change in wavenumber

k′nℓ − knℓ = −δℓL

(4.44)

Let us first consider the special case in which only δ0 is nonzero. The spacingof successive values of kn0 is π/L [see Eq. (4.43)]; if we now consider a givenvalue of k, say k = kF , and imagine turning on the potential, then eachallowed k value shifts, according to Eq. (4.44), by an amount −δ0(kF )/L.Then the number of states k with k < kF increases by an amount

Nimp =δ0L

· Lπ

=δ0π.

Now you are invited to prove the general Friedel sum rule (4.38), which reads,in terms of the phase shifts δℓ,

Nimp = π−1∑

ℓ

(2ℓ+ 1)δℓ(kF ).

53

Using these results, one can also show that, at zero temperature, the densityof electrons far from a small spherically symmetric impurity shows oscillations

nimp(r) ≈ − 1

4π2r3cos[2kF r + δ0(kF )] sin[δ0(kF )], (4.45)

where δ0 is a scattering phase shift evaluated at the Fermi energy.

4.3 Many-particle formulation

In this section, we’ll use the many-particle formalism to describe the electrongas in the presence of an impurity.

We are interested in the electron density n(~r) of a gas of noninteractingelectrons in the presence of a potential U(~r). The electron density can beexpressed in terms of the many-electron temperature Green function, using

n(~r) = 〈ψ†(~r)ψ(~r)〉= lim

τ↑0G(~r, ~r; τ), (4.46)

where the temperature Green function is defined with respect to the electroncreation and annihilation operators,

G(~r, ~r′; τ) = −〈Tτ [ψ(~r, τ)ψ†(~r′, 0)]〉. (4.47)

We now calculate the many-electron temperature Green function usingdiagrammatic perturbation theory. It is advantageous to change to the mo-mentum representation,

ψ~k =1

V 1/2

∫

d~rψ(~r)e−i~k·~r, ψ(~r) =1

V 1/2

∑

~k

ψ~kei~k·~r,

so that the temperature Green function in momentum representation reads

G~k;~k′(τ) = −〈Tτ [ψ~k(τ)ψ†~k′

(0)]〉

=1

V

∫

d~rd~r′G(~r, ~r′; τ)e−i(~k·~r−~k′·~r′). (4.48)

The Hamiltonian is separated into an “unperturbed” part H0 and the “per-turbation” H1,

H = H0 +H1, (4.49)

54

with

H0 =∑

~k

(ε~k − µ)ψ†~kψ~k, (4.50)

H1 =∑

~k

∑

~q

U~qψ†~k+~q

ψ~k, (4.51)

where U~q is the Fourier transform of the scattering potential,

U~q =1

V

∫

d~rU(~r)e−i~q·~r. (4.52)

Here, we have included the chemical potential into the unperturbed Hamil-tonian H0.

As in the phonon case, it is not clear a priori that the perturbation H1

is small. Still, in the method of diagrammatic perturbation theory, it isassumed that an expansion in H1 is possible, and that resummation of thatexpansion gives the correct results. Since we are going to expand in H1, itis useful to calculate the temperature Green function for the “unperturbed”Hamiltonian H0,

G0~k;~k′(τ) = −δ~k,~k′θ(τ)

e−(ε~k−µ)τ

1 + e−(ε~k−µ)/T

θ(τ) + δ~k;~k′θ(−τ)e−(ε~k

−µ)τ

1 + e(ε~k−µ)/T

. (4.53)

This result is obtained directly from the second-quantized form of the Hamil-tonian H0, using the Fermi-Dirac distribution function and the fact that theimaginary-time dependence of the operators ψ~k and ψ†

~kis

ψ(~k; τ) = e−(ε~k−µ)τψ(~k; 0), ψ†(~k; τ) = e(ε~k

−µ)τψ†(~k; 0), (4.54)

for −~/T < τ < ~/T . The Fourier transform of the temperature Greenfunction is

G0~k;~k′(iωn) =

δ~k,~k′

iωn − (ε~k − µ). (4.55)

We now calculate the Green function in the presence of the impurity usingdiagrammatic perturbation theory. The calculation is done along the lines ofthat of Sec. 3.5. We introduce the “interaction picture” evolution operator

U(τ, τ ′) = eτH0e−(τ−τ ′)(H0+H1)e−τ ′H0

= Tτ exp

(

−∫ τ

τ ′

dτ1H1(τ1)

)

, (4.56)

55

and use time-dependent operators for which the time-dependence stems fromH0 only,

H1(τ) = eτH0H1e−τH0 . (4.57)

Following the same steps as in Sec. 3.5, we then write the temperature Greenfunction as

G~k;~k′(τ) =〈Tτ [U(1/T, 0)ψ~k(τ)ψ

†~k′

(0)]〉0〈TτU(1/T, 0)〉0

. (4.58)

The brackets 〈. . .〉0 indicate an thermal average with respect to the unper-turbed Hamiltonian H0 only.

The next step is to expand the evolution operator in powers of H1 andcalculate averages using the Wick theorem. In the language of the generalformulation of Sec. 2, the Wick theorem allows one to calculate an averageof the form 〈A1(τ1) . . . A2n(τ2n)〉0, where all operators Ai, i = 1, . . . , 2n arelinear in fermion creation and annihilation operators and the average is takenwith respect to a Hamiltonian H0 that is quadratic in fermion creation andannihilation operators. Clearly, every order in the perturbation expansion ofEq. (4.58) satisfies these criteria. The Wick theorem states that

〈A1(τ1) . . .A2n(τ2n)〉0 =

(

−1

2

)n∑

P

(−1)PG0AP (1),AP (2)

(τP (1), τP (2)) × . . .

× G0AP (2n−1),AP (2n)

(τP (2n−1), τP (2n)), (4.59)

where the sum is over all permutations P of the numbers 1, 2, . . . , 2n and(−1)P is the sign of the permutation. In our case, only Green functionswith one creation operator and one annihilation operator are nonzero. Thatsimplifies the set of allowed permutations, and we can write the Wick theoremas

〈ψ†~k1

(τ1) . . . ψ†~kn

(τn)ψ~k′n(τ ′n) . . . ψ~k′

1(τ ′1)〉0

=∑

P

(−1)PG0~k1,~k′

P (1)

(τ1, τ′P (1)) . . .G0

~kn,~k′P (n)

(τn, τ′P (n)). (4.60)

With the help of the Wick theorem, we can now calculate the average(4.58) to every order in H1 by drawing all possible diagrams. As in the caseof phonons, all disconnected diagrams cancel between numerator and denom-inator, and one is left with connected diagrams only. Also, the combinatorial

56

+ ...

τ1

k1

τ1 τ2

+

τ

+

ττ’ τ’k k’k k’τ’ τ’=

k=k’k’k

ττ

Figure 7: Diagrammatic representation of Eq. (4.61). The double arrow cor-responds to the full Green function G~k,~k′(τ, τ

′), single arrows correspond to the

unperturbed Green function G0~k,~k′

(τ, τ ′), and the dashed lines with the stars cor-

respond to the impurity potential U~q. (The momentum argument of the impuritypotential is the difference in electron momenta at the scattering vertex.)

factor cancels the factorial in the Taylor expansion of the exponent in theevolution operator U(1/T, 0), so that one sums over topologically differentdiagrams only. We thus find

G~k;~k′(τ) = G0~k;~k

(τ)δ~k,~k +

∫ 1/T

0

dτ1U(~k − ~k′)G0~k;~k

(τ − τ1)G0~k′;~k′(τ1)

+

∫ 1/T

0

dτ1

∫ 1/T

0

dτ2∑

~k1

G0~k;~k

(τ − τ1)U(~k − ~k1)

× G0~k1;~k1

(τ1 − τ2)U(~k1 − ~k′)G0~k′;~k′(τ2) + . . . . (4.61)

Diagrammatically this result may be represented as in Fig. 7. The doublearrow represents the full Green function, while the single arrows correspondto the Green functions of the free electron Hamiltonian H0. The dashed lineswith the stars correspond to scattering from the impurity potential U . Theseries can be written as a self-consistent equation for G~k,~k′(τ),

G~k;~k′(τ) = G0~k;~k

(τ)δ~k,~k′ +∑

~k1

∫ 1/T

0

dτ1G0~k;~k

(τ − τ1)U(~k − ~k1)G~k1;~k′(τ1). (4.62)

This equation is called the “Dyson” equation. It is represented diagrammat-ically in Fig. 8. The convolution with respect to the imaginary time variableτ can be handled by Fourier transformation,

G~k;~k′(iωn) = G0~k;~k

(iωn)δ~k,~k′ +∑

~k1

U(~k − ~k1)G0~k;~k

(iωn)G~k1;~k′(iωn). (4.63)

The summation over the momentum ~k1 is more complicated. It can notbe carried out exactly for an arbitrary impurity potential. As a simple

57

+ k’

k’k

’ττkτ’ τ’=

k=k’

k

τ1

1ττ

Figure 8: Diagrammatic representation of the Dyson equation for impurity scat-tering, Eq. (4.62).

+ k’

k’k

n n

n n

k=k=k’

k1ω ω

ω ω

Figure 9: Diagrammatic representation of the Matsubara frequency representa-tion of the Dyson equation for impurity scattering, Eq. (4.63). Notice that theMatsubara frequency ωn is conserved at every scattering event.

example, we return to the case of a point scatterer, U(~r) = uδ(~r). In this

case, U(~k−~k1) = u/V does not depend on ~k and ~k1, see Eq. (4.30). Summing

Eq. (4.63) over ~k, we find

∑

~k

G~k;~k′(iωn) = G0~k′;~k′(iωn) +

u

V

∑

~k

G0~k;~k

(iωn)

∑

~k

G~k;~k′(iωn)

, (4.64)

from which we derive

∑

~k

G~k;~k′(iωn) =G0

~k′;~k′(iωn)

1 − uV

∑

~k G0~k;~k

(iωn), (4.65)

and, hence,

G~k;~k′(iωn) = G0~k;~k

(iωn)δ~k,~k′ +u

VG0

~k;~k(iωn)

1

1 − uG0(iωn)G0

~k′;~k′(iωn), (4.66)

where we abbreviated

G0(iωn) =1

V

∑

~k

G0~k;~k

(iωn). (4.67)

What remains is to perform the inverse Fourier transform,

G~k;~k′(τ) = T∑

n

e−iωnτG~k;~k′(iωn). (4.68)

58

C1 C1

C

C

2

2

Figure 10: Integration contours for the derivation of Eq. (4.69) and (4.70).

This is done with the help of the following trick. The summation in Eq.(4.68) is written as an integration over the contour C1, see Fig. 10,

G~k;~k′(τ) =1

4πi

∫

C1

dze−zτ [tanh(z/2T ) − 1]G~k;~k′(iωn → z). (4.69)

We are interested in the case of negative τ .14 Then we make use of the factthat G is analytic for all frequencies away from the real and imaginary axes,so that we may deform the contour C1 into the contour C2, see Fig. 10. Notethat there is no contribution from the segments near infinity by virtue ofthe factor [tanh(z/2T ) − 1] for Re z > 0 and by virtue of the factor e−zτ forRe z < 0. We thus obtain

G~k;~k′(τ) =1

4πi

∑

±

(±)

∫ ∞

−∞

dωe−ωτ [tanh(ω/2T )− 1]G~k;~k′(iωn → ω ± iη),

=e−(ε~k

−µ)τ

1 + e(ε~k−µ)/T

− u

V

∑

±

(±1

2πi

)∫ ∞

−∞

dωe−ωτ

1 + eω/T

1

ω± − (ε~k − µ)

× 1

1 − uG0(ω±)

1

ω± − (ε~k′ − µ), (4.70)

where ω± = ω ± iη and η is a positive infinitesimal. The first term in Eq.(4.70) is the temperature Green function in the absence of the potential,cf. Eq. (4.53). The second term is the correction that arises as a result ofscattering from the impurity at ~r = 0.

14If we would have wanted to calculate the temperature Green function for positive τ ,we would have used Eq. (4.69) with a factor [tanh(z/2T )+ 1] instead of [tanh(z/2T )− 1].

59

We first use this result to calculate the impurity contribution to the totalelectron density

Nimp = limτ↑0

∑

~k

(

G~k;~k(τ) − G0~k;~k

(τ))

= − u

V

∑

±

(±1

2πi

)

∑

~k

∫ ∞

−∞

dω1

1 + eω/T

1

(ω± − (ε~k − µ))2

1

1 − uG0(ω±)

= −∑

±

(±1

2πi

)∫ ∞

−∞

dω1

1 + eω/T

∂

∂ωlog[

1 − uG0(ω±)]

. (4.71)

This result can be rewritten in terms of the T matrix and the scatteringphase shifts. Using the relation (4.33) between G, the T -matrix, and thescattering phase shift, we recover the Friedel sum rule,

Nimp =1

π

∫

dω

(

− ∂

∂ω

1

1 + eω/T

)

δ0(ω + µ). (4.72)

At zero temperature, Eq. (4.72) simplifies to Eq. (4.29) above.In order to find how the density change depends on the distance r to the

impurity site, we have to change variables from the momenta ~k and ~k′ to thepositions ~r and ~r′. Hereto, we start again from the second term in Eq. (4.70)

and note that the ~k-dependence is through the factors 1/(ω+−(ε~k−µ)) only.The corresponding Fourier transform has been considered in Sec. 4.1.1 andthe result for the asymptotic behavior as r → ∞ is

nimp(~r) = limτ↑0

∑

±

(±1

2πi

)∫ ∞

−∞

dωm2v

2πr2k2~3

× sin δ(ω + µ)e−ωτ

1 + eω/Te±i[δ(ω+µ)+2(kF +ω/~vF )r].

In order to perform the ω-integration, we expand the phase shift δ(ω + µ)around ω = 0 and keep terms that contribute to leading order in 1/r only.The result is

nimp(~r) = −sin δ(µ)

4π2r3

∫

dω

(

− ∂

∂ω

1

1 + eω/T

)

cos[δ(µ) + 2(kF + ω/~vF )r].

(4.73)

60

At zero temperature, this simplifies to Eq. (4.45). These density oscillationsare known as “Friedel oscillations”. At zero temperature, Friedel oscillationsfall off algebraically, proportional to 1/r3. At finite temperatures, the maincause of decay are the thermal fluctuations of the phase shift 2ωr/~vF , whichare important for T & ~vF/r.

The final results (4.72) and (4.73) are to be multiplied by two if spindegeneracy is taken into account.

61

4.4 Exercises

Exercise 4.1: Scattering from a single impurity in one dimension

In this exercise, we consider scattering from a delta-function impurity in onedimension. The impurity has potential U(x) = uδ(x) and is located at theorigin.

It is our aim to calculate the full Green function G(x, x′; iωn) in coordinaterepresentation. In order to calculate G(x, x′; iωn), one may either use theGreen function in momentum space and Fourier transform back to coordinatespace, as we did in Sec. 4.3, or derive the Dyson equation in coordinaterepresentation. Following the latter method, you can verify that one finds

G(x, x′; iωn) = G0(x, x′; iωn) (4.74)

+ G0(x, 0; iωn)u

1 − G0(0, 0; iωn)uG0(0, x′; iωn).

(a) Calculate the temperture Green function in coordinate representation.

(b) Use your answer to (a) to calculate the retarded Green function. Ifx′ < 0, show that it can be written as

GR(x, x′;ω) = tG0R(x, x′;ω)θ(x)

+ [1 + reiφ(x,x′)]G0R(x, x′, ω)θ(−x). (4.75)

The complex numbers t and r are called “transmission” and “reflection”amplitude.

(c) Explain the origin of the phase shift φ(x, x′) in Eq. (4.75).

Exercise 4.2: Anderson model for impurity scattering

In some applications, a simple scattering potential is not a good descriptionfor an impurity. This is the case, for example, if the impurity introduces anextra localized state for electrons in an otherwise perfect lattice. In this case,one can use the following Hamiltonian to describe the impurity,

H =∑

~k,σ

ε~kσψ†~kσψ~kσ

+ εimp

∑

σ

ψ†iσψiσ

62

+∑

~kσ

(V~kψ†

~kσψiσ + V ∗

~kψ†

iσψ~kσ). (4.76)

Here εimp is the energy of the extra localized state. The second line of Eq.(4.76) represents “hybridization” of the impurity state with the Bloch states

|~k〉 of the lattice. The matrix element V~k describes the overlap of the impurity

state with the state |~k〉. For a strongly localized impurity orbital at the origin,

one can neglect the ~k-dependence of the matrix elements V~k.For this system, we define Green functions with respect to the wavevec-

tors of the Bloch states and with respect to the impurity site. Hence, thetemperatue Green function is denoted as G~k′,~k(iωn), Gi,~k(iωn), G~k′,i(iωn), andGi,i(iωn), etc.

(a) Show that the temperature Green function satisfies the equations

1 = (iωn − εimp)Gi,i(iωn) −∑

~k′

V ∗~k′G~k′,i(iωn),

0 = (iωn − ε~k′)G~k′,i(iωn) − V~k′Gi,i(iωn),

0 = (iωn − εimp)Gi,~k(iωn) −∑

~k′

V ∗~k′G~k′,~k(iωn),

δ~k′~k = (iωn − ε~k′)G~k′,~k(iωn) − V~k′Gi,~k(iωn).

Solve these equations and analytically continue your answer to find theretarded Green function GR.

(b) Show that the T -matrix is given by

T~k′~k(ε) = V~k′Gi,i(ε)V∗

~k.

Calculate the total scattering phase shift δ(ε), and the impurity densityof states νimp(ε).

(c) Evaluate your expressions for the special case of a “flat conductionband”: the conduction electron states have a constant density of statesν0 for energies −D < ε < D, where D is the bandwidth. For the hy-bridization matrix elements, you may set |V~k|2 = V 2/Ns, where Ns isthe total number of lattice sites in the sample. (For a lattice, the num-ber of available wavevectors also equals Ns.) The conduction electrondensity of states ν0 is defined as density of states per site.

63

Exercise 4.3: Resonant tunneling

In this exercise we consider a “quantum well”, which is formed between twotunnel barriers in a three-dimensional metal or semiconductor structure. Thetunnel barriers have a potential U(~r) = u(δ(x− a) + δ(x+ a)).

Since the system is translation invariant in the y and z directions, butnot in the x direction, we use a mixed notation, in which Green functionsare Fourier transformed with respect to y and z, but not with respect to x.The Green function is diagonal in the momenta ky and kz.

(a) Following the same method as in exercise 4.1, calculate the retardedGreen function GR

ky,kz;ky,kz(x, x′;ω). Express your answer in the same

form as Eq. (4.75) and calculate the transmission and reflection ampli-tudes tky ,kz and rky,kz .

(b) When is the transmission is unity? Find a simple physical argumentfor this relation. (Hint: use your answer to Ex. 4.1.)

64

5 Many impurities

We have seen that it was a difficult task to calculate the Green function fora single impurity. In the presence of many impurities, the calculation of theGreen function is as good as impossible.

One might ask whether that is a big problem. Do we really want tocalculate the Green function in the presence of many impurities? The answeris no, and the reason is very simple: Nobody knows the exact locations andpotentials of all impurities in a realistic sample! Of course, if we don’t knowthe exact potential U(~r), it does not make sense to try to calculate the exactGreen function in the presence of that potential.

The information one usually does know is the concentration of impurities,i.e., the number of impurities per unit volume. The impurity concentrationcan also be characterized by the mean time τ between collisions. Becausethat is all information one has, one considers an ensemble of different sam-ples, all with the same impurity concentration, but with different locations ofthe impurities. Such samples are called “macroscopically equivalent but mi-croscopically different”. We then calculate quantities that are averaged overthis ensemble. Such an average is called a disorder average, to distinguish itfrom the thermal average we have been considering so far.

There is an important difference between disorder averages and thermalaverages. In a thermal average, one averages over time-dependent fluctu-ations. According to the ergodic hypothesis, if one waits long enough, asample will access all accessible states. Hence, the thermal average equalsthe time average. In a disorder average, one averages over impurity configu-rations. For a given sample, the impurities do not move. Hence, in order toobtain a disorder average experimentally one has to, somehow, involve manysamples that have the same impurity concentration but different impuritylocations.15

One way of obtaining a disorder average experimentally is “thermal cy-cling”: to repeatedly heat up and cool the same sample. The reason whythis works is that, at sufficiently high temperatures, impurities are somewhatmobile. Hence, after heating up and cooling down the sample once, a differ-ent static impurity configuration is obtained. An example of this procedureis shown in Fig. 11. Here, the conductance of a metal wire is measured as

15Some observables may be “self-averaging”: all samples in the ensemble will behave inthe same way, irrespective of the microscopic details of impurity configuration. In the caseof a self-averaging observable, it is sufficient to measure one sample only.

65

Figure 11: Reproducible magnetoconductance curves measured at T = 45 mK.Different curves are measured for different cool downs of the same GaAs wire.Figure is taken from D. Mailly and M. Sanquer, J. Phys. (France) I 2, 357 (1992).

Figure 12: Average (a) and variance (b) of the magnetoconductance curves ofFig. 11, together with theoretical predictions. The average and variance weretaken over 50 different disorder configurations. The figure is taken from D. Maillyand M. Sanquer, J. Phys. (France) I 2, 357 (1992).

a function of the magnetic field. The different curves correspond to differ-ent cool downs of the same wire. Within one cool down, the conductancecurves are fully reproducible, as one would expect for a static impurity con-figuration. There are small differences between cool downs. Indeed, differentcool downs should correspond to slightly different impurity configurations,for which one expect slightly different Green functions, and, hence, a slightlydifferent conductance. From the experimental data of Fig. 11, one can calcu-late the average conductance for this “ensemble” of wires, the variance, andso on. The average and variance can then be compared to a theory. You cansee plots of the average conductance and of the variance in Fig. 12.

66

5.1 Disorder average

We look at an impurity potential

U(~r) =

Nimp∑

j=1

u(~r − ~rj), (5.1)

where ~rj is the location of the jth impurity. The Fourier transform of theimpurity potential is

U~q =1

V

∫

d~rU(~r)e−i~q·~r

= u~q

Nimp∑

j=1

e−i~q·~rj , (5.2)

where u~q is the Fourier transform of the scattering potential for a singleimpurity. For delta-function impurities with u(~r) = uδ(~r) one has u~q = u/V ,see Eq. (4.30).

In performing the disorder average for a Green function, we have to aver-age products of the disorder potential U~q over the positions of the impurities.The corresponding averages are

〈U~q〉 = u~qNimpδ~q,0,

〈U~q1U~q2〉 = u~q1u~q2[Nimp(Nimp − 1)δ~q1,0δ~q2,0 +Nimpδ~q1+~q2,0],

〈U~q1U~q2U~q3〉 = u~q1u~q2u~q3[Nimp(Nimp − 1)(Nimp − 2)δ~q1,0δ~q2,0δ~q3,0

+Nimp(Nimp − 1)(δ~q1,0δ~q2+~q3,0 + δ~q2,0δ~q1+~q3,0 + δ~q3,0δ~q1+~q2,0)

+Nimpδ~q1+~q2+~q3,0], (5.3)

and so on. We’ll consider the case when the sample is large and there aremany impurities. In that case, we can neglect the difference between Nimp−1or Nimp − 2 and Nimp in the above formulas. Studying the case of manyimpurities does not mean that one can neglect, e.g., the term Nimpδ~q1+~q2,0

in the second line of Eq. (5.3) with respect to the term N2impδ~q1,0δ~q2,0. The

reason is that the former term has one more summation over the momenta,which may bring in extra factors volume, etc.

The disorder average is taken after the thermal average. One can for-mulate a set of diagrammatic rules for the impurity average that should beimplemented on top of the diagrammatic rules for the thermal average. Theserules are

67

+ + +

+

+

+ +

+ + ...

n =kk

ω

Figure 13: Diagrammatic expansion of the disorder averaged single-particle Greenfunction 〈G~k,~k′(iωn)〉.

• Momentum has to be conserved at an impurity vertex (a star in thediagram),

• Every impurity vertex (represented by a star in the diagram) con-tributed a factor Nimp.

• An arbitrary number of dashed lines can meet at a single impurityvertex. Every dashed impurity line carrying a momentum ~q contributesa factor u~q.

5.2 Disorder average of single-particle Green function

We now use the diagrammatic rules derived in the previous section to calcu-late the disorder average of the single-particle Green function,

〈G~k,~k′(τ)〉 = −〈Tτψ~k(τ)ψ†~k′

(0)〉. (5.4)

Our first observation is that the disorder average restores translation in-variance. Hence, 〈G~k,~k′(τ)〉 is nonzero only if ~k = ~k′.

A diagrammatic expansion for 〈G~k,~k′(iωn)〉 is shown in Fig. 13. In thisfigure, all labels for the imaginary frequency and wavevector have been sup-pressed. You can include them in accordance with the diagrammatic rulesof the previous section. The diagrammatic expansion of Fig. 13 looks ratherwild. One can introduce some order in this expansion with the concept of“irreducible diagrams”. A diagram is called “irreducible” if it cannot be cutinto two pieces by cutting a single internal fermion line. For example, the

68

+ + + + ...

n

=

k

ω

Figure 14: Diagrammatic expansion of the self energy Σ~k for the disorder aver-aged single-particle Green function 〈G~k,~k′(iωn)〉. The self energy is denoted by theshadded bubble at the left hand side of the figure.

k

+ + ...

+

+n

n

n

kk=

=

n

k

n

n

k

n

n

k

n

k

n

n

k

n

k

n

k

n

k k kk

k

ω

ω

ω ωω

ωω

ω

ω ω

ω ω

ω

ω

Figure 15: Diagrammatic representation of the Dyson equation for the disorderaveraged Green function 〈G~k,~k′(iωn)〉.

first, second, third, fifth, and eigth diagram in the expansion of Fig. 13 areirreducible. The other diagrams in Fig. 13 are “reducible”. The sum of allirreducible diagrams, without the external fermion lines, is called the “selfenergy” Σ~k. Diagrammatically, the self energy is given by the expansion ofFig. 14. In terms of the self energy, the disorder averaged Green function isseen to obey a Dyson equation, see Fig. 15,

〈G~k,~k(iωn)〉 = G0~k,~k

(iωn) + G0~k,~k

(iωn)Σ~k(iωn)〈G~k,~k(iωn)〉. (5.5)

The solution of the Dyson equation is

〈G~k,~k(iωn)〉 =G0

~k,~k(iωn)

1 − G0~k,~k

(iωn)Σ~k(iωn)=

1

iωn − ε~k + µ− Σ~k(iωn). (5.6)

With this result, we have reduced the calculation of the disorder averagedsingle-particle Green function to that of the self energy Σ~k(iωn).

The reason why the quantity Σ~k is called “self energy” is clear from Eq.(5.6): the self energy provides a shift of the energy ε~k that appears in thedenominator of the single-particle Green function. The shift is caused by theaverage effect of scattering from all impurities.

69

+ + + ...

n

=

k

ω

Figure 16: Diagrammatic expansion of the Born approximatino for the self energyΣ~k for the disorder averaged single-particle Green function 〈G~k,~k′(iωn)〉.

Still, the diagrammatic expansion for the self energy is too complicatedto be solved exactly. In the limit of a low impurity concentration, one canrestrict the diagrammatic expansion to those diagrams that have one im-purity vertex. Indeed, according to the diagrammatic rules of the previoussection, every impurity vertex contributes a factor Nimp, so that diagramswith two or more impurity vertices contribute to higher order in the impu-rity concentration. (See also exercise 5.1).) The approximation in which oneconsiders diagrams with one impurity vertex is referred to as the “Born ap-proximation”. One refers to the “nth order Born approximation” as the Bornapproximation where only self energy diagrams with n + 1 or less impuritylines are kept.

Diagrammatically, the Born approximation corresponds to the expansionshown in Fig. 16. For the self energy this implies

Σ~k(iωn) = Nimp

u0 +∑

~q

u−~qG0~k+~q

(iωn)u~q

+∑

~q1,~q2

u−~q1G0~k+~q1

(iωn)u~q1−~q2G0~k+~q2

(iωn)u~q2 + . . .

. (5.7)

If we compare this with the expression for the T -matrix of a single impurity,Eq. (4.16), we arrive at the important conclusion

Σ~k(iωn) = NimpT~k,~k(iωn). (5.8)

In calculations, we’ll be interested in the analytical continuation of theself energy to real frequencies ω ± iη. After analytical continuation, the real

70

part of the self energy is nothing but a shift of the pole of the single-particleGreen function. It can be absorbed in a redefinition of the chemical potentialor in a redefinition of the energy ε~k, and will be neglected. The imaginarypart of the self energy is more important. It can be expressed in terms of thescattering time τ~k using the optical theorem, Eq. (4.26),

Im Σ~k(ω + iη) = −Im Σ~k(ω − iη)

= −Nimpπ∑

~k′

|T~k′~k(ω)|2δ(ω − ε~k1)

= − 1

2τ~k, (5.9)

where 1/τ~k is the rate for scattering out of the momentum eigenstate |~k〉.The scattering rate is Nimp times the scattering rate due to a single impurityonly, see Eq. (4.22). (It is only in the limit of a low impurity concentrationthat the scattering rate is proportional to the number of impurities.)

The imaginary part of the self energy gives the particle in a momentumeigenstate |~k〉 a finite lifetime, as you would expect for impurity scattering.You can see this explicitly if you use the result (5.6) for the disorder averagedtemperature Green function to calculate the retarded Green function,

〈GR~k,~k

(t)〉 = G0R~k~k

(t)e−t/2τ~k . (5.10)

5.3 Gaussian white noise

In the remainder of this course, we’ll use a simpler model for the impuritypotential than Eq. (5.1) above. In this model, both the positions and thepotentials of the impurities are random. The simplest way to characterizethe impurity potential U(~r) is to say that it is a random function, with aGaussian distribution of zero average,

〈U(~r)〉 = 0, (5.11)

and Gaussian delta-function correlations,

〈U(~r)U(~r′)〉 =1

2πντδ(~r − ~r′), (5.12)

where τ is the mean free scattering time (see Sec. 5.2) and ν is the densityof states at the Fermi level per unit volume,

ν =k2

2π2~vF. (5.13)

71

k−q

n n

=

k

ω ω

Figure 17: Self energy diagram for Gaussian white noise potential.

Higher moments of the impurity potential can be constructed using the rulesfor Gaussian averages. This random potential U(~r) is called “white noise”because it is delta-function correlated in space.

Performing the Fourier transform,

U~q =1

V

∫

d~rU(~r)e−i~q·~r,

we find that the potential U~q is still random and Gaussian. Its first twomoments are given by

〈U~q〉 = 0

〈U~qU~q′〉 =1

2πντVδ~q+~q′,0. (5.14)

For the diagrammatic rules for impurity average this means that only thoseimpurity vertices with two dashed lines contribute to the average.

To leading order in the impurity concentration, we then find only onediagram for the self energy, see Fig. 17. Hence, the self energy is given by

Σ~k =∑

~q

G~k−~q,~k−~q(iωn)1

2πντV

=1

2πντ

∫

d~q

(2π)3

1

iωn − ε~k−~q + µ

= − i

2τsign (ωn). (5.15)

This is the same answer as we obtained previously in Sec. 5.2. In fact, thenormalization (5.12) was chosen precisely to get this answer.

72

5.4 Electrical conductivity from Boltzmann equation

Impurities in a conductor form an obstacle for the flow of electrical current.In the next chapter, we will present a rigorous calculation of the electricalconductivity in the presence of many impurities. Here, we’ll give a simple,more phenomenological calculation of the conductivity that makes use of theBoltzmann equation.

The central object of the Boltzmann equation is a distribution functionf~k for electrons with momentum ~~k. In equilibrium, the distribution functionf~k is the Fermi function f 0,

f 0~k

=1

1 + e(ε~k−µ)/T

. (5.16)

The presence of an electric field causes the flow of electrical current, i.e., itchanges the distribution f .

The conductivity σ is defined as the proportionality constant between thecurrent density ~j and the electric field ~E for small electric fields,

~j = σ ~E. (5.17)

Since we are only interested in the response to small electric fields, we onlyconsider deviations from f 0 to linear order in the electric field,

f~k = f 0~k

+ f 1~k. (5.18)

The Boltzmann equation provides an expression for the time derivativedf/dt as a result of the electric field ~E and impurity scattering,

df~k

dt=

(

∂~k

∂t

)

· ~∇~kf~k = I~k. (5.19)

Here I is the so-called “collision integral”, which describes changes in f~k as a

result of impurity scattering. The derivative ∂~k/∂t is nothing but the forceexerted by the electric field,

~∂~k

∂t= −e ~E. (5.20)

The collision integral I can be expressed as

I~k = −Nimp

∑

~k

W~k~k′δ(ε~k − ε~k′)[

f~k(1 − f~k′) − f~k′(1 − f~k)]

, (5.21)

73

where W~k~k′ = W~k′~k is the transition probability from a state with wavevector~k to a state with wavevector ~k′ as a result of scattering from a single impurity,or vice versa, see Eq. (4.21). The first term in Eq. (5.21) represents scattering

out of the state |~k〉, the second term represents scattering into the state |~k〉.After some algebra, one can rewrite Eq. (5.21) into a simple form,

I~k = −f 1

~k

~τ~k,tr

, (5.22)

In general, this simple form is known as the “relaxation time approximation”,although it is exact for the problem we consider here. The time τ~k,tr is knownas the “transport mean free time”. It is expressed in terms of the transitionprobabilities as

1

τ~k,tr

= Nimp

∑

~k′

W~k~k′(1 − cos θ′)δ(ε~k − ε~k′), (5.23)

where θ′ is the angle between ~k and ~k′.16 For the white noise potential, thetransition rate W~k~k′ does not depend on the angle between ~k and ~k′, so thatone has τtr = τ .

Combining Eq. (5.22) with the Boltzmann equation (5.19), and, again,

linearizing in the electric field ~E, we find

f 1~k

= eτ~k,tr~E · ~∇~kf

0(~k). (5.24)

We can also express ~j in terms of the distribution function f and find

~j = −2e

V

∑

~k

f~k

~~k

m

= −2e

V

∑

~k

(

eτ~k,tr~E · ~∇~kf

0~k

)

~~k

m

= −2e2

V

∑

~k

~v~k

(

~E · ~v~k

) ∂f 0~k

∂ε~kτ~k,tr, (5.25)

16You can find a derivation of Eq. (5.23) in the lecture notes for P636, Advanced SolidState Physics.

74

where we introduced the velocity ~v~k = ~~k/m = ~−1∂ε~k/∂

~k. We then perform

the integration over ~k by first averaging over the angles and find

σ = −2e2

3V

∑

~k

v2~kτ~k,tr

∂f

∂ε~k,

which we can rewrite in terms of the density n of electrons and for lowtemperatures T as

σ =ne2τtr(kF )

m. (5.26)

This is nothing but the well-known Drude formula for the conductivity.

75

k−q

k−q

k−q

−q

’

k−q

’

k−q

k−q

−q

’n nnn nn ω ωωω ωω

Figure 18: Diagrams that were omitted in the calculation of the self energy.

5.5 Exercises

Exercise 5.1: Corrections to the self energy

For the Gaussian white noise potential, we neglected some self-energy dia-grams. The simplest diagrams we left out are shown in Fig. 18. Calculatethese contributions to the self energy and show that they can be omitted ifdisorder if the impurity concentration is small.

Exercise 5.2: Boltzmann Equation

Derive Eq. (5.22).

Exercise 5.3: Self-consistent Born approximation

In the self-consistent Born approximation, one substitutes the unperturbedGreen function G0 with the disorder averaged full Green function G in thecalculation of the self energy. Since G depends on Σ, the result of this sub-stitution is a self-consistent equation for the self energy Σ.

(a) Draw diagrams representing the equation for the disorder averagedGreen function G and the self energy Σ in the self-consistent Bornapproximation.

(b) Write down a self-consistency equation for the self energy in terms ofthe T -matrix. Hint: since the real part of Σ can be absorbed into thechemical potential, it is the imaginary part only that plays a role.

76

(c) Discuss the solution if the T -matrix has only a weak energy dependencefor energies within a distance ≪ εF from the Fermi level.

(d) Discuss the self-consistent Born approximation for the Gaussian whitenoise potential.

77

78

6 Linear Response

6.1 Kubo formula

Quite often, instead of calculating the expectation value of an operator inthermal equilibrium, one wants to see how an observable responds to a per-turbation of the system. For example, an electric field generates a current,or an external charge in the proximity of a metal generates a polarizationcloud.

If one is only interested in the linear response, the change in an observableA to first order in the perturbation, the equilibrium formalism outlined inthe previous sections can still be used.

We consider the time-dependent Hamiltonian

H = H0 +H1, (6.1)

where the unperturbed Hamiltonian H is time-independent and the pertur-bation is H1. We assume that the perturbation is switched on slowly, so thatH1 → 0 for t → −∞. We are still interested in the average of the operatorA, which now depends on time as a result of the perturbation H1. Using theSchrodinger picture, we have

〈A〉 = trAρ(t), (6.2)

where ρ(t) is the now time-dependent density matrix. In the presence of theperturbation H1, ρ(t) is different from the density matrix ρ0 in the absence ofthe perturbation, which is given by the standard expressions for the canonicalor grand canonical ensembles,

ρ0 =

Z−1e−H/T canonical,Z−1e−(H−µN)/T grand canonical,

(6.3)

see Sec. 1.1 The time-dependence of ρ is a consequence of the time-dependenceof the total Hamiltonian H . It is described by the equation

∂

∂tρ(t) = − i

~[H, ρ]−. (6.4)

Note that the sign in Eq. (6.4) differs from that of the Heisenberg pictureevolution equation (1.12) for the operator A. In principle, the density matrix

79

ρ(t) can be solved from Eq. (6.4) using the boundary condition ρ → ρ0 ift→ −∞.

In order to find the linear response to the perturbation H1, we expandthe density matrix in a power series in H1 and truncate the series after firstorder,

ρ(t) = ρ0 + ρ1(t) + . . . (6.5)

Truncating the evolution equation (6.4) for ρ(t) at first order, we find

∂

∂tρ1(t) = − i

~[H0, ρ1(t)]− − i

~[H1, ρ0]−, (6.6)

with the boundary condition ρ1(t) → 0 if t→ −∞. Equation (6.6) is a linearequation, so that it can be solved by standard integration. Its solution is

ρ1(t) = −ie−iH0t/~

[∫ t

−∞

dt′eiH0t′ [H1, ρ0]−e−iH0t′/~

]

eiH0t/~. (6.7)

Substituting this result into Eq. (6.2), we find the average 〈A〉 to first orderin the perturbation H1,

〈A〉 = trAρ0 + trAρ1(t) (6.8)

= 〈A〉0 − itrAe−iH0t/~

[∫ t

−∞

dt′eiH0t′/~[H1, ρ0]−e−iH0t′/~

]

eiH0t/~,

where the brackets 〈. . .〉0 indicate a thermal average with respect to theunperturbed Hamiltonian H0. Switching to the Heisenberg picture,

A(t) = eiH0t/~Ae−iH0t/~, H1(t) = eiH0t/~H1e−iH0t/~, (6.9)

and using the cyclic property of the trace, this can be rewritten as

〈A〉 − 〈A〉0 = −i∫ t

−∞

dt′〈[A(t), H1(t′)]−〉0

=

∫ t

∞

dt′GRA;H1

(t; t′). (6.10)

This result is known as the Kubo formula. Since the right hand side of Eq.(6.10) is a retarded Green function, defined with respect to a thermal averagewith the equilibrium Hamiltonian H0, Eq. (6.10) can be calculated with themethods described in the previous sections.

80

Quite often, we are interested in the response to a perturbation that isknown in the frequency domain,

H1 =1

2π

∫

dωe−i(ω+iη)tH1(ω). (6.11)

Here η is a positive infinitesimal and the factor exp(ηt) has been added toensure that H1 → 0 if t→ −∞. Then Eq. (6.10) gives

〈A(ω)〉 − 〈A(ω)〉0 =

∫

dteiωt(〈A〉 − 〈A〉0)

= GRA;H1

(ω + iη). (6.12)

Although the Kubo formula involves an integration over the real time t′,for actual calculations, we can still use the imaginary time formalism andcalculate the temperature Green function GA;H1(τ). The response 〈A(ω)〉 isthen found by Fourier transform of G to the Matsubara frequency domain,followed by analytical continuation iωn → ω.

6.2 Kubo formula for dielectric function

Consider a metal or other dielectric with an added charge. One example ofsuch a charge is an ion with a different valence in a metal. We’ll call thischarge the “external charge” and denote the corresponding charge densityby ρext.

The external charge determines the “electrical displacement” ~D, which isthe solution of the Maxwell equation

ǫ0~∇ · ~D = 4πρext. (6.13)

Here, ǫ0 is the permittivity of vacuum. In a dielectric, the external chargewill induce a screening charge ρind. The total charge, screening charge plusexternal charge, determines the electric field ~E,

ǫ0 ~∇ · ~E = 4π(ρext + ρind). (6.14)

In the language of “external” and “induced” quantities, you may consider~E as the “total electric field”, and ~D as the “external electric field”. Thedifference ~E − ~D then corresponds to the “induced electric field”.

81

In general, the electric field, the electric displacement, and their difference(or, more precisely, the longitudinal parts of these fields) can be writtenas gradients of electric potentials φtot, φext, and φind, respectively. Thesepotentials are expressed in terms of the corresponding charges as

φ(~r) =

∫

d~r′VC(~r − ~r′)ρ(~r′), (6.15)

where VC(~r − ~r′) = 1/ǫ0|~r − ~r′| is the Coulomb interaction.For a sufficiently small external charge, the induced and total potentials

are proportional to the external potential. This is the regime of “linearscreening”. The proportionality constant, which may be a non-local functionin space and time, is called the inverse “relative permittivity” of the dielectric,or the dielectric response function,

φtot(~r, t) =

∫

d~r′∫

dt′ǫ−1(~r, t;~r′, t′)φext(~r′, t′). (6.16)

Since ǫ is defined as the linear response to the external charge, we can usethe Kubo formula to calculate ǫ. We make use of the fact that the externalcharge causes a change of the Hamiltonian given by

H1 =

∫

d~rρe(~r)φext(~r, t), (6.17)

where ρe is the electronic charge density operator,

ρe = −e∑

σ

ψ†σ(~r)ψσ(~r). (6.18)

Then, according to the Kubo formula, we find that the induced charge densityis

ρind(~r, t) =

∫

d~r′∫

dt′GRρe,ρe

(~r, t;~r′, t′)φext(~r′, t′). (6.19)

As in any use of the Kubo formula, we assume that the external potentialφext was switched on at some time before t. The quantity

χRe (~r, t;~r′, t′) = GR

ρe,ρe(~r, t;~r′, t′) (6.20)

is known as the “polarizability function”.

82

From Eq. (6.19) we then conclude that

ǫ−1(~r, t;~r′, t′) = δ(~r − ~r′) δ(t− t′)

+

∫

d~r′′VC(~r − ~r′′)χR(~r′′, t;~r′, t′). (6.21)

In a translationally invariant medium, the dielectric response functionǫ−1(~r, t;~r′, t′) depends on the differences ~r − ~r′ and t − t′ only. Then it isadvantageous to perform a Fourier transform to position and time. Defining

ǫ−1(~q, ω) =1

V

∫

d~rd~r′∫

dte−i~q·(~r−~r′)+iωtǫ−1(~r, t;~r′, 0), (6.22)

φ(~q, ω) =

∫

d~r

∫

dteiωt−i~q·~rφ(~r, t), (6.23)

we findφtot(~q, ω) = ǫ−1(~q, ω)φext(~q, ω). (6.24)

Fourier transforming Eq. (6.21), we thus find

ǫ−1(~q, ω) = 1 + VC(~q)χRe (~q, ω), (6.25)

where VC(~q) = 4π/ǫ0q2 is the Fourier transform of the Coulomb potential.

As an application, we now calculate the dielectric response function for anoninteracting electron gas

χRe (~q; t) =

1

V

∫

d~rd~r′χRe (~r, t;~r′, 0)e−i~q·(~r−~r′)

= −iθ(t) 1

V〈[ρe,~q(t), ρe,−~q(0)]−〉, (6.26)

where the Fourier transform of the electron charge density operator followsfrom Eq. (6.18),

ρe(~q) = −e∑

σ

∫

d~re−i~q·~rψ†σ(~r)ψσ(~r)

= −e∑

~k,σ

ψ†~k,σψ~k+~q,σ. (6.27)

83

Inserting the proper time dependence of the operators ψ~k+~q,σ and ψ†~k,σ

, we

find

χRe (~q; t) = −ie2θ(t) 1

V

∑

~kσ

∑

~k′σ′

〈[ψ†~k,σψ~k+~q,σ, ψ

†~k′,σ′

ψ~k′−~q,σ′ ]−〉

× ei(ε~k−ε~k+~q

)t/~. (6.28)

The thermal average of the commutator is easily found to give [nF (ε~k −µ) − nF (ε~k+~q − µ)]δ~k,~k′−~qδσσ′ , where nF (ε) = 1/(1 + exp(ε/T )) is the Fermidistribution function. Finally, performing a Fourier transform to time, wefind the result

χRe (~q, ω) =

2e2

V

∑

~k

nF (ε~k − µ) − nF (ε~k+~q − µ)

ε~k − ε~k+~q + ω + iη, (6.29)

where η is a positive infinitesimal. Note that χRe satisfies the general integral

relation (2.42) that expresses the full retarded correlation function in termsof its imaginary part only.

An alternative method to calculate χRe is to use the imaginary time for-

malism. One first calculates

χe(~q, τ) = −e2

V

∑

~kσ

∑

~k′σ′

Tτ 〈ψ†~k,σ

(τ + η)ψ~k+~q,σ(τ)ψ†~k′,σ′

(η)ψ~k′−~q,σ′(0)〉

=e2

V

∑

~kσ

∑

~k′σ′

[

G~k+~q,σ;~k′,σ′(τ)G~k′−~q,σ′;~k,σ(−τ)

− G~k+~q,σ;~k,σ(−η)G~k′−~q′,σ′;~k′,σ′(−η)]

. (6.30)

The positive infinitesimal η has been added to ensure the placement of cre-ation operators in front of the annihilation operators at the same imaginarytime in the time-ordered product. The second term does not depend on τ .After Fourier transform with respect to τ it will give a contribution pro-portional to δΩn,0, which can be discarded for the analytical continuationiΩn → ω+ iη we need to do at the end of the calculation. Fourier transformof the first term and substitution of the exact results for the electron Greenfunctions gives

χe(~q, iΩn) =2Te2

V

∑

~k,m

1

[iωm − (ε~k − µ)][i(ωm + Ωn) − (ε~k+~q − µ)].

(6.31)

84

k iε −µ− Ωk+q nε −µ

C1 C1

C2

C2

Figure 19: Integration contours for the calculation of the polarizability of thenoninteracting electron gas.

Here Ωn is a bosonic Matsubara frequency (recall that ρ~q is quadratic infermion creation and annihilation operators) and ωm is a fermionic Matsubarafrequency. In order to perform the summation over ωm, the summation iswritten as an integral over a suitably chosen contour in the complex plane,see Fig. 19. The integrand is a function of the complex variable z and haspoles at the fermionic Matsubara frequencies z = iωm, at z = ε~k − µ, and atz = ε~k+~q − µ− iΩn, hence

χe(~q, iΩn) =e2

2πiV

∑

~k

∫

C1

dztanh(z/2T )

[z − (ε~k − µ)][z + iΩn − (ε~k+~q − µ)]

=e2

V

∑

~k

tanh[(ε~k+~q − µ)/2T ] − tanh[(ε~k − µ)/2T ]

ε~k − ε~k+~q + iΩn

. (6.32)

Now the analytical continuation Ωn → ω+ iη is easily done and one recoversthe result (6.29) upon substituting tanh(ε/2T ) = 1 − 2nF (ε).

Let us now consider the dielectric response at finite frequency in moredetail. The polarizability describes how the induced charge varies with thetime-dependent external charge. A time-dependent induced charge requiresa time-dependent current density ~j, which, by the continuity relation, isproportional to the time derivative of the induced charge. The real part of thepolarizability describes that part of the induced current that is out of phasewith respect to the external electric field. For such currents, the time-averageof ~j · ~E is zero, so that no energy is dissipated in one cycle. Hence, Reχe

describes the dissipationless response to an external potential. Similarly, theimaginary part of the polarizability describes dissipation: current and field

85

q/kF

ω/εF

30 4210

2

4

Figure 20: Support of the imaginary part of the polarizability χRe (~q, ω) for a

noninteracting electron gas.

are in phase, so that the time average of ~j · ~E is nonzero.For the noninteracting electron gas, the dissipative response is given by

the imaginary part of Eq. (6.29) above,

ImχRe (~q, ω) = −πe

2

V

∑

~kσ

[nF (ε~k − µ) − nF (ε~k+~q − µ)]

× δ(ε~k − ε~k+~q + ~ω). (6.33)

Hence, electromagnetic energy can be dissipated by the excitation of particle-hole pairs of energy ~ω and momentum ~q. Let us investigate Eq. (6.33) inmore detail for zero temperature and ω > 0.17 Then the difference nF (ε~k −µ) − nF (ε~k+~q − µ) is one if ~k is inside the Fermi sphere and ~k + ~q is outsidethe Fermi sphere, and zero otherwise. Taking a parabolic dispersion relation,ε~k = ~

2k2/2m, the delta function then implies ω = q2/2m~ + ~k · ~q/m~. In

general, one has |~k · ~q| ≤ mvq/~, where v = ~k/m is the velocity for an

electron with wavenumber ~k. Since ~k is inside the Fermi sphere, one hasv ≤ vF , so that the momentum and frequency range where ImχR

e is nonzerois given by

q2

2m~− vF q < ω <

q2

2m~+ vF q. (6.34)

The support of χRe is shown in Fig. 20.

17The case ω < 0 follows from the relation χRe (~q, ω) = χR

e (−~q,−ω).

86

6.3 Kubo formula for electrical conductivity

The most common transport measurement for a conducting material is thatof the conductivity, the current that flows in response to an external electricfield.

For a normal metal, this response takes the form

jeα(~r, t) =

∫

dt′∫

d~r′∑

β

σαβ(~r, t;~r′, t′)Eβ(~r′, t′). (6.35)

Here σ is the conductivity. Note that the conductivity is a tensor and thatit describes a response to the electric field that is non-local in time and inspace.

The electric field is related to the electric potential φ and the vectorpotential ~A,

~E = −~∇φ− ∂

∂t~A. (6.36)

This relation holds both for the external electric field and for the total electricfield. If the current is carried by electrons, the electrical current operator is asum of a “paramagnetic” contribution ~jp and a “diagmagnetic” contribution~jd,

~je(~r) = ~jpe (~r) +~jd

e (~r), (6.37)

where

~jpe (~r) = − ~e

2mi

∑

σ

[

ψ†σ(~r)(∂~rψσ(~r)) − (∂~rψ

†σ(~r))ψσ(~r)

]

, (6.38)

~jde (~r) =

e

m~A(~r)ρe(~r). (6.39)

Here ρe is the electronic charge density, see Eq. (6.18) above. To linear orderin the electric field, the electron’s Hamiltonian is H0 + H1, where H0 is theHamiltonian in the absence of the electric field and

H1 =

∫

d~rρe(~r)φext(~r, t) −∫

d~r~je(~r)Aext(~r, t). (6.40)

Now we’ll apply the general Kubo formula to find an expression for theelectrical current density ~j in the presence of the electric field. We’ll choosea gauge in which φext = 0. This would be the natural choice if, e.g., the

87

Φ(t)

E(t)

Figure 21: The gauge with zero scalar potential φ and a time-dependent vector po-tential ~A is the natural choice if the electric field is generated by a time-dependentflux through the sample.

sample is a cylinder and the electric field is generated by a time-dependentflux through the sample, see Fig. 21.

Fourier transforming Eq. (6.35), we find

jeα(~r, ω) =

∫

d~r′∑

β

σαβ(~r, ~r′;ω)Eβ(~r′, ω), α = 1, 2, 3, (6.41)

where, in frequency representation, the electric field is related to the vectorpotential as

~Aext(~r, ω) =1

iω~Eext(~r, ω) =

∫

dteiωtAext(~r, t) (6.42)

and

σαβ(~r, ~r′;ω) =

∫

dteiωtσαβ(~r, t;~r′, 0). (6.43)

Since we are interested in linear response, we only need to know H1 tolinear order in ~A, so that we can replace the electrical current density ~je inEq. (6.40) by the paramagnetic current density ~jp

e .18 Then the Kubo formula

18If the Hamiltonian H0 contains a vector potential ~A0 due to a magnetic field, weshould replace ~je by ~jp

e + (e/m) ~A0ρe.

88

gives

σαβ(~r, ~r′;ω) =i

ωΠR

αβ(~r, ~r′;ω) +eρe(~r)

iωmδ(~r − ~r′)δαβ. (6.44)

Here the function ΠRαβ is the retarded current-current correlation function,

ΠRαβ(~r, ~r′; t) = GR

jeα,jeβ(~r, ~r′; t) = −iθ(t)〈[jeα(~r, t); jeβ(~r′, 0)]−.〉 (6.45)

For a translationally invariant system, the conductivity depends on thedifference ~r−~r′ only. Performing Fourier transforms for the conductivity andfor the current density,

σαβ(~q, ω) =1

V

∫

d~rd~r′e−~q·(~r−~r′)σαβ(~r, ~r′;ω), (6.46)

~je,~q =

∫

d~re−i~q·~r~je(~r), (6.47)

we can write the Kubo formula for the conductivity as

σαβ(~q, ω) =i

ωΠR

αβ(~q, ω) +eρe

iωmδαβ . (6.48)

Here ρe is the electron charge density averaged over the entire sample (i.e.,the ~q = 0 component of the charge density divided by the volume), and

ΠRαβ(~q, t) = −iθ(t) 1

V〈[jeα,~q(t), jeβ,−~q(0)]−〉. (6.49)

In general, the conductivity is defined as the response to the actual elec-tric field in the sample, not as the response to the external field. If the electricfields are time dependent, the currents are time dependent and, thus, gener-ate their own magnetic and electric fields. In addition, as we discussed in theprevious section, an applied electric field may lead to a build-up of charge,which can screen the field. What we calculated above is the response to theexternal electric field. This procedure gives the correct result only if we canneglect the induced electric field. This is the case if, on the one hand, thefrequency of the signal is sufficiently low that no magnetic fields are induced,whereas, on the other hand, the frequency is large enough compared to thesize of the sample or the wavelength of the electric field that no screeningcharges are built up. In practice, both conditions are met if the electric field

89

is perpendicular to the wavevector ~q or if the wavevector ~q is taken to zerobefore the frequency ω is made small.

Let us now calculate the conductivity of a noninteracting electron gas.If there are no impurities, the conductivity is infinite (see exercise 6.3). So,in order to do a meaningful calculation, we consider the conductivity of anelectron gas with impurities.

The conductivity follows from the current-current correlation function(6.49). For our calculation, we consider the imaginary-time version of thatcorrelation function,

Πα,β(~q, τ − τ ′) = − 1

V〈Tτj~q,α(τ)j−~q,α(τ ′)〉, (6.50)

and perform an analytical continuation to real frequencies at the end of thecalculation. The current density operator ~j~q is expressed in terms of thecreation and annihilation operators for electrons in momentum eigenstates|~k〉 as

~j~q = − e

2m

∑

~kσ

(2~k + ~q)c†~k,σc~k+~q,σ

. (6.51)

Since we are dealing with a non-interacting electron gas, we can perform thethermal average in Eq. (6.50) using Wick’s theorem, with the result

Πα;β(τ, τ ′) =e2

4m2V

∑

~kσ

∑

~k′σ′

G~k+~q,σ;~k′,σ′(τ′ − τ)G~k′−~q,σ′;~k,σ(τ − τ ′)

× (2kα + qα)(2k′β − qβ), (6.52)

plus a term that does not depend on τ − τ ′ and can be discarded. It is im-portant to note that the Green function in Eq. (6.52) is the full temperatureGreen function, calculated in the presence of the exact impurity potential.Fourier transforming Eq. (6.52) and changing variables ~k′ → ~k′ + ~q, we find

Πα;β(iΩn) =e2T

2m2V

∑

~k,~k′,m

G~k+~q,~k′+~q(iωm + iΩn)G~k′;~k(iωm)

× (2kα + qα)(2k′β + qβ), (6.53)

where we summed over the spin index.In order to calculate the full temperature Green function, we use the

perturbative expansion of G in powers of the impurity potential, which is

90

++ + ...

m

m n

m

m n

m

m n

m

m n

m

m n

k k’k

k+q

k’

k’+q

k

k+q

k"

k"+qk+q k’+q

ω

ω +Ω

ω

ω +Ω

ω

ω +Ω

ω

ω +Ω

ω

ω +Ω

Figure 22: Diagrammatic representation of all diagrams without crossed lines thatcontribute to the conductivity. The double lines represent the disorder averagedGreen function.

shown in Fig. 7. Here, our aim is to calculate a disorder average. It isimportant to realize that, here, we calculate the impurity average of theproduct of two Green functions. That is not the same as the product of theaverages!

In order to keep the calculations simple, we’ll restrict our attention to thecase of a Gaussian white noise random potential. Then, diagrammatically,the impurity average means that we connect all pairs of impurity lines, asin Fig. 17. Still, it is an impossible task to sum all diagrams. We canfurther simplify to limit ourselves to those diagrams in which no impuritylines cross. This is a sensible approximation, since you have seen in exercise5.1 that diagrams in which impurity lines cross are a factor kF l smaller thandiagrams without crossed impurity lines.

The diagrams without crossed lines can be summed up as in Fig. 22.The building blocks are the impurity averaged Green functions, which arerepresented by double arrows. The open circles at the two ends are called“current” vertices. They represent the factors (2~k+ ~q)/2m and (2~k′ + ~q)/2mthat appear in the calculation of the current-current correlation function.These factors are referred to as the “bare current vertex”, and they aredenoted by the “bare vertex function” Γ0

α(~k, iωm;~k + ~q, iωm + iΩn),

Γ0α(~k, iωm;~k + ~q, iωm + iΩn) = −e(2kα + qα)

2m. (6.54)

The middle part of the diagram is a “ladder” which has the impurity averagedGreen functions and impurity lines as building blocks.

All but the rightmost part of the ladder diagram can be written as a soliddot, see Fig. 23. This part of the diagram is known as a vertex correction.It is described by the vertex function ~Γ(~k, iωm;~k + ~q, iωm + iΩn). A self-consistent equation for the vertex function can be found from the second lineof Fig. 23,

Γα(~k, iωm;~k + ~q, iωm + iΩn)

91

= +

+= + + ...

m n

m

m n

mm

m n

k

k+q

k’

k’+q

k

k+q

m n

mm

m n

k’

k’+q

k

k+q

m

m n

k

k+q

k"

k"+q

k

k+q

k

k+q

k’

k’+qω +Ω

ω

ω +Ω

ω ω

ω +Ω

ω +Ω

ω ω

ω +Ω

ω

ω +Ω

Figure 23: Diagrammatic representation of vertex correction Γ necessary for thecalculation of the current-current correlator. The double lines represent the disor-der averaged Green function.

= Γ0α(~k, iωm;~k + ~q, iωm + iΩn)

+1

2πντV

∑

~k′

〈G~k′,~k′(iωm)〉〈G~k′+~q,~k′+~q(iωm + iΩn)〉

× Γ(~k′, iωm;~k′ + ~q, iωm + iΩn), (6.55)

where Γ0 is the bare vertex function, see Eq. (6.54) above. In order tocalculate the vertex function Γ, note that the left hand side of Eq. (6.55)contains the quantity

∑

~k′

〈G~k′,~k′(iωm)〉〈G~k′+~q,~k′+~q(iωm + iΩn)〉Γ(~k′, iωm;~k′ + ~q, iωm + iΩn)

only. Hence, a self-consistent equation for the vertex function is found uponmultiplication of Eq. (6.55) with 〈G~k,~k(iωm)〉〈G~k+~q,~k+~q(iωm + iΩn)〉 and sum-

mation over ~k,

−e∑

~k

2kα + qα2m

〈G~k,~k(iωm)〉〈G~k+~q,~k+~q(iωm + iΩn)〉

=

∑

~k′

〈G~k′,~k′(iωm)〉〈G~k′+~q,~k′+~q(iωm + iΩn)〉Γ(~k′, iωm;~k′ + ~q, iωm + iΩn)

×

1 − 1

2πντV

∑

~k

〈G~k,~k(iωm)〉〈G~k+~q,~k+~q(iωm + iΩn)〉

, (6.56)

92

m

m n

k

k+q

ω

ω +Ω

Figure 24: Diagrammatic representation of the current-current correlator in termsof the vertex function Γ (solid dot), the bare vertex function Γ0 (open dot) andthe disorder averaged Green function (double lines).

from which we conclude

Γα(~k, iωm;~k + ~q, iωm + iΩn)

= −e(2kα + qα)

2m

−( e

2m

)1V

∑

~k′(2k′α + qα)〈G~k′,~k′(iωm)〉〈G~k′+~q,~k′+~q(iωm + iΩn)〉2πντ − 1

V

∑

~k′〈G~k′,~k′(iωm)〉〈G~k′+~q,~k′+~q(iωm + iΩn)〉 .

(6.57)

We are interested in the current-current correlator in the limit ~q → 0. Thenumerator of the last term of Eq. (6.57) goes to zero if ~q → 0. Hence, weconclude that, for our choice of the random potential, there is no vertexcorrection.19 (That is one of the reasons why we have chosen this model forthe random potential.)

Now the current-current correlation function is easily expressed in termsof the vertex function Γ and the bare vertex function Γ0, see Fig. 24,

Πα,β(iΩn) =2T

V

∑

~k,m

Γ(~k, iωm;~k + ~q, iωm + iΩn)Γ0(~k + ~q, iωm + iΩn)

× G~k,~k(iωm)G~k+~q,~k+~q(iωm + iΩn). (6.58)

It remains to perform the summations over ~k and m and do the analyticalcontinuation iΩn → ω + iη. We first do the summation over m. As before,

19For a different choice of the impurity potential Γ and Γ0 are different. This “vertexcorrection” amounts to the replacement of the mean free time τ with the “transport meanfree time” τtr, see Sec. 5.4.

93

C1 C1C

C

2

2C

C

2

2

Figure 25: Integration contours for the calculation of the current-current correla-tor.

this is done by representing the Matsubara summation as an integral in thecomplex plane,

Πα,β(iΩn) =1

2πiV

∫

C1

dz∑

~k

Γ(~k, z;~k + ~q, z + iΩn)Γ0(~k + ~q, z + iΩn)

× G~k,~k(z)G~k+~q,~k+~q(z + iΩn) tanh(z/2T ). (6.59)

The integrand has singularities at the real axis and at the axis z = x− iΩn,x real. Hence, we deform the contour as shown in Fig. 25.

Πα,β(iΩn) =1

2πiV

∫

dξ tanh(ξ/2T )∑

~k

∑

±

(±1)

×[

Γ(~k, ξ±;~k + ~q, ξ + iΩn)Γ0(~k + ~q, ξ + iΩn, ~k, ξ±)

× G~k,~k(ξ±)G~k+~q,~k+~q(ξ + iΩn)

+ Γ(~k, ξ − iΩn;~k + ~q, ξ±)Γ0(~k + ~q, ξ±, ~k, ξ − iΩn)

× G~k,~k(ξ − iΩn)G~k+~q,~k+~q(ξ±)]

, (6.60)

where we used tanh(z/2T ) = tanh[(z + iΩn)/2T ] and ξ± = ξ ± iη. It is notnecessary to add an infinitesimal ±iη to ξ±iΩn, since iΩn is imaginary itself.At this stage of the calculation we can perform the analytical continuation

94

iΩn → ω + iη,

Πα,β(ω + iη) =1

2πiV

∫

dξ tanh(ξ/2T )∑

~k

×[

ΓRR(~k, ξ;~k + ~q, ξ + ω)Γ0RR(~k + ~q, ξ + ω;~k, ξ)GR~k,~k

(ξ)GR~k+~q,~k+~q

(ξ + ω)

− ΓAR(~k, ξ;~k + ~q, ξ + ω)Γ0RA(~k + ~q, ξ + ω;~k, ξ)GA~k,~k

(ξ)GR~k+~q,~k+~q

(ξ + ω)

+ ΓAR(~k, ξ − ω;~k + ~q, ξ)Γ0RA(~k + ~q, ξ;~k, ξ − ω)GA~k,~k

(ξ − ω)GR~k+~q,~k+~q

(ξ)

− ΓAA(~k, ξ − ω;~k + ~q, ξ)Γ0AA(~k + ~q, ξ;~k, ξ − ω)GA~k,~k

(ξ − ω)GA~k+~q,~k+~q

(ξ)]

.

(6.61)

Sofar our result has been quite general. Equation (6.61) is valid for anyform of the vertex correction and even holds in the presence of electron-electron interactions (which will be discussed in a later chapter). We nowspecialize to the limit ~q → 0, which is needed to find the dc conductivity.As explained above, we should take the limit ~q → 0 before we take the limitω → 0. For zero wavevector ~q there is no vertex renormalization, and hence

Πα,β(ω + iη) =e2

2πiV

∑

~k

kαkβ

m2

∫

dξ tanh[(ξ − µ)/2T ]

×[

1

(ξ − ε~k + i/2τ)(ξ + ω − ε~k + i/2τ)

− 1

(ξ − ε~k − i/2τ)(ξ + ω − ε~k + i/2τ)

+1

(ξ − ω − ε~k − i/2τ)(ξ − ε~k + i/2τ)

− 1

(ξ − ω − ε~k − i/2τ)(ξ − ε~k − i/2τ)

]

, (6.62)

where we shifted the ξ-integration by the chemical potential µ. We firstperform a partial integration to ξ,

Πα,β(ω + iη) =e2

4πiV T

∫

dξ1

cosh2[(ξ − µ)/2T ]

×∑

~k

kαkβ

m2

(

1

ω− 1

ω + i/τ

)

ln(ξ − ε~k)

2 − (ω + i/2τ)2

(ξ − ε~k)2 + 1/4τ 2

.

95

With this manipulation the summation over the wavevector ~k is convergent.The summation over ~k is separated into an angular average over the directionof ~k and an integration over the magnitude. The latter integration can berewritten as an integral over the energy ε~k and yields

Πα,β(ω + iη) = − 2e2νk2F δα,β

3(1 − iωτ)m2. (6.63)

We use the relation ν = mkF/2π2 for the density of states per spin direction

and per unit volume, and the relation n = k3F/3π

2 = (−ρe/e) for the totalparticle density to rewrite this as20

ΠRα,β(ω) =

ρeeδα,β

m(1 − iωτ). (6.65)

20You can verify this result in the limit ω → 0 without expanding around the Fermisurface. Starting point is Eq. (6.62), which we rewrite as

ΠRα,β(0) =

e2

πV

∑

~k

kαkβ

m2

∫

dξ tanh[(ξ − µ)/2T ]Im 〈GR~k,~k

(ξ)〉2.

We make use of the fact that the integral of 〈GR(ξ)〉2 vanishes and perform the angular

average over the wavevector ~k,

ΠRα,β(0) = − 4e2

3πV m

∑

~k

k2

2m

∫

dξ1

1 + e(ξ−µ)/TIm 〈GR

~k,~k(ξ)〉2δαβ .

Using the explicit form of Im 〈GR~k,~k

(ξ)〉 one verifies that

Im 〈GR~k,~k

(ξ)〉2 = −1

2

∂

∂ε~k〈A~k,~k(ξ)〉. (6.64)

Performing a partial integration with respect to ε~k one thus finds

ΠRα,β(0) =

2e2

3πVm

∑

~k

ε~k

∫

dξ1

1 + e(ξ−µ)/T

∂

∂ε~k〈A~k,~k(ξ)〉δαβ

= − 2e2

V m

∑

~k

∫

dξ

2π

1

1 + e(ξ−µ)/T〈A~k,~k(ξ)〉δαβ

=ρee

mδαβ .

Note that the partial integration gives a factor 3/2 since the density of states is proportional

to ε1/2~k

. This is the desired result.

96

Substituting this in Eq. (6.48), we find that the conductivity at wavevector~q = 0 and frequency ω is given by

σαβ(0, ω) =e(−ρe)τ

m(1 − iωτ). (6.66)

In this limit ω → 0 this reproduces the well-known Drude formula.It may be somewhat disappointing to recover the Drude formula after so

much work. However, we could not have expected to find anything else! Infact, this whole calculation was not for nothing. First of all, we have founda firm microscopic derivation of the Drude formula. But more importantly,with this formalism we are ready to study corrections to the Drude formula.One of these corrections, the weak localization correction, will be studied atlength in a later chapter.

97

6.4 Exercises

Exercise 6.1: Tunneling density of states

In this exercise we consider two pieces of metal that are weakly coupled by atunnel barrier, see Fig. 26. The two pieces of metal are labeled A and B. Inthe tunnelling Hamiltonian formalism, the total Hamiltonian H is written as

H = HA +HB +HAB, (6.67)

where HA and HB operate on states of each system separately, with creationand annihilation operators c†Aµ, cAµ, and c†Bν , cBν for the single-electron statesin A and B, and HAB is a tunneling Hamiltonian HAB that describes thecoupling between the two systems,

HAB =∑

νµ

(Tµνc†AµcBν + T ∗

νµc†BνcAµ), (6.68)

where, in principle, the indices µ and ν refer to any choice of an orthogo-nal basis for the single-electron states in the two systems A and B. Thetunnel matrix elements Tµν are complex number numbers that depend onthe extension of the single-electron wavefunctions in A and B into the in-sulating region between them. In general, the tunnel matrix elements areexponentially small in the separation between A and B.

The current through the tunnel barrier follows from the rate of change ofthe charge QA in metal A (or metal B) as

I =∂QA

∂t= i[H,QA]−, QA = −e

∑

µ

c†A,µcA,µ. (6.69)

Substituting Eqs. (6.67) and (6.68) for H , one finds that only the tunnelingHamiltonian contributes to the commutator, and that

I = ie∑

µν

(Tµνc†AµcBν − T ∗

νµc†BνcAµ). (6.70)

The current through the tunnel barrier is linear in the tunneling matrixelements Tνµ. In order to calculate I to lowest (second) order in the tunnelingmatrix elements, it is sufficient to calculate the (change of the) current to

98

V

A B

I

Figure 26: Setup for tunnel spectroscopy. Two metals, labeled A and B, areseparated by an insulating material, e.g., an oxide or simply vacuum. A batterymaintains a finite difference between the chemical potentials of A and B.

linear response in the tunneling matrix elements. The Kubo formula thengives

I(t) =

∫ ∞

−∞

dt′CRI,HAB

(t, t′), (6.71)

where the correlation function CRI,HAB

is defined as

CRI,HAB

(t, t′) = −iθ(t− t′)〈[I(t), HAB(t′)]−〉. (6.72)

Here, the thermal equilibrium average 〈. . .〉 and the time-dependence of thetunneling Hamiltonian HAB are taken with respect to the Hamiltonian HA +HB only, i.e., without inclusion of the tunneling Hamiltonian HAB.

In the average (6.72), the two systems are completely separated. Hence,in thermal equilibrium, their chemical potentials µA and µB need not beequal. In fact, it is only if µB − µA = eV 6= 0 that a nonzero current flowsbetween the reservoirs. Even in the presence of the tunneling current, a quasiequilibrium in which µB − µA = eV 6= 0 can be maintained with the help ofa battery, as long as the contacts between the battery and the metals A andB have a much smaller resistance than the tunnel barrier between A and B.

(a) Calculate the correlator CI,HABand show that the tunneling current

99

can be written

I = −e∫

dω

2π

∑

µν

|Tµν |2AA,µ(ω)AB,ν(ω + eV ) (nF (ω + eV ) − nF (ω)) ,

(6.73)where nF (ω) = 1/(1 + exp((ω − µ)/T )) is the Fermi distribution func-tion. Be sure that your derivation makes no specific assumptions aboutthe Hamiltonians HA and HB! Equation (6.73) is valid both with andwithout electron-electron interactions.

(b) If metal B has a spectral density that is more or less constant,∑

ν

|Tµν |2AB,ν(ω) ≈ const., (6.74)

show that the tunneling current can be used as a direct measurement ofthe spectral density of metal A. This method of measuring the spectraldensity is known as “tunnel spectroscopy”.

Exercise 6.2: Polarizability function

In this exercise, we consider the polarizability function for the non-interactingelectron gas at zero temperature

χ0e(~q, ω) = −2e2

V

∑

~k

θ(µ− ε~k+~q) − θ(µ− ε~k)

ω − ε~k+~q + ε~k + iη(6.75)

=2e2

V

∑

~k<kF

[

1

ω − ε~k+~q + ε~k + iη− 1

ω − ε~k−~q + ε~k + iη

]

.

(a) Perform the summation over ~k to show that the real part of χe is

Reχe(~q, ω) = −2νe2 (f(x, x0) + f(x,−x0)) , (6.76)

where ν is the density of states at the Fermi level per spin directionand per unit volume, x = q/2kF , x0 = ω/4εF , and

f(x, x0) =1

4+

(1 − (x0/x− x)2)

8xln

∣

∣

∣

∣

x+ x2 − x0

x− x2 + x0

∣

∣

∣

∣

.

100

(b) The expressions for Imχe are somewhat more complicated. Show thatif q < 2kF and ω ≥ 0 one finds

Imχe(~q, ω) = −πνe2

8x

4x0/x if 0 < x0 < x− x2,(1 − (x0/x− x)2) if x− x2 < x0 < x+ x2,0 if x0 > x+ x2.

(6.77)

The dimensionless quantities x and x0 were defined above. On theother hand, if q > 2kF , one finds

Imχe(~q, ω) = −πνe2

8x

(1 − (x0/x− x)2) if x2 − x < x0 < x2 + x,0 if 0 < x0 < x2 − x

or if x0 > x+ x2.

(6.78)

Exercise 6.3: Conductivity without impurities

Calculate the retarded current density correlation function ΠRα;β(ω) for a non-

interacting electron gas without impurities. Is the result the same as the limitτ → ∞ of Eq. (6.65)? What does your result imply for the relation betweencurrent density and electric field?

Exercise 6.4: Kubo formula for conductance

The response of a bulk sample to an applied electric field is characterizedby the conductivity. However, for a sample of finite size, one measures theconductance, the coefficient of proportionality between the total current Iflowing through a cross section of the sample and the total voltage drop Vover the sample.

In order to derive a relation between the total current I and the voltagedrop V , we introduce a coordinate system (ξ, ξ⊥), such that ξ is parallel withthe electric field lines and ξ⊥ is parallel with the equipotential lines, see Fig.27. The current can then be written as an integral of the current density,

I(ξ) =

∫

dξ⊥ξ ·~j(ξ, ξ⊥), (6.79)

101

ξ

ξ

electric field linesequipotential lines

I

V

Figure 27: Schematic of a conductance measurement. In the derivation of theKubo formula for the conductance one uses a coordinate system where the coor-dinate ξ points along the electric field and the coordinate ξ⊥ points along equipo-tential lines.

where ξ is the unit vector in the ξ-direction. We are interested in the dcconductance only. Then, current conservation implies that I(ξ) does notdepend on the choice of the cross section, i.e., I(ξ) does not depend on ξ.

(a) Use the Kubo formula for the dc conductivity to express I(ξ) in terms

of an integral over the electric field ~E(ξ′, ξ′⊥).

(b) Use current conservation to show that the conductance G can be writ-ten as

G = limω→0

Rei

ωCR

II(ω), (6.80)

where CRII is the retarded current-current correlation function,

CRII(t) = −iθ(t)〈[I(t), I(0)]−〉. (6.81)

The current can be calcalated among an arbitrary cross section alongthe sample.

(c) Discuss when the relation G = σA/L is true for a sample of crosssection A and length L.

102

Exercise 6.5: Vertex renormalization

In this exercise, we address the vertex function Γ. First, we consider thenumerator of the last term in Eq. (6.57). We perform the analytical contin-uation iωm → ξ ± iη and iΩn → ω + iη.

(a) For the + sign, show that the numerator of the last term in Eq. (6.57)is zero.

(b) For the − sign, show that the numerator of the last term in Eq. (6.57)can be written as

1

V

∑

~k

2~k + ~q

(ξ − ε~k − i/2τ)(ξ + ω − ε~k+~q + i/2τ)

=2πνq

vF

[

2kl − i

ql

(

1 − i+ ωτ

2qllnωτ + ql + i

ωτ − ql + i

)

+i

2lnωτ + ql + i

ωτ − ql + i

]

≈ 4πνklq

vF ql

(

1 − i+ ωτ

2qllnωτ + ql + i

ωτ − ql + i

)

. (6.82)

where k = kF + ξ/vF , ν is the density of states per spin direction andper unit volume, and l = vF τ is the mean free path.

Now let us consider the denominator of the same term. Again, we considerthe analytical continuations iωm → ξ ± iη and iΩn → ω + iη.

(c) For the + sign, show that the denominator of the last term in Eq. (6.57)equals 2πντ .

(d) For the − sign, show that the denominator of the last term in Eq. (6.57)can be written as

2πντ − 1

V

∑

~k

1

(ξ − ε~k − i/2τ)(ξ + ω − ε~k+~q + i/2τ)

= 2πντ

(

1 +i

2qllnωτ + ql + i

ωτ − ql + i

)

. (6.83)

103

104

7 The interacting electron gas: Hartree-Fock

and random phase approximations

Sofar we have discussed electrons mainly as if they were non-interactingparticles. This is a huge simplification of reality that turns out to workremarkably well. In this chapter we include electron-electron interactions intoour theory and uncover part of the picture why it is that the approximationof noninteracting electrons is as good as it is.

In our theoretical description, we’ll consider a general electron-electroninteraction Hamiltonian of the form

H = H0 +H1, (7.1)

whereH0 is the “unperturbed” Hamiltonian, which we assume to be quadraticin fermion or boson creation and annihilation operators, whereas H1 is theperturbation. The Hamiltonian H1 will include the effect of electron-electroninteractions, which, in the most general case, read

H1 =1

2

∑

νν′,µµ′

Vνµ,ν′µ′ψ†νψ

†µψµ′ψν′ . (7.2)

Here the indices µ, ν, µ′, and ν ′ label a complete set of single-electron statesand Vνµ,ν′µ′ is the corresponding matrix element. For electrons, the fermionstatistics is taken care by the anticommutation relations of the creation andannihilation operators ψ† and ψ. Depending on the problem of interest, theeffects of impurity scattering may be included in H0 or as an additional termin H1.

Of course, the true microscopic electron-electron interaction is the Coulombinteraction. Using a representation in real space, the Coulomb interactionHamiltonian reads

H1 =1

2

∑

σ1,σ2

∫

d~r1

∫

d~r2e2

4πǫ0|~r1 − ~r2|ψ†

σ1(~r1)ψ

†σ2

(~r2)ψσ2(~r2)ψσ1(~r1). (7.3)

In momentum space, the Coulomb interaction reads

H1 =1

2V

∑

σ1,σ2

∑

~k1,~k2

∑

~q

V~qψ†~k1+~q,σ1

ψ†~k2−~q,σ2

ψ~k2,σ2ψ~k1,σ1

, (7.4)

105

where

V~q =

∫

d~re2

4πǫ0re−i~q·~r =

e2

ǫ0q2. (7.5)

One important aspect of the Coulomb interaction is its long range. Asa matter of fact, the energy required to charge a system is infinite! Thatis the reason why all objects are electrically neutral. In most cases, chargeneutrality is maintained by the positive charge density of the ion lattice, butcharge neutrality can also be maintained by the charge on objects that arenear the sample of interest, but that are not a part of it, such as nearbypieces of metal or condensator plates.

The goal of this chapter is to include the Coulomb interaction into thediagrammatic theory. However, first we briefly review other approaches totreat interactions between electrons.

7.1 Hartree Fock approximation

In the Hartree Fock approximation, the electron-electron is replaced by aneffective potential, which is then determined self-consistently. Replacing theinteraction by a potential is a very useful apprximation, because potentialscan be dealt with on the single-particle level.

If the interaction is replaced by an effective potential, the Hamiltonian isquadratic in electron creation and annihilation operators. This means thatWick’s theorem holds for this Hamiltonian. The principle of the HartreeFock approximation is that the interaction Hamiltonian is replaced with aHamiltonian that is quadratic in electron creation and annihilation operatorsin a way that respects Wick’s theorem,21

HHF1 =

1

2

∑

Vνµ,ν′µ′

(

ψ†νψν′〈ψ†

µψµ′〉 + ψ†µψµ′〈ψ†

νψν′〉 − 〈ψ†µψµ′〉〈ψ†

νψν′〉)

− 1

2

∑

Vνµ,ν′µ′

(

ψ†νψµ′〈ψ†

µψν′〉 + ψ†µψν′〈ψ†

νψµ′〉 − 〈ψ†µψν′〉〈ψ†

νψµ′〉)

.

(7.6)

Here, the first line corresponds to what is known as the “Hartree Hamilto-nian”, whereas the second line is the “Fock Hamiltonian”. If the labels µ andν refer to different particles, the Fock contribution is zero; for indistinguish-able particles, however, the Fock contribution is important. The averages

21You verify that any average of the Hamiltonian HHF1 is the same as that of the original

interaction Hamiltonian H1 if the Wick theorem were to be used.

106

〈ψ†µψµ′〉 are to be regarded as parameters in the Hartree-Fock Hamiltonian

that should be calculated self-consistently at the end of the calculation.In a non-magnetic translationally invariant system, any average of the

form 〈ψ†~kσψ~k′σ′〉 should be nonzero only if ~k = ~k′ and σ = σ′. Denoting

n~kσ = 〈ψ†~kσψ~kσ〉, (7.7)

the Hartree-Fock Hamiltonian in a translationally invariant system can bewritten

HHF = H0 +HHF1 =

∑

~kσ

εHF~kσψ†

~kσψ~kσ, (7.8)

where the Hartree Fock energy is given by

εHF~kσ

= ε~k +∑

~k′σ

(V0 − δσσ′V~k−~k′)n~k′σ′ . (7.9)

Note that the Hartree-Fock energies εHF~kσ

depend on the occupation numbersn~k′σ′ which, in turn, depend self-consistently on the Hartree-Fock energiesεHF~kσ

.In the next sections we return to the Hartree-Fock approximation and

give it a diagrammatic interpretation.

7.2 Diagrammatic Technique

The derivation of a diagrammatic technique for interacting electrons proceedsin exactly the same way as in Sec. 4. We present the rules for the specificcase of the calculation of a single-particle Green function. Their extensionto more complicated correlation functions is straightforward.

The single particle Green function can be expressed as a thermal averageover the non-interacting Hamiltonian H0,

Gσ1,σ2(~r1, τ1;~r2, τ2) = −〈Tτe−∫ 1/T0

H1(τ ′)dτ ′

ψσ1(~r1, τ1)ψ†σ2

(~r2, τ2)〉0〈Tτe

−∫ 1/T0 H1(τ ′)dτ ′〉0

. (7.10)

Here operators are represented in the interaction picture. As before, theGreen function is calculated in a series in H1. Each term in the series canbe represented by a Feynman diagram. The diagrammatic representationof the interaction (7.2) is as in figure 28: it consists of four fermion lines

107

µµ ’

’ν ν

Figure 28: Diagrammatic representation of an interaction vertex

connected by a dotted line.22 Each interaction line carries the weight −Vµν;µ′ν′

appropriate to the representation that is used, the minus sign arising from theminus sign in the exponent of the evolution operator in Eq. (7.10). For theCoulomb interaction in coordinate representation, the interaction involvesone particle at coordinate ~r and one particle at coordinate ~r′ and has weight is−e2/4πǫ0|~r−~r′|. For the Coulomb interaction in momentum representation,

the Coulomb interaction scatters electrons in momentum states ~k and ~k′ tomomentum states ~k+~q and ~k′−~q, respectively, and has weight −V~q/V . Thefactor 1/V follows from the normalization of the Fourier transform of theCoulomb interaction, see Eq. (7.4).

The diagrammatic rules for a perturbation H1 that includes interactionsare the same as those for the case of impurity scattering we discussed inchapter 4. There are two additional rules, specific for the case of interactions,

1. The interaction Hamiltonian H1 involves four creation and annihila-tions opertors at the same time. In order to ensure that creation op-erators are always kept in front of annihilation operators, one adds apositive infinitesimal η to their imaginary times,

H1(τ) =1

2

∑

νν′,µµ′

Vνµ,ν′µ′ψ†ν(τ + η)ψ†

µ(τ + η)ψµ′(τ)ψν′(τ). (7.11)

With this convention, the time ordering operator Tτ guarantees thatψ†(τ) always appears in front of ψ(τ).

2. With interactions, connected diagrams may include closed particle loops,see, e.g., Fig. 29. For fermions, each loop corresponds to an odd per-

22In the literature, the dotted line is often replaced by a wiggly line.

108

1

1+ +

+ + +

+

+

++ +

+ + + ...

k

k=

kk’k

k k

k k

Figure 29: Diagrammatic representation of all diagrams for the single particleGreen function, up to second order in the Coulomb interaction. The Coulombinteraction lines are drawn dotted.

mutation of the creation and annihilation operators in the perturbationexpansion, and hence carries a weight −1.

As before, all internal times are integrated over, all internal spin indicesare summed over, and all internal coordinates or momenta are integratedover or summed over, depending on the representation that is used in thecalculation. You verify that combinatorial factors conspire in such a way that(i) all disconnected diagrams cancel between numerator and denominator and(ii) one has to sum over topologically different diagrams only, with the sameweight for each diagram, up to a factor −1 for each fermion loop. You canfind a detailed proof of these rules in the book by Abrikosov, Gorkov, andDzyaloshinskii.

All diagrams for the single particle Green function, up to second orderin the Coulomb interaction, are shown in Fig. 29. The second, fourth, fifth,eleventh, twelfth, and thirteenth diagram in the figure have an odd numberof fermion loops and, hence, receive an extra factor −1. The first three termsin the expansion of the single-particle Green function read

Gσ,σ′(~r, τ ;~r′, τ ′) = G0σ,σ(~r, τ ;~r′, τ ′)δσ,σ′

−∫ 1/T

0

dτ1∑

σ1

∫

d~r1

∫

d~r2G0σ,σ(~r, τ ;~r1, τ1)

(−e2)4πǫ0|~r1 − ~r2|

109

× G0σ1,σ1

(~r2, τ1;~r2, τ1 + η)G0σ,σ(~r1, τ1;~r

′, τ ′)δσ,σ′

+

∫ 1/T

0

dτ1

∫

d~r1

∫

d~r2G0σ,σ(~r, τ ;~r1, τ1)

(−e2)4πǫ0|~r1 − ~r2|

× G0σ,σ(~r1, τ1;~r2, τ1 + η)G0

σ,σ(~r2, τ1;~r′, τ ′)δσ,σ′ , (7.12)

if the coordinate representation is used, whereas in momentum representationthe first three diagrams read

G~kσ;~k′σ′(τ ; τ′) = G0

~kσ;~kσ(τ ; τ ′)δσ,σ′δ~k,~k′

− 1

V

∫ 1/T

0

dτ1∑

σ1

∑

~k1

G0~kσ,~kσ

(τ ; τ1)(−V0)

× G0~k1σ1;~k1σ1

(τ1; τ1 + η)G0~kσ;~kσ

(τ1; τ′)δσ,σ′δ~k,~k′

+1

V

∫ 1/T

0

dτ1∑

~k1

G0~kσ,~kσ

(τ ; τ1)(−V~k−~k1)

× G0~k1σ,~k1σ

(τ1; τ1 + η)G0~kσ;~kσ

(τ1; τ′)δσ,σ′δ~k,~k′. (7.13)

Note that the interaction matrix element that appears in the integrationsdepends on the distance ~r1−~r2 only (if the coordinate representation is used)

or on the transferred momentum ~k − ~k1 (if the momentum representation isused). Also note that the Coulomb interaction does not flip spins, so thatspin needs to be conserved separately at each end of an interaction line.

In order to deal with the integrations over imaginary time, it is advanta-geous to make a Fourier transform to imaginary time, i.e., to write

Gσ,σ′(~r, τ ;~r′, τ ′) = T∑

n

Gσ,σ(~r, ~r′; iωn)e−iωn(τ−τ ′),

G~kσ;~k′σ′(τ ; τ′) = T

∑

n

G~kσ;~k′σ′(iωn)e−iωn(τ−τ ′), (7.14)

where the ωn are the fermionic of bosonic Matsubara frequencies. Each in-ternal integration over imaginary time for each interaction vertex gives 1/Ttimes a Kronecker delta for the Matsubara frequencies. Since an interactionvertex carries only one internal time, this implies that only the sum of thefour Matsubara frequencies is conserved at each interaction line, not the Mat-subara frequencies of the individual fermions or bosons participating in the

110

interaction.23 In other words, at interaction vertices a Matsubara frequencydifference can be “transferred” between particles. Note that a “transferred”Matsubara frequency difference is of the boson type if it is transferred be-tween bosons and bosons or fermions and fermions, and of fermion type if itis transferred between a boson and a fermion. Since the Coulomb interactionis instantaneous, the weight of the Coulomb interaction lines do not dependon the Matsubara frequency that is transferred at the interaction line. Allremaining intermediate Matsubara frequencies that are not fixed by the con-servation rules are summed over; normalization requires an extra factor T foreach summation over Matsubara frequencies, in accordance with Eq. (7.14).

To illustrate the use of Matsubara frequencies, we rewrite Eq. (7.13) usingMatsubara frequencies,

G~kσ;~k′σ′(iωn) = G0~kσ;~kσ

(iωn)δσ,σ′δ~k,~k′

− T

V

∑

σ1

∑

m

∑

~k1

G0~kσ,~kσ

(iωn)(−V0)G0~k1σ1;~k1σ1

(iωm)eiηωmG0~kσ;~kσ

(iωn)δσ,σ′δ~k,~k′

+T

V

∑

~k1

∑

m

G0~kσ,~kσ

(iωn)(−V~k−~k1)G0

~k1σ,~k1σ(iωm)eiηωmG0

~kσ;~kσ(iωn)δσ,σ′δ~k,~k′.

(7.15)

The factors exp(iηωm) were added to account for the infinitesimal time dif-ference in the intermediate Green functions, cf. Eqs. (7.13) and (7.14).

Looking at all the indices and summations that appear in Eq. (7.15), itbecomes useful to group all these indices together into one “four-vector” thatincludes the momentum, the spin, and the Matsubara frequency,

k = (~k, σ, iωn).

Then the symbol∑

k

=T

V

∑

~k

∑

σ

∑

n

will denote summation over the momentum ~k, the spin σ, and the Matsubarafrequency ωn and includes the appropriate normalization factors.

23The statement that the sum of the four Matsubara frequencies is conserved meansthat the sum of the Matsubara frequencies of the two outgoing particles equals the sumof the Matsubara frequences of the incoming particles at the interaction vertex.

111

+ + +

+ + + + + ...

=

Figure 30: Diagrammatic expansion of the self energy, up to second order in theCoulomb interaction.

7.3 Self energy

We saw that the concept of a “self energy” was very useful in order to organizethe summation of a diagrammatic perturbation series for a disorder-averagedGreen function. Here, we shall see that the same is true for a thermal averageof interacting electrons.

Again we call a diagram “irreducible” if it cannot be cut in two by removalof a single fermion line. In Fig. 29, the seventh, tenth, twelfth, and thirteenthdiagrams are not irreducible. The self-energy is defined as the sum over allall irreducible diagrams without the two external fermion lines, see Fig. 30.With this definition of the self energy, the single-particle Green functionobeys the Dyson equation

Gk,k′ = G0k,k′ +

∑

k1,k2

G0k,k1

Σk1,k2Gk2,k′. (7.16)

Note that, in contrast to the case of the disorder average we studied before,the self energy Σk,k′ does not need to be diagonal in the four-vectors k, k′.

A perturbative expansion for the self energy is shown in Fig. 29. The firsttwo diagrams of Fig. 29 have our special attention: The interaction line canbe seen as an effective “impurity line”. This is particularly clear for the firstdiagram, which is known as the “Hartree” diagram. For the second diagram,which is known as the “Fock diagram”, you should consider the fermion lineas being part of the “effective impurity”.

At first sight, the Hartree and Fock diagrams appear to be rather different.However, they are related by simple exchange of particles participating in theinteraction! To understand this, look at the effect of exchange on the basic

112

µµ

ν ν

’

’ ν ’

µ’

ν

µ

Figure 31: Interaction vertex and exchanged interaction vertex.

interaction vertex, see Fig. 31. It is precisely this exchange operation thatmaps the Hartree diagram into the Fock diagram and vice versa.

Let us calculate what these two contributions to the self energy are. Cal-culation of the Hartree diagram gives

ΣH~kσ

(iωn) = −TV

∑

~k′

∑

σ′

∑

m

(−V0)G0~k′σ′(iωm)eiηωm . (7.17)

The factor exp(iηωm) arose because we take the creation operator in theinteraction Hamiltonian at an infinitesimally larger imaginary time thatnthe annihilation operator. Performing the summation over the Matsubarafrequency ωm and the momentum ~k, we find

ΣH = V0

∑

~kσ

n0~kσ

= V0n, (7.18)

where n is the particle density. Similarly, calculation of the Fock diagramgives

ΣF~kσ

(iωn) =T

V

∑

~k′

∑

σ′

∑

m

(−V~k−~k′δσσ′)G~k′σ′(iωm)eiηωm

= − 1

V

∑

~k′

V~k−~k′n0~k′σ. (7.19)

Here, there is no summation over spin, because spin is conserved throughoutthe diagram.

Looking at these results, we first note that the Hartree and Fock correc-tions to the self energy are real: they correspond to a shift of the particle’senergy, but the particle’s lifetime remains infinite. The results (7.18) and(7.19) are almost identical to the correction to the single particle-energyin the Hartree-Fock approximation. The only difference is the absence of

113

+=

Figure 32: Diagrammatric representation of the Hartree-Fock approximation. Theself energy is expressed in terms of the full single-particle Green function, which,in turn, depends on the self energy as in Fig. 17.

self-consistency in Eqs. (7.18) and (7.19): in those equations, the occupa-tion numbers n~kσ are calculated in the non-interacting ground state. Self-consistency can be achieved if the single-particle Green function G0 in theHartree and Fock diagrams is replaced by the full Green function G, see Fig.32. In that case one recovers precisely the Hartree-Fock equation (7.9). Youeasily verify that, because of the self-consistency condition, the Hartree-Fockapproximation includes all diagrams of Fig. 30 with the exception of thefourth and the eigth diagrams of that figure.

Note that, formally, the Hartree self energy is divergent. In practice, thenegative charge of the electrons is balanced by the positive charge of the lat-tice ions, and there is no contribution from the ~q = 0 mode of the interaction.Hence, for a translationally invariant system, the Hartree contribution to theself energy can be neglected.

7.4 RPA approximation and screening

Of course, one can continue and calculate more and more diagrams thatcontribute to the self energy. As you can see from Fig. 30, this number ofdiagrams grows rather fast with the order of the perturbation theory. Hence,it would be useful if we had a method to pick those diagrams that have thelargest contributions.

In defining the relative importance of interactions, one introduces theso-called “gas parameter” rs, which is a dimensionless parameter measuringthe density of the electron gas. The parameter rs is defined in terms of theelectron number density n as

a0rs = (4πn/3)−1/3. (7.20)

A sphere of radius a0rs contains precisely one electron on average. Using the

114

relation n = k3F/3π

2, we can rewrite this as

rs = (9π/4)1/3(a0kF )−1. (7.21)

A low gas parameter corresponds to a high density of electrons.The importance of the electron density for a perturbation expansion in

the Coulomb interaction follows from the observation that the gas parameteris proportional to the ratio of typical Coulomb interaction for neighboringelectrons, which is of order e2/ǫ0kF , and the electronic kinetic energy, whichis ~

2k2F/2m. Hence, at high electron densities the main contribution to the

electron’s energy is kinetic.For realistic metals, rs is between 1.5 and 6. That is not really small.

Hence, our small-rs expansion is an idealized theory, and we should expectthat some of our quantitative conclusions cannot be trusted.

In order to see which diagrams are important in the limit of high density,we need two observations.

• First, we consider a diagram contribution to the self energy to nthorder in the Coulomb interaction. In order to estimate its dependenceon the gas parameter rs, we measure energy and temperature in unitsof the Fermi energy, which is ∝ k2

F . Then, noting that every nth orderdiagram has n integrations over internal momenta (there are 2n − 1fermion lines and n−1 constraints from the interaction lines) and thatit contains 2n− 1 fermion Green functions, we find that

Σ(n)/εF ∝ rns . (7.22)

(The summations over internal Matsubara frequencies do not contributeto this estimate, since, for each internal Matsubara frequency, the dia-gram carries a factor T .)

• Second, we note that the Coulomb potential is singular: for small trans-ferred momentum ~q, the Coulomb potential diverges Vq ∝ q−2. The roleof these divergencies is stronger if the same momentum ~q is transferredmore than once, i.e., if there is more than one interaction vertex withprecisely the same momentum transfer. Hence, we conclude that thosediagrams with the maximal number of identical transferred momentahave the largest contribution.

115

+ + + ...=

Figure 33: Diagrammatric representation of the random phase approximation forthe self energy in a translationally invariant system.

With these two observations, we can identify the leading diagrams in the limitof high electron density. Namely, for each set of diagrams with the samelargest number of identical transferred momenta, the diagram of minimalorder in the interaction potential dominates for small rs. Such diagrams areprecisely the diagrams shown in Fig. 33. The approximation in which onekeeps those diagrams only is known as the “random phase approximation”.

Note that the random phase approximation does not treat the directinteraction and the exchange interaction on the same footing: performingthe same transformation as in Fig. 31 on the diagrams of Fig. 32 leads todifferent contributions to the self energy. The reason that these contributionsare left out in the present calculation is that they are less singular. In thenext chapter we consider the effects of a short-range interaction, for which itis essential that exchange and direct interactions are treated equally.

Before we sum the entire series for the self energy, we note that thediagrams can be interpreted as an effective Coulomb interaction. In fact,what the series of Fig. 33 represents is that the Coulomb interaction polarizesthe electron gas, and that the total interaction is found as the sum of thedirect interaction and the interaction with the polarization cloud. This isnothing but “screening” of the Coulomb interaction! Of course, the randomphase approximation is only a subset of a more complete series of diagramsthat represents the modification of the Coulomb interaction by the electrongas.24 However, as we discussed above, the random phase approximation

24In a complete description of the effective Coulomb interaction, one introduces theconcept of an “irreducible” polarization propagator χirr, which is any part of the diagramfor the effective Coulomb interaction that cannot be cut into two parts by cutting a singleinternal interaction line. Then the effective interaction can be written as

V eff~q (iΩn) =

V~q

1 + V~qχirr(~q, iΩn)/e2.

The random phase approximation amounts to keeping only the zeroth-order in interactiondiagram for χirr, see Eq. (7.24) below.

116

+ + + ...

+

=

=

Figure 34: Diagrammatric representation of the random phase approximation forthe self energy in a translationally invariant system.

describes the dominant effect for an electron gas with a high density.A diagrammatic expression for the RPA screened Coulomb interaction is

shown in Fig. 34. The screened interaction is represented by a double dottedline. The central object in this expansion the “pair bubble”, which is nothingbut the retarded polarizability function of the non-interacting electrons gasχ0

e~q(Ωn), see Eq. (6.29),

χ0e~q(iΩn) =

2Te2

V

∑

~k

∑

m

G0~k+~q

(iωm + iΩn)G0~k(iωm). (7.23)

(We added the superscript “0” to indicate that χ0e is the polarizability func-

tion of non-interacting electrons. Later we will calculate the full polarizabilityfor interacting electrons.) Solving for the effective Coulomb interaction, wefind

V RPA~q (iΩn) =

V~q

1 − V~qe−2χ0Re~q (iΩn)

=e2

ǫ0q2 − χ0Re~q (iΩn)

. (7.24)

For small momentum and energy exchange, we can replace the polariz-ability χ0 by its value at ~q = 0. Since χ0R

e0 (0) is negative (the signs of inducedcharge and external charge are opposite), we write

ǫ0k2s = −χ0R

e,~q→0(iΩn → 0) = 2νe2, (7.25)

where ν = mkF/2π2 is the density of states at the Fermi level, per spin

direction and per unit volume. Equation (7.25) can be rewritten in terms ofthe Bohr radius a0 or the gas parameter rs,

k2s =

4

π

kF

a0

= k2F rs

(

16

3π2

)2/3

. (7.26)

117

Then the effective Coulomb interaction in the limit of small ~q and ω reads

V RPA~q (iωn) =

e2

ǫ0(q2 + k2s)

if ~q, ωn → 0. (7.27)

The quantity 1/ks is known as the “Thomas-Fermi screening length”. Thesingularity of the Coulomb interaction at large wavelengths is cut off bythe screening of the interaction by the electron gas. The Thomas-Fermiscreening potential is found when we use the limiting form of Eq. (7.27) forall momenta and frequencies. In that case, the screened Coulomb interactionis instantaneous and has the spatial dependence of the Yukawa potential,

V TF(~r) =e2

4πǫ0re−ksr. (7.28)

The true RPA screened interaction we derived above, however, has a differentfrom on short length scales and, more importantly, is retarded (i.e., it dependson frequency).

Returning to the self energy: if we sum the entire RPA series for the selfenergy, we find

ΣRPA~k

(iωn) = −TV

∑

~k′

∑

m

V RPA~k−~k′

(iωn − iωn)eiηωm

(iωm − ε~k′ + µ). (7.29)

7.5 Dielectric response

Sofar we have studied the behavior of the electron gas in thermodynamicequilibrium. Let us now apply our results to study the response of the elec-tron gas to a perturbation.

The response function we study first is the polarizability function χe,which was defined in Sec. 6.2. The polarizability function expresses theinduced charge density as a function of the external potential,

ρind(~q, ω) = χRe (~q, ω)φext(~q, ω). (7.30)

The polarizability function could be calculated as the retarded charge densityautocorrelation function, which is obtained as the analytical continuation ofthe temperature charge density autocorrelation function

χRe (~q, τ) = − 1

V〈Tτρe,~q(τ)ρe,−~q(0)〉 (7.31)

= −e2

V

∑

~kσ

∑

~k′σ′

〈Tτψ†~k,σ

(τ + η)ψ~k+~q,σ(τ)ψ†~k′+~q,σ′

(η)ψ~k′,σ′(0)〉.

118

eχ− +

++ +

+ +

+++

+

+ + ...

+

=k k’

k’+qk+q

=

Figure 35: Diagrammatic expansion of the polarizability function χe. The figurecontains all diagrams to first order in the interaction and a subset of the diagramsto second order in the interaction.

Diagrammatically, the polarizabibility function is obtained as in Fig. 35.We can organize the diagrammatic expansion as we did previously whenwe looked at the single-particle Green function. A polarization diagram iscalled “reducible” if the two end points can be separated by cutting a singleinteraction line. For example, of the diagrams in Fig. 35, the seventh, ninth,tenth, eleventh, and thirteenth diagrams are reducible. All other diagramsare irreducible in this sense. Defining the irredubile polarizability function as(minus) the sum over all irreducible diagrams, we can write the polarizabilityfunction as

χe(~q, iΩn) =χirr

e (~q, iΩn)

1 − V~qχirre (~q, iΩn)/e2

. (7.32)

The diagrammatic series for χirre is as in Fig. 36.

In the RPA approximation, one only keeps the zeroth order diagram forχirr

e , which is the polarizability function for the noninteracting electron gas.After analytical continuation to real frequencies, one finds

χRPAe (~q, ω) =

χ0Re (~q, ω)

1 − V~qχ0e(~q, iΩn)/e2

. (7.33)

You may find that the calculation of the polarizability function is verysimilar to that of the screened Coulomb interaction. That is no surprise.

119

eχ− irr

+ ...

+

+

+ +

+++

=k k’

k’+qk+q

=

Figure 36: Diagrammatic expansion of the irreducible part χirre of the polarizabil-

ity function. The figure contains all diagrams to first order in the interaction anda subset of the diagrams to second order in the interaction.

On the one hand, according to Eq. (7.30) above, the polarizabilitly functiondescribes what is the screening charge induced by an external potential. Onthe other hand, the screened Coulomb potential describes how the “internal”potential of a charged particle is modified by the charge density this particleinduces. As you see, one has solved the same problem twice!

We can now use our result for the polarizability function to discuss thedielectric response function

ǫ−1(~q, ω) = 1 + V~qχRe (~q, ω). (7.34)

Substituting the RPA result (7.33), we find

ǫ−1(~q, ω) =[

1 − χ0Re (~q, ω)V~q

]−1. (7.35)

The polarizability function of the non-interacting electron gas was calculatedin Ex. 6.2. In the static limit, ω → 0, the result is

χ0e(~q, 0) = −ν

[

1 +

(

kF

q− q

4kF

)

ln

∣

∣

∣

∣

2kF + q

2kF − q

∣

∣

∣

∣

]

, (7.36)

where ν = mkF/2π2 is the density of states at the Fermi level. Hence,

ǫ−1(~q, 0) =

[

1 +e2kFm

2ǫ0q2π2

[

1 +

(

kF

q− q

4kF

)

ln

∣

∣

∣

∣

2kF + q

2kF − q

∣

∣

∣

∣

]]−1

. (7.37)

The long wavelengths limit of Eq. (7.37) is the Thomas-Fermi dielectric func-tion,

ǫ(~q, 0) = 1 +k2

s

q2, (7.38)

where k2s = me2kF/π

2ǫ0 is the inverse Thomas-Fermi screening length.

120

V =q

U =qRPA

RPA

+ + + ...

+ + + ...

=

=

Figure 37: Diagrammatic representations of impurity potential and screenedCoulomb interaction in the RPA approximation.

An important feature of the RPA result for the static dielectric responsefunction is that it is singular at q = 2kF . This has a number of physicalconsequences, that are discussed in the book by Doniach and Sondheimer.

For large frequencies ω ≫ qvF , the polarizability function χ0 can beexpanded as

χR0 (~q, ω) =

k3F q

2

3π2mω2

(

1 +3

5

(qvF

ω

)2)

=nq2

mω2

(

1 +3

5

(qvF

ω

)2)

, (7.39)

where n = k3F/3π is the density of electrons. Substituting this into the

dielectric response function gives

ε−1(~q, ω) =

[

1 − ne2

mǫ0ω2

(

1 +3

5

(qvF

ω

)2)]−1

. (7.40)

Let us now consider the effect of impurities in the metal. The impuri-ties are charged objects — otherwise they would hardly interact with theelectrons. However, in our previous treatment, we have modeled impuriteswith a short-range potential. Now we see why that was correct: the impuritypotential is screened by the electron gas, so that

U~q → U~q + U~qχRe (~q, 0)V~q. (7.41)

Diagrammatically, the screening of the impurity potential and of the Coulombinteraction in the RPA approximation can be represented in striking similar-ity, see Fig. 37.

121

7.6 Plasmon modes

For some frequency/wavelength combinations, the dielectric response func-tion can vanish. Then, there is an induced charge density without an externalone! Such a situation is referred to as a “collective mode” of the electrongas. It is a motion that can sustain itself without external input.

In priciple, the dielectric response function is complex. In those cases,the occurence of zeroes is unlikely. However, there are regions of (q, ω) spacewhere ǫ is real. In such regions, long-lived collective excitations can be ex-pected. The support of the imaginary part of the polarizability functionof the noninteracting electron gas (which equals the support of Im ǫ−1) wasshown in Fig. 20. Based on that figure, we expect collective excitations in theregion of finite frequencies and long wavelengths. In this parameter regime,the dielectric response function is given by Eq. (7.40) above. We see that,indeed, there are zeroes, given by the dispersion relation

ω(~q) = ωp +3q2v2

F

10ωp, ω2

p =ne2

mǫ0. (7.42)

The frequency ωp is known as the “plasma frequency” and the correspond-ing collective modes are known as “plasma oscillations” or “plasmons”. Theplasmon modes are long-lived as long as the dielectric response function re-mains real. At the point when the dispersion curve (7.42) enters that regionof (q, ω) space where ǫ−1 is complex, the zeroes of ǫ at real frequency ωand wavevector q cease to exist. In this case, one can still find zeroes ofǫ at complex frequencies. These correspond to damped collective modes.The damping that arises from the fact that ǫ is complex is called “Landaudamping”.

The plasmon dispersion relation is shown schematically in Fig. 38, to-gether with the support of Im ǫ(q, ω).

7.7 Free energy

With all the diagrammatic rules we derived above, the easiest way to calculatethe free energy of the interacting electron gas is to use Eq. (3.116) of exercise3.2,

F − F0 =

∫ 1

0

dλ1

λ〈λH1〉λ, (7.43)

122

q/kF

ω/εF

30 4210

4

2

Figure 38: Plasmon dispersion relation (thick curve) together with the supportof Im ǫ. Plasmon modes are strongly damped (Landau damping) if Im ǫ 6= 0.

where F0 is the free energy of the noninteracting electron gas and the pa-rameter λ multiplies the Coulomb interaction. Rewriting H1 as

H1 =1

2V

∑

σ1,σ2

∑

~k1,~k2,~q

V~qψ†~k1+~q,σ1

ψ~k1,σ1ψ†

~k2−~q,σ2ψ~k2,σ2

− 1

2n∑

~q

V~q, (7.44)

we can express the average 〈H1〉 as

〈H1〉 = − 1

2e2limτ↑0

∑

~q

V~qχe(~q, τ) −1

2n∑

~q

V~q,

= − T

2e2

∑

~q

∑

n

e−iΩnηV~qχe(~q, iΩn) − 1

2n∑

~q

V~q, (7.45)

where χe is the polarizability function. Within the random phase approx-imation we can use Eq. (7.33) for χe, replace V~q by λV~q and perform theintegration to λ. The result is

F − F0 =T

2

∑

~q

∑

n

eiΩnη ln(

1 − V~qχ0e(~q, iΩn)/e2

)

− 1

2n∑

~q

V~q. (7.46)

We can perform the summation over n writing F − F0 as a contour integralin the complex plane and shifting the contours to the real axis,

F − F0 =∑

~q

∫ ∞

−∞

dξ

4πcoth(ξ/2T )Im ln (ǫ(~q, ξ)) − 1

2n∑

~q

V~q. (7.47)

Here we used Eq. (7.35) to express the argument of the logarithm in termsof the RPA dielectric response function.

123

Evaluation of this expression for the Free energy is not entirely straight-forward. Some insight into the physics of Eq. (7.47) is obtained if we lookat the contribution from the plasmons. Using Eq. (7.40) for the dielectricresponse function ǫ(~q, ω) near the plasmon branch ω ≈ ωp(~q) ≫ qvF at lowwavenumber, we find

Im ln ǫ(~q, ω) =

0 if |ω| > ωp(~q),π if qvF ≪ ω < ωp(~q),−π if −ωp(~q) < ω ≪ −qvF .

(7.48)

We then see that each plasmon mode with vF q ≪ ωp(~q) has a contribution

δF~q =1

2

∫ ωp(~q)

dξ coth(ξ/2T )

= T ln(sinh(ωp/2T )) − const, (7.49)

where the additional constant reflects the contribution from the integrationclose to ξ = 0 where the approximation (7.48) does not hold. The first termon the r.h.s. of Eq. (7.49) represents the free energy of a plasmon mode atwavevector ~q, while the additional constant represents the contribution fromlow energy particle-hole excitations to the free energy.

Doniach and Sondheimer report a result for the ground state energy asan expansion for small gas parameter rs,

E −E0 =

(

−0.916

rs

+ 0.0622 ln rs − 0.094

)

Ry. (7.50)

In the same units, the ground state energy of the non-interacting electrongas is (2.21/r2

s) Ry. The unit Ry, for Rydberg, corresponds to e2/2a0 ≈ 13.6eV. Higher-order terms in the small-rs expansion of the ground state havebeen calculated. However, for those terms, one needs to go beyond the RPAapproximations.

124

7.8 Exercises

Exercise 7.1: Coulomb interactions in two dimensions

In certain materials, such as a quantum well at the interface of the semicon-ductors GaAs and GaAlAs, the electrons are confined to two spatial dimen-sions only.

For such a system, the non-interacting part of the Hamiltonian can bewritten

H0 =∑

~kσ

ε~kσψ†~kσψ~kσ, (7.51)

where the summation over wavevectors ~k is restricted to wavevectors in theplane of the two-dimensional electron gas. The creation operator ψ†

~kσcreates

an electron with spin σ and orbital wavefunction

ψ~k(~r, z) =1

A1/2ei~k·~rζ(z), (7.52)

where ~r is the coordinate in the plane of the two-dimensional electron gas,and z is the transverse coordinate, and A is the area of the two-dimensionalelectron gas. For most practical purposes, the function ζ(z) can be approxi-mated by the delta function δ(z).

The interaction between the electrons in the two-dimensional electron gasis the three-dimensional Coulomb interactions,

V (~r) =1

4πǫ0ǫr|~r|. (7.53)

The relative permittivity ǫr appears because the two-dimensional electron gasis not surrounded by vacuum, but by a semiconducting material. For thissystem, write down the interaction Hamiltonian for the Coulomb interactionusing the momentum representation. Find an explicit expression for theCoulomb-interaction matrix element V~q in two dimensions.

Exercise 7.2: Four-vertex formalism

Abrikosov, Gorkov, and Dzyaloshinskii discuss a different diagrammatic for-malism to describe electron-electron interactions. The diagrams for the direct

125

µµ

ν ν

’

’ ν ’

µ’

ν

µ µ

ν ’ν

µ’

Figure 39: Diagrammatic representation of both the direct and exchange parts ofthe electron-electron interaction, following the notation of the book by Abrikosov,Gorkov, and Dzyaloshinskii.

and the exchange interaction are combined into one diagram, as shown in Fig.39.

Discuss the diagrams contributing to the self-energy to second order inthe interaction using this formalism. Also discuss the Hartree-Fock approxi-mation. Does the simplest four-vertex formulation of an effective interactionagree with the RPA approximation? Why (not)?

Exercise 7.3: Plasma oscillations

It is possible to find the plasma frequency ωp from simple classical consid-erations. Consider an electron gas in a rectangular box of length Lx andcross section A. Since the mass of the ions is much larger than the mass ofthe electrons, the ions can be treated as an inert positive background chargethat compensates the charge of the electrons. The electron gas can be set inoscillating motion by translating it a distance δx in the x-direction, leavingthe ion background fixed. Show that the frequency of this oscillation is pre-cisely the plasma frequency ωp. Can you generalize this argument to plasmaoscillations at finite wavevector ~q?

Exercise 7.4: Plasma oscillations in two dimensions

In this exercise we return to the case of a two-dimensional electron gas thatexists at a quantum well in a semiconductor heterostructure.

(a) Calculate the dielectric response function ǫ(~q, ω) for a two-dimensionalelectron gas in the RPA approximation.

(b) From the zeroes of the dielectric response function, calculate the dis-

126

persion relation of plasma oscillations in the two-dimensional electrongas. Discuss for what wavenumbers plasma oscillations are damped byLandau damping (excitation of particle/hole pairs).

(c) Calculate the screened Coulomb interaction in a two-dimensional elec-tron gas in the RPA approximation.

(d) Show that, at small wavevectors q ≪ kF and low frequencies ~ω ≪ εF

the screened interaction is given by

V RPA~q =

e2

2ǫ0ǫr(q + ks), (7.54)

where ks is the two-dimensional Thomas-Fermi screening wavenumber.Calculate ks for the case of a GaAs/GaAlAs heterostructure, for whichthe effective electron mass is m∗ = 0.067m, relative permittivity ǫr =13, and electron density n = 1015 m−2. How does it compare to theFermi wavelength for this system?

127

128

8 Magnetism

In this chapter we study the magnetic excitations in a solid. We start witha calculation of the magnetic susceptibility χ, which is the coefficient of pro-portionality between the spin polarization induced by a magnetic field andthe magnetic field ~h. The susceptibility not only tells us how easy it is tomagnetize a solid, it also points to the onset of a magnetic phase. Thishappens at the point where the susceptibility diverges. A divergent suscepti-bility signals an instability: a spin polarization is formed for arbitrarily weakmagnetic field.

The magnetic instability is governed by a competition between two ef-fects. On the one hand, the kinetic energy of the electrons is minimal ifthere is no net spin polarization, because then all energy levels ε~k can bedoubly occupied. On the other hand, the short-range part of the electron-electron interaction favors a magnetic state. This is because electrons withthe same spin have a strongly reduced repulsion since the Pauli principlealready guarantees that they are spatially separated.

We first consider the susceptibility for a gas of interacting electrons witha parabolic dispersion (“free electrons”). This is the example that we studiedin detail in the previous chapters. A calculation within the RPA approxima-tion shows that electron-electron interactions enhance χ, but also that theydo not lead to a divergence. In other words, for free electrons, the competi-tion between kinetic energy and interaction energy always favors the kineticenergy, and, hence, forbids a spontaneous spin polarization.

The picture is different once the ionic lattice is taken into account. Bandstructure effects turn out to be crucial for the occurence of a ferromagneticinstability. In a narrow band, the kinetic energy cost for polarizing elec-tron spins is strongly reduced, and a magnetic instability becomes possible.Depending on the material one looks at, one may have a ferromagnetic insta-bility (uniform magnetization) or a spin-wave material (nonuniform magne-tization). One example of a spin-wave material is an antiferromagnet, wherethe spin polarization has opposite direction on neighboring lattice sites.

8.1 Magnetic susceptibility

In order to make the above picture more quantitative, we consider an electrongas in the presence of a magnetic field ~h. We are interested in the spin density

129

~s(~r),

sα(~r) =1

2

∑

σ,σ′

ψ†σ(~r)(σα)σ,σ′ψσ′(~r), α = x, y, z, (8.1)

where σα is the Pauli matrix. The response of a magnetic material toan applied magnetic field is characterized by the magnetic susceptibilityχ(~r, t;~r′, t′), which relates the change in spin density at position at position~r and time t to the magnetic field at position ~r′ and time t′,

δsα(~r, t) =∑

β

∫

d~r′∫

dt′χαβ(~r, t;~r′, t′)hβ(~r′, t′). (8.2)

Note that χαβ is a tensor: in principle, a magnetic field ~h can cause a spin

density in a different direction than the direction of ~h. The magnetic field ~henters into the Hamiltonian through the Zeeman coupling,

H1 = −~

2µBg

∫

d~r~h(~r, t) · ~s(~r), (8.3)

where µB = e~/2mc is the Bohr magneton and g = 2 the electron g factor.According to the Kubo formula, the spin susceptibility is equal to the

retarded spin-spin correlation function,

χαβ(~r, t;~r′, t′) = iµBg

2θ(t− t′)〈[sα(~r, t), sβ(~r′, t′)]−〉. (8.4)

Note the absence of a minus sign on the r.h.s. of Eq. (8.4) because of theminus sign in front of the magnetic field in the expression (8.3) for the Zeemanenergy. We may perform a Fourier transform to find the susceptibility infrequency representation,

δsα(~r, ω) =∑

β

∫

d~r′χαβ(~r, ~r′;ω)hβ(~r′, ω), (8.5)

where

χαβ(~r, ~r′;ω) = iµBg

2

∫ ∞

0

dteiωt〈[sα(~r, t), sβ(~r′, 0)]−〉. (8.6)

For a spatially homogeneous system, we may perform one further Fouriertransform to the coordinate ~r, which gives a susceptibility χαβ(~q, ω),

χαβ(~q, ω) =1

V

∫

d~rd~r′e−i~q·(~r−~r′)χαβ(~r, ~r′;ω). (8.7)

130

For the calculation of χαβ(~q, ω) it is useful to write the spin density interms of the plane wave basis,

χαβ(~q; t, t′) = iθ(t− t′)µBg

2V〈[s~q,α(t), s−~q,β(t′)]−〉, (8.8)

where

s~q,α =1

2

∑

~k,σ,σ′

ψ†~k,σ

(σα)σ,σ′ψ~k+~q,σ′ . (8.9)

In practice, the x and y labels of the tensor χαβ are often replaced bylabels “+” and “−”, referring to the linear combinations

s± = s1 ± is2. (8.10)

These combinations are expressed in terms of raising and lowering operatorsfor the spin, which are much easier to calculate. In particular, we have thesusceptibility

χ−+(~r, t;~r′, t′) = iµBg

2θ(t− t′)〈[s−(~r, t), s+(~r′, t′)]−〉. (8.11)

The susceptibility χ−+ is known as the “transverse” susceptibility, while χzz

is known as the “longitudinal” susceptibility. For an isotropic system, thetensor χαβ is proportional to the unit tensor. In that case one has

χαβ =1

2δαβχ−+, α, β = x, y, z. (8.12)

A calculation of the susceptibility for non-interacting electrons is verysimilar to the calculation of the polarizability that we discussed in Sec. 6.2.One finds

χ−+(~q, t) = iµBg

2Vθ(t)

∑

~k,~k′,~q

〈[ψ†~k,↓

(t)ψ~k+~q,↑(t), ψ†~k′,↑

(0)ψ~k′−~q,↓(0)]−〉.(8.13)

Diagrammatically, the susceptibility for non-interacting electrons can be rep-resented by the diagram shown in Fig. 40. The white dots are spin vertices.For the transverse susceptibility χ−+, they enforce that the ingoing and out-going spins have different sign, see Fig. 40.

131

−+χk k

k+qk+q

=

Figure 40: Diagrammatic representation of the transverse spin susceptibility χ−+

for non-interacting electrons.

As a matter of fact, we can find χ−+ without doing a calculation if wecompare Eqs. (8.13) and the expression for the polarizability of the non-interacting electron gas, Eq. (6.29). We then immediately conclude

χ0−+(~q, ω) = −µBg

2V

∑

~k

nF (ε~k − µ) − nF (ε~k+~q − µ)

ε~k − ε~k+~q + ω + iη, (8.14)

where nF is the Fermi function and η is a positive infinitisimal.In the limit ~q → 0, ω → 0 of a static and uniform magnetic field, one

has χ−+ → νµBg/2, where ν is the density of states per unit volume and perspin direction. This result is known as the “Pauli susceptibility”. The Paulisusceptibility accounts for the fact that it costs a finite kinetic energy tohave a finite spin density. The value of the Pauli susceptibility then followsfrom the competition of the magnetic energy gain (proportional to the spindensity and the magnetic field) and the kinetic energy loss (proportional tothe square of the spin density).

The imaginary part of the susceptibility χ0(~q, ω) characterizes energydissipation through magnetic excitations. The support of the imaginary partof χ0 is the same as that of the imaginary part of the polarizability of thenon-interacting electron gas, see Fig. 20).

For the interacting electron gas, we can use the diagrammatic theory tocompute the susceptibility. Some diagrams are shown in Fig. 41. They canbe rearranged in three steps. First, we incorporate all interaction lines thatconnect the upper or lower fermion line to itself into single-particle Greenfunctions. Second, we replace the interaction by the screened interaction byinserting any number of “bubbles” into the interaction. Below, the single-electron Green function will be calcualted using the Hartree-Fock approxi-mation, the effective interaction in the RPA. Third, we write the diagram interms of a “vertex correction” as in Sec. 6.3, see Fig. 41. The renormalizedvertex, which is indicated by a black dot, then contains all contributions frominteraction lines that connect the upper and lower fermion lines.

132

−+χ

σ σ + ...

+

+

+ +

+++

=

m

m n

k k

k+qk+q

= ==

k

k+q

ω

ω +Ω

Figure 41: Diagrammatic representation of the contributions to the transversespin susceptibility χ−+ for interacting electrons, up to second order in the inter-action. The arrows indicate the spin direction, while “σ” denotes an additionalsummation over spin. Note that spin is conserved throughout the upper and lowerfermion lines.

In the spirit of the RPA approximation, we keep only “ladder” diagramsfor the vertex correction, see Fig. 42. In this approximation, the calculationhas become very similar to that of the current-current correlation functionof Sec. 6.3. Denoting the renormalized vertex by Γ(~k, iωm;~k + ~q, iωm + iΩn)and the screened interaction by V eff

~q (iΩn), we have

Γ(~k, iωm;~k + ~q, iωm + iΩn) = 1 − T

V

∑

~k′

∑

p

G~k′↑,~k′↑(iωp)

× G~k′+~q↓,~k′+~q↓(iωp + iΩn) − V eff~k−~k′(iωm − iωp)

× Γ(~k′, iωp;~k′ + ~q, iωp + iΩn). (8.15)

Note that the effective interaction depends on the Matsubara frequency dif-ference iωm − iωp, in contrast to the bare interaction which is frequencyindependent. Then, in terms of the renormalized vertex, the spin suscepti-bility is calculated as

χRPA−+ (~q, iΩn) = −TµBg

2V

∑

~k

∑

m

Γ(~k, iωm;~k + ~q, iωm + iΩn)

× G~k↑,~k↑(iωm)G~k+~q↓,~k+~q↓(iωm + iΩn). (8.16)

The analytical continuation iΩn → ω + iη is done as in Sec. 6.3. The result

133

= + ...++

= +

m n

m

m n

mp

m n

p

p n

m n

mp

p n

k

k+q

k

k+q

k’

k’+q

k’

k’+q

k

k+q

k"

k"+q

k

k+q

k’

k’+q

k

k+q

k

k+qω +Ω

ω

ω +Ω

ω ω

ω +Ω

ω

ω +Ω

ω +Ω

ω ω

ω +Ω

Figure 42: Diagrammatic representation of the RPA approximation for the trans-verse spin susceptibility χ−+.

is

χ−+(~q, ω + iη) = − µBg

8πiV

∫

dξ tanh(ξ/2T )∑

~k

×[

ΓRR(~k, ξ;~k + ~q, ξ + ω)GR~k↑,~k↑

(ξ)GR~k+~q↓,~k+~q↓

(ξ + ω)

− ΓAR(~k, ξ;~k + ~q, ξ + ω)GA~k↑,~k↑

(ξ)GR~k+~q↓,~k+~q↓

(ξ + ω)

+ ΓAR(~k, ξ − ω;~k + ~q, ξ)GA~k↑,~k↑

(ξ − ω)GR~k+~q↓,~k+~q↓

(ξ)

− ΓAA(~k, ξ − ω;~k + ~q, ξ)GA~k↑,~k↑

(ξ − ω)GA~k+~q↓,~k+~q↓

(ξ)]

.

(8.17)

For the full frequency and momentum dependent screened interaction,the RPA equation for the renormalized vertex cannot be solved in closedform. Here, we’ll adopt the Thomas-Fermi approximation and replace thescreened interaction by the interaction at ~q = 0, ω = 0,

V TF~q (iΩn) ≈ e2

ǫ0k2s

, (8.18)

where k2s = 2νe2/ǫ0 is the Thomas-Fermi wavenumber. This approxima-

tion corresponds to a delta-function interaction in real space, V TF(~r) ≈(2ν)−1δ(~r). With this approximation, calculation of the renormalized ver-tex is straightforward, and we find the simple result

Γ(~k, iωm;~k + ~q, iωm + iΩn)

134

=

1 +T

2νV

∑

~k′

∑

p

G~k′↑,~k′↑(iωp)G~k′+~q↓,~k′+~q↓(iωp + iΩn)

−1

. (8.19)

The summation in the denominator has the same expression as the suscepti-bility for non-interacting electrons, the only difference being that the single-electron Green functions have been replaced by their values in the interactingelectron liquid. In other words, this is the susceptibility for the interactingelectron liquid without vertex renormalization. Using the Hartree-Fock ap-proximation for the single-particle Green function, we write

χ0,HF−+ (~q, iΩn) = −µBgT

2V

∑

~k

∑

m

G~k↑,~k↑(iωm)G~k+~q↓,~k+~q↓(iωm + iΩn)

= −µBg

2V

∑

~k

nF (εHF~k,↑

− µ) − nF (εHF~k+~q,↓

− µ)

εHF~k,↓

− εHF~k+~q,↑

+ iΩn

, (8.20)

so that we can express the full transverse susceptibility in the Thomas-Fermiapproximation as

χTF−+(~q, ω) =

χ0,HF−+ (~q, ω)

1 − 1νµBg

χ0,HF−+ (~q, ω)

. (8.21)

For a uniform static magnetic field, one has χ0,HF−+ (0, 0) = νµBg/2, where

the density of states ν should be interpreted as the “density of poles” of thefull Green function, i.e., the density of Hartree-Fock energy levels. Since thisis the same density of states as the one that occurs in the screened interaction,one has

χTF−+(0, 0) = νµBg. (8.22)

We conclude that, according to the Thomas-Fermi approximation, in aninteracting electron gas the electron-electron interactions increase the spinsusceptibility by a factor two compared to the case of the non-interactingelectron gas.25

25The Thomas-Fermi approximation is not quantitatively correct. A more correct treat-ment of the RPA approximation which takes into account the momentum dependence ofthe effective interaction gives

χRPA−+ =

νµBg

2 − (2/3π2)2/3rs ln(1 + (3π2/2)2/3/rs),

see Ex. 8.1.

135

The enhancement of the susceptibility by the interactions follows fromthe fact that the repulsive interaction between the electrons is lowered ifa finite spin density is created. After all, the Pauli principle forbids thatelectrons of the same spin are at the same location, so that they have asmaller electrostatic repulsion. The interaction energy gain is proportionalto the square of the spin density, so that the net energy cost for polarizingthe system is reduced with respect to the case of non-interacting electrons.

If the interaction energy gain is larger than the kinetic energy cost, aspin polarization will form spontaneously. This instability is known as the“Stoner instability”. We see that for the standard case of a parabolic bandthe Stoner instability does not occur. The situation becomes different oncethe precise band structure of the material is taken into account. The nextsection discusses a simple model for this.

8.2 Hubbard model

The approximations of the previous subsection are appropriate for the stan-dard parabolic band of free electrons. In real materials, the electrons occupybands, and the band structure needs to be taken into account for a calculationof the magnetic susceptibility. A simple model to take the band structureinto account is the “Hubbard model”.

For the construction of the Hubbard model, we consider electrons in anenergy band with Bloch energies ε~k described by the Hamiltonian

H0 =∑

~k

ε~kψ†~kψ~k. (8.23)

For such a band, wavefunctions can be represented in the basis of Wannierwavefunctions at lattice site ~ri,

φi(~r) = N−1/2∑

~k

ei~k·~riψ~k(~r), (8.24)

where the summation is restricted to wavevectors in the first Brillouin zoneand N is the number of lattice sites. The Wannier wavefunctions at differentlattice sites are orthogonal.

The Coulomb interaction can be written in terms of the basis of Wannierwavefunctions. The corresponding matrix elements are

Vij,i′j′ =

∫

d~r1d~r2φ∗i (~r1)φ

∗j(~r2)V

eff(~r1 − ~r2)φj′(~r2)φi′(~r1), (8.25)

136

where V eff is the screened Coulomb interaction. The approximation of theHubbard model consists in neglecting all interaction matrix elements exceptfor the matrix element U = Viiii. The justification for this approximation isthat the screened interaction is short range, while the Wannier wavefunctionsare mainly localized at one lattice site. This approximation leads to theinteraction Hamiltonian

H1 = U/2∑

iσσ′

niσniσ′ , (8.26)

where niσ = ψ†iσψiσ is the Wannier number operator and

ψiσ = N−1/2∑

~k

ψ†~kσei~k·~ri (8.27)

is the Wannier annihilation operator.The interaction Hamiltonian can be further simplified if we use n2

iσ = niσ,so that

H1 = U∑

i

ni↑ni↓ + (U/2)∑

i

(ni↑ + ni↓). (8.28)

The second term is nothing but a shift of the chemical potential and will beneglected. Fourier transforming the first term in Eq. (8.28) we then arrive at

H1 =U

N

∑

~k,~k′,~q

ψ†~k,↑ψ~k+~q,↑ψ

†~k′,↓ψ~k′−~q,↓. (8.29)

Let us now calculate the spin susceptibility for the Hubbard model. Thecalculation proceeds along the same lines as that of the previous section. Theonly differences are that, for the Hubbard model, the wavevector summationis restricted to the first Brillouin zone, and that the strength of the interac-tion is set independently by the interaction constant U . The result for thetransverse susceptibility is

χ−+(~q, ω) =χ0,HF−+ (~q, ω)

1 − 2UµBg

χ0,HF−+ (~q, ω)

, (8.30)

where

χ0,HF−+ (~q, ω) =

µBg

2N

∑

~k

nF (εHF~k,↑

− µ) − nF (εHF~k+~q,↓

− µ)

εHF~k+~q,↓

− εHF~k,↑

+ ω + iη. (8.31)

137

For the Hubbard model, the Hartree-Fock energy levels have a very simpleconnection to the bare energy levels ε~k,

εHF~kσ

= ε~k + Un−σ, nσ =1

N

∑

~k

nF (εHF~kσ

)〉. (8.32)

In the absence of a spontaneous magnetization, the levels of electrons withspin up and spin down experience the same uniform shift, which can beincorporated into the chemical potential. In particular, this implies that thesupport of the imaginary part of χ0,HF

−+ and, hence, of the imaginary part ofχTF−+, is the same as that of χ0−+, which is shown in Fig. 20. In the presence

of a spontaneous magnezation, the Hartree-Fock energy levels for electronswith spin up and spin down experience different shifts. We return to thiscase in Sec. 8.5.

As before, the limit of large wavelength and low frequency has χ0,HF−+ (0, 0) =

νµBg/2, where ν is the density of states per spin direction and per latticesite. Hence

χTF−+(0, 0) =

νµBg/2

1 − Uν. (8.33)

We can use these results to find the interaction correction to the Freeenergy for the Hubbard model. Repeating the analysis of Sec. 7.7, we findthe result

F − F0 =

∫ ∞

−∞

dξ

4π

∑

~q

coth(ξ/2T )Im ln(1 − 2Uχ0,HF(~q, ξ)/µBg). (8.34)

This result may be analyzed using the detailed expressions for χ0,HF of Ex.6.2. For low temperatures, the main contribution comes from frequenciesaround ω = 0, and one finds that the specific heat is enhanced close to theStoner instability,

CV ∝ −T ln(1 − Uν). (8.35)

This is to be compared to the usual specific heat of the noninteracting electrongas, which is proportional to νT .

8.3 The Stoner instability

As long as the interaction constant U is small enough, the susceptibilityremains finite. A finite susceptibility implies that it requires a finite value of

138

the magnetic field ~h to create a spin polarization. However, if the interactionstrength increases, the susceptibility increases. If the static susceptibilityχ(~q, 0) diverges, an instability occurs, and a spontaneous magnetization willform.

From our general result (8.30) for the Hubbard model we find that theinstability criterion is26

U = Uc(~q) = 1/χ0,HF(~q, 0). (8.36)

This is the Stoner criterion.For zero wavevector ~q = 0, the Stoner criterion describes the ferromag-

netic instability, which occurs at critical interaction strength Uν = 1. Fora parabolic band, the instability occurs first at ~q = 0, since χ0,HF(~q, 0) is amonotonously decreasing function of |~q|, cf. Ex. 6.2. However, in the presenceof a more complicated band structure, instabilities at finite wavenumbers arealso possible. An instability at finite ~q leads to the formation of a so-called“spin density wave” state, as was first pointed out by Overhauser [Phys. Rev.128, 1437 (1962)]. As an example, we consider the band

ε~k = −2t∑

α=x,y,z

cos kαa, (8.37)

which corresponds to electrons on a simple cubic lattice with lattice constanta and hopping amplitude t, see Ex. 8.7. For this band, there is a competitionbetween the ferromagnetic instability at wavevector ~q = 0 and an antiferro-magnetic instability at wavevector ~q = ~qA = (π/a)(1, 1, 1) at the far cornerof the first Brillouin zone. You verify that one has

χ0,HF(~qA, 0) = −∫ µ

−6t

dξν(ξ)

ξ. (8.38)

For chemical potentials close to the band edges ±6t, the instability occurs atwavevector ~q = 0, while for a chemical potential µ close to the band center atε = 0, the instability occurs at ~q = ~qA. For this simple cubic band structure,

26For finite frequencies, the denominator of Eq. (8.30) is generally complex, so that onecan rule out the existence of zeroes. Physically, one expects the instability to happen atzero frequency, since an instability at nonzero frequency correspons to a time-dependentpolarization, which is damped. Response at nonzero frequencies is studied in a latersection.

139

one has χ0,HF(~qA, 0) → ∞ as µ→ 0, so that the antiferromagnetic instabilityoccurs already for arbitrarily small repulsive interaction!

Note that for interaction strength U > Uc, our expression (8.30) for thespin susceptibility in the Thomas Fermi approximation is, again, finite. How-ever, this result is unphysical, since, for U > Uc the ground state has a spon-taneous spin polarization (or a spin-density wave state). Hence, we concludethat Eq. (8.30) is valid for all U < min~q Uc(~q), whereas a state with sponta-neous magnetization is formed for U > Uc. A description of the ferromagneticstate is given in Sec. 8.5 below.

8.4 Neutron scattering

Neutrons have spin 1/2. The neutron spins interact with the spins of elec-trons in a solid through the magnetic dipole interaction. Hence, the neutronscattering cross section is sensitive to the magnetic structure of a solid.

In order to describe the scattering of slow neutrons, we return to the firstBorn approximation for the neutron scattering cross section, but take theneutron spin into account explicitly,

d2σ

dΩdω=

1

2

∑

f,σf ,σi

k′

k

(

M

2π

)2⟨

|〈~k′σf , f |H ′|~kσi, i〉|2⟩

δ(ω + Ei −Ef ), (8.39)

where σf and σi denote the outgoing and incoming spin states of the neutron.The extra factor 1/2 is added since we average over the possible incomingspin states. For the interaction between an electron and a neutron we takethe first quantization form27

H ′ = V0(~rn − ~r) + V1(~rn − ~r)~sn · ~s, (8.40)

where ~rn and ~sn are the position and spin of the neutron, while ~r and ~s arethe position and spin of the electron. Using the second quantization languagefor the electrons, H ′ can be rewritten as

H ′ =

∫

d~rV0(~rn − ~r)n(~r) + ~sn ·∫

d~rV1(~rn − ~r)~s(~r), (8.41)

27For more details: see L.D. Landau and E.M. Lifschitz, Quantum mechanics, non-

relativistic theory, Addison and Wesley (1958).

140

where n(~r) and ~s(~r) are the second quantization operators for the electron

number density and spin density. Writing ~q = ~k − ~k′, we then find

〈~k′σf , f |H ′|~kσi, i〉 =

∫

d~rnd~re−i~q·~rnV0(~rn − ~r)δσf ,σi

〈f |n(~r)|i〉

+1

2

∑

α

∫

d~rnd~re−i~q·~rnV1(~rn − ~r)(σα)σf σi

〈f |s(~r)|i〉

= V0~q〈f |ρ~q|i〉δσf ,σi+V1~q

2

∑

α

(σα)σf σi〈f |s~qα|i〉, (8.42)

where V0~q and V1~q are the Fourier transforms of V0 and V1, respectively. Next,we write the delta function in Eq. (8.39) as an integration over time, absorbthe factors exp(iEit) and exp(−iEf t) into a time-shift of the operators n and~s, and perform the summation over final states. The result is

d2σ

dΩdω=

(

M

2π

)2k′

2πk(8.43)

×∫

dteiωt

[

|V0~q|2〈n−~q(0)n~q(t)〉 +|V1~q|2

4〈~s−~q(0) · ~s~q(t)〉

]

.

For an isotropic system, the spin-spin correlation function appearing inthe spin-dependent part of the inelastic neutron scattering cross section isan example of a “greater Green function”, see Sec. 2.3. It can be written interms of the spin raising and lowering operators s+ and s−,

〈~s−~q(0) · ~s~q(t)〉 =3

2〈s−,−~q(0)s+,~q(t)〉. (8.44)

Using the general relation (2.46) between Green functions, the Fourier trans-form with respect to time can be written in terms of the imaginary part ofthe retarded spin density correlation function,

∫

dteiωt〈~s−~q(0) · ~s~q(t)〉 =3

2µBg

ImχR−+(~q, ω)

1 − e−~ω/T. (8.45)

The minus-sign in the denominator appears because χ is a correlation func-tion of spin densities, which are even in fermion creation and annihilationoperators.

141

The approach to the instability point leads to a strong enhancement ofthe neutron scattering cross section. For small ~q and ω and a parabolic band,one has (cf. Ex. 6.2)

ImχR−+(~q, ω) =

πµBgω/8qvF

(1 − Uν)2 + (Uπω/4qvF )2. (8.46)

Thus, as the ferromagnetic instability is approached, inelastic neutron scat-tering is strongly enhanced for low frequences. The physical origin of theenhanced neutron scattering rate at low temperatures is the slowing down ofspin relaxation at the approach of the Stoner instability.

8.5 The ferromagnetic state

Beyond the instability point a spontaneous magnetization forms. In princi-ple, the magnetization can point in any direction, although, for each sample,a specific direction will be selected. Here, we assume that the magnetizationpoints in the positive z-direction.

The main difference between the ferromagnetic state and the paramag-netic state is that, in a ferromagnet, the Hartree-Fock energy levels are dif-ferent for up spins and down spins, see Eq. (8.32). For the Hubbard model,the difference between levels for spin up and spin down does not depend onthe wavevector,

∆ = εHF~k↓

− εHF~k↑

= U(n↑ − n↓). (8.47)

The existence of the gap ∆ implies that electrons with spin up have a largerFermi momentum than electrons with spin down, see Fig. 43, which, in turnreflects the fact that there is a net spin polarization. The electrons withspin parallel to the magnetization direction are referred to as “majority elec-trons”, whereas those with opposite to the magnetization direction are called“minority electrons”.

One can calculate ∆ self-consistently from the requirement that the totalnumber of particles does not change upon the formation of the magneticmoment. Together with Eq. (8.32), one can then solve for the energy gap ∆.Nontrivial solutions exist for Uν > 1 only.

The difference between Hartree-Fock energy levels for different spin di-rections leads to a modification of the expression for the transverse spin

142

kFkF

Figure 43: Support of the imaginary part of χ−+ for a ferromagnet and spinwave dispersion.

susceptibility,

χ0,HF−+ (~q, ω) = −µBg

2N

∑

~k

nF (εHF~k,↑

− µ) − nF (εHF~k+~q,↓

− µ)

ε~k − ε~k+~q + ∆ + ω + iη. (8.48)

The support of the imaginary part of χ0,HF and, hence, of χ, is affectedby the appearance of the gap ∆: The imaginary part of χ−+ is nonzeroif there are excitations of pairs of a particle and a hole of opposite spin.Such excitations are called “Stoner excitations”. The existence of the gap ∆implies that Stoner excitations at ~q = 0 require an energy ∆. Zero frequencyStoner excitations require q to be at least equal to the difference betweenthe Fermi momenta of electrons with spin up and spin down. Repeating thearguments of Sec. 6.2, we can find an expression for the support of Imχ−+ fora ferromagnet, somewhat unrealistic case of a parabolic band. For positiveω one finds that, at zero temperature, Imχ−+ is nonzero for

−vF q +∆

~+

q2

2m~< ω < vF q +

∆

~+

q2

2m~. (8.49)

This support is shown in Fig. 44.Substitution into Eq. (8.30) then yields for ~q → 0

χ−+(~q → 0, ω) =(gµB

2

) ∆

Uω. (8.50)

143

q/kF

ω/εF

30 4210

4

2

∆

spin waves

Stonerexcitations

Figure 44: Support of the imaginary part of χ−+ for a ferromagnet and spinwave dispersion.

The pole of χ as ω → 0 corresponds to a uniform rotation of the magne-tization. This collective excitation has zero frequency because no energy isrequired for such a uniform rotation.28 At finite wavevector ~q, the transversespin susceptibility χ−+ has a pole at frequencies ω = Dq2, where D is a pro-portionality constant. Such an excitation is known as a “spin wave”. Spinwaves are virtually undamped for small wavenumbers. At larger wavenum-bers, the spin wave branch enters into the region of the (q, ω) plane whereχ−+ is complex. Then, spin waves can decay by excitation of “Stoner exci-tations”, a pair of a particle and a hole of opposite spin, see Fig. 44.

The fact that spin waves exist at low frequencies if ~q → 0 is guaranteedby the rotation symmetry of the ferromagnetic state. A uniform rotationof all spins does not cost energy, and, hence, a long-wavelength spin wavehas energy proportional to q2. Quite generally, low-lying collective modeswhose existence is guaranteed by a continuous symmetry of the problem arereferred to as “Goldstone modes”.

8.6 The transition to a magnetic insulator

In Sec. 8.2 we considered the interaction term in the Hubbard model as aperturbation. Without interactions, the electrons are delocalized. In ourtreatment, the interaction does not change that; the main effect of interac-tions is to change the spin susceptibility, and, eventually, beyond the Stoner

28Collective modes that do not require energy are known as Goldstone modes.

144

instability, to create a gap ∆ between the self-consistent energy levels ofelectrons with spin up and spin down.

This perturbative picture should remain correct as long as the interactionenergy is small compared to the band width. In this section, the opposite caseis investigated, when the band width is small compared to the interactionstrength U . Doniach and Sondheimer call this regime the “atomic limit” ofthe Hubbard model.

In the atomic limit, the coordinate representation is preferred over themomentum representation. Fourier transforming the non-interacting part ofthe Hamiltonian, Eq. (8.23) above, we find

H0 =∑

ij,σ

t(~ri − ~rj)ψ†iσψjσ, (8.51)

where ψiσ is the annihilation operator for an electron with spin σ in Wannierwavefunction φi(~r). The interaction Hamiltonian is

H1 = U∑

i

ni↑ni↓, (8.52)

which is the first term on the r.h.s. of Eq. (8.28) above. The indices “0” and“1” refer to the non-interacting and interacting parts of the Hamiltonian,respectively. As we discussed above, in the atomic limit the Hamiltonian H1

is, in fact, dominant.The problem of studying the full Hamiltonian H0 + H1 has proven ex-

tremely difficult. Whereas an exact solution has been obtained in one di-mension, in two and three dimensions no more than a few limiting are fullyunderstood. In order to make progress, we now make use of the fact that theinteraction U is much larger than the band width. Hereto, we write

H0 = H00 +H01, (8.53)

whereH00 =

∑

i,σ

t(0)ψ†iσψiσ, H01 =

∑

i6=j,σ

t(~ri − ~rj)ψ†iσψjσ, (8.54)

and consider H01 as a perturbation. (Note that H00 cannot be considered aperturbation, since t(0) determines the position of the band, not the bandwidth.)

145

Below, we are interested in the retarded single-particle Green function,

Gij,σ(t) = −iθ(t)〈[ψiσ(t), ψ†jσ(0)]+〉. (8.55)

Recall that the poles of the single-particle Green function contain informationabout the particle-like excitations of the system. Since the “unperturbed”Hamiltonian H00 +H1 is not quadratic in the fermion creation/annihilationoperators, the diagrammatic perturbation theory we used in the previouschapters cannot be employed to calculate Gij,σ. Instead, we have to calculateGreen functions using the equation of motion approach. An illustration of theequation of motion approach for the calculation of phonon Green functionswas given in Sec. 3.3.2 above.

Let us first calculate the single-particle Green function in the absence ofthe perturbation H01. In that case, one has

∂Gij,σ(t)

∂t= −iδ(t)〈[ψiσ(t), ψ†

jσ(0)]+〉 − iθ(t)〈[∂ψiσ(t)/∂t, ψ†jσ(0)]+〉. (8.56)

Using the equation of motion for the annihilation operator ψiσ(t),

∂ψiσ(t)

∂t= −it(0)ψiσ(t) − iUni,−σψiσ(t), (8.57)

we find(

−i ∂∂t

+ t(0)

)

Gij,σ(t) = −δ(t)δij (8.58)

− (−i)Uθ(t)〈[ni,−σ(t)ψiσ(t), ψ†jσ(0)]+〉.

Repeating the same procedure for the Green function that appeared at ther.h.s. of Eq. (8.58), we find

∂

∂t

[

−iθ(t)〈[ni,−σ(t)ψiσ(t), ψ†jσ(0)]+〉

]

=

= −iδ(t)〈[ni,−σ(t)ψiσ(t), ψ†jσ(0)]+〉 − t(0)θ(t)〈[ni,−σ(t)ψiσ(t), ψ†

jσ(0)]+〉− Uθ(t)〈[ni,−σ(t)2ψiσ(t), ψ†

jσ(0)]+〉= −iδ(t)δij〈ni,−σ〉 − i(t(0) + U)

[

−iθ(t)〈[ni,−σ(t)ψiσ(t), ψ†jσ(0)]+〉

]

.

After Fourier transforming and substitution into Eq. (8.58), we find

Gij,σ(ε) =(1 − 〈ni,−σ〉δijε− t(0) + iη

+〈ni,−σ〉δij

ε− t(0) − U + iη. (8.59)

146

In a paramagnetic state, the expectation value 〈ni,−σ〉 = n/2, where n is theelectron number density.

The result (8.59) has a very simple interpretation: If an electron of spin−σ is present at a site, an added electron with spin σ has energy t(0) + U ,so that its retarded Green function is Gii,σ(ε) = 1/(ε− t(0) − U + iη). If noelectron of spin −σ is present, the Green function is Gii,σ(ε) = 1/(ε− t(0) +iη). The probability that an electron of spin −σ is present is 〈ni,−σ〉, henceEq. (8.59).

Once the hopping term H01 is included, the equation of motion for theGreen function can no longer be truncated. However, following Hubbard,one can make the following “guess” for the result. It should be pointed out,however, that this guess has no formal justification; nevertheless, as you’llsee below, it conveys a compelling picture of how the system behaves. Thesingle-electron Green function (8.59) can be interpreted as the inverse ofε−Σi, where Σi is a measure of the energy of an electron at site i, includingthe interaction modifications. Without interactions, the Green function ofthe electron would be G~k,σ = 1/(ε− ε~k + iη), where

ε~k = t(0) +∑

i6=j

e−i~k·(~ri−~rj)t(~ri − ~rj). (8.60)

Note that we switched to the momentum representation. With interactions,we replace ε− t(0) by (Gii,σ(ε))−1. Hence, we find

GH~kσ

=1

(Giiσ(ε))−1 − (ε~k − t(0)) + iη. (8.61)

Let us now consider the positions of the poles of the Green function GH~kσ

.They are at the solutions of the equation

(ε− t(0))(ε− t(0) − U) = (ε~k − t(0))(ε− t(0) − U(1 − n/2)). (8.62)

You verify that for n→ 0 or U → 0 one recovers the Bloch band ε = ε~k. Youalso verify that for the limit of very small bandwidth max |ε~k − t(0)| ≪ Uthe solutions approach the on-site energies t(0) and t(0) +U of the Hubbardmodel in the atomic limit. The effect of the inter-site hopping is to changethe N -fold degenerate solutions ε = t(0) and ε = t(0)+U into narrow bandsaround t(0) and t(0) + U . For max |ε~k − t(0)| ≪ U these bands are given by

ε = t(0)(n/2) + ε~k(1 − n/2), ε = t(0)(1 − n/2) + U + ε~k(n/2). (8.63)

147

t

t +U0

0k

ε

Figure 45: Schematic picture of the two energy bands for the Hubbard model forthe case that the interaction U is much larger than the bandwidth.

A schematic picture of these bands is shown in Fig. 45.In general, the two bands may overlap. An exception is the case that n

is close to 1. The case n = 1 is known as “half filling”, since it amounts tohalf of the maximum possible number of electrons in the system. At n = 1,the two bands are always separated, although one may question whether thisresult is realistic for small U . Since the lowest band is fully filled at n = 1,there is an excitation gap, and the system becomes an insulator. In this case,the excitation gap is referred to as “Coulomb gap” and the insulating stateis known as a “Mott insulator”.

8.7 Heisenberg model

The Heisenberg model gives a description for an insulating magnet. Themagnetization arises from the alignment of local magnetic moments, notfrom the shift of Fermi levels for “majority” and “minority” electrons.29

The Heisenberg model is described by the Hamiltonian

HHeis =∑

ij nn

J~si · ~sj

=∑

ij nn

J(sxi s

xj + sy

i syj + sz

i szj ) (8.64)

=∑

ij nn

J

(

1

2s+

i s−j +

1

2s−i s

+j + sz

i szj

)

. (8.65)

29Parts of this section are based on lecture notes by W. van Saarloos, Leiden, 1996.

148

In writing (8.65), we consider a lattice whose sites are labeled with the indexi, the summation is over nearest neighbor pairs i and j on this lattice only,and ~si denotes the spin 1/2 operator at site i. In principle, one can includenext-nearest neighbor interactions, etc., in the Hamiltonian (8.65). The spinoperators commute for different sites, and obey the usual spin commutationrelations on the same sites. Thus, in terms of the spin raising and loweringoperators,

[s−i , szj ] = δijs

−i , [s+

i , szj ] = −δijs+

i [s+i , s

−j ] = 2δijs

zi (8.66)

[sxi , s

yj ] = iδijs

zi , and cyclic. (8.67)

We will take the quantum Heisenberg model as a starting point here; theHeisenberg model with antiferromagnetic coupling (J > 0) can be derivedfrom the Hubbard model at half filling, with nearest-neighbor hopping andlarge positive U , see exercise 8.7.

In the literature, sometimes an anisotropic version of the Heisenbergmodel (8.65) is studied,

HanisHeis =

∑

ij nn

[

J⊥(sxi s

xj + sy

i syj ) + Jzs

zi s

zj

]

=∑

ij nn

[

J⊥2

(s+i s

−j + s−i s

+j ) + Jzs

zi s

zj

]

. (8.68)

Clearly, in this case the model is not invariant under arbitrary rotations ofthe spins, but is invariant for rotations of the spins about the z-axis.30

Despite its simplicity, the Heisenberg model is difficult to study analyti-cally. One reason that is of interest to us, is that Wick’s theorem does nothold for the Heisenberg model. Hence, we cannot rely on the diagrammaticperturbation theory.

An important question of interest for the Heisenberg model is when andhow a magnetic phase is formed. This question is in the realm of the theory

30In passing, we note that for J⊥ = 0, HanisHeis reduces to the celebrated Ising model, which

plays a central role in the theory of critical phenomena. Unlike the Heisenberg model, theIsing model does not have a continuous symmetry, and, as a result, the Ising model hasno Goldstone modes. On a bipartite lattice, the antiferromagnetic Ising model at zeromagnetic field is equivalent to the ferromagnetic Ising model. This result does not extendto the Heisenberg model; the ferromagnetic and antiferromagnetic Heisenberg models arefundamentally different.

149

of phase transitions and critical phenomena, and we will not discuss it here.Instead, we will assume that a low-temperature magnetic phase exists, andaim at a description of the possible excitations of the system in the low-temperature magnetic phase and the high-temperature paramagnetic phase.We will carry out this program in detail for the ferromagnetic Heisenbergmodel, for which the low-temperature magnetic phase is the fully polarizedphase. For the antiferromagnetic Heisenberg model, this question is muchmore difficult, as the precise nature of the antiferromagnetic phase is notknown.

Let us now turn to the isotropic ferromagnetic (J < 0) Heisenberg model(8.65) in absence of a field to introduce the concept of spin-waves in moredetail. The case of the ferromagnetic Heisenberg model is particularly simple— but somewhat a-typical — in that the order parameter associated withthe broken symmetry, the total magnetization, is also a conserved quantity.(This is sometimes referred to as an exact spontaneously broken symmetry.)

Since the total spin ~S commutes with HHeis, we can diagonalize the Hamil-tonian in each subspace of eigenvalues of the operator Sz. Let |+〉 denotethe state with all the spins pointing up, so that Sz|+〉 = (N/2)|+〉, whereN is the total number of spins. The state |+〉 is the ground state of HHeis ifJ < 0, although it is not the only possible ground state. In fact, all statesobtained from |+〉 by multiple operation of the total spin lowering operator,

S− =∑

i

s−i (8.69)

is a ground state of the Heisenberg Hamiltonian as well. For a hypercubiclattice in d dimensions, the ground state energy is E0 = dNJ/4.

Since the ground state of the ferromagnetic Heisenberg model is so simple,some low-lying excitations, the spin waves — can be determined exactly. Aspin-wave state with wavevector ~q is created by the operator S−

~q operatingon the fully polarized state |+〉, with

S−~q = N−1/2

∑

i

ei~q·~ris−i . (8.70)

You verify that a state with a single spin wave is an exact eigenstate of theHeisenberg Hamiltonian HHeis, at energy E0 + ~ω~q, where E0 is the groundstate energy and

~ω~q =|J |2

∑

~δ

(1 − ei~q·~δ), (8.71)

150

u

e

u

e

u

e

u

e

u

e

u

e

u

e

u

u

e

: A: B

Figure 46: An example of a bipartite lattice. Every neighbor of the A sublatticeis a site on the B sublattice, and vice versa.

where the vector ~δ is summed over all nearest-neighbor directions in thelattice. For small ~q, the energy ω~q increases proportional to q2. Note that,for ~q → 0, the state S−

~q |+〉 = |~q〉 approaches the uniformly rotated stateS−|+〉 continuously, and therefore we expect the excitation energy ~ω~q ofthis mode to vanish continuously as ~q → 0. This again illustrates that theappearance of such “low lying modes” or “Goldstone modes” for ~q small is ageneral phenomenon when a continuous symmetry is broken.

The spin-wave state |~q〉 is a collective mode; it describes a delocalizedstate with many spin excitations involved. For small ~q, it describes a slowrotation of the spins about the z-axis. To make this more explicity, one cancalculate the expectation value of the transverse spin correlation in the state|~q〉,

〈~q|sxi s

xj + sy

i syj |~q〉 =

1

Ncos[~q · (~i−~j)]. (8.72)

Spin waves do interact: the two-spin wave state S−~q1S−

~q2|+〉 is not an exact

eigenstate of the Hamitonian (8.65). On the other hand, spin-wave interac-tions are small if the number of excited spin wave modes is small. Then,the spin waves can be considered independent boson modes, their distribu-tion being given by the Bose-Einstein distribution function. The picture ofindependent spin wave modes leads to the Bloch T 3/2 law for the tempera-ture dependence of the low-temperature magnetization in three dimensions,〈Sz(T )〉 − 〈Sz(T = 0)〉 ∝ T 3/2.31 For one and two dimensions, the density ofstates of low-energy spin waves is higher than in three dimensions; there isno regime of a small number of excited spin waves, leading to a breakdownof the ferromagnetic state in one and two dimensions.

31See, e.g., Solid State Physics, by N. W. Ashcroft and N. D. Mermin, Saunders (1976),and the discussion below.

151

Within the Green function approach, these conclusions can be arrived atif we look at the retarded transverse spin-spin correlation function,

RRij(t) = −iθ(t)〈[s−i (t), s+

j (0)]−〉, (8.73)

which is the Heisenberg-model equivalent of the transverse spin susceptibilityχ−+ we studied previously. (Note that the correlation function contains acommutator of spin operators, since spin operators satisfy commutation re-lations.) The function RR

ij(t) describes the response of the spin polarizationto a magnetic field, where both the response and the field are perpendicularto the z axis, which is presumed to be the direction of spontaneous magne-tization. The poles of the correlation function corresponds to the collectivemodes. Further, the spin-spin correlation function is related to the averagemagnetization through the relation

limt↓0

RRij(t) = 2i〈sz〉δij. (8.74)

In order to calculate the transverse spin-spin correlation function, weconsider its equation of motion

∂

∂tRR

ij(t) = 2iδ(t)〈szi 〉 − θ(t)

∑

l

〈Jli[s−l (t)sz

i (t) − szl (t)s

−i (t), s+

j (0)]−〉, (8.75)

where we defined Jli = J if l and i are nearest neighbor sites and Jli = 0otherwise. In order to close this equation of motion, we replace the operatorsz

i (t) by the average 〈sz〉 ≡ 〈szi 〉 in the ferromagnetic state,32

∂

∂tRR

ij(t) = 2iδ(t)〈szi 〉 − i〈sz〉

∑

l

Jli(RRlj(t) − RR

ij(t)). (8.76)

The justification for this approximation is that, in the ferromagnetic state,fluctuations of sz

i around its average are small. Moreover, since Eq. (8.76)contains a sum over nearest neighbors, only the sum of sz

i over nearest neigh-bors is important. In high dimensions, the summation over nearest neighborsleads to an even further reduction of fluctuations, and one expects the ap-proximation sz

i ≈ 〈sz〉 to continue to hold outside the almost fully polarized

32This approximation is the equivalent of the Weiss molecular field approximation, see,e.g, Statistical and Thermal Physics, F. Reif, Mc Graw Hill, 1965.

152

ferromagnetic state. Solving Eq. (8.76) by means of a Fourier transform, wefind

R~k(ω) = − 2〈sz〉ω + ω~k + iη

, (8.77)

whereω~k = 〈sz〉J

∑

~δ

(1 − e−i~k·~δ), (8.78)

the summation being over all nearest-neighbor vectors ~δ. Note that thissolution satisfies the relation (8.74).

For zero temperature, 〈sz〉 = s, and we recover the spin-wave dispersionrelation (8.71). For finite temperatures, the spin-wave frequencies dependon the average spin polarization 〈sz〉, which, in turn, is a function of thetemperature T . In order to determine 〈sz〉, we use the “greater” correlationfunction,

R>ij(t) = −i〈s−i (t)s+

j (0)〉, (8.79)

for which one has

R>ii (0) = −is(s + 1) + i〈sz(sz + 1)〉. (8.80)

In keeping with the approximations we made previously, we neglect fluctua-tions of sz and set

R>ii (0) ≈ −is(s + 1) + i〈sz〉(〈sz〉 + 1). (8.81)

We calculate R>ij(t) from the retarded correlation function (8.77) using the

general relation (2.46),

R>~k(ω) = 2i

ImRR~k(ω)

1 − e−~ω/T

= 4πi〈sz〉δ(ω + ω~k)

1 − e−~ωT. (8.82)

Performing the inverse Fourier transform, we then find

R>ii (0) = −2i

N

∑

~k

1

e~ω~k/T − 1

, (8.83)

from which we derive the self-consistency equation

s(s+ 1) − 〈sz〉(〈sz〉 + 1) = 〈sz〉 2

N

∑

~k

1

e~ω~k/T − 1

. (8.84)

153

In the limit T → 0 the r.h.s. vanishes in three dimensions and above, recov-ering the solution 〈sz〉 = s at T = 0. For one and two spatial dimensions,the r.h.s. does not vanish as T → 0, implying the breakdown of the ferro-magnetic state by spin wave generation, as we discussed previously. For lowtemperatures and three dimensions, one find 〈sz(T = 0)〉 − 〈sz(T )〉 ∝ T 3/2,which is the Bloch T 3/2 law mentioned previously.

The self-consistent relation could be used to find the Curie tempera-ture, the highest temperature at which the spontaneous magnetization exists.However, close to the Curie temperature, spin fluctuations are abundant, andthe approximation sz

i → 〈sz〉 we made in the derivation of Eq. (8.84) is notjustified in systems with low dimensionality.

We now turn to the antiferromagnetic Heisenberg model J > 0. Clas-sically, an antiferromagnetic state is possible only if the lattice is bipartite:it can be separated in sublattices A and B such that all nearest neighborsof a site in lattice A belong to lattice B and vice versa, see Fig. 46. If thelattice is not bipartite, the antiferromagnet is “frustrated”, and a completelydifferent physical picture arises, which we do not discuss here. The classicalantiferromagnetic ground state is a state in which spins on one sublatticepoint “up”, whereas spins on the other sublattice point “down”. This stateis known as the “Neel state”. Intuitively, we expect that the Neel state isthe ground state of the quantum antiferromagnetic Heisenberg model as well.However, the Neel state is not an exact eigenstate of HHeis!

33 Neverthelessthe true ground state has strong antiferromagnetic correlations reminiscentof this state.

Even though the classical Neel state is not the exact ground state, it doessuggest a definition of the antiferromagnetic “order parameter”, which takesthe role of the magnetization for the ferromagnetic Heisenberg model. Letus consider a hypercubic lattice with lattice spacing a in d dimensions andlet a reference spin on sublattice A point up; then the so called “staggeredmagnetization” is defined as

MAFi =

〈szi 〉 if ~ri in sublattice A,

−〈szi 〉 if ~ri in sublattice B

. (8.85)

In the classical Neel state, this order parameter is 1/2 for all sites.This choice of the order parameter may be justified for a Heisenberg model

with spin s≫ 1/2. The larger the spin s is, the more “classical” the behavior

33The classical Neel state is the ground state of HanisHeis with Jz > 0 and J⊥ = 0, but not

of the model with J⊥ 6= 0.

154

of the model becomes, as the commutation relations (and hence fluctuations)become less important. In the classical limit, we can represent the spins byvectors on a sphere of radius s. In this s → ∞ classical limit, the groundstate of the antiferromagnetic Heisenberg model consists of vectors pointingin opposite directions on the two sublattices. In this limit, (8.85) is a goodorder parameter, whose value approaches s for s large.

In recent years, there has been a lot of research on the properties of quan-tum antiferromagnets, in particular in d = 2. The upsurge of interest in thisproblem was stimulated to a large extent by the discovery of high temper-ature superconductors, which are obtained by doping compounds which areinsulating quantum antiferromagnets. The consensus, basis on many largescale numerical simulations, is that the ground state of the two dimensionalHeisenberg model does have antiferromagnetic order, but that the order pa-rameter (8.85) is reduced from the classical Neel value 1/2 to about 0.34.

In the ferromagnetic case, the order parameter was a conserved quantityand as a result a single spin wave state |~q〉 was an exact eigenstate of theHamiltonian. In the present case, the staggered order parameter (8.85) isnot a conserved quantity, and the spin wave analysis can not be performedexactly (after all, the ground state is not known exactly either!), so we haveto resort to an approximate treatment. In order to understand the natureof the approximation better, it will turn out to be useful to look at theHeisenberg model for the case of large spin s. Since we expect the groundstate to get closer to the classical Neel state for large s, increasing s gives usa way to get a controlled approximation.

As in the case of a ferromagnet, to analyze spin waves we need to considerstates where one spin is flipped. Since the ground state is not explicitlyknown, it is more convenient to use (in the Heisenberg picture) the equationof motion for the spin flip operator. Using the commutation relations (8.66)(which are also valid for arbitrary spin s), we get

∂s−~i∂t

=1

i~[s−~i ,HHeis] =

J

i~

∑

~δ

(

s−~i sz~i+~δ

− s−~i+~δsz~i

)

. (8.86)

As before, the sum over ~δ is a sum over the nearest neighbor vectors on thequadratic (d = 2) or cubic (d = 3) lattice.

For large s, spin waves consist of minor distortions of the Neel state,with the z-component of the spin slightly reduced from its maximal value+s on the A sublattice, and slightly increased from its minimal value −s

155

on the B sublattice. To a good approximation, we can then write in (8.86)sz~i|ground state〉 ≈ ±s|ground state〉 depending on whether ~ri is on the A or

B sublattice. Using the notation

|~i〉A = s−~i |ground state〉 for ~ri in sublattice A, (8.87)

|~i〉B = s+~i|ground state〉 for ~ri in sublattice B, (8.88)

we then get from (8.86)

∂

∂t|~i〉A =

J

i~

−2ds|~i〉A − s∑

~δ

|~i+ ~δ〉B

,

∂

∂t|~i〉B =

J

i~

2ds|~i〉B + s∑

~δ

|~i+ ~δ〉A

. (8.89)

Since the two sublattices are distinct, we now have two coupled equations,instead of one in the case of a ferromagnet. The equations for the spin wavescan be solved by introducing Fourier transforms,

|~q〉A = N−1/2A

∑

~i∈A

ei~q·~i|~i〉A, |~q〉B = N−1/2B

∑

~i∈B

ei~q·~i|~i〉B. (8.90)

Writing the temporal evolution of these modes as e−iω~qt, we then get from(8.89) the dispersion relation

∣

∣

∣

∣

~ω~q + 2dsJ 2dsJγ~q

−2dsJγ~q ~ω~q − 2dsJ

∣

∣

∣

∣

= 0, γ~q =1

2d

∑

~δ

ei~q·~δ. (8.91)

Solution of Eq. (8.91) yields

~ω~q = 2dsJ√

1 − γ2~q . (8.92)

By expanding γ~q for small q, we find that long wavelength antiferromagneticspin waves have a linear dispersion

ωq ≈ 2sJq√d, q ≪ 1. (8.93)

This linear dispersion is reminiscent of that of acoustic phonons in a crystal.The difference between the quadratic dispersion of ferromagnetic spin waves

156

and the linear dispersion of the antiferromagnetic spin waves — the Gold-stone modes of the antiferromagnet — can be traced to the fact that in thefirst case, the order parameter itself is conserved. This is rather exceptional,and indeed Goldstone modes usually have a linear dispersion.

Our earlier discussion implies that (8.93) is a good approximation forlarge s. Although the estimate of the spin wave velocity c = sJ

√2d/~ may

not be that accurate for spin 1/2, the linearity of the antiferromagnetic spinwave dispersion is a robust feature that is not affected by the approximationsmade.

In closing this section, we note that the present analysis may be extendedto a systematic 1/s expansion for quantities like the spin wave velocity andthe order parameter. Quantum mechanical fluctuations are more and moresuppressed as s increases. Intuitively, this becomes clear from the fact thatfor s = 1/2, the operators s+

i or s−i lead to a local reversion of the spin, andhence tends to locally make the order parameter of the wrong sign. For larges, these operators just reduce the z-component of the spin slightly — theydo not lead to a complete spin reversal.

157

8.8 Exercises

Exercise 8.1: Spin susceptibility

The Thomas-Fermi approximation (8.22) for the spin susceptibility in aninteracting electron gas is not correct. It neglects the structure of the inter-action on the scale of the Fermi wavelength. A better approximation is madewhen the full Thomas-Fermi screened interaction is used,

V RPA~q (iωn) =

e2

ǫ0(q2 + k2s)

if ~q, ωn → 0,

see Eq. (7.27). Hence, the momentum-dependence of the screened interactionis taken into account, but retardation effects are neglected.

(a) Argue that, in the limit ~q → 0, the renormalized vertex Γ(~k, iωm;~k +~q, iωm + iΩn) of Eq. (8.15) does not depend on the direction of the

wavevector ~k or on the energy ωm.

(b) Since Γ(~k, iωm;~k + ~q, iωm + iΩn) of Eq. (8.15) does not depend on

the direction of the wavevector ~k, the effective interaction V eff~k−vk′

in

the vertex equation (8.15) may be integrated over the directions of ~k′.Show that the integrated interaction is

V eff~k,~k′ =

e2

4ǫ0kk′lnk2

s + (k + k′)2

k2s + (k − k′)2

. (8.94)

(c) Argue that the renormalized vertex is determined by the effective in-teraction at the fermi level. From here, derive the result

χRPA−+ =

νµBg

2 − (2/3π2)2/3rs ln(1 + (3π2/2)2/3/rs). (8.95)

(d) Can you explain this result qualitatively in the limits rs ≪ 1− andrs ≫ 1?

158

Exercise 8.2: Specific heat

Verify Eq. (8.35) for the low-temperature specific heat of the Hubbard modelclose to the Stoner instability. For your verification is is sufficient if you findthe order of magnitude of the proportionality constant in Eq. (8.35). You donot need to find precise numerical coefficients.

Exercise 8.3: Ferromagnet with parabolic dispersion relation

In this exercise we consider the Hubbard model, with a parabolic dispersionof the band energies ε~k. This model is somewhat unrealistic, but it allows usto calculate many quantities in closed form.

(a) Using the Hartree-Fock energy levels, show that the spin polarization

ζ =n↑ − n↓

n↑ + n↓(8.96)

obeys the equation

U( n

9π4

)1/3 2mζ

~2= (1 + ζ)2/3 − (1 − ζ)2/3. (8.97)

Here n = n↑ + n↓ is the total electron density. Find a relation betweenζ and the energy gap ∆.

(b) Analyze Eq. (8.96) and find for what values of the Hubbard interactionstrength U a spontaneous spin polarization is formed.

(c) If the Hubbard interaction strength U is further increased, a full polar-ization is achieved, ζ = ±1. This situation is known as a half metal.

(d) Calculate the transverse spin susceptibility χ−+ for low frequencies andlong wavelengths. What is the spin-wave dispersion that you find fromyour answer?

159

Σ =P + + + ...

Figure 47: Singular contribution to the electron self energy at the Stoner insta-bility.

Exercise 8.4: Self-energy correction

Close to the Stoner instability point there are important corrections to theself energy that are not included in the Hartree-Fock approximation. One ofthe leading corrections is shown in Fig. 47.

(a) Calculate the contribution to the self energy shown in Fig. 47 for theHubbard model.

(b) Discuss how the self-energy correction you calculated under (a) affectsthe electron effective mass m∗ and the density of states at the Fermilevel.

Exercise 8.5: Coulomb gap

In this exercise, we consider the ansatz (8.59) for the single particle Greenfunction of the Hubbard model. The Green function not only contains infor-mation about the possible single-particle excitations through its poles, butalso through the spectral weight of each pole.

(a) Find the spectral weights for the case of large U , U ≫ max |ε~k − t(0)|,for which you can use Eqs. (8.63) for the poles of the Green function.

(b) Do the same for the case of small U , U ≪ max |ε~k − t(0)|. Formulateyour answer for the case n≪ 1 and n = 1 separately.

160

Exercise 8.6: Bloch T 3/2 law

Derive the Bloch T 3/2 law for the magnetization of a three-dimensional fer-romagnet from Eq. (8.84).

Exercise 8.7: Antiferromagnetic Heisenberg model from large-U Hubbard

model at half filling

In this exercise we consider the Hubbard model with nearest neighbor hop-ping,

HHub = −t∑

ij nn,σ

(

ψ†iσψjσ + ψ†

jσψiσ

)

+ U∑

i

ni↑ni↓. (8.98)

(a) When U = 0, the Hubbard model describes non-interacting electrons,and we can explicitly determine the single electron energies ε~k. Showthat these are given by Eq. (8.37) above.

Because of the Pauli principle, there can not be more than two electrons persite. Let us from now on consider half filling, i.e. the case where there isone electron per site. Moreover, we consider the limit of large positive U(U ≫ t). Then it is energetically very unfavorable to have two electrons onthe same site — in other words the low energy sector of the Hilbert space isthe one where all sites are occupied by one electron. Our strategy now willbe to project onto this part of Hilbert space of singly occupied sites.

What will the interactions look like? Well, the operator ni↑ni↑ only countswhether a site is doubly occupied, therefore it does not depend on the abso-lute spin direction. Therefore, the effective spin interaction will only dependon the relative spin orientation, in other words, it will be of the Heisenbergtype ~si · ~sj .

Furthermore the interaction will be antiferromagnetic. Physically, thiscan be seen as follows. Virtual excitations to states in which one site isdoubly occupied involve the hop of an electron to a neighboring site andback (see the figure); in the intermediate state, there is an extra energy Uassociated with the double occupancy.

161

6

?

s s

(a)initial configuration

→ hop t

6

?

s s

(b)

virtual state

→ hop t

6

?

s s

(c)

final state

6 6s s

hopping blocked

(b) Argue that, with these considerations, the large-U Hubbard model athalf filling reduces to an antiferromagnetic Heisenberg model with in-teraction J = 2t2/U .

The range over which the antiferromagnetic phase exists as a functionof the ratio U/t and doping of the Hubbard model, is still under activeinvestigation.

162

9 Superconductivity

Just like the presence of the electron gas modifies the electron-electron inter-action, and leads to an effective, or “screened” interaction, the presence ofthe ionic lattice and the interaction between electrons and phonons causes amodification of the electron-electron interaction. This phonon-modified in-teraction is attractive, and is the root cause for superconductivity. In thischapter we will briefly review the origin of the attractive interaction, theinstability that leads to the superconducting state, and the fundaments of aGreen function description of superconductors.

9.1 Phonon-mediated electron-electron interaction

For a discussion of the phonon-mediated electron-electron interaction we firstneed to describe the interaction between electrons and phonons.

Starting point is the potential the electrons feel from the lattice ions,

Hlattice =

∫

d~rn(~r)

N∑

j=1

Vion(~r − ~xj), (9.1)

where the vector ~xj runs over the positions of all ions in the lattice and nis the electron number density. Expanding Hlattice around the equilibriumpositions ~xj = ~rj , j = 1, . . . , N , we have

Hlattice =

∫

d~rn(~r)N∑

j=1

Vion(~r − ~rj) −∫

d~rn(~r)N∑

j=1

~uj · ~∇~rVion(~r − ~rj). (9.2)

Here ~uj = ~xj − ~rj is the displacement of the jth lattice ion. The first termrepresents a static potential and gives rise to the electronic band structure.The second term depends on the actual ionic positions. It is the second termthat leads to the electron-phonon interaction.

Fourier transforming the second term of Eq. (9.2), we find

Hep = − i

V

∑

~kσ

∑

~qλ

(√

N~

2Mω~qλ

~vλ~q · ~qV ie

~q

)

ψ†~k+~q,σ

ψ~k,σ(bλ~q + bλ†−~q), (9.3)

where ψ~kσ is the annihilation operator for an electron with wavevector ~kand spin σ, bλ~q the annihilation operator for a phonon with momentum ~q

163

and polarization ~vλ~q . In the derivation of Eq. (9.3) we used Eq. (3.21) to

express the displacement ~uj in terms of the phonon creation and annihilation

operators bλ†~q and bλ~q .At low temperatures, the phase space available for “umklapp processes”,

corresponding to a momentum transfer ~q outside the first Brillouin zone, issmall. Moreover, the weight of umklapp processes is suppressed because ofthe q-dependence of the potential V ie

~q for large q. Hence, we will restrict thesummation over ~q to the first Brillouin zone.

In an isotropic material, the polarization vector ~vλ~q is either perpendicular

or parallel to ~q. The corresponding phonon states are labeled “transverse”and “longitudinal”, respectively. From Eq. (9.3), we conclude that only lon-gitudinal phonons interact with electrons. If we want to simplify Eq. (9.3) tothe case that we keep longitudinal phonons only, we must remind ourselvesthat in our original formulation of the phonon modes, we have chosen thesame polarization vectors for opposite wavevectors, ~vλ

~q = ~vλ−~q. Hence, if the

longitudinal polarization ~v~q points along ~q, ~v−~q points opposite to −~q. We re-solve this problem by defining an arbitrary function sign (~q), which can takethe values 1 or −1 and which is such that sign (~q) = −sign (−~q). Keepinglongitudinal phonons only, the expression for the electron-phonon interactionthen simplifies to

Hep = − i

V

∑

~kσ

∑

~q

(

V ie~q q sign (~q)

√

N~

2Mω~q

)

ψ†~k+~q,σ

ψ~k,σ(b~q + b†−~q). (9.4)

where the creation and annihilation operators b† and b refer to longitudinalphonons only.

The calculation of Green functions for a system of electrons and phononsproceeds in a way that is very similar to what we have seen in the previouschapters. Since our discussion of phonon Green functions there focused onthe Green functions for the displacements rather than the phonon creationand annihilation operators, we rewrite the electron-phonon Hamiltonian inthe form

Hep =1

V

∑

~kσ

∑

~q

g~qψ†~k+~q,σ

ψ~k,σu~q, (9.5)

where u~q is a longitudinal displacement,

u~q =~q

q

√

~

2Mω~q(b~q + b†~q), (9.6)

164

and g~q describes the strength of the electron-phonon coupling,

g~q = −iN1/2qV ie~q sign (~q). (9.7)

The properties of lattice vibrations were discussed in chapter 3 we dis-cussed lattice dynamics. Lattice ions perform vibrations around their equi-librium positions, the frequency of the vibrations being determined by thesecond derivative of the interaction between ions. However, this interactionis an “effective interaction”, mediated by the electrons. Without electrons,the potential between lattice ions would be the Coulomb potential. As wehave seen in the electronic case, the long-range Coulomb potential does notallow for phonon-like low-frequency collective modes. Instead, the collectivemodes allowed by the Coulomb interaction are plasma modes, which have afinite frequency even for zero wavevector.

A description of the interaction between electrons and the lattice and thelattice dynamics itself is not straightforward. After all, in such a theory theelectrons interact with lattice vibrations, that themselves exist by virtue ofthe existence of the electrons. A theory of phonon-mediated electron-electroninteractions in which the phonon dispersion is, in turn, governed by theelectron-mediated interaction between the lattice ions is a risky enterprise.

In order to circumvent such a “chicken and egg” problem, it is preferableto start from first principles, and use a description in which the only interac-tion is the Coulomb interaction, just like we did for the case of the electrongas. As a complete “ab-initio” treatment is impossible here — for that we’dhave to find the lattice structure from first principles —, we use a simplifiedmodel for the ionic lattice. That simplified model is the “jellium model” ofEx. 3.1. In the jellium model, the lattice is replaced by a positively chargedion “fluid” (or “jellium”). In such a model, the only difference between theions and the electrons is the different ratios of mass density to charge density:the ion jellium has the same average charge density as the electron gas, butits mass density is a factor ∼ 105 higher.

In our theory we’ll consider the electron-phonon interaction as a pertur-bation. Without electron-phonon interaction, the phonons are the quantizedoscillations of the ion jellium without electron screening. Such oscillations,which are the equivalent of plasma oscillations in the interacting electron gas,have frequency ΩE

Ω2E =

Z2e2N

ǫ0MV=Ze2n

ǫ0M, (9.8)

165

k

k+q

q

Figure 48: Diagrammatic representation of phonon line and of electron-phononinteraction vertex.

where n is the electron density. The frequency does not depend on thewavelength of the phonons, as in the Einstein model of lattice vibrations.Also, in the jellium one has longitudinal phonons only; transverse phonons,which do not correspond to a change in ion density, can not be described ina jellium model. Thus, the phonon Hamiltonian in the jellium model is

Hphonon =∑

~q

~ΩE(b†~qb~q + 1/2). (9.9)

The relevant phonon Green function is defined with respect to the displace-ment operators u~q. It was calculated in Chapter 3,

D~q(τ) = −〈Tτu~q(τ)u−~q(0)〉, D0~q(iΩn) = − 1

M(Ω2n + Ω2

E). (9.10)

The superscript “0” indicates that this phonon Green function is calculatedwithout taking the electron gas into account.

In this first-principle jellium model, the interaction potential betweenelectrons and ions is the Coulomb interaction. We thus set V ie

~q = −Ze2/ǫ0q2

in Eq. (9.7),

g0~q =

Ze2i

ǫ0qN1/2 = i

√

ΩEMe2

ǫ0q2. (9.11)

Diagrammatic rules for a combined electron-phonon system are quite sim-ilar to those of the previous chapters for electronic systems. We denote acorrelation function of displacements with a “springy” line, see Fig. 48. Theelectron-phonon interaction is a vertex where a phonon line meets an incom-ing and outgoing electron line. Energy and momentum are conserved at theelectron-phonon vertex. Each phonon line is represented by minus a phononGreen function, −D0; each electron-phonon vertex has weight g0

~q . Note thatg0−~q = g0∗

~q .

166

−D 0

e−−D (−D ) χ( ) (−D)0

+=

=

Figure 49: Diagrams for the effective phonon Green function D.

The first effect of the electron-phonon interaction is to modify the phononGreen function. Diagrams for the phonon Green function in the randomphase approximation are shown in Fig. 49. In this diagram you recognize theelectron polarizability function χ, which we calculate in the RPA approxi-mation as well, see Eq. (7.33). We find

DRPA~q (iΩn) =

D0~q (iΩn)

1 − χRPA~q (iΩn)D0

~q(iΩn)|g0~q |2/V e2

= − 1

M(Ω2n + ω2

~q ), (9.12)

where

ω2~q =

Ω2E

ǫRPA~q (iΩn)

=Ze2n

ǫRPAǫ0M. (9.13)

Here we wrote the polarizability function in terms of the dielectric responsefunction ǫRPA = (1+V~qχ

RPA/e2)−1, where V~q = e2/ǫ0q2 denotes the electron-

electron interaction. Substituting the Thomas-Fermi approximation (7.38)of the dielectric response function, we find, for small q,

ω~q = vF q

√

Zm

3M, (9.14)

which is the Bohm-Staver expression for the phonon dispersion, see Ex. 3.1.We thus arrive at the satisfactory conclusion that, once the electron-phononinteraction is taken into account, the phonon frequencies are renormalizedfrom Ω to ω~q, where ω~q is proportional to q for large wavelengths. You hadreached the same conclusion in Ex. 3.1, based on macroscopic arguments.

Next, let us consider the electron-phonon interaction. This interaction isalso modified by the presence of the electron gas. Instead of exciting a phonon

167

g0 0 g

e−χ( )g V

+

+

=

=

Figure 50: Diagrams for the effective electron-phonon interaction g~q.

directly, as is described by the electron-phonon Hamiltonian (9.5), an electronmay excite an electron-hole pair, which, in turn excites a phonon. Againusing the random phase approximation, such processes can be described bythe diagrams listed in Fig. 50. We thus find

gRPA~q (iΩn) = g0

~q (1 + V~qχRPA~q /e2)

=g0

~q

ǫRPA(~q, iΩn). (9.15)

Let us now turn to the electron-electron interactions. A diagrammaticexpression for the effective electron-electron interaction is shown in Fig. 51.The left diagram in Fig. 51 corresponds to interactions that do not involve theexcitations of phonons. This is the RPA effective interaction we studied inSec. 7.4. The right diagram contains all processes where a phonon is excited.Notice that the electron-phonon vertices are renormalized vertices of Eq.(9.15) and that the phonon propagator is the full renormalized propagator ofEq. (9.12). Combining everything, we find that the effective electron-electroninteraction is

V eff~q (iΩn) =

V~q

ǫRPA~q (iΩn)

+ gRPA~q (iΩn)gRPA

−~q (iΩn)DRPA~q (iΩn)

=V~q

ǫRPA~q (iΩn)

(

1 −ω2

~q

Ω2n + ω2

~q

)

, (9.16)

where we have made use of the identity

g0~qg

0−~q = Mω2

~qεRPA~q (iΩn)V~q. (9.17)

168

VRPAV eff g D g

+

+=

=

Figure 51: Diagrams for the effective electron-electron interaction V eff~q .

Transforming to real frequencies, we finally obtain the phonon-mediatedeffective electron-electron interaction

V eff~q (ω) =

V~q

ǫRPA~q (ω)

ω2

ω2 − ω2~q

. (9.18)

Using the Thomas-Fermi approximation for the dielectric response function,this simplifies to

V eff, TF~q (ω) =

e2ω2

ǫ0(q2 + k2s)(ω

2 − ω2~q ). (9.19)

In contrast to the RPA screened interaction, which hardly depended on fre-quency as long as ω ≪ ǫF , the inclusion of electron-phonon interactions givesrise to a frequency dependence on a much smaller energy scale: V eff dependson frequency on the scale ω~q, as sketched in Fig. 52. For large frequenciesω~q ≪ ω the effect of the lattice is very small; V eff is very close to V RPA. Theeffect of the lattice is most dramatic, however, for small frequencies, ω ≪ ω~q,where the effective electron-electron interaction becomes attractive, ratherthan repulsive.

What is the effect of the attractive interaction? At first sight, one is in-clined to believe that the effect is small. After all, the interaction is attractiveonly in a very small frequency range, ω . ωD, where the Debije frequencyωD is the typical phonon frequency. Nevertheless, at low temperatures onlyelectrons close to the Fermi level play a role. For these electrons interactionswith small transferred energy are most important. It is precisely for thoseprocesses that the interaction is attractive.

9.2 Cooper instability

What is the effect of an attractive interaction? In order to investigate theeffect of a weak attractive interaction, we consider a cartoon version of the

169

V RPAq

V effq (ω)

ωωq

Figure 52: Sketch of the frequency dependence of the effective electron-electroninteraction.

effective interaction we derived in the previous section.The cartoon version, used since the original formulation of the theory of

superconductivity by Bardeen, Cooper, and Schrieffer, reflects the fact thatthe attractive interaction exists for small energy transfers only, and combinesit with our knowledge that only electrons near the Fermi surface play a role.With this motivation, we look at the interaction Hamiltonian

H1 = − λ

2V

∑

~k1,~k2,~k3,~k4

∑

σ1,σ2

ψ†~k1σ1

ψ†~k2σ2

ψ~k3σ2ψ~k4σ1

× δ~k1+~k2,~k3+~k4θ(~k1)θ(~k2)θ(~k3)θ(~k4), (9.20)

where the function θ(~k) is nonzero only for |ε~k − µ| < ωD and λ is a posi-tive constant. Note that the interaction (9.20) is not the result of a formalmanipulation starting from the effective phonon-mediated electron-electroninteraction. Instead, it is merely a cartoon that is used because of its techni-cal simplicity. In some applications, it is favorably to rewrite the interaction(9.20) in coordinate space. The interaction is local on lenght scales largecompared to vF/ωD,

H1 = −λ2

∑

σ1,σ2

∫

d~rψ†σ1

(~r)ψ†σ2

(~r)ψσ2(~r)ψσ1(~r). (9.21)

If the coordinate representation (9.21) is used, care must be taken that itis, in fact, the Fourier transformation of Eq. (9.20), and, thus, requires a

170

R =

+

− −+

− + ...

p

p nm

m n

k’

−k’+q

k

−k+q

m

m n

m

m n

m

m n

k’

−k’+q

p

p n

m

m n

p

p n

k

−k+q

k’

−k’+q

k’’

−k’’+q

m

m n

p

p n

k

−k+q

k

−k+q

k

−k+q

k’

−k’+q

k’

−k’+q

k

−k+q

k’’

−k’’+q

k’

−k’+q

ω

−ω +Ωω

−ω +Ω

ω

−ω +Ω

ω

−ω +Ω

ω

−ω +Ω

ω

−ω +Ω

ω

−ω +Ω

ω

−ω +Ω

ω

−ω +Ω

ω

−ω +Ω

Figure 53: Ladder diagrams contributing to the Cooper correlator R of Eq. (9.22).

momentum cut-off at momenta of order ωD/vF .34

For reasons that will become clear soon, we look at the correlator

Rσ1σ2,σ3σ4

~k,~k′,~q(τ) = −〈Tτψ−~k+~q,σ1

(τ)ψ~k,σ2(τ)ψ†

−~k′+~q,σ3(0)ψ†

~k′,σ4(0)〉. (9.22)

We calculate R using diagrammatic perturbation theory, taking the effectiveelectron-electron interaction we calculated in the previous section. The di-agrams contribution to R are very similar to the diagrams that contributedto the spin susceptibility in chapter 8. They are shown in Fig. 53. Note thatthere are contributions from the direct interaction and from the exchangeinteraction, which come with an extra minus sign. Summing the geometricseries in Fig. 53, we find

Rσ1σ2,σ3σ4

~k,~k′,~q(iΩn) = (δ~k,~k′δσ1σ3δσ2σ4 − δ~k,~q−~k′δσ1σ4δσ2σ3)

×(

T∑

m

G−~k+~q(iωm + iΩn)G~k(−iωm)

)

+1

Vλ(δσ1σ3δσ2σ4 − δσ1σ4δσ2σ3)

34The exact Fourier transform of Eq. (9.20) has the form

H1 =λ

2

∑

σ1,σ2

∫

d~r

∫

d~r1d~r2d~r3d~r4θ(~r − ~r1)θ(~r − ~r2)θ(~r − ~r3)θ(~r − ~r4)

× ψ†σ1

(~r1)ψ†σ2

(~r2)ψσ2(~r3)ψσ1

(~r4),

where

θ(~r) =1

V

∑

~k

ei~k·~rθ(~k).

171

×(

T∑

m

G−~k+~q(−iωm + iΩn)G~k(iωm)

)

×(

T∑

p

G−~k′+~q(−iωp + iΩn)G~k′(iωp)

)

. (9.23)

Here λ is a renormalized interaction,

λ = λ

1 +T

Vλ∑

~k′′

′∑

m

G~k′′,~k′′(iωm)G−~k′′+~q,−~k′′+~q(−iωm + iΩn)

−1

, (9.24)

where the symbol∑′

~k denotes∑

~k θ(~k)θ(−~k+~q). Performing the Matsubara

summations, we have

T∑

m

G−~k+~q(−iωm + iΩn)G~k(iωm)

= T∑

m

1

[−iωm + iΩn − (ε−~k+~q − µ][iωm − (ε~k − µ)]

= −1

2

tanh[(ε−~k+~q − µ)/2T ] + tanh[(ε~k − µ)/2T ]

ε~k + ε−~k+~q − 2µ− iΩn. (9.25)

We perform the analytical continuation iΩn → ω + iη and consider thelimit ~q → 0, ω → 0. Substituting Eq. (9.25) into the expression (9.24) forthe effective interaction strength, replacing the momentum summation by anintegration over energy, and integrating by parts, we find

1

2V

∑

~k

′ tanh[(ε~k − µ)/2T ]

ε~k − µ= ν

∫

~ωD

−~ωD

dξtanh(ξ/2T )

2ξ(9.26)

= ν tanh~ωD

2Tln

~ωD

2T−∫

~ωD/2T

0

dxln x

cosh2(x).

The second term is convergent and approaches the value −0.81 as T →0, whereas the first term diverges logarithmically. We thus see that thedenominator in Eq. (9.24) diverges if

1 = λν(ln(~ωD/2T ) + 0.81), or T = Tc = 1.14~ωDe−1/λν . (9.27)

172

It is important to realize that this divergence happens for an arbitrarily weakattractive interaction, provided the temperature is sufficiently low.

Of course, a divergence of λ corresponds to a divergence of the correlationfunction R. The reason why we studied the correlation function R was pre-cisely because of this divergence: a divergent correlation function is a signalof an instability. To explain this in more detail, let us describe the systemusing the grand canonical ensemble and calculate the response to a fictitiousperturbation of the Hamiltonian,

H1 =

∫

d~r(∆(~r)ψ†↑(~r)ψ

†↓(~r) + ∆∗(~r)ψ↓(~r)ψ↑(~r)). (9.28)

It is important that we use a grand canonical formulation, since the pertur-bation H1 does not conserve particle number. The response we look for is ofthe quantity

F ∗(~r) = 〈ψ†↑(~r)ψ

†↓(~r)〉. (9.29)

The response of F ∗ to the perturbing field ∆(~r) is easily found using theKubo formula,

F ∗(~r, t) =

∫

dt′∫

d~r′RR(~r − ~r′; t− t′)∆∗(~r′, t′), (9.30)

where

RR(~r − ~r′, t− t′) = −iθ(t− t′)〈[ψ†↑(~r, t)ψ

†↓(~r, t), ψ↓(~r

′, t′)ψ↑(~r′, t′)]−. (9.31)

Here we omitted a contribution to the response that involves a commutator offour creation operators, which is nonzero in the non-interacting ground state.Fourier transforming and switching from the retarded correlation function tothe temperature correlation function, we find

R~q(τ) = − 1

V

∑

~k,~k′

〈Tτψ~k+~q↓(τ)ψ−~k↑(τ)ψ~k′+~q↑(0)ψ−~k′(0)〉

=∑

~k,~k′

R↓↑↑↓~k~k′~q

(τ), (9.32)

where R↓↑↑↓~k~k′~q

is the correlator we defined previously in Eq. (9.22).

At high temperatures T ≫ Tc, the response function R~q is finite, indicat-ing a finite response to the perturbation ∆(~r). Lowering the temperature, we

173

see that the response to the fictitious field ∆(~r) diverges upon approachingthe critical temperature Tc. The divergence signals a phase transition to astate in which F ∗(~r) acquires a nonzero value spontaneously.35 This is thesuperconducting state. This important observation is the basis for the Greenfunction description of the superconducting state of the next section.

The fact that the quantity F ∗ becomes nonzero in the superconductingphase has two consequences: First, the absolute value |F ∗| becomes nonzero,and, second, F ∗ acquires a phase. Upon rethinking this, the second statementmay be more striking than the first one. After all, what determines the phaseof F ∗? For this, it is useful to compare the normal-metal-superconductorphase transition to the normal-metal–ferromagnet transition we studied inthe previous chapter. Upon passing the Stoner instability, the spin polar-ization acquired a finite magnitude, and a certain direction, although theHamiltonian did not contain any preferred direction. The spin-rotationalsymmetry is broken spontaneously upon entering the ferromagnetic phase.In principle, the direction of the magnetization could be fixed by the ad-dition of an infinitesimal magnetic field to the Hamiltonian. In the sameway, the superconducting phase is a “broken symmetry phase”. The quan-tity F ∗ acquires a phase, whereas the Hamiltonian does not contain a termthat determines the phase of F ∗. As in the case of the ferromagnet, thesymmetry may be broken by a small perturbation, which has the form of theperturbation of Eq. (9.28).

9.3 Green’s function description of a superconductor

In this section, we will derive an equation of motion for the superconductorGreen functions. Hereto, we need the equation of motion for the creationand annihilation operators. Using the coordinate representation (9.21) ofthe interaction Hamiltonian, we find

∂

∂τψσ(~r) = −(H0 − µ)ψσ(~r) + λ

[

∑

σ′

ψ†σ′(~r)ψσ′(~r)

]

ψσ(~r)

∂

∂τψ†

σ(~r) = (H∗0 − µ)ψσ(~r) − λψ†

σ(~r)

[

∑

σ′

ψ†σ′(~r)ψσ′(~r)

]

. (9.33)

35If one would insist on a canonical formulation, the quantity that turns nonzero at thenormal-metal–superconductor transition is 〈N + 2|ψ†

~k↑ψ†

~k↓|N〉.

174

Here H∗0 is the complex conjugate of the Hamiltonian H0. Eventually, the

repulsive part of the interaction at higher frequencies can be included intoH0 as a self-consistent Hartree-Fock potential.

Slightly generalizing the arguments of the preceding section, we charac-terize the superconducting state by means of the standard Green functionGσσ′(~r, τ ;~r′τ ′) and by the so-called “anomalous Green function”

Fσσ′(~r, τ ;~r′, τ ′) = 〈Tτψσ(~r, τ)ψσ′(~r′, τ ′)〉,F+

σσ′(~r, τ ;~r′, τ ′) = 〈Tτψ

†σ(~r, τ)ψ†

σ′(~r′, τ ′)〉, (9.34)

where H0 is the electron Hamiltonian without interactions. Note that theabsence of a minus sign in the definition of the anomalous Green functionsF and F+. The anomalous Green functions F and F+ are conjugates in thesense of Ch. 2,

F+σσ′(~r, τ ;~r

′, τ ′) = [Fσ′σ(~r′, τ ′;~r, τ)]∗. (9.35)

Writing down the equation of motion for G, we the find

− ∂

∂τGσσ′(~r, ~r′; τ) = δ(~r − ~r′)δ(τ)δσσ′ + (H0 − µ)Gσσ′(~r, ~r′; τ) (9.36)

+∑

σ′′

λ⟨

Tτψ†σ′′(~r, τ)ψσ′′(~r, τ)ψσ(~r, τ)ψ†

σ′(~r′, 0)⟩

.

The imaginary-time arguments of the creation and annihilation operators inthe last term on the r.h.s. of Eq. (9.37) should be increased by an infinitesimalamount in order to ensure the correct order under the time-ordering operationTτ . Anticipating a description in terms of a self-consistent field (as in theHartree-Fock approximation we discussed in Ch. 7), the average of a productof four creation and annihilation operators on the right hand side of Eq.(9.36) can be written as a product of two pair averages using Wick’s theorem.Application of Wick’s theorem results in three terms. Two of those are theHartree and Fock terms we encountered before in Chapter 7. These canbe absorbed into the Hartree and Fock self consistent potentials, which isincluded in the Hamiltonian H0. The third term involves the anomalousGreen function F ,

⟨

Tτψ†σ′′(~r, τ)ψσ′′(~r, τ)ψσ(~r, τ)ψ†

σ′(~r′, 0)⟩

→ Fσ′′σ(~r, τ + η;~r, τ)

× F+σ′′σ′(~r, τ ;~r

′, 0),

175

where η is a positive infinitesimal. Defining the field ∆σσ′(~r) as36

∆σσ′(~r) = −∆σ′σ(~r) = λFσσ′(~r, τ + η;~r, τ), (9.37)

we arrive at the equation of motion(

− ∂

∂τ− (H0 − µ)

)

Gσσ′(~r, τ ;~r′, τ ′)

+∑

σ′′

∆σσ′′(~r)F+σ′′σ′(~r, τ ;~r

′, τ ′) = δ(~r − ~r′)δ(τ − τ ′). (9.38)

Note that ∆(~r) is antisymmetric in the spin indices. This reflects the factthat the superconducting instability exists for pairs of electrons with zerospin only. Similarly, one derives an equation of motion for the anomalousGreen function,

(

∂

∂τ− (H∗

0 − µ)

)

F+σσ′(~r, τ ;~r

′, τ ′)

−∑

σ′′

∆∗σ′′σ(~r)Gσ′′σ′(~r, τ ;~r′, τ ′) = 0, (9.39)

(

− ∂

∂τ− (H0 − µ)

)

Fσσ′(~r, τ ;~r′, τ ′)

−∑

σ′′

∆σσ′′(~r)Gσ′σ′′(~r′, τ ′;~r, τ) = 0, (9.40)

where∆∗

σ′σ(~r) = λF+σσ′(~r, τ + η;~r, τ). (9.41)

These equations are known as Gorkov’s equations. Note that Eq. (9.39)contains the complex conjugate of the Hamiltonian H0.

It is the presence of the field ∆(~r) that makes the Gorkov equations dif-ferent from the equations for the Green function in a normal metal. The field∆ is related to the energy gap for quasiparticle excitations in a superconduc-tor (see below). It also serves as an “order parameter” that distinguishes

36Strictly speaking, one would need to take the finite range of the attractive interactioninto account. In accordance with the momentum cut off at |ε~k −µ| ≥ ~ωD, the interaction(9.21) has a range of order vF /ωD. However, as we’ll see below, the anomalous Greenfunction F(~r, ~r′) is smooth on the scale vF /ωD, its spatial variations occurring on themuch larger scale vF /Tc only. This implies that the approximation made in setting bothcoordinates equal in the argument of F is justified.

176

the superconducting state from the normal state. In that sense, it plays thesame role as the magnetization in the normal-metal–ferromagnet transition.

Solving these equations by means of a Fourier transform, we find

G~kσ,~k′σ′(iωn) = −(iωn + (ε~k − µ))δ~k~k′δσσ′

ω2n + (ε~k − µ)2 + |∆↑↓|2

F+~kσ,~k′σ′

(iωn) =∆∗

σ′σ

ω2n + (ε~k − µ)2 + |∆↑↓|2

. (9.42)

Self-consistency is achieved upon substitution of the solution for F into thedefinition of ∆,

1 = λT

V

∑

n

∑

~k

1

ω2n + (ε~k − µ)2 + |∆↑↓|2

. (9.43)

Performing the summation over Matsubara frequencies, replacing the sum-mation over momenta ~k by an integration over energies ε~k and remember-ing that only momenta with |ε~k − µ| < ~ωD play a role, we find the self-consistency condition

1 = λν

∫

~ωD

0

dξtanh[

√

ξ2 + |∆↑↓|2/2T ]√

ξ2 + |∆↑↓|2. (9.44)

This is the same equation as the “gap equation” from standard BCS theory.The highest temperature with a nontrivial solution of the self-consistency

equation is the point where the normal-metal–superconductor phase transi-tion occurs. Since ∆↑↓ = 0 at T = Tc, we find that the critical temperaturesatisfies the equation

1 = λν

∫

~ωD

0

dξtanh(ξ/2Tc)

ξ. (9.45)

This is precisely the same equation as we found for the divergence of thecorrelator R in the previous section, see Eq. (9.26) above. Hence

Tc = 1.14~ωDe−1/λν . (9.46)

Quasiparticle excitations of the superconductor correspond to poles ofthe retarded Green functions G and F . Analytical continuation of Eq. (9.42)

177

gives

GR~kσ,~k′σ′(ω) = − (ω + ε~k − µ)δ~k~k′δσσ′

(ε~k − µ)2 + |∆|2 − (ω + iη)2

F+R~kσ,~k′σ′

(ω) =∆∗

σ′σ

(ε~k − µ)2 + |∆|2 − (ω + iη)2. (9.47)

We thus conclude that the quasiparticle energies are

ξ~k = ±√

(ε~k − µ)2 + |∆|2. (9.48)

From here we conclude that the excitation spetrum has a gap of size |∆|.We close this section with a few remarks on the role of a vector poten-

tial ~A(~r) and gauge invariance. A vector potential can be included in theHamiltonian H0 by the replacement

~∇ → ~∇− ieA. (9.49)

In the complex conjugate Hamiltonian H∗0 this corresponds to the replace-

ment ~∇ → ~∇ + ie ~A. Gauge invariance requires that the theory is invariantunder transformations

~A→ ~A+ ~∇χ, (9.50)

where χ is an arbitrary function of the position ~r. Green functions are notgauge-invariant objects. Under the gauge transformation (9.50), the Greenfunctions transform as

G(~r, ~r′) → G(~r, ~r′)eie(χ(~r)−χ(~r′)),

F(~r, ~r′) → F(~r, ~r′)eie(χ(~r)+χ(~r′)), (9.51)

F+(~r, ~r′) → F+(~r, ~r′)e−ie(χ(~r)+χ(~r′)).

In particular, this implies that the field ∆(~r) transforms as

∆(~r) → ∆(~r)e2ieχ(~r). (9.52)

The Gorkov equations serve as the basis for further investigations of thesuperconducting state. Results obtained from the Gorkov equations agreewith those obtained from the mean-field BCS theory. For reviews, we refer

178

to the book by Tinkham.37 Here we forego a discussion of thermodynamicproperties and the Josephson effect, and limit ourselves to a discussion of theresponse of superconductors to electromagnetic radiation.

9.4 A superconductor in a weak electromagnetic field

Here we calculate the current density in a superconductor in response to anexternal vector potential ~A.

Starting point of the calculation is the Kubo formula. Repeating resultsof Sec. 6.3, we have

je,~qα(ω) = −∑

β

ΠRαβ(~q, ω)Aβ,~q(ω) +

eρe

mAα,~q(ω), (9.53)

where ΠR is the retarded current density autocorrelation function,

Παβ(~q, τ) = − 1

V〈Tτje,~qα(τ)je,−~qβ(0)〉, (9.54)

and~je,~q = − e

2m

∑

~kσ

(2~k + ~q)ψ†~k,σψ~k+~q,σ. (9.55)

Equation (9.53) describes the linear response of the current density to a

vector potential ~A. We used this relation in Sec. 6.3 to find the conductivityof a disordered normal metal, the coefficient of proportionality between thecurrent density and the electric field ~E(ω) = iω ~A(ω). In the normal metal,

the choice of the gauge for the vector potential ~A was not relevant. Thisis different for the case of a superconductor. The reason is that the vectorpotential ~A enters in the equation for the single-particle Green functions Gand F , and, hence, affects the value of the “order parameter” ∆. To linearorder in ~A, ∆ can only depend on ~∇ · ~A. Hence, if we choose the gauge suchthat ~∇ · ~A = 0, ∆ is unchanged to linear order in the vector potential. Thischoice of the gauge is known as the “London gauge”.

37See Introduction to Superconductivity, M. Tinkham (Mc Graw Hill College Div.,1995). Other useful references are Superconductivity of Metals and Alloys, P.G. deGennes (Perseus, 1999) and Methods of Quantum Field Theory in Statistical Physics,A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski (Dover 1977).

179

Let us now calculate the current density autocorrelation function for thecase of a clean superconductor. Using the Matsubara frequency language,we find

Παβ(~q, iΩn) =e2T

2m2V

∑

~k,m

(2kα + qα)(2kβ + qβ)

×[

G~k+~q,~k+~q(iωm + iΩn)G~k;~k(iωm)

+ F~k+~q,~k+~q(iωm + iΩn)F+~k,~k

(iωm)]

. (9.56)

Changing variables to ~k± = ~k ± (~q/2) and ωm± = ωm ± (Ωn/2), we rewritethis as

Παβ(~q, iΩn) =2e2T

m2V

∑

~k,m

kαkβ

[

G~k+,~k+(iωm+)G~k−,~k−

(iωm−)

+ F~k+,~k+(iωm+)F+

~k−,~k−(iωm−)

]

=2e2T

m2V

∑

~k,m

kαkβ (9.57)

×(iωm+ + ε~k+

− µ)(iωm− + ε~k−− µ) + |∆|2

(ω2m+ + (ε~k+

− µ)2 + |∆|2)(ω2m− + (ε~k−

− µ)2 + |∆|2) ,

where we substituted the solution (9.42) for the Green functions G and F .

Next, we replace the summation over ~k by an integration over ξ = ε~k − µ

and over the angle θ between ~q and ~k.Next we introduce polar coordinates for the vector ~k. We choose the di-

rection of ~q as the polar axis. The average over the azimuthal angle φ canbe performed directly. Noting that ~q and ~A are orthogonal in the Londongauge, upon performing the average over φ we find that Π is diagonal, andproportional to (1/2) sin2 θ, where θ is the angle between ~q and ~k. Further,since the main contribution comes from momenta close to the Fermi momen-tum, we set the magnitude of the factors kα and kβ equal to kF . Finally,using the equality νk2

F/m = 3n/2, where n is the electron density and ν thenormal-metal density of states at the Fermi level (per spin direction and per

unit volume), and replacing the summation over ~k by an integration over

180

ξ = ε~k − µ and an integration over θ, we find

Παβ(~q, iΩn) =3Te2n

4mδαβ

∑

m

∫

dξ

∫ π

0

dθ sin3 θ

× (iωm+ + ξ+)(iωm− + ξ−) + |∆|2(ω2

m+ + ξ2+ + |∆|2)(ω2

m− + ξ2− + |∆|2) ,

where ξ± = ξ ± (vF q/2) cos θ.Because of the approximations involved, the summation over Matsubara

frequencies should be performed before the integration over ξ. Performingthe integration over ξ first gives a different answer. The answer depends onthe order of the integrations because the integral (9.58) is formally divergent.Our formulas for the ξ-dependence of the single-electron Green functions arevalid close to the Fermi level only. After the summation over the Matsubarafrequencies, the ξ integration is, in fact, convergent, the main contributioncoming from momenta near the Fermi level.

For the case of the superconductor, there is a trick that allows us to dothe ξ integration first, nevertheless. The trick is to look at the differencebetween current density correlation functions for the normal metal and forthe superconductor. For this difference, the integrations and summationsover ξ and ωm are convergent, so that the order of integrations is no longerimportant. Hence, if we calculate the difference between current-currentcorrelation functions for the normal metal and superconducting states, we canperform the ξ integration before the summation over Matsubara frequencies.The current density autocorrelation function for the normal metal is

ΠNαβ(~q, iΩn) =

3Te2n

4mδαβ

∑

m

∫

dξ

∫ π

0

dθ sin3 θ1

(iωm+ − ξ+)(iωm− − ξ−)

=eρe

m, (9.58)

see Ex. 6.3. Integrating with respect to ξ we then obtain

Παβ(~q, iΩn) =eρe

m− 3πTeρe

4mδαβ

∑

m

∫ π

0

dθ sin3 θ

×[

(iωm+ + i√

ω2m+ + |∆|2)(iωm− − qvF cos θ + i

√

ω2m+ + |∆|2) + |∆|2

(ω2m− + |∆|2 + (qvF cos θ − i

√

ω2m+ + |∆|2)2)

√

ω2m+ + |∆|2

181

−∆− Ωi n ∆− Ωi n

∆−∆C1 C1

C2

C2C2

C2

Figure 54: Integration contour for the calculation of Eq. (9.61).

+(iωm− + i

√

ω2m− + |∆|2)(iωm+ − qvF cos θ + i

√

ω2m− + |∆|2) + |∆|2

(ω2m+ + |∆|2 + (qvF cos θ − i

√

ω2m− + |∆|2)2)

√

ω2m− + |∆|2

]

.

(9.59)

Next, we perform the summation over the Matsubara frequency ωm. Weundo the shift of variables for the Matsubara frequencies, i.e., we replaceωm+ → ωm + Ωn and ωm− → ωm. We then represent the summation overMatsubara frequencies as an integral in the complex plane and deform theintegration contour as shown in Fig. 54. The integrand has a square rootbranch cut. In identifying the correct sign of the square root, one needs therelations

√

|∆|2 − (ω ± iη)2 =

√

|∆|2 − ω2 if |ω| < |∆|,∓i sign(ω)

√

ω2 − |∆|2 if |ω| ≥ |∆|. (9.60)

After the integration contours have been deformed as in Fig. 54, we canperform the analytical continuation iΩn → ω + iη.

Obtaining the final equations for Παβ(~q, ω), which contain one integrationover a real energy variable and one integration over the angle θ is straightfor-ward, but tedious. Below, we’ll restrict ourselves to static response ω → 0.In this case, the integration above the lower branch cut cancels the integra-tion below the upper branch cut in Fig. 54. What remains is the integrationabove the upper branch cut and the integration below the lower branch cut,

ΠRαβ(~q, 0) =

eρe

m− 3πeρe

8πmδαβ

∫ π

0

dθ sin3 θ

∫ ∞

∆

dζ tanh(ζ/2T )

×∑

±

|∆|2√

ζ2 − |∆|2(|∆|2 − (ζ ± iη)2 + (v2F q

2/4) cos2 θ).

182

(9.61)

The easiest way to evaluate this integral is to rewrite it as a sum over Mat-subara frequencies,

ΠRαβ(~q, 0) =

eρe

mδαβ − 3πeρeT

4mδαβ

∑

m

∫ π

0

dθ sin3 θ

× |∆|2(ω2

m + |∆|2 + (v2F q

2/4) cos2 θ)√

ω2m + |∆|2

. (9.62)

Evalution of this expression is further simplified in the limits qvF ≪ max(|∆|, T )and qvF ≫ max(|∆|, T ).

If qvF ≪ max(|∆|, T ), we can neglect the q-dependence of the summandand find

ΠRαβ(~q, 0) =

eρe

mδαβ

(

1 − ns(T )

n

)

, (9.63)

where ns is the so-called “superconducting density”,

ns =nπT |∆|2

2

∑

m

1

(ω2m + |∆|2)3/2

. (9.64)

For zero temperature the superconducting density is equal to the electrondensity n. The superconducting density approaches zero as T approaches Tc.

If qvF ≫ max(|∆|, T ), the main contribution to the integral over the angleθ comes from angles close to θ = π/2. For those angles we may set sin θ = 1.Replacing the θ-integration by an integration over cos θ and extending theintegration interval to the entire real axis, we find

ΠRαβ(~q, 0) =

eρe

mδαβ − 3π2eρeT

2mqvFδαβ

∑

m

|∆|2ω2

m + |∆|2

=eρe

mδαβ − 3π2eρe|∆|

4mqvFδαβ tanh

|∆|2T

. (9.65)

Unlike in the case of a normal metal, we find that for a superconductorthe current density ~j(~q, ω) is proportional to the vector potential ~A itself,and not to its time-derivative. This dependence is the cauase of the Meissnereffect, the phenomenon that no magnetic field can exist in a superconductor,

183

except for a thin layer of thickness δ close to the superconductor surface. Fora superconductor for which vF/δ ≪ max(T,∆), the Meissner effect can bedescribed with the help of Eq. (9.63) above,

~j = −nse2

m~A. (9.66)

Combining Eq. (9.66) with the Maxwell equation ∇2 ~A = −~j for the London

gauge ~∇ · ~A = 0, we find δ =√

m/nse2. Equation (9.66) is known asthe London equation, and δ is known as the London penetration depth.A superconductor for which δ ≫ vF/max(T,∆) is said to be “of Londontype”. For a superconductor for which δ ≪ vF/max(T,∆), Eq. (9.65) hasto be used instead of Eq. (9.63). In this case, the mathematical solution ismore complicated, but the qualitative conclusion, no magnetic field insidethe superconductor, except for a surface layer of thickness δ, remains true.Superconductors for which δ ≪ vF/max(T,∆) are said to be “of Pippardtype”. Equation (9.65) is known as the Pippard equation.

Most pure superconductors at T ≪ Tc are of Pippard type. Close to Tc,δ increases, and a crossover to the London type of behavior is seen.

These calculations can be repeated for the case of a disordered supercon-ductor with elastic mean free time τ . The calculation proceeds along thelines of Sec. 6.3 and has the following results: First, the addition of impuri-ties does not affect the magnitude of the superconducting order parameter∆. Second, the effect of impurities is small if qvF τ ≫ 1, whereas, in theopposite limit, qvF τ ≪ 1, one recovers the London equation (9.66), with areduced superconducting density

ns =nπT |∆|2

2

∑

m

1

(ω2m + |∆|2)(

√

ω2m + |∆|2 + 1/2τ)

. (9.67)

As a result of the reduced superconducting density, the London penetrationdepth δ is strongly enhanced. As a result, for sufficiently strong disorder,disordered superconductors are always in the London limit.

184

Figure 55: Diagram contribution to the renormalization of electron-phonon vertexby phonon scattering.

9.5 Exercises

Exercise 9.1: Migdal’s theorem

In principle, the electron-phonon vertex can also be renormalized by therepeated exchange of phonons, as, e.g., in the diagram of Fig. 55. (Note thatthe phonon line exchanged between the electrons is a renormalized phononline.) Show that such diagrams are smaller than the diagrams without theexchange of a phonon by a large factor ∼

√

M/m, where M and m are ionand electron mass, respectively.

Exercise 9.2: Thermodynamic properties

One can calculate the effect of the superconductivity on the free energy Fusing the general relation between F and the interaction Hamiltonian H1 ofEq. (9.20).

F − F0 =

∫ λ

0

dλ′1

λ′〈H1(λ

′)〉. (9.68)

(a) Keeping contributions that are specific for a superconductor only, showthat the superconductivity contribution δF to the free energy reads

δF =

∫ λ

0

dλ′

λ′2|∆(λ′)|2, (9.69)

where ∆(λ′) is the superconducting order parameter at interactionstrength λ′.

185

(b) One can rewrite the gap equation (9.44) as

1 = λν

[

ln2ωD

|∆| − 2∞∑

n=1

(−1)n+1K0(n|∆|/T )

]

. (9.70)

Using this result, together with Eq. (9.69), show that

δF = −ν[

|∆|22

− 2∞∑

n=1

(−1)n+1T2

n2

∫ n|∆|/T

0

K1(x)x2dx

]

, (9.71)

where K1 is a Bessel function.

(c) Show that, at temperatures T ≪ Tc, the heat capacity of a supercon-ductor is

C = 2ν

√

2π∆50

T 3e−∆0/T , (9.72)

where ∆0 is the magnitude of the order parameter at zero temperature.

186

10 Diffusion modes

10.1 polarizability function for a dirty metal

Let us return to our calculation of the polarizability function of the non-interacting electron gas. In Sec. 6.2 we looked at the response of a cleansystem. What about a disordered metal?

First, at high frequencies or small wavelengths, we do not expect that thedisorder will play an important role. However, at low frequencies or largewavelengths, impurity scattering becomes important and should affect theway the electron density responds to a change of the potential.

Repeating the calculations of Sec. 6.2, we find that the polarizabilityfunction χe is equal to

χe(~q, iΩn) =2Te2

V

∑

~k,~k′,m

G~k+~q,~k′+~q(iωm + iΩn)G~k′,~k(iωm), (10.1)

where the Green function G is calculated in the presence of the exact impu-rity potential. We will be interested in the impurity averaged polarizabilityfunction 〈χe〉. For the disorder, we use the model of Gaussian white noise,see Sec. 5.3. Evaluating the average, we limit ourselves to the ladder dia-grams shown in Fig. 56. We rewrite the diagrams in terms of a propagatorD shown in Fig. 57,

D(~k, iωm;~k + ~q, iωm + iΩn)

=

1 − 1

2πντV

∑

~k′

〈G~k′+~q(iωm + iΩn)〉〈G~k′(iωm)〉

−1

. (10.2)

We can replace the summation over ~k′ by an integration over the energyξ = ε~k′−µ and the angle θ between ~k′ and ~q. We are interested in wavevectorsq ≪ kF , so that we can expand

ε~k+~q = ε~k + vF q cos θ. (10.3)

Since the main contribution of the ξ-integration comes from a window ofwidth ∼ ~/τ around the Fermi energy, we can assume a constant densityof states ν for the entire integration range. The result of the ξ-integrationdepends on the position of the Matsubara frequencies iωm + iΩn and iωm

187

++ + ...=

m

m n

m

m n

m

m n

m

m n

m

m n

k

k+q

m

m n

k k’k

k+q

k’

k’+q

k"

k"+qk+q k’+q

k

k+q

ω

ω +Ω

ω

ω +Ω

ω

ω +Ω

ω

ω +Ω

ω

ω +Ω

ω

ω +Ω

Figure 56: Diagrammatic representation of ladder series for the polarizabilityfunction is a disordered non-interacting electron gas.

with respect to the real axis. If they are on the same side of the real axis, theintegration over the energy ξ vanishes, and one finds D = 1. If ωm + Ωn > 0whereas ωm < 0, the ξ-integration is finite, and one has

1

2πντV

∑

~k′

〈G~k′+~q(iωm + iΩn)〉〈G~k′(iωm)〉

=1

2

∫ 1

−1

d cos θ1

1 + Ωnτ + ivF qτ cos θ. (10.4)

For low frequencies, iΩn → ω ≪ 1τ and for long wavelengths, we can expandthe fraction in the integrand and perform the θ-integration. The result is

1

2πντV

∑

~k′

〈G~k′+~q(iωm + iΩn)〉〈G~k′(iωm)〉 = 1 − Ωnτ −1

3v2

F q2τ 2.

Similarly, for ωm + Ωn < 0 and ωm > 0, one finds

1

2πντV

∑

~k′

〈G~k′+~q(iωm + iΩn)〉〈G~k′(iωm)〉 = 1 + Ωnτ −1

3v2

F q2τ 2.

We thus conclude that for |Ωn|τ ≪ 1, qvF τ ≪ 1, one has

D(~k, iωm;~k + ~q; iωm + iΩn)

=

1 if ωm + Ωn > 0, ωm > 0,1/[τ(Dq2 + Ωn)] if ωm + Ωn > 0, ωm < 0,1/[τ(Dq2 − Ωn)] if ωm + Ωn < 0, ωm > 0,1 if ωm + Ωn < 0, ωm < 0,

(10.5)

where we introduced the “diffusion coefficient” D = v2F τ/3. Because of the

appearance of the factor Dq2±Ωn in the denominator, which is the kernel ofthe diffusion equation, the propagator D is referred to as the “diffuson” or

188

1 += + ...+

Figure 57: Diagrammatic representation of the diffusion propagator.

“diffusion propagator”. Performing the analytical continuation iωm → ε±iη,iωm + iΩn → ε + ω ± iη, we refer to the four cases listed in Eq. (10.5)as retarded-retarded, retarded-advanced, advanced-retarded, and advanced-advanced,

DRR(~k, ε;~k + ~q; ε+ ω) = 1,

DAR(~k, ε;~k + ~q; ε+ ω) = 1/[τ(Dq2 − iω)],

DRA(~k, ε;~k + ~q; ε+ ω) = 1/[τ(Dq2 + iω)], (10.6)

DAA(~k, ε;~k + ~q; ε+ ω) = 1.

In terms of the diffusion propagator, the polarizability reads

χe(~q, iΩn) =2Te2

V

∑

~k,m

〈G~k+~q,~k+~q(iωm + iΩn)〉〈G~k,~k(iωm)〉

× D(~k, iωm;~k + ~q, iωm + iΩn). (10.7)

The calculation of χe differs in an essential way from the calculation of thediffusion propagator, since Eq. (10.7) involves both a summation over Mat-

subara frequencies ωm and a summation over wavevectors ~k. The summationover ωm, which needs to be performed first, can be done as in Sec. 6.3. Re-placing the summation over Matsubara frequencies by an integration in thecomplex plane and deforming the contours as in Fig. 25, and performing theanalytical continuation iΩn → ω + iη, we find

χRe (~q, ω) =

e2

2πiV

∫

dξ tanh(ξ/2T )∑

~k

×[

〈GR~k,~k

(ξ)〉〈GR~k+~q,~k+~q

(ξ + ω)〉DRR(~k, ξ;~k + ~q, ξ + ω)

− 〈GA~k,~k

(ξ)〉〈GR~k+~q,~k+~q

(ξ + ω)〉DAR(~k, ξ;~k + ~q, ξ + ω)

189

+ 〈GA~k,~k

(ξ − ω)〉〈GR~k+~q,~k+~q

(ξ)〉DAR(~k, ξ − ω;~k + ~q, ξ)

− 〈GA~k,~k

(ξ − ω)〉〈GA~k+~q,~k+~q

(ξ)〉DAA(~k, ξ − ω;~k + ~q, ξ)]

.

For the first and fourth terms between the brackets, the diffuson propagatorDRR = DAA = 1. For the second and third term, we write DAR = 1+(DAR−1). Then there is a contribution from all four terms between brackets, whichis identical to that calculated in Eq. (6.63), and a contribution from theremaining part of the second and third term between brackets. Adding bothterms and making use of the limits qvF τ ≪ 1, ωτ ≪ 1 and εF τ ≫ 1, we find

χRe (~q, ω) = −2e2ν − e2

2πiV τ(Dq2 − iω)

∫

dξ

(

tanhξ

2T− tanh

ξ + ω

2T

)

×∑

~k

〈GA~k,~k

(ξ)〉〈GR~k+~q,~k+~q

(ξ + ω)〉

= −2e2ν − 2e2νiω

Dq2 − iω

= − 2e2νDq2

Dq2 − iω. (10.8)

To see what this result means, it is useful to perform a Fourier transformto time t and coordinate ~r. For the density response to a sudden increase ofthe potential δφ(~r, t) = δ(~r − ~r′)θ(t)δφ we find from Eq. (10.8)

δρe(~r, t) = −2e2ν

[

δ(~r − ~r′) −( π

Dt

)3/2

e−|~r−~r′|2/4Dt

]

δφ. (10.9)

This result has a very simple interpretation: the change in the potential at~r = ~r′ creates a density profile that spreads diffusively. This result, which wasobtained by summation of a set of diagrams, thus reproduces what we knowbased on the picture of electron diffusion. Note that the above calculationis quite similar to that of the Drude formula. So it is no surprise that werecovered results from diffusion theory in both cases.38

The fact that dynamics in a dirty metal is diffusive, rather than ballistic,for time scales longer than the mean free time τ and length scales longer than

38Note that the Drude conductivity is a function of the diffusion constant as well,

σαβ(ω) = 2e2νDδαβ .

190

φ(r,t)

electron motionto screen potential

Figure 58: The electrons move periodically in order to screen an external time-dependent potential. It depends on the frequency of the potential whether theelectron gas can sustain this periodic motion in the presence of disorder.

the mean free path l = vF τ , has important consequences for the screeningof the Coulomb interaction: since it is more difficult to move charge around,the electron gas has a reduced capacity to screen time-dependent potentials.The relevant frequency for the time dependence is not the ballistic frequencyvF q (as it is in the response of a clean electron gas), but the diffusive one,ω ∼ Dq2, where 1/q ≫ l is the length scale that the system is probed at.Qualitatively, the effect of disorder on the screened Coulomb interaction canbe seen from the following picture, see Fig. 58. In order to screen a time-dependent potential with frequency ω and wavevector q, the electrons needto move a distance ∼ 1/q in a time ∼ 1/ω. As long as ω ≪ Dq2 the electrongas can accomodate such a displacement, and Thomas-Fermi screening theoryapplies. If, however, ω ≫ Dq2 the electron gas cannot follow the potential,and screening is absent.

The reduced screening capability is seen quantitatively in the RPA cal-culation of the dielectric function of the interacting electron gas, which nowbecomes

ǫRPA(~q, iΩn) = 1 +k2

s

q2

Dq2

|Ωn| +Dq2, (10.10)

where k2s = 2νe2/ǫ0 is the Thomas-Fermi wavenumber. Similarly, the effec-

tive electron-electron interaction becomes

V RPA~q (iΩn) =

e2

ǫ0q2ǫRPA(~q, iΩn)

=e2

ǫ0

(

q2 + k2s

Dq2

|Ωn| +Dq2

)−1

. (10.11)

For |ω| ≪ Dq2 the effect of the disorder is negligible, whereas for ω ≫ Dq2

the Coulomb interaction is essentially unscreened. This is different from the

191

V

electrodereference

conductordisordered

I

Figure 59: Tunnel spectroscopy setup for a measurement of the spectral densityof a disordered conductor.

case of a clean metal, where Thomas-Fermi screening breaks down at themuch higher frequency ω ∼ vF q.

One example where the modified screening capability of electrons playsa role is the so-called “zero bias anomaly” in the tunneling density of states.The density of states can be measured by a tunneling experiment, in whichelectrons from a reference conductor with known (and constant) spectraldensity tunnel into the sample, see Fig. 59. Indeed, as shown in Ex. 6.1, thecurrent I through a tunneling barrier contacting the sample at position ~r isproportional to the spectral density,

∂I(~r, eV )

∂V∝ 1

4T

∫ ∞

−∞

dξA(~r, ξ)∑

±

1

cosh2[(ξ ± eV )/2T ]. (10.12)

(At zero temperature, this simplifies to ∂I/∂V ∝ A(~r, eV ).)For a non-interacting electron gas, the spectral density is essentially en-

ergy independent, hence the tunnel current is proportional to the appliedbias voltage V and independent of the temperature T . For the disorderedinteracting electron gas, the situation is more complicated. When an electrontunnels into the sample, it charges the sample locally. For a cloud that isbigger than the mean free path l, dispersion of the charge cloud is a slow pro-cess, which is determined by the diffusive motion of the electrons. Until theenergy of the charge cloud, which is a sum of kinetic and the Coulomb energycontributions, has dropped below the bias voltage eV or the temperature T ,

192

Figure 60: Lowest order corrections to the single-particle Green function in adisordered interacting electron gas. Left: Hartree diagram, right: Fock diagram.The shaded blocks represent the diffusion propagator, see Fig. 57.

the electron that tunnels into the metal has to exist in a virtual state. Theprolonged existence in a virtual state suppresses the tunneling current or, inother words, the density of states. At low voltages and low temperatuers,the suppression should be maximal, the tunneling density of states being anincreasing function of V and T for small voltages and low temperatures.

A non-perturbative semiclassical theory of the zero bias anomaly hasbeen formulated by Shytov and Levitov [JETP Lett. 66, 214 (1997)]. Herewe present an argument based on perturbation theory in the electron-electroninteraction, following the original calculation of Altshuler and Aronov [Sov.Phys. JETP 50, 968 (1979)] and Altshuler, Aronov, and Lee [Phys. Rev.Lett. 44, 1288 (1980)]. To lowest order in the interaction, this correctionis given by Hartree and Fock diagrams. The diffusive electron motion isreflected in the presence of the diffusion ladders around the interaction line,which, in turn, represents the effective interaction of Eq. (10.11) above. Thetwo relevant diagrams for the correction to the single-particle Green functionare shown in Fig. 60.

We consider the Fock diagram first. The correction to the single-electronGreen function shown in Fig. 60 reads

δ〈G~k,~k(iωn)〉 =T

V

∑

~q

∑

m

〈G0~k,~k

(iωn)〉2D(~k, iωn;~k + ~q, iωm)2

× (−V~q(iωn − iωm))〈G0~k+~q,~k+~q

(iωm)〉. (10.13)

The correction to the spectral density is found as

δA(~r, ω) = −4Im δGR(~r, ~r;ω)

193

C1

C1

C2

Figure 61: Integration contour for the calculation of the zero-bias anomaly to thetunneling density of states.

= −4T

V 2Im∑

~k,~q

∑

m

〈G0~k,~k

(iωn)〉2D(~k, iωn;~k + ~q, iωm)2

× (−V~q(iωn − iωm))〈G0~k+~q,~k+~q

(iωm)〉∣

∣

∣

iωn→ω+iη. (10.14)

where the extra factor two accounts for spin degeneracy.Since the diffusion propagator is divergent for small momentum transfers,

the main contribution to the integration in Eq. (10.13) comes from small ~q.For these momenta, we can use the screened interaction of Eq. (10.11).

A singular contribution to the density of states can come from the poleof the diffusion propagator at small momentum and frequency. For the caseωn > 0, this pole exists for ωm < 0 only. Replacing the summation over ωm

by an integration in the complex plane, deforming the integration contour asin Fig. 61, and performing the analytical continuation iωn → ω + iη we find

δA(~r, ω) = Im1

V 2πi

∑

~k,~q

∫ ∞

−∞

dξ tanh(ξ/2T )〈G0R~k,~k

(ω)〉2 1

(Dq2 + iξ − iω)2τ 2

× (−V~q(ω − ξ + iη))〈G0A~k+~q,~k+~q

(ξ)〉. (10.15)

Since the main dependence on ω and ~q comes from the pole of the diffusionoperator, we set ~q = 0 and ω = ξ in the arguments of the Green functionsin Eq. (10.15). Performing the summation over ~k and shifting integrationvariables ξ → ω − ξ we then find

δA(~r, ω) = −Im2ν

V

∑

~q

∫ ∞

−∞

dξ tanh

(

ξ − ω

2T

)

V~q(ξ + iη)

(Dq2 − iξ)2. (10.16)

194

The summation over ~q and integration over ξ are, in fact, divergent. Thedivergence is not physical, since our approximations break down when vF qτand ωτ become of order unity. In order to arrive at a meaningful result thatdescribes the temperature and energy dependence of the density of states, wesubstract δA in the limit ω → 0, T → 0 (limits taken in this order). In Eq.(10.16) this amounts to the replacement of tanh[(ξ − ω)/2T ] by tanh[(ξ −ω)/2T ] − sign (ξ). For the interaction, we substitute the screened Coulombinteraction in the presence of disorder, Eq. (10.11) above. First performingthe remaining summation over ~q, one finds

δA(~r, ω) = −Im1

4πD3/2

∫ ∞

−∞

dξ

(

tanhξ − ω

2T− sign (ξ)

)

×[

1√−iξ − 1√

Dk2s − iξ

]

. (10.17)

We omit the second term between the square brackets, which gives a vanish-ing contribution to the spectral density if ω, T ≪ Dk2

s . For the remainingintegration, we find

δA(~r, ω) = − T 1/2

π√

2(~D)3/2f(ω/T ), (10.18)

where

f(x) =

∫ ∞

0

dy

2y1/2

(

sinh y

cosh y + cosh x− 1

)

. (10.19)

Asymptotically, the function f(x) behaves as

f(x) ≈ −√x− π2

24x−3/2 if x ≫ 1,

f(x) ≈ −1.072 if x≪ 1. (10.20)

You verify that, indeed, δA → 0 if we take the limit ω → 0, followed byT → 0.

For two dimensional samples and one dimensional samples, the calculationof the density of states is very similar. The first difference is that the Coulombinteraction has a different form in one and two dimensions,

V~q(iΩn) =

(e2/2ǫ0)[

q + (e2/ǫ0)νDq2

|Ωn|+Dq2

]−1

if d = 2,

(e2/ǫ0)[

− 4πln(q2a2)

+ 2(e2/ǫ0)νDq2

|Ωn|+Dq2

]−1

if d = 1.(10.21)

195

Here ν is the density of states per spin direction and per unit area (in twodimensions) or unit length (in one dimension), whereas a is the thicknessor radius of the sample. In quasi two-dimensional samples, ν = ν3a, whereν3 is the three-dimensional density of states. With this modification, thecalculation proceeds as in the three dimensional case.

In two dimensions, the final integration over ξ and summation over ~q islogarithmically divergent. Performing the summation over ~q first and trun-cating the ξ-integration at ξ ∼ 1/τ , we find

δA(~r, ω) =1

4πD

∫ ∞

0

dξ

ξln

ξ

De2ν/ǫ0

(

tanhξ + ω

2T+ tanh

ξ − ω

2T

)

≈ − 1

4πD

[

lnmax(ω, T )

D2(e2/ǫ0)ν2τ

]

ln [τ max(ω, T )] . (10.22)

Similarly, in the one-dimensional case one has

δA(~r, ω) = − aks

2√

2πTD

[

lnDk2

s

max(ω, T )

]−1/2 ∫ ∞

0

dy

y3/2

sinh y

cosh y + cosh(ω/T ),

(10.23)where ks is the three-dimensional Thomas-Fermi wavenumber.

The contribution from the Hartree diagram is similar. The transferredmomentum ~q in interaction factor of the Hartree diagram does not have tobe small for the momentum in the diffusion propagators to be small. Sincethe Hartree diagram involves large momentum exchanges in the interactionmatrix element, the precise magnitude depends on the details of the short-range part of the Coulomb interaction. In that case we cannot use the simplescreened form (10.11) above, but need information about the microscopicdetails of the sample. For the discussion here, we limit ourselves to thestatement that the energy and temperature dependence of δA is still the sameas that of Eq. (10.18), whereas the prefactor depends on sample details. Fordetails of this calculation we refer to the article Electron-electron interaction

in disordered conductors, by B.L. Altshuler and A.G. Aronov, in the bookElectron-electron interactions in disordered systems, edited by A.L. Efros andM. Pollak (North-Holland, 1985).

10.2 Weak localization

Our calculation of the conductivity in Sec. 6.3 was done to leading order inthe impurity concentration. The leading contribution consisted of diagrams

196

12πντ

2πντ + ... +

+ + ... =

=

m

m n

k

q−k

k

q−k

k’

q−k’

m

m n

m

m n

m

m n

k

k+q

k’

k’+q

k

k+q

ω

ω +Ω

ω

ω +Ω

ω

ω +Ω

ω

ω +Ω

Figure 62: Diagrammatic representation of maximally crossed diagrams that con-tribute to the conductivity (top). The double lines represent the disorder averagedGreen function; the solid dots refer to the renormalized current vertex, see Fig.23. The maximally crossed diagrams can be represented in terms of the cooperonpropagator (right and below).

without crossed impurity lines. Diagrams that have impurity lines are a fac-tor 1/ǫF τ smaller. However, as we’ll show here, such diagrams can have asingular temperature dependence in one and two dimensions, actually diverg-ing in the limit of zero temperature. The singularity arises from the diffusiveelectron motion we considered in the previous section.

The leading correction in powers of 1/ǫF τ is given by the so-called “max-imally crossed” diagrams.39 The maximally crossed diagrams are shown inFig. 62. They can be represented in terms of the renormalized current vertex(see Sec. 6.3) and the so-called cooperon propagator. The cooperon is a seriesof ladder diagrams, similar to the diffusion propegator we encountered in theprevious section. However, unlike for the diffuson, where the difference of themomenta is constant throughout the ladder, for the cooperon the sum of themomenta is constant throughout the ladder, see Fig. 62. The cooperon prop-agator derives its name from its close resemblance to the electron-electroncorrelation function that describes the superconducting instability.

39Don’t be confused by the word “maximally crossed”. These diagrams, in fact, have thesmallest number of crossed impurity lines of all the diagrams that remain after calculationof the Drude conductivity.

197

The cooperon propagator reads

C(~q; iωm, iωm + iΩn) =1

2πντV

∑

~k〈G~q−~k,~q−~k(iωm + iΩn)〉〈G~k,~k(iωm)〉1 − 1

2πντV

∑

~k〈G~q−~k,~q−~k(iωm + iΩn)〉〈G~k,~k(iωm)〉 .

(10.24)

Performing the momentum summations, we find for qvF τ ≪ 1 and |Ωn|τ ≪ 1

C(~q; iωm, iωm + iΩn) =1

τ(Dq2 + |Ωn|)θ[(−ωm)(ωm + Ωn)].

(10.25)

For the current-current correlator, the diagrams of Fig. 62 imply a correction

δΠαβ(~q → 0, iΩn) =2e2T

m2V 2

∑

~k,~k′

kαk′β

∑

m

〈G~k,~k(iωm)〉〈G~k,~k(iωm + iΩn)

× ~

2πντC(~k + ~k′; iωm, iωm + iΩn)

× 〈G~k′,~k′(iωm)〉〈G~k′,~k′(iωm + iΩn)〉, (10.26)

The summation over m and the integrations over ~k and ~k′ can be done as inthe previous sections. Alternatively, we note that the Green functions have avery weak dependence on ωm for |ωm| < 1/τ , so that we may set iωn → −iη inthe arguments of the Green functions. Then summation over m gives nothingbut a factor n = Ωn/2πT . Now the analytical continuation iΩn → ω + iη ispossible, followed by division by ω and the limit ω → 0. Also, if we write~k′ = −~k + ~Q and note that the mean contribution of the summation over~Q comes from ~Q close to 0, we can ignore the ~Q dependence of the single-electron Green functions in Eq. (10.26). The remaining integration over ~kthen gives

1

V

∑

~k

kα(−kβ)〈GR~k,~k

(0)〉2〈GA~k,~k

(0)〉2 = 4πντ 3k2F/d, (10.27)

where d is the dimensionality of the conductor. Combining everything, wefind

δΠRαβ(~q → 0, ω) =

2τe2k2F

πdm2V

∑

~Q

iω

DQ2. (10.28)

198

Hence, using the general relation D = v2F τ/d, we find that the correction to

the conductivity reads

δσαβ = − 1

iωδΠR(~q → 0, ω)

∣

∣

∣

∣

ω→0

= − 2τe2k2F

πdm2V

∑

~Q

1

DQ2

= −2e2

πV

∑

~Q

1

Q2. (10.29)

In three dimensions, the summation over ~Q is divergent for large Q. Thereason for the divergence is that the approximations we made cease to bevalid for Q larger than the inverse mean free path l = vF τ . Truncating thesummation for Ql ∼ 1, we find

δσ ∼ −e2

l. (10.30)

This correction to the conductivity is a factor kF l smaller than the Drudeconductrivity. Hence, for three dimensions, we conclude that the strategy ofSec. 6.3, where we kept diagrams without crossed lines only, is valid.

For two dimensions, the summation over ~Q is divergent both for large Qand for small Q. The divergence at large Q is removed by truncating theintegration at Q ∼ 1/l. The divergence at small Q, however, has an entirelydifferent origin. Small Q correspond to large length scales, and on largelength scales our description in terms of non-interacting electrons in a staticand unbounded environment breaks down. In a real sample, electrons exitthe sample through the contacts, or they loose their phase memory because ofscattering from photons or phonons, or because of electron-electron interac-tions. That means that our description, in which such processes are not takeninto account, breaks down at long time scales or long length scales. Here wedo not attempt to formulate a full microscopic theory that includes the con-tacts and decohering processes. Instead, we use a phenomenological solution.We cut off the momentum summation from below at Q ∼ max(1/L, 1/Lφ),where L is the system size and Lφ is the length scale at which the elec-trons loose their phase memory. Alternative, we keep the full momentumsummation but replace the frequency −iω in the Cooperon propagator by

199

max(1/τesc, 1/τφ), where τesc = L2/D is the time it takes to exit the samplethrough the contacts and τφ = L2

φ/D is the dephasing time. We then find40

δσ =

−(e2/π2) ln[min(L,Lφ)/l], if d = 2,−(e2/π)[min(L,Lφ) − l], if d = 1.

(10.31)

In three dimensions, the fact that we need to add a lower cut-off to themomentum integration has a minor effect. It amounts to the replacement ofEq. (10.30) by

δσ = − e2

2π2

(

1

l− 1

Lφ

)

. (10.32)

These results are very important for our understanding of electron trans-port through disordered conductors. On the one hand, you verify that δσis, at first sight, a factor kF l smaller than the Drude conductivity. However,δσ has an additional dependence on system size or temperature that leadsto a divergence at large system sizes and low temperatures. (The dephasinglength Lφ → ∞ if T → 0.) Hence, we see that these small corrections be-come large in large samples at low temperatures, and that the conductivitybecomes significantly smaller than the Drude conductivity. In one and twodimensions, the correction δσ we calculated is the first signature of the phe-nomonon of “Anderson localization”, the statement that all “metals” in oneand two dimensions become insulators at sufficiently low temperatures. Forthat reason, the correction δσ is called “weak localization correction”.

The weak localization correction has a simple physical explanation. In asemiclassical picture, the electrons are described as wave packets that movealong well defined trajectories, acquiring a phase as they move along. Hence,the electron propagation along a certain path is described by an amplitude,i.e., by an intensity and a phase. The probability for an electron to gofrom one location in the sample to another location is proportional to thesquare of the amplitudes for all trajectories linking start and finish. Withoutquantum phase coherence, one would add intensities of all trajectories, notamplitudes, i.e., one squares the amplitudes first and then adds them. Thedifference between the two descriptions — add squared amplitudes or squareadded amplitudes — is “interference”. Interference can be constructive ordestructive. In most cases, the interference correction contains a sum over

40In obtaining the numerical factor for d = 1 in Eq. (10.31) and for d = 3 in Eq. (10.32)below, we replaced ω by i/τφ in the diffuson propagator and used τφ = L2/D to expressτφ in terms of Lφ.

200

Figure 63: Constructive interference of two time-reversed trajectories returningto the point of departure.

many terms with different phases, and averages to zero. However there is oneexception: interference of trajectories that return to the point of departure.In this case, time-reversed trajectories acquire the same phase, and, hence,interfere constructively, see Fig. 63. Thus, the probability to return to thepoint of departure is enhanced by quantum interference. This also impliesthat the rate at which particles diffuse through the metal is decreased, which,in turn, leads to a reduction of the conductivity. This reduction is preciselywhat we calculated above!

The interference between time-reversed trajectories is described by theCooperon propagator. Recall that the Cooperon propagator was importantprecisely for the case of opposite momenta ~k and ~k′.

In this picture, the cut-offs needed for the calculation of the weak lo-calization correction are more transparent. The large momentum cut-offcorresponds to the break down of diffusion theory at length scales below themean free path. Indeed, since one needs impurity scattering in order to re-turn to the point of departure, there is a minimal length for such a trajectory.Also, the interference correction exists only if the electrons keep their phasememory along the trajectory. This explains the cut-off at low momenta.

In a magnetic field B, time-reversed trajectories acquire a phase differenceequal to 2π times the enclosed flux Φ divided by the flux quantum Φ0 = hc/e.Since the typical area swept out for a trajectory that ventures a distance Laway from the point of departure is of order L2, trajectories with L2B/Φ0 & 1

201

do not contribute to the interference. Hence, in a magnetic field, we shouldimpose a lower momentum cutoff at Q ∼ 1/lm, where lm = (B/Φ0)

−1/2 is theso-called magnetic length. If lm is smaller than the dephasing length Lφ andthe system size, the dephasing length Lφ should be replaced by LB in Eqs.(10.31) and (10.32). Hence, for large magnetic fields, the weak localizationcorrection to the conductivity is suppressed.

10.3 Formulation in real space

In Sec. 10.1 we saw that the diffuson is equal to the pole of the diffusionequation. We arrived at this conclusion from an analysis in momentumrepresententain, which, in principle, is valid for a bulk system only. Forfinite-sized systems or for inhomogeneous sytems, a description in real spaceis more useful. In this section, we rederive the results of Sec. 10.1 using thecoordinate representation.

Before embarking on a diagrammatic calculation in coordinate represen-tation, let us recall the expressions for the impurity averaged Green function,

〈G(~r, ~r′, iω1)〉 = − m

2π~2|~r − ~r′|e±ikF |~r−~r′|−|ω1||~r−~r′|/vF−|~r−~r′|/2l, (10.33)

where the + sign is for ω1 > 0 and the − sign is for ω1 < 0, and the averagingrule for the Gaussian white noise potential,

〈U(~r)U(~r′)〉 =~

2πντδ(~r − ~r′), (10.34)

where τ is the elastic mean free time and ν is the density of states per spindirection. Equation (10.33) is valid in three dimensions; similar results can beobtained in one and two dimensions. Throughout this discussion we assumeweak disorder, kF l ≫ 1, where l = vF/τ is the elastic mean free path.

Starting point is the diagrammatic expression for the diffuson operatorin Matsubara representation D(~r, ~r′; iω1, iω2), which we repeat in Fig. 64.Writing down the Dyson equation in real space, one finds

D(~r, ~r′; iω1; iω2) = δ(~r − ~r′) +~

2πντ

∫

d~r′′〈G(~r, ~r′′; iω1)〉

× 〈G(~r′′, ~r; iω2)〉D(~r′′, ~r′; iω1; iω2). (10.35)

In coordinate representation, the disorder-averaged Green functions are shortranged, see Eq. (10.33). Hence, anticipating that D will be a slowly varying

202

i ω2

i ω1

2πντh

2πντh

+= +=+ ...

Figure 64: Defining equation for the diffusion propagator.

function of ~r and ~r′, we replace the product of single-electron Green functionsby a delta function. If ω1ω2 > 0 (both Matsubara frequencies on the sameside of the real axis), we find

~

2πντ〈G(~r, ~r′′; iω1)〉〈G(~r′′, ~r; iω2)〉 = sign (ω1)

i

2kF lδ(~r − ~r′′), (10.36)

whereas

~

2πντ〈G(~r, ~r′′; iω1)〉〈G(~r′′, ~r; iω2)〉 = (1 − |ω1 − ω2|τ )δ(~r − ~r′′), (10.37)

if ω1ω2 < 0 (iω1 and iω2 on different sides of the real axis). When iω1 andiω2 are both on the same side of the real axis, the second term on the r.h.s.of Eq. (10.35) can be neglected with respect to the l.h.s. of Eq. (10.35). Wethus find

D(~r, ~r′; iω1, iω2) = δ(~r − ~r′) if ω1ω2 > 0. (10.38)

If ω1ω2 < 0, however, the l.h.s. and r.h.s. of Eq. (10.35) almost cancel, leavinga contribution proportional to |ω1−ω2|τ only. In this case, approximating theproduct of single-electron Green functions by a delta function is not correct,and we need to take into account the spatial dependence of the diffusonpropagator,

~

2πντ〈G(~r, ~r′′; iω1)〉〈G(~r′′, ~r; iω2)〉 = (1 − |ω1 − ω2|τ +Dτ∇2

~r)δ(~r − ~r′′),

(10.39)

where D = (1/3)v2F τ is the diffusion constant.41 Hence, if ω1ω2 < 0, the

diffuson propagator satisfies the diffusion equation

(

−D∇2 + |ω1 − ω2|)

D(~r, ~r′; iω1, iω2) =1

τδ(~r − ~r′). (10.40)

41Equation (10.39) is derived “under the integral sign”, considering the integral of〈G(~r, ~r′′; iω1)〉〈G(~r′′, ~r; iω2)〉 times an arbitrary function f(~r′′) over ~r′′. Expanding thefunction f in a Taylor series around ~r′′ = ~r then gives the desired result.

203

conducting

sample

rrr

insulator

Figure 65: Relative positions of a point ~r close to the sample boundary and itsmirror image ~rr. For the derivation of the boundary conditions for the diffusionpropagator, the point ~r is chosen well within a mean free path from the boundary,but much further away than a Fermi wavelength.

For a bulk system, Fourier transform of Eq. (10.40) gives D(~q; iω1, iω2) =[τ(Dq2 + |ω1 − ω2|)]−1, in agreement with the results of Sec. 10.1.

The boundary conditions for the diffuson propagator can be obtained ina similar way. If we fix the point ~r′ to be inside the sample, the bound-ary conditions are found by taking the correct form of the disorder averagedsingle-electron Green functions at the sample boundary. Since the wavefunc-tions have to vanish at the sample boundary, you verify that one has

〈G(~r, ~r′′, iω1)〉 = 〈G(~r, ~r′′, iω1)〉bulk − 〈G(~rr, ~r′′, iω1)〉bulk, (10.41)

where ~rr is the mirror image of ~r in the sample boundary, see Fig. 65, andthe subscript “bulk” indicates the expression for the average Green functiondeep inside the sample, see Eq. (10.33) above. Since we are interested inspatial variations of the diffusion propagator on length scales of the meanfree path l and beyond, we take the point ~r such that its distance from theboundary is much smaller than the mean free path l but much larger thanthe Fermi wavelength λF = 2π/kF . Then, using Eq. (10.33), we find

~

2πντ

∫

d~r′′〈G(~r, ~r′′; iω1)〉〈G(~r′′, ~r; iω2)〉 = (1 +Dv−1F n · ~∇~r)δ(~r − ~r′′),

(10.42)

where n is the outward oriented unit vector normal to the sample boundary.Hence, using Eq. (10.35), we conclude that, at the sample boundary, the dif-fusion propagator D(~r, ~r′; iω1, iω2) with ω2ω1 < 0 must satisfy the boundary

204

condition

n · ~∇~rD(~r, ~r′; iω1, iω2) = 0. (10.43)

Similarly, at points where the sample is connected to a perfect conductor,one has the boundary condition

D(~r, ~r′; iω1, iω2) = 0. (10.44)

The boundary condition (10.43) describes the fact that, at a boundary withan insulator, the current density is zero, whereas the boundary condition(10.44) describes the fact that, at a boundary with a perfect conductor, theparticle accumulation must vanish.

In the presence of a vector potential ~A that varies on a scale that isslow compared to the mean free path, the Green function (10.33) acquires

an extra factor exp[i(e ~A/c~) · (~r − ~r′)]. However, upon multiplication of〈G(~r, ~r′′; iω1)〉 and 〈G(~r′′, ~r; iω2)〉 on the r.h.s. of Eq. (10.35), this extra factordisappears from the equation for the diffuson. Similarly, an eventual slowlyvarying potential V (~r) in the Hamiltonian, which causes a small changekF → kF −V/~vF in the wavevector appearing in the exponent of the single-electron Green function of Eq. (10.33), also drops out from the equation forthe diffusion propagator.

As a slight generalization of the previous discussion, we may discuss a dif-fusion propagator for the case where the upper and lower lines in the diagramof Fig. 64 correspond to Hamiltonians with the same disorder configuration,but with a different magnetic field or with a different value of a slowly varyingbackground potential V (~r). In this case, the phase vectors from the vectorpotentials and from the orbital motion do not cancel exactly, and one arrivesat the following equation for the diffuson if ω1ω2 < 0,

−D[

~∇~r +ie

~c( ~A1 − ~A2)

]2

D(~r, ~r′; iω1, iω2)

± (ω1 − ω2 + iV1 − iV2)D(~r, ~r′; iω1, iω2) =1

τδ(~r − ~r′),

(10.45)

where the + sign is taken for the case ω1 − ω2 > 0 and the − sign is takenif ω1 − ω2 < 0. Similarly, in the boundary condition (10.43), the deriviative~∇~r is replaced by the covariant derivative ~∇~r + (ie/~c)( ~A1 − ~A2).

205

The same considerations apply to the Cooperon. One finds that theCooperon propagator is zero if ω1ω2 > 0, whereas it satisfies the differentialequation

−D[

~∇~r +ie

~c( ~A1 + ~A2)

]2

C(~r, ~r′; iω1, iω2)

± (ω1 − ω2 + iV1 − iV2) C(~r, ~r′; iω1, iω2) =1

τδ(~r − ~r′),

(10.46)

if ω1ω2 < 0. The boundary conditions are the same as those for the diffusonpropagator. In the presence of a magnetic field, the differential operator ~∇~r

is replaced by the covariant derivative ~∇~r + (ie/~c)( ~A1 + ~A2). Note that,unlike the diffuson, the cooperon propagator depends on the magnetic field,even if ~A1 = ~A2.

10.4 Random matrix theory

Now let us look at an isolated piece of a disordered conductor of finite size.We are interested in the density of states in the conductor. Since the samplehas a finite size, the density of states will be a sum of Dirac delta functions,

N (ω) = 2∑

µ

δ(ω − εµ), (10.47)

where µ labels the eigenvalues of the single-electron Hamiltonian (withoutspin) and the factor two accounts for spin degeneracy.

The positions of the energy levels εµ and, hence, the density of states,will depend on the precise impurity configuration. As before, we’ll take astatistical approach, and calculate the probability distribution of the densityof states.

Calculation of the average is straightforward, 〈N 〉 = 2νV , where V isthe volume of the sample. The more interesting question is that of thefluctuations of the density of states, which is described by the function

R(ω1 − ω2) = 〈N (ω1)N (ω2)〉 − 〈N (ω1)〉〈N (ω2)〉. (10.48)

In order to calculate R, we first express the density of states in terms of theelectron Green functions,

N (ω) =1

πi

∫

d~r[GA(~r, ~r;ω) −GR(~r, ~r;ω)]. (10.49)

206

i ω2

i ω1

1r r

2

i ω2

i ω1

i ω1

i ω2

1r r

2i ω1

i ω2

r’

r’

r’’

r’’

r’

r’

r’’

r’’

Figure 66: Diagrams contributing to the fluctuations of the density of states in adisordered metal grain.

With this expression, R(ω1−ω2) can be calculated with the help of diagram-matic perturbation theory. The relevant diagrams for R(ω1 − ω2) are shownin Fig. 66. (Note that the diagrams without any lines connecting the innerand outer electron lines contribute to the average of the density of states andnot to the fluctuations.)

After substitution of Eq. (10.49) into Eq. (10.48), we distinguish fourcontributions to the density of states fluctuations. Performing the samecalculation as the one leading to Eqs. (10.36) and (10.37), one finds that thecontributions involving two retarded Green functions or two advanced Greenfunctions are smaller than the contributions involving one retarded and oneadvanced Green function by a factor kF l ≫ 1. Hence, we conclude

R(ω1 − ω2) =2

π2Re

∫

d~r1d~r2〈GR(~r1, ~r1, ω1)GA(~r2, ~r2, ω2)〉. (10.50)

Making use of the auxiliary results

GR(~r1, ~r′, ω1)G

R(~r′′, ~r1, ω1)GA(~r′, ~r′′, ω2) = −2πiντ 2δ(~r′ − ~r1)δ(~r

′′ − ~r1)

GR(~r1, ~r′, ω1)G

R(~r′′, ~r1, ω2)GA(~r′′, ~r′, ω2) = −2πiντ 2δ(~r′ − ~r1)δ(~r

′′ − ~r1),

(10.51)

we find that, according to the diagrams of Fig. 66,

R(ω1 − ω2) =2τ 2

π2Re

∫

d~rd~r′[DRA(~r, ~r′;ω1, ω2)DRA(~r′, ~r;ω1, ω2)

207

+ CRA(~r, ~r′;ω1, ω2)CRA(~r′, ~r;ω1, ω2)]. (10.52)

We now calculate the density of states correlator at energy ω1 and vectorpotential ~A1 and at energy ω2 and vector potential ~A2. Hereto, we write thediffuson and cooperon propagators as a sum over eigenmodes of the diffusionequation,

−D[

~∇~r +ie

~c( ~A1 − (±) ~A2)

]2

φm±(~r) = γm±φm±(~r), (10.53)

where the φm are properly normalized eigenfunctions and the γm are thecorresponding eigenvalues,

DRA(~r, ~r′;ω1, ω2) =1

τ

∑

m

φm+(~r)φm+(~r′)∗

γm+ − i(ω1 − ω2), (10.54)

CRA(~r, ~r′;ω1, ω2) =1

τ

∑

m

φm−(~r)φm−(~r′)∗

γm− − i(ω1 − ω2). (10.55)

Substituting this into Eq. (10.52) we find

R(ω1 − ω2) =2

π2Re∑

m

(

1

(γm+ − i(ω1 − ω2))2+

1

(γm− − i(ω1 − ω2))2

)

.

(10.56)This result, which was first obtained by Altshuler and Shklovskii [Sov.

Phys. JETP 64, 127 (1986)], is a key result in the study of spectral statisticsin small disordered conductors. Let us look at it in more detail. First, weconsider the case in which there is no magnetic field. Then, the cooperon anddiffuson contributions are equal. Finding an exact expression for R(ω1 −ω2)requires solving the diffusion equation, which may be difficult in a samplewith an irregular shape. However, we can make three general statements:First, all γm are non-negative. Second the smallest eigenvalue of the diffusionequation is γ0 = 0. Third, the second smallest eigenvalue γ1 is of order D/L2,where L is the sample size. (The length L is the longest dimension of thesample for a sample with an anisotropic shape.) The smallest eigenvalue γ1

in the absence of a magnetic field is known as the “Thouless energy”, ETh.Hence, if |ω1 −ω2| ≪ ETh the sum in Eq. (10.56) is dominated by the m = 0term, and one has

R(ω1 − ω2) = − 4

π2(ω1 − ω2)2if no magnetic field. (10.57)

208

Notice that this result is completely universal: it does not depend on samplesize, sample shape, or disorder concentration.

If, on the other hand, there is a finite magnetic field, the lowest eigenvalueγ0− for the cooperon propagator is nonzero. For small magnetic fields, onecan find γ0− using perturbation theory,

γ0− =4De2

~2c2

∫

d~r| ~A|2 ∼(

Φe

hc

)2ETh

δ, (10.58)

where δ = 1/νV is the mean spacing between energy levels and the vector

potential ~A is taken in the London gauge (~∇ · ~A = 0 inside the sample and

n · ~∇ ~A = 0 on the sample boundary, where n is the unit vector perpendicularto the sample boundary). Equation (10.58) is valid as long as γ0− ≪ ETh.Hence, for magnetic fields for which γ0− ≫ |ω1 −ω2|, the cooperon contribu-tion to the density of states fluctuations is suppressed, and one finds

R(ω1 − ω2) = − 2

π2(ω1 − ω2)2with magnetic field. (10.59)

The results (10.57) and (10.59) are divergent if ω1 − ω2 → 0. This diver-gence appears because of a level’s “self correlation”: Looking at the defini-tions (10.47) and (10.48), one sees that the correlator R(ω1 − ω2) containsthe singular term

Rself(ω1 − ω2) = 4∑

µ

[〈δ(ω1 − εµ)δ(ω2 − εµ)〉 − 〈δ(ω1 − εµ)〉〈δ(ω2 − εµ)〉]

=4

δδ(ω1 − ω2) −

4

δ2. (10.60)

The origin of the divergence in our diagrammatic calculation is that thediagrammatic perturbation theory breaks down when the energy differenceω1 − ω2 becomes comparable to the spacing δ = V/ν between energy levelsin the sample and thus fails to resolve the delta-function correlations of Eq.(10.60). One way to cure this problem is to give all levels a finite widthγ, which regularizes the divergence arising from the self correlations. Thisamounts to replacing Eqs. (10.57) and (10.59) by

R(ω) =4

π2βRe

1

(γ − iω)2

=4

π2β

γ2 − ω2

(γ2 + ω2)2, (10.61)

209

where β = 2 with a magnetic field and β = 1 without a field. Using field-theoretic methods, Efetov has been able to calculate the exact correlationfunction R(ω) down to ω = 0 [Adv. Phys. 32, 53 (1983)]. Without magneticfield he found

R(ω) = − 4

π2

sin2(ωπ/δ)

ω2− 4

π2

∂

∂ω

[(

sin(ωπ/δ)

ω

)∫ ∞

δ/π

dxsin(ωx)

x

]

for ω 6= 0.

(10.62)With a magnetic field, only the first term on the r.h.s. of Eq. (10.62) is kept.Note that Eq. (10.62) simplifies to our results (10.57) and (10.59) if |ω| ≫ δ.

One striking consequence of the correlator (10.57) and (10.59) is thatthe spectrum is rigid. This means that the fluctuations of the number oflevels N(ω2, ω1) in a certain energy interval ω1 < ω < ω2 increases slowerwith |ω1 − ω2| than |ω1 − ω2|1/2. (Fluctuations proportional to |ω1 − ω2|1/2

are expected if the levels were distributed as uncorrelated random numbers.)Indeed, one finds

varN(ω2, ω1) =

∫ ω2

ω1

dωdω′R(ω − ω′)

∼ 4

π2βln

(ω1 − ω2)2

δ2, (10.63)

where we used the regularized result (10.61) with γ ∼ δ to cut off the diver-gence of R(ω) at small ω.

The universality of the spectral fluctuations for energies below the Thou-less energy is more general than the case of a disordered conductor of arbi-trary shape we considered here. In fact, one finds the same spectral fluctu-ations for the energy levels in a ballistic conductor with an irregular shape,for the resonances in a heavy nucleus, and for the eigenvalues of a hermitianmatrix with randomly chosen elements.42 In these cases, the only relevantinput is the average spacing between levels δ and the symmetry of the sys-tem: without a magnetic field, time-reversal symmetry is present, whereaswith a magnetic field, time-reversal symmetry is broken. For hermitian ma-trices, the presence or absence of time-reversal symmetry corresponds to real

42For a ballistic conductor, the Thouless energy is ETh ∼ vF /L, whereas for a randommatrix of size N one has ETh ∼ Nδ. For a ballistic conductor, R(ω1 −ω2) for |ω1−ω2| ≪vF /L does not depend on the precise sample shape as long as the shape is irregular,whereas for a random matrix R(ω1 − ω2) does not depend on the distribution of thematrix elements if |ω1 − ω2| ≪ Nδ. For a ballistic sample, the universality of spectralstatistics has the status of a conjecture, although there is extensive numerical evidence.

210

or complex matrix elements. Mathematically, random matrices are mucheasier to deal with than disordered conductors. The study of eigenvaluesand eigenvectors of random matrices has grown into a field of its own, called“random matrix theory”. In fact, the label “random matrix theory” is alsoused to describe the universal aspect of spectral statistics of disordered met-als and heavy nuclei within an energy window of width ≪ ETh. You can findmore about the mathematical aspects of the theory of random matrices inthe book Random matrices by M.L. Mehta (Academic, New York, 1991).

211

212

11 Beyond linear response

11.1 formalism

In Chapter 6 we discussed the response of a physical observable to a time-dependent perturbation to linear order in the perturbation. The centralresult of that chapter was the Kubo formula, which relates the expectationvalue of an observable A at time t to the strength of the perturbation atprevious times t′ < t. As we saw in Ch. 6, the linear response can becalculated in terms of an equilibrium correlation function.

If one wants to go beyond linear response, knowledge of equilibrium cor-relation functions only is not sufficient. However, there is a very powerfulextension of the equilibrium formalism to the case of nonlinear response.We’ll discuss that extension in this section.

The quantity of interest is, as always, a Green function. We start fromthe greater and lesser Green functions, since all other Green functions canbe built from these, see Eq. (2.19),

G>A;B(t, t′) = −i〈AH(t)BH(t′)〉, (11.1)

G<A;B(t, t′) = ±i〈BH(t′)AH(t)〉, (11.2)

where the + sign is for fermions and the − sign for bosons and the operatorsare taken in the Heisenberg picture (which is, for now, indicated explicitlyby the subscript H).

We write the Hamiltonian as

H = H0 +H1, (11.3)

where the perturbation H1 includes all interactions between the particles andtime-dependent external fields and the perturbation H1 is switched on slowlyafter a time t0 ≪ t, t′. Since the system is not in thermal equilibrium at thetimes t and t′, we cannot use a direct thermal average to calculate the Greenfunctions G< and G>. However, it is known that the system is in thermalequilibrium at times before the time t0 when H1 was switched on. Using theevolution operator, we can express the operators at times t and t′ in termsof the corresponding operators at times before t0,

AH(t) = U(t0, t)A(t)U(t, t0). (11.4)

Here we have used the interaction picture for the operator A(t) and theevolution operator U . In the interaction picture, the time evolution with

213

tt0

Figure 67: Contour used to calculate the operator A(t) in the interaction picture

the Hamiltonian H0 is included in the operators, e.g., A(t) = exp(iH0(t −t0)/~)A exp(−iH0(t−t0)/~), and excluded from the evolution operator U(t, t0).The evolution operator U(t, t0) can be written as a time-ordered integral overthe perturbation H1(t) (where H1(t) is taken in the interaction picture aswell),

U(t, t0) = U(t0, t)† = Tt exp

[

−i∫ t

t0

dt′H1(t′)

]

. (11.5)

We now rewrite Eq. (11.4) as

AH(t) =[

Ttei∫ t

t0dt′H1(t′)

]

A(t)[

Tte−i∫ tt0

dt′H1(t′)]

=[

Tte−i∫ t0t dt′H1(t′)

]

A(t)[

Tte−i∫ t

t0dt′H1(t′)

]

= Tce−i∫

cdt′H1(t′)A(t), (11.6)

where “c” indicates integration along a contour that starts at t0, goes throught, and returns to t0, see Fig. 67, and Tc implies time-ordering along thecontour.

Similarly, for a correlation function that invovles the product A(t)B(t′)one can use a representation in terms of a contour c that starts at t0, goesthrough both times t and t′, and returns to t0,

G(t, t′) = −i〈Tce−i∫

cdt1H1(t1)A(t)B(t′)〉0

〈Tce−i∫

cdt1H1(t1)〉0

. (11.7)

Here 〈. . .〉0 indicates a thermal average with respect to the Hamiltonian H0.43

The arguments t and t′ refer to a point on the contour c, instead of a time onthe time axis. The contour c can be chosen arbitrariliy, as long as it startsand ends in t0 and passes through the real times t and t′. Direct applicationof Eq. (11.4) for A(t) and a similar result for B(t′) would lead to the contourof Fig. 68a if one were to calculate G< and the contour of Fig. 68b if one

43Note that we assume that both the interactions and the perturbation are switched onafter the time t0, so that H0 is the full Hamiltonian at time t0.

214

tt0

t’

t0

(a) (b)t

t’

Figure 68: Contours used to calculate the greater and lesser Green functions G>

(b) and G< (a).

wants to calculate G>. The contour may be deformed, and parts that aretraversed in both directions may be canceled. You verify that one obtainsthe Kubo formula if Eq. (11.7) is expanded to first order in H1.

A convenient choice is the Keldysh contour, which starts at t0, goes to+∞ and then returns to t0, see Fig. 69. The arguments t and t′ can bechosen on the “upper” and “lower” branches of the contour, cf. Fig. 69. Ifboth arguments are on the upper branch, one has

G(t, t′) = −i〈TtA(t)B(t′)〉, t, t′ upper branch, (11.8)

where Tt indicates the standard time ordering. If both arguments are on thelower branch, G(t, t′) is equal to an “anti-time-ordered” Green function,

G(t, t′) = −iθ(t′ − t)〈A(t)B(t′)〉 ± iθ(t− t′)〈B(t′)A(t)〉≡ −i〈TtA(t)B(t′)〉, t, t′ lower branch. (11.9)

Finally, if t is on the upper branch and t′ is on the lower branch, or viceversa, one has

G(t, t′) = G<(t, t′) if t upper branch, t′ lower branch,

G(t, t′) = G>(t, t′) if t lower branch, t′ upper branch. (11.10)

The Green function G(t, t′) can be represented as a 2 × 2 matrix, where thematrix index indicates what branch of the Keldysh contour is referred to,

G(t, t′) =

(

G11(t, t′) G12(t, t

′)G21(t, t

′) G22(t, t′)

)

, (11.11)

where

G11(t, t′) = −i〈TtA(t)B(t′)〉,

G12(t, t′) = G<(t, t′),

G21(t, t′) = G>(t, t′),

G22(t, t′) = −i〈TtA(t)B(t′)〉. (11.12)

215

tt0 t’

Figure 69: Keldysh contour. The arguments t and t′ can be taken on each branchof the contour.

In the literature, one usually uses a different representation of the matrixGreen function (11.11),

G = Lτ3GL†, (11.13)

where

τ3 =

(

1 00 −1

)

, L =1√2

(

1 −11 1

)

. (11.14)

Using Eq. (11.12) you quickly verify that

G =

(

GR(t, t′) GK(t, t′)0 GA(t, t′)

)

, (11.15)

where GR(t, t′) and GA(t, t′) are the standard retarded and advanced Greenfunctions (but now calculated outside equilibrium) and

GK(t, t′) = G>(t, t′) +G<(t, t′) (11.16)

is the so-called Keldysh Green function. Note that the matrix form (11.15)is preserved under matrix multiplication.

A perturbation expansion in powers of H1 can be done with the helpof Wick’s theorem for the contour-ordered products. One difference withthe imaginary-time version of the diagrammatic theory we used before isthat electron-electron interaction matrix elements and phonon Green func-tion come with a factor i instead of a minus sign. Disconnected diagrams ofthe numerator of Eq. (11.7) cancel the contribution from the denominator.Integrations over the Keldysh contour may be replaced by integrations overthe real axis,

∫

c

dt . . .→∫ ∞

t0

dt[ upper branch ] −∫ ∞

t0

dt[ lower branch].

In the matrix notation of Eq. (11.11), this amounts to matrix multiplica-tion at each vertex (impurity vertex, phonon vertex, or interaction vertex),

216

(a) (b)

Figure 70: First-order diagrams for impurity scattering and phonon scattering.

combined with insertion of a minus sign at the lower branch argument. Forexample, the diagram of Fig. 70a for the single-electron Green function withone impurity scattering is evaluated as

G(1)ij (~r, t;~r′, t′) =

∫

d~r′′∫

dt′′∑

k

Gik(~r, t;~r′′, t′′)U(~r′′)σkGkj(~r, t

′′, ~r′, t′),

where σ1 = 1 and σ2 = −1. Similarly, the phonon diagram of Fig. 70b reads

G(1)ij (~r, t;~r′, t′) = g2

∫

d~r1d~r2

∫

dt1dt2∑

kl

Gik(~r, t;~r1, t′′)σk

× (iDkl(~r1, t1;~r2, t2))Gkl(~r1, t1;~r2, t2)σlGlj(~r2, t2;~r′, t′)

where Dkl(~r, t;~r′, t′) is the matrix phonon Green function and the electron-

phonon coupling constant g~q in the electron-phonon Hamiltonian (9.5) ischosen independent of ~q.

Upon transformation to the matrix representation of Eq. (11.15) thosediagrammatic rules are somewhat modified. Performing the transforma-tion (11.13), one finds that an impurity vertex carries no sign, whereas theelectron-phonon or Coulomb vertices are modified as in Fig. 71, where

γij1 = γij2 =1√2δij , γij2 = γij1 =

1√2(τ1)ij , (11.17)

where τ1 is the Pauli matrix. Note that, unlike in the equilibrium diagramswe considered previously and unlike in the matrix language of Eq. (11.11),phonon and interaction lines need to have a direction. However, the resultsdo not depend on the direction of the assigned direction and the direction ofthe phonon or interaction lines has no physical meaning.

For future use, we now write down the Dyson equation for the single-electron Green function. As before, calling all diagrams that can be cut in

217

absorption

emission

γijk

γijk

~

j

i

k

i

j

ki

i

j

jk

k

Figure 71: Diagrammatic rules for electron-phonon and Coulomb vertices.

two by deleting a single fermion line “reducible”, we can write

Gij(~r, t;~r′, t′) = (G0)ij(~r, t;~r

′, t′) +

∫

dt1dt2

∫

d~r1d~r2∑

kl

(G0)ik(~r, t;~r1, t1)

× Σkl(~r1, t1;~r2, t2)Glj(~r2, t2;~r′, t′), (11.18)

see Fig. 72. Here G0 is matrix Green function for the Hamiltonian H0 and Σis the self energy, the sum of all “irreducible” diagrams (excluding externalfermion lines, but including the matrices γ and γ if necessary). The selfenergy has matrix structure

Σ =

(

ΣR(~r, t;~r′, t′) ΣK(~r, t;~r′, t′)0 ΣA(~r, t;~r′, t′)

)

. (11.19)

Using the operator language for the dependence on the coordinates ~r, ~r′ andthe times t and t′ and matrix representation, we can rewrite the solution ofthe Dyson equation as

G =[

G−10 − Σ

]−1. (11.20)

In the operator language, the inverse Green function G−10 is equal to

G−10 =

[

i∂

∂t−H0

]

1, (11.21)

where H0 is the unperturbed Hamiltonian in first quantization form and 1 isthe 2 × 2 unit matrix. Note that adding an imaginary infinitesimal ±iη to

218

+ + ...

+

+=

=

Figure 72: Diagrammatic representation of the Dyson equation for the matrixGreen function G. The shaded circle represents the self energy Σ and the singleline represents the unperturbed Green function G0.

G−10 does not make a difference here, hence there is no need to distinguish

retarded and advanced components. Alternatively, using operator languagefor the dependence on ~r and t, the Dyson equation (11.20) can be rewrittenas

(G−10 − Σ)G = 1 = G(G−1

0 − Σ). (11.22)

In order to be a useful starting point for calculations, Eq. (11.20) shouldbe supplemented with a further theory or approximation scheme for the selfenergy

As all single-electron Green functions, the matrix Green function G canbe chosen to depend on two positions ~r and ~r′, on two momenta ~k and ~k′,or on the quantum numbers µ and µ′ of any other complete set of single-electron states. The above diagrammatic rules do not depend on the choiceof the representation as long as the first and second arguments of the Greenfunctions are not mixed. For some applications, it is favorable to use a“mixed” representation in which the Green function G is chosen to dependon sum and difference coordinates, followed by a Fourier transform to thedifference coordinate,

G(~R, T ;~k, ω) =

∫

d~r

∫

dte−i~k·~r+iωtG(~R+ ~r/2, T + t/2; ~R− ~r/2, T − t/2).

(11.23)

We’ll refer to ~R and T as “center coordinate” and “center time”, and to the ~kand ω as momentum and frequency. The “mixed representation” or “Wignerrepresentation” is particularly useful in a semiclassical approximation, whenelectrons are assigned both a momentum and a position. However, in themixed representation the convolution of Green functions is changed. UsingEq. (11.23) and its inverse to express the operator product [G1G2] in the

219

Wigner representation in terms of the Wigner representation of the two fac-tors G1 and G2, one finds that the convolution of Green functions G1 andG2 is given by

[G1G2](~R, T ;~k, ω) = ei2

(

∂1∂1

~R

∂2∂2

~k−

∂1∂1T

∂2∂2ω

−∂2

∂2~R

∂1∂1

~k+

∂2∂2T

∂1∂1ω

)

G1(~R, T ;~k, ω)

×G2(~R, T ;~k, ω), (11.24)

where ∂1 refers to a derivative with respect to an argument of the first Greenfunction and ∂2 refers to a derivative with respect to an argument of thesecond Green function.

In general, the advanced and retarded Green function contain the infor-mation on the available states (appropriately modified for the nonequilibriumstate of the system at times t and t′), whereas the Keldysh Green functioncontains information on the (nonequilibrium) occupation of these states. In-deed, in equilibrium, the greater and lesser Green functions are related to thespectral density via Eqs. (2.46) and (2.47). This relation implies a relationbetween the Keldysh component and the advanced/retarded components ofthe matrix Green function G,

GK0 = tanh(ω/2T )

(

GR0 −GA

0

)

. (11.25)

A similar relation holds for the self energy in equilibrium. (Note that, inequilibrium, the Green functions do not depend on the “center time” T , butthey may depend on both the center coordinate ~R and the momentum ~k.)

In the mixed representation, the unperturbed Hamiltonian H0 takes aparticularly simple form,

H0 = ε~k + U(~R), (11.26)

where U is the potential.

11.2 Boltzmann equation

The Boltzmann equation, which was used in Sec. 5.4 as an alternative methodto derive the Drude formula for the conductivity of a normal metal, is an ex-ample of a “kinetic equation”. In general, a “kinetic equation” is an equationthat describes the time-dependence of the occupation of quantum mechan-ical states. A time-evolution equation of the Keldysh Green function is anexample of a quantum kinetic equation, since the Keldysh Green functioncontains information about the quantum states and their occupation.

220

Physical observables are expressed in terms of the Green functions. Ex-amples are the electron charge density

ρe(~r, t) = −eG<(~r, t;~r, t), (11.27)

or the current density

~je(~r, t) = − ~e

2im(~∇~r − ~∇~r′)G

<(~r, t;~r′, t)

∣

∣

∣

∣

~r′→~r

− e2

m~A(~r)G<(~r, t;~r, t), (11.28)

where ~A is the vector potential and a summation over spin degrees of freedomis implied. In linear response, one can find the charge and current densitiesfrom the Kubo formula. Beyond linear response, one needs a solution of theDyson equation for the matrix Green function G. Here we’ll describe howsuch a solution can be shown to be equivalent to a solution of the Boltzmannequation.

Starting point of our discussion is the Dyson equation (11.22) in whichwe subtract the far left and far right sides. In operator language, one thusfinds

[G−10 − Σ, G]− = 0, (11.29)

where [·, ·]± denotes the commutator (− sign) or anticommutator (+ sign).The Keldysh component of this equation reads

G−10 GK −GKG−1

0 − ΣRGK +GKΣA − ΣKGA +GRΣK = 0. (11.30)

We rewrite this result in terms of commutators and anticommutators,

[G−10 − ℜΣ, GK]− − [ΣK,ℜG]− =

i

2[ΣK, A]+ − i

2[Γ, GK]+, (11.31)

where A = i(GR −GA) is the spectral weight of the full Green function and

Γ = i(ΣR − ΣA),

ℜΣ =1

2(ΣR + ΣA),

ℜG =1

2(GR +GA). (11.32)

221

The definition of “real” and “imaginary” parts is the same one as used inChapter 2. Note that in equilibrium one has GK = −i tanh(ω/2T )A andΣK = −i tanh(ω/2T )Γ, so that the right hand side of Eq. (11.31) vanishes.

We can make further progress if we assume that the Green functions in-volved are slowly varying functions of their arguments. Then, in taking theGreen function products according to the rule (11.24) we can expand theexponential and truncate the expansion after first order. For an anticommu-tator, the derivatives cancel, and one finds

[G1, G2]+ = 2G1G2 + higher order gradient terms. (11.33)

Similarly, one finds that a commutator becomes equal to a Poisson bracket,

−i[G1, G2]− = [G1, G2]Poisson (11.34)

=

(

∂1

∂1~R

∂2

∂2~k− ∂1

∂1T

∂2

∂2ω− ∂2

∂2~R

∂1

∂1~k

+∂2

∂2T

∂1

∂1ω

)

G1G2.

This approximation is known as the “gradient expansion”.Without further knowledge of the system under consideration this is as

far as one can go. We’ll now specialize to the case of a system with impurityscattering but without interactions between the electrons. Taking the casein which a slowly varying external potential Uext may be present, we have,in the Wigner representation,

G−10 = ω − (ε~k − µ) − Uext(~R, T ). (11.35)

Using the self-consistent Born approximation for the self energy, we find,in the Wigner representation,

Σ(~R, T ;ω,~k) =Nimp

V

∑

~k′

|u~k−~k′|2G(~R, T ;ω,~k′), (11.36)

where u~k is the Fourier transform of the impurity potential of a single impu-rity and Nimp is the number of impurities. In the dilute impurity limit andfor slowly varying external potentials, we can use the gradient expansion,neglect the ~R and T dependence of the self-energy, and drop the self energyfrom the l.h.s. of Eq. (11.31),

[G−10 , GK]Poisson = ΣKA− ΓGK. (11.37)

222

Also, in the dilute impurity limit and for slowly varying external potentials,the spectral density A approaches a delta function,

A = 2πδ[ω − (ε~k − µ) − U ]. (11.38)

As a result of the delta-function character of A, the Keldysh Green functionGK is also strongly peaked as a function of ω. Hence, we write

GK = −2πiδ[ω − (ε~k − µ) − U ](1 − 2f~k), (11.39)

where

f~k(~R, T ) =

1

2− 1

4πi

∫

dωGK(~R, T ;ω,~k). (11.40)

Then, writing the Poisson bracket explicitly in Eq. (11.37), integrating overω, and performing one partial integration, we find

(

∂

∂T+ ~v~k · ~∇~R − ~∇~RUext · ~∇~k

)

f~k(~R, T )

= −2πNimp

V

∑

~k′

|u~k−~k′|2δ(ε~k − ε~k′)(f~k − f~k′), (11.41)

where ~v~k = ~∇~kε~k.Equation (11.41) is the Boltzmann equation. In Sec. 5.4 you used it to

derive the Drude formula for the conductivity of a normal metal. Hence, thepresent discussion, together with that of Sec. 5.4 can be seen as an alternativederivation of that result.

The function f~k that appears in the Boltzmann equation plays the role ofan “occupation factor”. Note that it contains an integration over frequency ω,so that f~k is related to an equal-time Green function. Using the relationshipG< = (GK + iA)/2, together with Eqs. (11.38) and (11.39) and Eqs. (11.27)and (11.28), one quickly recovers the well-known formulas for charge densityand current density,

ρe(~R, T ) = − e

V

∑

~k

f~k(~R, T ), (11.42)

~je(~R, T ) = − e

V

∑

~k

~v~kf~k(~R, T ). (11.43)

Also note that, by Eqs. (11.25), (11.38) and (11.40), one has f~k = [1 +

exp((ε~k − µ)/T )]−1 in equilibrium, indepenent of ~R and T .

223

The Boltzmann equation is only a simple example of a kinetic equa-tion. Generalizations of the Boltzmann equation have been derived in theKeldysh formalism for electron-phonon interactions, for weak localizationand electron-electron corrections to the conductivity of a normal metal, andfor quasiparticle dynamics in superconductors. For more details, you are re-ferred to the review “Quantum field-theoretical methods in transport theoryof metals”, by J. Rammer and H. Smith, Rev. Mod. Phys. 58, 232 (1986).

224

12 Electrons in one dimension

12.1 Why is one dimension different?

With the exception of a short discussion in chapter 10 we have studied elec-trons in three dimensions. We encountered two possible excitations of anelectron liquid in three dimensions: particle-hole pairs and plasma modes.These excitations appeared as the support of the imaginary part of the po-larizability and susceptibility at finite frequency ω and finite wavevector ~q.The plasma modes have a high threshold frequency for excitation, leavingthe particle-hole pairs as the main type of excitations at long wavelengthsand low frequencies. This observation, together with the observation fromFermi liquid theory that particle-hole pairs are long-lived excitations if theirenergy is low, is the main justification of the use of the non-interacting elec-tron picture as the starting point for our description of the three-dimensionalelectron liquid.

In one dimension, the situation is completely different. Here, with “one di-mension” we mean a truly one dimensional system, i.e., a quantum wire witha width a so small that only one transverse mode is populated at the Fermienergy. In practice, this means a ∼ λF , where λF is the Fermi wavelength.44

In order to see why the one-dimensional electron liquid is so different, we re-examine the polarizability and spin susceptibility in a one-dimensional wire.

Before we look at the polarizability, we need to know the Coulomb inter-action in one dimension. For a wire of thickness a in vacuum, the Coulombinteraction is

V~q =e2

4πǫ0

∫

dxe−iqx

|x|

≈ e2

2πǫ0ln

1

qa, (12.1)

which is valid for qa ≪ 1. We have cut off the integral at x ∼ a, since forx . a a three-dimensional form of the Coulomb interaction should be used.If interactions in the one-dimensional wire are screened by a nearby metal atdistance b, the interaction V~q saturates at q ∼ 1/b.

44This use of the word “one dimensional” is different from that of Chapter 10, where“one dimensional” referred to a system that is much longer than it is wide, but that doesnot need to be as narrow as λF . In order to distinguish the two cases, the latter case issometimes referred to as “quasi one dimensional”.

225

In the random phase approximation, the polarizability reads

χe(~q, ω)RPA =χ0R

e (~q, ω)

1 − V~qχ0Re (~q, ω)/e2

, (12.2)

where V~q is given by Eq. (12.1) above and

χ0Re (~q, ω) =

e2

V

∑

~k

tanh[(ε~k+~q − µ)/2T ] − tanh[(ε~k − µ)/2T ]

ε~k − ε~k+~q + ω + iη(12.3)

is the polarizability of the one-dimensional non-interacting electron gas. Inone dimension, the ~k summation in Eq. (12.3) is easily done: the momentum~k is represented by a real number k, and one has εk+q = εk + qvF sign k.Then, replacing the summation over k by an integration over εk, one has

χ0Re (~q, ω) =

e2ν

2

(

2qvF

ω + iη − qvF− 2qvF

ω + iη + qvF

)

=2e2q2vF

π[(ω + iη)2 − q2v2F ], (12.4)

where we used the fact that, in one dimension, the density of states ν isrelated to the Fermi velocity vF as ν = 1/πvF . Substituting this into Eq.(12.2), one finds

χe(~q, ω)RPA =2e2q2vF

π(ω + iη)2 − πq2v2F − (e2/πǫ0) ln(1/qa)q2vF

. (12.5)

The one-dimensional electron liquid can dissipate energy only if the imag-inary part of χe(~q, ω) is nonzero. Inspection of Eq. (12.5) shows that that isthe case for

ω = qvc, (12.6)

where the velocity vc is

vc = vF

√

1 +e2

π2vF ǫ0ln

1

qa. (12.7)

In the presence of a metal gate at distance b from the wire that screenes theCoulomb interaction inside the wire, the momentum q in Eq. (12.7) shouldbe replaced by 1/b.

226

ω/εF

q/kF

30 4210

2

4

modesplasmon

spin modes

particle−holeexcitations

Figure 73: Support of the polarizability χe and the transverse spin susceptibilityχ−+ for a one-dimensional interacting electron liquid in the RPA approximation.

The result (12.6) describes the one-dimensional version of plasma oscil-lations. Unlike in three dimensions, where the plasma frequency remainedfinite (and large) if the wavevector q → 0, in one dimension ω → 0 if q → 0.However, the plasmon speed is larger than the Fermi velocity, since the plas-mon propagation speed is enhanced by the repulsive Coulomb interactions.Hence, in one dimension, plasmons are relevant perturbations in the lowfrequency, long wavelength limit.

However, this is not the entire story. The result (12.5) differs in onemore aspect from its three dimensional counterpart: There is no particle-hole continuum for excitations, not even a single particle-hole branch! Notethat at ω = ±qvF the polarizability of Eq. (12.5) is well behaved and has noimaginary part. Hence, we conclude that the only charged excitations in aone dimensional electron gas are the plasmon collective modes.

The situation is different if the wavevector q is of order kF , or if thetemperature or frequency are of order εF . In that case, one cannot ap-proximate the spectrum by a linear dispersion. Without interactions, thepolarizibility function χ0R

e (q, ω) becomes a truly complex function of q andω for qvF − q2/2m < ω < qvF + q2/2m, just like in three dimensions, and,hence, the full polarizability χRPA

e becomes complex as well for that wavevec-tor and frequency range. The support of Imχe for the entire frequency rangeis shown in Fig. 73.

If we want to investigate spin excitations, we should look at the spin sus-

227

ceptibility. Taking the Hubbard-model result for the transverse susceptibilityχ−+(~q, ω),

χR−+(~q, ω) =

χ0R−+(~q, ω)

1 − (2U/µBg)χ0R−+(~q, ω)

, (12.8)

where

χ0R−+(~q, ω) = −µGg

4V

∑

~k

tanh[(ε~k+~q − µ)/2T ] − tanh[(ε~k − µ)/2T ]

ε~k − ε~k+~q + ω + iη

= − µBgq2vF

2π[(ω + iη)2 − q2v2F ]

(12.9)

is the transverse susceptibility of the non-interacting electron gas, we findthat the imaginary part of χ−+(~q, ω) is nonzero if and only if

ω = vsq, (12.10)

wherevs = vF

√

1 − U/πvF . (12.11)

The spin excitations are collective excitations as well, moving at a speedthat is below the Fermi velocity. The reason why vs is smaller than vF isthat the propagating of spin excitations is slowed down by the ferromagneticexchange interaction.45 At the Stoner instability, Uν = 1, the propagationspeed has come to zero, and large-scale ferromagnetic fluctuations becomepossible. Note that, again, χ−+ is purely real at ω = vF q.

The phenomenon that, in general, vs is different from vc is known as“spin-charge” separation. In a one-dimensional electron liquid, excitationswith spin move at a different speed than excitations with charge. In view ofwhat we just discussed, this is not a big surprise. Since, in one dimension, theonly possible excitations are collective modes, there is no reason to expectthat spin modes and charge modes have the same velocity. You may recallthe calculations of the zero sound velocity in a Fermi liquid, where spin anddensity modes depend on different Fermi-liquid constants and, hence, havedifferent propagation velocities.

45In three dimensions, the fact that vs < vF implies that spin excitations are stronglydamped because they can decay into particle-hole pairs (Landau damping). In one dimen-sion, there are no particle-hole excitations, so that spin excitations are long-lived in spiteof the fact that their propagation speed is below the Fermi veloicty.

228

Quantitatively, our observations have been based on the random phaseapproximation. Going beyond the random phase approximation will mostcertainly change our estimates for the velocities vc and vs of collective chargeand spin excitations in the one-dimensional electron system, but not the qual-ititative conclusion that the collective modes are the only long-wavelengthexcitations of the one-dimensional electron system. We’ll see this in Sec.12.3, where we start from a microscopic model for electrons in one dimen-sion and show rigorously that all dynamics is collective. Another way to seethis is to note the similarity between electrons in one dimension and ions ina lattice. The similarity arises because, in one dimension, electrons cannotpass each other. In that sense, a description in terms of a “solid” wouldbe more appropriate than a description in terms of a “liquid”.46 Indeed,all long-wavelength dynamics of lattice ions is collective; it is only at shortwavelengths that the properties of individual ions become important.

We conclude that, for long wavelengths and low frequencies, all excita-tions of the one-dimensional metal are collective modes; there are no dissi-pative modes at |ω/q| = vF , which is the frequency-wavevector relation oneexpects for particle-hole excitations in a degenerate one-dimensional Fermigas. This observation makes us wonder whether a better description thana description in terms of particles is possible. Can one make a theory thatuses the collective modes as the building blocks, rather than the individualfermions?

Another reason to look for a theory that does not start from a system ofnon-interacting fermions is that, in one dimension, the quasiparticle lifetimescales inversely proportional to the excitation enery of the quasiparticle. Thisis different from three dimensions, where the life time is inversely proportionalto the square of the excitation energy. In three dimensions, this dependencewas the basis for “Fermi Liquid theory”, the statement that a picture basedon non-interacting electrons is a good starting point. In one dimension,quasiparticle decay is much faster, and Fermi Liquid theory is not valid.What description one used instead follows from the above considerations, aswe’ll see in the next sections.

46The difference between a “solid” and a “liquid” has to do with the presence of trans-verse rigidity (shear). In one dimension, there is no transverse direction, so that this formaldifference between “solid” and “liquid” disappears. We usually denote the one-dimensionalelectron system as a “liquid”, because of the lack of long-range order. However, as we seefrom the present discussion, one-dimensional electron systems have characteristics of bothsolids and liquids.

229

12.2 Effective Hamiltonian for a one-dimensional elec-

tron liquid

Based on the considerations of the previous section, we now write down aneffective Hamiltonian that has the same (collective) excitations as the one-dimensional electron liquid. For simplicity, we consider the case of spinlessfermions, so that the only collective excitations are the plasmon modes. Also,we assume that the long-range part of the Coulomb interaction in the wireis screened by a nearby piece of metal, and, hence, take the plasmon velocityvc to be independent of q.

With these observations, we are tempted to write down the followingHamiltonian as an effective Hamiltonian for the one-dimensional electronliquid,

H =∑

q

ωq(a†qaq + 1/2), (12.12)

where the summation over the wavenumber q extends over both positive andnegative q, with ωq = vc|q|, and where a†q and aq are boson creation andannihilation operators that obey commutation relations,

[

aq, aq′

]

−=

[

a†q, a†q′

]

−= 0

[

aq, a†q′

]

−= δq,q′.

As we discussed above, the Hamiltonian (12.12) is valid for long wavelengthsand low frequencies only. At high wavenumbers |q| ∼ kF , particle-hole exci-tations become important, and our description in terms of collective modesonly ceases to be valid.

Although there is nothing wrong with the Hamiltonian (12.12) as aneffective Hamiltonian for the spinless one-dimensional electron liquid, we cangain considerably more insight if we replace the creation and annihilationoperators a† and a by “momentum” and “displacement” operators Π and φ,where

φq =

√

~

2Kωq(aq + a†−q),

Πq =

√

K~ωq

2

(

ia†q − ia−q

)

. (12.13)

230

We will use the freedom of the arbitrary constant K later in order to givea physical interpretation of the operators Π and φ. You verify that these“momentum” and “displacement” operators satisfy the usual commutationrules for canonically conjugate variables,

[Πq,Πq′]− = [φq, φq′]− = 0

[Πq, φq′]− = −i~δqq′ . (12.14)

In terms of the new variables Π and φ we write the Hamiltonian as

H =∑

q

(

1

2KΠqΠ−q +

1

2Kω2

qφqφ−q

)

. (12.15)

Next, we perform a Fourier transform to real space,

Π(x) =

√

1

L

∑

q

Πqe−iqx,

φ(x) =

√

1

L

∑

q

φqeiqx. (12.16)

Since the effective Hamiltonian (12.12) is valid for low q only, the summationover q should be restricted to q ≪ kF only. This means that delta functionsand divergencies appearing in the real-space formulation should be cut off atdistances x ∼ 1/kF . In the real-space formulation, the effective Hamiltonianbecomes

H =

∫

dx

(

1

2KΠ(x)2 +

1

2Kv2

c (∂xφ(x))2

)

. (12.17)

The commutation relations of the fields φ(x) and Π(x) are

[φ(x), φ(x′)]− = [Π(x),Π(x′)]− = 0

[Π(x), φ(x′)]− = −i~δ(x− x′). (12.18)

Let us now calculate time-derivative of φ(x),

∂

∂tφ(x) =

i

~[H, φ(x)]− =

1

KΠ(x). (12.19)

We want the operators Π and φ to be related to the electron current densityand particle density, respectively. If such an identification is to hold, Eq.

231

(12.19) should represent the continuity equation ∂tn = −∂xj. Taking aderivative to x on both sides of the equation, we see that such an identificationholds, if we identify the (excess) particle density n(x) with the x-derivativeof φ(x) and the current with Π(x). In fact, the following identifications aremade,

n(x) =1

π∂xφ(x), (12.20)

j(x) = −vF Π(x), (12.21)

which implies K = 1/πvF . However, different identifications are possible,depending on the microscopic details of the one-dimensional electron liquid.

Combining everything, we see that we have arrived at the effective Hamil-tonian

H =~vF

2π

∫

dx

[

π2

~2Π(x)2 + g2

(

∂φ(x)

∂x

)2]

, (12.22)

where we introduced the dimensionless parameter

g =vc

vF. (12.23)

You may recognize Eq. (12.22) as the Hamiltonian of an elastic string, whereφ is the displacement and Π is the momentum density. The Hamiltonian(12.22) is quite different from the Hamiltonian you would write down for non-interacting fermions in one dimension. It reflects the fact that all excitationsare collective and, hence, bosonic, rather that quasi-particle like. In orderto stress the difference of the collective dynamics of the one-dimensionalelectron liquid and the quasi-particle dynamics of the Fermi liquid in higherdimensions, the former case is referred to as “Luttinger Liquid”.

The Hamiltonian (12.22) is a great starting point if one wants to studythe excitation spectrum and thermodynamic quantities of a one-dimensionalspinless electron liquid. One example is the calculation of the compressibilityκ, which is the derivative of the particle density to the chemical potentialµ = ∂E/∂n, where E is the energy of the electron liquid,

κ =∂n

∂µ=

(

∂2E

∂n2

)−1

. (12.24)

According to the effective Hamiltonian (12.22), one has κ = vF/πv2c . Without

interactions, when vc = vF , ths simplifies to κ = 1/πvF = ν, ν being the

232

density of states. With interactions, κ is decreased by a factor g2 = (vc/vF )2,reflecting the increased energy cost for addition of charge. Similarly, youverify that the specific heat is decreased by a factor 1/g2 with respect to thenon-interacting electron case.

For some applications it is important to be able to make reference toindividual electrons. An example is a tunneling experiment, where electronstunnel into a one-dimensional electron liquid from a weakly coupled elec-trode. Since we employ a long-wavelength description, we will not attemptto describe a process in which an electron that is created or annihilated is lo-calized within a wire segment of length comparable to the Fermi wavelength.Instead, we’ll want to write down an operator that creates an electron thatis delocalized over a piece of wire of length λ ∼ λF . This spatial “smear-ing” amounts to a momentum cut off factor exp(−|q|λ/2) in Eq. (12.16) or,equivalently, replacing the fields φ and Π by

φ(x) →∫

dx′2λ

π[λ2 + 4(x− x′)2]φ(x′),

Π(x) →∫

dx′2λ

π[λ2 + 4(x− x′)2]Π(x′). (12.25)

On the other hand, since we give up spatial resolution on length scales belowλF , we can specify the momentum of the electron to within kF , i.e., wecan specify whether the electron moves left (L) or right (R). The creationoperator of a right moving electron at position x then becomes47

ψ†R(x) =

1√2πλ

U †Re

−ikF x+iφ(x)−iθ(x). (12.26)

Similarly, for a left-moving particle one has

ψ†L(x) =

1√2πλ

U †Le

ikF x−iφ(x)−iθ(x). (12.27)

Here the auxiliary field θ(x) is defined as

θ(x) =π

2~

∫

dx′Π(x′)sign (x− x′). (12.28)

47In the next section, where we give a more formal derivation of the Hamiltonian (12.22),you can find more details of the formal relationship between the fields φ(x) and Π(x) andthe underlying electron creation and annihilation operators ψ† and ψ.

233

ρ

φ

x’x

Figure 74: Upon creation of an electron at position x, the field φ acquires a kink,corresponding to a peak in the excess density ρ.

You verify that θ commutes with itself, whereas the commutator with φ isgiven by

[θ(x), φ(x′)]− = [φ(x), θ(x′)]− = −iπ2sign (x− x′). (12.29)

In these equations, the fast exponential factor exp(±ikFx) corresponds tothe orbital phase of left and righ moving electrons. The operator U † isknown as the “Klein factor”. It is a formal operator that increases the totalnumber of left or right moving particles by one. It is necessary, because theoperators φ(x) and Π(x) conserve the total number of particles. The operatorexp[−iθ(x)] produces a shift of the field φ(x′) by −π/2 at x′ < x and by π/2at x′ > x. Hence, the derivative ∂xφ is peaked near x, corresponding toan excess particle density at that point, see Fig. 74.48 The operator eiφ(x)

changes sign each time φ is increased by π, i.e., each time a particle passesthrough the point x. This property enforces the fermion anticommutationrules for the ψ-operators. Furthermore, e±iφ(x) displaces the field Π(x) by one,corresponding to a current density ∓vF δ(x). Finally, the prefactor (2πλ)−1/2

is chosen in conjunction with the regularization (“smearing”) procecure forthe fields φ and Π. It is chosen such that a calculation of the fermion Greenfunction for non-interacting electrons gives the same result in boson and infermion language.

48This statement does not conflict with the fact that the operator φ conserves particlenumber. At the upper and lower ends of the wire there is an “antikink” in φ, correspondingto a decrease of the particle number by one (in total). It is the operator U † that keepstrack of the total particle number.

234

The information of the boson fields Π and φ is contained in their Greenfunctions. The calculation of these Green functions is quite similar to thatof the phonon Green functions of Ch. 3. For technical reasons, we calculateGreen functions for the fields φ and θ. In terms of the fields θ and φ, theHamiltonian reads

H =~vF

2π

∫

dx

[

(

∂θ(x)

∂x

)2

+ g2

(

∂φ(x)

∂x

)2]

. (12.30)

In order to calculate the Green functions of the fields θ and φ, we use theequation of motion approach. Hereto we need the imaginary time evolution,

∂

∂τθ(x, τ) = −ivF g

2∂φ(x, τ)

∂x,

∂

∂τφ(x, τ) = −ivF

∂θ(x, τ)

∂x. (12.31)

We define the temperature Green functions

Dθθ(x, τ) = −〈Tτθ(x, τ)θ(0, 0)〉,Dθφ(x, τ) = −〈Tτθ(x, τ)φ(0, 0)〉,Dφφ(x, τ) = −〈Tτφ(x, τ)φ(0, 0)〉,Dφθ(x, τ) = −〈Tτφ(x, τ)θ(0, 0)〉. (12.32)

Using Eq. (12.31) we find that these Green functions satisfy the equations ofmotion

∂

∂τDθθ(x, τ) = −ivF g

2∂Dφθ(x, τ)

∂x,

∂

∂τDθφ(x, τ) = i

π

2δ(τ) sign (x) − ivF g

2∂Dφφ(x, τ)

∂x,

∂

∂τDφθ(x, τ) = i

π

2δ(τ) sign (x) − ivF

∂Dθθ(x, τ)

∂x,

∂

∂τDφφ(x, τ) = −ivF

∂Dθφ(x, τ)

∂x. (12.33)

The solution to these equations is easily found by inspection,

Dθφ(x, τ) = Dφθ(x, τ)

=1

4ln sin[πT (τ + ix/vc)sign (τ)]

235

− 1

4ln sin[πT (τ − ix/vc)sign (τ)],

vF

vcDθθ(x, τ) =

vc

vFDφφ(x, τ)

=1

4ln sin[πT (τ + ix/vc)sign (τ)]

+1

4ln sin[πT (τ − ix/vc)sign (τ)]. (12.34)

Regularization amounts to smearing of the boson fields by a Lorentzian factor2λ/[π(λ2 + 4(x− x′)2], see Eq. (12.25) above. In the Green functions such asmearing amounts to the substitution x → x± iλsign (τ). This correspondsto the addition of λπT/vF to the argument of the sine function, so that thedivergence of the Green functions at x→ 0 and τ → 0 is cut off at distancesof order λ and times of order λ/vF .

As an example, let us now calculate a fermion Green function using theboson language. We consider the temperature Green function for right-moving electrons,

GRR(x, τ) = −〈TτψR(x, τ)ψ†R(0, 0)〉. (12.35)

Using Eq. (12.26), we write this as

GRR(x, τ) = −eikF x

2πλ〈TτUR(τ)U †

R(0)〉⟨

Tτe−iφ(x,τ)+iθ(x,τ)+iφR(0,0)−iθ(0,0)

⟩

.

(12.36)Since the total Green function needs to be antiperiodic in τ , whereas theboson part of the Green function is manifestly periodic in τ , we require thatthe time-ordering for the Klein factors UR and U †

R is that of fermions. Sincethe operators UR and U †

R are simply ladder operators, we have

〈TτUR(τ)U †R(0)〉 = sign (τ). (12.37)

Calculating the boson part of the Green function is easier than it seemsat first sight. Since the Hamiltonian of the boson fields is quadratic, we canuse the cumulant expansion, which, for 〈φ(x, τ)〉 = 〈θ(x, τ)〉 = 0, reads⟨

Tτe−iφ(x,τ)+iθ(x,τ)+iφ(0,0)−iθ(0,0)

⟩

= e12〈(iφ(0,0)−iθ(0,0)−iφ(x,τ)+iθ(x,τ))2〉

= eDφφ(0,0)−Dφθ(0,0)−Dθφ(0,0)+Dθθ(0,0)

× e−Dφφ(x,τ)+Dφθ(x,τ)+Dθφ(x,τ)−Dθθ(x,τ).

(12.38)

236

Without regularization, the equal-time and equal-position Green functionsDφφ(0, 0) + Dθθ(0, 0) are divergent (and negative). With regularization, thisdivergence is cut off at distances ∼ λ (see above). Substituting Eq. (12.34)and putting everything together, we find

GRR(x, τ) = − 1

(2πλ)1−νsign (τ)

(

T/2vc

sin[πT (τ + ix/vc)sign (τ)]

)ν/2−1/2

×(

T/2vc

sin[πT (τ − ix/vc)sign (τ)]

)ν/2+1/2

, (12.39)

where we abbreviated

ν =1

2

(

g +1

g

)

=1

2

(

vc

vF+vF

vc

)

. (12.40)

It is instructive to compare this result to the electron Green functioncalculated in the fermion language,

GRR(x, τ) = − T/2vF

sin[πT (τ − ix/vF )]. (12.41)

Taking the result obtained in the boson language, and setting g = 1, werecover the fermion Green function. This argument, a posteriori, fixes thecut-off dependent prefactor (2πλ)−1/2 in the relations (12.26) and (12.27)between the fermion creation/annihilation operators and the boson fields φand Π.

There is an important difference between the single-electron Green func-tion for g = 1 and for g 6= 1. In the absence of interactions, the single-electron Green function has a simple pole at −iτ = x/vF . With interactions,the location of the singularity shifts, and the singularity acquires a differentanalytical structure. This may be brought to light by a calculation of the re-tarded Green function, i.e., by performing a Fourier transform to Matsubarafrequencies followed by analytical continuation iωn → ω + iη. For simplicitywe look at the equal-position Green function,

GRR(0, iωn) =

∫ 1/T

0

GRR(0, τ)eiωnτ , (12.42)

which is related to the spectral density ARR for right-moving electrons. Wedeform the τ -integration as shown in Fig. 75, avoiding the branch cuts for

237

τ0 1/T

Figure 75: Integration contour for the calculation of the density of states in aLuttinger liquid.

Re τ = 0 and Re τ = 1/T . After the deformation of the contours, the integralreads

GRR(0, iωn) = − 2i

(2πλ)1−ν

∫ ∞

0

dte−ωntRe

(

T/2vF

sin[πT (it+ λ/vF )]

)ν

= − 2i

(2πλ)1−ν

∫ ∞

0

dte−ωnt

× Re e−iπν/2

(

T/2vF

sinh[πT (t− iλ/vF )]

)ν

. (12.43)

Now we can take the analytical continuation iωn → ω+ iη and calculate theretarded Green function,

GRRR(0, ω) = − 2i

(2πλ)1−ν

∫ ∞

0

dteiωtRe e−iπν/2

(

T/2vF

sinh[πT (t− iλ/vF )]

)ν

.

(12.44)

Evaluation of the remaining integral depends on whether the energy ω issmall or large in comparison to T . If ω ≪ T , we find

GRRR(0, ω) ≈ − 2i

2πvF

(

πTλ

vF

)ν−1

. (12.45)

238

(The numerical prefactor is valid for ν close to unity.) You verify that oneobtainsGR

RR(0, ω) = −i/2vF for the non-interacting case ν = 1, which impliesARR = 1/vF , the result that we expected for the spectral density of right-moving electrons.. If, on the other hand, T ≪ ω, one finds a result similarto Eq. (12.45) with πT replaced by ω.

We conclude that for an interacting one-dimensional electron liquid thedensity of states at the Fermi level has a power-law singularity, and vanishesproportional to max(ω, T )ν−1. This behavior is quite different from the caseof a Fermi liquid, where the density of states is non-singular at the Fermilevel. Also note that the cut-off length λ enters into the expression for thedensity of states if ν 6= 1. For a quantitative estimate, one should replace λby λF .

12.3 Luttinger’s model

12.3.1 Formulation of the model

Although the discussion of the previous section provided the physical moti-vation of the effective Hamiltonian (12.22), it might leave some uneasynesswith those of you who want a “constructive” description of a field theory ofinteracting electrons in one dimension. Therefore, we now return to a de-scription of a one-dimensional electron liquid in fermion language, and derivethe effective Hamiltonian (12.22) using formal manipulations. The discus-sion of this section follows that of F. D. M. Haldane, J. Phys. C 14, 2585(1981), although it is less rigorous at points. Those readers who want a fullyrigorous treatment are referred to Haldane’s article, or to the tutorial reviewby J. von Delft and H. Schoeller, Annalen Phys. 7, 225 (1998).

As a starting point, we use a description of the one-dimensional elec-tron system in which we have linearized the electron spectrum, see Fig.76. We shift momenta by an amount −kF (kF ) for right (left) movingelectrons, so that the Fermi points are shifted to k = 0, and denote thecreation/annihilation operators for right moving (left moving) electrons withshifted momentum k by c†kR and ckR (c†kL and ckL), respectively. We thenwrite the kinetic energy as

Hkin = vF

∑

k

(k :nkR : −k :nkL :), (12.46)

where nkR = c†kRckR is the density of right-moving electrons and nkL = c†kLckL

239

ε=ε +F v (k−k )F F

leftmovers

rightmovers

k

ε

k

ε

F F−k k

k=0 forleft movers

k=0 forright movers

ε=ε −F v (k−k )F F

Figure 76: Linearized spectrum for electrons in one dimension. In our description,we separate right and left moving electrons and shift momenta such that the Fermipoints correspond to k = 0.

is the density of left-moving electrons. The symbol : . . . : refers to normalordering, i.e., to subtraction of the densities in the (non-interacting) groundstate. The summations over k should be truncated at |k| . kF because thelinear dispersion is valid for a window of wavevectors of size . kF aroundk = 0 only and because the distinction between left movers and right moversimplies that the (translated) wavevector of a right mover cannot be smallerthan −kF , whereas the (translated) wavevector of a left mover cannot belarger than kF , see Fig. 76. In terms of the fields

ψR(x) =1√L

∑

k

eikxckR, ψL(x) =1√L

∑

k

eikxckL, (12.47)

the Hamiltonian (12.46) reads49

Hkin = vF

∫

dx[

:ψ†L(x)(i∂x)ψL(x) : − :ψ†

R(x)(i∂x)ψR(x) :]

. (12.48)

49The normal ordering in Eq. (12.48) means that ψ†ψ has to be replaced by −ψψ† forstates with momentum smaller than kF . This prescription is not transparent in a realspace formulation.

240

12.3.2 Bosonization

We’ll now show that Hkin can be written in terms of the densities ρqR andρqL of right and left moving electrons,

ρqR =∑

k

c†k,Rck+q,R, ρqL =∑

k

c†k,Lck+q,L. (12.49)

We consider the case q = 0 separately; note that we do not need to use normalordering for q 6= 0. The commutation relations of the density operators are

[ρq,R, ρ−q′,R]− =qL

2πδqq′,

[ρq,L, ρ−q′,L]− = −qL2πδqq′ ,

[ρq,R, ρ−q′,L]− = 0. (12.50)

To see how this result is obtained, let us look at the first line of Eq. (12.50)in detail,

[ρq,R, ρ−q′,R]− =∑

k,k′

[

c†k,Rck+q,R, c†k′,Rck′−q′,R

]

−

=∑

k

(

c†k,Rck+q−q′,R − c†k+q′,Rck+q,R

)

. (12.51)

If q 6= q′, this is zero, as one can see from relabeling the summation index k.Otherwise, if q = q′ we find

[ρq,R, ρ−q′,R]− =∑

k

(nk,R − nk+q,R) . (12.52)

The summation on the right hand side of Eq. (12.52) might appear ambigu-ous. It can be computed unambiguously by normal ordering of the summand.Normal ordering of the operators between brackets gives

[ρq,R, ρ−q,R]− =qL

2π−∑

k

:nk,R − nk+q,R :

=qL

2π. (12.53)

The summation of normal ordered terms gives zero after relabeling of thesummation index k. We had to normal order first, since relabelling thesummation index is allowed for normal-ordered summands only.

241

With these commutation relations, the density operators can serve asboson creation and annihilation operators. Operators ρqR with q < 0 play therole of creation operators, whereas operators ρqR with q > 0 are annihilationoperators. Similarly, for left-moving particles, the creation operators are theρqL with q > 0 and annihilation operators are ρqL with q < 0.50 Whenregarded as creation and annihilation operators, the density operators spanthe entire Hilbert space with fixed numbers NR and NL of left moving andright moving electrons. You can verify this statement by explicit constructionof the states, or by comparing the partition functions for the electron liquid infermion and boson representations, see footnote 51 below. Since the densityoperators span the entire Hilbert space, the kinetic energy can be representedin terms of the density operators, rather than the fermion operators. In orderto achieve this, we look at the time derivative of ρq,R and ρq,L, for which wefind

∂

∂tρq,R =

i

~vF

∑

k,k′

k[

c†k,Rck,R, c†k′,Rck′+q,R

]

−

=i

~vF

∑

k

k(

c†k,Rck+q,R − c†k−q,Rck,R

)

=−iqvF

~ρq,R. (12.54)

Similarly, for the left-moving particles we get

∂

∂tρq,L =

iqvF

~ρq,L. (12.55)

Utilizing the commutation relations of the density operators, we find that wecan write the kinetic energy in terms of the boson operators ρq,R and ρq,L,

Hkin =2πvF

L

∑

q>0

(ρ−q,Rρq,R + ρq,Lρ−q,L) +πvF

L(N2

R +N2L), (12.56)

where the additive constant reflects the additional cost of adding electronsto the system.51 Another way to obtain Eq. (12.56) is to note that, byEq. (12.49), the operator ρq creates a multitude of electron-hole pairs, all

50Formally, this identification requires that we multiply the density operators by(L|q|/2π)−1/2.

51 Now we are in a position to prove that the set of states that is spanned by the density

242

with the same energy vF q. Hence, the energy of the “boson” created by ρq

is vF q. Taking into account the normalization factor (2π/qL)1/2 from thecommutation relations (12.50), one arrives at Eq. (12.56).

The interaction Hamiltonian is, in fact, easier to deal with. Since itcontains four fermion creation or annihilation operators, it is quadratic inthe boson operators ρq,R and ρq,L. Using the observation that the interactioncouples to the total density only, one arrives at the Hamiltonian

Hint =1

2L(V0 − V2kF

)∑

q 6=0

(ρq,R + ρq,L)(ρ−q,R + ρ−q,L). (12.57)

The prefactor (V0 − V2kF) follows from the difference of Hartree and Fock

type interactions, assuming that the interaction Vq depends only weakly onq on scales ≪ kF . For a point-like interaction, one has V2kF

= V0, and Hint

vanishes, as is required by the Pauli principle.The above manipulations, in which the fermion operators in the Hamil-

tonian are replaced by boson operators, are known as “bosonization”.The total Hamiltonian H = Hkin+Hint can be rewritten in terms of boson

fields φ(x) and Π(x) that depend on the coordinate x only. These fields arerelated to the particle density and current density as in Eqs. (12.20) and(12.21). In terms of the density operators, the field φ(x) reads

φ(x) = π

∫ x

dx′ :ρ(x) :, (12.58)

operators is complete. Calculating the grand canonical partition function for the fermions,we find

Z =

[

∞∏

n=1

(1 + w2n−1)2

]2

,

where we abbreviated w = exp(−πvF /TL) and put the chemical potential precisely be-tween two successive energy levels. The square within the square brackets correspondsto states with different signs of k, i.e., to particle and hole-like excitations, whereas theoverall square reflects the identical contributions from left movers and right movers. Forthe boson basis, we find

Z =

∞∏

n=1

(1 − w2n)−2

(

∞∑

m=−∞

wm2

)2

.

Here, the first factor is the partition function corresponding to the boson degrees of free-dom, whereas the second factor gives the contribution of the total number of particles tothe energy. You verify that these two partition functions are, indeed, equal.

243

where ρ(x) is the total electron density,

ρ(x) =1

L

∑

q 6=0

eiqx(ρq,R + ρq,L)e−|q|λ/2, (12.59)

whereas the field Π(x) is given by

Π(x) = −∑

q 6=0

eiqx(ρq,R − ρq,L)e−|q|λ/2. (12.60)

(The exponential high-momentum cutoff is the same as in the previous sec-tion.) Using the commutation relations between the density operators ρq,R

and ρq,L, you verify that the fields φ(x) and Π(x) obey canonical bosoncommutation relations, cf. Eq. (12.18). With this change of variables, theHamiltonian H = Hkin +Hint acquires the form (12.22), with

vc = vF

√

1 +V0 − V2kF

πvF. (12.61)

Instead of a formulation in terms of the fields φ(x) and Π(x), which referto total electron density and current density (after normal ordering), oneoften uses fields φR and φL that are related to the densities of right and leftmoving electrons,

ρR(x) =1

L

∑

q 6=0

eiqxρq,Re−|q|λ/2, ρL(x) =

1

L

∑

q 6=0

eiqxρq,Le−|q|λ/2, (12.62)

as

φR(x) = 2π

∫ x

dx′ :ρR(x′) :, φL(x) = 2π

∫ x

dx′ :ρL(x′) : . (12.63)

These fields have commutation rules

[φR(x), φR(x′)]− = iπsign (x− x′),

[φL(x), φL(x′)]− = −iπsign (x− x′), (12.64)

[φR(x), φL(x′)]− = 0.

They are related to the original fields φ(x) and Π(x) as

φR(x) = φ(x) − π

2~

∫

dx′Π(x′)sign (x− x′), (12.65)

φL(x) = φ(x) +π

2~

∫

dx′Π(x′)sign (x− x′). (12.66)

244

In terms of the fields φR(x) and φL, the Hamitonian reads

H =vF

4π

∫

dx

[

(

∂φR

∂x

)2

+

(

∂φL

∂x

)2]

+V0 − V2kF

2π

∫

dx

(

∂φR

∂x+∂φL

∂x

)2

. (12.67)

12.3.3 Fermion creation and annihilation operators

Having formally rewritten the electron Hamiltonian in terms of boson fieldsφ(x) and Π(x), we still want to be able to express the creation and anni-hilation operators of a single electron in terms of the boson fields. Beforewe do this, let us recall the form of the Hilbert space in the boson repre-sentation. In the boson representation, any electronic state can be writtenas density operators ρq,R and ρq,L acting on the ground state with NR rightmoving electrons and NL left moving electrons. Hence, every electronic stateis represented by occupation numbers nq,L with q > 0 and nq,R with q < 0for the boson modes, and the numbers NR and NL,

|state〉 = |nq, NR, NL〉. (12.68)

What is the action of a single-fermion creation operator ψ†L(x) or ψ†

R(x) inthis Hilbert space? First of all, ψ†

R(x) (ψ†L(x)) increases NR (NL) by one.

Inside the boson Hilbert space, this is achieved by “ladder operators” U †R

and U †L,

U †R|nq, NR, NL〉 = |nq, NR + 1, NL〉,

UR|nq, NR, NL〉 = |nq, NR − 1, NL〉,U †

L|nq, NR, NL〉 = |nq, NR, NL + 1〉,UL|nq, NR, NL〉 = |nq, NR, NL − 1〉. (12.69)

In addition to this, the operator ψ†R(x) creates a multitude of electron-

hole pairs, so that, in the end, the change in particle and current densitiesis sharply peaked around x. In order to capture these particle hole pairs interms of the boson fields, the notation that uses the separate fields φR(x) andφL(x) for right and left moving electrons is particularly useful. Returningto the qualitative arguments of the previous section — the addition of an

245

electron corresponds to a “kink” in the fields φR(x) and φL(x) —, we nowmake an educated guess for ψ†

R(x) and ψ†L(x),

ψ†R(x) =

1√2πλ

U †Re

−iφR(x), ψ†L(x) =

1√2πλ

U †Le

iφL(x), (12.70)

where the cut-off dependent prefactor (2πλ)−1/2 has already been inserted.We verify that this gives, indeed, the correct commutator with the densityfields ρR(x) and ρL(x),

[

ψ†R(x), ρR(x′)

]

−=

1

2π√

2πλU †

R

[

e−iφR(x),∂

∂x′φR(x′)

]

−

=1

2√

2πλU †

Re−iφR(x) ∂

∂x′sign (x− x′)

= −ψ†R(x)δ(x− x′),

[

ψ†L(x), ρL(x′)

]

−= −ψ†

L(x)δ(x− x′). (12.71)

In deriving this result, we used the fact that the commutator [eX , Y ]− =eX [X, Y ]− if [X, Y ]− is proportional to the identity operator. Similarly, youverify that the ψ†

R(x) and ψ†R(x′) anticommute, as well as ψ†

L(x) and ψ†L(x′).

There is a problem, however, with the calculation of the anticommutator ofψ†

R(x) and ψR(x′), which becomes ambiguous when x → x′. The ambiguityis lifted once the regularizing momentum cut off in Eqs. (12.59) and (12.60)or Eq. (12.62) is taken into account. In coordinate representation, this mo-mentum cut-off amounts to the replacement of the boson fields φR(x) andφL(x) in the exponents in Eq. (12.70) by the “smeared” fields

φR(x) →∫

dx′2λφR(x′)

π[λ2 + 4(x− x′)2],

φL(x) →∫

dx′2λφL(x′)

π[λ2 + 4(x− x′)2], (12.72)

where λ ∼ λF is the length scale over which the electrons are delocalized.This way, the “kink” in the fields φR(x) and φL(x) that is created by theoperators ψ†

R(x) and ψ†L(x) is smeared out over a length λ, and, hence, the

added electron is delocalized over a segment of length ∼ λ.52 For the regu-

52We have chosen a regularization integral in the exponent of Eq. (12.70), instead of aregularization integral before the exponent. The latter choice would amount a superpo-sition of sharply localized electrons, which does not need to be the same as a delocalizedelectron.

246

larized fields, the sign-function in the commutator is smeared over a distance∼ λ,

[φR(x), φR(x′)]− = 2i arctan[(x− x′)/λ],

[φL(x), φL(x′)]− = −2i arctan[(x− x′)/λ]. (12.73)

With this smearing of the boson fields, it can be shown that the ansatz (12.70)provides the correct anticommutation relation for the fermion creation andannihilation operator. For details, see the original paper F. D. M. Haldane, J.Phys. C 14, 2585 (1981) or the tutorial review J. von Delft and H. Schoeller,Annalen Phys. 7, 225 (1998). Writing the fields φL(x) nd φR(x) in terms ofthe original fields φ(x) and Π(x) and undoing the momentum shift q → q±kF

we performed at the beginning of this section, you recover Eqs. (12.26) and(12.27) of the previous section.

Finally, we have to ensure that operators for left moving and right movingfermions anticommute. There are various ways to achieve this. The simplestmethod is to postulate that the ladder operators U †

R and U †L anticommute.

An alternative way is to postulate that the commutator between the left-moving and right-moving boson fields φL and φR is not zero, but iπ instead.You are referred to the specialized literature for details.

12.4 Electrons with spin

For electrons with spin, one can take the same route as we took in Sec.12.2, and write down an effective Hamiltonian based on the spin and chargevelocities in the one-dimensional electron liquid. Introducing fields Πc andφc that are related to the particle current density and particle density as

nc(x) = n↑(x) + n↓(x) =1

π∂xφc(x)

√2 (12.74)

jc(x) = j↑(x) + j↓(x) = −vF Πc(x)√

2, (12.75)

together with fields Πs and φs that are related to the spin current densityand spin density,

ns(x) = n↑(x) − n↓(x) =1

π∂xφs(x)

√2 (12.76)

js(x) = j↑(x) − j↓(x) = −vF Πs(x)√

2, (12.77)

247

we can write the Hamiltonian as

H =~vF

2π

∫

dx

[

π2

~2

(

Πc(x)2 + Πs(x)

2)

+ g2c (∂xφc(x))

2 + g2s (∂xφs(x))

2]

, (12.78)

where gc = vc/vF and gs = vs/vF . The boson fields Πc,s and φc,s are normal-ized such that they satisfy canonical commutation relations,

[φc,s(x), φc,s(x′)]− = [Πc,s(x),Πc,s(x

′)]− = 0

[Πc(x), φc(x′)]− = [Πs(x), φs(x

′)]− = −i~δ(x− x′), (12.79)

[Πc(x), φs(x′)]− = [Πs(x), φc(x

′)]− = 0.

The form of the fermion operators is found in the same way as for the spinlesscase,

ψ†Rσ(x) =

1√2πλ

U †Rσe

−ikF x−iφRσ(x),

ψ†Lσ(x) =

1√2πλ

U †Lσe

ikF x+iφLσ(x), (12.80)

where σ =↑, ↓ and where the fields φLσ and φRσ are defined as

φRσ(x) =1√2

(

φc(x) + σφs(x)

− π

2~

∫

dx′[Πc(x′) + σΠs(x

′)]sign (x− x′)

)

,

φLσ(x) =1√2

(

φc(x) + σφs(x)

+π

2~

∫

dx′[Πc(x′) + σΠs(x

′)]sign (x− x′)

)

.

(12.81)

Alternatively, one can find a description starting from a microscopic pic-ture, as in the previous section. In this case, bosonization of the kineticenergy is done as before. For this, we refer to the references cited above formore details.

One important aspect of the one-dimensional electron liquid with spin, isthat spin excitations and charge excitations travel at different speeds, see our

248

discussion in Sec. 12.1. We again emphasize that this phenomenon, which isknown as “spin-charge separation”, also exists for collective modes in higherdimensions. The difference between the higher-dimensional case and theone-dimensional case is that in higher dimensions there are quasiparticle ex-citations that carry both spin and charge, in addition to the spin-charge sep-arated collective excitations. These quasiparticles dominate the low-energyphysics of higher dimensional electron liquids, so that, in the end, spin andcharge always travel together. In one dimension, there are no quasiparticleexcitations, and the spin-charge separated collective modes dominate the lowenergy physics.

To illustrate the phenomenon of spin-charge separation, let us create aright-moving electron with spin up at position x = 0 at time t = 0. As wediscussed above, creating an electron corresponds to making a kink of size πin the field φR↑ around x = 0. The time-dependence of the charge densityand the spin density then follows from decomposing φR↑ in terms of the spinand charge fields, and solving for their time-evolution using the Hamiltonian(12.78). A kink in φR↑ corresponds to kinks in both φc and φs, but these twokinks travel at different velocities. Hence, although initially charge and spinwere located at the same point, they are separated after a finite time. Onlyfor a non-interacting electron gas, for which vc = vs = vF , spin and chargeare not separated.

12.5 Does a one-dimensional electron liquid exist?

Now that we have seen that electrons in one dimension behave quite differ-ently from electrons in three dimensions, it is natural to ask whether all ofthis was an academic exercise, or whether one-dimensional electron systemsexist in nature. Fortunately, the answer is positive.

One example of a one-dimensional electron liquid is realized in extremelythin and clean wires formed in the two-dimensional electron gas that existsat the interface in a GaAs/GaAlAs heterostructure. This is the examplethat comes closest to the physics we have discussed in this chapter. Spin-charge separation in these thin wires has been observed directly, in the formof interference between charge and spin excitations in a wire of finite length.You can find more about the experiment and the theory in Y. Tserkovnyak,B. I. Halperin, O. M. Auslaender, and A. Yacoby, cond-mat/0302274 andreferences therein.

Another example is that of a carbon nanotube. Nanotubes are very good

249

candidates to observe Luttinger Liquid behavior, since they are thin andclean by virtue of their chemistry. (The wires, on the other hand, have tobe carefully engineered.) However, carbon nanotubes have two propagatingmodes at the Fermi level (per spin direction), which slightly modifies some ofthe properties. The anomalous density of states has been observed in carbonnanotubes, see, e.g., M. Bockrath et al., Nature 397, 598 (1999).

The third example is that of edge states in the quantum Hall effect. Themain difference between edge states and the one-dimensional electron liquidwe considered here is that edge states are “chiral”: all electrons move in onedirection. For more details, you are referred to the original papers by Wen[Phys. Rev. B 41, 12838 (1990); 43, 11025 (1991)].

In addition to the question of real existence, one-dimensional electronsystems have been used as a model for quite a variety of problems in con-densed matter physics. One example is the Kondo problem with a point-likemagnetic impurity, where it can be argued that the impurity interacts onlywith s-wave states in the metal, which, in turn, can be modeled as a one-dimensional electron gas. Another example is that of a Coulomb blockadedquantum dot, where the electron gas in the point contacts between the dotand the bulk electrodes is modeled as a one-dimensional electron gas. Inthese applications, most of the electron liquid is non-interacting, the role ofthe impurity or of the quantum dot being to introduce a “localized interac-tion” in the Luttinger Liquid.

250