Introduction to Many Body Physics. Lecture Notes, University of Geneva, 2008-2013

0.1 Goal of the game......Page 7
0.3 Disclaimer......Page 8
I Electronic properties of metals: Fermi liquids......Page 9
1.1.1 Free electrons......Page 11
1.1.2 Electrons in periodic potentials: band theory......Page 13
1.1.3 Thermodynamic observables......Page 16
1.2 Coulomb interaction......Page 21
1.2.1 Coulomb interaction in a solid......Page 22
1.3 Importance of the interactions......Page 26
1.4 Theoretical assumptions and experimental realities......Page 27
2.1 Brief reminder of quantum mechanics......Page 35
2.2 Linear response......Page 37
2.3 Fluctuation dissipation theorem......Page 40
2.4 Spectral representation, Kramers-Kronig relations......Page 42
3.1 Why not Schrödinger......Page 45
3.2 Fock space......Page 47
3.3.1 Bosons......Page 48
3.3.2 Fermions......Page 51
3.4.1 Definition......Page 54
3.4.2 Examples......Page 56
3.5.1 Definition......Page 58
3.5.2 Examples......Page 60
3.6 Solving with second quantization......Page 62
3.6.1 Eigenvalues and Eigenstates......Page 63
3.6.2 Bogoliubov transformation......Page 65
4.1 Interaction Hamiltonian......Page 77
4.3.1 Single particle Green's function......Page 79
4.3.2 Properties and spectral function......Page 82
4.3.3 Connection with photoemission......Page 84
4.4 Phenomenology of the spectral function......Page 85
Imaginary part: lifetime......Page 86
Real part: effective mass and quasiparticle weight......Page 87
4.5.1 Landau quasiparticles......Page 89
4.5.2 Lifetime......Page 91
5.1 Mean field approximation......Page 97
5.1.1 Method......Page 98
5.1.2 Spin and charge......Page 100
5.1.3 Static fields: thermodynamic response......Page 101
5.2 Collective modes......Page 103
5.2.1 Short range interaction: zero sound......Page 105
5.2.2 Long range interations: Plasmon......Page 109
5.2.3 Landau Damping......Page 111
II Strongly correlated systems......Page 115
6.1 Susceptibilities and phase transitions......Page 117
6.2.1 Ferromagnetism......Page 119
6.2.2 Charge instabilities......Page 122
6.3.1 Basic concept......Page 123
6.3.2 Spin susceptibility......Page 125
6.3.3 Charge instability......Page 126
6.4.1 Susceptibility......Page 127
6.4.2 Attractive interaction, BCS instability......Page 128
6.5 Ordered state......Page 130
7.1 Basic idea......Page 135
7.2 Mott transition......Page 140
7.2.1 Gutzwiller wavefunction......Page 141
7.2.2 Gutzwiller approximation (Nozières formulation)......Page 142
7.2.3 Half-filling......Page 145
7.2.4 Doped case......Page 149
7.2.5 Summary of the properties......Page 150
8.1.1 Reminder of spin properties......Page 155
8.1.2 Superexchange......Page 156
8.2.1 Ground state......Page 159
8.2.2 Magnetic field......Page 161
8.3 Macroscopic number of spins......Page 163
8.3.1 Holstein-Primakoff representation......Page 164
8.3.2 Localized Ferromagnetism......Page 166
8.3.3 Localized antiferromagnetism......Page 171
8.4.1 Symetries and presence of long range order......Page 175
8.4.2 Link between classical and quantum systems......Page 179
8.5.1 Nuclear Magnetic Resonnance......Page 182
8.5.2 Neutron diffraction......Page 183
III Numerical methods......Page 191
9.1 Purpose and basic idea......Page 193
9.2 Monte-Carlo technique: basic principle......Page 194
9.3.1 Definition......Page 196
9.3.2 Steady state......Page 197
9.4 Metropolis algorithm and simple example......Page 198
9.5 Variational monte-Carlo......Page 200
A.1 Normalizations in a finite volume......Page 201
A.3 Complex plane integration......Page 202
A.4 Some properties of operators......Page 203
B Example of Monte Carlo program......Page 205