Many body quantum theory in condensed matter physics nodrm

Symbol Meaning Definitionˆ˙ |νi Dirac ket notation for a quantum state ν Section 1.1hν| Dirac bra notation for an adjoint quantum state ν Section 1.1 a annihilation operator for particle

Many-body Quantum Theory in Condensed Matter Physics

Many-body Quantum Theory

in Condensed Matter Physics

Niels Bohr Institute, University of Copenhagen

Copenhagen, 14 July 2004Corrected version: 14 January 2016

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14

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We have gone through the trouble of writing this textbook, because we felt thepedagogical need for putting an emphasis on the physical contents and applications ofthe machinery of quantum field theory without loosing mathematical rigor We hope

we have succeeded, at least to some extent, in reaching this goal

Since our main purpose is to provide a pedagogical introduction, and not to present

a review of the physical examples presented, we do not give comprehensive references

to these topics Instead, we refer the reader to the review papers and topical booksmentioned in the text and in the bibliography

We would like to thank our ever enthusiastic students for their valuable helpthroughout the years improving the notes preceding this book

Copenhagen, July 2004

Preface to corrected edition January 2016

The book has been corrected for an, unfortunately, rather large number of prints We would like to thank all the colleagues and readers who have sent corrections

mis-to us and in particular the many students and the teachers of the courses CondensedMatter Theory I at University of Copenhagen and Transport in nanostructures atthe Technical University of Denmark for helping in locating the misprints We havenot made major changes to the book other than Section 10.5 has been rewrittensomewhat

v

vii

3 Phonons; coupling to electrons 52

CONTENTS ix

10.3 Coherent many-body transport phenomena 158

CONTENTS xi

16 Impurity scattering and conductivity 285

16.6.1 Statistics of quantum conductance,

17.3.2 Jellium phonons and the effective

CONTENTS xiii

Symbol Meaning Definitionˆ

˙

|νi Dirac ket notation for a quantum state ν Section 1.1hν| Dirac bra notation for an adjoint quantum state ν Section 1.1

a annihilation operator for particle (fermion or boson) Section 1.3

a† creation operator for particle (fermion or boson) Section 1.3

aν, a†ν annihilation/creation operators (state ν) Section 1.3

a±n amplitudes of wavefunctions to the left Section 7.1

A(r, t) electromagnetic vector potential Section 1.4.2A(ν, ω) spectral function in frequency domain (state ν) Section 8.3.4A(r, ω), A(k, ω) spectral function (real space, Fourier space) Section 8.3.4

A0(r, ω), A0(k, ω) spectral function for free particles Section 8.3.4

A, A† phonon annihilation and creation operator Section 17.1

b annihilation operator for particle (boson, phonon) Section 1.3

b† creation operator for particle (boson, phonon) Section 1.3

b±n amplitudes of wavefunctions to the right Section 7.1

c annihilation operator for particle (fermion, electron) Section 1.3

c† creation operator for particle (fermion, electron) Section 1.3

cν, c†ν annihilation/creation operators (state ν) Section 1.3

II(ω) retarded current–current correlation function (frequency) Section 6.3

C(Q, ikn, ikn+ iqn) Cooperon in the Matsubara domain Section 16.5

CR(Q, ε, ε) Cooperon in the real time domain Section 16.5

Cion

V specific heat for ions (constant volume) Section 3.5

xiv

LIST OF SYMBOLS xv

d() density of states (including spin degeneracy) Eq (2.31)

DR(rt, rt0) retarded phonon propagator Chapter 17

DR(q, ω) retarded phonon propagator (Fourier space) Chapter 17

D(rτ, rτ0) Matsubara phonon propagator Chapter 17

D(q, iqn) Matsubara phonon propagator (Fourier space) Chapter 17

DR(νt, ν0t0) retarded many particle Green’s function Eq (9.9b)

Dαβ(r) phonon dynamical matrix Section 3.4)

∆k superconducting orderparameter Eq (18.11)

e elementary charge

e2 electron interaction strength Eq (1.100a)

E(r, t) electric field

E total energy of the electron gas Chapter 2

E(1) interaction energy, first-order perturbation Section 2.2.1

E(2) interaction energy, second-order perturbation Section 2.2.2

Ek dispersion relation for BCS quasiparticles Eq (18.14)

ε energy variable

0 the dielectric constant of vacuum

εk dispersion relation

εν energy of quantum state ν

ε(rt, rt0) dielectric function in real space Section 6.4

ε(k, ω) dielectric function in Fourier space Section 6.4

ε(k, ω) dielectric function in Fourier space Section 6.4

|FSi the filled Fermi sea N –particle quantum state Eq (2.22)

φ(x) displacement field operator Eq (19.49)

φext external electric potential Section 6.4

φind induced electric potential Section 6.4

φ, ˜φ wavefunctions with different normalizations Eq (7.4)

φ±LnE, φ±RnE wavefunctions in the left and right leads Section 7.1

gqλ electron–phonon coupling constant (lattice model) Eq (3.38)

gq electron–phonon coupling constant (jellium model) Eq (3.42)

G<0(rt, r0t0) free lesser Green’s function Section 8.3.1

G>

0(rt, r0t0) free grater Green’s function Section 8.3.1

GA0(rt, r0t0) free advanced Green’s function Section 8.3.1

GR

0(rt, r0t0) free retarded Green’s function Section 8.3.1

GR

0(k, ω) free retarded Green’s function (Fourier space) Section 8.3.1

G<(rt, r0t0) lesser Green’s function Section 8.3

G>(rt, r0t0) greater Green’s function Section 8.3

GA(rt, r0t0) advanced Green’s function Section 8.3

GR(rt, r0t0) retarded Green’s function (real space) Section 8.3

GR(k, ω) retarded Green’s function in Fourier space Section 8.3

GR(k, ω) retarded Green’s function (Fourier space) Section 8.3.1

GR(νt, ν0t0) retarded single–particle Green’s function ({ν} basis) Eq (8.34)

¯

G(rστ, r0σ0τ0) Matsubara Green’s function (real space) Section 11.3G(ντ, ν0τ0) Matsubara Green’s function ({ν} basis) Section 11.3G(1, 10) Matsubara Green’s function (real space four–vectors) Section 12.1G(˜k, ˜k0) Matsubara Green’s function (four–momentum notation) Section 13.4

G0(rστ, r0σ0τ0) Matsubara Green’s function (real space, free particles) Section 11.3.1

G0(ντ, ν0τ0) Matsubara Green’s function ({ν} basis, free particles) Section 11.3.1

G0(k, ikn) Matsubara Green’s function (Fourier space, free particles) Section 11.3

G0(ν, ikn) Matsubara Green’s function (free particles ) Section 11.3

G0(n) n–particle Green’s function (free particles) Section 11.6G(k, ikn) Matsubara Green’s function (Fourier space) Section 11.3G(ν, ikn) Matsubara Green’s function ({ν} basis, frequency domain) Section 11.3

Γ imaginary part of self–energy

Γx(˜k, ˜k + ˜q) vertex function (x–component, four vector notation) Eq (16.21b)

Γ0,x free (undressed) vertex function Eq (16.20)

~ Planck’s constant (h/2π),~ → 1 in Chap 5 and onwards

H a general Hamiltonian

H0 unperturbed part of an Hamiltonian

H0 perturbative part of an Hamiltonian

Hext external potential part of an Hamiltonian

Hint interaction part of an Hamiltonian

Hph phonon part of an Hamiltonian

Ie electrical current (charge current) Section 6.3

LIST OF SYMBOLS xvii

J∆

σ(r) current density operator, paramagnetic term Eq (1.98a)

JA

σ(r) current density operator, diamagnetic term Eq (1.98a)

Jσ(q) current density operator (momentum space) Eq (1.98a)

Je(r, t) electric current density operator Section 6.2

Jij interaction strength in the Heisenberg model Section 4.5.1

Jαβ interaction strength in the Kondo model Eq (10.91a)

kB Boltzmann’s constant

kn Matsubara frequency (fermions) Eq (11.42)

k general momentum or wave vector variable

`, `k mean free path or scattering length Chapter 12

`0 vkτ0mean free path (first Born approximation)

L normalization length or system size in 1D

Λirr irreducible four–point function Eq (16.18)

m mass (electrons and general particles)

m∗ effective interaction renormalized mass Section 15.4.1

µ general quantum number label

n particle density

nF(ε) Fermi–Dirac distribution function Section 1.5.1

nB(ε) Bose–Einstein distribution function Section 1.5.2

nimp impurity density

N number of particles

Nimp number of impurities

ν general quantum number label

ω frequency variable

ωq phonon dispersion relation

p general momentum or wave number variable

P principle part

ΠR

αβ(rt, r0t0) retarded current–current correlation function Eq (6.25)

ΠR

αβ(q, ω) retarded current–current correlation function

Παβ(q, iωn) Matsubara current–current correlation function Chapter 16

Π0(q, iqn) free pair–bubble diagram Eq (13.37)

q general momentum variable

r general space variable

r reflection matrix coming from left Section 7.1

r0 reflection matrix coming from right Section 7.1

ρσ(r) particle density operator (real space) Eq (1.94)

ρσ(q) particle density operator (momentum space) Eq (1.94)

ρL, ρR left and right mover density operators Eq (19.20)

σ general spin index

σαβ(rt, r0t0) conductivity tensor Section 6.2

ΣR(q, ω) retarded self–energy (Fourier space)

Σ(q, ikn) Matsubara self–energy

Σk impurity scattering self–energy Section 12.5

Σ1BA

ΣPσ(k, ikn) pair–bubble self–energy Section 13.5

ΣRPA

σ (k, ikn) RPA electron self–energy Eq (14.11)

t general time variable

t tranmission matrix coming from left Section 7.1

t0 transmission matrix coming from right Section 7.1

τ0, τk life–time in the first Born approximation Section 12.5.2

LIST OF SYMBOLS xix

V (r), V (q) general single impurity potential Eq (12.1)

Veff combined Coulomb and phonon–mediated interaction Section 14.2

WRPA RPA–screened Coulomb interaction Section 14.2

χ(q, iqn) Matsubara charge–charge correlation function Section 14.4

χRPA(q, iqn) RPA Matsubara charge–charge correlation function Section 14.4

χirr(q, iqn) irreducible Matsubara charge–charge correlation function Section 14.4

χ0(q, iqn) free Matsubara charge–charge correlation function Section 14.4

χR(rt, r0t0) retarded charge–charge correlation function Eq (6.37b)

χR(q, ω) retarded charge–charge correlation function (Fourier) Eq (8.81)

ψν(r) single–particle wave function, quantum number ν Section 1.1

ψ±

ψ(r1, r2, , rn) n–particle wave function (first quantization) Section 1.1

Ψσ(r) quantum field annihilation operator Section 1.3.6

Ψ†σ(r) quantum field creation operator Section 1.3.6

FIRST AND SECOND QUANTIZATION

Quantum theory is the most complete microscopic theory we have today describing thephysics of energy and matter It has successfully been applied to explain phenomenaranging over many orders of magnitude, from the study of elementary particles onthe sub-nucleonic scale to the study of neutron stars and other astrophysical objects

on the cosmological scale Only the inclusion of gravitation stands out as an unsolvedproblem in fundamental quantum theory

Historically, quantum physics first dealt only with the quantization of the motion

of particles, leaving the electromagnetic field classical, hence the name quantum chanics Later also the electromagnetic field was quantized, and even the particlesthemselves became represented by quantized fields, resulting in the development ofquantum electrodynamics (QED) and quantum field theory (QFT) in general By con-vention, the original form of quantum mechanics is denoted first quantization, whilequantum field theory is formulated in the language of second quantization

me-Regardless of the representation, be it first or second quantization, certain basicconcepts are always present in the formulation of quantum theory The starting point

is the notion of quantum states and the observables of the system under consideration.Quantum theory postulates that all quantum states are represented by state vectors in

a Hilbert space, and that all observables are represented by Hermitian operators acting

on that space Parallel state vectors represent the same physical state, and thereforeone mostly deals with normalized state vectors Any given Hermitian operator A has

a number of eigenstates |ψαi that are left invariant by the action of the operator

up to a real scale factor α, i.e., A|ψαi = α|ψαi The scale factors are denoted theeigenvalues of the operator It is a fundamental theorem of Hilbert space theory thatthe set of all eigenvectors of any given Hermitian operator forms a complete basisset of the Hilbert space In general, the eigenstates |ψαi and |φβi of two differentHermitian operators A and B are not the same By measurement of the type B thequantum state can be prepared to be in an eigenstate|φβi of the operator B Thisstate can also be expressed as a superposition of eigenstates |ψαi of the operator A

as |φβi =Pα|ψαiCαβ If one measures the dynamical variable associated with theoperator A in this state, one cannot in general predict the outcome with certainty

It is only described in probabilistic terms The probability of having any given|ψαi

as the outcome is given as the absolute square |Cαβ|2 of the associated expansioncoefficient This non-causal element of quantum theory is also known as the collapse

of the wavefunction However, between collapse events the time evolution of quantumstates is perfectly deterministic The time evolution of a state vector|ψ(t)i is governed

by the central operator in quantum mechanics, the Hamiltonian H (the operatorassociated with the total energy of the system), through Schr¨odinger’s equation

1

“ket” states |φi, and use standard geometrical terminology; e.g., the norm squared

of |ψi is given by hψ|ψi, and |ψi and |φi are said to be orthogonal if hψ|φi = 0

If {|ψαi} is an orthonormal basis of the Hilbert space, then the above-mentionedexpansion coefficient Cαβ is found by forming inner products: Cαβ = hψα|φβi Afurther connection between the direct and the adjoint Hilbert space is given by therelationhψ|φi = hφ|ψi∗, which also leads to the definition of adjoint operators For

a given operator A the adjoint operator A† is defined by demanding hψ|A†|φi =hφ|A|ψi∗ for any|ψi and |φi

In this chapter, we will briefly review standard first quantization for one- andmany-particle systems For more complete reviews the reader is referred to standardtextbooks by, for instance, Dirac (1989), Landau and Lifshitz (1977), and Merzbacher(1970) Based on this we will introduce second quantization This introduction, how-ever, is not complete in all details, and we refer the interested reader to the textbooks

by Mahan (1990), Fetter and Walecka (1971), and Abrikosov et al (1975)

1.1 First quantization, single-particle systems

For simplicity consider a non-relativistic particle, say an electron with charge −e,moving in an external electromagnetic field described by the potentials ϕ(r, t) andA(r, t) The corresponding Hamiltonian is

hr|n, l, m, σi = Rnl(r)Yl,m(θ, φ)χσ (hydrogen orbital), (1.4)where Rnl(r) is a radial Coulomb function with n− l nodes, while Yl,m(θ, φ) is aspherical harmonic representing angular momentum l with a z component m

A third example is an electron moving in a constant magnetic field B = B ez,which in the Landau gauge A = xB ey leads to the Landau eigenstates |n, ky, kz, σi,

FIRST QUANTIZATION, SINGLE-PARTICLE SYSTEMS 3

Fig 1.1 The probability density |hr|ψνi|2 in the xy plane for (a) any plane wave

ν = (kx, ky, kz, σ), (b) the hydrogen orbital ν = (4, 2, 0, σ), and (c) the Landauorbital ν = (3, ky, 0, σ)

where n is an integer, ky (kz) is the y (z) component of k, and σ the spin variable.Recall that

poly-in Fig 1.1

In general a complete set of quantum numbers is denoted ν The three examplesgiven above correspond to ν = (kx, ky, kz, σ), ν = (n, l, m, σ), and ν = (n, ky, kz, σ),each yielding a state function of the form ψν(r) =hr|νi The completeness of a basisstate as well as the normalization of the state vectors plays a central role in quan-tum theory Loosely speaking, the normalization condition means that with proba-bility unity a particle in a given quantum state ψν(r) must be somewhere in space:R

dr|ψν(r)|2 = 1, or in the Dirac notation: 1 =R

drhν|rihr|νi = hν| (Rdr|rihr|) |νi

Similarly, the completeness of a set of basis states ψν(r) means that if a particle is insome state ψ(r) it must be found with probability unity within the orbitals of the basisset: P

ν|hν|ψi|2 = 1 Again using the Dirac notation we find 1 = P

νhψ|νihν|ψi =hψ| (Pν|νihν|) |ψi, and we conclude

X

ν

We shall often use the completeness relation (1.7) A simple example is the expansion

of a state function in a given basis: ψ(r) = hr|ψi = hr|1|ψi = hr| (Pν|νihν|) |ψi =P

νhr|νihν|ψi, which can be expressed as

It should be noted that the quantum label ν can contain both discrete and tinuous quantum numbers In that case the symbolP

con-ν is to be interpreted as a bination of both summations and integrations For example, in the case in Eq (1.5)with Landau orbitals in a box with side lengths Lx, Ly and Lz, we have

+1, if (ijk) is an even permutation of (123) or (xyz),

−1, if (ijk) is an odd permutation of (123) or (xyz),

When turning to N-particle systems, i.e., systems containing N identical particles,say, electrons, three more assumptions are added to the basic assumptions definingquantum theory The first assumption is the natural extension of the single-particlestate function ψ(r), which (neglecting the spin degree of freedom for the time being)

is a complex wave function in 3-dimensional space, to the N-particle state functionψ(r1, r2, , rN), which is a complex function in the 3N-dimensional configurationspace As for one particle, this N-particle state function is interpreted as a probabilityamplitude such that its absolute square is related to a probability:

The probability for finding the N particles

in the 3N−dimensional volumeQNj=1drjsurrounding the point (r1, r2, , rN) inthe 3N−dimensional configuration space





.(1.14)

FIRST QUANTIZATION, MANY-PARTICLE SYSTEMS 5

1.2.1 Permutation symmetry and indistinguishability

A fundamental difference between classical and quantum mechanics concerns the cept of indistinguishability of identical particles In classical mechanics each particlecan be equipped with an identifying marker (e.g a colored spot on a billiard ball)without influencing its behavior, and moreover it follows its own continuous path inphase space Thus, in principle, each particle in a group of identical particles can beidentified This is not so in quantum mechanics Not even in principle is it possible

con-to mark a particle without influencing its physical state, and worse, if a number ofidentical particles are brought to the same region in space, their wavefunctions willrapidly spread out and overlap with one another, thereby soon rendering it impossible

to say which particle is where

The second fundamental assumption for N-particle systems is therefore that tical particles, i.e., particles characterized by the same quantum numbers such asmass, charge and spin, are in principle indistinguishable

iden-From the indistinguishability of particles it follows that if two coordinates in anN-particle state function are interchanged the same physical state results, and thecorresponding state function can at most differ from the original one by a simpleprefactor λ If the same two coordinates then are interchanged a second time, we end

up with the exact same state function,

1 This discrete permutation symmetry is always obeyed However, some quasiparticles in 2D hibit any phase e iφ , a so-called Berry phase, upon adiabatic interchange; they are therefore called anyons.

ex-1.2.2 The single-particle states as basis states

We now show that the basis states for the N-particle system can be built from anycomplete orthonormal single-particle basis{ψν(r)},

where Aν 1 ,ν 2 , ,ν N are just complex numbers Thus any N-particle state function can

be written as a (rather complicated) linear superposition of product states containing

N factors of single-particle basis states

Even though the product states QN

j=1ψνj(rj), in a mathematical sense, form aperfectly valid basis for the N-particle Hilbert space, we know from the discussion onindistinguishability that physically it is not a useful basis since the coordinates have toappear in a symmetric way No physical perturbation can ever break the fundamentalfermion or boson symmetry, which therefore ought to be explicitly incorporated inthe basis states The symmetry requirements from Eqs (1.16a) and (1.16b) are in

Eq (1.22) hidden in the coefficients Aν 1 , ,ν N A physical meaningful basis bringingthe N coordinates on equal footing in the products ψν1(r1)ψν2(r2) ψνN(rN) ofsingle-particle state functions is obtained by applying the bosonic symmetrization

FIRST QUANTIZATION, MANY-PARTICLE SYSTEMS 7

operator ˆS+or the fermionic anti-symmetrization operator ˆS−defined by the followingdeterminants and permanent:2

ψν1(r1) ψν1(r2) ψν1(rN)

ψν2(r1) ψν2(r2) ψν2(rN)

ψνN(r1) ψνN(r2) ψνN(rN)

−

p∈S N

YN j=1

p ∈S N

YN j=1

√

N ! , where nν0 is the number of times the state |ν 0 i appears in the set {|ν 1 i, |ν 2 i, |ν N i}, i.e 0 or 1 for fermions (which means n ν 0 ! = 1) and between 0 and N for bosons For fermions the prefactor thus reduces to √1

1.2.3 Operators in first quantization

We now turn to the third assumption needed to complete the quantum theory ofN-particle systems It states that single- and few-particle operators defined for single-and few-particle states remain unchanged when acting on N-particle states In thisbook we will only work with one- and two-particle operators

Let us begin with one-particle operators A given local one-particle operator T =

T (r,∇r), say the kinetic energy operator or an external potential, takes the followingform in the|νi-representation for a particle number j:

FIRST QUANTIZATION, MANY-PARTICLE SYSTEMS 9

In the N-particle system we must again take the symmetric combination of the dinates, i.e., introduce the operator of the total interaction energy Vtot,

1.3 Second quantization, basic concepts

Many-particle physics is formulated in terms of the so-called second quantizationrepresentation also known by the more descriptive name occupation number repre-sentation The starting point of this formalism is the notion of indistinguishability

of particles discussed in Section 1.2.1 combined with the observation in Section 1.2.2that determinants or permanents of single-particle states form a basis for the Hilbertspace of N-particle states As we shall see, quantum theory can be formulated in terms

of occupation numbers of these single-particle states

1.3.1 The occupation number representation

The first step in defining the occupation number representation is to choose anyordered and complete single-particle basis{|ν1i, |ν2i, |ν3i, }, the ordering being ofparamount importance for fermions From the form ˆS±ψν

The basis states for an N-particle system in the occupation number representationare obtained simply by listing the occupation numbers of each basis state,

N-particle basis states: |nν1, nν2, nν3, i, X

j

It is therefore natural to define occupation number operators ˆnν j which as states have the basis states|nν ji, and as eigenvalues have the number nν j of particlesoccupying the state νj,

In Table 1.1 some of the fermionic and bosonic basis states in the occupation numberrepresentation are shown Note how by virtue of the direct sum, states containing adifferent number of particles are defined to be orthogonal

1.3.2 The boson creation and annihilation operators

To connect first and second quantization we first treat bosons Given the occupationnumber operator it is natural to introduce the creation operator b†

νj that raises theoccupation number in the state|νji by 1,

SECOND QUANTIZATION, BASIC CONCEPTS 11

Table 1.1 Some occupation number basis states for N-particle systems

N Fermion basis states|nν1, nν2, nν3, i

0 |0, 0, 0, 0, i

1 |1, 0, 0, 0, i, |0, 1, 0, 0, i, |0, 0, 1, 0, i,

2 |1, 1, 0, 0, i, |0, 1, 1, 0, i, |1, 0, 1, 0, i, |0, 0, 1, 1, i, |0, 1, 0, 1, i, |1, 0, 0, 1, i,

N Boson basis states|nν 1, nν 2, nν 3, i

0 |0, 0, 0, 0, i

1 |1, 0, 0, 0, i, |0, 1, 0, 0, i, |0, 0, 1, 0, i,

2 |2, 0, 0, 0, i, |0, 2, 0, 0, i, |1, 1, 0, 0, i, |0, 0, 2, 0, i, |0, 1, 1, 0, i, |1, 0, 1, 0, i,

b†νj | , nν j−1, nν j, nν j+1, i = B+(nν j)| , nν j−1, nν j + 1, nν j+1, i, (1.38)where B+(nν j) is a normalization constant to be determined The only non-zero ma-trix elements of b†

νj and bνj are the fundamental operators inthe occupation number formalism As we will demonstrate later any operator can beexpressed in terms of them

Let us proceed by investigating the properties of b†

ν j and bνj further Since bosonsare symmetric in the single-particle state index νj we of course demand that b†

ν j

and b†νk must commute, and hence by Hermitian conjugation that also bνj and bνkcommute The commutator [A, B] for two operators A and B is defined as

We demand further that if j6= k then bν j and b†νkcommute However, if j = k we must

be careful It is evident that since an unoccupied state can not be emptied further

we must demand bνj| , 0, i = 0, i.e., B−(0) = 0 We also have the freedom tonormalize the operators by demanding b†νj| , 0, i = | , 1, i, i.e., B+(0) = 1.But sinceh1|b†

νj|0i∗ =h0|bνj|1i, it also follows that bνj| , 1, i = | , 0, i, i.e.,

B−(1) = 1

It is clear that bνj and b†

[b†νbν, bν] =−bν, [b†νbν, b†ν] = b†ν (1.42)Second, for any state|φi we note that hφ|b†

νbν|φi is the norm of the state bν|φi andhence a positive real number (unless|φi = |0i, for which bν|0i = 0) Let |φλi be anyeigenstate of b†νbν, i.e., b†νbν|φλi = λ|φλi with λ > 0 Now choose a particular λ0andstudy bν|φλ 0i We find that

(b†νbν)bν|φλ 0i = (bνb†ν− 1)bν|φλ 0i = bν(b†νbν− 1)|φλ 0i = bν(λ0− 1)|φλ 0i, (1.43)i.e., bν|φλ 0i is also an eigenstate of b†

νbν, but with the eigenvalue reduced by 1 to(λ0− 1) As illustrated in Fig 1.3, if λ0 is not a non-negative integer this loweringprocess can continue until a negative eigenvalue is encountered, but this violates thecondition λ0 > 0, and we conclude that λ = n = 0, 1, 2, Writing|φλi = |nνi wehave shown that b†

νbν|nνi = nν|nνi and bν|nνi ∝ |nν− 1i Analogously, we find that

(b†νbν)b†ν|nνi = (n + 1)b†ν|nνi, (1.44)i.e., b†

ν|nνi ∝ |nν+ 1i The normalization factors for b†

ν and bν are found from

kbν|nνik2= (bν|nνi)†(bν|nνi) = hnν|b†νbν|nνi = nν, (1.45a)

kb†ν|nνik2= (b†ν|nνi)†(b†ν|nνi) = hnν|bνb†ν|nνi = nν+ 1 (1.45b)Hence we arrive at

SECOND QUANTIZATION, BASIC CONCEPTS 13

νn j

1.3.3 The fermion creation and annihilation operators

Also for fermions it is natural to introduce creation and annihilation operators, nowdenoted c†νj and cνj, which are the Hermitian adjoints of each other:

c†νj| , nν j−1, nν j, nν j+1, i = C+(nν j)| , nν j−1, nν j+1, nν j+1, i, (1.48)

cνj| , nν j−1, nν j, nν j+1, i = C−(nν j)| , nν j−1, nν j−1, nν j+1, i (1.49)But to maintain the fundamental fermionic antisymmetry upon exchange of orbitals,apparent in Eq (1.24), it is not sufficient in the fermionic case just to list the occu-pation numbers of the states, also the order of the occupied states has a meaning Wemust therefore demand

| , nν j = 1, , nν k = 1, i = −| , nν k= 1, , nν j = 1, i (1.50)and consequently we must have that c†νj and c†νk anti-commute, and hence, by Hermi-tian conjugation, that also cνj and cνk anti-commute The anti-commutator{A, B}for two operators A and B is defined as

For j 6= k we also demand that cν j and c†

ν k anti-commute However, if j = k weagain must be careful It is evident that since an unoccupied state can not be emptiedfurther we must demand cνj| , 0, i = 0, i.e., C−(0) = 0 We also have the freedom

to normalize the operators by demanding c†

an operator identity in general and below derive the consequences of this construction

In summary, we define the operator algebra for the fermionic creation and annihilationoperators by the following three anti-commutation relations:

{c†νj, c†νk} = 0, {cνj, cνk} = 0, {cνj, c†νk} = δνjνk (1.52)

An immediate consequence of the anti-commutation relations (1.52) is

Now, as for bosons, we introduce the Hermitian operator c†νcν, and by using theoperator algebra Eq (1.52) we show below that this operator, in fact, is the occupationnumber operator ˆnν In analogy with Eq (1.42) we find

1.3.4 The general form for second quantization operators

In second quantization all operators can be expressed in terms of the fundamental ation and annihilation operators defined in the previous two sections This rewriting

cre-of the first quantized operators in Eqs (1.29) and (1.32) into their second quantizedform is achieved by using the basis state identities (1.47) and (1.57) linking the tworepresentations

For simplicity, let us first consider the single-particle operator Ttotfrom Eq (1.29)acting on a bosonic N-particle system In this equation we then act with the bosonicsymmetrization operator S+ on both sides Utilizing that Ttot and S+ commute andinvoking the basis state identity (1.47) we obtain

SECOND QUANTIZATION, BASIC CONCEPTS 15

where, on the right-hand side of the equation, the operator b†

ν b stands on the site

nj To make the kets on the two sides of the equation look alike, we would like toreinsert the operator b†

νnj at the site nj on the right To do this we focus on the state

ν ≡ νn j Originally, i.e., on the left-hand side, the state ν may appear, say, p timesleading to a contribution (b†ν)p|0i We have p > 0, since otherwise both sides wouldyield zero On the right-hand side the corresponding contribution has changed into

b†νb(b†ν)p−1|0i This is then rewritten by use of Eqs (1.41), (1.46a) and (1.46b) as

b†νb(b†ν)p−1|0i = b†νb

 1

pbνb

† ν

(b†ν)p−1|0i = 1

pb

†

νbbν(b†ν)p

Now, the p operators b†νcan be redistributed to their original places as they appear onthe left-hand side of Eq (1.58) The sum over j together with δνaνnj yields p identicalcontributions cancelling the factor 1/p in Eq (1.59), and we arrive at the simple result

ν i ,ν j

Vtot= 12

|νkνli to the final state |νiνji In second quantization, the initial state is annihilated

by first annihilating the state |νki and then the state |νli, while the final state iscreated by first creating the state|νji and then state |νii:

|0i = aνaν |νkνli, |νiνji = a†

Note how all the permutation symmetry properties are taken care of by the ator algebra of a†νand aν The matrix elements are all in the simple non-symmetrizedform of Eq (1.30b).

oper-1.3.5 Change of basis in second quantization

Different quantum operators are most naturally expressed in different representationsmaking basis changes a central issue in quantum physics In this section we give thegeneral transformation rules, which are to be exploited throughout this book.Let {|ν1i, |ν2i, } and {|µ1i, |µ2i, } be two different complete and orderedsingle-particle basis sets From the completeness condition (1.7) we have the basictransformation law for single-particle states:

In the case of single-particle systems we define, quite naturally, creation operators

a†µand a†ν corresponding to the two basis sets, and find directly from Eq (1.64) that

a†µ|0i = |µi =Pνhµ|νi∗a†ν|0i, which guides us to the transformation rules for creationand annihilation operators (see also Fig 1.6):

The general validity of Eq (1.65) follows when applying the first quantization result

Eq (1.64) to the N-particle first quantized basis states ˆS±|νn 1i1 .|νn NiN leading to

a†µn1a†µn2 a†µnN|0i =X

νn1

hµn 1|νn 1i∗a†νn1

 .X

νnN

hµn N|νn Ni∗a†νnN



|0i (1.66)The transformation rules Eq (1.65) lead to two very desirable results First, that thebasis transformation preserves the bosonic or fermionic particle statistics,

SECOND QUANTIZATION, BASIC CONCEPTS 17

hν|µi aµ

BBBBBBBBBM

Fig 1.6 The transformation rules for annihilation operators aνand aµupon change

of basis between{|νi} and {|µi}

1.3.6 Quantum field operators and their Fourier transforms

In particular, one second quantization representation requires special attention, namely,the real space representation leading to the definition of quantum field operators If

we in Section 1.3.5 let the transformed basis set{| ˜ψµi} be the continuous set of sition kets{|ri} and, suppressing the spin index, denote ˜a†

Note that Ψ†(r) and Ψ(r) are second quantization operators, while the coefficients

ψν∗(r) and ψν(r) are ordinary first quantization wavefunctions Loosely speaking,

Ψ†(r) is the sum of all possible ways to add a particle to the system at position

r through any of the basis states ψν(r) Since Ψ†(r) and Ψ(r) are second quantizationoperators defined in every point in space they are called quantum field operators From

Eq (1.67) it is straight forward to calculate the following fundamental commutatorand anti-commutator,

Ψ(r1), Ψ†(r2)

Ψ(r ), Ψ†(r )

In some sense the quantum field operators express the essence of the wave/particleduality in quantum physics On the one hand they are defined as fields, like waves, but

on the other hand they exhibit the commutator properties associated with particles.The introduction of quantum field operators makes it easy to write down opera-tors in the real space representation By applying the definition (1.69) to the secondquantized single-particle operator of Eq (1.61) one obtains

In this section we will use the general second quantization formalism to derive someexpressions for specific second quantization operators that we are going to use repeat-edly in this book

1.4.1 The harmonic oscillator in second quantization

The 1D harmonic oscillator in first quantization is characterized by two conjugatevariables appearing in the Hamiltonian: the position x and the momentum p,

This can be rewritten in second quantization by identifying two operators a† and

a satisfying the basic boson commutation relations (1.41) By inspection it can beverified that the following operators do the job,

a≡ √12

x

x

SECOND QUANTIZATION, SPECIFIC OPERATORS 19

2-normalized) complex number formed by the real part x/` and theimaginary part p/(~/`), while a†is found as the adjoint operator to a From Eq (1.75)

we obtain the Hamiltonian H and the eigenstates|ni:

H =~ωa†a +1

2

and |ni =(a†)

in the studies of the photon and phonon fields, as we shall see throughout the book

If we as a measure of the amplitude of the oscillator in the state|ni with n quantause the square-root of the expectation value of x2= `2(a†a†+ a†a + aa†+ aa)/2, wefindp

hn|x2|ni = pn + 1/2 ` Thus the width of the oscillator wavefunction scalesroughly with the square-root of the number of quanta in the oscillator, as sketched inFig 1.7

The creation operator can also be used to generate the specific form of the functions ψn(x) of the oscillator starting from the groundstate wavefunction ψ0(x):

√2`−i~p

`

√2

n

|0i = √1

2nn!

x

`−`dxd

n

ψ0(x).(1.77)1.4.2 The electromagnetic field in second quantization

Historically, the electromagnetic field was the first example of second quantization.The quantum nature of the radiation field, and the associated concept of photonsplay a crucial role in the theory of interactions between matter and light In theapplications in this book we shall, however, treat the electromagnetic field classically.The quantization of the electromagnetic field is based on the observation that theeigenmodes of the classical field can be thought of as a collection of harmonic os-cillators These are then quantized In the free field case the electromagnetic field iscompletely determined by the vector potential A(r, t) in a specific gauge Tradition-ally, the transversality condition ∇·A = 0 is chosen, in which case A is denoted theradiation field, and we have

E =−∂tA ∇2A−c12 ∂2

Different quantum operators are most naturally expressed in different representationsmaking basis changes a central issue in quantum physics In this section... formalism is the notion of indistinguishability

of particles discussed in Section 1.2.1 combined with the observation in Section 1.2.2that determinants or permanents of single-particle states... see, quantum theory can be formulated in terms

of occupation numbers of these single-particle states

1.3.1 The occupation number representation

The first step in defining