Problem N-ciał - Tremablay.pdf

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PHY-892 Problème à N-corps (notes de cours)
André-Marie Tremblay
Hiver 2011
2
CONTENTS
1Introduc ion
17
I A refresher in statistical mechanics and quantum me-
chanics
21
2 Statistical Physics and Density matrix 25
2.1 Density matrix in ordinary quantum mechanics ........... 25
2.2 Density Matrix in Statistical Physics ................. 26
2.3 Legendre transforms .......................... 26
2.4 Legendre transform from the statistical mechanics point of view .. 27
3 Second quantization 29
3.1 Describing symmetrized or antisymmetrized states ......... 29
3.2 Change of basis ............................. 30
3.3 Second quantized version of operators ................ 30
3.3.1 One-body operators ...................... 30
3.3.2 Two-body operators ...................... 31
4 Hartree-Fock approximation 33
4.1 The theory of everything ........................ 33
4.2 Variational theorem .......................... 33
4.3 Wick’s theorem ............................. 34
4.4 Minimization and Hartree-Fock equations .............. 35
5 Model Hamiltonians 37
5.1 The Hubbard model .......................... 37
5.2 Heisenberg and t-J model ....................... 38
5.3 Anderson lattice model ......................... 40
6 Broken symmetry and canonical transformations 43
6.1 The BCS Hamiltonian ......................... 43
7 Elementary quantum mechanics and path integrals 47
7.1 Coherent-state path integrals ..................... 47
II Correlation functions, general properties
49
8 Relation between correlation functions and experiments
53
8.1 Details of the derivation for the speci fi c case of electron scattering
55
9 Time-dependent perturbation theory 59
9.1 Schrödinger and Heisenberg pictures. ................. 59
9.2
Interaction picture and perturbation theory ............. 60
10 Linear-response theory 63
10.1 Exercices ................................ 66
10.1.1 Autre dérivation de la réponse linéaire. ............ 66
CONTENTS
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11 General properties of correlation functions 67
11.1 Notations and de fi nition of  00 ..................... 67
11.2 Symmetry properties of  and symmetry of the response functions 68
11.2.1 Translational invariance .................... 69
11.2.2 Parity .............................. 69
11.2.3 Time-reversal symmetry in the absence of spin ....... 70
11.2.4 Time-reversal symmetry in the presence of spin ....... 72
11.3 Properties that follow from the de fi nition and
00
00
q −q (−) 74
11.4 Kramers-Kronig relations and causality ............... 75
11.4.1 The straightforward manner: ................. 76
11.5 Spectral representation ......................... 77
11.6 Lehmann representation and spectral representation ........ 79
11.7 Positivity of  00 () and dissipation ................. 81
11.8 Fluctuation-dissipation theorem .................... 82
11.9 Imaginary time and Matsubara frequencies, a preview ....... 84
11.10Sum rules ................................ 86
11.10.1Thermodynamic sum-rules. .................. 86
11.10.2Order of limits ......................... 88
11.10.3Moments, sum rules, and high-frequency expansions. .... 88
11.10.4The f-sum rule as an example ................. 89
11.11Exercice ................................. 90
11.11.1Fonction de relaxation de Kubo. ............... 90
11.11.2Constante diélectrique et Kramers-Kronig. .......... 91
11.11.3Lien entre fonctions de réponses, constante de diffusion et
dérivées thermodynamiques. Rôle des règles de somme. .. 91
q −q ()=−
12 Kubo formula for the conductivity
95
12.1 Coupling between electromagnetic fi elds and matter, and gauge in-
variance ................................. 95
12.1.1 Invariant action, Lagrangian and coupling of matter and
electromagnetic fi eld[ 10 ] .................... 96
12.2 Response of the current to external vector and scalar potentials .. 98
12.3 Kubo formula for the transverse conductivity ............ 100
12.4 Kubo formula for the longitudinal conductivity and f-sum rule . . 101
12.4.1 Further consequences of gauge invariance and relation to f
sum-rule. ............................ 102
12.4.2 Longitudinal conductivity sum-rule and a useful expression
for the longitudinal conductivity. ............... 104
12.5 Exercices ................................ 105
12.5.1 Formule de Kubo pour la conductivité thermique ...... 105
13 Drude weight, metals, insulators and superconductors 107
13.1 The Drude weight ........................... 107
13.2 What is a metal ............................. 108
13.3 What is an insulator .......................... 109
13.4 What is a superconductor ....................... 109
13.5 Metal, insulator and superconductor ................. 111
13.6 Finding the London penetration depth from optical conductivity . 112
14 Relation between conductivity and dielectric constant 115
14.1 Transverse dielectric constant. ..................... 115
14.2 Longitudinal dielectric constant. ................... 116
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CONTENTS
III Introduction to Green’s functions. One-body Schrödinger
equation
121
15 De fi nition of the propagator, or Green’s function
125
16 Information contained in the one-body propagator 127
16.1 Operator representation ........................ 127
16.2 Relation to the density of states .................... 128
16.3 Spectral representation, sum rules and high frequency expansion . 128
16.3.1 Spectral representation and Kramers-Kronig relations. ... 129
16.3.2 Sum rules ............................ 130
16.3.3 High frequency expansion. ................... 130
16.4 Relation to transport and fl uctuations ................ 131
16.5 Green’s functions for differential equations .............. 132
16.6 Exercices ................................ 134
16.6.1 Fonctions de Green retardées, avancées et causales. ..... 134
17 A fi rst phenomenological encounter with self-energy
135
18 Perturbation theory for one-body propagator 137
18.1 General starting point for perturbation theory. ........... 137
18.2 Feynman diagrams for a one-body potential and their physical in-
terpretation. ............................... 138
18.2.1 Diagrams in position space .................. 138
18.2.2 Diagrams in momentum space ................ 140
18.3 Dyson’s equation, irreducible self-energy ............... 141
18.4 Exercices ................................ 144
18.4.1 Règles de somme dans les systèmes désordonnés. ...... 144
18.4.2 Développement du locateur dans les systèmes désordonnés. 144
18.4.3 Une impureté dans un réseau: état lié, résonnance, matrice . 145
19 Formal properties of the self-energy
147
20 Electrons in a random potential: Impurity averaging technique. 149
20.1 Impurity averaging ........................... 149
20.2 Averaging of the perturbation expansion for the propagator .... 150
20.3 Exercices ................................ 155
20.3.1 Diffusion sur des impuretés. Résistance résiduelle des métaux. 155
21 Other perturbation resummation techniques: a preview
157
22 Feynman path integral for the propagator, and alternate formu-
lation of quantum mechanics 161
22.1 Physical interpretation ......................... 161
22.2 Computing the propagator with the path integral .......... 162
IV Theone-particleGreen’sfunctionat fi nite tempera-
ture
167
23 Main results from second quantization 171
23.1 Fock space, creation and annihilation operators ........... 171
23.1.1 Creation-annihilation operators for fermion wave functions 172
23.1.2 Creation-annihilation operators for boson wave functions . 173
23.1.3 Number operator and normalization ............. 174
23.1.4 Wave function ......................... 175
CONTENTS
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