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Danhong Huang: Properties of InteractingLow-Dimensional Systems — 2011/3/31 — page I — le-tex

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Godfrey Gumbs andDanhong Huang

Properties of InteractingLow-Dimensional Systems

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Danhong Huang: Properties of InteractingLow-Dimensional Systems — 2011/3/31 — page II — le-tex

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Danhong Huang: Properties of InteractingLow-Dimensional Systems — 2011/3/31 — page III — le-tex

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Godfrey Gumbs and Danhong Huang

Properties of InteractingLow-Dimensional Systems

WILEY-VCH Verlag GmbH & Co. KGaA

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Danhong Huang: Properties of InteractingLow-Dimensional Systems — 2011/3/31 — page IV — le-tex

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The Authors

Dr. Godfrey GumbsUniversity of New YorkDepartment of Physics695, Park AvenueNew York, NY 10065USA

Dr. Danhong HuangUSAF Research Lab (AFRL/RVSS)Adv. E/O Space Sensors Group3550, Aberdeen Ave, SE Bldg 426Kirtland AFB, NM 87117USA

All books published by Wiley-VCH are carefullyproduced. Nevertheless, authors, editors, andpublisher do not warrant the informationcontained in these books, including this book, tobe free of errors. Readers are advised to keep inmind that statements, data, illustrations,procedural details or other items mayinadvertently be inaccurate.

Library of Congress Card No.:applied for

British Library Cataloguing-in-Publication Data:A catalogue record for this book is availablefrom the British Library.

Bibliographic information published by theDeutsche NationalbibliothekThe Deutsche Nationalbibliothek lists thispublication in the Deutsche Nationalbibliografie;detailed bibliographic data are available on theInternet at http://dnb.d-nb.de.

© 2011 WILEY-VCH Verlag GmbH & Co. KGaA,Boschstr. 12, 69469 Weinheim, Germany

All rights reserved (including those of translationinto other languages). No part of this book maybe reproduced in any form – by photoprinting,microfilm, or any other means – nor transmittedor translated into a machine language withoutwritten permission from the publishers. Regis-tered names, trademarks, etc. used in this book,even when not specifically marked as such, arenot to be considered unprotected by law.

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Printed in SingaporePrinted on acid-free paper

ISBN 978-3-527-40894-8

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Danhong Huang: Properties of InteractingLow-Dimensional Systems — 2011/3/31 — page V — le-tex

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V

Contents

Preface XIReferences XII

Part One Linear Response of Low Dimensional Quantum Systems 1

1 Introduction 31.1 Second-Quantized Representation for Electrons 31.2 Second Quantization and Fock States 61.3 The Boson Case 61.4 The Fermion Case 91.5 The Hamiltonian of Electrons 121.6 Electron–Phonon Interaction 131.7 Effective Electron–Electron Interaction 141.8 Degenerate Electron Gases 161.9 Ground-State Energy in the High Density Limit 181.10 Wigner Solid 201.11 The Chemical Potential of an Ideal Bose Gas and Bose–Einstein

Condensation 221.12 Problems 24

References 29

2 The Kubo–Greenwood Linear Response Theory 312.1 Fluctuations and Dissipation 312.2 Nyquist’s Relation 322.3 Linear Response Theory 332.3.1 Generalized Susceptibility 332.3.2 Kronig–Kramers Relations 352.3.3 Dielectric Function in Three Dimensions 362.4 The Density Matrix and Quantum Statistics 362.4.1 The von Neumann Density Matrix 362.4.2 Entropy 372.5 Kubo’s Theory 382.6 The Kubo Equation 402.7 Fluctuation–Dissipation Theorem 41

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VI Contents

2.8 Applications 432.8.1 Mobility and the Nernst–Einstein Relation 432.8.2 Electrical Conductivity and the Nyquist Relation 452.8.3 Magnetic Susceptibility 462.8.4 The Langevin Equation 462.8.5 Stochastic Model of Magnetic Resonance 472.8.6 Gaussian Process 482.9 Kinetic Equation for Elastic Processes 492.9.1 Boltzmann’s Transport Equation 492.9.2 The Collision Term 492.9.3 Solution in the Ohmic Regime 502.9.4 Conductivity and Mobility 522.10 Problems 52

References 55

3 Feynman Diagrammatic Expansion 573.1 General Formalism 573.2 Functional Derivative Techniques 633.3 Unrenormalized Expansion for G and Σ 673.4 Renormalized Expansion for Self-Energy Σ 703.5 The Schrödinger Equation in the Hartree–Fock Approximation 743.6 Screened External Potential 753.7 Retarded Polarization Function 763.8 RPA for the Polarization Function 773.9 Problems 78

References 81

4 Plasmon Excitations in Mesoscopic Structures 834.1 Linear Response Theory and Collective Excitations 834.1.1 Screening and the Self-Consistent Field Approximation 854.2 A Linear Array of Nanotubes 864.2.1 Tight-Binding Model 874.2.2 Numerical Results and Discussion 924.3 A Linear Array of Quantum Wires 934.4 Coupled Half-Plane Superlattices 954.4.1 Hydrodynamic Model 964.4.2 Numerical Results and Discussion 994.5 Problems 101

References 111

5 The Surface Response Function, Energy Loss and Plasma Instability 1135.1 Surface Response Function 1135.1.1 The Image Potential 1145.1.2 A Bi-Layer System 1155.1.3 A Dielectric Slab 1175.1.4 A Layered 2DEG System 118

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Contents VII

5.2 Electron Energy Loss for a Planar Surface 1195.2.1 Transfer-Matrix Method 1205.2.2 Motion Parallel to the Surface 1225.2.3 Motion Perpendicular to the Surface 1225.2.4 The Inverse Dielectric Function Formalism 1235.3 Plasma Instability for a Planar Surface 1255.4 Energy Transfer in Nanotubes 1325.4.1 Energy Loss on a Single Wall Nanotube 1325.5 Problems 141

References 145

6 The Rashba Spin–Orbit Interaction in 2DEG 1476.1 Introduction to Spin–Orbit Coupling 1476.2 Spin–Orbit Coupling in the Dirac Equation 1486.3 Rashba Spin–Orbit Coupling for a Quantum Wire 1516.4 SOI Effects on Conductance and Electron-Diffusion Thermoelectric

Power 1546.5 Problems 156

References 157

7 Electrical Conductivity: the Kubo and Landauer–Büttiker Formulas 1597.1 Quantum Mechanical Current 1597.2 The Statistical Current 1607.3 A Green’s Function Formalism 1617.4 The Static Limit 1637.5 Model and Single-Particle Eigenstates 1647.6 Averaged Conductivity 1677.7 Applications to One-Dimensional Density Modulated 2DEG 1717.8 Scattering Theory Formalism 1757.9 Quantum Hall Effect 1767.10 Problems 177

References 177

8 Nonlocal Conductivity for a Spin-Split Two-Dimensional ElectronLiquid 179

8.1 Introduction 1798.2 Kubo Formula for Conductivity 1808.3 The Self-Energy and Scattering Time 1828.4 Drude-Type Conductivity for Spin-Split Subband Model 1838.5 Vertex Corrections to the Local Conductivity 1858.6 Numerical Results for Scattering Times 1918.7 Related Results in 3D in the Absence of SOI 192

References 194

9 Integer Quantum Hall Effect 1979.1 Basic Principles of the Integer Quantum Hall Effect 1979.1.1 The Hall Effect 197

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VIII Contents

9.1.2 The Quantum Hall Effect 1989.1.3 An Idealized Model 1999.1.4 Effect of Finite Temperature 2019.1.5 Effect of Impurities 2029.1.6 Application of the Quantum Hall Effect 2029.2 Fundamental Theories of the IQHE 2039.2.1 Energy Spectrum and Wave functions 2039.2.2 Perturbation and Scattering Theory 2059.2.3 Gauge Symmetry Approach 2069.2.4 The QHE in a Periodic Potential 2079.2.5 Topological Equivalence of the Quantum Hall Conductance 2089.3 Corrections to the Quantization of the Hall Conductance 2109.3.1 Properties of the Green’s Function 210

References 212

10 Fractional Quantum Hall Effect 21510.1 The Laughlin Ground State 21510.1.1 The Lowest Landau Level 21510.1.2 Laughlin’s Wave Function 21610.1.3 Properties of the Laughlin Wave Function 21810.1.4 Justification of the Laughlin State 21910.2 Elementary Excitations 22010.2.1 Fractional Charge 22010.2.2 The Complete Set of Quasi-Hole States 22210.3 The Ground State: Degeneracy and Ginzburg–Landau Theory 22410.3.1 Ground State Degeneracy 22410.3.2 Ginzburg–Landau Theory of the Quantum Hall Effect 22510.4 Problems 228

References 229

11 Quantized Adiabatic Charge Transport in 2D Electron Systems andNanotubes 231

11.1 Introduction 23111.2 Theory for Current Quantization 23211.3 Tunneling Probability and Current Quantization for Interacting

Two-Electron System 23511.3.1 Spin Unpolarized Case 23611.4 Adiabatic Charge Transport in Carbon Nanotubes 23811.5 Summary and Remarks 240

References 241

12 Graphene 24312.1 Introduction 24312.2 Electronic Properties of Graphene 24512.3 Graphene Nanoribbons and Their Spectrum 24912.3.1 Zigzag Edge 251

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Danhong Huang: Properties of InteractingLow-Dimensional Systems — 2011/3/31 — page IX — le-tex

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Contents IX

12.3.2 Armchair Nanoribbon 25312.4 Valley-Valve Effect and Perfect Transmission in GNR’s 25512.5 GNR’s Electronic and Transport Properties in External Fields 26212.6 Problems 26712.A Energy Eigen States 27012.B The Conductance 271

References 273

13 Semiclassical Theory for Linear Transport of Electrons 27513.1 Roughness Scattering 27613.1.1 Model for Elastic Scattering 27713.1.2 Numerical Results for Roughness Scattering Effect 28013.2 Phonon Scattering 28213.2.1 Model for Inelastic Scattering 28313.2.2 Numerical Results for Phonon Scattering Effect 28513.3 Thermoelectric Power 28713.3.1 Model for Non-equilibrium Phonons 28813.3.2 Numerical Results for Thermoelectric Power 29113.4 Electron–Electron Scattering 29313.4.1 Model for Pair Scattering 29313.4.2 Numerical Results for Coulomb Scattering Effect 295

References 298

Part Two Nonlinear Response of Low Dimensional Quantum Systems 301

14 Theory for Nonlinear Electron Transport 30314.1 Semiclassical Theory 30314.1.1 Transient Boltzmann Equation 30314.1.2 Numerical Procedure 30614.1.3 Numerical Results for Bloch Oscillations and Dynamical

Localization 30914.2 Quantum Theory 31214.2.1 Force Balance Equation 31214.2.2 Boltzmann Scattering Equation 315

References 318

15 Spontaneous and Stimulated Nonlinear Wave Mixing of Multi-excitons 319

15.1 Spontaneous, Stimulated, Coherent and Incoherent Nonlinear WaveMixing 323

15.2 n C 1 Wave Mixing in QD Fluids and Polymer QDs MoleculeSolutions 328

15.2.1 Stimulated and Spontaneous Incoherent Signals 32915.2.2 Spontaneous Coherent Signal 33015.3 Application to Two-Photon-Induced Signals 33315.A Semiclassical vs. Quantum Field Derivation of Heterodyne Detected

Signals 337

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X Contents

15.B Generalized Susceptibility and Its CTPL Representation 340References 342

16 Probing Excitons and Biexcitons in Coupled QDs by Coherent OpticalSpectroscopy 345

16.1 Model Hamiltonian for Two Coupled Quantum Dots 34616.2 Single-exciton Manifold and the Absorption Spectrum 34816.3 Two-exciton Manifold and the 2D Spectra 35116.4 Summary 35716.A Transformation of the Electron–Hole Hamiltonian Using Excitonic

Variables 35716.B The Nonlinear Exciton Equations 35916.C The 2D Signals 360

References 361

17 Non-thermal Distribution of Hot Electrons 36317.1 Introduction 36317.2 Boltzmann Scattering Equation 36417.3 Numerical Results for Effective Electron Temperature 36717.4 Summary 369

References 370

Index 373

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Danhong Huang: Properties of InteractingLow-Dimensional Systems — Chap. gumbs8948f01 — 2011/3/31 — page XI — le-tex

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XI

Preface

There has been a considerable amount of research on “mesoscopic” structureswhose sizes are intermediate, that is, between the macroscopic and the atomicscale. These include semiconductor heterojunctions, quantum dots and wires aswell as carbon nanotubes and atomic layers of graphene. One is unable to explainthe properties of these systems simply in terms of a single-particle Schrödingerequation since many-body effects cannot be neglected. Therefore, there is a needto combine an introduction to some typical topics of interest and the methods andtechniques needed to handle them in a single volume. This book tries to achievethat goal by carefully presenting a number of topics concerned with the opticalresponse and transport properties of low-dimensional structures. This material issupplemented by a selection of problems at the end of each chapter to give the read-er a chance to apply the ideas and techniques in a challenging manner. There areseveral excellent textbooks which already deal with electron–electron interaction ef-fects. However, the material we cover supplements those publications by coveringmore recently studied topics, that is, especially in semiconductors.

Since the aim of the book is to be self-contained, we first present some back-ground diagrammatic methods. This is based on standard field theoretic tech-niques. (See, for example, the early work in [1–17].) However, here we give severalexamples which have useful applications to the topics covered in later chapters.We evaluate the Green’s function expansion and the polarization function whichis a necessary ingredient in our investigations employing the linear response for-malism which can then be used in the study of the collective plasma excitationsin quantum dots and wires, electron transport, light absorption, electron energyloss spectroscopy for nanotubes, graphene containing massless Dirac fermions orlayered semiconductor structures, just to name a few examples which are coveredin this book. We have developed some of the methods introduced here in collab-oration with co-workers. By learning the formalism and getting introduced to thenovel physical properties of low-dimensional systems, the reader should be able tounderstand scientific papers in condensed matter physics dealing with the effectsarising from many-particle interactions.

This book is inspired by a collection of lectures which were given over the yearsat the Graduate Center of the City University of New York. This course is usu-ally taken by graduate students who have had some exposure to basic quantum

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XII Preface

mechanics, statistical mechanics and introductory solid state physics at the under-graduate level. What makes this book different from previously published ones onthe many-body theory of solids is that it presents a variety of current topics of in-terest in the field of mesoscopic systems and it also provides basic formal theorywhich is relevant to these systems but not available in previously published books.This makes it suitable for either a “Special Topics” course in solid state physics inwhich a few of the chapters may be selected, or as a textbook for an advanced solidstate physics course in which the methodology is taught. In any event, problemsolving would be an integral part of the course. A number of problems and the ref-erences have been given at the end of each chapter for students wanting to becomemore familiar with the topics and their background.

We would like to express our gratitude to Dr. Oleksiy Roslyak for his generoustime in helping us write the chapters on graphene, nonlinear Green’s functiontheory as well as on excitons and biexcitons in quantum dots. This contributionas well as his helpful comments on the manuscript are gratefully acknowledged.Our thanks also to Dr. Paula Fekete, Dr. Tibab McNeish, Dr. Oleg Berman Dr. An-tonios Balassis, Andrii Lurov, Hira Ghumman, and Alisa Dearth for their helpfulcomments and criticisms of the manuscript.

Hunter College of the City University of New York Godfrey GumbsAFRL, New Mexico, March 2011 Danhong Huang

References

1 Martin, P.C. and Schwinger, J. (1959)Theory of Many-Particle Systems. Phys.Rev., 115, 1342.

2 Abraham, M. and Becker, R. (1949)Classical Theory of Magnetism, Hafner,NY.

3 Ambegaokar, V. and Baratoff, A. (1963)Tunneling Between Superconductors.Phys. Rev. Letts., 10, 486.

4 Anderson, P.W. (1964) Lectures on theMany-Body Problem, Vol. 2 (ed. E.R.Caianello), Academic Press.

5 Anderson, P.W. (1958) Random-PhaseApproximation in the Theory of Super-conductivity. Phys. Rev., 112, 1900.

6 Ashcroft, N., Mermin, D., and Bardeen,J. (1956) Theory of Superconductivity,in Handbuch der Phys., Vol. 15 (ed. S.Flugge), 274.

7 Bardeen, J. (1961) Tunnelling from aMany-Particle Point of View. Phys. Rev.Lett., 6, 57.

8 Bardeen, J. and Schrieffer, J.R. (1961) inProg. in Low Temp. Phys. (ed. C.J. Gorter),North Holland, Vol. B, p. 170.

9 Bardeen, J., Cooper, L.N., and Schrieffer,J.R. (1957) Theory of Superconductivity.Phys. Rev., 108, 1175.

10 Blatt, J.M. and Butler, S.T. (1954) Super-fluidity of a Boson Gas. Phys. Rev., 96,1149.

11 Bogoliubov, N.N. (1958) On a NewMethod in the Theory of Superconduc-tivity. Nuovo Cim., 7, 794.

12 Bogoliubov, N.N. (1959) Us. Fiz. Nauk,67, 549.

13 Cohen, M.H., Falicov, L.M., and Phillips,J.C. (1962) Superconductive Tunneling.Phys. Rev. Lett., 8, 316.

14 Ferrell, R.A. and Prange, R.E. (1963)Self-Field Limiting of Josephson Tunnel-ing of Superconducting Electron Pairs.Phys. Rev. Lett, 10, 479.

15 Feynman, R.P. (1972) Statistical Mechan-ics, W.A Benjamin, Reading, MA.

16 Fock, V. (1932) Konfigurationsraum undzweite Quantelung. Z. Phys., 75, 622.

17 Callaway, J. (1974) Quantum Theory ofthe Solid State, Ch. 7, 2nd edn, AcademicPress Inc., San Diego.

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Danhong Huang: Properties of InteractingLow-Dimensional Systems — Chap. gumbs8948c01 — 2011/3/31 — page 1 — le-tex

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Part One Linear Response of Low Dimensional QuantumSystems

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3

1Introduction

1.1Second-Quantized Representation for Electrons

The use of a Schrödinger equation to describe one or more electrons already treatsthe electron quantum mechanically and is sometimes referred to as first quanti-zation. As long as electrons are neither created nor destroyed, such a descriptionis complete. However, an electron that is transferred from state n to state m is of-ten described as the destruction of an electron in state n and creation in state mby an operator obeying an algebra of the form c†

m cn . It is convenient therefore tofurther refine the algebra of such operators analogous to the operators b†

q and bq

that create and destroy phonons of wave-vector q. However, electrons are fermionsrather than bosons and the state occupancy number c†

n cn should only be permit-ted to take the values zero or one. This aim is achieved by using anti-commutationrules fdescribed by bracesg or by square brackets with aC subscript, that is, [� � � ]C,instead of commutation rules described by brackets or square brackets with a �subscript, that is, [� � � ]�.

In this book, we will be primarily concerned with low-dimensional systems suchas quantum wells, dots and wires. A typical band structure of the valence and con-duction bands for a heterostructure like GaAs/AlGaAs is shown in Figure 1.1. How-ever, the formulation in this chapter and in some of the others is independent ofdimensionality.

The phrase “second quantization” is descriptive of the notion that the Schrödin-ger wave function Ψ (r) is to be quantized, that is, treated as an operator. In termsof any complete set of states φk (r), we can write:

Ψ (r) DX

k

ck φk (r) ,Z

d3rφ�k (r) φk 0 (r) D δk k 0 , (1.1)

where the anti-commutation rules are given byhck , c†

k 0i

CD δk k 0 , (1.2)

and

[ck , ck 0 ]C Dh

c†k , c†

k 0i

CD 0 . (1.3)

Properties of Interacting Low-Dimensional Systems, First Edition. G. Gumbs and D. Huang.© 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA.

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4 1 Introduction

z (D)0 200 400 600 800 1000 1200

Ene

rgy

(eV

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-2

-1

0

1

2T=5 K

Figure 1.1 Valence (lower curve) and conduction (upper curve) bands of electrons in a semi-conductor heterostructure.

For a single state, we can omit the subscripts and examine the consequences. Equa-tion (1.3) implies that

c2 D 0 ,�c†�2 D 0 . (1.4)

Let N D c†c, then

N 2 D �c†c�2 D c†cc† c D c† �1 � c†c

�c

D c†c � �c†�2c2 D c†c � 0 D N . (1.5)

Thus, N D 1 or 0.Consider the two eigenstates of N:

N Ψ0 D 0Ψ0 , and N Ψ1 D 1Ψ1 . (1.6)

Then, is c†Ψ0 also an eigenstate of N?�c†c�

c†Ψ0 D c† �1 � c†c�

Ψ0

D c† Ψ0 ��c†�2

cΨ0 D c† Ψ0 . (1.7)

Therefore,

N�c†Ψ0

� D 1�c† Ψ0

�, (1.8)

that is, c†Ψ0 is proportional to Ψ1. Evaluate the normalization:Zd3r

�c†Ψ0

�� c†Ψ0

� D Z d3r Ψ �0 cc†Ψ0

DZ

d3r Ψ �0

�1 � c†cΨ0

� D 1 . (1.9)

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1.1 Second-Quantized Representation for Electrons 5

Therefore, c† Ψ0 is normalized and we can simply choose

c†Ψ0 D Ψ1 . (1.10)

Similarly,

cc† (cΨ1) D c�1 � cc†�Ψ1 D cΨ1 . (1.11)

Therefore, cΨ1 is an eigenvector of cc† D 1�c† c with eigenvalue 1 or eigenvectorof c† c with eigenvector 0. By a similar procedure to Eq. (1.9), cΨ1 is normalized andwe can write

cΨ1 D Ψ0 . (1.12)

Note that

cΨ0 D c2Ψ1 D 0 , (1.13)

c†Ψ1 D�c†�2

Ψ0 D 0 . (1.14)

We begin by rewriting the Schrödinger equation in second quantized form. Inmost cases, the Hamiltonian has the form

H DNX

iD1

T (xi )C 12

NXi¤ j D1

V�xi , x j

�, (1.15)

where T is the kinetic energy and V is the potential energy of interaction of theparticles. Here, xi D (x i , ti , s i ) is the space-time-spin point. The potential energyterm represents the interaction between every pair of particles counted once, pre-cisely why we have the factor of 1/2. We will not give the details for reformulatingEq. (1.15) in second quantized form since it can be found in many textbooks onquantum mechanics and will simply quote the results along with some others. Forthe Hamiltonian in Eq. (1.15), we have

OH DZ

d3x , OΨ † (x ) T (x ) OΨ (x )

C 12

Zd3x

Zd3 x 0 OΨ † (x ) OΨ † �x 0�V

�x , x 0� OΨ �

x 0� OΨ (x ) . (1.16)

The single-particle operator

J DNX

iD1

J (x i ) (1.17)

in second quantized form becomes

OJ DZ

d3x OΨ † (x ) J (x ) OΨ (x ) , (1.18)

and, in particular, the number density n(x ) DPNiD1 δ(x � x i ) is given by

On (x ) D OΨ † (x ) OΨ (x ) , (1.19)

where OΨ †(x ) and OΨ (x ) are creation and annihilation operators, respectively.

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6 1 Introduction

1.2Second Quantization and Fock States

For a system in which the number of particles is variable, it is essential to introducecreation and destruction operators. However, it is also possible to do so when thenumber of particles is conserved. In that case, of course, the perturbation operatorswill contain an equal number of creation and destruction operators. In that case,it is customary to describe the procedure as “second quantization”. First quantiza-tion replaces classical mechanical equations of motion, and second quantizationreplaces a Hamiltonian containing one-body forces, two-body forces, and so on bya Hamiltonian that is bilinear in creation and destruction operators, quadratic increation and destruction operators, and so on. Nothing new is added, but the com-mutation rules of the creation and destruction operators make the bookkeepingof the states simpler than using permanents or determinants for the Schrödingerwave functions. For Bose particles, second quantization was developed by Dirac [1],and extended to Fermi particles by Wigner and Jordan [2]. A more detailed discus-sion is given by Fock [3] and by Landau and Lifshitz [4].

1.3The Boson Case

It is simplest to describe the relation between the Schrödinger description and thesecond quantized description by assuming that we have a set of N non-interactingparticles that can occupy any one of a set of orthonormal states φn(xi ). Besides, ifthe particles do not interact, the wave function can be a product of the φn that areoccupied. In addition, (for Boson statistics) the wave function must be symmetricwith respect to exchange of any two particles. If there are N1 particles in state φ1,N2 in φ2 for a total N D P

i Ni of particles, the wave function can be written inthe form

Ψ (N1, N2, � � � ) Dr

N1!N2! � � �N !

Xφ p1 (x1) φ p2 (x2) � � � φ pN (xN ) , (1.20)

where p1, p2, . . . , pN is any set of occupied states (such as p1 D 1, p2 D 3, p3 D 4,etc.). These indices must not all be different since some states can be multiplyoccupied. However, where they are distinct, the sum must be taken over all permu-tations of the distinct indices. Since the number of ways of placing N1 particles (outof N) in one box, N2 in a second and so on is given by N !/(N1!N2! � � � ). The prefac-tor in Eq. (1.20) is added to preserve normalization. We can refer to Ψ (N1, N2, � � � )as a Fock state with N1 particles in φ1(x ), N2 in φ2(x ), and so on.

We then introduce creation and destruction operators B†i and B j defined by

B†i Ψ (N1, � � � , Ni , � � � ) D

pNi C 1Ψ (N1, � � � , Ni C 1, � � � ) , (1.21)

and

B j Ψ�N1, � � � , N j , � � �

� D pN j Ψ�N1, � � � , N j � 1, � � � � . (1.22)

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1.3 The Boson Case 7

The principal simplification of second quantization is that a one-body operator

V DX

a

V (r a) , (1.23)

which can take Ψ (Ni � 1, Nk ) into Ψ (Ni , Nk � 1) by “destroying” a particle in statek and creating one in state i in the Schrödinger permanent wave functions can bemuch more easily calculated when the operator

OV D Vi k B†i Bk (1.24)

acts on the Fock states. In particular, the matrix elementZd3r Ψ � (Ni , Nk � 1) OV Ψ (Ni � 1, Nk )

DZ

d3r Ψ � (Ni , Nk � 1) Vi k B†i Bk Ψ (Ni � 1, Nk )

Dp

Ni Nk Vi k , (1.25)

where the first integral is over the Schrödinger space dxl dx2 � � � dxN . The secondone is thought of in terms of creation and destruction operators in a space de-scribed by the number set fN j g, and the matrix element

Vi k DZ

d3rφ�i (r) V (r) φk (r) (1.26)

is the usual one-body matrix element in the Schrödinger representation.Landau and Lifshitz [4] do not derive this result. They merely state that “The

calculation of these matrix elements is, in principle, very simple, it is easier to do itoneself than to follow an account of it.” It would be unfair to leave the matter there:Landau knows how to do it; let it be an exercise for the reader.

We can make the answer plausible by showing that the right-hand side ofEq. (1.25) is a product of four factors:

P ��

(Ni � 1)!Nk !(Ni C Nk � 1)!

�1/2 � Ni ! (Nk � 1)!(Ni C Nk � 1)!

�1/2

Vi k M . (1.27)

The first factor comes from the normalization factor of the initial state, and thesecond from the normalization of the final state. Factors involving N j for j ¤ i ork are ignored since they merely contribute to the normalization of the remainingstates. A single product of φs for the initial state, a product for the final state andone V(r a) give rise to a term Vi k or zero. The factor M is simply the number ofsuch non-vanishing terms. Since each V(r a) for a D 1, 2, . . . , N makes the samecontribution, M contains a factor N D (Ni � 1) C Nk D Ni C (Nk � 1), the totalnumber of particles in these two states (in either the final or the initial states). Fora given V(ra ), the factor

Rd3r a φ�

i (r a)V(ra )φk (ra ) appears and one particle is usedup. The remaining N � 1 D Ni C Nk � 2 particles must now be distributed withNi � 1 in the ith state and Nk � 1 in the kth state. Thus,

M D N(N � 1)!

(Ni � 1)! (Nk � 1)!, (1.28)

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8 1 Introduction

where the first factor, N D NiCNk�1, is the number of terms in the sumP

a V(ra )and the second factor is the number of ways of distributing N � 1 particles in twostates with Ni � 1 in the first state and Nk � 1 in the second state. Finally,

P Dp

Ni Nk(Ni � 1)! (Nk � 1)!

(Ni C Nk � 1)!Vi k

(Ni C Nk � 1)!(Ni � 1)! (Nk � 1)!

Dp

Ni Nk Vi k . (1.29)

The matrix elements in Eqs. (1.21) and (1.22) are such as to insure the commu-tations rules

[Bi , B j ]� D [B†i , B†

j ]� D 0 , [Bi , B†j ]� D δ i j . (1.30)

A natural generalization of Eq. (1.25) to two-body operators implies the replace-ment: X

a>b

W (ra , r b)!X

W i kl m B†

i B†k Bl Bm . (1.31)

A compact statement of these commutation rules in Eq. (1.30) can be obtained byintroducing an operator ψ(r) in the form

ψ (r) DX

i

Bi φ i (r) , ψ† (r) DX

i

B†i φ�

i (r) . (1.32)

Then, the “Schrödinger operators”, ψ(r) and ψ†(r0), obey the commutation rules�ψ (r) , ψ

�r 0��

� D�ψ† (r) , ψ† �r0��

� D 0 , (1.33)

�ψ (r) , ψ† �r 0��

� DXi, j

φ i (r) φ�j

�r 0� δ i j D δ

�r � r 0� , (1.34)

where the δ i j arises from Eq. (1.30) and the Dirac delta function follows from thecompleteness.

The second quantized Hamiltonian of a boson system with one and two-bodyforces can be written in the form

OH DZ

d3r� „2

2mrψ† (r) � rψ (r)C V (r) ψ† (r) ψ (r)

�

C 12

Zd3r

Zd3r 0ψ† (r) ψ† �r0� W

�r , r 0� ψ

�r 0� ψ (r) . (1.35)

In addition to the correspondenceXa

V (r a)!Xi,k

Vi k B†i Bk (1.36)

for one-body forces, we have a similar correspondence for two-body forces:Xa>b

W (ra , r b)!X

i,kIl,m

W i kl m B†

i B†k Bl Bm , (1.37)

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1.4 The Fermion Case 9

where

W i kl m D

Zd3r

Zd3r 0φ�

i (r)φ�k (r0)W(r, r 0)φ l(r)φm(r0) . (1.38)

We note that the commutation rules, Eqs. (1.30), (1.33), and (1.34) for bosons arethe same as the ones we are familiar with for harmonic oscillators and phonons,which are of course bosons.

1.4The Fermion Case

In the fermion case, the Pauli principle requires that the wave function be antisym-metric. The simplest example of a set of independent fermions is then describedby a determinant

Ψ (N1, N2, � � � ) D

ˇ̌̌ˇ̌̌ˇ̌φ p1 (r1) φ p1 (r2) � � � φ p1 (r N )φ p2 (r1) φ p2 (r2) � � � φ p2 (r N )� � � � � � � � � � � �

φ pN (r1) φ pN (r2) � � � φ pN (r N )

ˇ̌̌ˇ̌̌ˇ̌ (1.39)

in terms of the set of functions φ i (r). The latter are usually taken as members of acomplete set of eigen-functions of the one-body Hamiltonian. Here, N is the totalnumber of eigen-functions appearing in the determinant, that is, the total numberof occupied states. The set of numbers p1, p2, � � � , pN are some chosen orderingof the set fig. To make the sign of the determinant unique, a fixed order must bechosen. It is conventional to choose the ordering

p1 < p2 < p3 < � � � < pN . (1.40)

This is not necessary, but a fixed choice must be maintained in the ensuing discus-sion.

The result of second quantization for fermions will look similar to that for bosonsin the sense that Eq. (1.32) is replaced by

ψ (r) DX

i

Fi φ i (r) , ψ† (r) DX

i

F†i φ�

i (r) , (1.41)

where the boson operators Bi have been replaced by the fermion operators Fi .With this change, Eqs. (1.35)–(1.38) remain valid. However, fermion states can haveoccupancies only of Ni D 0 or Ni D 1. This is accomplished by the use of anti-commutation rulesh

Fi , F†j

iCD δ i j ,

�Fi , F j

�C D 0 ,

hF†

i , F†j

iCD 0 (1.42)

rather than the commutation rules used in the boson case. In particular, Eqs. (1.33)and (1.34) are replaced by

[ψ(r), ψ(r 0)]C D [ψ†(r), ψ†(r0)]C D 0 , (1.43)

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10 1 Introduction

[ψ(r), ψ†(r 0)]C D δ(r � r 0) , (1.44)

which follows directly from Eq. (1.42).To see that the anti-commutation rules, Eq. (1.42), accomplish the desired objec-

tives, we first consider a single state φ i(r) with operators Fi and F †i , and omit the

index i.

F 2 D 0 ,�F†�2 D 0 . (1.45)

Let N D F†F . Then,

N 2 D �F†F�2 D F† F F†F D F† �1� F†F

�F

D F† F � �F†�2F 2 D F†F � 0 D N (1.46)

so that

N D 1 or N D 0 . (1.47)

Consider the eigenstates of N:

N Ψ0 D 0Ψ0 or N Ψ1 D Ψ1 . (1.48)

Then, is F†Ψ0 also an eigenstate of N?�F †F

�F † Ψ0 D F† �1 � F†F

�Ψ0

D F† Ψ0 ��F†�2

F Ψ0 . (1.49)

Therefore,

N F†Ψ0 D 1�F† Ψ0

�, (1.50)

that is, F †Ψ0 is proportional to Ψ1. Evaluate the normalization:Zd3r

�F†Ψ0

�� F†Ψ0

� D Z d3r , Ψ �0 F F†Ψ0

DZ

d3r Ψ �0

�1� F†F

�Ψ0 D 1 . (1.51)

Therefore, F† Ψ0 is normalized and we can choose

F†Ψ0 D Ψ1 . (1.52)

Similarly,

F F † (F Ψ1) D F�1 � F F†�Ψ1 D F Ψ1 . (1.53)

Therefore, F Ψ1 is an eigenvector of F F† D 1� F†F with eigenvalue one or eigen-vector of F †F with eigenvector zero. By a similar procedure to Eq. (1.28), F Ψ1 isnormalized and we choose

F Ψ1 D Ψ0 . (1.54)

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1.4 The Fermion Case 11

Note that

F Ψ0 D F 2Ψ1 D 0 , (1.55)

F†Ψ1 D�F†�2

Ψ0 D 0 . (1.56)

The above discussion has established that the anti-commutation rules generate aset of states with occupancies zero and one. The full correspondence between firstand second quantization requires that we establish the analogue of Eq. (1.36):

V DX

a

V (r a)!X

i k

Vi,k F†i Fk . (1.57)

This involves the matrix element of the one-body operator V between two determi-nantal states. In effect, a transition in which a fermion in state k is destroyed andone in state i is created was found to have the matrix elementZ

d3r Ψ � (Ni , Nk � 1) V Ψ (Ni � 1, Nk ) D Vi k (1.58)

between determinantal states.[4, Eq. (61.3)] allege (without proof) that the result in Eq. (1.58) should instead beZ

d3r Ψ � (Ni , Nk � 1) V Ψ (Ni � 1, Nk ) D Vi k (�1)Σ , (1.59)

where the symbol

Σ Dk�1X

j DiC1

N j . (1.60)

This discrepancy can be resolved as follows. In our calculation, we obtained the fi-nal wave function (before anti-symmetrization) from the initial wave function sim-ply by replacing φk (r a) with φ i (r b). However, this procedure does not preserve thechosen ordering, Eq. (1.40). To restore the chosen ordering, one must interchangerow i and k in the final determinant. These gain a factor (�1)Σ where Σ is thenumber of occupied states between i and k.

To maintain the validity of Eq. (1.57), the operator Fk takes the state Nk D 1 intoNk D 0 with an extra factor

Fk Ψ (Nk D 1) D ηk Ψ (Nk D 0) , (1.61)

and

F†k Ψ (Nk D 0) D η�

k Ψ (Nk D 1) . (1.62)

The anti-commutator [Fk , F†k ]C is unchanged as long as jηk j2 D 1. If we define

ηk Dk�1Yj D1

(�1)N j , (1.63)

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12 1 Introduction

then F†i Fk acquires just the extra factor (�1)

P(iC1,k�1) demanded by Eq. (1.59).

Moreover, it is easy to see why

Fm Fn D �Fn Fm (1.64)

because one of m , n (say m) is higher in the sequence of states. Then, the matrixfor Fm in Fm Fn is reversed in sign because Fn has acted and eliminated the state nbelow m. In the reverse order, Fn Fm , Fn is unaffected by the elimination of statem above it. Hence, the two orders differ by a factor �1 to yield the desired anti-commutation rule. These remarks are stated clearly in [5].

1.5The Hamiltonian of Electrons

We first consider the case of a single electron, or of a set of non-interacting elec-trons. The Hamiltonian can be written in the form:

H DZ

d3r ψ† (r) U (r) ψ (r) . (1.65)

Here, ψ(r) and ψ†(r) are regarded as operators that can be expanded in an arbitraryorthonormal set φn(r):

ψ (r) DX

n

cn φn (r) , (1.66)

ψ† (r) DX

m

c†m φ�

m (r) . (1.67)

The Hamiltonian then takes the form

H DXm ,n

c†m

Zd3r φ�

m (r) U (r) φn (r) cn

DXm ,n

c†mhmjUjnicn , (1.68)

where hmjUjni represents the matrix for destruction of electrons in n and its cre-ation in m. If the original Schrödinger equation is�

� „2

2m�r2 C V (r)�

φ D E φ , (1.69)

where m� represents the mass of an electron, then U is the operator defined by

U D � „2

2mr2 C V (r) , (1.70)

whose matrix element is

hmjUjni DZ

d3r φ�m (r)

�� „

2

2m�r2 C V (r)�

φn (r) D Umn . (1.71)

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1.6 Electron–Phonon Interaction 13

To best understand the eigenstates of the operator

H DXm ,n

Umn c†m cn , (1.72)

we can choose the φn to be the eigenstates of U with eigenvalues En . Then,

Umn D Em δmn , (1.73)

H DX

n

En c†n cn . (1.74)

The Hilbert vector Ψ that is an eigenvector of H will then simply be described by aset of occupancies zero or one of each N D c†

n cn , for example, j0,1,0,0,1,1,0,1,. . .i.To solve the Schrödinger equation H Ψ D E Ψ , we can writeX

m ,n

�c†

m cnUmn�

Ψ D E Ψ . (1.75)

1.6Electron–Phonon Interaction

Following Callaway [6] the Hamiltonian is written as an electron energy, plus aphonon energy, plus an electron–phonon interaction:

H DXk ,σ

Ek c†k σck σ C

Xq

„ωq a†q aq

CXk ,q,σ

hD(q)c†

kCq,σck σaq C D(�q)c†k�q,σck σa†

q

i. (1.76)

Quasi-momentum conservation is built into the above expression and σ is the in-dex for electron spin. The original form for phonon absorption was

Vk 0k δk 0,kCq c†k 0σ ck σaq . (1.77)

We assume that the potential is local so that Vk 0k D Vk 0�k . There are also screenedCoulomb electron–electron terms that we ignore here. These have the form

Zd3r1 � � �

Zd3r n Ψ � (r1, � � � , r n)

12

nXi¤ j D1

W�r i � r j

�Ψ (r1, � � � , rn) ,

(1.78)

corresponding to the many-body wave-function energy. If we insert ψ (r) DPj c j φ j (r) which expresses the operators ψ(r) in terms of the c j operators,

we get

Htwo-body D 12

Xi,kIl,m

c†i c†

khk ljW jmlic l cm , (1.79)

which is a representation of two-body interactions in second quantized form.

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14 1 Introduction

1.7Effective Electron–Electron Interaction

In an electromagnetic field, the charge-1 acts on the field and the field acts oncharge-2. If we can eliminate the field, we obtain a direct interaction betweencharge-1 and charge-2. Here, the field is the phonon field. After we eliminate theelectron–phonon interaction, we should obtain an effective electron–electron inter-action.

Let

H D H0 C H1 , (1.80)

where H1 is the interaction Hamiltonian. The transformed Hamiltonian

HT D e�i S H ei S

D H C i [H, S ] � 12

[[H, S ], S ]C � � �

D H0 C H1 C i [H0, S ]C i [H1, S ] � 12

[[H0, S ], S ]C � � � (1.81)

to second order of S. To dispose of H1 to lowest order, we set

H1 C i [H0, S ] D 0 , i.e., [H0, S ] D i H1 . (1.82)

Then,

HT D H0 C i [H1, S ] � i2

[H1, S ] D H0 C i2

[H1, S ] . (1.83)

Let jmi and jni be energy eigenstates of the complete system of electrons andphonons. Then, we get

hmj[H0, S ]jni D ihmjH1jni , and hmjH1jmi D 0 ,

(Em � En) Smn D ihmjH1jni ,

Smn D hmjH1jniEm � En

, for m ¤ n . (1.84)

However,

hnq � 1jaq jnqi D n1/2q , hnq C 1ja†

q jnqi D�nq C 1

�1/2 . (1.85)

Writing S as operators cs (electronic part) and as a matrix in the vibrational part,we obtain

hnq � 1jS jnqi D iXk ,σ

D(q)c†kCq,σck σ n1/2

q

E (k C q)� E(k)� „ωq. (1.86)

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1.7 Effective Electron–Electron Interaction 15

Here, a phonon of wave vector q is absorbed and an electron is scattered from k tok C q at the same time, that is,

hnq C 1jS jnqi D iXk ,σ

D(�q)c†k�q,σck σ

�nq C 1

�1/2

E (k � q)� E(k)� „ωq. (1.87)

In Eq. (1.87), a phonon of wave vector q is created and an electron is scattered fromk to k � q at the same time.

We are concerned with the effective second-order interaction

i2

[H1, S ]

and by this we mean the part diagonal in the phonon numbers. (The off-diagonalelements can be transformed away to give still higher-order interactions). We canwrite

hnq j i2 [H1, S ]jnqi D i2

X˚hnq jH1jnq ˙ 1ihnq ˙ 1jS jnqi� hnq jS jnq ˙ 1ihnq ˙ 1jH1jnqi

�, (1.88)

where the intermediate states jnq ˙ 1i are summed over. We write out one termexplicitly:

i2

˝nq jH1j nq � 1

˛ ˝nq � 1 jS j nq

˛D i

2

Xk 0,σ0

D(�q)n1/2q c†

k 0�qσ0 ck 0σ0Xk ,σ

i D(q)c†kCq,σck σ n1/2

q

E(k C q)� E(k)� „ωq

D �12

Xk ,k 0Iσ,σ0

jD(q)j2c†k 0�q,σ0 ck 0σ0 c†

kCqσck σ

E(k C q)� E(k)� „ωq, (1.89)

where D(�q) D D�(q). Callaway’s Eq. (7.8.5) [6] states that the four terms combineto give

H1 DX

k ,k 0,qIσ,σ0

jD(q)j2„ωq

[E(k)� E(k � q)]2 � �„ωq�2

� c†k�qσ c†

k 0Cqσ0 ck 0σ0 ck σ . (1.90)

With the replacement k ! k C q, we get

H1 DX

k ,k 0,qIσ,σ0

jD(q)j2„ωq

[E(k C q)� E(k)]2 � („ωq)2

� c†k σ c†

k 0Cqσ0 ck 0σ0 ckCqσ . (1.91)

Note that the effective Hamiltonian is independent of temperature. For E(k)�E(k�q) < „ωq , the interaction becomes attractive.

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16 1 Introduction

An alternate derivation of the above H1 is obtained in a semiclassical way byRickayzen [7, p. 117–121], by considering an electron fluid and an ion fluid andretarded interactions between the two components.

We will be dealing with systems of many interacting particles and, as a result,we need to include the inter-particle potential in the Schrödinger equation. Thisproblem is the basis of the present book. The N-particle wave function in configu-ration space contains all the possible information. However, a direct solution of theSchrödinger equation is not practical. We therefore need other techniques whichinvolve (a) second quantization, (b) quantum field theory, and (c) Green’s functions.

Second quantization describes the creation and annihilation of particles andquantum statistics as well as simplifying the problem of many interacting parti-cles. This approach reformulates the Schrödinger equation. The advantage it hasis that we avoid the awkward use of symmetrized and anti-symmetrized productof single-particle wave functions. With the method of quantum field theory, weavoid dealing with the wave functions and thus the coordinates of all the particles –bosons and fermions.

Green’s functions can be used to calculate many physical quantities such as(1) the ground state energy, (2) thermodynamic functions, (3) the energy and life-time of excited states, and (4) linear response to external perturbations. The exactGreen’s functions are also difficult to calculate which means we must use pertur-bation theory. This is presented with the use of Feynman diagrams. This approachallows us to calculate physical quantities to any order of perturbation theory. Weuse functional derivative techniques in the perturbation expansion of the Green’sfunction determined by the Dyson equation and show that only linked diagramscontribute. Wick’s theorem which forms all possible pairs of the field operators isnot used in this approach.

1.8Degenerate Electron Gases

We now illustrate the usefulness of the second quantization representation by ap-plying it to obtain some qualitative results for a metal. The simple model we use isthat of an interacting electron gas with a uniform positive background so that thetotal system is neutral. We ignore the motion of the ions/positive charge. We donot consider any surface effects by restricting our attention to the bulk medium.We insert the system into a large box of side L and apply periodic boundary condi-tions; this ensures invariance under spatial translations of all physical quantities.The single-particle states are plane waves

'k ,λ (x ) D 1L3/2

e i k�x ηλ , η" D

10

, η# D

01

, (1.92)

where ki D 2πni /L, ni D 0,˙1,˙2, . . . The total Hamiltonian is

H D He C He�b C Hb , (1.93)

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