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Introduction
This monogram is written with the graduate student in mind. I had in mind to write a short, crisp book that would introduce my students to the basic ideas and concepts behind many body physics. At the same time, I felt very strongly that I should like to share my excitement with this eld, for without feeling the thrill of entering uncharted territory, I do not think one has the motivation to learn and to make the passage from learning to research.

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1 Introduction 5
2 Scales and Complexity 11
2.1 Time scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 L: Length scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 N: particle number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 C: Complexity and diversity. . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Quantum Fields: Overview 17
4 Collective Quantum Fields 27
4.1 Harmonic oscillator: a zero-dimensional eld theory . . . . . . . . . . . . . 27
4.2 Collective modes: phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 The Thermodynamic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Continuum Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5 Exercises for chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5 Conserved Particles 53
5.1 Commutation and Anticommutation Algebras . . . . . . . . . . . . . . . . . 54
5.1.1 Heuristic Derivation for Bosons . . . . . . . . . . . . . . . . . . . . . 55
5.2 What about Fermions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Field operators in dierent bases . . . . . . . . . . . . . . . . . . . . . . . . 57
5.4 Fields as particle creation and annihilation operators. . . . . . . . . . . . . 59
5.5 The vacuum and the many body wavefunction . . . . . . . . . . . . . . . . 62
5.6 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.7 Identical Conserved Particles in Thermal Equilibrium . . . . . . . . . . . . 68
5.7.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.7.2 Identication of the Free energy: Key Thermodynamic Properties . 71
5.7.3 Independent Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.8 Exercises for chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6 Simple Examples of Second-quantization 79
6.1 Jordan Wigner Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 The Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.3 Gas of charged particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3.1 Link with rst quantization . . . . . . . . . . . . . . . . . . . . . . . 88
6.4 Non-interacting particles in thermal equilibrium . . . . . . . . . . . . . . . . 90
6.4.1 Fluid of non-interacting Fermions . . . . . . . . . . . . . . . . . . . . 91
6.4.2 Fluid of Bosons: Bose Einstein Condensation . . . . . . . . . . . . . 94
6.5 Exercises for chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7 Greens Functions 105
7.1 Interaction representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.1.1 Driven Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . 111
7.2 Greens Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.2.1 Green's function for free Fermions . . . . . . . . . . . . . . . . . . . 117
7.2.2 Green's function for free Bosons . . . . . . . . . . . . . . . . . . . . 120
7.3 Adiabatic concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3.1 Gell-Man Low Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.3.2 Generating Function for Free fermions . . . . . . . . . . . . . . . . . 125
7.3.3 The Spectral Representation . . . . . . . . . . . . . . . . . . . . . . 127
7.4 Many particle Green's functions . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.5 Landau's Fermi Liquid Theory . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.6 Exercises for chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8 Feynman Diagrams: T=0 145
8.1 Heuristic Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.2 Developing the Feynman Diagram Expansion . . . . . . . . . . . . . . . . . 152
8.2.1 Symmetry factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.2.2 Linked Cluster Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.3 Feynman rules in momentum space . . . . . . . . . . . . . . . . . . . . . . . 163
8.3.1 Relationship between energy, and the S-matrix . . . . . . . . . . . . 164
8.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8.4.1 Hartree Fock Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8.4.2 Response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
8.4.3 Magnetic susceptibility of non-interacting electron gas . . . . . . . . 172
8.4.4 Electron in a scattering potential . . . . . . . . . . . . . . . . . . . . 177
8.5 The self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.5.1 Hartree-Fock Self-energy . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.6 Large-N electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.7 Exercises for chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
9 Finite Temperature Many Body Physics 193
9.1 Imaginary time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
9.1.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.2 Imaginary Time Green Functions . . . . . . . . . . . . . . . . . . . . . . . . 199
9.2.1 Periodicity and Antiperiodicity . . . . . . . . . . . . . . . . . . . . . 200
9.2.2 Matsubara Representation . . . . . . . . . . . . . . . . . . . . . . . . 201
9.3 The contour integral method . . . . . . . . . . . . . . . . . . . . . . . . . . 204
9.4 Generating Function and Wick's theorem . . . . . . . . . . . . . . . . . . . 208
9.5 Feynman diagram expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
9.5.1 Feynman rules from Functional Derivative . . . . . . . . . . . . . . . 212
9.5.2 Feynman rules in frequency/momentum space . . . . . . . . . . . . . 216
9.5.3 Linked Cluster Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 218
9.6 Examples of the application of the Matsubara Technique . . . . . . . . . . . 219
9.6.1 Hartree Fock at a nite temperature. . . . . . . . . . . . . . . . . . 220
9.6.2 Electron in a disordered potential . . . . . . . . . . . . . . . . . . . . 221
9.7 Interacting electrons and phonons . . . . . . . . . . . . . . . . . . . . . . . . 229
9.7.1 2F: the electron-phonon coupling function . . . . . . . . . . . . . . 237
9.7.2 Mass Renormalization by the electron phonon interaction . . . . . . 240
9.7.3 Migdal's theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
9.8 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
9.9 Exercises for chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
10 Fluctuation Dissipation Theorem and Linear Response Theory 253
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
10.2 Fluctuation dissipation theorem for a classical harmonic oscillator . . . . . 255
10.3 Quantum Mechanical Response Functions. . . . . . . . . . . . . . . . . . . . 257
10.4 Fluctuations and Dissipation in a quantum world . . . . . . . . . . . . . . . 259
10.4.1 Spectral decomposition I: the correlation function S(t 􀀀 t0) . . . . . 259
10.4.2 Spectral decomposition II: the response function (t 􀀀 t0) . . . . . . 260
10.4.3 Quantum Fluctuation dissipation Theorem . . . . . . . . . . . . . . 260
10.4.4 Spectral decomposition III:
uctuations in imaginary time . . . . . . 261
10.5 Calculation of response functions . . . . . . . . . . . . . . . . . . . . . . . . 261
10.6 Spectroscopy: linking measurement and correlation . . . . . . . . . . . . . . 265
10.7 Electron Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
10.7.1 Formal properties of the electron Green function . . . . . . . . . . . 269
10.7.2 Tunneling spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 270
10.7.3 ARPES, AIPES and inverse PES . . . . . . . . . . . . . . . . . . . . 273
10.8 Spin Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
10.8.1 D.C. magnetic susceptibility . . . . . . . . . . . . . . . . . . . . . . . 275
10.8.2 Neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
10.8.3 NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
10.9 Electron Transport spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 281
10.9.1 Resistivity and the transport relaxation rate . . . . . . . . . . . . . 281
10.9.2 Optical conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
10.9.3 The f-sum rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
10.10Exercises for chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
11 Electron transport Theory 293
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
11.2 The Kubo Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
11.3 Drude conductivity: diagramatic derivation . . . . . . . . . . . . . . . . . . 300
11.4 Electron Diusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
11.5 Weak Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
11.6 Exercises for chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
12 Path Integrals and Phase transitions 323
12.1 Introduction: Broken symmetry, coherent states and the path integral approach323
12.2 Coherent states and Grassman mathematics . . . . . . . . . . . . . . . . . . 328
12.2.1 Completeness and matrix elements . . . . . . . . . . . . . . . . . . . 330
12.3 Path integral for the partition function . . . . . . . . . . . . . . . . . . . . . 332
12.4 General evaluation of Path Integral for non-interacting Fermions . . . . . . 337
12.5 Hubbard Stratonovich transformation . . . . . . . . . . . . . . . . . . . . . 339
12.6 Superconductivity and BCS theory . . . . . . . . . . . . . . . . . . . . . . . 342
12.6.1 Introduction: Superconductivity pre-history . . . . . . . . . . . . . . 342
12.6.2 The BCS Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 343
12.6.3 Computing Tc carefully . . . . . . . . . . . . . . . . . . . . . . . . . 347
12.6.4 The Nambu Greens function . . . . . . . . . . . . . . . . . . . . . . 348
12.6.5 The structure of the Boguilubov quasiparticle and the BCS wavefunction348
12.6.6 Twisting the phase: the Anderson Higg's mechanism . . . . . . . . . 348
A Appendix: Grassman Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 348
A.1 Dierentiation and Integration . . . . . . . . . . . . . . . . . . . . . 348
A.2 Change of variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
A.3 Gaussian Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
B Exercises for chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

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