PHY-892 Problème à N-corps (notes de cours) André-Marie Tremblay Automne 2005 2 Contents I Introduction: Correlation functions and Green’s func- tions. 11 1Introduction 13 2 Correlation functions 15 2.1 Relation between correlation functions and experiments . 16 2.2Linear-responsetheory......................... 19 2.2.1 SchrödingerandHeisenbergpictures.............. 20 2.2.2 Interactionpictureandperturbationtheory......... 21 2.2.3 Linearresponse......................... 23 2.3Generalpropertiesofcorrelationfunctions.............. 24 2.3.1 Notations and definitions................... 25 2.3.2 Symmetrypropertiesoftheresponsefunctions....... 26 2.3.3 Kramers-Kronigrelationsandcausality........... 31 2.3.4 Positivity of ωχ00(ω) anddissipation............. 35 2.3.5 Fluctuation-dissipationtheorem................ 36 2.3.6 Sumrules............................ 38 2.4Kuboformulafortheconductivity.................. 41 2.4.1 Response of the current to external vector and scalar potentials 42 2.4.2 Kuboformulaforthetransverseresponse.......... 43 2.4.3 Kuboformulaforthelongitudinalresponse......... 44 2.4.4 Metals, insulators and superconductors . .......... 49 2.4.5 Conductivity sum rules at finite wave vector, transverse and longitudinal........................... 52 2.4.6 Relation between conductivity and dielectric constant . 54 3 Introduction to Green’s functions. One-body Schrödinger equa- tion 59 3.1 Definition of the propagator, or Green’s function .......... 59 3.2 Information contained in the one-body propagator ......... 60 3.2.1 Operatorrepresentation..................... 61 3.2.2 Relationtothedensityofstates............... 62 3.2.3 Spectral representation, sum rules and high frequency ex- pansion............................. 62 3.2.4 Relation to transport and fluctuations............ 64 3.2.5 Green’s functions for differentialequations.......... 65 3.3 Perturbation theory for one-body propagator . .......... 67 3.3.1 Generalstartingpointforperturbationtheory........ 67 3.3.2 Feynman diagrams for a one-body potential and their phys- icalinterpretation........................ 68 3.3.3 Dyson’sequation,irreducibleself-energy........... 71 3.4Formalpropertiesoftheself-energy.................. 73 3.5 Electrons in a random potential: Impurity averaging technique. 75 CONTENTS 3 3.5.1 Impurityaveraging....................... 75 3.5.2 Averaging of the perturbation expansion for the propagator 76 3.6 Other perturbation resummation techniques: a preview . 81 II Traditional approaches to the normal state d 3. 87 ≥ 4 Finite temperature formalism 89 4.1Mainresultsfromsecondquantization................ 90 4.1.1 One-bodyoperators...................... 91 4.1.2 Two-bodyoperators. ..................... 92 4.1.3 Second quantized operators in the Heisenberg picture . 94 4.2 Motivation of the definition of the second quantized Green’s func- tion GR ................................. 95 4.2.1 ExampleswithquadraticHamiltonians:........... 96 4.3Interactionrepresentationandtime-orderedproduct........ 98 4.4 Kadanoff-BaymandKeldysh-Schwingercontours.......... 102 4.5 Matsubara Green’s function and its relation to usual Green’s func- tions.(Thecaseoffermions)..................... 104 4.5.1 Definition............................ 105 4.5.2 Antiperiodicity and Fourier expansion (Matsubara frequen- cies)............................... 107 4.5.3 Spectral representation, relation between GR and and an- alyticcontinuation.......................G 108 4.5.4 Spectral weight and rules for analytical continuation . 111 4.5.5 Matsubara Green’s function in momentum space and non- interactingcase......................... 112 4.5.6 SumsoverMatsubarafrequencies............... 117 4.6 Physical meaning of the spectral weight: Quasiparticles, effective mass, wave function renormalization, momentum distribution. 119 4.6.1 Spectralweightfornon-interactingparticles......... 119 4.6.2 Lehmanrepresentation..................... 119 4.6.3 Probabilistic interpretation of the spectral weight . 121 4.6.4 Angle-resolved photoemission spectroscopy (ARPES) on a Fermi liquid compound. .................. 122 4.6.5 Quasiparticles[9]........................ 125 4.6.6 FermiliquidinterpretationofARPES............ 127 4.6.7 Momentum distribution in an interacting system . 130 4.7 A few more formal matters : asymptotic behavior and causality . 131 4.7.1 Asymptotic behavior of (k;ikn) and Σ (k;ikn) ....... 132 4.7.2 Implications of causalityG for GR and ΣR ........... 133 4.8Threegeneraltheorems........................ 134 4.8.1 Wick’stheorem......................... 135 4.8.2 Linkedclustertheorems.................... 139 4.8.3 Variational principle and application to Hartree-Fock theory 141 5 The Coulomb gas 149 5.1Feynmanrulesfortwo-bodyinteractions............... 149 5.1.1 Hamiltonianandnotation................... 150 5.1.2 Inpositionspace........................ 151 5.1.3 Inmomentumspace...................... 158 5.1.4 Feynmanrulesfortheirreducibleself-energy........ 160 5.1.5 FeynmandiagramsandthePauliprinciple.......... 161 5.2Collectivemodesanddielectricfunction............... 162 4 CONTENTS 5.2.1 Definitionsandanalyticcontinuation............. 162 5.2.2 Density response in the non-interacting limit: Lindhard func- tion............................... 163 5.2.3 Expansion parameter in the presence of interactions: rs . 169 5.2.4 Elementary approaches to screening and plasma oscillations 170 5.2.5 Densityresponseinthepresenceofinteractions....... 174 5.3 More formal matters: Consistency relations between single-particle self-energy, collective modes, potential energy and free energy . 182 5.3.1 Consistency between self-energy and density fluctuations . 182 5.3.2 Generaltheoremonfree-energycalculations......... 185 5.4Single-particleproperties........................ 186 5.4.1 Hartree-Focktheory...................... 187 5.4.2 Curing Hartree-Fock theory: screened interaction in the self- energy.............................. 192 5.5 General considerations on perturbation theory and asymptotic ex- pansions................................. 205 5.6 Beyond RPA: skeleton diagrams, vertex functions and associated difficulties................................. 207 6 Broken Symmetry, Ferromagnetism as an example 213 6.1TheHubbardmodel.......................... 213 6.1.1 The non-interacting limit U =0 ............... 214 6.1.2 The strongly interacting, atomic, limit t =0......... 214 6.2 Weak interactions at low filling, Stoner ferromagnetism and the BrokenSymmetryphase........................ 216 6.2.1 Simplearguments,theStonermodel............. 216 6.2.2 Variationalwavefunction................... 217 6.2.3 Feynman’s variational principle for variational Hamiltonian. Orderparameterandorderedstate.............. 217 6.2.4 The gap equation and Landau theory in the ordered state . 218 6.2.5 The Green function point of view (effective medium) . 219 6.2.6 Instabilityofthenormalstate................. 220 6.2.7 Magneticstructurefactorandparamagnons......... 222 6.2.8 Collective Goldstone mode, stability and the Mermin-Wagner theorem............................. 223 6.2.9 Kanamori-Brückner screening: Why Stoner ferromagnetism hasproblems.......................... 226 6.3Two-ParticleSelf-ConsistentApproach,motivation......... 227 6.4 Antiferromagnetism close to half-filling, nesting . .......... 228 6.5 Additional remarks: Hubbard-Stratonovich transformation and crit- icalphenomena............................. 228 CONTENTS 5 6 CONTENTS List of Figures 2-1Electronscatteringexperiment. ................... 17 2-2 Skin effect:transverseresponse..................... 44 2-3Penetrationdepthinasuperconductor................ 51 3-1 Diagrammatic representation of the Lippmann-Schwinger equation forscattering............................... 68 3-2 Iteration of the progagator for scattering off impurities. 69 3-3 Feynman diagrams for scattering off impurities in momentum space (beforeimpurityaveraging)....................... 70 3-4Dyson’sequationandirreducibleself-energy............. 72 3-5First-orderirreducibleself-energy.................... 72 3-6 Second order irreducible self-energy (before impurity averaging). 73 3-7 Direct iterated solution to the Lippmann-Schwinger equation after impurityaveraging............................ 77 3-8 Second-order irreducible self-energy in the impurity averaging tech- nique................................... 78 3-9 Taking into account multiple scattering from a single impurity. 80 3-10 Some diagrams contributing to the density-density correlation func- tionbeforeimpurityaveraging..................... 81 3-11 Some of the density-density diagrams after impurity averaging. 81 3-12 Ladder diagrams for T-matrix or Bethe-Salpeter equation. 82 3-13Bubblediagramsforparticle-holeexitations.............. 82 3-14 Diagrammatic representation of the Hartree-Fock approximation. 83 > 4-1 Kadanoff-Baym contour to compute G (t t0) . .......... 103 − 4-2Keldysh-Schwingercontour....................... 104 4-3Contourfortimeorderinginimaginarytime............. 107 4-4 Deformed contour used to relate the Matsubara and the retarded Green’sfunctions............................ 110 4-5 Analytical structure of G(z) in the complex frequency plane. G(z) R A reduces to either G (ω) ,G (ω) or (iωn) depending on the value of the complex frequency z. There isG a branch cut along the real axis.112 4-6 0 (p, τ ) for a value of momentum above the Fermi surface. 115 G 4-7 0 (p, τ ) foravalueofmomentumattheFermisurface....... 115 G 4-8 0 (p, τ ) for a value of momentum below the Fermi surface. 115 G 4-9 Evaluation of fermionic Matsubara