Many-Body Quantum Theory in Condensed Matter Physics

Höfundar: Henrik Bruus; Karsten Flensberg (Útgáfa: 0)
Many-Body Quantum Theory in Condensed Matter Physics

Kaup valmöguleikar

This book is an introduction to the techniques of many-body quantum theory with a large number of applications to condensed matter physics. The basic idea of the book is to provide a self-contained formulation of the theoretical framework without losing mathematical rigor, while at the same time providing physical motivation and examples. The examples are taken from applications in electron systems and transport theory.

On the formal side, the book covers an introduction to second quantization, many-body Green's function, finite temperature Feynman diagrams and bosonization. The applications include traditional transport theory in bulk as well as mesoscopic systems, where both the Landau-B├╝ttiker formalism and recent developments in correlated transport phenomena in mesoscopic systems and nano-structures are covered.

Other topics include interacting electron gases, plasmons, electron-phonon interactions, superconductivity and a final chapter on one-dimensional systems where a detailed treatment of Luttinger liquid theory and bosonization techniques is given. Having grown out of a set of lecture notes, and containing many pedagogical exercises, this book is designed as a textbook for an advanced undergraduate or graduate course, and is also well suited for self-study.

Nánar um bókina

Útgefandi
Oxford University Press Academic UK
ISBN
9780191057472
Print ISBN
9780198566335
Format
Page Fidelity (PDF)
Útgáfa
0
Höfundar
Henrik Bruus; Karsten Flensberg
Tungumál
English
Útgefið
2004-09-03
Prent takmörkun á líftíma
100
Prent takmörkun
2
Afritunar takmörkun
2

Kaflar

  • MANY-BODY QUANTUM THEORY IN CONDENSED MATTER PHYSICS AN INTRODUCTION
  • COPYRIGHT
  • PREFACE
  • CONTENTS
  • LIST OF SYMBOLS
  • 1: FIRST AND SECOND QUANTIZATION
  • 1.1 First quantization, single-particle systems
  • 1.2 First quantization, many-particle systems
  • 1.2.1 Permutation symmetry and indistinguishability
  • 1.2.2 The single-particle states as basis states
  • 1.2.3 Operators in �rst quantization
  • 1.3 Second quantization, basic concepts
  • 1.3.1 The occupation number representation
  • 1.3.2 The boson creation and annihilation operators
  • 1.3.3 The fermion creation and annihilation operators
  • 1.3.4 The general form for second quantization operators
  • 1.3.5 Change of basis in second quantization
  • 1.3.6 Quantum �eld operators and their Fourier transforms
  • 1.4 Second quantization, speci�c operators
  • 1.4.1 The harmonic oscillator in second quantization
  • 1.4.2 The electromagnetic �eld in second quantization
  • 1.4.3 Operators for kinetic energy, spin, density and current
  • 1.4.4 The Coulomb interaction in second quantization
  • 1.4.5 Basis states for systems with di�erent kinds of particles
  • 1.5 Second quantization and statistical mechanics
  • 1.5.1 The distribution function for non-interacting fermions
  • 1.5.2 The distribution function for non-interacting bosons
  • 1.6 Summary and outlook
  • 2: THE ELECTRON GAS
  • 2.1 The non-interacting electron gas
  • 2.1.1 Bloch theory of electrons in a static ion lattice
  • 2.1.2 Non-interacting electrons in the jellium model
  • 2.1.3 Non-interacting electrons at �nite temperature
  • 2.2 Electron interactions in perturbation theory
  • 2.2.1 Electron interactions in �rst-order perturbation theory
  • 2.2.2 Electron interactions in second-order perturbation theory
  • 2.3 Electron gases in 3, 2, 1 and 0 dimensions
  • 2.3.1 3D electron gases: metals and semiconductors
  • 2.3.2 2D electron gases: GaAs/GaAlAs heterostructures
  • 2.3.3 1D electron gases: carbon nanotubes
  • 2.3.4 0D electron gases: quantum dots
  • 2.4 Summary and outlook
  • 3: PHONONS; COUPLING TO ELECTRONS
  • 3.1 Jellium oscillations and Einstein phonons
  • 3.2 Electron-phonon interaction and the sound velocity
  • 3.3 Lattice vibrations and phonons in 1D
  • 3.4 Acoustical and optical phonons in 3D
  • 3.5 The speci�c heat of solids in the Debye model
  • 3.6 Electron-phonon interaction in the lattice model
  • 3.7 Electron-phonon interaction in the jellium model
  • 3.8 Summary and outlook
  • 4: MEAN-FIELD THEORY
  • 4.1 Basic concepts of mean-field theory
  • 4.2 The art of mean-�eld theory
  • 4.3 Hartree{Fock approximation
  • 4.3.1 Hartree-Fock approximation for the homogeneous electron gas
  • 4.4 Broken symmetry
  • 4.5 Ferromagnetism
  • 4.5.1 The Heisenberg model of ionic ferromagnets
  • 4.5.2 The Stoner model of metallic ferromagnets
  • 4.6 Summary and outlook
  • 5: TIME DEPENDENCE IN QUANTUM THEORY
  • 5.1 The SchrŁodinger picture
  • 5.2 The Heisenberg picture
  • 5.3 The interaction picture
  • 5.4 Time-evolution in linear response
  • 5.5 Time-dependent creation and annihilation operators
  • 5.6 Fermi's golden rule
  • 5.7 The T -matrix and the generalized Fermi's golden rule
  • 5.8 Fourier transforms of advanced and retarded functions
  • 5.9 Summary and outlook
  • 6: LINEAR RESPONSE THEORY
  • 6.1 The general Kubo formula
  • 6.2 Kubo formula for conductivity
  • 6.3 Kubo formula for conductance
  • 6.4 Kubo formula for the dielectric function
  • 6.4.1 Dielectric function for translation-invariant system
  • 6.4.2 Relation between dielectric function and conductivity
  • 6.5 Summary and outlook
  • 7: TRANSPORT IN MESOSCOPIC SYSTEMS
  • 7.1 The S-matrix and scattering states
  • 7.1.1 De�nition of the S-matrix
  • 7.1.2 De�nition of the scattering states
  • 7.1.3 Unitarity of the S-matrix
  • 7.1.4 Time-reversal symmetry
  • 7.2 Conductance and transmission coe�cients
  • 7.2.1 The Landauer formula, heuristic derivation
  • 7.2.2 The Landauer formula, linear response derivation
  • 7.2.3 The Landauer{BŁuttiker formalism for multiprobe systems
  • 7.3 Electron wave guides
  • 7.3.1 Quantum point contact and conductance quantization
  • 7.3.2 The Aharonov{Bohm e ect
  • 7.4 Summary and outlook
  • 8: GREEN'S FUNCTIONS
  • 8.1 "Classical" Green's functions
  • 8.2 Green's function for the one-particle SchrŁodinger equation
  • 8.2.1 Example: from the S-matrix to the Green's function
  • 8.3 Single-particle Green's functions of many-body systems
  • 8.3.1 Green's function of translation-invariant systems
  • 8.3.2 Green's function of free electrons
  • 8.3.3 The Lehmann representation
  • 8.3.4 The spectral function
  • 8.3.5 Broadening of the spectral function
  • 8.4 Measuring the single-particle spectral function
  • 8.4.1 Tunneling spectroscopy
  • 8.5 Two-particle correlation functions of many-body systems
  • 8.6 Summary and outlook
  • 9: EQUATION OF MOTION THEORY
  • 9.1 The single-particle Green's function
  • 9.2 Single level coupled to a continuum
  • 9.3 Anderson's model for magnetic impurities
  • 9.3.1 The equation of motion for the Anderson model
  • 9.3.2 Mean-�eld approximation for the Anderson model
  • 9.4 The two-particle correlation function
  • 9.4.1 The random phase approximation
  • 9.5 Summary and outlook
  • 10: TRANSPORT IN INTERACTING MESOSCOPIC SYSTEMS
  • 10.1 Model Hamiltonians
  • 10.2 Sequential tunneling: the Coulomb blockade regime
  • 10.2.1 Coulomb blockade for a metallic dot
  • 10.2.2 Coulomb blockade for a quantum dot
  • 10.3 Coherent many-body transport phenomena
  • 10.3.1 Cotunneling
  • 10.3.2 Inelastic cotunneling for a metallic dot
  • 10.3.3 Elastic cotunneling for a quantum dot
  • 10.4 The conductance for Anderson-type models
  • 10.4.1 The conductance in linear response
  • 10.4.2 Calculation of Coulomb blockade peaks
  • 10.5 The Kondo e�ect in quantum dots
  • 10.5.1 From the Anderson model to the Kondo model
  • 10.5.2 Comparing the Kondo e�ect in metals and quantum dots
  • 10.5.3 Kondo-model conductance to second order in H(2)
  • 10.5.4 Kondo-model conductance to third order in H(2)
  • 10.5.5 Origin of the logarithmic divergence
  • 10.6 Summary and outlook
  • 11: IMAGINARY-TIME GREEN'S FUNCTIONS
  • 11.1 De�nitions of Matsubara Green's functions
  • 11.1.1 Fourier transform of Matsubara Green's functions
  • 11.2 Connection between Matsubara and retarded functions
  • 11.2.1 Advanced functions
  • 11.3 Single-particle Matsubara Green's function
  • 11.3.1 Matsubara Green's function for non-interacting particles
  • 11.4 Evaluation of Matsubara sums
  • 11.4.1 Summations over functions with simple poles
  • 11.4.2 Summations over functions with known branch cuts
  • 11.5 Equation of motion
  • 11.6 Wick's theorem
  • 11.7 Example: polarizability of free electrons
  • 11.8 Summary and outlook
  • 12: FEYNMAN DIAGRAMS AND EXTERNAL POTENTIALS
  • 12.1 Non-interacting particles in external potentials
  • 12.2 Elastic scattering and Matsubara frequencies
  • 12.3 Random impurities in disordered metals
  • 12.4 Impurity self-average
  • 12.5 Self-energy for impurity scattered electrons
  • 12.5.1 Lowest-order approximation
  • 12.5.2 First-order Born approximation
  • 12.5.3 The full Born approximation
  • 12.5.4 The self-consistent full Born approximation and beyond
  • 12.6 Summary and outlook
  • 13: FEYNMAN DIAGRAMS AND PAIR INTERACTIONS
  • 13.1 The perturbation series for G
  • 13.2 The Feynman rules for pair interactions
  • 13.2.1 Feynman rules for the denominator of G(b; a)
  • 13.2.2 Feynman rules for the numerator of G(b; a)
  • 13.2.3 The cancellation of disconnected Feynman diagrams
  • 13.3 Self-energy and Dyson's equation
  • 13.4 The Feynman rules in Fourier space
  • 13.5 Examples of how to evaluate Feynman diagrams
  • 13.5.1 The Hartree self-energy diagram
  • 13.5.2 The Fock self-energy diagram
  • 13.5.3 The pair-bubble self-energy diagram
  • 13.6 Cancellation of disconnected diagrams, general case
  • 13.7 Feynman diagrams for the Kondo model
  • 13.7.1 Kondo model self-energy, second order in J
  • 13.8 Summary and outlook
  • 14: THE INTERACTING ELECTRON GAS
  • 14.1 The self-energy in the random phase approximation
  • 14.1.1 The density dependence of self-energy diagrams
  • 14.1.2 The divergence number of self-energy diagrams
  • 14.1.3 RPA resummation of the self-energy
  • 14.2 The renormalized Coulomb interaction in RPA
  • 14.2.1 Calculation of the pair-bubble
  • 14.3 The groundstate energy of the electron gas
  • 14.4 The dielectric function and screening
  • 14.5 Plasma oscillations and Landau damping
  • 14.5.1 Plasma oscillations and plasmons
  • 14.5.2 Landau damping
  • 14.6 Summary and outlook
  • 15: FERMI LIQUID THEORY
  • 15.1 Adiabatic continuity
  • 15.1.1 Example: one-dimensional well
  • 15.1.2 The quasiparticle concept and conserved quantities
  • 15.2 Semi-classical treatment of screening and plasmons
  • 15.2.1 Static screening
  • 15.2.2 Dynamical screening
  • 15.3 Semi-classical transport equation
  • 15.3.1 Finite lifetime of the quasiparticles
  • 15.4 Microscopic basis of the Fermi liquid theory
  • 15.4.1 Renormalization of the single particle Green's function
  • 15.4.2 Imaginary part of the single-particle Green's function
  • 15.4.3 Mass renormalization?
  • 15.5 Summary and outlook
  • 16: IMPURITY SCATTERING AND CONDUCTIVITY
  • 16.1 Vertex corrections and dressed Green's functions
  • 16.2 The conductivity in terms of a general vertex function
  • 16.3 The conductivity in the �rst Born approximation
  • 16.4 Conductivity from Born scattering with interactions
  • 16.5 The weak localization correction to the conductivity
  • 16.6 Disordered mesoscopic systems
  • 16.6.1 Statistics of quantum conductance, random matrix theory
  • 16.6.2 Weak localization in mesoscopic systems
  • 16.6.3 Universal conductance uctuations
  • 16.7 Summary and outlook
  • 17: GREEN'S FUNCTIONS AND PHONONS
  • 17.1 The Green's function for free phonons
  • 17.2 Electron-phonon interaction and Feynman diagrams
  • 17.3 Combining Coulomb and electron-phonon interactions
  • 17.3.1 Migdal's theorem
  • 17.3.2 Jellium phonons and the e ective electron-electron interaction
  • 17.4 Phonon renormalization by electron screening in RPA
  • 17.5 The Cooper instability and Feynman diagrams
  • 17.6 Summary and outlook
  • 18: SUPERCONDUCTIVITY
  • 18.1 The Cooper instability
  • 18.2 The BCS groundstate
  • 18.3 Microscopic BCS theory
  • 18.4 BCS theory with Matsubara Green's functions
  • 18.4.1 Self-consistent determination of the BCS order parameter �
  • 18.4.2 Determination of the critical temperature Tc
  • 18.4.3 Determination of the BCS quasiparticle density of states
  • 18.5 The Nambu formalism of the BCS theory
  • 18.5.1 Spinors and Green's functions in the Nambu formalism
  • 18.5.2 The Meissner e�ect and the London equation
  • 18.5.3 The vanishing paramagnetic current response in BCS theory
  • 18.6 Gauge symmetry breaking and zero resistivity
  • 18.6.1 Gauge transformations
  • 18.6.2 Broken gauge symmetry and dissipationless current
  • 18.7 The Josephson e�ect
  • 18.8 Summary and outlook
  • 19: 1D ELECTRON GASES AND LUTTINGER LIQUIDS
  • 19.1 What is a Luttinger liquid?
  • 19.2 Experimental realizations of Luttinger liquid physics
  • 19.2.1 Example: Carbon Nanotubes
  • 19.2.2 Example: semiconductor wires
  • 19.2.3 Example: quasi 1D materials
  • 19.2.4 Example: Edge states in the fractional quantum Hall e ect
  • 19.3 A �rst look at the theory of interacting electrons in 1D
  • 19.3.1 The \quasiparticles" in 1D
  • 19.3.2 The lifetime of the “quasiparticles” in 1D
  • 19.4 The spinless Luttinger{Tomonaga model
  • 19.4.1 The Luttinger{Tomonaga model Hamiltonian
  • 19.4.2 Inter-branch interaction
  • 19.4.3 Intra-branch interaction and charge conservation
  • 19.4.4 Umklapp processes in the half-�lled band case
  • 19.5 Bosonization of the Tomonaga model Hamiltonian
  • 19.5.1 Derivation of the bosonized Hamiltonian
  • 19.5.2 Diagonalization of the bosonized Hamiltonian
  • 19.5.3 Real space representation
  • 19.6 Electron operators in bosonized form
  • 19.7 Green's functions
  • 19.8 Measuring local density of states by tunneling
  • 19.9 Luttinger liquid with spin
  • 19.10 Summary and outlook
  • APPENDIX A: FOURIER TRANSFORMATIONS
  • A.1 Continuous functions in a �nite region
  • A.2 Continuous functions in an in�nite region
  • A.3 Time and frequency Fourier transforms
  • A.4 Some useful rules
  • A.5 Translation-invariant systems
  • EXERCISES
  • SELECTED BIBLIOGRAPHY
  • INDEX