Many Body Quantum Theory in Condensed Matter Physics An Introduction 1st edition by Henrik Bruus, Karsten Flensberg ISBN 0198566336 978-0198566335

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ISBN 10: 0198566336
ISBN 13: 978-0198566335
Author: Henrik Bruus, Karsten Flensberg

 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-Büttiker 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.

Many Body Quantum Theory in Condensed Matter Physics An Introduction 1st  Table of contents:

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

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